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0 ( 1 ) ; Bi u calc ulatea from Eqs . ( 91) and ( 9 2) . ==
8
P
26
Villadsen and Mic helsen
T he lower branch of the n versus ¢ curve is well approxi mated by one- point collocation ( curve marked " c " on Fig. 1 ) . For exam ple , at point A , w here ¢ 2 1 . 2 , one obtains ( ¢ * ) 2 :;;: 1 . 3 5 , and insertin g the modified 0 . 1 5 5 5 ] in Eq . ( 8 2) , one obtain s y 1 = 0 . 8 46 8 value of 6 [ S * :;;: 0 . 0 5 28 /9 an d n "' R ( y 1 ) = 1 . 7 0 2 , which may be compared w ith t h e exact result , 1 . 683. T he value of 0 1 = 1 . 0 2 3 8 , while 0 ( 1 ) is 1 . 0 1 7 0 . One- point colloca tion breaks down near the first bifurcation poin t , where the p arabolic ap p roximation for the concentration profile insi de t he pellet becomes very poor . There is another model app roximation t hat appears to give remarkably good res ults . T he energy balance ( 6 5) is inte grated over the pellet volume , and if 13 is small , one may ass um e that the pellet is nearly isothermal , in which case the first - order rate exp ression can be inte grated analytically to give the ri ght- han d si de of Eq . ( 8 9) : =
=
•
Bi [ 0 ( 1 ) - 1 ] = SBi H M
¢ 1 coth cp 1 - 1
'1' 1 coth A,
cp 1 + Bl.
M
_
1
( 8 9)
( 90) and e is a representative pellet temperature . p l f w e are on t he upper branch of the n v er s us cp c urve of Fi g . 1, the pellet temperature is so far above the bulk temperature 1 that the reaction is c ompl et e in a thin boundary la ye r . H ere the representative pellet tem perature 0 s hould be equated to 0 ( 1 ) and ( 8 9 ) is an equation to deter
p mine 0 ( 1 ) 0p. On the lower branch of t he curve , most of the reaction occurs in the 1 . 2 illus trates the difference b e twe e n bulk of the pellet . The case cp 2 0 ( 1 ) = 1 . 01 7 an d G p , which in the one- p oint collocation approximation is taken to be e 1 = 1 . 0 2 3 8 . I n Villadsen and M ic he l s en ( 1 978 , p . 2 8 and C hap . 6) it i s shown t hat the factor 0 { 1) - 1 on the left - h an d side of Eq . ( 8 9) can be replaced by the larger fact o r 0 p - 1 if at the same time BiM is replaced by a s maller , modified value BiH defined by =
=
Bi [ 0 ( 1 ) - 1 ] H
:::
( 91 )
Bi ( S - 1) H
where Bi H is foun d by pert urbation analysis t o be
1
- ex =
s + 3
s = 0 , 1 , or 2
( 9 2)
Thus , also for a modest reaction r ate , one obtains an algebraic equation for the representative pellet temperature 0 p . W hether 0 p is taken to be 0 ( 1 ) or the left -hand side of Eq . ( 89) is replaced by Eq . ( 9 1 ) to deter mine a 0 p > 0 ( 1 ) , the effectiveness factor is calculated by 3Bi
11 =
7M
cp
1
coth cp 1 - 1
cp 1 cot h cp 1
+
BiM - 1
( 93)
27
N umerical M e t ho ds in Reac tion E ngi neering For 0 p cot h
larger than "' 1 . 2 , one obtains values of
2 ( e . g . , < 5) Eq . ( 97 ) b y strai ghtfor ward method Every evaluation of the right - hand side of ( 1 0 1 ) i however , boun dary ( 97) transformed into N coupled algeb raic equations . After havin g the collocation e quation s , find the sensitivity respect xA in order good startin g values for the boun dary value the value xA . problem solved by collocation before x A chan ged : 10 -r R ) the concentration decreases We re fe r to t hese conditions e xponentially with distance from t he surface . as t h e diffusion-co n trolled or diffu s ion -li m i t e d regim e . H ere t he reac tion = 1) . The i mpact of the the r m al gradient s on the in t rin sic selectivity of any multireaction net work can be determin ed rapi dly by use of the external re sis t anc es model . When all the reac tion s are either e xot h er mic or en dothermic , 0. 2 ) the concentration p rofile within the pel le t cannot any more be neglected an d b o t h E q s . ( 1 0) and ( 1 1 ) need to be solved simultaneously . An especially important case from the practical standpoint is when the order with respect to the gaseous component is unity (n = 1) . In this situation it is possible t o combine E q s . ( 1 0 ) and ( 1 1 ) in to a sin gle equation by defining a cumulative gas concentration as K ( p ) + 1 ( p ) K ( p ) ] o o 1 f 1 2 1 " 2 P [ I o ( p x ) + 1 t ( px f ) k o ( p x f ) ) [ p l 1 ( p ) + E and t; . The limiting rays in t he
For
z
=
z
( 1) 0 at =
z;;
=
0
and z =
X --�--2 at B
(1
XA )
( 10 2) z;;
=
1
N umerical M e t ho ds in Reac tion Engineering
29
values of (cosh r;/cosh at the collocation the calculationsstored reactions. s for fastpoints, and different xA values. For xA the profile increases overall rate of productionanofinflectionzeropointin (where inner partlocalof pofleproduc then decreases tion of is zero) to long ,_!:!2 at part in the obtainspline an accurate , collocation points may notused. be enough collocationvalueor iterations somey other method may have to be ontwohigh-order collocation equationsandare( avoided problem treated as differential equations coupled to algebraic equations: [p.
T hroughout
abscissas are , of course
in a table .
cl>!
( h)
"Figure 2 show s 'Profiles N umerical difficulties ari e � 'L 1 0 , six collocation = 0 to a snarp m.ax\m'l:lm �\. \. "'" \.max (..'\.'-'.� B is the the el t ) , through the rate B a interior of the pellet . To
':1" <. 1:.) fo� � "-
=
,
=
of
an d
six
T he time- con s umin g i f the N
N+1 � .t...J C , . y. ]1 1 i=l
+
( 98 )
is
99)
1
]
+
-
a
( 103)
y. ( 1 ]
2 2
1 .6
1.2
.8
.4
.6
.4
.2
0
.8
profiles for component in k 1cA concentration Parameter on curves:for xA = fluid phase conversion of yield of is obtained the pellet profile has zero slope at FI GUR E 2
A
Pellet
-
A.
B
k 2cB
2
B
xA
C.
Maximum
r;
B
=
1.
"'
0 . 7 5 w here
30
Villadsen and Michelsen
0
=
dy
N+ 2 =
dt
( 1 04 )
1
( 1 0 5)
( 1 06 ) where
x XB N +3 N+2 A' t = dummy in t e g ra t ion variable Y
Y
=
p j
=
p ret ab u lat e d values of cos h
=
4>1 1 / cosh 4> 1
The boundary ordinate YN + 1 is e li mi n a ted an d the r e s ulti ng set o f N + 2 differential - al geb raic e q ua tion s is solved [ e. g . , by the S T I FFALG pac k a ge I n itial values of Yi • i = 1 , 2 , , N , are based on E q s . ( 3 6 ) an d ( 37 ) ] . W hen 1 0 or = 0 . YN + 3 fou n d by s ol utio n of Eq . ( 1 0 3) with Y N + 1 = YN + 2 1 2 collocation p oi nt s are used , the optimum ( xA , x B ) ( 0 . 7 5 1 4 , 0 . 3 2 3 2 ) for 4> 1 10 and a. = 1 is fo u n d wi t h four- t o fi ve - di gi t a c c u r ac y . T he advantages of the p rese nt method are easily summed up : no un necessary computation is spent in solving Eq. ( 97 ) " e x ac tly " at every in t e gr a tion step , an d all components Y 1 , y 2 , , YN + 2 • YN+ 3 are calculated with a balanced accuracy . For � 1 -+ oo at a fixed value of ex , the pellet equation cannot be solved An asymp totic solution i s , how by collocation a s i t stands i n Eq . ( 97 ) . eve r , easily derived u si ng the following transformation . Let u !f> 1 ( 1 - (;) and rewrite ( 97) as •
.
•
=
=
=
.
.
•
=
2
£....¥. d
du
- exp ( - u )
=
2
with y ( u
=
0)
=
- �� -
du
+
a
XB / (1
-
2
(1
-
x )y A
2
XA ) an d y ( u
( 1 0 7)
-+
oo
)
0,
( 1 08) u=O
The b o u n dary condition at " u -+ oo " can be applied at two different u values , say u 5 a n d 8 , and Eq . ( 1 0 7 ) is solved by collocation for each of these va l u es . If there is no apprecia ble change in dy I dul u = O , the solution i s ac ce p t ed ; otherwise, u > 8 i s tried . Alternatively , one may make a secon d transformation to obtain a finite in t e rval : =
v
=
( 1 + u)
-2
where v
=
1 at u = 0 and v = 0 at u
-+
co
( 1 0 9)
N umerical Methods
X
B
in
31
Reaction Engineering
=1 CA0
.8
.6
.4
.2
XA
= 1 - ..,.A. c
"Ao
k 1 /c A okz and of $ = FI G U RE 3 Maximum yield of B as a func tion of s 2 Lp ( k 1 / D }t . C urve s for 4> = 0 , 1 , an d oo ar e shown , and poin t s with s 2 0 , 1 , . . . , 10 are con nected . C om p ar e with Fi g . 7 . 7 of Leven spiel ( 1 9 7 2) for consecuti ve first - order reac tion s , $ 0. =
=
The tran sformation ( 109) is well c hosen in the present ex a mple sinc e fro m Eq . (10 7 ) it is e asily see n that y "' u - 2 fo r lar ge u . Fi gure 3 sho w s the boundary c urves $ 1 = 0 an d 4> 1 + for t he o p timum ( x A , xB ) as a fun c ti o n of 1Ja.2 [ w hic h is c alle d s 2 in the classic al p aper of A dashed line indi Wheeler ( 1 95 1 ) on first -or der consecutive re actions ] . cates the locu s of the $ 1 1 op timum solution s , an d at seven different value s o f 1 /a.2 ( 1 0 , 5 , 2 , 1 , 0 . 5 , 0 . 2 , and 0 . 1) other dashe d lines connect solutions with t he same value of 1 /a2 . The solution w as found by the technique ( 10 3 } to ( 1 06 ) , usin g E q . ( 97) to obtain the collocation equations T he asy mp totic solution
=
=
0
0
Villadsen and Michelsen
32
d 2z
Y
2 d 1;
with z
o
2¢>�ch 1
=
Y o < z;== o >
dz
1
=
( 1 10)
- x A ) y zY o Yo
d r,:
and
I
0
=
0
Havi n g obtained the correct value of Y o for a given ( xA , xB ) , a sin gle in t e gration of E q s . ( 97 ) and ( 1 1 1 ) from � "' 0 to � "' 1 gives the sensitivity
of y with respect to
(l
From the
x
A:
2 2
2
+
XA ) y zXA
2( 1
comp uted values of zx
]
z
X
A
( r,: = 1) and d z x
A
xA.
( 0)
dz
=
XA
� .,
0
=
/d � l _ 1 , one may obtain r,:A First , c alculate dy 0 /dxA ,
the second derivative of xB with respect to equivalent expressions for dy ( l ) :
using the followin g
ay l A
dy ( l ) = a x
1
dx
A
+
a
axy l
or
dy ( l)
=
[
z
x
A
( 1) +
z
Yo
( 1)
B
1
dx
dy 0 ]
dx
-
A
B
=
l
(1
xB
( 111)
0
- XA )
2
+
1 1 -x
dxB
]
--- --
A
d xA
dx
A
( 1 12a)
dx A
( 1 12b )
from which E q . ( 11 3 ) is obtain e d :
=
( 113)
Next ,
( 114) an d fin ally ,
( 1 15 )
33
N umerica l Met hods in Reac tion Engineering w here f ( xA ) 1 ( 114) .
=
- ( cp 1 t anh
cp 1)
-l
dy /d r,;
l
r,;= 1 • and f
A s a comment on t he p rocedure of E q s .
( 97)
1
< 1>
and (
< x ) is found from 1
1 10 ) to ( 1 1 5 ) , one
mi ght add that for explicit inte gration of the di fferential equation ( 97) , it is computationally si mpler to fin d
YO
by the sec ant met hod .
exp lici t inte gration it m ay be pre ferable to calculate z turbation o f E q .
( 97) rather than by E q .
( 111) .
XA
Also , for an
by numeric al per-
The comp utational cost is
the s ame and the computer program is simpler . Very difficult catalyst pellet proble m s have recently been solved by in genious forw ar d inte gration technique s : nonisothermal p ellet s ) ,
K aza et al . ( 1 980 ; methan ation on S un daresan and A m undson ( 1 980 ; diffusion an d re
action in a bou n d ary layer surroundin g a c arbon p article ) , Villad sen ( 1 9 8 3 ;
and Holk an d
absorp tion followed by e xothermic reaction in a liquid film ) .
Other examp le s are given in Villadsen and Michelsen ( 1 97 8 , C h ap s .
5 and 9) .
We m ay conclude that t h e model for a single reaction on a c at aly st pelle t can be treated by a standard numerical approach :
finite - difference methods ,
collocation , or for w ard inte gration from t he center of the particle , c ase may be .
as the
E xamp le s with several indepen dent reactions will frequently
require a numeric al technique w hic h is tuned to the specific p roblem .
Ex
ample 5 has shown some o f the methods that can be used to construct an efficient numerical solution , an d t he refe rences given above illustrate other techniq ues .
As a fin al comment it should be acknowled ged that the catalyst pellet model is virtually the s ame as the model used to calculate absorption with chemic al reaction .
I n particular , t he catalyst effectivene s s factor is closely
relate d to the enhancement factor w hich is used in the film model for chemi cal ab sorption .
The design of m ulticomponent countercurrent gas - liq uid absorbers [ o r
movin g- bed reactors , as i n Peyt z et al .
( 1 98 2 ) )
presents some nasty n umeri
c al proble m s - far more difficult than those encountered in the boun dary value problem s as sociated with catalyst pelle t s .
A set of exit concentrations
in the gas phase must be gue s sed , and after in te gration of the gas - phase
and liquid - p hase mass balances t hrou gh the column , one makes a comp arison with the inlet gas composition to o b t ain an iterative c alc ulation procedure .
The differenti al equations are often very stiff , an d it requires careful program min g to achieve a st able iteration .
S tead y - S tate T ubular or Fi xed - Bed R eactor Mod e l s Detaile d discussion
of
reactor models appears i n other chapters of thi s book ;
our task is to brin g up some of the common techniques w hich are applicable
in a numerical study of t he models . For this p urpose we need only one mass balance ( 1 1 6 ) and the energy balance ( 1 1 7) . A xial diffusion terms c an b e ne glected i n t he steady - st ate model since they are sm all comp ared to the convective term s . v v
z av
ay a z
=
LD 2 r v t av
_!X .1._ ax
( b:) - � x
ax
V
av
R
( 1 16)
34
Villadsen a n d Miche lsen
v v
a e az
z
-
av
where
2
=
r v pc t ac p
X ax
(x �!)
len gt h L
x
L ( - t. H ) c
+
ive
r a di al coordinate r el at
v
av
pc T p
r
R
( 1 1 7)
r
to t he tube radius rt
flui d - phase re actant concentration and temperature relative
y,e
t o a reference state with concentration c
T
R z
� 'D PC
a
1
axial di s t an c e in r e ac tor relative to the to tal re act or
z
v
Lk
r
and temperat ure
r
==
reaction rate divided by t he re ference concentration c
==
radial velocity distribution in t he reactor tube
==
radial diffu sivity of he at and mass
r
p
I n a " o o neo u s " reactor model R is a fu nct io n of fluid - p hase properties only ; the tubular reactor with or without an inert solid p ackin g is d e s c i bed by this typ e of model . T he c at aly tic fixed- bed reactor is describe d by a " hetero geneous" model in whic h R i s a fu n c t on of particle phase temp erature and concentration , but the se extra variables do not appear explicitly , and in t he numerical treatment of a c a t aly tic fixed- bed reactor mo d e l , the pellet p roblem is treat e d separately from the fluid-phase m od el [ E qs . ( 1 16 ) an d ( 1 1 7) ] - an i d e al case for solution by p artitio n in g . T he in fluence of pellet - p hase variable s on the tot al model appe ar s only in te r m s of the effectivene s s fac tor Tl , an d R i s an implicit function of the flui d - p ha s e variabla s y 9Jld e . C alculation of Tl m ay be more or le s s com p c at e d , r an gin g from the sol u t o n o f simp le al gebraic equations for a sur face reaction on impervious pe l e t to the solution of co u p le d nonlinear boundary value p roblem s , as in e a p e s 4 and 5 e ar er in this section . I n all ci rc u m st anc es one s hould try to d uc e the complexity of t he pellet model as far as possible ( Examp le 4 sho we d us how far one can get in this re sp e c t wi thout s a ri fic i n any i mp ortant feature of the pellet model) . The fluid - phase model is itself i t e complicated , an d the in e c e of re actor m acrovariable s (e . g . , the lar ge temp e r ature gradients which o cur radially reactor ) can be studied with sufficient accuracy in the u be s of a re fo r min without too many details in the pelle t - p hase desc ription . I f the r a e of e a t o n is moderate , the radial diffusion terms are small comp ared to the axial gradients , and one m ay r e u c e Eqs . ( 1 1 6 ) and ( 1 17) to a one - dimensional model . T here are c e rt ainly situation s w here this approximation may lead to loss o f major fe atures of the model , but the potential re d c tion in computer expenditure is so s i i ic an t that we feel it n ec s s a y to comment on p rop er aver a gi n techniques for t he radial gradients before we discuss suitab le numerical techniques for solution o f he fu ll od e l .
h m ge
r
i
li
i l s xm l
c
t
t
e
r
li
re
g
qu g
flu n
r ci
c
d
u
gn f
g
t
m
A v e raging o f t he S teady -State Mo de l over t he C ross Section of t he T u b e
r
D
ep
t
A ssume that t he p hy sical p rop e ti e s k I PCp and are i n d en d e n of x an d i nt e gra t e over t h e c ros s s t on of t he t u b e t o obtain t w o couple d ordin ary
ec i
N umerical Methods in Reac t ion E ngineering differential equations reactor position z :
� where R dO dz
L v av
==
dz
{ 1 18 )
a n d ( 1 1 9) for t h e average s of y a n d 0 at
35
( 1 1 8)
R
2 1 J R dx , which is approximated by R ( y , O ) : 0
=
{ 8w -
2LU r v pc t av p
==
0x==1)
L ( - ll H ) c +
v
r pc T av p r
R
( 1 19)
w here O w is t he value of 0 at t h e reactor w all and Ox:: 1 is the value of 8 just inside t he w all . In the same m anner in w hich we introduced a modified pellet heat tran s fer coe fficient in E q . ( 9 1 ) we shall define a modified w all heat tran s fer coe ffic i e n t U : a e ax
- k r t
I
x== 1 =
- 8 ) U{ O w x- 1
U{ Ow
_
( 1 20)
Expre ssion ( 1 2 0 ) is inserted in E q . ( 1 1 9) an d n o w only e and y appear i n the approxim ate reactor mo del , which c an be solved by inte gration from z = O to z = l . It remain s to calculate U in terms of U an d the radial heat conductivity k. In gene r al , U is obtained from ( 1 2 1)
where the con stant a depends on the radial velocity distribution . Villadsen and Michelsen ( 1 97 8 , C hap . 6) disc uss a pert urbation tec h n iq ue by w hich a m ay be c alculate d . T he value of a will fall between 1 / 4 for a flat velocity pro file V z = Vav and 1 1 / 2 4 0. 45 8 3 for the parabolic velocity p rofile of the Typic al result s between t hese values are laminar flow tubular reactor . =
19 48
a = - =
0 . 3958
for
�(1
v
_z_
v
=
2
av
4 - X )
an almost flat profile , and
a ::;
4 9 5b 2 + 2 3 4b + 31 1 2 0 ( 3b
+
1)
2
z
v for
v
=
6( 1 -
av
) (X 3b + 1 X
2
2
+ b)
For b = 1 / 4 this profile h a s a m1m m um at x = 0 an d a m aximum at x 'V 0 . 7 , a feature that is observed in m any experimental studies on packed - bed velocity distribution s . a = 0. 3277 for b 1/4. T he res ult ( 1 1 8) obtained by strai ght forw ar d averaging o f the m as s balance can be improved b y includin g one more t e r m i n pert urbation an aly sis from R 0. For a first -order , isothermal rea c tion ( R = k c /c ) R r =
=
Villadsen and Miche lsen
36
in a tubular re a c t or with p arabolic velocity distribution , one obt ai n s st ead o f E q . ( 1 1 8 ) ,
,
in
( 122) where D a = kRL /vav an d the numerical c ons t an t 1 / 48 is calculated by perturbation an a l y si s as discussed in Villadsen and Michelsen ( 1 97 8 , p . 2 7 1) . This result was first given in a famou s p aper b y Sir G eoffr e y Taylor ( 1 953) . Q ualit atively s p e akin g he in te rprete d the radial diffusion term in E q . ( 1 16) where D is now a true molecular di ffu sivi ty in terms of fic ti tiou s " axial dis p er s ion term" in a one - dimension al model : ,
-
� dz
=
_:£__
k L v
y
+
av
2d y 2 dz
1 __ Pe ef
( 1 2 3)
w h e re
Pe
1 92
=
ef
=
Equation ( 1 2 3 ) m� be so l ve d usin g the simple " semi - in finite" bound ary condition ( e . g . , y = 1 at z = 0 and finite for z -+ oo ) or with the more complicated boundary condition y ( O) +
l
1 dy Pe d z z= O f e
--
--
=
1
and
dy dz
l
=
0
z= 1
I n b ot h cases , one obtain s y ( z = 1 ) = exp
[ 0 p�:J J -Da
2
+
0 (p�:J
( 1 2 4)
w hic h is t he same result as that obtained by pert urbation analysis ( 1 2 2 ) . T he final result ( 1 2 4) is well known from e l em ent ary textbooks in re action en gin e e rin g [ e . g . , Levenspiel ( 1 97 2 , pp . 2 8 3 - 2 8 7 ) ] , where a " di spe r sion number" 0 v
ef = L av
1 192
=
is u sed to correct "near plug flow " data obtained in a tubular re actor . T he improvement is substantial , at least for sm all values o f t he disp e rsion Thus a t D a 1, w here Eq . ( 1 1 8) yie l d s the result y( z = 1) = number . exp ( - 1) 0 . 36 7 9 re gardles s of the contribution from t he radi al dispersion , one obtains the followin g results from Eq . ( 1 2 4) : =
=
37
N umerical M e thods in Reac tion Engineeri ng D 1 92 � v L
1
av
y by E q . ( 1 24)
y
by a " true model"
10
5
2
0 . 37 5 2
0. 3818
0 . 3892
0 . 417 9
0 . 3750
0 . 3809
0 . 3919
0 . 4038
""
( 0 . 443)
The bottom line of the table is calculated by hi gh - order collocation- as described belo w - applied to Eqs . ( 1 16 ) for a first - order isothermal reaction 2 2( 1 - x)
� az 1
y(z
=
1)
=
L av 1 1 9 2D f e
v =
a
; ax
2 4( 1 - x ) y ( z
0
=
( �) X ax
- Da y
( 12 5 )
1 , x ) x dx
The case Def -+ oo ( w hich m ay be interpreted as D "' 0 ) corresponds to a comp letely segre gated flow with no cross- sectional mixin g caused by radial gradient s .
N umerical Solution o f S teady -State Reactor Mo del
When E q s . ( 1 1 6 ) and ( 1 17 ) are discreti zed in the x direction , there appears a set of couple d ordinary differential equations (j = 1 , 2 , . . . , N) :
� = dz d6 j dz
T
M -J
a.. . C .
l. - - R ( y . , 6 . ) L
v
av
l
l
( fi T ) =
T
( 126)
m ax
( 1 2 7)
r
T he coefficients Cji are defined in E q . ( 7 0) . T he y have different values dependi n g on the method of discreti zation . Ordinary finite - difference methods , global collocation , G alerki n ' s method , or spline collocation have been used by various authors in numerous comp uter studies over the last 2 0 to 2 5 years . Equations ( 1 2 6 ) and ( 1 2 7 ) can be solved from z 0 by any of the standard p acka ge s for coupled initial value p roblems . A n imp licit or a semi -implicit method is preferable to an exp licit method because of the lar ge spread of the ei genvalue s of Q. T hi s is true in particular when ortho gonal collocation is used to discreti ze the radial derivative ; even for a relatively low order method ( N = 4 or 5) the ratio between the lar gest and the small est eigenvalue can be of t he order of 1 0 0 0 . T he structure on the right hand sides of Eqs . ( 1 2 6 ) and ( 1 2 7 ) is , howeve r , so simple that the Jacobian can be comp uted very easily . It is al so pos sible to disc reti ze E q s . ( 1 2 6 ) and ( 1 2 7) in the z direction to obtain a so -called doub le - collocation method . This method w as origi nally proposed by Villadsen and S�rensen ( 1 96 9 ) , but it w as only recently =
Vi l ladsen and Michelsen
38
elaborated into efficient computer codes by S�rensen ( 1 9 8 2 ) . T hese codes appear to be p articularly suitable w hen applied in a parameter estimation problem .
A sit uation that lends itself q uite n at urally to solution by double col location is t hat of a w all - c at aly zed c hemical reaction : the rate term appears in the boundary con dition of an otherwise linear partial differential equa tion [ Eq s . ( 1 1 6 ) and ( 1 1 7 ) without the rate terms] . Michelsen and Villadsen ( 1 98 1 ) give a det aile d discussion of this p articular type of chemical reactor model , w hich m ay be solved either by double collocation or by the S T IFFALG routine [ Eq s . ( 36) and ( 37) ] ; there is one nonlinear algebraic equation ( t he boundary con dition ) and N linear differential equations which can be trans formed into the diagonal form ( 18) and used in this form thro u ghout the calculation T he solution of E q s . ( 1 1 6 ) and ( 1 1 7) for a rate exp ression whic h is linear in y an d 9 ( or has been lineari zed from a given reference state) deserves p articular attention . C rank ( 1 95 7 ) has discussed the multitude o f practical proble m s in re action en gineerin g which are generated from t he same b asic equation , the linear " diffusion equation . " N umerically , all these different problem s are han dled by the same general techniq ue , b ut with sli ght modific ations for different boundary conditions . Let us use a linear mass balance as our " standard p roblem " : •
v( u )
�
_
az -
4 ( 1-s) /2
u
....£..._ ( au
u
( s+ 1 ) / 2
lz.) _ Da y
( 128)
au
where u W hen s = 1 an d v ( u ) i s x 2 and s = 0 , 1 , and 2 . velocity distribution , we have a steady - st ate tubular reactor When v ( u ) = 1 and z is interpreted as contact time , we have transient pellet proble m . Here we use $ 2 = kR rp 2 !D in stead vav as t he dimensionless group of parameters . Equation ( 1 2 8 ) is discreti zed by N t h - order collocation : =
the radial problem . a linear of Da = k L I R
( 129)
�
v ( llj ) , an d the ( C , b j ) are given in E q . ( 7 1) . ¥ is dia gonal with V H Di fferent side con ditions at u = U N + 1 = 1 lead to different problem modifications . _1 Diagonali z ation of M = Y ( Q * - D aD yields the standard form ( 1 8) . Michelsen and Villadseii ( 1 98 1 ) prove t hat all ei genvalues of M are real and distinct if collocation is m ade at the zeros of an d the boundary =
condition at u
( 1957) .
=
P�O , O) (u) ,
1 is any of the linear expressions discussed in C rank
T he solution appears as an N - term expansion in ei gen functions exp ( Aj z) , and t he modi fied Fourier coe fficient s are found by simple m atrix- vector m anip ulations [ see Villadsen and Michelsen ( 1 978 , C hap . 4 ) for the solution of a p articular example ] . For the case Da = 0 , Michelsen ( 1 97 9) proves t hat the ei genvalues and ei genfunctions of Eq . ( 1 2 8 ) with boundary condition
� d du
Bi
M + -2
y = constant at u
=
1
( 1 30 )
39
N umerica l Me t hods in React ion Engi neering
be derived for any value of BiM by a simple algorithm that utilizes t he eigenfunctions obtained at any specific reference value which , for example , may be either zero or infinite . This leads to substantial savings in computer time when estim atin g t he value of BiM ( and a diffusi vi t y ) from a series of measurements of y at different and x . W h en Da f. 0 , t h e diagonalization of M must , however , be performed for each value of Da. Another important speci al case , that of a finite exterior medium , i s t re ate d in Sotirchos an d Villadsen ( 1 9 8 1 ) . The boundary condition at u = 1 is given by an overall mass balance can
eigenvalues an d (BiNI >r ,
z
-
b z
q y ( )
+
l y( x , z)
s 1 dx +
0
=
a
-
(
Jz J1 0
0
R
dx
8+ 1
)
d z'
=
a -
f
( 131)
Outside the reaction medium of volume 1 (or the catalyst pellets as the case may be) there is a radially w ell mixed volume q into which the react ant m ay diffuse . The reactant concentration is Yb in the finite exterior medium at position (or at contact time t) . In Eq . ( 1 31 ) the total "mass" in the re actor and the outer medium is equated to the initial "mass" les s the total amount of reacted material. Concentrations in the two media are connected by a boundary condition of type ( 1 3 0 ) where the right - han d - side "con stant" = (BiM /2)yb ( z) . Let v( u) = 1 and write the jth collocation equation , z
a,
dy . J dz
N
I:
i=1
b.
]1
_l
c .. y.
q
1
where f is given in
f-
R(
y) ]
.
+ b.
Eq . ( 1 3 1 ) .
- b
J
1
0
For a eq
an
y dx
s+
1
N
=
L 1
�
( 1 32)
] q
j
( 2AN+1 ·) w. q
___!
+
1
J0
R dx
,1
w1. y1.
nonlinear reaction rate the N e q uations uation for t he average rate of reaction :
df dz
B\J
s+l
=
( 132)
are solved together with
N
L w.R(y ) 1 1.
for a first-order reaction the numerical approach Eq . ( 1 2 9) :
but in
( 1 3 3)
1
is j ust
like that used
Villadsen and Michelsen
40
dY =
dz
y
=
� -y
+
b*
(y 1 , Y2 ' ' ' ' , yN '
f)
T
,bN '
b*
�
(¥ =
- Da� T D aw
( 134)
0 }T
E q . ( 1 3 2 ) to ( 1 9 8 1 ) have been corrected . o ( 1 34 ) can
In
s
( 1 34) the unfortunate misp rint s in Sotirchos and Villadsen A number of important ap p li c ation s of Eq . ( 132) , ( 133) , r be easily listed : Ab sorption o f a reactant gas from a gas phase and reaction in the liquid phase [ gas flow rate va (m 3 /h) , liq uid flow rate vL( m 3 /h) , and q vL /vo l . 2 . Determination of reaction rate constant and pellet diffusivity by 1.
=
3.
concentration me asurements in a finite volume of well stirred liquid in w hich the c a taly st particles are sus pen d ed . All reactant is ori gin all y in the fluid outside the particles . Analysis of membrane reactors : a certain amount of reactant v( m 3 /h) flow s in a tub e coated with a catalyst l ayer of thickness o . R e ac t an t diffuses into the catalyst layer and reacts there . It is desired to c alculate t he len gth of t u b e required to obtain a c e rt ain average concentration in the flowin g liquid .
As a final illustration of the ge ner al numerical approach to linear steady state reactor models , one may consider Eqs . ( 128) and ( 130) with the addi tion of a s mall axial diffusion term ( 1 / Pe M ) ( d 2y / a z2) on the right - hand side of E q . ( 1 2 8 ) . T he p roble m has only ma rginal in terest in indus tri al reactor design ( radial diffusion may be accounted for by a fictitious axial dispersion term as discussed earlier , but an axial diffusion term per se is almost always insi gni fi c ant ) . A ddition of the second derivative with re spect to z in the p ar tial differential equation ( 128) does , however , lead to formidable numerical complications which have intrigued many authors [ see , e . g. , Papoutsakis et al . ( 1 980) or Michelsen and Vil lads en ( 1981) ] . It is interesting t h at wit h the exception of the inlet zone of the reactor , one may study the influence of an axial di ffusion term ( 1 /Pet\11 ) ( a 2y I a z 2 ) on the solution of E q . ( 128) , using only the eigenfunctions and eigenvalues o f the radial diffusion operator . The technique and its application to the "extended Graetz problem" discussed here and to the much more interesting asymptotic stability analysis of catalyst pellet mo dels are developed in Villadsen and Mic hels en ( 1978 , Chap . 9) . Every one of the numerical methods that have been described in this section have been based on an approximation of the x prof'J..le by a poly nomial . High -order methods such as collocation will not be able to handle th e near discontinuitie s t hat appear in t he profiles close to the reactor entrance ( t he inlet concentration m ay be 1 at all interior points an d zero at u 1 for z 0+ ) . Spline collocation with a spline point that gradually moves away from u = 1 as z incre as e s has b een successful in the computa tion of "penetration front s in the inlet zone of the re actor . One example =
=
"
N umerical Metho ds in Reaction Engineering
41
is given in Holk - Nielsen an d Villadsen ( 1 98 3 ) , and other exa mples are shown in Villadsen and Michelsen ( 1 9 7 8 , C hap . 7 ) . Another , qui te different but apparently versatile approach is to intro duce a variable trans formation n = f( z , x ) w hich as far as possible describes the combined influence of z and x on the solution near z = 0. T his is called " similarity transform ation" and i s treated in most textbooks on partial differential equations . T he reduction of the model into a problem with only one independent variable succeeds only in trivial cases , and one will usual ly en d up with two independent variables n an d z in the transformed equa tion as well as in the boundary conditions . I f , however , n has been judiciously chosen , a perturbation analysis of the transformed equation from z = 0 will give an accurate solution of the original problem in the inlet zone . T here are several examples in Villadsen and Michelsen ( 1 97 8 , C hap . 4) where this technique has been used to solve linear reactor problems , and a pellet model with a concentration dependent diffusivity is treated in the same re ference p . 3 3 9 ) . U n steady - S tate Fi xed- Bed Reac to r Mod e l s
Except for re gions of hot spots , the steady - st ate axial temperature and concentration p rofiles in a fixed bed are smooth functions of z , and the radial profiles can almost always be represented by low - order polynomials if they are not averaged aw ay as discussed in the preceedin g section . Thus ste ady- state reactor simulation is a rat her modest affair on a hi gh speed computer , requirin g at most a couple o f seconds comp utin g time to obt ain e ( x , z ) and y ( x , z ) . W hen a time derivative is included in Eq s . ( 116) and ( 1 17) there is a fundamental change in the nature of the solution . Sharp concentration and temper ature front s m ay be formed , and these front s move slowly through the reactor . To simulate the dynamic response of the reactor to a control action in the inlet ( e . g . , a jump in reactant concentration ) , one has to fol low the reactor profiles o f the dependent variables through many thousands This , of course , means t hat the amount of of fluid - p hase time constants . computation incre ases drastically , and 2 5 to 40 sec of computer time per simulation is not at all extravagant . Q uite apart from the cost of the com putation , the time it t akes to make a computer sim ulation of the unsteady state reactor makes it very difficult to use the result , for instance , in a computer control of the reactor . A compromise must be m ade bet ween the demand for accurate modelin g of the reactor and a reasonable computational effort . T he literat ure on un steady - st ate reactor sim ulation offers a bewilderin g array of model simplifi cations and numerical s hortcuts . Some of the simplifications are j ustified and should certainly be generally accepted , but others are downright silly and will lead to order-of-magnitude errors in , for example , the b reak through time of a cat alyst poison . T hus for gaseous reactants it is reasonable to ne glect accumulation terms in the reactor fluid-phase mass and energy balances and in the pellet mass balances . T he time constant for convective transport of mass through the reactor and for diffusion into t he pellet are order s - o f- m agnitude smaller than the velocity of t he reaction zone . T he pellet temperature is taken to be independent of position in the pellet ( as in Examp le 4 ) , and while the pellet time constant for ener gy transport is certainly larger than the time constant for mass transport , it is still much smaller than the thermal
42
Villadsen and
Michelsen
residence tim e for the bed , an ob servation that may justify a fur th er as sumption that Bp = e g at any time . To reduce the dimensionality o f the proble m , it is common practice to ne glect t he cros s - sectional conductive term in Eq . ( 1 1 7 ) w hen solvin g unsteady - state reactor models . This m ay be justifiable w hen the diameter -to-len gth ratio of the bed is larger than 2 0 , b ut a r adial gradient c an also be incorporated as a fictitious axial dispersion term and there is experi mental evidence that true axial conduc tive terms m ay have to be taken into account in modelin g of transient fixed bed reactor behavior . It is usually dan gerous to t amper with the rate expression to obtain comp utational advant ages . If it is reasonable to suspect a substantial re sistance to mass transfer in the pellet s , it will not do to ne glect this dis persin g effect , since breakthro u gh times that are much different from those observed experiment ally are likely to be c alc ulated . Our discussion of the pellet proble m for a sin gle reaction has also shown t hat a complicated rate expres sion is treated with almost t he sam e e ffor t as a simpler rate expres sion . T herefore , it is usually not advisable to j eopardi ze the trust wort hi ness of t he simulation b y ne glectin g the in fluence of one o r more reactants on the rate of reaction . Unstead y - state reactor models display such a variation in complexity that it b eco m es impossible to discuss s uitable solution techniques in general . At one end of t he spectrum one fin ds t he one - component isothermal gas adsorption unit w hich can be treate d analytically ( Aris and A m undson , 1 97 3 ) . At the other en d of t he spectrum there are studies of reactor con trol with several independent variables , steep temp erature front s , and with time - chan gin g kinetic and transport p arameters . Simulation of reactors for reduction o f mineral ore and simulation of explosion fronts are other com plex problems . An example that illustrates major feat ure s of unstead y - s t ate reactor behavior without bein g very complicated is discussed in Michelsen et al . ( 1 97 3 ) and Villadsen and Michelsen ( 1 978 , C hap . 9) : The reaction t akes place in t h e fluid phase , but the inert p ackin g m aterial of the reactor in troduces a l ar ge thermal residence time . A fter linearization t he model is analy zed in term s of transfer functions , u sin g collocation to account for the axial variation of the variables y and e . In the present chapter we use an example of moderate complexity to illustrate certain numerical techniques which we believe can also be applied in many other situations . Examp le 6: Simulation of co ke b urning in a fixe d - bed reac to r A re formin g cat alyst is slowly deactivated due to deposition of tarry m aterial in t he pore s . T his so -called " coke" has to be burned o ff periodic ally , and a reliab le sim ulation is important to minimize the len gth of the burn - off period without sinterin g the metal cryst allites by excessive overhe atin g of the pellets . O ri ginally , t he coke is homogeneously distributed on t he particles wi th concentration c00 • Nitro gen co n t aini n g a small proportion of oxy gen ( concentration Coi ) is fed to the reactor inlet at temperat ure T gi . A reaction zone develops and moves slowly throu gh t he reactor . When the reaction zone passes out of t he reactor the b urn is complete , an d it is succes s ful if only a sm all residual coke concent ration cc ( z) is left behind and if the temperature at no time has exceeded T max anyw here in the re actor . Durin g the operation of t he reactor a temperature wave p asses through t he b e d . I t m ay move faster or slower than t he reaction zone can be used to control t he relative the inlet oxygen concentration c oi
43
N umerical M e t ho d s in Reac tio n Engineeri n g
Both front s move exceedingly slowly comp ared velocities are 0 . 1 m / h fo r the r e ctio n zone and 0 . 9 m /h for the temperature front . A model with three equations [ ( 1 3 5 ) to ( 1 3 7 ) ] for the pellet p hase and two [ ( 1 3 8) and ( 1 3 9) ] for the phase w as proposed by Liaw et al . ( 1982) .
movement of t he t w o front s .
to the gas residence time in t he reactor .
a
Typical
fluid
y
c
c
-
=
c
c
co
=
1 at t
=
0
( 135)
( 1 36 )
1
t
ae ----2. ax
H
=
H <e - e ) + a Da n y p g p g
- .-£. p T . T
e
gl
e
T
g
=
_g T gi
( 1 37)
c
-Da ny ae
�z
g
y
g
..EK = c .
( 1 3 8)
01
H (9 - e g) p p
=
a
( 1 3 9)
The many in E q s . ( 1 3 5) to ( 1 39) ( t R c h r act eri s t ic reaction time , t H = therm al residence time , Hp = fluid -to-pellet heat tran s fer units ,
constants
=
a
burned)
is
z and
a also used
The inlet gas s
the e ing
c
n
T
the
in
gradient
kcc
phase ,
44
Villadsen and Michelsen
less justified assumptions t hat have been made in p revious s t u di es o f the p roblem . T he present model does not assume " a constant burnin g rate , " low -temperature bu rnin g " ( where cp � c g ) , or " di ffusion - controlled b urn in g" ( where t he rate is almost independent o f te mper at u re ) , and con se quently the p redicted b urn t is close to t hat o b t ained in lab or atory e xp e ri ment s . T he model has recently been critically reviewed from a n u m eri c al poi n t of view by F un der ( 1 9 8 2 ) , and as a result the computation time per burn out w s reduced from about 5 mi n in Liaw et al . ( 1 9 82 ) to a o ut 35 to 40 s . We shall re vi ew some of his conclusions . 1. It is me nin gle s s to si mp li f the model as sumption e g 9s = e . T his i s fre q ue n tl y done in n u meric al st udies o f u n stead - st te reactor models , an d the resultin g " h omo geneous " model with
ou history
simulation a
b
a
y
ae az in ste ad o f E q s .
=
1 t
H
by the
y
=
a
( 1 40)
SD a n y g
( 1 3 7 ) and ( 1 3 9} is amenable to s o lution by t he met ho d o f
char acteristics u s i n g increments related by
dt
T here
a
=
t
H
may
de
dz =
( 141)
SD a n y g
b e about 1 0 % s a vin g in comp uter time w hen this app roxim ation
r
is m de , but si gnificant errors in t he tem p e at u r e peak may
also by
occ ur .
2. T he implicit th re e - point inte gration scheme su g ge sted C hristian sen and An dersen ( 1 980) i s ve r s uit able for inte gration of the fluid b alances in a re c t an gul ar grid . T he pellet mass b alance is approximated 3 or 4] , an d the resultin g co llo c atio n [ usin g PN ( O , O ) ( x 2 ) wit h N equation s for Y g • an d { ( Yp • Y c > ; j = 1 , 2 , . . . , N } are solved a Newt o n method for each grid poin t . A ll z values at a given t are t r eate d before t i s increased . where 3. Fun der ( 1 9 8 2 ) poi nts to an i nte r esti g model approxim ation : as it i incorrect to a ss u me a c o n st ant the p ellets ( e xcept w hen the pellet is far from the react io n zone) , it is q uite reasonable to assume a const an t coke level or a coke profile t hat ch an ge s in a few step s . W hen Yc is assumed to be pie ce w i s e constant in t he pellet , it is possi ble to solve t he pellet model analytically , and in c e n is t he only "output" of the pellet m a s s balance , t he problem is reduced from one in 2N + 3 vari ables to ( ideally ) one in four or five variables . The resulting red u ction in c o mp t e r time is very si gni fic an t ( from 35 to 4 to 6 pe burnout simulatio n ) an d the result s are only a few percent age points different from those obtained by the full model , probably b e c a use the sharply decreasin g oxygen profile in the p e lle t i s very little influenced t he coke profile as lon g as t he correct average coke concentration i s u s e d . C haracteristically , the solution of the unsteady - st ate reactor model is smoot h except in t he reaction zone an d in the temperature front . In Exam ple 6 the oxygen concen t r ation is zero down stre am from the re ac t ion zone , and it incre ases rapidly to the inlet value c at po ition s upstream from the 0i reaction zone . Glob al colloc ati n doe s not give a satis facto r y represent ation of the p rofile when two , more o r less flat p art s are joined a st e e p front .
y
by
eg , � ·
by
=
n oxygen level in
s
s
u
s
by
s
o
by
r
N umerical M e t hods in Reaction Engi neering
45
Spline collocation is a possibility , but a suitable al gorithm must be con structed to move the zone with m any collocation points at the same speed as the reaction zone . With several reacti n g species an d an energy balance , the front s will move with different speeds , and unle s s the program auto matically follow s eac h o f the fronts , spline collocation will also be imp ractical . An ideal code should :
1. 2.
Ensure that eac h front i s followed by a sufficient number o f nodal points . Move nodal point s in and out of the fronts - not too fast ( since otherwise all nodes would be sucked into t he front s ) and not too slow ( the fronts will be inaccurately represente d )
W hen t he profile s are given i n terms of Yj ( t ) at positions � ( t ) [ e . g . , between ( Yj , Zj ) and ( Y + 1 • �+ 1 ) 1 , one j must req uire that ( Yj • Zj > quickly settle down to become smooth functions of t. This cert ainly requ1res that condition 1 is satisified . I f a number of nodes pass quickly throu gh the front , the profiles ( yj , Zj ) will oscillate wildly with t . Since the e quations that determine t he ( Yj , Zj ) are surely goin g to be "stiff , " an implicit or semi -implicit inte gration method in t must be used . T he most p romisin g recent develop ment tow ard a general al gorithm for solvin g problems wit h steep front s ( or schocks ) is due to Miller an d coworkers ( Miller an d Miller , 1 98 1 ) . T heir method employs a complicated system of " sprin g force s " and " viscosity forces" to move the node s - q uickly
as a set of piecewise linear functions
but not too quickly-in and out of the fronts . Comp utin g times ( with an implicit Gear m ethod to solve the differential equations for the time develop ment of the profile s ) appear to be rem arkably small . We have used the in formation given in t he published articles to test their method , both on B urgers' eq uation ( t heir test example ) and on a chrom atographic separation example . Our comp utation times are , how eve r , three to four times as hi gh as stated by Mille r , but undoubtedly the method deserve s further study , since the b asic ideas seem to be j ust right for solution of some extremely complicated proble ms of great practical interest . PA R AM E T E R F I T T I N G I N
R E A C T I O N E N G I N E E R I N G MO D E LS
Up to this point we have presented various n umerical technique s w hich are suitable for reactor sim ulation (i . e . , computation of reac t ant concentration , temperature , etc . ) as functions of the independent variables in the reactor model . We shall now apply these techniques to determine a set of parameters in a postulate d rate expression , heat transfer model , or reactor model . At n specified values of t he ( possibly multidimension al) independent variable x , there are measurements of some or all the components of the M - dimen Sional state variables in the postulated model . The experiment al values are ( y 1 , Y 2 • · . . , yM ) k for k T hey are collected into an 1, 2, . . . , n. experimental state vector � . T he experimental values of � are s upposed to be free of exp erimental error , w hile the experimental error of � e is sup posed to b e normally distributed with a m ean o f zero ( this i s possibly not true , but it is often as good as any other guess , and the bias in the estimated E. that results i f the assumption is incorrect will be difficult to assess ) . =
46
Villads en and Michelsen
T he postulated model contains m p arameters ( P 1 • P 2 • p ) E.• and for a give n value of 2. the model can be used to c alculate vr:fi ues r. s of the state vector at t he exp erimental positions x . T he parameters will be determined by minimi zation o f an object function
•
•
,
=
( 142) The (n x n) covariance m atrix 1; for t he experiment al data is assumed to be diagonal ( uncorrelate d dat a ) -and the diagon al elements are I:ii V ( yi ) = =
the vari ance of the ith experimental point . � is assumed to be known 2 except for a scalar cr Differentiation of .p with respect to 2. yields the followin g algorithm for improvement of an initial estimate e,o for e,:
cr�, 1
where
T his is t he G au ss - Newton algorithm , w hic h can be applied iteratively until
< E: and 2. �. the maxim um likelihood estimate for E.· T he variance -covariance matrix y is obtained from the estimate g ( � for the Hessian matrix that result s after conver gence has been reached in Eq . ( 1 43) :
l t� EI
=
¥
=
E [ ( E - e_)T(E_ -
2
£) ] = s g
-1
(
�)
( 1 44)
where
cr
2
1 2 "2 tv cr = s = -- n - m
"" <2.>
( 1 4 5)
Gaus s - N e wton ' s al gorith m is at least linearly convergent , but it may fail to Most stan d converge even w hen excellent st artin g values 2. o are provided . ard codes for parameter estim ation with a n e xplicit object function e mploy one of t he several variants of Levenberg-Marquardt' s restricted step T he simplest version is t hat in which g + v!_ is used instead al gorithms . of g in E q . ( 1 4 3 ) , but an arbitrary positive - definite matrix m ay be used inste ad of the identity m atrix !_. At the start of the ite ration a rather lar ge value is chosen for the Levenberg p a rameter v , and v is gradually reduced when ll £ + _1!. T hus the method start s as a steepest- descent method and it may en d as a Gaus s - Newton method in the last iterations . It is rob ust and often works well , although it is not spectacularly fast . A good , recent description of Levenber g-Marquardt methods is Fletcher
47
N umerical Methods in Reac t io n E ngineering
( 1 980 ,
Chap s .
In the followin g w e assume that one of t he
5 and 6) .
Levenber g-M arquardt
lib r ary
. .
codes ( e g
, IMSL or a I. I a �
calcu l ate d values of I. s an d of it s gradient
tt
T he parameters to be determined fall in t w
1. 2.
H arwell ) with externally is available .
group s :
( R TD ) ,
Kinetic parameters ( rate constants an d equilibrium constant s )
R e ac t or
bution
p arameters [ re sidence - time dist ri
i
sectional velocity - and heat cond uct vit
t
tran s fer coe fficients , pellet - o - fl
dispersion , etc .
)
uid
y
cross
distribution , w all heat
transfe r parameters , axial
The measurements can be of wildly differe n t co mplexity .
in clu d e :
T h ey m ay
1.
Direct me asurement of the reaction rate at given conversion and
2.
Me asurement o f the average con ve r sion and temperature p ro files
space time ( recirculation reactor )
h
in t e axial direction of a
fixe d - bed
reactor , or profiles of con
version versus time in a batch reactor
3.
Radial an d axial temperature and in a
fi xe d - be d
reactor at
x ri ments
T he e asie st e p e reactor .
( p o ssib ly )
di fferen t
concentration pro files
value s of time
to analy ze are those o b t ai ne d in a recirculation
The experimental procedure is slow comp ared to batch reactor
experiment s - only one se t of outlet conc e n t r ation and temperat ure is found for each ste ady - st ate experiment - but in principle it is possible to obtain
a
complete pict ure of the rate expres sion w it h o ut w or ryin g about inter
ference
fro m
E xp erimen t s over the whole range of i gui sh as far as possible betw een differ for o xi d ation of C O , t h e pure first - order
t he reactor parameters .
conversion s hould be used to dis t n
ent kinetic p arameters .
T hus ,
behavior is seen for me as u rem en t s t aken at conversion close to 1 , while the
" ne gative
order kinetics" ( i . e . , the in fluence of CO inhibition ) is observed
n e i lar ge inlet CO concentration . I n tr ap ar ticle mass tran s fer resi st anc e is measured on pellets of differ ent si ze . Jackson ( 1 9 7 7 , C h ap . 10) reviews t he subj ect of di ffu si vit y me asurement s by steady - st ate methods as w e l l as by t r an sie nt me t ho d s . It may be safer to meas ure t he transport parameters un der re ac tin g condi tions . O n e o r more standard reactions with known ( an d simple ) kinetics may be u se d , but in m any st udies it has also been proved t hat kinetic parameters can be fou n d simultaneously with the p e llet m ass tr ans fer re sistance . Similarly , the pellet he at co n d u c ti vity c an be m e as u re d independently , followed by measurements of t he pellet - to - fluid phase transfer properties , radial heat c on ducti vit y , w all heat transfer coefficient , an d so on , in the packed bed op erated as a heat rec up e rato r . U n fortunately , different ex for small co v r s on and
perimental methods may
gi ve
( see , e . g . , Hoffm ann et al . ,
very different values of these parameters
1 97 8) , an d many authors doubt that the
parameters can also be applied for sim ulation of the reactor u nde r reac
condition s .
tin g
A word of warnin g s hou ld fin al ly be given c onc e r nin g recirculation re ri ment s : w hen the conve rsion is very hi gh , the recirculation ratio is likely to be too sm all , and for m e asu r em en t s at small co nve r sio n t he amount of catalyst m ay be too small to give a homo geneous bed - or t here actor e xp e
48
Villadsen and Miche lsen
may be severe s hort - circ uitin g in the reactor itself. Wedel and Villadsen ( 1 98 3 ) show some nasty systematic error s that m ay arise in the in t e rp re ta tion of recirculation reactor data e rror s whic h could not be avoided by including R TD me as ure me nt s in the data analysis . Still , as far a s possible , one s hould try to proceed as in dicate d above , usi n g different equipment to extr ac t i n for m at ion re g ar di n g different groups It is far more difficult , from both an experiment al and from of par amet e r s , a sim u l ation point of view , to determine kinetic and t r an s p ort parameters simultaneously . The mo s t prob able outcome is that the p hy si c al p arameters ( which fr e q u en tly have a m u c h smaller in fluence on t he conversion than the kinetic pa rame t ers ) will be d ete r mine d very poorly . M any authors (see , e . g. , Clement and J�r ge n sen ( 1 98 3 ) ) have ob ser ve d that the par am eters obtained by t he "buildin g-block" app ro ac h fail to give s ati s fac tory prediction of the reactor pe r for m an c e unde r reactin g conditions . H e n c e they feel that it is nec e ss ary to d et e r mine all ( or most ) of the parameters si m ult an eo u sly in or d er to give a reactor model that can be used for inter polation and for control purposes . One m ay argue that the failure o f the b ui l di n g - block method m ust indicate t hat there is som e misconception of the kinetic model or of the flow p attern in the reactor , and t h at these un certainties had better be re solve d . I f the re are errors in the model , the predicted parameters are of dubious value fo r lar ge - scale extrapolation , and for control purposes a very simple b u t s e l f - ad ap t in g algorit hm is s at is fac to ry if the di s turb an c e s are small in magnitude . The fin al goal o f the model work is to understand the un derlyin g physics and c he m ist ry of t h e syste m , and t hi s goal will not he re ac h e d un l e ss the complete reactor performance can be patched to ge ther by the build ing block met ho d . N u m e r i ca l
Techniques T he lib rary rou tine s for be g en er a te d extemally ,
parameter e s ti m a tion r e q uire that
E.
=
•
=
•
=
.
•
•
.
49
N umerical M e t ho ds in Reac tio n Engineering
TAB LE
2
S teps lea din g to .12. ( or
Recirculation
el
Plu g- flow ( or batch) reactor mod
b ackmi x )
reactor model
Model :
E;,
� :::: TR ( � . J2.) T ::::
Algebrai c
F
equation :
i D e ri v ati ve
T. 1
=
1
-
d E;,
1
E;,
i
=
.-
�
1
T. 1
E;,i
-
i
n
-
i
0
R ( E;, i , E._) :::: 0
1, 2,
dF.
of F . :
Iteration :
E;,i :::: - -
1
�
d r,;
R ( l,; , J2.)
(
dF
dF
E;, 1.
.
dE;,:
r
d This leatheds
For each i repeat the conver gence . Next , u p at e .12. usin g matri x ( a �J a �) :
=
d l;
F
i
E;, .
1
1, 2,
.
•
.
, n
1.
i
1 R.
1
=
: =
dFi /dE;,i ,
E;. .
1
+
R. F . 1
1
of F i , and � E;, i until = � s at t he given £ :::: E o · followin g procedure wit h a diagonal
calculation
to
�
Gradient of z:
for bot h reactor m o de l s
C ! t �. "
Finally , use Eq . ( 1 43) or
a
similar for m ula
s 12 ( : � ) �¥_ R
to update .12..
d r,;
50
Vil ladsen and Miche lsen
S till , a close analo gy exists i f t he differential equation model is approx imated by a discrete model from T = 0 to T m ax · T his can be done , for example , by global colloc ation . I f the colloc ation order N is hi gh enough , t he di sc rete model is an adequate representation of the original model , and the solution :l. sc is c los� to :l_ s · Hence , w hen t here is t he us ual no_! se in t he dat a , the estimate £. is not statistic ally inferior to t he estimate :e_ which would be obtained by an "exact" solution of the model . When t he N t h - order collocation model is solved to m achine accuracy , the sensitivitie s are calculated wit h only a s mall extra effort , and they are based on t he exact solution of t he ( approximate ) model at T i • j ust as is t he case for solution of the ( exact ) algebraic equation model whic h describe s recirc ulation reactor dat a . T his is import an t : if there is an inconsistency bet ween 'l. s and a l. / a e_ , the parameter- fittin g algorithm converges slowly s or not at all . A remark on this p oin t was made earlier in the section on sensitivity function s . Most autom atic inte gration routines , whether they are b ased on explicit or semi -imp licit R u n ge - Ku tta procedures or on Gear ' s method , operate wit h a fixed accuracy i n each step or w i t h some other de vice that is outside t he u ser' s cont rol , and i t i s difficult to obtain cor respondin g value s of l. s and a l. s I a £.. A truly implicit , stepwise inte gration routine will give consistent sen sitivities when t he sensitivity equation s are solved one by one after con ver gence of :l_s has been reached in t he step [ see E q . ( 46) ] . When u sin g an exp licit or a semi - implicit inte gration method , one s hould solve the state equation s to gether with m numerically perturbed equations . ( £_ = P l . P 2 · , Pm > · T his is no more diffic ult t han the p rocedure ( 4 6 ) when t he method is explicit . For a semi -imp licit method where all ( m + l) M e quation s have to be solved sim ultaneously , it is hi ghly pro fit able to modi fy t he LU decomposition routine to utili ze the (M x M ) block diagonal structure of the [ M ( m + 1 ) ] - dimen sional J acob ian matrix . Collocation applied to p arameter estimation in di ffe rential equations was first proposed by Van den B o sc h an d H ellinckx ( 1 97 4 ) . As explained above , it m ay be an excellent app roac h to transform each differential equa tion into N collocation equation s , but u n fortun ately a n umber o f misconcep tions about the proper use of the method are app arent from the literat ure , the most misleadin g bei n g that " the experimental points should be chosen at value s o f the indepen dent v ariable ( axi al position , etc . ) whic h are zeros of a specific J acobi polynomial . " Obviously , the N collocation points s hould be chosen to m ake t he ap p ro xim ate solution l.sc a s close to :l. s as po ssible . T he collocation order N in one particular coordinate m ay have to be m uch hi gher than the available number of data points n in this direction - a situation t hat occurs i f Y s i s not a smoot h function o f t hi s particular in dependent variable . A fte r havin g solved t he model by a sufficiently hi gh order collocation method , one m ay ( without los s of acc urac y ) interpolate the solution to t he values of the independent variable w here the experi ment al value s X.e are available . I f , on t he other hand , there are m any more e xperiment s t han required for comp arison with a collocation solution of sufficiently hi gh order N ( t his may be t he case if t he experimental results are recorded autom atically ) , one applies N t h - order polynomial least - sq uare s fittin g o f the ori ginal data to obtain a synt hetic data base :X.ec at the same values o f t he independent variable where the collocation solution is constructed . B aden and Villadsen ( 1 98 2 ) give a c ritic al review of collocatio n - based T hey emp hasi ze t he i mportance of choosin g parameter estimation met hods . .
•
.
N umerical Me t ho ds in Reactio n Engineeri n g a co llocation met hod that give s t he b e s t po s sible fit between t he solution to t he ap p ro ximate model an d t he ( u n know n ) solution � s of the exact l.sc model . T heir model B is t he same as t hat proposed above : solve t he collocation e qu a t ion s , fi n d the gr adien t dr_ sc / de_ an d up d ate 2.· One m ay
51
,
either choose to solve t he collocation equations to m ac hine accuracy be fore
calculatin g l'l 2. or take j u s t on e step in t h e iteration towar d t he solution o f
t h e collocation equations be fore u p d at in g
method A ) eters .
only
E.·
A p articular variant ( their
m ay b e valu a b le to find good startin g value s
E_
for the p aram
If the differenti al eq u at ion model is line ar in t he parameters ( or
sli ghtly no n lin e ar ) , a sit uation t hat fre q u e n tl y occurs , one obtains a
very good e s timate for
£. b y
( 1 46 )
ON is t he re sidual of the differential equations when the dependent v ari ab le
l. is
rep l ac e d by �e c , t he " synt hetic" data b ase re ferred to the c ollocation points as described above , '!F 1 is an e stimate for the di spe r si on m atrix for
the re sid u al O N .
I t is only a weak function of p and may be evaluated at
any re aso n abl e point E. o .
( 147)
T h e g ene r al co llocation method m ay ,
however , b e ap p li e d for models that are nonlinear· in t he parameters as well as i n the state variab le �· I t is equally ap p li cab l e for p arameter estim ation in n onlinear coupled ordinary differen tial - eq u ation mode l s ( w here it competes wit h t h e e x plici t or semi
implicit initial value tec hniq ue s ) an d for p arameter estimation in boundary value pr o b l em s .
The best doc u m e nt e d general code for p a r amet e r e st i m at io n u sin g col
location i s developed b y S �rensen a n d S tewart .
in app en di x e s to SISren sen ( 1 982).
T h e soft w are i s shown
Special attention should be given to t he linear bo un dary value proble m s
which ar e the b a sis for Fourier expan sion o f lin e ar PD E [ e . g . , variant s o f Eq . ( 128) ] , since the se models occ u r so freq uently .
Michelsen ( 1 97 9) convincin gly demonstrated that the transport p ar am
eters ( radi al d isper si on and a wall transfer num be r )
can be e stimated with o n ly one di ago n ali z at ion o f the collocation matrix ( for B iM -+- oo ) and sen sitivi ties calculat e d analytically
from the e i ge nvalue s an d ei genvectors o f
C.
T he resultin g reduction o f computer time comp ared t o p re vious c alc ul a
one
W hen v z ( x ) = 1 , a situation that c ertainly occurs in pellet problems , m ay find the radial transport p arameter , t he wall transport parameter ,
tion s
on t he same dat a bas e w as at le ast by a factor of 2 0 .
and Da simultaneo u sly .
dure
of Michelsen
l_( T , D a )
=
T here are only trival modi fications in the proce
( 1 979) :
The solution with Da
exp ( - Da T ) �( T , D a
=
0)
f. 0
is si m p ly
( 1 4 8)
52
Villadsen a n d Michelsen
where dx_/d T = < Q - D a !) x_ an d x_( T , Da = 0) is found by the procedure ( 1 2 8 ) and ( 1 2 9) with D a = 0 . W hen V z ( x ) depends on x , a new diagonali zation i s required for each
value of Da but the sensitivity wit h respect to Da can still be de ri ve d C onsider the system quite inexpensively . 1
d dT ( x_)
Dap
\¥'X,
=
with solution
�
exp ( �
T )�
-1
x_ 0
( 1 49)
The sen sitivity zo a with respect to Da is found by solution o f the inhomo geneous linear differential equation d (! ) oa dT
=
Mz
= -D a
- V - 1v =
Jl...
w here z 0
- a
( T = 0)
=
0
( 150)
with solution
�
- ex p ( � h )
Da
J
T
0
1 exp ( - � t ) ;¥ - _l( t ) dt =
w here
s.. 1]
and
=
{ ()..�
Tr . 11
for i 4: j
r . . is an element of � = 1]
( 1 51)
exp O.. T ) 1
for i = j
� - 1;¥ - 1�
When the rate term is nonlinear i n x_ s , one p ro c ee ds more o r le s s a s described abo ve for the linear partial di fferential equation . Collocation i s used t o discreti ze t h e P D E i n the x direction b ut now t he collocation equa tions are in t e gr ated in the T di rec tio n , either by collocation as proposed by Sorensen ( 1 98 2 ) or by a step wise p rocedure . Sen sitivities can be c alculate d by numerical pert urb ation , but at least 1
for t he case of a sin gle reaction it is also pos sible to use a semi -implicit Ru n ge - K utta routine to inte gr a te the N s tate equations and the N x · m analy ti c ally derived sensitivity e quations . T he calculation of the two highet i n E q . ( 4 7 ) is not difficult since , in contrast to d eri v ati ve s fyy and fp y the case of M coupled mdependent reactions , the collocation differential equations have a very simp le structure . T he nonlinearities are found only in the m ain di a go n al of the Jacobian ;[ , w he r e as all the other ele m ent s in l are constant s . It is n ear l y impossible to give a balanced picture of the subject of parameter e st i mation within the framework o f one section of one chapter in a boo k . So many things are involved , and most o f t hem have more to do with common sense en gineerin g that with mathe m atics : the p roper choice
N umerical Metho ds in Reactio n E n gi neering
53
of experimental equipment an d analytical method , p lannin g of experiment s , and a gener al pessimism concerning the influence of all the variab les w hich for one re ason or another were not included in the investi gation .
T he methods t hat we have c hosen to discuss have been used by us and by others in many different situations , but still m any topics have been left out . It is hoped that E xample s 7 and 8 , which m ark the end of our review , will also emphasi ze some of the facets of p arameter estimation that could not be adequately covered in the p recedin g text . sion
Examp le 7 : Determination of a firs t -order ra te cons tant and a disper coefficient fro m exp eriments in a laminar flow t ub ular reactor Consider
the mass balance ( 1 2 5 ) with zero mass flux at t he wall , and consider a series of experiments in which t he velocity - average d outlet concentration y in Eq . ( 1 2 5 ) has been measured at di fferent axial positions z iri a reactor tube of given diameter . It is desired to estimate the rate constant kR and Experimental result s of t hi s kind were obtained the radi al diffusivity D . b y C leland and Wilhelm ( 1 9 56) , w ho determined D assumin g that kR was
known . T h e experimental setup is also described by Seinfeld ( 1 96 9 ) , Sein feld and Lapidus ( 1 97 4 ) , and Van den B osch and Hellinckx ( 1 9 7 4 ) , w here kR was to be determine d w hile the dispersion n umber w as assumed to be known from independent measurement s . We shall consider the possibility of simultaneous estimation of the two parameters . Write E q . ( 1 2 5 ) as
( 152) where 1/J gi ve
2 D /k R r . t
=
N
Y
=
L
The solution of Eq . ( 1 5 2 ) by collocation is averaged to
c. exp ( >. . Da z )
1
( 1 53)
1
1
w here the con stants Cj an d >. i are functions of 1jl only . A set of equivalent e xp eriments would comprise variation of va v at fixed tot al reactor len gth and diameter . Let Dar = k R L /vav r and , Da z
=
r
Da z'
where z '
z
v
av , r v av
---
( 154)
1 , the value of z' = vav r /v av . Numeric al values of Ci and of A i are c alc'ulated for 1jl = 0 . 1 , the value used by Seinfeld ( 1 96 9 ) and Lapidus and S ein feld ( 1 97 4 ) :
and the outlet
1
2 3
>. . 1
,
z=
1
c.
- 0 . 842 9
0 . 92 1 1
- 2 . 30
0. 069
- 5 . 26
0 . 007
54
Vi lladsen and Michelsen
A perturbation analysis for
l ar ge 1)1 [ see E q . ( 1 2 2 )
for the first term ]
yields A.
c
1
1 -1 + - 4 8 1j1
=
1
1 -
=
1 7681j1
2
1 1 9201)!
2
+ 0 ( 1)1 - 3 )
( 155)
3 O ( lji- )
+
2 while A. i 1\, O ( lji- 1 ) and � 1\, O ( f > for i > 1 . W hen ljJ is large enou gh to allow terms in ljJ- 2 to be neglected ( ljJ 0. 1 is barely large enou gh to permit this approximation ) , the expansion ( 1 5 3) can be truncated after the first t e r m : =
Y�
exp
[ ( Da
r
-1
+
�
4 1)1
) ]
( 156)
z'
and measurement s of y alone are not sufficient t o determine sep arate values of D� and lji. Only the product Dar ( - 1 + 1 / 4 8 \jJ) will be found by plotti n g In y versus z' We have c o n s t ruc ted the followin g " mea s u r e me n t s " of y for Dar = 2 and ljJ = 0 . 1 by high - order collocation of E q . ( 1 5 2 ) and truncatin g t he results after the third digit : •
z'
y
- In y
- In y/ z'
0. 1
0 . 826
0 . 1 912
1. 912
0. 2
0 . 687
0 . 3752
1 . 877
0. 3
0 . 555
0 . 5551
0. 4
0 . 48 1
0 . 7319
1 . 850 1 . 830
0. 5
0 . 404
0 . 90 6 3
1 . 813
0. 6
0 . 340
1 . 079
1 . 798
0. 7
0 . 286
1 . 252
1. 788
0. 8
0 . 2 41
1 . 42 3
1 . 779
0. 9
0. 203
1 . 5 95
1 . 772
1
0 . 172
1. 7 6 0
1 . 760
A c c o r di n g t o E q . ( 1 5 6 ) , t h e ratio ln y / z ' sh o u l d b e constant , an d since this is not quite so , we m ay be able to determine both parameters D ar and
lji.
Linear regres sion of t he data with z ' > 0 . 5 yields In y - 0 . 0 5 3 - 1 . 7 1 z' now , from Eq . ( 1 5 3 ) , truncate d after the first term ( w hich is certainly adequate fo r lar ge z ' , say z' > 0 . 5 ) , one obtains i ni tial estimates for Dar =
and
an d 1)!:
"Y �
(
l -
1
-
7 68lji
2
)
( 157)
55
N umerical Methods in Reac t ion Engine e ri n g
The re s ul t i s 2. o
( DRr , lji) o = ( 1 . 9 3 , 0 . 1 5 7 } . These values are used as initial values in a comp le t e least - squares pro ce dure b ased on Eq . ( 1 5 2) , which i� solved by t hre e - p oin t collocation , and on all the " experimental" p oin t s ( z' , y ) . T he sen sitivity with re sp e c t to DRr is found by differentiation of Eq . =
( 1 52) : N
>. . z ' c . 1 1
L:
d dD a
1
1'
1
exp ( >..Da
r
( 158)
z')
while the sen si ti vi ty with respect t o 1jJ i s found by numerical p e rt urbation ( although an a ly tical differentiation could also be used ) . C on ver gence is ob t ai n e d in a few iteration s , and total c om p uti n g time ( WATFlV compiler , H ar w e ll VB0 1AD p a rame t e r estim ation pro gram ) is 0 . 2 s.
The results are 1j!
1 . 994 ± 0 . 1 %
=
( 1 59 )
0 . 104 ± 2 %
1j! is su b s t an tial , con sidering that the data are accurate to di gits , an accuracy that is n o t likely to be found in practice . T he rate constant is , however , determined w i t h t he expected small s t an d ard deviation . The two parameters are s tro n gly correlated ( correlation coe f ficient 0 . 95) and t hi s , combined with the large standard deviation of lji , show s that t he use o f l a mi n ar flow reactor data for y are u n suit able for determination of lj! , b a s e d on a " known " value of D a . A small error in the rate constant will be stron gly m a gni fie d when 1/J is e sti mate d . Converse ly , an approXimate value for 1j! is all we need to determine a very good The
error o f
three
estimate
for t he rate constant , us in g exactly the same dat a . Finally , it sho ul d be m e ntio n e d that concentration measurements at t he e xi t of t h e tube , but at the centerline ( i . e . , m e as u r e m e n t s on samples
taken through a small-bore tube which is centrally placed in the reactor tube) will be much better suited for determination of the two p arameters . We have repeated t he estimation , but based on 10 " measurements" taken at x = 0 and with the same " ex p e rime n t al error . " Now 1jJ is found to 0 . 1 0 04 wi th a standard d eviat ion o f on ly 0 . 4% , while the value of D a is 1 . 997 and has the s a m e standard deviation ( 0 . 4 %) as be fore . The result , which is o bt ai n e d by a si m p le n u m e ric al analy sis of t he model from whi c h the parameters are to be extracted , illustrate s how an experimental program can be gui d e d by si m ul at ion studies .
Examp le 8: Kinetics of a homogeneous liq u i d -p h a se subs t i tutio n reactio n ( " the Dow C hemical Company tes t p ro blem " ) In 1 9 8 1 , B lau et al . from the Dow C he mi c al Company p ub li s he d the formulation of a multi-response parameter estimation problem . T he problem is co n c er n e d with an i nd u s t ri al reaction , " hidden" behind the followin g symbolic nomenclature :
HA + 2BM
=
It is a liquid - p hase re ac tion
M- an d Q+ .
( 160)
AB + HMBM c at al y z e d b y Q M ,
which i s fully
di s sociated to
56
Villadsen
and
Mic he lsen
T he problem w as given t o a number o f research group s in t he U nited State s and in E urope w ho were s upposed to cont ri bute with a solution . A com pari so n of t he solutions would , it w as hop e d , give a review of the state of the art in p ar ameter fit tin g for r eaction en gi n eerin g m o dels H ere w e sum m ari z e t he more ge n e ral conclu sions from the solution that w as sub mitt e d by o ur group ( S arup , 1 98 2 ) . T he postulated mechanism is ( B lau et al . , 1 98 1 ) : .
K
HA
2
-- A- +
H
+
HABM ( - H )
MBM
-
+
H
+
+
HMBM
T able 3 summari zes t he T he reaction is carried out in a b at c h re actor . co nc en t r ation - t i m e p r ofi le s for one of the t hree temperatures used in t he e x p er i me n tal w or k T here ar e three rate co nst an t s , k 1 , k _ 1 , a n d k2 , an d there are t hree equilibrium constant s , K 1 , K 2 , and K 3 . A ctivation energy and frequency fac t or for t he t h re e rate constants and value s o f t he three t e m p e r at ure i n depe n den t e quilibri u m constants are d esired . T he 10 s p ecies HA , B M , HABM , AB , HMBM , M-, A-, H+ , ABM , and MBM- are given by mass balances six ordinary differential e q uation s coupled w it h three eq uilibrium relation s and one condition that expresses electroneutrality . At t = 0 o n l y HA and B M are present with a s m all amount of c atalyst QM . - 17 I nitial estim ates for all p arameters are given by B lau et al . ( 1 9 8 1 ) . From the set of initial e st i m at es we shall mention only t h at K = K = 1 0 3 1 and K 2 10- 1 1 Our solution of t he problem p roc eeds in t he fo llowin g t hree step s : 1 . A na lysis of t he o ri gi nal dat a . A closer study of the dat a in T able 2 reveals an al ar min g fit o f the mass balan c e for B . Except for obvious misprints it is seen t hat .
-
,
=
c
BM
=
c
BM
-
c
H AB M
- 2c
AB
( 161)
T he data were an aly ze d by t h e method of B o x ( 1 9 7 3) an d i t w as found that t here is not o n ly one , but two linear relations between t he " ori ginal" dat a . C orr e s po nd en c e with B lau gave the following s upplementary information : a. b.
Cs M was not me asured , but back - c alc ulated from t he m ass balance ( 16 1 ) . T he concentrations of HA , H ABM , an d AB were all measured , but t heir s um was normali zed to a gree with a " known" initi al concentra
tion of HA :
57
N umerical Metho ds in R eactio n E ngi neering
Concentration Versus Time Profiles for Dow C hemic al Company Test Example ( T 4 00 C )
TAB LE 3
T he
=
Concentrations ( g mol /kg)
Ti m e
BM
HA
HABM
AB
0 . 00
8 . 3 20 0
1 . 7066
0 . 0000
0 . 0000
0 . 08
8 . 3065
1 . 6 96 0
0 . 0077
0 . 0029
0. 58
8 . 2 95 4
1 . 6 82 6
0 0 2 34
0 . 0 00 6
1 . 58
8 . 2 7 30
1 . 6 5 96
0 . 0 47 0
2 . 75
8 . 2 4 37
1 . 6 3 05
0 . 0763
3. 75
8 . 2277
1 . 6143
o . 0 92 3
4 . 75
8 . 2026
1 . 5 8 92
0. 1174
5 . 75
8. 1781
1 . 567 3
0 . 1 37 1
8 . 75
8 . 126 5
1 . 5133
0 1 93 5
23. 05
8. 0 167
1. 4075
0 . 2 94 9
21 . 75
7 . 8440
1 . 2308
0 . 4760
2 8 75 .
7 . 6 97 7
1. 0931
0 . 6047
0 . 0088
46 . 2 5
7 . 3234
0 . 7268
0 . 95 30
0 . 026 8
52 2 5 .
7 . 14 95
0 . 5773
0 . 0881
0. 0412
76 . 2 5
6 6123
0 . 206 5
1 . 2 92 9
0 20 7 4
106 . 2 5
6 2309
0 . 06 5 0
1 1 94 1
0 . 4475
124. 2 5
6 . 1220
0 . 0391
1 . 1 37 0
0 . 5 3 05
1 47 . 75
6 . 0084
0 02 44
1 . 0528
0 . 6 2 94
172 . 2 5
5 9 1 93
0 . 0 1 45
0 98 3 5
0 . 7086
1 96 . 2 5
5 . 8556
0 . 0083
0 . 9326
0 . 76 5 9
2 1 9 . 75
5 . 8037
0 0 074
0 . 882 1
0 . 81 71
2 40 . 2 5
5 . 7680
0 . 0050
0 . 8 4 92
0 . 8514
274. 25
5. 7222
0 0047
0 . 8064
0 . 8 957
2 92 . 2 5
5 . 702 1
0 . 0042
0 . 7869
0. 9155
3 16 . 2 5
5 6 72 2
0 . 00 1 5
0 76 2 8
0 . 94 2 5
340 . 75
5 6 5 93
0 . 0017
0 7 495
0 . 9556
36 4 2 5
5 . 6351
0 . 7263
0 . 97 9 3
3 86 . 7 5
5 . 6176
0 . 7112
0 . 9956
412 . 2 5
5. 6131
0 . 7063
1 . 0 00 3
4 4 2 . 75
5 . 5 99 1
0 6 92 7
1 . 0 141
460 . 7 5
5 . 5 95 9
0 . 6871
1 . 0 1 95
( hrs)
.
.
.
.
.
.
.
.
.
.
0 . 0024
.
.
.
.
.
.
0 . 0 0 42
.
58
Villadsen and Mic he lsen
TAB LE 2 ( C on ti n u e d ) C onc e nt r ati on s ( g mol /kg)
T ime ( hrs )
HA
BM
HA B M
AB
4 83 . 7 5
5 . 5 90 5
0 . 6837
1 . 0229
5 07 . 2 5
5 . 5 7 . 36
0 . 6672
1 . 0 3 96
55 3 . 7 5
5 . 5568
0 . 6 4 94
1 . 0574
580. 75
5 . 5631
0 . 6467
1 . 0551
651 . 25
5 . 5472
0 . 6408
1 . 0660
673. 25
5 . 5516
0 . 6452
1 . 0616
842 . 75
5 . 5 465
0 . 6 3 97
1 . 0669
So urce :
0 . 0 0 46
B lau ( 1 9 8 1 ) .
H
C A
=
CH
A
+
C HA BM
+
c
( 162)
AB
T his normali zation is common practice i n analysis of chromatographic dat a an d t he error is not as " crude" as that involved in ( a ) . Still , when normali zed m e as u r e me n ts are used , the error structure of t he true measurements is completely distorted and one m ay u se a stand ard le ast - sq uares app ro ach rather than one o f the more sop histicated statistical criteria .
.
T he comments re garding the data base should not be co nstr ue d as a criticism of the author of t he p ro bl em A preli minary analysis of the raw data is alw ays valuable , and t he flaws in the original data m aterial are p robably typical for what can be e xpe cted in pr actice . O u r conclusion has been to delete t he concentration meas u r em e n t s o f c from t he dat a b ase since they con t ain n o independent information . BM 2 . Preliminary analysis of the model . Total m ass b alances for A , B , and M and the three equilibrium relation s can be used to eliminate 6 out o f 10 dependent variable s to give a set of three coupled di fferential equations and one algebraic equation . T he postulate d values of the dissociation const ants K 1 , K 2 , and K 3 are so small t hat the acids are almost totally undissociated . One mi ght assume that the concentration of H+ is nearly zero , an d this reduces the model to thre e differential equations in t hree components , chosen as H A , B M , and M-. C - , CABM , and c MB M - are given by A c
A
_
=
(1
+
K
3
K2
c
HABM c HA
+
Kl
c M H BM K cH A 2
)
-1 (c
Q
+
_
c
M
-
)
( 163)
59
N umerical Metho ds in R eac tion E ngineeri ng K c
HBM-
1
c
K2
=
C HMB M Ac HA
while c A B , cA B , an d c H M B M are calculated from t he total m ass balan c e s M H for A , B , and M . E xc e p t for the ass u mp tion c 8 + "' 0 , t he r e are no ap proxim ations involved re l ati ve to the original model , and we have c he c k ed that our results for the parameters are virtually unaffected by t he assump tion c H+ "' 0 . With this assumption - which appears to b e q uite realistic-it i s impos sib le to find ab so lu te values for the equilibrium constants K 1 , K 2 , and K 3 . Only their ratios K 3 / K 2 an d K 1 /K 2 can be foun d . I f these ratios are as sm all ( c a . 10 - 6 ) a s in dicate d in t he p ro b le m de scription , the s ys t e m is extremely stiff : cH A will decrease al m o s t to zero , following p seudo - ze r o - o r der kinetic s . When cH A h as b e c o me of t he order of K 1 /K 2 , t he rate e xp re s sion chan ge s t o pseudo first or d e r in cHA . All H A would essentially have to react before the " dead- end" reaction to HABM rever ses direction an d st arts to s e n d A B M- b ack into t he main reac
tion sequence .
3. Sim u lation and parame ter es timatio n . To s im ul at e the solution � of the mode l , we used the IMSL ro ut in e D GEAR , which w as modified to include algebraic ( " e x act" ) evaluation o f the sen sitivitie s with respect to t he p aram e t er s T here are in all ei g h t parameters in k 1 , k_ 1 , k 2 , K 1 /K 2 , an d K 3 /K 2 , but s en si tiviti e s wit h respect to activation ener gy Ej and fre q ue n c y fac to r ex;, of t he rate con stants lq c an easily be co m p ut e d from t he sen sitivities with re spect to ki :
.
.£.1: a
a.
=
ll.
ll. a k
·a k a
��
aE
exp
=
a
ak aE
- �
T
=
(- � ) ll. (- � ) � T
exp
ak
( 164)
ok
T
2 4 s en siti vi t y e q u at io ns With thi s simple device the o ri gi n al 3( 2 • 3 + 2 ) are reduced to 15 e q uatio n s . Also , it is important to re sc ale the p arameters since the m a gnitude of a an d E are so diffe ren t t hat it become s difficult to har mo ni ze t he sensitivi ties . Instead of <X and E , we u s e k ( T o ) an d E / 0 d e fin e d by =
k
=
a
( �)
exp -
=
[
T
k ( T ) exp 0 T
E
(1
O
T :)]
( 16 5 )
w here T i s a sui t able re fe rence temperat ure ( chosen a s t h e middle o f the o t hree te m per at u r e s used in the e xp er i me n t a l investi gation ) . W it h t he re scaled p ar amet er s a lar ge part of the inherent correlation between t he two par am e t er s involved in k is removed . In ge ner al if a new p arameter P t f( p 0 ) is introduced inste ad of p 0 , one obtains the sensitivity wit h respect to p 1 by the simple al gorithm
,
=
Vi llads en and Mic he lsen
60
( 166)
T he Harwell code VB 01AD was used for the leas t - s quares minimization . The routine is b ased on the Levenberg- M arq uardt method , and it require s an alytic derivative s of the error vector with respect to the p ar ameters . T here w as no diffic ulty in obtainin g the minimum of the sum of least square s - about 1 0 s of CPU ti me to determine all eight parameters ( w hic h corresponds to ca . 0 . 5 s per inte gration of t he model and the 1 5 sensitivity differential equation s ) .
It i s remarkable that the inte gration of t he three equations for C H A , CS M , an d eM - t o get her with all the sensitivity equations - requires only H ere it is twice t he time used for the three quation s o f the state vector . m uc h better to use exactly derived s ensi t i vit y equations than to use numeri cal perturb ation of the parameters , w hich is consider ably slower ( by a fac tor of 6) and requires a m uc h hi gher inte gration acc uracy to give meaning ful re sults . A lso , a comment on t he initial values of K 3 /K 1 an d K 1 /K 2 i s necessary . To obt ai n the best e stim ate s for these p arameters ( 1 . 43 and 0 . 0 6 7 2 , re
spectively) it w as necess ary to start with an initial estim ate of K 1 /K 2 which w as decidedly larger than the small value 1 0 - 6 su gge sted by B lau
( 1981) .
It is not difficult to se e from T able 3 t hat c H A B M starts to decrease much before H A has been consumed , and consequently that K 1 /K 2 has to be lar ger th an w - 6 . It is , however , not at all obvious that the routine K 1 /K 2 and q 3 = K 3 /K 2 if the fails to predict the correct values of q 1 ori gi n al e stimate o f the parameters is used . T he reason is that the object function is affected only by t he ratio q 3 /q 1 , not by their absolute values when they are as small as 1 0 - 6 , and it becomes impos sible for the routine to drag the parameter values all t he w ay from 10- 6 to t heir correct values 1 of about 1 o - . =
The integration routine is much faster w hen ( K 1 /K 2 , K 3 / K 2 ) are of the T he reason is the sudden increase order of 1 t han w hen they are c a . w - 6 . of stiffness that occurs when c H A "' 0 , and t he apparent reaction order o f the reaction A- + B M -+- A B M - c han ges from zero to 1 . T he problem m ay occur q uite frequently in t he app lic ation of standard " kinetic codes , " an d it m ay be illustrated by the sim p le rate exp re ssio n ( 1 6 7 ) :
with k
"'
1 an d K
"'
10
-6
( 167)
Even a good numerical inte gration routine ( e . g . , DGEAR ) will fail to notice the rapid chan ge in the structure o f the solution when yA "' 1 0 - 6 and will happily i n t e gr at e to ne gative concentrations of A . This m ay have a disas terous e ffect on the computation of other reaction species , w hich m ay be heavily influenced by the concentration level o f component A .
R E F E R E N C ES Aris , R .
T he Mathema t ical T heory of Diffusio n and R eac tion in Permeab le C a t aly s t s , C l are n do n P re s s , Oxford ( 1 97 5 ) .
N umerical Metho ds in Reac tio n Engineering
61
Aris , R . , and N . R . Amun dson , Firs t O rder Partial Differen tial Eq uatio ns with A p p lications , Prentice - H all , En glewood Cliffs , N . J . ( 1 97 3) . B aden , N . , and J . Villadsen , A Family of collocation based methods for parameter estimation in differential equations , C hern . En g . J . , 23 , 1 ( 1 982 ) . B lau , G . , L . Kirkby , and M . M arks , An industrial kinetics problem for testin g nonlinear p arameter estim ation al gorithm s , The D ow C hemical Company , Midlan d , Mich . ( 1 98 1 ) . Box , G . E . P . , W . G . H unter , J . McGre gor , and J . Erj avec , Some problem s associated with the analy si s of multiresponse data , Techno metrics , 1 5 , 33 ( 1 9 7 3) . B yrne , C . D . , Some software for solvin g ordinary differential equations , in Fo u ndatio n s of Comp uter-A ided C hemical Process Design , Vol . 1 , S . H . M ah an d W . D . Seider , eds . , E n gineering Foundation , New York , 1981 , p . 403 . C aillaud , J . B . and L . Padmanha.b an , A n improved semi-implicit Run ge Kutt a met ho d for stiff system s , C hern . En g . J. , 2 , 2 2 7 ( 1 97 1 ) . Christian sen , L . J . and S . L . Andersen , T ransient profiles in sulphur poi sonin g of steam re formers , C hern . En g . Sci . , 3 5 , 3 1 4 ( 1 9 8 0 ) . Cleland , F . A . and R . H . Wilhelm , Diffusion and re action in viscous flow tubular reactor , A I C hE J . , 2 , 4 8 9 ( 1 9 5 6 ) . Clement , K . and S . B . Jor ge n sen , E xperimental investi gation of axial and radial t herm al dispersion in a p acked bed , C hern . E n g . Sci . , 38 , 8 3 5 ( 1 98 3 ) .
Crank , J . , Mathemat ics of Diffusio n , Clarendon Press , Oxfor d ( 1 95 7 ) . Finlayson , B . A . , T he Met ho d of W ei gh ted Residuals and Variat io nal Princ i p l e s Academic Press , New Y ork ( 1 97 2 ) . Finlay son , B . A . , Orthogonal collocation in c hemic al reaction en gineerin g , C atal . Rev . Sci . E n g . , 1 0 , 6 9 ( 1 97 4 } . Finlayson , B . A . , N o n linear A na l y s i s in C hemical E n gi neering, McG raw -Hill , New York ( 1 980 } . Fletcher , R . , U nco ns trained O p timizatio n , Wiley , New York , ( 1 980) . Funder , C . R . , M . Sc . thesis (in D ani sh ) , I n stituttet for Kemiteknik ( 1 98 2 ) . Gear , C . W . , N umeri cal Initial V alue Prob lems i n O rdi nary D i fferential Eq uations , Prentice - H all , En gle wood Cliffs , N . J . ( 1 97 1) . Guertin , E . W . , J . P . Soren sen , and W . E . Stewart , E xponenti al colloca tion of stiff reactor models , Comp . C hern . En g . , 1 , 1 97 ( 1 97 7) . Hoffman , U . G . Emi g , an d H . Hofmann , C omp arison of different me thods for determination of e ffective thermal con ductivity of porous catalysts , A C S Symp . S er . , 65 , 1 8 9 ( 1 97 8 ) . Holk-Nielsen , P . , and J . Villadsen , Ab sorption with exothermic reaction in a fallin g film column , C hern . En g . Sci . , 3 8 , 1 4 3 9 ( 1 98 3 ) . Jackson , R . , T ransport in Poro us Catalysts , Elsevier , Amsterdam ( 1 97 7 ) . K aza , K . R . , Villadsen , J. , and R . J ackson , Intraparticle diffusion effect s i n the methan ation reaction , C hern . En g . S ci . , 35 , 1 7 ( 1 980) . Levenspiel , 0 . , C hemi c al Reac t io n Engineering, 2nd ed . , Wiley , New York , ( 1972) . Liaw , W . K . , J . Villadsen , and R . J ackson , A sim ulation of coke burnin g in a fixed bed reactor , A C S Symp . Ser . , 1 96 , 3 9 ( 1 982 ) . Michelsen , M . L . A fast solution technique for a class of linear PDE , and Estimation of he at transfer parameter s in packed beds from radial temperature measurements , C hern . En g . J . , 1 8 , 59 , 67 ( 1 97 9} . Michelsen , M . L . and J . V . Villadsen , Polynomial solution of differential equations , in Fo u ndatio ns of Comp uter-A i ded C hemical Process Design , ,
,
,
62
Villadsen and Mic he lsen
Vol . 1 , S . H . M ah an d W . D . S eider , eds . , En gi neering Fou nd atio n , New York , 1 9 8 1 , p . 3 4 1 . Mic helsen , M . L . , H . B . Vakil , and A . S . Foss , S tate space form ulation of fixed bed reacto r d ynam ic s , Ind . En g . C hern . Fundam . , 1 2 , 3 2 3 ( 1 97 3) . Miller , K . and R . Miller , Movin g finite element s , Siam J . N umer . Anal . , 1 8 , 1 0 1 0 , 10 3 3 ( 1 9 8 1 ) . Papout saki s , E . , D . R a mkris h na , and H . C . Lim , The extended Graetz problem with p r e s cri be d w all flux , A I C h E J . , 2 6 , 779 ( 1 980) . Patter son , W . R . , and D . L . Cresswell , A S imple method for the calcula tion of effectiveness factors , C hern . E n g . Sci . , 2 6 , 605 ( 1 97 1 ) . Peyt z , M . J . , H . Livbjerg , and J . Villadsen , S t eady state operation of a dual reactor system with liquid catalyst , C hern . E n g . S ci . , 3 7 , 1 0 9 5 ( 1 98 2 ) . S ar u p , B . , M . S c . th e si s (in D anish) , I n s tituttet for K e mit e kni k ( 1 98 2 ) . S e i n feld , J . H . , I dentification of p arameters in PDE , C hern . E n gr . S ci . , 24 , 6 5 ( 1 96 9 ) . S ein fe ld , J . H . an d L . L a pi d u s , Process Mo deling, Es timatio n , and Identi ficatio n , Prentice - H all , En glewood C li ffs , N . J . ( 1 97 4 ) . Sotircho s , S . V . an d J . Vill a d s e n , Diffusion and reaction with a limited amount of re actant , C he rn . E n g . Comm un . , 1 3 , 1 4 5 ( 1 9 8 1 ) . S un dare san , S . and N . R . A mu n d son , Diffusion and reaction in a stagnant boun dary laye r about a c arbon particle , I nd . En g . C hern . Fundam . , 1 9, 3 4 4 , 3 5 1 ( 1 9 8 1 ) . S�rensen , J . P . , S im u l a t io n , Regressio n , and C o n t ro l of Chem i c al Reac tors by Co llocatio n Techniq ues , Polyteknisk For lag , Lyn gby , Denmar k ( 1982) . S� r e n s e n , J . P . and W . E . S tewart , Collocation analy sis of m ulticomponent di ffu si on and re a ctions in porous c at alysts , C hern . En g . S ci . , 3 7 , 1 1 0 3 ( 1 98 2 ) . T aylor , G . , Di s p e r si o n of s ol uble m atter in so lve nt flo w i n g slowly through a tube , Proc . R . Soc . , A 21 9 , 1 86 ( 1 95 3 ) . V an den Bosch , B . , an d L . H e llin c k x , A ne w m e t ho d for the estimation o f p arameters i n differential e q uat i on s , A I C hE J . , 20, 2 5 0 ( 1 97 4 ) . Villadsen , J . , Se lec ted A p p rox im a t io n Methods fo r C he m i ca l Engineering Prob lems , I n s t itu t t et for Kemiteknik , Lyn gby , D enmark ( 1 97 0 ) . V i llad se n , J . and M . L . M i c he l s e n , So lution of Diffe ren tial E q uation Models by Po lynomial A p p roximation , Prentice - H all , E n gl ew ood Cli ffs , N . J . ( 1978) . V illad s e n , J . and W . E . S tew art , Solution of boundary - value problems by ort hogonal collocation , C hern . En g . S ci . , 22 , 1 48 3 ( 1 9 6 7 ) . Vi ll a dsen , J . and J . P . S � rensen , S olutio n of parabolic PDE by a doubl e collocation metho d , C hern . E n g . Sci . , 2 4 , 1 3 37 ( 1 9 6 9 ) . Wedel , S . and D . L us s , A r ational approxim ation of t he e ffectiveness fac tor , C hern . E n g . Commun . , 7, 2 4 5 ( 1 980) . Wedel , S . and J . Vi l l adsen , Falsification of kinetic p arameters by i nc orr ect treatment of recirculation reactor data , C he rn . E n g . S ci . , 24 , 1 34 6 ( 1 98 3 ) . W heeler , A . , Reaction rates and selec ti vit y in c at aly st pores , Adv . C atal . , 3 , 316 ( 1951) .
2 Use of Residence- and Contact-Time
Distributions in Reactor Design REUEL S H I N N AR
N ew
The C ity Col lege of t h e City U ntversity of New York , York , N ew Y o rk
I N T RO D U C T I O N
Tracer experiments an d residenc e - time di s trib u ti on s m eas u rem en t s have been used b y reac tion en gineers for 5 0 y e a r s . They are a p ow erful tool with an i n cr e asi n g number of application s . T he fi r s t ri go ro u s p re s e n tation was given by Danckwerts ( 19 5 3) . S i n c e then a large b o dy of literature has accumulated an d the topics are discussed in several textbooks (Froment and Bischoff , 1979 ; Himmelblau and Bischoff , 1968 ; Sein fel d and L api d u s , 1 9 7 4 ; Nauman , 198 3 ; Le v e n s p iel and B i s c ho f f , 1 9 6 3 ) an d revie w articles ( S hin n a r 1 977 ; P e t ho and N oble , 1 98 2 ; N auman , 1 9 8 1 ; Weinstein and Adler , 1967) . T racer experiments and residence- time d i st ri b u ti on s are useful to re a ction en gineers in several ways . T hey p r o vi de a di a gn os tic tool for t h e detection of m aldistrib utions and the flow pattern in side a react or . T hey a re useful in experim entally measuri n g the parameters of simplified flow models and p rovi d e i d e a s an d gui d e lin e s for creatin g and testin g such models . Finally , u n d e r s t an din g the concept of residence - time distrib ution gr e at ly sim plifie s the solution of many p roblems in reactor desi gn , incl u din g the choice of reactor confi guration , op ti mi z a ti on , and scaleup , and allows one to solve t hem without complex and ti m e c on sumin g c o mp u t a tion s . One must , however , be c ar e fu l not to expect more from the t echn iq ue s than they can deliver . It is extremely rare that one can eit her construct a reactor m o d e l o r p re di c t a r e act or p erfo r m an c e b ased solely on a residence time di s t rib u tion ( S hinnar , 1 97 8 ) in a s it uation involvin g c o m p lex flo w s . Most reac tor s are heterogeneous , and the theory as p re sented in most text books is rigorously a p p li c ab l e only to ho m o ge neou s reactors . It is t herefore important to understand the a d van t a ge s as well as the li m i tati on s of the method . This writer has been in volved both in d evel opi n g som e of t he theoretic al c on ce p t s related to their u se and in their p r ac tic a l ap plic at ion s ( N aor an d Shinnar , 1 9 6 3 ; S hinnar and N aor , 1 9 67 ; K rambeck et al . , 1 969; Shinnar et al . , 1 97 2 ; N ao r et al . , 1 97 2 ; Zvirin and S hinnar , 1 9 7 5 ; Glasser et al . , 1 9 7 3 ; K rambeck et al . , 1 9 6 7 ; Silverstein and S hinnar , 1 9 7 5 ; S h in n ar et al . , 1 973) . I n the followin g sections I t ry to p re s e n t the subjec t in a way t h a t ,
-
63
64
S h innar
s e es a al pplicatuson discussing ad ta h sorand fal s e t c al work which the future u presents a personal viewpo applicability int r c c and in that sense chapter th h the re r with based on my re n original no historical mp t be m full Theng theoretical concepts for h mog n us systems are de d in the t c u the r in the measurements and their use and a li a ns in reactor mod of industrial reactors. Next, we deal with a t on he s n ofttepletih moge ous ac o s w h m ti has e tracers. Following a i ro u contact-time distributions, a e residence-time h is useful catalytic re actors. cwat summari c re zce thedeuse of residence-time co ta t me
t r ss p r ctic a i s both van ge pi t l . T hus , by necessity , I m t exclude a large amount of t e i may b ecome useful in b t has yet to p rove its in p a ti e , this ad e own experience . Al o ug I try to p rovide fere c e s to the literature , at te will a d e to give proper credits to p ap ers or to develop ment . o e eo rive fo l lo wi sec tion . Then we discuss experimental e hn iq e s for i pp c ti o eli n g and s ng p p lic a i s to t de i g o ne re ti n a n d it ul p systems and the use of m ul i t h t we n t d c e c on c p t similar to distribution t at in Finally e and n c ti di stributions in a ly t i a tor si gn ,
,
-
.
D E F I N I T I O N S O F T H E R E S I D E N C E- T I M E D I S T R I B U T I O N , AG E , A N D F U T U R E L I F ET I M E D I ST R I B U T I O N
a homogeneous rat i outlet streams are completely t the s stable and steady and that the inlet and a timer mixed. Let us now assume t a each fluid a e e that is activated when it t rs or a t ne a i gl exit from w h nn return to s i e with the ac tor. Each r e at th exit has a e Thisscalar property i , one the rfinesi thecedistribution timetimefor thatofp t As with any b ac o rti l This h less h a given t (see Fig. fr cti F(t) eis the s s rib atige as t e im orl arl F(O) aha m i ice with Res idence- T i me D i s t r i bution
e c or as shown in F g u re 1.
Consider flow i
Ass ume tha
p r ti c l is quip p e d with the reactor and stops when it exits . I t is i mp t n to d e fi s n e hic the particle c a ot pa tic l e tim t a s oc a t d it , w hich re measures the it s en in the reactor . q uan t ty is defined as e de n p article . can de this p roperty y the fr ti n of the p a c e s for which t e residence time is t an value 2) . a on defined a the residence- time distribution . [ Alternatively , one can d e fin residence - time di t u on h fraction of p articles molecules which s a residence time l r r than t , s ply 1 F ( t) . ] C e y 0 n d F ( t ) m us t b e a on o to n c ally n r a si n g function F ( oo ) = 1 . h t
·
en e
-
,
=
ro a dF(t) d
D e n s i ty Function For any p f(t)
I n le t f l ow
F I G U RE 1
b b ili ty
=
distribution , F(t) , the
derivative t a
h t is defined by
( 1)
t
R E AC TOR volu m e V
Q
e
S t ad y s ta -
t
Out
let
f l ow Q
e homogeneous reactor.
65
Residence- and Con tact- T ime Dis trib u t ions 1.0
u. z
0
1:::> IIl a:: 1(/)
0.8
0.6
F - F r a c t i on o f p a r t i c l e s w i t h re s i d e n ce t i m e in the r e a c tor less thon (t)
Cl
::!; j:::
w
w (.) z w 0 (/) w a::
0. 4
0. 2
0.0
0.0
1.0
2.0
TIME ( I )
(a) 0. 8
---
3.0
4.0
5.0
-- -- ----
0.7
0.6 0
::!; 0.5 !E
:::>
�
)(
0.4
0.3
0. 2 0.1
0.0
IDEAL S T I R R E D TA N K 1.0
2 .0
3.0
4.0
(b) FI G U R E 2
D e finition o f resi dence - time di st rib ution .
5.0
66
Shin nar
is called the den sity fun c t ion and is often of great in te r est .
In the case where F ( t ) is a residence - time distribution , f( t ) dt corresponds to the frac tion of fluid p ar tic l es whose resi dence time is bet w een t an d t + dt . S ome typical distrib utions of F ( t ) and f( t ) are given in Fi gure 3 . The way such distribution s are measured an d comp uted will be shown in the next s ectio n Some properties of the function will be discussed next . One should note that in the chemical en gineerin g literat ure the density function f( t ) is often designated E ( t ) and is sometimes called the res idence time dis t rib u tion . [1 F( t ) ] is often d esi gnated as l ( t ) . As there is a large literature on the mathematical p rop er ti e s of re si den ce time distribution outside the chemical en gineerin g literature , the standard notation F ( t ) and f( t ) will be used in this chapter . .
-
-
M ea n , Va ria nce , a n d Momen ts of f ( t )
The expected value or mean of t he den sity function f(t ) i s defined by 1"
=
Il l
f
00
00
=
E (t)
f
=
tf( t ) dt
=
0
f0
[1
- F ( t ) ] dt
( 2)
T his is also equal to the first moment of f(t) and is a widely accep ted meas ure of the central value of a di s t ri b ution In react or systems with con stant density , it is equal to the volume or liquid hol d up of the reaction , divided by the flow rate : .
1"
=
v
( 2a )
Q
and is also known as the time c o n s t a n t of the syste m . For purposes of com parison , it i s often useful to n or mali ze the residence- time distribution s uc h Therefore , we define that the time con stan t or mean value is unity .
e
=
t
( 2b )
w here e i s a dimensionless normali zed variable . Most of the graphs in this chapter use e . Another importan t measure i s the variance o f a distrib ution , V f ( t ) , which is defined by 00
=
f
[t
0
-
2 E r < t ) l f( t ) dt
=
2 Er
( 3)
V f ( t ) can also be c o m p ute d directly from F ( t ) by
{
00
2
t [ 1 - F ( t ) ] dt
( 3a )
The variance ( or the second moment wit h respec t t o the average ) meas ures dis t rib u tion Relative disperison is also frequently characteri zed by the coefficient of variation , the dispersion of a
.
67
Residence- a n d Con tac t- Tim e Dis trib u tio ns 1.0
2 .
/ ._ _ _ _ _ _ // /
v�
0. 8
'I
0.6
F(8 )
-- S t i r r e d To n k
0. 4
- · -
�
0.2 0.0
1 0 S t i r r e d To n k s i n S e r i e s
- - - - One Dim. D i f f u s ion
2
0
e
(a)
=
10
3
4
3
4
1 . 50
·'\
1. 25 1 . 00
I (8)
,1\ \ '/ \ .\ \ '\�
0.75
I I
0. 5 0
I I ,
0. 2 5 0.0
0
II ;
/,,
I '
./ '\
I.
1
\
\
\ .
·
(b) ( a ) Residence- time distribution , ( b ) residence- time den sity function for some simple theoretical flow models .
FI GURE 3
S h in nar
68
( 4)
y (t) W hen needed , hi gher momen t s can b e com p u t e d u si n g stan dard methods . T he rth moment of the density function f ( t ) is defined as 00
=
J
O
t
r
f( t ) dt = y
00
J
0
ty- l[ 1
-
( 5)
F ( t ) ] dt
I t sh o u l d be stressed that the function s f , F , and ( 1 - F ) refer to t he distrib ution of the resi dence times of particles as they app ear at t he exi t of the flow s y s tem . Li n ea r P r o pe rti e s of Resi dence- T i m e D i s t r i b u t i o n s Residence- time distrib ution s have t h e p rop er tie s o f linear function s . T hus if two systems are put in series ( see Fi g . 4a ) , the den sity function of t he total system is t he convolution of the two sy stem s , an d if they are par a llel , it is t he weighted average ( Fi g . 4b ) . T h us for mathe matical operations it is usef ul to ap ply the Laplace tran sform of f ( t ) , L [ f( t ) 1
00
f(s)
J 0
e
- st
( 6)
f ( t ) dt
T he Laplace transform can also be used as a moment - generatin g function . T he rth moment of f( t ) , ll r , is then given by
( 6a ) Flow sy stems are nonlinear s yst e m s in sofar a s the Navier- S tokes equa
tion i s nonlinear . T he fact that re si den ce - ti me dis trib u tion s have the prop erties of a linear system in dic ates t hat they provide an incomplete descrip
tion of the flow p roce ss in the reactor . T herefore , they cannot completely describe t he flo w , b u t represent only some importan t p roperties of it . If
one desi gn a tes t h e complete hydrodynamic descrip tion of all t h e fl o w s an d m a s s transfer processes in the reactor as M , then F is a linear property
as sociated with M . W hi le the tran sform ation of M into F i s unique , the op p osite is not tr ue . A s F is an incomplete description o f M , a sin gle F can have an in finite n umber of func tions M as sociated wit h it . In other words , there can b e a large n umber of comp le tely differen t hy drody nam ic
flow sys tems wi th the
same residence- time dis trib u tio n .
I n ten s i t y Function or E scape P ro ba bi l i ty
A residence - time di s t rib ution is a deterministic property of the flow sy stem . I t is sometimes advan t a geous to interpret it probabili stically . One can also look at F ( t ) as the probability that a sin gle molecule will have a residence
69
Residence- and Con tac t- Time Dis t rib u t io n s RE ACTOR I
Q
R E A C T O R II
Q
VOLU M E V1
V O L U M E v2
R T D - F1 ( t )
Q
RTD - F (1) 2
DE N S I T Y F U N C T I O N :
DEN S I!Y F U N C T ION :
f2 ( I )
t1 ( I )
R T D OF TOTA L S Y S T E M : f ( 1 + 2 ) (t )
I
=
0
/tl ( I -T) t2 ( •) d T
OR I N L A P L A C E D O M A I N :
f( l + 2 ) ( s )
=
t1 ( s )
*
t2 ( s )
AV E R A G E R E S I D E N C E T I M E :
=
(a) Ql
R EACTOR I
Ql
R E AC TOR II
Q2
1 1 ( 1 ) , v,
Q
Q2 = Q - Q l
Q -i
1 2 1 1 l , v2
vl
+ v2 Q
F1 ( I )
AV E R A G E R E S I DE N C E T I M E :
T
(b)
Residence - time distribution of two reactors : in p arallel .
FI G U RE 4
(b)
( a ) in series ;
Sh innar
70
T hi s p resents the in formation contained in C onsider a system characteri zed by the re sidence- time den sity f ( t ) . In probability term s , f ( t ) can be interpreted as follow s . If a p article enters the sy stem at time t 0 , the p robability of its leavin g the system within t h e time interval (t to t + d t ) is equal to A particle f ( t ) dt . However , a sli ghtly di fferent proble m may be posed . has already stayed in the sy s tem for a time t ; one wis hes to know the p robability of its leavin g the system within the next time element dt . Let time les s than
t in
the
reactor .
F ( t ) in a more acce ssible w ay .
=
this probability be denoted by A ( t ) dt . T h e func tion A ( t ) may be evaluated as follo w s . T he p robability of a particle leavin g the system between time t an d t + dt e quals f( t ) dt . T his p robability is the product of t wo terms : T he p robability of the particle not leavin g before t , which is 1 F ( t ) , an d the probability of its leavin g between t an d t + dt , as sumin g that it is still in t he system an d has not left before- this was define d as A ( t ) dt . T hus we obtain -
f ( t ) dt
=
[ 1 - F ( t ) ] A ( t ) dt
( 7)
an d by simp le rearran ge ment A ( t ) is derived as A ( t)
=
1
F ( t) F(t) -
-
=
d ln [ 1 - F ( t ) ]
( 8)
dt
A ( t ) i s t he con di tional p robability distribution an d is also known as the in t en si t y function or the escape probability . l t must be stress e d that there exists a one - to - one corre spon dence bet ween A ( t ) and f( t ) . In deed , the tran sformation of A ( t ) into f ( t ) is governed by
[{
t
f( t ) = A ( t ) exp
-
A ( t' )
dt
]
( Sa )
Physically , A ( t ) dt is a meas ure o f t h e prob ability of escape for a particle that has s tayed in t h e sy stem for a period t durin g the time in ter val t and t + dt . T hus , in th e case of an ideally ( exponentially ) mixed ve s sel , one expects the i nte n s it y function A to b e a constan t , as the c hance of escape d u ri n g any ti me interval dt i s independen t of p revious hi story an d is e qual to dt / 1" . T his is in deed t he case . A l t h ou gh there i s a one to - one correspondence bet ween f ( t ) , F ( t ) , an d A ( t ) , information about the prop erties of the residence - time distrib ution and the nature of the flow system is more easily acces sible from A ( t ) . U tili zin g the inten sity function , it is easy to demon strate a previously mentioned p roblem with re sidence - time distrib ution s , namely , that the rela T his is illu strated in Fi gure 5 , whic h tion b et ween F an d M is not unique . describes the properties of a n um b e r of theoretical flow models . Thus , for an ideal uniformly s tirred tank , A ( t ) m us t b e a constant ( see Fi g . 5a , curve A ) . However , the converse is not t r u e A A ( t ) that is con stant im plies t hat the escape prob ability of a particle i s in depen den t of its p revious history . In other words , t he information as to how lon g a particle has been in the sy stem tells us nothin g about i t s future behavior . T he fact t hat a system has this p roperty does no t p rove or in dic at e that it is an ideally s tirred reac tor . T he latter also implies that the concentration o f .
71
Residence- a n d C o n tact- Time Dis t rib u tions
1 02
I
lc
��-----------------------------------.
I I 1
B � -- -
- - - - - - -
: /
I '
y
A!8l
,J
10
n
' I
°
Ii I I I
10,
;
0.50 ,
A
I
I
2.50
8
(a)
4. 5 0
6 .50
I
I '
i\ c! \a l \
1 -F ( 9 )
I
I
0 1
.
A
.
! \
�------�--�
I
2 0.0
2
e
{b) FI G U R E 5
3
4
( a ) Intensity function for some simple flow mo dels :
A,
s tirred
tank ; B , approach to plu g flow ( estimated by 30 tanks in series ) ; C , ideal
plug flow
(o
function at e
for the models
in
part
(a) .
=
1) .
( b ) C umulative residen c e - time distrib ution
Shin nar
72
all
differen t sp ecies in the reactor is completely uniform at all poin t s , which is not a necessary condition for A ( t ) to be constant . In many reactors , such as p acked beds or baffled fluid- bed reactors , one wi shes to app roach plug flow as c losely as p os sible . In reality , one cannot achieve ideal p lu g flow an d the escape p robability will resemble the one shown in Fi g . 5a , curve B . This con dition will be called near plug flow . Such a flow can be modeled by either a series of s tirred tanks or by a one - dimen sional di ffusion mode l . B oth of these share one property , which is p hy sically very well represented by the escape p robability . For values of t t hat are s m all compared to the average resi dence ti me ( t << T ) , the escape p robability i s close to zero . As t app roaches T the escape probability rises un til it reaches a final constan t value . Physically , that means that for t » T the p rob ability of t he molecule b ein g near the exit if i t is still in the sy stem is hi gh . I n a series of s tirred tanks one can therefore assume that for t » T the particle is probably in the last tank and its escape probability is the probability of escapin g from the last tank ( see Fi g . 6a , curve A ) . It c an be shown ( see Shinnar and N aor , 1 96 7 ) , that in a series of s tirred tanks wi th forward and backward flow between The same the tanks ( Fi g . 7 ) , the escape p robability has a similar form . applies to the one- di men sional di ffusion model ( Fi g . 6a , curve B ) . I n a real p acked b e d the flow is not neces sarily one - dimensional . T here mi ght be a flow maldistribution due to a bad distributor . If there are re gion s in w hich the fluid flows faster , then to main tain the same average re sidence time , t here must also be re gions of slower flow . T hree such models are gi ven in Fi gure S . I n such a case , i f t » T , the par ticl e i s probably in t he slow re gion and has a lower escape p robability . T he s ame is true if we consider a system in which there are stagnant region s which particles can diffuse in and out of , or if there are adsorption p henomena ( Fi g . Sa , c urve B ) . Whenever any A ( t ) has a maximu m an d then decreases , it implies that the system either has region s of fast an d slo w flows or stagnant re gion s in w hich a particle c an get trapped . C urve C in Fi g . Sa shows that adsorp tion p henomena do not neces sarily lead to decreasin g values of A I f the mass transfer processes are rapid compared to the overall flow velocity , no stagnation will be occur an d the escape probabilities will not b e significantly affected . T he exact value of the inten sity func tion is often les s i mportan t than its form , which p rovides p hy sical insi ght into the nature of the flow . T he critical features of A ( t ) are recogni zable from the form of ln ( l F ) versus t as A ( t ) i s the derivative of - ln ( l F) . This can be seen from Fi gs . 6b an d Sb , w here ln ( l - F ) plotted versus t for the flow models given in Figs . 6a an d Sa . Fi gure 8c illustrates f( t ) for cases with stagnancy . Note that it i s much harder to interpret f ( t ) then ln ( l F) or A ( t ) . •
-
-
-
Age Di s t ri butio n Consider the thought experiment where a timer is attached to all the parti enterin g the system . Instead of collectin g them at the outlet , the whole sys tem i s stopped at some time t o to collect all t he p articles in side the reac tor . One can then con struct a distribution G ( t ) s uc h that G ( t ) measures the fraction o f p articles inside the reactor that entered after to - t . G ( t ) expresses the probability that a particle c hosen at ran dom inside the reactor has an age less t han t . It is also called the age dis t rib u tion . G ( t ) is related to F ( t ) by the relation
cles
73
Residence- a n d C o n t ac t- T ime Dis trib utions 3.0
2 .0 Al B ) 1 .0
0.0
0
2
e
(a) 10°
�
4
3
6
5
'
1- F (B )
10 2
L.._____. -----'-
0.0
(b)
0.5
1.0
1.5
e
2 .0
2.5
( a ) I n tensity function for some simple flow models : A , three stirred tanks in serie s ; B , one - dimensional diffusion model . ( b ) C u m ulative residence- time distrib ution for the models in p art ( a ) .
FI G URE 6
74
Shinnar
u,
v,
v
FI G URE 7
Schematic dia gram for a series o f stirred tanks in series with
forward and
g(t) =
bac k wa rd flow between the tanks .
1 - F(t)
E (t )
( 9)
f
w here g ( t ) is t he density func tion of G ( t ) . In a multistage system , one c an examine a sin gle stage Z an d define a local age distrib ution G z with respect to the inlet in the same way as one defines a local residence- time distribution F z ( s ee S hinnar and Naor , 1 9 6 7 ; Zvirin and S hinnar , 1 97 5) . T he same c a n b e done for a local zone i n a re actor if the flow i s n ot turbulent and the p roperties of the reactor are steady in this zone . A related fu n c tion is the
again
future
lifetime
the thought experiment where all
distrib ution A ( t ) . Consider mole c u les in side the re act o r are
1 .7 5 1.50 1.25
A<9J
1.00
0 .7 5 0.50 0.25 0.0
(a)
0
2
8
3
4
FI G URE 8 ( a ) Intensity function for three simple flow models described above . { b ) C umulative residence - time dist ribution for the models in part ( a ) . ( c ) Residence- time density function for t he models in part ( a ) •
Residence- a n d C o n tact- T i me Dis t ribu tions 10 °
75
r-��----�
1- F ( 8 )
2 1 00.0
2
3
8
(b)
4
0.6 0.5 0.4 0.3
f (8)
0.2
0. 1 0.0
0
2
8
(c) FI GURE 8
( Con tinued )
3
4
76
Shinnar
Thi s time , at t = t 0 set all t he timers to zero . T hen record t he time t for each molecule as it exits an d construct the dis t ri b u ti on A ( t ) . T hus A ( t ) is t h e probability that a particle which is in t he reactor at time to has exit�d before t 0 + t . One can also define an o t h er related distribution in side the reac tor . C on sider t he thought e x p e ri ment where all t h e particle s inside t he reactor are again provided wi th a timer . At a set ti me t 0 all t he clocks are set to 0 . One can t hen record the clock time s on the p ar tic le s at t he exit and con struct a di s tri b ution A ( t ) . A ( t ) is called the future lifetime distribution . I t can be s ho w n that A (t ) is e qual to G ( t ) ( s e e N oar and Shinnar , 1 9 6 3 ) . Once again , one can define a local future li fetime distribution A z ( t ) . How ever , A z ( t ) =1- Gz ( t ) . B oth Az ( t } and G z ( t ) can be measured and are sometimes useful in m o de lin g ( Zvirin and S hinnar , 197 5) , but this is outside the scope of t hi s chap ter . provided with a timer .
=
M u l ti p ha s e S y s tems
All t he definitions giv en until now refer to a sin gle fl ui d . I f a fluid is made up of different components , eac h of them may have a different resi dence - time distrib ution . I n a ho m ogeneou s reac tor , the differences between the residence- ti me di strib ution s are in general n egligib l e . However , in he t e ro gen eou s reactors , t he differences may be quite lar ge , as differen t compounds m ay adsorb differently or distribute t he mselve s differently be tween t he p hases in mul ti p h a s e reactor s . Here t he initial hy p�t he tical The system has definition can be very helpful in keepin g thin gs s t raight . to be steady state an d one must be able to tag a particle with a clock such that it retains its identity in passin g t hrou gh the reactor . If the com poun ds reac t , one may have to tag an element that does not change and ob tain a set of residence - time dis t rib ution s Fi ( t } for each s uch element . For each Fi ( t ) the definitions of fi ( t ) and A i ( t ) do not c han ge . T he only relation that does not hold is that 'i , the first moment of f( t ) , is no lon ge r equal to V /Q . T he holdup or average residence time ' i is now different for each el e m e n t and not known a priori . A detailed discus sion of this problem is given in a later section .
T R AC ER E X PE R I M E N T S : M E AS U R EM E N T O F R E S I D E N C E- T I M E D I S T R I B U T I O N S M ethods o f M ea s u rement
S tep C hanges in T racer Concen tra tio n
T he residence - time distrib ution discussed in the p recedin g section can be meas u re d in any s t eady - state sy s te m by use of a s uitable t rac e r . A ssume that at time t = 1 on e starts labelin g all the particles in the fe e d . One then measures the fraction of l ab el ed particles at the exit from the reactor at time t . Any labeled p a r ti c l e that appears at the exit at time t must have entered the reactor after time t = 0 s in ce before that time , no p articles were labeled . Similarly , any unlabeled particle must have entered the re 0 since after that time all particles were labele d . actor p ri o r to time t T herefore , the fraction o f labeled particles i s , b y definition , equal t o F ( t ) , which is the p robability of s tayin g in the reactor less than time t an d t h e fraction of unlabeled particles is equal to ( 1 - F ) . For s mall time intervals =
Residence- and
77
C o n t ac t- T i m e Dis trib u t io n s
it is easier to measure F , w hereas for lon g time span s it is more convenient to measure 1 F. I t is not neces sary to label all the particles . A ssume t hat at some in stant in time t :::; 0 , instead of labelin g all the p articles , one s tarts labelin g a con stant fraction C f of the particles in t he feed and then measures the fraction of labeled particles C ( t) at the exit . C learly , -
C ( t ) :::; F ( t ) C
f
Since one is u sually more interested in estimatin g F { t ) , this is usually expressed as
( 10)
In this case , t he fraction of unlabeled particles is no lon ger directly re lated to ( 1 F ) , as some unlabeled particles have entered the reactor after time t :::; 0 . I f one waits lon g enou gh , then a s t -+, F ( t ) -+- 1 an d C ( t ) -+- C r . I n practice , equili b ri u m i s reac hed w hen t » T , the ti m e con stant of the sy s te m . A t that point the fraction of labeled particles a t the exit will equal C f , the fraction of labeled particle s at the feed . O nc e the sys tem reaches equilibrium it is u se ful to stop suddenly the flow of labeled partic les (tracer ) . F rom this point on , the fraction of labeled particles at the exit will decrease monotonically . T he fraction of labeled particles , C ( t) at the 0 is then given by exit , measured from the new t -
""
:::;
[1
C (t)
-
F(t) ] C
f
or
1
-
F(t)
C (t)
:::; c;-
This is identical to the case in the precedi n g paragraph except that now it is the labeled particles that have entered t he system prior to time t :::; 0 . A step c han ge i n tracer i s the most acc urate way o f measurin g a residence - time distrib ution . C ( t ) / C r should be meas ured after the tracer is introduced until equi libri um is achieved and then after the tracer is stopped . I f one could measure the average concent ration < C / C r> inside the reactor after a s t ep input at time zero ( av eraged over the total volume at each instant of ti me ) , this yields an e s ti mate of G ( t ) , the age distribu tion o f the system . T hi s c a n be achieved experi mentally b y usi n g a radio active tracer and measurin g the total radiation emanatin g from t he reactor .
Pulse
Expe rime n t s
The step input experiments de scrib ed measure F ( t ) an d 1 F(t) . Another tion of a fixed amount m of tracer at of labeled particles C ( t ) at the exit . of the density function f( t ) given by -
in the precedin g section directly more commonly used tool is the injec t 0 an d then measuring the fraction T hi s leads to a direct measurement :::;
Shinnar
78
mf( t )
=
QC (t)
or ( 11)
QC (t) m
f( t )
w here m i s t h e amount o f tracer injected a t t = 0 a n d Q i s the volumetric flow rate . To have reliable estimates of f( t ) , one needs in dependent meas urements of Q and m . The relation
( 1 2)
C ( t ) dt
m
allows a n estimate of t he consistency o f available estimate s . I n t h e absence of acc urate knowled ge of eit her Q or m , one can use Eq . ( 1 2 ) to reformu late E q . ( 1 1 ) as
( 13)
C(t)
f( t )
foo
C ( t ) dt
0
T hi s allo w s an esti mate of f ( t ) from a p ulse injec tion in any steady - state system directly from the meas ured fraction C ( t ) . It is the most common w ay in which tracer p ulse experi ments are utilized .
Sin usoi dal Inp u ts
Consi der the case where the frac tion of labeled particles C f at the inlet varied as a func tion of time . I t can be s ho wn that f ( t ) is equal to the linear response function for t he tracer , or C (t)
=
f( t ) * C < t ) r
is
( 14)
where C ( t ) i s t h e outlet concent ration , C f ( t ) i s the inlet concentration , an d * denotes the convolution inte gral . I f one writes Eq . ( 1 4 ) in t he Fourier tran sform , then A
C (jw)
A
=
f(jw )
x
C (j w ) f
( 1 5)
E quation ( 1 5 ) is we ll known from c on t rol theory . One can therefore evalu ate f( t ) in t he same way that one evaluates the process transfer function in con troller desi gn . I n t hi s case C f ( t ) i s varied in a sinusoidal form . After waitin g for all transients to die out and for steady state oscillation in the output to be establishe d , the amplitude ratio C ( t ) / C f ( t ) and the p hase angle are found by recordin g i np u t and outp ut data ( see Fi g . 9a ) . T h e frequency of the input si gnal is c han ged and a new ampli t ude ratio and p hase an gle are de termined . T hus the complete frequency - re sponse c urve is determined ex perimentally by varyin g t he inp ut frequency over the ran ge of in terest . T hese measuremen ts are direct measuremen t s of the Fourier tran sform of
Residence- an d C o n tact- T ime Dis t rib u t io n s
Q(t) q( t )
(a) 101
=
=
Q sin
(w t )
q
(wt + B )
sin
w
8
79
=
=
PHASE ANGLE
�-----
------.
=- - - - - - - - ::..:� 1 0° -::1 ------r.::-:---
------,\...
AR
\
\.
I
. . ' '
'.
\
'
\
1 6 2 ��-rrn�r-.-rrn.mr-oL,rnmm-.-rrl nn�,-TTonm 1� 1 62 ------
S I N G LE C S T R
- ------ --- 5 - C S T R
IN SERIES
s
W T
- PA:a�LE� 7 --� --- S ING L E PFR
C ST R
Q .1
(b)
FIGURE 9
50 %
1
· - -- PA R A L L E L C S T R TRAINS N
7 B� %1 1 1 20%
--
Q•1
4
4
4
( a ) Sinusoidal inp ut - output c urves , w is frequency of inp ut (b ) Frequency response for some simple flow models- amplitude ( c ) Frequency response for the flow models in part ( b ) - p hase lag . ratio . (d) Laplace transform of the density function for the flow models in part signal .
(b) .
80
4 5.0
W T
{c)
4.0 (d) FIG URE 9
( Contin ued )
8.0
--.--
-----.----:-: 1 2�.� 0 1 0. 0
114.0
Residence- and C o n tact- Time Dis tribu t io n s
the
81
ty function :f(jw ) . One can look at f(ju' ) as a vector in polar co Th e amplitude ratio defines t he m a gni tud e of the v ec to r for eac h w , and the phase lag defines cj) . I f one can measure both the amplitude ratio and the phase lag over a wide range of fre quen ci e s , one has a good estimate of the Fourier transform of f(t ) . Some typical forms of these functions are given in Fi g . 9b and c . den si
ordinates ( r , cj) ) . r
Measuremen t of the Lap l ace T ransfo rm from
A rb i trary Inp u t s
In
some cases it is hard to perform an accurate step or p ulse exp e ri ment . might not be able to feed the tracer directly to the r e ac tor , or one might have a two-stage system where the inlet concentration to the second stage can b e measured , but the tracer cannot be fed directly to it ( 0ster gaard and M ic he l sen , 1 969) . For "all such cases , one can utilize Eq. ( 1 4 ) and obtain the Laplace transform f(s) of t he den sity func tion f(t) by One
C(s)output f( s) :::
( 16)
. put C(s) m
As
C(t) is a m easure d function , C (s ) must be obtained by numerical integra This is illustrated in Fjg. l Ob fo r several values of s . Measurement of f( j w) and f( s ) poses a problem . Although in theory it is
tion .
possible
f(t) f(t)
to obtain
=
=
1
2 1Tj
2 11 j 1
a+ o
fa- o J°
- cS
e
by inversion ,
" e st f{s ) dt f( j w )
jwt "
( 17)
d(jw)
depe n d s t ron gly on t he value� of f ( s ) ass -+ 0 or f ( j w ) as which are hard to me as ure . As f ( s ) is hard to invert , either one can try an d fit f( s ) with a reasonable mo d el and t hen invert it , or one can use the measurement of f( s ) to evaluate directly the properties of the residence-time distribution . Some typical forms of f( s ) are shown in Fig. the
re s ul t s
f( t)
w + 0,
9d .
In direct E s t imates o f f( t ) an d F ( t )
Indirect way s of measurin g residence-time distribution are discuss ed later . Thes e are based on t h e effect of residence- time distribution on conversion , especially on product distrib ution in complex reaction s .
Measuremen t of Moments of Residence- Time Dis trib u t ion The
f(t)
t h e residence-time di st ri b u tion can be computed from either F(t) by using Eq. ( 2) or
moments of or
82
0
8 UNIT:
z
�
0: � z ILl (.) z 0 (.) 1/) 1/) ILl _J z
Q
1/) z ILl
� 0
Shin nar
ECLP
6 INLET D E T E CT O R
4
2
0 0.0
(a} 10
0. 5
1 .0
1.5
D I M E N SION L E S S T I M E
2.0
2.5
°
C (s ) C0 (s )
(b)
s
( a ) T racer response o f a radioactive gas tracer i n a n Exxon Donor Solvent coal li q uefic ation reac tor . ( b ) Laplace transform of the re sidence- time distribution from part ( a ) ( top o f reactor ) obtained by FI G U RE 1 0
Residence- and C o n tac t- Time D is tribu tions lJ (f)
=
r
r
J
00
t
r- 1
83
( 1 - F ) dt
0
The comments on accuracy made before apply al so here and the step re sponse has a si gni ficant advantage . In homogeneous systems , the fir st moment can also often be estimated in depen dently from lJ
1
v = T = -
( 18)
Q
where V i s the volume of the fluid in the reactor . T his is an important check on the acc uracy of the measurement of f( t O . Relation ( 1 8) can only be used in homogeneous system s , as in a system where there is adsorption (or sol ution in another phase) indepen dent knowledge or estimate of the hol d up is difficult . In the case w here , instead of directly meas urin g f ( t ) or F ( t ) , the
Laplace transform f( s ) is me asured , there are two way s to obtain the mo ments . When the Laplace transform is obtained from meas urin g the response C 2(t) to an arbitrary input si gnal C 1 ( t ) ( see Fi g . 1 0) , one can get the moment s from the fact that the moments for t wo systems in series are addi tive . Recall that f(t) =
fi
( 11)
C ( t)
m
Thus if one now writes
lJ
1 = T1
.
=
and
w here l..l
Q_
!""
m 0
t C ( t ) dt
( 1 9a )
1 and V 1 ar e the mean an d variance , then ( 1 9b )
00
f( s )
=
f
l
-
st
C ( t ) dt
0
� 00------
J 0
1
- st
C ( t ) dt 0
where C i s the concentration a t the top and c 0 t he con ce n t rati on a t the 1 . 18 and inlet . T he curve can be fitted by [ 1/ ( 1 + T s /N ) ] where T N = 17. ( F rom T army , 1 983 . ) ,
=
S h innar
84
where V 2 an d T 2 are defined analogously to E q . ( 19a) . There is another way to extract the moments from l ( s ) , whic h is e s p e cially important for cases where f( s ) is obtained direct ly , as will be di sc ussed later . One can obtain the moments of f ( t ) directly from f ( s ) as shown in E q . ( 17) . Unfortunately , i t is hard to estimate acc urately the highe r derivatives near s + 0 , but one can us ually get a good estimate for the fi r st derivative by usin g a Taylor expansion , get a reasonable estimate for V an d y , the co e ffi ci e nt of variation . I f y is small ( t he flow is close to plug flow ) , V and y can b e estimated by a filterin g mo del such as f( s )
=
--"1'--- (1
T s /n )
+
( 1 9c )
n
which i s a good approximation for flows that are close t o plug flow . Equa tion ( 1 9c ) has the form of transfer function for s tirred tanks in series , b ut here n is used as a fittin g parameter . I f one can fit f ( s ) w i t h Eq . ( 1 9c ) , then y
2
=
1
( 19d)
-
n
varianc e from the Laplace tran sfer for ·cases close to plug flow can be derived as follows . T he measured q�ntitie s in Fi g . 1 0 From t hi s one can derive not only f( s ) b u t also a re C 1 ( t ) and C 2 ( t ) . d f/ ds :
Another way to get the
1
df( s )
f( s )
�
1
A--
=
A
1
( 2 0)
w h er e dC 2(s)
( 20a)
ds
Evaluation of the derivative of f ( s ) t herefore does not require any numerical di f ferentiation , but can be evaluated from the concen tration c urve s by in te gration . I f f( s ) can be approximated by E q . ( 1 9c ) ,
1 A
f( s )
df( s ) ds
=
-T (1 + :s )
-1
( 2 0b)
and f(s)
df( s ) /ds
=
s n
-
1 T
Plottin g the left side of Eq . ( 2 0c ) v e rs us s gives a strai ght line , the s lope of which is equal to the coefficient of variation .
( 2 0c )
Resi denc�:r"
an d C o n t a c t- T ime Dis tri b u ti o n s
85
Requi red P ro p e r t i e s of R eac to r s a n d T racers for
Mea s u reme n t s of R e s i dence- T i m e D i s t r i buti o n s
The most fre quen t p roblem t he author has encountered in the practical measurement of residence - time distribution is that not enough though is given to verify tha t C ( t ) /C f is really an esti mate of the resi dence - time distrib ution desired . T here are several i mportan t con ditions that need to be checke d . The c hoice of the tracer is very important . It s hould be acc urately measurable over a wide ran ge of concen tration s . T here are several other properties that also affect its c hoice . Furthermore , there are a number of other conditions t hat have to be fulfilled in order that the tracer experi ments mentioned earlier will lead to a proper estimate of the residence- time distribution . 1.
T he s y s tem m u s t b e a s teady s ta t e and s ta t io n ary in t he s e n s e t h a t
C ( t ) /C f is i n de p e n d e n t of the time a t which t h e trace r is in tro d uc e d i n to the fe e d As most flow s are turb ulent , this is not necessarily true for all
reactors . However , if the len gth - to - diameter ratio is large , the turb ulence effects are averaged out and at the outlet , this condition is satis fied at least to a first ap p roxi mation . T he same applies to systems equipped with mechanical agitators . I n that case the si ze of the large eddies is much s maller t han t he diameter of the reactor an d therefore are often averaged out in depen dent of the len gth- to- diameter ratio . This can be tested in t wo way s . First , t he s tep res ponse m us t be monotonous . A respon se s uch as the one s hown in Fi g . 11 cannot be considered as a residence- time distribu tion . The ability to apply this criterion directly i s a significant advanta ge Furthermore , two s uccessive tracer experiments in usi n g the s tep resonse . should gi ve t he same res ult w ithin t he expected exp erimental error . T hi s condition is not just a question of measurement . T he hypothetical experi ment of equippin g each molecule wi t h a clock had exactly the s ame require m en t It was ass umed t hat i t di d not matter when the particle entered . .
.
cleo
f
FOR S T E P I N PUT
, _
1 1 R e s p on s e to step inp ut for a system in whic h the response to tracer is a function of the time of injection . In thi s case , C ( t ) /C f does not give the residence- time distrib ution of the system . T his may be due to turbulent mixin g , fluctuation in t he flow rate , or large b ubbles , or other factors . FIGURE
a
86
S hinnar
In K ra m b e c k e t al . ( 1 96 9) the concept of residence - time dis t ribution is ex but t his is out si d e the scope of this c hapter . 2. T h e tracer response must be linear in t h e ra nge of t he s tep change . I f t he tracer particles behave e x ac t ly as one of the molecular compounds in t he steady - state system , t hi s condit ion will be fulfilled automatically . I f one can really tag particles i n a s teady - state flow , i t cannot m ake an y dif ference how many p artic le s are ta g ged . As lon g as the system is in st e ady s tate , it also does not m atter i f the transport processes are nonlinear . T hi s also applie s t o m ultiphase systems in which one of the compounds is eit her adsorbed on a solid s urface or dissolved in the liq ui d phase . As lon g as t he s y s tem is at s te a d y state in the sence defined in t he first criterion , each compoun d has a re s i de n ce - tim e distrib ution , an d if one finds a real tracer for t hi s compound , its response wi ll be linear , as it cannot However , t he only way to obtain matter how many particles are tagged . tracers wit h s uch p ro pertie s ( at least appro x imate ly ) is to take the real reactant feed and mark it b y exchan gi n g one of its atoms wit h an isotop e . S uch t racers a r e s eldom used and i n most c a s e s o n e uses tracers that are inherently different from t he fluid a t s teady state . There is no thin g w ron g with t hi s method as lon g as one recogni zes its s hortcomin gs an d takes them into account . I f the tracer is different from the steady state fee d , it may u p s et the s t e a d y - stat e condition s , and t here is no a p riori reason for a linear response. I f one w ants to obtain a residence- time distrib ution , one m ust verify ex perimentally that t he respon se i s linear . This is ac com pli s he d ' b y two suc ces sive tracer exp eri ment s wit h two differen t step magni tud e s which s hould gi ve t he same re spon se . W hat one ob t ain s in this case is a residence- time distribution of the tracer compoun d , w hi ch is not n eces sarily i d en tical with t he residenc e - time di st rib ution of the fluid in t he reactor . Therefore , one m ust add yet another criterion . T he tracer should be similar in b ehav ior to t he fl u i d in the rea c tor . 3. I t is hard to specify t hi s condition in a general way , as it dep en ds strongly on the specific sit uation . I n a homo ge n e o us stirred tank , almost any tracer will do , as diffusion proces ses have n e gli gi b le effects on F a n d there is no potential for adsorption . The tracer is c ar ri e d alon g wit h the liquid . This ap plies to most hom oge n eo u s reactors . I n s uch cases one seldom has to worry about linearity . On t he ot her han d , in a fluidized - b e d reactor , dif f u s ion an d adsorption are very important factors , and t he results depen d s tro n gly on t he tracer c hosen . W hic h tracer to choo s e is t herefore a ques tion not only of availability b ut also of what inform ation is s o u ght from the ex p e rimen t . In m ulti p h a se systems it is often ap p ro p ri ate to u s e several t racers , e ac h si m ulatin g a di fferent compoun d , or fluid p ro pe r ty . T his is d i sc us se d in detail later . t e n ded to s y st ems t hat lack this p rop e r t y ,
R e l a ti v e Acc uracy of T racer E x peri m en ts U ti l i z i n g S tep a n d P u l se I n puts
In principle , f( t ) and F ( t ) co n t ai n the same information and e a c h one can be d e ri v e d from the other . I n pr a c tic e , t he experimental errors are di fferent for pulse i n put s and s tep i n p uts . S tep inputs tracer experi ments have s i gni ficant advantages over pulse experiments , p rovide d that they are carried out in both direction s . T hese advantages include :
87
Resi dence- and Contac t - Time Dis t ri b u tions 1.
2.
I mmediate r eco gnition of some unsteady phenomena . T hese are illustrated in Fi g . 1 1 . I n a step experiment a ll one needs i s a kno wl e d ge of C f a n d C ( t ) . As they are meas ured by the same instruments , any bias in measur in g C f will also appear in C ( t ) and will ten d to cancel . However , it is important t hat t he step be carried out in both directions , as it is di f fi c ul t to measure C ( t ) / C f accurately i f it is close to unity ( lar ge t for F an d s mall t for 1 - F ) . "
I f one . can do this , a direct estimate of the other han d , in a p ulse expe ri ment C ( t ) of f(t ) . One must derive it utili zin g eit her Eq. ( 1 3) , t he problem of bi a s in measurin g
- F is obtained . On does not give a di r ec t estimate E q . ( 1 1) or ( 13 ) . I f one uses C ( t ) i s eliminate d , but one
F and 1
faces the problem of estimatin g the inte gral . If the residence - time distrib u tion is si mple and well b ehaved and has an expon en t ial t ail , a fairly ac cura t e estimate of f ( t ) is easily obtained . Other wise , t he overall form of f( t ) can us ually be derived , but t he relative magnit ude may not be accurate . One way to v e rify the accuracy of the me asurements utilizes Eq . ( 1 2) and com-
pares 1000
C ( t ) dt t o m / Q .
If possible . this s hould always be done , as it
greatly i mp rove s the reli abili t y of the measurement . However , in in dustrial reac tor s it is often impossible or very diffic ult to meas ure either Q or m ac c ura t e ly A step experiment measures F and 1 - F di rec t ly and therefore also allows a direct me asurement of t h e escape prob abili t y . I f one plots ln ( l - F ) versus time , all the in t erestin g feat ures of the escape probability (or intensity function ) will be directly apparent from the plot . On the other hand , to derive the intensity f u n c t io n h ( t ) from f( t ) usin g Eq . ( 7 ) , one must first compute [ 1 - F ( t ) ] by i n t e gr ati o n of f( t ) , w hich introduces in acc uracies . I n order T o minimize the error in es t ima ti n g A ( t ) , one should .
use the relation A (t)
C (t)
=
fa> t
The v al ue of 10
00
( 21)
C ( t ) dt
C ( t ) dt is es timated by plotti n g C ( t ) on a lo gari t hm i c plot
and ap p ro xi m at in g the tail by an exponential function . T he ti me constan t o f t h e exponent i s then estimated from t h e tan gent o f l o g C ( t ) for the largest values of t for w hic h the mea s u remen ts are s till reliable . In ge ne r al , the author does not recom mend t he use of frequency- respon se techniques or sin usoidal in p ut s for measurin g residence- time dis tribution s . T he only reas o n they are disc ussed i s that t hey mi ght b e available from control studies an d can t herefore serve as a startin g poin t . The reason for this broad s tatement is that whenever s uch experi ments are feasible , a proper step response in bot h direction s is also feasible , is less time consum in g , and pr ovi des far b etter information . I n control desi gn the oppo si t e is true . It is always best to get a direct estimate of t he property of the func1ion re quired for design . I n chemical controller desi gn one needs to know f(jw) and it i s pre ferable to measure it direc t ly , compared to es timatin g it from a step respon se . On t he ot her hand , to evaluate the flow p a t ter n , one
Shinnar
88
needs to know the escape probabi li ty and therefore a direct estimate of
[ 1 - F ( t ) ] i s p refer ab le .
Furthermore , the accuracy and the r an ge of w [( j w ) is neede d is different in t h e two c ases . I n controller desi gn , one needs t he overall form of f(j w ) an d accurate values n ear the phase lag of - 18 00 . Acc urate estimates of f(jw) for very hi gh an d very s mall w are of little i nte re s t in con t ro lle r desi gn , but are im p or t an t in eval u atin g at which
residence - ti me distrib ution .
in p u t that reason ably app ro xi or a pulse cannot be pe rfo rm ed , and one m us t often compro mize and utili ze other inputs . A pra c tical example is s ho wn in Fi g . 1 0 . In this sett in g one cannot directly meas ure F ( t ) of f ( t ) b ut c an still m easure an esti mate of t he Laplace tran s form . If o n e has an ap p roxi m ate flow model , one can fit it direc t ly to f( s ) as s hown in Fi g . 9d . This is less accurate than es ti ma ti n g [ 1 - F ( t ) ] by a step or p ulse e x p erime n t . With the latter on e can often get a goo d es ti m ate of the tail of t he function w hich determines the moments . On t he ot he r hand , to obtain reliable estimates of the tails from f( s ) , one n eed s to have acc urate meas urements of f( s ) for values of s w he re s -+- 0 . E s ti matin g t he s e co n d derivative near s -+- 0 from experi t m e n al ly determined f ( s ) is much more difficul t , and therefore estimates of variance are more diffic ult and have to be obtained by first fittin g a two parameter model of f( s ) and then es ti m ati n g the variance from t hi s model . T hi s p ro cedu r e involves some mistakes , but if t he variance is s mall , the re sul t s are often reasonably accurate [ 0st er gaard and Michels en ( 1 9 6 9) ] . An example i s shown in Fi g . 1 0 . I f one could p lot f( s ) i n a w ay equivalent to an escape probability , i t wo u l d a l low b e tte r in sigh t in to the meanin g of any deviation s of f( s ) from t he simplified flow models u se d to fit the experimen t al res ults . However , the author i s not aware of any met hod t h at would make thi s po s sib le . I t is sometimes p os si b le to use E q . ( 16 ) , In cases w here an e x p e ri m e nt wi t h an
mates a s tep
C(s) f(s)
=
output
C ( s) . t mp u
( 16)
to filter out the effects of meas urement noise , e s p eci ally on t he tai l . It sho uld b e s tressed t h a t t his m ethod i s alway s one of l a s t resort a n d that one alway s loses si gnificant information . In many cases the m o s t important information of t he t racer e x pe ri m e n t is c o n tai n ed in the tail ( t -+- oo ) and t h er e is no way to replace gettin g an ac c u rat e measurement of i t . U s in g E q . ( 1 6 ) merely o b s c u r e s t he fac t that the me as u r em e n t noi s e prevented an acc urate measurement of the t ai l of f ( t ) . The me thod is sometimes useful w hen one knows t ha t the contrib ution of t he tail i s small or when one is interested in b y p a s s p henomena [ which dep en d on t he pr op e rt ie s of f( t ) n ea r zero ] . In all s uc h cases it is importan t to us e an adequate number of values of s in t he ran ge 0 < s < T , where T is the exp e c t ed average residence time . B ut one s hould always be aware of the s hor tc om in gs of this m et ho d .
Residence - and C o n t a c t - Time Dis trib u t ions U S E O F R ES I D E N C E- T I M E D I S T R I B U T I O N S I N R E A C TO R MO D E LI N G A N D T E ST I N G O F I N D U ST R I A L R E A C T O R S
89
Testi n g fo r M a l d i s t r i bu tion i n I n d u s t r i a l Reacto r s Residence - time dis trib ution s a re a n i m p ort an t di a gno s tic tool in e valuatin g the performance of i n d us tria l reactors as we ll as in s ettin g up si mplifie d
flow
T hey c a n be utili zed fo r m o de li n g both homogeneous and T he most i mport a nt a pp lic a tion i s p robably chec kin g for m al distrib ution in flo w s . A la r g e number of industrial reac tors a re de In the signed to a pp roxim at e either a s tir r ed tank or a plug- flow reac tor . first case good mixin g i s essential . I n the second case , s uch as a packe d b e d re ac tor , i t is very i m p o rt ant t hat t h e fee d i s w e ll distributed over the c ro ss section and t hat t her e be n o b y p as sin g due to fa ul ty in ternals or de fec tive cataly s t lo a din g . I f a reac to r does no t p erform as p re dic ted , it is importan t to find out w hether t he poor pe r forman ce is due to an in he ren t scaleup prob le m , a faulty cat aly st , or a wron g mechanical d e si gn . A trac e r experi me n t can be a va luab le tool in distin guishin g between t he se pos s ib ili ties . If there are similar reac tors in op er ation which are acces s ib le , one can mode ls .
he t eroge neous reac tors .
con duct a t ra ce r experiment on bot h r e acto r s an d compare t he res ult s .
Oth e rwise , one can c o m p a re the results of the trac er e xp erim ent to those
p redic t e d by an ideali zed mode l .
T hey key q uestion i s how one in t erp ret s any devi ation s between the re sults of t wo r eacto r s 01' bet ween t he actual per for manc e and the mode l . Here t he escape p rob ab ility o r in t en si t y func tion , A ( t ) , i s a u s e fu l tool , a s i t pr ovi de s o n e with immediate in si ght as to
what
or
t he
deviation s m ean .
comp u te A ( t ) .
One does not even always have to actually plot
Wi t h some e xp erienc e on e c an recog ni z e t he i mp o rtant
features of the e sc ape probability from a plot of ln ( 1 - F ) versus t . T her e fore , w hat is really ne eded i s a reliable estimator of ( 1 - F ) whic h ca n usu ally be obtained from tracer experi m e n t s . T h e followin g two examples will
illustrate the method .
Imperfect Mixin g and Bypassing in S tirred- Tank Reac tors There are several problems that c an occur in the scaleup of stirred- tank reactor s .
One of t he m is t h a t the mixin g is in s ufficient to en sure that the
enterin g flui d mixes s uffic ien tly r ap i d ly with the contents of the reac tor
[ (Evan gelista et al . , 1 9 6 9b ; S hinnar , 1 9 6 1 ) .
In hetero ge n e ou s stirred- tank
t r an s fe r bet ween the p h a ses , uniformity o f turb ulence , and the other complicatin g fac tor s whic h A l t h ough these p rob are di scussed in S hinnar ( 1 96 1 ) an d T army ( 1 98 3) . lems can be studied u sin g tracer experiment s , they involve methods tha t are outside t he scope of this c hapter . T here is on e p roblem that commonly occurs in the sc ale u p of a s tir red tank reactor and can be effe c ti v ely dealt with u sin g the residenc e - time distrib ution . T his is t he case of a d e si gn where t he escape probability is to remain con stan t and is discussed belo w . When try in g to analy ze w hether the e sc ap e p robability is re al ly constant , consider the type of deviations to be e xp e c ted . One would always expec t some de vi ation a s t + 0 , as there is a minimum fin ite time for t he p ar tic l e reactors there are the additional p roblem s of mass
90
Shinnar
2
�'"'\
A( 8 )
I
\, o
IY"' \
1
\
....._
r'--A �-/...I. 0
J
__
//
I ;, 18
?----------------- - - - - - -- - - -
...... ....... .... -- -c- - - - - - - - - - -...... --::-. -.::::::: .::::: .. -
e
FI G URE 1 2 Inten sity function for i m p er fe ctly mixed stirred- tank reactors : A , short delay between inlet and outlet ( normal cas e ) ; B , delay between inlet and outlet due to insufficient sti r rin g ; C , b yp as s between inlet and outlet ; D , bypass b etwe en inlet an d outlet and stagnant r e gions due to in sufficient s t irri n g .
Howeve r , this time is usually to reach the exit ( s e e Fig . 1 2 , c u rve A) . small comp are d wit h the av e ra ge re s idence tim e T , and if T is large , on e would not notice it . I f the mixin g in te n si t y is too low , this in itial deviation might be quite la r ge , e s p ecially in tall reactors with viscous fluids . In that case one would e xpec t A (t) to look like curve B i n Fi g . 1 2 . This could be re m e died by i n c rea sin g the agitator sp eed or the agitator desi gn . Another com mon type of faulty pe r form ance is shown by c urve C in Fig . 1 2 . It occ urs i f the in l et and outlet pipes are close t o each other and mixin g i n te n si ty is insufficient to distribute the incomin g fluid sufficiently fast over the total reactor . A fraction of the liquid will reach the outlet dir ec tly creatin g a bypas s . For small t on e could then expect A { t ) to be significant ly hi gher than the average expected value , as enterin g p a rtic le s have a chance to be in that bypass . I f a particle has s t aye d in the reactor lon ger than it takes to reach the exit via t he bypas s , its c hance to exit is n ow lower t han in the be ginnin g and A ( t ) will exhibit a maximum at low values of t . I f the vessel is really badly mixed , A ( t ) wi ll even have a decreasin g tail ( Fi g . 1 2 , curve D ) . A si gn i fic an t a dv an t a ge of r e p r e s e n tin g the information con tain e d in the residence- time di stribution b y plo t tin g A ( t ) is that one can plot the expected form of A ( t ) without any comp utation s , a s A ( t ) co rr e s p on ds to the p hy sical concept of the sys tem . An actual example taken from an industrial reac tor is shown in Fi g . 1 3 ( Murphree et al . , 1 964) . T he conversion in the re actor was m uch lower t han expect ed from the pilot plant res ults . A step in p ut t racer experi m en t was performed an d the result is replotted in Fi g . 1 3a , curve E ( one i mpeller) . Fi gure 1 3b gives ln ( l - F ) a n d Fi g . 1 3c the c o r re sp on di n g curve . T he escape p rob ability is similar to c urve B , in Fi g. 12. I t in dicates i mperfect mixin g . The mixer was modified an d e q uip p ed with t wo impellers and a gain tested by a t racer step inp ut . T he results
Res i de n c e - a n d Con tac t - T ime Di s t rib u tions
91
are also given in Fi g . 1 3 , c urve G , and t he es c ape probabili ty is similar to curve A , in di catin g goo d mi xin g . T he new mixer al s o increased the conver sion to that expected from t he pilot plant . T he tra c er result s in Fi g . 1 3 , illu s t r a t e another aspect of the use of re si d en ce - tim e distributions t hat is not e vide n t from t h e escape probability B oth curves refer to a reactor with t he same volume and volu as plo t t e d . me tric feed rate , and therefo re the same avera ge re si d en ce time T . O n e can in bot h cases es ti ma te T from !0
oo
(1 -
F) dt u sin g an exponential tail
and c o m pare it to i t s known value . For curve G the estimate i s correc t , whereas c urve E u n de re s tim at e s T by 30%. T he fac t that c urve E under estimates T show s that the reactor is not well p erf us e d and has a stron gly stagnant re gion w hic h does not appear in t he plot . I f one could measure F ( t ) acc urately for lon g ti me s , curve E must cross curve G for the t wo in tegrals to b e equal . The true escape probability would look like c u rve D in
Fig . 12 . I f T i s eq ual for the t w o case s , then curves of F ( t ) m u s t cross ea c h other . Therefore , in d epen d ent knowledge of T provi d e s some i m po r t an t checks on the acc urac y of a t rac e r experimen t . A fairly constan t es c a p e p ro b a bility , in depen den t of history , which is characteristic of a w e ll - stir re d tank , occ u r s in quite a num b er of other situ ations , such as solids reacted in flui d beds , an d t he test describ ed here appli e s to all these si t uati on s . Packe d -B e d a n d T ri c k l e - B e d R eac to rs
In the desi gn of p ac k e d - b e d and t ric kle - b e d reactors , it i s i mportant to en s ur e t hat the feed is we ll distributed and th a t th e cat aly s t is a rran ge d in s uc h a w ay that t here are no p r e fe r r e d p a t h w ay s wi t h lower pre ssure drop s
F
ON E IMPELLER 0
1 .0
(t l
o.s
o
0. 5
(a )
0 0
o o o
0 0
o o 0 o o o
TWO I M P E LL E R S
a a 0
0
1 .0
°
o o
1.5
o@
a@
2.0
T I ME -
FI GURE 1 3 Residence- time distribution for reactors with i m per fe c t m1xm g . ( a ) Experimental residence - time distrib ution for a s ti rr ed - t ank re actor ( from ) Woo dr o w , 1 97 8 ) . The reactor did not matc h the pi lot plant and gave low conversion . A t rac er experiment showed a s ever e maldi strib ution of the flow . A s eco n d impeller wa s install ed , whic h im p rove d the flow pattern as m e a s u r e d by a residenc e - time d is t rib ut io n a p p roac hi n g an i deally s ti rred tank . T he reactor then matched the pilo t plant . Here the i mp e r fe ct mixin g is not directly recogni zable fro m curve E . I t can only b e recogni zed by c o mp a rin g the estimate of f ( 1 - F ) wi t h the kn own value of T e s tim a t e d from volume divided by flow rate ( see the t e xt ) . T his indicates that in case E t h e reactor is not w e l l p e r fu se d . ( b ) C um ulative re sidenc e - time di s t rib u tio n . ( c ) E scape p rob abilit y (in t e n si ty func ti on ) .
92
Shinnar
Gli
B
D 0
0 D
D
D
D
0
0
0 0
0
T W O I MP E LL E R S
D
0
0
0
0
0
0
0
0
D
0
1 - F (I)
0
0 0 0
ONE IMPELLER
o
0
0
TI M E -
(b) 3.0
ON E IMPE LLER
..1\ ( t }
2 .0 1.0
(c)
0 0 0 0 0 0
0
0
o o 0 0
0 0 0 0 0
T W O I M PE LL E R S
D 0 0 0 0 D D 0 0 0 0 0 0 0 0 0 0
0. 5
FI G U RE 1 3 ( Continue d )
1.0
2 .0 1.5 TIME -
Res idence- and Contac t - Time Dis tribu tions
93
which create a bypass . B efore proceedin g with analytical examples , one must first define the concept of bypass and stagnant regions . A well- designed packed bed approaches plug flow and will lead to a A ( t) function s uch as curve B in Fi g. Sa . Most of this is due to the fact that in a randomly packed bed , the path len gth for different molecules is unequal and part of it is due to mixin g in the interstices between the par ticles . W henever f( t) is not a delta function , one can in theory represent the pathways of the individual molecules by an organ pipe model such as the one shown in Fig . 14 . As the pathways are of unequal len gth , one can conceive of the shorter one as a bypas s . In general , this is not the prime concern of the diagnos tic test . One is lookin g for m uch stron ger differences caused by maldistribution of flow . One can utilize the residence- time dis trib ution in this settin g because the typical flow distribution in a packed bed has a A function similar to that of curve B in Fi g . 5a with no peaks . If the maldistrib ution is stron g enough to cause a peak in the A function , it indicates that one i s dealing with maldistribution in flow and not simply a backmixin g p rocess . There are some exceptions to t he foregoin g s tatement . I f a tracer is adsorbed , it could lead to a peak in 1\ The same applies if there is a stron g diffusional resistance inside the catalyst particle or a mass transfer resistance to the particle . Theoretical examples of such cases are illustrated in Fi g . 1 5 , which shows two examples of a packed bed with a significant chromatographic effect . The residence- time distribution s in Fig . 1 5 are based on the standard equation used in chromatography or ion exchan ge ( S hinnar et al . , 1 97 2 ; Hiester and Vermeulen , 1 9 5 2 ) . T hese c a ses demon strate that to interpret a residence- time distrib ution properly , one needs ei t her a good understandin g of the system or a good data base , such as a tr acer experimen t from a pilot plant or a similar in dustrial reactor . Exam ples are illustrated in Fi g . 1 6 , which shows tracer studies from an indus trial tric kle · bed used for hydrotreatin g fuel oil . C u rv e A is from a pilot plant and is similar to curve B in Fi g Sa . C urve B is from an industrial reactor t hat performs well an d shows a small peak in A ( t ) . Curve C is from an industrial reactor that di d not perform well . I t shows a strong peak in A and indicates a strong bypas s . Shinnar ( 1 9 7 8 ) reports that this bypass was correctable b y a change in the design which led to better reactor performance . I f the bypass is really stron g , one would expect f( t ) to be double peaked. This occ urs rather infrequently and a stron g peak in 1\ is a much more sensitive test . For a packed bed one can also check the flow distribution from the Laplace transform . For near- plu g- flow conditions the Laplace transfer {unction should be oJ the for m exp [ - 1: i' ( s ) ] . One can check the form of P ( s ) by plottin g ln [ f( s ) versus s . For one- dimensional diffusion with very large Peclet numbers , •
.
FI G U R E 1 4
Alternative flow model for a packed bed .
94
f(s)
Shin nar
-
exp ( Pe / 2 ) [ 1
( 1 + 4 -r s / P e )
1/2
( 22)
]
Equation ( 2 2 ) is approxi mate on ly for hi gh values o f P e ( Pe > 2 0 ) . On e can ( 2 2 ) by plottin g ln [ 1 /f( s ) ] - 1 vers us -r s {ln [ 1 / f ( s ) ] } - 2
check the fit to E q .
·-
which s hould give a s t rai ght line with s lope -rs and i n te rc e p t 1 / Pe ( 0s ter gaard and Mi c he l s en , 1 9 6 9 ) . However , plottin g ln [ f ( s ) ] ve r s us s might be s u ffi ci e n t an d should also be close to a strai ght line with s l op e - 1 n e a r s + 0 . I f in this plot the slope near zero is less than -r or there are 1 / Pe] ,
s t r on g c han ges in t he slope ( see Fi g .
9 ) , it in dic ates a s tron g deviation It is extremely diffic ult to dis tin gui s h sole ly by mean s of a tracer exp eriment bet ween maldistribution s ca u s e d by a ba d di strib utor and those c aused either by di ffu sio n proces ses in side the particle or by adsorption p roc e s s e s ( see Fi gs . an d 2 6 ) . For t un ately , in an industrial case , one often has a reference case from another we ll - p er formin g reactor which can be used for compari son . One s hould , fro m a well- dis trib uted plu g - flow reac tor .
15
2 .0 ,...-----,
1 .6
f {I)
1-\ I \
I I I I
1 .2
\
I I I I I
0.8
\ \ \ \
\
I
0.4
\ \ \
).. = 1
\ ).. \
= 10
\
\
'
...... ""!- _ ��--...... 0.0 L____L____,.L_____J____-===-3.0 2.0 1 .0 0.0
(a} FI G U R E 1 5
( a ) Resi dence - time distrib ution of a chromato grap hic
( F rom S hi n n a r
f
=
et al . ,
e->-• [o(t -
( b ) C u mul a tive
-r)
1 9 7 2 ; H i e st e r and V e r me u l e n +
; e - A. ( t - -r ) / r�
t
1� ,
, 1952 . )
I � >. � l( t r
re sidenc e - time distrib ution for the column
( c ) E scape p robability
( inten si ty
in
c olu m n :
•>)]• = 0. 5 part ( a ) .
func tion ) for the colu mn in part ( a ) .
r
=
1
95
Residence- and Contact - Time Dis trib u t ions
' 1-F ( t )
'
'
'
'\
'
'
'\
\
\>.
\
=
10
\
\
\
\
(b) 6
r-------� _ .... ...... / ,..,
4
A(t)
2
I
I
I
I
I
I I
I
I
I
/�
/ =
/
/
//
10
>.
v
=
1
0 L-----L-----�--�--� 0.0
1.0
(c) FI GURE 1 5
( C ontin ued )
3.0
\
S hinnar
96 3
z 0
� 2
1-
P I LOT P LANT
:J l.L. >1-
(/) z w 1z H
INDUSTRIAL R E A CTOR ( C )
0
0
10
FI GURE 16
et al . ,
20
TIME
( min )
T racer experiments in a
30
40
trickle -bed
1 978 . )
reactor .
( From Shinnar
however , be careful when usin g pilot p lant re sults as a comparison unless
the pilot plan t has the same linear velocity .
A ch an ge in linear velocity
c han ges the m ass transfer coefficient s an d t herefore also the residence time di strib u tio n in a packed bed .
trickle bed .
It also c han ges t he flow re gi me in a Often this can be predicted by fluid dynamic condition s .
U se of RT D i n Reactor Mode l i n g
T racer experi men ts a n d resi dence - time distribution s are just one of many One seldom starts in a vacuum tools in s tudyin g flow systems in reactors . and us ually has other information about the r e actor In principle , one As F ( t ) meas ures a linear p roperty wan ts to un ders tan d its fluid dynamic s . of a n onlin e a r system , the information obtainable from it is alway s incom p lete , although it can be quite importan t . I f a reactor model is available , one c an compute the expected res ults of a tracer expe ri me n t and comp are it with the actual experimental resul t s . I f one w ants only t o con firm t he model or meas ure its parameters , doint it via a re si d ence - time distrib ution is an unneces sarily comp licated te c hnique However , the use of residence- time distrib ution can be of benefit in two important ways . Firstly , one n ever exp ec t s models to be completely accurate , so t h a t re sults which deviate sli ghtly from t he prediction s of t he model are not sur U sin g I\ ( t ) gi ve s a p hysical i n si ght a s to t h e meanin g o f the de p ri sin g . S econ d , h ( t ) gi ve s in formation in a form th at i s a t leas t partially viation s . .
.
One still has a p hysical model o f s t e a dy st a te and tracer be havior b ut does not need any a p ri o ri assumption s as the the exact flow model . I nterp retin g the res ults in such a way forces one to consider what model free .
-
Residence- and Contact-Time Dis t rib u t ions
97
other explanations or alternative flow models may be consistent wit h the observed data . T his type of approach is very important in any reactor desi gn p roblem ( Overcashier et al . , 1 959) . As a n example , one might cite some o f the early tracer experiments in fluid beds , w hic h are represented in Fig . 1 7 . A low fluid b ed has a tracer response quite similar to a stirred tank , b ut representin g it in the A ( t ) domain im mediately indicates that one i s dealin g with a bypass p roblem , which was s ubsequen tly confirmed in other ways . It is often useful to have a knowledge of the p roperties of residence time distrib utions o f different flow models . These are shown i n T able 1 . One must always remember that transport processes are nonlinear and cannot be described adequately by simplified models . F ( t ) is a linear property of such a model and the only time one can use it to p redict re actor performance is in first- order reactions in homogeneous system s , a problem that is not often encountered by the reaction en gin eer . However , simplified models are a very valuable tool . I n S hinnar ( 1 978) the term learning models was introduced to distin guish them from models used in actual desi gn predictions . T hey provide an un derstandin g of how the tran s port processes might affect the chemical reactions an d give some guidance for de si gnin g the scaleup . Fortunately , many reaction s are not very sen sitive to scaleup . N evertheless , it is important to be able to recogni ze those cases where significant scaleup problem s may be expec ted . There is one important case where models derived from t racer experiment s are di rectly useful in reactor desi gn . In many reactor p roblems , a plug - flow re actor is t he op timal configuration or if not optimal , is the only design that is safe for scaleup . As real reactors are seldom true plu g- flow reactors , one wants to know how closely the desi gn approaches plug- flow and how the deviations could affect reactor performance . Here one utilizes the fact that if deviations from plu g flow are small , one s hould get a reasonable estimate about their impact from any model that has a similar residence- time distrib ution . T his is equivalent to an asy mp totic expansion around a solu tion retainin g only the first -order term s . In such a case , one could use either a model based on one- dimen sional diffusion or a stirred tank followed by plug flow or a series of stirred tanks . The latter is preferred as it is easier to compute , and t he additional complexity of a diffusion model is not justified for cases where the real phy sical transport processes are not molecular diffusion . It is also more similar in its form to act ual measure tracer responses , as compared to a sin gle stirred tank followed by a plug flow reactor .
It is common to derive t hese simplified asymptotic models by demandin g the variance of the residence- time distribution of the model is equal to that of the actual measured residence- time distribution . The variance is then expressed as an equivalent P eclet number by the relation Pe : 2/y 2 , where is t he coefficient of variation as defined earlier . For a series of stirred tanks the equivalent P eclet number is approximately 2n . It will be shown later that such models give similar kinetic performance as long as the deviation from plug flow is small . These rela tio n s make sense for Peclet numbers larger than 10 ( pre fe rab ly 20) . For reactors in whic h the P eclet numb er is s maller , simplified models based on one- dimensional diffusion or series of stirred tanks make no p hysic a l sense ( unless , of course , one is really dealin g with three stirred tanks hooked up in series ) . that
y2
98
Shinnar
1 .0
0. 8
0.6
�
0. 4
"
1 - F (BJ
""
"
0. 2
0.1
"
�
"
e
(a )
"
"
3
2
0.0
"
4
2.0 1 .5
A
1 .0 0.5
0.0
(b)
0.0
0.5
1.0
1 .5
e
2.0
2.5
3. 0
3.5
FI GURE 1 7 ( a ) Experimental cumulative residence - time distribution for a fluidized bed . ( b ) Experimental inten sity func tion for a fluidi zed bed . Solid line : flow of gas in fluidi zed-bed reactor ; dashed line : completely mixed reactor ( for comparison ) .
99
Residence- a n d C o n tac t - Time D is trib u tion s TAB LE 1
P rope r ties
o f Some T heor e ti c al Residence- Time Distributions
I deally mixe d vessel Density :
f(t)
= .!T e
C umulative di st ri b uti on :
1
F (t )
-
-t / T
=
e
-t h
Expected value :
= 1
Coe ffic ien t of va ria tion :
y
Laplace trans for m :
L(f ,s) = E(e
I ntensity func tion :
f
1\ ( t )
-st
)
1-
-
-
-
1 +
TS
1
= T
I deally mixed vessels in serie s ( T is the parameter o f eac h sin gle vessel ) n - 1 - t /T e 1) ! T (n
D e n si t y :
f( t )
C u m ulative distrib ution :
1 - F(t)
(t / T )
-
=
e
E (t) f
Coefficien t of variation :
y
Laplace trans for m :
L (f , s) - E(e
Intensity function :
1\
Special values :
f
=
(t)
A ( 0)
i
i!
i=O
Expected value :
=
( t} T )
� 1 L.J
-t /T
nT
1
,;n
--
=
=
- st
1
-
) -
1
+
TS
n- 1 /( n - 1) ! (t /T) ,.. n - 1 )i 1 . , 1. T L. i=O ( t I T
0
1
fl ( oo ) = T
}
(n > 1)
Two nonidentical ideally mixed ve ssels in serie s ( T and T 2 are the 1 parameters of the t w o ves sels )
D en si t y :
f( t )
=
'1
1
-t / T
'2
(e
1
e
-t /T
2)
S hinnar
1 00 TABLE
1
( C on tin ue d )
Cum ulative distrib ution :
1 - F ( t)
=
1 _ ;:.._ ..:;
_
1
1
Expected value :
Coefficient of variation :
2
T1 Laplace tran sform :
In ten si ty func tion :
Special values :
L( f , s )
1\ ( t )
1\ ( 0)
E(e
=
11 2
- st
T
+
) =
2 2
�---7-:-.,...-----:(1
1 1 1s ) ( 1
+
-1
+
1 2s )
-t /1 t/ 2 1 e __--,-___e'----:--
-t/1 1
=
J\ ( oo )
2T
+
-t/1
- 1 e '/,
2
0
=
min (;1 ' ) 1
=
T2
Plug - flow ve s se l Den si ty :
f( t )
C um ulative distrib utuon
1
=
unit Dirac function
F(t)
=
Expected value
f
t �
1 1
= 0
Coefficien t of variation :
y
Laplace trans form :
L ( f , s)
I ntensi ty func tion :
1\ ( t ) =
T wo
t <
{�
=
E (e
{�
- st
)
t :f t =
=
e
- sT
1 1
-
parallel , n oniden tical , i d ea l ly mixed ve ssels [ fraction II passes through T 1 ; frac tion ( 1 II) p a s s e s t hrou gh second vessel wit h parameter T 2 1 one ve ssel wi t h p arameter
D e n sity :
f(t)
1 - II
+ --- e
T
2
-t / 1 2
1 01
Reside nce- and Con tac t- Time Dis trib u ti o n s TAB LE 1
( Contin ued )
C um ulative distrib ution :
1 - F(t)
lie
=
-t/T
1
( 1 - II ) e
+
-t/T
2
Expected value :
Coefficient of variatio n :
Laplace tran sfor m :
I n ten sity function :
y
f
=
L(f , s) = E(e
A (t)
-st
( II / T ) e 1
II
..,..1-s --
) =
+
-t/T
l
A
( 0)
+
1 - II 1 + T s 2 - t/ T
+
( 1 - II h 2 ) e
-t/ T
2
= ------,--------:---
-t/T
II e
Special values :
T1
II
= -
+
1
-
1
II
+ ( 1 - II ) e
2
---
'
2
T w o parallel vessels , one ideally mi xe d , one plug flow [ fraction II passes through plug flow v e ss e l with pa r a me t e r T 1 ; frac tion ( 1 - II ) passes
t hro u gh i deally mixe d vessel with parameter < 2 1
Density :
f(t)
=
1 -
f( t ) = II
Cu m ul ative di s t rib ut ion :
II
---
1 - F(t)
'
2
x
=
{
- th e
Coefficient of variation :
T
1)
unit Dirac function ( t
II + e
e Expected value :
( t :f.
2
-tJ,
-t/ T
2
( 1 - II )
2 ( 1 - II )
=
T ) 1 t <
t ;
'1 '1
1 02
TABLE 1
Shinnar
( C ontinued )
L ap l ac e t ran s form :
L( f , s )
I n te n sity
A (t)
fu nc ti on :
=
E(e
=
S pecial value :
A ( O)
)
=
ITe
-t !c (1
A ( t)
- st
-
IT ) e
2
- s -r
1
+
(1
-
IT ) -:-= 1 +'--' s
1
2
+ IT
1
=
-
=
---
1
-
rr
E s ti mation of Model Pa rameters f rom R e s i dence- T i me D i st r i buti o n A s di scusse d in the p receding section , to esti mate the kinetic impact of s mall de v i ations from p l u g flow , one e stimates an equivalent P eclet number from the coefficient of variation of the residence - time distrib ution , One can direc tly use either f( t ) or F ( t ) [ s e e E qs . ( 1 ) to ( 3 ) ] . In the litera ture , re sidence - time distributions are often used to estimate simultaneously several param eters of a complex mocel . The author doe s not recommend this . Resi dence - ti me distributions a r e hard to measure accurately and are quite constrained . In a homo geneous flow , the first moment is fixed by V /Q . T he limit s of experimental accur acy seldom allow one to measure more than t wo additional momen t s . Often only one additional moment , namely the variance , is all one can reasonably expect to obtain . Therefore , one can not estimate more t han one or two param eters from such experiments . Fur thermore , the models them selves are not accurate , and estimating multiple param eters for approximate models from a si n gl e exp eriment i s a doubtful procedure at best ( Overcashier et al . , 1 9 5 9) ,
U s e of T racer E x pe ri ments i n Col d- Flow Mod e l s and D e si g n of R ea c to r I n te rna l s T racer experi ments and residence- time distrib ution s are becomin g a widely tool in this area . In principle , t here is very little difference in the approach from t hat w hich was disc ussed in the p reviou s sections . One w an t s to make sure that one of the limitin g models , such as a stirred tank or plug flow , is app roxi mated as closely as possibl e . Q uite often one uses multiple tracers ( which will be discussed later ) , b ut the p rinciples remain the same . The followin g two exam ples may help illu strate this point , Baf fles , especially horizontal baffles , allow one to modify the complex flow in a fluid bed such that it approaches plug flow quite closely . Properly de signed baffl es achieve this by breakin g up the bubbles and stagin g the bed ,
used
1 03
Residence- a n d C o n tact- Time Dis trib u t io n s
reducing solid mixing bet w een top and bottom and th e reby also reducin g recirculation of gas either adsorbed on the catalyst or entrained in the dense p ha s e inside p ar ti cle clusters . An example of this effect and its me as u re m ent by a tracer expe riment is gi ven in Fig . 1 8 [ for details , see Ov er ca s hie r et al , , 1 9 59] . Fi gure 1 9 gives an example of the study of gas dis t ribu tor s in fluid beds. In Fi g . 19 some of th e f( t ) plots ar e double peaked , w hich clearly indicates a maldistribution . Here both f(t) and the e sc ape probability A ( t) clearly indicate that the desi gn of the dis t rib uto r is unsatisfactory and leads to a maldistrib ution in t h e overall reactor , w hich can be corrected by a proper design . This example ill u st r a te s the importance of a p rop e r base case in comparing al t ern ativ e s Fluid beds , especially short fluid beds , have maldistributions of flow caused by the formation of larger bubbles . These are due to inherent hy drody n ami c ins tabilities and c a nno t be pre vented by the desi gn of the distributor . In the cases shown in Fi g. 1 9 , some o f t he distributor d e si gn s give maldistributions which are much more p r on ou n ced c om pa r e d to the best achievable case , which is a p oro u s plate . R e siden ce ti me distributions cannot distinguish between t h es e two ph enomen a and should always be considered only an additional , al bei t important dia gn os tic tool . .
-
BAFFLED
UNRAF F L E D
2
(a )
(8 )
4
FIGU RE 1 8 ( a ) E xp e ri mental residence - time distribution for gas fl ow in a fluidi zed - be d reactor ( gas velocity = 1 . 6 ft / s ec ) - inten si ty function . ( From Overca shier et al . , 1 9 5 9 . ) ( b ) Cumulative residence- time distrib ution for the flow in part ( a ) .
1 . 0 ,...---..;;;;::-----,
o.e
0.6
0.4
0.2 1·F( 8 )
0. 1
0.0 8
0.0 6 0.04
0.0 2
0.01
0
2
9
( b)
3
4
( Continue d )
FI G U RE 18
f (t)
s s no z z l e p l a t e s an dw i c h g a s d i s t r i b u t o r
poro u s p l a te
c
Umf
_ u_
Umf u
=
=
1 .7
2 .0
F I G U R E 1 9 Residence- time distribution measured wi th di ffe re n t gas distribu tors in a fluid bed , as a fun c tion of fluidi zation indexes u / u ( From Woodrow , 1 9 7 8 . )
mf
"
1 05
Res idence- and Con tac t - T ime D is tri b u tions Mea s u remen t of Flow R a tes a nd R ea c tor
Volume o r Hol dup
In hydrology and physiology , tracer exp eriments are wide ly used to measure
flow rates and volumes [ Wein stein and Du dukovi c ( 197 5) ] . In reacti on en gin eerin g it is possible in most cases to measure flow rat e s directly . Excep tion s are reactors with internal circulation caused by density an d pressure differences , where t racer experiments are sometimes useful to measure flow It is not a very ac c urate method and should b e used rates u si n g E q . ( 2 ) . only as a last resort . In homogeneous reactors , the volume or h ol d up is almost always know n . One can then compare t h e volume es ti mate from t he residence- time di stribu tion with the known volume . To do thi s , one has to esti mate the contribu
tion of the tail o f either F ( t ) or f ( t ) to the first moment . If the volume estimated from the first moment i s si gnificantly smaller than the know n volume , the ves s el is not w ell p er fu sed and has stron gly stagnant region s . An example o f this was shown i n Fi g . 1 3 . In multi phase systems the rela tive volume and holdup of solid liquid and gas are sometimes hard to measure , and he re tracer experiments can be valuable , despi te the s ho r tcomi n gs men tioned above . An example of t hi s use is p resented later .
U S E O F R E S I D E N C E- T I M E D I ST R I B U T I O N S I N T H E
D ES I G N O F H OMO G E N EO U S R E A C T O R S
One advantage of u sin g t h e concept o f a residence- time distribution i n eval
uating tracer ex p e ri men t s has already b een discussed , namely , the p hysi cal insi ght in to the p roperties of t he system t hat is gained from the u s e of the escape probability II ( t ) . Just as important are the direct conceptual in sight s , w hich. can h e l p in un d er s tandin g reac tor desi gn problem s . T racer experiments are very useful i n organi zin g and evaluating alter native reactor de si gn s , a s well as in desi gni n g and evaluating pilot plant experi ment s . T hi s requires m ore than simple al go ri th mi c u s e s of the di s tribution in w hic h o n e tries t o comp ute performance directly from a resi dence time distribution . However , the conceptual framework w hich offers quide lines for scaleup and scale- down p roblem s is generally worth the effo rt . T he results in this chapter can be derive d ri gorously only for homogeneous re a c tor s , w hich are a s m all fraction of the cases met by the de si gn en gi neer . However , the gui deli nes for reac t or design derived from the m ( i n contrast to desi gn al gorithm s ) apply to het ero geneou s reactors , as lon g as one un der stan ds the basic principles . The differences are di scu ssed in a later section . Fi rst-O rder I sotherm a l Reac tions
Consider the case of a first- order i s ot her m al reaction system . set of co mpounds A j ( j
T here is a
1, 2, , n ) which can undergo composi tion c han ges such that the rate of formation of a compoun d A j is given by n
r(A.) ]
- L
-
i=1
=
( - k1]. . A ].
.
+
]1
k..A.) 1
•
.
( 23)
In other words , all reaction s between t wo compounds A an d B can be ex pressed by a set of reaction s
Shinnar
1 06
( 2 4) w here K A B i s t h e equilibrium con stant of t h e reaction A <::::;> B . I f one s tarts wit h an inlet c o m po site C j o , th e ou tle t c o m p osition of a plu g - flo w reactor is given ( Wei an d P rate r , 1962) by the equation n-1
C . (t) J
=
a. .
JO
'E
+
r=l
a. .
Jr
( 25)
exp ( - l. t ) r
where Cj is the co n c e n t ratio n
of
componen t j
ex p r e s s e d as a mole fraction
tt o and tl r an d A. r a r e the (n - 1) nonzero eigenvalues of the s y s te m . j j are c o n s t a n t s that depend on the initial and e quili b ri u m composition of the system , an d t is the residence time in the ho mo gen eou s isothermal plug- flow r e acto r ( or the r eac ti on tim e in a homogeneous b at c h r ea c to r ) . E quation ( 23) an d i t s solution [ Eq . ( 2 5) ] are co m p let ely equivalent to a Consider a pro b le m in probability theory describin g stochastic s y st ems . lar ge set of particle s , w hich can ch a n ge their state in a discrete w ay . E ach particle is a s s oci a ted with a state Aj . The p ro b a bilit y of a sin gle p a r ti c le b ei n g in a s pe ci fic state A i i s Pj . I f the number of p a rtic l e s in the se t i s very lar ge , the p robability of a p a rt iqle to be in Aj i s exactly equiva lent to the mole fraction s of partic le s Aj in the sy stem , w hich in a first or d e r system i s p ro p o r tio n a l t o t h e co n c en tr at io n Cj . One can therefo r e look at Cj ei ther as a concent ration or as the p robability of a sin gl e particle ( o r molec ule ) t o be in s t a te j . T h e mathematical proof is gi ven in Shinnar et al . ( 1 97 3 ) . T h e reader unfamiliar with probability theory can i n terp ret E q . ( 2 5) as follow s . The concentration C jo of the p a rti c les entering the r ea cto r expresses , for a si n gle particle , the probabilit y of bein g in state j . T h e reactor changes this p rob ability into C j ( t ) . Each p a rti c le behaves here independently from its nei ghb o r and can u n dergo chan ges completely independent of an y other molecule in the reactor . This is t h e p hysical meani n g of a fi r st - or der reaction syste m . Once thi s is understood , some important results in re actio n engineering become clear wit hout further mathematical derivation . Consider a homo gen eo u s isothermal reac t o r w hich has a s t ea dy - st at e flow that can be charac te ri z ed by a residence - time di st rib u ti on f( t ) . T he inlet con c en tr ation is One can now p er fo r m t he follo win g t ho u ght experiment . An observer C jo stan ds at the outlet of t he reactor an d collects a ll the particles and sort s ·
t he m int o bi n s acc o r di n g to t h eir residence time . To d o s o , o n e has to divide t he residence time into intervals ll t . If ll t is small , all particles in t he bin , t to t + 6. t , e s s entiall y have the same re sidence time , t , in the As each molecule in t he bin behaves in dependently of i t s nei ghbors , sy s te m . the concentration in t he bin would be the same as at the outlet of a homo gen eous i sother mal plu g- flow re ac tor with residence time t . n- 1
J
p.(t)
=
C . (t) l
=
JO
c:t .
+
L
= r 1
a. .
Jr
( 26)
exp ( - A. t ) r
T he average concentration at the outlet of the reactor < c .> can b e obtained
by avera gi n g ov e r all th e
bins :
1
107
Residence - a n d C o n t ac t - Time Dis trib u tions
j'0 f( t ) 0
[ a.
+
JO
�1
r= l
exp ( - A. t ) r
a. .
Jr
]
dt
( 27)
which is equal to
10
c.
n- 1
+
:E
a.
r=l
Jr
"'
f( A. ) r
( 28)
where f( A. r ) is t h e L apl ac e transform o f f ( t ) wi t h A. r sub stituted for the t rans for m variable ( s ) . E q u a tio n ( 2 8) is equivalent to the outlet of a hypothetical reactor described in Fi g . 2 0a , w hi c h con si sts of a set of paral lel tubular fl o w reactors with different residence times t . T he fr ac ti on of fluid e nt e rin g a tube with a r e si d e n c e time b e t w e en t an d t + A t i s f { t ) A t . In practice , a real reactor wo uld h ar dly look like Fi g . 2 0a , but t h e re is one case that is equivalen t . C onsi der t he case w here th e reactor feed consists of separate dro p l e t s encap sulated by im p e rm e abl e membranes which have a residence - time distrib ution F ( t) in the reactor {it i s equi valent to the re actor in Fi g . 2 0a ) . The in divi du al droplet s in suspension polymeri zation In a mixed reactor this is t r ue only in the li mi ti n g case behav e t hi s way . called se gr e ga t e d flow . In a firs t- order system , by definition , each mole cule can be considered s uch a sep arate dro p le t . T he well- known case of an ideally stirre d tank will illustrate E q . ( 2 7 ) . Consider the irreversible reaction A ->- B . In the case of a plug- flow re actor , C A ( t ) C A 0 e- k t and for a sti r re d - t an k reactor , a mass balance will giv e =
l
-
I �--------��--�, ------� ------------
------�1
L-
-
(a)
(b) Alternative flow models for Zwieterin g method of bou nding the conversion for a re action of nth or d e r : ( a) segregated flow ; ( b ) maximum mixedness . F I G URE 2 0
1 08
Shinnar
0
( 2 9)
or
For a stirred tank , f ( t ) i s here ( l l l ) e
E q . ( 27 ) ,
- tiT
and t herefore acc o rdin g to
( 30) In the mass balance [ Eq . ( 2 6) ] , i t was assumed that the concentration in the stirred tank i s uniform at any point in t he tank . For a linear system , a much w eaker assumption is s ufficien t namely that for residence - time dis Residenc4 tribution is exponen tial or that the escape probability is constant . time di s t ri bu ti o n s are linear p rope rty of the nonlinear model of the trans port p roc e s s in the reactor , and first- order reactions depend only on this linear p ropert y of the flow . In reality very few reaction s are truly linear , but it is a sufficiently good app roximation for m any system s . Many react io n s are also p seudolinear in the sam e sense that w hile the reac tio n rates contain nonlinear terms such as a Langmuir expres sion , they can be w ritten in a form ,
,
·
( 31) where Ki j is a mat ri x of linear reaction rate coefficients an d <j) ( Cjo ) is a non linear function of inlet con c en t ratio n an d pressure that can be taken out in front of the reaction rate m atrix . Although this is seldom completely cor For all those syste m s residence- time rec t it is often a good approximation . distributions are an imp o r t ant modelin g tool . ,
Dev i a tion of the R e s i dence- T i me D i s t r i b u ti on from K i netic E x periments
If one has a complex reac One can also use E q . ( 2 8) in t h e reverse way . tion system in which the ei genvalues A r chan ge over a si gnificantly wide ran ge , we can use it to recon struct f( s) . A s sume th at the values of A r hav e been measured from a batch e x p e ri men t or a plu g flow reactor , and one now has experimental values for all the concentrations C j at t he outlet o f an industrial reactor . One can then compute the v alue s of A r and use them to e s ti m a t e f( A r ) for the in dustrial reactor . One then has sev �ral values of t he Laplace tran sform f( s) from which a reconstruction of f( s ) can be attemp ted . I f the A r are wi dely spaced , the reconstruction i s quite goo d . T hi s approach will b e discussed in more detail later when we deal with contact - time distrib ution in hetero geneous reactors . -
...
109
Residence - a n d Con ta c t - Time Dis tri b u tions Opti m a l R eactor C oncepts fo r Fi r s t- O rde r R eact ion s
Iso thermal Reac to rs T h e conce p t s derived in the p recedin g s ection lead to some important con Sp ecifically , t hey clu sions with regard to choosi n g a reac tor confi gura ti on . provide gUi delines for cases in w h i c h a plug- flow reactor is th e preferred choice and for cases in w hich other reactor confi guration s will be p refera b l e . The p rincip l e that a first - order reaction system is c om pl e t ely equivalent to the case w here sin gle m ol ecu le s under go c han ge s from one state to othe r states , with defined probabilities , si m p li fie s many desi gn p ro b lems . Many op timal p ro b l e m s that are v ery difficult to deal wit h analy tically become in tuitively clear if one understan d s the fo regoin g p rin ci p le and applies it approp riat e l y . In the p recedi n g section it was s how n t hat in th e case of a fi r s t - order isothermal reaction system of arb i t rary complexity , the h y p o t he tical reactor in Fi g . 2 0a w ill give identical re sul t s t o a ny reac t o r havi n g the same residence- time dis tribution . T he outlet concentration is therefore t h e average o b t ain e d from several plug- flow reactors with di ffe r en t residence O nce o n e re co gni zes t hat all one needs t o d o is ave ra ge over a num time s . ber of p l u g - flo w re acto rs , o p ti mi zation is si m p le . C o nsider , for example , the case w here < C j ( t ) > for a specific v alue of t has a de sirable o p ti m um minimum or m axi m u m concentration of < C j> · Th e n assume that one has found a re ac to r wi t h a re si denc e - time di strib ution f ( t ) that gives an o p ti m u m
yield of Cj · In this reactor t he o u t l et concentration < c > is si m p l y the j average of all t h e different plu g - flow r ea ct ors . There i s a simp le al geb raic theorem that say s t hat in any s e t of n u mb er s with average X , there i s at least one member of the set e q u al or large r than X and a.t least one member equal o r smaller t han X . I f f( t) is a plug flo w , one has only a sin gle plug flow rea cto r wit h t = E f ( t ) = T an d which has a u ni q u e C j < Cj > . B ut if f( t ) is not a p l u g flow , o ne has in Fi g . 2 0a diffe ren t tubes with different outlet concentration . One o f these reactors has an o u tle t concentration equal or l a r ge r than t he average concentration < C j > and in one of these re actors C j is s m al l er ( o r l ar ge r ) than < c j > . T herefore , depending on whether one wishe s to maxi mi ze or mini mi ze < C j > , on e can alw ays fin d a tubular reac tor in Fi g . 2 0 wi t h a uni form resi den ce ti me w hich is as good or better than the reactor wi t h the a rbi t rary residence time f( t ) . T hi s is a si mp le proof that for first - order systems a p lu g - fl o w reactor is optimal i n the sense t h at o n e c a n d e si gn a plug- flow reactor wi t h a yi el d or selectively as high or hi gh er than any i sothermal reactor with ar bi t r ary residence- time distribution . It results in a very sim p le b ut powerful con c e p tual tool fo r reactor desi gn . I n all cases w here t he s y s te m is such t hat one can o b t ain the outlet co n c e ntrati on si mply b y av e r agin g over the bin s of differen t re sidence times , t hen , for any reaction system which has the p rop erty that there is an o p ti m um yield of one co mpoun d , a p l u g - flow reactor is t he best choice . I f , fo r other reasons a simple plug- flow reactor is no t fe asible or desirable , one w an t s a desi gn in w hi c h the resi dence - time di stribution ap proaches plug flow as cl o s ely as possible . For example , F ( t ) for a series of s tirr e d tanks approaches plug flow if the number of stirred tanks is reasonably lar ge . One can j u dge t h e re q ui r e d number of tanks si mpl y by =
110
Shinnar
the allowable deviation from plug flow .
S ta ge d systems in general h av e a
similar property and one can use the s a m e approach to estimate the number of stages re quired . For av e r a gi n g to apply , the reaction does not have to be s t ri ctl y first order . A ll one re qui re s is that t he reaction behavior of a mo le cu le is not affected by t he nature of i t s ne i g hb o r s , as di s cu s s ed in the p r e c e di n g sec tion . O therwis e , one needs to know not only how lon g a mo l ec u l e has been in t he system , but al s o who its nei ghbors were . If the outcome i s s t ron gly affected by the nature of its n ei ghbors , a residence- time di s t ri b ution can not p rovi de an answer for the o ut co m e . N or can one show that plug flow is op timum , although one can still get some g ui de li n e s w hich will be di scus s e in a la t e r s e c tion . In a si mi l a r way , if one looks at t he conversion of a first - order simple re ac ti o n A <=> B wit h reaction ra t e
( 32) it is given by
a - a
eq
=
(a
0
- a
eq
)e
- ( k +k ) t l 2
( 3 2a)
A gain , the outlet concentration of a reactor wit h resi dence - time distrib ution f( t ) is obtained by av e r a gin g over the bins , - (kl k 2 ) t over f ( t ) . It is easy to see + w hich really me an s averagi n g e t hat a p l u g - fl o w reactor with the same average residence time t will h av e a low er a - ae an d hence hi gher co nv e r si o n t han a reactor in w hich the bin s have di f eren t t . ( Had the function been concav e , the opposite would For hi gh co n v e ri so n in a fir s t - o r d e r isothermal system , a hav e been true . ) plu g- flow reactor will h av e t he lowest res i d en c e time . A gain , o n e can evalu ate the permissible d e vi a tio n by some very si mp l e models ( such as a s e ri e s of s ti r r e d tanks ) an d jud ge the desi gn by measurin g the deviation from p l u g flow via a tracer ex p e ri m e n t . For small deviations the co effi ci en t of v a ri at ion y 2 of f( t ) is a good criterion . In the li tera t u re the P e clet number is oft e n used in stead and is app roximately equal to 2 f y 2 . M ore accurately , it is given by the expression ( see Shinnar a n d N aor , 1 9 6 7 ) .
T his is a convex function of t .
�
y
2
=
2[e
- Pe
- ( 1 - Pe) ] 2 Pe
( 33)
w hich for large Pe ap proaches
y
2
2 Pe
( Pe
+
00
)
( 3 3a)
B ut o n e should be careful not to use P eclet numbers smaller th a n 1 0 , be cause in m o s t c ases this is meanin gless , as t h e residence - time distribution will not fit a o n e - dimensional diffusion mo d el . I n fact that the re sults are not s e n s it i ve to the form of the deviation from plug flow is illustrated i n Fi g . 2 1 , w h e r e two cases are plot t e d . T he
111
Residenc e - and C o n tact- Time Dis t rib u tions 100
"I 9 0
. .q .... 0 z 0 Ul a:
� 80
'- I D E A L STIRRED
z 0 0
70
(a)
A ....'!.. a 1.0
2 .0
3.0
�
:I
X .q ::E
K
5.0
6.0
k
k
A _).. B � C
0.7
::t
4.0
-'---'---'--- ·_l __!.__L___L_--L._-.J
O. B
0 .q ..... CD
TA NK
0.6 0. 5 0.4
0.3 0. 2 0.1
0.0
(b)
IDEAL STIR R E D TA N K 1.0
2 .0
� kl k
3.0
4.0
5.0
( a ) Conv er si on o f a first- order irreversible reaction for differ residence - time distributions . ( b ) Selectivity of a con s ec u ti ve first order reaction for different residence- time distributuons . 1 . Mixed flow ; 2 , five C STRs in series ; 3 , one- dimen sional flow , Pe = 8 . 87 ; 4 , 1 0 C STRs in series ; 5, one- dimensional flow , Pe = 20; 6 , plug flow . FIGURE 2 1
ent
Shinnar
112
first illustrates an irreversible fi rst - order reaction at hi gh conversion , and For the second s how s the selectivity of a consecutive fir s t - order reaction . both case s , the limitin g cases of plu g flow an d a stirred tank are given and two models with equal Peclet number are given . O n e is a network of equal stirred tanks in series an d the other is a one- dimensional di ffusion system with a Peclet n umber chosen such t hat y 2 is equal for both cases . For y 2 = 0. 1 , the re sults are so close that it is impossible to plot the difference . For y 2 = 0 . 2 ( Peclet number 8 . 9 or 5 s tirred tanks in serie s ) the difference is already recogni z able b ut still s mall . One should not mi sinterp ret this statem ent . The results are alw ays sensitive only to the variance of f( t ) . In this case only models for which f( t ) has a reasonable similar form were compared . For such cases all models with eq ual y 2 will give si milar results , For computation it is therefore ad provided that y 2 is sufficiently small . vantageou s to use simpler models based on stirred tanks instead of diffusion , esp ecially since none of the models rep resent s the actual physics accurately . N o n iso thermal R eac tors
T he arguments in t he precedin g section can be easily extended to a noniso thermal reactor , p rovided that t he temperature profile is i mpose d from the outsi de and i s independent of the reactions . For example , consider an a rbitr a ry network of stirred tanks , each of which is homo geneous and has a uniform temp rature imposed from the outsi de , altho u gh eac h reactor may have a different temperature . The reactors are interconnected in an A gain , arbitrary manner . T here could be s everal inlets and outlets.. each enterin g molecule is equipped with a clock , but this time a more com plex clock i s c hosen that not only records t h e time spent i n a specific t ank , but also records each s tep in a specific tank in the order in which it occurs . ) If one looks at each clock , one finds a set of ti mes < t t . t 2 , t s , t l , t 2 , i dentifyin g each successive stay in tank i until it leaves the syste m . One again collects t hem into bins , but this time one insi s ts that not only the total residence time t is constant in each bin , but that all molecules in a bin have an i dentical clock history . T hus in each b i n not only are t he separate stays in each tank of equal len gth , but they occur in the same sequence . In each stirred tan k the reaction s are first order wit h fixed reaction rate , determin e d by t he temperature of the tank . This means that if one know s the state of a molecule ( or the concentration of each compound ) enterin g the tank and the len gth o f stay i n the tank , the p robability dis tribution at the time it leav e s the tank is known or can b e computed . T hi s means that if one knows t he whole history of a molecule in a bin , one kno w s the p robability of it bei n g in each one of the permissible states , w hich in reaction term s means that one kno w s the concentration of the dif For t hose w ho are somew hat uncom fortable ferent co mpounds in the bin . with p robability theory , thi s concept can be t ranslated a s follow s . Follow a blob of molecules that has an equal history . As t hey travel through each tank , the concentration of the b lo b changes in each tank in the same way it would in a plug- flow reactor wit h the same temperature and residence time . T here is now only a cascade of plug- flow reactors with different resi dence times and temperatures ( or reaction rates ) . Each bin represents a unique sectional plu g- flow reactor o f this type ( see Fi g . 2 2 ) . It was shown pre vio usly t hat each stirred tank can be presented in such a w ay that mole c ule s with residence time t can be lumped to get her and treated as i f they would be in a p lu g - flo w reactor of t he same residence time t . The avera ge outlet .
.
•
Residence- a n d
FIG U R E 2 2
113
C o n t ac t - T im e Dis tri b u tions
E qui v a l en t representation of a
particle
S ec t io n ally uniform plug- flow reactor . each section ; ti residence ti me of each s ec tio n
mal reactor .
,
history in a noni so the r
T i , t emperat ure of
.
is now the average of all those sectional plug- flow reactors clear that if one is all ow e d to average them , one can always fin d a single bin (or a plug- flow reaction with a s p e cial temperature and tim e sequence ) , which yi e l d s a result as good or better than t he network of stirred tanks . This p roof i s not limited to n e t wo r ks of stirred t an k s One can take any arbit ra ry flow and represent it by such a n etwork if one makes the stirred tanks small en ough This i s therefore t r u e for any arbitrary flow
concentration
and it is
.
.
system provided that : 1.
2.
T he re ac tion rates are locally first order in concentration . The temperature profile i s independent o f the reactions and i mpos ed from the ou tsi d e .
If co n di tion 2 is not fulfilled , as for example in an adiabatic reactor , one can no longer average . T he temperature of a tank will depend on its con centration , and therefore the r e ac ti o n rates affecting a sin gle molecule will depen d on the n at u r e of t he other molecules in the tank , which violate one Howeve r , as many r e a c ti ons are approximately first condition for a v er a gi n g order , ave r a gi n g p rovides a s i mp le guideline for reactor desi gn . E ven more important , it lets one understan d and reco gni ze the e x c ep ti o n s to the rule , in the cases w here plug- flow reactors are not optimal . T hese inglude , for example , all cases in w hich it is d e sir abl e to maximize the contact between produc t s and reactants . .
Bounding
M e t ho ds for N o nlinear
Reac tions
For a fi r st or d e r system , knowled ge of the r e si d e n ce tim e distribution co m pletely determine s t he pro d uc t co m p o s ition in a premixed i sothe r m al homo For nonlinear re ac tio ns this is not e nou gh as one n ee d s geneou s reactor . to know n ot only the history of a molecule , but al so the property and s tat e However , for a certain class of of the molecules in i t s close neighborhood . nonlinear r e ac ti o ns wehre the reaction rate is of the type -
-
,
rA
=
- kan
( 3 4)
Zwi et erin g ( 1 9 5 9) has show n t hat o n e can get ri gorous bounds on t he pos sible conversions directly from f ( t ) . C on side r again a s tirr e d tank w it h the simple irreversible re action A + B but t hi s time let the re ac tio n rate be second order. If the tank is ideally sti rr e d , a mas s bal anc e will give ,
- QC
Z
- VkC
2 A
=
0
( 3 5)
114
S h i n nar
and
( 3 6)
One can also namely , that
look
at the other extreme i de ali zed case
mentioned
stay together ( w hich Zwieteri n g term s se gre gated flow ) .
equivalent to the re actio n c on centratio n will be
in Fi g .
13.
T hi s case i s
I n each tub ular reactor the outlet
1
=
an d t h e
before ,
the enterin g flui d enters as separate encapsulated drop s that
( 37)
average
outlet concen tration will b e
>
S exp (
th e average
simply
over f( t ) ,
or
A
=
C
AO
S ) Ei ( a >
a
Ei i s t he exponential inte gral . ferent results . However , if k C A O t small ( see Fi g . 2 3 ) .
00
1
=
kC
w here
t
AO
not
T h e reason for the difference is that
1
=f
-
very large ,
for a
e
0
that the two
N ote is
E.
-u
u
0 /u
lead t o difference
cases the
( 38)
dif is
s e c o n d o r der reaction the
p robability of a molecule of substance A r eac tin g depends on its p robability
molecule of A . In the segregated case , t his is larger than in t he mixed case , a s in the latter t h e enterin g molecule is immediately exposed to p roducts t hat reduce C A in its surround in gs . K nowle d ge of the resi denc e - time di strib ution alone is here insufficient to comp ute C A , as one needs to know w hat happened not only to the mole cule but to its neighbors . H o we ve r , for this simple cas e of secon d - order reaction , Zwietering was able to show that ideal mixing a n d s e gregated flow of
bein g close to another
probability
condition s are extremes and that all other possible mixin g sit uations for
f ( t ) is the
w hic h
C
A
For n
r
A
(ideal
=
=
s am e will res ult
mixi n g )
> C
A
in an
( s e gregated
outl et concentration such that
( 3 9)
flow )
1 / 2 or - kC
A
1/2
( 40 )
the limits become
2
( � ) (� � kt
2
C
AO
2
kt
2
2
C
AO
( 41)
115
Residence- and Con tac t - T ime Dis t ri b u tions
0.8
0.4 0.2 0.0
0
2
4
3
5
-
-
F I G U R E 23 C o nversio n of a sec on d o r der irreversible reaction ( A B) in a sti rre d tank as a function of time for segregated an d completely mixed flo w .
an d
C
[ (� )
l ( s e gre ga ted flow ) = C Ao A
+
kt
2
2
C
2
-
( 42)
AO
Note that C A for t he i deal stirred tank is l e s s than C A for se gregated flow . Here the presence of molecules of A reduces the r eact ion r a t e a n d mi x in g However , one has to be v e ry careful wit h products improves conver sion . with such arguments w hen applie d to reaction s of fr ac tional order . T h ey arise nor m ally from Langmuir- ty p e re ac tion r ate expres sions . Expressin g them as a fractional order is correct only over a lim it ed ran ge of con c entra tion s . T here is a high probability t hat in in t e gr ati n g the results for Eq . ( 40) , one m ay go o u tside the ran ge w here this is applicable , an d t herefore T he one must verify that the final result is not se n siti ve to this mistake . results for a n on line a r reaction t herefore dep end not only on f ( t ) but on a mo re complete descrip tion of the mixin g phenomena which Zwieterin g t e rms
micromixin g .
Swieterin g has also s hown that these results can b e extended t o any arbitrary re si dence - time di stribution as lon g as the r e ac ti o n is a si mple re In that case C A I C A o will al way s be bounded by two ex tre me cases : the case of segre ga ted flow given in Fi g . 2 0a and another Where case defined as maxi mum mixedne s s , w hic h i s illu strated in Fi g . 20b . as in Fi g . 2 0a ea c h tubular re ac t or is s epa r a t e in Fi g . 2 0b , t here i s inter mixing betw een them and the feed inl e t s are arranged such that t he expected action of order n .
,
S h in nar
116
future life is constant for all molecules in a given cross section . P h ysi c al ly , one could reali ze t he a r r an gem en t of F i g . 2 0b b y d e si gnin g a pl u g flow r e actor in w hich additional feed is i ntr o d u ce d at different poi nt s along the t ub e . For a r ea c tio n of order n the two limi t s or bounds on conversion are se g r e ga t e d flow and maximum mixedness . For the case of segre gated flow , 00
J
[1
+
( n - 1)RJ
111
-
n
f ( t ) dt
( 43)
0
- kt R = -�1-n
C
AO
For the case of m aximum mix ed n e ss , < C A > / C A O h a s to be o b t ai n e d by solv
i n g the differential equation
( 4 4) w here A is an i nt e gr a ti o n variable and A ( A ) is the escape p robability a s de fined by E q . ( 8 ) . E q ua ti on ( 44) can be int e grated numerically , although for certai n valu e s of n and certain values of 1\ ( :>.. ) it has analytical solutions . Here one sees another useful p roperty of the escape p robability and why one w a nt s to m e a s ur e the re sidence- time distribution in such a w ay that A ( t) can be comp uted reliably . For reactors close to plug- flow reactors , the bounds are very close and one does not have to worry about micro mixi n g w hi ch i s im p ortant mainly in stirred- tank reactors . S t ri c tl y speakin g , the results of E q . ( 43) apply only to si mple reaction of o r der n , in homogeneous ideally stirred reactors , w hich is a rather rare desi gn p roblem . However , the concepts of boun di n g t he outlet co m position of a reactor is an e ssential tool of reaction engineerin g . I t i s really a fo rm of p e rfor min g a s e n siti vi t y analysis on a de si gn . Consider the example o f a A priori one knowns that this is an i deali zed model t ha t can sti rre d tank . H o wever , o n e c a n approach i t in t he sense that the not b e fully reali z e d . time a droplet of feed spends in the reactor before it becomes t ot all y mixed i s very s mall com p ared to the tot al residence time ( Ev an geli s t a et al . , 1 9 69b ; S hinnar , 1 9 6 1 ) . O ne also kno w s t hat durin g scaleup t he residence time s t a y s constant , w hereas the mi xi n g ti me always i nc re a se s ( Evan gelista et al . , 1 96 9a , b ; Shinnar , 1 9 6 1 ; W ei n s t ei n and Dudkovic , 1 97 5) . For a se co n d - o r d e r sin gle reaction , Z wieterin g' s results show that one p robab ly does not need to worry about this , a s the completely mixed case is t he worst ( least de T here are many ot her cases for which this i s not true . One si rable ) cas e . wants to know how important s uch deviation s from complete mixin g are . To answer t hi s , the two limitin g cases for any arbitrary reac ti o n system are d e ter m i n e d . If the reactions are complex an d nonlinear , t he se two limiting cases ( s e gr e g at ed flow and complete mixing ) will not give rigorous bounds b u t will indicate i f t he system is sensitive to mixin g and w hat p enalties are i n c ur re d for in com p l e t e mixin g . In scaleup , one cannot achieve se gregated flow , so the case of complete mixi n g must be s ati s fac tor y or the desi gn must be chan ged . If the s y s t e m i s sen sitive t o mixin g , one can estimate t he per missible mixin g tim e by u si n g models for incomplete mixin g that si m ul at e
Res idence- and C o n t ac t - Time Distrib u tio n s
117
small deviation s from in co m p le t e mi xin g ( E v an geli s t a et al . , 1 9 6 9a , b ; B elevi
et al . ,
1981 ) .
I f , in a sti rr ed - t an k reactor , s e gr e ga t e d flow leads to improved re sult s ,
a plug- flow reactor will n orm ally be even better . If one r e qui r e s the stir rin g , one can a ppro ach plug flow by putting several such s ti rr ed tanks in se ries . E xperiments that de m ons t r a t e the s e n si tivit y to d e vi ation s in mixin g can al so be performe d in a pilot p l an t . For ex amp le , one can desi gn a
pilot plan t similar to the one shown in Fi g . 2 4
des pi t e the fact that the T here hav e been a number o f models propo se d that look at m icromixin g for more comp lex flow co n di t ion s over a w i d e r ran ge of mic ro mixi n g [ see , e . g . , Weinstein and Adler ( 1 9 67 ) ; N g a n d Rippin ( 1 96 5 ) ] , T he s e models , which are reviewed in detail in N auman ( 1 98 1 ) , are more in t he nature of l e arnin g model s than de si gn models , an d are th e r e fo re outsi de t he scope of this ch a p t er . Si milar c o nsi d er atio n s app ly to p ack e d - bed and fluid- bed reactors . For nonlinear r eac ti o n s the only safe w ay to sc al e them up is to approac h plug flow as clo s ely a s p o s si b l e . A gain , one wants to know how closely plug flow m u s t be appro ac hed and w hat the p e n al t y is for s mall deviations . One can again p e r form e x p e rim e nt s w hic h ex p e rim en t all y check the sensitivity of the reaction system to d e v i ati ons from plu g flow an d desi gn the reactor accordingly . H e re re si dence- time di s t ri b u tio n s are directly u seful , since for small de v i atio n s from plu g flow , one can estimate the deviation fro m A si mple model will p l u g flow fro m the tr ace r response by m ea s uri n g y 2 . gi ve reasonably reliable bounds as to the possible impact of such small de viation s , even i f t he reactions are hi ghly nonlinear . If the sen sitivi t y i s large , o n e s hould take a safety coefficient with re s p e c t to t he allowable co efficient of v ari a t ion . T hi s is i l lu st r a t e d in Fi g . 2 5 , w here the effect of small dev ia tio ns fr o m p l u g flow on s e v eral reactions are comp ared . One i s simply the irreversible r e act io n A + B of se co nd ord e r , and t h e other is a nonlinear ca s e very sen sitive to mixin g . In both cases th e re sult s a re in distin gui s hable , and for y 2 = 0 . 2 ( five stirred tanks in series , or a Pecl e t number of 1 0 ) the results are r ea so nab ly close . In both cases we also giv e the limiti n g cases of p l u g flow an d a stirred tank to indicate the sensitivity to back mixi n g . T he comments made with re ga r d to Fi g . 2 2 apply here final reactor i s c omp l etel y mixe d .
FIGURE 24
Si m ula ti o n of finite mi xi n g in a p i lo t p la n t .
118
Shinnar
equally w ell .
T he results are not unique functions of y 2 .
How ever , for
s m all deviations from plug flow , t he results are not model sensitive as lon g
as the residence - time distri butions are rea sonably similar . For simulation p urposes , a model based on stirred tanks is therefore preferable , as it is T he fact that y 2 alone i s not s ufficient t o charac much easier to compute . teri ze the system unless y 2 i s very s m all is illustrate d in Fi g . 2 5c w here two ot her models with y 2 0 . 1 are compared to a serie s of stirred tanks . For a reaction sen sitive to mixin g , such as t he one show n in Fi g . 2 5b , the differences are very lar ge and a m uch smaller y 2 is necessary for them to become ne gli gible . Such reaction s t he refo r e require extra care in scaleup and a close app roach to plug flow ( or a stirred t an k ) . =
80
70 <[
60
LL.
0 z 0
u; 0:: ILl >
� 50 u
40
30
3.0
2.0
1.0
4.0
(a) F I G U RE 2 5 ( a ) C onversion of a second - order irreversible reaction for dif ferent flow models . ( b ) C o m p a ri so n of p roduct yield in a consecutiv e non Reaction A + B + C fir s t order ; linear reaction for different flow models .
reaction A + C + D second orde r . Inlet concentration of A , 1 . 0 ( di m ensio n less unit s ) . 1 , Mixed flow ; 2 , five C S T R s i n series ; 3 , one- dime nsional flow , Pe 8. 8 7 ; 4 , 1 0 C S TRs in series ; 5 , one- dimensional flow , Pe = 2 0 ; ( c ) Same as in p art ( b ) for different flow models : 6 , p l u g flow . =
(I )
VPF
+ =
[=:J
+
Ll..J
0. 69 V , V ST
=
+
(II )
0. 31.
1 0 stirred tanks in serie s ( I I I )
....
l.LJ
....
c:=J+
119
Residence - a n d C o ntac t- Time Dis t ribu tions 0. 0 5
0
A .!. e.!.c A + C 0"1
0.04
w w u.
z c(
D
0.0 3
w ..J
0 ::iE 0
......
0.02
0 u.
0 ::iE
0.01
/
;y
/:/_ .
0. 0 0
...... -
- -��
-- - �
-·
1 .0
2.0
3.0
4.0
L-----'-----'----.I..... ---'---___.___.J
(b) 0.04
-
/ �Pl.� ./
II) w ..J
--
- '"4
TIME -
----- -------�
A 2. e ....!. c A +
c ..QJ.. o
0.0:3
0 IL 0 Ul w ..J 0
::!: 0. 0 2
0.01
0.00
(c)
1 .0
2.0
F I G U RE 2 5 ( Continued)
3.0
T I ME -
4.0
120
S h in nar
T here is another area in which the concep t of a residence - time distribu tion is useful in o r ganizi n g ones thinkin g and experimentation . T h inking in terms of residence times and joint p robabilitie s is helpful i n de ci di n g what reactor types t o choose . For all linear and p s e u do lin e a r system s , iso thermal as well as nonisothermal , plu g- flow reactors are alw ay s p re fer able . The case w here bac kmi xin g or a stirred tank has a po si ti ve influence is Ex w hen there are complex interaction s betw een p roducts and reactants . amples a r e autocatalytic reaction s , autothermic reac tions ( hea t c a n be con sidered a prod uct ) , or reactions of the types A + B
+
C
A + C + D If D is d e si r abl e , a mixed reactor is p r e fe r abl e , as can b e seen from Fi g .
2 5 . I f D is undesirable , a plu g- flow r eacto r is p re ferable . of thi s reaction di scussed in R enick et al . ( 1 9 8 2 ) : Methanol
+
olefines
+
aromatics
A r om a tic s + m e t hanol + durene
+
An example
p araffin s
I f D ur en e form ation i s to be avoi de d , the reactor should b e as close as possi bl e to plug flow . T hi s re s ult can be derived wit hout reference to residenc e - time distribu tion , but thinki n g in term s of resi dence time p ro vides a useful fram e w ork . T here are a num b er of ca s e s in w hich the desirable confi gur ati on i s neither p l u g flo w nor a sti rr e d tank . Combustion is an example . B ut here one can often b reak dow n the optimal or desi r e d flow sys t e m into confi gura tions composed of w el l - d efi ne d flow s . I f , o n the other hand , the flow i s complex , residence- time distributions may pr ovi de dia gnostic test s , b ut one can neither model nor scale such cases sol e ly on t h e basis of residence- time distributions . It s hould be emphasi z e d that at the p resent state of knowled ge , t here is no w ay that one can reliably scale up a reactor if the reactor is sen s itiv e to the form of f ( t ) and the r e ac tor does not approach one of the well defined i deal cases . ·
H E T E R O G E N EO U S R E A C T O R S A N D
M U LT l P H A S E S Y S T E M S
Prope rties o f T racer E x peri m e n t s i n M u l ti p hase S y s tem s
Residenc e - time distrib utions p r ovi d e a powe rful method for the a n al ysi s of ho mo g e ne o us i sothermal reactors . Unfortunately , very few real reactors used are t r uly ho mo ge n eous . Those few that are seldom p rovide se riou s difficultie s in desi gn or scaleup . The majority of t he re act ors used are heterogeneous , eit her gas - solid , gas - liqui d , or triple phase . Catalytic re actors are almost alway s heterogeneous , even in homo geneous catalysis , w he re most of ind ustriral reactors deal with gas - liquid reactions . If t h e use of re si den c e - tim e distrib utions was restricted to homo ge n eo u s Luckily , m o s t of t he reactors , t heir usefulness woul d b e rather limited . concepts derived in the pr evio us sections can with some care , be applied to hetero geneous reactors . One must , ho wever , be careful to take i nt o account the hetero geneous nature of the system . T his rules out many of
1 21
Residence- a n d C o n tact- Time D is t ri b u tions
the algorithmic uses of residence - tim e di st rib utions . One cannot simply m easure a residence- time di stribution and predict the performance of a com Even t he concept of a residence- time di s t rib u ti on is not sim plex reactor . ple any more , as different reactants m a y have totally different residenc e time di stribution s . I n a hetero geneous reaction one cannot measure reaction rates and re sidence - time distributions i ndependent ly and reliably predict re actor performance even for first - order system s . B ut the main uses focu sed on in the precedin g section remain .
1.
T he intensity function is still a useful tool to dia gno s e maldistribu tions in t he flow and provide clues to the nature of the flows in Of special importance is t h at it allows one to check how t he system . closely a desi gn approaches plug flow (or an ideally stirred tank ) It is also usefu l in cold- flow models , w here it c an be used to test the efficiency of flow distrib utions or baffle s . T he examples given earlier were fro m hetero geneous reactor s Residence- time distributions still provide a powerful conceptual tool for desi gn and scaleup . A lthou gh often one cannot get accurate p rediction s , one can derive a better un d erstandin g of the scaleup p roblem . Resi denc e - time distrib utions i n multiphase systems allow m easurement of t he hol dup o f each phase . .
.
2.
3.
T here are some imp o rtant differences between tracer exp eri m en t s in homo geneous and he t ero ge n eou s sy stem s . It is i mportant to understan d these differences not only in evaluating tracer experiments , but also because they are related to desi gn and scaleup . In a hetero geneous system , di fferent compounds m ay have stron gly different resi dence - time distribution s . In a homo geneous reactor t he difference in residence- time distrib utions between different compounds i s , i n most cases , ne gli gible . T his is especially so if T he only exceptions are slow laminar flow s , the mixing process is turbulent . w here mo lecu l ar diffusion will be important . T hu s in most homo geneous re actors the nature of t he t racer compounds is of small importance . On the other hand , in heterogeneous system s , there can b e lar ge differences be tween the response of different tracers , and therefore between the respon s e o f a tracer and that of a reactant . Even different compounds in t he feed or different products may have different residence- time distribution s . An example o f this is shown in Fi g . 2 6 ( see also O rth and S c h ii ge rl , 1 9 7 2 ) . Fi gure 2 6 i l l u st rates data taken from a pack e d - b e d Fi scher- T ropsch reactor in which CO an d H 2 were reac te d over a Ni c ataly s t . T he reactor w as operated at s t eady state and a step in p u t of CO containin g 1 3 c w as intro T he helium concentration w as duced to gether wi th a step input of helium . T he re sponse o f kept very small so as not to di sturb the steady state . the 1 3 c in t he C O is plug flow with a lo n ger residence time , which show s that the CO is adsorbed but t hat the interchan ge of the adsorbed CO with the CO in t he gas p hase is very fast . On the other han d , both the C H 4 and C 3 H 6 p roduct b ehave like a stirred tank i n series with a p lu g flow . This indicated not only stron g adsorp tion but that , in this case , the inter change b etw een the adsorbed phase and the gas p hase for t he pro d uc t i s T he re slow and has a characteristic time similar to t he residence time . sult s could also have been the reverse ( slow i n t erchan ge of the reactant and fast interchange of the p roduct ) . In a reactin g system the concept of a residence- time distribution itself b ecomes rather c om ple x to define .
1 22
Shinnar
,/H e
1.0
I I , I I r I r r I
0.5
0
r I I I I I
0
/
100
200
30 0
400 I ,
sec
FI G U RE 2 6 Response of a packed- bed Fi scher-Tropsch reactor , fed with an d H 2 , to a step input of 1 3 C O and H e . Reactor kept at steady state . ( From Biloen et al . , 1 98 3 . ) CO
One can always define it with respect to specific elements such as C in Fi g . 2 6 ( o r molecular combination s that d o not chan ge such a s benzene rings ) , but the only way to measure this would be by means of proper isotope markers . In Fi g . 2 6 the tracer response is given separately for 1 3 C O and 13c H 4 . From the ori ginal data one could have measured a tracer response for 1 3c , which would give a residenc e - time distribution for carbon atoms . I t i s a proper residence - time di stribution , containin g valuable information , but its interpretation is more difficult . It is not only a function of the flow pattern but also a function of conversion ( or catalyst reactivity ). To measure maldistribution in flow , a nonreactive tracer is often p referable . T he average residence time of a sin gle tracer is not necessarily equal to total reactor volume divided by the flow rate . I t is therefore hard to estimate independently the average residence time , which due to adsorption or similar phenomena , can vary by a factor of 5 or more betw een different tracers . Furthermore , not all tracers give linear responses . This makes the choice of a proper tracer much more critical than in a homo geneous system . Residence - time distributions are by definition linear func tion s . At steady state all compounds in a reactor have a measurable resi dence-time distribution re gardless of the nonlinearity of t he flow , nonlinear adsorp tion phenomena , and so on . If one can properly mark a compound ( e . g . , usin g an i sotope ) , the tracer experiment will result in a residence time distribution . B ut if one introduces a tracer that is not present in the system , it could behave in a nonlinear way , and the result could be very dependent on the amount of tracer introduced . Furthermore , one does not always want a tracer that behaves similar to the reactant . For example , in some cases w here one is studying reactor internals in a packed bed , non adsorbing tracers are preferable , as they are more sensitive to a maldistribu tion in the flow . This can be seen from Fi g . 2 6 . A heli um tracer would have given a much better indication of a maldistribution in gas flow than using 1 3 c o and measuring the total amount of 1 3 c leavin g . The proper choice of t racers is discussed in more detail in the next section . Residence- time distributions cannot be used directly to predict outlet composition even for linear reaction systems . Reaction rates in catalytic reactors are measured and reported on the basis of either reactor volume ,
1 23
Residence- a n d C o n tact- Time Distrib u tions catalyst volume , or catalyst w ei gh t . same time scale as t he re si de n c e ti me tem t hi s re la ti o n s hip i s not v ali d ( due solubilitie s , etc . ) , and the di ffe re n c e
In a h o mo ge n e o u s system this has the
distribution . In a hetero geneous sys to adsorption , ab s orp tion , different in time sc ale can be a fac tor of 5 or more . T he r at i o is di ffe r e n t for different reactants . Consi der , for exam ple , a p a cke d b e d reactor . T he estimat e s of ki ne tics are based in most cases on overall re si d e n c e time s of m ole reactant}lb catalyst p e r hour . T he real residence - time di st ri b u ti o n of the gas i s in most cases unknown b e c aus e t h e ho l d up of t he gas depends on adsorption . If one could really tag a re actant one would not nec e s s a ril y get p l u g flow in a tubular laboratory re actor ( see Fi g . 2 6 ) . However , t h e ki n e ti c d a t a u n d e r l yi n g t h e s cal e u p assumed plug flow in t h e lab reactor . T he rate ex p re ss ion s are really not molecular kinetic e x p r e s s i o n s d e s c ri bi n g real molecular events but r ep r es ent overall re l a ti o n s allow i n g one to scale up a lab reactor to an industrial packed b e d . T his ap proa c h works very well even if the reactant is stron gly adsorb ing and t he reactions are h i gh ly nonlinear . The fact that for a hi gh l y ad so rbi n g reactant t he real residence time of a molecule is , in reality , far from plug flow , a n d different for all react ant s , does not affect the u seful ness of s u c h overall kinetic relations for steady - st ate desi gn . I t m ay how ever , s t ro n gl y affect t h e d y n a m i c behavior of the reactor and one has to be ca r e fu l w hen u sin g r at e e x p re ssio n s o b t ai n e d fro m s t ea d y s tat e e x p eri men ts to de scribe the dy n am ic b ehavior o f a reactor ( S hinnar , 1 97 8 ; B i loe n et al . , 1 98 3) . T his al so p revents one from u si n g such methods as t he boundary methods of Z w i e t e ri n g unless once a gain o n e i s d eali n g with s m all p e rt u r ba tions . In c a t al y ti c reactors , reactivity can c ha n ge wi t h p o si tio n even i n i so t her m al rea c to r s I n so me c at al y t ic re ac ti on s e s p e ci ally at hi gh t em p e r a tures , reaction s occur in the g a s p h as e as well as at the ca t al yti c s urface . T here is t h e re fo re no unique co r re latio n between residence time and t he state of t he molec ul e , w hich i s a p roblem already en co u n t ere d in t he case of the nonisot herm al reactor . While a residence- tim e di strib ution c an gi ve i m port an t information about t he flow , one w o u l d need ad ditional information to p r e di c t t h e o u t l et co m pos i t i on , even if the reactor is i so t h er m al and first ord er T he co n ce p t of a re si d e n c e t i me distribution is s till hel p fu l in the sense outlined in t h e b e gi n ni n g , as lon g as on e is careful to reco gni ze the di ffe r ences . This wi l l be di s cu sse d in detail at t h e end of t he s ection . Firs t , some techniques t hat a re e s p e ci ally applicable to h et e ro ge n eo u s reactors will be discusse d , n a m e ly m u lti ple tracer e x p eri m e n t s and contact- time distributions . -
-
,
-
,
.
,
.
-
,
M u l t i p l e T racer E x pe r i ments A s in hetero geneous s y s t em s , different compounds may have di ffe r en t re
sidenc e times . I n de p e n d ent knowled ge of the re si d e n ce - ti m e di stribution of sev er al tracer compounds gives a b e tt e r description of the system . I f the feed cont ains different compounds , one may wish to use m ar k e rs for e ac h of t he m to study t h e system . One can also learn a consi derable amount about the s y st e m by u si n g a n u m b e r of different t r ac ers even if t h e y are completely di ffe re n t from the re ac t a nt flui d ( Shinnar et al . , 1 97 2 ; N auman , H o w e v e r , one is really measurin g the 1 9 8 1 ; Ort h and S c hli g er l 1972) . residence- time di s t ri b u tio n of the s p e ci fic tracer , which is not neces sarily ,
,
Shinnar
1 24
representative of the system .
If several t racers of stron gly varyin g
p rop er ties are utili zed , the residence- time distributions
of t he comp ounds in the system can be bracket e d . One should note , how e ver , t hat even i f one uses a n i sotope of a reactant , o n e still cannot call thi s a resi d ence - tim e distrib ution of t he system , as in a hete ro geneou s system different reactants may have totally di fferent residence - time distributions . One does not alw ay s want a trac er that behaves like the reactant flui d . Consider , for e xamp le , a p acke d - bed reactor . To measure maldistributions of the gas flow , one w ant s a tracer that does not di ffu se into the cataly st and d o e s not get adsorbed . I n this case the t ransport pr o cesses i n si de the cat al ys t parti cl e s an d a d so rp t ion p rocesses are irrelevant fo r scaleup or reactor performance in the s e n se that one deals with s c ale u p of a p i lot plant The main p ro b lem o n e n eeds to co n s i der with i dentical cat alyst p a rticl e s . is t hat mass t ran s p o r t processes to and from the particle to no t chan ge as the linear velocity in the large reactor increases . T racer experiments are not sensitive enough to check this p rop e r t y , but t here are other m etho ds to e n s u re t hi s . The main use o f a tracer experi men t is to test that t here are no maldistributions in the overall flow . An exam pl e of t hi s was shown i n Fi g . 1 6 . I n a packed bed , i f the re are maldistributions , one can use the result of a t racer exp eriment with a no n ad sorbing tracer to e s ti m at e their impact on c o n v e rsion and p ro d uct distribution . In t rickle beds one must worry about both liqui d and gas distribution , and would there for e w ant two t racers , one for each p hase . In all these c a ses , A ( t ) is the most sensitive way to c heck for a mal dist ri b u tion , but in a trickle b�d one m u s t Stron g d evi a ti o n s from pl u g flow can occur due to fluid dy n ami c be careful . effects an d o n e t here fo re needs a goo d reference case ( Mi l ls and Dudukovic , 1981) . In fl ui d - bed reactors the situation is more complex . If th e solid p hase is not s t atio na ry b ut is fed an d removed from the bed ( as in the treatment of solids in an F C C cracke r ) , one may w ant to use a solid tracer to check if the solid has the desired residence- time di st ri b utio n . In a sin gle stage one pro b ab l y want s complete mixin g of the soli d s or a constant A ( t ) , w h e rea s in a m ul tis t a ge bed and for t reatment of solids , one w ant s the residenc e - time di st rib ution for the s oli d to corr es pon d to a s e ries of stirred tanks , or to ap p ro ach plug flow . I n some t ric kl e beds , part of th e feed ev ap or ates . In t his case , to study the flow p r o p erl y , it is recommended that b ot h an evaporatin g and nonevapo rating fracti o n of t he liq uid be tagged . An e x am p le of such a multiple t racer experiment is shown in Fi g . 27 , which illustrates r e s ul t s ob tained in a hydro t re at e r for fuel oi l ( Snow , 1 9 8 3 ) . T h r e e tracers a re used . T h e C 1 6 compound rep resents The C g co mp o un d enters in the gas phase . a comp ound that under t he conditions of t his h y dr ot reat er i s p resent in both T racer phases , whereas the C 3 2 c o m po u n d stays m ai n l y in the li qu i d phase . Re results are gi ve n for two react o r s op erat i n g under similar conditions . actor A co ntai n s a 1 / 8 - in . e x t r u d at e hyd ro t r e atin g cataly s t , w hereas react o r B op e rat e s with a 1 / 1 6- in . e x t r uda t e . Reactor B s how ed a better perform ance at hi gh conversion , allo wi n g a 5 0 % hi ghe r through p u t at 99 % c o nver sio n . T he results gi v e a good i ll us t r a tion of t he advanta ge of u sin g proper multiple tracers . T he most i nte re s ti n g information in Fi g . 2 7 is not the for m of the individual t racer respon se but the relative p o si tio n of the i n divi d u al tracer responses . In a t rickle bed , o n e would expect that t he residence tim e of t he l qi ui d is larger t han t hat o f t he gas . The results bear t hi s out , as the average residence time of th e C 32 response is larger in b o t h reactors then in th e C 8 tracer . In r e act or B the C 32 tracer appears 3 min a fte r the C g t race r . However , looking at Fig . 2 7a one notices imme d iat ely
125
Residence- a n d C o n tact - T ime Dis t rib u tions . 1 6 .------
-- - -- · -
Unit A
f(!)
.1 2
I I I •
"'
lj /�
.0 8
I '
�
.04
I•
0 0
4
8
12
20
16
. 3 2 .-------,
f(t)
Uni t B
.24 .1 6
.08
0 0
(a)
4
8
12
16
20
F I G URE 27 Multiple tracer experime n t s i n t w o industrial fuel oil hydro treaters . ( Private communication with A . I . S now , Arco Petroleum C o . , 1 9 8 3 . ) T hree compounds tagged w i t h 1 3 c introduced simultaneously in feed
(pulse exp eriment ) and measured at reactor outlet . ( a ) R esidenc e - time density function ; ( b ) inten sity function ( escape probability ) of unit A ; ( c )
intensity function ( escape probability ) o f unit B ; ( d ) cumulative residence time distribution ( unit A ) ; ( e ) cumulative residen ce - time distribution ( unit
B).
1 26
Shinnar
0. 3 0
U nit A
ACtl
0 .1 5
0.00 6
0
12
18
(b) ACt>
Unit B
0.60
0. 3 0
./· - -- ------�.!2_
tl
�--
_ __
.l.._ _J._____JI... .l.._ ...- --1---...J o.o o L-�i_____.J.___!__ _L_...._....
(c )
0
FIG URE 27
6
( Continued )
12
18
Res idence- and C o n t ac t - T ime Dis t ri b u tions
10°
1- F
127
r---��----� Unit A
(d )
10 3 �--L_ 0
6
(e) FI G U R E 2 7
12
18
L___L___L___L___ L---L---L---L-�
__
( Continued )
128
S h i n nar
that the c 32 t rac e r in reactor A ap p ea r s about 1 mi n before the C s t rac e r . This s how s a si gni fican t difference between the two reactors an d s how s th at so m e t hi n g is w ro n g with reactor A w hich explains its lower p e r fo rm a n c e . If one ex ami n es in detail at t he e sc ap e p r ob a bility ( Fi g . 27c and d) , one also notes that the C 3 2 re s po ns e shows a s tron g maldistribution in reactor A and good liquid di stribution in re ac to r B . I nterestin gly , react or B has a m al di s t rib utio n in the gas flow ( C s ) which is n o t evi dent in reactor A . B u t it is ap p ar en t ly not si gni fi c an t eno u g h to cause p rob l e m s . T hese tracer results contain some very i n t eres t in g information about what h ap pe ns i nside the se re ac t o r s , an d h avi n g sim ultaneous information from both a liquid and gas t racer gre atly adds to t heir value . However , the i n t e rp re t ati on of these resul t s is in no w ay uni q ue , as the mal di stribu tion of t he C 3 2 re s p o n s e in reactor A co ul d be due to s e v e r al causes . It co ul d be c a u s e d b y a mechanical p robl e m re s u l ti n g in a nonuniform liquid di s t ri b u tio n , or b y inherent flow problems , or by the di ffere n t catalyst si ze . Some reactor m od eli n g could indicate w hi c h of these e x p l an at ion s i s mo re likely . It is i m p o rt an t to note that in such a m ul ti phase system , there is n o w ay to deduce from a single t racer experiment if a. maldistrib ution s uch as the C 3 2 response in reactor A ( Fi g . 2 7 ) is caused by n bad di s t ri b u tor or by b y pa s s p henomena inherent in the flow regime of t hi s s p e ci fi c r ea c t o r or by ph e no m en a inside the catalyst particle . T rickle beds , by their n at u re , do not sc al e equally well in different flow r e gim e s , which are d ete rmi n e d by various flow p ar a me t e r s , such as gas v e lo ci t y and density , liquid p roper t i e s , a nd liqui d - to - ga s ratio . This problem w a s alr e a d y e n co un t e re d in the disc u s sion of the fluid bed , and it is i m p o r t an t to reali ze this . However , often on e has two reactors o p e rat i n g under similar con di tions and , in s uc h a case , a difference in tracer re s po n s e cle ar l y indicates a mechanical problem . I n s t u dyin g t he gas flow in a fluid bed , one faces a p roblem not en co untered in pa c k e d beds . If the r e a c t an t is a d s o r be d , it will move wit h t he s oli d . I n an i so t he r m al or adiabatic packed bed , i f a pilo t plant and a large p l an t have the same residence- time di st ri b ut ion for a non ad so r bi n g t racer , one would expect them to behave si mi la rl y . I n a fl uid bed one can no t make t h e same as s u m p tio n , a s the mixi n g p rocesses o f t h e solid and the a d so r b e d reactant with it are di ffe re n t for t he t w o c as e s . T h e only ti m e that one can r eli a b l y scale a fluid b e d an d p r edi c t i ts p er for m an c e from a pi lot p l ant i s w here in bo t h cases the g a s flow approaches p l u g flow as closely as pos sible . One way to a c hi ev e this is by u si n g baf fle s in t he fluid bed . B affles here have a double purpose . T hey prevent by p as s in g of gas d ue to lar ge b ubbles by breakin g up the b ub bl e s , and re duce solid circulation , t hereby also r e d u c i n g the b ac k mixi n g of gas by moment of reactants adsorbed on the c a t aly st . In cases where one w ants to test w h et h e r t he flo w distributor op erates w ell , a no n a d s o rbi n g t racer is a more sensitive tool ( see Fi g . 19) . When there are b a ffl e s in a fluid bed , the problem becomes more complex . A good case can b e ma d e t ha t an ad s o r bi n g t racer is more s ensi ti ve , as it shows if a d s o r b e d tracer is recycled by the catalyst to the bottom of the T herefore , i f a baffled fluid bed ap p r oa ch e s plug fl o w wit h a r ea c to r . strongly a d so r bi n g rracer , it giv e s one better confidence in the d e si gn . However , one has to b e careful to c h ec k t hat the t r a ce r be ha v e s li n e a rly . To check t hi s , one alw ay s needs co m p a r ativ e result s from a p ack e d- b e d
1 29
Residence- and C o n tac t - Time Dis t ri b u tions
SF 6 has been reported to hav e desirable properties , and its ad reactor . sorption can be chan ged in a cold- flow model over a wide ran ge , as the adsorption equilibrium i s a function of the w ater content of t he air . If a properly baffled desi gn gives a residence- time distribution with a hi gh Peclet number ( low coefficient of variation ) for both a nonadsorbing and an adsorbin g tracer , thi s improves the confi dence in the scalability of the desi gn .
Un derstandi n g the effec t of adsorption on the residence - time distribu tion i s he lp f ul i n an ot he r w ay . I t explains why di ffe r en t reactions may have different sensitivity to scaleup in the same fl ui d bed . To understand this concept , one must first consider the evaluation of tracer experiments wit h adsorptive t racers in a more ri gorous way . T he approach u sed i s de sc ri b e d in detail in Shinnar et al . ( 197 2 ) . Consider a pack ed b e d or a chromato graphic column in w hic h a t racer can be adsorbed inside a porous p a rtic le . One can now perform a t racer experiment with a nonadsorbing tracer w hich can also diffuse into t he porous particles an d has t he same co effi ci en t of diffusion inside the particles . For simplicity a p ulse tracer ex periment will be used . T he density function of the nonadsorbing tracer i s desi gnated fo and that of the adsorbin g tracer f . Also a s s u me that the a adsorption equilibrium is linear and the amo unt of t racer adsorbed at equilibrium per unit volume of C o gas phase is dc 0 , w here Co is the con centration in the gas phase . Consider the case where ideal plug- flow con ditiond exist . T hen fo( t ) for t h e nonadsorbin g tracer is a delta function at t T. One can then measure fa ( t ) for different value s of T an d define a f unc t i on IJ.In ( t ) =
•
ljJ ( t ) a
=
f ( t - -c ) a
( 4 5a )
w hic h gi v e s the so jo u rn - ti m e distribution of the adsorbi n g t racer in the adsorbed state . The simplest case is if the tracer adsorb s in a way that the ra t e of adsorption and de sorp tion is very fas t . In this case ljJ ( t ) a
=
o
(en)
( 4 5b )
and for th e general case w here a is a cons tant , that gives the ratio of t h e residence time in the adsorbed p h a s e to that in t he contin uous p hase . ( 4 5c )
fini t e rat es , E q ( 4 5c ) will not hold . t he b e d has uniform prop erties and the flow in the gas phase is plug flow , lji ( t ) must b e a unique function of T in the continuou s ph a se . a One can f u r th er show that for vario us values of -c If ad sorp tion and desorp tion have
However , if
ljJ ( t / -c 1 a
+
• 2>
t
=
f
0
ljJ
ap
c e / , 1 ) \ji
ap
( ( 1 - e] / ;: ) d e 2
( 4 6a )
w here 1/Ja. p ( t / 1" 1 ) i s the function tfu p ( t ) given that T for the continuous phase is '1 · T he sub script p was added to in dicate that the flow in the continuou s phase is plu g flow . I f E q . ( 46) holds , one can al so show that the Laplace t ran sfer of IJ.Ia ( t ) must have the form ( S hin n ar et al . , 1 9 7 2 )
1 30
Shinnar
�ap and
(�)
e
=
- T P (s)
( 4 6b )
therefore ( 46c )
p ( s ) there fo re c haracterizes the adsorption process or in a more general w ay the transport processes in an outer p hase not reachable for the tracer use for f0 • In E q . ( 4 5c ) it was as s um ed that p ( s ) = as . If one as sumes a finite ra t e of adsorption an d desorption , then in the example given p ( s ) becomes P ( s)
a.s
A
=
7 1-+-=:-(a-:-: /}_,...,.)s
( 47a)
w here A i s a normali zed exchange coefficient . For understanding the gen eral concepts , one does not need more complex form s of p ( s ) . The re a de r should consult Shinnar et al . ( 19 7 2 ) for more details . If p ( s ) is given by E q . ( 47a) , then 1jJ A
ap
- e - CXTS / [ l+ ( CX/-A ) S )
( s)
( 47b )
_
is a well- known function used in chromato graphy . To d erive it , one must not only assume that fo( t ) is a plu g flow , but al so assume t ha t the catalyst bed has uniform properties and t hat t he c a t aly s t is s t ation ary T hi s mea n s that any molecule that got a d sorbe d returned to the same place it left . In the limitin g case w here A -+ oo [ E q . ( 4 5c ) ] , t he last assumption is not needed. H ow e v e r the assump tion of uniformity i s s till n ece ss ary If the bed is uniform in its properties and each particle returns to the same place in the gas phase , Eq . ( 46) can be generalized to the case where fo ( t ) is not plug flow . For any such c a se with arbitrary fo ( t) , which
.
,
$a C t )
t
ap
t
f (t)
a
and
$a ( s )
f 0 ( T ) 1J!
f0
=
f0
=
=
where p ( s )
f ( -r ) lji 0 ap
-t -r
(
d -r
t --' ,
( 48a)
)
( 48b )
d -r
r0 ( p ( s ) )
has
.
( 4 9a )
been substituted for the variable s in
f( s ) .
Si milarly ,
( 49b ) Note that in
the
previous simplified
case ,
1 31
Residenc e - a n d C o n tac t - T ime Dis t rib u tions
( 4 9c )
T he re s ults o f
Eq.
( 49c ) are experimentally testable , as one can me as u re
a s / ( 1 + as / A.) in a p acked- bed reactor under conditions approaching p l u g
Howe v er , t here are stron g re stri c tions on its applicability . E qu ation ( 49) also allow s computation of �a ( s ) an d therefore of 1/J ( t ) from measure a ments of fa ( t ) and f 0 ( t ) in o t h er cases w here the adsorbed particle returns to t he same place . T he only other case w here this direct c o mp utatio n is po ssi b le is w hen t here is n o correlation between the total residence time of a particle in the adsorb e d p hase and i t s residence time in t he ga s p h a s e . In this case flow .
( 50 ) w here * i s agai n t he convolution i n t e gral . Unfortunately , thi s i s a rather unre ali s tic case for a catalytic reactor . In real cases w hen can Eq . ( 49) b e use d ? One case is a packed bed with back diffusion . The other i s a stirred tank reactor in which bot h the continuous and gas p hase are mixed ( such as in a B erty reactor ) . I n that case all molecules in t he continuous phase have equivalent location an d the r e fore E q . ( 4 9 ) ho l ds . There are additional i m portan t i mplic a tion s that can be deduced from E qs . ( 48) and ( 4 9) . Consider the special case of a pa c k ed - b ed reactor in w hich the flow
o . Further assume that flow or r0 ( s ) = e all reactants and products have the same rate of ad sorption and d es o rp ti on and also the same adsorption equilibrium . T herefore , lJla ( t } is con stant for all compounds . Finally , assume t hat a firs t - order complex reaction oc cu r s In t hat case on e can w ri t e an equivalent of Eq . in the adsor�ed phase . ( 28) fo r the product distribution :
in the internal p h a s e
< c .> J
=
a.0 + J
n- 1
'E
r=1
-• s
is p l u g
1/J ( t.. ) ,..
a.
Jr
a
( 51 )
r
which for a packed - bed reactor can be w ritten
=
10
a.
n- 1 +
'E
( 5 1a )
r=l
In standard kinetic p r ac tic e one neglects adsorption processes an d simply assumes that t h e rea c to r is plug flow and o n e writes an equivalent of Eq . ( 2 6 ) :
P . < t > = c < t>
J
where /.. �
1
.
=
a. 0 + J
n- 1
E
* - T t.. r p
( 52)
r= l
are ei genvalues based on p henomenolo gical r at e constants . T is p time constant that has t h e di m en sions of time but does not refer to any real measurable time . It is exp ressed in lb fee d /lb catalyst per ho u r (or lb mo l feed /lb cataly st per hour ) . N ote t ha t
a
Sh.innar
1 32 1:
0
p( A ) r
=
1:
A*
( 5 3)
p r
Equation ( 5 3 ) implies t hat it is alw ays possible to describe a first -order re action occurring in t he adsorbed phase of a cat alyst by assumin g t h at t he reactor is a plu g- flow reactor . T he reaction rates will not relate to any real reaction rates at the catalyst surface , but this is not import ant in phenomenolo gical rate equations in reactor desi gn . Norm ally , tha kineticist is interested in eliminati n g from transport resistances that deal with transport to and from t he catalyst particles and di ffusion in the m acropores , as w ell as b ack di ffusion in the reactor . T hi s can be achieved by as suming that the cat alyst p articles are sm all enou gh and t hat the microreactor used is lon g enou gh [ Silverstein and S hinnar , 1 97 5 ) . I n thi s special case , one c an t hen estim ate t he conversion of an iso therm al packed - bed reactor i n w hich fo ( t ) deviate s from plug flow ( either due to di ffusional resistances or due to m aldi strib utions) , by meas uring f0 ( t ) with an inert tracer and then comp uti n g the effect of the deviation from p lu g flow j ust as one di d in a homo geneous reactor . T hi s is the re sult of derivin g E q . ( 5 3 ) from Eq . ( 5 1 ) . One must , how ever , note that fo( t ) has a different time scale t han E q . ( 5 3 ) . One m ust therefore non dimensionali ze f0 ( t ) and scale it such that the new first moment is the same as that u sed in comp uti n g T hi s is p articularly important , as kineticists often use di fferent definitions for ' P in comp uti n g Equation ( 4 9) to ( 53) also imply that for a first -order reaction , one can predict t he behavior of a p acked bed by getti n g kinetic measllrements in a completely mixed reactor such as a B erty reactor , and vice vers a . One cannot , how ever , predict t he outcome of any reactor for which E q . ( 4 9) does not hold , such as a fluid bed in which t he catalyst move s . If t he movement of the catalyst is slow comp ared to t he adsorption - desorption processes of all reactants , one can get approximate predictions . For exam ple , if the fluid bed is st a ged by vertical baffles , it approaches a series of stirred t anks or , for practical purposes , the react ants return to the same place . T he outcome c an then be predicted using a packe d - bed reactor ( or another fluid bed close to plug flow ) as a model . T he relationship between E q s . ( 5 1 ) and ( 5 2 ) w as derive d for the case w here 1Jia ( t ) is equal for all react ants . T his is not a very likely case but w as used for didactic reasons . How ever , the principle that one can pre dict t he outcome o f a first -order reaction in a reactor for w hich E q . ( 4 9) holds by me asurin g fo ( t ) and the overall kinetics in a packed - bed reactor is valid for arbitrary soj ourn -time distributions in t he adsorbed p hase , even when t hey are different for each compound . T he concept therefore has To prove t he validity of t he more general much bro ader applicability . principle , one can use an argument similar to t he one used earlier when w e discussed op tim al reactor concepts for nonisothermal reactors . For any first - order reaction one can derive a phenomenolo gical equation such as Eq . ( 5 2 ) . Kinetici sts define a first -order reaction by thi s condition as they do not measure real reaction rates in the adsorbed phas e . Consider such a case in w hich 1/Ja. ( t ) is different for all compoun ds . T he average sojourn - time di stribution of the molecule s will now depend on their st ate and therefore on the reactivity of the cat alyst . T hi s is included in :>.. * . Once again , t he hypothetical observer stands at t he outlet o f a plu g- flow reactor and collects the molecule s . T he molecules are now sorted into bins . T he molecules in each bin have a unique so journ time e in the continuous
:>.. ;
:>.. ; .
:>..; .
Residence- a n d C o n t ac t - Time Dis trib u t io ns
phase . I n each bi n , gi ven B one can comp ute P j ( B) , t he probability o f bein g in s t ate j given B. As the number of molecule s is large , p · ( B) i s eq ual t o Cj ( B) . Averagin g over all t h e histories yields < p or or the outlet c o n c en t r at ion . Note that if t he conditions leadin g to Eq . ( 49) hold [ nam ely , t he catalyst has uniform p rop e rti es and a molecule alw ays
?
1 33
�Ci>
re turns to the s ame place in the space of fo ( t ) ] , t hen
tool to test w hether t he system approaches plug flow or not . In a packed- bed reactor one can measure .\ . In fact , if one uses tracers with low er values of A and still gets a-good approximation to plug flow , one can then get an es ti m ate of the solid backmixin g . To esti m ate its imp act on the reaction , one would have to bound A fo r all the re ac t ant s and products , which can in p rinciple be done by usin g tracer methods similar to those used in Fi g . 26. There is one essential point to remember in testi n g for maldistribution of flow or for the e ffectivenes s of internals in fluid beds and trickle beds . This is the need for a goo d comp arison case of a w ell - per for min g reactor . If the residence -time distribution obtained by a tracer experiment deviates stron gly from p lu g flow , it does not necessarily indicate a maldi stribution of flow due to b ad desi gn . It could be a fluid dynamic instability p henom e non which is common in fluid b e d s and which can also occur i n lar ge
1 34
Shinnar
trickle - bed re actor s .
In
trickle - bed
rea ct or s it can be due to di ffusion and
adsorption p henomena in t he liquid p hase . cannot di sti n gui s h
between
I nput output tracer experiment s
t he two cases .
W hat t hey can do is p rovide com
p ari sons bet w een two reactors and give clue s t o t he
tive
w hich is
m e asure s s uch as baffles ,
effectiveness
o f correc
a very valuab le tool p roperly used .
Mea s u rement of H o l d u p i n M u l t i phase S y s t e m s I n m ultiphase
If
systems
the sep arate holdup o f each p h a s e m ay be unknow n .
t he system i s si m p ly soli d gas or liq uid gas , t he holdup o f t he solid or
li q ui d m i ght be obt ained by a pressure difference . T he holdup of the gas is more di ffi cult to measure , as it m ay dissolve in t he liquid or adsorb on a solid . H e re tracer experiment s can be very help ful . One has to be careful wit h the choice of t he tr acer . For a solid tracer it is im po rt an t to find p articles that do not sep arate and have the same si ze distribution and den sity as t he solid s t u died . For best results t he solid s hould be impre gn ate d with t he tracer when possible . I f t he liquid doe s not adsorb or evaporate almost an y tracer will do , b ut i f it does , it i s best to tag the liquid or p art of it . C hoice of a proper tracer from t he gas phase presen t s similar prob lem s . I f t he gas adsor b s or is soluble in t he liquid , one needs a tracer wit h similar p ropert ies or preferably an isotop e of the real compound . T hi s When a suit able tracer is sometimes the only direct w ay to measure holdup . is available , two further requirements m ust be met .
1.
Accurate measurement of
total feed
rate
for the
flow
of the p hase
me asure d
Accurate measurement of the amount of tracer introduced
2.
In a
p u l se experiment
!0
Q C ( t ) is equal to
gral
oo
better than
one can use !0
t he holdup obt aine d
if the
the
always
amount of
care fully
tracer
check t hat the inte
introduced
wit h
an
acc ur acy
95%. In this case , one s hould never use Eq . ( 1 3) . T he first will then give an estim ate of t he holdup . In a step experi -
moment o f f( t ) ment
one should
tracer is
oo
is
( 1 - F)
dt
to
e stimate the holdup .
T he e s timate o f
includes adsorp tion p henomena . T herefore , of the real flow , compoun d , one has to be
one that
not an isotope
very c are ful in i t s interpretation . T he example given in Fi g .
It
i s based
on
dat a
supplied
1 0 c an
1 98 1 ) from a study of the E xxon Donor men t s here
were
problems . by E xxon ( Mills an d D udukovic , Solvent Reactor . T he tracer experi
serve to illustrate the se
t o the author
performed t o me asure holdup o f t he t hree p hase s .
T he
system . T he reactin g gas is a mixture o f hydro gen and hydroc arbons , and t he li qui d is a mixture of dis solved coal liquids and recycled solvent . T he solid p hase i s ash and coal particle s . The tracer used to measure gas holdup was an isotope o f ar gon w hic h dis solves in the coal liquid , and it is t hi s experiment t hat is shown in Fi g . 10 . The solu bility of the tracer at the reactor temperature an d pre s s ure can be me asured and , at equilibrium , the mole fraction of ar gon in the gas is five time s t he m ole fraction of ar go n in t he liqui d . In Fig . 10 t he Laplace transform of t he residence- time distribution com puted from the tracer response was plotte d . T he first moment can be ob t ained from it by estimatin g the tan gent of f( s ) as s + 0 . T his i s sy stem illustrated is a three - p hase
1 35
Residence- and C o n tact - Time Dis t ribu tions A
e as the flow is close to plug flow and therefore ln[ f( s) ] s line ( see Fig . l O b ) , the slope of whic h is to the first moment . The variance has to be es tim at ed from the second derivative near zero . This is not easy , but if the coefficient of variation is sm all , one can do it by model fit t i n g usin g relatively easy h re versus ( s ) i almost a straight equal ,
=
where
1 (1
+
T S /n )
( 1 9c )
n
This pp a several problems the fact cer sol What y the of the tracer , estimate l m r r e ce t lar there no that esti and the gas holdup . Fortunately , in this the es i t ev r reasons . Either gas-liquid mass r s or t the s small , to the has only very small im in the range of accurate measurements because it s The coefficient then in this is ap ec t then error the st the example ve , the mass e c e t could and this alternative could ruled It therefore a o e that mass transfer fast , hic that the the liquid phase the phase . the liquid h l u the of a o is No that if have to be difficult to e i at volume of the phase the reactor .
n in here is a c urve - fittin g p arameter . a ro ch was discussed earlier . are caused by that the gas tra is uble in the liquid . one reall wants to know is not holdup but an of t he gas vo u e in the eacto If t he co ffi i n of variation y is ge is w ay one can reliably particular case , y is small mate t m a e could be obtained . The sm all y could be due to s e al the t an fe r coe fficient is small i is If m as transfer coefficient is adsorp i n in very lar ge . liquid phase a p ac t on C 2 ( t ) , m ain ly affects C 2 ( t ) for t + co ( or + 0) . of variation is reality lar ge , but not p r ia e d when comp uted from the measured form o f f( s ) . One can get a lar ge 2 in e imate of ll and y . In gi n trans fer co ffi i n be measured indep endently be out . is re s n abl to assume the is w h allows one to assume gas adsorbed in is in equilibrium with gas I f one knows o d p and the solubility of gas , a condition for the e ffects b s rp t ion can no reliable way to be made . te y 2 was not very sm all , there make any such correction . and it would s t m e t he gas in
There
.
,
C O N T A C T -T I M E D I S T R I B U T I O N S
has in t n a sojourn-type in active phase ob by s tracer as d t s ds r tracer of i then , for u flow , can n y in the adsorbed phase
The term con tac t - t ime dis trib u tio n been used he literature in two O e is distribution t he which is tained u ing a sli ghtly adsorbin g well as a non a sorbin g inert tracer . I f one de si gn a e f( t ) of the a o bin g as fa ( t ) and the n na d sorb n g tracer as fo ( t ) , pure pl g one define de si t function of a sojourn - time distribution
ways .
o the as
w here < o is the first momen t of f0 ( t ) . This distribution was discussed in the prec e din g section , w here it w as shown t hat the function lj!a ( t ) can be measured not only for plug flow but also for a limited number of other cases . However , alt h ou gh this distribution is useful for reactor modelin g and for dia gn osing maldistributions , it is not directly useful for e sti matin g the behavior of a first -order reaction . The kinetic relations assume p lug flow in a packe d bed , whereas tracer experiments with a proper isotope tracer will not necessarily give plug flow
1 36
S ht n nar
( see Fig . 2 6 ) . A c lo se examin ation of Fi g . 2 6 not use the real residence- time distrib ution o f dict the behavior of a he te ro ge n eo u s reactor .
w il l demonstrate that one can the r e ac t a n t s t o re li ab ly pre In Fi g . 2 6 t he overall flow is plu g flow . However , the re sidence time of the 1 3 c tr ac er is not plug flow an d has a lar ge coe fficient o f v ari ation For ot h er reactants the dif fe re n c e could even be l ar ge r If one applies t he kin e tic rel ation s w hic h are b as e d on t h e assumption of p l u g flo w to a model b ase d on the act ual re si d en ce - t im e distribution , o n e would obtain very erroneous results . More me ani n gful results can be obtained by u si n g the data from t he h e li u m tracer . T he helium tracer has a re side nc e - t i m e distribution similar to fo ( t ) .
.
true overall residence-time distribution of any of the actual reactor is a fluid bed , one can no lon ge r rely on the re sults from a helium t r ac er , nor can one rely on the t ru e residence time distribution . However , if the true residence- time distribution for both the fluid bed and t h e packed be d are available an d t he y are very close to each other , one could deduce t hat the re ac tio n should scale up safely . In b u t dis similar to t he
If
re ac t an t s .
t he
other cases , one cannot directly apply the in formation . A m o re useful definition of contact - time distrib ution is b a s e d on t he re lation bet w een the product di s t ri b u t ion of first-order reac tio n s and t h e E arlier it w a s shown that knowl residence-time dis t ri b u ti o n of the s y s t e m edge of the p ro du ct distrib ution defines the Laplace tr an s for m o f the residence-time distribution density function [ Eq . ( 2 8 ) ] . F or a fi r s t or der irreversible re ac tion A -+ B with a re act io n rate r ( A ) = - ka , the outlet con c ent r ation in a hom o gen eo u s s y s te m all
.
-
"
a
-
a
==
( 5 4)
f( k )
f "
f{ k ) is t he Laplace tran s form of f( t ) w ith the reaction rate k sub stituted for s . For a first -order re act ion o cc ur rin g in a he t e ro ge n eo us re actor , one can use Eq . ( 28) to d e fin e the di s t rib u tion l!_ by w here
==
a. 0 J
n-1 +
I;
r==1
a.
( 55 )
$c >.. r >
J r-
equivalent of Eq . ( 2 8 ) . This distrib ution , o ri gi n ally pro po s e d et al . , 1 96 2 ) an d Glasser ( Glasser et al . , 1 9 7 3 ) , is not a real distribution of sojourn times b ut has the s ame dimensions and p rop e r ties . Re gar d le ss of the rea� s oj o urn ti me distrib ution , A. r is defined and m e as ure d by a s s u m in g that .!J:( A.r ) is plug flow for a p acked - bed r e ac t or . For a complex first - order reaction o cc ur ri n g in a p ac k e d be d reactor , one can determine A. r for each r . I f the reaction then occurs in a different reactor s u ch as a fluid bed , one c an obtain me asurements of C j · The i n for m at i on from Cj an d the Ar fro m t he packe d - bed reactor giv�s estimates for several values o f j:( A r ) . T o ob t ain the complete fun ctio n j:( >.. r > , one would h ave to use cu r ve - fittin g functions . H owever , in p rac tic e this is o fte n not ne e d e d , as sufficient information is available from the linear v al ue s of _i{ A. r ) In this case , A r c orre s p ond s to the L ap l ace transform variable o f 1/J ( t ) even t h o u gh 1/J ( t ) is an artificial constraint defined by Eq . ( 5 5 ) with no physical me ani n g: Another w ay of m e asurin g iOr ) is t o vary k in Eq . ( 5 4) . In a cat aly t ic reactor , this can be d on e by eit h e r changing the te m p e r at ur e of the r eac to r or the re ac tivi ty of the c at aly s t . One can then use an equivalent of Eq . ( 5 4) which is an
by Orcutt ( O rc utt
-
-
"
•
137
Residenc e - and C o n t ac t - T ime Dis t rib u t ions
( 5 6) to define �.
I t h a s the same me anin g as the definition in Eq . ( 5 5) with k re placing B ut it may n o t have the identical numerical form . Chan ging temperature or reactivity m ay change the complex t ran sport proce ss as well a s the ad sorption and de sorption rate s in side t h e cataly st p article s . Therefore , the values of < k ) obtained in this w ay will not be identical to t hose obtained usin g W < A r ) as de fined in Eq . ( 55) . However , the form will be similar . As ume that one measures ( k ) or �( "- r > in a p acked bed and then re peats the s e experiments in a fluidized be d keepin g the sp ace velocity constant , and measures the conversion for each ( k ) . T he highest conversion possible at eac h ( k ) is the same as in a plu g - flow reactor . I f , for any ( k ) , the con version is higher [ or f(k) smaller] , it follows that either the reaction is not first order , or the reactor is not isotherm al , or t he initial small packed -bed reactor w as diffusion controlled . If the conversions for all values of ( k ) are equal to that of the packed bed , the fluid is essentially described by a p l u g flow . If not , the conversion at each point is a m e asure of the w ay the fluid bed deviates from the packed bed . If a /ao for the two cases is plotted vs . ( k ) , one can interpret the curves as plots of ( k ) for the two reactors . (k) for the p acked b e d is by definition plu g flow . Much c an be learned from the nature o f the plot . In Fi g 2 8 an experim ent al contact -time distribution for a relatively short bu bb ling fluid bed is ve n and compared with a plu g- flow re
t..;.
i
s
_t
_t
.
_i
gi
actor and a stirred t ank . Note that $ ( k ) of the fluid bed is lar ger than $(k ) o f the packed - bed reactor ( plu g- flow case ) for all values o f ( k ) . A t lar e values of ( k ) , < k ) , for t he fluid bed , is very flat and , within exp P.rime n tal accuracy , almost p arallel . This indicates that part of the material es sentially bypasses the reactor . In fact , a s im p le bypass model
g
i
i< k )
=
a exp ( - k T) + b
( 5 7)
fits the experimental result s quit e well . Fi gu re 28 also gives tank . Note that it fit s t he experimental data les s well . (k)
_t
$( k ) for a stirred is larger than that
1.0 .B
.6
t( k )
.4
.2 5
10
15
20
k
25
E x peri m e nt al cont act - time distribution for a flui d - bed ( Orcu tt
FIGURE 2 8
et al . , 1 9 6 2 ) . Comp arison with variou s models ; - - - - - , stirred - t ank plu g- flow re actor allowin g reactor ; - - - - - plu g - flow reactor ; for bypass and stagn ancy ( experimental) . ( F rom Glasser et al . , 1 9 7 3 . ) �(k) is measured here by conver sion of ozone in fluid be d over an iron catalyst .
k
is varied by dilatin g the catalyst with inert s .
1 38
Shin nar
of a s tirre d tank for sm all and l a r g e values of ( k ) . For intermediate range i< k ) is smaller than for a stirred t ank , w hi ch is typic al o f s o m e byp as s p h eno m en a . The $(k ) for the fluid bed shown i n F i g . 2 7 has anoth e r int er es t in g proper ty th at is common to most fluid beds . The s lo p e of $(k ) n e ar k -+ 0 is small e r than for a plug- flow reactor . ( The only w ay to k now this is to measure the reaction in a packed bed with the same cat aly s t under isothermal conditions . ) This indicates that not only does the bed h ave a bypass in the bubble phase , but that part of the c at aly s t p h as e is not w ell contacted . In the author's ex p e ri en c e properly baffled (or turbulent) fluid b e d s do no t s ho w any by p as s in the gas p h as e but will s h a re the p ro b l e m that the dense catalyst phase is not perfused . Such baffled beds be h ave like ne ar - p lug - flow reactors with a s m all e r e ffect ive cat aly s t volume ( about 50 to 70% of real c at alyst holdup) . ( A possible e x p lan at io n i s give n in Krambeck et al . ( 1 967) . ] I f one c o ul d re a lly m e a s ure t he slope of ji ( k ) for very small ( k ) , it would app roach T as k + 0. T his i s the s a me p ro b le m t h at w as i llu s t r at ed in Fig . 13 w hen m e a s urin g the tail of F(t) . I f a region of t he reactor i s n o t well perfused , T will be underestimated . If an i n d e p e n d e nt estim ate of T is available , underestimatin g T usin g t h i s technique is evidence that the reactor is n ot well p e r fu se d . For the case shown in Fig . 1 3 , one could estimate T from the volume of the reac t or and the fe ed rate . In the c a se of � ( k ) , the on ly way to e s ti m a t e the first moment is by usi n g a packed bed reactor for comparison . Note t hat $( k ) here yie lds some v al u ab le in formation directly ap p lic able to reactor de si gn . One can compare the infor m ation obtained from Fi g . 27 with that obtained by tracer exp erim en t s in similar fluid beds ( see Fi g . 1 7 ) . Fi gure 17 for a similar be d also clearly in dica t e s R by p ass phenomenon . T he e scap e probability A ( t ) m eas u re d from the tr ac er response of a nonadsorbing tracer show s a c le ar m aximum , indic at in g a b y p a s s or stagnancy . The method shown in Fi g . 17 has one significant a d v an t a ge . It a l lo w s one to p lot A ( t ) w hic h gives dire c t p hysical in s i gh t into the nature of the transport p roc e s s . T here is as yet no e q ui v alen t way to extract e qui valen t information directly fro m in s pe cti on of $ ( k ) . I f f( t ) is measu r ed , compu t a tion of f( s) or f'( k ) i s s t r ai gh t fo r w ar d an d relatively e asy . T he op p o si t e is not true . i:< k ) i s hard to in ver t accurately , a n d therefore it Js h ard to obtain an acc ur ate estimate of the escape p ro b a b ility A ( t) from ljl( k ) . There are way s of invertin g � ( k ) eit he r by n u m e ric al inverSion or by fittin g it wit h a theoretical mo del :- but bot h methods are inacc urate . Ho w ever , i:
1 39
Residence- and C o n tac t - T ime Dis trib u t ions hand , lJia < t > is
u niq u ely measurable only in
t he s e cases where the ab sorbed
in other word s , wh e r e ad s o rp t ion c han ge t he residence - time distribution in the continuous p h ase
tracer p articles return to the sam e pl ace , does
not
.
i&_( k ) provid es a useful alternative . Unfortunately , �( k ) s u ffe rs from t he same problem as lJia ( t ) . T h at is , although it is u s e fu l in mo d e lin g , it cannot b e used for scaleup , as the � ( k ) d e p e n ds on a d so rp tion behavior and would be di ffere n t for different fU'st - o rd e r reaction s . One must t herefore be careful w he n ap p lyin g $ ( k ) t o ac t u al s c aleup problem s . I t will , i n a c ol d flo w model , give good in dica tion s o f bypas s p he n om e n a . T here i s , ho w e v e r , no guar an t e e t h at t he As t hi s condition is oft e n not met ,
real re acj ant will h ave model . lji ( k ) is u s e ful trial reactor s .
one and
If
an
the same
behavior as reactions used in t he cold flow
in stu dyin g m aldistrib utions of flow in act ual indus
do e s no t pe r for m up to expect ations , � ( k ) by chan gin g the t e m p erat ure t he n c o m p are the effect of c h an gi n g in the
in d ustrial reactor
c an o ften obtain sever al va l u e s t h e r eb y
(k) .
One can
industrial reactor to
of
(k)
pi lo t plant re s ult s or similar indu strial reactors .
From
one can then infer � ( k ) and possible re a son s for the poor p er fo rm anc e . This is a power ful di a gno s t i c tool , complementin g or s u b s ti t u ti n g for tracer studie s in indu stri al reactor s . One can get similar re sult s even if the re ac tion is n on li n e ar and complex in volvin g several co m p o u n d s . One starts with the a s s u m pt ion of a plu g- flow re acto r , estimates a s e t o f rate const ant s , and t he n obtains e xpre ssio n s comp letely eq uivalent to E q s . ( 1 6 ) and ( 20) . this
In this c ase , unlike the homogeneo u s c a se , one
comp ute t he outlet composition .
cannot use f( t ) or lji ( t ) to -
o u tlet com p o si tion can still be used to e sti m at e $(k) , as � ( k ) is an operator t r an s for min g the results of a plug flow re ac t o r into our r e ac to r . Note t h at r eli ab le d e si gn requires that $ ( k ) be close to p lu g flo w . An especially i nt er e s tin g case is one of consecutive re ac tion s with equal
reaction r at e s .
T he
Consider , for example , t h e
ca se
The solution of this c ase is gi ve n in Table 2 . I f the values of the r e ac tion rates are equal , one gets an e s ti m ate n o t only of � ( k) b u t also of the hi gh e r derivatives of _i ( k ) or ( dn� ( k) /dkn ) k =k . i Even if the v alue s
of
( k ) are di ffe re n t ,
the values
of
C j give a goo d approximation of the are small . If esti mat es of the
derivative s as lon g a s the differences derivatives are availab le ,
expansion around ( k ) re actio n in
sec u ti ve
a
$(k)
can be recon structed from a
( see G la s s e t et
cold flow model .
al . ,
1 97 3) .
However , if
reaction , one c an comp are t he
It is
h ar d
T ay l o r
to
series
u se such a
an industrial reactor has a con distribution to that of a
p ro d u c t
If signi fic an t deviations are foun d , this allo w s so me nature of t he flow . A good b as e case , preferab ly an isot herm al packe d - be d reac t o r , is es s e n ti al as well as a well-per formin g re actor of a similar type . One doe s not al w ay s w an t to rec o n s t r u c t $ ( k ) ( alt h ou gh this is pos sible ) . T he nature and m a gnitu d e of t he deviations give some i mme di at e qualitative infor m ation ab o u t t he n at u r e an d m a gnit ud e of the deviation from plu g flo w , which c an b e i n t e r p r e t e d by looki n g at some sim p le flow models an d st y din g t heir effect on p ro d uct dis tri b ution . packe d - bed
conc l usio n s
r e ac to r .
as
to the
gi ve n in Table 3 . In t h e se examples , the overall resi dence time T is kept c on s t a n t for all c ases . The re ac t io n rat e k i s varied for each flow model so as to keep the outlet concentration of A ( A /Ao ) con stant . T hi s is don e to make t he e x amp le more consistent with act ual practic e . In m any c a se s , c atalyst activity c han ges and is c o m p e n s ate d for
An examp le is
1 40
Shinnar
TAB LE 2 Dep en denc e of t he Product Distribution of a Fir s t - Order Consecutive Reaction on the Residence - Time or Contact - Time
Distribution8
Reaction :
Reactor with residence time distrib ution f(t)
Plu g- flow reactor
A " A f(k 1 ) 0
-
)!_ A = k1 0
2
E ._1
1-
_Q_ = k k A 1 2 0
D
TI
e
'J-1' 2
- k, T
( k. - k . )
j :# 1
L
. _1 3 1- '
B A0
1
J
TI
J- '
j:# i
- =
1
-k.T
e 1 ( k . - k. )
'- 1 3
J
=
1
1
2
E
._1
1-
c
- =
A0
k k
1 2
f( k ) 1 ( k . - k. )
--::-:-=--,-
II
j =1 ' 2
J
j:#i
3
E ._1
1-
1
f(k ) 1 __.:;_ "
TI
j=1 ' 3
__
]
(k. .
j :# i
-
k.)
__
1
1-A-B -C
=1-A-B -C k.
]
a
k
For contac t - time di stribution , sub stitute
=
3
1
( k ) for
-
E
i= l
f(k ) 1
IT
j=1 , 3 j :# i
k.
k. - k . ]
1
f( k ) .
by changing the re ac to r temperature . In practice one would therefore comp are the results of the industrial reactor to th e pilot plant at constant I A o . One does the same if the conversion is reduced du e to a 'naldistribution i n the flow . H ere the c o n c e nt ration of variou s products of a consecutive re action , w hich are a function of t he residence-time di s t rib ut ion , can clearly show if the reduced conversion is due to a change in catalyst acti v ity or due to a m aldistribution in flow ( provided , of course , that one can confirm in the p ilot plant that catalyst reactivity does not strongly affect selectivity ) . The results for the case of co n s ec u tive reactions A -+ .... C -+ D w he r e all reac tion rates are equal are given in T able 3. The results in t hi s example are very sensitive to the residence-time and contact - time distributions .
A
B
1 41
Residence- and Con t ac t - T ime Dis t ribu tions
k 1 = k 2 = k3 = k Plu g - flow reactor e
-kT
Reactor with residence time distribution f( t )
AAo B
Ao
=
=
2
2
T �
k
e
Ao c
-kT
1 - A - B - C
1 -
f( k1) A
f( k ) = -k 1 ddk
=
I k=k
1
k 21 d 2f( k) 2! dk 2 k=k 1 1 - A - B - C
( if=O ( k.� ) i] e -k -r 1'
A P P L I C A T I O N S O F CO N T A C T - A N D R ES I D E N C E - T I M E
D I ST R I B U T I O N S T O T H E D E S I G N O F H E T E R OG E N E O U S R E AC T O R S
Earlier , rigorous concepts as to how residence - time distributions affect the
conversion and contact - time distribution in i sothermal homo geneous reactor s were discus se d . U n fortunately , very few reactions are truly homogeneous and fewer are isotherm al . In the previou s sections it was shown that t he se principles cannot be applied to hetero geneous reactors in strai ght forward way . However , the m ain applic ations are still valid if one is care ful to recogni ze t he special fe atures o f residence -time distributions in hetero geneous
...... ""' ""
TABLE 3 Depen dence of the Product Distribution of a First - Order Consecutive Reaction on Residence- Time ( Contact - Time) Distribution :
Numerical Example k
k
1
k
3
2 A --- B --e -D k
Flow model
1
= k
2
= k
3
= k
Average residence time ( h )
K
( h- 1 )
A JA
0
B /B O
C tC
0
D !D
0
1.
Plug flow
1. 0
1.0
0 . 368
0 . 368
0 . 1 84
0. 08
2.
Ten stirred tanks in series
1. 0
1 . 05
0 . 36 8
0 . 35
0. 183
0 . 099
1.0
1 . 06 8
0 . 368
0. 336
0 . 1 92
0 . 104
4.
1. 0
1 . 36
0. 368
0. 236
0 . 1 91
0 . 205
5.
1.0
4. 03
0. 368
0. 33
0 . 147
0 . 156
3.
1.0
2: CJ) ;:s ;:s Q ....
143
Residence - and Contact- Time Dis tri b u tions
reactors . I n this section t he concepts presented earlier w i ll be extended to the more general case o f hetero geneous reactors . I n the pre sent state o f knowle d ge , ri gorous al gorit h mic use o f residence time distributions to predict product distrib ution in heterogeneous reactors is very re st ric t e d , even for first -order reactions . T here is no real goo d algo rithmic equivalent of Zwieterin g' s method . Fortunately , t he c oncept of boundin g the outlet composition is still valid . U tili zin g different models , one can still evaluate w hether t he reaction performan c e is sensitive to the act ual flo w model . T he resi den c e - time and contact - time distributions can then be used to place c onstraints on the range of flow models that one should consider . If the re ac t or is close to plug flow , one c an evaluate bounds by me asurin g the deviation from plu g flow by m e an s of a tracer experiment . T hi s works well only if t he devia tion from plug flow is small . The c on c ep t u al uses of residence - time distributions given earlier are still valid . Plu g- flow reactor s or staged reactors approachin g plug flow are optimal for all first - order reaction s and for all reactions in which contact between product s and reactants should be minimi zed . In fact , the example mentioned earlier was from c atalytic heterogeneo us reactors ( B e le vi et al . , 198 1 ) . Also noteworthy is t he con c ep tual use of t he contact - time distribu tion � ( k ) in react or design and scaleup . Recall that ( k ) was really an oper ator that showe d quantitatively the w ay the trans po rt process in the reactor modi fy the p roduct distribution as comp ared to a p acked - be d reactor . Its time scale was the reaction rate used to pre dict the results of a pac ked bed reactor , w hich is simply the overall reaction rate p e r unit volume or per unit wei ght of catalyst . It therefore has the same time scale as the in verse of the space velocity . In a packed-bed reactor with a good flow distribution , it is possible to keep both ( k ) and fo ( t ) , the residence-time distrib ution in t he gas p hase , constant . H owever , durin g sc ale up , t hi s is no t automatically guaranteed . fo ( t ) will norm ally stay con stant and close to the plug flow if the microre · actor or p�ot plant is su ffici e nt ly lon g [ Silverstein and s hinnar , 1 97 5 ) . However , lj! ( k ) will stay constant only if there is no mass transfer resistance between t he gas p hase and the catalyst parti c le . Otherwise , increasin g re actor len gth will chan ge the mass tran s fer coefficient , which will chan ge ( t ) and t herefore (k) . Another interestin g con c ep t u a l conclu sion can be derived from the con cept of a contact - time distribution . I t can be seen from Eqs . ( 4 8 ) to ( 50) that , for certain classes of systems , tPa ( t ) and a ( s ) can be uniquely d e rived from two s e t s of information .
�
i
i
_i
�
1. 2.
T he residence - time distri b ution fo ( t ) in the continuous p hase T he residence -time distribution fa ( t ) of an adsorbin g tracer in a plu g - flow reac tor
Consider a first - order reaction that occurs in the adsorbed phase .
As
each of the different reactants has a di fferent adsorption behavior ( its adsorptive e q ui li b ri um may be different , as well as the rate at w hi c h it
adsorbs or desorbs ) , e ac h species has a different i ( t ) or (s) . To proper ly model the system , one would need an exact tracer (isotope) for each re actant . However , for t he special case described , one could predict lji ( t ) or fo r each of the spec i es , given t he tracer results from a plu g- flow reactor and fo ( t ) , which is common to all species . I f one follow s a molecule
_i
_i(s)
Shinnar
1 44
un der goin g a first -order reaction , one c an concep tually measure for each molecule a total s oj o u r n time i n the adsorbed p hase . To comp ute its residence - time distribution , one must also know the reaction rates in order to determine how much time it s p e n d s in e ac h of the various states it can assume . T his is si mi l ar to t he concept u sed earlier to deal with fi rst - order reactions in n o ni sot h e rmal homo geneous system T he above is equivalent to c han gi n g t he n at ure of p( s ) in E q . ( 4 9a ) . Al t ho u gh one can n ot make this computation without detailed knowledge of a d sor p tion kinetics , one can use the concept to derive an im p or t an cri terion in reaction engineerin g : n am e ly , that for all case s for which Eqs . ( 4 8 ) to ( 50) ap p l y , one can u se the p se udokinetics o b t ain ed from a packe d bed reac to r and ap p ly it to all case s , u sin g fo( t ) to comp ute the result s . T his allows one to d eal w i t h di s p ersion and recyclin g in p ac k e d beds . I t also show s that for a fi r s t - or der reaction , t h e result o f a stirre d - t ank catalytic reactor is direct ly scalable to a pac k e d - b e d reactor , and vice vers a , as a stirred tank is o n e of t he few cases wit h movin g c at al y s t for which Eq . ( 48) app li e s . T his s c a li n g i s po s silb e despite the fact that one is re ally m e as u ri n g a p s e udokin eti c first - order reaction rate b as e d on space ve lo ci ty and not the ac t ual kinetics based on r e al re sid ence t i mes . T his con c ept can be extended q u ali t at i ve l y to s ec on d - or d e r r e ac t i ons in a catalytic reactor . I n reactors o f thi s type , mic r o m i xi n g is no t impor tant . However , it is o ft e n important to know w hether t he p ro d uc t distribu ti on can b e p re di c t e d reasonalby from fo ( t ) . For cases w here Eq . ( 48) applies , the Berty re ac t or p rovides a simple test . If a s ti rr e d - t ank cat aly ti c r e ac tor y i e lds kinetic r el ation s that p redict well the behavior of a packed - bed reactor , t hen nonlinear interactions between products and react ants are not i m p or t ant an d fo ( t ) can be used to p r e di ct the effect of dispersion in scaleup . The comp arison above is often i mp or tant in the d e si gn of complex re It is t herefore important to understan d the p ri n cip le s by w hich actor s . E q s . ( 4 8 ) an d ( 4 9 ) w e r e derived . T he m ain assum p t i on s were : •
1. 2.
The system is sp atially uniform . A n adsorbed p a rtic le (or a p artic le diffu sin g i n t o a c at aly s t particle ) always r e tur n s to the s ame place [ or to a place equivalent to it in terms of fo( t ) 1 .
T he se assump tions are not alw ays valid . N everthele s s , t her e are many cases where one w an t s to predict the p er for m an ce of a hete rogeneous re actor w hich i s not a p acke d bed from eit her a s mall p acke d - b ed reactor or a p ilot plant . This applies to fluid beds as well as tran s p ort reactors such as a riser cracke r . I f o n e comp ares s u c h a reactor t o a p ac k e d bed , it is im p o s sible to keep b ot h fo ( t ) an d i_ 0(k) con s t an t . For example , the first I f t h e reaction occurs moment of f o ( t ) will be m u c h larger for the riser . only at the solid surface of this c atalyst , this s hould not matter . B ut in m any comp lex reaction systems c at alytic surface reactions compete and inter act with gas - p hase reactions . T he cat alyst or solid may also act a s an in hibitor for the g as - p hase reactions . In som e catalytic reactions , intermedi ates forme d at t he surface will either diffuse through the gas phase or re ac t in the gas p hase . One s hould therefore be careful in such cases , and evaluate the po s si b le impact of c h an ges in fo ( t ) relative to l/Ja ( t ) . Ri gorously , it is n ot enou gh that f o ( t ) and ljia ( t ) stay constant . T he distribution of sin gle stays in eit her p h a s e should also s t ay constant . I f
Residence- an d
Contac t - Time Distributions
1 45
one cannot assume this to be true , one can experimentally test t he sen sitivity of t he reaction s t o variation s in fo ( t ) , k eepi n g 1/!a ( t ) const ant . T his technique is a simple and power ful conceptual tool for or gani zing one ' s t hinkin g about the pit falls of a given scaleup problem . C onsider , for ex am p le , a catalytic fluid - bed cracker in which the crackin g really oc c u r s in the dilute phase riser . A t low t e m perat u re s , gas - p hase reactions are slow
and the res u l t s correlate very well with t ho se obtained in a dense - p hase bed at the same space ve loci ty . At hi gher temperatures , the gas -phase re actions become si gni ficant and the correlation is poorer . As t he se gas phase reactions are un d e si rab le , it is imp or t an t to reduce t he average residence time o f fo ( t ) , keeping the space velocity const an t . Anot her import ant example is t he c ase of a dens e - p h ase catalytic fluid bed . E q u ation s ( 48) and ( 49) do not apply ri gorously in a fluid bed for two r eas on s : 1. 2.
The gas phase is nonuniform , as part of the gas i s in bubbles and part in a dense phase around the catalyst . T he T he solid circulation and ad sorbe d reactants circulate w ith it . sojourn - time distribution at e ac h reactant depends on t he rate at which it adsorbs and desorb s from t he cataly s t .
One can , however , design a fluid bed s uch that Eq . ( 4 8) ap p ro xi mate l y For example , one can see to it that fo ( t ) ap pr o ac h es plu g flow . One can al so use an adsorbi n g t rac er with fast e x ch an ge an d desi gn the re actor suc h that fa ( t ) app ro aches plug flow . That would occur on ly if all gas particles are adsorbed and desorbed many ti mes . T hi s then ap p roac hes the condition of spatial uni for mi ty despite the pressure of bubbles . One can achie ve this either by w o r kin g in the tur bulent fluidi zation regime or by usin g properly desi gne d baffles . Designin g the tu rb u lent flow regime only help s condition 1 . The second condition c an b e approached only by b a ffli ng or stagin g t he be d . One can al w ay s approach plug flo w by a s u f ficient num ber of sta ges . I f the catalyst circulation between the st ages is slow comp ar e d to the rate o f a ds orp tion and desorption of reactants , one ap p ro ach e s condition 2 , na m e ly , that each molecule always returns to a place su c h that its r e side n c e time in t he continuous p has e is not c h an ged . These ar gu ments allow one to underst and on t h eo r etical grounds the B affl es give added confidence to t he scale importance of ba ffl es in scaleup . up even if the fluidi zation is t urb ule n t and a t rac e r experiment shows that T his is so b ec au se solid mixing one ap p ro ac h e s plug flow wit hou t baffle s . in a fluid bed alw ays incre ases d urin g scaleup even when t he pilot plant is reasonably l ar ge . It is i m port an t t hat t he b affles really reduce solid mixin g and divide the solid cat aly st phase into stage s . Vertical b a ffles , w hich are often used , do not achieve this goal and t h e refor e do not provide the needed safety net for scaleup which can be ac hieve d u si n g ho ri zon tal b a ffles . We will conclude with on e final e x ampl e . R e c entl y , fl ash hydro ge natio n of coal , in w hich coal i s devolatili zed and p art i ally gasi fi e d in t he p res e nce of hydrogen at hi gh pressure , has received consi d erable attention . A simplified kin etic scheme of the reaction is ap p li e s .
Coal
H2 H2 H2 - heavy liq ui ds - B T X -- - methane ( 1)
( 2)
( 3)
1 46
Shinnar
are also c at aly zed by the con sec utive reaction , plug flo w of t he gas is req uire d for good selectivity , and it is also important to control the re sid en ce time in a narrow optimal ran ge . Kinetic an d the rmo dy n am ic con si de r atio n s r equi r e t he reactor to be as isot herm al as po s si b l e It has and the e asi est w ay to achieve this is by backmixin g of solids . the re for e been su ggested that flui d - bed reactor s or risers with hi gh solid circulation would have si gni fic an t advantages for s uc h a de si gn . However , all pre sent experimental data were obt ained in c urrent dilute phase reactors in which t he solid / gas ratio is an order of m agnitude low er t h an would pre vail in a riser with solid recirculation . T here is no w ay to estimate reliably from data in a d il ut e p ha se cocurrent reactor w hat would happen in riser w it h hi gher concentration of solids . O ne cannot keep residence time and con t ac t time constant sim ultaneously . For any de si gn estimates one would need data fro m a system in w hic h both con ta c t time and residence time are similar to t he desired desi gn . Or one would need evidence th at one of these t wo has no i m p ac t on the re act io n T hese simp le examples should be sufficient to illustrate the varied ap p lica ti on s of contact - and resi dence - time di stributions to the desi gn of hete ro geneo us reactor . They provide guideline s for d e ali n g with th e diffi c ult proble m s of re ac t or scaleup i n complex reactors . Steps
2
and 3 can occur
coal ash as
w e ll as
by
in
t he
t he gas p h as e an d
char .
A s t hi s
is
a
,
-
.
N O T AT I O N
a J
A.
A(t)
concentration of compound A name of co mp o un d j inside the reactor fraction of artic le s inside t h e reactor that will le ave it be fore time t concentration of compound A in prod uct st re am fut ure life distrib ution of p article
p
C C
A Af
concentration of product f in feed stream
a
s trea m
C(t)
concentration of t r c er in exit
C (t) f
concentration of tracer in feed stream
E f
F(t)
expected value of t
n
n or J0"" t f( t ) dt
Laplace transform of f( t ) re si den c e - ti me distrib ution :
raction o f p artic le
f
enterin g at time
zero that leave t he reactor before time t
f( t )
g( t )
G ( t) k
kij
i i fr ac ti o n o f c s exitin g between t and t l1 t = f( t ) L1 t den sit y function of G (t) internal age distrib ution : fraction of p artic le s which are in t he reactor at a gi ven time of ob se rvation that have been inside the reactor less t han time t kinetic rate c on s t ant den s ty function o f re s d e n ce time distribution : p ar ti l e
rate constant of reaction A
+
i
,... A
j
147
Residence - and Contac t - Time Dis trib u t ions K
equilib rium constant of reaction A
+
Aj
i amount of tracer introduced in pulse experiment
ij
m
pi
probability of molecule or particle to be in state j
Q
volumetric flow r ate
t
time
Peclet n umber
Pe
V
volume of reactor variance of density function f( t )
G reek Lett e r s
constants in Eq . ( 2 5 )
�r
{V;
y = A
coefficient of variation of f( t )
T
ei genvalue s of firs t - order kinetic systems [ see Eq . ( 2 5) ]
r
escape prob ability o r inten sity function of residenc e - time distrib ution r E C t ) = rth moment of f( t ) r
A (t) ll T
r
=
average residence time in homo geneous reactor
=
V /Q
Laplace tran sform o f contact - time distribution sojourn - time distribution density function o f stay in the adsorbed p hase [ see Eq . ( 4 6 ) ] R E FE R E N C ES
Belevi , H . , J . R . Bourne , and P . Rys , C hemi c al selectivities dis guised by mass diffusion , Helv . C him . Acta , 64 , 1 6 1 8 ( 1 98 1 ) . Biloen , P . J . N . Helle , F . G . A . van der Berg , and W . M . H . S achfler , I sotopic transients in Fischer T rop sch reactions , J . C atal . , ( 1 9 8 3 , in press ) . D anckwerts , P . V . , C ontinuous flow systems , C hern . En g . S ci . , 2 , 1 ( 1953) . Evan gelista , J . J . , S . K at z , and R . S hinnar , T he effect of imperfect mix ing on stirred combustion reactor s , 1 2th Symp . ( Int . ) Combust . , Combu stion I nstitute , Pittsburg , P a . , ( 1 96 9a ) , p . 90 1 . Evan gelista , J . J . , R . Shinnar , and S . K at z , Scale - up criteria for stirred tank reactors , A I C hE J. , 1 5( 6 ) , 8 4 3 ( 1 96 9b ) . Froment , G . F . and K . B . Bischoff , C hemical Reac tor A nalysis and Design , Wiley , New York , ( 1 97 9) . Gilliland , E . R . and E . A . Mason , Gas mixin g in beds of fluidi zed solids , Ind . En g . C hern . , 4 4 , 2 1 8 ( 1 95 2 ) . Glasser , D . , S . K at z , and R . S hinnar , T he measurement and interpretation of cont act time distributions for catalytic reactor characteri zation , I n d . Eng. C hern . Fundam . , 1 2 , 1 6 5 ( 1 9 7 3 ) . ,
1 48
Shinnar
Hiester , N . K an d T . V erme ulen , S at u ration p e r fo r m ance of ion -e xchan ge and a d so r p ti o n columns , C hern . En g . P ro g . , 4 8 , 505 ( 1 9 5 2 ) . Himmelblau , B . M . an d K . B . B ischoff , Process A nalysis and Sim u latio n , Wiley , N e w Y ork ( 1 96 8 ) . K rambeck , F . J. , S . K at z , an d R . Shinnar , Stochastic models for fluidized beds , C hern . E n g . Sci . , 2 4 , 1 4 97 ( 1 9 6 7 ) .
S hinnar , Interpretation of tracer experi me n t s in sys te m s with fl uc t u a ti n g throughput , I n d . En g . C hern . Fundam . 8 , 4 3 1 ( 1 969) . Le hman , J . and K . Sch ii gerl , I n ve s ti g atio n of gas mixin g and gas di strib u tor pe r for m an c e in fluidi zed beds , C hern . Eng. J . , 1 5 , 91 ( 1 97 8) . L e ve n s p ie l , 0 . , and K . B . B i sc ho ff , Patterns of flow in chemical process vessels , Adv . C h ern . Eng . , 4, 95 ( 1 963) . Mills , P . L . , an d M . P . D u d u kovic , Evaluation o f Liquid - solid con t ac t i n g in t ri c k le b e d reactors by tr ac e r methods , AIChE J . , 2 7 , 8 9 3 ( 1 9 8 1 ) . M u rp hree E . V . , J . V oo rhies , Jr . , and F . X . Mayer , A pp lic atio n of cont ac t i n g studies to the an aly s is o f reac to r per form anc e , I nd . Eng. C hern . Process Des . Dev . , 3 , 3 8 1 ( 1 96 4 ) . N aor , P . and R . S hin n ar , Repr e sent atio n and evaluation of resi denc e time distrib ution s , I n d . En g . C hern . F un d am . , 2 , 2 7 8 ( 1 96 3 ) ; N aor , P . , R . S hinnar , and S . Kat z , I n dete r min ac y in the estimation of flowrate and transport functions from tracer e xp e ri m en t s in closed circ u l ati on , Int . J . En g . S ci . , 1 0 , 1 1 5 3 ( 1 9 7 2 ) . Naum an , E . B . , R e s i dence time distrib utions an d micromixing , C hern . E n g . C ommun . , 8 ( 5 3 ) ( 1 9 8 1 ) Nauman , E . B . a nd B . A . Buffham , Mixing in Co n t i n uo u s Flow Systems , Wile y , New Y or k ( 1 98 3 ) . N g , D . Y . C . , an d D . W . T . Ri p pi n The e ffe c t o f incomp lete mixin g on conversion in hom o ge n eo u s reactions , Proc . 3rd Eur . Symp . C hern . R eac t . En g . , P e r ga m on Pres s , Oxford ( 1 96 5 ) , pp . 1 6 1 - 16 5 . Orcutt , J . C . , J . F . D avi d s o n and R . L . Pi gfor d , Reaction time dis t rib u tion s in fluidi zed c ata ly tic reactors , A I C hE C hern . Eng. Pro g . S ym p . S e r . , 38 , 1 ( 1 96 2 ) . Orth , P . an d K . S c h ii ger l Distribution of re sidence times and cont act times in p acked bed re ac t or s : influence of t he c hemical reaction , C hern . E n g . S ci . , 2 7 , 4 97 ( 1 9 7 2 ) . Overcas hier , R . H . , D . B . T odd , and R . B . O lne y Some effects of baf fles on a fl ui di ze d system , A I C hE J. , 5, 5 4 ( 1 95 9) . 0 s t er gaar d , K . an d M . L . Mi c he lse n On the use of the i m p e r fe c t tracer p ulse method for determination of ho l d - up and axial m ixi n g , C an . J . C he rn . E n g . , 4 7 , 10 7 ( 1 96 9) . Pennick , J . E . W . Lee , a n d J . Maziuk , D e ve lo p m e n t of t he methanol to gasoline ( M T G ) P ro c e ss , 7th I nt . Meet . R eac t . E n g , Boston ( 1 982) . Petho , A . and R . D . N o b l e , eds . , Residence Time Dis trib u tio n T h eo ry i n C hemical E n gineeri n g , V e r l a g C h e m ie W ei n h eim , W e s t Germany ( 1 98 2 ) . S ei n fe l d , J . an d L . L ap i d u s Process Modeling, Es t imatio n , and Iden t ifica tio n , Prentice - H all , E n glewood C liffs , N . J . ( 1 9 7 4 ) . Shinnar , R . , On t he behavior of liquid di sp e rsi on s in mi xi n g vessels , J . Fluid Dyn . , 1 0 , Pt . 2 , 2 5 9 ( 1 9 6 1 ) . S hinnar , R . and P . Naor , R esi d en ce time di s t ri b utio n s in systems with i n t er nal reflu x , C he rn . E n g . S ci . , 22 , 1 3 6 9 ( 1 9 6 7 ) . Shinnar , R . , P . N aor , an d S . K at z , I nterpretation and e v al ua tio n of multi ple tracer exp e ri men t s , C hern . E n g . S ci . , 2 7 , 16 27 ( 1 97 2 ) . Krambec k , F . J . , S . Kat z , an d R .
-
,
.
,
,
,
,
,
,
.
,
,
149
Residence - and Contac t - T ime Dis t ri b u t ions S hi n n ar , R . , D .
Glasser , a n d S . K at z , First order kinetics in c o n tinuous
C h e rn . En g . S c i 28, 6 1 7 ( 1 973) . S hinn ar , R . , Tr a c e r ex pe rim e nts in re ac t or desi gn , 2nd C on f . Physico c he m . H y d ro d y n . , Oxford ( 1 9 7 7 ) . S hinn ar , R . P ro c e ss cont rol rese arc h : an evaluation of present st a t us and research needs , A C S Symp . Ser . , 72 , 1 ( 1 97 8 ) . Si lve rst ei n , J . , and R . S hinn ar , D e si gn of fixed bed cat alytic microreactors , Ind . En g . C hern . Proce s s D e s . Dev . , 1 4 , 1 2 7 ( 1 9 7 5 ) . Snow , A . I . , p riv ate comm unication , A R C O Petrole um C o . ( M ay 1 98 3 ) . T army , B . , private communication , Tracer experiments performed on the reac tor s ,
.
,
Exxon ED S Pilot Plant (March 1 98 3 ) . J . an d C . D . P r a te r S tructure an d analysis of complex chemical re act io ns , Adv . C atal . , 1 3 , ( 1 96 2 ) . Wein stein , H . and R . J . A dler , Mi c ro m i xi n g e ffects in continuous chemical reac to r s C hern . E n g . Sci . , 22 , 65 ( 1 96 7 ) . Wein stein , H . an d M . P . Dudukovic , Tracer methods in the c i rc u l at io n in Topics in T ransport P he n om e n a R . Gutfin ger , ed . , H e mis p he r e , New York ( 1 9 7 5 ) . Woodrow , P . T . , U se o f intensity function rep r e s e ntatio n of residence time variability to understand and impro ve p er for m anc e of industrial reactors , ACS Symp . S e r . , 6 5 , 57 1 ( 1 9 7 8 ) . Z virin , Y . an d R . Shinnar , Int . J . Multip hase Flow . , 2( 5 / 6 ) , 4 9 5 ( 1 97 5 ) . Z wieterin g , T . N . , The d e gr e e of mixin g in cont i n uo u s flow systems , C h e rn . E n g . S ci 1 1 , 1 ( 1 95 9 ) . Wei ,
,
,
,
,
.
,
3
Catalytic Surfaces and Catalyst Characterization Methods W . N I C H O LAS D E LG ASS
E D U A R D O E . WO L F
Purdue U nivers i ty ,
Wes t Lafay e t te ,
Univers i t y o f No t re Dam e ,
No t re Dame ,
In diana In diana
I N T RO D U C T I O N
C atalysi s i s a complex p henomenon i n which a multit ude o f i nt er e st in g
A review of the prin c i p l e s an d factors involve d in variables intervene . determinin g c a taly ti c ac t i vity is b e y o n d the scope of t h i s chapter . We have
focused on the characteri zation of cat aly t ic propert ies to enhance our un Emphasis is given to t he modern s u rface a n aly si s tec hniques , which can p ro vi de new in sights for interp ret ing catalyt ic phenomena . Such methods , whic h were often first applied to st u dy model surface s , are n ow b ei n g use d to analy ze the more complex sur faces foun d . on t ec hnical heterogeneous catalysts . T wo ty pes of c harac t e ri zation techniques are di scuss e d in this chapter : ( a ) p hy sical charac teriza tion , and (b ) physicochemic al c haracterization . T h e first type refers to total surface are a and p ore size distribution . Total area is re l evan t in de ter m inin g the contact between t h e catalytic a gent and reac t an ts . P oro s i ty is si gni fican t in c on t r o l li n g the tran sport of the c atalytic a gen t in to the support durin g catalyst p r ep ar ation as well as t he transport of r eac t ants and p r o duc t s between t he b ulk fluid p hase and the active sites durin g re acti on Since surface area and pore vol u me are r elat e d a b alance b etween these two variables is a key factor in cataly st de s i gn . The B ET a d s orp ti on method to me a sure total surface area a n d metho ds to m e a s u re pore s i ze dis trib ution are disc u s se d in t he follow in g two section s . O t h er imp o rtan t variables t hat cha r ac t e ri z e a su pp or te d cataly st are the area of the a c tiv e cataly tic component ( i . e . , metal area in t he case of a s up po r te d metal catalyst ) and the c ry s t alli t e size distribution of t he active Many reac tion s are sensi t ive to the crystallite size , and t hey are p hase . termed s t ruc t u re -sen s i tive or deman d i n g reac tions . Va ri o us factors are in volved in this structure sen siti v i ty The occu r re nce of a partic ular crystal phase , the ratio of edge to corner sites , the st r uctur e of the surface , and th e s tab ility of o v e rl ay e r s can all depend on c ry s t alli te si ze . A di s t in ct iv e chemical feature of c ryst all it e s in t he size re gion 0 . 5 to 5 nm is the low c oor dination n umber of the s urface atoms . Later s ec tion s de sc rib e sel ec tiv e de r s tan din g o f the underlyin g processes .
,
.
,
.
1 51
1 52
Delgass and Wo lf
chemisorption of gases as a method to measure total metal area an d x - ray diffraction an d tran s mission electron mic roscopy as methods to determine avera ge crys tallite si ze and size distrib ution . Although the methods mentioned previously have been known for many years , their development for chemic al analy sis of surfaces is more recent . As know ledge of the principles of op eration , capabilities , an d limitations of surfac e analysis methods expands , the in terpretation of catalytic p henomena will become accordin gly more sop hi sticated . Ultimately , thi s new knowledge of surface chemical behavior will support the search for and desi gn of new catalysts . Knowled ge of the composition of a surface is a prerequisite for understan din g its c hemistry . X - ray p hotoelectron spectroscopy ( X P S ) , Au ger electron spectroscopy ( A E S ) , an d ion scatterin g spectroscopy ( I S S ) p rovide quantitative analysis o f surfaces o r thin surface layers . This in formation can reveal surface en richmen t in alloy s , the surface concentration of promoters in multicomponent supported c atalyst s , and the composition of the s urface of mixed oxides . These methods can also detect imp urities and poison s an d , in general , follow c han ges in the cataly s t surface after expo sure to the reaction environ ment . Oxidation state an d p hase iden tification come most readily from XPS an d , for selec ted elements , from t he Mossbauer effect . T he c urren t frontier in surface analysis is the meas urement of local s urface composition an d structure by application of ( a ) extended x -ray ab sorption of fine struc ture ( E X AF S ) , an d (b ) secon dary ion mas s spectros copy ( S IM S ) . We disc uss the se analy tic al tools and magnetic resonance in relation to the determination of surface composition an d structure . T he distrib ution of a catalytic material , or a poison within ·a support , can be a key variable in c atalyst performance . Slight chan ges in p rocedure durin g cataly st preparation may in troduce significant c han ges in t he distri b u tion of cataly tic materials inside the pellets . We disc us s the use of elec tron microprobes and electron microscopes equipped with analytical capabili ties to assess the distrib ution of materials in side p ellets . To complete the chemical c haracteri zation of a surface , one must examine its interac tion with adsorbin g an d reac tin g gases . We discuss methods by whic h the effec ts of gas - solid contac t on the composition an d chemistry of the soli d surface can be examined . T he influence of the surface on the adsorbed molec ules is usually probed with vibrational spec troscopy ( I R or Raman ) an d can be investigated with a magnetic resonance ( N M R or E S R ) . Another measure of gas - surface interac tion is the stren gth of the adsorbate bon d . Sabatier suggested lon g a go that this bon d must be stron g enou gh to perturb the molec ule b u t not so stron g as to form an un reac tive s urface compoun d . T emperature- p rogramme d desorption and reaction methods for studyin g surface bon ding are discusse d , alon g with vibrational spec troscopic tec hnique s for examinin g the adsorbate molecules themselve s . T hese adsor bate studies can serve either as a direct probe of the reaction it self or as an additional mean s of chemical c harac terization of the catalytic surface . T he tec hniques revie wed in this c hapter are those best developed and most commonly applie d or most promisin g at this time . T he presentation emphasi zes the es sential conc epts an d features of eac h technique and major application s in cataly sis . We have attempted to avoid excessive use of jargon to keep the disc ussion s un derstan dable to the non specialist . Refer ences to more detailed descriptions are given in each section . We note , finally , that thoro u gh characteri zation of catalysts often requires use of combination s of techniques chosen to give complementary information , and
1 53
Cataly tic S urfaces
that characteri zation o f the catalyst surface encompasses only half of the information neces s ary to fu lly un derstand a catalytic system . T he other half is the evaluation of t he element ary kinetic steps of the reaction w hich are the m anifestations of the influence of the c ataly s t surface on the react
in g molecules .
TO T A L S U R FA C E A R E A :
BET METHOD
There are several technique s t o e sti m at e the tot al surface area an d pore size distribution of porous m aterials . Various review article s concerning such measuremen t s have been published ( E m met t , 195 4 ; Innes , 1 968) . The B E T method is , however , the most com mon t e ch niq u e employed today for such measurements ; conse q uent ly , it is the only t e chnique di scussed here . The reader is directed elsewhere in reference to other techniques ( I nn e s , 1 96 8) . BE T A dsorp tio n
Many physical adsorption isotherms exhibit S shapes which are inconsistent with the Lan gmuir i sot herm b ased on monolayer adsorption - de sorption e quilibri u m . B runaue r , Em mett , and Teller ( 1 93 8) p roposed an explanation base d on the assump tion that m ulti l ayer adsorption could occ ur . T he B E T
model also pos tulates s ur fa ce h omo gen ei ty and no intermolecular interaction The most common form of the B E T a ds ortp ion isotherm also assumes that an infinite n umber o f multilayers can be ad sorbed . T he resu l tin g lineari zed B E T i sotherm has the form
between adsorbed s p e ci e s .
V d (P a s
p
O
·
- P)
=
1
V C m
+
c - 1 V C m
p P 0
( 1)
where Vads is the volume at STP occupied by molecules adsorbed at a pres sure , P , Vm is the volume corre spon din g to monolayer coverage , C i s a constant , and P o is t he saturation vapor pressure of t he adsorbate over a plane surface . A plot of P / V ads ( P o - P ) versus P /P o is a s t r ai gh t line with slope S = ( C - 1) /Vm C and i n t erc ep t I = 1 /V m C . Knowin g S and I p e rmits c alc ulation of V m and t here fore the number of gas molecules ad sorbed in a mono l ayer , w hich , when m ultiplied by the cross- section al area of the ad sorb ate , give s the total surface are a o f the solid . T he followin g cross - sectional areas (in square angstroms per mole cule ) give reasonably self-consistent total are as : N2 16. 2 , 02 1 4 . 1 , Ar = 1 3 . 8 , Kr 19. 4 ( Adamson , 1 97 6) . N 2 is the p re ferre d adsorbate , but Ar and Kr are also commonly used . =
=
=
T he most common e xperimental con fi guration s used to measure B E T iso therms ( Innes , 1 96 8 ; E vere t t and Otterwill , 1 97 0 ) are ( a ) the classic al
volumetric static app aratus used by E m m e t t , in w hich pressure associated with increm en t al volumes of gas is measured ; ( b ) a recirc ul ati n g batch flow system with helium dilutent ; and ( c ) a pulse chrom atographic dyn amic
flow system .
Several commercial s y s te m s b ase d on s t atic or d y n amic techniques are available ( Everett and Ott er will , 1 97 0 ) . T he expe ri m e nt al obje c ti ve in every These case is to determine V ads as a func tio n of the adsorb ate pressure . values are then sub stituted into Eq . ( 1 ) so that the intercept and the
1 54
Delgass and Wo lf
slope of the lineari zed BET isotherm c an be determined . T he re gion of best fit for the isotherm is in the P /P o r an ge 0. 0 5 to 0. 3 . C ontinuous an d autom atic systems have also been de si gned and differ from batch systems in that the gas is added continuously to the system at a very low rate instead of in lar ge increment s . Contin uous a d dit ion per m i t s automation , w hich , when coupled with microcomp uters , provides di rec t c alculation of surface area an d pore si ze distribution . M ajor factor s in all B E T measurements are p roper de gassin g of the samp le and e q uili bration . O t her t e c hni q ues used for c alc ul ation of total surface area , such as gravi metric techniques , se le c tive a d so r p t ion from liquids , and small - an gle x - ray scatterin g , are briefly summarized by I nnes ( 1 96 8 ) . PO R E S l Z E A N D D I S T R I B U T I O N
T he distribution o f pore si zes i s one o f the i m p o rt an t characteristics o f s up ported c atalysts since i t i s related to the value of the effective transport coefficients , and consequently it can affect activity , selectivity , and rates of de activation . C at alyst pellet s have a comple x p ore structure which pre sents a wide distribution of si zes . Pores b et ween 10 and 1 0 0 A are re ferred to a s mtcropores , those between 1 0 0 0 A and 10 �m are called mac ro pores , and those of intermediate size are c alled mesopores . T he micropores are us ua lly characteristic of the s upport p o ros i t y , whereas macropores can ori ginate in the interp ar ticle s p ac e crea t e d d urin g formation of the pellets . A m ateri al e x hi bi t in g both micropores and macropores is described as having a bimodal pore di s t ri b u t io n . Detailed disc us sions of pore structure and its determin ation have been presented in various textbooks and mo no grap h s ( E verett and O t terwill , 1 97 0 ; Gregg an d S i n gh , 1 96 7 ; Parfitt and Sin gh , 1 976) .
A first - order approximation or estimate of the ave r a ge p o re si ze can be pe r fo rm ed by ass umin g that all pores are nonintersectin g c ylin de r s of uniform len gth L an d radius rp ; t he n
Pore s u r fac e Pore volume
=
s
v
p
=
2
r
( 2)
p
or
2V
r
p
___E. s
w here V p is t he t ot a l pore volume and S the total surface area . For materials with a distribution of pore si zes , it is necessary to kno w
the pore volume at various pore si zes . The derivative of the pore volume curve wit h respect to the radius give s the pore si ze distribution . T he two m os t common tec hniques used to determine pore volumes are the BJH method ( B ar re t t et al . , 1 9 5 1 ) , base d on t he use of either a d so rption iso therms or porosimetry . Pore Size Dis tribu tion from Phy s ical A ds o r'p t i o n Isother'ms
If a physical adsorption isotherm is e x t e nd e d to the re gion in which P /Po about 1 , a rapid increase of V occurs due to condens ation of the ads
is
1 55
C a taly tic S urfaces
ad s o rb at e o n ! he pore walls . Conversely , a de c r e a s e in pressure res ults in e vap o ration of the adsorbed o r condensed liquid . Kelvin derived t he followin g relation between the vapor pressure reduction over a liquid con tained in a c y lind rical c api llar y and the radius r : =
where P o
2V o
rRT
cos ¢
( 3)
the sat u r ation pressure , a t he surface tension o f t he liq ui d , simplicity ass um ed to be 0° ) , and V the molar volume of the liquid . T he K elvin equatio n c an be u s e d in pri n ciple to calculate r fo r any c orresp on din g value of P /P o . However , sever al com plic at ions arise : ( a ) a hysteresis loop is often observed i n ad sorp t ion isotherms ; ( b ) the pores are not cylindric al and have v aryin g radii ; an d ( c ) an adsorbate film of varyin g t hickne s s decreases the e ffec tive Kelvin radius . A disc u ssion o f the various theories that have been p ro p o s e d to account for the factors cited above is given by G r e g g a n d S i n gh ( 1 967 ) . A detailed c alculation p rocedure , known as the B J H me t ho d , has been reported by B arrett , Joyn er , and H alenda ( 1 95 1 ) . The m e t hod c on sists o f step w i s e c alculation s based on t he dat a obtained from the de s orp tio n branch of the is other m abo ut the re gion where P /P o :::: 1 . T he c alc ula tio n s yield a cum ulati ve pore volume versus po re radius whic h is then differentiated with respect to the radius to obtain t he pore si ze distribution . is
Mercury Porosimeter
In this case the pore volume is measured di r ec tly by forcin g a nonwetting liquid , mercury , into the pores . A force balance between external pres sure and surface ten sion in a cylindric al pore gives r
or for r
=
2 o cos
p
e
( 4)
Hg , =
7500 p
E q u a ti on ( 4 ) allo w s direct calculation of the pore radius at a given pres s ure . A fter plottin g the cumulative pore vo l u m e , which equals the volume
of H g force d into the pore s , versus p ore radius , one can o btain the p o re si ze distribution by di f fer en tiatio n with re s pe c t to r . With proper experimen t ation , both N 2 de sorption an d H g p orosi m etr y ge n era ll y a gree well . T he N 2 deso rptio n met hod is adeq uate for small pores ( 2 to 30 n m ) , whereas lar ge p ore s are p re ferenti a ll y measured by low - pre ssure H g p oro si m e t ry . With so m e materials hi gh - p re ssure H g pene tration modifies the pore structure b y br e akin g pores , w hich can lead to spurio us re sult s .
D ET E RM I N A T I O N O F M E T A L A R E A
BY
S E LE C T I V E
C H EM I S O R P T I O N O F G A S E S
The m os t common m et ho d employed to deter m in e metal area is selective che mi sorption of ga ses . T he m e th o d has been discussed in review articles an d
D e l gass and Wo lf
1 56
te xtbooks ( Whyte , 1 9 7 3 ; Anderson , 1 9 7 5 ) . It con sists o f me as uri n g the volume of gas adsorbed at monolayer coverage as in the case of B E T ad sorption ; however , gase s that c hem iso rb selectively on the surface under study are used instead of those that physisorb . T able 1 lists a number of metals and gases that form c hemisor bed monolayers . Since for selected gases chemi sorption is irreversible , fast , and usually sin gle layer , mono lay e r coverage is de t e rm in ed from the plateau of t he i s ot herm , extrapolated to zero pressure . A simple stoichiometry is required to relate the number of gas molecules adsorbe d w i t h the num ber of surface atoms . For a given met al , an average site density can be c alculate d for t he cry stallo grap hic planes exposed . From the site density , an area per surface atom ( Sj ) is calculated . For examp le , for Pt the densities of the ( 1 0 0) and ( 1 10) planes are 1 . 3 1 x 1 0 15 2 and 0. 93 x 10 1 5 sites /cm 2 or 1 . 1 2 x 10 1 5 sit e s /c m 2 on the site s / c m average ( 5 0 - 5 0 distribution ) . T hi s c orre sp o n d s to an average area o f 8 . 9 A 2 f atom . T he total metal area per gram of catalyst ( SM ) is then calculated as
=
s . vv 1
ads
( 6 . 023
x
10
23
> ( 5)
22 , 400 m
adsor ption stoichiometry ( s urface atom s /n um ber of adsorbed , V ads the volume of gas adsorbed at monolayer coverage and standard temperature p ressure , and m the mas s of catalyst used .
w here v is the molecules
TAB LE 1
S elective
C hemisorption
of G ase s
on Metals
a
G ases Metals
Ti , Z r , H f , V , N b , T a , C r , Mo , w , Fe , Ru , O s Ni , C o Rh , P d ,
Pt , I r
02
C H 2 2
C 2H 4
co
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Mn , C u
+
AI , A u
+
+
Li , N a , K
+
+
M g , A g , Zn , C d ,
+
I n , Si , G e , S n , Pb , A s , S b , Bi
a + , S tron g chemi sorption occur s ; ± tion is unob servable .
c hemisorption
H
is weak ;
2
C02
N
2
+
+
±
c he mi s o r p -
1 57
Cataly tic Surfaces T he metal fraction expose d , or dispersion ,
D
::
metal surface atom s tot al met al atom s
=
V lJ
.M
1
--
mw
X
10
-6
is c alc ul ated as ( 6)
where M is t he molecular wei ght of t h e metal , w t h e fr ac ti on al metal load up t a ke in micromoles . T he average c ry stal lit e si ze is c al culated by assumin g that crystallites are of t he same r e gul a r uniform s h ape ( i . e . , crystallite s are often ass umed to b e c ube s with five faces ex posed and the sixth bein g in contact wit h the support ) . For this
in g , an d ll i t he
geometry d
( 7)
wher e P M is the density and d the ave r ag e c r ys t alli t e s i z e . Spenadel and B o u d ar t ( 1 96 0 ) de m on st r at ed that H 2 c h e mi s orptio n on Pt black ( 10 torr , 2 5° C ) a gree d well with B E T d at a . A greement b et w ee n H 2 c h e m i sorp t ion and x - ray d i ffraction re sults was also obtained in t he case of a Pt supported on alumin a c ata ly s t . A variation of t he direct che m i so rp t ion techniq ue is titration of pre adsorbed 0 2 w i th H2 ( Gruber , 1 96 2 ) . B en son and B oudart ( 1 9 6 5 ) propose d that the ti t ra ti o n stoichiom e try is
w hic h allows for the up t ake of three h ydro ge n atom s per Pt at om , thus in t he· se n si tivit y of t he techniq ue . Mears and H ansford ( 1 9 6 7 ) found that a di ffe ren t Pt - H 2 stoichiometry appeared to be ap plicable to their result s . The dile mma w as resolve d by Wilson and H all ( 1 97 0 ) , w ho found that the s to i c hio m et r y of oxy gen chemisorption d ep en d e d on c r y s t al lite si ze . For this reason , titration t e c hn i q u e s are not used wit h s am p le s havin g very small si zes . T he u se of CO c he m isorpt io n has also been pro p osed to d e t e r m i n e H ow ever , the relative c on t ri b u metal surface areas ( H u ghe s et al . , 1 96 2 ) . tion of t he linear and br i d ge d form s of CO m u st first be assessed to e s tablish t h e proper stoichiometry . O t her gases , such as NO , have been used to evaluate the metal surface are a of R u ( R amamoorthy and Go n z ale z , c re asin g
1 97 9 ) .
The experimental te ch niqu e s to meas ure V a ds have not been mentioned above , but in m o st references a st atic or volumetric te c hni q ue similar to the one use d in BET ab s orp tion has been used . A flow pulse tech ni uq e t ha t p er mi t s rapid determination of V ad s has been described by several authors ( Gruber , 1 96 2 ; B en e si et al . , 1 97 1 ; H uas en and Gr ube r , 1 9 7 1 ) . T he effect of s am p le volume , amount of catalyst , pretreatment , an d so on , on t he value obtained for V ads usin g t he pulse technique has been rep ort e d . P ulse results agree well with the static te chnique s if ad so rption is very fas t and irreve rsib le . Reversible ad s orp t io n introd uces tailin g on t h e emer gi n g peaks , which can lead to error s ( W anke et al . , 1 9 7 9 ) . A s t rin ge n t re q uirem en t of the pulse t echni q ue is the use of clean
1 58
D e l gass and Wo lf
ultrahi gh - p urity gases and various trap s to eliminate trace imp uritie s . S arkany and Gon zale z ( 1 982b ) have discusse d conditions for proper use of the p ulse technique , in p artic ular , linearity of t he T C cell respon se and t he effect of reversible a dsor p tion Approp ri at e ly used , the flow technique gives good and expedient re sult s and the app aratus is easy to build . For a new catalyst it is advisable first to establish the operatin g conditions in w hi c h reproducibility with the static or other techniq ue s is attained . .
D E T E RM I N A T I O N O F AV E R AG E C R Y ST A L L I T E S I Z E A N D C R Y S T A L L I T E S I Z E D I ST R I B U T I O N X - ray D i ffract ion X ray s have often been used in c at alytic work in relation to the e stimation of cryst allite si ze and to obtain information on t he b ulk crystallograp hic st ructu re of catalytic m ate ri al s A n u m ber of mono grap hs exist on x - r ay diffraction ( Klug an d Alexander , 1 97 4 ; Cohen an d Schwart z , 1 97 7 ; C ullity , 1 9 7 8 ; Schult z , 1 98 2 ) and the subject is so exten sive t hat it can be treated here only in basic ter m s .
.
N ature of X rays X r ay s are electromagnetic r adi at io n with w avelen gths in the an g str om ran ge . Con sequently , they can penetrate m atter and therefore are especially well suited to probe the structure of solids . X rays are c om monly gen e r at ed by bom bardin g a solid ( tar ge t ) with hi gh- energy e le c tr on s to create inner - shell electron vac ancie s . T w o types of x rays are thus generated : ( a ) a c on tin uo us spectrum , and ( b ) a characteristic line spectrum . T he con ti n u ous spectrum arises from t he deceleration of t he incoming hi gh- speed e le ct ro n s by t h e tar get . Li n e s pectra occur when an electron from a high-energy orbital fi lls a low - ener gy v ac anc y in th e inner electron orbitals . The line spectrum is characteristic of the emitting m at eri al an d use used for x -ray spectroscopy , w hich will be dis c us se d in detail later . The p eaks of t h e line spectra are desi gnated by the orbital ( shell ) into w hich the electrons fall ( K , L , M , N ) and a Greek letter indicatin g the ori ginal orbital ( a , S , etc ) . Due to th e characteristic well - defined ener gy , the x - ray lines ar e us e d as a quasimonochromatic x - ray source in di ffraction
equipment .
X -ray D i ffrac t io n
Diffraction occurs when a w ave interfere s with an array of sc atterin g cen ters , causin g the ou tc o m in g wave s to reinforce e ach other ( constructive interference ) or to be out of phase and cancel e ac h other ( de structive inter ferenc e ) ( see Fi g . 1a ) X rays are scattered by the electrons of the irradiate d m atter . Elastically sc a tt ere d x rays have t he same freq uency as the inco min g x ray s ( coherent scat t eri n g ) ; t he opposite is true for in elastic scatte ri n g ( incoherent scatterin g ) . Elast i c ally scattered x rays , which are imp or t an t in di ffr action can be depicted as reflected by the scat t e ri n g ato m . Fi g B ragg used the reflection an alogy to explain x -ray di ffr action u r e 1b s ho w s a mon oc hro m atic x - ray beam of w avelength A directed on t wo succes sive planes of a crystal . T he reflected beam ( i . e . , scattered ) •
,
.
1 59
C ataly tic S urfaces
SOURCE
(a)
SOURCE
CONSTRUCT I V E
DES T R U C T I V E
(b) FIGURE 1
Schematic representation o f ( a) constructive (in p hase ) and de structive ( out of phase ) inter ference w hen a w ave motion i s scattered by slits , and ( b ) B ragg' s reflection analogy for x - ray diffraction showin g con structive interference .
and associated wavelen gths are also shown .
I f the reflected r ays are to be in p hase , the path di fference of the two successive incomin g rays , CB + BD = 2d si n a , m ust be equal to an inte gral n u m be r of w avele n gt h s , that is , 2d sin 9
=
N >.
The equation above , known as B ra gg ' s law ,
( 8) relates the d s p acin g o f the
crystal wit h t he an gle of inci dence an d wavelen gt h o f the i nco min g x rays . N is known as t he order of the reflection . E quation ( 8 ) indicate s t hat measurin g the intensity of the di ffracted beam as a function of the incident an gle gives a diffraction p attern which is characteristic of the crystallo grap hic str ucture of the irradiated s ample . S uc h measurement s , w hic h con stitute o n e of t h e m any m e t hods of X R D analysis , are most easily at tained with an x - ray diffractometer .
1 60
Delgass and Wo lf
An x - ray diffractometer con sists of a cir c u l ar table with a stationary x - ray source an d a movin g detector locate d in t he circ umference of t he t a b le with t he sample set at the center . T he moving detector , u s ually a p roportion al counter , records t he inten sity of t he reflected beam as a
function of the reflected an gle 2 9 . A strai ghtforwar d app lication of XRD is t he u se of the X RD p attern s to identify t he various phases existin g in a m aterial . Diffraction patterns of powders are compi l ed in t he Pow der Diffraction Fi le Searc h Manual ( B erry , 1 97 5 ) .
C rys tallite Size D e terminatio n fro m XRD Line B roadening Diffraction line s should , in p rinciple , be very narro w ; however , when the
size of a polyc ry st alline material falls below 1000 A broadeni n g of t h e diffraction lines is observed . I n strumental limitations and lattice strain c an also c ause line broadenin g . Line broadenin g due t o p artic le si ze only arise s bec ause of incomplete de str uctive interfe rence . When the incident be am is slightly o ff the B ragg an gle , the reflected ray which would nullify it ori ginates from N p lane s in side the c rystal . I f the crystal is sm aller than about 1 0 0 0 A most crystalli t e
,
,
planes do not have their destr uctive counterparts N p lanes aw ay , so that finite diffracted lines will be ob served at t he off- B r a g g an gle se t tin g ,
Various measures o f the peak broadenin g are causin g line bro adenin g . u sed , the most common bein g t he width at half peak i n t e n sit y 8 , and t he ,
I d 9 /I w here I is the ob serve d inten sity 13 . fe 1 max of the diffrac ted bearJ . T he relation between line b ro a d eni n g and crystallite size for a stre ss free materi al , known as t he Scherrer form u l a , is
inte grate d breadth ,
=
K A.
8
hkl
cos
.,.
92
e
( 9)
where K is a const an t that depends on the de finition of 8 used and the crystal geometry . T he con stan t K v arie s from 0 . 98 to 1 . 3 9 , b ut because of e xp erimental uncertaintie s , the con stant i s o ften set equal to 1 . 13 is de fined as t he inte gr al breadth ,
W hen
si ze .
In the ab sence o f lattic e stre s s , b r o a denin g also occurs due to experi mental limit ation s such as nonpar allel and nonmonochro m atic irradi ation .
Other experimental factors also contrib ute to b roadenin g , even if the
s p eci men is of "in finite si ze" (i . e . , l a rg er than 1 0 0 0 A ) . C onseq uently , for t he correct e s ti m at io n of cryst allite si ze u sin g Scherrer , s form ula , the experimental contri bution to line broadenin g must be separ ated to obtain the pure diffr action broadenin g . Alth o ugh various procedures exist , two are o utlined here :
(a) a simp lified p rocedure , an d ( b ) an accurate
Fourier tran s form met hod .
Simplified Procedure :
In this c ase t he sample is mixed wit h a standard
that has p ar ticle si zes greater t han 1000
A and p roduces a di ffraction line
near the line o f the sample un der analy si s .
Altern atively , di ffraction p at
tern s o f t he samp le and o f another sample o f the same m ateri al containin g cryst allites lar ger than 1 0 0 0
A c an
be u se d .
In either c ase , if B is t he
width of t he samp le containi n g sm all crystallites an d b the width of the
1 61
Cataly tic Surfaces stan dard s amp le wit h lar ge cryst allites ( > 1000
f3
in g of the pure diffraction p a ttern
A),
the breadt h or bro ade n
i s e s tim ated by
( 10) or
S
=
B - b
( 1 1)
T he first correction i s d eri ved by ass umin g that t he shape of t h e peak is gau ssian , whereas t he secon d one corre spon d s to a C auchy p rofile .
B would be either
13t
o r Bi .
H ere
O ver t he years , the gau ssi an approximation
has been wi dely use d , altho u gh some evidence indicates that the p ure dif fraction p rofile is better ap pro xi m ated by the C auchy fo rm when a di stribu tion of p article si ze exist s ( K lug an d Alexan der , correction c an be used .
1 97 4 ) .
H owever , either
I t must be emp hasi zed that because the shape of
the pro file s m u st be assumed , the simplified procedures gi ve speedily re sults of good relative accurac y b ut have restricted ab solute signi fic ance . Fourier T ran s form Method :
T he t heory of thi s method is beyond the
scope o f this outline an d the reader is referred to the speciali zed literature ( K l u g and Alexander , 1 97 4 ; C o he n and S c h w ar t z , 1 97 7 ) . T he experimen tal determinations are s i milar to those outlined above , but t he p ure di ffrac tion p ro file is c alculate d from t he Fourier synthesis of the sample p rofile ,
h( £) ,
si ze ,
an d a re ference profile obtained wi t h a reference sample of in finite g(
£) ,
whe re
£
i s t he an gular chan ge throu gh the p e ak .
T he Fourier tran s form coefficients o f the true diffraction profile s are calculated from the real and imagin ary Fourier serie s c alculate d from t he inten sitie s of sample an d re fe rence profile s meas ured at various an gular in tervals o f the broadened peaks .
Computation procedures are gi ven
Gane son et al .
( 1 97 8) and Sashital et al . ( 1 9 7 7 ) . T he crys t allite size c an be c alculated u si n g t he breadth of the recon structed p rofile an d S c herrer' s form ula or directly u sin g t he Fourier coefficie n t s . W arren and A verbac h ( 1 950) relate d t he crystallit e si ze to t he fi r s t derivative of the Fourier coefficie n t s wit h respect to the Fourier harmonic . Furthermore , a plot o f the second derivative o f the Fourier coefficie nt is p roportional to t he dis tribution of c ry s t allite si ze s . Fi gure 2 sh ow s re sults of Sas hit al et al . ( 1 97 7 ) o b t ained u sin g the Fourier tran s form metho d , includin g strain and imperfection correction s .
T able
2
s u m m arizes t he average crystallite si ze
for the various c rystal planes alon g wit h dispersion result s obtained from hydro gen chemisorption technique s .
I t c an be seen t hat t he Fourier
tran s form c alculation s agree w ell wit h the chemi sorption res ult s .
T ra n sm i s si o n E l e ct ro n M i c rosco p y
T h e most com mon use o f an elect ron micro scopy ( EM ) in cat aly si s is to meas ure t he dist rib ution of cryst allite si zes as well as the m o rp holo gy of supported or un supported active m at e ri als . tron microscop e s , w hich combine
EM
Howeve r , new an alytical elec
w it h spectroscopic analysis , also re
veal in form ation about the chemical nat ure of the c rystallit e s .
To under
s t and t he pos sibilitie s of EM beyond t h at of takin g pic t ure s , it is
nec e s s ary to have a basic understandin g of E M .
A detailed description o f
t h e s ubj ect i s presented in various textbooks ( H eidenreich ,
1 97 2 ) and review article s ( B eer , 1 9 80 ;
S chmid t
et al . ,
1 96 4 ; H awkes ,
198 2 ) .
1 62
De lgass and
�
3
' ' ' 'I '
�
( u,,
'
: I
Q 2
I
!' I
..
N c
�
' ' '
11,
' '
.., .... 4.c I N ..,
0
Wolf
0
10
.d
tJ
)>
b,
//
20
'!�..
"·
30
b- • • ..,
40
50
(a) 8
2
-0 ..
;,
c
4. "'
'0
0
b
"' " '0 ....
0
tr
rl
'
10
r
sJ sJ
P, : · I o ' ' ' ''
I
20 L =
(b)
7
( L ) I OO
�·
b. .
.
'q
30
no� , X
N
5
"'
4
c
"1.
6
0 .. .., ....
( l l�,
.:f .., 2
3
· q_
'Q.
0
'•
40
50
� I
N
,o
A ...._� ..._ __JL._____L__
0
10
20
0
30
40
L = no 3 , A
(c)
F I G U R E 2 Particle si ze distribution for a catalyst containin g 40% Pt on Si0 2 prepared by impre gnation of chloroplatinic acid : ( a ) < 1 1 1> direction ; ( b ) < 1 00> direction ; ( c ) < 3 1 1> direction .
An electron microscope differs from an optic al microscope in t hat the electron beam rep laces t he li ght be am an d electrom agnetic lenses replace T he use of an electron beam imp rove s t he resolution the op tical len se s . of the microscope , which is p roportional to the wavele n gt h >.. of the inci dent beam , in accord with ( Heidenreich , 1 964 ) ( 12)
w here x is the resolution ( i . e . , the m1mmum si ze t hat can be distin guished in t he microscope ) and c8 is the sp heric al aberration of the incident beam . R elation ( 1 2) illu strate s the advantage of usin g an electron beam ( A 0 . 5 to 10 A ) in ste ad of li ght ( A = 40 0 0 to 7000 A ) . =
1 63
C ataly tic S urfaces TABLE 2
C a talyst
P ar ticle Si z e s
a
7 . 1- SiO - PtC l - 8 2 ( 1)
< L > hkl
a n d D i s p ers io n
< L > 1 00 (A)
Dx
Dh
10 12
7. 1
21. 5
( %)
13 1
82
111
75
( 1)
49 48
43 43
45
47
25
( 2)
( 1)
43 40
39 39
42 39
31 39
27 29
22
20
44
( 2)
2 1 . 5- S i0 2 - I onX - L 2 7 - Sio 2 - IonX - s
( 2) 40- Si0 2 - Pt C l - S
( 1)
30
26
( 2)
25
23
108
45
81
47
( %)
7. 1
25
27 . 3
39. 8
47
a
7 . 1 , 2 1 . 5 , 2 7 , an d 40 indicate percent of P t loadin g o n Si0 2 . PtCl and IonX in dicate preparation vi a c hloroplatinic acid im p re gn ation and ion e x c han ge , S and L refer to Si0 2 w i t h sm al l ( 1 2 0 t o 1 40 ) and lar ge ( 7 0 to 8 0 ) mesh si ze . D x i s t h e di sp er s ion obtained from x - r ay line broadenin g , and Dh is t he value obtained from hyd ro ge n c hem i sor p tio n .
Two fundament al models of operation are u se d in EM : transmi ssion electron microscopy ( TEM ) and sc ann i n g electron microscopy ( S EM ) . A t hird , hybri d mode , scannin g transmission ele c t ron m i c ro s co p y ( S TEM ) , is also used . Fi gure 3 d e p ic t s the schem atic s of the various modes of oper a t ion t o ge t h e r with the fundament al component s involved in each case . In t r an sm i s sio n electron microscopy ( TEM ) the elec t ro n beam , ge n era te d by a heated filament ( electron gun ) , p as s e s t hrough two electrodes and a con denser len s . P arallel r ay s thus cre ate d impin ge on t he spe c i m e n , where they are scattere d as a re sult o f t he spatially v ari ab le refractive inde x . Scattered ray s from t he same point in t he s p ecim en are b ro u gh t to t he T he overall effect same p oi n t in t he i m a ge formed by t he o b j ec ti ve lens . is eq uivalent to transmisssion o f primary electrons t h ro u gh t he sample . Electrons are also diffracted ( as in the case of x ray s ) , thus producing rays which are s li ghtly off the angle of r ay s formed by t r an s mi t t ed elec trons . Thus , by selectin g the p r op e r ap e rt ur e , an i m a ge of the trans mitted ( b ri gh t field ) or di ffr act e d electrons ( d ark field ) is obtained after the electrons pas s through intermediate a n d p roj e ct o r lenses and i mpin ge on a fluorescent screen or a p ho to graphi c plate . A typi cal con ventional electron m icr o s cop e operates at 1 0 0 kV , a vacuum of to 1 0- 6 torr , and is c ap ab le of 4 . 5-A point - t o - p oin t resolution and man gific a tion of 3 0 0 , 00 0 . D edicated hi gh - re soluti on electron microscopes are c ap able of re soluti on of 2 . 5 A with magnifications of 800 , 0 00 to 1 , 0 0 0 , 000 . An im p ort an t aspect of TEM is that a three - dimensional s ample yi elds a two- dimensional TEM i m a ge w hi ch m i gh t be difficult to interpret . Further more , the contrast between the s u pp o rt an d t he active component mi gh t
1o- 5
164
De l gass and Wo lf E l ECT RON S O U II C E
�
�"""""'
S P E CIMEN
� AMPLIFIER
OPTICAL
- - - - - )>
M I C II O S C O PE
A P E II T U II E
� I:8J
SCANNING COILS
fV'V"" I N T E II II E D I ATE LENS �
�
SPE C I M E N
�
� FINAL LENS
---- �
�::;TOll
E N E R G Y DI SPEIISI VE SPECTROMETER
(0 )
TEM
( b)
SEM
(c) EPMA
Schematic s of ( a) transmis sion electron microscope ( TEM ) : ( b ) sc annin g electron micro scope ( S EM ) , an d ( c ) electron probe microanaly zer ( EPMA ) .
F I GURE 3
not be sufficient to distin guish the m etal crystallite s . In this case , dark field images are useful since electron s diffract p referenti ally from the cryst allit e s , thus producin g b ri ghter spots from them com p ared to the weak diffraction from the nearly amorp hous s upport .
S up ported Catalysts Fi gure 4 shows bright- and dark - field i m a ge s of a supported 5% Pt / Si0 2 c at alyst . T he s ample is prep ared by mortar grindin g the catalyst particles ( 2 0 0 m e s h ) to ultrafine pow der , w hich is then dispersed in a liquid ( w ater , alcohol ) . A droplet of the solution is then deposited on an EM grid . The liquid evaporates , leavi n g on the grid microscopic p articles which are thin enough for electron tran s mission . T he micro grap hs shown in Fi g . 4a were obtained wit h a 100 JEOL S TEM operating at a m a gni fication of 4 0 0 , 00 0 . I n this c ase , the contrast i s relatively low because o f the presence of the support . Counti n g and si zin g the crystallites in both t he bri ght - an d dark - field im a ge s is o ften diffic ult , and m an y micro grap h s are requi red to obt ain a s t ati sti cally meanin gful si ze distribution . A m ajor limit ation here can be the p re sence of s m all crystallites that cannot be seen under the Furthermore , in some cases the diffraction p attern of con ditions used . the metal c annot be distin guishe d from that of the s up port ( S chmidt et al . ,
1 65
Cataly tic S urfaces
(a) 25 0 A : FIGURE 4 TEM photograp hs , all at magnification 400 , 00 0 ; 1 e m ( a) 5% Pt /Si0 2 ; ( b ) model sample , Pt sputtered on Si02 film ; and ( c ) model sample , Pt impregnated on Si0 2 film with chloroplatinic aci d . All catalysts were reduced in H 2 at 300° C . =
1 66
Delgass and Wolf
(b) FIGURE 4
( Continued)
Catalytic Surfaces
1 67
(c) FIGURE 4
( Continued)
1 982) . It is always recommended that one check the electron diffraction pattern to determine the relative role of the support and active metal in dark- field studies . C har gin g of the sample by the electron beam is a com Chargin g mon problem with catalysts in which the support is an insulator . can be reduced by coatin g the sample with a carbon or gold film . Higher metal loadings ( 5 to 1 0%) and higher beam voltages result in better con Sam trasts ; however , sample damage can be caused by the electron beam . ple preparation procedures are given by B aker et al . ( 1 979) Glass! et al . ( 1 98 0 ) report artifacts introduced due to sample preparation . The particle size distribution is obtained by the method described by Underwood ( 1 97 0 ) o r b y means of a n automatic sizing and counting device . Model Samples Schmidt and coworkers ( Schmidt et al . , 1 982) , among others , have dis cussed the advantages of usin g model samples in which the catalyst is
1 68
De lgass and Wo lf
d ep osi t e d on a planar nonporous substrate of a morp ho u s silic a . A thin flake of sili ca is p rep ar ed by vacuum de po sition of silica fo llo we d by vac u um deposition of th e met als ( e . g . , Ni , Rh ) onto the sub str ate . This t e c hni q u e avoi d s p ar t icle over l ap an d p ro duce s hi gh co ntr a s t between the met al p ar tic le s and the s u pp o rt Figure 4b show s TEM r e s u l t s obt ained by s p ut terin g Pt onto a model silicon grid and Fi g . 4c sho w s similar re s ult s obtained by i m p r e gn ati n g t he gri d with H 2P t C L s an d then re d u cin g it in H2 at 4 0 0° C . T he di s adv an ta ge of t he technique is that the to t al active area is lo w for re action studie s . F u r t her m or e the morp holo gy of t he p ar ticle s de p o si t e d on the model , nonporous support , m ay be di ffe rent t han w hen t he metal i s de po site d on p o rou s si lic a . A l t h o u gh TEM has been used extensively to stu dy cataly sts , in conven t io n al op e rati on it views s am pl e s in a vacuum environment , pre or p os t re action . A te c h niq ue that allow s o n e to use TEM in a reactive environment is controlled at mosp he re transmission electron microscopy ( C ATEM ) , De t ails of the CATEM te ch nique have been do c um e n ted in a review arti cle by R . T , K . B ake r ( 1 97 9) , w h o has intro d uced t he te c h niq ue to c atalytic re s ear c h ers In CAT EM , t h e sp eci me n is m ounted in a gas re action cell w hich is fi xe d to the translation al m an ip ula tor of the sample . With this arran gement it is p o ssi b le to operate at gas pr e s s u re s of t he order of s e ve ral torr within t he cell and at t e m p er at ur e s up to 1 300° C . T hi s con trol of sample environment re qui r es some sacri fice of resolution . C h an ge s occ u rrin g durin g reac ti on are recorde d b y a TV video sy st em B aker ' s e xce l len t paper ( 1 97 9) i ll u s t r at e s the use of CATEM i n s t u dyi n g gas - soli d systems w here the reaction prod uce s c arbo n de p o s its o r d uri n g gasification of t he deposit s , which induces motion of small metal p arti c les and c han ges in t he structure of the spe ci me n . E xt r aor di nary movies have been pro duced s how in g the growth of carbon filame nts as well as the mo ve ments of met al p ar tic le s d urin g c arb on gas ifi c atio n S uch i n fo r m ation has reve aled the com p le xit y of t he dy n am ic s of gas - solid i n t e r ac tio ns ·.
,
.
.
.
,
COM PO S I T I O N A N D S T R U C T U R E O F S U R FA C E S A N D S O LI DS
Since the 1 960s , several techniques h av e been de ve l ope d to c har ac t eri ze A gr e at deal o f effort has b e en spent on the measurement of surfaces . basic prop e r tie s o f model s u r face s ( such as single crystals ) to establish t he theory an d va li di t y of each surface an al y si s met hod . T h es e st udies have le d , in some cases , to int e rp r e t ation of c at al yti c act i vi ty in t er m s of fun dament al p roper t i e s and h ave demonst r ate d the potential of these tech T he di s c u ssi on t hat follow s is niques in s t u die s of t e c h n ol o gi c al interest . p r es en te d in t he context of t he app lic ation of t he se techniq ues in the i nt e r p r e t atio n of s ur face p rocesses relate d to catalysis . T heoretical a s pe c t s are treated only on the m o st b asic term s needed for i n t e r p re tat i on of dat a . T he reader i s re fer r e d t o t he s p e ciali z e d literature for fu rt he r de t ail conc e r nin g theory . an d for det ailed discussion of e q ui p me n t and opera t io n Most of the te c hni q u e s no w com m e r ci ally available ar e b ased on the e x cit ation o f t he sample by a n in co mi n g beam o f p article s or electromagnetic energy . Followi n g such e xcit atio n are d e - excitation p rocesses , w hic h re sult .
in t he emission of e le c t ro ns or p hotons of di s c ret e energy w hich can t hen be me asu red by a suit able ene r gy analy zer . F i gu re 5 s ho w s the s c he mat ic s
1 69
Ca taly t ic Surfaces Visible or U V Source
e�. fiV//?'flZ' ( �- -'fff - - - -------- FL
oy:
/'@///21/7/7; ( b • •
X- r a y Sou rce
(a)
)
L3
• •
----+--- L 2 l1
�
------
� .. --__ -;) • K
EXC I T E D
/
P h o t o E l e c t r on
IONIZED
Photon De- e •c i t a t ion
e-
X- r o y P ho t o elect ron ( X PS)
ATOM
� E l ec t ro n De - e x c i t a t i o n
K 011 R a d 1 o t i o n
'ZVTfl/ < a l
V/J2'//L C a l
K L 2 ,L3 Tr a n s 1 t 1 on
V22?JZ ( by-
/ZZZ?Z ( b )
/.
•
• • • •
•
•
II
L3
--<--�- L 2
/
I I f
----+�--- K
K L L Auger E m 1 s S 1 0 n
K oc 1 X - r a y F i o u r e s c e n r: e
(b)
F I G U RE 5 Sche m atics o f ( a ) p hoton excitation and de - excitation by p hoto electron emis sio n , and ( b ) d e - e x ci t at io n p roces ses l e a di n g to x - ray and A u ge r elect ron e mission
of t he two excitation and d e - excit ation p rocesses most com monly u s e d for
surface analysi s , toge t he r w i t h the name and ac rony m o f t he an al ysi s tech
nique
( XP S -AES ) .
W hile several o f the tec hni q u e s discussed in t his sec tion
yield similar information , t he t yp e of instrumentation and data interp ret a tion v arie s .
For
t he s ake of
clarity , each technique is di sc usse d separately ;
how ever , t here are m any a sp ec t s t hat are co m m on or complementary amon g
them .
( U HV ) .
One such common fac t o r
is t he need
for
usin g ultrahi gh v ac u u m
At 1 0- 6 t orr , about 1 o 1 5 molecule s of gas strike 1 c m 2 of the sur sec o n d . At 1o- 9 t or r , monolayer coverage occurs in about 15 min
fac e per if every. molecule that hit s sticks ; t h u s a clean s urface can be att ained only at '!J H V con ditions . T he first t e c h ni qu e described below is X P S ( also known as E S C A ,
electron s p e ct rosc o p y for c he m ic al analysis ) w hic h cor res p on d s to t he
Delgass and Wo lf
1 70
case w here the excitation source is x-ray photons , and the result is elec tron emission . X - ra y Photoe l ec t ron S pec t ro scop y
X -ray photoelectron spectroscopy ( X PS ) w as developed in the early 1 96 0s by Kai Siegbahn ( Siegbahn et al . , 1 9 6 7 ) to characterize solids , liquids , and gases . B ecause the technique probes the top atomic layers of a solid , it has become one of the m ain tools for surface analysis . Several books , monographs ( Sie gbahn et al . , 1 9 6 7 ; C arlson , 1 9 67 ; Ertl and Kuppers , 1 97 9 ; B ri ggs , 1 97 7 ; Delgass e t al . , 1 97 9) , and review articles ( Palmber g , 1 97 5 ; Hercule s and C arver , 1 97 4 ; Delgass et al . , 1 97 0 ; Hercules and Hercules , 1 97 4) have been p ub lished describin g theory and applications , with a few article s focusin g on applied hetero geneous catalysts ( Hercules and C arver , 1 97 4 ; Del gass et al . , 1 97 0 ; B hasin , 1 98 1 ) .
Pho toelect ro n Emissio n ( PE ) or Pho toionizatio n While the detaile d p hysics of electron emission b y p hotoexcitation , is quite comple x , the basic p rinciple is straightforw ard and has been k nown for decades . Emission of electrons occurs w hen matter is exposed to or ex cited by electromagnetic radiation or photons . If t he photons have energies in the ultraviolet ran ge , they cause ejection of the les s ti ghtly bound valence electrons that emerge at discrete energies [ ultraviolet photoelectron spectroscopy ( U PS ) ] . I f t he incomin g photons are x rays of . suitable en er gies ( 1 to 3 keV or soft x rays ) , ejection of inner or core electrons occ urs . If the ener gy of t he incoming photons is known , analysis of the energy of t he emitted electron s permit s calculation of core electron binding energies . Since the electron bindin g energies are quanti zed , the emitted electron s have discrete energies t hat reflect the electronic st ructure of the parent atom and thus identify the irradiated element .
Energy Relationships T he basic energy relationship occurring durin g photoionization w as first proposed by Einstein . It states t hat the ener gy of the incomin g photons , hv , equals the kinetic energy of the ejected electron ( E k ) plus it s bindin g energy , E b ( or photoioni zation ener gy ) :
( 1 3) In practice , relation ( 1 3) must be modified to account for additional potenti als existin g in t he spectrometer . Fi gure 6 show s the ener gies of electron bindi n g meas urements in solids . Once the electron is free from its core level ( E b ) and reaches the Fermi level , it must spend some energy to free itself from t he solid m atrix ( work function cl> s > . T he rem ainder energy appears as a kinetic ener gy E k w hich is sli ghtly chan ged to E by t he cont act potenti al between t he sam ple and the spectromete r . Since the Fermi level of t he spectrometer ( cl> sp > is equal to that of t he sample be cause t hey are in electrical contact , one can w rite
k
hv
=
E
b
+ cj>
sp
+
E'
k
( 1 4)
1 71
C ataly tic S urfaces E hv E XC I T I N G
_____..__ r
PHOTONS
(a)
CORE E L E C T RON LE VEL
Al z p ( a 3 , 4 l
�
0 �
en '
Cl:l.
So zp(a 1,2 l -
..
a:
(Mg Ka)
� 7'
l
e- "
R E TA R D I N G LENS
( b)
So 2 p ( a 3•4 l
� 6
X- RAY
u
SAMPLE
A\ z p ! a 1 2 ' > -No
G-:-: YR-:C-::: E-:: ::-1-::: N-:: E-:-: ""'T -E L--K-IN
zs
(a1 , 2 )
Al z p ( a 3 4 > •
J
.
of electron bindi n g in so li d s h \i , excitation energy ; cJ>s , s ample work function ; Ek , ki n etic energy as ejected from the sam p le ; Ek , kinetic ener gy as meas ured by the s p ect rome te r ; FL , Fermi level ; CB and VB , co n d uc tion and valence band s . ( b ) S c hem atic of basi c components of an x - ray p hoto electron spectrometer wit h d at a for a Y zeo li t e . ( From D e l gas s et al . , 1 97 9 . ) FI G U R E 6
( a)
p hot on en er gy
;
E ne rge ti c s Eb
,
In most app li c ations ,
electron bindin g
rel ative
are the relevant i n formatio n .
.
bindin g en e r gi e s or chan ges in t hei r values For each me asurement , it is only necessary
t hat cJ> sp re m ai n constant ; co n v ers ely , t he i n s tr um e nt c an be calibrated u si n g as a re fe rence an elem ent of known bin din g ener gy ( A u or C , for example ) .
i rr adi ate d electrons emit ted fro m various levels , v ario u s kinetic ener gies , are emi tt e d . A photo e le ctro n spectrum consists of the number of p hotoelectrons e mi t te d at a given ener gy level , N ( E ) , ver sus t heir kinetic or bind i n g ene r gy ( see Fi g . 6b ) . F r o m t h e spectral lines or peaks , w hi c h are desi gnated in accord W hen t he sample is
,
and t h ere fo r e possessin g
wit h t he ener gy level from w hi c h ele ctrons are ejecte d , E b is c alc u lat ed
( 1 4) and the values co mpared w i t h t heoreti c al or ex pe ri men t al 1 97 8 ; Padley et al . , 1 968) . Thus X P S can be used for c h e mi c al identi fi c ation o f elements , b ut more si gni fi c antly it can be u se d for chemical analysis of surfaces because of its surface sensitivity . Although , in principle , a sin gle s pectral line would be observed w hen a usi n g E q .
values ( W a gner et al . ,
De lgass and Wo lf
1 72
core e le c t ro n i s e j e c t e d , o ft e n one fi nd s additional li nes o f lower i n t e n s i ty kno w n as
satellites .
1
O n e type of s at e lli t e is t he
" s h ak e - u p '
line .
T hi s
occurs w hen the energy o f the x - ray photon causes b o t h p hot oe mi s si on and exci t atio n of a vale n c e electron to a low - lyi n g bo u n d state .
A s e co n d com
mon s ate lli t e li n e i s a dis cret e ener gy los s line caused , for e x am p l e , w he n t he photoelectron excites a plasmon oscillation i n t he electrons i n t he s am p le
.
S urface Sensi tivity
A l t ho u g h x rays c an penet r ate solids far b e yo n d t h e s u r face , t he mean
free p at h of ele ct r o n s with energies b e t w ee n 10 and 1000 eV i s o f t he order of 1 0 to 30 A . C on s e q u e n t l y , only electrons o f ato m s locat e d near the surface obey t he energy conversion equation [ E q . ( 1 4) ] and are anal y zed by t he s pec t rom e t e r . T he relation betw een mean free p a t h ( or e s cap e dep t h ) of e le c t ro n s as a function of ene r gy is s ho w n in Fi g . 7 . It c an b e seen t hat t he m e an free p at h i s affected b y the sub s tr at e , and for
most elem ent s the escape dep t h o f p hotoelectrons co mes fro m t he top one
to t hree atomic layers n e ar t he s u r face .
T he ability to se le c t electrons
m akes XPS characteristic of X P S
ejected n e ar the s u r face is w h a t
A nother i mport ant
a surface - sensitive techniq ue . i s i t s abi li t y to p roduce an
adequate sp ectrum even t ho u gh the absolute amo unt of m ateri al p r ese n t i s o f t he order o f 1 0 - 6 t o 1 0 - 8 g .
Fractions o f monolayer a s s m all a s 0 .
01
c an b e detected in favorable cases .
C hemical Shift s W hile t he electron bin din g en er gi e s o f p ar t i c ul ar elements are s u ffi cie n t ly invariant to provide c he mical i d e n tific at io n of t he element , s m all but measur able shifts in bi ndi n g energies occur w hen t he X P S s p e c t r a of v ar io u s com pounds o f the s am e e le m e n t are meas u re d ( Siegbahn e t al . , 1 9 6 7 ) .
100
� ....... z � ll. loJ 0
loJ ll. <( u (f) loJ
�
Au
'
Au
\.
10
a: � u loJ ...J loJ
Fe I
I
10
Ag
Au
/
The observe<
�
--Mo
•....._ w
Ni
100 ELECT RON E NE R G Y ( eV )
1000
1 0, 000
FI G U R E 7 M e an free path or e s c ap e depth o f electron s as o b t ai n e d from A u ger an d p hotoelectron spectroscopy d at a . ( From D e l g as s et al . , 1 9 7 9 . )
1 73
Cataly tic S urfaces
shifts in E b , or c he m ic al s hi fts , can be explained i n term s of c h an ge s in e le c tro static forces whi c h p articip at e in electron binding ( Siegbahn et al . , 1 967 ; 1 96 8) . A core electron is subjected to at t r ac ti ve n uclear Fadley et al . forces w hich are p roportion al to t he atomic n umber and are also affec ted by the p re s en c e of the o uter v al e n ce electrons . I f an e le ct ron is re move d from t he outer shell , t he core electrons feel an increased attr active forc e from t he nucleus and therefore E b increases . C onve r sel y a gain of an outer e lect ron has the opposite effect . A lt ho u gh various theories have been proposed to o b t ai n a prio ri infor mation on chemical shifts , not all t he contributions to shift s on solid s am p le s can be evaluated accurately . However , sufficient data h a ve now been com pi le d to make use of chemical shi fts as an in dic at o r of t he oxidation st ate of many elements ( Wagner et al . , 1 97 8) . ,
,
I n s t rume n tation
The m ajo rit y of the x - ray p hotoelectron spectrometers consist of t he follow i n g five basic component s : ( a ) source , ( b ) sample mount , ( c ) ele ct ron ene r gy analy zer , ( d ) detec t or , and ( e ) data ac quisit io n system . The source i s an x - r ay tube of conventional de si gn c on si s tin g of a heate d cathode or electron gun from which electron s are acce le rate d to a cooled anode or t ar get . AI and M g with K a. lines located at 1 4 86 . 6 and 1 2 5 3 . 6 eV are the most comm on ly u sed anode s . T he sample re gion m ust be m aintained at pres sures lower t han 1 0 - 9 tor r to avoid contamination of t he sample . This restriction is more severe t han the 1 0 - 5 torr maximum pressure in t he spectrometer required to minimi ze electron energy loss on colli sion wit h gaseous molecules . I mprovements in electron ener gy an aly ze r s led to the development of X P S . A l t ho u gh various types of analy zers have been u sed , the most com mon types in use today are electrostatic an aly zers ( also known as a - spec trometers or monochromators) . T hey co n sis t of two concentric bodies ( sp heres , c ylinde r s ) onto which a voltage is superimposed . Electrons en te rin g the sp ace between t he b o die s fo llo w t r aj ec tori es t hat depend on electron ener gy . T h us for a gi ve n voltage only t ho s e e l ectron s w hic h h ave a specific kinetic ener gy will re ac h the detector . Fi gure 8 shows the sc h e m atics of an X P S sp e c t rom e te r with a cy li ndri c al m ir ro r analy z er ( C MA ) . T he CMA consists of two concentric c y li n de r s wit h a slit p e rmi tt in g elec trons with ener gies proportion al to the ap p li ed vo ltage to reach t he d e t ector Ge n er al ly the analy zer is s e t a t 10 0 e V and t he p hotoelectrons are retarded by a ne gati v e potential ap p lied to a lens system in front of the ener gy analy zer . T he comp lete spectrum is pro duced by sweepin g the ret ar din g volta ge . A d at a acquisition system is used to collect electron counts at t he detector as a function of electron kinetic ener gy . Accumula tion of count s d urin g repetitive sc anni n g imp ro ve s the signal-to- noise ratio . T he spec t r al lines are often su p e ri m p o s e d on a raising b ack grou nd w hic h occurs d u e t o secondary an d ine la stic al ly scattered electrons . Pow dered samp les can be sp re ad uni form ly into a self- adhesive mat e rial which is attached to the mount or pressed into se l f s up port i n g w afers . C harging of t he s ample is not as severe as in the case of excitation by electron i m p act and is us ually compen sated by floo di n g t he sample r e gi on wit h low - ener gy electrons . For t hi s reason , X P S is the preferred surface analysis tech ni q ue for supported catalysts . .
,
-
1 74
D e l gass and Wolf COM P U TE R SYS TE M
ANALYZER CON TROL
X - RAY SOURCE
\ F I RST APER T U R E
\' S E C O N D
APE R T U R E
F I G U R E 8 Schematic representation of a com bined E S C A - A u ger - S AM sys tem usin g a double - p ass cylindrical mirror analy zer ( PHI model 5 5 0 ) . ( From W a gner et al . , 1 97 8 . )
A p p lica t io n s
Applications of X P S are numerous , and m any article s have been published in various areas of m aterials science , surface science , and cat alysis . The X P S chap ter by Delgass et al ( 1 97 9) pre sents and discusses several exam
ple s of XPS applications . Rather t han enumeratin g the many examples p ublished in the literature , the m ajor areas of application related to cataly tic activity are discussed below in terms of qualitative and q uantit ative analysis Q ua l i t ative A nalysis A strai ghtforw ard application is a s urve y of elements existin g on a catalys t
This is particularly i mportant in connection with catalyst poisoning by imp urities contained in the feed , for example , sulfur compounds or met als ( B hasin , 1 98 1 ) . C hemical shifts p rovide an important clue about the oxidation state of the active component o f a cat alyst and its modific ation by pretreatment , reaction environment , an d support . S ever al st udies are reported by Del gass et al . ( 1 97 9) in w hic h chemical s hi ft s have been used to determine the oxidation state of W supported on Si0 2 and Al 2 0 3 , Cr supported on Si0 2 , and Co M o / A l 20 3 hydrodesulfurization c at alyst s . Other st udies on supported metals include Pt /Al20 3 , Pt Au / C , NiO , FeV 20 3 , C uO /Al 2 0 3 , and R h / C , among many other examp le s . T he spectra o f a 2 % P d s up ported on a Z SM - 5 zeolite illustratin g the use o f chemical shift s arisin g from different oxidation states of Pd are shown in Fi g . 9 . -
Q ua n t itative A n alysis
The areas under the XPS peaks carry quantitative information ; however , accurate interpretation requires a str uctural model of t he surface layer .
1 75
C a taly tic S urfaces
337 2 "
335.4
II I I I I I
I I I I I I I I I I I I I
356
350
344 338 B. E . ( e V )
332
326
FI G U R E 9 XPS spe ctra of Pd 3d transition of a 2% Pd /Na- ZSM- 5 c at al y s t . D ashe d line : oxidi ze d ; solid line : fre shly reduced . ( From S ah a and Wolf , 1 98 4 . )
For a homogeneous solid wit h a fl at surface , the followin g relation has been derived ( C arlson , 1 97 5 ) : I
=
( 1 5)
K on A F
· where K is a s p e c t rom e t er con stant , o t he p hotoelectron cro ss section for a given s hell n the c oncen t r ation of atoms under goin g photoel ectro n ej ec tion , F t he x - ray flux , and A the electron mean free path for inelastic col lision . Other models con siderin g an gular ener gy depen d e nc e have also been propo se d ( Fraser et al . 1 97 3 ) . In the case of porous m aterials such as s up po rte d metal catalysts , the model leadin g t o t he in t e gr at e d in ten sity s h o uld con sider met al c ry s talli te si ze distribution , distribution o f the metal within t he pore structure , a n d t h e effect of surface roughnes s o n an an gle dependence . S o m e sirn p li :li e d mo de l s for q uan tit ati ve an alysis of s up ported cat alyst have been proposed ( A n ge vin e et al . , 1 97 7 ) . M any of t he various factors involved in t he expression of t he i nt e gr at e d inte n si t y cancel out w hen t he ratios of intensitie s of two key components are co m p are d For e xam p le for a c omp le t e l y uniform ly and mo n at om i c dis derived from the i n te gr a te d persed m aterial , the ratio I ,
,
.
,
me t al /1 suppor t areas corrected for cross section should be p roportion al to t he ratio of the number of metal atom s per unit area of support ( Fraser et al . , 1 97 3 ) . I f the in t en s i t y ratio is higher t han expected from t he met al surfac e density , it indicat e s m i gr at ion of the m e t al to the exte rnal s u r fac e , and i f it is lower , it s uggests partic le growth or preferential deposi tion in small pore s not easily seen from t he exterior of the particle . Anot he r case w herein relative comp arison can give quan ti t at i ve information is i n analysis of metal lic or polymetallic c atalyst s w here the int e grated inte n si t y r atios , cor re ct e d for phot oe lectron c ross section , can be c o m p are d wi t h th e atom ic ,
1 76
D elgass and Wolf
ratio of t he met als as l o a d e d into the catalyst s . D e viation from a 1 : 1 cor res po n den ce would s u ggest surface enrichment of one metal in preference to t he other ( Liao and Wolf , 1 98 2 ) . In a few c ases it has been possible to c o r re l at e the c at alytic act i vity with the intensity o f a p articular XPS si gn al . B rinen and Melera ( 1 97 2 ) correlated t he acti vi t y of a R h /C c at alyst wit h t he ratio o f R h i n oxi d i ze d to metal form . Se ve r al studies of a d so rb at e metal interaction in clean m e t al s ur face s have been published in c onn e ct ion with this application . In c lo sin g thi s brief introduction of th e u tili zation of X P S to s tud y c atalysis , the m ain applic ation s are summari ze d . X P S is a useful tool in qualitative anal y si s of su r faces to identify cont aminants and promoters , a nd to de t ermin e t h e o xi d at ion s t at e o f m etal s and e x am i ne m e tal - s upp or t interactions in su ppo r ted c at alysts . Accurate absolute q u antitative analysis requires a m o d e l of t he surface l ay e r , a di ffi c u l t task wit h s up p o r t e d cat alyst s . However , re l ati ve corrected inte grated i nt e n sit y ratios are a useful clue to under s t an din g s u rfa c e enrichment and s urface composition in p ol yme t allic c at aly s t s . Nearly all c at alyst applications require careful handlin g of t he sa m ple , in cl u di n g in si t u pretreatment in a p re p ar a tio n chamber attached to the spec t r om e t e r . A com p ar iso n of X PS with o t h e r s u r face an aly s i s t e c hn iq ue s ( Powell , 1 98 2 ) concludes t hat X P S an alysis c auses t he least d am age to the
sample durin g a n al ysi s and is t he tec hniq ue for which t he most e xt e n si ve
p roced u res for quali t at ive and quantitative analys i s have been reported .
A u ger E l ectron S p ect roscopy
Another too l now routinely used for surface an aly si s is A u ger e l e c t ro n s p ect ros c op y ( A E S ) . While the app li c atio n of X P S to c har a c t e ri ze cat alysts has now become well e s t a b li s hed , A E S has been a useful tool in ap pli c ations F o r t his re aso n , le s s i n vo lvi n g flat s u r fac e s such as sin gle c r yst als .
space will be d e voted to A E S than to X P S . H o w e ver , since most of t hese tec hni q ue s are in const ant d e ve lo p ment , the po tenti al ap p lic ations of AE S to c at aly sis are still q u i t e open . As in t he c a s e of X P S , t he principles o f A E S a re briefly described , b u t emphasis is gi ven to t he pos sible applica tions of A E S to hetero geneous ca t a ly si s . The reader is re fe rr e d to the various reView article s an d books ( P al m b er g , 1 97 5 ; H e rc u l e s , 1 97 8 ; Chan g , 1 97 4 ; D avis e t al . , 1 97 6 ; McGuire , 1 97 9 ; B hasin , 1 97 6 ) on the s ubj ec t for a more det ailed t heoretical di sc ussio n .
A uger Effect A u ge r electron spectroscopy ( AE S ) is n a me d after it s discovere r , Pierre A u ge r , w ho in 1 92 5 di sco ve r e d in a W i l son chamber that a two- electron emis si on occ urred w hen ar gon ato m s were excited by x r ay s . One of t he elec t rons was a p ho toele c tr on ej ected as described in Fi g . 5a in t he case of p hotoe lectron emis sion . T he secon d e lect r o n , k nown as t he A u ger electron , o ri gi n ate s from anot he r dee xcitation process , w hic h competes wit h photo
e lec t ron emission . W hen a primary exci t ation process , such as x ray or e le ctron bombardment , causes the c reation of a v acan c y in an inner e lec t ro n shell , for ex a m p le , t he K shell , deexcit ation of t he ioni zed atom can occur via t he dro ppi n g of an electron from an o ut er level ( i . e . , L 1 ) to the The e x ces s ener gy is equilibrated by ejec tio n o f t h e vacated inner leve l . s eco nd e le c t ro n fro m an upper en e r gy level ( i . e . , L 2 ) ; t hi s A u ge r
1 77
Cataly tic S urfaces
transition i s then known as KL 1 L 2 • T he kinetic ener gy o f t he ejec te d A u ger electron is ap p ro xi m at e ly equal to EK EL 1 E L and is character 2 istic of t he i r r adi at e d ele ment . Moreover , the ener gy of the A u ger transi tions is inde p en d e n t of t he energy of t he i n ci de n t beam . The number of A u ger transitions possessed by an atom i nc rease s wit h t he a t o mic nu m be r of the element . An atom generally possesses one or two stron g A u ger tran sition s . Whereas A u ger electrons c an be generate d by u sin g bot h x r ays and e le c t r on be ams , the l at t e r mode of op eration has become t he most commo n type used in A E S bec ause the cap ability of e le c t ro n be ams to be focused in a very sm all surface area ( 10 to 100 lJ m in di ame t e r ) provides better sp atial resolution t h a r. do x rays . X - r ay e xc ita t ion is only u s e d in con j1mction wi t h X P S analy sis . An Au ge r spectrum con sists of a plot o f the number of e le c t rons emitted , N ( E ) , as a fun c t io n of energy . Auger emis sion occurs ac c om p ani ed by secondary and sc attered prim ary e lec t r on s ; conseque ntly , A u ge r transition s are not very p romi n e n t in s uch N ( E ) ve r s u s E p lo t s T o e n h an c e t he A u ger s i gn al and remove the b ackground c au s e d by ot he r electrons , the derivative dN ( E ) /dE is u s ually plot ted as a function of E . Fi gu re 1 0 sho w s both i nt e gr al and differential spectr a for Ag. T he minim um o f t he differenti al si gn a l is usually used to characteri ze t he energy of t he A u ger t r an si t io n s T he A u ger spectra of all t he elements ( except H 2 and He ) have been obtained an d co mpi l e d (Davis et al . , 1 97 6 ) . -
-
.
.
S urface Sens itivity Similar ar g u m e n t s gi ven for X P S p hotoelectron escape depth are also vali d for AES . T he literat ure indicates t h at for A u ge r electrons with ener gies between 50 and 2 0 0 0 eV , t h e escape depth is of the or d e r of 4 to 2 0 A ( Palm be r g 1 97 5 ) . For most elements with A u ge r transitions below 1 0 0 0 eV the esc ape d e p t h is only a few atomic layer s . .T he greater surface sen si tivit y sometimes ci t e d for A E S ver s us X P S results because m an y ele m e n t s h a ve A u ge r lines at 1 0 0 eV kinetic ener gy , the minimum in Fi g 7. ,
.
N !E )
dN(El dE or
F I G U RE 10 s a mp le :
ver s u s E .
400 600 800 ELEC TRON ENERGY ( eV)
1000
Secondary electron e n e rgy distribution curves from a silver ( b ) 10 x N ( e ) ve r s us E : ( c ) dN ( E ) /dE ( From B hasin , 1 97 6 . )
( A ) N ( E ) ve r s u s E ;
,
1 78
Delgas s and Wo lf
C hemical S hifts A s in t he case o f X P S , chemical shi ft s have also been ob served in A E S . H o we ver , t heir interpret ation is not well understood at present . C hemical shi fts of many li gh t elements ( Z < 40) have been observed and have been relate d to oxidation state . I n general , A u ge r li n e s in vo lvi n g only core elec trons are e asiest to int e r p ret . I n so m e cases t he shape of the A u ger peaks is also affected by str uct ural factor s ; for example , carbide an d gr ap hi ti c carbon p e aks d ep osited on st ainles s s tee l are strildn gly different ( Siegbahn et al . , 1 967 ) . S uch lines are co uple d to the valence level and t h us carry de tailed bonding information . Complex Auger line shapes can also be used to fi n ge rpri n t the identity of adsorbed mole c ul e s ( Grant and Hooker , 1 976) .
Scanning A uger
The ability to focus an electron beam to a very fine di am e t e r ( ca . 5 )JID ) which can b e scanned across a surface a s i n scanning electron microscopy is the b asis for sc annin g A u ger electron spectroscopy . Most AES instru ments are equipped wit h lo w - res o l ution S EM ( 2 0 0 0 x magnification) cap abilitief to select an area for an aly s i s . Fur thermore , by locking the vo lt age of the energy analy zer ( w hich is of t he same type as in X P S ) , the detector only receives electrons comin g from t he selected e l e m e n t . Thi s cap ability per mits one to obtain a m ap of the e le m e n t as t h e electron beam scans a selected area on the sample un der analysis . T his cap ability is referred to as a s c an ni n g auger microprobe ( S AM ) and is parti cul ar ly useful in micro electronics .
Dep t h Profiling
S AM s y s te m s can be e q ui pped with an Ar ion gun that can be used to im pact ( s p u t te r ) the surface with high -ener gy ions that remove atoms from the top surface layer . I f t he ele c tr on beam is also di rect e d to the same area , it is pos s i b le to analy ze the composition of the freshly exposed s u b layer as a function of dep th ( depth profilin g) . T hi s c ap abilit y is important for detection of selective d eposi tion of imp urities , fo rm ation of sur face layers , and , in general , to detect compositional c han ges with depth i n t o
the m aterial ( sub surface an alysis ) .
Ins trumen tatio n
T h e b asic components o f an A u ge r electron spectrometer ar e those d e s c r ib e d for X P S except t hat an electron gun replaces the x - r ay source wit h the option al addition of the ion gun for sputterin g or e t chi n g , and an ad di tio nal electron gun to o b t ain S EM images of the sample . Since t he acquisition of a spectrum requires ener gy analysis of e le ct rons , the s ame type o f energy analy zer used in X P S ( cylindrical mirror ) can be used for AES . In fact , some m an u fact urers offer systems t hat can do both A E S and XPS as well as other types of an aly si s ( U P S , I S S , etc . ) . A s pointed out earlier , solid flat s u r fac e s are ideal for AES an a lysis . In the case of p ow d er s , they must first be pressed into the shape of a w afer or a flat plate . De gassin g and drying the sample is necessary to ret ain the U H V ( lo- 8 to 10 - 9 torr) v ac u um req uired in the analysis cham be r . U n fortun ately , most c at alysts are supported on i n s ul atin g oxides ( A l 20 3 , Si0 2 ) which re s u lt in se ve re s ur face ch ar gi n g . Althou gh some remedies for sur face chargin g can be adopte d , the problem is o ften so se ver e as to limit t he applicability of t his techniq ue with technical catalyst s .
Ca taly tic Surfaces
1 79
A p p lica tions In qualitative analysis , A E S is used to survey possible contaminants and poison s that might affect the catalyst activity . The poisonin g of a copper catalyst by lead , and of a P d /y -Al 20 3 c atalyst by iron , has been reported by B hasin ( 1 9 7 6 ) . A E S is u sed routinely in sin gle - c ryst al studies to measure the cle anlines s o f such surfaces , in p articular , to determine car bon and oxy gen cont amination , surface composition , and the presence of adsorbed layers ( C han g , 1 97 4) . As in X P S , q u anti tat ive analy sis of powders is comp lic ate d by surface rou ghness and hetero geneity . Further comp lications arise from crystallite si ze distribution and nonuni form distribution of t he metals in the support . Approxim ate an alysis c an be done in terms of relative intensities between t he sample and a known st andard . The atomic concentration of an element c i in t he sample is given approximately by
( 16)
are t he peak - to-peak A u ge r current s of the sample and and I i s td i the surface concentration of the standard . standard , respectively , and Ci s td Relation ( 1 6 ) assumes that escape depth from t he samp le and standard are similar and t hat the standard is homo geneou sly distributed in t he s ample . Auger sen sitivity factor s with respect to the sensitivity to oxy gen have been given in the literature ( Palmberg , 1 97 5) . These factors must be used when the concentrations of different elements are comp ared . Although many papers describing the use of AES in single -crystal studie s have been pub lished [ about 3 0 0 are li sted by Chan g ( 1 97 4) ] , papers in relation to applica tions of AES to hetero geneous catalysis are limited , pres u m ably due to char gi n g problem s . A comp arison of the various spectro s copic techniques for surface analysis and result s com piled by the A S TM com mittee for sur face analysis , have recently been published ( Powell et al . , 1 97 9 ; 1 98 2 ) . I n s ummary , A E S renders in formation similar to X P S , and similar prob lem s can be studied with this technique . A E S ha s a m ajor advantage in lateral spatial resolution and sublayer analysis that is not yet available in X P S . As a disadvantage , AE S present s serious char gi n g problems wit h hetero geneous catalysts supported on insulatin g materials and chemical shi fts are not as easily interpreted in AES as in X P S . w here I
J o n Scatte r i ng S pect rosco p y
I n ion scatterin g spectroscopy ( I S S ) , me asurement of the ener gy o f pri mary ions sc attered by the solid s ample provides t he data from which sample composition is calculated . At medium and hi gh ener gies ( 1 02 to 10 3 ke V ) , ion s penet r ate deep into t he solid . T he elastic collision with an atomic core is governed by the mass of that atom , while interactions be tween the ion and electrons in the sample cause a loss of ion ener gy which i s p rop o rt io n al to the dist ance traveled in the solid . At these ener gi e s , the scatterin g cross section is governed by simple nuclear potentials and the experiment is called R u therford s cat teri n g . At ener gies from 0 . 5 to 10 keV , scatterin g cross sections and ne utrali zation probabilities are so
1 80
De lgass and Wo lf
high that ions are sc attered only by the first one or t wo atomic layers . Thus low - energy ion scat terin g i s a true surface technique . With proper c alibration , peaks in the sc attered ion spectrum identify m as ses and am o unts of elements present i n the surface . T his is a relatively new tech niq ue to c atalysis , but in combin ation wit h ion etchin g o ffers both quantita tive and st r uc t u r al a n alysi s of comple x surface s .
Princip les T he two im p ortant parameters in low -energy ISS (LEI S S ) are the ener gy and inten sity of the scattered ions . C alculation of the mass from the energy of t h e p e ak is strai ghtforward . Quantit ative interp ret ation of in ten sity is more difficult . T he ener gy of the scattered primary ion is governed by mo me ntu m and kine t ic ener gy conservation in t he t w o - body collision between t he p rimary ion and a sin gle a t o m in t he solid sample . T he energy E 1 of t he ion scat tered at an gle eL is given by
( 1 7) w here m 2 i s t he tar get atom m a s s , E 0 t he initial primary ion ener gy , and m 1 its m ass ( B uck , 1 97 5 ) . At 9r, 90° , a common expe ri me nt al value , E q . ( 1 7 ) reduces to =
=
( 1 8)
T he simp lici ty of this an a ly si s is possible , in p art , because of the relative ly hi gh ener gy of the collision . M ultiple sc atterin g effects do occur , b ut t h eor y an d experiment show t hat the binary collision mechanism dominates t he s c at te rin g of the p ar ti cle s that remain ionized ( Poate and B uck , 1 97 6 ) an d , except for possible resolution p roblems , Eq. ( 1 7 ) o r ( 1 8) p rovides assi gnment of masses to t he corre spondin g peaks . As i s cle ar fr o m these equations , m 1 c an app roac h m 2 to improve resolutio n . To be q u an ti t ative about s urface composition , one needs either a theoretic al description of t he collision cross section and the ion neutrali za tion p roba bi li t y alon g t he ion p at h or a metho d to c alibr ate the se effects em p i ri cally . T he well - de fined Rutherford sc atterin g law , w hich describes the hi gher - ener gy cros s sections , is comp licated in the 1 - keV re gion by t he fact that the n ucle ar potentials are partially screened by the electrons . A T ho m a s - Fermi potential is oft en used to account for this screening ef fect ( Po ate an d B uck , 1 97 6 ) , but ot hers are also used ( Taglauer and H eiland , 1 97 6 ) . T he ion n e utrali zation p rocess is e q ual l y import ant and difficult to model . T he extra ti m e spent by a low - ener gy ion in scat t eri n g from sub s ur face levels app e ar s to be s u ffici ent to allo w subst antial neutrali zation , p reventin g its detection . T he charge transfe r p rocess depends on ion ene r gy an d i de ntity as well as on sub strate m aterial and t ar get atom . At
Cataly tic S urfaces
1 81
not a theoretical method that p re di ct s t he se effect s re T h u s one m ust turn to e xp e ri m e n t al c ali bration methods .
present t here is
liably .
sur fac e c o m po sitio n and I S S signal is linear for p rim ary ion and energy , c ali b ratio n is st r ai ght for w ard . Linear re lationship s have been found for allo y s ( Smith , 1 97 1 ) and fo r sulfur and oxy gen o n nickel ( T a gl a ue r an d Heiland , 1 97 4 , 1 9 7 5 ) . S ur face p roper tie s such as work function can be e x p e c ted to affect ion neut r ali z ation , ho w ever , and nonlinearities in c alibration fu n c tion s have been observe d ( Niehus and B aue r , 1 97 5 ) . I n p r ac ti c e , some of the most useful informa tion is obtained b y o b se r vi n g ion ratios as a function o f t he n u m ber of surface layers removed by ion etchin g . C h an ge s in ion r atio are t aken to i ndi cat e chan ges in relative concentration ( Taglauer and Heiland , 1 9 7 5 ) . T his approach assumes that any chan ges in c ros s s e ct io n or n e ut r ali zat io n p ro b abilit y acco m p anyi n g c hemical ch an ge s from layer to laye r c an c e l out in t he ion yield ratio . If t he re l a tio n b e t w e e n
a give n
Experimen t
The e s se n ti al
components of t h e I S S in s trum en t , shown in Fi g. 1 1 , ar e t he sample , and the energy analy zer , all e n c lo se d in a UHV sc at t e ri n g chamber . T he 0 . 5- t o 5-keV H e , Ne , or A r b e am s used for low - e ne r gy I S S are t y pi c al ly formed by electron imp ac t on the gas at 5 x l 0 - 6 t o 1 0 - 3 t orr ( B uck , 1 97 5 ) . T he ion gun is di ffe r e nti ally p umped , delivers a b ea m with a stable io n c urrent of 10 to 2 0 0 nA , a s p ot si ze on the order of 1 m m in di am e te r , and an ene r gy spread of the or de r of 1 eV . The be am may also be mass filtered . The s c at te ri n g c hamber is a t ypical ultrahi gh - vacuum bell jar with a base p ressure of " 10- 10 torr an d oper at in g pr e s s ure < l o - B t o r r . A s with ot he r particle s p e c t ro sco p i e s , an at tached cat alyst p re t r e a t m e n t chamber is especially u s e fu l . T he sample is mounted in t he s c at t e rin g chamber on a m anip ulator to facilit ate ot he r the
ion be am ,
analyti c al experiments in the same instrument and possible an gular - de pend ent studie s . Ins ulator samp le s can accumulate a po si ti ve c harge durin g
ION GUN
EL£C'mOSTAnc
EtERGY ANALYZER
DETECTOR FIGURE
11
S c hem ati c
of an ion
s c a t t e ri n g spectrometer .
Delgass and Wo lf
1 82
T his char ge is remo ve d by the electron floodin g t e c hnique I on et c hin g of the surface c an be done wit h the p ri m ary be am itself or , preferably , with a seco n d hi ghe r c urrent beam . I n any case it is i m p o r t an t to avoid ed ge effects w hen subst anti al e t c hin g i s done . G ener ally , one monitor s only a sm all ce nt r al portion of t he etched re gi on . E lectros t atic ion energy an aly zer s are used in low -energy sc at t eri n g . Ener gy resolution b. E /E on t he order of 1 % is t ypic al . T h u s A u ger or X P S e le c t ron monochrometers are c andidat e s for this t ask , p r o vi d e d t h at collima tion c an e s t abli s h an unique s c at t e rin g an gle � .
an e xp er i me n t .
familiar in XPS and SIM S .
A p p lications
T he power of t hi s m et ho d is it s ability to give q u an t it at i v e or at lea s t semiq uantitative an alysis of t he uppermost laye r of a s ur face .
s en sit i vity s urp asses bot h
XPS
an d A E S ,
s ur face
I ts
and t hi s m akes the tool especially
we l l s uited to st udies of p henomena such as s ur face enrichment in alloys an d the structure of the upper layers of more co mp le x s u r fac es .
work of B ron gers m a et al .
T he ( 1 97 8b ) nicely illustrates the use of low - energy
I S S fo r surface enrichment studie s .
T hey show stron g surface enrichment
in Cu for bot h C uN i and C uPt and i l lu s t r at e e ffects o f te mp e r at ur e in n e alin g and quenc hi n g surface co mpo sit io ns
.
an
A naly si s of the d at a con firms
t he e ffic acy o f c ali br ation by t he p u r e com po un d s and show s t he n e c e s si t y for truly clean surfac e s since adsorbed layers c an differenti ally s h a do w
me t al ato ms .
More direct ap p lic at i o n s in catalysis have e m p h asi zed mixed
o xi des an d supported oxide s .
ISS
an aly s i s of the t r an s i t io n met al c ation
content of spinel surfaces has be e n found by S he l e f et al .
( 1 97 5 ) to be in
goo d q uantit ative a gr e eme nt with tit r ation o f the s ur face by NO ad sorption .
St u die s of s up p or t e d C oMo ( D el ann ay et al . ,
et
et
1 98 0a , b )
and NiMo ( Kn o zinger
al . , 1 98 1 ; Can o s a Rodri go et al . , 1 98 1 ; Ahart et al . , 1 98 2 ; al . , 1 98 3 ) oxides on alumina offer excellent example s of t he
ISS ,
Jeziorowski ability of
co up le d wit h ion e t c hi n g , to examine surface struct ure in these hydro
s ulfuri zation c at alyst precurso r s .
For the
N iMo
system , Mo /Al and
Ni /Mo
i n t en si ty ratios as a function of e t c hi n g ti m e i n di c at e e s s e n t i ally sin gle lay er coverage of A l 20 3 by Mo with Ni c o nc e n t ra te d at t he M o - A l interface
after calcin ation at 8 7 0 K ( Je zioro w s ki et al . , 1 98 3 ) . T he I S S spectra 12 show a low AI in t e ns i ty at t he outset , indicatin g t hat
shown in Fi g .
t he A I 20 3 surface i s covered , and a m a xi m um in t he Ni /Mo i nt e n sit y ratio wit h s p u t teri n g time , in di c ati n g t he enhanced concentration of Ni below
the sur face .
T he order of addition of N i a nd Mo w as also fo un d to affect 1 98 2 ) . I f Ni w a s i mp re gn ate d on the
the po sition of t he Ni ( A b art et al . ,
A l 2 0 3 first , then the Ni /Mo in te n s it y increased with etchi n g ti m e Ni t o be enhanced near t he Mo / A I 2 0 3 int e r fac e .
,
s howin g
I f Ni w as added after
M o , it s concentration w as m aximum at t he outermost surface o f t he cat alyst .
( B ron ge r s m a et al . , 1 97 8a ) an d individual metals of Al 20 3 ( W u and Hercules , 1 97 9 ; Chin and Hercule s , 1 9 8 2a , b ; Zin gg e t al . , 1 98 0 ) . The t ec hniq u e is r e l ativ e ly new , but p r o gre s s is s u fficient to m ar k t he m et h od as worthy o f c o nti nued at te n tion . T he fe at ure s of t hi s met hod t hat m ake it att r acti ve for c at alysis also H ere ordered or even j u st flat surface s allow apply in s urface sc ie n c e . a more s ui t able use of t he s h a d owin g or blocking e ffe c t s caused by t he relative p o s i tio n s of atom s on t he s ur fac e . T he re l at i ve intensities of 0 and C , for example , s how CO to bon d ca r bo n down on Ni ( S mit h , 1 96 7 ) . Applications to catalysis also include studies of B i 2MoO a
1 83
Cataly tic Surfaces
600 ° C
5 0 0 eV H e• 0 I
AI I
N1 Mel
I
I
0�
110S s
240s1
1J45s I
15100s
o.s
F I G U RE 12
E / E0
1.0
I o n scatterin g sp ec t r a of Ni 3M o1 2 /Al20 3 as a function of increas ( From Jeziorowski et al . , 1 98 3 . )
ing bom b ardme n t time .
Quantitative an alysis of s ha dow in g effect s on in t e n s ity as
azimut hal angle have also b een used to a dl ayers ( Heiland and Taglauer , 1 97 2 ) .
a func tio n of
determine t he st r uct u r e o f ordered
S eco n d a ry l o n M a s s S pect rometry
S e con d ary ion mass spectrometry ( SIM S ) is a p ar ti cle spectroscopy usi n g 2 0 0 to 5 0 0 0 - eV Ar+ p ri m ary ions to bombard t he s ample s u r face an d c a u s e ej ection o f s econdary i o n s which are detected by m as s spectrometry . Fi g ure 13 is a schem atic represent ation of the experiment . The pres s ure i n t he spectrometer must be low enough to avoid si gnificant io n /molec ule col li sion s Norm al op e r ati n g pressures vary from 1 0 - 6 to l o- l l torr . .
Delgass and Wo lf
1 84
F I G U R E 13
S chem atic of a secon d ary ion mass spectrometer .
M aterials scie ntist s and e n gineers have for some time used dynamic ( hi gh p rimary ion flu x ) S I M S for dept h pro filin g in met als and semiconductors ( M c H u gh ,
1 97 5 ; Werner , 1 97 5 ) .
I ntroduction of t he s t atic mode ( B ennin g
hoven , 1 9 7 3) , in which low primary ion fluxes c ause removal of only a frac tion of a monolayer durin g the experiment , has m ade S I M S a relatively re T he unique feat ure cent entrant as a surface science and c at alytic t ool .
of t his met hod is t hat t he combinations of elements t hat appear in t he clus ter ion s recorded in t he spectra are representative of loc al environments on t he s ur face . Thus SIM S stands with EXAFS as one o f t he· few met hods with the potential to provide proximity , local compo sition , and ultim ate ly , local s tructure information .
Princip les T he primary ion brin gs a lar ge amount o f energy to a smal l s ur face region and starts a collision cascade t hat affect s hundreds of atoms and c an cause ejection of as m any as 1 0 or more atom s from the surface .
T he ener gy spectrum of the secondary ion peaks at about 2 eV , so only a sm all frac tion of the p rimary ener gy finds its w ay to t he secon dary ions that com
prise t he S I M S spectrum .
T here are , of course , some high- energy
secondary and p rimary ions , b ut t hese are filtered out an d not detected . A t he rmodynamic model of t he S I M S p rocess appears to be s uccess ful in quantit ative an alysis of dyn amic S I M S ( Anderson an d Hinthorne , 1 97 3) . A complete , q uantitative found ation for st atic S I M S has not yet emerged , but many characteri stics of t he process are know n .
T he second ary ion
yield has bot h kinematic components , w hic h account for ene r gy tran s fer in t he colli sion cascade , an d electronic co mponent s , w hich govern the p robability t hat an ejected particle will be ioni zed or rem ai n ionized until detected . T he kine matic effects h ave been modeled by classical dynamic c alcula tions which include all p airwise interactions in an inte gration of t he equa tion s of motion describin g the collision of t he p ri m ary ion with the solid ( H arrison et al . ,
1 97 8 ; Winograd and G ar rison , 1 98 0 ) .
T hese c alc ulation s
show that for met als and m e t al /adsorbate syste m s , cluster ions form pri m arily over t he surface b y recom bin ation o f atoms t h at were close nei ghbors but not nece s s arily conti guou s ( G arrison et al . ,
1 97 8) .
S tron gly bonded
species such as CO or C sH s are s hown to be ej ected int act , ho wever ( G arrison et al . , 1 980 ; G arrison ,
1980) .
T he clearest demon stration of
1 85
Cat aly t ic S urfaces
stron g electronic influences on t he S IM S sp ec tr u m is t he observation that positive mo no m e r ion yi el d s from a m e t al can chan ge by more than three orders of m a gni t u de w hen oxy gen is c hemi sorbed on t he surface . The favo rin g positive char ge emis presence of o xy gen raises t he w o rk fu n c tio n sion an d ret ar din g electron s . T he importance of ini ti al ion char ge ( D ay et al . , 1 9 8 0 ) and electron exchan ge in the near-surface region ( Lin and Garri s on , 1 9 8 2 ) has al s o been demonstrated . Pe rhap s t he most important issue c on c e r nin g S I M S is the degree to which s t r u c t ur al in formation is preseq-v e d in t he s pe c t r u m Extensive work ,
.
on molecular solids sho w s t hat w he n intramolecular bondin g is s tro n ge r
t h an bon di n g between molecule s , emission of intact molecules i s t he rule rather t h an the exception ( Day et al . , 1 9 8 0 ) . S p ec t r a of m aterials suc h as q uaternary amines s ho w intact emis sion and some gas - p hase ion fr a g
men t at io n
b u t li tt le evid e n c e fo r m o le c u l ar d a m a ge d urin g emission ( D ay Hi ghly ionic solids c an give l ar ge ions ( B arlak et al . , 1 98 2 ) , while re co m bin at io n over t he s urface is t he dominant cluster ion T he utility of t h e SIMS mec hanism for met als ( G arrison et al . , 1 97 8) . method for i d e n ti fic ati on o f molecular s p e cie s an d element p ro xi mit y on s ur face w il l continue to increase as t he r e li abilit y o f s t r uct ur al i nt e rp ret ation advances . Si gnificant ad v anc e s in un der s t an di n g and control of t he ioni za t io n p rocess will be needed before t he method can b e truly q uantitative .
et
al . ,
1 97 9) .
Exp erim e n t
A SIMS spe c tro m e t e r consists o f an ion
gun s a m ple , s econ d ary ion e n e r gy of which are enclosed in an ultra high- vacuum ch a m b er an d co n trol le d by a di git al data acq uisition sy s t e m T he ion gun is di fferentially pumped and c an deliver ions at e n er gie s from a fe w hundred eV to about 5 0 0 0 eV at c urre n ts ran ging from l o - 1 0 to Io- 5 A. T he bea)ll diameter at the s am p le is of the order o f 1 mm . In s o me c ases t he beam is m a s s filtered and m ay be rastered over the s urface . The sample is mounted on a m anip ulator . It is n o t uncom mon to have other an al y tic al devices , ( e . g . AES , XPS ) i n t he c h am ber and the sample is m ove d fro m o n e anal y si s p o si t i o n to another on t he m anip ulator . A bility to heat and coo l t he s am p le in the analysis position and to dose adsor bin g gases on it is hi ghly desirable . For catalyst studies , a pretreatment cham ber de si gn e d for c lea n tran s fer to and from the m ai n chamber is a neces s it y ( e . g . , O tt e t al . , 1 97 9) . C at al y st s can t he n be pretre ated o r re acte d j ust p rior to analysis . I ns u l a ti n g samp les such as s up porte d cat alysts often char ge up durin g an experime n t . S uch char gin g serious l y de grades filter , mass analy zer ,
,
and detector , all
.
,
t he spectrum and must be avoi ded . E lectron - beam char ge compens ation is o ften an acceptable solution ( W it m aac k 1 97 9 ; Re uter e t al . , 1 98 0 ) . In some c as e s c har gi n g can also be avoi ded by proper s ample p re par atio n Burnishing a very thi n sample layer onto a metal foil , mixin g the s am p le or with a cond uctor such as metal powder or graphite , s c r at c hin g the surface of a thick coating to expose some of the metal bacldn g p l a te will have all ,
,
b een succes s ful ( D ay et al . , 1 98 0 ) . T he p refilter condition s t he i ons for optimum m ass analysi s , usu al ly by a qu ad ru po le filter . In fast ion b o m b ar d m e nt ( FAB ) , the p rim ary ion beam is n e utr ali ze d so t h at the s am p le can ride at a high potential an d a hi gh r e solut io n m a gn etic mass analy ze r can b e used ( B arber et al . , 1 98 1 ) . S e co nd ar y i on m a s s s p e ct r a are oft e n int e n se en o u gh to be collec t e d in an analo g mode , b u t c ap ab ili t y for p ulse counting i s necessary for the st u d y -
1 86
D e lgass and Wo lf
of weak sign als . T he time n ee d e d to colle c t a spectrum d epe n d s on many factors b u t is ge n er ally of order 1 m in to 1 hr . A d d e d feat u re s e speci a lly useful in surface s cience app li c ation s are e ne r gy s e lection o f t he se con d ar y i on s and c ap a bility of a zimut hal and po l ar an gle re solution of the s e condary io n s ( W i n o gr ad and G ar ri s o n , 1 980) . In cat alytic exp eriment s , good char ge com p en sation and c onve ni e nce and e ffectiveness o f samp le p re t r e at m ent are p ri m e consi de r ations .
A p p licatio ns
Since t hi s tec hni q u e is still developin g , S I M S applications are s eldo m routine . The st ren gt h s of the method are ( a) hi gh sensitivity ( c a . 10 - 6 monolayer in t he hi ghly favorable c as e of alkalai metal s ) a n d restriction to the upper most surface layer , ( b ) abili t y to detect h y d ro ge n , ( c ) s e n si ti vi ty to iso topes , ( d ) detection o f a ds o r b e d molecu lar s p eci es , ( e ) indication of element p roximity on a multicomponent surface , and ( f) t he p os si bilit y of s truc t u ral analysi s . T he main w e ak n e s s e s of the met h o d are t he di ffic ulty of q u an titi zation and the c har act e ri sti c of most p article spectroscopies t hat t hey req ui re low ambient pressure and see on ly the external surface of a porous catalyst . Ap p li c ations of S I M S to cat alysis have been relatively few , but FeRu+ ions h ave in dic ate d the for m ation of PeRu alloys o n S i0 2 ( Delgass et al . , 1982) . For cobalt on alumina catalysts , Alco+ ions have shown the inti m at e m ixin g and low co ve ra ge of Co on the Al20 3 at low Co loadi n gs ( C hin and H e r c u le s , 1 98 2 b ) . T hese ion s were re plac e d by Coo 2+ , characteristic of Co30 4 , w hen the Al 2 0 3 sur face was he a vi ly loaded with Co . Molecular sp e ci es on catalyst s ur faces have shown up as a va ri e t y of C 2 to c 6 c ar boxylate s in the n e gative - io n spectrum of a commercial silver catalyst used for ethylene oxidation ( B e n nin g hove n , 1 97 3) and as the p roton ated molecu lar ion for isobutene and pyridine ad sorb ed o n S i and A I oxides ( S chu m ache r , 1 98 3 ) . Structural a n aly si s of o xi d e s , potentially a very interes t in g are a , has be gun with e valu at ion of
M gC u A l oxides ( B arber e t al . , The spectra have 1 97 6 ) an d Fe aluminates ( B uhl and P r ei si n ge r , 1 97 5 ) . been i nt e r p r ete d as in dic atin g oxide struct ure . The potential of t he met hod for an alysis of he te ro geni zed hom o ge n e o u s cat alysts have been demon strated by the observation o f s t r u c t u r ally si gn i fi cant second ary ions from transition met al complexes ( Pierce et al . , 1 9 8 2 ) . S tudies on well- defined me tal s ur fa ce s s ho w t h e u tility of t he met hod for s urfac e reaction studies . D rechsler et al . ( 1 97 9) have used S I M S to i d en ti fy N H as the most abundant s urface intermediate in N H 3 decomposi tion on Fe ( 1 1 0 ) . Other s t u die s i d e nt i fy s ur fac e s p ecie s p r e s ent d uri n g d ecom p o sition of formic aci d on nickel ( Mohri et al . , 1 97 8) and during flash d e sorp t ion of C 2 H 4 from R u ( 0 0 1 ) ( Lauderback and D e l gas s , 1 984) . S I M S has also con firmed t he e th y li d y n e struct ure for et h yle n e adsorbed on Pt ( 1 1 1) ( C rei ght on and W hite , 1 9 8 3 ) . Fi gu r e 1 4 s hows direct S I M S o b servation of a ss oci at ive and dis sociative a d so r ptio n o f CO o n R u ( 0 0 1) T he m ol e cu lar st ate is i n di cated by ( L au d e r bac k and D e l ga ss , 1 98 3 ) . + Ruco+ and Ru 2 co ion s . Fin ally , t he an gular - dependent SIMS me as ur e ments of Wi n o gr ad and coworkers for adsorbate / metal c r y s t al systems s ho w stron g an gle - dependent variations in SIMS intensity c haracteristic T hese me asurements , t o ge t he r o f s u r face symmetry ( Gibbs et al . , 1 98 2 ) . with t he c las sical dyn ami c c alculations , s u g gest new S I M S m e thod s for de t ailed surface structure dete rmin ations ( Wi no gr ad , 1 98 1 ) .
1 87
Cataly tic Surfaces
Ru RuC 1 3
>1-
Ul z I.LJ 1z
Ru2 C• l iiJ)
I
�• lSID
Ru2C 1 3
<• l SS)
I
8
520 K
�
1-1
Ru
RuC
<• l S�> I
RuC 1 80 <• u rn
Ru2
Ru2C 1 BO
<• l S>
<• 1 111 111>
Ru2C
"
C• l iiJS>
18 . 1 4 Positive -Ion SIM S spectra of CO o n R u ( 0 0 1 ) after ( A ) 9 . O L C 0 1 3 2 at 2 0 K , and ( B ) 4 80L 3co at 5 0 K . ( From Lauderback and D e l gass , 1983 . ) FIGURE
E xtended X - ra y A bso rption F i n e St ructure
Active sites on a c at alyst surface are defined by local structure . T he average s ur face compo sition of a c at alyst , as determined by X P S , Au ger electron spectroscopy , or I S S , is inform ation important in differentiation of catalysts and in followin g c h an ge s occurrin g duri n g activation and d e activation . T hese tools can only imp ly local structure , however . A more direct me asure is needed . E xtended x - r ay absorption fine structure ( EXAFS ) , alt hough not the complete answer to this loc al struct ure need , offers information on atomic arran gements in catalys t s . T he method has several import ant characteristics . It is element specific ( i . e . , one c an individually examine x - r ay ab sorption e d ge s characteristic of t he various elements in a m ulticom ponent cat alyst ) and does not requi re cryst allinity . Analysis of t he extended fine structure of these ed ges gives t he inter atomic distances , to ca . 0. 01 A for sub sequent shells ; the occup ation of the shells ( coordin ation number ) , with a current accuracy not better than ± 2 0 % ; and the mean - square displacement of t he atom in question ( Joyner , 1980) . Alt hou gh a relatively new and not yet fully developed technique , EXAFS has s ho w n low coordination numbers but relatively unchan ged bond distances for hi ghly dispersed s upporte d met als ( Via et al . , 1 97 9) , con firmed the C u surface coatin g model for supported C uR u clusters ( Sinfelt et al . , 1 98 0 ) , and s hown sulfided Mo ( A l 20 3 and CoMoA1 2 0 3 ) to cont ain small dom ain Mos 2 structures ( C lausen et al . , 1 9 8 1a ) . In the U nited States , experiment s are best done at the synchrotron radiation facilities at Stan for d , Cornell , or B rookhaven ( Rowe , 1 9 8 1 ) .
1 88
D e l gass and Wo lf
Princip les
EXAFS be gins wit h the m e asuremen t of transmitted x - r ay intensity as a function of frequency . From these data o ne c an calculate the x - r ay ab sorption c ross section as a fu nct io n of ener gy . E ach t i m e t he ene r gy rea c he s the threshold for p hotoemis sion of a core electron in the sample m aterial , the absorption cross sect io n increas es dram atically . T hu s a broad scan of ener gy , as shown in Fi g . 1 5 , p rod u c es an absorp tion spec trum with step s or 11edge s 11 characteristic of t he bindi n g ener gi es of core electrons in t he sample . Except for accident al overlaps , each ed ge cor - , responds to a p articular element . C lose examination o f a given ed ge s hows t hat for several hundred eV to the hi gh -ener gy side of t he steps , the ab s o r p tion initially oscillates with ener gy . T he oscillations be gin n in g 5 0 to 1 00 e V above t he edge , t he exte n d e d fine structure , res ult from t he con str u ctive and destructive interference bet ween t he o ut goin g photoelectron wave and its component backscattered from near nei ghbors . A pebble dropped a mon g reeds in a still p o n d produces a classical an alo g o f t his e ffect . B e c ause of its origin , the det aile d structure of t he EXAFS spectrum cQntains information on the number and lo c ation of t he neighbors surround in g a p articular type of atom . Extraction of this i n formation is com p lex but tract able an d still developi ng ( Teo and Joy , 1 9 8 1 ; H ayes and Boyce , 1 9 8 2 ; Einstein , 1 9 82 ; S andstrom and Lytle , 1 9 7 9) . First the nonoscillatory part of the absorp tio n is removed by cu rve fittin g and the EXAFS function x ( k ) is fo r me d from the eq u ation
x( k )
11
( 1 9)
12
ENERGY, KeV
13
14
F I G U R E 1 5 X - ray ab sorption spectrum at 100 K in the re gi on of t he L absorp tion e d ge s of Ir and Pt for a Ptl r - Si0 2 catalyst cont ai nin g 10 wt % of eac h metal . ( F rom Sin felt et al . 1 982 . ) ,
1 89
Ca taly tic S urfaces
l where k = ( 2mE ) J n , m is t he mass of an electron h is Planck' s constant by 2 n , E is t he kinetic ener gy o f the p hot oe lect ron , JJ the nat ural lo g of the ratio of the i n ci d en t and transmitted x - r ay p hoton flux , and ll 0 t he abs orp tion coe fficient of t he free atom . T heories based on sin gle scatter ing of the p ho toelec tro n by atom s in the coordin ation shells surroundin g the ab sorbin g atom give x(k)
=
L: A . ( k ) . 1
J
sin [ 2kr . + o . ( k ) ] ]
]
( 20)
w here t h e sum m ation i s over t he coordination shells , rj is the radial dis tance from t he a b s o r b i n g atom t o atoms i n the j shell and 6 j ( k ) the total phase shift for emission and sc atterin g , Aj ( k ) is an amplitude function that depends on t he n umber of atoms in a given shell ; rj , a b ackscattering amplitude ; and the root - m e an - square deviation about rj . W i t h proper ac countin g for p hase shift s and t he k 0 point , Fourier tran s form ation of a kn wei ghtin g of x ( k ) ( S tern et al . , 1 97 5 ) , produces a radial s tr u ct ur e function t hat reflect s the positions and number of atoms for each coordina tion shell as shown in Fi g . 1 6 . T he se are the d at a from which initial structural assignment s are made . One metho d o f refinement of structural parameters is to retr an s form a small region of radial space ( say , the T he process act s as a Fourier nearest - nei ghbor- shell ) b ack into k s p ac e . =
filter w hich r e mo ve s all but the nearest - nei ghbor contri bution to t he x ( k ) Nonlinear le ast - square s techniques c an t hen be used to fit the function . struct ural p arameters to t hi s function ( Via et al . , 1 98 1 ) . This di s cuss io n sho w s t he o rigin of the E X A F S function and the radial structure function . Further details of t he t heory an d dat a red uction pro cedure are given in t he re ferences cited a b o ve , but a few additional com ments abo ut t he techni q ue are included here . T he EXAFs p he nom eno n re s u lt s from b acksc atterin g of t h e excited p hotoelectron . X - ray absorption is one of several ways to detect the occurrence of thi s e ffect . A lt e r nati v e w ays of detectin g x - ray ab sorption i n clude x - r ay fluorescence and A u ger electron measurement s . S ti m ulation of core electron emis sion m ay also be by el ec t ron bom bardment ( H ayes and B oyce , 1 982 ; Ein stein , 1 98 2 ) . T he low mean free p at h of electrons in solids offers opport unity for EXAFS analysis accent uatin g surface struct ure ( Landm an and A dams , 1 97 6 ; Einstein , 1 982) . A mon g t he aven ue s opened b y t he use of electron sti m u l atio n i s the o bse r v ation of extended electron ener gy lo s s fine struct ure ( E X ELFS ) in an electron m ic ro s cop e . T his experiment give s both local struct ure and high spatial resolution over a s ample ( Le ap m an et al . , 1 9 8 1 ) . Exp erimen t a l Me t ho ds
The alternative app roaches mentioned above m ay eventually provide X - r ay ab so r p t io n measurement s can be m ade wit h conventional x - ray sources ( K n ap p et al . , 1 97 8 ; del C ueto and S he v chi k , 1 97 7 ) but countin g time s are generally lon g comp ared to those obtained wit h the intense b e am s available from synchroton radia tion ( H ayes and B oyce , 1 98 2 ) . Since absorption m us t be scanned as a function of ene r gy , a broad continuous band of radiation is needed and the sharp emission lines of the conventional x - r ay so ur c e must be avoided . A sy n ch ro t ro n or storage rin g p ro duces this r a di at io n by t he curve d orbit of a beam of hi gh- energy electron s . T his , or t he B remstrahlung
routine access to EXAFS - type information .
1 90
Delgas s a n d Wo lf
10
20
5
10
20 0
1"'1 0 X
1
0 20
+
R u - C u/ S i 0
0 2
10
10
0
0
:::!! 0:: 0
.....
VI z oct oc: �
.....
.....
0 c :::1 1-
2
t!l oct
:::!!
40
C u/ S i 0 2
0
2
2
C u/S i 0
20
+
2
2
10
20
0 20
R u -C u/S i 0
0 C u - R u/S i 0
C u -Ru/S i 0
10
2
10
+
0
2
2
5
R, A 0
2
4
6
8
ruthenium
F I G U RE 1 6 Effect of exposure of silica- supported r u t hen i u m , copper , and copper c atalysts to oxy gen ( 1 % ox y ge in at roo m temperature on the properties of the catalysts as shown by Fourier trans E XA F S d ata at 100 K o r the c atalysts in the ab sence and presence of oxygen . T he tran sforms in the upper half o f t he i g re are the K absorption edge of T ho s e in t he lo w e half for the C u K edge . ( From Sinfelt et al . , 1 9 80 . )
forms of for
f ruthenium.
n helium) r
f u are
Catalytic Surfaces
1 91
b ack gro u n d radiation of an x - r ay t u b e , i s p asse d t hro u gh a monoc hrom ator and then t hro u gh an incident flux detector , the sample , and a t r an smit t ed fl ux d e te c to r . X - r ay en er gy i s s c an ne d w it h t he monoc hro m ato r . T he raw d at a , tran smitted int e n sit y ratio as a function of energy , are t hen treated as d e sc rib e d in the pre c edi n g section . A s with any work o n c at alys t s , proper cells for p re tre atin g t he s a m p le an d m ai n t ainin g its en viron ment d uri n g t he me asurement are e s se n ti al ( Via e t al . , 1 97 9) . S am ples are o ft e n m aintained at lo w temper atures ( 80 to 1 0 0 K ) to d e c r e as e th e m ea n - s qu a r e d i s p lac e m e n t of t he atom s and i m p r o ve r e so l utio n .
A p p lications EXAFS s t r uc tur a l studies of in t er e st in c at al y si s have r an ged from examina tion of s upp orte d m e tal c ar bon y l clusters ( B e s son et al . , 1 98 0 ) to analy si s of t r an sition metal coordination in ze oli te s ( M orri son et al . , 1 980) . To i llu st r at e the method , we foc u s on ap p li c atio n s to supported met als and
HDS c atalyst s . In a series of papers , Sin fe lt , Via , and L y tl e report E X A F S o f sup ported metals and b im e t alli c clusters . Studies o f O S , Ir , an d Pt on silica and a l u mi n a ( Via et al . , 1 97 9 ) u ti li ze d s am p le s wit h 1 w t % metal lo adin gs and p e r c en t metal exposed ( dispersion) of 70 to 1 0 0 % an d com pared res ult s to values for t he pure metals . N e arest - nei ghbor di s ta n ce s for t h e supported metals were within 0 . 02 A of the pure me t al values . Co ordination n umbers for the neare s t - n ei gh b or s hell w e r e 7 to 10 for the catalys ts , lower than the bulk value of 12 because of the hi g h di s p e r sion of the metals . T he high s ur face - to - volume ratio of t he metals also caused hi gh root - mean - sq uare di sp l ac e m en t values fo r t he catalyst s c ompar e d to the b ulk metals . In general , t he close a gr e em en t between the i n t e r at o mi c distances for t he catalysts and t h e bulk metal speaks a gain s t struct ural re a rr an geme nt o f t he sm all m e t al clusters ( B urton , 1 97 4) . Di s tor t io n s in s t ruc t ure have been observed , however . Moraweck et al . ( 1 97 9 ) re p ort Pt near est - n e i gh bo r distances shortened by 0 . 1 2 A for 1 2 - A Pt p art ic les in Y ze olit e . Interestin gly , t hi s val ue returned to the Pt - Pt b ul k distance after adsorption of hydro gen . J o yn e r ( 1 980) , in di s c u s si n g the d at a of Via et al . ( 1 97 9) , notes that the radial s tr uc t ure functions for Pt and I r on sili c a are consis tent wit h t hree - dimensional face - centered - cubic p ar ti c les , but Ir and e sp e ci ally Pt on A l 2 0 3 appear to have v ari at io n s in peak in ten si tie s beyond t he first n e i g h bo r distance t hat are b e st described by raft s with ( 1 0 0) orientation . EXAFS of s upported bi me t alli c clusters has i nc l u d e d t he RuCu ( S i n fe lt et al . , 1 9 80) O s C u ( S infelt et al . , 1 9 8 1 ) , Ptlr ( Si n felt et al . , 1 9 8 2 ) , and Here a n ew dimension is adde d to RhCu ( M eit zner et al . , 1 98 3 ) sys tems . t he po w e r o f t h e te c h n iq ue because EXAFS of ed ges cor re s p on di n g to each element c an b e comp ared to analy ze element p roxi m it y . For both RuCu and O s C u , w he re t he met als are essentially i m m i s ci ble , copper coat s t he group VIII metal . R u( O s ) i n t he se catalysts has interactomic di s t ance s co r r e s po n di n g p ri m ari ly to R u ( O s ) n ei gh b or s . T he C u , o n t he other hand , is coordin ated ex t e nsi ve l y to Ru( O s ) as well as to C u . F i gu r e 16 shows t he effects of 0 2 on t h e trans form of t he wei ghted E X A F S fu nct io n for both C u and R u . Note t hat t he dat a are t aken at lo w temp erature to improve re solution . T he inten sities of li n e s below 2 A ar e suppressed in this dat a b ec a u s e of t he restricted re gion o f k s p ace used for the trans form . Thus the i n t e r p ret atio n o f t h e results is based m ainly on t he m aj or m e t al - met al
D e l gass
1 92
and Wo lf
n eare s t n ei ghbor peak . For R u / S i0 2 oxygen exposure at room tempera t ure si gnificantly re d uces t he number o f R u - R u n ei gh bor s . T his pe ak is essentially unch an ge d after oxygen exposure o f t he R uC u /Si0 2 cat alysts . T hu s Cu shields R u from oxidation as expected from t he C u coat ing model . T he copper dat a in the bo ttom four spectra of Fi g . 16 s how that C u in C u / S i 0 2 is only partially oxidi zed by 0 2 ( t he core of t he particle i s pro tected by an oxide skin ) but that C u i n the t hin layer on Ru i n C uR u / S i0 2 i s completely altered b y oxy gen expos ure . T he R h C u and Ptlr sys tems also show surface se gre gation p henomena but indicate more intermix in g of t he elements , in accord wit h t he greate r miscibility of these systems . For a 50- A met al p articles in Ptlr / SiO 2 , x - ray di ffr action is consistent wit h homo ge n eo u s alloy formation , b ut EXAFS show s t he part i cl e s to consist of P t - ric h an d l r ric h re gi o n s T he structure of s upported cobalt - molybdenum hydrodesulfuri zation cat alysts has been a topic of interest approache d by many an alyti c al tech nique s . Recent E X A F S stud ie s have contributed significant new informa tion . C l ause n et al . ( 1981b ) have used the Mo K ed ge to ex am i ne the local structure around Mo in sulfided Mo / A l0 3 and CoMo /AI 2 0 3 catalysts . Com p arison of result s for well - crystalli zed MoS 2 and the t wo c at alysts s hows t hat Co has litt le e ffect on t he Mo environment , the first coordination s hell of Mo is essentially the s am e in all t hree m aterials , and t he second coord ination s hell occurs at the s ame distance for t he t hree s ample s but its pop ulation is si gn i fican t ly lower for t he cat alyst s . Mo S 2 s hows struc t ure at hi gh interatomic distances , b ut t he catalys t s do not . The authors interp ret t he se dat a to indic ate t hat Mo has the s ame s ulfur coo rdin ation in the c at aly sts as in MoS 2 and t hat t he catalytic s pecies is not an oxysul fide . The low a mpli t u d e of the peaks cor r e spon din g to lar ger interatomic dis t ances is interpreted as indicat in g very s m all MoS 2 cryst allites ( on the order o f 10 A ) , S t u dies of EXAFS follo win g the Co K edge for t he sul fided CoMo /Al 2 0 3 catalysts s how that C o is surrounded by s ulfur , but t he lack o f any more distant peaks s u ggests a hi ghly di so r dere d environment and that Co is located at t he s ur face o f t he s mall MoS 2 crystallites ( Clau s en et al . , 1 981a) . Finally , we note that EXA FS of Mo has als o been used to confirm an MoS 2 structure for a propene methat hesis catalyst pre pared by reaction of Mo 2 ( rr - C 3 H 5 ) 4 with t he O H groups of y A l 2 o 3 ( S ato et al . , 1 9 8 2 ) . At this time , c at aly tic m at e ri als most amenable to E X A F S study are t he ones in w hic h the element of intest is in a sin gle environment . T he struc ture and composition of t he first coordination sphere i s t he inform ation most readily att ained . D eve lopi n g experime n t al met hods will u n doubtedly widen the scope o f this technique , w hi le impro ved analysis increases t he sop histication o f i n te r p ret atio n and allows solution of i n cre as i n gl y comple x structural p roble m s . -
,
-
.
-
Mos sba uer S pec t ro scop y
T he Mossbauer effect is a n uclear gam m a - r ay resonance with s uch hi gh precision t hat spectral fe atures reflect the chemical st ate of t he correspond ing atom . T he requirement of a relatively low - lyin g n uclear excited state restricts t he number o f i so top e s convenient and possible for us e in cataly tic experiments to those s hown in T able 3 . I so t o p ic abundances indicate t he fraction o f t he naturally occurrin g element w hic h is t he Mos sbauer active is o t o p e W hen the techniq ue app lie s , it is a powerful metho d for .
1 93
Cataly tic S urfaces TAB L E 3
M oss baue r
M o ss bauer i so top e
57
Fe 119 Sn
151
Eu
121
Sb
125
127 1 97 99
Te
1
Au
Ru
1 9 3l 1 95
83
r
Pt
Kr
1 81
Ta
P arent a
57
co
1 1 9m 8 151
Sm
1 2 1m
125
8n
1
1 2 7m
Te
1 97
Pt 99 Rh 1 93 0s 1 95 Au
83
Rb
181
n
w
Parent
half-life b
>. - ray en e r gy
( ke V )
N at ur al
a bun d an �
( %)
2
2 7 0d
14. 4
2.
2 4 5d
23. 9
8. 6
9 3y 7 5y
5 7d
21. 6
47 . 8
37 . 2
57 . 2
35 . 5
7. 0
1 0 5d
57 . 6
100
20h
77 . 3
100
1 6d
90
12 . 7
3 2h
73
62 . 7
98 . 8
33. 8
9. 3
11. 6
1 9 2d
8 3d 1 40d
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lite si ze effect s for specific M o ssbauer element s bound to or within s oli d For isotopes not r e q ui ri n g low temperat ure in order to observe m at ric e s . a s p e c t r u m , the technique c an be ap plied in sit u to catalysts operati n g in Applications have in c lud ed use of 5 7 c o , l 1 9 s n , the reaction environment . 1 5 1 E u , 1 2 1 s b , 1 97 A u , and 9 9Ru , but b far t he lar ge s t number of st udies y 5 7 Fe . T he reader is referred to D u m e sic and Top s(j e ( 1 976 ) have been on an d D e l g a ss et al . ( 1 9 7 9 ) for detailed reviews of catalytic ap p li c at ion s , and t o Wert heim ( 1 96 4 ) an d G reenwood and Gib b ( 1 97 1 ) for r e fe re n ce books on t he M os s b aue r e ffe c t i t se l f . Pri ncip les
The M os sbauer effect can be thou ght of as the nuclear equivalent of op ti cal spectroscopy . T he source of radi ation is ge ne r ally a m et as t ab le iso tope which decays to the nuclear e x c i t ed state of the M os sbauer isotop e . Dec ay from this excited state t o the ground state emit s the M ossbauer y ray . Re s on an t ab sorp tion can t hen occur i f this y ray i m p i n ge s on a Since the y - r ay n uc le u s of the M o s s bauer isotope in it s ground state . ener gies are hi gh ( 1 0 4 to 1 0 5 eV ) , the M ossbauer atoms must be bound to a so li d to p r e ve n t the recoil " kick" from destroying t he resonance . The area under a M o s s b au e r ab sorption line is a m e as u r e of the p rob abili t y of "recoil- free" e m ission an d ab so rp t i on , w hich in t urn is re l at ed to the This relation is complex coup li n g of t he M o ss b aue r atom to the so li d . ( Debrunner and Fraue n felder , 1 97 1 ) , bu t spectral areas c an be used as
1 94
Delgas s and Wolf
indicators of changes in bondin g of a Moss bauer atom to a cat alyst matrix ( G arten et al . , 1 9 7 0) . Since the width o f the Mossbauer line is o f order .: 1 0 9 eV , the precision of t he measurement is so hi gh that it reveals t he very small perturb ations m ade on the n uclear ener gy levels by the atomic electrons . These nuclear hyperfine interactions project chemical informa tion to the nucleus and hence to the Moss bauer spectrum . T hree spectral parameters are of p articular interest . T he isomer shift ( I S ) , me asured in simple cases by t he centroid of t he spectrum ( K alvius , 1 9 7 1 ) , indic ates relative electron density at t he nucleus . Only s and relativistic p electrons contribute directly to this term ; other electrons affect it s value because of shieldin g . For example , isomer shift between hi gh spin Fe 3+ and Fe 2+ is about 0 . 8 mm /s or about t hree line widths . Fe 3+ has a hi gher electron density at the nucleu s t han does Fe 2+ because it has only five 3d electrons while Fe 2+ has six to shield t he 1s , 2s , and 3 s electrons . Covalent bondin g can comp licate interpretation of isomer shift s of 5 7 pe because the 4s character of the bond increases t he electron density at the nucleus w hile non - s character decreases it . Nevertheless , assi gnment of oxidation state for iron in oxides and ot her ionic compounds is usually straight forward . The second spectral parameter is q uadrupole splittin g ( Q S ) . This e ffect remove s t he magnitude but not the sign degeneracy o f the nucle ar spin st ate s . I n 5 7 Fe , for example , the I 3 / 2 excited state is split into 1 / 2 , is un lz ±3 / 2 and l z ±1 / 2 states . The ground state , with I split . A quadrupole split spectrum for 57 Fe in an isotropic powered sam ple has t wo lines of equal density ( Greenwood and Gibb , 1 97 1 ) . T he si ze of the splittin g depends on the n uclear q uadrupole moment ( a constant ) and the electric field gradient ( EFG ) at t he nucleus . The EFG is a meas ure o f the sym metry o f t he electronic environment of t h e ato m . I t is zero for cubic and tetrahedral symmetry , b ut increases in magnitude for more hi ghly asymmetric arran ge ments . There are two important contributions to the EFG . T he lattice component arises from t he surroundings of t he atom . The electronic component is a result of t he valence election config uration o f t he atom itself. Hi gh - spin Fe 3+ has an electron EFG of zero . T he extra d electron on Fe 2+ causes a stron g , temperat ure -dependent quadrupole splittin g w hen 57 Fe is in a low - symmetry environment ( Wertheim , 1 96 4) . T he combined effects of isomer shift and quad rupole splittin g are shown in Fi g . 1 7 , w hich illustrate s oxidation /reduction of Fe / Y - zeolite ( G arten et al . , 1 97 0) . S pectr um 17a shows two ferrous states , with spectra correspondin g to a lar ge o uter and sm aller inner doublet . Spec trum 1 7b corresponds to Fe 3+ and spectrum 1 7d to ferrous ions whose bonding to the zeolite has been w eakened by hydration . When the net electron spin o f an atom is greater t han zero and is fixed in space lon ger than 1 0- 7 s , the nucleus feels an effective magnetic field and all t he ( 21 + 1) nucle ar sublevels are split . For 57 Fe , the selection rules allow six transitions with relative intensity 3 : 2 : 1 : 1 : 2 : 3 for an isotopic powder samp le ( Greenwood and Gibb , 1 971) . The six- line , magnetic dipole splittin g is characteristic of iron in any m a gneticall y ordered material . T he sp littin g of t he outer lines is proportion al to the e ffective magnetic field at the nucleus , a parameter often useful for phase identification ( Tops�e et al . , 1 97 3) . B ecause the effective magnetic field is meas ured at the atomic site , anti ferro and ferri - as well as ferro magnetic states are detected . Di fferent crystallographic site s in a given =
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Cataly tic Surfaces 1 . 02 1 ,0
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FI GURE
2+
1 96
Delgas s and Wolf
m aterial can also have different effective m a gnetic fields . T his is the case for x - carbide ( F e 2 . sC ) w here three different sites produce 1 8 overlapping lines . W hen t he volume of a m a gnetically ordered crystallite becomes s m all enou gh , t herm al ener gy , kT , m ay be sufficient to flip t he entire coupled spin system fast enou gh to cancel out t he m a gnetic field at t he nucleu s . T he Mossbauer effect shows this conversion from a magnetic to a sup e r p aramagnetic state w hen p article di ameters are of t he order 30 to 1 50 A , dep ending on the m aterial and the temperature o f t he measurement ( Morup et al . , 1 980) .
Exp eriment
In the typical Mossbauer experiment , y rays pass from the source , t hrou gh the catalyst absorber wafer , to the detector . T he source is mounted on an electromechanical drive w hich imparts a back - an d - forth motion with a carefully programmed velocity alon g the y - r ay p at h . The velocity c auses a Doppler shift in the ener gy of t he y ray an d is thus the means of scan nin g ener gy . A periodic t rian gular velocity versus time function is com mon . T he Doppler shift correspondin g to velocitie s ran ging from - 10 to + 1 0 m m / s is sufficient to scan t he full ran ge of 5 7Fe s hi ft s and split tin gs . T he detected count rate ( num ber of counts in a fixed dwell time ) is collected in a multichannel analy zer or minicomputer w hose repetitive sweep is synchro nize d so that each channel corresponds to a given source velocity . B y convention , spectra are plotte d as p ercent transmission vers us velocity , with positive velocity indicating motion of the source toward t he absorber . Sources must have a stron g , narrow , sin gle emission line . T hey con sist of the parent of t he Moss bauer isotope well bonded in a cubic m atrix and are available commercially . 5 7 co diffu sed into a Rh foil is typical for 57Fe e xp eriment s . T he detector s are usually proportional or scintillation counters , but solid - state devices are also used . F ull systems , including· the drive , ali gnment track , detector , data accum ulation , and output device , and all associated electronics are com merci ally available . A b sorber cells for control of t he cat alyst en vironment are also available , b ut are o ften custom m ade ( Delgass et al . , 1 97 6 ; S hen et al . , 1 9 8 1 ; C lausen et al . , 1 97 9) . T he cell i s a vacuum - ti ght enclo sure with facility to he at and cool the catalyst w afer in a selected gas environment and B e windows for transmis sion of the y rays . Special dewars are necessary if both source and ab sorber must be cooled to liquid - helium temperature , as for 9 9Ru ( C lausen and Good , 1 9 7 7 ) . In the transmission mode discussed above , the optimum sample si ze is 10 1 8 atoms o f 5 7 Fe atoms per cm2 for e ach line in the spectrum . A 5 / 8in . - diameter wafer typically contains 0 . 1 to 0 . 8 g of c at alys t . T hi s value gives signals in the transmission ran ge 90 to 99% , bi g enough to see , b ut not so big as to c ause excessive broadenin g due to reabsorp tion (Mar gulies and Ehrm an , 1 96 1 ) . I ron loadi n g below 1 wt % usually require use of i ron enriched in 5 7 Fe , available from O ak Rid ge N ational Laboratories . T he time required to obtain a s pectrum depends on the stren gt h of the source and varies from 10 min to 2 4 h , with 4 to 6 h bein g common . Transient effects are best followed by op er atin g t he spectrom eter in the constan t velocity mode and followin g the intensity o f a given peak a s a function o f time ( R aupp and Delgas s , 1 97 9) . An alternative to the approach j ust described is t he source mode . Here the catalyst m ust be made with t he radioactive p arent isotope . It is
C ataly tic S urfaces
197
t hen placed in a reac tion /pretreatment cell and used as the y - r ay source while t he drive move s a standard sin gle -line absorber to scan energy and generate the spectrum ( Tops�e et al . , 1 981 ) . S urface sensitivity of the Mossbauer ef fect can also be enhanced by usin g the backscattering geom etry. Phillips et al . ( 1 9 8 0 ) have used this m ethod to follow the decomposi tion of Fe ( C0 ) 5 on oriented graphite foils .
Applicatio ns
T he Mossb auer effect is a seasoned tool in c at alysis research and accounts for advances in t he understanding of bulk solids ( Top�e et al . , 1 97 3 ) , sup porte d metals an d alloys ( G arten , 1976), oxides (Lund and D umesic, 1982), sulfides (Top s�e et al . , 1 981 ), carbides ( A melse et al., 1978; Niem ant sverdriet et al . , 1 980), catalyst genesis and activation (Boudart et al . , 1977), oxidation-reduction behavior ( Garten et al . , 1 97 0) , support interactions ( T at archuk and Dumesic, 1 981 ) , and the dyn amics of some catalytic processes ( R aupp and Delgass, 1979). I n decidi n g what inform ation the Mossbauer effect can provide about a c atalyst , one m ust bear in mind that the radiation penetrates the sample completely and, generally , all Moss bauer atom s in t he sample contribute to the spectrum . As the fraction exposed ( dispersion ) o f the p hase contain in g the Mossbauer atom increases, the Mos sbauer effect becomes more sen sitive to the surface . S urface sensitivity is prob ably best c hecke d experi ment ally by observin g t he spectral perturbations caused by adsorbed gases. A few speci fi c examp les of Moss bauer studies are outlined here . G ar ten ( 1 97 6 ) , Garten and Sin felt (1980), an d Vannice e t al . ( 1 97 8 ) , have shown that iron form s bimet allic clusters on Si02 and Al20 3 with a variety of met als, includin g P d , Pt , Ir, and R u . Lund an d Dumesic ( 1 982 ) , have o bserved a sup port interaction between Fe and Si02 that involves Si02 migration over t he iron p hase , and Tatarchuk and D umesic ( 1 981 ) , have reported one between Fe an d Ti02 involvin g dissolution of Fe in t he Ti02 p hase . T he role of S n in a s upported PtS n steam dehydrogenation catalyst has been carefully detailed by G ray and Farha ( 1 97 6 ) . The recent 57 Co source experiment s of Top s� et al. ( 1 98 1 ) use t he Mossbauer p arameters to identify a mixed CoMo sulfide as the active site for hydrodesul furi zation . It is o f interest in this study t hat 57 Fe - doped into CoMo cat alysts doe s not probe the correct cobalt chemistry ( C lausen et al . , 1978). Fi gure 1 8 illustrates in formation obtained on sup ported iron - cobalt alloy cat alyst s by S t anfield and Delgass ( 1 9 8 1 ) . Spectra A to C show nearly complete reduction o f iron since paramagnetic Fe 2 + or Fe 3+ states would contribute extra spectral intensity in t he re gion 0 to 2 mm . The chan ge in splittin g of the o uter lines s how s t hat t he e ffective magnetic field fol lows t he expected composition variation and t hat iron-cobalt alloys wit h t he nomin al composition have been formed successfully. Other spectra show that improper pretreatment can comp letely prevent alloy formation . Spectra D to F show t hat increasin g Co concentration supresses carb uri zation of t he catalyst durin g Fisc her-Trop sch synt hesi s . The comp lex s pectra of the carbides lie bet ween -4 and +4 mm/s . Constant - velocity d ata taken in situ at the velocity of the rightmost Fe O line can give a dynamic trace of the amount o f carburization as a function of time .
Delgass and Wolf
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FIGURE 18
Room-temperat ure Mossbauer spectra of re duced and reacted vacuum - d ried c atalysts: (A) lOFe/Si0 2 , reduced 8 h; {B) 8. 95 Fel. 0 5Co/ Si02, reduced 8 h; ( C ) 7 . 4Fe 2 . 6Co /Si0 2 , reduced 8 h; (D) I OFe / Si0 2 , re acted 6 h; ( E ) 8.95Fel. 0 5Co/S i0 2 , reacted 6 h; ( F ) 7 . 4Fe 2 . 6Co/Si0 2 , re acted 6 h. ( From Stanfield and Del gass , 1 981 . )
M a gnetic Resonance
N ucle ar magnetic resonance ( NMR ) accurately me asures the properties of a nuclear spin system . To an even gr e at e r extent t h an with the Mos s bauer effect , the se nuclear properties c an re flect detailed chemical structure and b on ding in form ation. Electron spin resonance ( ES R ) m e as ures the energies of the valence electron spin s y stem directly and th us provides a sensitive meas ure of the chemic al s tat e of species having a net electron spin. In bot h meas urements an applied m agnetic field orients the spins, and transi tions between spin states are ob served by absorption of electro m a gnetic
radiation . NMR ab so rpt ion s are in the radio- frequency region, w hile E S R is i n t he microw ave .
Catalytic Surfaces
199
The high sensitivity of these two magnetic resonance techniques to subtle chemical changes makes the methods at onc e potentially powerful probes of chemical reactivity and techniques demanding thorough under standing of the underlying quantum mechanical effects in order to extract the full measure of information available. ESR has a long history of ap plications in catalysis. It has been successful in studies of the environ ment of paramagnetic transition metal cations in catalys ts, in observing the chemical properties of electrons trapped in defects in oxide catalysts, and using the paramagnetic probe molecule NO, in examining the interactions of adsorbed molecules with surfaces (Lunsford, 1972, 197 9a ) . Recently, ESR has been particularly important in identifying and characterizing o- and other oxygen anions on catalytic surfaces (Che an d T enc h , 1982) and in examining mobility of adsorbed species (Howe, 1982). NMR has made significant c ontributions to catalysis research both in the identification of chemical species and -in measurements of species mobility (Lunsford, 1979b). M agic angle spinning (Andrew, 1981) and other recent advances in solid-state NMR make this technique one of the most promising for future catalysis research. High-resolution 29si NMR, for example, can quantitatively measure the n umbers of Si ions with zero, one, two, t hree, or four AI next -nearest neighbors and thus has provided new understanding of Si-Al ordering in zeolites (Ramdas et al., 1981; Magi et al., 1981; Nagy et al., 1981). A complete review of magnetic resonance is well beyond the scope of this chapter . Our main purpose is to point out some of the most recent developments in NMR that suggest that an investment of time to understand magnetic resonance tools may be worthwhile.
Principles
The magnetic resonance effect is eas iest to vi s uali z e for a spin-1/2 s ystem. In the presence of a magnetic field , the magnetic moment associated with the spin will precess about the magnetic field direction. This motion is quantized an d for a spin 1/2 system only two z components of the spin vec tor are allowed, the +1/2 and the -1/2 states. The energy of the spin s ystem in the presence of the magnetic field is proportional to the product of the size of the field and the z component of the spin. Thus, as the magnetic field increases, t he energy separation between the +1/2 and -1/2 states increases proportionately. At a given magnetic field s trength, resonant absorption of energy will occur when t he frequency v of the ap plied elec tromagnetic radiation is s uch that hv is equal to the energy sepa ration between the +1/2 and -1/2 states, as shown in Fig. 19. In real systems spins are not isolated in free space, but reside in atomic or nu clear environments which are generally not isotropic and can provide a variety of perturbations that affect the energy of the system. In general, one mus t write down a spin Hamiltonian that describes the appropriate interactions and use it in fitting the data obtained . In ESR the proportionality between the electron s pin and the magnetic field is indicated by the g tensor . This tensor has three principal values, which deviate from the free electron value because of the interaction be tween the electron spin and orbital motion. Account must als o be taken of the fact t hat the powder samples typical of catalysts have crystallites at all possible orientations with respect to the applied magnetic field (Lunsford, 1972). Hyperfine interactions, the coupling of electron and nuclear s pins, caus e additional splittings in ESR spectra and can be
Delgass and Wolf
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FIGURE 20 First derivative ESR spectrum of 0 2 on t he surface of hi gh surface-area MgO . T he field increases from left to ri ght and the gain has been reduced five fold for t he central line ; a small portion of the high field spectrum is overmodulated to show t he outermost lines . For clarity no attempt has been made in this diagram to insert levels for t he lines aroun d g1 . ( From T ench and H olroyd , 1968.)
201
Catalytic Sur'faces
help ful in d efinin g the en vir onme nt of an electr on spin. As shown in Fig . 20, the i s ot ope 17Q, w ith a nucl e ar spin of 5/2 shows 6 line spli ttin g s for 17Ql6o b u t 11 line s pli t t in gs for 17o2 ( Ten ch and Holroyd, 1968). Note that the s p e c tr um is di s p layed as the derivative of the energy ab sorption curve, as is standard in ESR spectroscopy. NMR has s en si tivi ty to chemical b ondin g through shieldin g effects. The more free electr ons around the nucleus of i nt erest are to circ u late in response t o t he app li ed m agnetic field, the more they decrease th e st ren gth of the field felt by the nuclear spin. The ran ge of chemical shifts in duced by this shie ldi ng is ge n erally large with respect to NMR line widths in liquids , giving t he method high chemical s e nsitivity. Spin -spin cou pl in gs betw ee n n onequivalent nuclei als o occur, giving NMR furt her c ap abil i t y for de fi nin g the chemic al en vir on ment of a give n type of nuc leus . In li qui ds , many interactions are a vera ged to zero because of the rapid m o tion of the molecules. In solids, however, rel axation times are long and direct - dipo le co up ling of the s pin s and anis ot rop y of the chemical shift t e n s or both tend to broaden lines to such an extent that meanin gful chemical s hi ft information An alysi s of t hese two in cannot be obtained from a standard s pect rum . ter actions s hows that if t he sample is s p in nin g on an axis 54°44' to the ma gneti c field direction, t he static components of the dipole-dipole inter action and the c he mical shift an i sot r op y both b ec o me zero (Andrew, 1981).
S i d eba n d s , whose p osi t i ons depend on rotational frequen c y , still appear but do not obscure t he s pectrum . T his tec h ni q ue , called magic angle
spinning, drama t i ca lly improves the resolution of NMR of solids (Andrew,
1981). Further impr ovemen t s can be made wit h spin decouplin g , c ro ss polarization (Yannoni, 1982), and m u lti p le p ulse te c h ni ques ( Tay lo r et al., 1980; Ryan et al., 1980). In the p ast, NMR ha s been a p o we rful tool for c h e mi c al st ru ct u re a n alys is . The potent ial of the new NMR method s is to bring s u c h detailed s tr uc t ural information to the study of catalytic solids. NMR App licatio ns
Magic angle s pi n nin g NMR has provided a vari ety of new o p p o rtunities for stu d yi n g the d et ai le d structure of zeolites. As shown in Fig. 21 , 29Si NMR cle a rly resolves Si with zero to four Al n eigh b ors bo n ded th r ou gh oxyge n bri d ges . In this work by R amd as et al . ( 1981 ) , the distribution of AI neig hb or s a round Si wa s st u di ed as a function of Si/Al r at i o for synthetic faujasite s . They found that Loe w e nstei n ' s rule forbi d din g AI atoms from occupying nei g h bori ng tetrahed ral sites was obeyed . As t he Si/Al ratio approached 1, the NMR spectrum became a single line cor res p ondi n g four AI nei g h bo rs around Si. By analyzin g t he relative in tensit ie s of lines as a fun c tion Si/Al ratio , the authors const r u c te d an or d erin g scheme for Al pl aceme n t in the fa uj a si te lattice . The on se t o f pa r a p osi tionin g for Al in the hexagonal ri ngs facing the superc ages was found to begin at Si/Al r at io of a b out 1 . 4. The simulation in Fi gu re 21 is based on thi s structural m od el. It is inte r esting to note that Loewenstien's rule appears to be broken for zeolite A ( B ursi ll et al., 1980). T ho mas and c ow orkers have also used solid - st ate NMR to show the s i mi la ri ty of ZSM-5 and Silicalite (Fyfe et al., 1982). Nagy et al. ( 1981) have observed that the chemical shifts for ZSM- 5 and sim ila r zeolites are more ne gative t han those o f the fauj as i tes and have ascribed that dif f e re n ce to the presence of the fi ve - membered rings. In another study,
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FIGURE 21
li[p.p.m.l
100
110
Observed ( at 79.6 MHz) ( upper) and simulated (lower) high resolution 29si NMR spectra with magic an gle spinnin g for a Si/Al ratio of 1. 35 in synthetic fauj asite . ( From Ramdas et al . , 1 98 1 . ) Nagy et al. (1982), u sed cross polari zation ( couplin g of t he proton and 29Si resonances ) to identify silanol groups in the spectrum o f ZSM- 5. Alt hou gh magic an gle spinnin g is certainly important t o applications o f solid- state NMR to catalysts , it is not always necessary. Stichter ( 1981 ) has reporte d a very interestin g set of 195pt NMR experiment s on Pt/Al203 samples supplied by Sinfelt . T he data were obtained by t he spin echo technique. T he lines observe d were very broad , about 4.5 kG, the ran ge of che mical shift between diamagnetic Pt compounds and Pt metal . T he large shift o f metallic Pt , called the Knight shift, arises from the polari zation o f t he spins o f t he conduction electrons . Analysis of the peak shape as a function of Pt dispersion ( fraction exposed) shows t hat the amount of diam agnetic Pt is directly related to dispersion . T he spectra were interpreted to indicate that adsorption of gases on the sur face ti e s up the otherwise free electrons of. t he metal and e ffectively demetalli zes the s urface atoms. Direct observation of such fundamentally important p henomena can h ave a great imp ac t on our understandin g of catalysis . Fourier trans form 13c-NMR o f species adsorbed on catalyst surfaces has also been effective . Nagy et al . (1982) have been able to follow the isomerization of 1- butene to 2- butenes on a tin- antimony oxide catalyst. Gay ( 1 974) ob served a variety of molecules adsorbed on silic a and ex amined their mobility . Cirillo et al. ( 1980) h a ve used proton NMR to
203
Catalytic S urfaces identify three types of low- te mperat ure molybdenum/alumina catalysts.
H2
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on alumina and
DISTRIBUTION OF CATALYTIC MATERIALS WITHIN CATALYST PELLETS:
ELECTRON MICROPROBE, SCANNING ELECTRON
MICROSCOPY,
AND ANALYTICAL MICROSCOPY
n
tant y i and no n main tools
of materials ins t poisoning due to metal such analysis are the l ng
A impor variable in the performance a catalyst is the distribution of catal t c nc atalytic ide the pellets. The latter is significa t in the case of ca talys deposit o . The now available to provide e e ctr n probe microanalyzer (EPMA), scan ni electron microscope ( SEM) , and scanning transmission electron microscope (STEM). The principle of oper tio n in all three cases is the but the EPMA is designed to ve best resolution for chemical a alys s , whereas the electron microscopes are designed to high and good p a and depth resolution, with chemi cal analysis b i g added capability. The SEM p r od es two-dimensional e ect on backscattered from the material instead of from elec trons transmitted through the material, as is the case in T EM . A STEM w th sca ni g capabilities and also be used for chemical yno p i that follows will briefly describe the principle of emission for chemical i these instruments and then present description of scanning electron microscopy. A brief co pari the EPMA SEM presented. Fi ally, applications t cat i
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n
o
Chemical mi c roa al y is is carried out by an y is of the energy x rays emitted when a sample is excited by electron i pact or x rays. The phenomenon has already been referred to p re vio s sections in connec tion with sources for x-ray diffraction and photoelectron e s io . x rays are used as the xcitati n so rce , the ec niq e is .r-ray fl uores ce nce analysis. S uch instruments are pr ril y for chemical l ys s s mples because x ys are if icu lt to and there fore the spatial e l tio as good w he electron impact is the excitation source. Electron beams can be focused to diameters fro 25 1000 A. F rtherm re , since electron beams are the primary excitation in e lectron icros op y, x-ray analysis constitutes a natural additional capability for such i stru e ts . Figure 5b shows that w he e lectr is remo e inner shell of an be filled by an electron an outer shell. The energy be emitted as an x-ray phot . Deexcitation by A ger electron emission can also occur. Since levels are quantized, it fo l ow that the energy of the emitted x-ray pho t n is discrete and is characteristic of the e le me t which it e it ed . Consequently, n w e ge the emitted x-ray energy be used for chemical characterization of the ele en . If in the K the L level , photon is designated as Ka (Ka Ka2 exist
Principles of X -ray Emission or X -ray Fluorescence
n
s
al s m in
e
ana
o
u
i of bulk a r so u n is not
ra
as ranging
m
to
u
source
of
u
mis n t h u called used ima d f focus as n
o m
When
used
c
n on
m n n an v d from an atom , creating a vacancy , it may from released by the transition may on u the energy l s o also a quantity n from was m t k o l d of can m t the vacancy occurred level and the transition occurs from the and due 1 to differences in spin ). Such emission gives rise to characteristic lines
Delgass and Wo lf
204 s up erimp os ed on a co ntinuo us spectrum.
The characteristic lines produce quasi-monochromatic x rays which are us ed as the source in XRD . W hen x ra ys are used as an ex citation source (x-ray flu or es cence ) , no con ti nuou s spectrum is observed, and the background is very low, which is esp ecia l ly suited for energy ana lysis . The additional pro cess es res ultin g from e le ctro n impact are shown schematically in Figure 22. S eco n dary and transmitted electrons are used to produce im ages in SEM and TEM, respectively. Auger electr ons are used to obtain s urfa ce chemical information. Energy analysis of the char acteristic x rays also provides such information. Also shown in Fig. 22 is the depth of escape of the v ario us deexcitation processes. It can be seen
that the escape de pth of x rays is on the or der of micrometers ; there fo re , they do not con t ain surface information (as in the case of Auger elect ro n s) . X -ray em iss ion is genera ted in a v o lume of about 1 \1m3 , w hich makes x-ray analysis a m ic ro analytical tool . (a) w ave le ngt h dispersive spectrometry
Two methods are us ed to analyze the energy of the emitted x rays: (WDS), and (b) energy dispersive
spectrome t ry (EDS). In the former met hod , the energy of the x ray is obtained from knowing its wavelength (E = hv hc/A). A is determined usi ng a crystal of known crys tal spa ci n g which diffracts (disperses) the inc oming x rays, allowing only one wavelength element to reach the de The pri nci ple is the s ame as d es cri bed in XRD, a nd Bragg's law tector. is used to determine A . Energy dispersive analysis (EDS) i s a cco m plishe d by u si ng a semiconductor crystal that absorbs x rays, thereby exciting the crys t al ' s electrons into the c ond ucti on band, producin g ari electr ic pulse proportional to the energy of the x ra ys . These pulses are then counted and sorted in a multichannel a nalyz er , and a displa y of in te nsity of the signal versus its energy gives the desired spectrum (see Fig. 23a). The advantages and di sa dva nta ge s of these two· types of x-ray energy analyzers are s umm ari ze d in Ta b le 4. =
Incident Beom BSE
1!t
CXR
Surface
AUQer Electrons (AE)
lf""�+-B -+--�1"-- ock Scollered Electrons (8SE) lmicron
Characteristic X- roy (CXR) X- roy Continuum (XRC)
FIGURE 22
S c he m at ic of various signals originated by electron impact,
showing the co rrespo ndin g volume of p rimary e xc it a tio n .
205
Catalytic Surfaces
CoKa
Si Ka
> t-
ii) z I.LI t z
ENERGY
�
(o I
jj;
0 0
.., ....
.., ..... 0'
e
z Q
� a:
tz w (.)
z 0
LARGE PORE, 8v
= =
0.15 0.20
0.08
i5
0
�
SMALL PORE, 8y
0.12
0.04
;:::)
•-
0.16
(.)
�
o-
0.20
0.6
0.8
1.0
FRACTIONAL RADIUS
0.4
( bl
X-ray spectrum
FIGURE 23 (a) of a sample containing Si, K , C o, and Ti obtained by EDS. (b) Vanadium concentration profiles on a 1/16" extrudate catalyst during hydrotreating of a heavy oil residuum. Effect of pore di ameter. (From Tam m et al., 1979.)
TABLE 4
C omparison of EDS and WDS
C omp act : low cost Rapid (qualitative analysis) Simultaneous multielement analysis of the fu ll x -ray spectrum (Na upward) Display of the entire spectrum in digital format
A dvantages of energy dispersive spectroscopy
Low sensitivity to geometric effects High collection efficiency
Lack of higher-order lines which are generated in crystal diffraction Digitally produced outputs for element line scans and distribution maps Advantages of wavelength-dispersive spectroscopy Higher inherent element separation (resolution ) High count rate on individual elements Analysis can be highly quantitative B etter peak-to-background ratios An alysis of wide range of elements (Be to U)
a
Higher sensitivity Operates at room temperature
206
Delgass and Wolf
Instrumentation
A brief discussion of the major components of the various instruments used in electron probe microanalysis and SEM is given below; several books (Postek et al., 1980; Goldstein and Y ako w it z, 1975; Thornton,
1974)
1968) and
review articles (Sargent and Embury, 1976; Johari and Samudra,
on
SEM, and EPMA (Heinrich, 1980; Purdy and Anderson, 1976) are available for a more detailed description. Fi gure 3 shows the schematics of a TEM, an electron probe microanalyzer (EPMA), and a SEM. These instruments are equipped with a system to produce and focus an electron beam and with a system for detection and analysis of x rays.
The EPMA is designed primarily for x-ray analysis, with
an electron-beam intensi ty which optimizes the x -ray yield and a low-magnifi cation optical microscope for viewing the sample.
100, O O Ox
The SEM, on the other
A.
hand, is designed to obtain images which, under ideal conditions, can be magnified in the 20x to Because
of
tail in the
range with resolutions of 70 to 100
the greater ve r sat ility
followin g
of
the SEM, it is described in more de
paragraphs.
The SEM differs from the TEM in that ima ges are produced from second ary electrons generated by the
electron
beam.
Thus the sample need not
be transparent to electron transmission as in TEM. a
volume
of primary excitation (see Fig.
Electron impact creates
22) from which elastically scat
tered ( backscattered) a nd inelastically scattered electrons or secondary electrons are emitted.
Secondary electrons have low energy (less than
50 eV) , and therefore only those electrons near the surface can be emitted and detected.
Secondary and backscattered electrons are emitted outward
in all directions.
An electron detector system is used to collect both
types of emitted electrons and produce a signal proportional to their in The detection of secondary electrons depends on the sample
tensity.
topography.
Electrons emitted from the specimen surfaces facing the de
tector will show a bright face, whereas those emitted from surfaces facing away from the detector will not be detected, creatin g a shadowin g effect. Furthermore, as the name indicates, an SEM is equipped with deflecting coils which permit scanning the electron beam over the sample in a raster mode.
The raster pattern of the primary electron beam is synchronized
with the scannin g pattern of a cathode ray tube (CRT) used to display the detector signal so that a
produced .
t wo-dim ensional
Images are produced in the CRT
records can be
obtained
image of the scanned area is
or
a TV, and permanent
directly from the CRT usin g an instant camera or
by videotape recording from a TV.
Of the two types of x-ray energy analyzers, WDS is the preferred mode of analysis from the standpoint
ly
of
x -ray spectroscopy and is used exclusive
in x -ray spectrometers as well as on EPMA instruments.
WDS analyzes
one wavelength element at a time and requires mechanical components to rotate the crystal as
well as
electrical components.
Crystal alignment is
quite important, and WDS is difficult to use with rough surfaces. analyzers are used in most SEMs and in some electron probes.
of and
EDS
EDS pro
vides a faster and easier survey analysis
the elements but at lower
energy resolution and lower sensitivity,
it does not cover the entire
spectrum, especially of light elements such as carbon.
Clearly, for
versatility one would like to have an SEM capable of usin g hi gh beam intensities such as in EPMA an alyzers.
and
equipped with both WDS
and
EDS
For hi gh-resolution microscopy work or x-ray spectroscopy
work, separate instruments are preferred.
Figure 23a shows the x-ray
207
Cat alytic S urfaces
spectra of one spot on the sample. However, the be a m can be scanned over the sample with the analyzer fixed on a certain energy so that only x rays cor respondin g to a sp eci fi c element are disp l ayed This mode o f .
oper ation pr o duces an x-ray m ap of the element in ques t io n over a line Another w ay of o b t ai ni n g elemental ( line scan) or area under an a lysis distribution is to move t he sample under the fixed electron beam. This technique is particularly useful when analyzin g catalyst pellets because it per mits one to scan t he entire width of the pellet. For electron m i c ro analy si s the sample m ust have a flat s ur fa c e . For powders, the sample is first embedded onto a p last ic medium, w hich , afte r drying , is ground to reveal the cross section of its i nter io r . Con duct ive coatin gs ( C, A u) are evaporated onto the surface of nonconductive materials (Purdy and Anderson, 1976) . For SEM, the sample is mounted on a small stub coated with adh e siv e wh i ch ho l ds th e par t ic les as received .
,
or after fracturin g to re v eal t he interior.
Speci fic applications of SEM and E PMA to ca t al ysis are di scussed by
Applicatio ns
Sargent and E m b ury ( 1976) and Purdy and Anderson ( 1976) . SEM is used in general for ph ysica l c haracterization of c at al yst particles, s uch as particle size, sh a p e s urface s tr ucture and poro s it y makeup of aggregates , a nd dispe rsio n of one phase into ano ther The practical m a gnif i c ation and ,
,
.
resolution of the SEM is r el a t iv ely low compared w ith the TEM and STEM, which are pref erentially used for e x am i nin g crys tallit es in the size range 1 to 10 mm. X-ray em issio n cap abilities of the electron microprobe or SEM instru ments are used to obtain bulk chemical information about the materials present in th e s ample . The most import ant ap pli cat ion is to me asure the distribution .of catalytic materials or i mpuriti es over the cross s ectio n of c at al yst pellets. A cl assi ca l example is t he effect of catalyst p rep a r a tion in the dist ri but ion of the active catalyst inside pellets . The metal profiles insi d e the pellets can be followed as a func ti on of preparatio n c onditions Dep en di n g on the m etho d used t h e c at alyst can be depo sit ed u ni for ml y inside the pellet , or it can concentrate toward the external face of the p el let Extension of this application will increase o ur understanding of catalyst p repa rat ion. Another classical ap p li catio n is the st udy of metal d eposition oc c u rri ng d ur in g dehydrodes ulfurization of he a vy residuum (Tamm et al., 1979). Fi g u r e 23b shows the V cont ent of a catalyst as a fu nction of po sitio n in side th e p el le t , a s o bt ain ed us in g an EPMA . The p o is on i n g of catalyst s due to impurities is an area of study in which EPMA and SEM are pa rti c ular ly useful tools . .
,
.
ADSORBED SPECIES ON SURFACES Infrared Spectroscopy
Unlike the surface a n a lysis t e c hniq ues based on electron spectroscopy, i n fr ar e d (IR) s p ec troscop y is a well se asone d tool for catalytic research, as witnessed by the v a rio u s books (Little, 1976; Hair, 1967; Clark and -
Hester
P lis k in
,
,
1974-1982) and review articles (Del gass et al . ,
1958) published on the s ubj e ct
.
Eichens
IR spectroscopy was firs t 1979;
and
Delgass and Wolf
208
developed to study t he chemical structure of bulk compounds, but it was soon re alize d that it can also be used to gain s tru ctura l in fo rmation abo ut s p eci es adsorbed on solid surfaces. The books by Hair ( 1967) and Little ( 1 976) contain a good introductory description of IR to get he r with m any ap The review article by Eichens and Pliskin ( 1958) has withstood plications. the test of time and remains one of t he most quoted re f erenc esThe more recent review arti cle s (Delgass et al., 1 979; Basila, 1 96 8; B lyhold er , 1968; Pritchard and C atte rick , 1976) e mp h asi ze newer developments. The descrip tion give n here cov ers only the most fundamental aspect s of IR spectroscopy and its maj or applicatio n s to cataly ti c research . IR spectrosc opy gives qualitative information abo u t the way adsorbed molecules are bonded to sur fa ces , as well as structural information on solids. It can also be used to measure the amount of materials some case s it can be used to stud y the ra te at which certain surface proce sses occur.
adsorbed, and in
Principles of Vibrational Spectroscopy When an atom or molecule is p laced in an elec tro ma gn et ic field , there is a tr ans f e r of ene rgy between the field and the quantized energy leve ls of the mo l ecu le s . When energy is adsorbed , me asurement of the transmitted energy versus the frequency of the radiation produces a spectrum uniq ue to the excited molecule. If the frequency of th e incoming radiation is in the IR range , the spectrum is due to t ran siti ons in vi b ration al and rotation al energy
levels. In each electronic energy level there is a series of vibrational transitions, and for each vibrational level there is a series of rotational Transition in electronic ener gy levels occurs when t he excita transitions. tions have fre quenci e s in the UV or visible rangeSelection rules govern t he perm i ssibl e transitions amo ng the various energy levels.
Diatomic Molecule: Vibrational spectra arise fro m the motion of one atom relative to another. While a detailed group the or y has been developed to describe the phenomena, a simple explanation in whic h a diatomic mo lecule is mod ele d as an harmonic o s c illator is often adequate to describe the first ord er processes involved. The energy and f re quenc y of an ele ct r omagne ti c wave are related by E = hv, where his Planck's c o nst an t . Since AV c, where c is the s peed =
of li ght , it has be co m e a common p r acti ce to r ep ort spectra in terms of frequency i ns te ad of AThe unit most commonly used is the wa v e n umber 104/).. (micrometers ) , which has units of cm-1. The IR spectrum is divided in t o near IR, 10,000 to 5, 000 cm- 1 (A: 1 to 2 ]..lm); mid IR 5000 to 200 cm- 1 (A: 2 to 50 11m), and far IR, 200 to 10 cm-1 (A: 50 to 1000 IJID). Solving the classical mec h anical eq ua tions descri bin g the oscillations of two masses, m 1 • m2 , attached to a perfect spring which obe ys Hooke's law, one can calc ul ate the change in potential en ergy associated with the oscilla t io n s as =
u
( 21)
where k is the force c onst an t that measures the stiffness of the spring and r is the displacement from the center of mass. The motion of this system is sinusoidal and the fre quency is given by v = ( 21T)-lv'k/ll, where 11 = m m /Cm + m ) is the reduced massThe simple result of the harmonic 2 1 2 1
209
Catalytic Surfaces
oscillator has immediate practical applications in the interpretation of spectra. Knowing v, one can calculate the force constant k; conversely, changes in v upon adsorption reflect weakening (lower v) or strengthening (higher v) of the bond. Another application of the result obtained for the simple diatomic oscil lator is in connection with changes in frequency due to isotopic substitution. If the force constant remains the same, then upon isotopic substitution of species with reduced mass � 1 and a frequency v1 by an isotope of reduced mass �2, a frequency change predicted by
( 22)
should be observed. Failure to observe a frequency change indicates that the assumed species or group is not involved in the vibration being ob served, and another structure must be postulated [see Delgass et al. (1979) The potential energy and Hair ( 1967) for illustrations of this application]. diagram described by Eq. ( 2 1) is similar to the lower region of the energy diagram of a real diatomic molecule. For this reason, the quantized form of the harmonic oscillator analogy is adequate for a simple description of vibrational spectroscopy. To account for quantum effects, the potential energy function of the harmonic oscillator is substituted into Schrodinger's wave equation. The solution of this equation gives the energy of the quantized vibration as ( 23)
E
where v is the vibrational quantum number. Certain selection rules are used to ascertain the probability of transitions between energy levels. For ±1 are permissible. harmonic oscillators, only transitions between 11v Since at room temperature most molecules are in the ground state, vibration al transitions between ground state and the first energy level are usually observed. Another important selection rule requires that a change in dipole moment with vibration must exist in order to couple the vibration to the electromagnetic radiation. In the absence of such a dipole moment change, as in the case of linear diatomic molecules containing the same atom (i.e. , N 2> , no vibrational energy transitions are observed. A corollary of the foregoing selection rule is that IR inactive molecules may give rise to dipoles when they are adsorbed on a surface and therefore exhibit an IR spectrum. Rotational transitions often occur simultaneously with vibrational transi tions. The combined vibrational-rotational energy relationship is of the form =
E
=
( 24)
where I is the moment of inertia of the molecule and m is a rotational quan tum number that can have values ± 1, ±2, and so on. The center of the vibrational-rotational band is at v0; thus the second term in Eq. (24)
210
Delgass and Wolf
defines t he rotational fine structure . Dependin g on w hethe r m i s positive or ne gative , two branches, known as P an d R br an c hes , are produced in the hi gh - and low - frequency sides of vo. Another branch , known as the Q branch, may app ear in t he vicinity of v 0 due to changes in angular momentum about the axis j oini n g the nuclei or d ue to unpaired electrons . Fi gure 24 s hows , in addition to adsorbed CO, t h e hi gh-resolution absorb ance spectrum of gaseous CO, depicting the R and P branches resolved into specific lines due to rotational transitions. Lower resistance spectra will sh ow only two broad bands about vo. For CO ad s orbe d on a m e tal s urface, the rotational fine structure disappears and give rise to a single band.
Poly atomic Molecules: Diatomic molecule s exhibit only one vibrational mode , but as more atom s ar e added to a molecule , additional vi br ations are p ossi ble . For a n onli n e ar poly atomic molecule co n t ai ning n atoms, 3n - 6 vibrations are possible ( 3n - 5 f or linear molecules). For these , n - 1 can be defined as stretching (opposite movement of atoms) and 2n - 5 are bending or deform ation vibrations ( atoms move at an an gle to each other ) . In some cases, the vibrations correspond to a similar type
90 r----.--�---.---�---.
78 66 0
Q
54
�
42
"
w
C( ro a: 0
� C(
30
-5 L---�----�---L--__J 2400 2250 2100 1950 1800 1650 1500 WAVE NUMBERS ( cm-1)
I n frared spectrum of CO in the gas plase and adsorbed on a c atalysts . T he d ata were collected at a resolution o f 2 em - 1 w ave number in a single scan in a F T I R instrument . T he fine structure superimposed into the broad peaks (P , R branches ) represent s rot ation al transitions . The sharp peak at 2070 cm- 1 cor resp on d s to adsorbed CO. FIGURE 2 4
5% Pt /Si0 2
Ca talytic Surfaces
211
of motion and can be distinguished only because they are perpendicular to eac h other ( degenerate vibration s ) . Hair ( 1 9 6 7 ) discu sses in detail t he number o f possible vibrations in polyatomic molecules .
Group Freq uencies: While any vibrational mode involves all atoms in a molecule, much of t he motion is dominated by a few atoms or group of atoms . Thus, whenever a characteristic combin ation of atoms. is p resent on a com pound , t he same char acteristic frequency appears more or less independent of the structur e of the rest of the molecule . Group frequencies permit one to identify struct ural g roups from t he IR s p ectra . Group frequencies often involve t erminal atoms of sm all mass such as C-H, O- H, N-H, or internal atom s connected by bonds which have a distinctive force constant . E xtensive tabulations and disc ussions regarding the use of group frequen cies for chemic al identification are available ( Alpert et al . , 1 97 0 ) . T he identification of the spectra of adsorbe d species often consists of matching t he sur face spectra with t hose of known compound s . Furthermore , infrared bands that are forbidden for molecules in the gas p hase on grounds of symmetry may be possible w hen t he molecule is adsorbed and p erturbed by surface forces . The frequency of the adsorbed s pecies shifts wit h surface coverage , co adsorption of other molecule s , particle si ze , or , in general , wit h chan ges in t he electronic environment o f t he metal-adsorbate interaction . B and Intensities: I f radiation of intensity I o imp inges on a media of thicknes s d containing a concentration of molecule s C , t he intensity of t he trans mitted radiation is given by the Beer-Lamb ert law, A =In
10 = ECd I
( 25)
where E is the extinction coefficient and A the absorbance. For adsorbed molecules , the extinction coefficient can be larger or smaller than in the gas p hase (Little , 1976). I f E is constant with coverage , there is a linear re lation between absorb ance and covera ge which can be used for quantitative analysis . T he best approac h to measuring extinction coefficients is t<..· meas ure independently the amount of gas adsorbed along with the IR absorbance. Recent approaches have used a pulse technique (S arkany and Gonz alez , 1982a) and mass s p ectroscopy ( S avat sky and B e ll , 1 9 82) for evaluation of the amount adsorbed . If the band width of the adsorption peak changes , it is recommended to use an i ntegrate d absorbance over the full frequency
B an d intensity increases with increasing metal loading and sample thick ness. H owever , scattering and ab sorption by t he support sets an upper limit to the amount of met al and m aterial that c an be used . Scatterin g is p roportional to the cube o f particle size and inversely proportional to the fourth power of the w avelength . It h as been found that good transmis sion res ults are obtained if p article sizes are less than 1 ]Jm . Thus , a choice of support wit h a p article si ze of < 1 ]Jm will facilit ate IR studies .
range.
Experimen t al
In the past, t he m ajority of IR studies have been c arried out usi ng disper sive IR instruments operating in a transmission mod e . However , Fourier transform infr ared ( F T IR) spect roscopy o ffe rs a number of advantages over
Delgass and Wolf
21 2
dis pe r s ive instruments , in p a r tic ular in c at aly ti c applications .
R e flection
spectroscop y ( specular , diffuse ) i s also avai lable to st udy o p aque samples
and i s es p eci ally suited to use w it h flat , op aque surfaces ( foil s , sin gle crystal s ) in c o m bin ation with other spectro scopic techniques . T he d es c rip tion t hat follows com p ar es the various modes of IR u tili z atio n and summari zes briefly adv ant a ge s an d disadvantage s . Dispersive Versus Fourier T r an s form I R : T h e classical IR instrument is a disper sive instrument consistin g of a source p r o ducin g a re fe rence and a sample beam , a sample c hamber , an op t ic al system to com bine t he b e am s and s ep arat e the vario u s fr e qu e ncie s , and a detector . F requency discrimin ation d ep en ds on dispersion of t he radiation by a gr atin g system . A brief b ut cle ar description of dispersion i n s t rument s is p resented by B ell ( 1 9 80 ) and is available from t he tec hnical literat ure for com mercial instrument s . Fourier tran s form instr umen t s operate on an entirely di fferent p rinciple to se par ate the frequencies . A sin gle beam of I R r adi atio n from the source is sent into a Michelson interferometer , w hic h , by t he di splacement of a mir r or , pro duces an int er fero gr am containing all t he frequencies within t he de si re d IR spec trum . T he details and pri n cip le s of F T IR s p e ct ro scop y are p resented in Gri ffith' s ( 1 97 5 ) book on t he subject and are also summarized in Bell's ( 1 980) review article . The information con t ain e d i n th e interfe ro gr am is con ver te d into a frequency versus absorbance or transmittance s pe ctrum by using a Fourier trans form inversion . T he t r an sfo r mation is carried out by a dedicated mi ni c om pute r which also operates the spectrometre and is used to display spec tra . FTIR has two major advantages over gr ati n g spectrometer£! . First , the interferometer o b se rves all sp ectral re s olut ion elem ents si multaneou s ly , where as a monochrometer spends only a fraction of the tot al time obs er vi n g each resolution element . Consequently , under id en tical conditions , the FT spectrom eter can acqui re sp ectra m uch fas ter than the dispersive spectrom eter ( this is kno wn as Felge t t 's a dvan tage ) . Con vers ely , for the same time of d at a collec ti on , the FT spectrometer can produce spectra of m uch better s i gnal - to - n ois e ratio . The second fundamental advantage o f an F T spectrometer over a grat in g spectrometer is the increas ed op tical tho ugh p u t of an in te rferom e t er com pared to a grating monochrometer (Jac q uino t ' s advan tage ) . Griffith ( 1 9 7 5 ) has presented a quantitative comparison between dispersive and FTS instru ments whic h has been sum m ari zed by B ell ( 1 9 8 0 ) . Furthermore , FTS data ac quisition and computer data storage capability provide an a d di tional advan tage over disp er siv e IR instruments , alt hough i t must be pointe d out that si milar computer capabilities are now bein g added to dispersive units . The main disadvantage o f FTS instrum ents i s their hi ghe r cost ; however , new models with r e s t ri c t ed capabilities are now being introduced in the market at prices that are comparable to those of di spersive un it s . In general , the hi gher IR th ro ughp u t and smaller data c oll ection tim e required for F T I R i ns t r u m ent s make them e s pecially suited for c ataly ti c studie s .
T rans mi s sion Versu s R e flection an d E mission :
One of the most widely
used methods of I R s p ectroscopy is t he transmis sion mode , where in t he I R
beam i s transmitted t hro u gh t he samp le , w hich must b e partially transp arent to IR radiation . M a n y oxides , s uc h as silic a , alumin a , zeolites , and stoichio T he sample is pressed in t he form met ric zinc oxi d e , are I R trans p arent . o f a t hi n di sk ( 0. 3 to 0 . 1 mm t hi ck ) with a de n s ity between 10 and 100 m g / cm 2 . The preparation of t he wafer is o ften t he key to s uccess ful experimen t ation . Although s p eci fi c values depend on t he m aterial in q ue stion , t here
Catalytic S u rfaces
21 3
are certain general guidelines . T he disk is pressed in a die which must have hi ghly polished surfaces to prevent the disk from adhering to t he die surfaces . I n some c as es it is convenient to condition t he pow ders with a certain de gree of moisture and temperature prior to pressin g . Transparency increases when the crystallites of the pow ders are very small ( silica and alumina prepared by hydrolysis of t he chlori des in a H 2 - 0 2 flame produces s m all particles ) . An open pore struct ure ( i . e . , low die pressure ) is de sirable to facilitate transport of reactant s inside the disk . I n the case of supported catalysts , metal loadi n g s hould be such that crystallite si ze should not exceed 100 A. Pt can be loaded from 5 to 1 0 % on silica ( Cabosil) or alumina aerogel and still give a goo d absorbance s pectra . T he support s hould not absorb in t he same frequency as t he adsorbate . Other methods of s ample prep aration include evaporation of the metal onto an IR window , formin g a t hin film ; however , the sensitivity is much le ss than w hen the met al is hi ghly dispersed in a porous support ( Pritchard and C atterick , 1 9 6 8 ) . T he books of Little ( 1 9 7 6 ) and H ai r ( 1 96 7 ) contain details of disk prep aration , and the IR chapter in the Del gass et al . ( 1 97 9) book also con tains useful in formation in this re gard . In transmission experiment s , t he disk is placed in a cell containing sp ecial IR win dows ( C aF 2 , s app hire , etc . ) and equipped to meet the pres s ure and temperat ure requirements of the experiment . S everal cells de Newer designs si gns are reproduced by Little ( 1 97 6 ) and H air ( 1 96 7 ) . for transient experiments consist of small - volume cells cap able of rapid response to chan ges in feed concentration and temperature ( S avat sky and B ell , 1 9 8 2 ; K aul and Wolf , 1 984) . When t he solid is comp letely opaque to IR , the pressed s alt method can be used . T he opaque solid is first ground and mixed with a halide tran s p arent to I R ( KB r , C sl ) and can then b e pressed into a self- sup portin g di sk . D epending on t he m aterial , dilutions of 1 / 10 0 to 1 / 400 sample per halide are used . T he disadvant age of this technique is that access to t he s urface by gases is lost and therefore adsorption or desorption st udies are no lon ger possible . However , t he techniq ue is useful when analysi s of the structure of the catalysts rather t h an t he adsorbate - surface interaction is sou ght . Other m ethods that have been used are deposition on a liqui d m atrix a n d in a solid argon matrix ( Pritchard and C atterick , 1 9 7 6 ) . Reflectance techniq ues have been used with op aq ue s amples that do not permit tran s mission . Specular reflection has been used in t he case of metal mirrors such as sin gle cryst als or foils and , i n principle , p rovides an attrac tive combination with other spectroscopic techni ques involvin g flat sur faces . D iffuse reflectance spectroscop y ( D R S ) is used w hen the reflectin g surface is rou gh and the incident radiation is reflected ( scattered ) di ffusely . S uch radiation must be collected by a hemispherical or elliptical reflector and focused into t he detector . K lier ( 1 980) summari zes the literature , theory , and application of D R S to t he study of adsorbed species on c atalysts . N ew developments in vibrational spectroscopy ( excludin g I R excitation ) are reviewed in the monograp h by B ell and H air ( 1 980) . T hey include electron energy loss spectroscopy (EELS ), w hich is useful with single crys t als or metallic foils ; inelastic electron t unnelin g spectroscopy ( l ET S ) , used to st udy met al - i n s ulator junction s ; and neutron scatterin g s pectroscopy (NSS ) .
A p p lica t ions Rat her than an extensive listin g of t he many IR studies p ublished in the literat ure , the m ajor classes of app lic ations in connection with cat alysts are
21 4 summarized below .
De!gass
and Wolf
T he reader is referred to the several review papers for
speci fic app li cation s . B ulk Struct ures : Studies of t he structure o f silica surfaces and acid oxide surfaces , includin g r\lumin a , silica- alumina , and m ole c ular sieves , are s ummari ze d by H air ( 1 9 6 7 ) . A classical example of t he use of I R in struc tural analysis is t he work o f Flanigan et al . ( 1 97 1 ) , who , usin g K B r - z eo lit e disks , compared the spectra of various A , X , and Y zeolites and hydroxy sodalite and assigned I R bands to specific struct ural groups on each zeolite I n more recent studies ( Pichat et al . , 1 9 74 ) , t he degree of crystallinity of
the zeolite has been relat ed to the IR bands occ ur rin g near 5 6 0 to 3 80 cm- 1 A relatio n between degree of cryst allinity ( as obt ained from I R ) and crystal si ze ( obt ained from X RD ) has also been attained .
I R Studies of Adsorbed Molecule s : A great deal of t he I R wo r k relevan to catalysis has been done with adsorbed molecules . T he review of Ei c hens an d Pliskin ( 1 95 8) s u mmari zes st udies of hydrocarbons ( ethylene , ethane , acetylene , olefins) on nickel and CO chemisorption on Pd , Pt , iron , and rhodium . M uch of this work consisted of structural assi gment of t he ob serve d I R bands to elucid ate t h e type of bondi n g involved . In the case of CO chemisorption , IR can be used to detect the number of met al atom s bonded to CO by the observation of frequencies co rrespondin g to linear CO ( C O /M ::: 1 ) . brid ged CO ( CO /M = 1 / 2 , or two CO per metal ( CO /Rh ::: 2 ) . O bservation of frequency shifts wit h s urface coverage , crystallite s i z e , and so on , are also important in t he elucidation of the C O - metal interaction . I R studies o f N O ad sorbed o n metals an d oxides have also been used as a prob to elucidate t he type of sites existin g on supported catalysts ( Peri , 1 97 4) . S i gnificant freq uency shifts have been observed when h yd ro ge n , from hydro gen - containin g mo lec u les , is bonded to the OH gro ups of sili ca . S uch frequency shifts are in terp rete d in terms of charge transfer or changes in the OH dipole due to the presence of t he adsorbed molecule ( B asila , 1968) . One of the classical examples in the use of I R spectroscopy to examine the nat ure of active sites is the differentiation of B r!jnsted or proton donor acid sit es and Lewis or electron acceptor acid sites . Am monia ( or nitrogen containing molecules gives absorption bands near 3 3 3 3 and 1 6 3 9 cm - 1 if in the N H 3 con fi guration ( Lewis acid ) and near 3 1 2 5 and 1 4 2 8 cm - 1 if t he form ( B rljnsted acid ) . The frequencies of t he coordinatively bonded mole cules usually shift wit h coverage , which is interpreted in terms of the stren gth of the acid sites . B asila ( 1 9 6 8) presents a review of I R st udies of the acidity of silica - alumina and zeolites , t o get her wit h frequency assign ments for ammonia and pyridine . It has already been mentioned that isotopic substitution can be used to el uci d ate the structure of surface complexes ; H 2 on ZnO , ethylene- oxygen on Ag , and propylene on ZnO are a few examples discus sed in some detail in the book by Delgass et al . ( 1 97 9) .
NH�
In frared Kinetic Studies : I f t he rate of reaction can be correlated with the rate of form ation or disappearance of an I R band , it can be concluded that s uc h species is a reaction i n termedi at e . I f t he reaction is carried out in a batch c e ll under transient c o n di tions , and if the I R spect rometer is capable of measuring spectra in the time scale of the reaction experi ments , transient I R e xp eri m en t s p remit one to determine the kinetics of s u r face species an d the reaction mechanis m . T he use of transient techniques in cataly sis , although proposed some time ago by Tamaru ( 1 9 6 4) , has been t he s ubj ect of various review articles ( T am a ru , 1 96 4 , 1 97 8 ; Bennett , 1 97 6 ; Kobayas hi and K obayashi , 1 97 4 ) .
Cataly tic Surfaces
215
A combination o f I R transient st udies with isotopic labelin g has been re ported by Kokes et al . ( 1 97 2 ) for 1- butene isomeri zation on ZnO . B and as si gnment w as first resolved by isotopic substitution ; then t he I R bands were followed a s a function of time and compared with gas - p hase kinetic dat a . Simultaneous me a s u rement of surface and gas phase composition , together with i so t opic substit ution , have also been c arried out py T am aru ( 1 9 7 8 ) durin g t he w ater - gas s hift reaction c at alyzed by ZnO . T he use of fast F T I R spectrometers has expanded the use of transient studies to c at alytic systems more active than ZnO . B ennett ( 1 982 ) describes examples o f t he use of pulses , steps , and cycled feed to st udy transient cat alysis usin g IR spect roscopy . T he reactions of carbon monoxide on Pt and iron have been studied u si n g t ransient I R . T he effect of transport p rocesses and dynamics of chemisorption have been analy zed by Ho and Hegedus ( 1 98 2 ) . The dynamics of three - w ay catalysts have been studied by Herz ( 1 9 8 2 ) using IR diode laser spectroscopy . Studies of self- s ustained oscillations using FTIR have been carried out by K aul and Wolf ( 1 985) . In most applic ations , the time constant associated wit h t he t ransient surface proce sses m ust be of the order of t hat fo r I R dat a collection ( about 0 . 1 to 1 s for FT I R spectrometer s ) . Time -resolved FTI R spectroscopy has also been used to meas ure fast chemical reactions ( D urana and M ant z , 1 969) ; ho wever , the techniq ue has not yet b een app lied to surface st udies . In closing t he synop sis on t h e fun dament als and applications of I R s p ec troscopy to cat alysis , it is clear t hat this is a mat ure and necessary tool New frontiers are now bei n g opened with t he use of in catalytic researc h . T he development of reflection and new vibrational transient techniques . spectroscopies will be useful in fundamental studies usin g flat or opaque surfaces . R a ma n
S pec� rosco py
Raman spectroscopy has much in common with in frared spectroscopy , b ut since it records vibrational spectra from scattered rather t han absorbed radiation , it gives sli ghtly di fferent and o ften comple ment ary information . T he availability of intense, monoc hrom atic laser light sources compensates for t he low scatterin g cross section s , and co mmercial spectrometers are capab le of recordi n g Raman spectra of catalytic materials and adsorbed species . T he main a dv a n t a ge of t his technique is that absorption by the solid matrix does not interfere with t he low - frequency bands . T hus vibra tional modes of t he catalyst it self and of t he bond bet ween the ad sorbate and t he cat alyst are ac c e ssible by R aman spectroscopy . Since t he e ffect requires a chan ge in polarizability wit h vibration rather that a c h an ge in dipole moment , spectra show stron g peaks fo r nonionic bonds and sym metric vibrations . T he method has been p articularly succes s ful for t he It has st udy of hydrodesulfuri zation catalysts cont ainin g molybden u m . also been used to st udy ad sorbed aromatic molecules and electrochemic al sur fa ces . T heory , experimental considerations , and examples o f applications are disc ussed below . Review articles by Hendra ( 1 9 7 4) , Cooney et al . ( 1 9 7 5 ) , E gerton and H ardin ( 1 97 5 ) , and H aller ( 1 9 7 9 ) provide further information .
Princip les
Electroma gnetic radiation induces a di p o le moment in a molecule . T he si ze of the dipole moment is proportional to t he magnit ude o f the electric field . T he p roportionality constant is t he molecular polari zability . I f t he
Delgass and Wo lf
21 6
distribution of charge in the molecule chan ges d uring a particular vibration , t he polari zability will also vary wit h t he vibrational m otion , and t he vibration will be Raman active. Analysis of the interaction between the oscillating elect ric field of the exciting radi ation and t he induced dipo le oscillating w ith the vibrational motion reveals t hree main scatterin g terms ( Haller , 1 9 7 9 ) . T he first is the elastic or R aylei gh scatterin g , which occ urs at t he fre quency o f t he excitin g radiation . T he second and t hi rd terms , 10- 9 times the inte nsity of t he R aylei gh scatterin g , represent inelastic scatteri n g in w hich ener gy is exc han ged bet w een t he radiation and the molecule . T hese terms have frequencies v o v m and v o + V m , w here v o is the freq uency o f the excitin g radiation and v m is t he freq uency o f t he molecular vibration . T hey are called the Sto kes an d anti-Stokes R aman lines , respectively ( C hantry , 1 97 1 ) . T he intensities of these lines depend on the change in polari zability wit h the vibration and , for symmetric vibrations , can be re late d to bond order ( Hendra and Stratton , 1 96 9) . For molecules with a center of sym metry , vibrations that are symmetric about t he center of sym metry chan ge t he polari zability and are R aman active , as s hown for C0 2 in Fig . 2 5 . A symmetric vibrations of s uc h molec ules are R am an inactive . T hese sensitivities are exactly opposite to those governin g IR activity for such molecule s , but IR and Raman activity need not be m utually exclusive for more complex molecular symmetries . T he vibrational selection rules are t he s ame for Raman an d IR (i . e . , .6. U = ±1 for harmonic oscillators ) . A s pecial case of Raman spectroscopy must be mentioned in a discus sion of catalyst surfaces . Jeanmarie and V an Duyne ( 1 97 7 ) and Alb rech t and C reigh ton ( 1 97 7 ) reporte d that pyridine ad s orb e d on silver in an electrochemical cell s howed scattering cross sections 1 0 5 to 1 0 6 hi gher than for pyridine in solu tion . This p henomenon of s urface - enhanced Raman spectroscopy ( SERS) is still not c ompletely unders tood ( Furtak and Rey e s 1 98 0 ; Van Duyne , 1 97 9 ; Aravind and Metiu , 1 9 8 3 ) but for A g , C u , and Au , the m etals for which it has been ob served , it offers a si gnal enhancement w hich makes study of molecules adsor bed on foils and ot her low - surface - are a s amples straight App lications have been made in electroc hemistry ( Bi rke et al . , forw ard . 1 98 3 ) a nd s hould also be possible in catalysis . T he technique is particularly attractive because , with an optic al multichannel an aly zer , s urface spectra can be recorded e ve ry 2 0 msec ( Weaver et al . , 1 9 83) . -
,
,
Experime n tal M e t ho ds A Raman spectrometer consists of a li ght source , t he s ample scattering chamber , a monoc hrometer , and a detector . A laser li ght source is essential for surface work . A r gon ion ( 5 1 4 . 5 nm and 4 8 8 . 0 nm ) , helium - neon ( 6 32 . 8 nm ) , and krypton ion ( 6 47 . 1 n m ) lasers are in current use . T his li ght is hi ghly collimated or passed t hrou gh an inter ference filter to remove back ground radiation accomp anyin g t he laser line . T he li ght scattered from the sample can be viewed at any an gle , b ut 0° , 90° , or 1 80° are commonly used . Cells are typically glass with an optically flat window . B ecause of the relative hi gh power of the laser li ght , care must be taken to avoid s ample decomposition or excessive heat in g by t he excitin g radiation . T he li ght c an be scanned across t he s ample ( Zimmerer and Kiefer , 1 97 4) or t he sample can be rot ated ( C he n g et al . , 1 98 0 ) to avoid t his p roble m . To obtain the R am an spectrum , the scattered intensity must be measured as a function of freq uency . A monochrometer accomplishes this
Cataly tic Surfaces
21 7
0
0
i
c
� �
�
0
(") \) {)
Ant 1 sym met r�c a 1 St retc hi ng 0
0
\ I
I \
c
c
0
0
Bend i ng ,
, 112
113
Polari zability change s during the vibration of carbon dioxide ( not to scale ) . ( From Tobias , 1 9 6 7 . )
FIGURE 25
Delgass and Wolf
21 8 t as k .
T he li ght passin g throu gh t he monochrometer system at a given
energy i s recorded by countin g p hoton s with a p hotoelectric det ector .
In some cases , species p r e s e nt in cat alyst samples are excited by t he laser li ght an d emit a fluorescence spectrum t hat can obscure weak R am an
lines . W hen or ganic imp urities are the source of t his p roblem , t he fluores cence b ackground can ofte n be remove d by heatin g t he s ample in O z 1 97 6 )
.
( E gerton et al . , 1 97 4) or tre atin g it with hy drogen peroxide or concentrated
nitric acid ( A dams et al . ,
Other fluorescence rej ection schemes use
is
time resolution to eli min ate t he relatively lon g lifetime fluorescence si gnal ( V an D uyne et al . ,
1 97 4) .
Note that fluorescence radiation
easily i denti
fied by t he lack of dependence of its freq ue ncy on t he frequency of t he excitin g radiation . A p p lications
R am an spec troscopy applies to t he study of vibrations within t he catalyst
itself an d to t hose o f adsorbed molecules at t he s urface .
T he utility o f t he
m etho d for st udyi n g c at alys t s them selves is well illust rated by the growing li ter at ure on t he genesis , p recursors ,
an d s ul fided forms of molybdenum
cont ainin g hy d rodesulfuri zation c at alysts .
Molybdenum is usually added to the alumina s upp o rt by i mp re gn ati on
with an aq ueous solution of ( NH4 ) 5M o
�24
to t he point of i ncipient wetness .
As summari zed by W an g and H all ( 1 98 0 ) , the met al deposition p rocess is complex . At hi gh pH , t he molybdenum species in solution is monometric
- , w hile t he polymolyb d ate s pecies Mo 7 0 2 4 6 - predomin ate� at low pH . Furt hermore , the local pH in t he pores can be modi fi e d by t he b u ffe r capacity o f the alumin a a n d b y the evo l u tio n o f ammoni a . S pectra of bot h wet ( Fi g . 2 6 ) and d ried c at alyst s p re p ar e d at carefully cont rolled pH o f
M oo
;
8 . 6 , 3 . 9 , an d 1 . 0 s how t he ability o f t he Raman spectroscop y t o follow t he molybdenum chemistry ( W an g and H all , 1 980) . At pH 8 . 6 , peaks at 3 2 6 c m - 1 and 90 5 c m - 1 iden ti fy t he m ajor species a s Mo04 2 - . B an d s a t 2 1 5 cm - 1 and 3 6 5 c m - 1 show M o 7 o 2 4 6 - to dominate at p H 3 . 9 and 1 . 0 . R am an spectra also yield much inform ation about calcined or oxidi zed
Zin gg et al . ( 1 980) have used t he chracteristic spectra of Mo0 3 catalyst s . an d A l 2 ( Mo0 4 ) 3 to show t he app e arance of Mo0 3 when m oly b den um s up
ported on alumina exceeds monolayer coverage and t hat increased c alcination temperat ure or duration convert s Mo0 3 to Alz ( Mo0 4) or another t etrahe d ral
Mo species .
A n umber of aut hor s have reported on the interactions of M o
with alumina ( B rown et al . , 1 97 7a , b ; Jeziorowsld and Knozin ger , 1 97 9 ; and
Dufresne et al . , 1 9 81 ) . In general increasin g Mo coverage of the surface in creases the two- dimensional aggregation of o c t ah e dral Mo sp eci e s , as i ndi c ate d by increasing fre qu enc y of the Raman bands . T he aggregated Mo species in teracting with alumin a support has a characteristic R am an li ne in the 950-cm - 1 re gion . Details of the effects of ad di tion of Ni a n d Co to MoO / AlzO a catalysts have also been elaborated ( Chen g and Schrader , 1979a ; Je ziorowos ld and Knozin ger , 1 980 ; Kno zi n ger et al . , 1 9 8 1 ) . Schrader and C he n g ( 1 98 2 ) reported t hat s ulfidin g of Mo /AI 20 3 in 10% H zS / H 2 at 400°C p roduces a s am p le with a Raman spectrum characteristic
of MoS z , althou gh s li ghtly broadened and s hifted . Studies b y these authors of the effects of Mo loa di n g , s ulfidin g temperature , and s usceptibility
of the s ulfide to reo xi d ati on s how MoO a to be difficult to s ulfide at low
temperature , i s o late d Mo species to be the most di fficult to sulfide , and polymolybdate s pecies to s ul fi de most easily b ut to produce vacan ci e s c aus e d by reduced Mo . Brown e t al . ( 1 9 7 7b ) and Payen et al , ( 1 98 2 ) also
21 9
Catalytic Surfaces 955
100 1000 900 BOO
700 600 Ct./1-1
F I G U R E 2 6 Raman s pe c t r a of w et c at aly sts prepared by im p r e gn at io n of a l um in a with ( N H 4 ) 6 Mo7o 2 4 at p H ( a) 8 . 6 , { b ) 3 9, and ( c ) 1 . 0 . ( From Wang and H all , 1 9 80 . ) •.
report Mos 2 s p e c t r a from sulfided cat alyst s . T he latter paper demonstrates t he spatial resolution of the R a m an mic rop ro b e , which t akes advantage of the ca . 1 �m s po t si ze of the laser radiation . T he R aman literat ure on adsorbed molecules is no t lar ge , but a nu m be r of interestin g fi ndin gs have been reporte d . S t udies of molecules with arom atic rin gs offer good e x am p les of t he u t ilit y of t h e Raman measurement . Ram an spectra of b e n ze n e adsorbed on silica s how only s li ght peak shift s from t he spectrum for li q ui d ben zene , b u t t h e in te n si t y o f t he 9 94 - c m - 1 rin g b r e at hi n g band is si gni ficantly red uc e d . E gerton et al . ( 1 97 4 ) in te rp ret t hi s result as a decrease in polarization of t he n elect r on s and an indication of n b o ndi n g of the silica s urfac e . S urp risin gly , t hio p hen e show s similar 1T i n t e rac ti on with surface SiOH gro up s by loss of i n te n si t y in the rin g b r e at hi n g vibration at 8 3 4 c m - 1 ( T am et al . , 1 97 5 ) . Py ri dine , on the ot her hand , indicates heteroatPm interaction with a sili c a surface by a fre quency shift in t he rin g b re at hin g mode from 9 2 2 c m - 1 in t he liquid to 1 0 0 7 cm - 1 at low co ve ra ge , w here hydro gen bondin g domin ate s ( Hendra et al . , 197 1 ; Ege r t on et al . , 1 9 7 4 ) . In addition , these studies rep ort
D e l gass and Wo lf
220
coordinative bondin g of pyridine to Lewis sites on alumina . Pyridine adsorp tion on hydrodesulfuri zation cat alysts gives R am an spectra characteristic of physisorbed species and a specific chemisorbed species with a peak in t he re gion 1 008 to 1 0 1 4 cm - 1 ( C heng and Schrader , 1 97 9b ; P ayen et al . , 1982). Ram an spectroscopy has also elucidated surface reactions of adsorbed molecules . K uiper et al . ( 1 973) have combined Raman and I R result s to show that ben zaldehyde is converted to ben zoate and ben zyl alcoholate on an alumina surface . T urner et al . ( 1 976 ) used Ram an s p ectroscopy to ob serve t he isomeri zation of 1 - hexene to 2 - hexene on alumina activated at 12 0 0 K . Most of the applications of surface- enhanced R aman spectroscopy have been in an electrochemical environment , but it is clear that the technique does apply to the interaction of gases with clean metals w hich show the effect . SERS spectra of carbon monoxide ( Wood and Klein , 1 980) , ethylene (Moskovits and Dilella , 1 980) , and water ( Pockrand , 1 982 ) adsorbed on sil ver at low temperature have all been reported . For ethylene on silver , the Raman spectra cont ain bands for vibrational modes that are Raman inactive in the gas p hase . T hus adsorption alters the symmetry of t he molecule . Moscovits and Dilella interpret the band intensities and shi fts as suggestin g that bot h ethylene and propylene are 11 bonded to the silver surface . Thus we see that , in general , Raman spectroscopy has been e ffective in identifyin g surface phases and coordination . It can reflect details of ad sorb ate bondin g and gas -s urface reaction , and in t he s urface -enhanced mode , it can give this inform ation on submonolayers of adsorbates on flat surfaces . Applications to catalysis have been significant , b ut " t his is a technique in the process of bein g developed for catalysis research . I n creasin g u s e can b e expected . Temperat ure- P ro g r a m m ed
D eso rption
and
Tem pera t u re- P ro g r a m med Reaction
Althou gh many experiments in catalysis demand isothermal conditions , t he unsteady state produced by a rapid , pro grammed chan ge in temperat ure can yield a wealt h of information about a catalytic system . Perhaps t he simplest version of this experiment is w hat is o ften called flash desorption spectroscopy ( FD S ) or thermal desorption spectroscopy in t he s ur face science literat ure ( King , 197 5 ) . In this exp eriment , gas is irreversibly adsorbed on a foil or sin gle -crystal surface at low temperat ure . T he temp erat ure of the sample is then raised linearly at a typical rate of 1 0 K I sec . The desorption spectrum , usually recorded by a m ass spectrometer in the vacuum syste m , shows peaks at temperat ures related to t he heats of adsorption of t he various adsorbed states . Temperature -program med de sorption ( TD S ) ( C vetanovic and Amenomiya , 1 97 2) is essentially the same experiment on a cat alyst sample , but extra care must be taken to account for possible influence of heat transfer , di ffusion , or readsorption of the gas in the porous sample . Temperature -programmin g rates are usually slo wer in TPD an d most experiments are done in a carrier gas instead of a vac uu m . T he concept o f using temperature to scan t he energies of in teraction of molecules with surfaces is a useful one and has appeared in many variations , including temperature -programmed reduction ( H urst et al . , 1 982) , temperature - pro gram med oxidation ( Miura et al . , 1 97 5 ) , and temperature- programmed reaction ( TPR ) ( Falconer and S chwarz , 1 98 3 ) .
221
Cataly tic S urfaces I n all o f t h e s e t e c hn iq u e s
spectrum
,
t he appearance o f more than one peak i n t he
in dic ate s h e t e ro gen ei t y in the reaction ener getics of t he surface or the adlayer . Q u antitative i nter p r e t at ion of t hese results can be difficult ( Gorte , 1 982 ) , but detailed analysis of data is pos sible ( Falconer and S chwar z , 1 98 3) and even fingerprinting of suc h complex p h eno m en a on c at aly s t s can be very help ful in di fferenti atin g and developin g new cat alysts .
Princip les I n flash desorption spectroscopy , molecules desorb from t he s u r face directly and are p u mped from t he vacuum sy st e m I f the ratio o f p u mp i n g s p e e d to syste m volume is hi g h and t he heat in g rate i s not e xc ess i ve t he desorption rate is p roportio n al to t he press ure in t he sy st e m or to t he m ass spectrome ter si gn al above b ac k gro un d In the ab sence of readsorption .
,
.
( 26) w here 6 i s the fractional coverage o f t he surfac e , n the order o f t h e desorp tion , S the h e at i n g rate d T /dt , k d , 0 the preexponential fac tor o f the de sorption rate constant , E d the ac tiv at io n ener gy for desorption , T t he ab solute temperat ure , and R the gas c o n s t an t In ge n e ral kd , o , E d , and even n can be a function of coverage . I f these factors are const ant and t he meas ured si gn al is p t"opo r tio n al to d 6 /dt , then settin g t he derivative of .
( 26 ) eq ual m aximum :
Eq .
.
to zero p roduces
,
t he followin g relation d es cri bi ng the peak
( 27 )
Here 9m an d T m refer t o values a t the peak m aximum . Note that for first order reactions T m does not depend on 6 , an d E d can be calculated if 13 and k d , O are k n ow n A rough estimate for E d ( kcal / g mol) is given by T m / 16 ( M adix , 1 980 ) . A t hi gh c arrier gas flow s and in t he absence of re adso r p tion and dif fusion limitations , TPD is e quiv a len t to F D S . I ncreasi n g degrees of so p hi stic ation must be a d d e d to t he an alysis when coverage dep e n de nc e of the p arameters , re adsorpt io n , or di ffu sion limitations apply . A s o u t li n e d succinctly by Falconer an d S c h w art z ( 1 983) , variation of he ati n g rate and initial coverage ; p artial in te gr atio n s of c urves to produce desorption rate isotherm s ; det ailed analyses of peak widt h , shape , and skewness ; a n d direct curve fitting provide a variety of ap proaches for m aximi zin g the re .
t urn from T PD experiments . I n trusion of tran sport and re adsorp tio n problem s c annot alw ays be avoided , but r e cent analyses have provided q uan ti t ati ve g uideli n es for op ti m i zin g the desi gn of t h e experiment ( Herz et al . , 1 982 ; Lee and Schw artz , 1 98 2 ; Gorte , 1 9 8 2 ) . Even under ideal experiment al conditions , fast interconversion between v ariou s s ur fac e states can hide in fo rmatio n from view by temperat ure - programmed methods . Nevertheles s , this approach creates a u s eful fa mily of tool s fo r characteri z in g gas - surface energetics and surface reactivity .
D e l gass a n d Wo lf
222 Experime n t a l
M e t hods
TPD /T P R be gan as a simple and inexpensive experiment . P uri fied carrier gas or reactant passes throu gh one side of a t hermal conductivity cell , over the catalyst , and throu gh the other side of the cell . In the absence of desorption or reaction , t he t hermal conductivity cell is b alanced and gives W hen a linear temperat ure pro gram i s applied t o t he fur a zero readin g . nace around the sample , the temperature of t he small catalyst char ge is re corded by a delic ate t hermocouple placed in t he bed and a desorp tion or reaction event is recorded by t he imbalance in t he t hermal conductivity cell , which is mounted near t he reactor exit to minimi ze the delay time . Higher sensitivity can be gained by usin g a flame ioni zation detector , but a more flexible app aratus is created w hen the detector can identify t he composition of the reactor effluent . Since analysis must be done at the speed of t he te mperat ure p ro gram , a mass spectrometer is the most versatile detector and q uadrupole mass spectrometers are often used . Accuracy and fast re sponse req uire a well - d esi gn e d inlet system connectin g the atmospheric pressure reactor effluent to the low -pressure environment needed for ioni za tion and mass detection . Further det ails of experimental desi gn and opera tion are discussed by F alconer and Schwarz ( 1 9 8 3 ) . A p p lications
Temperat ure -pro grammed techniques have been widely applied to catalyst s . Falconer and S chwarz ( 1 9 8 3 ) cite over 1 00 references . Our p urpose here is to illustrate temperature-programmed desorption , reduction , · reoxidation , and reaction with a few representative examples .
Temperat ure Pro grammed Desorption : Anderson et al . ( 1 9 7 9) have ex amined H 2 desorption from Pt as a function of metal dispersion ( fraction exposed ) on different s upport s and of alloyin g with Au on silica . B roaden in g of the H 2 desorption p rofile with increasin g dispersion w as attributed to an increase in t he number of hi gh- energy sites at hi gh dispersion . Al loying of Pt with Au decreased the amount of H 2 desorption but not t he peak position , showing that A u acted only as an inert diluent in the Pt sur face . For some metals , adsorption of hydro gen can be activated . H 2 TPD as a function of adsorption temperature has revealed strong activated ad sorption effect s for a variety of cobalt catalysts ( Zowtiak et al . , 1 98 3 ) . CO s hows complex , multi peaked desorption spectra for m any metals . In studies of the effects of sulfur poisoning on CO desorption from Ni ! Si0 2 , Zagli and Falconer ( 1 983) s howed that s ulfur preferentially removes t he highest - energy CO adsorption sites . I n an interestin g contrast to the find in gs for H 2 desorption , Fa ger and Anderson ( 1 97 9 ) showed that alloyin g Au with Pt chan ged t he CO desorption spectrum si gnificantly even though pure Au does not ad sorb C O ( see Fi g . 2 7 ) .
Temperature -Pro grammed Reduction : This experiment is based on the premise that the reducibility of metals on a c at alyst s urface can be a sen sitive function of the surface chemical environment ( Holm and Clark , 1 96 8 ) . Reducibility is typically m easured by H 2 consump tion d urin g a linear tem perat ure pro gram . In the fi rst application of this method to s up ported cat alysts , Robertson et al . ( 1 97 5) elucidated the conditions favoring alloying of C u and Ni on Si0 2 . Precalcination tended to maintain separate p hases with identifiable reduction peaks for each metal . After direct reduction and intervenin g calcination , temperature - p ro grammed reduction showed one
Catalytic S urfaces 39!)K •
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FIGURE 2 7
Temperat ure - p ro grammed desorption profiles for carbon monoxide C atalyst Pt 1 5 , A u 8 5 : , CO ; - - - - , C0 2 , both components by mas s spectrometry . Catalyst Pt 76 Au 2 4 CO ; . . . , C0 2 ; both components by m ass spectrometry . Samples : platinum gold /aerosol ( Pt 1 5 , A u 8 5 ) , 0 . 4 3 g . Platinum - gold /aerosol ( Pt 7 6 , Au 2 4 ) , 0 . 43 g . ( From Fager and Anderson , 1 9 7 9 . )
from platinum - gold /aerosil catalysts .
__
• - ·
,
main reduction peak , correspondi n g to rereduction of both C u and N i. I n teractions o f P t with G e and R e o n y - Al 20 3 have also been studied ( M c Nicol , 1980 ) . A calcined 0 . 3 7 5 wt % P t 0 . 2 5 wt % Ge / y - A I 2 0 3 reformin g catalyst showed Pt 4+ to Pt O reduction at 55 3 K and Ge 4+ to Ge 2+ reduction at 87 3 K . Oxidation - s t ate chan ges were calculated from t he H 2 uptake per metal atom . Rereduction after reoxidation s howed both Pt and Ge reduction occurrin g below 5 5 3 K and only a s m all residual peak at 87 3 K . H 2 c hemi sorption after interruption o f t he T P R cycle show ed t hat the H 2 upt ake on Pt w as diminished in concert with the onset o f Ge reduction . T his res ult was interpreted as a coating of the Pt surface by Ge . Temperat ure - P ro grammed O xidation : P arti al oxidation over mixe d - oxide catalysts often involves an o xidation - re d uction sequence i n the surface layer . Temperat ure - pro gram med reaction is applicable to t he study of t he surface redox properties and has been used to advantage to study reoxidation of prereduced surfaces . Miura et al . ( 197 5 ) and U d a et al . ( 1 9 8 0 ) found two reoxidation peaks for bis muth molybdate and as sociated t he hi gher te mpera ture ( 600 K) peak with c at alyst activity . In the l atter st udy , A u ger electron spectroscop y w as used to assi gn the lo w - temperat ure peak to Mo 4+ to Mo6+ conversion and a partial reoxidation of t he bis m ut h . T he hi gh temperature reoxidation , neces s ary for conversion of propylene to acrolein , completed t he conversion of bismut h to Bi 3+ . T he lability of s ulfur in hydrodes ulfuri zation cat alyst s is also amenable to temperat ure - pro grammed st udies . N ag et al . ( 1980 ) have s ho w n a
224
D e l gass and Wolf
an d de c reasi n g temperat ure of H 2 S appear ance i n a n H 2 - He stre am p as si n g over a sulfided c at alyst .
correspondence b e t w ee n ac ti vit y
Reaction : T hese st udi e s can include decompo surface complexe s , reaction of adsorbates or reactive surface layers wit h gas - p hase react ant s , and react ion of coadsorbed s pe c i e s . Falconer and Wise ( 1 97 6 ) used temperature - pro grammed me t ho d s to st udy hydra zi n e and ammonia decompo sition on Ir/Al z o a . T he similarities in ni t ro gen evolution from t he N z H 2 and N H 3 dosed su r fac e s s u ggested t hat t he N - N bonds are broken in N zH z d eco m po si tion . Pread sorbed oxygen w as found to shift t he N 2 deso rption peak d ramat ic ally . C O !H z re acti o n s have been a focal poin t for m any TPR s t u di e s . Z agli et al . ( 1 97 9) e x am in e d t he temperat ure response of coadsorbed C O and H z ; and H z reac tio n with preadsorbe d CO on s upported nickel c at alyst s . Meth ane d e so r b e d in a temperature ran ge m uch n arrower than that for C O de so rpt ion . W ater d e s o rb e d in t he s am e temperat ure ran ge . C O z adsorption was found to be activated and dis soci at i ve on Ni / Si0 2 . Temperat ure program mi n g of adsorbed CO z in hydro gen ga ve m e t h an at ion beh avio r i d en ti c al t o that for CO ( Falconer an d Z a gli , 1 980). M c C arty a n d W i s e ( 1 97 9) deposited c arbon on Ni / Al 2 0 3 by C O di s p ro portionation at 5 5 0 K an d studied its temperat ure - pro grammed reac t io n in hy d r o ge n . This m ethod is termed temperature -program me d s urface re ac tio n ( T P S R ) . T hey observe d me t hane production at roo m temperat ure as well as in the re gion expected from t he C0 / 1-l z result s . E ven .more carbon reactive at room temperat ure w as fo und in T P S R on R u / S iO z ( Low and B ell , 1 97 9 ) . Both studies s howed that t herm al aging of t he c arbon red uces its re ac tivit y . Wolf and K aul ( 1 985) have developed a tech niq ue t hat com bine s tempera ture and concent r ation p ro gramming with tran sient F T I R spectroscop y . With its added co ntro l and observation o f i m p o rt ant vari able s , t his technique has proven to be a powerful tool fo r s t u d yin g c at aly t i c re ac t i o n s un d e r t ran si ent co n dition s . Tempe rat ure - Pro gram med
sition o f adsorbed gases or
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4 Diffusion-Reaction Interactions in Catalyst Pellets DA N L U SS
U niv ersi ty of H o u s to n , H o us to n , Texas
I N T RO D U C T I O N
T he reac tants within a porous catalytic pellet a r e con s u me d by c h emical re actions an d replenished by transport p roc e s se s .
Competitive kinetic an d
transport rate p rocesses affect the ob servable r at e an d selectivity of a
c at alys t
.
The en gi n ee ri n g implication s of this interaction betw een the diffu
sional transport an d chemical reaction s were poin ted out first by Dam kobler
( 1 9 3 7) , Z eldovic h ( 1 9 3 9) , T hiele ( 1 9 3 9 ) , Wheeler ( 1 9 5 1 ) , an d W ei s z and Prate r ( 1 9 5 4 ) .
T he m os t co mprehe n s iv e review of the s ubject is the mon o
graph by Aris ( 1 97 5 ) . A porous medium c on sists of a complex m atrix of m any interconnected c hannels of varyin g diameters and len gths , throu gh whic h the reac tant specie s have to diffu se in or d e r to reach the active c atalytic site s .
Ob
viously , ac c urat e geom etric models of a p oro u s structure cannot be devel oped .
A p roper an aly sis of the p e rform ance of a cataly s t re q uire s use of
a reasonable ,
simplified model whic h incorporates the e s sen tial chemic al an d
phy sical propertie s of the p ellet .
T his is us ually accomp lished by u s e of
a single - p hase pseudohomogeneous model .
T his c rude represen tation elimi
nates the di stin c t ion b etween the solid an d fl ui d p hases within t he p ellet
an d si mplifie s considerably t h e p r edic t i on of most propertie s w hic h are of en gineerin g importance . We describe here m ethods of p redic t in g the impac t of the couplin g b e
t ween t h e chemical a n d tran sport rate p roc es s es o n t h e activity an d selectiv
ity of porou s c at alytic p ellets . selection , testin g ,
T his information is essential for the ration al
developmen t an d application o f c a taly st s .
We start wit h
a review of the rate of t ran sp or t within an d to porous materials .
T RA N S P O R T W I T H I N A N D TO PO R O U S M E D I A T ra n s port of Gases W i t h i n Poro u s M ed ia
T here exist s at p re s ent no flux rel at ion with a comp letely sound t heoreti cal basis capable of de sc ribin g transport of gases in porous me dia over a wide ran ge of pore sizes an d pressures . A c om p r ehensive review of the s ub ject
23 9
L uss
240
can be foun d in t he on gra h b y J ac k son ( 197 7) . E xi s i n g rela t e the gra en ts in the m p siti n s , pres s r e , an d temperature , assume t hat a p oro s mediu m can be des c rib e d eit her as a m atrix of in te r n n e c t e d channels or as an assemb ly of d s t p articles dispersed in the gas . Flux models for c apillary n e w r k s are n aturally based on known tran s la s for a si n gl e ca p a y i s the mechanism res p s ib le for t r a s or t wit hin isothermal , isobasic , narrow capillary , whose diameter is all n co m ari so with the mean free ath the gaseous s p ecie s The flux relation for this Knudsen flow is
m o p t transport flux to local di co o o u co station t o ill r M olecular- wall collisions np an sm i p n p of
models , which u ary u port w on
.
.
D
KA
( 1)
rad P g A
RT
( 2) where p A p a rtial pressure average molecular ve loci ty
is the of sp ecies A and uA is the correspondin g Gaseous diffusion is the dominant transport mechanism in a wi de capil lary wit h a diameter much l a r ge r than the m ean fr e e pat h len gth . The correspon din g i so b a ric i so t h e r m a l fl u x es sati sfy the S tefan - Maxwell relations .
-
-
,
..L grad RT
x
i
=
n
L:
i= 1
i�
where Dij
si mp le
x.N .
l
1
1
( 3)
- x. N .
D.. 1]
l
is the b in ary ga s e o u s di ffu s i n e ffic ie n t A cc r din g to the of gases , the self- diffusion c e f cien is b a n e d
o co
kinetic t h eo r y
.
o fi
o t
o ti
as ( 4)
Very accu rate prediction s of binary diffu sion coefficien t s can b e obtained the Chapman-Enskog formula , which can b e found in st an dard text an d refe rence b ooks ( e . g . , B ird et al . , 1 9 6 0 , pp . 5 0 4 - 5 1 3 ; Perry , 1 9 7 3 , Chap . 3 , pp . 23 0- 2 34 ; Reid et al . , 197 7 , pp . 5 4 8 - 564) . n S tefan -Max well rela t ion s m ay be w ritten for an n -componen t mixture , but only n - 1 of these are in d ep en d en t
from
.
The viscous flow of a p ure component s ubstance through a circular capillary satisfies the relation
N
=
;
r p
�
8�RT
dz
( 5)
where J.l is t h e visc si ty of A orou s be c on id e e d to be a complicated matrix channel s . It is r easo ab e spec ulate if all the
p connected
o media may
the gas . s r n l to
that
of inter channels
Diffus ion -Reaction In teractions
241
are s mall c ompared with the m ean - free -p ath len gth , the local averaged flux vector , which desc rib es the flow per unit total cros s section , is related to the int erstitial local average c oncentration gradien t in the followin g way :
( 6) T he effec tive K n u d sen diffu sivity is defined as
( 7)
£p is t h e pellet void fraction , and < rp > • the mean pore diameter , i s com puted as
::
J"" 0
r f[ E: ( r ) ] dr p p p
�
( 8)
where f [ E:( rp ) ] drp is the fraction of the void volume with pores of radius + dr The tortuosity factor , ', accounts for the tortous path p of the molecules in the porous medium and the varying pore cross section . T his em pirical fac tor depends mainly on the porou s material , it s porosity , an d the method of preparation . The value of ' is us ually in the ran ge 2 to 7 . Wohlfahrt ( 1 98 2 ) presente d correlation s of ' for several types of porous cataly s t s . T he data for cataly sts prepared from compacted powders were c orrelated as rp to rp
•
( 9) where m dep end s on the porous m aterial . Typical data and values of the parameter m for several cataly sts are described in Fig . 1 . A d ditional data were report ed by Prob st an d Wholfahrt ( 1 97 9 ) . Feng and Stewart ( 197 3) pre dicted tha t ' s hould be about branches . It is reasonable to assume that are wide compared to the Stefan - Maxwell relations if the effec t iv e diffusion coefficien t s
3 for isotropic sy stem s with no dead - end that the fluxes in a porous media , with pores mean free path len gth , still satisfy the bin ary diffusion coefficient s are replaced by
( 1 0) Ornata and B rown ( 1 97 2 ) pointed out that the tort uosity factor is not nece s H oweve r , with sarily the same for both Knudsen an d molecular diffusion . out detailed measurements , this is a n ec essary assumption . Similarly , it is reasonable to expect that the forc ed flow throu gh a porous media satisfies the relation
N =
B oP � ll R T
dz
( 11)
242
1::. v
•
m
�.(
� �
0.70 V-oxide col. (S02 - oxid ) 0.85 NiO - col. unsintered
/
x 0.95 corbonyl - iron pellets. 5 p.m
+
L uss
�/ � ;{
1.05 porous Ag . 65 p.m
o 1.10 boehemi te , 90 p. m
0 1.60 porous boehmite 70p.m
/:
�#')
2 �n (E l"r )
�/
i
0
-I -2 -3 -4 -5
-
-6
/
A/ /
-
l
�n
-I
-2
-3
-7
( E /(2 - E ) )
FI G U R E 1
Dependence o f the tortuosity factor on the p or o si ty for several
materials .
( From Wohlfahrt ,
1 98 2 . )
w he re the p ermeability of the m ediu m , B o , depen ds on the geometry and structure of the pores . A method of fin din g the flux relation s in the intermediate situation s , where the three tran sport m ec hani sms m ay be of comparable importance , was p roposed by M a son and E vans ( 1 9 6 9 ) . T hey s uggested that the rate of m omentum transfe r by colli sion with the wall s an d a m on g molecules s hould be combined additively . This leads to a set of n in dependent equaD
tion s for the diffu sive fluxes Ni :
1 grad RT
N?
pi
1 = --
D
In the s peci a l case of
D
e
n
+ I:
ki
j=1 i=j
D
x.N. J 1
RT
ij e
( 12)
a bin ary mixture , Eq . ( 1 2 ) reduces to
( 1 3)
e A
D
D
x.N. 1 J
grad p
A
w he re
1 e D A
-
=
1
-- +
e
DK
A
1 - ( 1 + a) x
e D AB
A
( 14 )
24 3
Diffusion -React ion In teractions
an d =
a
( 1 5)
Solvin g th e n equation s E q . ( 1 2) gi v e s
N�
1 RT
=
1
n
G.
L .
where the elements of are 1]
F
1
.
.
G1J
are those of the matrix F
for
D l�.J +
1
--
=
ii
gra d p .
x.
=
F..
D
e
ki
n
-1
.
The entrie s of F
;t j ( 1 7)
x.
I: .....L D�.1]
j=1 i 4j
( 16)
1
.
lJ
J
for i =
M ason an d E vans a s s u m ed that the t o t a l fl u x is the su m of t h e diffu sive flux and viscous f low : N
i
=
N� 1
where
N:'1
=
+
-
N:'
( 18)
1
x B oP i
11 R T
( 1 9)
grad p
S u b stitution of E qs . ( 1 8 ) and ( 1 9 ) i n E q . ( 1 2 ) gives a relation fo r the total fluxe s :
N.
1 1 grad p = -- + e i RT
Dk
i
n
�
j=1
i :#j
x.N.1 ]
-
D�.
lJ
x.N . 1 ]
+
xB p i O
D k iJ.J R T e
gra d p
( 20)
An elec trical analog of the ad ditivity p rocedure p roposed by Mason an d E vans is shown in Fig . 2 . It is remarkable that identical additivity r ul e s are obtained by a ri gorous analysis b ased on the dusty - gas m odel (Mason et al . , 1 9 6 7 ) . Figure 3 enables a r apid ro u g h estimate of the dependence of the dif fusional regime on the average pore si ze and operatin g pressure . For example, when the pore size is ab ou t 1 0 0 A , K n u d sen flow is a dominan t me c h a ni s m for pressures b elow 5 x 1 0 5 Pa , while gaseous diffusion is the main contrib utor for p res s u re s above 5 x 10 6 Pa .
244
Luss
�� . �?+ �r (ADO FLOWS)
No
N' - I
- I
:s
a;_� -,N .., ..., ---
BULK DIFFUSION
+z
�
�
� !6l-s ---
dZ
c:...,- � ...,N
VISCOUS FLOW
KNUDSEN
DIFFUSION
DIFFUSIVE FLOW
FIG U RE 2 Electrical analog for the additivity rule s for the gaseous fluxes ( F rom Mason et al . , 1967. ) accordin g to the dusty - gas model .
10.000
... ...
o<(
Bu l k d i f f USIOn
1 ,000
Q; E
0
...
0 a..
0
1 00 10
'
T ypi c a l
. mo l e c u l a r • SIZe
Pressure
Pascals
T ran sition between b ul k and K n u d sen diffu sion for typical cata ( F rom Andrew , 1 98 1 . )
FI G U RE 3
lyst s .
1
24 5
Diffusion -R eac t ion I n t e ractions
T he i sob aric flux relations are usually not applicable under r eac tion con dition s . T he reason is that the set of isobaric fluxe s t hr o u gh a poro u s medium has to sati s fy t he Gr a hm r e lat ion ,
n
L
i=1
N . 1M." = 0
1
1
( 21)
which i s usually incon sisten t with the stoichiometric requiremen t of n
:E
i= l
N .M . 1 1
=
( 2 2)
0
C onsequently , a pressure g r adien t u sually exists in a porous pellet in which a c h e m ical re ac ti on occurs . Althou gh thi s p re ssu r e gradient may be sm all , it may have an im p ort an t in fluence on the specie s fluxe s . A very ge n e ral transport model was formulated by Feng and S tewart ( 1 9 7 3 ) , w h o assumed that a porous medium may be rep resented by a c lo s e ly in t e rcon n ected network of cylin drical pores and that the dusty - gas model describes the flux in each pore w it h D
�A , D�B ,
an d B 0 h a vin g values ap
p ropri a te for a s t r ai g h t cy li n d rica l pore of r a di us r p . T h e ir model predicts i sot ropic a lly orien ted , the to r t uo sit y factor is i n de pen dent of pore size and surface diffusion is n e gli gible , then that if the pores are
N.
1
=
n
1
:E
_ _£
-r R T
.
J=
x.
2
g ( r ) grad p . . .
1]
p
1
-
1
BllkT
grad p
( 2 3)
where
G ( r ) f [ £ ( r ) ] dr p 1] p p . .
( 24 ) 2
r f [ £ ( r ) ] dr p p p
radi u s rp .
and Gi j ( r p ) are the en tries of the matrix G for a cylin drical c ap ill a ry of F en g et al . ( 1 9 7 4 ) have s hown t h a t the in t e gr al s in E q . ( 24 ) may b e des cri b ed rather closely b y a tw o - t e rm approximation even for p ellets with a w ide pore size distrib ution . T his avoids t he need to c o m pute G ij for a wide r an ge of r and s i m p lifi e s c on siderably the app lic at ion p of t hi s model . A m ajor a d v an t a ge of this model is t hat it in c orp or ates in formation about the structure of the porou s medium into the de t e r min a tion of the emp i ric al tran sport c oefficien t s . A c om p a ri son of the predic tion s of this model with exp erimental data w a s p r e s en te d by F en g et al . ( 1 974 ) . For i sobaric b i n a ry system s E q . ( 2 3 ) reduces to Eq . ( 13 ) with
1
-
( 1 + a) x
D
AB
A
]
-1
f[ £( r ) ] dr p
p
( 2 5)
Luss
24 6
is sometimes desired to compare the flux relations of the dusty-gas with those obtained u sin g a pseudo-Knudsen diffusion coefficient. Schneider ( 1976) suggested that when a single chemical reaction k ( 26) 0 :E i=1 occ urs in a porous pellet , in which the total pressure gradient is negligible, the following approximation should be used : k v.x. - vjxi v n :E 1 i -- + + i 1 , 2, . . k :E e j = 1 D �. D� D j =k+ 1 D�. It
model
=
v .A.
1 1
1
1
-
D 1�
ki
=
-
x :Ek .
X. 1
v. 1
1 ]
=
1]
•
1]
( 27 )
i k+1 , . . . , n
l
=
1 j = 1 D�. lJ
Wakao and Smith ( 96 2 ) developed a completely predictive model for the binary , isobaric flux in1 compressed powder catalysts which have a bimodal pore size distrib ution . They refer to the small pores within the powder particles as micropores and to the interstices between the particles as macropores . Wakao and Smith assume that the one-dimensional flux con sists of three contributions : (a) flux through macropores having a radius of ra , { b ) flux through micropores which have a radius of ri , and (c) flux through a series of micropores and macropores. They denoted the Knudsen coefficie n t and void fraction corresponding to the macro an d micro pores by D K a , D KA . , a , and q , respectively . Their model predicts that D Ae = a2D a + ( 1 - a ) 2D . + 4 e:a ( 1 - a >[n1 + D1. J - 1 ( 28) a where A
e:
e:
1
£
1
£
D
1
i --
1 - cocA + D A B / D KA .
1
( 29)
The main advantage o f this model is that it contains no adjustable param eters . Its main deficiency is that it is not clear how to incorporate into the model pressure gradients (which are usually present in reactive sys tems} or how to generalize it for multicomponent mixtures . Several steady-state and t ransient experimental techniques have been developed for determining the effective diffu siv ity of porous pellets and of the parameters appearing the transport models . The most commonly used steady-state method is that in which either a cylindrical (Wicke and Kallenbach , 1941) or spherical (Weisz , 1957} pellet in a special cell is exposed to two gas streams having the same pressure and temperature but different compositions. The net diffusive flux is determined from the change in the composition of the gas streams . in
24 7
Diffus ion -Reaction In terac t ions
m u e the on e - dime s ion al p.
For a
binary ix t r ( Satterfield , 1 97 0 , 44) . e
p D AB
=
where
R TL ( 1
+
a.)
n
In
1
( 1 + o. ) x
1
A
( 0)
flux satisfies the relation
+ +
e e
e
0 AB 1°KA
e
( 30)
0AB 1°KA
a " :: " -(:�r
( 31)
Noting that e 0AB
--
=
( 3 2)
Hp
e are determined by measuring the flux at the values of D Ke A an d D AB several pressures. This e hni ue is very sensitive to leaks next to the wall and t his pr even t s its u se at high temperatures . T he most widely used t ransien t t e chni ue is the pulse response of either a chromatographic column (Cerro an d S mit h , 19 7 0 ; H ayn es and Sarma , 1 97 3) or a sin gle pellet ( S u zuki and S mith , 1 97 2 ; Dogu an d Smith , 197 6 ) . T h e main advantages o f this et hod are t h a t it can be ap plie d to particles of arbitrary shapes , it accounts for the con trib u tion of dead-end pores , and it can be used to meas u re t h e diffu siv at high tem eratu es and pr es s re s . I t s main di s a d van age is th at the interpretation of the data i s based on sim p lif in ass umption s whic h are not alw ay s valid . The chromatographic tec hnique consists of measuring the first an d secon d moments of a t racer that has been in ecte in to a packed bed of p o rous p ellets at s e ve ra different flow r a t e s of the inert carrier gas . The theory used to inte p ret the measurements is de sc ib e d in detail by C e rro and S m ith ( 1 970) . The accuracy of this method is h igher for adsorb able gases and improves with increasing particle size an d bed length and with dec r e a s ing effective dispersion in t h e bed . Experimental measurements and theoretical prediction s of gaseous tran s port through porous media are usually dominate d by t h e fl u x th o gh large diameter pores. However, most c ataly tic sit e s are present in mic ropo es , and it is questionable et er the effective diffusivity is ade uate for characterizing the t ransport under reaction conditions . t c
q
q
m
ity
p
r
u
t
y
g
j
d
l
r
r
r u
r
wh
h
q
S u rfa c e D i ffu s i on
Most gases are adsorb ed to some extent on solid surfaces . The migration of the adsorb ed species is referred to a s s urfa c e diffus ion and is pa rti la rl y i m p ort a n t for porous materials with large s r a e area and for strongly adsorbed sp e s . The surface diffusion fluxes combine additively with the gaseous di fu sive fluxes an d th ere ore can b e computed separately . T he surfac e diffusive flux may be d ete r mi e d from a Wicke-Kallenbach di fusio n cell operatin g at low pressures , so that the gaseous dif sion is of the cu
u f c
eci
f
f
n
f
fu
L uss
248
type . A separate easurem en with a nonadsorbing gas is o determine the contribution of this Knudsen flow to the total flux . that the surface and bulk concen tration satisf the equilib rium rela tion s s C T ) C f( c C ( 33) Knudsen t
m
t
used
A s sumin g
A'
y
B '
'
fusi s bs n�
the su rface dif
=
on flux
1"
...
5.
=
n'
' ' '
A
may be expressed as
grad C
3 fc
s A
a
For sufficien tly low gas concentration , t ( 34) to
linearity and Eq . =
simplies
D sA p
b Sg K
L
grad
A(T)
C
1
.
grad
ci
( 34)
h adsorption equilib rium approaches e
( 3 5)
A
where the adsorption equilibrium con stan t , K A ( T ) , us ually decreases with i rea temperatures accordin g to an Arrhenius temperature dependence nc
s in g
( 36)
For low surface coverage the pin echani s hop
g m
m
Ds � T* A
=
2
=
,
A 2v
2
diffusivity m ay
( E1)
exp -
RT
,
be
p
c om uted
from the site
( 3 7)
where A is the aver g hopping d stance the lin gerin g time , and t he vibrational e enc of the adsorbed mol ecules . C ombin g E qs . ( 3 5) , ( 36) , and ( 3 7 ) gives fr q u
a e
y
i
L*
v
in
( 38)
Thus
surface diffusive inc rea e with decreasing pore size (increas either inc reas e or decrease with temperature , depending on h sign of E� - E! . The surface d f si ity usually depen d s on su rface covera ge and hence on the pressure an d c o po on of the gas phase . Higashi et ( 1 96 3 ) ed c ed that in g S
the ) an d may
flux
s s
g
t e
if u
pr
i t
v
m
siti
al .
( 3 9)
Diffus ion-R eac t ion In terac tions
24 9
where e is the fractional surface coverage . This relation properly p redicts an inc rease in t he diffusivity wit h surface coverage , but is not expected to be valid for e > 0. 9. Y an g et al . ( 197 3 ) derived a modified relation for high values of e . Sladek et al . ( 1974) have shown that surface diffusivity depen ds on the stren gth of the adsorption an d can be correlated wit h the differential h eat of ad sorption . The surface diffusivity of physically adsorbed molecules at room tempera ture is i n t h e range t o - 3 t o 1 0- 6 cm 2 J s . Measured values were reported by B arrer ( 19 6 7 , p . 5 57) , Reed and B utt ( 19 71) , Schneider an d S mith ( 1 968) , and Sladek et al . ( 1 9 74) . In most gas - solid reaction surface diffusion is not an important fac tor . D i ffu s i on i n Z eol i te s
Many import ant in dus trial catalysts c on sist o f small aluminosilicate ( zeolite ) crystals incorporate d i n a porous matrix . T hese c rystals have narrow ( 5 to 1 0 A diam eter) structured c hannels throu gh whic h the reactants and products have to migrate . T he diameter of the pore s is often close to the molecular dimen sion of the reactin g species . T herefore , small differences in the shape of a molecule can have a lar ge imp act on the migration rate , an d this can be utili zed to create large differences in the catalytic selectivity of the catalyst . The diffusional processes in zeolites can affect signific antly the selectivity of cat alysts as only molecules which are smaller than a critical si ze can enter t he pores and only products smaller than this size can leave t he pore s . Moreove r , the restricte d si ze of the cavities prevent forma tion of lar ge intermediate s . Consequently , by a proper desi gn of the pore si ze it is possible to engin eer a shape selective catalyst whic h will produce a desired specie s at a concen tration whic h excee ds significantly the equilib rium concen tration . This will occur if the desire d product can leave the cavity , at which it is bein g formed , at a much faster rate than the other products . For example , when xylene is p roduced by the alkylation of toluen e wit h methanol on a conventional catalyst , three different isomers of xylene are obtained . When the same reaction is carried out over certain zeolites , the product is es sentially pure para - xylene , as the effective dif fusivities of the ortho an d met a isomers are smaller than those of the para isomer by three orders of magnitude . It is c ustomary to a ssu me that the rate of migration in zeolites can be described by an activate d diffu sional process so that ( 40)
even though t hi s is p rob ably not the p roper functional dependence . The reason is that the movem ent in the narrow pores of the zeolite is definitely not ran dom , a s it should be for a diffusional process , as the potential energy of a mi gratin g molecule must be influenced b y the attractive forces of the lattice at leas t when they pass t hrou gh the openin gs b etween the cavities . Moreover , in the case of c ounterdiffusion the flux in one direction may be stron gly hindered by that in the opposite direction , even though this effect is not p redicted by E q . ( 40) . The values of the effective diffusivity are usually determined by gravi metric sorption experiments [ see , e . g . , Gorin g ( 1 973) ] . Riekart ( 1 9 7 1) presen ted a theoretical analysis of this met hod and it s pitfalls . Eberly
250
Luss
( 1 969) used t he c hromatographic technique to meas ure the diffusivities of
various gas e s in zeolites . A review of the various methods used to de termine the effective diffusivity in zeolites w as presented by Eberley ( 1976) . The reported values o f the effective diffusivitie s are in the wide range 10- 6 to 10- 1 4 c m 2 /s . T he temperature depen dence of these diffusivities is stron ger than that for b ulk diffusion an d is usually described by an Arrhen ius temperature dependence . Gorin g ( 19 7 3) found that the activation ener� for diffusion of various hydrocarb on species varied in t he ran ge 10 to 60
e
There exists a t pr sent n o reliable method for p redictin g the effective diffusivity in zeolites . T he diffic ultie s in volved in the development of such a theory are b rought out by Fi g . 4 , w hic h describ es the depen denc e of the diffusivit y in a potas siu m T zeolite on the carb on chain len gth of normal paraffin s . The ex p ri men t s by Goring ( 1973) show that an n - dodec ane molecule m oves throu gh the zeolite lattice at about 1 4 0 times the velocity of an n - octane molecule , despite the difference in si ze . Thus the in teractions of the molecules with the zeolite lattice and cation s m us t be the m ajor fact or a ffec tin g the diffusion , while molecule- molecule in terac tions are likely to be weak and determine mainly if a given p ath is free or b locked .
kJ /mol .
e
r--
u en
0
I
"'Eu
-;;:..: ....-.
\ \
\y
c .!! u
0 u c 0 "'
· ;n
-
10- 12
i:5
10- 13
I
I
10- 11
0
I
1\ ··--
I
2
-. 3
I
i
1\
\
I
5
6
I
� · -
I
\H
:L
\
4
\
L
I
[ I
\
I
\
1\_l
1\ !
Jf_
�
7
):
8
9
10
II
12
13
14
Carbon Number of n - Al kane
FI G URE
3 00°C .
4
Diffusion coefficient s of n - alkanes in p o t a ssiu m T zeolite at ( F rom Gorin g , 1 97 3 . )
D i ffus ion -R eaction I n teractions
251
D i ffu s ion in Li q u i d - F i l l ed Po res
The tr an s p ort in liqui d - filled pore s is us ually described by a Fickian diff u s ion re la t ion w ith an e ffe c ti ve diffu s ion c oe ffic ie n t =
ED AB
( 4 1)
T
The bulk binary diffu sion coefficient is c o mp ut ed by on e o f the s t an dard availab le c orrela tion s ( Reid et al . , 197 7 , p p . 5 6 7- 5 9 5 ) . S u rfac e diffusion may b ec o m e i mport an t wit h stron gly ad sorbin g molec ules , and t hi s contribu t ion has then to b e added to t ha t in th e pores . K omiyama an d S m it h ( 1 9 7 4 ) de scribed a c a s e i n whic h surface diffu sion was the m ai n c ontrib u tion t o
t he s p ec ie s flux .
In certain application s the si zes of the diffu si n g m olecules an d the pores are of the same order of m a gni t ud e . The in terac tion b etween the walls and the mol ec ules ten ds t o decrease the e ffec tiv e diffusion coefficient , w hic h has t o be c ompu t ed by the e quation
=
( 42)
T h e equilib rium p art it ion fac tor , K p , i s t h e ratio bet ween the concen tration of sp ecies A in side an d ou tside the pore a nd d e p en d s on A , t h e ratio of pore to molec ule diameter . For small solv ent molecules , K p satisfie s the relation ( Ferry , 1 9 3 6 )
K
p
= ( 1 - A) .
2
( 43)
When eithe r the so lut e o r t h e solvent adso rb on t h e su rface K p may b e much s m aller t han t h e v al ue p redicted b y E q . ( 4 3) . For example , S atter field et al . ( 19 7 3) m eas u re d a p a rti tion factor of 1 / 1 8 3 for A of 0 . 38 . The r e si s t an c e caused b y dra g exerted on the movin g molecules b y the walls is e x p re ss ed throu gh the drag coefficien t ,
K
r
=
( 1 - A)
m+ 2
where m is bet ween
e
0 AB
=
0 an d 2 .
( 4 4)
C om b inin g E qs . ( 4 2 ) , ( 4 3) , and ( 4 4 ) gives
( 45)
T he values of K p an d K r are st ron gl y affec ted by t h e s ha pe o f the molec ule and any in t e rac t i on b etw een the surfac e an d e it h e r the solute or Thus E q . ( 4 5) is jus t a rou gh approximation for the in fluence the solven t . of A on the re s t ric te d diffusion coefficien t . R ecent expe rimental measu re m en ts and c orrelation s of r e s t ric t ed diffusion c oe ffi c i e n t s were pre sen ted by C han t on g an d Massoth ( 1 98 3 ) an d B altus an d A n d e rson ( 1 9 8 3 ) .
252
Luss
I nt r ap art icle temperatur e gr a di en t s may have an i m por t an t influence on the ac tivity an d s elec tivity o f poro u s cataly st s . A small number of experimental m ea s u re ments of the e ffe c t i v e t hermal c onductivity of p ellets have b een re porte d in the literature an d a com pila tion of mos t of the m can b e found in S atterfield ( 1 980 , p p . 344- 3 4 8 ) . T he d a ta indicate that the effective con duc tivity is usually rather insensitive to c h ang e s i n t h e porosity , p o r e size distrib u tion , and the i m p regn a te d metal. A comparison of di ffe r e n t methods for de t e rm inin g the e ffe c tiv e thermal c on d uc tiv it y w a s p r e sen te d by Hoffman et al . ( 1 97 8 ) At p re s en t there ex i st s no gen e ral t he ory c ap ab le of p redic tin g the ef fe c ti ve c on du ctivi ty o f porous p ellet s a s the m ec hanism of heat tr an s port is rather complex . Butt ( 1 96 5 ) generali zed the random pore model of Wakao and S mith ( 1 9 6 2 ) and deve loped a t heory t h at is ca p a b le of c orrel atin g the e ffec t of chan ge s in the gas t e mp eratu re an d po rosity on t he t h e rmal con ductivi ty Fortunately , the t e mp e ratur e p rofile within most c atalytic pellets is ra th e r uniform , so that the u nc er t ai n ty c o nc e rn in g the value of the con ductivity doe s not h ave an i m p o rt an t impact on our a b ility to p redic t the E ffec t i v e T he rm a l C ond ucti v i ty
•
.
performance o f porous c a talys t s .
A
M a s s a n d H ea t T ra n s fe r C oe ffi c i e n t m as
s
e
o ute the flux o f sp ecie s b e t w een c ataly st . For a b in ary sy stem
the
tran sfer c oefficien t i s u s d to c m p
t he ambien t fl ui d an d the s ur ac e of
f
k
c
( 46)
X
where is the the convective flux . This is equal dilute e be unity . Its exact value is bulk - flow c orrec tion fac tor ,
x
of
diffusion and is close to unity for
to unity
mix tu r
is often as sumed to
(Geankopolis ,
1972 ,
p.
e
for e quimolar cou n ter s. I n application s t he factor giv en by the relation
w hic h acc ounts for t h e i n fl u nce
fac to r
254)
( 47)
X =
where
( 48 )
( RN )LM the case of diffusion through and
-
is
XA
The p r e di c tion
In mean of RN
-
to
X As an d R N - XA b . ( 4 7 ) re duce s
a stagnant film E q .
The special
c oe ffici en t s depen d on the c oncentration s of
the
of m a s s transfe r coefficien t s for m ulticomponent mix tures is
quite complex , as
the
vari -
253
D iffu s ion - R eac t ion In t e rac t i o n s
in this case , an d a s urvey of the subject is p re se n ted in th e m on o g r aph ous species in t h e mix ture .
by C ussler ( 1 9 7 6 , C hap .
In gen eral , approximate met hods must b e used
9) .
The m a s s and h eat t ran sfer coefficien t s between the pellet an d the
ambient flui d are determined by empiric al correla tion s , as the p roc e s ses are too c omplex to analy ze in detail . is customary to correlat e the data in
It
terms of j fac tors , w hic h depen d on the Reynol d s n umber , t he geometry of the pellet , an d the b oun dary c on di t ion s . T hese factors a re defined a s
in
2/3
: (on:B )
k =
(c A�
h
jH
=
\l P CP
( 50)
� )2 / 3
( 5 1)
D wive di a n d U p adhay ( 1 97 7 ) fou n d that t h e followin g relation s r epr e sen te d
well m os t of the available data ab out mas s transfer between a fluid and pel lets in packed beds :
=
� jD w here Re
{
=
1 . 1 068Re 0 . 4 548Re
- O. 72
- 0 · 4069
for Re <
( 52 )
10
( 53)
fo r Re > 1 0
� upd
( 54 )
\l
The heat flux to a p article satisfie s t h e e quation
Q
=
h(T
b
- T ) s
( 5 5)
T his relation is useful for predic tin g the temperature difference bet ween
the surfa c e of a pellet and the ambien t
fluid .
T he value of the hea t tran s
fer coefficien t i s c omputed from t h e j H fac tor for all cases i n which the heat exchan ge b y radiation c an be n e glected . Experimental data show that
the values of
jH an d j0 are essen tially equal ( G upta an d Thodos , 1 9 6 2) .
I M PAC T O F M A S S T RA N S PO R T L I M I T A T I O N S D ete r m i n a t ion o f t h e Effec t i v en e s s F a c tor
T he reactan t s within a porous ca taly tic pellet are con sumed by the reac tion
When the time c on s t an t of t h e diffusion is and replenished by diffusion . s m all relative to that of the reac tion , the c oncen tra tion of the reac tan t is rather uniform throughout the p ellet . However , when the time c on stan t of the chemical r eac tion is small c ompared to that of diffusion , the c oncen t r at i on of t h e reactant w it hin t he pellet is expec ted to be lower than that
254
Luss
close to the exterior sur fac e so that t h e average r eact ion rate ( per un it T hus volume of cataly s t ) depends on the diffu sion - reaction interac tion . we expect the ob servable ( or net ) reaction rate to depend on the ratio b e t ween t h e t i m e con stants of the c hemical reaction a n d t he diffusion within the pellet . The ove rall impac t of t he diffusional resistances is usually assessed by c om p uti n g the effec tiven ess fac to r n , which is defined as the ratio between the ob served reaction rate to that obtained w hen neither concentration nor temperature gr a di e n t s exist between t he ambient p hase an d any p oin t wit h i n t h e pellet . C learly , t he lar g er t h e de vi at ion o f the effectivene s s fac tor
,
,
from unity , the m or e i m p o r t an t i s the in fl u e n ce of t h e tra n s p o r t re sistances . A comprehen sive survey of th e l i tera tu re ab out the effectiveness factor can be fo un d in the m o nogr aph by A ri s ( 1 9 7 5) . We illus trate the p rocedure of c omp u tin g the e ffec ti ve n e s s fac tor b y con siderin g the case of a first -orde r , isothermal , irre ve rsib le A -+- B reaction For the sake of co mpac tn ess , oc c u rrin g within a s p heric al c a t aly ti c p e l l e t . we denote from now on by C and n e the c onc en t rat ion and effective dif fusivity of specie s A . The concentration of the reac tant satisfie s the equation e D 2 r
� (r2 ) dC
dr
=
c
dr dC
=
dr
c
s
at r
=
at r =
0
S ol vin g E q .
- kC
=
1
0
( 56 )
R
( 57)
0
( 58 )
( 56 ) subjec t to
Eqs .
( 5 7) and ( 58 ) we ge t
sinh cp �;; t;; s in h cp w he r e r;
( 5 9)
= R !:..
( 60)
T he p ar a m e t er
=
=
rate in the in te ri or of th e cataly st is rather l o w compared to that at t he exterior shell . T he region close to cp = 1 or ' D ' i s referred to as R t h e in ter m edi ate regi on . =
Diffusion -Reac tion In t e rac tions
255
1.0 �----05
0.2
1J
0. 1 0 05
0.02 0.01
I
Ql
� = R�
5
100
10
D ependence of the effec tiven ess fact or on the T hiele modulus for I n set de sc ribes correspon din g c oncentrati on profiles . FI GURE
a first - order reaction in a sp herical pellet .
The effectivene s s factor for this c ase is given by the expression n =
IP 2
1.-
( 4> coth 4>
-
1)
( 6 1)
The logarithmic grap h of n ve rsus $ is shown in Fi g 5 . As expected , fo r small 4> the impac t of diffusion is sm all an d n is close to unity . For la rge $ diffus ion al limitations reduce signific antly the observed reaction rate and n app roache s asymptotically 3 / $ . The effectiven ess factor has b een com p uted for p ellets of various geo metrical shapes ( Aris , 1 97 5 , C hap . 3 ) . Fortunately , a rapid e stimate of the value of n fo r partic les of c om plex geometrie s can be ob tained by capit ali zin g on the fact tha t if V p /Sx is used a s the characteristic len gth of a generali zed T hiel e modulus A , t h e n the graph of n vers us A i s rath er in sen sitive to c han ges in the geometry ( A ri s , 1 9 57) . Thus the graph for th e s p h erical pellet can be u se d to predic t t he value of n for pellets of othe r geomet ries . The adequacy of this approximation is illus trated by Fi g . 6 , which c om pare s the n versus A grap h for a first -order re action i n a sphere with thos e for a semi -infinite slab and cylin der . Similarly , by use of a p roperly n ormalized Thiele modulus , .
A
=
V r( C ) s P s x
[
2D
e
�q
C
s
]-0•5
( 6 2)
r ( C ) dC
one ob tain s a grap h of n versus $ which is in depen den t of the form of the kin etic s rate expre ssion for either large or s m all A ( A ris , 1 96 5 ; B isc hoff , 1 96 5 ; Pete rsen , 1 96 5) . For an n th - order reaction the normali ze d T hiele ,
modulus is
L uss
256 1.0
�
f:"
If ti .. .. ., c: .,
-.;: u
0.8
0.6
Slob
0.4
Spher
lnfini
>
� w
0.2
0. 1
0.4
0. 2
cal
c
0. 6
pellel
ind r
1.0
2.0
4.0
A = Vp� s. oe
of
60
10 0
for slab , cylin drical , modulus . ( F rom Petersen ,
D epen denc e t h e effectiveness fac tors and spherical pellet s on the generalized T hiele
F I G U RE 6 1965. )
=
II
vP
S
X
(n
+ l)kC
n- 1 s
( 6 3)
and Fig . 7 the l'l versus
of t
shows that
a reactant , such as h
first-order of
o r
A = Vp
�c,n - 1 s.V "'2[;e- -
s fac tor on the o ul s for several nth-order reactions .
F I G URE 7 D e pen de nc e of the e ffec tiv en es T hiele m d u
gen e rali zed
s
257
Diffus ion -R ea c t ion Interactions r( C )
kC
=
(1
+
KC )
( 64 )
2
the T hiele modulus ( Roberts an d S atterfield , 1966) . This occurs w hen t h e surface concentration excee ds the value c o rr e s p on din g to the maximum rate . In these c a s e s a re duc tion in the c on c ent r at ion of the reactant b y the dif th e effec tive n e s s factor m ay exceed unity for s ome in termediate values of
fu sional limitation s can increase t h e r eac t ion ra te o ver t h at c o rres p on din g to the surface concen tration an d the c or r es p on din g n may exceed unity .
F or c ert ain rate expres sion s the in teraction between transpor t and che mic al rate processes can lead to s t e a dy - state multiplicity ov er a boun ded ra n ge of the Thiele modulu s , so that s e ve r al values of the e ffec tivene ss factor may be ob tain ed for certain kin e t ic p aram ete rs and p ellet dimen sion s . Thi s b ehavior is r a th er uncommon an d i t can b e p roven ( Lu s s , 1 9 7 1 ) that i t can not occur for any rate expression that s a ti sfi e s the con dition (C
s
- C)
-
( ��)
< r( C )
for all
0 <
C < C
s
( 6 5)
w here r ( C ) is the rat e of con sum ption of the reactan t A . For exa m ple , c on dition ( 6 5) is satisfied by the rate e xp r e s sion defin ed by Eq . ( 6 4 ) for all C < C s if an d only if K C s < 8 . C learly , con dition ( 6 5) is satisfied for any rate expression that is mon otonically in c r e a sin g function of t h e con c en t ra t i on o f the re acta n t M any r e p e t it ive calculation s of the eff ecti v ene s s factor need to be c a r ried out in c om p ut e r sim ultation s of p ack ed - b e d reactors and param eter e s timation o f rival rate e x pre s si on s S everal methods have b een developed T h e se in c l u de the sin gl e - point for get t in g a ra p id a ppr ox i mation of n . collocat ion m � thod of S te wart and Villadsen ( 1 96 9 ) an d several met h o d s that ca pitali ze on kn ow ledge of t he asymptotic value of n ( C hurchill , 1 9 7 7 ; W edel and I u s s , 1 9 8 0 ; G o ttifre di et al . , 1981a ,b ) .
.
.
.
S everal inve stigators computed the effectivene s s factor u sin g the dusty ga s model to describe t he tran sport within t he p ellet ( e . g . , Abed an d
1 9 7 3 ; W on g an d D en ny , 1 9 7 5 ) . I t is of in terest to k n ow the diffe r t h e results ob taine d u sin g this model a n d that obtained usin g a p seudo - K n u d s en diffusion model . Unfortunately , this difference depen d s o n t he value chosen for t h e effective diffusion co e ffi ci e nt a n d there exists no unambigious w ay of predictin g its value . One pos sible scheme is the use of E q . ( 2 7 ) . T h e few available com p u t a tion a for gaseous mixtures seem Rinker ,
ence b etween
to indic ate that us e of an effec tive K n u dsen diffu sion coefficient u sually gives a reaso n a b le approximation to the effec tiv e ne s s facto r co m p u te d by the dusty - gas m odel . T hi s is prob ably due to the fact t hat diffusion co efficien t s of most gases are of si m i la r magnitude . However , t his point re
.
quires furthe r a n aly si s Many catalytic re ac t ion s involve a c han ge in the number of moles a n d henc e in volume . Rigorous calculation s of t he e ffec ti v e n ess factor for s uc h cases are rat her cumbersome , as t h e y require use of the dusty - gas
flux relation s to account for the c hanges in b oth the composition and total C alcula tion s by K e ho e an d A ris ( 1 97 3 ) in dicate p r es su re within the pellet . that i gn ori n g the momentum ba lan c e an d t ot al p r e s sure g ra die n t does not usually introduce a lar ge error in the value of the e ffe c ti v en e ss fac tor . Weekm an an d G ori n g ( 1 9 6 5) c om p ut ed the effectiveness factor for sym metric pellet s in whic h the reac tion A + mB occurs , ign ori n g th e in flue n c e
Luss
258
of the press ure gradien t on t he tran sport wit hin the pellet . T his simplified model enables a rapid computation of the ratio b etween the effectivene s s factor that accoun t s for the v ol u m e c hanges ( n ' ) to t h a t w hic h i gnores i t
( n ) . Typical results for a secon d - order reaction are sho wn in Fig . 8 . T hey indicate that when the reac tion decreas e s the n umber of moles ( m < 1 ) , the bulk flow tow ard the c enter of the pellet increases t he effec tiveness fac tor above that obtain ed w hen the volum e c han ge is n ot taken in to account (i e n ' > n ) . On the ot her han d , when the reac tion increases t he n u m b e r of moles ( m > 1 ) , the outw ard b ulk flow causes n ' to be smaller t h an n . The influence o f t h e c han ge volum e i s decreased by t h e presence of
. .,
in ert s (i . e . , by a dec rease in
in
xAs ,
the m ole fraction of the reac t an t at the
surface ) . When the influence of ex ternal mass transfer is n ot n e gli gible , the boundary con dition on the s u rface of the c a talys t becomes
c
s
)
=
n
e
(�)
( 66)
an
- s
For a sym metric pellet the sur where n is a vec tor normal t o t h e surface . face concentration is uniform and it can b e s ho wn that for a first - order reac tion ( Aris , 1 97 5)
1 n
=
1 nD ( A )
+
A2
-
Bi
1.6
A - mB
Second-order reaction spherical pellet
1.4
Tj
TJ '
( 67)
m
1. 2 4> • 0
0.5
1.0 0.8 0.6
10
0.4 -I Vo lume
Change
Modulus , ( m - 1 ) X As
FI GURE 8 D ependence of t h e ra t io n ' / n o n t h e volu me c hange modulus for a secon d-order reaction . ( From Weekman an d G orin g , 1 9 6 5 . )
2 59
Diffusion -R eac t ion In te rac tions
where Bi
( 68) m
and nn < A ) is t h e
effec tivene s s factor fo r t h e c a s e o f n e gl i gi b l e external
mass transfer resistance ( i . e . , w he n B i m i s in finite ) .
effectivene s s factor is the
It follow s t hat th e
,
whic h is the re absence o f an ex tern al m as s tran sfe r r es i s t a n c e an d an exte rn al transport resistance of A 2 / B i m · N ot e that in t he pre sence of 2 a mass tran s fe r resistance , th e as y m pt o tic v alu e of n for large A i s Bi m / A r at he r than 1 / fl.
ciprocal of n in
the
s u m of an in tern al resistance ,
•
Aris ( 1 97 5 , S ec . 3 . 9) h a s p roven t h e re markable res ult that E q . ( 6 7 ) i s valid for re a c tion s wit h a rb it r a ry ki n e t ic e x pres s i o n s for large valu e s of the T hiele mod ul us Y ortsos an d T s ot si s ( 1 98 2 ) have generali zed t his
.
m o s t p ractic al prob l e m s B im e xceeds 10 . T h u s , a c co r din g to Eq . the value of n m ay b e a f fec t ed by the extern al m a s s tran s fe r r e
r e s ul t for p e ll e t s w i t h n onu ni fo r m c ata ly ti c ac tivity .
In
( 6 7) ,
sistance only w hen n n i s con side rably sm alle r t h an unity . T hi s ob servation led to the d e ve lop m en t of a rule of t h u mb that it is possible to encounter cases in w hi c h the e x t e rn a l mass transfer resist ance is n e gli gible w hile the internal di ffu s i onal resistanc e is a p p r eci b le but t he c onverse does n ot oc cur . Note that thi s c on c lu s i on w ould not be valid if B i m could attain s mall
a ,
-
values .
a
In tric k le b e d reactors one h as often to c om p ute the effec tiveness fac tor of a li q ui d - fill e d ca t aly t i c pellet , of whic h only a fr c t ion f of the ex terior su rface is wetted by the li qu i d
an d D u dukovic ( 1 98 0 )
Tan and S mit h ( 1 9 8 0) an d Mills
have s hown that t he co r re s p on din g effec tiveness fac
tor for a first - order r e ac t i on
can
b e approxim at ed by the relation
+
n =
.
1
-
f ( 6 9)
,
*
w here CL is t h e concen t ration of the rea c ta n t in the b ulk flui d CL the li qui d phase c oncentration of the re ac t an t in equilib riu m wit h the b ulk gas
concentration , and B i w an d B in th e m ass t ran s fe r B iot n u mbers for the wet and dry p art s of the p ellet , respec tively . H e r skowit z an d S mith ( 1 9 8 3 ) pre
-
sen t e d a c o m p r e h en siv e review of t h e in fl uence of
ness factor of li qui d fil led ca t a lyst s .
w ettin g
on the
e ffec tive
D i ffu s i on D i s g u i se of I n t ri n s i c K i n et i c s
K inetic ex pressions a re b a s e d
.
on e x p e rim e n t s i n w hic h the ra t e is meas ured
as a fu nc t i on of t h e c onc entration o f the reactan ts an d the t e m p e ra t u re If the in fluence of t h e diffusional r esis t an c e on the observed rate i s not t a k en
,
ac c o un t erroneous or dis guised rate e x p r es sion s are obtaine d . The t e m p e ra t u re de p en d e n c e of t h e rate c on stant can u s u ally be de sc ri b e d by the A rrheni u s relation
into
k ( T ) = k ( T 0 ) ex p
[� ( - �)] 1 T 0
( 7 0)
Luss
260
When the rate expres sion is of a separable form , that is ,
r( T , C )
=
( 71)
k ( T ) f( C )
the apparen t or ob serve d activation energy is dete rmin e d from the slope of the natural logarit hm of the meas ure d rate at a specific c onversion again st t he reciprocal of the absolute temperature , that is , d ln r E
a
=
-
R
( 72)
m
T he measured reaction rate satisfies the relation
r
m
=
k ( T ) f( C ) n
Differentiation of E q . E
a
= E + O 5 ·
( 73) ( 7 3 ) and s u b s titution in E q . ( 7 2 ) gives
d In 11 d ln ¢
E + R
(
d ln dT
_?e ) l
( 7 4)
I n the kin etic -controll ed regim e ( s mall ¢> ) , the effectiven e s s factor is e s
sen tially in depen den t o f t h e T hiele modulus a n d accor din g t o Eq . ( 7 4 ) the apparen t and in trin sic activation energies are equal . In th e diffusion c on trolled re gime ( ¢> > 3 ) , d In 11 /d In ¢> + - 1 , so that e A -l
d ln D E
a
=
0 . 5E - 0. 5R dT
( 75)
m The effec tive diffusivity is p roportional to T , so that the last term on the left - hand si de of E q . ( 7 5 ) is eq ual to 0 . 5mRT , w hich is us ually very s mall We c onclude that the apparen t activation in in compari son to 0 , 5E . When the the diffusion -con trolled regime is ab out half it s in trin sic value . external m as s tran sfer resistanc e is not n e gli gible , d In n /d In ¢> + - 2 for
energy
la rge ¢> , so t hat
( 7 6) I n this case t h e apparen t activation en ergy is v e ry s m all an d is in depen den t of t h e in trins ic activation energy . We c onclude that tran sport in trusion s mask an d dec rease the observed ac tivation ene r gy . T h e reason for this effec t is t hat inc reasin g the tempera ture decreases the time con stan t for t h e c h e mical reaction an d t h e effec tive n e s s factor . T hi s , in t urn , dec reases the sen sitivity of the reac tion rate to c han ge s in t he temperatu re . This b ehavior is c learly illu strated in Fi g . 9 , w hic h describes t he logari t h m o f the rate of c u mene crackin g a t various temperatures for c ataly sts of different sizes ( W ei s z an d Prater , 1 9 5 4 ) . The
261
Diffus ion -R eaction In teractions
Q)
a::
0 1:
I
-9
... :: ' E ...
�� 10 - 6 .
0
,�8�� \�
· O
��
Values
E�peromental
I
0
19.� --�------t""'"--". --t------j ·05: '"a
.2 t>
c Q) a::
' "'s
Cit)
..... ....
- --
1o- 7
...... ..... ....... '
I
o .....
420 " C
1. 3
1 .4
' 0' -..
..... ,
...... ......
... .....
1.5
Te m perature
-
-1T° K
,
I � --- ....
x
1.6 10 3
FI G U RE 9 I n fluence of temperat ure an d p article si ze on the c rackin g rate of cumene on Si0 2 - A1 20 3 catalys t . ( F rom W ei s z an d Prater , 1 9 54 . )
data show t hat the s lope of the g r aphs , or e q uivalen tly the a p p a re n t activa 2 a s the p a rt ic le ' s ra dius in c re a ses from · o . 0056 em to 0 . 1 7 5 e m . Diffusional intrusion s may also di s gui se the functional form of the rate expres sion . C on s i de r as an illustration the case of an nth -order reaction for w hich the e x t ern al resistanc e is ne gligible . H ere
tion en ergy , decreases by a factor of about
( 7 7) The slope of the loga rithm ic graph of the o b s e r ve d rate again st t he concen tr a t ion of the r eac t a n t is d e fi n e d as the apparen t reac tion order n ' : n' =
d log r
m
d log C
=
n
+
.!.
d log n 2 d log A
{n �
_
1
_
d log D
e
d l og C
)
( 78 )
I t follow s that the a p p a re n t a n d true or d er s are eq ual i f the experi men t s are c a r rie d o u t i n the kinetic - controll e d regime i n w hic h 11 is essen
I n the diffusion -con trolled re gi me the asy mptotic tially in depen dent of A . slope of d lo g n / d log A is - 1 and the app arent order d e p en ds on the re lation b e t w een the c on c e n tration of the lim it in g reactant an d the effective diffusivity . When the chan ges in the c oncentration of the reac tant do not affec t n e , E q . ( 7 8 ) p r e dic t s that
n
,
=
n + 1 --2
( 7 9)
Luss
262
Thus t he apparent an d true orders are differen t for all but first -order re act ion s . The diffusion al in trusion s disguise the true order an d s hift it toward unity . F or example , a secon d - order reaction behaves as if it is of order 1 . 5 . When the change s in the reactant' s c oncen tration are accomplished by vary in g the total pressure of a gas eou s mixture and if gaseous b ulk diffu sion is the dominan t mechani s m , then d log D
e
d log C
=
e d log D d log p
=
_1
Sub stit utin g E q . ( 8 0 ) in to E q . ( 7 8 ) we get that for large A , n'
=
!!..
( 80)
( 81)
2
T h u s in t hi s case diffusional in tru sion s dis guise the o rder o f all b u t zero order reaction s . We c onclude that it is e s sen tial to account p roperly for the in fluence of diffusion to avoid pitfalls in the analysis and m o d elin g of kinetic data . E x p e r i m en ta l D i ff u s ion a l
of
D etec t i on a nd D ete rm i n a t i on
Li m i ta t i on s
S everal experimental methods exist for detectin g diffu sional limitation s an d p redictin g the effectiven ess fac t o r . T h e m ost direct ( b u t len gthy ) method con sist s of determinin g the isothermal reaction rate for pellets of differen t
si ze s at a fixed mixture composition . The experiment s a r e contin ued un til pellets of two differen t sizes have the same activity , which is a s s umed to be the intrin sic one . The effectivene s s factor for lar ger pelle t s is the ratio between t heir activity and the intrinsic one . T h e experimen t s can b e carried out either in a diffe ren tial reac tor o r a w ell - stirred hetero geneous reactor or a recycle reac t or . T his test is m eanin gful only if all the pellets have the same intrin sic activity . T hi s can be accom plished by crushin g lar g e p articles to smaller sizes . T his m et hod may lead to pitfalls w hen the p ellets have a non unifor m activity p rofile . For example , if a c atalyst has a very narrow active e x te rior shell an d a n in ert core , c u ttin g t h e p ellets doe s n o t affect the ob serve d activity even if diffusion limit s the r ate in the exterior shell . A more e fficient method of dete rmin in g TJ is b ased on the fac t that if an isothermal reaction is c arried out in sp herical pelle ts of two differen t sizes , then
( 82)
It follow s fro m E q . ( 6 1 ) that for a firs t - order r eaction the ratio between the two ob served rates satisfie s the rela tion 3A 1 coth 3 A 1
3A
2
coth 3A 2
1 1
( 83)
263
D iffusion -Reac t ion I n teractions
T he t w o e q ua ti on s ( 8 2 ) an d ( 8 3 ) can be solved for A 1 an d A 2 , w hic h in turn can be u s ed to c ompute 11 1 and 11 2 . The method is very sen sitive t o e xp e ri m e n t al errors if b o t h experiment s are conducted in t he di ffus io n controlled re gime [ i . e . , w hen ( rm ) l / ( rm ) 2 "' R 2 /R 1 l . In this case addi ti on al ex pe ri m en t s with smaller pellets need to b e carried out . T h e ad vantage of t his met hod is t h at , in prin ci p le , two e xp eriments suffice to determine t he H ow e ver , to min i mi z e the influence of inheren t ex p eri e ffec ti v en e s s factor . men t a l errors , it is p referable to con duc t ex p e ri men t s with three or four di ffe r e n t particle si ze s and determine the Thiele m od ulu s an d effectivene s s factor t ha t fit b e s t all t he ex p e ri men tal data . is ri go r o u s ly correct only for a fi r s t - or d e r reaction . H ow ever , w hen t h e ge n era li ze d definition o f t h e T hiele m odulus i s us ed [ E q .
Equation ( 83)
( 6 2 ) ] , the effec tiv en e ss factor is a r a the r in sen sitive function of th e kinetic expression if the rate is a monotonic function of the r e acta n t ' s concentration . Thus this met hod gi ve s a r ea s on ab l e estimate of 11 fo r many non - first -order r eac t i on s . Weis z an d P r at e r ( 1 9 5 4 , p . 1 6 7) have shown t hat the effecti v en e s s fac tor for an n t h - o rd e r reac tion could always b e related to the p ar a me t er f1
(3V )2 _____E_ s
( 84)
X
which con tain s only measurable quan tities . T hey n oted that for rea c ti on s of order zero , one , an d two , 11 exceeds 0 . 9 5 if
in C s . U sin g s in gul a r p e r t u rbat ion s ( Wedel an d L uss , 1 98 0 ) , it can be shown t h a t 11 devia t e s fr o m un ity b y less than 5% for a s p h e ric al pellet if
where R ( C A / C A ) is a di m en s ion l e s s n or mali ze d rate [ i . e . , R ( l ) s
( 8 5) =
1] a n d
R' i s its derivative with respect t o the dimen sionles s c oncentration C N ote t h a t for a fi r s t - o rd e r reaction , R' ( l ) equal to the W eis z - Pr a t e r c ri t e rion . H orak ( 1 967 ) su gge sted that 11 > 0 . 9 if
=
1 and E q .
A
/C
A
( 8 5) is e s se n t i al ly
s
.
( 86 )
w he re l;, is the c on ve r sion for whic h the r eac t i on rate is 0 . 9 of r ( C s ) . For exam p le , E,: is 0 . 1 for a fir st - orde r reaction , so that accordin g to E q . ( 86 ) , 11 > 0 . 9 if
2 + C
-+
2CO
L uss
264
the rate of w hi ch satisfies
r
=
1 + 4P
CO
t he
+ 5 0 , O O OP
e q u a ti on
( 87 ) CO
For a fee d of pure carbon dioxide a t 1 atm , E;. is equal t o 0 . 5 5 x 1 0- 5 and Eq . ( 8 6 ) predic t s that n > 0 . 9 on ly if � < 3 . x 1o- 5 , while Eq . ( 8 5) pre dicts t hat n > 0 . 95 only if x l o- o . B ot h predictions are much s m aller than the val u e of unity p redic te d by t he W eis z - P rater diagnostic . We conclude t h a t ( a ) if � is lar ger than unity , diffu sional li mit a tion s de fin i t ely exist ; ( b ) if � is m u c h s m aller t han unity , diffusional in trusion s
2
� < 3. 75
3
ar e absen t ; an d ( c ) if � is sli ghtly smaller tha n unity , knowledge of the for m of t he rat e expression is needed to dec ide if d iffu s ion al limitations affect t he ob se r v ed ra t e . The p r es en ce of d i ffu s ion al intrusion s may som etimes be detected by This may be ac c o m p lis he d c han gin g the e ffec t i v e diffu sion coefficien t . when gaseous diffusion domin ates by u s in g reac tion m ix t u r e s c on t ai n in g a large e xces s of an in ert diluent . If the same rate is obtained for mixtures with in e rt s of w idely different m olec ula r wei ghts , an d henc e binary diffu sion coeffi c ie nt s , it c an be assumed that in trapa rtic le diffu sion is not lim it This test cannot be u s ed for nonisothermal r e act ion s since in g the rate . the difference in the inerts leads to differences in t h e h eat transfer co efficient s , and the difference in the p ell e t tem p e ratures may mask the res ults . Koros and N o w ak ( 1 96 7 ) p ropos e d to use as a dia gn o s tic test for the pre s en ce of diffusion al limitations the fact that t he reaction rat e can be p ro They port ional to the number of ca t a ly tic sites only in the kinetic re gime . sug ges t ed to p repare c a ta ly s t s with diffe rent n umber of ac t ive sites but similar diffu sional properti e s either b y p ell e t i zin g mixtures of cataly s t p a r ti cles with inert powder or by inp r e gn at in g the cataly st with different amount of the active componen t . C ertain reaction s are s tructure sen sitive , so t h a t their in t rin sic activity depen ds on the c ry s talli te size s . Thus a failure to satisfy this test doe s not al w ay s imply the p resence of d i ffu s ion al limitations .
A det aile d disc us sion and illustration of t he a p p lic a tion of this test was pre sented by Ma do n and B oudart ( 1 98 2 ) . The external mass tran sfer resistance i s ass u med to be ne gli gi b l e if t he conversion does not chan ge upon an increase in t h e lin ear velocity in an is ot herma l p acked - bed reactor the len gth of w h ic h i s adjusted to give a con stant residence time . In n on i s o t h e r m al cases thermal gradient s m ay mask the results of thi s t est . In ga s - li q uid slu r ry re ac t or s the test c on si st s of c heckin g the con s t an cy of the rate as the a gitation is increased . C hambers an d B o u dar t ( 1 966) poin ted out that at the low Reynol d s numbers e n cou n te r ed in many labo rato ry reactors , the transport coefficien t s may be rather in sen sitive to changes in the flow rate , and this diagnostic test m ay lead to erroneous con clusion s . Fi gure 1 0 ill u s t rat e s the ap plication of t hi s diagnostic test t o the oxid ati on o f S0 2 u sin g cataly s t p articles of t wo different sizes an d beds of two different le n gt h s . The external mass transfer re s ist anc e is n e gligible in t h ese case s , as chan ges in the velocity do not affect t h e conversion . Pore diffusion limitation s are p re s e n t at 4 7 0° C , as the con ver sion depen d s on t he s i z e o f the pellets .
Diffus ion -Reac tion I n terac tions
265
l UI ..J
IS
�
�
(k: IJJ > z 0 u
iii
W/F, g CAT /( M OL 5 02 /HR)
F I G URE 1 0 I n fl u e nc e of lin ea r velocity and particle size on the c on ver sion of S 0 2 . ( F rom Dow den an d B ri d ge r , 1 9 5 7 . )
Mears ( 1 97 lb ) has shown that for an nth- order reaction the in te rph a se c on c en t r a t ion gradie n t can be i gn ored if v
_E._ < 0 . 05 s x
( 88)
I t is sometimes d iffic ul t to predict t h e value of t h e mas s transfe r coefficien t at t he low v e lociti e s used in l a bora t o ry reactors . This value may be deter mined from a mass - trans fer- limite d reac tion rate m e as u red at elevated
t emperatures .
I n fl u ence of D i ff u s i on on Mu l t i rea c t i o n N etwo r k s
A c o mp re hen s iv e a n aly si s of t h e diffu sion - reaction in teraction for al l possible
reaction networks and kin etic s is obviously a n im p o s sib l e task . Followin g ( 1 9 5 1 ) , we examine the impact of diffu s ion limitation s on the y iel d of a de sired product for three t y p ic al isothermal two- reaction networks an d illustrate how the knowledge of the sin gle - reaction case may be exploited to p r edi c t the q u a li t a tive features of more complex reaction n et wo rks . Con sider first two parallel , in depen dent , first - order r eac tion s of whic h only t h e first is desirable : Wheeler
Luss
266
An example of s uc h a case is t he hydrogenation of ol efin s in t he p resence of a romatic com poun ds We define the selectivity a s .
( 89)
In a ny useful catalyst k 1 » k 2 , so that the tim e c on stant for the fast re action is much s maller than that for t he secon d reac tion and the rate of the desired reaction is affe c ted by t h e diffusional limitation s m uc h more than is the rate of the un desired reaction . In the a b se nc e of di ffu sional limita t ion s n 1 = n = 1 , so that
2
( 90)
When diffus ion limit s both reaction s ,
( 91)
and the s electivity is given b y the relation
( 9 2)
Thus the diffusional resistance taxes the desired reaction at a hi gh er rate than it does the un desired reac tion . We con cl u de that it is e ssen tial to minimi ze t he diffusion al resistance to get a high selectivity for this reaction n et work . N ext consider t wo con secutive fi rst - order reac ti on s
The species b alances are de scrib e d b y the diffe rential e q u a tion s
( 93)
2
'17 C whe r e 1
1\ .
=
2
=
_£. v
s
X
2 "- 1c 1
2c A 2 2
( 94 )
0
# i
i
=
1, 2
( 9 5)
267
Diffusion -R eaction Interactions
0
=
The corresp ondin g b oun dary con dition s are 1
1S
i
c. = c.
=
( 9 6)
1, 2
Sol ution of E q s . ( 9 3 ) an d ( 94 ) gives the poin t ratio of the two fluxes ( i . e . , the differe ntial yield of c omponent 2 ) as
( 97)
where
( 98 )
an d n . is the effectivene s s factor for a sin gle first - order reaction havin g a 1 T hiele mod ulus of A i , When diffu sional resistances are negli gible , n i -+- 1 an d E q . ( 9 7 ) p redicts that
( 9 9)
and as expected the differential yield is in depen den t of the diffu sional processe s . H owever , in the diffu sion -controlled regime , n i A ? + A i ( for 1 , 2 ) and the p oin t yield approaches asymptotic ally the .Jalue i =
dC dC
2s 1s
=
( 100)
For all useful c ataly sts s » 1 an d t h e a symptotic yield in t he diffusion controlled regime is m uc h lower than that ob tained in the kin etic regime , Equation ( 97 ) in dicates that the differen tial yie ld depen ds on the local C 2s /C 1s ratio . The concen tration of the desired intermediate at the exit of an i sothermal p lu g - flow reactor is foun d by inte gration of E q . ( 9 7 ) to b e
L u ss
268
w he r e
( 102)
p
P i s equal to 1 / s when the diffusion al resistance s are ne gli gib le a n d to
� in t h e diffus ion - c on trolled regim e . An exp erimen tal ill u stration of the s hift wit h in c reas in g diffu sion al resistance of the in t e gra l yield ( at a fixed conversion level ) from on e a sy m p to t e to an o t he r is s hown in Fig . for the oxidehy drogenation reac tion
11
B u tene
-+
b ut a dien e
+
CO
By differen tiation of E q . obtain e d w hen
P( o s
+
2
( 1 0 1 ) we fin d that the maxi mal yield of C 2 i s
s
1)
--- +
cracked p roduc ts
C ( 0) 2
p p
s
1 p
os
c ( 0) 1
P ;__ _ o.,.-1
( 103)
1
N ote th a t w hen 6 the expre s sion s for C 2 ( L ) / C 1 ( 0 ) in Eqs . ( 1 0 1 ) and ( 1 0 3 ) si m p li fy con side rably . An in c rease in the diffusional re si st a n c e s ( in creasin g A i ) decrease s t he c onve rsion a t whic h the maximal yie l d i s ob tain ed an d reduces its valu e , as ill u s trat e d by the example s hown in Fig . T hus it is not pos sible to c omp en sate for los s of yield due to diff u sional limitation s by in cr e asin g the conversion or equivalen tly the reactor =
1,
12.
�. o 0::: 0 iii ... Cll > c
90
8
80
0
70
m
0
60
'ii >-
50
� 0
I() f() Cll c Cll
'6 "5
0 "0
0
Butene - Buta d i ene - C02 j_ _ _
2
_
4
6
P a r t i c l e D i a m eter ,
FI G U RE 1 1
mm
8
10
D e p e n d e n c e of the yield of b u ta die n e on iron oxi de c ataly s t ( F rom Voge a n d Morgan , 1 9 7 2 . )
p article si ze at 6 2 0 ° C .
Diffus ion -R ea c t ion I n terac tions
§-
.8
u
.6
u
.4
.....
..J
..
269
A , � A2!'A 3 s 16 8=I =
C2 (0)
.2
=
0
''DIFFUSION CONTROL
0 0
.2
.4
.6
.8
Conversion
FI G U R E 1 2 Dependence of the yield of the desired produc t on the con version in the kinetic an d diffu sion -con trolle d regimes . (From C arberry ,
1 96 2 . ) selec tivity m ay be m uch more c ritical than its effect on the conversion . When the interparticle ma s s tran sfer resistance is not n e gli gible , the yield for sy m m etric p ellets may be expre ssed in t e r m s of the effectiven ess factor for a sin gle reaction . T he yield i s foun d by in te g ra t ion of E q . ( 104), which is a s im ple modification o f E q . ( 97) ( C arb erry , 1 96 2 ) : len gth .
W e conclude t h at the imp ac t of mass tran sport r e s i st an c e on the
( 104) Van d e V u s se ( 1 96 6 ) h a s presented a similar analysis for case s i n w h ich the first reaction i s o f orde r n w hile t he secon d is of order m , where n , m = 0, 1, 2. The analy sis ab ove in dicates that the yield for two con secutive first order reaction s c an be expressed as a function of the effectiveness factor for a sin gl e first - order reaction . Wei ( 1 96 2 ) w as the first to develop a scheme for predictin g the yield fo r a general network of first - order reactions in terms of the e ffectiveness factor for a sin gle reaction . A different approach to the solution of t h e same problem w as presented by Aris ( 1 97 5 , p . 3 4 5 ) . Consider now the two simultaneo u s reactions ( de sired , of order n )
( un desired , of order m )
Lu.ss
2 70
occurrin g in a porous c a talys t . A s an an alog to E q . ( 90) , we de fine the se lec t i vi t y to b e the ratio b etween t he rate of forma tion of the desire d to the un desire d pro duc t :
/
s
=
v
n
k 2A
=
m
J
dv
p
J v
k A l
dv
n
dv
v
so
J v
p
( A /A ) s
( 105) ( A /A ) s
m
dv
p
where s 0 t he in trin sic selec tivity is defined as
so
=
k n-m _!_ (A ) k2 s
( 1 06 )
W h e n both reaction s a r e of t h e same order , the ratio b e t w een t he rate of formation of P1 an d P 2 i s in dep en den t of t h e c oncentration of A , a n d the sel ec tivi ty is e q ual to k 1 /k 2 • In this case the diffu sional resistances affect only the rate of c on s u m p tion of the reactan t . When the two reaction s are of different orders , the selectivity is equal to the in trin sic one only when no diffusional limitation s exi st ( i . e . , when the concentration o f t h e reactan t is u ni for m throu ghout the p elle t ) . How ever , w hen diffu sional limitation s exist , we expect them to tax at a higher rate the reaction with the hi gh er o rder . T hus diffu sional re si s t anc es can either increase or decrease the s e lec tiv ity dep en din g on t h e sign of m n. T he in tuitive gu ess is ve rifie d by E q . ( 10 5) , whic h show s that the selec ,
-
tivity is en hanced by diffusional limi t a t i o n s if the order of the desired re
ac tion is less than that of t h e un desired one . The converse i s true w hen n is larger than m . I t is not possib le to ob tain a general , explicit ex p res si on for the selec tivity of this r eac t i on n etwork . Robert s ( 1 9 7 2 ) derived some e x p licit asy mptotic expres sion s for the selec tivity an d the int e rested reader is re ferred to that p ap er The an al ys is of the three two- reac tion net works illustrates that it is often possible to predic t the qualitative influ ence of diffusion on the yie l d of a desired p roduct from knowle d ge of its impact in the sin gle - reaction case . This ability to p re dic t the qualitative in fluence of various variables on the yield is very use fu l . Ob viously , a quan titative prediction of th e yield requires a n umerical solu tion o f t h e g ov ern in g sp ecies balances . N u m erical p roce dures for solvin g these equation s are desc ribed in C hapter 1 . In c ertain molecular sieve cataly sts ( zeolite s) the e ffec t ive diffusion coefficients of so m e sp ecies may di ffe r by s everal orders of m agnit u d e from t hose of the othe r . I n these cases di ff u si on al limitation s may have a p ro foun d in fluence on t h e product d i st ri b u ti on an d on the yield o f t h e desired produc t . For example , t h e xylene mixture p roduced by alk yl at ion of toluene on a zeolite Z C M - 5 c ataly s t can have para - xylene far in e xc e s s of the bulk gas - phase equilibrium c oncentration . Wei ( 19 8 2 ) has anal y zed this s e l ec ti v i ty enhancement an d showed that it was caused by the large difference in the diffusivity between the para - xylene and the o r t ho - an d me ta - xylenes . .
2 71
Diffusion -Reac t ion In terac tions I N F L U E N C E O F T EM P E R A T U R E E F F E C T S
H eat release o r consumption b y chemic al reaction s may lead t o intraparticle and interphase temperatu re gradients , w hich can have an important impact
The impact of these temperature gradien t s m ay be more p ronounced than that of the concentra tion gradien t s . W hen a sin gle chemical reac tion occu rs in a symmetric catalytic pellet , the maximal intraparticle temperature M ax ( T ) and the surface temperature T s m ust satisfy the relation s (Lee an d Luss , 1 9 6 9 ; C arberry , 1 9 7 6 , p . 2 3 1)
on the ac tivity an d selectivity of cataly tic p ellets .
Max ( T ) - T s ---=----=- < B ( 1 - 4l * ) b T b T
T
s
b
B * *
Tb where
B
r
=
b
=
B* B b
( 107)
( - ll H ) D C
( 108)
e
b
B*
e
:ll. T
=
h
k
c
b
A D
Bi
e e
=
Bi
=
( - ll H )k C C b
hT
b
( 10 9)
m h
' N ote that ali the term s on the right - hand side of Eqs . ( 1 0 7) an d ( 1 08) are observable quantitie s . When * = 1 , Max( T ) - T s and T s � T b ( 1 + B *) ( i . e . , the pellet has a uniform temperat ure which exceeds by B*T b that of the bulk phase ) . When 4l * � 0 , a x ( T ) < T b ( l + B b ) and T s - T b and most of the temp erat ure ris e is within the pellet . Di vid in g E q . ( 1 0 8) by E q . ( 1 07) gives a bound on the ratio between t he inter - and i n t r apha se temperature gradien t s ,
M
- T b s r <�> * -,(;-:;T:-,),-----: T::: ;> 1 - * =M ax s T
( 1 10)
In practice the anticip ated values of B b are in the ran ge 10 - 3 to 0 . 3 , while those of r are in the range 10 to 10 4 for gas - solid systems and 1 0 - 4 to 1 0 - 1 for liquid- solid systems ( C arberry , 1 97 6 , p . 2 30) . I t follow s from Eqs ( 1 1 0) and ( 1 0 9) that in gas- solid systems a large temperature difference may exi s t b etween the su rface of th e pellet and the b ulk gas . For example , Maymo an d S mith ( 1 96 6 ) measured a 3 0 0° C in te rphase temperature gradient durin g the oxidation of hydrogen over a platinum catalyst . In liquid- solid sy stems the interphase temperature gradient is usually s mall b ecause of the low value of r , and most of the the rmal resistance is within the catalytic pellet . T his intrapellet temperature gradien t is u sually small , as the value of B b i s rather s mall for liquid- solid sy stem s .
Luss
272
The foregoin g c riteria are mos t useful for estimatin g w hether any of the temperature gradient s are negli gible . T his information is very useful , as the analysis of a limitin g case in which either the in trapellet or the in ter phase temperature gradients are n egli gible is much si mp ler than that of the gen eral case . S i ngl e - R ea c t i on C a s e
Interphase temperature gradients are usually the dominant ones for gas solid reac tions . We examine here the impact of the interphase gradients on an nth - order catalytic reaction A + B occ urrin g in a porous p ellet in w hich the intraparticle gradien ts are ne gligible . T he correspon din g species and energy b alances are s k (c X C
b
- c )
=
S
v
p
r( c
S
,
( 1 1 1)
T ) S
( 112)
Combinin g the t w o equation s gives c
-
c
s
b
=
+
1
B*
-
y
B*
s
( 1 1 3)
where T
y
s
=
s
Tb
( 1 14)
Substitution o f E q . ( 1 1 3) into E q . ( 1 1 2 ) gives a sin gle equation describ ing the steady - state temperature . For example , for a first - order reaction with an A rrhenius te mperature depen dence the dimen sionless steady - state temperature satisfie s the equation y
8
-
1 = Da( l
+
B*
-
y )X (y ) s s
�
( 115)
f(y ) s
where Da
=
y = RT
k(T )V b
E
___
C
k S
X
_.P ._
-
( 1 16)
b
We define the effectiven ess factor to be the ratio between the ob served rate an d that obtained at b ulk fluid con dition s . Thus if intraparticle gra dients are absen t , r =
m
( 117)
To illustrate the calc ulation o f n , we consider t h e c a s e of an irrever sible nth - order reaction for which E q . ( 1 1 7 ) becomes
273
Diffus ion -Reac tion In terac tions
n
=
( c s ) X (y s ( >
n
cb
+
1
=
S*
S*
-y) s
n ( 118)
X(y ) s
E quation ( 1 1 1 ) may be rewritten a s
cs
1
-
4> *
( 1 19)
Sub s titution of this result an d E q . ( 1 0 8 ) in to E q . ( 1 18 ) gives ( 1 20)
Fi gure 1 3 describes the depen dence of the effectiveness factor for a first -order reaction on the ob servable p a rame te r 4> * . I t is apparent th a t in certain cases the effectiveness factor exceeds unity , as the in cr eas e in the reaction rate constant due to the interphase temperature gradien t over compen sates for t h e decrease in the rate due to the concentration gradient . This surprisin g effect is encountered m ain ly in highly exothermic reactions T he external transport resistances in laboratory an d pilot plant packed b e d reactors are larger than thos e in in dustrial reactors , as t he former operate u sually at a muc h lower linear velocity . T hus it is e s sent ial to account properly for t he impact of temperatu re gradien ts in the analysis of kinetic data and r eac tor desi gn .
.
1 0 0 �------.
Y= ER Tb
g
S>
-..... .. ...
tE
...:
10
=
20
LINEAR K I N E T I C S
0 II
E="
. I L-� .I .00 1 .Of
clS* --
FI G U R E 1 3
- ( VP ) rm
C b kc
D e p en d en ce
-
����--��_u�
of the external effectiveness factor on � * an d S * . Sx
(From C arberry and K ulkarni ,
1 9 73 . )
2 74
L uss
The interaction b e tw een t h e ex t ernal gradie n t s an d the intrin sic kin etic s may lead to the existen ce of m ul t ip l e steady - state solution s for the same s et of parameters . For ex am p le , if for a fi rs t - or de r reaction
( 121)
three steady - state solution s exist for all Damk ohler n u mbers in t he re gion �--�
Y
min - 1 >
f(y
min
w he re
)
Ym ax , m in
Da
>
Y max - 1 f( y
m ax
( 1 22)
>
y ( 2 + 13 *) ±
/ y 13 * [ y f3 * +
2( 1
13 * )
4( 1 + 13 * ) ]
( 1 2 3)
Similar criteria may be de riv e d for r e ac tion s with different rate exp ression s ( C han g and C alc , 1 979 ; T sotsis and Schmit z , 1 97 9 ; Leib and Luss , 1 981 ; Luss , 1 98 0 ) . The existence of s teady - state multiplicity may lead to a hy s t ere s is in the reaction rate or surface temperature when one of the ope rat in g condi t ion s , such as the ambient concen tration or temperatu re , is slowly c hanged , F i p:u.re 14 illus trates this b ehavior for the oxidation of b u tane on a platin um wire ( C ardos o an d Lus s , 1 96 9 ) . The gas temperatures at which a sudden inc rea s e or decrease in the wire ' s temperature occurs upon a sli ght change
l
700
�
ci
t!· E
.. u a
500
�1
:::1 (/)
't:
en 1-
300
G
100
0
1.5 %
% B utane
900
100
•
25.6 o m / e m m • n
200
300
Tb , Ambient Gas Tempe rature
FI GURE 1 4 Dependence of catalyst wire surfac e t e m p erat ures on gas t e mp e ra tures an d b utane concentration s . ( From C ardoso and Luss , 1 96 9 . )
275
Diffusion -R eac tion Interactions
in the ambien t c on ditions are r efe r re d to as t he ignition and ex tinc tion temperatures , respectively . We exa mine n ext the more c ommon case in which the pellet temperature is uniform b ut di fferen t from the ambien t one , while the intraparticle c on centration gradi en ts are not n egligible . The sp ec i e s and e n e r gy b alances for an n th - order reaction are
( 1 24 )
X
S h(T
S
=
- T ) b
Combinin g E q s .
-
J
ll H )
v
k(T )C s
n
( 1 25)
dv
p
( 1 2 4 ) and ( 1 25) gives
1 + B*
=
(
-
y
B*
s
( 1 26 )
Sub stitution o f E q . ( 1 2 6) in to E q . ( 1 2 5 ) an d rearran gement gives the followin g equ ation for the dim en si on l ess surface temperature :
- 1
Ys
w here =
Da
11
2
=
_£_
V s
(y ) s
X
=
S *DaX ( y ) s
k(T k
b
)C
c+
n- 1 b
c
11 2 ( l ) X (y )
s
i\
(
8* - y
2( 1)
B*
)
n
s
r) __E. S
=
n . [ A (y s ) l 1
2
k(T
b D
x
1 + 8* -
Ys
8*
)n
lC
e
1
n
b
( 127)
-1
( 1 28 )
an d ni ( 1\ ) is the e ffec tiven e s s factor for an nth-order isothermal reaction . Equation ( 1 2 7) can be solved for y s and the correspon din g 1\ an d n . . An analysis of this m odel in d icat e s that most of it s qualitative fe1 tures are similar to those of the lumped - parameter model E q . ( 1 1 5) , T he overall effectiven es s factor is the ratio b et w een the observed to a m bi ent reaction rates ,
n
n =
k(T )C n. 1 s s
k( T )C b b
n
=
B* B*
( 1 2 9)
and its value may exceed unity . T h e main influence of the p ore di ffu s ion to reduce ni fr om the value p redic ted in their a b sen c e . The model predic ts that steady - state m ultiplicity may exist i n certain cases . I t is diffic ult to derive s impl e explicit criteria pre dic tin g the limitation s is
,
L uss
2 76
parameters for which the uniqueness - multiplicity transition occurs , but it is relatively easy to obtain bounds on t he parameters for whic h either uniqueness or multiplicity exists . For example , for a first-order reaction a unique solution exists for all Da and A ( if ( Van den B osch and Luss , 1 977)
1)
y < 4 ( 1 + s�)
while multiplicity exists for some D a an d
1\
( 1) if
(130) ( 131)
Sharper b ounds may be obtain ed for specific 1\ ( 1) and similar criteria may be derived for nth - order reaction s . S tudies by Nielsen and Villadsen ( 1 984) and Hu et al . ( 1985a , b ) have revealed that this model predicts that up to five steady - state solutions exist for some nth - order reactions an d set of parameters . In contrast , the lumped model p redicts that at most three solutions can exist for any nth - order reaction . When the interphase concentration and temperature gradients are negli gible , the steady- state species and energy balances are
2 l.e'i7 2T
D e ll C - r( C , T ) +
0
=
( - il H ) r ( C , T )
=
0
( 132)
( 133)
subject to the boun dary con dition s =
C
C
and
s
T
=
T
on the s urface
s
( 1 34)
Combinin g the t wo equation s gives c
1
c
+ B
=
s
-
( 135)
y
B
where B
=
2
y
where
s
e D _
____
e
,_ T
s
y
=
T
T
( 136)
s
the steady state is the solution of a sin gle equation of the form
Thus
'i7
( - ll H ) C
2
=
+ R
2 B IP f( y ) = 0
2
r(C
s
C
D
s
,
Ts)
e
( 137) (
138)
277
Diffus ion -R eac tion In terac tions 1 0 0 0 �----�---.--,
�
2 u
� "' "' "' c: .,
-�
100
10
u
.2! w
I
10
r/J = R� N onisot he rmal effectiven ess fac to r s ( From W ei s z an d Hick s ,
FI GURE 15 in
a spherical pellet .
f( y )
The correspon din g
11 =
1 v
p
J
v
s
r( l , T )
s
effectiveness
f(y)
fi r s t - o r d e r
reaction
r(y , T )
r( C , T ) r( C , T ) s s
=
for a
196 2 . )
( 1 39) fac t or is ( 140)
dv
p
N umeric al calculation s o f the effec tiven ess factor for a first - order re act ion in a sphere by W eisz an d Hicks ( 1 96 2) ( Fi g . 1 5 ) indicate that t he effectiven ess fac t o r may exceed uni ty , due to the competin g influenc es of the intraparticle temperature and the concentration gra dien t s Moreover , st ea dy st at e multiplicity may be caused by the in t e rac t ion of the diffusion reaction r a t e processes . Luss ( 1 9 6 8 ) has s hown t ha t a sufficien t condition for the exi stence of a uniq ue solution for all cj> 2 is t h a t .
-
(y
_
1)
d In f( y ) dy
<
1
for
all
1< y<
1 + f3
F or t h e c ase of a first- order reac tion , crite rion n ess for all
( 14 1 )
( 14 1 )
guarantees unique ( 14 2)
Luss
2 78
This c rit erion is us ually s atisfied , as 13 i s most cases rather small . Methods of boundin g the values of
hi gh values of y and 13 w hich are n ot enc oun tered in pr ac tic e ( Co pelowit z W hen both the in terp has e and in trap ar tic l e gradien ts an d A ri s , 1 97 0 ) . have to be ac c o un t ed for , on e needs to solve t he two equation s ( 13 2) an d ( 1 3 3 ) subject to the boundary con dition s k (C c
h( T
b
b
- C ) = D n -V' C e
( 14 3)
s
- T ) s
=
e t.. n -'V T
( 144)
T he b ehavioral features of t his c a se are similar to those found for the limit in g models disc ussed b e fo r e ( i . e . , the effec tiven ess fac t o r may exceed un i ty and m ultiplicity occurs for certain values of the parame t e rs ) . The in teraction bet ween the in terp h ase and in t rap a rtic l e gradien t s may c au s e the appearance of two dis t in c t re gion s of mul ti p lic i ty , as shown in Fig . The t w o multiplicity region s m ay overlap , givin g five steady - state s ol u t ion s .
16 .
1 0 0 0 �------. Y = 27
!='"
s
100
u
�
"' "' .. "
-�
t; � w
10
T h iele Modulus -
Vp -
Sa
{-f; o•
F I G U RE 1 6 Effectiven ess fac to r as a func tion of the Thiele modulus for a first - orde r react ion in a slab ca t aly s t . ( F rom A ri s and Hatfield , 1 96 8 . )
279
Diffus ion -R eac t ion Interac tions
It is rather difficult to derive exact criteria predicting explicitly the s et of parameters for which multiplicity exists in this case . S ufficient con ditions for uniquen ess w e re presented b y Jackson ( 1 9 7 2 ) . T he analysis above in dic ates that either in tra- or interphase tempera ture gradient s m ay cause the effectiveness factor to exceed unity and lead to steady - state multiplicity . E stimation of the magnitude of these gradients is use ful for decidin g w hic h of · th e limitin g models s hould be used in a specific application . M u l t i rea c t i on
N etworks
Temperature gradients may have a signific an t in fluence on the yield of a desired p roduct in a multireaction network . Whenever the activation ener gies of the various reaction s are not equal , thermal gradients affect the ratio amon g the rates of the differen t reaction s and hence the intrinsic yield . The temperature gradient s also affect the ratio amon g the time con stants for diffusion an d reaction and hence the in fluence of diffusion on the yield . It is impossible to analy ze the behavior of all reaction networks and rate expressions . Thu s we shall discuss ju st two typical two- reaction net works an d comment on t h e behavior of more in tricate networks . We start by examinin g a case of two consecutive reactions , A t -+ A 2 -+ A 3 , T o simplify the an alysis , we ignore intraparticle concentration and temperat ure gradients an d assume that only in terphase gradients exist . The correspon din g steady - state species and energy b alances are ( 14 5 ) ( 14 6 ) ( 14 7)
T he three equations may be combined to give the followin g equation for the dimensionle s s temperature :
( 1 4 8) where _
kiVp
Da i k .S
C1 X
(3 !" 1
1
< - t. H . )k . c . =
1
hT
C1
b
b ( 1 49)
Equation ( 1 4 8 ) is a highly nonlinear function of Y s , as each o f the Damkohler numbers inclu des a temperat ure- depen dent rate constant , The corre spon din g yield i s given by the equation ( C assiere an d Carb erry 1 973)
Luss
280 dB
dA
=
k C1 - k C2 2 s s 1
k C 1 1s
=
n n* 2
�2
n !k 1
c2 b
c 1b
( 1 50)
where n !" 1
=
1
1 + Da.
=
i
1
1, 2
When b oth D amkohler n umbers are very small ( D � << 1 ) , the e x t e rnal c lo se t o u n ity an d E q . ( 1 50) b ecome s
l
effectiveness factors n!" are
dB dA
=
( 151)
I n this case the t e m p e rat u re gradi en t s affec t the yiel d on ly b y c h an gi n g the ratio k 1 /k 2 . • When E 1 > E 2 the t e m p er a tu re gra d i en ts enhance the yield , w hile the c on v e r se occurs if E 2 > E W he n the Damkohler numbers r are not s m all , temperature gr adi en t s affect the rate constants as well as the
diffusional limita tion s . An an alys i s of E q . ( 1 5 0) in dic a t es t hat it m ay h ave up to five different s ol ut ion s , even though at mo s t th ree exist w he n o n ly one reac tion occurs . B alak otaiah and Luss ( 1 9 8 2a ) have proven that m ul tip lic i ty can exist for some Damkohler numbers if and only if at l e a s t on e of the t w o . reactions is exo t h e rm ic an d the s e t Yi , 13 , a n d a s a tis fi e s at least on e of five con dition s i given in t ha t work . Moreover , they s howed how to determine t h e Damkohler number for whic h m ult ip lic i t y exist s . The in tere sted reader is r eferred to t ha t work . W he n multip le s t eady states exist in t his system , the bi furcation diagrams , w hic h d e sc r ib e t h e y iel d or surface temperature as a fu n c t ion of a slowly c han gin g state variable , may be muc h more in tricate than those found for Moreover , the n u m b er of different types of bi the sin gle - reac tion c a s e . fu rcation d iagr a m s ten ds to inc rease rapi d ly wit h the number of t he re ac tion s ( B alakotaiah an d Luss , 1 9 8 2b , 1 9 8 3b ; H arold and L u s s , 1 9 8 5 ) . F i gu r e 1 7 describes a case in which two hystere sis loops are c omp l e t ely nested within a la r ge one . Obviously , it would b e easy to mi ss t he internal nested loop in an experimental study , un l e ss there exists some t heoretical g ui dan c e ab out its existence . T he existence of these " e xot ic " b i fu rc ation dia gra m s points out the need for a t horough an aly sis of the impact of ther mal gr adien ts , t o av oid p it fall s in the de s i gn a n d o p eration of t h e reactor . When the in fluence of in traparticle concentration gradients is taken into ac cou nt , the eq ua t ion s t hat d e s c rib e t he dim en s ionle s s steady - state tempera ture becomes mo re in t ricat e ( M c G r eav y an d T ho r n to n , 1 9 7 0 ) a n d their analy sis is more tedious . We shall n ot discu ss t hi s ca se h e re . Intraparticle temperature and c on c e n tra tion gradients m ay have an im p o rtan t influence on the yi e l d of the desired in term ediate A 2 for this re ac t ion net work . B utt ( 1 9 6 6 ) has shown that even w hen the activation en er gi es of both re ac tion s are equal , the in t r apa r ticl e g ra dien t s may affect the yield when b oth rea c tion s are exot h e r m ic . In s ome extreme cases , t h e t e mp erat u r e gradi en t s may even c h a n ge t he sign of t h e yield , so that in st e a d of producin g th e desired sp ecie s the reaction c on s umes it . When E1 E 2 the t em p e ra tu re gr adi en t s do not affect the ratio between the two =
281
Diffus ion-R eact ion I n terac tions 3.9 "/o co 4.2 "'o C 2 H6
220
T9 - GAS TEMPERATURE (°C)
FI G U RE 1 7 D epen dence of the temperature of a catalytic pellet on the gas temperature durin g the simultaneous oxidation of CO an d C 2H . ( From 6 Harold an d Luss , 1 98 2 . )
rate c onstants k 1 an d k 2 . However , the increase in the values of t hese rate con stants with risin g t emperatu res increases the deleterious impact of the concentration gradients which tax t he desired fast reaction ( A 1 + A 2 ) ,
On the other more than they do the slower undesired reaction (A 2 + A 3 ) . han d , the yield may be improved by thermal g radien t s when E t = E 2 for en dothermic r eaction s . T his is , however , accompanie d by a large decrease in the reac tion rate . When the ac tivation energy of the de sired reaction is larger than that of the un desired one ( E t > E 2 ) thermal gradien t s ten d to improve the yield for exothermic reac tion s , as the increased ratio of k 2 / k 1 overcom pen sates for t he larger diffusional resistances . The inverse occurs when E 2 > E 1 . C onsider now a sin gle reactant that is c on s umed by two simultaneou s first - order reaction s . When intraparticle gradien ts c an be ignored , the local selec tivity is defined a s ,
( 152)
so that the impact of the temperature gradien ts d epen ds on i t s influence on the ratio k t < T s ) /k 2( T s > · T hus the selectivity is im proved by thermal gra di en t s if the r e act ion i s exothermic an d E > E 2 or i f the reaction is en do 1 the rmic and E 2 > E 1 . The interaction between the chemical r eac tion an d the in terpellet tempera t ure gradien ts leads to steady - state m ultiplicity for some Damkobler numbers in this case if at least on e of the followin g three c ondition s is satisfied ( Michelsen , 1 97 7 ) :
( 1 53 )
Luss
282
( 1 54)
(y
2
-
y ) ( 13 *
1
2
- 13 *) > 4 ( 1 + 13 * ) ( 1 + 13 * )
1
( 1 5 5)
2
1
The last con dition [ E q . ( 1 5 5 ) ) may be s a tis fie d even if b oth reaction s are endothermic - a rather surprisin g result in view of the fact that m u l tip lic ity cannot occur for a sin gle , fi rst - order en dothermic r eact ion . W hen intrap article c oncentration an d te mp e rat u r e gradients exis t , the species an d energy b alances can be solv ed n u m e ric al ly and the selectivity is computed fro m
s
=
( 1 56 )
When the in traparticle resistances are small (
E 1 /E 2
)I
I
)2
=
1.3
5
FI G U R E 1 8 D ependence of t he yield for two simultaneous first- order re ac tion s in a slab on the Thiele modulus . ( F ro m 0 s te r gaa r d , 1 96 5 . )
Diffus ion -Reac tion In terac t ions
283
the q uali tati ve effect of the in tr ap article temperature an d concen tration gradi en t s can us ua lly be p redic ted from knowled ge of the influence of the diffu sional resistance in the sin gle - r eaction case . The in formatio n about the i mpact of the inte rp hase thermal gradients on the intrin sic se lec t ivity an d of the in traparticle diffusional limitations can be used to es t i m ate the qualitative effect o f temperature gradients in multireaction sy ste m s Ob viously , the qu anti t ative p re dic t ion of th e in fluence of thermal gradien t s re q u ir e s a n u me rical solution of a set of nonlinear differential e q uati on s that describe the system . B al akot ai ah and Luss ( 1 9 8 3a ) have devised a procedure for get t in g sufficien t multiplicity c rite ria for networks of many irreversible fir st order reaction s . .
-
D i a g n o s t i c T es ts fo r Tem pe ra tu re G ra d i e n t s
C riteria ( 1 0 7) and ( 108) enable a rapid prediction of the magnitude of the gr adie n t s b ased on observable quantitie s . S everal c ri teria have been propos ed for p re dict in g w hether thermal resistances have an im portan t in fluence on the in trin sic reaction rat e . Obv iously , the i m p act of t empe rature gradien t s de p en d s on t he e xotherm ici t y of the reaction the sen sitivity of the re action rate to changes in the tem p e ra t u r e , the rate and the t ra nspo r t c oe ffic i en ts . A nderson ( 1 96 3 ) p redic t ed that intra particle temperature gradien ts cause the observed rate to deviate by l e s s than 5% from the value obtain ed un de r isothe rmal c o n diti on s i f
temperature
,
4l
8y < 1
( 1 57 )
Kubota a n d Y amanaka ( 1 969) p redic ted t hat for positive -order reac tion s the combined effect of concent rat i on an d in trap ar ticle temperat ure gra dien ts i s less than - 5 % if 4> <
1
n
Sy
( 1 58)
W hen n 13 y i s c lo se t o zero , t h e p e rt u rbation app roach used to derive Eq . ( 1 5 8) is no lon g e r applicable . In this case the a sy mptotic expansion of the effectiven e s s factor for large valu es of cp ( P etersen 1 96 5) can be used to der iv e a d i ffe ren t criterion . M ea r s ( 197 1b ) s u ggeste d u sin g a value of 1 3 fo r the ri gh t - hand side of E q . ( 1 5 8 ) when b oth n an d y B are close to C riterion ( 8 5) can also be used in thi s case to p re dic t when the unity . in t rap a r tic l e gradien t s do n ot a ffect the rate by mo r e t han 5 % . T h e m aj or resistance to heat transfer is i n the bound a ry layer for g a s solid reaction s . M ears ( 1 9 7 1a) has shown that in te rp has e temperature gra dien ts cause les s than a 5% d evia tion in t he rate if -
,
4> * Y l s * l b
< o . o5
A comparison o f c riteria ( 1 5 7 ) and ( 1 5 9 ) in dic at es limit the rate b e fore t h e in t rap a rticle gradien ts do
Bi
a
=
h
-
>.. e
_.E_ s < 2. 2 v
x
( 1 59 )
that if
i n te rp hase gradien t s
( 160)
L uss
284
This con dition i s u s u a lly satisfie d in la b ora to ry reactors due to the low flow rates and heat transfer coefficients . A c om p ari s on of criteria ( 8 8 ) an d ( 159) in dic a te s that the ex ternal
t
temperature gra dien s does if
affect the rate before the e x t e rn al mass trans fer
( 16 1 ) T hi s c on dit i on reac ti on s .
is satisfied by many highly exothermic or endothermic
the m ed
Mears ( 197 1b ) h as s hown that for an n t h - order reaction co bin in trapa rticle - in terphase gr a di en t s affec t the rat e by less than 5% if ,
( 16 2 )
T his c riterion reduces to E q . ( 1 5 8) w hen the external resistances are negli gible ( 41 « 1 ) . Alt ho u g h t he diagn o s tic t e s t s are very useful , t h e e st i ma tion of som e of relevant para meters may be difficult at times . In such cases direct e xperimen tal t ests may have t o be c arried out to c h ec k for the i m pac t of th e thermal gradien ts . A c om mon test for extern al gradient is c heckin g for the in t'luenc e of the flow rate on the conve rsion at a fi x e d space velocity . If no e ffect is found , it i s concluded t h a t the exte rn al r e s i s t a nc es are not t e m p e ra u r e gradien t s usually affect t h e conversion in an in tegral pac k e d bed reactor more t han do in terp hase g rad ien t s Thus t his t es t does n ot provide a conclusive an swer as to w he t h e r the interphase resistance i s i m p or t an . A d irec t examin ation of the in fluence o f the in te rp has e gra dients can be c a r rie d out eithe r in a well - stirred h et e ro gen eou s catalytic r eac to r suc h as the B e rty reactor , or in a recycle reactor in which r adial t em p e ra t ur e gra d ien t s can be eliminated .
the
radial
t
-
important . Unfortunately
the
t
.
,
A C K N O W L E D GM E N T S
t
I wish t o thank John Villadsen , Arvind Varma , c om m en s
and Sti g Wedel for he lp ful
.
N O T AT I ON
i B a Bi
B O , B 1, B 2
c
m
p
c
eq
c.
)
CL
e
Biot n umber for mass transfer k c V / ( D e s ) X p s tructural p ara m e t e r s Biot n umber for
heat tran sfer kVp / P.. S ) x
specific h eat
to chemical e q uilib riu m
molar concentration of s ec ie s j
concentration correspondin g
p
concen tration of reactant in bulk fluid
Diffusion -R eac tion I n terac tions
285
liquid-phase c oncentration in equilibrium with b ulk gas concentration
C*
L
diamete r of particle binary molecular diffusivity Knudsen diffusion coefficient effective diffusivity surface diffusivity Da
Damk()hler number , defin ed by E q . ( 1 16 )
E
activation en ergy
E
apparent activation en ergy
a a
EA
activation energy for adsorp tion of species A
Ed
activation energy for desportion of species A
f
fraction of wetted exterior surface of pellet
A
f(C
A f(r) F. . ll
)
surfac e adsorption isotherms element of a matrix defin ed by E q . ( 1 7) pore si ze density function
element of the matrix F - 1
g ij G ij h
element of matrix defined by E q . ( 2 4 )
H
in
proportionality coefficient , defined by E q . ( 3 2 )
k
reaction rate c onstant
jH
heat transfer coefficien t
factor for m a s s transfer , defined by Eq . ( 5 0 ) factor for heat transfer , defin ed b y Eq .
mas s transfer coefficient adsorption equilib rium con stant equilibrium partition fac tor increased drag factor L
half
m
exponent in c orrelation
M
molecular wei ght
n
reaction order ; number of moles
n'
ap parent reaction order
n
normal outward unit vector
thickness of slab
total molar flux of species diffusive flux of species i viscous molar flux of species i
( 51)
286
L us s
p res su r e
p
p
quan tity defined by E q . ( 10 2 )
r
radial po sit ion
r
po re d i am et e r
measured reaction rate
m
r
p r( C )
reaction rate
R RN
p article radius ; universal gas constant ratio of molar fluxes , d efin ed b y Eq . ( 4 8 ) ratio o f rate c onstants k /k
R eynolds number defin ed by E q . ( 54 )
Re s
s e lec t ivit y
s s
exte rnal su rf ac e area of cataly st
x
t
time
T
t em pera t ure
T u
bulk fluid velocity
8
surface temperature
avera ge mole c u la r velocity of species
1
u.
volu me of catalyst
vc
volu me of p ellet
vp
mole fract ion of A
XA
quan tity de fin ed by E q . ( 1 23 )
dimen sionless temperature , defin ed by
y Y y
z
Y
2
surface area per unit w ei ght of catalys t
g
s
1
m ax
quantity defined b y E q . ( 12 3 )
min
dimen sionless surface temp erature , defin e d by E q .
s
len gth coordinate
C reek Letters
E q . ( 1 36 )
ratio of molar fluxes , defin ed by E q . ( 1 5 )
Prater number , d e fin e d by
Prat e r number , defined by E q . ( 1 0 9)
Prater numbe r , defined by E q . ( 1 0 9)
y
A rrh en i u s n umber , defined by E q . ( 1 1 6 )
r
ratio o f B iot n umbers , defined b y E q . ( 1 09)
0
ratio of effective diffu sivitie s , D /D
- l\ H
E q . ( 1 36 )
heat of reaction
e 2
e 1
( 114)
287
Diffusion -Reac tion I n teract ions €:
porosity of p ellet
fb r;
dimensionless len gth , r /R
bed
voi da ge fr ac ti on
n
effectiv ene s s fac tor
n*
external effectiveness factor
e
fraction of surface covered
Af
ratio of molecule to pore diameter ; average hoppin g distance
>.
==
1 /( 1 + Da)
t h e rm al conductivity of flui d
,_ e
effec tive t he rmal co n duc t ivi ty
A
normali zed T hiele modulu s , defin ed by E q . ( 6 2 )
]..1
viscosity
v
!;
v
vib rational fr e q uen c y
i
con ve r sion at w h ic h rate is D . 9 of initial ra t e den sity
p
P
stoichiometric coefficien t
b
b ul k c at aly s t de n si t y ratio of heat s of reaction
<
t or t u o si t y factor for
q,
<*
�
� X
lin gerin g time T hiele modulus
*
with R as c haracteristic len gth
ob s ervable modulu s defin ed by E q . ( 8 8 )
ob servable modulu s defin ed by Eq . ( 8 4 )
bulk - flow correction fac to r
R E F E R E N C ES
cataly sts in the m olecular , transition , and Knudsen re gi m e s , AIChE , J . , 1 9 , 6 1 8 ( 1 97 3 ) .
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5
Gas-Solid Noncatalytic Reactions Pune , India L.
K.
D O R A I SWAMY a n d B .
D.
KULKARN I
N a t io n a l C h emica l Laborato ry ,
I N T R O D U C T I ON
R eaction s bet ween gases and solid s p lay a major role in the materials processin g in dustrie s , enc om p as sin g a b road range of operation s , such as extractive met all u r gy , combustion of s oli d fuels , environ mental control , ener gy generation , and catalyst manufacture and r e gen eration Several ex a mp les of gas - solid reaction s of in dustrial importance are cited in Table 1 . The breadth and diversity of sys tem s encountered in gas - solid reaction s , as evi dent from t he table , suggests that any attempt at formulatin g a unifyin g theme for the desi gn of gas - solid r e acto rs would not meet w i t h much success . However , on� can recogni ze a nu mber of common steps in the overall desi gn of ga s - soli d reaction s , a c o m bi n at ion of w hic h may be used in the develop ment of models and form ulation of design s t r at egies for these sy s tem s It i s i n this spirit that the p resent chapte r attempts to thread together the otherwise diverse system s . The foremost consideration in the analy sis and design of these systems is th e mode of contact . The n umber of reactor types ( or modes of contac t ) is very large i n t h e chemic al in dustry . E ven for the same operation , for example , calcination of lime , different types of reactors ( e . g , , rot ary kiln , fla t hearth , flui dized bed , movin g b e d ) are used . A similar situation pre vails in many other gas - solid reaction s involvin g calcination , decomposition , roastin g , and so on . The fixed - , fluidi zed- , and movin g-bed techniques appear to be most commonly employed and have received gr e at e r attention from both the theoretical and p rac ti c al poin t s of view . The other modes of c on t ac t less com monly employe d , include hori zontal movin g bed , pneum atic conveyor , rotatin g cylin der , and flat hearth furnace . A lt hou gh a priori selection of the mode of contact is difficult , a c hoice will ha ve to be made a t as early a stage as pos sible- certainly before any si gnific ant effort is p u t into the detailed desi gn . This m akes it necessary to undert ake an ex pe rim ental progra m with a view to ob t ainin g some q u ali t a ti ve an d s emiquantitative in fo r mation regardin g the system . The tools of mathematical modelin g should be invoked at this .
.
,
293
t-.o (0 ol>o.
T ABLE 1
E xamples of Gas - Soli d Reaction s
S ystem
1.
Red uc tion of hematite
Ty pe of reac tor
Fi xe d b e d
Fixed b e d
Con s ec utive reaction of solids
3.
C oal gasification
S el ec tive chlorination of i l m enite Regeneration of coked
catalyst
al.
( 1 9 6 8)
( 1978) ( 1976)
Ahner and Feinm an ( 1 9 6 4 )
bed
M ovin g b e d
Ohmi et
et al .
Doheim ( 1 9 7 3)
Multistage fluid bed
2.
Spit zer
T say et al .
Movin g bed
Fluid
References
R e marks
Complex reaction
Y oon et al .
( 1 979)
Fluidi ze d bed
U b hayakar et al .
( 1977) ; Yo shida and Kunii ( 1 974)
Fluidized b e d
Doraiswamy e t al . ( 1 9 59) ; F u w a et al . ( 1 97 8 )
Reaction of two solids
with t h e same gas
Ramachan dran et al
.
( 1 9 7 5)
0
tl
a
�· � !:)
'<
3
!:) ;:J Q.
� s::
>:: Q
a.
4.
5.
6.
7.
Reduction of iron ore with CO a n d /or H 2
Oxidation of ZnS
Reduction o f metal chlorides with hy d ro gen
Reduction of cobalt sulfide in presenc e of C aO
Decomposition of potassium ben zoate
8.
Simultaneous oxidation and
sulfation of c uprous oxide
M ovin g b e d
Fixed bed
F l ui d b ed
R eaction of two gases with s am e solid
Simple reaction
T say et al . ( 19 76) Osman et al . ( 1 96 6 ) ; B arner et al . ( 1963)
Natesan an d Philb rook ( 1969) ; Y o shi d a and Wen ( 1 97 2)
Multistage fl ui d bed
Fukunaka et al . ( 1 976)
Fixed bed
B everidge an d K a w am ura ( 1974) Homogeneous - hetero geneous reaction Gaseous product of first reaction reac t in g with a secon d soli d Solid product of first re a c tion r eac tin g wit h a sec o n d ga s
R ao ( 1 9 8 1 ) Fahim e t al .
!:l (I) I til 0
Q
-·
(:l.
� 0 ;::s
(") p
S' '<' ...
c:;·
�
::0
( 1 978)
s·
0 ... ;::s C'l)
Gokhale et al . ( 1 9 75) ; K ulkarni
and Doraiswamy ( 1 980a)
B ourgeois et al . ( 19 74)
t-.:1
296
Doraiswamy
and K u l karni
stage to fu rther substantiate th e final choice . Focusin g on t h e p henomena takin g pla ce in the reac tor ( i . e . , the reac tion , an d transfer of heat , m as s , and m ome nt um ) , one can reduce th e ap parent diversity to a small n umbe r of m odels or basic reactor types . T he modelin g and desi gn of reactors i s then b a s e d on the equation s describin g these phen omen a . After a t ent ative c hoice of the mode of contact is m ade , the next i m me diate p roblem concerns the particles - their p hy sic al and c hemical inter action s with the surroun din gs . The variab ili ty in volved here c o uld be from on e extreme set of condition s to another . The particles may be p acked , as in a fixed-bed reactor , or completely mixed or movin g in plu g flow , as in a fluidi zed - or movin g- bed reactor . The gas phase could b e i n plug flow o r i n a state o f complete mixin g . I nte rme di ate stages o f mix in g are also possible . A dditionally , the chemical in teraction s could involve si m p le sin gle - step reaction s or m ultistep reaction s wit h or w it hou t c han ges in t h e volume of the phases an d heat effects . The p articles them selves could b e porous or nonporous or may become po ro u s durin g the reaction ; t hey m ay also chan ge their si ze , an d si gn i fi ca n t alteration s in the structure No sin gle model can account for might accompany t he chemic al reaction . In fact , different classes of models , with all t he possible events listed . one more suitable th an t h e others for a given situation , can be developed . C le arly , the first st ep in t he ration al synthesis of a desi gn strate gy is a full understan din g of the particle beha vior . S I N G LE - P A R T I C LE S T U D I E S B a s i c S teps in G a s - So l i d R eac t i on s
D espite the b ewilderin g complexities that can occ ur in the study o f a sin gle p arti c l e , the formulation an d accoun tin g of these complexities can be sim pli fi e d if one adheres to an d understands the b asic step s involved . F i gu r e 1 sketches a solid particle reactin g wit h a gas specie s to give product s of reaction . T he typic al elemental step s involved include gas - phase mass trans fer , diffusion in side the pores , adsorption of reactan t s , and reaction A lt ho u gh these steps occur in succes sion , and desorption of p ro ducts . any one or more of these might be rate li m itin g . The rate of reaction for the particle as a w hole for a slowly re actin g system will b e in fluenced b y the surface kinetic s ( the intrin sic rate ) . For somewhat faster reaction s , pore diffusion intrudes to limit t h e rate . For s t ill fas ter reaction s wit h exothermic effects , th e temperature gradient across the particle dimen sion or the gas film may b ecom e t he li mi tin g factor. I n the e xt re me case of very rapid reac tion s , mass transfer across t he fluid interface m ay becom e the controllin g fa ctor . It is necessary to demarc ate these Cf\ntrollin g regimes - for in a given situation when , say , gas film mass tran s fer is c on trollin g , it would be w aste ful to show un d u e concern for t h e detailed reaction mechanism . T hese con sideration s are com m on to catalytic and noncatalytic gas - solid reaction s . An important point that differentiates gas - solid reactions from their catalytic counterparts is the fact that here the solid i s also involved as a reactan t . The c on tin uo u s con sumption of the solid re ac t an t leads to in evitable struc t u ral chan ge s durin g the reaction , and the system as a w hole is always in a transient state . An important implication of t his is that even t h e co n t ro llin g re gime migh t continually c han ge w it h time for the
same particle .
29 7
G as- So l i d Nonca taly tic R eac t ions
P OS I T I O N O F R EA C T I O N Z O N E WITH
TIME
ASH UN RE ACTED SO L I D I I I
G AS CO N C ENTRA T I O N PROFILE
I
I 0 D I S T A NC E
FIGURE 1
B a sic
FROM
CENTER ,
r
step s w hen a soli d p ar ticl e reacts with a gas sp ecie s
.
The problem , complex as it is in th e case of a si n gle p a rticle assumes further dimen sions w hen a re ac tor is involved . T he particle , in fluenced by its own mic roen viron men t , determines the controllin g regime ( an d there fo r e the net rate of production per pellet ) , w hic h at a giv en instant of time is different for different particles . T his problem althou gh con sidered in greater detai l later , here follow s logically from a con sideration of the b a sic step s in ga s soli d reaction s . The rates of reaction w hen gas fi l m mass tran sfer is con t rollin g can e asily be obtained from a knowle d ge of the transport coefficient across the fluid - solid interface . S eve ra l correlation s for predic t in g the transport co efficients for a fixed or movin g solids and a movin g gas stream have been reported in the literature . These c orrelations s hould be used with some c a ution , for many of them are valid for restricted operatin g conditions . T he recommended e quat ions for e s ti m atin g the mas s transfer coefficient are given in a later section . In the p res enc e of pore diffusion e ffec ts it is u s t o m a ry to invoke the laws of molecular di ffu s ion and assume that they are app lic able to a porous medium with an e ffec ti ve diffusion p aramete r . T his equivalence is at best an app roxi mati on , and t he total flux balance should include , b esi de s the diffusive flux , the con trib ution s from the pressure gradient an d b ulk flow d ue to di ffus ion . C onve n tion ally these c on trib u tion s are neglected and perhaps justifiably so . However , in certain in stances , s uc h as when m ulti c o m p onent diffusion is involved , they s hould be taken into account . The concept of effective diffusivity as ap p lie d here ( as di stin c t from gas - solid ,
,
-
,
,
c
Dora iswamy and
298
Kulkarni
catalytic reac tion s ) also needs recon sideration . I n most gas - soli d systems , the morphology of the solid chan ges due to the p re sence of reaction , tem perature gradien t , and so on . The typical c hanges that may accompany a reaction inclu de pore s hrinkage or closure , s wellin g , softenin g , or crackin g of the p article . The temperature may affect the sint erin g of particles , whereby the density of contact is increased . In view of these in fluences , the effective diffusivity w ould be modified c ontinuously wit h the p rogress of reaction . Basic
Model s
The relevance of the b asic steps in gas - solid reaction s has brought out a n umber of features tha t are more or less uniq ue to these types of reac tions . I t i s n ecessary now to model these features such that a further in si ght into the b ehavior of these sy stem s can be gained usin g numerical experi ments . As already men tioned , it is obviously not possible for a sin gle model to incorporate all the features of gas - solid reaction s . I n fact , dif fe rent clas ses of models have been pos tulated an d used to desc ribe these sy stem s . I n the p resent section w e briefly summarize these basic models , an d in the n ext pre s en t a s tate -of- the - art disc ussion of the extension of these basic models to different situations of greater p ractical relevance . A comprehensive review has recently been published on the modelin g of gas - soli d reactions by Ramachan dran and D oraiswamy ( 1 9 8 2b ) ( see also Dorais wamy and S harma , 1 98 3 ) .
Sharp I n t e rface Model
T he sharp in terface m odel ( S IM ) is on e of the earliest models used an d is well described in the standard textbooks on chemical reaction en gin eerin g . T he model i s mainly applicable t o nonporous o r relatively less porous solids an d assu mes that the reaction occurs at a s harp in terface that separates the reacted outer shell (or ash layer) an d the un reac ted inner core of solid . The steps involved are diffu sion throu gh the gas film , diffusion through the ash layer , an d reaction at the in terface . The respec tive rates of these steps in mol ls ( for a s in gle p elle t ) are : Diffusion t hrough gas film : ( 1)
Diffusion through ash layer :
( 2)
C hemical reaction at the in terface : R
Ac
=
1 s A1
2 4 1r R . k C
.
The rate of reac tion per pellet at any time t accountin g for all the three resistances is obtained simply by summin g Eqs . ( 1) to ( 3) as
( 3)
299
G as- Solid N onca ta ly tic Reac t ions =
or
(
1 1 + -- + -R R Ad Ae
1
4 rr R
�
+
g
1
R - R.
4rr R R D i e
+
1
)
( 4a )
-1
4rrR�k 1 s
a stoichiometric b alance on soli d B as
c
Ab
( 4b )
This rate of c on s u m p t ion of A can be related to t h e conversion of B throu gh
( 5)
I n t e grati on o f t h is e qu at ion s ub j ec t to the initial radius ( =R ) at t = 0 le a d s to the conversion - time rel a ti on ship , _g_ Rk
X +
+
2D
e
___g_ [ 1 3k
k
s
[ 1 - 3( 1 - x )
- (1
x)
1 /3
2/3
+
2( 1
J
-
x) ]
( 6)
T h e time required for c omplete con version c an be ob taine d by settin g T he eq ua ti on c an also be w ri t t en in terms of the time required for complete con v e r si on if any one of the three b asic step s were to control the rate as x = 1.
t
+
t
a sh
+ t
. re ac t ion ( 7)
where Tf • Ta , a n d Tr • r e p re s en t , respectively , the ti me require d for com plete c onversion if film transfer , ash diffusion , or reaction alon e were to con trol the rate . The function s fo , f 1 , an d f 2 a n d the various T' s as sume differen t forms for different geometries of the particle an d are defined in Table 2 together with o t he r rele van t pa ra met ers . It is eviden t from the r ela t ion s h ip s present in the table that the time required for a given c on ver sion x de p e nds on T , whic h in turn depends on ( a) the radius of the pa rtic le R , ( b ) the concentration in the gas p ha se CA g • ( c ) the m ol al density of solid P B , an d ( d ) the stoic hiometric c oeffi cien t v 8 , in a d d i t i on to k g , De • o r k8 , de pe n din g on w he t h er fil m , ash diffusion , or r eac tion con t rols . I t may be n ot ed that the dep en denc e of T o n pellet s i ze is differen t for controlli n g r e gime s . T hus it is fir s t orde r in R for reaction control , second order for ash diffusion con trol , and 1 . 5 to 2 order for fi l m diffusion con trol . By varyin g the si z e of the particle it is t herefore easy to delineate the c on tr ollin g regime .
to> c c
T A B LE 2
regime
C ontrollin g
Fil m diffusion , f0 ( x ) ==
A sh diffusion , f (x ) 1
==
a
A ( g) +
Flat plate
X
==
T he c onversion x :::: 1
v
+
R ( g}
+
Geometriesa S (s)
X
2
X
- (r/R)
B+ l
1 - ( 1 - x)
1 - (1
1 - (1
, where B
==
2 x) 1 /
VB ( 8
X
x + ( 1 - x ) ln ( l - x )
_
T
S p h e re
Cylin der
X
R eac t ion , f2 ( x )
BB (s)
Time - Conversion R elation ships for SIM for Di fferent Particle
_
2/3
X) 1
+
2( 1 - x )
2v ( 6 B
B
+
l)k C g Ag
R
PB +
R
pB R
/3
0 , 1 , an d 2 for flat plate , cylin de r , and
p
n v B k SCA g
sphere ,
respec tively .
2
l) D e C A g ;3 iii'
0
tl
3
�
'<
1:1 ;::s Q.. 1:::
�
� �.
Gas- So l i d N onca taly tic
R ea c t ions
301
Further , for pellets of diffe ren t sizes , different controllin g mechani s m s A t the start of the reaction there is no as h layer present . can prevail . Also , unless the reaction is extre mely rapid , film diffusion is normally not limitin g . T he system therefore begin s wit h the rate controlled by the sur face reac tion . Wit h p rogres s in time , sufficien t ash layer b uilds up and the system passes to ash diffusion c ontrol . The depen denc e of ' on the gas - p has e c oncentration , while not very important for a sin gle particle , as sumes great si gnificanc e for particles in a reactor . As already men tioned , in a reactor the gas - p hase c oncen tration is u s ually not c onstant but varies with position . Thus , even for particles of the same dimen sion s , we can have differen t controllin g regimes when they are placed in a reactor . This poin t is discu ssed fu rther later . An alternative approach to the analysis of gas - solid reac tion s is to ex pres s the results in analogy wit h cataly tic sy st em s , in term s of an effec tive n es s factor . S uch an approach has been used by I shida an d Wen ( 1968) , and for a general n th - order reaction , the effectiven e s s factor can be simply expres sed as
( 8)
I n view of the c han gin g rate R A , the effectiveness factor i s also chan gin g with time and the followin g general implicit equation can be derived :
n = 1
- Ri D a 2
1
S
h
+
1
1 - R.
R.
1
( 9)
n- 1 where Sh kgR IDe an d Da = k s R C Ab The movement of the interI De . face Rj with time appearin g in Eq . ( 9) can be obt ain e d by integratin g Eq . ( 5 ) , where th e term RA on the right -han d side can be replaced by RA given by Eq . ( 8 ) . The case con sidere d thus far assumes linear kinetics . I t is pos sib le for a reac tion to follow fractional - order kinetic s with respec t to the gaseous re actant . T ypical exa mples are the reduc tion of C r 20 3 by hy dro gen ( C hu and Rahmel , 1 97 9 ) and oxi dation of ZnS ( Cannon and D enbigh , 1 9 5 7 ) . In many in stances the reaction rate may also be rep resented by Haugen - W atson ( H - W ) In gen eral , in the p resence of s uc h nonlinear kinetics , no analy ti kinetic s . c al expres sion for the conversion - tim e relations can be obtained . Numerical integration procedures for power law kinetic s usin g S I M have been proposed by S ohn and S zekely ( 1 9 7 2) an d for H -W kinetic s by Ramachandran ( 1 9 8 2 ) . The case of zero- order reac tion has b een reported by Simonsson ( 1 9 7 9) . A zero - order re action typic ally ex hibits sharp transition between the kinetic T h e combination of first - an d zero- order an d diffusion control re gime s . kinetic s has b een analy zed by Kam et al . ( 19 7 6 ) for the case for oxidation of carb on . S I M is a phenomenolo gical model , an d in the extreme case of reaction control or ash diffusion control requires only one parameter - t he relevant time factor T- to describ e the syste m . The model p redic t s total con version of s olid in a finite time an d is well suited for many p rac tical systems . But i t c annot accoun t for s uc h features a s the l evelin g off o f conversion o f the S - shaped nature of the conversion - time curve . =
Do raiswamy and K u l karni
302 V o l ume R eac t io n M o d e l
When the soli d is porous , the gas can penet rate the solid and the reac tion may now be assumed to take place all over the volum e of the pellet rather than at a s harp interface . In general , the rate of reaction at the interior points would be lower than at the surface due to the concentration gradient . I n a special situation w hen no diffusion resistance exist s , the reaction occurs uniformly all throu gh the particle , leadin g to the so-called homogeneous model . The rate of reaction per unit volume of the pellet is represented as
where m and n refer to the orders wit h respect to the gas and solid , re spectively . A distinctive feature of the volume reaction model is that the rate of chemical reaction depend s on the activity and concentration of the soli d reactant . T his means that only for certain values of n (n < 1) can the soli d reactant be completely reacted at any p oint in a particle an d its concentration reduced to zero . T he general conservation equation s for the gaseous an d solid reactant species for the volume reaction model illustrated in Fi g . 2 can be written as ( 1 0)
( 1 1)
where
and
T he boun dary conditions required for the solution of the model are t
=
0 , ft
=
A
t > 0, R
=
CB A
1:
1:
dC
dR
A
A
t > 0, R
0:
A
( 1 2)
= 1
dR
A
= Sh( 1 - C
=
0
A
)
or
A
C
A
1
( 13)
( 14)
�
s:l
c;')
GA g
t = O ---+
'
�I I
I
1
8o
I
GASFij R
0 LOW
GA
til
&
g
I
�I RL
cb
I
LARGE
Q.
8o
z 0 ;:s
(') s:l s:l
«" ..... c;· (I)
::tl
0 cl>
( ·- - -
8o L a r g e t---+
FIGURE 2
R
Zon e I Zone II
8
LOW
CZ,
0
, - m e
m•l
Zon e Ill R LARGE
CZ,
( I N D ICAT I N G 0
OF
THE
THR E E
-
�
;:s en
I )
Ash
s·
layer
- Re ac t ion zone - Z o n e of
un reacted 8
D E V E L O PM E N T ZO N E S )
Concentration profiles for the homo geneous model for vario us T hiele moduli . w c w
304
Doraiswamy a n d K u l karni
No a n a ly tic al solution to t his s e t of equations for arbitrary values of m and n seems pos sible , and recourse to numerical methods is nec essa ry How e ve r , for certain simplified cases analytical sol u t i on s can b e ob tained . T h u s for the case of low values of
x
{;
"
- exp ( -fl
( 1 5)
for n = 1 for n
=
( 16)
0
At hi gh e r values of
1jJ
=
ff A
C
0
A
A
dt
( 1 7)
T he trans formed equation for the special ca s e of m = n = 1 can be written as 'V
2
tjJ
=
2
( 18)
with boundary con di tion s 1/J{ l) = f and ( d \jJ / d R ) ft =O 0. T his transformation , s u g gested by D el B o r gh i et al . ( 1976) an d Dudukovic and Lamba ( 1 9 76) , is =
extreme ly useful for system s with no structural variations and for power la w t y p e kinetics . This case ( m = 1 , n = 1) h as also been analy zed by Dudukovic an d Lamba ( 1 97 8a ) ; approximate analytical solutions to t h e prob lem have been prop os e d by Ramachan dran an d K u lka rni ( 1 980) and by W en an d W u ( 1 97 6 ) . S ome of the u seful time - conversion relations a re presented in T able 3 . The case of m = 0 , n = 1 h as b een analy zed by D udukovic and Lamba ( 1 978b ) ; m bei n g equal to zero in this case , the gas concentration can dr op to zero within the p ellet , dependin g on the value of
=
•
=
.
1:1 CIJ I
<;)
1·0 IC
2
z
en a: Iii > z 0 u
Q
0•8
-
- - --- -
- -- -
o . s r-----,_
m
1
1 0
'
n
0 , 1
,_-+-+-+,_rr����--,_����t++---��--,_��-t-r�
__
g 0·4 �-----1--t--+-1-1���---b��--���� :J
..J cr
� cr a: �
i 1·0
I
� � 0 ;:s (") 1:1
z
s -<' .... c;· «> 1:1 0 ...
:::0 ;:s CIJ
6'
I I I I I II
10·0
r
FI GURE 3 Effect of reaction orders on x versus f plots b ased on volume reaction model for slab geometry ; Bi m = co . ( From Dudukovic and Lamba , 1 978b . ) D I M E N S IO N L E S S
T I ME ,
c.:o Q Cl1
w c Ol
TABLE
3
C onservation E quations for Heat and Mass for Volume Reaction and Particle-Pellet Models
Model
Volume reaction model
Governin g equation s
N
1
N2
ac
R
A
1
"2
""""31 a
:
at
a cB
=
aT
&
a
=
1:
+
""""
("2 --) R
R aR 2
A
--:;::-
aT
"' m "' n C C exp
=
B
A
at
R
2
oR a
ac
=
aR
A
A
-
ac
A
+ s
m
¢
.
c c A
[y (1 - � )] "
Sh( 1 - C
A
R
=
0:
t = 0: where
N
l
=
aR
a
.....
C
A
=
aR
aT
--;:::-
aR aT
= 0, T
=
1, C
n- 1 2 m C f R t k e v Ab B s e D
eB
e xp
Nu ( l
T)
a 0
t:l
0
=
B
"""" =
);
[y (l - � ) ] [y (1 - � )]
2" m " n
2 m n
"
ac
- cp C A C B exp
B
=
1
c;; · � p
3 '<
p
;:s Q.
� s:::
"" p
�.
2
N
8
( - ll H ) D
--
1
n- 1
Q rn
2
A b c B s f OR t k'
m
� I
E:
�
eB
C
at
=
2
a cA 2 aR
ac
-
-- + -
2
i
"
A
i
" 2 " a' � 2r ic A G
� -_ 2"2 a 2i � N " " + ! + 13 m ¢ rGi C A 2 2 1
a;
ai
0 ;:s (') Q -
2:
Ab
k' T e b
a cA
1
e
e
=
m
N
Particle- pellet model
s p sk v
P c
=
A
a'
R
-< ... Q
5'
� m Q (') ...
rn
s· ;:s
"
a rG
"
i
,.. -
=
-c
at
a' A
w here exp [ y ( l
a'
N
N a
T he conversi on x = 1 -
( r /R ) 13 + 1 , where B
=
1
1
c
R \i
-
1 /T ) ]
k c B Ab s
D e p erG O
vR 2 p C MB CAb k s s ps
1 2
_
_
k' r =
=
1 /T ) ]
+ D a 0 r Gi ( l - rGi ) e xp [ y ( l vf
1
-
e G0
0 , 1 , and 2 for flat plate , cylinder , an d sp here , respectively .
Coo 0 ""'
308
Dora i swamy and K u l karni
compri sin g a produc t zone n earer the surface an d a r e ac tion zone wit hi n the pellet . I shi da and W en ( 1 96 8 ) divi d e d t he t ot al reaction time into two periods (i . e . , the constan t - rate period and fallin g- rate period ) , in analogy wit h the d ryin g process . T he ti me-conversion relation s for t he two periods are
X
=
3 ( cp coth cp
----���--�----�--------
<1> [ 1 2
-
1)t
+ ( 1 / S h ) ( cp c o t h
cp
( con stan t - rate period )
1) ]
-
( 1 9)
and 3R
cp
"3
i
"
x = 1 - R1• + -- [ ¢ R . cot h ( cj> R1. ) ] 2
( fallin g- rate period )
1
( 2 0)
T he c onv er si on at the en d of c on stan t ra te period is obtained by s ub s tit ut in g f r in E q . ( 1 9) , where fe r i s t he time required for completion of t he constan t - rate period is given by =
cr
A
t
-
fe
2 cp = 1 + 6
(
2 1 + -
Sh
)
-
-
( 21)
E quation ( 2 0) relates the conversion in the falli n g - rat e period to the dimensionle s s interface position in t h e particle . I t i s pos sib l e to obtain a relation ship betw een t h e in te r fac e p o sit i on an d time , which is given i m plicitly a s .
In
R . sin h cp 1
sin h "
"
1
)
+
cp6 ( 1
R . ) ( 1 + 2R . ) ,..
2
-
1
2
,..
1
( 22)
In t h e t w o - zone models describe d above , when the rate of reac tion i s very rapi d , the concentration of t h e gas s p e ci es drop s very sharply i n the reac tion z on e The reaction zone c an then be further sub divided into two zones , co m p r i si n g the actual reaction zone w he re the b ulk of the reaction occurs and a core of completely unreacted solid . T his has led to the de v elop men t of s o - calle d z o ne mo dels ( B owen and C hen g , 1 96 9 ; M antri et al A com mon feature of these models i s the existence of a reaction 1976) . zone of t hi ck n e ss 11 Y which is a fu n c t ion of the T hiele modulus . Mantri et al . ( 1 97 6 ) h ave p r ep a r e d a plot of /1 Y versus cp , which is reproduced in Such a plot i s useful in p a r am ete r estimation . T h e width of the Fi g . 4 . reaction zone ( /1 Y ) c an be in depen dently determined from measurements usin g electron p rob e microanalysis ( see Prasannan and D oraiswamy , 1 9 8 2 ) . For extremely large values of cp , t he reaction zone wi dth become s very nar row , l e a di n g to t he s harp interface model . Ramachandran and D orai swamy ( 1 98 2a) considered t he cas e w he re the reac ti on is zero order with resp ec t to both gaseous and solid r eac tan t .
.
,
309
Gas- So l i d N o ncatal:y tic R ea c t ions 1·0
\ \
......
\
[\
<1
'\
>-
.
� z 0 N
.'\
� 0
C/) C/) � z lo:: u
i
1\[\. \I !
I
z 0
1u ct � a:
""'
1\ 0·1
r-- -·-
! --
- -- -· -
---1
...J ct
t: z
I
( O udu k ov i c
I
i
j_
I I et
.....-- m = 1 ( M a n t r i
"" �[\.
\. '\
· - �- A
�
�
I I !
I ! I
�
··- - ·
--- -
!
I
�
'
!
!
I
"\
"
- -- - --
I
I
II
2
10
THI E L E
1\
M O DU L US ,
q,
i
--
I I
I
! I 'II ' I I ! . I
L -t-�
\
I
' 1\.
· -
Ii
�
:
\.
i
I
al , 1 9 7 6 )
and Lomba, 1978 b�i\, t m = O
I
0 · 01
\.
r - r---
I
::z: 1-
1\
i
r\ - -
I
I I
;
-·-
� ;
-
-- - - -
· --
�
"\
100
FI GURE 4 Reaction zone thickness as a function of the T hiele modulus for the volume reaction model . ( F rom Mantri et al . , 1 97 6 . )
specie s . For finite concentration s of ga s wit hin the pellet ( no severe dif fusional li mitation s ) , this case lead s to the simple solution X
=
t
( 2 3)
When si gni ficant resistance to the di ffusion of A exist s , a case ( depen din g on the value of cp ) may arise w here the concent ration of A actually drops to zero within t he p ellet . A simple c riterion for this sit uation to occur ( as already mentioned) is given by
cp c r �
6 ( 1 + 2 /S h )
1/2
( 24)
F o r values of cp >
31 0
Doraiswamy and K u l karni
shift s w i t h time . Once the zone is formed , the reaction p rocess occurs within this zon e un til the solid there get s completely exhausted . The re action then jumps to anot her zone , w here the phenomenon continues . In this case , therefore , we have a p hysi cal picture of w here the reaction zone jum p s from one position to anothe r , and Ramac han dran and Dorai s w amy ( 1 98 2a , b ) have termed this the jump ing zone model . The jum pin g zone con cept is ill us t r ate d in Fi g . 5 .
G rain ( Part icle- Pel let ) Model
I n the b asic grain model , also referred to as the partic le- pellet mo del , it is as sumed that eac h pe l l et c onsists of s everal grain s of soli d . T he grains are spherical and of the same si ze . I t i s further assumed that each grain re acts accordin g to S I M and that the si ze of the grain does not c h an ge wit h reac tion ( thereby implyi n g no voi da ge change in the pellet ) . A diffus ion resi stance for the gaseous species exists in the interparticle spaces within the p ellet . The basic features of t he model are sketched in Fi g . 6 . The m at he ma tic al formulation of the model requires consideration of the rate p rocesses w it hin an in dividual grain , and the overall mass balance for the gaseous reactant in the pellet and its stoic hiometric relationship with the extent o f soli d consumed . As far as the individual grain is concerned , the rates of diffusion t hrough the reacted portion of the grain an d of reac tion at the in terface can be obtained in an alo gy with E q s . ( 2) and ( 3) as
If) I&J :z
2 :z 0 1u
IL. 0
...J
:: 0 1-
TH I ELE
MODU L US ,
If>
FI G U R E 5 N umber of reaction zone t hat develop in the p e lle t for various values of cp .
31 1
Gas-Solid Nonca taly tic Reac tions
rG i
-
r G
rGo
PA R T I ALLY
-
-
RADIUS
OF
UN R EA C T E D
CORE
R AD I U S O F GR A I N SOME R E A C T I O N IN I T I A L
GRA IN
AFT E R
SIZE
R E AC T E D
G R A IN
SCHE MAT I C
OF
A PA R T I A LLY R E A C T E D GRA I N
GAS
S C H E MA T I C
A
PA R T IA L L Y
FI G U R E 6
OF
REACT E D
PELLET
S.chematic representation of the grain si ze model .
( 2 5)
( 26 ) Eli mi n ati n g the unknown concentration C i a t t h e interface , we ob t ain the A overall rate per unit grain as
( 2 7)
Once the rate of reaction fo r the in dividual grain is known , we can proceed to write the overall pellet equation as n v
2c
A
=
R
GA
3( 1
-
3
4rr rG O
f)
( 28 )
Doraiswam:y and Kul karni
31 2
where the term in pa re n the ses refers to the total number of gr ain s in the p ell e t volume . T he te rm Ra A i n v olv e s a k n o w l e d ge of the interfacial posi t i on rm w i t hi n each grain , which is a fu n c t i on of both the time and p o si tion To evaluate t hi s , a s t oic hi o m et ric balance on the solid wit hin the pellet . r e a c t an t B c an b e written in an alo gy with Eq . ( 5) for an in dividual grain as ( 29 ) The term C A a p p e arin g in R a A in t hi s equation fixes the p o si tio n of t h e individual grain in t he p ellet . T he a pp r op riat e b oun dary con dition s to the problem are
r r
t
= R:
D
dC A � e
dC
=
0:
= 0:
A
-- =
=
k (C - C ) g As Ab
0
rGi = rG O dr
( 30)
In general , this set of equation s re q ui r e s n umerical solution . A mathematical an al y s i s of the p article - pellet m od e l has been presented by C alve lo an d S mith ( 1 9 7 0 ) and S zekely and Ev an s ( 1 9 7 1a , b ) for a s i mp le isothermal fir s t - o r d er reaction . An approximate analytical s o lu tion has al so been p rop o s e d by Evans and Ranade ( 1 9 8 0 ) . T he c on s t an t - si ze grain m0del h a s been generali zed by S zekely an d Propster ( 1 9 7 5) t o include grain si ze dist ri b u t i on . T he case of nonlinear kinetics has been proposed by S ohn an d S ze ke ly ( 1 9 7 3a , b ) for r e ac tion s followin g the H - W rate law . T he microscopic visuali zation of t h e solid as consistin g of grains requires two p aram et e rs to be known for describin g the model . Considerin g t he
physical features of t he model , the t wo characteristic times involved are
T
g,
the tim e required for complete conversion o f t h e grain in t h e C A environ ment , and T a , the time for c o m pl e t e conversion of the particle by di ffusion if the g ra in conversion process i s extremely fa s t . I n the li mitin g case of T g c on trollin g , the diffu sional resistance for t h e gas within the particle is negli gible an d t h e simple homogeneous model res ult s . Th e in divi dual grains coul d follow the shrinkin g core model with ash di ffu si on or reac tion con trol lin g . I n view o f t h e p rocesses within the grain determinin g t h e system be havior , the c onversion - time re lati on s hi p will be indepen dent of the p ellet di m en si on s . On the other extreme , w hen diffusion within the pellet controls , one would ob serve s hrinkin g core behavior with ash diffusion control , and typica lly ( s ee T able 2 ) the system behavior ( T a.R 2 ) will be de p e n de n t on the p el le t di m e n si on s . In th e intermediate re gion w here both < g and T d are of the same order of magnitude ( note t hat T g is n ot a specific number but h as a di st rib u t ion t hat is co mm en s ur at e wit h t he C A p ro file in the pellet , which in t ur n is dict a t e d by T d ) , one could expect t h e pellet behavior to lie within the con fines of the limitin g cases of s h r in kin g core with reaction and a s h diffusion c on t r ol li n g p rocesse s .
31 3
Gas- Solid No ncataly tic Reac t ions
T he model is useful in cases whe re pellets are formed by compaction of particles of very fine si zes . T hi s is not so in naturally occurring minerals . in w hich case fictitious grains will have to be invoked to apply the model . Also the model , in its simple form , does not explain the earlier quoted S - s haped behavior an d levelin g off of conversion .
C rac k ling C o re Model Park and Leven spiel ( 1 9 7 5) proposed the so -called c rackling core model ( Fi g . 7 ) to account for the S - shaped b ehavior of x versus t plots ob serve d in some system s . I n this model t he reaction is assumed to occur in two step s A ( nonporous ) A ( porous )
-+
+
A ( porous )
Product
T he first stage could represent a simple p hysical process or may involve a porous interm e di ate , formed out of reaction , t hat further undergoes reaction T he reduction of hematite an d man ganese accordin g to the second stage . oxide (De B ruijin et al . , 1 9 8 0) and uranium oxide ( Le Page and Fane , 1 9 7 4) with hydrogen follow s such a sequence :
FeO
Fe
T he crackli n g core model has b een used successfully in the interp retation of experimen tal data for these sy stem s . T he model envisage s two stages , each of which is characterized by a · certain c haracteristic time , a n d t h e model therefore involves a t least t w o parameters . A s before , ' g represent s t h e t i m e required for complete con version o f t h e grain , and T c represents the time required for the original I n the limiting case of nonporous particle to b ecome completely porous . fast c racklin g ( -r g- << -r c ) , the controlli n g proces ses shift over to in dividual T he in dividual grain s grains and one oB serves a simple homo geneous - model .
FR ESH P E L LET
FI G U RE 7
C RA C K L E D
P ELLET
PE LLE T
P E LL E T
B asic stages in a c racklin g core model .
Do raiswam:y an d K ul karni
31 4
follow t h e s h ri n ki n g c ore model with either reaction or as h diffusion control lin g . I n view of the p roces s e s within the grai n c ontro lli n g the conversion time rel a tion ship b ecomes independent of the p ellet dimension s . In the other e x treme of slow crac k lin g , the p roces se s within the grain (in the porous part of the pellet ) get over muc h faster , and for the pe l l e t as a w hole one observes shrinkin g core behavior with r e ac tion control . Typical of such sit uations ( see T able 2) , the system time constant s hows linear dependence on the size of the pellet . The s tudy of the limi tin g cases as observed here makes pos si b le some compari sons with the earlier particle - p ellet mode l , T he case of < g »
,
co
,
N ucleation Model An altern at i ve approac h to p ro vi de an explanation for the ob served si gmoi d behavior in so m e gas - solid systems has been a t t emp t e d in a clas s of models which are gene rally referred to a s nuclea tion model s . N ucleation effect s are often signi ficant in systems such as reduction of metallic oxides , In these sys tems the p rocess proceeds with the g enera tion of metallic n uclei , whic h s ubsequently grow an d finally over l ap . T he rates of these in dividual processes stron gly affect the conversion - time b ehavior of the system . I n t h e event t h a t the n uclei gener at ion rate is faster , the whole surface gets covered w ith the metallic p hase and the reaction proceeds in a topochemical On the other hand , for a slow generation rate , the metal - oxide manner . interface is irre gu la r and differen t c on sideration s prevail in e stimatin g the con v ersion - time relation s . An empiri c a l model that relates the c onversion to t he ti me has been p ro posed by A vrami ( 1 9 40) : x
=
1
-
exp ( - C t
N
)
( 31)
C an d N in t his e q uat ion are con stants w hic h can be obtained by fittin g experime n t al dat a . The foll o win g three - p arameter equation in differential form has b ee n proposed by Ero feev ( 1 9 6 1 ) :
31 5
Gas-Solid N o n ca taly tic R eactions
( 3 2)
dx dt
an d seems to provide a dequate fit to expe ri mental data ( N eubur g , 1 9 7 0 ) . In a mo r e rational approach to nucleation , Ruckenstein and Vavanell o s ( 1 9 7 5) consi de red the existence of ger m n uclei in t he form of i mp u rities present and the embryos of the new solid phas e . In t he process the germ nu c leu s is t r an s form ed into a g rowth n ucleus , leadin g finally to p rodu c t formation , The growth p ro ces s is treated as an activated proces s and e m pl oyi n g the mo di fi ed A v ra mi mo d e l , t h e fol lowin g conversion - time relation may be ob tai ne d : ,
x
=
� Qk
3 N k growt h n , o n f
1
1
0
exp ( - k
t
n f nf
3 ) ( t - t ) [ 1 - x ( t ) ] dt nf nf nf
( 33)
w here N n 0 i s the number of germ n u c lei per un i t solid volu me at the begin ' nin g , an d kn f and tn f are the rat e con stant and time required for nucleus formation , respe c ti vely B hatia and Perlmutter ( 1 98 0) hav e also analy ze d this prob l em usin g the popula tion balance approach . T he pos sib ili t y of both hom o ge n e o u s and hetero geneous nucleation aroun d a germ nucleus is considered an d the b a s ic Avarami model has been modified to i n c lu de a term for the initial volume of a growth n ucleu s . S ome e xp erim en tal dat a on t he form at i on an d gro w t h of nuc l ei for the oxida tion of copper iodate and reduc tion o f wustite have been provided by Neub u r g ( 1 9 70) and E l - R ahaiby and Rao ( 1 97 9 ) . S ome aspects of the theory of nuclea tion have also b e en discussed ( R ao , 1 97 9) . .
E x te n s i on s of t h e B a s i c Mod e l
The p rec eding section dealt wit h simple b a si c model s . In the present section we exten d these models by c on si de ri n g several complexities that are likely to be p resent i n practic al situation s . The c omp l ex it i e s that can be considered are those due to the effect of bulk di ffu sion , exis t e n ce o f pressure gradien t s in the pellet s , nonisot hermal effects , and e ffe c t s due to s tr uc t u ral chan ges . The effect of structural chan ge s , b ein g of much gr e a te r prac tical relevance , w i ll be considered at some len gth a t the end of the s ec t ion .
B u l k- Flow
Effects
Detailed formulation s accountin g for t h e effect of b ulk fl o w have b e e n p r e s en t e d by a n umbe r of aut ho rs ( e . g . , B everidge and Goldie 1 96 8 ; Gower , 1 97 1 ; Sohn and S ohn 1 9 8 0) for the sharp interface model . The b ulk flow within the solid assumes greater i m portanc e for reaction s with volume c han ge and typical examples o f in dustrial gas - solid reaction s wit h volume change are ,
,
C ( s ) + C 0 2 ( g)
Ni ( s )
+
4C O ( g)
FeC1 ( s ) 2
+
= =
H 2 ( g)
2C O ( g) N i ( C O ) 4 ( g) =
F e ( s ) + 2HC1
,
Doraiswamy a n d K u l karni
316
T he c on ser vati on e q ua t ion for the mass of react an t in situation s of this type should include a t e r m for b ulk flo w and can b e represented in anal o gy with th e gas - solid catalytic sy stems ( W e ek man an d Gor rin g 1 96 5 ) in dimen sionless form as ,
( 34)
wit h th e b o un d a ry c on di t ion s R
= 1:
C
( 3 5)
= 1
A
( 36) 13 is a c on s t ant and cp an d 6 re p re sent , r e s p ecti vely the T hiele modulus and volume chan ge modulus in thi s e quati on , and are defin ed as ,
,
¢ = 2 ( 13 + 1 ) D
( 13 + l ) V
k
p
e
( 37)
( 3 8)
T he c on ce n tra tion p rofi le for t he gaseous reactant as obtained from the solu tion of E q . ( 3 4 ) can be related to the c on s u mp t i o n o f solid as follow s :
( 3 9)
where t
=
( 13
A
p
+ 1)V
t
( 4 0)
p
B earin g in mind the relation ship between conversion and inte rfacial p osit i on ,
( 41)
we can calculate t he conversion - time relation ship . S ohn an d S ohn ( 1 9 8 0 ) have obtained the f ollo win g a sy m p to t ic solution for the conversion - time behavior :
Gas- Solid N o nca taly tic R eactions ln ( l + 8 ) 8
:: =
1
_
( S + 1) ( 1
x) s
p(x)
2/( S+1) -
1
31 7
- 2( 1 - x )
( 4 2)
In gen eral , the n et effect of vo lu me cha n ge is to cha n ge the time re qui red to attain a given ex te n t of conversion by a fac to r 8 / l n ( l + 8) . I t is al s o po s sib l e t o inco rpora t e the e ffec t of e x ternal mass tr an s fe r in t he presence o f volume c hange e f fec t s . T he c onversion - time relationship c an now be ob t ai ned as t
[1
_
(1
_
X)
1 / ( 8+ 1 )
1 +
2
[
8 ) 2 ( 1 + 8) x + p( In ( 1 + 8) X Sh
]
( 4 3)
In the even t t h at n on is oth erm al c on di ti ons p r e v ai l , si gn i fic ant pr es su re gradien t s can exist within t he pellet and the simp le an a lys is above is no lon ger ap p li cab le . I n fact , there is a need now to mo di fy the conventional Fickian law of diffu si on , and more s o p his t ic at e d models for diffu si on in p or o us media ( s uc h as the du s ty - ga s model of Mason et al . ( 1 9 6 9 ) m ay have to be invoked . ( T his has not been a tt empte d so far . ) Even in i n st an ce s w hen the b ulk - flow e ffect s a re not pre sent ( no mole c h an ge ) , pres sure gradient s may arise due to temperature gr adie n t s . In the presence o f p re s sure gradients it has b een s ho w n ( Won g e t al . , 1 9 7 6 ; Hite and Jac k s on , 1 9 7 7 ) that t he c onven t ion al Weis z - P rater t em p e rature holds true only in the case of K n u dsen diffusion . Si gnifican t alteration s occur
2 I n fac t , an additional term ( B t 4) i s inc luded m in the e qu at ion for maximum temperature r i se where B m refers to the ther micit y factor . P r es s u r e g r a dients can thus have a significant effect on the c on v ersion - time relation s , de p end in g on the regime of di ffus ion . In the case of gas - solid reaction s , the p resence of stru c t u ral effect s con tinual ly alters the d i ffu sion al regime s . The effe c t s of p ressure gra dient s cannot therefore b e i gn o re d all throu gh the course of c on v e r sion . The in fluence o f p r e s s ur e g r adie n t s in ga s - solid reac tions followin g SIM has been examined by Deb Roy and Abraham ( 1 9 7 4 ) an d Turkdogan et al . ( 1 9 7 3 ) . D e t ai l ed numerical solution s have been ob t ain e d to the pressure gradie n t s that arise because of Knudsen flow in the ash layer with s mall pores and due to the re ac tant gas havin g a di ffu s i vity different from that of t he p roduc t . w hen bulk diffusion prevails .
N o ntsot hermal Effects
W hen th e chemical reac tion is ac com p ani e d by evol ut i on of heat , t e mp er at ure gra d ien ts may exist within the pellet . The local reaction rat e in the i n terior of the p article can thus b e hi gher , and the hi gher in s id e temperature can c aus e some diffic ultie s , such as s in t e rin g . A d ditionally , dep endin g on the heat ge neration /removal rate , the sy s t em may exhibit multiplicity behavior . In the analy sis of n oni s ot h ermal S IM , equation s d e scrib in g heat trans fer t h ro u g h the gas film an d ash layer a n d heat ge nerati on d u e t o reac ti on at the i nt e rfac e can be w ritt en in a manner analogous to the mas s tran sfer p roc e s s . S uch a fo rmula t ion a fford s an eas y ana ly s i s of the nonisothermal ef fec t s . T he ap proach , however , p resumes the validity of the p seudo s te a d y - s t at e a s s u mp t ion , w hic h while justifiable for mass tran s fer p rocesses , is not a goo d appro ximat ion for heat t ran s fe r proce s s e s .
31 8
Doraiswamy and K u l karni
Lus s an d A m undson ( 1 96 9) have provide d an analysis that incorporates the transient heat accum ulation term an d their solution can be re p res e nted as
1
1
T.
3
�p
=
1
f:
sin "'
n=1
R
�
i
1
f( � )
sin
R.
where � ' s are the positive roots of the tran scen den tal e quation 1J
cot �
+
Nu - 1 = 0
( 4 5)
and the various othe r parameters are defined as follow s :
� - R.
--------�1�-
f( � ) =
1
Da exp [ y ( l - 1 /T . ) ]
P 2 ( N u - 1 ) 2 + 1J
Nu(Nu - 1)
f ( ].J ) =
+
1J
( 46 ) 2
=
M
p k' B e p C D C
( 4 7)
B s ps e Ab
2
( 4 8)
E quation ( 44) gives the interface temperature in terms of the interfacial which can be obtained fro m
position � , 1 -,..=
dR . dt
( 4 9)
n
w he re n refers to t h e effectiveness factor un der nonisothermal con ditions . For t he s pecific case s of c ok e regeneration an d reaction of carbon with oxy gen an d steam , model e quation s based on SIM have also been developed by S hetti gar and H u ghes ( 1 97 2 ) and Rehmat and S axena ( 1 980) . T h e p ro blem of t h e r m al instability in gas - solid systems has been discussed by C annon and D enbi gh ( 1 9 5 7) . Shen and S mith ( 1 96 3 ) , Aris ( 1 9 6 7 ) , Is hi d a an d Wen ( 1 9 6 8) , an d Wen an d W an g ( 1 97 0 ) fo r S IM . T he phenomenon of thermal in stability leads to sudden transition s in t he rate -controllin g step s an d has an important bearin g on re actor op e rati on , as runaway conditions may occur . T he nonisothermal S I M is well suited for de scribin g decomposition re action s ( N arsim han , 1 9 6 1 ; C ampbell et al . , 1 97 0 ; Hills , 1 9 6 8 ) . For a de c ompo sition reaction A(s)
-+
B ( s ) + P ( g)
E ach the application of the phase rule s u ggest s one de gree of freedom . temperat ure T d therefore has a fixed value of p artial p ressure P d of the
Gas- Solid
31 9
N onca taly tic Reactions
product gas P , an d once t hi s value is reached , the react ion starts and the T he process is controlled either by heat or gas diffu front moves inside . sion through the p roduct layer and t y p ically yields S I M b ehavior . In the event that heat transfer through the ash layer controls , the re action interface stays isothermal and t h e equation for SIM wi t h ash diffusion control a n d D e replaced by k � ( in the definition of T in T able 2) represen t s the conversion - time behavior . W here gas diffusion t hrou g h the ash layer is con trollin g , t he variation of D e with temperature ( D e T 1 / 2 in t he kn udsen regi me : D e <X T L 5 - 2 in t h e b ulk diffusion regime ) should be ap propriately accoun ted for (Mu an d Perlmutter , 1 9 8 0) . Nonisothermal effects have also been analy zed for the volume reaction model by Shetti gar and H u ghes ( 1 9 7 2) , and for the particle - pellet model by C alv elo and S mith ( 1 9 7 0) . The latter analysis is based on the p seudo - steady The governin g state hypothesis , w hich is not v alid for heat t ran sfer . transient equation s for heat and mass for the volume and grain models are presented in T able 4 together with the relevant boundary con ditions . An a n aly si s of the complete transients for the p a rticle - pellet model indicates a shi ft in the temperature maximum from the surface to the center with progress of time ( S am p ath et al . , 1 97 5 ) . N onisothermal effec ts alon g wit h structural variation s have not yet been analyzed . <X
Effects of S t ruc t ural Varia t ions
T he earlie r parts o f this section were concerned with t h e exten sion of the b as ic models to incorporate ce rtain modi fic ation s res ultin g from the nonlinear nature of chemical reacti on , p re s sure gradie n t s an d b ulk flow within the H ere we shall reexamine these models wit h a view to incorporati n g pellet . The the structu ral chan ges t hat occur within the soli d durin g reaction . main structural chan ges are due to chemical reaction and sin t e rin g . T he differences .in the molal volumes of the reactant and product lead to varia tion s i n t he p ar ticle voida ge an d e ffec t i ve diffusivity durin g t h e course of reaction . T o account for these effec ts i t i s necessary to fo r mulat e a rela tions hip bet ween the void age , diffus ivi t y , and extent of conversion . A large number of such r elation sh i p s to account for these effects have been proposed u sin g different models , and the more important o f t hese are sum marized in T able 5. T he incorporation of such e f fec t s in th e b asic models gi ve rise to conversion - time rel ations hip s which show a si gmoi d behavior A lso , w here t he porosity at the s urface of the observed experimentally . pellet becomes zero ( pore closure ) , t he governi n g e qu ation s pre dict incom plete conversion . S uch b e h avior ha s , i n fac t , b een observed i n some e x peri mental system s . Hartman a n d Coughlin ( 1 97 4 , 1 9 7 6 ) and Georgakis et al . ( 1 9 7 9) have used a modi fied grain model to de scribe pore closure due to grain swellin g in the sulfation of lime . I n a departure from models based on pore behavior , Garza an d D uduko vic ( 1 9 8 2a , b ) have proposed a variable - si ze gr ain model for sy stem s with I n corporatin g the diffusivity variation s with poro s ity structural chan ges . for sy stems w i th no pore closure , the mo del first calculates t h e time re quired for c omplete conversion of t he s urface layer of the solid . T his pro d uc t layer then move s in side , toward the cente r . In system s where pore clos ure occ urs , the model evaluates the time required for pore closure and obtains the concentrat i on profile of the unreacted solid at this in stant . Further processes in side the pellet are described by S I M with the incorpora tion of a soli d concent r ation p r o file . A pp roxim ate analytical solution s have
w "" 0
TABLE 4
Typical Diffusivity and Porosity Variations due to S tructural
N umber
1
=
f c
e De o D
2
co
f
=
ex
+
c
De
f co
ex
(
+
f =
D
0
� [1 - � J
r
=
f
n
2
( B /ll ) cx
, an d
1-
ex
co
c
)
+ ex 2 -
11
Model
References
Volume reaction model
Fan et al . ( 19 77 )
Volume reaction
Fan et al . ( 1 977)
-1
model
3 are constants and B refers to solid concentration
V f
in Pellet
Relations hip
(1 - __.!!_ )
cx 2 ex p
w he e a: 1 ,
3
1
Chan ges
V
B
B
-
Leven spiel ( 1 979)
+ VI v
sV s
eo f co
c
t:1
1 4
1 D D
-...., e
eo
f c f co =
=
f c
f
co
( ::0)
3 where r
G
refers to the radius of the
grain
Particle - pellet model
r;; · � Q 3 '<: Q ;:I l=l. ::0:: £.
a
where V B and V s refer to molar volumes of species an d B and s per mole of solid . VI refers to molar volume of inert p er mole of solid B .
Ramachandran an d Smith ( 1 9 7 7a)
.... Q
2.
p "' I
:;)
Eventually , with p rogress of time , the pore will be completely blocked . T he time required for pore closure is obtained as 1
=
Bi
p
-
1 z
v
l- - zv
D
e
-
D
=
1
f ) 2/ co +
1
1
where Z
5
(1
1
zv
co ·
v ==
1
-
v
v
1-
p
(1
co (
f
co
eo
f ) MB c
Y
_
1) ( 1
( - ) - t 1 +
1 -
M C pB C
B C
f
-]
3
-
1
1
(1
f
co
3
r a· A
zv
1
)
z
f
v
co
113
z +
v
1 -z v
)
where y refers to the ratio of the volume of solid product formed p er unit volume of reactant consumed and rm refers to the dimensionless position of the shrinking core in the grain . T he time required for p ore closure i s obtained as Da T == 1 - n + ___s: 2
where
n
=
[1 +
<1
{1 -
n2
+
- ::� - ·y)] f
o
3 213 rY + 0 Y>n 1 1 - y _
;?.. i5: til
)
Particle -pellet model
G eorgakis et al .
( 1 9 7 9)
0 ;::l 0 Q
........ ....
:z:
� c:;· Q
' co
(I) p 0
� ;::l "'
s·
Variable grain model
Gar za and D udukovic
( 1 982)
}
1/3 c..:o t\:1 ...
TABLE
5
M o del
Volum e reaction
m o de l
S om e U s eful T i me - C onversion Relation s hips S ystem
de scription
Reaction first order in gas
R e fe renc e s
Time conversion
In
and solid ;
[
A
�
_
exp ( - t )
2. 33
0 . 699
]
+ tA
_
-
¢_
_
2
10. 5
[
_ 1 - x 1 0 . 6 99
e xp ( -t ) A
+
2 . 33
J
spheric al
K ulkarni an d R a m ac handran ( 1 98 0)
pellet
Particle pellet
Firs t - order reaction
t
=
-
1
model +
( 1 - x)
¢
1/3
18
2
( 0 . 2 1x - 0 . 31x )
t = Uniform grains
of unchan gin g si ze ; both
pellets and
grain s c an b e
of arbitrary geometry
B
tM p
B
g
A k r
B G
t - g F g( x )
O
s
( T 0)
2F
1
- - [1
w here
v
2
Da
g
3( 1
( :n . +
an d
Fg(x)
+ ____g f
- ¢
1
x)
+
2(
1
-
x) ]
[-o. ( .:.�J
exp
=
2/ 3
9
[
3( 1
m
- fc O) k ( T O) R s r
G OD e
R a nade ( 1 9 8 0 )
]1/2
2
2
+
¢p
� fF ( x ) p g
S zekely e t al .
( 1 976)
p
w here subs c ripts g and p re fer to gr a in and pellet resp e c t i ve ly and
,
Evan s and
F to the shape factor an d takes values of 1 , 2, cyli nde r an d sp h e re respec tively .
or 3 fo r in finite slab ,
f (X) = . F. f
1
1
1/F [ -
( 1 - X)
-
1 gF g ( X ) =
F.
-1
2
1
where i = g or p
Uni form grains of varyin g sizes ; both pellet and
,. t
=
gF ( x ) + ___..8: 2F
- I
grain can have
Ran dom pore model
T akes account of structural variations throu gh population balance approach ; kin etic re gime
p
F (x) g
+
p
g
x = 1 -
1jJ
-
F (x)
g
F
2F
g
(
m
R
s
2 0
p
F
) [3
exp
t
=
g
p
2 /F
1 - f
cO
k A s 0t �
-
(1
cO
T hi s expression can als o b e expressed as
dx """"A = dt
( 1 - x ) [ 1 - 1/J ln ( 1 - x ) ]
1/2
-
-
- x) ] g] + 1 - [ y + ( l - y ) ( 1 2) ( y 1) (F g
m
f
Garza and Dudukovic
f (X)
m k A S t s g 0
1
x
1
....:..£__ f 2
1 - 2 [ 1 - ( 1 - x)
k A t 1 - �
4L o ( l - fc o >
where
=
F
2 F.
1 -
4>
g
- J
2 /F .
w here gF g( x ) as above and
arbitrary geometry
x)
(1
Da
g
g
- )]
� k s A S Ot g m
+
4
1
fc O
2 /F
g
l
( 1 98 2 )
B hatia and
Perlmutter ( 1 980)
TABLE 5
( Con tinued)
Model
System description Diffusion control regime
S CM
Gasification
reaction
d
a( l -
x
--;:;;- = dt
1
27 c O
Random capillary model
Gasification reaction
v
fc 0
f
a• 3 -
G'
is
G'
-
fc o)
B MB D e G S O
[(
-
1
where _!_
+
x) [ l - 1/J ln ( l - x) ]
2ks pb ( l
f cO
x =
Time-conversion
1 +
)
mt
k A s �
r-p 0
1/J
z \)
References
11 2
[ II - 1/J ln ( l - lj!) - 1 )
G ' - 1 - ( k A mt ) / r s g po G' - 1
a s olution of +
1
=
Bhatia and Perlmutter ( 1 98 1a , b )
- 1]
S zekely
et ( 1976)
0
2
exp [ - 2TI( N v t 2 + 2N 1 v t ) ] 0 where v is the reaction velocity defined as the rate of change radius with time , and N 0 and N are struc tu ral parameters 1
x
=
N0
1 -
=
l sm1nmax S
.
where p(S 0 )
p ( S O ) dS
O
N1
s
r m ax
=
the density dist ribu tion is the pore size
is dist rib ution , and S
al .
)S .
mm based on
of p ore
S Op ( S O ) d S O the initial
pore size
Gavalas ( 1 98 0 )
325
Gas- So l i d N o ncataly tic R eac t ions
been developed by Gar za an d D udukovic ( 1 98 2a , b ) and are included in T able 5 t o gether other models . A ne clas s o f models i n cor o r a t i n g as o e si ze distribution have come up in rec e n t years . T h e simplest of such mo de l m o de l o f Ramachandran and S mith ( 1 9 7 7a) and C h rostowski and G eor gaki s ( 1 97 8) . The model of R am a an dr n 8 , a n aly z e t h e h n e s t a n g in le pore is s u p e d to b e representative of overall p ellet . The s i n gl pore is as s um e d to be cylin drical with t h e once tric in g of solid B a s soci D ep en din on the molal volumes of the re ac an t an d produc t , ated with i t . the o r e si ze increas e s , decreases , or stays cons t ant with progre ss of re action . T he model yields a s i m ple conversion - time relation ship an d requires a k n o w le d ge of t e average structural properti e s of the solid ( pore radius , len gth , r ad i u of a s o i a t e d solid ) . T h e si m ple model has been ex te n ded to incorporate the effect of b ulk flow an d re v er s ibili y of reaction ( Ulrichson an d Mahoney , 1 9 8 0 ) . T he u s e of average s t r uc tural p r o p e r t t o mathematically t ra t M ore able models t hat are n o rm a lly a d e q u a te for most gas - solid system s . ri rou s s ho uld include the di ri b u tion a n d v a at io n s in r o p e i e s with reac tion . T hus most solids have an ini tial pore i z e di t rib utio t h at e v o lve s d u in g the reaction due to pore in t e r
with the solutions for p structural features such
w
p r
s is the single- pore
sketched in Fig. which pos
s
ch a and Smith , ki place in a s g the e c n r t
c a g
g
p
h
s
sc
t
ies leads
go models , however , the structural p rt s s n
section and coalescence .
an d
a ny ri go ro u
s
c
st
ri
r
T he net reaction surface is t herefore a variable ,
struct ural model should accoun t
for this
variation .
S ome
c
of t h e recent models t hat a co u n t for this will now b e disc ussed .
pp t rate of r c t w
T he model of C hri stman and E d gar ( 1 9 8 0 ) as s umes the a licab ili y of a s in gl e ore model at a local point r in t e pellet . T he e a tion t hus ob t ai n e d i s ave ra ge d ove r t h e e x i s t in g pore si ze di s t r ib u i o n , hic h is then u sed in t he mass b a l an e eq u at ion together with the average values of other parameters ( effec tive di f siv t y rate constant , etc . ) . T he mass b alanc e eq uation is s u p p e m e n ted by an e qu a t i on describin g t h e evolution of pore size dis t ribution wit h the p ro gre s s of re a c t i on and C h ristman and E d gar ( 1 9 8 0 ) modeled t his p rocess usin g a o u l a t i on balance ap proac h . T he key feat ure of t his model is t a t it gi ves the v ari a t ion of pore s z e d s t ri b u ti o n i t h t im e an d posi ti on T his addition al inform ation can be A si milar ap ro ac h based on u s e d for model verific ation an d s ele tion p o p u ati on balance has b e e n introduc ed by Gavalas ( 1 9 8 0 ) . T h e model of Si mon s and Rawlins ( 1 98 0 ) a ssumes that each pore within t he solid e n d s up at the surface of the pellet , where a di s t ri b u ti n of pore exist . T he model calculates the flux o a o e of radi us r p and r di averages it over the s urface pore size dist rib ution t o obtain an average flux of the gaseous re ac a at t h e su rface . T hi s i s t hen related to t h e conver sion of solid B . The model , ho u gh si m l e doe s not account for he varia t i n of reaction surface wit h time . Perhaps the most realistic model is t he an do m pore mo de l of B h atia and Perlm utter ( 1 9 8 0 , 1 9 8 1b ) , which assumes that the actual reaction s ur fac e of soli d B is t h e res ult of the r a n do m o e l a ppin g a set o f y li n d ri de ve lop m e t of t he reac tion surface as en vi sa ge d in t i s model is sketched in Fig . 9 . T he mod el oc eeds c a l u la ti on of the r ea c t ion su rface an d then the conversion - time r e l a t ion s hi p in term s of the i n rin s c s t uc u r a ro p e r t i e s the solid . T he follo win g equ ation s for the conversion - time behavior for the cases o f kinetic s an d as h di ffu sion on ro l have been obtained : -
p
h
c
fu
i
,
l
,
pp
h
i
w
i
.
c
p
.
l
o
a i can
fr
p r
t nt
t
p ,
t
o
r
v r
cal surfaces . The actual
i
r t
c
h
pr
t
of
n
l p
of
c t
with
c
t.> t\:1 0:.
� SS /:0 1
PRODUCT C
f ll ;-- SOLID
�
�� l (a I
-
a: 0 Q.
AT
FI G U R E 8
SOLID B
B A S SOC IATED WITH THE PO R E
(cl
(bl Zv < I
T I M E ZE R O
A FTER
Schematic
representation of
the
sin gle - pore
model .
SOME
R E ACT I O N
Zv
>
t
a
0
t7
;;·
�3
'< Q ;:s Q.
�
>:: s:: Q ,
2.
Gas-Solid N o ncataly tic Reac t io n s
(a )
\ D I F F US I O N L EN G T H FOR GA S A \ A R E A OF UN I FO R M G A S CONC ENTRATION
( c )
( b )
FI GURE 9
327
D evelopment of reaction surface accordin g t9 the random pore The shaded area represents unreacted solid B , the dashed area t he mode . produc t layer . ( a ) Early stage showin g product layer ar o un d each pore ; ( b ) intermediate stage , showin g some overlappin g reaction s urfaces ; (c ) later stage , showing full development of p roduct l ay er and reaction s urface for the partic ular view chosen . ( From B hatia an d Perlm utter , 1 9 8 1b . ) dx ,..
dt
=
( 1 - x ) [ l - ljJ ln ( l - x ) ]
and dx ,..
dt
where
,.. C
=
1
+
( l - x ) [ l - ljJ ln ( l - x ) ]
S ' Z /Iji { [ l - ljJ ln ( l - x ) ] A
and
( 50)
l/ 2
1 /2
1/2
- 1}
( 51 ) ( 52)
t
T he parameter S ' characterizes the diffusional resi stanc e to the flow of the ga s e o u s s p ec ies in the p roduct lay er . F or ljJ 0 , E q . ( 5 0) reduces to the volume reaction model with n 1 , and for ljJ -+- 1 the behavior closely ap I t is apparent , therefore , that the concept of proaches the grain model . reaction order with respect to solids is closely related to t he str u c ture of the solid , an d the ra n do m pore model for the first time provide s a rational meanin g to the order of reaction with respect to solids . The relation be tween tjJ and n is graphically illustrated i n Fi g . 1 0 . S everal distributions for the parameter tjJ can b e as s umed , and B hatia and Perlmutter ( 1 9 81a) have s hown that a uniform pore si ze leads to the lowest reactivity of solid . Also , for a given p ore si ze di s trib u ti on , an op timal structure exi s t s for hi ghest activity of soli d . =
=
328
Doraiswamy a n d K u l karni
0·8 0:
z 0
u c 1&1 a::
I=
...
0
a:: .... Q a:: 0
0·4
o �------�-0 0·5 1 ·0 HI 2·0 P O R E S T R U C T U R E PA R A M E T E R , 't'
FI G U R E 10 Relations hip between the struct ural parameter of the ran dom pore model and the order of reaction with respect to solid in the volu me reaction model .
( F rom B hatia and Perlm utter , 1 98 0 . )
The model of B hatia an d Perlm utter ( 1 9 8 0 , 1 9 8 1a , b ) , discussed above , considers the pore s an d reac tion surfac e areas to be e qual . Although thi s seem s to be a good approximation at l o w levels of con version , it is less accurate at hi gh c on version levels . In a subsequent p aper , B hatia and Perlmut ter ( 1 9 83) provided generali zation in term s of movin g pore an d re action surface , an d for the case of no in trap article and boundary layer diffusional resistances , obtained the followin g rel ation :
t
( 53)
+
For S ' + 0 and '"' , t his equation reduces to the cases of kinetic control an d product layer diffusion con trol . In this equation S * and S refer to the dimen sionless reaction surface area an d pore surface area , resp ectively , an d these quantities can be app ropriately evaluated for differen t models of t he intern al structure of the soli d . Thus for the grain model ,
�
329
Gas- So lid N o ncataly tic Reac tions S*(x)
S* ( x) p
==
==
(
1 -
[1
+
x)
(Z
m
v
- 1) x]
( 54) m
( 55)
while for the ran d o m por e m o d el , S * ( x ) ==
S * (x)
p
==
(1 [1
- x) •ll - 1jJ ln ( l - x ) +
(Z
v
- 1) x ] v' 1 - 1jJ ln [ 1
Sub stitutin g these expression s for S * and
+
s;
( Z v - 1) x ]
( 56 )
( 5 7)
( 53 ) an d in te gratin g
in Eq .
leads to the desired conversion - time relation ship . A c om p ari so n of pre dic tion s ob tained usin g the grain m o d e l an d the ran dom po re model for a
set of p a ra m ete r values is ill u st rate d in Fi g .
11.
Sin teri n g
The temperature vari at ion s w it hin t he p e ll et i n t h e p re sen ce o f a n e xo t h e r
mic reac tion result in sinterin g , whic h causes a de c r ea s e in t he effective
diffu sivity of the pellet . A lte rn at ively , the grai n si ze increases an d the s p ecific surfac e area of the grain d e c re a se s . S imple empirical laws s uch as e xponential decay of di ffu s i vi ty with reaction ( Evan s et al . , 1 97 3) or first - order dec ay in s u r face area ( R an ad e and H a rr i son , 1 9 7 9 , 1 98 1) can be used to account for sin terin g . T he primary in fl uence of sinterin g is to de cre a s e pore in t e r s ec t ion . T hi s causes a decrease in p o ro si ty an d an in cr e ase in t he tortuosity factor of the p ellet . T hese effec t s have been taken into accoun t by K im and S mi t h ( 1 97 4 ) . C han and S m i t h ( 1976) , and Ramac han dran and S mith ( 1 9 7 7b ) . The combined effec t of c he m ic al reac tion an d sin t e ri n g on the di ffusi vi t y in the pellet has been expressed by the r el a tion r
3
2
( 1 - f > ( Gi ) J o - f ) r p GO
0
( 58)
w here fp re p re s e n t s the fraction of pore s removed . A s sumin g the rate o f rem oval of p ore s to be por p ort i on al t o their availability , the fo llowin g simple
e quati on for th e variation of fp with time r e s u l t s : df _g_ = k (l - f ) p dt p
( 59)
w here k p , analo gou s to the conven tional rate con st an t , represents the rate const ant for pore removal .
G en era l Fo rm u la t io n o f M o del Equa tions
I n t he previous section s w e described the basic models and considered t heir exten sion s t o i ncl u d e more c omplex cases , such as b ulk flow , non
isothermicity , an d so on . I n the presen t s e c t i on we develop a general mathematical for mu l at ion of these mo d el s , includin g a few case studie s .
� � c
a fii" �
0
t:l
0
S' .
-
0
FI G U R E 1 1 of
II
TIME , t
10
i2
14
Compari son of conversion predictions for s everal models for vario us values
Parameters are 1jJ = Z v
==
1 , cp
=
0.
!:)
3 '< Q ;::s Q,
!:. �
;>;"' Q --s
2.
331
Gas- Solid N o ncataly tic Reac t ions I t i s p refe rable to begin with t he volume reaction model , whic h is
describ e d by E qs .
component ,
(a: RB a�A)
��
RB
F or a reac tion first order in t he gaseous
=
( 60)
ax at
CA
( 61)
F(x)
==
dt
aR
aR
dx
( 1 0 ) to ( 1 4 ) .
these eq uation s c an be re written as
w h e re a: refe rs t o t h e diffu sivity ratio D e I D e o , x the conversion , ¢ the T hiele modulus based on the initial diffus ivity , and 8 t he geome tric fac tor . Follow in g the gene ral tec hnique developed by D el Borghi et al . ( 1 9 7 6 ) , t he
cumul ative gas concen tration c an be w ritten as
1/!
=
and E qs .
f
0
t
C A dt h
h
=
f
x
f ( x ) dx
( 62)
0
( 6 0 ) an d ( 6 1 ) t hen tran sformed into
( 63) S u b s tit utin g for
1/! from E q . ( 6 2 ) we obt ain 13
f x)
d
�
.:C
R
R
+ f' ( x )
( �) d
dR
2
- k(x)
=
0
( 6 4)
E quation ( 6 4 ) is a n ordinary secon d - orde r differen tial eq uation of secon d de gree an d i t s solu tion directly gi v e s t h e conve rsion p rofiles . T he equa
tion is s ufficiently gen eral , since several functional forms [ f( x ) ] suitable to di fferent models can be c hosen . T ypical forms of f( x ) for some of the
more frequen tly used models are presented in T able 6. P rasannan et al . ( 1 9 8 2 ) h a s e mployed a collocation procedure to solve t his equation , and the results obtain ed by them usin g t h e fun ctional fo rm s of f { x ) corre spon din g
to the particle - pellet model , the particle - p ellet model with structural varia
tion s , an d t he nucleation model are reproduced in Fig.
12 .
A N A LY S I S O F R E A C T O R S W e have s o far been concerned wi th the desc ription o f the sin gle - p article
system , and s everal models attemptin g to describe qualitatively and semi quantitatively the c om plexities in real systems have been disc ussed . N ewer
models - m any o f them extension s of the b asic model- are con stantly appear in g in the literature and e fforts to quantitatively and realistically d escrib e the systems continue .
Despite these efforts , the sin gle - p artic le beh avior
may con tinue to involve uncertainties , and the sit uation becomes even more
332
Doraiswam:y and Kulkarni
TABLE
6
Function al Form s for F ( x ) for Different Gas - S olid Reaction Models
N umber
F unctional form
1
(1
2
-1 -
-
x)
-- n
(1
_
x)
4
..
n
+
1/3
(1
Volume reaction model _
Sh [ Z
x)
2/3 Grain model
2/3
...;,__='"-- + -----�-_:.;;..:._ .. _-=-
Sh
-1 -
1
1/ 3
Sh
( 1 - x)
3
Reaction model
f( x )
1
1 - x
In
( 1 � x)
( 1 - x)
\)
+ ( 1 - Z ) ( 1 - x) ] \)
1 /3
G rain model with structur al variation s
( 1 - n ) /n N ucleation model
0· 6
z 0
Cll a:: ""
> z 0
u
0· 4
0- 2
n=3
1 �--L--O L---�--�-=��--�----L---� �--�----�--�--� 0
0·4
0·6
D I M EN S I O N L E S S
FI G URE 1 2 S oli d conversion ve rs us dimen sionless reaction time for particle pellet model wit h and without structural c han ges and for nucleation model .
1·2
Gas- So l i d N oncataly tic R ea c t ions
333
di ffic ult w hen p a rticle assemblies , a s in t he case of reactor s , are concerned . The common types of reactors employe d in ga s - solid noncatalytic systems are packed - b ed , flui di ze d - b e d , and transport - line reactor s . T he ad ditional complexitie s that need to be accounted for in the analysis of reactors ari se out of the flui d flow asp ect of the problem .
I n a ddition
to t he heat and m a s s trans port p roces s e s within t he pellet , one needs now Also , specific to to con si der the tran sport p roc esses b etw een t he pellet s . each mode of operation , the desi gn c ri terion may depen d quite critically on the pres sure drop - ga s flow relation ship as in t he packe d - bed , or on the minim u m flui di zation velocity as in the fluidi ze d - b e d , or on the terminal T he soli d s fee d to velocitie s of particles as in the tran sport - line reactor s . these reactors may be of con s t ant size an d s hape or may involve a si ze dis trib ution . N on e of t h e se e ffect s m anifest them selves in sin gle - p article studies but become import an t con side ration s in the design of reac tors . present b elow a di sc u s sion o f eac h o f t h e se reactor types .
We
Pac ked - Bed R ea c to r s M any gas - solid noncatalytic reac tion s are carried o u t in packed - b e d reactors , some important examples b ein g the roastin g an d sinterin g of ores , reduction of metal oxide s , production of li ght wei ght ag gregate s , an d incin eration of solid wastes . A typical sketch of a packe d - be d reactor is shown in Fi g . 1 3 . T h e conven tional packed b e d contain s a charge o f solid reactant and the gas passes t hrough the voids in the b e d .
T he reac tion between t h e gas
and the soli d leads to the formation of a reaction front that gradually move s T his mode of contact i s batch with respect to alon g the p ac k e d hei gh t . the solid and is unsuitable for large - s cale p a rticle applications . In many in s t ances it is p re ferable to h ave solid s on a movin g grate supplied with gas blo w n throu gh t h e bed on t h e grat e ( Fi g .
1 3b ) .
T h e principal correspond
ence between the s e two modes of c on tact i s that the time for t he reaction front in Fi g .
1 3a can be i d entified with L / u in Fi g .
1 3b , providin g j u stific a
tion for the work d o n e o n conventional transient fixe d - be d sy stems ( S zekely , et al . , 1 9 7 6) .
GAS IN
U N R E A CT E D R EGION
R E AC T I O N
F RONT
R E ACTE D
,..__ R E A C T I O N
REGION
F RONT
t
(a)
GAS I N
(b)
FI G U R E 1 3 G as - solid con t ac tin g in a fixed - b e d arran gement : bed ; ( b ) p acked b e d wit h c ro s s - flow contactin g .
( a ) p acked
334
Doraiswamy and K u l karni I n the p acked bed as sketched in Fig.
1 3a , several nonidealitie s can
T he flow c an deviate from the i deal because of radial variation in exist . velocity and mixin g effects b ro u ght about by the p resence of pellets . A ls o , the temperature across the cross section of t he packed bed in practical
sy stem s may not be u niform . These problem s lead to a consideration of the variation of system p roperties in the axial and radial direction s . One w ay of accountin g for these situation s is by superimposin g an effective trans port mechani sm on the overall transport by plug flow . The fluxes due to these additional trans port mechanisms are described by the Fic k ' s law of
diffusion for mass and the Fourier law of conduction for heat transfer with effec tive s y stem propertie s . Althou gh this way of repre sentin g disper sion in p acked reactors appears quite sound , the res ultin g equations are in variably rather c umbersome in t hat t hey involve solution of t w o - point boun dary val ue p roblem s . Also ; this class of models p re dict s that the in jected pulse of tracer s p reads at an infinite velocity in all direction s , w hich has never been ob served experiment ally . C hiefly due to these disadvantages , t wo other classes of models have been p roposed for desc ribin g the packed bed . In so- called fi n i t e- s tage models the bed is consi dered as bein g c omposed of a sy stem of interc onnected mixin g cells of roughly pellet diemnsion s . The first quantitative st udy usin g s uc h a model ap peared in a paper by K ramers and Alberda ( 1 953) . D e an s an d Lapidus ( 1 9 6 0a , b) developed a mathematical model desc ribin g mixin g c haracteristics in fixed beds for both reactive an d non reactive system s . McGuire and Lapidus ( 1 96 5 ) analy zed t he transient behavior of packed - bed reactors . T he review by Hlava6ek and Vot r ub a ( 1 9 7 7 ) covers most of t h e application s of finite - stage m odels , a n d it appears that practically no application to gas - soli d n oncataly tic systems exist s . The finite - stage models are attractive if only a few stages are involved ; however , for a ri goro us descrip tion , it is often necessary to consider a large number of s t a ges - w hen si gnificant computation al advan tage is lost . To overcome this diffic ulty Hin duja et al . ( 1 9 8 0 ) have developed a cros s flow model , w hich divi des the void re gion i n t h e b e d in to stagnant and flow The s t a gnant region corresponds to the wakes of packin g ele in g parts . men t s , and di sper sion is introduced by assumin g t hat there is exc han ge of fluid bet ween t he s tagnant and flowing regions throughout the bed . T he p hy sical situ ation gives ri se to fi rst - order initial value differential equations , as again st recurrence relations in the mixi n g cell models and secon d - order differential e q uation s in the dispersion model s . None of these alternative models have yet been applied to ga s - soli d noncatalytic system s . In the p resen t section , there fore , w e s hall be mainly concerned wit h dispersion models . B e fore presen tin g details of the dispersion model s , some comments on their suitability for noncatalytic reaction s are in order . T he section on sin gle - pellet st udies has amply b rought out the fact that the behavior of a sin gl e pellet is c h aracteri zed by chan ges in t h e controllin g regime durin g the course of its conversion . In t he simplest case of S I M , even if one starts with a reaction - con trolled system , the behavior event ually becomes di ffu sion cont rolled , su ggestin g that unlike in the case o f nondecayin g catalytic system s , the effectiveness factor for the pellet is a func tion of time . This m akes it neces s ary to consider individual pellets in the reactor explicitly .
I n ot h e r words , the use of pseudohomogeneou s models becomes
335
G as- Sol i d N o n cataly tic R eac tions inap prop riate .
O f necessity , t herefore , we have to use the one - or t wo
dimen sion al hetero geneous models .
T hese models require a knowled ge of
intraparticle an d interphase transport p roperties .
In view of the noniso
tropic n ature of the p acked bed , the interp hase transport prop ertie s are generally different in the axial and radial direction s . T he models , in gen eral , are therefore more comp lex and do not yield analytical solutions
even in si mple case s .
In w hat follows we p resent a ge n eral fram e work for the develop ment of these models by considerin g the one- dimen sional hetero geneous model . As far as the sin gle p ellet i s considered , it will be a s sumed that the proce s s
can be described by t h e grain model .
Formulation of the Model
T he first step in the formulation of the model i s w ritin g t he mas s and en ergy b alance equation s for the extern al field s urroun din g the p ellet s .
T h e se equation s are coupled to the mass and ene r gy balance e quation s for the in divi dual p ell ets , where the general che mical reaction A ( g)
occ urs .
+
B (s) �
C ( g} + D ( s )
( t. H < 0)
I t is seen from the reaction stoic hiometry that there is no c han ge
in moles durin g the reaction , and therefore bulk- flow effec ts wit hin the To simplify the analy sis , it is fu rther assumed that pellet can be i gnore d . the phys ic al p ropertie s of the solid , such as t hermal conductivity , heat
capacity , molar density , an d so on , are constant .
T he chemical reaction is
a simple first - order reversible reac tion occ urrin g at a s harp interface within each grain in the pellet and transp ort resi stance throu gh the ash layer in the in dividual grains is n e gli gible . T he nonisothermal nature of the re action does not b ring about sinterin g or a g glomeration of soli d p articles an d
the initial p hy sical structure rem ains un altered . Also , the solid tempera ture is assumed to be uniform t hrou ghout the p ellet . A rigorous p roof for the i sothermality of p ellets has been given by Carberry ( 1 9 7 6 ) . In the external field it is assumed that axial di ffusion terms are
n e gli gible comp ared to convective flow t erms , the concentration an d tempera t ure are constant in the radial direction , an d the press ure drop across t he bed is ne gli gible . These assumption s allow t h e use of the si m pl e one T he nonisot her micity of reac tion affect s dimen sional hetero geneou s model . the linear velocity an d density of gas alon g the bed len gth , a n d these Also , the heat generated in the reactor is
variation s cannot be i gnored .
lost throu gh the sensible heat carried by the gas leavin g the system and by convection at the out side s urface of the reac tor wall .
These assumptions simplify t he formulation of the mathem atical model althou g h the model b ecome s somew hat more re strictive . Omittin g the de t ailed m ate rial an d energy b alances , one could w rite the di men sionless equations a s follow s : Fluid Phas e : a
a T
Reactant :
Doraiswamy a n d K u l karni
336
Product :
+
c:c) - - fB)A 4w 2 ( :A - :� I R=1) A
aaz
3( 1
The initial and boundary conditions associated with these in dimen sionless form are :
= 0
( 66)
equations
pressed
ex
Initial conditions : Y_A ( z ,
1 " 0)
Yc < z .
1 " 0)
a:
( z,
S( z ,
1
1
"
Boundary y' ( z A
=
( 6 8)
1. 0
( 6 9)
1. 0
( 70)
=
condition s :
0'
y' ( 0 , 1 ) c
=
( 67)
0
0) = 1 . 0
0)
"
=
=
o: ( 0 , 1 )
1
> 0)
1
-
=
[
Ao
y'
+
3A 4 (1
,.
)y fB
YA ( 0 , 1 )
6( 0 , 1 )
-
S *A ( 1 3 1 + A (1 3
/
A I R= l] [1 3A 4 ( 1 - ] +
'
fB )
( 71)
( 72)
( 7 3)
-
f ) B
+
1
( 74)
fB )
Equations ( 7 1 ) to ( 7 4 ) are obtained by carryin g out a molar b alance of gaseous reactant A , an overall molar b alanc e , and an energy balance of the gaseous mixture at the entrance to the packe d - bed reactor . Pellet Equations :
0
Product : R
v'
a
"2 aR
ye) (a R by A 2
aR
A -
+
3( 1
where the dimen sionless radial position grain is given
( 7 5)
( 7 6) of the
reaction fron t inside each
337
Gas- Soli d N o ncataly t ic Reac tions
( 77 )
and the initial an d boundary condition s as sociated with the internal field problem are given in dimensionless form as follows : I nitial con ditions : yA ( z ,
R,
Ye ( z , R ,
' " 0)
T
" 0)
r Gi ( z , R , ' " 0 )
=
0
( 7 8)
=
0
( 7 9)
=
( 80)
1
B oundary condition s : ay A -�- ( z , O , T )
0
( 81)
0
( 8 2)
Cl R
ay
e
-�- ( z , O , T ) Cl R
-�- ( z , l , -r ) ay
e
aR
=
=
v , -v
( 83)
a yA
�- ( z , l , -r ) Cl R
Fluid p ha se :
Energy B alance Equation s : f .£..! B Cl T
=
6 a "'
Cl z
-
( 84 )
-
"' .£..!
( 85)
Cl z
The chan ge of superficial gas velocity with res pec t to the axial position is ob tained as
az
() a:
( 86 )
where
a: ( O , T )
=
8 ( 0 , -r )
( 8 7)
The chan ge of temperature of the reac tin g pellets a t any p articular axial position with respec t to time in dimen sionless form can be obtained as A ( 6*
7
-
6)
( 8 8)
338
Doraiswamy and K u l karni
with the initial con dition 8* ( z ,
T
�
0) = 1
T h e p arameter R
R
-9
=
il O
h
( 89)
defines A
[Rr
the extent of reaction and i s given by
Gi
. ] -- d R Gl a T 2
d
r
A
( 90)
Equation s ( S5) to ( 9 0 ) completely define the sy stem subject to the as sumptions stated earlier . N o an alytic al solution to t hi s set of e q ua tion s exists and numerical methods have to be employed . T he model involves formulation of several dimen sionless group s w hic h express the rate of one p rocess in relation to another . For lar ge and small values of these param eters , the s y stem approaches one or the other asymptotic limit . In w ha t follows we s hall q uali tatively an aly ze the in fluence of these dimensionless g-roup s .
, The dimen sionle s s group A s ( - t. H / C p s T o > in volve s the heat of re action . I n the even t that As approaches zero , the system behavior ten ds =
to b e isothermal .
Even for finite values
of A s . isot hermal behavior can be heat trans fer through the wall of transfer through the gas phase in the bed . T h e parameter A4 is a measure of the relative rate of m as s tran s fer across t h e gas - solid interface to the rat e of m a s s t ran sfer through the gas phase in t he bed . For large val ues of A 4 , the flow t h r ou gh the bed b ecomes cont rollin g , an d in the limitin g case , the ga seous reac tant gets completely consumed at some axial di stance in the bed . T he s harp inter face thus formed moves alon g the axial coordinate with time . In the other extreme of very low values of A 4 , the tran s fer acro s s t he flui d - solid inter face takes con t rol . More stric tly , at low values of A 4 , t he cont rolli n g regime is shifted from the external field t o the internal field . The internal field is specified in terms of two other dimensionless approached i f A5 defines the con vective the reactor in relation to the axial heat
group s . T he group A 1 measures the relative rate of the c hemical reaction in rela tion to the internal gas diffusion , while the group A 2 measures the rela tive importance o f external gas film mas s transfer to internal diffusion .
For a specified value of A 4 depen din g on the value of A 2 , the b e d b ehavior app roaches either ga s - fil m control ( A 2 -+ 0) or a stage where the entire bed behave s as a sin gle pellet with a uniform environ ment of gas aroun d ( A 2 -+ lar ge value ) . T h e parameter A t , for a specified value of A 4 , d e
termines whether system .
T he
reaction
cont rol or diffusion con trol would p re vai l in
the
advan tages of a skeleton analysis of this type is t hat it provides
asymp totic solutions to t he general
fo r the be 0 ) with a very small A 4 , it is seen that any of the three mechanisms ( ga s film , in ternal diffusion , or reac tion ) in the intern al fiel d will govern the reactor behavior . T hese equation s can be solved analytically to obtain the followin g expression s for t he extent of reaction (
the model . T h u s , value of the parameter
model as
well as gui deline s
for isot hermal sy stems ( A s
D iffus ion Con trol
=
I n view of con s tant concentration of gas every w here in the b e d , the pellet and the bed becomes indistin guishable from e ac h other and the extent of reaction in the bed
is
ob t ain e d as
Gas - So l i d N o ncataly tic R eac tions
339
( 91)
where z 1 refers to t he dimensionless position of the reaction front within the pellet and is obtained as
( 92)
T he mole fraction of gaseous reactant within the pellet is given by
( 9 3)
Reaction control :
T he extent of reaction in the beti is obtaine d as
w here rm re fe rs to the dimensionless position of the reaction front within each grain and is obtained as ( 95 ) Gas film control :
$*
=
T he extent of reaction in the bed is given by ( 96)
In the other extreme case , where the gas flow through the bed b ecomes controllin g in relation to its transfer across the gas - solid interface , the con trollin g regime lie s in the external field . T his was the case analy z ed by Evan s and Son g ( 1 974) , who con sidered isothermal an d nondispersive con di tions i n the reactor a n d spherical pellets with no external mass transfer resistance . For this case the gas concentration varies with axial position in the bed and each p ellet is subjected to a different environment . A re action modulus a analogous to the reaction modulus A 1 was defined , and their numerical results in dic nte that after an initial p eriod the p lot of the extent of reaction a gain st position in the bed shows a si gmoid shape which sub sequen tly travels throu gh the bed at constant velocity . The si gm oid shape correspon ds to the reaction zone width , and the material lyin g outside this zone is either compl etely reacted or unreacted . For higher values of a the reaction width increases and for sufficiently large values can cover the entire bed length . T he simple model of Evans and S on g ( 1 9 7 4 ) has been extended further b y Ranade and Evans ( 1 9 80) , w ho additionally considered the nonisothermal nature of the bed . The variation of velocity and gas properties with the tem p erature p rofile in the bed , as formulated in the gen eral equations here , has , however , not b e accounted for . The authors used the model to describe the reduc tion of commercial iron ore ( taconite) with hydrogen , and generally satisfactory simulation was reported . A
340
Doraiswamy and Kulka rni
noni sothermal packe d bed has also b een analy zed by B everidge and Kawam ura ( 1 9 74 ) . T hese authors also n e glected the axial and radial miXm g in the bed ; the behavior of the pellet was accounted for by definin g an effectivene s s factor . The model has been ap plied to the roastin g of zinc sulfide , an d the experimen tal and c alculated results s how reasonable suc cess of the model . The incorporation of axial an d radial dispersion of heat and mas s in the system leads to c on siderable complication s . While the case of axial dis
persion of heat and mass has been treated by S ampath et al . ( 1 9 7 5 ) , the case of radial variation s doe s not seem to have been attempted in the litera ture . In the mo del of S am path et al . ( 1 975) the governin g equation s , w hic h now include axial Peclet n umbers for heat and mas s , were simplified usin g ortho gonal c ollocation . T he authors applied this model to the regeneration of coked catalyst in a fixed bed . The results of this analysi s predicted the
maximum t em perature that could be attained in the reactor , and a n umber of operatin g strategies to keep this temperature within permissible limits were s uggested as a consequence of model simulation s . The p recedin g sec tion discussed some of the basic models for the packe d b e d noncatalytic reac tor . D es pite the simplifyin g assumptions made , the models are unable to p resen t a general qualitative pict ure o f the bed ; indeed , most of these models were develope d wit h a view to obtainin g additional in formation for specific syste m s . In real p acked beds there exist ad ditional comp lexitie s , such as porosity and velocity dist rib utions . In a ran domly packed bed it has been experimentally ob s e rved that the porosity near the w all differs from the average value ( Pillai , 1 9 7 7 ; Korolev et al'. , 1 9 7 1 ; All these authors report an oscillatory be B enenati and B rosilow , 1 9 6 2 ) . h avior of porosity near th e w all . T he significant data on porosity varia tions have been correlated by C handrasekhar and Vortmeyer ( 1 9 7 9) as
( 97)
T hi s equation is adequate for most p ractical purpose s . T he variation of void fraction near the w alls leads to a lower conver sion in the wall re gion t han wit hin t he bed . In the presence of an exother mic reaction , w here the local heat liberated depends on the extent of con version , the u s e of average voi d fraction i n the b e d can therefore lead to I t is desirable results significantly different from t hose actually ob served . in such a situation to e mploy the initial void distribution as given by E q . ( 97) . I n many gas - solid noncatalytic reaction s where c han ges i n particle si ze occ ur as a result of reaction , it is evident that this initial distrib ution will vary and more sop hi sticated models employin g the evolution of porosity distrib ution in packe d beds with p ro gre s s of reac tion would become n eces sary . Although no at tempts to incorporate any of these featu res in models for �as - soli d noncatalytic reactors are evident as of now , they s ho uld form part of any fu rther develop ment p ro gram s in t his area . Anot her area of researc h , w he re practically no in formation exist s , re lates to the dynamic respon se of packe d - bed reactors to minor disturb ances in the feed com position , temperatures , and so on . It i s known t hat in a reactor t he h eat flow , fluid flow , an d mas s flow tran sien t s are coupled an d are interactin g wit h each ot her . A small disturbance someti mes can cause significant variation s in the speeds of p ropa gation of concent ration
Gas - So l i d N o nca tal:y tic Reac t ions
341
an d te m p e ra t ure front s in the b ed A s is known for c a t a ly st reactors ( Mehta et al . , 1 9 8 1 ) , this can lead to t ransient high-temperature p eak s I t is necessary to know a priori w hich c an sinter or damage the bed . w hen such a p h e n o men o n can occur , and p re dic t ive criteria for the purpose are essential . Admittedly , the heterogeneous nature of the packe d - bed non catalytic reactor does not make it possible to obtain such a criterion for quick use in practice . However , attempt s to obtain si m ple criteria usin g alternative d e s c rip tion s ( s uch a s c ell models or cros s - flow models ) for packed bed would be notewort hy . Conventional p acked- bed reactors , although s uitable fo r most isothermal system s , pose problems for h igh ly exothermic or en dothermic r e ac tion s Additionally , these r e ac to r s do n o t offer any advantage from the viewpoint I t i s preferable in suc h instances t o employ fluidized of solids handlin g- . bed reactors , w here a great e r case in the han dlin g of solids and uniform bed temperat ure can be reali zed . The fluid - b ed re ac t or s , however , are un suitable for s y s t em s where the p os s i bi l it y of agglom e r a t i on of bed material exist s . Al so , in view o f t h e n earl y c omp le t e mi xin g o f soli d s in t h e se re actors , high c on v e r s i on o f solids would require longer residence times . The movin g- bed technique has an a d v an tage here , since solids flow m or e or less in a plu g - flow m an n e r However , for hi ghl y exothermic reactions the occurrence of hot spots , and so on , could be a drawback to movin g bed op erat ions . I n a s ub se quen t two section s we briefly describe the m o deli n g aspects o f these reactors . .
.
.
Movi n g - B ed R eactors
n um b e r of c on t act in g system s :
A
1. 2.
3.
arran gements can b e envi s a g ed for movin g bed
C ocu rre n t flow of gas and solid C o u n t ercu r ren t flow of gas and solid , as in a shaft furnace C ross flow of gas and solid , as in si n t e r bed operations
In most of these modes of contact th e gas and the solids flow pat te rn is closer to plu g flow ; however , deviations from ideality are expected in c e r t ai n situation s and can easi ly be accommodated . In many si tua t ion s the uptake rate of gas by the solids is marginal in c o m pari son with the net flow , r e s u l t in g in an uniform environ ment of ga s all aroun d the bed . In such situation s the conversion of solids leavin g the reactor i s t he same as that of a sin gl e particle exposed to a gas for a time e qual to the r e si d en c e time of t he particle in the re ac t o r For the case of varyi n g gas c om posi t ion s , the mo d e l equations under isothermal conditions can be form ulated as follow s : .
F p ( d x / dl )
s s
M
s
D A p f (d C g
c g B
M
2
A
2
/dl )
F p ( dC
g
g
M
A
- R
A
A
c
==
0
( 98)
/dl} ( 9 9)
Doraiswamy and K ul karni
342
where D s and D g refers to the effective axial dispersion coefficients of the solid gas phases , F s and F g the volumetric flow rates , Ac t he c ross - section al area , x the conversion of solid , an d CA the gas - phase concentration . The equations c an be put in the dimen sionless form as Pe
s
f B
-+ -
2 d x
1
dx dz
dz2
2 d C Pe 2 g dz
1
A Z(l c
dC dz
1
f
F
B
-
g
J Ps
fB )
F (__g_
s
2 d C Pe 2 g dz
�
dx d-r
F
Ps
1
=
_
0
dC dz
)
=
0
( 1 00 )
( 101)
These equations are accompanied b y inlet a n d outlet boundary conditions given by X = x
0
+
c co + =
dx dz
=
0 =
1
Pe
dx dz
s
1 Pe
g
dC dz
de dz
I
I
l
( 1 0 2)
z= O
( 1 0 3)
z=O
z=1
-
( 104)
The model as formulated assumes isothermal st e ady state condition s , uniform con ditions in th e radial direction , negli gible mass transfer ac ross gas - solid interface , and cocurrent movement of the gas and solid phases . With ap propriate changes in the boundary con dit i ons and sign of the flow term accordin g to t he direction of flow , the system can be generali zed to include countercurrent operation T he set of equation s ( 1 0 0) to ( 1 0 4 ) has been solved by Hartman et al . ( 1 9 7 9 ) to obtain the concentration profiles for the gas and solid reactants . T he specific system con sidered by the authors was removal of S0 2 from flue gases using calcium s ulfate and sodium s ulfide particles . The case of countercurrent movement of gas an d solid p hases w i th no dispersion in the phases has been analyzed by Evans and Son g ( 1 9 7 4 ) , and illustrative r es ul t s for conversion as a function of various operatin g param eters were presen ted . The consideration of nonisothermality of operation requires additional e q u a tion s describin g heat balances in the gas and solid phases . Assumin g plug- flow con ditions , these equation s can be easily formulated as
.
F p C g g
MA
c
pg
( 10 5)
The equation suggests that the temperature change in the axial direction is due only to the effect of heat transfer from the pellets . T he heat balance on the solid phase yields
343
Gas- Solid N onca taly tic Reactions
dT s dl
=
ha( T g - T s )
-
+
( 106)
{ - I'. H ) R A
where p C refers to the combined p rop e rties of soli d reactants and product � afA� R A refers to the reaction rate in appropriate unit s The e qua tion assumes that t he t e m p eratu r e ch an ge in the axial direction occurs as a result of heat generated in the pellet due to c hemi c al reaction . In view of the large thermal conductivity of solids , isothermal conditions with in the pellet can be assumed . The set of equations needs additional condi tion s , whic h can be defined as .
Tg
=
1
Ts O
Ts Ts
T g-0
=
Ts O
1
( 107)
=
0
=
0
( cocurren t )
=
L
(countercurrent)
( 1 0 8) ( 1 09)
of nonisothermal countercurrent moving-bed reactions has been by Tsay et al . ( 1 976) , who applied this model to the reduction of hematite with hydrogen and carbon monoxide . The simulation results indi cate that an optimum gaseous fee d c om po siti on exists for certain range of operatin g variables . It is possible to inc or porat e additional features in the model , such as a side s t ream inj ec ti on ( to adjust temperature and co mposi tion in the bed) corrections for maldistribution of gas an d solid flo w ( Yagi and S zekely , 1 9 7 9) , and so on . It is necessary for larger p articl e s to incorporate t h e diffusive processes wit hin the particle . C onventional analyses i gnore the mass and heat t r an s fer processes within the particle . Also , for exothermic reactions the axial and m ore particularly radial dis p e r sion effects should be included . A com p le t e analysis inco r por ati n g these effect s is lacking. The di s p e rsion pheno m enon present in the bed is especially i m p ort an t for exothermic or more specifically n on lin e a r processes , as it establishes the feedback neces sary for the occurrence of non unique behavio r of the bed . The existence of multiplic ity an d instability has been established for catalytic reactors . The literature on noncatalytic systems is rel ativel y scarce an d more effort s in this direction would be fruitful . T he case analy zed
,
F l u i d i zed - Bed R eacto r s
Fluidized-bed reactors re p re s ent a m ul tip h ase system consistin g of two , or more correctly , three phases . As a prerequisite to the design of t hes e re actors , it is necessary to know the ga s flow t hro u gh the constituent pha se s as well as their volume fraction s , e x ten t of mixing , an d rates of in t e r pha se mass transfer . Several models depictin g the behavior of the flui d bed are available in the literature , and the p re s en t treatment presupposes a knowl edge of these essentials . R elations hip s exist between the prop erties of the bed and the diameter of bubbles (which grow alon g the bed due to coales cence) , expansion of the bed , and gas transfer fro m the em ulsion phase . The predi c tive equations for obtaining these properties of the bed are listed in t he section on estimation of parameter values ;
344
Doraiswamy and K u l ka rni
The chief features of the fluid bed as used for the noncat alytic reactor differ from those for the conven tional catalytic reactor . The main depar tures arise due to different consideration s as sociated with the solid phase . In noncatalytic sy stems the solid act s as a reactant and therefore its con sumption durin g the reaction s hould be accounted for i n contrast t o cataly tic systems . The reaction brin gs about c han ges in the size and shape of these p articles w hich affect the fluidization behavior . Nondecayin g catalytic sy stems can be operated in the batch mode with respec t to the solid , bu t the consequences of reaction and solids consumption o ften require that the noncatalytic systems be operated on a continuous basis . T he solids fee d to these reactors would generally contain a si ze distribution . Besides its in fluence on the fluidi zation behavior , the si ze distribution has a direct effect on the residence - time distribution and therefore on the conversion of the solid reactant . In catalytic sy stem s , the gas mixing and interchan ge co efficients play an important role in decidin g the conversion in the reactor . In noncatalytic reactions , the degree of backmixin g and residence time of particles are additional important factors . Conventional catalytic reactors operate wit h sufficiently fine p articles so that processes wit hin the p articles can often be ignored . In noncatalytic reactions wit h si ze distribution of solids , pseudohomogeneous conditions cannot always be assumed and explicit recognition of t he processes within the catalyst particle becomes necessary . I n the discus sion that follow s we attempt to progres sively incorporate some of these complexities . Co n s t an t
Gas -Phase
Env i ro n m e n t
In this section we analy ze t he case w here the gas - p hase concentration is constant throughout the reactor . S uch a case arises for situations w here a large excess of reactant gas is available in the system . The analysis for conversion now depends mainly on the effect of solids mixin g . The mixing pattern of the solids in most reactors can be assumed to be close to back mixin g , with the residence - time distribution of the solids given by the expression E(t)
=
1
tc R . )
[- �] . t(
exp
]
R )
( 1 1 0)
]
w here t ( R j ) is the avera ge residence time for p articles of size R j and E ( t ) dt represent s the fraction of the particles that have a residence time between t an d t + dt . T his in formation on residence - time distribu tion can be com bined with a suitable model for computin g the conversion in a sin gle parti cle , to yield the avera ge conversion of solids in the reactor : x
R
=
M F (R) .
F
s
Fs
J
(
JO
'"
x(R)
j t(R)
1
j
exp
[
t - t(R)
j
J
dt
( 1 1 1)
where RM i s the maximum size in the feed and F s ( R ) j i s t he q uantity of R particle s of size j enterin g wit h the feed . However , the analysis i gnores any los s of particle s due to elutriation . We shall now con sider this effect .
34 5
Gas- S o l i d N oncataly tic R eac tions Effec t of El u t ria t ion
Elutriation rates from fluidized beds depend on many factors : size distribu of particles ; particle characteristic s , s uc h as shape and a t tri t ion be h avio r ; mode of op e ra t ion of the reac tor ; location and type of internals ; fee dboard hei ght ; and so on . O ften , many of these factors are interrelated , and it is customary to express the rate of elutriation on the basis of an elutriation constant E* as follow s : tion
or
(
E*
Rate of removal of solids of si ze
�
W ei ght of t h a t size of solid in the bed
Rate of removal of solid s of si ze R . per unit bed area J
or d R) (
1 A c
dt
i
=
E*
s
=
E
(
*
s
)
Fraction of bed wei ght consi stin g of si z e R . J
W(R). ___l
( 1 1 2)
w
where may now be viewed a s a s p eci fic elutriation c ons t an t . q ua n ti ti es E* an d E: are related simply as follow s :
E;
A
E*
=
E*
s
)
The two
c
( 11 3)
w
, (u - ) ( ) (
T he followin g general e qua t i on , based on the experimental data for batch operation s with binary m i xt u re s ha s b een s u g g e ste d by Wen an d Hashin ger ( 1960) :
E*s
= (1. 7
X
10
-5
)
gd
u
p
t
0 5 .
d u
t ___2_.!. ]..1
0 . 725
)
P - P 1. 15 s g --pg
( 1 14)
T he avera ge residence time o f solids of si ze R; in a fluidi zed bed can be predicted in terms of the e l utriation c on s t an t ( t even spie l , 1 9 7 9 ) :
( 1 1 5)
Knowin g t( R j ) , we can use Eq . ( 1 1 5 ) to predict the solids conversion in the p resence of elu triation . Other s t udi e s on elutriation i n c lude those of S y c h e v a and D on a t ( 1 9 7 4 ) , T a n ak a and Shinohara ( 1 97 2 ) , on t a n e v an d Paulin ( 1 97 5 ) , L an ge et al . ( 1 9 7 7 ) and Horio et al . ( 198 0 ) .
,
G
Doraiswamy a n d K u l karni
346
A s has been point e d out earlier , chan ge in p article si ze is a basic feat ure of certain gas - solid reaction s . In the previous section s , the effect of parti cle size distribu tion was included by defining an average residence time for each p article size , leadin g finally to E q . ( 1 1 1 ) . An alternative approach is to accoun t for chan ge in particle size by assumin g a simple topochemical model for p article growth or shrinkage as a first approximation . Equations can then be developed for the quantity of material leavin g the reactor as well as for particle size distribution in t he existin g stream . The rate of chan ge of particle radius can be represented for most practical cases by a si mple lin ear model or an inverse shrinka ge model as
Models Based on Particle G ro w t h or S hrin kage
dR dt
=
dR dt
=
-k = constant
( 1 16 )
k
( 1 1 7)
R
T h e parameter k in t he first case is related to the rate constant k s for the S IM by
( 1 1 8) The latter case arise s for a gas - film -con trolled reaction ( e . g . , oxidation of carbon ) . The equa tion s to p redic t c han gin g particle si ze can be utili zed to calcu late the quantity of material of a given size (F s t > leavin g the reactor . , For a system with feed F s 0 havin g a size distribution � ( R ) , an exit stream F s l with a si ze dist ributi�n h < R ) , an elutriation stream F s 2 , and an , arbitrary rat e law for the changin g particle size , a material balance on solids of size between R an d (R + dR ) yield s Solids in feed ( k g /s )
solids in out flow (kg/s)
-
soli ds leaving in elutriation stream ( kg /s )
growth of mass solids in increase an d out of solid + of the w ithin the interval in terval ( kg /s ) ( k g /s )
=
0
( 1 1 9)
On the assumption of complete b ackmixin g of the solid , this equation can b e written as
( 1 20)
T he terms E * ( R ) , ( S F ) , a n d r ( R ) represen t , respec tively , the elutria tion c on s t ant defin ed by E q . ( 1 1 3 ) the shape fac tor of the particles , an d ,
34 7
Gas- So l i d N o nca t a ly tic R eac t io n s
Eq ua tion ( 1 2 0 ) , valid for a pa rtic ul ar the rate of chan ge of p article size . si ze r an ge , can be s up p le men t ed by a balance over all sizes to giv e
F
s,2
+
F
s,1
- F
s,O
=
( SF ) W
J
R � ( R)r(R)
RM
R
dR
( 1. 2 1)
E q u a tion ( 1 2 1 ) assumes positi ve or negative values dependin g on whether the particle gro w s or shrinks . Equation s ( 1 2 0 ) and ( 1 2 1) can be rear
]
r an ge d to obtain t he outflow stream in terms of the in p u t stream as
�O
( R ) dR
3
R I
dR
( 122)
The correspon din g outflow size distrib ution is giv e n by R3 s,O Wr( R ) F
( 1 23)
In t h e s e e quation s RM represen t s the sm allest feed si ze for a gro win g p a rticl e or largest si ze for a s h rinki n g p a r ti c l e , and I is an inte gral de fined as F
s,1
/W
+
r(R)
E *( R )
dR
J
( 1 2 4)
The set of E qs . ( 122) to ( 1 2 4 ) ca nno t i n general be solve d analytically . How ever , they represent a total gen erali zati o n with re spect to feed si ze distrib u t ion , reaction kin etics , and p re se n ce of an elut riation stream . For sp ec ific situation s , such as con stant size feed , linear kin etic s , or no el ut ria t i on , thes e equ ation s can be simplified to obt ain analytical solution s . Leven spiel ( 1 9 7 9 ) has su m m ari z ed some of these analytical solutions .
A na l y s i s fo r
S y s t e ms with Varying G as- Phase
C o m p os i t ion
The assumption of kno wn and cons t ant gas environ m en t in the re acto r simplified t he treatment con siderably , since an an alys i s based on the solid phase alone could be used to desc ribe the bed b ehavior . Many practical systems usin g large particle beds fluidi zed wi t h a large excess o f gas con form to thi s situation . In man y other sys te ms ( such as vi gorou s l y fluidized beds of fine p art ic les ) , however , the com p osit ion of gas vari e s wi t h p o s ition In in t he reac tor . A simple analysis as above would then be inapplicable . the se instanc e s , it is n ec e ssary to include se p ar a te e quation s for the b ubble and cloud - w ak e p hase s . A number of models to d epict the behavior of the bed are available , and some of these have b een e xten d ed to gas - solid non c atal ytic reactors . T hus the tw o - phase model of Da vid so n and H arrison ( 1 9 6 3) has been used by C ampbell and D avidson ( 1 9 7 5 ) to analy z e the data on the com bustion of carbon p articles for short p eriods of combustion in a
Doraiswamy an d K ulkarni
34 8
batch reactor . T he model has also been used and c onsiderably extended by Am un dson ( see B ukur et al . , 1 9 7 7 ) . Tigrel and Pyle ( 1 97 1) have used thi s model for the not - too - different problem of catalyst deactivation . Kunii and Leven spiel ( 1 9 6 9 ) and Kato and Wen ( 1 9 6 9) have extended their models to gas - solid noncatalytic systems . A general model for noncatalytic beds shoul d take account of variation in both size and den sity of particles . No model that takes account of all the realities in t he bed is however available . A particularly useful model t hat takes account of some of the complexities in p ractical sy stems has been s uggested by Chen and Saxena ( 1 9 7 8 ) . B ased on a more ri gorous description of the bed , C hen and S axena have shown that the solids in the emulsion phase may have a net downward , stationary , or net upward movement , dependin g on the rate of solids feed . As a result , other parameters , suc h as bubble volume fraction and rise velocity , b ecome depen dent on the solids feed rate . A detailed solids popu lation balance has been formulated an d a computer simulation pro gram has been p resented by these authors . The results show that the extent of gas bypassin g throu gh b ubbles is reduced when the solids feed rate is in creased . These authors have also proposed a criterion to predict the direction of solids movement in the emulsion phase . An illustrative plot of this criterion i s reporduced in Fig . 14 . T he quantities F c 1 and F c 2 i n this figure are defined as follows : )s s c ( 1 - fm f w b 1 + s wb f mf s o dW r wb dt 1 - 0
p A
u - u mf
p s A c s wb ( 1 - fm f ) ( l � 1 + s wb fmf
o
- s wb o )
( 125)
and dW
dt
r
( 126)
w here W r is t h e weight o f t h e solids i n the reactor a n d p t h e average particle density over the entire bed . If F 0 < Fe ! > the solids move down in the emulsion , whereas for F 0 F c 2 , them move upward .
>
Modeling of Shal low- B e d R eac to rs
T he modelin g of gas - solid nonc atalytic fluid - bed reactors followe d t he de velop ments in catalytic systems and in most cases these models were de rived as their extension s . A dequate for qualitative purposes , these models are really applicable only for tall beds (L /dt > 1 ) . Most industrial fluid beds , however , operate wit h small L /dt ( < 1 ) , and the use of these models becomes restrictive . The es sential differences b etween the two types of beds arise due to the chan ged flow profiles . Thus in tall beds it is j usti fiable to assume no radial variations of concentration a n d temperature . The measurements on lateral mixin g of solids in shallow beds , however , indicate a muc h lower value than the axial mixin g coefficient s . This implies that the lateral variation s cannot generally be i gnore d . T hus it is likely that stron g temperature and concentration gradien ts in the radial direction
349
Gas- So lid Noncataly t ic R eactions
�
....
u
E
LLI en
z 0
en u ::>
::E LLI
>1u 0 ...J LLI >
0 ...J 0 en
A I D O W N FLOW
OF SO L I D S I
8 I NO N E T
SOL I D
FI GURE 14
FLOW OF
SOLI DS I
FEED
R A T E , g / s ec
- Classification of fluidi zation s according to the solids feed rate .
exist in shallow fluidi zed beds ( Highley and Merrick , 1 9 7 1 ; Chavarie , 1 97 3 ; Fan e t al . , 1 9 7 9 ; F an an d Chang , 1 9 8 0 ; Grace , 1 9 80) . The remarks , gen erally applicable to gas - solid catalytic an d noncatalytic systems , are much more pertinent to t he latter case in view of t he solids bein g involved as a reactant . A rigorous model for t he gas - solid fluid- bed reactor therefore shoul d account for the radial dispersion of gas an d solid reactant and for temperature distribution . T h e s e considerations would lead to a far more complex formulation , w hich may be difficult to use practically . However , simpli fyin g a ssumption s , suitable to each c a s e , can be made to ge t the qualitative behavior of the bed , which will be far more realistic than that obtained u s in g t he models for tall beds . Thus one might consider the applicability of the simple two- phase theory with t he emulsion phase at minimum fluidization conditions an d com pletely mixed -in this case , only in the axial direction . The bubble phase would be considered to move in plug flow . T he solids could be introduced at t he center or at the bottom and overflow occurs at the farther end . The solid and ga s feed temperatures can be identical or might differ , dependin g on the specific case . The formulation of the model requires accountin g of the gas in the bu b b le phase . In the interest of generality , an overall heat balance to take care of nonisothermal situation s can also be readily written .
Doraiswamy a n d K u l karni
350
Em ulsion Phase
Sol i d r e a c t a n t : The material balance on soli d S in the emulsion phase over the control volume can be written as
Accumulation of solid B
==
net rate of B in by convection
n et rate of B in by dispersion
rate of disappearance of solid B + due to reaction
solid B in feed stream
or in equivalent mathematical form as
w here C B is the concen tration of solid B , D eB the effective diffusivity of solid species B , rB the rate of con sumption of solid B F the molal feed rate of solid per unit area of feeder , H the height of bed , and 6 t he bub ble fraction i n the bed . ,
Gaseous reac tant : T he material balance on the gaseous component A in the emul&ion phase can be written as
Accumulation of gaseous species A
=
net rate net rate of of A + A in by in by convection diffusion
+
exchan ge of gas with t he b ubble phase
consumption due to reac tion
or in equivalent mathematical form as
+
1� 0-o-
1 H
fH 0
B ubble Phase : A material balance on the gaseous component A in the bubble phase can be written in accordance with D avidson ' s model as Accumulation of A
net rate of A by convection
exchan ge to emulsion phase
or in equivalent mathematical form as ( 1 27)
as
T he overall energy balance for the con trolled volume can be written
351
Gas -So l i d N o ncataly tic Reac tions
Accumulation energy
net rate of heat energy from solids and gas
n et rate of heat energy in by conduction
=
or in equiva l en t mathematical notation a s
+
FM C B ps ( T H
heat generation + due to r eac tion
u pC + --� ( T -- T ) gO H
B0
- T)
+
( 1 - o ) ( - t. H ) r
( 1 2 8)
A
The set of E q s . ( 1 27) and ( 1 2 8) de sc ribe the fluid - bed behavior for a shallow bed . In the form gi ven , these equations are quite complex and no analytical solution is pos sible . C han g et al . ( 1 9 8 2 ) have solve d these equations un der steady -state condition s on t he a s s umption of isothermality of operation an d no r a di al variation of concentration of gaseous component . A lso , the reaction rates r A and rB ar e r el at ed through stoichiometry , and zero o r der an d first - order dep e n denc e on soli d concentration was considered . T he analytical results ob tained by the authors are sum mari zed in T able 7 an d shoul d be useful in a quick es ti m a t i on of reactor perform ance . The general model deve l ope d above has been ap plied t o the case of roastin g of zinc sulfi de , with the followin g general conclusion s : The lateral -
TABLE 7 Reaction
Exit C oncen tration of S olid from the B ed
S olid concen tration
Zero order Fi rst order
(C ) B exit
"
=
"
w here n is order with respect to gaseous reactant , � f the dimen sionless posi tion of the fee der poin t , and p ,
=
( <jl
c
1 Ae
)
1/2
q, 3
o)
------- · · - · -- ----- -----
352
Doraiswamy and K ul karni
mixin g of solid s has a p rofoun d influence on the c onversion of soli ds for beds wit h smaller bubble dimension s . As such a situation is common in shallow fluidi zed beds , the in fluence of late ral mixin g should not b e over I n beds with lar ge - diameter b ubble s - where the controllin g mec h looke d . ani sm lie s across the bubble /emulsion in terface -lateral mixin g o f solids has relatively little influence on bed p erformance .
S t a ge d Fl uidized - B e d Reac tors Frequen tly , from t he standpoint of b etter c ontac t b etween t he gas an d the solid , it becomes n ecessary to stage the fluid - be d reactor . A three - stage fluidi zed b e d together with relevant stream s is shown in Fi g . 1 5 . T he solids in each bed can b e con sidered to be completely mixed . The b ed weight , average residence time of solids , and residence - time distrib ution ( R T D ) of soli ds may differ from bed to b e d .
E I u t r ia t e d SOlid S
P a r t i all y spent gases
f
I Cgl
C yclon e s
S tage
I
St age 2
li]JRj§{]
__ .
-F 3
Product
Stage 3
Ga seous reduc tant FI G U R E 1 5
unit .
Schem atic representation of a three - stage fluidi zed reduction
353
Gas-So l i d N oncataly tic Reactions
Eq .
by
The R T D func tion for any p a rtic ular si ze of fee d solids d p is given b y ( 1 1 0 ) , where t he avera ge re sidence time o f solids of si ze dp is given [{ d )
( 1 2 9)
p
E ( dp ) in this equation is the elutriation constant appropriately defin e d tak in g account of t h e p arameters i n Fi gure 1 5 a s
( 1 3 0) a n d S 1 an d S
e
are define d b y t h e equation s
( 13 1) and
s0(D ) d(d ) p
p
sin ce /(d ) p max (d
n(
�)
S(d ) d(d ) p p
. p mm
)
=
1
( 132)
in E q . ( 1 2 9 ) represents t h e cyclone efficiency .
For an ass um e d conversion - time relation s hip , the unconverted fraction of soli ds of size d p is given by E q . ( 1 1 1 ) . Since the bed is composed of
many size fraction s , this equation has to be inte grated over t he size dis trib u tion to obtain mass conve rsion of solid in the stage . T h e mas s con ver sion in t h e stage can be ob tain e d as
1 - X
1
=
w here
1
y.
_!_E_ T. ( d )
t. ( d ) =
1
p
-
3 y . + 6y.
2
1
1
( 133)
( 134)
and an d ' i repre sen t , respec tively , the average residence time of soli d of si ze d an d the time required for its complete conversion in the ith
tj_
stage .
I� is
important to bear in mind that the value of T i will depend on
T he ga s p hase could either the local c oncen tration of ga s in the sta ge . have a con stant composition through t he stage or may b e varyin g . Suitable balanc e equation s in the ga s phase s houl d b e written for this purpose .
�
(4
TABLE 8 N u mber
1
Recommen ded Equation s for E ffective Diffusivity an d Flow Permeability Parameter
Recommended e quation
References
Effective diffusivity Molecular diffusivity
D
Ae
=
....£ f
D
AB
'f
wh ere
D.. 1]
M
cr , n
T s
f
g
=
::
=
=
=
0. 001858 3T 3 1 2 P o� n . . 1] lJ
(
M
_! i
+
.!.
M. J
)
0
1/2
molecular w ei ght of sp ecies i ,
l;j
c.;;· �
a
j
constants in L ennard - Jones p otential ener gy function tortuosity factor total surface area at porous solid
�
Q
;:s �
� s::
i �-
1 9 , 4 0 0f Knud sen diffusivity
E ffective diffusivity
in transition region
2
1 0 e
Flow permeability
DA
=
km o k
1
m
=
+
(- � t ' 2
2 c
D
� Q C'l.l I
� E:
en
1
--
0
!<'!
" " s ] / 6 ) ! � � [r • ( " e
Ake
«" Asaeda and Toei ( 1 9 7 9 )
where KN , the modifie d Knudsen number, is defin e d a s
('IT R ) ( 1 )
K N = _K_ T
SM
A
112
_
f(!
fc
... ... ;:s 0 Q Q
c;·
�
�0
c;· ;:s C'l.l
us p
where u refers to the viscosity of the
gas an d S is the specific sur face
w Cll Cll
Co) "" Ol
TABLE
9
Recommended Equations for Estimatin g Transport Coefficients Recommen ded equation
Coefficient Fluid-particle transport coefficient Sh
transfer coefficient
Nu
=
Nu
=
General correlation for fluidized beds
f(J
Bed-to-wall heat transfer coefficient
h
Particle -to- hori zontal stages an d tubes
=
h
bw
h
=
Wakao and Funazkri ( 1 978)
2 + 1 . 1S c 1 1 3Re 0 ' 6 3 < Re < 1 0 , 0 0 0
M ass transfer coefficient
Heat
Reference
2
+
1 . 8Re 0 ' 5Pr1 1 3
Re
0 0 . 3 6Re · 9 4 =
=
Re
> <
Glidden and C rafield
100
( 1 9 7 0)
100
Gupta et al . ( 1 9 7 4 )
2 . 87 � + 0 . 3 0 2 3 . 35 Re Re 0 3 5 . 8( k '
g
) 0 ' 6d- 0 · 36 p 0 ' 2
s
p
2/3 7 . 2 k' ( 1 - ff ) g d p
+
( M KS unit s)
0 2 2 s . su · c
P
pg g
d
p
al .
t:J
c;;· �
'< Q ;:s Q. ::0: E.
�
B otterill ( 1 9 75)
Zabrodsky et
0 a
( 1 98 1)
�
�.
I n terchan g e coefficien t s in flui d beds
B ubble - emulsion m a s s and heat
be
k
transfer coefficient
_ 4 5 -
·
u
d
0
�e
C lou d - e m ulsion mas s and
k
heat transfer coefficient
h
=
4.
5
mf b
-�
+
5
. 85
(D ) b
g
1/2
Q 1:1.1
C'l I
1/2
E:
�
2 d5/ b
:.::
ce
=
6 . 75
m fg p g
K unii and Levenspiel ( 1969)
�
(
mf
f
·�
k'
c
'<' c;· Q
c
( 1 97 5 )
Mo ri and Wen ce
-... ... 0 ;;J (') Q
�gu b
)
(11
::a
§
;;J "'
s·
1/2
� CTc ...,.
c.o trc 00
T ABLE 1 0
Recommended Equations for Computin g B asic Parameters of the Fluidized B ed
Parameter
1.
Minimum fluidi zation velocity
Equation
g( ps
- P
p u
g
2
g
2.
Bubble diameter
d
o
p - 2. 4
mf
0 . 0 1 < Re
mf
<
s
- p
ps g( p
1 <
)d
l.l = d
om
where
�m
=
2
- (d
X
bm
[
p gg ( p s
5 00 <
1 00 0 ,
) g
10
5
p
d3
<
bO
- d
)
p
s
/p
g
ll
<
,
5 0 , 00 0
10 ex pL
\
0 0 . 6 5 2 [ A c ( U O - Um f) ] . 4
0· 3 dt
1 0 . 8 5 0 . 13 + ) ; ( o , > o!J p
2
Z)
Reference 33. ,
B roadhurst and
B ecker ( 1 97 5 ) a
I:! 0 ;;· �
�
'
Q :::s Q,
� s::
[ �.
oO
d
3.
4.
B ubble fraction in bed
5.
Clou d frac tion in
6.
Solid
7.
u
B ubble velocity
mixin!!"
rate
Attrition rate
bed
[Ac
= u0 - u f m
b
(U
c =
=
-
o
U
nA +
=
0
0
=
0 . 347
0.711
10. 4
mf ):J
c:'.l � til I til
./ gdb
0 ;:s 0 �
� s:
z
� �
....
....
c;·
=
o "'
b
"'
a: �1 (U
0
K (U
-
o
U
-
--
M
mf
=
b
umf)
c
f m fu b
m (1
)A f
�
::a � ....
;:s til
s·
u
mf
-
fmf ) s
2 . 73
X
Rajan
and Wen ( 19 8 0 )
0 . 067 < f < 0. 3 m
10
-8
< K < 9. 1 1
X
10
-s
Merrick and Highley ( 1974)
c.:. en co
Doraiswamy and K u l karni
360 PA R AM E T E R EST I M AT I O N OF T H E MO D E LS
The precedin g section was concerned with models for gas - solid noncatalytic reactors . These models were formulated in terms of dimensionless groups which consist of the physical quantities an d structural parameters of the system . The present section is devoted to the estimation of these proper ties . Some of the physical quantities can b e measured directly or are readily available in the literature . On the other hand , certain other proper ties , suc h as effective diffusivity an d heat and mass transfer coefficients , have to be obtained usin g proven relationship s . S everal of these param eters are not specific to gas - solid noncatalytic reactors , but occur very often in operations of gaseous mas s transfer in porous materials such as catalytic reaction , adsorption - desorption , dryin g , and so on . Consequently , a large volume of literature exist s on the estimation of these propertie s un der a variety of con ditions . We present in T ables 8 to 1 0 the recommended equations for these properties for use in design . N O T AT I O N
a
interfacial area for transfe r , cm 2 /cm 3
A
general notation for gaseous reactant species
Al
A2 A3
A
A
4 5
A6 A7
A A
c p
dimen sionless group defined as k R /D s e dimensionles s group defined as k R /D g e dimen sionless group defined as hR T /PC U g O p O dimen sionless group defined as k t u 0 g dimensionless group defined as h w R T 0 ;u 0P C g
dimensionless group defined as - ll H /C 1 T 0 ps dimen sionless group defined as hR /D C 1 T e ps O cross section al area o f the bed , cm 2 area of pellet , em
P
2
B
general notation for the solid reactant sp ecies
c
dimen sionless concentration of gas in the bed
C , C A B C Ab
concentration of sp ecies A , B , mol /cm 3
concentration of gaseous species A in the bulk or in the b ubble phase , mol /cm3 concentration of gaseous species A in emulsion phase , mol /cm 3 concentration of gaseous species A at the interface , mol /cm 3 concentration of gaseous species A at the surface , mol /cm 3 initial concentration of species b , mol /cm3
Gas-So l i d c
Nonca i'aly tic
sp e cific heat of gaseous sp e c ies , c al / g • s
p
cps C'
specific heat of
c
effect i ve diffusivity solid , c m 2 f s
D .. e
D
D
D
of
ga seou s stream ,
of ga s e o u s species A in the
effective di ff u si vi ty of gaseous species A solid B , c m 2 t s
eB
effective diffusivity o f grain , cm 2 /s
eG
effective diffusivity s ol i d S , c m 2 / s
eS
effective axial bed , c m 2 /s
g
the
ga seo us species A in
the
gaseous species A in
cm
2t
D amkohler n umber
E
activation energy , c al / mol
E * , E*
elutriation c on stan t
E(t)
residence - time ( 1 1 0)
s
f (x) , f1(x) , f (x)
initial p orosity
f B f mf
void frac tion in
0
2
F
p arameters
c2
s
Fs , 0 '
H
w
bed
p
)
Fs , 1 '
de fi ne d
by E q .
volum et ric flow rate o f
Fg
h
by E q .
functions of conversion defined in Table 2
frac tion of pores remove d
c1 '
h
u
voi dag e at minimum fluidi zation
f p
F(F
solid in
di s tri b ti on function define d
p e ll e t voi d a ge
f, f c f o
F
s
effective axial dispersion coefficient of bed , cm 2 fs
s
the
dispersion coefficient of gas in the
Knudsen diffusivity ,
k
of
in
Da
F
cal / g • s
dimen sionle s s c oncentration o f species A , B
cA . CB
D
of solid , cal / mol • s
average s p ecific hea t
Ap
D
solid , cal /g · s
molar sp ecific heat
ps
D
361
Reactions
Fs , 2
( 1 2 5)
ga s p h a se ,
3 em / s
volumetric flow rat e o f solid , em 3 1 s
of the en t e rin g , leavin g , and elutriatin g streams in fluid-bed reactor , c m 3 / s
volumetric feed rat e
of frac ti on o f pores removed 2 heat t ran s fe r coefficient , cal /s •cm • °C func tion
heat transfer coefficient at the
total h ei gh t
of
t h e bed ,
em
wall , cal / s •cm
2
•°C
362
Doraiswamy a n d K u lkarni
general notation for rate constant , (cm 3 /mol ) n - 1 s- 1
k k'
e
g
k k
p
k
s
k* K
effec tive thermal c ond uc ti vi ty of solid ,
p henomenological mass transfer coefficient , rate c onstant for pore surface
reac tion
removal
rate constant ,
equilibrium constant
in te rc han ge coefficient across pha s e cm /s
general notation for orde r to gaseous species
p
partial pressure
p
p r r
d
G Gi
rG O R
R
R
R
A
Ae
Ad
p res sure :
of
of re a c tion
decomposition pressure
radial
position in the grain , em
radial position of interface in the grain ,
initial radius o f
the
grain ,
radius of pellet , em
gen eral notation for rate of reaction , rate
of che mical
R
1
em
em
mol l s
reac tion at the in terface ,
rate of diffusion t hrough ash layer , mol l s rate
mol l s
o f diffusion through gas film , mol l s
dimensionless radial position
R.
with respect
also used to denote product species
R
�l
respect
wit h
species
rate of
R.
bubble - e mulsion
B
RA g R g R GA R. 1
/s
general notation for order of reaction to solid species
molecular wei gh t of specie s
n
em
em / s
dimensionless reaction rate cons tant , k lk ( T 0 )
,
m
cal ls • e m • ° C
gas constant
reac tion p e r
radius of
the
grain ,
interface ,
mol lcm
3
em
radius of the ith particle
dimen sionless radial position of the interface
extent o f
reac tion
parameter defined
by E q .
( 90)
volume fraction of solids in wake per unit volume of bubble S* p S*
dimen sionless pore surface area dimen sionless reaction surface area
G as -So l i d Nonca t a l y t ic Reac t ions
shape fac tor of solid
SF
time
t
average residenc e time for partic le of size R .
t(R.) J
f t
cr T
T
b'
TB O '
T
g
dimen sionles s time as variedly defined in text
To
T
dimen sionless time d e fine d by E q . ( 2 1 )
gO
1
,... s
T. u
u
u
1
o
b
u mf
v w
p
w1 w2
w
3
YA A
y'
Yc c
y'
z
z
l
G reek Letters a.
f3
J
temp erature variable
T. T
363
temperature in the b ulk
inlet temperature of solid and gaseous streams
temperature in the ga s phase temperature at the interface
temperature at t he solid surface
dimen sionless temperat ure at the in terface
superficial velocity in the b e d , em /s
initial velocity in t he bed , em /s
b ubble rise velocity , e m I s
minimum fluidi zation velocity , c m /s 3 volume of the pellet , c m weight of feed
dimen sionless ratio of the pellet size to grain size , R /r
GO
dimen sionless ratio o f reactor len gth to pellet size ,
L /R
dimen sionless ratio of reactor len gth to reactor
radiu s , 2L / d
b mole fraction of gaseous reactant within the pellet
mole frac tion of gas eou s reactant in t he gas mixtu re
enterin g the bed
mole fraction of gaseous produc t within the pellet
mole fraction of gas eous produc t in the gas mixture enterin g t he bed dimension les s axial dis tance in the reactor
defin e d by Eq .
( 92 )
diffusivity ratio , D e /D e o : also re fers to dimensionles s superficial gas velocity in E q . ( 65 )
shape fac tor equal to 0 , 1 , and 2 for slab , cylinder , and sphere , respec tively
364
�
Doraiswamy and K u lkarni
exothermicity factor Arrhenius parameter , d e fi n e d as E /R Tb g molar d en sity of gas to that o f the solid volume fraction of bubbles in t he bed heat of reaction , cal l g mol
m
y yl
6
effec tiveness fac tor as defined by E q . ( 8)
Ll H n
dimensionless gas temperature in the bed , T / T 0 g dimen si n ess coolent temperature , T /T 0
e e
ol
c
dimen sionles s solid temp e rature in the bed , T I T 0 s
9*
root s of transcen dental equation E q . ( 4 5 )
]..1 \) \) A '
\) ,
PB
\) B
a T
T T
T
T T
T T
c
dimen sionless diffusivity o f
ga seou s
D
eA
/D
eA ,
0
stoichiometric coefficien t s dimen sionle s s diffusivit y of gaseous p roduct D / D 0 e e 3 density of solid species B g /em ,
reac tion modulus
dimensionless time relative to the proc e s s in · the pellet , D et fR 2 defined
as
a
time required for complete conversion if ash diffusion alon e controls
c
the ori ginal nonporous material to become completely porous
d
time required for complete conversion for diffu sion c on trolled processes
D
dimen sionless time relative to the in ternal diffusion con trolle d process
g
f r
¢ o( R ) , ¢ 1( R )
time required
for
complete conversion of grain
time required for complete conversion if film diffusion cont rols
alone
time required for complete conversion if reaction alone
n Thiele modulus defined in Eq . ( 1 1) ext en t of reaction in the bed para met e rs defined in T able 7 critical valu e s of T hiele modulus size distribution of p art ic les in the entering stream s , re s p ec ti vel y co t ro l s
and leavin g
Gas -Soli d
N oncataly tic Reac tio ns
365
REFERENCES
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( 1 9 8 1b ) .
K . and D . D . P er mutt e r , Effect of the p ro du ct layer on the kinetics of the C0 2 - lime reaction , AIChE J . , 29, 7 9 ( 1 9 8 3 ) . B ot t e rill , J . S . M . , Fluid Bed H eat T ransfer, Academic Press , London ( 1975) . B our geois , S . V . , J r . , F . R . G roves , J r . , and A . H . Weke , Analysis of flue gas desulfu ri zation , A I C h E J . , 2 0 , 93 ( 1 974) . fixed bed sorption : Bowen , J . H . and C . K . C hen , A diffuse interface model for fluid - solid reaction , Chern . E n g . Sci . , 24 , 1 8 2 9 ( 1 9 6 9 ) . B roadhurst , T . E . and H . A . B ecker , On se t of fluidization and sluggin g in beds of uniform p artic le s , A I C hE J . , 21 , 2 3 8 ( 1 9 7 5) . B uku r , D . , H . S . C ar am , and N . R . Amu n d s on , Some model studies of fluidi zed bed reactors , in C hemical R eac tor T heo ry , L . Lapi dus and N . R . Amundson , eds . , Prentice- H all , Englewood Cliffs , N . J . ( 1 9 7 7 ) . C alvelo , A . and J . M . S mith , I ntrapellet t ransport in gas - solid non c atalytic reactions , Chern . Proc . , N o . 3 , 1 ( 1 9 7 0 ) . C ampbell , E . K . and J . F . D avidson , in Fluidizatio n T echnology , D . F . K eairns , ed . , Vol . 2 , H emisphere , Washin gton , 28 5 ( 1 975) . C ampbell , R . R . , A . W . D . Hill s , and A . Pau lin , T ransport p rop e r ti e s of porous lime and their influence on the decomposition of porous co mpac t s of calcium c arb on a te , Chern . E n g . S ci . , 2 5 , 929 ( 1 9 70) . B hat i a , s .
366
Do raiswamy
and K u l karni
C annon , K . J .
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D el B or ghi , M . , J . C . D unn . an d K . B . B i shoff , A technique for s oluti on of the equation s for flui d - soli d reaction s with diffu sion , C he rn . En g . S ci . , 3 1 , 1 0 6 5 ( 1 9 7 6 ) . D oheim , M . A . , A mathematical model for iron ore reduc tion : c on t inu ous m u l tis t age fluidi ze d sy stems with solid if size di st ribution an d elutriation occurs , J . A ppl . C he rn . B iotechnol . , 2 3 , 3 7 5 ( 1 9 7 5 ) . D oraiswamy , L . K . an d M . M . S harma , H e te rogeneous Reac tions -A nalysis Examples a n d R eac tor D es i gn , Vol . 1 , Wiley , New Y ork ( 1 983) . Dorai swamy , L . K . , H . C . Bijawat , and M . V . K unte , C hl o rin ation of ilme nite in a fl ui di zed bed , C he rn . E n g . Prog . , 55 , 8 0 ( 1 959) . Dudukovic , M . P . an d H . S . Lamba , S ol u ti on of movin g boun dary problems for ga s - so lid noncataly tic reaction s by orthogonal c ollocation , C hern . En g . Sci . , 3 3 , 3 0 3 ( 1 9 7 8a ) . Dudkovic , M . P . an d H . S . Lamba , A zone m od e l for reactions of soli d p a r ticles with stron gly adsorbin g species , C hern . En g . Sci . , 3 3 , 4 7 1 ( 1 97 4 ) .
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36 7
Gas -So li d N oncataly tic Reac tions
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6
Design of Fixed-Bed Gas-Solid Catalytic Reactors R ijksunivers i teit ,
G I L B E R T F. FROM E N T
G en t , Belgium
H A N N S P . K . H O F MA N N U niversitiit Erlangen-N ii.rnb erg , Erlangen , Federal Rep u b lic of G e rmany
I N T RO D U C T I O N
A t p rese n t the majority o f commercial gas -p hase c at alytic p roce s se s are I m p o rt an t e x c e ptio n s fo r which fl uidi zed carried out in fixed - be d re a c t ors bed o p e rat i on is prefe rre d- eit her to continuously r ege n era t e t he catal y s t or to im p r ov e heat removal - are the catalytic crac k in g of ga s oil , the syn th e s e s of a c ry lo ni t ri le an d ethylene dic h l o r ide and naph thalene o xi da t i on i n to p h th alic a n hyd ri d e Fi x e d be d r e a c t o r s a r e prefe r r e d because of simpler technology an d ease of op er a tion T hi s i s p art ic u l a rly t ru e fo r a d iabatic re ac t o rs W h e n the p roc es s h a s a pronounced heat effec t , s i m ple adiab atic op eration i s not al way s applicable , howeve r . W i t h very exothermic reaction s t he t e m p e rat ure rise may b ecome exces sive and harm the se l ec t ivity or t h e catalyst , shift Very en do the e q u ilib ri u m to the w ron g side , or ca u s e s af e ty prob lem s . th e rmic reaction s , on the other hand , may b e ex t i n gui s h e d when carried out in a sin gle a d iab atic rea ctor . Thi s is w hy multib e d adiab atic reac tors with in terstage heat e x c h an ge direct or i n di r ec t , internal or external , are frequently en co un tered Examples are ammonia , met hanol , and sulfuric acid syn t h e s i s ; carbon monoxide con version ; an d c ataly tic re formin g o f n aphtha . The flow in such reacto rs is generally axial, b u t when a frac tio n of the ef fluent is re c y cle d it m ay be a dv an t a geo us to use radial flow a n d to spread the c a ta ly s t out over a l a r ger surf ace to r e d uc e the p res sure dro p ,
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,
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Wit h multibed a di ab at i c reac to r s
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in te rst a ge in j e ct ion with the r e ac tin g
gases i s possible only b et w een the beds. T h e te mp erat ure profile i s o f the sa w too t h type a n d u sually s uboptimal . Theoretic ally , only c ontin uo u s heat e x c ha n ge permit s a close a pp ro x im ation of the o pti m u m tempe rat u r e p ro fi l e but the h e a t e x c han ge coe fficie n t s required to obtain t hi s often t ake values beyon d t hose that can be reached in p ract ic e C on ti n u o u s h ea t e x c h an ge is ach i eve d in t w o types of fi xe d - b ed reac tor . In the fi r s t , t he catal y s t is packed in si de a n u mb e r of pa ra llel tubes s urro u n de d by a heat - exchan gin g fluid . In the second , t he tubes t h r oug h whic h the heat - exchan gin g fluid flows are in ser t ed in to a massive bed of ,
.
373
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Fro m e n t a n d Hofmann
T h e first type i s referred to in thi s c hl'lpter as a m u l ti t ub ul a r catalyst . fixe d -b e d reac to r and i s encoun tered , for example , in o - xylene oxidation
to phthalic anhydride , s team reformin g , and Fische r - T ropsch syn thesis . T he secon d ty pe is referred to as a fixed-bed reac to r wi t h i n t e rnal hea t exchange and is encountered in ammo n i a and methanol synthesis . In the s t ron gly exothermic o - xylene oxidation , the tub e s have an inter nal diameter of only 2 . 5 e m . T hey are filled wit h catalyst and cooled by One ves sel may con tain 2 0 , 0 0 0 tubes . means of c i rc ulat in g molten salt . For endot hermic steam re formi n g t he tubes are inserted into a gas fir e d
furnace . In am monia synt he s i s a bed of 1 . 8 m internal di ameter for a daily p roduction o f 2 0 0 met ri c tons , w hich is rat he r low by to day' s s t andards , -
may con tain 7 5 0 coo li n g tubes . T h e feed flo w s t h r ough these tub e s an d is p r e h e ated by the re act in g ga ses before enterin g the catalyst bed . The process c a n b e operated i n a n autothermal way . E x o the r m ic equ ili b ri u m reaction s c an re ac h higher conversion s at lower temperature levels , becaus e of more favorable equilibrium con dition s . At lo w te mperatures the reaction rate is too low , howe ver , so that exc e ssive amount s of catalyst would b e required . N onisothermal operation is pre ferred in this case an d the optimum temperature p rofile decreases toward the exit . S uc h an optimum profile is better approximated in a m ul ti t ubul a r reactor t han in a m ul t i bed reactor , at the expense of a considerably more complicated internal design and flow patterns , however . The effluent t e mperature may even be s u ffi ci ent for partially preheatin g the feed an d for p roducin g high - pressure steam , as i s done in am monia sy nt h esi s . To avoid heat los s e s , but also for safety reasons , the b oile r is p refera bl y in serted into t he high p re s s ure vessel . H i gh - p ressure reactors m ay reach a con siderable level of sophistication . -
Another reason for i mposi n g a ce rtain temp erature p ro file may be t h e reaction sequence . T ake , for example , a consecutive reaction sc heme con sistin g of two step s . When t he s econd step has an activation ene r gy t h a t is hir-her than that of the first step an d the in te rmediate pro d uc t is the desired one , the selectivity for this intermediate is favo re d by a decreasin g temperature profile .
Fixed - b ed reactors can be s c ale d up in several ways . Similarity t heory I t is ge n e ral p rac t ic e to p roc eed from t he is of l it tl e use in t hi s matter . laboratory to the commercial scale gra dually , involvin g at least two inte r mediate scale s : t he pilot an d demons t ration units . Even lar ge pilots c annot be expected to behave exactly like the ultimate comm ercial reactor , so that further i n s i ght is sometimes require d . Mockups are useful for provi din g in formation on hy d rod ynamic s , for e x a mple , on the redistribution over a do wnstream st age of a m ul t i b e d reactor of gas es comin g from an external heat exc h an ge r a p roblem encount ered in s ul fu ric acid synthesis . To date it has been inconceivable to proceed with reactor sc aleup wit hout at least a cer t ain de gree of modelin g . The mo d eli n g can b e fun damental and rely on separate observation of various aspec t s of t he p roces s , ti e d to ge th e r an d cast into a s e t of equation s sim ulating the reactor , or it can be semi em pirical , thereby relyin g o n pilot experimentation for t uni n g so - c alled " effective" parameters of the model . Obstacles e n The approach follow e d in t his ch apter is fun damental . ,
countered i n such a n approac h are complex hydrodynamic s an d t h e l ack of sufficien tly detaile d and acc urate ch emical kin etics ( a c hronic p roblem wit h
complex re ac tio n s and complex feedstocks ) .
Modern aspects of kinetic analysi s of c omplex porcesses have been disc u s s e d in recent years by
3 75
Fixed - B e d Gas -Soli d C a taly tic R eac to rs
Froment ( 1 975 ,
1 9 7 6b )
,
( 1 98 1 ) ,
Froment and Hosten
( 1 984) , and Hoffm ann and H o fm an n ( 1 97 6a , b ) . when internal diffusion is limitin g .
Hosten and Froment
Additional di fficulties arise
I t i s ge n erally too complicate d - if not
impossibl e - to derive b oth the kinetic equation and the diffusional p arameters simultaneously from the us ual kin etic e xp e rimentation .
The intrin sic kinet
ic s should b e obtaine d fir st and in the absence of any diffu sional limitation s . The diffu sion al p arameters are determin e d in a secon d st a ge by experimen ta
1 9 7 6 ) or by p urely
tion wit h p articles of larger s i ze ( D u m e z and Froment , physic al meas urements usin g , for example ,
the Wic ke - K allenb ac h cell or the
1 9 8 2) .
gas c hromatograp hic t ec hnique ( D e D eken et al . ,
arise from c a t aly st deactivation by c okin g , poisonin g ,
Other complication s
an d sin terin g .
Sev
1 9 8 2 ; B utt , 1 97 8 , 1 9 8 0 , 1 9 8 2 ; D elmon and G r an g e , 1 9 8 0 ; Pom mer sheim and Dixit , 1 9 8 1 ) . It should be e m p hasized here that the main deficiencies in t he mo delin g
eral re •riew s deal wit h this phenomenon
( F roment ,
1 976a , 1 9 8 0a ,
of fixe d - bed catalytic p roce s s e s originate fro m incomplete and inaccurate c h e mical kin etic s .
U n fortunately , this i s in s u fficiently rec o gnized by t hose
who have the b ack groun d to contrib ute to t his area :
the chemical en gineers .
A fter t his brief discus sion of qualitative desi gn aspec t s and of p roblem areas encoun tered in t h e quantitative d e si gn b as e d on the fun damental approach , the latter will now be dealt with in detail .
C L A S S I F I C A T I O N O F M O D E LS Table
1 shows the clas sification of models for fixed - b ed tubular reactor 1 97 2b , 1 9 7 9) an d widely accep t e d
models p roposed by F roment ( 1 97 2a , sinc e .
The clas sification con tain s t w o categori e s :
an d het ero gen eous model s .
the p s e u dohomo gen eous
T he latter di stin gui s h b etween temperatures and
con c en tration s in the b ulk ga s phase an d those inside , or at the su rfac e of
The former do not .
t h e soli d .
E ach cate gory con tain s two-
and one
dimen sional models , to accoun t , in more or l e s s detail , for gradien t s on t he reactor sc ale .
The basic model , A I , post ualtes p lu g flow t hrough the bed .
It is one - dimen sional because it a s s u m e s uniformity in a s ec tion perpendic u
lar t o flow , except i n a very thin layer near the wall , when heat i s ex
c hanged with the s u rroun din gs .
A first c omplication , leadin g to model A l i ,
is to account for de viation s from plug flow by superimposin g mixin g or " di s pe rsion" i n t h e axial direction ( i . e . , i n t h e flow direction ) . sion i s al so c on sidered i n t he radial direction
When disp e r
( i . e . , in a c r o s s section ) ,
because of the occurrence of te m p erature gradient s ,
t he model beco m e s t w o
di m en sion al .
TABLE 1
Classification of M odels for Fixed - B ed T ubular R eactors P s e udohomo geneous T = T
One - dimen sion al
AI Al l
T w o - dime nsio n al
Al i i
·
S'
C = C
S
B a sic , ideal
+ axia l
di s pe rsion
+ radial di spersion
-:{;
4
H eterogeneo u s T
T8;
C
CS
BI
+
BII
+ int raparticle
in t erfaci al gr adie nt s gradien t s
Bill
+
r a dia l di spersion
3 76
Fromen t a n d Hofmann
T he first model of the heterogeneous category i s derived from t he basic pseudohomogeneous model AI by considerin g temperature and concentration gradient s over the film surroundin g the catalyst p articles . T he second model , B I I , s till one - dimen sional , adds gradients inside t he catalyst particle Finally model B i l l considers axial and radial gra to the precedin g model . dient s in t he reac tor as well as inter- an d in t raparticle local gradien ts . The problem is now t hat of quantitatively expre ssin g these phenomena , partic ularly the dispersion . T his is generally done in terms of a continuum typ e model , but s o -called cell m o d e l s , accountin g for the t wo -p hase nature In c ontin u u m models the overall flow of the reactor , have als o b een used . is c on sidered to follow parallel s treamlines , although not necessarily at a uniform velocity . Any mixin g c aused by t he p resence of the packin g i s modeled i n terms of a Fickian , diffusion - type mechanis m . T hi s requires the flux of a component between two point s to result from a large number of steps , eac h in dependent of the p recedin g one , as in a random walk . T he flux res ultin g from s uc h a mechanism is proportional to the gradient , just as in molec ular or t urbulent diffusion . T he proportionality factor between the flux and the gradient is lumped with t he molec ular and turbulent dif fusivities in to an " e ffective" diffusivity or dispersion coefficien t . W hereas the basic model AI leads to first - order differen tial equation s ( ordin ary for steady stat e , partial for transient operation ) , the models inclu din g disper sion lead to secon d - order diffe rential equation s . For n onisothermal situation s , h eat tran s fer has to be included in t he model . In the basic p se udohomogeneous model , AI , heat is considered to When axial and / or radial be transferred by the overall c on vection only . dispersion is added , yieldin g models A l l or Al i i , heat is also trans ferred by effective con duc tion . T his is a mode of heat transfer occurrin g in a pseudosolid that lump s th e fluid an d solid an d therefore the mechanisms as sociated with each pha s e . In hetero geneous models , heat and mass transfer from the bulk to the catalyst surfac e is described in terms of c onvective mec hanis m s . T ran sport mechanisms in side the particle are expressed in terms of effec tive diffusion and conduction , wit h the approp riate effective diffu sivitie s and conductivities . In model B i l l t he dispersion in the bed is accounted for in a way analogous to that used in the pseudoho mo geneous e q uivalent , A l i i , but t he mechanisms
are s eparated into t hose occ urrin g in t h e fluid and solid p has e s , re spectively . B ecause of t heir nature , contin uum models , based on effective transport mechanisms an d leading to differential equation s , are adequate only from a certain bed depth and a certain ratio of bed to particle diameter onward . The first cell models were one - dimension al and consi sted simply of a series of completely mi xed tanks , each associated with a sin gle particle . T hey lead to algeb raic equation s for the steady state an d to ordinary dif feren tial equation s for transien t sit uation s . Unlike continuum models , these cell models cannot generate b ackmixin g . T his feature was only in troduced later by the backflow cell models . Althou gh the properties of this type of cell model were thorou ghly analy zed ( Roemer and D u rbin , 1 9 6 7 ) , t heir ap plication seems to be laggin g behin d . However , D eck wer ( 1 9 7 4 ) has ap plied them to noni sot hermal reac tors . C ell models c on tain the property of in finite speed of propagation of a sign al . T o avoid this , a time delay would have to be introduced bet ween t he cells , b u t this has not been done so far ( S undaresan et al . , 1 9 80 ) . Any sophistication of t he model , s uc h as allowin g for a couplin g bet ween the particles by radiation or by considerin g a c ell to contain the b ac k half
Fixed-Bed Gas - Soli d C a ta ly t i c R eac tors
377
of a p article a n d t h e front half o f a particle in the next row , dest roys the simplicity of the comp utation . C ell models can also be u sed for heterogeneous situation s , when , for example , h eat tran s fer re sistance in the external film an d mass t ransfer resistance in side the particle have to b e accounted for . T w o - dimen sional c ell models were in troduc e d by D eans and Lapidus ( 1 96 0 ) . In these models the cells o f a given row are connected to the next , down st ream row , which is offset one - half cell widt h , thus c au sin g radial mixin g . Generally , heat is con sidered to be transferre d through the fluid phase only . This has b een shown to lead to serious errors , however (De W asch an d Froment , K unii an d Furusawa ( 1 9 7 2 ) h ave acc ounted for all pos sible heat 1971) . tran s fer mechani sms in a fixed bed , accordin g to the lines set by Y a gi an d K unii ( 1 9 5 7 ) and K unii and S mith ( 1 96 0 ) , to be discussed further in this c hapter , b ut the computational advant age of the original cell models with re spect to the continuum models is t hen completely lost . The eq uivalence bet ween the p redictions of continuum an d cell models both one - and two- dimensional cases ( A ri s has b een s t u died exten sively an d A mundson , 1 9 5 7 ; Olbric h et al . , 1 966 ; S un daresan et al . , 1 980 ; Varma , 1 980) . It s hould not be forgotten that the cell models were developed for mathe matical convenience . In spite of their seemin gly closer relation to reality , they are s till far from true hydrodynamic models . The c ell si ze , for example , is not the void fraction between p articles : it is calculated to match the results of residenc e - time - distribution experiments . To achieve this , the c e l l si ze in the r adi al direction has to be 0 . 9 times t he p article
for
diameter . The models used in this chapter are exclusively of the continuum type . With p resent - day n u meric al t echniques and computers , inte gration of differ ential e quation s no lon ger presen t s great challen ges . T he methods avail able for t his will b e men tioned briefly in t he approp riate sec tion s .
M O D E L E Q U A T I O N S A N D T H E I R A PP L I C A T I O N S Pseudohomogeneo u s Model s B a s ic
Model wi t h P l u g Flow
G en eral Model E quation s : I n this and the followin g sec tions the model equation s will b e written for constant density of the fluid p has e , constant number of moles , and for one reaction only , to avoid obscurin g the essen tial features . More complicated features will be dealt with , when necessary, in specific examples . T he model equation s for a nonisothermal , noni sobaric reactor with axial flow , ope ratin g in the s teady state , may be written u
dC -+ p s dz
B
r
A
=
0 a. 1
4 dt ( T - T ) w
2 fp u g s gd p
( 1)
(-llH)pB rA
=
0
( 2)
( 3)
From en t an d Hofmann
3 78
with initial condition s :
at z = 0 , C = C o , T T o , and P t T he Pt O · ( 1 ) and ( 2 ) is eviden t . T h e c ouplin g of t hese with E q .
couplin g of E q s .
=
=
( 3 ) res ults from the tot al press ure dependence o f the rate and the variation
of P g wit h temperature and the number of moles . T h e set of ( 1 ) to ( 3 ) is ea sily in tegrated u sin g R un ge - Kutta t ec hnique s .
equations
In E q . ( 2 ) T w is t he internal wall temperature an d t1j_ is a heat trans fer coefficient b e t w een the bed an d the intern al wall for w hic h Leva ( 1 948)
proposed t he followin g correlation s . from the wall , =
a. . d
g
� d a
0 . 813
when it is c ooled
T
( )· 0
J.l
e
/d >
P
t
( 4}
the wall
from
(d G)0·7 _g_
= 3 . 50
9 - 6( d
When the reac tion mixt ure is heate d
J.l
e
-4. 6(d
p
/d ) t
( 5)
The ratio dp /dt determines t he void fr action , p articularly in t he vicinity of
the w all and there fore t he flow conditions in t h at re gion and is introduced
in an e mpirical w ay .
however , sin ce
value of Clj_
T he ran ge of application of these equatio n s is
t hey predict a z e ro
zero
at
dp G / J.i , thereby i gn orin g the contribu tion of the solid p hase and con vection . De Wasch and F roment ( 1 9 7 2 ) ,
limited ,
R eynolds n umber
of
natural
who measured heat tran s fer over a ran ge
of R eynolds n um b e rs e x t en din g from 3 0 to 1 0 00 , c orrelated their results
a s follows :
=
a. ? d
�
+ 0 . 0 33
g
c
� J.l
g
d G _p_
( 6)
J.l
and � r is c alc u l ated from a cor relation to be presented in t he sect ion on the t w o - dimension al pseudohomo geneous model .
where a.:
=
2 . 44
E quation ( 3 )
� r /dt 4 1 3 is
the pres s ure - drop equation for turbule n t flow ,
to
be
u s e d in conjunc tion wit h t h e friction fac tor correlation s disc u s se d b elow . T he friction factor f is general ly c orrelated in the followin g form versus the void fraction an d the Reynolds n u mb e r :
( 7)
f = � (a + b �) Re 3 £
Hic k s ( 1 9 7 0 ) p roposed a differen t form of correlation for sphere s : f
=
.. ._ :; ) 1_ �;. .___: 1 < ::;_. 6. 8 � 2
e:
Re
.
-0 2
for 3 00
<
Re
1 -
e:
which implies t hat a an d b are not true constan t s ,
T allmadge ( 1 9 7 0 ) .
< 6 0 , 000
in
a greement with
( 8)
�·
'rJ
Cl) 0. I
Cl) 0.
to
TABLE 2
0 Cl) I
"
Values of Parameters in Ergun - Type Friction Factor Correlations
R e ference
a
b
Ergun ( 1 9 5 2 )
1 . 75
150
Handley an d Heggs ( 1 96 8 )
1. 24
368
1 . 8 ( smooth particles ) 4 . 0 ( rough )
180
McDonald et al . ( 1 9 7 9 )
M ehta and H awley ( 1 9 6 9) Tallmadg-e ( 1 9 7 0 )
1 . 75
1 . 75
[
1 +
t ( 1 �pe;) dtJ
til
Flow ran ge
Re 1 -
1000 < Re 1 4 1 + - _____£ 6 ( 1 - E) d
4. 2
5 6 Re 1
£
< 500 1
�e
£
C"'l 0 -
Q r;· 0
< 5000
1 0 , 000
<
0.
Cl) p 0
::;:,
;;i
0
-
2
d
1 50
£
&
t
0. 1 < 1 -- < 1 0 , 000 Re
-
£
w � (0
Fromen t an d Hofmann
380
For nonspherical geometries the equivalent diameter , d' p
=
particle
diameter is replaced by
an
6 _E. v s
p
where V p and S p are the volume and surface area of the particle . With hollow rin gs , however , this correction factor is not accurate enough . B rauer ( 1 9 5 7 ) introduced the followin g groups to relate t he results for hol low rin gs to cylinders : ( 9)
with E
=
v s P cyl
s
and
p cyl v
accordin g to Reichelt and Blas z ( 1 97 1 ) . T he void fraction £ in packed beds of spheres has been correlated by Haughey and Beverid ge ( 1 96 9 ) in the followin g way :
( 1 0)
The basic pse udohomogeneous model is applied in the followin g sections to a number of typical fixe d - bed reactor problems encountered in industrial practice . These application s are intended as illustrations of the fundamen tal approach advocated in this chapter . The various aspects of reactor design an d operation illustrated here may require , in specific cases , more appropriate models . Even then the simulation by means of t hi s relatively simple model will be a valuable guide for the selection of the ultimate higher-level model , because it will reveal essential features about the process and the reactor behavior . The criteria for the escalation to the higher-level models li sted in Table 1 are given in each correspon din g section .
Application to Adiabatic and Multibed A diaba tic R ea ct ors : Equations to ( 3) are also applicable to adiabatic reactors by putting a.i equal to zero in Eq . ( 2 ) . A use ful relatio n , valid in isobaric adiabatic reactors , is obtained by combinin g Eqs . ( 1 ) and ( 2 ) : ( 1)
ll C
=
__!L.2_ p
c
- li H
ll
T
( 11)
Dou glas and Eagleton ( 1 96 2 ) have integrated Eqs . ( 1 ) and ( 2 ) ( with a. i 0 ) for zero- , first - , and secon d - order kinetic s an d expressed the results in terms of exponential integrals , which are tabulated functions . =
Fixed-B e d G as -So l i d C a taly tic Reac tors
381
A s mentioned earlier , it is of importance to reduce the pres sure drop through the cataly st bed , in p artic ular when a fraction of the effluent has to be recycled , as in catalytic reformin g or ammonia synthesi s . This re quires rather shallow bed arran gements , b u t care has to be taken to achieve a good distribution of the feed . W hen this con dition is satisfied , it is evi dent from the continuity equation t hat for i sothermal con ditions at least and for equal total p ressure , the same conversion will b e ob tained w hen W IF A o
in the shallow bed equals that in the deep bed . To avoid impractical re ac tor confi guration s an d dimension s , radial flow is sometimes substituted for axial flow . The question t hen rises if centrifu gal or centripetal flow is preferable . I t follow s from the detailed theoretical st udy of B alakotaiah and Luss ( 1 9 8 1 ) that , for plug- flow , isothermal , and isob aric operation , the con version is indepen dent of the flow direc tion . Marin an d F romen t ( 1 98 2 ) recently compare d axial with both cen trifu gal and centripetal radial flow in the cataly tic reformin g of a c 6 stream , un der con ditions and in con fi guration s suggested by in dustrial cataly tic reformin g . For radial flo w the con tin uity equation s for the componen ts are written as follows . T he c ontinuity equation s for a feed componen t i , when the axis is oriented in the flow direction an d the origin of r is at the inlet of the cataly st bed , are written d(u C .r) s 1 dr
==
� a. 1) • . pBr. l
•
( 1 2)
J
where j is one of the reaction s in w hic h component i is involved . in g the conversion
Introduc
F' R - F!r 1 i. 0 0
dx.
1
dr
for b oth centrifu gal and centripetal flow . The radial axis is directed in the flow direc tion an d R o is the radial distance , measured from the reactor vertical axis on ward , where the feed en ters the bed . The energy equation c an be written
dT dr
( 13)
The p res s ure drop is still calculated from E q . ( 3) with f as given in E q . ( 7 ) , b ut with u8 in t he R eynolds number varyin g with position . T he boun dary con dition s are obvious .
382
Fromen t a n d Hofmann
TABLE 3 Sim ulation of Reac tors with A xial and with Radial Flow for the Reformin g of a c 6 S tream Makeup feed rates ( k g /h )
C CC 4
Dimethylbutanes Methylpentanes n-
H exan e
Feed rate recycle
6 7 , 808 1 7 2 79
10, 164 ,
4 0 , 6 56
Methylcyclopentane
20 , 84 5
B en zene
1 1 , 0 56 9 , 986
Hy drogen
1 0 0 , 000
77, 794
1 6 7 , 8 0 8 kg /h T otal hydrocarbon feed rate : 6 . 00 ratio recycle /HC feed : Molar ratio H 2 / H C feed : 4 . 20 Weight o f catalyst : 6 3 , 000 k g
Mol ar
Axial flow
Centripetal
B ed diameter in ternal ( m )
8
2 . 00
B ed diameter external ( m )
8
5 . 04
B ed height ( m )
2. 02
6 . 03
I nlet temperature ( ° C )
495
495
Exit temp erature ( ° C )
460. 5
4 59 . 0 1
I nlet pressure ( bar )
15
15
Exit pressure ( bar )
1 4 . 96
1 4 98
Effluent flow rates , c - c 4 1 (kg /h)
.
70 , 343
70 , 192
7 , 599
7 , 6 59
26 , 763
26 , 4 54
34 , 2 7 5
34 , 6 1 1
6 , 300
6 , 672
B en zen e
2 1 , 76 5
2 1 , 478
Hydrogen
10 , 684
10 , 66 7
Dimethylb utanes Methylpentanes
n
-
H exane
M ethylcyclopen tane
Fixe d-Bed G as -Solid Ca taly tic Reac tors
383
T he simulation re s ul ts are given in Table 3.
The selec ted radial con
fi guration leads to a pressure drop w hic h i s s li gh tly lower t han the axial ,
yie l d s the same product di stribution , and looks acceptable fro m a practical point of view . For several reasons already referred to , multi bed operation is required w hen the temperature effects of the process are too p ronounc e d . T he dis tribution o f t h e cataly s t over the various beds an d the inlet temperatures to each of these are desi gn variables . Well - established ri gorou s techniques suc h as dynamic p ro gram min g ( B ellman , 1 9 5 7 ; A ris , 1 96 0) c an be used to derive optimal values for these variables . E xamples h a ve b een published for a mmonia and sulfuric acid synthesis an d CO conversion by the wat er gas shift reac t i on . Fi gu re 1 s how s t he result for S 0 3 syn thesis u sin g Capelli ' s rat e equation ( F roment an d B i sc hoff , 1 9 7 9) . Analogous results and a comparison of seve ral optimi zation t echniques w e re p resented by Chartran d and C rowe ( 1969) . A conversion versus temp erature d i a gram is a convenient way to rep re sent the operation of a reactor for the case of a sin gle reaction . It does not , of cours e , contain any infor mation as to where in t he reactor a given For an adiab atic b e d the relation ( x , T ) is obtained set ( x , T ) is r each e d . from the e q u i v alent of Eq . ( 1 1 ) . I n Fig . 1 , the c u rve r e represen t s the equilib rium lin e , the c u rve r >. m the l oc u s of opti m u m reaction rates in an adiabatic bed , an d the c u rve s r 1 , r 2 , r 3 an d r r are loci of
f i_ . 2· 3
X
--------
0.8
"-,
'
/
/
,
'
0.4
'
/
·
>-."'
/
/
'
'
�
---....._
--'*----
/
'
,
'
0.2
'
'
FI GURE 1
oxi da tion .
650
700
7 50
800
850
'
�
.K '
600
o
-- .-<..._� r; • •
. - · -' - - - � -
0.6
'
900
r.m
I
r
)km ol fkg cat hr
'l m
T C"K )
O ptimal con version versus temperature trajectories in SO 2 ( A ft e r F romen t and B i sc hoff ,
1979 . )
384
Fromen t an d
Hofmann
optimum inlet and outlet c on dition s for each of the t hree beds . The p ro fi t function include s the benefit resultin g from t he c on ve r s ion and t he penalty associated with the cost of catalyst and op e rat i on . To achieve op timal oper ation w i t h an exothermic e q uilib r iu m reac tion , the temperature has to be lo wered as t h e conversion increases . The multib ed ad i abatic re acto r per m it s only a s a wtoot h temperature p rofile . Optimal reac t ion rate in eac h p oin t c an only be ob taine d by fol l ow in g the rm curve , an d this c an only b e a t t e mp ted in a rea c t or with contin uous heat exc ha n ge , either o f the m ulti tub ular typ e or of the internal h e at exchan ger type , as m en t ioned ea r li e r . Tubular Reactors w i th C on t in u o u s H ea t E xch an g e : M ultitubular r eac t ors with c ataly s t in sid e the tubes a r e ge n er a lly modeled by considerin g on ly one tube , s u p p ose d to be representative for all of t h em . This is ge nerally j us tifie d when the heat - e xchan gi n g me di u m is ei t her well s t i r r e d or flows alon g or across the tubes in a sin gl e pas s . When , how e ve r , the shell side of t he b undle is b a ffl e d and t he heat trans fer medium flow s t h r ou gh it in seve ral passe s , the complete b u n dl e has to be considered , to acc ou n t for the di ffe re nt condition s in eac h tube an d for th e interaction between tubes . The l a t te r case will not be c ons i de r e d here ; let it suffic e to refer t o work by McG reavy an d Dun b o bb in ( 1 9 7 6 , 1 9 7 8) . One prob lem that is of p a rt ic u l ar im p o r t anc e w hen e x ot h e rm ic reactions are carried out in reactors of the type con sidered here is t he occurrence of temperature peaks , c all e d ho t s p o t s . T he op e r a t i n g condition s , s uch as the feed concentration an d t e m pe ra tu r e , or the coolant t e m p e rat u r e an d the t ube diameter , h ave to be c hosen j u dici o us l y to keep the hot spot within limit s . What this limit is can be d ec i d ed on by external constrain t s s uc h as s a fe t y or c a tal y s t stability , o r b y w ha t Van W elsenaere a n d F r om ent ( 1 97 0) called i n t rinsic constra i n t s or c r ite ria , related to the n a ture an d t he p a ramet e r s of the r e ac ti on an d to the operat in g condition s . Bilous a nd A m undson ( 1 956) were amon g t he first t o draw atten tion to the s e n s iti vi ty of temp erature p rofiles to the p r op er t ie s of the r e ac tio n or to the op e r at in g con ditions . B arkelew ( 1 9 5 9) d eriv e d , from an extensive set of simulati on s , simple int rin sic rules th at are us eful in ear ly s t a ges of d esi gn to s elect ran ges of safe o p e ration an d to set li mit s on the geometry . U n fortunately , t h e results were ob t ai n e d for a distorted temperature de The p robl e m was taken up later , wi t h pendence of the r a t e coefficient . t h e A r rh e ni us dep en d e nce of t he rate coefficient , by De n t e and C ollin a ( 1 96 4 ) , M arek et al . ( 1 9 6 9 ) ( also H la v ace k , 1 97 0 ) , and Van W e l s e n a e r e and The latter compared the various results in a diagram rep re F rom en t ( 1 97 0 ) . sented in Fi g . 2 . T he approac h follo wed by Van Welsen aere an d Froment has been ex ten de d to fi rs t - or d e r reactions carrie d o u t in th e p resence of interfacial temperature gr adient s an d intraparticle concen tration gr ad i en t s ( model B I I ) by Rayadhyaksha e t al . ( 1 97 5 ) . Recently , M o r b ide lli and Varma ( 1 9 8 2 ) derived s i mp le c riteria for all positive - order r e ac t ion s . The proc edu r e for o b t ainin g the critical values is exp lic i t b u t r e quires a ( s in gl e ) numerical inte gration of a differen tial e q uation . All other p a ra m ete r s b ein g e q ual , runa way is m o re likely the lower the reaction or d e r . T he de t aile d de si gn or s i m ul a t ion of a reactor re q uir e s n u m e ric a l i n te gra tion to gen erate the various p rofi le s in side the tubes . An example of one aspect of t h e desi gn i s given in Fi g . 3 , whic h show s reactor temperature p ro fil es in axial direc tion an d t heir sen si ti vi t y with respect to the salt bath temperature in o - xylene o x i da t ion over a V 2o 5 c a t a ly s t ( Froment , 1 96 7 ) .
385
Fixe d -B e d G as -So l i d C a taly tic Reactors
5 N
2,5
2.0 I NSENSITI VE 1.5
R U NAWAY A - 8 1 5 t o rd er
1,0
To
=
Tw
0 · 52�---�,�--� 6---,,�, o----:! z� o -....'-...1 � ,o ,_...L-.,.. 6o ,._-�o 1.o..-----
FI G URE 2
Comp arison of c riteria for parametric sen sitivity and runaway
b a s e d on one - dimen sional pseudohomogeneous model wit h plug flo w . F romen t ,
1974 . )
( A fter
60
40
20
IL-------=-�-----=�------;� - - --- ·---·· - J.---- - ··-
100
FI GURE 3
200
300
400
z
C u rve at 3 7 4°C with diluted bed .
=
z
'/d
p
Parametric sensitivity in o - xylene air oxidation .
bath te mperature . 1 96 7 . )
Effect of salt
( After Froment ,
386
Fromen t and Hofmann
T he figure s ho w s how , b eyon d a certain salt bath temperature , runaway
leadin g- to an almost adiab atic temperature rise occ urs .
illustrated here is parametric sensitivity , not instability
The p henomenon
The
sensu s tric t u .
latter cannot b e generated b y t his type of reactor model w hen the wall temperature is con s t an t .
T he p aramet ric sensitivity of a multitub ular reactor with coolan t in co c urren t flow an d varyin g in temp erature was recen tly studied by S oria In c ertain cases it is importan t to optimi ze the temper Lop e z et al . ( 1 9 8 1 ) . ature profile when a seq uence of reac tion s wit h different temperature de pen dencie s is t akin g place or w hen the proces s is reve rsible an d exothermic , as in am monia , met hanol , and S 0 3 syn t hesis . A gain t his is b e s t illus trated in a conversion ver s us temperature d iag ram s uch as the one presented in I n this diagra m the r Fig . 1 for S 0 3 synthesi s . m c urve represent s the sets ( x , T ) a s s ociated with maxim u m rates , so that follo win g this set ( x , T ) in the reac tor e n s u res maxim um c onversion with a m ini mu m amount of cata lyst . The temp erature profile that ac hieve s this c an be obtained through the use of P on t ryagin ' s ( 1 96 9 ) maximum p rinciple . Murase et al . ( 1 970) performed s uc h calcula tion s for an am monia - synthesi s reac tor , b u t t he p ractical reali zation is not so simple , because unrealistic heat trans fer rates O p timal t e mperature p rofiles may also be obtained by appro are re quired . priate dilution of the catalyst b e d with inert particles ( B utt and W en g , 1 97 6 ; F romen t , 1 97 1 ; N arsim han , 1 97 6 ) . Autothermal operation becomes possible w hen the heat of reaction i s
s ufficien t to h e a t the fee d to t he reac tion temperature .
I n modern multi
bed adiab atic am monia syn t he s i s reactors the e ffluent and the ' feed exchan ge
heat in an external heat exchan ger .
Internal heat exchan gers whereby
heat is exc han ged b etween fee d and reactin g gas es are also encountered ,
T h e equation s for the simulation and design
again in ammonia synthesi s .
of s uch reactors are given b elow .
For autother mal ope ration in a m ultib e d adiabatic reactor for a sin gle ( Fi g . 4 ) : Catalyst b e d :
reaction with key component A
( 14) dT dz
=
External heat exchanger :
=
B oun dary condition s : Cataly s t b e d :
x
A
( 0)
=
x
i
8" 'll
� 0. I
t:c � 0.
<;) Q C/)
I Cl)
T ( Z ) : T2 ( Z ' )
�
0.
C"l Q .... Q
z:O
z:Z
r,
I
T.
FI GURE 4
T._.
' zlz''==zO
T2 ( 0) : Te 1.--- ------l r 1 c z• > = r 0
T( O)
r1 co> = Ti
::0 � Q () .... 0 "':: C/)
xA (O) : x i �c=�---��= 0 ---�z "'
0
z
=
z
z' = z•
z'
Autothermal operation of an adiabatic reactor with external heat exchan ger .
t.l Oo �
388
T1( 0)
Heat exc han ger :
T ( 0) 2
T 2 ( Z' )
Fromen t and Hofmann =
T
=
T
=
i e
T(Z)
For autothermal operation in a multitub ular reac tor ( Fi g . 5) : 1T
d
2
t
pB r
-
4F
dT
AO
A
dz
( 1 5)
B oun dary condition s : X
A
-
-
X
i
at
Z
= 0
z =
0
z = z
with a i
N otice t hat U is an overall heat transfer coefficient , not to be confused in Eq . ( 2 ) . The fee dback of h eat in troduces a feature hitherto not encountered in this chapter , namely the possibility of multiple steady states . This
z=O
z :Z
xA( O ) = x i ""'----z: O
FIGURE 5
Autothermal operation of a m ultitub ular reac tor .
·---....1..-
z:Z
Fix e d - B e d Gas -Soli d
x (Z)
Cataly tic R eac tors
389
III
T
Steady - state operatin g points in an autothermal multibed adiabatic reactor .
FI G URE 6
phenomenon is re a dil y un derstood on the ba sis of a now clas sical dia gram ( Li lj enro t h 1 9 1 8 ; Van Heerden , 1 9 5 3) shown in Fig . 6 . In an x - T plane the sigmoidal curve i s a meas ure of the heat produced ,
by an exothermic equilibrium reaction , while the straight line is a measure of the amount of heat exchanged between , for example , the feed to an adiabatic bed and the effluen t . T he slope of this st raight line i s given b y the ratio 1. / [ 1 + ( U 7T d t /mc p ) L ] . For a certain r an ge of inlet temperatures an d for certain values of thi s slope , therefore of operatin g condition s and confi guration , three intersections (i . e . , three steady states are po ssib l e ) instead of one . The inter section leadin g to an intermediate temperature and conversion corresponds to an uns table steady state , the upper and the lower to stable steady states . The u s u al operation of an ammonia- synthesis reactor is represented by a strai ght line nearly tan gent to t he sigmoi d al curve . S li ght var iation s in the feed c on di t ion s may therefore cause extinc tion of the reactor , with the associated drastic coolin g and possible mech anical damage to the catalyst an d reactor . Consequently , autothermal re actor s c an run away either because of parametric s ens i ti vi t y already encountered in t he simple tubular reactor without feedback , or because of instabilities s e n s u stric t u , as s oc ia t ed with the existence of multiple steady states . Eviden tly , computer simulation s permit locatin g ranges of operation in whic h s uch undesirable si tuation s could develop . Yet it is al ways conven ient to have an analytical criterion for this purpose . S uch a criterion was de ve lop ed by I n o u e ( 1 978 ) for nth - order reactions : ,
N > S y r
for irreversible reaction s -
Kn
with
K
=
1
;
K
for
reversible reac tions
where
( 1 6)
390
Fromen t and H o fmann
and 13 r is the dimen si on l es s adiabatic te mperature ris e in the r e ac tor There is no risk of i nst ab ili ty with n t h - o rde r irreversible reactions when 8 and for first - order re v ers ib le reac tion s when B r y < 8 K . Bry The opti mal o peratin g c ondition s for auto ther m al reaction s are close to the b low - off c on ditions , beyon d which an extern al heat sup ply is require d . For first - order reac tion s , irreversible or reversible , Ampaya and Rinker ( 1 97 7a , b) related the c ri tical fee d temperature , T F , le ad in g to blow- off and the c ritical be d in let temperature , T 0 , to t he operatin g a n d de sign para m eters . A t b l o w o ff t h e followin g equation is valid for a firs t o rder irrever sible reaction :
.
<
(y
where
-
-
l ) T e xp ( y )
-
=
1
( 1 7)
T he required de gree of preheatin g then follow s from
=
-
1
4 UL
u c p d s p g t
y
( 18)
1
When transients have t o b e accounted for , t h e time deriv a tive al so en ters into the models . Most dynamic st udies have been c arrie d ou t by means of the one - dimen sional mo de l with plug flow [ E qs . ( 1) to ( 3 ) ] . Even then The d yn a m ic s of fixed the comp utational requiremen t s become imp o rtan t bed reactors were disc u s se d by Ray ( 1 9 7 2 ) . A n in terestin g case o f t ran sien t s was st udied analytically by Mehta et al . ( 19 8 1 ) -- so - called wron g- way behavior - a reduction of the feed temp erat ure may lead to a transien t temperature r ise because of the different speeds of propagation of conc en The cold feed cools the tration an d temperature dist urbances in the bed . up stream sec tion of the react o r an d decr e a se s the reaction rates an d there The less con verted cold fluid then con fore the c o nversi on in t h a t re gion . tac t s hot cataly st , thus leadin g to a sub s tantial temperat ure rise . A gain , t he key parameters determinin g the m agnitude of t he respon se are t h e adiab atic tem p erature rise , t h e activation ener gy , the h e a t t rans fer capacity , the coolant temperatures , the ma gn i t ude of the disturbanc e , and the reac t or len gth . For a first - order reaction carried out in an a diabatic reac tor
.
and for a zero - order rea c ti on in a cooled reac t o r simple expres sion s c an be d eri ve d for the maximu m temperature rise . For the fir st case the exp res sion is as follow s :
,
( 13
r
+
w)
ex p
(
s
r
y +
w)
-
w
( 1 9)
where w represen t s the r a ti o of the t emp e ra t ure of the feed after and b e fore the dist urbance .
Fixed-Bed
Gas -Soli d Ca taly t ic R eac tors
3 91
For dy n a m ic situations , howeve r , it may be n ece ssa ry to distin guish b et ween ga s and solid , b ec au se of th e difference in h e at capacity of the T his w ill b e ill us trat e d later when we deal wi t h the one t wo p hase s . dimen sional heterogeneous model w it h in t erfacial gra dien t s . T he cyclic variation of the feed concentration or temp erature ha s been shown , in cer tain c a ses , to be ben eficial for the conversion and s elec tivit y but has not b een adopted yet in industrial p rac tice . Reviews on t his s ubj ect have b ee n pre sen te d by S chadlich et al . ( 1 98 3 ) and Renken ( 1 98 2) .
A xial Dispersion Mo del
I t i s c l ea r that t h e flow con dition s i n fix ed - b e d reactors deviate to a c e r tain ex ten t from plug flow . S inc e it is imp o s sib le - or too complic ated-to ri g-oro u s ly describe the true flo w p attern , the d evia tio n in mass a n d heat tran sport is described in terms of di ffu s ion - an d conduction -like mec h anisms that op e ra te in s up e r pos i tion o n transport by p l u g flow . T h e fluxes of these mechanisms are expressed by F ic k - and F ourie r - type equations . C on s e qu en t ly , a temp erat u re peak will generat e a h eat flux in the ups tream direction in a d dit ion to t hat c a u s e d by ra dia t ion . The m a gni tud e of th i s flux d e p en d s on the gra d ien t and on the effec tive c on d uctivity appearin g in th e Fourier- type law . Hiby ( 1 9 6 3 ) has shown that this effec t would extend to o nl y about one particle d ia m ete r and ac c o r d in g to Wicke ( 196 0 , 1 9 75) an d S u n daresa n et al . ( 1 9 8 0 ) , it w o ul d b e ove re s t i m a t e d by the cor relation s that are p re s ently available for th e m odel par a m e t ers . The react or model eq u a t ion s are as fol l o w s , with the assumption s an d re stric t ion s al rea d y m a d e : 0
4
a.
d
.2.
(T - T
t
an d the pressure - drop equation ( 3 ) . dC u s ( c 0 - C ) - ( D ea ) s dz
dT u p c (T - T ) = - >. 0 s g p ea d z dC dz
=
dZ dT
=
( 20 )
0
W
) + ( - ll H ) p
r B A
=
0
( 21)
T he b oun dary con dition s are
at z = 0
at z = L
These b o u n dary con dition s have been e x t e ns i ve ly disc ussed ( Wicke , 1 9 7 5) . Mathematically , they lead to a t w o - p oin t boun dary value proble m , for which an ap pr opria t e in t e grat i on algorithm is recommended . E ac h t erm in E q . ( 20 ) has dimen sion s kmol / m 3 · h an d ( D ) is bas ed r ea s on the s u p e r fic i al flow velocity . N otic e , h o w e ver , that the effective axial dispersion c o e fficien t is obtained from c orr ela tion s P e rna versus Re in whic h T he se c or r el a t ion s therefore yield t h e Peclet group u s u ally contains u. . 1
3 92
Fromen t an d Hofmann
( D e a > s l e: . For gase s , Pe rna approaches a value of 2 at the Re ( D ea > i numbers usually encountered in commercial fixed - b ed reac tors ( F roment and B i schoff , 1 97 9 , p . 5 2 7 ) . T he effective axial thermal conductivity , A ea • is obtained from correlation s Peha versus Re in whic h the former contains us . T hese correlation s s how con side rable sp read , however ( Dixon an d C re s swell , 1 9 7 9 ) . Y agi et al . ( 1 9 6 0 ) derived the followin g correlation from their experimen tal data at Re < 2 0 : =
A
::
ea
A
g
A eoa Ag
+ 1jJ RePr a
( 22)
1 /Pe with Wa ma of the order of 0 . 7 5 and A� a I A g of the order of 8 to 1 0 , whic h is close to values measured for A � r / A g , t o b e di sc ussed later . T his would lead to a value for =
Pe
ha
=
u p c d s g p p A ea
of the order of 2 . A theoretical derivation by B i schoff ( 1 9 6 2 ) , based on axial mass dispersion an d radial velocity p rofiles in packed beds , leads to a depen denc e on Reynolds , which is more complicated than that of E q . ( 2 2 ) . Extensive experimental data of Votruba et al . ( 1 9 7 2 ) , exten din g to Re 1 50 an d hi ghe r , lead to conside rably lower Pe ha , of the order of ·o . 5 ( i . e . , A ea rou ghly four time s hi gher than those of Yagi et a l . ) . Axial disper sion effec ts may be neglec ted when the followin g criteria , derived by Young and Fi nl ayson ( 1 97 3 ) , are s atisfied . For situations in w hich the rate is monotonically decreasin g wit h bed le n gt h (isot herm al opera tion , adiabatic operation of an endothermic reaction , exothermic reaction with excessive coo li n g) , the criteria are set up for inlet conditions . Axial dispersion effec ts are ne gli gible at the in let w hen =
« Pe
( 23)
rna
and
( T 0 - T )u w
<< p c s g p
Pe
ha
Even w h e n effec ts a r e noticeable at the in l e t t h e y a r e not neces sarily felt at the exit w hen total or equilibri um con version is achieved in side the re ac tor , of course , but intermediate states may be affected . W hen the rate of reac tion is maximum at some in termediate p osition ( e . g . , because of a hot spot ) the criteria above are inadequate . Evidently , the s teepn e s s of the temperature peak or of the concentration gradient determines whether or not axial dispersion has to be accounted for , but generally , a priori criteria are not available . A gain accordin g to Y oun g an d Finlay son ( 1 97 3 ) , axial dispersion is ne gligible when
393
Fixe d - B e d Gas - S o l i d C a taly tic Reac tors
max I max l
d ( z /d ) p dx
« Pe
rna ( 24 )
d( z / d ) dT
«
p
Pe
ha
The gradien t s can be approximated from a simulation based on the one dimen sion al pse udohomogeneous mo d el with p l u g flow , AI . I n in dus t ri al r e ac tor s wit h moderate temp erature p eaks , t h e flow velocitie s an d r e ac t or In t h ei r len gth a r e s uc h that axial dispersion effec ts may be neglected . simulation of t h e T VA ammonium syn thesis reac tor , Baddour et al . ( 1 9 6 5 ) ob served t ha t ax i a l dis p ers ion o f h e a t altered t h e s t ea dy st a t e temperature p rofile by less than 0 . 6°C fro m that c omputed by mean s of the basic model , AI . V an Welsenaere an d Froment ( 1970) s i m u l a t e d hydrocarbon o xidat ion close to runaway con dition s , with a hot s p ot o f 50°C and a c onver s ion of some 7 0 % in the fir st meter . E ven for such a severe sit uation the criteria ( 2 3 ) and ( 2 4 ) w ere s ati s fie d . More drastic case s are generally not realistic an d have to be a v oi d e d for reason s of selectivi ty . The s tr uctur e of the model e q u ati ons ( 2 0 ) and ( 2 1 ) introduces the pos sibility of m u l t ip le s t ea d y states . Consider , for ill ustrative p urposes , the adiabatic ver sion of these equation s an d let X ea = ( D ea > s . C o m b in in g the equation s leads to
-
Pe
dT
:Ua
+
dz'
f(T)
0
( 25)
w he re
z'
f( T )
z
d
Pe'
rna
p k(T
-( D
0) L )
ea s
=
2 Ps
( T a d - T ) exp
( ) (1 - ) E R'T
T
T
o
O
T hi s equation may lead to a monotonic relation bet ween T ( Z ) an d T 0 , yi e ld in g unique s teady - state p rofiles or , for a certain ran ge of parameter values , to a relation as shown in Fi g . 7 . With suc h a relation t h ree steady - st ate p rofile s are possi b le for inlet temperat ures ( T o) 3 < To < ( T o> l · O f t he three valu e s of e xi t temperatures , T ( Z ) , co r respon di n g to a T o in side that ran ge , only the outer t wo are stable and can be ob serve d without e t e rnal action . The run away as soci ated wit h multiplicity of steady states has different propertie s from that generated by p ara metric sen sitivity , discus sed unde r the basic plu g-flow m ode l , AI . Indeed , as T o is in c rease d al on g b ranch A , t he exit t e m p era t u re T ( Z ) , increases progres sively up to T { Z ) 1 . For a value s li gh t ly exceedin g ( T o h • the exit temperature T ( Z ) jumps to T ( Z ) 2 . Notic e t he discon tin uity in ' T ( Z ) : w h ereas in the p aramet ric sen sitivity case any exit temperature can be achieved , in the pr e s ent case the range T ( Z ) 1 < T ( Z ) < T ( Z ) 2 is
x
,
Fromen t and Hofmann
3 94
I I B I 41 I -- - - - - - -+ - - - - - - - 1 I
I I I I I I
I I �
I I I
I +---+---+----1f- t--- T ( Z ) 2 T!Z l
FI G U R E 7 Relation of feed an d exit temperatures in an a diabatic fixed -bed re ac t o r wit h axial mixin g leadin g to the possibility of three steady - state profiles . ( F rom F romen t , 1 9 8 0b . )
excluded for the C o an d flow rate that were s el e ct e d . T his is clearly an in st ab ility in t he res tric ted sense . I n deed , if the inlet temperature is lowered from ( T Q) 2 onward , T ( Z ) will progressively decrease along branch C until a value T ( Z ) 3 is reac he d . I f T ( Z ) is further reduced , T ( Z ) will T he temperature jump down ward to T ( Z ) 4 , a value t h a t differs from T ( Z ) 3 . history exhibit s a hysteresis , w hic h is typical for an unstable system . Several c rit e r i a for detectin g mu l tip lic ity have been derived , either th rou gh an analytical a pp roach or throu gh nu merical computation s . I t is clear from Fig . 7 that a n eces sary and sufficien t condition for uniqueness is the ab The sence of a hu m p or in ma t hem a tic al terms t hat no bifurcation occ urs hump i s not a s ufficien t con dition for m ultiplicity , however , since this is possible for a certain range of T o valu e s only . In s hort , m ultiplicity would be pos sible only for s tron gly exothermic p rocesses with a very high activation ener gy , carried out in r e ac tor s with considerable in flu e nc e of axial mixin g . Acc or din g to Hlavacek and H ofm ann ( 1 9 7 0 ) , for fir st - order reaction s multiplicity woul d b e po s sib l e as soon a s ,
Q > �-'r Y
.
4y y-::-4
where S r is t he dimen sionless adiaba tic temperature rise in the r eactor ( - ll H ) C o / P gC p T o , an d y t he dimen sionless acti v atio n energy , E /R T o . Fir.-ure 8 s ho w s that even w hen this con dition i s satisfie d , m ultiplicity can occ ur only when Pe rna is smaller than a certain value , dependin g on the ratio ,
Fixed-Bed Gas -Sol i d Ca taly tic Reac tors
395
t: 8
4 0 0 r-------�
300
Bo II
Uj l
2 00
Dea
2
100
0.05
k ( T0 ) C0
0.1
n -1
Do =
FI GURE 8
l
Areas of m ultip licity of steady - state p rofiles in an adiabatic tubular reactor with axial mixin g . 1 0 ; y = 2 0 ; t = X ea /D ea P gc p . 8 y r ( F rom HlavaC"ek an d Vot ruba , 1 9 7 7 . )
Pe Pe
rna
=
=
ha
an d b e t ween two values of the fir st D amkobler number , k ( T o ) L / ui . The v alue of 8 r o f 0. 5 is exceeded in ammonia- met hanol and oxy syn
thesis . Values of X e a i ( Dea > s P gc p ran ge from 1 to 4. Even for the maxi mu m value , a value of Peln a of 2 0 0 , correspon din g to a bed depth of rou ghly 1 0 0 particles , would exclude m ul tiplicity . I n dustrial reactors typically lead to Pe :n a of 6 0 0 to 2 0 0 0 . These values were c on firmed in a v ery extensive analysis of this p roblem by Jensen an d Ray ( 1 9 8 2 ) , w ho concluded further that oscillation s caused by axial dispersion are totally unlikely in fixed - bed reactors . I t s houl d b e added that multiplicity o f s teady states has b een observed in laboratory tubular reactors , for some extreme con dition s . Even if axial dispersion is relatively more important in laboratory reactor s , it now looks as if other phenomena are respon sible for this behavior : intraparticle gra dien t s , a partic ular form of rate equation , o r surfac e p henomena o n the c atalys t . T hese causes will be dealt with in later sec tion s .
Fromen t and Hofmann
396
Two -Dimens ional Pse udo homogeneous Mode l When reac tions wi t h a p ronounced heat effect , such as ethylene , ben zene , xylene , and naphthalene oxidation , are c arried out in m ultitub ular reactors , radial temperat ure an d cor.sequent c oncentration gradients develop . I t then becomes necessary to extend t he one - dimensional model AI to account for these gradien t s , caused by finite rates of heat and mass tran s fer in radial direction . I n fixed b e ds the resistance to heat transfe r is not limited to a very t hin layer in the immediate vicinity of the wall , but exten ds through out the c ro s s section . T he flux in radial direction of heat and mass is super imposed on the flux in axial direction by plu g flow an d is again expres sed in terms of e ffective diffu sion and conduc tion , obeyin g F ic k ' s and Fourie r ' s law . T he effective dispersion coefficient c on t ains contributions from molecu lar and turbulen t diffusion and from t he dispersion caused by the cat alyst particle s . T he effec tive thermal con d uctivity contain s vario us con tributions , s uc h as c on d uc tion in the solid an d in the fluid , radiation between gas filled voids and b e t w e en solids , c on vec tion from t he fluid to the solid , and conduction through st agnan t fluid in the vicinity of contact s urfaces be t ween solid particles . It has been derived from experiments that the effec tive thermal con duc tivity A. er is relatively constant in the core of the bed , but rapidly decrease s toward the wall - as if s upple men tary resistanc e operates c lo s e to t his boun dary . This is not s u rp ri sin g : it is known that the packin g - an d therefore t he flow c haracteristic s - in t hat re gion are quite differen t from t hose in the core . R ather than usin g point values , A. e r has generally been considered to be con s t ant up to the wall an d a second p arameter , accountin g for the heat trans fer at the wall , o. has w been introduc e d . It is defined by - A.
aT er Cl r
o. ( T w
=
-
,
T
w
at r = R
)
( 26 )
-
T hi s wall heat tran s fer coefficient should n o t b e confused w i t h t h e heat transfer coe fficient , o. i , of t he one dim e nsi o n al model , which lu m p s t he re sistance to heat transfer completely in t he film near the w all . S ince in in dustrial reactors the flow rate i s generally high , the contri
bu tion of axial effec tive diffusion can be n e glected . T he model e quation s can then be written , for s teady state , in an isobaric reactor and no change in number of moles :
(D
A.
>
er s
( a 2c + _! a c) ar
(·Cl + 2
er
Cl r
T 2
2
=
r Cl r
.!
Cl T
r ar
)
0
( 27)
( 28)
with b oun dary con dition s C = C0, T = T0
at z
ac
at r
3r
= 0
=
0, 0 < r < R 0 and r = R
( 29)
397
Fixed-Bed Gas-Soli d Cataly tic Reac to rs
CI T Cl r
O
=
at r
-
____!!!_ ( T Aer ct
T
w
at r
)
=
0
=
R
( 2 9)
Notice t hat axial effective diffusion has been ne glected in this mod el for the reasons explain ed in the precedin g section . Like D ea , the effective radial d iffus i vi ty is obtain ed from correlations Pe mr versus Re . In Pe m r the velocity is the in terstitial , so that ( D er > i is derived from this . A gain , ( D er > s = £ ( D er > i · Correlations for D er have been derived from experiments . Accordin g to Fa hien and S mith ( 19 5 5 ) , for Re > 10 ,
Pe ....:m =r =-- -=-
_ _ _
+
1
19. 4(d /d ) p t
2
=
10
.
The denominator con tain s the ratio d p l d t , to accoun t for the rad i al non homogeneity of t he p ac kin g For he at transfer De Wasch and Fro m e nt ( 1 97 2 ) o bt ai ne d the following correlation for air and a V 2 0 5 - Ti0 2 catalyst : ( 30)
o . 0 1 1 5d
ct w
d
t
Re
( 3 1)
p
Yagi and K unii ( 1 9 5 7 ) and K unii a n d S mith ( 1 96 0 ) developed models for c alc u la ti n g A er an d
er
= Ao
er
+ A
,
t
er
( 3 2)
:
The model for t h e calc ulation of A r contains t he mechanis m s men t io n e d above . Zehner and Schliin der ( 1 9 7 0 ) develope d a sli ghtly different con cept yieldin g the followin g e qu a tion :
��
r
g
=
(1
-
11="£>
(1
+
£
ct s dp
�g
)
+
1
+
[ (ct
d
rs p
��\ g
-_
e:B ] ( A g / A s ) e
( 3 3)
Fromen t a n d Hofmann
398 with e =
j { {1 +
{ 1 + [ ( a.
1
+
[ ( a.
d /A ) g rs p
1J ( A
d /A ) rs p g
B] ( A
g
The dynamic contribution A
r
/A ) } s
B - 1 [ ( a. d / A ) - B J ( A / A ) rs p g g s
where B = b [ ( 1 - £ ) / £] 1 0 / 9 wit h b ders and R as c hig rings .
1j!
g
t
er
=
d /A rs p s B(A /A ) g s
1 + a.
/ A ) }B s
In
2
B + 1
+
m-
(a.
d
� A
g
1 . 2 5 for sphe res an d 2 . 5 for cylin
is obtain ed from
( 3 4)
P rRe
where 1JT = 1 / P e m r · From the heat tran sfer experiments of D e Wasch and F roment ( 1 9 7 2 ) , the followin g correlation for 1J!r can b e derive d :
( 35)
whereas Y a gi an d K u n ii ( 1 9 5 7 ) ob tain e d values between 0 . 1 0 and 0 . 14 Y a gi and Wakao ( 1 9 5 9) and Y a gi an d K unii ( 1 960) al so derive d the followin g c orrelation for CXw a n d developed a model alon g the lines adopted for t he modelin g of A e r to p r e dict this parameter : a.
d
� A
( 36)
g
V alues of a. � d i A in cylindrical beds are of the order g p O f 3 . 0 , dep endin g on t he so lid , its si ze , and t h e ratio d t /dp . H e n ne c ke and Schliinder ( 1 97 3) also p r e s ent ed a correlation for a. w . based on an e x
w here lfJ w = 0 . 0 41 .
tensive set of experimen tal dat a .
data and correlations , both for A
S pec hia et al .
t er
and
a.
w
.
( 1 9 8 0 ) have presented new
The numerical p roblems asso-
ciated wit h the solution of se t s of secon d - order nonlinear partial differential equation s have been thorou ghly investigated and various technique s , such as the i m plici t C ran k - Nicolson scheme ( G rosjean and Froment , 1 9 6 2 ) and orthogonal collocation ( Finlayson , 1 97 1 ) , have been used and compared with o ne another (Mihail and Jor dac he 1 97 6 ) . Comp ut e r pro gram s for t heir application are available ( Ho ffm ann et al . ( 1 9 7 7 ) ; Lerou and Froment , 1 982 ; D e nte et al . ( 1 972 ) . Yet for sit u ations that are very demanding in comp ut ational e fforts s uch as those involving transients , i t m ay still be of interest to use a model with a s truc ture requirin g les s c o m p utat ion b u t containin g , n everthele s s , certain feat ures of the two- dimen sion al mode l . An "equivalen t" one - dimen sional model has been developed that aim s at re p roducin g- the radially avera ged c oncentrations and temperatures generated by the two- dimen sion al model . By matchin g temperature p rofiles in the ,
,
,
399
Fixe d -B e d Gas -Solid Cataly tic Reactors
absenc e of reaction , F rome n t ( 1 96 1 ) derive d t h e followin g relation from whic h t h e p a ra m eter s a . of the e qu ivalent one- dimensional model can be 1 calculated : a1 .
-;;:1
1
=
w
+
d 8 ::\.
t
( 37)
er
S li f!ht l y different b ut analogous relations were d er ive d b y Crider and Foss ( 1 96 5) , H laval?ek ( 1 97 0 ) , and Finlayson ( 1 97 1 ) . A comparison of b o t h model p re dic t ion s was made by Fro ment ( 1 96 7 ) for o xyl en e oxidation . For -
severe b ut still realistic operatin g con dition s , the deviations b et w een both models become important . And , of course , the values in the axis may still si gnifican tly differ from the radially averaged value , n e cess itatin g a more
detailed mode l . Examples o f sim ulation s b a s e d o n th e two- dimensional pseudohomogeneous mo d e l have been p re s e n te d by Froment ( 1 9 6 7 ) , Hawthorn et al . ( 1 9 6 8 ) , C arberry and White ( 1 96 9) and Garcia- Ochoa et al . ( 1 9 8 1 ) , amon g others . These simulations reveal that the radial profiles are not very sensitive to the Peclet number for radial mass trans fer , but that , on the other hand , for severe conditions , A er and
30 25
T - To (°C )
20 15 10 5 - -
- --
- - -
___ _
I I I
_. ..J
R r
FI GURE 9 M ea s u r e d and computed radial temperature pro file s in a m e t hyl cyclohexane d e hy dr o gen a tion reac tor . Solid line : I . D . tube , 1 . 6 5 e m ; 1. 1. c:xwd t i 2 Aer = 2 . 9 ; dashed line : I . D . t ube , 0 . 7 1 em ; o: w dt / 2 ::\. er ( From H aw thor n et al . , 1 968 . ) =
Frome n t an d H o fmann
4 00
nearly equal parts and the reac tion rate is si gnificantly varyin g with radial position , while the temperature profile is deviatin g from parabolic . In the reactor with intern al tube diameter 0 . 7 1 e m , the 6 T is 1 3 ° C , with nearly t w o - third accoun ted for by the wall film resistance ; the reac tion rate is nearly uniform in t he cross section and the temperature profile is very nearly parabolic . I n this case a one - di men sional model would be suf ficien t to successfully simulate t he reactor , provided that the he at trans fer coefficient a;. woul d be adap ted to accoun t for the in fluence of the radial temperature p rofile , as is don e in E q . ( 3 6 ) . Hofmann an d coworkers ( Hofmann , 1 97 9) s t udied reactions with widely differen t heat effec ts in a sophi stic ated ben c h - scale tub ular reactor allowin g axial and radial profile s of c onversion and temperature to be measured . T his work again emp h asi ze s the hi gh de gree of acc uracy required in the modeli n g of reaction and heat tran s fer under severe condition s . The ex pres sions for the s tatic con tributions to A er and ow contain a number of parameters w hic h are not always available with a sufficien t accuracy : even the apparen tly trivial solid con d uc tivity , A s , is a p roblem , in partic ular with complex cataly s t s ( Hoffmann et al . , 1 97 8) . Di sc repancies between observed an d predic ted parameter values are m ore likely also at low d t / dp , for whic h t he reported data s how a wide scatter and for w hich deviation s from plug flow should no lon ger be described by Recently , a non - Fic kian model was developed by a Fickian mechanism . C han g ( 1 9 8 2 ) , b u t much remain s to be done in this area . Uttle has been don e also to account for t he radial variation in the The latter has a value of velocity caused by the nonuniform void fraction . 1 at the wall and decreases in an oscillatin g w ay to a rather constant value T h e followin g empirical correlation was derived around the cen tral axi s . for the void fraction in a bed of sp heres by Martin ( 1 9 7 8 ) from the experi men tal data of B en enati an d B ro silow ( 1 96 2 ) :
{
£( r' )
£
£
min
�
£
+ (1 -
+ (£ . mm
-
•
mm £
oo
) r'
)e
2
-- 0 . 2 5r '
- 1 " r' < 0 TT
cos - r' a
r' :;;, 0
2 ( r /dp ) - 1 and a = 0 . 8 1 6 for d t /dp "" and 0 . 8 7 6 for dt /dp = £min occurs at a distance of ( 1 / 2 ) dp fro m the w all , w hi le £ oo re 20 . 3 . fers to the value in the axis . For bed s of spheres £m in eq uals 0 . 2 and e:oo roughly 0 . 4 , depe n din g on dt / d p . For a dt /d p of 8 , Schwart z and Smith ( 1 95 3 ) m ea s ure d velocities near the w all t wice as hi gh as those in the axis . Accountin g for the poin t values of the flow velocity in to the treatment of their radial heat tran s fer data , Lerou and F romen t ( 1 97 7 ) came to values for A er very close to those predicted by the c orrelation ( 3 0) , in whic h the dependence on the Reynolds number w as arrived at t hrou gh the variation of the global flo w rate . Fi gure 1 0 s how s a comp arison of radial temperature profile s in an o - xylene oxida tion reactor , comp uted on the basis of uniform flow velocity and a flow velocity distrib ution induced by the local void fraction ( Lerou an d F romen t , 1 97 7 ) . A gain it is clear that c on side rable sophistication is require d in the modelin g of demandin g situation s . I t is unrealistic to believe t hat gross
where r'
=
=
Gas -Solid Ca taly tic Reac tors
Fixed - B e d
401
1 00
. u is
rad ial wa l l
m•an
- - - - - - -
- - - - - - - - - -
1.4
- - - -
R E A C TO R L E N G T H ( m ) 1.6
1 .8
2 .0
F I G U R E 1 0 E ffect of flow velocity profile on radially a v era ge d temperature , dp and conversion p rofil e s in a tubular r eact or for o - x y l e n e oxidation . 0 . 0 0 4 5 m ; d t = 0 . 02 5 m ; T = 3 6 1° C . Dott e d lin e , model with radial velocity ( After Lerou and Froment , p rofile ; solid lin e , m ode l with uniform ve locity =
1977. )
.
models would be s u ffic ie nt for the modelin g of a reactor , just because it To decide whether or n o t is in dustrial and geomet ric ally relatively simp le . a two- dimensional model is required , Mears ( 1 97 1 a ) p ropo s e d to evaluate the group
I t. H I
r
4A
A
(T) p
er
R'T 2
a
d
.?
(1
w
+
8 d /A w p er
0:
�)
at the hot spot p r edic t e d by the one - dimensional simulation .
( 3 7)
If the value
of the group exc eeds 0 . 4 , the simulation should be extended to the p seudo
The idea behin d t he criterion is that homogeneou s two- dimen sional model . if the cross sec tion is to be un ifo r m in temperature , the radially avera ged rate in the hot - spot location should not deviate from the rate taken at the wall temperature by more than 5%. For exothermic reaction s the criterion is some what op t i mi stic since the temperature predicted by the one dimen sional model at a given b e d de p t h i s lower t han the radial average of the t wo- dimensional tem p erature p ro file ,
.
402
Frome n t
and Hofmann
C l early , there are cases where there is no j us ti fic ation for not usin g a two- dimensional model in des ignin g or sim ul a tin g a reactor . And fro m the work of Hofm ann and c o w orkers ( Hofmann , 1 9 7 9 ) , t his would seem to be true for kin etic analysis also , when pronounced hot spots are ine vi t able . Finally , it s ho uld be added t hat the m athe matical struc ture of the reactor model as prese n te d here cannot generate m ultiple steady states ( Y oun g and T he param etric s e n si tivi t y of t h e model has been investi Fin l ay s on , 1 9 7 4 ) . gated numerically alon g the B arkelew approach by A gnew an d Pot t e r ( 1 96 6 ) . H et e rogeneous
Model s
Pl u g -Flow Mo de l w i t h In te rfacial G ra dien ts
M o de l Eq u ation s an d Correlations for the Parameters : When the tran s fer o f hea t a n d m a s s from t h e b ulk gas p ha se to the c ataly s t involves si gn i fic an t gradient s in the vicinity of the surface of the p ar tic le , the model commonly used to d e sc r ib e such a si t u a tion in a fixed-bed reac t or is of the h e t ero gene o us type . I t is written , in the steady state , for a sin gle reaction , constant de n sity , and isob aric operation : Flui d p ha se : dC -u s dz
( 3 8) (T - T
W
)
=
I
V
h ..a ( T
s S
- T)
Solid phase :
( 3 9)
( 4 0) ( 41)
with initial con dition s C C o . T = T o at z = 0 . The alge b r aic eq ua t ion s · s ( 4 0 ) an d ( 4 1 ) are fir st iteratively s olved , yiel din g the values of C and T s s s w hic h enter in t o t h e rig h t - han d side s of t h e di fferenti al e q ua tion s for the fluid . T hese are then solved by a Run ge - K utta ro utine . N otice that this model does not explicity c on t ain any axial couplin g between t he solid phase ; heat an d mass are t ransferred in the axial direc tion th r ough the gas p ha se only . T he correlation s for the mass an d heat trans fer coefficien t s have been reviewed by S c h l iln d er ( 1 9 7 8) . A frequently used co rr el a tion for k g is t hat of Petrovic an d T hodos ( 1 9 6 8 ) : ==
E
Sh =
0 0 64 1 sc · 3 3 0 . 3 5 7R e ·
3 .;;; R e .;;; 2 0 0 0
( 4 2)
403
Fixe d - B e d Gas -Soli d C a taly tic Rea c to rs
whereas for heat transfer Whitaker ( 1 9 7 2) p roposed
( 4 3) and Gelbi n et al . ( 1 97 6 ) : e:
=
Nu
1 . 00
(� ) e
0 . 563
0 33 Pr •
( 44)
Evidently , these transfer coefficients between bulk gas and a bed of parti cles differ from those b e t we en a gas and a sin gle particle , but Martin ( 1 97 8 ) and Gnielin ski ( 1 98 2 ) expressed Nu in terms of that for a sin gle particle , N u s p , w hic h is in turn first related to laminar and turb ulent con tribution s : Nu
= 2
sp
with =
+�
o . 664
Nu
�
+
Nu
( 45)
/Pr IRer
3
O
an d
;
0 . 0 3 7Re' . 8Pr c---7'-=-----=---=::c---0 1 0 · 6 6 - 1) 1 + 2 . 4 4 3Re • - · ( P r
an d
Re'
u p ISTi s g p ]..I E:
The N u s selt n u mber for the p ack e d b e d is t hen calculated from Nu
=
f Nu a sp
( 46)
with t h e shape factor , fa = 1 . 0 for spheres , 1 . 6 for cylin ders , 2 . 3 for Relation ( 46) and the s hape fac B erl - S addles , and 2 . 1 for Raschi g rings . Gnielin ski ' s correlation fits a tors also hol d for the S herwood n umber . large n u mber of experimen tal data of various authors , e xcept at l o w values of Pe h = P rRe . Martin ( 1 9 7 8 ) and S c hliinder ( 19 7 8 ) explained this by the error in troduce d by a s s um in g uniform radial flow velocity in the deviation of hf from the e xp e ri m ental data and derived an asymptotic relation vali d when Pe h + 0 . The fluid - solid heat tran s fer c oefficien t derived from experiment s in packed beds heated from t he wall is really a conglomerate res ultin g from several mechani sm s . Three c on tributions are considered by the true fluid to solid heat tran sfer coeffi B alak rishnan and P ei ( 1 97 9 ) : cien t , t he p article - t o - p article heat trans fer coefficient , an d the coefficient for heat trans fer throu gh the soli d . Usin g microwave s to heat the solid enabled Bhatt acharyya and Pei ( 1 9 7 5 ) to measure the true fluid to solid heat tran sfer coefficien t . As mentioned al r e a dy , the model e q uations ( 3 8 )
Frome n t an d Hofman n
404
to ( 4 1 ) do not contain any couplin g between partic les . Therefore , for fixed beds exchan gin g heat throu gh the wall , the u se of the global hf , ob tained from the correlation s ( 4 3 ) , ( 4 4 ) or ( 46) is consistent with the
model , although it is not neces sarily the most acc urate way of dealin g with I nterfacial gradients develop w hen the rate of reaction the p henomenon . is high , the heat effect pronounc ed , and the flow velocity relatively low T he most likely gradient is a temperature gradient . Mears ( 1 9 7 1b ) derived t he followin g c riterion for neglectin g in terfacial concentration gradients in t he presence of an irreversible reaction with order n :
( 4 7) For a less than 5% deviation in the rate equation , due to in terfacial tempera ture gradien t s alone , h e derived ( 1 9 7 1a )
( 48)
Interfacial gradients are generally not of concern i n in dustrial reac tors , In am monia synthesis B addour et al . since the flow velocity is high . ( 1 9 6 5 ) simulated a t. T of 2 . :JO C b e tween gas and solid at the top of the re actor , where the rate i s maximum an d 0 . 4°C at the outlet . I n methanol synthesi s , C appelli et al . ( 1 9 7 2 ) calculated a t. T of maxi m um 1 . 5°C . I n phthalic anhydride synthesis . Froment an d Bischoff ( 1 97 9 ) calculated a � T of m aximum 3°C for norm al operatin g conditions and in ste am reforming o f natural gas , De Deken e t al . ( 1 982) obt ained less t h an 4° C . There are excep tions , however , for example , w hen a component of the c at alyst itself is involved in t he reaction , as in catalyst reoxidation or c at alyst regeneration by burning off the coke formed by side reactions .
Exam p le of A p plication : An illustration of the need for heterogeneous The models with interfacial gradient s i s p rovide d by H at c her et al . ( 1 9 7 8 ) . case dealt with i s the reoxidation of a nickel cataly st in the secondary re former of an ammonia plant , p rior to openin g the reactor for maintenance . D urin g normal operation the gases comin g from the p rimary reformer are fed at 1 1 0 0° C and exit at 900°C . The reac tor is adiabatic . When it has to b e opene d , the first step in stabili zin g the cataly st is to circ ulate steam throu gh t he bed until the exit has cooled down to 2 5 00 C . D urin g the coolin g period the steam strip s hydrogen from the c at alyst support , b ut it does not remove hydro gen adsorb e d on t he nickel s urface . At this point , nitrogen , containin g small amoun t s of oxygen only , is fed to the reac tor . The oxy gen reacts with the adsorbed hydrogen an d also partially oxi dizes the nickel cat aly s t . B oth reaction s are very exothermic an d a thermal T he reac tor sim ulation eq uation s have to wave t ravels t hrou gh t he bed . account for the non - steady - state nature of the operation . The contin uity equation for the oxy gen in the gas phase may be written GR' T M p a k m t v g
(
ap
A i3 z
+
� G
i3 p
)
A at
=
pA
( 4 9) s
Fix e d - B e d Gas -Solid C a taly tic Reac tors
405
The energy e quation for the gas phase is
T ...K + ....:. at G EP
a
)
= T
- T
s
( 50 )
The c ontinuity eq uation for oxy gen in t he soli d phase is
( 5 1)
whe re the te r m in b rac ke t s is simply ro
-
2
·
S ince the rate of hy d ro g en oxidation is in s tan ta n e ou s and the p h enome non is t herefore m a s s tr an s fe r limited , the oxy gen p r e sent on the surface is e s sen ti ally zero as lon g as there is hy d r o g en on the surface , so that
( 52)
with c H 2 is t he concentration of adsorbed hyd ro ge n , i n k mol / k g cat .
T he
energy equation for the solid phase is writt e n
The initial and b oun dary c on di tion s are t
=
0 , z � 0:
t =f. 0 , z = 0 :
T
P
s
A
= T = T =
P
A0
•
o
T
,
PA =
T
=
s
PA
=
0,
cH
=
2
C
H2.
C
Ni
=
c �a
0
The rate equation for n ic k e l reoxidation , de riv e d from bench - scale experi mentation , was writt en as
rN i
=
�
k 'K p A AC i 1 + K p A A
( 54 )
,
w it h �i in kmol /k g c at •h , C N i in kmol /kg c at al y s t and k' in kmol /kg cat alyst · h . The simulation w as carried out for a reactor with a diameter of 1 . 8 m an d a l en gth of 2 . 5 m , op erat e d adiabatically . The reoxidation gas was fed at 2 50 ° C an d at an ab solute pre s s ure of 1 . 5 bar . T he oxygen I t s mas s flow ve lo c i ty amounted partial pressure in the feed wa s 0 . 0 7 5 atm . to 1 3 4 2 . 8 k g f m 2 • h . The e q uival en t cataly st diameter was 0 . 0 14 m an d the ca t aly st bed d en s it y 950 k g /m 3 reactor . T h e initial Ni concentration was 0 . 0 0 6 2 4 kmol /kg cat , and the hydro gen concentration , 0 . 00 0 9 3 k mol /k g cat . T h e set of eq uations w as inte grated alo n g the characteristics by
Frome n t an d H o fmann
406
means of a R un ge - K u tta- Gill routine . The results are rep resen ted in t he Fi gs . 1 1 an d 1 2 . T h e sim ula t ion ill ustrates the c ombin ed effect of too hi gh an oxy gen content an d too low a flow velocity , whic h led to the fusion of the catalyst from a bed depth of 1 . 4 6 m on ward . This requires a temperature excee d i n g 1 4 00° C . Fi gure 1 1 s hows h o w t he simula tion p redic t s a temperature of 1 4 000 C , r each ed bet ween 1 . 2 an d 1 . 3 m , in excellent a greement with the Figure 1 2 ill u s t r ates that the differenc e bet ween gas and solid observation . temperatures can reach 1 0 0° C . C lea rly , a reactor model accoun tin g for interfacial gradients is re qui r ed here . N otic e from Fig . 12 how the solid temperature exc eeds t he ga s temperature at positions already passed by the thermal wave an d how the difference rapidly inc reas e s on the back slope of the wave . Near the peak of the wave the difference b ecomes smaller . In front of the wave the ga s temperat ure exceeds the catalyst tempera ture . It is also observe d that afte r a cer tain time the wave travels at a ne a rly c on stant rate . Analogous e quation s are used to simulate the re gene ration of coked c atalys t s and similar results are ob tained ( O lson et al . , 1 96 8 ) .
1 4 00
1 200 1 000 800
7
T I N E HRS
CURVE 1
0. 2 O lo
2 3
06
6
1 2
'
0 8 1 0
5
1 '
7
lb2 G
=
=
0
075 atm.
1 3 4 2 8 k g / m2
hr
600
400
200 0 +-------��---r--r-� 05 1 .0 1 5 z,m
(a)
FIGU R E 1 1 Reoxidation of a steam r efo r min g cataly st in an a diabatic secondary reformer , simulated by means of the heterogeneous one - dimen sion al m o d e l with interfacial gradien t s . ( a ) Solid temperatu re p ro fil es ; ( b ) Nickel and hydrogen concen tration p rofil e s . ( A fter Hatcher e t al . , 1 9 7 8 . )
407
Fixe d - B e d Gas - Solid C a ta ly tic R eac tors
0.5
0
m ·� 5 �� , .� .o----------� ,� .5-----------o------------o� �
Po2
0
=
0 . 0 7 5 aim
G
=
1 342 .8 kgtm2h r
.5
(b)
0
0.5
1 .0
1 . 5 Z,m
FI G U RE 1 1 ( C on tinue d )
of
Multiplicity o f S t eady States : Ever since t he work Wicke ( 1 9 6 0 ) , Liu and A m u n d son ( 1 962 ) and Liu et al . ( 1 9 6 3 ) , t he re has been intensive academic interest in m ult iplici ty of steady state s generated by the existence of gr adi ent s at the fluid - solid interface in exothermic processes . T he p ur pose of the p resent section i s to d e s c ri be this p henomenon i n s im p le term s and to evaluate whether it is likely to occur or not in i ndus t rial tubular reactions . Consider a si n gle reaction and a catalyst particle w ith uniform tempera ture T s and concentration C s of a key component . The bulk fluid s urround ing it is at a temperat ure T and concentration C . At ste ady state the heat
Fromen t an d Hofmann
408
1 00
0. 4 H O U R S 0
· 1 00
----------�--- ----�!--::-=0 0 .5 1 .0 1 .5 Z , m
1 00
0. 8 H OU R S o +-------�
· 1 00 -r-----�--r-0
0.5
1 .0
1 .5
Z,m
F I G U R E 1 2 Simulated te mperature difference bet ween solid an d gas i n the reoxidation of a s team reformin g catalyst . ( A fter Hatc her et al . , 1 97 8 . ) generat ed by reac tion Q R = ( - il H ) rA PB h a s to equal the h eat transferred from t he p article to the bulk fluid , Q T = h fa v < T s - T ) . I t can be seen from Fig . 13 that Q R has a sigmoidal shape when plotted versus T s , where a s Q T leads t o a straight line . W hen the intersection of Q r an d Q T occu rs in A , the correspon din g values of T s are low , whereas intersection in C require s a hi gher fluid temperature and leads to a hi gh temperature for the solid . For a certain range of T , three inter section s are possible , the outer two correspon din g to s table , the intermediate to unst ab l e operation . Which one o f the two st able steady st ate s is selected by the s y stem depends on the fluid temperat ure and on the previous history .
S uppose that the sy st em operates in a point on branch A . A s soon as the gas temperat ure is increased above a value leadin g to the solid tempera t ure ( T s ) l ' the latter will jump to a v al u e ( T ) . For t he hi story just s 2
Fixed -Bed Gas -Solid Cataly tic Reac tors
409
FI G U RE 1 3 Heat production an d tran s fer rates for various gas an d cataly s t temperatures .
de scribed , the poin t s in the ran ge ( T s ) 1 , ( T s ) 2 do not correspond to a steady s tate for T s . I f the particle is initially at ( T s ) 2 an d it s tempera ture is slightly lowered below ( T s ) 3 , the temperature will drop to ( T s ) 4 an d extinct ion will occur . With this history the range ( T s ) 3 to ( T s > 4 does not correspon d to a s teady state for the solid te mperature . The hy steresis involve d in the description above is shown in Fig . 1 4 . Exten din g this reasonin g from a sin gle p article to a fixed-bed reactor leads to the conclusion that the c oncen tration and temp erature profile in a reac tor - even without fee d - e ffluent heat exchange - would not only depend on the feed con dition s an d the w all temperature , but also on the tempera ture profile at t he time the feed i s initially admitted to the reactor . Figure 1 5 shows simulated temperature p rofiles in an adiabatic reac tor with interfacial gradien ts ( Liu an d Amundson , 1 96 2a , b ) . Figure 1 5a relates to a sit uation in which multiplicity is n ot pos sible and in which runaway would res ult solely from parametric sen s i tivit y . Upon removal of the per turbation causin g the runaway , the ori ginal profile would b e recovere d . Any intermediate temperature profile would b e possible wit h an appropriate selection of operatin g con dition s . Figure 1 5b , on the other han d , relates to multiplicity of steady states and true instability . Notice the much hi gher feed c oncentration required to achieve t his . In c as e A the initial bed temperature is lower t han 3 93 ° C . U p to 0. 2 7 m t he particles are in the lower steady state . T he temperature then rises very steeply over a few
Fromen t an d H ofmann
41 0
T
FI G U RE 14 T emperature hysteresis in c a t a lys t temperature upon variations in b ulk gas te mperature for the case of mul tiple steady stat e .
1200 . 08
8 00
T
p
T .06
Ps
l at ml
Ts
I I B I I I I I
r,
1000
!° C I BOO
.04
600
600
I
'00
(a)
FI G U RE 1 5
.2
;r
lml
.3
.4
(b)
..... - r
I
.02
.I
/
0
.I
.2 z lm l
.3
Simulate d tempera ture p rofiles in a n adiabatic tubular reactor
.�
( a ) Nonuniq ue steady with in terfacial gradien ts for a first - order reaction . 393°C ; for B : P O = 0 . 1 5 at m , T o = 3 9 3° C ; initial T s for A : state case : ( A fter C. 9° 4 4 = o T , m at 07 0 (b ) Unique s teady s t a te ; P O = 0 . 56 0°C . ) . 96 L ee and A m un dson , 1 2
Fixed -Bed G as -Solid Ca taly tic Reac to rs layers of catalyst to reach bed temperature is 560°C .
41 1
u ppe r s teady state . In case B the in itial The feed is r a pi dly heated from 3 93°C onward
the
an d it takes only a few centimeters to i gnite the reac tor to the upper steady state . Experimentally , t he characteristic s as s o cia ted with Fi g . 1 5 would have parametric to be checked to d i stin guish bet ween the causes of runaway : sen sitivity or multiplic i ty of s teady states . S i milar , although les s dras tic behavior would be calculated for orders higher t h an one . For the more gen eral Houge n - W atson- type rate eq u ati o n s , which account for adsorption of t he reactin g components , the results may di ffe r in some aspects ( Cardoso an d L us s , 1 9 6 9) . The analysis of the multiplicity of steady states caused by interfacial gradients procee ds alon g the lines described earlier . A priori cri teria for testin g multiplicity associated w it h in t erfac i al gr a dient s were re vie w ed by Lus s ( 1 97 6 ) . B riefly , for exothermic reaction s uniq uen e s s is guaranteed in a c ross section in the reactor with b ulk c on di tions ( C , T) when
For orders 0 E;; n E;; 2 , the function f( 8 f , n ) is boun de d b e t w een 1 ( for 0 ) and 5. 82 ( for n = 2 an d Sf = 0 ) . For lar ge S f t he n = 0 and S f function value asymptotically approaches 4 , for all n . For n 1 t he fun c t ions f ( Sr . n ) e q u al s 4 , for al l S f · When Sn / 1 + Br exceeds the value o f f ( Br , n ) , m ulti p licit y is possible for some values of t he dimensionle s s parameter =
=
a
=
n- 1 v k(T) c p Ps �----------p g f
8 k
e
n- 1
( Van den B osc h and Luss , 1 9 7 7 ) . Recently , T sotsis et al . ( 1 9 8 2 ) p resen ted c riteria for Hou gen - Wat son type rate equa tion s . In th i s case m ul t i pl ic i ty may also occur w i t h endo 0) . In ammonia , t he r mic reac tion s and for i sot h e r mal situation s ( S f methanol , and oxo s yn t he s i s , values of 8ty / 1 + S f exceedin g 10 may be en coun tered . Y et for realistic operat in g con dition s , A T o f 2 . 5°C and less were simulated by B ad dour et al . ( 1 96 5 ) and Cap elli et al . ( 1 9 7 2 ) in ammonia and methanol synthesi s . Such low li T values cannot generate multiplic ity . It has already been men tioned that the model [ E qs . ( 3 8 ) to ( 4 1) ] do e s not contain any axial couplin g between t he particles , a serious simpli fic ation w hen steep gradien t s oc c ur over a few layers of cataly st particles only . Eige n be r ger ( 1 9 7 2a , b ) exten d e d the model to account for thi s an d found a The backward significan tly different b ehav ior in the ra n ge of multiplic ity . flow of heat preven t s some part s of t he reactor to be in t he lower steady state an d others in the hi gh steady state , w hich would b e pos s ible if there were no couplin g . I f the reactor is in the h i gh s teady state at the exit , it will be so at the inlet , too , so that the temperature peak is ri gh t at t he en t ran c e . Vanderveen et al . ( 1 96 8 ) an al y z e d s uch a situa tion by means of a cell model . =
41 2
Frome n t a n d
Hofmann
Pl ug-Flow Mo del w i t h I n t e rfacial and I n t raparticle G radien t s
Model Equations an d Parameters : Many in dustrial p rocesses make use of catalysts and operatin g condition s le a d in g to intraparticle g r a di en t s In t h e p resent section a rat her general model will be given first , b e fore enter in g in t o the pos sibility of simplifying it . T he eq u a t ion s can b e written accountin g for t h e already mention e d assumption s : Fluid phase : .
-u
dC s dz
( 5 5)
( 56 )
S olid p hase , wit h sp herical particles :
( 5 7)
( 5 8) with b oun dary con d iti o n s z =
0:
E; = 0 : E;
_E_ : d
=
2
C
=
dC
c0, T
=
TO
dT
s
� ::
s d E;
=
s k (C - C ) g s s - T) h (T f s
0 dC =
=
-D
e
d E; dT
- ;I.
( 5 9)
s
s
e d E;
The e ffe c t i ve diffusivity D e is usually related to the molecular and K n u dsen diffusion in side the pores of the catalyst by means of the B o sanquet relation :
D
e
=
e:
s ---,---'::.._--,T ( l /D m + 1 /DK )
__
( 60)
which is strictly sp e aki n g vali d for equimolar countercurrent di ffu sion i n a binary mixt ure only . B ulk flow due to a p r e s sur e gradient or surfac e diffusion can generally be neglected . The i nternal void fraction of the catalyst , E: s , enters into thi s formula because t he diffusion evidently occurs in the pores o n ly and t he
Fixe d-B e d Gas - Solid C a taly tic Reac tors
41 3
to rt uosity , T, because t hes e are randomly oriented (i . e . , not alon g the D e can be meas ured u sin g the normal fr om the s urface to the center ) . s t e a dy - s tat e Wicke - K allenbach me t ho d ( 1 94 1 ) or the transient gas chroma togra p hic tec hnique , which also accoun ts for the contribution of dead an d pores ( Van D eemter et al . , 1 9 5 6 ) . Prob s t and W ohlfahrt ( 1 97 9) presented e mp iri cal me t hods for e s t ima tin g De . S e veral pore struc ture models have been proposed for c alc ulatin g D e ( or T ) , sin c e reliable correlations are available for D m and DK an d since Es is easily meas ured . N o s uc h models are available for X e , except t h at of B utt ( 1 9 6 5 ) , based on the sim p l e pore model of W akao and S mith ( 1 9 6 2 ) , but te m p era t u r e effects in side the p article are generally n e gligible w hen the reac tion does not involve any component o f the soli d . The n umerical effort introduced by the in t e gr a t ion of the system of secon d - order ( non linear) differen tial equations for the solid p hase in eac h of t h e nodes used in the in t e gra ti on of the flu id - p h ase equation s is s ub s t an tial , in particular for complex reac tion s , even with modern c ollocation methods . I t has been s hown , both by calc ulation s an d exp erimen tally , that intra p article t emp era t u r e gra dien t s are generally ne gli gi ble , s o t ha t Eq . ( 5 8 ) can be droppe d . Also , when there are exte rn al gradients , these are gen erally only te mperat ure gradien t s . The sy stem of equations then reduces to : Fluid p has e : -u
-�
dC s dz
dC I -D a e v d E,;
( 6 1) E,;=d
p
12 s h a (T - T) f v s
( 6 2)
S olid phase :
( 6 3)
wit h boun dary c on ditions for the solid : A t t,;
-f: d
=
A t t,; =
0:
h ( Ts f s
dC
s
(1'["
=
T)
0
-
(
-
� H)D
dC
-
e d E,;
( 6 4)
an d the u s ual b o un d a ry con diti on s for t h e fluid p hase . C l e arl y , the problem is c on siderably fu r t he r simplifie d for first - order reactions , for which an analytical solution for E q . ( 63 ) is possible . The result i s gen erally cast in t o t h e follo win g convenient for m :
41 4
n
=
1 3 Q> coth 3 Q>
�
3 q>
- 1
Frome n t a n d Hofmann
( 65)
where n , the effec tiveness factor , is the ratio of t h e actual rate to that in the ab sence of diffusional limitation s , that i s , when the surface con dition s p revail throughout the particle and
is a modified T hiele modulu s , w hic h , like the original T hiele modulus , con tain s the rate coe fficient and the effec tive diffusivity an d therefore varie s with position in t he reac tor w henever there is a temp erat ure profile . The use of q> leads to a unique relation bet ween n and Q> , whate ver the shape of the particle s , for both low and hi gh values of the modulus and to dif ferences between cylin ders an d spheres smaller than 1 5 % for intermediate value s , as shown in Fig . 16 . The n c oncept allows replacin g E q . ( 5 7) [ or E q . ( 6 3 ) ] , and when it is inc luded , E q . ( 5 8) , by the algeb raic equation s ( 66)
( 67)
When t he particle is nonisot her mal , n may exceed 1 . I t cannot be obtained analytically , b u t Eq . ( 7 1 ) can still be used for the sake of uniformity an d abb reviation of t he equation s .
FI G U RE 1 6
E ffectiven ess fac tor versus Thiele modulus q> C , cylinder .
S , slab ; P , sphere ;
=
( Vp / S p ) / k p /D e . s
Fixe d -B e d G as - Sol i d C a taly tic
41 5
Reac tors
n ii
In the treatment above , the r ate rA is taken at surfac e co d t on
s s (C ,T ) .
Sometimes it
s
s
is
referred t o the b u l k - p ha s e condition s , r
A
(C , T) .
r words , t h e ext ernal conc en t ration gra die n t is included into the value . T he an aly t ic al expression for the isot hermal effectiveness factor for fi rst - or der reaction then b e c om es In o the n
n
� c o h � - sinh � � c o s h � + ( S h - 1 ) s in h �
s
3 Sh
G 7 =
is
( 68)
where Sh k g cS / D e * and cS t he film thickne s s . The u s e o f n o furt her reduc e s t h e s y s t e m ( 6 7) into =
Eqs .
( 5 5 ) , ( 56 ) , ( 6 6 ) , an d
( 6 9)
( 70) The structure of these model e q ua tion s is i dentical to that of the basic p seudohomogeneous m o d e l , AI . For orders differe n t fr om one , n a n d n a c an n o t be obtai n e d an aly t ic ally . In th o se case s E q s . ( 7 1 ) a n d ( 7 4) c a n still be u s e d , b u t t h e n c once p t doe s not in troduc e any ga in any more , since ob taini n g n n u m e ricall y in each integration node is n ot any diffe rent from solving Eq s . ( 5 7 ) an d ( 58 ) . Thi s is w hy a p p r o x imate solution s for the rela t ion n versus � h av e been
worked out .bY A ris ( 1 96 5 ) , B ischoff ( 1 9 6 5 ) , on the in troduction of
� =
and Petersen
( 1 965b ) , b ased
V rA ( C , T ) p s s s P s
12 s
s
( 71)
p
The use of this " generalized
pi
i n
modulus" exten ds the ap l c a t o of Eq . ( 6 5 ) t o a n y shape a n d t o a n y t y p e of r a t e equation , even the Hou gen - Wat on form , b u t still only for a g"le r e a c t on . E quation ( 7 1 ) also sho w s t h at for or de r different from one , the surfac e concentration enters into the modulu s , t h a t n is a oin t val ue , ary n g in the re a c to r , even w h e n l a tt e r is isot hermal . Evidently , t h use of n yiel d the con centration r o fil e particles , b ut this i g e r ly not o f concern in t he reac tor de s i gn stage . When it is o f i m po r a c to hav e information on in rap a r icle j!ra d ie t s - fo r e x a m l , when coke or /and poison are
sin
s so
the
t
p
p
inside the
t
n
i
v
pe
e
i
s
does not s en al t n e
ratio of a convec tive coefficien t to an inte rna l diffusivity or con duct i vit y is more p r o p e rly termed a B iot n u mbe r for mass an d heat .
*The
Frome n t an d Hofmann
41 6
deposited inside the c atalyst an d an insight i s required into the relation between t he catalyst st r uct u re and penetr ation depth of coke or poison , as e xe m p li fi ed by wor k of S ummers and H e ge dus and co- workers in au to mobile exhaust purification ( S ummers and H egedus , 1 97 9) -the required calculations can be disconnected from the r e act or desi gn and c an be c ar rie d out for s ur face values generated in the reactor sim ulatio n . Various crit e ria have been derived for evaluating , on t he basis of observables , w hether or not it is neces s ary to account for intrap article gradient s . A r at her extensive se t of proposed crite ri a is given by B utt ( 1 980 ) . A few only are given here . For the absence of concentration gradients in an is ot her m al particle -in ot her words , to ha ve a v alue o f n > 0 . 95-Weis z and P r at e r ( 1 95 9 ) proposed 2 r p d A s p s 4D C
6. 0 <
0. 6 0. 3
e s
for zero- order reac tion s for first -order reac tion s for secon d -order re act ion s
( 7 2)
in whic h rA is t he ob serve d rate . For fir st - order reac tion s , this c riterion is slightly m or e conservative t han that p re viou sly proposed by Weis z and Prater ( 1 95 3 ) an d requirin g the left - han d side to be s maller than 1 . For t h e ab sence of temperature gradients ( i . e . , the observed rate differs by no more t han 5 % from the rate prevailin g in an isothermal p article ) , Ande rson ( 1 9 6 3) derived
( 7 3) or 2 p d A s 2 s D C e s
r
<
1
!3 p y
s s and T . F or the absence of s s combined in tra pa rticle and in terfacial gradient s ( n = 1 ± 0 . 0 5) , Mears ( 1 9 7 1a , b ) p roposed with 13 p an d y taken at
<
surface
c on di t ion s C
1 + 0 . 3 3 y [ ( - LI. H ) r p d / 2h T J f A s 2 n - 13 y [ 1 + 0 . 3 3n ( r p d / 2k C ) J A s
p
with Bp and y evaluated at
b ulk
p
( 7 4)
g
gas con dition s , C an d T .
Multiplicity of Steady S tates : In recent years great attention has been paid to t he mathe matical features of the soli d - phase eq uation s ( 57) and ( 58 ) . For c ertain ranges of the parameter value s , more than one steady - state profile would be possible for given geometry and op eratin g con dition s . A gain the approach used to derive c riteria for uniquene s s or multiplicity in a cross section of the bed is the one mentioned earlie r . The condition for uniquene s s arrived at by Luss ( 1 977) a n d by Van der Bosch an d Luss ( 1 97 7 ) for a fi rst - or der irreversible reaction a s re gards a particle with
Fi:x:e d -B e d G a s - S o l i d C a taly tic
Reac tors
41 7
concentration an d tem p erature gra dien t s and a pa r tic l e h avin g internal concentration gr a d i en t s and i n t er fac i a l concentration an d temperature g ra d ien t s can be written in the same form , S y / ( 1 + 13) < 4 , wit h 13 = 13 in the first an d 13 £ s 13 f in the sec on d case . I f t h is con dition is viola e d , multiplicity may exist for a c ertain range of val ues of th e T hiele mo d ulu s in the first case an d of the Sherwood n umber , s s ( k /D e H V p / S p ) , in t he g secon d case . In t he second case no simple analytical expre s si on s can be given for these ran ges . Internal concentration gradient s reduce the region in parameter sp ace for w hic h mul t ip lici t y of steady states is possib le as c o m pared to the s i tuati on in whic h on ly interfacial gradient s occur . I n iso the rmal p ar tic l es mul t ip licity would still b e p ossib l e with rate equation s that are not monotonic wit h res p ec t to the con version ( e . g , H au gen -Watson type rate e qu a ti on s with very hi gh va l ues for the a d sorption c oefficients of one of t he re ac t an t s ) . The rate e q u a t ion s for air ox id ation of C O or H 2 on platin um / alumina are ex a mp l es of t hi s . Multiplicity h a s in deed b een observed in laboratory stu die s of th ese reactions ( S mith , 1 9 7 5 ; Hegedus et al . , 1 9 7 7 ) . W hether this is really caused by the in te r ac tion between rea c t ion and interfacial an d in tra partic l e gradients or by s om e different phenomenon has not b een s et t le d unequivocally , howeve r . Accordin g to K eil and Wicke ( 1 980) , t he m u l tiplic i ty observed in their work on CO oxi dation is kinetically i n d uced ( i . e . , would result from an in t e r ac ti on between reaction and ad s orp tion ) . E i genberger ( 1 9 7 6 , 1 9 7 8 ) has modeled this i nterp retation an d its con se quences . A realistic review o f the area o f m ultiplicity of steady state s has been presen ted b y Sc h m i t z ( 1 9 7 5) . I t s hou l d be stressed that m ul ti p l icity has not been re po r ted yet for in d u st rial op e r at ion of t u bular reactors wit h ou t fe e db ac k
r
=
,
,
.
Examples of Appl ication : T here are very fe w examples of design studie s whic h explicitly include in tern al diffusion limitation s into t he model . T he fi r st reason for this is t h a t very few kin etic studies of c at alytic processes went into sufficient de p th t o per m it a q uantitative exp re ssion for the in t ra p a rt icle diffusional limitation s to b e derived . T he s econ d reason is the computational effort involved in the in tegra tion of the full set of e qu a t ion s ( 5 5) to ( 58) . What has generally been done i s to use an " effec tive" rate e quation , based on bulk gas phase con dition s only an d assumed to accoun t implicit ly for these effects , so t h a t the reac tor model is re d uced to the pseudohomogeneous base case AI . T he ac cur ac y of such an " effec tive" - in r e ali ty dis tor t e d r a te equa tion may be seriously q ue s tion e d , -
however . In an optimization study of 1 bute n e dehydrogenation on a chromi a alumina catalyst , requi rin g a la rge number of reac to r simulations , Dume z an d Fro ment ( 1 976 , 1 9 7 7 ) circ umvented t he n eces sity of in t e g ra tin g the soli d - p hase e q ua tion s ( 5 7 ) and ( 58 ) in a far more rigorous way . Havin g determined actual reaction rates , rE , for various par tic le si zes , they plotted these versus the true , in t rin s ic rates , r H , obtain ed wit h finely crushed cataly s t un der c orresp on d in g b ulk gas phase composition s . A uniq ue curve was derive d whic h c ould be expre s sed as -
rE
=
0 . 0 0 3 4 + v' 0 . 006 8 5( r
H
+
0. 0017)
( 7 5)
W hen both sides of th i s equation are divided by rH , the resultin g equ ation asymp totically behave s like t h e rel ation bet ween n an d
Fromen t a n d H o fman n
418
0.5
0. 1 +-----r---�--�--,--r-r-r�r--r�,-����-- - --- ---, - -oo 0.1 o.o5 0.01 o.oos 0.001 rH
1 7 E ffectiven e s s fac tor versus in trin sic rate of 1 - butene dehydro genation . ( A fter Dume z and Fromen t , 1 9 7 7 . )
FI GURE
rr;
r E tends toward rH as t he latter goes to zero and b ecome s p rop o r t ion al to as r H ten ds to in finity . The relation n = r E / r H ver sus rH is shown in Fi g . 1 7 . D u me z an d F romen t ( 1 9 7 6 ) also con sidere d the full set of e q uation s ( 5 5) to ( 58 ) . T hey d e ri v e d a value of 5 for T from rate measurement s at va rio u s si zes an d the kn owledge of the in trin sic rate . With D m an d D K
calculated from publi shed correlation s and E: s determin e d by phy sical measuremen t s , D e was c alculated from E q . ( 6 0) . T he model e qu a tion s ( 5 7) an d ( 58 ) were in tegrated in each node used in the in te gration of th e fluid field equa tion by collocat ion . In their work on s team reformin g of natural ga s on a nickel - alumina catalyst , De Deken et al . ( 1 9 8 2 ) obtained a T of 4 . 5 from the gas c hromatographic t ec hnique , u s in g finely crushed c at a ly s t as a colu m n material . T he effec tiveness fac tor was very l o w with this very ac tive c ataly s t : on t he order o f 0 . 0 2 to 0 . 04 . T he reaction was con fin e d t o a ve ry t hin layer n ear t h e surface , b u t neverthele s s still repre sentin g an i n t e rn al area of about 1 0 0 0 times t h e ext ernal s u r face area . The model used in t h e reac tor simulation con side re d interfacial tempera ture and in traparticle concen tration gra di en ts , l eadi n g to equation s ( 6 1 ) to ( 63) . Two con tinuity eq u a tion s were req uired to define the co m p o si t ion o f the reac tion mixture . The rate equation s were of the H ou gen - Wat son type and t h e g e neralize d modulus c oncep t [ Eq . ( 7 1 ) ] was used . St ric t ly speak in g , this concept is limited to a sin gle reaction , but an approximate relation could be established b et w e en the t wo in dependent reaction s determinin g the proce s s . Figure 1 8 s hows the varia t i on of n c H an d 11 c o alon g the 4
reactor for typical operatin g condition . T he discontin uity at a bed depth of 6 m result s from the use of a different catalyst si ze farther down stream . Notice t h a t both 11 are es sen tially zero at the inlet , where th e r eac t ion rates are extremely hi gh . The t:, T over t he external fi lm s urroun d in g the cataly st is very s mall : 4°C u p to 6 m , 3°C from t h e re onward , and 2 ° C
41 9
Fixe d - B e d G as -Solid Cataly tic Reac to rs
007
.,. ...,...
0 u
0.06
Cl 2
:I: u
en a: 0 fu
u UJ u.. u.. UJ
j::
0. 01
0 �------�---; 0 2 4 6 B 10 12 A X I A L R E A CTO R COO R D I N A T E
FI GURE 1 8
[m]
Ste am re forming o f natural gas . Variation o f the effectivenes s C factor for methane conversion and for O production with b e d depth .
( A fter D e Deken et al . , 198 2 . )
an d less from 1 0 rn onward . Neglectin g thi s t. T w o u l d further sim p lify t he set of model e q u a t ion s without si gnific an tly modifyin g the results .
Two -Di mens ional H e t e rogeneous Model T he model equations , con s i de rin g axial and r a dial gradien ts in the reac tor ,
in terfacial and in tra p article concentration , and temperature gradien t s , c an be w ri t t e n , when use is made of the e f fec tiveness factor n , ei t h e r in its
a n a ly tic al expression or just formally , to abbreviate the system of e quation s ( D e Wasch an d F romen t , 1 97 1 ) : Fluid p hase :
(D
)
er s
/er
(a
( a zc ar
2T
ar
2
+
+
2
!.
r
!.
a
c)
u
r ar
a)
.!!. r
_
u p gc s
ac
s az
CI T p Cl z
k
a (C -
g v
Cs) s
s = h �v ( T - T ) s
( 76 )
( 7 7)
Fromen t a n d H o fmann
420
Soli d p hase : ( 7 8)
( 79 )
wit h boun dary con dition s c
T
:::
ac a-r
ac
c
o
at z
=
0 , all r
To 0
=
::: =
--
s Cl r
0
:::
clw ( T w
T)
clw ( T w
- Ts )
at r = 0 , all z
0
;/
at r
er a r
=
A
s er
=
R , all
z
aT
s ar
The secon d term in the righ t - hand si ze of Eq . ( 7 9 ) expresses heat tran s fer i n radial direction involvin g both the solid and fluid p hase s . T h e t wo phase character o f the model is also reflected in the b oun dary con dition for A. s f heat transfer at the wall . T he model parameters are now D , A er er ' er ' s f a , and a . w w
The effec tive con d uctivity and wall heat transfe r coefficient concept , used in the two - dim en sional p seu dohomogeneou s model ( Y agi and K unii , 1 9 5 7 ; Kunii and Smit h , 1 960 ; Zehner and S chli.in de r , 1 970 ) , is not directly applicable here . In thi s concep t the heat tran sfer mec hani sms are lumped accordin g to static an d dynamic contrib ution s ( Re 0 and :/: 0 , resp ec tively ) an d not accordin g to phase , as was done by Sin ger and Wilhelm ( 1 9 50) . However , Y a gi an d K un ii ( 1 95 7 ) considered the followin g heat t ransfe r mechani s m s , group e d here accordin g to phase : Fluid p hase : =
1.
2. 3. 4.
C on d uc tion C onvection T ransfer from fluid to solid Radiation from void to void
4 21
Fixe d -B e d G as - Soli d Ca taly tic R eac tors
Solid p hase :
1. 2. 3. 4.
C on d uction C on duction fro m p ar tic le t o part ic le t h rou gh co n tac t s ur fac e s C on duction th ro u gh the st a gn an t film surroun din g the c on t ac t s u r fac es Radiation from par ticle to pa rticle
C ombinin g these mechanisms in the ap p ro pr i a te way and expressin g them in terms of the heat t r a ns fe r for mulas lead to ( D e W asch and F ro ment 1971) ,
=
€( A
g
+ (3 d
a
p rv
+ p c D
g p
er
( 80)
)
( 8 1)
where 8 i s a c oeffic ie n t c o mp r i se d between 0 . 9 and 1 . 0 , et rv and a rs a re radiation coefficien t s between voids and solids , respectively , an d cp can be calculated from a fo r m ul a p resented by K u nii an d Smith ( 1 9 6 0) . B a se d on t he model of Yagi an d K unii ( 1 9 6 0 ) for wa l l heat t ran s fe r the followin g re s lation s c an be w ri t t en for a n d a , p rovid e d t hat T an d T are n ot too s w w different : ,
rl
f a = w
A
f
-- a
A
er
er
( 8 2)
w
( 83)
Dixon and Cresswell ( 1 97 9) d isc us se d experiment al heat transfer correlations and de rived rel a tion s between the p seudohomogeneous p ara mete r s A er and f s s f a and the parameters of the heterogeneous m o de l , A , A a , and a . w er er , w w De Wasc h and F roment ( 1 9 7 1 ) simulated a hy drocarb on oxi dation reactor and obtain e d the radial temperature p rofile s shown in Fi p; . 1 9 , for condition s whereby T) = 1 . T h e t. T b e t ween solid an d fluid doe s not exceed 2° C , al though the reac tion is ve ry e xoth ermic . Fi gure 2 0 compares radial mean temperatures as a func tion of bed d e p th , calculated by v a ri ou s models dis cu s se d in this c hapter an d for typical i n du s tri al o pe rat in g c on di tio n s C u rves 1 an d 2 were obtaine d by means of the one - dimen sional models , pseudohomo geneous ( A I ) and h e te ro ge n eou s , but with T) = 1 ( B I ) , curve s 3 an d 4 , by the two - dimensional models Al i i an d B i l l , respec tively . Curve 5 , w hic h predic t s an exc es sive h o t spot , was obtain e d by means of a model For t h e condition s of the exam n e glec tin g heat t ransfer through t h e solid . ple , some 2 5% of the radial heat flux occurs through the solid p hase . .
T
('IC)
400
3 8 0 L------L------�----��----� 0. '15 0 . 50 0. 25
R r
FI G URE 1 9 S imulation of hy drocarbon oxidation by mean s of two- dimen sion al hetero gen eous model . Gas and solid radial temperature p rofile s . ( A fter De Wasc h an d F romen t , 1 97 1 . )
FI G URE 2 0
0.2 5
0,50
0.75
z
<ml
Hy drocarbon oxidation . Simulation by means o f various models : 1 , b asic p se udohomogeneous one - dimen sion al model ; 2 , on e - dimen sional
hetero geneous model wit h interfacial gradient s ; 3 , pseudohomogeneous two dimen sional model ; 4 , two- dimen sional h eterogeneous model ; 5 , two dimen sional heterogeneous model with heat tran s fer in radial direction t hrou gh ga s phase only . ( After De Wasch and Fromen t , 1 97 1 . ) 422
423
Fixed - B e d G a s - So l i d C a ta ly t ic R eac tors
The h e t e ro ge n eo u s nature of the case dealt with here results from in t erfa ci al gr a di ent s , but a gain these were shown to be negligible . De D e ken et al . ( 1 9 8 2 ) u s e d a two- dimen sional m o de l t o si m u la te a process in
whic h the interfacial gradients c o ul d be n e glec te d , b ut not the intra p article c on c e n tration gr adie n t - t h e steam r e fo r m in g dealt with earlier . Fi gure 2 1 shows radial t emp er at ur e p r o fil e s at various b e d depths . T he m aximum difference in proce s s gas temperature bet ween the centerline and the inn e r t ube wall amoun ts t o 3 3°C , whic h is not exces sive an d per mits the radially averaged temp erature to b e simulated accurately by mean s of the correspondin g on e - dime n si on a l model with a heat transfer coefficient calculated from Eq . ( 37 ) . N ote that the model is rea lly a hetero g en e ou s model , but that since there are no in terfacial gradient s and sin ce the in t r ap artic le con c en t ra tio n gradient is accoun ted for by means of the effectivene s s factor concept , its struc ture is that o f the pseu dohomogeneous two- dimen sional m od e l A l i i ,
z =
9m
:.:.::M W" ' " •• • �
u
�
�-
....
w a: ::>
....
w a: ::>
....
....
a: w <1.
z =
w
::!:
6m
....
r.n
t.:J r.n r.n w u
0
w
::!:
....
....1 ....1
a: Q..
a: w Q..
w m ::>
110
....
a: w z z
eoo �--z
0
= 3m
� � � M � NORMALIZED D I STANCE F ROM R E ACTO R C E N TE R L I N E
FI GURE 2 1 S team refo r min g o f n a tu ral gas . Simulation o f radial t e m p era t ur e profil es a t v a ri o us b e d d e pths by mean s of two- dimen sional hetero Comp arison geneous model with i nt r ap article concentration gradient only . of r ad i ally aver aged values with pre diction of temperature by me ans o f one - di m e n s ion al model w it h in traparticle concen tration gr a d ient . ( A fter De D eken et al . , 1 9 8 2 . )
424
Frome n t a n d H o fmann
A g ain becau se there are n o inter wi t h n m ultiplyin g the rate term rA PB . facial gradien ts an d because the particles are isothermal , the radial heat t ransfer parameters are those of the pseudohomogeneous mode l , A. er and a w . SPEC I A L CASE :
T H E MONOLI T H CONVERTER
T h e monolith converter consists o f an array o f duc t s with catalyst coatin g on the w all s . I t is used for con trol of automotive e mission and is bein g con si de red for waste abatemen t . The discus sion o f reactor design and opera tion in the previous sections was oriented exc l u sive ly toward reactors for p roduc tion proce s se s . The c at alytic muffler does not fall in this c ategory , but since there may b e a potential use for monolith converters in other areas and since its simulation makes use of reactor models described in this chapter , it is discussed briefly below , mainly from t he sim ulation point of view . More detailed discus sion s were p ublished by Wei ( 1 9 7 5 ) , who reviewed some of the reac tion en gineerin g asp ect s an d early models for CO oxidation . Lee an d A ris ( 1 97 6 ) also reviewed the features of possible models , wh e r ea s H e gedus and coworkers ( H e gedus et al . , 1 9 8 0) significantly contrib uted to the mathe matical modelin g of the phenomena at the level of the catalyst particle , in p artic ular of ins t abilitie s an d oscillations . B elow , models are given for the comparison of t h e performance of a mon o li t h converter , alon g the lines set by Youn g and Finlayson ( 1 9 7 4 ) . T hey are written , for the sake of generality , for transien t operation , whic h is also the most typical type of operation for the catalytic muffle r . S ince the transient re s pon s e is determined main ly by the thermal response of the solid , all other time derivatives are set equal to zero . The reaction considered i s C O oxidation . Diffu sion al limitations in the thin catalytic layer is ne glected , although it could - formally , at leas t -easily be incorporated in to the model e quation s by mean s of t he effectiven e s s factor n . A c ell comprisin g a cy lindrical duc t an d surroundin g solid is chosen . The operation of the cell is ad iabatic ; that is , the flo w rate i s t he same over all duc ts an d the solid temperature at any cross section of the monolith is uniform . A xial conduction in the solid is neglected , as is axial diffusion in the fluid since the Peclet number , based on t he length of t he duct , exceeds 500 . The first model dealt with by Y oung an d Finlayson is a one - dimen sional heterogeneous model with plug The c on flow an d in terfacial gradient s ( B I in the terminology used here ) . tinuit y equation for CO is w r it t en , in terms of mol fraction s ,
u
rz
( 84)
an d the energy equation for the fluid ,
u p c
s g p
g
Cl T Cl z
=
4
h
_}_ d
t
(T
s
- T)
( 85)
A t the wall s urface , where t h e reac t ion occ urs ( and with t h e rate o f disap pearance of CO taken to be positive ) ,
425
Fixe d - B e d Gas - S o l i d C a taly tic R eac tors
( 86 )
aT
s
( 87 )
at
w here S s represent s the surface area pe r volume of solid in the monolith c ell . The boun dary and initial c ondition s are
T = T 0
y
y
T
s
at z = 0
=
T
1
s.
( z)
at t
=
( 88)
0
The sec on d model is of the two- dimen sional type , but with laminar flow , since the Reynol ds n umber in the duct ran ges from 2 0 to 4 0 0 . Con tinuity equation for C O :
( 8 9) Energy e quation for the fluid :
( 9 0)
an d at the wall s urfac e :
� � Dm M Cl r m
A
ar
Cl T
g
I
r=R
=
r=R
( - l\. H ) r
CO
+
p c
s
s p s
s
aT
s at
w here r e o is expres sed in k mol / h an d unit wall sur face area . and initial con dition s are
T T
s s
T y
=
=
at r = R , all z
T T
1
s.
To
at t
=
0
at z = 0
( 9 1)
The b oun dary
426
Fromen t a n d Hofmann
The one - dimen sional model m ay h a v e m u lt ip l e s t ea dy state s , but t he t wo - dimensional m ay not . Fi gu r e 2 2 s h ow s that the lower t eady - s t at e pro
s
file s of T s and T of the one - dimen sional model are obtained when the mono
lith is h ea t ed from an ini ti a l low t e m p e rature , while t he hi gher steady state
p rofi e s are reac hed when the converter is c ooled from some initial high t e m pe rat u re . For interme diate initial temp erat ures the final st e ady state
l
may lie between those s ho n . T h e u i que Ts and T p r o fil e s of the two dimen s ion al model ar e closer to those c or re spon din g to the lower steady state of the one - dimen sion al model . The t w o - dimensional model i s d e fin i te ly a more re ali st ic model of the
w
n
m on o l i th , but it is computationally much more de andin g . This is w hy it is of in tere st to in ve sti ga t e whether or not it is possible to approximate it by me ans of the on e - di men s i on al model - overlookin g for a w hile the multi plicity of s t e a d y states a s sociated with the latter . T he curves related to the one - dimen sional model Fi g . 2 2 we r e calculated for c o n s t ant values of the S herwood an d N u s selt n umbers , k g d t / D m and hf dt / A g , respectively . To come to an exact fit , however , these n umbers have to continuously vary alon g the reac t or . T he Nu number , for ex a m p l e , would have to b e calculated , for each axial p o s it ion , from the temperature gra d ien t at the w all , generated b y t he t wo - d i m e n i o al ode l :
m
in
s n
m
T (°C)
N u , Sh
s.o
"ff
3.0 1 .0 0
2.5
Z (c m ) 50
7. 5
10
FIGURE 2 2 O n e - and t wo - di m en s ion al simulation of m on o li t h converter . S olid line , on e - dimen sional heterogeneous dashed m od el . ( A fter Y oun g and F inl ay s on , 1 97 4 . )
model ;
line , two-dimensional
427
Fixe d-B e d Gas -Solid C a taly tic Reac tors 2 (
Nu
T
l r=R
( 92)
- T
s
The S herwood num where T is the c up m 1 xm g temperature o f t h e fluid . ber woul d be obtain e d in an analogous way . Figure 2 2 al so shows how Nu and Sh , t aken to be identic al in this case , vary with axial p osit ion Also shown is the ratio of the reaction rate at wall condition s to that at mixin g c up c on dition s . T his rat i o varie s stron gly w i t h p os i tio n an d b e .
comes in fin i t e w hen t h e gas re ac h e s the w all temperature , in the reaction zon e , after light off .
T o avoid the two- dimen sional simulation a s a prerequisite for on e dimen sional c aJ c u l a t ion s , H eck et al . ( 1 97 6 ) appro xi mated the Nu p rofile before the discon tin uity by the value for p ur e heat transfer with cons tant heat flux an d t hat b eyon d t he discontin uity by the value c orrespondi n g to c on st an t wall t e mp er at u r e S un daram an d Froment ( 1 97 9 , 1 9 8 0) derived
t he local Nu for laminar flow and in t he ab sen c e of rea c tion from a l a r ge n u mber of two dime n s i on al simulation s . T hey came to the followin g correla tion s : .
-
Nu
+
3 . 655
=
0 . 1 4 50Z
- 0. 778
exp ( -7 . 1 5 8Z )
( 93 )
T he a sy mp t otic value of N u of 3 . 6 5 5 co m pl e t e ly a gree s wit h t h e experimen tal v alu e . For t u rb ulent flow , Whitaker ( 1 97 2 ) ob t a in ed the followin g correlation : ( 94 ) with
a
2
0 8 -2 Re · , in agree where Nuoo is the asymp totic Nu num ber ( e . g . , 2 . 2 8 x 1 0 ment with the Dittus - Boelter correlation ) . S undaram and F romo nt ( 1 97 9 , 1 98 0 ) also s ho we d t ha t N u depen ds stron gly on t h e p ropertie s of t h e reac tion . For en dot hermic reaction s and both laminar an d t urbulent flow they correlated the N u sselt profile ver s us the dime n s ionle ss adiabatic tempera -
ture rise , the L ewis n u mber , and the dimen sionles s activation energy . T he disc on tin uity in the ratio ( 9 2 ) for exothermic reaction s at T = T s p re c ludes a correlation t hat would b e as accurate as that given for en dothermic r e ac t ion s T he prec e din g disc u s sion once a gain illustrates the impor t a n ce of a re p re sentative model not only to come to the hi ghest pos sib le ac c uracy 1 b ut also to a voi d t h e p re d ic tion of a behavior t ha t may not be real at all . The question of w h e t h er or not multiplicity is possible in a monol it h converter re main s largely unan s w e red , even if Hlavacek an d Votruba ( 1 9 7 4 ) pre T his has been exp l ai n e d in t e rms of sented ex p e ri me n t al evidence for it . axial h e a t transfer in t h e wall . It is more likely , however , to result from .
1
428
From e n t a n d Hofmann
phenomena at t he c atalyst surface in stead of tran sport phenomena . S u r fa c e phenomena c a u s in g m ult ip licity would b e re fl ec t e d by more t han one rate - determinin g s tep and rate equation s more c omp licated than those use d in th e work of Y o u n g and Finlay son . M u l t ip licity would t he n , of course , also b ecome pos sible with the two- dimen sional mo del . C O N C LU S I O N
I n this c hap te r the design or s i m ul a t ion of fixed -bed reactors i s based on fun damen tal models . These combine the hy drodynamic a n d heat tran sfer phenom ena with the chemical reaction aspects by means of a set of m at he matical equations of varyin g c om pl exi ty . With the pre s e n t computational facilities and n umerical tec hniques , the solution of these e q uat ion s is no lon ger an ob stacle to the application of these models to the real problems and c on fi guration s e ncounte r ed in in dustrial operation . E vi de n tly , no model will ever represent the reactor behavior in an ab s ol ute ly rigorous w ay . The equa t ions c ontain a n u mb e r of p a ram ete rs , some of w hic h are " effec tive" in the sense that t hey lump the effect s of p hen o m e na that would otherwise have to be exp re s sed by means of a d d i t ion al equa tions , to b e solved sim ultaneously in the reactor sim ula tion . A dmittedly , considerable improvement an d exten sion of exi stin g correlation s is r e q ui re d for severe reaction con di tion s or unusual reactor con fi guration s . I n deed , for convenience , correlation s for these p arameters were generally obtained from p urely p hy sical experiments , possibly without sufficien t · te stin g under con dition s similar t o those encountered in the p resence of reaction . Under r e ac t ion c on dition s the p aram e t er values may then t urn out to b e inaccurate , even when the un de r ly in g description of the ph e n om en on is correct . B ut the pa ra m ete r s s hould not be tuned a rbit r a rily on the basis of experimen tal testin g of t he reac tor mode l , to avoid inc on si st ency and violation of their physicoc hemical back groun d . T he models develope d here are all of the continuum type . This di s p ute bet ween the advocates of c on tin u u m and cell m od e ls is mainly of acade mic in teres t , except perhaps for very severe reaction conditions . An e s c a l a tion of model sop histication is p r e s e n t e d and c ri t e ria are given for t he selection of the a pp ro p riate model . Whic h model is " appropriat e " depen ds in the first p l ac e on t he reactor , the o p e ra tin g conditions , and t he r eacti on s , b ut als o on t he required ac c u r a cy and on the a vail ab i li t y an d p r eci sion of the S ome of the models p re sented here are re qui r e d fun damental in formation . a bare minimum for a r ealis tic description of the re ac to r o perat ion , others m ay look too refined and of academic interest only . N evertheless , it is safe to p r e dic t t hat w hat may seem to be too refined t o day may .well be the minimum level of mo d e lin g tomorrow , pa rtic ularly for d e man din g con ditions . Models will certainly continue to b e refin e d , but reali sm s ho uld be a guide in t his development . It s hould never be for gotten , however , that advanced re ac to r m ode li n g requires first , accurate kinetic s , not only on fres h , b ut also on gradually deactivatin g catalysts . I n the past this has b een t h e m ain obs ta c le to the fu nd a m en t al approach in reactor desi gn . I t will c on tin ue to be a serious challen ge in the fut u re , because it re q u ire s a co ns i d e r ab le experimental effort o f a team with both b ro a d scientific and technic al bac k g ro un d which is seldom p ut together at the ri ght time (i . e . , at an ea r l y s t a ge of d e v e lo p m e n t ) .
429
G as - So li d Cataly tic Reactors
Fixe d - B e d NOTAT I O N
a
2 3 external particle surface ar e a pe r unit re actor volume , m tm P r 3 gas phase concentration , k mol tm g 3 concentration of A in feed , kmol / m
v
c
co c p
specific heat of gas , kJ /kg • K
specific he at o f gas in sid e t h e solid , k J /kg • K
�s
c
speci fic he at o f solid , k J / k g • K
s
3 concentration in side catalys t , kmol /m g 3 s urface concentration , kmol / m
c
s cs s d. d d
g
1
internal diameter of c ata ly s t rin g , m particle diameter , m
p
in ternal t ube diameter , m
t D e
D
D
r 3 e ffective d i ffusi v ity inside catalyst , m /m • s g p 2 Knu d sen diffusivity , m / s 2 molecular diffusivity , m J s
K m
(D
p
p
ea
)s
effective diffusivity i n a xi a l bed direction based o n superfacial flow velocity , m 3 /m • s g r e ffective diffu sivity in radial b e d direction based on superfacial flow velocity , m 3 /m • s g r 3 activation e n e r gy , kJ /k mol or m •atm /kmol •K ex p on e nt ial in t e gra l friction factor molar feed rate of component A , kmol • A / s acceleration of gravity , m t s
2
2 superficial mass flow velocity , kg /m s r
2 s u perfic ial mas s flow velocity of component i , kg i / m · s r 2 gas - to - solid heat tran s fe r coefficien t in a fixed bed , kJ /m • s · K p 3 reaction rate coeffi c i ent ; for fir st order , e . g . , m g / k g cat • s 2 3 ga s - to - solid mas s tran s fe r coefficien t in a fixed b e d , m /m •s g p equilibrium constan t , adsorp tion con stant of compo nent A , 1 /atm total len gth of reactor , m m
M n
mas s flow rate , kg /s m
Nu
r
molecular weight of mixture , kg /kmol order o f reaction N u sselt number C h
flt / A g )
4 30
Fro m e n t a n d H o fmann
PA
pt Pehr Pe rna
1
1
' Pe rna Pr
r r
A
rE rH R R' Re 8cyl
s
P
s
T, T
T
2 o
T
s s u ax ui us TS
u v
v X
cyl p
y
z,
1
,
s Sh t T
partial pressure of component A , atm total pressure , atm Peclet number for radial effective diffusion , [ dp u. /D er ] Peclet number for axial e ffecti ve diffusion in fixed bed , [ dp u. / ( D ea ) . ] Peclet number for axial e ffe ct i ve diffusion in fixed bed , [ d /L / { D )1. ] p ea Prandtl number ( c p )1 I >.. g ) radial position , m rate of reaction of A per unit catalyst weight , kmol /kg cat •s effective rate o f dehydrogenation of 1-butene , kmol /kg cat •s intrinsic rate of dehydrogenation o f 1 - butene , kmol /kg cat •s tub e radius , m gas constant , kJ /kmol • K or m 3 •atm /kmol • K Reynolds number ( d p G I ll ) external surface area of cylinder with same external diameter and height as the ring- shaped catalyst , mp2 surface area of a particle m 2 surface area per volume of solid in monolith cell , m. 2 / m 3 Sherwood number ( k gdt /D m ) time , s process gas temperature , K coolant temperature , K feed temperature of gas , K solid temperature , K solid surface temperature , K velocity in the axis of monolith duct , m / s in ter stitial flow velocity , m /s r superficial flow velocity , m g3 t mr2 •s overall heat transfer coefficient , kJ /m 2 •s •K volume of cylinder with s ame external diameter and height as ring shaped catalyst , m p3 particle volume , mp3 conversion mole fraction axial distance in reactor ( respectively , heat exchanger) , mr
z'
z , Z'
1
length of reactor { respectively , heat exchanger) ,
m
r
4 31
Fix e d -B e d Gas - S o l i d C a taly tic R eac t o rs G reek Let t e r s a. 1
a ..
a
lJ
rs
be d - to - wall heat transfer coefficien t in one - dimen sional model , kJ t m 2 · s • K
2
stoic hiometric coefficient o f component i in reac tion radiation coefficient b etween particles , kJ /m • s • K 2 radiation coefficient between void s , k J /m • s • K wall heat trans fer coefficient i n two- dimen sional model , kJ / m 2 • s• K 2 st atic contrib ut ion to a ' kJ / m • s • K w dimensionless T a d in a film surroun din g a particle [ k ( - ll H ) C / g h Tl f
Ts]
in a particle [ D C 8( - A H ) I A ad e s e s dimen sionless adiabatic temperature rise in a reactor [ ( -A H)C / p c T ] 0 g p O dimen sionless activation energy ( E / R T , E / R T , E / R T ) 0 heat of reaction , kJ / kmol dimen sionless T
:
e:
s
n nG ).
e
>..
ea
a diabatic temperature rise , K 3 3 void fraction of bed , m /m g r void fraction of particle , m 3 J m 3 g p 3 a group ( M L /u p ) , m · s /k mol m s g r catalyst effec tiveness factor based on surface condition s
(Cs s
,
catalyst effec tiveness fac tor based on b ulk gas con dition s ( C , T ) effective con d uc tivity in side a particle , kJ / m • s K •
effec tive axial conductivity of bed
+
gas , kJ / m •s • K
),_ 0
static contribution t o t he effective axial conductivity kJ / m s • K
>..
effective radial con duc tivity o f bed + gas , kJ / m • s •K
ea
er
),_ 0
er ;l.f er
•
static contrib ution to the effective radial con ductivity , kJ /m • s • K effective radial con ductivity for gas phase , kJ / m • s • K
),_S
effec tive radial c onduc tivity for solid p hase , k J / m • s • K
>..
gas c on d uctivity , kJ / m • s • K
er g
dynamic viscosity , kg /m • s i nside p article , oriented tow ard s ur face coor di nate , m bulk density of bed , k g cat tm 3 r 3 gas phase den sity , kg / m g 3 / c atalys t density , k g cat m p
T
s T ) s
tortuosity Thiele modulus [ = ( V IS ) ( k iD e p p
)]
p
From e n t a n d Ho fmann
4 32
1 / Pe 1 /Pe
rn a
mr
R E F E R E N C ES
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Gas -Soli d
Ca taly t ic
Reac to rs
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4,
and R . Aris , Modeling t h e
268 ( 1 97 2 ) .
monolith : some methodolo gical con sider a
Heidelberg , D ec hema VI - 2 3 2 . and J . B utt , H et erogeneou s c a t alytic reac tors under goin g chemi cal deac tivation , A I C h E J . , 28 , 4 0 5 , 4 1 1 ( 1 9 8 2 ) . Lerou , J . J . a n d G . F . Fromen t , V el oc it y , temperature an d c on ver sion p ro files in fixed bed catalytic reac tors , C hern . En g . Sci . , 32 , 853 ( 1 9 7 7 ) . Lerou , J . J . and G . F . Fro men t , Fix e d bed reactor de sign and simulation , in Comp u ter-A ided Process Plan t D e s i gn , M . E . Leesley , ed . , Gulf Publishin g Comp any , Houston , T e x . ( 1 9 8 2 ) . tion s , C hern . R eac t . En g . , Proc .
( 1 976) , Lee , H . H .
p.
4th I n t . Symp . ,
437
Fixe d -B e d G a s - S o l i d C a taly t i c Reac t o rs
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.
-
.
,
-
-
,
,
.
.
-
-
,
,
,
,
,
438
Fromen t and Ho fmann
Olbric h , W . E . , J . B . A gn e w , and 0 . E . Potter , Dispersion in packed beds and the cell model , T ran s . I ns t . C h e rn . En g . , 4 4 , T 2 0 7 ( 1 9 6 6 ) .
Olson , K . E . , D . L u s s , and N .
R.
A m u n d son , R e generation of adiabatic
fixed b e ds , I n d . E n g . C he rn . P roces s D e s . Dev . ,
Petersen , E . E . , C hemical Reac tion A nalysis , C liffs , N . J .
7,
96 ( 1 96 8 ) .
Prentic e - H all , Englewood
( 1 96 5a ) .
P etersen , E . E . , A general c riterion for diffusion in fluenced c hemic al reac tions in p orous solids , C h e rn . E n g . Sci . , 20 , 5 87 ( 1 96 5b ) .
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Pom mersheim , J . M . an d R . S . Dixit , Fixed bed reactors wit h deactivatin g cataly s t s , A C S Symp . S er . , 1 68 , 36 7 ( 1 9 8 1 ) . S . , V . G . B ol trya n s k i , V . G a mkrelid ze , and E . I . M r sc hen ko , Ma t hematische T heorie q)p timaler Prozesse , R . O l den bourg ,
Pontrya gin , L .
Munic h , W e s t G ermany ( 1 9 6 9) .
Probst , K . and K . Wo hl fah rt , E mp iri s c he Abs c hiit z u n g effektiver Diffusions koeffi zienten in porose n S ystemen , C he rn . In g . Tech . , 5 1 , 7 37 ( 1 97 9) . Ray , H . W . , Cataly st p article and fixed bed reac tor dyn amic s . A review , P roc . 5th Eur . S ymp . Chern . Reac t . E n g . , Elsevier , A m sterdam ( 1 9 7 2 ) . Rayadhyaksha , R . A . , R . Vas udeva , an d L . K . Doraiswamy , Parametric sen sitivity in fixed bed reactors , C he rn . E n g . Sci . ,
30,
1399 ( 1 975) .
Reichelt , W . an d E . B l as z , S t romun gstechni sche U n tersuc h u n gen an mit Rashig-ringen ge fiillte n Fiillkorperrohren und - S iiulen , Chern . In g . Tech . , 4 3 , 949 ( 1 97 1 ) . Renken , A . , Instationiire P ro ze s fii h r u n g kontinuierlicher Reakt or e n , Chern . Ing . Tech . , 5 4 , 5 7 1 ( 1 982 ) . Roemer , H . H . and L . D . D urbin , Tran sien t response an d moment analy sis for b ackflow cell models for sy stems with longit udinal mixin g , I n d . En g . Chern . Fundam . , 6 , 1 2 0 ( 1 96 7 ) . Schiidlich , K . , U . Hoffma nn , and H . Hofmann , Periodical operation of
processes and evaluation o f conversion improvements , Chern . Eng . Sci . , 38 , 1 37 5 ( 1 98 3 ) . Schliin der , E . U . , T ran sport Phenomena in Packed B ed ·Reac tol' S , C hern . Reac t . E n g . Rev . - Houston , A C S S y mp . S e r . , 72 ( 1 9 7 8 ) . S c h mit z , R . , Multiplicity , Stability an d S en sitivity , C hern . R eac t . En g . Rev . , A dv . C hern . S e r . , 1 4 8 , H . Hulbur t , e d . , A C S , Was hin gton , D . C . ( 1 9 7 5) , p . 156 . S c h wart z , C . E . an d J . M . S mith , Flow distrib ution in p acke d b e d s , Ind . En g . C he rn . , 4 5 , 1 2 0 9 ( 1 9 53 ) . Sin ge r , E . , an d E . H . Wi l h e lm , H e at transfer in p ac ke d beds . Analytical solution s and design methods , C h ern . En g . Pro g . , 4 6 , 34 3 ( 1 9 5 0 ) . Smith , T . G . , J . Z ahr a dn i k , an d J . J . C a r b e r ry , N on - isothermal inter in traphase e ffec tive n e s s fac tors for negative order kin etic s - C O oxidation over Pt , C h e rn . E n g . Sc i . , 3 0 , 7 6 3 ( 1 97 5 ) . Soria L ope z , A . , H . I . d e La sa , an d J . A . Porras , Parametric sen sitivity of a fixed bed cataly tic reactor . Coolin g fluid in fluence , C h e rn . E n g . ·
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36 ,
S um mers , J . C . an d L . L . Hegedus , Mode s of catalyst deactivation in stoic hio metric automobile exhaust , I n d . En g . Chern . P rod . Res . Dev . ,
1 8 ' 31 8 ( 19 7 9) .
Fi:x:e d -B e d Gas -Soli d Ca taly tic R eac tors
4 39
Sundaram , K . M . a n d G . F . Fromen t , A comparison of simulation models for empty tubular reactors , Chern . En g . S c i . , 34 , 1 1 7 ( 1 97 9 ) . S un d a ram , K . M . an d G . F . Fromen t , T wo dimen sional model for the simu lation o f t u b u lar reactors for thermal c rac k in g , C hern . En g . S c i . , 3 5 , 364 ( 1 9 8 0 ) . S u n d a resan , S . , N . R . Amundson , an d R . Aris , O b se rva t ions on fixed bed dispersion models . T he role of the in terstitial fl ui d , AI C hE J. , 26 , 5 2 9 ( 1 9 8 0 ) . Tallm ad ge , J . A . , Packed bed pressure drop - An exten sion to higher Reynolds n umbers , A I C hE J . , 1 6 , 1 0 9 2 ( 1 9 70) . Tsotsis , T . T . , A . E . H ade ri , and R . A . Sch mit z , Exact un iqu en e ss an d multip licity c riteria for a class o f l u m p e d reac tion system s , Chern . En g . Sci . , 3 7 , _ 1 2 3 5 ( 1 9 8 2 ) . Van D eemter , J . J . , F . J . Zuiderwe g , and A . Klinkenberg , Lon gitudinal diffusion and resi s t an c e to mass transfer as causes of nonideality in chromatography , C he rn . Eng . Sci . , 5 , 2 7 1 ( 1 9 5 6 ) . V an den B osc h , B . , and D . L u s s , Uniqueness and multiplic ity criteria for an n th orde r c he m ic al reaction , C hern . En g . S ci . , 32 , 2 0 3 ( 1 97 7 ) . V an d er veen , J . W . , D . Luss , and N . R . A mun dson , S tability of packed
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28,
Votruba , J . , V . Hlava6ek , and M . Marek , Packed bed axial the rmal con d ucti vit y , C hern . En g . S ci , 2 7 , 1 8 4 5 ( 1 9 7 2 ) . Wakao , N . and J . M . S mit h , Diffus ion in cataly s t p e lle t s , Chern . Eng . Sci . , .
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1 48 , H .
440
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Y agi , S . an d D . K unii , S tu die s on heat transfer near wall surface i n packed beds , AI C hE J . , 6 , 97 ( 1960) . Y a gi , S . and N . Wakao , H eat and mass tran sfer from wall to fluid in packed beds , AI C h E J . , 5, 7 9 ( 1 9 5 9 ) . Y a gi , S . , D . K unii , and N . Wakao , Studies on axial effective thermal con duc tivity in packed beds , A I C h E J . , 6 , 543 ( 1960) . Youn g , L . C . an d B . A . Finlayson , Axial dispersion in non - isothermal packed bed chemical reac tors , Ind . En g . Chern . Fun dam . , 1 2 , 4 1 2 ( 1973) . Youn g , L . C . an d B . A . Finlayson , Mathematical modelin g of the monolith convertor , C hern . React . Eng . I I , Adv . C hern . Ser . , 1 33 , A C S , Washin gton , D . C . ( 1 9 7 4 ) . Zehner , P . and E . U . Schliin der , Die effektive Warmeleitfahi gkeit durchstromter K u gelsch iittun gen b ei mas zi gen un d hohen Temperaturen , Che rn . I n g . T ec h . , 4 2 , 9 3 3 ( 1 97 0 ) ; 4 5 , 2 7 2 ( 1 9 7 3 ) .
7
Fluidized-Bed Reactors PET E R N .
Londo n ,
ROWE a nd J O H N C ,
Y AT E S
U niversity College Londo n ,
E ngland
I N T RO D U C T I O N
Industrial reactors in
w hic h solid particles are maintain ed in a fluidized state by an up ward - flowin g gas stream were first applied on a wide sc al e in the 1 940s in t he fluidized c a ta ly tic c r a ck in g ( F C C ) p roce s s for the pro duction of gasoline , althou gh a more locali zed use of the t e c hn i q ue ha d been p r ac ti ce d in G e r m an y 10 y e a r s earlier in t h e Winkler coal gasification proc e s s . The m ain aim of those who developed t he F C C process was to devise a method of en gineerin g the catalytic crackin g reac tion of gas oil , discovered by Houdry , without usi n g fixed -bed rea ctor s that not only suf fered from considerable o p e r a t ion al proble m s but al s o were stron gly pro tec t ed by p a t e n t s ( Jahni g et al . , 1 9 8 0 ) . T he s uccess of the develop men t work on t he F C C p rocess may be j udged from the s hort time span of fo u r years be t w een its initial c onc e ption and the startup of full - scale plant and it is apparent with hin d si gh t that a m aj o r reason for this s uc c es s was the eminent s uit ab ili t y of the c ra c kin g process fo r operation in fl ui di zed- b ed reactors . For example , the relative ease of t r an s fe r of the finely p o wde r e d catalyst bet ween reactor and regenerator overcame t h e difficulty of cyclic op e rat ion of fixed beds an d at t h e same time solved the p roblem of tran s ferrin g heat from the exothermic r e genera tion to the en d o t h er m ic c r acki n g reaction . Furthermore , t h e hi gh th ro u gh p u t s that it was p o ss ib l e to main tain gave the F C C p roc e s s an enormous ad van t a g e over its c om p e ti to r s an d it is no t s urprisin g t h at some 4 0 years later it remains one of the most impo rta n t and ele gant cataly tic processes in all indu s try . Sub s e q uen t develop ments in the field of industrial fluidi ze d - bed re actors , howeve r , did not always res ul t in t he same de gree of s ucc e s s for the processes to w hic h they were applied and it r ap i dly became clear t ha t more needed to be understood about the basic p hysic s of ga s s oli d fluidi za tion before l ar ge sc al e r ea c to r s c oul d w it h c on fi d e nc e be de si gn e d or scaled u p from smaller unit s , and durin g the last thre e decades with this ai m in view a great deal of re s e a rc h has been carrie d out on the fun da me n tal s of the s ub j ec t . -
-
441
442
Rowe and Yates
The ty p e of information required for the design of a fluidi zed -bed r e actor is not mark edly different from that needed for any other gas - solid reac tor an d relates to chemical kine tic s and the mode and extent of contact
b etween the gas and solid p hases .
The way in w hic h this in formation is
as sembled constit utes the reactor model . Con sider the general scheme shown in Fi g . 1 , where gas enters at the base and rises t hrough th e bed a t a s uperficial ve loc it y U , in excess of the minim um , U m f , r e q u ir ed just
to fluidize the p articles . Under these con dition s it is gene r al ly observed that the bed divide s into two phases an d each accommodates a proportion of the total flow ( Fi g . lb ) . T h e in terstitial emulsion - p hase flow Q i passes at a velocity c lose to U mf through the space between t h e fluidized particles, while the exces s , Q b , flow s in t he form of rap idly movin g voids or " bub bles" which are essentially free of particles an d grow by coalescence as they rise through the bed . Gas flowin g interstitially is in intimate contact with particles an d , if the chemic al kinetics are favorable , the efficiency of the c hemical conve rsion in t his p hase will be h i g h B ub ble - phase gas how .
,
ever , does not make good contact with solids , so reaction here will be in efficient . T he re i s s o m e exchan ge between t h e t w o phases , b ut t h e extent of this , Q E , is d et e r mi ned mainly by t h e s i ze of the b u bb le s and decreases as they become larger . The exten t to whic h a reac tant ga s is con verted to prod ucts as it pas ses throu gh the bed is therefore determine d b y the followin g factors :
1.
2.
T h e division o f flow between the t wo p hases T h e contac tin g pattern of ga s an d solids wit hin e ach phase
Products Cyclone
separator
Freeboard space Intersti t i a l
phase
Gas bubbles Gr i d
Tota l flow= Ot (a ) F I G U RE 1 a fluidi zed
bed .
Flow
(b)
distribution in a fluidi zed bed . ( a) Cross section through ( b ) S chematic of gas flow division .
443
Fluidized -Bed Reac tors
4.
3·.
I n t erp h ase e x chan ge C hemical ldnetics
The three hydrodynamic fac tors are disc us sed i n detail below , b ut fi rst it will be useful to con sider some of the more basic features of gas - fluidi zed beds that have important implication s for their use as c hemic al reactors .
B A S I C F E A T U R E S O F G A S - SO L I D F L U I D I Z A T I O N M i n i m u m F l u i d i zation V elocity A b e d of p articles is n ormally load bearin g because any impre ssed force i s
ultimately resisted by static friction between grain s in con t act and is spread over a volume of the w hole . Thus one can walk over dry s an d , al thou gh sinkin g a little at each step as t he grain s rearran ge to provide frictional resistance . If a fluid , liquid , or gas , i s pas sed upward t hrough a particle bed , relative motion c auses an upward drag force on each p arti cle , effec tively reducin g its apparen t weight . At a c ritical veloci ty drag force equals gravitational force , each particle is in dividually supported and they no lon ger rest upon one other . T hi s condition ca n b e written
6 1f
3 d ( p 1\
-
Pf
)
g
( 1)
for uniformly sized spheric al particles . In this con dition there is no static friction between p articles an d the whole ceases to be load bearin g . I t behave s like a fluid with shear forces resisted essentially b y the vis cosity of the interstitial flui d - hence t h e origin o f the descriptive name fl ui dized b e d . T he fluid velocity at whic h this con dition is first reac hed is called the minimum fluidi zation velocity U mf · I t is convention al and con venien t to expres s t his as a s uperficial velocity ,
Q u
mf
=
mf
-
( 2)
A
where A is the c ros s - sectional area of the b ed .
Ligh t bodies will float o n fluidi zed particles a n d heavy one s sink . T he bed s u r fac e remain s level althou gh subject to wave motion an d splashin g , just a s a liquid . The whole can b e readily stirred and particles will flow through orifices in the side or bottom of the c ontainin g ve ssel . The above applies whether the fluid is a liquid or a gas , but the effec ts are more dramatic and initially s urpri sin g in the latte r c as e . T he e ssential c on dition of E q . ( 1 ) can be expres se d differen tly by con side rin g the bed as a whol e rather than in dividual p article s . The bed b e comes fl uidi zed when t h e upward flow o f fluid produces a pressure drop equivalen t to the bed wei ght . In a parallel - sided vessel , .
( 3) is t h e nece ssary con dition for fluidi zation . Generally , in gas - fluidized sy s tem s P s >> P f , a n d the latter c a n ususally b e n e glected . T he b ulk den sity of dry powder in a ve ssel of cons tant cros s - sec tion al area is given by
444
Rowe
a n d Y ates
( 4)
an d b ulk de n s ity is rel ated
so t hat
H( l
- E) =
to the in divi dua l particle density by
( 5} ( 6)
M Ap
s
from w hic h it i s seen that the p rod uc t H ( 1
- E)
in Eq .
( 3)
is constant .
E q u a ti on ( 6 ) also shows how b e d voidage an d its va riat ion is c alculated A difficulty arise s in m eas urin g the particle s are porous , as is com monly the case with materials such as catalysts ( K ni ght e t al . , 1 980 ; K night and Rowe , 1 9 80 ) . T h e density requi red for this p urpo se is the ratio of p article mass to the vol u me enclosin g all s olid material together with gas contain e d in all pore s and cavities ( i . e . , gas that is not part of the flowin g s t r e a m ) .
from t he measurement of bed height . particle density P s w hen
Up to the p oint of fluidi zation t he bed b ehave s a s a packed bed and
p re s sure drop can be c alc u l a t ed from well - known se mie mpiric al e quation s . For fine partic les w he re the loc al Reynolds number i s s m all , th e Carman K o zeny equation is suitab le an d can be written for minimum fluidi zation con dition s ( Ric hard son ,
u
mf
where
197 1 )
( 7)
tl p
=
5( 1
S is the p article specific surfac e .
Combinin g this wit h E q .
( 3)
yields
u
mf
=
( 8)
3
---25( 1 - Em f ) E
mf
For uniformly sized s p h eric al particles S
u
mf
=
2 o . o oosd < P p
1.1
s
-
Pf> g
=
6 / dp and Emf
=
0 . 4 , so
( 9)
I t i s useful f rom this to re m ember the way in w hich U mf varies wit h si ze , den sity , and vi sc o si ty , but it is not an accurate predictor for most pow ders of prac tical in terest . I t should be noted that when c on s i de rin g a fluidize d - b e d chemic al re ac to r , fl uid den sity and viscosity an d the particle d en s ity will gene rally be fixed by chemical and thermodynamic con sidera tions . U mf wi l l t h en b e d e te r m ined solely by part ic le si ze to whic h i t is quite sen sitive , at leas t with fin ely divided p a r t icl e s .
445
Fl uidize d -B e d R eac tors
A more ge n e ral pressure- drop equation a pp lic ab l e t o larger p a rtic l e s an d Reynolds n umbers is t hat due to E r gun ( 1 952) , w hi ch can be written for the presen t p urp o se
,
( 1 0)
The two te r m s on t he ri gh t - h an d side c o r re s p on d re spec tively , to t h e viscous or skin friction component an d the turbulent or form dra g con t ri b u t i on an d thus span a wide Reynolds number ran ge . The first term i s t h e same as Eq . ( 7) , al t h o u gh with a sli ghtly differen t constant . Sub stitutin g fro m Eq . ( 3) and re ar r an gi n g y i e l d s
=
150
D e fi n in g
[
(1
( 11)
( 1 2)
an d Galileo n umber ,
( 1 3)
we have Ga , 1 5 0
(�) '
mf
Re m
'
f
+ 1 . 75
(,L)
!
Re r
( 1 4)
Thus R e mf ( containin g the required U m f ) is quadratic in Ga , whic h is a For the
co m bination of the phy sical prop ertie s of p a r t ic l e s and fl ui d . partic ular case of uniform sp heres , Em f 0. 4 : Re
2 mf
+
or Re
mf
=
5 1 . 4 Re m f
-
=
0 . 0 36 6 Ga
25. 7[ ( 1 + 5. 5 3
x
10
-5
=
( 1 5)
0
Ga)
1/2
-
1]
( 16)
Equation ( 1 4 ) i s q uite a good an d reliable predictor , althou gh a little cl u m sy to use . M os t powders of p r a c t ic al intere st inclu de a ran ge of particle s i ze s in which case the s urface mean diameter should b e used . Nonsp herical particles can b e c haracteri zed by the diameter of a sp here wit h the s a m e s p e ci fic surface , but it m u s t b e kept in mind that particles
Rowe and Yates
446
diffe rin g gr ea t ly from sp heres in ove r all s hape ( e . g , p lates a n d needle s ) may n o t be readily fluidi zed . A p urely e mpirical correlation due to Leva ( 1 9 59 ) bec o mes , in S I units ,
( 1 7)
an d is vali d for Remf < 1 0 . A t hi gher values the result of E q . ( 1 7) i s m ultip lied b y the correction factor shown i n Fi g . 2 . Leva' s equation is simple to u s e and reasonably reliable but not as good a s E q . ( 1 4 ) . A little in si gh t come s from con side rin g the e xp eri m en t s of Rowe ( 1 96 1 ) . The d r a g force w as meas ured on an isolated spherical particle hel d in a
fluid s t rea m and the result ex p r e s s e d as drag coe ffi c i ent varyin g with
R eynol ds n umber . Similar meas urements were m ade un der identical condi tion s except that the observed s phe rical particle was c los e ly surrounded by s i mi la r particles also held stationary . T he results are s ummari zed in Fig . 3 , from which it is seen t hat at a given Reynolds n u mber drag co efficient is inc rease d by a fac tor of about 70 w h en an isola te d pa r ticl e is surroun de d by close nei gh b or s . If a particle is to be held stationary in an upward - flowin g stream , the con dition s of E q . ( 1 ) m u st apply w he t he r the p article is i solated or one wit hin a fl u idi ze d bed . The p roduct C o u 2 must be the sa m e for these t wo case s :
( 18)
ex: 0 I (.) <( u. z
0 i=
u w ex: cx:
8
(I)
< > w -'
02
��0------------------� , o�o�----------------,o�oo
F I G U RE 2 Leva ' s correction factor a s a function o f Reynolds nu m b e r at minimum fluidi zation .
447
Fluidized-Bed Reac t o rs
CLOSE PACKE D 10 u -
10
1 0-1
1o2 103 Re -
104
105
NEWTON, Co=0.44
D rag coefficient as a function of
FIGURE 3
Reynolds
n umber .
refer to the drag coefficien t s under isolated and mini where C n 1 an d C D mf mum fluidi zation ( close d p acked) con dition s , re spectively . The relation ship bet ween these two is revealed by Fig . 3 . A s i s further seen from Fig. 3 , ( 19)
an d =
( 20)
T h e form o f this functional relations hip is n o t easy t o exp re ss mathematical ly because of the varyin g contrib ution s o f skin friction and form drag as Reynolds number c han ges . T here are various empirical equation s to de scribe it , b u t they are not helpful for the present p urpose . T he two end con dition s can be treated analytically . S tokes' re gime lie s at Re < 0 . 2 when form dra g can be neglec ted and in this case =
24
( 2 1)
Re
I t follo ws from E q .
( 1 ) that
whic h is a well - kn own expression . 70
X
U
mf
From
Eqs . ( 1 8) to ( 2 0 ) , ( 2 3)
448
Rowe a n d Yates
and the refore , 0 . 0 0 0 8g ( p
0
mf
p )d f
s
( 2 4)
2
----�----p .._
=
w hich is similar to E q . ( 9) but with a sli ghtly different const ant . Newton ' s re gi me applies at R e > 1 0 3 when CD
( 25)
0 . 44
=
I
From E q . ( 1 ) it follow s that
( 26) and again from E q s . ( 1 8 ) to ( 2 0) that
UT
=
=
170
8 . 4U
u
( 27)
( 28)
mf
mf
Thus
( 2 9)
It is now seen that , excepti n g those expres sions that treat skin friction and from d r a g as additive , e q ua t i on s for p redic tin g minimum fluidi zation
velocity are of the form u
mf
=
kg( p ·
( 30)
s
and Table 1 compares the various equation s in this way . In partic ular it s hows that Leva ' s e mpirical e q ua tion is a c ompromise at a R ey n o l d s n u mber that is evidently near t he average of p rac tic al sy stems examined
TABLE 1 a
K
S tokes C arman -Ko zeny L ev a
0 . 0008 0 . 0006 ( 0 . 805)
Ergun N ew ton
0 . 066
b
c
d
1
2
0
-1
1
2
0
-1
!
�
0. 94
1
1. 82
t 2
-i 0
-!
- 0 . 88 0 0
I n c reasin g Re mf
j
Fl uidize d - B e d Reac tors
449
T ABLE 2 Re
m
T
0. 1
4 . 65
1
4. 35
10
3. 53
100
2 . 81
1 000
2 . 39
U /U T mf
=
53
71
e
-m mf
7 0 from E q . ( 23)
13
26
8. 9
8 . 4 from E q . ( 2 8 )
A part from t he in sight given by the approach above , i t shows t h e re lations hip between U mf and U T , whic h is o f c onsiderable p rac tical i m p ortanc e . I t is fu n dam e n t al to any c on sideration of a fluidized - b e d c he mic al r e a c t o r t hat it should b e o p era t ed at a fluid velocity greater than U mf b ut not in excess of U T if partic les are to be re t ain e d in the bed . P ractical pow ders a re gen e rally made up of a ran ge of pa rticle si ze s , so the r e is a correspon d The above is the r efore on ly a gu id e to in g range of U T ( but n o t o f U mf ) . an e s ti m ate of elutria t ion , but it is a useful o n e . I n terme diate valu es of the rat io s given by E qs . ( 23) and ( 2 8 ) can b e derived from t h e w ork of Ric hardson an d Zaki ( 1 9 5 4 ) . As is mentioned in the next section , liquid fluidi zed beds e x p an d unifor m ly as liquid ve locity inc reas e s , and the r el a t ion sh i p bet ween voidage and v e loc i t y can b e described b y u
where m is constant for a given system b u t varie s with Re T . case of E q . ( 3 1) i s
( 31) A p artic u l a r
an d this h a s b e e n evaluated in Table 2 for t h e c ase o f unifor m sp he re s , w he re E:m f = 0 . 4 an d u sin g values of m foun d by Richardson an d Zaki ( 1 9 54) . The end conditions a gree well with the previously described and quite in d e pen dent ex p er i m en tal meas u remen t . Pa r t i c ulate F l u i d i z a t i on
The s ubject of t h i s c hapter is gas fluidi ze d b e d s , b ut it is necessary b riefly to r e fer to the sp ecial case of li q u i d fluidi zation because u n der c er t ain c on dition s d ry beds can behave as if liquid fluidize d . Once t h ey have been flui di zed , particles generally b ehave differen tly depen din g on wh et h e r the fluid stream is a l i q ui d or a gas . Li q u i d flui di zed beds expand uniformly as velocity is increased . The particles se p arat e to accommodate mo re l i q uid flow between them while r e tain in g t he same drag force . As is im p lie d by Fig . 3 , the dra g coefficien t dimini shes
Rowe an d Yates
4 50
as ti
if
p article
on s hip
spacin g
in c r e a s e s b ut cnu 2 re m ain s constant . T h e sa m e rela i e ren tly by E q . ( 3 1 ) . B et ween E qs . ( 6) an d ( 3) ,
is exp ress e d d ff
fol l o w s
that
u log U T
=
m
log
[
1
-
(H�p }]
( 33)
s
fro m whic h it is seen how bed hei ght varies with velocity w hen plotted
a gain st l o ga rit hm ic coordinates . Uniform e x p an s ion of t h is kind
is generally referred to as particulate
uni for m particle mix fairly rapidly by a process akin to e d dy diffusion . The ( modifie d ) eddies in the liquid carry particles wit h them c ausin g both lateral and lon gi t u di n al disper sion . fl ui diza tion and for
c on cen t rat ion
most purposes it can
t hroughout .
be regarded a s of
The particles are not stationary b ut
Sol i d s C l a s s i fi c a tion T he ob servation th a t certain p o wde r s undergo c on siderable expan sion b e yond t h e poin t of minim u m fluidization an d only b e gin to bubble when an other c ritical gas velocity , U mb , is exceeded has led to the acceptance of t h e p rinciple t hat s uc h p o w ders should be grouped together and t heir fluidization p rope rties c on sidered sep arately from those materials t hat show more "normal" b ehavior and b e gin to b u bb le at U mf · Moveover , experience with a wide ran ge of particulate s ol i d s in gas fluidi zed beds has shown there to be other mode s of behavior that are different from t hose of the foregoing material s , and as a result it has been foun d con venient to divide all pow ders in to four group s acc o r d ing to c riteria of particle size an d den sity . Such a sc heme of clas sification was s u g gested or igin ally by Geldart ( 1 9 7 3) and it has now bec om e firmly established . I t clas sifie s p owde r s as belon g in g to G roup s A , B , C , or D as given in Table 3. Although in practice the division s between t he groups are somewhat diffuse , makin g it diffic ult in some cases to a s si gn a p o w der firmly to o ne gro up or another , the clas sific ation does nevertheless p rovide a broad guide to the type of behavior that may be expected from a given p o w de r . I t is useful to repre sen t the fou r !?'roups of materials on a " p hase diagram" s ho w in g the densi ty differ ence ( P s - P g ) p l o t ted a ga in st particle diam e te r d p ( F i g . 4 ) . Here the b o u n darie s between groups C an d A and gr o u p s B and D are entirely empirical . The A /B boun dary , however , may be defined on the basis of w he the r or not a po w d er begins to form bubbles at it s minimum fluid zation
For group A m aterials , U m f / U m b � 1 , w hile for group B , m b > 1 ; a convenient dividing line w ould thus be one for which U m f / U m b 1. and G eldart ( 19 7 3 ) correlated U m f and U mb with p ar ic le
U m f/
velocity . U
B aeyens gas p ro p ertie s
t
and
n
used powder s and showed that y is given approximately by
1 05
( 34)
for a wide range of com monly
for air at ambient con d i tio s the A /B b ou ndar the
line
d (p p
s
where (\,
- p ) g
=
2
X
( JJ m ) is the su rfac e mean diameter of the p ar tic l es . ( 3 4 ) i s p l o t t ed as lin e XY on Fig . 4 .
n
E q u atio
=
:::! 5.
�
(II Q. I
i::j •
(II �
b:J
TAB LE 3
Ran ge o f particle size Group
A
(II l:l (')
!:l;j
Solids C lassification for Fluidi zed B ed s
( 1-I m )
30- 2 0 0
Range of particle den sity ( k g fm 3 )
5000- 8 0 0
0
B ehavior when fluidized "Abnor m al" bed exp ansion between
Examples
"""l "'
C rackin g c ataly st
U mf and Um b followed by bed collaps e U mb > U mf , Ub > U j ,
E1 > E mf
B
50- 1 0 0 0
4 00 0 - 1 0 0 0
c
< 30
Any value
C ohesive pow ders , diffic ult to fluidi ze , prone to c hannelin g
D
>400
) 10 0 0
Prone to slu g gin g , bubbles
Source :
B aeyen s and G eldart ( 1973) .
Normal b ehavior U mb = U mf • E ;; E mf i
ill - defin e d , u b <
ui
S and , glas s ballotini Talc , flour , cement
S teel shot , dried
peas
�
of:>.
4 52
Rowe an d Y a tes
f: 5000 0.2ooo q. "'
� 1000 u c
�
.'!:: lJ
500
F I G U RE 4
conditions .
Powd er clas sification diagram for fluidi zation with air at am bient ( B ased on B aeyen s and G eldart , 1 9 7 3 . )
G a s F l u i d i z a t i on a n d t h e T w o - P ha s e
T heo ry
When the fluid velocity is increase d above U m f , gas fluidized · beds begin to bubbl e and look remarkably like b oilin g opaque liquids . T hey are sometimes called boiling b e ds , w hic h is certainly de scrip tive . In contrast to " particu late , " this h a s been referred to a s aggre ga ti v e fluidi zation , b u t that is not a very de sc riptive name and b o i l i n g or b u b b l i n g is to be preferred . T he two - p hase theory states essentially that the gas flow divides into that which flows in terstitially amon g the closely packed particles an d that whic h takes part in bubble flow . That is , ( 3 5)
As gen erally expres se d , the two-phase theory further assumes that Q i :::: T his Q mf , which here w ill be referred to as t h e simple two- phase theory . can convenien tly be written ( 36)
whic h gives an im mediate estimate of the bubble flow for any operatin g velocity . The two- p hase theory is fun damental to most models of fluidi zed-bed chemical rea c tors . Gas in each of the t wo phases has a different residence time an d pattern of contactin g with the p a rticles . In realistic situation s there is generally some exchange between the phases . T wo obvious correc tion s are nec e ssary to Eq . ( 3 6 ) , although they are often small enough to be n e p,-lected . Fir st , as the b e d fills with bubbles the overall average height inc reases b u t t he c ross- sec tional area available for interstitial flow is reduced . Sec on d , there is necessarily a pressure
4 53
F lu i dized - B ed R eac tors
d r op ac ros s t he b e d and therefore gas must e xp a n d as it rise s . Volumetric flow an d therefore gas ve locity c on se q u e n t ly inc re a se s wi t h hei gh t As T a b l e 2 s h ow s U m f is largely independent of gas density ( or v ar ie s in versely as its square root for large p artic l es ) , so that t h e excess flow avai l able for bubbles ( U - U m f ) increase s wit h height . I t can be s ho wn ( Rowe , 1 9 7 8b ) th a t these t w o c orrection s l e a d t o .
,
( 37 ) where the s uffice s D and H refer , re s p ec t ivel y , to the distributor p l a n e an d a plane h eig h t H ab o v e it . T he exp ans ion correc tion is u s u ally neces sary only for deep beds o f den se material o r when op e ra t in g at s ub a t mo s p heric p re s s u r e . Even in a vi go ro u s ly fluidi zed bed , fB is rarely gr ea t e r than about 3 0 % and , e s p e cia l ly when Un » U m f , this correc tion also is com monly very s m a l l T he s i m p l e t w o p h a s e theory is ess e n t ial ly correct for many p o w de r s partic ularly fo r those with mean particle size greater t h an about 1 0 0 �m . F r om con side ration of t he minimum fluidi zation condition s as i m pli e d for example , in Eq . ( 1 1 ) , it foll o ws that the interstitial ga s velocity must remain I t is di ffic ult to at U m f if the voidage of the dense p h a se remain s at E mf . i m a gi n e th at p a r t ic l es may become closer sp aced afte r fluidi zation , b ut they may s eparat e mor e widely , thu s c au sin g t he den se phase to be more per meabl e . It i s thus to be expec ted an d there are in dee d cases when in t e r s ti t ial flow i s gre at er t h a n U m f and the two - p hase t heory n e e d s modification . T h ree s i t u a t i on s have been o b s e r ved experimen tally where U i > U mf ·
.
-
,
,
1.
2. 3.
B ed s of fine p artic l es
o f group A ( Geldart 1 97 3) ( typically the pa r ticl e si ze is l e s s than 1 0 0 ].l m ) . When o p era tin g at h i gh gas press ure I n the b o t t o m seve ral centimeters of t h e bed , w here it appears to take m e as u r abl e time for the flo w s to come to e q ui li b ri u m ,
These special c a s e s will be di s c u s s e d in the foll o w in g s ec tion s . P roperti e s of B ub b l es
In many w a y s the b u bb l e s in ga s fl uidi zed beds have s i mil a r p rop er t ie s to b ub b l e s in a true l i q ui d T hey have a ro u ghly similar s ha p e they rise at about the same v e lo cit y di s t urb t he s u r ro undin g me dium in much the same way , an d coalesc e and b reak up . However , they differ in several p oin t s o f .
,
,
detail and t he analogy must be drawn with c are . B ubbles are fundamental to the behavior of gas fl uidi ze d - be d r ea c to rs so it is i m p or t ant p rop e rl y to ,
unders tand their p ro p e r t ie s as muc h is now known ab o u t them from re search experimen ts . In any re a li s t ic reactor there will be many b u b b l e s , but it i s h el pf ul fir st to exa mine the p ro p e rtie s of sin gle b u b b le s in i solation . B ecause the dens e p h a s e of c lo s e ly sp ac e d p o w d e r pa r ti c le s is opaque , b ubbles cannot n o r m a lly be s een wit hin the bed and s p ec ia l experimen tal techniques are re q uir e d to re ve al them . One such t echn i q u e i s the so - c alle d two- dime n s i o n a l b e d , w h e r e t h e po wder is c onfi ned bet ween gla ss plates p e r h aps 1 em apart . This effectively e x p o s e s a t hin slice o f t h e bed an d the b ubbles
Rowe a n d Y a t es
4 54
w '
.._ _ _..
FI G U RE 5
/
U n di sturbed gas b ubble in a fluidi zed bed .
can be seen by suitable li ghtin g and recorded photograp hically ( Rowe , B ubble s hape , rise velocity , an d some other propertie s are change d 1971) . a little by this c on st rain t wit hin t w o dimen sion s ; neverthele s s , t h e tech nique reveals many es sen tial features an d gives m uc h insigh t . Another tec hnique with differen t limitation s is x - ray cin e photo graphy . Un disturbed bubbles are spherical with an in den ted base , as shown in Fi g . 5 . T he boun dary i s quite definite an d t he interior is empty of parti cles . They rise by p articles flowin g a roun d t hem in streamlines rat her as a bluff body move s t hrough a true fluid at low Reynol ds number . B ubble rise velocity can be desc rib e d by t he Davies - T aylor expression ( D avies an d T aylor , 1 9 5 0 ) , w hic h can be w ritten
( 3 8)
The standard de viation of a sin gle observation is very high and even un der ideal experimental con dition s p rediction of velocity for a partic ular b ubb l e is subject to much error . Even with large numbers of ob servation s it is not possible to confirm that E q . ( 3 8 ) is the c orrec t form and it is chosen simply on the b asis of analogy with bub b les in liqui d s . From many data it is clear that the con s tant of proportionality in E q . ( 3 8 ) is unity within the limit s of experimental error . For the purpose of reac tion en gi neerin g it is behavior of the average b ubble that is of in terest , and for this , Eq . ( 3 8 ) is ade quate an d as good as any o ther From this eq uation it is seen tha t bubbles rise at a few ten s of e m Is and the prac tic al ran ge of v eloc i ti es is les s than one order of magnitude . T he in den tation at the bottom of a b ubble is a wake of particles travel in g with it and roughly circul atin g as in dicated in Fig . 6 . I t i s very like the attached vortic e s behin d a bluff body movin g through a liquid at The wake grows as further p a rticles are capture d un til it 10 < Re < 1 0 0 . b ecomes un stably l arge w he n a l a r ge portion is she d - a gain c losely ana l o gous to b ehavior in a true liquid . Pursuin g this analo gy s u g gests that the fluidized p articles have a viscosity of 1 0- 2 N • s /m 2 , b ut it c an be mis lea din g to attrib ute a true N ewtonian viscosity to t he fluidized system . A t hi gh shear rat es particles will c o m e in c on t a c t with each other , a n d apparent vi s c os i ty will c han ge ab ruptly to ela sticity . T he system has been treated as a rheolo �ic al fluid ( Rietema , 1 96 7 ) , but such an ap p roac h has not a d vance d o u r un de rstan din g v e ry far . The c on tin uous growth and disc ontin uous s h e d d in g of wakes accoun t s I t also accounts for the fac t that in part fo r t h e v a ria b ility of velocity . in s tantan eou s pictures of b ubbles s how a variety of wake si ze s . T he .
,
·
4 55
Fl ui dized-Bed Reac tors
FI G UR E 6
P a r ticl e c irculation in bubble wake .
average wake volume is ab out one - third of the b ubble v ol um e or one - fourth of the whole sp h e re volu me .
v
w
=
=
V
B
( 3 9)
3 Ti d
3
B
( 40)
24
A dditionally , the wake fraction inc reases with the ab solute size of the b ub ble so that bi g ge r bubbles appear flatter , but this is a fairly small e ffec t an d fo r m o st p r ac t ic al p urposes E q . ( 39 ) is a de q ua t e B oth wake frac tion an d vel oci ty appear to be i n depen d e n t of particl e p roperti e s . .
G a s Flow W i t h i n t h e B ed
Gas flow wit hin the interstitial p h ase is e sse n tially streamlin e because the suppress turbulenc e and prevent the d e v e l o p m e n t T his is at first surprisin g of ed di e s any lar ge r than interstitial spac e . b ec au se t he im m e dia t e im p r e ssion on seein g a bubblin g fluidi zed bed is that it i s well m i xe d As far as gas is c on c e rn e d this is quite w r on g and with n on porou s particles there is very li t t le gas mixin g , a n d reactors that feed r eac t an t ga ses sep a r a tely usually p e rfo r m very b a dly b ecause of this . A limited de gree of gas m ixin g occu r s w ith porous particle s simply ·because they ab sorb gas , move , and release it elsewhere . I t has been mentioned above that gas v e loci t y in the interstitial p h a s e U i , remain s at minimum fluidization value U m f excep t in some special circum stances w h en it may be exceeded to a limited d e gre e R ey n ol d s n u mber base d on p a r ticle diam e t e r an d in te rstiti al ve loc i ty i s usually quite small . For example , for particles of dia m e t er , dp = 1 0 0 ]..l m fl u idi ze d by gas at This s upports the ob s e rvat ion that gas mixin g NTP Rei is of order 1 0- 1 . is poor an d also in dic ate s that the heat an d mas s t ran s fer c oe ffic i en t b e t w e en p a rt ic l es an d ga s will be near their lower lim i tin g values . c los e l y space d p a rtic l e s
.
,
,
.
Rowe and Y a tes
4 56
As i s fairly obvious an d is explain e d fully b elow , particles move in the bed as bubbles pass amon g them . T he particles move as a w hole in a fairly orde rly w ay and in so doin g move gas with t hem while the relative velocity or slip remain s at U i . T h'.ls ga s may follow a tortuous path from bottom to top but wit hout mixin g . There i s a p res sure drop from bottom to top o f a fluidized bed a s given by Eq . ( 3 ) , and c on sequen tly the p res sure just below a bubble is greater than that just above i t . This pressure difference is the hy drostatic pres sure chan ge over t he b ubble height , just as for a b ub ble in a liqui d . T here is no material r esistance to gas flow through the empty space of the bubble where the p res s u re is c on stant an d therefore ga s must flow upward through it . W hen b ubbles rise slowly relative to in terstitial gas , they of fer an easy fl ow route an d gas conver ge s on these regions of hi gh permeability and by passes some dense phase as it flow s through the bed . Figu re 7 illustrates qualitatively the typ e of path ga s may follow in these circ um stances . T he streamlin es are distorted by the b ubbles in much the same way that magnetic lin e s of force are distorted by soft - iron bodie s in the magnetic field an d for much t he same reason . For the p urpose of reaction en gineerin g it is neces sary to know the residence time of elements of gas in the den se in terstitial p hase an d in t he interior of bubb les . W hen bubbles rise quickly relative to in terstitial ga s , a quite different situation deve lop s . Gas must still flow upward t hrough the emp ty space because o f the pressure diffe rence , but as it flow s out through t he b ubble roof , it enters a re gion of fa s t - movin g particles flowin g round the bubble as in Fi g . 6. In consequence , gas is sw ep t around t he top of the bub ble , down the sides , and near the low er half it is p ushed b ack into the empty sp ace by exce s s p ressure in the dense phas e . Hence gas i s trap ped and form s a s p h e rical vortex or cloud around the bubble as it rises , as illus T he overall result under these circumst ances i s t hat each trated in Fi g . 8 . T he gas b ub ble becomes surrounded by a cloud , as indicated in Fi g . 9 . has how di vi ded i n t o two phases , that flowin g interstitially and that per ment aly as sociated with the bubbles . Con si d e ring the ga s enclosed by the outer bound ary show n in Fi g . 9 , only that p art i n the annular region bet ween outer and bubble boundaries
FI G U RE 7
Gas flow through slow - movin g b ubbles .
Fl ui dized-Bed Reac tors
457
•
:- FLOWING
• • •
PRESSURE DIFFERENCE CAU SES UPWARD FLOW TO COM PLETE THE CIRCU LATIO N PAT H O F T RAPPED GAS
• • • • • • •
•
•
PART ICLES
/ DRAG GAS WITH
•
•
THEM
- AND EXCESS PRESSURE RETURNS IT TO THE BU BBLE
"
FI G URE 8
Fo r m a tion of gas cloud aroun d fas t - movin g b ubble .
is actually i n contact with particles . Only this fraction of ga s can react at any in stant , althou gh sin c e the whole is circulatin g , each element will have a limited p e r iod of reaction time . Fu rthermore , since a necessary c on di t ion for thi s hydrody n a m ic sit uation is that UB > U i , ga s formin g the cloud h as a shorter residence time than in terstit i al gas . Plainly these li mit a ti on s have im p o r t an t c on sequences in reaction engineerin g and the mechani sm de scribed ab ov e form s a basis for reactor modelin g . I t i s evident t hat the two v eloc i ti e s , UB and U i , are impor t an t para m They are eters i n determinin g t h e n ature of gas flow through t h e bed . U B de p en d s only on b u b b l e size , as shown by i n d e p end ent of each other . Eq . ( 38) . Ui is p ro portional to U m f , w hich in turn depends on the gas and particle p rop e rti e s as gi ven by Eq . ( 9) , ( 1 4 ) , or ( 1 7) For small values of U mf , w hen cloud s are lik ely to for m , in ter stitial ga s velocity de pen ds primarily on particle si ze , rou ghly as T h e important ratio U /U B i It s hould be noted that conventionally , U is then proportional to mf is expressed as a superficial velocity •
d�1 2 !d�.
F I G U RE 9
B ubble with c a p t i ve cloud .
458
Rowe an d Ya tes
Q
u
mf
mf
-
:::
( 4 1)
A
and therefore when the simple two-phase theory applie s ,
( 4 2)
C louds form aroun d the b ubbles whenever U B > Ui ' that is , when the ratio :::
CL
( 43)
> 1
A simple b u t elegant theory ( D avidson a n d Harrison , 1 9 6 3 ) treats t h e b ub bles a s perfect circles aroun d which c oncentric clo u d s form the diameter , b ein g given by d c
(�) 1
:::
et
1/ 2
( 44)
-
for a two- dimen sional or di sklike bubble ( the only c a s e ob servable experi mentally ) or d c
�
( �)1/ CL
=
-
1
3
( 4 5)
for thre e - dimen sional or spherical bubbles . T hese equation s are shown graphically in Fi g . 1 0 . From t his i t i s seen that when a < 1 , the cloud is of in fin it e extent , or in other words , is open to through - flow .
16 15 lXI
u -a
.:!:!
1 4
13 12 11 10
10
2
3
4
5
""'-
6
7
8
9
10
FI G U RE 1 0 Ratio of cloud to b ubble diameter as a function of the dimen sion les s parameter a .
Fl uidized-Bed R eac tors
4 59
E quation ( 4 5 ) gi v e s an estimate that i s u s e ful in cal c u l atin g c on tac ti n g effic ie n cy in reactor modelin g , but a b et t e r e stimate is gi ven by the semi
em p iric al e q uati on ( Partri d ge a n d Rowe,
a
1966)
.
1 17 -
( 46)
1
�
I n ga s fluidi ze d b e d s of particles where dp 100 � m , values o f the velocity ratio a re c ommonly in the ra ge 1 0 1 00 . From Eq . ( 46 ) it i s t hen seen that the cloud volume i s b etween 1 3 and 1% greater than b u b bl e volume . That is , on ly bet ween 13 an d 1 % of gas in the b ubble or clou d p ha s e is in c on tac t with solid p a rtic les at any in sta n t . This is very poor c on t ac t in g an d explains the performance of reactors that operate un de r suc h ill c ho se n c on dit ion s . B ubbles u sually form in i ti ally at the distributor plane an d un l ess there i s c h an ge in the v ol u m e t ri c gas flow with hei gh t , the b ubble p op ula ti on T here is , however, c h an ges therea fter only by coa l e sc en c e an d sp li t t in g an initial or fo r mati on zone in w hich thi s is not t r ue
< a<
-
.
.
D i s t ri buto r a n d B u b b l e Fo rma tion
I n an in d u st ria l reactor the di st ri bu t o r plate us ually c on t ain s a n umber of regularly spaced hol e s which may or may not be c a p p e d in some way t o prevent s ol i d s fallin g t h r ou g h on shut down an d to h el p di strib ute the flow uniformly . Gas velocity th rou gh in dividual holes can be c on sid e ra b le ( typically ten s of m /sec ) , y e t under all re ali st ic c on dition s gas enters t he bed from the m in the fo rm of b ubbles ( Rowe et al . , 1 9 7 9 ) . The ga s momen tum is e ffec tiv ely zero an d e v en with hori zontal entry , b ubbles ri se ver t ic al l y from the source . The mode of en try is very similar to the way b ub bles fo r m w hen gas i s blown into a li q uid . I n one i m p or t an t respect gas enterin g a fluidized bed differs from that en t e rin g a liquid . Wi th increasin g flow rate the freq uency r e main s ap pro x im at e ly c on st an t a t ei gh t per sec on d and t he si ze o f b ubble a t detac hment increase s . I nto a l i qui d t he r e v e r s e i s t ru e at least at mode st flow ra t es T his is an i m p ortant an d u seful fact , as it enables in it i al b ubble si ze to be calculated . I f it i s a s sumed that the total gas flow is equally divide d bet ween the ho l e s the bubble v ol u me at d e t a c h m en t follow s from the ob served fr e qu e ncy However , t hi s is subject to t he correction mentioned .
,
.
below .
From x - ray observation it is seen that the size of bubble at detachment mu l t i p li e d b y observe d f r e quen c y acc oun ts for only about on e h al f of the known flo w t h rou gh the di st ri b utor orifice . T hat i s the bubbles are only about half the e x p ec t e d volum e . T hese n ewly formed bubbles continue to grow a ft e r detachment an d by the time their c en t e r s are about two di a m e ters above the hole , t h ey are of a size a n d fr equ e ncy to account for all the flow as depic ted in F i g 1 1 . I t i s now s e e n that the initial b ubble size -
,
.
i s given by
� ( 0)
( 47)
460
F I G U RE 1 1
Rowe an d Yates
B ubble for mation a t an orifice .
3 where Q o is the flow t h rou gh a sin gle orifice in m / s . At the moment of detachmen t the bubble is sphe rical but quic kly develops a wake , as shown in Fig . 1 1 . I f for a p ropo sed de sign of distributor t he calc ulation of Eq . ( 4 7) gives an impossibly small diameter ( e . g . , less than the hole si ze or markedly less than 1 0 0 particle diam eters ) . it is evident that the de sign i s un satisfactory and not all holes will operate . Similarly , if the bubble si ze an d hole spac in g is such that gross overlap occurs , the holes will not operate in depend ently . These con sideration s , together with the well - known empiric al rule that p ressure drop across the plate s houl d be about on e - t hird of that across the b e d to en sure stability , are es sential guide s to successful dis trib utor de si gn . The facts de sc ribed above can be explain ed in a semiqualitative way that is importan t in modelin g chemical reactor b ehavior in the shallow dis tributor zone . Unlike a b ubble formin g in a liqui d , one in a fluidi zed bed can grow only by gas flowin g perpen dicularly across the bubble boundary . This is the only way in whic h a force c an be applie d to in dividual particles to cause t he m to move an d so exten d the boun dary . A fter time t t he still attac hed b ub ble may have grown to diameter dB , as shown in Fi gu re 1 2 ,
' /
Fl G URE 1 2
/
Re gion o f gas penetration durin g b ubble formation .
4 61
Fl ui dize d -B e d Reac tors
but d urin g this time the gas has advanced as far as the dash e d line in d icated . The volume w it hin the outer b oun dary is Qt plus the volume occupied by particles in the eccentric annular reg1on . This annular region now con tain s excess gas , and therefore the local powder voi dage , c , must inc rease . Such in crease is un stable and e vi de n t ly when the bubble detac h e s and b e gin s to rise t hrough the un stable ex pan ded region , it relaxes and the excess gas pours back in to the bubble , almost certainly throu gh its base . T he experimental evidence sho w s that this is completed by t he t ime the b ubble has travele d little more than one Since ei ght b ubbles form each second , it is a very rapid se diameter . quence of e v en t s and hy drodynamic equilibri um is achieved in a time o f order 1 0- 1 s . R eferrin g a gain to Fig . 1 1 , the upper b ubble i s now stable an d will have formed a clo u d aroun d it self if the velocity ratio is appropriate . Half the gas in this cloud and b ubble first passe d into the dense p hase and then r e t urned to form a stable bubble . This brief b ut efficient c on tact between ga s an d partic les can lead to good reactor p erformance in this shallow layer , and a model for this is developed later in this chap te r . B u b b l e Coa l e scence a n d Av e ra g e Si z e
B ubbles c o ale sce rea dily fo r althou gh t h e fluidi zed powder has a property equivalent to s urface tension , it is small . T he rate of coal�scence depends
essentially on the closen e s s or c oncentration of bubbles . From this it is pos s ible to postulate the way in wh ic h b u b b l e concentration will c han g e with hei ght , and c ombinin g this wit h the as sumption t hat the bubble flow re main s c on stan t , b ubble size can be e stimated . The e mpirical constant is fitted from t he da ta , le a din g to ( Rowe , 1 9 7 6 ) + h ) 0
=
g
1/4
3/4 ( 48)
h o i n E q . ( 4 8 ) arise s a s a con stant o f in te gration an d i t s p hysical mean in g is clear by reference to Fi g . 1 3 . I t is that imaginary distance beneath the distrib utor where the bubbles woul d b e of zero si ze . It is related to initial bubble diameter by
(U
-
U
mf
)
2/3
( 4 9)
and so can b e estimated from E q . ( 4 7 ) . In p ractice , ho is us ually only a few c entimeters and can often be neglected w hen estimatin g b ubble size in beds of rea s on a ble depth . For b ubbles to achieve an average size at a given height an d flow rate , it is obvi ous that the conta inin g vessel m ust be big enou gh . I f not , the bed will slu g and E q . ( 4 8) i s u seful in in dicatin g when th i s con dition will begin , an important point in reac tor modelin g ( Rowe , 1 9 7 8a ) . Another expres sion for bubble si z e that is based on slightly different reasonin g is ( D arton et al . , 1 9 7 7 )
Rowe and Ya tes
4 62
i
!D u a:: w 1w <(
:L
0 w ....J CD CD ::::> CD z <(
\
3 de = ( U - Umf )1'2 ( h • ho) /4
w
g 1J4
:L
,- - - - , d e
FI G U R E
d
e
13
=
BED
H E I GHT
h
---+
Variation of b ubble diameter with b e d height .
g
0. 2
u
mf
0 ) .4
( 5 0)
113 where de is t he e q ui val e n t s p here diamete r [ d e ( 3 /4) dB ] a n d A o is the c a tc hm ent area fo r a bubble s tr ea m at the distrib utor plate . The two e qu a t ion s ( 48) and ( 5 0) are very si m ila r n um e rica l ly and , within the li m i ts ==
of ob s e rv ation al e rror , desc ribe t he data e q u a l ly well . B ubbles also split by ins tabilities that de ve l op as " fin gers" from the roof whic h m ay g-row rapidly enough to divide t he bubble usually into two I t i s q ui t e common for the t w o parts s ub s e q uen tly to unequal p arts . On average , s m al l d a u ght e r b ub bles p e r si s t as in dividuals for coa l e s c e . some t im e , so t hat the si ze d i s t ri b u t ion b ec omes positive s k ew ed with an e x c e s s of s m all size s . A w e ll - de s i gn e d distrib utor wit h uniform holes p ro d u c e s initially a more o r les s mono- s i zed distrib ution , b u t b y c oal e s c e nc e and s p lit t i n g this develop s toward an exponential di stribution with hei ght w hi c h is e q uiv a le n t to time .
Within a re gion where there are m any b ub bles an d fr e q uen t coalescence , average b ubble v e loci t y is inc reased by the ac c el e ratio n s accompanyin g coalescence . It has already b een poi n t e d out that even with i so la ted an d undist urb ed b ubbles , there is a high error variance in e stimatin g ve l ocity from bubble si ze . This variance is increased w hen many b ubbles are p re s en t , so that from t he data it is not c l e a r t hat there is any c or relation at all . I t i s , h o w e ve r , apparent that velocity increas e s with gas flow rate , an d a c om m on ly used ex p ress i on t hat t a k e s thi s into ac c o u n t is
( 51) whic h i s probably sati s fac t o ry for most reac tion en gin e e rin g purpose s .
463
F l ui d iz e d -B e d Reac to rs
E qu a t ion bubble shape phase t heo ry an d this can
( 4 8 ) c a n be exp re ssed a s av er age bubble volume usin g t he and p roportions gi ve n by E q . ( 3 9) an d ( 4 0 ) . If the two a p p lie s as given by E q . ( 3 6 ) , the volumetric flow is c on s t an t be written
( 52 )
where n is the fr e q u ency with whic h b ubbles cros s a hori zontal plane thr o u gh t he b e d . C o m b in in g these eq uation s gives an e xp r e s sion for this frequency ,
A n
=
+
1T ( U -
h )
9/4
( 53 )
0
B ubble concentration is related to frequency by =
N
=
n
( 54 )
8 3/4
( 5 5)
from whic h it is seen that bubble concen tration falls rapi d ly wit h increas in g height . A l ar ge industrial - scale reactor ope rated at a fairly high gas velocity may bubble furiously near the b ottom b u t appear almost quiescent a t the top . · Under these c on dition s bubbles can be t r eat e d as isolated in m u c h of the upper part of the reactor . The question is commonly a ske d : Is t he re a ma xi m u m bubble si ze ? B ut it is an acade mic q u e s ti on without much p rac tical importance . E q ua tion ( 4 8) i mp lie s growth without li mit and , w hile it is ba s e d on coal e s c ence fre quen c y observe d on a fairly small scale , on an industrial scale it does fit the few data there a r e . C ertainly , b ubbl e s of diam e t er in the re gion of 1 m have been reported from m o r e than on e sourc e . Rate of c o al es c e n c e falls o ff ra pi d ly with c on ce ntra t ion s o that rate of growth declines with h ei gh t , and it re q ui re s an i m pos si b ly la r ge and deep bed to ob se rve bub bl es much more than 1 m in diameter . It is inconceivable that close bubbles shoul d fail to coalesce just beca use the dis t rib uto r plane is far below . C e r t ain ly , b i g b ubbles ten d to break up , but th en th e fra gmen ts all too com monly recoalesce . Evidence and lo gic suggest there is no u p p e r limit , an d the po s sibi lity of ex p e ri men tal p roof seems r e mot e . On the co n tr a ry , there is surely a lower size limit . A bubble of p a rt i cle si ze is me anin gless . T he b ubble has reality only when p articles can flow around is as a p s e u d ofluid . T his su ggests a lo w er limit of or de r 1 0 2 x dp . There i s no experimen tal evidence for b ubbles less than ab ou t 1 e m in di a m ete r , b ut observation would be diffic ult . Pa rticle M i x i ng a n d Seg regation
T he transport of a wake an d streamline motion of p a rtic l e s cause s a la rgel y predic table displace ment of m a te ri al . This can be demonstrated by p re parin g
464
R o w e an d Y a tes
a bed init ial ly with , s ay , black p article s formin g the lower half and white one s above with a hori zontal interface bet ween ( Rowe et al . , 196 5b ) . Fi g ure 14 in dicates the kind of displacement that occurs as first one and t hen s uccessive bubbles di sturb the initial arran ge ment . E ac h b ubble carries wake material to the su rfac e , s p i llin g some enroute w hile streamlin e motion draws up a spou t or drift profil e . Repe t i tion complicates the pat tern an d the whole quickly b ecome s well mixed . If bubbles tend to rise in the center , the overall motion is a convection p attern , up t he center and down the side s . T hi s reverses if b ubbles rise near the walls and more complic a ted pattern s occur in la r ge b eds . I t i s i mpor t ant to reali ze from these demon s tr ati ons that pa r ticle mi xin g i s c aused solely by displacement by the bubbles . Wit hout b ub bles t here can b e no mixin g T here i s no mec hani s m e q uival en t to molec ular diffusion wh e reb y n eighbo rin g p a rtic les may jostle and exchan ge positions - excep t in the wake . I n spite of appearances to the contrary , particle move ment is rela t ively orderly and predicable to a limited de gree . I t follo w s from t hi s that t he rate of p ar t icle mixin g depends on the U mf ) . b u b b l in g ra t e It is directly related to the exces s gas flow rate ( U as seen from Eq . ( 3 5) . In most realistic circum stanc e s , the b ubblin g rate i s s uch that mixin g occ urs rap i d ly and in t he m a j orit y o f cases it is appro p riate to treat the particles a s perfec tly mi x ed The circ ulation rate can be e s timated roughly ( Rowe , 1 9 7 3 ) from .
.
-
,
.
t
( 56 )
c
where t c is a measure o f t he time taken for a p article to make a ci rc ui t of the bed . I f in a chemical reactor where the particles are chan gin g composi tion t he rate of chemical change i s small compared wit h c i rc ul a tion rate , the solids sy stem c an justifiably b e treated as perfectly mixed , an d the above is a simple test to check the correctness of the assump tion . The over si mp li fie d nature of Eq . ( 56 ) s ti ll leaves it quite suitable for t his p u rpose . Uneven ga s distribution quickly l eads to c onvec tion patterns of solid s motion , and the type of pa t te rn is not diffic ult to im a gi n e with the fore goin g mechanism in min d . S ometimes this is done deliberat e ly to in duc e a
a) In itia lly FI G U RE 14
b)
After 1 b u b b l e
c)
A f t e r seve ra l b u bb l es
Particle mixin g cause d by risin g b ubble s .
465
Fl uidized-B e d Reac tors
A s it is not easy to en sure uniform di stri b ut ion ov er desired c irc ul a tion . a la r ge area , it m ay occur involuntarily an d un wan ted in in dustrial reactors . B ulk m ove me n t of particles in some kin d of c ir c u la ti on pattern will af Equation s s uc h a s fec t bubble ve l oc i ty an d p r oce s ses such as coalescence . ( 4 8) and ( 51 ) m u st be used wit h ca u tio n i n reactors w here a p p r ecia b le con vection is known or s u spe ct e d to occur . I f the po wd e r to b e fluidized c ont ain s grain s of different typ e s , segre gation may occur . S hape difference s s eem to be unimportant and the only relevant differences between particles are size an d den sity . Of these , den sity is m uc h t he mor e im p or t an t . I n con trast with m ec h anic al powder mixers , fluidi zed beds w ill to l e r a t e a very wide ran g e in particle size wit h o u t appreciable se gr e gat ion . Minimum fluidi zation velocity varies with mean particle size , so that if the si ze composition of a m ix t u r e changes , so d o e s U m f · If fines a re elutriated or if agglomeration oc c u r s , U m f in c r e as es a n d a t a constant gas flow rate the vigor of fl ui di za tion d ec rease s . This may even tually cause se gre gation becau s e , a s explain e d above , t h e rate of mixi n g depen ds on the b ubblin g rate , w hich declin es with in c r e asin g U mf ac cor d in g to E q . ( 3 6) . A gain in contrast with m echanical mixers , p articles of di ffe ren t den sity can readily s e gre gate in a fluidi zed b e d . While there is usually a spectrum of particle si z es in p o w de r s pf indus trial in terest , there are rarely more t han two or three different den sities and most commonly only two . The m ajority of research s t u di e s are c on fin ed t o binary sy stem s . I f two po w de r s made up of p a rticl e s differin g only in den sity are well mixed and then gently fluidi zed , the dense component settles to t he bottom as je t s a m . I t form s a p u r e layer , b ut a small portion remain s uniformly mixed in the less d e ns e flotsam above it . With increasin g gas velocity ( b ubblin g rate ) , the p roportion of jetsam di s p e r s e d in the flotsam in c r e a s e s , an d e ven t u al ly it may b e possible to achieve an overall u ni for m mix . T his is ill u s trat e d graphically in Fig . 1 5 . The r at i o M x / x a s defin e d b y Fi g . 1 5 i s a s i m p l e mixin g in dex , zero for c omplete se gregation an d u n i t y for p e rf ect mixin g . It is a fun c t ion of v e loc ity , of den si ty ratio , an d to a s m all exten t , of si ze rat io . A simple empirical relation s hip is ( Rowe an d N ienow , 1 97 6 ) =
I INC REAS I NG J < U - U m f. l
TOP
i
BED H E I GHT h
BOTTOM +--+-- · 0 x
I
1 0 X FRACTION OF JETSAM --
FI G U RE 1 5
( U - Umt ) - SMALL
X
( U -Umt) - LARGE ( U - Um.�J - 0 I PERFECT SEGREGAT I O N I PARTIAL SEGREGATION I PERFECT MIXING 1
S e gregation pattern s for binary mixtures of solids .
4 66
=
M
f(U)
( :; )
( )
5/2
d
d
1
Rowe a n d Ya tes
1/2
( 5 7)
2
w here ( d 1 /d 2 ) > 1 . T hi s shows q uantitatively the relative importance of den sity and size differenc e . The variation with velocity , f( U ) , is compli cated by the above-mentioned fact that U m f varie s with composition . I t is of the double exponential form shown in Fig . 16 . For each bin ary system there is a critical velocity U T o below which the system rap i dly se gre gates and above which it rapidly mixes . An empirical equation from which this can be estimated is
[(U ]1.2 )
mf J ( Umf)F
where
H * is
H*
=
a
reduced bed
1 - exp
(- 5)
+
)1.1 )0. o.9 (p; - ( � p
d
1
7
d
2 . 2x 1 1 2< H * > ( 58>
h e i gh t defined by
( 59)
Like mixin g s e gr e gati on is brought about by t h e bubbles . A b ubble through a re gion of j e t sa m enables it to d e sc en d throu gh n e i gh bo r S imilarly , the b ubble s cause mixin g by liftin g j e t s a m in their in g flot sam . w a kes carryin g it to t h e upper part of the bed . The two processes are in c om p et ition but mixin g increases more rapidly than doe s segregation . From the poin t of view of chemical reac tion e n gin e e rin g in terest lie s in kno win g the distribution of material in cases where segregation may occur . C omplete mixin g is often desired an d the equation s above give a guide as to whether or not thi s will occur .
passing
,
,
,
T wo quite di ffe r en t and independent heat transfer coefficien ts fl uidi zed bed . h refers to heat trans fer bet ween gas and G /S
H ea t a n d M a s s T ra n s fer
PERF�'f�1
i
M
M I X I NG I NDEX
FI GURE 16
Variation o
f
mi xin g in dex with
gas
velocity .
apply to a solid particles
467
Fl uidize d -B ed Reac tors
an d hB /W re fe r s t o heat t ran s f e r b etween t h e b e d and i t s c on taini n g walls or i m m e r s e d s u r fa ce s . H eat t r an s fe r between ga s a n d p artic le s is u s uall y very rapid b u t only because of th e enormous s urfac e area p re s en te d by finely divided solid s and because of t h eir large heat c ap a c i ty on a vol u m e basis compared with t h a t of the gas . T h e tran s fer coe fficient h a IS is small , near its lo w er li mi t i n g value , because t he local R ey no lds number is small , as exp l ained earlier . T he local n u m b er can be e s t i m at ed from ( Rowe an d C l ax ton , 196 5) Nu
h
=
A
w h e re
A =
fd
=
G /S +
B Pr
2
2
1 - (1 -
B and
=
( 6 0)
113
Re
n
( 61)
e:) 1 / 3
( 6 2)
( 6 3)
3 e:
( 2 - 3n ) / ( 3n - 1 )
=
4 . 6 5Re
- 0 28 ·
( 6 4)
C onsider the case w here flui dizin g gas en t e r s the bed at a different temperature from the particles . A s s u m e t h a t t h e p ar ticles are uniform spheres and are perfec tly mixed by b ubble di s t urb an c e a s described above . Further a s s umin g no heat loss a n d that the solids have a much gr ea t e r h e a t c apac i ty per unit volume than gas , it is rea dily shown that the depth of bed over whic h 9 0 % of the gas c h an ge in t emperature occurs is given by
( 65) I f t h is is evaluated for a ty p ic al ga s - soli d flui di zed sy stem with h a IS c alc u ( 6 1 ) , i t is seen that this layer is only a fe w p ar ticl e diam
l a t e d from E q .
e t e r s thick . T hat i s , virtually all h e a t transfer occ urs in a very s hallow b o tt o m laye r . E x p e ri m en t al att e mp t s to detec t a temperature gradient at the b o t t o m of t h e bed invariably fail , w hic h is in accordance w it h E q . ( 6 5) . This foll ow s from t hree simple facts : the particles are well mixed , t he y h ave much larger volu metric h eat c apac i ty than gas , an d they offer a lot of surfac e area for heat tr an s fe r . I t is w el l known that a vi gorou sly b ub blin g ga s fl ui d i zed bed is re markably uniform in t e m p erat u r e even when hi ghly e n e r ge t ic reaction s are T his all follows from what i s w ri t ten above an d is no t a con occurrin g . s q ue nc e of a hi gh t ra n s fe r coefficien t w hen this is con sidered on an in dividual partic l e ba s i s . Heat t ra n s fe r from bed to w all s is also generally ve ry high , and this is expres sed by a high valu e of the coefficient hB /W w h en , as is logical , it is define d in t e rm s of wall area . The magnit ude of the c oeffic ie nt is u s u ally comparable wit h that for hea t transfer to a boilin g liq uid , and often t h e
468
Rowe a n d Y a tes
overall coefficient is li mited by c on dition s on the other side of t he wall . Numeric al values depend on pa rticle and gas prop e r ties in a co m p l ic a ted way that has been well doc umented ( B ott erill , 1 97 5) . The reason for hi gh heat transfer rates is e asily ex pl ain e d . It foll ow s from t he fac t tha t particles are well mixed and of un i for m temperat ure . The temperature p ro fi le across a section of the bed is flat , and therefore there is a very steep gradien t or drivin g force a gain s t the c on tain in g walls . Another way of s ayin g this is that hot part icle s from the center are contin ua lly brought to the cold wall s an d then moved on before they have ti m e to cool app reciably . T he re q uirement for good heat tr an sfe r is si mply that the particles keep movin g . T he fore goin g also applies t o s ur fac e s immersed in t he bed , but diffi culties can arise when t hese interfere with fluidi zation , the ri se of b ubbles , and the consequen tial movemen t of p article s . I t is e a s y in these general terms to un de rs tand why a s ub m erge d heat exchan ger m ay n o t perfo r m well , but improvemen t depends on en gineerin g e m piricis m with this model kept in min d . There is no realistic equivalent of mas s transfer between surfaces and t he bed but an ob vio us on e b et ween the p a rtic l es and the g as . On an in dividual pa rticle basi s this is closely analogous to heat transfer and E qs . ( 6 0 ) and ( 6 1 ) s i m ply b e co m e Sh
=
=
kG / S +
A
d
i
BSc
( 66)
11 3
Re
n
( 6 7)
with A , B , and n defined as before by E q s . ( 6 2 ) to ( 6 4 ) . There is very little more to s ay about mass transfer , although the simpleminded often seek an overall c o effic i en t that will yield conversion in a reactor from fac t s suc h as the inlet condition s and the bed t e m p e r ature . There is no si m p le answer of this kin d . Overall c on ve r s ion follo ws from an in tegration of all the hi storie s ex p e rienc e d by elemen t s of gas in their p as sage throu gh the bed , in bubbles , in in ters titial flow , or as p art of a cloud . T h e local r e ac tion rate w ill depend on local concen tration and on t he transfer coefficient given by E q . ( 6 7 ) . B ut thi s is the basis of the rea c t o r models dealt with l ater in this chapter . Fine p owde r s behave differently from coarser ones in ways that are important for reaction en gineerin g , and so m e of the previous description r eq ui re s modification . Fine pow der in t hi s context means Geldart 's group A powders ( Geldart , 1 9 7 3 ) , w hich are roughl y de fi n ed as l yi n g in the ran ge Fi n e Pow d e r s
(p (p
p )d f
s s
--
p
f
)d
�
p
>
10
< 2
6 X
10
5
3 when de n s it y is e x p r e s sed in kg /m and p article diameter in ].l m . T h u s 3 3 solids of density 1 0 k g / m s ho w group A beh a vio r when in the ran ge
( 68)
469
F l u i d i ze d -B e d Reac to rs
30 < dp < 2 0 0 J.! m . B elow t his size they are very difficult to fluidi ze unifor mly and ten d to c hann el ( group C ) , an d above it they b ehave as The classification is approximate at best , hitherto desc rib ed ( group B ) . depen ds also on other poorly understood p article propertie s , and there is a g ra dation of behavior and not abrupt change . Most powders clearly s how group A b e havior when dp < 1 0 0 JJ m . T he dis tingui shin g feature of group A powders is t hat they expand uni formly w hen fluidized at low gas velocities and do not begin to bubble un til a higher velocity , U mb , is reached . For group A pow der s U mb / U mf > 1 . T his i s illus trated b y Fig . 1 7 , which show s how bed hei ght chan ges with gas ve locit y . The uniformly e x pan d e d state is quite stable . There is no hy stere sis and the same height is obs erved whether gas velocity has inc reas ed or de creased . T he first appearance of b u b b l e s is a b reakdown of this st ab il i t y , and obse rvations aroun d U mb are not v e ry reproducible , m akin g it diffi c ult to dete rmine wit h p rec i si on . B ehavior in the region U m f < U < U mb i s closely akin to that of liquid fluidized beds as de scribed briefly ea rlie r . T he hei ght or bed voidage varies with velocity as E q . ( 3 1 ) e x c e p t that the c ons t an t of proportion ality i s less t han U T . A s a corollary , the expon en t m is greater than ob se rve d with liqu id s at t h e same Reynol ds n umber . With d ec r e asin g average particle si ze this group A behavior becomes more marke d . B ed hei p-ht can double before b ubblin g begin s w hen U mb / Um f 3. Expan sion is inc reased by the addition of fin es to the powd er , whic h has the e ffect of red ucin g mean particle si ze dp · Many cataly s t powders p repare d for u se in fluidi ze d - bed reactors b e have as group A an d fluid - b e d c rackin g cataly st ( F C C ) i s a good example . Expan sion of the dense p hase an d increa sed in terstitial flow above U m f has important consequences in reactor behavior and modelin g . Since this oc curs on l y wit h fine powders where in te rs t i t ia l flow i s necessarily small , in crease in t his very effec tive p hase can inc rease reactor performance m a rk e d l y .
�
� 0
---IUII' NIFORML� GJ I hoi0
�
PACK ED
�LUI DISE�� I BED
EXPANDE D
I
I I BUBBLING �FLUIDISED BlD I
*---"I UNSTABLE R EGIO�
0 +------+--4---r-0 Umb Umt GAS VELOC ITY u -
p o w de rs .
FI GURE 1 7
Variation o f b e d height with gas velocity for group A
4 70
Rowe and Yates
i
}
0
9 <( f= LL _J
�
0::: w ! z
I I I I
I NCREASING FINES CONTENT I D EAL 2 PHASE T H EORY
O m .f. o �--------�L0
( G ROU P
B
GROUP A POWDERS
POWDER )
-------
TOTAL FLOW Q
.,
FI G U R E 1 8 Variation of in t e r s titial flo w with total flow as a func tion of fin es content for group A powders .
Although observation is difficult , t here is evidence that the dense phase remain s e x p an ded a fter bubblin g begin s an d more gas than Q mf flow s in terstitially in a vi gorou sly fluidi zed bed of group A powder ( Rowe et al . , 1 97 8 ) . Fi gure 18 s ho w s how interstitial flow increases for g roup A pow ders as the fin es con tent is inc reased ( dp decrease d ) . Only limited data are so far a vailabl e and it is not yet pos sible to quantify t he in forma tion of Fi!r. 1 8 for a previously untested powde r . I t should b e noted that in terstitial flow will decrease an d reac tor p e rformance will consequently fall as fines are elutriated from t he bed . S uc h is com mon ly ob served by plant operator s . ,
E ffec t o f I nc rea s i n q P r es s u re
Most fluidi ze d - b e d reactors operate at or a little above a t mosp heric pr'e ssure , but there a re cases w hen it is required to op erate at hi gher pressures . Fluidi zed coal combustion un der p ressure , for exa mple , gives improved e f ficiency b e c a u s e of t he reduced volume o f the plant for a given d u t y an d because of t he opportunity to recover further ene r gy by expan din g hot gases through a turbine ( Roberts et al . , 1 98 0 ) . For this application , pre s sures up t o 20 b a r are feasib le . There are , additionally , che mical proce sses where it is thermodynamically desirable to operate at hi gh pressure . At the same mass flow rate it is obvious t hat velocity will dec rease wit h increasin g p r es s ure , so t h e re will be fewer an d s maller bubbles with an increasing p roportion of interstitial flow . It is of more interest to compare
4 71
Fl uidize d-B e d Reac tors
behavior at the same volumetric flow rate as p re ss u r e increas e s and this is t he c on dit ion in all t hat follo w s . I t is commonly reported by p la n t operators that fluidi zation is " smoother" a s pre s s ure increas e s , and this is said to be b ec a u s e the bubbles are smaller . T he e vid e nc e for t h i s is very slen der b ec a u s e of the diffic ulty of observation , but the c oncl usion is w i dely acc epted . Data from re search ob ser vat ion s un der pressure are scarc e , again because of the d iffic ul ty of observation , but w h a t little evidence there is p oin t s to the conclusion that proportionately more ga s flows interstitially as p re s s u re inc reases ( Rowe and M ac Gi lli vr ay , 1 9 8 0 ) . The ra t io U m b / U mf for clas s A powders does i nc r e a se w it h in cr e a sin g pres sure b ut not dramatically , about 2 5% for a fiv e fo ld increase in p r e s s u r e ( R owe et al . , 1 9 8 2 ) . C e r t ai n ly , some group B po wde r s show group A b e havior as p r e s s ur e inc rease s , and this is not p re dicted by t h e criteria of Eq . ( 6 8 ) . An alumina p o w de r of dp 4 50 �m that is ty pic all y group B at N T P b e g a n to s how g rou p A b e hav i o r at pres sures as low as bar an d showed it ma rk ed ly at hi gh er pressure s ( R owe et al . , 1 98 3) . X - ray cine photographs of thi s p o w de r fl ui di ze d w ith n i t ro gen show that as e x p ec t ed , b ubble si ze dec rea se s with p res sure and t h e avera ge vol u me is hal ve d at aroun d 6 0 bar . A gain , t h i s is n o t a d ramat ic c hange . B ubble frequency increases an d the visible b ubble flow varie s ap p r o xi ma t e l y in p r op or t ion with A ( U - U m b ) . T he bubble velocity c oef fic i en t / u B / ( 1 / 2) gd B inc r ea se s by about 50 % at 30 bar an d the reafter c han ge s li t t le . I t should be noted that U m f falls with increasin g pre ssure as p r e di c t e d by E q . ( 1 4 ) . T e r min al falling v e l oc i t y , U T , also falls wi t h pres sure . The chan ge can be c on si d er ab le , an d for the 4 50 � m al u m in a referred to above , U mf falls to on e - t h ird o f i ts a t mo sp heric value at ab o u t 6 0 b ar . U T falls to ab o u t one - fifth over the same ran ge . T he change inc rea s e s with parti cle s i ze . Equ a t i on ( 1 4 ) fits the p re sen tly available data very well up to p r ess u re s of 1 0 0 bar . Chan ge of U m f w it h pressure means that at con s ta n t volumetric flow ra t e the bed will be fl ui d i z ed at a g re a t e r m ultiple of U mf a s p r e s s u r e in c reases . The develop ment of group A b eh a vio r mean s that a larger pro por tion of g a s wi ll flow in ter stitially , a n d t h e n e t result m a y be a reduction in bubble flow wit h c on s e q u en t ly s maller bubbles . These changes would im p ro ve reac t o r p e rformanc e . With the 4 50 - IJ m alumina ( the only m ate rial so far to be s t u die d s y st em atically in t h i s way ) , dramatic c han ges do occur at p r e s s ur e s greater than about 8 0 b ar . As t h i s pressure is approached , the wake obtrudes more de e p l y into t he bubble and they begin to break up b y wake reachin g to the roof , a mode of b re aku p n ever observed at lower pressures . At 8 0 bar the b ub b l e s are no l o n ge r clearly observable , and it may be that the s y s t e m h a s c han ge d toward partic ulate fluidi zation . =
2
E n t ra i n m e n t a n d E l u t ri a t i on
W hen fl uidi zed b e d s are operated at gas velocities in excess of th e mm1mum fluidization value . the eruption of ga s b ub b le s a t the bed s urface c auses I t is generally p articles to be ca r ri e d up w ard into the freeboard region . accepted that these p articles ori ginate from the bubble wake re gion ( G eorge and G race , 1 9 7 8 ) a nd that the h e i ght to whic h they are carried above the surface b efore fal li n g back i s a fu n c t i on of their s i z e and den sity as well as of the ve loc ity of t he fluidizin g gas .
Rowe a n d Y a tes
4 72
U n der con dition s o f op e ration in w hic h the s u p er fic i al gas ve locity exceeds the te rminal fall velocity of the bed particles , ent r ain men t leads to elutriation an d p artic l e s are carrie d out of the sy s t e m completely . Thus th ere is usually a conc entration gradie n t in t h e freeboard , wit h the lar ger d e n se r p a r tic l e s bein g carried so m e way above the surface before returnin g to it , and the s m al l e r lighter p ar tic l e s b ein g removed from the system . T h e hei gh t above the s u rface at whic h the p a rtic l e concentration b ecomes c on stant is known as the transport disengaging heigh t ( T D H ) . I t is clearly im port an t from a d e s i gn point of view to b e able to p re dict how m uch bed material will be elutriated for a g i v en set of op eratin g con dition s , since t hi s has a strong b e a ri n g o n cyclone and filter specification . A b a s ic assumption generally made is that entrain ment is a fi r st o rd er process and t h e rate of el u triation of p a r tic l e s in a p articula r size range is direc tly p roport ion al to t h e mass fraction x of those p a rtic les in the i be d :
,
-
-
( 6 9)
where A is the bed area , M the bed mass , an d K * ( k g f m 2 • s } a fi r s t o r der T hen for an ap p rox im a t e ly constant bed mas s ,
rate constant .
( 70)
where Xi O is the initial m a s s fraction of e lut riable p artic les . A gr e a t deal of work has be en carried out to e s t a b li s h relation ship s b e t w e e n K * a n d the p ro pert ie s of flui di zin g ga s and bed p artic les , and as a resul t a nu mber of correlation s have been devised an d some are listed in T a b l e 4 . I t must be
TAB LE 4
S ome
Published C orrelations for E l ut ri at ion from Fluidi zed B e ds
Yagi and Aochi ( 19 5 5) :
1 2 0 = 0 . 0 0 1 5R e t · 6 + 0 . 0 1R e t ·
Zen z an d Weil ( 1 9 5 8 ) :
( u� )
1.
39 . 1
K* p U g O
x
10
2
2 gd p p s
=
(�) 2
7. 02
X
10
3
gd p p s
87
1 . 15
oe;;
.
58 1 8
� 58 1 . 8
X
X
10
10
-3
-3
4 73
Fluidize d - B e d Reac to rs
TA B LE
4
( Contin ued )
Wen and Hasin ger ( 1 96 0 ) :
K* Tanaka et al . ( 1 9 7 2 ) :
= 4.6
K*
10-
x
2
Merrick an d Highley ( 1 9 7 4 ) :
P
K
�
g 0
= A + 1 3 0 ex p
G el dart et al . ( 1 9 7 9) :
P
P:�o
=
o
gd
U
p
ts
(u )o . 5 ( ds
0
0
)2
J
0
o. s
_m�
0
�
3 Re .
mf
( s g) p
_
P
P
g
) 0 . 2 5]
[- s. 4( 0d: )]
2 3 . 7 e xp
p + L: p . ( p. e = g 1 1
[
- 10. 4
�
(U
_
solid loadin g o f ith size frac t ion in exit ga s )
=
Colakyan et al . ( 1 979) : K* =
33
� - J:) '
B achovchin et al . ( 1 9 7 9 ) :
3. 35
K* =
where
dp X8
=
=
x
10
_
5
(;au -0
p
g
)4 .
67
( P )1 . ) ( D __g P
s
6
2
(l
average si ze of p article elutriate d
d
p
fraction of fin es at the bed surface
Lin et al . ( 1 98 0 ) : =
9. 43
X
10
-4
2
( :: ) d
1 . 65
.fX 1 . 1 5 � ) d p
0. 15
Row e
474
an d Y a t es
TABLE 5 Experimen t al C onditions for En train ment or Elutriation of F in e s from F l ui d i z e d B eds ( from C h en an d W en , 1 9 8 2 )
I n vest ir,a to r
Yagi an d A oc hi
( 1 9 55 )
C olu mn diam e t er , De bed heigh t , L ; fr e e b oa r d , H ( m )
Experiments
D
E l u t ri ation two and multicomponent s , batch op e ration s Entrain men t of F C C c at aly s t , s t e a dy st ate o pe ra tion
D
W e n an d H ashin ger
Elutriation , two and m ul ti componen t s , batch operation
D
( 1 960)
0 . 051
=
c
X
0. 61
= 0. 0 6 7
c
L + H
Elutriation , t wo components , batch an d c on tin uo u s
M errick and Highley ( 1 9 74 )
Elutriation , s t ea dy sta t e op e ra tion
D
+
L
D
-
=
c
c
=
L
Air
0 . 0 3 1 -- 0 . 1 2
H = 0 . 2 54 - 2 . 8 0
-
Tanaka et al .
( 197 2 )
=
L
Zen z and W eil ( 1 96 0 )
0 . 0 52- 0 . 071
=
c
Gas
=
Di s t ri bu t or Fixed bed of steel balls
Air , mean grid pres su re 8 atm Air , He
Filter cloth
Air
Perforated plate
2 . 08 , 1 . 83
0 . 067
H = 1 . 80
= 0 . 90 x 0 . 4 5
Air
0 . 61- 1 . 22
L + H = 3 . 96 G el dart et al . ( 1 9 7 9}
Elutriatio n ,
m ulti c omp on e nts b atch o pera tion
,
DC = 0 . 0 7 6 ==
L
A ir
0 . 3 5- 0 . 4 5
L + H = 3 . 80
o . 90
X
Colakyan et al . ( 1979)
Elutriation , multicompon ents , batch o pera tion
L + H = 6 . 30
B ac h o vc h in et al .
E n t r a in m e n t ,
D
( 1979)
Lin et al . ( 1 98 0)
m ultic o m p on en t s ste a dy s tat e op e ra tion
,
-
E n train m e n t a n d el u tria tion
L
C
c =
=
=
o . 90
0 . 1 52 4
Air
Perforated p late
Air
Perforated plate
Air
G rid
0 . 23 - 0 . 2 5
H = 0 . 7 5- 4 . 0 0
D ,
m ulticomponent s ,
steady - state ope rati on
D
Filter paper covered by wire mesh
c
L
=
H
=
=
0 . 60
0. 25
X
0 . 60
0 . 6 3- 3 . 2 7
4 75
Fluidi zed-Bed R eac tors
Particles San d , glass seed , refractory
FCC catalyst
with size distribution
Glass spheres , coal powder
Glass beads , sand , stainless balls , lead b alls , N eobeads Coal ash with size distribution
I n itial
Size of coarse solids in parent bed
weight fraction of fines
( �m )
(�m)
( %)
8 5 - 50 0
3 1 0- 1 6 4 0
Size of fine s to be entrained
5- 2 0
20-- 1 5 0
S uperficial gas veloc ity ( m / sec )
In ternals
o . 92- 1 . 6 2
N one
0 . 30- 0 . 72
None
o . 2 2- 1 . 3 2
None
40- 140
1 00- 2 8 0
6 0 - 8 00
141- 2300
1. 28- 2 . 70
None
1 4 0 0- 3 1 7 0
0 . 6 1- 2 . 44
N one
0 . 60 - 3 . 0 0
None
0-- 1 4 0 0
San d , shot , alumina
38- 327
1 50 - 35 5
S and with si ze distribution
3 7 - 3 56
3 56 - 2 36 0
San d
22- 180
C oarse : 4 4 3- 1 0 9 5 ; Medium :
6- 1 0
5- 7 5 10
0 . 9 0- 3 . 6 0
10- 2 0
0 . 61- 1 . 25
N one
0 . 1 0- 0 . 3 0
None
With o r without immerse d heat transfer tubes
2 2- 1 0 9 5
Sand /char with size distribution
0
125 ( char) --
1 2 5- 4 1 9
o . 01- 1
4 76
Rowe a n d Y a tes
! I/i
��·
II
/ '
1 -
ii ,. // I
if / /
0.5
02L. ! i 6 /
03
05
19
FI G U R E
6 4- llm
size
1
- YAGI AND AOC H I
'-- G E LDART ET Al . .
1 I I
j
I
-�
0.1
I
/ /
WEN AND HASHI NGE R AND TANAKA ET AL .
--L--
1
-L - _t__ j 2 5
U ( m/s)
S peci fic elu triation rate con stan t versus gas velocity for fraction . ( From eo g an d Grace , 1 9 8 1 . )
G r e
of are frequently ina li b e sd e t hey were determined . Furthermore , there
emphasize d , ho wever , that o win g to t h e physical complexity the elutria tion proces s , these c orrela tion s pp ca l ou t i e t h ran ge of condition s
for
w hic h
is evidence that wall effec t s can have an in fluenc e on particle carryover , so that bed diam eter s ho uld be i nc l u de d in a correlation as an im p o rtan t Table 5 indic ates the range of con variable ; t his is seldom don e , however . dition s , particle size , and so on , for w hich t h e pr e vi ou s ly listed correlation s
were ob tained . A n m er inve sti gators have elutriation wit h their own experimen t al results an d in many have foun d poor agreement ( s e , G u gnoni an d 1980) . G eor ge and Grace ( 1 9 8 1 ) , however , correlation to fit their data quite well a con ditions , whereas the relation ship s of Merrick and Hi ghley ( 1972) C olakyan al . ( 1 9 7 9 ) gen erally poor ( Fi g . 1 9) . would
u b
of
compared the
correlation s
cases
e e . g. , found the Zenz-Weil range of and et
Zen z ,
over
were
It
of
t hus seem that a mechanistic m odel in dependent empirical measuremen t s would preferable for predic tin g elutriation rates un der a n y circum stanc e s , but some attemp t s to e stablish such a C h en and W en , 1 9 8 2 ; G eorge an d 1 9 8 1 ) , progress so far has limited .
be
despite
Grace ,
foreseeable model ( see been
I N TERPHAS E GAS EX C H A N G E
�ene rally accep ted bubbles in fluidiz e d b e d s are of particles and t hat except in t h e cloud wake region s , the opportunity
It
is
that gas
devoid
and
4 77
Fl ui d i z e d - B e d Reac t o rs
for gas - soli d in teraction is limited . A s a result of the " porous" nature of the bubble -cloud boun dary , howeve r , ga s exchan ge can occur b et ween the unreac tive b ubble phase an d the reactive dense phase , and this can lead to a significant inc rease in the extent of con ersion in a fl i di e d bed reac tor . The precise m ec hanism of the gas - e xchan ge process has been the ob j ect of much speculation , and a n mber of methods have been propos e d for calculat in g its ma gn i t u de under given condition s . There is however , a s ec on d mec hanism by mean of which b ubble gas can contac t bed p a rticles , and this is during the coalescence p roce s s when gas " le aks" from a t railing to a lead in g bubble thro ugh the in tervenin g dense p hase . It will be convenient first to examin e the p roposed tran sfer mechanisms for sin gle b ubbles an d then to consider what modific ation s need to be made to take accoun t of b ubble c oalescence .
v
u
s
u z
-
,
G a s E x c h a n ge from Si ng l e B ub b l es
C havarie and Grace ( 1 9 7 6 ) ident i fied three "classical" models of mas s trans fer from sin gle bubbles that have been discussed in the literat ure . The first is b a se d on the ass umption that transfer i s c ontrolle d by di ffus ion across the boun dary bet ween the cloud and in terstit ial phase , an d was developed principally by Rowe et al ( 1 966 , 1 9 7 1 ) , Toei et al . ( 1 96 9) , and C hi ba an d Kobayas hi ( 1 9 7 0 ) . T he Rowe approach , for exam p le , assumes gas in the b ubb l e and c loud to be completely mixe d and the tran s fer rate to be a function of a sin gle coefficien t k gc ( m I s ) : .
mol t
ls
( 7 1)
where Vb c i.s the volume of h e bubble plus cloud , S ci the surface area of the cloud - dense phase interface , an d C bc and Ci t he concentrations of gas in the bubble - cloud and interstitial phases , re s p ecti ely The gas oli d system is then modeled on the basis of transfer f rom a drop of one im miscible liquid risin g hrough another , the drop dia m e t e r bein g equal to the cloud diameter de . T h e Sherwood number , S he , for mass transfer f o m a sp here of uniform c omposition in the appropriate range of Rey nolds number ( 3 0 < Rec < 2 0 0 0 ) is given , accordin g to Rowe et al . ( 1 96 5a ) by the expres sion
v
t
.
-
s
r
,
c
=
d
� D
k Sh
=
2
+
0 . 6 9 Sc
11 3
Re 1 1 2 c
( 7 2)
where the Schmidt n u mber , Sc = ...£...
( 73)
pD
and t he R eynolds n u m ber , Re
( 74 )
c
u r
with D b ein g the mole c l a diffusivity of the gas and U R the relative veloc ity b e t ween b ubble an d interstiti al ga s : UR = U U i . T he b ub b l e
b
-
Rowe an d Y a tes
4 78
cloud - to - in terstitial p hase exchan ge rate ( volume of gas per unit volume of b ub b le per secon d ) is then
Q
=
bci
=
k
gc v
nd
2 c
£
3. 9£.0 Sh bc
v 2/ 3
c
( 7 5)
be
A secon d approach to in terphase ga s exc han ge was explored by D avid
son and H arrison ( 1 96 3 ) , who di sregarded the b ubble cloud an d con sidered transfer to occur directly between b ubble and interstitial phases by a com T he molar gas exchan ge rate is bination of convective an d diffusive flow .
( 76 ) w here the c onvective term , q , is given for a s pherical b ubble by
( 7 7) db b ein g the b ubble diamete r , V b the bub b le volume , and S b i t he inter p hase s urface a rea . T he m as s transfer coefficien t kbi is evaluated on t he basis of the analogy with diffusion from a gas b ubble ris i n g in a liquid : ( 78) T he overall rate of exchan ge is then given by
( 7 9)
=
( 80)
The thir d " classical" model is that of K unii an d Leven spiel ( 1 96 9) , w ho combined elements of the fore goin g t wo methods and con side red the e x T he first stage involve s c han ge p rocess to take place i n two stage s . tran s fer from t he b ub b le to the cloud re gion an d is desc ribed in term s of a coefficient Q b c identical w i t h t he bub b le - to - interstitial phase coefficient Q bi of D avid son and H a rrison . In the secon d s t a ge gas is considered to be transferred from c loud to in terstitial phase by a diffusive process , described by the coefficient Q ci , w hich may be evaluated on the basis of the penetration theory of Higbie ( 1 9 35 ) :
4 79
Fl uidize d -Bed Reac to rs
Q
.
Cl
=
6 . 78
overall flow
(
e m
�u
�
b
)
112
( 81)
from
T he t w o tran sfer processe s are
Qb i 1
=
rate
of
--
+
1
Q
bc
ga s
assumed t o occur i n serie s , so that the in te r s titia l phase , Qbi o is given by
bubble t o
( 82)
Although the K unii - Leven spiel m o d e l assumes se p a r a t e re sistances to flow at t h e b ubble - c loud in terfac e and a t the cloud- den se p hase boundary , in p r a c t ic e the latter i s n orm ally the dominant factor , so the model prediction s a re not very di ffere n t from those derived from the purely diffusive models . T his is shown by the val ues given in T able 6 , where e xchange c oe ffic i en t s from t h e three models a r e calculate d for a system i n whic h a s in gle spheri cal b ubble of ozone ( D o = 2 x 1 0- 5 m 2 / s ) wit h a dia me te r of 0 . 1 0 m i s a s s u m e d to be injecte d i n t o a b e d of par ticle s o f U m f = 0 . 0 5 3 m / s in c ip ien tly fluidi ze d w i t h air . It is apparent that t h e model of D avi dson a n d H arrison in w hic h the con vec tive through - flow term is domin ant ove r the diffusion term give s a value an order of m a gnit u de greater than those derive d on the basis of the m ain ly di ffu si v e m o de l s C harvarie an d G race ( 1 97 6 ) an d Sit and G race ( 1 9 7 8 ) c a r ri e d out ga s e xc h an ge experimen t s in a tw o d i men s ion a l b ed into which b ubb l e s of air con t ainin g lo w concentration s of ozone w e r e injected . T hey foun d value s of ove rall transfer coefficien t s , co r r ec t e d for un s t e a dy bubble growth an d entrance ef(ect s , that w e r e som e w h e re b e t ween those predicted by the thro u gh fl o w and diffusion - dominated models . T hey conc lude d that for
.
- -
TABLE 6
B ubbles
-
in
Predicted In terphase Gas Exchan ge Coefficien t s for O zone an A ir - Fluidi ze d B ed
C alculated
Model
P artrid ge an d Rowe ( 1966)
an d H arrison ( 1 96 3)
K u n ii a n d
Leven spiel
So urce :
=
Qbci Q
David son
( 1 96 9)
coefficient
Exchange
bi
Q bi
=
=
3 . 9eD v
4. 5 Q
Sh
c
0 . 24 5
2/3
be
e ) mf
d
b
+
(D 1 / 2 l / 4 g d
Q bc ci
Q bc + Qci
A fter C harvarie an d
value 1 (s- )
G ra c e (
b
5 /4
)
3. 21
0 . 466
1 9 76 )
.
480
Rowe an d Ya tes
sin gle b ub b l e s t he b e st a gree men t with experimen t was shown by the
model of M urray ( 1 96 5 , 1 96 6 ) ( which asc ribes a modest value to the through- flow ) , modified on the b asis of p enetration t heory to give an over all tran sfer coefficien t for s p her ic al b ubbles of
+
___!!___ d
3/2 b
(D Em f U b )
1/ 2
1T
( 83)
The value of Q b i from E q . ( 8 3) c alc ulat e d for the ozon e - air sy stem con sidered in T able 6 is 1 . 34 s· 1 . G a s E x c h a n g e i n S y s tems of M u l t i p l e B ub b l e s As already mention e d , in t er p ha s e g a s exc h an ge c a n in c re a se con s i de rably
durin g the p r oc e s s of b ubble coalescence , so that in vigorously bubblin g b e d s the extent of gas - solid contac t l ead in g to chemical reac tion can be expec t e d t o b e greater t han would b e predicted on t he b asis of the sin gle Sin gle - b u b b le models bubble models di sc ussed in the precedin g sec tion . are n�verthele ss u s e fu l in su gge stin g a basic mechanism o f tran s fer into whic h the e ffects of coalescenc e may subsequen tly be incorporated . There is as yet no con vincin g theory to account for all the c omplexities I t is known , however , that the proc es s nor mally of b ubble coalescence . proceeds by a trailin g bubble accelera tin g into t he rear of a leadin g one an d prod ucin g a s in gl e b ub b le whose volume i s greater than the sum of the volum e s of the original p air In a contin uation of the st udy men tion ed earlier , Sit an d G race ( 1 98 1 ) inve s t i gat e d the interphase transfe r of o zone durin g coalescence in a two - dimen sional b e d . From cine film of t he p roces s , they divided coalescence in to four perio d s : .
1.
2.
3. 4.
A n a p p ro a c h pe rio d between t he time of b u b ble formation and the poin t of c on tac t of the nose of the trailer with the wake re gion of the leade r An enc roac hment period when the .10se of the trailer passed through t he wake re gion an d touched the lower e d ge of t he leader A period d urin g whic h c oalescence p roc e e d ed A postcoalesccnce pe riod
T h ey foun d t he greatest c ha n ge in o zone c oncen tration to oc c ur durin g the a pp ro ac h an d enc roachment p eriod s and that 7 7 % of the gas tran s fer occ urred wit hin the fi r s t 50% of the total time span . Furthermore , they es timated the m a ss transfer coe fficient of the leadin g bub ble to be t wo to three tim e s great er durin g the approach period than when the b ub b le was
in isolation . On the basis of these observation s , S it and G race propose d that d u rin g coalesc ence the convective through - flow p rocess is increased w hile the dif fusive p roce s se s are li ttle affected . T hey thus proposed applyi n g an e mpiri cal enhancement factor of 1 . 8 to the through - fl ow term in the expres sion H owever , since from for Qbi based on t he Murray theory [ E q . ( 8 3) ] . visual observation 4 0 % of t h e b ub bles are at any in stant in t he early s tages of coalescenc e , a more realistic e stimate of the enhance ment fac tor was
Fl uidized-B e d R eac to rs
sugge sted to be ( 1 . 8 x 0 . 4 + 0 . 6 ) for the exc hange coefficien t : ( D e:
u
mf b 1T
4 81
1.
32 ,
givin g the modifie d expression
> 1/ 2
( 84)
where ub is the mean velocity of bubbles in the bed an d db their mean diamete r . Equation ( 8 4 ) appears t o give the best e stimate o f the inter phase trans fer coefficient in freely bubblin g , t hree - dimen sional b eds of nonab sorbin g particles . R EA C T O R M O D E LS
A large number of fluidi zed -bed reactor models have been de sc ribed in
the cterature durin g the last three decade s ' an d although most of them are designed for application to gas - solid catalytic reac tion s , s uitable modifi cations for noncatalytic p rocesses are relatively easily incorporated ( K unii and Leven spiel , 1 9 6 9 ) . T h e e s sen tial feature of all the models is the desc ription given of the bed hydro dynamic s , and since this involves the interplay of a number of phy sic al variables , the nu mber of possible permu tations and combination s an d hence the n u mber of models , is quite lar ge . Seve ral reviews have been published and these should be referred to for detailed compari son s ( Pyle , 1 9 7 0 ; G race , 1 97 1 ; Rowe , 1 97 2 ; Y ates , 1 97 5 ; Horio and Wen , 1 97 7 ; van S waaij , 1 97 8) . I n general , t he startin g poin t of any model is a consideration of the division of flow bet ween the den se phase an d the lean or b ubble p hase , an d in the majority of cases the two phase theory · is assumed to apply ( i . e . , it is a ssumed that all gas flowin g in to the bed in excess of that required for minimum fluidi zation passes through in the for m of b ubbles ) . I t is then neces sary to consider the e x tent o f conve rsion in each p hase , t h e a moun t o f tran sfer bet ween phases as ga s flow s t hrough , and finally how the two stream s are combin e d at the In addition to t he bed surface to give the outlet reac tant conc entration . even ts takin g place in the bed itself under con dition s of steady two - p hase flow it is neces sary for c ompleteness to examine the pos siblity of chemical reaction occurrin g in both the gri d zone , where fresh ga s enters the b e d and bubbles are in embryonic form , and in the freeboard re gion above the bed surfac e , where high concen tration s of ac tive entrained particles can be found . We will , howeve r , exa mine first some of the p urely two phase models an d defer a con side ration of the effect of these end zones . The mo dels to be conside red h ave been chosen to rep resent three approaches to the problem of reactor design . The first ( D avidson and Harrison , 1 9 6 3 ) models bed performanc e from fir st principles and uses t he minim u m amount of empirical inform ation . T he secon d ( K unii and Leven spiel , 1969) , while based on t wo- phase theory , imposes an op eratin g con dition on t he b e d ( a n artific ally controlled bubble si ze ) that enables a re la tiv ely simple solution to be ob tain ed . T he third model ( due to the S hell company ) incorporates theory with a lar ge amoun t of empiric al in formation obtained from pilot plant experimen t s and is a fairly typical example of in dustrial methodology . ,
482 Mod e l of D av i d son a n d H a rri son
( 1 963)
Rowe and Y a tes
This w a s one of t h e earliest models t o be b a s e d o n t h e two-phase theory , an d alt hough its basic assumptions repre sent an oversimplification of the physic s of fluidi zation , it has been successfully applie d in a num ber of case s . T he assumption s on w hich the model is based are as follows :
1. 2. 3.
Gas bubbles are evenly distributed throughout the b e d and are of e qual size . Reac tion occurs only in the dense phase , where gas flowin g a t U f m is either completely m ixe d or in plu g flow . Gas is exchanged between the bubble and dense phases by the convection - diffusion mechanism discussed earlier .
A material balance on reactant A , wit h initial concentration C A • leav o in g the interstitial p hase in the case w here this is assumed to be comp letely mixe d gi ves the concentration , C A . , in terms of three dimen sionles s param 1 eters , X the in terphase exchange term , S the bubble fraction , and k' the reaction kinetic term : C
A
(l - e
)
0
-
1
-X
where
Se
-X
+
( 8 5)
k'
( 86)
X
( 8 7) and kH k'
=
mf
( 8 8)
u
H ere H i s the expan ded bed hei ght , H m f the height at m1mmum fluidi zation , U b the bubble velocity , Vb the b ubble volum e , and k a fir st -order reaction rate cons tant . T he in terphase trans fer coefficien t Q bi is gi ven by E q . ( 8 0 ) and this l eads t o an expres sion for X in term s o f the b ubble diameter db :
X =
H
+
( 8 9)
For reac tion s involvin g mixtures of gase s , D avidson an d H arrison rec om men ded t he use of the methods of A rnold or Hir sc hfelder ( Reid an d Sher wood , 1 95 8 ) for the e s tim ation of the gas - p hase diffusion coefficie n t s , D .
483
Fluidized-Bed Reac to rs
A se p a ra te mass balance o n t he bubble phase lea ds t o
( 9 0) and the ov e ral l outlet concentration of A , C A , is then fou n d b y sub stitut from E qs . ( 8 5 ) an d ( 9 0 ) in b
in g for C A i and C A
( 91) givin g the overall frac tion of reactant u n con ve rt e d : Se
-X
+
(1 1
-
-
Be
Se -X
-X 2 )
+ k'
( 9 2)
Inspection of E q . ( 92) shows t hat for large values of the inte r p h a s e ex c h an ge term X the model , as would b e e x p ec t e d reduces to t ha t of a C S T R . F urthe rmo r e , E q . ( 92 ) p r e dic t s that c onversion can never be com plete even for very fast values of k' : ,
Se
,
-X
as a result of the bypassin g action o f b ubb l e g a s I f the den s e - phase ga s is assumed to be in plug flow , m a s s balance e q u at ion s on b o t h pha se s lead to the followin g e x p re s sion for the fraction of react an t leavin g the bed unconverted : .
( 9 3) m
1
an d m
2
bein g found from
2H ( l - 8 ) m
=
(X
+
k' ) :!: [ ( X + k ' )
2
-
4k' X ( 1
-
S)]
112
( 94)
w he r e m m 1 with the p o s iti v e sign an d m = m 2 with the ne gative sign In both for m s of t h e model t he variable playin g the mos t important role i s t h e b ubble diameter , db , since it determin e s , a mon g ot her thin gs , t h e ex te n t of in terp hase ga s tran sfer . In the earlier p ublication s an average value of db fo r the whole b ed was estimated from t he reac tion data of other workers , but more recently , Darton ( 1 9 7 9) has p roposed a method whereby the v ar ia t ion in b ubble diameter with be d hei ght may be allowed for . The me t ho d is based on a deterministic mo de l of b ubble g row t h by stage wis e c oalescence an d lead s to t he f ol lo w in g expression for the in ter p hase transfer term : =
.
484
X
32" O
==
{ 1/2D l/2 e:
mf
(U
- u
g
mf
>
1/2 )2/ 5 ] �
[
1 /10
1/5 Ao
7/10
R o w e and Yates
-c :::,1/.
,
(9 ) 5
j
In this e q u at ion A 0 is the a rea of dist rib utor plate per orifice and H is the expanded bed height est i mated by iteration from
-
( 92) (9 5)
)3/ 5] l)
(4A v� i
m
< 96)
229-m
T he use of Eq . in c onjunc tion with the " dense phase completely mixed" e q uati on was shown by Darton to give reasonably good predic tion s of the conversion s measured by Fryer an d Potter ( 1 9 7 6 ) u sin g a 0 . diameter reactor fitted with a bubble c ap distrib utor with 6 1 caps , givin g 2 Ao 6 7 5 mm . =
Mod e l
of
K un i i a n d Lev en s pi el
( 1 96 9 )
T his model i s designed specifically for beds that are vi gorously bubblin g but not sluggin g and in whic h the b ubble si ze is c ontrolled at a un i fo rm value throughout by the u se of internals . As w i th the fore goin'g model of Davidson and H arrison , the average b u b b le diameter is the sin gle most important para meter since it determines the bubble velocity by Eq . ( 5 1 ) and the exten t o f interphase gas exchan ge b y Eq . ( 80 ) . Conversion in the interstitial phase at ga s velocities in excess of about 3 U mf is considered to be negli gible , the great majority of gas flowin g as bubbles . T he bubble ga s is assumed to be in plu g flow and the frac tion of reactant emergin g un converted at the bed surface is
( 7) 9 where k
f
:
H k
f
=
{
i s an effective overall rate con stant given r
b
Yb +
k
Q
1
r
bc
+
/Q . )
1 y
c
+ [ 1 /(k
r
C1
+
1 /y. ] 1
by
J
( 9 8)
Here k r is the first -order reaction rate constant , Q bc and Q ci are the interphase exchange rates defined in Eqs . ( 8) ( Q bi :: Qb c > and ( 8 1) , and the y term s represent the ratios of solids dispersed in bubbles , yb , c louds , yc , and in terstitial phase , Yi , to the total volume of b ubbles in the bed :
(99)
Flui dize d - B e d Reac tors
4 85
f is the fraction of bed occupied by bubbles , a term also dependent on b bubble diamete r : u - u
mf
( 1 0 0)
The de gree of gas - solid con tac t , and hence the extent of reaction , depen ds stron gly on the value s o f the y term s , and the model in it s simplest form takes Y b 0 w hile y c is calc ulated from =
( 101)
w here the ratio o f b ubble volume V b t o wake volume V w i s found from the empiric al correla tion o f Rowe an d Partridge ( 1 9 6 5) . The value of Y i is found by sub stit ution of Y b and Y c in E q . ( 99) . T he K unii- Levenspiel model was foun d by Chavar i e an d Grace ( 1 97 5 ) to show the best agree ment of all t he seve ral models t hey tested against exp erimental re s ults of the o zon e decomposition reac tion in a la rge two- dimensional fluidi zed bed , al though they found some of its si mplifyin g assumption s to b e self-compensat in g . A m uc h more sop hi sticated treatment o f the same basic model has been developed by Potter and his co - workers w ho included a consideration of reaction in the den se phase of the bed and incorporated equation s to allow for the chan ge in bubble si ze with bed height . T heir so -called coun ter c urren t b ac k mixi n g model has b een well reviewed by Potter ( 19 7 8 ) . S hel l Model
The t wo models disc ussed earlier althou gh firmly based on the t wo - p hase theory o f fl uidi zation , do incorporate e mpirical relation s hip s derived from experimen t s with laboratory - scale e quip ment , and this has c aused the validity o f their applicat ion to large in dustrial units to be questioned . A number of di ffe rent approache s to the modelin g o f large beds have there fore been explored , a n d althou gh most r emain hidden behin d t he cloak o f industrial sec recy , o n e suc h app roach h a s b een disc us sed fairly fully in the open literat u re ( van S w aaij and Z uiderwe g , 1 9 7 2 , 1 9 7 3 ; de Vries et al . , 1 97 2 ) . The S hell company , faced with de signin g of fluidized bed reactor for their solid-cataly s e d hydrogen chlori de oxidation process , adopted a combined theoretic al an d empirical attac k on the problem in whic h data from a number of beds of in creasin g d iameter were used to obtain scaleup corre lat i ons . T he t heoretical b asis w as t he t w o - p hase theory model o f van Deemter ( 1 9 6 7 ) , w hich as sum es that : 1.
2. 3.
B ubble g a s i s in plug flow . Gas th ro u gh - flow i n the dense p hase i s ne gli gi ble . Gas mixe s i n t he dense p hase b y a p roces s o f eddy di ffusion .
A s with the David son - Harrison mode l , mass b alances on reactant gas flow in g into t he two phase s leads to an equation for the bubble phase :
Rowe and Ya tes
4 86
( 1 0 2)
( 1 0 3)
Here l; is the dimen sionless bed height ( h / H ) , dimen sionless groups defined as follow s : 1.
2.
3.
T he is a T he fi is is a T he rate
an d
N � , N E , and N r are
n umber of in terphase transfer units , N � == k' H /U , where k' g g mass tran sfer coefficient number o f interstitial phase mixin g unit s , N E = U H /fi E where the volume fraction of bed occupied by interstitial gas a n d E
mixin g coefficient
number of reaction units , N r c onstant for dense p h as e
=
k'H / U , where k' is an effective
The t wo hyd r od y n a mic groups N � and N E were found from the results of tracer experiments in beds 0 . 3 , 0 . 6 , an d 1 . 5 m in diameter . T hus from the residenc e- time distribution of tracer gas , the follo w in g expression for the height of a transfer unit , H � , was obtained : ( 1 04)
Then N� H /Ho.. E q uation ( 1 04) was predic ted to be valid for the micro sp heroidal catalyst in beds with heights and diameters of up to 10 m . T h e term N E was found from values of E mea sured in experiment s with labeled solid p articles in pilot - scale beds up to 3 m in dia m e te r ; E is plotted as a function of bed diameter in Fi g . 2 0 , and values of Nr and N � are given as functions of solids fines content and gas velocity in T able 7 . The dim e n s i on s of the full- scale reactor were calculated in the following w ay . B ed diameter was found in a strai ghtforw ard way from known stoichiometry of the process and a reali stic value of the fluidizin g gas velocity . An estimated figure for bed height was put into E q . ( 1 04) and a value of H � , a n d hence N o. • calculated . T h e latter , alon g with t h e e m p iric al N E an d N r term s , were then substituted into the solution s of the two simultaneous equations ( 1 0 2 ) an d ( 1 0 3) and an overall reactant conver sion calculated . T he difference bet ween t his calculated conversion an d the required value w a s t hen adjusted by c hoosin g a second value of the bed height and re peat1n p,- the c alculation . The method , a lthou gh costly in terms of pilot plant work was claimed to re s ul t in a s a tisfactory final de si gn . =
Model fo r R ea c t i on i n
t h e D i s t ri b utor Region
As mentioned earlier , there is experimental evidence to show that up to the point of detac hmen t of a b ubble from an orific e in t h e distributor of a fluidized bed , t he volumetric flow of gas throu gh the orifice is greater than that leadin g to the fo r m ati on of a visible bubble , an d furthermore
487
Flui dize d - B e d Reac tors
1 .0
( SPRAY · DRIED S I L I C A , 24"1o FINES )
( CRACKING CATALYST MAY )
8
6
4
2
2
•
0.01 !:----!------L---±---';-----::! 0 1 2 3 4 5
D(m)
FI G U RE 2 0 Mixin g coefficie n t a s a fu nc t ion o f bed diameter . de V rie s et al . 1972 . )
( F rom
,
that e qualization of flow is observed only after t he bubble has risen some distance into t he bed . The sequence of events i s illustrated in Figs . 1 1 and 1 2 , and the conclusion that i s drawn from the visual evidence i s that durin g the p e r iod up to the ti m e of detachment of the b ubble , t d , the un accounted or· invisible gas has p a s s e d into the den se phase , causin g the particle s to expand to a void fraction e: in e xc e ss of Emf . The time t d is clearly re l at e d to the fr equ enc y o f b ubb l e formation n ( H z ) by =
TABLE 7
Percent
1
( 1 0 5)
n
E mpiric al Parameters Used in Shell Chlorine Process Model
u
C on ve rsion relative to e quili brium
( m)
Nr
Na
( m)
H
Ha
fines
( m /s )
20
0. 2
95. 7
10. 0
15
4. 0
2 . 50
17
0. 22
95. 0
10. 4
20
3. 5
2 97
12
0. 2
93 . 5
9. 7
33
3. 0
3 . 26
7
0. 2
91 . 0
10. 0
34
2 . 56
3 . 86
So u rce :
de V rie s e t al . ( 1 9 7 2 ) .
.
4 88
Rowe a n d Yates
and during this period the volume of gas en terin g the den se phase and re acting is Q ot d ( l - x o ) , where x o is the fraction of flow that forms the visible bubble . Further, if we assume that the gas enterin g the dense phase undergoes a pseudohomogeneous first-order reaction with the particle s at a rate given by ( 1 06)
where k s is the first -order rate constant , the fraction of reactant unconverted after time t i s
A
re
mainin g
= ex
p [ - ( 1 - £)k s t ]
( 107)
Now if the time taken for the bubble gas to reach equilibrium (i. e . , time for the bubble to form completely ) is t e and if the mean residence time of t he ga s in th e expanded dense phase is 0 . 5t e , t hen E q . ( 1 07) becomes the
CA
CA O
=
exp [ - 0 . 5 ( 1
- £) k
s
te ]
( 108)
When the bubble has formed completely , t h e gas i t contains will be made up partly of that which has been in contact wit h the dense phase and partly of that which has not , so that a t equilibrium the fraction of A re maining unreacted in side the bubble will be C Ab CA O
=
=
Q Ot d x O
xo +
+
Q Ot d ( l Q�d
- x O ) ( CA / C A O )
( 1 - x o ) exp [ - 0 .
5( 1
( 109)
- £)k s te ]
Assuming that a value of the rate constant k s is available from auxiliary experiments in , for example , a fixed-bed reactor , the b ubble - p hase concen tration of reactant gas may be calculated once the three remaining parame ters x o , £, and t e are known . In the present work the proportion of incoming gas appearin g as a visible bubble at the orifice ( x o 0 . 36) and at a position 2 5 em above the orifice ( X 2 5 0. 79) have been measured , so by a lin ear extrapolation from these two values the height at which x = 1 . 0 , and hence at which the bub ble gas has reached equilibrium , may be found . I f this height is H e , the time for the bubble to reach equilibrium is clearly =
=
( 1 1 0)
so that if the b ub b le velocity is known , the residence time of gas in the expanded p ha s e may be calculated . Now the velocity of a bubble in a
489
Reac t ors
Fluidize d-Bed
fluidi zed b e d is normally expressed in term s of its diameter , d b , by the D avie s - T aylor relation ship ( 1 1 1)
the observe d coefficient of p roportionality bein g unity . As approximately one - fourth of the volume of the sp here centered on t he bubble is occ upie d by b ubble wake , the volume of gas for min g the bubble is v
b
=
ub
6
d
( 1 12)
3 b
Q Or
=
X
--
( 1 1 3)
=
(�)1/3v�/3
( 1 14)
H ence
�
�4 2!:
= =
0
n
b
(ft'2(� 0.
t
259V�/ 6
'
\
�1
an d combinin g Eqs . ( 1 1 0) . ( 1 1 3) . and t
3 . 8 6H
e
e
( ::)
1 /6
( 1 1 5)
6
( 11 6) ,
( 1 16)
w e have
( 1 1 7)
T he value of the third parameter i n E q . ( 1 0 9) , t he voidage of the expa n de d den se p hase , e: , is at present unknown , but reason able values in the ran ge 0 . 4 to 0 . 7 may b e assumed for the purpose of an illustrative calculation . C ombinin g E q s . ( 1 09) and 1 1 7 ) , the frac tion of reactan t A re mainin g un reacted in the b ubble phase at a hei ght H e above the distrib u tor i s
(
( 118) Exam p l e T he value of H e c alculated from t he foregoin g measurements of visible bubble volume at two bed heigh ts ( 0 an d 0. m above the orifice ) i s 0. 3 7 5 m . O ther values used in the calc ula tion are as follow s :
x
o
=
o . 36
7 Hz
25
4 90
Qo
k
=
5
£
10
-4
m3 /s
1, 5, 10 s
=
s
X
Rowe and Y a tes
-1
0 . 4 , o. 5 , 0. 6, 0 . 7
T able 8 give s t he calc ulated values of C A b /C A O at hei ght H e above the distrib utor an d it i s clear that , pa rticularly in the case of fast reaction s , the mode l predicts c on side rable conversion in the distributor zone . R e a c t i on s in t h e F reeboa rd
When b ubbles burst at the surface of a fluidi zed bed , the soli d particles released into the freeboard can cause fu rther reac tion to take place with any gas that has passed t hrough the bed un reac te d . Reaction in the free board region is not s ub ject to the same c on straints as that which occ urs in the bed itself , and it can give rise to severe temperature excursion s in the gas space above the s urface , l eadin g to damage of both bed structures an d catalyst particles . I t is well known , for example , that uncontrolled reaction s in the free board of fluidi ze d - bed pht halic anhydride plants were the cause of con side rab le problems , as also were afterburnin g reac tion s i n catalytic cracker re generators . Yates and Rowe ( 1 97 7 ) developed a simple model of a catalytic reaction b ased on the assumption that the freeboard contained p erfec tly mixed , equally disp ersed particles derived from b u b ble wakes . T he p hysical form of the model is illustrated in Fig . 2 1 . A ss umin g the two-phase theory to apply to the fluidized b e d an d the frac tion of wake particles ejec ted above the surface to be f , the rate of pa rticle ejec tion is =
f( U - u A ( l - e: ) mf mf )
( 1 19)
3
TABLE 8 Calculated Fraction Unconverte d Leavin g Distrib utor Zone
e:
0 . 40
k
s
=
1 s
0 . 876
-1
k
s
=
5 s
0 . 577
-1
k
s
=
10 s
0 . 4 34
0 . 50
0 . 8 95
0 . 620
0 . 60
0 . 914
0 . 672
0 . 51 2
0 . 70
0 . 934
0. 733
0 . 577
0 . 46 6
-1
4 91
Fl ui dize d-Bed R eac tors
Bed surface
Particle vol u me
B asis of freeboard reaction model .
FI G U R E 2 1
1 97 7 . )
( From Y ates and Rowe ,
w here Q B is the volumetric flow rate of b ubble ga s and A is the bed cros s sec tional area . T h e total holdup o f particles i n the freeboard , V , i s v
=
( 1 2 0)
where Hp is the freeboard height and UT is the particle terminal fall velocity . The freeboard voidage is then e:f
=
1
_
[f(
1
3
-
e:mf)
(u
u
-
umf)� u
T
Now if the particles a re equally sp aced , the v ol ume Vc of the gas cell surroun din g each particle is given by
( 121)
4 92
Rowe and Y a tes
v
3V
c
p
(�
( 1 22)
T he rate o f chan ge o f conc entration of where V p is t h e partic le vol u me reac tan t A within a cell is a fu n c t ion of the particle surfac e area A p , the concentration drivin g forc e ( C A h - C A p ) , an d a ma ss tran sfer coefficient hM :
h A M
- - � (C dC
A
-
dt
V
c
Ah
.
- C
Ap
( 1 2 3)
)
while the rat e of c han ge of C A at the p art icle surface i s determined by th e rate of chem ica l reac tion . I f the freeboard ga s is a ss umed to be in plu g flow , the e quation s lead to the followin g exp ression for t he fraction of re actant unconverted at any hei ght h above the bed surface :
CAh {
C
As
exp
:
-
3( U
fu
-
UT )
[ .., ( :•_ •
'm r >
+
�]
}
w here C A s is t he r e ac t an t conc entration at the bed s urface an d k is an assumed firs t - order reaction rate constant . The correspon din g equation for mixed flow of gas in the freeboard is
1
( 1 2 5)
3(U -
The mas s transfer coefficient h may be foun d from the expression of Rowe M et al . ( 1 9 6 5a ) :
( 1 2 6) where t he S h erwood n umber , Sh the
=
h d M p D
( 1 2 7)
Schmidt n u mber , Sc
= l pD
( 128)
4 93
Fl uidize d - B e d Reac tors F l u r d Bed
�
0. 8 �:00. 6 0.4 02 0o 0. 2 0. 4 0 6 08 1 0 Complete
- Plug Flow
- - - -
M rx r ng
H
�
FI G U R E 2 2
Entrainment Fractron, f
1. 2
1. 4 1.6 1.8 2.0
Conve rsion versus b e d height for k
=
5 s
-1
an d the Reynolds n u mber , U =
d p T p
( 1 2 9)
T he terminal fall ve locity may be obtained from graphical correlation s be t ween Re T and the velocit y in depen dent group 2dp p g ( P s - P g ) / 3 ]..1 2 ( Kunii and Levenspiel , 1 9 6 9 ) , an d hence the Sherwood number an d h M may be c alc ulated .from E q s . ( 1 2 8 ) and ( 1 2 9 ) . S olution s to the equation s were obtained for various combination s of the parameters , and it was shown that under some circ um stance s a greater level o f conversion can b e obtain ed in the freeboard than in the fluidi zed bed it self ( Fi g . 22) . Experimen tal sup port for these conclu sion s ha s b een p ro vide d by M iyauchi an d Furusaki ( 1 97 4 ) an d Furusaki et al . ( 1 9 7 6 ) , who have also develope d theoretical mocels of the effec t . Conc l us i on s I t would s e e m from t h e disc ussion presen t ed i n t hi s section that t hree zones First , the need to be considered in the desi gn of a fluidized-bed reac tor : entry zone n ear the di stributor where fresh reactant gas fir st comes into c on tac t with bed p articles an d where b ubbles are in the early stages of developmen t ; sec on d , th e fl ui dized bed it self , w here in general reaction is controlled by mass t ran s fer between the b ubble and in terstitial phases ; and thi rd , the freeboard zone above the bed surface . The necessary de sign equation s for e ach zone are q uite different and must be examined thorou ghly for the full ran ge of potenti al operating variables be fore the overall per formance of a reactor can be p redicted with con fi dence . NOTA T I ON
c i s area o f bed , m
2
concentration of r eactant A , mol / m
3
494
Rowe and Ya tes
specific heat of gas , J /mol • K 2 molecular diffusivity , m t s
drag coefficient ( sin gle particle )
b e d diameter , m
�
f B F
particle diameter , ll m volume fraction b ubble holdup
g
drag force on a sin gle particle , N 2 acceleration of gravity , m t s
h
height above distributor , m
h h
B /w G /s
2 bed to wall , W / m • K 2 gas to solids W /m K
heat transfer coefficien t :
,
heat tran sfer coefficien t :
H
total bed height above distributor , m
K
thermal con ductivity , W /m K
K
a constant
K
•
•
G /s
mas s transfer coefficient , m /s
m
exponent in E q . ( 3 1 )
M
total mass of bed , k g
M
simple mixin g in dex
n
bubble frequency crossin g horizontal plane , s
n
exponent in E q . ( 4 5 )
N
bubble number concentration , m absolute press ure , N t m 2
Q p
X / x ( Fig . 1 3 )
=
volumetric flow rate ,
-3
rn 2ts
s
pa rticle external surface per unit volume , m
u
s uperficial fluid velocity , m / s
VB
bubble volume , m
3
G re e k Let te rs
velocity ratio ( UB / U i ) L'l p
pressure drop across the b e d , N t m bed voida ge
\..1
fl uid viscosity , N • s tm
2
powder b ulk den sity , k g / m fluid den sity , k g / m
3
particle density , kg /m
3
3
-
2
-1
1
495
Fl ui dize d -B e d Reac to rs S pec i a l N u m be rs Nu
Pr
Re
Sc
Sh
(h
d /K ) G /s p Pran dtl n um b e r ( C J.l / K )
N u sselt number
G R ey no ld s n umber ( U d p / ).1 ) p Schmi dt n u m b e r ( ).1 / p D ) Sherwood n umber
(K
S u bscri p t s
B
D
F
flotsam
f
G
H i I
flu i d ga s
plane a t height h
in te r s tit ial isolated
mf mb
minimum b ubblin g
T
t e r min al
s
w
/D )
distributor plane
jetsam
J
d
bubble
clou d
c
G /s p
min im um fluidi zation
solid ( particle)
wake
fallin g
R E FE R E N C E S
B achovchin , D . M . , J . M . B eer , and A . F . S arofim , Elutriation from fluidi zed beds , AI ChE 7 2n d Meet . , S an F rancisco ( 1 979) . B aeyen s , J . an d D . G e ld a r t Predic tive c alc ulation s of flow parameters in ga s fluidi zed b e d s and fluidi zation b ehavi o ur of various powders , Int . Symp . Fluidi zation Appl . , Toul o us e ( 1 97 3 ) , p . 2 6 3 . B o t t e rill , J . S . M . F l ui d -B e d H ea t T rans fe r , Ac ade mic Press , London ,
,
( 1 9 7 5) .
and J . R . Grace , Perfor mance analysis of a fl ui di z e d bed reac tor , Ind . En g . C hern . Fun da m . , 1 4 , 7 5 , 79 , 8 6 ( 1 9 7 5) . Chavarie , C . and J . R . G race , Interphase mass transfer in a ga s flui di z e d bed , C hern . Eng . Sci . , 31 , 7 4 1 ( 1 976) . C hiba , T . an d H . K obayashi , G a s exchan ge between the b ubble and emul sion p ha s e s in ga s s o l id fluidi zed beds , Chern . Eng . Sci . , 2 5 , 1 3 7 5 ( 1 970) . -
Chavarie , C .
-
-
4 96
Rowe and Y a tes
Colakyan , M . , N . C atipovic , G . Jov anovic , and T . Fit z gerald , Elutriation from fluidi zed beds with immersed heat tran s fer t ubes , AI ChE 7 2nd M eet . , S an Franci s co ( 1 9 7 9 ) . Darton , R . C . , A bubble growth t heory of fluidized bed reactors , Trans . Inst . C hern . En g . , 57, 1 34 ( 1 97 9) . D arton , R . C . , R . D . La Nauze , J . F . Davidson , and D . Harrison , Bub ble growth d ue to coalescence in fluidi zed beds , T r ans . I nst . C hern . Eng . , 5 5 , 2 7 4 ( 1 97 7 ) . D avidson , J . F . and D . H arrison , Fl uidized Particles , C am brid ge U nive rsity Pres s , Cambridge ( 1 9 6 3 ) . D avidson , J . F . , D . H arrison , R . C . D arton , an d R . D . La Nauze , C hemi ca l Reactor T heo ry , Prentice - H all , Englewood Cli ffs , N . J . ( 1 97 7 ) . D avies , R . M . a n d G . I . T aylor , The mechanics o f lar ge bubbles rising thro u gh extended liquids and throu gh liquids in t ubes , Proc . R . Soc . , A 200 ,
37 5 ( 1 9 5 0 ) .
de Vrie s , R . J . , W . P . M . van S w aaij , G . Mantovani , and A . Heijkoop , Desi gn cri teri a and performance of the commercial reactor for t he S hell chlorine proces s , Proc . 2nd I nt . S ymp . C hern . React . Eng . , Am sterd am ( 1 97 2 ) , p . B 9- 5 9 . Ergun , S . , Fluid flow t hrou gh p acked columns , Chern . Eng . Pro g . , 4 8 , 89 ( 1 9 52 ) .
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.
( 1 969) .
Leva , M . , Flui diza tion ,
M c G r aw - Hill ,
New
York ,
( 1959) .
Fl ui di ze d -B e d
Reac tors
497
, and C . Y . We n , Elutriation and attrition of char bed , Powder Techno! . , 2 7 , 1 0 5 ( 1 98 0 ) . Merrick , D . , and J . Hi ghley , Particle si ze r e du c ti on and elutriation in a fluidized - bed proces s , A I ChE , Sy m p . Ser . , 70( 1 37 ) , 36t3 ( 1 97 4 ) . M iyauc hi , T . an d S . F u ru sa k i , Relative contribution of variables a ffe c tin g the reaction in fluid bed contactors , AIChE J . , 20 , 1 0 8 7 ( 1 9 7 4 ) . M urray , J . D . , O n the m athematics of fluidi zation , J . Fluid Mech . , 22 , 5 7 ( 1965) . M u rray , J . D . , Mathemati cal as p ects of bubble motion in fluidi z ed beds , Chern . Eng . Pro g . Symp . Ser . , 62( 6 2 ) , 57 ( 1 966) . P art ri d g e , B . A . and P . N . R o w e , Chemical reaction in a bubbli n g gas fluidized bed , T r ans . Inst . Chern . Eng . , 4 4 , 3 3 5 ( 1 9 6 6 ) . Potter , 0 . E . , Mo d e li n g fluidi zed- bed reactors , C at a! . Rev . - S ci . Eng . , 1 7 , 1 5 5 ( 1 97 8 ) . Pyle , D . L . , Fl ui di z ed - bed reactors : a review , Proc . 1st I nt . S ym p . Chern . Re act . E n g . , W as hi n g to n , D . C . ( 1 9 7 0 ) , p . 1 0 6 . Reid , R . C . and T . K . S h er w oo d , The Prop erties of L i q ui d s a n d Gases , M c Graw - Hi ll , New York , ( 1 9 5 8 ) . Ri c h ar ds o n , J . F . , Incipient fl ui di z atio n and p articulate s ys t em s , in Fl ui diza tio n , J . F . D avi d s on and D . H arrison , e d s . , Academic Press , London ( 1 97 1 ) . Rich ardson , J . F . a n d W . N . Zaki , S e dimentation and fluidization , T rans . I ns t . Chern . E n g . , 32 , 35 ( 1 9 54 ) . Ri e t e m a , K . , Appli c a ti on of mechanical s t res s t h eory to fl ui di z ati on , I nt . Sym p . Fluidi z ation , Eindhoven ( 1 9 6 7 ) , p . 1 5 4 . Roberts , A . G . , P . Raven , R . N . Phillips , S . N . B arker , K . K . Pillai , and P . Wood , PFB C - a 1 0 0 0 h o u r p rogram me , in F l u i diz e d Comb us tion , I ns t . Energy S y m p . Ser . , No . 4 , 3 . 1 . 1 ( 1 9 8 0 ) . Rowe , P . N . , D r a g forces in a hy d r auli c model of a fluidi zed bed : Part I I , T r an s , I nst . C hern . En g . , 3 9 , 17 8 ( 1 96 1 ) . Row e , P . N . Experimental p roper ti e s of bubbles , in Fluidiza tion , J . F . David son and D . H ar ri so n , eds . , Academic Press , London , ( 1 9 7 1 ) , C h ap . 4 . Row e , P . N . , Fluidi sed bed reactors , Proc . 2nd Int . S y m p . C h e rn . R e act . En g . , Ams terdam ( 1 97 2 ) , p . A 9 . Row e , P . N . Estimation of so lids circulation rate in a b ubbling fluidised bed , Lin , L .
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28 ,
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·
•
498
Rowe and
Yates
Rowe , P. N . and B . A. Partrid ge , An x - ray study of bubbles in fluidised beds , Trans. Ins t . C hern . E ng . 43, 157 (1965). Rowe , P. N . , K . T . Claxton , and J. B . Lewis , H eat and mass transfer from a single sphere in an extensive flowing fluid , Trans , Inst . Chern . Eng . , 43, 14 (19 6 5a ) . Rowe , P . N. , B. A. Partridge , A. G . C heyney , G . A. Henwood , and E. Lyall , The mechanism of solids mixin g in fluidised beds, T rans . Inst. Chern. Eng . , 43, 271 (1965b). Rowe , P . N. , T . J . Evans , and J. C. Middleton, Transfer of gas between bubbles and dens e phase in a two-dimensional fluidised bed , Chern . Eng . Sci . , 26, 1943 (1971). Rowe, P . N. , L . Santoro , and J. G . Yates , Division of gas between bubble and interstitial phases in fluidised beds of fine powders , Chern . Eng . Sci., 33, 133 (1978). Rowe , P . N., H . J . M acGillvray , and D. J. C heesman , Gas dischar ge from an orifice into a gas fluidised bed , Trans . Inst. Chern . Eng . , ,
57, 194 (1979).
Rowe , P . N. , P. U Foscolo , A. C . Hoffman , and J. G . Yates , Fine powders fluidised at low velocity at pressures up to 20 bar with gases of different viscosity , C hern . En g . S ci . , 37, 1115 (1982). Rowe , P . N . , P . U. Foscolo , A. C. Hoffm an, and J . G . Yates , X- ray observations of gas fluidised beds under pressure , in Fluidization IV, D. Kunii and R. T oei , eds. , Engineerin g Foundation , New York (1984), p. 5 3. Sit, S . P . and J . R. Grace , Interphase mass transfer in an a:ggregative fluidised bed, Chern . Eng . Sci . , 33, 1115 (1978). Sit , S . P. and J. R. Grace , Effect of bubble interaction on interphase mass transfer in gas -fluidized beds, Chern . Eng . Sci . , 36, 3 2 7 (1981). Tanaka , I., H . Shinohara , H. Hirousue , and Y . T anaka , Elutriation of fines from fluidi zed bed , J. Ch e rn E n g . Jpn . , 5, 5 1 (1972). Toei , R., R. Matsuno , H . Miyigaw a , K . Nishitani , and Y. K omagawa , Gas interchange between a bubble and the continuous phase in gas solid fluidized bed s , Int. C hern . Eng . , 9, 358 (1969). van Sw aaij, W . P. M . , T he design of gas - solids and related reactors , Proc . 5th Int . S ymp. C hern . React . Eng. , Houston (1978). van Swaaij, W . P, M, and F. J. Zuiderweg , I nvesti gation of o zone decom position in fluidized beds on the basis of a two-phase model , Proc. 2nd Int. Symp . C hern . React . Eng . , Amsterdam (1972), p . B9-25. van Sw aaij, W. P. M . and F . J. Zuiderweg , The design of gas-solids fluidized beds : prediction of chemical conversion, Proc, Int . Symp . Fluidization Appl . , Toulouse (1973), p. 454. van Deemter, J . J. , T he countercurrent flow model of a gas - solids fluidiz_ed bed , Int. Symp . Fluidization , Eindhoven (1967), p . 334. Wen , C . Y . and L. H . Chen , Fluidized bed phenomena-entrainment and elutriation , AIChE J . , 28, 117 ( 1 98 2) . Wen , C . Y . and R. F. H ashinger , Elutriation of solid p articles from a dense phase fluidi zed bed , AIChE J. , 6, 220 (1960). Yagi , S. and T. Aochi , Elutriation of particles f rom a batch fluidi zed bed , Soc . Chern. En g . Japan, Spring M ee t . (1955). Yates , J. G., Fluidi zed bed reactors , Chern . E n gr. , 671 ( 1 975) . Yates , J . G. and P . N. R owe , A model for chemical reaction in the free board re gion above a fluidized bed , T rans . I nst . Chern . Eng . , 55, 137 •
.
(1977).
Zenz , F . A. and N . A . Weil , A theoretical-empirical approach to the mechan ism of p article entrainment of fluidi zed beds , A I ChE J. 4, 472 (1958). ,
8 Coal Gasification Reactors MORTON M. DENN
University
REUEL SHJNNAR
New
of California, Berkeley, California
The City College of the City University of New York,
York, New York
INTRODUCTION C o al gasific atio n reactors are devices that c onta ct coal with r ea ctive gases,
usually oxygen and ste am
h ydrogen or me thane
,
to prod uce gaseous prod u ct s , generally CO and
Proposed or existing con figu ra ti o ns include fluidized
and countercurrent mov ing (nonfluidized) beds and ent rai n ed flow reactors; .
pro d u ct c omp os iti ons depend on the coal type and the reactor configuration. The
dry ash free com po si tio n
of a typical eastern United States bituminous
coal is ro ughly CHo. gO o .l· In excess of 40 wt% of this material will volatilize with slow h eatin g , yi eldi ng a p roduct cont aini ng tars, oils, water, methane, hydrogen, CO, CO2, a nd other gase ou s products. More of the coal may
volatilize with very rapid heating.·
mos tly of carbon and in or ganic ash.
The porous residue, or
char,
consists
Typical proximate and ultimate anal yse s of Illinois No. 6 bituminous coal,
an eastern U.S. coal which is u sed subsequently for illustrative purposes,
are sh own in Table 1.
The following pri mary reactions can be expected to
take place between the carbon in the char and the gaseous rea ct ant s : O xi dati on :
c
+
21 02
C + 0
AH
t co
t C02
2
26 4
(1)
-94.2
( 2)
liH =
+32.2
( 3)
+41. 4
( 4)
liH
=
=
-
.
Gasification: C
+
H20
� CO
C
+
C02
� 2CO
liH
�
2 2 liH=- 0.
(5)
t.H=+l.4
( 6)
C + 2H
2
CH
4
+
H2
=
499
Denn and Shinnar
500 TABLE 1
Typical An al y sis of Illinois No. 6 Coal
Wt % Proximate Ash Moisture Vola tile matte r
9. 6 4.2
34.2 52.0
Fixed carbon
Wt % dry ash free ba si s
Ultimate c
77.3
H
5.9 1. 4 4.3
N
s
0
11.1
In a ddi tion, three important gas-phase r eactions may take place: W at e r - gas shift:
Me tha nati on
H20 3H 2
O xi d a ti on :
+
CO
+
1
H2 +
CO
2
°
2
+ + + + + +
H 2
C0
+
H20
+
CH
=
-9.2
(7)
=
-52.4
(8)
�H = -58.6
( 9)
�H
2 4
H 0 2
�H
Reactions (7) and ( 8 ) can also take plac e be tween the sa me sp ecie s in the
d evolatilization p ro duc t s ; in addition, the vol atile s are combu st ibl e , a nd the
The catalytic activi ty of the as h which differs from c oal to coal, i s li ke ly to affect the ra te s of both the h e te rogeneo u s and gas-phase reactions. Many othe r re actions be tween the major species mi ght als o occur, but othe r r ea cti o ns of poss ibl e intere st are derivable fro m thi s basic set. The me thanati on reaction
higher-molecular-weight consti tue nts might crack. ,
+ +
!cH 4 +!co 3
3
i s si mpl y the sum of o ne t hi rd reaction (5) e xample. Only four reactions are, in fact, cal species made up of thr ee ele me nts, but here because of the ir possible importance . (2) to (5) and (7) fo rm useful in de pe nden t -
an d two-thirds reaction (3), for in de pe ndent for t he seve n ch emi the re mainin g five are included Reactions ( 1 ) to ( 3) and (5) a nd sets.
USES OF MEDIUM-BTU GAS
Thi s chap te r is c onc e rned main ly wi th me di um- btu g asifi ers
a g eneri c
name for gasifiers in which the gasification me dium is a mixture of steam ,
Coal Gasification Reactors and oxygen.
501
Air- blown g asi fiers , which are normally an i ntegrate d par
of power plants, will not be treated here.
t
The gas obtained from medium
n
btu gasifiers has several uses, each of which requires a diff ere t gas com position an d hence affects des g n considerations.
i
Fuel Gas Production
l
Medium-btu gas is a c ean fuel that can directly substitute for natural gas
in
existing boilers and many other industrial applications and is also suitable
i
The dis advanta ge of medium-btu gas for
for combined-cycle power p ants.
fuel gas use is that it is expensive to pump over long distances. For fuel gas uses the major criterion is the lower heating value, which is approximately p roport ional to the n umbe r of moles of CO + 0.85H2 + A high CO conte nt is there 2.85CH4 produced per mole of carbon or coal. (It is still com fore preferable, and the shift reaction (7) is undesirable. mon practice to use the higher hea ng value, which is proportional to CO + Hz + 3CH 4, and we sh al sometimes follow this practice. Effi ciency based on lower heating value is close in value for coal conversion processes to the e fficiency based on available free energy. )
ti
l
SNG
r
Altho ugh interest in pu e SNG production h as presently subsided, it was the main driving force for many recent developments in gasifi ca tio n tech no o gy There are two criteria that d st in s h SNG p rod uc o from fuel gas and syng a s g asi er s , based on the fact that the total yiel d of methane per mole of coal is p orti nal to CO + Hz + 4CH 4:
l
.
fi rop
i
gui
ti n
o
CH4 in the outlet of
1.
Maximizing
Z.
and · methanation cost.
the gasifier
redu ces
gas cleanup
The ratio of H 2/CO has to be larger than unity and higher ratios Otherwise, the gas has to be shifted. are preferable. ,
Syngas
CO a nd H 2 are used in the man uf ac ture of many chemicals. They can also be converted to transportation fuels, either via a Fischer-Tropsch proc es s or via m e tha nol and further conversion of methanol by zeolites. For each In all such cases any methane use a special Hz/CO ratio is required. formed in the g a i er is a by-product that h a s to be separated from the product. This by-product mP.thane can either be sold or reformed to syngas.
s fi
The need to separate methane does not necessarily imply that for syn gas it is always desirable to minimize methane production. As we will show
fi
later, gasifiers with high thermal efficiency always produce si gni c a nt amounts of methane, and el m na ting methane production imposes severe penalties on the design. These pe na lties therefore have to be weighe d against the cost of separating the methane from the final produc t The penalty is e ry low in the pro duct on of met hano and in some Fischer Tropsch proc esses but is significant for many chemicals. The sep a ration
i i
i
v
l
.
,
penalty also depends on the scale of the plant and the value of the methane by-product.
In a synfuel plant that is coproducing S NG and liquid fuels, the total clean fuel has a higher heating value that is proportio nal to CO + H 2 + 4C H 4 ,
502
Derm a n d Shinnar
and it is i nt e re s ti ng tha t this i s almost i ndependen t of the fuel c ons u mption as lon g as we de al w ith pure h ydro car b on s . GASIFICATION AND PARTIAL COMBUSTION
dis ti ncti on between gasificatio n and partial combus si n ce they correspond to di ffere nt mode s of re actor op erati on . By p art ial combustion we mean a proce ss in whi ch the oxygen supplied to the gasifier i s sufficie nt to conv ert a ll c arbon to CO . In that case , the role of s team in t he feed is simply to shift a po rtio n of the product to hydrogen. In co n tr as t steam is re qui re d for conversion of at l e as t a portion of the carb o n in gas ifi c at ion . Conside r a coal with a com po si ti on CHaOb; a is typically 0.8; b is ty :.>i c ally 0.1 for a n e as te rn U.S. b it um i no u s coal a nd 0.2 f or a we st ern U. S. s ub bituminous coal. Linear com bi n atio n of the basi s reacti o ns (1) to ( 3) and (5) lea ds to t he overall equation It is use ful to draw a
tion,
,
S uR - b - 2m ) C O
+
+
( 2R
+
Su R + b + m -l) C0
2
+ (suR+�- )
2m H
2
+ mCH
S pRH o 2
4 (10)
- -,
R is the molar ratio of oxyge n in the fe ed to c arb on in the coal. Su + Sp is the feed s te a m t o oxy gen ratio; Su denotes steam utilized for gas ifi c a tio n or t he shift rea c ti on while Sp de note s unre acte d ste am that leav e s with the product ga s .
m is the molar production rate of m et hane
reaction or t h ro ugh volatile ev olu ti o n .
,
e ith e r t hro ugh
Thi s overall re acti on syste m has an imp o r tant invariant: the tota l H 2) pl us four ti m es the methane pro duction per mole of carbon feed. We de note this i nvaria nt by I:
molar produc ti on of syngas ( C O plus
CO + H + 4CH 2 4 c
I
a = 2 + --b- 2 R 2
(11)
For a given coal (a and b fixe d) a nd e sse nti ally complete carbon conversion, this combination o f pro du cts is uniquely fixe d by the oxyge n - to - c arbo n fee d ra ti o a nd i s in de pen de n t of ga s ifier type . An alterna ti v e form whe n there is no CO 2 i n the feed , based only on pr od uc t ga s mole fracti ons , i s
CO + H
I
=
2 CO + C 0
+ 4CH
4 + CH 2 4
(12)
Thi s relation m us t be modified slig htly if carbon co nv e rsio n is not comp l e t e or if the feed contains C0 2 .
The reactor ope rate s as a partial comb us ter when Eq. ( 10) can b e ba la nced without the nee d to utilize ste am or fe e d C0 2 to e ffec t complete carbon conver s io n . For p urposes of our discussi on he re , it is mo s t
503
Coal Gasification Reactors convenient to neglect methane formation.
In t hat case the criterion for
partial combustion is
The minimum oxygen rate for partial combustion, then
R c
which we denote Rc, is
1 - b
= -2--
(13)
[We could attempt to be mo re pr ecis e in the de finitio n of Rc by acc ountin g for methane produced by devol atiliz ati on and considering the maximum amount of carbon that could be gasified by devolatilization in the p resence of H2. To do so would make the di sc ussio n depend ent o n reactor co nfi gura ti on . Equation (13) is conservative in that a react or that operat es as a partial combuster under this definition is a partial combuster under any m ore strin gen t definition, and it is thus useful for reac t or classification.] We wish t o emphasize that the defini tion of Rc is strict ly stoi chio m etri c . If R > Rc, we could in principle reach the final composi tion by first totally gasifying the coal with oxygen an d t hen rea c tin g the gaseous products with steam. The fact that in re ality the ox y gen- co al reaction may lead to C02 The and the coal will later rea ct with C02 and H20 is not relev an t here. value of Rc will typically be between 0. 40 and 0. 45. The kinetic processes occurrin g in the reactor are fundamentally different depending on whether R is greater or less than R c; in t he former case, carbon conversion will be l ar ge ly by o xid a ti on , and the steam is simply used to shift the product, while direct steam (and perhaps C0 2 ) gas ific ati on of car bon must occur in the latter case. Combination of Eqs. (11) and (13) leads to the following equality, bas ed solely on c o a l and product gas compositions:
(14)
It
Rc
is particularly impo rtant that pil ot studies be carried ou t in the same -
R re gi me as the planned commercial process, to ensure that scaleup
o ccurs under comparable ki n eti c conditions.
We shall retur n to this poi nt
in the discussion of specific ga sifers . The gross molar heating values of hyd ro ge n and CO are nearly the same, and approxi m at el y one-third that of m e t hane •
.
Thus the gross
pro duct h eatin g value per mole of carbon fed is approximat e ly proportional to 3m plus the sum of the coefficients of CO and H2 in Eq.
(10}; we donate
this sum as h:
h
=
1
+
2 (R ! 2 c
- R)
=
I
-
(15)
m
The significance of methane formation in a rea ctor for the production of syngas is illustrated in Eq.
carbon ratio.
(15 ) .
I i s fixed enti rely by the oxygen-to
For a given amount of oxygen, p roduction of methane is
detrimental to the he ati n g value of fuel or syngas, and the m ethane forma tion is of valu e only if it red u c es the amount of oxygen required.
As we
Denn
504
and
Shinnar
shall see when we di scuss st oichiom etry in more detail, formation of
me thane significantly reduces oxygen requirements because re action (6) h as a much lower heat of reaction than rea ction (3).
Finally, as we s hall discus s further below, overall effici ency is greatly
affecte d by the cost of feed steam and oxygen,
in g about four times as much as 1 mol of s team . to R ( 4
+
Su
+
with 1 mol of oxy ge n cost This cost is propor tional
Sp), and economic opti m ization of a syngas proces s in clu des
m aki ng t hi s quantity small whi le making Rc - R lar g e. M ini m izi ng R might be of less importance in the overall optimization than minimizin g S p ,
the unused s team appearing in the product.
It readily follows from the re
qui reme nt that CO and C02 have positive stoichiometric coefficients in Eq.
(10) that the ste am requi r ement for gasification and shift ( wi thout methane
formation) is bounded as follows: 2
- (R
R
- R) � Su � R 2
c
(
R
c
- R
+
1)
(16)
-
2
Comparison of the actual feed steam-to-oxygen ratio to the upper bound in
Eq.
(16) provides a minim um estimate of the excess steam.
Excess steam
is required ei the r to moderate temperature or du e to inherent thermody namic limitations.
GASI Fl ER TYPES There are four broad classes of reactor configu r ations : bed, entrained flow, fluidized bed, and molten bath.
moving (or "fixed")
The last of t he s e is
fundamentally different from the first three and is not discussed here .
Numerous d es i gns have been proposed within each class, a nd we will focus
here on generally applicable principles and a few representative designs
that are in or near commercial use. actor descriptions ,
There is a complete c ompilation of re
operating ranges , and oper ati n g histories in Nowacki
(1981), and v er y useful descriptions in Biss ett ( 1978) and the proceedings We focus here on reactor 1.981). design principle s as applicable to coal gasifiers. of an EPRI c onfe rence on synfuels (EPRI,
Countercurrent Moving Bed The prototypical movin g - bed gasifier is the Lur gi dry ash pressurized gasi fier, as shown schematically in Fig. operation in 1936.
1, which was first put int o commercial
The antecedents of this reactor can be traced to non
p r ess u rized town gas producers that have long been in use.
Coal with a
parti cle size range of about 6 to 30 mm (the reactor cannot accept coal fines that are formed during grinding) is fed through a lock hopper and moves
under gr avity countercurrent to the ri sin g gas stream. Steam and oxygen, air, or enriched air are fed through the bot tom and di s tributed throu g h the rotating grate, which is used t o remove powdered ash. A similar reactor configuration developed at the U.S. Department of Energy Morgantown Energy Tec hnology Center and imp lem ented as well in the General Electric GEGAS r eactor does not have a water-cooled wall but does have a stirrer that c an be moved thr ough the coal b e d . A ty pi cal effluent com position fr om gasification of Illinois No. 6 bituminou s coal with steam and oxygen in a Lurgi reactor is shown in Table 2 . Illinois No. 6 is a low reactivity,
505
C oal G asifica tion Reactors
�
Feed Coo I
Tor
Distributor
Gas
�
Grote Grote Drive
FI G U R E 1
Schematic of dry ash L urgi press urized gasifier.
2
TABLE Effluent Composition for Oxygen-Blown Pres s u ri zed Gasification o f Illinois No. 6 C oal
D ry ash Lurgi (Woodall-Duckham Lt d . ,
H2 co
C0
CH 4
2
C H 2 6
H S + cos
s Inert
39. 1 17.3 31.2 9.4 0.7 1.1
1.2
1974)
1 98 1 )
Texaco (McDaniel,
37. 5 39.5 21.5 1.1 0. 4
506
Den n
caking coal,
and Shinnar
and the choice of this particular example is for comparative
purposes only.
The current a ppli cations of this type of
gasifie r, such as
at SASOL, are limited to reactive, noncakin g coals, although successful
gasification of caking coals has been reported in stirrer-equipped reactors. The moving- bed gasifier can
be conceived as operating as shown in
The coal first contacts hot product gases and is dried.
Fig. 2.
volatiles are then driven off by relatively slow heating.
The
Next, the char
contacts a hot, o xygen - free gas, enabling the endothermic gasification re
actions ( 3) and ( 4 ) to occur. Finally, the remaining char enters a region of oxygen-rich gas, allowing combustion to take place. There are two important consequences of this reactor configuration.
First, the volatiles are released in such a way that they cannot react with
oxygen.
Thus the methane,
CO , and hydrogen from the volatiles appears
as do e s the tar.
in the product gas,
The condensible
product gas
coal
driven
Drying
(moisture
Zone
off)
Endothermic
(
)
Oevolatilization Zone coal gas, tar, and oil driven off
Thermally Neutral
Gasification Zone •
�
(little or no oxygen) Endothermic
N
c 0
u 0 •
0:
Combustion Zone (oxygen rich
g as)
Exothermic
ash
steam
+
+
unreacted carbon
FIGURE 2
air or oxygen
Schematic of moving-bed gasifier.
liquids are gener ally
Coal
507
Gasification Reactors
undesirable in the produc t gas and must be separa t ed by q ue nch i ng . (The was t ewa ter treatment can be quite e x pen s ive , particularly in a sm all installation. In larger ins tallations the tars could in principle be r e c ov e red as a valuabl e byproduct. One of the m ost i m port ant c ha l le nges in ga si fica tion res ea rch i s to overcome this pr o bl e m and devolatilize the coal withou t tar formation. This mi ght be done by dec ompo sing the tar in a cat al y tic reactor before cond ensi ng the steam.) D ev o latiliza tion is the primary source of methane in this r eac tor , since direct methanation of char is re la tively slow, although some pro d uc t methane undoubtedly c om e s from one or more of the me than e - fo rmi ng reactions. Second, go od control of the maxi mum t emperat ure in the bed is essential to ens ur e that the inorgani c ash remains in a dr y state for removal by the grate and d oes no t become sticky and a gglomerate. The ash softening te mp era tur e is ap proxi ma te ly 1200°C ( but highly d ep e n dent on coal type ) , and large amounts of excess steam ( typica l ly 7 m ol of steam per mole of ox ygen ) a r e requir ed to keep the temperature in t he combustion region below this value . The exces s steam is the primary reason for the relative thermal i ne ffi c i e ncy of t his type of gasifi e r . T he exce ss steam re qui re m ent for high-reactivity coals is less than for l ow - reac tivi ty coals, which par tl y exp lai ns why t he gasi f ier is used mainly for the former. The da ta in T abl es 1 a nd 2 le ad to a value of Rc - R of 0. 13 for the Lurgi with eastern c oal (an d a hi gher val ue if methane formation is inc lu ded in the defini ti on of Rc ) ; thus this reactor clearly operates in the gasific a tion regime.
The excess steam can be reduced in pri nciple if the grate is r ep lace d hearth and t he steam-to-oxygen ratio is reduced to a level t ha t al lows t he maxi mum t em pe rat ur e in t he combustion region to rise above the point at whi c h ash is a free ly run ning liquid; this temperature is typically about 1450°C but i s highly dep e ndent on c oal ty pe a nd fl uxing a gent . A sch em ati c of the British Gas Corporation/Lurgi sta ggi n g gasifi e r is shown in Fig. 3. ·This s ystem has not yet been commercialized. R e p or ted data (Scott, 1981) for the slagger for gas ificat ion of a Pitt sburg No. b it umin ous coal with a m ol a r steam-to-oxygen ratio of between 1. 1 an d 1. 2 show co nsi de ra bly more CO and less hydrogen t h a n in the dry ash Lurgi at nearly the same pres sure (Woodall- D uck ham Ltd. , 197 4), and lit tl e CO2. This is to be ex p ec te d from the equilibrium of the water-gas shift r ea c ti on ( 7 ), i n view of the su bstantia l ly reduced steam content. Less m e than e was The sta g ge r has a value of Rc -- R t hat is c lo se pr o duce d in the stagger. to zero, but slightly in the gasification regi m e. T h e mov i n g beds are the only reactors in c omm ercial or near-commercial ope ra ti o n that are tr ue gasifiers in the sense defined above (Rc - R > 0). with a
8
Entrained Flow Reactor
Entrained flow reactors are devices in whic h coal par ti cl es th a t are ty pic ally about 100 1Jm in diameter are fed cocurrently with the st eam and the oxidant. The coal may be tr ans p orted either in a gas str ea m or in a water slurry; in the latter case the s u sp e n ding water provides the steam for gasifica tion. B isse tt (1978) has pr ovi ded a c ompr ehe nsiv e review of the ch a ra ct eristic s of entrained flow reactors. Two entra ine d flow gasifiers have reache d major com mercial st at us. The Koppers- Totzek a t m osp h eric reactor, shown schematically in Fig. 4, was first put into operation in 1952. The coal is fed th r ough oppo sin g j et s . The stea m-to-oxygen ratio is less than unity , and the combustion zone is
Denn and Shinnar
508
Cool lock
Cool
Distributor Drive
----- Dusty tor recycle
'-....
+r-- Steam
Coal Distributor and Stirrer ----·I�TT1r::":r "' rlr iff-,
� (-Gas
Wo>h Cool"
Water Jacket -----
Tuyere
Gasification Medium -Steam and Oxygen
---�
----- Quench Water
Slog Lock -----
FIGURE
3
Schematic of slagging pressurized gasifier.
above the slagging temperature.
A pressurized gasifier of this type is
under development by Shell and Koppers. The Texaco pressurized gasification reactor, shown schematically in Fig. 5, is a modification of an established reactor for the partial combustion of crude oil residual. The first major commercial installations were under way at the end of 1 9 8 2 , although Bissett (1978) gives a detailed description of a semicommercial-scale Texaco gasifier that operated from 1 956 to 195 8 . The coal is fed in
a water slurry, which has been reported to contain up
to 70% solids by weight.
The result is a steam-to-oxygen ratio of close to
2.
unity, and the ash slags. coal are shown in Table
Pilot plant data for gasification of Illinois '' N o.
6
These data illustrate the fundamental difference
The value of Rc - R for the Texaco gasifier is - 0. 09 , and the reactor operates as a partial combustor.
between the Texaco and Lurgi gasifiers.
al. (1982). In 11) and (12) is 0. 52, indi 1 The Koppers-Totzek and
[The same result is obtained for the data of Dillingham et
both cases the value of R computed from Eqs.
(
cating excellent carbon material balance closure.
Shell- Koppers gasifiers both also operate with slightly negative values of Rc
-
R, based on data compiled in Shinnar and Kuo
( 1 97 8 ) .
Devolatilization takes place in these entrained flow reactors in
that is oxygen -rich.
region
Thus combustion of the volatiles occurs, and there is a
Coal Gasification Reactors
509
Gas Outlet
H.P. Steam Drum
Boiler Coal Feed
Cool Feed
Rotary Lock
A;, y�
S c rew
Coal Feed
(M����==�==��
Water
Boiler Feed
FIGURE 4
Quench Tank
Schematic of Koppers-Totzek reactor.
Coal-Water Slurry
--Cooling Water Out Synthesis Gas Generator
FIGURE 5
Schematic of Texaco pressurized gasifier.
Denn and Shinnar
510
little or no m e th ane or condensible liq u i d in t h e
product gas. The gas need not be quenched prior to heat exchange. An entrained flow reactor co nfigurati o n that ca u s e s devolatilization to occur after combustion, and hence maximizes methane pr od uct ion is emb odi ed in t h e design of the high p r es s ure Bi-Gas reac to r but it has n ever been successfully carried beyond t he pil o t stage. ,
,
Fluidized Bed
No
pres s urized fluidized-bed gasifier is as well developed as
the examples is therfore appro priate
to discuss here some of the problems faced in the design and scaleup of a cited above of other gasifier configurations , and it
fluid - be d ga si fi e r .
The only economically available fluid-bed gasifi er is the atm os p he ric pressure Winkler, which was developed in the 1920s for German Bra unk o hle The reactor operates o nly with very reactive coals, and carbon co nvers ion is limited to 90%. Typical effluent da ta are shown in Tabl e 3. Ruhr Chemie h as developed a p ress urized version that w ill operate at 30 atm, and a d e mon s t ration plant is planned. The published estimates claim t hat overall p erformance characteristics will be identical to the old Winkler, but with 95% c a rb o n conversion. A fluidized bed can in prin cip l e be staged, and it th er e fore has a po
tential advantage over a true countercurrent reactor. Conditions in the zone can be ad j usted independently of those in t h e reaction zone in order to crack tars and maximize methane production; the tempera ture profile in a countercurrent reactor cannot be controlled, and the residence time in the devolatilization zone is quite short. There are also several in h e r ent disadvantages relative to a r eac t or like the slagger. On e is that t h e g as ificat io n zone temperature is limited by the need to prevent agglomeration. No good data exi st o n the maximum permissible t e mp er at ure The second is t hat a significant am o un t of carbon must be main tained in the ga si ficat i on zone in order t o achieve reasonable reaction rates. If this
.
devolatilization
TABLE 3
Ef flu ent Data for Oxygen-Blown Gasification of German Braukohle in a Winkl er Gasifier at 2 a t m and 1700°C
38.44
H 2
co
33.36
C0 2
21.4
CH4
1.8 0 /C 2
0.96
Steam/0 2
1. 52
R
=
Cold gas efficiency
74.7
Net efficiency
56.8
.
Coal Gasification Reactors
511
zone is mixed, the ash leaving the reactor will contain excess carbon; this for the limited c on vers io n in the Winkler. There are two further problems that affect the Winkler and all other f lui d - b ed gasifiers under development: 1. Some of the partially converted coal disintegrates into fines. Fines are difficult to convert, as shown in Chapter 5 , and they are swept from the g·asifier and tend to adhere to the wall of the cyclone. If fed back to the bed, buildup of fines could reduce the bed density. Reducing the load on the cyclone requires a lo w bed velocity that is high enough to prevent ash agglomeration. This problem is solved in the Winkler by addin g oxygen near the top of the bed, raising the temperature of the dilute phase and (There are no detailed data to indicate preferentially com bus ti n g the fines. if the fines really combust, or if the combustion is in the gas phase and the fines convert because the increased temperature.) The net result is that the Winkler operates as a partial combuster, with R < Rc. 2. The combustion reaction near the o xygen inlet is much faster than can be dissipated either by the gasification reaction or by the mixing of the fluid bed. The met hod used to date to overcome this fast reaction and prevent agglomeration and high temperatures is to dilute the oxygen with either steam or cold recycle gas to act as a heat sink. The Higas and Syn thane reactors developed in the United States required as much steam as a diluent as the dry as h Lurgi, eliminating most p ote n tial economic advantage. An alternative way to overcome the problem is to increase the mixing in tensity near the no zzle . Recent American development work has concentrated on two pilot plants, the UGas gasifier of IGT and the W est inghouse fluid-bed reactor. Both are cu rren tly conceived as single-stage gasifiers (e.g., Schwartz et al., 1982) in tha t the co al is fed directly to the fluid bed and there is no separate zone for devolatilization. They therefore produce less methane th an a gasi fier designed to have a separate devolatilization zone with independent temperature control, although not all of the methan e formed by devolatiliza tion is reformed. The major achievement of these programs has been the devel opm e n t of an ash aggl om er ati n g z one at the bottom of the gasifier. The ash agglomerates grow much larger than coal particles and hence separate out at the bottom of the bed and can be removed selectively. A high fluidization velocity is required to prevent agglome ra tion of the coal parti cles, especially in caking coals; this permits a reasonably high coal concen tration to b e maintained in the bed, while removing ash at the bottom having a much lower carbon content. Both the UGas and W es t i nghouse reactors have demonstrated the feasibil ity of ash agglomeration and segregation. The Westinghouse has operated thus far with coal conversions below 90%, and usually below 85%, mainly because of carbon loss due to imperfect fines recovery. The Westinghouse has also oper ated with large heat losses, both because of nonadiabatic oper ation and heat losses in the fines recycle. Pilot data for R - Rc are therefore quite negative ( -0. 2), although Westinghouse claims that the com mercial gasifier is projected to operate with a positive value of R - Rc comparable to that of the dry ash Lurgi. The UGas has operated until now only at low pressure (ca. 3 atm), but claims to have achi e ve d 95% coal con version, Results of both gasifiers were significantly better for reactive c oals than for bituminous coals, as would be expected from simple kinetic considerations.
is the reason
512
Denn a n d Shinnar
STOICHIOMETRIC ANALYSIS
Some useful p rinciples rega rdin g gasifier operation can be obtained by examinin g overall stoichiometric constraints and energy balances under idealized conditions. We will consider only the gasification of char, which is taken as consistin g entirely of carbon. Any effect of the presence of volatiles can be ad de d afterward in this analysis . Reactions of char to form methane in t he absence of a suitable catalyst are generally much slower kinetically than combustion or steam and C02 gasification. It therefore suffices in the first approximation to neglect any methanation and to remove reaction ( 5 ) from the set of reactions un der con sideration . Reactions ( 1 ) to (3) then form a basis from which to construct all other reactions to form CO, C02 , an d hydrogen from carbon, water, and oxygen. The three species fe d to the reactor are conveniently represente d in Fig. 6 as the vertices of a trian gular diagram on which each point repre sents the mole fraction of reactants. The feed compositions for reac tion s ( 1 ) to (3) are sho wn on the axes . It is readily established that incomplete carbon conversion will occur for startin g compositions that lie above the line connectin g C + ( 1/ 2)0 2 and C + H 20, while incomplete + oxygen conversion will occur for compositions below the line connectin g C 0 2 to the H20 vertex. A ga sifier would operate ide all y with no excess steam. It c a n be s hown by linear combination of the basis reactions [ t wice (3) plus (2 ) minus twice ( 1)] that steam must appear in the effluent for starting compositions that lie below the line connecting C + 02 with the reaction LlH
=
+23.0
(17)
c
FIGURE 6
Triangular diagram for reactor
feed
con dition s .
513
Coal Gasification Reactors [Some steam will, of course,
appear in the effluent for starting formulations
above this line, since the an alys is assumes complete conversion and neglects any possible chemical equilibria, including that of the water-gas shift re action (7) . ]
T he shaded trapezium area thus represents the most desirable
area for ga si fier operation. The operating range can be further restricted by the condition of auto
thermal ope ra ti on (overall thermal neutrality):
the exothermic reactions
must produce just sufficient reaction enthalpy to drive the endothermic re actions (assuming that an external heating source, such as a nuclear re
actor, is not to be u se d ) .
We suppose that the inlet and outlet streams for
which we will take to be 700°C X 2 , and 1 - X1 - X 2 represent the fractional conversion of carbon through each of t h e basis reac tion s ( 1) , ( 2 ) , and ( 3), char g'asification are a t the same temperature,
for this discussion.
Let XI,
Thermal neutrality is then represented by
respectively.
(-26.4)X 1
+
( - 94 . 2) X
2
(+3 2 .2)(1- x
+
- X ) 2
1
0
=
and the thermal efficiency based on the lower heating value (L HV) of the feed char at 7 00°F will be 1 00%.
The overall reaction for thermal neutrality
is the one-parameter family
C
+
( 0. 2 55
( 0.74 5 or,
in
0.036X1)o 2
+ +
0. 464X
1
)CO
+ +
(0.7 45
0,536X ) H 0 2 1
-+
( 0. 2 5 5
0.46 4X ) C 0 2 1
+
(0.795- 0 . 536X )H 1 2
terms of the bas is reactions,
(0.255
0. 4 6 4X ) ( C + o -+ C 0 ) 2 2 1
( 0 . 7 45
0. 53 6 X ) (C + H 0 1 2
-+
CO
+
H2 )
6. Point A corresponds to 0 . 55, with react ions ( 1) and (3) in the following propor t ion s :
This reaction family lies along line AB in Fig.
X1
=
c
+
1
202 -+ co
0 . 82C
+
0 . 8 2H 0 2
0.82CO
+ 0 . 82H2
X 1 cannot exceed 0 . 55, or else CO2 will be required as a feed. Point B -1.61 and represents the minimum value for which it It is most is p o ssibl e in p rinc iple to produce a product gas without steam. c onve nient to e x pre s s point B in terms of reactions ( 2 ) and ( 17) as c or res pon ds to X1
C
+
02
4 . 0 9C
+
-+
C0
2
=
8 . 17H 0 -+ 0 . 8 2 C O + 0. 82H 2 2
Denn and Shinnar
51 4
Values of X 1 b elow - 1 . 6 1 woul d r e q u ire CO as a feed in this simplified The H 2 / CO ratio in the product ga s varies f rom 0 . 45 at p oin t A anal ysis . to infinity (no CO ) at p oi n t B . Con ve r s e l y , p oin t A has no C 0 2 in the pro du c t , while th e H 2 / C 0 2 rat io at point B is 2 . This s i mp l e stoichiometric an a ly s is , which n e gl e ct s all kinetic an d e q uilib riu m limitation s , is gene rally in s truc tive but is mi s l e a din g in on e i mport ant r e gar d : the thermal e ffici ency of all p oin t s alon g line AB based on the lower heatin g value at 700°F i s t h e sam e . A more realistic picture is ob taine d by accountin g for the en e rgy require d to produc t the feed steam and oxy ge n at the assumed pressure of 400 p si . It can be shown ( S hinna r and K u o , 1978) that production of 1 mol of oxygen at 700° F and 4 0 0 p s i requires the same ene r gy as 4 . 1 mol of steam at t he same con dition s . The overall reaction s a t poi nt s A an d B are , respectively , A:
C
B :
C + 0 . 1950
0 . 2 7 50 2
+
2
+ 0 . 4 5H 0 +
2
+
CO +
2
+
C0
1 . 61H 0
2
+
0 . 45H2
1 . 16H 2
Point A r e q ui r es the energy e quivale n t of 1 . 58 mol of steam to c onv er t one mol of carbon , while poin t B r e q u ires the e q uiv al en t of 2 . 4 1 mol of ste am . There is t h u s a net energy l oss equal to the p roduction of 0 . 83 mol of steam in pa ssin g from p oint A to point B , w hich is equivalent to 9% based on the lower h e atin g value of reactan t s . When the ene rgy required to p re pare steam an d oxy gen is t ake n into accou n t , the efficiencies . for the idealized processes at p oint s A and B drop to 8 1 % an d 7 2 % , respectively . Lin e AB can be p aram eterize d by the feed steam - to - oxy gen ratio ; the computed ef ficiency as a function of steam - to - oxygen ratio for autothermal operation is s ho w n in F i g . 7 . A n equivalent analy s i s c an b e carried out b y a s sumin g that methanation will take p l ac e an d that t he reaction product s are only C O , C 0 2 , and C H 4 . This limit i s unrealistic in t h e ab sence of t h e approp riate cataly st s , b ut help s in definin g the performance that might b e obtained . Combinations of the
1 00
�
... u c: . !!! �
�-----,--�--,------, •-----,
90
A
w 80 0
�.,
�
1-
e...._ ......_
A
dL ' -.... ...._ Ji Loss
+
to s t e a m f e e d B
..._
-- -- ..._ -- .
L o s s d u e to o • y Q e n feed
70
60
B
B
0
2
a 6 4 S t e a m /Oxygen ( M o l ar)
10
Effic ie n cy a s a fu n c ti on of m olar steam - t o - oxy gen ratio . Shinnar a n d Kuo , 1 978 . )
FI G URE 7
( From
T A B LE 4
Thermal E fficiency
o f Limiting C ase s in C har
Gasifiers with Stoic hiometric an d Energy Constraints
Point A Overall reaction
C
+
0 . 2 7 50
2
+ 0 . 4 5H 0 2
CO + 0 . 45H Steam -to -oxy gen ratio
Steam requirements
+
C
+
0 . 1 9 50
C0
2
2
2
+
+ 1 . 6 1H
1 . 61H 0 2 2
Point F
Point E
Point B +
C
+
0 . 0 1 50
2
0 . 4 9 2 5C H
4
+ 0 . 985H 0 +
8
2
0 . 5 0 7 5C 0
+
2
C + 0 . 180
2
0 . 2 1 3C H
4
+
0 . 4 2 8H 0 2
+ 0 . 788CO
1. 6
8. 2
64 . 5
2. 4
47. 8
170 , 9
10 4 . 7
45. 4
9. 2
32. 7
20 . 0
8.7
[ lb /MMB tu ( L H V ) gas ]
Steam (MBtu /lb
carb on )
mol
Oxy gen requirements [ scf/MMBtu gas ( sc f I mscf) ) Oxy g en (MBtu /lb mol
6 11 ( 19 0)
4 1 1 ( 124)
23 . 0
14 . 9
81
72
3 5 ( 1 0)
1. 3
carbon )
Therm al efficiency
8Con dition s :
fe e d an d outlet tem p erat u r e ,
700°F ; no
87
4 0 5 ( 1 26 )
15. 0 86
h eat recovery from products below 70 !P F ; B tu requirements of feeds ,
B tu /scf oxy g en , 1 1 30 B tu /lb steam , both at 700°F a n d 400 psia . So u rce : S hin n ar an d K u o ( 1978) .
220
+
51 6
D e n n and S hinnar
c
FI G U R E 8
Region of autothermal op eration with methane formation .
two limitin g analy sis leads to the operating diagram shown in Fi g .
quadrilateral AB EF bounds the au to t herm al re gion an d of p ossible c omplete utili za tion of carb on and steam .
EF .
Kin etic constraints ,
this o p t ion
however .
,
8.
The
the re gion
T he c harac te ris ti c s of
If m e t han a tio n coul d b e
the c or n er point s are s u m marized in T ab l e 4 . rie d ou t wit hin the g a si fi e r
li es within
it is c l e ar t h at operation s hould be alon g line
car
to be discusse d subseq uently , curren tly preclude
T hu s it is c le ar that the desirable re gion for opera
tion of a ga si fic at ion reactor is close to p oin t A . ,
The as sumption in thi s a n al y s is that all reactions go to c omple t io n is
not as re s t ric tiv e a s it migh t appear .
The oxygen an d steam utilized to con
vert carbon must satisfy t h e c on s tr aint s outlined here , w hic h thus define
upper bounds on efficienc y .
E quilibrium ,
k in et ic
p rovi de information r ega r din g excess steam .
,
a n d t ransport limitation s
Oxygen u tili zation will al ways
be e ssen tially complete .
T H ERMODYNAM I C AND PROC E SS CON ST RA I N T S
The net e fficiency of a
g as ific at ion
re quired to prepare the fee d ,
p roc ess depen d s stron gly o n the energy
which i s ap p ro x i mat e ly p roportion al to th�.
moles of steam pl us 4 . 1 tim e s t he moles of oxy gen ( S hinnar an d K uo ,
1978) .
The ratio of irrecove rable e n ergy u s e d in pre p ari n g the fee d to the lower heatin g value of the product gas is therefore a useful measure of gasi fie r
e ffi ciency ;
we den ote this r a t io as E L :
(18) The lower E L , the higher the net e fficiency . replace the denominator with CO
+
H 2
+
( For an SN G plant we w oul d Poin t s E an d F in T abl e 4
4C H . ) 4
51 7
Coal Gasifica tion R eac tors
are clearly s upe rio r to po in ts A and B in ha vin g lower values of E L , but the r e are several reason s why this stoic hiometric limit cannot be app roac h e d . The fi r s t is a p roc e ss constraint . T he calculation s in T ab le 4 a s s um e that p ro duc t ga s c a n b e h e at e x c h a n ge d with the feeds and on ly con sider T he actual heat requirements t he heat re qui r e men t s of chemical reactions. are g rea te r Gasification requires high t em pe ra tu re s (>1500°F ) w ith o u t a c a ta l y s t I t is impractical to h eat e x c hang e a high - temperature p r o d u c t st ream , for t w o rea s on s : .
.
1. 2.
H e a t t ran sfer a t high p res sure from stream s c on t aini n g H 2S is limited by mat e ri al constraints to t e m p e ratu re s b e low 1 1 0 0°F . G a s i fie r p roduct st ream s o ften c ont ain tars and soli ds w hich will coke a n d co ve r h e at ex ch an ge r surfaces .
g a sifie r s the only w ay in whic h heat is fed back gasifier feed is b y c oun t erc u r rent ga sifie rs Thi s ex c h an ge recovers a large fraction of the s en sibl e heat of the p ro d uc t gase s an d hence gi v e s an inherent ad v an ta ge to coun tercurrent gas i fie r s , since they require l e s s heat to be supplie d to the g a si fica t ion zone an d t h e refore less oxygen . In a c ocu r re n t or w ell - m i x e d sin gle - stage gasi fie r the o n l y way in w h ic h the sen sible heat of the pro duct gases can be re covered i s a process steam . S ince t h e steam can also b e p roduce d directly from a comb ustion of c oal , w e suffer the p enalty incu rre d in separa ti n g the incremental oxygen that is re quir e d to su ppl y the heat to raise t he p roduct to g a si fi e r tempe r a ture The secon d rea son t h a t poin t s E a n d F a r e inaccessible with c urrent ga s ifi e r s i s the t h e r m o dy n a mic consequence of the kin etic rates of t h e dif ferent ga s i fi cat ion reac tion s . T hese point s re q ui re methane formation by reac t ion ( 6 ) , w he re no H 2 is p roduc e d . Reaction s ( 3 ) , ( 4 ) , and ( 8) are al l faster t han reaction ( 6) at normal gasification con dition s , and re ac ti on ( 7) is gen era ll y muc h f a s te r than all t he others . T h u s , while the e quilibri urn c on ver sion of reaction ( 6) is hi gh at hi gh t e mpera t ures the other product gase s will also be present in the reacto r an d t he gl obal equilibrium must be taken into ac c ount M ethane yield at g lob a l equilibrium i s low at t empe r a t u r e s above 1 1 0 0 °F , e x c ept at very hi gh pressures ( Shinnar et al . , 1 98 2 ) . A t lo w t emperat ure s , w h ere m et ha n e is the dominant product , steam conversion is approximately 50% , and t he increased s t ea m r e q ui re m en t re moves any advantage over p o in t A ( at least for fuel gas) . I t is important to em p h a s i ze that the inab ility to ob tAin a high methane yield at hi g h temperature s is not a ri gorou s t her m odyn amic con straint but i s a c on se qu e n c e of th e fact that th e primary gasi fic a ti on reac tion is C + H 20 � CO + H 2 . M et han e is formed m ainly b y c on sec utive p roc e s se s in vol vin g reaction s ( 5) an d ( 8 ) ; i n n eit h e r case could t h e product exceed the equilibrium c o m posit ion . I f , on the other hand, one could find a cataly st that p r o m o t e d reaction ( 6 ) or so m e other reac tion ( e g , 1 . 5C + H20 + 0 . SC H 4 + C O ) via a s u r fa ce m ec hani sm that doe s not first requi re formation of molecular h y d r o ge n there is no thermodynamic reason w hy met h an e yie l d c on sistent with the equilibrium y i e l d s of these l a t t er reac tion s Present catalysts , s uc h as p o t a s si um carbonate , could not be a pp r o ach e d do not allow t hi s , at least not a s p r e s en t l y app lie d an d very high p ressure ope ra tion w ould only p re sent mean s of o b t ain in g a hi gh me t h an e yie ld wit h hi gh steam c on version . EL is shown as a func ti on of t e m perat u r e in Figs . 9 and 1 0 under e q u ilib ri um condit ion s fo r an eastern an d w e stern U . S. coal , re spec tively , In present commerc i a l
from the p ro d u ct gas to th e
.
.
,
.
.
,
.
,
.
51 8
Denn a n d Shinnar
T (K )
1 800 1 600 3 . 00 �.,;-;.--,...-___,. ...;--___,.---,----,
\ \ \
J: u
� :£
+ 0
t\i
•
2 . 00 -
+
:z:"' 10
�d
8 +
1 . 00
0
0
\ . \\ A
--�
1400
1 200
1 000
L urg i
Dry Ash
Tel!OCO
x Sl ogger
6 [J +
\ '\ • \ �B , \ \
Koppers - Totzek S h e l l - Koppers
•
c
·, ' "-, ' .......
[J
-
+
,=- · - · --:"'"" �----� ·�������---------� 0 -. -:...:-o..::=-....=�---·- -l'- � �
-
-
..:::::::-
.J UJ
0 �--�---��--�---J 1 200
1 800
2400
3 000
FI G U RE 9 EL as a function of gasifier temperature for an eastern U . S . coal . A , well - mixed reactor , n o methane formation ; B , countercurrent reactor , no methane formation ; C , well - mixed reactor , methane at global equilibrium ; D , coun tercurren t reactor , methane at global equilibri um .
T (K)
\
,.,-... � J:
o"' �
Cl)
+
+
2 .00
�
0
+ 0 u .___....
I .OO
\
\
o.., :z:"' J:
__
\
\A \
'' B
0
\ '
+
____
Winkler Dry A s h
Lurg i
S h e l l - Koppers
\ X '\ '
....... .... ' ' "";::..::. ---=-= c-. -.
o· - . ::::..,:::_ ::.::...-,. _ r _.
- · -
...1 UJ
0
x
, 8�o_ o_,
6o l1 _o
� �o 4o o--�12�o�o�-�� o� o� ��M 3.00
',
___
O L---�--�--�----�--��� 1 2 00
T ( OF )
2400
3000
EL a s a function of gasifier temperature for a Western U . s . Symbols as in Figure 9 .
FI G U R E 1 0
coal .
1 80 0
51 9
Coal Gasifica tion Reac tors at 400 p si with steam and oxy gen fed at
1 100°F and coal at ambien t tempera ture , There are four calculation s : the reactor is assumed to be either well mixed or countercurrent , and methane is either formed and is at the global equilibrium composition or is not formed at all . Gases formed during devolatili zation in the countercurrent gasifier do not affect the equilibrium , while in the well - mixed or cocurrent gasifier they reduce the maximum steam con ver sion . There appears to be a broad minimum in all cases aroun d 200°F , correspon din g to optimal thermal efficiency for fuel gas . The pre sence or ab sence of m ethane has little effect on the location of the mini mum , nor doe s the gasifier configuration . EL increa se s at lower tempera ture s , but the increase is much le ss steep for the cases with methane pro duction . If one had to operate at lower te mperatures for proces s reasons , methane formation would b e e ssential for high efficiency . The actual values of EL for several gasifiers operatin g on eastern coal are also plotted in Fi g 9 . W e note that the slagger approaches to theoreti cal minimum quite closely . ( T he devolatili zation com p o sition s used for the countercurrent calculation s were derived from data on the slagger , so the countercurrent reactor curve must p ass near the slagger point . T his has no bearin g on the discussion of relative reactor efficiencies . ) T he Texaco gasifier , by contrast , lie s well above the thermodynamic equilibrium line, i n part b ecause t h e reactor uses a water slurry as a feed and requires The Shell -Koppers an d excess oxygen to supply heat for vap ori za tion Koppers - Tot zek cocurrent gasifiers , which use steam in stead of water , also lie above the optimum ; the short residence time apparently p recludes attain ment of equilibrium . EL for the dry ash Lur gi is also very l a r ge , de spite the fact that it is c ountercurrent . This reactor is not suitable for eastern U . S . coal because of the high steam -to- oxygen ratio required to keep the ash below the fusion temperature . As shown in F i g . 10, the dry ash L urgi perform s much better on western coal , where the more rapid en dothermic gasification reactions aid in moderatin g the temperature and reduce the excess steam re quirement . There are no data available for the slaggin g movin g b e d , and none of the commercial or semicommercial reactors have achieve d results a s good as those of the slagger for eastern coal . The Texaco gasifier is not suitable for these coals at p resent because the heatin g value is too low to permit efficient slurryin g with w ater . C alculation s of R under the same four assumptions are shown in Figs . 1 1 an d 1 2 for eastern and western coal s , respectively , together with R c an d data for the same gasifiers . R never reaches R c for any of the calcu lation s , regardless of C H 4 formation , an d there is no opti m u m near the mini mum of EL . The minimum of E L is a consequence of the fact that R increases with temperature , whereas steam requirements decrease . R is approximately eq ual to Rc for the Texaco , Shell - Koppers , K oppers - T ot ze k , , and Winkler gasifiers . The reason for this high value of R is kinetic and is not a thermodynamic constraint ; these reactors use oxygen for partial combustion because steam gasification is too slow under their process conditions . This is easiest to see for the T exaco an d rather p u z zlin g for the fluid bed , b u t we lack any reliable data to explain the latter . R is close to the equilibrium limit for the slagge r and is still low for the dry ash Lurgi ; the movin g beds are therefore the only gasifiers that operate presently at true gasification con dition s , and the slagger is the only gasifier that operates close to the thermodynamic op tim um .
.
.
Den n and Shinnar
520
T (Kl
1 8 00 CJ
+
0 Dry Ash Lurgi x S l ogger
6.
+ Te x a c o
S he l l - Koppers
CJ K o ppe r s - Tot z e k
O L---�--��--�--3000 2400 1 800 1 200
FI G URE coal.
11 R as a function of gasifier temperature for an eastern U . S . S ymbols as in Fig. 9 .
0.50
1 000
T (K) 1 4 00
1 2 00 0
/
/
//
1 800 +
c
- - =-=....,._...- ---
.. - -- · -x--..,:.: . -- /o
1 600
0 X
+
Winkler D r y A s h Lurg i S h e l l - Koppe r s
O L---�---L--�----L--_j 1 200
FI GURE coal .
1 800
2400
3000
1 2 R as a function of gasifier temperature for a western U. S . Symbols as in F ig . 9 .
Coal G asifi cation R eac tors
521
It is i m p ort an t to note that methane formation by it sel f is n ot always beneficial , b ut depen ds on where it occurs . I f methane is formed in the gasification zon e it re d uc e s the heat requirement , and t h er e fo r e R , and also i m p ro ves steam conversion . If it is formed in the low - temp erature zone of a cou ntercurrent ga s ifie r , th e valu e de pen d s on the m ec h anis m . M ethane formation by devolatili zation or by reaction of H 2 an d steam with tar an d coal are b eneficial , since t hey reduce the char feed to the ga si fi er . The heatin g value of methane formed from these reactions is larger than the heatin g value of the hydrogen formed in the ga sific a t ion zone ( S hinnar et al . , 1 9 8 2) . O n the othe r han d , the impact is nega ti ve if me t h an e is for m e d by r e acti on of CO or C 0 2 with H 2 , sinc e the reaction is hi ghl y exo thermic an d the heat of reaction i s not fed b ack to the gasification zone . The heatin g value of t h e methane formed in this way is lower than that of the syngas reacted . Such methane formation can occur in the lower tem perature zone since fres h iron oxide and other ash constituents are methana t ion ca talysts which are lat er p oison e d by t h e H 2 S . T his p roble m also c au s e s diffic ultie s in the evaluation of pilot plant results . S teel and stain les s steel cataly ze m e t h anation as well as shift , an d can therefore lead to o utl et composition s w hich contain more C H 4 an d H 2 t han would b e ob tain e d in a commercial g as i fie r w h e re su rface effects are much smaller . K I N E T I C S - FR E E C A LC U LA T I O N S
I t i s often p o s sib le t o estimate reactor performance b y t h e u se o f kine tic s free c al c ula t ion s ; such calculation s assume that k e y rea c ti on s occur in finit e ly fas t , and h ence go to c om p l e t ion or to thermodynamic equili b riu m , or t h a t they do not occur at all . ( A s su m p tion s a re thus made both about relative kinetic rates and ab o ut res i de nc e t im es , so s u c h calculation s are only apparently free of kinetic s . ) The p roc e d u re is t h e same for all r e actor types , b u t the detailed calculation s differ slightly depen din g on whether or not the volatiles are re l ea s e d in an oxy gen - rich environment . Thus it is convenien t to con side r co- and coun tercurrent reactors separately for illustrative purp ose s . C o u n te rc u r rent Mov i n g Bed
Con sider t he movin g -b e d reactor , shown schematically in Figs . 1 to 3 . We will foc u s on the reaction zon e , b elow the r e gi on s where devolatili zation an d drying take place . Direct hy d roge n a t i on to form methane is u sually slow and will ini ti al ly be n e gle cted in this elementary analysis . Oxygen reactivity is assumed to be in fini t ely fast , an d oxy gen attack on the fixed c a rbon is ir r e ve rs ib l e and i s co m p le t e d before any other reaction be gins . Steam ga sification , reaction ( 3 ) , is t ak e n to be sufficiently fast to reac h equilib ri um or completion at t h e top of the reaction zon e . The w ater - gas shift , reaction ( 7) , is taken to be in stantaneous an d alw ays at equilibrium . Reac tion s ( 1 ) , ( 3) , an d ( 7) form a fundamental set for overall b al anc es without methanation , and are t h e r e for e the only reaction s that need be considered . T wo general type s of beh avior can exist , depen din g on re l a tiv e feed rates . If the coal feed rate is too large to utili ze al l the steam , reaction s ( 3 ) and ( 7) [ and hence also reac ti on ( 4 ) ] will be in eq uilib ri u m at the top of the reaction zone an d there will b e unreacted carbon in t h e a sh at the bottom of the reactor . Reduction in the carbon feed rate relative to steam
D e n n a n d Shinnar
522
will lead to reduced amounts of carb on in t he ash , while maintainin g the
equilibria at the top . The carbon in the ash will reach zero at a critical At carbon fee d rates b elow t his critical value there will be carbon rate . an excess of steam and the carb on - steam reaction can no longer be at e quilibrium at the top of the gasification zone , although the equilibrium of reaction ( 7) will be maintained . ( T his behavior is contrary to the intuition of many people and is a consequence of the countercurrent operation . ) The overall mass and energy b alance s and the e quilibrium equations are sufficient to define the composition s an d temperatures of the effluents ( with the additional assumption t hat the solid and gas temperatures are equal ) . The calculation therefore requires only the solution of a set of al geb raic equation s . The fixed c arbon - to- ash ratio must be given , since the ash is inclu de d in the energy balanc e , and we u se a ratio of 5 : 1 as typical of Illinois N o . 6 coal . The calculation s shown in Fig . 1 3 from Yoon et al . ( 1 9 7 9a) are for a gas feed temp erature of 700°F and an operatin g pres sure of 2 5 atm . T he solid line s are lines of con stant fraction of un reacted carb on in the ash . The line of zero unreac ted carbon represents the case in whic h feed rates of carbon , steam , an d oxygen are suc h that the carbon - steam reaction reaches e quilibrium at the top of the reaction zone without unreacted carbon at the b otto m ; this line , and all points below , are characteri zed by complete utili zation of carbon , first by oxy gen an d then by steam There is a noticeable change in slope of the lines of con stant unreacted carbon at a steam - to - oxygen molar ratio of about 1. 2 . To the left of the change in slope the equilibrium of the c arbon - steam reaction is far to the •
c
FI G UR E 1 3 Lines of constant fraction of un reacted fixed ca rbon ( solid ) , and increase in gas sen sible h eat ( dashed , kcal /mol fixed carbon ) for a counterc u rrent movin g - b e d reactor . ( A fter Y oon et al . , 1 9 7 9a. )
523
Coal Gasifica tion Reac tors
ri ght , and this reaction as well as oxidation c a n be taken to be e ssen ti ally irreversible . The line of complete carbon conversion in this re gion is thu s very close t o th e li n e connectin g r eact ion s ( 1) an d ( 3) i n Fi gs . 6 and B . The region to t h e ri ght of the b reak represen t s an exces s of steam . The exc e s s steam lowers the temp erature , an d su fficient e x c e s s steams brin g s the peak temperature b elow the ash melting point an d allow s operation of a dry ash reactor . The line s of constant carb on in the ash in this r e gion approximate lin e s of c on stant oxy g en - to-carbon molar fee d ratios , The dashed lines in Fi g . 13 are lines of constant inc rease in g as sen sib le heat in kc a l / g mol o f fixed c a rb on . Autothermal o p e ration for char gasific a tion lies to the right of the lin e of complete carb on c on ve r s ion . A coal con tainin g 1 0 % moist u re will require abou t 5 kcal / g mol of carb on for heatin g an d dryin g ; if we p re s u m e t hat the ga s exit temperature must be at least 700°F to prevent con den s at ion of tars , op eration in the r e gion to the ri ght of the 5 -kcal / mol lin e is exclu de d because of the need to add h eat to the gasifie r . O per atin g p oin t s to the left of the 5- k c al line will represent an app reciab le los s of energy unl e s s the s e n sib l e heat can b e conveniently re covered down stream , whic h will b e diffic ult in a gas stre a m containin g con den sible s that will foul a heat e x ch an g e r . The intersection of the 5 - kcal / mol sensible heat lin e wit h the c o m p l ete carb on con version lin e thus rep r e ( T h i s is sli gh tly to t h e le ft of sen t s the most efficient op e ra t in g p oin t . point A in Fig . 6 , w hic h was de t e rmin e d on the b asi s of a dry , hot char fee d . ) This intersection occ urs n ear a steam - to -oxy gen ratio of 1 . 2 , which , as n ot e d above , is typic al of the fee d to t h e B C G /Lur gi sla gg er . The com plete carb on conversion line is close to the 1 0 -kcal /mol sen sible heat line at a stea m - t o - oxy gen ratio of a b ou t 7 , whic h is typic al of a dry ash Lurgi re actor ; this represen t s an a d ditional los s of efficiency of about 5 % . T h e net thermal e fficiency ( S hinnar and Kuo , 1 9 7 8 ) is shown as a fu nc tion of stea m t o - oxygen ratio at c o mp l e t e carb on conve r sion in Fig . 14. ( T h i s c alculation
90
�
>.
..,
c: Q)
.., -
80
Constra i n t s
- - K i n et i c
- E q u il i b riu m
w
E
0
....
70
Q) .c I-
�
2
60
5 0 �-------�-------�------��------� 8 4 2 6 0 St e a m / O xygen ( M o l a r )
FI G U R E 1 4
Thermal efficiency for a
countercurrent movin g -bed
a function of molar steam - t o - oxygen ratio .
reac t or a s
( F rom Shinnar and K uo ,
1 978. )
524
Denn an d S hinnar
is at 400 p si . ) T h e dashe d line is for an e quilib rium c on stant that is only 10% of the t r ue value as a means of a t te m p tin g to account for kinetic effects . This approach to kinetiP. an d tran sport con s train t s in a m ovin g b e d ga si fie r has been used by G um z ( 1 9 50) an d Woodmansee ( 1 97 6 ) . One fu rther u seful conclusion can be drawn from this anal y sis . The steam ga si ficat i on reaction ( 3) is not e q uimolar in gaseous specie s , so the equili b ri um will be p res su re depen dent . The e ffec t of de c re a sin g the pres sure is to m ake the reverse reaction m ore unfavorable , and h ence to move the b reak point on the lin e of c omplete carbon conversion to t he ri ght along the lin e c onnectin g the reaction s ( 1 ) and ( 3 ) in Fig . 6 . Since all points to the left of the b reak correspond to es sentially complete utili zation of steam , i t c a n be c on c lu de d t h a t slaggin g mo vi n g b e d ga s ific a t ion will be in sen sitive to dec r ea s in g p re s sure ; this p h e n omenon has been ob served by Ellman et al . ( 1 9 7 7 ) in pilot plant stu die s . C onversely , at p r e ssure s sub stantially hi gher th an 3 7 0 p si , the breakpoint would move sufficiently far to the left to cau se exces sive unreacted carb on . T here is no c on cept ual or c omp utational difficulty in repeatin g these c alculation s w it h reac tion ( 5) inclu ded in the b a si s in order to account for me t h an a tion . The line of complete carbon conversion with m et h ana tion equilibrium is shown in Fig . 1 5 , to get h e r with the correspon din g line with out methanation . The t wo line s c onverge at stea m - t o - oxygen ratios of o rder unity , w hich is the region of stagger o p e ra tion The high te mperatures here drive t he e q u ili b ri u m far t o the left and suppress methane formation . Thus , for sla g gin g op eration the met hanation reaction s need not be con sidered . At the hi gh e r steam -to- oxygen ratios and lo w e r temperatures c haracteristic of dry ash m o vin g b e d reactors , however , me t hana ti on is As noted in Table 4 an d in the p recedin g sec thermodynamically fa vora b le tion , for m ati on of m ethane in the reaction z on e in creases t h e e ffic i e n c y . This is becau se methanation is exothermic and enables gasific ation to be carried out with les s utili zation of oxygen . -
-
.
-
.
c
Methana l 1 o n E qu i l i br i u m
FI GURE 1 5 Lines o f c omplete c a rb on c onve rsion w it h an d without methanation for a countercurrent movin g - bed reactor . ( A fter Y oon et al . , 1 9 7 9a ) .
525
Coal Gasifica tion R eac tors
Ther e ar e c om p et i n g t her m od y n am ic effects with pressure at the high steam - to- oxygen r ation s in dry ash o p e r ation . Methanation is fa v o r e d by in creased p r e s sur e , but we h ave alr e ady noted that increased press ure will lo w e r t h e e ffic ie n c y of steam gasification . T his observation is consistent with the p relim inary report from the Ruhr 100 dry a sh Lurgi gasi fier ( Lohm an and Langho ff , 1 9 8 1 ; Loh m ann , 1 98 2 ) , which is intended for ope r ati on at up to 1 0 0 at m . A p re ss u re increase from 25 to 50 atm ( with a ch an ge in ash content of the coal ) sh owed an increase in raw ga s methane from 9 . 1 to 1 3 . 5 % ( but little c h an g e in o t he r hydrocarbon s) , with d e cr ea se s in hydroto 1 5 . 0 % , respectively , and an in gen and CO from 3 9 . 1 to 3 7 . 9% and crease in raw gas h e a tin g value of 1 3 %. There w a s also a 5% inc rea s e in c ar b on conversion an d a 5% d ec rea s e in oxygen con s u mption . Thus the op ti m a l pressure will depend on the reac ti vit y of the coal. Re ac tivity of methana tion re ac tion s seems ge n eral ly to be low at in ter m e d iate pressure ; the methane yields shown in Table 5 for movi n g - b ed g a s ifi e r s operat in g at 30 atm a n d less are not si gnificantly greater than w oul d b e e x p e c t ed solely from t he volatiles . The m e t han e yie l d from the ga si fic ation of coke p robably rep re s en t s t he maximum m eth ane obtainable in the reac t ion zon e ( adj ust e d for p res s u re if c omp ar e d to the other da t a ) . The slightly hi ghe r yields in the dry ash re a c t o r relative to the slagger may reflect the lower te m p er a tures in the ga sific at ion zone of the forme r , w hic h are thus m or e fa vor ab l e to me t han e formation . The inc rease d m e t ha n e p r o d u c t i on at 50 atm is ac co m p an ie d by a decrease in oxy gen utili zation correspon din g r oughl y to a t ra de - o ff between the exothermic reaction ( 5 ) an d reac tion s ( 1) an d ( 2) in a 3 : 1 r a ti o , and methane formation in the reaction zone ( rather than a c han ge in the devolatilization p roce s s ) is a plausible e xp lanat i on for the inc rea se d methane p roduc tion . The t r ap e zemiu m of autothermal o p e ratio n in the reac tio n zon e ( A B E F ) is also inc lude d in Fig . 1 5 . The ne i gh b orhoo d of line EF , which is c arbon con ve rsion to met han e , C O , a n d C 0 2 , an d contains the hi ghest efficiencie s , is quite far removed from the line of complete carbon conversion with met hane equ ili b ri um . T hi s region around line EF the re for e appears not to be obtain able . S elective c a ta ly s t s t hat p rovide a kin etic pathway ap p ro a chin g global equilibrium by a t raj ec tory p as si n g t hro u gh the n ei ghb or hood of line EF woul d be required, as di s cu s se d by S hi n n ar et al . ( 1 9 8 2 ) . The a naly si s above also pi n poin t s the de fic ien cie s of a kinetic s - free a n al y s i s . I t fits the sl a gger quite well and expl ai n s certain tren ds for t he dry a sh Lurgi , but it m isses t he m ain feature that control s the d e si g n of the dry ash r eac t or . The steam-to-oxygen ratio in a dry ash Lurgi is con trolle d by the relative selection rates of combustion ( which is di f fu sion con trolled ) and r ea cti on ( 3 ) . Nonreactive c oal s t he re fo re re q uire a much h i gh er ste a m - to - oxyg e n ratio , as discussed below in the se ction on m od elin g . In
16. 1
the sl a g ger thi s is of no consequence ' and therefore the a s s um p tion that co m b u s tion is in fin i t el y fast has no e ffe c t on the steam requirements .
I t is
i m p o rt an t in any de si gn or de velop m e n t effort to un de r s ta n d clearly what is t he critical c on st r a in t , and the case above is a good example of how const raints can shi ft by chan ge s in desi gn . always
C oc u r ren t En t ra i n ed Flow
A n an alo g o u s fi guration .
"kin etic s - free " analy s i s can b e carried out for an y reactor con with less detail , for the cocu r re n t entrained flow
We d o s o here ,
� Ol
T AB LE 5
Methane Production of Various Movin g - Be d G as i fier s scf Methane per lb
Gasi fi er
C oa l
S team / oxy gen
Pres sure ( atm )
dry ash fr ee coal a
7. 5
27
4. 2
8, 5
30
3. 85
German Long Flame
7, 7
25
4. 0
Dry ash Ruhr - 1 0 0 Lurgi
G erman Lon g Flame
6.5
50
5. 4
Gran d Fork s Slagger
Lignite
1. 0
27
2. 1
B C G / Lurgi Slagger
Scotti sh Frances
1. 3
30
2 . 65
S olihull Slagger
Avenu e N o .
1 . 17
20
0. 3
Dry ash Lurgi
Wyoming ( w e stern U . S . )
Dry a s h Lurgi
Illinois N o .
Dry ash Ruhr - 1 00 Lurgi
a
6 ( eastern U . S . )
2 coke
1 scf C n H m converte d to equivalent scf C H 4 by ratio of heats of c ombu stion .
So u rc e :
C omp ile d by Shin n a r and K uo
( 1 978 ) ,
excep t data on the S olihull slagger , which are from L a ce y ( 1 96 9) , an d
the Ruhr - 1 0 0 gasifier , which are from L oh man n an d Lan ghoff ( 1 9 8 1 ) .
t:l
;:s ;:s � ;:s Q, �
{I) ;:,o
s· ;:s � .,
527
Coal Gasification R eactors
reactor without char recycle , s uch a s the Texaco gasifier ( Fi g . 5) . T he volatiles are driven off in an ox y gen - ric h environ ment in thi s type of re
actor , an d comb u stion of the volatiles i s expected to oc cur even b efore The volatiles m u s t therefore be included in the m aterial an d energy balan ces, and t h e gra phi cal con struction s will be specific to each coal . It is ea sie st for c ompa ri son t o use the fixe d carbon in the c oal as the The lines de fini n g the basis for calcul ation s on the trian g ul a r dia gram . broad boun ds o f th e op era tin g re gion are th u s sh i fte d somewhat from th ose in Fig . 6 . The c or resp on din g line s c on n ec tin g reaction s ( 1) a n d (3) an d reaction ( 2) an d the H 2 o ve rt e x are drawn in Fi g . 16 for a coal with t he analysis in Tab l e 1 . The pul v eri ze d coal is fed to the T exaco ga si fie r in a water sl u rry . The m aximum s lurry concentration is limited by current technology to about 70% solids by weigh t . This limit is s hown as a st rai gh t line e ma natin g from the 0 2 vertex . The gasifier m u st operate to the ri gh t of t his line. A gasifier in w h ic h the c oal feed is transported solely by gas c oul d operate The 50% solids slurry li n e , w hich an y wh e r e to the right of the C - o 2 axis . is pr ob a b ly the prac tic al lower limit for o perati on , is al so shown in Fi g . 16. The effluent temperature a n d c o mposit ion can be e st ima t e d by makin g ass umption s about r elative kinetic rates a n d t h e available residenc e ti m e . Reaction ( 1) is a ssume d to occur first , an d to go to completion . Since such rea ctor s alway s operate with s li gh tly more oxygen th an r e q ui r e d to utili ze all of the carb on throu gh re acti on ( 1) , the excess o x y gen is a ssu me d to re act to form C 0 2 . T he resi den ce time is t hen as sumed to be suffic iently long to ac hieve equilib rium in the ga s phase. The reactor o perates in t he s la g gin g r e gim e , and met hanation is n ot ex pected at t he hi gh temperatures , so oxy gen atta ck on the re m ainin g fixed carbon .
FI G U RE
16
N o . 6 c oal .
O pe ratin g
diag r a m for a Texaco en t rain e d flow g asifi e r , 11linois
( A ft er D en n an d Wei ,
1982 . )
528
Denn and Shinnar
reaction ( 5) need not be included in the basis . The calculations shown D here , from enn and Wei ( 1 98 2 ) , were carried out for an operatin g pressure of 6 2 0 p si and an oxygen feed temperature of 3 0 0°F . Lines o f constant effluent temperatures of 2 2 8 0 and 3 0 00°F are shown on T h e former is t h e mini m u m temperature t h a t will ensure slaggin g , Fi g . 1 6 . while the latter probably represents a n ab solute upper boun d on the system temperature because of materials limitation s . The effluent composition is summarized in lines of constant efficiency , from 6 0 to 9 0 % . Efficiency in this figure is defined as the rati o of the hi gher, or gross heatin g val ue of the p roduct gas to the higher heating value of the coal . A s shown in E q . ( 1 5 ) ( with m = 0) , li n es o f constant higher heatin g value are lines of con stant carbon - to - oxy gen ratio , increasin g i n efficiency with decreasing oxy gen ( an d hence wit h decreasin g effluent C 0 2 ) . It is clear that for a given slurry concentration , the maximum efficiency will be obtained by operatin g at the lowest permissible effluent temperature . The maximum obtainable gros s efficiency , without takin g any kinetic con straints into account , is slightly over 0 . 8 with the limitation of a 7 0 % slurry . This mu st be reduced when the maximum temperature in the reaction zone is incorporated in the analysis . I t is important to note that all p resent cocu rrent gasifiers operate in the partial combustion regime ( R - R c > 0 ) . This can be predicte d from thermodynamic consideration s for a Texaco gasifier with a coal - water slurry containin g 30% water if we a ssume a minimum operatin g temperature of 2 7 00°F . I t follows from Figs 1 1 and 1 2 , on th e other hand , that R Rc n ee d not be positive for a cocurrent gasifier with dry feed , and yet all p ub lish e d data for such gasifiers are also in the partial combustion re gime . This is a con sequence of the use of sufficient steam with the coal feed to req uire a high oxy gen - to -coal ratio in order to reach t he operatin g tempera t ure . The steam is u sed here only as a coolant and not as a reactant for the gasi fica tion reaction s , and it serves only to s hi ft the product gas . This seems to in dicate that the steam-carbon reaction is too slow even at these high temperatures for significant reaction to occur in the short residence time in such reactors . The assumption of equilibrium in a cocurrent reactor se e m s therefore to be justified only if R - R c > 0 , and even then it is questionable since it gives pos sible operatin g regimes b ut does not predict the op e ra ti n g condition s . It is very importan t to understand this limitation when we deal with kin etic models . -
DETA I LED REACTO R MO D E L I N G
The stoichiometric and kin etic s - free calculation s define use ful b oun ds on reactor op eratin g re gio n s , but reactor operability i s ultimately _d etermined by the detailed te mperatu re , composition , and ash- state profiles . We b riefly describe in this section the approaches to such detailed modelin g and the problems that must be dealt with . D etailed models do not seem to have been utili ze d in the initial design of any of the existin g op eratin g systems, and their major use will undoubtedly lie in t he development of p rocess un der st a n din g for control , fee d st ock change , and perhap s design modification s for increased efficiency ( e . g . , Lu and Denn , 1984) . B efore un dertakin g a discussion of detailed reactor modelin g , it is well to say a few words about mode l validation . M ost data have been obtained on reactor effluents , an d effluent condition s are extremely insen sitive to the D details of the mode l . ata are generally obtained under successful operatin g
Coal G asifica tion Reac tors
529
con dition s , which mean s nearly complete carbon c on ve r sion . Given the fact s that combustion an d water - gas shift are fast an d that com ple t e carbon conversion implies a s uffic i e n t residence time for any n ec e ss a ry steam or c o2 gas i fication , th e effluent is e s sentially fixed by feed c on d iti on s . ( I t is only a slight over s t atem e nt to in sist that it is difficult to c on st r uc t a model of a succes sful g as i fi e r that will n o t p re dict the effluent correctly . M any of the models in the literature , e sp e c ially for hi gh - t emp e rature gasifiers , are simply c omplex ways to do t h e rmo dy namic calculations . ) T emperature an d composition profiles woul d be s en sitive tests , but only limite d and in compl e t e data are available from pilot - an d full - scale unit s , and these do not seem to have been used in any m o deli ng study . R ea c t i on Rates
All detailed reactor models r e qui r e a description of the soli d - ga s reaction s , u si n g models o f var yin g degrees of detail to describe the physic ochemical phenomena in t he n ei ghb orh ood of a c oal or c h a r p article . T he ki n e tic data in the li t e r atu re in dicate con si derable variation with coal rank in r e ac tivit y
to steam and C 0 2 , b u t less variability in the rate of oxy gen attack . It is ge n e rally agree d that reaction s ( 3) and ( 4) , steam and C 0 2 gasification , proc ee d at rou g h ly the same rate , and c onsi de r ably s lower than oxidation . S ome p ubli she d data on steam gasification rates are shown in Fi g . 1 7 . The data represent a range o f coals , p res s ure s , and steam a n d hy dr ogen mol e frac t ion s , so ab s ol u t e rat e m a gn itu de s are not directly comparable , but e a ch line is at c on st an t ga s composition a n d pres sure . The va riability in reporte d activation en e r gi e s is a cau s e for concern and must affect any modelin g stu dy . Lupa and Kliesch ( 1 9 7 9) have compil ed and evaluated the available rate data . Typic al d a ta showin g rel a ti ve magnitudes of the se veral gas - solid reaction rates in a constant environment are plotte d in F i g . 1 8 ; the combu stion data are from S ergent and S mith ( 1 9 7 3) and the gasification data from Gib son and E uker ( 1 9 7 5 ) . The di ffu sion limits are based on a shell pro gre s si ve model for a 10- m m - diameter particle at 2 00 0°F and 50% carbon conversion , which is typical of c o n di tion s in the combu stion zone of a m o vin g bed gasifier . G asification may be comparable in rate to combustion under
these c ondition s . One of the im port an t , still unre solved d i ffe ren ce s in the kinetic exp re s sion s employed in m ovin g - b ed models is whether to inclu de reaction ( 9) , t h e forma tion of w ate r from hydrogen and oxy ge n . This free - radical reaction would be e x pe c t e d to oc c u r very rap i d ly in t h e combustion zone , but there may be sufficient c har an d ash su rface p resent to quench the reaction . The p r ese nc e or ab sence of ga s - phase oxidation has li t tl e effect on reactor efflu ent p rop e rtie s , but it will greatly a ffect the computed maximum temperature and hence the range of op erability . Movi n g - B ed Model s
M o vin g - b e d reactor models are of two basic types : h omo geneou s and heterogeneou s . The e q uat ion s for the former take the gas and solid tempera ture s to be equal , an d average over the two p hases , w hil e the latter treat gas an d solid phase s separately . S teady - state m ul tip lic ity and limit c ycles have b een ob served only with heterogeneous models . The major di fferen c e s
bet ween models are in t h e choice of reactions to be considere d , sources of kinetic da t a , details of sin gle - p a r ticl e models , treatment of devolatilization ,
D e n n a n d S hin nar
5 30
1 500
......
.r:.
c:
_g
8
:e
1 0-
1
...... c: 0 .D
u �
0
B l ackwood a n d Gra y ( 197 8)
e
.E
0
.D
G i bson and Euke r ( 1 9 7 5 )
Cl)
0
a::
Johnson ( 1 974 )
l o-4 L-------��--�� 0 . 50 0.45 0.4 1 [ l i T (0R}] X 10- 3
FI G URE 17
Published data on steam gasification rates .
an d inclusion of radial variations an d heat transfer . All models as s u me a sin gle p article size an d plu g flow of the gas . The published movin g - bed models are summari zed in T able 6 . * A one - dimensional , adiabatic analysis is ade quate to establi s h the important op eratin g features of a full - scale gasifier with wall cooling at normal th roughp ut , since the penetration dis tance of the wall thermal b oun dary layer can be estimated to be only a small *S till m an ' s ( 1 97 9 , 1 9 8 1 ) model contain s an error in t he energy balance w hich results in computation s equivalent to replacin g the heat c apacity cp by cp ( l + d In cp / d ln T ) . This introduces a position - dependent error in t he heat capacity that can exceed 50% in the gas p hase in the region of steepest The error is unlikely to chan ge the qualitative re tem p erature profiles . sults p ublishe d usin g the model , but detailed steady - st ate an d tran sient profiles are unreliable .
531
Coal Gasification Reac t o rs
1 0- 2
rf) E
u
4000
L
lo - 3
2 I
Q) 0 a:: c: 0 � u 0 Q) a::
(OF )
:t�:i�d:: �i�u:�
Carbon - Steam React ian
Limited /_ � �f�i: {Wyamin'il) Combust i on
"' ......
0 E
T
2000 1 500
_
_
C a r bon - Stea m
10-4
10-5
Ca rbon -Steam ( I l l i no i s )
I 0- 6 ��--_.����--._--�� 0. 2
FI G U R E 1 8
0. 4
0.6
1.2
( 1 / T ) x iOOO , T ( K }
1 .6
Rates of combustion an d gasification reactions .
fraction of the diameter ( Y oon et al . , 1 9 7 8 ) , Wall cooling and radial varia tion will be important in a s mall - diameter pilot reactor or in a full - scale reactor at low throughput or in a banke d ( hot standby ) state . All movin g-bed models that have been compared with plant effluent data have shown generally good agreement . T able 7 show results of two simula tions of the Westfield test on Illinois No . 6 coal in an oxy gen -blown Lurgi gasifier . The difference between the two simulation s seem s to be the result of different means of treatin g volatile evolution and reaction , which is the source of nearly all the methane in both cases , and is not a consequence of the fact that one model is homogeneous and one is heterogeneous . The effluents all satisfy the invariant relation defined by Eq . ( 1 2 ) ( adjusted Yoon ' s calculations agree with the plant to account for the H 2S an d C 2H s ) . data to wit hin the uncertainty of the oxygen plant purity , but we have already note d that effluent p roperties are not a reliable means of model validation . The compute d maximum temperatures are at the upper end of the dry ash range , which is where the plant would b e expected to operate , b ut the maximum temperature is very sen sitive to the assumptions made re gardin g the C O /C0 2 selectivity of oxidation (D enn et al , , 197 9 ) , while the effluent is in sen sitive to this parameter . Reliable model validation would require detailed axial p rofiles , which are extremely difficult to obtain in a pressuri ze d movin g bed . Profile data of Heb den et al . ( 1 9 5 4 ) are notable , and have apparently not b een u sed . Woodmansee and Floes s ( 1 9 75) have reporte d data on the in side wall temperature profile in the GEGAS reactor , but these data may not be representative of the interior .
(J1 "' �
Published Models of Movin g- B e d G asifiers
TABLE 6
H omo geneous
One - dimen sional or radial
Detailed
Computer c o de available
R efe ren c e s
S teady - state o r transient
A mun dson and Arri ( 1 978)
s-s
Hetero
1-D
Yes
Biba e t al .
s-s
H etero
1-D
No
C aram an d Fuentes ( 1 9 8 2)
s-s
H etero
1-D
No
C ho an d Joseph ( 1 9 8 1 )
s-s
H etero
1-D
No
C wiklinski et al . ( 19 8 1 )
s-s
Hetero
1-D
Yes
D aniel ( 1 98 2 )
T
H omo
1-D
No
Desai an d Wen ( 1 9 7 8 )
s-s
H om o
1-D
No
Kim and Joseph ( 1 98 2 )
T
H e t e ro
1-D
No
Stillman ( 1 979 , 1 98 1 )
s - s and T
H etero
1-D
No
IBM
Yoon et al .
s - s an d p seudo -
Homo
1-D + core / b ou n dary layer
No
EPRI
s - s and T
Homo
2- D
No
EPRI
( 1976 , 1978)
( 19 7 8 , 1 979b )
or
hetero geneou s
s-s
Yu et al .
( 1 98 2 , 1 983 )
depen dence
sin gle -particle
model
DOE
tl
;:s ;:s !:l ;:s � Cf.l ;:r (\)
s· ;:s p -s
533
Coal Gasifica t ion R eac to rs
TABLE 7 S i mul ati on of P ressu ri z e d Lurgi Gasification of I llinois N o . 6 C oal at Westfield Usin g Models of Y oon et al . ( 1 9 7 8 ) an d C ho and Joseph ( 1 9 8 1 ) 8 Yoon et al .
Cho an d Jose p h
W e stfield pl ant datab
17 . 3
co
13 . 8
27. 5
19. 5
31. 2
H
42 . 4
44 . 4
39. 1
C0
2 CH
2
4
C H 2 6 H s 2
I nert
32. 9
6. 5
8. 6
9. 4
0. 7
0. 7 1. 0
2. 1
0,5
1. 1
1. 2
a
S tea m /oxy gen molar ratio = 9 . 6 , fixed carbon / oxygen molar ratio 2. 7 3 . b Plant data are from Woodall - D uckham L td . ( 1 974 ) . =
T ypical te m p er a t ur e p rofiles comp ute d with the model of Yu et al. ( 1 983) , w hich inclu des Y oon 's model as a special case , are sh own in Fi g . 19 for gasification of I llin oi s N o . 6 coal in an air - blown 1 2 - ft - di a m e ter dry ash Lurgi ga si fier . The c ol d zone near the wall is the source of n early all the unreacted c a rb on The effect of blast t e m p e rat u re is show n in Fi g . 2 0 . There i s a critical blast te mp er atu r e b elow whic h carb on conve rsion effec tively cease s . The model is probably inaccurate in p redicting the p recise •
E I O�------���-� G)
L__--- 1 5 2 5 -----
G) >
1 5 70 -----
0
� .8 <(
G) <J c: 0
-; 0
1 54 5 -----
1 60 5 ----1 6 60 -----
0 J(
<(
Radi a l Posi t i on ( f l )
C omputed te mp e r a t ur e contours for a dry ash Lur gi gasifier . ( F rom Yu et al . , 1 983 . )
FI G U RE 1 9
Denn and S hin nar
534
1 .0 -e
0. 8
u
.,
0. 6
c ;:)
0. 4
c 0
0 u "0
�
0
0 -� u
2 3 I
5
4
Rad • a l Pos i t i o n
1 - D - bound a r y 2 5 . 3 "4 88 . 5 "4 96 . 5 "4
1 - 0 , a d i a b a t i c core
Ioyer
c
�
0
B l ast Tem p e r a t u r e ( ° F )
1 000
FI GURE 20 Comp ute d radial distribution of unreacted carbon at the bottom of a dry ash Lurgi gasifier as a function of blast temp erature . (F rom Yu e t al . , 1 98 3 . )
value of this critical temperature , since t he finite heat transfer rate be tween the blast gas and the ash is not taken into account in the homo There is a geneous mode l , but the effect i s c ertainly real and important . detaile d di sc ussion of the in teraction s bet ween wall coolin g and blast temperature in Yu et al . ( 1 9 8 2 , 1 9 8 3 ) . The dashed lin e in Fig . 2 1 i s the locus of optimal fee d con ditions com puted by Y oon et al . for Illinois N o . 6 coal ; it lies very close to the line in Fig . 1 3 computed with the kinetic s - free approach for complete carbon conve rsion and no methanation . Computed lin es of maximum temperature for oxy gen - blown gasification of I llinois N o . 6 coal are also plotted in Fi g . 21. This c ompletes the definition of operability region s . The dry ash reactor must operate to the right of the 1 1 7f)° C contour , but a s close to that line a s possible . T he point denoted " L " on Fig . 21 represent s the operating point of the Westfield test of Illinois N o . 6 coal in a dry ash Lurgi reactor ( W oo dall - D uckham Ltd . , 1974) , an d it is con sisten t with the reactor model calcul ation s . A sla ggin g reactor must operate to the left of the ash meltin g temperature and , as noted p reviously , ideally close to point Point S is the reporte d operatin g point in a B C G /Lurgi slagger at A. Westfield ( S cott , 198 1 ) fo r a B ritish Rossin gton coal that is b elieve d to be similar to I llinois N o . 6 . The steam -to-oxy gen ratio is essentially the same as at point A , an d correspon ds to complete steam utilization , b ut the carbon to - blast gas ratio is lower than would be expected from the model . T he mo del predic t s more unreacted carbon than is observed in the slagger , probably becau s e of an inadequate description of the re gion n ear t he hearth , an d thi s is one possible explanation of the difference in feed ratios . A s a final note on t his aspect of modelin g , it is useful to return briefly to the kinetics - free method of analysis . Table 8 shows a comp arison of the gas p ropertie s above the reaction zone , but p rior to devolatilization , as computed with the detailed model an d by the kinetic s - free met ho d , ass umin g complete carbon conversion an d no methanation with the latter . The
Coal
535
Gasification Reac tors
FI GU R E 21 O p e r a ti n g region and locus of optimal feed condition s computed for dry ash Lurgi reactor . ( A fter D enn an d Wei , 1 9 8 2 . )
TABLE 8 C omparison of D etailed an d Kinetic s -Free M odel Calculation s for Oxy gen -Blown Gasification of I llinois an d Wyomin g C oal s in a D ry Ash Lurgi Ga sifier
W y omin g coal
I llinois coal
H 2o
Detailed model
Kinetic s free
Detailed m odel
Ki n etic s free
0 . 68
0 . 68
0 . 49
0 . 45
0.32
0. 45
0 . 55 0 25
2
o. 32
co
0 . 09
0 . 09
0. 22
0. 2 2
0 . 22
0. 002
0
0 30
0 . 03
0
144 1
1437
1383
1320
3 . 52
2 . 07
2. 21
2361
2362
H
co CH
2 4
Exit
temperatu re ( OF )
H 2 /CO
Maximum
temperature ( O F )
Sou rce :
Denn et al . ( 1 9 7 9 ) .
3 . 43 20 7 2
20 04
.
.
0 . 30
Denn a n d Shinnar
536
m aximum temperature is e stimated from the adiabatic temp erature rise of
c ombustion followed by water- ga s s hift equilibrium , ass umin g complete utili zation of oxy gen an d equal a moun t s of
p roduct s .
CO and C 0 2 as c ombustion
The detailed m odel calculation s ass ume the same selectivity and
an initial particle size of 2 0 mm .
All of the results , includin g the estimate
of maximum temperature , are close . activity Illinois coal then
A gree men t is b etter with the low
with the hi gh - activity Wyomin g coal ,
latter allo w s for more methanation ,
since the
The kin etic s - free app roach i s clearly
an e ffective means of estimatin g p roc e s s performance if the feed con ditions
n ece ssary for complete carb on conversion are known .
( T he maximum tem
perature es timated in t his way will differ significantly from the kinetic s - free calculation only if the particle size is sufficiently large that the oxidation
rate is limite d by mass tran sfer an d b ecomes comparable to the steam gasifi
cation rate , or if large amounts of m et hane are formed . )
There are two time scales that are relevan t in the transient re sponse of
a movin g - b e d gasifier .
T he first is
of
the order of the gas residence time ,
which is approxi mately 1 min in the dry ash reactor and
10
s in t he slagger .
T he respon se to gas load chan ge s over this time scale is only a con sequence
of chan gin g amounts of the h ot gas effectin g devolatili zation an d dryin g .
T h e temperature profile i n the reaction zone i s unchanged , so t he kinetic
proce sses remain the same an d the st eady - state mode l of the reaction zone suffices .
This short time scale is the important one for chan ges in gas
de mand in a power cycle , an d the resp onse can be computed from an overall
energy an d mass b alance on the dryin g and de volatilization zones . There is a sec on d time scale , of order hou rs ,
move ment of a thermal wave through the bed .
which correspon ds to the
A pseudo- steady - state analy
sis can be used to follow the wave p ropa gation t hrough the b e d following a
load c hange ,
and to c ompute the effluen t p rop ertie s from the steady - state
model for the given p rofile ,
as done by Y oon et al .
( 1 9 7 9b ) .
T hi s approach
is not ade quate for the large chan ges a s sociated with startup from , or turn down to, a significantly reduced throughput or a hot banked state .
The m o st important p roce s sin g result from the full transient model of Y u
e t al .
( 1 98 2 ) is that t h e lon g - term tran sient respon se o f a dry ash gasifier
is approxi mately first orde r , with a time con stant that depen ds on the op
eratin g level b u t ( based on limited calculation s ) seem s to be in dependent of the magnit u de of the chan ge .
dry ash Lurgi with Illinois N o .
Typical c omputed transien t s for an air- blown
6 coal are shown in Fig . 2 2 .
The time con
stant for turn down is about 6 h , w hile the time c on stant for a load increase
is about 3 h .
The tran sient time for startup is shorter than for turndown P ublished
becau se an increase in the gas flux increases the flame velocity .
experimental data on tran sients in dry a sh gasifie rs are not available , but
the lon g time con stan t s are c on sistent with the known operating characteri s tic s of these reactors .
Yu et al . used the dynamic model to study feed
policie s durin g turn down to maintain the comb ustion zone
in
a fixed position ,
an d c onclu ded that a step chan ge to the new steady - state ratio is effective . The nature of the t ran sient in a slagger is fundamentally different ,
since the low s t eam- to- oxy gen ratio and h ot slag tap burner gas always keep
the combustion zon e at the heart h .
Thus a throu ghput c hange doe s not
c ause a t hermal wave , an d the major dynamic resp on se i s in the region close
to the wall , where the greater relative importance of heat los s decreases
the conversion .
R es p on se times for load increase s and decreases are about
six times as fast as in the dry a sh reactor , an d effluent compositions are
essentially unchanged .
T he major transient is t he h eat transfer between
537
Coal Gasification Reac tors Cl) c 0 N
80% of f u l l l oad • • 50% /:;. • • 30 % o , •
c
0 .2 -�:::) :=Ill c
c
E a. 0 Cl)
.c
O
u c; _
....
O (/)
.C >. U -c
g... ....g
0
8: (/)
<:t �
c z
-
c:: 0
CII 0
£0 ._ ...
u..
0
4
8
T i me
12 (h)
16
20
FI G URE 22 Computed tran sient respon se of a dry ash L urgi ga sifier to load c han ge s . ( A fter Yu et al . , 1 9 8 2 . )
the bed an d the water jacket . Slagger tests at Westfield ( Scott , 1 9 8 1 ) show stable operation wit h l arge load chan ges effecte d over minutes , but the in sen sitivity of the effluent , even un der transient operation , make s such experimen ts inadequate for testin g a dynamic model . E n tra i ned F l o w M odels
Entraine d flow reactor models fall in to three general cate gorie s : two compartmen t , m ulticompartment , an d two - dimen sional turbulence . The Most models represen t publi shed descriptions are summarized in T able 9 . commercial prop rietary codes an d are noted by the corporate or in stit utional name as well as the authors of de sc riptive papers . The t wo compartment models simply assume that the gas hydrodynamic s can be approximat ed by a we ll - stirred region immediately downstream of the nozzle , followed by a plu g - flow section , with plug flow t hroughout of the solid particles . Volatiles are as sumed to evolve an d burl) , in the well stirred region . These models show good agreement with published conver sion data for the Texaco gasifier , but such agreement is to b e e xpected in
light of the commen t s made p reviously about the factors that determine reac tor effluents . There is little methane produced , so total syn gas ( C O + H 2 ) is fixed entirely by the feed oxygen - t o - c arbon ratio . The steam serve s only to shift the p roduct an d determine the CO /H 2 ratio ; prediction of the effluent temp e rature in order to compute the equilibrium of reac tion ( 7) is thu s the main re sult of the model , and the temperature prediction can be " tuned" by small changes in the assumption s about the si ze of the well stirred re gion an d the heat loss in the plu g - flow region .
D e n n a n d Shinnar
53 8 T A B LE 9
E n t rain e d Flow Gasifier Mo de ls T yp e
Reference
M o de l
A vco - Everett
Ubhayakar et al .
B rin gham Y oun g 1-DI C O G
S m oot e t al.
B ri gha m Youn g P C G C - 2
H eb den e t al .
( 1 977)
( 1979) , ( 19 8 1 )
Multicompartment
Heb den et al .
( 19 8 1 ) ,
M artin ( 1 98 2 ) Delaware
Denn and W ei ( 1 98 2 ) ,
J aycor
Ghate a n d Martin
Ph y sic al S ciences
Ghate
S - Cub ed
Schneyer et al .
T exaco
Multicompartment
Olsen ( 1 98 1 ) and
( 1982)
Martin ( 1 9 8 2 )
( 19 8 2 )
L u p a a n d Kliesch ( 19 7 9 ) ,
T w o - dim e n sion al Two - compartment
T wo - dimensional M ul ticompartment
Two- dimen sional T wo-compartment
Lupa ( 1 9 8 2 )
West Vi r ginia
W en
an d C h u an g ( 1 9 7 9 )
T wo -co mpartment
The p redi c tion of the maximum temperature is an important model out p ut , since this va lue m u s t lie within the b ound s established by the ash sla g gin g temperature an d the materials of construction . The region compris in g maximum temperatures in the ra n ge 2 8 0 00 to 3 0 0 0°F computed from the Delaware model ( D enn an d Wei , 1982) i s adde d to the trian g ul ar o p er atin g d ia gra m in Fi g . 2 3 . T h e shaded region is the feasible operati n g ra n ge for a slurry - fed , oxy g en - b l own ga sifier with I llin oi s No . 6 coal , accordin g to the mode l calcul ation s , b o un de d by a maximum temperature of 3 0 00°F , an effluent gas t e mpe r ature of 2 28 0°F , a n d a s lu rry in the ran ge 50 to 7 0 % solids by wei ght . T h e maximum temperature p rediction is very sen sitive to the assumed si ze of the rec i rc u la tin g zone , howeve r , which depends on the unknown jet c on fi gu rat ion and velocity . In ad di tion , the as sumption of a sin gle well - stirred re gion must b e a poor approximation to the tempera ture p rofile from the jet core to the wall an d should b e expecte d to predict too low a maxi mum . L u pa an d K li e sc h ( 1 9 7 9 ) note that reactor t empe rature s ( at an u n s tat ed location ) p redicte d by their model differ from plant measure men t s by s everal hundred d egrees an d are sy st e m atically low . ( Th e raw data are p ro p rietary and have not been p ublishe d . ) The tran s ie n t model predictions ( L upa , 1 9 8 2 ) a re ge n er ally good with regard to total syn ga s pr o duc t ion , b u t they deviate from the component data and fail to predict the " wron g - w ay " dynamics ob s erved in some of the reported run s . The m ulticom p artment mo dels attempt to de scrib e th e p hysic al p roces ses on a fin e r spatial scale , includin g kin etic model s of devolatili zation , bu t still use macroscopic approximation s to t he gas and solid dynamic s . The t wo - dimen sional models solve full c on tinuum de s c rip ti on s of the momentum , mas s , and energy t ran s p ort , incl u din g detaile d kinetic an d turbulenc e A con siderable d a t a ba se , includin g detailed ( m ixin g - len gth or k - e:) model s . p r ofile s of the typ e com p u ted in the two - dimen sional models , is not being develop ed for model testing on sm all entrained reactors ( H ebden et al . , 1 98 1 ;
539
Coal G asification Reac tors
FI GURE 23 O pe rability region comput e d ga s ifie r . ( A fter D enn an d Wei , 1 9 8 2 . )
for a Texaco entrain e d flow
S moot , 1 9 8 2 ; Solomon an d H a m b len , 1 9 8 2 ) , b u t a p plic ab ility of these com
plex mode l s to the analy sis of full - scale reactors h a s not been demon strated
in th e open literature . F l u i d i zed - B ed Model s
Several steady - st ate models o f fluidi ze d c har gasific ation have been p ub li s h e d usin g conventional flui dization m o de ls : pe rfectly mixed , D avidson - H arrison , an d K unii- Leven spiel . The most complete such model is that of Caram an d A mun dson ( 1 9 7 9 ) ; ot he r , less general t rea t m en t s inclu de those of Yoshida and K unii ( 1 9 74 ) an d W eimer an d C lou gh ( 1 9 8 0 ) . A model briefly describ ed by Fin s on ( 19 8 2 ) seems similar in c on c e p t to that of C aram an d A m un d son . Caram and Amundson s how good a greement w it h effluent data from a small char ga sifier , but t he m od el is not app lic able to an in d u s tri al or demon stra tion scale confi gu ration . Finson s hows poor agreement with U Gas an d W e s tin ghou s e reactor data . A n u mb e r of " fir s t p rinciple" m ode l s to follow de t ai le d two - p hase fluid mec hanic s , as well as the c he mic al reaction s , are under develo p m en t and are de s c rib e d in G hat e an d Martin ( 1 9 8 2 ) . Although these detailed hydro dyn a mic models seem to show the flow characteri stic s as sociated with flui di z e d beds , the p ubli shed de sc rip tion s in dicate that they ar e still far from p rovid in g an en gineerin g tool for d eal in g with the p robl em s describ e d earlier re ga rd in g fluidized gasifier design . The most diffic ult prob l e m is near the oxygen noz zle . T his is a highly nonisothermal zone in which pe r formanc e depen d s s tr on gly on the relative rates of comb u s ti on and solid mixin g arou n d the noz zle . B ot h ra t e s depe n d on the p roperties of the coal and the
Denn and Shinnar
540
state of t h e char , an d an ade q uate des c rip t i on of this re gi on is b e yon d c u rr e n t mo de lin g c a p abilitie s . T he only reliable w ay to develop a ga s i fi er is therefore to h a v e a pilot pl ant in whic h th e mixin g z on e i s lar ge enou gh to be du plic at e d in th e commercial plant by multiplication . H ence b oth the nozzle a nd the re gion around each oxy gen nozzle m us t b e kept c on st an t to
allow reliable scaleup . A secon d p r ob l e m is t hat the kinetic p rop e rtie s of the c ha r dep en d on p a rticle history , and
t he kinetics measured in a fixed or small flui d bed
might not b e applicable to a l ar ge b e d ; t he r e ac tivity of a pa r tic l e alter n a t e ly exposed to oxi di zin g a n d re du ci n g atmospheres is not nece s s arily the Finally , the a s h agglomeration same a s one kept in a con st an t er �viron ment . p roc es s is not sufficien tly u n d e r s t co d to be modele d reliably or scaled to differe n t o p e ra tin g c on dit ion s . REFERENCES
A mundson , N . R . an d L . E . A rri , C har ga s i fica t ion in a countercurrent reactor , A I C h E J . , 24 , 8 7 ( 1 9 7 8 ) . Biba , V . , E . K lo s e , J. Mal e c h a , an d J . Macak , M athematische s M o d e ll zur K oh lever ga sun g unter D ruck , Ener gietechnik , 26( 1 ) , 2 8 ; 2 6( 2 ) , 26
( 19 7 6 ) .
Biba , V . , J . Macak , E . Klose , an d E . Malecha , Mathematical m o de l for the gasification of coal under p re s s u re , I n d . En g . Chern . Process Des . Dev . 17.
92 ( 1 97 8 ) .
Bissett , L . A . , A n Engi n e e ri n g A s se s smen t of E n t rain men t G as ifica tion , MERC / RI - 7 8 / 2 , U . S . Department of Energy , M or ga ntown , W . Va . ( 1978) , C aram , H . an d N . R . A mu n d son , Fluidi ze d bed gasification reactor mo deling , I nd . En g . C hern . Pro ces s D e s . D e v . , 1 8, 8 0 , 96 ( 19 7 9) , C aram , H . and H . S . F u en te s , Simplified m o d e l for a countercurrent char ga s ifie r , I n d . E n g . C hern . Fun dam . , 21 , 4 6 4 ( 1 9 8 2 ) . C ho . Y . S . an d B , Joseph , H e tero ge n eo u s model for moving-bed coal gasifi c a t ion reactors , I n d . E n g . C hern . Proc e s s Des . De v . , 20 , 314 ( 1 98 1 ) . C wiklin s ki , R . R . , J . W ei , an d M , M . Denn , A h e te ro ge n e ou s movin g bed S ee also D enn ga si fi e r mo del , A I C hE D etroit N atl . Meet . (Au g . 1 9 8 1 ) . ( 198 2 ) .
Daniel , K . J . , Fixe d b e d gas i fier m o d e lin g , a p p e n di x to R . R . P rie st l e y , Movin g -B e d G asifica tio n - Comb in e d -Cycle Con t ro l S t u dy , Vol . 2 , EPRI A P - 1 8 4 0 , Vol . 2 , Elect ric Power Research I n s t i t u t e , Palo A lt o , Cali f . ( 1982) .
D enn , M . M . , S teady - state and dy n am ic simulation of movin g -bed gasifie r s , in Coal G asifica tion Mo deling Wo rks hop Procee dings , M . G hate and D J . W . M a rti n , eds . , OE /M E T C / 8 2 - 24 / UC - 90c , U . S . D epartment of E n e r gy , M or gan t o wn , W . Va . ( 1 98 2 ) , p . 3 9 1 . D enn , M . M . and J . W ei , Operab ility c on s t r ain t s for coal ga s i fic a tion re actors , P roc . Joint M eet . C hern . En g . , C I E S C /AI C hE , B e ijin g , C hina
( 1 98 2 ) . p . 6 3 1 .
D enn , M . M . , W . - c . Y u , a n d J. Wei , P ara m e ter se n s it i vit y and kinetics free modelin g of movin g b e d c oal gasifiers , Ind . En g . C h e rn . F un dam . , 18,
Desai,
2 8 6 ( 1 97 9) .
P . R . an d C . Y . Wen , C o m p uter Mo de ling of Morgan town Energy
Research C e n ter's Fixed B e d Gasifi e r ,
Energy , M or gan t o w n
, W . Va . ( 1 9 7 8 ) .
MERC /C R - 7 9 / 3 , U . S . Dept . of
Coal
G a s ifi c a t io n
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Dillingham , E . W. G . N . Richter , and W. G . Schlinger , Enriched air and oxy gen gasific ation of Illinois No . 6 coal in a Texaco coal gasification unit , in Proceedi ngs of t he Firs t A n nual EPRI Co n t rac to rs ' Co n fe rence o n Coal G a s i fi c a t io n , EPRI A P - 2 3 9 4 , Electric Power Research I nstitute Palo Alto , C alif. ( 1 982 ) , p . 1 - 1 . Ellm an , R . C . , B . C . Johnson , H . H . Schobert , L . E . Paulson , and M . M . Fe gley , C urrent status of studies in slaggin g fixe d - bed gasi fi cation at t he Grand Forks Energy Research Center , 1 97 7 Li gnite Symp . G r and Forks , N . D ak . ( 1 97 7 ) . EPRI , Co n feref!Ce Proceedings : Sy n t h e tic Fue ls - S t a t us an d Directio ns , 2 vol . , E P R I W S - 7 9- 2 3 8 , Electric Power Research Institute , P alo Alto , ,
Calif. ( 1 98 1 ) . EPRI , Proceedings o f t h e Fi rs t A n nual EPRI C o n t ractors ' C o n fe re nce o n C o a l Gas ificatio n , EPRI AP - 2 3 94 , Electric Power Research Instit ute , Palo Alto , C alif . ( 1 9 8 2 ) . Finson , M . L . P S I m odel for fluidized - bed coal gas ifi cat ion , in Coal Gasificat ion M o d e l i n g Wo rks ho p Proceedings , M . G h at e and J . W . Martin , eds . , D O E /MET C / 8 2 - 2 4 / U C - 90c , U . S . D e p art ment of Energy , Mor gan t ow n W . Va . ( 1 98 2 ) , p . 5 8 . Ghate , M . and J . W . Martin , Coal G a s i fication Mo deling Wo rksho p Proceed ings , DOE / M E T C / 8 2 - 2 4 / U C - 90c , U . S . Dept . of Ener gy , Morgantow n , W . Va . ( 1 9 8 2 ) . G ibson , M . A . and C . A . Buker , Jr . , M athem atical modelin g of fludized bed coal gasification , AIChE Annu . Meet . , Los Angeles ( Nov . 1 97 5 ) . G u m z , W . , G a s Pro du c e rs an d B l as t Furnaces , W il e y , N e w York , ( 1 95 0 ) . Hebde n , D . R . F. E d ge , and K . W . Foley , Inve s t ig a ti o n s w i t h a Sma l l ,
Pre s s u re G as i fi e r : Part I . ,
The S t u dy o f t h e Reac t io n s Occurri ng i n
G as Council Research C o mmu nication G C 1 4 , T he Gas Council , Birmingham , U nited Kingdom ( 1 9 5 4) . H eb den , P . ·0 . L . D . S moot , and P . J . Smith , Predic t io n a n d M eas ureme nt t he Fu e l Be d ,
,
o f O p timum O p erating C o n di t io n s fo r E n t rained Flow Coal G as i fic a t io n
Processes , 3 vols . , DOE /M C / 1 43 8 0 - 1 2 1 0 , U . S . Dep t . o f Ener gy , Mor gantown , W . Va . ( 1 98 1 ) . Kim , M . and B . Joseph , The dynamic behavior of moving- bed coal gasifiers , in Coal G a s i fi c a t io n M o d e li n g Wo rks hop Procee dings , M . Ghate and J . W . M arti n , eds . , DOE /ME T C / 82- 2 4 /U C - 90c , U . S . Dept . of Energy , Mor gantown , W . V a . ( 1 98 2 ) , p . 352 . Lacey , J . A . , G asification of coal in a slagging pressure gasi fier , Adv . Chern . , 6 9 , 31 ( 1 96 9) . Lohm ann , C . , Kohlevergasun g- Erfahrun gen mit der versuchsanlage Dorsten ( LU R G I - R U H R 1 0 0 ) , Gas - Wasserfach , G as - Erd gas , 1 2 3 , 287 ( 1 982 ) . Lohmann , C . and J . Langhoff , "Ruhr 1 0 011- develop ment work on t he pres surized gasi fier in Dorsten , in Co n fe re n ce P ro c e e d i n gs : s y n t h e ti c Fue ls - S t a t u s and D i re c t io ns , 2 vol . , E P R I W S - 7 9 - 2 38 , Electric Power Research Institute , Palo Alto , C ali f . ( 1 9 8 1 ) , p . 2 3- 1 . Lu , C . H . and M . M . Denn , Enhanced efficiency in dry ash movin g bed gasifiers throu gh a distributed feed , in Fro n t i e rs i n C hemical Reaction E n ig i n e e ri n g Wiley Eastern , New D elhi ( 1 98 3 ) , p . 2 9 9 . Lup a , A . J . , S i m u l a t io n o f a T exaco Gasifier , vol . 2 : A n U ns teady S tat e Mode l , EPR I A F - 1 1 7 9 , Vol . 2 , Electric Power Research I nstitute , Palo Alto , C ali f . ( 1 9 8 2 ) . Lup a , A . J . and H . C . Kliesch , Simulatio n o f a Texaco G asifier , Vol . 1 : A S t e ady - S t a t e Mode l , EPRI A F - 1 1 7 9 , Vol . 1 , Electric Power Research Institute , P alo Alto , C alif . ( 1 97 9) . -
,
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McDaniel , J . , EPRI te sts on Rukrkohle /Ruhrchemie ' s 1 6 5 ton per day T exaco coal gasification plant , in Proceedings o f the Fi rs t A n nual EPR I Co n t rac to rs ' Co n ference on Coa l G asifica t io n , E PR I AP- 2 3 94 , E lectric Power Research In stitute , Palo Alto , Calif . ( 1 9 8 2 ) , p . 2- 1 . Nowacki , P . , Coal G as ification Process es , N oye s D ata C orp . , Park Ridge , N . J . ( 1 981) . O l sen , K . Co m p u t e r Mo de ling of t h e Texaco E n train e d Flow Coa l Gasifier, B . Ch . E. ( with Distinction ) thesis , U niversity of D elaware ( 1 98 1 ) . Schneyer , G . P . , J . L . Cook , D . H . B rownell , Jr . , and T . R . Blake , Dy namic Sim u l a tio n of a Singl e - S t age E n t rain e d Flow Co al G asifier ,
EPRI AP - 27 40 , Electric Power Research Institute , Palo Alto , C alif. ( 1 982 ) . Schw art z , C . W . , L . K . R at h , and M . D . Frier , The Westinghouse Gasifi cation process , C he rn . Eng . Pro g . , 78 ( 5) , 55 ( 1 98 2 ) . Scott , J . E . , U . S . Coal Tes t Program on B C G !L urgi S laggin g Gas i fi er , EPRI AP - 1 92 2 , Electric Power Rese arch Institute , Palo A lto , Calif. ( 1 98 1 ) . Sergent , G . D . and I . W . S mith , Com bustion rate of bituminous coal char in the temperature ran ge 800 to 1 7 00°K , Fuel , 5 2 , 52 ( 1 97 3 ) . S hinnar , R . and J . C . K uo , Gasifier S t udy fo r Mobi l Coal to Gaso line Process es , Final Report , FE - 27 6 6 - 1 3 / U C - 90D , U . S . D ept . of Energy , Germantown , Md . ( 1 978 ) . S hinnar , R . , G . Fortuna , and D . S hapira , Thermodynamic and kinetic con straints of catalytic synthetic natural gas processes , Ind . Eng . Chern . Process Des . D ev . , 2 1 , 7 2 8 ( 1 9 82 ) . Smoot , L . D . , Experimental dat a for code evaluation , in Coal Gasification Mo deling Wo rks hop Pro ceedings , M . Ghate and J. W . Martin , eds . , D OE /M E T C / 82 - 2 4 / U C - 90c , U . S . D ept . of Energy , Mor gantown , W . V a. ( 1 982 ) . p . 2 92 . Smoot , L . D . , P . 0 . Hedman , and P . J . Smit h , Mixing and Kene tic Processes in Pu lverize d Coal Com b us tors , 2 vol . , EPRI FP- 1 1 99 , Elec tric Power Research Institute , Palo Alto , C alif. ( 1 97 9 ) . Solomon , P . R . and D . G . H am blen , Measurement of coal gasification reac tions with on- line in - sit u F T I R analysis , in Coal Gasification Modeling Wo rks hop Proceedings , M . Ghate and J . W . Martin , eds . , D OE /ME TC / 82 - 2 4 /U C - 90c , U . S . Dept . of Energy , Morgantown , W . Va . ( 1 9 82 ) , p . 275 . Stillman , R . , Simulation of a moving be d gasifier for a western coal , IBM J . Res . Dev . , 2 3 , 2 49 ( 1 97 9) . Stillm an , R . , Movin g bed coal gasi fier dynamics using M O C and MOL tech niques , AC S Symp . Ser . , 1 8 6 , 331 ( 1 98 1 ) . Ubhayakar , S . K . , D . B . Stickler , and R . E . G annon , Modeling of entrained - bed p ulveri zed coal gasifiers , Fuel , 5 6 , 2 8 1 ( 1 9 7 7 ) . Weimer , A . W . and D . E . Clo u gh , Modeling of char particle /size conversion distributions in a fluidi zed bed gasifier : non - isot herm al e ffects , Powder Techno I . , 2 7 , 85 ( 1 98 0 ) . Wen , C . Y . and T . Z . Chaung , Entrainment coal gasification modeling , Ind . Eng . Chern . Process Des . Dev . , 1 8 , 6 8 4 ( 1 97 9) . Woodall - Duckhan , Ltd . , T rials of A me rican Coals in a L urgi G as ifie r at Westfield , Scot land , Res . and D ev . Rep . F E - 105 , ERDA , Washin gton , D . C . ( 1 97 4 ) . Woodmansee , D . E . , Modelin g of fixed bed gas producer perform ance , Energy Commun . , 2 , 13 ( 1 976) , "
Coal Gasification Reactors
543
Woodm ansee , D . E . and J . K . Floe s s , Coal gasi tiability evaluations in a o n e - foot diameter , fixe d - bed ga s producer , AIChE Natl . Meet. , Houston , Tex . ( 1 97 5 ) . Yoon , H . , J . Wei a n d M . M . De n n , A Model for mo vin g b e d coal gasifica tio n re actors , A I ChE J . , 2 4 , 885 ( 1 97 8 ) . Yoon , H . , J . Wei , and M . M . Denn , Feasible operating re gio ns for moving bed coal gasi fic at ion reactors , Ind . Eng. Chern . Proces s De s . Dev . , 1 8 , 3 0 6 ( 1 9 7 9a ) . Yoon , H . , J . Wei , and M . M . Denn , Transient behavior of moving bed co al gasifiers , A I ChE J . , 2 5 , 4 2 9 ( 1 9 7 9b ) . Yoshida , K . and D . K u nii , Complex r eac tio n s in fluidized bed s - simulation of ga si fi c at io n , J . Chern . En g . J p n . , 7 , 3 4 ( 1 97 4 ) . Y u , W . C . , M . M . D e n n , and J . Wei , Transient sim ulation of moving- bed coal gasifiers , A C S Symp . S eri e s , 1 96 , 35 9 ( 1 9 8 2 ) . Y u , W . C . , M . M . Denn , an d J . Wei, R adial effect s in moving bed coal ga si fie r s , C he rn . En g . Sci . , 3 8 , 1 46 7 ( 1 9 8 3) .
9 Gas-Liquid Reactors S E R G I O C A R R A and M A S S I M O MO R B I D E L L I
Po litecnico di Mi lano ,
Milan , Italy
I N D US T R I A L R E A C T I O N S A N D T Y P E S O F R E A C T O R S C hemical reac tion s between a ga s an d a solute dis solved i n a liquid are very
common in i n d u s t ry .
In Table 1
ex a m p l e s of s o m e i m p or tan t p rocesses per
for m ed in gas - liquid re ac tors are gi ven .
In s uch re actors the gaseous c om i n the liqui d , w he re t he y react with t he other The fun damental a n aly sis of t h ese units is very c o m p lex as a
ponen ts are dissolve d
components .
con sequence of the simultan eous occurrence of diffusion and c hemical reac tion .
The hy drodynamic c on ditions of th e sy s tem are also difficult to define .
the rate s of mass tran s phase s and by the rate of c h e mic a l reaction . Actually , the diffusivity in the gas p hase is some orde r s of magnitude high I t follow s t hat g a s - side mass tran s fer er th an the diffu s ivity in li q ui d s . The overall rate of the p roce s s is affecte d by
fer in the g�s an d li q uid
resistance becom e s signific ant only in the c a se of v ery fas t chemic al r e a c
tion s .
Accordin g t o the relative m a gn i t u d e of mass tran s fer rate with respect
to re act i on rate , t wo extreme s itu a tion s c an b e evidenc e d :
1.
I nstantaneou s
2.
vol u m e of liquid ,
irreversible re a c tion
in
whic h the ra t e of
t he overall
proce s s is c on t rol le d by t h e diffusion rates of the re act an t s . Slo w reaction s with u ni for m concentration o f reactants in the whole whi l e the concentration of the dis solve d gas is
determin e d by phase equilib rium .
The rate of the overall proce s s
i s controlled b y the rate o f the chemical reaction .
Some cases of p rac t ical im por t anc e may be ap proxi m ate d by the men
H owever , for m any importan t in dustrial processes t he rate of mas s tran sfer and chemical reaction s are co m p ara ble . Therefore , b ot h effec t s must be accounted for in t he de sign of c he mic al tione d asymptotic sit u ation s .
r e ac t o rs .
In a gas - li quid reac tion , yield and selec tivity are also affecte d by mass
tran s fer , the n a tu re of gas - li qui d con tactin g , and residence - time distrib ution
545
Carra a n d Morbi dellt
546
TAB LE 1
Examp le s of P roc es se s Performed in G as Li qu id R ea c t or s -
Absorption of acid gase s
Ab s orp t ion of S 0 3 in dilute s ul furic acid Ab sorption of N 0 2 in dilu t e nitric acid
Removal of c o 2 and H S by ab so r p tion in 2
alkaline liq uids
Oxidation of organic compoun d s by oxy gen or air
Oxidation of paraffin s to aci d s
Oxidation of p -xylene to te reph t ha lic acid Oxida t ion of cyclohexane to cycloheanone Oxidation of acetal dehyde to acetic acid Oxi da tion of e t hyl en e to acetaldehyde C hlorination of benzene to chlorobenzene
C hl orina t ion
C hlorination of do decane Chlorination of toluene to chlorot o luene A ddition of chlorin e t o e t hylene
Hydro genation of o rganic c omp oun ds
H ydrogen ation of olefins
Other re ac tion s
Ab s or p tion of isobutylene by sulfuric acid
Hydro genation of esters of fatty acids
S ul fati on of al c ohol s by sulfur trioxide
Polymeri zation of olefins in or ganic solvents
For in stance , in a con secutive re a cti on network , the in the t wo phases . in sufficient rate of mas s t ransfer in the li qu i d may gi ve rise to a de crea se in selec tivity with respec t to an in te r m ediate desired product . This situa t i on may occur for example , in t he c hlorin a tion of orga nic c ompoun d s ( van de Vusse , 1 966a , b ) . The oxi da tion of hydrocarb ons is an ot h e r gas -liquid reac tion in whi ch m a s s transport may plan an important role in definin g t h e se le ct ivi ty toward t h e de sired product ( C arra and S an tac es a ria 1 98 0 ) . T he de sign of a gas - liquid reactor proceeds in two step s : ,
1. 2.
The c hoice o f a suitable type of reactor The de fin ition of reaction condition s an d the de si gn of the geometry
par a m e t e r s To b e gi n we c on sider the first point . S ome typical exa mpl es of gas liquid reactors are illu strate d in Fi g . 1 , an d each will be considered in t u rn ,
.
S ti rre d - Tan k Reac tors
T he se reactor s which are agit ate d mechanically , re p r e sen t ver satile equip ment for d i s pe r sin g gase s . These unit s are the best for sy s te m s with ,
547
Gas -Li q ui d Reactors
c o i l cooled
J a cke t e d
L
G
G
G
G
.
L
.
. . . . . . . . . . . . .
�
L
Bubble c o l u mn
.
I
.
. ·. .
G
m u lt i s t a g e d
FI G U R E 1
p a c k e d colu mn
. .
G S pray colu mn
L
G
G
L
. . ..
• • • .
G
e x t e r nal c oo l i ng
. . . . . .
·:
fL
ej e c t or rea c t o r
e j e c tor
Ve n t u r i
S chematic representation o f some industrial ga s - liquid reactors .
lar ge heat effects an d are particularly u seful for performin g slow reaction s req uirin g high liquid holdup . B ub b l e C o l um n s
I n these column s t he g a s is supplied at t h e bottom o f a tower whose height is at least three times it s diameter . G as bubbles rise throu gh the liquid ; the bubblin g r eac ti on creates mixin g . These unit s are employed for relative ly slow reaction s in whic h the key componen t is u su all y in the liquid phase . They are more economical than stirred tanks , Pac k e d C o l um n s
In these column s liqu i d an d vapor p a s s c o - o r countercurrent t o each other They are very good for
throu gh the passages c reated by the packin g .
548
C arrd a n d Morb i de lli
ser vice t h a t re q u i r e s lo w p res s u r e d rop and for corrosive service because These units are de sign e d as ab sorbers an d the reaction is vie we d as an accelerated ab so rp ti o n .
of the wide se l ec t ion o f m a te rial of c ons t r uct ion .
Pla t e C o l um n s
( Pla tes :
Cap s o r Valv e s )
Perfo ra te d ,
w ith B u b b l e
L i q ui d and vapor follow separate paths between stages , an d they contact each ot her on plates . In these u nits liquid holdup on plate s is a d va nt a geous for slow reaction s requiring lon g contact ti m e . S p ray Columns
Liquid is sprayed with n o z zles from the top of a column and the gas flows
upward . The unit is e mpty , and it is re qui re d in the c a s e of gas streams c on tain in g solid particles . A hi gh liq ui d area is c r ea t ed near the n o z z le s , but dr op s coale sce an d the area is rap id ly reduced . For this reason these uni t s are ge ne ra l ly used for relatively easy ab s o rp t ion d u ty .
PH A S E E Q U I L I B R I A A N D D I F F U S I V I T Y O F GASES
IN
LI Q U I DS
The de sign of ga s - liq uid reactors requires a knowledge of the distribution of in volv e d c ompon en t s b etween phases in a state of thermodynamic e q uilibri
A l th ou g h complete theo retical prediction s of the mention ed effect s are not pos sible at p resent , some correlation s have been p rop os e d to e valu a t e their role in gas - liquid
um an d the rate at whic h mass transfer occurs .
c o nta c t ors . T hey are u s e d only in the absence of ex p eri m ental data an d after careful analysis of t h e condition s of validity . Pha se E q u i l i b r i a T hermodyna mic e q u ili b riu m con dition s bet ween the gas an d liquid p h as e are co n ve ni en t ly exp res sed t hrough a vaporization fac t or as y i
-
X. 1
=
K. ( x ,y , P , T ) 1
( 1)
S tandard thermody n amic s ( P rausnit z , 1 9 6 9 ) yields suitable e x p re s sion s for the v apori za tion fac tor Ki · T wo si t u a tion s must be con sid e r e d . The fi r s t refers to the case in whic h s ome c omponents are p r e sen t in the vapor p h ase at a tempe ra t u re higher than their c ritical temperature ( superc rl tical c omp on e n t s ) . I t oc c u rs , for in s t an c e , when hy drogen or o xy ge n fee d the ga s - liqui d r eac to r . In this case
•
( 2)
where
549
Gas -Li q ui d Reac tors
- In
(�) RT p
( 3)
g
where B ij is the second virial coefficient o f the mixture components i and j , H i the Henry constant , and Yi the activity coefficient of superficial It must s ati sfy the unsumm etrical nor component i in the liquid p hase . mali zation con dition lim y� 1
x.+ O 1
=
( 4)
1
T he limitin g Henry ' s constant can b e predicted from nonideal solution behavior , in agreement with the theory of regular solutions ( Hildebrand et al . , 1 9 7 0 ) , through the relation
[ :� - 8> 2]
Hi = P i exp
(o
( 5)
i
where is the extrapolated vapor pres sure of component i , and Vi is the molar volume of the solute ga s , often assumed equal to the value of the liquid volume at 2 5°C ; Oi an d 'f are the solubility p arameters of the solute i and of the bulk liquid , re spec tively . The solubility parameter reflects the in fluence of forces b etween the molecule s , through the cohesive energy density expre s sed by
pf
o. 1
t. H ev , 1
.
=
-RT ( 6)
-
where t. H ev , i is the molar enthalpy of vaporization an d V i is the molar volume of component i . I f the size of the molecules of the solute i are greatly different from those of the solvent B , E q . ( 5 ) m ust be modi fied as follow s : H.
1
=
1
p? exp
[v.
-1
RT
(o.
1
-
"6)
2
+
In
v.
-1 -
V
B
+
(
1
v.
)]
-1 -
V
B
( 7)
The p rece din g equations are satisfactory for solutions w here both s ol ven t an � solute are nonpolar or for sli ghtly polar solven t s . The values o f O i an d 8 can b e obtained either from Reid et al . ( 1 9 7 7) or from Wilhem an d Battino ( 1 97 3 ) . The solubility of some ga ses in water as a function of temperature i s sum mari zed in Fi g . 2 . An empi rical correlation for nonpolar gas solubilities in alcohol - water solution s has been proposed by Tokunaga ( 1 97 5 ) in terms of the alcohol volume fraction
550
C arrd and Morb i de lli
�
"' 10 z 0
1u ct 0:: ......
w ..J 0
�
>1-
z 10
103 104
..J
iii
J ..J 0 en
105
L O G T E M PE R AT U R E T IN K
Solubility of gase s in water .
FI G U RE 2
( F rom H ay duk and Laudie , 1 9 74 . )
where the e mpirical coefficien t s depen d only on the nature of the alcohol an d temperature , n ot on the nature of the solube gas ( 0 2 , C 0 2 , N 2 or even CH 4 ) . At mo d erate p ressure ,
K. =
( 9)
1
where
fj
is the fu gacity of p ur e component i and Yi is the activity coefficient It s a ti sfie s the symmetric normali zation con dition :
in the liquid p hase .
li m 1
x.+ 1
y. = 1 1
( 10)
The in flu en c e of electrolytes on the solub ili ty of ga ses can be estim ated through th e followin g equation p rop o se d by van K revelen and H oftij zer ( 1 9 4 8b ) :
lo g
H
10
H B
= hi
( 11)
5 51
Gas -Li q u i d Reac tors
where Hs is the value of H en ry ' s con stant in pure solvent an d I the ionic stren gth of the solution : I
=
1� 2 2 .1....1 C . Z . -
( 1 2)
1 1
. 1
C i bein g the concentration of ions of valency Zi ; h is a saltin g coefficient that can be expressed through the sum of different contributions as
h = h
g
+ h
+
( 1 3)
+ h
h g refers to the specie s of gas , h+ and h to the species of positive and negative ion s , respec tively . The numerical value s of these coefficients for some gases an d ions are given in Tables 2 an d 3 . I n mixe d electrolytes the following equation , suggested b y On da e t al . ( 1 9 7 0 ) , can be applied : _
log
H 10 H
=
B
'E . h . I . ] ] ]
( 14 )
TABLE 2 S altin g Coefficients for Inorganic Ions ( L /g mol ) I on
H
+
Na K ·
h
+
+
NH
Li+
+
4
2+
Mg 2+ Zn 2+ Ca 2+ Ba 2+ Mn 2+
Fe
Co Ni
2+
2+
Cr
3+
Source :
+
I on
h
- 0 . 1110
Cl
0 . 34 16
- 0 . 0183
Br
0 . 3310
- 0 . 0362
N0
- 0 . 0737
OH
- 0 . 04 1 6
HS0
- 0 . 0 56 8
HS
- 0 . 0590
HC0
- 0 . 0547 - 0 . 04 7 3 - 0 . 06 2 4 - 0 . 0602 - 0 . 0532 - 0 . 0520
0 . 3230
3
0 . 3875 3
0. 3 7 1 8 3
co z3
so so
PO
0 . 3869
i
3
-
2-
3-
4 C 6H 0 5 Mn0 4
- 0 . 0986 C harpentier ( 19 8 1 ) .
0 . 4286 0 . 3 7 54 0 . 3446 0. 3275 0 . 3265 0 . 4084 0 . 2600
en en .,..,
T AB LE 3
S altin g Coefficient h
H
T ( °C )
2
-
0
0
g
2
- 0 . 2222
15
- 0 . 2 1 97
- 0 . 1 78 6
20
- 0 . 2 132
- 0 . 1771
25
-
- 0 . 1892
- 0 . 2277
H S 2
- 0 . 2551
NH
3
- 0 . 2 3 94
c
�2
C H4 2
- 0 . 2124
- 0 . 2003
- 0 . 2240
- 0 . 1951
80
2
- 0 . 3 1 54
N
2
- 0 . 1 90 4
- 0 . 3122 - 0 . 2327
40 So u rc e :
2
- 0 . 2 1 10
- 0 . 2 1 70
35
Co
- 0 . 1653
10
o . 2 1 15
for Gases ( L / g mol )
C harpentier ( 1 9 8 1 ) .
n 1:1
i
1:1 ;;:s Q.
E: 0
t}
i
553
Gas -Liq ui d Reac tors
where I j is the ionic s t ren gth attributable to the jth electrolytic s p ec i es ri stic coefficient of that electrolyte .
and hj 1s a cha rac te
D i ffu s i v i ty i n Li q u i d s
D es pit e different theoretical tr eatm ent s available for the desc ription of diffu
sion in liquids , there are no satisfactory met h od s to predict the diffusivity coefficien t s , T herefore , p re dic t ion proc e d ur e s must be applied only when exp e rimen t al data are not a v ailab le . Such prediction procedures can be divi de d in t ho s e suitable for nonelec t rolyt e s an d t h o s e s uitable for elec tro lytes . M ost studies h av e foc u se d o n the estimation of diffusivities in dilute solution s ( S kellan d , 1 97 4 ) , T h ere are two theoretical a pp ro ac h e s to the di ffusion al t h eo ry of non elec t rolyte s , T he first is known as hy drody namical t heory , an d in it t h e drag of a large s p he ric al molecule i, movin g throu gh a con tinuu m of small solvent molecules j , is evaluated . The followin g equa tion is derive d : D. .�.
�
( 1 5)
=
T
where ri is the radius of the molecule i . In the latter app roach , the liqui d is t r eate d as a quasi - crystalli ne lat tice in which holes are scatte red . T h e followin g relation was proposed b y Jost ( 1 9 52 ) :
D � ij j T
-
f ( molecular volum e s of mixture )
( 16 )
ri gh t - han d side of E q . ( 1 6 ) h a s b een semiempirically e valua t e d a n d a set of p ro po s e d relation ship s are su mmarized in T able 4 , T he molecular volum e s can be estimated by addin g up t h e contrib u tion s of the at o m s in the molecule as reporte d by Liley an d G ambill ( 1 9 7 3 ) , pp . 232- 2 3 4 ) . The p a r am et e r l',; w hich appears in t h e Wilke - C hang e q u a tion (Eq . 1 in T a b l e 4 ) , is an a s sociation factor for the solvent . It i s equal to 2 . 6 for water, 1 . 9 for m ethanol , 1 . 5 for ethanol , an d 1 for unas sociated solvents such as hydr oc arb on s , ethers , an d so on . The uncertainity in volve d in a s si gnin g values to l',; for the new solvents has resulted in efforts to eliminate this factor , as in t he other equation s reported in t he table . The effect of c on c en t r ation on diffusion coefficients can be accounted by mean s o f the followin g e qua tion , propos e d by Vi gn e s ( 1 96 6 ) : The
( D . �. . ) 1
1J cone
=
x.
( D':'. � . ) l ( D � � . ) 1J
J
J1
1
x.
1
1 +
(
d
ln y 1. )
d 1n x i
( 17)
bein g t h e di f fu sion coefficien t s in dilute solution ; ll i · � j . an d a n d of t h e solu tion , re spec tively ; and Y i t he activity c o e ffic i en t of the solute . E qu ation ( 1 7) can be a p p lie d t o n ona s s oci a te d solu tion s or to an a s sociate d solution wit h a c on s t ant de gree of association , In an electrolytic solution both cation s an d an ion s diffuse at the sam e rate , so that el e c tric al neutrality of the solution is p re serve d . If complete
njj
an d
nfi
� ij the visc o sitie s of pure compon e n t s
Carrll an d Mo rb i deUi
554 TAB LE 4
D
S emiem pirical Relationship s for
Solution of Nonelectrolytes
Equation
Restriction
1
E xclu de
D
water a s
iffu sivity in Dilute Binary
E quation
a
i J..i ii
T
solute
D j..l i ij
Exclu de
2
T
water as
solute 3
:::: 8 . 2
D
General
i
]..l
T
D . ]..l .
5
A q ueous
l . _.!..J. T
solvents
V ./ V. ) 213
] 1 ----:-�___:_ :. _
1 + (3
-8
- 1/ 3 v i
=
]
1
< 1. 5
1 0 for ( V . / V . )
=
a
Organic
10
ij
bein g a
4
X
=
- 8 . 5 for ( V. / V . ) > 1 ; 5
8 . 52
X
10
-
8
1
J
/ - - 1/3 1 . 40( V . V . )
-
-
l _...;l�___....l _...;i;... __..
__
- 11 3 v ,
+
(V./V )
solution s
only
dis sociation takes place ,
the diffusion coefficient can be p re dicted very
accurately at infinite dilution by usin g the followin g equation p roposed by Nerst
D'? l where
( 1888) : =
Df
8. 931
X
10-
1
4
T
JI. O JI.O
R.: : 'i'O +
z+
z_
+
z z +
-
i s the diffusivity of electrolyte ;
conductances at infinite dilution ,
R.:
( 1 8) and R. / the cation an d anion
�
respectively ( mhos equivalent ) ;
the ab solute values of cation an d anion valenc e s .
dfrf .11.0 , are given by Liley and Gambill
( 1973 ,
usivitie s at other than infinite dilution ,
recommen ded b y G or don
( 1 9 3 7) :
p.
235) .
Some values of
Z+ an d Z
!I.�
and
For evaluatin g the
-
the followin g equation has been
555
Gas-Li q u i d R eac t o rs
A verage error Or
gan ic t
s ol ven s
Water as s olven t
27
( %)
Water as solute
up 2 50
11
Reference
Overall
10
Wilke and C han g
( 1 955)
up 2 50
25
15 18
S cheibel
9
13 . 5 18
26
16
Reddy a n d D o rais w amy ( 19 6 7 )
Othmer and Thakar
( 195 3)
5
Lusis an
( 1 96 8 )
(D )
i cone
m
( 1 95 4 )
=
D ? -11
C
.V .
)
)
� ll ij
1 + m _ a_In _ a ___
(
Y
m
+
)
d
R atcliff
( 1 9)
bein g t he molalit y of t he �ectrolyte , y + the m ean activity c oe f
ion s based on molality , a n d V . t h e p artial m ol ar . ) MASS T R A N S F E R
volu me o f
fici ent
of
solven t .
IN G A S - L I Q U I D CON TACTORS
B a l a nce o f a R ea c t i n g S y stem
The mass b ala n ce eq uations of a ga s -liquid re a c ti n g sy stem are written in terms of phase equilibria , in terphase mass tran sfer , and reac tion rate da ti� .
556
Carra
and
Morbi delli
ga s o u t l i qu id i n
gas FIGURE 3
S chematic r ep r e se n t ation of a well - mixed gas -liquid unit .
T h e role of these fac tors m ay be evidenced by con s i de rin g a n i de ali z e d pe rfe c tly mixed , t wo - p h a se system , as illu s tr ate d in Fi g . 3 , in w hich a reaction occurs in the liq ui d phase . T he mass tr an s fe r of t h e volatile com T he first s tep is the tr an s fer from the bulk ponents in volve s two step s . gas p h ase to the ga s - liquid in terface ; t h e second involves t h e t ran sfer E quilib r iu m condi from the ga s - li qui d interface to t h e bulk li q ui d phase . The mass transfer rate per tions are a s s u m e d to exist a t the in terface unit interfacial area , Nj , of e ac h component i is e xpressed as a function o f the bulk - phase concentration s : .
where
H.1 =
p K. �
( 2 1a)
p R.
( 2 lb ) The asteris k in dic a t e s the ga s - liquid in te rfac e ; E i is an e n hanc emen t
of chemic al reaction on mass trans fe r , as wi ll b e explaine d late r . Althou gh more ri go ro u s treatment s are available ( K i n g 1 96 4 ) , the usual combined resistance a p p ro ac h is e mployed to de im e the overall mass tran s fer coefficient K gi in terms of gas-phase and liquid - p hase re sistanc e s a s follows :
fac tor whic h accounts for the in fl u e nc e
,
( 22) I n " b ub ble- type" gas - liquid rea c tion syste m , 1/ k� is gen e r ally s mall enough com a e d T he p o s sib l e exception 1s the case in which k 0 . E.
p r
0 .E. . with H./k 1 .<.1 1
.<.1 1
Gas -Li q ui d Reactors
557
is very lar ge because of a fast reaction i n t h e liquid phase . The m a s s balance o f componen t i in the unit under consideration may b e stated a s
( 2 3) In ad dition , the mass b alance in the liquid phase is given by
( 2 4) where
�
is the overall rate of con sumption of the ith compon ent .
Effects of C hemical R eac t i on s on M a s s T ra n s fe r
The differential equation of chan ge i n t h e liquid p hase arise s from a mass balance on eac h component ( D anckwert s ,
1 9 7 0 ; A starita , 1 96 7 ) .
I t ex
presses the phenomenon of si multaneous diffusion snd c hemical reaction and may be written as ac
i
at accum ulation
=
-u
•
II C .
1
+
convection
D . \1
1
2
R.
C.
( 25)
1
1
molec ular transport
reaction rate
I f the b ulk of the liqui d is well mixed , the p recedin g equation needs to b e inte grated ove r the region n ear the gas - liquid interface . This im plies an un derstan din g of the flui d dyn a mic s in such a re gion . The sim plest model re lie s on the a s sumption that the velocity vector � is parallel
to the in terface . In a ddition , the concentration gradient vector I! Ci is assumed to be perp en dic ular to the in terface . A s a c onsequence of these T hi s p rocedure is accom a ssumption s , the convective term is dropped , plished in c la s sical film an d p enetration mo dels . The models contain one parameter each , which can not be sp ecified theoretically an d whic h reflect the in fluence of the prevailin g flow p attern ( S herwood et al . , 1 97 5 ) . In the former model the p aramete r is the film depth o in which the resistance to
mass tran s fer J s c on fin e d .
,
In the p en etration mode l , the p arameter is the
mean lifeti me s of the elements of fluid whic h are randomly replaced at the
surface . B oth p a rameters are se miempirically evaluated from experimental values
of the mass transfer c oefficient k t without chemical reaction ,
_
�s
( 26)
Attemp t s have b e e n made to iden tify t h e more reliable of t h e mentioned The general conclu sion is that for describin g the rate of gas ab models . sorption in the presence of chemical reaction , only small differences exist bet ween the two models once the same value for the purely phy sical mass
transfer coefficien t k t is assumed .
In fact , the accuracy of available
C arra an d Morbtdelli
558 experi men tal da t a is frequently in s u fficie nt
to di scrimin ate between the two differ by more than a few p erc en t . Signifi c an t differences appear only when the gaseous and liquid reactant s have gre a tly different diffusivitie s ( B rian et al . , 1 9 6 1 ) or w h en a c o mp l ex re action sc heme i s involved ( T avares da Silva , 1 9 74 ; H u a n g et al . , 1 9 8 0 ) . It follow s that t he choice b e t w e e n the models is usually made on the b as i s of si m p lici ty in de sc ribi n g t he r ea c ting system . For these reasons the film mo de l has been more wide ly u se d an d will be ap p lied in t he followin g . The dependence of the mass tran s fer coe fficient on the diffusion c o e ffici en t pre dicted by the t w o models i s different ; the power 0 . 5 , given b y the pe n e t ra t ion t heory is recommended . The occurrence of a chemic al re ac tion in the liquid ph ase affects the value of the mass t ra n s fe r coefficient . This e ffec t can be accounted for throu gh the int ro du c tion of an en han cemen t fac to r E or a reac tion fac tor T h e former is gi ven by the ratio b etw een the m a s s tran sfer fluxes E* . with an d wi t hout reaction , in the s a m e hydrodynamic con dition s ; mode l s ,
whic h do not usually
,
/ ds )s
- D ( dC i f k
R-i cc i
1
=
0
( 2 7)
- c u>
factor is defined as t he ratio betw ee n the total flux of the c o m p onent i u n de r examination in t h e presence of a reaction an d the hypothetical ma s s flux in the absence of reaction and with ze ro con centration of i in the li q u i d b ulk : Inste ad , the re ac t io n
ma ss
( 2 8} T h e m at h em atic al analy si s of gas -liquid reactions i s complex as a con s e qu e n c e of the interference of hydrodynamic an d chemical fa c tor s It follow s that t h e possibility of obtainin g analytical expressions of the men tioned factors is li mi t e d to cases of si mp lifi ed reac tion kinetic s only . The type of chemical sy st e m which has received t he most attention is that in whic h the di s s olve d gas A un der goes an irreversible reaction with a reactant B di ssolve d in the li q uid . The reaction is de sc rib e d by th e fol lowin g stoichiometric equation : .
A + vB B + p rod uc t s
( 2 9}
The film model as sume s t hat even on a mic ro scop ic scale t h e process occurs in steady state , so t ha t the term ( a C i / < H ) in Eq . ( 25 ) is dropped . I t follo ws that the material b al a n ce s for t h e two components are
DA D
d
2
c fA
� 2
B
d C ds
m
2
- r
- v
=
B
r
0
=
0 � s � 0 0
( 30}
Gas -Li q ui d Reactors
559
r bein g the reaction rate an d s the distance in the liquid from the inter face . D A an d D B are the diffu sivities of A an d B in the liquid .
The boun dary con dition s are
( 3 1)
( 3 2) To obtain t he b ulk concentration of component A (i . e . , C R.A ) , it must b e taken in to accoun t that some o f it reacts within the film an d the rest i s tran s ferred across t h e film . For in stanc e , in a well - mixed reactor , the following m as s b alance equation is obtained :
- aD
A
d C fA
( 3 3)
-
ds
Thi s equation states that the transfer rate of dissolved gas to the b ulk liquid must b e e qual to the amount reacted in the bulk plus that which leave s the reactor in the effluent stream . B y in te gratin g the precedin g differential equation s , the concentration profiles in the liquid film are ob tained , and from there the values of the enhancement and reaction factors can be evaluated throu gh E qs . ( 2 7) and ( 28 ) , respectively . E n ha n cemen t Facto r fo r Fa s t R ea c t i on If the reaction rate is high , it is justifiable to conside r t he reaction to be completed in the film . In this case the concentration of the ab sorbed gas Therefore , the boun dary condition ( 3 2 ) in the bulk of the liquid is zero . can be simplified as follows :
at s
==
o
( 3 4)
and the enhancement and reaction factors are then identical . Althou gh this situation is not fulfilled for many important . gas - liquid re " actions , it is worthwhile to consider it in some detail . C ase s in which the reaction also occurs in t he liquid b ul k are discussed extensively in the next section . The typical form of the concentration p rofiles of the reagents A an d B T he reaction is fast an d it occurs of reaction ( 2 9 ) is illu strated in Fig . 4a . completely in the liquid film durin g the diffusion of componen t A . F o r re actors in whic h this situation occurs , hi gh interfacial area is required , while the liquid holdup is not important . The use of packed column s is thus recommen ded . Let u s conside r , first , the case in which reaction ( 29) take s palce th rough a bimolecular reaction of first order with respect to both
Carra and Morb i delli
560
s
(a)
s
C1s
s
(C)
FI G U R E 4 Concentration profile of gaseous and liquid reactants in th e liquid phase : ( a ) fast reaction ; ( b ) fast reaction with C R.B » ( c ) instantaneous reaction .
C !A ;
reagents . In this case the enhancement factor can be expressed as a func t ion of a di m e n s ion l ess p arameter known as the Ha t ta n u mb e r :
M
l D A kC R.B
=
k
( 35)
R.A
an d of the concentration diffusion parameter :
( 36)
=
A n appro xi mate expres sion of the enhanc e me nt factor E has been pro posed by van K revelen and H oftij z e r ( 1 9 48a ) . I t w a s obtained throu gh a lineari zation tec hnique based on the assumption that the concentration value of B throu gh ou t the film is c on s t a n t and e qua l to its val ue at the interface . Wit h this assump tion the film model equation s can be integra te d and the followin g implicit expre ssion for the enhan cement factor is obtained : M [ (E
.
A ,1n
tanh {M [ (E
- EA ) / ( EA - 1) ] , tn
1/2
.
. - E ) / (E . A A ,1n a ,tn
-
1) ]
1/2
}
( 3 7)
EA , in is the enhancement factor correspon din g to an i n s tantaneous reaction an d is given by
561
Gas -L i q ui d Reac tors
E
( 38)
A ,In .
The behavior of the enhancement factor as a function of the param et e r s M and Zn is given in Fig . 5 . T wo p articula r cases merit attention . The first oc c ur s if t h e concentration of component B in t he bulk liquid is muc h g r eate r than C A , a s shown in Fig . 4b . In this case the kinetic s of the reaction becomes p seudo - first orde r with an app ar e nt reaction rate An e xact solution of E q . ( 3 0) can thus be ob constant given by kC tB . taine d , which give s the followin g expression of the enhancement factor :
!
E
A
==
M tanh M
( 3 9)
If 4 < M < E A , in / 2 , the p oints re p re s entin g the enhancement factor are very close to the diagonal in Fig . 5 . The secon d c a se is the one in whic h the reac tion rate is so high that the concen tration of the reactant B is less than the solubility of the gas A , beca u s e it react s in stan tan eo usly wit h compon ent B . T hi s situation occurs when M > l OE A , in . In fact , it can b e seen from Fi g . 5 that w hen t h e preceding con dition occurs , an in Since crease of the H atta number leads to a limitin g value of E A E A , in • the reaction is in stantaneous , the liquid phase may b e con side re d to be =
made u p of two sub sequent layers in which the resistance to the transforma tion is gi ven by the diffusion of the reagents ( Fig . 4c ) . I n this case ex pres sion ( 3 8 ) for the enhancement factor is obtaine d .
1 � �----� c w 0::: 0 1u � "" 1z w
�
w u z � ::t: z w
FI G U RE 5 reaction .
10
M ::
10
�DA k C11{ ktA
E n h anc e m en t factor for a fast bimolecular secon d - order
Carra an d Morb i delli
562
A more gene ral case is that in w hich reaction ( 2 9 ) tak e s p lace through to A a n d n with re spect to B. This p rob l e m was st u died by Hikita and A sai ( 1 963) , whose basic idea was to linearize the reaction rate with respect to the gaseous re actan t con c ent r a tion t hrou gh the approximation a bimolecular r eac ti on of order m with r e s p e c t
( 40) whic h is ob vio u s ly exact in
t h e case of
m
=
I ntroducin g this ap p roxima
1.
tion , t he p reviously mentioned lineari zation technique can readily be ap
plied ( i . e . , a s s u min g C s a s c on stant within the film ) . This app roach leads t o the followin g analytic al imp lic it expres sion for the enhancement factor :
E
M[ (E
A
=
. - E ) / {E . - 1) ] A A , 1n A , 1n
n/ 2
----���--�--���------n -/�2�
tanh { [ ( E
.
A , 1n
- E
) (E . A / A
, 1n
- 1)]
}
( 4 1)
where t h e Hatta number in this c a s e is defined as
M =
1_
__
k
tA
[(-2-1Jhn m
+
A
m C * - 1cn 9.-A
] 1/ 2
2B
( 42)
T he behavior of the e n ha n ce men t factor E A in is still given by Eq. ( 3 8 ) . , as a f un c t ion of the p aramete r s is gi ven in Fi g . 6 for va ri o u s values of n , with fixed m = 1 . The accu racy of ex p r e s s ion ( 4 0 ) is s uffici en t for en gi n e erin g calculation s . In Fig . 7, for a p seudo-mth - order reaction ( i . e . , 2 �------�--,
c
w
f
-M
Enhanc e ment fac t or for a fast bimolecular reaction of first order with respect to t h e ga seous reac tan t and nth order with re sp e c t to the liquid reactant .
FI G URE 6
563
Gas -Li q ui d Reac tors
l: .c r: "'
' l: .....
c
.....
f
106 102 0.9 8 0.,
0.4 0.6 0.8 1.0
0.2
-M
2
4
6 8 10
20
FI G URE 7 Ratio of the true en h a n ceme n t factor numerically calculate d , EA to the app r oxi m at e factor calc ulated by Eqs . ( 3 9) an d ( 4 2 ) , in the case of a pseudo mth -order r e a c t i on .
where CB is c on stant ) the ratio b et we e n t he a p p roxi m a t e and the numerically calcul a t e d enhancement factors is reported for variou s values of m as a func tion of the H atta nu mb e r . The same kin d o f accuracy is also exhibited by E q . ( 40) in t h e case of reaction ( 29) of mt h or d e r with r e sp ec t to A and nth or de r with respect to B . I t is worth m en t ioni n g here the large amount of literature devoted to the de finition of ex p lici t expression s of the e nh a nc e m en t factor un der the a s su mption of complete gas eo u s r e ac t an t de pl e tion in t h e liquid b ulk , C R.A 0. For r e ac tion ( 29) , first order with respect to both reactants , it has b ee n reviewed by Wellek et al . ( 1 9 7 8 ) , w ho recommen de d the use of the expression p rop o s e d by Yeramian et al . ( 1 9 70) : =
M2
2(E
A , 1n .
- l ) tanh
2 {rll
+
4 ( EA , ln -
M
1)
1 ]
\, E A , ID tanh
M2
2 -
·1
c 4a )
where M and EA , in a r e given by Eqs . ( 3 5 ) an d ( 3 8 ) , respectively . In t h e p r e s e n t s e c tion we have con sidered a particular situation for the gas -liquid reactor which allow s t h e n e gllilctin g of th e reaction in the liquid b ulk ,
this
is n ot
reach e d by the gaseous reactan t .
req uires t he c oupli n g of th e film equa t ion s with
tor behavior .
sinc e
Dee p e r a n aly si s of the p roblem
t ho se
describin g the reac
This is developed l at e r and more general approrimate exp re s
sion s of the reaction factor are also given .
M E A S U R EM E N T S OF I N T E R FAC I A L A R EAS A N D
MASS T R A N S FE R C O E F F I C I E N T S
Gas hol d up e:g . in t e rfac ial area a , and mass t ran s fe r coefficients are imp or tant parameters determinin g the mass t rans fe r rates in gas - li qu i d reactor s . Different procedures have been p r op ose d for their measurement , which can local measurements wit h p hys i c al techniques be c las sifi e d in two cate gories :
564
C arra a n d M o rb i d e ! l i
A detailed an a ly si s of the p roc e d ur e s h a s b een gi v en by C ha rp e nti e r ( 1 9 8 1) . and glob al measure ments w i t h che mic al p r oc edu r e s .
P h y s ic a l M ethod s The ga s holdup i s di re c t ly determined by measurin g the height of the aerate d liquid a n d that of the clear liquid without reaction ( Fair et al . , 1 9 6 2 ; Marrucci and N icodemo, 1 96 7 ; H u ghmark , 1967 ; Y os hi da et al . , 1 9 7 0 ; Eissa an d S c hii gerl , 1 9 7 5 ; Y os hi d a and Akit a , 1 96 5 ; Miller , 1 97 4 ) . T his proce dure is rap i d b u t not very ac cu r a t e . A more p recise t ec hni q ue , in which the ga s h ol du p i s ob tain e d from measurements of the clear liquid hei gh t in the dis p e r s ion at s uc ces si v e manometric tap s on the side o f the froth container , has been p roposed an d employed by B ur ge s s an d C alder b an k ( 1 97 5a , b ) . An electrical technique , p roposed by Linek an d Mayrhoferova ( 1 96 9) , can also be used . I t involve s t he m eas u r e m e n t s of the surface elevation at c er t ai n selected p oint s by means o f an electrically c on d uc tin g p rob e . The h e i ght i s d et e r mi n e d by the ve rtic al position of the tip at which th e s um of the c on tact times equals one - half of the measurement p e riod T h e y ray t r a n s mi s sion t echn i q ue has al s o been employed to determine the local gas holdup in gas - li q ui d cont actors ( C alderbank , 1 95 8 ; C a l d er b a nk and Rennie , 1 96 2 ; B e r n ar d and S arge nt 1 966 ; H w a and B e ckmann , 1 9 60) . .
-
,
From t h e gas holdup , the sp ec ific in terfacial area can be calculated as
( 4 4) where d s v i s t he volume surface mean diameter or S au t e r mean diameter , de
fin ed as
d
sv
( 4 5)
=
db is the diameter of the sin gle bubble ( or drop ) and ni the number of bub
bles ( or d rop s ) of diam et e r db · The value of dsv can be direc t ly e valuate d t h ro u gh a statistical a n a l y si s of high - speed photomic rographs p e rfo r m e d in
1 9 6 2 ; Vermeulen et al . , 1 9 5 5 ; Porter 1 9 6 6 ; Akita and Y os hi da 1 9 7 3 ; A s hl ey and H aselden , 1 9 7 2 ) . The local int er fa cial contact area is direc tly e v al uat e d by li ght - transmis sion an d r e fl e ction t ec h n iq ue s A p a ra ll el beam of li ght is passed through the d isp e r s ion and a p hotocell is placed at some distance from it ( Ve r m e ul en Only th e un scattered part of the inciden t p a rall el beam i s recorded 1955) . by t h e photocell . It has b een shown by Cal derbank ( 1 9 58) that for bubbles the sc at te rin g cross se ct i on is e q u al t o it s p roje c t e d area . The total inter facial area p e r unit volu me of the dispersion is r e la t e d to the in t e n si ty ratio of t h e ra diation through the followin g eq uati on : t he di s p e r sion ( C alderb an k and R ennie ,
et al . ,
,
.
,
i
= a 4
( 46)
565
Gas -Liq u i d Reac tors
where R. is the optical path len gth an d I o/ 1 the intensity ratio between the incident beam of radiation and the transmitted beam . C he m i c a l Met hods
The chemical met hods are derived from the theory of mass transfer with che mical reaction , previously illustrated . A gas is ab sorb ed into a liquid , where it reacts with a dissolved species . A n excellent review of such met hods ha s been published by Sharma and D anchw ert s ( 1 970) . By choos in g a reactant with a dequate solubility and a suitable reaction rate , either the products k R.a , k g a , or the interfacial area can be deduced from the overall rate of gas ab sorption . The determination of the in terfacial area relies on the fact that when the reaction bet ween the two components in the liquid phase is mth order in A and nth order in B if EA .
4 < M < �
( 47)
2
where M is defin ed by E q . ( 4 2 ) , the rate of ab sorption is given by (
2 r - a ___ m + 1 _
* m+ 1 n k D A C R.A C R.B
)1/ 2
( 48 )
This equation shows t hat the specific absorption rate is in depen dent o f hy drodynamic c on dition s . Therefore , i f the solubility , the diffusivity , and the reaction kinetic s are known , the sp ecific interfacial area can be deter min ed from the experimentally determined total absorption rate . Interfacial areas determined from the physical and chemical methods may differ by more than 1 0 0 % . A careful analysis reveals that the results of the photographic method and the sulfite oxidation method (a specific c hemi cal method disc ussed later) show large variations at high gas velocities . For homogeneous b ubbly flow , Schumpe an d Deckwer ( 1 9 8 0 ) have found that a
photo
=
1 . 3 5a
sulfite
( 4 9)
For churn - turbulent flow , the difference is even larger . These results re veal that the p hotographic method is not very reliable . The analytic determination of the absorption rate s can be performed in two limitin g situations , the transport or reaction - controlled cases . In the for mer case , which correspon ds to instantaneou s reaction regime , the chemical systems e mployed are in dicated in Table 5 . In the latter case , which correspon ds to slow reaction regime , the employed chemic al systems are s ho wn in Table 6 . O f course , the interpretation o f experimental data relies on the model employed to desc ribe the ga s - liquid dispersion , an d customarily it is as sumed that the liquid phase is perfectly mixed , while the gas phase is either in plug flow or perfectly mixed . This subject has been critically examined by Midoux et al . ( 1 9 8 0) , who proposed a flow model for gas dis persion into liquids inhibitin g coalescence that seem s more realistic than the plug - flow model .
C arra and Morbi delli
566
T AB LE 5 C h e m ical S y stems E m p loy ed to D etermine k t a I nstantaneous Reaction Regime
S ol u te gas so 2 ' C I 2
,
H so 2 4 N aO H
HCl
H S , HCI , C 0 2 2 0 2
the
Reference
Reactant
NH3
in
S harma and Danckwerts ( 1 97 0 ) Sharma and Danckwerts ( 1 9 7 0 )
A mines
S harma and Danck werts ( 1 97 0 )
N S 0 2 2 4
Jhaveri and S harma ( 1 9 6 8 )
Oxygen ab sorption into sodium sulfite in the presence of copper ions catalyst has been widely used for the determination of absorption rates in packed colu mn s , bubble c ol u mn s an d mechanically agitate d vessels . The ab sorption oc c u rs through the reaction as
,
( 50)
An e x t en ded revie w on its u se as a model reaction of known k in e tic s in determinin g accurately m ass trans fer characteri stics of gas -liq uid contactors has been p u b li shed by Linek an d Vacek ( 1 9 8 1 ) .
TABLE 6 R e gi m e
S olute gas
0
2
diluted with air
C hemic al Syste ms E mployed
Reactant
Cu Cl
NaClO
Cu S O 4
0 2 diluted with air
N a 2s o 3
C oS O 4
B utadiene
Maleic -anhy dride
k ta
in t he Slow Reaction
Reference s Danckwerts and S harma ( 1 9 6 6 ) Danckwerts an d Gillham ( 1 966) J haveri and S harma ( 1 96 7)'
Na so3
2
Determine
C atalyst
0 2 dilute d
with air
to
De Waal and Okeson ( 1 96 6 ) Wes selin gh and Van 't Hoog ( 1 970) 4 A st arita et al. ( 1 9 6 )
Linek ( 1 9 6 6 ) and Mayrhoferova ( 1 9 7 0) Onda et al . ( 1 9 7 2 ) N eelakantan and G ehlawat ( 1982) Linek
Gas-Liq ui d Reac tors
567
M O D E LI N G OF I SO T H E R M A L G A S - L I Q U I D R E A C TO RS
Similarly to most c hemical reactors , reliable sizin g and selection of optimal operatin g c on dition s for gas -liquid reactors require the use of suitable mathematical models . T he hydrodynamics of industrial gas -liquid reactors is quite complex , due to the p resence of two phases , one dispersed and the other continuou s , in relative motion , in some cases through a stationary solid packin g . In p articular , mixing in both phases can significantly affect the reactor performance in terms of reactant conversion and p roduct selec tivity . W en and Fan ( 19 7 5 ) and S hah et al . ( 19 78) have reviewed back mixin g in empty an d p acked gas -liquid reactors . The most common technique used to define the extent of backmixin g in a given unit is first to measure the residence - time distribution ( R T D ) of a suitable tracer in each phase and then to fit the obtained data with an ap propriate model . A number of such models have been proposed in the litera ture , and some of these are sum mari zed in Table 7 . They can b e divided into two main group s : p seudocontinuou s models and cell models , leadin g to differential and algeb raic equation s , respectively . Each model is character ized by one or more p arameters , which are usually tuned in order to re produce the experimental RTD data . The simplest p seu docontinuous model is the axial dispersion model , where backmixin g is characterized by a one - dimen sional diffusional term , which also includes the effects of both radial dispersion an d nonuniform TABLE 7
B ackmixin g M odels A dj ustable parameters
Reference
Pseudocontinuous models Axial
dispersion model
Multistaged dispersion model Modified mixin g cell model C ross -flo w model Piston diffusion exchange model Cell models Series of stirred - tank model Modified series of stirred -tank model
1
2
Levenspiel ( 1 97 2 )
Nishiwaki and K ato ( 1 9 7 4 )
2
Deans ( 1 963)
3
van S waaij et al . ( 1969)
1
Dean and Lapidus ( 19 6 0a , b )
3
B ang an d C holette ( 1 97 3 )
2
B ackflow model
Variable
Dispersion backflow model
Variable
Hochman an d Effron ( 196 9)
Miyauchi an d Vermeulen ( 1963) Nishiwaki
et
al . ( 1 97 3 )
568
Carra an d Morbi delli
velocity profile . It is equivalent to the series of C S TRs models , where back mixin g is again characterized by only one parameter : the number of tanks in the series . In the other models listed in Table 7 , some more parameters are introduced in order into account the possible simultaneou s presence of flow with axial dispersion , bypass , and a fraction of stagnant fluid . Althou gh they give a better representation of experimental RTD data , when used to describe the kinetic performance of a gas-liquid reactor , they lead to complex mathematical equation s , whic h can be of limited use for reactor simulation or design . Therefore , in the following we con sider only the axial dispersion model for both the gas and liquid phase s . Ob viously , a model of this type can de scribe only macromixing situation s inter mediate between plug flow and perfect mixin g. More complex situations can be examined by modelin g the unit under examination as a combination of suitable s maller uni t s , each desc ribed by the axial dispersion model , with different values of the dispersion coefficient s . This model can then be re garded as the fundamental unit for buildin g more complex models , such as those presented by W olf and Resnick ( 1 96 3 ) or those mentioned in Table 7 . W e will first outline the most general form o f the axial dispersion model , whic h can be used to simulate most gas - liquid reactors , provided that suitable values are given for the involved parameters . The C S TR model , whic h is al so the limitin g situation for infinite axial dispersion , is examined in detail due to it s very frequent application . T wo mode s of operation will be considered : both phases continuous and batch for the liquid p hase . As in most in dustrial application s , the chemical reaction s are assumed to occur only in the liquid phase . The mathematical model of a gas -liquid reactor is constituted of two interacting portion s . The first describes the diffu sion and reaction process within the liquid film , depen din g on the concentration values in the bulk of the two phases in relative motion . The second describes the time or space evolution of the bulk concentration s , takin g into account the operatin g mode , hy drodynamic s , and macromixin g in each phase . Li q u i d Fi l m Mod e l
Considerin g a generic ith component , its dimensionless steady - state mass balance in the liquid film , where N R reactions occur , is written as follows : 2 d cp l;; i
dx
fi 2
�
- Ha
2
L
NR j= l
vij P /j
wit h b oun dary con ditions (B C s ) x = 0: X
=
1:
(j>
fi
cp fi
=
=
X
E
( 0 , 1)
( 51)
- '
4> Ii
( 5 1a )
<�> .u
( 5lb )
where
569
G as -Liq ui d Reac t o rs
cp fi
=
s X : I)
( 5 2)
r
f
j
:J.
=
J
r?
where the sub script A in dic ates the gaseous reactant an d the superscript 0 indicates a reference con dition , usually c orrespon din g to the feed stream or the initial condition for continuous or b atch ope ration s , respectively . It is worthwhile to recall that the last of equation s ( 5 2 ) is a result of the film t heory , but it is not veri fi e d in practice since from experim e n tal measurements the purely p hysic al mass transfer coefficient k R-i appears to be p rop o rtional to the square root of the diffu sion coefficient Di . In the solution of the liquid film model , it is on ly necessary to calculate the mass flux of the ith component across t he gas-liquid film (x 0 ) and the liquid fil m - liqui d bulk ( x 1) in terfaces . The first is given by =
=
( 53 )
where
the re action
to 1
The
factor E;t' , d e fi ne d b y Eq . ( 2 8 ) , u sin g Eq s 1
( �) d
.
( 5 2 ) reduces
( 54)
x=O
last e q uality i n Eq , ( 5 3 ) i s based o n also usin g t h e film t heory for ma ss tran sfer process , and in dimen sionless form leads to
gas - side
g'l *
•
=
1 + y . E ;t' / H ;I'
( 55)
--""" � -.=-:-
1 1
the
1
w here H !" 1
and
the ga s li q ui d equilib rium "' *
"' gi
=
-
H * "' *
i "' R-i
( 56 )
at
the
interface [ ( Eq . ( 2 1b ) ] re duces to ( 5 7)
The material fluxe s at the liq ui d boundaries can be calculated, usin g Eqs . ( 5 1 ) , ( 5 3 ) , an d ( 5 5) , through the relation ship s
C arra and Morbi delli
570
( 58) "'
11
]=1
cpgi
( lJo
v . .p .
�
1
a s a function of the b ulk concentration s ated through the
NR
2
T. Ha
+
an d
1
cp .u .
J
( 59)
f. dx 1
T hese can be evalu
balances examined later , dep en din g on the
macroscopic
actor hy drodynamic s and operation mode .
re
A x i a l Di s pe r s i on Mod e l
U sin g t h e axial disp ersion model for b oth phase s , t h e i sothermal , steady dimen sionless material b alances for the ith component are , in t h e
state ,
liquid p hase ,
1
Pe
2 d cj> ii
-- +
1
dz
2
d(v i
n*
cp i i )
+
dz
( .J. )
1 x=
1
+
StU
a c- .,
i
Ha
2 NR
""' '-t j =1
z £ ( 0 , 1)
and in
) cj> g gi dz -
- ( J1. )
x=
0
where besides those defined in Eqs .
tities ha ve been introduc e d :
g1 •
st
Pe
n*
=
c .
- � co
z = ¥... L
g
Lak • .
"'1
g
=
u
u0 L
g g
a =
=
gi
=
__g!_ o u
1 - £
0
( 52 ) ,
v
Lak St
i
__JL_ E £
1, -1
( 59) ,
ii
-0
=
=
0 ( 6 0)
t
ao
g:
ao
the
followin g
by
interfacial
dimensionless quan
g
Pe
g
z £ ( 0 , 1)
=
!!.
E (1
U0L i £
g
in dicates countercurrent an d cocurrent flow ,
dimen sionless
an d
. . p .f.
1] J 1
the gas p hase, d(v
A. 'I'
v
resp ectively .
material fluxes a r e given , usin g Eqs .
=
St
.
g1
1 1
y.E�
( 6 1)
ao )
The
( 5 5) , ( 5 8 ) ,
( 63)
+
Ha
2
-1:"
"' i
NR
L j=1
v1. . p . ] l
(1
L0
f. ]
5 71
]
Gas-Li q ui d Reac to rs
dx
( 6 4)
When , as in Eqs . ( 6 0 ) , the variation of the super ficial ve loc it ie s v R. an d equation o f state H owever , for most usual app lications of each phase shoul d be in troduced , the liquid s uperficial velocity can b e assumed con stant along the reactor axis , w hile for the gas p hase the i de al gas law can be u se d : V g along the reactor axis m u s t be tak en into account , the
( 65) which introduced i n Eqs . ( 6 1 ) and ( 6 3 ) leads t o d ( v P /P 0 ) g
dz
yiEf/Hf
NC
+
L
St
i =1
gi
=
0
( 66 )
where p o i s the pressure value a t t h e reference con dition s , which i n thi s
case are those relative to the feed stream ( i . e . , po /RT
=
co
g
=
E� Cl 1=
Cf . ) gl
,
Accordin g to Dec k wer ( 1 976) , the pres sure gradien t c an si gnificantly affect reactor behavior and should then be taken into a ccou n t through the relation ship ( 6 7) whe re a. = p R. gL( l E:g ) /P T and PT is t he pres sure at the reactor top . N ote that , u sin g E q . ( 6 7 ) , the first term in Eq . ( 66) vanishe s , T he boundary conditions for E q s . ( 6 0) and ( 6 1) d e p en d on the re actor operatio n mode : cocurrent ( n * - 1) or countercurrent ( n * = 1 ) : Liquid phase : -
=
z
=
z
=
1 + n*
V R,
2
1
-
n* :
f
d
=
=
v
-
1
Pe R.
d
V R,
R.i dz
=
0
f v R,
( 6 8a )
( 68b )
Gas phase :
z
z
=
=
0 : vf
g
1:
g1
d
dz
g
:; 0
g1 •
-
1
Pe
g
� dz
v
g
=
v
f
g
( 6 9a )
( 6 9b )
5 72
C arro a n d Mo rb i delli
TABLE 8
Mass B al an ce s for t h e C S TR M o del
Liquid phase :
Ha
2
�
NR
j =l
\J l• J•
p f J. J.
-
( Jl. ) x==l
Gas phase :
v
f == v g g
NC
+
� i =l
(J ) i x==O
S p e ci al cases of t his model , particularly used in practic e , are the
C S T R and the PFR models , whic h are obtaine d assumin g infinite ( P e
� 0) and zero ( Pe + "" ) axial di s p e r s ion coefficient s , respectively . The equa tion s for the C S T R model are sum mari ze d in Table 8 , where con st an t pres sure i s as s um e d in the re actor . For t h e PFR mo de l , the equation s readily derive from th o se of the axial di s p e r si on m o de l , by simply de letin g all the terms c on t ain in g the Peclet numbers and the b oun dary con dition s at the reactor outlet .
Semi flow Ba tc h Model
A common op e r at ion mode for gas -liquid
r e ac t o rs is to c on tin uo u sly flow the gas p ha se th ro u gh the liqui d pha se hel d in side the re ac tor . The liquid con centration s are then time dependent , but not space de p e n de n t , sin c e t h e li q uid c an u s u al ly be con sidered well mixed . For the li q ui d p hase , the fol lowin g material b alances can be written
e > o with the initial c on di tion s , e
=
1
tk .R.
A
cp ti
=
in
.t i
at a
=
0� and w here
a
--�-"--E:
g
( 70)
( 71)
ao
T h e t ot al mass fl u x of t he i s given by
ith c omponent en t e rin g the liquid b ulk ,
(Ji)x = l ' ( 72)
5 73
Gas -Li q ui d Reac t o rs
where ( Ji > x=1 is given by E q . ( 6 4 ) as a f unction of the gas - p hase composi tion , whic h for Pe g =f. 0 depend s on the reactor axis z accordin g to E q . ( 6 1) . U sually , p se udo- steady - state con dition s c an be assumed for the ga seou s phase ( Sc haftlein and Russel , 1 96 8 ) , so that the mas s balances in this phase are identical t o t hose previously de scrib ed for the axial dispersion model . V a r i a t i o n of I n te rfac i a l A rea a n d G a s
Holdup
In all previous models i t is a s sume d that t h e gas -liqui d interfacial area an d the g as holdup are constant along the reactor axis . A lthou gh experimental
results do not fully con firm this a s sumption ( D eckwer , 1 9 7 7 ; D eckwer et al . 1 9 7 8 ; Kolbe! et al . , 1 9 7 2 ) , sufficient data have not yet b een collecte d to deriv e reliable correlation s . Howeve r , in some applicat ion s ( i . e . , dea d - end reactors ) th e variation of the gas flow rate due to ab sorp tion is such that chan ges in interfacial area and gas h oldu p cannot be neglecte d . T o this en d , Schaftlein an d Rus sel ( 1 9 6 8 ) an d Shaikh and Varma ( 1 9 8 3b ) have sug gested t he followin g re lation sh ip s :
( 7 3a )
£
_g:_ £
f
g
v
-
_
_K v
( 7 3b )
f g
where the ratio b etween the value of a ( or £ g ) with an d without (i . e . , at ab so rp tion is related to t he superficial gas veloc b ased on the as sum p t ion that the di spersed gas swarm of spherical bubbles flowin g with no b reak shape variation durin g shrinka ge .
v�)
feed c on dition s , v g ity . These equation s are phase is c on stituted by a age or coalescence an d no =
ES T I MA T I O N
O F C H A RAC T E R I ST I C PARAM E T E R S
The u s e o f math ematical models for the simulation o f variou s types of in dus trial gas -liquid reactors r equi re s a suitable evaluation procedure o f all the parameters appearin g in the model equation s . In this section recommen de d semie mpirical relation ships for the three most widely u sed reactor typ e s spar g e d , p acke d , an d mec hanically agitated reactor s - are reported . S pa r ged Reac t o r s
T h e s e are empty reactors w h e re the gas -liquid m1xmg e ner gy i s p redomi nantly introduced with the gas p hase , which c an be di spersed into the liquid p hase t hrough variou s sparger types : sin gle orific e s , pip es with holes , sieve an d p orous plates , and two-phase no z zle spargers of injec tor an d ejector typ e ( M e r smann , 1 97 8 ; Fair , 1 9 6 7a ) . Main adv an ta ge s of the se units are the absence of movin g parts , elimi m i tin g the need for seals ( le s s maint enance an d cost ) ; soli d s han dlin g ability ; la r ge heat tran sfer rate p er
C a rra a n d Morbidelli
574
unit reactor volume ; an d l ar ge values of the ga s - liqui d mass t r an sfe r co
e fficient s . The disadvantages are b ac k m i xin g , p a r tic u l a rly in the li q ui d p h ase ; hig h p re s s u r e d rop ( si gn ific an t p articularly at low p re s s u re ) ; and
the redu c tion of in t e r fac ial area due to bubble c oal e sc enc e , si gni fic an t for L /dc > 12 . Vertical bubble column s are the s par ged ga s - liq ui d reactors most widely used in in d u s try . S ome modifications have b e en introduced to improve their sec tionali ze d b u b b l e colu mn s ( low b a ck mixi n g and b ubble p er for ma n c e : coalescence ) , hori zontal bubble columns ( lo w p re s s ure dr op ) , downflow bub ble c ol u m n s ( l ar g e residenc e time ) , and plate colu mn s (large n u m b er of t ran s fer unit s ) .
General review articles on hy drodynamic s an d mass and
( 1 9 7 3 ) , Pohorecki ( 1 976) , S tichlmair an d Mersmann ( 1 9 7 7 ) , Charpentier ( 1 9 8 1 ) , and Z ah r a dn ik et al . ( 1 98 2 ) for p l a t e colu mns , and Joshi an d Sharma ( 1 9 7 6 , 1 9 7 8 ) for hori zontal b ubble c ol u m n s . heat transport have b een r e p or t e d by Fair et al.
Various studie s on t he design an d simulation of b ubble columns have 1 95 0 ; Sideman et al . , 1966 ; F air , 1967a , b ; S harma and Mashelkar , 1968 ; Cic hy et al . , 1 9 6 9 ; Cichy a n d Russel , 1 96 9 ; M a s he lk ar , 1 9 7 0 ; D eckwer , 1 9 7 7 ; M er s m ann , 1 9 78 ) . Shah et al . ( 1 9 8 2 ) h a v e p u b li s he d an extensive survey on t he c h ara ct e ristic parameters estimation for b ub b l e column s . These p a ram e t e r s are affected by the colu mn regime , which chan ges by increasin g the upflow ga s velocity a s foll o w s : been r e p ort e d in the literature ( S hulman an d M o l s t a d ,
1.
2.
3.
B ubble flow , c ha ra c te ri ze d
by u n ifo r m
flow of e q ually sized bub bles ;
it occurs a p p r oxi m at ely at U g < 0 . 0 5 m / s ( F air , 1 9 6 7a) . C hurn turbulent flow , where l a r g e an d s m all b ubbles are present
with mo s t Slug tion ; d0 �
di ffe re n t rise velocities l e a din g to an u n st e a dy flow ; it is the common in in dustrial u ni t s .
flow , where the bubble oc c u p ie s the entire c olumn c ross sec it oc c u r s at l a rge u p fl o w ga s ve loci t y , with col u mn s diameter ,
0 . 15 m .
The a priori de fin i t ion of t h e flow re gime is difficult , since it is affected b y many parameters , suc h a s Sp a r ger type , p hy sic al p rop e rtie s , an d velocity of the liquid p ha se . An ap p ro x imat e map of flow re gi mes is shown in Fi g _ 8 . The recom mended e m p i ric al e q ua tion s for t he e valuation of the character i s tic pa ra m e t e r s o f a b u b b le column are reported in T able 9 . T he volum e surface mean diameter of th e s w ar m of b ubb le s , defined by Eq . ( 4 5 ) can be e v al u a ted through Cal derbank' s ( 1 9 6 7 ) correlation w hen gas i s s p a rge d by t w o - p h as e noz zles an d the energy dissip ation rate is gi ve n by Pt /V d U g P R, O - Eg ) g . For less effective gas spargers , such as perfo ra te d plate s and sin gle orific e s , the correlation p r o p o sed by Akita an d Y o s hi da ( 1 974) c an be used . I n s p a r ge d r eactors the bubble si ze i s controlled by the e q uilib r i um between b re a k u p an d coalescence rates . For lo w - vi sc ou s aqueous solution s , in t he bubbly flo w regim e , the bubble si ze s t ab ili ty di a gr a m shown in Fi g . 9 has b een p rop o se d by B erghams ( 1 9 7 3 ) . In it the critical Weber n u m b e r , whic h correspon d s to the men tione d equilibrium , se p ara tin g the stable and u n s t a b le regions , is shown . Pressure values up to 1 . 6 MPa do not affect the b u b b l e diameter if th e gas ve locity is corrected to take into ac co un t the p ressure in t he column ( K o1bel et al . , 1 9 6 1 ) . =
5 75
Gas -Li q ui d Reac tors
C h u r n Tu r b u l e n t f l ow
.. :I
I
- de
(a)
B U B B LY FLOW oo o o 0 00 0 0 00 0 0 0 0 0 o0 0 00 oo 0 o 0 0 0 0 0 () 0 0
g
o0o0o0 o0o
C H URN TURB U L E N T FLOW 0
•
0
. O" o · 0
. 0
' •Do 0 0 . 0 °
0
(b)
, oo
0
••
0 0
0
•
·0a 0 0.
•
S L U G FLOW
0
•
0
0
0
0
( a ) App roximate depen dency of flow regime on gas velocity u g and column ·diameter de . ( b ) S chematic representation of flow re gi mes in
FI GURE 8
bubble colum n s .
FI GURE 9 S t abilit y diagram for low -viscous aqueous solution s bly flow regime .
in
t he bub
S emiempirical Expres sion s for B ubble Columns
TABLE 9
B ubble diamete r d
C
C
1
c1 d
=
vs
1
c
=
=
n =
n = n :::
0. 2
g
Pi
g
<
<
0. 3
0.6
0
=
0
(J) . 25 lJ Q,
b co J
=
=
gdc
:::
J :::
H
7
2 p
2 ) ll 9, 0. 6
1 0. 2
3
2
) (_g__ �
m , use d
:J . 8 5 7)
7 0 . 94H 0 " 4 1 :. ( 2 < H < 5 9 . 3 ) 3 , 42H 0 • 44 ( H > 5 9 . 3)
i
C
m for p ure li qu id s
-
t --
m ; if d > c
M - 0 . 149(J -
+
( 1 967)
0 for aqueous solution s of alcohols
=
Ter minal bub b le ri se velocity Re
C alderbank
for aqueous solutions of electrolytes
3
09,
C
c
0 . 40; C 2 0 . 65 ; C 2 c2
gdc p i --
26d
< d E
0. 4
c:n
90;5 0.50; 0.0009 \ 2 1 - 0 .5 (
=
0 . 07
( P T /V d )
2. 25 ; 1 . 4. 1 ;
=
VS
1
0. 6
0 9,
tit � C)
) - 0 . 14 E M - 0 . 1 4 9( 1-l R, I 1-l w 0
u
- 0 . 12
Akita an d Yoshida ( 1 974)
g c
c
=
0.6
m; u
g
<
0. 42
m/s ;
Clift et al .
( 1978)
(") �
i !=) ;:s Q.
� �
� ::::
M -
E
=
0
g
4
- p g: ) f,l £( p .ll.
< 10
2 3
p Jl.a £
g( P .II. - P g ) d s
!
all.
-3
Re oo b
=
d
VS
u
boo
f.! £
p
£
) 0. 1
::a
Ill Q 0
....
< 40
0
(1
-
C C
=
=
Akita an d Y o s hi da ( 1 9 73 ) 1/ 8
g £
g
)
4
c
=
=
f
=
=
0 . 6 7 2f
=
c
d
p
'
u
g
<
-0 1 31 --) • (P
0 . 4 2 m/ s
f.I JI.) 0 . 5 7 8 ( f.l ! g
___IL.!:_ a ll.
P £ all.
3
£
.
< 0 3
g
)0 . 0 6 2 (
_K P JI.
1 for pure li quid or nonelectrolyte solu tion s 10° ·
04 4 1
f = l.1
I
·
< 0. 6 m; u (
f
( :: ) (:�! ) (�) 1 / 12
0 . 25 for elec trolyte solution s
c
g
�
0 . 2 0 for nonelectrolyte solution s
0 . 15 < d
£
.E" E. �
Gas holdup
£
c;) Q C'l.l I I:"'
for i < 1
kg
0. 1 m ;
\l n "
7 ) 0 . 10
H ikita et al .
( 1 98 0 )
ion / m 3
for i > 1 k g ion / m 3
ionic stren gth of the solution =
f.l
.:_g_
0 . 04 2 < u
g
< 0 . 38 m / s
""' Cll ""'
u
e:*
Fr
u
g
+
u _��,
2
- -- 2 gdc p .1', F
=
F 2d
a
We F
g
z
c
p £ p .Q,u .Q,
=
+
p gu g
148 < F < 3 3 6 kg /m 2 • s ; 0 . 0 0 3 < e:* < O . Z4 ; 0 . 004 < e: 184 < p
g
< 0.3
J < 5 3 4 0 ; 3 7 < J.l £ / J.l < 2 2 20 ; 0 . 0 5 5 < o2 < 0 . 0 7 4 N /m 9 pg g C)
Interfacial area per reactor volume
a = 3d
1
c
0 . 07 < d
(
c
( ·:: ·) :? ) 2
0. 5
3
'
< 0 . 6 m ; if d
c
>
$:) "'S
�
0. 1 ,� . 1
0.6
m ,
3
Akita an d Y oshida ( 1974) use d
c
=
0.6 m;
u
g
< 0 . 4 m /s :
e:
g
< 0 . 14
$:) ;:l Q.
;;::: 0
;}
�
;:;;
4 8 . 7( u
a =
dc
=
i i
eff
a =
<
a =
0.6
d
bu
>
g
0 8 ·
0 . 02 m / s
p
pi
i i
i
m;
0. 6
if d > c
ll i
m ; u se
d
c
=
k a 1
ug =
=
0. 2
m;
D
i
dc
0 . 003 < u
g
<
<
0 . 0 8 m /s ;
k i dv s
10 - 3 m) ,
0 . 1 5 m ; 0 . 003 0 . 0 0315u � · 5 9 ll ��· 84
Liquid - side
(l)
g ....
g
m; u
g
mass
-2 _
s:: ...
!:).
Akita and
< 0 . 33
Liq uid p hase
c Sintered plates 0. 1 <
0,6
� t. .o·
m /s ;
e:
g
Y oshida ( 1 9 7 3 )
0
;a
< 0. 3
Deckwer et al . ( 19 7 4 )
C ross of noz zles ( =
e:
<;)
::tl
g
Sparger
d
( 1981 )
transfer coefficient '
0. 6
<
Schu mpe and D eck wer
51
0 : 31 0 gd 2 0 . 62 gdc3 2i 1. 1 ( ) JJ i c ) 5 (-) (-2 cr p D
R. 2 d c
c
) 0.
mas s
liquid
D
0. 15
k
I JJ
0 . 14 m ; u
Volumetric k
g
+
c
d
0 . 08
< u
= 0 . 14 m
trans fe r coefficient
g
<
0 . 08
D
i
0 . 467
T ap water
0 . 46 0
m /s
1 . 174
S alt solutions
1. 445
solu tions
Dekwer e t
1/3
3 (p i - pg) g ] . [dvs 0 . 31 ]..l i
water
Tap S alt
m/s
b --
, d
vs
< 2. 5
x
10
-3
Calderbank
al .
( 19 8 2)
and
Moo-Young ( 1 96 1 )
m <:n "' co
TABLE 9
[
( C ontinued )
] � )0. ( ) ( 1/3)0.116
( �)
11 2
k d Q. vs
D Q.
0 . 42
=
p Q. D Q.
2 + a --
v;:- =
3
vs p Q. ( p Q. -
2
]l Q.
5 46
l-1 i
k d t vs
d
d
vs
a = 0 . 06 1 for sin gle bubbles
0 . 025
< d
c
< 1.1 m;
of
U 0 �
s
p
1-I Q.
p iD Q.
a = 0 . 0 1 8 7 for s warms
u
Q.
Pg) g
0 . 779
1/3
d
g
T g y
::
=
=
d T
g( 1
<
0 . 12
m /s ; u
s
= u
=
x
10
-3
g
IE
g
-
m
H u gh m ark ( 1 9 6 7)
U 0 / C l - E: g ) � Gedde s ( 1 9 4 6 )
6 . 58
(":) Q
bubble velocity
gas mole fraction of diffusin g component
� g
+
y)
L/ ub ; ub
k d
D
-
> 2. 5
g
bubbles
(···" :!:g)
vs
vs
vs
D Q.2 / 3
Gas - side mass tran sfer coefficient
k
; d
tn Oo 0
=
3 . 29
x
1 0- 4
0 . 0 1 < d < 0 . 02 m ; c
(
d
v su
1 < u
g g
J.l ,...
g
p
)
O . 7 56
< 5 m/s
l
S hilimkan a n d S tepanek
( 1 97 8)
Q ;:I Q.
iii:: 0
�
� Q.
�
D eckwer et al . ( 1 9 7 4 )
Liquid -phase axial dispersion coefficient E 0 = 0 . 6 78d 1 . 4u · 3
0
0. ::tl
. d 1 4u 0 · 3 < 4 0 0 c g
E .Q,
=
0 . 3 3dc ( uc + u ) .Q,
uc = 1 . 3 1 0. 1 < d
[ ( g dc
Joshi ( 1 9 8 0 )
�g
ug - 1
s
)]
u .Q, - s g� oo
g
< 1 . 1 m ; 0 . 003 < u
c
g
1 13
u
E E
g
=
r
=
g
g
=
r relative veloc ity between the gas an d li qui d p hase 1.
c
5
(u I s ) g
1 33
= 56 . 4 d . c
(u
g
g
3
Is )3 g
•
0
;;j
Dib oun an d S chii gerl ( 1967)
5u dc 50d
� Q (")
< 0 . 4 5 ml s ; 0 . 12 < u £ < 0 . 002 m ls
Gas -pha se axial dispersion coefficient
E
I t"'
.o· E.
g
c
Yv
c;') Q Cl)
56
Man gartz an d Pilhofer ( 1 9 8 0 ) Field and Davidson ( 1 9 8 0 )
0 . 08 < d < 3 m ; 0 . 0 0 8 < u < 0 . 1 m/ s g c Heat transfer ccsfucient
h
w
P "'n C p Q,u Re
=
g
p 2 f "R .,,, = O . l , - · � -"' r r )
p " I J.l ; Fr u d £ g vs "'
=
Deckwer ( 1 9 80 ) 0 . 25
2 u I gd ; Pr VS g
=
C
p £ J.l .Q,
lk
T£
en Oo ....
"' Oo t\)
TABLE
9 ( C ontin ued )
0 . 0 3 < ( R eFrPr 2 ) < 7 ; 6 < h
_!!__
_
5. 4
x
10
=
-4
0 . 411
(u
g
X
10
- 12
< 98 5
0 3 08 4 ) • (k g l-l Ti _£_ 3
) - 0 . 8 5 1(
__g__! OQ,
< u u Q. / oQ, < 7 , 6
4 . 9 < C p Q, 11 Q, /k T £ < 7,7
11
Pr
9
3
x
3 4 < ].1 g / p 0 < 1 . 6 Q, £ £
10
X
P Q, O Q,
C p Q, ]J Q,
)2/3
Hikita e t al . ( 1 98 1a)
-2
10
-6
(') � "'S
�
p ;::s J:).
s
(5.
�
E
Gas -Li q ui d
583
Reac t o rs
The choice of one expres sion for the evaluation of the gas holdup is q uite diffic ult , due to its sen sitivity to the material sy stem , includin g trace impuritie s . The gas holdup depen ds on the superficial gas velocity to the power 0 . 7 to 1 . 2 in the b ubble flow re gime , and 0 . 4 to 0 . 7 in the churn
turbulent one . On the other han d , it is almost indepen dent of pres s ure value s up to 1 . 6 MPa ( Ko1bel et al . , 1 9 6 1 ) , of the presence of internals ( m ainly for heat tran sfe r ) , and of the colu mn diameter for de > 0 . 15 m . It is then convenient , p articularly for noncoalescin g or non - N ewtonian liquids , to
evaluate E g through a simple experimental measurement in a laboratory - scale column < de > 0 . 1 5 m ) , wit h the de sired material system and gas velocity . T he most accu rate correlation s seem those proposed by Akita and Yoshida ( 1 97 3 ) and Hikita et al . ( 1 9 8 0 ) , valid for sin gle - or multinoz zle spargers with diameter of about 1 mm or larger an d for low - viscous ( 11 .1', < 0 . 0 2 Pa · s ) and coale scin g liquids . Such systems are characterized by the followin g con dition ( Mers mann , 19 7 8 ) : /J. a
1
( 74 )
where x in dicate s the mole fraction of the componen t of lower surface ten
sion , an d 11 o i s the difference in the surface ten sion of the two component s of the binary m i xture . I f condition ( 7 4 ) is violated , foamin g occurs in the column and t he gas holdup inc rease s , a s in the case of aqueou s solution s of For downflow bubble column s the rela electrolyte s ( Deckwer et al . , 1 97 4 ) . tion s hip s p ropose d by Friedel et al . ( 1 9 8 0 ) are recommen ded . The gas -liquid interfacial area per unit reactor volume , a , can be calc u late d from E q . ( 4 4 ) usin g the correlation s mentioned for gas holdup a n d b ub ble s i ze .
I n partic ular , the correlation p roposed by Akita and Yoshida ( 1 9 7 4 ) is recommen de d , which has been derived for less effective spargers ( wit h one or more distributors ) and can then b e u sed in general at least for a con s e rvative e stimation . I n the c ase of 0 . 1 4 < E g < 0 . 3 , the value of a s hould be calculated from E q . ( 4 4 ) usin g the expressions for s g an d d vs p ropose d by the same authors and reporte d in T able 9 . A more general correlation spec i fically tested for t wo - p hase n o z zle spargers is shown in Fig . 10 ( N a gel e t al . , 1 97 8 ) . T h e direction o f flow , u p o r down does not
1000
500
..
t
200
1 00 50
20
10 0.1
0. 2
FI G U RE 1 0
0. 5
50
100 200
500
S pecific in terfacial area s in bubble column s ;
x , hori zontal pip e .
• ,
vertical pipe ;
584
C a r'ril a n d Mo r'bidel!!
subs tantially a ffe ct the interfacial area in c o c u r ren t flow ( S hili mkan an d Fina lly , for non - N ewtonian hi ghly vi sc o u s media and S tepanek , 1 9 7 8 ) . colu mn s operatin g in t h e slu g re gime , as it is usually th e case for such me dia at u g > 0 . 02 m / s , Schumpe and Deckwer ( 19 8 1 ) d e ve lop ed a u seful The effective li q u i d viscosity � e ff is calculated a c c o r din g to c o r r ela ti on . N is hik a w a et al . ( 1 9 7 7 ) . The volumetric liqui d - side m a s s t r an sfe r coefficient depen ds on gas velocity , sparger typ e , an d liquid p h y sic oc h emic al p rop erti e s , while they are un a ffe ct ed by the liquid velocity and the c ol um n di a m e te r , above de = 0 . 1 5 m . Also , in t his c a s e the correlation p ro p o se d by Akita and Yoshida ( 1 97 3 ) ap p lie s to less effective sp a r g er s , an d t h e one p roposed by Deckwer et al . ( 1 9 8 2 ) to n on - Newtonian highly vi s c o u s media . The k � a valu es ob t aine d from Akita an d Y o sh i d a ' s e qua tion are conservative values ; much hi gh e r values can be ob t ain e d u s in g sintered plates or two-phase noz zles for gas s p a r gin g [ Deck wer ( 1 9 7 7 ) s u g ge s ts , as a rule of th um b , a correc tion fact or of about 4 to 5] . T his s am e equation can also be used for multi stage colu m n s . A m o r e general expres sion is gi ve n b y D eckwer et al . ( 1 97 4 ) , where t h e con stant b is q uite sen sitive to sp a rg e r type an d material system . T hi s same e x p r e s sion can be used for downflow bubble colu mns ( H erbrechts meier and S t e i ne r , 1 9 7 8 ) . Gas - side mass transfer resistance is us ually ne gli gib l e , and it becomes i mp o r t ant only in the case of fast or in stantaneou s ch e mic al reaction s . Ac c u r a t e correlation s a re not available ; t h ose reported in T a b le 9 are the most widely use d . It is worthwhile r eit e ratin g that for all t he parameters de sc ribed up to this point , the effect of the colu mn d i a me te r de b ec om e s in significant for de valu e s larger th a n 0 . 1 5 m for l es s vi sc o u s li q u i ds , and larger than 0. 30 m for hi ghly viscou s liqui ds (J.l 1 > 0. 0 2 Pa • s ) . The liquid - p h a s e axial di s p ers ion coefficient E 1 depends on t he gas velocity a n d column diameter an d geometry , whereas it is in depen dent of the liquid veloc ity , at least up to u 1 = 0 . 03 m / s . Rectangular c olu m n s ex hibit larger b ac kmi xin g than cy lin dric al ones ( S t rie g el and Shah , 1 9 7 7a ) , In a coiled c ol u m n , E t de p en ds on liquid ve l ocity an d approaches the same de gree of axial mixing as in a sin gle liquid -phase colu mn for Reynol ds num b er of the o r d e r of 4 0 0 to 2 0 0 0 . Info rmat ion about back mixing in hori zon tal column s , in multistage colu mn s p a r ti tion e d with orific e an d p e rfo ra t e d plates , and in sp r ay columns can be fou n d in t he review of S h ah et al . ( 1 9 78) . In nonisothe rmal bubble c ol u mn s it is n ec es s a ry to take in to a c c ou n t the heat e xc h an ge with the s u r r o un din g s . I t is rema rkab le that for N ewtonian liq uids the heat transfer coefficien ts from t he reactor wall and from immersed c o ils are al most identical . For non - Newtonian li q ui d s , the l a tt e r is lar ger , an d then i t is a dv ant a ge ou s to use c oils . A l s o , the l oc ation of the heat t r a n s fe r section alon g the c olumn axis doe s not ap p ea r to affect the heat transfer c oefficient ( Hikita et al . , 1 98 1a ) . Pac ked C o l u m n s
T h e p r e se nc e of p a ckin g s in side t h e c olu mn limits b ubbles coalescence , l e adin g to s maller bubble s , which rise with lower velocity . A s a con sequence , back mixin g dec reases in b ot h phase s , and interfacial area and gas holdup inc rease with respect to empty b u b b l e column s . The p re s s u r e drop in crea s e s , but usually not significantly ( C a rlet on et al . , 1 9 6 7 ) . High - porosity p ack in g s , s uch a s screen p a c kin g s ( e; p � 0 . 9 0 ) , are partic ularly efficient
585
Gas -Liq ui d Reac tors ( C hen and Vallabh , 1970) .
Packed colu mn s are specifically in dicated when
h an dlin g corrosive materials an d when no si gnific an t required .
heat
exchange
T wo main flow pattern s can b e operated in a packed column :
flow ,
gas form s a continuous
is
trickle
phase while the liqui d p ha se flows down ward as a film over the solid p acki n g and its level is b elow the pack in g ; an d bubble flow , w here the ga s flows as bubbles t hrough the li q ui d , whose level i s now above t h e p ac kin g . A detailed de scription of t he column hydrodynami c s and the p acking characteristics i s reported by Fair et al . ( 19 71) and T reybal ( 1 9 6 8 ) . In T able 10 approximate v alu e s of t h e void fr ac where the
tion e: p and the tot al d ry s urface area p er unit p acked volume at are reported for some p ackin gs . T he characteristic pac kin dimension is defined as
g
follo w s :
d
___..._ P_
6( 1
p
which
=
is
not
-
e:
)
( 75)
nominal dia m e t er dn . the e v al u ation of the characteristic term s of the operation mode an d the
normally th e sam e as the
T h e recommen ded relation s hip s
for
para mete r will be examine d b elow in
specific flow p atterns .
U se ful survey s have rec en tly been reported by
Charp entier ( 1976 , 1 9 7 8 ) on mass transfer , by S hah
mixin g in
packed
et
al .
( 1 9 7 8 ) on back
colum n s , an d by Satte rfield ( 1 9 7 5 ) on trickle - b ed reactor s .
Coun t e rcurre n t Pac k e d Columns
These uni t s are us ually o pera ted in the trickle - flow re gime , the loadin g zone . F or such sit uation s the various involved
that i s ,
p ara
b elo w
m et ers
can Little in formation be evaluated with the expression s reported in Table 1 1 . is available on column s operated c oun tercurrently in the b u b ble fl ow regime ( C arleton et al . , 1 967 ; Sahay a n d S harma , 197 3 ) . -
e:� ;
T h e liquid holdup is given t he static
in gs ,
eters
an d
holdup ,
e:�,
the operation al
by
the sum of two
given by hol dup ,
the
e:�.
liquid
contribution s :
retain ed by the
T h e values
of t h e
s
e: .t = e:.t
+
dr aine d pack
adju stable
para m
appearin g i n the correlation develope d by Shulman et al . ( 19 5 5 ) are T ab le 1 2 for some packin gs . For other types of packin g the
reporte d in
p rocedure p roposed by Dombrow ski and B row nell ( 1 9 5 4) c an be applied The evaluation of the interfacial area per unit of pac k ed volume ,
and the
liqui d - side mass
transfer
coe fficient k ,t c an
be
a , p ob taine d , for tfie
summari zed in T able 1 0 , from Figs . 11 function of the liquid superficial velocity . T h e ga s velocity For situation s n o t include d in t h e se does not affect the reported c urve s . figu re s , the relation s hi p s rep ort e d in Table 11 can be u se d . The value s of the critical surface t e n sion a0 for various material s , in c l u din g polyp ropylene and p olyethylene hy drophili zed by t r ea t men t with chromosulfuric acid
packin g s and operatin g condition s an d 1 2 a s a
,
( Linek et al . ,
1 9 7 7 ) , are reported in Table 13 . N ote that the expression et al . ( 1 9 6 8 ) a s s u mes that the ga s - liquid interfacial area
propose d by O n da
o
ap identifies t h e pac k in g wetted surface area a w , w hic h is n t portion of the li q ui d is disperse d as small droplet s or rivulets ,
true when a as for Pall
rin gs (in this case the interfacial are a m ay be u nderesti m ated by 5 0 %) .
tn Co Q)
TABLE 1 0
Packing C haracteristic s an d O p eratin g C on ditions for t h e Data S hown in Fi gs . Packin g
Type A
C eramic Intalox saddles
B
Ceramic Pall rin gs
c
Steel Pall rings
D
C eramic Rasc hi g rin gs
E
C era mic Rasc hig rin gs
F
G Hl
H2
N ominal diameter , dn ( in . )
1/ 2
1/2
T emperature (OC)
Column diameter de
25
4 in .
25
4 in .
5/8
25
6 in .
30
4 . 3 7 em
C eramic I ntalox rin gs-
3 /8
Ceramic Rasc hig rin gs
1/2
25
4 in .
C eramic Pall rin gs
I nox steel Pall rin gs
3/8 1/2 1
1
30
30
25
25
10 an d 20 em 10
em
9 in . 20
em
N u mber of particles p er u nit of
packed column ( m- 3 )
6 3 0 , 000
1 1 a n d 12
D ry packin g area . at ( c m- 1 ) 4. 7
360 , 000
4. 2
1 , 0 70 , 000
5. 1
22 0 , 0 0 0
980 , 000
3 85 , 000
3 70 , 0 0 0 4 9 , 000
49 , 000
3. 5
-
3.8 2. 2
2. 0
Void
fraction , s
0 . 78 0 . 93
0 . 68
-
-
0 . 64
-
0 . 94
C) � -s
a, �
� Q.
iS: 0 ""S
9: Q.
H3
1
12 13 1
Polypropylene Pall rin gs
Ceramic lntalox saddles
Ceramic lntalox saddles
Polypropylene I ntalox
J
Ceramic Rasc hi g rin gs
K
Ceramic R a schi g rings
Ceramic Ra sc hi g rin gs
PVC
L M N
0 p
Raschi g rin gs
C e ramic R a sch i g rin gs
C eramic lntalox saddles C eramic Ra schig rin gs
Polypropylene Pall rings Polypropylene Intalox saddles
1
1
1 1 1 1
1 1
1
/2 1 1/ 2 1 1/ 2 1 1/ 2 2 1
25 25 25 25 25 25 25 25 20 11- 34
11- 34
1 1 -34 11- 34
20 em 9 in . 20 e m 20 em 12 in . 9 in .
20 em 20 em 18 in . 0. 5 m 0. 5 m 0. 5 m 0. 5 m
51, 000 84, 000 7 5 , 300 53 , 5 00 -
48 , 000 50 , 6 00 51 , 400 1 4 , 000 21 , 000 1 3, 000 15 , 500 6 , 2 00
2. 0 2. 5 2. 5 2. 0 2. 03 1. 8 1. 9 1. 9 1 .3 1. 6 1.3 1. 3 1. 1
0 . 90 0 . 77 0. 7 7 -
0 . 74
!;') Q
0) I !:"' .0
E.
Q.
::a
�(')
... 0
;;J
0 . 71 0. 8 0
0 . 71
0 . 91
til OQ ""''
TABLE
11
Semiempirical Expre ssions for C ountercurrent Packed C olumn s
-0 · 44 � gp 6. 28 '.; e.) ( p";')
Otake a n d Ku!li git a ( 19 58)
Liquid hol dup
p
E: .Q,
s
E:
.Q,
.Q,
Eo
n
d
u d
0 . 676
d
2
3
+
3 8 .
�pl0-4
c2 c3 C 1).1 .Q. a .Q. o. 3 7
=
Shulman e t al ·
P �.
1/ 1-l ) (; . - 2. 2 . "'-£ n U
_
gp
.Q, n d
2
3 +
1.8
�_u 2 )1 / 2 £
gd
= nominal packin g si ze ( Table 10)
a
..E.
a
t
t
=
1
] ) 1 ( ) u .Q, p .Q, - [ ( )0 . 7 5 (-) 0. (( -�0. 041 (-2 )0 · 133( - )0 . 182
a
a£
u2
£
lt
at1J .Q,
g
- 0 05 .
2
O n d a et al . ( 1968)
0· 2
p£
a f.l t
Jl..
u .Q, p
a JI.. a
£ -
t
a c a£
C'l
Q '1: '1: Qt
;:
cr a .Q. t
p
s:
0.
= total dry area of the p ackin g per unit of packe d volume u
t
u .Q. p
cr c - 1. 45 -
exp
.Q, ..£. ::: 1 . 04 5 -
a
( 1955)
B uchanan ( 1 967)
n
Interfacial area per unit of packed volume
a
� Oo Oo
0
pohl Puranik and vo gel ( 1974)
� E: Q,
0 . 2 5 < P ,q, U .Q. < 12 k g / m 2 • s ; 1 0
0. 3
<
.
-2
cr.Q. crc < 1 . 3 ; 0 0 8 < a p / a t < 0 . 8 ; /
k ap t 0. 1 <
=
gP
)2 /
,q, 0 , 0025 a t ll t
u,q,P g,
3 ( 2 g p
ll t
g,
/9
/ ( ll g, U � a : )1 4
< 4 2 k g /m • s ; 0 . 0 1 5 < u
2
0 , 006 < d
2 g P ,q,
p < g g
1
< 0 . 0 5 m ; 0 . 06 < d < 0 . 5 m c
p
cl
(-P J ll g_
tD t
k
R-
=
0 , 00 5 1
1/3
g
p R.
a
=
1<
w ll R.
0 . 0 0 6 3 < dp < 0 , 0 5 1 m ;
aw
2 /3
- 1/ 2
( ) (U,q, P g, ) (�) D
pt t
( 1 969)
- 1/ 2
::tl
Cl)
�
0 �
Onda et al . ( 1 96 8 ) 0 ( a dp ) · 4 t
( U P la ll ) < 500 ,q, g, w g,
a p ( as given b y the same authors in t his table )
Onda e t al. ( 1 968)
Gas - side mass transfer c oefficient
ak _..s:_ D t g
Mohunta et al ,
s::: �
2 . 2 2 kg / m • s ;
Liquid- side mass transfer coefficient
ll g,
...
1:1 I'll
.o·
Volume t ric liquid mass t ransfer coefficien t 1
�- --)
C)
< dp < 0 . 0375 m
--
p U 0 7 f �
( ), ( a ll t g
f = 2 . 0 for d
p
ll
__:_g_ D pg g
< 0. 013 m
)1 / 3
(a d t p)
n · - �.
C.ll Co co
tn � c
TABLE 1 1
f
( C ontinued )
5 . 2 for d > 0 . 0 1 3 p
=
0 . 006 3 < d
m
< 0 . 0 5 1 m ; 5 < u p / a ]..! < 500 g g t g
p
Liquid -phase axial dispersion coefficient
�
( Pe / e ) 1
=
0 . 00 7 < d Pe 2
� x
p
�
703 0 . 0 0 758 R e ·
< 0 . 016 m
= 8 3 6R e 10
-5
S ater an d Leven spi el ( 1 966)
; 0 • 31 7Re ; 2 · 0 1
< Re
- 0 · 31 7 g
Re
�
Lin ek et al. ( 1 978)
- 2 • 01 t
< 2
x
10
-3
P e = u d /E ; Re = p R.u tdp / ]..1 R e = p u d / ]..l R. ; 1 1 R. p g g p g g Gas -phase axial disp ersion coefficient
�
Pe / ( £ P
_
£1)
=
2 . 4R e
;0. 2
0 1 3 9R e 0 . 1 6 Pe ' = 1 . 1SRe . R. g g
Pe' = u d /E g ; 5 · < Re g
g p
..
g
x
<
( 0 . 0 1 3 - 0 . 08 8 d /d ) R e
p
x
10-
10
- 0 . 1 3 1 Re 0 . 3 8 5 R-
50 ; Re n < 1 0 0 J<.
c
D e Maria and Whit e
R.
( 19 6 0 )
C)
1:1
Linek et al . ( 1 978)
l 1:1 ;:s 1:1. :::: 0 ,
!2:
�
TABLE 1 2 Valu e s of the Adjust able Parameter s A pp e a rin g in S hul m an ' s Equation , R ep o rte d in Table 1 1 , for V ariou s Packin gs
P a ckin g
N o minal diameter , d
Type
C arbon Rashi g rin gs
C er a m ic R ashig rin gs
C e ra m ic B erl saddles
n
(in . ) 1
1
1
c3
1 . 35
c2 0 . 02
0. 23
0 . 9 04
0 . 02 0 . 04
0 . 99
c1
2. 75
-Ut
0. 55
FI GURE 1 1 I nte r faci al area per unit p acked volu me, a p in countercurren t p ac k e d column s for the p ackin g illustrate d in Table 1 0 .
_u,
F I G U R E 1 2 L i q ui d sid e m a s s tran sfer coe fficient k R. in cou n t e r c u rr en t p acked columns for the p ackin g i llu strate d in T able 1 0 . -
C arra an d Morbidelli
592
C
TAB LE 13 ri tic al S u rface Tension for Some Packin g Materials
O"C X 103
Ma te ri al C arb on
(N /m)
56 61
C eramic
33 73
G la s s P a ra ffin
20
Polye t h yl e n e
Poly vi n y l c hlori de
40
S t e el
75
Polypropylene an d polyethylene
54
hydrophylyzed Axial dis p ersion is usu ally p resent in both p hase s, alt hou gh it is larger in the li qui d phase [ rou ghly one o r d e r of m a gni t u de in the Peclet nu m b e r ; Dunn et al . ( 1 9 7 7 ) ] . R eli ab l e se miempirical e x p re s si on s are not available for this p a rame te r , due to di sc r e p an cies obtained usin g different e x pe ri m en t al a p p roac he s ( steady - state or t ra n si ent ) or different evaluation me t ho d s of the e x pe ri m en tal respon se curves . In Table 1 1 th e most widely used expression s are reporte d ; the correlations p rop o s e d by Linek et al . ( 1 9 7 8 ) app ear p a rtic ul arly accurate , althou gh they are limited to a 0 . 1 3 9 - m diameter column p ack e d with 1 5 - m m ceramic Raschig rin gs . Cocurre n t Packed Columns
I n t h e se unit s , since floodin g cannot occur , un li mi t e d flow r ate s can be u se d . So , unlike countercu rrently op e r a te d column s , they do not exhibit
given gas and li q uid flow rate , the pres sure dr op is lo w er . Obviously , due to reduced ave ra ge in te rp has e mass tran s fer drivin g force , their application is li mi t e d primarily t o c ol umn s with fast ch e m ic a l reaction s . For small gas flow r a t e s in a down w a r d coc urrent column , the trickle or b u bb ly flow c an be presen t , dep e n din g on the m a gn it u de of For in c r e a s in g ga s vel ocity , in both cases a transi the liquid flow rate . c ap acity limitation s an d at a
tion to pulse flow , an d s ub se qu en tly to spray flow , take s place . Flow pat terns and tran sitions from one form to anothe r , as a fu nc tion of the gas and liquid flow rate s , ar e re p o rted gra p hi c a lly by various a u thor s , with reference to specific sy s t e m s ( h a rp e n ti e r et al . , 1 96 9 ; B ei m e s c h and Kessler , 1 97 1 ; C harp en ti e r and Fa vier , 1 9 7 5 ) . T rickle flow cannot be rea li ze d in up flow c olumns .
C
T h e recommen ded expres sions for t he evaluation of characteristic param e t e rs are s um mari ze d in T ab le 1 4 . In t h e expression developed by L a rkins
et al . ( 1 9 6 1 ) , w hic h i s not valid in the trickle flow re gi m e , t he pres s ure drop for each p hase is c alc u lat e d t h rou gh t he followi n g Ergun-type
equation :
TAB LE 14
Semiemp irical Expression s for C ocu r re n t Packed Columns
Liquid holdup and pressure dr op log ( E.R,/ EP ) log
ll P
LI P L
=
- 0 . 744 + 0 . 5 2 5 log x - 0 . 1 0 9 ( log x )
LG
+ ll P G
0. 05 < x
=
( )1/2 ( --u .R, p
Q,
u--
gpg
::0
0 �
0 . 4 16 2 ( log x ) + 0 . 6 6 6
a
Rashi g rin gs
.a· E.
Q.
__
112
Q C'l.> I t"
< 30 B akos an d Charpen ti er ( 1 9 7 0 )
Liquid holdup +
Larkin s e t al . ( 1961)
.:::.:. ..:....; .=..:..
=
0 . 0 5 < X = ( ll P /L!. P ) G L
log ( E.R,/ EP ) = a 1
2
C)
)
- 2 log x + a ( log x ) 2 3 li P
LG --- + 1 L g pg
-1/2
< 10 0
Spheres
Pellets
.
a1
- 0 . 570
- 0 . 280
- 0 . 363
a2 a 3
0 . 16 5
0 . 175
- 0 . 095
- 0 . 047
0 16 8
d > 0. 002 m p
- 0 . 043
Inter f acial area p er unit of packed volume a
..£. = at
a - 2 a 1y
Cl1 (0 w
tn <:o ....
T A B LE
14
( C ontin ue d )
L'> P
LG
E
_£.
y
=
y
>
12 Pa , a
y
<
12 Pa , a
<
0. 5
p
E
L
a
Pa
t
1
0. 25, a
=
2
= 0. 05, a2
1
Gianetto et al .
= 0. 5 =
( 1 970)
C harpentier ( 1 9 76 )
1. 2
Volumetric liquid mass transfer coefficient k a i p
El
Ei
a 1 ( E iD i /D
=
a2
W/m ; Dw 3
=
>
5 < E
w)
80 W/ m , : a 3
i
<
1
=
1 0 0 W/ m , a
3 E £ < 5 W/ m ,
a
1
2.4
0 . 0 1 73 , a 2
3
=
=
1
= 0 . 001 1 ,
0 . 0 0 8 , a2
=
0
x
10
-9
Reiss
= 0. 5 a2
=
C) �
2 m Is
1 0 .
�
( 1967)
C h arp entier
� ;::s 0.
PI
( 1976)
a:
0
�
0.
S;
Volumetric gas mass transfer coefficient
k a g p E
g
=
2
=
u
g
+
li P
0. 07 E g
Reiss ( 1 967)
2/3
.Q ;::
E:
3 W /m
LG
L
::tl
Liquid -phase axial dispersion coefficient
0 . 0 165
u d l', p £ £E £
u £dp
0 . 128
=
£ E £ £
(
ll o
p)
d
0 . 24 5
(
(s p
g p
-
£
g
)E
g
=
1.8
[
)l
jl g
ug dp p g
g
(1
_
P)
p u d g g
Gas - phase axial dispersion c oefficient u d
Co an d Bib au d ( 1 97 1)
1 33 0 4 . p d u £ ( p£ ) p £u £ )1 £
0
Q "' I t""
£ ) p
J
-
-0. 7
0
x
.
16
(a
d ) t p
0 · 53
1 0 0 . 00 5 p £u £ dp 1 \1 £ ( 1 - £p )
....
(I) Q 0
0 ., Cl.l
S tiegel and Shah ( 1 97 7b ) Hochman and Effron ( 1 9 6 9 )
tn tO tn
C a rra and Morbi delli
596
(
t. P
h
L
1
+
h
pu 2 a f1 t
)
( 76)
.
where h 1 an d h 2 are given in Table 15 for va r io us p ac ki n gs In t h e case of t ric klin g li q ui d , £Q, i s gi ven by the e q ua tion d e velop ed by B ak o s an d C h a rp en t ier ( 1 970) , an d the pres sure d rop c a n b e estimated from Eq . ( 76) by rep lac in g t h e pa cki n g porosity En w it h ( £ p £ Q, ) t o account for t he liquid film . N ote that the relation sfup s r ep o rt e d in Table 1 5 for t he evalua· tion of pre s s u r e dr op an d li q ui d h ol du p are not valid for sa d dl e s an d Pall rin gs ; for these p a ckin g s useful data are r e p or t ed by Dodds et al . ( 1960) , Exten sive data on down flow p acked Wen et al . ( 196 3 ) , and Reiss ( 1 96 7 ) . beds wit h foamin g and nonfoamin g materials are rep ort e d b y Midoux et al . ( 1 9 76) . The same relation ship s p rop o se d for down flow colu m n s al so give rea son abl y accurate results for up flow c olumn s ( C arleton et al . 1 9 6 7 ) . The evaluation of mass tran sfer coefficient s and effective in terfacial areas i s hi gh l y affecte d by the flow regime , which in turn dep en ds on the ener gy di s sipation rate . In the relation ship p r op o s e d by R ei s s ( 1 9 6 7 ) for the e valuation of k tap in liq ui d s more viscous than water , the complementary
-
,
correc tion D Q, \.l
�· 8
=
c on s t a nt for constant
temperature , should be ap p lie d
.
In t h e case of k g a p , the expression d e velop e d by Reiss ( 1 96 7 ) is not valid for the tric kle flow re gime , where the same eq ua tion r epo r te d in Table 11 for the countercurrent case gives c on se rvative results . Mit chell and Perona ( 1 97 9) found , for hi gh - porosity p ackin gs ( E p > 0 , 7 ) , in terfacial are a values significantly la r ge r than those giv e n by the r ela tion shi p p ro pose d by Gianetto et al . ( 1970) an d C h a rp en ti e r ( 19 7 6 ) . F o r upflow
TAB LE 1 5 Values o f Paramete r s h 1 and h 2 in E q . ( 76 ) for Various P a ckin gs Pa ck i n g
N o minal diamete r , � ( in . )
T yp e
/
D u mp e d
D um pe d
S ta c k ed
I n tal ox
Sphere
Rasc h i g r in g
Sourc e :
0 . 3 40
1
2. 56
0 . 918
R ei s s ( 1 9 6 7 ) .
2 . 33
3 . 58
3
3 . 39
1/8
4 . 17
3/8
7 . 39
1
C ylin de r
2
3 . 06
1
S tac k e d
h
1 2
1/2
S tacke d
h1
3/8
0 . 039
0 . 0 195
0. 018
3. 75
0 . 333
3. 28
0 . 167
0 . 292
0 . 38 8
597
Gas-Li q ui d Reac tors
columns the k R.ap values are rou ghly 1 0 0 % l a r ge r t han tho se calculate d u sin g the relati on s hips reporte d in Table 14 ( S pecchia et al . , 1 974) ; in the b ubbly regime a c on servative value, k R. ap 0 . 1 5 s- 1 is recommen de d ( C h ar pen tie r , 1 97 6) . Axial disp ersion in the liquid phase can be de sc ribed using the relation ships developed by Co a n d B ibaud ( 1 9 7 1 ) an d S tiegel and Shah ( 1 97 7b ) for trickle b eds an d b ub b le- p acke d c olumn s , r espective ly . For the gas phase , the rela t i on shi p developed by H ochman an d E ffron ( 1 9 6 9 ) ap pli es only to trickle flow [ for R e R. > 6 50 the disper sion coefficient is in dependent of the liqui d flow ra t e ; Woodburn ( 1 974) ] . For bubble -packed column s n o data are availab le on ga s p hase axial di sp er sion which can be expected to be smaller than in upac ke d bubble c olumn s un der equivalent flow con dition s . =
-
,
Mec ha n i ca l l y Agi tated R eactors
Stirre d t an k s a r e particularly suitable for viscous liquid s , la r ge li q ui d volume s , an d low gas flo w rates . N oticeable is the fle xibili ty of t h ese units wit h r espec t to the heat transfer rate and the liquid r e si de n c e time . Princi pal d r a w b a c ks are the large bac k mi xin g in both phases and the hi gh cost of the mec hanical agi t a to r , pa rticu la rly in t he case of highly corrosive li quids . A s a rule of t h u m b with nonco rro siv e liquids, the mec hanically a gitat e d reactor is most economical when the overall reaction r a t e is five times l ar ge r than the mass tran sfer rate in a b u bbl e column , If a l a r g e fraction of ga s needs to be ab sorbed , stirre d - tank r eac to r s are not rec o m R ecent reviews on these unit s have been reported by Calderbank men de d . ( 1 96 7 ) , C harpentier ( 1 98 1 ) , and Joshi et al . ( 1982) . In Table 16 a li st of the most i mpor t an t agit a tor types is given , together wit h t hei r main characteristics [ uA M an d ll R.M in dicate the maxi mum all o w able values of the a gi t ato r t ip velocity an d of t h e li q uid vi sc os it y respectively ( S tei ff et al . , 1 9 8 1 ) ] . The mo s t common a gitator for dispersin g gas into liq ui ds is the si x fla t blade t urbine, w hile a pitche d b lade turbine is t o be p refe rre d when suspe n sion of soli d particles is requir e d . B affles in t h e ve s s el c h an ge the liquid flow pattern s , imp rovin g t h e gas -liquid m a s s tran s fer . The bafflin g r ec ommen de d is four 9 0° baffle s , d R / 1 2 in w id t h , o ffse t fro m the v essel wall on e - sixth of the baffle widt h . I n a sin gle i m p e ll er unit , values such as H /dR = 1 , dA idR 0 . 4 to 0 . 5 , and hA i dR = 0 . 3 3 to 0 . 5 a r e recommen de d , w here dA an d hA in dicate th e a gitator diameter and height off t h e tank bottom , re spec tively . Gas spargin g can be reali z e d w i t h a n in e xp en s iv e p e r for a t e d p l at e ( at l ea s t in the t u rbul en t re gi me where the bubbl e size is regu l a t ed by the coalescence an d breakup phenomen a ) . The spar ger i s locate d ju s t below the agitator , with a diameter 0. 8 to 1 of t h e impeller diameter . An even gas distribution , in the case of a sp arge rin g , can b e ob tain e d by arrangin g the size an d number of holes s'O that the gas vel oc ity through t he holes is at least three times that through t he pipe use d as spargin g rin g . The use of a disk - mounted turbine is s ui t ab l e , pa rtic ula rly at low agitation spee d in o r der to p reven t bubbles c r os sin g a hub - mounted im p elle r in stead of bein g dispersed . Further information on reac to r design is given by B a te s et al . ( 1 96 6 ) and N a gat a ( 1 97 5 ) . I n these uni t s t he ga s - li qui d disp ersion is re al i zed usin g not only the energy as sociated with the gas inlet flow rat e , as in sp a rge d reactor s , but also a s igni fic an t con tribution is by mechanical agitation . The latter is lar gely predominant when th e agitation speed exceeds a critical value N 0 , which correspon ds ro u ghly to a tip ve locit y of ab o u t 2 . 2 5 m / s and is gi ven
,
-
-
-
=
,
,
by
c:n co Oo
TABLE
16
Qualitative I n formation on t he Applic ation of t he Most C ommon A gi tat or s in Multip ha se S y stem s u
Stirrer design
P rop ell e r
dA / dR
0 . 1 5- 0 . 4
AM
( m /s )
J..l LM
( Pa s ) •
Heat trans fer B affled
Wall
C oil
G a s - li quid
disp ersion
a Li qui d - s oli d d isp er sion
15
5
Yes
gg
g
gg
gg
Disk flat - blade turbin e
0 . 2- 0 . 45
15
10
Yes
gg
g
gg
g
Pitc he d - bla e turbine d
0 . 2- 0 . 45
12
20
Yes
gg
gg
gg
I mpeller
o . 5- 0 . 7
12
20
Yes
g
g
Paddle mixer
0 . 4- 0 . 5
5
50
gg
gg
g
bg
Anchor impeller
Ye s
gg
-
gg
o . 9- 0 . 98
5
50
No
gg
-
g
bg
Helic al mixer
0 . 9- 0 . 98
1000
No
gg
a
1
gg , Highly s uit ab le ; g , s uitable ; b g , con ditionally suitable .
C"l l:l ""$
a,
;:s � l:l
::::
0 ""$ 0'
E:
�
599
Gas-Liquid Reactors
N 0d
A
=
g o1 --
1/4
d R a +bd A
( ) ( pi
)
(77)
1.22
5
and b 1. 2 for where a and b dep end on t h e agitator ty p e: a = turbine i mpellers; a = 2 5 and b for t wo- and four-blade a gi tato rs (Westerterp et al., 1963). F urther details on the h ydro dy namic s and flow patter n s of agitated gas-liquid reactors, including t he estimation of the agitator flooding conditions, a re reported by Rushton and i mb inet ( 1968), John (1971). Nag at a (1975), Hicks and Gates (1976), Joshi et al. (1982), and Ri ete ma ( 1982). Another c ritical agitation speed, Nc, can be defined bey on d which gas above the free liquid surfac e is entrappe d by the liquid. It can be evalu ated from the following relationship (Joshi et al., 1982):
2.
N c
=
1.65
=
0.68
=
B
0.031 0.625 0.19 ;0.125 (::g) (:�) (:�) N
(78)
where N p is th e po wer number a n d WA th e i mpeller bl ade width. Due to the strong dependence of mass and heat transfer parameters on
liquid visc osity, su rf ac e tension, coalescen�e characteristics (such as solu tion ionic stren gth ), and foam ability, it is very difficult to predict a priori these pa ra m eter s (Elstner and Onken, 1 9 81) Therefore, the semiempirical expressions reported in Tab l e 17, developed primarily for the air-water sys tem, should be used with caution. The power ab sorbed by a mechanical s ti r rer in an unae r ated liquid, P , can easily be calculated (Bates et al., 1966; Nagata, 1975). The re duced power consu mpti on i n gas liq uid systems can be evaluat ed accordin g to the relation ships report ed i n Table 17. T he correlation developed by Hugh mark ( 1980) is b ased o n experimental data of various authors; it refers to flat blade tu rbines, Newtonian fluids, and does not account for the power con tribution of gas sparging. Further experimental results are reported by Hassan an d Robin so n (1977). The correlations developed by Sridhar and Potter (1980a,b) fo r inter facial a rea and gas holdup constitute extensions of the cor resp on din g cor relation s by Cald erbank ( 1 9 5 8) , to account for larger gas flows and operating p ressures b eyon d atm ospheric. In such cases the c o nt rib ution s of the kinetic power Pk and the gas exp ansion po wer P q are comparable to the At atmospheric press ure, Pt .::_ P g and mechanical agitation power Pg. P g .::_ Pa• and th es e equations reduce to those dev elop e d by <_::,alderbank ( 1958). Mean bubbl e diameters can be esti ma t ed using E q . ( 44) a nd suitable correlations are r epo r ted in Table 17. Specific interfacial a rea values are highly sensitive to the solution ionic streng t h In fact, the inhibition of bubble coalescence by ionic solutes, even if present in very small amount s (Elstner and On ken, 1981), leads to l ar g er values of a. The ex pression s propose d by Van't Riet (1979), valid for aqueou s solution and for agitation speed beyond the critical value N 0• account for this effect. The volumetric liquid mass transfer coefficient is not affe c ted by the agitator type, locati on, and number a n d by the spa rg er type and posi tion
.
-
0
.
,
Semiempirical Expressions for Mechanically Agitated Reactors
TABLE 17
Impeller power consumption
f
0.10
=
0
(Q#)-1/4( 2 �3)-1/5 N d
t
0.87 < pi< 1.6
0.1 < d
R
<1m;
gwAVt
3 kg/m ; 0.0008
< llt
0.25 < d /d A
R
Total power consumption p
T
5
=
3 N d Apt =
0.15
g
�
=
=
0, 75
[
g -hA d R
(u IE ) r:r
r:r
� 2dA2
0.025
< 0.46; 0.31 < P tP
g
1/4
2..
< 0.028 Pa•s;
0
<
< at< 0.072.N/m;
0.8 Gray et al. (1982)
1/4
t ] (a) a
w
agitator height off the tank bottom;
.4 � ) /2( 4 1/2 (�N2d!pt)1/4 V�/3)
< '1t < 0
Gas holdup
c
Hughmark (1980)
4
p
h A
0) Q Q
Q
0
•
2.5
74 ___g_ NV '-
x
10
-3
5 m; 0.25
1
'
m;
<
dA/d < 0.45; 0.75 < h /d < 1.5 A A R
2 N d A
g wA
10
-2
at
< V Jl, < 51
m
3;
V
2/3 t
ug < 0.053 m /s
Hughmark (1980)
a
n Q
a, Q ;:s s:l.
� d-
�
5!
(£ )112 0. pk u
£
u
g
b
=
=
PT
=
P
=
q
__K__£ o
265 m/s; p
p
g
+ p
+
o. 000216
+
u.
=
a
q
P " gh Q ; u L g s �
g
]( ) ( ) (P )0.16
0 0 IV > .4P .2 R,
R,
0.6
a1
u
_K
o.s
u b
P
...1.
P
g
=
a
<;, a
=
=
2
t
x
10
R.
-2
m;
0 0 / V ) .4P .2
g
44
1
1
0.6
g
0
5 2 9
2 N d
(1980)a
p
atvi
g
__!
P
u b
o1
�
<51 m; u
u
E.
�
gwAVR.
J.
�
-
..c
::0
Hughmark
<
Cj) Q tn
l:l.
gas velocity at spar ger exit
2 N d
NV
0a
198 )
g g s
! 1)0.187 3 ! )O. ( 1/ � 2 / l ) i (gp ) ( ( 1.38 2/3 2/3 v1 3 0.053 . 5 -3 10 . 0 0 1( ) [( P ) (p 16 . 5� p ) 1. a
Sridhar and Potter (
2 0, 5Q P u ;
Interfacial area per reactor volum e
=
:....K p a
den sity of air at operatin: g conditions
; pk =
[
g
J p
a
<
m/s
Sridhar an d Potter
(
1980b)
symbols as for gas holdup correlation Volumetric liquid mas s t ran sfer coefficient
k a= i
, 1 5 551/v (p ) -0. 0248 d4
R
0
...K
vi
5
Q
0.
g
�
d R
Chandrase kharan an d Calderbank
(1981)
C) 0 ...
TABLE 17 90 < P
g
IV 9.. < 170 0 W
( )m
-
k a- b _[ 9.. 0 v 9..
P
3 Im ; 2 . 6
-3 10 <
10
-3
< u g < 1. 8
-
E: =
¢
=
10
-2
ml s ;
b = 0. 026; m 0
=
b = 0.002; 0 3
v!l, < 4.4 m ;
(eptlf.l9..)
0.592Dt
1/2 -
SN3d5 ¢ A 2 h d R L
0.
3
< d
R
<
1. 2 m
Van't Riet (1979)
0. 4; n = 0. 5
m= 0.7; n=
500 < P /V g
!l,
Liquid-side mass transfer coefficient k = 9..
x
n
For ionic solutions: X
x
ug
For pure water:
2
0) 0 ""'
(Continued)
0.2
< 10,000
Wlm3 Prasher and Willis ( 1973)
1/4
Q q=�
3 NdA
1 - 1.26q for q < 0.035
C":! !:)
¢ = 0.62 - 1.85q for q > 0.035
d /d = R A
1/3; 0.006
<
s
Gas-side mass transfer coefficient k
g
=
6.
ss[d � ---
<
g
_
y
>
J
�
2 3 < 0.4 m Is Newman
( 1931)
!:) ;:l � i5:: 0
�
�
y = gas mole fraction of diffusing component
Joshi (1980)
Liquid-phase axial dispersion coefficient ER.
H
2 N R 0.116 (d + H) 2 (d /d ) R A
l J
[
=
Heat transfer coefficient h d
� k R. T
b
=
b
=
Re
_
b Re
=
3
1. 5;
0.87;
=
10
3
= 0.59
m =
dApR. \lR.
A_ ; 11
__
g
< Re < 5
1. 02 <
Fr
-0.1
(11
__:!!..
for jacket
llR.
0.64 for coil
-- (d N
N2d Fr =
m
m Pro.
33
A
w x
+
4u ) g
t"'
..o· s::
E:
�
clear liquid height; valid for disk turbine impeller
=
c;'l !:l I
Cll
; Pr
=
....
0
3
Rao and Murti ( 197
)-0.14
C
«< p 0
;;j
)
R.\lR.
� TR.
(\l ) = liquid viscosity at inside vessel (coil) wall temperature c 5 10 ; 2 <
Pr
< 266;
0.008 < Fr <
11w/ 11 R. < 1. 50 for coil; 0.18 <
11w/ 11 R.
0.53
< 0. 98 for jacket
aThese relationships have been corrected with respect to the original version according to Hughmark ( 1982).
0:. <::> w
604
Carra and Morbidelli
sp ar ger diameter and agitator loca 1979). When o pera ting at lar ge flow rates, t he gas sp argin g power contr ib u tio n should also be con sidere d together with mechanical power Pg in the evaluation of k R, a (Botton et al. , 1980). Extensions to sy stems other than 02-water can be made as s u ggest ed by Fair ( 1967a): as lon g as the indicative conditions on
tion p revi ously mentioned are satisfied (Van't Riet,
-
(k a) 0 t 2 wa te r where n0 t 2-wa er
=
2.4
x
(
D0
D
)
-
1/2
(79)
2 water
2 9 10- m / s
.
No general correlati on is available
for non-Newtonian liquids ; experimental data reported by Hooker et al. ( 1981) can be useful. The correlations for gas- and liquid-side mass trans fer coefficients are not very accurate; for the l a tt er the relationship pro po sed by Calderbank and Moo-Young (1 961) re por te d in Table 9, can b e used The axial d is p er sion coe fficients in b ot h phases, Eg and ER., are greatly enhanced by me ch anical agitation, so that the expr ess ions re p o r te d for spa r g ed reactors give in this cas e conservative estimates. Perfect mix ing can rea s ona bl y be assumed at stirring speeds above No. Fu r the r in formation ab o ut liquid-phase b ackmixin g in mech a nical l y agitated contactors is reported by Sideman et al. (1966) for sin gle-sta ge syste ms and by Sul li va n and Treybal ( 1 97 0) and Joshi ( 1980) for multistage systems. In Ta b l e 17 semiempirical correlations for the evaluation of the heat tr an sfer coefficients, for both the coil- aer ated liqu id and th e wall aer ated liqu id heat transfer, are repor ted These relationships, ob t ain ed using a fl at bl ad e turbine as agi t ato r , are of general validity, since the type of
.
.
-
-
Re9= ugdAP1IIl1 ; Pr1 = 111 Cp1/ kn
Frg=u � /gdA; St=h/u9PtCpt
VI
I
System· water- air Baffled tank: 1
10"
-
FIGURE 13
H =dR
° 10
(Reg Frg
C orrel ation
Pdl113
101
of h ea t transfer data in aerated stirred ve sse ls.
605
Gas-Li q ui d Reactors
impeller has little effect on heat transfer (Joshi et al 1982). As in the tran s fe r , the effect of gas sparging on the heat transfer co e fficient decreases at higher agitation speed; in any case, it never causes variations with respect to the ungassed case (Uhl, 1966; N agat a, 1975) larger than about 2 5 to 30%. In Fig. 13 the data obtained by various 2 authors for the air-water system at high stirring intensity (NdApR./\.IR. > 10 5 ) are shown (Steiff et al., 1981). ••
case of mass
APPROXIMATE SOLUTIONS FOR ISOTHERMAL MODELS
The complete model of a gas-liquid. reactor is obtained by coupling the suitable macroscopic materi al balances with the liquid film model. Th e lat ter must then be solved for different values of the bulk concentrations cJlgi and cp ii• w hic h vary along the reactor axis (steady-state axial dispersion model) or during time (se miflow ba tch model). The solution in general re quires numerical techniques. Since boundary value differential equations are involved, the weighted residual methods (Finlayson, 1972 ; Villadsen and Michelsen, 1978) and the implicit difference methods , such as the one p ropose d by Lee ( 1968) , are particularly convenient. Comments on their application to problems of this type are reported by Huang et al. (1980) , Shah and Paraskos (1975) , and He and Lee (1980) for the orthogonal col location method; by Szeri et al. (1976) for the Galerkin method; and by Deckwer ( 1976) for the Lee method. It is apparent that numerical solution of the complete models although always feasible, is quite complex and time consuming. It is therefore use ful to develop approxi mate solutions, valid only in limited situations, which can be applied for firs t - t rial reac tor design and optimization. Such solu t ions are developed below for the film and the reactor models separately. ,
Approximate Solutions of the Liquid Film Model
As already discussed, the only information relative to the film model that is required for the solution of the reactor model are the material fluxes at the interfaces (Ji>x=O (or Ef) and (Ji h=1 as a function of the bulk concen trations cJlgi and cp J1.i. Approximate relationships for (Ji)x=O and (Ji>x=1 as functions of cJlgi an d cp ii for various k in etic schemes are reported in the following. Some relationships of this type have already been reported, but they were. all obtained assuming zero concentration in the liquid phase for the gaseous reactant (i.e. , cp R.A 0) , and therefore their use is confined to cases where the reactor is operating in the fast reaction regime, as dis cussed later. It is evident that this assumption depends on the reactor performance, which is obviously not known a priori, and therefore, if not used with caut ion it can lead to serious p itfalls in the simulation of gas liquid reactors as shown by Shaikh and Varma (1983c). Teramoto et al. (1969, 1973) modifi ed the original expression of van Krevelen and Hoftijzer (1948a) for consecutive reactions to account for cp R.A if. 0 and the general reaction order, using the approximation ( 40) de veloped by Hikita and Asai ( 1963) . This has been extended by Morbidelli et al. ( 1984b ) to a complex scheme of consecutive reactions and compared with the exact numerical solution. The approximate solution is reported in Table 18 and applies to the following scheme of multiple irreversible reactions: =
606
j
=
1, 2,
.
•
•
, NR
Carra and Morbidelli
( 80)
where the rate of the generic jth reaction is given by r. l
m. n.
(81)
k c 1c 1 j fA fj
=
The isothermal liquid film model equations are used, the only exception being the boundary condition ( 51a) for the liquid reactants and Pj), which, since only the gaseous reactant A is considered to be volatile, is replaced by ( d
(Bj
=
=
15
10 •ct LoJ
f
a=10;
511-oo. P,=l
g, = gz=g3 •1 -3 �tz=�t3•10 ;�n·1
�n���A=4 p2:10.�=10 3
3
�
5
. \
-Ha
FIGURE 14 Behavior of the reaction factor, EA. as a function of the Hatta number Ha in the case of three consecutive bimolecular reactions of first order with respect to both reactants.
607
Gas-Liquid Reactors
TABLE 18 Scheme (80)
Approximate Liquid Film Model Solution for
Volatile reactant, =
A
E
ta
=
=
-
*
A: R.A I
y nh' y'
(JA)x=O
the Kinetic
<Jl
1
StgA
y'
yAEAIHA * /H*A
+
1
StR.A ¢1A
y'
tanh y'
1
(
cosh y'
¢gA/H_A y E* /H* A A A
¢ R.A- 1
+
Nonvolatile reactant, B : j 0 0; (Jj)x=O
Ej
=
(Jj)x=l =
-StR.j
y'
:�
<
t h y'
¢!A -y'
where 2 y'
r
j
=
NR
�
H a2
=
2 Ha
"'* '�'
R.A
j= 1
-- ( --!;. ]
2
P j m.+ J
1
m.-1
,
m.-1 ]
+
,., * 'I'
R.j
J -p
-
y' shy'
]
].
' 2
n.
(1 - co!h y')
¢M)
+ 1 j-1 m. ]-1
mj_1-1
n._1
¢*R.j
J
)
( 1972), Hikita et al. (1977), and a ppr oximate solutions for the kinetic schemes reported in Ta ble 1 9. All the reactin g components are,considered nonvola tile the only exception being the gaseous reactants A and A'; the boundar y condition ( 5 lb) has also been used at the liquid film-liquid bulk interface (i.e., ¢ R.A o/: 0). The approximate solutions are repo rted in Table 20 in t he form of a sy st e m of nonlinear algebraic equations. The unknowns are, in a ddi tion to the r eac tion factors E_A and EA_, (if A' is present), t he dimensionless concentrations of all the nonvolatile reactants at the gas-liquid in te rfa ce
Using a
similar t c n q
Zarzycki et al.
,
,
e h i ue
( 1981)
Onda et al.
have developed
Carra and Morbidelli
608
TABLE 19
i Kinetic Schc mes Con sidered in Table 20 Reaction rate
Reaction
( a)
(b)
r
m
1
r 2
(c)
v
(d )
+
A
A'
vBB
k
n
= klCACB =
P q
k c c 2 E F
-- v C +product C 1
k 2 A' + v' C --
C
v
BB
A+ vA,A' --- prod uct
q r = k Cp C 2 2 A, C r
l
=
k C C 1 A A'
(e)
+
A
k2 produ ct vCC -
(f)
Thi s confirms the substantial equivalence of models (Onda et al., 1972). the three models, a lre a dy discus sed. Finally, it is w o rt h w hile mentioning the stu dy of P an g arka r ( 1974) and Shaikh and Var ma ( 1983c) on t he effect of liquid reactant vo latilit y , of Karlsson an d Bjerle ( 1980) on the effect of the gas - s i de mass tran s fe r re si sta nc e , and variou s a naly ses of ma s s transfer enhanceme n t in the c ase of c ompl ex kinetic schemes (Chandhari et al . , 1975; Ca r mic hae l and Chang , 1980; Zarzycki et a l ., 1981; A starita and S av age , 1982). Anot her way to obtain approximate sol ution s for the liquid fil m model i s to a s su me a specific ope ra t i n g regi me for the reactor. Accordin g to A s ta rita ( 1967) and Danck werts ( 1970) , an d considerin g, for ex a mple , the (m,n)th-order re a c tion (i.e. , mth o r der with respect to A and nth order w ith respect to B) : ·•
A
+
vB
+
li q ui d products
(82)
characteri zed by the following kine tic expression: r
=
m n kC C A B
(83)
TABLE 20
Approximate
Solution
Reaction (a)
Approximate
E*
A
E* A
(b)
E* A
=
{c)
E*
A
-- --
q (1 B
�B q + A V B
1
=
--
EA_ -
(1
q )(l
i;F
--
B
�
=
q,.' E*
=
A'
.A,
=
E
i;E
= � qB(¢!E E
-
M¢*tB
.0 E.
and M'1
1
P.
�
Cl)
n/2
g
0
v
n/2
(
T
1+�
)1/
"l (/)
2
i;E
¢tE)
q, I
q, QA I
AI
� = -
cosh M
1
sin h
-
M1
q A
-tA - ¢
vAI
[
E
qB(1- <�'!B)
-- ----
� til I 1::""
qB(4>1f- ¢tF)
=
F
m
!j>* ) 9-B
����
i;B = v
( 1- q ) A
-
M
M¢*
�E/vE T) + (� /vE)[(q /T) - qB !j> E](l- sech M l) 9E A � --��--------�� ��� 1 + (tanh M1/M 1)1; /vET
+
A ��
q, - (1- q) A
(d)
solution
q,
=
Q
of the Film Model for the Kinetic Schemes Reported in Table 19
Hal
(1
[
AI
i;B )] +-q(l- ¢* ) .Q.B V
M¢*tB n /2
and M'!j>*
tC
q/2
B
sc v qB<
- q,1c> M"' S
v
A'
C*
9-A
i;A,CiA'
O'l c co
0) .... .
0
TABLE 20
(Continued)
Approximate
Reaction E* A' (e)
E* A
1
=
=
+
-- =
=
E* A
=
+
- 1)
qA
and
M].
'C'i A C* - qA' "A' 1A'
v t"
A
�
E* = 1 A -
(f )
(E* A
M1
solution
qA
(1+
�B
v8
vc�B --
VB sc
+
=
qA
+
qB (1 u
1 -
+
�c
v
C
* ) B (
sa
M¢* 1B
v qB(1 B
A'
=
gaseous
sA' VA'
(1
- qA,)]
=
=
C£AIC£A; qA'
n/2
and
iB
M"
q/2
a. Q ;:I �
=
/C'iA
1C'
V¢*'!)
(') Q �
reactant
B , C, :E, F liquid reactant, nonvolatile
+
where
A,
n M(
q
=
C£A,IC£A,;
� �
�
M
1
tanh
=
M1
(1
__! m + 1
k _
T k
1
V
=
M2
M12
p
+
-
qA sech M1);
C*
(p+ q -m-n )
'lA
1
¢1
qB
k
·
\
(1
-
q A1
sech M1 1)
(p - 1) * q/ *n
t;)
1:1 (I) I t'"'
.a· E. �
:tJ
(I) 1:1 ()
....
0 -s (I)
1
1
2 -2- k D C *( m - 1 ) Cn I k Q,A m + 1 1 A £A m
-p+ -
tanh M
( p+ q -n-1)
__! C * ( p+q-m-n) q ( q-n ) m + k1 £A B p +
=
=
�1
2
( p - 1) q
* D 1 vA1k2 A1C£A
2 CtC/k£A1
( p - 1)c q /k2 M"2- -2k D C* Q,B tA I p + 1 \)A I 2 A I ,R,A I
M•rr2 - k D C* /k2 1 A iA' ,R,A U =
V¢* ( q- 1) /
�A,C!A•
S = C!A v A'
£B
q A' ) - qA (1
1
3EA
1
+
3 E* A
EA
....
Cl) .._
612
C arri1 and Morbidelli
t he follow ing parameter can be introduced: _2_ 2 M =
m
+
2
=
m
1
+
1
Ha
(84)
2 *m-1,.n
¢ R.A
'�'
R.B
which is the Hatta number de fined at the liquid bulk conditions and repre
sents the ratio of the ch arac te r istic times of the diffusion and
processes.
reaction
Its value can then be used to define the reactor regime.
Slow Reaction Regime
In th is c ase the reaction in the liquid film is negligible, and t h e reaction factor identifies with the purely phy s ical diffusion value: E!l' 1
(85)
=
The material fluxes at the interfaces can then be calculated as follows:
(86) ( 87) The gas-liquid interface concentration values are given by Eq. (57) and 1 d• * - 4> gl + Y·4>o <jlgi- 1 + y./H�
(88)
•
1
1
E quat ion s (86) to ( 88) represent the liquid condition
film
model solution under the
(89)
Such conditions has been further specified by Barona ( 1979) fOl.i a CSTR model with a (1, 1)-th order reaction of type (82) as M < 0.1. It is worth men t ion in g that two subregimes can be iden tified with the slow reacti on regime depending , in add iti on to Eq. (89), on the following conditions: Kinetic Subregime 2 aM « 1
(90)
In this case t he chemical reaction is the rate-determining step, so that the gaseous reactant concentration is
¢ R.A
= rp iA ,
independent of the reaction
61 3
Gas-Liquid Reactors
-
rate. T he gas li qui d reactor then behaves as a homogeneous reactor, where only the li quid reactant concentration s change with time or p o siti on
.
Diffusional Sub re gi me
(91) In t his case the gas-liquid mass tran s fer is the rate-determining step, so that the li q ui d concentration of the gaseous reactant vanishes. The total amount absorbed from the gas stream can then be calc ulated by solving t he The interfacial material fluxes are easily only gaseous phase m as s balance. evaluated from Eqs. ( 86) an d ( 8 7 ) w i t h � R.A 0, leadin g to a lin ear expres f sion for the sink term in the gas-phase mass balance o the ga seous reactant. =
Fast Reaction Regime
In this case it is assumed that the reaction is c o mple ted within the liquid
film, so that the liquid bulk concentration of the gaseous reactant van i shes, hA 0. This simplification is of great utility in pr actice , since the liquid phase mass bal an ce of th e gaseous reactant is not needed, and the approxi mate exp res si on s for the reaction facto r rep ort ed earlier can now be used. This regime is established for =
EA.
M2
» 1
In Fi gs
.
(92)
15 and 16, a comparison
between the two approximate solutions
developed above and the comple te liquid film model solution for two reactions of type ( 80) of first order w i t h respect to both rea c t a nts (i.e. , NR = 2 ,
mj
=
nj
=
1) in a CSTR is shown.
It is apparent that the two approximations
--��----, ---------complete film model --------
3r-
cl)t1/4l[A=4
---slow reaction approximation
P,=P2•1
�, =�2 =1
cl)ll.,;$12"10 2 Sh =oo
-3
---las! reaction approximation
*c 1&.1
'
-M
2
3 f
A
FIGURE 15 C omp arison of t he reaction actor, E calculated with the com plete model and the slow and fast reaction approximations.
CarrO. and Morbidelli
614
...
-&- 0.5 St l
"
=00
•�, I ��i. :4 P, :p2:1 g, :g2=1 Clll, =1: .12•10"3
o����uw��--�--��W7--�� 0.01 M
*
FIGURE 16 Comparison of the concentration ratio !j> R.A /!j> 9-A calculated with the complete model and the slow reaction approximation. are quite accurate in the appropriate intervals of M values, while a lack of ac curacy appears for both of them for intermediate M values, depending on th e liquid hold up value a., defined by E q . ( 62 ) . It should be remarked that the value of M is not known a priori, since in gener al it depends on the reactor op eratin g conditions through the bulk reactant concentrations (and then it is a fu nctio n of the reactor axis in the steady-state axial dispersion model, and of time in the semi flow bat ch model). Therefore, care must be exercised to en sure that all possible values of M along the reactor axis or during time are such as to satisfy conditions ( 8 9) or ( 92 ). Obviously, this is not true for the approx· imate solutions reported in Tables 18 and 2 0 , which can be applied for all re actor regimes. The available solutions, valid for a specific reactor regime, and the possible model simplifications arising from li mit in g values of the dimensionless charac teristic parameters are reviewed below for each reactor mo del reported earlier. Approximate Solutions of
the
Axial
D i s pe r s io n Model
The macromixing effects on reactor performance are taken into account in each phase separately through the introduction of a Peclet number. In Fig. 17 the e ffe ct of Peg on the reactor outlet conversion, in the case of a pseudo first-order reaction in the slow regime, is shown. Extensive calculations for this system, performed by Pavlica and Olson ( 1970 ) , have shown that tne Peclet number affects the reactor conversion by no more than 1 0%, except in the interval 0.1 < Pe < 2 0 . Such a conclusion, valid for each phase independently , has been also verified in the fast reaction regime by Deckwer ( 1 976). For Pe < 0.1 and Pe > 2 0 , the CSTR and PFR models can be ap A lso, a PFR model can generally be assumed for the plied, respectively. gaseous phase for small bubbles columns (de < 0. 5 m) Frequently, gas-liquid reactors can be described using a CSTR model for the liquid phase and a PFR or an axial dispersion model for the gaseous phase. In this case the reactor model can be obtained by suitably combin ing the appropriate equations of the models presented earlier. The only difference is that in the CSTR model the local material flux entering the
Gas-Liquid
Reactors
615
-
Peg
FIGURE 17 Eff ect of the g aseous phase Peclet number Pe g on the outlet reactor conversion, in the case of a pseudo-first-order re ac tion in the slow reaction re gi m e .
must be replaced by the avera ge one, i x=1 by Eq. (72), as done in the semiflow b at ch m odel.
liquid bulk (J )
(Ji)x=1'
de fined
A useful sim plification of the axial dispersion term has been pro p osed by
Hlavacek and H ofm an n ( 1970 ) . 1 Pe
2 d
where
tgA
::::
�
dz
(A p) 2
-
-�
(93)
A
Pe 2 A2 - Pe /4
(94)
Since such an approximation correctly approaches the CSTR model as Pe + 0, while obviously it does not a p p ro ach the PFR model as Pe + oo, its applica tion should be limited to situa tions with rel a tively small Pe values. Another i m p ort ant asp ect of the axial d is persion model is the variation of the superficial gas velocity along the reactor axis. Gas e x p a nsion due to reduced pressure and g as shri n k a ge due to gas absor ption c aus e o pposi te variations on the gas velocity and thus on the reactor perfor manc e This is quantified in Fi g 18, where the effect of 0: and the inlet gaseous reacta n t .
mole fraction y
�
.
on the reactor conversion for a first-order re action in the
fast reaction re gime is s hown From the results re p orted by 'Deckwer ( 1976), it can be concluded that the hydrostatic head variation can be n eglec te d for Ci < 0. 1 (in particular, for large pressure values, P > 20 atm), while the f gas ab sor p t ion effect can be neglected when y < 0. 2.
Slow Reaction Regime
.
A
The model is given by Eqs. (60), (61). (66) to (69), (86), and (87), where the only n o nlin e arity is due to the gas velocity v ariation al on g the reactor axis and to the chemical reaction in the liquid bulk.
Carra and Morbtdelli
61 6 0 0.1 0.2 0.5
2
a 5
c:
.2
"' .... .. > c: 0 0
'
FIGURE 18
Influence of the hydrostatic head variation
mole fraction in the feed y
� on its conversion.
a
and the reactant
The dashed line indicates
the solution where the gas velocity is assumed constan t .
For a reaction type (82) with the reaction rate given by Eq. (83), if f 1) and the kinetics of the gas velocity is assumed constant (v = v g g pseudo first order, the model becomes linear. The analytical solution can then be obtained by standard techniques (Pavlica and Olson, 1970) In Table 21 the solution obtained assuming a PFR model for both phases (Pe , g Pe Jl, + co ) is reported (Cichy et al. , 1969). In the s ame table the solution in the case of a gaseous phase constituted of pure reactant A is also re ported. Of particular practical interest is the case where the liquid phase is described by the CSTR model and the gaseous phase by the axial dispersion model . Then cjl li is independent of the axial position , so the mass balances in the gaseous phase can be solved analytically. Those in the liquid phase require the introduction of the average fluxes ( > defined by Eq. ( 72). Ji x=1 Since t he reaction in the liquid film is negligible, (N ) s= IS (N ) • which 0 i i in dimensionless form can be rewritten as =
•
=
-
(Ji)x=l = (Ji)x=O
St •.
"'1
y1 St
gi
�
(95)
where (J.) 0 can be evaluated from Eq. (72), using Eqs. (61) and ( 6 9) as 1 x= follows: (96)
617
Gas -Li q uid Reactors
TABLE 21 PFR Model with the Bimolecular Reaction (82) of Pseudo 1, vII. = 1) First Ord er (v =
=
4>gA
A
±
=
c e 1
d). z +
+
c e 2
dA
z
I=
-1
for n*
1
=
2 11 2 2 ) _ 2 �*�p�q�� 2�n�*g�_ [ q� n_*(�p�+�1�)�±�(q� _+�p _+�2�p +�1�+�n --�� �2q __
__
Case of gase o u s phase of pure A: =
for n* =
A
I"z
d(p + 1) q
+ bq
p
+
1
I"z
)
618
C arrd and Morbidelli
TABLE (v
==
2 2
Liquid-Phase CSTR,
1)
If> gi(l)
-
4q(lj/.
=
(1+
q)
b/d)e
Pe
Gaseous-Phase Axial Dispersion Model
g
Pe q/2 2 g - (1 e stu
a--Ha �i For Pe g
-+-
d
==
b
=
=
q
- q) 2
2e
-Pe q/2
g
+�
d
� £..J j=l NR
oo:
lf>gi(1) =(If>�- �)e-
d
/2
St gi
1
+
Y/ H { + YiH{
�
dH{If>_u
(
1
1 '2 + 4d ) Pe g
Assuming constant gas velocity (i.e., vg = 1), the model reduces to the sys tem of nonlinear algebraic equations reported in Table 2 , which can be 2 solved by standard techniques. Note that this procedure applies to all kinetic schemes and rate expressions, including reversible reactions. In the case of a nonvolatile component, the equations in Table 22 still hold with H!" 1
=
0 and
l.
gi
==
0, which lead to lf>r:n(1) .,.
=
0.
Fast Reaction Regime
In this case the mass balance of the gaseous reactant in the liquid phase is not necessary, since it is assumed that lj> 0, Under the conditJpns of constant velocity, v g == 1, and reaction factor independent of the axial posi tion, the mass balance of the gaseous reactant in the gas phase can be solved analytically as follows:
R.A
If> gA
_
� -
cp A
2
==
Pe e 2 Pe e 1 _1 < 1_+_q_) _z _2 ..__ ...; (_1_-_ q_>_ e_x_p_[_-_ _-2-q _0_+_q_>_ _x_p_[_ _.R-(1--_q_) z_1_ _1 _-_2 g _+_ P _,g g
_
(1
+
q)2-
(1-
.
q)2 exp(-Pegq)
(97)
619
Gas-Liquid Reactors
where
q=
(
YAE*A /H*A 1 y E* /H* AA A
A 1+ 4� St
Pe
+
g
0.
)
5
(98)
The assumption EA. cans t . is quite restrictive in practice, However, it is valid in the case of a reaction of type ( 82), first order with respect and nth order with respect to with component B nonvolatile and Thus, according to Eq. ( 39), if 4 < M < the li qu id phase well mixed. EA in/2, in the reactor factor is given by , =
B,
to A
EA_
(kD =
M
=
A e Q.B ) n
0.5
(99)
k_Q.A _
____
_____
which is con st ant along the reactor axis, because the liquid phase is well mixed, (i.e., C canst.). In the fast reaction regime, the gaseous re actant is depleted within the li q uid film and therefore at steady state it follows that vB(N )s= = -(N ) , which in dimensionless form leads to
Q.B
A
=
O
B s=o
(100)
where (JA)x=O follows:
ca n readily be evaluated, using Eqs.
( 61)
and (69), as
( 101) The final system of nonlinear algebraic eq uation s which constitutes the model is reported in Table 23. Wit h reference to the same reacting system, in the case of PFR model for the gaseous phase, the reactor model can be greatly simplified, although taking into account the variation of the superficial gas velocity (Deckwer, 1976). The final version of the model is reported in Table 24. Moreover, extensi ve calculations performed by Deckwer ( 1977) e nable us to account for the effect of the gas-phase axial dispersion on the outlet reactor conversion. In Fig. 19 the ratio of the reactor length s calculated using the axial disper sion model, Lpe (Pe = finite), and the PFR model, Leo (Peg -+ ) to ob g tain a given value of the outlet conversion is shown. The res'ults are shown for various values of the Pec.!_et number, Peg. It is remarkable that the hydrostatic head parameter, a, and the gaseous reactant mole fr ac tion in the co
�
,
feed stream, y , although affecting the gas velocity appreciably, do not
significantly alter the results of Fig. 19. That is, the effect of axial dis persion seems almost independent of the variation of the gas velocity along the reactor axis.
620
Carra and Morbidel!i
-
T ABLE 23 Liquid-Phase CS TR, G aseous Ph ase Axial Dispersion Model, with the Bi molecular Reaction (82) of (l,n)-th Order ( v g 1; Nonvolatile) B Pe /2 f 4cj> qe g A 1 cpgA ( ) -Pe q/2 Pe q/2 2 g ( 1 + q) 2e g (1 q) e =
- -
=
= 0
cp R.A
-1-
For Pe g
oo:
q
E* A
.�o.
Ha '�'
=
=
)
( + 112 1 4d Pe g
n/2 tB
TABLE 24 Liquid-Phase CS TR, Gaseous - Pha se Model with the Bimolecular R ea ct ion ( 82) of (l,n)th-Order (B Nonvola tile ) f
f
- (1 - cl>gA )
cl>gA t; s
f
.....
_c�> gA
ln(l - l;)
<-:v:-"g'-cl> _..g"""A_>;:. z =...:; .. 1 ;.
__
=
v (l) g
f cp gA
-
1
=
-
c�>f
v
f
g
=
1
A>
(a. + l)cp : (1) g
cp iA = 0 f
v R. cp R.B E* A
=
=
H a '�'.�o.
vi cp n/2 iB
R.B
+
v St f B iA t; cp y St gA A gA
St
gA
PFR
E yA A/HA
1 + y EA /H A A
1 + 0.5a.
1
+
a.
621
Gas-Liquid Reactors
L=����
1 0.01
FIGURE 19
0.5
0.1 0.05 -conversion
In flu en ce of the gaseous-phase Pec let number Pe g on the
reac tor length necessary to obtain a
�
an d y
is e x am ine d at Pe
g
given
conversion .
The effect of
_ a
= 2.
The effect of the gaseous reactant order on reac tor perfor m ance can be They concern an (m,n}th-order reaction of type (8 2), with c o mponent B non volatile and plu g flow for b o th ga se o us an d l iquid phases. Also in t his c ase, . / 2, the reaction is given b y assu m ing that 4 < M < ,1n in ve s t iga te d throu gh the relations hips developed by Mhaskar (1974}.
EA
(102) 25
In Table
the analytical solutions obtained by Mhaskar ( 197 4 ) for
various value s of the rea c tio n orde rs m an d n are repo rte d , a s s u ming plug
flow in b oth p h a s e s (i.e . , Pe , PeR. -+- co), ne g i gibl e gas-side mass transfer g resistance (i . e . , « 1), and constant superficial velocities (i.e., VR, vg 1). Four situations have been con sidered: c ocur re nt flow (n* -1), c o u ntercur re n t flow ( n* 1), the case where the gase o us reactant con centration «P A does not vary appreciably alon g the reac tor ( IPgA = 1) , and g finall y the case where the liquid reactant c on cent rati on cp R.B doe s not vary appreciably along the reactor ( cp R.B 1) . It is worth mention in g also the app roxi mate solutions obtained by Szeri et al. (1976) for a (1, 1)th-order Such solutions are r eac tion in the fa st and instantaneous reaction re gime s. valid in the ran ge of lar ge and small Peclet number and have been obtained using the p e r t u rb a t i on and Galerkin methods, resp e ct ively .
l
YAEAIHA
=
=
=
=
=
Approximate Solutions of the CSTR Model The equations that constitute the model are su m m a ri ze d in Table 8.
S ince
these are algebraic equations, the problem here is only to simplify the l iqui d film mode l .
In the case of con secu tive reac t i on s ( 80) of general order , t he
reported in Table 18 can be use d. The resu ltin g n online ar algebraic system can be solved through v ar ious methods, as repo r t ed by Has him o to et al. (1968), Teramoto et al. (1973), M orb ide lli et al. ( 198 4 ), and Shaikh and Varma (1983b). In the last reference, a (1,1)th-order reaction is co nsi dere d , taking into account the gas holdup and interfacial area variation approx imation
622
Carrlt and Morbi delli
TABLE 25
(B
PFR M odel with the Bi m olecular Reaction ( 8 2) of ( m , n ) -th Order v v R. = 1 , y EA IH.A « 1 )
Nonvolatile ,
C ocurrent ( n *
n
m 1
2
1
2
1
1
1
1
1
1
1
A
=
g
1/ 2
gA gA
gA gA
gA
ln
=
d/
- (1
d ) exp
-
1 / (b z + 1)
=
=
[1
d
+
d
=
1 /{ 1 +
=
d
v' q(
_
gA
v' q (
+ 2 arctan
d
(q
b z /2)
- 1)
2
(q
+
- 1) + 1
[ 4(q -
+
1
1/2
[ q ( ,j,"' A g ln
1
0
gA
q) 1/ 4 =
=
(1 -
11=9>
- 1 ) ( q
gA
gA
1 ) ( q
- q
- 1) +
- 1)
+
(q
- q - q
=
+ +
1) ] 1) ]
-bz
1/ 4. +
1/4
r::v :t:
(q
1] 1 / 4 1/ 4 + 1)
exp ( -b z )
-
< 1)
r--:
v q
1) / 2]
- 1
l(i""-=!
+
(q
- 1) 11 4
1/4 10" + g - 1 "=1" 1 - [ 4 ( g - l) ] + ln 1 + [ 4 ( q - 1 ) 1 1 1 4 + v'q=1
11 4
2
exp ( -b z �) ] 2
( q > 1)
= 1)
(1
-·
(1 -
q ) 1 /4
q) 1/4
+
2 arctan
[ q ( .t. "'
1 1 1 - ( 1 - g) 1 4 + ln + 2 arctan 11 4 11 4 ( 1 - q) 1 + (1 - q)
[ 1 + bz(m
> )
exp ( - b zv'l= 1
4 + 1) ] 1/
+-1 -A--.--,-,1),_<1> g /"q=1 - /,...q""'"(-
[ ( b z + 4) / 4] 4
[ q (
- -b z ( 1 0
=
+
[ 4( q -
1 ) ( q
(q
1)
==
[ 4(q
1 -
1
I= 1 )
tan ( bz/i:l=l/ 2) ] 2
+ l"1='"q - ( 1 - li"""=q)
lq=""T - 1
1/2
+
f1"=(i'"
+
+ arc t a n [ 4 ( g - 1 ) ]
1
(q
d - 1
1)
=
[ /"q=l
[1
( db z )]
;q=--1 tan ( b z ;q::-i / 2} 1 2
[1 -
[1
- 1)
=
2 / { 1 -m )
A
g -
-:- , 1 )
+
1] 1 /4
1 ( 1 - q) / 4
( q < 1)
623
Gas -Liquid Reac tors
TABLE 25 m
n
0
2
( C ontinue d ) C oc urrent ( n *
¢1
0
2
¢1
0
2
¢
= gA = gA
gA
=
d
{1 {1 +
+
(1 + -d
[
0
1
l¢ A= ( 1 g
0
1
¢ A g
=
1
rcf +
(1 -
ld - ( 1 -
+
l=d tan
-
./ - d
m
n
1
2
1
1
1
1/ 2
=
/d) exp [ -bzld/ ( 1 -
+
tan
]
(b qz v'-d./2 2 (bqz.r=ci/ 2 )
/l=d) exp ( - b z ( /q / 2 )
exp( -bz)
1 - q( 1
-
(q
=
¢
¢ A g
In
=
q
{1 -
/! 1
0
1
0
cf> A g
0
2
cp
gA
=
[1
+
- l ¢ gA -
[ v'h -
(cpoB t'
=
q
zh
1 /4
� exp 1 2 � + (Ill + I¢B ) exp (b lh z ) J 2 �s -
< lil
_
2
4 +
In
arctan
( h - gA)1 / 4 - b q cp
h
_
¢0 1 / 4
h1/4
+
h
-
1 14
J
+
B
"'8
11 4
b z(m - 1) / 2 ] 2 1 ( 1- m )
rv'hl
1)
1)
( b )
�
I=
(q
d
) gA
r lil -
h '/
1)
= e xp - z
=
> 1)
by
gA
arctan
(q
d) ] }
1)
h 1/ 4 + ( h - q ¢gA ) 1 / 4 1 1-11.,-4----�=--h - (h - q <jl ) 4
-2
<
(q
C ountercurrent (n *
h
:
-
rcf) e xp [ b z /d / ( 1 - d) ] }
The liquid concentration of B is given
- 1)
( q = 1)
bz / 2 ) 2
1
=
+ < /h / q
+
1)
1)
exp( -bz/hq) ] 2
exp ( - b zlh q ) ]
2
624
Carra
TABLE
Cocurrent (n* = - 1)
n
0
1
gA
2 = q!!_ [ v'q / h cos(bz ./q/2) - v' 1 - q /h sin(bz ./q/ 2) ] of B is
The liquid concentration
A ny
A ny
2
m
n
=
[1
+
cp gA
cp A g
A ny
St gA y A H a H * ( m+ 1) /2
= ___..,___,..,.�
d
=
=
-
1
1 ) /q ;
;
=
= =
q
A
(1
=
- 1; cp
bqz(n - 2 ) / 2 1 2 1 ( 2-n)
consumption (
Negli gible
b
A (n*
exp ( -bq z)
=
A ny
1
given by
Cocurrent and pure
n
:/:1
Mo rb i delli
25 ( Continued)
m
m
and
h 1
= q
+
exp ( - b z) =
$}3 ;
\! B St R. A St gAYA =
A
=
1)
625
Gas-Li q uid Reac tors
due to gas absorption according to Eqs . ( 73) . In Figs . 20 and 21 the con versions of the gaseous and the liquid reactants , X A and X B , are shown as a function of the residence time , operating with fixed gas and liquid flow rate , re spec tively . The behavior represe n te d c an b e explaine d con sidering the actual reactor regime , which is defined by the value of the parameter M defined by Eq . ( 84 ) ( not by M f , which is the M value calculated usin g the feed concentration s ) . Similarly , for the kin et ic schemes reported in Table 19 , t h e approximate solution s of the film model listed in Table 20 can be used , thus r educin g the reactor model to a sy st e m of algebraic equations . Some useful approxi mations for the model solution , based on the specific reactor regime , are give n n e x t .
Slow Rea c t ton Regime
Assuming constant gas velocity ( vg = 1 ) , the model equations reduce to those reported in Table 2 6. In general , this is a system of nonlinear equa tions with the liquid bulk concentrations
In the case of a ( m , n ) th-order irreversible reaction of type ( 8 2 ) , assuming component B nonvolatile and 4 < M < EA , in / 2 , the reaction factor is given by Eq . ( 10 2 ) . In this case the model equation s can be rewritten as shown in Table 2 7 . Approxi mate Sol u t i on s of the Semi flow B a tc h Model The mass b alance in the liquid phase for these models i s given by Eqs . ( 70) to ( 7 2 ) . From c o mparison with the ma ss balance equations in the liquid phase of the ·C STR model reported in Table 8 , it appears that the two set of e q uation s are identical , the only exception being that the accumulation term d
-
=
(103 )
Carra and Mo rb i delli
626 1 . 0 r---r---, l: g = 5 s
0' 2 �-'---'-' -' - '--4 o 3 4 2
c X
- t t !m i n )
f
0 �--��--��--��--��--� 100 200 300 0 400 500
(a )
0.6
Ill · X
t
0 .4
(b)
o �---�---�-----L---�----�. 500 0 100 200 300 400
F I G U RE
20
Conversion of the gaseous re actant X
liquid residence time
T
£
.
(b)
. .
A
as a function o f the
C onversion of t he li qui d reactant x8 as a
function of the liquid residence time
T
£.
Mf
=
f
(kD C £ ) B B
1/2
/k £ A .
627
Gas -Liq ui d Reac t o rs
..
T
(a)
•
T
M r = 0. 0 2 O L-----L---� 0. 2 0. 3 0.5 0 01 0.4
(b)
· ( a ) C onversion of t h e gaseous reactant X A as a function o f t h e gas residence tim e , -r g . (b) C onversion o f the liquid reactant x 8 as f 1/2 a function o f the g ase residence tim e -r . Mf ( kD C ) /k � F I G U RE 2 1
g
=
A
R.B
A·
Carra an d Morbf delli
628
CSTR Model in the Slow Reaction TABLE with Constant Velocitie s (v = v ll. = 1 )
26
-
-
Regime
q/ . + a . H:"
St ll.i
y. St 1
�
v�
=
=
1;
.
Y/Hf
.. - 1 + y
S t r.ri
gi
vi =
i
/H *
i
1
TAB LE 27 CSTR Model in the Fast Reaction Regime Bimolecular Reaction ( 8 2 ) of ( m , n ) th Order
with the
629
Gas - Liqui d R eac tors
S emiflow B atch Model in t h e S low Reaction Re gi me , with the TABLE 28 Bimolecular Reaction ( 8 2) of Pseudo Fir st O r e r (v = 1 )
d
w her e
h = 4q e [ for Pe
g
-+
Pe / 2 g
h
co ,
or. H a
2
d
S t gA
q
(1
+
I [ (1 + q) e 2
=
+
Pe q / 2
g
-
(1
-
2 q) e
-
Pe q /2 g
]
exp ( - d ) ]
a
1 +
HA*
0
-f� gA
yA /HA yA / H A
4 d / Pe )
112
g
( 10 4 ) while t h o s e
for
t h e ga s m a s s b alance are unchange d .
(i . e . , m = and i s reporte d in Ta bl e 2 8 . fi rs t - o r d e r kinetics
1, n
=
In the c as e
of
u
p s e do
0) , t h e analytical solution is pos sible
approximation h a s been p ropose d b y B arona and Prengle ( 1973) , vali d for slow reaction s , such that a.M 2 < 0 . 2 5 . I t con sist s of evalu atin g lj> ,tB ( 6) from E q . ( 1 0 4 ) , as su min g that � and then calculatin g � R. = � A f o m E q . ( 1 0 3 ) wh e r e neglected ( i . e . , 0) .
A useful
r
R. A the time derivative is
tA ,
d � ,e A /d 6
=
C a rra and Morbi delli
630 TABLE 29 Semiflow B atch Model in the Slow Reac tion Regime , with the B i molecular Reaction ( 8 2 ) of ( m , n ) th Order ( v 1 ; a.M 2 < 0. 2 5) . A p p roxi m ation of g B arona and Pren gle ( 1 9 7 3 ) =
(n � 1) (n
a. H a
2
m
=
1)
n
<j) R-A <jl 9-B
where the parameters h an d a are de fine d a s in 2 T able 28 .
The gas -liqui d interface mass flux can b e obtained b y n otin g from E q s . ( 8 7 ) , ( 9 5 ) , and ( 96 ) that
( 105) U sin g these app roxi mation s , the model reduce s to the sy stem of non linear algebraic equation s report e d in Table 2 9 . This solution i s adequate for case s s uch as the chlorination of b en zene to monoc hlorobenzene at 8 0°C an d the auto - oxidation of p - toluic acid in acetic acid solution , cataly z e d by cobalt an d manganese b romide , where the reaction s are ( 1 , 1 ) th order ( A gnello and Williams , 1 96 0 ; Groggin s , 1 9 58) an d ( 0 . 5 , 1 ) t h order (J;.tavens , 1 9 5 9 ) , resp ectively . In these cases the equation s reported in T able 29 can easily be solved analytic ally .
N O N I SO T H E RM A L G A S - L I Q U I D R EA C T O R S Model i n g N o n i sot hermal G a s - L i q u i d R eacto rs
Gas - Li qui d reactors are widely u s e d for temperature - sen sitive p roces ses , s uch as oxidation , hy drogenation , alkylation , c hlorination , sulfonation , and so on , whic h are all accompanied by lar ge heat effects . I n fact , owin g to the hi gh heat capac ity of the liquid phase and the large dispersion prevailing
631
Gas -Li q ui d Reac tors
in gas - li q ui d reactors , the temperature is almost con stant in the entire re actor volume and c an easi ly be controlled . To this en d , heat is u s u ally re move d t h r ou g h the ve s se l w all or t h r o u gh a coil immerse d in the liquid vol u m e . I n both c a s e s , th e p roc e s s - si de heat transfer coefficient is m uch lar ge r than in the absence of b ubbles , due to the increased turbulence pro voked by bu b b le s in the liquid film at the heatin g surface . The j oin t effect of the heat of ab s orp tion and the heat of reaction can lead to significant temperature gra die nts within t he liquid film , whic h can la r gely affect the total mass ab sor ption rate an d , in the case of a complex kinetic scheme , also th e p r oc e s s se le ctivity . Thermal e ffec t s in gas ab s or pt ion without reaction have b e en studied by Chiang and Toor ( 1 964) and V e rm a and Delancey ( 1 97 5) by couplin g the energy and m a s s balances in t he surface renewal model . The latter con sidered the in fl uen c e of heat of abs orp tion , vol um e c h an ge s in the li q uid p ha s e , variable liqui d density , and the pseudo -Dofour effect ; they con eluded that while for the ac c ur ate p re diction of the s ur fac e temperature rise the first two mentioned phenom en a must be taken into accoun t , if only the in terp h a s e mass flow is to be evaluate d , excellent estimates ar e obtaine d usin g the isothermal model. This is la r gely due to t h e equivalence between the enhancing e ffe c t of the liquid volume c h an ge and the de p ressin g effect of the heat of absorption , which lead s to larger s up er ficial temperature and t hen to lower solubility of the gaseous reactant . T hi s study , where gas si de r e si s tan c e s are alway s as sumed ne gli gi b le , has been veri fie d experi mentally for the systems ammonia- water and p ropane - decane . I f a chemical reaction is present , its thermal contrib ution must also be taken into account . While the heat of ab sorption is released only at the gas - liquid interface , t h e heat of re ac tion is released in t he entire liquid film , an d in th e case of slow reac tion , also in the liq uid bulk . Thus the e sti m ation of t he temp erature profile re q uire s the solution of both energy and mass b alances in the li q ui d film . An upper limit of the s u p er ficial temperature values has been r epor t ed by D an ck we rt s ( 1 9 7 0 ) , assumin g that all the heat of reaction , as well as the heat of absorp tion , is liberated at th e interface : ( 106) Accurate analyses have b e e n made in t h e case o f a first -order reaction using both the p en e tra tion theory model [ Danc k werts ( 1 952 , 1 96 7 ) and Shah ( 1 972) examined the effect of the temperature ri se on reactant solu bili ty , diffusion coefficien t , an d re acti on rate c on s t an t ] and t h e film . p10del (Mann an d M oy e s , 1 9 7 7) . In the latter it was c onclude d th at for p seudo - first o r de r kinetic s in the fast reaction r e gime (i . e . , C �A :::: 0) , the nonisothermal reaction fac t o r , defined a s the ratio between the ab s o rp ti on rate with reac ti on heat e ffect s and the p urely p h ysic al ab sorption rate with zero heat of solution , is given by Ha *
2
D k ( T*) A.::. .,- = ....::. k2
R. A
( 107)
632 where T R. an d
T*
Carril and Morbide lli in dicate the liquid b ulk an d th e gas -liquid in terface
temperature , respectively ,
The latter can be evaluate d through the
equation
( 108) which must be coupled with a suitable expres sion of the ga seous reactant solubility a s a function of temperature C *( T )
=
f( T )
.
A s poin ted out by C le gg an d Mann ( 1 96 9 ) , from Eq .
( 1 0 7 ) it appears
that the solubility decrease due to the interfacial temperat ure rise c an re
duce the in terface absorption flows even below those ob tained under con di
tion s of purely physic al isothermal ab s o rp tion (i . e .
,
E,A
< 1) .
I n these
approaches the only heat effect on the s urface temperature , and cons equent}j
on the gaseou s reac t an t solubility , is taken into accoun t .
Therefore ,
such
res ults c an be applied with confi dence to c as e s w here heat effect s are small
or ab sorption rat e s as well as solubilities are low . a deeper analysis of the p roble m is neces sary .
If the converse is true ,
The most complete treat ment of nonisothermal gas - li q ui d mass transfer has b een given by T a m ir et al . ( 1 9 7 5 ) an d T amir an d Taitel ( 1 9 7 5 ) u sin g a penetration - type model for gas absorption into a laminar liquid stream . In i t t h e solvent volatility is taken into accoun t , together with t h e gas - side A b oun dary layer model is a s s umed for the fluid mas s tran s fe r resistance . mec h anical behavior o f t h e gas p hase , and i n t h e li q ui d phase t h e transverse b ulk - flow c ontrib ution , i gnore d in p revious works , is taken into account . This is foun d to
a ffec t the interface ab sorption ( i . e . , when the reaction
rate s or contact times
flow only for low reaction
contribution is s mall ) ; under
these con dition s the dec rease wit h respect to the isothermal case i s limited ,
and t h e same situation depic ted p reviously for p urely p hysical ab sorption is obtain e d .
At lar ge reaction rates or contact time s ,
the solvent evaporation
and the gaseous environment effect s a re of the same order of m a gnit ude as the heat effec t , leadin g to serious decreases of the mass flux with respect to the isothermal case .
The
effect of heat conduction from the liquid film
to the gaseou s phase is ne gligible compared to other heat contribution s .
It
is wort h p oin tin g out that for inc reasin g reaction rates the inter f acial ab sorption flow decreases , whic h is c ontrary to the tren d obtaine d under iso t hermal con dition s .
I n the case of ab sorp tion with chemical reac tion of
c hlorine in toluen e , experimentally investigated by M an n and C l eg g ( 1 97 5) ,
t hi s analy sis sho w s t hat the effect of the gaseou s environment i s to dec rea s e t he total ab sorption flux of chlorine by 2 5% more than what is predic te d us
in g the simplified models men tioned previously , whic h take into account only the h eat e ffect on the interfacial temperature .
The complete mode l of a nonisothermal gas -liquid reactor , inolu ding the woul d be q uite com plicate d , so , us ually , seYeral simpli fic ati on s are introduced in or de r to model detaile d de sc ription of the n onisothe rmal liquid film ,
such unit s .
I n the fi r st p lace , the gas - and liq uid-phase temperatures
generally assumed to be identic al
at any
given axial position .
are
This as sump
tion can be verified using the relationship for the h e a t trans fer coefficient develope d by C al derti ank an d M oo - Y oun g ( 19 6 1 ) for small b ubbles ( dv s < 0 . 0 0 2 5 m , as is t rue in almost all in dustrial gas -liquid contactors ) :
=
( 109)
633
Gas -Li q ui d R eac tors
Only one energy balance is then require d for these reactors , which us
ing t h e axial disp e r sion model ( a s s um in g the ga s and liq uid velocities con
stant alon g the reactor axi s , an d the p h ys ic al p rop e rtie s of t h e gas an d
liquid and the heats of reaction , ab sorp tion , an d evaporation in de pen de n t
of t e m p e rat u r e ) is given by
( 110)
NC
+
a
L
i =l
( - li H ) (N ) i s= O i
-
qR
where on t h e right - h and side three contributions are p re sent : he at exchange with he at in g or cooli n g devices ( ac is the heat e xc h an ge surface are a of coils or jacke t , p e r u n it re ac t o r vol u m e ) , h ea t lib er a te d .2.r ab �orbed by evap o rati on or solution of th e liq ui d [ ( N i > s = O < 0 an d li H = ll H ev l or the gaseous [ ( N i > s = O > 0 and li H = li H a l reactants , r e s pective ly , and finally , t h e heat o f reaction per unit reactor volume . T his la st term , in the case of on l y one reaction wit h one gaseou s reactant [ see Eq . ( 8 2 ) ] , can be evalu ate d as foll o w s :
( 1 11) The overall heat tran sfer coefficient c a n be calculated a s usual through the followin g relation ship :
( 112)
w h e re th e p roc e s s - si de heat t ran s fe r coefficient h w can b e evaluated as si de heat tran sfer coefficient he usin g standard t ec h ni que s ( K ern , 1 950) . Little in formation is available on the e valu a t ion of e ffe c tive axial thermal con ductivity E T . H owever , since the mec hanis m s for a x i al heat transfer ar e wide ly accepted to be r e spon sib le also for axial mass t ran s fer (i . e . , turbulent mixin g d ue to li q ui d circulation , natural convection , and con duction ) , it is reasonable to assume equal effective axial diffusivity for mass and heat : shown earlier , an d the coolin g ( or heatin g)
( 1 13)
where the index d refers to the gas -liquid dispersion .
C h en and McMi llian ( 1 9 8 2 ) have recently s h o wn that E q . ( 1 1 3) is reasonably accurate for batch bubble c olu mn s , while for p a ck e d colu mn s it give s sli gh t ly lower value s for the effective axial t h e r m al con ductivity , d u e to t h e additional heat c on duc tion t hrough the solid p ackin g s . In many case s o f p rac tic al intere st , as shown by Fair e t al . ( 196 2 ) usin g a 1 - m - diameter column wit h sup erfic i al gas velocities o ve r 0, 006 m /s , liq ui d
Carra an d Morbidelli
634
b ack mi xi n g is quite exten sive , lea din g to t e m p e ra t u r e uniformity . In this case ( E T + co ) the C S TR model can b e applie d for the energy b alanc e : (Q p C g g p g + Q Q. p Q. C p Q. ) ( T NC
+
aV
R
L:
( Ni )
i =1
f
-
T)
:::
Ua V ( T - T ) c c R
( 114)
H ( = - 6 i)
s 0
where U i s gi v e n by Eq. ( 1 1 2 ) , an d q R , in t h e c a s e of o n e r eac tion involv in g th e only gaseous reactant A , c a n be evaluate d a s
( 1 15) Models of t h i s type have been used in several theoretical studie s of n oni so thermal ga s-liquid reactors ( Hoff m an et al . , 1 9 7 5 ; S harma et al . , 1 97 6 ; Ra ghur a m e t al . , 1 9 7 9 ; H u an g an d Varma , 1 981c , d ) . Finally , it is worth m e nt i on in g the shortcut and rigorou s design calcula tion procedures for packed gas ab sorbers in volvin g large h e a t effects pro p o s e d by Stockar an d Wilke ( 1 97 7a , b ) an d Fein tuch an d T r eyb al ( 1 978) and appli e d to c a s e s such a s the a b s o rp tion in aqueous s ol u t i on s of air gaseous streams c on t ainin g am monia or a cet one or hydrogen chloride . Steady-State M u l t i pl i c i ty
T he stea dy state obtain ed in isothermal b ack m ix e d ga s - li qui d reac tors is Thi s is n o t true for noni sothermal reactors ,
u sually unique an d stable .
w here t he interaction between the chemical reaction , the h eat an d mass tran s p ort re s i stan ces , and the gaseous reactant solubility causes the oc c urre nce of m ul tip le s t ea dy states . Experimentally , u p to three s t e a dy state s have been foun d by Din g et al . ( 1 9 7 4 ) in a C S T R durin g chlorination of n - decane , while Hancock a n d K enney ( 197 2 , 1 9 7 7 ) showed unusual dynamic phenomen a , such as sus tained periodic oxcillation , in a se miflow sti r re d - t ank column durin g m et hanol chlorination , with hy d roc hloric acid in t he p resence of Zn C l 2 catalyst , to methyl c hl o ri de . Earlier theoretical analyses of steady - state m ultiplici ty an d s t ab ility were presented by Schmit z and A m un d son ( 1 96 3a - d ) and Luss and Amund son ( 1 96 7 ) u s in g reactor models which assume that the ch e mic al reaction and the int er p hase mass transfer are two in de pe n de n t , noninteractin g p roce s ses . Thus these analyses are re stricted to the case of slow reaction . H offm an et al . ( 1 9 7 5) and H uan g a n d Varma ( 1 9 8 1 d ) have e xam in e d the steady - state multiplicity for the bimolecular reaction ( 8 2 ) of ( 1 , 1 ) th order in an adiabatic In both case s the CS TR model an d a nonadiabatic reactor , re spectively . for both phase s an d the energy balance ( 1 14 ) ha s been u s e d , and the agree ment with th e ex pe ri me n t al re sul t s rep orte d by D in g et al . ( 1 97 4 ) is satis factory . I t i s conclu ded that an isola with up to five steady state s c an occur in an adiabatic CS T R , while in t he corre s p on din g nonadiabatic case ( even wit h a sm all heat los s ) only three s t e a dy states can exist . In th e case of two consecutive re action s of the s a m e kin d , S harma et al , ( 1 97 6 ) have shown
635
Gas -Liquid Reac tors
the p o s sib l e existence of up to s even s t e a dy s t at e s . N ote tha t th e se ar e u niq u e features of gas-liquid reactors , as in a s in gl e - p h a s e adiabatic C S TR a ma ximu m of t h r ee an d five steady states c an occur for one or two con secutive re ac tion s , re sp e c tive ly . A lso t he occ u r renc e of an i sola is n ot pos sible in adiabatic r e ac t o r s . A n aly tically necessary an d sufficient criteria for steady - state multi plicity , uni queness , and local stability have b een developed for C S T R re actor s with p seudo - first - order re ac tio n s in the fast reaction re gime (R a ghu r a m and S hah , 1977 ; Raghuram et al . , 1979 ; Huan g an d Varma , 1 98 1b ; Si n g h et al . , 1 982) . Under these c on dition s , the occurrence of I t is wor t hw hile mentionin g the signi fi limit cycl es is us ually not p os sib le. cant usefuln e s s o f st e a dy - st at e m ultip licity a n aly sis as a tool for discriminat in g amon g rival models an d fo r the accurate evaluation of some parameters of t h e s e models ( Hoffman et al . , 1975 ; Huang an d Varma , 1 9 8 1e ) . In t h e latter work surprisin g a greement b e tw e en the multiplicity region s p redic t e d by the pseudo - first -order re a ction model in the fast r eac tion regime an d the complete ( 1 , 1) th - order mo del wa s fou n d . Fi nal ly , Huang an d Varma ( 1 98 1c ) and Aluko and C ha n g ( 198 2) have exa mined the steady - state multip licity of nonadiabatic b ubble colu m n s with fast p seudo-nth - order reactions , a s s u min g th e li q ui d phase well mixed , the gas phase in pl u g flow , and the energy b ala n ce in the form o f Eq . ( 114) . Every multiplicity patt e r n observe d in a gas -liquid C S T R is also p o s sibl e in a bubble column , an d the p o s s ibility of multiplicity in t he bubble column is always higher than that in the C STR for e q ui val en t system parameters . A n aly tic criteria are developed for the p re dic tion of u ni q u en e s s an d multi p li c ity of the s t e ad y states as a fu n c tion of the sy s t e m p hy sicochemical
parameters .
Stagew i se Noni sot h e r m a l Mod e l s
Stagewise m o de ls are widely u s e d in the simulation of nonisothermal gas
Resortin g to a stage wise , tray - tower analogy , they can
liquid c ontactor s .
describe not only plate column s but also packed colum n s . Followin g Carra et al . ( 1 979b ) , the model e q u ation s are summarize d in T ab le 3 0 . I n e ac h stage the ga s - li q uid system is as su m e d to be m o r e or less close to the e q ui librium con ditions , d e p e n din g o n t h e value o f th e Murp hree e fficienc y EM . T hi s parameter takes in to account t h e p o s s ible influence of m as s tran sfer re si s t a n c e s for the va rio u s components . I n some c as e s , a similar approach can be use d to de scribe th e effect of t h e heat tran sfer re si s t an c e le a din g to diffe ren t t e mp eratu re values for the gas and liq ui d s t re am s (Bou rn e et al . , 1974a) on the jth stage : T
g , ).
==
T
g , ). - 1
+
E
MT
(T
- T ..." , ). g , ).+ 1)
· I
( 1 16 )
efficiencies c an b e e sitmated followin g several correlation s , a s ai r and Mersmann ( 1 977) and Pohorecki ( 1 9 76 ) . How sin c e t h e m o de l re s ul t s c a n b e qui te sen sitive to the v alue of s uc h
Murphree
reported by S tic hlm eve r ,
B ou rn e et al .
( 1974b )
suggested de te rminin g the Murphree c on di tion s t hro u gh suitable meas urements con ducted on a s in gle plate of the same de sign as t hose of the unit under c on sideration . As it appears from Table 3 0 , st a ge w i se models lead to a sys t e m of non linear al geb rai c e q ua tion s . S eve ral numerical t e c h niq ue s are available for quantities ,
c
c
e ffi ien y values as a fu n ct ion of the op e ra tin g
Carra and Mo rbidellt
636 T A B L E 30
S ta ge wise N onisot hermal Model Equation s
Material balances of the ith co m p on en t on the kth s ta g e :
NR + V
n k "' '
L
j == l
V• 1
r
, ] j •k •
Equilibrium c on dition s for the ith compon ent on t he kth stage :
y. k 1,
=
E
M
•
k
K
1. , k
x
. ,k 1
+
(1
-
E
M k •
)y
1. ,
k+ 1
E nthalpy balance on the kth stage : · F
�f
k
H
k
+
0
k+ 1
-
H
G
g , k+ l
�
k
H
g ,k
NR + V
R. , k
L ( - tdiR , j )
j == 1
r
j ,k
L H
k
- q
k
Overall m a te ri al b al ance on the kt h st ag e
=
R. , k +
NC
.E ( x1. , k
j == 1
-
y
1. ,
k)
==
k-l
H
R. , k - 1
0
:
NC
Stoichiometric e q u ati on on the kth st ag e :
L
NR
I:
.E
i == 1
j == l
v. 1
.t. k
,) ]
0
•
0
the solution of th ese sy ste m s , an d most of the m have b een reviewed b y H u nek et al . ( 1 9 7 6 ) and Holland ( 1 9 8 1 ) . The gl ob al N ewton - Raphson me t ho d has been succ e s s fu lly u se d for the case of distillation columns with chemical reaction s ( C arr a et al . , 1 979b , d ) . On the other han d , a dynamic simulation method was found by B ou rne et al . ( 1 9 74a ) to be p ar tic u la rly efficient in the case of gas ab sorbers . The use of distillation column s wi t h chemical reaction s is q uite common in chemical in du stry . Mayer and Worz ( 1 980) give u seful suggestion s on the optimal selection , from the stan dpoint of energy savin gs , of eit her a re ac tion column or a batch reactor mounte d by a column , de p en din g on the relative volatilitie s an d the reaction rate s of the c ompoun d s involved . Couplin g distillation with chemical reaction is pa rtic u larly useful when it is neces sary to displace the c h emic al equilibrium in order to increase the de sired p roduct yield , as in esterification and transesteriilCation reaction s (B aratella et al. , 1 974 ) . In the case of two con secutive reactions in the
63 7
Gas -Li quid Reac to rs Xw • Yw · Yepy
FI G U R E 2 2 Profiles of some liquid-phase mole fractions Xi ( solid lines) an d some vapor-phase mole fraction s Yi ( dashed lines) along a column for the synthesis of epichlorohydrin .
li q uid p hase , where the intermediate desired product is highly volatile , the use of a distillation column reactor can significantly improve the process selecti vity . This is the case in epichlorohydrin ( C arra et al . , 1 97 9a ) and propylene oxide ( Carra et al . , 1 97 9c ) synthesis by dehydrochlorination , which both follow the following simplified kinetic scheme : chlorohydrins _.. epoxide + hydrolysis products . In the case of epichlorohydrin synthesis 0.2 0
>Q. UJ
1
0.9 5
0.1 5
0.90
0.10
0.8 5
0.05
0. 8 0 0
50
100
1 50
2 00
- Vj ( t m3)
250
300
0 350
�
C)
f
FI G U R E 23 Yields of epichlorohydrin and glicidol as a function of the liquid volume of the column plate V R, , EPY , ( distilled epychlorohydrin) / (reacted dichlorohydrins) ; GLY , ( glycidol produced) / ( fed dichlorohydrins) .
638
C arri1 a n d Morb i delli
from a mixture of a(:l - an d ay - dichlorohydrin s , the calculate d conce nt r a tion
profile s alon g the c ol umn and yields to the main product and the by
product ( glyci dol ) as a function of the liquid volu m e on the plate of a p i lot column are s ho wn in Fi gs . 22 and 23 , respectively ( C arra et al . , 1 9 7 9b ) . Fin ally , the application of stage wise models to the simulation of gas absorption in nonisothermal plate column s is worth mentionin g . The model re sults have been verifie d accurately throu gh co m p arison with e x p e rime n tal concentration an d temperature profiles (in clu din g the column startup an d shutdown tran sient s ) for the case of the air - am monia- water systems in two different column s : one with five b ubble cap plates an d the other one with 11 sieve plates ( B ourne et al . , 1974a , b ) .
R EA C T O R C H O I C E A N D SC A L E U P
Reactor selection for a given p roces s is usually p e r for m ed on the basis of p reviou s experience or analogies with similar existin g processe s . In fact , especially in the first stages of the research , limited information is available ; gas - liq ui d reactor s , several basic operations which could require differ en t reactor types must often be performed simultaneously . Thus a ri gorous selection proc edure is not available , al thou gh some useful gui delines can be given . In Table 31 t he characteristic values of the hy drodynamic and mass transfer parameters of various types of in dustrial reactors are reported . S uch value s , coupled with ph y sical and kinetic in formation about the system un der consideration , u sin g the reactor models reported earlier , can lead to reactor selection . A procedure of this kin d has been propose d by Nagel et al . ( 19 7 8 ) .
in
The main scop e of a gas -liquid contactor is to produce suitably large values of the mas s ( and /or heat ) t r ans fe r parameters k g , k � , and a , usin g some kind of e nergy ( gas sp ar gi n g or mechanical agit atio n ) . A mon g these , only the mass t ran sfer area can vary throu gh several orders of m agnit ude dependin g on the c hoice of the engineerin g parameters , thus allowin g the ac hieve ment of the de sired reactor p erformance . T hus once the process re q uirements , in terms of mas s transfer surface area , are known , the reactor choice can be made wit h the use of Fi g . 2 4 , w here the surface area values attainable with various reactors , an d the relative ene r gy cost , are shown . The values in the figu re are referred to the gas flow rate Q g , since for a given reactin g syste m , the depletion of the gaseous reactant c oncentration wit hin t he reactor is basically define d by the only parameter ( a /Q g ) ( N a gel et al . , 1 97 3 , 1 97 8 ) . In the case of mechanically a gitate d reactors , a useful procedure for t he selection of c ommercially available turbine a git a tor s has b een p_r,esented by Hicks an d Gates ( 1 9 76 ) . In Tables 32 and 3 3 , the turbine prime- mover power and shaft speed , which uniquely identify a commercial turbine , are reporte d as a function of three parameters c haracterizin g the ga s - l iquid dispersion proces s . These are the equivalent liquid volume Ve q _ ( P i I P w ) V i , the superficial gas velocity u , an d the scale of ag1tation in dex . The latter is evaluated usin g the gas - dispersion agitation scale reported in T ab l e 3 4 ; w here the index ran ges from 1 to 1 0 , and 0 i n di c at es any con dition producin g a flooded impeller , never u se d in chemical processe s . This proce dure is valid for baffled reactor s , de signed accordin g to the su g gestion s given earlier an d equipped with a sparge rin g an d a six- flat -blade turbine whose prime - mover power is fully inve sted in t he gassed con dition s .
g.
=
TABLE 3 1
t;j Q 0 I
Hy drodynamic and Mas s Transfer Parameters Values in Industrial Reactors E Q_
Type of r e act o r
1 04 k g
( mol /c m 2 • s •atm )
1 0 2k t ( e m/ s )
a
( c m- 1 )
1 0 2 k 1a
( s- 1 )
Packed columns
C ountercurrent
0 , 4- 2 0 . 4- 6
2- 2 5 2- 95
0 , 03- 2
B ubble cap
10- 95
Sieve plates
10- 9 5
0 . 5- 2 0 . 5- 6
1- 5 1 - 20
B ubble column s
60-98
o . 5- 2
Packed bubble columns
6 0 - 98
C ocurrent
0 , 1- 3
0 . 1- 3 . 5 0 . 1- 1 7
0. 04- 7 0 . 04- 1 0 2
t:
.Q
s. Q.
� Q
�
(') .... 0
�
Plate colu mn s
1- 4 1- 2
1- 2 0 1- 40
1- 4
0. 5-6
0. 5- 2 4
0 . 5- 2
1- 4
o . 5- 3
0 . 5- 1 2
5- 9 5 5- 9 5
0 . 5- 4 0 . 5- 8
1- 10 2- 5
0 . 5- 7 1- 1 0
2- 2 0
0 . 5- 2
0 . 7- 1 . 5
Tube reactors
Horizontal and coiled
Vertical Spray c olumn s Mechanically agitated bubble
20- 9 5
-
reactors S ub merge d an d plungin g:. Hydrocyclone
Ejector reactor Venturi
jet
94- 9 9 70- 9 3 -
5- 30
-
-
2- 10
o . 3- 4
0 . 15- 0 . 5 10- 30 -
5- 1 0
0 . 1- 1 1- 2 0
0 . 2- 1 . 2
o. 2 - 0 . 5
1 - 20
1 . 6- 25
o . 5- 7 0
2- 100
0 . 0 7- 1 . 5 o . 3- 8 0
0 . 0 3- 0 . 6 2- 1 5 8- 25 Q) t.) 0:0
Carro
640
...
c:l
� > "'
a n d Morb i de lli
1 0 ° 1----'-/
I
10
�--�--�---�---�
1 0 -1
--
FI GURE
24
10 1 10 2 Pr / Q 9 ! K W s m-3>
103
104
A pproximate depen dency of in terfacial area on the power
dis sipation den sity for various types of reactors .
Therefore , other s ,
for sy st e m s that are gas se d at some times and ungassed at
the agitator must b e equipped wit h a t wo- spee d motor to prevent
dan gerous ove rload when the gas flow cease s .
Finally , the a gitator diam
eter , expres sed in meters , can b e evaluate d a s
( 1 17)
where Hp an d N are t h e p rime - mover powe r an d t h e agitation spee d r e ported in Tables 3 2 an d 3 3 a n d expre ssed in hp an d rpm , resp ectively ; P / P o is evaluated as follow s :
In
:
.
=
In
[
1
+
235. 8
] � d!) Q
2 . 667
9. 155
N
Q
(4
)
o . 833
( 1 18)
Nd A
I n t h e case o f multiple impeller sy stems the same proc e dure can b e ap plie d , as reported by Hicks an d Gates ( 1976).
I t i s worthwhile mentionin g
the alte rnative approach based on the P fau dler agitation B arona
( 1979) .
scale de scribed by
The re actor dimen sionin g can be performed usin g either a modelin g or an experimental approach .
In the first case , a first trial value of the geo
met ric dimen sion s an d the operatin g p arameters ( such
as
p re s s ure ,
tempera
ture , and feed stream c haracteristic s ) must be selected to initiate the p ro ce dure .
The process specific p arameters ( i . e . , physical an d kinetic data
of the system u n de r examin ation ) are then calculate d or m e asured independent ly , an d the self- a dj ustin g p a rameters ( relative to the hydrodynamic s ,
mass ,
TAB LE 3 2
Prime -Mover Power ( hp ) an d A git ator Speed ( rp m ) for Superficial Gas Velocity , u 0
Scale of a gitation
750
1 , 5 00
1
1 . 5I 1 2 5
2 / 56
2
1 . 5/84
2 / 45
3
2 / 45
4
3 / 84 3/ 68
5
3 / 56
6
5 / 12 5 5 /84 5 / 100
7
3 / 84 3 / 68 3 / 56
2/84 2 / 125
7. 5 / 1 55
5 / 125 5/84 5 / 10 0 5/45 7. 7. 7. 7.
5 / 1 25 5/ 155 5/68 5/84
1 0 / 84 10/ 100
1 0 / 56
E quivalent volume , V eq
3 , 000 5/ 84 7. 5/125
7 . 5/ 68
5 / 45
10/84
10/45 1 0 / 56
15 / 155 1 5 / 68 1 5 / 84 1 5 / 45
2 0 / 100 2 0 / 68 20/45 2 5/ 125 25/ 84
Q 1::1 (I) I
0 . 021 m /s
t"'
.E'
( gal)
5 0 , 000
7 5, 000
5 , 00 0
10 , 00 0
2 0 , 000
7. 5/68 15/155
1 5 / 56 2 5 / 100
3 0 /45 5 0 / 84
100 / 56 125/68
2 0 / 68 14 / 45 30/ 125
4 0 / 56
1 54 / 84
2 00 / 1 2 5
10/45 1 0 / 56
2 5 / 84 2 5 / 56 20 / 4 5
5 0 / 68 75 / 12 5 50 { 4 5 5 0 / 56
2 50 / 1 5 5 125/45
300 / 1 0 0 3 50 / 1 2 5 200 / 4 5
15 / 68 20/ 100 15 / 84 20/68
3 0 / 100 40/155 3 0 /68 40/84
7 5 / 1 00
150/45 200 / 68 1 50 / 56 2 00 / 4 5
2 50 / 5 6 3 00 / 68
25/125 2 5 / 84 2 5 / 1 00 2 5 / 56
50 / 1 0 0 50/68 50/84 5 0 / 45
1 0 0 / 1 55 1 00 / 1 00 1 00 / 56 100 / 6 8
250/84 2 50 / 56
3 50 / 8 4 4 00 / 1 0 0 3 50 / 45 4 00 / 5 6
30/ 155 30/ 100 30/ 125 3 0 / 68
60/ 12 5 60/ 15 5 6 0 / 84 6 0 / 56
125 /125 12 5 / 68 125 / 45
3 00 / 1 0 0 3 50 / 1 2 5 3 00 / 68
50 0 / 68
40 / 155 4 0/ 8 4
75/ 190 7 5 / 1 00
1 50 / 15 5 150/84
1 0 / 84 7 . 5 / 45
7. 5/84 5 / 56
10 / 1 0 0
=
6 0 / 84
6 0 / 56 75/ 68
150/ 4 5 2 00/ 6 8
2 50 / 84
� ... . l:l.
::tl
(I)
g
..... Q
�
300 / 4 5
3 5 0 / 84 400 / 100 4 00 / 5 6
6 00 / 84
0) 11:>. ...
0:. ol:o. ""'
T A B LE 3 2
Scale of
agitation
( Continue d ) Equivalent volun:e , Ve q
750
1 , 500
( gal)
3 , 00 0
5 , 000
1 0 , 000
20 , 000
25/ 100 2 5 / 56
40/ 100 4 0 / 56
75 / 12 5 7 5 / 68
1 5 0 / 1 00 1 50 / 4 5
15/ 155 15/84
30/155 3 0 / 10 0 30 / 1 25
5 0 / 100 5 0 / 68 50 / 8 4 50/45
1 00 / 1 5 5 100 / 1 0 0 1 00 / 56 10 0 / 6 8
2 00 / 1 2 5 200 / 6 8 200 / 4 5
500 / 6 8
6 00 / 8 4
7 (cont . ) 8
7 . 5 / 1 25 7 . 5 / 68 7 . 5/84
9
1 0 / 1 00
1 5 / 68
30 / 68
60/ 125 60/ 155 60/84 6 0 / 56
125 / 12 5 1 2 5 / 68 125/45
2 50 / 1 5 5 2 50 / 84 250/ 56
10
10/84 15/155 20/ 100
20/ 100 2 0 / 68
40 / 1 56 40/84
75/ 190 7 5 / 100 7 5 / 12 5
150/155 150 / 84 1 5 0 / 100
3 00 / 1 0 0 3 50 / 1 2 5 30 0 / 6 8
5 0 , 000
7 5 , 00 0
()
Q ., .,
r:u Q ;:s l:l. S::::
0 .,
9:
�
TAB LE 3 3
Prime -Mover Power ( hp ) and A gitator Speed ( rp m ) for Superficial Gas V elocity , u 0 E quivalent volume ,
S cale of agitation
750
1 , 50 0
V eq
( gal)
3 , 00 0
5 , 000
1 0 , 0 00
2 0 , 00 0
10 / 4 5
30/68 60/155
50/45 6 0 / 56
1 5 / 68 2 0 / 1 00
40 / 8 4
7 5 / 68 1 00 / 100
=
!;')
s:l 0 I t'"' ...
0 . 0 6 1 m /s
,Q $:: ...
5 0 , 000
7 5 , 000
1
1. 5/84
3 / 56
7 . 5 / 68 10/ 100
2
1 . 5 / 56
3 / 45
15/155
3
2/84
5 / 100
10/84 7. 5/45
25/ 125
7 5 / 19 0
125/ 125
4
3/84 3 / 68
5 / 84 7 . 5/ 155 5 / 56
1 0 / 45 10 / 56
3 0 / 15 5 20/68 1 5 / 45 1 5 / 56
50/ 100 4 0 / 56 3 0 / 45
150/ 155 75/45
2 00 / 45
3 00 / 4 5
5
3 / 56
7 . 5/ 125 7 . 5 / 68 7. 5/84
15/68 1 5 / 84 1 5 / 45 1 5 / 56
2 5 / 84 2 5 / 1 00 2 5 / 56
6 0 / 1 25 50/68 50/84 50/ 4 5
1 00 / 56 1 00 / 6 8
300/68
350/45 40 0 / 5 6
6
5/ 125 5/84 5 /100
1 0 / 84 1 0 / 100
2 0 / 1 00 20/68
3 0 / 1 00 30/125 3 0 / 68 30/45
60 /84 60/56 60/68
1 25 / 68 125/45
3 50 / 8 4 3 50 / 45 3 00 / 4 5
500 / 6 8
7
7 . 5/ 155
1 0 / 56
2 5 / 1 25 2 5 / 84 2 5 / 100 2 5 / 56
40/ 155 25/84
7 5 / 100 7 5 / 125 75/68 75/84
1 5 0 / 84 1 5 0 / 100 1 50 / 4 5 150/56
400 / 100 400 / 56
6 00 / 84
40/ 100 4 0 / 56
150/45 2 50 / 8 4
Q.
::a
Ill s:l (') .... 0 , 0
3 0 0 / 100 3 50 / 1 2 5 2 0 0 / 56
� oQo. c.:.
� � �
T AB LE 3 3
S cale of agitation 8
9
10
( C ontinued) E quivalent volum e , Ve q ( gal) 750 7 . 5 / 1 25 7 . 5/84
1 0 / 100
1 5 / 1 55
1 , 500
15/ 155 15/84
1 5 / 68
25 / 1 25 25/84
3 , 000 3 0 / 1 55 30/ 100 30/ 125 3 0 / 68
40/155 40 / 8 4
5 0 00 ,
5 0 / 100
5 0 / 68
50/84 5 0 / 56
6 0 / 125 60/155 60/84 6 0 / 56
7 5 / 19 0
7 5 / 1 00
7 5 / 12 5
1 0 , 000 1 00 / 1 5 5
100 / 100 1 0 0 / 56 1 0 0 / 68
2 0 , 000 200 / 1 2 5 2 0 0 / 68 2 0 0 / 45
125/ 1 2 5 125/68
250 / 84 250/56
150 / 1 5 5
3 0 0 / 1 00 300 / 68
1 2 5 / 45
150 / 8 4
1 5 0 / 1 00
250 / 155
50, 000
7 5 , 000
5 00 / 6 8
6 00 / 84
n Q
i Q ;:l Q.
s:
� Q. (l)
E
645
Ga� -Liquid Reac tors
TAB LE 3 4
Sc al e of A gitation for Di spe rsin g G a s into Liq uids
Scale of agitation
0 1- 2
3- 5
6- 10
D escription A gitation level of zero in dicates a flooded impeller . recommen de d in C PI service .
Not
A gitation levels 1 - 2 c haracterize applic ation s in w hic h the degree of gas dispersion is not critical . A gitators c apable of scale level 2 will : Provide nonfloode d impeller conditions for coarse dispersion of gas in the liquid sy ste m . B e typic al of application s that are not mass transfer limited .
A gitation le vel s 3 - 5 characteri ze app lic a tions in which the degree of gas dispersion is moderate . A gitators capable of scale level 5 will : Drive fi ne bubbles completely to ve s sel wall . Provide recirc ulation of dispersed bubbles back into impeller .
A gitation levels 6 - 1 0 characterize critical gas-liquid reactors w here rapid mass transfer is required . A gitators capable of scale level 10 will : Maximi ze in te r fa ci al area an d recirculation of disp er sed bubbles b ack in to impeller .
and heat tran sfe r ) are calculated as sh own earlier . All the values so ob tained are fed to the s uitable mathematical model , whose solution leads to the e stimation of the r e ac t o r performance and then to its dimensionin g an d optimi zation ( Deckwer , 1 977) . The most uncertain step s of this proce dure concern the correct evaluation of the self- adjustin g p arameters , p articularly for bubble c olu m n s an d mechanically stirre d contactors , an d the related problem of the correc t reactor model selection . Also , the complete physical ( solubility an d diffusivity of t he solute in the r e ac ti n g mixture ) and kinetic ( rate constant of all the involve d reaction s ) characterization of the sy s t e m can be difficult an d uncertain . An experimental procedure can overcome these difficulties , leadin g to more accurate results . In this case the equation s reporte d earlier can be very useful to define the kin d of depen dence amon g the variou s p ara me t e rs S mall - scale experiments are performed on t he same gas-liquid system usin g a laboratory apparatu s similar in shape , agitation , and contact time to the chosen in dustrial reactor . The sca lin g rule , whic h ensure s the same specific interfacial area , is to maintain con stant the t ot al power inp ut per unit liquid volume ; that is , P t /V t a n d uf are constant in a mechanically g agitated contactor , and P t gu g is c on st ant in a tubular s p ar ged re actor _ g this scaleup criterion , the heat transfer ( Reith , 1970) . N ote that usm coefficient h w has rou ghly the same values in the model an d in the prototype , .
9 s1nc . e 1•t decreases for 1nc reas1ng . reactor d1ameter . as d - l/ T here fore , on R scaleup of agitated reactors with significant heat tran sfer , installation o f .
•
Carra an d Morb i delli
646
a dditional h e a tin g or coolin g d evic e s i s u s u ally necessary , since the con t en t s increase with the cube and the heat exchange with the s q uare of the Since the in t e r facial area ( an d then r e act or diameter ( S teiff et al . , 198 1) , the reactor p e rfo r m an c e ) inc rease s for increasin g valu es of t h e power in p u t per uni t li q ui d volume , and the o p e ratin g costs a re direc t ly p rop o rt ion al to the p o we r inp ut , the reactor volume selection should be based on th e minimum total c o s t . A useful scal eu p pro ce d ure , b a se d on c on st an t de pletion of the re actant gase o u s concentration , has been proposed by N a gel et al , ( 1 9 7 3 , 1978) . I t allows , t h r ou gh the use of suitable w orkin g diagram s , de finin g the op ti mal reactor vol u me u sin g as a s caleup criterion a co n s t ant value of the q uan tity a /Q g · An alternative scaleup p roc e dur e for b ubble co lu mn s has been p ro p os e d by K a s tan e k et al . ( 1980) . An alternative e xperi m e n t al ap p roach is the simulati on o f in d us t ri al unit s u sin g l ab ora tory - scal e ap p ara tu se s , wit h th e s a m e ga s - li qui d sy ste m ( s o avoidin g de t e r min ation o f kin e tic , s olub ility , and di ffu sivity p ar am et e rs ) It is b a se d on t he ob servation an d a di fferen t mean o f li quid agita tion . tha t if the p urely p hysical mass t ra ns fe r coefficients in the two u ni t s are e qu al , t h e ab so rp t i on rates per uni t interfacial area , also in t h e p re s e n c e of This fact appears q ui te reason ab l e n otic i n g that the reaction , are identical . the p re dic te d effect of chemic al r e ac tion on t h e rat e of ab so r p tion differs lit tle whether the film or surfac e ren e w al - t y p e models are used as a basi s fo r ca lcu la ti o ns , p r o vi de d that th e same value of the p urely physical mass t ran s fe r coefficient k £ is assumed in all mode ls ( Danckwerts , 1 9 7 0 ) . When the ab sorption p roc e s s invo lve s a dilu te s olute in t h e gas phase (i . e . , si gni fican t gas - p hase tran sfer resi stance is p resen t ) , the ga seou s mass transfer coe ffic ie n t k g must be the same in b oth unit s . Finally , for slow reaction s that do n o t completely occur in the liquid film , t h e ratio a / £2 mu st be t h e same in b o t h unit s , to ac c ou n t for the li q uid bulk reaction contribu
tion ,
S o , in general , the three c riteria for simulation ( i . e . , to reproduce
the same ab sorp tion rate per unit in terfacial area ) are iden tical valu e s of
k 2 , k g . and a l E £ in the in dustrial an d laboratory ab sorber .
The p roce d u r e c an t he n be outlined as follows : 1.
M ea sure m e n t or unit
2. 3.
c alcul at i on o f k £ , k g , and a / e- 2 in t he in du s t ri al
Reproduction of t h e same valu es in the laboratory unit
M e a sur e m en t of t he interfacial mass flux in t h e laboratory unit,
which i s t he same i n the in du s trial unit
To accomplish the last two st e p s , it i s necessary to u s e labor at or y - s c a l e
ab sorbers w here the int e r faci al area is known with good accuracy and the
values of th e three p ar amet e rs k t , k g , an d a / e. R. can be varied over a reason T he first one is u s u ally v arie d by me c h ani c al liquid a git at io n , the second one by independent me ch ani c al gas agitation , or sim ply by chan ging t h e relative gas - liquid velocity . T h e control of the third p aram eter r equire s more com p lex app aratus , such as a string o f sph ere s wit h po o ls at t he t o p ( Alper and D anc k w e r t s , 1 97 6 ) or t he modified stirred cell p roposed by Lev e n spi el an d G o d fr e y ( 1 9 7 4 ) . H ow e v e r , few a pp lic ati o n s of t he third crit e rio n are k now n i n the lit erat u re , since usually only fas t re actio n s have be e n e x am p lin e d . In T ab le 35 and Fi g . 2 5 , t he most import an t typ e s of laboratory equipment have been summ ari zed to gether w it h t he re lative values of the ably wide r ange of values .
T A B LE 3 5
I;') Q (ll I t:"' ... ..0 I: ... �
Values of the C haracteri stic Parameters of the Laborabory Models S ketched in Fi g . 2 5
� (I) Q
Model Laminar jet
Parameter C ontact time ( s) Interfacial area , A ( c m 2 ) k
2
1 ( cm • s - )
10
10 5 k g ( mol •cm - 2 -1 -1 s atm ) -1
)
C ylinder
-3
10
-1
-1
S p here 10
2
1
0. 3
10
10
1. 0
100
40
10
1.6 1. 6
A IV 2 (em
Wetted w all column
10
X
10
X
10
-2 -1
3, 6 1.6 1
X X
10 - 3 -2 10
1.6
1
C one
-1
5 X
S trin g of disks
2
X
1 0- 1
10 -
1
80
10 - 3 X
10
-2
5 X 10- 3
1. 1 1
X
10
3
Rotatin g drum
c ell
6
1
X
10
-2
2
X
10
c ell
-4
2
-1
X
10
10
30
4
2
2
80
1 00
80
60
1, 6
1
X X
10
-3
10-
2
1. 6 2. 1
X X
10
-3
10-
2
1. 6 3 . 56
X
10 X
-2
10
-1
2
X
2
X
-
1
-
15
9
9
9
25
15
20
25
20
20
2
80
60
40
60
70
60
X
1 0- 3
5 , 4 X 1 0- 1
100
1250
1 0- 2
10
1
40
;A
stirred
2
3, 6 -2
Stirred
Modifie d
C'l .... Q
2
10 -
X
1
10- 3 2 10 -
10
-3
Cl ""' "'l
0) ""' co
MODH
SCHEME
I
I
r
JET
I
FI G URE 2 5
WE T T E D
CY L I N D E R
I
I,
SPHE R E
-
B J E R L E E T AL
l iou
11972 1
I
R O B E R T S AND DANCK WERTS < 1 96 2 1
S T R I NG
CONE
DI SKS
r
�·
l
-
R E F E R ENCE
CO L U M N
WA L L
[IQU;�. ' ,1i'�a�'- 1] gaS m , �_., �"ll] gas [iQUid 1Q�S, , gaLLP LAMINAR
D AV I D S O N
ANO
CULLE N
( 1 957 1
I.
�
!
I
��-�.::. :-:=..
DANC
K
WERTS
(1
970
I
or
Q
S TE P H E N
AND MORRIS
(1951 1
S chematic representation of some laboratory equip ment s .
I
STIRR E D
CE LL
�-
DANCKWERTS AND A L PE R
(1 9 7 5 )
I
R O TATING DRUM
� �--=:��:� - - · "--'-
DAN C K W E R T S AND
K E N N E DY
( 1 958 1
I
:
I
MOD t r i E D S T I R R E D CEL L
i
=I LEVENSPl £ L
A N D G O DFREY
( 1 9 74 1
(") Q
� �
;:I �
i:::
0
&
� :::;:
649
Gas-Liquid Reactors
relevant parameters .
These , coupled with
T able
31, can lead to the choice
of the most suitable laboratory unit for the simulation of a given industrial
unit (C harpentier and Laurent, 197 4) . Note that this equipment i s usually employed for the determination of kinetic rate constant s , di ffusivities , and solubili ti es ( Lauret et al., 197 5 ) . A classic example of the procedure described above is the simulation of a packed column with a stirred cell for reactions occurrin g in liq.uid film ( Danckwerts an d Gillh am , 1 966; C harpen tier , 1978; and references therein ) . In this case a mutual correspondence can be determined between the agita tor speeds in t he gas and liquid phases i n t h e model u nit ( whic h define kg an d kR,, respectively ) and the gas and liquid superfic i al velocities in the packed column, respectively . The interfacial m ass flow values measured in the laboratory unit can be used to perfor m a differential or an integral simulation of an in dustrial tubular reactor. In th e first case, the material flux is evaluated at the various concen tration values present alon g the column , and reactor be havior is then predicted through a numerical in te gration (Danckwerts and Alp e r 1 975; Laurent an d Charpentier, 197 7 ; and re fere nces therein ) . In the case of m ultiple liquid or gaseous r ea ct ant s or when the reaction con tribution in the liquid bulk is si gnificant, the differential simulation is not feasible, an d only the integral one can be p u rs ue d In this case the column is simulated as a w hole , using the laboratory model unit de scribed above . The global ab sorption rate in the model is measured at the same operating conditions, inc lu din g the outlet an d inlet concentration values , at which the indu strial u nit is operated. The global absorption r at e in the industrial unit is t he n given by the product of the glo b al absorption rate in the labora tory unit and the ratio bet ween the liquid flow rates in the indust ri al and model unit s , respectively (Alper and Danckwerts, 1 976). It is noticeable t hat both the se types of sim ulation procedures exhibit, in all literature app lications (Laurent and C harpentier , 1977), errors that never exceed 20%. ,
.
NOTATION a
interfacial area p er unit reactor volume
a p
interfacial area per unit packed vo lu me
a t
total dry surface area of packin g per unit packed volume
A, A'
gaseous reactant
B
liquid re actan t
a
heat exchan ge surface per unit reactor volume
c
c
liquid reactant
c.
c
d
concentration of the ith c om pon ent
1
p
A
specific heat
,
d , p
d
D
vs
d , d R e
diameter of agitator , columns, reactor
d
characteristic an d nominal packing diameter
n
Sauter mean b ubble diameter, defined by Eq. ( 4 5) di ffu sion coefficient
Carra and Morbidelli
650 E, E* E E R. g' E. 1n EM,
EMT
ET
f
enhancement and reaction factor,
defined
( 28), respectively
by Eqs . (27) and
axial effective mass dispersion in the gas and liq uid phases
r eac t ion factor for an instantaneous reaction
Murphree efficiency for mass
axial effective h eat di sp e rsi on
a nd he a t tra n sfer
fugacit y
r/r0
f
molar flow rate
F
feed
g
gravitational acceleration
G
ga s molar flow rate
h A
agit a tor height off the reactor bot tom
h
coil and wall bed heat transfer coefficients
h c, w
h R.
gas -liquid hea t transfer c oeffi cien t
H
interfacial equilibrium
H
liquid height
H*
-
H
H Ha
ratio, defined by Eq. ( 21a)
dimen sionless Henry's constant , defined by Eq .. (56) molar
enthalpy
Henry's con st ant
Hatta n u mb er , defin ed by Eq. ( 52)
I
solution ionic strength, defined by Eq.
k
reaction rate constant
J
kB
k , kR. g
kT
K
K
g
L
di mens ionles s mass flux
B oltz m ann ' s
gas-side
c on s t ant
and li quid - sid e mass transfer coefficients
the rm al conductivity
equilib rium ratio, defined
by Eq . ( 1)
overall mas s t r an sfer coefficient, defined by Eq. (22) reac t or length
L
liquid molar flow rate
m
reaction order
m.
1
M n n*
N
( 12)
molality of the ith c om pon e nt
H atta n u m ber , defined by Eq. (84)
re a ction order
flow in de x ;
1
=
agitation speed
countercurrent,
-1 = c oc urren t
Gas-Liquid 1
phase) power number
p
NC
component number
NR
reaction number r /r 1
j
Pj
pO
vapor pressure
p
Po
Pt,
pressure
power input in unaerated reactor p
g
total and mechanical power input
Pe
Peclet number , defined by Eq. (6 2)
Q
volumetric flow rate
r
reaction rate
R
universal gas constant
Re
Reynolds number
R. 1
overall rate of consumption of the ith component
s
distance into the liquid phase
s c s g St
wall or coil width
t
time
T
temperature
u
superficial veloci ty
specific gravity Stanton number, defined by Eq. (62)
u
overall heat transfer coefficient
v
volume
v
VR WA
X
x. 1
dimensionless sup erficial ve locity
reactor volume agitator blade width s/o mole fraction of the ith component in the liquid phase
y
axial coordinate
y l
mole fraction of the it h component in the gas phase
z
z.
1
ZD
651
mass flux of the ith component ( positive if leaving the gas
N.
N
Reactors
y/L mole fraction of the ith component in the feed stream concentration diffusion p arameter, defined by Eq. ( 3 6)
652
Carra an d Morb idelli
Greek Letters
ratio of the bulk liquid volume to film volume, defined by Eq. ( 62 ) dimensionless parameter defined b y Eq. ( 67 ) y
activity coefficient
y
dimensionless parameter defined by Eq. ( 56)
c
fil m thickness
t.H a ,t.ii ev
,
t. ii
R
e: ' g e:JI,
e:
e
p
molar heat of absorption, evaporation and reaction (negative if exothermic)
fractional gas and liquid holdup packing void fraction
dimensionless time, defined by Eq. ( 7 1)
Jl
viscosity
vB
stoichiometric coefficient of reactant
B,
positive
\) ..
stoichiometric coefficient of the ith component in the jth reaction
�
DID
lJ
p
A
density
(J
surface tension dimensionless concentration
IP
fugacity coefficien t
4>
Special Operators
gradient operator
"
\/2
Laplacian operator
Subscripts
d
dispersion
f
film
g
gas
i
component index reaction index
k
stage in dex
11.
liquid
s
soli d
w
water vector
653
Gas-Liqutd Reac tors Supersc ripts
f
feed
in
initial
0
ref e rence condition (feed for continuous and initial for batch re actors) molar
*
surface pa rti al molar
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a
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655
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55 ,
1768 ( 1959) .
Doraiswamy , E stimatin g li q uid diffusivity , Ind . Eng . Chern . Fu n dam . , 6 , 77 ( 1 96 7 ) . Reid , R . C . , J. M. P rau snit z , an d T . K. S h e r wood , The Prop erties of Gases and Liq ui ds , Third ed , , McGraw - Hill , N e w Y o rk ( 1 9 7 7 ) . Reiss , L. P . , Cocurrent gas - liquid c ontactin g in p acked colu mn s , I n d . Eng . Chern , P rocess Des . Dev . , 6 , 486 ( 1 96 7 ) . Reith , T . , I n te rfacial area and sc alin g up of ga s - liqui d contactors , B r . Che rn . E n g. , 1 5, 1559 ( 19 7 0) , Riete ma , K . , Science an d t e c h n olo gy of di sp e r se d t wo - p h as e sy s te m s : I and I I , C he rn . E n g . S ci . , 3 7 , 1 1 2 5 ( 1 98 2 ) . R ob e rt s , D . an d P , V. Danck wert s , Kineti c s of C 0 2 ab sorption in al k alin e solution s : I . T ran sien t absorption rate s an d c atalys i s by arsenite, Chern . Eng . Sci . , 1 7 , 961 ( 1 9 62} . Rushton , J . H . and J . J . B imbinet , H ol dup and flooding in air li q uid mixin g , Can , J. Chern . E n g . , 46 , 16 ( 1 96 8 ) . S ahay , B. N . and M. M. S h a rma , Effective interfacial area an d liquid and gas side mas s tran s fer c oefficients in a packed column , Chern . En g , Sci . , 28 , 4 1 ( 1 97 3 ) . Sater , V , E . and P , Le ve n spiel , T wo -p hase flow in p acked be ds , I n d . En g . Chern . Fun dam . , 5, 86 ( 1966} . Satterfield, C . N . , Trickle - b e d reactor s , AIChE J. , 21 , 209 ( 197 5 ) , S c h a ftlein , R . W . a n d T. W . F . Russel , T wo - phase re ac tor design : t an k type reactor s , Ind . En g . Chern . , 60( 5 ) , 12 ( 1968) . Scheibel , E . G . , Li q ui d diffu si vitie s , In d . Eng , Chern . , 46 , 2 0 0 7 ( 1 954) . Sch m it z , R. A . an d N . R , A mund son , An analy sis of ch e mic al reactor Va . Two-phase sy s te m s in p hy s ic al equilibriu m - ! , stability an d control : C h e rn . Eng . Sci . , 18, 26 5 ( 1963a} . R d dy , K . A. an d
L . K.
Gas-Ltquid
663
Reacto rs
Schmit z , R . A . and N. R . Amundson , An analysis of chemical reactor Vb . T wo-p hase gas -liquid an d concentrated stability an d c ontrol : liquid - li quid reactin g sy stems in physical equilibrium - 2 , C hern . Eng . Sci . , 1 8 , 391 ( 19 6 3b ) . Schmit z , R . A . an d N. R . A mun dson , An analysis of chemical reactor stability an d c ontrol : VI . T wo-phase chemical reactin g systems with heat and mass transfer resistances , C hern . En g. Sci . , 1 8, 4 1 5 ( 1 96 3c ) . Schmit z , R . A . an d N . R . A mun dson , An analysis of chemical reactor VI I . T wo-phase chemical reactin g systems with stability and control : fast reaction s , C h ern . En g . S ci . , 1 8 , 447 ( 1 96 3d ) . Schumpe , A . , and W . D . D eck wer , Vergleic h der Fotografischen un d der S ulfit - Oxidation -Methode z ur Phasen grenz fliic hen -Bestimmun g in Blasen saulen , Chern . - I ng . - T ech . , 52 , 468 ( 19 8 0 ) . Schumpe , A . and W . D . Deckwer , D etermination of interfacial areas in non - N ewtonian ga s - liquid flow , A C S Meet . , Atlanta , C a . March 3 0 , ( 1 9 8 1) . Shah , Y. T . , Gas-liquid interface temperature rise in the case of tempera ture depen dent p hy sical , tran sport and reaction properties , C hern . En g . Sci . , 2 7 , 1469 ( 1972) . Shah , Y . T . and J . A . Paraskas , Approximate solutions to single an d t wo phases axial dispersion p roble ms in isothermal tub ular fl o w reactors , Chern . En g . S ci . , 30 , 4 6 5 ( 1 9 7 5 ) . Shah , Y . T . , G . J . Stiegel , and M . M . Sharma , B ackmixin g in gas -liquid reactors , AI C hE J . , 24 , 369 ( 1 97 8 ) . Shah , Y. T. , B . G. Kelkar , S . P . Godbole , and W . D . Deckwer , Design para meters e stimation for bubble column reactors , AI ChE J . , 28 , 3 5 3 ( 1 982) . Shaikh , A . A . an d A . Varma , Gas ab sorption with chemical reaction : the case involvin g a volatile liquid reactant , C hern . En g . Sci . ( 1 9 83a , in press . ) · Shaikh , A . A . an d A . Varma , Studies in ga s -liquid reactors : I . Steady state behavior of i sothermal C S T R s ( 1 983b , s ub mitted for public ation . ) Shaikh , A . A . and A . Varma , Studie s in gas -liq uid reactors : II. On the use of explicit versus implicit enhancement factors . ( 1 98 3c , sub mitted for publication . ) Sharma , M . M . an d P . V. Danck wert s , C hemical methods of measurin g interfacial area and mass tran sfer coefficient s in two-fluid sy stems , B r . C hern . Eng . , 1 5 , 5 2 2 ( 1 970) . Sharma , M . M . an d R . A . Mashelkar , Ab sorption with reaction in bubble colu mn s , Inst . C hern . Eng . Symp . S er . , 28 , 10 ( 1 968) . Sharma , S . , L . A . H offman , an d D . Lus s , Steady state multiplicity of adiabatic gas-liquid reactors : I I . The two con sec utive reaction s case , A I C hE J 22, 3 24 ( 1976 ) . Sherwood , T . K. , R . L . Pi gford , an d C . R . Wilke , Mas s T ransfer, McGraw Hill , New Y ork ( 19 7 5 ) . Shilimkan , R . V. and J . B . S tepanek , Interfacial area in cocurrent gas liquid up ward flow i n tub es o f various size , C hern . Eng . Sci . , 32 , 149 •
•
- I
( 1 977) .
S hilim kan , R . V . and J . B . Stepanek , Mas s tran sfer in coc urrent gas- liquid flow : gas side mass transfer coefficients in upflow , interfacial areas an d ma s s tran sfer coefficient s in gas and liquid in downflow , C hern . En g . S ci . , 33 , 16 7 5 ( 1 9 7 8 ) .
Carrli an d Morbidelli
664
Shulman , H . L . and M . C . Molstad , Gas -bubble column s for gas - liquid con tactin g , Ind . Eng . Chern . , 46( 6) , 1 0 5 8 ( 1 950) . S hulm an , H . L . , C . F . U llrich , N. W ells , and A . Z. P roul x , Holdup for aqueous and nonaqueous systems , AI C hE J . , 1 , 2 5 9 ( 1 9 5 5 ) . Sidem an , S . , 0 . Hortac su , and J . W . Fulton , Mas s transfer in gas - liquid contacting sy stems , Ind . Eng . Chern . , 58 , 3 2 ( 1 966) . Singh , C . P . P . , Y . T . Shah , and N . L . Carr , The effect of gas feed temperature on the steady state multiplicity o f an adiabatic C S T R with a fast p seudo -first order reaction , Chern . En g . J. , 23 , 1 0 1 ( 19 8 2 ) . S kellan d , A. H . P . , Diffusional Mass Transfer , W ile y , N e w York ( 1 974) . Specchia , V . , S . Sicardi , and A . Gianetto , A b sorp t ion in packed towers with cocurrent upward flow , AI ChE J . , 20, 646 ( 1 9 74 ) . S rid h ar , T . , and 0 . E , P otte r , Gas holdup an d b ubble diameters in pre ssuri ze d gas - li q ui d stirred v es s els , In d . E ng . C hern . Fundam . , 1 9 , 21 ( 1 980a ) . S ri dhar , T . an d 0 . E . Potter , I nterfacial areas in gas -liquid s ti rre d ves sels , Chern . Eng . Sci . , 35 , 6 8 3 ( 1980b ) . S teiff , A . , R . Poggemann , and P . M . Weinspach , H eat transfer in agitated vessels with rnultiphase sy ste ms , Ger . C hern . En g . , 4, 30 ( 1 98 1 ) . S tep he ns , E . J . an d G . A . Morris , Determination of liqui d fil m absorption coefficients , Chern . Eng . Pro g . , 4 7, 2 3 2 ( 1 95 1 ) . S tichlmair , J. an d A . Mersmann , Dimen sioning plate columns for absorption an d rectifications , Chem . - I n g . - Tech . , 4 9 , 1 0 6 ( 1 97 7 ) , or I n t Chern . E n g . , 1 8 , 2 2 3 ( 1 978) . S tie gel , G . J. an d Y . T . S hah , Axial dispersion in a r e c t an gu la r bubble column , Can . J . C hern . Eng. , 55, 3 ( 1 97 7a ) . Stiegel , G . J . and Y . T . Shah , B ackmixin g and liquid holdup in a gas liq uid cocurrent upflow packed colu mn , Ind . Eng . Chern . Process Des . Dev . , 1 6 , 37 ( 1 97 7b ) . S t oc kar , U . V. a n d C . R . Wilke , Ri gorous an d short -cut design calculations 1. A ne w computational for ga s absorp tion involving large heat effects : method for packed gas ab sorbers , I n d . Eng . Chern . Process Des. Dev. , 1 6, 88 ( 1 97 7a ) . Stocka r , U. V . an d C . R . Wilke , Rigoro us and short -cut design calculations for gas ab sorption involvin g large h eat effects : 2. Rapid short cut de sign procedure for p acke d gas ab sorbers , Ind . En g . Chern . Process Des . Dev . , 1 6 , 9 4 ( 1 9 7 7b ) . S ullivan , G . A . and R . E . Treyb al , A xial mixin g and gas absorption in a mechanically agitated ab sorption tower , Che rn . Eng . J . , 1 , 3 0 2 ( 1970) . S zeri , A . , Y . T . Shah , an d A . Madgavkar , Axial dispersion in t wo phase co-current flow with fast and in stantaneous reactions , C hern . Eng . S ci . 31 2 2 5 ( 1976) . T amir , A . , an d Y . Taitel , T he effect of the gas side resist ance on absorp ti on with chemical reaction from binary mixtures , C hern . Eng . Sci . , 30 , 1 4 7 7 ( 1 97 5 ) . Tamir , A . , P . V . Danckwert s , an d P. D . Virkar , Penetration model fo r ab sorp ti on with chemical reaction in the p resence of hea t generation , bulk flow an d effects of the gaseou s environ ment , Chern . Eng . S ci , 30, 1 2 43 ( 1 97 5 ) . Tavares Da Silva , The penetration theory for gas absorption accompanie d by a first order c hemic al reaction with a di ff usin g cataly st , C he rn . Eng . S ci . , 29, 2 7 5 ( 197 4 ) . -
.
,
'
.
G as -Li q ui d R eac to rs
665
Teramoto , M., T . N agaya s u , T . M a t s ui , K . Hashimoto , and s. N a gat a , S el ec t ivity of c on sec utive gas - li qui d r eacti on s , J . C he rn . Eng . Jpn . ,
2 , 1 86 ( 1 96 9) .
Te ram oto , M . , K . Hashi moto , an d S . N a gata , Effect of mas s transfer on the selecti vity of ( m , n ) - ( p , q) or d e r c onse cu tive ga s - liq ui d r eacti ons ,
J. Chern . En g . Jpn . , 6 , 5 22 ( 1 973) . T okun a ga , J . , General correlation for n on p ol ar gas solub ilit ie s in aqueous alc o hol s olu tion s , J . Che rn . Eng . Jp n. , 8 , 7 ( 1 97 5) . Treybal , R . E . , Mas s - T ransfer O p e rations, 2n d ed . , Mc Graw - Hill , New Y ork ( 1 96 8) . Uhl , V . W . , Mechanically aided heat transfer , in Mixi ng , Vol . 1 , V . W . U h l and J. B . Gray , e ds . , Acade mic Pre s s , N e w Y ork ( 1 96 6 ) , p. 2 79. Van de V u s se , J. G . , Con secutive react ion s in h et er o gen e ou s systems : I . The effect of mass tr an s fer on sele c tivit y , Chern . E n g . Sci . , 21 , 6 3 1 ( 1 966a ) . Van de Vusse , J . G . , C on sec u ti ve reaction s in h e t er oge ne ous system s : I . Influence of order of r eac tion r at es o n sel ectivity , C hern . Eng . Sci. , 21 , 6 4 5 ( 1966b ) . van K revelen , D . W . , and P . J . H of tij zer , Kinetic s of gas -liquid reaction s . General theory , Reel . Trav . C hi m . , 67 , 56 3 ( 1 9 4 8a ) . van K revelen , D . W . and P . J . Hoftij zer , S ur la solubilite des ga z dan s le s solutions aqueouses , I nt . C ongr . C him . Ind . , 2 1st , B ru s sels , Special N o . , p. 1 6 8 ( 1 9 4 8b ) . van S waaij , W . P . M . , J . C . C h a rp en ti er , an d J. V iller m aux , Residence time distribut ion in t he liq uid p ha s e of trickle flow in packed columns , C hern . E n g . Sci . , 24 , 1 0 8 3 ( 1 96 9 ) . Van ' t Riet , K . , R e vi e w of m ea s urin g methods and result s in nonviscous ga s -liquid mas s tr ans fer in s tirr ed vessels , I n d . Eng . Chern. Process Des . Dev . , 1 8 , 357 ( 1 979) , Verma , S . L . an d G . B . D elan cey , Thermal effects in gas ab sorption , AI C hE J . , 21 , 96 ( 1 9 7 5 ) . Verm eule n , T . , G . M. Willia m s , and G . E . L an glo is , I nterfacial area in li q ui d - li q ui d an d g as - liq ui d a gi tat io n , C h e rn . E n g . P r og . , 51 , 8 5F ( 1 9 5 5) . V i gn es , A . , Diffusion in b inary sol utio ns , I n d . E n g . C h ern . F un dam . , 5 , 1 8 9 ( 1 96 6 ) . Villa ds e n , J. , and M . L . Michelsen, Solution of differen tial eq uation mo de l s by Poly nomial App roximations , Prentice-Hall , Englewood C li ff s , N . J . ( 1978) . W ell ek , R . M . , R . J . B runson , and F . H . Law , En hanc emen t factors for ga s - a bs or ptio n with sec on d - or d er irreversible chemical reaction, C an. J . Chern . E n g . , 56 , 1 8 1 ( 1 9 7 8 ) . W en , C . Y . and L . T . Fan , Mo dels for flow sys tems a n d chemical reac tors , Chemical P roce s s in g an d En gi neerin g Monograph Series, Marcel D ekk er , New York ( 1 975) . Wen, C . Y . , W . S . O ' B rien , and L . T . F an , P ressure d rop through p acked beds operated cocurrently , J . C he rn. Eng . D a ta , 8 , 47 ( 1 96 3 ) . W esselin g h , J . A . an d A . C . Van ' t H o og , Oxidation of aq ue o us sulfite solution s : model reaction for measurements in gas -liquid di s p ersion s , T rans . Inst . C he rn . E ng . , T 69, 48 ( 1970) . W est er terp , K . R . , L. L . van Dier en donc k , and J . A . de K raa , I nterfacial areas in agitated ga s- li q ui d contactors, Chern. En g . S c i . , 1 8 , 1 5 7 ( 196 3 ) .
666
Carra a n d Morbi delli
Wilhem , E . and R . B attino , Thermodynamic functions of solubilities of gase s in liquids at 25°C , Chern . Rev . , 73 , 1 ( 1 9 7 3 ) . Wilk e , C . R . an d P . C han g , C orrelation of diffusion coefficients in dilute solutions , A I C hE J. , 1 , 264 ( 1955) . Wolf , D . an d W . Resnick , Re s i denc e time distribution in real systems , I n d . E n g . Chern . Fun dam . , 2 , 2 8 7 ( 196 3 ) . Woodburn , E . T . , Gas phase axial mixin g at extremely high ir rigation rates in a large packed absorption tower , A I C hE J . , 20, 1 003 ( 1974) . Yeramian , A . A . , J . C . Gottifredi, and J . J . Ronc o , Mas s tran sfer with homo geneous s econd order irreversible reaction . A note for an explicit expression for the reaction factor , Chern . Eng . S ci . , 25, 16 2 2 ( 1 9 7 0 ) . Yoshida , F . and K . Akita , Performance of gas bubble columns : volumetric liquid -p hase mass transfer coefficient and gas holdup , AI C hE J . , 1 1 ,
9 ( 1 965) .
Y oshida , F . , T . Y amane, an d Y . Miyamoto , O xygen absorption into oil- in water emulsion s , I nd . Eng . Chern . Process Des . Dev . , 9, 5 7 0 ( 1 970) . Z ahradnik , J . , F . Kastlfnek , an d J . Kratochvil , H ydrodynamic s an d mass transfer in uniform ly aerated bubble columns reactors , Collect C zech . Chern . C ommun . , 4 7, 262 ( 1 98 2 ) . Zar zycki , R . , S . Le dakowic z , an d M . S tarzak , Simultaneous ab s orp tion of two gase s reactin g between themselve s in a liquid : I I . Approximate solutions , C hern . En g . Sci . , 36, 1 1 3 ( 1 98 1 ) .
10
Gas-Liquid-Solid Reactors YAT I S H T . S H A H
U niversi ty o f Pit tsb urgh , Pit tsburgh , Pennsy lvania
MAN MO H A N S H A R M A
I NTRODUCT I ON :
U niversity o f Bombay , Bom bay , India
C LA S S I F I C A T I O N S :
EXAMPLES
O F I N D U S T R I A L I M PO R T A N C E
Three - phase reactor s , involving gas , liquid , and solid phases , are widely encountered in industrial practice . Several examples are illustrate d in Table 1 . Other example s are given by 0 ster gaard ( 1 9 6 8 ) , S hah ( 1 97 9 ) , Shah et al . ( 1 9 82 ) , and Doraiswamy and Sharm a ( 1 98 3 ) . As can be seen from this t able , solid phase can be either reactant or product or it could be catalyst . In some cases we m ay have four phases , either throu gh t wo immiscible liquid p hases ( either as react ants and /or products ) or t hrou gh two solid p h ases , one reactant or product and ot her cat alyst . Altho u gh , three - phase· systems are found in chemical , polymer , biochemical , and petroleum industries , they are best known in hydroproce ssin g operations . Two m ain c ate gories of three -phase reactors are fixed- bed reactors , which are norm ally used when cat alyst p articles are lar ger than 10- 3 m , and slurry reactors , wherein t he solid phase is in suspension . The most common type o f fixed - bed reactor used in industry is the trickle - bed reac tor , wherein liquid trickles over a fixed bed of cat alyst while gas flows co currently downward as a continuous p hase . Fixed - bed reactors are often operated in the bubble flow re gime , w herein both gas and liquid flow cocur In some processes , such rently upward and gas is the dispersed phase . as for hydrodemet alli z ation of high - met al- content he avy crude , the cat alyst needs to be replaced frequently , and a movin g fixe d - bed reactor i s used . In s uch reactors , a p ulsating flow regime m ay prevail in the bottom part of the reactor . As pointed out by 0stergaard ( 1 9 6 8 ) , in true slurry reactor s , solid particles are very fine ( generally less than 1 o- 3 m ) and gas flow s m ainly in an d out of the reactor . I n m any cases , slurry reactors can b e mechanically agit ated . I f solid p articles are lar ge s uch that t hey form a discrete p hase within the reactor and if the liquid - solid mixt ure flows in and out o f the reactor , t he reactor is o ften called a t hree -p hase fluidize d In the present chapter we concentrate on slurry reactors ( with fine catalyst p article s ) with some allow ance for slurry flow in and out of the reactor . Several type s of fixe d - bed and slurry reactors used in industrial practice are illustrated in Fi g . 1 .
b e d re ac to r .
667
Shah and S harma
668 T AB L E 1 ( a) Examples of Fluid - Fluid - Solid Systems : Sparingly Soluble Absorption of Co
2
Absorption of co
2
Solids
in aqueous suspension of lime and B a ( O H ) 2 in a suspension of B aS
Absorption of lean so
2
in a slurry of CaCo
3
Absorption of lean H s , C O C1 , Cl , etc . , in an aqueous suspension of 2 2 2 lime Absorption of C O in a suspension of lime in the m anufacture o f ( H C00) Ca 2 Solid Particles Insoluble in
( b ) Examples o f Fluid - Fluid - Solid Systems : Medium
The
Absorption of co
2
in a suspension of M gO
Absorption of c o
2
in a s uspension of C aS
Absorption of so
2
in a suspension of C aC0
3
Chlorination of wood p ulp Hydro genation of styrene - b ut adiene rubber latex and othe r polymeric substances s uspended in solvents Hydro genation of sodium p articles suspended in mineral oil to give NaH Chlorination o f
suspended polyethylene ( and PVC ) p articles in w ater
M anufacture of zinc dithionite by reaction between so and zinc particles 2 suspended in w ater (c)
Examples of Fluid -Fluid - Solid ( C atalytic) Reactions
System
C atalyst
T ype of reactor
Hydro genation of uns aturated fats
Raney nickel
SR
Hydro genation of nitrobenzene sulfonic acid ( potassium salt ) , nitrophenols , etc . , in aqueous solutions
Pd , Pt , Ni , etc . , based cat alyst
SR
Hydro genation and hydrocracking o f a-cellulo se ( also po wdered coal ) to liquid and gaseous fuels
Ni
Hydrogenation of nitriles
Ni- based
Hydroisomeri zation of butene - 1 to butene - 2
Undisclosed
T B R : p acked bubble column ( up- flow )
Hydroformylation and hydrogenation of monoole fins , methyl sorb ate , etc .
Nobel met al ( Rh ) - based cat alyst
SR
c atalyst
SR _ ,
catalyst
SR
Gas-Liqui d -So ltd Reactors
T AB LE 1
669
( Continued )
System
C at alys t
Type of reactor
Hydrogenation of xylose ; hydro genolysis of s accharides
Raney Nickel ; Ru
SR
Fischer- Trop sh synthesis : conversion of C O + H 2 to C H 4 , .p araffins , etc .
Ni - M gO
SR
Cu - Zn oxide
( Three -phase ) fluidi zed - bed reactor
Noble met al ( Pd ) o n carbon
SR
Conversion of CO +H 2 to methanol
Reaction bet ween nitrite and H 2 to give hydroxylamine salts ; reaction between N H 4N 0 3 , H aP0 4 , and R 'l.
to �ve hydroxylamine phosphate
Reduction of sodium chlorate in ( diaphragm cell ) aq ueo u s caustic liquor with H 2
Not disclosed
TBR
Hydrogenation of iodine dissolved in aqueous I I
Ru or Rh
SR
Reduction o f uranyl ( V I ) to uranyl ( I V ) in aqueous solutions with H 2
PD based
TBR :
Reduction of UF 5 dissolved in molten LiF - BeF - Th F 2 m e lt with 2
Pt- based catalyst
SR
Deuterium exchan ge between H and water
Pt on carbon
TBR
Air oxidation of ethanol to acetic
Pd- based catalyst , CdO , ZnO , etc .
TBR ; S R
Oxidation of isobutylene glycol to a- hydroxyisobutyric acid
Pt on carbon
SR
Amination of monoethanolamine to ethylene diamine
U ndisclosed
T B R•
Isomerization of propylene oxide to allyl alcohol ; rearran gement of oxirane compounds
Li 3P0 4 s uspended in diphenyl ; Si0 2 - Al 2 0 3
SR
Oxidation of glucose to K- gluconate
Pt on carbon
SR
Hydrogenation of 1 - bromo , 2 , 4- dichloroben zene to 1 , a- dichlorobenzene
Pd on carbon
SR
H2 2
acid ; conversion of primary alcohols to the correspondin g sodium salt of the acid
p acked
col
umn reactor
bubble
Shah and Sharma
6 70
TABLE 1
( Continued )
System Alkylation of ben zene wit h propylene
C at alyst Phosp horic acid
Typ e of reactor TBR TBR
Reaction between isobutylene and aqueous form aldehyde ( a step in the m anufacture of synthetic isoprene ) Reaction between isobutylene and m ethanol to give methyl - tert butyl ether
Ion - exchange resins
Multitubular
Epoxidation of ethylene to ethylene oxide
Silver oxide on silica gel suspended in dibutyl phthalate
SR
Epoxidation of propylene with tert - butyl hydroperoxide , ethyl ben zene hydroperoxide , etc . ; Ep oxidation of styrene and a-ole flns with cumene hydroper oxide
Mo sulfide /oxides
SR
Slurry polymeri zation
Organometallic compounds
SR
Oxidation of S 0 2 to H 2 S 0 4 in w ater ; ab sorp tion of H 2 S from stack gases in alkaline solutions ; osication of aqueous sodium sulfide ( black liquor )
Active carbon
TBR , S R
Air oxidation of dilute formic acid and acetic acid solutions ( treatment of waste stre am s )
Copper oxide - zinc TBR oxide ; ferric oxide
Hydrodesulphuri zation and hydrocrackin g of petroleum fractions
Molybdenum - and tungsten - based cat alyst
TBR
Conversion of p ropylene to isop rop anol
T un gstic oxide based catalyst
TBR
Acrylamide from acrylonitrile
Copper chromite
TBR
Dehydrogenation of secondary alcohols to ketones
Ni or copper chrome
SR
Dehydrogenation o f isoborneol to camphor
C u - Ni -Mn
SR
Hydro genation of polymeric s ub st an ces
Raney Ni ; Pd / C
SR
p acked- bed reactor
_ ,
671
Gas-Liq ui d-So lid Reactors
TABLE 1
( Continued ) C atalyst
System
Type of reactor
Hydrogenation of epoxides of styrene , a-olefins , etc . , in the presence of w ater as a second liquid p hase
Raney Ni
SR
Hydro genation of 2 -methyl nitro benzene in cone . H 2 S0 4 medium to 4- amino - 3- methyl p henol
Pt / C
SR
Absorption of NOx ( at low ppm ) in water
Patented alloy
TBR
Coal liquefaction
Ni - Co - Mo
SR
Organic substit ution reactions via phase transfer catalysis
Quarternary am monium or phosphenium salts bound to
SR
gau ge packing
cross - linked p o ly s t yre ne resins , resinbound linear poly e thers , re sins - sup ported oli. gooxyethl e n e s p oly ( vinyl pyridin e ) bound polymers
,
The nature of the gas - liquid solid reaction m ay often be complex . The product of t he reaction is usually soluble in the liquid phase but in several inst ances the product may be insoluble in the liquid p hase . In slurry and fixed- bed reactors the liquid phase m ay be a reactant or may simply be a solvent . In a special case of t he slurry reactor , the particles m ay simply be adsorbent s . In some uncommon cases , such as demet alli zation of heavier petroleum fr actions , t he catalyst gets laden with the product ( metals in the case of heavier petroleum fractions ) . Solid - bound cat alyst s and react ant s have gained importance in recent year s and here both slurry and fixed - bed reactors are used . There is an incentive to use heterogeni zed - homo ge neous catalysts , as t he p roble m
of the recovery of the cat alyst can e asily be handled using such cat alysts . There are special reasons to immobilize a soluble re act ant ; t he objective m ay be to achieve high selectivity or to avoid product cont amination . In fact , ionic react ant s exchanted on ion- exchan ge resins have also been considered The solid p articles m ay be reactant , which m ay be soluble I in t he literat ure . sparingly soluble /insoluble in the liquid phase , and if insoluble , may exist on the solid reactant as a product layer or as a discrete phase . A variety o f t wo - phase reactions involving a sparingly soluble react ant can be carried out successfully using t he technique of phase trans fer cat alysis .
Here the cat alyst ( P T C ) forms a complex with the reactive
672
S hah and Sharma Ga s
!
i
i
l
Gas
i
L i qu i d
T r i c k l e- b e d
r e a c tor
Pa c k e d
r
bubble
c o lumn rea c tor
Liquid
�
Ga s
�
So l i d c a t a l y s t
(a) FI G U R E 1
re actors .
Mov i n g
t i x e d-b e d r e a c t or
Various types of ( a ) fixe d - bed reactors, and ( b ) slurry -,
G as-Liq uid-Solid
Go
S i n g l e S t a ge
673
Re actors
Multi
Go
Go
M u l t i Cha n n e l
Staged
W i t h Stat i c M i xe r s
G + L,.. S
.. . .. . l . .. . .
.
:;rr·�:o ·:. :. ; ... :/1} .
.
.. : . ._ :
r
.
..
:_ .. . .
. .
.
•. .
:: ::
..
···· · • ······ · ·• <
G+L+ S
Loop R e a c t o r ( External )
Loop Reacto r ( Interna l )
J e t R tt a c t o r
D o wn f l o w B u b b I tt Column
G,
Pi pe L i n e S l u r r y r e a c to r " '
3-Pha se Be d
Go
F l u i d i zed R ea c t o r
( b) FI G U R E 1
( Continue d )
Slurry
Go
Rwa c t o r
Shah and Sharma
674
species from t he aqueous or solid phase and transports it across t he inter face into the or gani c phase , w here the reaction takes place . The main drawbacks o f t he technique involve t he sep aration and recovery of the costly P T C , but these can probably be overcome by employing a solid supported P T C . This type of catalysis was termed trip hase catalysis b y Re gen ( 1 97 5 ) . The special feat ure of triphase c atalysis is that the two re acting species are locate d in two different p hases and the solid cat alyst forms the third phas e . Finally , in recent years , three- phase electrochemical reactions have also gained some import ance . Both slurry and fixed - bed reactors have been considered where gas -liquid - solid and liquid - liquid - solid systems are In t hi s c h ap t e r encountered , and some new desi gns have been su ggested . we attempt to cover briefly the m ain features of gas - liquid- solid re actor s , with the principal emphasis on conventional fixed and slurry- bed reactors . H Y D R O D Y N AM I C S , M I X I N G ,
A N D T R A N S PO R T
C H AR AC T E R I ST I C S O F F I X E D - B E D R E AC T O R S
Three - p hase cat alytic fixed - bed reactors are generally operated with T he most p re downw ard or up ward cocurrent flow of liquid and gas . dominant flow regime encou n t ere d in co mm e rci al operations is the trickle flow regime , in w hich gas and liquid flow cocurrently downward ov e r a fixed bed of cat alyst . As shown by S at t er fie ld ( 1 97 5 ) , in m any hydro processing operations , flow conditions are such that the react or is operated at the boundary of trickle flow and puls ating flow re gimes . T he pulsating flow regime can also be achieved when gas an d liquid flow cocurrently up w ard at high velocities . Some commercial reactors are operated under a bubble flow re gime wherein gas and li q uid flow cocurrently upw ard through a fixed bed of cat alyst . Finally , hydrodemet allization of he avy crude is carried out in moving fixed - be d reactors with cocurrent downw ard gas liquid flow d ue to r ap i d agin g of the catalyst . The flow re gime in such reactor can change from trickle flow at the top to pulsating flow at t he bot tom . T he di scu s sion presented below is therefore restricted to ( a) trickle flow , ( b ) pulsatin g flow , ( c ) p acked bubble flow , and ( d ) moving fixed - bed reactors .
F low Regi me Bounda r i es In all work on two -phase downflow over p acked beds before 1 9 7 5 , flow re gimes were considered to be functions only of gas and liquid flow rates . The flow re gime boundaries have been obtained in terms of superficial m as s flow rates o f air and w ater . The coordinate s for t he flow maps , showing flow regime boundaries , have been either G 0 versus G L ( Weekm an and Myers , 1 96 4 , 1 965 ; Bei mes c h and Ke ssler 1 97 1 ; S ato et al . , 1 97 3b ; Satter field , 1 97 5 ; C hou et al . , 1 9 7 7 ; C harpentier et al . 1 96 9 ; Hoffm ann , 1 97 8 ; Sicardi e t al . , 1 9 7 9 ) o r G L versus G o ( Turpin and Huntington , 1 9 67 ) . S ato et al . ( 1 97 3b ) noted that the boundary between continuous and pulse gas flow moves to low liquid flow rates and returns to hi gh liquid flow rates as t he p article si ze is increased . Charpentier and Favier ( 197 5 ) introduced the e ffects of density , viscosity , and sur face tension and the foaming nature of liquid of the flow re gime boundaries and presente d a flow map usi n g B ake r ' s coordi nat e s for gas - liquid flow in an empty tube . Their results are shown in Fi g . 2 .
675
Gas-Li q ui d -Soli d R eactors
10 ..
.c
f l ow
....... .... !!)•
Wa t e r - a i r
S p h e r i c a l ca ta l y s t
Cyc l o hexane - a i r
S p her i ca l c a t a l y s t
Wa t e r -a i r
C y l i n d r i c a l ca t a l y st
C y c l o h exa n e - n i t r o g e n
Tr i c k l e
!!)
S p ra y f l ow
0-1
Key
Pa c k i n g
Sys tem flow
Cy l i n d r i c a l cata l y s t
G a s o l i n e -c a r b o n d i o x i d e
Cy l i n d r i ca l c a t a l y s t
Gaso l l n e - n i t rogen
Cy l i n d r i ca l cata l y s t
G a so l i n e - h e l i u m
C y l i n d r i ca l ca t a l yst
Pe t r o l eu m e t h e r - n i t ro g en
Cy l i n d r i ca l cata l y s t
Pet ro l e u m e t h e r- c a r bon- d i o x i de Cy l i n d r i c a l cata l y s t
0
1
() c
2 t.
2 •
2
•
2 £ 2
2
�
(a)
P u l se d f low
.. I!) 1 0
...... _, I!)
K e r o s e ne -a i r
S ph e r i c al c a t a l yst
De su l fu r i z e d g a s o i l - c a r bo n d i oxi de
Cy l i n d r i c a l ca t a l ys t
De s u l f u r i z e d g a s o i l - a i r
C y l i n d r i c a l ca t a l yst
Desu l fu r i z e d
C y l i n d r i c a l ca ta l yst
gas o i l-hel i u m
N o n de s u lfu r i ze d dioxide
gas
o i l - ca r bo n Cy l i ndr i ca l ca t a l yst
Nonde s u l f u r l ze d g a s o i l - a i r
C y l i n d r i c a l ca ta l yst
Nondes u l tu r l zed g a s o i l-hel i um Cy l i n d r i c a l c ata l yst K e r o s e n e -a ! r
Cy l i n d r i ca l ca t a l y st
Kerosene- n i t rogen
C y l i nd r i c a l c a taly s t
).. :
[ A r� G
-
Pwat
·
Pa i r -
, y- :
awa t
O" L
[
�L
J.l wa t
(P;:tnO·U
- t
(b) FI G U R E liq ui ds .
2
Flow - regime boun daries for ( a ) foamin g an d ( b ) nonfoamin g ( A fter C harpen tier an d Favier ,
Key
Pa c k I n g
Syste m
1 97 5 . )
2
2
2
2
2
2
(J +
l( A.
*
II J.
2 0 2 •
676
S hah and Sharma
GA S CO N T I N UO U S
(B l u r r i n g f l o w , Sp ray F l o w ) Uppe r Bou n d a ry F o r Pu l s i n g
-J I
0
I
tl 0 Ill
TWO CON T I N UO U S PH A S E S
u
·-
.. -
Gl
::J
f
0 >
-
I n e rt i a + G ra v i t y fo rces
I n t e r fa c e + V i s c o u s
R e s i s t a nces
1 000
=
G A S CO N T I N U O U S
( B lu r r i n g f l o w , S p r a y f l ow )
U p p e r b o u n d a ry � f..:: ....�� o r p u l s i ng
__
-J 0 I
I Ill
FOA M I N G
cB
AN D
u
PU L S I N G
·-
Low e r b o
2
u�
fo r pul s in g
FO A M I N G 1 �-L--L-J��-L--����-��-+� ·1
FIGURE
3
Flow
maps :
two - p hase
downfl.ow t hroug h p acked beds :
non foam i n g liq uid s ; ( bottom ) foaming liq uid s .
( F rom S hah
et al . ,
( top) 1981 . )
677
Gas-Liqui d-Soli d Reactors
Talmor ( 1 97 7 ) p re s en te d flo w m ap s ( F i g . 3) for foa min g and nonfoamin g liqui d s u sin g a sup e r fi cial volumetric gas -to -liquid ratio versus a ratio of Clements an d H alfac re ( 1 97 9 ) sh o wed drivin g force to resistance force . that the data of C hou et al . ( 1 9 7 7 ) for aqueous solution of alcohol , an d their own data fo r air -isopropanol aqueous solu tion ( 0 to 48 wt %) , were correlated by th e flow map s of Talmor ( 1 9 7 7 ) but not by the flow m ap s of C h arp en tie r an d Favier ( 1 97 5) , T he se w or ker s also concluded that surface tension exert s the dominant influence over flow pattern transitions an d foam in g in the packed bed . A Talmor flow map has been used by Patil and Sh arma ( 1 98 1 ) for a tube packed with Pall rin gs an d multifilament wire gau ge pac kin g s . T he flow regime boundaries for coc urre n t up fl o w over a fixed bed in terms of gas an d li q uid mass flow r ate s have been re p or t e d by s everal workers ( Ford , 1 9 6 0 ; T ur pin and Huntin gton , 1 96 7 ; S p ec c hi a et al . , 1 97 4 ; S at o et al . , 1 9 74 ; Pitt sbur gh Energy Research Center , 1 976) . The results of the latter study are shown in Fi g . 4. S aada ( 1 9 7 5 ) obtained data ch a rac teri zin g the flow re gi me for s mall p ar ticle s ( d < 0 . 2 e m ) . He also modifie d l> the coordinates of flow map s to include the effects of densitie s and viscos itie s of gas an d liquid and bed poros ity . In S aa da ' s work , flow is divided into two r e gi me s . At relati vely hi gh gas flow rates , the pores of the pack in g are con sidere d to be traversed by both p h a se s together , whereas at low
"i'.,.
Go J - c o n t l nuou s or s p � f l ow , ,
'E
N
u
01
""
,
"'
I
I
c,!)
CD ..
� u 0
... >
Ill Ill 0
E
"' 0 c,!)
0· 1
S u r g i n g f l ow
0·1
Bubbl e f l o w 1 ·0
L l q u l d mo s s v e l o c i t y ,Gt
--- P i t t s bu r g h e n e rg y r e sear c h c e n t e r ( 1 9 7 6 ) dp - 1 · 9 e m x 1·9 e m cy l i n d r i c a l p a r t i c l e
shape
10
_ ,
( g c: m 2 .-t ) e t al - ( 1 9 7 4 ) dp - 1 • 2 2 c m s p h e r i c a l
- - - - Sato
pa r t i c l e s h a p e
FI G U RE 4 F lo w re gi me b o un darie s of water -air upflow t h rou gh a p ac k ed ( F rom Pitt sb urgh Ener gy Research C enter , 1 9 7 6 . )
bed .
Shah and Sharma
678
gas flow rates each p ha se is c onsidered to flow seperately through the The former is referred to as a two -p hase pore -flow regime and the
p or e s .
latter as a single-p hase p o re-flow regime . This flow map is shown in Fig. 5. Generali zed flow map s such as those of Talmor ( 1 97 7 ) , for downflow
ove r p acked beds , are not available for the cocurrent upflow case . P ressu re D rop
Reported data on pressure drop in two-phase flow over a fixed bed cover various flow re gimes and a large ran ge of de sign and operatin g variables . Howe ver , most of these studies are for air - water sy ste m s . Other gas liqui d sy ste m s have also been in ve stigated ( Clement s an d Schmidt , 19 76 ,
198 0b ; Larkins et al . , 1 961 ; S pe cchia and B aldi, 1 977) , but more data on hydrocarbon an d foamin g sy stems are neede d . The data b a se of Larkin
Reg i m e o f t wo - p h a s e pore f l ow
N
o,_____,
\ Re g i m e of \ s i n g l e-phase
\ P O r e f l ow \
1',... U»
�0
w Cl
FI G U RE 5
\
\ \ \ \ \ \ \ \ \ \ \ \ \
C orrelation of flow tran sition by Saada
_ ,
( 1975) .
679
Gas-Li q uid-Soli d Reactors
et al . 's ( 1 96 1 ) c o rrela tion is relati ve ly large and close to the conditions in a c o m m e rc ial reactor and th e refore is more commonly used . In t h e correlation of pressure - drop data for two-phase flow in pac ked bed s , basically two approaches have been used , In one approach , the vari ables are lumped into vario us dimensionle ss groups , such as Reynolds num b e rs for gas an d liquid, Weber numbers , an d so on , and the pressure drop or friction factor is expressed in term s of these dimensionless groups , The use of Tallmadge or Lockhart -Martinelli paramete rs is inclu de d in this ap proach ( Turpin and H u n tin gton 1 967 ; S pecchia and B aldi , 1 97 7) , T he other approach is to correlate the two -phase pressure drop with the pres sure drops of energy losse s for individual phases in sin gle p ha se flows
,
-
unde r identical condition s .
Ford ( 1 96 0 ) m ea s ured pressure drop in packed beds for c oc urrent up re gim e s ba se d on the flow t ha t takes place in side the pores : n a mely , single-p hase pore flow and two-p hase p ore flow , He p r e se n ted correlati on s havi n g th e form
flo w an d identified the flow
( � pz )
1
--
g
--
�
pL
The value s
LG
a( R e
=
L
X ) ( Re
G
)
y
(- ) ].J
1J
z
L
G
( 1)
dep en d on the t yp e of flow r egi me . Different for the two re gimes . Saada ( 1 97 5 ) modified t h e Ford c orrelation an d rep lac ed term by dp / de (i . e . , the ratio of the particle and column diameters) , A large number of data points obtained by t h e Pittsburgh E n e r gy Re search C enter ( PERC ) ( 1 976) in hi ghly p ulse d and spray flow regimes have value s of
a,
of
a,
x , y , an d z
x , y , an d z have accordingly been sugge st ed
the ll L ilJ G
been correlated by Tallmad ge ' s correlation ( 1 970) , Sato et al . ( 1974) , on the other han d , correlated their data with the Lockhard - Martinelli type of relation in all t h r ee flow regimes (i . e . , bubble , pulsed , and sp ray ) , The more widely used correlation of Larkins et al , ( 1 9 6 1 ) , for cocurrent downflow is base d on the secon d approach , It c o rre lates the overall t wo phase energy loss for t he gas and liquid passing through the r eac t o r to the two individual single -phase ene r gy l o s se s as follows :
( 2)
,
-
-
where o g is the ga s phase energy los s o L the liqui d p hase energy lo ss O L G the two - p hase energy loss , an d X = ( c L/ o 0 ) 1/ 2 , T he c on st an ts x 1 an d K 2 have t he values 0 . 4 1 6 and 0 . 6 6 , r esp ecti ve ly Larkin s et al . ( 1 96 1) defined the two-phase p r es s ure g r adie n t (. � P n Z ) a s follows :
.
,
_ ,
LG
( !� )
LG
=
o LG
-
P
M
( 3)
where
( 4)
Shah and S harma
680
S everal other workers have use d Eq . ( 3 ) directly ( Abbot et al . , 1 967) or in modified form ( Reiss , 1 96 7 ; Sato et al . , 1 97 3a) for corre l a tion of their data . C harpen tier et al . ( 1 969) derived p ressure- drop correlations by c omparing t wo-phase pres sure loss l! PL G with that for sin gle - phase flow , in terms of the energy-based dimen sionless group s . Clements an d Sch midt ( 1 9 7 6 , 1980a , b ) combined the foregoin g two app roaches and gave the cor relation for p ressure drop in two -phase cocurrent down flow in terms of Reynolds n umbers of ga s an d liqui d phases , gas -p hase Weber number, and pressure drop for the gas p hase alone . C l ement s ( 1 97 8 ) also gave a theo retical justification for this approach . Theoretical correlations for pres s ure drop under tric kle flow con dition s have been proposed by H utton and Leung ( 1 974) and S pecchia and B aldi ( 1 97 7 ) . I t is important to note that in most of the studies reported, Reynolds numbers for liquid and gas phases are considered to be the controllin g dimen sionless group s . However , those s tu dies that use systems other than air - water reco gn i ze d the role of surface tension on the pres sure drop and used either an empirical correction factor ( S pecchia and B aldi , 1 9 7 7 ) or the dimensionless Weber number ( C le ments an d Schmidt , 1 97 6 , 1 98 0a , b ) . More data ove r a larger range of Weber numbers are needed . Phase Holdups
Liquid Hol dup Total liquid holdup in a fixe d -bed column is the sum of the operatin g an d static holdup s . S tatic holdup , the amount of liquid in the bed after the liquid inlet is shutt off and the column is allowed to drain , represents the liquid retained in t he pore volume of the catalyst and its pac kin g . Operat ing holdup , the liquid external to the catalyst particles , depends on the liquid and gas flow rates , their properties , and the nature of catalyst packin g . 2 S tatic holdup is often correlated by Eotvos number , Eo ( = p L gd / crL , p where dp is the nominal particle diameter and g the gravitational accelera tion ) . The correlation of Van S waaij et al . ( 19 6 9) in dicate s that smaller particle diameter an d fluid density and larger surface tension give larger static li q ui d holdup . The correlation al so indicates that a porous material give s a larger static liquid holdup than does a nonporous material . Basically , two types of correlation s for the dynamic or total liquid holdup are reported in the literature . Some inve stigators have correlated the liquid hol dup directly with the liquid velocity an d fluid properties by either dimensional or dimensionless relation s . In more recent in ve stigation s , liquid holdup i s correlated t o the Lockhart - Martinelli parameter l! PL /l! P a - ' or its equivalent . For coc u rren t down flow , the dimensional correla tion s ( S atterfield and Way , 1 97 2 ; Jesser and E lgin , 1 943) are very few . Most of the correlations are in the followin g dimen sionless form ( C lement s and Schmidt , 1 9 8 0a ; Otake an d Okada , 1 96 3 ; Mic hell an d Furze r , 1 97 2 ; Buchanan , 1 967 ; G elbe , 1 96 8 ; Mohunta and Laddha , 1 96 5 ; Van S waaij et al . , 1 96 9 ; Specchia and B aldi , 1 97 7 ; Davidson et al . , 1959) :
(5) The values of
a,
13 ,
y , and
n obtaine d by different workers differ .
6 81
Gas -Liq uid -Solid Reac tors
,
Many correlation s u sin g the Lockhart -Martinelli parameter or its equivalent are al s o reported in the literature ( Abbott et al . , 196 7 ; Larkins et al . , 1961 ; S a t o et al . , 1 973a ; B akos and C h arp en tie r 1 970 ; Midoux et al . 1976) . T he correlation s of Midoux et al . ( 1976) , in terms of energy based dimen sionle ss group s , for n onfoam in g and foaming liquids are recommen ded for u s e in t he case of hydrocarbon liquid s . For cocurrent upflow a large amount of e xp e ri m en t al data were ob tain e d by the P itt s b u r gh Energy Research C enter ( 1 976 ) . The average liqui d holdup was found to be a very st r on g function of ac tual gas velocity and a relatively mild function of liquid velocity . Various dimensionless c orrelation s The modified correlation of are av ailab l e for cocurrent upflow reactors . Ford ( 1 960) ha s been given by Saada ( 1 9 7 5 ) in the for m
( )13 Re
EJ. = a
Re
�
( 6)
where a an d 13 d e p en d on the flow re gime . S tie gel and Shah ( 1 97 7b ) p ro pose d a correlation which is in a more gen e ral form :
( 7)
S ato et al . ( 1974) corr elat e d their data in term s of the L ockhart Martinelli parameter . A th eor etic al model of the liquid h ol dup for coc ur re nt upflow through a p acked column was propose d by Hutton and L eun g ( 1 974) . With the help of t h eir model , these workers s h o we d that cocurrent up flow operation gi ve s higher liquid holdup than cocurrent dow nflow operation under the same gas and li qui d flow con dition s , a fact demon strated experi mentally by Turpin and H un tin gton ( 1 967) , and shown in Fi g . 6. The ran ge of variables an d ph y sic al p ro per ti e s e xamined in the litera ture ca n be roughly stated as 0 . 3 Re < 3 0 0 0 , 0. 4 mm 50 mm ,
<
Tu r p i n a n d
Q.
H u n t i n g ton
L
< � <
(1 967 )
�
0 0·5
.c:.
FIG U R E 6 Li q u i d hol dup for cocurrent upflow versus cocurrent downflow through a packed b e d .
682
S hah and Sharma
2. 6 < as d < 6.0 2, and 14 3 20 for particle s hap e s consis ti n g of G aL p Ra s c hi g nn gs . B erl saddles , spheres , and i rr e gular granules . Both foam
<
<
in g and nonfoaming liquids have been examined. Although there are signifi cant di s crep ancie s in the pre di ctions of various correlations , quali tatively they all i n dic ate that the li quid holdup under trickle flow conditions in creases with liqui d velocity and is es s entially independent of gas flow rate . A lthou gh not completely cle ar , an increase in particle size appears to de crease the liquid holdup . An increase in G alile e number for the liquid also de creas e s the li quid holdup . These statements are generally ve ri fie d by three theoretical models : those of Hutton and Lu e n g ( 1 974) , Reynier and C harp en ti er ( 1 97 1 ) , and Clements ( 1 97 8 ) . It is im por tant to note that none of the co nt rolli n g dimensionle ss groups ( R e L , R ea, GaL , and as dp ) in liquid holdup correlations include a s urface tension term (i.e. , the liquid hol du p is considered to be independent of surface tension ) . This conclu sion needs to be verified in view of the observed dep en de nc e of flo w re gimes and pressure drop on surface tension .
Holdup
Gas
,
In a pac k ed be d column , the gas holdup can be e v alu ate d if the total liquid Ho w ev er in cases where th e gas holdup is small, it is more desirable to evaluate it directly rather th an through the difference . For cocurrent upflow in a bubble flow r e gi me Ford ( 1 9 6 0 ) s u gge s t e d t h e re lat ion -
holdup and void volume of the column are known .
e:
=
G
2 1. 2
'
( )0.2( ) Re L
--
Re
G
ll
0 . 24
L
( 8)
--
ll
,
G
whereas
Achwal and S tep anek ( 1 976) p rop os e d two correlations for the one ba s e d on the homogeneous flow model and t he other based on force b alance on the column. bubble flow regime :
A xial
Li q ui d
D i s persi on
Phase
While in most commercial trickle- bed operations , the liquid p ha s e is believed
t o move in p lu g flow , in p ilot s c ale operations , t he liq ui d phase can be backmixed . As s ho w n in Fi g . 7 , t h e axial disper sion in the liquid phase under trickle flow conditions can be an order of magnitude hi gh e r than t hat in the sin gle phase . For trickle flow , the correlation of H oc h ma n and -
E ffron ( 1 96 9 ) ,
Pe
L
=
0 . 04 2Re
�· 5
( 9)
where Re L = UoL P Ldp /J.l L ( l - e:) , is recommended. For the cocu rrent up flow packed bubble column , the correlati on shown in Fi g . 8 is re commende d No similar correlation for the pulsating flow regime is presently av ai lable
.
.
Gas Phase
Just a s for th e liquid phase , the axial dispersion in the gas phase under trickle flo w conditions is hi gher than that u nde r sin gle p has e flow conditions . The correlati on of Ho chman an d Effron ( 1969) , -
683
Gas-Li q ui d -Soli d Reactors
,
0.... � ....J
FI GURE 7 Hofm ann ,
P e an d Re L relations for single - p hase and trickle 1 97 7 . )
1 10
.... II
Q. .......
....J
(&)
1 0°
a-,
0c:P
�a'�6 o
0 0
10
%0
'6 6 a'...
- - - - H e l l m a n n and
...
ll
1 0-1
10
0
ReL
3
t 4', • ' '
rote
}
f€G
,
Shoh (1977o, �
Rec t a n g u la r co l u m n 10
0
0
Hofmo nn(lm>'
Mo x . gas f low rote
M i n . ga s flow
0 0
( A fter
/!�� ·
A , . ,O Data of Sti egel and ll
o4 6
flow .
3 ·3 dp
cyl i nd rical '
c o lu m n
10'
F I G U R E 8 H eilm ann - H ofmann correlation for b ackmixing in a coc urrent up
flow p acked bubble colum n .
( A fter S tie gel and S hah ,
1 97 7a . )
684
Shah and Sharma
( 10)
where Pe G = U Gdp /D G , Re G = G G dp / � g ( l - £) , and ReL = G Ldp / � L (l - E) for trickle flow conditions , is recommended . No similar correlations for the bubble flow and pulsating flow regimes are presently available . Mas s Tra n s fer
Gas -Liqui d
Just as for liquid holdup , the correlations for the mass transfer coefficient ( K LaL ) are reported in two ways . Some investigators correlate KLaL with liquid and gas velocities in dimensional ( Gianetto et al. , 1 9 70 ; Goto and Smith , 197 5 ; Goto et al . , 19 7 6 ) or dimensionless (Goto and Smith , 197 5 ) forms , whereas others ( C harpentier , 1 976 ; Reiss , 1967 ; Satterfield , 1 9 7 5 ) have presented energy correlations . The dimensional correlations assume that K LaL U�U� , where the value of r and s depend on the types of packings ( Shah , 19 79). For very small gas and liquid velocities Goto and Smith ( 1 97 5 ) and Goto et al . ( 1 97 6) gave more accurate predictions . C harpentier ( 1976 ) , Reiss ( 1967), Gian netto et al . ( 1970) , and Specchia et al . ( 1974) gave relations between KtaL , aL , and KL and the energy parameter . Reiss ( 1 9 6 7 ) gave a correlation K LaL 0 . 173 [ ( I'. P / f.. Z ) LG UoL J 0 . 5 , which was subsequently modified by Satterfield ( 19 7 5) and later by C harpentier ( 1976) to account for the liquid properties . Specchia et al . ( 1 97 4 ) gave energy correlations for both up flow and downflow , as shown in Figs . 9 to 1 1 , and indicated that for the same value of energy parameters , up flow gives a better transfer coefficient and interfacial area than does downflow . Better values of KL are obtained for slower liquid velocities in upflow compared to downflow , presumably due to an increase in circulation inside ex:
=
FI GURE
9
Energy correlations for k L .
(After Specchia et al. ,
197 4.)
685
Gas -Liq ui d-Soli d R eactors
( AP). AZ
--
FI G U R E
10
L
G
2
3
� ( k g 1 m m-2 ) -
a,
Energy correlations for a /a • L 8
( A fter Specchia et al . , 1 9 7 4 . )
the liquid drops , caused among other things , by the greater slip velocity Specchia et al. ( 197 4 ) also showed that the upflow values of K L a L are , on the average , two times that of down flow values in pulsed and spray flow re gi m es because the gravitational force leads to higher liquid holdup and pressure drop . between the liquid and gas phases .
(
-
FI G U R E
11
AP
AZ
)
LG
UOL
(
kgt
-3
m m
Energy correlations for k
-1 )
s
L
aL .
( A fter Specchia et al . , 197 4 . )
686
S ha h and Sharma
da t a or correlations for conservative app ro xi m a tion , KLaL can be taken to be 0 . 15 s- 1 in th e bubble flow regime . He also s u g ge s t ed that the gas -liquid interfacial area varies wit h about 0 . 5 power of th e superficial gas ve locity regardless of t h e p acking si ze an d type , column diameter , and s uperficial liqui d velocity . Morsi and C harpen tier ( 1 9 8 1 ) indicated that the mass t ran sfe r pr op e rtie s of hydrocarbon system s are s ubs t an ti al l y different from those of air - w ater system s . Also , the interfacial area ( aL ) for hydro carbon syste m s is much le ss than that for air - water system s . The effective interfacial area an d the li q ui d - side mass trans fer coefficient for trickle-bed reactors p acke d with a variety o f packings an d for organic liquids have also been measured by M ahajani an d Sharma ( 197 9 , 1 9 8 0 ) . Fu ture work must be directed toward system s other than ai r w at e r and under high - temperature and hi g h - p r e s s ure c on di tion s . Charpentier ( 1 97 6 ) su gges t e d that if no reliable
a given p ackin g are available , as a
,
-
Liqui d-Soli d Under trickle flow condition s , t h e
liquid- solid ma s s transfer coefficient can of Goto et al . ( 1 9 7 5 ) ( see Fig . 1 2) or that of Dharwadkar and S y lve s t e r ( 1 97 7 ) , In the p ul s a ti n g flow re gime the correlation of Lemay et al . ( 1 97 5 ) is r ec om m en de d . For the p acked bubble flow colu mn , the correlation of Kirillov and N asam anyan ( 1 97 6) is recom mended for large p article s ( dp > 3 mm ) and t h e correlati on pres e nte d in Fi g . 12 i s recommended for s m all particle s ( dp < 3 m m ) .
be estimated using t h e
correlation
10
data
Da ta w e r e tak e n for
di" = 0 · 24 1 0 · 1 08
�2
em , em
O • O S 4 1 c: m
s
20
0· 3 ��������----_.--�--�--���--� l i qu i d
Rey n o l d s n u m be r , Re i.
FI G U RE 1 2 Jn versus Re correlation for cocurrent upflow . L 1 97 5 . )
et al . ,
( A fter Goto
687
Gas -Li q ui d -Solid R eac tors
H ea t
T ra n sfe r
The data reported on heat transfer in trickle - bed reactors are sparse . Weekman and Myers ( 1 96 5 ) m e as ured wall - to-bed heat transfer coefficients in a cocu rrent air- water downward flow through packed colum n . Their re sults are discussed by Shah ( 197 9) . A m or e recent work on heat transfer in trickle-bed reactors is that by S pe cchia and Baldi ( 1 97 9) . In the low interaction regime , t hey prop o s e d a correlation between N usselt number and Prandtl number and a modified li qui d R eynol ds number . In the hi gh - inter action r egi m e , the heat transfer coefficient was found to be independent of ga s flow rate and packing si ze and s hap e . S pecch ai and Baldi ( 1 97 9 ) also obtaine d some results for the effective thermal cond uc tivity of the bed in both low- and hig h - in te rac tion regimes . Sca l e u p C o n s i d e ra t i on s
Experiments on the laboratory or pilot scale are invariably conducted with 25 - m m - I . D . a n d 0 . 5- to 2 - m -lon g reactors at about the s a me li quid hourly space velocity as in the industrial reactor . However , the latter reactor Thus the superficial liquid velocity in may have a height of 20 to 25 m . the p ilot reactor may be merely about 1 0 % of the industrial uni t an d one may have different flow regimes in the two cases ; if this is the cas e , the reactor scaleup may be seriously jeopardized . We have already emphasi zed the i mpo rt ance of different flow regimes , in p a rti cular the di ff e ren t func tional dep e n de nce of various de si gn variables , on op erati n g pa ra met e r s . In fact , if the proper precautions are not t aken , one may get a situation where external mass transfer resistance may become i mp ort an t or dominan t in the pilot reactor but may be uni m port ant in the industrial reactor . There may also be different residence-time distributions for gas and liquid p has es in t h e two cases � One may often benefit by replacing ex p e n sive catalyst with inexpensive support material of e qui v ale n t dimension when s oli d - li qui d or gas-liquid film resistance is important . G en e ra l Rema r k s
Very little is kn o w n on the hydro dyn ami c , mixing , and transport character istics of m ovin g fixed -bed reactors . It is suggested that such reactors should be divided on the basis of the prevailing flow regime and approximate correlations for each flow regime should then be applied .
H Y D R O DY N AM I C ,
M I X I N G , A N D T R A N S PO R T
C H A R A C T E R I S T I C S O F S L U R RY R E A C T O R S
_ ,
As s hown in Fig . 1 , a v arie ty of s lurry reactors ar e used in industrial practice . Some of these are : 1. 2.
Gas sparged (including vibra tin g ) reac t or M ec ha nic ally agit a t ed (including gas inducin g) reac t or
3. 5.
Three- phase fluidi zed reactor T rans p or t reactor Multitubular reactor
6.
Loop reactor
4.
Shah and Sharma
688
7. 8. 9.
Plate colum n re ac t or Pipe reactor Spouted bed re ac tor
Some novel distillation column reactors for MTBE and for some hydro genation reactions , s uch as the conversion of isophorone to trimethylcyclo h exan one and trimethylcyclohexanol , have also been report e d ( Smith and H u ddle s ton , 1 9 8 2 ; Schmitt , 1 960) . In the following we di s c us s primarily reactor types 1 and 2 . Some de t ails are , however , applicable to other types of reactors , and these are n ot e d . Flow Regi me B o u n d a r i e s
In gas sp ar ge d , vertical b ubble columns , flow regime maps have bee n described by a number of researchers and are discussed by Sh�h et al . ( 1 98 2 ) . A s an ap p roxi m at e evaluation o f the flow regime in a vertical slurry reactor , Fig . 13 can be us e d . It s hould be , noted , however , that the typ e of sparger used , liquid velocity , physicoc he mical p roperties of the liquid , and the p resence of solid may also affect the flow regi me boundaries. Walli s ( 1 962 , 1 969 ) has sug geste d the use of flow re gime charts b as e d on Kara ( 1 981 ) and Kelkar ( 1 98 2 ) h av e recently the drift flux approach . shown that this approach can be app lie d to three-phase slurry re actor sys tems . The flow regime for gas-liquid flow in hori zontal pipe has b e en di s c us s ed at great length by Govier a n d Aziz ( 1 97 2 ) . N o similar flow maps for th ree - p has e flow are p resently available . 0 ·1 5
0
tO
Slu g flow
I
C h u • n - T u , b u l e nt Fl o w H eto•o g e n e o u s
I ��
!jo·"OS� Homogeneous ( B u b b l y ) F l ow
FI G URE 13 Approximate dep en denc y of flow regime on gas velocity and column diameter ( water and dilute aqueous solutions ) . ( F rom Shah et al . , 1982 . )
689
Gas -Liq ui d-Solt d R eac tors Bubble Dy n a m i cs
Bubble size and its distribution and b ubble rise velocity have a direct bear the performance of the bubble columns . A brief review of this sub ject has been given by Shah et al . ( 1 981, 1 982) . When gas is spar ged by single orifices or perforated or sintered plates , the followin g correlation of Akita and Y os hi da ( 1 97 4 ) c an be used as first app roxi mation for the estima tion of average bubbl e size : ing on
Relation ( 1 1 ) has not been tested for three-phase systems , vessel the corre la tion of Calderbank ( 1 967) ,
may be
us e d as a first approximation .
For a stirred
Here constants C and n d ep end on P /Vn represen ts the energy
the stirrer type and the type of liquid phase .
unit volum e o f di sp ersion .
The bubble si ze di s t r ib ution
dis sip ation rate per
is often giv en in terms of large b ubb les and small bubbles . The definition of large and small bubbles has been based on the dynamic bubble disengagement t echni qu e described by Sriram and Mann ( 1 97 7) , S c hu m p e ( 1 9 8 1 ) , and Vermeer and Krishna ( 1 98 1 ) , amon g others . B ase d on this technique , the fraction o f gas th r o ughp u t carried by large bub b les as a function of gas velocity obtained by B einhauer ( 197 1) is s hown in Fig . 14 . For slurry reactors , such a plot should depend on the
FI GURE 14 Fraction of from B einhauer , 1 9 7 1 . )
volum e
throughput of
large bubbles .
( Calculated
Shah and S harma
690
p hy si c al pr op erti es of liquid , solids parti cl e si ze , nature , and concentration , a relati on s hip not yet been de v elop ed i n t he li t er at ure . Kara ( 1 98 1 ) and Kelkar ( 1 9 8 2 ) found that Zuber and Findlay' s modifica
tion of drift flux theory p r e di ct ed m eani ng ful bubble size , bubble rise velocity , and radial distribution of gas holdup in three -phase slurry re The bubble si ze evaluation for the gas -liquid flow in hori zontal actors . pip e s is given by Govier and Azi z ( 1 97 2 ) , A similar evaluation of the mec h anic ally a git at e d contactor is given by Joshi et al . ( 1 9 8 2 ) . M i ni m u m Ga s a n d Li qui d V e l ociti es Req u i re d fo r F l uidi zati on
In a t hre e - p h as e system the particles form the fluidized phase , with the liquid as the fluidi zing medium . The gas p as si n g t h rou gh t he s ys te m im p ar t s t h e requisite ener gy to the li qui d to keep the p arti cl e s s uspended . The bed is flui di zed wh en the s up erficial velocity p a s t t he p article s is greater than t heir s e t t lin g velocity . K ato (1 963), Roy et al . ( 1 964), I mafuku et al . ( 1 968) , N arayanan et al . ( 1 9 6 9 ) , an d B e go vich and W at s on ( 1 9 7 8 ) studie d the minimum gas velocity required to suspend particles in a stagnant liquid medium . These works are revi e we d by Shah (1 9 7 9 ) and S h ah et al , ( 1 982 ). B e govic h and Watson ( 1 97 8 ) m easu red the minimum gas and li q ui d vel ocitie s required to fluidi ze various types of solids as a fun ct ion of p ar ticl e si ze , particle de n si t y , and liquid vi sco si t y . No e ffec t of the initial bed hei gh t or column diam eter was found . I n m echanically a gi t ate d contactors , some minimum impeller speed is required to kee p the par ticle s in suspen sion . For li qui d - s oli d s u sp e n si on s (in the absence of gas ) , Zwietering ( 1 958) p rop os e d s uch a correlation . Arbiter et al . ( 1 9 76) and S u b b ar ao an d Taneja ( 1979) have shown that the value of m ini m u m speed is hi gher in the presence of gas then in the absence of gas . Smith an d coworkers (Smith , 1 9 80 ) have st udied the effect of c avi ty formation , behind the i m peller bl a de s , o n t h e impeller sp eeds re qui r e d for s u s pe ns ion of solid particles in gas -li q ui d - s oli d system s . More discus sion of this subject i s given by Joshi e t al . ( 1 98 2 ) . Phase H ol d u p
The available correlations for t h e gas hold u p in t w o - and three-phase bub ble columns are reviewed by S hah et al . ( 1 98 2 ) . For slurry reactors , the gas holdup dep en d s m ain ly on t h e s up erficial gas v eloci ty and i s often very s en sitive to the p hysical p rop er ti es of the li quid . T he dependen ce of the gas holdup on gas velocity is gen er ally of the fonn
( 13)
where 0 . 7 n 1 . 2 for b ubbly flo w and 0 . 4 n < 0 . 7 for ch urn tu rb u lent flow . A l ar ge n um ber of co rre lation s for gas h ol dup have been reported in the literature , however , the large scatter in the re p orte d data does not allow a sin gle correlation . Some of the i m po rtan t correlations are reported by Shah et al . ( 1 982 ). The la r ge s catte r ( S ch ii gerl et al . , 1 9 7 7 ; B ach an d Pilhofer , 1 978) is due mainly to t he extreme s en sitivity of the holdup to
<
<
<
Gas -Liquid-So li d
691
Reac tors
the m aterial system and to the trace impurities , which is not well under stood . The easily available physical properties , such as density , viscosity , and surface tension , are not nece ssarily sufficient to expalin the scatter observe d . The effect of gas properties on the gas holdup is s hown to be small ( Hikita et al . , 1 9 8 0 ) . Kblbel et al . ( 1 9 6 1 ) and Deckwer e t al . ( 1 980a ) have also shown a very small effect o f p ressure on the gas holdup . Jeffrey and A criv o s ( 1 97 6 ) found that for particles as small as a few ( < 1 0 ) micrometers , s uspensions behave macroscopicall y as non - N ewtonian fluids . T herefore- gas holdup values i nv olvin g non- N ewtonian liquids might be useful in analyzin g gas - slurry systems . Gas holdup data in non- Newtonian media have been rep orte d by Nishikawa et al . ( 1 97 7 ) , B uchholz et al . ( 1 97 8) , N ak ano h an d Yoshida ( 1 98 0 ) , and Schumpe et al. ( 1 98 1 ) . Gas holdup values hi gher than pre dicted by the correlation of Akita and Yoshida ( 1 97 3) w er e obtained . The effect of spar ger s was found to be insi gnificant in t he slug flow re gime Hikita et al . ( 1 98 0 ) com p ared their data with t ho s e predicted from dif ferent correlations and attributed the di sa greem en t between theories partly to the differences in the design of gas sparges . They conclu ded that at low gas velocities , the single- noz zle gas sparger gives lower values of gas hold up than does a multino z zle or a porous plate sparger . Zahradnik and Kastanek ( 1 97 9) carried out expe ri me n t s with sieve plates of different hole diameters an d observed that the critical gas velocity corresponding to the onset of stable perform anc e of a plate depends on the plate hole diameter . N umerous data for different kinds of spargers and various liquid media ( synthetic fermentation media ) have been reported by Schiigerl et al . ( 1 97 7 ) Oels e t al . ( 1 97 8 ) . Hills ( 1 974) and U e y ama e t al . ( 1 980) reported pro nounced radial holdup p rofiles , and Kobayashi et al . ( 1 9 7 0 ) propose d an empirical correlation to evaluate such a radial distribution . Shah ( 1 97 9 ) has critically reviewed the published information on individu al phase holdups and has recommended the correlations of Kim et al ( 1 97 2 , 1 97 5 ) and H ovm an d an d Davidson ( 1 96 8 ) . More re cen t ly Dhanuka and Step anek ( 1 97 8 ) m e as ur ed and theoretically predicted the individual p hase holdup s in t he 0 . 0 5- m - I . D . column . Be govich and Watson ( 1 97 8 ) determined the individual phase holdups from 0 . 0 7 6- and 0 . 1 5 2- m - I . D . colu mns and developed cor relati on s which cover their own data a n d thos e reported by 0stergaard ( 1 9 65 ) , 0stergaard and T hi ese n ( 1 966) , 0 s t ergaard and Michel sen ( 1 9 6 8 ) , Michelsen and 0stergaard ( 197 0 ) , Rigby and Capes ( 1 97 0 ) , Efremov an d Vakhrusev ( 1 97 0 ) , Dakshinam urthy et al . ( 19 7 1 ) , B hatia and Epstein ( 1 97 4 ) , and Kim et al . ( 1 97 5 ) . Kato et al . ( 1 97 2 ) and Kato and Nishiwaki. ( 1 97 2 ) , Ying e t al . ( 1 98 0a) , and Vasalos et al . ( 1 9 8 0) i nv e s ti gated the effect of solids concentration and concluded t h at an in c re as e in solids concentration generally decreases the gas holdup but that the effect becomes insignificant at hi gh -gas velocitie s ( >0 . 1 m /s ) . Yin g et al . ( 1 980a ) also ap p li ed the correlation of Akita and Y oshida ( 1 97 3) to their data and con cluded that the correlation of Akita and Y oshida is equally a d e q uat e for the t hree phas e system . Kojima and Asano ( 1 98 0 ) meas ured the fractional gas holdup and average bubble diam eter from 0 . 0 5 5 - and 0 . 0 9 5- m - I . D . contactors . T he particle si ze was varied in the ran ge of 1 0 5 to 2 000 ]JJJI . The liquid viscosity was varied from 1 x 10- 3 to 1 8 . 8 x 1 0- 3 Pa •s . The results were empirically corre l at ed Liquid fluidized beds of small particles have been found to contract u p on t h e introduction of gas ( Stewart and D avi dso n 1 964 ; 0 stergaard and T hi es an , 1966 ; Ki m et al . , 1 97 2 ) , w hile the reverse trend occurs with large particle s . The contraction can be explained by considering liquid wakes .
.
,
-
.
,
Shah
692
and S harma
behind the gas bubbles . The liquid in the wake moved at a much faster rate than the continuous liquid phase . As a res ult , the average velocity of the bulk liquid phase decreases and the bed contracts , causing an in crease in the solid holdup . The large particles can cause a bubble break up by virtue of their inertia and hence result in an expansion in fluidi zed beds . Bhatia et al . ( 197 2 ) and Arm stron g et al . ( 1 97 6a ) studied the effect of solids wettability on the behavior of three -phase fluidi zed beds of small and large particles , respectively . They have also qualitatively described the role of wakes and bubbles in such bed . Kim et al . ( 1 9 7 5 ) have re ported the existence of two distinct types of three -phase fluidization , which may be termed as bub b l e coalescing and bubble disin tegrating. The former occurs when the particles are smaller than a critical size and the latter oc curs when they are larger . Axial solid concentration has been studied by Cova ( 1 9 6 6 ) , I mafuku et al ( 1 9 6 8 ) , Farkas and Leblond ( 1 96 9 ) , N arayanan et al . ( 1 9 69 ) , Yaman aka et al. ( 1 97 0 ) , and G ovi nd rao ( 1 97 5 ) . Cova ( 1 966) reported that high gas and liquid velocities and high viscosities tend to give more uniform solid distribution . Imafuku et al . ( 1 968) observed that the critical gas velocity for complete suspen s ion depends mostly on the liquid flow near the gas distributor , w hile Govindrao ( 1 97 5 ) pointed out that particle diameter and bed volume have a stron g influence on the axial distribution of solids . As a rule of thumb , in a cocurrent column , p articles less than 1 0 0 JJm in diameter can form a p seudohomogeneous slurry , while particles greater than that will result in some axial solids distribution . In m ultistage columns Schiigerl et al . ( 1 9 7 7 ) observed higher values of gas holdup than in a single- stage column . They also noted a strong effect of the distributor design in m ultistage columns for a noncoalescing medium . However , in an air-water medium , Zahradnik et al . ( 197 4) and Zahradnik and Kastanek ( 1 97 4) observed very little effect of the distributor design and the liquid velocity in multistage columns . Freedman and Davidson ( 1 969) and B otton et al . ( 1 97 8 ) carried out experiments with ·draught tubes and observed insi gni ficant effects of the draught tube on the holdup values . Wieland ( 1 9 7 8 ) reported that in a bubble column with an external loop , the holdup values are comparable to the values in a single-stage bubble column . As a rule of thumb , it can be concluded that the effect o f internals on the gas holdup is ne gligible , an d any correlation that fairly represents the values of holdup in a single - stage bubble column can be used to calculate the values in the presence of internals . As a first approximation the same conclusion should apply to three -p hase system s . The phase holdups in mechanically agitated contactor is discus sed by J o s hi et al . ( 1 98 2 ) . .
A x i a l Di spe rsi on
Liquid Phase
The effects of suspended solid particles on liquid-phase backmixing have been studied by Schiigerl ( 1 967) , lmafuku et al . ( 1 968) , Vail et al . ( 1 968) , Michelsen and 0stergaard ( 1 97 0 ) , Kato et al . ( 1 97 2 ) , ()stergaard ( 1 97 8 ) , and El- Temtamy et al . ( 19 7 9 ) . Shah et al . ( 1 97 8 ) have critically reviewed the literature and outlined the present state of the art . The correlation reported by Kato et al ( 1 97 2 ) is suitable for cases where the particle si ze is relatively small . The experimental observations may be summari zed as follow s : .
693
Gas -Li q u i d -Soli d R eac to rs
parti cle s are very small and /or the difference between the densities of li qui d and solid is small an d /or the solid loading is relatively small , the extent of li qui d - p has e axial mixin g in three phase systems is practically the same as that for gas -liquid systems . The liquid -phase axial dispersion coefficient varies with colum n diameter and gas velocity , and under turbulent conditions it is practicall y i n d ep e n de nt of liquid v elocity and particle diameter . When t he p arti cle si ze i s large an d the density difference i s hi gh , t he di sp e rs ion coefficient depends on the particle si ze an d t he s up er ficial gas and liquid velocities (Michelsen and 0 stergaard , ( 1 97 0 ) . W hen t he
1.
2.
Joshi ( 1 980 ) has shown that the observation of Michelsen and 0st er ga ar d ( 1 97 0 ) c an be explained if t he average li q ui d circulation velocity ( Vc ) i s selected as the c orre latin g p aramete r . When the rate o f energy in p ut is higher than that dissipated at the ga s - li qui d and li quid - soli d interfaces , recirculation of liquid occurs . The data r ep ort ed by K ato et al . ( 1 9 7 2 ) , Michelsen and 0s te r gaa r d ( 1 970) , 0s ter ga ar d ( 1 97 8 ) and Vail et al . ( 1 96 8 ) were analyzed by Joshi ( 1 98 0 ) . From an accuracy paint of view , only those data were analy zed for which at leas t 20% of the in put energy is dis sip ated in the liquid motion . B ased on the analysis , the follo win g equa tion with a s t an d a r d deviation of 1 6 % was ob taine d :
( 14) Riquarts ( 1 9 8 1 ) made us e of another theoretical approach . In an alogy to the well - known Pe = 2 relation for fixed beds , Ri q uarts ass umed t hat this relation is also valid for bubble columns . The li qui d - p h as e dispersion coefficient i s . given by ( 15) and intro d ucin g app rop riate expressions for the bubble ri se v e locit y an d bubble diameter , he arri v ed at the equation
( � )1/8 Fr
Pe
L
=
14. 7
Re
( 16 )
L
whe re
( 17)
Equation ( 1 6 ) de sc ribe s meas ured data with ac c ur acy similar to that of other correlations r ep ort e d in the literature . The effect o f s olid p arti cle s on the liquid disp er sion coefficient i s not clearly understood due to the lack of enough experimental evidence . At hi gh solid c on c en t r ati on s , Y in g et al . ( 1980b ) observe d lar ge discr ep anci e s between actual an d theoretical values of di s p ersion c oe ffi cien t s p re di c te d
Shah and Sharma
694
with the he lp o f di ffe ren t exi s ti n g corr elations . T hey also observed the ve loci ty o n di s p e r sion coefficients . The observations are s upp o r te d by K ar a et al . ( 1 98 2 ) and Kelkar ( 1 9 8 2 ) , who noted that t he di s p er s ion coefficients are lower at high liquid velocities than those ob served in th e absence of s o lids .
effect of li qui d
Soli d Phase
al. ( 1 97 2 ) studied soli d - p has e backmixin g under a wid e ran ge of conditions . T hey found that for s m all particle sizes in s mall - diameter col umns , the axial dis persion c oe ffi ci en t for solids is the same as that for the li quid . For relatively large p article si zes ( up to 0 . 177 mm ) the dispe r sion coefficient depends on th e p artic le Reynolds number ( Shah , 1 97 9 ) . More experimental investi gations p ert ai ni ng to s oli d - p has e bac k mixin g in the presence of large p arti cl es ( > 0 . 2 m m ) are neede d . Kato et
Gas Phase
The ga s -p h a se backmixing in g a s - liq ui d - soli d sys tem s have been measured S ch ii ge rl ( 1 967 ) , 0stergaard and Michelsen ( 1 96 8 ) , and Michelsen and 0stergaar d ( 1 9 7 0 ) . Schiigerl ( 1 967 ) reported that at low liquid velocities , the gas - p has e Peclet number increases with the gas rate , but at high liquid veloci tie s , the Peclet number shows a m aximum with respect to gas rate . From S chii ge rl ' s data , the effect of liquid velocity on the gas - p has e Peclet number is unclear , al t hou gh at low velocities , the ga s - p h as e Peclet number by
app e ars to decrease with increase in liquid velocity . B ased on s t u di es of many nonaqueous and aqueous systems , in 0 . 1- and 0 . 1 5 - m -I . D . bu b ble columns , Mangartz and Pilh o fer ( 1 9 8 0 ) reported the follo win g equation for the gas - p hase disp e rsion c o e f fi ci en t :
( 18) is in m 2 fs , de is in meters , Uoo in m /s , and E(J i s fractional hold up in t he presence of solids . Field and Davidson ( 1 980 ) , who c on ducte d meas urements on gas - p hase dispersion in a large- diameter ( 3 2 times that used by Pilhofer et al . ( 1 97 8 ) ] c olum n , sli ghtly m odi fi ed E q . ( 18 ) and proposed the empirical correlation
where D o
gas
D
G
=
1 . 33 5 6 . 4d c
(u ) O
G
3 56 .
( 19)
e:0
Al t ho u gh the pre dictions of Eqs .
( 1 8 ) and ( 19) may differ up to 50 % , both lack of sufficient literature data . More
correlations are recommended due to
dat a
on three -phase
sys t e m s are needed .
Ma s s T ra n s fe r
Gas -Li q ui d G a s - li quid mas s
tr a ns fe r i n a b ubble column h as been revie w ed e xtensively by Shah et al . ( 1 98 2 ) . In t hre e - p hase bubble c ol u m n reactors , K a can
L L
Gas -Li q ui d-Soli d Reac tors
695
be affected by t he presence of solids . Investigations of various a uthors ( Voyer and Miller , 1 968 ; Sles ser et al . , 1 9 6 8 ; S harma and Mashelkar , 1 96 8 ; K ato et al . , 1 97 3 ; J uv e kar and Sharma , 1 97 3 ; Si t ti g , 1 97 7 ; Zlokarnik , 1 97 9) indicate that the de gree of influence of sus pe n ded p artic le s on k L a L de p e n d s on the p arti cl e concentration , the particle si ze , the liquid solid d en sit y difference , the geo m e trical si zes , and the op e r ati n g conditions of the r eac to r (i . e . , gas and liquid ve locitie s ) . N guyen - Tien and D ec k we r ( 1 98 1 ) showed that at high liquid velocities ( U o L = 0 . 093 m /s ) an d low gas ve loci tie s , t he kLa L values are sli ght ly hi gh er in the pre s e n c e of solids . A small increase in kL aL was also reported by Si t ti g ( 1 97 7 ) for the solids ( v arious plastic par ticles ) concentrations below 15 wt % and p arti cle si zes between 50 and 200 j..lm . Wit h risin g gas velocities and dec reasi n g liquid velocitie s , the p ar tic le di s t rib ution b e c om e s incr ea sin gly nonuniform an d the kL aL value s are lower than the ones without the solids . The influence of the solid co n c ent r ation on k L a L dep en ds lar g ely on the liquid and gas velociti es . Kato et al . ( 1 97 3) have also shown that for higher solid concentration a steep de c reas e i n kL aL is found , which is c au se d by a decrease in aL . It has been shown by Dh a n uka and Stepanek ( 1 9 8 0 ) that with an i n c re as e in parti cl e size , kLaL de c reas e s because of a decrease in a . For the typical operating conditions p rev aili n g in Fischer- T rop s ch s y nth esi s in t he slurry phase , Zaidi et al . ( 1 9 7 9 ) and Deckwer et al . ( 1 98 0 ) have shown that the pre se n c e of solid p ar tic le s ( diameter less than 5 0 J..l m , con ce n t rations of s oli d s < 16 w t %) has a negligible effect on kL a L · Weiland ( 1 9 7 8 ) investigated m a s s transfer in a b u bble column with an external loop . He r ep o rt e d an a g r ee m en t of kLaL data for w at e r - 0 2 with the data reported b y Y oshida and Akita ( 1 965 ) an d Lin et al . ( 19 7 6 ) . In t he absence of other data , thes e d at a should be used as a first approxima tion for three -p hase systems . Voi gt et al . ( 1 9 7 9 , 1 9 8 0 ) an d H ec ht et al . ( 1 98 0 ) have studied mas s transfer in a · m ultistage bubble column for electrolyte solutions and hi ghly viscous New t oni a n an d n on - Newt oni an ( carb oxyme t hyl cellulose an d poly acrylate ) s olutio n s . Th ey have recommended us e of m ul tis t a ge colum n s for a coalescence p ro moti n g medium , alth ou gh for non - Newtonian s olu ti on s bet ter mas s trans fer can be ac hi eve d in si n gle - sta ge bubble columns ( Voigt et al . , 1 9 8 0 ) . H y dr o gena tion of dissolved elastomeric materials may show pronounced non - N ewtonian be h avi or ( Falk , 1 9 7 6 ) . Mas s t ran s fer in such systems has been reviewed by K u m ar et al . ( 1 9 7 8 ) and M a s h elkar and Chavan ( 1 97 3 ) . Fan et al . ( 1 97 6) and Wang and Fan ( 1 97 8 ) st u die s oxygen tran s fer in a bubble column packed wit h Koch m otion le ss mixers . They reported a significant in c r e a s e ( as hi gh as twofol d ) in kLaL an d de pendence of k L a L values with Koch mixers t han the one s with sieve trays in bu bble columns . Orazen and Erickson ( 1 97 9 ) reported significantly larger k L a L in the two- s ta ge airlift tower for a gas ve locity between 0. 2 0 No di ffe ren ce in kL aL was apparent at superficial gas vel and 0 . 4 5 m / s . ocities from 0 . 1 2 to 0 . 2 0 m /s . Lin et al . ( 1976) have shown that in tower cycling fermenters for volati le substrates , higher mass tran s fe r r at e s can be ac hi ev ed . F ujie et al . ( 1 98 0 ) observed hi gher volu metric oxygen trans fer coefficients in a downflow bubble column compare d to those in conven tional upflow system s . The av e ra ge liquid- side volu m e t ric mas s transfer coefficient depends on th e average bubble diameter and their rise velocity . The bubble dyn a m ic s in a cocurrent- upflow , three -phase fluidi zed-bed system have been studied by Massimilla e t al . ( 1 95 9 , 1 9 6 1 ) , �st e r gaard ( 1 96 6 ) , and L ee ( 1 965 ) .
6 96
Shah and Sharma
Massimilla et al . ( 1 96 1 ) foun d that the bubble si ze and the rate of bubble co ale sce n ce d ecre a sed wit h i n creasin g fluidi zation intensity . T hese result s were in agreement with a su b s e quent study of 0 ster gaard ( 1 96 6 ) . Adlin g ton and T homp son ( 1 965) found t hat t h e gas - li q ui d interfacial area de creased with de creasin g bed porosity an d w as les s sensitive to ch an ge s in p ar ticle si ze . However , for lar ge partic le s and large bubble s , Lee ( 1 96 5) showed t h at th e ga s - li q ui d interfacial area increased with increasin g dis tance a way from the gas di stributor . T he a d osrp tion rate dep ends on t he gas -liquid interfacial area , the gas residence time , and the g a s - li qui d mass transfer coefficient . It is the first factor that has an imp or t an t an d complex effect on the m ass t r an s fer rate in th ree - p hase fluidi zed - bed sy s t ems . The study of 0 s te r g aard an d Sucho z eb r s ki ( 1 96 8 ) s howe d that t h e volumetric m ass t ransfer coefficient { i . e . , KLaL ) i n crease d with inc re a sin g gas flow rat e b ut w as unin fluenced by variations in the li q ui d ve locity . Particle size w as observed to have a pron ou n ced e ffect on the adsorption rate . 0 ster gaar d and cow or ker s also showed that an increase in t he superfici al liqui d velo cit y had no effect on the ab sorption rate in beds of 6 - m m p artic les and in solid - free bubble col u mn s . In b e d s of 1 - m m particle s , however , an increas e in li q ui d velocity c ause d a m a r ke d increase in the volumetric mass transfer coeffi cient . T hese res ult s agree d well with the fact that both the g a s holdup and b ubble size , in be ds of 6 - m m particles an d in the so li d - free b ubb le colu mn , w e re in de pen d e nt of liquid ve locity . whereas in beds of 1 - mm p article s ( � st ergaard . 1 96 6 , 1 97 1b ) an increase in t he liquid velocity caus ed a marked r e d uction in b ub ble si ze and an i n c re as e in gas hol dup . T he rate of b u bble coales cence w as hi gher at low li q uid velocities than at hi gh liquid velocities in beds of 1 - mm p artic le s ( 0 s t er gaar d , 1 96 6 ) . An increase in the gas velocity increase s the vol umet ric mass transfer coefficient . Shah ( 1 97 9 ) reviewed the studie s on mass t ran s fer across the gas - liquid interface . The best u p - t o - dat e work is reported by J oos t en et al . ( 1 97 7 ) . T hey showe d that in the ab sence of solids , the vo lu me t ric mass tran s fer coefficient can be well corr elate d to the total power , di s sip at e d by stirrer an d gas , per unit volume . However , the correlation with power di ssip ated by the stirrer alone w as poor . Joos t en et al . ( 1 97 7 ) also fou n d th a t for a gi ven stirrer po wer i nput an d s uperfici al ga s veloci t y , the vo l um e t ric mass transfer coefficient first incre ased with the volume o f solids ad d ed . Al tho u gh this is in q u alit ative agreement wit h the o b se r v ation s of Slesser et al . ( 1 96 8 ) and C han drase kar an and Sharma ( 1 97 7 ) , u n like these authors , Joosten et al . { 1 97 7 ) , fou n d the incre ase to be rather small ( bet ween 10 and 2 0 %) . F u r t hermo re , as more solids were added , the volumetric mass transfer coefficient remained constant at first and then started to decline at a s p e ci fic concentration t hat depended on so lid t yp e an d p a rti cle si ze . Joosten et al . { 1 97 7 ) exp l ai n ed the data based on the increase in t he ap p arent vi scosity of the s lurry by the addi ti on of soli d s . Mass transfer in mechanically a gi tate d contactor is discussed further in Joshi et al . ( 1 982) . Liq uid- So li d In three - ph a se bubble column slurry reactors , the mass t r an s fer from bulk li q ui d to t h e solid surface can play an important role in th e overall
app arent reaction rate ( C houdhari and R am c han d ran , 1 980 ) . Li q ui d - solid m as s t r an s fer is based on the slip velocity { boundary layer ) between t wo phases , and in ge n e r al , leads to s uch expressions as
G as- Liq uid- So l i d R eac tors
697 ( 20)
Frosslin g ( 1 938) an d Ran z and M ar s hal l ( 1 95 2 ) showed that for large values of Re , a = 0 . 6 , m = 1 / 3 , and n = 1 / 2 . S hah ( 1 979) has discussed in detail the recommen dations of various investigators ( Friedlander , 1 95 7 , 1 96 1 ; H arriot , 1 96 2 ; Levich , 1 96 2 ; B rian and Hales , 196 9) for evaluatin g the particle Reynolds number
u d Re = _!_E.
( 2 1)
\)
where u r is the relative velocity between two phases . In the case of bub ble columns where the solid p articles are su spended , this velocity is diffi cult to determine . Kolmogoroff' s theory of isotropic turbulence avoids this problem , as discussed by Shah et al . ( 1 982 ) . Only a few experimental studies ( K a ma wura and S asano , 1 96 5 ; Sano et al . , 1 97 4 ; S an ger and Deckwer , 1 98 1 ) have been carried out in three phase vertical b ubble columns . T he se investi gations and the proposed correlations usin g Kolmogoroff' s characteristic parameters are outlined by Shah et al . ( 1 98 2 ) . The data of S ano et al . ( 1 9 7 4 ) covers a range of S chmidt numbers from 200 to 1400 . S an ger an d Deckwer ( 1 98 1 ) have pro vided t he data for Schmidt numbers ran gin g from 1 37 to 5 0 , 000 . The measured data d evi at e d at low Reynolds n um b e r s because the as s u mp tio n o f isotropic turbulence is not justified , as disc ussed by Levins and Glaston bury ( 1 97 2 ) . Figure 1 5 i ll u s t r ate s that the data of the se three in ve stiga tors are consistent . T he difference in three curves is due p artly to
1 00
10
0 · 1 �'--------:!-----=':�---�,..---� 0 ·1 1 10 1 00 1 00 0 3 1 /3 4
( e dp
/vL )
F I G U RE 15 Correlations of liquid - solid mass transfer coefficients in bubble columns . ( From S hah et al . , 1 9 82 . )
698
Shah and S harma
uncertaintie s in the estimation of diffusivitie s . T he fi gure also demon strates the usefulness an d applicability of the theory of isotropic t urb u lence . This is advantageous , as the energy dis sip ation can be easily measured in b ubble column s . T he liquid - solid mass tran s fer in bubble columns with and without internals have also been measured by Sharma and coworkers ( Sharm a , 1 98 2 ) usin g C u - dic hromate system . The equation by San ger an d Deckwer ( 1 9 8 1) is recom men ded for the calculation of liquid - solid mas s tran s fer since it covers a wide r an ge o f experiment al parameters . More recommendations for mechanically agitated cont actors are given by Joshi et al . ( 1 9 8 2 ) . H ea t T r a n s fe r
Heat transfer from t h e reactor wall and inserted rods and coils t o the vertical flowin g gas in liquid dispersion has been the subject of many in vesti gations , and suitable correlation s have been reported ( K o1bel et al . , 1 958 , 1 960 ; Mulle r , 1 958 ; Martins , 1 95 9 ; Kast , 1 96 3 ; Shaykhutdinov et al. , 1 97 1 ; B urkel , 1 97 2 ; Mersmann , 1 97 5 ; H art , 1 9 7 6 ; Steiff and Weinsp ac h , 1 97 8 ; D eckwer , 1 980 ; Deckwer et al . , 1 980 ; Joshi et al . , 1 98 0 ; K ato et al . , A summary of the p ublishe d data collected on bubble columns and 1 98 1) . stirred ve ssels is presented by Steiff and Weinspach ( 1 9 7 8 ) . Muller ( 1 95 8 ) , K olbe! et al . ( 1 9 5 8 , 1 96 0 ) , Viswanathan et al . ( 1 96 5 ) , Armstron g et al . ( 1 97 6 b ) , and Zaidi et al . ( 1 9 7 9 ) measured heat transfer coefficients from t h e inserted heat source to the bed ( gas - liquid - solid dispersion ) or between the column w all and the bed . Analysis of the re ported data indicate the followin g behavior of gas - liquid- solid system s : 1.
2. 3.
T he values of the heat trans fer coefficient h w are at least 1 0 to 1 0 0 times higher then that for sin gle - p hase ( gas or liquid ) flow , 0 . 9 to 1 . 3 time s higher than for bubble column s , and 1 . 2 to 2 5 times hi gher than for gas - solid fluidi zed beds . T he values of h w are practic ally indepen dent o f t he column diam eter and dimensions o f heat transfer element . In general , h w increases with a n i ncre as e i n particle diameter up to 3 mm . However , for p article si zes lar ger than 3 m m , h w be comes essentially indepen dent of the particle diameter ( Armstron g et al . 1 97 6b ) . When the solid concentration e:s is increased from zero , at the be ginnin g h w increases w ith an increase in e:g , reache s a m aximum value at a certain value of e: s , and then falls . 0 25 . h w var1es as u · z m' d'1 et al . ( 1 97 9) have reported that above OG a Uoo of 0 . 1 m /s , h w is independent of Uo o in a 0 . 1 - m -I . D . column . T his be havior was not observed by Armstron g et al . of 0 . 2 3 m /s in a 0 . 2 4- m - I . D . column . ( 1 976b ) up to a The design o f gas sprager has no effect o n h w · ,
4.
5•
6.
.
Uoo
For suspensions of Kieselguhr in w ater and oil , K o1 bel et al . ( 1 9 5 8 , 1 960) reporte d separate correlations for laminar and turbulent re gions . T heir results are disc ussed by S hah ( 1 97 9 ) . Vishw anathan et al . ( 1 96 5 ) and Arstron g e t al . ( 1 97 6 b ) reported their experimental data i n t h e form of graphs and correlations are not proposed .
699
G as- L i q u i d- So li d R e actors
Deckwer ( 1 98 0 ) has analy zed the problem of heat transfer on t he basis of Higbie ' s penetration theory together with Kolmogoroff' s t heory o f i sot ro pi c t urbulence an d p rop o s ed the cor re latio n St
=
2 - 1/4 a( ReFrPr )
( 22)
where 2 OG F r = - gd U
h S t = ----'-w'---_ pC U p OG
b
c
\l
Pr = �
,
s
K
where k , p , C p , and J.l ar e the t hermal con ductivity , de n sity specific heat , and viscosity of the ga s - liquid - solid system . T hese are calculated by the followin g eq uation s : k = k
P =
L
8s P s
+ EJ.PL
( 2 3)
1 + 4. 1-1 = J.l L c 5 e:8 >
Zaidi et al . ( 1 97 9 ) have s hown t hat the data re p ort e d by Muller ( 1 95 8 ) , Ko1 be l et al . ( 1 9 58 , 1 96 0 ) , an d Zaidi et al . ( 1 9 7 9 ) can be correlated with E q . ( 2 2 ) within 2 0 % , and the value of a i s reported to be 0 . 1 . For de si gn purposes , Eq . ( 2 2 ) is recom me nded . However , a pre cau tion is desirable for cases w here no more than 8 0 % of t h e input energy is di ssip ate d at t h e gas - li q uid and liq uid - solid interface s and at the column w all . Further Eq . ( 2 2 ) w as p roposed for slurry re actors , and th e re for e for t he case of t h r e e - p h a s e fluidi zed beds , U o o may be replaced to ge t the form
( 2 4)
Joshi et al . ( 1 980 ) have proposed a new correlation based on the circula T hey reported that in some gas - liqui d tion cell model ( Joshi , 1 9 8 0 ) . solid systems , all t he i n p ut ener gy d u e t o the gas flow is di ssip ated a t the gas - liquid and soli d liqui d interface rather t han in the liquid motion . B y usin g the method o f avera ge re ci rculati on ve locit y ( Joshi , 1 98 0 ) , they pro po s ed the fol lo w in g e q uation to c alc ul at e the w all b e d heat trah sfer co efficient :
-
h
d
w c
k
=
J.l_;:::__-e;..;G:..u; ..;b:_; oo_P
l . 33 0 33 (U g .
dC ( 0 . 4 8) �
_____
)
1/3
__
( � L) ( � �) C
x
0 . 14
( 2 5)
This e q uat ion is based on the analo gy between mec hanic ally agitat e d con tactors and bubble columns . Joshi e t al . ( 1 980) s how e d that the apparent
Shah and Sharma
700 disc rep ancy between the deterministic ( Joshi ,
1 9 8 0 ) an d the stochastic ( D ec k w e r , 1 9 8 0 ) ap p r oa c h is immaterial and that bot h p hy si c al concepts are equally c ap able o f re p re se nt i n g transfer p henomen a in the liquid p hase o f b ubble columns and an be re gar de d as limiti n g cases o f a more ge n er al model . This s ubj e c t is disc ussed in more detail by Jo shi e t al . ( 1982) . Scaleup C o n s i derations and in t h e case of gas s p ar ged re actors , the h ei gh t - to - diamet e r ratio m ay
L ab or a tory e xperi m e nt s are invariably m ade in 7 0 - to 1 5 0 -mm re actors ,
be 3 : 10 .
In l ar ge i n dustri al reactors w e o ften h a v e a low h ei gh t - t o
diameter ratio of 1 : 2 , an d in t he case of gas s p ar ge d reactors we also
norm ally have lower values of s uperfici a l gas velocity . Even t ho u gh the hei ght - to - diameter r a t io i s low in industrial reactors , one m ay en co unte r p rob le m s associated with t he hydros tatic head of the liqui d and lo w values of superficial gas velocity . I n p articular , we have to ens ure tha t the solid particle s concentration is unifor m t hro u gho ut the reactor . In some cases we m ay encounter a substantial c h an ge in t he s u pe r fici a l gas velocity from b o tto m to top , b ut such cases have received very little attention . W hen back mixi n g in the g as p hase is relev an t , one needs to as c er tain thi s more c lo s el y . This problem m ay be p artic ularly impor t ant i n m ec han i c a ll y a git at ed re ac tor s . One m u st bear in mind that ju st as for the fi xed - bed re ac �or , if ex p en s ive catalysts are used ( say , those based on noble m e t al s ) and the re sistance is e s sen ti ally located in t he solid - liquid film o r gas - liquid film , one m ay benefit s ub s t an ti all y by re p lacin g a p art of the cat aly s t with an equi v ale n t support material ( say , alu mina or zeolite or activated carbon ) but without the noble metal .
C A LC U LA T I O N S O F A B SO R PT I O N / D ES O R PT I O N R AT E S
T he calculation o f ab sorp tion / de sorption rate s for a t hree - p hase system w hen the solid p h ase is a c at a l y st is discussed at great len gth by S hah
( 1 9 7 9) , S hah and S harm a ( 1 97 6 ) , and C h a ud har y an d R amchandran ( 1 9 80) T he fi l m t h eor y anal ysi s illustrated in and will not be repeated here . Here t hese references c an be applied to more comp lex reaction syste m s . we restric t o ur discussion to novel reactio n systems such as micellar cata ly sis used in li qui d - liq uid re ac tio n s , c at al ytic reactions in which catalyst p artic le s are sm aller than di ffusion film thickness , an d slurry reactors in volvi ng reactive solids . S ome det ailed m athematics of the last two case s is gi ven by C h au dh ar y and R amchandran ( 1 9 8 0 ) . M i ce l l a r C ata l y s i s
Micellar catalysis has be e n s ho w n t o have a remarkable effect o n the rate Micelles are aggre gates of surfac o f a v arie t y of liquid -liquid re action s . tan t s formed in a qu e ou s solutions , where the ap ola r group s extend into the interior of t he aggre gate and the head groups are loc ated at the Typ ic all y , micelles have an average radii of 1 2 - 30 interface with w ater . A and co n t ain 20 to 1 0 0 m onome r s . T hese mic e lle s c an so l u bili ze large quan tities of m a t erial s w hich otherwise have a low solubility in the aqueous p ha s e .
G as- Liquid- So lid Reactors
701
Rate acceleration or inhibition of or ganic reactions in micellar solutions arises from t he different rate s of reaction of t he sub strate in the micellar medium and in the b ulk phase and the distribution of substrates between T y pical reactions that have been studied in presence these two p hases . of micelles include hydrolysis reaction s and nucleophilic substitutions . Fen dler and Fendler ( 1 97 5 ) give a good summary of the work done on micellar systems up to 1 97 5 . Most of these systems involved slow reactions w here t he mass tran s fer of the reactant s is fast comp ared to the reaction rate and hence need not be con sidered . Menger and Portnoy ( 1 96 7 ) de veloped a pse udop h ase model for unimolecular reaction system s . By assum ing t he total rate to be the s um of t he rates of reactions occurring in the micellar and the aqueous p s eudophases , they obtained an exp ression for t he overall ob served first -order rate constant : k k ob s
+ k K (D )
1 + k (D ) s n
w
m
s
n
( 2 6)
w here k w and km are the true first -order rate constants in the aqueous and .nicellar p seudophases , K s t he bin din g con stant of the react ant to the micelle , and ( D n ) the concentration of micelli zed surfactant . The p seudop hase model has also been applied to bimolecular reactions by considerin g the distribution of both re actants bet ween t he micellar and water phases ( B u n to n , 1 97 9) . For these systems the micellar effects m ay be due either to the increas ed concentration of reactants in t he micellar medium ( p roximity effect ) or to incre ased re activity in the micellar medium over that in the aqueo u s p hase ( medium effect ) . Dilution effect s m ay come into play w hen micellar concentrations are m ade very high because of t he decreased probability of findin g bot h the reactants in t he same micelle , and this leads to· reduced reaction rate s . Recently , Janakiram and S harm a ( 1 98 2 ) have considered the case of slow and fast liq ui d - liquid reaction s w hen m ass tran s fer of reactant is important . T he film theory ap plied to mass tran sfer with chemical reaction w as used , wherein it w as as sumed that all the resistance to mass tran s fer The re exi sted in a thin film near the interface ( called diffusio n films ) . action was assumed to occur in t he aqueous phase only , w hich contained the micelle s and the reactant B . T he hydrophobic reactant A was assumed to have a low solubility in t he aqueous phase . B ecause of their small si ze , the mic elle s are pre sent in t he film and as A diffuses throu gh the film , it is t aken up sim ultaneously by t he micelle s , w here it reacts with the micellar T he mic elle s th u s steepen the concentration profile of A in the bound B . film and lead to hi gher mass transfer rate s . By writin g down transport eq uations for A and B in t he film and considerin g micellar incor p oration to be a parallel step , equation s have been derived to predict mas s trans fer rates of A in micellar syste m s for different re gimes ( Janakiraman and Sharm a , 1 9 82 ) .
C ata l y s t Pa r t i c l e s Smal l e r t h a n F i l m T h i c kness : Reaction Occ u r s i n the D i ffu s i o n F i l m and I s Cata l y zed by So l i d Pa r t i c les We can conceive i f situations where catalyst particles are s ufficiently small in si ze ( s ay , below 20 ]..l m ) to be s maller than the gas - liquid diffusion film
Shah and Sharma
702
,
thic kness . I n such cases a parallel model of tran sport has to be considered . I n the e s t abli shed practice of slurry reac t or s the step s as sociated with gas - liquid an d li q ui d so li d mass transfer are taken in series . Further , t he in di ge nou s reaction m ay be sufficiently fast to occur partially or wholly in the diffusion film , yet it is further amenable t o c at alysis by fine particle s . Pal et al . ( have considered this situation . Earlier work in this area has been carried out by D ec k wer , A lper , and coworkers . T here is con siderable p ractic al signi fic ance for such cases , as not only m ay one get marke d inc rease in the specific rate of absorption but one m ay also reali ze the desired selectivity . In t he case of magnetic catalyst p articles s uch as Fe , N i , an d CO , or those that can be magneti zed , we have a powerful method of recovering fine catalyst p article s efficiently via hi gh - gradient magnetic separation . T his subj ect merits further attention . A brief m athematical analysi s is pre sented below . Since t he reaction is fast enough to occur in t he diffusion film wihtout a catalyst , it is logical to ass ume that in the pr esence of fine c atalyst particle s , t he tran sport of dissolved gas to the cat aly s t p ar ti c le s will control the rate due to c at aly s t . For a reaction that is first order in t he dis solved gas and di s solved reactive , species , th e go ve rn in g equation s are
-
1982)
( 2 7)
where
27)
account s for the T he second term on t he right - hand side of E q . ( the film due to c at alyst p article s . A solution for the absorp tion rate of A with t he boundary con dit ion s reac tion in
z = z =
0: oL:
CA
= C
CA
=
_A ,
dC
B
dZ
(28)
= 0
0
can be given as
=
_A I D MAk 1 D MAk s ap �-:-;; ::=;k==;== �-=tanh /D a D C
(
+
MA 1
:
1
M A ks p
)
(29)
H ere t he contribution due to the p ar all e l reaction in the film is unimportant as t he particle is relatively very coarse , leading to low values of ap and k s (in the particle si ze r an ge relevant here , k8 will be inversely propor tional to the particle si ze ) . When D MA k 1 + D M A ks 8p lk1 > 3 , Eq . ( 2 9) c an be exp re s sed as 1
(Y�)A_ 2 1
D A C M
k
=
k a
s p
=
k
6w s p d p p
( 30)
703
Gas-Li q uid-Soli d R eac tors For s m all d
, where k d s p
p
against D MAw for th e origin h avin g
/DMA
=
2,
2 . a p lot of [ ( 1 /D MA ) ( N A / C A )
a sp eci fie d particle size a slope of 1 2 / p
temperatures can be
d2 •
should
T hu s
p p
k11
-
be a strai gh t line t hrou gh
data pertainin g to different
accommodated in a si n gle plot .
Pal et al .
n
( 1 98 2 ) have
aqueous solutions of sodium verified this theory for absorption of 02 i it fin e ac tiv at ed car on s ulfide at ele va t ed te m p era ure a d p e s ur
t
r
n
b
r s e w h
pa ticle s as catalyst .
S l u rry R eactors I n v o l v i n g Rea c t i v e Sol i ds
( a ) p articl e s ar e In this c ate gory we can visualize two distinct situations : soluble in the medium and th e reaction occurs between t he dis solved gas and the dissolved r e activ e species obtained from the dissolution
spa rin gly
parti c le s ( the product of the re ac tion may be soluble or insol uble or ( b ) p arti cle s are insoluble in the medium (the product reaction m ay be soluble or in solub le i n the medium ) .
of solid
in th e of the
medium ) ,
Soli d Particles Sparingly in the Medi wn
This situation is commonly encountered in practice an d we can t hink of two s inc tly different sit uations : ( a ) solid p articles are lar ger than the thick ness o f t h e li q ui d film (this t hickn e s s is that of diffusion film wit hi n w hich the resistance to mass transfer is confined in the absence of chemical re di t
actions , or
( b ) soli d p articles are smaller than the liquid film thickne s s .
Particles Larger Than the deal wit h thi
s
pr o
bl e m
as
li q ui d Film
the c on s tituen t
Thic kn e ss It is quite e asy to namely , di ffu si on of dis .
step s ;
solve d gas from th e interface into the liquid p hase and dissolution o f soli d se rie s ( Fi g . 1 6 ) , We can p r e dic t the sp eci fic rate of m ass trans fer for any of the controlling regimes . It m ay well be t h at in view of the p artic l e si ze an d l oa din g , the bulk li quid p hase may be s at u rated with re spect to the di s s olve d s oli d s pe cie s . However , even if we have a " finite" slurry th at causes bulk concentration to b e lower than the saturation con particles in
centration , we can solve the problem analytically .
h De pe ndin g on the o f gas liqui d contactor , we may have a liquid film thickness ( diffusion Thus if we hav e p articl e s smaller film thickness ) in the range 2 t o 40 llm . t han the film thickne s s , we c an conceivably have the step of dissolution of solid particles in paralle l to t h at of diffusion of dissolved gas from the ga s li qui d interface to the bulk liquid phase . De pendin g on the r elative Particles Smaller Than t e Liquid Film Thicknes s .
typ e
-
-
rates of diffusion and chemical reaction , the entire amount of dissolv e d gas may be consum e d in the film . Further , dep e n din g on the c ont rollin g re gi m e , the concentration of dissolved reactive species in the liquid film may be uniform or m ay be zero close to the g as li q uid interface ( see Fi gs . 17 and 1 8 ) . When the c onc en t ration of dissolved s oli d sp e ci e s is uniform in the film , t he s i m ultaneou s di ss oluti on of soli d p article s in t he film will not affect the sp eci fic rate of m as s transfer , However , if we hav e depletion of the re active sp ecies , the sim ul t aneo us dissolution of solid p articles can lead to a situ ati on where the de p l e tion i s en ti re ly overcome and c onc entr a ti on becomes -
uniform in the film .
The
m os t excitin g situ ati on arises when the r eac tion is instantaneous
and the concentration of both speci e s , namely
the dissolved gas and
L I QU I D
�
GAS F IL M
LIQU I D FILM
L I QU I D
F I LM
SU RROU N D I N G
IS
SOL I D
TH E
- - - P S E U D O - F I RST- OR D E R REACT ION -
D E P L E T ION
REG IME
L I QU I D
I
GAS FILM
I
L I QU ID F I L M
I
BU L K
L I O.U I D
I
t
I
L l QU I D - F I L M S U RROUN D I N G THE SOL I D
FI GURE 1 6 Concentration p rofiles based on the film theory for two gas liquid - solid systems when the soli d dissolution in the film is unimportant .
NO
- - - - C O N C E N T R A T I ON
CO NTAIN I N G
S U SPEND E D
- - C O N C E N T RA T IO N C � HA I N I N G
6
PROF I L ES FOR SOLU T I O N , S U S P E N D E D SOL I DS PROF I LE S
FOR
F IN E
SOLU T ION PA RT IC L ES
FI G U RE 1 7 Concentration profiles for an in s t an ta n eou s reaction w he n the solid dissolution in the film is important .
70 5
Gas-Liq ui d -Soli d Reac tors
m a:
o c* A
MODEL BY UC H I D A tl Q1 RAMACHA N D RAN AN D S HARMA' S M O D E L FILM MODEL F O R NO SOL I D
<
l!;
SUSP E N S I O N
�
�a:
r z w u
�
u
G
0
FI G U RE 1 8 Concentration profiles of A and B ( A , A1 , A11 ar e the main re action planes for these models ) : ( a ) in liquid film near gas -liquid interface ; 0 to ( b ) in the nei ghborhood of solid p ar ti cle s present in the re gion z z = A.
=
dissolved solid , is zero at a reaction plane that is very close to the gas liqui d interface . H e re the concentration of dissolved species drops to zero from the b ulk concentration , which may w ell be the same as th e saturation concentration . The sim ultaneous dis solution of the solid particle s in the film will obviously au gment the flux of the dissolved solid species and hence the sp ecific rate of ab sorption will increase . Ultimately , the reaction plane may coincide with the gas -liquid interface and we will get a very interesting situation w here the s pecific rate of absorption will be proportional to the Since for very small particles the solid square root of the particle area . liquid mass transfer coefficient is inversely proportional to the particle si ze ( Sherwood nu mb e r 2 ) , we can conceivably get a situation where the specific rate of mass trans fer will be inversely p roportional to the particle size . The simultaneous dissolution of very fine soli d particles in t he film may lead to a situation where the " in stan t an e ou s " reaction-controlled mechanism (but havin g known value of rate constant ) chan ges over to the case of de pletion of reactants and eventually to fast p seudo-roth- order reaction- con trolled mechanism . We also think of situations where the interfacial concentration of t he dissolved gas is comp arable or even high e r t han the saturation concentration of the solid p articles , an d in such cases we will have si gnifican t enhance ment in the rate of di ssolution of solid particles in the film due to the in stantaneous reaction ( Fig . 1 8 ) . The mathematics associated with those cases is evaluated by Ramachan dran and Sharm a ( 1 969) . Sada an d coworkers ( 1 97 7 , 1 9 8 0 ) and Uchida and co workers ( 1 97 5 ) have also applied the theory to the absorption of s o 2 in slurries of CaC0 3 an d C a ( O H ) 2 . =
Soli d Particles Insolub le in the Medium
Particles of Unchan ging Si ze . H ere the p roduct is insoluble in the medium and stays with th e p article , and the reactant has to diffuse through the product layer . Fi gure 19 shows typical concentration profiles .
706
Shah and S harma
9- ::i
::J ..J 0 -
:i lL
0
..J o
:,s: :J
::> j:D ..J
CA
�
0
-
...J 0 lll
::i ::::! w
LL I 1-
0 - a:: => ..:r
� w
..J Z
� CAS
_J
�
a: w t-
� �
O UJ �--u � W ct:
::r: l/l �
Cl:: O
�
0 w 1u � W C::: z ::>
UJ ct: o u
CAR
C oncentration p rofiles for gas - li quid - solid systems where
FI G URE 19
soli ds are reacting but insoluble in liquid ; solids of constant si ze .
TABLE 2
Controllin g M echani s m s for C onstant Si ze Particles
Controlling mechanism
C ontrollin g Remarks
rate term
Mass transfer from phase A
k a L L
XB o:t
Hydrodynamic factors are very important
Solid -liquid mass transfer
k
x
Hydrodynamic factors
to phase B
s
B
at
have some effect in the turbulent regime
Diffusion through solid
D
ms
ta[ ( l
+ (1
-
-
x
B
)2 1 3
XB ) J
Hydrodynamic factors are li kely to be unimportant , effect of temperature should be insignificant
Che mical reaction
Effect of temperature very significant ; no effect of hydrodynamic factors
Gas -Li q uid-So li d R eac tors
707 0
__, 0
::::! w :::E V'I
U. ::I: 1-
9 o:: Q w
=> <
o _ _, O ln
w u
< u. a: ::> V'I
--J Z
FI GU R E 2 0 Conce ntration profiles for gas -liquid- solid systems where solids are reacting but insoluble in the liquid ; solids of chan gin g si ze .
I t i s possible to follow systematic procedures which will allow the con trolling mechanism to be clearly discerned . Table 2 brings out the salient features of different controlling regimes . Particles of C hanging Si ze . Here the product of the reaction is soluble i n the medium and hence the r eact an t s hrinks . Figure 20 shows the conc en tration profiles and Table 3 shows the pertinent features of different con trolling regimes . Hydrometallurgy provides a variety of examples which fall in this category of reac tions . Oxidative leaching of copper , cobalt , and silver are A novel case is that of the man u fac t u re of alkali some typical examples . hydrides by hydrogenation of alkali me tals suspended in a hydrocarbon sol vent . Chlorination of wood pulp provides another example .
Controlling Mechanisms for Particles of Changing Relation between time and fractional Controlling Controlling rate t erm conversion mechanism TAB LE 3
kLaL
xB at
Mass
k
s
x
Chemical reaction at the surface
k
R
x 8 at
Mas s transfer p has e A to phase B
from
transfer to solid par ticle s
B
Si ze
R em arks
Hydrodyn amic factors are very important
at
y rodynamic factors ha v e so m e e ffe c t i n the turbulent regime Effect of temperature signific ant ; no effect of hydrodynamic H d
3
factors
708
Shah an d S harma
MO D E LS FO R F I X E D - BE D CA T A LY T I C R E ACT O RS I sot hermal Models
generalized isothermal actor can be written as
A
d
D. . lJ
2
steady- state
model
for the fixe d - bed
ell
catalytic re
( 31)
dz 2
where the subscript i refers to the phase and j refers to the species . E qua ti on ( 3 1 ) is a one-dimensional model w hich is generally applicable to the fixed -bed systems . If the reactor i s to be operated under tr an sient condi tions , the ri gh t - hand side of E q . ( 3 1 ) will have a time- dependent concen tration term . In Eq. ( 31 ) , Di j is the axial disp e rs i on coefficient for sp ecies j in phase i , Ui the superficial velocity of phase i , and Tij the transfer in and /or out of s peci e s j from phase i to ot h e r phases . For example , for a gaseous reactant j in the liqui d p ha s e Tij can be expressed as ,
lJ
=
T. .
K a ( C . - C j ) - K .a ( C L L L GJ L SJ S
j
- C SJ. )
( 32)
Here i = L and the species is transferred in from the gas phase and trans ferred out to the solid catalyst phase . Rft is the reaction rate of s p ecie s in the phase i . The + sign indicates generatlon and the - si gn -indicates de pletion . For the fixed-bed reactor , the first two terms on the left-hand side of Eq . ( 31) drop out . The first term on the left -hand side also drops out for the gas phase . If the gas velocities of each phase vary along the length of the reactor , this can be evaluated by taking a total material balance on each phase . In gene ral , the total number of equations that need to be solved simultaneously equal the sum of component balances for each p h as e plus the total number of overall material balances . It should be noted that Eq . ( 3 1 ) assumes tha t the dispersion model applies to the reac tor , an as sum p tion generally found to be reasonable . Equation ( 3 1 ) i s subjected to the boundary condition s dC . .
and
D.j1 _2l. d z dC . .
___.!l
dz
=
0
1]
- U.(C. . 1
at
z =
C .?. ) 1]
L
at
z = 0
( 33)
( 34)
where L is the length of the reactor . Various cases of this generalized model have been examined in the literature , and these are reviewed by Shah ( 1 97 9 ) . In an evaluation of p ilot - scale trickle-bed reactors the m ode l described above is often not applicable due to flow anamolies such as incomplete catalyst wetting , insufficient li qui d holdup , backmixing , and so on . Shah ( 1 9 7 9) has reviewed at great length the reported modeling of pilo t - s cale hydroprocessing trickle- bed reactors .
709
Gas -Liqui d -Soli d Reac tors Non l sothermal Mode l s
Many in d u s tri al hydroproces sin g reactors such a s those for hydrodesulfuriza tion and hy dr oc r acking operate under adiabatic or close to adiabatic condi
tions . Since hydrodesulfuri zation , hy dro c rackin g , an d hy d ro gen ati on reactions are highly exothermic , the tempe ratur e rise in such reactors is fre q u en tly controlled by the injection of quench streams at one or more positions alon g the length of the reactor . These fixe d-bed ( under trickle flow conditions ) reactors are usually operated under pl u g - flow condit ions and the reactor temperature profile exhibits discontinuities at the position of the qu e nch e s . The maximum temperature rise in commercial trickle -bed reactors i s controlled by the amount of nature of the quench and its p osi
tion along the length of the reactor . In com mercial hydrodesulfuri zation re act or the catalyst bed deactivates
nonu nifor m ly due to the dep osi tion of metals . The catalyst deactivation is usually counterbalanced by the in cr e ase in feed te mperature in order to maintain t h e con stant quality of the product . This process is continued until the temperature in any part of the reactor reache s the maximum all o w able temperature based on the metallurgical limit . At this point , t he re actor is shut down and the catalyst is regenerate d .
The time required to achieve this maximum allowable temperature is c alle d the reac tor cycle life . Shah et al . ( 19 7 6 ) an d Mhaskar et al . ( 1 97 8) have p ointe d out that the re actor cycle life depends on the locations of quench pos ition s ( for one and two quenches ) and that by properly choosing the quench positions , one can significantly improve the reactor cycle life . A bi c han d ani et al . ( 1 98 0) evalu ate d the problem for a more generali zed nonadiabatic reactor . These studies illustrate that in industrial fixed- bed hy dropro c es sing reactors , catalyst activity can be maintained at a higher level over a lon ger period of time by properly controlling the temperature distribution in the reactor . The thermal .behavior of a fixe d-bed hy d ro c racki n g reactor with one or more quenches has been evaluated by Yan ( 1 98 0 ) . Generally , commercial fix e d-b e d reactors have a large catalyst bed length -to-particle diameter ratio an d they are operated under plu g- flow con dition s . The criteria for axial dispersion effe c ts in adiabatic trickle - bed hydr op r oc e ssin g reactors for ( a) residual hydrodesulfuri zation , ( b ) hydro cracking of gas oils , and ( c ) denitrogenation of shale oils were derived by Shah an d P arasko s ( 1 97 5 ) . These criteria indicate t hat at high c onvers ions , an adiabatic r e ac t or p ro du c es a lar ge r axial dispersion effect than the iso thermal operation . At low conversions , the opposite results are obtained . Some exp eri ment al verification s of these analytical criteria ar e neede d . M O D E LS FO R S LU R R Y R EA C T O R S
.,
I sotherma l Mode ls
A gen erali zed model for t he fi xed - b ed catalytic reactor described in the p rece din g section is applicable for both the gas and liquid p hases in slurry reactors . If t he soli d p article s are very fine (less than 1 0 0 �m ) , the li q ui d - soli d slurry is often assumed t o form pseudohomogeneous slurry and the separate material balances for the solid p hase may not b e required . A model of t he slurry reactor wherein solid concentration gradient pr evails within the reactor has b e en evalu ate d by ovind a rao ( 19 7 5 ) and discussed by Shah ( 1 97 9 ) . Some aspects o f s tea dy - s t at e behavior of three -phase
G
71 0
Shah and Sharma
fluidi ze d - bed reactor as applied to coal li q u e fac tion are discus sed by Parulekar and Shah ( 1 9 80) and Shah ( 1 981 ) . Thi s model should be , in general , applicable to other three - p hase systems . reactant is T he modelin g of a three -phase slurry reactor with solid as Joshi e t al . ( 1 98 1 ) modeled t he pe r or m an ce o f an a m re difficult tas k . reactor whe rein the sulfur in coal p articles is oxi di z e d to form sulfuric acid . W hen the solid phase di ssolves or r eac ts substantially , its volume fraction lo n g the reactor chan ge s . This pro le m has not been evaluated in the literature . The selectivity p roblem in the operati on slurry reactor has also
f
o oxydesulfuri zation
a
a
b
of a
For example , the liqui d - p hase half not been yet completely evaluated . hydro ge n at on of 2 , 4- dinitrotoluene ( DN T ) can lead to different isomer dis tribution de pe n di n g on the p res ence or absence of mass t ran s fer resistances ( Acres and C ooper , 197 2 ) . Even more interestin g is the case of obtainin g s t er e spe cifi c product depending on the presence /absence of pore diffusional resistance ( Woer de et al . , 1 982) . S ele c t vi ty is also very important in coal T he se types o f problems need to be evalu li q u e fac tion an d F T synthesis . a ed the future .
i
o
t
i
in
N on isotherma l Model s
The steady -state thermal behavior of an adiabatic three -phase coal liquefac tion reactor operated un der slow an d fast hydrogen consumption reaction re gi mes have been analy zed by Parulekar and Shah ( 1980 , 1 9 8 2 ) . The theoretical calculations were based on the axial dispers i on mo del , an d the predictions of the model for steady - state temperature distribution in the presence as well as the absence o f q uen ch were found to agree well with those m eas ur e d experimentally ( S hah , 1 98 1 ) . It shou l d be noted that un like in plu g - flow reactor , in a backmix r e ac t or the temperature continuity at t he maintaine d ; and as shown by Krishnamurthy and
quench location is
Shah ( 1 9 7 9 ) an d Singh et al . ( 1 98 1 ) , the quench gas flattens the tempera ture profile in the entire backmixed reactor . The m ultiple steady states in adiabatic reactors have been investigated by Sin gh et al. ( 1 98 0 ) . Nunez et al . ( 1 98 2 ) studied the effects of preheater variables on the m ultiple steady state in an adiabatic coal li qu efac t on re actor . Shah an d Sin gh ( 1 9 8 1 ) p resen t e d some e xp e ri m e n tal data in adiabatic r eactor and s howed that significant temperature excur sions could occur un der practical ranges of ope ration s . Parulekar et ( 1 98 0 ) examined a fir s t - or der reaction in a three - p hase bubble column reactor wit h the help of an axial dispersion model . For the
i
laboratory
a
al .
ex tr e m e case of a continuous - stirre d - t ank reactor , these authors identifie d a fairly large re gi on five steady states for a non zero concentration of the gas eo s reactant the liquid phase , a fact p r e vi us ly observe d for an For the case ga s - li q ui d C S T R ( R a gh ram Shah , 1 97 7 ) . partially column reactors , the authors derived criteria for the uniquene s s of steady states .
u adiabatic
in
of
backmixed bubble
u
and
o
of
S E L EC T I O N O F R EA C T O R S
of
A variety reactors have been used for ga s - li q ui d - soli d system s , and Satterfield ( 1 97 0 ) , Shah ( 1 9 7 9 ) , and Doraiswamy and Sharma ( 1 983) have consi dered several a sp ects of these reactors . As in dicated elsewhere , the m ain clas sification of r eactors may be based on solid particles b ein g very
7l l
Gas -Liq ui d-Soli d Reac tors
fine when we have slurry reactors and solid particles being coarse w hen we usually adopt fixed-bed reactors . The selection of a reactor for a specified duty will depend on a number of factors , and Table 4 lists the important points . S l u r ry
V ers us
F i xed - B ed R ea c to r s
The relative merits of slurry and fixed-bed reactors have been reported by several of the authors listed above . Slurry reactors operate as thermal fly wheels and maintenance of isothermal conditions is relatively easy . The slurry can be pumped and regeneration can be conveniently accomplished . Further powdere d catalysts are generally cheaper and fine catalyst particles whould show a negli gible to small resistance to isothermal diffusion . Slurry reactors measure up very well with re spect to the important points listed in Table 4 . The main disadvantage of slurry reactors is associated with re moval of the catalyst particles from the product stream . In case the prod ucts are volatile and are stripped under operatin g conditions ( e . g . , in Fischer - Tropsch operations for C H 4 and lower hydrocarbons ) , the question of catalyst removal does not arise except for purposes of catalyst regenera tion /replacement . This disadvantage is liely to be relatively unimportant for catalyst p articles , suc h as those based on Fe , Ni , Cr , and Co , which
TAB LE 4 Gas - Liquid Solid Reactors : Checklist for Selection of Reactor Residence time of liquid phase Allowable press ure drop Relative flow rates of gas and liquid Scale of operation Recycle of gas Changes in type of feedstock Countercurrent or cocurrent mode of operation Corrosion Foaming behavior Rheological behavior Micromixing Selectivity Heat removal /supply Solid p articles as reactant or catalyst Particles fine or coarse Product of reaction soluble in insoluble Specification of solid product Fouling of catalyst
_ ,
Shah an d S harma
71 2
are magnetic , or particles that can be m a gneti z ed , as the catalyst separa tion ( HGMS ) . Recent reports indicate that many noble metal catalysts can be prepared on mangetized s upports an d that HGMS can even deal with sub micron particles For very large scale operation for relatively cheap materials such as in hydroprocessing of petroleum fractions , slurry reactors m ay prove to be unwiedly and expensive . However , for vacuum residua or bottom fractions of heavy crude oils , w hich contain relatively very hi gh amounts of organo metalli c compounds , foulin g may be severe and slurry reactors have a clear edge over fixed -bed reactors . In slurry reactors the liquid phase is essentially backmixed ; the gas phase may be p lu g flow or partially backmixe d . In fixed-bed reactors , depending on th e regime of operation , w e may approach plug-flow behavior for both phases . In fixed-bed reactors , the trickle flow giv e s a relatively very low residence time for liquid , and slow to very slow reactions , which require a long resi de nc e time , cannot be conviently conducted . It i s possible to over come this problem to some extent by using packed bubble column reactors . Fixed-bed reactors are not generally suited for application , where the feed consists of a slurry , or one of the products of reaction is solid , as this may be lead to excessive pressure drop and maloperation of the column . •
Gas S parged Versus Mec ha n i ca l l y Ag i tated S l urry Reac tors
H aving decided to u s e slurry reactors , we can adopt different · versions de pen din g on circu m s t ance s . The p rocess may call for relatively small amounts of gas , and in case gas sparged reactors are considered , the problem of catalyst suspension may become severe and we may be forced to opt for a mechanically agitated version . It is possible to use much higher flow rates of gas and to recycle the unreacted gas in case we are dealin g with pure gas , which is the case in a variety of hydrogenations . In large reactors there is definite merit in optin g for gas spar ged reactors , as the problems of shaft seal , shaft stability , and s o on , can be avoide d . Values of the heat transfer coefficient are comparable in the two types of contactors . However , when the heat load is very high w e can have a p ump-around system as in a loop reactor . We also have the option of a heat exchanger of the m ultitubular reactor type , where the slurry passes through tubes together with the gas ; heat transfer fluid i s circulated through the shell . In gas sparged reactors , depending on the solids loadin g , particle si ze , and density , we may app roach three-phase fluidi zed- bed behavior . Mechanically agitated contactors can be operated u sin g the " dead-end" system . Here , p ure gas is introduced in the head sp ace and the pressure is maintained at a speci fi e d level ; no reaction product is volatile . Such a system is used extensively in hydrogenation of a variety of edible and non edible unsaturated oils . This operation is very lar ge worldwide ; a few mil lion tons per year are processed in this way . A variety of fee d s tock s are used and frequent change s in feedstock become necessary . Further , in the case of edible fats , the product has to meet fairly strin gent require ments . For the foregoing reasons , the mechanically agitated contactor has remained a "workhorse" in this in dustry de spite large - scale operation and other benefits of continuous operation in fixed -bed reactors with respect to heat recovery . Some improvement may be possible in mechanically agitate d contactors by usin g a smaller impeller ( si ze about 4 0 % of that of the main impeller ) at a distance from the interface roughly equal to its own diameter .
71 3
Gas -Liquid-Solid Reactors
This arrangement should improve gas disper sion and hence gas- liquid inter facial area . Recent work indicates that with better control s t ra te gies and fa s te r analytical te c hnique s it may become possible to use fixed- bed reactors for the hyd rogenati on of edible and nonedible oils .
WO R K E D E X AM P L E :
M A N U FAC T U R E O F H Y D R O XY LAM I N E
PH OSP H A T E I N A S LU R RY
REACTO R
Desi gn a continuous bubble column ( slurry ) reactor for the production of 100 metric ton per day of hydroxylamine phosp hate ( H Y AM ) based on the reduction of aqueous ammonium nitrate -phosphate buffer . A palladium -based c at alyst wi ll b e employed . Da ta
Temperature Press ure
=
=
7 0°C
4 atm at the top
( solutions for pressures of 8 , 1 0 , 1 5 , and 20 atm at the top will also be considered ) S uperficial gas velocity at the top = 1 e m / s (solutions for s uperficial gas velocities at the top of 0 . 5 and 2 c m /s will also be considered ) in feed = 20 wt % 3 4 Concentration of H Po 4 in feed = 2 3 wt % 3 De sir e d conversion of NH 4 No to H Y AM = 80% 3 Hydrogen rate at t h e bottom 1 0 % excess o ve r the theoretical requirements
Concentration of N H NO
=
Henry' s coe ffi ci en t for hydrogen , H C at alys t
=
=
4. 63
x
3 1 0- 7 mol /cm •atm
x
10
2 % palladium on carbon = 1 pd 1 0 - 3 em
Concentration of active palladium , C Particle size of the catalyst D en si t y of catalyst
=
=
0. 6 g / e m
8 3
x
Diffusivity of dissolved hy d ro gen
=
5 . 24
x
-3
g /em
3
of solution
2
1 0- 5 cm /s
Reaction • I
So l u tion
Rate of p ro d uction of hydroxylamine phosphate =
24
6 100 X 10 360 0 X 1 31
X
S h a h and Sharma
71 4 =
8 . 8 4 mol ls
0
:
8 8
=
N H N o required in the feed 3 4
.
=
1 1 . 0 5 mol /s or 8 8 4 g /s
Concentration of N H N o in the feed = 2 0 wt % 4 3 Feed rate 8 8 4. / 0 . 2 = 4 4 2 0 g /s �
Hydrogen requirements
�
8 . 84
x
3
=
2 6 . 52 mol ls
Hydrogen supplied is 1 0 % over the theoretical requirements : 1. 1 2 9 . 17 mol ls
�
2 6 . 52
x
�
Volumetric flow rate of hydrogen at the top �
2 , 652
�
1 . 86
X x
0, 1
10
4
X
22, 4
X
10
3
X
2 7 3 + 70 273
X
1
-
4
3 em /s
Superficial gas velocity at the top
�
1 em / s
Cross sectional area of the colum n
�
1 . 86
Diameter of the column
�
Superficial liqui d velocity
x
1 0 4 em
2
1 . 54 m �
4420
___;:;.;;.;:;:__ :. __...,..
__
1 . 22 X 1 . 86
X
10
4
=
0 . 1 94 cm /s
Determ i n ation of R a te- C on t rol l i n g S tep i n the HY A M Forma tion
The followin g steps are involved in the reaction between hydrogen an d monium nitrate :
1. 2. 3.
am
Transfer of hydrogen t o the bulk liquid Transfer of hydrogen and ammonium nitrate from b ulk liquid to the catalyst surface Reaction o f hydrogen and ammonium nitrate o n the catalyst surface
The relative importance of different resistances will be considered at the top of the column . The followin g assumptions have been made to simplify the problem : 1.
2. 3.
4.
The gas holdup and the ( volumetric ) liquid- side mass transfer co efficient are uniform throu ghout the reactor . (the value of E is calculated by using an average value of V G at the top and the bot� tom of the column . ) The vapor pressure o f water over the solution has been ignored , as this is insignificant in practically all cases . The cataly st particle s are homogeneously dispersed throughout the reactor . The liquid p hase i s completely backmixed .
Transfer of H2 to the B ulk Liq ui d
I f this is the rate -controllin g s tep , the rate o f absorption o f hydrogen will be given by the equation
Gas -Ltq ui d-Solt d Reac tors
R'A
=
( units of �
kL a [ A * ] -
=
2 3 cm / c m
71 5
of solution)
From the data of de Rooij et al. ( 1 97 7 ) , we have
kL
( at v 0
�
d
==
0. 1
=
1 . 0 cm / s ) = 2 or
d
B
B
=
-�
x
==
0. 9
10
-2
2.22
cm / s x
10
-2
em
.
de Rooij et al . ( 1 97 7 ) have reported some d ata on fractional gas holdup for the same system under consideration and the data can be correlated by the followi n g equation in the range 1 0 < v 0 < 1 0 , cm /s : e:
:::: .,.....,. �,...30 + 2 V 0 G
1 . 5V0
= 0 . 0 4 8 5 ( at
( 35 )
V
= 1 . 0 cm / s )
G
Effective interfacial area : a
::: :::
dB ( 1
=
. 3 2 1 3 . 77 e m /em of solution
6
2 . 22
e: G )
6 X 0. 0485 -2 10 X ( 1 - 0 . 0 48 5 )
X
( 3 6)
( This value of � is high , but for the specific system under consideration it appears to be reasonable . )
RA
:::
k L� J A * ]
::: 2 ==
X
5. 1
10
x
-2
10
X
-7
1 3. 77 mol /cm
X
3
4
,
X
4 63
X
10-
{ 3 7)
of solution
Transfer of Hydrogen from B ul k Liquid to the C a taly s t Surface
-
7
• I
Sano et al . ( 1 97 4 ) have given the followin g equation for the calculation of a true solid liquid mas s transfer coefficient .
Sh
::
=
k
d SL p D
[
A
(:f ) d
2 + 0.4
4
1 /4 sc
1'3
]
{ 38)
¢0
S ha h a n d
71 6
Sharma
where =
E
rate of ener gy dis sipation per unit mass of liquid
VG
=
u
of
=
diameter
=
shap e factor ;
d
g
2 3 98 1 cm t s , at the top
=
p
x
catalyst p article here
its
kinematic viscosity =
=
=
8
x
10
-3
em
value is assumed to be equal to 1 -2 2 4 . 2 x 10 cm ts
Substitution in Eq . ( 3 8 ) gives
Sh
2 . 02
::::
and k
=
SL
x
1. 32
-2
10
cm /s
Thus the Sherwood number is very close to 2 and this situation will be valid for other operating conditions considered in this example . Concentration of catalyst :
w
=
=
1
x
-3 3 g active Pd /cm of solution 10 2 g active Pd / 1 0 0 g catalyst 3
0. 05 g /cm
solution
wt % of Pd on C is 2 and concentration of active Pd is 1 g /c m 3 of solution . ) Solid -Jiq uid effective interfacial area :
( Note :
a
-p
=
=
k
� SL p
6
X
0 . 05
0. 6
X
8
X
10
=
3
( 3 9)
-3
2 3 6 2 . 5 cm /cm solution ::::
0 . 825 s
-1
k SL�p [ A *] =
1 0-
6w pd p
For this step to be rate controlling ,
RA
x
0 . 82 5 X 4 1 5 . 28
x
X
will be given by
-7 10 3 mol / ( em of solution ) •s
4. 63
1 0- 7
RA_
X
{ 40)
Gas -Lt q ut d - Solid R eac tors
71 7
Reac tion of Hy drogen an d A m moni um Ni trate on the C a t a ly s t Surface
de Rooij et al . ( 1 97 7 ) have given the following empirical equation for the rate of reaction of hydrogen in aqueous N H 4NOa- H aP0 4 buffer : =
R'
A
c pd [A*] O .
ka C
34
( 41 )
where k is the reaction rate constant and is given by the following equation : k
=
1 . 696 X 1 0
=
1 . 696
=
x
4
10 4
RT ! ) (=-!!! exp [ 8 exp
- 7674 7 1. 9
1 . 02 0 . 66 1) ( em ) 0 218 ( mo (g active Pd) •s
x
( 27 3
+
70)
]
·
where ac c
pd
R'
A
=
= = =
relative activity of the catalyst and is assumed to be equal to 0. 8 concentration of active palladium , 1 x 1 0 3 g active Pd /cm 3 of solution - 7 0 34 3 0 . 2 1 8 X 0 8 X 1 X 10- ( 4 X 4 . 6 3 X 1 0 ) • 3 7 19 . 6 1 x 10- mol / ( cm solution) •s --
.
From the. individual values of R_A above it can be seen that at the top of the column , all three steps contribute to the overall resistance . As we go toward the bottom of the column , the value of [A*] increases because of the increase in the static head of the liquid . Further , the absorption of gas is accompanied by considerable bubble shrinkage and the value of a is likely to decrease along the height of the column (as we go up the column ) . For the calculation of a it is assumed that Eqs . ( 35) and ( 36 ) are valid . Due to the two factors-above , the rate of gas -liquid mass transfer increases as we go toward the bottom because of the increase in both [A *1 and a. For the case under consideration we have assumed , as stated earlier , that we have uniform values of E and kL !! throu ghout the reactor . Since , in bubble column slurry reactors the liquid phase is expected to be completely backmixed , the values of dissolved hydrogen concentration and the hydrogen concentration at the catalyst surface will be uniform throughout the contactor . As a result , the rate of solid-liquid ·m ass trans fer will be constant (the variations in the value of ks L in the range of V G encountered in the reactor were found to be small and hence the value of ks L !!. p has been assumed to be constant ) . When all the three steps contribute to the overall resistance , the rate of dissolution of hydrogen in the liquid bulk is given by the equation ( 42) The rate of transfer of hydrogen from bulk liquid to the catalyst sur face is given by the equation
71 8
Shah and Sharma
( 43) The rate of reaction of ammonium nitrate and hydrogen on the catalyst surface is given by t he equation
R'A
34 ka C [A 1 °• c pd s
=
( 44)
( The relative importance of s urface reaction in relation to gas - liquid and liquid -solid mas s transfer resistance will depend on the press ure , as the intrinsic kinetics shows 0 . 3 4 order in dissolved hydrogen . ) Since the liquid phase has been assumed to be completely backmixed , the values of [ A ol and [ A s l remain uniform throughout the contactor . Elimination of [ A s ] from Eqs . ( 4 3 ) an d ( 44) gives [A l = o
R' A k a
( 4 5)
SL-p
The overall m aterial balance across the c on t ac tor gi ves R'
A
G.1 -
=
G
0
( 46)
Sh ( l - £) t
where h t is the height of dispersion . give s [ A ol
S ub stitution of Eq . ( 46) i n Eq . ( 45)
( 4 7)
The material balance with respect to hydro gen over a differential height d h d gives the followin g ordi nar y differe n ti al equation :
dG = dh d
R' A
=
where
[ A *]
1jJ
X
S ( l - £)
k !!( [ A *l - ( A ] S ( l - £0 ) 0 L =
=
H ( P T + ljlh d ) (1 dP dh
=
d
EG ) p 1033
( 48 )
( 4 9)
( 50)
Substitution of Eq . ( 50 ) in E q . ( 4 9) takes the form dG = dh d
ex. +
Sh
(5 1)
71 9
Gas -Liqui d-Solid Reac to rs
-
where k �S ( 1 L
s
-
EG ) {HP
k L �SHljJ( 1
-
T
EG )
( 5 3) The boundary conditions are
an d [ A0] i s given by Eq . ( 4 7 ) . At the top :
h
h
At the bottom :
G
0,
=
G
h , t
=
= 2 . 6 52
G0
=
( 5 2)
[A0] }
= G.
1
=
(5 4)
29. 17
( 5 5)
Inte gration of Eq . ( 5 1 ) gives
( 5 6) or
h
- a.
=
t
2 +_/ 1 a. +
1
2 S ( G.
-
G0 )
( 57)
P roced u re fo r t h e C a l c u l a tion of H ei g ht
1.
The procedure is essentially trial
an d
error :
A ssume the height o f the column and calculate [ A0 ] using E q . ( 4 7 ) . C alculate the value o f KL� from Eqs . ( 35) and ( 3 6) based o n the average value of V a . K now i n g [ A 0 ] and k � ' calculate a. and S using Eqs . (52) and L ( 5 3) . Calculate h t from Eq . ( 5 7 ) If the calculated value o f ht i s greater /smalle r than the initial gues s , th e next guess should be in between these two values . The numerical method of partial sub stitution may be used for quick convergence .
2.
3. 4. 5.
In case the reaction follows first-order kinetics , the height of dispersion can be given by the following analytical expres sion :
-a. ±� where a.
S
y
k
=
k L �HP
=
k
=
k �S ( l L
a.
2
1 a (-
L-
T
S(l
kS � L p
+
-
-
2 y ( Gi y
EG ) +
1
ka c c pd
EG ) H lji
3 cm / ( g active Pd ) •s
G0) ( 1
)
+
S)
720
Shah an d Sharma
TAB LE 5
Hydroxylamine
Phosphate :
Slurry Reactor Details for Different
Op er atin g Conditions Total
pressure at the
No .
top ( atm )
Su perfi cial gas velocity at the top ( cm / s )
Volume of dispersion
Column diam et er
Hei gh t of dispersion
(em )
( em )
( m 3)
1
4. 0
0. 5
218
1106
41. 27
2
4. 0
1. 0
154
1878
33. 35
3
4. 0
2. 0
109
3200
2 9 . 85
4
8. 0
0. 5
154
1 2 41
23. 15
5
8. 0
1. 0
109
2202
20 . 5 4
6
8. 0
2.0
4 30 3
2 0 . 07
7
10. 0
0. 5
1 31 2
1 9 . 58
8
10 . 0
1.0
97 . 5
2 3 97
1 7 . 90
9
10 . 0
2. 0
68. 9
4793
17 . 8 9
10
15. 0
0. 5
112 . 6
1493
1 4 . 86
11
15. 0
1.0
79. 6
28 5 6
14. 21
12
15. 0
2.0
56. 3
5913
1 4 . 71
77 . 1 138
13
20 . 0
0. 5
97 . 5
1669
i 2 . 46
14
20 . 0
2.0
68 . 9
3 2 80
1 2 . 24
15
20 . 0
2. 0
48 . 7
6923
1 2 . 92
Height of dispersion = 1 7 8 7 em
Vol um e of di sp e rsion
=
33 . 35 m 3
The problem was reworked for several values o f superficial gas velocity and total press ure at the top . The results are given in Table 5 . Due to complete backmixing in the liquid phase , t he concentration of dissolved hydrogen remains uniform throughout the contactor . Therefore , for cases where th ere is s ub s tan ti al diffe rence in the top and b o ttom p re s s ur e s , de so rp ti on is likely to occur near the top of the column . It was foun d that deso rption occurs in all ca se s for valu es of s uperficial gas velocity at t he top of the column greater than 2 em /s . We should avoid , as far as possible , d e si gni n g columns for conditions that are conducive for desorption . There is a remarkable effect of pressure on the volume of the contactor . Since hy dr o gen is usually available from steam reformers at higher pre s sures in the range 1 5 to 2 5 at m , i t is useful t o employ a pre s s u r e of 20 atm at the top of the contactor . The case of a superficial gas vel oci ty of 0 . 5 em /s at the op appear s to be most reasonable . We m ay therefore opt for a 9 7 . S - cm - diameter ( say , 1 . 0 m ) column with an effective dispersion hei ght of 1 6 . 6 9 m ( dispersion volume = 1 2 . 4 6 m 3 ) . T he case of a pressure
Gas -Liq uid -Solid R eacto rs
72 1
10 atm at t he top of the c on tac t or with a superficial gas velocity of 0 . 5 cm / s at the top also appears to be a t tr ac t i ve with a column diameter of 1 . 38 m and an effective dispersion hei ght of 1 3 . 1 2 m ( dispersion vol u m e = 1 9 . 5 8 m 3 ) . N O T AT I O N gas - liq uid in ter faci al area
!
a p a s
p article surface area
p article surface are a per unit volume in E q . ( 5} and Figs . 9 t o 1 1 . gas - liq uid interface , bulk liquid and p article s ur fac e concentrations in w orke d example concentration s pe cific heat solubility
�
d
c
b u b ble diam ete r column diameter
d p
particle diameter
d
average bubble si ze
vs D
axial dispersion coefficient
D
molecular di ffu sivity
D D
m ms
n
diffusion through solid product layer concentration of micellized surfactant [ Eq . ( 26} ]
e
energy consumption in Fi g . 1 5
F
Froude number , dimensionless [ Eq . ( 1 7 ) ]
r g
gravitati on al constant
G
mass velocity
h
slurry wall he at transfer coefficient
w
j
D k s
Kl ' K r
j factor in Fi g . 1 2 liquid- solid mass transfer coefficient _ ,
reactor rate constants
KL
liquid- side gas -liquid mass transfer coe ffi ci ent
K overall L K s K K K w' m, obs
gas - liquid mass transfer coe fficient
K
binding constant in Eq .
( 26)
various ki netic constants in Eq . { 26) thermal conductivity
L
length of reactor
N8
ab s orp tion rate of s p e ci e s
A
Shah
722
p
Pressure
Pe
Peclet number , dimensionless
Pr
Prandtl number [ Eq . ( 22 ) ]
an d S harma
power per unit volume of disp ersion volumetric gas flow rate
radius of p article
radius of unreacted core
r
c Re
Reynolds number , dimensionless Schmidt num ber ( ]..1 p /Dm )
Sc
n um be r ( k
St
d /D ) m Stanton number [ E q . ( 22 ) ]
t
ti m e
Sherwood
Sh
u
s p
velocity bubble rise velocity relative velocity between two phases circulation velocity
w
p article loading
We
Weber number in Fi g . 3 con ve r sion in Tables 2 and 3 axial di st ance in the reactor or within liquid film
G reek
Lette r s
energy los s thickness of liquid film
viscosity pressure drop pressure gradient e: e:
vo l u m e t rac tion
void fraction in the fixed bed dynamic liquid holdup
parameters in Fig . v
p
cr
2
reaction planes in Figs . 1 7 and 1 8 kinematic viscosity
de nsity surface tension
Gas -Liq ui d-Soli d Reac tors
723
Subsc r i pts
A
refers to sp ecies A
air
refers to air
B
refers to sp e cies B
G
gas phase
L
li quid phase
M
mixture
OL
superficial liquid velocity
OG
sup er ficial gas velocity
p
p article
LG
s
w at
two -phase
soli d pha se
refers to wate r in Fig . 2
S upersc r i pt s o *
b ulk con di t ion
refers to interface condition
R E FE R E N C ES
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( 1 967 ) , '
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( 1972) . A dli n gton , D . and E . Thomp son , Desulfurization i n fixed and fluidi zed bed catalyst systems , Proc . 3rd Eur . Symp . Chern . React . Eng . , Pe r gamon Press , Oxford ( 1 96 5 ) , p . 2 0 3 . Akita , K . an d F . Yos hi da , Gas holdup an d volumetric mass transfer co efficient in bubble columns , Proces s Des . Dev . , 1 2 , 76 ( 19 7 3 ) . Akita , K . an d F . Y oshida , B ubble si ze , interfacial area Wtd liquid phase mass transfer coefficients in bubble columns , Ind . En g . Chern . Proce ss D es . Dev . , 1 3 , 84 ( 1 9 7 4 ) . Arbiter , N . , c . C . H ar ri s , and R . F . Yap , Int . J . Min . Proces s . , 3 ,
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257 ( 1 9 7 6 ) .
Shah and Sharma
724 B ac h , H . F .
and T . Pilhofe r , Variation o f gas holdup in bubble colu mn s
wit h physic al p roperties of liq ui d s and operating p arameters of columns ,
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( 1 96 9 ) .
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C h audhari , R . V . and P . A Ramchandran , Three phas e slurry reactors , A I C hE J . , 26 , 1 7 7 ( 1 980) . Chan dr ase ka ran , D . and M . M . Sh ar m a , Absorption of oxygen in aqueous sol utions of sodium sulfide in the p res ence of activated carbon as c ataly st , Chem . E n g . Sci . , 3 2 , 6 6 9 ( 1 977) . C harpe ntie r , J . C . , Recent progress in gas - li q ui d mass t ran s fer in packed CE209 ( 1 96 7 ) .
•
beds , C hern . Eng . J . , 1 1 , 1 6 1 ( 1 97 6 ) . Charpentier , J . C . and M . Favier , Some liq uid holdup experimental data in trickle bed reacto rs for fo am ing and nonfoaming hyd roc arbo ns , AI ChE J . , 21 , 1 2 1 3 ( 1 9 75 ) . C harp e ntie r , J . C . , C . Prost , and P . Legoff , Chute de pression pou r des eco ule men ts a co- courrant dans les colonn e s a ga rni s s a ge arros e : C om p ari so n av e c l e garnissa ge noye , Cham . En g . Sci . , 24 , 1 7 7 7 ( 1 969) . Chou , T . S . , F . L . Worley , Jr . , and D . Luss , Transition to pulsed flow in mix e d - p hase cocurrent downflow thr ou gh a fixed bed , Ind . Eng . Chern . P rocess Des . Dev . , 1 6 , 4 2 4 ( 1 9 7 7 ) .
Gas -Liquid-Solid
725
Reac tors
Clements , L . D . , Dynamic li q ui d holdup in cocurrent gas - li q ui d downflow in pack e d beds , in Two p hase transpo rt and Reactor Sa fe ty , Vol . 1 , T . N . V e ziro glu and S . Kakac , eds . , H emisphere , Was hi n gt on , D . C . ( 19 7 8 ) , p . 6 9 .
D . and G . Halfacre , Liquid composition an d flow regime effects in gas -liquid downflow in packed beds , in Two Phase T ransport an d R e actor Safe ty , Vol . 1 , T . N . V e ziro glu and S . K akac , e d s . , Hemis p he re , W as hi n gt on , D . C . ( 1 9 7 8 ) , p . 6 9 . Clements , L . D . and G . H alfacre , Liquid co mpo sition and flow re gim e effects in gas - liq ui d downflow in pack ed beds , AI ChE 72nd An nu . Meet . , Hous ton , T ex . ( 1 97 9 ) . Clements , L . D . and P . C . Schmidt , Two-phase pressure drop and dynamic liquid holdup in cocu r re nt downflow in p acke d beds , 69th Annu . AI ChE Meet . , C hic a go ( 1 9 7 6 ) . Clements , L . D . and P . C Schmidt , D y n ami c liquid holdup in two p has e downflow in packed beds ; air- silicone oil s yst e m , AI ChE J . , 26 , 3 1 7 ( 1 9 8 0a ) . Clem ents , L . D . and P . C Schmidt , Two-phase pressure drop in cocurrent downflow i n p acke d beds ; air- silicone oil systems , AIChE J . , 26 , 3 1 4 ( 1 980b ) . Cova , D . R . , Catalyst suppression in gas - a git ate d tubular reactors , Ind . En g . Chern . Process . Des . Dev . , 5( 1 ) , 20 ( 1 966) . Dakshin am u r t hy , P . , V . Sub r ah m an y am , and J . N . Rao , B ed poro si ti e s in gas -liquid flui di zation , Ind . Eng . C h ern . Process Des . Dev . , 1 0 , 3 2 2 ( 1 97 1 ) . Davidson , J . F . , E . J . C ullen , D . H an s on , and D . Roberts , The holdu p an d liquid film c oe ffi cie n t of packed towers , Trans . I ns t . Chern . E n g . , 37 , 1 2 2 ( 1 95 9 ) . Clements , L .
•
•
Deckwer , W . - D . , On m echanism of heat trans fer in bubble column reactors , Chern . Eng . Sci . , 35( 7 ) , 1 341 ( 1 98 0 ) , Deckwer , W . - D , Y . Louisi , A . Zaidi , an d M . Ralek , Hy d rodynamic proper ties of the Fi s c her - T rop sc h slurry p ro c e s s , I nd . En g . C he rn . Process Des . Dev . , 1 9 , 699 ( 1980) . Dhanuka , V . R . and J . B . S te pa ne k , Gas and liquid hold u p an d p re s s ur e drop measurements in three phase fluidi zed bed , in Flui diza tion , J . F . Davidson , e d . , Cambridge University Press , C amb rid ge ( 1 97 8 ) , p . 1 7 9 . Dhanuka , V . R . and J . B . S tepanek , Simultaneous measurement of inter facial area and mass transfer coefficient in three phase fluidized beds , AI ChE J . , 26 , 1 0 2 9 ( 1 98 0 ) . Dharwadkar , A . and N . D . Sylve s te r , Liquid soli d mass transfer in trickle b e d s , AIChE J . , 23( 3 ) , 3 7 6 ( 1 97 7 ) . Efremov , G . E . and I . A . Vakhurshev , A s tu dy of t he hydrodynamics of three pha s e fluidi zed beds , I nt . Chern . En g . , 1 0 , 37 ( 1 910 ) . El - T emt am y , A . A . , Y . o . Ed - Sharn oubi , and M . M . El- Halw agi , Liquid di s p ersio n in gas - liquid fluidi zed beds ; P art I . Axial dispersion , The Axially Disp erse d Plug- Flow Model , Chern . Eng . J . , 1 8 , 151 ( 1 9 7 9 ) . Falk , J . C . i n C a talysis in Organic Syn thesis , P . N . R yl an der and H . Greenfield , e d s . , Academic Pres s , New Y ork ( 1 976) . W an g , M as s transfer coefficient an d Fan , L . T . , H . H . H s u , and K . B pressure drop data of two- p hase oxygen- water flow in bubble column packed with static mixers , J . Chern . Eng . Data , 2 0 , 26 ( 1 97 6 ) . Farkas , E . J . an d P . F . L e b lon d , Solid concentration profile in the bubble column slurry reactor , Can . J . Chern . E n g . , 4 7 , 360 ( 1 969) . •
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( 1 97 7 ) . S tei ff , A . and P . M . W einsp a c h , Heat transfer in stirred and non- stirred gas -liquid reactors , Ger . Chern . Eng . , 1 , 1 5 0 ( 1 978) . Stewart , P. B . and F . D avi d s o n Three phase fluidization : w at er particle air , Ch e rn . En g . S ci . , 1 9, 31 9 ( 1 964) . S tie gel , G . J . and Y . T . S h a h , C an . J . C hern . Eng . , 5 5 , 3 ( 19 7 7 a ) . S tie gel , G . J . and Y . T . Shah , B ackmixi n g and li q ui d h ol d up in a gas liquid cocurrent upflow p acked colum n , Ind . Eng . Chern . Process D es . Dev . , 1 6( 1 ) , 37 ( 1 97 7b ) . Subbarao , D . and V . K . T anej a 3rd Eur . Conf. Mixing , Institute of Chemical Engineers ( U . K . ) ( 1979) , p . 2 2 9 . T all m ad e ge , J . A . , Packed bed pressure drop - an extension t o hi gher Reynolds numbers , AI ChE J . , 1 5 , 1 0 9 2 ( 1 97 0 ) . Talmor , E . , Two p has e d ownflow t h rou gh catalyst b e ds : Part I . Flow map s , AIChE J . , 2396 ) , 868 ( 1977 ) . T urpin , J . L . and R. L . Huntington , Prediction of pressu re drop for two p has e two- component concurrent flow in p acke d beds , A I C hE J . , 1 3 ,
,
,
( 6) , 1 1 96 ( 1 967 ) .
Uchida , S . , K . Koide , and M . Shind o , Gas abs or p tion with fast reaction into a slurry con t aini n g fine p article s , Chern . En g . Sci . , 30 , 644 ( 1975) , Ueyama , K . , S. Morooka , K . Koide , H . K aji and T. Miyauchi , Behavior of gas bubbles in b ub b le columns , Ind . En g . Chern . Process Des . Dev . ,
14,
492 ( 1980) .
Vail , Y . K . , N . K . Manakov , and V . V . Mans hilin , Turbulent mixing in three phase fluidized bed , Int . C hern . E ng . , 8 , 2 9 3 ( 1 96 8 ) . Van S waaij , W . P . M . , J . c . C harpentier , and J . Villermaux , Residence time dis t rib uti on in t he liquid p hase of trickle flow in packed columns , Chern . Eng . S ci . , 24 , 1083 ( 1 969) . Vas alus , I . A . , E . M . Bil d D . N . Ru ndell , and D . F . T atterson , Experi mental techniques for s tu d ying thi rd dyn am ics of H - COAL Reactor , in Coal Proce s s i n g Technolo gy , CEP Technical M an u al 6 , 226 ( 1 980) . ,
Gas- L iq u id- So lid Reacto rs
733
Vermeer , D . J . and R . K ris hna , Hyd rodynami cs and mas s tr ans fe r i n b ubble colu m n s operatin g in the turbulent re gim e , Ind . Eng . C hern . Proces s Des . D ev . , ZO , 475 ( 1 9 8 1 ) . Vis wanathan , S . , A . S . Kakar , and P . S . Murti , Effec t o f di s p e rs in g bub � bles in to liquid fluidized b eds on heat trans fe r and hold up at cons t ant b e d e xpan sion , Chern . E n g . Sci . , 20 , 903 ( 1 965) . V oi gt , J . and K . S chii ge rl , Ab s orp ti on of oxy gen in countercurrent multi I . Aqueous solution with low vi s co si ty , C hern . s t a ge bubble colum ns : Eng . Sci . , 34 , 1 2 2 1 ( 1 979) . Voi gt , J . , V . Hecht , and K . S chiigerl , Absorption of oxygen in c o un t e r current m ulti s ta ge bubble columns : I I . Aq ueous solution with high vis � cosit y , C he rn . E n g . S ci . , 35 , 1 3 1 7 ( 1980) . Voyer , R . D . and A . I . Miller , Im proved gas �liquid contacting in cocurrent flo w , Can . J . Chern . Eng . , 46 , 3 3 5 ( 1 9 6 8 ) . Wallis , G . B . , I n t e rac tion between flui d s and particles , I ns t . Chern . Eng . ( London ) , 9 ( 1 9 6 2 ) . W alli s , G . B . , One-Dimensional Two Phase Flow , M c G raw - Hill , New Y ork
( 1 96 9 ) . Wang , K . B . an d L . T . Fan , M as s transfer in bubble colu m ns packe d with motionless mixers , Chern . E n g . Sci . , 33 , 945 ( 1 9 7 8 ) . Weekman , V . W . , Jr . , and J. E . Myers , Fluid flow c h ar ac te ristic s of co current gas � liqui d flow in p ac ke d beds , AI ChE J . , 1 0 , 9 5 1 ( 1 9 6 4 ) . Weekman , V . W . , Jr . , and J . E . M yers , Heat transfer ch aracteristics of concurrent g as - li q uid flow in p ack e d beds , A I C h E J . , 1 1 , 1 3 ( 1 965 ) . W eiland , P . , Unters uchuunj g ei n s Airli ft - Reaktors mit au b e rem Umtauf in Hin b li k and seine An wen dum g als B io re akt or , Dr . Inj . thesis , Univer� s ity of D o r tm un d ( 1 97 8 ) . W oe rde , H . M . , L . J . Bostelaar , A . Hoek , and W . M . H . Sachtler , The effect of p o re diffusion in enantioielective hydro ge natio n with s uppor t e d nickel catalys ts , J . C atal . , 76( 2 ) , 316 ( 1 9 8 2 ) .
T . S eki zawa , an d H . Kubota , Age distribution of suspended solid particles in a bubble column , J. Chern . En g . Jpn . , 3, 2 64 ( 1 97 0 ) . Y an , T . Y . , D yn ami c s of t ri ckle bed hydrocracker with que n ch stream , Can . J . Chern . Process De s . Dev . , 1 9 , 6 3 5 ( 1 980) . Ying , D . H . , E . N . Giv e n s , an d R . F . W eim er , Gas holdu p in gas -li qui d and gas -liqui d solid flow reactors , I nd . En g . Chern . Process Des . Dev . , Y am anaka , Y . ,
1 9 , 6 3 5 ( 1 980a) . Ying , D . H . , R . Sivas ubram anian , and E . N . Given s , Gas /slurry flow in coal li qu e fa c tion process , DOE Re p or t FE- 1 48 0 1 - 3 ( Jan . 1 980b ) . Yos hida , F . , and K . A ki t a , Performance of gas bubble columns : volumetric liquid-phase m as s t r ans fer coe fficient and gas hodlup , AI ChE J . , 1 1 , 9 ( 1965) . Z ah r ad nik , J . and F . Kastanek , T he effect of pl at es on some •parameters of a h et e ro ge n eou s bed in b ubble - type flow reactors , Collect . Chech . Chern . Commun . , 3 9 , 1 4 1 9 ( 1 9 7 4 ) . Z a hr ad nik , J . an d F . K a s tan e k , Gas hold up in uni for m ly aerated bubble column reactors , Chern . Eng . Com m u n . , 3 , 4 1 3 ( 1 97 9 ) . Z ahradnik , J . , F . K as tan ek , and M . Rylek , P o rp osi t y of t he heterogeneous bed and liquid ci rc ulation in m ultistage bubble - type column reactors , Collect C zechos lov . Chern . Commun . , 3 9 , 1 4 0 3 ( 1 97 4 ) . Z ai di , A . , Y . Louisi , M . Ralek , and W . - D . Deckwer , Ger . C hern . Eng . , 2, 94 ( 1 97 9) .
734
Shah and
S h arma
Zlokarnik , M . , Sorption characteristics of slot injectors and their dependence of the coalescence behavior of the system , Chern . Eng . S ci . , 34 , 1 2 65 ( 1 97 9 ) . Z wietering , T . N . , Suspension of solid particles in liquid by agitators , Chern. En g . Sci . , 8, 2 4 4 ( 1 958) .
11
Polymerization Reactors M AT T H EW T I R R E L L a nd R A FA E L G A LV A N
U niversity of Minneso ta ,
Minneapolis , Minneso ta RO B E R T L .
LA U R E N C E
University of Mas s achusett s , A m hers t ,
Massachus e t ts
I N TROD UCT I O N
A separate chapter on polymerization reactors in any treatise on chemical reaction engineerin g is required for several reasons . One is the steady and considerable glowth in production of p olymeric p roducts . I n 1 985, it is expecte d that production of the five leadin g thermoplastic polymers alone (polystyrene , polyvinyl chloride , polypropylene , and high- and low- density polyethylene ) will exceed 1 0 0 billion pounds worldwide ( Webber , 1 985) , This total doubles the p roduction of 19 7 5 and amounts to about $ 4 0 billion At this scale , even relatively minor efficiencies created worth of product . by good engineering are very significant . Another · challen ge lies in the unique difficulties of performance optimi za tion of polymeri zation reactors . To optimize a _ system , we need to know w ha t optimal i s-to define the objective . Polymeri zation en gineerin g is com plicated in this regard since most polymeric products en d up as fabricated materials , so that criteria such as processibility and en d - use ( mechanical , etc . ) properties are the real reactor performance measures . No truly analogous situation exists in the desi gn of a reactor for a monomeric prod uct .
We can contrast the poly mer reactor with a reactor makin g am monia , for example . W e know the p roperties of ammonia . The only design problem is to make as much of it as fast and economically as possible . We want to maximi ze conversion and if there are parallel or sequential reactions in the synthetic scheme , maximize selectivity . The first point of contrast is that simply to specify the generic name of a polymer ( e . g . , poly styrene) is in adequate to specify its properties . The secon d point of contrast is that polymerization reactions p re sent selectivity problems that are orders of magnitude more complex than those for the monomeric reactor . An elaborate set of sequential and parallel reaction s occurs . The re sultin g distribution of polymeric product , which composes any com mercial polymeric material , is of comparable importance to the c hemical structure of the polymer in deter mining product performance in end- use properties . The product distribu tion , as well as the generic chem ical formula , m u st therefore be de scribed in any complete specification of a polymer prod uct . 735
736
Tirrell , G alvan , and Laurence
As in other reactor desi gn situation s for optimal yield an d selectivity , the factors of reactor configuration -batch or contin uous ; mixin g pattern ; re sidence - time distribution ; recycle ; heat tran sfer- all play role s in deter minin g the product distrib ution . The sensitivity of the product distribution to these factors is enormous . Polymers produced in a batch reactor can be very different from those produced in a continuous reactor under other wise substantially identic al chemical con dition s . A final point to be made re gardin g the distin guishin g features of p oly mer reactors pertain s to poly m ers at the other end of the production scale from the commodity polymers mentioned in the first paragraph . Those products not withstandin g , many more polymeric products are low - volume specialty materials , frequently copolymers , de signed to perform a specified function . Efficiency is not the p rincipal desi gn objective for these reactors ( frequently , there is a high margin of profitability ) . Rather , the need is for reproducible production of a complex product distribution so that the polymer performs precisely as required . Product distribution analysis and control i s , therefore , the p rincipal novel feature an d objective of polymerization reactor en gineering relative to other chemical reactors . T his c hapter is informative b ut not comprehen sive . It is desi gned to provide an entry into the field by pre sentin g the main ideas an d techniques , by way of some examples an d placin g the em phasis where possible on the most recent ideas , techniques , problems , and c hallen ges . This is made pos sible by referrin g the reader in tere ste d in more comprehensive information on this subject to several other excellent sources on polymeri zation reactor en gineerin g ( Ray , 1 9 7 2 ; Ray an d Laurence , 1 9 7 7 ; Bie senberger and S eb astian , 1 98 3 ; Plat zer , 1 9 7 0 ; Gerrens , 1 9 8 2) .
M A T H EM AT I C A L T EC H N I Q U E S I f the elementary reaction steps of a polymeri zation mechanism are known or can be postulated with an a dequate de gree of certainty , mathematical analysis is a ( some what un derutilized) tool available to the en gineer for makin g prediction s about product distribution s . There are two alternative generic startin g points for the employment of mathematical analysis in p olymeri zation . One can start from t he mass balance equation s on the reactor of interest . This leads to differential or algeb raic difference equations , dependin g on reactor type , which must be solved to get the de sired information on product distribution . D ata are needed on the rate constants of the mode l . This i s a systematic approach , readily a dapted to the analysis of different reactor types . T his method produces an infinite set of mass balance equations , one for each possible polymer chain length , 1 to oo . The solution of any particular set o,� such equation s may be diffic ult an d involve d , but there is a catalo g of solution methods available ( Ray an d Laurence , 1 97 7 ) . Alternatively , statistical methods can be useful ( Flory , 1 9 3 6 ; Schulz , 1935 ; Lowry , 197 0 ) . These rely on picturin g the polymer chain as growin g by a process of selectin g monomers from the reaction mixt ure accordin g to some statistical distribution . The success of this approach depen ds on a correct , and frequently intuitively based , correspon dence of the polymeri za tion process with a postulated stochastic process . It is therefore less easily adaptable to different reactor confi gu ration s since the appropriate cor respon dence may be difficult to fin d or, in fact , may not exist . On the
Polymerization Reac tors
other han d , wh en in a comp a rativ ely
statistical
me tho ds do work ,
they us uall
y
737
lead to results
simple way and can sometimes reveal features of a poly
n
en gineer s hould be ab le to take a d v an t a ge of both mass b al anc e and statistical techniques . Table 1 lays out several approac h e s to solvin g infinite sets of mass balance equation s , illustrated with an example from anionic p oly m e ri z a ion . All the tec hniq ue s in that table have general utility beyond this p articular mec hanism . The b a si c example equation of T ab le 1 has been simp lified throu gh the use of s e ve ral generally p r ac tic al m ane u ver s . To ill u s tr ate these generic tricks, we go back to the mec hanism of anionic polymeri zation with no t e r mi ation an d instantaneous initiati n ( S z warc , 1 968 ; N an da an d Jain , 1 96 4 ) . In this case , only the propagation reaction is of interest . meri zation system that a mas s b al an c e ap pr oac h does not .
equipped p oly me riz a tio
T he fully
t
n
k Pj
+
M
p
..
p
j+l
o
( 1)
t
(concentration I o) a certain num are establishe d . They add mono mer with rate constant k p until monomer is e xh au s t e d . T he polymerization en ginee r ' s problem is to determine the distribution of polymer c hain lengths at any sp ecified instant of the reaction . S olutio of the mass b alan c e Pj , for arbitrary j , gives the answer : At the in stant of intro duc ion of i niti at or ber of growin g chains ( concentration
lo)
n
dP . J dt
::: k M P j- 1 p
on
( 2)
ly
In equation s like ( 2) , it is fr eq ue n t useful to scale time with the rate constant and monomer c oncentration in order to elimin ate kp from explicit consideration temp orari ly an d to d ecoup l e E q . ( 2) from the monomer balance : dM dt
� -k M P. p LJ J
=
M ( O)
j= l
=
M
( 3)
0
Thi s sc ali n g is done by definin g a new time variable ( Dostal and Mark ,
1 936) :
T
=
ft 0
k p ( t 1 ) M ( t' ) dt'
_ ,
(4)
or ( 5)
n
Thus te mperature variatio s an d monomer depletion are eliminated from explicit consideration in calculation of the poly mer c hain -len gth distrib utio n . Usin g E q . ( 5 ) , E q . ( 2) becomes
738
Tirre l l , Galvan , and Laurence
TABLE 1
T echniques for Solvin g Polymerization Rate Equation s ,
with Anionic Polymeriz ation
I llustrated
Techniq ues
1.
.
substitution :
Analytical , s uccessive
p
=
1
-
p
=>
1
=>
1
1) ! 2.
N umerical :
1.
conceptually identical to
l OT
j-1 -T
U se d when
are either impossible to inte grate analytic ally or form for
3.
Pj
cannot be found by
conve rts the in finite se t of differential - differ
Laplace transform :
J
=
P. ( A ) J
=
]
=
P.( A )
ia> 0
i<X>
P ( T )e j •
P e
0
T he rate equation b ecome s ]
AP. :::
pj
=
A
-P. J
+
P]. - 1
_!- 1 �-1
j
=>
From a compilatio n
T
(n
rate equations a general
when
succes sive sub stitution .
ence equations to a sin gle difference e quation
L [P.( T ) ]
e
�
2
pj
=
j
10
-AT
-AT
dT
dT
=
J
AP.
(,. � s
of Laplace transforms ,
n-1 _
1) !
e
-T
e
- AT
u
Th s
e
-T
dT
- I . e.
739
Poly meriz ation Reac tors
( Continued )
T A B LE 1
4.
Generatin g
function transformation : converts the infinite set to a differential equation (ODE ) . In principle , can pro M oments are easily obtained o t herwise . vi de the total M W D . Definition : ordinary
single
co
L
G (s , T ) =
sjPj ( -r )
j=l
Moments are
a
k
(a
obtained
G(s , T )
In
s)
s= l co
j s P.
J-
1
= sG ( s , T ) , the rate
G ( s , T ) = - ( 1 - s) G (s , T ) = si e 0
G ( s , -r )
-
the
Expandin g
(1
-
10
C omparin g
5.
r; 1
=
si
0
s) T
� j ,JLJ S (j 1) ! .
1
e
_
power series in s yields -T
j= l
-T ,j - 1 l) ! e _
generation :
\l k = momen ts .
e q uation becomes
with the definition , we obtain
10 (j
M o men t
G ( s , O)
exponential in
00
G ( s, T ) -
P ( T ) :::; j
technique 5 b elow)
= ll k
k
E. 1 J=
U sin g
by notin g ( see
jk P j ,
to
uses the
definition
of
convert the infinite set
the moments
of the MWD ,
to a fe w ODEs
for the
Used when no analytical solution is possible or when tak derivative of G ( s , T ) is c umbersome , an d the in formation given by moments s uffices . The rate equation become s in g
�
logarithmic
the
00
0
k
=
E
j= 1
• I
J
.k p j - 1 - \l k
� ( 0)
Lettin g k :::; 0 , 1 , 2 , . . . )l
.
0
ll 1
=
=
0
ll o
> ll
:::; I
0
gives
0( T )
=> )l l( T )
= l =
0
1 0( 1
+
T)
Tirrell , Galvan , and La urence
740
TAB LE 1
( C ontinued)
Frequently , only the three leadin g moments are needed . 6.
Continuous variable approximation : assumes the chain sufficiently lon g such that the chain len gth j can be considered to be a con tinuous variab le . Exp an din g P . _ 1 in a T aylor's series about P . , 1 J the followin g is obtained :
3 aj
+
1 2
-
2 a P. ___I 2 + aj
T runcating after the quadratic term , the rate equation ( an set ) is converted to a p artial differential equation for P . : a P.
a P.
_l + _J. d T aj
1 a
2
J
P.
- ___]_ 2 a j2
with solution J
P.( T )
=
10 ( 21r T )
in finite
1/2
exp
r( lj
- 1 - T) 2-r
2)
This approximation is valid only for reasonably high molecular weight, 1 / j small , for the T aylor' s expansion to be vali d . Equiva lent to Pois son distribution for large T . 7,
S tatistical techniques : historically , the first techniques used in polymerization problem s . Can be classified as follows : Direct approach Markov c hain theory approach Recursive approach
Pj
= P. 1 J-
-
P. J
Flory ( 1 953) Lowry ( 1 97 0 ) Lop e z - S errano e t al . ( 1 980)
( 6)
where the dot signifies a derivative with respec t to T . Techniques for solvin g this equation are enumerate d in Table 1 . Some additional results on the resultin g p roduct distribution are given by Peebles ( 1 97 1 ) . I mplicit in thi s reaction scheme , and necessary for this time - scalin g procedure to work , is kp indepen dent of j . This is known as the eq ual reac tivity as s ump tion , meaning that the reactivity of a chain end is un influenced by the len gth of c hain to which it is attached ( Flory , 1 9 53) . T his is an assumption which is very often , but not alway s , correct . I t
Po lym eriza tio n
74 1
R e ac tors
has been examined for se veral different mechanism s and certain exception s have been noted , the most important of which are the termination reaction in free radical p olymerization ( C oyle e t al . , 1 98 5 ) and m onomer addition at high conversion in p olyc onde nsation ( G upta and Kumar , 1 9 8 5 ) . C ertain solution techniques for p olymerization equations work better in certain situation s . O nly experience teache s this , but some guidelines can be laid down . Analytical solution s ( technique 1 , T able 1) are the excep tion , not the rule , and are more the province of the academician than of the engine e rin g p ractitioner . However , this does not mean that sequential com p uter n u m e ric al solution ( tec hnique 2 , Table 1) should be called in im mediately by those with pragmatic views . Moment generation via techniques 4 and 5 of Table 1 will lead to a set of moment equations that may be solva ble analytically , even when the mass balance cannot , an d may provide adequate , though incomplete p redictive information about the MWD in the polymeri zation reactor . It is important for the en gine e r to decide in ad vance what level of analy si s is appropriate , based on the level of informa tion required from t he analysis . It is frequently a good idea to set the simplest in formation first (i . e . , m o ments ) and then move on to attempt more involved analytical or numerical solutions . T he level and varieti es of mathematics required for polymerization analysis are shown in T ab le 1 . Analy sis i s not a s arcane as it is some times made to seem . Solution of ordinary differential equations is fun da mental . Partial differential equation s arise in several ways : ( a ) when the continuous variable approximation ( Bamford and T o m p a 1954 ; Bam ford et al . , 1 95 8 ; Zeman and A mundson , 1 963 , 1 965a , b ) is made (technique 6 , T ab le 1 ) ; ( b ) w hen diffusion limitations are considered ( N agel e t al . , 1 980) ; ( c ) in some applications of generating functions . D i fferen ce equations , particularly of the form Pj = a.Pj - l o ari se frequently . It is useful to reali ze that the solution to thi s first-order difference equation is Pj = K all . I n other w ord s 1 this difference equation h a s a geometric dist ri b ution a s its solution Clearly , a sine q ua non is facility with the ideas of probability and distrib ution function s . Serie s of various types , and their infinite ( an d , les s often , partial ) sum s , arise frequently . Familiarity with the short list of mathematical areas above will provide the engineer with tools neede d to employ modelin g profitably . A final c aveat on mathematical an alysi s is never to let it become more complex than neces sary , nor let it become an end in itself . T wo rules help to avoid this . One , men tioned above , is to have a good idea in advance of the kin d of information you want out of an alysi s before wadin g into it too deeply . The second rule is to look always for a phy sical interpretation of any derived eq uation . F re quent ly b earin g in min d the two generic and complementary modeling ap proaches , mass balance and statistical , is h elp ful in this regard . For example , the difference equation mentioned above ; ,
.
,
( 7)
arises in many polymerization ma ss balances . In the anionic polymeri zation mechanism of T ab le 1 , but occurrin g in a steady - state continuous stirred tank reactor of average residence time 8 , a balance on Pj with no Pj in the feed gives
( 8)
Tirrell , Galvan , an d Laurence
742
[ compare E q . ( 2) ] , whic h i s exactly the form o f E q . ( 7) with
ex =
k M P 1/6 + k M
( 9)
p
Alternatively , we could arrive at E q .
( 7) by
an eq uivalen t ,
but at first
si gh t perhap s different , statistical argument .
We can re gard the concen tration Pj as the p rob ab ili ty o f for min g a poly m er of length j . Pj is t hen the product of two ot h er p rob ab ilitie s : Pj - 1 • the p rob ab ility of forming a p ol y m e r of len gth j - 1 time s , an d ex , t lie reac ti on p rob abili ty for addin g a monomer to the g ro wi n g polymer . E quation ( 9) then giv e s an e x p li cit The gro w in g c hm n has p hysic al interp re t ation of thi s reac tion p rob abili ty . two c hoices : ( a ) add a m on o m e r with rat e kpM , or ( b ) wash out of the reac to r with rate 1 / e . Equation ( 9 ) give s tlie probability t h at it reacts . Very similar stati stical ar gum en ts app ly to several ot her p oly m eri za ti on
situations ( Lowry , 1 970) . PO LY M E R I ZA T I O N R E A C T I O N S
all p olym e ri zation reactions as eit he r step growth o r chain di sc u sse d by Len z ( 1 96 8 ) . I t is i mp o rt an t to n ote that this is a c l as si fic ation of reaction s , not of the polymers themselves , since there are in stances of poly mers that c an be made via either step - growth or chain - growth rou t e s . G enerally speakin g , however , most i m p ort an t poly mer s are made e xc l u si vely by one or the other . The p rin cip al characteristic d i s tinguishin g ste p - gr o wt h an d chain - growth reac tion s is that each poly mer chain grow s at a relatively slow rate over a much lo n ge r p eriod of time for the former than for the latter . In step grow th p oly meri zation there is usually a single type of chemical reaction that link s p oly m er molecules of all sizes . The re ac tion s are of th e same mechanism that link mono m ers , oli go mer s , an d poly mer s . I n contrast , a chain - gro wt h polym e ri z ation con sists of di stinc tly different chemical steps : initiation , which creates a hi ghly reactive p oly me ri zation locu s ; p rop a ga tion in which m on ome r (only ) adds to t he ac tive c e n ter , w hic h gr o w s in len gth ; ( freq uen t ly ) t er min ati on , in which the ac tive center is extin guished an d this polymer c hain n o lon ge r p arti ci p ates in t h e reaction ; (some time s ) tr an s fer , where the active locus is not e xtin guis he d p ermanent ly but rat her transferred in location from one growin g c hain ( whic h becomes dead) to another species in the me dium ( monom er , so lven t , a gen t , impurity , etc . ) , whic h t h en p rop a ga te s . T he existence of the active center in c hain growth p olym e ri za tion , in that not every c hain has one , is another i m p ortant distinction between step gr owt h and chain gr o w t h . In the rem ain in g p arts of this section we disc u s s the characteristic molecular weight di s t ribu tion s that arise in each of t h e se typ e s of polymer i zation in a b a tc h reactor . This will fu rth er differentiate the two types of p olym e ri zation . S ub sequent sections will deal with the mo dified distribu tions p roduced i n other ty pe s of reactors . Clas sification o f
growth has been
S tep-G ro w t h Po l y m e ri z a t i o n
The
m o s t important chemical reaction use d for th e p r eparation
via step - growth
poly meri za tion
is t hat
of poly m e r of addition and elimination at the
74 3
Poly merization Reac tors
carbonyl double bond of carboxylic acids and their derivatives (Len z , 1968) . T hese reaction s are the route s to most nylon s and polyesters . Other im portant step - growth reaction s are the urethane - formin g double -bon d addi tion and carbonyl addition - sub stitution for polyacetals and p henol-formalde hyde . Many , but not all step - growth reaction s produce a con densation product ( e . g . , water or glycol ) that must be remove d . Schematically , for the p urposes o f molecular weight distrib ution analy sis , the general step- growth reaction in its least encumbered form can be represente d :
pn
+
p
k
p n+m
m
n, m
=
1,
.
.
.
,
00
( 10)
In step - growth polymerization attention is focuse d o n the e n d groups of the polymer chain . Restrictin g our attention for the moment to bifunctional monomers , every polymer has exactly one end group of each chemical type involved in the linkage . Thus fractional conver sion of end groups is directly related to the number of polymer molecules , an d therefore also to the molecular weight and molecular weight distribution of the polymer . In the case of exact stoic hiometric equality of the concentration s of the t wo functional group s , we expect that a secon d-order rate law will govern the increase in fractional con version , p , of c 0 initial functional groups :
� --
( 1 1)
dt
which inte grates to
� = c 0kt + 1 1
(12)
Kinetic data from many step - growth polymerizations fit the rate equation ( 1 2) ( Flory , 1 95 3) . One exception occ urs in polye sterification with no added acid catalyst ; in this c ase , acid functional group s participate also as catalysts an d thus the rate law is third order . As we shall show below , the number - average de gree of polymerization in this reaction is 1 /( 1 - p ) . So a result of this analysis in Eq . ( 1 2 ) is that the number - average molecular weight builds linearly with time over t he course of the reaction . T he analysis of the molecular weight distribution in the simplest step growth situation of Eq . ( 10) with equal stoichiometry of the two en d group s is most easily done with statistical arguments . The fractional conversion p is alternatively interpreted as the p robability that a functional group selected at random has reacted to form a linkage . If we inq uire after the probability of findin g one molecule containing n monomer unit s ; we realize that this is a molecule havin g (n - 1) linkages and one free unreacted functional group of a partic ular type . The probability of having ( n - 1) linkages is the product of (n - 1 ) separate probabilities that an ester link age would have been formed , pn - 1 . So the probability of fin din g the de sired molec ule with ( n - 1) linkages and one unlinked end is pn
=
( 1 - p)p
n- 1
which is the geometric or " most probable " distrib ution . T he average molecular weights and chain lengths resulting from this distribution are
( 13)
744
Tirre l l , Galvan , and L aurence
given in Ray an d Laurence ( 1 97 7 ) . T hese authors , foll owin g Abraham ( 1 963) and ot hers ( Kilkson , 1 96 4 ; S zabo an d Leathrum , 1 96 9) , h av e shown how the mass balance a p p roach for t h e mechanism of E q . ( 1 0 ) leads to the same result as E q . ( 1 3 ) . From an en gineerin g viewpoint one of the most i m p ort ant c haracteristics of E q . ( 1 3 ) is that fractional conversion completely specifies the entire molecular weight distribution an d therefore all the aver a ge molecular weights . Conversion control is the only means of molecular wei ght control for this system available to the reactor en gineer . The nu mber- an d wei ght-average de grees of polymerization resultin g from the distribution equation ( 13) are 1/ ( 1 - p ) an d (1 + p ) / ( 1 - p ) , resp ectively , whi c h show that only as p nears 1 is a hi gh - molecular- weight product produced . There a re two signific ant exception s to this absolute control of the molecular di stribution by conversion . These are interfacial polymeri zation an d nonlinear , that is, b ran c hi n g or c ross - linkin g , step - growth poly m eri za tion both of whic h are disc ussed later . W hile E q . ( 1 3) gives an acc urate global description of many linear step growth polymerization processe s , in p rac tice t here are i mp or t an t additional effects an d c om pli ca ti on s that often arise . S t oic hio m etry is sometimes in advertently or intention ally unb alanced . Monofun c tional species (chain stoppers ) may be adde d to gain an indep enden t means of control over the MWD . Equal reactivity may not strictly apply ; this is , monomer functional groups may react more r apidly than identical groups on longer chain s , Re versibi lity of reaction ( 10) can be particularly as conversion gets hi gh . an important consideration . Diffusion limitations on the rate of removal of con densation product or the rate of poly mer c ouplin g may dominate the kinetics at hi gh conversion . Two of th e s e effects will be considered further here to give a more complete picture of step - growth polymeriz ati on and to ill u st r ate the use of some of the mathematical modelin g techniques of the p recedin g section . W e shall examine , in t urn , the effect of addition of monofunction al agents and t he effect of reversibility of reaction ( 10) , both for the c ase of the batch reactor . R eacto r configu rati on effects are con sidered in a subsequent sec tion . This section conclu de s with brief di sc ussion s of interfacial step gro w t h polyme ri zati on an d nonlinear step - growth polymerization .
Effec ts of C hain S toppers ( G up t a an d K u m ar , 198 5 ;
Kilkson , 1964)
The practical sit uation is on e in which a monofunctional compoun d such as ace tic acid is added to a mixture of difunctional a mines an d acids , or amino acids , a common p ractice in ny lon - 6 manufacture . We now have two dis tin guishable kinds of polyme rs , those wit h two reactable ends , ( 14) and those with one end blocked by t he monofunctional agen t ,
p
( 1 5)
nx
We have , therefore, two kin ds of p oly me r lin ka ge reac tion s :
__:k"--1_ p
n+m
( 16 )
745
Polymerization Reac tors
p
n
p
+
k �� P
mx
( 1 7}
( n+m ) x
X - terminated molecules cannot react with one another . the two kin ds of polymeric species read : n- 1 " P.P . - kP ( 2P + P ) x /..J J n -J n 1 = j
dP
__!!. = k dt
n- 1
dP
nx = dt
P. p JX
2:
k
j= 1
.
n-J
kP
nx
P
p ( 0)
n
p
nx
( 0)
=
Mass balances on
( 1 8)
0
=
0
( 1 9)
= r . ( The x in parentheses is used to denote situaP ( x) n 1 n (x ) tion s where the same equation applies for P n or Pnx . ) The first terms on the ri ght -hand sides of each equation give the rate of formation of Pn ( x ) by linkages between smalle r molecules . The remainin g terms give the rates of dissappearance of Pn ( x ) by reaction will all other species in the mixture . The factor of 2 in the second term of E q . ( 1 8) is due to the fact that a Pn h a s two distinct ways to couple with any other P m . Equation s ( 18) an d ( 1 9 ) apply for n ;;;, 2 . For n = 1 , we have no formation reactions ; thus 00
where P
dP
1 dt =
dP � 1x
:::
- kP 1( 2P
=
- kP
1x
+
( 20)
P ) x
p
P 1 ( O) x
=
p 1x0
( 2 1)
It is convenient to scale time in a manner similar to Eq . ( 4) :
'[
=
it
k dt
0
( 22)
For the solution of the set of equations ( 1 8 ) to ( 2 2 ) we illustrate the generatin g function tec hnique , as in technique 4 , Table 1. We will need generatin g functions for the two distrib utions : ,
co
G ( s , -r )
=
- I
�
n= 1
00
G (s , T ) x
=
l:
n= 1
n s P
( 2 3)
n
n s P
nx
( 24)
Tirrell , Galvan , an d Laurence
746
A ddin g E q . ( 1 8) to E q . ( 2 0 ) and Eq . ( 1 9) to E q . ( 21 ) , m ulti p lyi n g the t wo results by sn , and summing a s in dic ated in E q s . ( 2 3 ) an d ( 24) gives two di ffer enti al equations for t he generatin g function s :
aG =
a -r
G
2
- G ( 2P
+
Px )
Cl G � = G XG - GX P 3 1"
G ( O)
=
sP 1 0
( 25)
G ( O)
=
sP
( 26)
Partial derivatives are in dic ate d 00 1 N ote that G ( x ) ( ) = E n= 1 P n ( x ) ( 26 ) give s dP d -r dP
=
-(P
=
X
d-r
2
+
pp X )
P ( O)
1xO
since G ( x ) is = P
=
p
(x)
·
a function of both
s
an d
-r .
Setting s = 1 in E q s . ( 2 5) and
( 27)
10
( 28)
0
illus t ra tin g that as molecular weight b ui l d s , the total n u m b e r , P , of difunc tional monomer s dec rease s but , as expected , the total number , P x , of mono functional species remains constant, although these chains do _ grow in length . Kilkson ( 1 964) s ho w e d that these equations are easily solved using normali ze d generatin g functions : =
y
G
p
yX = to
( 29)
GX p
( 3 0)
X
give :
Cl y
=
h Cl
y
x
"1T
Py (y - 1)
( 3 1)
X
( 3 2)
=
Py ( y - 1 )
from which it is clear that the two generating two distrib ution s , s a ti s fy the proportionality G
=
p
p X
Finally , to
G
make
and compare with t h e statistically derived r e s u lt of no monofunctional a g en t , it is con in terms of con version o f end grou p s Conversion , p , in this case is de fin e d as
contact
( 1 3) for the simpler situation vien t to solve E q s . ( 3 1 ) an d ( 3 2 )
rather t han
time .
t he re fore the
( 33)
x
Eq .
functions , and
747
Po lymerization Reac tors
( 3 4)
as t h e fractional conversion of all initial group s of the , say , acid type , includin g t ho s e on the monofunctional agent . The time rate of c on ve r sion can easily be obtai n ed from Eq . ( 2 7 ) : ( 35) whe r e r = 1 + ( P 1 x o i P 1o ) . always gre a te r than 1 , gives the of the initial addition of monofunctional agent . Dividing E q . ( 3 5) gi ves
� dp le a din g to
y (y - 1 )
1 -
y =
p
s at p
=
s toichiom e t ry ( 3 1) by E q .
( 3 6)
0
s( l - p) y -
( 3 7)
ps and by Eq . ( 3 3) . s( 1 - p) Y x - 1 - ps 1 -
_
( 38)
that the nor mali z e d distribution s o f monofunctional an d difunctional chains oo n - 1 xn , we see by are ide n tic al . R ec o gn i z in g t h at ax / ( 1 - b x ) = a E = 1 b n rewritin g Eqs . ( 37) an d ( 3 8 ) as power series in s that so
p
nx :: p X
(1 -
p )p n - 1
( 3 9)
In other words, even in th e presence of monofunctional a gen ts , we ob t ai n a p ro duc t with a " most probable" or geometric distribution like Eq . ( 1 3) . The distribution is still uniquely de t e r min e d by p , but in t hi s c ase p is not ex clusively a fu n c tion of time as it i s in E q . ( 1 2) . Here p is also dependent on the initial stoichiometry , r . For the si m p le st e p - gro w t h ca se leading to
Eq . ( 1 3 ) , p
With
=
( 4 0)
1
monofunctional agents ,
P =
(1 L) ! -
The e ffe c t of =
P 10
m olec ula r 1
1 - p
< 41)
r
weight is i m p o rtan t .
In b ot h cases ( 42)
Tirrell , Galvan , and Laurence
748
which for the case of added
chain
stoppers
gives
1
In
the high conversion limit
�- J DP
n P= O
=
( 4 3)
P /P 1 0 ) ( 1 /r)
1 - (1
P
+
0
and
w e fin d
r r - 1
--
( 44)
Thus final molecular weight can be influenced not only by reaction time an d conversion b ut also by addition of monofunctional agent . However, this addition does not c han ge the overall character or shape of the distri
bution very much , except to shift the mean to lower degree of polymeriza
tion .
The similarity and implications of these results for another case of practica l importance , that of unbalanced stoichiometry of the two difunc tion al monomers , should also b e c le ar . Effects of Reversibili ty and I n te rchange Reactions In practice , many step - gr o w th p oly m erizations , polyesterification and poly amidation include d , are run at elevated temperatures ( 25 0 °C or higher) in
order to force the reaction to completion and to drive off and . extract low molecular- weight condensation products ( e . g . , water or glycol) . The same high temperatures also p romote rever sibility of the p oly merization reaction as well as in terc hange reactions , such as transamidation and transesterifica tion , between the polymer molecules formed ( Lenz , 196 8 ) . It is of interest then to know what effect rever sibility has on th e molecular weight distribu tion . This example will also afford the opportunity to illustrate some additional useful techniques in polymerization analy sis . T he reversible st e p - growth mechani s m is written p
n
+ p
m
k k'
p
n+m
( 45 )
with the u n de rst an din g that the lar ger s p ecies Pn +m may unlink any of its n + m - 1 links . The in te rchan g e reaction is a sim ultan eou s unlinking an d relinkin g b y two polymers : ( 46)
Both reactions should be expected to have a similar effect on the molec u lar weight dis tribu t ion . The effect of both rever sibility an d in terc han ge is to unlink a poly mer at a randomly cho se n link . Statistical arguments have been used to show that a most probable distribution of the form of Eqs . ( 1 3 ) and ( 3 9) is obtained when reversibility is an important factor . This is, in fact , an important gene ral conclusion . Random scission processes , such as thermal or radiation degradation will also lead ultimately to a most probable distribution , eve n startin g from a monodisperse product .
74 9
Polymerization Reac tors
Reversibility presents some special difficulties to the mass balance ap proach to m o delin g ( G upta and Kum ar , 1 98 5) . The mas s balance on Pn re s ul tin g from the mechanism ( 4 2 ) is
dP
n- 1 k
n
Cit = 2
:E
.
P P J n-J
j= 1
kP P n
•
-
k' ( n
-
00
1) P
n
+ 2k'
:E
j=n+ 1
( 4 7)
- -
T he fi r st two t e r m s of this equation are identical ( aside from the factor of 2 difference in t he definition of the rate constants here ) to the first two terms in E q . ( 18) . T he thi r d term gives the rat e of dissappearance of P n The fourth term give s the by a Pn unlinkin g any of its ( n 1) b on ds rate of appearance of P n by larger molecule s unlinkin g , the fac to r of 2 accountin g for the fact that any polymer has two site s at w hich it can un link to give a polymer of some specified shorter len gth . Mathematically , th e nPn term an d the partial su m term due to reversibility pose the dif ficulties . Applyin g the generating function operation of Eq . ( 23 ) yields
!9_ at
=
� G(G 2
-
2P )
-
Is �
k'
-
.
�
- - -- )
as
G
0 1
sP
s
{ 4 8)
Rather than solve t hi s nonlinear first-order partial differential e quation , we shall use it to d e riv e moment equations ( see T able 1 ) by s uccessive
differentiation of E q . ( 4 8 ) with re spec t to In s an d takin g the limit as s + 1.
{ 4 9)
0
{ 50)
with initial con ditions ll ( 0 ) k
=
P
. Equation ( 50 ) expresses the fact that 10 00 the total number of monomer units , E nP , in the reac tion mixture ren n= 1 E vi den t ly Eq . ( 4 9) and ( 50 ) can be solve d together to mains con stant . give c on ve r sion p and de gree of poly meri zation DPn as fun c tion s� �me . It is foun d that the most probable di s trib ution relation between DPn and p , Eq . ( 4 2 ) , is obeyed .
,
on
Higher moments present the mathematical difficultie s . The set of moment e q u a tion s is not close in th at '�-� 2 depends '�-� 3 • the e q u ation for ll 3 would involve ll 4 and so on . This arise s from the thi rd term of E q . ( 48 ) . T e r m s like this also arise in analysis of other chain scission reaction s . One rou te to analysis of this type of problem is t he moment e xpan s ion p ro cedure disc ussed b y B amford and Tompa ( 1954) and Hulburt and Katz ( 1 964 ) . T his is an approximation procedure which for the purposes of ob tainin g relations amon g the moments, replaces t h e unknown true distribu tion by an expan sion in Laguerre polynomials about a stan dard di st rib ution
750
Tirrell , Galvan , and Laurence
for which the relations amon g the moments can b e calculated exactly . T hese known re lation s enable one to express , for e xamp le � 3 • in term s of the lower moments . Application of the Hulburt - Katz procedure gives
( 52) fact , this moment closure approximation has been examined in se veral reversible p olymerization problems and foun d to give very accurate n umeri cal re sults , 1% error w hen three to five terms are use d ( B amford an d T ampa , 1954 ; Gupta an d Kumar , 1 98 5 ) , for the moments w hen compared with the m uc h more laborious complete numerical solution . It cou l d there fore be recommen ded as a generally useful approximation provided that there is no a p riori reason to believe that the distribution will have a very un u s ual shap e . In summary , the effect of reversibility can be treated by the means de scribed here . Reversibility does n ot chan ge the conclusion reached from an alysis of irrever sible linear step - growth polymeri zation that the product MWD is most p robable with the mean dictated completely by fr ac tion al con version o f en d group s . Reversibility does, however , alter t h e fractional conversion itself . Means of drivin g the eq ui li b ri u m toward hi gher conver sion an d therefore hi gher molecular weight are frequently employed . In
Interfacial Po lycon densation T he precee din g discussion has illu strate d the intimate connection between conver sion , stoichiometry , an d molecular wei ght di stributio n in linear step growth poly meri zation . Reaction en gineers should always seek additional control handles on MWD . In ste p - gro w th polymeri zation , in terfacial reac tion provide s an altern ative to the strict con version an d stoichiometry con straints . T his has been exploited in polyamide p roduction . The principles of the tec hnique have been e xp lain ed by Kilkson ( 1 9 6 8 ) . It relies on w hat he has termed the concep t of s low influx . Con sider polymerization of two difunctional monomers of the AA an d B B types ( e . g . , hexamethylene diamine an d adipic acid or its derivative s ) , where we start with exact stoichiometric equivalent amounts of the t wo monomers . A " normal11 polycon den sation would be done homo geneously , ad din g all of both the monomers a t the begi n nin g . This is the sit uat ion of strict con version an d stoichiometry control , where most of the analysis of this section applies . I magine no w t he followin g alternative . All of the B B monomer is placed in the reactor a n d slow addition of AA is begun . As each increment of AA enters the large exces s of BB , i t will b e immediately reac ted on both ends by B B . This process c an be contin ued until the full stoic hiometric equivalent amount of AA has been adde d , cons umin g the last I f the addition has been very slow , the polymer at that point, B group . will b e of essen tially infinite molecular weight since all the AA a n d B B units have been linked into the polymer . A ddition of further AA , in stoichiometric exces s , now serves only to dilute this hi gh - molecular - weight product , not decrease molec ular wei ght as it would in the case of the homo geneous reaction with unbalance d stoichiometry [ see Eq . ( 4 4 ) ] . The poin t is that molecular weight is now in fluenc ed stron gly by the monomer addition rate , not e xclusively by conversion and stoichiometry . In interfacial polycondensation , the slow in flux of monomer is acc omp lished
7 51
Po lymeriza tion Reac tors by imposin g diffusion limitations .
Each monomer is dissolve d in one of a pair of m utually immiscible solvents . Reaction occurs at the interface , where t he monomer supply is limited by diffusion . In p rac tic e , interfacial p oly m e riza tion is don e under mild reaction con di tion s . T herefore , reactive monomers must b e used . Diacid chlorides are used instead of diacids . S tirrin g an d addition of emulsifier to create an d stabili ze large interfacial areas are also important considerations in this type of re ac tor .
Nonlinear S tep-Growth Poly merization N ot a great deal of literature is available on reactor de si gn for nonlinear step - growth polymerization . T wo factors probably account for this . One i s that n on linear s t e p - grow t h poly m ers have never been produced in very lar ge volu m e s . The other is that many of these materials are thermosetting materials , neither soluble nor fusible once made , so that the " reactor" is frequently simply a m old of the desire d sha p e . Examples of im po rtant non linear step - growth poly m e ri za tions include p henolic resin m a nufac t u re , urea formal dehy de resin manufacture , epoxies , and many p olyu re th anes . Nonlinear polymeri zation require s the presence of some m on o m ers with fun c tion ality greater than 2 . The am oun t of multifunctional monomer is another variable that influences MWD . For example , the n umber - average de gree of p oly me ri z ation is given by ( Flory , 1 95 3 ) DP
n
2
=
2 -
( 53)
pf
where f is the average functionality of the monomers containin g the branch in g group . For example , consider a polymerization of AB , B B , and a tri functional A monomer , A 3 . If there is one A 3 in every 10 ori ginal mon o m ers · con tainin g A fu nct ion al group s , then f = 2 . 1 . For linear step growth f = 2 an d E q . ( 5 3) reverts to the linear result : 1 / ( 1 - p ) , N otice that for f > 2 in E q . ( 5 3 ) DPn becomes infinite at conversions less than 1 . For e x am ple , if f = 2 . 1 , DY"n dive r ges at p 0 . 9 5 2 . Thus even a small amount of multifunctional monomer increases the molecular wei ght greatly at conversions les s t han 1 . Di ve r gen c e of the molecul a r wei gh t ( i . e . , gelation ) is the most spectacu lar feature of nonlinear p olymeri zation . The " g el " poin t where this occ urs i s most succinctly expressed in terms of the branchin g probability a., de fined by Flory ( 1 9 5 3 ) and discussed in detail by B ill mey er ( 1 971) . a. is the probability in a reaction mix ture of linkin g two m ulti func ti onal monomers . This will depend in gen eral upon conversion an d stoichiometry of b ranc hed an d linear monomers . Consult Flory ( 1 953) or Billmeyer ( 1 97).) for de tailed calculations for specific case s . Gelation occurs at a critical value of a. : =
a.
c
=
1
f
-
1
( 54)
whic h corre spon ds to th e divergence of the weigh t - average mole c ula r weight . Analy sis of the MWD up to an d after gelation is beyon d the sc ope o f this cha p t er . We simp ly conclu de wit h a few i mp ortan t observations . Weight - average molecular weight goes to infinity at the gel poin t , not the
Ti rrell , Galvan , and Laurence
752
numb er -ave ra ge molecular w ei ght . T his means that there is a " s ol " frac tion of finite - sized molecules p re sent e ve n after gelat ion . Historically , statistical methods of calculating MWD , s uc h as those of Flory ( 1953) and S tockm ayer ( 1 943) , have been employed . These methods neglect the pos sibili ty of cycli zation , that i s , different branched units b ein g c onnected by two or more chain s , w hich i s redundant from the point of view of the connectivity require d for gelation . T here is much interest currently in determinin g how im p o r t ant this process i s in t he structure of nonlinear poly mers ( F amily and Landau , 1 984) . Certainly as a first app ro xi ma tion the Flory - S tockmayer method i s excellent . However, it remains to be decided if th er e are sub tle but important in flu ence s of cy cli zation on t he gel point , mechanical propertie s , an d sc at te rin g . Fin ally , the more chemical en gineerin g- oriented mas s balance approach has only recently b ee n a pp lied to nonlinear polymerization ( Ver S trate et al . , 1 98 0 ; H en driks et al . , 1 98 3 ) . T hi s has t he advantage of more ready ge n e rali zation t o diffe rent reac tor c on fi gu r a tion s , such as continuous reactors . In ad diti on , the time evolution of th e MWD is more e xp lici t in t he mass balance treatment . Incorporation of cycliz ation in t he se methods is also p rob le matic . C ha i n -G rowth Pol y m e r i z a t ion
Chain - growth p olym eri z ation is base d on the application of a free radical or ionic re ac tion to a poly meri za ti on chain or m ac r omole cule whic h results from the propagation of one kinetic c hain reaction ( Len z , 1 96 8 ) . C hain growth polymerization is initiated by a r e active sp ec ie s produced from an initi at or or cat aly s t . A chain reaction can be initiate d by a radical producin g initiator such as azobisbutyronitrile or benzoyl pe r o xi de . It can be initiated thermally , p hot oc h e mic ally , wit h cationic and anionic initiators , or with true cataly s t s that are n ot consumed by the reaction
R
I - nR
1
( I n chain - growth p olym eri z at ion we m us t distinguish between grow in g chain s , whic h w e designate R , a n d dead chains with no active center attached , which we de signate P . ) The reaction c hain propagates by reac tion of a mon o m e r with the ac tiv e sp ec ie s or radical . T he propagation re action is simply represented by
T hi s repre sentati on does not detail the various methods or mechani sm s · of p ropa gation or growth of the active chain . Lenz ( 1 968 ) treats this q ues tion in con siderable detail . The kinetic chain can be tran s fe rre d by several reactions
Rj + M - Pj
R.
J
Rj
R
j
+
J
S - P. +
+ A --- P +
+
pk
j
- pj
R1
R1
+ R1 +
R
k
transfer to monomer
solv en t age n t p olyme r
753
Po ly meriza tion Reac tors Tran s fe r reac tion s terminate the active chain and initiate another one ,
We
will find that transfer reaction s do not affect the shape of the distrib ution ,
althou gh they affect stron gly the average de gree of polymeri zation .
T he
polymer tran s fer reaction in free radical polymeri zation s leads to branchin g ( B amford an d Tampa ,
1 9 54 ; G raessley et al . ,
1 96 9 ; N a gasub ramanian an d
G rae s s ley , 1 9 7 0a , b ) . Sooner or later the p ropa gatin g polymer chain is terminated not by a tran sfer reaction , b u t by annihilation of ac tive group s .
T wo ac tive groups
react with each other in radical polymeri zation by ( a ) combination , and ( b ) di sp roportionation .
is
A s horthand representation o f the termin ation reactions
combin ation disp roportiona tion The termination reaction s occur bet ween t wo large macromolecular active
Such a reaction runs the risk of bein g stron gly diffu sion con
specie s .
This leads to a phenomenon termed the ge l effec t or Trommsdo rff A n exten sive discu ssion of the ramifica ( T ulig an d Tirrell , 1 9 8 1 ) ,
trolle d .
effec t
tion s of this p henomenon is particularly important in the design of poly meri zation reactors involvin g chain - growth polymeri zation taken to
c onver sion s .
high
I onic and heterogeneous cataly z e d poly meri zations suffer
termination by small molec ule poison s present accidentally or p urposefully to c ontrol molecular weight .
Free Radical Of the several typ e s of polymeri zation s classed as chain growth , free
radical initiation has foun d more general in dus trial application .
This m ay
be largely hi s toric al , but it remains that a significant fraction of polymer p roduced today are made by radic al mec hani s m s .
F or the sake of simplicity , consider a b atch c hain - growth polymerization
involvin g only initiation , p ropagation , an d combin ation termination .
We
must de scribe the evolution of monomer , M ; active polymer sp ecies of chain
len gth j , Rj ; dead or in active p olymer of chain len gth
I.
The reac tion sequence is given as k
I
1
-
M + ]
R.
2R k
R.
1
2
R. 1 ]+
]
+ R
k3 k
P. k ]+
j,
Pj ; an d initiator ,
initiation
( 5 5)
p ropagation
( 56)
combination termin ation
( 57 )
T he material b alance on the specie s of interest are di
dt
=
- k
1I
( 5 8)
Tirrell , Galvan , and Laure nce
754 co
( 59)
dRj
= 2fk 1 o ( j 1
dt
-
k :!1 < Rj - l
+
1)
(j
>
-
Rj )
-
co
k R 3 j
}:
n= l
Rn
( 60)
2)
(61)
N ote that Pj = 0 for j < 2 . W e use t h e Dirac function o ( j 1 ) to de signate a term that occurs only when the argument is zero , in this c ase when j = 1 . The ass umption s implicit in this formulation are that :
delta
-
The reaction veloci ty con stants are in depen den t of molec ula r A fraction f of the initiator form active radicals . I sothermal reaction con ditions prevail .
1.
2. 3.
The generatin g function can be use d to collap se the in finite set of ential equation s to a finite set :
size .
differ
dl - = -k I dt 1
( 6 2)
elM
( 63)
-
dt
=
- k M G ( l) 2
Cl G ( s ) at
=
a H(s) at
=
2 k 3G ( s )
=
2fk11 -
a o( l) at
2fk 1 s - k 2M ( l 1
-
s)G(s)
-
k G ( s ) G ( l) 3
( 64)
( 65) k
3
G 2( 1 )
( 66)
where G an d H are t he generatin g functions for the R and P distributions, re spectively . These equation s can b e solve d formally s ubject to initial conditi on s at t 0: =
G(s, 0) M ( O)
=
=
H ( s, 0) = 0
M , 0
1 ( 0)
=
( 6 7)
1
0
The procedure is to solve fir st for I , then for G ( l ) , M , an d finally for G ( s ) an d H ( s ) . In doin g so , most analyses of free radical p olymeri zation use an assumption known as the quasi-steady - state app roximation ( QSSA) .
755
Poly merization Reac tors
Bamford et al . ( 1 9 5 8 ) s tate this fun dam e ntal a ss ump tion of radical chain - growth p olymeri zation as " t he assumption that the concen tration of radic al intermediates remain s c on st an t d urin g the polymeri zation . " More specifically , 00
I:
d dt
J
R.
j= 1
or
=
00
��
«
J
R.
d
j=1
( 6 8)
0
dM
( 6 9)
Cit
or that the reaction has proceeded so far that the net rate of c han ge of c on c en tra tion of radical in t e r me diat e s is very much le ss than both thei r rate of p roduc tion an d their rate of destruction . One can examine more rigorou s ly what the Q S SA really means by at te mptin g a rigorous solution of some of the equation s an d co mparin g with the re sult of assumin g Eq . ( 6 8 ) T he equation s governin g the evolution of total polymer radical [Eq . ( 6 4) with s 1] , t he initiator [ E q . ( 6 2) ] , and the monomer [ E q . ( 6 3 ) ] can be s olve d for an isothermal b atch p olym e ri zation star tin g with only monomer and initiator . The equations are •
=
( 70) dG ( 1 )
� dM dt
-
At t
=
=
2fk I
=
1
2 k G ( 1) 3
-
( 7 1)
z·-
- k -M G ( 1 )
0, M
=
Mo, I
( 72) =
I o , and G ( l )
0,
T h e solution t o Eq . ( 7 0) can b e
obtaine d re a dily :
( 73) The equation for total polymer radical can b e tran sfermed into a classical Riccatti eq u a ti on with the followin g defini tion s : y
=
G ( l)
10
--
( 74)
756 E quation ( 7 1 )
,
Tirre l l , Galvan , an d Laurence then , becomes
( 7 5) Solution methods for Riccatti equation s are well p re sented in Davis ( 1962) . The sub stitution x dv y = - dx o.v
( 76)
into Eq . ( 7 5) leads to
( 7 7) The solution is strai ghtforward an d leads to the following expression for y :
-
ll K l ( y ) I l ( "y x ) - K 1 ( Y X ) I 1 ( y ) a K1 ( y )I ( Y X ) K 0( y x ) I 1 ( y ) 0
y -
( 78)
where K i an d It are the modified B essel functions of order i . . We c an make a comparison of t hi s e xact solution to that solution re sultin g from the q uasi steady - state approximation (Q S S A ) . If the assumption characteri ze d by E q . ( 6 8 ) i s applie d to E qs . ( 7 0) and ( 7 1 ) , the result is
( 7 9) This suggest s that Q S S con ditions p revail if y
;;!
y ' which requires s ( 80)
Saidel and K atz ( 1 96 7 ) an d B amford ( 1 96 5 ) have shown this to be a reasonable ass umption for value s that can be as signed to y for typical radical chain - growth polymeri zation . The Q S S A leads to the followin g representation of the kinetic s : dl dt
- =
dM
dt
-k
1
I
= -k _M G ( 1 ) z-
=
-k
2
(2fk ) 1 -
k3
( 8 1) 1/ 2
1 1/ �
( 8 2)
( 8 3)
Po lymeriza tion Reac tors d
�; s ) ;;:
dH ( s ) dt
=
2 fk. I s
=
0 k
f
1
G ( s)
-
757
k t'1( 1 - s ) G { s ) - k 3G ( s ) G ( l)
( 84)
2
( 85 )
T he quasi - s teady - state assumption yields
G ( l)
=
G(s)
=
( ) 2fk 1
-
( 86)
I
k3
an d
1/ 2
as
1
where
( 8 7)
bs
1 2 a = ( 2fk 1I ) [ k :f1 + ( 2fk k l ) / ] - 1 1 3
b
=
+
k :f1 [k :zM
( 8 8)
( 2fk 1k 31 ) 1 / 2] - 1
N otice that this is exac t ly the same form of the generatin g function as Eq . ( 37 ) found in step - growth polymeri zation . This is an i mp ortan t general con clusion concernin g free radical polymeri zation : the distribution of growin g chain s is most prob able . This is the same di stribution as is foun d in step growth polymeri zations , but the param eters of t h e distribution ar e con trolled by entirely different factors . H ere the mean of the MWD chan ges with conversion since a and b depen d on M an d I . T he distribution of the terminated chains is sli ghtly more complex and is disc us sed b elow . Monomer consump tion is given by the solution of E q . ( 82) which can be rewritten as
d - (ln M ) dt
The
(--)
2 fk. I 1 0 -k k3 2
=
,
1/ 2
exp
( 8 9)
solution of the monomer equation can be exp ressed as
ln
..!Y!_
M
0
=
- ( )
Note that as t M
00
1n M o
2k
2
k1
+
2fk 1 1 0
k3
1/2
[1 -
( 1t )] k
exp -
( 90)
2
oo ,
=
T his i s sometimes termed dead-end polymerization . initiator type an d amount will avoid this .
( 91)
Appropriate
choice
of
Tirrell , Galvan , and Laurence
758
The rate dM
-k
=
dt
of poly m e ri z ati on
( ) 2fk
2
is
given
by
1/2
1
1 11 �
k3
( 9 2)
Here we see the various influences on the kinetics . In c h e mi c ally initiated ra dic al chain poly m e ri za tion the rate is prop o r tion al to the 1 / 2 power of the initiator c onc e n tra ti on and the first power in mon om e r concentration . T he te rm in a tion rate c on s tan t k 3 , oc cu rs in the expres sion as a - 1 / 2 power term . Should for some reason the termination reaction velocity con stant decreas e , the polymerization rat e will in c re a s e . This is t he effect p re v i ous ly mentioned as th e Tromms dorff or ge l effect . T he reaction be have s muc h in t he fashion of an autocatalytic reaction . R e gar din g t h e chain le n gth distribution of the dea d chains , we get t ha t from the distribution of gr owin g chains via E q . ( 8 5 ) . We saw that
,
G(s)
=
so that for
--=-
the combination =
dH ( s ) dt
:-:-::
2fk I s 1 ---11 2 C k :zM l - s ) + ( 2fk 1 k 1 ) 3 ----
k3 2
(1 - bs) as
t ermin a ti on
( 9 3)
sc h e me
2
( 94 )
-
with a and b defined by E q . ( 8 8 ) . E q uati on ( 94 ) can be re a dily inverted by e xp an d in g t he ri gh t han d side as a po w e r series in s in a fa s hion similar to what was don e to E qs . ( 3 7 ) an d ( 3 8 ) with the re sul t
dP -"-1 dt
=
2 k a j 3 -- b 2
- 2
(J'
( 95)
1)
Since a and b a r e function s of time , readily d e t e r mine d from the solu tion s to Eqs . ( 8 1 ) and ( 8 2 ) , the entire distribution for Pj can be di rec t ly ob tained by inte gration of E q . ( 9 5 ) . Our discu ssion s e a rli e r suggested that many p oly m e ri z a tion schemes T he are best un derstood in terms of the m o m en t s of the di s t rib u tion m omen t equation s can be readily ob tained from the eq u ation for the tran s form o f t h e c h ain l en gth distrib ution . T h e moment- generatin g property l..l k _
1.
lm
.
-
k d G(s)
s-+ 1 d(ln s )
k
( 96)
gives the follo w in g se t of moment equation s from t h e o ri gi n al rate e q u a ti o ns :
2( 1
-
b)
2
( 97 )
759
Po lymeriza tion R eac tors
( 9 8)
k
=
3
a
2
( 1 - b)
4
( 2 + b)
( 99)
One can see that the instantaneous n umber- an d wei ght - aver a ge de gree of polymeriz ation have the si m p le form ( for b near 1 , which is necessary for high - molec ular- wei ght p o ly m er ) d
]J 1
d JJ d
o
=
J..l 2
d J..l
1
2
( 1 0 0)
1 - b
3
( 10 1 )
1 - b
a n instant [ the ratio o f E qs . for the m echani s m of combina tion termination used in t hi s discus sion of radical chain - growth p oly m e ri z a tion . Notice t hat b is very similar in its definition to the reac tion probability p of s te p - grow t h polymeri zation [ se e E q . ( 1 3 ) ) , or the a of Eq . ( 9) . W e have disc us s e d a very simple mec hani s m for radical chain p oly m e ri z a tion . I t is worth e x ami nin g b riefly w hat h ap p en s when o th e r reaction s occur , c hain .transfer an d branchin g , for e xample . Con sider the followin g mechanisms : T he poly dispersity o f any polymer forme d i n ( 100) and ( 1 0 1) ] is constant and always 3 / 2
k1
I ]
R.
]
l
+ A
R. + R J
initiation
k2 + M -- Rj + l
R. + M
R.
2R 1
k3
k4
pj
- Pj
k5
+ Rl
+ Rl
- P. k ] +k
p ro p a gati on
tran sfer to monomer
transfer to solvent or agent
_ ,
combination termination
The material balances for such a polymerization in a batch re ac tion are given by
( 102)
Tirrell , Galvan , and Laurence
760 00
:
- (k
=
+
2
k )M 3
L
j:: 1
dR,
=
_j_ dt
Rj
( 103)
(�1· t, k4A E R< )6(j k5 E==1 Rk)Rj Rj l - ( k5 E +
00
k�
Rj +
1)
R-= 1
k:f'
+ k � ( Rj _ 1 -
00
+
k 4A
( 104)
+
n
00
(
"
2 1
-k
- 1)
k== 1
( 105)
These same eq ua ti on s can be pu t in the generatin g function format : dl -
dt
=
�
d (s) t
=
I
( 106)
1
[ 2fk 1I
-
+
( k aM
k aMG ( 1)
+
k4
( 107) AG ( l ) ] s - k MG ( s ) ( l - s ) 2
( 1 08)
+ k A ) G ( s) - k G ( 1)G( s) 5 4 ( 109)
( 110)
(2fk 1) 1 /2
The QS SA can lead to c on siderable simplification since
0 ( 1)
=
k5
1
( 111)
( 1 1 2)
The distrib ution produced by this mechanism is not significantly different from that pre sc ribed by the simpler mechanism . Only the mean value s are different . G ( s ) can a gain be written in t he form
761
Poly meriza tion Reac tors G ( s)
-:-=1
where a =
�-
as
=
( 1 13)
bs
2fk I + ( k 3M + k 4A ) G ( l ) 1 k 5 G ( l) + k -}'1 + k 4A + k M 2
( 1 14 )
E qu ation ( 1 1 0 ) can be greatly simplified to yield =
dH ( s )
dt
(
k __§_
2
- ) as
1
2
bs
+ (k 3M + k A ) 4 1
-
as
( 1 1 5)
bs
The e q uation for H ( s ) can be i n ve rt e d readily to give t he equation for Pj :
:j
=
k 5a
where
2
(
;
j
1
b
j
-
2
+ Sb
j- 1
)
( 1 16)
( 1 1 7) A gain we should note that t h e quantitie s a , b , an d 8 are function s of time , readily evaluated from the con servation equations for M , S , and I . N ow it mi ght help to e x amine the moment equation s . Let the moments of the radical an d polymer distribution s be de fin e d as
]Jj
-
00
00
:E n=1
and
( 118)
W e have seen t hat the distribution for polymer radicals has a most pr obab le distrib ution [ i . e . , E q . ( 1 1 3 ) ] , so that the momen ts of the polymer radical distribution s are • I
cr = 1 (1 0
2
=
a
-
( 1 2 0)
b) 2
( 1;.._. +....b ;;..:) .; :;... a ,.; ;;;;.
(1
-
b)
( 119)
3
( 1 2 1)
The equations for the momen t s o f t he distribution can be readily derive d from E q . ( 1 1 0) in terms of the moments of the radical poly mer distribution
Tirrell , Galvan ,
762
an d Laurence
( 1 2 2)
( 1 23)
( 1 24)
T h e in s tan t aneous wei ght - and n umbe r - aver a ge de grees of polymeri zation then assume the form
d� 1
d 1l 0 =
o ( o + Sa ) 1 0 o0 ( o / 2 + S a ) 0
( 125)
( 1 26)
T he re lation s for o 0 , cr , an d cr 2 1 1
1
d
].l 1
-
[
1 + 13 ( 1
b 1/ 2 + 13 ( 1
-
-
afford considerable simplification :
b)
b)
]
( 1 2 7)
=
( 1 28)
will show th at as 8 + 0 ( no tran sfer reactions) the in stan taneous poly disp ersity has the same value , 3 / 2, as the simp le mechanism and th at in the limit of dominant tran sfer 13 -+- oo , the polydispersity a p
A little exercise
proaches 2 from below . V o l ume C hange
in Radical
Po lymerization
V ol e han g e is c n effect in un dilute d or b ulk the case of styrene , H ui an H amielec ( 1 97 2 ) used linear tween vo m e and con version ,
um c
a ommo
lu
d
a
polymerization . cor relation
For
be
V = V ( 1 + e:X ) 0
w he re e: is the fractional c h an ge in volume of the system between 0 an d 1 0 0 % c on v e r sion . e: de p en d s also on te m p e ra tu re an d may be calculated reasonably accurately from the indivi d ual properties of mon o m e r and poly mer , by a s s umin g volume additivity and dis r e gardin g molecular weight dependency . For the case of styrene , v alu es for e: ra n ge app ro xi m ate ly from - 0 . 1 6 at 6 0°C to - 0 . 2 2 at 1 80° C ( volu me con t r ac t ion ) . The effect of in c orp o ratin g volume change in the kinetic model can be evaluate d quantiatively . While there are n o appreciable changes in the conversion time relation , it does affect the p re dic tion of average molecular
163
Poly merization Reac tors
weights . E rrors due to neglect of contraction in the poly dispersity may app roach 50% at hi gher conversion s .
A nionic Po lymerization Anionic p olymeri zation is important in dustrially for various types of co polymers and also b ecause the same b asic kinetic scheme is foun d in many heterogeneously catalyzed polymerization s . In the course of illu s trati ng many of the techniques of molecular weight di stribution analysis , the simplest ver sion of anionic polymerization ( instantaneous initiation , no termination ) has already been examined . T his mechanism , which can be closely approximated in p ractice with very pure sy stems , produces the narrowest possible MWD for any as -polymerized , unfractionate d material . I t is in struc tive to examine what can happen if some of the condition s of simple ideal anionic polymeri zation are not achieved . W e look b riefly here at the case of slow initiation . The kinetic scheme for this mechanism might be expressed as
T his is still a simple mechanism readily de scribed as before . balance s are
The species
( 1 31 ) 00
dM
=
dt
R
dR .
( 132)
n
1 ) - k 2M ( Rj - Rj _ 1 )
( 1 33)
This set of equation s can be reformulate d usin g b ot h a gen eratin g function an d the tran sformation : T
=
f
t
0
k
�
. ,
dt
( 1 3 4)
an d E q s . ( 1 3 1 ) to ( 1 3 4 ) become dl dT
dM
CIT
=
=
-al
-al
( 1 3 5)
-
G ( l)
( 1 36 )
76 4
Tirrell , Galvan , an d Laurence
a. l
dG ( 1 )
=
( 13 7 ) ( 138)
a. I s - ( 1 - s ) G ( s )
The initial con dition s are tran sformed to 1 ( 0 ) = l o , M ( O ) 0. A solution to Eq . ( 138) c a n b e readily written as G(s, T)
=
a. I o s
1
-
a.
- s
(e
-
a.T
-
e
= Mo .
and G ( s , 0)
- ( 1 - s ) T)
T he inversion of G ( s ) is straightforward , if lengthy , and has been pre sented by N an da an d Jain ( 1 96 4 ) :
Rj
00
=
a. I e - T( l - a. ) 0
-j� k=j
[(1 -
k!
a.) T) k
( 1 39)
This distrib ution is often termed the Gold dis trib u tion an d for values assigned to the parameter a. the distribution can vary from very narrow ( Poisson distrib ute d ) to very b road at large a. for small a. .
Cationic Polymerization ( Plesc h , 1 96 3) As in most chain - growth polym eri zation s , in c ationic reaction s the chain must be initiated
( 1 40)
This sc hematic repre sentation is typical of initiation by proton donor ( p rotonic acids ) . Lewis acids form another class of initiators . For such sy stems one generally needs a coc atalyst . The initiation is very rapid and reversible : ( 141) There are, of c ourse, o th e r means of ionic catalysis , but we will , in our disc ussion s , restrict ourselves to these two . The active chain propagates
R. J
+
M
( 14 2 )
T h e propa gation reaction c a n b e comple x as a result o f many intramolecular rearran gements . In our di scussion we avoid these but maintain an aware ness of their e xi stenc e .
Polymerization Reac tors
765
Transfer reaction s occur com monly in cationic polymerizations . They are often calle d termination reaction s . We avoid that usage since the kinetic chain is n ot destroye d .
( 143) Spontaneous transfer termination can also occur and is de scribed by
( 1 44 ) T he kinetic c hain can be terminated ( by combination of the carbonium ion with the gegenion ) and is best represen ted by
( 145) Other tran sfer reaction s are important and should be considered in any model . T he analysis of cationic polymeri zation is much les s complete than for anionic polymerization due to the more complex kinetic sc heme . E qua tion s ( 1 4 0) to ( 1 45 ) represent a typical minimum set of reaction equations t hat should b e analy zed .
Heat and Mass T ra n sfe r Effec ts in C hain -G rowth Po lymerization Poly merization reactions are typically stron gly exothermic . Moreover, poly mer s are typically poor thermal conductors . Mass transport by diffusion is also a slow p rocess in polymer systems . H eats of reaction of 1 5 to 2 0 kcal/ m ol are com mon , as are thermal diffusivities of 1 0 - 4 cm 2 /s an d mass diffusivities , for small molecules in polymers , of 10- 8 c m 2 /s . For poly mers diffusin g in polymer , l 0 - 1 4 cm 2 / s is not an unus ually low value of the diffusion coefficient . T wo e xamples of tran sport effects that stand out for their practical im portance are (a) the autoacceleration or Tromm sdorff effect in hi gh conver sion -bulk free radical polymerization , and ( b ) the heterogeneous catalyzed polyme rization of ethylene , propylene , an d a.- oleim s . These will be di sc ussed b riefly in turn . Tromms dorff Effec t ( C ardenas and O 'D riscoll , 1 97 6 , 1 97 7a , b ; Ross an d Laurence , 1 97 6 ; Dionisio an d O ' D riscoll , 1 9 7 9 ; Marten an d H amielec , 1 9 7 9 ; Tulig an d Tirrell , 1 9 8 1 ; B alke ., et al . , 1 98 1 ; S oh an d S undberg, 1 981a - d ; C hiu et al . , 1 98 3 )
I n a free radical polymeri zation conducte d in bulk o r solution , i t is found that monomer consumption follows the exponential approach toward complete conversion suggested by E q . ( 9 2) . However, when the reaction has pro ceede d to the point that the medium contain s 20 to 40% polymer ( i . e . , 20 to 4 0 % fractional conversion of monomer in bulk polymerization , later in solution poly meri zation ) , there is , in many free radical poly meri zations , a n abrupt increase in the rate of polymeri zation . T his occurs , a s su ggested under E q . ( 92) , becau se the termination rate constant takes a drastic d rop .
Tirrell , Galvan , and Laurence
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The prac tic al implication s of this phenomenon are e n or mous . D ue to the hi gh reaction e x ot h e r mic ity an d the relatively low thermal con ductivity of poly me ri c media, di spe n sin g with thi s adde d heat production is diffic ult an d can lead to si gn i fic an t , even dangerous temp erature excursions . The b ook by B iesenberger and S eb a s ti an ( 1 9 8 3) disc usses thermal runaway p henomena exten sively . Even without large temperature chan ges , the highly nonlinear kinetic b ehavior can lead to multiple s t eady state s an d other uncommon dynamic phenomena otherwise normally associate d with In addition to these op erati onal control problems , nonisothermal op e r ation . this a u t oac cele ra ti on chan ges t he nature of the polymeric p roduc t signifi cantly . Reduction i n t h e termination r ate produces lon g-lived free radicals an d therefore hi gher- molecular- weight poly mer . W hat is found a s the acc e le ration sets in is that the molecular weight distribution of the polymer b ein g p roduced at that in st a n t of time shifts ab ruptly to higher average molecular weight . A s this higher - molecular- weight p oly mer accumulate s , the ag gre ga te mole c u lar wei ght distrib ution b ecomes quite broad an d bimodal . The p hy sical basis of this phenomenon is well understood q ualitatively . It clearly r es u l t s from an ab rup t slowin g of the polymer diffusion process . As polymer c oncentration builds in the reaction medium , diffusion of large molecule s , like the growin g radicals , is goin g to be retarde d more than that of monomers . Therefore , the termination reaction , which involve s two large molecules , is in hib i te d more s tron gly than the propa ga ti on reaction an d the overall rate of polymerization goes up . C onv e r tin g this qualitative understan din g to practical , acc urate , quan titative , p r e dic ti ve modelin g is difficult , but also very import an t and is t he subject of m uch cur ren t activi ty and recent progres s . T he key p oints of any model of this effect are { a ) how the onset of the au toacc eleration is handle d , and { b ) h ow t he rate constants are s uppos e d to var y with polymer concentration and molecular weight in the reaction medium . I t has b een clear for some time that the onset of autoacceleration is no t the onset of diffusion control of the termination reaction . Even at zero polymer concen tration (i . e . , zero c onversion of monomer to polymer) , it has b een demon strated that the termination reaction rate is diffusion controlled , as it sc ales directly with monomer or s olven t vi sc osit y . What happens at the on se t , then , is an ab r u p t change in the charac ter of the diffusion control . Re cent work has ga t he re d convincin g evidence that the on set of acceleration corresponds to the on set of entan glement in the polymeri zatin g mixture . A fun damental approach to the modelin g of diffu sion - controlled reaction in thi s onset re gi me is very difficult sin ce t he b ehavior of the poly mer diffusion coefficient it self is not clearly un derstood here with respect to it s depen dence on c oncentration an d molecular weight . S everal authors have proposed similar , ad hoc means of dealin g with the ons et , which is to treat it as a transition , m uch like the transition from unentan gled to · E m tan gled b e havior seen in rheological prop e r ti e s with increasin g conc en tra tion . With minor variation s amon g them , all s u g ge s t that the acceleration onset occ urs at a concentration , c o , that varie s with molecular we i ght of the polymer like M- a , where the exponent a is between 1 / 2 an d 1 { Tulig an d T i r r e ll 1 98 2 ) . Exponen t s in this regime are found for the appearance of U l entan glement in rheological properties with increasin g concent ration timately , a smooth tran sition function from dilute to en tangle d behavior i s needed , b ut this will require advances i n o u r fun damental un derstan din g o f entan glement . T he re is a wider div er si ty of approac h to how the variation of kt with conversion an d molecular wei ght is handled after the on set . This pa rt of ,
,
.
Poly meriza tion Reac tors
76 7
the mo de l is the important p art for prediction of molecular weight distribu tion and the resultant polymer properties . The on set part of the model by contrast has more b earin g on the con trol of the reactor . There are se veral effective e mpirical or semiempirical approaches to treatin g the kt variation ( Ross and Laurenc e , 1 9 76 ; C ardenas and O ' Driscoll , 1976 , 1 9 7 7 a , b ; Marten an d Hamielec , 1 979) . Amon g these several have been developed from the notion that the termination rate constant should be inversely proportional to the viscosity of the reaction me dium (Marten an d H amielec , 1 979) . This hypothesis leads to a p articular dependence of kt on polymer concentration ( conversion ) an d molecular wei ght . Given that a number of adjustable p arameters are unavoidable in the construction of models like thi s , these models can be made to fit e xperimental data well . Recently , this kin d of model has been criticized on theoretical grounds and an alternative hypothesis p rop ose d , namely that kt should be propor tional to the diffusion coefficient ( D ) of the free radicals rather than the vi sc o sity of the entire reaction medium ( Tu li g and Tirrell , 1 98 2 ) . T his model implies that t he concentration an d molecular weight dependence of k t should follow that of D . Experimentally , these depen denc es of D are currently in the process of examination . The reptation model ( de Gennes , 1 976a , b ) of p olymer diffusion suggests that D varie s with the inverse square power of molecular weight an d with concentration to some power be t ween - 3 and - 1 . 7 5 , dependin g on the quality of the solvent . Experi mental evidence to date s upports the predicted molecular weight depen dence of D b u t s uggest s that the conc entration dependence is much stronger ( Tirrell , 1 9 84) . A model with k t depen din g on molecular wei ght an d con version as su ggested by the reptation model has been develop ed and shown to fit kinetic , molecular wei ght , and molecular wei ght distribution data well ( C oyle e t al . , 1 9 8 5) . H owever , this model , like its predecessors , con tain s adjustable parameter s , so that it is difficult to make a convincin g argument for its superiority based on it s predictive power . T he bot tom line is that any model w here there i s an abrupt s witc h in kt from a mild to a stron g depen dence on molecular wei ght and concentration can be made to fit the data reason ably well . T he behavior , especially con version ver sus time , is simply not very sen sitive to t he exact form s chosen for these depen denc e s . On the other hand , there is some recent evidence to s uggest that the model b ase d on the reptation theory is more fundamentally soun d . E xperi mental measuremen t s of kt have been made in solution s of polymethyl methacrylate in methyl methacrylate monomer where the molecular weight and concentration of p olymer in the reac tion medium can be varied inde pendently ( Tirrell an d T ulig , 1983 ) . In this way , one can examine kt at fixe d concentration an d vary the molecular wei ght of the added polymer and therefore the viscosity of the reaction medium . On dOin g this , one sees that k t is only very weakly depen dent on vi scosity an d molecular weight of added p olymer . T his is in contrast with early models but ex actly what is expected from the reptation theory , whic h focuses on diffu sion of the radicals . T h i s diffu sion rate is expec t e d to be in depen dent of the molecular weight of surroundin g chain s as lon g as they are sufficiently lon g and concentrated to be entan gle d . Only the molecular weight of the radic als plays a role in the diffusivity . T hi s area is one of active research curren tly and should therefore be watched closely by the en gineer who wishes to keep abreast . On the practical side , it should be mentione d that there are also several completely
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e mpirical models of autoacceleration cited ab ove that have been use d suc cessfully for certain modelin g purpose s . A fundamentally based theory for the reaction of two macromolecules is clearly more de sirable in the lon g run . Such a theory would fin d application not only to the problem of autoaccelera tion in free radical polymeri zation but also pos sibly to hi gh-conversion step growth polymeri zation and certain branchin g an d cross - linkin g reactions .
Heterogeneo us Polymerization I n several kinds of chain - growth polymeri zations the active center may be in a separate phas e , frequently in the form of droplets or particles , from other components of the reaction medium . E m ulsion ( Penlidis et al . , 1 98 5) , precipitation , and heterogeneously catalyzed polymeri zation s are all exam ( S uspen sion polymeri zation , while m ultiphasic , contains all ples of thi s . reac tive c omponents in one p hase an d therefore has more in common with b ulk an d solution polymeri zation s t han with heterogeneous polymerization . ) T hese poly meri zations s hare several features . One is that the active center b ecomes isolated somewhat by t hi s p hase separation an d therefore very lon g-lived . Very hi gh molecular weights can b e produced an d fre quently chain tran s fer additive s are used to con trol molecular weight . A secon d common feature is that reactants have to diffu se into droplet or particle form in order to participate in the reaction , openin g up the pos sibility of diffusion limitation s on the rate . Also, heat has to diffu se out of and be carried away from the reaction particles, presen tin g the pos sibility of local temperature excursion s . These problems have received the most attention in heterogeneously catalyzed polymeri zations , an important class of which are the Ziegler-Natta ( Z - N ) catalyzed polymerizations ( and copolymerizations ) of ethylene , propy lene , other a.-oleim s , and sometime s , dienes . T he class of Z - N cataly sts includes transition metal halide s (Ti, V , Z r , etc . ) complexed with alumin um alkyls or alkyl halides (Boor, 1 9 7 9) . T hese catalysts have been developed to the poin t that very hi gh rates an d yield of polymer ( per wei ght of catalyst) are obtained ( Goodall , 1 9 8 1 ) . This means that the initial porous c atalyst partic le is rapidly fragmented by an d buried in side the evolvin g polymer . Interestin g m orphologies ( B oor , 1 97 9) of poly mer particles are formed in this p rocess . How they are formed is incompletely understood , but it is known that some morp hologie s are more desirable than others for sub seq uent p rocessin g of t he polymer . The morphology of the polymer p article is obviously very important in regulatin g diffusion of monomer to the active site . A very common kind of morphology is one w here the poly mer particle is composed of an agglomeration of smaller microparticles , each of which presumably surroun ds a fragment of the original catalyst particle . • Diffu sion from outside therefore involves movin g throu gh the pores of the macroparticle ( b etween the microparticles ) and penetration throu gh the poly mer coatin g of the mic roparticle to reach the active site . S everal groups have developed diffusion - reaction models for Z - N poly meri zation s , some of whic h have attempte d to account explicitly for the morphology describ e d above . S timulus for this modelin g has been p rovided by the experimental fact that Z - N polymeri zations , almost without exception , produce a broad distrib ution of molecular wei ghts , even thou gh the chemis try at the active site resembles in many way s the anionic polymerization mechanism and therefore should produce a relatively narrow molecular weight distribution . D eb ate has develope d about the origin of the broad
Po lymeriza tion R eac tors
769
molecular weight dis t ribution . Possibilities are that (a) diffusion limitations produce a variable supply of monomer to different active site s ( Schmeal and S tree t , 1 97 1 ; N agel e t al . , 1 98 0 ; L au r enc e an d Chiavetta , 1 98 3 ) ; ( b ) multiplicity o f types an d ac tivity of ac tive site s ( C aunt , 1981 ; Keii e t al . , 198 4 ; Barbe et al . , 19 8 3 ) ; an d ( c ) time variation of site ac tivity (i . e . , deactivation ) . Diffusion limitation s certainly occur , but w heth e r they are stron g en o ug h alone to produce the b ro a d molecular wei ght distributions seen is doubtful . Active site h ete ro gen eity and deactivation have both been documented an d are very likely partly responsible for the b road mol ec ula r w ei g h t distribution s . For e xam p le , a rec en t an aly s i s ( Galvan and Tirrell , 1 985) ha s shown that the T hie le m o d ulu s , a =
a
�
D
( 146 )
a characteristic size of the polymer p a rtic le , k is the turnover active site , and D i s t he di ffu sion coefficient for mon o m e r , must exceed 10 for there to be significant molecular wei gh t distribution broaden in g by di ffu sion . D ata are limited an d t he model does not correspond ex ac tly to reality in that the chemical mechanism is o ve r simp lifi e d but , speak ing approximately , thi s seems to be on the hi g h end of w ha t might be foun d in p rac tic e . Clearly , the r ela tive i m p or tan c e of diffusion limitation s will dep e n d on whether t h e reaction is o cc u r ri n g in a slurry in liquid di l u en t or in a gas phase . F u rth e r m or e , Z - N copolymerization , such as t h at e mp loye d to pro duce EPDM ru bbe r or linear low - density polyet hylen e , may experience diffu si on e ffects on the c omposition an d sequence distribution s , even when the effects on the molecular wei gh t distribution are not particularly impor where a is rate at the
tant .
O PT I M I Z A T I ON A N D CO N T R O L
S ev e r al uniq ue problem s are encountere d in t he op ti mal desi gn an d control of polymeri zation reactor s , as disc ussed b rie fly earlier . One set of prob lems arises from the fact that t he perfor m an ce criteria for s u c h re ac t or s
are not readily defined . Further , such criteria are not universal for a class of poly m eri z a tion reactors . Poly m eri z ation re ac t or s pro d uc e mate ri al s w ho se q u ali ty is assessed in terms of s t r en gt h , stiffness , pro c e s s ab ility , an d so on . Even if th ese ul ti m ate qu ali ti e s can be reduced to more ab stract an d re a dily q u anti fi ab l e measures ( molecular w ei ght di st ri bution , cop oly me r c o mpo sition distribution , seq u ence distribution ) , it_ �s unavoidable that ( a ) these measures are distributions (or characteristic values thereof) themselves , and (b) different me a s ures react di ffere n t ly to chan ge s in t he reaction environment . Thus it is n ot at all apparent at first si gh t how to formulate properly a m e anin gful optimi zation p rob le m . This problem i s e s peci ally severe in the case of copolymerization (Ray and Gall , 1 96 9 ; Tirrell an d Gramley , 1 98 1 ; B ej ger et al . , 1 9 8 1 ; Garcia-Rubio et al . , 1 982) , D is tribution s of molecul a r w ei gh t , composition , an d seq uenc e are all i mp ort an t , and what op tim i ze s on e m ay de gr ad e an ot h e r . O b vi o u s ly , t he in for m e d ju d gmen t of a de s i gn e r is required at some stage in the op ti m i z ation p roce s s in or de r to set priorities a mong ob j ec ti ve s that may be in conflict . The abili ty to do this re lie s on accurate information
770
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Galv an ,
an d Laurence
about the process from data and modelin g . T he procedure that has been used virt ually exclusively in polym eri zation reactor optimi zation problems is to formulate a sin gle objective functional which combines all identifiable performance measures ( some of which may be in conflic t in the sense that they react in opposite direction s to manip ulations of the control variables ) with weightin g factors on each measure chosen a priori . This c hoosin g of weightin g factors is one means of exercisin g jud gment . S everal more restricted (i . e . , sin gle obj ective ) opti mi zation problems in polymerization have been treated ( Ray and Gall , 1 96 9 ; Hicks et al , , 1 969 ; Wu et al . , 1 98 2) . Examination of these cas e s help s build experience and an intuitive basis for polymeri zation optimization . Minimi zation of the b readth of the molec ular wei ght distribution in free radical homopolymeriza tion has been treated several times . Effective manip ulate d variables are addition rate of monomer or initiator and temperature . All of these studies reach the conclusion that the narrowest pos sible distribution of molecular wei ghts is obtained when the reactor produces a constant average molecular weight at every in stant in time . Examination of E qs . ( 1 2 5 ) and ( 1 26) shows that if initiator concentration is approximately con stant , average molecular weigh t will ten d to decline with increasin g conversion because of declinin g monomer concentration . A d din g monomer can the refore be used to keep molecular wei ght constant . Increasin g temperature will re duce the in stantaneous average molecular weight , so a decreasin g temperature policy could be used to control a down ward drift in molecular weight . Even this simple example , however , de mon strates the types of conflicts inherent in polym eri zation reactor optimi zation : decreasin g temperature produces a bet ter molecular wei ght distrib ution , but at the e xpense of a slower reaction an d therefore slower production of polymer . Initiator addition can be effective if con dition s are suc h that initiator concentration is declinin g ( dead - end polymeri zation ) , thereby driftin g molecular weight upward . Copolymeri zation problems enc ompass an even larger set of objective s than homopolymeri zation . In optimal design an d control o f copolymerization reactors , it is useful , indeed important , to un derstand the trade -offs amon g the objective s quantitatively b efore selectin g an " optimum" solution . T soukas et al , ( 1 98 2 ) emphasi zed this in their st udy of se mibatch copoly meri zation . They examined the objective s of simultaneous minimization of the breadths of the composition an d molecular wei ght distribution usin g temp erature and monomer feed rate as manipulated variables . The p rincipal conclusion of their study was that under no circumstances would it be wise to optimi ze wit h the weight very heavily on one or the other of these ob jectives . In every copolymeri zation situation that they examined there was a policy of adjustment of the manipulated variable ( temperature and mono mer flow rate) away from the optimum for either individual objective , which would produce si gnificant improvement in the other objective with very ll ttle damage to the former . T he practical le sson is that even when a p articular one of all the possible objectives seems to be dominan t , it pays to examine other relevant objectives to see if some imp rovement in these can be ob tained in addition , at little or no extra cost . This is even more valid if there are more control variab les available to manip ulate . The examples above have b een of op timal control desi gned to combat drift dispersion , that is , p roduct nonuniformity due to reaction conditions that change in time . T his kin d of dispersion occ urs in batch reactors ( or with axial position in tubular continuous reactors ) . T here is another type of dispersion , termed statistical dispersion , about w hich little can be
Poly merization Reac tors
7 71
done via optimal control . This kind of disp er sion re sults from the poly me ri za ti on mec hanism itself an d from the reactor in which t h e polymer is made . For ex a m p le , E qs . ( 1 27) a n d ( 1 2 8 ) s ho w that 1 . 5 is the mini m u m p o s s ible value of p oly di sp er sity for a free radical polymerization . This coul d be produced in a C S T R op er ati n g at steady state or in a b atch r e actor un der time-in v arian t conditions . Drift b ro ade n s t he di strib u tion . Choice of reactor is , of course , another element of polymeri zation op timi zation . I dentical reaction s run in different reactor confi guration s will p r o d uc e p roducts with different molecular w e i ght distributions . T hi s i s easily seen o n c o m p arin g the results o f Table 1 for batch an ion ic p oly m eri z a tion with those of Eqs . ( 7) t o ( 9) for a nion ic polymerization in a C S T R . T h e former produces a P oi s son distribution , the latter a most probable di s tri b uti on . T he relative b r oa den in g in the C S T R is due to the fact that flow out of the reactor introduces an effec tive termination in t he reaction scheme t h at is abs e n t in ideal batch anionic polymeri zation . This is one e xamp le of what might b e c alled Denbigh 's rule ( D enbi gh , 1 94 7 ) . T hi s rule c o u l d be ph ra se d in t h e followin g way : when the mean lifetime of a growin g chain is lon g compared to t he mean re si d en c e time in the C S T R , the C STR will pr o duc e a b roader MWD than the b atch reactor . This rule applies an d is born e out in p rac tice in anion ic c hain - growth and in st ep growth polymeri zation s . T he conver se o f Denbigh ' s rule i s not al way s true . Free radical chain - growth p oly m e ri zati on is an e xam p l e where the gro win g chain s are very s ho r t - li ve d compare d to the mean residence time . Here the shift from b atc h t o C S T R u s ually p r o duc e s narrowin g due to the eli minati on of drift dis p e r sion . In the ab sence of si gnific an t drift in t he bat c h s i tua ti on , n a rro win g cannot b e achieved in the C S T R since w it h such short li feti me s , t he gr o win g chains do not have the opp or t uni ty to sen s e w hether they have been made in b a t c h or C S T R . M uc h work on other ide al re ac tor configuration s , s uch as se gregated c ontin uou s stirre d tank reac tors a n d re cy c le reactors , h a s shown h o w these mo di fy mo l ec ula r wei gh t dis tribution as well ( M ecklenb urgh , 1 9 7 0 ; N auman , 1974) . Real reactor b ehavior can sometimes be represented ap p ro xi m at ely by these ideal reac t o r s . O n - line control an d op ti mi z a tion is a topic of si gni fic a nt current interest , facilitated greatly by the r apid evolution of c omp utin g equipment . P ar ticu lar interest is in a d aptive , se lf- t uni n g controllers . The i de a is to use a model of the proces s in t h e controller . A re d u c e d , ap hy sic al model is us ed w i t h p aram e t e r s t hat d o not n ec e ss arily have any physic al si gnificance th e m s e lv e s . H owever , algorithms are used for c on tinu ally revisin g the parameter values based on c om p a ri s on of process data with the de si gn e d control trajectory . In this w ay the parameters of t he con troller model are tun e d to give op ti m al p e rfor m an c e . This has been shown to be e s p eci ally effective for n on - s te a dy - s ta t e situations w here t u n in g of PID · con trollers i s difficult . Self-tunin g re gula tor s can , however , also be used t o s e t the gains of PID c on t rolle r s ( B ej ger , 1 9 8 2 ; S um mers , 1985) . On - line opti mi zation requires more c om p u ti ng power an d has not yet been accomplis hed to any si gnific an t e xtent . This requires a b e tt e r model of t he process in t he on - line c omputer that can c om p u t e updated op timal control policies b a se d on n e w data . Th e problem with this sort of scheme i s n o t t h e av ailab l e c omp utin g power , it is the availability of sensors to mon it or the p rocess in an in formative way . C urrently , on - lin e data on poly meri zation r e ac tor s is limited largely to temperatures , crude estimates of me di u m viscosity ( e . g . , agit at o r motor t or q ue ) an d perhaps some m e as u r e s
7 72
Tirrell , Galvan , and Laurence
of polymer content or conversion ( Del Pino et al . , 1 98 2 ) . Q uantities such as molec ular w ei ght an d its distrib ution , copolymer composition , and se quence di s trib u tion an d par ticl e size distribution in em ulsion p oly me rization must in turn b e estimated from these . Work is p ro gr e ssi n g on all of the se fronts : on- line light scattering and visco sity for molecular weight , surface tension and dielectric rela xa ti on for particle size distribution , an d gas c h ro m ato g raphy for copolymer compositi on . On -line gel perme ation chroma tography for molecular wei ght distribution re main s a desirable but difficult and dis tant goal .
REACTOR SELEC T I O N
The polymer " p roperty " t hat al l prac ticin g engineers really strive to opti mize is the bottom- lin e cost per unit p roduction , subject to perfor m an ce con straints . Meetin g these p erformance constraints has been the s ubjec t of most of the first p art of this c hapter ; however , cost can h av e important in fluenc e on the choice of polymeriza tion reactor . Simplicity an d en ergy efficiency are major goal s alon g this route . C on sideration s b eyon d cost an d polymer product performance en ter in as well . Viscosity of the reaction mixture is a primary c onsi der at ion . Ef ficient mixin g depen ds in p a rt on power input , which in t urn depends on viscosity ( Oldshue , et al . , 1 9 8 2 ) . An important caveat here is to take proper account of possible n on - N ew t on ian rheolo gic al properties of t he re action mixture ( M i d dleman , 1 9 77 , Chap . 13) . Flow p rope r tie s lar gely deter mine agitator size , po w e r , and design . Motionles s mixers are a l so an im portant possibility to c on sider ( M id dle m an , 1 977 , p . 327) . C onv eyin g an d p umpin g equip ment must be designed aroun d the flow properties of the re action mixture . E x t ru de r s can be u sed as pumps an d mixers . I n many case s , the flow properties in the reactor have determined historic ally the c h oice of r eac tor for particular purpose . The rationale for t h e se choices is foun d in the p aten t literature , which has been revie wed nicely up to the late 1 970s by Gerrens ( 1 9 8 2 ) . S eparation p rocesses t h a t may have to be acco mp li s hed after the p oly meri zation reaction can in fluence the choice of reactor . T wo principal separation operation s follow p oly me ri za tion p rocesses : elimination of mono mers or diluents an d separation of solid p olymer . Devolatili zation , to reduce residual monomer content to n e gli gible levels , is an important example of the former . Extruders , tower, and wip e d film reactors are all u s e d in processes where devolatili zation is a chief concern . T here are also many common design concerns between devolatilization and reactor desi gns for d rivin g certain step - growth polymeri zations to very hi gh conversion by re moval of con densa tion p roduct ( pres sure r e duc ti on , surface area regula tion ) . Effective se p a ra tion of solid product has b een one of the important innovation s in certain reactors , such as the Union C arbide fluidized-bed reactor process for p olyol efins . Loop reactors in co rporatin g a settlin g portion are also useful for separatin g solid polymer from slurrie s . Temperature and its control are always considerations . H ea t s of poly meri zation are typic ally hi gh , as mentioned earlier , so that maintainin g de sired temperature is not alw ay s a si m ple matter . Temperature can b ecome spatially nonuniform an d globally out of control . The typical consistency
Polyme riza tion Reactors
7 73
of the reac ti on medium is a gain a fac t or . Normal good heat t ra n s fer desi gn for lar ge heat transfer surface , with coils or corrugated surfaces , is not effective in polymerization reactors sinc e these surfaces c reate dead zones an d acc um ulate material . This can pro d uc e nonuniform molecular weight distribution ; foul th e h e at t r an s fe r e quip m en t , e xacerb atin g the heat re moval diffic ultie s ; an d make the reactor very difficult to c le an . For these reasons , s mooth heat t r an s fe r s urfaces are us ually p re ferab le , wit h the be s t possible agitation to sweep the fluid clearly over t he s ur fac e . An example of t he s e con sideration s can be seen in hi gh - p r e s s ur e ethyl ene p olyme ri z a ti on . Thi s reaction is done c om m ercially in bot h tubular and stirred autoclave reactors . T ubular reactors have a high surface- to-volume ratio , w hich is goo d for heat tran s fe r ; t hey have , howe ve r , no m ixin g , w hich may p ro d uc e s e gregati on an d in h om o gen eity . S tirre d-tank r e ac t or s have a comparatively low surface -to -volume ratio , al t ho u gh t emper a tu re can b e ma nipulated somewhat u sin g the feed temperature . Gen erally , these re ac tor s have better mixin g .
de sired form (pellet , powder , b ead , etc . ) can in fluence the re For examp le , su spen sion po ly m eri zer s p ro duc e beads that may be directly useful in proc e s sin g . On t he other hand , rou n d b ea d s can be dan gerous if s p ill e d and may not have sui tab le bulk flow prop erties . E xt ru de r reactors { S tub er and Tirrell , 1 9 8 5 ) are able to produce pellets , sheets , or c oa tin gs qui te e a sily an d direc t ly . S a fety con siderations always place inviolable c on straint s on p oly m e riz ation process de si gn . Ventin g to t h e atmosp here in a safe an d e nvi ron men tally sound m ann e r m u s t al w ay s be de si gn e d for . C le a r ly , the entire p roc e s s , n o t e xclu sively the reaction , must b e considere d in any u se ful polymerization p roces s optimi zation . The articles by P la t ze r ( 1 97 0 ) an d Gerrens { 1 9 8 2 ) give a d dition al useful in forma tion and in si ght into the p r ac ti c al aspects of reactor selec tion . T he
actor desi gn .
C O N C LU S I O N P olyme ri za tion reactors are rich in c h allen g es an d opp or t un iti e s for the chemical en gi n ee r . This ric hness derive s from t he co mp le xity of p ro d uc t distribution s in even t he si m ple s t polym er reactor , the diffic ulties of handl in g viscous , exothermic re ac tion masses where informative measuremen ts are extremely limite d , an d finally , fr om t h e fact t hat a polymer reactor is de si gne d to follow a molecular a rch itect ural b lu e p rin t for a ma terial . D e vi a tion from this blu e p rint is reflecte d dir e c tly n ot in the b ehavior of the re actor but in the p er fo rm an ce of th e material . This is c u rr en tly a charac teris tic n e a rly unique to p oly m e r reac tors , but may b ec o me an inc re sin gly i m p ort ant matter to de al with in other materials fab ric at ion r e e.c tors , s uc h as c he mic al v apo r d e p o sition reactors for microelectronics app lic ation s . This c h apter , while coverin g a fair amount of groun d , is at b e s t an intro du c tion to the t ec h ni q u e s of analy sis an d experience on the b e hav ior of p oly m e ri zat ion reactors . In p artic ula r , the very important topics of copolymerization , emulsion p oly m e ri za tion , an d varieties of con ti nu o u s Our hope is polymeri zation have re c eiv e d only supe r ficial atten tion here . that in th e techniques and e xam ple s we have p resente d , the reader will find the motivation an d m e an s to a t tac k n ew a rea s of p olym e ri za tion en gineerin g .
Tirrell , Galvan , and Laurence
7 74
N OTAT I O N initial c oncentration of functional group s in polycondensation number- ave rage de gree of polymeri zation
D
diffusion coefficient of monomer in polymer
f
func tionality of reactin g mixture (in polycon den sation ) ; initiation efficiency (in free radical polymeri zation )
G ( s, T )
generatin g function for gro win g c hains ( see T able 1 for definition )
H (s, T)
gen eratin g function for " dead" chain s
I
initiator concentration
k
reaction rate constant
M
monomer concentration
p
conversion of function al group s
pj r
concentration of growin g c hain s of len gth j ( in polycondensation and anionic p olymeriz ation ) ; concentration of " dead11 chains of length j (in free radical p olym eri zation )
stoichiometric ratio concentration of growin g c hain s of len gth j (in free radical
p olymeri zation )
time volume of polymerizin g media conversion in free radical polymeri zation normalized generatin g function , defined in E q . ( 2 9 ) normalized generating func tion for chain s with a stopper , in E q . ( 3 0 )
defined
G ree k Lett e rs a
y 6
e:
e
f.l k
p robability of p ropa gation ; p arameter in G old distribution ; Thiele modulus , dependin g on the context defin e d in E q . ( 7 4) Dirac delta function fraction al chan ge in volume due to reac tion residence time kth moment of the MWD of 11 dead" chain s (in free radical polymer iz ation ) or growin g chain s (in anionic polymerization ) ( see T able 1 for definition ) kth moment of the MWD of growin g radicals in free radical poly meri zation
T
X
,,
re duc e d time , defined in Eq . ( 4 ) defined i n E q . ( 7 4 )
7 75
Polymerization R eac tors R E F E R E N C ES
Abraham , W , H . , Path depen dent di stribution of molecular weight in linear polymers , I n d . En g . Chern . Fun dam . , 2, 2 2 1 ( 1 96 3 ) , B alke , S , , L . Garcia , an d R . P atel , C onversion pre diction in high conver sion in free radical polymerization s , AI C hE Meet . , New Orleans ( N ov . 1 98 1 ) . B amford , C , H . , F ree radical poly merization with rapidly decayin g initiators , Polymer , 6 , 6 3 ( 1 96 5 ) , B amford , C . H . and H . T ampa, The calc ulations of molecular weight dis tribution from kinetic schem es , T ran s . Faraday Soc . , 50 , 1 0 9 7 ( 1 9 54 ) . B amford , C . H . , W . G . B arb , A . D . Jenkin s , an d P . F . Onyon , The Kine tics of Viny l Polymerization by Radical Mechanis ms , B utterworth , Lon don ( 1 9 5 8 ) . B arbe , P . C . , L . N oristi , G . B aruz zi , an d E . Marchetti , Microscopic analysi s of polyolefin initial forma tion on TiC1 4 /M gC1 2 supported cata ly st s , Makromol . Chern . Rapid C ommun . , 4 , 249 ( 1 9 8 3 ) . B ej ger , T . P . , Real- time control of polymerization reactors , Ph . D . thesis , University of Minnesota ( 1 98 2) B ej ger , T . P . , M . Tirrell , an d G . S tep hanopoulos , B atch cop oly merization self-tunin g optimization and control , AIC hE Meet . , N ew reactors : Orlean s ( N ov . 1 9 8 1 ) . Biesenberger , J . A . an d D . H . S e basti an Princip les of Polymerization Engineerin g , Wiley - Interscience , N ew Y ork ( 1 983) . B illm eyer , F . W . , T ex tbook of Polymer Science , 2nd e d . , Wiley - Inte rscience , N ew Y ork ( 1 97 1 ) . B oor , J . , Ziegler N a tta C a taly s ts and Polymeriza tions , Acade mic Press , N e w Y ork ( 1 97 9 ) . C ardenas , J . N . and K . F . O 'D ri scoll , High conversion polymeri zati on : I . T heory an d applic ation to methyl m e thac rylat e J . Poly m . S ci . , C hern . E d . , 1 4 , 8 8 3 ( 1 976 ) . Cardenas, J . N . an d K . F . O ' D riscoll , Hi gh conversion poly m e ri za ti on : I I . Influence of chain tran sfer on the gel effect, J . Poly m . S ci . C hern , E d . , 1 5, 1 8 8 3 ( 1 97 7a ) . C ardenas , J . N . an d K . F . O ' D ri scoll , High conversion poly meri zation : I II . Kinetic b ehavior of ethyl methac rylate , J . Polym . S ci C he rn . E d . , 1 5 , 2097 ( 1 97 7b ) . Caun t , A . D . , The de termination of active centres in olefin polymerisation , B r . Poly m . J . , 1 3 , 2 2 ( 1 98 1 ) , Chiu , W . Y . , G . M . C arratt , and D . S . Soon g , A comp uter m odel for the gel effect in fre e -radical polymeri zation , Macromolecules , 1 6 , 348 ( 198 3) . C oyle D . J . , T . J . T uli g , an d M . Tirrell , C omputer analysis ' of high con version free ra dical polymerization on models usin g the finite element method , I n d . Eng . C hern . Fun dam . , ( 1 98 5 , in press ) . D avis , H . T . , I n tro duc tion to N onlinear Differen tial and In tegral E q uation s Dover , N e w York ( 1 96 2) . de Gennes , P . G . , Dynamic s of en tan gle d polymer solution s : I . The Rouse model , M acromolec ule s , 9 , 5 8 7 ( 1 976a) , de Gennes , P . G . , Dynamics of entan gle d polymer solutions : I I . Inclusion of hy drodynamic inte raction s , Macromolecules , 9 , 5 9 4 ( 1 97 6b ) , Del Pino , 0 . , J . A . Romagnoli, an d J . M . Castro , Predicting conversion usin g temperature as the meas ured variable in the reaction injection .
,
,
.
,
,
776
Ti rrell , Galvan , and Laurence
moldin g process : Part II I m p l e m e nta tion of the on - line c or rec tive algorithm , Polym . Eng . Sci . , 23 , 8 9 5 ( 1 9 8 3) D enb igh , K . G . , C on tinu o u s reaction s : Part I I . The kinetics of steady sta t e polymeri za ti on , T ran s . F a ra day , S oc . , 4 3 , 6 4 3 ( 1 947 ) . Dioni sio , J . M . and K . F . O ' D riscoll , H i gh c on ver sion c op o ly me rization of s ty ren e and me t hy l methacrylate , J. P oly m . Sci . Polym . Lett . E d . , 1 7 ,
7 0 1 ( 1 97 9) . F am ily , F . and D . P . Lan dau , e d s . , Kine tics of A ggrega tion and Gelation , North - Hollan d , New Y ork ( 1 98 4 ) . Flory , P . J . , Molecular si ze distrib ution in linear c on den s a tion polymers , J . A m . C h e rn . S oc . , 5 8 , 1 8 7 7 ( 1 93 6 ) . Flo ry , P . J . , Princip les of Polymer C hemis try , C ornell U ni ve r si ty Press , Ithaca , N . Y . ( 1 9 5 3 ) . Galvan , R . an d M . Tirrell , O rt ho gon a l collocation applied to a n a ly sis of heter o geneo u s Zie gler-Natta p olymeri zation , C omp . Che rn . En g . , ( 1 985 , in p re ss ) . Ga re is - R ubi o , L . H. , J . F . M ac G rego r , an d A . E . H am ie lec , M o de li n g and con trol of c op olyme ri za ti on reactors , A C S S y mp . Series , 1 97 , 8 7 ( 1 982) . G erren s , H . , How to select p olym e ri za ti on reactors : Part I I , Chemtec h , 434 (July 1 9 8 2 ) . Goodall , B . L . , S up er hi g h ac tivity supported c at a ly s t s for the stereo sp ecific polymeri zation of a-olefin s : hi story , de ve lop m en t , mechanistic aspects and characteri zation , lnd . Symp . on Transition M etal C atalyzed Polymerization : U n solve d P roblems , Mic h . ( Au g . 1 9 8 1 ) . G r ae s sley , W . W . , W . C . Uy , and A . Gandhi , Molecular wei ght distribution at hi gh c on ve r si on s in free radical poly mers , In d . E n g . Chern . Funda m , 8 , 6 96 ( 1 9 6 9 ) . Gupta , S . K . an d A . Kumar , Reac tion Engineering of S tep G row t h Po lymer ization , Pergamon Press, E l m s for d , N . Y , ( 1 9 8 5 ) . Hendricks , E . M . , M . H . E rn st , an d R . M . Ziff , C oa gula tion e qu a tion s with g e la ti on , J . S ta t . P hy s . , 31 , 5 1 9 ( 1 9 8 3 ) . Hicks , J . , A . Mohan , an d W . H. Ray , The optimal control of polymerization reac tors , C an . J . C hern . En g . , 4 7 , 5 9 0 ( 1 96 9 ) . Hui , A . W. an d A . E . Hamielec , T h erm al p oly m e ri zation of styrene at high c on version s an d te m p er at ur e s . An experimental study , J . Appl . P oly m . S ci . , 1 6 , 7 4 9 ( 1 97 2 ) . Hulb urt , H . M . and S . K at z , S ome p rob le m s in p ar t ic le te c h n ol o gy . A s t atist ical mec hanical form ulation , C hern . En g . S ci . , 1 9 , 5 5 5 ( 1964) . Keii , T . , Y . D oi , E . S u z uld. , M . T am ur a , M . Murata , and K . Saga , P rop en e p oly me riz a ti on with a m agnesi um c hlori d e - sup p o rt e d Z i e gle r C ataly s t : 2 . Molec ular w ei ght distrib ution , Makromol . C h e rn . , 1 85 , 1 53 7 ( 1 984) . Kilkson , H . , Effect of reaction path and initial distribution on molecular w ei g ht distribution of irreversible condensation p oly m e r s , I n d . En g . Chern . Fun dam . , 3 , 2 8 1 ( 1 96 4 ) . Kilk son , H . , Generalization of v a rious polycon densation p roble m s . C onc ep t of slow influx , Ind . En g . C he rn . Fundam . , 7, 3 5 4 ( 1 96 8 ) , L au r en c e , R . L . , and M . G . C hiavetta , He a t and mass tran s fer du rin g olefin poly m eri z ation from the gas phase , in Procee di n gs of B erlt n Workshop on Polymer Reac tor Engineering, K . H . R eic her t , ed . , Carl Han ser Verlag , M unich ( 1 98 3 ) . Len z , R . W . , O rganic C hemis try of Sy n t hetic High Poly m ers , Wiley I n te r sc i en c e , N ew Y ork ( 1 96 8 ) .
Polymeriza tion Reac tors
777
Lope z - S erran o , F . , J . M . C astro , C . W . Macosko , and M . Tirrell , A recursive approach to c opoly m e ri z ati on statistic s , Polymer, 21 , 2 6 3 ( 1 980) . Lowry , G . G . , Markov C hains an d Monte Carlo C alculations in Polymer Science , M arc e l D ekker, N e w Y ork ( 1 970) . Marten , F . L . an d A . E . H a mie lec , High conversion diffusion -controlle d p oly m eri zation , A C S Symp . Serie s , 1 04, 4 3 ( 1 97 9) . M ec kle nb u rgh , J . C . , The influence of mixin g on the distribution of co polymerization compositions , C an . J . C he rn . Eng . , 48 , 27 9 ( 1 9 70 ) . Middleman , S . , Fun damen tals of Poly mer Processing, McGraw -Hill , N ew Y ork ( 1 97 7) . N agasub ramanian , K . an d W . W . G raessley , Continuous reactors in free radical p oly me ri z a tion with b ranchin g : I . Theoretic al asp ect s an d preliminary con siderations , C hern . En g . S ci . , 25 , 1 54 9 ( 1 970a ) . N agasub ramanian , K . an d W . W . G rae ss ley , Continuous r eac tor s in free ra dical p oly m eri zation with b ranchin g : I I . E xpe rim en t al re sults on viny l ac e ta t e p oly m e rization , C hern . Eng . Sci . , 2 5 , 1 5 5 9 ( 1 970b ) . Nagel , E . J . , V . A . Kirillov , and W . H . Ray , P re dic tion of molecular weight distribution s for high- density polyol efin s , Ind . En g . C h e rn . Prod . Res . D e v . , 1 9, 3 7 2 ( 1 980) . N an da , V . S . and R . K . Jain , The statistical character of anionic poly mers , J . P oly m . Sci . , A 2, 4 5 8 3 ( 1 964) . Nauman , E . B . , Mixin g in p olyme r reactors, J . M ac r o m . S ci . Rev . , C 1 0, 74 ( 1 97 4 ) . O ld s h ue , J . Y . , D . 0 , Mechler , and D . W . Grinnell , Fluid mixin g variab l es in suspen sion and e m ulsi on poly m e ri zation , C hern . En g . Prog . , 6 8 ( M ay 1 98 2 ) . Peeble s , L . H . , Molecular Weigh t Dis trib utions in Po lymers , Wiley , N ew Y ork ( 1 97 1) . Penli di s , A . ; J . F . M ac G re gor , an d A . E . Hamielec , Dyn a mic m o d e ling of em ulsion p olymerization reactor s , AIChE J . , 31 , 8 8 1 ( 1 98 5 ) . Platzer , N . , Desi gn of continuo u s an d batch p oly me ri za tion process , Ind . Eng . Chern . , 62 , 6 ( 1 970) . Plesc h , P . H . , T he C hemistry of C a tionic Polymerization , Macmillan , New York ( 1 96 3 ) . Ray , W . H . , On the mathematical m o de lin g of polymerization reactors , J . Macromol . Sci . Rev . , C 8, 1 ( 1 9 7 2 ) , R ay , W . H . and C . E . Gall , The control of copolymer composition distribu ti on s in batch an d tubular re actor s , M ac romolecule s , 2, 4 2 5 ( 1 969 ) . Ray , W . H . an d R . L . L au ren ce , Polymerization reaction e n gin e erin g , in C hemical Reac tor Theory , N . R . Amundson and L . Lapidus, eds . , Pren tice - H all , En glewood Cliffs , N . J . ( 1 97 7 ) . Ros s , R . T . and R . L . L auren c e , Gel effect and free volume 'in the b ulk p olyme ri zati on of methyl met h ac rylat e , AI C hE S ymp . S e r . , 72( 160) , 74 ( 1 976) . Saidel , G . M . an d S . Katz , M om en t s of t he size distribution in radical poly merization , AI C hE J . , 1 3 , 3 1 9 ( 1 9 6 7) . S c hm e al , W . R . an d J . R . S treet , P oly m e ri z ation in e xp andin g catalyst particles , A I C hE J . , 1 7, 1188 ( 1 97 1 ) . Schulz , G . V . , Uber die B eziehun g z wischen Reaktion s geschwin digkeit un d Z usammentet zun g des Reaktionsprodkte s bei M akropolymerisation s vor giin gen 1 2 2 . M it te ilun g iiber hochpolymere V erbin d u ge n , z . Phys . C hern . ( Leip zi g ) , B 30, 3 7 9 ( 1 935) .
778
Tirre ll , Ga lv a n ,
and Laurence
S oh , S . K . an d D . C . S un dberg , Diffusion -controlled vinyl polymeri zation : I . The gel effec t , J . Poly m . Sci . Chern . E d . , 20, 1 2 99 ( 1 98 2a ) . S oh , S . K . and D . c . S un dberg , Diffusion -con trolled vinyl polymerization : II . Limitation s on the gel effect , J . Polym . S ci . Chern . Ed . , 2 0 , 1 3 1 5 ( 1 98 2b ) . S oh , S . K . an d D , C . S undberg , Diffusion -controlled vinyl polymeri zation : I I I . F ree volume p arameters an d diffusion -controlle d p ropagation , J . Polym . S ci . C hern . E d . , 20, 1 3 3 1 ( 1 98 2c ) . S oh , S . K . an d D . C . S undber g , Diffusion -controlle d vinyl p olymeri zation : I V . C omparison of theory and e xperiment , J . Poly m . Sci . Chern . Ed . , 20 , 1 3 4 5 ( 1 98 2 d ) . S tockmayer , W . H . , Theory of molecular si ze distribu tion an d gel formation in branche d - chain polymers , J . C he rn . Phys . , 1 1 , 4 5 ( 1 94 3) . S tuber , N . P . an d M . Tirrell , S tu die s of continuous methylmethacrylate polymerization in a twin - screw extruder , Poly . Proces s En g . , 3 , 7 1 ( 1985) . S ummers , L . , C omposition c on trol of suspen sion copolymerization at high con version , M . S . thesi s , U iversity of Minne sota ( 1 98 5 ) . S zabo , T . T . , and J . F . Leathrum , Analysi s of con den sation polymeri zation reac tors, J . Appl . Poly m . Sci . , 1 3 , 4 7 7 ( 1 96 9) . S z warc , M . , Carbanions , Liv i n g Po lymers an d Elec tron T ransfer Processes , Wiley - I n terscienc e , N e w Y ork ( 1 968) . Tirrell , M . , Polymer self- diffusion in entan gled systems , Rubber Chern . T echno! . , 56, 5 2 3 ( 1 98 4 ) . Tirrell , M . an d K . Gram ley , C omposition control of batch copolymerization reactors , C hern . En g . Sci . , 36, 3 6 7 ( 1 9 8 1 ) . Tirrell , M . an d T . J . Tuli g , Diffu sion -con trolled reaction s in polymers : the influence of poly mer concentration an d molec ular weight in the re action en vironmen t , in Procee dings of B erli n Works hop o n Po lymer Reaction E n gi n e e rin g , K . H , Reic hert , ed , , C arl H anser Verlag , M unic h ( 1 98 3 ) , T soukas , A . , M . Tirrell , an d G . S tephanopoulos, Multiobjective dynamic optimization of semibatch copolymerization reactors , C hern . Eng . Sci , , 37, 1 78 5 ( 1 9 8 2 ) . Tuli g , T . J , an d M , Tirrell , Toward a molecular theory of the Tromm sdorff effect , M acromolecules , 1 4 , 1 5 0 1 { 1 98 1 ) , Tuli g , T . J . an d M . T irrell , On the on set of the Trommsdorff effect , M acromolecules , 1 5 , 4 5 9 { 1 9 8 2 ) . Ver S trate , G . , C . C o ze with , an d W . W . Graessley , B ranchin g by copoly meri zation of monovinyl an d divinyl monomers in continuous - flow stirred reactor s , J . Appl . Polym , S ci . , 25, 59 ( 1 9 8 0 ) . Webber , D . , Poly mer output reac hed a new hi gh in 1 984 , C he rn . En g , News, 63, 1 2 ( 1 98 5 ) . W u , G . Z . A . , L . A . Denton , and R . L . Laurence , B atch polymerization of styrene -optimal temperature historie s , Polym . En g . Sci . , 22, 1 ( 1 982) . Z eman , R . J . an d N . R . Amundson , C ontinuous models for polymeri zation , AI C hE J . , 9 , 2 9 7 ( 1 96 3 ) . Z eman , R . J . an d N . R . A m un d son , C on tinuou s polymeri zation models , I , Chern . En g , Sci . , 20, 3 3 1 ( 1 96 5a ) . Z eman , R . J . an d N , R . A m un d son , C ontinuous polymeri zation models , I I , C hern . En g . Sci . , 20, 6 37 ( 1 96 5b ) . . \
12
Biological Reactors
LA R R Y E .
E R I C KS O N
Kan sas S tate University , Manhattan , Kan sas
G R EG O R Y S T E P H A N O PO U LOS
*
California Ins ti tute of Techno logy ,
Pas aden a , C aliforni a
I N T RO D UC T I O N
Accordin g to recorded hi story , some of the fir st re acto r s were biological reactors . F e r me n to rs fo r th e prod uction of alcoholic beverages such as beer an d wine w e re de si gn e d , con structe d , an d use d more than 5 0 0 0 y ears ago ( P ederson , 1 9 7 9 , pp . 1 - 2 4 ; Rose , 1 9 5 9 ; Amerin e , 1 9 6 4 ) . Accordin g to Pede rson ( 1 97 9, p . 1 4 ) , t h e B ab ylonians made beer as e arly as 5 0 0 0 The b akin g of leavened bread may have started in E gypt some 700 0 B . C . 3 50 0 years ago ( P e de r s on , 1 97 9 , p . 4 ) . C heese m an u fact u r e also appears to have b een carried out by the E gyptians ab out 3000 B . C . (Pederson , T he science of ferm en t ation advanced significantly in 1 8 5 7 1 97 9 , p . 6 ) . when Pasteu r showed that microor gani sm s were present an d active i n fer mentations ( Aiba et al . , 1 97 3 , p . 4 ) . A n o th e r m ajor advanc e was the pro duc tion of p enicillin in deep aer ated tanks durin g W orld War II . It was d urin g this p eriod that Elmer Gaden , the father of modern b io c h e m ic al en gin eerin g , did his Ph . D . researc h on oxy gen tran sfer to aerobic c ultures ( Hix on an d Gaden , 1 9 5 0 ) . Most of the science an d e n gi n eerin g advances in b iolo gic al reactor design have appeared durin g the p e rio d from World W a r II to the presen t . D u rin g this p eriod several books have appeare d in which biolo gic al reactor design is an important consideration (Aiba et al . , 1 97 3 ; B ail ey and O lli s , 1 9 77 ; Pirt , 1 9 7 5 ; W an g et al . , 197 9 ; Ai b a an d N a g ai , 1 97 5 ; an d B ekers , 1 98 0 ) , Journals have starte d whic h are devoted to bioc hemical en gineerin g , such as Bio t ec hnology an d Bioengi neering, and many articles on b io lo gic al reactors are contained in the volumes of A dvances ., in Bioc hemical Engineering. B iolo gical reactors include a wide variety of type s , both hum an - made Thus on e can fin d aerobic and anaerobic reactor s , well- mixed an d natural . or stratified reactors , en zymatic reactors with soluble or i m mob ili z e d en zymes , immobilized c ell reactors of various c on fi gurati on s , an d other s that will b e di sc u s se d i n this c ha p te r . B iological reactor design h as b een influenced by several i m por t an t factors . S train se lec tion s an d mutations placed time va ryin g specifications on the biological reactors while promotin g batch or
*Curre nt affili a t io n : D epartment
of C hemical E n gineerin g , Mas s achu setts
I n stitute o f T echnolo gy , C am b ri d ge , Mas s ach u s e t t s
779
7 80
Eric kson and S tep hanopoulos
fe d - ba tc h o p e r a tion , Rheo lo gic al con ditions may chan ge s i gni fic an tly durin g the c o u rs e of a b a t c h fermentation ( van S uij dam et al . , 1 9 8 2 ) . Furthermore , ma r ket force s frequently dictate that the same fermentor be u s e d for the production of different p roduct s by different microor ganism s during it s oper ational life . All the s e considerations make the p roblem of optimal desi gn and operation of b ior e ac t or s very important indeed . Anaerobic reac tors are used in the production of e th anol , acetone , b u t an ol , m any food p roducts , and anaerobic waste treatment . T emperature and pH control are us ually important c onsideration s . Reactor de si gn has re c eive d limited a tt e n ti on e xcept for the case of a n a erobic digestion , w here mixed c ulture in t e r ac tions have b een considered extensively ( B ai ley an d Ollis , 1 97 7 ; Gould , 1 97 1 ; S un d strom and Klei , 1 9 7 9 ) . A c om p le te ly mixed reactor is frequently employed . Th e de si gn of aerobic reactors , frequently em p l oyed for anti b iotic pro duction , h a s p ri m arily considered oxy gen mass transfer and oxy gen limita tion on reactor p erformanc e . Gas , liquid , and s o li d p hase s are present . In hy drocarbon fermentations , hy drocar b on transport may also limit gro wth ( E rickson et al . , 1 96 9 , 1 97 0 , 1 975 ; S hah et al . , 1 9 72a , b ; E rickson and H umphrey , 1 96 9a , b ; Gutierre z and Erickson , 1 97 8) . In ae robi c ferementa tions in whic h the organic substrate is in soluble , four ph as es are important for bi olo gi c al reactor design . The utili zation of solid s ub strate s , such as c e ll u los e , by microorganisms als o r eq ui re s t hat tran s port limitation s to growth be considered ( Lee e t al , , 1 9 8 0 ; Av ge rinos et al . , 1 98 1 ) . Trans port and s urface area limitation s are also i m p ortant in the bacterial leachin g of ores ( Gormely et al . , 1 97 5 ; C h an g an d Myerson , 1 98 2 ) . In the modeling of t hes e multip hase growth p r oc e s s e s , basic c oncepts of re action en gi n e e rin g , such as r esi de n ce time , s e gre gation , surface adsorption , transport li mit a tion , an d reaction kinetic s , have b een a p p li e d . En z y m atic reactors is anot her importan t cate gory of b ior eactor s . En zym e s are pre sen t ly used and hold a stron g promise for fu tur e applications as catalysts of biochemical rea ction s , In soluble or immobilized form they are anal o gou s to other homogeneous or heterogeneous catalysts and p rin c ip le s of r e action an d diffusion theorie s can be applied to op timi z e bioreactor op er a ti on . Enzyme reaction en gin eerin g studies have resulted in the develop ment of mem b ra n e reactors to retain the en zymes ( D eeslie an d C h e ryan , 1 9 8 2 ; C heryan and Deeslie , 1 9 8 0 ; an d Vieth et al . , 1 97 6 ) , fi x ed - b ed re actors with immobilized en zy mes or i m mob ili ze d whole cells ( Lee et al . , 19 7 6 ; Vieth et al . , 1 97 6 ; an d T ya gi and G hose , 1 98 2) , an d fluidized- and semifluidized- bed re act or s ( Vieth et al . , 1 976 ; B arker et al . , 1 98 1 ; Fan and H su , 1 9 8 1 ) , Im mobilized enzymes have also been u sed to de vel op en zym e elec trodes an d ot h e r e n z y me - ba s e d sen sors in which re ac tor s are used for i n st r u men t a tion ( C hen et al . , 1 9 8 2 ; B arker an d Somers , 1 97 8 ) . Several c on ferenc e s and books have been devoted to en zy m e reaction en gineeriri g and t he s upp ortin g science of en zym e an d w hol e cell immobili zation ( W in gard , 1 9 7 2 ; Pye and Win g ar d , 1 97 4 ; Pye and Weetall , 1 9 7 8 ; W e e t all and Royer , 1 980 ; W in gar d et al . , 1 97 6 , 1 98 0 ; Zaborsky , 1 97 3 ) . Media sterili zation , enzyme deactivation , and the thermal processin g of foods in c annin g is another i m por t an t application of biolo gic al reaction engineerin g , The ac ti va tion en ergy in the A rr he ni us expre ssion for temper at ur e dep en denc e of t he rate of sterili zation is very hi gh ( Aiba et al . , 1 97 3 , pp . 2 4 2- 2 4 3 ) . Thu s hi gh - t e m p e ratu r e short - time processin g of foods op timi zes food quality . T he des i gn of continuous p roc e s s e s which closely ap p r o xi m a t e plug flow requir6S the applic ation of re s i den ce - ti m e distribution analy s i s (Aiba et al . , 1 97 3 , p p . 2 5 8- 2 6 4 ; B ail e y and Ollis , 1 9 7 7 , pp . 5 5 1- 554 ) .
7 81
Biolo gical Reac tors
There are many different types o f biolo gic a l reactors and bi ochemic al reaction s ; howeve r , in many c a s es , t h e b asic p ri n ci p les of reaction en gin eer ing desc ribed in earlier c hapters may be ap pli e d in desi gnin g the reactor system . T o avoid unnecessary duplication , at te ntion will foc us on those aspect s that are c h a racte ri stic of biolo gi c al systems an d p r oc es se s The stoichiometry , ldnetic s , and energe tic s of these sys tem s will fi rst be re vi e wed , for they appear to differ from thos e of si mp le chemical system s . S ubl!.�t\\l.�"t\\\':J , t\w. 'l a"ri..Q\3£ �'i)es of bioreactor configurations will be pre sented along with the d e sign equations for the simple ones . Mass transfer con si derati on s especially as they app ly to oxy gen tran sfer in aerob ic processes and p roc e d ur es for bioreactor de si gn will occupy the last two sections of thi s c hapter . Much of the disc us sion will be as soc i ated with aerobic microbial growth and product formation , for most o f the current biological reaction en gineerin g research is foc us e d on the op timi z a tion of aerobic ferme n ta tion p r oc e s s e s
.
,
,
.
M I C RO B I A L G ROWT H
The stoichiometry , ldnetic s , and energetic s of a bioc hemic al p roc es s must b e known an d understood in or de r to d e s i gn the reactor system , T hese b asic conc epts as app lie d to microbial growth p rocesses with p ro d uc t forma tion are reviewed in this section . Re gu la ri ti e s are introduced because of the simplification ac hieved throu gh their app lic ation in p roce ss analysis and desi gn .
S toi c hiometry a n d N u tri t i on
In direct an alogy to the case of a sin gle chemical reaction oc c ur rin g in a chemic al r e ac to r t he p roc e s se s of growth and p roduct formation are rep re se nte d by a chemical reaction in which am monia ( or another nitrogen source) with a c arb on -energy so u rce is con verted into biomass , product , carb on dioxide , an d water : ,
+
carbon -ener gy source
y CH O N + b p n q am moni a ( nitro gen sourc e )
c ell biomass
( 1)
+ y CH 0 N
p
+ y co + cH 0 2 2 d r s t
product Reaction ( 1 ) is true for the c ase of a single carbon - energy sdurc e ( or I f complex mixture s of organic s ub s tr ate s substrate ) and a sin gle p ro duct and products are involve d , ap p r oxi m ate formulas of the type in dic ated above are written for the sub strate and product . It is fu r t he r assumed that cell b iom ass can be re pre se n te d by the form ula C H p On N and that the q bio mass composition r e main s con stant durin g the course of the fermentation . A l th ou gh deviation s occ ur , the ab ove is a rather reasonable as s um p tion In addition to carb on , hy dro gen , oxy gen , and nitrogen , on a dry basi s , there is al s o 3 wt % p hos p hor us , 1 % s ul fu r 1% p ota ssi u m , 1% sodiu m less than 1% calcium , ma gn e s i um chlorine , and iron , an d trace amounts of other elements , such as c o p p er an d zinc . All the latter are n ot re p r e sent e d in formula ( 1 ) .
,
,
,
•
,
Erickson an d S tep hanopoulos
782
It should be n ote d that reac ti on ( 1 ) is carbon atom of substrate . Therefore , Y b • o f or ganic substrate c onverted to biomas s , respectively ; that is , y
y b + p
+
y
w rit te n on the b asis o f one Yp . and Y d gi v e t he frac ti on product , and c arbon dioxide ,
1. 0 d =
is a carbon b alance for the p roces s .
( 2)
Si milarly , a
nitro gen
balan ce
give s
One c an also write balances for hydrogen an d oxy gen involvin g all stoichiometric coefficients of reaction ( 1 ) . Si nc e the water involved in the p rocess is not measurable , it is , in general , de s i rab le to eliminate the stoichiometric coefficient of water c between the hy d rogen and oxygen balances . The result is Eq. ( 4) , which may also be in te rpre t e d as an available electron balance (Minkevich and E roshin , 1 9 73 ; E ric k s on et al . , 1 9 7 8a ) :
the
( 4)
E q u a tion ( 4) in tro d uc e s the c on c ep t of t he de gree of re d uct an ce of biomass , produc t , and s ub s tr a t e , Yb • Yp • an d Y s • respectively . The degree of reductance of a s ub st anc e is defined as the number of e q uiv alen t s of avail able electrons per gr a m atom of carbon of the substance which would be transfered to oxy gen upon oxi dation of the substance to C 0 2 , H 2o , and N H 3 . Thus
y
y
y
b
p
s
::;: 4 +
p
2n - 3q
( 5)
::;: 4 + r - 2s - 3t
( 6)
::;: 4 + m - 2R.
( 7)
where the valences C ::;: 4 , H ::;: 1 , 0 = - 2 , an d N ::;: - 3 have been used. The c on c e p t o f the de gree of re ductance is a very us e fu l tool in establish in g a uni fyi n g treatment of the bi olo gic al processes of microbial me tab oli s m . F urth e r more , s o m e re gu la riti e s c on c e rni n g the degree of reductance of b io mass have been ob s er ved (Minkevich and E roshin , 1 973 ; Mikevich et al . , 1977 ; E ric k son , 1 9 8 0b ) . By examinin g various b i o m as s c om p o sition s , these authors found that Yb is relatively c on st an t at an average value of ii'. 291 with a coefficient va ria tion o f 4% . Another re gula rity was also ob served concernin g t h e weight fra c ti on of carb on in bioma s s , crb . The sa m e authors ob serve d that cr b has an average value of 0 . 4 6 2 wi t h a coe ffic ien t of variation of 5%. In a typical bioreactor situation the chemical for m ula s of the substrate and product are known an d that of biomass can be determined by an elem en t al analy sis . The only re mainin g unknowns, then , are the stoichio metric coefficients of reac tion ( 1 ) , n am e ly a an d b an d Yb • Yp • an d Y d · N otic e that these coefficients are direc tly related to the yie l d s of the process . The available equations are the carb on balance [ E q . ( 2 ) ] , the
783
Biological Reac to rs
ni tr o gen balance [ E q. ( 3) ] , an d the e le ctron bal an ce [ E q . ( 4 ) ] . By me as uri n g th e r espir at ory q uoti en t (i . e . , t he r ati o y d / b ) an additional equation becomes available and the system can be deter mined in the absence of product formation . If p ro duct ( s ) is bein g for me d an a d dition al mea.surement ( s ) is needed to close th e system . In s um m ary , the b alances above can be use d ( a ) to de t er m in e the s t oichio metry of a p ro c e s s , in con junc tion w it h the necessary measurements ; ( b ) to m onito r on-line the state of a b i or e ac t or to provide the ap p r op ri at e control action (Wang et al . , 197 7 ; Stephanopoulos an d S an , 1 9 8 2 ) ; an d (c ) to test t he con si stency of measurements ( F errer and Erickson , 1 9 7 9 ) and identify th e measurement that m o s t likely introduces large error ( W an g and S te p han op o ulo s , 1 98 2 ) . The stoichiometry of a bi ologic al p rocess is an i mportan t element in wri tin g the p roper macrosc opic material b alances over a biore ac tor , which H owever , in ad dit ion to mas s , is the subject of the followin g section . heat is al so e xc h an ged in mo s t of the aerobic fermentations . One c an write the following enthalpy b al anc e for th e heat of reaction /:, H r :
!;, HO = ( !;, HO )
c s ub s t rat e
r
- y ( !;, HO )
b
c bio m a ss
p
- y ( !;, H O )
c produ c t
( 8)
where the pro d uc t s of combustion are t ak n to b e C 0 2 ( g ) , H 20 ( R. ) , an d dilut aqueous solution , and eac h of the values of t;, Hg are for 1 atom o f c rbon . Equation ( 8 ) utilized in two diffe ren t way s . If th e stoic hiomet
N H 3 in a
a
e
e
can b e and Yp are k no w n, t he he a t rele ase d durin g growth can be calculate d and th e cor r e s p on din g c oolin g requirements determined . C on ve r sely , if the reaction heat can be m e a su r e d , E q . ( 8 ) p r ovi d e s an addi tional b ala n c e fo r th e determination of the unknown yields . The heat of gro w t h can be measured on-line by u sin g flow microcalorimeters (Eriksson and Holme, � 97 3) . Alternatively , a correlation may be u se d t hat re lat es the he a t evolved to the amount of oxygen consumed . Thus , a p proxim at ely , as s hown in Fig . 1, ric c oe ffi c ient s Y b
l!. H0r
=
( 9)
- 2 7 ( 4b )
where l!. H� i s in kc al / g a t om of carbon in the sub st r at e . This approximation is based on t h e early work of Thornton ( 1 9 1 7 ) , K harasch ( 1929), an d Kharasch and Sher ( 1 92 5 ) an d the more re c e n t work of Minkevich an
d Eroshin ( 1973) , Roels ( 1 9 8 1 ) , C ooney et al . ( 1 969), Minkevich et al . ( 19 7 7) , E ric k son et al . ( 1 97 8b ) , and Patel and E rickson ( 1 9 8 1 ) . For aerobic fermentation s the che mic al ene r gy per e q uiv al en t of avail ab le electrons is about th e s am e for t h e or ganic substrate , the biomas s , an d the extracellular product . Minkevic h a n d E ro s hi n ( 1 973)· 'and Minkevic h et al . ( 1 9 7 7 ) foun d a valu e of 26 . 95 kcal / g e qui v of available elec trons ( 1 1 2 . 8 kJ /Q equiv ) for th e quantity of energy given up when dry biomass is oxidized to C 0 2 ( g) , H 2 0 ( R. ) and N H 3 in a dilute aq u eous solution .
Per gram atom of carbon , t hi s yield s
( l!. H0 ) c bi
o mas s
N ow , E q . ( 4) can £ + n + E; P
=
-
-
- 26 • 9 5y b
be 1.0
divi d d by y 8
e
( 1 0)
to yield ( 11)
Erickson an d Step hanopoulos
784
2400 0 ::!! E
Ql
0 ... (!) '
0
0 -"'
iii
a
E
�
0 u
0
0 Ql J:
Equivalents of Available Electrons/Gram Mole
1 00
FI GURE 1 Line ar variation of h eat of combustion of or ganic compounds with the number of equi v alen ts of available electrons . ( From P a tel and Eric kson , 1 981 . ) where the th ree q uantitie s e: ::; 4b / y s , n ::; Yb Y b / y s , an d i;:p ::; Y p Yp i Y s are fractions of the availab le elec tron s transferred to oxy gen , mcorporated into
biomas s , an d incorporated into extracellular p roduct , respectively . I n view o f t h e regularity menti one d earlier , accordin g to whic h the c hemical ener gy per equivalen t of available elec tr on s is the s am e , the quantities above may also be interpreted as the fraction s of s ub st rate energy evolved as heat , incorporate d int o biomass , and incorporated into extracell u lar products , resp ectively . In this re gard, the electron balance Eq . ( 4 ) ob .:. ' t ain s the meanin g of an energy balance . It s hould be noted that the subject of stoichiometry and yield s is e s s ential in bioreactor design . In biological p roces s e s the stoichiometry is unknown and , unlike chemical system s , m ay vary during t he course of the process . Consequently , th e establishment of the proper stoichiometry , and the ability to follow any v ari ation s on- line , cons titute the first objec tive in the an aly si s of a biolo gi c al process . The second is the determination of the process kineti c s , and thi s is the s ubj e c t of the next s ection .
Biological Reactors
785
K i n eti c s
The kinetics o f biological processes deals with the subject o f the determina tion of th e rates of substrate cons umption , biomass growth , and product formation , Typically , in a chemical reaction , these rates are interrelated through the reaction s t oic hiom et ry . However , the variation of yields in bio logic al processes complicates the situation . T he s tartin g point is the specific rate of biom ass growth , � . A variety of mo dels exists of varying de gree of complexity . Thus one can find lumpe d-parameter m odel s which treat the biomass as one entity , structured models that distinguish the various types of biomas s , se gregated models that recogni ze the individual nature of each mi c roo rgani s m , and the p os sible combinations amon g them . This discussion focuses on lumped parameter models . Various structured and s e gre gate d models can be fou n d in Fredrickson and Tsuchiya ( 1 97 7 ) and Bailey and Ollis ( 1 9 7 7 ) . The typical trade-off with these models is that the number of p aram eter s in volved and the mathematical complexit y increases dramatically , but more information becomes available and better dynamic response is achieved . T he lumped-parameter models express the s p e ci fi c growth rate as a func tion of key substances of the abiotic medium . To be s ure , a large number of factors affect the growth of a c ultu re . By ke epin g the concentrations of all b ut one above the saturation level , the spe ci fi c growth rate b ecomes a function of the concentrations of this ra te - li miting sub s t anc e . Fi gu r es 2 an d 3 depict the two basic functional relationships be t w e en 1.1 and S . The characteristics of the function shown in Fi g . 2 can be re p re s ented by the rectangular hyperbola of Eq . ( 1 2 ) :
f'mox
--- - - -- - -- - - - - - - - --
'..:::
• I
S ( mg / 1 )
FIGU RE 2 Specific growth r ate � plotted as a function of s ubstrate con c en tration S according t o t h e Monod model , where � ax = 1 . 0 h- 1 an d K = s m 10 m g /L . Note that � = ll max when S = K8 • ( From Pirt , 1 97 5 , p . 9. )
0. 5
Eric kson and Step hanopoulos
786
Q5
"j.....
0.4
•
..t::. -
::l.
0. 1
0 0
FI G U R E acetate .
4
8
12
16
20
( S Hg acetate/ I )
24
Substrate inhibition for gro w t h of
3
Candi da uti lis on sodium
( F r om Aiba and Nagai , 1 97 5 , p . 1 8 8 ; Cama and Edward s , 1 97 0 . )
( 1 2)
The model is the c e lebr ate d Monod model . It is an empirical model , pat te rne d after the Michaelis-Menten kinetics o f en zymatic reactions , which
was found to describe adequately the growth kinetics of several microbial 1 94 9 ) . This model has also b ee n used re peat e d ly in a variety of applications rangi ng from activated slud ge kinetics to the analy sis of dynamics of mixed culture systems . It contains the basic features exhibited by most microbial system s , nam ely linear dependence at low S values and saturation at high S values . Furthermore , it is a si mp le , two parameter model and in the absence of indications to t he opp osite , it is used to fit data or to study the general dynamic characteristics of a reac system s ( M onod ,
tor system . The meaning of the two parameters 1-l max and K s is shown in Fi g . 2 . us ually estimated by plottin g available data in the
These parameters are
form 1 / 11 versus 1 / S . For many organic subs trates the s aturation constant K s is between 0 . 1 and 2 0 mg /L ( B ailey and Ollis , 1 977 , pp . 346 , 3 47 ; Pirt , 1 97 5 , pp . 1 0- 1 2 ) . Maxi m um s p eci fic growth rates are usually larger for bacteria t han yeast . Values betw een 0 . 3 and 1 . 0 h - 1 are frequently re ported ; however , both lower and higher values are found ( Pirt , 1 97 5 , p . 145) . pH .
The maximum sp ecific growth rate , 1-l m ax , depends on temperatull'e and Microor gani sms are clas sified as thermop hiles , mesophiles , and
p sychrophiles with resp ect to their optimum temperature for growth
( B ai le y and Ollis , 1 9 7 7 , p .
344) .
mesophylic organisms is about
The optimum temperature for most
37°C ; as shown in Fi g . 4 , at temperatures
lower than about 37° C , t he com mon Arrhenius te mperature dependence is ( Bailey and Ollis , 1 97 7 , pp . 1 3 2- 135 , 344 ; Prokop and Erickson , 1 97 2 ) . Values of the ac tiv ation energy have been found to be si gni ficantly different for growth ( 8 . 2 kcal /mol ) , respiration ( 1 7 . 8 kcal /mol ) , and penicil lin p rodu ction ( 2 6 . 8 kcal /mol) for Penicillin chry soge num ( Aiba et al . , 1 9 73 , pp . 108- 1 10 ; C alam et al . , 1 97 1 ) .
observed
787
Biological Reac to rs Temperature , •c
1.0
0
..
�
:t.
c Clll
- 3.0
A
- 4. 0
- 5.0 0.0030
0.0032
0.0034
1 / T , •K- 1
0.0036
FIGURE 4 Arrhenius plot for the effect of temperature on the maximum specific growth rate of a mesop hile ( E . coli , curve A , E 2 5 . 5 k c al / g mol ) and a psychrophile ( Pseudomonas strain 2 1 - 3C , curve B , E = 1 6 . 2 kcal / g mol ) . (Data o f Ingraham , 1 9 58 . ) =
As shown in Fi g . 5 , the op tim um pH for grow t h depends on the or gani sm . B acteria frequently grow in the pH ran ge 4 to 8 , yeasts grow in the pH range 3 to 6 , molds from 3 to 7 , and hi gher eucaryotic cells from 6 . 5 t o 7 . 5 ( Wang et al . , 1 97 9 , pp . 8 9- 90 ) . A rather narrow window of optimal growth exists around t h e opti m al pH with the growth r ate drop pi n g rapidly outside this window . However , there exist or gani sm s , such as yeas t , Esc herichia coli , and other s , which m aintain optimal growth over a w id er pH v a ria t ion . If it c a n be a s su m ed that pH affects growth by in activ a tin g some rate -limiting en zymes thro u gh protonation - deprotonation re actions , then the p rim a ry effect of p H variations i s t h e decrease of ll m ax . C han ge s in Ks m ay al so be pre s e n t , but t h ey are secondary im portan c e . T emperature and pH c ont rol are co mm o nly included in the de si gn o f fermentor s . Some system s exhibit r at e inhibition at high sub strate concentrations . T his situation is d e pic t e d in Fig . 3 an d de scribed by the m o d el
( 1 3)
Equation ( 1 3) has a maximum and exhibits s te ad y - s tate multiplicity and very intere s tin g dynamic features when considered in a CSTR situation ( San and Stephanopoulos , 1 98 3 ) .
788
Erickson an d S tep hanopoulos
T..c
-
e ::1..
0
0
I I I I
u
LLI
0. 2
0
I
s:r -Q \8
1
E
0
\ \ I I
.;
::1..
0.05
0
0 4
-
'.c -IC
pH
9
0
::I u u 0 u
.2
>. ..c Cll
-
2
FI G U R E 5 Effects of medium p H on maxi m um bacterial specific growth rates : A , E . coli in anaerobic casein hyd roly sat e medium ( from data of Gale and Epps , 1 94 2 ) ; B , Methy lococcus caps ula tus grown on methane ( from data of H arwood and Pirt , 1 972 ) . ( F ro m Pirt , 1 975 , p . · l 45 . )
Other models of microbial kinetics have also appeared in the literature . These models contain various features that occasionally were found to be important , such as multiple s ubs trate limitation , product inhibition , de pendence of � on biom as s concentration , and others . A detailed de sc ription of these models is given by Fredrickson and Tsuchiya ( 1 97 7 ) and in tabular It should be also pointed out that even though S form by Ollis ( 1 9 7 7 ) . usually stands for an organic limitin g s ub s tr ate , it can represent any limit ing compound . Johnson ( 1967) , for example , fou nd a value of Ks = 0. 0 2 34 m g /L for a Monod model with oxygen rate limi tin g . Also , expressions com pletely analogous to thos e p r es en te d above have been developed for the velocity of en zymatic reactions . The rate of nutrient consumption is modeled by the product of the specific grow t h r ate and t he bioma ss concentration divided by an appropriate yield . Various possibilities re gardin g these yi eld s and their variation with time will be presented when the bioreact or desi gn equations are deriwd in a later se ction . Product formation i s often t h e objective of carrying out a bi olo gic al proces s . The kinetics of p roduct formation depend on the product or products t hat are produced . Some products are as sociated with the process of growth , whereas some others are not . The specific ( ::: per unit amount of biom as s ) product formation rate in the first case is proportional to the speci fic growth rate ; it is constant in the second case of non- growth- asso ciated produc t . Intermediate situations also exist . Le u de kin g and Piret ( 1 95 9 ) introduced a s uccess ful mathematical model which includes all the fo regoin g possibilities :
Biolo gical Reac tors 1 dP X dt
-
-
=
a +
f3 Jl
789 ( 1 4)
The m odel above has been
applied to a number of ferm ent ati ons w here and a ' determined experimentally ' indicated the extent of growth association . Some flexibility can be introduced in the model by p rovidin g different parameter values for t he different phases of growth . This ce rt ainly exten ds the ap pli c abi lity of the model but does not alter the fact th at it is an e m pi ri cal one and re s ul t s based on this model cannot be extrapolated in gen er al . In c on s tr uc tin g product formation m o de ls with a mechanistic basis , the classification of the various products in rational c ate go ries can be u s e ful . Gaden ( 1 95 5 , 1 9 5 9 ) has p rop o sed a classification s cheme which includes type I p rodu cts , which are direct ly as sociated with primary ener gy metabo lism , s uch as ethanol ; type l l prod ucts , which are indire c tly as s oci at e d with primary energy metabolism , s uch as citric acid ; and type I I I products , which are complex molecules that are not directly associated with primary ener gy metabolis m , such as penicillin (B ailey and Ollis , 1 9 7 7 , pp . 37137 5 ) . Wang et al . ( 1 97 9 , pp . 1- 45 ) , divide microbial p roduct form ati on into biosyn thesis of primary metabolites , secondary metabolites , and en zymes Primary metabolites are end with a fourth cate gory of bioconversions . products of low moelcular weight that are us ed as b uildi n g blocks by cells . Secondary metabolites are p rod uc ts s uch as a ntibioti cs which are not re quired for growth ; they are us ually s ynthes i z e d late in the grow t h cycle . S econdary metabolites are us ually type Ill fermentations accordin g to G aden ' s classification . Finally , Deindoerfer ( 1 96 0 ) , ha s organi zed fermentations p at terns into four c ate gorie s , according to the simi l ari ty of t h e formati on of fe rment ation products to those produced b y simple , simultaneous , consecutive , the
p ar am ete r s
a
and stepwise reactions .
I n type I fermentations , th e last term in Eq . ( 14) i s usually im p ortan t and product formation i s primarily grow th a s s oci ated . In type II ferm enta tions , product formation kinetics are fr eq u en tly complex , with m axi m um prod uct form ation rate s often occurrin g late in the ferm en t ation at low speci fic growth rate s . The rate of pro duc t formation , the prod uct yield , and the concen trati on of the product in the fin al b roth are important econo mic considerations . Product inhibition is an important consideration in many fermentations be ca u s e this li mi ts the final p roduct concentration . Pirt ( 1 97 5 , pp . 1 701 8 5 ) has consi d er ed the effect of p rod u ct inhibition on growth and has presented the exp re s sion s
( 1 5)
for competitive product inhibition and
( 16)
in hib ition , respectively . T h e concentration o f which inhibits growth is I , and Ki i s the kinetic saturation
for noncompetitive p rod uct the product
Eric kson an d S tep hanopoulos
790
constant . For the p rodu c ti on of ethanol , Aiba et al . ( 1 9 6 8 ) found that ethanol inhibits growth by noncompetitive inhi biti on . Product formation ldnetics have been developed and used in reactor de sign and operation for m any indus trial fe rm e ntations . C on stanti ni de s et al . ( 1 9 7 0a , b ) have used kinetic data for penicillin production to optimize temp erature pro fi le s in batch fe r m e n tation s to ac hieve i mp rov ed yie ld s of ab o ut 1 5 %. Kine ti c models for the p rod ucti on of gluconic acid from glucose have been developed and used to opti mi z e pH and te mp erat ure ( Constantin ides and 1 97 4) T h e opti m i z a tion resulted in im p roved yields and rates . Rai an d Constantinides ( 19 7 3 ) foun d a time s avin g of up to 6 0 % by conducting experiments near the optim al temperature and p H . In these examples . the ldnetic models are first developed and then employed to optimize b atch op er ati on . One of t he deficiencies of the Monad model is its inability to model the dyn amic s of grow t h processes . Structured models in which microbial com position is considered are needed to p roperly represent the dynamic varia ti on s associated with growth which occur because of s udde n changes in environm ental conditions . Harder and Roels ( 1 98 2 ) h ave recently reviewed st r uct ured models ; in much of the work they review . RNA is con si der ed as a variable in the model. While structured models may be extensively used in future reactor desi gn work . they have not yet b ecom e wi dely accepted in industrial prac ti ce (Harder and Roels . 1 98 2 ) . Another developing area which may b e much more important in the fut u re is that of growth kinetic mod els for mi xe d cultures . Fredrickson ( 1 9 7 7 ) has reviewed the t yp es of int er ac ti on s an d as s oci at ed dynam ics in mixed cu lt u res . A mon g others the interaction of com p e ti tion is receiving increasing attention and related w or k is s ummari zed in a rece nt review ( Fredrickson and S tephanopoulos . 1981 ) . One of the useful kinetic models of mixed cultures is t hat of A ndre w s and Graef ( 1 97 1 ) , which has b een used in anaerobic di ges to r control . B ailey and Ollis ( 1 977 , pp . 6 3 5 - 7 4 3 ) present several kinetic models for mixed cultures and di s c us s mixed cul ture app lic ation s . Harrison ( 1 9 7 8 ) point o u t that t h e number of mixed c ult ures in com mercial use which are well studied and understood . and deliberately co n s tit ute d , is extrem ely sm all . Fredrickson and T su c hiya ( 1 97 7 ) have reviewed mixed cu ltu re kinetic models in cl u di n g much of their own work .
Rai
•
•
Energetics In the desi gn of a fermentation proces s , t h e ener getics must be considered b al an ce s must be made for the p roce s s . Aerobic proces ses may be limited by the availability of carbon in the or gani c s ubstrate or by; the quantity of energy in the or ganic s ubstrate ( Erickson , 1 981 ; Linton and S tep he n s on , 1 9 7 8 ) . Figures 6 and 7 s how the variation of the bio m as s carbon yield , Yb , and the biomass ener getic yi el d , n , as a fu n ction of the re ductan c e de gree of the or ganic substrate . For low v alu e s of y s , these re s ult s s how t ha t the growth process is ene rgy limited ; however . at a value of Ys ab ov e 4 , the results indicate that the availability of c arb on becomes For example , Veselov et al . ( 1 9 7 4 ) reported subs tantially larger important . yi eld s with C 0 2 in the gas fed to the fermentor where n - alkanes were used as the or ganic substrate < Ys = 6 . 1 ) . Figures 6 and 7 are for grow th proc es s es wit ho ut prod uc t formation ; the p roc es s of product form ation m ay als o be carbon limited or energy limited ( Ericks on , 1 9 81 ) . and energy
791
Biolo gical Reactors
0.7
00
FI G U RE 6 V ariation o f biomass carbon yield with re d u ct organic substrate . ( From Erickson , 1 98 1 . )
ance
de gree of
0.7 0. 6 0.5 &=""
0. 3 .
0,2
I
0. 1 o. o
0
2
3
4
5
6
7
8
ys FIGURE 7
Variation of biomas s energetic yield with reductance degree of ( From Erickson , 1 9 8 1 . )
organic substrate .
Eric kson and S tephanopoulos
792
The energetics of growth and product formation may be viewed as a process in which available e lectrons in the or g ani c s ubstrate are rearr anged into available electrons in biomass and product s with similar energy p er available elect ron . Energy in the form or A T P is used to achieve the rem ax m ax are m easures o f the arr an g emen t The parameters Y A T an d Y / P P ATP quantity of biomass and product , respectively , that can be formed from or ganic substrates per mole or ATP . T h us under energy-limited growth conditi ons , the maximum bio mass energetic yield ( Erickson , 1 980c ) , n , max and the m a ximum p rod u ct ener geti c yield , are related to Y and .
,
Y
�;ax ,
��
and the effi cien cy by which ATP is fo rme d ( P /0 ) by
;�TP
( 17 )
( 1 8)
( P /0) is the gram moles of ATP high - ener gy b on ds formed per gram atom of oxygen con s umed and the sub scripts s and o re fe r to sub strate - level p hos p horyla tion and oxidative p hosphorylation , respectively . For glucose as the or ganic sub strate , the theoretically e s ti m at ed value of Y
��
=
28 . 8 g
of cells /mol A T P may be used with the value of 3 . 1 67 m ol of ATP /g atom of oxygen for (P / O ) s + ( P /0)0 to obtain nmax = 0. 8 8 as the theor etic al maxi m um biomass energetic yield for aerobi c growth with glucose as the organic s ubs trate (Erickson , 1 980c ; Stouthamer , 1979) . For polysaccharide p roduc tion from glucose , Erickson ( 1 980c ) has found a m aximum product ener getic
�;ax
yield of
= 0 . 95 .
Actu al yields s hould be s maller than the theoretical
maximum yield . By usin g available electron c on c ep ts Erickson ( 1 9 7 9 ) obtained the fol lowin g e xp res sions for the e ner geti c yields in terms of nm ax ' me and ,
.
�
max .
p
•
=
1
n
n
£
d n
m -1- + �
nmax 1 n ---'m"""ax_ -
=
=
1
-
J.l
+
+� max
-
n ..,P
�
I
( 1 9)
m
J.l
� +
(ys /yb ) nmax nmax
( 20)
+
m
e
J.I
+� n
1
_
( y /y s
p
)�
max
p
( 21 )
793
Biological Reac to rs
These equations m ay be use d with parameter estimation methods to estimate max , is the from experimental data . The parameter , n n , m , and t; max p e max true growth yield b ased on that fraction of the available electrons in the max is the true product organic substrate that are as sociated with growth , t; p yield based on that fraction of the available electrons that are as sociated with product formation , and me is the maintenance param eter . Values of the parameters n max and m e have been estimated for micro bial growth processes without product formation ( Solomon et al . , 1 9 8 1 , 1 98 2 ; Ferrer and Erickson , 1 97 9 , 1 980) . Equivalent parameters have also been estimated by other s ( Pirt , 1 9 7 5 , p . 6 9 ; S touthamer , 1979 ; Heijnen and Roels , 1981 ) . Erickson ( 1 980a ,c ) , Erickson and Hess ( 1 9 8 1 ) , and Oner et al . ( 1 9 8 3 ) h ave considered the true product yield
t;�ax in
estimat
in g yield parameters . Solomon et al . ( 1 98 2 ) and Oner et al . ( 1 98 3 ) have developed parameter estim ation methods in which all the data are used sim ultaneously in estimating p arameters .
T Y P E S O F B I O LO G I C A L R EA C T O R S S ti rred Ta n ks
Well- stirred reactor system s have traditionally been employed for carrying out the great m ajority of fermentation processes . Providin g a homogeneous reactor environment has not been always an easy tas k . Large residence times in continuous operation or batch operation are certainly positive fac tors in maintaining a liquid p hase that is well mixed with respect to the nutrients . On the other hand , the large size of these reactors and the complex and time-varying rheological behavior of a multiphase reaction sys tem make homogeni zation in these reactors an energy consumin g and , in general , difficult proposition . Nevertheless , very large ( 2 00 m 3 ) fermenta tion vessels have been built and operated in the pharmaceutical , food , and chemical industries mainly because of the relative simplicity in their desi gn and operation . Because of the need to avoid m utations and maintain the superior qualities of the genetically developed strain , b atch or fed- batch operations are the rule in most applications . The intrinsic property of continuous reactor systems to select for the best grower and not for the best p roduct producer has hindered the expansion of contin uous operation . Neverthe less , continuous culture operations are of great interest in research laboratories , for t hey provide a time-invariant environment at s teady state FUrthermore , which facilitates greatly the s tudy of a biological process . some industrial operations employ continuous reactors , such as the sin gle cell p rotein facility of I C I in Billingham (total reactor volume of about It should be noted 2300 m 3 ) , all waste treatment processes , and others . that it is relatively common to follow a b atch proces s with a period of fed batch or continuous operation . Also , batch cultivation i s the optimal start up p rocedure for continuous cultivation or fed - b atch cultivation in mos t cases ( Y am ane et al . , 197 9 ) . Recently , many batch operations have been transformed into fed -batch ( semicontinuous ) by introducing the nutrient gradually into the reactor in stead of chargin g everything at the be ginning of the operation . T he ration ale is to optimally control the feed to m aximize a composite performance index .
Erickson an d S tep hanopoulos
794
F(t)
/X=X
F max
max
s
s*
(b)
(a)
S c he m a tic representation of an optim al fed-batch operation : ( a ) m utual di s posi ti on of specific growth and product formation rates ; (b ) feeding s chedule . S max is the maximum substrate c on c ent ra ti on allowed to prevent undesirable side re ac tions . Xm a x is the maximum cell concentra tion allowed by mi xin g and oxy gen tr an s fer re q ui re ment s . FIGURE 8
For the case of penicillin fermentation , for e xample , for which the specific grow th rate and the specific p eni cilli n formation rate are mutually disposed as s hown in Fi g . Sa , t he optimal feed policy is s hown schematically in Fi g . 8b . Accordin g to t hi s strate gy , the fermentation is carried out i n two During the first , cell biomas s is built up to the maximum level al phases . lowed as quickly as pos sible . The second is a p roduct fo rm ation phase durin g which the feed is controlled as an optimal level of hi gh p enicillin production and s ufficient biomass growth to make up for the dilution effects . Other advantages of fed - batch ope ra tion include flexibility in the introduc ti on of growth factors , deliberate variation of the feed con centr ation , and similar control actions . The m ass balance eq u ati on s for all three modes of operation are as follows : dX dt dS dt
dP dt
=
=
=
llX - D X
D(S
o
-DP
( 2 2)
- S) -
y
+ (a +
ll X
--
m ax
s
S lJ ) X
- m X s
(a
+ S ll ) X
y
m ax P IS
( 2 3)
( 24 )
w here D is the dilu tion rate -equal t o z ero for batch operation and t o F /V i for c on tinuou s o pe rati on and for fed - b atch operation ( notice that in the latter case , F dV i /dt ) . =
795
Bio logical Reac tors
In aerobic fermentations a continuous airstream flows through the re actor to provide the necess ary oxy gen for growth . Balances for t he liquid and gas -phas e concentrations of the two main gaseous components , oxygen and carbon dioxide , are as follow s :
dC
0
=
dt
(k a' ) ( C * L o
c > o
-
-
Q
0
x
2
DC
0
+
DC
f
( 2 5)
0
( 2 6)
v
dp
o dt
J. RT
v
dp
c dt
J.
RT
in Q P in o RT
=
Q
in Q p in c RT
=
Q
ou tP o RT
- ( k a' ) ( C * L o o
-
C )V o R.
( 2 7)
outP c + (k a ' ) ( C - C *) V c c c R. RT L
( 2 8)
w here Q 0
an d Q c 0 2 are the specified oxygen upt ake and carbon dioxide 2 evolution rates respectively , defined by ,
=
ll _
__
y
max
0
-
+ m
max
__!!__ Y
D
-'-' a_+ --'-' -'13 l.L"
o
+
+ -
Y
m
D
( 29)
max P /0
+
a
+
13 l.L
( 30 )
max
Y P /D
The dilution rate D in Eqs . ( 2 5 ) and ( 2 6) is define d as before . The values of the non - growth- and growth-associated p roduct formation coef fi ci en t s , a and 13 , respectively , depend on nutrient concentration , tempera.
max
max max , and Y , 0 D are related to th e true bi om as s energetic yield parameter , Tlm ax , as has b e en shown by Eri c ks on ( 1 97 9) . Th e maintenance param eters , m s , m 0 , and m D are also related because of t he s toichiometry of the maintenance proces s ( E rickson , 1 9 7 9 ; Erickson et al . , 1 97 9 ) . Erickson ( 1 97 9 ) has also
ture , and pH
•
Th e true growth yteld p ar amet er s
,
Y
s
, Y
,
shown ho w the true product yield p ar amet e r s
,
max
max
max
Y /S , Y /O , ·and Y / , are P P P D
m ax . p A s toic hio m e t ry difficulty associated with the use of the Luedeldn g and Piret m ode l for p roduct formation in the previous equations has been dis cus sed by Fredrickson and T s uchiy a ( 1 9 7 7 ) . These aut hors point out that If the p r ec ur s or is the growth- limit products come from some p recursors . in g s ubstrate , then , a cc ordin g to E q . ( 2 4 ) , product is for m e d at a finite rate even after the precursor has been exhausted and is not present in the s ys tem I f the precursor is the biomass itself , this should be accounted for by addin g a sink term in Eq . ( 2 2 ) . In either case , some modification related to the true product energetic yield , �
.
Erickson an d S tep hanopo ulos
7 96
is n ee de d in the equations or the parameters if the models above are going to be valid d uri n g all p h as e s of grow t h and prod uct formation . Steady - state operation can be achieve d in a conti n uo us culture system . The s te a dy - st a te concentrations of biomass , s ubstr at e , pro d uct , 0 2 , and C 0 2 in t h e liquid phase are obtained by s e tti n g t he left- hand sides of E qs . ( 1 6 ) , ( 2 2 ) , and ( 2 6) equal to zero . The pos sibility of multiple steady s tates exists , especially for the more com p lica t e d mod el s o f t he s peci fic growth rate . Fredrickson and T s uchiya ( 1 9 7 7 ) have analy zed the dynamics o f various p ure and mixed culture systems . S tephanopoulos ( 1 980) p re sented a m e t hod for q uickly analyzing t he dynamics of two-pop ulation mixed This me t hod makes use of top olo gi c al ar gum e nt s and culture s ys t em s .
Poincare inde x . Various mo di fi c ati on s an d extensions o f the p re vio us model equations have appeared in the literature . The purpose of this work has been two fold : fi rst , to model various departures from the ideal sit uation of a per fectly mixed v es sel , suc h as wall growth , film formation , and incomplete mi xin g . Second , to describe some in teres tin g process or hardware co n fi gura tion s , s uch as cell recycle , vessels in se ri es , film reactors , and so on . Some of this work has been reviewed by Ollis ( 1 97 7 ) •
Other D e s i g n Con s i dera ti on s The desi gn e q u ation s of the p re ce din g section were d ev e lop e d with t he assumption of well- mixed li q uid and gas phase s . Even though m ac ros copic mi xin g is a goo d as s um p ti o n , m ic ros c opi c mixin g is oft en hard to achieve . Dunlop and Ye ( 1 98 2 ) hav e p oint e d out t hat in many fermentors the smallest turbulent eddy has a size o f about 5 0 \.liD . This , con tras t e d with an average si ze of about 1 t o 2 ).lm fo r a b ac t e ri um , indicates that various cells may be enclosed in an environ m e n t where nutrients are rapidly depleted . In c r e asi n g t he p o w e r in p ut h a s a minim al effect , for the size of t he smallest eddy is inversely p ro po r ti on al to the on e - fo urt h power of the ener gy input . Th e s am e authors conclude that the len g t h scale of turbulence and the rates at w hich eddies a r e b ein g created and destroyed m us t be q u an tifi e d . S u p p lyin g sufficient oxygen and removin g the p roduced carbo n dioxide in th e case of aerobic fermentation is probably the si n gl e m os t i mpo rt an t issue in fermentor d e si gn . Various air s up p ly m echanism s , i mp e lle r de si gn s and m o dific ati on s ·of the bioreactor interior (baffles , etc . ) have been p rop osed to facilitate t h e transport of these gases . Much of the d ev el op ment in thi s area is p rop rie tary _ The equilibrium con c e n t r ati on C is us u ally provided by applyin g H e n ry ' s law as th e p h as e e q uili b ri um r e lati on s hip :
�
P0
=
0
0
H C*
( 31 )
wi t h H 0 b ei n g
the Henry1 s law con st an t for oxygen (Perry and Chilton , 1 97 3 , pp . 3- 98 ; S chumpe et al . , 1 97 8 ) . Similarly , for carbon dioxide , p
c
:::
H C* c c
where H e is t he Henry' s law constant for carbon dioxide ( Stumm and Morgan , 1 97 0 , p . 1 48 ; Perry and Chilton , 1 97 3 , pp . 3- 96) .
( 32 )
797
Biological Reac to rs
Carbon dioxi de in the li q ui d phase is pre sent as dissolved c o 2 , H 2 co 3 , HC0 . The fraction of bicarbonate and carbonate ions which , and co 3
;
are present at e qui li b rium depends on pH ( Stumm and Morgan , 1 9 7 0 , pp . 1 1 8- 1 2 9 ) . At low p H and up to pH 5 , m os t of the C 0 2 is present as dis solved C0 2 ; for the pH ran ge 7 to 9 , the bicarbonate ion is strongly favored , an d at p H 11 and above the carbonate ion is strongly favored , as c an be deduced by consi derin g the equilibrium constants :
( 33 )
=
10
- 10 3 .
( 34)
at 3 0°C ( Pirt , 1 97 5 , pp . 7 6- 7 7 ; Stu m m and Morgan , 1 97 0 , pp . 1 4 8- 1 5 0 ) . Under gas /liquid - phase equilibrium conditions , the total quantity of carbon in the liquid phase is given by
( 35 )
increas es with pH above 5 for fixed concentration of CO 2 in the gas phase as shown in Fi g . 9 . Under co.nditions where the bicarbonate ion i n the broth i s being con verted t o H 2 C 0 3 and t h e H 2 C0 3 t o C 0 2 + H 2 0 , the rates o f chemical reac tion should be considered . The reversible chemical reaction
is very fas t and may be at equilibrium .
Kern ( 1 960 ) gives a value of
( 3 6)
for the e quili b rium constant at 2 5°C .
T h e chemical reaction
- •
2 0 s - 1 and is much slower . Kern ( 1 96 0 ) reports average values of k 1 1 2 = 0 . 0 3 s - at 5 ° C , with activation ener gies o f 1 6 . 5 and 15 kcal /mol , k2 respectively ( Stum m and Morgan , 1 97 0 , p . 1 5 3 ) . If both dissolved C 0 2 and bicarbonate ion are present within the cell and bo t h diffuse throu gh =
Eric kson an d S tep hanopoulos
7 98
a::
�
0 ::!!
z 0
�
a::
z 1&.1 u z 0 u
....
E
01
FIGURE
9
A queous carbonate equilibrium constant
equilibrated with t he atmosphere ( P c o
2
j uste d with stron g base or s tron g aci d . p . 12 7 . )
Pco 2 •
Water is
= 1 0- 3 " 5 atm ) and the p H is ad
( F rom Stum m an d Morgan , 1970 ,
the membrane rapid ly , the bicarbonate ion will need to be converted to H 2 C 0 3 and the H 2 C 0 3 to C0 2 in the bulk liquid as a part of the C0 2 r em oval process . It is clear from the publishe d literature that rate pr oc es s es as sociated with C0 2 removal from the broth may be of intere st in some indus t rial ferm entations (Ishi zaki et al . , 1 97 l a- c , 1 97 3 ; H arfor d and H all , 1 979 ; Ya gi and Yoshida , 197 7) . B ecause of rate limitations , concentrations of C0 2 , H 2 C 0 3 , an d H C 0 3 may substantially exceed the values that would be in eq ui libri u m with the exhaust gas from the fermentor . As microbial cultures are very sensitive systems , they pose . hi gh de mands on the m oni tori n g and control c ap abi li ti e s of a bi ore ac t or . A schematic of the instrumentation av ai lab le for on -line bioreactor m onitoring is shown in Fi g . 10 . Off- line an alysi s by liquid chromatography or en zymatic as say methods can also be employed for the measurement of s ub s tances associated with the process of growth as substrates or products . Presently , on - line meas urement is basically limited to off- gas analysis ( 0 2 and C0 2) by m as s spectrometer or paramagnetic 0 2 and infrared C 0 2 analyzers , and t o the concentrations of dissolved oxy gen and carbon dioxide . These measurements can be utili zed toge t he r with the elemental balances developed earlier to provide on -line estimates for the reactor state variables , w hic h , in turn , can be us ed for con tr ol or other purposes .
799
Bio logical Reac t o rs
! BIOCHEMICAL
R E AC T O R I NS T R UM E NTAT I ON
SHAFT POWE R
I
IMPE L LE R SPEED
[ Tor sion dynamometer (externa l ) ] [Generator - type tachometer
[
Strain gauge measur . ( i n tern a l ) D r op count
FEED RAT E
Addi t i o n vessel on load ce l l Elec trom a gne t i c f lowmeter
l
PRESSUR E
CONT ROL :
Silicone emulsion an ti foa m
Mechanical system
O i l based (ster i l ity problems)
[
[ G ia;-referen ce ] E lect rode
[Simple d i a phragm gauge
l
]
I
[
l
� as
pH
FOAM D E T E C T I O N
C apac i t ance Conductance
V f but impeller tip speed i s the some 7T D ( RPM )
RPM
R E D OX
Combined P I and reference elec trode ( measurement hard to inter p ret) Intera ction w i th DO.
DI SSOLV E D OXYGEN Polarographic sterilizable elect rode ]
02+ 4 H + + 4 e- - 2 H 20
TEMP E R AT URE
[Thermometer bul b ] Thermocouple
PRODUCT REMOVAL
Thermistors
ihafl 'ower
[
[Ove r f low , leve l contr o l , ] load cell .
V IS CO SI T Y
]
FG - GAS F LOW RATE Rota met e rs
Mass f l ow m ete r s
p0
2
..,.. - - .]
[ Paramagnet i c 0 2 an a l Elect rochemica l cel l
[IR
pC02 ana lyzer ]
-
T URBIDI T Y
[ Spec t rophotometer
l
Biomass measurement (not for de nse cultures) SENSORS UNDER DEVELOPMENT
Glucose Ethanol
NH 4 , Mg +
,
..
+
K , No
AT P, A D P , AMP
DN A , R N A
+
,
..
:.
Cu , P 0 4
N AD H _ ,
FI GURE 1 0 Schematic of biochemical reactor instrumentation . Temperature and pH controls are not shown . Viscosity measurements can be obtained from the slope of the shaft pow er versus time graph . Redox meas urements are usually hard to interpret .
]
BOO
Eric kson and S tep hanopo ulos
On- line , automated microanalytical methods , as well as various enzyme prob e s , are pr esent ly bein g developed for the p urpose of provi ding the m eans for m oni torin g m any oth er reactor variables on- line . Bioreactor control is p resently concerned with the control of the abiotic culture e n vi ronm ent as the latter i s described by the temperature , the pH , the dissolved 0 2 concentration , and the presence of foam . All the fore goin g parameters are m aintained at set p oint s which are optimal for the Manipulative variables are the coolant flow rate for the tempera proces s . t u re , the rate o f an acid or alkali addition for p H , agitation speed and gas flow rat e for dissolved 0 2 , and m echanic al foam breakers or agent addition for foam . In fed-batch op e rati on the feed rate of nutrients is also controlled accordin g to a predetermined schedule or on- line meas urement or Certain interaction exists between the various control actions , es tim at e b ut the multivariable control characteristics of these systems has not been studied . Mos t of the control is p re se ntly carried out by local analog con However , as the reactor systems and p roces se s grow more com trollers . plex and the monitorin g-estimation capabilities increase , more sophisticated control p olici e s are expected to be introduced in bio r eac tor op erati on .
.
O t h e r B i o reactor D e s i g n s
Alt hou gh the s ti rred tank has been employed extensively in the past , it is consi dere d inadequate for more demandin g applications . The main fact ors against it are technical , but there are also economic and biolo gical reasons for developin g novel bioreactor schemes ( Sch iigerl , 1 98 2b ) . Schiigerl ( 1 9 8 2b ) s um m ari ze s the main reasons for the introduction of new bioreactors as ( a ) the in abili ty of s ti rre d tank scaleup to ve ry large si ze because of desi gn di fficulties and heat removal problems , often accomp anied by hi gh power requirements and hi gh energy and utility c os t s ; ( b ) reduction of specific en er gy and capi t al costs ; ( c ) avoidance of cell dam age ; and ( d ) increase o f yield s and reduction o f s ubstrate loss e s . Information on these new systems is scarce . Most of the available knowled ge comes from laboratory and pilot plant units and is s um m ari zed in a co mp rehensive review by S ch ii ge rl ( 1 982b ) . The di sc us sion below follows his paper . The variou s types of bioreactors are classified into the followin g cat egories accordin g to the m e t ho d of energy introduction : -
1. 2. 3.
Reactors with mechanically moved internals External li qui d p u m pin g Reactors wi th gas comp res sion
The t ran di ti on al submerged-culture s tirred tank s b elon g to th� first They can have various types of agi ta tion without cate gory (Fig. 1 1 ) . ( 1 . 1 ) and wit h ( 1 . 2 ) l oop . Other types are mechanically stirred loop reac tors ( 1 . 3 ) ; self-aspirated aerated reactors without ( 1 . 4 ) and with ( 1 . 5 ) loop ; m echanically s tirred , self-aspirated , ae rate d hori zontal loop reactors ( 1 . 6) ; cascade reactor s with rotatin g mixing elements ( 1 . 7 ) , with axially moving mixing elements ( 1 . 8 ) , or with p ulsating li qui d ( 1 . 9) ; and fin ally surface reactors s uc h as t hi n lay e r reactors ( 1 . 1 0 ) , disk ( 1 . 1 1 ) , and paddlew heel reactors ( 1 . 1 2 ) Types of reactors in which the energy input is by means of an ext ernal pump are shown in Fi g . 1 2 . Such s ub mer ged ty p e reactors i nclu de plun gin g ,
-
•
-
Bio lo gical Reac tors
801
G
G
LA
LA
LA
M-Motor G-Gas ( Ai r ) G
G
- - -'o - ·
i 1 .1 2
/
'
- - �-
G 1.8
'
1.3 Overfilled
\
.
52
..'Sof;I .. (3 H
d-G G ,
1. 5
- i-
G
I
SZ-baffles 1.6 L R-c ond u it tu be M Motor SZ-Foa m b reaker G Gas ( A i r ) G
::t G
G
G
1.3 With overflow
1.2
1.1
F
-6
G
L R-conduit 1 -Self-ga s agitator C.
� � ® G -oP u l sa t i o n 1.9
M-Orive motor W-Ro ller 1.10
1. 1 1
FI G URE 11 Biochemical reactors in which energy input is by mechanically moved internals . ( From Schii gerl , 1 982b . ) .
I
80 2
Erickson a n d S tep hanopo ulos
' G G
F G
F
G I
23 G
F u
G
'
&
G AS SE PA R ATO R
1.6
SK· F i oat
I D · Injector nozzl e
2.7
2.1
G
FIGURE 1 2 Biochemical reactors in which energy input is by circulation using an external pum p . ( From Schii gerl , 1 9 8 2b . )
jet reactors ( 2 . 1 ) , jet -loop reactors ( 2 . 2 ) , plunging- channel reactors ( 2 . 3) , noz zle loop reactors ( 2 . 4) , perforated -tray or sieve -tray cascade reactors ( 2 . 5 ) , tubular-loop -reactors ( 2 . 6) , reactors with rotatin g injector noz zles . ( 2 . 7 ) , packed -bed (percolating ) reactors with counterflow of the phases ( 2 . 8) , and bubble column , downflow reactors ( 2 . 9) . Packed -bed reactors can also be classified as surface reactors . T he third cate gory , shown in Fi g . 1 3 , includes submer ged - type reac tors in which the energy is introduced by means of compression . In this category , along with the sim ple single -stage bubble column ( 3 . 1 ) , are the single - stage m am moth loops with centrally arranged draft tubes ( 3 . 2 ) , ex ternally attached tubular loops ( 3 . 3) , vertical partition wall ( 3 . 4) , and downflow loop ( 3 . 5 ) . B etween the loop reactors ( 3 . 2 and 3 . 4 ) and the down flow loop reactors ( 3 . 5) there i s , of course , a certain overlappi n g . B ubble columns wi t h s tatic internals belong t o t hi s group , includin g con current bubble columns with stage-sep arating trays ( 3 . 6) , concurrent
803
Biological Reac tors
G
G
G
G
3.3
G
G
G
¥ I
G
;
I
tW
F F 3.4 G
2
J.S G
G 3.9 1 . Gas valve 2. Static m i•er Valve 1 is periodic ally opened and closed
F
3.8
3.7 FI G U R E 13
t
Biochemical reactors in w hich ener gy input is by compression . 1 98 2b . )
( From Sch iigerl ,
bubble columns with s tatic mixers ( 3 . 7 ) and mammoth loop reactors with stage - s eparating trays ( 3 . 8 ) , and bubble columns with stage- separating trays with external tubular loops filled with static mixers and• with pneu matically imposed li quid p ulsation ( 3. 9 ) . Schii gerl ( 1 98 2 b ) also provides tables with the characteristics of these reactors and references for more details . MASS T R A N S F E R I N A E R O B I C F E R M E N T AT I O N S
I n m ultiphase processes with chemical reactions , i t i s common for mas s trans The transfer of oxygen to aerobic fer to be an important consideration . cultures and the trans fer of carbon dioxide from the culture to the exhaust gas are prob ably the two m os t important mass trans fer considerations in
804
Erickson an d S tep hanopoulos
biochemical engineerin g . Oxygen trans fer from the gas phase into the liquid phase has been the subject of much research (Moo Young and Blanch , 1 981 ; Schiigerl , 1 98 1 ; Aiba et al . , 1 97 3 , pp . 1 6 3- 2 1 7 ; Bailey and Ollis , 1 97 7 , pp . 4 11- 496) . T he high viscosity of many fermentation broths has been one of the important reasons for this research . With bacteria and yeasts , oxygen transfer from the gas phase to the liquid phase is often the most important mass transfer consideration ; however , it is relatively easy to design industrial equipment in which the desired mass transfer rates are achieve d . As the size of the microorganism increases , oxygen trans fer from the liquid phase to the microorganism becomes more important . For mold fermentations , it is important to c onsid e r oxygen transfer both from the gas to the liquid and from the liquid to the microbial biomass . For mold pellets , transport within the pellet is also important . The transport of C 0 2 from the broth and the transport of 0 2 to the broth s hould both be considered in the an alysis of the mass tr ans fer associated with an aerobic fermentation . The sa.me gas flow rate is common to both of these processes . Furthermore , the pressure in the fermentor affects the dissolved 0 2 and C 0 2 concentration s , but in opposite directions . Increasing the pressure enhances oxygen tran sfer , but it negatively influ ences the rate o f C0 2 release . Coefficient Mass trans fer in gas -liquid system s is treated in detail in Chap t er 10 . In addition , mass transfer in biochemical reactors is reviewed by. Moo Young and Blanch ( 1 98 1 ) . In most fermentors , the mass transfer coefficient in the liquid phase depends on the Sher woo d , Gras hof , and Schmidt numbers because the density difference is the dominant force that causes fluid motion between the bubble and the liquid . For small rigid bubbles less than 2 . 5 mm diameter (Calderbank and Moo Youn g , 1 96 1 ; Calderbank , M a s s T ra n sfer
1 95 8 , 1 95 9 , 1 96 7 ) , Sh
=
2. 0
+
0.
3 1 Gr 1 1 3s c 1 1 3
and for bubbles greater rigid ,
than 2 . 5
( 37 ) mm
in
diameter which
are
much less ( 38)
where .
'
( 39 ) ( 40 )
( 41 )
805
Biological Reac tors
Equations ( 37 ) and ( 3 8 ) appear to be ap p li c ab le to a wide variety of geom mixin g c on fi gu ration s because the den sity difference cau s es fluid motion near th e ga s liquid interface . Fukuda et al . ( 19 7 8 , 1 9 7 9) have m e as ured kL in an airlift ferm en t or and a p er for ate d plate colum n and foun d that Eq . ( 3 8 ) with a cons tant o f 0 . 5 0 in s t ead of 0 4 2 fits their data This small difference may be due to s ome pr op ertie s of the ve ry well . solutions which are not considered in the correlations and ex p eri m ent al error . Equation ( 37 ) may also be u sed to e sti m ate mass trans fer coefficients to mi c robi al biomass . For bacteria and yeast , the expres sion S h = 2 . 0 p rovi des a useful estimate o f the mas s transfer coefficient . This expres sion show s that mass trans fer coefficie nt s are relatively lar ge for very small objects such as bacteria and yeasts . For m old pellets and other s uspen d ed p ar ti cles with densities close to t ha t of the continuous p h a se whe re agit ation in a s ti rred mixin g ve ssel creat es the dominant force for relative fluid motion between the phases (Calderbank , 1 967 )
etries and
-
.
( 42)
the power p er unit volum e s upplie d by t h e agitator . frequently present in fermentor s , Surfactants provide a b arri er for diffusion through the film a t the interface ; t h ey also act to give the bubble more rigidit y . The m as s transfer coefficient may decrease si gnificantly becaus e of surfactants { Aiba e t al . , 1 97 3 , p p . 1 8 4- 1 8 5 ) . Swarms of b ub b le s and non-Newtonian liq ui d phase flow behavior als o may affect the value of the liqui d - p hase mass transfer coefficient . Correc tions for these factors have been revi e w e d by Moo Young and B lanch ( 1 98 1 ) w here P /V is Surfactants are
and Schii gerl ( 1 9 8 1 ) .
I n terfaci al A rea The gas-liquid interfacial area is an i mpor tant variable in aerobic microbial growth proces ses . B ecause the interfacial area is affecte d by a large num ber of phys i c al properties and other variable s , very few correlations that accurately p r edi c t the interfacial area und e r fermentation conditions are available . However , knowle d ge of the e ffe ct of v arious p hysic al properties , equipment designs , and op e rati ng conditions on the interfacial area is of considerable importance . The bubble size dis tribution and the gas -phase holdup or volum e fraction are directly re late d to the i n ter faci al area per uni t volume of dispersion ; that is ,
( 4 3)
where e:g is th e gas phase volum e fraction and d 8
diameter ;
the Sauter mean bubble
Erickson and
806
d
s
=
S t e p h anopoulos
( 44)
The bubble s i z e frequency di s t ri b ution , fi , depends on the physical proper ties of the broth , equipment d esi gn , and op eratin g conditions . Equation ( 4 3 ) shows that change s in gas holdup as well as changes in t he bubble si ze distribution will r e sult in changes in the interfacial area. The bubble si ze distribution depends on the g as - liqui d interfacial ten sion ; liquid -phase density , vi sco sit y , and ionic strength ; gas flow rate and hold up ; gas distr i b utor ; r ate of t ur bule nt energy dissip ation ; and e q uip ment design . The chemical composition of the liquid p h as e also appears to C al de rb ank ( 1 96 7 ) m e as ure d bubble si ze distributions and be import ant . inter facial areas in stirred t anks and sieve tray columns . Recently , Bhavaraj u et al . ( 1 97 8) i nvestigate d the effects of viscosity and turbulence on bubble si ze and interfacial area . Schiigerl ( 1 98 1 ) has revie wed t he liter at ure with re spect to bubble si ze interfacial area and m ass trans fer in hi ghly viscous broths . The sparger or gas distributor desi gn and the gas flow rate throu gh the sp ar ger determine t he initi al bubble si ze distribution in the re gio n of the sp ar ger . For moderately high gas flow r ates , B havaraju et al . ( 1 9 7 8 ) found that =
( 45 )
where d 0 is the no z zle diameter and
( 46)
is the ori fice Reynolds n um b e r based on the volumetric flow rate through the orifice , Q0 , an d ori fice di am e t er , do · T he orifice Froude number is
Fr 0
( 47 )
The rate o f co ale s ce nc e influences the bubble si ze distribution . T�e ionic strengt h and che mic al composition of the broth significant ly influence the coalescence frequency and t he bubble si ze di stri butio n ( Schii gerl , 1 981 ; Schiigerl et al . , 1 97 8 ; Zlokarnik , 1 97 8 ) ; however , equations that properly incorpor ate these variables into expressions for the interfacial area or b u b ble si ze distribution for ferm e nt atio n broths are lacking . T his is especially true for hi gh - viscosity broths . T he level of t urbulence greatly influences bubble br e akup t hrou gho ut the fermentation broth . As viscosity increases a point is reached where bubble breakup due to t ur bu le nce is unimportant . B h av ar aj u ( 1 97 8 ) have used the criterion that the scale of the ener gy containing eddies should be
80 7
Bio logical Reac tors
200 times larger t h an the sc ale of t he energy dissipatin g eddies for turbu lent breakup to be important . They found that
d
- 0 7
s
'
0. 6 a 0. 4 2 p . ( P /V )
�
(llL)0.1 ll a
( 48 )
-
fo r the Sauter m e an diameter i n a g as sp ar ged vessel w hen turbulent break up w as import ant . For mixing vessels with a git ator s , C alderbank { 1 96 7 ) found t hat
d
= 2 . 25
s
0 . 6 0. 4 p g
(J
( 49)
Schiigerl ( 1 9 8 1 ) fo u nd that in viscous fermentation brot hs s m all bub bles have too lon g a mean re sidence time i n the broth . As e q uili bri um con ditions are approached , the concentration driving force decreases and oxygen tr an s fe r fro m t he sm all bubbles to the liquid s low s con siderably . Rene w al of t hese sm all bubbles is ve ry import ant i n highly viscous bro t h s The vertic al circulation p attern of t he airli ft fe rm e ntor and various multistage column desi gns m ay help achieve improved small bubble renew al rates ; how ever , furt he r in ve st i gat io n is needed . B ecause some of the interfacial area is not effe c t i ve , Sch ii gerl has chosen to show c or r e l atio ns in term s o f t he co m bine d vol um e t ric m as s t r an s fe r coe fficie nt A n u m be r o f exp ressio n s have been presented for t he gas holdup in aer ate d mixing vessels ( C alder b ank , 1 967 ; H u ghm ark , 1 980 ; B ailey and Ollis , 1 97 7 , p . 4 3 6 ; Aib a et al . , 1 97 3 , p . 1 7 8) . For bubble columns , Hikit a et al . ( 1 9 8 0 ) an d Moo Yo u n g and B lanch ( 1 9 8 1 ) h ave recently re vi e w e d the literature o n g as ho ld up S c h ii r gel ( 1 98 1 ) has divi de d the hold up in viscous media into that as sociated with very small b u b ble s and t h at associated with int e r me diat e and lar ge bubble s . He has presented t he re su lt s in a series of graphs . The interfacial are a m ay be obtained using e xp res sio n s for t he gas hol d up e: g , and the S auter mean bubble diameter , ds , i n Eq . ( 43) or by u sing correlations for interfacial area . Calder bank ( 1 96 7 ) , Hughm ark ( 1 980 ) , Moo Youn g and Blanch ( 1 98 1 ) , and Joshi e t al . ( 1 982 ) have pre se nt e d correlations for interfaci al area , for example . H o w ever , because of the difficulty o f obt aining correlations in which all the import ant variables ar e included , most fermentor design is c arried out using co r rel atio ns in volving the volumetric mass transfer coefficient . .
.
.
,
Vol umet r i c M a s s T ra n s fe r
Coefficient
It is imp o r t ant to reali ze that t wo different volumetric mass transfe r co e f fi cie nt s are c om mo nly found in the bioche m ic al engineering literature for gas -liquid m as s trans fer . The i nte r faci al area per unit volume of gas liq�d dis p er s io n , a , and the interfacial area per unit volume of liquid , a' , ar e related ; that is , a' ( l
-
e:
G
) = a
( 5 0)
80 8
Eric kson
and Step hanopo ulos
Equation ( 4 3 ) leads directly to the interfacial area per unit volume o f di spersion , a ; however , in studies where t h e liquid p hase is completely mixed and the quantity of oxy gen added to the liquid phase is measured , In reviewing kL a' arises naturally , as shown in Eq . ( 2 5 ) , for example . the literature , one m ust be very careful to determine if t he results pre sented are for kL a or kL a' . The flow models selected for dat a analysis should also be examined in The correctness of estim ates of the reviewing d at a in the literat ure . volumetric mass transfer coe fficient depends on the selection o f proper flow models w hich appropriately fit t he experiment al system w hen si gnificant variations in concentration occur with position .
B ub b le
Co l umns
In bubble columns [ Fi g . 1 3 , ( 3 . 1 ) 1 the energy to provide turbulence and bubble breakup is contained in the entering gas ; t hat is , the power
P
=
QP
G
RT P1 In M p2
( 51)
-
T he volumetric mass is directly related t o t h e volumetric gas flow rate Q . trans fer coe fficient k a has been correlated by Akita and Yoshida ( 1 97 3 , L 1 97 4 ) , who found that
n- ad t k d L t
_ -
5 O . SS c 0 . B o 0 . 62 0 a 0 . 3 1 e: 1 . 1 0
( 52 )
w here t he B ond num ber is
Bo and the G alileo number is
These authors also correlated t he gas holdup and found e:
G
(1
-
e:G )
4
=
0 . 20 Bo
0 . 12 5
where the Froude num ber is
0 083 Fr Ga .
( 5 3)
80 9
Biolo gical Reactors
wi th UQ th e s u p e r ficial gas velocity . These two correlations were t e st e d in l ar ge - di am e t e r bubble columns usin g C0 2 an d w ater and it w as found that the effect o f col um n diame t e r di s appe ared above about 0 . 60 m . Fo r diameters above 0 . 60 m , the authors and Kataoka et al . ( 1 97 9 ) re com m e n ded usin g 0 . 6 m in Eqs . ( 5 2 ) and ( 5 3 ) . The authors recom mend u sin g these correlations for superficial gas velocities up to 1500 m /h and gas holdup up to 30%. T hese equations ar e for di s persions i n which the vi s co sity is sufficiently low for t urbu le nce to play a domin ant role in d e ter m i nin g t he b ub ble size distribution and interfacial area . Liquids up to 0 . 2 1 p oi s e were con sidered b y the authors . Deckwer ( 1 9 81 ) has t e s t e d Eq . ( 52 ) u si ng some of his d at a and other literat ure d at a ( see Fi g . 1 4) . H e concludes that the correlation is good in so me c ases , but that with some sp ar ge r s lar ger values of kL a are foun d . The effects of salts and or g anic substances on coalescence are not included
in the correlation . Schiigerl et al . ( 1 9 7 7 ) have investi gate d gas ho ldup , interfacial are a , an d m as s transfer i n bubble columns for a wide variet y of model fermenta tion media u sin g p orou s pl ate , p er for ate d plate , injector no z zle , and ej e ctor no z zle gas distributors . T hey point out that higher volumetric mass trans fer coefficients than t hose pre dicte d by E q . ( 52 ) can be achieved by using distributors that p rod uc e a b u bble si ze distribution in the e nt r an ce zone which is sm aller than t he eq ui libri um bubble si ze distribution asso ciat e d with the turbulence due to the ener gy dis sip at ion from Eq . ( 5 1 ) . Additional ene r gy is req uired at t he d i stri buto r to produce these small bubbles . Un d er the condition s of t heir exp eriments , the holdup , interfacial area , and
0
....
.Ill:
1 0- 1
.31 D ( S c ) 0 .5( B o )0.62 ( Go ) 0 2"" dt
E"
1. 1
G
FIGURE 14 Volumetric m as s tr ans fer dat a o f To w e ll et al . ( 1 965 ) for � 40 . 6 em , � ; Deckwer et al . ( 1 97 4) fo r dt 20 em , o ; K ast ane k ( 1 97 7 ) for d t = 1 0 0 em , o ; an d K at aoka e t al . ( 1 97 9 ) for dt = 5 5 0 em , 0 plo tt e d against t he correlatio n of Akit a and Yoshida ( 1 97 3 ) u si n g dt 60 em for t he l ar ger - d i am e t e r co lu m ns . =
=
81 0
Erickson and S tep hano p o u los
volumetric m as s transfer coe fficients varied si gnifi c ant ly with the various Deckwer ( 1 98 1 ) p oi nts out that these resu lt s are no t well correlated by Eq ( 5 2 ) Moo Yo u n g and Blanch ( 1 98 1 ) have revie wed mass trans fer correlations for b ub ble columns and found that volume t r ic m as s tr ans fe r coe fficient s are di re ct ly proportional to the s uper ficial gas velocity . These results are for low viscosity broths in which t ur b ulenc e p l ays an i m p or t an t role in de t er mining t he bubble si ze distribution . As viscos it y increases , a point is reached where t he behavior at t he gas distributor become s more import ant ( B havaraju et al . , 1978) . T hi s occurs as the scale of t he energy dissip at i n g eddies becomes lar ger . A t higher viscosities , the re sults for bubble colum ns are not in goo d a gree m e nt Hen zler ( 1 98 1 ) cor related the d ata of B uchhol z et al . ( 1 97 8a , b ) for a 3 . 9 m hi gh bubble column wit h 0 . 1 4-m di ameter for app arent viscositie s r an ging from 4 6 to 2 1 7 m P a • s . The re
. .
m edi a select ed .
sult is ( see Fi g .
( )
k a'
_2
_!._ U
�
0.
g
G
. --
15)
333
Sc 0 . 5
=
0 . 06
[
u
-
( g\J )
�
.
]
333
-0. 9
( 5 4)
Deckwer et al . ( 1 982 ) point out that values of kL a' predicted by this ex r e s sion are at least five times lar ger than those predict e d by N akanoh and Y o s hid a ( 1 98 0 ) . The results of Deckwer et al . ( 1 982 ) for aqueous carboxy me thyl cellulose ( CM C ) solutions are correlated by t he eq uatipn
p
kLa
=
2 . 08
x
. 1 0 - 4u 0 5 9 1.1
G
O.D l
- 0 . 84
( 55)
0.1
(
FIGURE 1 5 B ubble colum n by H e n z le r ( 1 98 1 ) to o b t ai n lo wi n g ap p are nt vi scos itie s : to 1 40 mPa • s ; .&, 1 0 9 to 1 7 2
uG
0. 3 3 3
( g v)
)2
dat a o f B uc hho l z et al . ( 1 97 8a , b ) as correlated Eq . ( 5 4 ) . For aqueo us solutions with the fol 0, 46 to 6 5 m P a • s ; • , 6 4 to 9 5 mPa • s ; v, 92 m Pa • s ; an d o , 1 30 to 2 1 7 mPa • s .
81 1
Bio logical Reactors
where ua i s me an su p e rfi cial gas velocity in em /s and "il is the app arent vi scos ity in P a • s . This correlation predicts res ult s which are s m aller t han those of N akanoh and Yoshida ( 1 98 0 ) . T here are si gni fic ant differences which m ay exp lain these res u l ts . B uchholz et al. ( 1 97 8 a ) used a sintered plate sp ar ger wit h a mean hole diameter of 0 . 0 1 7 5 mm , while Deckwer e t al . ( 1982 ) used a sintered p late with 0 . 20-mm holes . The results of Deckwer et al . ( 1 9 82 ) and the results of Voi gt et al . ( 1 980 ) for CM C so lu tio ns are in fairly good agreement for g as velocities above 2 em /s . Plates of 0 . 51 . 0- , and 3 . 0 m m hole diameter were used by Voigt et al . ( 1 980) i n a multist age colum n ; however , only one stage was used for t he e xp eriments wit h C M C . Henzler ( 1 98 0 ) correlated the results of Voigt et al . ( 1 980 ) to o bt ai n : ,
-
k a'
=
L
o Gv - 1 . 0 3 cu . G
( 5 6)
or
0. 075
[
u
G
( gv)
O.
33 31 �
-0. 4 ( 57 )
The results of D eckwer et al . ( 1 982 ) and t hose of B uchholz et al . ( 1 97 8a) show that t he des i gn o f the gas di st ri b ut or m ay be im p or t ant for visco u s so lutio n s T he me asure of B h av ar aj u et al . ( 1 97 8 ) for t he existence of an internal subran ge in the ener gy spectrum of t urbulence i s .
=
d po. t L
0 . 75
ll L
5
(�)
0. 25 ( 58 )
where 1 / n T s ho u l d b e lar ger than 2 0 0 bas e d o n their dat a . T able 1 show s 1 / ll T fo r some of the experiments of Deckw er et al . ( 1 982 ) . S o me of t he values o f 1 / n T at hi gh gas s uperfici al velocity and low CMC concen T hi s s ug gests that turbulence should be tr at ion are gre ater t han 200 . However , at 1 . 3 and 1 . 6 wt % CMC , l i n T is le ss than 2 0 0 at import ant . The form of Eq . ( 5 5 ) s u ggests that all s up er fi ci al velocities investigated . turbulent fo r c es are i mp or t ant in the experiment s of D e ckw er et al . ( 1 9 82 ) . The exponent o f 0 . 5 9 for s up er ficial velocity s ho w s that the power input is import ant ; ho w e v e r the value of the exponent is sm aller than the value The dependence on vi s co sit y differs of about 1 . 0 for low vi s cosi ty broths . from that found by Akit a and Y oshida in Eq . ( 5 2 ) . Equation ( 5 5 ) m ay be cont rasted with Eq . ( 5 4) w here t he distributor appears to have si gni ficant ly affecte d the b u bb le size distribution . In Eq . ( 54) , kL a' is proportional to the 0 . 1 power of super ficial v elocity Viscoelastic fluid s in bubble columns have been con sidere d by H e c h t et al . ( 1 980) an d N okanoh and Y osh i d a ( 1 98 0 ) . T h e correlation of N akanoh and Yoshid a is value s of
,
-
.
( 5 9)
81 2
Eric kso n an d S tephanopo ulos
TABLE 1 Ratio of t he Scale of E n ergy Co nt ainin g E ddies to Energy - D i s sip ati n g E ddies for B ubb le Column Experiment s o f Deckwer e t al . ( 1 982 ) -
S up e r fi cial gas veloci ty ( em /s )
CMC ( wt %)
R at io
1 / nT
0. 4
1
1.0
1
1 85
2.0
1
229
3. 0
1
2 60
4. 0
1.3
167
0. 4
1.6
55
2.0
1.6
103
4. 0
1.6
1 35
1 39
where t he Deborah nu m b er is D e = ub >. / d s where ub is t he bubble ri s e ve lo cit y , >. the characteri stic time of the viscoelastic m ateri al , and d8 t h e ,
S auter me an bubble diameter .
In E q .
( 5 9) ,
kL a is p roportional t o the
s up er fi ci al gas velocity to t he 1 . 0 power . This m ay be due to the fact that N akanoh and Yoshida used a sin gle no z zle that w as 4 mm in diameter in their 1 4 . 5 5 e m di am e t er colum n . U nder these co n dit io n s almost all o f the sur face area must be ge ne r at e d by t u rb u l en t forces . -
-
,
A irlift Columns Airli ft tower fe r me ntor s ( s e ve r al differ e nt d es i g ns are s ho w n in Fi g . 1 3 ) T he vertical h ave received considerable attention d uri n g t he last 1 5 years . liquid circulation t h at occurs because of t he up flo w and downflow sections of the airlift co lu m n helps to keep sur face - active a ge nt s distributed through out the b ro t h T he cocurrent upflow s e ct io n is si m il ar t o a bubble column in some respects ; however , t h e cocurrent li q ui d flow influences the two T he transition from bubble flow occurs at a m �(j!h phase flow behavior . l ar ger s upe r fici al gas ve lo ci t y in an airlift fe r me nto r then in a bubble column ( Hatch , 1 97 3 ) B e c au s e of this , the p o w er dissip ated per unit volume in t he up flow section of the airlift can be considerably greater under bubble re gi me op er at i n g conditions compared to an ae ratio n tower . B lenke ( 1979) has reviewed his work and that o f o t he r s in airli ft reactors , j et loop reactors , and propeller loop re ac t o r s ; he has shown that fo r c e d liquid circulation m ay be u s e d t o i n c r ease s u b st anti ally gas holdup and Correlations for liquid circulation rate , gas holdup , and interfaci al area . volumetric o x yge n trans fer coe fficient have been presented by Bello et al . ( 1 981) ; for oxy gen tran sfer , .
•
B i o logical Reactors
81 3
( 60 )
where b is 0 . 000507 for w ater an d 0 . 0 0 0 5 02 for 0 . 15 M N aCl solut ion , P a is the power ( watts ) , V i s the tot al reactor dispersion volume ( m 3 ) , and Ad I A r is t h e ratio of t he are a of t he down come r to the ris e r . Blenke ( 1 97 9 ) found that the interfacial area per unit volume of li qui d , a' , in cre as ed with power per unit volume of liquid in the bubble flow re gim e ; to correlate his data in the bubble flow re gi m e , expon e nts r an gi n g from 0 . 85 to 1 . 0 are needed on power per unit volume . At superficial gas ve locities above about 30 em /s , the dependence on power per u ni t volume decreases ; this m ay be due to s lu g flow . Orazem and Eri ck so n ( 1 97 9) fo un d that the oxygen trans fer coeffi cient increased linearly with s up er fi ci al gas velocit y in a t wo - st a ge airlift tower . H atch ( 1 97 3) cor relat e d the volumetric m as s trans fer coe fficient for the up flo w re gion , head re gion , and downflow re gion . At the same sparger gas flow rate , the hi ghest volumetric mass transfer coefficients are in t h e head re gion ; those in the draft tube are slightly higher than t ho se in the downflow re gion . Moo Youn g and Blanch ( 1 981 ) have reviewed some of their own work and that of Lin ( 1 97 6 ) and found a s li ght ly gre ater dependence on gas sup er ficial velocity wit h values o f the exponent ranging from 1 . 06 to 1 . 2 9 . H atch ( 1 97 3 ) has s hown that volumetric mass t r ans fer coefficients in the head re gion are app ro xi m ately twice as large as those for an aeration to wer under similar s p arger gas flow rate s . In the downflow section the gas holdup must be les s than that in the up flow s ection bec ause the liquid cir c ul at ion r ate depends on the difference in dispersion density between the two re gions . The most co mmon op e r atin g condition results in a liquid circu lation rate sufficient to entrain small bubbles but not lar ge ones in the liquid downflow . Under ferment ation conditions where coalescence rates are re lat i ve l y s mall , the bubbles in t he downflow appear to be segre gated with little coalescence occurring . Very little attention has been directed t o re search in the downflow section of the airlift fermentor ( Orazem et al . , Blenke ( 1 97 9 ) and Fuj ui et al . ( 1 980) have in ves ti gate d some 1 97 9 , 1 980) . related cocurrent down flow phenomena . The effect of vi s co sit y on oxygen transfer in airlift biolo gi cal reactors has received very little study . Several in ve s ti g ators have used airlift fermentors and obtained good result s u nder vis co u s condition s ( K o ni g et al . , 1 9 82 ; Malfait et al . , 1 9 8 1 ; B arker and Wo r g an , 1 98 1 ) ; however , furt h er re search is needed to better appreciate the o xy ge n transfer limit ations as sociated wit h hi gh vis co si t y in an airlift fermentor . Bec a u se hi ghe r values of power pe r unit volume may be o b t ai ne d in an airlift co m p are d to a bubble column , t he airlift column s ho u l d be s uperior to a bubble co l u mn u n der hi gh - vi scosity co n dition s . The effect of the ratio of the downflow area to up flow area on oxygen transfer has been in ve st i g ate d by Hatch ( 1 9 7 3 , 1 9 7 5 ) . B ased on the per for m an ce ratio , the optimum area ratio is about 0 . 8 .
C o lumns with Mo tion less Mixers As shown in Fig . 1 3 , m ult is t a ge columns with sieve plates , co l um n s with Koch motionless mixers , and col um n s with other stationary i nte rnals that
Erickson an d
81 4
Step hanopoulos
promote mixing have been investigated ( S chii gerl , 1982b ) . Kit ai et al. ( 1 969 , 1 97 2 ) showed that volumetric m as s transfer coefficients in multist age perforated plate columns were larger than t hose in aeration towers at high superficial gas velocitie s . The column internals help to m aintain bubble flow conditions by removin g lar ge bubble s and slu gs . K . Hsu et al . ( 1 97 5 , 1 97 7 ) investi gated oxygen tran s fer i n m ultistage columns with sieve trays and w it h Koch motionles s mixers ; they found that Koch mixers and perfor ated plates gave fairly similar results . For both , the perform ance ratio w as hi gher when column internals were employed comp ared to an aeration tower . These result s are very much in agreement with those given above for bu bble columns , where it w as pointed out that the gas distributor could positively influence t he volumetric m as s transfer coefficient . Columns wit h Koch motionles s mixers and liquid circulation have been considered by Fan et al . ( 1 97 5 ) , H . Hsu et al . , ( 1 9 7 5 a ) , and Wang ( 1 977) . Wan g ( 1 97 7 ) has developed correlations for the volumetric m as s trans fer coefficient which includes the superficial liquid velocity UL , as well as the superficial gas velocity uo ; he found for 7 A Y who le Koch mixers that 0. k L a = 0 . 1 2 0u L
UG
62 4 1 . 9 9u
G
+ 47 . 1
( 61)
1 and t he superficial velocities are in em /s . Vertical circulation within the fermentor is o ften needed to distribute sur face - active agents which are carried to t he liquid s ur face by air bubbles . Foamin g problems were encountered in early desi gns which did not allow for vertic al circulation . Considerable research and the commerci al applica tion of colum ns with motionles s mixers has been reported in Latvia . This work has been reviewed by Viesturs et al . ( 1 980 , 1 98 1 ) . Better oxygen trans fer efficiencies are reported for 1 0 0 - m 3 colum ns with st ationary inter nals . Schiigerl and coworkers h ave investigated oxygen transfer in multi stage columns with p erforated plates at low viscosity ( Voigt and Schii gerl , 1 97 9 , 1 9 8 1 ) and high viscosity ( Schii gerl , 1 9 8 1 ; S chii gerl et al . , 1 982a ; Voigt et al . , 1 9 80 ; Hecht et al . , 1 98 0 ) . The optimum spacing bet ween st ages depends on the coalescence rate of the bubbles . For rapid rates of coalescence , the bubble si ze distribution generated by the distributor needs to be renewed more frequently , and a shorter distance between plates is su ggested ( Voigt and Schii gerl , 1 97 9) . For slow rate s of coalescence , bub ble si ze distributions generated by the distributor are retained for a longer time ; under t hese conditions t he contribution of the distributor to oxygen trans fer c an be greater . Under actual ferment ation conditions larger volumetric mass trans fe r coe fficients are found for the m ultist a ge 'colum n ( Voi gt and Schii gerl , 1 9 8 1 ) . Appropriate correlations that account for the contributions of both the distributor and bulk turbulence t o the interfacial area and vo lume tric mass transfer coefficient are lackin g . The effect of coalescence m ust be con sidered in order to relate the bubble breakup at a gas distributor to the volumetric mass tr ansfer coefficient throughout the fermentor . The power dissip ated at the gas distributor should also be considered . Zlok ar nik ( 1 97 8 , 1 97 9 ) has correlated the volumetric m as s trans fer coefficient with the power of a liquid jet of an injector no z zle . He has found ( Zlokarnik , 1 9 7 8 ) that
where k L a is in units of s -
B io logical Reac tors
k a' L
( �)
0 . 33
g
=
0 . 0002 3
[
]
P /V
0" 8
L 4 0 . 33 3
p ( vg )
81 5
( 62 )
(1979)
or that the volumetric m as s t r an s fe r coefficient is proportional to the power per unit volume t o the 0. 8 p ow er . Blenke has also s ho w n that p ow e r supplied at t he no z zle of his j et loop reactor should be con sidered in cor rel ati n g interfacial are a ; that is , a = 5. 4
x
3
1 0 u�· 4
(p
;
)0 . 6 6
( 63)
·
3
where a is in m - 1 , u 0 is in m /s , and PL /V is k W /m o f dis p er sion volume . Zlokarnik ( 1 97 9) has shown how the coalescence rate influences the volumet ric m as s trans fer coefficient in lar ge - s cale s ystem s w it h i nje ctor no z zles .
Mechanically A gi tated Sy s tems
Mechanically agit ated fermentors ( Fi g . 1 1 ) wi t h air sparged into the ves sel below the a gitator are widely used in in d us t r y . Volumetric mass transfer coefficients for s uch s y s te m s have been reviewed by Moo Young and Blanch ( 1981 ) , Joshi et al . ( 1 9 82 ) , and Sc h ii gerl ( 1981) . Moo Yo u n g and B lanch
( 1 98 1 ) and Joshi et al . ( 1 9 8 2 ) have presented tables of correlations for the volumetric mass t r ans fe r coefficient from the liter at ure . The agitation power p e r unit vo lum e , the superficial gas velocity , and t he p hy si c al p rop e rtie s of the broth affect the volumetric mass transfer coefficient . Miller ( 1 97 4) , Zlokarnik ( 1 9 7 8 ) , Yagi and Yoshida ( 1 97 5 ) , and Henzler ( 1 980 ) , for example , have inve st i gat e d volumetric mass transfer coefficients in agitated ve sse ls . Yagi and Yoshida developed the correlation
=
0 . 060
x
( : t {diN�r1 9 ( 5
d N pL
(� �
�a
uG
)0. 6
(
N
d
I UG
\
)0.32 g
[1
+
\0 . 5 pD/L
�a
( 64)
2 ( AN ) 0 . 5 1 - 0 . 6 7
using both Newtoni an and non - N ewtonian fluids . The di me n sio n le ss groups are t he modifie d S herwood number , impeller Reynolds number , Froude num ber , Sc hmi dt number , gas flow number , aeration num ber , and ' Deborah number . Henzler ( 1 98 0 ) developed the correlation k a' V L L
Q
Sc
0. 3
=
0 . 045
[ p
G
Q p ( gv ) L
]
0. 5
0 _ 667
usin g d at a fo r w ate r , gluco s e , and gly ce rin . he found that
( 65 )
For aqueous CM C solution s ,
81 6
Erickson and S tep hanopoulos k a'V L L
Q
= 0 . 0 82
[
Q pL
p
G
1 J
0. 6
( 66)
0 667 ( gv ) .
Zlokarnik ( 1 97 8 ) has examined the effect of coalescence on t he volumetric m ass transfer coe fficient in agit ated vessels .
D ES I G N O F B I O LO G I C A L R E A C T O R S
There are m any factors that must be considered i n the desi gn of biological reactors . Providing t he desired environment for the microbial pop ulation at reasonable cost m ay be accomplished with an available fermentor or by desi gn of a new fermentor . When a new fermentor is designed , m any more decisions m ust be m ade . Decisions that may be m ade with an existin g mechanically agitated fermentor include the agitation rate , the gas flow rate , t he operating pressure in the ferme ntor , the liquid volume , the nutrient composition of the feed , pH , temperature , method of foam control , and mode of operation ( batch , fed batch , or continuous ) . For a new fer mentor , equipment design must also be considered . In both cases , the rate of oxygen t r an s fe r in aerobic fermentations is usually one of the im port ant concerns . Economic considerations m ust be carefully considered in the design process . Some of t he important costs include t he capital costs for equip ment , the co st of or ganic s ubstr ates and other nutrients , energy cost for aeration , cooling w ater or re frigeration costs , product separation costs , waste treatment costs , and personnel costs . An economic optimum exists because of the followin g considerations : The rate o f t he fermentation process should be hi gh to reduce fermentor cap it al costs ; however , aera tion and coolin g costs increase as the rate increases . Microbial growth is usually most e fficient at relatively high specific growth rates . Cell and product concentrations should be as large as possible to reduce separation costs ; however , aeration , cooling , and nutrient costs ( yield ) must also be considered . An optimum gas flow rate and operating pressure need to be selected in the desi gn proces s . For a given production rate of the product , t})e net present value of the capital and operating costs should be minimized . M echa n i ca l l y Agi tated Fermentors
The desi gn of mechanically agitated fermentors is treated by Aiba et al . ( 1 97 3 , p . 1 7 1 ) , B ailey and Ollis ( 19 7 7 , p . 443) , and Wang et al . ( 1 97 9 , p . 3 5 = PIp N d , 1 5 7 ) . Relationships between the power num ber , P the 1 c no impeller Reynolds number for agitated liquids are presented in Fig . 1 6 . Recently , S ch iigerl ( 1 98 1 ) h as presented similar graphs for non - Newtonian viscous broths . H ughmark ( 1 980) correlate d the gassed to un gassed power ratio and found that for flat - blade turbine impellers ,
��
( 67)
B io lo gical R e ac tors
81 7
1 00
S IX-BLADE TURBINE :
,;o
II H
RUSHTON
z
� • 0:: ILl
BATES
iS:
S IX-BLADE PAODLE
I I
FOUR-BLADE PADDLE :
0::
PROPELLER :
FIGURE 1 6 Correlation of impeller Reynolds number and p o w er number for different imp ellers in ungassed liquids . ( From Wang et al . , 1 97 9 , p.
1 60 . )
corre l atio n s have been p re s e nte d by Luong and Volesky ( 1 9 7 9 ) , H assan and Robinson ( 1 97 7 ) , and Oyama and Endoh ( 1 95 5 ) . Luo n g and Vo le s ky ( 1 97 9) and S c h ii ge rl ( 1 9 8 1 ) have investigated non - Newtonian CMC so lutions , while H aas an and Robinson in ve s ti g ate d ele ct rolytic solutions . Joshi et al . ( 1 9 82 ) has re ce ntly reviewed the av ailable correl at io ns and has recom mended the cor r e l at o n of H u gh m ark ( 1980) , w hi ch is pre s ent e d Other
i
above .
For a mechanically agit ated fermentor , the op t i m al design for continuous production of microbial bio m as s is con sidere d to illustrate t he econom ic de cisions that m ust be made . The speci fic gro w t h rate ll , the biomass concen tr atio n in t he fe r me ntor X , t he a git at io n s p e e d N , and the gas flow rat e Q may be considered as independent d ecisio n variables whi ch are to be selected to minimize t he net present value o f capital and o p erati n g costs p er unit of The biomass e n er getic yield n is related to t he specific biomas s p ro duce d gro w th rate , ,
.
1 Tl
=
+ ll m
e
( 6 8)
and the m ass yi eld is Y s = crs y8 n / cr b y b in units of gr am s of biom ass per gram o f organic substrate . The oxygen transfer rate m ay be found from the available e le ctro n b alance to be
81 8
Erickso n and S tep hanopo ulos y
�X 0
=
( 6 9)
w here Y 0 is g ram s of biom ass produced per gram mole of oxygen cons umed and � X /Y 0 is gram moles o f 0 2 trans fe rr ed per unit volume of liquid per unit time . The heat that needs to be removed is directly related to the oxygen transfer rate ; that is Qo = 1 1 2 . 8 kJ /equivalent of oxygen or 45 1 . 2 kJ /g mole of oxygen ; thus
1
I
�X
45 1 . 2
=
( 70)
1 2n
0
For Monod kinetics , the exit or ganic substr ate concentration is �K s S = ----'J.l m ax
11
( 71)
__
-
and from the yield equation X s
X
0 = y s
=
Ys ( S
0
- S)
1
+ s
(72)
The computational sequence i s as follows for
a
desired productivity rate
P 1 1 where
FX = P
( 7 3)
1
where F is the liquid volumetric flow rate First values of the independent decision variables j.l and X are selected . T hen n and Ys are found using Eq . ( 6 8 ) . Next the oxy gen transfer rate .
1
I
L
k a' ( C * - C )
o
o
=
J.l X
( 7 4)
Y0
i s c alculated using Eq . ( 6 9) . E q u ation ( 7 0 ) is used to estim ate t he heat to be removed per unit volume per unit time . The volumetric liquid flow r ate m ay be found using Eq . ( 7 3 ) . Since the specific growth r ate equals t he dilution rate t he required liquid volume is 1
v
F
L
11
=
( 75)
E quat ions ( 7 1 ) and ( 72 ) m ay be used to find the or ganic s ubstrate concen trations S and S0 • From the required liquid volume the tot al dispersion volume m ay be estim ated using the relationship 1
( 76)
Bio lo gical Reactors
81 9
where the gas holdup e:a is to be estim ated initially and calculated more precisely later from the expres sion ( Hughmark , 1 9 8 0 )
( 77 )
whe re the bubb le di ameter d b m ay be estim ated from available correlations , such as Eq . ( 4 9) . The dimensions of t he fermentor m ay now be selected ; the height -to diameter ratio , the num b er o f impellers , and the type of impeller are in dependent variables . The gas flow rate and t he agitation speed may also be vie w e d as indep endent variables which m ay be optimally sele c te d based on eco nom i cs . The gas flow rate m ust be more than sufficient to supply the o xy gen needs of the ferment ation ; that is , for air ,
For 3 0 % consumption of oxy gen , for example , RT ].I XV
Q
in
=
L ( 0 . 3) ( 0 . 2 l)p. Y 1n o
( 78)
The op timi z at�on of oxygen tr ansfer efficiency in the a git at e d ves sel for the required oxy gen transfer rate involves selecting the height - to - diameter ratio , number of impellers , type of imp eller , gas flow r ate , and agitation speed . For st andard imp eller desi gns , t his fixes the impeller diameter and allows one to find the po wer number using Fi g . 3 and the calculated impeller Reynolds number . The ungassed power P m ay then be found from the power number . The gassed power may be found from Eq . ( 6 7 ) . Equa tion ( 6 4 ) or ( 65) m ay be used to c alc ulate the volumetric m ass trans fer coefficient , kL a' . Equation ( 7 4) m ay now be used to calculate the driving force ( C - C0 ) , where the dissolved oxygen concentration C0 is usually chosen to be small ; that is , about 0 . 0 3 m mol / L or less . Henry' s law re l ates C� to the t otal pressure p and the gas - phase oxygen mole fraction y ; that is ,
�
yp
=
HC * 0
( 7 9)
The mean mole fr action of oxygen in the exit gas is found from the gas phase oxygen balance ,
( 80 ) For a comp let e ly mixed g as phase , the mole fract ion oxygen i n the fermentor equals the exit mole fraction , and the required tot al pressure m ay be
820
E ri c kson
and Stephanopo ulos
calculated using Eq . ( 7 9) . T his is the avera ge p ressure within the dis persion . The capital costs , op e ratin g costs , and net present value of the cost s per unit of producti vit y m ay be calculated . The sep aration costs depend primarily on t he li q uid flow rate F and biomass concentrat ion X . T he fer me ntor costs depend o n the total dispersion vol ume V , the pressure p , m d the number o f agit ators . The size of the agitator motor an d t h e po w er costs associated with it may be determined . The si ze of the gas compressor and the power re quire ments m ay be determined . The cost of heat transfer m ay be determined from the coolin g r e q ui rements . The cos t of organic substrate m ay be det e rmined using the quantity of s u b st r at e F So pe r unit time . The optimization problem involv e s s ele cting the values of the independent variables so as to minimi ze the net present value of the costs p e r unit of productivity . Thus values of s p e ci fic growth rate , biomass conc en trat ion height - to- diameter ratio number of im p e lle r s , type of impeller , impeller speed , an d gas flow rate m ay be vari ed in the search for an e co nom ic optimum . Of these variables , th e hei ght -to diameter ratio , numbe r of im pellers , and type of impeller would probably be selected based on prior e xp e rie nce . T he e x am p le above is pres e n t ed to illustrate how the m any variables are related to each other in fermentor design . The simplest flow mo d els have been used by as sumin g complete m ixin g in both phases . For the gas phase a flow model composed of two completely mixe d tanks in series may be more approp ri ate ( Joshi et al . , 1 9 82 ) . The ass umption of complete mix ing of the liquid phase is app ropriate . For s implicit y , the possibility of co 2 inhibition is not included . ,
,
-
B ub b l e
Co l u m n s
When a bub ble colum n is d esigned t o b e used a s an aero bic fermentor , several import ant considerations should be t ake n into account . T he sup e r ficial gas velocit y should be selected such t h at the bubble t wo -phase flow regime is p res e nt . The pressure gradi e nt due to the liquid head should be taken into account in the design . Axial dispersion s hould be conside re d The effect of s e gr egat ion m ay be impo r t ant because s m all bubbles have longer residence times than larger bubbles and they also h ave greater inter facial area per unit volume than large bubbles . The gas -p hase molar flow rate m ay change with position due to gas absorption ; howe vel- , in c ar bo hydrate fermentations the respir atory q uotient is nearly unity and the molar flow r ate is approxim at ely constant . he desi gn of bubble columns has been consid e red by D ec kw er ( 1 97 6 , 1 97 7 ) and Deckwer e t al . ( 1 97 4 , 1 97 7 ) . Shah et al . ( 1 978) have revie w ed axial dispersion or backmixing in bu b ble columns and other gas - liquid re actors . Deckwer ( 1 97 6 , 1 97 7 ) has p resented m athematical models which take into account axial dispersion and the pressure gradie nt . Deckwer reco m men d s i ncluding t he e ffect of the pressure gradient w he n the p ressure at the bottom of the column is more t han 30% larger than the pressure at t he top above the liquid phase . Deckwer ( 1 97 7 ) found that the effective value of the vo lum e t ric mass trans fer coefficient was smaller in large indus trial- scale bubble co lum ns comp ared to sm aller columns . S e gre gation could account for this . Despite the limit atio n s of Deckwer's succe s s the approach used has considerable merit ; however , comput ationally , numerical solution of the s yst e m of differential equations is required .
.
T
,
821
B io logical Reactors
A simpler , less exact appro ach is to consider dispersion of oxygen in the axial direc tio n in the liquid phase and eit he r a complete mi xin g or plug flow model for t he gas phase . The complete - mixing model is a conservative one since the exit gas -phase m ole fraction of oxy gen is used for the entire vessel . I f gas - p hase se gregation is import ant , some bubb les will have a lower mole fraction of oxy gen than the value predicted by the complete mixin g model . Since the liquid -p hase superficial velocity is zero (or close to zero ) , the oxygen b alance is t aken to be
( 81 )
w here X is the hei ght above the bottom of the column :
c�
P TY
� [1
=
0
+
a.( 1
( 82 )
z)]
-
and
( 83) is the ratio of the liquid head t o t h e pressure at the top of the co l umn T he boundary conditions are PT .
,
dC
0 = dx
at x = 0 and x = L
0.
A dimensionle ss distance x
=
x /L and dimensionless concentration c
=
C /C *
o oT is the liquid -phase oxy ge n con p y/H 0 T centration i n equilibrium w ith the g as phase at the top o f t he column . The dimensionless groups m ay be introduced , where C
�T
=
( 84)
and
B2
=
J..I X
( 85 )
Y k a' C * o L T
appear in t he dimension le s s exp ression
B 1 -22d c
dx
c
=
B2
-
1
-
a. +
a. x
( 86)
Eric kson and Step hanopoulos
822
which h as the solution -M
c
=
A e 1
1
X
( 87)
where
a(e
M
1
-M
a( e
- 1)
( 88)
1 1)
( 89)
and 1
( 90)
�
Figure 1 7 shows the effect of dispersion in the axial direction o n the dimensionless dis solved oxygen profile along the length of the co lum n for a = 1 and B 2 = 1 . Fi gure 1 7 is also a plot of =
c
a
-M e
1
X
( 91 )
for all values of a and B 2 in which c > 0 for ?E_ = 1 . Note that zero -:?rder kinetics are used for the dependence of � on c ; that is , as long as c > 0 , the specific growth rate does not depend o n the dissolved oxygen concen tration. The column should be designed such that c > 0 at the top of the colum n . Values o f the axial dispersion coe fficient D L have been correlated by Deckwer et al . ( 1 97 4 , 1 982 ) and others ( Shah e t al . , 1 97 8 ) . Deckwer et al . ( 1 97 4) found that ( 92 )
w here unit s o f centimeters and seconds are used . A desi gn methodology analo gous to that used for mechanic ally agit ated vessels m ay now be visuali zed . Four independent variables m ay be optimally selected b ased on economic considerations . The specific growth rate , l.l and biomass concentration , X , m ay be selected as independent variables if continuous biom ass p roduction is again considered as t he exam ple of interest . Complete mixing is assumed for the organic substrate and biomass in the column ; this should be appropriate unless it is necessary to consider the disp ersion of the or ganic substrate fed to the column . After values o f 1.1 and X are assumed , values o f n an d l.I X /Y0 m ay be found from Eqs . ( 68 ) and ( 6 9 ) . The rate of heat removal m ay be found ,
823
B io lo gical R eac tors
m
N
+
lu
F I G U R E 17 E ffect of axi al dispersion and pressure gradient on dimension less dissolved oxygen concentration in a bubble column . The parameter is L 2kLa' /D L . . Results are limited to conditions where c ;;;?; 0 .
�
Eq . ( 70 ) and the organic substrate concentrations from Eqs . ( 7 1 ) and ( 7 2 ) . The liq ui d flow rate may be found from Eq . ( 7 3 ) and the liquid vol ume from Eq . ( 7 5 ) . The fraction of oxygen in the g as phase which is con sumed and the superficial gas velocity m ay be selected as the other two i ndepe nd ent variables . The oxygen balance for a constant molar flow rate , using
nT '
=
\.! XV L y
0
( 93 )
may be used to find nT . A trail- and -error iteration is required at this point since the pressure , column diameter , and gas superficial velocity are all re l ate d to the molar gas flow rate . Assuming a pressure , the colum n diameter may be found from the relationship
The gas holdup may now be estimated from an appropriate correlation such as Eq . ( 5 3 ) . The total dispersion volume may be found from Eq . ( 7 6) and
Erickson and S tep hanopoulos
82 4
the h eight L of the column m ay also be found . The volumetric mass trans fer coefficient kL a' m ay be fou n d from an appropriate correlation , such as Eq . ( 5 2 ) . T he axial dispersion coe fficient DL m ay be found using Eq . ( 92 ) . The value o f c at x 1 m ay be found from Eq . ( 87 ) an d comp ared to the desired value . The tri al - and -error iteration involves adj ustin g the pressure until a desirable value of c is obt ained . The optim al values o f \1 , X , u a , and y are to be selected based on economic considerations . =
A i r l i ft Fermento r s
Airlift fermentors have a well - defined loop for liquid circulation which is driven by the gas flow . The liquid circulation rate depends on the d esi gn of the airlift colu m n . The extent to w hich the gas is sep arated from the liquid at the top of the colum n p rior to e nt eri n g the downflow section can signi ficantly affe c t the liquid circulation rate . In the work of H at c h ( 1 97 3) and G ut ie r re z and Erickson ( 1 97 7 ) , for e x amp le the dispersion enters the downflow section with more gas than can be carried downward by the liquid . Some of the lar ge gas bubbles m u st rise against the dow n flow of t he disper sion and escap e . It is necessary for this to happen because the average gas ve locit y is m uch sm aller than the liquid velocity in the downflow section , and only a fraction of t he gas in the dispersion entering the do wnflow re gion can be carried downward . For this disen gagement proces s to occur , the actual liquid velocity in the downflow section must be larger than the rise velocity of small bubbles and sm aller than the rise velocity of l ar ge bubble s ; that is , it should be about 1 0 to 30 cm /s . H atch ( '1 97 3 ) reported super fici al liquid velocities r an gi n g from 10 to 30 em /s and Orazem et al . ( 1 97 9 ) reported values ran ging from 17 to 2 1 cm /s . H at c h ( 1 9 7 3 ) point s out that the liquid circulation ve locity increases with gas superficial velocity in the upflo w section until t he liquid circulation velocity is limited by the gas bubble dis e ngage m e n t process at the top of the downflow section . Fur ther increases in gas superficial velocity lead to liquid recirculation wit hin the u p flo w section similar to that found in bubble columns . Hsu and Dudukovic ( 1 98 0 ) and Merchuk and Stein ( 1 9 8 lb ) have investi gate d the liquid velocity in airli ft columns , but in both c ases the desi gn was such th at the down flow s ection was almost free of gas . Merchuk and Stein found t hat the act u al liquid velocity could be correlated with t he They observed liquid velocities superficial gas velocity to t he 0 . 41 power . above 1 0 0 em /s at superficial gas velocities above 20 em /s . They found that t heir dat a could be correlated by the expression of Zuber and Findley ( 1 965 ) ; that is , ,
( 94)
0.
3 3 m /s . That is , vn w here C A = 1 . 0 3 and vn = 0 . 33 m /s is approx im ately the gas velocity relative to the superficial velocity o f the mixture , UM ·
=
Hsu and Duduk.ovic ( 1 980 ) presented an overall momentum balance for the airlift loop and presented a correlation for the two-phase friction factor . Their result s do not correlate well with the pop ular Lockhart -Martinelli cor relation ( Perry an d Chilton , 1 97 3 , pp . 5 - 4 1 ) .
825
Bio logical Reactors
Hatch ( 1 969) fo u nd that the optimum ratio of upflow to downflow area Since the up flow liquid velocity increases with s up e r fici al about 0 . 83. velocity , one w o uld expect the optimum area ratio to d e cr e as e as super ficial gas ve lo c ity i ncreas e s . H atch ( 1973) has measured the axial dispersion coe ffi cient D L in the air lift up flow section and found that D L is constant at about 58 cm 2 /s for low super ficial gas velocities where liq ui d recirculation within the up flow section is small . H at ch concl ud e s th at t he li q uid phase may be modeled usi n g a plug- flow model for both the up flow and do wn flo w sections . H atch reco m me nd s a complete - mixing model for t he head regio n . Merchuk and Stein ( 1981a) have m odele d the airlift ferme ntor using plug flow in the up flow and down flow re gio n s and complete mi xi ng in the head re gion . H o et al . ( 197 7 ) have modeled the airlift fermentor using a tanks- in-series model in the upflow and downflow regions and a completely mixe d tank for the head region . The design methodology for the airlift fermentor is analogous to that described above for bubble columns . T he fraction of o xygen co nsum e d , the feed gas superficial ve lo cit y in the draft tube , and the ratio of up flow to dow n flow area may be s elected as i nd epen dent variables to get he r w it h ].l and X . A t ri al - and - error calculation is required to find t he op eratin g pres sure that results in an ac ceptab le dissolved oxygen concentration . A sim ulatio n p rocedure an alo gous to that of Ho et al . ( 1 977) or Merchuk and Stein ( 1 981a) is needed to determine the dissolved oxygen level in the vari ous parts of the airlift fermentor for a p art ic u l ar op erat i ng pressure . The available correl at io n s for use in t he d e si gn of airlift fermentors are no t as co m p le t e and extensive as t ho s e for agitated tanks and bubble co lum n s . Further research is needed to develop such correlations . Airlift fermentors have been compared to bubble columns and agitated tanks and the mass transfer efficiency o f airlift columns has been demonstrated ( O r a z e m and Erickson , 1 97·9 ; H atch , 1 9 7 5 ) . A variety of e q ui pment arrangements invo lvin g vertical cir cu l ation due to airlift designs have been reported . The len gt h of the vertic al divider range s from the short downcomers of Viesturs et al . ( 1 980 , 1981) to the very long downcomers of the deep shaft fermentors ( Takamatsu et al . , 1981a , b ; Kubot a et al . , 1 97 8 ; Hosono et al . , 197 9 ; Hines et al . , 1975 ) . Fur ther research is needed to develop optimal designs for biolo gical applica tions of these systems . was gas
Tower Reacto r s w i t h Mo t i o n less M i xe rs
The de si gn methods of t hi s chapter can be ap plie d to various tower fermen tors with internals such as Koch motionless mixers , sieve trays , and down comers . App ro p ri at e flow models should be selected for the Jjrop osed system and the effe ct of p re s sure should be co n sid ere d . Correlations for the volumetric mass transfer coe fficient and the gas ho ldup are needed . The work of Viesturs et al . ( 1 980 , 1 98 1 ) has resulted in the ind us tri al - sc ale ap plic ation of such sy s t em s ; t he y h ave comp are d the pe rfor m ance of several large -scale s y ste m s exp eriment ally . Further research is nee de d to optimi ze m at hem ati c ally such desi gn s .
A C K N OW LE D GM E N T
This work w as
partially
supported by
t he National Science Fou ndatio n .
Erickso n an d S tep hanopoulos
826 N O T AT I O N
a
stoichiometric coefficient in Eq .
( 1) ; inter faci al are a
a'
interfaci al ar e a p e r unit volume of liquid are a stoichiometric coefficient i n E q .
( 1)
di m e nsion less group de fined b y Eq . ( 8 4 ) di m e ns ion le s s group defined
2 Bond num ber ( gd p / cr) t L
b y Eq . ( 85 )
c
dimensionless dissolved oxy gen , C
c
molar car bon dio xide concentr ation in liquid
c C d
c
0
d
d d
d
b
bubble diam eter
s t
De
F Fr 0
Ga Gr
Hc
H
width of
at the top of
impeller blade
orifice diameter S auter mean diameter
0 a+
[ see Eq . ( 4 4 ) ]
tower diameter
dilution rate F /V
L
Fr
the column
im p e lle r di ameter
l
D
g
equilibrium oxygen concentration in liquid
P T Y /Ho ;
0
D
phase
m o lar oxygen concentration in liquid phase
()T
d i
0 / CO T
i
1 (hr - )
; molecular diffusivity
di s p er s ion coefficient for liquid phase Debor ah num ber
( u .A /ds ) b
volumetric flow rate of liquid p h as e
u0 N gd ; Froude number t Q 2 td 5 g ; ori fic e Froude num ber 0
0
gr avit ation al acceler atio n
: �;
gd /u
3
Galilee n umber
db p L g ( p L - p g ) H enry' s law
lf.! L 2
con stant for co 2
H enry ' s law con stant for oxygen hydrogen ion concentration
inh ib it or
concentration
liquid - p hase m a ss tran sfer coefficient saturation constant for
produc t inhibition
s a tu r ation constant of Monod kinetic model
maintenance c oe ffi c ie nt based on C 0 2 , g mol C 0 2 ( g biom a s s ) • h
maintenance
coefficient , g eq uiv available electrons substrage / g
82 7
Biological Reacto rs
m
maintenance coefficient based on oxy gen , g mol 0 < g biomass 2 h- 1
0
-1
•
maintenance coefficient , g sub strate / ( g biomass) •h molecular weigh t molar flow rate o f gas impeller speed PC
partial pressure of co
2 partial pressure of oxygen
Po
PT
tot al pressure
p p p1
product concentration , g /L power productivity in Eq . ( 7 3)
PG
gas sed power
p
no Q Qco2
Qo Q
2
o
R Re Sc Sh T
3 3
power number , P / p N d 1 c volumetric gas flow rate
evolution , g mol / ( g biom ass) •h 2 specific rate of oxygen utili zed , g mol 0 / ( g biomass ) •h 2 heat evolved per equivalent o f available electrons trans ferred to oxygen , kJ /equiv specific rate of co
gas const ant OL
4 pL Q 0 /1r d 0 1J L ; ori fice Reynolds lJ L I PL D ; Schmidt number
number
k d /D ; S herwood number ; use d for bubble si ze distribution L b s absolute temperature bubble rise velocity gas -·phase superficial velocity liquid-p hase superficial velocity
superficial velocity of gas - liquid mixture dispersion volume gas volume liquid volume height from bottom of the column biomass concentration , g /L mole fraction oxy gen in gas phase biomass carbon yield in Eq . ( 1 ) c arbon dioxide yield in Eq . ( 1 ) product yield in Eq . ( 1 )
828
Erick so n and S tep hanop o ulo s
true biom ass yield
g biom as s I g mol A TP
,
biom ass yield b as ed on co 2 , g biom a s s / g mol co 2 y
0 m ax Y 0
m ax P /A T P m ax Y P /D max Y P /0 m ax Y P /S Y Q Y
y
y
s max s
true biomass yield based on C0 2 , g biomass / g mol co 2 biomass yield based on oxygen , g bio m as s /g mol oxygen
tr u e bi omass yi el d based on oxyge n , g biom ass / g mol oxy gen true product yield b ase d on ATP , g product / g mol A T P true p roduct yield b as e d on c o 2 , g product / g mol c o 2 true pro duct yie l d based on oxygen , g pro d uc t / g mol oxy gen true product yi e ld based on substrate , g product /g substrate
biomass yield b as e d on heat evolved , g biomass /kJ bio m as s yield based on or ganic s u b s t r ate , g bi o m a s s /g substrate true biom as s yield based on substrat e , g biom as s / g substrate
G reek Let te rs
pressure ratio defined by Eq . a.
( 83)
rate coefficient for p ro d uc t form ation , g product / ( g biomass ) •h rate coefficient for product formation , g p ro d u ct / g bio m ass
re d uct an ce d e gre e of biomas s [ see E q
.
( 4) ]
reductance de gree of p ro d uct [ see Eq . ( 7 ) ] reductance degree o f s u b s tr at e [ see Eq .
( 6) ]
standard heat o f com bustion per gram atom o f carbon e:
he at of reaction per gram atom of substrate carbon fraction of available electrons trans ferred to oxygen volume fr action of dispersed gas phase biomas s energetic yield
true biomas s ener getic yi el d Kolmo goroff scale [ see Eq . ( 58 ) ] ].1
characteristic time of viscoe lastic m aterial specific growth r at e
app arent viscosity vi s cosi t y of gas viscosity of li q uid
,
h- l
829
B io logical Reactors
maximum spe ci fic growth rate kinematic viscosity apparent kinematic viscosity product energetic yield
ll max
\) \)
E; p E;max
true product energetic density surface tension weight fraction carbon weight fraction carbon weight fraction carbon
p
p
cr
yield
in biomass in product in substrate
S u b sc ri pts
b c g G
in L II, 0
p
s
biomass continuous phase ; CO 2 gas phase gas phase inlet condition liquid phase liquid phase oxy gen ; orifice product organic substrate
S u pe rsc r i pt *
equilibrium concentration
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830
Erickson and Step hanopoulos
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831
Biological Reactors
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13
Analysi s and Des ign of Photoreactors ELI ANA R .
De sarro llo Tecno logico pa r a
DE B E R N A RD E Z , M A R I A A .
A L B E RTO E .
CASSANO
Indu s t ria Q u { m ica- IN T E C ,
Institu to
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Cien t (ficas y Tec n icas ,
la
San t a Fe , A rgen t i na
U n iv e rs i dad N acional de l L i t o ra i - Co nsejo
Nacional de Inves t igacio n e s ,
S I G N I F I C A N C E AN D SCOPE
The advant ages of photochemical reactions over thermal or catalytic reac tions have frequently been quoted : selectivity , negligible side reactions , low operating temperature , possibility of carrying out some reactions that would be thermodynamically infeasible by thermal means due to unfavorable equilibrium conditions , liquid-phase operation , and so on . Despite these advantages , photoreactors have been employed industrial ly only when either therm al or conventional catalytic reaction paths have proved to be very difficult . T he reasons are : lack of suit able reactor models and design methods , lack of quantitative information about the per tinent physical and chemical parameters , limited variety of equipment avail able , and the understandable reluctance of successful industrial companies to publish design and scaleup procedures . However , many reactions are suitable for photochemical production . The most common industrial photoreactions are those that proceed via chain mechanisms . Among these , p hotoch lorinatio ns are undoubtedly the most important processes due to their high-energy yields . Thermal clorina tions , carried out at high temperatures , produce a variety of different molecular species which have to be sep arated ; this results from the broad energy distribution of the thermal field and its effects on reactions wit h different activation energies . I n contrast t o this , i n m any cases , the photochemical process m ay produce only a single product . The synthesis of y - hexachlorocyclohexane has been one of the photochlorinations of greatest economic significance . An industrial process for the preparation of volatile primary mercaptans , which are commonly used as additives to natural gas , is based on the anti Markovnikov addition of hydrogen sulfide to terminal olefins , a free radical chain reaction initiated by li ght . Photo po lymerizations are another example . Such chains , started by light , have high energy yields ; in addition , no fragments of initiator molecules remain and production rates by means of the flow process are extremely fast .
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De Bernardez , C laria , and Cassano
840
The p ho tooximatio n of cyclohexane , a reaction with low energy yield , is currently used to prepare cap rolactam , which in its turn is utili zed for the production of nylon - 6 ; nylon - 1 2 is prepared by a similar route . This reaction involve s a free radical mech anism but is not a chain reaction . Several companies have experimented with this p rocess ; however , the first lar ge - scale use h as been reported by t he Toyo Rayon Co . of Jap an . The p ho tooxidatio n of ole fins has been investigated extensively in rela tion to the " smo g" problem that appears in the atmosphere by the p hoto chemical reaction bet ween several atmosp heric contaminants . Also , a num ber of or ganic compounds , epoxides , aldehydes , and ketones have been produced in the labor atory by reaction of oxy gen atom s (photochemically produced ) with ole fins . Generally , these reactions have hi gh yields and side reactions are minor ( at least for terminal olefins ) . There are photoche mical reactions with low ener gy yield s that are generally limited in application to the synthesis of hi gh-priced , low - volume chemicals such as p harm aceuticals or perfumes . T he photosynthesis of vit amin D 2 is the most import ant example in the p harm aceutical industry . It is based on the ergosterin photolysis and is t he only w ay to obt ain this vit amin . We should add that , very often , photochemical p rocesses have been proposed with the following objectives : 1.
2. 3. 4.
Decomposition of w ater contaminants refractory to other w ays of destruction Sterili z ation processes B acteriological treatment o f drinkable water Destruction o f very dangerous chemicals that m ay have been pro d uced eit her voluntarily or by error , p articularly when there are no other w ays of m aking them harmless
Several more applications can be included , but perhaps t he most im port ant develop ment in the field of photochemistry is the recent emphasis on p ho to -assisted cat alysis ( Childs and Ollis , 1 980) . Hetero geneous photo assisted cat alysts are subst ances that require photo -excited electrons or holes in the course of a hetero geneously cat alyzed reaction . As an example we can mention several semiconducting oxides , such as Ti0 2 and Z nO ; the former , when irt•adiated , becomes active in the selective oxidation of alkanes to ketone s and aldehydes as well as in the dechlorination of polychlorinated bip henyls ; on the other hand , carbon monoxide oxidation occur s over illuminated ZnO . Several p hoto- assisted semiconductor electrodes , such as crystals of SrTi0 3 and the perovskite KTa0 3 , can electrolyze w ater without an applied electromotive force . M any other applications could be listed . Photo - as sisted cat alysis m ay become practically useful if a relatively che ap source of light with appropriate w avelength is available , the sun being the obvious source . T he p rospects for solar energy conversion follo wing this p ath are p articularly appealing . Despite all these potentialities , photoreactor design is still a novel field in re action engineerin g , with m any import ant problems still not p roper ly solved . In fact , up to the present , no reliable desi gn methods have been made available to the practicing engineer . We believe that a signifi cant part of t he responsibility for this lack of know - ho w lies mostly in considering its analysis as a kinetic variation of a conventional reactor ,
841
A nalysis and Design of Pho toreactors
when in fact it must nece s s ari ly include the radiation field as the key view point . Consequently , if some pro gress i s to be m ade , the d e si gn of a photor e actor cannot avoid all the complexities that r adi ation problems in volve . First , we present a definition o f the problem and the requisite theory . Later , the de sign methods are shown t hrou gh four applications , three of which have already been tested i n be nch- s cale exp erim e nt s with excellent agreement . Finally , in two s hort sections we propose some new reactors not yet theoretically mo dele d and we end with what we believe are the most i m p ortant problems still unsolved in t he field . ,
T Y PES O F R E A C T O R S
When dealing with the design of the equipment for carrying out a photo chemical reaction , several aspects must be considered . Some of them are common to the d e si gn of conventional therm al re act o r s such as t he kinetic characteristics of the reactions involved , the phases of the system , the necessity of tempe rature control , the re q uirem ents about the material of construction , and so on . Others arise specifically from the selection of the appropriate radiation source for the re act io n under study (i . e . , the spectral distribution of the emitted light and the geom e trical con fi gu ration of the reactor - lamp arrangement ) . In addition , a certain number of im port ant photochemical reactions share the common characteristic of present i n g products and reactants with highly corrosive or di s solving p rop ertie s ; a typical example is the chlorination of hydrocarbons where the presence of chlorine , hydrochloric acid , and ch lo rin ate d solvents creates difficulties in the selection of materials for equipments and s eals . In the case of hetero geneous systems , the mixing pattern of the reactor is extremely im port ant . Another important aspect in the design is the requirement for temperature control in strongly exothermic or endothermic reactions . We m ay add t hat in some p articular cases the design can be improved by add ing a re flecting device ; its use comp licate s the analysis of the radiation field . Many of these difficulties c an be ove r come by choosing an ap prop ri at e confi guration o f the p ho t or e act or system . Among a wide ran ge of possible reactor- lamp - reflector confi gur ation s , t hose m o s t wi d ely studied in the lite r ature are the annular reactor , the cylindrical reactor with elliptical reflector , and the cylindrical reactor with p arabolic reflector ; t he y are described briefly below . Other types of configurations can be used in com mercia! applications ; nevertheless , they arise from different combinations of the three mentioned above . ,
A n n ular Reacto r
Annular photoreactors are an excellent approach to what is perhaps the most practical type of photochemical reactor to be used for commercial purposes . The utili zation of energy can be the maximum expected and , moreover , they re pr e s ent the common case of a reaction vessel with a tubu lar lamp placed at its axis by m e ans of an immersion well . Fi gure 1 shows the main feat ures of the system . The reacting stream is contained in the annular space that surroun ds the lamp . If the reactor
84 2
De Be rna rdez , Claria, an d Ca ssano
I I
I
:
,
!
' I
I I
:
!' S
I
: :
·"
I I
I
: � .... ......
,. ... .... �---:----L ....- ·r· --r- 1 r t_· ·· \. _�-��: ..
....
... ...
.. ... ..
FI GURE 1
R
__ _ ,,,
\
... .
Annular reactor .
ve s sel is separated from the l a mp , the a nnu la r space bet ween th em can be u se d to cool the la mp and / or c on trol the op e ratin g te mperature of the re
H owever , it must be n oted that the re quire men t of light tran s mis sion intro duce s some limitations in the heat tran sfer p o ssibilitie s becau se
actor .
at least the inner re actor wall must be made of qua r t z or gl a s s . Since li ght tran sp arency is not re q uired in t he ou ter reac tor w al l i� provide s muc h more free dom for de sign purpose s , such a s coolin g , re flec tion of nonab sorb e d light , an d so on . Hence if one de sire s to improve t he ab sorp tion e fficie n cy of th e reac t or , a re flector sur face cover ing th e ou ter reac tor ,
wall can b e u se d .
In dealing wi th laboratory re search , the annular photoreac tor cannot reco m me n de d for continuou s - fl ow operation s due to it s una voidably large
cro s s - sectional area , wh ich re q uire s the h a n dlin g of lar ge volu metric flow rat e s . On the con trary , th is charac ter i stic make s it very ap p r op ri ate for co m mercial - scale ope ration s .
FI G U R E 2
Cylin dric al reac tor with e ll ip tical t•e flec tor .
be
843
A n aly s is a n d D e s ign of Pho toreac tors
F I G U RE
3
Cylindrical re ac tor ir radiat e d fro m t he bottom .
C y l i n d rical R ea c to r w i th E l l i pt i c a l R efl ec tor This sy ste m con s i s t s of a reactor ma de wi th a cylin dric al tube plac e d at on e of the focu se s of a cylin dr ical reflec tor of ellip tical cross se ction . A tubular radiatio n source i s p la c e d at the o ther focu s .
This p articular
arrange me n t i s often c alle d an e l l ip t ical p h o toreac tor ( Fi g .
2) .
The pos sibility of u si n g cylindrical tubular reac tor s with
a
s mall c ro s s
se c tional area to g e ther w i t h t h e gen erally acc e p te d ( b ut only partially tru e ) concep t of the existence of a unifor m irr adiation from out side h a s r e sulte d in a r ather ex ten sive ap plication of t h i s reactor for laboratory an d bench scale re se arch work . On th e other h an d ,
s ince t he incidence e fficiency of this typ e of re
actor - de fin� d as the cap acity of conce ntrati n g the energy with in the reac tor bo u n darie s - i s not very high ,
th is is not an advisable de vice for
co m mercial - sc ale operation unl e s s other en e r ge tic con siderat ions comp e l i t s use . C y l i n d r ic a l R e a c to r
w i th Pa ra bol ic
Refl ec tor
Another p o ssibility of isolat in g the reac tion sy ste m from the ra diation source ( wh ich coul d al so simplify the solu tion of the well - known p roble m of wall
wall )
o f t h e u se o f a cylin drical re ac tor irradiate d fro m t h e b ottom by a tubular dep o site s ,
gen erally more se vere at the ra diation entrance
con si s t s
source locate d a t the focal ax i s of a cylin drical r eflec tor of parab ol ic cross
tank reac tor , thi s device is e specially se c tion ( Fig .
3) .
S ince t he cylin dric al reactor may be a per fec tly st irr e d
app rop riate
t o c arry o u t l iqui d o r Th i s typ e of re
ga s - liqui d re actio n s where vi gorou s stirring is require d .
ac t ion is indicate d for both l ab ora tory an d c o m mercial scale work a n d can be u se d in batc h ,
se mibatc h ,
or con tinuou s operatio n s .
Proble m s of cor
rosion a n d se aling c an b e more e asily han dl e d in this sy s te m .
PHOTO R E A C TO R D ES I G N : T o d e sign
a
S T A T EM E N T O F T H E PRO B LEM
photoche mical reac tor ,
infor mat ion ( a )
the reac tion ; ( b )
an en gineer shou l d ha ve the foll o win g
the ra diation energy deman d ( gen e rally
84 4
De B e rn a r de z,
C laria ,
an d C a s sano
qualitatively define d) , wh ich can b e sati sfie d with a ra diation source ( l a mp ) an d p erhap s a re flec tor ; an d ( c ) a choice of possible reac tor con figuration s ( sometime s with some restric tio n s ) . Then the followin g que s tio n s must be an s were d :
1. 2.
3.
4. 5.
6.
Whic h i s t he b e st source - reactor -re flec tor arran ge men t ? Whic h is the op timal value of the spectral distribu tion an d over all ra diation p o wer outp u t to con ve rt reactants into t he de sire d produc t ? Which is the optimal fee d co mposition ? How much con ver sion can be ach ie ve d ? Which i s the op timal size of the reac tor? How muc h selec tivity can be obtain e d ?
T h e an swe rs to the se qu e st ion s m u s t b e treate d in a logical sequence which sta r t s with the knowle dge of the rate of a photoche mical reac tion . The descrip tio n of the rate make s it nece ssary to distingu ish b e t ween dark an d ra dia tio n -ac tiva te d s tep s . To treat dark reactions one u se s the same me thodology as for ther mal reac tion s ; the main hindrance appe ar s when evalua ting the rate of the initiation step . It is known t hat the rate of this step is relate d to the local vo l u me t ric ra te of e n e rgy ab sorp tion ( L VREA) . The e valuation of the LVREA is perfor me d stating fir st a b alance of ra diant e ner gy at steady state for a c on trol volu me . For simp licity , a nonemit tin g mediu m is generally a s su me d . Afterwar d , it is nece ssary to incorporate a radiation source model an d a c on stitutive equation in the ab sorp tion ter m . The sou rce models ha ve b een treate d by se ve r al research teams , which have develop e d differen t p roposals to p r e dict radiation profile s insi de photo reactors . C oncep tual app roache s may be cla ssifie d int o t wo well - de fine d group s : ( a ) th ose which a s su me a given r adiation distribution in the vicinity of the re ac tor ( in c i dence mo del s ) an d ( b ) th ose which start from a propo sal of a model for the source ( e missio n mo del s ) . There is no way of u si n g incidence model s with ou t exp er ime ntally a dju s table para meter s . The rea der intere ste d in thi s ap proach can fin d a critical re view in a p re viou s wo rk ( Al fano e t al . , 1985) . However , since w e are looking for an a priori de sign of p h otoreac tor s , atten tion will only b e paid to the e mission Their for mulation will b e pre sen te d later . model s . Once we know how to for mulate the ra te of a phot oche mical reac tion , we can p rogre ss further ahe a d with the de si gn p rob le m . Then the mass , energy an d mome n t u m balan c e s must be e stablishe d . The expression for the c on servation of momen tu m is similar to tha t of conventional reac tors . Due to the energetic al featu re s of photochemical reac tio n s ( very low an d sometimes null "equ ivale n t " acti vatio n ene r gy for the global reac tion ) , it is very frequen t to find that the reac tion be havior can be accura tel y de sc ribed with the mass balance and the ener gy balance re duce d excl u sively to ra diation . H owever , in so me case s , when hi gh ly exother mic or e n do the rmic reactions a re present , the de sign o f the reactor will una voidable require the complete ener gy e quation of c han ge . In any e vent , even for the simp le st case , the mass and radiation balance s are nor mally coupled throu gh concen tra tions of the ra diation - ab sorbin g spe c ie s . The n atu re of this co upling is well understood . The local volu metric rate of ene r gy ab sorption ( L VREA ) is gen e rally a func tion of the radiation fiel d in side the sp ace of reac tion , the c oncentration of the ab sorbing specie s , an d other phy sicochemical parame ter s . If species i which i s being con sider e d in the ma s s balance is an ab sorb ing s p ec ie s , a c oupling b e t ween the
A naly s i s an d De sign of Pho toreac tors
84 5
LVREA an d the co rre sp o n din g mass balance e quation will take p l ac e . Simul taneously , the concentration of the ab sorb i ng s p eci e s ( i) goe s i n s ide an in te gr al ex p re s s io n , a c c oun ti n g for t h e ra diation attenuation by ab sorp tion . Thu s t he general problem of modeling a p h o t oreac tor p re se n t s an i n t e gra differen tial mathematical nature . Thi s degree of co mp lex i t y of the p r oble m shoul d n ot b e su r p ri si n g since the phy sical situation in dicate s that the pr ogr es s of the reac tion at a p oint depends on the L VR E A ( which is , a mong othe r variable s , a fu nctio n of t h e ra diation attenuation suffere d by each bu n dle alon g it s p ath in side the reac tor ) ; in its turn , the a ttenuat ion suf fer ed by each p encil of ra dia t io n ( du e t o it s conce ntration depen dence ) is a fu n c tion of t h e reaction exte n t reache d at each one of the point s c on s t i t ut in g the t ra j e c t o ry of the b un dl e . We di sc u s s later how t o mo del the ab sorp tion proce s s in the re ac t or by on e or more specie s , as we ll as e valua ting the L VREA an d p re se n t in g se veral ap plic ation s u sin g the t hree main re a c tor configurations d e scr ibe d p re viously . To sol ve t he de si gn p roble m , a d ditional in for m a t ion about the l a mp , the react or , the reaction , an d perhap s , if it is u se d , the re fl ec tor will be We sh oul d st a te beforehan d that we a s su me tha t the reac tion is neede d . alrea dy known . T hi s means that kine tic dat a are a va i la b l e a s well a s qual Because of this , the itative in form ation re garding radiation req uirem e n t . sy ste m will i mp o s e some con s t r ai n t s on the selec tion of radi atio n source s an d re ac tor c on figu ration s . A t the sa me time t he market a v ailab ility of ul tra violet - and visible -ligh t l a mp s an d qua r t z ( or Pyrex gl a s s ) tub e s an d p l ate s , if t hey are neede d , will p r o vi de d e fin ite gui d el i ne s abo u t the type of re ac tor - la mp c ombin a tio n better suit e d for each proce s s . In any c a se , a de si gn will generally start with a te n t at i vely ch o sen ra diation source . More often than not , the c hoice of the lamp con siderably r e duces the freedom concern ing most reactor dimension s . T h u s the pr ob l e m m u s t b egin with a ten t a ti ve list o f neede d infor mation , gener ally oriente d by one ' s qu al i t at i ve kn o wledge of the reaction an d pos E ve n if t h e infor mation is complete , ver y sible re a c tor c on figu r a t io n s . often the de sign will h a ve to be d on e in an it e rat i ve way beca u se of the li m it at ion s i m po se d on t he c on s t ru c tion of t h e reac tor by the above m e nt io n e d market restric tion s . Let u s list wha t is nee ded :
1.
ten ta t ive r a diation so ur c e : I t s p o we r con sump tion an d power outpu t . b . The sp e c t ral distribution of it s powe r outpu t . c. I t s ge o met ri c al di m ension s . d. I t s op e rat in g con di tio n s . About the ten ta t ive reactor ( a ssumin g that some d e c i sion has been t ak en a bo u t i t s sh ap e and ty p e of opera tio n ) : a . I t s ge o me t rical di me n sio n s . b . In so me de r ' gn s , the t r an s mi s sio n charac te ristic of the wall t h ro u gh which th e ra diation shou l d en ter th e re a c to r . c. The spe c t ra l distrib ution of the wall tran s m is s io n charac teris Abou t the
a.
2.
tics .
I n for ma tion about ran ge s of operat in g c on d itio n s ( e . g . , pre s sure , temperature , require d pr o duc t ion , e tc . ) . A b ou t th e re ac t io n : a. A k inetic se quence wi th the foll o win g c h a r ac te r i s tic s : ( 1) A n initiation step tha t is ac tivate d by ab sorp t i o n o f r ad ia ti on .
d.
3.
846
De Bernardez , C laria, and Cassano
(2)
4.
One or more " dark" reactions . From the industrial point of view the most import ant reactions are t hose recogni zed by t he name of chain reactions . Starting from an atomic species or a free radical formed in t he initiation step , they involve the following constituents : ( a) One or more prop agation steps . ( b ) One or more termination steps . In some cases it is possible to work with a global kinetics that relates the absorption o f radiation to t he formation of products . Nevertheless , if it is available , it is always better to work with microkinetic information such as that mentioned in ( 1 ) and ( 2 ) . b . The radiation absorption characteristics o f reactants , inerts , and p roducts . T he p ri m ary q uantum yield of t he initiation step (i . e . , how c. m any atomic or free radical s pecies are e ffectively produced for e ach absorbed photon by the appropriate react ant in the first reaction of the kinetic sequence ) . I f only global kinetic expressions are available , an overall q uantum yield m ay be necessary , which is defined according to t he reaction scheme (moles of product formed or reactant consumed /ener gy absorbed ) . About the tentative re flector (if it is used and assuming that its s hape is given ) : a . Its geometrical dimensions . b . Its reflectivity The spectral distribution of its reflectivity . c.
Here we have pointed out mainly the p articular desi gn variables and the parameters of p hotochemical reactors . In addition , we m ust know all other p hysicochemical properties or characteristic p arameters usually needed for therm al re actors ( densitie s , diffusivities , thermal properties , etc . ) . In the next sections we derive t he theory and show the methods used to reach an a p riori desi gn of a photoreactor , w hen the problem is posed as indicated above .
EM I S S I O N MO D E LS
We begin our analysis of the emission mode;s by defining the fundament al properties of the radiation field . Then the b alance of radiant energy for an absorbing m edium will be derived with which we will finally be able to obtain an expression for the r ate of t he initiation step for a photochemical reaction . The concepts introduced here are based on well - accepted defini tions in the are as of radiation engineering and radiation gas dynamics . However , we shall point out some differences bet ween the conventional t herm al applications of radiation ( e . g . , furnaces ) and those that involve the use of visible , ultraviolet , and even s horter w avelength radiations . D e fi n i t i o n s
Sp ecific Intensity Radiation en gineerin g generally studies the energy exchange between sur faces ( Fi g . 4 ) . To characteri ze the amount of energy departin g from a
847
A nalysis and Design of Pho toreac tors
FI G U R E 4
Geometric represent ation of the S E E S model .
surface , t h e concept of specific intensity of radi atio n is used . specific intensity of r adiation is defined frequency ran ge bet ween v an d
v
as
The
the amount of energy in the
+ d v , leaving a s urface per unit area
norm al to the pencil or rays , per u nit solid angle and unit tim e .
From
now on we will simpli fy our not ation and call energy in the frequency in terval between v
and
v + dv
simply energy of frequency
v.
To illustrate
the rise of t he specific intensity concept , let d l/Jv be the radiant energy of frequency v per unit time and u nit area , leaving a given surface dAe
i n the direction e e and contained wit hi n a solid an gle d w e ( Fi g .
I
4) ; then
( 1)
\)
where f ( e ,
We m ust extend this concept to photoreac
tor en gi neerin g , w here t he energy exchan ge m ay involve not only surfaces but volumes
as well .
When t he radiation source is a fluorescent lamp , only the external sur face of the lamp p articip ate s in the emission p rocess .
Hence the e ne rgy
exch an ge phenomenon involves two sur face s , one for emission and the other for reception ( Fi g .
4) .
On the other hand , if the radiation source is a
nonfluorescent arc lamp , the emission is volum etric ( i . e . , the entire volume of
the lamp p ar ticipates in the emission process ) .
In this case ( Fi g .
5)
t h e energy exchange phe nomenon involves an emitting volume ( dVe > an d a
receiving s urface ( dAr ) . T he first p ropos al of an extense model was pub li she d by Ira zoqui et al . ( 1 97 3) . It w as applied to a nonflucir escent arc lamp and it is kno w n as the volumetric emission extense source model ( V E E S ) . Strami gioli et al . ( 1 975 , 1 9 7 7 ) presented the s uper fi cial emission extense source model ( SEES ) to describe t he performance o f fluorescent lam p s . T heir form ulation was an extension of the VEE S model . In spite of the fact that radiation sources have finite spatial dime nsio ns many authors have used a third model , which considers the lamp as a line ( Figure
6) .
T his model , first proposed by J acob and Dranoff
is known as t he s p herical emission linear source model ( S E LS ) .
( 1966 , 1 968)
As we shall
see later , t he S E L S model can be used only in some speci al - but very im port ant -reactor desi gn s .
,
De Bernardez , Claria, and Cassano
848
FIGURE
5
Geometric representation of the VEES model .
We can now derive a formulation of the specific intensity of radiation which will account for the type of energy exchange involved ; then we have : 1.
2. 3.
Emitting surface- receiving surface ( S E E S model) Emittin g volume - receiving surface ( VE E S model) Emitting line - receiving surface ( S E L S model )
When formulating t he equations we make the following assumptions : 1. 2.
3.
T he emitters of the radiation source are uniformly distributed in side its extension ( be it a volume , a surface , or a line ) . Any elementary extension of the source emits per unit time , at any frequency , an amount of energy proportional to its extension . The emission characteristics are constant along the z direction ( this assumption does not imply a constant radiation field in the z direc tion ) .
n.
dZ
FI GURE
6
Geometric representation of the
SELS
model .
A nalysis and Design o f Pho toreactors
849
Maint aining the expression ( 1) for the de fi nition of l v , w e may intro the differences in the energy exchange p roc e ss in the expression for dW v . Le t us use a ge n e ri c e xpre s sio n for an "energy flux" ( d '!' v ) which includes t he po ssi bilit y of emi ssio n by either line s , surfaces , or volumes . The generic extent of the lamp will be indicated by d r e • which may h ave dim ensions of length , are a , or volu m e ; dependin g on the case . T he n
duce
d '!'
dE
=
\)
\)
( 2)
df e
where d 'l' v is the amount of energy of fre quen cy v per unit time t h at leaves an infinitesim al extent of the lamp , ch ar act eri z e d by d f e . Thi s distinction is the definition of t he specific intensity of r adi at io n allo ws us to achieve complete co nsi s tency in a u ni fie d theory . Con sid e ri n g Eq . ( 2 ) the equ ation for l v becomes
,
( 3)
I \)
Here a supe r s cript in p arentheses indicates t he order of the infintesi mal quantity . Values of ( d ) and the exp res sio n s of d ( d ) r e for each emis sion model are given in T able 1 . I t i s obvious that b = d + 2 . At this point it should be noted th at in a diactinic m edi um ( a ho mo geneous nonabsorbing medium ) the specific intensity is constant alon g the propagation path , and he nce independent of distance . On the other hand , in ab s or bi n g and /or nonhomogeneous media the s peci fi c inte nsit y will change ( d ue . to absorption , scattering , reflection , and r e fr action ) accord ing t o the physical p rop erti e s o f the medium . In t he following analysis w e consider three s p ecial cases : W hen the emission is i n d e p e nde n t o f the direction , it is cal le d iso t ropic emis sion ; then f( 8 , (jl ) = 1 . When the emission is diffuse , it follow s what is known as the "cosine law ; " t he n f( 8 , (jl ) = cos Se ·
1. 2.
W he n the directional characteristics of the emission are different from cases 1 and 2 , t he model needs the sp e ci fication of function
3.
f( 8 , cp ) .
From now on we con si de r only the fi rs t two cases , isot ropic and diffuse emission . To simplify the treatment of the equations , we can ,reduce the two e mis sion c ase s in to a u niq ue exp re s sion : \)
I'
For
( 4)
isotropic v
I'
=
I
v
emission ,
cos
ee
I'
\)
= I . \)
In
the case of diffuse emission , ( 5)
850
De Bernardez , C larUi , and Cas sano
TABIB 1
dl
Mo e
d
S ELS
1
Final Equations
d
(
d
)r
for
the Three Emission Models
f*
dE L
dz
dw
L
d SEES
2
dA
p2
e
A
3
dV
dw r
e
r
p 2 d wr d p
dw
2
d
r
dE
2
v
p
dA
cos e r v
I'
dw
I'
dV
v
p
dV
2
e
:=
r
e
=
e
dE
dA
v
v
v
dV
I'
=
r
p A
2
dE
co s e
r
e
p
r
dE
1T r L2 L L
v
v cos a d z
p
=
2
v dw dA e e
dA
VEE S
p
dz
I'
=
dE
2 1rr L L L
=
v dz
e
[ Eq . ( 9 ) ]
( 4) ]
[E q .
e
e
l d ( d )<:!v l
I 'v
I'
v
e
=
2
dw r
dp
e
of direction , However , since the product l v cos 9e is T he distinction between the two types bot h cases I� is always isotropic . of emission will be introduced in the following derivations only when neces sary . for e ach emission T able 1 summ ari zes the final expressions of model . Finally , for polychromatic source s , the radi ation intensity is obt ained as inte gratin g the specific intensity over the follows :
independent
in
I�
whole frequency range ,
•
( 6)
Sp ecific Energy Flux Density
energy of
Vec to r
frequency The amount of 4 , 5 , and 6 ) ,
dAr ( Figs .
v
passing t hrough a s ur face of m ay be exp ressed as
per unit time
extent ( 7)
851
A nalysis and Design of Pho to reac tors
Isotropic emission [ Eq . ( 2 9) ] K ,J l-1
<
\)
exp
I
L
L
e
\)
K
(- (
K v ll v f �
)
x
si n 8
(- {
K V l-I V
�� �
2
)
-V
L
LL
x
E
-
K V 1-1
L,v
2 2 4TI r L L L X
d p de d;
X
COS
::::;
d
For both
(c)
J J8 ¢
*
Po
1-1
dA
r
cos 8 r
c I d( )
q _
v
TI L
\)
d p'
cjl
-V
L
dp'
E
2 sin 8 cos cp
sin
V
c between the definition of d ( \
E
�
L
,
2TI r L L L
)
d 8 d<jl
2
8
E
L,v
2 2 TI r L
L L
"vdp
We do not need to
for each model because it
From E q . ( 7 ) i f follows that
I
( B)
1s th e roo d u I us or th e spec1" fi1c energy f1ux dens1"t y veetor .
9v I
·
_
:::::
-
:::::
Eqs . ( 7 ) and ( B) it is clear that c b 2 d. The relationship bet ween the specific inte nsi t y and the m o d ulus of the sp ecific energy flux de n s it y vector can be derived relating E q s . ( 4) and ( 8) :
dw
( 9) e
p
v
2 2
��� ' exp �{ ) Po
d p d 8 d <jl
i s a properly re lat e d to the receiving surface .
I
1-1 v
fp *
\)
si n 8 d z 2 p
is th e specific energy flux density vector .
m ake a distinction
W here
L
(-J:* Po )
\)
(
L
0
exp
K \) 1-1
dp de d;
sin 8
where d ( \
L,v
B 7r r
)
"v
"v dp
c
E
J
K V l-I V
4n L L
�
" v dp
' exp
\)
E
p
D
i ffuse emission (K )
a Diffuse emission , e \) [ Eq . ( 33 ) ]
e mi ssion
dz 2
0
+(
,
I sotropic
a
2
De Bernardez , C laria, an d Cassano
852
where we used the geometrical relationship dw
=
e
dA
r
8
p2
cos
r
( 10 )
(d) In spite of the fact that I d q� V I and I '\) have different physical meanings , they are often confused in the photoreactor engineering litera ture . In a diactinic medium while I� remains constant , Eq . ( 9) shows that the modulus of the specific energy flux density vector changes with dis tance . T able 1 summarizes the expressions of Eq . ( 9) for each emission model . A t tenuatio n Coefficien t
The attenuation coefficient is a property of the substance through which the radiation travels and is related to the process of radiation absorption . It is defined through a constitutive equation commonly referred to as Lam bert's "law , " which can be derived from molecular theory . The con stitutive equation states that for an absorbing and homogeneous medium , the contribution of absorption to the gradient of the specific intensity is
div dp
I
a =
- Jl
( 1 1)
I
\) \)
Equation ( 1 1 ) is valid for any type of emission model with proper use of the definition of I v . The left - hand side of Eq . ( 1 1 ) is the directional derivative of Iv (i . e . , the component of the gradient of the scalar function I v along the direction of propagation of the ray ) . Hence Eq . ( 1 1 ) may be rewritten j
•
V I \)
_
=
- Jl
\)
I
( 12 )
\)
where j i s a unit vector i n the direction of each ray . dependent of distance
p,
Eq . ( 12 ) is also valid for I' .
Since f( 8 , $ ) is in
\)
R a d i a t i o n B a la nce
We shall state the steady-state radiation balance for a nonemitting , non dispersin g homogeneous control volume ( Fi g . 7 ) . The difference between the energy flux density of frequency v impinging on dAr at position p with direction ( 8 , $ ) , and the energy flux density of frequency \) leaving the fixed control volume with the same direction , at position p + d p , is due to the absorption of radiation : ( 13)
where on the right -hand side the subscript respect to p .
p
indicates differentiation with
85 3
A nalysis an d Design of P ho toreactors
FI G U R E 7
Radiation balance in an elementary absorbing volume .
I ntroducing Eqs . ( 8) and ( 10 ) into Eq . ( 1 3) , we have ( 1 4) Combining Eqs . ( 1 4) and ( 9) yields
( 15) Substituting the constitutive equation ( 1 1 ) into Eq . ( 1 5 ) gives us
( 16) Finally , considering Eqs . ( 9 ) , ( 1 4) , and ( 1 6) , the steady - state radi ation balance becomes
( 17 )
Rate of the A c t i vation Step
As mentioned e arlier , the r ate of the activation step is proportional to the local volumetric rate of ener gy absorption ( LVREA ) . The LVREA is defined as follows :
dV
r
( 1 8)
with f = b + 1 - 3 b - 2 = d. Com bining Eq . ( 1 4) with Eq . ( 1 8) and considering that dVr = p 2 d w e d p , we obtain =
De
854
Bernardez , C laria,
and C assano
( 1 9)
Fin ally , introducin g the radiation b alance ( 1 7) into E q . ( 19) , w e h ave
To evaluate d
(d) a e we need to know V
I
d
(d)
q
-V
I
.
( 2 0) From the radiation balance
and inte grating inside the reacting space ( where the absorption of the radiation process t ake s place ) , we obtain ( Fi g . 8 )
2 P 1
In
d
�v
Eq . ( 2 1 ) ,
I
I
d
(d)
=
P
q
I
-V
2 o
I
d
(d)
q _v
I
o
e xp
( J P*< -
a, ¢)
p�( S , <j>)
(21)
is the modulus o f the specific ener gy flux density
vector for a point W ( p * , 9 , ¢ ) at the surface of entrance of radiation to the 0
0
reactor volume . To evaluate it , assuming no absorption bet ween t he source s ur face and point W , we use the equations valid for diactinic media , that is ,
P
2
1 d
FI G U R E 8
< ct >
c:! v I
Schem atic represent ation of the attenuation process .
( 22)
A nalysis
855
and Design of Pho toreactors
I�
B ecause written
rem ains constant in a di actinic medium , Eq . ( 22 ) m ay be
( 2 3)
,
Combinin g Eqs . ( 2 0 ) , ( 2 1 ) , and ( 2 3 ) , we obtain I' o v � \1 2
p
( 2 4)
Finally , the local volumetric rate of ener gy absorption at fre quency v is obt ained by inte grating Eq . ( 2 4) :
1�;"
=
exp
t( ) "v
dp
d
(d)
r
( 25)
e
In Eq . ( 2 5 ) , I� , o must be relate d to the power output of the radiation source . From Eq . ( 4) we have
=
dw
I'
v ,o
e
d
(d)
r
( 26)
e
for a diactinic medium . Inte gr ating Eq . ( 2 6 ) for the whole range o f solid an gle s and for the whole extension of the emission source , we obtain the lamp power outp ut at frequency v : E
I'v
L,v
,o
dw
( 27)
e
where we have m ade the following assumptions : 1. 2.
In the case of volumetric emission , e ach element ary volume is transparent to the emission of its surroundings . I n the case o f three - dimensional emission so urces , they are s up posed t o b e bounded b y m athematical surfaces o f zero thickness . Thus any bundle of radiation coming out of the lamp does not change its intensity or direction when it crosses this P<>undary .
Eq .
To perform the inte gration in emission cases : 1.
Iso t ro p i c
emissio n .
L,v
\1 , 0
=
I
,
\1 , 0
we m ay distinguish the two , which is con stant (independent
T hen E q . ( 2 7) become s
of distance ) . E
H ere I'
( 27)
= I'
v ,o
41f r *
e
( 28)
856
De Be rnardez ,
Clarta ,
and Cas sano
where the expression of r * for each emission model is condensed e in T able 1 . Finally , we have a e \) 2.
�
=
exp
p
(- J P* ) P0*
ll v
dp
( 2 9)
In Eq . ( 2 7 ) we must replace I v' by I cos e '0 v •0 e and then perform the integration :
Diffuse emissio n .
E
L,v
=
1 \) 0 ,
1
e
cos
dw
e
e
e
[ d( d )r e r
( 30)
e
where l v 0 is constant (independent of distance ) . Here , since the ' intensity of radiatio n follows the cosine law , the emission solid angle is 2 rr . T hen E
L,v
=
I
v
,o
rr f *
( 31 )
e
Finally , we have I'
v ,o
=
1
v , o cos ee
=
E
and e
a
\)
=
�
cos e
p
L,v
1T
e
e
2
r*
e
cos e e
( 32 )
( 3 3)
The final expressions of e� v alid for diffuse emission for each model can be found in T ab le 1 , w h er e cos e e has been replaced by the appropriate expression for each model . E q u ations ( 2 9) and ( 33) are valid for mono chromatic light if we deal with polychromatic radiation , these expressions must be int e gr at ed over the entire frequency spectrum , that is ,
( '3 4)
M O D E L I N G T H E A B S O R PT I O N P R O C E S S
As mentioned in preceding sections , the m ain difference between a photo chemical and a therm al reaction is due to the presence of a radiation activated step . T he rate of reaction of t his step is proportio nal to the local volumetric rate of energy absorption ( LVREA ) . Hence the evaluation of the L VREA constitutes the most important stage in the analysis and design of a p hotochemical reactor .
A nalysis and Design of Photoreactors
85 7
Table 1 illu s tr ates that for each emission model , the LVREA is a func tion of the spatial variables , of the physic al properties and geometrical ch ar acteri s tics o f the lamp - reactor system , and some physicochemical proper ties of the reacting mixture . T he L VREA dependence on the geometrical characteristics of t he lamp -reactor system will be studied in d et ail in t he next sectio n , and it come s from the limits of inte gr ation of t he LVREA ex pression for each emission model . The physi cal properties of the l amp are represented by the p ar ame t er K v in the LVREA equ ations ; both are de scribed in T able 1 for each model . Fi n ally , the attenuation coe fficient ].l v involves the dependence on the physicochemical properties of the reacting system . This section is devoted to the evaluation of the att enu ation coef ficient and t he local volumetric rate of energy absorption . Attenuation C oeffi c i ent
In the evaluation o f this coefficient it is necessary to make a dis tinct ion between homogeneous and heterogeneous media. The latter are encountered when one attempt s to model systems in which gas bubbles are dispersed in a liquid , or solid particles are incorporated into a gas or liquid phase as in photocatalytic applications . The distinction arises from t he fact that the presence of hetero geneities i ntroduc e s unavoidable distortions in the r adia tion field . Ho moge neous Media
When the medium is homo ge neo us , the at te n u at ion of radiation changes con tin uou sly . A s me nt io ned before , thi s variation i n t he specific inte nsit y due to absorption inside an elementary volume of homogeneous media may be ex pressed by a constitutive equation which involves an attenuation coefficient It is a function of the frequency of r adi ation v and the state variables ].l v . such as temperature , composition , and in some cases pressure . The exact form of the attenuation coe fficient function can be dete rm i ne d from the micro scop ic theory by s t ud yi n g t he absor p tion of a photon by atom s or molecules exposed to a radiation field . The reader is referred to the specific litera ture [ e . g . , Pai ( 1 966) or Vi n cent i and Kruger ( 1 96 5 ) ] . Macroscopically , the at te nu atio n coefficient is considered to be a linear functio n of the concentration of the absorbing species : Cl \)
c
( 35)
where Cl is the absorption coe fficient or ab sorp tivit y . When more than one v p articip ate in the a b sorp tio n process , Eq . ( 35 ) may be w rit te n as
sp e cies
= L; i
ll i , v
= � i
a.
1 ' \)
c1
( 36}
Both Eqs . ( 35) an d ( 3 6 ) are s t ric t ly valid for dilute solutions . The range of validity of these equations can be experimentally tested by st u dy ing t he v ari atio n o f ].l v wit h concentration by means of a spectrophotometer . This type of experiment also allo w s us to obt ain values of the absorption coefficient necessary to model the attenuation process . On t he other hand ,
De B ernardez , C larta, and Cas sano
858
for m any chemical compounds , v al ue s of the ab sorption coefficient and the ran ge o f validity of Eqs . ( 35 ) and ( 36) can generally be found in st and ard handbooks of physics and c hem i s try . The absorption characteristics of many other species and mixtures of di ffe re nt compounds m ay be found in speciali zed p ub li c atio n s in t he field of s p e ct ro s copy . Later we will see an examp le o f a p hoto c he mi c al reaction with an inor ganic complex in which t h e char acteristics of the ab so r bin g s pe cies ch an ge with co m p o sitio n and t he d at a re q uir e d b y Eq . ( 35 ) are not available . I nstead , t he attenuation co efficient i s e v al u at e d as a function of the com positi on o f the absor bi ng m e di um .
Heterogeneous Media S i nce ho mo geneou s s y stems have been t he m ain o bjective of the re s earch performed so far in photochemical reactor analysis , the information available in the literature about m o d eli n g t he attenuation in hetero geneous media is p ar ticul ar ly scarce . A critical review about this subj ect can be fo un d in Alfano et al . ( 1 98 5 ) . T he problem can be ap p r oac he d in two di fferen t ( a ) by modelin g the attenuation process taking into acco unt the w ays : s c at te rin g of radiation produced by the hete ro ge neitie s , or ( b ) by using t he e xi st in g formulation of ho m oge n eou s media with an e ffe cti ve attenuation About t he first str ate gy , it c an be ascer t aine d th at t he u se coe fficient . o f a ri gorous ra di ation source model having no adj ust ab le parameters , to gether wit h t he comp lexity provided by t he heterogeneities of the medium , have not yet been attempted . On t h e other hand , the simplified method u si ng an e ffe ctive attenuation coe fficient is appropriate to carry out the calculation of the rate of radiant energy absorption in hetero geneous sys tem s with a good de gree of approxim ation . B ased on these considerations , if one attempts to d esi gn and /or analy ze an industrial p ho t o c he m ic al reactor , the second st r ate gy m ust be followed for the time be ing . At prese nt , several correlations have bee n p rop o s ed to evaluate the e ffective attenuation coe fficient in gas -liquid dispersions . Amon g them , we can particularly m e ntion t wo : 1 . T he first one , prop o s e d by Ot ake et al . ( 1 98 1 ) , is a simple em piri cal expression w hich accounts for t he absorption e ffects produced by the liquid phase and t he re fle ctio n , refraction , and t r ansm i s sion effect s pro voked by t he gaseous p hase . Considering that the la tter are proportional to the specific sur face area ( a ) , \J e ff v m ay be represented by t he follow -
�
ing correlation : =
\.1
v
(1
E:
g
)
+
'
ka
I
( 37 )
v
where \.l v i s the attenuation coe ffi cie nt of the liq ui d p hase , E: g the - ho ld up of t he di spersed phase , and k is an e m piri c al coeffi cie nt d ep e ndi n g o n th e opti c al properties of t he s yste m . Otake et al . ( 1 9 8 1 ) plot t ed the exp erime n t al i n for m ation obt ained by them and ot he r authors , finding that the values can be well co r rel at e d by u si ng k = 0. 12 5 for \J e ff v , \.1 v , and a in the ' same units ( cm - 1 ) . 2 . T he other correlation proposed by Yokota et al . ( 1 9 81 ) , in which t he effective attenuation coefficient becomes a fu n c tio n of t h e attenuation coefficient in the liquid phase , the bubble di ameter , and the gas holdup . The expression is
�
859
A nalysis and Design of Photoreactors =
f1
\)
(1
+
( 3 8)
h e: )
g
and ( 3 9) where d b is the bubble di ameter in mm for f.! v and fl eff , v expressed in m - 1 , Both correlations m ay be used in the case of fl v s m al le r than 40 m - 1 . W h en fl v is gre ater than 40 m - 1 , the dispersing effect of radi ation due to the pre sence of bubble s m ay nor m ally be ne glected . Later we will see an example of the design o f a two-phase photochemi cal reactor where the absorption process is modeled using an e ffective at tenuation coe fficient . However , no reli able desi gn procedures have yet been developed for photocatalytic reactors ( solid - fluid photoreactors ) . Loca l Vo l u met r i c Rate of E n ergy A b sorption
As derived in t he preceding section , the inte grated expression for the local volumetric rate of ener gy absorption may be expressed as =
for
( 40 )
monochromatic
radiation for each point I ( rz , Sr , z1 ) inside the re actor In Eq . ( 40) the mod ulu s of the energy fl u x density vector I q
volume .
_ \)
I
repre sent s the ener gy flux arriving from all directions i n sp ace an d from the whole volume of the lamp ; f1 v is the attenuation coefficient , which is a function of the concentr ation of the ab s orl>i n g species at point I . Figure 8 i l l ustr ates the attenuation p rocess undergone by each radiation bundle be fore reaching point I inside the reactor volume . To evaluate I q I at point I , every direction ( 6 ,
tion ( 6 ,
source without changing it s speci fic intensity until it reaches the reactor
wall , represented by point
W ( p� , 6 , $ )
in
0 and l ( p * , 6 , <j> ) of Fig . 8 ,
�J p*(6 , <)1 ) )
absorption .
exp
W ( p * , 6 ,
-
in
t he
This chan ge is represented
P�( e ,
the exp re s sion of
f1
I
"'
q
- 'J
Fi g .
8.
In t he
specific intensity ch anges due to
by t he
exponential factor
( 4 1)
d p'
I
trajectory between
for e ach model .
Due
to
the co ncentration de -
pendence of the attenuation coefficient , the modulus of the radiation flux den sity vector , and hence the LVREA , become s a functional of the con centration field inside the reactor . T his functional characteristic of the LVREA make s the problem of modeling a photochemical reactor more complex than e xp ecte d . T he complexity is due t o the couplin g between the radia tio n field evaluation and the resolution of the mass balance equations . The
De B e rnardez ,
860
C larUi and Cass ano
the reaction at each point I d e p en ds on the local value of t he LVREA , a function of the extent of the re ac tion reached at all points in sp ace t hrou gh which every r ay arriving at poi nt I has pre vio u sly t ravele d . There are some speci al situations whe r e this comple xity is reduced . Working with a p erfe ctly mixe d reactor , the absence of st able s p e cies con centration grad ient s inside the reactor volume m ake s the attenu ation co In this case t he L VREA is no lo n ge r a fu nction e fficient uniform i n sp ace . of the concentr ation field but is still spatially dependent t hrough its de p en de nce with I q I .
of
but the LVREA value i s
_v
This condition m ay also b e att ai ned by working wi t h a p ho tosensi tized reaction ind e p e nd e nt l y o f the mixing con di t io ns in t he reactor . In this special kind of react ion , the absorbing sp ecies does not ch an ge its concen tration ( i . e . , it i s not cons umed ) . In this case , due to t he s p ati al and t e m p oral uniformity of t he conc e ntr atio n field o f the absor bing specie s , the link " p ro gress of the re action attenuation of r adiation" is destroyed . In the liquid p h as e , photosensiti zed re actio n s generally involve lar ge con j u g at e molecules such as ben zophenone . In gas - p h a se work , sulfur dioxide , b e n ze n e , an d o ther or gani c molecule s are used . In addition , me t al v apors h ave be en p ar ticul arly useful in g as - p h as e photose n si ti zed reactions , such as N a , Zn , C d , and H g , of w hic h t he latter has been applied mo st e xte ns ively . Anot her situ ation where the d egre e o f co mplexi t y i s lessened is in the modeling of b lack body reacto rs . In t hi s type of reactor , almost all the abs o rption o f radi ation occurs in a t hi n layer close to the entrance of li ght . T his characte ristic m ay be att ained by wo rkin g with a s t ron gly absorbing m e di u m . In this case the at t e n u atio n can be mod e le d as occurring on t he reactor wall (i . e . , at the " black" surface ) . Consequently , t he evaluation
of the radiant field is reduced to estim ate the energy flux d en sit y on the reactor wall .
E V A L U A T I O N O F T H E LV R E A FO R D I F F E R E N T R E A C T O R S
The rate of initiation o f a p hot or e ac tion is proportional t o t he LVREA ( i . e . , q, e a ) ; t he porportio nality co nst an t is t he p rimary q uantum y i e l d . Q act The definition of this yield will depend on the nature o f t h e p rocess The prim ary process m ay be d e fi ne d , i ncludi ng both the initial concerned . act of ab so rptio n and those im m e di at e ly following processes which are de t e rm i ne d by the prop erties of the initially excited electronic state . In almost all cases the prim ary p roces s le ads to a dissociation of the abs orbing molecule . For such cases the prim ary q uantum yield m ay be defined ' as =
=
nu m ber of molecules dissociated by the primary process
numbe r of quanta o f radiation ab s orbed
q, cannot be me asured e asily ; to d e t erm in e
it s value we m ust gain informa other possibilities mo re li ke ly i nvolve d in the p r imary process ( recombination , physical quenchin g , et c . ) . Some times , these rates m ay be o btai ned by i ns p ectio n o f t he absorption spectrum . T hen we may be able to e s tim at e the value of q, To illustrate this we can co n s i d e r the p hotochemical dissociation o f chlorine molecules : referring to t h e potential curves for chlorine ( Herzber g , 1 95 0 ) , t he inequivocally show tion about the relative r ates o f the
•
861
A n alysis and Design of Pho to reac to rs
that upper state s are unstable and the molecule dissociate s very rapidly . this p ro cess will p re do minate over all others and we can surely say that , almost independently of the frequency co n si dered , � = 1 . Fo llo wing the statement of the second photochemical law ( T urro , 196 7) , the m axim um pos si ble value of Ill i s unity ; t herefor e , for t he great m aj orit y of cases its value m ay be taken as unity . In addition , we may use the overall q uantum yield as Then
num ber of molecule s of a given reac t ant o r or formed o d uc t fi n� ally decomposed �� ���� � �pr ����
� ov = --
�
number
-
� of quanta
�
��
absorbed
This property will generally give an indication of t he n ature of the re action mechanism . If
.
-V
the radi at ion source K v ( see T able 1 ) . sim pli fy the t re at m e nt , we will derive the values of j q I
s t ant o f
To
for mono-
chromatic radiation and isotropic emission . T he extensio n t o polychromatic radiation an d diffuse emission is str ai ghtforw ard . In addition to the assumption alr e ad y described for the emission m od el and t he lamp , we must add the following ones about the reactor ( Irazoq ui et al . , 1 97 3 ) : 1.
2.
.
The inner wall of the reactor is bounded by a cylindric al mathemati cal surface without thickness (i . e . , no re fle ct io n or re fraction o cc ur s ) T he op aque zones at t he top and bottom parts of the re actor do not re fle ct or e m it radiation .
The mo d ulu s of the radiation flux d e n sit y vector at any point in the re act i n g space , I ( r , B , z ) , is t h e energy flux arriving fro m all dire cti o ns in sp ace and from th e whole extension of the l amp . It can be obt ai ne d afte r i ntegr atio n o f
I
qI
=
K
j r
e
-\P
e xp
P*
o
1-1
dp
d
(d) r
( 42 )
e
.
be present along this se ctio n is complexity of the limits for the inte grals of Eq . ( 4 2 )
T he problem that will always
t he
tJP * )
ce ntered on
De Bernardez , C laria,
862
and
Cass ano
A n n u l a r Photoreactor VEES
Mo del
I n this case Eq . ( 4 2 ) is written as fo llo w s ( see T able 1 ) :
=
K
ja J4> Jp
da
dp
d
(- Jp*( 'd,<j>) )
sin a e xp
P�( a ,
1.1
dp
( 43)
To obt ain t he i nte gr atio n limits , one starts from the point of reception l ( r , a , z ) and investi gates all the arriving r ays w it h directions ( a , <j> ) . One must fi n d those that intersect the bound aries of the surface of the cyli nder representin g t he lamp ( Fi g . 9) T he limits for variable p are the intersections of t his coordinate with the front and r e ar parts of the l am p in any a ,
=
r co s
2 cos
sin
r
2
+
2 1 /2 r ) L
( 4 4)
a
The limits for variable a are obt ained considering that the limiting rays coming from the lamp and reaching the generic point (r , a , z ) m ust satisfy two restrictive conditions : ( a ) a ray limits the e angle when its equation has a common solution with the e q uat ion of the circum ference . that defines t he nonemitti ng zones at the upper and lower parts of the lamp ; and ( b ) the ray will b e a limit w he n t he common sol utio n abo ve corr esp o nd s t o the i ntersectio n of t he ray with that po rtio n of the circum ference w hi c h limited by two ge neratri x lines corresponding to the cp lim iti n g an gles , is closer to the generi c p oi nt ( r , a , z ) . T he second condition is needed because for any plane at co n s t an t
,
FIGURE 9
Geometry and coordinate systems for
dire ct
radiat ion
A nalysis and Design of Photoreactors
FI GURE 1 0
Limits of
In this w ay , for
e 1 ( 4> )
e2 ( 4> )
= tan
= t an
-
1
e1
e and 4> for direct radiation . and
e 2 we
obtain t he following
2 r cos 4> - [ r ( cos 2 4> L
-1
863
r
COB 4> -
L
-
z
1) +
[ r 2 (cos 2 q, - 1) -z
+
�
r ]
r
expressions
1 /2 ( 45a)
2 ] 1 /2
L
( 45b )
approach should re quire comp utation of the reactor The three - dimensional characteristic of the emission proces s c auses the reactor to be p artially irradiated in the incoming and This fact increases t he e ffective re action out goin g regions { Fig. l la) . volume . T hi s effect is more noticeable when the inner reactor wall is close to the lamp . T he e xp re s s ions for the a-limiting an gles should be m odifie d in the partially irradiated re gions ; the reader is referred to Romero et al . ( 1 983) for a det ai le d analysis of the reactor wedge comp ut ation . On the other hand , Fig . l lb shows that the calculations should be corrected to account for the effective lamp len gth . If no corrections were introduced , we would be computing as emission volume s portions of t he cylinder that correspond to regions in the s p ace where t her e is no emission , at all . T his e ffe ct is also signifi c ant when the inner reactor w all is very close to the lamp ; for a more det ailed analysis , the re ader is referred to De Bernardez and C assano { 1985) . The limitin g rays in the 4> direction can be o bt ained by imposing the r est rictio n that both intersections of t he p coordinate with the lamp boundary
A
more rigorous
and t he
lamp wedges .
must coincide , that i s , =
( 46)
864
De B e rnardez ,
C laric'i,
and C as sano
IRRADIATED VOLUME
I t r./3, Z l
(a) FIGURE
[ - lT
( b)
11
( a) Reactor wedges ; ( b ) lamp wedges .
With Eq . ( 44) and since cP can take value s only in the closed interval
/ 2 /2] . , lT
4>2
- 4> 1 =
= co s
-1
(r2
With these li mits , E q . ( 45 )
- 2 ) 1 /2 rL
( 47 )
r
can
be inte grated to o btain
" ··) ( 48 )
SEL S Mo del In this case Eq .
( 42 )
�- J
becomes exp
)
( see T able 1 ) :
P *( 8)
p� ( 8 )
11 d p '
( 49 )
Let us call
p ( 8)
cos e
p ( 8) sin 8
= r = z
( 50)
(51 )
Then we have
dz 2 p
de
r
( 52 )
Design
A naly sis and
865
of Pho toreactors
and finally , de
exp
(-J
p * ( 8)
( 5 3)
P6 ( 8)
where e 1 and 82 c an be obt ai n e d from Eqs . ( 44) and ( 45) , m aking rL = 0 and cP = 0 . A gai n , a more ri goro us appro ach should consider the reactor wed ges ( see Fi g . l la) , as it was m entio ne d when the VEES m o del was dis cu s s e d . Cy l i n d rical Photoreactor w i t h E l l i ptical Reflector
Four additional assumptions m ay be added
to tho s e gi ve n earlier ( Cerda
et al . , 1 97 3 ) : 1.
2. 3.
4.
T h e reflector is a p e r fe ct ellipti c al cylinder . T he lamp is located in such a w ay that its centerline p asses through one of the foc u se s o f t he ellip tic al re flector ( F l ) . S p e c ul ar reflection occurs w it h an average reflection coefficient that is independent of w ave le n gt h and di re ctio n . ( These restric tions could be easily relaxed an d are used here for simplicity . ) T he refle cted radiation come s only from t he elli p tic al re flector (i . e . , the top and bo t to m parts of the cylinder do not re flect r adi at io n ) .
T he radi atio n i m p i n gi n g at any point in si d e the elliptical cylinder is of two parts : direc t radia t io n and reflected radiatio n . E xcl u din g direct radiation , a r ay could arrive at a point of reception with a give n direction by a single re flectio n process or as a re su lt of successive con t act s with the elliptical mirror . In t he second case , for points located close to the focus F 2 , the ray will describe a broken line touching the surface of t he reflector at le ast three times . T he question is whether it is always possible for a ray comin g out of t he lamp to und e r go so m any re flections and still arrive at t he point of interest . It can be answered by fo llowi n g the r ay p ath in the reve rse w ay ( i . e . , starting from the point o f re c ep t io n ) A m ultip ly reflected ray w il l be e ffective at a gi ve n point in s p ace i f and only i f , tr ave li n g the reverse t r aj ector y p re vio usly de scribed and in sp it e of its pro gr es si ve vertical displacement , it sti ll inter sects t he sur face o f the lamp bo u nd ar y that is s ee n from the p oi n t s of incidence . T h rou gh t he s ame reaso ni n g , a m ultip ly reflected ray would have no re al existe nce if this i nt e r section falls outside the length of t he lamp . The problem consists in fi n din g the m athematical re l atio ns hip bet ween the incident radiant energy flux an d t he reactor , lamp and re fle c t or dimen sions , the o p ti c al prop erti e s of the r e fle ct or and t he reactor wall , an d the p hysical properties of the r adi atio n com ing out of t he so urce T he p oint of receptio n ( I ) on the reactor volume is referred to a cylindrical coordinate m ade u p
.
the centerline of the cylindrical reactor with origin at r e flect or bottom ( r 1 , 81 , � ) .
system located at
the
.
866
De Bernardez ,
V EES Model
Direct Radiation :
C laria,
For this case , the limits of integration for
p may be obtained in a similar way as for the annular reactor .
are
( see Fi g . 1 2 )
6.
where
and Cassano
�
2 CO S 4J ± [ 6. 2 CO S 2 4J - ( 6. 2 - r ) ] l / sin e
variable
The
results
( 5 4)
( 55 }
( 56)
( 57)
( 58)
4>
2
= 2 7f
F I GURE 12 integration .
-
4>
( 5 9)
1
Variables used in the determination of the limits
of
86 7
A nalysis and Design of Pho toreac tors For direct radi ation the final expression after i nt e gratio n of
E q . ( 4 3 ) is :
( 60 ) The methodolo gy of I ndirect Radiation with Only One Reflection : an alysis for reflected radiation is to follow a ray starting from t he p oi n t of reception in or der to identify the point of re flection , and from this ( apply ing the laws of optical p hysics ) to search in the direction of emission . To understand the analysis it is necessary to locate several key points ( Fig . 13) : first , the point of incidence of the re flect e d ray ( point of reception I ) ; second , the point of reflection ( P ) ; and finally , the t wo points o f intersection of the dire c tio n o f emission with the lamp cylinder ( E 1 and E ) . 2 Any ray comin g from the lamp will be indicated by an incidence position vector at the point of re flection ( P E ) and a reflection position vector at the point of reception ( P I ) . The former has its origin at P and the latter at I . The intersections o f P E with the cylinder of the lamp tube define two vectors , PE 1 and PE 2 . T he details of the geometrical derivations and , , definitions of the variables are explained in the appendix to this chapter . The re sults are
( 61 )
e1
=
t an
....:. 1
Pj
+ D cos ( 62 )
X
FI G U RE 1 3 Vari able s involved in the analysis o f indirect radiation with only one reflection .
868
e
2
= tan
_1
P}
De Bernardez , C lariCi, and Cassano
---==-
+ D co s E; - ( r
-!-------
�
- D
2
2 12 sin o 1
-----
L ) 2 L ( 1 / ) ( LR f -
( 63)
- r;I
( 6 4)
T hen the inte gration o f Eq . ( 4 3 ) in the p coordinate gives us the following exp re ssion for the modulus of the radiation flux density due to indirect r adiation with only one reflection :
=
2Kf
Rf
i
d
2 L
-
D
( lp *(B,cj> )
1 / 2 ( 62 ( <1> ) 2 d e exp si n E; )
2
e1<
J�
p�( e , cp )
J.l
( 65 )
where the attenuation of t h e radian t beam s at t he mirror surface was t aken into account u sin g the reflection coe fficient of the elliptical mirror rRf·
Indirect Radi ation with Two o r More Reflections :
As explained earlier ,
the m at h e m atical analysis is accomplished by considering t he broken
trajec
torie s of the rays arriving at the reaction surface after more than one re flection . T he derivation will not be presented he re . Details m ay be found in Cerda et al . 1977) . ( Asymmetries : An additional p roblem with this system is originated by the a zim ut hal asym metry . This is due to the overlapping of two effects : the arrival of direct radiation (only on the points situated in front of the lamp) and t he existence of umbra and penumbra zones , givin g rise to non uni formities in the indirect radiation . When the characteristics of t he re actor generate import ant asymmetries , the addition o f an gular terms to the m ass balances transform s all problems from t wo - to three - dimensional . A pre vious work ( De Bernardez and C as s ano , 1 98 2 ) showed that for this system the uniform angular distribution will be obt ained only if one works with a reactor di am e t e r /lamp di ameter ratio lower than 0 . 5 and with an eccentricity as low as possible ( about 0 . 4) . If one wishes to work with higher eccentricities , the reactor diameter /lamp di ameter ratio should be lower than 0 . 2 . It is recommended t hat values of the distance between focuses of the ellipse be as large as possible . S EL S Mo del
,
Direct R adiation : The exp res sion o f the modulus of the energy flux density vector for this model is similar to that obtained for the annular re actor , but r m u st be re p l ac e d by 6 with 6 given b y Eq . ( 55 )
=
i
1e2 e1
d B exp
(- J
P *< B>
P�( e)
)
1-1 d p '
T he limit s i n t he e coordinate can be o bt ai ne d from Eqs . ( 5 6) and m aking
L
( 66 )
( 57 ) ,
A nal:ysis
and
Design of Pho tore ac tors
2c -----1
f-----
20 ---
---FI G URE
869
Proj ections of reflected ray trajectories .
14
Indirect Radiation with Only One Reflection : All reflected rays must lie on planes that also contain the reactor axis . A r ay which after reflec tion reaches the reactor centerline may be projected on a plane that goes through t he point of emission and is perpendicular to the lamp axis . The lengt h of this projection is equal to 2a ( Fi g . 1 4) . Each one of these planes allows us to define a circumference con structed with the locus of all the end points of rectified rays ( Fig . 1 5 ) . This can be done in this w ay because the reflector is a cylinder and the lamp is assumed to be a line paralle l to the generatrix line of the mirror . This circumference m ay be repeated for each point of em is sion of the lamp , giving rise to a cylinder of circular cross section of radius equal to 2a . This cylinder m ay be thought like a virtual surface of emission that acts as a substitute for the linear lamp . T aking a planar cross section going through t he reactor focal axis of the elliptical cylinder , we finally obtain a situation like the one depicted in Fig. 1 6 . Points such as those
\
\
\
\
\
\
I
I
I
I
I
I
I
D
: ·�·�f----2a--�.-j ',
// /
//
/
I
FIGURE
15
I
I
I
I
/
\
I
I
I
\
\
', \
\ \
\
\
Locus of end points of rectified rays .
De B ernardez ,
870
C lari a ,
and
C ass ano
I I I 1
�---2 a
2a ----�
Limiting an gle s for t he a coordinate .
FI G U R E 1 6
desi gnated in Fi g . 1 5 with letters A , B , C , D , and so on , when converted into straight line s as is done for one c ro ss section in Fi g . 1 6 , rep resent the w ay in which the linear lamp would be s ee n by ob ser ver s located at the reactor centerline if they looked in the indi c ate d directions . The pro j ection o f t he actual dist ance travelled by any ray with di re ctio n a on the l amp centerline is 2a -.-- cos a sm a
=
2a cot e
The key poin t , a s
( 67 )
in any
emiss io n
model , is to know the limit s o f the arriving at the reactor axis "line " sour ce ) . I f the ray e ffectively
e an gles (i . e . , out of all those e dir ectio ns
which are re ally comin g out of the comes out of t he lamp (i . e . , if it is a direction that transports energy ) , its proj ec tion give n by Eq . ( 67 ) must lie wit hin the lamp lon gitudinal dimen sions . This allow s us to define the limiting angles for e in the fo llowi n g m anner :
a 1 = tan
a2
= t an
a
= t an
a
3
4
= tan
-
1
2a - r
I
( 1 / 2 ) ( LRf + LL ) -1
2a - r
I
-
( 1 /2) ( LRf -1
2a + r ( 1 /2) ( L
-1
Rf
2a + r ( 1 /2) (L
Rf
LL )
-
-
1;; 1
1;; 1
1
+ LL ) - 1;; 1
1
- LL)
-
1;; 1
( 6 8)
( 6 9)
( 7 0)
( 7 1)
To comp ute the reflected radiation at point I , we have split the inte gral to account for energy contributions from the whole clo s e d re flectin g surface of t he elliptical mirror . The final exp r es sion for the mo d ulu s of the r adi at io n flux d en sit y of the re flected radiation with the linear model is
into two parts
A nalysis a n d Design of Pho t o reac to rs
=
K
r
+
[
1
RF 2a - r
1
2a +
r
1
1
i84 83
fe2 e1
� f p *(
de exp -
d e exp
871
e)
p�( e)
�-J P*( 8) d p)]' p�(
8)
fl
( 72)
I t should b e stressed that since it i s a line ar model , we have lost t he integration in the cp coordinate , which must always be taken into account when an extense model is used ( with finite dimensions for rL ) . C y l i nd ri c a l Photoreactor w i t h Parabo l ic Reflector
Here again we m ay have direct and re flected radiation . For the analysis we will require two coor dinate systems . The first one ( x F - y F - ZF ) is defined with the origin at point F located at the center of the bottom of the empty cylindrical re actor , with YF , consequently , being its centerline ( Fi g . 1 7 ) . An auxiliary coordinate system p ar allel to the former is also needed . Its center is located at the point of incidence designated by I in Fig . 1 7 and it can be move d all over the space under analysis . VEES Mo del
Direct Radiation : For this c ase , the limits of inte gration m ay be ob tained in a similar w ay as for the annular and elliptical reactors . The results are /:;. 1
sin ct> + /:;. 2 cos cp sin e
( 7 3)
y
F I G U RE 1 7
Geometry of the tubular source with parabolic reflector .
8 72
where 81
�2
e 1 < IP >
::: :::
YI
+ t
r
sin
l
tan
=
-
De Bernarde z , C laria, and Cas sano ( 7 3)
a
13 1
p'
-1
( 7 4)
1
L /2 L
+
z1
( 7 5)
( 7 6)
( 7 7) T he limits for cp are obtained from the solution of the implicit equation
( 78) and the final expression for
x
(
exp -
I
qI
P*(6,cp) JP6(6,cp)
is
ll d p '
( 79)
)
Indirect R adiation : T he methodology of analy sis for this type of radia tion is similar to that em p loyed with th e elliptical photoreactor and will not be repeated here . T he expre ssion re sultin g for the modulu s of the radia tion flux den sity due to indirect radiation is identical to Eq . ( 6 5 ) . T he limits for the p v a riab le are the same as those which were obtained for the elliptical photoreactor , that is ,
P2 , 1 B ut
=
D cos E; ±
(ri
sin 9
-
D2
sin 2 t:; > 1 12
( 80)
the meaning of t h e variables in this case is D
sin E;
=
=
(X2
L D
1
+
2 1/2 y )
L
( XL
sin cp E
-
( 81) yL cos
cp E )
( 82)
873
A nalysis and Design of Pho t o reactors
=
cos E;
( 83)
where
( 8 4) ( 8 5)
=
E
t an
=
G
-1 (1
2 G ) sin
-
ca 2
-
1 ) co s
2G c o s
-
2G
-
- a
2
The limits for expressions :
=
=
tan
tan
( 87 )
the
-1
_1
e variable are
Pi -
+
D
PI
+
cos E;
D
-
cos E;
represented (r
-
2
L
of
2a tan
2
-
(r
D2
2 L
by the followin g
sm 2 E;, ) 1 / 2 .
D2
.
s1n
( 88 )
2c-7 ) 1 / 2
-=---��,----=-------+ z1
LL / 2
an d
PI
( 8 6)
sin
-
2 [ a 2( 1
+
tan 2 1jl )
cos
+ a ( t.
t. tan
( 89)
112 ( 90)
To ob t ain t h e limits for the
the
( 91) Finally , it s hou l d be mentioned that in certain cases it m a y be neces sary to put a protecting cover over the emission system w hich can also be used to support the cylindrical reactor ( see Fi gs . 18 and 1 9 ) . I f this is the case ' there will be totally and p artially irradiated zones w ithin the re action space . Con sequently , for each incidence point I , w e have to deter mine which portion of the p arabolic reflector or source is capable of illuminating the point . From the mathematical point of view , the int e gration interval for each point I must be adjusted . To do this , the limiting an gles determined by th e circular hole of the lamp - reflector cover system must be calculated ( see Figs . 18 and 1 9 ) , and afterward co m p are d with the limit in g an gles defined for the inte gration of the source volume [Eqs . ( 8 8) to ( 9 1 ) ] . For more detailed information , the reader is referred to Alfano et al . ( 1 98 4b ) .
8 74
De B ernard e z ,
C larid,
and Cas sano
y
B
Illustrative rep resentation of chan ges needed in the limit s for due to the shape of the emitting system .
F I G U R E 18
SELS Mo del
Direct R adiation : The expres sion for this case is t he same as the one given for the annular reactor , but replacin g r by t:. with 1'1 given by ( 9 2)
:::
T he limits in the B coordinate can be obtained from Eq s . ( 7 5) and ( 7 6) making r 0 and L :::
y
X
FIGURE 1 9 I llustrative representation of chan ges needed in the limit s for
A nalysis and D esign of Pho toreactors
875
=
tan c/J
( 9 3)
Indirect R adiation : Following the same approach as the one derived to obtain I q I for the elliptical re fl ec tor , w e get
I <1 1
=
Kf
Rf
where el
92 p' I
-1 tan
=
tan
=
-1
4a ( Ll.
=
1
E
I na b i l i t y
e
I
I
+
LL / 2 p'
I
+
-
+ a)
[ ( L\. 1 -
of
p'I ) 2
1
+
p'
ex p
E
�-r·<•> ) p 0* ( 9)
" dp
' pE
LL / 2 + 4a
=
p'
d9
1
p'
abd p'
i e2
p'
z1
( 9 5)
E
zi
( 9 6)
L\. 2 2 +
Ll.
( 94)
( 9 7)
2 1 /2 2]
( 98)
the L i n ea r M od e l s to P r ed i ct t h e R a d i a t i o n
of the E m i tt i n g S y s tem
F i e l d W he n C u rved R eflec t i n g S u rfaces A re Pa rt
From previous theoretical developm ents it is clear that the linear models are much si mp l er to use t han the extense ones . The computing time is con siderably reduced , and consequently they should always be the starting equations for p hotoreactor design . Cerdli' et al . ( 1 978) and Romero et al . showed that for certain cases the S E L S model provides good results . For the annular type of reactor , w here only direct radiation is involved , errors are never larger than 1 5 % . A r e cen t work ( Clarili et al . , 1 98 6 b showed that under no conditions can line ar models be u sed if curved re flect ing surfaces combined with tubular lamps are employed . Thi s work con firmed experimental results o f Alfano e t al . ( 1984) , regardin g the validity of linear models when a reflector is u sed . We will not reproduce here the details of these papers , b ut we shall present a brief summary of their main results as well as of their conclusions . Fi gures an d 2 1 s ho w computed and experimental results for two reactions (chlorination of ethane and photodecomposition of uranyl oxalate ) carried out in an elliptical p hotoreactor . Plots include predictions with the S E L S model and the VEES mo del It is obvious that the error of the linear model is i ntolerab l e for any p ractical purpose . [ The details are given in Clarili' et al . ( 1 9 8 5b ; 1 98 6a , b ) . ] Table clearly shows that when only direct radiation is considered the differences betw een the linear model and the extense one are negligible .
( 1 983)
)
20
2
.
De Bernardez , C la ria, and C assano
876
'· .
...... .
LR
'·
....... .
30
% CL2 = 2 =
% EtH
-
•
.... .
'·Ill
. ....... . _
· -· - . •
-·-
em
4
S E LS
M odel
-· -· \l E E S
Mo del
/200
·-
.
1600
2000
F I G U RE 20 VEES m od el and SELS model t h eo re ti cal p re d ic t ion s comp ared with expe ri m ent al results for ethane photochlorination .
E ven though there exist con dit ion s w here this approximation is not good , it can be safely used for most engineering purposes (see C laria et al . , 1986b ) . Table 3 s how s experimental verifications p e rfo r m ed in a system which has a cylindrical reflector of parabolic c ro s s s e ct ion u n der t h e follo wi n g e x perimental con di ti on s : measurem ents of total and only direct r ad i ation w ere performed w ith a p r eci s e t h e r m opil e ; and they were done with the radiation probe receiving the sum of both d i rec t and indirect radiation and with the same sensin g device but t he reflector covered wi th a copper sheet completely masked with carbon deposition . The data were obtained for dif ferent positions in space and the results compared with predictions made the SELS model and the VEES m o d e l . In all c a se s previsions were taken in the com p u t ation of the radiation field to acc ou n t for the orientation and finite si ze of the sensor (thermopile) . C learly , in c ase s like this , t he linear models are u seless . [The det ails may be found in Alfano et al. ( 1 984) . ] Without going into the det ail s of the theor etic al proof of this in ability of the linear models to p redi ct reflected radiation , let us explain the reasons in an intuitive way . When the tubular lamp is placed p ar allel to the generatrix straight line of the cylindrical reflector and it i s modeled as a s t rai gh t line , the source does not " reco gni z e " the curvature of the reflect ing surfac e . H e nc e the con cent rati n g effect of the curved mirror , be it elliptical or p a rabo lic , cannot be predicted .
877
A nalysis and Design of Pho toreactors rR
% X ox
LR
c;x
20 10
0.2 •
12
em
em
= 2 .63
Cur = 0. 5 1 7
40 30
=
- SE L S
- · - VE E S
'
\
10- s M
1 0- J M
Mode l
Model
.
\ \
\
�
·,,
,........
· � . ......
·- · - . -- · ---
0 ���----�--_. 160 120 80 40 0
Q
FIGURE 21
VEES
model
with experimental results
( em1 min- • )
an d SELS model t h eo re tic al predictions for uranyl oxalate photodecomposition .
T AB LE 2 Comparison of the SELS and VEES Models for Direct Radiation ( Elliptical R e fle ct o r S ystem ) a
l gl
D
x
10 7
(einstein /cm 2 ·s )
rL / c
SELS
VEES
0. 018
0 . 031
0. 03 1
0 . 022
0. 047
0 . 046
0. 025
0 . 058
0 . 058
0 . 030
0 . 083
0 . 082
0 . 033
0 . 102
0 . 100
0 . 037
0 . 127
0. 123
0. 040
0 . 143
0 . 1 39
0. 05 0
0 . 2 15
0. 205
compared
De Bernardez . C laria , and
8 78
C as sano
T A BLE 3 Direct - to - T otal R adiation Ratio : C o mp ari s on of T he SELS and V E E S Models ( C ylindrical Reactor Irradiated from the B ottom ) a
C ontribution
C yli n d ri cal coordinate y r
=
1
=
1
13 1
y r
1
1
= =
y r
1
1 2 . 0 2 em
=
131 y
ri
1
=
=
aI
a
a,
( %)
Experimental values ( %)
69. 0
27. 7
29. 2
69. 3
28. 1
30 . 8
69. 2
28 . 0
31. 2
69 . 1
27 . 7
31. 6
1 2 . 0 2 em
= =
( %)
/4
1 . 69 em
1
Prediction of VEE S mo de l
2 . 80 e m
=
131
direct radiation to total r adi ation
Prediction of S E L S model
1 2 . 02 e m
Tr
of
0
2 . 99
em
0 1 2 . 02 4 . 35
= o
2. 1
em
em
em ;
R. , 8 . 4
em ; r
L
, 0 . 95
em ;
L
L
, 1 5 . 2 em ; lamp , GE UA 3 - 3 6 0 .
Depending on t h e significance of direct radiation on the total radiation flux density , the error m ay go from more than 1 0 0% , as in the case of the p arabolic reflector , to up to more than two orders of magnitude for the case of the elliptical one . In conclusion : for most practical p u rposes , when only direct radiation is involved , the linear models can be used . When reflectin g curved surfaces are p resent , only the extense source type of model can be used . The finite si ze of the radius of the tubular l am p must be incorporated in the model , and in this way the extent of the emis sion source will be able to recogni ze t h e curvature of the mirror and its concentratin g effect s .
P H O TO R EA C T O R D ES I G N :
For
APPLI CAT IONS
the photoreactor design problem stated earlier , all the theory has now been develop ed . In this section we show how it is possible to perform the design of a p hotoreacto r . T o do so , we have chosen four applications . The fi rs t deals with the d esi gn of an annular photoreactor with simple kinetics . The methodology is applied to a bench - s cale reactor and the
A naly sis and D esign
-
879
of Photo reac tors
theoretical results are comp ared with experimental data . T he second is a gas ph a se photochlorination , a typical chain reaction . Again the t heo ry is p rese nt ed and app li ed to a b en ch sc a le photoreactor inside an e llip tica l R es ults are als o comp are d with experimental d ata A t hi rd refle ctor p roblem d e al s with a heterogeneous reaction . Problems related to the gas absorption , t he reaction rates , the radiation field i n a t wo p has e s y s t em and the mixin g states must be tak e n into account to formulate a ri gorous model . T he method is ap p li e d to a chain reaction and one must search for simplifications that could reduce th e d e si gn p roblem to a tractable one . Once more , p re di ct io n s for a b e nch s ca l e system are com p ar e d with experi mental results . Finally , the op ti m al design of an an nu la r reactor for con secuti ve p hotochlori nation s is illu st r at ed in the fou rth ap p lication . Here the
-
.
-
.
-
whole p roce s s of desi gnin g a p hot o reac to r is present e d in de t ail
,
,
starting
with the information that a p racticing en gin ee r would have at hand to begin work . with A l mos t Mon oc h romatic R ad i a tion
Simple K i netics i n a C on t i n uous A n n u l a r Photorea cto r
uranyl oxalate flowing through an annular reactor irradiated by an ultra
Consider the isothermal steady - state p hotodecomposition of a solution o f
violet arc lamp located at it s
axis .
-
For the b e n ch s cale lamp and reactor
.
characteristics listed below , we wish to study the incidence of th e volumetric flow rate on the ox ali c acid exit con version Reactor characteristics : rin = 2 . 3 em ; r0u = 2. 7 e m ; L R = 3 0 e m . Lamp characteristics : GE G 1 5T 8 ( 15 no mi n al watts) ; L L = 43 e m ; rL = 1 x I0- 6 m ol / cm 3 , 1. 25 e m Ur anyl oxalate initial com p ositio n ( U0 2 S0 4 ) :
.
.
-
H 2C 2 0 4 : 5 x 1 0- 6 m ol / cm 3 The u ranyl oxalate photochemical decomposition is a we ll know n chemical actinometer . c hem i s ts have found that the rate of oxalic acid d ecomposition is p rop ortion al to the local volumetric rate of energy absorption , w h ere the proportionality factor is the overall quantum yield , It re prese n t s a good example of a p hot o c he mi cal reaction that can be m odele d using an overall kinetics . To find the relationship between the oxalic acid exit con version and t he volumetric flow r at e , we must solve the mass b al an c e equation for the oxalic acid .
·
We can introduce t he follo w in g assumptions and /or si m plifi cation s in the general eq uation : ( a) n e gligib le thermal effect s ; (b ) steady state ( 3 C 0x / i H = 0 ) ; ( c ) physical properties such as velocities , diffusivities , and d en sities are constant th rou gh the reaction volume ; ( d ) axial l amin ar flow ; (e) N e w to ni an fluid ; ( f) azimuthal symmetry ; and ( g) axial diffusion negli gible w hen co m pa red with the convective flux . If the hypoth e si s above are fulfilled , t he mass balance equ ation for the oxalic acid red uce s to ac
v ( r) � Z
a
Z
=
D
OX
- ( 1
... " -r ar
ac r � 3r
)
+
n
OX
( 99)
with the following boundary conditions : v- , z
ac
ox ar-
=
0
( 1 0 0)
De B ernardez , C laria, and
880
r
=
r
and
z
ou '
z =
V. r,
ac
ox ar
v- ,
c
0,
=
0
( 1 0 1)
= c ox o
ox
Cassano
( 102)
We assume that the velocity profile is fully developed ; for example , allowing the existence of an entrance region , we can express z
v (r)
=
2 < v>
1 - ( r / r. > 1n
2
+
-
[( 1
2 x >
/ln ( 1 / x ) J ln ( r /r. ) 1n
__:;;o"-------------'---
---
1 + x
2
-
[(1
-
i > tln ( l / x ) J
( 1 0 3)
w here x = rin / rou · Finally , to solve the mass balance equation , we need an expression for the reaction rate nox · We know that n0x can be expressed by n
=
OX
- 4l
OV
e
a
( 104)
Now we need some data about the absorption characteristics of the uranyl oxalate solution . Light is absorbed by the uranyl oxalate complexes , which are decomposed into w ater , carbon dioxide , carbon monoxide , and formic acid , w hile the uranyl ion remains unchanged . T he overall reactions are uo
2+ 2 2+
U0 2
+
H
+
H 2c p 4
2
c 2o 4
+ hv
+ hv
-
uo
---
uo
2+ 2
2+
2
+ co
+ C0
2
2
+ co + H o 2 +
HCOOH
T he relative significance of each reaction path depends on the p H of the solution . Nevertheless , the overall rate of oxalic acid decomposition is p ro portional to the LVREA and independent of the reaction p ath . Ei ghty - five p ercent of the radiation power outp ut of the lamp falls into a single w avelength ( A 2537 A ) T he absorption characteristics of the uranyl -oxalate complex show the following significant variations =
A (A)
lJ
( em
-1
)
4l
ov
•
( mol /einstein )
2250
16. 71
0 . 56
2950
1 . 96
0. 57
3450
0 . 17
0. 51
I t is then clear that the reactor can be safely modeled as operating with monochromatic radiation . The chemical nature of the absorbing species changes with the com position of the solution . However , a useful data for solving the modeling p roblem would be a plot of the attenuation coefficient versus the ox alic acid concentration , at constant uranyl ion concentration ( Fi g . 2 2 ) . The
881
A n alysis and Design of Photoreacto rs
8
4
0
0
2
4
Cox (mole
cm- 1 ) x 1 0•
6
Attenuation coefficient as a function of the oxalic acid concentration . FIGURE 2 2
plot show s that the attenuation coe fficient is not a linear function of the oxalic acid concentration , w hich agrees with the fact that the oxalic acid is not the actual absorbing species . T he exp erimental data were fitted to give an analytic al expression for the concentration dependence of the attenu
ation coefficient : J.l ( e m
-1
)
valid for C ur
mol /cm 3
=
=
2 . 362 + 3 . 3 7 2
1
x
•
10- 6
X
mol /cm 3 ,
to 6 c ox A
=
( 1 05)
2 5 3 7 A , and C ox expressed in
Figure 22 show s that when the molar ratio of oxalic acid to uranyl ion is 4 or greater , t he attenuation coefficient is almost constant (i . e . , practical ly independent of the oxalic acid concentration ) . Hence , starting with a solution of co mp o si tion uo 2 so 4 1 x mol /cm 3 and H 2 C 20 4 5 x mol / c m 3 if t h e reaction proceeds wi t hou t exceeding a maximum p ermi ssible conversion value of 2 0%, the attenuation coe ffi ci e nt can be considered to be cons t an t durin g the reaction . This characteristic of the actinom eter is generally pointed out in the literature without further analysis . Although a constant value of the at ten u atio n coefficient allows us to si m plify calcula tion of the local volumetric rate of energy absorption , we are not able to use this p rop ert y in modeling a continuous , se gregated flow reactor . In the s e reactors , as a result of the unavoidable concentration gradients , there are regions where the requirement of maximum 2 0% conversion may be violated al t ho u gh the average exit conversion does not exceed that value . Fi gure 2 2 also show s that t he value of the attenuation coefficient is greater than 0 . 4 cm- 1 for the w hole ran ge of oxalic acid concentration . Hence , co n si de ri n g the criterion d raw n earlier for the absorption p rocess in heterogeneou s media , we can n e glect the distortion effect produced by the p r ese n ce of co and CO bubbles .
to- 6
,
2
to- 6
882
De B e rnardez ,
C lari.a,
and Cassano
T he oxalic acid mass balance rep resented by Eq . ( 99) to ( 1 0 2) cannot be solved analytically . Any of the numerical methods recommended for parabolic differential equations can be used , com bined with an appropriate integration technique ( Gauss , Simpson , etc . ) for calculating ea . Since the GE G 1 5T 8 is a nonfluorescent ultraviolet arc lamp , the ap propriate model to describe the radiation field inside the reactor is the VEES model with isotropic emission . However , as pointed out earlier , the SELS model can also be used as a good app roxim ation to p redict the radia tion field inside an annular reactor because only direct radiation is present . Using the SELS model with isotropic emission , t he computing time required to evaluate the LVREA is lessened . Either with the VEES model or with the SELS model the solution strategy must account for the functional characteristics of the L VREA (i . e . , in order to evaluate it we need to know the oxalic acid concentration field inside the whole reactor volume ) . Generally , the numerical procedure follow s the direction of t h e flow ; in this case , when solvin g each point , we should com pute bundles of radiation t raveling through regions already calculated ( up stream section) togethe r with bundles coming from regions w hose concentra tion field has not been calculated yet ( dow nstream section ) . Then , to solve the p roblem , we must follow an iterative scheme , until reaching a " steady state" oxalic acid concentration field . Finally , after finding the " steady state" concentration field , we estimate the oxalic acid exit conversion as follows :
X
ox
1 -
ox
>
---
co
( 1 06a)
ox
w here ( 1 06b )
Figure 2 3 show s a p lot of the oxalic acid exit conversion as a function of the volumetric flow rate calculated using the S E L S model together with Eq . ( 1 0 5 ) for the attenuation coefficient JJ ( solid line ) . In the same figure experimental results obtained in our laboratory are depicted . T his example show s that it is possible to perform an a p riori design of a photochemical reactor if w e use an app rop ri ate emission model to estimate the radiation field . The agreement between the experimental results and the theoretical p redictions show n in Fig . 2 3 corroborates what we have said . Later we will see that an a priori design of more complex reactions and reactors can be p erformed wit h a similar degree of accuracy . For more detailed information about the example , the reader may refer to De B ernardez an d C assano ( 1 9 8 5) . C o m p l e x K i netics i n a n E l l i ptica l P hotoreactor w i t h Polych romatic R a d i a tion
A gaseous mixture of N 2 , Cl 2 , and C 2 H s is fed into a bench -scale elliptical photoreactor (described in Fig . 2) whose characteristics are listed below . For the same assumptions made in the p recedin g example ( steady state ,
883
A nalysis and Design of Pho toreactors
�0 "'o X ox
25
0
/00
0
300
200
400
FIG URE 2 3 Comparison between theoretical p redictions and exp erimental results for the p hotodecomposition of oxalic acid . E ffect o f the volumetric flow rate .
laminar flow i sothermal operation , etc . ) w e wish to study the in ci d ence of the feed composition on the chlorine conversion for low - percental values of the chlorine frac tion i n the feed . ( T hese conditions may be convenient to overcome undesirable thermal effects and avoid side r eactions and conse quent ly imp rove p roduct selectivity . ) R e actor characteristics : LR 3 0 em , r R = 0 . 2 em ; reflector character istics : L R f = 5 9 em , a = 5 3 . 5 em e = 0 . 4 ; lamp characteristics : GE G 30T 8 ( 30 nomi n al w atts ) L L = 8 1 . 3 em , rL 1 . 25 em . Under unrestricted design con di tions the existence of di rec t radiation and umbra and penumbra zones for the reflected one originat es an gular asym metries in the radiation field . They induce the sam e problem in the concentration field . How ever , it has been shown ( De B ernarde z and C assano , 1 9 8 2 ) that w hen the ellipse e cc entri cit y is low ( :t;;; O . 4) and t he ratio of r R / rL is small ( < 0 . 5) azimuthal asym m etries inside the reactor can be safely neglected . T hi s condition implies that a three - di mensional mass To balance can be avoided , m aking computations considerably simpler . take advant age o f this convenience , t h e conditions above have been fulfilled in this example . R e and C as sano ( 1 986) showed that under normal practical conditio n s ( r R > rL ) , the three - dimensional modeling of the reactor is almos t indespensable With this exception only , the desi gn method here described can be applied to any reactor si ze an d confi guration . T he p roposed low chlorine concentration in the feed , to gether with the presence of an inert gas , allow us to safely assume that : ,
=
,
,
=
,
.
1.
2.
The m ain reaction p roducts will b e hydrochloric acid an d mono chloroethane . T his ass um ption can be easily confirmed with experi mental results . U nder the stated operatin g conditions thermal e ffects will be small . Hence isothermal behavior is applicable .
De B ernarde z , C !aria, and C assano
884 T he overall reaction i s
S everal research groups have made detailed studies of the chlorination kinetics of saturated hydrocarbons . T h e y have p roposed similar reaction mechanisms . Gosselain et al . ( 1 956) i n dica ted a general m echanism for the gas -phase halo genation of saturated hydrocarbons for both thermal and p h o toche mi ca l reactions that in c lu d e the possibility of usin g nonpolar sol vent s . T he re ac tion m echansim has all t h e possible reaction steps that are feasible u n d er low chlorine concentrations and temper at u r es below 600K . a
Cl 2 � Cl •
+
EtH
I nit iatio n
2Cl •
�
k
CIH +
Et •
p- 2 Propagation step s
C1
2
+
Et
•
� k
cmt
+
Cl ·
p-3
Cl •
+
Cl • + M
Cl •
+
Et •
Et •
+
Et •
k
___!!_._
Cl
2
+
M
k
__!!_. k
Homo geneous termination step s
C lEt
t6
- c 4H to
The author s also report ed that hetero ge neous term inatio n reactio ns ( at the wall) may be safely ne gl ecte d . T his was also precisely shown by C laria et al . ( 1 985a) . U nder these conditions , ei ght species are present , w hich will be identi fied , for simplicity , in t h e following m anner : Species
Cl
Number
1
2
Cl
EtH
EtCl
Et
HCl
2
3
4
5
6
C H 4 10
N2
7
8
From the momentum balance under the described op erating conditions ( fully develop ed laminar , Newtonian and imcompressible flow ) , we have
( 1 07) N eglectin g thermal effects , we o nl y need the radiation balance for the and t he m as s balance equation s . T hey will be derived under the
L VREA
885
A nalysis and Design of Pho toreactors assu mpt
io n s
T he m as
v
with
an d si mplifi cations described s balance for s p ecies i is
a c. z (r) a z
1
z =
Dim -- (r -- ) --
=
r
¥- ,
0,
r
=
v.- ,
r
=
z
z
a
ar
c
ar
i
+ n. 1
( 10 8)
( 1 0 8a)
C. = C?
r
V- ,
a
in the p revious application .
a
0,
1
1
=
c.
1
ar
a
rR '
( 108b )
0
c.
1
=
ar
( 1 08c)
0
If heterogeneous reactions were p resent , at r = rR should be ch an ged to
the boundary
the specie s V- ,
where the
all the heterogeneous st ep s w here is the a he te neo us termination j in the h het ero c coefficient
sp ecie s j p artici p at es ; fl h
summation m ust include r te of t h h e ro ge step and the s t oi hiome t ri c of species geneous step . Eq . ( 1 0 8 ) for eight homogeneous step s i lu d mechanism we have
Vj h is
In
the
=
Hence the
n n
1 2
n3
=
=
=
nc
ed
in t he reaction
r= 8
L
( 1 0 9)
virnr
r=1
rates of -
2
a
a
-
formation
k
of
s p e cie
s
c c + k 4c c t 8 P3 1 5
kP 2c 2c 3
+
i
are
; + kP _ 3 c 2c 4
k _ c 5c + k c c P3 1 5 6 p 2
2 2k c c - k c c -k c c t5 2 5 t4 2 s p-3 2 4 -k c c p2 2 3
+
-3
k - c sc s p 2
=
k
c c p3 1 5
k
ns
=
k
c c p2 2 3
k - c c p 2 5 s
2 2k s c s + t
p
kp - 3c 2c 4
( 1 1 0)
( 1 1 1)
( 1 1 2)
c c 4 + k 5c c 2 5 t 2
4
n
for
( l OBe ' )
z
ni
co nditio n s
- kp 3 c lc 5 - kt 5c 2c 5
( 1 1 3)
( 1 1 4)
886
De B e m ardez , C laria, and C assano na
= =
( 1 15)
kp 2 c 2 c 3 k
2
c ts s
( 1 16)
De B ernarde z and C assano ( 1 986) an d C laria et al . ( 1 985a) showed that in these system s the local or microscopic steady - state approxim ation for intermedi ate unstable species can be applied wit hout error . Hence Eq . ( 1 1 1) and ( 1 1 4 ) can be set equal to zero : n.
1
=
o
for
i
=
2 and 5
( 1 17)
T hese two algebraic equations are still coup led with the mass balance equations throu gh the reaction rate term . In addition , Eq . ( 1 1 0 ) and ( 1 1 1) clearly show the coupling of the mass balance equations with the radiation balance throu gh the rate of the activation s tep . Since a curved reflectin g surface is present , extense models (VEES or SEES ) must nece ssarily be used . On the other hand , the GE G 3 0T 8 is a n on fl uorescent ultraviolet ( U V ) arc lamp , so the app rop riate model to describe the emission p rocess is the VEES with isotropic emission . T he reacting system is homogeneous ( gas phase ) and chlorine is the only ab sorbing species in the w avelength of lamp the emission of the lamp . Hence =
( 1 1 8)
w here a. v is the absorption coefficient of chlorine . From the emission characteristics of the lamp ( see T able 4) app roxi mately 90% of the output energy falls into the UV re gion . Out of this amount , about 95% is emitted in a single w avelength ( 2 5 3 7 A ) . However , we will not consider monochrom atic radiation because the ab sorption by chlorine in this line is low , having instead much hi gher values at lon ger w avelengths . T hu s
e
a
=
=
( 1 1 9)
Since the tot al energy flux density vector results from the contribu tions of direct a n d indirect radiation ( the latter with o n e o r more reflec' tions ) , in Eq . ( 1 1 9) we have ( 1 20) When the hypothesi s of azimuthal sim m etry is used , it can also be show n that if c is sufficiently lar ge , only in di rect radiation w i t h one re flection is significant H ence we can safely consider only this cont ribution . Its expression w as derived earlier [Eqs . ( 40 ) and ( 65 ) ] : .
887
A nalysis an d D e s ign of Pho toreactors
T AB LE
4
S pectral Distribution of Power
Output for the GE
Wavelength
G 30T 8 E). X 1 0 7 ( einstein / s )
E ). (W)
interval ( A ) 2200- 2400
174. 20
2400- 2 600
8 . 34
2600- 2 8 0 0
0 . 01
0. 226
2800 - 2 90 0
0 . 02
0 . 476
2900- 3000
0. 03
0 . 7 39
3000- 3 1 0 0
0 . 02
0 . 510
3100- 3200
0 . 16
4. 2 1 1
3200- 3400
0 . 01
0 . 276
3400- 3600
0 . 01
0 . 292
3600- 3800
0 . 17
5 . 255
3800- 4 0 0 0
0. 01
0 . 326
4000 - 5 0 0 0
0 . 66
24. 814
500 0 - 6 0 0 0
0 . 33
15 . 164
600 0 - 7 0 0 0
G en er al Electric
Source :
e
a
( 1967) .
Co .
= ( 1 2 1)
For the w avelengt h ran ge of this application , the quart z transmittance , average reflection coefficient and the p rim ary quantum yield are con sidered independent of v . For the latter , in a ddition its value is t aken equal to 1 . In Eq . ( 1 2 1 ) , only E L \1 ( lamp sp eci fic p ow er output ) and ex \I (chlorine , absorptivity ) are w avelen gth de p e n d e nt . Hence we may w rite
,
the
X
f \)::oo v=O
dv E
L , v \1 ex
exp
(-
p * ( S , cj> )
Jp�( S , ¢ )
a C1
v
d p'
)
( 122)
De Bernardez , C larUi , and Cassano
888
(
Let us separately analyze the wavelengt h - dep endent inte gral :
A
=
=
v =oo fv=O v L,v v
exp -
f
exp
d
).=oo
). = 0
a
E
d ). EL ).
'
v
a
(-
fp*(S,
p (
p * ( S ,
Ip�( S ,
O' C d p ' v 1
a). c
1
dp'
) )
( 12 3)
( 1 24)
( 1 25) In Eq . ( 1 2 5 ) the w avelength interval has been reduced to the region in which we simultaneously have si gnificant absorption by chlorine and emission from the lamp ( see T ables 4 and 5 ) . F u rthermore , con sid ering that the absorption by choroine is continuous ( H erzberg , 1 95 0 ) and the lamp emission is discrete in very well defined lines , E q . ( 1 2 5 ) can be written
T AB LE 5 ).
{A )
C hlorine
A b so rption Coefficient
[ ac 1 1 J1.n X 1 0 2 ( c m 2 /mol)
-3
).
)( 1 0 -
[ aCl ] Jl.n 2 2 ( cm /mol) 41 . 1 3
2 450
1 . 08
3750
2550
3 . 33
3850
2 3 . 49
3950
14. 23
2 65 0
4 . 96
2750
10. 62
4050
9 . 89
28 5 0
2 8 . 78
4150
7 . 22
2 95 0
5 5 , 98
4250
5 . 73
3 0 50
92 . 3 3
4350
4 . 09
3150
12 7 . 33
4450
32 5 0
145 . 46
4550
3350
146. 59
4650
1. 34
3450
123. 11
4570
0 . 88
3 5 50
9 3 . 47
4850
0. 66
4950
21
36 5 0
Source :
64 . 5 1
C alvert and Pitts ( 1 96 6 ) .
2 . 94 2 . 02
o.
3
A nalysis and D esign o f Photoreactors
n A
=
I:
f d A.
E
i=l
. o ( A.
L ,1
n =
I:
E
L
i= 1
.a. ,1 1
exp
-
(- i
A.. ) a. exp 1
1
P *< e , .P >
P *< o
8 , .P >
(- J
889 d p ' aie l
p *( 8 , cjl )
P6( 8, cjl )
dp' a.c 1
1
)
)
( 126)
( 1 2 7)
where n are all the lamp sp ectral line s of emission and the summation must taken over all the n values were a i '/ 0. o is the Dirac function ; E L , i and a i are the values of EL and a at A. = A. i . In goin g from Eq . ( 1 2 6 ) t o Eq . ( 1 2 7 ) we h ave m ade be
=
E
. -
L ,1
cS
(A
A.)
( 128)
1
T his is equivalent to substitute a polychromatic radiation source b y n hypothetical monochromatic devices , which simult aneously occupy the s ame physical space . Each o f them emits a single frequency w hich corre s p o nds to one of the emis sion lines of the p olychro m atic source . The mass balance equations are solved in a r - z plane . However to evaluate the L V R E A we must consider bundles of radiation coming from every direction in space and from the w hole volume of the lamp . Con sequently , rays not belonging t o the r - z plane should al so be considered in the calculation , T his makes it necessary to evaluate the real di stance traveled by each energy bundle and the sequence of concentrations found along its p ath ; considerin g the a zimuthal sym metry , it is possible to find the required. information by performin g a cylindrical p rojection on the rays on the plane r- z ( Fi g . 2 4 ) . [ For further details see Romero et al . ( 1 9 8 3 ) . ] Substituting E q . ( 1 2 7 ) into Eq . ( 1 2 2 ) and s olving E q . ( 1 08) with Eqs . ( 1 10) to ( 1 1 7) w e can treat the p roblem with any conventional numeri cal procedure . We should only recall that to obt ain the I q I value at
_v
each point , we need to u se the m ethodolo gy indicated in Fi g . 2 4 and the computation of t he radiation field as described earlier . T he solution pro vides the concentration field for each sp ecies i. Finally , the averaged
exit conversion is defined as
X
( 1 2 9)
=
and the value of < C 1 > m ay be obtained as it w as illustrated in the pre ceding application [ Eq . ( 1 0 6b ) ] , Clarili et al . ( 1 9 8 5a , b ) solved this system and compared the predicted values with experiment al results obtained i n a b e nc h sc ale reactor . Some of t heir results are show n in Fi gs . 25 and 2 6 , w here t he incidence of the feed composition upon the chlorine conversion is analyzed . Con si d erin g that no adjustable p arameters have been used and the < T > ' and E L , v a re taken from the specifications p rovided values of < r R by the manufacturin g companies , we think that t he results are extremely
-
Rr '
good .
z
''
'
'
''
''
r
Cylindrical projection to compute the three-dimensional attenuation in a two - dimensional grid .
FI GURE 24
80
%X
Re = 600
"to E t H
-- = 2 "'o CL2
60
40
20 �------�--�L---� 0 4 2
Comparison of theoretical predictions and exp erimental re sult s for the chlorination of ethane . Effect of the chlorine inlet con centration . FIGURE 2 5
890
891
Analysis and Design of Pho toreactors 80
%X
Re . 6 00 % C l2 = 3
60
40
20 L-------�---L--��6 0 2 4
0/o
Et H
FIGURE
C omp arison of theoretical predictions and experimental results 26 for the chlorination of ethane . Effect of the ethane inlet concentration .
Reacto r I r rad i a t ed from t he Bottom
C o m plex K i n e t i c s i n a H et e rogen eo u s S em i b a t c h
The photochlorination of trichloroethylene is carried out in a semibatch stirred tank reactor irradiated from the bottom by a tubular source located at the focal axis of a cylindrical reflector of p arabolic cross section . The liquid phase is a homogeneous solution of trichloroethylene and carbon tetrachloride , while the gas phase is a mixture of chlorine and nitrogen . The overall reaction is
( 1 30) We wish to analy ze ( a) t he possibility of transforming all the C 2H c 1 3 initi ally p resent in the liquid phase into C 2 H C l 5 only , and ( b ) the effect of the inlet nitrogen concentration upon selectivity , conversion , and re action time . The modeling problem must be consider the following aspects : the radiation ener gy distribution in side the reactor , the reaction rate , the mixing pattern , and the gas absorption rat e . Since the reaction is con sidered to occur only in a p erfectly stirred liquid phase , the overall phenomenon involves the following elementary steps :
De
892
1. 2. 3.
B ernarde z , C larid, and
Diffusion o f chlorine from the bulk of the gas
p hase to
face , w here physical equilibrium is assumed Diffu sion o f chlorine from the interface to the p h as e C hemical reaction wi thi n the liquid phase
Cassano
the
inter
bulk of the liq uid
It i s useful to point out here that the chemical reaction p rocee ds through through a chain mechanism ; hence not only are stable species p resent but highly reactive intermediates as well . Since t h e liquid phase is considered to b e perfectly stirred , we can vi s uali z e the followin g mixing characteristics: 1. 2.
3.
T he concentration field of the stable species is uniform i n si de the whole reactor volume . T h e rate of initiation s t ep (e a ) i s no longer a functional o f the concentration field a s in the previous applications b ut is still spatially dependent T he spatial distribution o f the radiation field together with the short lifetime of the highly reactive intermediate species suggest that concentration gradients of these species should be expected . Moreover , they can be considered to be born , live , and die in the same place . Alfano an d C assano ( 1986a , b ) show ed that this characteristic corres pon d s to reality , and p e r fect mixing for the short -lived intermediat es c an not be assumed . Even thou gh the liquid phase is p erfectly sti rr ed and concentration gradi ent s of t he stable s p e ci e s are not p r e se nt the reaction rate is a fu n cti on of p o sitio n due to the s p ati al distribution of the radiation field and the short-lived intermediate concentration field . H ence the mass t ra n s fer (with chemical reaction ) coefficient may be spatially d epe n d e n t ,
.
For
si mplicity , the s p eci e s ill
be
identified in th e
followi n g
w ay :
Cl · 2
1
4
3
5
6
7
8
9
An order-of-magnitude an a ly si s f the charac te ris ti c reaction time and the characteristic ab s o rp tion time in the liquid film show s that
( 1 31)
and
t
i cf
rc
1
= ----7��-------if ) nl(C 1 , X = X �
=
l Os
( 1 3 2)
�max
w he r e we have estimated trc at position where e a is maximum .
the interface
conditions
and i n
the
spatial
A nalysis and Design of Pho to reactors
893
This means that the reaction is slow compared wit h the diffusional rate . T his physical situation was characteri zed in the literature as the slow reaction regime (Astarit a , 1 96 7 ) and in this case k
L
: k0
( 1 3 3)
L
That is , the mass transfer with chemical reaction coefficient is equal to the physical mass transfer absorption coefficient and hence independent of the spatial position . The chlorine ab sorption rate is
( 1 34)
where t
- -1-
ed - k0 a L v
cii is
=
effective diffusion time
the chlorine concentration
at t h e
( 1 35)
gas -liquid phas e , and c1 is the
chlorine concentration in the bulk of the liquid phase . The mass balance in the gas p hase is posed under the followin g assump tions : ( a) p seudo steady state , ( b ) no reaction in the gas phase , ( c ) negligible gas -phase resistance t o mass transfer , ( d ) well- mixed gas phase , and ( e ) no accumulation or si gnificant reaction in the liquid film . Hence we can w rite i
F 0 - F ::: N V = k 0 a vV ( C f 1 L 1 1 1
C )
1
=
l ( Cif 1
t
ed
C ) 1
( 1 36)
Assumption c must be made mainly because no reliable correlations can used to p redict the m as s transfer rates . I t is usually made in problems related to gas absorption with chemical reaction in spite of the fact that we known that even i f the gas - side resistance to mass transfer may occasionally be negligible , it is never zero . Since the inlet gas stream is a mixture of chlorine and nitrogen , be
=
If
=
( 1 37 a )
we assume equilibrium conditions at the interface ,
( 1 3 7b)
De B e rnard e z ,
894
C larid,
and Cassano
S olvin g Eq . ( 136) together with E q . ( 1 3 7) we obtain
( 1 38 )
Finally , th e mas s b alance eq u a tio n s in the bulk of th e liquid phase are
( 1 39a) dC.
1
(t)
dt
=
1 -
< n. ( x , t ) > ( t )
i =
2,
•
•
•
•
5
j = 6, 7, 8
( 1 3 9b )
( 1 39c)
where we have considered p e r fe c t mixin g for t he stable species and non mixing for the highly reactive intermediate species . The s ys t e m of represented by Eqs . ( 1 3 9) must be solved · together with the following initial conditions : 1
C . ( O) ]
C . ( O)
1
=
C�
i
=
1,
=
0
j
=
6, 7. 8
•
5
T he reaction scheme involves the I nitiation : a
CI 2 �
2Cl •
( 1 40a) ( 140b )
following step s :
( 1 4 la)
Propagation : k 2 C HCI + Cl • - C HCI4 • 2 3 2 k"' 2
( 1 41b)
( 1 4 lc)
( 1 4 ld )
( 1 4 1e)
A nalysis and D esign of Pho to reactors
895
Termination :
( 1 4 1 f)
( 1 4 1g)
( 1 4 1h )
products
Cl •
+
C
k lO
2c 15 - p roduct s
( 1 4 1i )
( 1 41j )
( 1 41k ) where t h e possibility of w all reactions has been disre garded . Considering that the rate of each step of the mechanism has the form of the mass action law , and ap p lyi n g the local or microscopic steady -state approximation for the highly reactive intermediate species together with the long- chain approximation , we obtain
( 1 42) ( 1 4 3)
( 1 44) ( 1 45)
n
6
=
n
7
=
n
( 1 46 ) 8
=
o
( 1 47)
and ( 1 48)
( 1 49)
896
De Bernardez, C Zari
( 1 5 0)
To complete the reaction rate expressions , the rate of the initiation step (
v
+ Cl
B
A
2 - H Cl
C
t
w here the first path ( A -+ B ) is an ad di ion reaction and the second path I t i s known that the characteristic ( B -+ C ) is a substitution reaction . time for both reactions should be different , that is , t << t -+ " Under A -+ B B C this condition it would be possible to obtain hi gh selectivity values for the intermediate product B ( C 2 H Cl 5 ) , as it is observed in homogeneous con secutive reactions . However , with single- phase system s , only· in exceptional situations could we expect to achieve 1 0 0 % selectivity . In a heterogeneous medium it is possible to improve selectivity by operatin g on the process con ditions so as to ch a nge the rat e - controllin g step as follow s . Let t A -+ b e the characteristic r ac ion time for the kinetic path A -+ B , B and t + be the one correspondin g to the path B -+ C . Let t be the B ed -1 e ffecti diffusion time calculated as (kL ay ) an d t D the characteristic diffusion time estimated with Eq . ( 1 3 1 ) . Un de r the slow reaction regime (t << t ) , the followin g possibilities are present ( A starita << t B -+ C Dif A -+ B et al . , 1 983) :
e t
C ve
1.
2.
if
<< t << t B -+ " B oth reactions are under the kinetic sub ed A-+ B C regime an d the overall expected behavior would be similar to that observed in the homogeneous system A + B + C . t << t << t + . The first p a A + B is under the diffu B C ed A+B sional subregime w hile the second is under the kinetic subregime . In this case it would be possible to obtain 1 0 0 % selectivity of C 2 H C l 5 l 3 because in p ractical terms with almost 1 0 0 % conversion of C 2 reactions A + B and B + C do not take place simultaneously . t A -+ << t " Both paths are under the diffusional sub << t B B -+ C ed regim e . However , for the reacting system under analysis and the type of reactor bein g od ele d (perfectly mixed stirred tank reactor ) , condition ( 3 ) cannot b e practically achieved . Hence this situation will not be analyzed here . t
th
HC
3.
m
897
A nalysis and Design of Photoreac to rs tank reactor under analysis .
Let us suppose that condition ( 2) is fulfilled in the semibatch s tirred Hence the reaction would proceed under the diffusional sub r egi m e until almost 100% s elec tivity of C 2 H C l 5 ( B ) to gether with almost 1 0 0 % conversion of C 2 H C l 3 ( A ) are attained . After that , the overall behavior c ha n ge s from the diffu si nal to the kinetic subregime and reaction B � C becom es more signi fi cant . If B is the desired p roduct the reaction operation s hould be frozen (by tu rning off the lamp) b e fore the transition from the diffusional the kinetic subregime takes place. U nd e r the diffusional subregime we expect the following cha ac teristics for t he concentration dist ri b utio n :
o
to may
1. 2. 3. 4.
r
Dependent o n the intensity o f m i xi n g o f the liquid phase Pr portional t o the interfacial area I ndependent o f the liquid volume Independent o f the emission system used to activate the reaction
o
T he latter characteristic allow s us to use a low - energy emis sion source , decreasi n g t h e op e rati n g cost s . th ere is a minimum permissible value of the ener gy ou tput below which t h e characteristic reaction time of the p ath A � B becom es greater than the e ffective diffusion time and the overall b e h avi or cha ge s the diffusional sub r e gi m e to the kinetic one [ i . e . , from c a s e ( 2) to case ( 1) ] , reduc in g t h e pos sib ilit y of achievin g 1 00% selectivity of C 2 H C l 5 with 100% conversion o f C 2 HCl 3 . To ill u st ate q u al itative analy sis we will report some of the re sults obtaine d by Alfano and C a s san o ( 19 8 6a , b ) , wh b e ch scale e xp eriments in a photore actor with the fo llo wi n g characteristics . Re act or : radiu s , 0 . 1 4 5 m ; height , 0 . 2 0 m ; bottom material , or Pyrex R eflector : glass . L amp : Ge U A 3- 3 60 ; radiu s , 0 . 0 0 95 m ; len gth , 0 . 1 5 2 m . characteristic const an of the parabola , 0 . 0 2 1 m ; distance from ve rtex p a ab oli c reflector to bottom of reactor , 0 . 084 cover radius , 0 . 0 1 3 m . Op e rati n g condition s : t em pe ratu re , 2 9 8 K ; p es sure , 1 0 1 . 3 kPa ; volume liquid , 0 . 002 m 3 ; stirrin g speed , 800 rpm ( no mi n al ) Theoretical p r e di cti n s were performed with the VEES m o del with iso tropic emission ( the GE U A 3 - 360 emis sion source is a nonfluorescent UV arc lamp ) a n d u sin g Y okot a et al . 's ( 1981) c r elati on [ Eqs . ( 38) and ( 39) ] to evaluate the effective attenuation coefficient . It is interesting to point out that any emission model can be u sed w hen the reaction is under the dif fu sional s u bregim e but only extens e s ou rc e mo dels ( VEES or SEE S ) can be safely u se d under the kinetic su b re gime . The exi stence of both subregimes needs to be investi gated to find the real possibilitie s of a ve ry hi gh selectivity . T o d o so , it i s necessary to c h an ge the m ass transfer characteristics of the system ( to search for the diffusional s u b regim e ) and the kinetics o f the reaction by changing the radiation field , for example (to search for the kinetic sub re gim e ) B oth t est s can be computationally and experimentally p erformed and w ere carried out by Alfano and Cassano ( 1 98 6a , b ) . T heir results showed that the first reaction w as sensitive to the stirring speed , w hile the second one ( starting from pent achloroethane ) was almost insensitive to changes in the mass trans fer characteristics , but indi cat ed a n oticea ble response to chan ge s in t he li gh t trans mi s sion properties of the r ea ct r b otto m C omputa tio n al and e x perimental re sults clearly support t h e i d e a of two different reacting condi tions . The d et ails can be fo u n d in t he papers cited .
However ,
n
r
r
from
the previous
o carried out quartz
t
r
of
m;
of
.
o
or
.
o
n
.
898 ..
Q JC -.., I
D e B e rn ard e z ,
C larid ,
and Cassano
1.0
e
u ..
0 e
(.)
0. 5
rr .. 0. 1 4
. --4�:l!o...--__J 0.0 .,___.�-.-___..._ 30 0.0 60 120 90
t
( min )
FI GURE 2 7 Comparison of theoretical predictions and experiment al results for the chlorination of trichloroethylene (chlorine fraction in the feed , 14%) .
Fi gure 2 7 show s an example obtained for the case in w hich the reaction is conducted using 8 6 % of nitrogen in the feed . Since the first reaction is in the diffusional subre gime the good correspondence bet w een computed and experimental values only indicates that t he mass flow rates of react ant gases have been p roperly modeled and the energy i n put to the reactor is suf ficient . In the second reaction we know n t hat it is kinetically controlled and the agreement must be ascribed to the quality of the radiation field p rediction as w ell as the kinetics and reactor modelin g . It can be observed that C 2HCl5 selectivity is alway s 1 0 0 % until almost all C 2 H C l 3 has been con sumed ( t = 90 min) ; then C 2 HCl5 concentration goes down becau se the sub stitution reaction ( C 2HCl5 + C l 2 -+ C 2 C l 5 + HCl) become s significan t . For this particular reactor- source- reflector configuration , condition 2 is certainly fulfilled and , in p ractice , both reaction paths A + B and B -+ C do not take place simultaneously . Stoppin g the reaction at t 90 min ( by turning off the lamp ) , it is possible to transform all the C 2 H C l3 initially present in the liquid phase into C 2HCl5 only . Figure 2 8 show s the results for the case in which 60% of nitrogen is used in the feed . Similar conclusions can be drawn from the selectivity and conversion viewpoints . However , the time necessary to achieve 100% C 2HCl5 selectivity together with almost 100% C 2 H C l 3 conversion has been reduced to 35 min . These results clearly show that it is possible to achieve 100% selectivity for the intermediate species to gether with almost 100% conversion of C 2 H C l 3 and that the presence of nitrogen does not affect the final p roduct distribu tion , but the time needed to reach a given value of conversion . The time of reaction can be sharply reduced by u sing p ure chlorine in the feed but , obviously , it is possible that under such conditions the reaction control will be much more difficult . =
899
Analysis and D es i gn of Pho to reactors 0
•
•
.. .-.... ,.,
•e u
0 e
...
-
u
05
0.0 0. 0
20
40
60
80
t
( m in)
•
100
28 Comparison of theoretical p redictions and exp eri mental results for the chlorination of trichloroethylene (chlorine fraction in the feed , 40%) .
F I G U RE
A n n u l a r Photo reactor
Consec u t i ve R eact i o n s in a Conti n uous
A gaseou s mixture of C l 2 , C H 4 , and N 2 is flowi n g through an annular photoreactor w hose inner and outer w alls are cooled to avoid large tempera ture rises , p recludin g undesirable overheating and by - prod uct formation . Moreover , with the s ame purpose , it is recommended that the chlorine inlet The aim i s to perform a molar fraction be m aintained no hi gh er preliminary design of the reactor , including the selection of t he emittin g For safety reason s , the system for an optimal p roduction of chloroform . maximum reaction temperature and p ressure are T = 3 6 3 K and P = 1 0 1 . 3 kPa .
than 5 0%.
In dealin g w it h the design of this photochemical reactor we mu st answer the following questions : (a) W hat is the si ze of the reactor ? ( For the annular reactor , this means the values of rin , r0u , and LR . ) ( b ) What are the operating conditions ? ( T hese include inlet composition , flow rate , etc . ) ( c ) Which is the approp riate emission system to carry out the reactio n ? ( T hi s involves the spectral distribution of p o w e r output , overall power out put , and values of rL and LL . ) The first two que stions are common in conventional therm al reactor design ; the last arises only in p hotochemical reactor desi gn and constitutes one of the most important stages in the design strategy . To provide an answer to the fore goin g questions , we have to solve the modeling equations several times because the number of variables involved in the desi gn p roblem is large . H ence it would be recommended to find a simple , even t hou gh only approximat e model to describe the performance of the p hotoreactor . This si m pl e model will allow us to ( a ) choose an emission system , and ( b ) find a starting set of values for the main variables to carry
De B e rnardez , C larili, and C assano
900
out the final d e si gn u sin g a reliable ( but more complex ) model . model should have the followin g characteri stics :
A simplified kinetic scheme with reaction rate expressions wi d e range of operatif!g conditions A simplified model to describe the reactor operation A simplified model for the emission process
T he simple
valid for a
Simplified K ine tic Scheme We can use the entire reaction m echanism for w hich the rate expressions for t he individual step s are known and obtain global kinetic expressions by introducing some typical sim p lification s such as t he local steady state ap p roximation for highly reactive intermediate species . The reaction mechanism for the chlorination of methane can be summari zed as follow s : Initiation step : ,
Cl 2 + hv
___!___
2
Cl •
( 1 5 1)
Prop agation step s :
2
+
C H 4 + Cl • � CH 3 • CH 3 •
+
.:__ C H 3C I
Cl2
HCl
( 1 5 2)
+ Cl •
( 1 53)
� C H 2Cl ·
C H 3 Cl + Cl •
+ HCl
5
+
C H 2Cl • + C l 2 � C H 2c12 C H 2c12 + Cl •
CHC1 • 2
+
� C HC12 •
7 CI2 � C HC 1 2
CHC13 + Cl • CC13 • + Cl2
8
:::::;::::=::
CC13 •
--=-- C C 1
4
+
( 155)
Cl •
+ HCl
( 156)
Cl •
( 157)
+
+
( 154)
( 1 5 8)
HCl
Cl ·
( 1 5 9) - I
Homogeneou s termination step s : CH3 •
+
Cl •
__!Q____
C H 3C l
11
C H C1 2 • C CI 3
•
+
+
( 1 6 1)
12
C l • - -- C H C 1 3
Cl•
( 1 60)
1
3
( 162) ( 1 63)
A n alysis an d Design of Pho toreac tors +
Cl ·
14 C l • - ct2
+
CH • 3
901
( 164)
15
CH • - CH - CH 3 3 3
( 165 )
16 C H 3 • + C H 2Cl • - C H 3 - C H 2CI
( 1 6 6)
+
17 C H C 1 2 • -- C H 3 - C H C 12
( 167)
+
18 C C I 3 • ---- C H
( 1 6 8)
CH •
3
CH • 3
C H 2 CI • + C H 2 C l •
.....!!.__
3
- CC13
C H Cl 2
( 169) ( 1 7 0)
21
C H CI • + C C I 3 • ---- C H CI - C C I 3 2 2 C H C 1 • + C H C I2 2
C HCI2 •
C C1
3
•
+
•
_E._
C H C I2 - C H C I 2
23 C C I 3 • --- C H C I2 - C CI 3 24
+ C C1 • - C C I 3 - C C I 3 3
( 1 7 1) 2 ( 17 ) 3 ( 17 )
( 1 74)
H etero geneou s termination steps : 2 5 C H • + w all -- p rod u cts 3 C H 2Cl •
+
•
+
CHC1
2
C C13 •
+
26
( 1 75)
w all -- p roducts
( 1 76)
2 7 w all --
( 177)
products
28
w all -- p roducts
29
C l · + w all -
p rod u ct s
( 17 8 ) ( 17 9 )
We introduce the followin g sim plifications :
1. 2.
D i s r e ga r d
the heterogeneous termination steps [ E q s . ( 17 5 ) t o ( 179) ] as show n by De Bernarde z and C a ss ano ( 1 986) . D is re ga r d the h omo ge n eo us termination step s that lead to form chlorinated derivatives of ethane [ E q s . ( 1 6 5 ) to ( 1 7 4 ) ] , w hich are 2 kn ow n to be ne gli gi ble ( Kurt z , 1 9 7 ; De B ernarde z and C assano , 1986) .
3.
the local steady-state app r oxim ati on t o the unstable intermediate s p e ci e s to get h er with the lon g-chain ap proximation in the re m ai ni n g reaction system [ E q s . ( 1 5 1) to ( 1 6 4) ] .
Apply
90 2
D e Bernardez, Thu s the c o mplex
scheme :
CI
+
CH4
C H 3 Cl
+
CH 2c1 2
+
a
mechan i s m i s r e duce d to the foll o win g
-
R 1 � C H 3 CI
2
+
H
R-2 CI 2 ::;--- C H 2c1 2
C1
R -3 � C H CI 3
2
C H C ia + CI 2
The r e
chain
C l a rili, and C a s sano
� CCI 4
+
CI
k inetic
( 180)
+
H CI
( 18 1)
+
HCl
( 1S 2)
H CI
( 183 )
individualnl
in e a r function of the con k in e tic c on stan t s of each step of the mechan i s m , an d the local volu m e tr ic rate of en ergy ab sorp tion . C on si de rin g th at the k in etic exp r e s sion for e ac h step of t he c t i on rate expre s sion s become a
th e
centration of the stab le spec ie s ,
no
mechan ism has the for m of the ma s s ac tion l a w ,
n1
n2 na n4
=
( k 2C 2
-
k 2C -f< 1>
=
( k 4C3
-
k 4 C .f 2>
(k 6 C 4
=
= ( k8C 5
- k6C .fa>
elementary
the final ex pre ssion s are
/2 1 a (� � ) /2 . J a (�� ) (!�a f
( 184 )
( 18 5)
/2
- k sC .f4 ) (
�
�) a
( 186 )
1/ 2
( 18 7)
where we h a ve iden tified e a ch species in the foll owin g way :
C1 2 1
C
H
4
2
an d in E q s .
C H3 Cl
C H 2c 1 2
C H CI3
4
5
( 18 4 ) to ( 18 7)
we h ve
3
C C1
6
4
H Cl
N2
7
8
a
( 188 )
( 18 9)
903
A naly s is and Design of Ph o toreac tors k C 6 4
S
k C
7
k 7C 5
+ +
( 190)
k C 7 1
( 19 1 ) an d ( 19 2) It shoul d be n otic e d that we ha ve ob tained kin etic expression s for the overall reaction s [ E q s . ( 18 4 ) to ( 19 2)] which do not need any adju stable parameter if an appropriate emission model is u se d for evaluating the radia tion field . Generall y , the kinetic expres sion s found in the literature make use of phenomenological rate con stants ob tained by fitting experimental data . If appropriate microkinetic s data are availabl e , empirical rates do not seem appropriat e for modern de sign pu rposes , since they are valid only for restric ted range s of operating con dition s .
S imp l ifie d Model for th e R eac tor Opera tion
The simplest model to be u sed for the prediction of the performance of a con tinuou s reactor is undoub tedly the plu g -flow model . H o wever , Romero et al . ( 1983 ) have shown that thi s model is a very poor app roximation when strong radial gradien t s in the radiation field are p resen t , which could be the case of the annular photoreactor if we have a mediu m - to -l arge op tical den sity . Moreover , it may provide totally incorrect predic tion s of the global kinetic behavior . Neverthel e s s , we can u se the maximu m gradient model u nder the followin g a ssumption s : ( a ) isothermal c on dition s , ( b ) flat velocity p rofile , ( c ) negligible axial dispersion , and ( d ) negligible radial dispersion . For steady - state flow , the species i mass balance e quation becomes
v
=
a
!1 . [ C ( r , z) , e ( C , r, z)] 1
-
i
=
1,
•
•
•
'
7
( 1 93 )
and C = (C
1
• • • C ) 7
T
( 194)
with i
=
1,
.
.
•
'
7
( 19 5 )
where v . .n . 1] ]
and
v
1]
. . is the stoich iometric c oefficient of species i in reac tion j .
( 196 )
904
De B e rna rde z ,
C l a ri6. ,
and Ca ssano
Simpl ifie d E m i s s ion Model T h e fir s t hi storic al s tep in the de velop men t of emission models con sidered a line source with emission in p a rallel pl anes perpen dicular to th e lamp axis ( PE L S model ) . This model is very attrac tive b ecau se of it s simplic ity but it has very little p hy s ic al mean in g s in ce it requ ires that all poin t s of the source e mit radia tion in pl an es perp en dicular to the lamp axis , with zero probability of emission in other direc tion s . A s for any type of lin ear model , it is not recom mended for evalu a tin g the radiation field when reflec t in g su rfac e s a re pre sen t . B u t , o n t h e o th e r han d , i t i s b elieve d to give an a p p r ox i ma te e stimation of the radiation field when only direc t radiation is pre sent ( a s in the c a se of annular photoreac t or s ) . H o wever , this state men t is partially true . A t poin t I ( r , 13 , z ) in side the reac tor volu m e ( Fig . 8 ) the ra d iat ion arrives from all direction s in sp ac e an d from the wh ole vol um e of the lamp . The attenuation path len gth changes with the direc tion an d may be much lon ger than the charac teri stic radiation path len gth ( r0 u - rin ) . B e sides , the radia tion field at point l ( r , 13 , z) in the annular reac tor resul t s from the c on tribu tion s of all ray s comin g fr o m the hemisphere containin g the l am p volume . A s can be ob served in Fig . 2 9 , thi s im portan t p hy sical situation is not accounted for when the PELS model i s u sed . In fac t , at point l ( r , 13 , z) in side the reac tor , the ra diation field resul ts from the con tribution of only one ray de p a rting from the lamp ; i n addition , the a t tenuation p a th len gth is a representation of ju st the minimu m l en gth existin g in the real process . Another failu re of the P E L S model to p redict the real radiation field arises from the fact tha t it d oe s not acc ount for the reactor wedge s ( see Fig . l la ) wh ich may be very si gnifican t in an annular reac tor whose inner wall is cl o s e to the lamp wall . With the se disa dvan tage s the PELS model is in ap p rop ria te to c arry ou t the design of a photoreac tor particularly when the characteristic radiation path len gth is one of the design para m eters . H ence the simplest model to predic t the radiation field is the SELS model , wh ich at least c on si der s the three - dimen sional charac teristic of th e emission an d attenuation proces se s .
r �-- - - - - - --I I
.----.
I
I I I I
I
1 c r,J3, z ) 1-t---+-I I I I I I ' '
M
---------- ..__�
5
F I G U RE
29
PELS model .
R
Schematic represen tation of the at ten u ation proce ss fo r the
A na l y s i s an d
905
Design o f Photoreac tors
R igorou s Mo del
For the final
c ompu tation
follo wi n g improvements : 1.
Regarding the
not u sed .
the model wa s perfec t e d
k in e t ic sc h e me :
by incorporating
the
The lon g -c hain approximation was
R egarding th e reac tor model : ( a ) The parabolic velocity profile was in cl u ded , ( b ) radial diffu sion was not ne gl ected , and ( c ) the
2.
multicomponent diffu sion coefficient s were calculated with the Stefan - Maxwell relation ship s . Regarding the e m i s s ion model : Ex ten s e models ( VE E S and SEES models) were u sed in stead of the SELS model .
3.
Des ign Prob l e m
Althou gh p rodu c t ion is generally the g oal of an optimal design , when deal ing with consecutive r eac tion s , the recommended c rit erion is s elec tivi ty . H oweve r , in order to decrease the cost o f the p r od u c t s ep a rat ion it may be a d vi sab l e to operate the reac tor with maximum conversion for at lea st one of the reactants . A s u itab le strategy for s ol vin g the design problem c on sists in s ta tin g an op timi zation p rogramme in wh ich the independent variables a re the de sign parameters an d the ob j ecti ve function resul t s from a combination of both c r ite r ia : s electivity an d con version . The des ign p roblem , which is de sc rib e d by Eqs . ( 19 3 ) to ( 196) , init iall y has the following degrees of freedom : inlet composition , cl., c2, and C� ; volu metric flow ra te , Q ; inne r an d outer reac tor radiu ses , rin an d r0u ; reac tor l en g t h , LR ; an d lamp charac teristic s : radiu s , le n gth , power ou t put , and spectral distribu tion of it s power ou tpu t , However ,. on ly some of the se variables are appropriate to p e rfor m an optimization problem . Alth ough th ey are extremely important in the d e s ign of a photochemical reactor, the emission system charac teristic s can n ot be varied c on tinuou sly becau se only a limited nu mber of lamp types are avail able in the market . H ence , in spite of the fact that the parameters rela ted to the lamp enter into the mas s balance equation s , they cannot be u sed as optimization v a riable s . N evertheles s , the problem may b e sol ve d for differ ent lamp s in order to select the most appropria te one . On the other han d , operating con dition s and ge o met r ical characteristic s of th e reactor c an b e continuou sly varied . Thu s they can be inclu ded in the optimization . Operatin g the reactor at c on s tan t pressure an d tem pera tu r e , the total inlet c oncentration is con stan t . Hence
co
1
+
co2
+
co
8
=
C0
=
con s tan t
and the total nu mb er of indepen dent variables is redu ce d . A t the same time , one coul d set an upper limit to the chl orine inlet fraction : for exa mple, not greater tha n 50 % to a vo i d exces sive heatin g . Thu s if we adop t a c on stant chlorine inl e t concen tration ,
C0
1
=
0 . 5C0
the only in dependent variable representing th e inlet composition
will
be
c2 .
De
9 06
B e rnarde z , Claria , an d
Cassano
In order to make u se of th e maximu m amou nt of energy coming out of the lamp , the reac tor length is generally taken equal to the lamp length . The inner reac tor radiu s is adopted as close to the lamp radiu s as possible , leaving a small distance for c ooling the lamp an d c on trollin g th e operating reactor temperature if neede d . Then , in our preliminary de sign p roblem we will adop t a con stant value for the inner ra diu s , an d equ al values for th e lamp an d reactor l eng th s Finally , the in depen dent variables are : me thane inlet concentration , C ; volu metric flow rate , Q ; an d outer reactor radiu s , rou . In the prec edin g paragraph s we mentione d the pos sibility of solving th e design problem by mean s of an op timi zation technique , The following statemen t of this op timi zation probl e m results : .
Maximi ze S
i
�
--:6--""--''-'---
=
�
j= 3
l
L
R
i
)>
=
3,
.
•
•
'
6
( 19 7)
Subject to
� � X
h
R,
� bk
vk
> p
R.
R.
( v , Q_,
m -
where
k
s. 1
C < i (LR ) >
X
!.)
0
m
2, 3
=
1, 2
=
1, .
( 198)
( 199 )
.
.
'
7
( 200)
selectivity of the desired chloromethane derivative =
v
=
a
=
b
=
R.
=
1,
=
P R.
=
t
=
c
average exit c oncen tration of species i calculated with E q . ( 1 06b ) in dependen t variable array ( Q , r
, e tc . ) ou lower bou n d for the indepen den t variable array upper bound for th e independent variable array R. - reactan t c on version minimum permissible value for R. - reac tan t con version chloromethane c oncentration array array that accounts for the fixed de sign an d operatin g parameters ( temperature , pressure , rin , C0 , lamp characteristic s , etc . )
( 198) 199)
define a feasible region for the optimi zation p roblem C on strain ts ( i . e . , a ran ge of variation for the in dependent variables ) . On the other han d , con s train ts ( allow us to establish a minimu m permissible v alu e for both reactant con version s so as to a ssure appreciable exit produc tion . Finall y , e quality con strain t s ( 2 00) represent the solution of the species mass b alance equation s which give a relationship bet ween the in dependent
A naly sis a n d Design
of Photoreactors
90 7
variables v and the dependent variables C necessary to evaluate Sj . Equa tions ( 1 9 7) to ( 200) define a nonlinear m Ui tivariable constrained optimization problem which can be solved using any of the several algorithm s available in the literature . To illu strate the design methodology , we solved the optimization problem for chloroform production u sing a modified Powell algorithm . The statemen t of the optimi zation problem become s : ( 2 0 1)
Maximi ze
sub ject to o "
c2 "
( 2 0 2a )
o . 5C 0
3 0 . 0000 5 " Q " 0 . 0 0 1 00 m /s
( 20 2b )
0 . 04 " r " 0. 12 m ou
( 20 2c ) ( 20 2d )
X
2
( 2 0 2e )
� 0 . 99 5
an d h ( !_, C , !) k
=
0
k
=
1,
.
•
.
'
7
( 203 )
an d t inclu des P T C0
ri n
=
=
1 0 1 . 3 kPa
3 63
K
3 3 . 6 mol / m 3 0 . 03 m
( 204a ) ( 204b ) ( 204c ) ( 204d)
an d lamp data . Th e upper an d lower bounds u sed in con straint s ( 202b ) and ( 20 2c ) are typical value s for annul ar photoreactor operation . Finall y , to complete the required information we mu st decide about the characteristic s of the lamp . Since methane photochlorination is a chain -type reaction , it should be expec ted that low -energy radiation sources woul d be require d . This statement was corroborate d by De B e rnarde z and Cassano who studied the incidence of lamp parameters on selectivity and con version in methane photochlorination . The au thors found that the re action can b e carried out u sin g sources of low - energy outpu t such as germicidal or black -light lamp s . B oth types of lamp s represent a very attractive solution from an economic point of view . In addition , they are available in the market in a variety of si ze s and ou tpu t power s . To illu s trate the lamp selection procedure , w e will solve the optimi zation problem for two different sources which are amenable to comparison :
( 1986) ,
9 08
De B e rnardez,
E mis sion model
Lamp
Po wer inpu t ( W)
Power ou tput
(W)
Clari6, a n d Ca ssano
Spec tral range of emis sion
Length
( m)
Germicidal
VEE S isotropic
30
8.3
2 53 7
0.91
B lack light
SEES diffu se
30
5.4
3 20 0 - 4 20 0
0 . 91
B oth lamp s have iden tical value s of en ergy input an d length , but the The germicidal lamp ou tput power an d spec tral distribu tion are different . emits at a wa velen gth ran ge where chlorine ab sorption is poor ( see Table 5 ) , while the e mission of the black lamp falls in a wavelength ran ge where the ab sorption by chlorine is significantly stron ger . H ence a h igher re actan t con version should be expec ted when the black ligh t l amp is u sed . However , the amount of energy emitted by the germ icidal lamp is greater and con sequen tly there exists th e p o ssibility of ob ta in in g a larger p roduc t out p u t for the same energy c on su mp tion . Th e final resul t will show which of the t wo characteristic s c auses a better imp rove ment in con version . In other word s , we wish to ob tain in formation ab ou t the relative sign ificance of the quality and the quan tity of the ene rgy emitte d . Similarly , it is im pos sible to an ticipate the effec t of both p rop erties ( power an d spec tral dis tribution ) on chlorofor m selectivity . · T able 6 summari zes the result s of the design problem for both lamps . Due to the characteristics of chlorine li ght absorption combined with the spectral distribution o f the energy output of both lamp s , we can state the following conclusions :
TAB L E 6 Op timi zation of Chloroform Production : Design Problem Resul t s Ger micidal 8
s
8 8 X
X P
c
C H3 C l
C H c1 2 2 c HC 1 3 c c1 4 C1 2
CH4 c H c i3 ( g/ s)
2
10
3 ( mol / m )
( m) ou 3 Q( m / s)
r
X
3
X
10
5
Black ligh t
11.3
3 .9
25. 0
18 . 9
40. 4
47. 0
23 . 3
3 0. 2
21 . 8
60.3
89 . 8
95.8
7.4
12 . 8
1.32
2 . 83
0 . 050
0 . 044
13 . 0
8 5 .
A nalysts and Design of
1.
2.
909
Photo reac tors
T he g e rmi cidal lamp allows us t o use greater equivalent di ameters , but t o obtain hi gh m e th a n e conversions , the system is for c e d to use lo w hydrocarbon concentration in the feed . T hi s consi de r ably reduces the out p u t p roduction . With germicidal lamps , chloroform s e le cti vit y would be approximately 4 0%. C h loro fo r m production and selectivity are gr e at er in t h e case of b l ack - li gh t lamp , indicatin g that in spi t e of the di fferen ce s in po w e r output ( more than 5 0%) , w hat really cou n t s is t he q u ali ty
the supplied radiation . With this lamp we get hi gher selectivity ( c lo se to 5 0 %) , very hi gh methane conversion ( almost 96%) , and can obtain 7 3 % mo r e output p ro duc tion of ch loro for m for the s ame energy i np ut and reactor len gth . Perhaps what is even more sur p ri si n g is that all these adv antages are obtained with an equivalent diameter that is 3 0% s maller. of
T able 6 also show s that the minimum p e r mi s si ble value of m et h an e con version could not be r e ac he d . However , by workin g with la mp s of greater e ne r gy i n p u t we can inc r e as e the le n gt h of the lamp ( and that of t h e re actor) , the amount of energy p er unit le n gt h of the lamp , and obviously , the mean residence tim e . This possibility could ce rtai n ly allow one to fu l fill the re qui r ed e xi t con versi o n . To increase the total amount of chloroform p ro d u ce d , several reactors (in p aralle l ) s hould be used . Other p o s si biliti e s are the multitube and the multitube - multilamp reactors . T hese result s c le arly illustrate that a photochemical re act or p rovi d e s new vari ab les for op ti mi z atio n . A lt hou gh r e ac t an t feed compo siti on ratio and mean re si d e n c e time are still very i m po rt ant p ar a met er s , the opti c al thickness ( c o m bi n atio n of absorptivity , characteristic radiation p ath , and absorbi n g sp e ci e s concentration ) as well as t h e output c h a rac te ri stics of the radiation . source have a si gnifi can t effect not only in the exit conversion but also in the overall p rod uctio n an d se l e c ti vi t y .
R E AC T O R C O N F I G U R A T I O N S FO R I N D U S T R I A L A P PL I C A T I O N S S i n g l e L a m p -M u l t i t u be R eacto r
T his arran geme n t ( Fi g . 3 0 ) is a suitable reactor for medium -pressure , hi ghly exothermic or endothermic ga s - p h a s e reactions . It pro vi d es good heat transfer c h ar ac teri s ti cs wi t h t he s ur rou n din g fluid . T he individual reactor s m ay have rather small diam ete r s and consequently , w ell -tested Pyrex glass or q uart z t u b e s can stand p r e s s u r e s up to 1 5 0 0 kPa. It could be u s ed for liq ui d - p h a se reactors as w ell . In any case , ap plications should be r e s t ri c t e d to rat he r fast re ac tio n s ( or equivalently low flow rates) bec aus e the reactor t ub e s will n e c e s s arily h av e len gth limitations (mainly motivated by m ark e t restrictions in the la mp s and safe op e rati n g conditions for the re ac to r ) .
M u l t i l a m p R eactor
This r e a c to r ( Fi g . 3 1 ) can be re com m e n ded for li q ui d - p h as e reactions with sm all - to - m ediu m heat transfer requirements , which could be i mp ro ve d by adding cooling coils inside the vessel .
The
w alls of the tank could have
D e Bernarde z , C laria , and
91 0
F I G U R E 30
C assano
Sin gle lamp - multitube reactor .
reflectin g properties , increasing the efficiency of the radiation emitted by the lamp s . A gai n , it can be used for low and medium reaction pressures , since restrictions are placed in the diam eter of the tubes s urroundin g the radiation source s . T his reactor is very app ropriate for reactions w here ener gy requirem ent s are rather hi gh or i f , in spite of having low -energy requirement s , the optical thickness of the m edium is large . In the second case the same input ener gy could be distributed in several radiation sources , obtainin g a more useful radiation field distribution inside the tank . It i"s also suitable for multiphase reactions since very good stirrin g is easily achieved . It is not appropriate for gas - phase reactions . Pseudoa n n u la r R eactor
T his reactor ( Fi g . 32 ) is a modified version of an annular photoreactor with the purpose of increasin g its heat transfer pos sibilities and its capa It is also very suitable bilities for low and medium op erating pressures . for reactions that m ay need medium -to - long m ean residence times . It could equally well be used for gas - or liquid -phase reactions .
A naly sis
an d
r--
-
. •
�
tt.
r
....
I
-
f-
r-
31
I i
911
-
��
-
I
� FIGURE
D esign of Photoreactors
�
. M u ltil amp
�
react or .
M u l ti l a m p-M u l t i t u be Reacto r
This reactor (Fig. 33) is perhaps the be st arr an gem e nt for large-scale The system h as good pressure and heat tran s fer capabilities , large flexibility re gar di n g energy requirements or op tical properties of the r eacti ng medium , and if there is ade q u at e connection between tubes , good fle xibilit y to overcome large residence-time requirements . It could not b e used efficiently for multi phase reactions , this limitation bein g its mai n hi n dran c e . production .
F l a t - Plate R eactor
This reactor (Fig . 34) is s uit able for reacti n g media o f very high optical thickness . Its main app li c atio n could be for decont amination of hi ghly re fra c tory m aterials to other tec h nologi es . In this case t he " clea nin g" of the reac tor feed ( to increase transparency from the op ti cal point of view ) could be infeasible for economi c or technical reasons . Hence i m p urities will greatly increase the ab sorp tion characteristics of " re ac t an t s . " A re d uc tio n in the radiation p at h len gt h w ould be the only solutio n and the fl at - p lat e reactor m ay be t he most practical w ay to achieve it . Both plates ( upper and bottom p art s ) s hould be made with practical p rovisions for
FI G U R E
32
Ps eu do an n
3 FI GU RE 3 91 2
r. ul ar re acto
re ac tor . multilam p M ultit ub e -
A nalysis
91 3
and Design o f Photoreacto rs
F lat plat e reactor .
FIG U RE 34
-
assembling and reas s e m bling purposes to facilitate unavoidable clean may be p hoto - assisted easy extension of our re actor irradiated from the bottom ( Fig . 34) . Finally , it s hou ld be noted re gar din g wall depo site difficulties that mechanical cleaning would b e difficult only in the pseudoannular re actor If chemi cal m ethods are used this reactor is one of the easiest to clean . fast
A second i m p or tant app lication i n g routines . catalysis . T he modeling of this reactor is an
.
U N S O L V E D PRO B LEMS
With re gard to photoreactor design , assu min g that the reactions are well known , w e recognize s ever al p roblems w h ere lack of information or reliable ways of hand li n g them can be observed . T he main ones are : modeling the ab s orp tion in h etero ge neous systems , dark deposits at the reactor w all t hrou gh which the radiation must enter into the reacting space ( which re duces li ght t r ans mi s sion and affects yield ) , and problems related to pre s su ri zed ga s p h ase r eactors Several research group s have made impor t an t contributions to the area 1 98 1 ; of m ode li n g the absorption in h etero gen e o u s media (Otake et al . Yokota et al . , 1 9 8 1 ) . However , we should p oint out that the only duly p roved co ntri buti o n s related to the modeling of hetero geneous systems still req ui re experimentally ad ju s tab le parameters . The use of ri gorous models combined with the mod elin g of the distortions produced by the hetero geneities constitutes a parti c ularly attractive problem where additional re search work should be p erfor m ed T hese problems have great significance in the field of p h otocatal ysis where an opaque solid may be pre s e nt . The reactions at the w all of the reactor are a serious disadvantage for the use of photochemical reactors in industrial processes and the main cause of freq u en tly shutdowns for cleaning routines . It is known that some of these difficulties have already been overcome by t he Toyo Rayon Co . of Japan in their cyclohexane photonitrosation commercial - scale process . S om e additional work on the s u bj e ct proposing novel reactor designs not yet under the stage of providing results of economic si gni fic ance has been proposed by L uc as ( 1 9 7 1 ) . The author proposed a "two- zone s e gregate d reactor" w here all dep o site s are avoided since all secondary reactions after the initiation step occu r in a place different from the one where the radia tion is supp li ed to the system . N evertheless , app rop riat e design of the reactor c o u ld p ro vi de contin u ou s cle aning by a mechanical scraping of t he w all where the r adiatio n enters the reactor , thus s u pp lyi n g a practical -
.
,
.
,
,
De B ernardez ,
91 4
C larid,
u
and
C assano
fundamental
solution to the problem . H op e f lly , the ti m e will come when research on surface che mi s t ry will be able to p rovide a definite answer . Since p hotochemical gas - p hase r e actors must have at least one quart z or Pyrex gla s s w all , there exist s om e di fficulties in working under p re ssure . could be thought as a m aterials science p roblem but mainly a challenge for new reactor d si gn s , which could ran ge from mul tit u bu lar reactors of small i am et e r (often used in industry for heat -release reason s ) to the in troduction of new technologies such as lasers co m bi ned wit h optical fibers ( see Marinangeli O lli s , 1 977 , 1 980) .
It
e
d
and
A PPE N D I X :
C Y L I N D R I CA L R EACTOR W I T H
ELLI PT I C AL
L I M I T I N G V A L U E S O F T H E C O O R D I N A T ES
RE FLECTOR .
F O R I N D I R E C T R A D I A T I O N W I T H O N LY O N E R E F LE C T I O N
The limiting values for t h e p coordinate can be as follow s ( see Fig . 13) . Let 5 t h unit v cto r repre se ti n g the direction of a ray
be e
e
I
that i m p i n ge s at any point by
:;
I
sin e cos
;;::
n
obtained
on the reaction sp ace with direction 9 , ¢ given
sin e si n cpj + cos ek
+
(Al)
T his ray has previously rebounded o n a point P o f the elliptical mirror , whose coordinates r f r re to point are :
I
ee d
x
=
Yp zp
=
e
w h re
PI
sin 9 COB ljl
PI
p
=
P I sin e sin
PI
- ( hb
2
cos ¢ + k a
2
sin ¢ )
and 0"
can
2
;;::
b
2
co s ¢ + a 2
h
;;::
c + r
k
;;::
r
i
1
sin
c os
2
cr2
+
ab [ cr
2
sin 9
- ( h sin ¢ - k cos
cp ) 2 ] l / 2 ( A 2)
. 2 sm ¢ ( A 3)
81
8I
T he expression for the unit norm al vector t o the reflector at point P w ritten as follows :
be
x
�p
=
p
+
a
2
h
.
1 +
Yp
b
+
2
k
j
(A4)
91 5
A nalysis a n d D esign of Pho to reac t o rs
T he unit vector
representing the direction
:E
of
the ray before reflec
tion (i . e . , emerging from the lamp ) can be obtained from the law of reflec tion : ( A 5)
Solving this vectorial equation , the following expression for obtained :
:E
=
sin 8 cos
where
and
= t an
lj> E
2
a (Y m
=
2
sin 8 sin
lj> E � +
:E
is
( A 6)
cos 8k
-
1) sin if> + 2m cos if> 2 2m sin if> - ( m - 1) cos if>
- 1 (m
p
2 b (x
p
( A 7)
+ k)
(A S)
+ h)
The two intersections of direction 8 , lj> E with the boundin g surface of the radiation source are p recisely the limits of p . T hey are : p
E2,
=
( A 9)
sin 8
1
where =
D sin E;,
[ (x
1
=
D
+
+ h [ (x
p
c)
+
2
+
(yp
+
h + c) sin
2 1 /2 k) )
- (y
(A 10)
p
+ k ) cos lj> E )
1 [ ( x + h + c ) cos E + (y + k ) cos p p D
=
cos E;.
p
( A l l) ( A 12)
T he limitin g values of the 8 coordinate can be defined by means of ( Fig. 1 2 )
( pi
+
P
E,1
) cos 8 1
1
=
2( L R f + L L )
-
z; I
(A 1 3)
1 8 = 2( L - L ) 2 L - z;I Rf
U sin g Eq s . ( A 2 ) and ( A 9) , the integration limits a s a function of
e1
=
tan
_1
Pi
2
2
.
1/
- D sm 2 E;, ) 2 L ------::-=----: ----:--:-:-::-:-::=--=L ) ( 1 / 2) (L R f + L - z; I +
D cos E;.
-
(r
( A 1 4)
De Bernarde z , C larid, and
91 6
PI
+
D
cos t;
-
( 1 / 2 ) (LRf
(r
2
-
L LL )
-
2 1/2
D 2 sm /; ) .
-
( A 15)
r,; I
w here =
- (hb 2 cos
Cas sano
ka2 sin ) + ab [ cr 2 - (h sin
p1 I
-
k
co s
t>2l 1 1 2
depends only on
,
( A 1 6)
With Eq . ( A 9) t he following expression is obtained : ( A 17)
Solving t hi s implicit equation numerically , the limiti n g values of
This work reports t he p hotoreactor d esi gn methodology developed from re search done with my former students , some of them now full professors , s uch as Horacio A . Ira zoqui , Jaime Cerda, and J acin to L . M arc he t ti and others that are still on t hei r w ay to get their doctor's d e gre e s , including Orlando M . Alfano , Roberto L . Romero , Roberto M . Re , and my two co authors , Eliana R . De B ernarde z and M arfa A . Claria. The theory has been verified with bench - scale experiments . T hanks must be given to Pedro G . G u ari no Mario J . J . Didier , Juan C . Garcia , A n t oni o c . Negro , Omar B ri zuela , Jose L . Gim en e z and Jose L . Giombi for their contributions to this part of the w ork C redit must also b e given to Yolanda P ereyr a for her excellent typing of this m o nograp hy and to Elsa I . Grimaldi for the thoroughness o f he r review of the English used in the manuscript . ,
,
,
.
N O T AT I O N
a a
v
a'v
ellip se semimajor axis or characteristic con s t an t of the interfacial area per unit liquid volume , em
-1
interfacial area per unit liquid plus gas volume , em 1 -
A
area , cm 2
b
ellipse semiminor axis ,
c
hal f distance between focuses of the e llip s e , em
c
concentration , mol /em 3
em
p arabo l a , em
91 7
A nalysis an d Design o f Photoreactors
bubble diam et er 2
,
em
diffusivity , cm /s
ellip se eccentricity , dimensionless 3
rate of ener gy ab sorption , ein stein /cm · s
local volumetric
F
radiant energy flow rate , einstein /a 1 molar flow rate , m ol l s
G
value
h
Planck's constant , J •s. or value defined by Eq . ( A 3 ) 3 Henry ' s constant , kPa •cm /mol
E
H
defined by Eq . ( 87) dimensionless ,
em
intensity of radiation ( dim ensions are source- model dependent )
I'
isotropic intensity of radi ation (dimensions are source - model dependent unit vector
i
k 1
k.
value defined by Eq . ( A 3 ) , em
reaction rate con s t ant for step i , s - 1 ( first order) or em tmol • s 3
( second order)
m ass transfer with chemical reaction ab sorption coefficient , cm /s
p hy si c al m ass transfer absorption
coe fficient ,
em /s
length , em
m
distance from acto r ,_ em
vertex
of
p arabolic r e flect or to th e bottom of the re
value defined by Eq . ( A S) , dimensionless
�
unit normal vector
p
p artial pressure , kPa , or p roductio n
q
radiation flux d e nsit y vector , einstein /em • s
Q
volumetric flow
r
r adi al coor di n at e , em
R
indicates reactor
Re
R e y nold s number , dimensionles s
N
P
mass ab sorption rate , mol /cm 3 •s
total
p r e s s ure , k P a
r ate
,
em 3 /s
,
g/s
2
S
selectivity , dimensionless ; also i ndicates radiation sour c e
t
time ,
v
velocity , em / s
V
volume ,
x
rectangular coordinate , em
X
s
em
3
fractional conversion , di mensionless
De
91 8 y
z
rec t an gul a r coordinate , em rec t ang u lar
or
B e rnardez ,
C laria,
and Cassano
m olar fraction
gas -phase
coordinate , em
G reek Lette rs
ab s o rp tion co e ffi cient , c m 2 / mol
cylindrical coordinate , rad
emitting exten sion ( dimensions reflection coefficient ,
are source - model
dependent )
d im e n sion le ss
Dirac function value
d e fi ne d by Eq s .
values defined
( 55) and ( 92) , em
( 7 4) , em
by E q .
dispersed -phase holdup , dimensionless axial coordinate , em
c oor di n at e , r ad
s ph eri c a l
e mi ssion p roperty of the lamp (dimensions are source model dependent )
K
characteristic w avelengt h ,
attenuation coefficient , em - l
frequency , s - l ;
\)
angle
d e fi n ed by
s p h e rical
p
(J
value
T
also
Eqs . ( A l l ) and (A 1 2 )
coordinate ,
defined by
st oi ch iom etri c coefficient
Eq . ( A 3 ) , e m
sp herical coordinate ,
p rim ary OV
w
dimensionless
rad
quantum yield ,
overall q uantum yi el d ,
mol /ein stein
m ol / ei n s t ei n
inner radius /outer r adiu s , dimensionless energy
rad
em
reactor w all tran smittance ,
em
flow rate p er
unit area ,
ein s t ein /s
•cm
2
radiant energy flow per unit of source extent ( di m ensi ons are source m o d e l de p e nde nt ) solid an gle , S r 3
reaction rate , mol /em • s S u bsc ri pts
act
relat i ve to
D
d e note s direct radiation
Dif
denotes a p rop erty re lat e d
an
activation step to diffusion
A n alysis an d Des ign of Photoreac t o rs
e
relati ve to emission
ed
denote s a property related to effective diffusion
E
denotes a property of an em er ging ray from the source
F
relative to a fixed coordinate s y stem
eff
d enote s an effective prop erty
d e notes speci es i
in
relative to the inner w all of the an n ular re act or
denotes an incident p oint or an in ciden t ray property
In
denot es indirect radiation
m
r elati ve
L
max
denot es a lam p property
to a multicomponent mi xture
denotes maximum value
o
denotes a w all p roperty
ox
relative to oxalic acid
r
relative to reception
ou P
rc
R
relative to t h e outer w all of the a nn ular reac tor
denotes point of reflection
relative to chemical reac t ion
denotes a reactor p rope rt y
Rf
denotes a reflector p rope rty
T
d en ot e s tot al value
ur
relative to the uranyl ion
z
relative to
1
d enot es a low er
2
denotes an upper limit of int e gration
the z
axi s
limit of int e g r at ion
G reek Lette rs
A.
v
p
d enot es a w avel ength d ep en de nce
d e note s a fre qu ency d epen d ence
rel ative to the p coordinate
S u persc ri pts
a if o
r elativ e to absorption
denotes i nter face prop erty
initial value
value p roject ed on the x - y plane
*
relative to the at t e nu ation p ath
91 9
920
R E F E R EN C E S
De B ernardez , Claria, an d Cassano
M . and A . E . C assano , Modelin g o f a two- phase photore actor irradiated from the bottom : I . T heory ( 1 986a , submitted for publication . ) Alfano , 0 . M . an d A . E . C a ssano , Modelin g of a two -phase photoreactor ir radi ate d from the bottom : I I . Experiments ( 1 986b , submitted for publica Alfano , 0 .
tion . ) Alfano , 0 .
M . , R . L . Romero , and A . E . C assano , R ad iation field in side a cylindrical photoreactor irradiated from the bottom : T heory an d experi ments , in Fro ntiers in C h emical Reac t io n Enginee ring, Vol . 1 , Doraiswamy and R . A . M a sh e lk ar , ed s . , Wiley E astern , N e w Delhi ( 1 98 4 ) , p . 506 . Al fano , 0 . M . , R . L . Romero , and A . E . Cas sano , Modeling of radiation tran sport and ener gy ab sorption in photoreactor s , Adv . Tran sport Proce s s , 4 , 2 0 1 ( 1 98 5) . A starita , G . , Mass T ran sfer with C hemical R e act io n , E l se v ie r , Am st erd am ( 1 96 7 ) . A starita , G . , D . W . S av a ge , and A . B i sio , Gas T reating with Chemical So l vents , Wiley , New Y ork ( 1 983) . C alvert , J . G . and J . N . Pitt s , Pho tochemistry , W ile y , New Y ork ( 1 966) . Cerda , J . , H . A . I r a zoq ui , and A . E . Cassano , Radiation field s in side an ellip t ic al photoreflector with a source of finite spatial dimen sion s , AIChE
Cerd a , J . , J . L . M arch et t i , and A . E . C assano , R adiation e ffic ie n c ie s in elliptical photore actor s , Lat . Am . J . H eat Mass T ran sfer , 1 , 33 ( 1 977) . C erda , J . , J . L . M archetti , and A . E . C assano , T he u se of simple radia tion m o del s for the case of direct irradiation of p h o t oc h e m ical reactors , Lat . Am . J . C hern . Eng. Appl . C hern . , 8 , 1 5 ( 1 978) . C hild s , L . P . and D . F . Ollis , I s photocatalysis catalytic ? J . C atal . , 68 , 38 3 ( 1 9 8 0 ) . C laria , M . A . , H . A . Irazqui , and A . E . C assano , A p r ior i de sign of the photochemical monochlorination of ethane : I . T heory ( 1 985 , submitted
J . , 1 9 , 9 6 3 ( 19 7 3 ) .
for publication . )
Clarta , M . A . , H . A . l ra z oq ui , and A . E . C assano , A priori design of the photochemical m onochlorination of ethane : II . C omputed and experimen tal r e s ul t s ( 1 98 5b , subm itted for publication . ) Claria , H . , A . I r a zoq u i , and A . E . C as sano , Modelin g and experimental validation of the ra di ation field in side an elliptical photoreactor ( 1 986a , submitted for publication . ) Claria , M . A . , H . A . Irazoqui , and A . E . C as sano , T h e u se of linear and extense source m o de l s in p hotore ac to r desi gn ( 1 986b , submitte d for p ubli
cation . ) D e B ernarde z ,
E . R . and A . E . C as sano , A z im u th al asymmetries in tubular ' photoreactor s , Lat . Am . J . H e at Mass T ran sfer , 6 , 3 3 3 ( 1 98 2) . De B ernarde z , E . R . and A . E . C as sano , A priori design of a continuou s annular photochem ical reactor . E x p er im e nt al validation , J . Photochem .
30 , 28 5 ( 198 5) .
De B ernarde z , E . R . an d A . E . Cassano , Optimal selection of the radiation source for a consecutive - chain type - reaction s in a continuou s photo reactor Ind . E n g . C hern . Process D e s . Dev . ( 1 986 , in press . ) General E lectric C o . , T ech . B ull . T . P . 1 2 2 ( 1 96 7 ) . Gosselain , P . A . , J . Adam , and P . Goldfinter , La specificite des halogena tions atomiques : II . Les rE) gles generales qui determinant le m ecanisme et a. sp ec ifici te des halogenation s a tom i q u e s , B ull . S oc . C hern . B el g . ,
65 , 5 3 3 ( 1 956) .
921
A nalysis an d Design of Photo reactors
H e r zbe r g , G . , Molecular S p e c t ra and Molecular Struct ure , Vol . I : Sp ec tra of D iatomic Molecules , 2nd ed . , V an N o st r an d Reinhold , New Y o rk
( 1 950) .
Irazoqui , H .
a . , J.
Cerd a ,
and A . E .
C assano , R adi atio n p ro file s in an
em pty annular p hotoreactor wit h a source of finite s p atial
AI C hE J . , 1 9 , 4 6 0 ( 1 9 7 3 ) .
dimensions ,
Jacob , S . M . and J . S . Dranoff , R adial s c ale - up of p erfectly mi x e d photo che mi ca l reactor s , C hern . E n g . Prog . Symp . S er . , 6 2 , 47 ( 1 9 6 6 ) . Jacob , S . M . and J . S . D ranoff , D esi gn an d an aly sis of p e rfe c t ly mixed p hotochemical re act ors , C hern . E n g . P ro g . Symp . S e r . , 64 , 54 ( 1 968) . K urt z , B . E . , Hom o ge n e ou s kinetics of m ethyl chloride c hl or ation , Ind . E n g . C hern . Process Des . Dev . , 1 1 , 3 3 2 ( 1 9 7 2 ) . Lucas , G . , A ne w concept in o xim ati o n by the nitrosyl c hlo ri d e molecule , Inf. C him . , 1 6 , 33 ( 1 97 1 ) . Marinan geli , R . E . and D . F . O llis , Photoassisted he t e ro ge n e ou s cat aly si s wit h optical fibers : I . I solated si n gle fiber , A I C hE J . , 2 3 , 4 1 5 ( 1 9 7 7 ) . Marinageli , R . E . and D . F . O lli s , Photoassisted hetero geneous catalysis II . Nonisotherm al si n gle fiber and fiber bundle , with optical fibers : A I C hE J . , 2 6 , 1 0 0 0 ( 1 9 8 0 ) . otake , T. , S . T o n e , K . H i guchi , and K . N akao , Li ght intensity p rofile in gas - liquid dispersion . Applicabilit y o f e ffe cti v e absorption coefficient , K agaku K o gaku Ronbun shu , 7 , 5 7 ( 1981 ) . Pai , S . , R adia tio n Gas Dy namics , S p ri n ger-V erla g , B e rli n ( 1 9 6 6 ) . Re , R . M . and A . E . Cassano , T hree - dimensional m o deli n g of a tubular photoreactor when th e radiation field has no angular sy m m e t ry ( 1 986 ,
s ubmitted for publication . ) Romero , R . L . , 0 . M . Alfano , J . L . M archetti , and A . E . C assano , Model ing and p aram et ric se nsi t i vi t y for an annular photoreactor with com p lex ki n e tic s , C hern . E n g . Sci . , 38, 1 5 9 3 ( 1 9 83) . S trami gioli , c ·. , F . S ant arelH , and F . P . For ab o s c hi , Photosensiti zed reac tions in an annular p h otoreactor , I n g . C him . Ital . , 1 1 , 1 4 3 ( 1 975) . Stramigioli , C . , F . S an t arelli , and F . P . Foraboschi , Photo s en siti z ed reac tions in a n annular continuous p ho to re ac tor , App l . S ci . R e s . , 33 , 2 3 ( 1 9 7 7) . Turro , N . J . , Molecu lar Pho toch emis try , W . A . B enjami n , New Y ork ( 1 965) . Vincenti , W . G . and c . H . Kruger , Jr . , In tro duction to Phy sical G as
Dy namics , Wiley , N ew Y ork ( 1 9 6 5 ) . Yokota , T . , T . I w ano , H . D e guchi , and T . T ad aki , Li ght absorption rate in a bubble colu m n p hotochemical reactor , K agaku K ogak u Ronbunshu ,
156 ( 19 8 1 ) .
7,
14
Electrochemical Reaction Engineering
C A R Y G . T R O S T , * V I C T O R I A E D W A R D S , and J O H N S . N E W M A N U nivers i t y
of Califro nia a n d Lawrence B e rkeley Labora t o ry , B e rkeley ,
Cal ifornia
I N T RO D U C T I O N
Electrochemistry i s involved t o a si gnificant e xtent i n the pre sent - d ay in d u strial economy . Examples are found in p rim a r y and secondary b atteries and fuel cells ; i n the p rodu ction of aluminum , chlorine , caustic soda , and other chemicals ; in electroplating , electromachinin g , and electrorefining ; and in corrosion . T he electrochemic al industries presently consume 6% of the total electric energy generated in the U nited States ( D arlin gton and Woo , 1 9 8 2 ; M antell , 1 960) . T he b attery i n du stry amounts to 0 . 2% of the total U . S . manufacturing ( U . S Departm ent of C o m m erc e 1 980 , p . 1 4 ; N atio n al R e se arch Cou nc i l 1 982 , p . 1 8 3) . Over 3% of the total U . S . ele c tric energy is c on sum ed b y t h e aluminum indust ry ( D arlin gton and W oo , 1982 ; M antell , 1 960) , and the chlorine industry con s u m es 2 % of the total U . S . electric ener gy ( D arlin gton and Woo , 1 9 8 2 ; Mantell , 1 960 ; B eck , 1 976 ; B oc kris et al . 1 98 1 ) . T he economic importance of corrosion is evident from the amount of money spent to p rotect what has been built and to re p lace w h at has been destroyed . It has been estimated ( W en glow ski , 1 966) that corro sion p revention products amount to 3 % of t h e U . S . electrochemical indu stry . T he q u an tity of iron destroyed by corrosion is between 25 and 33% of the total p rod u ctio n ( V an Muylder , 1 98 1 ) . In spite of the economic si gnificance of elect rochemi stry , en gineering desi gn procedures for electrochemical syste m s have not been developed as thoroughly as for the mass tran sfer operations such as distillation . Neverthe •
,
,
,
less , the fundamental law s governin g electrochemi cal systems. are know n . , T he purpose o f this ch apte r i s to review the d e si gn and analy sis o f cert ain electrochemical systems in relation to these fundam ental law s . One o f the difficulties i n de si gni n g an electrochemical p rocess i s t hat there are several different typ e s of reactors that c a n be used .
T hese re act ors can be b roadly cla s si fied as either b atch or con ti n uou s reactors . For example , many b atteries are b atch reactors , w hile fuel cells and many electrolysis cells are continuous reactors . N aturally , there are several different reactor geometries within each cate gory . For example , chlorine
* C u rren t affiliatio n :
R ay che m Corporation , Menlo Park
,
C alifornia
923
T ros t , E dward s , and N ewman
924
al . ,
and cau stic soda are p roduced in any o f three types of cells ( Bockris et
1981) .
T he di aphragm cell cont ains a vertical dimensionally stable anode ,
w here chlorine is evolved ,
and steel screen cathodes , w here hydro gen
and hydroxyl ions are p roduce d .
gas
T he elect rode compart m ents are s eparated
by a porous di ap hrag m t h at allow s passage of elect ric current . T he second type of chlor - alkali cell is the membrane cell . T hi s cell is similar to the diap hragm cell with the exception of the sep arator , w hich , in this case , is a catio n - exchan ge membrane , w hich inhibits transfer of c hloride and hydroxyl ions , and does not p er m it bulk flow of electrolyte .
T he third
type of cell , the mercury cell , t>ontains no diap hragm , b ut achieves separa tion of electrode p roducts by the use of a m ercury cathode , w here ion is reduced and forms a sodium - m ercury am algam .
sodium
The am algam is then
p assed to a s econd cell , w here the amalgam is reacted with w ater to form sodium hydroxide , hydro gen ,
and p ure m ercury ,
w hich is recycled to the
electrolyzer . A nother economically significant electrochemical reactor is t he Herault cell for the p roduction o f alu minu m .
H all
T his p rocess i s operated i n
t h e batch mode a t hi gh temperature ( app roximately
1000° C )
in a bath o f
aluminum o xi d e or alu mina ( A l zO s ) dissolved in fused cryolite ( N a sAlF s) , to which aluminu m fluoride is added .
T he therm al insulation is adjusted to
provide su fficient heat los s to free ze a p rote ctive co ati n g of electrolyte on
the inner w alls , but not on the bottom , w hich m u st m ake electrical contact A crust of fro zen electrolyte and alumina covers the molten electrolyte . Electric current enters t he cell
with the molten aluminum cat h od e .
throu gh carbon anodes and flow s t hrough the electrolyte , the anode and aluminum at t he cathode .
forming
COz
at
S teel collector bars joined to the
carbon lining at the bottom conduct electric current from the cell .
Some
review s of these and other industrial electrochemical proce sses can be found in B ockris et
al . ( 1 98 1 ) ,
K uhn
( 1971) ,
and H oughton and Kuhn ( 1 9 7 4) .
I N T RO D U C T I O N TO D ES I G N Any chemical p roce s s req uires the evalu ation o f various alternative routes ,
an elect roc he m i c al reactor are fi rst to n arrow dow n the c hoices of reactor whether they be chemical or electrochemical .
T w o m ajor t asks in desi gnin g
confi gurations and then to devise an economically optimum final design .
As
an e xam p le of eliminati n g som e reactor geo m etries , conside r metal-ion re moval from dilute w aste stream s .
In this case it would be i m p ractic al to
use two p lane p arallel electrode s because the reactor volum e would be very hi gh , whereas a porous electrode , with its high s urface area p er unit volume , w ould be a good candid ate for this ap p lication .
are also more suitable for gas - evolvin g reaction s .
Porous electrodes
O n the other hand , the
channel con figuration might be very use ful for removing valuable m etals , such as silver ,
from concentrated solutions because the m et al would be re
covered directly as a sheet , rather t han bei n g i ncorporated in a p orous structure . A fter choosi n g some electrode confi gurations to e valuate , one would w ant to estim ate the c apital and operating costs and the p roduct composition , or , in the case of a battery ,
the ene rgy and power out p u t
of t he feed conditions , the cell dimension s ,
,
as a function
and t he operatin g conditions .
T hi s can sometimes be a very difficult task because there are so m any p henomena that can int eract , s uch as m ass transfer , ohmic potential drop
,
925
Electrochemical R eaction Engineering thermodynamic s , and electrode kinetic s .
In m an y
cases , how ever , the
problem can be greatly simplified by reali zing w hich p henomena domin ate the system behavior . T his approach is exp edient in c hoo si n g d e si gn alt e r n ati v es . In
either the
c ru de d e sign or the final d esi gn
mi ze p rofit or minimi ze cost .
Thu s one should
,
it is desirable to maxi
be aware of the various
economic trade -offs that are im portant in electrochemical system s . the most obvious trade -off is betw een capit al and
o p erati n g
cost s .
Perhap s U su ally ,
the de gree of co n ve r s ion is specified , and one can m eet that spec ificatio n either by b ui ldi n g a la r ge cell , w i th a high c a pi t al cost , or by usin g m ore T here are al so ot he r t rad e - o ffs ; for ex am p le , power wit h a s maller cell . in a ch annel flow cell , one can reduce the ohmic drop and incre ase the mass transfer rate by m aki n g the i n t ere lec t rode gap very t hi n , but this will increase the p ressure drop and hence t he p ump ing cost . In addition , it will le ad to fab ri cat ion proble ms an d the po s sibilit y of s hort s .
G OV E R N I N G PH E N OM E N A I f a n ele c t roc h e mi c a l reactor i s t o b e compared t o a n alternative , b e i t chemical or electrochemical , detailed know led ge for the d esi gn and s c aleup To d e si gn an electrochemical system , one needs to consi der many is n eed e d . effects , such as m i gr ation and diffusion of charged species , fluid d y na mi cs , ther mo dyn amics , and reaction kinetics on electrode s ur fac e s . An ad dition al system variable , not pr es en t in analyses of most chemical reactors , is the electric potential . It is the difference in p ot en tial b e t w ee n t he e lect ro d e
and the solution that governs w hi ch electrochemical reactions will occur . In addition , tween
p ha s e s
grad i e n t s of electri c potential and differences in potential be cons t itut e m aj o r d rivi n g forces for m ass tr an s fer and for
electrochemical reactions . T h u s a know led ge of the c u rren t and potential distribution is useful in d e si gni n g a reactor to carry out a desired e lec t ro ch emi c al r e ac ti on . To de s c ri b e mass transfer in an electrochemical system , one must con sider no t only ordinary di ffu sion , but also migr ation o f c h arged species i n an electric field . B ecause it is t he flux of c h ar ged species that p roduces an e lectric current , one must include the contribution of mi gra t io n to the
speci e s flux to p redict correctly the current flow . In ad di tio n to m a s s tran s fer , one m ust consider thermodyn amics in elec troc he m i cal system s . T he rmodynamics can be us e d to d esc ri b e t he p roperties of electrolytic solutions and their dependence on co m p o sitio n , temperature , and pressure . T h e r mod y nami c s al so pro vi de s a fr am ew or k for d es c ribi n g reaction equilibria , w hich m ani fe st themselves in equilibrium cell potentials . Furthermore , the driving forces for irreversible processes are co nv e nien tly e xp re s s ed in t he rm od y n am i c t er m s . Departures from equilibrium co ndi ti ons are inherent in electrochemical ap plic ation s . E lectrod e kineti cs conce rn s t he non eq ui lib riu m d ri ving force , called surface overpotential , necessary to m ake het ero geneou s electrode re
actions proceed at appreciable rates . In ad dition to h et e ro ge n eou s reactions , there are o th e r interfacial phenomena at e l ect rode s urfaces that can affect the current - potential b e havior . T he m o s t p rominent of these i s the fo rm atio n of a thin double lay e r near electrode surfaces . T his double layer ma y be on t he order of 1 0 to 100 A ( 1 to 10 nm ) in thickness .
T rost , Edwards , and N ewman
926
Becau se there are so m any interactin g phenomena , it can sometimes be very difficult to desi gn an electrochemical system .
In t he present age of
computers , however , many prob lems that w ere intractable
30
years ago can
be handled today . In spite of the power of computers , however , it is still useful to plify the p roblem if certain p henomena dominate the system behavior . example , t here are some systems w here it tion variations near t he electrodes .
is
sim For
possible to neglect concentra
T he current distribution is t hen de
t ermined by t he ohmic potenti al drop in the solution and by electrode over · potentials .
( T he ohmic d rop is si m ply the voltage difference that arises
w hen c urrent flows throu gh the resistive solution . )
M athematically , this
means that the potential satisfies Laplace ' s equation , and m any results of potential theory , developed in electrostatics , the flow of inviscid fluids , and s teady heat conduction in solid s , are directly applicable . be c alled "potential- theory problem s " kinetics
provide s
( Newman ,
1968) .
T hese can
The electrode
boundary conditions which are u sually different from those
encountered in other applications of potential theory . T here are also systems where the ohmic potential drop can be neglected . T he cu rrent di stribution is t hen determined by the same principles which apply to heat trans fer and nonelectrolytic mass tran s fer . called " convective - transport problems " ( N ewman ,
1968 ) .
These can be
Some systems do not fall into either of the categories mentioned above. Fortunately ,
however , some of these complex systems can be treated realis
tically by using the computer to model t he interactin g phenomena .
F U N D AM E N T A L E Q U A T I ON S T o calculate the current and potential
distribution ,
which is necessary to
desi gn an electrochemical system , one needs a set of fundamental equations applicable to electrochemic al systems .
T hese equations include a description
of the movement of mobi le ionic species , an exp ression for c urrent flow , m aterial b alances , elect roneutrality ,
and equations from fluid mechanics .
In
additio n , t he differential eq uations describin g the electrolytic solution re q uire boundary conditions describing the thermodynamics and kinetics of electrode reactions . In dilute system s , the flux of each species is given by ( Newman ,
1 97 3a)
( 1) where ci> is t he electrostatic potential and Zj is the number of proton char ges carried by ion i .
T hi s movement is due ,
the fluid wit h the bulk velocity v .
However ,
first , to the motion of
the movement of the species
II Ci , or by mi gration i f there is an electric field , - II ci> , and the species is char ged . The mi gration term is peculiar to electrochemical
can deviate from this average velocity by di ffusion if there is a concentra tion gradient ,
systems or systems cont aining c h ar ged species . T he second and third terms on the ri ght side of E q .
( 1)
are the u sual
terms required to describe diffusion and convection in nonelectrolytic sys tems .
The species will diffuse from regions of hi gh concentration to regions
of low er concentration .
T he three term s on the right in Eq .
represent three mechanisms of m ass t ransfer :
( 1 ) t hus
mi gration o f a charged species
927
Electrochemical Reaction Engineering
in
an electric field , molecular diffusion due to a co ncen tratio n gradient , and convection due to the bulk motion of the mediu m Note that Eq . ( 1) only applies to dilute solutions . A more general form of Eq . ( 1) is ( New man , 1 973a) .
1
1
c. V lJ .
c. c.
=
RT
( v. � D___!_j_ c ..
1] T
j
-]
-1
( 2)
- v. )
where ll i is t he electrochemical potential o f sp eci e s i . T his equation must be applied to systems , such as molten salt s ys t e m s , where the ac ti vit y co efficient s are not close to unity , or to systems where the diffu sion flux of each species can be affected by more than one concentration gr adien t . The driving forces for diffusion and mi gr ation are both includ ed in the gradient of the electrochemical potential in Eq . ( 2) Fu rt h e r details of concentrated solution theory can be found in Newman ( 197 3a) . After writing the appropriate flux equation , one can write an exp re s sion for the current , which is , of course , due to the motion of char ged particles in an electrolytic solution . •
( 3) Here i is the current density expressed in amperes per square centi meter , is the charge per mole . Next we need to state a m at e ri al balance for any component i :
and zjF
=
- 'V
•
N.
-1
+
Since reactions are frequently restricted to the surfaces of elec trod es often zero in elec t ro c he mi cal systems . Finally , we can say that the solution i s electrically n e utral :
bulk reaction term Ri is 1 1
z.c. i
=
0
( 4)
R. 1 ,
the
( 5)
S uch electroneutrality is observed in all solutions e x c ep t in the t hi n double charge layer n ear electrodes and other boundaries . T hese equations p ro vi d e a consistent d e scription o f tran spo r t processes in electrolytic solu tion s . E quation s ( 1) and ( 2 ) s t at e t hat species in the so luti o n can move by migration , diffusion , and con ve ction . Equation ( 3) merely says that th e sum of fluxes of charged species con stit ut e s an e lectri c current . Equation ( 4) is a material balance for a species , and Eq . ( 5 ) i s the condition of electroneutrality . Although the specific d e sc riptio n m ay be refined any theory of elec t rolytic solutions will need to con si d er these physical phenomena . Note t h at in order to solve the e q u a tions describin g the mass transfer , it is necessary to know the c on vectiv e velocity v . T his velocity can be ,
Trost , E dwards , and
928
N ewman
found from the equations of flui d mechanics , such as the N avier - Stokes
e q uatio n
( 6) a n d th e c on ti n ui t y e q uat io n =
V •v
( 7)
0
Next one
p o t enti al
needs b ou n d ary con dition s de scribin g the differences in
b etween
.
T hese bound ary conditions describe the processes
phase s .
o c cu rrin g at the ele c t rod e i nt e r fac e
.
-
the ad s orp ti on of ions on to the surface . ( U s ually , anions are s p e ci fi c ally a d so rbe d ) A dou b le ch ar ge layer is form e d becau s e the ions of one si gn are attracted from t he solution to the adsorbed ions of the op p o si te sign . T hi s attractive force , however , is One p roc ess that c an occur is
,
balanced by t he rm al agit ati o n w hich tends to make t he ions w ander . T herefore , the double - char ge layer has a diffuse part in the so lu ti on is ,
the charge d e nsit y
t hi c kne s s of this
the
diffuse De b ye length ,
(
=
2
)
Lz:c. 1 1 100
T he De bye le n gt h
is
( 8)
t ypically on the order of 10
A . T h e major e ffe ct of interface is that it s up er electrode kinetic s of the electrode
the double layer on the over all be havi o r of
imposes a cap acitive e ffect on top reac tio n itself. This means that varied , the cu r r e nt that flow s is
cap acity ,
That
1 /2
e::R T F
.
T he layer of excess char ge density is c h ara c t e ri z ed by
decreases with dist ance from the interface .
of t he
t he
w hen t he p ot e n ti al of t he electrode is p artly due to ch ar gi n g t he do u b le - l ayer
and p artly d ue to the char ge - transfer reaction .
measure the cap aci t ance of
the do u bl e layer in
One can
several w ays ( Newman , 1 9 7 3a ; G rahame , 1 9 47 ) . Typically , t he double - layer c ap acity is 10 to 40 J.1 F /cm 2 ( B ar d an d Faulkner , 1 9 80) , but this capacit ance is a fu nc tio n of T he double layer is di scu s se d furt her in N e w m an ( 1 9 7 3a) , p ote n tia l
.
G rah am e ( 1 9 4 7 ) , ( 1 95 4) .
B ard an d
,
Faulkner ( 1 9 80) , D elahay ( 1 96 5 ) , and Parsons
A lth ou gh the e xi s t ence of the double layer should not be n e glec te d it charge - transfer reac ti ons at the e lect ro de s w hich are usually of p ri m a ry im port an c e in electrochemical systems of practical interest . T he kinetics of these reactions relate the potential driving for c e to the rate of reaction . T hus a kn ow le d ge of the e lec t rod e kinetics is nece s sary to formu late b o u n d ary conditions for the differential e q u at io n s describing t he move is the
ment of ionic sp ecies . For the p urpose of a s s e s sin g potential vari ation s in a solutio n , it is convenient to co n ceiv e of p l acin g reference electrodes into the solution at
approp ri at e locations , usually just outside the di ffus e Fi gure 1 illu s t r at e s the p la c e m e nt
the bulk sol u t i o n .
electrodes ( W hit e ,
1 97 7 ) .
the sam e kind as the
double layer or
in
of these reference
T he letter s desi gnates r e fe r enc e electrodes of
workin g
ele c tro d e , and the
l et t e r
g
r efe r s
to re fe re nce
929
Electrochemical Reac tio n E ngineering
D i ff u se
Wo r k i n g
Do uble
D i f fu s ion
L a ye r
Elec tro d e
L o yer
Su l k
Soluti o n
® r ls
® r 2s
® r ig
®
r 2g
F I G U RE 1 R e fe re nc e electrodes , which may be im aginary , positioned in the bulk solution and within the diffusion layer . ( From White , 1 9 7 7 . )
It is important to reali ze t hat a reference electrodes of a gi ven kind . electrode of a " gi ve n kind" cannot be a re fe re nce electrode of any kind . Instead , the reference electrode must be carefully selected to be as re versible and reproducible as possible . The s election of reference ele ct rode s is di sc u ss e in N ewman ( 1 97 3a) . One m ust also realize that a m easurement made by a re fe renc e electrode of a given kind must be co r re cte d for any liquid -junction potentials that might exist between the solution in question and that within the reference - electrode comp ar t m e n t ( see N ew m an , 1 97 3a ,
d
Sec . 40) . In the ab sence of concentration variations , the two reference electrodes show n in Fi g . 1 would measure the ohmic potential d rop between the two point s l ab e le d 1 and 2 . In the presence of concent r ation variations , the p ote nti al difference between the t w o reference electrodes will be composed of an o h mi c portion and what is called a concentration overpotential . The conc ent ratio n overpotential , nc , reflects the thermodynamic effect of the difference in concentration between the interface and the bulk solution . An add iti on al contribution to the o ve r all cell po t e nti al is t he driving force required to make the e le ctro d e reactions p roc eed at app re ci ab le rates . This surface overpotential , n s , is defined as the pot enti al of the working elec trode relative to a reference electrode of the same kind , placed in the solution adjacent to the surface of the working electrode , but just outside the diffu se double layer . In term s of the notation of Fig . 1 , t hi s surface overpotential for a re a cti o n j w o u ld be w ritten n
sj
= V -
V
v
41
or =
-
=
r 1s
0
-
V -
u.
v
V r 1 g - ( r ls
.,
] ,0
( 9)
( 1 0)
where � 0 = V r l g is the potenti al just outside the double layer , as me as u re d by a reference electrode of a gi ven kind , and u. J ,o
=
v
r ls
v
r lg
( 1 1)
930 u.
] ,0
=
(u�
RT ""
_
S,j
n.F LJ J i
J
::) - (u:.
In
T ro s t , E dward s , an d Newman
-
n::F �
•
; , re
In
!:
c ••
)
( 1 1)
Here the subscript re denotes t h e reference electrode reaction . T h e stoichiom et ric coe ffi cient , Sij , refers to species i in a reaction j , w ritten in t he form ( 1 2) i
where nj denotes the num ber of electrons transferred , i , and Zi is the charge number of sp ecies i .
Mi
refers to
species
For a large class of electrochemical reactions , the current density de
p e nd s exponentially on the surface overpotential ac cor di n g to the B utler Volmer equation =
i
[exp (a:� .
0j
F
) - exp (-aR� )] .
ns
j
F
( 1 3)
n sj
where ioj is the exchange current density for reaction j and depends on concentrations Ci . Usually , this dependence is e xpr e ssed as 1o!i'
.
=
.
1OJ.
, re f
rr (--)yij .
1
c io
the
( 1 4)
c 1. , re f
In Eq . ( 1 3 ) , the first exponential te r m corresponds to t he forward reaction rate for an anodic process , and the second term rep resents the reverse re action rate . Note that if the anodic surface overpotential is large , the re verse reaction term can be ne glected . This approximation , known as the T afel ap p roxim ati on , produces a linear plot of surface overpotential vs . the logarithm of current d ensity . A more thorou gh discus sion of ove rpot e nti al s can be found in Newman ( 1 97 3a) and White ( 1 977 ) . We have now introduced e nou gh of the elem ents to be in a posi t io n to di s c u s s the composition of the overall cell potential . T his is due partly to the ohmic potential drop i n the solution . In addition , there is a p ot e nti al los s as sociated with the c oncent ration variations in the solution near elec trodes , w hi ch we have termed the concentration overpotential . Fin ally , there is the surface overpotential due to the limited rates of the electrode reactions . T he sum of these is the cell potential , w hi ch can be w ritten V
= =
=
il>( cathode )
( anode)
( cat hod e ) + <1> 2
-
( ano de ) n s ( an od e )
+
ll ( ano d e ) c
+
fj
oh m
- ll ( c at h od e )
c
- ll ( c at hod e) s
( 15 )
where .P 1 i s the value that the potential adjacent to the anode would h ave if there w ere no co n ce nt r ation variations in the solution . The term s ll c and
E lectrochemical Reac tion Engineering Anode
Dif fus
D i f f us i o n Loyer
L oyer
Double
Bu l k S o l u t ion
Di f f usion Diffuse Loyer
----r.
(anode !
'l s
Double La ye r
Ca t h o d e
931
+ 'lc (ano d e )
--=,--I
I
Curr e n t ( p o s i ti v e charg e )
-+
- 'lc (cathode}
-t1
- ') 5 (cat hode) '
I
_j
FIGURE 2
___ _
Overpotentials in an electrolytic cell . T he solid line the absence of concentration variations .
p otenti al distribution in
represent s
n s for the cathode enter with negative signs because of the convention s that have been adopted . Since they are generally ne gati v e they make a p o siti ve contribution to the cell poten tial T hus none of these terms , ohmic drop or overpotentials , represents a source of energy . An example of how the vari ous overpotentials contribute to the overall cell potential is shown in Fig . 2. T he dashed line rep re se nt s what the potential i n the so l ution would be if all concentrations were at their bulk values . Note that this decomposition p resum es a b ulk solu tio n and a diffusion layer adjacent to each electrode . In sit uations where this does not prevail , such as a cell with ov erlappin g diffusion layers , or a porou s electrode , it is necessary to start over from the basic equations , but the fundamentals re ,
.
main the same .
C H A N N E L F LOW C E L LS
Channel flow between two p arallel planar electrodes is used in m any indus trial electrochemical processes , such as met al refining, energy storage , and electro -or ganic synthesis . Specific examples include copper refining , some zinc h alo gen energy storage cells , and the Monsanto process for conversion of acrylonitrile to adip onitrile -
.
T ro s t , E dwards , and N ewman
932
Channel flow cells are very useful, since they p rovide continuous p ro duction , they are simple to operate and maintain , and they do not require a high capital investment ( Fit zjoh n , 1 9 7 5 ) . In addition , the analysis of channel flow cells has been relatively well developed (Sakellaropoulos, 1979 ; Sakellaropoulos and Francis , 1979, 1980; Parrish and Newman , 1969, 1970 ; Caban and Chapman , 1 976 ; Lee and Selm an , 1 9 8 2 ; Jorne, 1982) . A ch anne l flow cell consists of two parallel plates , which serve as the anode and the cathode for electrochemical reactions . The electrolyte flows past the electrodes , and the current flow is perpendicular to the fluid flow . In general , a thin-gap cell with multiple reactions occurring on each electrode is difficult to analyze . The electrodes cannot be treated separately and mass t ran s fer , thermodynamics , and the kinetics of more than one re action must be considered simultaneously . This problem can be solved , but for de sign purposes , it is useful to see if any simplifications can be made. One assumption that is commonly u sed is that the diffusion layers are thin. In this case , the electrodes can be treated separately. This is known as the Lev&}ue ap proxim ation (Newman , 1973a ,b) , and it is valid as long as L h
( 1 6)
O . OlReSc
<
where Re =
2 < v>h / v v
(about 1000 for most aqueous systems) L length of electrodes spacing between electrodes h Thus we see that the thin-diffusion-layer app roximatio n breaks down only if the electrodes are very long , the gap is very thin , or the velocity is very low . Parrish and Newman ( 1970 ) used the Lev@que approxim ation to obtain the cu rr ent and concent ration distributions in a channel flow cell. T hey exami ned the case of a metal deposition and dissolution reaction. Intuitively , one would expect that mass transfer would not be important near the fro nt of each electrode , where the ions in so l u tion are still unreacted . Farther down the channel , however , the solution becomes depleted of metal ions. Mass transfer then begins to affect the cathodic deposition reaction The anodic dissolution reaction, however , is not affected by the concentration of ions in the solution . Therefore , we would not expect the anodic reaction to become mass-transfer limited . In the ab sence of mass transfer, the current distribution is governed by the ohmic potential drop and the reaction kinetics . This current dis trib u tion is called the secondary current distribution . I f there are no kinetic li mit a t ion s however , the current di s t ri b uti on is calle d the primary distribution. To calculate the p rim ary current distribution , one would solve Laplace's equation for the p otential , assuming that the potential ad jacent to each electrode is held at a constant value. For two elect rodes embedded in planar channel walls , the primary current density is infinite at the electrode edges . The secondary current distribution is s mi lar to the primary distribution , bu t it is more uni form because there are kinetic limitations . Therefore , the current density is not infinite anywhere , b ut it is still higher at the electrode edges . Sc
/D
=
=
.
,
i
Elec trochemical Reaction Engineering
933
I f mass transfer dominates the current distribution , on the other h an d , the current density is high only at the front of the electrode , and it de creases with distance down the channel . For an anodic dissolution reaction , one would expect that mass transfer would not be important . Therefore , the current distribution should re semble the secondary c u r ren t distribution . Parrish and Newman found that this is indeed the c as e . T he cat hodic current distribution , how ever , should behave like a secondary current distribution near the front of the electrode , but like a mass- transfer - limited current distribution near the b ack . In fact , Parrish and N ewman's numerical results show maxima in the cathodic current densities . These m axima ari se from a compromise between the secondary current distribution and the mass-transfer-limited current dis tribution . Throughout their analysis , Parrish and N ewman assumed a single reaction at each electrode . Very often , this simplifyin g assumption is valid ( Pickett , 1 9 7 9 ) . O ccasionally , however , multiple reactions must be considered ( Sakellaropoulos , 1 979 ; S akellaropoulos and Franci s , 1 97 9 , 1980 ; White and New m an , 1 97 7 ) . T he assumption of a si n gle reaction may be checked by identifying a sin gle p arameter which characteri zes the manner in which a side reaction tends to obscure the limitin g - current plateau for a main , or de si red reaction ( White and N ewm an , 1 9 77 ) . It is t he magnitude of the side reaction relative to the main reaction at potentials in the nei ghborhood of the limitin g -current plateau which is import ant . Fi gure 3 illustrates this concept for the example of copper deposition as the m ain reaction and hydrogen evolution as the side reaction . The ,
M a in Anodic React ion � � -�
--- ---
�
-!!
�0 Q)
w -o. - o.
F I G U RE 3 Q ualitative sketch of the current - potential curves for a main and a side re action showing some of t he param eters defined in t he text . { From Whit e ,
1977 . )
T ro st ,
934
Edwards , and Newman
solid line , labeled " main c at hodic reaction , " is the cu r re nt - p ote nti al curve
of inte re st .
are plots of the potential di ffere n ce , lo g �t hm of the current density . where V is the ele c t ro de po te ntial and
V
The curves in the figure
- � 0 , versus t h e
Tl
sj
=
v -
:0.
... o
-
u.
( 10)
] ,O
where U j , o is defined by E q . ( 11) . Figure 3 sho w s the followin g four curve s : 1. 2.
3. 4.
T afel approximation o f T afel ap pro xi mation o f ge n io n s ) T afel ap proxim atio n o f M ain cathodic re ac tio n
m ain anodic re actio n ( c opp er di s solu tion ) side anodic re ac t io n ( p ro d u ction of hydro
side cathodic r e acti on ( hy d ro ge n ev olutio n ) ( copper depo sition)
T he heavy line for at high I V - Cli 0 I ,
the m ain cathodic reaction ( curve 4) has t wo asymptotes : the re action i s mass- tran sfer controlled ; at low I V - Cli 0 j , the reaction is kinetically controlled . T he Tafel approximation is app lie d to the cathodic reactions , since it is unlikely that the b ack w ar d terms in the B utler - Volmer e qu atio n are i m po r t ant in t he neighborhood of the li miti n g - c u rrent plateau . w here - Tl s , m i s lar ge. T he anodic re ac tio n s are p lotted as T afel lines to illustrate that the limiting-current plateau is not a ffe cte d by either an odi c reaction . A lt hou gh
the T afel approximation is not valid near i = io , we are interested in the neighborhood of the lim it in g - cu rre n t plateau , which is far fr om fo . The r e fore , we only examine the p ortio n of the T afel curve ne ar the limitin g current plateau , w h er e the b ac k w a r d ter m s in the B u t le r - V olm e r equation
can be neglected . As long as there are no m a s s - t r an s fe r limitations , the plot of V - i v er s us log I i I will re m ai n linear . N o te that the so li d line for the main cathodic reaction does start to be n d over as m ass tr ans fe r comes into play . T he side c athodic reaction , hydro gen e volu tion . however , has no mass t r an s fe r li mit ati o n s b e c a u s e hyd ro ge n evolution o ccu r s from a so lu tio n with excess hy d ro gen ion in t he s up p ortin g elect rolyt e . To p lot a T afel line , one n e e d s not only the s lop e , but al s o the v alu e of V - � 0 at a sin gle current density , i . If this cur re nt density is chosen as the exc hange current den sity at the reference concentration s , then
V -
m , re
m , re f
f is gi ven
:::
e u me - u re
( 1 7) by
RT
n F m
� i
c s. In 1m
1. , re f
P
o
+
RT
n
re
F
� i
c. s.
1 , re
In
1 , re
P
o
( 1 8)
935
E lec trochemical Reac tion Engineering
-
,
T h e T afel app ro xi m ati on can also b e w ritten for the m ain reaction an o dic d irec t ion . A gain , at i io m re f I V ¢ 0 1 i s U m re f In ,
the
=
•
in
·
fact , the anodic and cathodic T afel lines , for any reaction j , int e r sect at ioj , re f and U j , r e f if the surface concentrations are at their reference values . T herefore , for any reac tion j , if o ne know s the exchan ge c ur rent densities ioj re f , and the anodic and c athodi c transfer coefficient s , ex cj and
,
exaj ,
•
on e can plot lines similar to t ho se show n in Fi g . 3 , T he parameters A m and A s , also shown in Fig . 3 , char act e ri ze the magnitude of the anodic p art of a reaction , at the electrode p ot ent ial for which the main reaction i s be ginning to reach the limiting current , T his defined (White
p ar ameter is
)
=
A.
.
CJ
(
1+ ex . / ex .
.
1o •
a1
, re f
1m ,
lim
exp
a
•
)
1 97 7 ) , for reaction j , as
and Newman ,
F
l
....!J_ l\ U . RT
( 1 9)
It s ho u l d be noted that i f the surface concentrations are not at the
refer en ce values , the Tafel lines will be shifted . For example , if the con c en t r atio n of dissol v ed hy d ro gen is les s than c H ref • the T afel line for 2,
- .
T h e exact placement of the an odi c side reaction will b e shifted to the left . case , however , does not affect the len gth of t he limitin g current plateau for th e m ain re action In fact , neither anodic reaction affects the length of the li miti n g c ur r e nt plateau . T hus , since t he li mit ing cu rren t re gion is of g reat e st interest , the anodic reactions are not i mp ort ant T he electrode p ote ntial , V m , for w hich the m ai n reaction is b e gi n ni n g t o re ac h t he li mi tin g current is , in the absence of an ohmic p ote nti al d rop ap pro xi m at e ly equal to thi s line , in t hi s
=
U
m
, r ef
RT
ex
em
F
In
-i
,
m , lim
( 20 )
i om , ref
( 20 ) , the cu r re n t d en sit y for the side re action divided by the limi ti n g current d e n sit y for the m ain reaction is pro portional t o exp ( - a F l\ U / RT) , where cs s
At the potential defined by Eq .
l\ U
s
=
U
m , ref
. -
-
U
RT
s , ref
ex
cs F
i
In
o s , ref
-i
m , lim
+
-RT
ex
em
F
In
i om , ref -i m , lim
�_:...-=..
corresponds to the le n gt h of t h e limitin g- current p lateau , as s hown in 3 . The p ar a m et e r l\ U s s how s that both the exchange c urre nt density
t he op e n ci rcuit
-
( 21)
Fi g and
.
potential determine the relative si gnifi c an c e of a si d e reac If the exchan ge current densi t y is very sm all or the open ci r c ui t ti o n p ote nti al is quite ne gative , l\ U s will be large , and the side reaction will not obscure t he limiting-current plateau .
C E LL O PT I M I Z AT I O N The goal in d esi gni n g any rea ct or i s to m ax1m1 ze the p rofit or to mi ni mi ze the total cost by making an ap p ropri at e choice of operating variable s and
.
T ros t ,
936
E dwards ,
an d
N ewman
cell dimensions . N atur al ly , there are constraint s , such as m ass transfer and cu r rent - pot ent i al relationships , as well as feed and p rod uct speci fica tions . U sually , how ever , there are few e nough constraints t hat there i s eno u gh flexi bili t y to perform a cost mini mi zation . An example of t h e trade -off betw een capi t al and oper ating costs in a fuel cell has been discussed by Newman ( 1 9 7 9 ) . Ibl and cow or k e rs ( lbl , 1 97 7 ; lbl and A dam , 1 96 5 ; l bl and Robertson , 1 973) have also treated the desi gn and optimization of various electrochemical system s . Alkire et al . ( 1 9 8 2) have dis cu s s e d t he o ptimi z ation of electrochemical systems having m any variables . We s hall discuss here the example of optimi zation of pot ential and hyd rogen utili zation in an acid fuel cell . A fuel cell for c o m m e r ci al pow er ge ner ation consists of a device to re form a hydrocarbon fuel to yield hydro gen and an electrochemical reactor , where this hyd ro ge n is com bined with oxygen from the air to produce w ater vapor and electric power. In this example , w e shall perform a sim plifie d cost analysis , w he re only the capital costs and t he fuel costs will T he fuel cost , denoted C f , w i ll include all op erati n g cost s be included . which can be associated with or set proportional to the co n sump tion o f fuel . The trade- off between T hus the reformer costs will be included in C f . capital and operati ng costs results because there is an op timum fue l utili za tion beyond w hich t h e increase in electrode area costs more than can be jus tified by t he increase in electric powe r p rodu c ed . T o p er for m the optimi zation , one m ust first w rite an expres sion for t he c os t to be minimi zed . In this ex am ple , we shall write the cost in terms of the hyd ro gen utili z ation u and the cell p otential V , where u tiliz ation is de fined as the fraction of the hydrogen in t he feed that is consumed in the elec t ro ch emi c al reactor . To write the cost in ter ms of only the variables u and V , one need s to know how the cell potential depends on current den sity and gas consumption . O nce the cost is w ritten in t erm s of u and V the optimi z at ion is p erformed by simply s etti ng the de ri v ati ves of th e cost with re spect to u and V e q u al to zero and solving for u and V . To c alc ulate the relation ship amo n g cell potential , current den sity and gas co m p o sition , one can a s sum e that the air cath ode obeys T afel kinetics . At t he hydroge n electrode , on the other hand , the surface overpotential is negligible and the Nernst equation applie s , w he re the anode potential di ffe rs from the potential of the adj ace nt hydrogen re fer en ce electrode mainly be cause the hy dro ge n partial pre ssure differs from the r e fer en c e pre ssure of 1 atm . Finally , one can calculate th e oh m i c potential drop within the cell and com bine the three equations to obtain the r elatio n ship amon g cell po tential , curren t d e n sity , and ga s co mpo sition ,
v
=
U' +
_ RFT _
P o PH
1/2
;: 1 --=2'-In _...;2'-
- R'I
( 2 2)
The p artial p ressures of oxy gen and hydro gen appearin g in Eq . ( 2 2 ) are at the catalyst l ayers and must be obtained from the values in the flowing stre am s by allowanc e for t he gas -p hase m a s s - t ran sfe r resistance through the inert gas i n the ele ct rod e sub strates . These p artial p ressures are then substituted into Eq . ( 2 2 ) to obtain the desired current - potential relation
ship . N ext , one need s an expres sion for t he total cost . In t hi s example , the to tal cost is the fuel cost plus the c apit al cost . Let C f be th e value of a
937
Elec trochemical Reaction En gineering
unit of hydrogen in the feed , and K u C f be the value of hydrogen in the exit fuel stream after a utili zation u , where C f is expressed in $ / C . Let the capital costs be expressed by C a , having unit s of $ /s •cm 2 , based on superficial electrode area . C a is obtained by multiplying the total capital cost ( $ /cm 2 ) by a factor such as 0 . 3 yr- 1 , representing the effect of in terest , depreciation , and t axe s . Components with a 5-year life and a 10% interest rate mi ght give rise to this factor . The total cost can now be ex pressed as cost
C 2FF X0 f a H
=
time where
F
F X
a
exit H
2
-
C ..K
r-u
2FF X xi a H 2
e
t
+ C A
( 2 3)
a tot
F arad ay ' s constant , 96 , 487 C /equiv
= =
flow rate of inert s on anode side , mol ls
=
ratio of moles of H
=
ratio of mole s of H
2
2
to mole s of inerts in the exit stre am
to moles of inerts in t he feed stre am 2 2 superficial electrode are a , em
T he fi rst term on the right is the cost of the hydrogen feed stream , the second term is the resale value of the hydrogen exit stream , and the last term is the capital cost . We are concerned with minimi zing the cost per unit of electric energy , which is t he ratio of the cost per unit tim e to the electric power p roduced , w here
Pow er
=
=
P
[A
tot
( 2 4)
VIdA
0
2
Here I is the s uperficial current density , expres sed in A /cm • As a step toward w riting the total cost in term s of u and V , w e can relate the flow rate and consumption of hydrogen to the electric current by Faraday ' s law ,
2F F ( X 0
H
a
- X
2
H2
)
0 = 2FF aX H
u' 2
=
(OA } n
I dA
( 25 )
where u' denotes t he value of the utili zation u for a local mole. ratio x 8 o f 2 hydro gen . N ow w e can write t h e total area as
A
tot
=
(u H 2 J0
2FF x o a
du '
and the electri c power from E q .
P
=
( 2 6)
I
( 2 4 ) becomes
u
( H 2 Jo
2FF X 0 a
V du'
( 27)
T ros t , E dwards ,
938
and
Newman
B y m eans of these relationship s , the pow er cost take s the form
c
C
fo
=
e
+
C � ( l - U) f U
C
fo
U ( du' / 1 )
a
u
( 28 )
V du'
T hi s equation h a s been w ritten so that the exten sive q uantities - power re quired , electrode area A tot , and flow r ate of the fuel stream - have been elimin ated in favor of the utili zation u , an intensive variable .
We restri ct ourselves here to the case w he re the cell potential V is a
cons tant for
all
values of the utili zation ,
of the power cost in Eq .
differentiation with respect to u gives ,
c
I
I
=
-
K
u
+
u
C
u
Now w e can set the derivatives
( 28 ) with re spect to u and V equal to zero .
i
0
u
du'
- + (1
I
T he
after some manipulation , dK U u ) -du
-
( 2 9)
where C
= C a / C f is t he ratio of the capital cost to t he fuel cost and has the units of A /cm 2 . The differentiation with respect to V gives
r Jo
u
-vc
=
(1L) av
du ' u
1
( 30)
2
T he result s of t his optimi z ation show that w hen depleted hydrogen can be sold at cost , that is ,
Ku 1 , it is economical to operate with the richest fuel and airstream s available , as expected . T hu s the optimum utili zation obtained from E q .
_ rr (� L) = +
() I
C
( 2 9) is u = 0.
Substituti n g t hi s into E q .
( 3 0 ) gives
Y
( 3 1)
I
T his equ ation , w hen coupled with the curren t - potential relationship gives the optimum current density or cell potential . to fuel cost , app roachin g infinity , E q . w here dVI /dl
=
0 , is the optimum .
For
C,
the ratio of capit al
( 3 1 ) s ays that the power m aximum ,
As fuel costs become more important
and C decre ase s , optimum operation is at a current density below potential above ) the power m aximum . favorable power co st .
c
e
=
c
f V
(1 £) +
T he li mit K u =
( and a
1 will give the most
( 32)
I
=
T he other extreme is K u 0 , w here w aste hydro gen h as no value . T his will be the worst case and will yield the highest op timum utili zation , for a given cost ratio
C.
Electrochemical Reaction E ngineering
939
PO RO U S E LE C T R O D ES In t he discussion ab o v e we have con sidered a flow c hanne l as a possible re act or con fi gu r ati o n . We wish now to focus attention on porou s e le c tro de s as another class of elect rochemical reactors . Porou s electrodes can be u se d in flow - th rou gh configurations as in fuel cells , re d ox energy st o r a ge systems , and c h e mic al reactors or in closed con fi gu ratio n s as in many p ri m ary and seco nda ry batteries . T hus p o rou s electrodes find potential ap p li c atio n s as electrochemical reactors in a variety of areas . Flow -through porous e le ct rod es are reviewed by N e w m an and Tiedemann ( 1 9 78) . T he m as s tran s fer of reactin g sp ecies within t he fixed bed and the ohmic potential variation throu ghout the bed are t reat ed in de tail . A s econd review article treats porous e lectrodes with re gard to bat tery ap p lic ations ( N ew m an and Tiedemann , 1 9 7 5 ) . A liter at u re s urvey is gi ven in each review . T wo distin g·uishin g featu re s of p o rous electrodes are the intimate con tact of the electrode with the solution ( and possibly a ga seo u s phase ) and the high surface are a- t o - volu m e ratio that can be obtaine d . The high sur fac e area- to-volume ratio is important to application s where the intrinsic rate o f the he t ero geneo us , electrochemical re action is slow , In processes u si n g dou b le - l ay er ad s or p tio n , t h e hi g h surface area is again import ant . Dilute reactants in solution re qu i re the close proximity of solution and elec t ro d e to e n ha n c e mass t r an s fe r to t h e electrode surface . In battery or fuel-cell ap p li c ation s , porous electrodes offer a means for st ori ng the soluble reactant s in c lo s e p roximity to the ele c t rod e surface . For no n con duc t in g reactants of low solubility , another solid p h a se ( as in batterie s) or gas phase ( as in fuel cells) may be incorp orate d into the sy stem .
Flow - throu gh porou s electrodes cou ld find ap p li c ations in the area of This includes the purification , e le c t rowi n ni n g , and possibly e lec tr opl ati n g of aluminum , copp e r , m a gn e si u m , sodium , m an ganese , metal p roce s sin g .
nickel , gold , silver , and chromium . T h e electrore fining of aluminum from an aluminum - man ganese alloy mi ght involve a flow -through porous anode in order to p re vent t he di s sol ved m an gan e se fro m re a c hi n g the c a thod e . Solu tions too dilute to treat with e l e c tro wi n ni n g p rocesses may be treated w i t h Dilute aqueous metal removal and recovery a flow - t h ro u gh porous electrode . of copper ( B e nnio n and Newman , 1 97 2 ) , s i lv e r ( V an Z ee and N ewman , 1 9 7 7 ) , and m ercury ( M atlos z an d Newman , 1 9 8 2 ) have been studied in fi xed - bed , flow - t hrough porous electrodes . C opper w as recovered from feed st r eam s of 667 ppm with e ffl u en t concentrations les s than 1 pp m , wi t h si m u lt aneou s p rodu ctio n of a con c ent r at e d stream of 47 , 66 0 ppm . Copper recovery h as also been investi gated in fl ui di z ed -b ed reactors ( Fleischm ann et al . , 1 97 1 ; Germain and G ood ri d ge , 1 976) . G old can re adi ly be recovered from p l ati n g baths which h av e d et e rio rat ed , an d s t ream s fro m p roce ss i n g photographic e m ul s i o n s can be r ed u c ed to less than 1 ppm silver. M e r c u ry contamination in brine solutions has been reduced to co n cent r ation s as low as 5 ppb from feed streams of 5 0 ppm . R ed uction of con t a mi nants us ually re q ui re s pro c es s in g of large volumes of dilute so l u tions . For m e t al removal , con sidera tion must be m ade for the periodic removal of material from the matrix . O xi d atio n of o r ga ni c contaminants , however , c a n p roceed with no r e t en tio n of solids .
94 0
T ros t , E dwards , and N ewman
T he u se of porous electrodes in electro-organic syntheses m ay provide an �conomical alternative to other chemical routes . T he electrochemical synthesis m ay give hi gher yields under less severe operatin g constraints . The ability to control the electrode potential allow s optimi zation for a p articu lar reaction product , w hile minimi zin g side reactions or multiple products . A bibliography of electro-organic syntheses h as been recently compiled by S w ann and Alkire ( 1 9 8 0) . E nergy storage and conversion system s m ay use flow -through porou s electrodes . Flow - redox system s , zin c - chloride hydrate storage systems , and fuel cells are som e examples . It has also been shown that the per formance o f some primary or s econdary batteries could be enhanced by a fres h supply of electrolyte (Wierschem and Tiedemann , 1980 ; Liebenow , 1 8 9 7) . A p romising zinc-bromine secondary battery system uses a flowing electrolyte . Flow -throu gh porous electrodes also find applications in fundamental studies . Appel and Newman ( 1976) apply a limiting-current method for the measurem ent o f m as s -transfer coefficients at very low Reynolds numbers . Fedkiw and N ew m an ( 1 9 8 2 ) summarize mas s transfer results of several workers . T he results are correlated by a dual- si zed , straight -pore model for t he bed ' s pore volume . Gover n i ng Eq u a t ions
T he m any potential applications o f porou s electrode s w arrant the need for O ne can t hen scaleup a system a m athem atical description o f the system . or predict the res ult of a chan ge in operatin g p arameters . T hi s modelin g can lead to designs t hat op timi ze or m aximi ze the desired process . Less de tailed modeling can be a guide in screening various alternatives such as newly proposed battery system s . I t i s important , then , to develop guide lines as to the behavior of the porous electrode s . Porou s electrodes are inherently different from planar electrodes due to the intim ate contact of the solution and m atrix phase s . Here the current flow s within the m atrix and the solution phase s and exchange s between the matrix and solution nonuniformly throu ghout the bed . An electrical analog that can help picture t he inhe rent complications is seen in Fig . 4 . T his figure show s two porous electrodes operatin g as an electrolytic cell . The subscript s 1 and 2 refer to the m atrix and solution p hases respectively . S ubscripts a and c refer to the anode and cathode , and R s represents the resistance due to the separator . When current flow s through the porous electrode , the electrical double - layer capacity ( represented by C d ) is T his is a net flow of current throu gh the solution causing a charged . In parallel to this process , chan ge in solution composition near the interface . net current flow s via a Faradaic reaction w here R F represents 1lhe charge trans fer resistance of the electrochemical reaction . T hese processes occur nonuniformly t hroughout the volume of the porous bed . It is cautioned that t his figure is m eant to serve as a guide in thinking about the distribu tion of reactions in p orous electrodes , but is not a substitute for modeling the porous electrode with the app ropriate governin g equations . Mass trans fer of reacting species , for example , is not considered in the electrical analo g . T he m athem atical de scription of porous electrodes assumes a macrohomo geneou s system . T h e porous bed is repre sented as the sup erposition of two continua , a m atrix and a solution . T he actual geometric detail of the
Electrochemical Reaction Engineering
941
FIGURE 4
S i m p li fi e d electrical analog of a porous anode and cathod e ( sub and c , respectively ) showin g t h e resi stance in t he m at rix p ha se s ( s ubs crip t 1) and solution phases { su b s cri p t 2 ) , Also show n are th e Farad aic ch ar ge - tr an s fe r resistance ( R F ) , electrical doub le - layer capacit y ( C d ) , and sep ar at or resistance ( Rs ) . scrip t s a
bed is i gn ore d . A ve r a ge p hysi c al parameters such as po rosi t y and s ur face T his type of an alysi s leads to a volume-averaged ap p ro ach are a are used . in the governing differential equations ( Gray , 1 9 7 5 ; D unning , 1 97 1 ) . A s ch em at i c of a s e cti on of a poro u s bed is shown in Fig. 5 . T he vo lu me - av era ged m at erial b al an ce ( D u n ni n g , 1 97 1 ) of a species i within a flooded porous bed in the absence of homo geneous chemical reactions takes the form [ c om p are Eq . ( 4) ] a ( E: c . ) 1
=
aj .
1n
- V •N . -1
( 3 3)
Here three different average s are used . E is the porosity or void volume fraction . a is the specific interfacial area ( surface area of pore walls per c is the concentration averaged over the volume unit volume of bed) . 1 Porous e l e ct rod e
- I
i1 : I
M e tal bocking plate X =L
X= O FI GURE 5
Schem atic
o f a on e - d i m e n sion al porous
electrode .
T ro st ,
942
Edwards ,
and N ewman
of solution in the pores . e: c i is then the superficial concentration averaged over the bed volume ( m atrix plus pores ) . i in is the normal component of the pore wall flux of species i into the solution relative to the velocity of the pore wall , averaged over the interfacial area . tl'i is the average flux of species i in the p ore solution averaged over the cross- sectional area of the pore plus matrix . With the flux � i referenced to the cross - sectional area of the pore p lus matrix , the superficial current density ,! 2 in the solution p hase is given as
h
=
L zi t:Ii
F
( 34 )
i
S imilar ly , .! 1 , the current density in the matrix phase , is defined to re fer to the superficial area and not to the area of an individual phase . T he m atrix and solution p hases are taken to be electrically neutral . For the solution , then ,
z.c. 1
i
I
0
=
( 35)
We have assumed that the electrical double layer at the matrix pore inter face is a small volume comp ared to our averaging volume . This assumption m ay break down for very dilute solutions and high - surface - area electrodes . Electroneutrality requires that the divergence of the total current density
1
+ 'ii' • i �
::
2
0
A com bi n ation of Eqs . ( 33 ) to ( 3 5 ) gives
'i7 • .! 2
=
aF
L: zijin
=
ai
n
( 3 6)
( 37 )
i
w here in is the average tran sfer current density from the matrix t o the 'i7 i 2 is the transfer current per unit volume of the elec solution phase . trode (A /cm 3 ) and is positive for anodic currents . F or a single electrode reaction represented as •
z. ""' s.M. I L-1 1 I i
-
ne
( 38 )
Faraday ' s law becomes as. aj .
In
1 . -I
nF n
=
( 39)
94 3
Electrochemical Reaction Engineering Substitution into Eq . ( 3 3) an d a ( E C. )
1
ne glect of
double - layer ch ar gi n g yi e ld
- v ·N.
=
( 40)
-I
, ( :; nJ - ( ;; nJ]
A kinetic polarization equation relatin g the local r at e of r e ac tion ( transfer current density ) to the surface concentrations and interfacial p ot en ti a l drop is needed . F or the porous electrode E q . ( 1 3) becomes
V •i 2
[
ai0 exp
=
exp
-
( 41)
Porosity chan ge s can be taken into accou n t b y a solid -p hase material balance for a single electrode reaction :
a c: at
=
- A V • i2
w h e re
A
o
}:
=
(42)
-
0
solid
1
1
s.M.
( 4 3)
p nF i
p hases
Transport p ro ce ss es are ne e d ed to com p le te our description . the m atrix phase is
Ohm's l aw
for
a is an effective m at ri x cond ucti vit y dependent on the co m p o si tion o f solid p h as e s . t he manner in w hich the gr an u le s of the conductin g phases are connected , an d t he volu m e fraction of conducting p h a s e or phases . For a dilute elect rolyti c solution in the pores , the flux of a m o bi le solute can be attributed to diffu si on , di sp e r sio n mi gration , an d convection
where
[ compare Eq .
N.
-1
-
E
=
,
( 1) 1 :
vc.
- ( D . + D ) 'i/ c. 1 a 1
c
- z .u.Fc. 'i/ 4> 2 1 1 1
- 1 + E
an
( 45 )
where � is a or rec t e d ionic mobility and Di is ionic di ffu �ion coefficient correct ed for the tortuosity of the pores . D a r ep re s e nt s the effect of axial di s pe r sio n . A di s cu ssi o n of t he effect of axial dis p e r si on on the average m as s transfer coe ffi ci en t is found in the review article by N e w m an and Tiedemann ( 1 97 8 ) ( see als o Fedkiw and Newman , 1 97 8) . The current den si ty in the solution phase can now be represented by 12
=
- K'i/ 4> 2
-
EF
L i
zi D i 'i/ ci
( 46 )
T ro s t , E dward s , and N ewman
944
where K = e: F 2 I:i 21 211j,Ci · T he second term in Eq . ( 46 ) re p re se nt s the diffusion pot e nti al . As a c on s eq ue nc e of electroneutrality , c o nve ction and dispersion make no contribution to the current d ensit y . We s hould also reco gni ze here th at other forms of the t r ansp ort equa tions m ay be n e ce ss ary to describe a system . The appropriate equations for a co nce ntrated binary electrolyte are given in New m an and Tiedemann ( 1975) . Two binary molten salt s are treated by Pollard and Newman ( 1 979) . When ne c es sary , the full m u lticom ponent transport equations can be used . [ C ompare the di scus si o n of Eq . ( 2 ) . 1 In summ ary , the equations presented above have been found to describ e ade q u at e ly porous electrodes in many case s . A certain level of comp le xity is neces s ar y in order to treat the si m ult aneo u s interaction o f the p hy si c al p ro ces s es . E q u atio ns ( 3 3) to ( 35 ) and { 45 ) govern the tran sport , conserva tion , and electrical n e utralit y in the solution phase . Equation ( 4 4) covers the t r an s port in the matrix phase . Eq u ation s { 3 6) and ( 4 1) coup le the species in the bulk phas e to the electrochemical p ro c ess es occurring at the interface . These eq uations are nor mally considered boundary conditions in systems not i nvolving porous electrodes , but here they are ap pli ed throu gh out the volume of the bed . Let us now illustrate an im p orta nt design principle for flow - t h ro u gh porous electrodes ( N e w m an and Ti e de m ann , 1 97 8 ; B e n nio n and New man , C onsider t he c a s e of re du ci n g a species i at the limiting cu r re nt 1972) . in an excess of supporting ele ctrolyt e . Equation ( 33 ) becomes dN .
1
=
dx
( 47)
aj . 1n
for a one- dimensional , s tead y - s tate m aterial balance . In the absence of mi gration in a n electric field for the reactin g species , t he sup er fici al flux of spe ci es i in the direction o f t he sup er fici al fluid velocity is ( from Eq . ( 45) ]
1
N.
=
- e: (Di +
de.
D ) a d
�
+ c. v 1
( 48)
T he local flux to t he w all i s given by a local mass transfer coefficient k m such t h at =
-k c. m 1
( 49)
where the w all c on c e ntr atio n q0 has been set eq u al to zero at limiti n g • current . Substitution of Eqs . ( 48) and ( 49) into t he c ontinuit y equation ( 4 7) gives
v
de .
1
dx
=
e: ( D .
1
+
1
2 d c. D )
a
dx
2
ak c. m 1
( 50)
a n ( 50) governs the concentration distribution o f species i th ro u ghou t the re actor . It is solved subject to the Danckwerts ( 1 9 5 3 ) and W e h n er and Wilhelm ( 1 9 5 6) boundary conditions for the co nc e ntr atio n of species i at the E qu tio
945
Electrochemical Reaction Engineering
inlet (x 0 ) and the outlet (x = L) of t h e reactor . This formulation assumes that an inert p ac king extends from t he active portion of the bed in both t he upstre am and dow nstream di re ction s . A result of the con straints that t he concentration of species i c annot increase without bounds and that the co nc en tration and fl u x should be continuou s is that =
0
at x
=
L
( 5 1)
For the upstream boundary condition , an nuou s concentration and flux give
conti
at
x
=
inlet
conc e nt ration of c
0
0
and
( 52)
Under this condition , the concentration at the in let of the bed will be less c0 b ec au se some of the reactant will hav e diffu sed ah e ad to the active portion of the bed . The solu tion to E q . ( 50) subject to boundary con dition s
than
( 5 1 ) and ( 5 2 ) is e
e - y /B B +
where w e e =
B
1
h qu an titie s
c.
ak X m
= --
y
__!.._ , c0
v
c
t:
ak
m
-2v
a
2
( 53)
=
D'
1 D = 1 + 1 + 4 '
=
( 5 3)
have introduced t e
Note that E q .
c.
D' ( D ' /B 2 ) e B y / exp [ -aL( l /B + B /D' )] (D ' / B 2 ) ( 1 - B ) exp [ - aL ( l /B + B /D' ) ] +
=
ak
(Di
+
D ) a
( 5 4) m
v
sim p ly reduces to
e - ax
0
= e -y
e
or
( 55)
when the effects of axial dispersion and di ffu sio n are i gno re d . Substitution of Faraday 's law for the loc al flux to t he w all in Eq . ( 49) govern s the behavior of the s up er fi ci al curr e nt den sit y in the solution . Equation ( 5 0) becomes v
de.
1
dx
=
2 d c.
s ( D1. +
D )
a
1
dx
( 5 6)
2
For an up st re am counterelectrode , the current de nsity i 2 must go to zero at the back of the electrode as all the cu r re nt has been tr an s fe r red to t he matrix and c u rre n t collector .
Eq .
( 5 6)
give
U sing
this boundary condition
and inte grating
Trost , E dwards , and N ewman
946
( 57) where the subsc ri pt L refers to the exit of the reactor . solution [ E q . ( 46 ) ] is taken as
O hm ' s law for the
-K
( 58)
T he diffusion potential has been i gnored here , a good assumption w hen an excess of supporting electrolyte is present . Inte gration of Eq . ( 5 8 ) sub ject to Eqs . ( 5 7 ) and ( 5 4 ) gives the local variation of potential throu gh the bed . T hi s result is expressed as
D' B
- (a L + 1 + D ' ) e
L
]
( 5 9)
Figure 6 shows the nature of the potential variation through the electrode bed . The potential in the m atrix is constant throu gh th e length of the bed if the matrix conductivity is very high . T he ohmic potential d rop in t he solution causes the vari ation in solution potential . T he potential driving force at the back of the electrode � 1 - cf> 2 ( L ) m ust be lar ge enough to ensure limiting current , w hile t h e potential difference at the · front of the electrode m ust not be lar ge enou gh to have secondary reactions , such as hydro gen evolution . T hu s we have a maximum allowable ohmic potential drop in our reactor . As the bracketed q uantity in E q . ( 5 9) is of order unity , the coefficient of t his q uantity rep re sents the magnitude of the potential variation in the porous bed . T hus we see that , for given values of c and K , side reactions limit the m aximum flow rate throu gh the bed . 0
VI ...J �
... z w ... 0 Q.
w 0 0 J:
� u
�z ( L l
MATRI X POT E N T I A L , � , �. �------------------------�--�
CAT HODE PO S I T I O N ,
x
x•
FIGURE 6 Variation of solution and m atrix potentials as a function of position through a porous cathode . M atrix conductivity i s infinite .
E lectrochemtcal
R eac t io n E ngineerin g
94 7
Since v lakm represents the order of magnitude of the distance through the reactor where the reaction occurs to an app re ciable extent , a limit on the velocity therefore limits the thickne s s of the bed . t11 2 ( L) - t11 2 ( 0) can be assigned a maximum value b ased on data that might be taken on a rotating- disk electrode . Eq uation ( 5 9) can then be solved for the velocity v . What is le ft to be specified , then , is the len gt h of t he electrode . T he len gth of t h e electrode is governed by the desired de gree of conversion . Equation ( 5 3) can be solved for L if y is replaced by aL and 9 is replaced In the case of ne glecting the effects of axial by 9L , the desired conversion . dispersion and diffusion ,
useful
L
=
v
ak
m
ln
( 60)
A s mentioned above , other confi gurations of anode placem ent and current collector p lacem ent can be considered ( T rainham and New man , 1 97 8 ) . Q ual itatively , an upstream counterelectrode will yield the hi ghest reactant con centration and the hi ghest potential d rivin g force at the front of the electrode . A dow nstream counterelectrode will have the maximum reactant concentration w here the potential driving force is smallest . Thus the latter confi guration mi gh t appear to give a more uniform reaction distribution throughout the bed . It is important then to consider the best con fi guration to achieve a given objective . C alculation s show that the best confi guration for achieving low effluent concentrations is with an u p st ream counterelectrode . T hi s rule applies for both high or low ratios of a I K . I n fact , a li mitin g current distribution cannot be achieved for a system utili zin g a down s t rea m counterelectrode ( except for short reactor s ) . For very high values of a I K , the m atrix potential is constant , and the p lacement of the current collector is not import ant . For moderate matrix conductivities t he optimum placement of the current collector depends on the particular chemical system being in vesti gated and the actual value of a K . W e have con sidered above that often a flow - throu gh porous electrode may be ohmically limit e d . T his su gge s t s an alternative confi guration seen at the bottom of Fi g . 7 . The flow - b y porous electrode configuration h as the advantage that the flow of current is perpendicular to the fluid flow . T he electrode s can be m ade thin to minimi ze t he ohmic potential drop in the elec trodes , and they can be m ade lon g to achieve high conversion s . The dis advantage of the flow - by con figuration i s the necessity of including a sep arator to avoid anolyte and catholyte mixin g . T he mathematical analysis is also inherently more complicated due to t he two - dimensional nat ure o f the problem . Two - dim ensional modeling work has been done by Alkire and N g ( 1974) . T hey treat a cylindrical p acked -bed electrode surrounded b y a con centric counterelectrode . T rainham and N ew m an ( 1 9 8 1 ) p re sent an engineer i n g model that compares the perform ance on an economic b a si s of the two porous electrode configuration s for redox energy storage . T he result s of the computer optimi zation show that the flow - by con fi guration is superior in this case , w here the dim ensionle ss q u a ntity e: nF D 0c o /sR K t. tll 2 i s lar ge . For dilute solution s ( sm all value s of e: nFDoc o / s R K t. t11 2) the flow -throu gh con fi guration m ay continue to have m erit . Fe dki w ( 1 9 8 1 ) presents a comparison of the performance of flow -through and flow - by electrode s operated at limit ing current . For a given m aximum ohmic potential drop and desi re d conver sion , he concludes that a flow -by electrode with a len gth - to - width aspect
I
T ros t , Edwards , and N ewman
948
�Separato r or Gap Porous
(a)
����
E l e c t rodes
Current v
lbl v{] [}�""'"' t
D i r e c t i o n of F l o w
�
D i r e c t i o n of F l o w
{c )
-{][}Coneot f
t
D i re c t i o n of Flow
FIGURE 7 Various confi gurations of electrode placement relative to the direction of fluid flow . ( a ) and (b ) are flow -through configurations (current and fluid flow are p arallel ) , and ( c ) is the flow - by configuration (current flows perpendicular to fluid flow ) .
ratio greater than 5 will have a hi gher processing rate than a flow -through electrode . An ultimate design of a flow -by electrode must recogni ze that it can be run below the limiting current . It is clear that the flow -by system holds sufficient promise that more detailed mathematical modeling and scaleup criteria are needed . The zinc bromine secondary battery system as well as flow - redox energy storage systems have evolved to flow - by con figurations . B a ttery A p p l i cations
Let us now focus attention on the use of porous electrodes for battery appli cations . Several levels of mathematical sophistication can be used to examine battery systems . Initially we should like to develop guidelines for screenin g prospective systems . A s development of a new or old system continues , we then resort to more sophisticated mathematical models for our design and scaleup . Before going into details , let us consider some general aspects of battery system s ( Tiedemann , 1 97 8 ) . We wish to examine desirable characteristics for batteries so that we may define i nherent limitations or areas needing development for new and existing systems . One recogni zes that listing a set of desirable
Elec t ro c h em ical Re action
949
Engineering
characteristics implies a p arti c u lar
application .
In examining a n e w o r old
sy s te m we wish t o look at advantage s and disadvantages in t erm s of cost
(perhaps amorti zed cost /cycle ) , ener gy and pow er density , shelf life , re versibility , ener gy and coulombic efficiencies , m aterial u tili zation e ffici en cy , cycle life , reliability , portability , safety , and availability of materi als . One of the fir st characteristics of a battery to b e considered is it s voltage and specific ener gy . K now le d ge of the overall reaction allow s calc u lation of the open- circuit p o te n t ial from thermodynamic data. Dividing t he potential by the equivalent wei ght ( k g / C ) yie l d s the theoretical specific energy of the electrode pair . T hus more ener geti c electrode couple s with low er equivalent w ei ght will increase the t h e ore tical sp eci fic ener gy . From a practical viewpoint , t he advantage of more ener getic couples must be wei ghed a gai ns t the more severe m aterials compatibility requirements in the more corrosive environm ent . For n ew b atteries the specific ener gy quoted is often a theoretical energy b ased only on the mas s of active materials . The theoretical specific energy of the lead - acid battery is 2 1 8 W • h /kg b a s ed on the active material s Pb0 2 , Pb , and H 2 S 0 4 . I n clu d in g the w e i gh t of th e solvent ( for an initial concentration of 5 molal H 2 S04) de cre ase s the ener gy W • h /kg . Addition o f the wei gh t s of current collector s , to approximately exce s s active m aterials , sep arator , container , and connecting po st s de creases t hi s value further t o ap p roximat e ly 4 0 W • h / kg . A r ule o f thumb that the final specific energy is 2 5 to 30% o f the theoretical specific ener gy has been found to be app roxim ately true for several systems . This rule s ho u ld be used with c aution , but it does illustrate the performance p enaltie s associated with b attery p ackaging and s cale u p . K now led ge of the specific energy of the system can be used to assess the p ro m ise o f a given system . D at a on the specific p owe r of the system are also needed to determine the ability of t he b at t e ry to d e live r ener gy at different rates . T he m aximum specific po w e r that a battery can deliver can be app roximately calculated by ( U / 2 ) 2 / Ri w h ere U is the op e n - circuit potenti al and Ri i s the sum of t h e internal ar e a specific re si stance s ( !1 • cm 2 ) times the loadi n g d ensity of material ( k g / cm 2 ) . T he are a s p e ci fi c resistance can be estim ated on t he b asi s of electrolyte co nd uc ti vi t y and se parato r thick ne s s . B attery de si gn s with large current densities should keep overall area specific resistanc e s down to app ro xi mat ely 1 !1 • c m 2 and keep area s p ecifi c resistances of the separator itself down to about 0 . 2 !1 •cm 2 . These calcula tions become more refined as we obtain more in formatio n on a system . Higher specific energies and sp e ci fic powers are import ant in applications w h er e total w ei ght or volum e ar e imp ort ant . In e le c t ric vehicle applications , for exam ple , a bat t ery m ay not have enou gh ener gy or power to c arry i t s elf and its s upp ort s t ruc t u re . T hree system e ffi ci e nci e s char acte ri ze t he performance of a battery . Coulombic e fficiency is a me a s u re of t he reversibility o f the electrodes or the p re sence o f si d e reactions . It is gi ven by the ratio of the number of cou lo mbs released durin g dischar ge t o t he n u m be r of coulombs required to char ge the s ystem b ack to its initial state . Energy efficiency is c alculated by m u ltip lyi n g the coulombic e ffi ciency by the ratio of avera ge di s c har ge voltage to average c h a r gi n g volt age . D iffer e n ce s in char ge and dischar ge voltage s are due to irreversibilities associated with t he ohmic potential drop in the solution and m atrix , and overpotenti als as sociated wit h electrode
100
kinetic s an d m as s transfer resistances .
Coulombic efficienci es can often
approach 1 0 0 % , w hile energy efficiencies are typically 50 to 8 0 % .
method of char gin g and di s c h a r gi n g the battery will a f fec t both
numbers .
The
of these
950
Trost , E dwards , and N ewman
T he m aterial utili zation efficiency is the ratio of the actual coulombs p assed during di schar ge to the theoretical amount of active material avail able . Utili zations of 7 0 to 80% are typical , and again are dependent on the method of dischar ge as w ell as any imposed cutoff volt age . Material utiliza tions often decrease with cyclin g . Amon g the factors that can lead to m aterial utili zations les s than 100% are the isolation of active m aterials , solubility p roblems with reactants and product s , and , as mentioned above , any imposed voltage cutoff . I solation of active m aterial can occur w hen an insoluble , insulating reaction p roduct covers the active m aterial . Hi ghly nonuniform reaction distributions can lead to pore blockage . Differences in the molar volume of reactants and products cause porosity chan ges which c an lead to m at rix fracture . The solubility of reactants and products is important to m aterial utili za tion as w ell as cycle life and s helf life . D unning et al . ( 1 9 7 1) identify a range of approximately 5 x I 0- 5 to 1 1 . 4 x l 0- 5 molar for the desirable solubility ran ge of a sparin gly soluble reactant . T he lower limit of solu bility is based on the need to diffuse the reactant to the active sites from sparin gly soluble crystallites . T he ability to store t he reactant in close p roximity to t he active site was listed as one of the advantages of porous electrodes . T he upper limit of solubility can determine the shelf life of In general , a soluble reactant the cell because of self- discharge processes . on discharge will react if it can diffuse to the other electrode , w hi le a soluble react ant on charge will be subst antially inert if it diffuses to t he other electrode . A soluble discharge reactant can be permanently incor porated into the other electrode and not be recoverable on charge . T he m agnitude of solubility also influences t he redistribution of active m aterial . T he relatively hi gh solubility of Z nO in KOH lead s to m arked concentra tion chan ges on cyclin g , and these can couple with the fluid flow to p ro duce zinc shape change ( Choi e t al . , 1 9 7 6a , b ) . We have mentioned above that t he m ethod of charging can affect the efficiencies of a given syst em . I deally , the char ging p rocess will re store the battery to i t s state at the onset of the discharge cycle . In general , the charging proces s p lays an important role in t he cycle life and per formance of a b attery . T wo common charging method s that m ay be used are constant current and constant voltage . T he se can be com bined with volt a ge or current lim its to help minimize side reaction s , or for therm al m anagement . Side reactions , however , often do occur on charge . If these reactions occur p re ferentially o n one electrode , a n imbalance in the The ability of the system to accept overchar ge state of charge occurs . then becomes important . More sophisticated char gi n g p rocedures can be developed , but m ay be p rohibitively expensive dependin g on specific applica tions . In some redox energy storage system s , for example , a third electrode ' is u sed to correct for im balance s in the state of charge . Porous b attery electrodes c an be constructed in a variety of con figura tions . O ften individual electrodes are constructed with a highly conducting , inert substrate ( grid) that mechanically holds the porous m atrix in p lace . It s hould be strong enou gh to withst and volume changes on cycling and be For a battery inert over the op erati n g temperat ure and voltage ran ges . p late with poorly conductin g active m aterials , the grid is necess ary to act as a current collector to conduct electrons to or from t he reaction site . For moderat ely conducting active m aterials it can act as a secondary current collector . In electrode confi gurations using grid s , the current is collected from each plate and is connected in p arallel with an interconnectin g bus .
Elec trochemical
R e ac t io n
Engineering
951
The optimal desi gn of the current collecting grid and intercell connectors Another battery con figuration that can be considered is a bipolar arran gem ent . Such an ar rangem ent eliminate s the need for a separate current collecting grid for each electrode and m ay possibly minimi ze the voltage and weight perfor mance penalties associated with the grid . Here the positive and ne gative active m aterials are p ut on opposite sides of an inert , conducting sub strate . C urrent flow s straight throu gh the cell stack and is collected at the ends . T he m aterials const raints for the conducting substrate are severe ; it must be subst antially inert to both the oxidi zing and reducing environment . Corrosion of the subst rate will lead to short circuits . Materials compatibility is often a m ajor problem in new or old battery systems . Materials constraints may limit cycle life or prohibitively increase the cost of a system . For example , corrosion of the current collecting grid is cited as a major failure mechanism for the lead - acid battery . Active material in ambient t emperat ure Li electrodes becomes electronically isolated from its substrate , presum ably due to reaction with impurities and the electrolyte itself ( B rummer et al . , 1980) . In general , we look for a bat tery system w here the active materials are compatible with each other and the other support materials required in the battery . We require the elec trolyte , separator , battery container , interconnectin g bus and post , current collectors , and so on , to be st able over the operating voltage and tem pera ture ran ges of our system w hile not catalyzing side reactions or otherwise reacting adversely with the system . A sep arator is a major component of the battery package . It is re quired to separate the po sitive and ne gative electrodes in the battery , so that they may be in close p roximity but not short together . A common failure mechanism in early battery development is shorting of the positive and negative electrodes by dendritic growth of material through the separa tor ( C hoi et al . , 1 97 6a , b ) . Sep arators must be electronic insulators , but have relatively high ionic conductivities . A s eparator with a hi gh area specific resistance ( Q •cm 2 ) leads to a hi gh ohmic potential drop with sub sequent poor performance . S ep arators may also be required to have other desirable properties . O ften they should be specifically conductin g to only certain ionic species ( C hoi et al . , 1 9 7 6a , b ) or must contain additives that increase electrolyte w etti n g . C osts , of course , must alw ays be considered within a particular applica tion . B esides the costs of raw m aterials , some other factors can be con sidered . C osts as sociated with safety , recycling ( particularly if availability of m aterials is low ) and associated environmental factors need to be considered . A common b asis for comparison of various alternatives of secondary batteries is the amorti zed cost o f the system in $ /W • h ·cycle . Scaleup of porous electrodes is not straightforward . Simply increasin g the thickness of an electrode , for example , does not necessarily bring about proportional increases in p erform ance . We wish to develop a micromodel of the porous - electrode system so that we can p redict the effects of changin g parameters such as electrode thicknesses on the behavior of the system . H aving this information in hand then leads us to couple the micromodelin g to further scaleup considerations of plate area and lengths of intercell con nectors . In examining the micromodelin g of porous electrodes let us first con sider the zero -time behavior ( N ewman and Tobias , 1 96 2 ) where concentra tions are ass umed to be uniform throu ghout the pore volume . Further , is a scaleup p roblem that will be considered later .
,
T ro s t ,
952
Edwards ,
and N ewman
let us i gnore double -layer c har gi n g effects . Four dimensionless ratios govern t he current distribution . T hes e can be state d as a dim ensionless current density ,
{ 61)
a dimensionle ss exchange \) 2 =
( C! + a
C!
c
)
Fai L o 2
---
RT
c u rre nt ,
(1 1 ) -
K
+
cr
{ 6 2)
the ratio of transfer coefficients in the polarization e q uation , a a I r:J. c , and the ratio of effective solution and matrix conductivities K I a . o an d v 2 are ratios d es cribin g the competing e ffec t s of ohmic p ote ntial drop and slow e lect ro de kinetics . For large values of o or v 2 , the oh mi c effect dominates with a n on uni form reaction distribution . For small v alu e s of K I cr , the reaction occ urs p referentially near the electrode- sep arator b ou nd a ry at the expense o f t he re gio n near the backin g plate . The non unifo rmity of the reaction distribution for T afel ki n eti c s can be seen in Fi g . 8 . For T afel kin eti c s the current distribution depends only
Ordi n a t e y = 0.5 y= l 0. 9 5 9 4 1 . 0 8 4 ' 1.943 10 0.693 1 00 0. 1 6 9 3 1 2 . 6 9
8
" ;< I
N
[>
y = -x. / L
R e d uc ed current distributions for T afel kinetics with equal F I G U RE 8 m atrix and so lu tio n conductivities .
953
Electrochemical Reactio n Engineering
on the p arameter o and the ra ti o K / o. Curves for lin ear polari zation would exhibit similar behavior as in Fi g . 8 . F or linear behavior , the di stribution becomes no nu ni form for large v and is i n d epend ent of t he total current . For both cases the ratio o f K I a serves to shift th e reaction distribution from one face to the other . T he distance to which the reaction can penetrate the electrode deter mine s how thick an electrode can be utili zed . This penetration depth is characteri zed by
[
L
v = ( a.a
+
a.
RT K o
c
) ai F( K
o
+
o)
]l/ 2
( 63)
Electrodes much thinner than the penetration depth behave like plane elec trodes with an enhanced surface are a . Electrodes much thicker are not fully utili zed . For hi gh current levels , in the T afel range , the ratio L / o will be more characteristic o f the penetration of the reactio n . To continue to follow t h e discharge behavior of a p oro us electrode throu gh the transient behavior , we need to consider the time derivative in Eq . ( 3 3 ) . Porous electrodes used in primary and secondary batteries in variably involve solid reactants and p roducts , and the m atrix is chan ge d during di sch ar ge . Consequently , n o st e ad y state i s strictly possible . We may nonetheless examine a steady- st ate operation of a porous e lect rod e . Just above w e have considered the irreversibilities associated with electrode kinetics an d ohmic potential drop . As the reaction proceed s , reactant is depleted at the p ore - solution interface . Thi s then represents an additional irreversibility . Newman and T obias ( 1 9 6 2 ) also t reat a redox reaction in a porous electrode . Convection is assumed to be absent , and mi gration is neglected due to an exces s o f supportin g electrolyte . T he stoichiometric coefficients of the reactant and p roduct species are taken to be + 1 and - 1 . For a redox reac tio n E q . ( 4 1 ) is often written as
( 6 4) N ow i 0 i s a constant representing the exchange current density at the com position c , c . T he potential di ffer ence c!l 1 is equal to l'l s plus an
�
2
c!l 2
additive term w hich depends on the local solution composition . [ Comp are this with Eqs . ( 10) and ( 1 1 ) . ] A new dimensionless group Yi SilL / can be formed due to the introduction of the diffusion coefficient nF £ Di c of each s p ecie s and a characteristic concentration . Another � pecial case that can be treated is deposition fro m a binary electrolyte . T he binary electrolyte formulation can be applied to sulfuric acid in lead - acid batteries or to the polysulfide in the sodium - sulfur cell if the melt is t ake n to be composed o f N a 2 S and s . T hi s formulation also a p p li e s to systems with con centrated KOH electrolyte , such as in Ni - Fe and Ni - Zn cells , althou gh t h e solubility of ZnO must be ignored . O ften a system cannot be app roximated by one of t he li miti n g cases pre sented above . Full treatment of t he complicated factors governing the behavior of the porous electrode require s the use of hi gh - speed digital
:
=
T ros t , E dwards , and N ewman
954
com p utation Newman and Tiedemann ( 1975) suggest a comp u t atio n al method for battery e lec t rodes involving a binary electrolyte . In gen e ral reactant species are depleted d uri n g the course of dischar ge , and time must be in cluded as a variable . T hus the cou p led eq uations are solved simultaneously at each time step . Pollard and N e w m an ( 19 81) t reat the transient behavior of the lithium - aluminum , iron sulfide hi gh - t e m per atur e b attery for a con s t a nt current di sc h ar ge C oncentration distributions across the cell sandwich are p re s en t e d at various times throu ghout the di schar ge . I n summary w e can list a n umber o f factors that can affect the per form ance of porou s electrode s : .
,
.
1. 2. 3.
4.
C h a r ge a n d dischar ge method s affect battery efficiencies and cycle li fe
.
T he solubility o f re act ant s and p roducts c an limit cy cle life and shelf life . Hi gher current densities yi el d hi gher overpotentials , and thus a given cuttoff potential is re ac he d sooner . T he pores m ay become constricted or even plu gged wit h solid re action produ ct s A nonuniform reactio n distribution will accentuate this problem at the mouth of th e p ore s . Utili z ation of the solid fuel can be limited b y co ve ring o f the reac tion surface w it h re ac tio n produc t s . Rates of mass t ran s fe r betw een crystallites and the reaction surface m ay become more limiti n g as the di sc h arge exhau sts the front part of t he electrode . This could account fo r changes in t h e apparent limit of utili z ati on with current density . .
5. 6.
U ntil now we h ave considered the mathematical modelin g of porous bat tery electrodes . E xp erim ent al dat a are needed , of course , to ensure that our u n der st an din g of the system is substantially correct . In constru cti n g an experimental cell we w ant to eliminate any scaleup effect s not included in t he m at hem atic al m o d e lin g so that w e can dire ctly compare e xp e ri me nt al and theoretical results . The scaleup effects of current collectors an d i n t e rco nn ec tin g bus and post will then be considered s ep arate ly . T he e xperiment al system can b e arran ged in a monocell con fi gur atio n with one positive and one negative plate , or as a bicell with a si n gle posi tive electrode and tw o ne gative electrodes . The bicell arran gem ent repre sents a " section" of a p o sitive and two half n e gative s that would be found in the scaled - up battery . T he construction and sym m et ry of discharge of this cell would be similar to the scale d - up version . The monocell's main advant age over the bice ll m ay be in the ease and cost of construction . A schematic of a bicell is shown in Fi g . 9 . This fi gure shows heavy , ' hi ghly cond ucti n g c u rre n t collec tin g s heets in the center of t he positive electrode and at the back of the two half negative electrodes . T hese cur re n t collectors p ro mote a uniform current distribution across the face of the electrode by minimizi n g the ohmic potential drop in the current collect ing s he et T his is i m p ortant for comparison to one - dimensional micromodel ing results or for use as dat a in subsequent s caleup calculations . Sep arate voltage and curre n t taps should be used to elimi nate any error in volt a ge readings d ue to ohmic potential drop in the cell lead s Reference electrodes should be used in the exp eriment al cell so th at the tot al cell potential can b e decomposed into contributions associated with the positive and ne gative electrodes . Althou gh w e will see that this decomposition is not necess ary .
.
Elec trochemical R eactio n Engineering
955
S. S. HOU S I N G FIGURE 9
Bicell desi gn of a p ositive and t w o half ne gative electrodes desi gned to p romote a uniform current distribution across the electrode face .
for our scaleup calculations . research efforts at improving the battery need to be largely di re c t ed to the limiting electrode . B atteries are often d e si gn ed so that the positive electrode limit s the battery cap acity . T hi s should then be the case with the experimental cell a s w ell . In general i t is important t o u s e the same electrode thi c kn es s , amount of active material , exce ss electrolyte , temperature , separator m aterial , and so on , that is b e i n g considered for t h e scaled - up version . Microm o deling result s can be used as a guide in the s election of some of these p arameters . T he cross se ction al area of the test cell is not im po rt a n t since our exp erim ent is de si gned to be one dimensional . T he app arent o pe n - circuit potential is measured during discharging or char gin g with a current -interruption t echniq u e . T he cell potential d uri n g
T ro s t , E dwards , and N ewman
956
current flow ( � p - � n ) and a ft er 15 s e co nd s * of interruption is inter preted according to the equation i
=
Y(U - �
p
+
� ) n
( 65)
where i is t h e current de nsity from negative t o p ositive plate ( A /cm 2 ) , Y is the conductance of a cell element ( n - l . cm - 2 ) , U is the apparent open circuit potential ( V ) , � p is the potential of t he positive plate ( V ) , and � n is the p ote nt i al of the n e gative plate ( V ) . This relationship a ss umes a linear p ol ariz ation curve ; however it can also be re garded as a step in the lineari zation of a nonlinear p roblem . Values of U and Y can b e determined as a function of the state of char ge for a given constant current den sity . D at a for a lithium - aluminum , iron sulfide hi gh- temperature cell t aken at Argonne N ation al Laboratory ( B arney et al . , 1 98 1 ) are shown in Fi g . 1 0 . The use of reference electrodes allow s t he decomposition o f the cell poten tial and s pe ci fi c conductance into values for the i n d i vi d u al electrodes . These values are relate d to t h e cell values by u
=
u
:::
-
and
1
y
1
y
u
+
+
+
( 66)
1 y
( 67 )
T he se results form a b asi s both for comp arison t o mi cromod eli n g re sult s and no w for scalin g up the p late si ze . C ost , w ei ght , and volume considerations dict ate that the current col lectors will not be the heavy plates us e d in our test cell . T he m as s of the current collector t h at s hould be used for a gi ve n plate area is a scaleup con sideration that is s ubj e ct to optimi zation . Our goal is to develop a dis char ge curve for the p late as a w hole ( with current collector s } based on individual cell element s show n in Fi gure 1 0 . T w o common con fi gur ati on s for the current collector are the sheet current collector and a grid cu rre nt collector . Tiedemann an d Newman treat the nonuniform current and poten tial di stributions in composite sheet electrodes (Tiedem ann and New m an , 1979c} and in battery p la tes with grid configurations ( Tiedem ann and
N ew m an , 1 9 7 9a) . T he lead - acid b attery uses a current collectin g grid with the active material pressed betw een the ribs . A honeycomb grid has b e en used in
*When the total external current is in t e r rup ted , we can id ent i fy three transient s : relaxation of the double- layer cap acity , a local equili zation of charge an d concentration from fro nt to back of t he electrode , and a reduc tion of concentration gradient s in the whole cell by diffusion across the separator . In the current -interruption technique , we wish to w ait lon g en ou g h for double - layer charging to relax . A characteristic time for thi s i s L 2 ac / K . T he app arent op en- circuit potential will continue to rise as the ot h e r transients con ti n ue . We choose 15 s here so that we m ay more c los ely approximate t he r e s ults that would be ob t ained with a 1 5 - s power test .
95 7
E lectrochemical Reac t io n E ngineering
1 . 36
1 .3 4
>
1""'::::��----
1 .32
; 1 . 30
:::1
C :ss 39 . 4 m A /c m 2
o
e
N
u .
�
>
....
C 3 1 74 .9 C 55 39 .4
o
1 .0
> ' ::I
o
0.8 0. 6
0.4 0. 2
0
1 00
0
32
FIG URE 10
Lp
=
0.
0
0
2 00
300
DISCHARGE C /
400
cm2
c
50 0
600
Electrochemical characteristics (U an d Y ) on disch ar ge for
em an d 3 0 % exces s capacit y in the ne gative electrode .
e xp e rim ents with the lithium - aluminum , iron sul fi d e battery . We ch oo se as an operational current - collector model a rectilinear grid with hori zontal and vertical ele m e nt s . A one - dimensional micromodel ( Tiedemann an d N ew m an , 1 97 9b ) or data as in Fi g . 1 0 is coupled to a two- dimensional model of the grid . Equation ( 65 ) can still be used w here the current , area , and pote n tials are now local values for node point s o n t h e gri d . T he polari zation parameters , U and Y , are curve fit as functions of depth of discharge an d local current density . Kirchhoff' s law is used with Eq . ( 65) to solve for the loc al potential distribution across the face of the plate during discharge . Results of this analysis give the overall plate b e h avior as a function of
state of charge . T he current distribution across the face of the electrode is nonuniform at the be ginning of dischar ge and becomes more nearly uniform
T rost , E dwards , and N ewman
958
as dischar ge p roceeds because of t h e dependence of the electrochemical re sist ance and apparent open - circuit potential on the state of char ge . The overall behavior can now be represented by
( 6 8) w here 6.
V
is the voltage displacement from open circuit , and R g is the re grid . Since 1 /Y and R g v ary t hrough discharge in w ays t hat depend on the specific system , a general form ulation of results cannot
sistance of the
be m ad e . However , we can consider the zero - time behavior of a system For the with constant p olari zation p arameters and formulate t h e problem .
p ri m ary variables fj the voltage displacem ent from open circuit , I t he total current leavi n g the grid , A t h e area of t he p late , Y t he conductance of the cell ele m e nt , M the m ass of the grid , a the grid conductivity , p the grid density , and L p the po sitive electrode thickness , four dimensionless does not have direct grou p s govern the system . One of t h ese , p relevance to the problem . We are left wi t h
V
L 2 JA ,
IT
t
=
I
2AY6.V '
rr
2
=
M er
pYA
-2
'
lit represent s a ratio of overall conductance I / 2 6. V to electrochemical con d uctance AY . S tated another way , ITt is the ratio of the actual current . leavin g the t ab to the current that w ould leave t he tab if the re w ere no ohmic re sist ance in the gri d . T he fac tor of 2 in ITt re flect s t he fact that the tot al current leavi n g the grid tab is being collected from both sides of the plate . represent s a ratio of grid conductance Mo/ p A to electro chemical conduct ance A Y . II 3 is t he volume fraction of grid m aterial . Other minor dimensionles s p arameters must now be chosen before the result s of t h e grid model c a n be p lotted . T hese include t he ratio of the t ab width to plate width , relative t ab po sition , aspect ratio of plate hei ght to p late w i dt h , ratio of total grid m ateri al on hori zontal elements to that in vertical elements , and the number of hori zontal and vertical elem ent s . F ur t h e r , results can be p resented for an infinitely conducting ne gati v e grid , a symmetric ne gative gri d with equal co n d uctivit y , or a complete descrip tion of a posi ti v e and n eg ative with different conductivities . Fi gure 1 1 is a plot of dimensionless p late current versus dimensionless plate area for t h e be ginni n g of di scharge ( Tiedem an n and N ew m an , t 9 7 9a) . T he volume fraction of grid m aterial has been t aken to be zero so that L p does not enter as a p arameter in this fi gure . Here the hori zont al and vertical grid elements have the same amount of grid m aterial , and a sym m etric ne gative grid with equal co n d uc tivity is used . Fi gure 1 2 is an ex amp le of an imp roved grid . E xtra c ond uc ti n g m aterial has been added to the two columns of vertical elements below t he tab and the hori zontal ele m ents acro s s the top of t h e g rid . T he two center vertical ele ments are heavier by a factor o f 1 1 , a n d the top hori zontal elements have four times the m as s of the base element s . Figure 1 3 is a dimensionless graph of plate current versus p late area for this grid design . Here the ne g ati ve grid has been assumed to be infinitely conductin g . A com parison of Fi gs . 1 1 a nd 1 3 s hows the imp rovem ent in overall conductance for the improved grid desi gn . This improvement is actually due to two m aj or effects , the
II2
959
Electrochemical R eac tio n Engineering
,.-..l>
..,� 0.2
...._
0. 4
0.2
0.8
A /PYo Mu
F I G U RE 1 1 Dim ensionless plate current a s a function of the dimensionless plate area . Hei ght- to -width ratio equ al s 0 . 8 . Ten percent tab is located ( F rom Tiedemann 30% from the edge of the plate . Lp is taken to be and N ewman , 1 9 7 9a . ) oo
reduction of ohmi c potential drop in the improved grid desi gn and the neglect of ohmic potential drop in the ne gative grid . Figure 1 4 is a dimensionless correlation of the same data as i n Fig . 1 3 . The dimensionless group s have been adjusted s o that all the d ata closely follow the same curve . Here it is recogni zed that the grid necessarily Tab
....... �---
f
, , , ,.....
,,_
FIG U RE 1 2 Improved grid design with extra conductin g material two vertical elements and top hori zontal elements .
in
the
Trost , E dwards , and
960
1 .0
tf
0
No g r i d resi stanc e , ,.3 • 0
o. e
0.6
� ....... 0
N ewman
4
I
I
I
I
I
I
I
I
I
7 , ,.,
/ ';//, /
v, / I /7 1 I
'' '/ /
'I l l
lt /1 1 1
"'
(em l /7'�- f--.1.. __ ---- 1 5 !-tif'-· I I I,_.;-�--._ _-:---- 12 - --
I
P l a t e S i ze
-. l "' '- 1 If, , r- · - - -. -- - o 8 ,, I 6 -· · ··· - ,',' ,r· I II / I 1 1/ I I
I
I
/t
0 .4
F I G U RE 13 Dimensionless p late current as a function of the dimensionless p late area for the improved grid desi gn .
X
..PI >
0 >'
�
Cc m l
Pla l e S i z e D • 0
..
" v
15
18
10 12
8 6
0
FIGURE 14 Dimensionles s plate c onductance versus dimensionless plate area based on a reduced area for current flow throu gh the m atrix bed .
,
961
Electrochemical Reac tion Engineering
displaces some active m ate ri al re d ucin g the area for current within the bed . The active bed cross- sectional area is therefore A - M / pLp . The area for current flow through the sep ar a tor is A . T hus we co rre ct the electrochemical conductance AY0 by an additive contribution of separator re sistance b ased on area A , and the re st of the electrochemical re sistance ba sed on area A M I pL . Thus 1 / Y 0 is replaced by p
_!_ Y
0
(
-
-
1 - Y R
1
O
S
M / pL A P
+ Y R o s
)
.
-1
-2 ,
in p lot tin g Fi g . 1 4 . T he value of t he sep arator resistance , R s H l •em ) can be e sti m at e d on the b asis of e le c t r ol y te con d u c ti vit y T he data p lotted in thi s m anner fall very closely on the s am e curve and h ave unity int er
cept s on both the ordinate and abscis sa . As pointe d out above , Fi g . 14 is s tri ctly valid for zero -time behavior . Our al ter n ati ve to p e r for mi n g complex grid c a lcu l atio n s to m ap out the overall conductance d u ri n g di sc h ar ge is to assume Fi g . 14 is valid throu gh out d i s c h ar ge , re p l aci n g Y 0 with Y and ( I /fl V) 0 b y I / !J V . H er e we are recognizing the dependence of the polari zation parameters on the average depth of di s c har ge but still assume that they are constant across the ele ct rod e face . Y and U act ually d epend on t he local current d ensity and local d e p t h of dischar ge . T his m ethod should give a goo d ap p roxi m atio n to the actu al behavior of the p late and reduces to an e x ac t solution to the complete grid fo rm ulation as the current densities become more uniform . A comp uter p ro gram can now be deve loped t hat uses data as in Fi gs . 1 0 an d 1 4 to generate scaleup p re di cti on s for energy and p ow e r For a given p late si ze and grid w ei ght we c an now calculate the specific energy and sp e ci fi c p ow e r t h r o u gh discharge . The calculations vary the av e r a ge d epth of discha r ge to a specified cutoff p otenti al for a 4- h di s c h ar g e rate . T he results of t he s e calculations , u si n g the correlation in Fi g . 14 and the data in Fig . 1 0 , are given in Fi g . 1 5 . We im m e di at ely see a conflict in t r yi n g to op timi ze sim ultaneously the b attery system for maximum energy and maximum power . I nstead , w e choose an i n t e rme di ate grid w ei ght that gives a ratio of p ow er to ener gy equal to the ratio of maximum power to m aximum ener gy . Alt e r n a ti ve co m p ro mi s es could be selected based on the p arti c u l ar ap p li c at i o n s at h an d . H avi n g selected a natural system ratio for d et e r mi ni n g the grid com p romise , w e now re co gni z e that our small p l at e s must still be connected in p ar alle l with an i nt ercon necti n g bus , and that the cell mu st be con nected in series to anot h er cell with an in te r con n e ct i n g post . T w o impor tant new param et e r s the len gth of inte rconnec t in g bus and length of interconnecting post - m u st now be con sidered . Figure 1 6 sh49w s a sym metric arr an gem e nt of interconnecti n g buses for three po sit i v e electrodes . Lbus is ap p roxi m at e ly eq ual to the sum of the w id t h s of the p o si ti ve elec trod e , ne gati ve electrode , and two separators . The post rep resen t s the cell terminal plus intercell connector t hat extends from one cell to the next cell . In our calculations w e t ake Lp o st to represent on ly that p art of the post as sociated with one cell (i . e . , the len gt h of the t e rm i n al plus half of the len gt h of i nt e rce ll connector ) . T he len gt h s of i nt e rco nnecti n g buses for the p o sit iv e a n d n e ga ti ve p lates are t aken to be the sam e , as are the post l e n gth s . Given the post and bus len gths needed to co n n ec t our cells , an op ti mi z atio n of the distribution of co n d u cti n g m aterial a mon g the p arallel
,
,
-
,
T ro s t , E dwards , and N ewman
962
L b u s = L post = 0 . 0.
1 3.6 u lL.
u w a. Vl
P l a l e S iz e 0 o 6
� 18 rz 8
60 L-��--���---L--��-L--� 120 80 160 200
S P E C I F I C P O W E R (W/k g l
FIGURE 1 5 M a xi m u m specific w ay through di s c har ge .
energy versus maximum specific power half
bus connections and the po s t connections can be made independently of The important quan tit y that results from other scaleup considerations . this optimi zation is
+
L
(Ppost)
1/2
post o pos t
I f the density of the bus and post are t aken to be equal , we see only the
sum of the bus and post len gth is import ant to the scaleup p re di c tio ns
.
We
also note that the number of positive electrodes per cell is subject to varia T he n u mbe r used will affect the final results for the power and ener gy , but doe s not affect the op ti mum distribution of material over the bus and post per cell " section . "
tion .
F I G URE 1 6 conn ectin g
Symmetric arrangement of interconnecting buses and inter posts for three positive e lect rod e s .
E lectrochemical Reac t io n Engineering
963
For a given plate si ze and grid wei ght , the calculated sp eci fic energies versus sp ecific powers will form a loop in the sam e manner as in Fi g . 1 5 a s the m ass o f the b u s and post i s varied from very small values t o lar ger T hus , for each plate area and grid w ei gh t w e can find a mass o f value s . b u s an d post that m aximi zes either the specific energy or sp ecific power . A s we investi gate different grid weights ( and thus different optimum post and bus w ei ght s ) we c an find values of the m ass of grid , bus , and post that yield the m aximum power . Similarly , w e search for values of grid Now w ei ghts and bus and post w ei ghts that yield the maxim um ener gy . our compromise ratio is t he value of the m aximum pow er to this maximum ener gy . Thus for eac h plate si ze our comp romise desi gn i s the values of the m ass of grid , bus , and post that give a p ow er-to- energy ratio equal to the com p ro mise ratio . More than one combination of grid and bus and post w ei ght will yield a p ow er - to - energy ratio equal to the comp romise value . T he largest values of energy and power are selected from this col lection of results to rep re s e n t the best compromise d esi gn for a given plate area . Fi gure 1 7 represents the composite prediction of sp ecific energy versus
specific power as they relate to the si ze of p lates and len gths of bus and Recall that each point give n represents the best compromise of specific energy and speci fic power for each plate area as just discussed above . T he perform ance penalty ( w ei ght and voltage ) of including the T his graph can now b e interconnecting bus and post is re adily seen . used to select the p late area w hich will give adequate perform ance at a suit able cost . T he grid wei ght s and plate areas used to generate Fi g . 1 7 are plotted in Fig. 1 8 . Also show n are the grid wei ghts that correspond to m aximum power and m aximum ener gy . T hi s fi gu re , then , yield s the grid wei ght that gives the m aximum power , m aximum ener gy , or com p romise for
post .
1 3 0 ,------.-,---.,--,---,
D' .>&.
' 7 1 1 0 1-
� > (.!) n:: w z w
8
;:;:: 9 0 ru
u w a. VI
�8 ls;m 1�·� 0· ·� , 6 cm
6 cm
A
0 o
(cml (em) (i'O "'(j"() 2 .0 1 .0 I 0 5.0 L po s t
Lbua
' 7 0 �----�·�----�� · ----�� ----= 1 70 150 130 110 90
S PEC I F I C POWER
(W/kg l
FIG U RE 1 7 Composite , comp romise p rediction s of the specific ener gy and s pecific pow e r for various post and bus lengths and electrode areas .
964
0. 20 .---r-r----, - - �--, ·- ..,.---,--..,.-.,..--,
0.0 I '---'---'--- -'-----'----.J'-- 1.-L 0 .01
FIGUR E 18
Tro s t , E dward s , and N ewman
0.10
Dim ensionle ss grid w ei ght versus dim en sionles s plate area
alon g the m axi m u m specific ener gy , m aximum sp ecific power , or the com p romise d e si g n .
e ach p late are a a n d len gth o f bus and post . grid w ei ght
I t i s seen that the optimum
is not greatly depen d ent on the len gth of the bus and post ,
and that this dependence is actually undetectable for m axi m um specific
T hese results s u g ge s t t h at we can improve our grid desi gn with out concern over interaction wit h the op tim u m post and bus len gth s , and We that improve m ents m ay be independent of depth of dischar ge as well . also see that the optimum grid wei ght s hould be rou ghly increased in pro portion t o A 3 / 2 . We should also note here that no consideration has been given to any minimum current -collector w ei ght nece s s ary to support the active m aterials t hrou gh c ycli n g . general , for s m aller cells it may be s t ated that a cur rent collector that is lar ge enou gh to contai n the active m aterial will have a w eight large enough to ap p roach or exceed the value corre sponding to More d e t ai le d s caleup considerations are t he d e si gn for m aximum pow e r . needed for bi gger plat e si zes . Fin ally , we s hould note that t he value of the grid w ei ght corre spondi n g to maximum specific e ner gy will depend on As we decrease the dischar ge tim e from the 4 - h value t he discharge ti m e . used here , the grid w ei ght will move u p w ar d tow ard the value for m aximum power .
In
specific power . the
In the results p re s ented above , the delivered ampere - hour capacity o f
battery
h a s been a depend ent variable .
An alternative desi gn sequence
for the tot al deli vered capacity . I n thi s case the area of t he plate i s varied until the opti mum values of grid w ei ght and bus and post w ei ght are found so t hat the specified capacity is reached at the
is to s p e cify a value
cu toff volt age . I n summ ary , w e h ave show n how to account for the perform ance p enal ties associated with the grid and interconnecting bus and post .
965
Elec trochemical Reac tion E nginee ring N O T AT I ON
a
specific interfacial area per unit volum e of porous electrode , cm- 1
A
cros s - sectional area of
]
A.
the plate , em 2
p ar am e t e r c har a cte ri zi n g magnitude of anodic reaction at the p ot e n tial where the main reac tion is be ginning to reach li miti n g current
[ see Eq .
( 1 9) ]
con stant defined by E q . ( 4 )
3
total superficial e l e ct rode area , em 2
quantity defined in
Eq .
( 54 )
concentration of species i p er unit volume o f local surface concentration of species i ,
concentration of species
mol /cm 3
0i , ref cico CL c
0
c
i
reference concentration of s p e ci e s i ,
exit concentration of reactant ,
3
mol /cm 3
feed concentration to flow - t h ro u gh el ec trod e , m o l / c m 3 2 ratio of c ap i t al cost to fuel co s t , A /cm 2 capital cost co e ffi ci en t , $ /em • s
fuel
a
3
mol /cm 3
co n cen t r ati on of s p eci es i in t he b ulk , mol /cm
electric ener gy co st ,
Di
mol /em
in the reference e l ect r o d e compartm ent ,
electrical double -layer capacity ,
D
sol uti o n , mol /em 3
$ !J
F / cm 2
co st coefficient , $ /C
di m en sio nle s s di s p e r sio n coefficient d efined in Eq . ( 5 4) 2 di s p e r si o n coefficient , em 1 s 2 diffusion coefficient of s p ecie s i , em / s
.. D 1]
cm 2 fs
F
F ar ad ay ' s
diffusion coe ffi ci en t
for i nt e rac tio n of s p eci e s i
with sp e ci e s j ,
co n st an t , 9 6 4 8 7 C /equiv
flow rate of inerts on anode side , mol ls acceleration o f gravity , cm / s 2 sp acin g between
current i �
i j i
1 2
m , li m
d e n s it y ,
electrodes , em A
/em 2
superficial current density
i n the matrix , A /cm 2
pore phase , A /em 2 j , A /cm 2
s u pe r fi ci al current density in t he
current density
for reactio n
limitin g cu r re n t
density for the main
re action , A / cm 2
Trost ,
966
i n i 0 i OJ , re f .
I
Edwards , and Newman
transfer current per unit of interfacial are a , A /cm 2 exchan ge current den sity , A / cm
2
exchange current density for reaction j with re ference concentra tions , A /cm 2 2 current d en sity , A /cm , or tot al current leaving t ab , A 2 pore - w all flux of species i , mol /cm coefficient of mass transfer between flowing solution and electrode sur face , cm /s
K
u
L
re s ale value of unused but depleted hydrogen , as a fraction of value in fee d len gth o f planar electrodes , o r length of porous electrod e , e m len gt h of interconnectin g b us , em thickness of positive electrod e , em lengt h of interconnectin g post , em current collector w ei ght , g molecular w ei ght of species i , g /mol number of electrons tran s ferred in electrode reaction , equiv /mol
J
n.
number of electrons transferred in reaction j , equi v /mol
tl i
flux of species i , or superficial flux of species i , mol /em 2 p res sure , dyne /em
PH 2
p artial p ressure of hydrogen at the anode , bar
Po 2
p artial pressure of oxy gen at the cathode , bar
p
p
R
a, l
c,l c, 2
RF
R
g
R.
1
R
8 . 3143
universal gas constant ,
Ra , 2
R
•s
electric power produced , W
R R
2
s
R'
Re
anode m atrix resistance , Q • cm
J / mol • K
2
anolyte solution resistance , n •em 2 cat hode m atrix resistance , n e m
2
•
catholyte solution resist ance , n • c m
2
Faradaic charge - transfer resistance , :! •em
resistance of grid , n · cm
2
2
rate of homo geneous p roduction of species i , mol /em 3 • s , or the sum o f internal area specific resistances , n •cm 2 sep arator resistance , n e m •
2
effective ohmic re si stance , n •em
2
Reynolds number
stoichiom etric coefficient of species i in electrode reaction stoichiom etric coe fficient of species i in reaction j
Electrochemical Reaction Engineering Sc
t
Schmidt number time , s
T
absolute temperature , K
u
fractional utilization of hydrogen 2 mobility of species i , cm ·mol /J •s
u.
u
1
U'
u. ] ,0 uJ. , re f
967
apparent open - circuit potential , V const ant in Eq . ( 2 2)
,
V
theoretical open- circuit potential for reaction j at the composition p revailin g locally at the electrod e surface relative to a re ference electrode of a given kind , V theoretical open - circuit potential evaluated with re ference con centrations , V standard electrode potential of reaction j
V
,
standard potential for the reference electrode reaction , V v
solution velocity or superficial fluid velocity , em Is average solution velocity , em Is electrode p otential or cell potential , V electrode potential w here m ain reaction begins to re ach limiting current , V
X
dist ance throu gh porous electrode , em ratio of moles of H to moles o f inerts 2 _ = ak x /v , dimensionless distance m -1 -2 electrochemical conductance , Q •em
z.
1
G reek
electrochemical conductance at start of discharge ,
valence or char ge number of species i
aJ a.c
-1
•em
-2
Letters ak /v , reciprocal reaction distance , em m transfer coefficient in anodic direction =
ll a ll .
!1
-1
anodic transfer coefficient for reaction j transfer coefficient in cathodic direction
a. cj y ij
cathodic transfer coefficient for reaction j
0
dimensionless c urrent density defined in Eq . ( 6 1)
t. us
approximate len gth of the limiting current plateau for the m ain reaction before the onset of the side reaction , V
t. V
voltage displacement from open circuit , V
t.
41 ohm
exponent in compo sition dependence of exchange current density
ohmic potential drop in the ab sence of concentration variations , V
T rost , Edwards , and N ewman
968 E
porosity F /cm
or voi d volume fraction ,
concentration overpotential for
sur fac e overpotential , V
sur face overpotential for
di mensio n le s s , or p e rmit ti vity ,
re a ctio n j , V
reaction j , V
=
c . /c , d im en sionle ss co n cent ration
=
c
1
L
0
I c0 , dim e n sionle s s exit co nc e nt ratio n
effective conductivity of De b y e
le n gt h , e m
s o lu tion ,
Q
-l
·cm - 1
viscosity , g /em • s
electrochemical potential of species i , J / m ol 2 kinem atic viscosity of the solution , em t s dim ensionless exchan ge current gi ven by Eq .
( 6 2)
= I / 2AY IJ. V , dimen sionless p late current 2 = M o / pYA , ratio of grid to electrochemical conductance , dimen sionle ss
=
M I pLP A ,
volume fraction of grid , dimensionless
density , g /cm p0
3
3 pure solvent density , k g /c m
e ffe ctive conductivity of t he matrix p h a se , or the of the c u rr e n t collec tin g grid , n - 1 ·cm- 1
a
conductivity
e lec t ro s t ati c p ot e n ti al , V
elec tri c p ot e ntial in t he matrix , V
electric potential in the solution , V
potential of potential in cp
face , V
p
the negative pl at e ,
t he
potential of the
V
bulk solution extrapolated to the electrode
positive
p l ate , V
S u bsc r i pts
g i
reference electrode of a
given
kind
species i
reaction j m
m ain reac ti on
o
electrode surface , solvent , reactor inlet ,
ohm
ohmic
discharge
or be gi nnin g
of
sur
969
E lectrochemical R eaction Engineering
reference elect rod e loc at e d j u s t outside the double layer
r1
re fere n c e electrode in th e bulk solution
r2 re
reference electrode
ref
reference concentrations sid e reaction , or re feren ce electrode of the same kind as the
s
working electrode co
bulk solution
1
porous
2
porous electrode so lu tio n phase
e lect ro d e m at ri x p h as e
S up e r s c ri p t s
exit
exit stream
o
fe e d stream
R E F E R E N C ES
Alkire , R .
an d P . K . N g, Two-dimensional curre nt distribution within a p acked - bed elect roc he mi c al flow reactor , J . Electrochem . Soc . , 1 21 ,
95 ( 1 974) . Alki r e , R . c . , G . D . C era , and M . A . Stadtherr , O ptim i z ation of ele ctro lytic cells havi n g lar ge numbers of v ari ables , J . E lectrochem . Soc . , 1 2 9 , 1 2 2 5 { 1 9 82) . App el , P . W , and J . New man , A p p lic ation of th e li m i ti ng current m eth o d to mass transfer in packed beds at very low Reynolds numbers , A I C hE J . , 2 2 , 979 ( 19 7 6 ) . B ard , A . J . and L . R . F au lk n e r , E lec t ro chemical Methods , Fu n d amen tals and A p p l ic ation s , Wil ey , N e w Yor k ( 1 9 80 ) . B ar ney , D . L . and others , Lithium /Iron Sulfide B a t teries for E lec t ric Vehic le Prop ulsion an d O ther A p p li c atio n s , Progress Report for Oct obe r 1 979- September 1 9 8 0 , Argonne N ational Lab or atory , A r gonne , Ill . , ANL- 80- 1 2 8 ( Feb . 1 98 1 ) , p . 7 1 . B eck , T . R . , Proceedings o f the Workshop o n Ene rgy Conservatio n in Indus t rial Electroc hemical Processe s , Argonne N ational Laboratory Report ANL / O E PM - 7 7 - 1 (Aug . 1 97 6 ) , p . 37 . B ennion , D . N . and J . N ew m an , E lectrochemical removal of copper ions from very d ilute solu ti on s , J . Appl . El ec t roch e m . , 2, 1 1 3 ( 1 972) .
B . E . Conway , E . Yeager , and R . E . White , e d s . , Comprehensive T reati se of Electrochemis t ry , Vol . 2 : Electrochemical Processin g , Plenum Press , New Y ork ( 1 9 8 1 ) . B rummer , S . B . , K . M . Abraham , V . R . Koch , and G . L . Holleck , Review of t he S tatus of A mb ien t - Temperature Seco n dary Li t h i u m B at t e ries fo r A p p licability to Load L evelling and E lectric Vehicle s , E I C Corporation , N e wton , Mass . ( July , 1 980) . C ab�n , R . and T . W . Chap m an , R apid computation of current distribution by or tho gon al c ollocation , J. E lectroch e m . Soc . , 1 2 3 , 1 0 3 6 ( 1 976) . Choi , K . W . , D . N . Bennion , and J . Newman , Engineering analysis of shape chan ge in zinc secondary electrodes : I . T heoretical , J . Electro chem . Soc . , 1 2 3 , 1 6 1 6 ( 1 9 7 6a ) .
Bockris , J . O ' M . ,
Tros t , E dwards , and Newman
9 70
D. shape change in P.
n
g
Choi , K . W . , Hamby , D . N . Bennion , and J . Newman , E gineerin analysis of zinc secondary electrodes : II . Ex eri mental , J . Electrochem . Soc . , 1 23 , 1 628 ( 1976b) . V . , C ontinuous flow system s . Distribution of residence D anckwerts , times , C hern . Eng . S ci . , 2 , 1 ( 1 953) . Darlington , W . B . and M . Y . C . Woo , R eport of the Electroly t ic Indus tries 1 981 , sponsored the Industrial Electrolytic Division of the Electrochemical Society 1 2 , 1 9 82) . Doub le L ayer and Elec tro de K i n e t i c s , Interscience , New York ( 1 96 5 ) . unnin , J . S . , Analysis of Porous Electrodes Soluble Reactants , Dissertation , Unive s ty of California , Los Angeles ( 1 97 1) . D u nin , J . S . , N . Bennion , and J . Newman , Analysis of porous electrodes with s ring y J . Electrochem . Soc . , 1 1 8 , 1 2 5 1 ( 1 971) . Fedkiw , P . S . , Ohmic potential drop in flow -through and flow -by porous electrodes , J. Electrochem . Soc . , 1 28 , 831 ( 1 98 1 ) . J . Newman , Low Peclet number of rate in p acked beds , C hern . Eng. S ci . , 3 3 , 1 0 43 ( 1 978) . Fedkiw , P . S . and J . Newman , beds at low Reynolds numbers , I nt . J . Heat Mass Transfer , 25 , 9 3 5 ( 1 98 2 ) . Fi zj o h , J . L . , E l c t o o r g ani synthesis , Chern . Eng. P ro . , 71 , 85 ( 1 975) . Fleischm an n , M . , J . W . Oldfield , and L . F lui di e d bed electrodes : IV . of copper of copper- coated spheres , J . Appl . 1 , 103 ( 1 9 7 1 ) . S. a F . Goodridge , Copper deposition in a fluidised bed cell , Electrochim . Acta , 21 , 5 4 5 { 1 976) . C . , The electrical double layer and the theory of electro capillarity , C hern . Rev . , 41 , 441 ( 194 7 ) . G y W . G . , A derivation of the equations for multi -phase C hern . Eng. Sci . , 30 , 2 2 9 ( 1 975) . Hou ghton , R . W . and A . T . K uhn , Mass-transport p roblems and some concepts of electrochemical reactors , J . Appl . Electro 1 7 3 ( 1974) . Ibl , N . , copper refining , Electrochim . Acta , 2 2 , 4 6 5 ( 1977) . N . and E . elektrochemischen Verfahrenstech nik , C hem . - In g . - T ech . , 3 7 , 573 ( 1 965) . Ibl , N . and P . M . Robertson , of use of surp lus electricity in electrochemical processes , Elec oc m Acta , 1 8 , 89 7 ( 1 9 7 3 ) . Jorne , J . , ra s fer in laminar flow channel with porous wall , J . Electrochem . Soc . , 1 2 9 , 1 7 2 7 ( 1 982) . Kuhn , A . T . , ed . , Indus trial Electrochemical Processes , Els vie , New York ( 1 97 1 ) . J . and J . R . Selman , E ffects of separator and terminal on the current in parallel- plate electrochemical flow reactors , J . Electrochem . Soc . , 1 29 , 16 7 0 ( 1 98 2 ) . Liebenow , C . , U ber die Berechnung der eines Str ark e , Z . E le t che . , 4 , 58 p . 63) ( 189 7 ) .
Report
by (May
D.
ri l soluble reactants ,
Delahay , P . , D g n g
pa
Fedkiw , P. and n
e r
Germain , Grahame , D . ra ,
with Sparingly
behavior the transfer Mass-transfer coefficients in packed
very
t
p
Part nd
-
c
g Tennakoon , z Electrodeposition in a fluidized bed Electrochem . ,
transport ,
design chem . , 4 , Optimization of lbl , Adam , Optimierung in der Optimisation
tr hi
night
.
Mass t n
e r
Lee ,
distribution
bei variabeler
omst
Kapazitat k ro m
Bleiakkumulators (particularly
971
Electrochemical Reactio n Engineering
Mantell , C . L. , Electroc hemical E ngineering, 4 th , McGraw - Hill , Ne w York ( 1 96 0 ) , p . 3 9 1 . Matlos z·, M. and J . N e w m an , U se of a flow - th r ou gh porou s electrode for r moval of m er ur from co nt am nated brine solutions , P roc . S ymp . T ransport Processes T he E l ect ro he ical Society , Vol . 8 2 - 1 0 ( 1 9 8 2 ) , p . 5 3 . ounci l , A ssessment o f Re search N ee ds for A dv anced Bat tery System s , NMAB - 3 90 , N ational M at ri a s Advisory Board , C ommission on n gi ne rin g and T echnical Systems ( 1 9 8 2) , p . 1 8 3 . Newman , J . , E n gineering desi gn of e l ctroch mic al systems , I nd . E n g . C hern . , 60( 4 ) , 1 2 ( 1 9 68) . New m an , J . , Elec t rochemical Sy s tems , H all , n gle w ood Cliffs , N . J . ( 1 9 7 3a) . N w m an , J . , T he fundamental p rinciple s of current di s t ribu on and mass transport in le ct oc h em c al cells , in Elec t roanaly tical C hemis t ry , Vol . 6 , A. J . B ard , ed . , Marcel Dekker , New Y ork ( 1 97 3b ) . N wm an , J . , Optimi zation of p otential and hydrogen utili z ation in an acid fuel c ell , Electrochim . Acta , 24 , 2 2 3 ( 1 9 7 9) . Newman , J . and W . Tiedemann , Porous- electrode theory with b at t r applications , AlC hE J . , 21 , 25 ( 1 97 5 ) . Newman , J . W . T i e dem ann , Flow - throu gh porous electrode s , in A dvanced Electrochemistry and Electrochem ical Engineering , Vol . 1 1 ( 1 9 78) , p . 3 5 3 . N e w m an , J . S . and C . W . T obias , T h eor e ti c l a n al si o f current distribu tion i n porous e l ct ro e s , J . Electrochem . Soc . , 1 09 , 1 1 8 3 ( 1 962) . Parrish , W . R . and J . N ew an , Current distribution on a plane electrode below th li mi ting current , J . E le t r oc e . Soc . , 1 1 6 , 1 6 9 ( 1 9 6 9 ) . P a rris , W . R . and J . N e w a n , C urrent distributions on p l ane , p aralle l channel flow , J. El ect ro . Soc . , 1 1 7 , 4 3 ( 1 97 0 ) . Parson s , R . , E quilib riu properties of le c t ri ied in Modern A spects o f Electroch emis try , Vol . 1 ( 1 9 5 4 ) , p . 1 0 3 . Pickett , J . , Electrochemical Reactor Design , 2nd ed . , E se vi er , New York ( 1 9 7 9 ) . Pollard , R . and J . N w m an , T ransport equations for a mixture of two bin ary molten salts in a po ou s electrode , J. lec t roc h m . Soc . , 1 2 6 , 1 7 1 3 ( 1 979) . Pollard , R . and J . N e w man , M athem atical mo d elin g of lithium - aluminum , iron sulfide bat er : I . G a v ano s t at i c discharge behavior , J . E lect ro chem . Soc . , 1 28 , 4 9 1 ( 1 9 8 1 ) . , Criteria for selective p ath promotion in S ak ll ar opo u lo , G . h mi al reaction s eq uence s , A I C hE J . , 25 , 7 81 ( 1 9 7 9) . Sakellaropoulos , G . P . and G . A . F ran ci , Electrochemical re ac t or analysis : se lec ti vit y of multiple competing reactions , J . Electro che m . Soc . , 1 2 6 , 1 9 28 ( 1 9 7 9) . S ak el ar opoulos , G . P . and G . A . Francis , S electivity and reactor anal sis for p arallel el t ro e ical r ac tion . J . T e c hnol . Bio technol . , 30 , 1 0 2 ( 1 98 0 ) . S w a nn , S . , Jr . R . Alkire , Bib liogra p hy of Ele c t ro -Orga nic Sy n theses 1 801 - 1 9 75, T he Electrochemical Society , Princeton , N . J . ( 1 980) . Ti ed m ann , W . , Electrochemical En gineering of B atteries , S hort Course Lecture N o e , University of C alifor ni a , L os A gele s ( M a 1 978) . T ie d an n , W . H . and J . N e w an , Current and potential distribution in l ad - aci battery plates , P roc . S mp . B attery Des . Op timi zation , The Electrochemical Society , Vol . 7 9- 1 ( 1 97 9a) , p . 3 9 .
ed.
e
c y
i Electrochem . Syst. ,
National Research C E
e l e e Prentice-
e
e
e
r
c m
E
ti
i
e
ey
and
e
e h electrodes in
d
a
m m
c
m
y s
hm chem e f
interfaces , l
D.
e
e ce c l
t y P.
s
r
E
e
the
l
electro
s
y
ec ch m
e
s
Chern .
and
e em
e
ts d
m
n
y
y
T ros t , Edwards , and N ewman
9 72
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R.
•
15
Reactor Steady-State Multiplicity and Stability MASS I MO MO R B J D E L L I
A RV I N D V A RM A
Po l i tec nico di M i l an o , Mi lan , Italy
U niversity o f Not re Dame , Notre Dame , Indiana
RUTHERFORD ARIS
U niversity o f Minneso t a , Minneapo lis , Minneso t a
I NTROD UCT I ON
A lthou gh it is titled broadly , the scope of this c hap ter is more limited . We attemp t to provide a review of concepts and results relate d to ste ad y - state multiplicity and dynamic beh avior in two types of re actor s :
the continuous
flow stirred - t ank re actor ( C S T R ) and the empty tubular re actor ( note that in qualifyi n g it as " empty , " w e m e an t h at there is no special re ference m ade to packing found in a p acked - bed reactor ; see C h apter 6 for the design of s uch re actors ) . I n thus limiti n g the scope , for reasons of brevity , we therefore make no direct mention of similar proble m s t ha t ari se and that h ave been at
tacked for otber typ e s of re actor s and for si n gle cat aly st pelle t s . sults are di scu s s e d in other chap ter s . -
T hese re
D espit e the limit ation in scope , the two re actor types that are covered are the most widely t reate d in the literature and offer a wide spectrum of re sults . M any of these results also carry over to other reactors . The C S TR and t he empty tub ular reactor thus serve as p aradi gm s for understanding steady s t at e m ultiplicity and d y n amic beh avior o f reactor s in general . I n its simple s t form , the C S T R is perfectly w ell - mixed ; thus there i s no In the spati al di stribution of reagent concentrations or of tempe rature . simple st model for the tubular reac tor , a plug of fluid flow s through the tube and reaction occurs alon g the le n gt h of the tube ; there i s no fluid mi xi n g i n the axi al direction w hile there is p er fect mixin g in the radial -
direction - the so - calle d plu g- flow model . Since the reaction doe s occur alon g the le n gth of the tube , there i s a di st ri b u ti on of rea ge nt concentra tion s - and also o f temperature for noni sother m al cases direction .
-
alo n g ' the axial
Fluid mixi n g can be taken into account by includi n g axial di s
persion superimposed on t h e p l u g flow , and a s noted e arlier ( Varma and Aris , 1 9 7 7 ) , the axial disper sion model i n t he limits of s m all and large dis
persion doe s indeed yield plu g - flow and C S T R behavior , re spectively . T his chapter may be considered an update of our p revious review ( V arm a In the interim period , numerous additional re sults have a n d Aris , 1 97 7 ) . appeared in the literature , and t hi s chap ter provides us with an op portunity to collect them in one p lace . 9 73
Morbidelli, V arma , and A ris
974 ST E A D Y -ST AT E M U LT I P L I C I T Y Cont i nuou s-Sti r red- T a n k R eactor
The m ass and ener gy balances for a C S T R under steady- state conditions can be written as follow s ( see Varma and Aris , 1 9 77) : q (C q pC
O
- C ) - Vf( C , T ) = 0
p
(T 0 - T )
+
( 1)
( - t. H ) Vf( C , T ) - U a ( T
-
T ) c
=
0
where all the symbols are explained in the notation section . ( 1 ) and ( 2 ) , i t h a s been assumed for simplicity that :
( 2)
In deriving E q s .
T he flow rate q and the reactor volume V are constant . T he fluid therm al and physical p arameter s , p , C , and li H are p independent of temperature and composition . T he temperature o f the cooling medium , T , is constant .
1. 2.
c
3.
Equations ( 1) and ( 2) c an be m ade dimen sionless by introducing the followin g quantities : E - y = RT
v = -
T
T
Da
V f( C 0 ,
=
(3
=
qC
m
m
0
Tm )
O
( - li H ) C
0 pC T ( 1 + o ) p m
Ua
=
r
q pC
=
( 3) P
f( C , T ) f( C 0 , T )
m
T he reference temperature T m is gi ven by the wei ghted - average value be tween the inlet and the coolin g temperatures :
T
( 4)
m
A s shown by K ausch us et al . ( 1 9 78) , this choice allow s the sim ultaneous treatment o f the adiabatic and the nonadiabatic case and reduce s by one the number o f dimensionless parameters . S ub stitutin g Eq s . ( 3 ) and ( 4) , E q s , ( 1) and ( 2) reduce to 1 - u - D ar ( u , v ) 1
-
v +
SD ar ( u , v )
= =
0 0
(5) ( 6)
T hese can be suitably combined to give the dimensionle ss concentration u
=
1
+a-
v
( 6)
9 75
Reactor Steady - S tate M ultiplicity /S tabi lity which substituted in Eq . ( 6 ) lead s to 1 - v
(
S D ar 1 +
+
!-
v
,v
)
=
( 8)
0
T he problem has then been reduced to the sol utio n of the nonlinear alge braic equation ( 8) in t h e only unknow n v . A p rio ri bounds of v , in t he case of an exothermic r e a cti o n ( S > 0) , can readily be derived from Eq . ( 7 ) , noting that from phy sical ar guments that 0 � u � 1 and _
( 9)
w hile for an e ndothermic reaction (j3 < 0 ) t he same inequality hold s with the si gn s reversed . I n the followi n g sections , Eq . ( 8) i s examined for various expre ssions of the di mensionle s s reaction rate r , to define , a priori , the number of all possible steady states for any given set of p ar ameter value s .
Posi tive n t h- O rder Kine tics U sin g the reaction r ate expression f k ( T ) C n (in dimensionle ss form , r = 1 /v ) ] ) , Eq . ( 8 ) can be re w rit te n as follows : un exp [ y ( l
-
n- 1 a Da
--
==
=
C 1 + S - v>
v
-
n
exp
1
[r (1 - -)� 1
:: F ( v )
v
( 1 0)
Since in the case of endothermic reaction s E q . ( 1 0) admits a u ni que solu tion for all D a , in the foll o w in g we shall con sider only exothermic reactions ( i . e . , 1f > 0)· . From in spection of Eq . ( 1 0) , F ( l ) +oo , F ( l + B) = 0 , and F ( v ) > 0 for v E: [ 1 , 1 + B) so that uniqueness is guaranteed for all Da if and only if ( Luss , 1 9 71)
,
dF dv
�
O
=
for all v
e:
BJ
[1, 1 +
( 1 1)
If condition ( 1 1) is violated , multiple steady states will occur for some value of D a . B y differentiating F ( v) it can b e seen that E q . ( 1 1 ) leads to
(n
-
1)v3 +
(B
+ y + 1
-
n)v
2
- y(2
+
S)v
+
y(1
+ B)
� 0
( 12)
( 0, 1)
( 1 3)
'
B ased on this relation , Van den B osch a n d Luss ( 1 97 7 ) derived stron g sufficient criteria for both uniqueness and m ultiplicity , and sub sequently , T sotsis and S chmit z ( 1 979) defined the c o rre sp ondin g exact necessary and sufficient conditions . T hese can be obtained in a more com act form by re w riti ng Eq . ( 1 2 ) , accordin g to Leib and Luss ( 1 9 8 1 ) , as follow s :
p
y �
(1
+
2
SZ ) [ 1 + Z( n az ( l - Z )
- 1) ]
-
G(Z ; n ,
B)
for all Z
e:
M o rbide lli ,
976
Varm a , and A ris
where the q u antit y Z (v - 1 ) / S has been introduced . From in sp ectio n of Eq . ( 1 3 ) it appears t h at G ( O) = G ( 1 ) = +oo , G ( Z ) > 0 for all Z E ( 0 , 1) , an d G ' = 0 for v al ue s of Z satisfying =
- S( n -
1 ) Z 3 + z 2 ( n - 1 ) ( 2S + 1 ) +
(2
+
i3"> z
-
1
=
o
( 1 4)
Since E q . ( 14 ) e x hibit s one and only one root Z in the interval Z for any value of a and n > 0 . it can be concluded t h at w h en
y
:!ii; G
::
€
( 0 , 1)
( 1 5)
G(Z)
there is uniquene s s for all value s of D a . On the other hand , if condition ( 1 5 ) is v iolated , the system admits multiple solutions in some range of D a . T he boundaries of such range are c alle d lower an d up p e r b ifurcatio n points : D a * and D a * . T hey can be calculated by s u b sti t ut in g i n E q . ( 1 0) the two roots Z and Z + (i . e . , v and v ) of Eq . ( 1 3) [ i . e . , y = G ( Z ) ] , t h u s ob + _ _ taining t h e following relationships :
Da
*
=
- n- 1 13- F(v > +
D a*
( 1 6)
T hu s for y > G the sy stem admits multiplicity for D a * < D a < D a * , and uniq ueness for Da < D a * or Da > D a * . T he fu n ction s F ( v ) and G ( Z ) are sketched in Fig . 1 , where all the above - mentioned e at u re s are in di c at e d . In F i g . 2 the bifurcation curves in t he B - y plane are shown for kinetic s of various p o si ti v e orders .
f
First - O rder Kinetic s : In the classical case of first - order re actio n , Eq . ( 1 4) c an be solved analytically , leading to Z = 1 / ( 2 + a) . T hen , u sing E q s . ( 1 3 ) and ( 1 5 ) , the w e ll - k n ow n nece ssary and sufficient condition for u n iq ue n e s s can be derived : ( 1 7) If co n dit ion ( 1 7 ) is violated , then multiplicity o c c ur s for Da where the bifurcation points are given by E q . ( 1 6) with
v
±
=
r< 2 +
ih
±
{yS£r'S 2(y
+
73>
- 4( 1
+
"S> l }
E
1'2 ( 18)
K au sch us et al . ( 1 978) have obtained equivalent conditions in te rm s of only two p ar am eter s : a = ( 1 + S) /yS a n d b = y S D a . This allow s u s to characterize the entire steady - state behavior of the system with a sin gle plot in the a- b plane , as shown in Fig . 3 . P aramet er values inside the re gio n enclosed by the two curves representin g the bifurcation poin t s lead to three st e ad y states , while values ou t s id e this re gion c an lead to unique ness . T he point w here the two curves mer ge and v ani s h is c alle d t he cusp po i n t ( a = 1 / 4 ; b = 4e - 2 ) . It is worth stressing that the sh ap e of the multiplicity pattern , such as that depicted in Fig . 1a , i s strongly dependent o n the d efini tion of the
Reactor
F
S teady-S tate Mul tip licity /S tab ili ty
FI XED : � .
977
n
(v_)
v_ -
v
(a) FIXE D :
C)
n
J2 G
Y,
(b)
0
=
z
z
--- z = C v - 1 ) 1 /3
FIGURE 1 y
i§,
z+
Schematic representation o f function s ( a) F ( v) and ( b ) G ( z) . = y 2 , three or one steady state .
y 1 , uniquene ss ; y
to study the multiplicity p attern with respect to the reactor residence time ,
dimensionles s param eters used .
For practical applications it is convenient
which can easily be varied in a continuou s fashion by varying the reactor inlet flow rate . After the first works of FUrusawa and Nishifn ura ( 1 9 68) and Hlav,cek et al . ( 1 970) , Uppal et al. ( 1 976) reported a detailed study of the multiplicity pattern in term s of the reactor residence time . Golubitsky and K ey fit z ( 1 98 0) showed through singularity theory that the m ul tip li city pattern , in term s of conversion /residence- time plots , may assume one of the. six different shapes shown in Fi g . 4 , in addition to the obvious uniqueness case . K nowledge of the se m ultiplicity patterns is
very important when discussing the p roblem of attainability of a given steady For exam ple , stable steady states on an isola ( see Fig . 4c) that lies state . entirely along with a stable continuou s b ranch cannot be reached by chang ing only the reactor inlet flow rate ( see Huang and V arm a , 1 9 80) .
Morbidelli, Varm a , and A ris
978 8
I C!l_
6
y FIGURE 2
Bifurcation curves for reactions of various positive order : Below the curve , uniquene s s ; above t he curve , three or one steady state .
and
B alakotaiah Luss ( 1 9 81) developed an a priori proced ure for the definition of the multiplicity pattern . Althou gh it requires different definition of p arameters o and i3 such that the reactor flow rate appears only in the spirit of this te c h n iq ue can be illustrated briefly u sing E q s . ( 1 0) and ( 1 1 ) wit h n = 1 . A s mentioned above , these equation s
a
Da ,
can
0 . 6 r---.----.---.---,
0 .4 .0
0.3
0.2 0.1
- o
F I G U RE 3 b =
y'i3 Da. state .
0. 3
Multiplicity behavior for a first - order reaction :
a = ( 1 + if) / ya, I n side the curves , three steady states ; out side , one steady
Reactor S teady - S tate M u l tip licity /S tability
X
979
( d )
2 0
If) 0::: w > 2 0 u
R E AC TO R
R ES I D E N C E T I M E ,
0
T
FIGURE 4 S c he m a tic representation of t h e m ultiplicity patterns in ter m s of outlet conversion versus residence time for a first- order reac tio n .
be solved simultaneously to give the bifurcation values D a * and D a * . Re gardin g th e function F = F ( v , D a , e_) , where E. rep resent s the vector of all parameters but D a , it can be st ated on the basis of the implicit function theorem that the transition from zero to two or four bifurcation points can occur only when
Cl F ov
oF o Da =
0
( 1 9)
which , u sing E q s . ( 1 0) and ( 1 1) , can be re p l a ced by either one of the two conditions =
0
or
Cl F Cl D a
0
( 2 0)
Each of these conditions identifies a surface in the parameter space , named "hysteresis variety" and " isola variety , " re spectively . Accordin g to G olub it s ky and K eyfit z ( 1 980) , cro s sin g the first s u r face through a contin uou s change of one of the p aram eters causes the appearance or disappear ance of an S ( Fig . 4a) or inverse ( Fi g . 4b ) pattern ; similar behavior ,
S
but relative to the isola p at t e rn (Fig. 4c ) , is observed for the second sur face . T he s e two surfaces are represented in the ct- B plane shown in Fig . 5 for the p articular case y = co and T c = T 0 = T m . They divide the plane in five region s , each correspondin g to a specific m ultiplicity pattern .
Morbide l li ,
980
Varma , and A ris
U N I QUE SOLU T I O N
c3 �
S PA T T E R N
B = r ii < l + 8 >
FI GURE 5 Sc he m at ic of the re gio n s in the parameter plane a. o /D a ve rs u s B which lead to the different multiplicity pattern s of Fig . 4 . =
For example , the loc ation o f the m u shroom p att ern re gion i s quite obvious
sin ce by crossin g the isola variety it m u st degenerate into an isola , while crossin g the hystere sis variety it must degenerate i nto a S - s ha pe pattern .
Note that the i n v ers e S - shape pattern never o c curs in the case illu strated in Fi g . 5 , sin ce it req uires very strong coolin g ( i . e . , T c < T o ) .
T his particular c a s e differs from the general Zeroth- Order K in e t ic s : one in that G ( l ) is positive but finite , which reflect s the well - known peculiarity of these reactions that they reach complete conversion with a p o sitive reaction rate . S in c e E q . ( 14) ad mit s the solu tion Z = 1 /S, it can be seen from Eq . ( 1 3 ) that th e uniquene s s condi tion for all Da is given by y
Y
<
for
4
< <1
+
s> 2
if
S ;;;. 1
( 2 1a)
1
( 2 1b )
for i <
No nmono to n e K ine tics
An unu sual feature of som e reactions in catalysis , as well as in o t h e r fields , such as biocataly sis , is to e xhib it kinetics which are not m o no t on ically in
creasin g with the reactant con c e n tr ati on . T his situ ation occurs w he n the reactant can somehow inhibit the rate of reaction . For example , in the case of o xid atio n of CO or various hydrocarbons ( see Voltz et al . , 1 9 7 3) or hydrogenation of olefin s ( see Sinfelt , 1 96 4 ; R o ge rs et al . , 1966) ov er noble m etal s , the reactant can be strongly adsorbed on the catalyst active si t e s , thus inhibiting the development of the re a c tio n . Similar behavior is en
coun ter ed in biocatalysis for certain subst rate -inhibited en zymatic reactions
( see Laidler and B untin g , 1 973) .
981
Reactor S teady - S tate Mul tip licity /S tability
Of particular interest is the case of oxidation ( hydrogenation) reactions , where oxy gen ( hydrogen) is p re sent in large excess . T his correspond s to the w ell - known Langmuir- Hin shelwood kinetics f = kC / ( 1 + K C ) 2 , where k is the reaction rate con stant and K the reactant ad sorption constant . In dimen sionle s s form
( 22) T sotsis et al . ( 1 98 2 ) have examined in detail the followin g general form of nonmonotone kinetic s :
( 2 3) which reduce s to E q . ( 2 2) for p = 1 and q = 2 . Under similar condition s , the intrin sic high nonlinearity o f nonmonotone kinetics leads in general to more complex multiplicity pattern s than those characteristic of monotone kinetics . In fact , in the latter case , the only source of nonlinearity is in the Arrheniu s temperature dependence of the reaction rate constant , w hile in the former m ultiplicity can occur even under isotherm al condition s . T hus for this type of reaction we shall examine separately the isotherm al and the nonisothermal cases . I sotherm al Reactor :
In this case v = 1 , and only the mass balance Eq .
( 5) nee d s to be con sidered . I ntroducing the kinetic expression ( 23) , Eq . ( 5) can be rewritten in the form 1 Da
=
q ( 1 + cr) ( 1 - u)
- F ( u)
{ 2 4)
Since F ( O ) = 0 and F ( l ) = , it is seen that the necessary and sufficient condition uniqueness is ( L uss , 1 9 7 1 ) ""
for
dF du
;;> O
for all u
p(q
p
E
[0, 1]
( 25)
T sotsis et al . ( 1 98 2 ) found that condition ( 2 5 ) is satisfied if an d only i f +
1)
+
q + 2 [ pq ( q
p
+
1) ] 1 / 2
( 2 6)
w hich in the case of a bimolecular Lan gmuir - H in shelwood reaction (p = 1 , 2 ) reduce s to
q
=
a � 8
( 2 7)
As expecte d , it appears that multiplicity under isothermal conditions occurs only for sufficiently stron g inhibition . I f Eq . ( 27) is violated , then in the interval
Morbidelli , Varm a , and A ris
982 -
1
F ( u _)
< Da < Da*
1 - F(u ) +
( 2 8)
w he r e u + and u
are the roots o f dF / d u 0, the reactor exhibits t hree steady states . O utside this interval the ste ady state is unique . T hus the most com plex p attern of m ulti pli c it y is again of the type 1 - 3 - 1 . =
N onisot hermal Reactor : Similar to the nth - order kinetics case , the nece ssary and sufficient condition for uniqueness is gi ven by E q . ( 1 1 ) , with F ( v ) defined as follow s : -p - 1
8 IJa -
( 1 + O) q v
-
1
(1 + 8
[1
+
o( 1 +
-
$
v)P
-
v)]
q
exp
[y ( �)] 1
= F(v)
( 2 9)
T he critical c on dit io n dF /dv = 0 leads in this case to a quartic eq uation in v , which im plies that F ( v ) can have none , one , or two pairs of extreme point s ( one m axim um and one minimum ) . T h ese three pos sible sit uations are illu strated in Fig . 6 . I t i s apparent that b y continuou sly varying the D amkholer number D a , the followin g m ultiplicity pattern s can be obtained : 1 - 3- 1 ( b ) , 1 - 3 - 1 - 3 - 1 ( c 1 ) , or 1 - 3 - 5 - 3 - 1 ( c 2) . As e xp e cted this case ex hibits the most complex m ultiplicity pattern among those e xam i ne d up to now . This is due to superimposition of the temperature nonlinearity with the nonmonotonic concentration dependence of the reaction rate . In fact , when 1 , t he system again exhibit s at the latter tends to vanish , as for q <; p most the multiplicity pattern 1 - 3- 1 .
,
-
>
(C2)
t-3-5-3-t
Sketches of function F ( v) correspondin g to any possible FI G U RE 6 m ultiplicity p attern for a nonisothermal C S T R with nonmonotone kinetics [Eq . ( 23 ) ] .
Reactor
983
S teady -State Multip lic ity /Stabil ity
T sot sis et al . ( 1 S82) have examined in detail this q uartic equation usin g theorem to obtain a priori criteria for the number of roots in the interval v e: ( 1 , 1 + 61 . Their procedure giv e s the exact nu mb e r of steady states of t h e reactor , but it is too complex to be reported here in det ail . Some simple sufficient conditions for uniquene ss by the se authors are : Sturm ' s
For an ov e r all negative order ( i . e . , q > p) and endothermic reaction , the condition is given by Eq . ( 2 6) . For an overall positive order ( p > q ) , t h e condition is given b y y ll6; 4 or e < 0 ( endothermic reaction ) .
2 , B rown et al . ( 1984) u se d singu For the spe ci fic case of p = 1 and q larity the ory to loc ate two b u tt erfly sin gularit ies , one in t he exothermic case =
an d the o th e r in the endothermic case . Around the se singular point s , a maximum of five steady states exist . In Fig . 7a , uniquene ss and m ultiplicity re gion s in the e- y p l ane are shown for variou s value s of the inhibition param eter a . T h er e is multiplicity , for some D a , to the right of the curve s lying on the a- p o siti ve half plane and to the left of t h os e lyi n g on the a- negative half plane . For the particular case a = 7 , all the possible m ultiplicity pattern s are indicated in Fi g . 7b .
More
Comp lex Configuratio n s
For all sy st em s examined above , a complete characterization of the mul ti plicity p at ter n has been ob tai ne d . Following an increasin g complexity order , the next system s to be examined w ould include either several reactor s in se rie s with a single reaction ( so a s to approach the behavior of tubular 4
3
20
2 I Ql.
16
0
I� 14
- 0 . 50
10
1
- 0. 2 5
- 0.75 - 1 .00
rr= 7
18
0
(a )
4
8
12
-- Y
16
20
24
28
12 8
6
( b)
I
2
3
4
, t: 2: 3' 4=
- "Y
I 1 -3 - t t - 3- t - 3 - 1 1 - 3 - 5- 3 - 1
5
6
7
( a) U niquene ss and multiplicity re gion s in the B versus y p lane cr . ( b ) Multiplicity p attern s in the case a = 7 .
for v a rio u s value s of the inhibition p ar am et e r FI GURE 7
Morb idelli , Varma , and A ris
984
reactors ) or a sin gle reactor with a more complex kinetic scheme- say , a series and /or parallel sequence of firs t - order reaction s . In both cases the number of equations of the model and the number of characteristic dimen This leads to a rapid sionle ss parameters of the system increase rapidly . increase in the complexity of the multiplicity st udy . In p articular , we now have to deal w ith a m ultidimensional p arameter space , w hich prevent s an exhaustive rep resentation of the system behavior in terms of p hase p lane . T he work in this area is then oriented tow ard the analysis of some sections of the p arameter space , which are u sually selected either on the basis of some specific practical interest or to identify all the pos sible qualitative be havior of the system . In the followin g , two cases of thi s typ e t hat have received particular attention in the literature are examined . T he aim is to give the reader a feeling for the rapid ly increasin g complexity and to indicate some of the techniq ues u sed to attack these proble m s . S eries of Continuou s Stirred - T ank Reactors : T h e interest in examinin g a series of C S T R relies on it s w ell - known use for modelin g tub ular reactors . T he steady - state mas s and energy balance for the generic ith reactor of the series can be w ritten as follow s : 1 1
u. - D a. r . 1
v. - v. 1_ 1 1
=
1 = 1,
0
S D a.r. - o . ( v. - v . ) = 0 1 1 1 1 C1
+
i
=
2,
., N
( 3 0)
1, 2,
. . ., N
( 31)
w here the sam e param eters defined by E q . ( 3) have been u sed , b u t takin g as reference temperature T o instead of Tm , so that uo = 1 , v o = 1 . More over ,
S =
i3. ( 1 1
+
o .)
1
and
v
ci
=
C1
T .
�
( 32)
In the case of N nonisotherm al C S T R s in serie s with a iirst - order reaction , Varm a ( 1 9 80) has shown t hat at most ( 2 N + 1 - 1) steady states are pos sible , It is worth noting in passing of which however at most ( N + 1 ) are st able .
that this same conclu sion applies also to this s ame system configuration in the case of a nonmonotone isothermal reaction of the type in E q . ( 22) .
T his result can be intuitively understood , notin g that each reactor in the cascade can exhibit three steady states i n correspondence to each steady state exhibited by the previous one . To the number thus obtained , we should substract all those steady states that imply physically unrealistic extinction p henom ena along the sequence of reactors (i . e . , the t ransition from a high - conversion steady state to a low - conversion steady state ) . A detailed st udy of the behavior of this system for the particular case of only two re actors in serie s has been performed independently by Kubicek et al . ( 1 9 8 0 ) and S voronos et al. ( 1 982) . Even for t his si mple case the total number of p arameters is quite elevated ( D ai , o i , S , y , V ci ei gh t indepen dent p arameters ) , so that most of the analysis has been dedicated to the case of two identical C S T R s ( i . e . , D a = D ai and o i = o ) in the =
lim it of the Frank - K amenetskii approximation of the Arrehnius temperature
dependence of the rate constant , which allows us to eliminate the dimension les s activation energy p arameter y . T o sim plify the treatment , in the
R eactor Steady -S tate M u l tip licity /S tab ili ty
985
following we treat only the case Vci = ( i . e . , cooling temperature equal to 1 for both reactors) . After all these approxim ation s , Eq s . ( 30) and ( 3 1 ) can be solved , leading to To
Da
=
=
Da
-
r [u ,u ;
cp , o 1 -
=
2
1
whe re
lj>
1 - u
ll [ u ; ¢ ] 1
4( 1
+
yS
u
(�
1
exp -
1
u
l
- u
u2
2
(1 - u ) 1
exp
{-4-- [ lj> / u
l
]
u
( 3 3a ) l u
u
2
1
+
1 - u
u1( 1 + o )
1
]}
( 3 3b)
0)
( 34)
A particularly im po rt an t feature of thi s system is that the behavior of the second reactor is dictated by the same equation as the first one , but with proper scaling of the unknow n and t he involved p arameters . This is ap p arent , noting from E q s . ( 3 3a) and ( 3 3b ) t hat ( 3 5) Since the reactor eq ua tio n s c an be solve d in c a sc a de , this allows u s im mediately to characterize the behavior of the second reactor once t h at of the fir st is known ( i . e . , the fun c tion !J has been investigated fully ) . N amely r exhibits th,e same behavior as 1:!. , but scale d in the interval 0 � u 2 � u 1 and with increased value o f the pa r am e t e r lj> /u 1 . T he m u ltiplicity pattern of the first r e a cto r is obt ai ned as in the pre
cedin g section :
For cp > 1 : For <1> < 1 ·
where Da*
m
±
=
=
{
uniq uene s s for all D a D a * < D a < Da *
m ultiplicity
D a < Da * or D a > D a *
uniq uen e s s
6 ( m ) , D a * = I:J. ( m_) , and
+
1 ± (1
2
-
<jl )
1 /2
( 3 6 a)
Note that Eqs . ( 36) an d ( 36a ) correspond to E q s . ( 1 7) and ( 1 8) limit of very large activation energy . Similarly , for th e second reactor it can be shown that For
1
( 3 6)
> 1:
For <j> / u 1 < 1 :
in
the
uniqueness for all D a
{
D a **
< D a < Da * *
D a < D a * * or Da > D a * *
multiplicity
u n ique n e s s
( 3 7)
0 0
f
0 0
b2
0
0 0
I
-x
0
t
I
- x
l
0
@
I
- x
0
-x
I
0
-x
0
I
0.5
0
1 .5
1.0
m,
0 0
0
,flJ
in f i g. 8 b 2.0
m.)
-
0 0
0
-x
2.5
8
,f\
- x
0 0
3. 5
3.0
0
0
j jj ,
mb
0 0
A ris
T�
I
~
0
8� f� �
Morbidelli , Varma , and
986
-x
n@ -x
I
u
-x
(a) R e gions in the
FIGURE 8
==
==
w here D a **
==
=
=
r
( u 1 , n+) ,
D a **
=
r
( u 1 , n _) , and ( 3 7a)
The im pact of the above - mentioned scaling property of the equations ex amined is evident by noting that E q s . ( 3 7) and ( 3 7a) reduce to E q s . ( 36) and ( 3 6a) , respectively , as u 1 = u o = 1 . Since s pace p revents u s from analy zin g thi s p roblem in further detail , we shall limit our discu ssion to the final answers for the specific case under examination . In Fig . 8 the multiplicity patt ern is reported through a proper subdivision of the
987
R eac tor S teady- S tate Mul tip licity /S tab i l i ty
0 7 0 � � ·· 0
0
c 0
0
-x
I
-x
�v - X
o
0.6 0
1.6
r;.0
0 0
\.
. ,
I t
"'
2�
2.4
8
2.8
o
-x
0 0
�
0
0 - x
- x
-- x
(b) FI G U R E 8
( C on tin ued )
(of w hi c h no more t han three can be stable ) , and the total number of po s sible m ultiplicity p attern s is m uch l a r ger than th at in the ca se of a sin gle reactor .
A m ore detailed treatment of this problem , includin g the ease Ve l :f. vc 2 :f. 1 , h as been reported by Svoronos et al . ( 1 982) . Other interestin g features , such as t he effect of two different Da valu es for the reac tors or the influence of the activation energy , h ave been reporte d by K ubicek et al . ( 1 98 0 ) . Moreover , in the latter work the e ffe ct o f re cir c ul atio n from the second to the first reactor is also examined . I t is found that for increasing recirculation ratios , the m ultiplicity re gion t en d s to disapp e ar .
W he n more th an one reacti on is considered ,
the variety of pos sible configurations is eno rm o u sly - in fact , infinitely - en larged . T he only way i n which t o pre sent some of t h e modes of attackin g Com plex Kinetic Schem es :
Morbidelli , Varm a , and A ris
988
the p roblem of multiplicity in these cases is to take a simple example , and the obvious exten sions of the sin gle reaction are the parallel reactions , A + B , A + C , and the sequential A + B + C ; we shall concen t rat e on
the latter . Since there are three reaction parameters ( the Damkobler num ber , the heat of reaction or Prater number , an d the A r rh e niu s number) for a single reaction , we should expect six param eters for A + B + C . In addition , there is in both cases the cooling rate parameter and the dimensionles s coolant temperature . We shall see that the latter m ay often be ignored , and that in s tead y s t at e problem s the coolin g rate parameter can be combined with the heat of reaction , but this leaves six for t he steady state and seven for s tabilit y q ue stio n s In the face of so m any p arameters we can hardly hope to explore every nook and cranny of param eter space , but som e p ro gress m ay b e made through the use of singularity theory . The work of B alakotaiah and Luss ( 1 9 8 1 - 1 98 4 ) has been outstand in g in this area , and one cannot get a better acc o u n t than is given in their ori ginal p aper s . I f C A , C B , and T d en ot e the two concentrations and t he temperature respectively , t he steady - state mass and heat b alances give the three equations :
-
.
( 38a)
( 38b)
q pC
P +
(T
--
- T)
0
UA ( T
( - l'> H 2 ) Vk 2 ( T ) C B
- T c) ::: 0
+
( - l'> H ) Vk ( T ) C 1 1 A ( 3 8c)
It is cus
where all t h e symbols are as defined in the notation section . to mary to take C B O = 0 and u se the dimensionles s q u antiti es
u
1
y
a
=
=
CA CA E
l'> H =
O
l'> H
CB
2
C Vk
l
RT
u
Da
=
v =
1
AO (T
0)
=
B
q
0
2
1
e
k (T 2 o> k
0
y (T
-
T
=
T ) 0
o
( - l'> H l ) C A OY pC
p
TO
E I.
=
2
-
E
l
( 3 9)
UA
= --
q p Cp
Then 1
-
u1
-
D au
1
(
exp ___r§_ e + y
)
=
0
( 40a)
( 4 0b )
989
Reactor S te ady -S tate M ultip licity /S tability
Replacin g To by ( T o + tS T c ) / ( 1 + tS ) . we have t he same equations but T hi s i s s atisfactory unle ss either q or U is t aken to b e the with e c 0. distin guis h ed p arameter . If T c = T o , this difficulty disappears , and t his =
assumption is com monly m ade . Since the equations are linear in U 1 and u2 the steady state is u sually found by solving E q . ( 4 0a) for u 1 in term s of e , sub stituting in Eq . ( 4 0b ) an d solving for u 2 to give , o n sub stitution i n E q . ( 4 0c ) , a sin gle equation in 0 . This m ay be writte n
1
(4 ) where = Da exp
H 1( 0 )
H 2 (e)
= v D a exp
[(1 ":ey) ((1
Y0 +
(y
(e
e
)
+
I
I
/! e )]
Da exp
8 8 ": Y )]
+ v D a exp
( 4 2)
T his i s the form which Bilou s and Amundson ( 1 9 5 5 ) u sed and it has the simple p hy sical interpretation of a heat balance ' ( 1 + tS ) e bein g propor tional to the rate of coolin g , an d H1 and to the two rates of heat generation . Of course , a. and B or both can be ne gative if one step is endothermic and the other exo thermic . A rguin g that H 1 and are curves with a steplike transition between v alues near zero and value s near 1 , most workers in thi s field have assumed that at most five steady states can be obtained . Actually , the sub tleties of sh ape of the ri ght side allow seven , but only in a very restricted region o f p arameter s pace ( Farr , 1 9 8 5 ) . T his phenomenon is lost w hen y is allowed to become large and the exponential app roxim ation is u se d ( Jorgen sen et al . , 1 9 84) . Form ally , we let y -+ oo in the two ex-p onential s , simplifyin g them to e0 and e "A G , but this i gnores the presence of y in 8 and B . T he true n ature of this so - called " positive exponential approxim ation" i s ob scure , but it is probably b e st to think of it as im plying that the deviations of t emperature from T o are not large . T hough E q . ( 4 1 ) a phy sically appealinf>.' form of the eq uation that has to be solved for the steady state , it is in better shape for com putation when regarded as a quadratic for Da given t h e steady - stat e temperature 0 . Thus the m ultiplicit y q ue stion is to be investigated b y exploring the nature of the equation
H1H 2
H2
is
+ B *D ae where B * can take
=
e: = D ae
e
- 8(1
B / ( 1 + tS ) . e
A
=
(1
+
D ae
+
)
=
"A 8
(1
+
D ae
0
) ( 4 3)
0
C ertain combinations emerge from tis form and we
+
a.) v B *D a
givin g
A £ 1 + "A
e
vD a0 e
B * e: - r e e:'- ( 1
+
1-"A
e:) - 0 ( 1
r +
e:)
=
vDa
=
0
1-
"A
( 44 )
Morbide lli , Varma, and A rts
990
If ,\ and
1 and 0 is specified , this is a quadratic for given A , B * , and r , its positive root will give the correspondin g Da, The usual form of multiplicity diagram is 0 versus Da, and this may take one of a number of forms shown in the bottom row of diagrams of Fig . 9 . In the up p er part of that figure all parameters except B * and Da are fixed , an d the c urve PQRST is the locus of turning points of the 0 - Da curve [ i . e . , the points where d ( D a) / d 0 0] . The b ranche s of this curve are numbered for the order in w hich they are encountered in trav ersin g the Da- 0 curves shown below in the dire c tio n of increasing D a . The form of the Da- 0 curve is shown in the lower part of the fi gure for the appropriate ran ge of B *. For e x amp le over the square C , any vertical with increasing Da encounters the b r anc he s of the tu rni n g p oi n t curve in the order 2 1 4 3 . Th ere are thus two re gions of three possible steady states . For D the se overlap and there is an interval of five states with three on eithe r side . The D a - 0 diagrams for the transitions are shown as vixillae attached to the vertical. Thus the B /C transition comes about by a so-called "hy steresis point where the tangent at a poi nt of inflection is vertical . On the other hand , the =
E:
=
,
-
,"
Do
s
e� Do
Multiplicity diagram for the reaction s hr a
FI G U RE 9
noni ot e m l C S T R .
A
-+
B
-+
C occurrin g in
a
991
Reactor S teady -S tate M u l tip licity /S tability
C /D t ran sit io n is m ade when t w o t u r nin g p oi nt s are vertically above one
D a - e dia gram is give n by a solu tion of F = 0 ac cordin g given by F = F e = 0 , an d w hen B * and the other p ar amet er s are held constant , this p air of eq u atio n s has from zero to four solutions for Da and e . F or e xam ple , F i g . 9 show s that for sufficiently sm all B * th e re are no roots , w hile for sufficiently large B * there are j u st two . T he transitions occur at p art ic u l ar value s of B * . Those that in volv e vertical t an gents at inflation poin t s Q , R , S are ob t ain e d b y solvi n g F = F e = F e e 0 fo r e , D a , and B * . T he double limit poin t s U , V , and W are sol u tion s for e 1 , e 2 , D a , and B * of the four e q u ation s F ( e t) F e ( e t) = F ( e 2 ) = F e ( e 2) = 0, where F ( e 1 ) = F ( e , D a , B * ; a , y , A , V ) with e 1 an d e 2 are t he ordinates of the turning p oi nt s that sh are a 0 0 1. common D a . T he sur face in par ame te r s p ac e giv e n b y elim i n at in g 8 and Da between F = F e = Fe e = 0 is called t h e " hy steresi s variety . " I f e 1 , 0 2 , and Da are eliminate d be t we e n F ( e 1) = F e ( e 1 ) = F ( e 2 ) F e ( e 2 ) = 0 the " double limit variety" is o btain e d . I solas arise when ei t he r the curve loops back on itself or an entirely new branch is born from an isolated poi nt . 0 is indet erm in at e , so both In both cases the dire ction of the curve F F e and F n a vanish sim ultaneou sly . T his does not happen in this p art ic ular case , but if 0 an d Da can be eliminated b et w ee n F Fe = FD a 0 , we have th e " isol a varie ty . " Follo win g the p recepts of sin gu lar ity theory we look for t h e points w he r e as m any a s po s sitlle of th e derivatives vanish . B alakotaiah and Luss ( 1 982a , b ) showed that there are points at w hi ch F and its first fou r deriva tives with re sp e c t to 0 v ani s h . For A = 1 , y = t he se are (8 , Da , B * , a , v) = ( 3 , 0 . 1 8 58 , 3 . 2 1 5 4 , 0 . 86 6 0 , 0 . 0 7 1 8 ) a n d ( 3 , 0 . 0 1 3 3 , 4 4 . 7846 , - 0 . 8 6 6 0 , 13 . 928 2 ) . A point w h er e F a nd it s fir s t four derivatives vanish is known as a " b utt e rfly p oint" and F is there " contact eq uivalent" ( see the A p pendix to t hi s ch ap te r ) t o x 5 - A 0. T h i s means that x is some tran s formation of e and A of t h e D amkcihler number D a, and that a full unfolding is given by three oth er param eters a1 , a 2 , and a 3 in the for m another .
A p oint on t he
to E q . ( 4 3) .
A t ur nin g p oin t is
=
=
=
=
=
co
=
,
=
( 4 5) S in gul a ri ty theory ( subject to app r opriate te st s ) s how s that a 1 , a 2 , a 3 w ill be smooth function s of B * , a , and v , alth ou gh it woul d be a work of some m a gnit u de to find them . H owever , the qualitative picture in the n ei ghbo rhoo d of the " o r gani z in g center" m u st be that of the butterfly . Thus , for a 3 < 0 t he re are only two possible e - D a di a gra m s , namely A and B . The former is ob t ain e d when the p aram et e r s a 1 an d a 2 lie beneath the par ab olic curve shown in the left h alf of t he top p art of Fig . 1 0 . O ther wise , the common p h e n o me no n of three s t ead y state s in the intermediate interval of the D amkob l er number is obtained . T he situation with a3 > 0 is more complicated and is shown on the r i gh t at the top of Fig . 1 0 . The variou s Da- 0 d i a gra m s sho w n b e lo w by the l et t er s A , . . . , G are obtained for a 3 > 0 when a. l an d a. 2 lie in the co r re s po n din gly marked re gion . Re fer rin g back to Fig . 9, wh er e the seq uence A , B , C , D , F , G , B is ob tained for in c rea sin g B * , w e see that it wo ul d be q ui t e con sistent with this if were a monotone - increasing fu n c tio n o f B * , an d that a and v were such t ha t a. 2 and a 3 w ere p o si tive , for t he n the same sequence would be obtained p a s sin g up ward alon g the dashed curve , PQ . an d
a1
992
Morbidelli ,
4
B
B
A
8
Varma , and A ris
B
D
c
E
F
G
Do
FIGURE 1 0 Qualitative bifu r ca t ion diagram s for the re ac tio n . A -+ B -+ C occurrin g i n a no n i soth e r m al C ST R , in the n ei gh b o rh oo d of the or gani zin g ce n te r .
B y w orkin g away from the singularity the various r e gion s of multiplicity as < 0 are obtained by generatin g the hy ster
can be determined computationally .
a s > 0 and the p arabolic for
T hus the c urv e 1 , 9 , 2 , S , 4 , 9 , 5 for
T he loci on esis vari ety , while 2 , 8 , 7 , 6 , 8 , 4 is the double - limit variety . which the p oint s 2 , 4 , an d 9 lie ari se when two h y st ere sis point s coincide ( i . e . , F = Fe = Fe e = F e e e = 0 ) , w hile 6 , 7 , and 8 are w he re a tu r nin g point lies vertically above or below a hy s t e r e s i s point [ i . e . , F ( e 1 ) = Fe ( e 1 ) = F e e < e t ) = F ( e 2 ) = Fe ( e 2 ) = 0 ] . In thi s way the m ultiplicity re gion s can be built up with som e confidence that we have a comprehen sive picture .
Further examples are give n in the p apers of B alakotaiah and Luss already re fe rred to , and no doubt w e shall see m any more st u d ie s u sing sin gularity
theory ( see Poli zopoulo s and T akoudi s , in p re s s ) . T he sequence of two st ir r e d tank s , for example , has been con sidered by D an ge l m ay r a n d S tewart ( 1 984) . '
T u b u l a r Reacto r T he steady - state m a ss and ener gy balances in a tubular reactor can be
written as follow s ( V arm a and Aris , 1 9 7 7 ) :
v
:;
- f( C , T )
=
0
( 46)
993
Reac to r S teady -S tate M u l t ip licity /S tab ility
( 47 )
together with D anckwerts' boun d ary conditions ( B C s) at z = 0
k
dT e dz
= vp C p ( C
at z
- C ) 0
=
( 48a)
( 4 8b )
0
In derivin g the se equations the followin g assumptions hav e been im plicit ly made , in addition to assumptions 2 and 3 reported earlier for the C S T R : Radial gr a di ent s of temperature and concentration are ne gligible . T h e fluid sp atial velocity i s const an t in the axial direc tion T he dispersive fluxes o f m a s s and heat can be described t h rou gh Fick and Fourier law , re spectively , with effective mass and heat diffusion coefficients .
1.
2. 3.
.
We shall n ot p ursue any further the cl a s si cal question of relaxin g the above mentioned restrictio ns in order to improve the model c ap abi lit y of simul ati n g experimental r eact or behavior ( see C hapter 6) O ur aim i s instead to indic ate the multiplicity p a tt ern of this model , u sually referred to as the •
" axial dispersion model , " in connection with various reactin g systems . Note in passing that the occurrence of multiple s t e a dy states in t u bul ar reactors has been experimentally verified in several works , which have been exten sively reviewed by Jen sen and Ray ( 1 982) . I n t ro d uc in g the dimensionle ss quantities
u
c
=
v
co
Da
8
::
=
Pe
m
=
T T o
0
Lf( C 0 , T 0 )
vC
0
< - t. H) c o Pe
=
c
=
T
z s = L
c
To
UPL
A vp C
P
{ 49) r
pC T p O vL D e
v
-
= h
f( C , T ) f(C , T ) 0 0
=
vL pC
� k
e
y = --
E RT
0
the model equations reduce to
1 Pe
m
2 d u 2 ds
du
ds
- D ar ( u , v )
=
0
( 50)
(0 (0 II>..
T AB LE 1
A Priori B ounds on the Steady - S tate T emperature of the Axial Disperison Model with Irreversible Reaction Adiabatic
Exothermic reaction
< a > 0)
=
{
0)
N onadiabatic v
c
Varma ( 1 977) .
max [ 0 , 1
-
I BI 1
� v � 1
{
v
v
c
c
c
(o
>
0)
� 1
1 � v � 1 + f3 + o (v
< 1
v
� 1
Max [ O , 1 -
I aI ]
� v � v
< 1
max [ 0 , 1 -
I
- o (l - v ) � v � l c
1 � v � 1 + a v
Endothermic reaction < a < 0)
So urce :
(o
c
� v �
c
- 1)
1 + a
B
I
c a:: 0
&
� -
< t:l
�t:l
t:l ;:s Q. > :l.
..
995
Reac to r S teady -S tate Mul tip licity /S tability
v
ds
d
+
�Dar (u ,v) - S (v - vc )
( 5 1)
= 0
with B C s
du
ds
du ds
Pe m (U
=
dv
0
=
dv ds
- 1) =
ds
0
P
at
s
e
=
h
at
(v - 1)
s
=
( 5 2a)
0
( 52b)
1
priori bound for the steady-state reactant concentration is obvious m p hy sical ar gum ent s : 0 ( u( s) " 1 for all s [ 0 , 1 ] , while for the T his has been developed , temperature a more detailed analysis is required . based on the st ron g m aximum principle for elliptic equation s , by Varma ( 1 9 7 7 ) and the obtained bounds are summarized in T able 1 . The a
e:
fro
U niqueness
A s a direct
a n d M u l tip licity C ri t e ria
con equ e nce of the Danckwerts B C s , e axial dispersion model to the CSTR and the plug-flow models for very large and very isp ers io n (i. e . , Pe 0 and Pe ) respectively . T he multiplicity pattern of the first limit has been fully characterized in the pr ece d in g section , at least for sim ple reacting systems . On the other hand , the plu g- flow m od e l does not admit any multiple steady state , since it is constituted by initial value ordinary diffe renti al e q u atio n s . However , while the multiplicity behavior of the axial d i spe r s ion model is well characterized for limitin g Pe values , it is still not so for inter mediate situations (i. e. , finite P eclet number v alue s ) . I n p articular , there is a lack of � priori criteria , based on the values of the relevant p aram eter s , for t h e definition of the reac t o r un iq uen e s s or multiplicity . Al though several attempts have been made in the literature , based on com parison th eo r e ms , fixed -point arguments , p erturb ation m ethods , lum pin g proced ure s , and other tec hniq e s , a necessary and sufficient criterion for uniq ue ne s s has flnot been found . ext , we summ ari z e brie y the m ost widely u sed sufficient or necessary , N or neither but simply ap p roxim te , bounds of t e multiplicity region which have been reported in the literature . Since no proo fs will be reported , the interested reader should refer to original papers or to a previous review ( V a rm a and Aris , 1 97 7 ) . S e a of A diabatic Reactor: of adiabatic reactor (o 0) with equal mass and heat Peclet (Pem Peh Pe) is of particular mathematical interest , since it allows to combine E q s . ( 5 0) and ( 51) [in same w ay we did for the C S T R case and leadin g to the same u -v relation given by E q . ( 7 ) ] , reducing the problem to the single ordinary differential eq uation (ODE) s
reduces small axial d
th
+
+
co
,
u
steady - state
a
h
the
p ci l C ase
The case
number s
=
=
=
the
d
2
ds
v 2
with B C s
- Pe
given
dv
ds
+
DaPe�r ( 1
by the
+ a
jJ
�
v
v)
0
second of Eqs . ( 52a) and
( 53) ( 5 2b ) .
Mo rb idelli ,
996
Varm a ,
and A ris
Lu ss ( 1 9 7 1 ) showed that the sam e c on di tion [ E q . ( 1 1 ) 1 which is neces sary and su ffi cie n t for u n iquene ss in lumped system s gives a sufficient Since condition in distri bute d system s of the single - cat alyst - particle type . Eq . ( 5 3} can b e reduced to t he latter form ( V a rm a an d A m un d son , 1 9 7 3c ) , t he followin g sufficient condition for st ab ility is obtained : dF dv
for any v
E
[ 1 , 1 + 81
( 54)
w here F ( v ) r ( v ) / ( v - 1 ) , an d the tem perature boun d for an exothermic r eaction from T able 1 has been u sed . An earlier sufficient crit e ri on developed b y Luss and Am un d so n ( 1 967b) u sin g com pari son methods w hich en sure s uniqueness i s =
Da
tor any v
__
wher e 1.1 tan
A
=
1
fi"
/ Pe
+
[1, 1
+
SJ
( 55}
Pe / 4 , an d A 1 is the smallest root of the equation
Pe
=
e:
fi"
( 5 5a)
Similarly , two ad ditional su fficien t uniquenes s criteria developed b y Luss ( 1 9 7 1 ) for a sl ab cataly st p articl e can readily be applied· to the case under exam ination : ( 5 6}
Da �
1.1
( 5 7)
F ( v_}
w here v+ and v r ep re se n t the local maximum and mm 1m um of F ( v) . In the case of an irreversible first - o rder reaction , F ( v } and v + are given by Eqs . ( 1 0) ( wit h n = 1 ) and ( 1 8 ) respectively . Depen d i n g on the v al ue of the Peclet num be r , eit he r condition ( 5 5 ) or con dition ( 5 6} is more sharp . General N onadiab atic C ase : U sin g comparison method s , Varm a and Am u n d s on ( 1 9 7 2b ) h ave developed the following two sufficient criteria for uniqueness , w hic h are restricted to reaction rates of positive order . The = Pe ; it req uires that first applies only t o the case P e = Pe h m for any v
e:
[ 1 , 1+ 13 ]
( 5 8)
where JJ i s th e sam e p arameter a s in Eq . ( 55) , and
a
1
=
ar
Sup � > 0
a
2
=
ar
Inf au
( 5 8a)
� 0
Note that E q . ( 5 8) reduce s to Eq . ( 5 5 } for o 0. is more general since it ap plies to P e =F Pe also : h m =
T h e second criterion
99 7
Reactor S teady - S ta te Multip licity /S tab ility +
t. 1 / 2
Pe m
( 5 9)
I )]
where t.
=
[ f3JJ m a
+ 6) ( JJ 3 + h
2
+ 2 f3 J.l
m
( J.l
m
+ 6 ) a 1 a 3 e x p ( j Pe
h
- Pe
m
l )]
( 5 9a )
and J.l m an d J.l h a r e ev alua t e d as J.l i n the p rec e din g criterion , b ut u sing Pem an d P� in Eq . ( 5 5} , re spectively . Note that for the nonad iab a ti c case , no u pp e r bound to t h e re gion of u n iq u e n ess i s av ail ab l e . From the results r e p ort ed above , at least som e qualitative con cl u sion s can be drawn : fo r su ffici e n tly l arge Peclet number Pe a nd heat transfer coefficient 6 , a n d for sufficiently sm all Damkholer n um be r Da and reactor len g t h L , the axial di sp er s ion model exhibits a unique steady state for any value of all the other p arameters .
Approxima te A nalysis o f Mul tip licity Pa ttern A s shown above , no exact n ece ss ary and sufficient criteria for the defini tion of the m ultiplicity p attern of the axial di s p e rsio n model are available . Hlav ace k and Hofmann ( 1 97 0a , b ) d evl eop ed a l u m p i n g p rocedure t h at allows u s to ap p r oxim at e the ori gi n al set of nonlinear ODE s , with a set of non linear al geb rai c equation s , whose multiplicity p at t e rn can be determined u sin g the methods described in the C ST R section . The l um p in g p rocedure co n si st s of ap prox im at in g the linear sp ace dif ferential o perator , say in Eq . ( 5 3 ) , with t h e first t e rm of the s e ri e s solu tion of the correspondin g tran sient problem where the nonlinear reaction term has been neglected , that is ,
( 60) w here A. 1 is given a s the smallest root of Eq . ( 55a} . lar ca se of E q . ( 5 3} , the p rob l em reduces to
(
A.
1
+
2
-4-
Pe
)
v
- D aPe f3r
{1 +
13 -
f3
v
, v
)
= 0
T h u s for the p articu-
( 61)
whose multiplicity pattern can b e e a sily determ ined . N ote that this procedure can also be a pp lie d readily to the more general case [ i . e . , E q s . ( 5 0) and ( 5 1 ) ] , and it is independent of the for m of the re action rate expression r(u , v) . It m ay be ob served that as expected from t h e intrinsic nature of the l u m p ing app roxim ation , E q . ( 6 1 ) reduces to the cor r ec t C S T R model as Pe + 0 . How eve r , t his is not t ru e for the other lim it condition ( i . e . , Pe + co ) , w here the characteristic u ni q ue ne s s of the plug- flow model i s not co rrectl y p redicted . Therefore , this p roc ed ure should be u sed with care , sin ce a s we will see in more detail in the case of a first -order
Morbidelli , Varma, and A ris
998
reaction , it can lead to erron eou s conclu sions about the reactor steady s t ate m ultiplicity . A different lumped model can be obtained by ap proximating the sp ace differential o p e ra to r throu gh a o n e - poi nt collocation m ethod . U sing L e g e ndre polynomials , so that t h e collocation point is loc at ed at the reactor center , we obtain
Pe
dv ds
=
2
2
+
+
Pe Pev 3Pe / 4
( 6 2)
w hich then lead s to al ge b r ai c nonlinear equations o f the same form as Eq . ( 6 1 ) . Jen sen an d Ray ( 1 98 2 ) h ave noted t ha t this lum pin g p rocedure exhibits similar limitin g behavior in term s of Pe as the p re viou s one , and thus it basically suffers from the same s hort c om in g s .
Irrev ersib le Firs t-- O rder Reaction in a N onadiabatic Reac tor
T his p a rt ic u lar case has been examined numerically in d e t ail by Varma and A m und so n ( 1 9 7 3c ) , Hlavacek et al . ( 1 97 3) , an d K ubicek et al . ( 1 97 9) , who found m ul tip l icit y p attern s of the type 1 - 3 - 1 , 1- 3- 1 - 3 - 1 , and 1 - 3- 5- 3 - 1 . More recently , K ap il a and Poore ( 1 98 2 ) h av e d eveloped an an aly tic al asy m p toti c solution of the model in t he limit of very large activation energy (i . e . , y + ) T his solution is achieved b y dividing the re�ctor into zones of various reaction in ten sit y , where a sim p le analytic rep resentation of the sol u tion can be gi v e n , then co mp l eti n g the solution throu gh the asy m p t oti c m atc hi n g technique . It has been conc lu d e d that whereas the adiabatic m ode l can exhibit at most three s te a d y states , the nonadiabatic one can yie ld up to s e ve n . T hese fin din g s have been confirmed by t h e numerical results re po rt e d by H ei n e m ann and Poore ( 1 98 2 ) . Some of the obtained res ult s a r e show n in Fig . 1 1 , in t e r m s of the m axim um tempera ture alo n g the reactor as a function of t h e D am kobler number . I t app e ars that wi th all other p arameters fixed , for in cre a sin g values of the dim ension less ac tiv at io n energy y , the model yield s u n i qu e solution (y 2 5 ) , 1 - 3- 1 m ul tip l i city ( y 40) , 1 - 3- 5 - 3- 1 m ultiplicity ( y ::: 5 0 , 6 0 , 7 5 ) , and finally 1 - 3- 5- 7 - 5 - 3- 1 multiplicity ( y "' 1 00 , 1 2 5 ) . Note t h at in all cases t he inter m ediate steady s t at e s are all unstab le ; only t h e lower and uppe r ones are oo
.
=
=
a symptotically stable . The in acc u r acy of the l u m p i n g ap proximation [ E q . ( 60) ] b e com e s evi dent in t h e case of a first -order r e ac t ion , since the lum ped m o d e l can , at Moreover , it has been shown most , p r e d ic t a. 1 - 3 - 1 m ultiplicity p att er n ( V arm a an d Amun d son , 1 9 7 3 c ) to predict uni q u en ess w hen the distributed no n adiab a ti c model exhibits multiplicity , and vice versa . H ow ever , in the adiabatic case wit h equal Peclet n u m b er s , numerical comparisons seem to in dic at e that the lum p ed model y iel d s a tr uly sufficient con dit ion for unique ness , althou gh no p roof is available . Jen sen an d R ay ( 1 982 ) have com pared t h e lum ping approxim ation ( 6 0 ) and ( 62) with the exact numerical calculation s for adiab atic and nonadiab atic reactors . As shown in Fi g . 1 2 , w h e n the m ul ti p li ci t y re gions are com p ared , both approximation m ay lead to e rro neo u s conclu sion s . In general , Eq . ( 6 2 ) seems to be more .
accurate .
1 20
.,
b l
(
y
2 �
E
�
=
125
1 .1 2
� E
E
:>
1 04
" 0
:2:
I
- D o m k o h l e r n u m be r , D e
1 2 0 r----r---,---. ( 0 l �
Cll Q
E
{!!.
E
E
:>
:2:
0
11 2
�
1 .04 10 · 1
I0 - 3
- -- D o m k o h l e r n um b e r , D o
Maximum dim ensionles s t emp erature a s a function o f D a for FIGURE 1 1 variou s activation energy values . ( a ) y = 2 5 , 4 0 , 5 0 , 6 0 , 75 ; ( b ) y = 0 . 5 , vc = 1 . 0) 1 . 0 , iS = 4 . 0 , (3 = Pe 1 0 0 , 1 2 5 ( p ar8m eter values : P e h m =
I 5
=
.
r-----.---r--r---.
10 5
0
b
,_- - -- - --
- -- - -- -
b - - -- -- - - - \. �
0
20
40
-
Pe h
60
80
100
Comparison of exact and app roximate regions of m ultiple steady FIG URE 1 2 states in the a y versus P eh plane ; exact ( - ) , E q . ( 62) ( - • - • - ) , E q . ( 6 0) ( - - - ) ; iS = 0 , P em = Peh ( curve a ) ; o = 0 , Pem = 3Pe h ( curve b ) ; o
=
2 , P em
=
Peh ( curve c ) .
999
997
R eac tor S teady - S tate M u l t ip licity /S tab i lity +
11
Pe
1/2
( 5 9)
I )]
m
w h e re 6 = [ � ll a
+ 6) a + ( ll h m 3
2
+
2 S ll
m
( ll
m
+
o ) a a e xp ( l P e - Pe 1 3 m h
l
)) ( 5 9a)
ar
3
= S up av
and J.l m and ll h are ev alu ate d as ]J in t he p recedin g criterion , but u sing Pem an d P� in Eq . ( 55 ) , re spectively . N ote that for the nonadiabatic case , no upper boun d to the region of uniqueness is available . From the result s re port ed above , at lea st some q u alit at iv e conclusion s can be drawn : for sufficiently large Peclet number Pe and heat tran s fe r coefficient o and for s u fficie ntly sm all D amkholer num ber Da and reactor
,
len g th L , the axial di sp e r sion model exhibits a u n iq u e steady state for any v al u e of all the other p arameters . A p p roxima te A na l y s i s of Mul tip lici t y Pa t t e rn
As shown above , no e x ac t necessary and suffici ent criteria for the defini tion of t h e multiplicit y p attern of t h e axial disp e rsion model are av ailable . Hlavacek and H ofmann ( 1 97 0a , b ) d e v l eop e d a lumpin g p rocedure that allows u s to a pp roxi m at e the original set of nonlinear ODE s , with a set of non linear algebraic equation s , whose m ultiplicity p attern can be d e t e rm ine d u sin g the m ethods described in t h e C S T R section . The lumpin g p rocedure con si st s of approxim atin g the linear sp ace dif ferential operator , say in Eq . ( 5 3 ) with the first term of the series solu tion of the correspondin g t r an sie n t p roblem where the nonlinear reaction term has been neglecte d , th at is ,
,
-
2 d v 2 ds
Pe
dv
ds
s
-(>.. 1
+
Pe
2
4
)
v
( 60)
w here A. 1 is given as the sm allest root of E q . lar case of E q . ( 5 3) , the problem reduces to
(
>..
1
+
2
Pe -4-
)-
v
D aPe Sr
(1
+
� S
-
v
,
v)
=
( 5 5a) .
0
Thus for the p articu-
{ 61)
whose m ultiplicity p attern can b e easily determ ine d . Note that t h i s procedure can also be applied r eadily to the more general ca se [ i . e . , E q s . ( 50) and ( 5 1 ) ] , and it is independent of the form of the reaction rate expression r(u , v) . It m ay be ob se rve d that as expected from the intrin sic nature of th e lumping ap p roxim ation E q . ( 6 1) reduc es to th e correct C S T R m odel as Pe -+- 0. However , this is not true for the ot he r limit condition ( i . e . , P e -+- "" ) w here the characteristic uniqueness of the plu g - flow m od el is not correctly p redicted . Therefore , this p rocedure s ho u ld be u sed with care , since as we will see in more detail in the case of a first -order
,
,
Morbidelli , V arma, and A ris
998
reaction , it can le a d to erroneou s conclu sions about the reactor steady state m ultiplicity . A different lum p e d model can be obtained by app r oxi matin g the space
differential operator throu gh a one- point collocation m ethod , U sin g Legendre polynom i al s , so that the collocation point is lo c ate d at the reactor
center , we obtain Pe
dv ds
=
2
2 + Pe Pev + 3Pe / 4
( 62)
which then leads to algebraic nonlinear eq uations of the same form as E q . Jen sen and Ray ( 1 982 ) h ave noted t ha t thi s lum pin g p rocedure ( 61) , exhi b it s si milar li m iting beh avior in term s o f Pe as t he previous one , and thus it b a sically suffers f:rom the same shortcomin gs .
Irrev ers i b le First·- O rder React ion in a No nadiabatic Reac tor T his pa rt ic ul ar case has c been examined numerically in detail by Varma and Am undson ( 1 97 3c ) , Hlava ek et al . ( 1 9 7 3 ) , an d K ubicek et al . ( 1 97 9) , who found m ultiplicity p atterns of the type 1 - 3- 1 , 1 - 3 - 1 - 3- 1 , and 1- 3- 5 - 3- 1 . More recently , K apila an d Poore ( 1 98 2 ) h ave developed an an aly ti c al asymptotic solution of the m odel in the limit of very large activation energy (i . e . , y + ) This s olut ion is achieved by dividing the reactor i nto zone s of v ariou s reaction in ten sity , where a s im p le analytic representation of the solution can be given , then com p leting the solution through the asymptotic m atching technique . I t has b een concluded that w hereas the adiabatic model c a n exhibit at most three s teady states , the nonadiabatic one can yi eld up to seven . T hese findin gs h av e been confirmed by the n um erical result s reported by Heinemann and Poore ( 1 98 2 ) Some of the co
,
•
obtained result s are show n in Fig . 1 1 , in term s of the maxim um tempera ture alon g the rea c t or as a function of the D amkobler number . It appears
that with all other p aram eters fixed , for increasin g value s of the dim ension less activation energy y , the model yiel ds unique sol u t ion ( y = 2 5 ) , 1- 3- 1 m ultip licit y ( y 40) , 1 - 3- 5- 3- 1 multiplicity ( y 5 0 , 6 0 , 7 5 ) , and finally Note that in all cases the inter 1 - 3- 5- 7 - 5- 3- 1 m ulti plici t y ( y = 1 00 , 1 2 5 ) . m ediate steady states are all unstab le ; only the lower and upper ones are asymptotically stable . T he inaccuracy of the lum pin g ap pro xi m ation [ E q . ( 60 ) ] becomes evi dent in the case of a firs t - o r d e r reaction , s i n c e the lumped m odel can , at Moreover , it has been shown most , p redict a 1 - 3 - 1 m ultiplicity p attern . ( V arma an d Amun dson , 1 97 3c ) to predict uniqueness w hen the distributed non ad iabatic model exhi b its multiplicit y , an d vice vers a . H owever , in the a diab ati c case with equal Peclet num bers , numerical com parison s seem to indic at e that t he lu m p ed model y ield s a t r uly sufficient condition for uniq ue Jensen and R ay ( 1 98 2 ) have com ness , although no p roof is available . p ared the lum pin g approxim ation ( 60) a n d ( 6 2 ) with the exact numerical calculation s for adiab atic and no nadi ab ati c reactors . A s s hown in Fig . 1 2 , when the m ultiplicity region s are com p ared , both app roxi mation may lead to erroneous conclu sion s . In ge ne r al , Eq . ( 6 2 ) seems to be more =
accurate .
=
1 20
E
E
::3
( b )
y
=
I
125
-------'---"10 0
104
·;c 0
�
I
- D o m ko h l e r n u m be r , D o
1 20 �-----.--� ( 0 ) 1.1 2
" 0
:::!
104 I0 - 3
-
- Domkohler num ber , Do
FIGURE 1 1 M aximum dimensionless temperature a s a function of D a for various activation energy values . { a) y = 2 5 , 40 , 5 0 , 6 0 , 7 5 ; ( b ) y = 1 0 0 , 1 2 5 ( p ar ameter values : Pe = Pe = 1 . 0 , 8 4 . 0 , a = 0 . 5 , Vc = 1 . 0) . h m =
10
)... C!l.
5
+
� - -- - ----=--=--::.-:.=..._ b
\. � b - - - - -- - - - -
O L--___J___,L___J___.J.___J 0
20
40
--
Pe h
60
80
100
FIG URE 1 2 Com parison o f exact an d approxim ate region s of m ultiple steady states in the By versu s Peh plane ; exact ( - ) , Eq . ( 62 ) ( • - ) , Eq . ( 60) ( - - - ) ; o = 0 , Pem = P� ( curve a) ; o 0 , Pem = 3P� ( curve b ) ; = Pe 2 , Pe o h ( curve c) . m •-
=
=
999
1 000
Morb idelli ,
N o nmo n o t o ne I so therm a l Reaction
Varm a ,
and A ris
Nonmonotone kinetics were examined in an earlier section . We refer here to the case of a substrate-inhibited enzyme reaction , of some importance in biocataly sis , examined in detail by DeV era an d Varm a ( 1 97 9) . T he iso thermal axial dispersion model in this case reduces to Eq . ( 5 0) , with boundary conditions given by the first of E q s . ( 5 2a) and ( 5 2b ) , with
r(u) where K
==
+ 1 /K ) u s 2 K + u + u /K s (1 + K
( 63)
and Ks rep re sent the ad sorption and substrate inhibition dimen sionles s constants , respectively . Note that Eq . ( 63) is quite similar to the biomolecular Lan gmuir- Hinshelwood reaction rate ( 2 2 ) examined pre viou sly in the context of a C S T R . In Fig . 1 3 the reactor outlet concentration u ( 1) is shown as a function of the D am kohler number Da for various Pe . It appear s that the reactor exhibits at most a 1 - 3 - 1 m ultiplicity pattern . In Fig . 1 4 the region of mul tiple steady states in the Pe - D a plane is shown for fixed values of the kinetic con stants K and Ks · T he calculated bifurcation values of Da have been compared with the uniqueness criteria reported above . It is found that E q s . ( 56) and ( 5 7) ( rewritten for the isotherm al model under examination ) give conservative bounds by a factor ran ging between 1 . 6 to 3 . 3 and 1 to 2 . 1 ; respectively . On the other hand , unlike the case of the adiabatic axial disperison model with first -order reaction , the lumpill g approximation fails in predicting the region of steady- state uniqueness close to the cusp point .
1.0
::J
c
·� � c � u c 0 u
:!i
:::J
0
0.8
0 .6
0 .4
0.2 0
D a m k o h le r n u m be r , Do
Outlet concentration as a function of Damkobler number for FI G URE 13 various Pem values in the case of isot hermal substrate -inhibited enzyme reaction ( 63) ( parameter values : K 0. 01, K 0 . 1) . ==
s
==
R eacto r S teady -State M u ltip licity /Stability
1 001
14
E
..
a..
1 2
10 8
6
4
2
0
I
2
3
4
- - Do
5
6
FIG URE 14 Region of multip le steady state s in the Pe m vers u s Da plane . Parameter values as in Fig . 1 3 .
D Y N AM I C B E H AV I O R In the preceding section w e examined several reactin g system s and have shown that under spec i fic condition s these m ay e xhibit one or more steady states . It is now important to characteri ze the st ability character of these steady states , which de term in es w hether or not t hey can be reac he d in practice by the r eac tor , through a suitable st arting p rocedure . How ever , in general , the an aly si s of all t he reactor steady stat e s and their s t ability character does no t exhau stively describe the reactor dynamic behavior . In p arti c ul ar , the occurrence of periodic or aperiodic solution s has to be investigated separately . In this section we focus on steady - state stability and p e riodi c solution s . Significant contributions h ave been made to the solution of the se prob lems in recent years , so that at least from a methodological point of view , definitive an sw ers are available in som e instances . How ever , a complete picture of reactor dynamic s has been obtained in only a few very simple c a se s . Maintaining the same structure as the precedin g section , we examine first the C ST R and then the axial dispersion model tub ular reactor ( i . e . , a lumped - parameter model and a distributed - p arameter model) . S ub se quently , examples o f ap e rio dic solutions will b e give n through the analysis of somew hat more complex r eactin g system s .
Conti n uous
S t i r red -T a n k
Reactor
Stability
T he transient mass and energy b alances can be readily derived from E q s . ( 1 ) an d ( 2 ) b y adding on th e ri ght - hand side the m ass and energy accumu lation term s , respectively . How ever , in m akin g the re sulting equation s dimensionless , it is convenient to u se the quantit ie s most widely u sed in the literature , w hic h are somewhat di ffere n t from tho se given by Eq . ( 3 ) . In p articular , u sin g the inlet tem perature as reference temperature T m = T 0 , and
1 002
t
e =
=
X
=
T
=
v -
Morb ide lli ,
Varma ,
and A ris
q
( 64)
t'
B =
T
we obtain dx dt
=
Le
d El dt
- x + D ar
=
+
-e
( 65) B D ar - 6 ( e - e > c
( 66 )
with initial conditions at t = 0
( 6 7)
w here D a , 6 , and r are as defined in Eq . ( 3) , w hile the Lewis number Le represents t he ratio betw een the tim e con stant s of the thermal an d m aterial tran sport , respectively . As discu ssed in detail by Ray an d H astings ( 1 98 0) , Le is defined as ---'P�::...
(VpC )
Le
=
V pC m
T
( 6 8)
p
where V represent t he total mass and thermal capacitances m and (V p C p ) T of the reaction unit , re spectively . T he first criterion for the stability of the steady state of a C S T R was developed in a pioneerin g p ap er by v an Heerden in 1 9 5 3 . It can be ob tained from the steady- state version of E q s . ( 6 5) and ( 66) ( i . e . , settin g the left - hand side equal to zero ) by solvin g explicitly for x , X
=
e + s ce - e > c ---�B�---
( 6 9)
and sub stitutin g back in Eq . ( 66 ) to ob tain an equation in the only un known e : R ( El ) - e
+
o (e - e ) c
=
B D ar
(e
+
a ce B
e
c
)
\ /
,e
: G ( El )
( 70)
The left - and right- hand sides of Eq . ( 70) rep resent the heat removed from th e reactor through flow and the cooling medium , and the heat p ro duced inside t h e reactor by t h e exothermic reaction , respectively . T hese two function s are shown in Fig . 1 5 , for those cases where the reactor exhibit s one ( curve 1) and t hree ( curve 2) steady states . On physical ground s , it can be argued that the stability of each steady state depends on the relative slopes o f these two curves . If at steady state t he slope
1 00 3
Reactor S teady - S ta t e M u l t ip lici t y /S t ab i lity
ct>
0:::
f) FIGURE 15 S chem atic representation of the heat removal R ( G ) an d the heat gene r a tion function s G ( G ) , defined by E q . ( 7 0) in a C ST R . of the heat generation curve is lar ger than that of the heat removal curve ,
the steady stat e is un stable , because for sm all pe rtu rb at ion of the tempera ture v alue the sy st em evolves aw ay fr om the steady state i t s el f B ased on this ob serv at ion van H erdeen concluded that the middle steady state in the case of multiplicity is unstable , while all the others , in c ludin g the case of a u n iq u e steady state , are st able The fi rst ri gorou s analy sis of this problem w as presented by B ilou s and A m u n d so n ( 1 95 5 ) , who ap plie d the first m ethod of Liapunov . It con sists of lineari zing E q s . ( 65 ) an d ( 6 6) aro un d the steady state , i n t r od ucin g the two deviation variables .
,
.
=
X
-
X
y
s
2
=
e - e
( 71)
s
and a n aly z in g the s t ability character o f the lineari zed system .
form t his can be w ritten as follows :
In m atrix
d�
2 dt = !! �
( 7 2)
where � = (y 1 y 2 ) T
•2 is
the capacitance matrix given by ( 73 a )
and
!! �
i s t h e Jacobian evaluated at st eady state defined as =
[
D ar
x
B D ar
-
x
1
D ar -1
8 -
o
+
B Dar
e
]
( 73b )
where rx and re indicate the p artial derivatives of the reaction rate , r at s t e ady st ate w it h respect to con version x and temp erat ure e , respectively .
M o rbide l l i ,
1 004
]
In t he case o f an irreversible first - order reaction r ( 1 + 8 / y ) ] , t he Jacobian reduce s to
J =
[�
=
Bx
X
- 1
X
+
(1
-1 (1
6 /y )
+
o
2
+ Bx
0 /y )
=
Varm a ,
an d A ris
( 1 - x) ex p [ 0 /
( 7 4)
2
T h e stab ilit y of the linear system ( 7 2 ) is dic t at ed by the ei genv al u es of
matrix �- 1 �:"
A ss um in g that Le
�,
eigenvalues of A±
=
tr
� ±
=
1 (i.e. ,
which are give n by 6. 1 / 2
1:!.
2
=
( tr �)
2
�
=
- 4 d et
p,
the s e coincide with the
i);
( 75)
The effect of t h e Lewis number Le will be examine d in detail later . T he n e ce s s a ry and su fficient condition for a symptotic st ability ( i . e . , s t abilit y to " smal111 perturbations from the steady state ) is that both eignevalue s have · n e gati v e real p art , that is , ( 76a)
det � > 0 and tr
( 7 6b )
1!: < 0
From standard linear differential e qu ation t h eo ry the following c las sific a tion of the s te ad y state of th e linear sy stem ( 7 2 ) is ob t ained : < 0:
1.
det �
2.
det
3.
de t
4.
det 1!: >
�
u n st abl e s ad d le p oint
> 0 , tr J: > 0 :
� >
0,
tr � <
0:
�
0:
0 , tr
=
u n st able node ( 6. ( !:!. < 0 )
( 77a)
>
0) or focus
stable node ( 6. > 0) or focus center
( 77b )
( 6. < 0) ( 7 7c ) ( 77d )
Liapunov' s theorem a s sures u s t hat the s t ab ilit y character of the n o nli n e a r system ( 65 ) an d ( 6 6) is the s am e as that of the linear o n e for the first three cases . For the fou r th , further analy sis is req uired . ' A few com ments on t he com parison bet w een t he rigo rou s stability con dit io n ( 76) an d the s lo p e con dition p revio u sly re p o r te d are now in or d er . It should first be noted t h at condition ( 76a ) , using E q s . ( 70) and ( 7 3 ) , is actually i d e n tic al to t h e slope condition , since it re qu i r e s the slope of the heat removal fu nction to be lar ge r than that of the heat gene ration function . T hus van Heerden criterion does ac tu ally give only a n e ce ss ary condition for stabili t y . T his implies that w hen thr ee steady state s occur , the middle one is certainly un s t able , w hile the lower and up p e r one s can b e stable . T hese two , as w ell as the si n gle steady state in the case of u niq ue n e s s , are in fact asym ptotically st ab l e if c ond it ion ( 76b ) , u sually referred as the dynamic c on di t io n , is also satisfied .
1 005
Reactor S teady - S tate M u l t ip licity !S tab ility of an a di abatic reactor ( i . e . , Finally , it can be
s how n ( see Varm a and Aris , 1 9 7 7 ) t h a t in
the
case
0) condition ( 7 6a ) im p lie s condition ( 7 6b ) automatically . In t his case the slope condition b e com e s neces sary and suf ficient for asymptotic s t ab ili t y .
o
=
Periodic Sol u t ions
com p le t e t h e p ic t u r e of the dy n am i c behavior of the reactor , it is nec e s sary to characterize fully the oc curr ence o f p e riodic solution s . A very lar ge num ber of p ap e r s have appeared in the lit e r at u re u sin g several m ath e m ati c al methods , such as Li ap u nov ' s direct m et ho d , av e ra gin g tech niques , Fou ri er analysis , an d perturbation m et ho d s [ see H lavacek et al . ( 1 970) and Uppal et al . ( 1 97 4 ) fo r det ailed reference s ] . The most con venient approach , which allow s u s to answer the problem exhau stively , has been p r e s en t e d by Poo r e ( 1 973) . T his ap p ro a c h is b a s e d on t h e Ho p i - F ri e d richs bifurcation t h eorem ( F riedrichs , 1 96 5) , which st ate s that bifurcation to a periodic solution occurs only from t he center of t h e linearized syst em , and under t he restriction that the trace of the m at rix D = di!:fd e: ( where e: is the bifurcation param eter) does not vanish . M o reove r , -thi s theorem allows the con struction of a fun ctio n o ( x ) whose sign at t h e b i fu rc a ti on p oin t allow s to estimate bo th the direction of the p e ri o d ic solution and i t s s t ab ili t y character . I t s hou l d be m en tion e d that in this type of bi fu rc at ion , often referred to as Hop f bifUrcation , a p air o f complex conju gate eigenvalues ch an ge their real p art sign by c r o s s in g the im aginary axi s , t h u s imp ly in g a ch an ge in the system stability ch aracter , since all the o t h e r eigenvalues are r eq uir ed to have neg ative real part . T his should not be confu sed with t h e so - calle d s tatic bifUrcat ion , w here one r e al ei genvalue chan ge s it s si gn by passing t h ro u gh the origin . T his is the bifurcation we have been referring to in the sec tion on st e ad y. - s t at e m u ltip li c it y . Poore ( 1 97 3) applied this th eory to the case of a C S T R with fi r s t - ord e r irreversible reaction an d Le 1 , in the limit of infinite ac tiv a ti on en e r g y . To in v e sti ga te all p os s ib le qualitative phase - plane behavior for the system , he c on sid ered the p aram eter p l an e B ve r sus o ( with fix ed y = "" , O c 0, and Le 1 ) takin g D a a s the b i fu r ca tion parameter (i . e . , ev en t u al ly th e only param eter t hat c h an ge s , w hile all the others are k ep t constant , to ex plo re the b ifu rc a tio n s and /or jumps of the sy ste m ) . The p h a se plane is then divided i n t o six r e gi on s by th r ee curves , whi c h are defined con c ept u ally To
=
=
=
as follow s : 1.
C urv e M . B elow this curve m 1 and m 2 [ defined as conversion v alue s c orr e s p on di n g to the Da bifurcation values shown in Fig . 1 6 (i. e . , 0) ] are com pl ex , thus det � > 0 1 above t h e
where det �
=
; < 0 for m 1 < x < m 2 an d det ; > ot h erwi s e . This curve t h u s represents the u ni qu en e s s condition for any D a , and i s ana ly tic ally give n , for a fir s t -order reaction , by E q . ( 1 7 ) Cui'V e S . I t is b a s e d on t he same concepts a s curve M , b ut with respect to s t ab ilit y in st ea dy of m u l tiplicit y . In dic atin g with s1 an d s 2 the conversion values w here the stability c h ara c t e r chan ges ( i . e . , tr � 0) ( see Fig . 1 6 ) , it follow s that below the curve s and 1 s 2 are complex , t hu s tr � < 0 ; ab o ve the curve for s 1 < x < s 2 , tr ; > 0 ; while fo r x < s and x > s , tr � < 0 . 2 1 curve , det
•
2.
=
Mo rb ide lli ,
1 006
Varma ,
and A ris
X
Do "
- Do
FIGURE 1 6
S ketch of outlet conver sion as a function of Da at steady
state .
solution s m 1 , m 2 , and s 1 , s 2 . In p artic ular , it rep re sent s the locus o f point s w here the m ' s and s ' s have t he sam e · value s ( i . e . , det � = o and tr � = 0 ) . T his curve e stablishes the relative location o f the
C u rv e SM .
3.
For the case examined by Poore ( 1 9 7 3 ) these curves are described by particularly simple expres sion s : C urve M :
B
C urve S :
B
Curve SM :
B
=
=
=
4( 1 + o )
3 + 0 + 2
(1 + o )
( 78a)
.f2+'T
( 7 8b )
3
( 78c)
0
Also , for the above - m entioned function o ( x) a simple expre s sion is derived for x s 1 and s 2 : =
1
o . = ( - l) 2
w. 1
i+ 1
�
1
1
[ w (b.
2 = b . ( B s.
-
1
1) + 2b.
-
i =
B s. 1
1 , 2 ( 7 9)
b . ) and b . = B s. 1 (3 . It is found that for bi 1 1 1 furcation at s 1 ( s z ) , o 1 > 0( o 2 > 0) indicate s bifurcation to a stable ( un stable ) p eriodic solution for D a values lar ger than the bifurcation value , and vice versa for 8 1 < 0( o 2 < 0 ) . The six region s in the (3 - o plane obtained u sing the three c urves given by Eq . ( 7 8 ) are s how n in Fi g . 1 7 . T he bifurcation behavior correspondin g to each region is illu strated as a function of Da in the side figure s , w here the capital letters refer to the various shap e s of p hase portrait s s hown in Fig . 1 8 ( here the symbols filled and open circ les indicate stable and un stable limit cycles , respectively , and their distance to the stead y - stat e where
1
1
-
-
-
a , �· X � ::� Da , Da E
A
A
X
I U•
·� .:. .:.. ·-
mb
r=:=1 ... I
A
r
I" I
52
•
Da
X m
l
51
I
era
::>;) (1) 1:1 (')
.... 0 "'S
h I i' I I •
"
I " I �u
3 0 �----�--��-r��---,
26
m
J:l. "< I
til ... p ... (1)
s:
1:
22
... -o·
14
� Ci) ...
1 8
§:
• : hhzJi,.__ � 1.5
X
sl
til .... (1) 1:1
��---1 I
I
52
oa s l
l
2.5
20
Va
. · · · ;�
bfoi/ a l_A . oa
X
52 51
11' b
( I �8
. �'; B
3.0
)
m2
. . �'' '
A
'�
A
I
Da
ml
4. 0
3.5
Uki '
1
4.5
-·
"<
-·
...
E VAPOR A T I O N )( or I
STAB L E ss
U N S TA B L E s o
"' � .. ,
A
s
p 0"
Do
STABLE LIMIT CYCLE
U N S TABL E LIMIT CYCLE
1 7 R e gions in the p arameter plane B ver sus ( 1 + 8 ) , surrounded b y the sketches of the correspondin g bifurcation diagrams in the X versus Da plane 1 ) . Irreversible first -order reaction in a 0 , Le ( param eter value s : y = oo , 80
FI GURE
=
C STR .
=
...... 0 0 "'I
Mo rbidelli ,
1 008
Varm a , and A ris
T Y PIC A L PHASE PLOTS
[F8cus [ J
CASE
NODE
A
J � J J S TA B L E
U N S TA B L E
SADDLE POINT
G
u M rr
STA B L E
CYCLES
UN STABLE
1
0
a
c
D
E
1
1
1
0
2
1
0
2
3
0
0
0
0
0
0
TOTAL INVARIANT S
1
0
0 1
0
3
3
0
18
1
0
1 1
0
4
2
J
H
G
2
1
1
0
1
0
1
4
4
5
0
ORDINATE-TEMPERATURE
ABSC I S SA-CONC ENTRATION
�
1
0
F
A
B
G
H
J
Phase portrait s characteristic of various regions in the param F I G U RE eter plane show n in Fig . 1 7 .
Reacto r S teady - S t a t e Multip l icity /S tab ility
1 009
cur v e re pre s e nt s the m ax im u m amplitude of the o scillation ) . T he se fi gure s have been taken from U ppal et al . ( 19 74) , who e x ten d e d Poore { 1 9 73) re s ult s to t h e case of finite activation energy , and co m p lete ly define d t he C S T R d y n am i c s by follo w in g the branch e s of p e rio dic solutions up to com pletion t h rough acc u r ate n u m e ric al in te grat io n . T he corre spondence between t h e re gion s in t h e parameter plane and the cor re sp ond in g b ifurc a tio n di a gram foll ow s d ire ct ly from the relative p o si ti on of each re gion with re spect to the above-mentioned curve s . For e x amp le , since bifurcation to a p e riodic solution occurs only for tr i! = 0 and det l > 0 , it follow s th at bifurcation at X = 5 1 occurs only in re gion s-IV ( 0 < S l < m < l m 2 < s 2 < 1) , V ( 0 < s1 < 82 < 1 , m , an d m z are complex) , and VI ( 0 < m 1 < m 2 < 81 < s 2 < 1 ) , w hile bifu rcatio n at x = s 2 can occur also in re gion I II ( 0 < m 1 < s 1 < m 2 < sz < 1 ) . S everal int e re st in g dy n am ic features of th e C S T R hav e been e vi d en ce d by thi s analysis . For e x amp le , for a s e t of p ar am ete r values inside region I I Ib in Fig . 1 7 , it app e ar s that starting with lar ge D a a n d d ec r e a sin g it , the sy st e m goes from a high- conversion st ead y state to a limit cycle ( tran s ition A - B ) , whose amplitude in cr e a ses up t o a D a value , w he r e the limit cycle e v ap o r at es and the system precipitates to a low - conversion st e ady state ( transition F - C ) . T his b eh avior has be e n likened by Poore ( 1 97 3) to the flic ker in g of the flame w h en t u rnin g off a bunsen burner . A noth e r i nt e re stin g ob servation concern s the m e ch ani s m s for the limit cycle di sappe ar ance pointed out by Uppal et al . ( 1 974) . T hre e of these have been ob served : shrinking into a s t e ady state , in cr ea sin g amplitude up to m e r gin g with a separatrix line , and coalescence of a stable and un stab le lim it cycle . Also , ab out the relative p os it ion of li m i t cycles and ste ady states some gen e ral conclusions can be d ra w n . Poore ( 1 97 3 ) has shown t h at any steady state in the uniquene ss region has index + 1 , w hil e in the m ultiplicity re gion the middle steady st ate h a s index - 1 , and the lower and uppe r ones have again index +1 ( see G avalas , 1 9 68) . S in c e any pe rio dic solutions must encircle a number of st e ad y states such that the sum of their i n di c e s gives + 1 , it can b e concluded that a lim i t cycle can either in clu d e th e s t e a dy state in the case of u niq ue n e s s , or in c lu de only t he upper steady state or only th e lower or all the t h ree ste ady states in t he case o f multiplicity. This conclusion is con fir m ed by t h e phase- plane portraits shown in Fi g . 1 8 . Alon g this same line , Poore ( 1 97 3 ) also p rove d that if a u niq ue unstable steady state exists , the n a st able p er io dic orb it m u st also ex i st s urr oun din g the steady st ate . As we h av e alr e ad y discussed in the multiplic ity section , the most con venient way to chan ge the D a value in p rac tic e is to c h an ge the feed H owever , this also a ffect s the cooling p ar ame t er o , de stream flo w rate . fined by Eq . ( 3) . So Uppal et al . ( 1 97 6 ) re d e fined the dimensionles s quantities , i solatin g t h e feed stream flow r at e in th e residenc e - time p aram eter , and e xp licitly accounting for it as follo w s :
Da
=
Da0
T
( 80)
b ifu rcation an alysi s h a s then to b e r ep e ated , now taking T in D a as a b ifurc a tio n parameter . A s m entioned e arlie r , the multi p lici ty p attern with re sp ect to t he reactor re sidence tim e is quite c om pl ex , giving rise to the isola an d mushroom s h ape s , which do not appear w hen the D amkholer number is taken as a b ifurc ation parameter . T herefore , it T he en tire stead of
Morb idell i ,
1 01 0
Varma , and A rls
i s not s urp nsmg th at more com plex features and up t o 1 7 different types of dynamic behavior have been found by Uppal et al . ( 1 976) . It should be noted that the number of po ssible phase portrait s is un c han ge d , since these depend only on the p aram eter s te ad y st ate value s . Thus i n t his case also , the nine portrait s shown i n Fi g . 1 8 cover all pos -
sible p hase - p lane shapes .
Effect of t h e Lewis Numb er T he Lewis number has been defined
in Eq . ( 68) a s t h e ratio bet ween the
thermal and the m ass cap acitance s of the reactor . Up to this point it has been assumed that Le = 1 ; however , w hile for homogeneou s tubular reac
tors the Lewis number is very clo se to 1 , for packed -bed reactors this is
not true , an d in general it tends to be m uch larger than unity . C learly , the Lewis number doe s n ot affect the value of t he re acto r steady state , only its stability character . In fact , Le appears only as a m ultip lie r of the t herm al accum ulation term in the tran sient model ( 6 5 ) and ( 66) . T he lineari zed s y st em ( 7 2 ) c an b e rewritten as dx_
( 81)
dt
w here
ff
1 � - !1
involves the lewis number .
In general , decreasing values of Le enlar ge the instability re gion of the reactor . For example , in t he case of a first - order r eac tion with y = oo , t he conve rsion v alues s , where tr � = 0 , can be readily calculated from 1,2 E q s . ( 7 4 ) and ( 8 1) as follow s :
B
+
o
1 + 2B
±
;B
where it app ears t ha t as Le tr
� >
0 for s 1
<
x
[ (B + 1 +
< s 2 , it
+
o )2 -
4BLe - 4B ( l
( 1 976)
o ) ] l/2
0 , s 1 decrea ses while s 2 increase s .
( 8 2) Since
follow s that decreasing value s of Le expand
t h e region where the dynamic stab ility condition num erical findin g s of variou s author s also go in that models o f this type tend t o exhibit periodic low Le values [ see Aris ( 19 7 5 ) for the case of a
Perel son
+
( 7 6b) is violated .
The
this direction , sho w i n g solution s for sufficiently cat aly st p article ] .
ex t end in g a p revious work by Luss ( 1 974) , has
examined the effect of the capacitance m atrix on the st ability o f the linear
s y s t em ( 8 1 ) in a more general w ay .
tive de finite sym m etric matrix , if has all real eigen va lue s
.
If
li,
�
It was ob served that
Q bein g a posi -
is nonsin gular a n d symmetric , then
�
is non singular and nonsymmetric , as in the
case under examination , the eigenvalues of
ff
can be either real o r com plex
T hu s conj u gate , dep endin g on the value of t he c ap acitance p arameter . takin g one of the elem ents o f Q a s bifurcation p aram eter (in our case , Le ) , and applying the above - m entioned Hopf- Friedrich s bifurcation theorem , for any given steady st ate of the reactor the value of Le at which periodic orbits bifurcate can be identified . In particular , for a sy st em eq uivalent to the adiabatic C S T R m odel , Perelson ( 1 9 7 6 ) found that such a critical value of Le is gi ve n by
1011
Reacto r S t eady -State Multip licity /S tab ility Le
c
(x
=
1)
-
h [
-
(1
Bx 2]
+
( 83)
0 /y )
w h e r e x and 8 a r e evaluated at steady- state condition s , and periodic solu tions can occur only for Le < Le0 • Si n ce the occurrence of H op f bifurca tion r equi re s det � > 0 , it can b e show n that Lee < 1 . T his im p li e s that for adi ab ati c system s of the type examine d , o scillation s can occur only for Le < 1 . T his :r: e s u lt has been p r eviou s ly reported by Ray et al . ( 1 974) , in th e context of m ultip li cit y an d st ab ilit y of c at aly tic wires . Since wires are characteri zed by Lewis number v al ue s m uch larger than 1 , it app ear s that they c an no t oscillate on t hei r own . These findin gs agree wit h the analy sis of Edwards e t al . ( 1 974) on fli ck e rin g o f c at aly t ic w ire s . T his ph en om en on , characterized by the occu rre n c e of a locali zed re gion of very hi g h luminosity w ho se intensity c h an ge s with tim e , h a s been at t ri b u t e d to flu ct u ation s of the concentration s in the ga seou s stream , which can be signi fic ant l y en h anced by the catalytic w ire R ay and H a stin gs ( 1 980) have applied bifurcation analysis to the sys t em of E q s . ( 6 5) and ( 66) t akin g as bifurcation p arameter the L e w i s number Le . T h e proce dure is identical to th at applied by Uppal e t al . ( 1 974 , 1 976) taki n g as bifurcation parameters t h e Dam kobler n um b e r a n d t h e reactor residence tim e , r e sp ectively . B y inspection of E q s . ( 6 5 ) and ( 6 6) it is re ad ily seen that by r epl acin g the p a r a m e t e r s , B , o , and 8 0 with the new qua n tit ies .
ex
1 + 0
=
"
B
Le
=
B
8
=
Le
c
( 8 4)
Le
Eqs . ( 65 ) and" ( 66 ) reduce to the same form u sed by Uppal e t al . ( 1 974) in the case of Le 1. Thu s , with these p aram et e r chan ges , the re sult s ob tained p re v iou s ly s till hol d . I n p arti c ul ar the B - 0. p ar am e te r plane [ pre viously , B ( 1 + 8 ) ] is a g ain divided into t h e same six region s , each leading to a di ffe re nt shape of the conversion ve r s u s D amkoble r n u m b e r b ifurca tion diagr am . T h is d ia g ram is show n in Fig . 1 9 for the case y = and 0 0 = 0 , in t e r m s of the new dim e n sio n le s s parameters accountin g for the effect of Lewis number . A q ui t e i nt e r e s t i n g feature of the dynamic s of t hi s sy st e m can b e ob 0 is affected served in Fig . 1 9 . Since the reactor steady state for 8 0 only by t h e r atio B / ( 1 + o ) , it foll o w s t hat the same x - D a bifurcation diagram c or re sp o n ds to all the p oi n t s on a ny st r ai ght line goin g throu gh the o ri gin For a given straight line , by decreasin g Le we move toward the upper - right re gion of the di a g r am So , dep endin g on t h e giv e n value of the r atio I ( 1 + 8 ) , the curve m ay enter r e gi on III ( curve 1) V ( cu r ve 2 ) , or even the small r e gion VI . In any event , in either one of these re gion s , the sy st em can e x hibit stable periodic orbits . Thus it ap p e ar s that for d e c r ea sin g values of Le , lower than the critical value Lee w here t he st r ai gh t line enters one of the above - mentioned r e gion s we have p e rsi st e n t stable periodic solution s . U sin g the sam e p r o c e dur e a dop t e d for de r ivin g E q . ( 7 8b ) , the followin g e xp r e s si on for Le0 , valid for y = oo and 0 0 = 0 , i s obtained : =
,
-
oo
=
.
B
.
,
,
1 01 2
Morb ide lli ,
26
Varma, and A rts
24
cv ...J ......
CD
2 0 1 6 1 2 8
4
@ 1.0
0. 5
FIG URE 1 9 E ffect o f Lewis number o n reactor dynamics ( p arameter values as in Fig . 1 7) •
Le
c
=
[< t [
+
1/1) 2
( 85)
4 \ji
where 1/J = B I ( 1 + o ) . It should be noted that the specific form of the bifurcation diagram x versus Le depends on the particular choice of the value s for B I ( 1 + o ) and D a . A det ailed analysis o f al l pos sible situation s h a s been reported by Ray and H astin gs ( 1 980) . These authors have also pointed out that in some cases the periodic orbits asymptotically approach relaxation oscillations as Le + 0 . I n this case the characteristic therm al time is so m uch smaller than that for m ass that the ener gy balance ( 66 ) can be assum ed in pseudo- steady - st ate con ditions with resp ect to the m ass b alance ( 6 5 ) . U nder the se conditions , relaxation oscillation s can occ ur only if the function e e (x) given by the p seudo - steady - state approxim ation of E q . ( 6 6) is m ultivalued ( see Gilles et al . , 1 9 7 8) . For example , t his is not the ca se for y = oo , which there fore does not exhibit this peculiar type of dynamic behavior . =
,
Some Comparison with Exp erime n t al Data
As a conclusion of the analysis of the dynamic behavior of a C S T R , it is convenient to report some experimental verification of the presented re sult s . I t should be understood that what we are really verifying with �uch a comparison is the C S T R model and its ability to simulate real system s . Seve ral experiments have been reported in the literature of multiplicity , oscillation s , and sen sitivity in C S TR s , for b ot h homogeneou s and hetero geneous reaction s . E xhaustive reviews have been presented by Schmit z ( 1 975) and S heintuch and Schmit z ( 1 977) . We shall here content with few references , specifically concerning homogeneo u s reactions in a C ST R , which explicitly prove some of the feat ures p reviou sly described . I n p articular , we refer t o t h e experim ents reported by Schmit z and coworkers u sin g a laboratory well - mixed continuou s reactor w here the second - order reaction between sodium thiosulfate and hydrogen peroxide occurs .
Reactor Steady -State Multip l icity /Stab ility
1013
Several o f the phase portraits shown i n Fig . 16 have been ex p erim e n tally foun d u sin g this sy stem . Vejtasa and S ch m it z ( 1 970) u se d an adiabatic reactor , which we have shown ( at least for a first-order reaction) to belong either to re gion I or II in Fig . 1 7 . A cco r din gly , no oscillations were found and only phase p or tr ait s of type A and E were reported . C han g and Schmit z ( 1 975a , b ) u sed a nonadiabatic unit , which allowed them to ob ser ve o scillatory behavior . B e side s t y p e A and E , a phase portrait of type B was also reported . Sub sequently , the au tho r s of the most si gni ficant l a te st t heo r e ti ca l and ex p erimen tal contributions in t hi s area joined together with the aim of experim en tally verifying additional dynamic features of the above - mentioned reaction system ( Schmit z et al . , 1 9 79) . Applyin g the Hopf- Friedrichs bi fur cation theorem , the dynamic b e h avior of a C S T R w ith a se cond - o rd e r re action was ex plo red . A gain , the same six region s characteristic of the first-order r e act ion were foun d . T he q uali t at ive agreement between the above- mentioned previou s experimental result s and the model prediction s was complete . M oreover , a fourth type of phase portrait , C was experi mentally found , w here three ste ad y state s are p re sent , b u t o nly t h e low conversion one is s t ab le . T his result w as achieved thanks to the dual nature , th eor e tical and experim ental , of this work . U n fo rt un at e ly , due to li m ita tion s intrin sic of any e x p er im ent al app aratus , not all the dyn a mic feature s found theoretically w ere po s sible to be rep roduced experimentally . An in t ere s tin g re sul t r e g ardin g the effect of the Lewis n um b e r has also been reported by these authors . T he experimental u nit was , i n fact , eq uipped with a j acket , s u r ro u ndin g the reactor wall , which could be filled wit h w ater in order to increase the t hermal cap acity of the reactor and thus also the corresponding value of the Lewis number . It was shown that repeating t h e e x p e ri m e nt relative to the p h as e portrait of type C , but now fillin g the j acket of the reactor with water , a stab le hi gh conversion steady state was obtain ed . T his co nfirm s the p reviously discussed stabilizing effect of incr e a s i n g value s of the L ewis number . T ub u l a r R eactor
Stability
The tran sient behavior of a t ub ular reactor i s d e s cr ib e d by the axial dis persion m odel t h rou gh the followin g s et of p arabolic p artial differential equation s ( PD E s ) :
au at
Le
=
av at
1
Pe
=
m 1
--
Pe
au
2 a u 2 as
h
as av
a 2v 2 as
as
- Dar ( u , v)
+ S D ar ( u , v ) - o ( v - v ) c
( 86)
( 8 7)
where the dimensionles s p aram e t e r s
t
_ tg_
-
v
Le
=
EpC
p
+ (1
-
£PC
p
£)
p C
s ps
( 88 )
1014
Mo rb i d e l l i ,
Varm a ,
a n d A ris
have been added to those p re v iou s ly defined in E q . ( 4 9) . The Lewis num ber is de fin e d as the ratio between the total h e a t cap acitance of the reactin g In the case of h omog ene o u s syste m ( fluid + solid) and that of t h e fluid . reactor s Le = 1 , w hile it rise s up to a fe w hundred in t he case of catalytic
-
reactor s . B oundary conditions of Eq s . ( 86) and ( 87 ) are the same reported for the st e ad y s t a t e model [ E q . ( 5 2 ) ] , while t he initial conditions are u = u
in
( s)
v
=
v
in
( s)
( 89)
at t = 0
It is w o r th point out that the B C s ( 5 2 ) p r op o s e d by D anckw e rt s in 1 9 5 3
a re valid also u nde r tran sient con d iti on s only when n o m a ss o r hea t dis
persion occur s in the inlet and outlet tube s to and from the react o r ( Wehner and Wilhelm , 1 95 6 ; van C auwenberghe , 1 96 6 ) . In the followin g we shall alw ays refer to the D an c k we rt s bound ary condition s , and eventual l y discu s s t h e effect on r e a c t o r st ab i l it y of the d e gre e of mixin g in the r e gio n s be fore and after the reacting re gion . In the p re cedin g se ction it w a s shown that E q s ( 86) and ( 8 7 ) can admit up to seven s t e a d y s t at e solutions . It is then nece ssary to dis c u ss numerical p roced ures and a priori crit e ri a to define the st ab ilit y character of a s t e ady state . U n fort unately , t h e m athem atical a pp a r a t u s for dea l in g with stability of p arabolic P D E s is not as well assessed as for ODE s . T herefore , excep t for ve ry special situation s , stron g statements about ste ad y state stability , such as t h o s e r e p o r t ed for the C S T R , cannot be m ad e . Even th o u gh no ri gor ou s m athematical proof is a v ailab le (of the typ e of Liap unov ' s first t heore m for ODE s) , it seem s quite r eas o n ab le to r el at e the local stability character of a giv e n steady s t a te of nonlinear sy stem of p arabolic PDE s to that of the c orr e s p on din g sy st em ob t ain e d t h r o u gh linearization about the steady st at e . I n trod uci n g the perturbation variables
.
-
-
u( s , t) = u ( s , t )
u ( s)
( 90a)
v( s , t)
v ( s)
( 90b)
v(s , t)
=
s
s
w h e re th e inde x s r e fe r s to steady - state c on d it ion , and performing t h e u sual chan ge of v ariab le s ( A m un dson , 1 96 5 ; V arm a an d A ri s , 1 9 77) , y1( s , t)
=
u ( s , t) e x p
y2 ( s , t)
=
v ( s , t ) exp
) (- � ) (Pe
m 2
Pe
s
s
the fol lo win g linearized vers io n of Eq s . obtained :
=
1
Pe
m
( 9 1a)
( 9 1b ) ( 86) , ( 8 7) , ( 52 ) , and ( 8 9 ) can be
1 01 5
Reacto r S t eady -State M u l t ip licity /S tab ility
( 93)
with B C s Pe
ay
1 as
m
-- y = 0 2 1
a y1
Pe
m
-- + --
2
as
and I C s
in
y1 = y1
Pe
ay 2
h -2 y2
as ay
y1
y2
Pe
=
0
2 as
=
y2
in
at t = 0
+
h
-- y 2 2
=
=
0
0
at s :::: 0
( 94a)
1
( 94b )
at s
=
( 95 )
Equation s ( 9 2 ) to ( 94) can be rew ritten in the followin g m atrix form :
( 96)
wit h B C s
ax as
+ 1:? x. = o
ax
- + D :x_ =
as
=
0
at s at s
=
0
( 97a)
=
1
( 9 7b )
and I C
l.
in
=
( 9 8)
X.
Its solution is g iv en by an infinite series expan sion of the form 00
c lJ. . Y lJ. . ( s) where
>...!
l
>.. j are ei ge nvalu e s of the ass oci at e d ei ge nv al u e p rob lem =
( 99)
exp ( A .t )
'
( 1 00)
that is , the v alues of >.. for which Eq . ( 1 00) admits a nontrivial solution . From E q . ( 99) it is seen that if all eigenv alue s h ave negative real p arts , the stead y state is stable ; if one of the ei genvalues h as positive real part , the steady s ta t e is un stab le . This is a q uite strai ghtforward
Mo rbidelli , Varma, and A ria
1 01 6
conclusion , b ut its ap p lic at ion can be q ui t e d iffic ul t , since in ge ne ra l the they can of cour se be either real or c om plex . T h e refor e , i ge ne ral it i s necessary to resort to suitable num erical techniques . T h e stan dard proced ure is to discretize Eq . ( 1 0 0) alon g the space variable s an d then comp ute with a suitable al gorithm the ei gen values associated with t h e obtained system of algebraic eq uation s . ei gen v al ue s thus obtained app roximate the correspondin g large st eigen values of the or i gin al infinite set , t he y allow us to id entify the stability c har acte r of the ste a dy state under examination . Particularly fort un ate is the case of an adiabatic reactor with Pe m = Pe h , for it leads t o a lin ea ri z d system ( 96 ) w hose linear sp a tial operator i s self- adjoint . In the follo w in g we first analy ze t h i s last case , w hich h a s received a great deal of attention in the lit erat u re . T hen we consider the general c a se of a non ad ia b ati c reactor and discuss some of the stability conditions which have been obtained u sin g various methods m ai n ly based on compari son theorems and Liapunov direct m et hod . Finally , s om e numerical tech nique s for solvin g the eigenvalue p roblem ( 1 0 0 ) are indicated . number of eigenvalues is infinite , and
n
Since
the
e
1
Sp e ci a l C ase of A diab atic R eactor : A s m ent ion e d above , in the case of a diab atic axial dis pe r s io n model with Pem Pe h an d Le = 1 , the s t ab i li ty A standard p ro c e d u r e used in t h e problem can be greatly simplified . earlier studies is to i n trod uce the so - c alled " residual enthalpy , " defined as =
( 1 01) B y s u itab ly ab ov e
combinin g E q s . ( 92 ) and ( 93 ) , un de r the assum ption s not e d
1 it is found t h at
dW
at
=
2 a w
1 Pe
as
2
Pe
4
( 1 02)
w
with B C s
Pe
aw dB aw
as
2
+
Pe
2
w
= 0
at s = 0
( 102a)
w
= 0
at s
1
( 102b)
=
which , u sin g th e IC w
in
=
( 1 0 3)
0
admits only the t ri vial solution w = 0 .
im p lie s that i f we limit the ( i . e . , w in = 0 lead s to v + 0 ) , then during t h e transient the follo w in g relation hold s : T his
an aly si s to s o - c alle d adiabatic p e rturbations
Su
=
v
+ Su
=
1
T his allow s u s to PDE
+
S
r e d u c e the ori gi nal
( 1 04) sy st e m
of eq uation s to the single
Reac tor S t eady -State Multiplicity /S tab ility
1 01 7
( 105)
.
B y linearizin g E q ( 1 0 5) a n d introducin g t h e chan ge of variable ( 91b) , a single linear PDE is obtained w hose space operator is self- adjoint . T he stability of the sy stem is then related to the ei genvalues of the problem :
A.ljJ
=
1 Pe
2 d w - 2 ds
(Pe 4
+ D ar
u
- D a Br
v
) ljJ
( 1 06)
with the B C s ( 1 0 2 ) with w replaced by ljJ . In this case it is known that the eigenv alue s A. n form an infinite set of real , discret e values . U sing the se propertie s A m und son ( 1 96 5 ) derived a nece ssary and suf ficient condition for stability w hich requires the solution of an initial value ODE whose coefficients depend only on the steady - state profiles . The final an swer to the stab ility question was given by Luss an d A m und son ( 1 9 6 7a) , w ho exploite d the possibility of reducin g the problem to a single PDE . U sin g topological concept s they p roved that in the case of m ultiple steady states the low - and high -conversion steady states are asymptotically stable , while the middle one is unstable . M oreover , usin g the m axim um principle , i t was also prove d that if a unique steady state exists , it is alw ay s globally stable . A gain u sing the m axim um principle , Lu ss and Lee ( 1 96 8 ) deter m ined finite region s of stability for each steady state . T hese allow u s a priori to determine the response of the reactor to large p ertu rbations and even the final steady state reached from a given initial condition , without per formin g any tran sient calculation . I n p articular , it was found that the un stable - steady - state p rofile separate s the regions of stability of the two stable steady states . For cases where the initial p rofile does not entirely lie in one or the other of the two st ability re gion s , it is necessary to in te grate the transient m odel . If at any tim e during the tran sient the pro file is · found to entirely lie inside one stability region , the above mentioned criterion can be applied again . It is worth pointin g out that in the p articular case under exam ination the axial dispersion model is very similar to the cataly st p article model in the case of Le = 1 . T hu s the re sult s obtained for one model c an be directly extended to the other . I n fact , some of the stability conditions reported below , as well as for the above - m entioned work by Luss and Lee ( 1 968) , w ere ori ginally derived for the catalyst p article . Even though the stability picture for thi s special case of adiabatic re actor is quite clear at this point , there are two reasons for reportin g re sults obtained w ith other techniques . T he first is that the above - mentioned stability conditions were obtained con siderin g only adiabatic perturbation s , which gives a character of conditional stability to the final an swer . Luss and Lee ( 1 96 8 ) have show n u sing Eq . ( 102) that the residual enthalpy monotonically decreases in tim e , thus indicatin g a more general validity of N evertheless , other techniq ues are avail the stability criteria developed . able which do not require the u se of the residual enthalpy concept at all ( Padmanabhan et al . , 1 97 1 ) . T he second reason is that since the above m entioned methods apply only to a sin gle PDE , they can be u sed only in this particular case .
Mo rb idelli, Varm a , and A ris
1 01 8
Wei ( 1 96 5 ) w a s the first to apply the direct method of Liapunov to the c at aly st p arti c le B er ger and Lapidus ( 1 96 8 ) an d P admanabhan et al . ( 1 97 1 ) e x t e n s iv ely u se d t hi s m eth o d , in clu d i n g more general situations such as t he cataly st p article with Le =f. 1 . V arma and Am undson ( 1 97 2 a ) applied the m ethod of comp arison function s . B oth these two last app ro ac he s do not require any restriction on the re sidual enthalpy . A s uffi cient condition for stability was first developed by Luss an d A m undson ( 1 96 7a ) ; it is identical to the uniquenes s con di t ion ( 55 ) . T hi s relation has also b e en derived by s ev e ral of the ab ov e m e nt io n e d authors usin g variou s m eth o ds For ex a mp l e , Aris ( 1 9 6 9) sh ow ed that it could be derived u sin g either the m axim um p ri n cip le or th e first or second Li apu no v m e thod s . Fin al ly t hi s e x p r es si o n was also ob tained by Gup alo and Ryaz ant sev ( 1 9 6 9) u sin g the S turm - Liouville eigenvalue problem p ro p e r ti e s again lim i te d to adiabatic p er tur b a tio ns T he i s s u e of t h e ap prop ri atene s s of t he D anckwert s b o u n d a ry condition s ( 5 2 ) in the tran si e n t case has been addressed in detail by Parulekar and R am k ri s h n a ( 1 98 4 ) for t he cases of both zero an d nonzero residual e nth alp y E ven t hough t he se boundary conditions are valid u nd e r transient conditions only in the c a se of no mass or heat dispersion outside the reactor , the stability r e s ult s were found to be q u alit ative ly q uit e sim ilar . B e sides the m ul t ip lici t y characteristic s , which are obviou sly u ncha n ge d Parulekar and R amkrishna ( 1 984) h ave p roved ag ain that a u nique steady state is glo b ally asym ptotically s t ab le A sufficient st abili ty condition very similar to E q . ( 55 ) w as al so obtained in thi s case , the only difference ari sin g from the effect of t he initial state outside the r eac to r on the re a ctor tr ansient which is obviously non zero only in t h e p resence of d i s p e rs ion . A d e tailed analysis of th e effects of v ar iou s mi xi n g inten sities before an d after the reaction region , as w el l as of the stability re gion s , is als o p re s e nt ed .
-
.
,
,
.
.
,
.
,
.
General Nonadiab atic C ase
A p rio ri co ndi tions : In the m o st gener al case of no n ad iab a ti c tubular re ac to r , or a di ab ati c with either Le :I 1 or Pem =f. P � , t he heat and energy b alance s cannot be u n co up led T hu s we have to deal w it h a s y s t e m of PDE s which , when line a riz e d , p roduces a n o n s el f ad joi nt space op erator . U n d e r these conditions most of the above-mentioned me t ho d s either fail or become exceedin gly complicated , w ith o nly the p ar ti al excep tion of the direct Liap un ov m e t ho d One of the very few d ire ct extension s of t h e re sult s in the p r e c edin g section is in t h e case of an a di ab atic reactor with Pem = Peh and Le :/: 1 . G av ala s ( 1 968) has shown that if a s t e ad y state is unstable for Le = 1 , it i s so al so for any other Le :I 1 . Nishimura and M at subara ( 1 9 6 9 ) dev e lo ped a suffi ci en t condition w hich r eq uir e s the int e gr atio n o f a set of init ial value li nea r ODE s w ho se coef ficients depend on the st e ad y st ate p rofiles under examination . Varm a and Amund son ( 1 972b ) , u si n g the method for op ti m i zi n g the Liapunov functional proposed by P ad m anab han et al . ( 1 97 1 ) have shown that the sa m e conditions w hich gu ar an t e e steady - state uniqueness also guarantee a sy m p toti c st ability . T hus E q s . ( 5 8 ) and ( 5 9 ) are su ffici e n t conditions for a sym p t o tic s t ability for the no n a diab ati c reactor with Pem = Pe h Le 1 an d Pem :/: Pe h Le I 1 , r e sp e c ti vely ( note t h at for a h om o ge n eou s reactor Le = 1 implies t hat P em = P eh ) More conservative s u ffi ent condition s for st ability , o b t ai n e d u sing different choices of the Liapunov functional , have been reported by various .
-
-
.
-
,
,
.
ci
=
,
1019
Reac tor Steady -State Multiplicity /S tab ility
a th or s ( C lou gh and R amire z , 1 97 2 ; Liou e t al . 1 97 2a , b ; Varma and Amundson , 1 9 73a ; McGreavy and Solim an , 1 9 7 3 ) The effect o f recycle on stabilit y has been studied by Matsuyama ( 1 9 7 0 ) for the adiab atic t ubu l ar reactor , and by McGowin and Perl m u tt e r ( 1 9 7 1b ) and Liou et al . ( 1 9 7 4 ) for
u
.
the
nonadiab atic case .
Numerica l metho ds : D ue to the lack of sharp sufficient conditions for stability , it is often nece s sary to re sort to num erical techniques . T h e first approac h is the direct int e gr ation of th e complete tran sient model . It was used in t h e earliest work of R aymon d and A m und son ( 1 9 6 4 ) and H lavacek
S i n ce then , m any sophisticated numerical methods for solvin g Som e of them have be e n used in g ene r al computer codes , such as PDEPAC K ( M ad s e n and Sincovec , 1 9 75 ) and PDEL ( C ardenas and K arp l u s , 1 9 7 0) . Most of th e s e t ec hnique s et al .
( 1 9 7 1) .
system of p ar abolic PD E s h ave been developed .
are b as e d on the m e tho d of line s ,
using
where the space variable is discretized
a suitable technique such as finite diffe rence or w ei ght ed residuals .
The obtained system of O D E s is th e n integrated in t i me using a suitable
marching t ech niq u e .
Since the system of O D E s can be stiff for c e rt ain
range of p ar am e t er values , implicit int e gration methods ( at lea st partly ) are usu ally recom mended . More detailed info rm a tion about these numerical techniques is r e port e d by Finlay son ( 1 9 7 2 ) and by M i c h el s e n and Villadsen ( 1 97 8 ) , w hile a detailed a n aly si s o f the a pli c a tion of the orthogon al colloca method to t h e p articular model under examination h a s b e e n repo r t e d by Georgakis et al . ( 1 9 7 7 a ) . Another approach c on si s t s o f evaluatin g the ei ge n v al u e s for the linear i ze d system of PDE s ( 9 2 ) to ( 95 ) given by the n o n s e l f ad j oi nt ei ge nva lue pro b l em ( 1 00) . I t sh ou l d be noted t h at al t hou gh E q . ( 1 0 0 ) is linear , it can only be solved num erically , becau se some of the elements of matrix � involve the p artial de ri v ativ e s of the re ac t ion rate at steady s t at e alo n g the reactor , w hich in ge neral do n o t admit analytical re pr e sent atio n U sually , a di s c re ti ation m ethod is u sed to t r an sform the p ar ab oli c PD E s in to a sy stem o f ODE s . T he most com mon t echniq ue s u sed for this are : finite d i ffe re n ce s ( K ir kby and Schmit z , 1 96 6 ; H e in e m ann et al . , 197 9 , 1980; Hei n em ann and P oo re , 1 9 8 1 ) , G al e rkin m ethod ( K uo and Amundson , 1 9 6 7 ; McG owin and P e rl m u tt e r , 1 9 7 1a , b ; Varm a and A m undso n , 1 97 3b ; Jen sen and Ray , 1 98 2 ) , and orthogonal collocation ( McGowin and Perlmutter , 1 9 7 1 a ; Georgakis e t al . , 1 9 7 7a ; Jensen and R a y , 1 9 8 2 ) . T he si ze N of t h e final system of ODE s depends on the n um b e r of te rm s in the trial solution in th e G alerkin m ethod , or of collo c at i on point s in th e ortho gonal collocation method . T he final step of the p roced ure i nv ol v e s evaluation of all the
p
tion
-
-
.
z
eige n v alu es o f th e discreti zed system , w hic h can be ac com pli s he d th rough standard routines ( M ichelsen and Villad sen ,
1 978) .
B ased on r e gu larit y con siderations of t h e solution , it is e xp ect ed that · by increasing the number of discreti zation elem ent s , N su c h that N + oo ,
O DE s gives a p erfect d e scrip t ion of the original system o f I n p articular , t h e N eigenvalu e s obtained a p p roxim at e the first N lar gest ei gen v alu e of the non - self- adjoint p ro b l em T his p roce s s is illu s trated b y t h e re s ult s report ed in T abl es 2 and 3 . I n the first on e are shown the sy stem of PDE s .
.
the estim ates o f th e largest ei genvalue as a fu n c tio n o f the number of trial
function s u sed i n the G alerkin' s method ( M c G owin and Perlmutter , 1 9 7 1a) . The orthogonal set o f trial functions u sed i s based on the solution of the sam e model without chemical reaction . T his e x lain s w hy convergence b e com es slower for incr eas i n g values of the o u t l e t conversion . I n T able 3
p
1 020
Mo rbidel li ,
Varma ,
and A ris
E stim ate of the Largest Eigenvalue as a T A B LE 2 Function of the N umber of T ri al Function s in 1 0 ; tS :: 0 . 1 ) Galerkin' s Method , n ( P em = Pe h =
X
0. 10
n = 1
- 2 . 237
n
::
2
- 2 . 227
n
=
3
- 2 . 227
n
=
4
- 2 . 227
n
=
n
=
0 . 35
- 0 . 02909
0 . 995
0 . 90
- 0 . 2 4 98
0 . 4212
0 . 34 04
- 1 . 413
6
0. 4537
0 . 5 38 2
- 0 . 6879
8
0. 4553
0. 9819
- 0 . 5102
=
10
0 . 4554
1 . 275
- 0 . 1721
=
12
0. 4554
1 . 3 86
- 0. 1277
n = 14
0 . 4554
1 . 412
- 0. 1144
1. 415
- 0 . 1 00 2
1. 415
- 0 . 0991
n n
n = 16 n
=
18 20
- 0 . 0986
n = 22
- 0 . 0 9 82
n = 24
- 0 . 0 98 2
n
=
are shown the first 1 5 ei genv al u e s estim ated u sin g the orthogonal colloca tion 3 1 ) collocation points re methods with 7 (i . e . , N = 1 5 ) and 15 ( i . e . , N s p e ctively ( G eorgakis , et al . , 1 97 7b ) . It ap pears that while convergence already seem s to have been achieved for t h e l arge st ei genvalues ( which are those that determine t he reactor dynamics) this is not true yet for the smaller one s About the discreti zation procedure , McGowin and P erlm utter ( 1 9 7 1a) have compared the G alerkin and the o r tho go n al collocation m et h od s . T hey found that the first is more rapidly convergent , and also conver ge s mono tonically , while the second converge s in a damped oscillatory m anner . In general , they recommended Galerkin' s m et hod , un le ss the model contains mixed boundary condition s , as in the presence of recycle . T hese .observa tions have been con firmed by J en se n and Ray { 1 982 ) . Finally , it is worth commentin g on the correctness of this entire proce dure , since as men tion e d above , a rigorou s mathem atical relation between the st abili ty of the origi n al nonlinear sy stem and that of t he lineari zed sys Varma and A mu n dson tem is not available for distributed p arameter models . ( 1 97 3b ) successfully checked the stability of the system p redicted through the linearization method in several cases by direct integr atio n of the original model . Since these re sults have been analyzed in detail by Varma and Aris ( 1 9 7 7) , this ar g u m ent is not pur sued further here . Som e of the re sult s ob taine d re g ardin g the stability of m ultiple steady states are discu ssed in =
.
,
1 02 1
Reactor Steady -State Multiplicity /S tability T A B LE 3 E stimates of the First 1 5 Eigenvalues U sin g 7 and 15 C ollocation Points 7 collocation point s 1
15 collocation points
]
+3. 001
]
A.
A. +3 .
2
- 0. 972
3
- 4 . 2 7 3 + 3 . 3 2 2i
00 7
- 0 . 9 77 -
4 292 .
+
3 . 3 2 1i
4
- 4 . 2 7 3 - 3 . 3 2 2i
- 4 . 2 92 - 3 . 321i
5
- 6. 793
- 6 . 766
6
- 1 1 . 02
7
- 1 1 . 02 - 1 . 937i
8 9
+
1 . 93 7i
- 1 9 . 5 1 + 2 . 097i - 1 9 . 5 1 - 2 . 097i
10
- 2 9 . 14 + 1 . 2 1 0i
11
- 2 9 . 1 4 - 1 . 2 1 0i
12
- 7 1 . 8 4 + 3 . 6 0 2i
13
- 7 1 . 8 4 - 3 . 602i
14
- 85 . 4 2 + 1 . 35 2i
15
- 85 . 4 2 - 1 . 3 5 2i
- 1 1 . 03
+
1 . 98li
- 1 1 . 0 3 - 1 . 981i - 19. 28
- 1 9 . 28 - 33 . 2 0
+
-
+
1 . 8 5 7i
1 . 8 5 7i
1 . 969i
- 3 3 . 2 0 - 1 . 969i
- 5 0 . 9 5 + 1 . 933i
- 5 0 . 9 5 - 1 . 933i - 7 2 . 66
+
1 . 9 3 1i
- 7 2 . 66 - 1 . 93 1i
the next section , where the occurrence of periodic solution s is also taken into account .
Bifu rca tio n and Dynamic Behavior A s we have noted in the case of a C ST R , the most efficient and sharp est techniq ue available for investigatin g m ultiplicity and dyn amic behavior is bifurcation analy si s . A rigorous m athematical treatment of the exten sion of this method to distributed syste m s can be found in C randall ( 1 9 7 7) an d Iooss and Joseph ( 1 980) . F rom the application point of view , the m ain conclu sion is that H op f bifurcation theorem can be extended to the case of p ar abolic PDE s , by approximatin g these with an infinite set of ODEs u sin g one of the discretization t echniques discussed a b ove . In other words , the bifurcation behavior of the system of Eqs . ( 86) , ( 8 7 ) , ( 5 2 ) , and ( 89) is determined by the eigenvalues of the discretized system of lineari zed ODE s given by E q s . ( 96 ) to ( 98 ) [i . e . , the solution of the non - self- adjoint eigenvalue p roblem ( 1 00) ] . A s in the case of lum ped system s , two t yp e s of bifurcation are possible : a static bifurcation , w hich occurs w hen one real eigenvalue is zero , and a Hopf bifurcation , which occurs w hen a pair of complex eigenvalues has zero real p art ( i . e . , they are purely imaginary ) . A gain , in order to have bifurcation to periodic solution it is necessary that
� x lZ_ X L::_
X
20 '
L
11 �
I
,.. .
I'
-
X� � Da lfa: l$ X� ·� l:;,-:n Do ·:rx� :.::·.. lZlli b
I
x
.
Do
X
�
I
::
Do
�Do I ... �Do � -;;X I
X
X X
.-
LL -.:-:: ;;D o
� .
1ZIJl o
/.
'··
Da
Da
..· " X b
· ·.
�
x
10
..
.
:Jll
...-
I
L2:_ -:-:-- Da
xo
. ..
,
m
1
o,
,..-
x
5 0
3
2
8
4
.... c t-.:1 t-.:1
Do
Da
l�=:, X lrf·� X I .: � a ... ..
:.. :liD
b
�
Da
� a.
E:
&.
-::::
�
�
�
§
�
).. J. ""
1 023
Reactor S teady -S tate Multip licity /Stab ility all the o th e r eig en v alues
,
besides the pair o f purely imaginary ones , have
a negative real p art , so that the sy stem actually undergoes a chan ge in
its stability charac ter . T wo different types of num e ric al techniques are u sually employed for For st atic b ifur cat ion point s one need s locat in g such bifurcation poin t s to solve the steady- st at e problem and calculate the value of the bifu r c ation One way is to solve p arameter at which on e of the eigenvalue s is zero . the steady st at e with the u sual techniques for any given value of the bi fu r cation p arameter , and u se the E uler - Ne w to n continuation m ethod to sati sfy the b i fu rc a t ion condition . T o avoid nume rical instabilities near the bi fu rca tion poi nt s where the Jacobian is sin gular , it is necessary to a dop t the solution norm ali zation m et hod p ropo se d by Keller ( 1 9 7 7 ) . A de t ail e d descrip tion of this m eth od , with specific r eference to pr e mi x e d laminar flam e models , is re p or t e d by H einemann et al . ( 1 979) . A di ffe rent met hod has been u sed su cce s sfully by Jensen and R ay ( 1 98 2 ) in the bifurcation an aly sis of tub u l ar reactor models . T his method is b ased s u b st ant ially on t he same idea proposed pr evious ly by Weis z and Hicks ( 196 2 ) for one - dimension al problem s , and then ap plied to multi dimensional p roblem s by Sorensen et al . ( 1 97 3 ) . It con sists of fixin g one v alue of t h e unknown solution ( e . g . , at one collocation point ) and t aking in its place the D am khole r number ( or T hiele mod ulu s for catalyst p article s ) as unknown . The p roblem s o formulated u sually admits a unique solution , and it is t h en pos sible to solve it for any given D a , e ve n in the m ulti p li cit y r e gion T he H opf bifurcation p oin t s can be obtained by ev al u atin g the ei gen values correspondin g to a given value of t h e bifurcation p arameter through the m et hod indicated in the precedin g section , and then sear ch in g for t h e bifurcation p aram eter value that satis fie s the bifuraction condition throu gh standard techniques , such as section , N ewton - R aph son , and so on . K ubicek ( 1 9J9) has su g g e st ed includ in g the H op f b ifurcation condition in the nonlinear sy stem of algebraic equation s w hich solve the steady - st ate p ro b l e m ( afte r di s c ret i z ati on ) and then solving it all to gether with a N ew to n R aph son m ethod . Jen sen and Ray ( 1 9 8 2 ) have su ggested usin g necessary con dition s for t h e characteristic polynomial to have purely im magin ary roots to improve the e ffi ciency o f t h e b ifu rcation poi n t s ear c h T he bifurcation t h e o ry h as recently been applied to the axial disp e r sion model i n the case o f a fi r s t order reaction by Heinem ann an d Poore ( 1 98 1 ) an d Jen sen and R ay ( 1 98 2 ) . Some of t he above - mentioned num erical method s have been applied in these two works , and the reported agreem ent between the obtained Hop f bifurcation poi nt s constitutes a check o n t heir correctne s s . I n the fi rst o f these two p apers t h e problem of st ability o f the periodic solution w a s also con sidered . Formulas of t he sam e type as -those illus t rate d for t h e C S T R [ see Eq . ( 7 9) ] were reported T hey allow us , on the basis of on ly stea d y st ate in form ation , to derive b o t h the direction and the stability character of the bifurcatin g p e riodic solution . .
.
.
-
.
-
FIG URE 2 0 R e gion s in the p aram eter plane B v er s u s o , s u rroun d e d by the sk e tc h e s of the corre spondin g bifurcation diagram s in t h e x ver su s D a plane (p arameter value s : Pe m = P8b = 5 , Le = 1 , 0 c = 0 , y = 20) . Ir r eve r s ib le first - o rder r eac ti on in a tub ular reactor with axial di s persion .
Morb idelli , Varma, and A ris
1 024
D ue to t h e larger number of p aram eters involved , a p ic tu re of dy namic b e hav ior a s c om p le t e a s t h at rep or t e d for t h e C S T R is now unattainable . We sh all t herefore be con tent with ex am i nin g some o f t h e nov e l feat ur e s ex hibited by thi s sy stem . For th e sp e ci al case of adi ab atic reactors the sit u ation is again q uit e clear . N am ely , sin ce the eigenvalue p roblem in t hi s case is self- adjoint , the ei ge n v al u e s are all real an d no bifurcation to p e rio d i c solutions car. occur . To sum m ari z e : this r e a c to r can exhibit at most t h re e steady states , with alte r n a te st ab ili ty character ( i . e . , stable un stable - st ab le ) , a n d can n e v e r exhibit s elf- sustained oscillation s . O n t h e other h and , when the adi ab aticit y condition is removed , t he U p to seven s t e ad y st ates d y n amic behavior b ecom es m uch m ore complex . with vario u s com binations of s t ab ilit y character can be found , i ncl u d i n g the case w here all st eady state s are unstable , w hich obviou sly implies the Fig . 2 0 th e sam e type of re p r e s en t a t ion o f th e p arameter space as in 1 5 for the C ST R h as been adopted to ill u st r ate t he dynamic behavior of the axial dispersion m o d e l . In p art icu la r the case w here Pe m 5 P eh is exam in ed . This situation is thu s i n t e r m e d iat e between the t w o w ell - known ideal c a s e s of the C ST R and the plug flow reactor ( C oh e n an d Poore , 1974) : t he latter alw ay s ad mit s a u n iq u e gl ob al l y st able steady s t at e since it is described by a system of initial value ODE s . In Fig . 2 0 , i t appears that besides curves M 1 o S 1 , an d S 2 , which correspond to c u rv e s M , S , and SM , re s p e c t ive ly , for the C S T R , one e x tra m ult ip li ci t y curve M 2 appears to get h e r wit h t h re e more H op f bifurcation curves : s3 , Sg , and 8 4 . C urve M 2 bounds t h e r e gion w here two additional steady s t ate s occur· ( the fourth an d fifth ) and curves S 3 and S envelope t he re gion w here two more Hop f bifurcation p oi n t s appear . C urve S 4 r ep res e n t s the locus w here two Hopf bifurcation p oin t s coalesce . On t h e w hole 1 4 re gio ns characteri zed by qualitatively different bi fu rc ati on dia gr am s of outlet conve rsion v e rs u s Da T he first six of t h e s e are t h e same as t hose of the C S T R , while are found . the o t he r s a re q uit e different , as is e vi d e nt from the sketches shown in the side of Fi g . 2 0 . I t i s noti ce ab le th at , b e s i d e s the occ urrence o f five s t e a d y st ates in r e gio ns VII and X I II the bifurcation d i a gr a m ex hi bit s three H op f bifurcation p oi n t s . A s a con s e q u en c e quite different phase port raits from any of those found for the C S T R ( see F i g 1 8) are occurrence of p eriodic solutions .
Fi g .
In
,
=
=
-
,
a
,
,
generated .
.
A simi lar analy sis has been performed t aki n g the Lewis number as the a s R ay an d H a s tin gs ( 1 980) did
b ifu r c a ti o n p aramete r , in the s am e m a nn e r
T his lead s to t he discovery o f more exotic bifurcation for the C S T R . di a g ram s the most peculiar being p e r h ap s those sho w n in Fig . 2 1 , w hic h include for a v e r y sm all Da r an ge ( on the ri gh t of t h e lower H op f bifurca tion point ) t h e pos sibility of h avin g five unstable steady states surrounded by a stable limit cyc le More i n te re s t i n g on practical groun d s ar e the re ported critical values o f the L e w i s num ber ( i . e . , the Le value ab ov e which no os cillation s are possible ) . Jen sen and R ay ( 1 98 2 ) h ave sho w n that this v al ue is less than 1 for any Pem P� > 1 4 , which in di c at e s that even i n h om o ge n eou s reactors ( Le ::! 1 ) o scilla tion s can occur only in v er y short reactors . O scillation s in a p acked b e d where the Lewis nu mbe r is of the order of few hundre d s , are then q uite u nlike ly at l ea s t as far as the phe no m en a described by t h e p seudohomogen eou s axial d i s p e r s i on m o d e l with a simple rea c t ion rate expression are concerned . ,
.
=
,
,
1 025
Reacto r Steady -State Multiplicity /Stab ility 1 .0 X
. � �::::::j : ...r. _. : :-:-: : _.-:: : : -:-: : : -:-:, , :� . . :-:-: r� . ..
0. 5 0
. . ....
6 <0
0. 1 5
Do
0.16
2
0
L..___.J...___�--....1
0. 1 4
4
0.17
0. 1 4
0. 1 5
Do
0. 1 6
0.17
( b)
(a)
FI G URE 2 1 B i fu rca tion diagr am s ( a) x v e r s u s D a , and ( b ) 0 ve r s u s D a Le 0. 8 , 0 = 3 , B 15 , characteristic o f re gion X I V ( p aram e t e r values : ot h er p arameters as in Fig . 2 0) . =
==
COM P L E X T R AN S I E N T S I N CST R s A s w e have already ob se rve d from the steady- state m ultiplicity poin t of , the com pl e xit y o f be h avi or o f a C S T R incre a se s significantly for even a sl i gh t increase in th e reac ti n g system complexity . T his was ob served earlier in the cases of an irreversible first -order reaction in two C ST R s in serie s , and of two such r e ac tio n s o cc urrin g in series in a C S T R . T h e aim of this s e c tio n is t o indicate so m e of the exotic dy n amic behavior which can occur in t hese system s . For completene s s , a few com ments on t he st abilit y character of the m u lt i p l e steady s t ates discussed earlier will be made firs t . vie w
F i r s t - O rder Reaction i n
a
Series of T wo CST R s
W e have already mentioned that a serie s o f N C S T R s , wi t h a fir st -order irreversible reac tio n , can exhibit at most ( 2N +l - 1) st e ad y state s , of T he st ab ility of two reactor s w hic h no more than ( N + 1) c a n be stable . in series h a s been examined by K ubicek et al . ( 1 98 0 ) and Svoronos e t al . ( 1 98 2 ) , by applying Liap unov ' s fir st m et hod to the tran sient mass and energy b alance s , w hich c an be re ad ily derived from E q s . ( 30) an d ( 3 1 ) . S in ce the Jacobian of this sy st em is eq ual to the product of the J acobian of the two single reactors , it follow s that a n ece s s ary and sufficient condi tion for stab ilit y is t hat e ach of the reactor s is it self stable . T his conclu sion doe s not hold if a recycle stream is p re sen t . Th ro u gh a detailed an alysi s o f the eigenvalues of t h e se two Jacobian s , Svoronos · et al . ( 1 9 8 2 ) determined the st ab ilit y character of each p o s sible steady st at e , as shown in Fig . 8 . O ther novel features have b een shown by K ubicek et al . ( 1 980) , who e xpl or e d different re gion s of the p arameter space and took th e reactor re sidence time as a bifurcation parameter . A s an example , two bifurcation di agram s ( ou tlet temp erature ve r s u s residence tim e ) obtained for differe n t set of h e at transfer p arameter value s o and c5 ar e shown in Fig . 2 2 . The first one ( Fig . 2 2 a ) exhibit s t h r ee i gnition and three extinction poi n t s according to the o cc u r re nce of five steady state s . In the second one
1.
z
Morbide lli , Varma, and A ris
1 026
(0 )
8
"'
<:t>
6
8?
\
\
4
\
2
0
\
\
=
8 � = 0.5
8
"'
<:t>
\ \
8 ? = I ; 8�
6
=
2
4 2
0
-2
- 2
-4
( b )
0.0 1
0 1
-4 T
0 1
0 .0 1
T
FIG URE 2 2 C h ar ac te rist ic bifurcation diagram (outlet temperature versus residence time) for two C ST R s in se ri e s with a first -order re c tion ( p ar am = 0 . 5 , y = 20 , B 10 , v eter v alues : Da = v 75
!
a
Da 2
==
,
01
=
02
= 0.
).
n
( Fig . 2 2b ) two isolas are p re s en t le adi g to a total of three . separate on the same b ifurc a tion diagram . Since the smaller isola lies en tirely in the shadow of the larger one , it can never be reached from any of the other t wo curve s simply by ch n gin g the reactor re side nce time . S voronos et al . ( 1 98 2 ) have app li e d Hop f- Friedrichs bifurcation t heory to t his model and reported anal ytical relation ship for the direction and s t ability of the bifurcatin g p e riodic orbits . In the case o f a si n gle reactor , this reduces to the same equation [ E q . ( 7 9) ] as t hat de ve lop e d by Poore ( 1 973) It h a s been fo un d that while in this case al so , p erio di c cannot occur in the adiabatic system , the dynamic behavior is in general more co mple x than in a sin gle C STR . F or example , it is p o s sib le that in the first reactor m ay force oscillations in the self- sustained o s cill a tion second , w hich , per se , would have been at a st ab l e s t ead y s ta t e curves
a
an
.
solutions
s
.
T wo Co n secu ti ve I rre ve r s i ble F i rst -O rder React i o n s in a CSTR
us
T he m ultiplicity p attern of this reacting sy stem w as di s c s e d earlier . Since st ability an aly si s o f these steady states can b e p erformed with the u s ual m ethod s , we shall concentrate immediately on the dynamic behavior . First note that the tran sient model is given by three nonlinear O DE s which can readily be obtained from the s tead y s t ate m ass and e nergy b alances ( 40) by addin g on the ri gh t h and sides the accum ulation term s for the mass of A , m a ss of B , and energy , respectively . We have already dis cussed the extension of b i fur cation theory from two to N dim en sion s in the context of it s application to a sy st e m of N O D E s arising from the dis c re t i z a t on of on e or more p arabolic PDE s . From t his p oi nt of view , the three- dimensional system now under examination can be re garded as the sim pl est case (i . e . , N 3) .
-
i
=
-
,
1 02 7
Reactor S t eady -State Mu ltiplici t y /S tab ility fi r s t reported oscillation s in a
exothermic fir st - order reactions , by direct inte gration of t h e transient H lava�ek et
model .
al . ( 1 97 2 )
H albe an d Poore
and p erfo rm e d
C ST R with two
app lied Hop f- Friedrichs bifurcation th eo ry
an exten sive search in the p arameter sp ace to i dent i fy all
( 1 98 1 )
p os sibl e shap e s of the first reactant conver sion versus D a m kob l er number
T en qualit atively different b ifu r cat io n d ia gr am s were found , w hic h included situation s w here up to fiv e st eady states a n d four Hopf bifurcation p oi nt s w ere p resent simultaneously . R e latio n s hip s w ere also re por te d for the a p rio ri definition of the direction and stability of the b i fu r c at in g periodic orbit s . T he d et er m ination of the H op f bifurcation poin t s is only th e fir st step t ow ard definin g the dynam ic beh avior of this sy stem , w ho se detailed under bifurcation diagram .
an identi fication of the struc t u re of the sy stem d yn amic s as the b ifu r c ation standin g req uire s a det ailed analy sis of each p e riodic b ranch .
parameter c han ge s away from the Hop f bifurcation point .
T his involves
type u su ally req ui res t e dio u s i n t e grat ion of the tran si e nt m o d e l eq u atio n s .
Analy sis of this
Doe del and Heinem ann ( 1 98 3 ) h ave ap plie d a continu ation method for the
calculation of the entire b r anch of periodic solution s , both stable an d un
s tab le .
value prob lem into a boundary value p roblem , where the peri o d i s t aken
T his method is b a sed on a transform ation of the original initial
as u nknow n an d correspondin gly the periodicity condition is added . T he final equation s are then disc retized u sin g o rth o go n al collocation , and the re sultin g system of nonline ar al gebraic equations is solved through N ewton ' s met hod . T hi s procedure has b e e n i m pl e m en t ed b y D oedel ( 1 9 8 1 ) o n a com puter c ode w hic h al so p r ovid e s the s tead y - state solu tion b ranches using K ell e r ' s ( 1 9 7 7 ) arc - len gt h m et hod for avoidin g sin gularity problem s in the Jacobian at the static bifurcation point , a s w ell a s the Hopf bifurcation poin t s and the bifurcatin g periodic solution , u sin g the u sual p erturb ation technique s . T hi s appears to be q uite a u se ful too l for the complete study of the dynamics o f a giv e n system , since it gives the entire bifurcation di a g r am ( i . e . , both steady state an d p e riodic s olution branches , both stable an d unstable ) . T he stability character of p e riod i c solution s is de termined t hrough th e c orre s p o ndi n g Floq uet m ultipliers . 1 0 bifu r ca ti on diagrams p reviou sly reported by H albe an d Poore ( 1 9 8 1 ) by U sing this p ro gram , D oedel an d H einem ann
( 1983)
h ave com pleted the
computin g the e nti r e p eriodic solution branche s . One o f th e most peculiar bifu rc a tion dia gr am s so obtained is sket c hed in Fi g . 2 3a , w h ere four Hop f
bifurcation p oints occur :
in all of them bifurcation occur s to the left ; the
while the second and fo u rth are st able . Now , star tin g at large D a an d decreasin g it , t h e system undergoes a so ft bifurca tion into a p eriodic stable orbit and follow s t hi s b ranch u ntil it becom es unstable . H ere a hard bifurcation occ ur s to another p e riodic solution , with smalle r oscillation s , and ev e nt u ally the system reaches the lowest stable steady state through one mor e hard b ifu rcation . It m ay be no ted that in t his d i a gr am the classic S - shaped m ultiplicity p at t e rn is re p roduced with p e ri od ic in stead of steady - st ate solutions . A not h er in te res ti n g and peculiar bifurcation di a gram i s s ho wn i n Fi g . 23b . H e re the periodic solution branche s em an atin g from the third and fou rth H op f b ifurcation p oi n t s are no t connecte d , but terminate at an infinite - pe riod b ifurcatio n point ( indicated by the sy mb o l +) . Note t h at due to the stability character of th e s e t w o b r anches and the relative posi t io n of the resp ective termination p oint s , there is alw ay s a stable steady st at e or pe r io dic solution available for the system for a ny D amkobler n u mb e r fi r st and t hird are un stable ,
value .
Morb i d e l li ,
1 028
. . . . ·00·700. .
( a ) X
•
•
Varma , and A ris
•
o
"o 00
. . .
"
•
"
., I
., ,. .,. . ,, . :. .
;
• •
)/
"
-
oa -. .... / : I . ..: ' .. .. .. ,
( b)
X .
,0 .
-
..
/··�)
. // /
-
Do
FIGURE 23 Q ualitative sketches of two bifurcation diagram s ( outlet con version of the first reactant versus D amkobler number) characteristic of a C S T R with two con secutive exothermic reactions .
T he detailed analy sis of periodic solutions does not exhau st all possible feature s of the dynamics of the system under examination , becau se it can also exhibit nonp eriodic solution s (i . e . , chaotic behavior ) . T his peculiar behavior , which accordin g to Poincare- B endix son theorem cannot occur in a two - dim ensional sy stem ( cf. a C S T R with one reaction ) , is instead quite com mon in models of larger dimensionality , just as periodic oscillations are common in two- dim en sional system s . K ahlert et al . ( 1 98 1 ) found chaos in a C S T R with two consecutive reaction s : the first exothermic and the second endothermic . T he direct integration of the tran sient model equa tion leads , for a certain set of p arameters , to the trajectory in the p hase sp ace shown in Fig . 2 4 . The p resence o f a well- defined attraction re gion , shap ed as a folded -over rubber strip , is an indication of the chaotic be havior of the system . T he way the sy stem approache s this situation is through a typical Feigenbaum ( 1 979) c ascade of periodic solution bifurcations . I n p articular , with reference to Fig . 25 and for the p arameter valu e s reported in the caption , it is seen that the system admits two Hop f bifurcation points , both bifurcating to the ri ght : the first stable and t he second unstable . The periodic solution b ranch can be con structed throu gh sub sequent
Reacto r Steady -State Multiplicity /St ab i lity
1 0 29
e
S hap e of the attraction re gion in the chaotic regim e in the
FIG URE 2 4
case of one exo and one endothermic reaction ( p aram eter v alues :
0c
=
0,
v =
0. 5 ,
o: =
•
+
y
- 0 . 4 2 6 , A = 1 , D a = 0 . 2 6 , B = 5 7 . 7 7 , c = 7 . 995)
+
=
oo , •
•
e
1. 9 5 6 0 5 - -
»
7 . 9 6205 __} 1 7. 9 6 4 8 5
F I G UR E 2 5
1;
_)
B ifurcation diagram :
outlet con version o f t h e first reactant
versus heat tran sfer p aram ete r , other p aram eters as in Fi g .
24 .
Mo rb idelli ,
1 030
Varma , and A ris
numerical i n t e g r atio n s s t ar tin g from t h e left Hop f bifurcation p oin t and in creasin g t h e value of o . It is found t h at by in c r e a sin g o the p e ri od of the stable oscillation s chan ge s continuou sly up to a critical o value w h ere su dd enly a d oub li n g in the oscillation p eriod o c c u r s . In ot he r words , the p er i o di c orbit becomes un st ab le and simult aneou sly a new one , st able and w it h doubled p eriod , app ear s . T his new typ e of bifurcation , whic h is o ften referred to as Po incare b ifurca t io n , o cc u r s a gain and a gain at suc c e s siv e ly closer o in te rvals , up to a n e w critical o value where chaotic behavior begin s . For a p articularly sim p l e one - dim ensional c as e , Feigen baum h a s shown that the o values at w hich period doubling occurs tend to a p p r oac h a convergent ge om et ri c serie s , such that the r ati o p =
( 107)
c on st an t an d equal to the universal v al u e 4. 6 6 4 2 0 1 6 . Chemburkar et al . ( in pre s s ) have further in v e s ti gat e d thi s sy stem by analy zin g in detail the periodic orbit s that b ranch off from the hard H op f bifurcation point o * ( all the numerical values r eporte d here are taken from this last source ) . T h e sam e type of bifurcation p a t te r n has also been found in this region , a gain e n d i n g i n an in fi nite p e rio d t e rm in at io n poi nt , as quali ta tive ly sketched in Fig . 2 5 . B y p lottin g t h e number of p eak s p er p e riod ( w hich is p r op o rt ion al to the oscillation period ) as a fu n c t io n of the bifurcation p aram ete r , as s h o w n in Fig . 2 6 , it clearly appears that there exists a w in d o w of o values where th e system ex h ibits chaotic beh avior . It should be mentioned t h at the w ay this system ap p r o ache s the chaotic re gion is q ualit atively sim ilar to that described by F ei g e nbau m ( 1 979) , but the geom etric series of the bifurcation o v al u es is not p e r fe c tly rep roduced ( at l e a st after the fe w do ub li n g s investigated ) , and the com p u t e d values of the ratio p [ E q . ( 107) ] are somewhat sm aller than the n at u r al F e i ge nb a u m ' s con stant ( rangin g from 3 to 3 . 5 ) . is
"0 0
0;
D..
"' a.
�
1/)
-""
c Ql D..
2
4
8
16
32
F I G U RE 2 6
7. 8
7. 9
Win dow o f chaotic
8 .0
b e h avior .
8. 1
1031
Reacto r S t eady - S t a t e Multip licity /Stability
Similar b ehavior has been fo u n d by Jor gen sen and A r i s ( 1 9 8 3 ) , w ho studied the same reac ti n g system , but con si deri n g two exothermic re ac tion s In t hi s case some more complex dynamic b ehavior has b e e n identified . A u se ful tool for e n li ghte nin g the bifurcation pattern of the sy s t e m is offered b y F loq u et multi p lie r s Let us de fin e � as t h e three - dimen sional vector oJ periodic solution s of the tran sient model characteri zed by a p eriod v alue T. Its stability can be d e t e rm i ne d by con si derin g the solution of the p e rtu rb a tio n p rob l e m .
dg
dt
=
( 108)
�g
with the initial co n ditio n cr = I the i den tity matrix , w her e Q is a 3 x 3 m atrix and J i s the J ac obi an of th e ;riginal system evaluated alon g the p er iodi c solu tion � . I n trod u cin g no w th e e i genval ue s 1l of the matrix g. it can be shown that one of these i s eq u al to unity ( thi s giv e s a u se ful check for the acc uracy of the a d op t e d n um eri c al int e gr ation technique) and the system is asymptot ically st ab le if and only if all the ot he r e i ge nv alu e s are such that ,
I ll l
<
1
( 1 0 9)
In the bifurcation study of p eriodic solution s it is t h e n quite c on v e ni e nt also to c om p u te such eigenvalue s- the so c alle d Floquet multipliers . In t h e system u n de r examination the doublin g of a p eriod ic solution cor r e spon d s to t h e situation where one of t h e m ultipliers crosses t h e unit circle in t he com plex plane at - 1 , and then imm ediately r e app e ar s in side the cir c l e ( i . e . , the orb it become s un stable an d a new one s tab le and with d oub le period e m e r ge s ) Some of the results obtained by J or gens e n an d Aria ( 1 98 3 ) are shown in Fi g 2 7 , where th e heat tran sfer p aram eter o is u sed as a bifurcation paramete r . The various t yp e s o f d y n amic re gi m e s have been indicated by a c apital letter . R e gi m e A co rr e s p on d s to the c a s e where one un st ab le steady state is p resent to ge t h e r with a stable limit cycle . Regimes B and C c orr e sp on d to the case w h e re d o u bli n g of the oscillation p e ri od occurs ( B ) u p to the emergence o f a chaotic re gio n ( C ) . A ll these features are q uite similar to t ho s e described above for the c a se of one endothermic and Regimes D , E , an d F correspond to in te r lu d e one exothermic reaction . sit u at io n s sep ara tin g the occurrence of a chaotic behavior . E ven t ho u gh some bifurcation s of periodic orbit s , in vol vin g doublin g of the p e rio d have been detected in t h e s e re gimes , the connection w i th t h e F e i ge n b au m " route to chaos" appears to be quite weak . Finally , re gim e G is characterized by quite complex behavior , w h e re p eriodi c solution s and chaC E! seem to co e xi s t It sh ould be ob served that at th is poin t the numerical accuracy of the inte gration t e c h niq ue becomes critical , and only some s i gnific an t ad vances in the th e ore t ic al insi ght can le a d to a b e tt e r understanding of sy ste m behavior . For illustrative purposes , some of the p e rio dic solution s characteristic of re gi m e B and the attraction re gion c h a r ac te ri zin g re gime C are show n in F i g s 28 and 2 9 , re sp e c t iv e ly In conclu sion , t h e d y n am i c behavior of a C S T R app ro ac h in g chaos is not yet w ell defined and m or e research in this are a is n ee de d For example , c on si de rin g a d i ffe rent kinetic scheme , constituted by two in d ep e n d e n t first -order exothermic re ac tion s : -
.
.
,
.
.
,
A
-+
B
C -+ D
.
.
.
Morb i d e l l i ,
1 032
V a rm a ,
and A ris
F I G U R E 27 B ifurcation p attern , wit h o as a bifurcation parameter , in the case of two exothermic reaction s (p arameter values : y = oo , G c 0, v = 0. 01 , a = 1 , A 1 , Da 0 . 55 , B = 1 7 . 5) . =
=
=
L y nch et al . ( 1 98 2 ) h ave defined the occurrence of chaos , which is reached b y the system t hrou gh a w ay different from that discu ssed so far . More over , a u seful numerical technique for the iden tification of the occurrence of chao s , based on the autocorrelation analysis , has been ap plied in this work .
(a )
8
8
(b)
8
(c)
FIGURE 2 8 Sequence o f trajectories in the phase space showin g the first t wo doublin g of the periodic orbit in re gim e B . (a) o = 6 . 94938 ; ( b ) o 7 . 2 82 9 4 ; ( c ) o = 7 . 2 8665 ( other p arameters as in Fi g . 2 7 ) . =
1 033
Reactor S teady -State Multip licity /S tability
8
FIGURE 2 9 Chaotic t r aj e c t or in r e gi m e C ; as in F ig . 2 7 ) .
o
u,
= 7 . 286670
( ot h e r p arameters
PA R AM E T R I C S E N S I T I V I T Y A N D R U N AW A Y I N
T U B U L A R R EA C T O R S
q uite common p ro bl em in m athem atical modelin g , as w ell a s in the experi
mental an aly si s o f any r eal - life device , is to establish t h e effect of each A
input p arameter on any of the m odel outputs . Quite often there exists regions in the p aram eter space where sm all variations of the inp ut param eters cau se large variation s of th e output s ; t h es e are u sually c alled regions of parame t ric sens i t i vi t y . T his c onc ep t was fi r s t introduced in the study of t ub ular r ea cto r s by Bilou s and Amundson ( 1 956) . Wh en a sufficiently exothermic reaction occurs within nonadiabatic tubular re a c to rs , the temperature p rofile at steady state exhibi t s a m a x im u m , usually called a hot spot . In the region of p aram e tric sensitivity the m a gnit u d e of t h e hot spot can undergo very large increases even for very small variations of one or more of the input p aramete rs ; this p h e no m e no n is u sually referred to as t emp e ra t ure ru naway . T his is illu st r ate d qualitatively in Fig . 30 , w he re for a p articu l ar example ( see van Welsenaere and Fro m e n t , 1 970) the t e m pe r at ur e profile variation as a fu nct io n of eq ual changes of t h e in let temperature is shown be fore and after the run away boundary . The p revention of exces sive t e m p erat u r e p e ak s alo n g the re act o r is very imp or t an t in p rac ti c al application s no t only for safety re a so n s but al so
1 03 4
800 r------r---.
Mo rbide l li , Varma , and A ris
UJ 0:: ::I
�
0:: w Q. :;; LLI 1-
5
0
..J IL
6 00 L------L--� 0 0. 5 1.0 R E ACTOR A X I A L L E N G T H , Z ( m )
F I G U RE 3 0 Examples of temperature profiles along the reactor axis for various inlet temperature values inside and outside the runaway region.
becau se these may adversely affect the conversion of exothermic equilibrium reactions , selectivity of the process , and in the case of catalytic reactors , the catalyst activity and durability. Therefore , most of the research in this area has been devoted to providing graphical or analytical representa tions of the region of runaw ay , so that such behavior could be avoided immediately at the earlier stage of reaction design . T o attain such a repre sentation , two problems need to be solved : 1. Formulation of an a priori criterion of parametric sensitivity , which has to be based on intrinsic reactor behavior with no reference to any specific situation Implementation of the criterion to produce a simple representation , 2. either graphical or analytical , of the re gion of runaway Most c ont rib ution s in the literature have examined the case of a plug- flow reactor with an irreversible nth -order reaction , whose mathematical model is given by du - Daun exp ( 1 + 00 /y ) ( 1 1 0) ds n d0 BDau exp ( l +00 / y ) - 6 0 ( 111) dS with I C u uo 0 0 0 at s = 0 ( 1 1 2) =
=
=
=
where in the definition of the quantities 0 , Da , y , and B the cooling temperature T c has been taken as reference temperature T r (i. e . , 0 = (T - T c ) E /RT c2 , and so on ] .
1 035
Reac tor S t eady - S t a t e M u l t i p licity /Stab ili t y
T he first a p riori criterion for runaway was p rop o s ed by B arkelew ( 1 9 5 9) as a res u l t of an em pirical an alysis of a large number of temp erature profiles alon g t h e reactor axis . D ente an d Collina ( 1 96 4 ) defined as run
away the occurrence of a region alon g the reactor w here the second deriva
tive of temperature with respect to distance is posi tive . T his i s physically so ul d since it identifies runaw ay with an " acceleration " in the temperature reactor axis plane . Accordin g to this criterion , re gion s in the param ete r space w here run aw ay occurs can be identified , a n d the boundaries o f this are give n by the locu s of p ara met er values w here t h e third derivative of temp erature versu s re ac to r axis is zero . Van Wel senaere and From ent ( 1 970) h ave adop ted this same definition of runaw ay . They p ro d u ce d simple explicit analytical exp re s sion s for t he boundaries of the runaway re gion by introd ucing a second criterion , b a sed on the occurrence of a maxim um in t h e loc u s of the maxim a in the temperature- conversion pl a n e , which accordin g to them is s ub s t an tially equivalent to the p revious c ri t erion . Som e additional in sight into t his p roblem is given by studie s reported
earlier in the context of thermal explo sion theory .
H lavacek et al .
p oint ed out that the lumped m ass and heat b alances in a symmetrically
( 1 96 9)
heated reactive m ate rial are , in fac t , identical t o E q s . { 1 1 0 ) and ( 1 1 1 ) , where the reactor len gth coordinate is replaced by tim e . T h e earlier criteria for ignition or explo s ion w ere based on the occurrence of a region with po sitive second - order derivative in the tem perature ver s u s tim e p ro
file ( see S em e no v ,
1 9 2 8) , w hich coincides w ith t h e crite rion p roposed by S ub sequently , A dler
Dente and C ollin a ( 1 96 4 ) in the r eac t or context .
an d E ni g { 1 96 4 ) p ropo sed a som ewhat more intrin sic de finition of i gni tion
or explosion b ased again on the occurrence of a positive second - order
derivative b ut in the temperature- conversion ( instead of temp erature - tim e )
I t w a s also shown that runaway i n the temperature- conversion plan e im plies runaway in t he temperat ure - time plane , but not vice v er sa .
plane .
T he relevant equation , which c a n easily be derived from E q s .
( 1 1 2 ) by introd ucing the conversion x =
1
- u,
( 1 1 0) an d
is simply given by
d0 dx
( 1 1 3)
with IC
e
=
e
0
at x = 0
{ 1 1 4)
T his criterion h a s b een u sed in the context of tubular reactors by
Morbidelli and Varma
( 1 9 8 2) ,
w ho con sidered reactions of any po sit i v e order
with no limitation on the activation ener gy or inlet t e m p er atu re v alue s . The runaway region was explored u sin g the isocline method , originally intro
duced by C ham bre { 1 956) , which p r ovi de s q uite an efficient p rocedure for calculatin g the region boundarie s , av oi ding tedious trial an d error .
T he runaw ay region in t h e parameter plane B versus c /Da (i . e . , heat
of reac tion ve rs u s heat tran sfer p arameter) i s shown in Figs .
31 and 3 2
for various value s of t he reaction order , activation energy , and inlet
temperature .
For p arameter value s above the line ( i . e . , lar ge values of
the he at of reaction p aram eter B and low value s of the heat t r an s fer p aram
eter c /Da) , the reactor un der goes run away , w hile it does not runaw ay for p arameter values below the line .
A s expected on phy sical ground s , it i s
.... c w 0)
[IJ .
a: w I-
�
40
'ft.
z 0 t=
30
0
10
(.)
� w J:
I
I
I y = 2 0 , 80 = 0
I
I
� 401
(ol
I
I
y = I 00 , 80 = 0
I
I
( b)
1 40 1
I
I
I
I
10
20
30
y = ao , B0 = 0
(c)
I
20
O L-�--�--�---L--� o e-�--�---L--�
10
20
30
40
-
50
10
20
30
40
HE AT T R A N SFER PAR A M E T E R , (
8 / 0a )
FIGURE 3 1 E ffect o f the reaction order on the runaway region for the activation energy ; the runaway re gion is above the curve .
� 0
t hr ee
value s of
ti
E:
E
�
Q
�Q
;:s $:). � :l. "' Q
::a
(l) !:) C'l
0 .,
�
!:) P. 'C (l)
C/l I
lD
a: w 11.1.1
::!:
Cl: a: <( Cl.
(0 )
�
ol
��
: _ "'
z 0
40 30
1u <( w a:
0 1<( w :t
J
["'
20 1-
lJ..
10
20
30
40
-----
FIGURE 3 2
y • 20 , n = l
Q:l
t
40
• -
30
20
I
f y• 20,, .: (
'
Q:l
1 0 .5
r
..... !:) ..... (l)
eo = - 3
o ��--�--�--�� o ��--�--�--�� 10 20 30 40 10 2 0 30 40
:::: s::
...
-6' -
§: .....
<.::
t;:j
�
<.::
.....
H E AT T RA N S FE R PA RAM E TE R , ( 8 / D o )
E ffect of the inlet temperature on the runaway re gion for three
reaction order s ; the runaway re gion is above the curve .
..... c c.> "'I
Morb idell i , Varma , an d A ris
1 03 8
al so app arent that runaway i s more likely for l ar ger value s of the activa tion energy y and the inlet temperature e i an d for smal ler reaction order n. A detailed comparison between the runaway region predicted by the lat t e r criterion and those predicted by the earlier criteria p r o p o se d by B a rk el e w ( 1 95 9) , D ente and Collina ( 1 964) , and van Welsenaere and Fro m e n t ( 1 9 70) h a s been reported by Morbidelli and Varm a ( 1 985) . It w as found that in ge n er al the criterion based on the tem perature conversion plane leads to the less con servative result s , and that the di s crep ancy be c ome s more si gnificant for low value s of the activation energy y and the heat transfer p arameter o /Da . This is app arent from Fig . 3 3 , w her e the runaw ay region s calcul ated by van Wel senaere and F rom ent ( 1 970) ( dashed line) an d Morbidelli an d V arma ( 1 9 8 2 ) ( solid line ) are com pared . On the other hand , t his criterion provide s estim ates of t h e nonrun away region which in fac t corre spond to safe ope ratin g co n d itio n for the re ac to r . For example , from i n sp e ct ion of the temperature ver su s conver sion profiles show n in Fig . 34 for inc reasin g v alue s of the heat of reaction parameter , it appears that the critical value B e = 1 4 . 3 1 inde e d represent s a re a sonab le boundary b etween safe and runaw ay op er a tin g conditions . Note that t he critical value of B e i s defined as the b ou n d ary between run away and no - r u n aw ay , an d thu s correspon d s to the situation where the third as w ell as the second derivative of te m p er a t ure with resp ect to con version vanish . In conclu sion , this c r it e ri on can be r e com me n de d for de si gnin g tub ular reactors operatin g outside the runaway region ·, without requiring serious l imi t at ion s on the ope rati n g condition s . 50 m
-
a:: w 1w
�
<( a:: <( a.
z 0
i=
u <( w a:: I.L. 0
t<( w I
n
40
= I ; 80 = 0
30 20 10 0
0
10
20
30
40
H E AT T R A N S F E R PA R A M E T E R , (
8 /Do )
50
F I G U R E 33 Comparison between the runaway region calc ulate d ac co r ding to the criterion of Morbidelli and Varma ( 1 982) ( soli d line ) , and the one by van We l se n a e re and Froment ( 1 970) (dashed line ) .
1 039
Reacto r S teady -State Mul tip licity /S tability
ILl a: ::l 1-
1 2
1 2
B = 16 ,
10
�
8
(l:)
ILl 1--
6
f
(/) (/) ILl ...J z 0
i
(/) z ILl
z
�
0
0.2
0.4
0.6
0
0.8
C O N V E RSION , X
....____.______,___,
0.99
- x
FIGU RE 34 Temperature versus conversion p rofiles in side and out side n = 1, y 2 0 , o /Da 20, e 0 = the runaway region ( p arameter value s : 0) . =
==
Following a proced ure adopted by Gray and Le e ( 1 96 5 ) and Hlavacek et al . ( 1 969) , Morbidelli and Varm a ( 1985) have propose d the following analytic repre sentation of the runaway region boundaries :
�a
AeB[l
=
where X =
-
e:)
213]
( 115)
___:8:... :. . .:._. 7. °:.., 6 -::-
_
7 . 66
+
( 1 1 6)
n '
is an e m p iric al factor introduced to i m p rov e accuracy . critical B value in the case of an adiabatic re ac t or (i . e . ,
B�
repre sents the o = 0) . U sin g the isocline method , the followin g explicit expression can be derived : so
c
=
e
-
e
0
o
ne o + -
-
etr > 2
e ty )
2
e
( 1 1 7)
-
-
where 8 is the unique solution of the equation (n
-
+
1)8
4
2 [ 2(n
+ -
2y ( n
-
1) ( 2
-
y) 8
1) + y ] y 0 + ( n 3-
-
3+ 1) y
[ 2(n 4
=
-
0
1) ( 3
-
y)
-
y ( y - 2) J y 2'82 ( 118)
Morb idelli , Varm a , a n d A ris
1 04 0
in the in tern al 0 e: [ 0 _ , 0 +] , where 0 ± = [ ( y -0 2 ) ± ./ y ( y - 4) ] y / 2 . T o facilit a te the u se of E q . ( 1 1 5) , the values of B e • calculated from Eq s . ( 1 1 7) an d ( 1 1 8 ) as a function o f y , ar e shown in Fig . 3 5 for variou s re action orders . In Fig . 36 a comparison between the runa w ay re gion predicted by E q . ( 1 1 5 ) an d that numerically calculated by Morbidelli and Varma ( 1982) i s shown . The agreement seem s t o be quite satisfactorly for any value of the r e ac tion order and the activation ener gy . I t s h ould b e m entioned that le s s satisfactory results are ob tained for inlet t emper ature value s 0 0 significantly However , since this situation is quit e unlikely in in different from zero . dustrial ap plication s , this point has not been pur sued furt her . Several attempts have been made in the literature to extend t h i s s en sit iv ity an aly sis to more complex m ode l s to account for ot h er phenomena which may al so be of im p ort an c e in e st abli s h in g run aw ay op e r ati n g condi tions of industrial reactor s . In particular , A gne w and P otte r ( 1966) have ex ten d ed B arkele w ' s ( 1 95 9) c rit e rion to account for heat and mass axial as well as radial di sp er s ion ; Soria Lo pe z et al . ( 1 9 8 1 ) extended van Wel senaere and F r omen t ( 1 9 7 0 ) c rit erion to the case where the cooling fluid temperature c h an ge s along the reactor ; fin ally , the effect of side reaction s o n the runaw ay region has been investigated by Dente et al . ( 1 96 6a) in the case of con secu tive reactions and by D ente et al . ( 1 96 6b ) and Weste rt e rp and Ptasinski ( 1 984) in the c a se of parallel re act io n s . It s houl d be mention ed that ru n aw ay behavior is most lik ely to occur in catalytic re ac tor s , where the int rin sic h e tero ge neo u s nature of the re act in g system can play a very impo rt an t role . However , t he effect of inter and in t rap ar ticle m a s s and heat transfer r e sis t an c e s on runaway has not received much att e n t io n so far in the literature . Mc G re avy and Add e rley ( 1973) have analy zed t he effect of t r ansp or t limitations on a sin gle c a t aly st p article ;
20
ou lD
12
0
5
10
- D I M E N S I ON L E SS
100
AC T I VA T I O N
500
1000
E NERGY . y
5000 0
F I G U R E 35 Value s of the adiaba tic critical p arameter B 0 from Eq . ( 1 17) as a function of y for various o r de r s o f reaction a n d 0 0.
0
=
III
!l? LLJ
r LLJ
:IE � 0:: � D.. z 0
ru Cl LLJ 0::
"0 rex
Reac to r S t eady -State Mu l t ip licity /S tab i lity
4 0 .----,--.---�--...,---. 30
( 0 ) y = 20
80
=
0
( b ) "f
:
CIO
;
80 : Q
20 10
LLJ �
t
;
1 04 1
10
20
30
40
50
- -- H E AT TRANS F E R PA RA M E T E R , (
8 / 0o )
FIGURE 36 C om pari son of the runaway region predicted by Eq . ( 1 1 5 ) ( dot t e d line) and the ex act one calculate d b y M o rbidelli and Varma ( 1 98 2 ) ( solid line) •
R aj ad hyak sh a et al .
( 1 97 5 ) h av e extended van Wel senaere and From ent ( 1 97 0) c r ite rion to account for t r a nsp o rt limitations in fo u r p articular regimes of re ac tor operation ; finally , the case of multi tubular reactor s has b ee n investi gated by McGreavy and D unobbin ( 1 98 4 ) . A PP E N D I X :
S I N G U LA R I T Y T H E O R Y
Singularity
theory p rovi de s a systematic way of tacklin g the multiplicity It is co n ce rne d with three basic ideas about m ap s or functions which bear directly on the s e arc h for the p attern of solution s of a non li n e ar equation . T he first of t h es e is stability and we say , that a map is s tab le if small p e r turb ation s do not r e ally chan ge it s c harac te r . T hu s the graph of u = x 2 is m erely shifted up to down by a p e r tu r b atio n u = x 2 + a. ; i t is distorted b y the perturbation u = x 2 + 2 (3 x , b u t b y writin g this as U = ( X + 13 ) 2 - 13 2 We see that its q u alita tiv e shape is in no way Changed . On the other hand , u = x 3 is a monotonic c urve which a p erturbation u = x 3 - ax will change int o a curve with a maximum and minimum ( i . e . , qualitatively d iffere n t ) if a > 0 , h ow e ve r small it m ay be . S in gularity t he ory provide s a w ay of de t e rm inin g when a map is stable . , For e x ample , the two dimen sional mapping u = x 3 - xy , v = y is stable , for it s cusp like c h ar acte r is unchanged by sm all perturbation s s u c h a s u = x 3 + 3a.x 2 - xy , v = y + a , and so on . N ow if y = a. is fixed , u = x3 - xa. and for a = 0 , u = x 3 , So f ( x ) = x 3 is embedded in F ( x , a.) x 3 - ax que stion .
=
or F is an unfolding of f . The second great topic of singularity theory is the question of how to u nfold an un stable m ap such as f ( x ) into a st ab le map such as F ( x , a.) u sin g the least number of p arameter s . For exam ple , x 3 - a. will not d o because it is stil l unstable and x 3 + 3 13 x 2 + ax is un economical , for it may be written as (x + 13 ) 3 + ( a. - 3 13 ) ( x + 13) - s 3 + 3 13 2 aS and 13 is o tio se . The third que stion is one of generfcity , that is , of
whether there are e nou gh stable m ap s to approxim ate
an arbitrary map .
Morbide l l i ,
1042
Varma , and A ri s
For m ul t iplic i ty problem s we need a sy st e m atic w ay o f fi n din g the boundaries of p arameter space across w hich the number of roots of a non l in e ar equ ation chan ge s . T h e se are sometim es called cata s t roph e su rfaces or s e t s after Rene T horn , a pioneer of singularity th eory , classified the elementary catastrophes and gave t h ei r canonical unfoldin gs . In th e litera t ur e there are several term s t h at it will be h elp ful to explain first . Sup p o se that A i s a s ub set o f R n , the E uclidean sp ac e of n dim en sion s , U an open set in R n , and V is any set ; t h e n if f is a function f : A n u + V w e denote the pair ( U , f) by F . f m ay be sp e cifie d further as continuous , as differentiable , or in w hate v e r way we wish , but t wo elem ent s of F
[e . g . , ( U t , ft ) an d ( U 2 , f 2 ) ] will be c on side r e d equivalent if in som e sub som e sub set of Ut n U 2 they define the sam e m ap (i . e . , ft l u = f2 j u > for som e U E: U t n U 2 . T he equivalence class defined by this relation is calle d a germ and this is clearly a way o f m akin g p r e ci se the fact that we are goin g to concentrate on local p roperties an d so do not need to con sider the function outside the immediate neighborhood of a p articular point . Any smooth function can be exp an d e d in a formal power series ab ou t a particular point ( say , the ori gin ) and this is c all e d it s T ay lo r series :
r=O
( B y fo rmal we mean that we are not saying anythin g abo u t the convergence of t he series . ) I f this s erie s is tru n c at ed at the kth term we have the k jet o f the function k
i
k
f ( x)
=
E
r:::: O which is a p ol y nom i al that b eh ave s like f " to or d e r k" in t he neighborhood of the ori gin . Clearly , t his may be a local approxim ation at best and may not " fit" v e ry well . For example , j l sin x = x quickly dive r ge s from sin x and it takes j 9 to giv e a d e c e nt approximation over the interval ( O , 'IT) , or mor e generally , j 8n+ l to app roxim ate over ( O , n 'IT) after which the k jet [ I t should be remembered that the formal T aylor series goe s o ff to ±oo . may not a l w ay s converge to the function , as , for ex am pl e , with f( x ) = O , x � 0 , f ( x ) = e xp ( - 1 /x ) , x > 0 , w hos e formal T aylor se rie s is identically z e r o . ] If t h e k jet of a fun c t ion vanishe s , it is said to be of o rder (k + 1) at the o ri gin and the function f - j k f , which h a s been deliberately m ad e of order (k + 1) , is so m e ti m e s called it s T ay l . The term contact eq uivalent also occurs frequently and i s a p reci se way of s ay in g that two function s are loc ally of e s s e nt ially the " s am e s hape , " so that w e can learn a lot a b ou t the one by st ud yi n g the o t h er . I n p art ic u Formally , lar , we can rely on a can o nic al form to give us a corre c t picture . F ( x , A. ) an d G ( y , J..l ) are co ntact eq uivalent if t h e r e is a smooth and in vertible change of c oo rd i n at e s x ,.. y given by y = Y ( x , A. ) , x = X ( y , J..l ) , a rep arameteri zation J..l = M ( A. ) , A. = L ( J..l ) , and a p osit iv e scale chan ge S ( x , :>.. ) or T (y , ].l ) such th at
R eacto r S teady -State Mul tiplicity /S tab ility or
F(X , A) =
1 04 3
S ( X , A ) G ( Y ( X , A) ,M( A) )
x 2 ± AX is contac t equivalent to y 2 - ).J 2 , for we have only L = - 2 J.l , T = 1 to m ake the tran sformation . Golubit sky [ in two striking p ap ers , one with Ke y fit z ( 1 980) and one with Schaeffer ( 1 9 7 9) ] has given a very u seful t e st for the contact equiv ale n c e of an equation F ( x , A ) to one of a number of canonical form s , and his method has been fruitfully ap plied by B alakotaiah and Luss ( 1 9 8 1 , 198 2a , b , 1 98 3 , 1984) and other s . It will be best explained by takin g a y ± ).J
For example ,
to put X
=
,
concre te example .
We will use the autocatalytic scheme that Gray an d
S cott ( 19 8 3 , 1 9 8 4) hav e
shown to have m any of the fe ature s
of
the si n gle
exothermic reaction without the algebraic complexity that the Arrhenius temperature dependence induce s .
W e consider the reactions A + 2 B + 3B , B + C takin g place i n a stirred tank and so , wi t h obvious notation , have
ea
9b
=
a
b
=
- a - ek 1ab
f
f
- b + e k ab 1
2 2
- 6k b 2
With a vie w to u sin g the residence time e to vary
as
a
parameter , we sh all w ant
we set
T he n
O'. X =
.
aY •
1
= y - y
+
2
al3x y
-
a
y
st at e is ob t ained w he n x = a(3y 2 ) - 1 and s u bstit u tin g this in y
The steady (1
+
+
F ( y ; a , l3 , y ) : a( l
a) Sy 3 - al3 ( 1
y
=
+
0. From x 0 we have x = 0 gives t he cubic
=
y)y
=
2
+
(1
+
a) y - y = 0
T his appear s to be the n eate st form , although cu b ic s can also be
for
x and z =
1 -
Thus the steady
S ince
F ( 0)
..;;; 0
( 1 + a) y . s t at e s will be the
x =
and F
+
+ oo
as y
there m ay also be m o re than one .
+
oo
root s of this
, there i s
cubic
found
equ ation F = 0 .
alway s one real root but
We shall have two steady state s when = 0 ) , and this re p re s ent s the
there is a double root (i . e . , F = Fy
Morb idelli ,
1 04 4
between one and three . There will be a unique t riple t : yy 0 . Let us record the derivatives for future use
tran sition F F F - y -
-
F =
F F
y
= =
YYY
F F
F
F
a ay
aa 8
y
+
a ) (3y
3
=
=
2
+
y)y +
- 2 a8 ( 1
6 a( l + a) 8y =
a8 ( 1
-
3 a( l + a) 8y
yy
root if
-
a( l
F
F
Varma , a n d A ris
-
2a8 ( 1
-
8( 1
+
2
+ ( 1 + a) y
y)y
+
-
y
( 1 + a)
y)
6a( l + a) 8
( 1 + 2a) (3y
3
3 ( 1 + 2a) 8y
2
-
+
28( 1
y)y +
2
+ y
y)y + 1
3
=
2 8y
=
a( l
=
- a8y
+
a) y 2
3
- a( l
+
y) y
2
- 1
Were we to eliminate y be t w e e n F = 0 and Fy = 0 we would have a com plicated equation between a , (3 , and y representing the surfaces in a- 8 - y sp ac e that separate regions o f one steady state from r egion s o f three . Steady - state di a gr am s giving y as a fu n cti on of , say , a could be con stru c ted by p a ssin g t hrou gh this space on a line of con st an t 8 and y . H ow e ve r , G olub itsky ' s metho d calls for going to the h ighe st order singularity we can find . C learly , we co uld have F = Fy = Fyy = 0 We cannot make simultaneously and this would be a li n e in a- 8 - y space . Fyyy 0 except by putting a or 8 = 0 , and t his is pointless . B ut it is p o s sible to find a point where F = Fy Fyy = Fo: = 0 , n am ely o: 1 , 8 = 256 / 2 7 , y = 1 / 8 , y = 3 /16 , and we find tliat although we had no right to expect it and for reasons w hi c h will become clear later , F ay al so vani she s here but F a a does not . These are p r eci sely the conditions for t h e "winged cusp" singularity , which Golubit sky and Keyfitz found to be the organizing center for the s tirred tank with a sin gle exothermic reaction and thi s ac coun t s for the similarity of behavior noted by Gray and Sco tt ( 1 98 4 ) . The 3 m canonical form of the winged cusp is x ± ;:�. 2 and Golbuit sky ' s theore s imply that there is a contact transformation th at will put it in this form . He al so gives the universal unfolding as x 3 + ( a2 >. + o:3 ) x + o: 1 + >. 2 • This canonical form allows us to d es c ribe every possible qualitative shape of the x- >. c urve This is shown in Fig. 3 7 , where in the c en te r a surface in a 1 - a 2 - a 3 space has been drawn . It really con sists of two parts : the so- calle d h y s t e resi s v ariet y , H , which is like a trough lyin g on its side and pinched into a line , OA , the a 1 axis in the plane a 3 = 0 ; an d the b ifu rca tion surface , B which is like a b re akin g wave with a cusp line COD . It is clear that 0 , the or ganizin g center , is a v e ry singular point , for the hysteresis trough is pinched down to a cusp , w hile the cu sp o f the -
=
=
.
=
1 04 5
Reac to r S teady -S tat e Multtplicit:y /S tability X
X
2
FIGURE 3 7
7
3
6
Q ualit ative bifurcation diagr am s for a win ged-cusp
sin gularity .
bifurcation w ave p asse s through it. T h e seven regions into which the se surface s divi de the a. 1 - a. 2 - a. 3 ) space corre spond to the seven types of bi furcation diagram s in the x- A p l an e T he hy stere si s variety is the locu s of triples ( a. 1 , a. 2 , a 3 ) for whic h the x - A diagram has a poi nt of in flection with vertic al t an gent , as in the 2 / 7 and 6 / 3 diagram s at the bottom . The bifurcation variety is the locu s of ( a 1 , a 2 , et 3 ) p oints for which a third solution is just be in g born by the vanishin g of the im aginary part of a pair of complex root s . T his is represented by the dot in the 7 / 6 ' diagram . The p oin t E rep re s en t s a p oint at which both the 2 / 7 and the 7 / 6 transitions take place at once ( i . e . , the dot is vertically above t he poi nt of vertical inflection ) ; the locus of p oint s li ke E might be d e no t ed by 2 / 7 / 6 . Noti ce t h at d ia gram s w ith one number, suc h as 2 , 7 , or 6 , b elon g to a three dimensional region of et sp ac e ; those like 2 / 7 , 7 /6 are two- dimensional sur face s , while those like 2 / 7 / 6 are one - dim en sional lin e s . T he or gani zin g center , 0 , i s a point . T he strength of the th eo ry i s that it exhau stively catalo gs the v arious bifurcation di agr am s that can arise . It is s y stem ati c and Golubit sky and K eyfit z ( 1 980 , pp . 3 26- 3 2 7 ) give a u seful t able of conditions for fi ndi ng .
Mo rb idelli ,
1 046
Varm a ,
and A rts
the variou s organizin g centers . Its weakne s s is that it demands consider able al gebraic labor in most cases , and an excessive amount if the contact tran sformation that leads to the canonical form is sought . In the case of Gray' s autocatalytic reaction there are only two addition al p aramet ers , for a plays the role of ;1. and this leaves only S and y to be related to a 1 , a 2 , a 3 . T his implies that the system moves in a two dimensional sub sp ace of a 1 - a 2 - a 3 space and it could be that one or more of the sev en possible diagram s is ab sent . Gray and S cott ( 1 984) give mush roomlike curve s such as 3 in the figure , an d these seem to be the most complicated form their system can attain , for there is an inherent degeneracy of the h ysteresis variety that does not allow one side of the mushroom to straighten out without the other following suit . Thus there is never the breaking wave of 4 , 5 , 6 , or 7 . A n attempt has been m ade to u se three p aram eters by takin g the order of the autocatalytic reaction as a further p arameter , but aside from the difficulty of re gardin g it a s continuous , this does not seem t o avoid t h e de generacy . In contrast to the autocatalytic schem e , which has two param eters for its p artial unfolding , the stirred tank with the irreversible , first - order reaction has a universal unfoldin g th rough the dim ensionle ss parameters representin g the ad iab atic temperature rise , coolant tem perature , and heat tran sfer coefficient . T his is shown by B alakotaiah an d Luss ( 1 984) , who give a diagram of the isola and hy steresis ·varieties in the neighborhood of the organizing center .
that
NOTA T I O N
a
external heat tran sfer area
A
cross- sectional area of the tubular reactor
�
matrix ,
�
-1
�
B
dimen sionless heat transfer coefficient ,
c
concentration
cp
specific heat
D
axial dispersion coefficient of m a s s
�
e Da
cap acitance m atrix defined by
D amkobler number , V ( tubular reactor)
( - li H ) yC 0 / p C p T c
E q . ( 7 3a)
f( C 0 , Tr) /qC 0( C STR)
Da/L E
reaction activation energy
f
reaction rate
�
Jacobian m atrix defined by E q . ( 7 3b )
k
Arrheniu s reaction rate constan t
ke
axial dispersion coefficient of heat
K
reactant ad sorp tion constant
L
reactor len gth
or
Lf(C 0 , T r ) /;c0
1 04 7
Reacto r Steady -State M u l t ip licity /Stab ility
Le n
Lewis number , defined by Eq . ( 6 8) ( C S T R ) or ( 88 ) ( tubular reac tor ) reaction order
p
perim e t er of tubular reactor =
/ k , Pe
= :;L /D
Pe
Peclet number ; P�
q
volumetric flow rate
r
dimensionle s s reaction rate , f ( C , T ) /f( C , T )
R
m
e
r
dimen sionle s s axial coordinate , z /L dimen sionle ss time , qt' /V
t'
time
T
temperature
u
e
0
t
T
p
universal gas con stant
s
T
:;L p C
m
r
reference temperature , defined by
T m , T , or T c 0 dimen sionless concentration , C /C 0
reference temperature :
u
overall heat tran sfer coefficient
v
average fluid velocity
v
X
Eq .
dimen sionle ss temp erature , T /T
( 4)
r
superficial velocity
reactor volume conversion ,
1
-
z
axial coordinate
z
(v - 1)
u
l"i3
G re e k Lette r s Cl.
B
B y 0
oo L'! H
8 X
p
ll H 2 /IlH 1
dimen sionless heat of reaction , ( - ll H ) C / p C T 0
p
O
dimensionle s s heat of reaction , ( - ll H ) C / p C T ( 1 + o ) P m 0 dimensionle s s activation energy , /R T r dimen s�nless heat transfer coefficient , U a /q p C ( C S ± R ) or P UP L /Av p C ( tub ular reactor )
E
0
P
/ -r
enthalpy c h an ge o f reaction
dimensionle s s temperature , (T - T )y / T r r
E 2/E1
k 2 ( T o>
/k 1 CT o >
density
1 048
Morbidelli, Varma , and
cr
A ris
it 0 re actor residence time , V /q ( C S T R ) or L/v (tubular reactor) dime n sionle s s parameter defined by Eq . ( 34)
dimensionles s in h ib ion p arameter , KC
T
<1>
S u bscripts
A
re actant
c
coolin
0
feed
g
AC K N O W LE D G EM E N T
Massimo Morbidelli was the recip ient o f a NATO - C N R Fellowship ( B ando
e
n u m b r 2 1 5 . 1 5 / 3) during the w riting of t his chapter .
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S ta b ilit y condition s for a class of distribu ted - p arameter system s and their app lication s to chemical re action system s , C hern . E n g . Sci . , 24 , 1 4 2 7 ( 1 96 9 ) .
Padm anab h an , L . , R . Y . K .
Y an g , an d L . Lap id u s , On the analy sis of the stab ility of distributed reaction system s by Liap unov ' s direct m ethod , C hern . En g . S ci . , 2 6 , 1 8 5 7 ( 1 9 7 1 ) .
1 053
R eac tor S teady - S t a t e M u l t i p l ici ty /Stab ility
n
Parulekar , S . J . and D . R amkri sh a , T ubular reactor stability revisited without the D an k w ert s boundary condition s , C hern . En g . Sci . , 3 9 ,
c
455 ( 1 984) .
the u
v
S . , A note on q a litat i e theory of lumped parameter sy stem s , C hern . E n g . Sci . , 3 1 , 1 7 0 ( 1 9 76) . P li zo po u lo s , I . an d C . G . T akou di s , On the steady - state solutions of chemically r eac tin g sy stem s : I . T heory and ex am e s , C hern . E n g . S ci . ( 1 98 5 , i n press) . Poore , A . B . , A model eq u at i on ari s in g from chemical reactor theory , A n al . , 52 , 3 5 8 ( 1 97 3) . R aj a d h y ak s h , R . A . , K . Vasudev a , and L . K . D or ais w am y , se s t ty fixed bed reactors , E n g . Sci . , 30 , 1 3 9 9 ( 1 9 7 5 ) . R ay , W . H . and S . T . H astings , of the Lewis number on the dy n am ic s of che m i c al y r act in g sy stem s , Chern . E n g . Sci. , 3 5 , Perel son , A .
o
pl
Arch . Rat. Mech. a n i ivi in
l e
Parametric
Chern . The influence
W. H , A. and A . B . On the dy n a m i behavior of catalytic wires , Chern . E n g . Sci . , 29, 1 3 30 ( 1 974) . R aymond , L . R . and N . R . A m n d so , S om e ob servation s on tubular re actor s abilit y , C an . J . C hern . E n g . , 4 2 , 1 7 3 ( 1 964) . G . B . R og er s , M . M . Lih , and 0 . A . H o u ge n , C ataly tic h y d ro gen atio n of propylene a d i ob y len e over l a ti n u . E ffect of ad s orpt on , A I C hE J . , 1 2 , 3 6 9 ( 1 966) . Schmit z , R . A . st ab ili y and s si tivi y of st a e s in chemi cally rea ti g sy stem s - a review , Adv . C ern . Ser . , 1 4 8 , 1 5 6 ( 1 97 5 ) . Schmit z , R . A . , R . R . B autz , W . H . R ay , A . U p al , behavior of a C S T R : some com parison s of t heor y experiment , A I C hE J . , 2 5 , 2 8 9 ( 1 979) . N . N . , Z ur Theorie des Verbrenn ungsproze s se s , Z . hy s . , 4 8 , 5 7 1 ( 1 928) . m it z , 0 scillations in eac t n s , S heintuch , )\11 . , and R . A . S C atal . Rev . - S ci . E n g . , 1 5 , 1 0 7 ( 1 97 7 ) . S i fel t , J . H . , H y droge noly si s of ethane over supported latin m , J . hy s . C hern . , 6 8 , 3 4 4 ( 1 96 4 ) . Sore sen , J . B . , E . W . G u e rti n , and W . E . Stewart , C om ut atio al models for y l n d ri al cataly st particles , A I C hE J . , 1 9 , 96 9 , 1 2 8 6 ( 19 7 3 ) . Soria Lop e z , A . , H . I . D e L a s a , and J . A . Porras , Parametric sensitivity of a fix d bed catalytic Chern . E n g . Sci . , 3 6 , 3 8 5 ( 1 9 8 1 ) . Svoronos , S . , R . A ri s , an d G . S teph anopoulo s , On the be havio r of two stirred t ank s in C hern . E n g . Sci . , 3 7 , 3 5 7 ( 1 9 8 2 ) . T sotsis , T . T . and R . A . Schmit z , E x ac niq n s s and c rit e ri a for a o s t ve - o rd er Arrhenius reaction in a l u m e d C hern . E n g . Sci . , 34 , 1 3 5 ( 1 9 7 9) . T sotsis , T . T . , A . E . H aderi , and R . A . Schmit z , E x a t uniq uene ss and multiplicity c ri t ri for a c l a s s of l m pe r ea t on s y s t e m s , C hern . Eng . S ci . , 3 7 , 1 2 3 5 ( 1 98 2) . Upp al , A . , W . H . R ay , and A . B . Poore , On the dyn amic behavior of continuou s stirred tank reactors , C hern . E ng . Sci . , 2 9 , 9 6 7 ( 1 9 7 4 ) . Uppal , A . , W . H . R ay , and A . B . oore , The classification of the dynamic behavior of continuou s stirred tank reacto r s - influence of t h e reactor residence time , C hern . E n g . Sci . , 3 1 , 2 0 5 ( 1 9 7 6 ) . C au wenberghe , A . R . , Further n o t e bou n d ary conditions for flow rea t o rs , C hern . E n g . Sci . , 2 1 , 2 0 3 ( 1 9 6 6) . 5 8 9 ( 1 98 0 ) .
R ay ,
Uppal ,
Poore , u n
n s ut , Multiplicity , c n
p t
•
t
i
c
m
noncompetitive t t p The dynamic and P catalytic r io p u p n
en h and
Semenov ,
ch
n
P n
c i
c
e
reactor ,
series , p ii
t u ue e
e a
u
d
ci
multiplicity p system ,
c
P
van
c
on Danckwerts'
Morb idell i ,
1 05 4
Varm a ,
and A ris
van den B o s c h , B . an d D . Luss , U niq u en e s s an d m ultip lici ty criteria for an n- th order chemical reaction , C hern . E n g . S ci . , 32 , 2 0 3 ( 1 9 7 7 ) .
van Heerden , C . , Autothermic p roces ses . Properties and reactor desi gn , In d . E n g . Chern . , 45 , 1 24 2 { 1 95 3 ) . van Welsenaere , R . J . an d G . Froment , P ar am e t ric s e n s itivi t y and runaway in fixed bed catalytic reactors , C hern . E n g . Sci . , 2 5 , 1 5 0 3 ( 1 9 7 0) . V arm a , A . , B oun d s on the s te ad y state concentration and temperature in a tubular reactor , C an . J . Chern . E n g . , 55 , 6 2 9 ( 1 9 7 7 ) . V arm a , A . , On the num ber an d s tab ility of ste ady states of a se qu e nc e of con tinuous- flow st irre d tank reactors , Ind . E n g . C hern . Fundam . , 1 9 , 3 1 6 ( 1 98 0 ) .
V arm a , A . and N . R . Amundson , Global a s ym ptotic stability in distributed
p a r am e t e r system s :
comp arison function approach , Chern . E n g . S c i . , 2 7 , 907 ( 1 9 72a) . Varma , A . an d N . R . Amundson , Som e problem s co n ce r nin g the non adi ab a tic tub ular reactor , C an . J . C hern . E n g . , 50 , 4 7 0 ( 1 97 2b ) . Varm a , A . and N . R . A m u n d so n , Local stability of tu b u l ar reactors , AIChE J . , 1 9 , 3 9 5 ( 1 9 7 3a ) . Varm a , A . and N . R . Am u n d s on , The nonadiabatic tu b ula r r e ac t o r : s t a b ili t y con si d e r ation s , C an . J . C hern . E n g . , 5 1 , 4 5 9 { 1 9 7 3b ) . Varm a , A . and N . R . Am u nd so n , S om e observations of u ni q u e n e s s an d m ultiplicity of s t e a d y states in non - adiabatic c h e m ically re ac t in g sys tem s , C an . J . C hern . En g . , 51 , 2 0 7 ( 1 9 7 3 c ) . Varm a , A . and R . A ris , S tirred pot s and e m p ty tube s , i n C h emical Reactor T h eo ry . A R eview , L . Lapidu s an d N . R . A m u nds on , eds P rentice H all , E n gle w ood C liffs , N . J . ( 1 9 7 7 ) . V ejtasa , S . A . and R . A . Schmit z , An e x p e rim en t al study of steady- state m u l tip lic it y an d s t ab ility in an ad iab ati c stirred t ank , AIChE J . , 1 6 , 410 ( 1 970) . Voltz , S . E . , C . R . M or gan , D . Liederman , and S . M . Jacob , Kin eti c st ud y of carbon monoxide an d propylene oxidation on platinum catalyst s , Ind . E n g . C hern . P rod . Res . Dev . , 1 2 , 2 9 4 ( 1 9 7 3 ) . Wehner , J . F . and R . H . Wilehlm , B o un d ar y conditions for flow reactor , Chern . Eng . S c i . , 6 , 89 ( 1 95 6 ) Wei , J . , The stability of a r e ac ti on wit h i n t r a - p a rt icl e diffusion of m a s s and heat : the Liap unov method s in a m etric fu n c t io n s p ac e , C h e rn . E n g . Sci . , 2 0 , 7 2 9 ( 1 9 6 5 ) . Weis z , P . B . and J . S . H icks , The b e h av io r of p o ro u s cataly st p a rt ic l es in view o f int e rn al m ass and heat diffu sion e ffects , C hern . E n g . S ci . , 1 7, 2 6 5 ( 1 962) . W e s te rte rp , K . R . and K . J . Ptasinski , S afe de si gn of cooled tubular re actors for e x o t hermi c , m ultiple reaction s : parallel reaction s : I . D e v e lop m e n t of criteria , C hern . E n g . S c i . , 3 9 , 2 3 5 ( 1 9 8 4 ) . .
•
•
Index
Absorbance , 2 1 1 Absorbed molecules , 1 5 2 Ab sorbed species , 20 2 Absorption /desorption rates , calculations of , 7 00 Absorption rate , 5 6 5 Accuracy of tracer exp eriments , 86 Activation ener gy for desorption from T D S , 2 2 1 Adiabatic reactors , 3 8 0 Adiabatic perturbations , 1 0 1 7 A d sorbed species o n surfaces , 207- 224 Adsorbing , 1 2 3 trace r , 1 2 8 , 1 3 3 , 1 4 3 Adsorption , 1 2 9 , 9 2 8 , 9 3 9 processes , 12 4 Aerobi c fer ment ations , m ass transfer in , 8 0 3 Aerobi c reactors , 7 8 0 A ge distributions , 7 2 A ggre gative fluidi zation , 4 5 2 Agitation for disper si n g gas i nto liquid s , 645 speed , 5 97 , 6 4 1 A gi t ator diameter , 640
types , 5 97 Airlift columns , 8 1 2 fermentor , desi gn methodology , 825 Alloys , 1 8 6 , 2 2 2
Aluminum , 923 , 9 2 4 Ammoni a , 2 1 4 decomposition , 1 86 s y nt hesis re actor , 3 8 9 Anaerobic reactors , 7 8 0 Analysis of reactors , 3 3 1 Angle re solution , 1 8 6 An gular- dependent S I M S m e as ure ments , 1 8 6
Anionic polymeri zation , 737 , 7 63 Anti - S tokes r am an lines , 2 1 6 Ap eriod i c solutions , 1 0 0 1 Are a speci fi c resist ance , 945 , 94 9 A re a surface , 1 5 1 Arrhe nius number , 9 8 8 A symp totic stability , 1 0 0 4 , 1 0 1 4 Auger e ffect , 1 7 6 - 1 7 7 Auger electron spectrometer , 1 7 8 A u ger electron spectroscopy ( A E S ) , 1 6 9 , 1 7 6- 179 applications , 1 7 9 char gin g , 1 7 8 chemical s hi ft s , 1 7 8 depth p rofilin g , 1 7 8 di sadvantage , 1 7 9 electron kinetic ene r gy , 1 7 7 focused electron beam s , 1 7 7 surface sensitivity , 1 7 7 Auger spectrum , 1 7 7 Autoacceleration , 7 6 5 Autocatalytic reaction , 1 2 0 , 1 0 4 6 Autothermal ope r ation , 3 8 6 Autothermic re actio n s , 1 20 Available electron balance , 78 2 Average cry stallite si ze , 1 5 8- 1 6 8 1 05 5
Index
1 056 Averaging of the steady state
model , 3 4 Axial di sp e rsi o n , 3 6 , 943- 945 , 94 7 coe ffici e nt , 5 8 1 , 595 , 6 0 4 gas phase , 6 8 2 , 6 9 4 liquid phase , 6 8 2 , 6 9 2 solid phase , 694 Axial di spersion model , 3 91 , 5 7 0 , 973 , 993 approxim ate solutions o f , 6 1 4 Axial therm al cond u c ti vi ty , 6 3 3
B ackscattering geometry , 1 97 102 , 145 B alance of a re acting system , 5 5 5 B arrett , Joyner , and H ale nda ( B J H ) method , 1 5 5 B asic models , 2 9 8 with plu g flow , 3 7 7 B atc h , 9 2 3 , 9 2 4 B atteries , 9 2 3 , 92 4 , 93 9 , 940 , 94 8 - 964 Bed voidage , 44 4 B eer - Lamber L aw , 2 1 1 B erty reactor , 1 3 2 , 1 4 4 Bifurcation , 1005 , 10 2 1 diagram , 1 0 0 9 , 1 0 2 4 p aram et e r , 1 0 0 5 , 1 0 2 3 points , 9 76 variety , 1 045 B im etallic clusters , 1 9 1 , 1 9 7 Bimolecular second -order reaction , 561 Binary , 9 4 4 , 9 5 3 , 9 5 4 Binding ener gy , 1 7 1 Biological reactors , desi gn of , 8 1 6 B iom a ss energetic yield , 7 90 Biore actors , types of , 800- 8 0 3 B ip ol ar 9 5 1 Bismuth molybdate , 2 2 3 B lo w - o ff conditions , 3 90 B oiling beds , 4 5 2 B ound ary value problem s , 1 8 B oundin g , 1 4 3 methods for nonlinear reactions , 113 B ragg' s Law , 1 5 9 B r anching , i n ste p growth polymeri zation , 7 4 4 B right field , 1 6 3 B ronsted acid , 2 1 4 B affle s ,
,
B runauer , Emmett , and T eller (B E T ) A d sorption Method , 1 5 1 , 1 5 3 Iotherm , 1 5 3 , 1 5 6 B ubble- cloud , 47 7 B ubble coalescence , 480 columns , 547 , 5 7 4 , 808 colum ns , desi gn of , 8 20 concentration , 46 3 diameter , 4 8 3 , 48 4 , 485 , 5 7 6 dynamics , 689 flow , 574 fre q uency , 471 ph ase , 4 7 7 - 478 , 4 8 1 rise velocity , 4 5 4 , 576 si ze , 4 5 7 , 460 , 461 , 46 2 , 471 velocity , 4 6 5 , 484, 4 8 8 volume , 4 6 3 , 4 8 5 w ake , 47 1 , 490 B ubbles , 1 4 5 B ulk density , 444 B ulk - flow effects , 3 1 5 B us , 950 , 951 , 95 4 , 956 , 958-- 964 B utler - V olmer eq u ation , 93 0 , 943 B u tt e r fly po i nt , 991 B y p as sing , 8 9 B yp as s p he nom e n a , 1 2 8
C alibr ation m ethods , 1 8 1 C ap acitance m atrix , 1 0 0 3 C ap acit y , 9 2 8 , 95 5 , 95 7 , 9 6 4 C ar bon dioxide , 7 97 C arbon monoxide chemi sorption , 1 5 7 , 2 1 4 desorption , 2 2 2 hydrogen re actions , 2 2 4 C arbon reaction , 2 24 C at alyst genesis , 1 97 particles smaller than diffusion film thickness , 70 1 pellet s , reaction and diffusion on , 18 p oiso ni n g , 1 7 9 , 2 0 3 pow d e rs , 4 6 9 surfaces physical char acteri zation , 1 5 1 physicochemical characterization of , 15 1 C at alysts , 5 1 7 distribution inside pellet s , 2 0 7
1 05 7
Index Colum n s bubble ,
( C atalysts ]
mixed
oxide ,
223
oxid ation st ate
o f active
co m -
p o n e nt , 1 7 4 C at alyti c re actor s , 1 2 3 C at alyti c wire s , 1 0 1 1 C at a s t roph e surfaces , 1 0 4 2 C ationic polymeri zation , 7 6 4
Caustic soda , 9 2 3 , 9 2 4 C e ll mode l s , 3 7 6 C hain grow t h polymeri zation ,
752
spray , p artial ,
effe cts ,
Cloud , 4 5 6 - 4 5 9 , 46 1 , 4 7 6 - 4 7 8
C loud - dense p hase i nt er face , 4 7 7 Cloud -to -inter stitial phase e x -
487
184
C o al ,
499 gasification re actors , 4 9 9 C o ale s ce n c e , 4 5 9 , 4 6 2 , 463 , 465 , 4774 4 8 0 , 483 Cocurrent p acked column s , 5 9 2 Coke bu rning , 4 2 Cold - flow mod el s , 1 0 2 Collision cross s ection ,
C olloc atio n , 1 9 hyperbolic , 2 2 one- poi nt , 2 2 spine ,
719
470
50 2
C om p artme n t m od e ls , 53 7
742 ,
C hemic al s hi fts , 201 of o xidation st ate , 1 7 3 C hemisorp tion titr ation t e chni q ues , 1 5 7 u s e of carbon monoxide , 1 5 7 Chlor- alkali , 9 2 3- 9 2 4 Chlorination o f n - dec ane , 634 C hlorine , 9 2 3 , 924 Churn turbulent flow , 57 4 C ir cul atio n rate , 4 6 4
180
Complex transient s ,
1 02 5
calculation
faces and
sur solids , 1 68 - 2 0 3
C o mpo sitio n of a s ur face , 1 5 2
solution theory , 927 ,
Conce ntrated
944
C onc entr atio n
gradie nt s , intrap ar ti cle , 384 overpotential , 9 2 9 , 930 programming ,
224
1 39 , 1 4 6 , 2 2 6 , 9 8 8 , 10 26 Conservation equations for heat and m as s for volume re action and Consecutive re action ,
models , 306
particle -pellet
Co n s t r ai n ts kinetic , 5 1 6 process ,
516
stoichiometric ,
512
ther modynami c ,
C ont act
516
e q uivalen t ,
Cont act p ote nti al ,
991 , 1043
1 70 - 1 71
Contact - ti m e di st ri bution , 1 3 5 ,
1 43
Conti nuous ,
92 3 , 932
approxim ate solutions of ,
6 1 8 , 620 mass b alanc e s for , 572 model , 616 liq uid
phas e ,
D am kohler number ,
D ark field ,
p roce dure ,
136 ,
Continuous - flow stirre d -t ank reac t or , 973 , 974 , 1001 C onti n uo u s phase , 1 43 C o nti n uo usly stirred tank re actor
988 ,
D anckwerts , 63 bo u nd ary conditions ,
20
C olum n height
574
548
Combu stion ,
7 65 effects , 7 6 5 Chain stoppers , 7 44 Channel , 9 2 4 , 925 , 93 1 - 933 flow cells , 9 2 4 , 9 2 5 , 9 3 1 - 9 3 3 Ch aoti c behavior , 1 0 30 C h ar ge , 9 4 9 , 950 , 954 , 955 Char ge transfer , 928 , 940 , 941 C h argin g , 1 7 3 C h em i cal mi croan aly si s , 2 0 3
Cluster ion s ,
548 ,
plate ,
Compo sition and structure o f
heat transfer mass transfer
chan ge ,
547 , 574
cocurrent p acked , 592 countercurrent p ac ked , 5 8 5 p acked , 5 47
163
621
997
101 4 ,
Dead end polymeri zation , 75 7 ,
770
D e b y e len gth ,
928
1018
Index
1 05 8 Definitions of the re si d en ce - ti me
and future lifetime di stribution , 6 4 D e gree o f reductance , 7 8 2 dehydrochlorination , 637 D e h y d ro d e s ulfu ri z ati o n , 20 7 D enbi gh ' s rule , 7 7 1 Dense phase , 4 7 7 , 4 81 , 488 D ense -phase c at a lyti c fluid bed , 1 45 D ense - p hase gas , 4 8 3 D e n si t y function , 64 De p th profiling , 1 8 4 Derivative s , 1 3 9 De s i g n , 9 2 3 , 9 2 5 , 9 2 6 , 9 3 2 , 935 , 936 , 9 4 4 , 948 , 951 , 9 5 8 , 95 9 , 963 , 96 4 o f hetero geneous re ac to r s , 1 4 1 o f re actor internals , 10 2 distrib ution age ,
o f carbon monoxide ,
Desorption
of h ydro ge n , 2 22
22 2
Determination of rate con t ro lli n g step in the hydroxylamine fo rm atio n , 7 1 4 Devolatili zation Di a gno sti c tests
for diffusional limitations , 2 6 4 for temperature gradi en t s ,
283
Diagonali zation of A , 6 924
Di ap hr a gm cell ,
Diatomic molecule : vibrational spectra , 20 8 Di ffer e n ti al s pe ct r a , 1 7 7
Diffuse re flectance spe ctroscopy , (DRS ) , 213 D i ffusion , 9 2 5 - 9 2 7 , 9 4 3 - 945 , 9 47 ,
95 3 , 943 ,
741
251
potential , 9 4 6 and re ac ti o n on c at alys t pellets ,
in liquid - filled por es ,
18
reactor in Zie gle r - N att a polymeri za
768
Dimensionless m aterial fluxes , 570 Disch ar ging , 9 4 9 , 95 0 , 95 3-- 958 ,
961 , 964 Dispersion , 375 , 943- 945 , 947 models , 33 4 D i spe rsive ver sus Fourier transform IR , 212 Distillation columns with chemical reac ti on s , 636
of acti ve cat alyst i n side p e llets , 207 of catalytic m ater i als , 20 3- 2 0 7 o f m ateJ;'i al s , 1 5 2 Distributor , 45 9 , 460 , 4 6 2 , 4 8 6 desi gn , 460
orifice , 4 5 9 Double layer , 9 2 5 , 927- 9 2 9 , 9 3 4 , 9 3 9 , 9 40- 9 4 2 , 956 D o u ble limit variety , 991 D r a g c oe fficie n t , 4 46 , 4 4 9 Drag force , 4 4 3 , 446 , 4 4 9 Dri ft disper sion , 7 7 0 D ri ft p r o file , 4 6 4 Dusty- gas m od e l , 243 D ynamic be h avior , 1001 , 1021 D y nam i c condition , 1 0 0 4
E conomy , 9 2 3
i n slow influx , 7 5 1 i n Zie gler- N att a polymeri zations , 769
tions ,
solutions , 927 syste m s , 926 , 92 7 , 9 4 2 , 943
953
2 40 layer , 9 3 1 , 9 3 2 gaseous ,
limit ations ,
changes in pellet , 32 0 Dilu te , 943
Di s t ri b ution
Detachment , 459
coe fficie nt
[Di ffusion ] surface , 2 47 in zeolites , 2 4 9 Diffusional re sistances , 1 3 2 Diffusivity , 5 5 4 effective , 4 1 2 i n liqui d s , 5 5 3 variations , d ue to structural
Efficiencie s , 949 , 9 5 4 E ffective di ffu si vity , 4 1 2 recommended equations , 3 5 4 E ffe ctive ne ss factor , 253 , 4 1 4 E ffe ctive ther m al cond uc tivit y , 2 5 2 ,
E ffects
396 o f chemical
r eactio n s on m as s t r an s fer , 557 E fficie ncy , 5 1 6
Electri c ve hi cle ,
thermal , 5 1 5
949
1 059
Index
Electrochemical potential , Electrochemical re actor ,
927 923- 925 ,
E q ui librium par tition factor ,
398
936 , 9 3 9 E lectroche mistry ,
9 23- 964 Electrode are a , 936- 938 , 956 , 958960 , 963 , 964
E r gun- type eq uation , E scape probability ,
solubility of gases ,
592
90
E s timati n g transpor t coefficients , recommended equations ,
Electrolyte influence o n the
Ethylene o n silver , Ethylidene ,
213
E lectron microan alysi s of a sample ,
207
220
1 86
E xchange coe fficients ,
479 , 481 , 930 , 934 , 935 , 952 , 953
E xchange curre nt density
E lectron microscope reso lution ,
162
E xtended electron ener gy loss fine
Electron probe microanaly zer
(E PMA ) ,
206
struct ure applications ,
E lectro n spin re son ance ( E S R ) ,
(EXELFS ) ,
Faraday ' s Law ,
E lectrostatic ion ener gy analy zer s ,
Fast bimolecular reaction ,
Element specific , 1 87 Elem ent m ap ,
178
4 4 9 , 47 1 , 47 2 , 474-
476 768 923 9 4 9 , 961 , 963 balances , 5 1 2 '·
dispersive spectrometry ( E D S ) ,
204 vantages of ,
1 8 4 , 187- 192 21 1
E xxon donor solvent reacto r ,
134
942 , 9 4 5 562
Fast ion bombardment ( FAB ) , Fast re action ,
559 re gime , 6 1 3 , 618
Felgett ' s Advantage ,
185
212
1 70
Flash desorp tion spectroscopy ,
220
Film depth ,
557
model , ap proxim ate solutions o f ,
609
dispersive spectroscop y , ad -
205
efficiency ,
949 of mass , 180 of s c attered primary ion vs . m as s ,
structure ,
E xtinction coe fficient ,
Fer mi Level ,
Em ulsion , E nergy ,
E x t e nd e d x - ray absorp tion fine
,
182
189
1 9 1 - 192
198 1 99 E lectroneutrality , 926 , 927 , 942 , 944 Electro -or ganic synthesi s , 9 3 1 , 9 40 E lectrost atic analyzer s , 1 7 3 applications
E lutriation ,
35 6 102
E stim ation o f model p ar ameter s ,
550
E lectron ener gy los s spectroscop y ( eels ) ,
251
E quivale nt model , one - dime nsional ,
180 selection , 1 86 storage , 93 1 , 939 , 940 , 94 7 , 948 E nhancement factor , 5 5 6 , 5 5 8 explicit expressions of , 5 6 3 implicit expre ssions o f , 5 6 2 E n trained flow , 5 25 gasi fier models , 5 3 8 re actor , 50 7 models , 5 3 7 E ntrainment , 471 , 472 , 4 7 4- 476 En zym atic reactor s , 780 Equal re activity ass ump tion , 740 E quations , 9 2 6- 931 , 940- 948
theory ,
569
Fine powders ,
468 , 469 1019 Finite - st age models , 3 3 4 Finite di ffere nce s ,
First - order chemical re actions ,
136,
138 First order differenti al equations ,
7 Fir st -order i sother m al re actions ,
105 Fis cher - T rop sch ,
122
Fixed - bed catalytic re actor s , models for ,
70 8
Fixed - bed re actor s hydrodyn ami c s , mixin g , and transport ch aracteristics of ,
674 models ,
33
unsteady s t ate ,
41
Index
1 060 Fixe d - b ed tubular re actor model s ,
3 75
Fu gacity coefficient ,
Flash desorption spectroscopy ( FD S ) ,
74 , 7 6
Flash hydrogenation ,
1031
Floquet m ultiplier s , Flotsam ,
465 , 466 94 7 , 9 4 8
G alerkin met ho d ,
Galileo number , Gas
Flow distribution ,
143
e xchan ge ,
permeability , recommended equations ,
674 , 688 velocity distribution , 4 0 0 Flow - redox , 9 4 0 , 9 4 8 Flow - t hrou gh , 939 , 940 , 9 4 4 , 9 4 7 , 9 48 Fluid bed , 1 0 2 , 1 2 8 crackin g catalyst , 469 Fluid dynamic inst ability , 1 3 3 liquid velocities required ,
Fluidi zed bed
,
690
145
dry ash pre ssurized , ugas ,
511
dispersion , 5 9 7
Gas -liquid
Gas-liquid
interface t em perature ,
G a s - liquid m a s s t rans fer ,
axial
Gas - p hase
coefficient ,
5 90
G a s - side mass transfer coefficient ,
Fluorescence background ,
Flux , 925 , 9 26 , 9 4 1 - 9 45 Form drag , 4 45 , 4 4 7 , 44 8 Foulin g , 7 7 3
58 0
441
Fluidi zed cat alytic cracking ,
Fluid- solid heat transfe r co efficient , 4 0 3
Gas- solid noncat alytic re actions ,
293
218
Gas - solid re actions basic ste p s , e xam ple s
2 96
o f , 2 94
Gas spar ged versus mechanically
189
Fourier transform , method :
6 8 4 , 69 4 682 ,
di spersion ,
6 94
510
539 reactors , 3 4 3 models ,
Fourier filter ,
5 04
632
358
gasi fi er ,
50 4
moving bed ,
44 3
equatio n s for b asic p ar ameter s ,
agitate d slurry reactors , 78
712
c rystallite si ze ,
1 3 C - NMR , 2 0 2
161
Fraction expo se d or dispersion
,
157
Gaussi an
elimination ,
Gelation ,
5
50 4
GE GAS reactor ,
75 1 753 , 7 5 8
Gel effect ,
Frank - Kamenet skii approxim ation , 984
Gene rali zed modulu s ,
415
Generati n g function
4 8 1 , 4 90 471 , 4 72 Free radical , 753 Frequency , 4 5 9 , 463 Freeboard ,
tran sformation ,
u se
re gion ,
o f b ubble formation , re sponse
techniques ,
7 4 5 , 754
739
Geometric di st ribution , 7 4 1 , 743, 7 4 7
487
9 2 3 , 924 764 Good selectivity , 1 4 6 Grain model , 3 1 0 Gold dist ributio n ,
378
9 2 3 , 9 36 - 9 3 9 , 94 0
501
of,
Geometrie s ,
F riction factor correlations , Fuel cells ,
5 7 3 , 5 7 7 , 60 0 , 682 240 G asificatio n , 50 2 Gasifier , 5 0 4 fluidi zed-bed , 5 1 0 lurgi , 50 8 , 5 3 1 G a seous diffusio n ,
Gasifier , Winkler fluid - bed , 5 1 0
Fluidi zation minimum gas and turbulent ,
477
holdup ,
355
re gime boundarie s ,
F luidi z ation ,
1019 445
464
distribution ,
525
entrained ,
F uel gas ,
751
aver age ,
Fut ure lifetime di st ribution A ( t ) ,
220
Flow - by ,
548
751
Functionality ,
reactors ,
gas -liquid -solid
722
Greek le tters for
1 061
In dex
Grid ,
950
9 5 1 , 956- 9 6 1 , 96 3 ,
,
96 4
Ideal p l u g flow ,
72 9
Implicit inte gration ,
zone , 4 8 1 Group freque ncie s and identi fication of structural group s ,
211
Im plicit methods , I ndex ,
1009
9
Indirect e stimate s of f( t ) and F ( t ) ,
81 Industrial re actor ,
G te nsor , 1 0 0
1. 3 9
l nealistic electron tunneling spect roscopy ( lET S ) , H all - H erault ,
213
Infrared
924
H armonic oscillator ,
208- 20 9
kinetic st udies ,
of ab sorption , 6 3 1
214 20 7 - 2 1 5 applications , 2 1 3- 2 1 5 b ulk structure s , 2 1 4
removal ,
dispersive versus fourier trans-
Hatta number , 5 6 0
spectroscopy ,
Heat
tran s fer ,
773 467 , 46 8 , 5 1 7 , 6 8 7 ,
698
form ,
212
instrument s ,
2 1 1- 2 1 3 211
fluid - solid , 4 0 3
isotopic sub stitution effect on
modified ,
prep ar ation of the w ater ,
coefficients ,
46 6 , 5 8 1 , 6 0 3
dat a , 6 0 4
fre quency ,
35
H e n gy constant ,
549 H eterogeneous , 5 2 9 models , 376
reactors ,
selection rule s , 2 0 9
2 14
768
window s ,
Hi ghly nonline ar ,
I nitiation , 7 4 2 , 7 5 2 , 7 5 9
Instantaneous nu m ber - aver age
763
High - conversion , 7 4 4
degree of polymerization ,
75 9 , 762
123
Instantaneou s re action ,
121
in m ultiphase system s , Homogeneou s ,
134
529
560
I nstantaneous wei ght- aver a ge
de gree of polymerization
isotherm al re actor , model ,
213
Initial bubble size , 4 5 9
H eterogeneou sly cataly zed polymer -
Holdup ,
2 1 2-
studies of adsor bed molecules ,
12 0
izations ,
20 9
213
two - dimensional , 419 polymerization ,
intensities ,
Intact emission , 1 84
106
44
I ntensity function or escape
Hop f bifurcation ,
pro babilit y ,
1005 , 102 1 , 1 0 2 7 Hori zontal baffle s , 10 2 Hot spot s , 3 8 4
I nteractions between products and
Hydrazine and ammonia decomposi
Intercell connectors , 951 ,
tion ,
224
Inter faci al are a ,
Hydrod esulfuri zation catalyst s ,
1 74 , 1 8 2 , 1 9 1 , 1 92 223
218 ,
,
port characteristics of fixed -bed reactors , o f slurry reactors , Hydrogen de sorption , Hydrostatic head ,
1 044
674
687
222
616
Hyperbolic collocation ,
961 563 , 5 7 3 , 578 ,
583 , 5 8 8 , 5 9 1 , 5 9 3 , 8 0 5
402, 412
I nt er facial polymeri zation ,
744 , 7 5 0
Interfacial temper ature gradie nt s ,
124
Hysteresis variety ,
120
Interfacial gradients ,' in plu g- flow model ,
Hydrodynamic , mixing , and trans
H ydrotreater ,
re actants ,
68
22 9 7 9 , 99 1 ,
384 Intern ac s in fluid beds ,
133 I nterphase exchan ge , 4 8 3 I nterp hase gas e x ch ange 442 , 4 78 , 480 , 4 8 4 Inter phase g a s transfer , 4 8 3 ,
Interphase transfer coefficient , I nterstitial flow ,
470
482
Index
1 062
Interstitial gas velocity , 4 5 3 , 4 5 7 Inter stitial phase , 4 7 7 , 4 7 8 , 4 7 9 , 484 Intraparticle concentration gradient s , 384 I ntr ap ar ti cle gradie nt s , i n plugflow m odel , 4 1 2 Invert , 1 3 8 Ion etchin g , 1 8 2 Ion gun , 1 85 Ionic mobility , 9 4 3 Ionic reaction , 7 5 2 Ionic s t re n gth , 5 5 3 I o n neutrali zatio n , 1 8 0 Ion scatterin g sp e c t ro s co py ( I S S ) , 1 79- 1 8 3 app lic atio n s , 1 82- 1 8 3 char ging , 1 8 1 - 1 8 2 instrum ent , 1 8 1 surface composition and calibration , 1 8 1 s u rface s e n sit i vity , 1 8 2 I ron- cobalt alloy cat alyst s , 1 9 7 l solas , 1 0 2 6 variety , 9 7 9 , 9 9 1 , 1 046 I som er shift , 1 9 4 I sotherm al gas - liquid reactor mo d eli n g , 5 6 7 I sothermal models
for fixed bed c at alytic reactors , 70 8 for slurry reactor s , 7 0 9
J acobian m atrix , 1 0 0 3 , 1 0 2 3 J ac q ui n ot ' s ad vantage , 2 1 2 Jetsam , 46 5 , 4 6 6 J u m pin g zone model , 3 1 0
Kelvin E q u ation , 1 5 5 Kinetic constraint s , 5 1 6 Kinetic s , 9 1 5 , 91 6 , 93 2 , 9 3 4 , 9 4 3 , 949 , 95 1 , 95 3 of b io lo gic al processes , 7 8 5 Kinetics - free calculations , 5 2 1 Kirchhoff' s Law , 9 5 7 Knudsen flow , 2 40 Koppers- T ot zek atmospheric reactor , 507
Laboratory e q ui p m en t , 6 4 8 Laboratory models , 6 4 7
Laboratory - scale ab s or b ers , 6 4 6 Langm uir - Hin shelwood kinetics , 981 , 1 00 0 Lapl ace ' s equation Laplace transform , 79 , 81 , 1 3 4 Laquerre polynomials , 7 4 9 Leveque ap proxi m ati on , 9 32 Lewi s acid , 2 1 4
Lewis number , 1 0 0 2 , 1 0 1 0 , 1 0 1 3 , 1024 Lew i s sites , 2 2 0 Liapun gy metho d , 1 0 0 3 , 1 0 1 8
Limiting- current plateau ,
9 3 3- 935 ,
940 . 944 ' 946 Linear , 86 Linear algebraic equations , 5 Lin ear polari zation , 9 5 3 , 9 5 6 Lineari zed - system , 1 003 , 1 0 14 Line scan , 2 0 7 Liquid film model , 5 6 8 ap proxim ate solution o f , 60 5 s olu tio n , 6 0 7 Liquid holdup , 5 8 5 , 5 8 8 , 5 9 3 , 680 Liquid m a ss tran sfer coefficients . 579 Liq uid - p hase axial d i s p er s ion , 68 2 , 69 2 coefficient , 5 9 0 Liq uid phase continuou sly s tir red tank re actor , 6 1 8 , 6 2 0 Liqui d - side m as s t r an s fe r coefficient 591 Liq uid - solid m as s transfer , 6 8 6 , 6 9 6 Lithium - aluminum , iron sulfide , 954 ,
95 6 , 9 5 7
Local age distribution G z , 7 4 Local composition , 1 8 4 Local pow der voidage , 4 6 1 Local st ru c t u re , 1 87 , 1 92 Local s urface composition , 1 5 2 LU d ecompo sition , 6 Lumping app roximation , 997 Lurgi dry ash p re s suri zed gasifier , 504 Lur gi g a si fier , 50 8 , 5 3 1
Macrohomo geneous , 9 40 Macropore s , 1 5 4 Magic angle s p inning , 1 9 9 , 2 0 1 Magnetic ally ordered cryst allite ,
1 96
M a gnetic dipole splittin g ,
1 94
1 063
Index
Magnetic re sonance , 1 98- 203 Maldistribution , 1 3 2 , 1 3 3 in industrial re ac to rs , 8 9 M arkov chain theory , 740 M as s balances for the continuously sti rre d tank reactor model , 572
M as s , ener gy of , 1 8 0 Mass spectrometer , 2 2 2 Mass tr an s fe r , 4 6 8 , 9 2 4 , 925 , 932-
934 , 936 , 939 , 940 , 943 , 949 , 954 i n aerobic fermentations , 80 3 coefficie nt , 4 5 5 , 4 7 8 , 480 , 4 9 2 , 5 6 3 , 8 0 4 , 940 , 9 4 3 , 944 effects of chemical reactions on , 557 gas - liquid , 6 8 4 , 6 94 liq ui d - so li d , 6 8 6 , 6 9 6 p ar amet e r s , 6 3 9 Material balance s , 9 2 6 , 9 2 7 , 9 4 1 , 94 3 , 94 4 Material utili zation effici en cy , 949 , 950 Maximum bubble si ze , 4 6 3 M aximum principle , 99 5 , 1 0 1 7 Mean free path of e lect rons , 1 7 2 Mean li fetime , 5 5 7 Mean - square displacement , 1 9 1 Mean , varia�ce , and m om e nt s of f(t ) Measure m aldist ributions , 1 2 4 Meas ure ment o f Laplace transform from arbitrary input s , 81 Measurem ent of re sidence - time distributions , 76 Mechanically agit ated fermentors , 815
Mechanically agitated re actors , 5 97 Mechanically a git at e d system s , 8 1 5 Medium -B T U , 500 Membrane cell , 9 2 4 Mercury cell , 9 2 4 Mercury porosimetry , 1 5 5 Mesopores , 1 5 4 M etal carbonyl cluster s , 1 91 Metal depositio n proce sses of Mo /A 1 2 o 3 , 2 1 8 Metal-ion removal , 9 2 4 , 9 3 9 Metal processin g , 9 3 1 , 9 3 9 M et al re fining , 9 3 1 , 9 3 9 Met al removal , 939 M eth anatio n , 5 0 0
Methane production , 5 26 Micellar catalysis , 70 0 Microbial growth , 7 8 1 Microp or e s , 1 5 4 Microre actor , 1 3 2 , 1 43 Migration , 925- 92 7 , 943 , 944 , 95 3 Minimum fluidization value , 4 7 1 Minimum flui di zation velocity , 443 , 448 ,
465
Minimum gas an d liquid velocities req uired for fluidi zation , 6 90 Mixed -oxide catalysts , 2 2 3 Mixed oxides , 1 8 2 Mixed reactor , 1 3 3
Mixing , 4 6 6 index , 46 5 proce sse s , 1 3 8 unit s , 4 8 6 Mobility , 9 4 3 Modelin g , 8 9 , 9 4 0 , 94 7 of isothermal gas -liquid reactors , 567
Models compart ment , 5 3 7 entrained flow g a sifi e r , 5 3 8 entrained flow reactor , 5 37 for fixed -bed cat aly ti c reactors , 708 fluidi zed -bed , 5 3 9 multi compartment ,
538
two - dimensional , 5 3 8 Model validation , 5 2 8 Modified heat t ransfer coeffi cie nt , 35
Modulus , gener ali zed , 4 1 5 T hi ele ,
254 ,
414
Molecular solids ,
185
Moment closure approximation , 7 5 0 Mo ment of detachment , 4 6 0 Moment generating , property of the generati ng transform ation , 758 Moment s of re sidence -time distribution , 8 1 Monod model , 7 86 Monolith converter , 424 Mossbauer applications , 1 97 experiments , 1 96 source mode , 1 96 transmis sion mo de , 1 9 6 sp e ct roscop y , 1 9 2 - 1 9 8 surface sensitivity , 197
Index
1 064
Mos t p ro bab le distribution , 7 4 3 ,
747 ,
771
Motionles s mi xer s ,
813 M ovin g bed g asi fi er , 5 0 4 models , 5 3 2 Movin g b e d re acto r s , 341 , 5 2 1 models , 5 2 9 Multib e d adiabatic reactors , 3 8 0 M ultico mp ar tm ent models , 5 3 8 M u lti ph ase system , 7 6 , 86 , 1 2 0 , 128
M ultiple re actions , 9 3 3 - 9 3 5 Multiple steady state s , 3 9 3 Multiple tracer e xp e ri ment s , 12 2 Multiplicity p attern , 9 7 7 , 998 Multiplicity , s t eady- st at e , 2 7 4 Multist age system , 7 4 Multitubular re actors , 3 8 4 , 1 0 4 1 Multi value methods , 1 1
Nearest - nei ghbor di st ance s , 1 9 1 N e ar p l u g flow , 7 2 Network o f stirred tanks , 1 1 2 Neutron scat t eri n g spe ctro s co py (NSS) ,
213
Newton ' s method , 6 Nonad sorbing t r acer , 1 3 3 No nc atalyti c re actions , gas - so li d ,
Newton' s re gi me ,
448
293
Nonisothermal effects , 3 1 7 Nonisotherm al gas - liquid reactor , 630 , 6 3 2 Nonisotherm al models for fixe d - bed catalytic re actors , for slurry re ac to r s ,
708
N onli n e ar step gro wth polym eri za tion , 7 5 1 112
Nonline ar system s , 6 , 6 8 Notation for g as - liq uid - solid 721
Not stationary , 1 2 4 N uc le ar hy p e r fine interactio n s ,
Nuclear m agnetic re so nance (NMR ) , 194
198 ,
199
plati n u m 1 9 5 e xp e ri m ent s ( 1 9 5 p t ) 202 shieldin g effect s , o f solids ,
20 1
spi n echo te chnique s , 2 0 2 o f re o lite s , 2 0 1
Nucleation model , 3 1 4 Numerical instability , 9 Numerical methods , 1 0 19
O hmic pote nti al drop , 9 2 4- 9 2 6 , 9 2 9- 9 3 2 , 9 3 5 , 9 3 6 , 9 3 9 , 946 , 9 4 9 , 9 5 1 - 95 4 , 959 Ohm's Law , 9 4 3 , 946 One - dimensional di ffusion model , 72 O ne - di men sio n al m od els , fixe d - bed tubular re actor s , 3 7 5 One-point co llocation , 2 2 O n - line bioreactor monitori n g , On- line cont ro l , 7 7 1 Optimal re actor co ncept s , 1 0 9 Optimi z ation , 7 3 5 , 7 6 9 , 935- 9 3 8 , 940 ' 9 6 3 - 964 . 9 5 6 ' 9 6 1 , 962 O r gani zin g center , 9 91 , 1 044 O r thogo n al collocation , 998 , 1 01 9 O stergaard and Mi ch el sen ( 1 9 6 9 ) , 88 O utlet react ant concentration , 4 8 1 Overall m as s t ransfer coefficient Kgi , 5 5 6 Overall rate of exc han ge , 4 7 8 Overall transfer coefficient s , 47 9 , 480
Oxidation , 50 0 Oxidation - red uction behavior ,
197 223 of active co m po ne nt of catalyst , 174
O xidation st ate , 1 5 2 ,
193 ,
194 ,
Oxide analysis b y secondary ion 710
Nonisothermal reactors ,
reactors ,
[ N u cle ar magne ti c re sonan c e ]
201
mass s pe ctroscopy , 1 8 6 Oxy gen ab s orpti on into sodium sulfite , 5 6 6
Packed - bed , 1 2 4 re actors , 3 3 3 and trickle -bed re acto rs , 9 1 Packed columns , 5 4 7 , 5 8 4 Packing c har act er is ti c s , 5 8 6 Packin g , d ry are a o f , 5 8 8 P and R branches , 2 1 0 Parallel electrodes , 9 2 4 , 9 3 1 - 9 3 3 Paralle l plates , 9 3 2 Parallel reactions , 2 6 5 , 9 8 8 Parameter e s ti m at io n , 360
1 065
Index Pore size ,
Parameter fittin g , 4 5 3 8 6 , 40 9 ,
Parametric sensitivity ,
1 54
distribution ,
151
macropore s ,
1033 Parti al combu stion , Particle density ,
mesopores ,
502
structural chan ge s in pellet , 320
conservation equations for heat 30 7
Porous electrode ,
167
Particle si ze distribution ,
Post s ,
Particle terminal fall velocity , Particulate fluidi zation ,
450 ,
491
949 ,
Pote ntial ,
471
221
Peclet number ,
995
95 1 ,
925 ,
963 ,
Penetration model ,
55 7
consumption ,
Penetration theory ,
478 ,
480
60 0
Prob abilit y theory ,
26
213
106
Product distrib ution , 1 36 ,
640
equation ,
6 90
189
378
Primary distribution , 1 32
171
490
Physical characteri zation of
reactors ,
Physicochemical character! z ation of
932
Probe molecule nitric o xi de , 1 9 9 Process constraints , 5 1 6 P roc es se s performed i n gas - liquid Prim - mover power ,
Photoelectron cross section , 1 75
Phthalic anhydride ,
641
546
catalyst surfaces ,
151
Product ener getic yield , 7 9 2 Product formation ,
788
catalyst surfaces ,
151
Product inhibition ,
789
Plate area ,
172
956 ,
Plate columns , Plu g flow ,
Propagation , 7 4 2 , 752 , 7 5 9 9 5 8- 960 ,
548 ,
96 3 ,
964
574
Pro ximit y ,
375
two - dimensional ,
w i t h interfacial gradient s , 4 0 2 wit h int erfacial and intrap article 41 2
re actor
Pulse experiment s ,
Pul se te ch niq ues ,
Pyridine ,
617,
120
77 157
21 4
622
Poincare bifurcatio n ,
1030
Poi sson di stribution ,
76 4 , 7 7 1
Polarizability ,
396
Pseudolinear system s ,
model with t he bimolecular reactio n ,
184
Pseudohomogeneous model , 376 basic , with plu g flow , 377
model , 9 7 3
gradient s ,
957 ,
Quadrupole splitting ,
1 94
Quantit ative an aly sis ,
179
temper ature dependent ,
216
Polari zation p ar ameters ,
958 ,
194
961 Polyatomic molecules : 210
Polymer di ffusion , 7 6 6
Polydispersity , 7 5 9
144
Pressure gradient , 5 7 1
Phenomenological equation , P hotoelect ron spectrum ,
139 ,
P re ssure drop , 6 7 8
548
Phase holdup , 6 80 ,
5 99
Pre ssed salt method ,
1005 , 1 0 2 3
Perturbation analysis ,
Plasmon line ,
936 ,
number , 988
Prater
1030
Pfaudler agitator scale , Phase shi ft s ,
929- 932 ,
964
absorbed ,
Phase equilibria ,
9 64
926
953
Periodic solution ,
926 ,
-
9 6 1 - 964
Power , 9 2 4 , 925- 9 3 6 , 9 3 8 , 9 4 9 , 9 6 1
Penetration dept h ,
Period doubling ,
924 , 9 3 1 , 93 9 954 ,
938 , 940 , 946 theory ,
Par titio ni ng , 1 5 Peak m aximum ,
1 54
Porosity variations , due to
Particle -pellet model , 3 1 0 and mass ,
154
micropores ,
444 46 4
Particle mixin g ,
154
vibrations ,
R adial structure function , 1 8 9 21 9
R am an adsorbed molecules , Raman app licat io n s , 2 1 8 - 2 2 0
In dex
1 066
Ram an electrochemical environment , 220
Raman selection r ules , 2 1 6 Raman spectrometer , 2 1 6 - 2 1 8 R am an spectroscopy , 2 1 5 - 2 2 0 R andom pore model , 3 2 5 Rate of absorption , 5 6 5 R ate of coalescence , 461 Rate controlling step , determination of in hydroxylamine formation , 714
R ayleigh scattering , 2 1 6 Reaction and diffusion on catalyst pellets , 18 factor , 5 5 8 o f hydrogen and am monium nitrate on t he c atalyst surface , 7 17 rates , 5 2 9 unit s , 4 8 6 Reactions consecutive , 2 6 6 parallel , 2 6 5 scheme of consecutive reactions , 6 05
simultaneous , 2 6 9 solid- gas , 5 2 9 with volum e chan ge , 3 1 5 Reactor ammonia- synthesis , 3 8 9 choice , 6 3 8 confi guration , 736 , 77 1 , 9 2 4 , 947 , 948 ,
[ Reactors ] coal gasification , 4 9 9 enzymatic , 78 0 mech anically agit ated , 5 9 7 " Recoil- free" emis sion and absorp tion , 1 93 Recombination , 1 8 4 Recovery , 9 3 9 Recursive approach , 7 4 0 Redox energy stor age , 94 7 , 9 5 0 Reductan ce , d e gre e of, 7 8 2 Reference electrodes , 9 2 8- 9 3 0 , 934 , 95 4 ,
956
References for gas - liquid - solid reacto rs , 7 2 3 Reflectance techniques , 2 1 3 Re gular solutions , 5 4 9 Relative concentration , 1 8 1 Relaxatio n ti me s , 2 0 1 Reliability of the measurement , 8 7 Reptation , 7 6 7 Residence -time distributions ( RTD ) , 63 ,
567
Residual enthalpy , 1 0 1 7 Reversibility , 7 4 4 , 748 Review s , 9 2 4 , 939 Reynolds number , 493 Riser , 1 4 6 Rotational fine structure , 2 1 0 Runaw ay , 3 8 4 , 3 8 5 , 1 0 3 3 Runge - Kutta method s , 8 Rutherford scattering , 1 7 9
950
control , 7 6 7 design and scaleup , 1 4 3 dynamics , 1 0 0 1 entrained flow , 5 0 7 fixed - b ed tubular models , 3 7 5 GEGAS , 5 0 4 geometries , 9 2 4 Kopper s - Totzek atmospheric , 5 0 7 modeling , 9 6 , 5 2 8 models , 5 2 9 movin g - bed , 5 2 9 moving bed , 5 2 1 selection , 7 1 0 T exaco pressuri zed gasification , 508
Westinghouse fluid -bed , 5 1 1 Reactors , 6 8 7 aerobic , 7 8 0 anaerobic , 7 8 0 classification : exam ples of indus trial importance , 6 6 7
S alting coefficient s , 5 5 1 - 5 5 2 S ame fluid bed , 1 2 9 S am ple heating , 2 1 6 S ample prep aration , 2 1 3 S atellite lines , 1 7 2 S auter me an diameter , 5 6 4 Scale up , 1 2 4 , 1 2 9 , 1 33 , 6 3 8 , 64 6 , 925 ,
9 40 ,
954- 956 ,
948 , 961 ,
949, 962 .
'
951 , 964
considerations , 6 8 7 , 70 0 Scanning auger , 1 7 8 microprobe (S � ) . 1 78 S c anning electron microscope ( SEM ) , 206
applications , 2 0 7 S cattering , 2 1 1 , 2 15 Scheme of consecutive reactions , 605 S cherrer Formula , 1 6 0 Schmidt number , 4 9 2
Index
1 06 7
S econdary curre nt distribution , 9 3 2
S e con d ary ion mass spectrometry a c
h ydr xyl ami e
pp li c atio ns , h ar gin g , 1 8 5
465 ,
71 3 184
466
ch emis o rption , 1 5 5 of g as e s on metals , 1 5 6 S e mi flow batch m od el , 6 3 0 solutions of, 6 25 S elective
S hi n nar Side
et
950 ,
74
( 1 97 3 ) , 1 0 6
al .
re actio n ,
9 3 3- 9 3 5 , 9 4 0 , 9 4 6 ,
951
industrial u nit s , 6 4 6 S im ult aneo u s reactions , 2 6 9 Si n gle bubbles in isolation , 4 5 3 S in gle crystal studies , 1 7 9 Sin gle - p arti cle st udies , 2 96 Si n gle - po re model , 3 2 5 S i n gle tracer , 1 2 2 S i n gu larity theory , 9 7 7 , 988 , 1 0 4 1 Simulation o f
Sintering , 3 2 9 Sinusoidal inputs , 78 Si ze of b ub b le at d et ach me nt , Skin friction , 445 , 4 4 7 , 4 4 8 Slope condition , 1004 Slow influx , 750 Slow re action re gim e ,
Slug flow , 5 7 4 Slurry reactors
612 ,
hydrodynamic , mixtng , t
45 9
615
models for ,
709
and reactive solids , 7 0 3
Solid p articles insoluble ,
on s ,
and
154
529
i n the
medium
705
s p arin gly , 703 Solid phase axial di s p er si o n , S olubilit y of gases in w ater ,
6 94 550
549
Sp ar ged reactor s , 5 7 3 S p e cifi c energy , 9 4 9 , 9 6 1 , 963 , 964 Specific i nt er faci al area , 5 6 4 S pe cifi c power , 949 , 961 , 96 3 , 9 6 4
S pine colloc ation , 2 0 Spray co lum ns , 5 4 8
Stability ,
1001 , 1013
diagr am ,
St age d S t aged
5 75
fluidi zed -bed
re actors ,
r eactor s ,
143
S t agewise nonisother m al
S t a gn ant re gi ons ,
352
models ,
635
72
S t atic bifurcation , 1 0 05 , 1 0 2 1 S t atistical ar gum e nt s , 7 43 S t ati stical dispersion , 7 7 0 Statistical m et hod s , 736 , 7 4 0
S te ady - state
CSTR
(anionic poly-
meri zation in A ) , 7 4 1 S t e ady - s t at e model , avera gi ng of, 34
S t e ad y - st at e multiplicity ,
274 , 634 , 97 3 , 97 4 , 995 S te ad y - st at e s , multiple , 393 S tream gasi fic ation rates , 5 2 9 d ata ,
530
S t e am re for mi n g , 4 1 8 S te p chan ge s i n tracer co nc ent r a tion , 76 Step growth poly m eri z ation , 742 S tirre d tank re acto r s , 546 Stochastic sy st em s , 106 S t oi c hiom e tric constrains , 5 12 Stoke s R am an Line s ,
ran s po rt characteristics of ,
687
Solid - gas re acti
Solubility p ar ameter ,
40 9 Separator , 9 2 4 , 94 0 , 9 4 1 , 9 4 7 , 9 4 9952 , 955 , 9 5 6 , 9 6 1 Series of C S T R s , 984 , 1 0 2 5 S eries of stirred t anks , 7 2 Sh ado wi n g , 1 8 2 " Shake - up" line , 1 7 2 Shallow -bed reactors , 3 4 8 Sharp int e r face model , 2 9 8 Sher woo d number , 4 7 7 , 4 9 2 , 4 9 3 ( 1967) ,
50 1
S ource mode , 1 96
Semi-implicit method , 1 0 S e nsitivity functions , 1 2 S e nsitivity , p ar am e tric , 3 86 ,
Shinnar an d Naor
x-ray sc att e ri n g ,
S m all- an gle
SNG ,
711
m anufacture of
n p hosph ate ,
o
classical dyn amic c alcul at ions , sample p re p ar atio n , 1 8 5 s pe ct romet er , 184 , 185 t he r mo d y n ami c model , 184 S egre g ated , 115 S e gre g ate d flow , 1 15 S e gre gation ,
re ac tor s ] versus fixed-bed reactors ,
w orked example :
1 83- 1 8 7 1 86- 18 7
( S IMS ) ,
[ S lurry
216
Stokes' re gime , 4 4 7 S t ro n gly ad sorbi ng , 12 3 S t ruc t ur al variations , e ffects 319
of ,
Structure sensitive or d em andi n g re actions , 1 5 1
Index
1 068
S ub script s for gas - liquid - solid reactors , 7 23 S ub strate- inhibited enzyme reac tion s , 980 , 1 0 0 0 Substrate inhibition , 7 8 6 S ulfiding of Mo /Al 2() 3 , 2 1 8 S uperp aramagnetism , 1 96 S uperscripts for gas- liquid- solid reactors , 7 2 3 S upported catalysts : T E M / S T EM , 164
S upported m et al s , 1 7 4 , 1 9 1 S upporting electrolyte , 944 , 946 , 953
S upport interactions , 1 97 S urface composition , 1 7 9 diffusion , 2 4 7 enhanced Ram an spectroscopy (SERS ) , 216
enrichment , 1 8 2 overpotenti al , 925 , 9 2 6 , 9 3 0 , 934
re action studie s , 1 8 6 se gre gation , 1 92 structure , 1 8 6 S wieterin g , 1 1 5 S ymmetry , 1 9 3 , 1 9 4 Synchroton radiation , 1 8 7 , 1 8 9 Syngas , 5 0 1
T afel approxim ation , 930 , 934- 936 , T afel kinetics , 9 3 6 Technique s for solving polymeriza tion rate equations , 7 3 8 T emperature gradient s , 2 7 1 interfacial , 384 T emperat ure -pro grammed desorp tion , 2 2 0 - 2 2 4 activation energy for desorption , 952
221
and temperat ure p rogrammed re
action , experiment al method s , 2 2 2 Temperature programmed oxidation , 2 23 - 2 2 4
Temperature program med re action , 2 2 0- 2 2 4
Temperature programmed red·u c tion , 2 2 2 - 2 2 3 Temperature- p ro grammed tech niques , applications of, 2 2 2
T erminal fallin g velocity , 4 7 1 , 4 72 4 93
T ermination , 7 4 2 , 7 5 9 combination , 7 5 3 disproportionation , 7 5 3 T e sting , 8 9 Texaco p re ssuri zed gasification reactor , 50 8 T hermal conductivity , effective , 396
T hermal de sorption spectroscopy , Thermal efficiency , 5 1 5 T hermal re sidence time , 4 2 T herm al runaw ay , 766 Thermodynamic s , 9 2 5 , 9 2 6 , 949 constraint s , 5 16 Thiele modulus , 2 5 4 , 4 1 4 Tim e - conversion relationship s , 3 2 2 Time scale , 1 38 T ortuosity , 4 1 3 T otal surface are a , 1 5 3 T racer experiment s , 6 3 T rade -offs , 925 , 936 Transfer , 7 4 2 , 75 2 , 7 5 9 curre nt , 942 , 94 3 o f hydro gen from bulk liquid to t he catalytic surface , 7 1 5 t o t h e bulk liquid , 7 1 4 unit s , 486 T ransient re sponse , 5 3 6 T ransient studie s , 2 1 4 , 2 1 5 T ran smis sion electron microscopy ( T EM ) , 1 6 3 model catalyst s , 1 6 7 - 1 6 8 T ran smis sion mode , 1 96 T r ansmission vs . re flection and emission 2 1 2 T ransport , 9 2 7 , 943 , 944 disengaging hei ght , 4 7 2 and re adsorption problem s , 2 2 1 T rickle beds , 1 2 4 , 1 3 3 T rickle - flow re gime , 5 8 5 T rommsdorff effect , 7 5 3 , 7 5 8 , 7 6 5 Tubular reactor , 9 7 3 , 99 2 , 1 0 1 3 models , fixed - bed , 3 7 5 T urbulent flow , 1 4 5 T u rb ulent fluidization , 1 45 T wo- dimensional bed , 4 5 3 T wo - dimensional models , 5 3 8 fixed -bed t ubular reactors , 220
,
375
heter o geneous , 4 1 9
1 069
Index
[ T wo-dimensional models ] p seudohomogeneous , 3 96 T wo -p hase theory , 4 5 2 , 4 5 3 , 481- 482 ,
485 ,
45 8 ,
4 90
U gas gasifier , 5 1 1 Ultrahigh vac u um , 1 6 9 Ultraviolet photoelectron spectros copy ( UPS ) , 1 7 0 U nb alanced stoichiometry , i n step growth polymeri zation , 7 4 8 Unsteady- state fixed -bed re actor models , 4 1 Up stream counterelectrode , 945 , 9 4 7 Ultili zation 9 36- 9 3 8 , 950- 9 5 4 ,
Vapori zation factor Ki , 5 4 8 Vibrational- rotational energy , 20 9
Volume chan ge in radical polymerization , 7 6 2 reactions with , 3 1 5 Volume reaction model , 3 0 2 conservation equations for heat and m ass , 3 0 6
Wake , 4 6 3 , 4 6 4 , 4 66 , 4 7 1 , 4 7 6 fraction , 4 5 5 of particle s , 454 volume , 4 5 5 , 485 Waste streams , 92 4 , 93 9 Water- gas shift , 5 0 0 W avelen gth dispersive spectrometry , (WD S ) , 2 0 4 Wavelength dispersive spectroscopy , advantages of, 205
Westinghouse fluid - bed re actor , 5 1 1 Wilke- C hange equation , 5 5 3 " Winged cusp" singularity , 1 0 4 4 Winkler fluid - bed gasifier , 510 Worked example : manufacture of hydroxylamine phosphate in a slurry reactor , 7 1 3 Work function , 1 7 0 , 1 8 5 X - ray adsorption cross section , 1 88 diffraction , 1 5 8 line broadening , 1 6 0 diffractometer , 1 6 0 emitted , 2 0 3 extended absorption fine struct ure , 1 87
map , 2 0 7 photoelectron spectrometers , 1 7 3 photoelectron spectroscopy , 1 6 9 , 1 7 0 - 176
applications , 1 74 - 176 cat alyst : 2% Pd /Na- ZSM - 5 , 1 7 5 catalytic activity , 176 chemical shifts , 1 7 2- 1 7 3 contaminants and p romotons , 176
qualitative analysis , 1 7 4 surface sensitivity , 1 7 2
Zeolites , 1 91 Ziegler -Natta polymeri zations , 768
Zone models , 30 8 Zinietering' s method , 1 4 3 Z yi rin and S hinnar ( 1 97 5 ) , 7 4
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This extr aordinary volume provides comptehensive analyses of dive rse aspec ts of chemical reaction
••
and reactor engineering, offering expert insights and ,experiences from recognized authorities in the field.
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Chemical R eoction and Ret�ctor Engineering presents mathematical modeling and numerical methods essential to the engineer . . . elucidates the residence-time distribution concept . . . focuses on catalyt ic property characterization . . . p1edicts coupling impacts of chemical and transport rate processes on the activity and selectivity of porous catalytic pe11ets, . . . cites examples of important indu strial gas solid catalytic and noncatalytic feactions . .
�
I
examines fixed-bed gas-soUd catalytic reactor design as
pects ., . .. di�usses proposed and existing �coaJ gasification reacto-r configu rations . . . gives examples of important gas-liquid and gas-liqu id-solid reactor processes . .. ., ,contains nume_rro us tables and iJ1us trations and over
1 300 bibliographic
citations to primary sources . . . and much more,.
Providing an incisive view of tiLe current status of chemica] reaction. and reactor engineering, this time-saving, easy-access sourcebook is, essential reading fo r chemical, petroleum, plant, and design engineeFs, and sewes as an excellent reference fo r chemical reaetion engineering courses.
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JAMES J . C A R B E R R Y is Professor of Chemical Engineering at the University of Notre Dame, where he has taught since
1 96 1 .
Previously he was Senior Research Engineer, du Pont Experimental Station,
Wilmington, Delaware . The author of the text Chemical and Catalytic Reaction Engineering and 'CO ,editor of the journal Catalysis Reviews : Science and Engineering (Marcel Dekker Inc.) Dr. Carberry., among numerous distinctions has been Churchill Fellow and NSF Senior Scholar at Camb ridge (U.K.), and Fulbright Senior Scholar (Italy). He is an elected member of the Advisory Council of the Chemi cal Engineering Depart ment at Princeton University, and was recipient of the Yale Engineering Asso ciation Award for Advancement o f Puce and Applied Seience and the American Institute of Chemical Engineers' R . H. Wilhelm Award in Chemical Reaction Engineering. Dr., Carberry received the B.S.
( 1 950) and M .S. ( 1 95 J ) degrees in chemical engineering from the Ph.D. ( 1 957) degree i n chemical engineering from Yale University.
University of Notre Dame,, and
A R VlND VA R M A ts Professor and Department Chairman of Chemical Engineering at the University of Notre Dam e , where he has been afftliated since l 97 S . Prior to this be was Senior Research Engineer with Union Carbide Corpofation. He has, been Visiting Professor at the University of Wisconsin-Madi son and Chevron Visiting, Professor at the ,california Institute of T�echnology. Dr. Varma has published over
75
rese_arch papers and is a member o f the editorial board o f catalysis Reviews: Science and En
gineering (Marcel Dekker, Inc.). He received the B.S. M .S.
( 1 968)
( 1 966}
degree from Panjab U,niversity�, India,,
degree from the University of New Brunswick,, Canada and Ph.D.
the University of M innesota all in chemical engineering.
Printed in the United States
of America
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( 1 972)
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