11
SERIES IN THEORETICAL AND APPLIED MECHANICS Edited by RKTHsieh
SERIES IN THEORETICAL AND APPLIED MECHANICS Editor: R. K. T. Hsieh Published Volume 1: Nonlinear Electromechanical Effects and Applications by G. A. Maugin Volume 2: Lattice Dynamical Foundations of Continuum Theories by A. Askar Volume 3: Heat and Mass Transfer in MHD Flows by E. Blums, Yu. Mikhailov, and R. Ozols Volume 5: Inelastic Mesomechanics by V. Kafka Volume 9: Aspects of Non-Equilibrium Thermodynamics by W. Muschik Forthcoming Volume 4: Mechanics of Continuous Media by L. Sedov Volume 6: Design Technology of Fusion Reactors edited by M. Akiyama Volume 8: Mechanics of Porous and Fractured Media by V. N. Nikolaevskij Volume 10: Fragments of the Theory of Anisotropic Shells by S. A. Ambartsumian Volume 12: Inhomogeneous Waves in Solids and Fluids by G. Caviglia and A. Morro
Diffusion Processes During Drying of Solids
K. PL Shukla
World Scientific Singapore • New Jersey • Hong Kong
Author K. N. Shukla Vikram Sarabhi Space Centre Trivandrum 695 022, India Series Editor-in-Chief R. K. T. Hsieh Department of Mechanics, Royal Institute of Technology S-10044 Stockholm, Sweden
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: 687 HaitweU Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication Data is available. DIFFUSION PROCESSES DURING DRYING OF SOLIDS Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without permission from the Publisher. ISSN 0218-0235 ISBN 981-02-0278-4
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
V
Preface Modelling heat and mass transfer in porous media is an area of great
importance.
transfer
in
moisture
transfer
the
moisture
transfer. the
temperature
body.
The
itself
and
gradient
mechanism
is
drives
the
complex
is
a coupled
process of
that
heat
Also the coefficients of heat and mass diffusion and
moisture
content
making
the
moisture
because
a l t e r s the temperature gradient
hence it
temperature
nonlinear. of
The
the
the
drives
and
mass
v a r i e s with
process
highly
An attempt has been made to develop the basic equations
heat and
moisture transfer
in porous body with reference to the
drying of m a t e r i a l . The monograph begins with a brief comment on the laws of
the mutually
boundary
value
obtained
in
connected
problems
Chapter
analytical e x p r e s s i o n s phase
and
chemical
for
transfer
phenomena. Solutions of
axisymmetric third
and
chapter
spherical
2.
The
for
heat and moisture transfer
transformations
fundamental
in
is
spherical
the
cases
are
to
the
devoted
in presence of
body.
Chapter
k
considers the intensive drying of an infinite p l a t e . Besides molecular transfer,
the
conjugate
process
of
filtration
problem of interacting
reference
to
freeze
drying
is
also
included.
Finally
a
porous solid with a fluid stream in
is
analysed
in
Chapter
5.
A
short
description of the integral transforms is provided in the a p p e n d i x . The whole analysis i s presented in the dimensionless form with the
help
Integral
of
dimensionless
transform
variables
technique
is
the
and basic
the tool
similarity for
numbers.
the solutions of
the boundary value problems. The monograph scientists
of
is designed
applied
for the graduate students,
mathematics
and
engineering
research
sciences and
the
practising engineers in m a t e r i a l s , energy and s p a c e . The
material
from the a u t h o r s '
of
the
present
monograph
is
developed
r e s e a r c h e s on Heat and Mass Diffusion
mainly
c a r r i e d out
vi in Banaras Hindu University Professor
R.N.Pandey,
University I
thank
for
for
calculations Institute
of
reading
suggestions, in Chapter
my 5.
Technology,
suggestions
on
the
R.S.Pandey
for
typing
final
and
draft
Director,
Dr. V.Swaminathan, carefully
constructive
of
I am indebted
Technology,
Banaras
introducing me the subject and guiding the
Dr.S.C.Gupta,
publication, VSSC
during mid s e v e n t i e s .
Institute
Head, the
the first
World
and of
the
the
draft,
Scientific
Richard reviewer
text. Shri
I
for
some Royal
for
also Co.
offering
Hsieh,
valuable
thank
T.Thankappan
Publishing
its
Group,
and
M.J.Chacko
Professor
for
Dynamics
manuscript Mr.
research.
permission
Aero-Flight
colleague
Sweden
kind
entire
I thank
presentation
the
VSSC for
to
Hindu
Pvt.
Shri
Nair
for
Ltd.
for
publishing the book. The utmost care has been taken in checking the calculations
but it is
quite
possible that
some of them might
have
gone unnoticed. I e x p r e s s my gratitude to the r e a d e r s in advance for all suggestions for further improvement of the monograph.
K.N.Shukla
I
vii
Contents
Chapter
Chapter
1
Phenomenological
Laws of D i f f u s i o n
1.1
Phenomenological
Laws
1.2
Transfer
1.3
I n i t i a l and Boundary
I.*
Dimensionless Quantities
18
References
25
2
Integral
1 I
of Heat and M o i s t u r e i n Porous B o d i e s
Equation
Conditions
Approach
17
to Heat and
Mass
Transfer
Problems
28
2.1
The I n f i n i t e
Circular
2.2
Solution for
a Sphere
Cylinder
31 37
References Chapter
45
3
Heat and Mass T r a n s f e r
3.1
Statement
with
Chemical Transformations
of t h e P r o b l e m
S o l u t i o n of t h e P r o b l e m
51
3.3
A n a l y s i s of
66
Analysis
t h e Result
of t h e S o l u t i o n
75
References
Chapter
Appendix
46 47
3.2
3.U
Chapter
6
90
<*
Heat and Mass T r a n s f e r
during Intensive
Drying
Statement of
4.2
Solution of t h e Problem
4.3
Analysis of t h e Result
118
References
127
the Problem
5
Heat and Mass T r a n s f e r
91 92
95
d u r i n g Freeze D r y i n g
128
5.1
Mathematical Model
129
5.2
S o l u t i o n of t h e P r o b l e m
132
5.3
D e t e r m i n a t i o n of t h e C h a r a c t e r i s t i c
5.1*
Asymptotic References
145
1
The Integral Transforms
146
AI
Laplace Transform
146
A2
Finite
150
A3
Solution
Fourier
Sine T r a n s f o r m
Roots
140 142
F i n i t e Hankel Transform
151
References
157
N omenclature
15 8
1
Chapter 1 PHENOMENOLOGICAL LAWS OF DIFFUSION
We shall begin our study with brief comments on the fundamental principles of the mutually connected transfer pehnomena to be followed by a short review of the work done in the field. I. I
Phenomenological Laws When two points of a system are at different
potential flow
difference
of entity
in the system is established
from
the
point of
higher
potentials, a
which causes
potential
lower p o t e n t i a l . The entity may be e i t h e r of e l e c t r i c i t y matter.
The
experimental
researches
the
towards that of
describing
or heat or
the
transfer
processes of these entities have been developed over the years and it
has
been
analytically
shown
that
all
of
these
transfer
phenomena
are
d e s c r i b a b l e by the law of transfer of energy, v i z . the
specific flux of the energy is d i r e c t l y
proportional to the gradient
of the energy. For example, F o u r i e r ' s law of heat conduction in an isotropic the
medium
gradient
conduction
the
states
proportional diffusion
s t a t e s that
of
thermal
that
the
heat flux electric
in a binary
concentration.
proportionality
directly
Ohm's flux
to the gradient of e l e c t r i c a l
proportional
law
for
(current)
potential.
to
electrical
is
Fick's
directly law for
mixture s t a t e s that the concentration flux of
e i t h e r of the components is d i r e c t l y the
is
potential.
Many
between
more
the
proportional to the gradient of
examples
fluxes
and
law
of
the potential gradients
of
the
linear
can
be given. Further above linear at
low
developments
law describing
intensity.
When
two
in the or
the
theory
have
process of transfer more
irreversible
shown is
that valid
only
processes
take
place simultaneously in a thermodynamic system, they may with each other and produce a c r o s s effect.
the
interfere
For the illustration of
2 the
cross
phenomena
diffusion
(Soret
phenomena
effects
be
and
cited.
diffusional
Many
effect),
thermal
(Dufour
effect)
thermo
researchers
in
the
past
have
to d e s c r i b e such t y p e of phenomena i n w h i c h t h e c o u p l i n g "><* 55 The e a r l i e s t of them was Thomson ' who the
velocities
thermoelectric
are
Thomson's electric
connected
reciprocal
current
-+
X.
be
phenomenon
by
law,
consider
and heat f l o w
the
is
the
the
an e l e c t r i c
forces
To
circuit
and
describe
in w h i c h
->
the
-»
force,
the
'
corresponding
be g i v e n
force
9
.
A,
xl
by
H =
A2
T
which
conditions.
are denoted by the f l u x e s j . and j
electromotive
?
in
symmetry
w h i c h causes the f l o w of heat w i l l
where
(Peltier
are s i g n i f i c a n t .
described
Let
thermo-electric
effect)
may
attempted
the
,
T
temperature
in
the
Kelvin
and j _ are independent, we d i r e c t l y
scale.
If
the
fluxes j .
have
x, = R, f, and
(1.1.1.) X
where
R.
and
respectively. they
R? But
interfere
forces
are
(1.1.1).
not
We
2 -*2
are
the
resistances
when
the
two
with
each
simply
have
R
2 "
other.
phenomena In
described
rather
of
such
by
complicated
a
electricity occur
heat,
simultaneously,
case,
simple
and
the
linear
fluxes
laws
phenomenological
as
and Eq.
expressions
as X,
R,,
j , <■ R ] 2 j
2
, (1.1.2)
X
2
=
R
2I
J
1
According to Thomson's R, 2 This
assumption
confirmed tentatively;
by
has
R
22
h-
hypothesis,
R2)-
been accepted
experiments.
however,
+
(1.1.3) generally,
Thomson
no fundamental
arrived principles
because at
this were
it
has
been
assumption known at
the
3 time of Thomson from w h i c h t h i s r e c i p r o c a l law could be deduced. The
relations
irreversible They 1,2)
expressed
process
in
by
Eq.
(1.1.2)
are
the
law
of
terms
of the resistances R.. ( i . j = 1,2). IJ i n terms of the conductances L.. ( i , j
can also be e x p r e s s e d as f j.
-/ where
-* X
L
+ L
"J
j2
L2]
X!
-» X?
U
J
f L22
(1.1.4)
X2,
a R
^22
12 11 " R,, R 2 2 R | 2 R 21 ' 12 R | 2 R2) R,, R 2 2 and so o n . The r e c i p r o c a l r e l a t i o n g i v e n by E q . ( 1 . 1 . 3 ) becomes L,2 L2, (1.1.5) L
Like conduction, electric and
the
there
of
are
solute
of
heat
also an i n t e r a c t i o n and
vice
versa.
expressed
relative
thermodynamical similar
is
conduction
velocities
flow
interaction
to
by the
with
electric
between mass d i f f u s i o n
In
Eq.
conduction
this
case a l s o ,
(1.1.4),
solvent
and
the
the
flux
force
the
and
forces
j _ is X?
the
is
the
p o t e n t i a l g r a d i e n t of the s o l u t e . F o l l o w i n g a method 19 20 ' has d e r i v e d the r e c i p r o c a l law
to Thomson, Heimholtz L
Heimholtz's fundamental
L
12
2T
assumption
was
principle;
however,
12 13 '
investigated
also
the
not
assumption
based was
on
any
confirmed
by
experiments. Estman conduction
and
phenomenon
in
phenomena caused heat
by
Dufour
in
of
caused Dufour gases.
also
solutions
gradient
by ' For
the
is
interaction studied
thermodynamic
a temperature
effect.
He
electrolytic
interact
diffusion
mixture
diffusion.
the
the
the
where
heat
thermo-electric
the
three
equilibrium.
The
mutual diffusion
known as Soret e f f e c t and
concentration
has
between
gradient
demonstrated
this
thermoelectric
is
known
effect
for
phenomenon
the as a in
4
electrolytes,
an electromotive
the flow of e l e c t r i c current
-» X.
force
is generated which causes -» j . . The processes of the t r a n s p o r t of ->
matter
and
heat
respectively,
will
the
occur
simultaneously.
fluxes
of
heat
If
and
j
■->
and
2
matter,
j ,
are,
then
the
phenomenological equations can be written as 71
]
-»
\
~
L
X
1I
7*
~~>
-»
+
1
L
X
I2
-+
+
2
L
X
I3
3'
-»
(1.1.6)
*-»
- L 2 1 X, + L 2 2 X2 + L 2 3 X 3
J2
and j 3 = L 3 ) X, ♦ L 3 2 X2 ♦ L 3 3 X 3 . The reciprocal relations L
L
.2
L
2.'
L
.3
L
3P
L
23
(
32
'-K7)
are to be s u s p e c t e d . The diffusion another
phenomenon
of s e v e r a l substances in the same medium which
may
be
expected
to
follow
a
is
similar
pattern and thus Eq. ( 1 . 1 . 6 ) may be written as
7,
I Lik V
where the reciprocal Lik
assumption equality set
of
to h o l d . the
of
Onsager
regression the
by
relation.
'
of
property
of
Eq.
arrived
fluctuations microscopic
(1.1.9)
is
at
t h i s conclusion
from
equilibrium
reversibility
known
as
the
by with
and
the
Onsager's
It s t a t e s that if linear r e l a t i o n s e x i s t among a
conjugate
phenomenological these
(,.,.9)
expressed
reciprocal
relations
Lki
are expected considering
(1.1.8)
fluxes
and
coefficients
phenomenological
forces
a r e well
and
defined,
coefficients
L .
if
the
resulting
then the matrix of possesses
invariant
symmetry. However, circumstances. magnetic
field,
the
For the
matrix
example, reciprocal
L., under
is
not the
symmetric influence
relations e x p r e s s e d
under of
an
by Eq.
certain external (1.1.9)
5 become Lik
(B)
Lk.
(-B),
where B is the magnetic induction.
(1.1.10)
The reciprocal
relations for a
rotating system are L
(CJ)
ik
L
(
ki
"U)'
(1.1.11)
where 63 is the angular velocity of rotation. Equation homogeneous Currie ' differ
(1.1.8)
functions
imposed by
an
of
states the
a restriction
odd
that
forces.
integer
the
fluxes
But for
are
an isotropic
linear system,
that entities whose t e n s o n a l
never
interact.
Thus
the
ranks
effects
of
coupling of heat flow and viscous flow e t c . are prohibited from the 29 31 isotropic system. However, the r e s e a r c h e s of Luikov ' have proved action
that the transfer of
of heat and matter are influenced
hydrodynamic
forces.
the influence of viscosity But by a consideration
It is thus a p p r o p r i a t e
by
the
to consider
in the process of the transfer of energy.
of these
viscous forces,
a contradiction
to
C u r r i e ' s theorem a r i s e s because only the tensors of the same rank, or
differing
by
an
even
entity,
can
be
included
conjugate fluxes and forces as defined in Eq. In heatfiow, order
actual
practice,
the
effects
of
in
the
set
of
(1.1.8). the
coupling
between
chemical reactions, viscous flow, e t c . are only of a small
and
so
generality,
they
the
are
neglected
for
transfer
of
molecular
all
practical
energy
is
purposes.
described
by
For the
system of Onsager's linear equations -» " -»
k
I Lik
k The Onsager theory take the
place nature
between of
the
is useful
processes. flux
V
vector
in systems where
It also gives (flow
of
entity
information
interactions regarding
across unit
surface
per unit time) which is unknown in the case of l i q u i d s , s o l i d s and dense g a s e s .
For
are inconsistent gases.
However,
dilute g a s e s , the r e s u l t s obtained
with the r e s u l t s obtained the theory
by the
theory
by the kinetic theory of
does not give us information
regarding
6 the
validity
of
experiment. coefficients eliminate
laws gives
of
motion.
the
It
reciprocal
be
verified
by
relationship
must
between
the
if the law i s c o r r e c t . The reciprocal laws are used to the
predict
the
It only
coefficients
certain
L.,
L, .
and
ik
relationski
physical
thus
that
it
enables
can
be
us
to
tested
by
experiments. In the
present context,
we are mainly
interaction of the process of transfer
concerned
with
the
of heat and that of moisture
in porous b o d i e s . 1.2
Transfer of Heat and Moisture in Porous Bodies Modelling heat and moisture transfer
in porous bodies is an
area of great importance. The s t r u c t u r e of the solid widely
in
fibres
shape.
or g r a v e s .
void s i z e s . porous
attempt
may,
for
instance,
be
matrix
composed
There i s , in general, no uniform processes through
bodies,
here
to
the
subject
develop
the i r r e g u l a r
is
the
not
yet
equations
cells,
d i s t r i b u t i o n of
void
fully
for
varies
of
Because of the complexities of the mechanism
in the diffusion in
It
involved
configuration
developed.
heat
and
We
moisture
transfer in porous media by assuming it as a continuum. Consider a porous body voids
in
bound
matter
an
temperature gas.
In the
with
unsaturated
state.
consists
liquid,
below
of
At
the
bound
matter filling
temperatures
vapour
and
above
inert
273K,
gas,
273K, it consists of i c e , supercooled
pores and c a p i l l a r i e s of the porous body,
the
and
the at
liquid
a
and
vapour and
air are p r e s e n t . Thus the c a p i l l a r i e s of the porous body are p a r t l y filled gas
with vapour, water or ice and p a r t l y filled
mixture.
multicomponent vapour
way,
system
(subscript
moisture body
In t h i s
(subscript
I), 3),
the
moist
consisting solid inert
in gas
of
capillary bound
the
form
matter of
(subscript
with the
porous ice
in
vapour
body the the
a
form
(subscript
<*) and
is
of 2),
skeleton
( s u b s c r i p t 0 ) . The volume concentration of the bound matter
(w) is defined
as the product of the specific mass content
(u) and
7 the density
of t h e medium ( y n ) ; thus u
ID
u
Yn '
The quantity
4 [ di| > i = 1
=
4 | UAi =I
u
(1.2.1)
u. is defined in terms of the porosity
of the
body 1
"l
1
where
P. is the density
factor,
defined as the pore volume per unit body
a factor
related
to the
of the bound substance, IT is the
volume concentration
per unit
small,
vapour
the
and dry
specific
air
is
mass of the
bound substance varying in the process of mass transfer. masses of
porosity
volume and b
Since the
in the pores of the capillaries are
mass content
of
the bound matter
is equal to
the sum of the mass content of the ice and moisture i.e. 4
u
y i=1
ux . = u20 + u,3 .
Conservation of mass and energy: volume in the system. mass in any
Let us consider a small control
The differential
equation of the transfer of
phase in presence of sources or sinks may be written
by the continuity equation as 3(YQ u ) 3t
where
j
is
divj.
the density
of
+
Q.
,
the flux
(1.2.2)
of
l-phase
the strength of the source or sink of the i-th Eq.
(1.2.2) with respect to i ( i
3(y 0 u) £r— d t
i
since the sum of all
i.e.
I i
Q.
and Q
is
1,2,3,4), we obtain
4 I
div
matter
component. Summing
j
,
(1.2.3)
,
the mass sources or sinks is equal to zero,
0.
To obtain the differential equation of the heat transfer, we
8 consider
the transfer
rate
change
of
of
of e n t h a l p y . enthalpy
concentration
divergence of the enthalpy flux; £
(h0TQ
*
l
h
pressure,
'h'
is
the
equal
to
local the
thus, u.) =
i Y o
div (4 + l h
L
y ,
(1.2.*)
1
I
where the heat flux q* is defined q>
At constant
by the Fourier heat equation
AVT.
Let us denote the specific heat at constant p r e s s u r e by c . , dh„
JU an I -j=—
C
,
U -p=—
c
/ 1 -> <;\ (1.2.5)
I dT ' o dT The Eq. (1.2.<*) changes to
(c
o Yo
+
I, c i Yo V 3T J , ht — n — 1=1
1=1 ,
div (q
4
+
.
T
. *• , 1 = 1
h. j )
(1.2.6)
1 1
With the help of Eq. ( 1 . 2 . 2 ) , Eq. ( 1 . 2 . 6 ) reduces to c
q
Y
3T 0 3T
r I
div q +
^
CK h.
l - l it
1
c. T . v f
,
(1.2.7)
l - l
where c
is the total specific heat defined
by
H
c q c + y c. 0 u. The
first
represents
term
of
pure
heat
heat
transfer
the
convective
convective
due
term
the
to phase be
of
^
hand
conduction,
transfer may
.
right
the
change and heat.
small
,
1
side
of
second the
term third
In the absence of in
comparison
with
1
Eq.
(1.2.7)
represents term
represents
filtration, the
the the
conductive
9 term and (1.2.3) with
therefore and
it may be neglected.
(1.2.7)
phase
are
the
transformation
transfer
in
the
The set of E q s .
equations
presence
of
(1.2.2),
of mass and source
or
heat
sink
Q..
The p h y s i c a l aspect of these equations will be described after
the
determination of source or sink functions and the flows of heat and matter. Determination 273K,
the
of
mass
bound
sources,
matter
inside
Q.: the
At
the
system
temperatures
is composed
above
mainly
of
two p h a s e s ; vapour and moisture. In t h i s c a s e , we have Q, -
Q3
(1.2.8)
and from E q . ( 1 . 2 . 2 ) , we have 3(
But the very
Y0 3t
u ) d i v j ,
vapour
small
portion
of
the
in comparison with
+
Q
(1.2.9)
r
bound
matter
the moisture
in the c a p i l l a r i e s (u,
>>
u.).
is
We may
set the r i g h t hand s i d e of equation equal to zero so that we get Q,
=
At
the
composed
of
Q3
=
div j *
temperature
vapour
and
(1.2.10)
r
below
ice
or
273K,
saturated
the
gas. -t
transfer in the solid s t a t e , therefore we have j 3(Y0 u 2 ) y——
=
is defined
For K
(1.2.11)
here the notion K as coefficient
of
icing
u2 (1.2.12) U-,
+
u^
0, it signifies that the d i s p e r s e d
and when K changed
is no mass i.e.
i.e.
u? — U
0,
is
by the ratio of the mass of the ice to the total
mass of the matter, K
There
matter
Q,.
Let us introduce which
bound
medium contains no ice,
1 , it signifies that the whole of the liquid has been
into i c e . Thus the
value of K varies in the range of 0 to
10
1 (0 < K < I ) . Keeping K independent of time, we have 3u,
„
-if
3u,
.
K
£c ~d
(, 2J3)
ff-
-
From Eq. ( 1 . 2 . 1 1 ) , the source Q? will be
a A
similar
temperature du
relation
above
273K.
is
obtained
A local
at any phase l may be caused
change
for in
the the
source
Q,
associated
by the phase transfer
at
matter du.
and
by the moisture transfer du ,. Thus '
if
du.
du.
+ du , .
lj
I
if
For a local change in the l i q u i d , we have du,
du,. + du,,
3
3j
3f
(1.2.15)
(1 + B ) d u 3 {
where
B
(0
B < "°). When g
<
6
du,./du,f.
,
The value of B ranges from zero to
infinity
0, t h e r e is no phase conversion and when
<*> , all the changes of mass are caused
by phase conversion.
Now defining again a new coefficient
a as
a
phase
conversion
coefficient,
where
the
value
of
varies
£
between 0 and 1 (0 < e < 1), we have du 3
- j ~ du3f.
(1.2.16)
The diffusion equation of the moisture t r a n s p o r t will therefore dU
v Y
3f
o ST
A-
--
dlv
b>
be
11 8u3 0r
'
3u3 div
0 FT
J3 +
(K2 ,7)
^ o i r -
On a comparison of the E q .
-
(1.2.17)
with
Eq.
(1.2.2)
at
i
2 , we have 3u, Q
where
e
3 -
„ K
^o l t ~
e
From E q s .
(1.2.14)
Qi
Y
^
e?
=
K,
ff
Q
e,
^0
37'
u
»<
( K 2
the
, 8 )
and ( 1 . 2 . 1 8 ) , we can summarise
;
i
2 , 3,
(1.2.19)
= e
Determination of the fluxes of heat and moisture: conditions,
-
moisture
flux
is
directly
Under
isothermal
proportional
to
the
g r a d i e n t of mass content.
"*
*
j3
a2
-*
,
y0 V u ,
(1.2.20)
where a - i s t h e c o e f f i c i e n t of p r o p o r t i o n a l i t y . is
the
analogue
to
the
g r a d i e n t of heat content
?
The
aq
'h'
c
capillary specific
is
proportional
to
c
porous
the
i.e.
a~
has
the
same
role
as
thermal
a and t h e r e f o r e i t i s c a l l e d c o e f f i c i e n t The qp o t e n t i a l of moisture t r a n s f e r 0 mass content u,
of is
i.e.
6,
m is the
3
(1.2.21)
coefficient
p r o p o r t i o n a l to t h e s p e c i f i c
where
q which
coefficient
diffusivity.
u
flux,
(enthalpy),
YQ V h .
proportionality
diffusivity mass
=
heat
The moisture f l u x j
(1.2.22) mean
body
and
mass content
with
constant t e m p e r a t u r e ,
i.e.
isothermal it
is
specific
defined
as
r e s p e c t to t r a n s f e r
the
mass rate
capacity
of
of change of
p o t e n t i a l of l i q u i d
at
12
c
( | H )_ m
d6
(1.2.23)
T
With the help of Eq.
(1.2.23),
Eq.
( 1 . 2 . 2 0 ) is
renarrated
as j, '3
-a' T „ c m '0 m
where A = a m
c
X V6 , m '
(1.2.24)
Y,, I S the mass conductivity coefficient analogous to
m m
0
heat conductivity Under influenced
V9
'
°
coefficient. nonisothermal
state,
the
moisture
transfer
is
by the gradients of heat and mass, which can be written
from Eq. ( 1 . 1 . 8 ) as j3
where
X V 9 m
X
X
VT
,
(1.2.25)
mt
is the coefficient of thermal mass c o n d u c t i v i t y .
mt
Since
'
the
transfer
potential
of
liquid
is
a
function
of
specific mass content u and temperature T, we can write 6
6(u,T).
and a small variation
7e By
vG
iTi
Eqs.
(1.2.23)
mav
8
( § 1 ) T v"u
(1.2.26) De
described
( | 1 )u
+
and
as
"T
(1.2.27),
(1.2.27) Eq.
(1.2.25)
can
be
rewritten as a
h where &
c
[( m
a
Y
0 ^u
a
m Y0 « ^ -
is the thermal gradient 6
With
m
similar
vapour as
| | ) + r2£ dI u X m
analogy,
we
can
C-2-28)
coefficient ]
write
(1.2.29)
for
the
flux
density
of
13
a where
Pk
5 P is the
partial
f(u,T), k_
is
1
" I TV T a
is
the
vapour
pressure and i s
the
thermal
of
the
ratio
i ' < I? >T defined
diffusivity
coefficient
of
by the function
coefficient thermal
of
P
vapour
diffusion
to
and mass
diffusion. Equations source or sink
matter
In t h i s c a s e ,
now
the t r a n s f e r
TQ M a
An
our
and
(1.2.30)
consideration
and l i q u i d )
determine
or
sink
of
equivalent
to two
phases of
at a t e m p e r a t u r e above
=
Similarly be obtained
C
q
Using t h e
Qi
div
u < < u3
strength
equation
the
(j^
+
(1.2.7)
0 f
dlV
Y
1
Q,
is
influences
obtained
eY
the
the
from
transfer Eq.
the
as
+ Q
o IT•
l
of
(1.2.3) i.e.
(1.2.32)
describing
expression
Q3
by
(1.2.31)
j*3).
equation
from E q .
the
273K.
i.e.
w i t h o u t t h e c o n s i d e r a t i o n of the influence of source or s i n k ,
YQ | ^
the
respectively.
p o t e n t i a l of moisture i s d e s c r i b e d
(1.2.2),
d i v T3 + Q3 ,
source
moisture.
limit
(vapour
r e v i s e d form of E q .
where
(1.2.28)
f u n c t i o n , f l u x e s of heat and m o i s t u r e ,
We s h a l l associated
(1.2.19),
h
l
+ Q
3
h
transfer
of
heat
may
14 from
Eq.
(1.2.19), C
whert
Y
q
the
=
0 I?
p
h.
convective is
the above equation is simplified div
h,
heat by
the
highest
rate
of
body
d
the
of
number,
with
Re,
for
has
and , where
for
and
place
transfer
be
way
of
is
taken
as
3.10
the De
Eqs.
i^ ot
div
(a
a
used of
heat
q
for
flux
r o
The
diameter Reynolds
as
'
f l u x e s of h e a t , m o i s t u r e 8 9 Vries ' substituting the and
(1.2.30)
in
Eqs.
Eq. q
e | i gt
(1.2.34)
6 VT) m a ' .' + a" , " mom mom ; a + a m n
(1.2.35)
the
chemical
where there
Similarly,
C
+ a
/ a , + a and m m
presence
conduction
^u m
m
of
This
(1.2.32) ( a ' v^u + a ' 6 m VT) m m m
in
to
convection.
(1.2.28)
div
a
(1.2.33).
m.
5
any
due
convective d r i e r s , 2 40 k g / m h . F o r a
, , ,n-2 -2J0
s
term
equivalent
determined
3
by
from
about the
is
The
Eq.
the
0.7,
deriving
given
fluxes
of
IT a
j£ 3t
process
value
in w r i t i n g most
10.3 x IP" 0.7 x 2.88
been
the
Bq. (1.2.34) may be
the
for
(1.2.33)
evaporation.
small to i n i t i a t e
An a l t e r n a t e vapour
of
t h e m o i s t u r e flow
is considerably
(1.2.31)
can
d
0 3T '
neglected that
fraction,
3 e "Try-
expressions
T
heat
moisture
void
j
Re
and
is
fact
capillaries
D
which
the
transfer
explained
porous
is
£P
^ '
as
process
of
transformation
i s no s o u r c e o r (1.2.33) -
+
moisture while
VT
Eq.
taking
(1.2.35)
for
substituting
the
sink.
can b e m o d i f i e d X
transfer
by
by
Fourier's
law
of
heat
as Y
0 ^
dlV
(X
q
VT)
+ CPY
0 I?
<'-2-36>
15
Equation (1.2.36)
is the usual heat conduction equation with a heat
E
source
PYn 3 u / 3 t due to moisture evaporation in the pores of the
porous body. For an intensive temperature
above
evaporation transport
there
of the moisture. phenomenon.
the
processes.
a pressure
This
of
presence
pressure
pressure
of
porous body at a
gradient
pressure gradient the
below 373K. Therefore,
influence
The
is
However,
occur at temperatures consider
heating of the c a p i l l a r y
373K,
influences
gradient it
gradient
a pressure
due to the the
may
also
is a p p r o p r i a t e on
gradient
the
to
transport
inside a
porous
body causes hydrodynamical motion (filtration) of vapour and liquid which a r e d e s c r i b e d by the Dercy law: I
=
X VP P
P where
X
is the coefficient
(1.2.37)
of filtration conductivity analogous to
X . q
The
system
of
differential
equations
describing
heat
and
mass
- ^
I c J ^ T
(,.2.38)
Vp)
(1.2.39)
transfer thus becomes fl=
div ( a q
v^T)
P3t
div (a
|£
d > v ( a
V~u m
+
+
a
c
6VT <• a
m
P
and
The the transfer
system
v 1 ) -
Eqs.
£
l ^ .
(1.2.35)
and
(,.2.,0) (1.2.36)
are
describing
of heat and mass without filtration and the systems of
Eqs.
(1.2.38),
heat
and
expressed
of
p
mass
(1.2.39) and (1.2.40) are describing the transfer of in
the
presence
of
filtration.
by a single generalised system as
They
can
also
be
16
i. =
gt where
I
V.(L.,
V6, ) ,
r
ik
k' '
(1.2.41)
6. are the c o r r e s p o n d i n g t r a n s f e r
Q . and
g •> are
used
and the c o e f f i c i e n t s
L., II
a
for
heat,
L . ik
and
the symbols
pressure,
9.,
respectively
are
+ — a o L,_, c m ' 1 2 q
q
potentials,
moisture
, L-, ' 2 1
c
a
° m
q
a L „ 22
a
L
a
, L,, m ' U
e p — - 5 Kp , c ' P
L.,,. 23
a 33
p
a 6
c
y
a 6p m K
p
'
v
P
L
31
e a_ 6
c
'
L
32
c
P
P
These equations have been d e r i v e d by L u i k o v and M i k h a i l o v 26 and Narang have further modified these equations to
Kumar include
the
Inspite
of
hydrodynamical mass
hydrodynamical velocity
v\
diffusion,
motion
the E q .
~
div
DT Dt
dlV
of
if the
(1.2.35)
(am vu
effect
+
in
on the
moisture
and ( 1 . 2 . 3 6 )
the
transfer
phenomenon.
capillary
porous
occurs
some
at
body, average
become
a m 6V~T)
(1.2.42)
and ,
U
-t\
VT)
q
+e
P "c-
3u IT '
, , , ,,-> '-"3)
( K
where the s y m b o l ■=— stands for the s u b s t a n t i a l
m In Eqs.
sT the
(1.2.38)
derivative,
+ v v
'
same
i.e.
(1.2.W) way,
(1.2.40)
the
diffusion
can be m o d i f i e d
equations as
with
filtration
33
17
§1=
div (a
7"T)
|£.
+
|f
I
q
^ ut
- div (a
Vu + a m
cx J V T
(..2.45)
7p)
(1.2.46)
l
VT + a m
P
and
5? = div 1.3
(a v P ^
e
(K2 47)
f If
-
Initial and Boundary Condition In order
to make the differential
equations
for
the transfer of heat and mass physically sound, we need some laws which may d e s c r i b e the interaction between the surface of the body and the surrounding: (a) the
Initial conditions; system
potential
at
of
start
the
Initial
of the
system
conditions state the
process.
is
At t h i s
supposed
to
be
instant,
behaviour of the
arbitrary
transfer
and
is
a
function of the space coordinates only. Thus
jTI
f (I?)
.
I u J where r is the position (b)
Boundary
transferred
a
f 2 (?) vector.
conditions:
At
the
surfaces,
the
moisture
is
under the influence of potential gradient of moisture and
heat. Applying t h e mass balance at the surface, we have X
The
m
(V
"*u)s
quantity
utilized
partly
of
+ X
heat
m
6
(
^ s
transferred
* V(t)
to
the
(, 3 2
= °-
surface
- - >
of
the
body
is
in the evaporation of the l i q u i d . Applying the heat
balance at the surface, we have X (v"Hs + q (t)
( l - e ) p qm(t)
= 0
(1.3.3)
18 In and
the
the
case of
system,
the
convective law,
the
interaction
exchange
of
between
heat
and
the
gaseous
mass
takes
medium
place
by
i.e. q
a
(Tc
Ts)
and q ^m w h e r e the s u b s c r i p t transfer heat
potentials.
and
mass
6Y n (U '0 s
U ) , c '
s stands f o r
the surface and c f o r t h e
The c o e f f i c i e n t s
transfer,
ambient
a and 6 are the c o e f f i c i e n t s
respectively,
thus
Eqs.
(1.3.2
of
1.3.3)
become,
X
m(Vs
+
X
m
6
+
<*%
8
V
U
U
s
(,
c> = °
'3-*>
and Xq(VT)s + Equations
a (Tc -Ts)
(1.3.4
+ BY0(US
1.3.5)
UC)
can also
0
(1.3.5)
be e x p r e s s e d
in
general
form as
-»
—»
(Vu)s + a 2 ( V T ) s
+ B
U
2
s
*2
( t )
°
(1.3.6)
and (VT)
where
+ a,T + B|U
a.,
a2>
thermophysicai
8|
82
coefficients
and
(1.3.7)
tne
4> ( t )
aggregates are
of
the
fluxes
a
process
the of
known
heat
and
reflect
the
experiments.
Dimensionless Quantities Differential
physical equation the
by
0;
are
and
moisture to be determined 1. k
#t(t)
picture is
change
amount
equations of
the
a consequence in
energy
liberated
from
of
dealing
process. of
the
energy
t h e system the
with For
system.
example, equation
as the Thus
the
diffusion
which
describes
equivalent a
f o r m of
differential
the
equation
19 occurring
in t h e
formulation
of a problem
describes
the
physical
laws which govern the system. The c h a r a c t e r i s t i c v a r i a b l e s the relation to
between the s e p a r a t e terms of the equations. We have
establish
similarity
define
such
relationship
theory
gives
a
among
method
the
to
different
transform
terms.
the
The
expressions
having differential operator into the simplest algebraic form. Now medium;
consider
the
the transfer
interaction
of
solid
with
the
phenomenon in t h i s case is governed
gaseous by
the
convective law X (-ajj) 3X s where T
<MT s
T ), a
(1.0.1)
is the ambient t e m p e r a t u r e . Let us further
form of an infinite
assume that the solid
body is taken in the
plate of finite thickness R and the temperature
drop over the t h i c k n e s s R is p r e s c r i b e d to T then for ( 3 T / 3 x )
a
constant, we have 3J 6T * 3x ~ R If the temperature difference T §1
_
AT
h being t h e heat transfer The r i g h t
(1.0.2)
-
T
is denoted by AT, then
*S
(1 4,3)
X
Ki.t.JJ
coefficient.
hand s i d e of Eq.
(1.5.1)
becomes
dimensioniess
and for t h i s we call 3iot number ' B i ' a t t r i b u t e d to the name of the s c i e n t i s t who has made a significant contribution to the development of t h i s
field. In
defining
the
variables,
a ,R
into
introducing
this
number
Biot
one is
number,
variable that
once
we
Bi. we
The fix
have
reduced
other a,R
benefit as
the
two in basic
20 quantities,
we
obtain
an
infinite
number
of
other
sets
quantities and thus an infinite number of phenomena are Now consider t h e basic equation of heat 2
of
these
determined.
diffusion
a
The term
3T/ 3t r e p r e s e n t s the rate of change of temperature
with
respect
to time and can be replaced by 6T / t . Similarly the term 2 2 hand s i d e 3 T/3x is the square r a t e of change of T 2 r e s p e c t to x and it can be replaced by 6I* R /x , where t h e
of the r i g h t with suffixes
t
and
temperature T,
R denote
the
time r a t e
and
space
rate
change
in
therefore 6 T
6T
t
- ^
=
R
a —|
,
(1.0.5)
x
For a plate of t h i c k n e s s R, the Eq. ( 1 . 5 . 5 )
!Ii
at
6T
R2
R
becomes
'
the r i g h t hand s i d e of t h i s equation being dimensionless This
represents
a
generalised
variable
which
we
quantity. call
Fourier
number ^-4 R Thus
the
Fourier
Fo.
number
is
(1.0.6) defined
as
the
ratio
of
time
rate
change in temperature with the space r a t e change in t e m p e r a t u r e . It has also t h e importance that it first reduces the t h r e e v a r i a b l e s a, t,
R into one and second once a set of a,
obtain
an infinite
different
set
of other
t,
combinations
R is e s t a b l i s h e d , each
characterising
we a
process. In
this
way,
we
see
that
the
introduction
of
these
dimensionless v a r i a b l e s not only reduces the number of v a r i a b l e but it generalises the r e s u l t s and c r e a t e s a firm scientific base for
the
21 mutually connected transfer analysis
has
transport
phenomena. A list
wide
of
phenomena. This is why the dimensional
acceptance
the
various
in
tackling
the
dimensioniess
problems
variables
of
the
which
are
often used in the transfer of energy and mass are given below. Biot number, Bi
= T— q
Eckert number, E Fedorov number, Fe
2 V 0 —T^F c AT q
EKO Pn
'
c q
Fourier number, Fo
a t ~~j
Kirpichev number, Ki
T^AT
R q
v ■u u v Kossovich number, Ko
pAu = -—TJ q
Luikov number, Lu '
a — a q
Peclet number, Pe Prandtl number, Pr Predvoditelev number, Po
V0R a
q — V
¥ q 2.~ OR q
Posnov number, Pn
Reynolds number, Re
6AT —g^j jd ——e
22 The multitude
growing
importance
applications
transmitting
such
energy,
in
supersonic
generated
tremendous
processes
in
the
as
of
interest
porous
the
heat
and
in
porous pipes,
cryogenic
hypersonic
modelling
media.
materials
Also
wind
heat
and
is
lack
there
for
lines
for
tunnels mass of
a
has
transfer
a
unified
theory
to d e s c r i b e the t o t a l process i n v o l v e d i n the v a r i e t y of t h e 29-32 porous b o d i e s . L u i k o v developed the equations f o r i n t e r n a l and
external
heat
solutions
of
presented
and
mass
the
heat
transfer and
for
mass
porous
transfer
bodies.
problems
Analytical have
been
under
Luikov
and
types
of
different boundary c o n d i t i o n s i n a t r e a t i s e of 33 Mikhailov . M i k h a i l o v and O z i s i k
capillary
differential
porous
equations
for
bodies
and
them.
Eckert
derived
the
with
system
of
Pfender
and
Faghari p r o v i d e d a general a n a l y s i s of heat and mass t r a n s f e r i n an unsaturated porous m e d i a . Huang et a i 21 ' 22 i n v e s t i g a t e d the 52 m u l t i p h a s e moisture t r a n s f e r in porous m e d i a . Saito and Seki considered materials
the
subjected
investigated porous
mass
transfer to
problems
bodies.
Liquid
sudden
condensation
investigated
the
There
in the
v o i d s of
process
the
solutions
cylinder.
of
the
Mikhailov
Entor
filtration
the of
in
moist
and
and
porous
porous heat
Shyganakov
heat
transfer
in
bodies evaporates or <*5 Tien investigated
insulation.
and
porous
mass
Shukla
transfer
during
processes.
been an attempt
described
rise
Ogniewiez and
in
phenomena
has
heating.
with
combustion and e v a p o r a t i o n
of equations
pressure
related
condenses and mixes w i t h a i r . the
and
to
obtain
in the p r e c e d i n g t e x t . transfer has
potentials
obtained
the
solutions Ply at
for
an
solutions
for
has
the
infinite for
set
presented
an
hollow infinite
plate and s p h e r e under the c o n v e c t i v e t y p e of boundary c o n d i t i o n s . 25 Kumar has a p p l i e d an extended variational p r i n c i p l e for the nonlinear applied mass
heat the
and
mass
boundary
transfer
layer
problem.
approach
for
Rai the
and
Pandey
solution
of
heat
have and
t r a n s f e r p r o b l e m of a c y l i n d r i c a l porous b o d y . Gomini and 18 Lewis have a p p l i e d the f i n i t e - e l e m e n t technique f o r o b t a i n i n g the
23
numerical
solution
solution
for
porous
body. The
role
in
of
heat
the
and
to
mean t h e
solid
by
evaporation.
the
drying
solid and
occurs
of
transfer of
been made
a
'
with
problems
have
an
of
Krischer
important
solids
generally
accepted
is
water
and
usually
from
approaches
'
the
capillary
the
to study
Luikov
.
In
deriving
made
by
and
by
in the
mass
been
f l o w of
second,
of
Fulford
the
liquid
moisture
external
heat and mass
research
on the
of
subject
was
"
approach for the d r y i n g 34 et a l i n v e s t i g a t e d the heat
equations
the
drying
importance
considered
generation. in
. Berger of
and
a
has Pei
the hygroscopic
drying
Kuts
capillary
of
as
the proceedings of w h i c h
Company
Lyons
transfer
and
for
Because
internal
The
Zagoruyko
Soviet
.
Hill
due to
processes.
by
the
symposium,
McGraw
transfer
extended
equations
A survey
industries,
the
while
of
a
porous 27 Grinchik
and porous
media
for
of
Matsumoti pneumatic
nonisothermal d r y i n g in unsaturated porous media. 38 39 and Pei ' developed a mathematical analysis of
drying
al
of
grains
investigated
processes
in
Shishedjiev temperature moist
in
drying.
the
the
r e s u l t of a temperature g r a d i e n t
has
solid.
to
vapour
thermodynamics
theoretical
considered
et
the
an i n t e r n a t i o n a l
capillary medium
due
heat
published
drying
in
by a combination of the c a p i l l a r y of
chemical
theme f o r
The
a liquid,
of
was
in
processes
of
as a d i r e c t
during
scrlids
drying
transfer
drying.
are
obtained
distribution
The two w i d e l y
coupling
Sokolavskaya
Mikhailov
moisture
i s assumed t h a t the i n t e r n a l moisture t r a n s f e r of
diffusion
thermodynamic method
of
mechanism
is c o n t r o l l e d
transfer
mass
removal
mode, i t
the
problem.
and
technology
taken
the f i r s t
the
temperature
drying presented
and
moisture
during the
various
unsteady
by
reduced
a
detailed
distribution
drying heat
and
pressure. numerical during
periods. mass
Labutin transfer
Mikhailov solution
contact
and
of
drying
the of
a
solid. In a d d i t i o n
to the technology
of
d r y i n g , the t h e o r y
of
heat
24
and mass transfer in porous media has been widely § q (19
science. DeVries and Philip ' '
studied in soil
developed the theory of heat and
mass transfer with reference to s o i l . A good exposition of the theory 2 can also be found in the treatise of Bear . The work on shrinking and swelling soils has been reviewed by Philip
• The permeability
water
was
of porous material with reference to
s t u
|
3
considered the transient heat and mass transfer in soil surrounding a buried
heat
source.
They
developed
a
unidimensional
model
to
predict the coupled heat and moisture migration phenomena. Dinulescu ana
Eckert
have
obtained
analytical
expressions
for
the
moisture migration in a slab of porous material. In
the
equations (1.2.35 under the
following
chapters,
1.2.36) and (1.2.38
various cases for
the basic
the
system \.2.W)
of
differential
have been studied
bcGies l i k e
plate,
cylinder
and sphere under the generalised boundary conditions. These studies present a detailed mathematical analysis of the generalised laws of diffusion as they apply to particular geometric situations of interest in the technology of drying.
25 REFERENCES
1.
Berger,
D
and
(1973)
293-302.
2.
Bear,
3.,
3.
Beladi,
Pei,
"Dynamics
Elsevier,
Childs,
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and
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392-405.
5.
Curie,
P..
6.
Curie,
Mass
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Porous
D.L.
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Dinulescu,
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D.A.,
Eckert,
E.R.G.,
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12.
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13.
Eastman, E . D . ,
3 . A m . Chem. Soc. 50 (1928)
14.
Eckert,
E.R.G.,
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Transfer
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15.
Eckert,
E.R.G.,
16.
Entov, (1983)
Mass
1069-78.
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Pfender, 6 (1978)
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1482. 283-92.
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of
the
VI
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1-12. M.,
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1613-23.
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Shyganakov,
N.,
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17.
Fulford G.D.,
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19.
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20.
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Huang, C . L . ,
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10.
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1975).
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7.
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G.
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R.W.,
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3.
Heat
Mass
379-91.
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1387-92.
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Heat T r a n s f e r
840.
22 (1979)
1295-1305.
19
26 22.
Huang, C . L . D . , Transfer
23.
24.
Siang, H.H.
3.
Heat Mass
0 . , "Die Wessenschaftiichen Grundiagen der
Trocknung
22 (1979)
Krischer,
and B e s t ,
C.H.,Int.
257-66.
Technik"
2nd ed ( S p r i n g e r ,
Krischer,
0 . ; and M a h l e ,
Beriin K.,
1963).
VDI Forschungscheft
(Dusseldorf
1959) 437. 25.
Kumar,
1.3., I n t .
3. Heat Mass T r a n s f e r
26.
Kumar,
1.3.
Narang,
27.
28.
and
(1965)
567-73.
Kuts,
D.S.
(1985)
843-47.
Labutin, V.P.,
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H.N.,
Grinchik,
V.A.,
Goluber,
Int.
N.N.,
L.G.,
14 (1971)
3.
Heat
1759-70.
Mass T r a n s f e r
3.Engineering
Safin,
3. Engineering P h y s i c s 45 (1983)
R.G.
and
Physics
Andrianov,
Luikov,
A.V.,
Int.
3. Heat Mass T r a n s f e r
1 (1961)
167-74.
30.
Luikov,
A.V.,
Int.
3.
Heat Mass T r a n s f e r
3 (1963)
559-70.
and
in
Luikov,
A.V.,
Bodies"
in
Advances
1964)
123-84.
York, 32. 33.
Luikov,
"Heat
A.V.,Int.
Luikov,
A.V.
in
Mass
Heat
Transfer
Transfer
Mikhailov,
(Pergamon P r e s s , O x f o r d ,
34.
Lyons,
Hatcher,
35.
Majumdar,
36.
Majumdar,
37.
Majumdar,
Heat Mass Transfer
15 (1972)
(Hemisphere/McGraw
Hill,
A.S.,
New Y o r k ,
International
3.D.
A.S. , Drying
A.S.,
and
Hill,
of
Porous
Press,New
1-14. Energy
and
1965).
Sunderland,
3.E.,
Int.
3.
897-905.
'80
I
:
New Y o r k ,
Advances
18 (1975)
Yu.A., "Theory
Mass T r a n s f e r " D.W.,
Capillary
1 (Academic
3. Heat Mass T r a n s f e r
and
49
911-14.
29.
31.
8
in
Development
in
Drying
1980).
Drying
1
(Hemisphere/McGraw
1980). Drying
Symposium
'80
2:
Proceedings
(Hemisphere/McGraw
of Hill,
the New
Second York,
1980). 38.
Matsumoto,
39.
Matsumoto, S. and P e i ,
(1984)
(1984)
S and
Pei,
D.C.T.,
Int.
3.
Heat Mass T r a n s f e r
27
Int.
3. Heat Mass T r a n s f e r
27
843-50.
851-56.
D.C.T.,
27 40.
Mikhaiiov,
M.D.,
Int.
J.
M.D.,
Int.
3.
Heat Mass T r a n s f e r
15 (1972)
1963-
65. 41.
Mikhaiiov,
Heat
Mass
Transfer
18 (1975)
797-
804. 42.
Mikhaiiov, Transfer
43.
M.D.
and
18 (1975)
Shishedjiev,
B.K.,
Mikhaiiov,
M.D.
Solutions of
Heat and Mass D i f f u s i o n "
York,
Int.
3.
Heat
Mass
15-24 and
Ozisik,
M.N.,
"Unified
Analysis
and
(John Wiley & Sons, New
1984).
44.
Mikhaiiov,
45.
Ogniewicz,
Yu.
A.,
Int.
3.
Heat
Mass
Transfer
1 (1960)
37-
45.
(1981)
Y and
Tien,
C.L.,
Int.
3.
Heat Mass T r a n s f e r
421-30.
46.
Onsager,
L.,
P h y s . Rev.
137 (1931)
405.
47.
Onsager,
L.,
Phys.
138 (1931)
2265.
48.
Philip, Fluid
49.
24
3.R.,
Rev.
"Flow
in
Porous
Mechanics 2 (1970)
Philip,
3.R.
38 (1957)
and
Med i a "
in
Annusl
Review
of
177-204.
DeVries,
D.A.,
Trans.
Am.
Geophys.
Union
222-32.
50.
Plyat,
51.
Rai,
S.N., Int.
52.
S a i t o , H. and S e k i , N, 3., Heat Transfer 99 (1977)
K.N.
3.
Heat Mass Transfer
and Pandey,
R.N.,
Int.
3.
5 (1962)
59.
Engg. Science,
14 (1976)
975-90.
53.
Shukia,
K.N.
Evaporation 3.C. 5k.
"Heat
Processes"
and in
Transfer
Future
Energy
Denton and N . H . Afghan (Hemisphere
Thomson, W . ,
P r o c . Roy. S o c ,
55.
Thomson, W . , M a t h . P h y s .
56.
Zagoruyko, Research
57.
Mass
V.A.
16 (1984)
Zagoruyko,
V.A.
Research 20 (1988)
and
in
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and
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by
1976) 2 523-36.
Edinburgh
21 (1857)
1 (1882)
Sokolovskaya,
105-112.
123.
232.
Heat
Transfer,
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Heat
Transfer,
Soviet
93-104. and
Sokolovskaya,
275-87.
28 Chapter
2
INTEGRAL EQUATION APPROACH TO HEAT AND MASS TRANSFER PROBLEMS
Expressions for temperature and moisture distributions are derived for bodies of axisymmetric and spherical geometries by integral equations method. Approximate solutions, applicable for small values of generalised time, have also been worked out for the system. There place
are
through
immobilise
a
some
absorption
of
number
the
pores
of
heat
the at
of of
processes solid
diffusing
times
body
in
matter
accompanied
which
which with by
diffusion may
the
takes
absorb evolution
heat transfer
medium
diffusing different
and
matter
produces through
a
the
cross-effect body.
Thus
from the individual transfer
on the
the
and
Mikhailov
to
through
absorption
phenomenon
of
becomes
phenomenon of heat or moisture
and needs a simultaneous consideration of both transfer 1 Crank , Luikov
or
due
change of s t a t e . The heat of evolution or absorption diffuses the
and
2
have considered
phenomena. a number of
phenomena of t h i s t y p e , but t h e i r considerations were limited to the simple type of the boundary conditions between the surface of the solid body and the medium. However, Luikov and Mikhailov have also considered a general type of the boundary conditions for an infinite p l a t e . They have developed an integral equation to solve the problem by considering first an a u x i l i a r y problem in which the transfer potentials are supposed to be p r e s c r i b e d . The solutions for heat and mass transfer problems in the axisymmetric and s p h e r i c a l bodies have been obtained by T r i p a t h i , Shukia and Pandey ' . In t h i s
chapter,
have been determined
the transfer for an infinite
by constructing an auxiliary
potentials of heat and circular
moisture
c y l i n d e r and a s p h e r e
problem in which the heat and moisture
fluxes at the surface are p r e s c r i b e d . An approximate
solution
has
29 also been worked out for small values of generalised time. The
transfer
preceding text,
of
heat
and
moisture,
as
described
in
the
can be described by the following set of differen
t i a l equations.
= dlv(x
Vo I T
srad
q
T) + E P Y
0 Ft
(I)
+ 6 grad T ) ] ,
(II)
and c
mY0 I t
where T
=
div[Xm(gradu
T ( r , t ) and u = u ( r , t ) are the transfer potentials of heat
and moisture. Now we shall define some dimensioniess variables: x
r/R,
Fo
a t/R2, q
6
= T/T ° ,
9 =u/u° m
q
and the similarity numbers: (i)
the
Luikov
number
of
the
field
of
bound
matter
in
relation to the temperature field Lu
am/ a q „ ,
( i i ) the Posnov number for bound matter
P
6T°/u° , and
n ( i n ) the Kossovich number for bound matter c nu Ko
=
—T"o-
c T q where the superscript 0 denotes the characteristic entity. With the assumption of constant thermophysical
properties,
the Eqs. (I) and ( I I ) can be written in the dimensioniess form with the aid of the dimensioniess
variables and numbers, defined above
30 for the unidimensional body as
32e1(x,Fo)
aOjU.Fo) Tc 3Fo
-
3e.(x,Fo)
r
+
o
~~ x
^ 2
ae2(x,Fo)
^ 3x
+ e Ko
5-5T 3 ro
(HI)
d X
a and
862(x,Fo) 3Fo
=
Lu
+
Lu
Pn
3262(x,Fo) + r~2 x x2 3 6«, ( x , F o ) 3x
0< x < 1,
392(x,Fo) iTx
+
L
_ 36 < (x,Fo)
X
2
(IV)
3 x
Fo > 0;
where T is a form constant, which is 0 for an infinite plate I for a c y l i n d e r and 2 for a s p h e r e . The
boundary
conditions
for
the
system
of
differential
equations (III) and (IV) are 9,
x
d,Fo)
92
xO,Fo)
+ Ajejd.Fo) + A 2 6,
+ B^U.Fo)
x(l,Fo)
*,(Fo),
+ B292(l,Fo)
(V)
* 2
(VI)
and e1;X where
=
0 , A.
(vii) and
thermophysical
B.
are
co-efficients
aggregate and
of
. (Fo)
known
are
dimensionless
prescribed
fluxes
to
to
be
be determined by the e x p e r i m e n t s . Initially, prescribed
the
transfer
SjU.o) where
potentials
are
supposed
as
f ( x ) a r e the given
f,(x) functions.
VIII
31 2.1
The Infinite Circular Cylinder Let
us
take
an auxiliary
problem
conditions (V) and (VI) are replaced 61
xU,Fo)
We problem the
shall
which
the
boundary
X^Fo) now
(2.1.1)
determine
the
solution
for an infinite circular cylinder
conditions
in
by
(VII)
and
(VIII).
For
of
( T = this
this
auxiliary
1) with the aid of
purpose,
we
define
a
Hankel transform with r e s p e c t to space variable x : 1 J
e^p.Fo) where p
p
x 8 x (x,Fo) J 0 ( p x ) d x ,
(2.1.2)
are the roots of the c h a r a c t e r i s t i c equation J,(p)
0
(2.1.3)
and a Laplace transform with respect to time variable Fo, 9 (p,s)
/
1
-sFo 9 . ( p , F o ) d F o .
(2.1.4)
e
o
The inversion formulae for ( 2 . 1 . 2 ) and ( 2 . 1 . 4 ) a r e :
9.(x,Fo)
: 1 °° JPnx> [ x 6,(x,Fo)dFo + 2 \ -^ 0 n= 1 3 Q (p n )
2
e^P^Fo) (2.1.5)
and i
e,(p,Fo)
where
~ -
the
0+ i°°
cc.
] o-i00
eir0
integration
A
61(p,s)ds,
is c a r r i e d
over
i
7-1,
the s t r a i g h t
(2.1.6)
line s
a
in
the complex plane. The transforms equations
simultaneous to E q s .
(Ill)
applications and
(IV)
of
yield
the
Hankel
a set of two
and
Laplace
simultaneous
32
si,
7,(p)
p2e, + eKo[s62
x,(s) 3Q(p)
f"2(p)]
(2.1.7)
and s |
f 2 ( p ) = Lu[3 Q (p) x 2 ( s ) + Lu P n [ a Q ( p )
p
2
^! p29,J
x,(s)
(2.1.8)
for r = 1 . On s o l v i n g these two e q u a t i o n s , we f i n d
z 9
2 .j ,
1 ( P' S ) -
r. L
j,k=l
'
1
fk(P) , 2 2
k
s+Lu p V.
* k
+ M
k '
( p )
~
-—r-r'o^
(2 K9)
•
-
s+Lu p v J
—'
where
v 2 =
j
*(
[1+
Uu"+ eKo
Pn]
* ( - , ) j ( [ l + Eu"+ e KoPn J 2
nr> i} (2.1.10)
and t h e constant c o e f f i c i e n t s a r e g i v e n by
L]
(-l)J ^ V ;
(-,) J ^H-
L.J
v, - v 2 , r * L ,
(-1) v u
, ■
J
2
Pn fn,
L L
v2 - v ,
] 2
2 . v -(-l) —! V W 2 v2
M
I -v V
i
E KoPn •
2
v,
2 EKO
(-I)J — L _
1/Lu
J
2 .
° '
v2 - V | ,
2
eK
V
2
2
Pn L U V .
i-
.
,
MJ
.
(-I)J
v
' 2
v
x 1
E Ko Lu v . , J
33 2
?
]l
(
"')J
.
v
J
Lu
2
2
Pn
M
'
Applying for
the
inversion
temperature
(
2 J
v2 -v,
expressions
2
i /i ?
l/Lu-v ~2 2 ^ v2 - v ,
"I)J
formulae
and
Lu
(2.1.5)
moisture
Pn
and
distributions
-
(2.1.6), are
the
obtained
as
e^x.Fo) -
l
[L;k
P;k 0 j k
] ,
(2...,,)
) , K- I
where
P..
2[
1 f x f, ( x ) d x 0
°° 3Q(pnx) ; —= exp(-Lu n=l J Q ( p n )
+
/ i
x f, ( x ) k
p
v
Fo)
Jn(p x ) d x ] u *n
(2.1.12)
and
Fo Qjk
jk
2[
j
Xk(u)du k
0
+
°° I
3Q(pnx) 1 (o ) —
n=l
W
Fo 2 2 f Xk(u)exp(-p v Lu F o - u ) | , 0
k
n
J
(2.1.13) provided
the c o n v o l u t i o n The
problem.
expressions
In o r d e r
determine
exists.
x .(Fo)
(2.1.11)
to s o l v e from
the
the
are
the
present
original
solutions
problem, boundary
it
to is
the
auxiliary
necessary
conditions
(V)
(VI). S u b s t i t u t i n g the
36 values of = 1 and 6 . at x
( 2 . 1 . 1 ) and ( 2 . 1 . 1 1 ) , we obtain
1 from E q s .
to and
34
X.(Fo)
2 I
+
i
, (A
.>k=,
M I
Fo Mf ) [ / X k (u)du
+ B i
JK
JK
0
00
+
r J n ( p x ) F o . — -— / " 30(pn) 0
7
L
9
exp(-p z v / n ■ J
Lu Fo-u) j ( . ( u )
= V F O ) J , (A, Lk + B. L V
du]
p Jk
(2 , , )
--"
j , k- I
and A
00 +
^o^Pn y ^ l
nil
*2(Fo)
B
1
The values
2 I j,k=1
x,(Fo) + X7(Fo) + 2 B Z l * '
of
l
integral
the
X
Fo
'
exp(
I
1 j,k =l
equations
functions
Xi(Fo)
"pn
A*
Jk
2
P
J
FO ( / Xk(u)du k 0
2 j
Lu Fo-u) X k ( u )
(2.1.15)
by
and
(2.1.15)
substituting
X. (u) in ( 2 . 1 . 1 3 ) , we obtain the e x p r e s s i o n s for Q. expressions
for
the
temperature
du)
Jk
(2.1.14) and
v
M*
and
moisture
determine the
values
the of
and hence the
distributions
are
determined^ In order to simplify
the calculations, we s h a l l determine an
approximate solution of Xi(Fo) for small values of generalised Under
the
Laplace
transformation,
the
integral
time.
equations
(2.1.14) and (2.1.15) take the form
2
A X(s) 1
+
2
I J,k=l
1
(A M '
7
1 ° °
♦ B, M k ) ( l '
t
^(p
x n
)
I
° i L _
n=l
0
K
n
35
*
)
4 s+Lu v . J
P
Xk(s)
= g* (s)
(2.1.16)
n
and A
2 _ J Mk J j,k=l
A
X2(s)
+
2
(s)
X|
2B2
+
}
4 ~ s+Lu v p J "
Xk(s)
. (I
°° I n=l
+
Jn(p x) ° ^ _ 0 Kn
= gj(s),
x
(2.1.17)
where
g^s)
e-sFo
|
p,(Fo)
.|=](A,
Ljk+B,
LJk)PjkjdFo
(2.1.18) and
g^s)
s F
/e-
°
^2(Fo)
L2,
B 2 [ k __,
PjkldFo, -*
where
it
is
assumed
that
<J>.(Fo)
e x p r e s s i b l e i n the e x p o n e n t i a l l y
Equations
(2.1.16)
converges u l t i m a t e l y .
in
Xi(s)
ar|
function
(2.1.17)
are
two
ordinary
A
d
)(,(s)
and contain a s e r i e s
which
Now a p p r o x i m a t i n g the term
2 s + Lu v .
~
\_ s '
p
have
X,(s)
1
continuous
decreasing f o r m .
I
we
a piecewise
and
A
simultaneous equations
is
(2.1.19)
♦ 2
I
j,k=1
L(A
M
'
♦ B
Jk
M2 ) ( J +
'
JK
J
n=l
°
W
n
) 1 J
s
(s)J
k
36
g,(s)
(2.1.20)
and A
2
A
X2(s)
+
A2 X , ( s )
-,
2B2 I
+
[M
»
k(U
J
j ,k= 1
JQ(p
n=I
r
0
A
2
. ( 2B [g ( s ) l W - ^ l s '
A
X,fs) 1
- i M2
1
,
A
X?(s)
d
(2.1.21)
anQl
restric-
3
°°
JZ
j=1
ar,
(1+
.
n
g2(s)
Solving these two equations f o r X i ( s ) I t i n g to the terms of o r d e r — o n l y , we o b t a i n
x)
T T ^ l - ' s >= k ( s ) ]
1
x) n(p On n=l J 0 k F V
i A
I
J
2
f g2(s)
I
1:0 (l
+
(A, M ] 2 ♦ B jM 2 2 )
3 n ( p x) TT77T— > 0ypn'
I n=1
x
]/d*^)
(2.1.22)
s
and r
A
x2(s)
2B
2
[-g/sifv-^ + g 2 (s){l +
f
oo
J
( D
I
3°(n
(,t
n=l
n
,
*
^ MJ, (U
^
J
n^P
x
*
1
3^-)}
I (A, Mj |+B| M*,) x X )
)
0lFn
> \ J/ U + 2|) ,
(2.1.23)
s
'
where a
2 £
(A,
M.,
♦ B 2 M^ 2 - A 2 A ,
M^
» (1
+
A^,
J0(pn
i . 57(71 n=1
0
r
n
M^)
x)
>■
x
37 The
inversion
can
be
c a r r i e d by expanding (1+2^- ) and s 1 0 order s and s only. The inverted
considering the term of e x p r e s s i o n s of X i ( s ) are
X,(Fo)
g (Fo)
3 n ( p x)
°° ( , +
«>
3_(p
I n=1
Fo }
I 3 (p ) n=1 0 V F V
( , +
Fo J g . ( u ) d u + 2B 0
2a
?
M%
2
x
J
2 8)(u)du '
i 0
x)
.
1 (A,M j=l ' J
. +B
M< )
x
JZ
'
Fo }
3 (p ) CTpn;
2 V j=l
&2(u)du
/ 0
(2.1.2*)
z
and
X 2 (Fo)
A2 g,(Fo)
+ g2(Fo)
Fo
2
j g (u)du
+2a
2B
Fo
the
g.(u)du + 2
(I
+
functions 2.2
y,(Fo)
values
?
I
M< ( U
Fo j
g,(u)du
co
3n(p
J
(
u
x)
"
)
x
2
/ 0
co I n=l
Expressions functions
2 a A2
of
3.(p , | n W
(2.1.2*)
which, the
V j=l
after
function
(A
M
+ B. M j . )
J
x) )
and
Fo / 0
x
J
g?(u)du.
(2.1.25)
(2.1.25)
determine
substitution
in E q .
Q., JK
ultimately
which
the
(2.1.13),
auxiliary give
determine
out the
6.(x,Fo).
Solution for a Sphere We s h a l l
moisture
now determine the solutions for the temperature and
distributions
in
a
sphere
with
the
differential
equations
38 (III) and (V
(IV) for
VIII).
problem
For t h i s
defined
equations
f = 2 under the boundary and i n i t i a l
is
in
purpose we shall Eq.
solved
(2.1.1).
by
the
also consider
The
system
simultaneous
of
the
the
applications
conditions
of
auxiliary
differential finite
sine
transform and Laplace transform. The is defined
finite
sine transform
with
respect
to space
variable
as
~ 6(p,Fo)
I f x 6 (x,Fo) 0
1
where p
p
SIR
pX
dx,
P
(2.2.1)
are the roots of the c h a r a c t e r i s t i c equation tan p
p.
(2.2.2)
The inversion formula of ( 2 . 2 . 1 ) can be obtained by 00
Vx.Fo)
where
x
aQ .
I n= I
sin p x an — - - = - ,
defining (2.2.3)
a
(n = 0, I , 2 , . . . , °° ) are the constant coefficients. The n 2 coefficient a„ is determined by multiplying equation ( 2 . 2 . 3 ) by x and integrating
between the limits
x
0 and x
1 . Thus we find
that 1 , a. - 3 | x ' e,(x,Fo)dx 0 The
other
coefficients
equation ( 2 . 2 . 3 )
a.,
a.,,...
by x sin p
(2.2.f) are
determined
by
multiplying
x and integrating within limit 0 to 1,
which gives
1
2p
an
2~sin
p n
r
/ 0
x
^(x.Fo)
sln
P n x dx.
(2.2.5)
39 Thus, the inversion formula e^x.Fo)
°° + 2 I
3 e^o.Fo)
becomes
p
s in p ^-2
2
n=1
sin
x
_^ 6j(pn,Fo)
p
(2.2.6) The
simultaneous
application
of
( 2 . 1 . * ) to E q s . ( I l l ) and (IV) for T
the
(2.2.1)
A
Tjtp)
S1
- x,(s)
p
"
and
2 gives
A
sT,
transforms
^
p2 6 , ( p , s )
+e K o ( s 9 2 - F 2 ( p ) ) (2.2.7)
and
s^
p 2 e 2 (p,s)]
f"2(P) - Lu Lx 2 (s) 5in_E + Lu PnLx,(s)
S1
"
P
p2 ^ ( p . s )
(2.2.8)
A
These
two equations
are
solved
for
A
6.
and
6,.
After
the
application of the inversion formula ( 2 . 2 . 6 ) , we obtain
^Cp.Fo)
I
[LJk (Vjk)
MJ k ( W j k ) j ,
+
(2.2.9)
j, k - 1
where "~ V,
jk
_ jk
and
the
e x vp ( - v sin
symbols
2 j
Lu rp
p
Fo j
L ,
and
p
Jk
^
A
k
2
Fo) f. ( p ) , k
(u)
M,
Jk
e x p (
have
.v/
7
j
Lu
been
7 p^
Fo-u)du,
defined
in
the
preceding
section.
Applying
the
inversion
formula
(2.1.6),
the
expression
40 (2.2.10) i s reduced to 6l (x,Fo)
J
[L^ Vjk
= )
Mjk Wjk]
+
(2.2.10)
where 1 3 f x i. (x) dx + 2 I k 0
V.. jk '
9
exp(-p
1 /
7
v
n
Lu Fo) J
oo . n=i LI
p ^ . 2 sin Kp
sin
sin p
x — dx
EL_
x x
n
x f. (x) k
0
p x
(2.2.11)
%
and
W
J
Fo °o f X k (u)du ♦ 2 I 0 n=1
= 3
Fo 0
ex
P
x "
sin
XL-( U )
J
p - p
sin p r
x n
(" v -
2 J
Lu p
2
Fo-u)
n
du,
(2.2.12)
a r e , again, the roots of the c h a r a c t e r i s t i c equation tan p = p . In order
the function
to solve our main problem, u
X|( )
from
the original
we have to
determine
boundary conditions (V) and ( VI).
Now substituting the values of X(Fo)
+2
+
2 I
(A
o o s i n p x F o I x sin p I Xk n=1 ^n 0
$,(Fo)
I j , k- I
and
1_ and in Eqs.(V) and (VI), we get 3x . , F M + B M ) [3 J ° xk(u)du
(A, L j k
+
_ ex
P
B, l]k)
<"Vj '
Vjk
Lu p
Fo-u)duj
(2.2.13)
41
X2(Fo) + A
+
<° s i n p x I, x sin "p K n=l n
2
2 I
x,(Fo) + B
*2(Fo)
B
2
Fo j. *k(u) 0
L
.1
Fo j Xk
7
M{
[3 _
ex
P("vj
? Lu
Pn F°-u)du]
'
(2 2
- '">)
jk V
J, K- I To solve these equations, we shall again apply the Laplace transform
with
respect
to
variable
Fo.
The
integral
equations
(2.1.13) and ( 2 . 1 . H ) then become A X,(s) X|
+
2 . _ , ) ( A , M1.. + B. M Z . ) [^ t 2 j,k = 1 ' Jk ' Jk s
]
4—2 s+Lu v
i(s)
8*
o o s i n p x ) —x sin pn n A,
(s)
*
(2.2.15)
p
and A
A0
2
v,(s) A
? ) . /• .
A
+ Ax-,<s) + B_
2
2
j,k=1
1
sin p x V ; — 2 — *■ . x sin Kp
3
[-
+ 2
s
A
A
1 s+Lu v .
2
Bz. jk '
2~ ] xk(s)
n=1
x
n
,
g2(s),
(2.2.16)
p
where g?(s)
I
/
e-sF°
[ ♦ (Fo)
I
0
V
JdFo Jk
I
j
)
(A
k
=
, I
L1
Jk
♦ B
L2 ) *
I
Jk (2.2.17)
42 and 2
g*(s)
e"sFo
J
1
[* ( F o ) - B ,
0
2
Equations
(2.2.15)
j)k=,
and
(2.2.15)
and
(2.2.16)
in
(2.2.18)
Jk
are
two
ordinary
A
X.(s)
are
jk
(2.2.16)
A
simultaneous equations
L2.. V - . ] dFo.
I
2
and
X7(s).
convergent
and
The s e r i e s may
terms of
Eqs.
be a p p r o x i m a t e d
for
large value of s by !
\_
. 2 2 s+ Lu v . p J n as a r e s u l t of w h i c h E q s .
(2.2.15)
~ s
and ( 2 . 2 . 1 6 )
<*, M!k ♦ B, ^
x,<*> ♦ \ , j , k = lx
~
X^(s) J
'
+
=
^ ( 's )
I
M
2
reduce to
I x'sY; X) n=1
r
n( 2 . 2 . 1 9 )
and 2
X
,(s)
+
A
X 2 (s)
+
B2
2
A
(3 + 2
I
9
°°
'
n=1
"
)
si n p
x
x
A j,k=1
J Xk(s)
Equations functions
X((s)
(2.2.19)
and X 2 ( s ) .
and
=
K
g2(s)
(2.2.20)
are
now
solved
The values of these functions are
as
/■y B?A A X,(s) = [ g 1 ( s ) + ^ g I ( s )
2 ? I M2(3 J j=1
n
(2.2.20)
°° s i n ♦ 2 I n=1
p
x n K
n
for
the
obtained
43 g7(s) ~ s
2 I .^
. _ (A. M _ + M._ B . ) ( 3 + 2 I j2 j2 1
°° I n ^
p x — — x sin pn s
in
)]/()■£-) s'
(2.2.21) and X2(s)
(3+2
(3+2
[-A2 ™ S ntl
g (s)
sin x
p x g (s) r—5— ) + - < p s n
00
sin
I
3 T i I n - T ) ] / ( , + f> -
- (A,
M
J,
r
B, M 2 ,
(Fo)
°° (3+2
(3+2
and
x
sin
p
A^A,!^
+
B,
M22))
x
2
Fo a J 0
g.(Fo) '
1
( A . M . + B, M . , ) ' J' ' J'
<2-2-22*
B2 M 2 2
+
T r^—). x Sln ntl Pn Now t a k i n g (1 + —) » (1 0 s e x p r e s s i o n s of X | ( s ) a s
x
x
J
n have been negiected and
<» +
M2
p x
+
(3
? I j=]
2 V j=1
sin
n=1 where t e r m s of o r d e r s
a
R -pg.(s)
+ g2(s)
si n p
x
00
sin
p
I, n=l
x s i n "p K n
obtain
2 £ Mz J j=l 2
J0 « 1 ( u ) d u
x
we
+ B
Fo )
I, x ,in "*pn n=l
g.(u)du
—), s
.X. j=l
the
inverted
x
. (A M
+B M 1 J2 , ;2 } J
Fo >
( 0
M
u ) d u
(2
-2-23^
44
X2(Fo)
A2
+a
g)(Fo)
Fo f g,(u)du ■L °2
Fo I g,(u)du 0
Equations
values
x,(Fo)
of
the
The heat
which,
2 V (A, j=l
and
after
functions
.
<=° J i
(3 + 2
sin p x . " x sin p
_ ♦ B, U2)i3*2
M -1
J
(2.2.24)
substitution
W,
which
determine in
boundary mass
properly
)
*
a= s i n p I . n=l
conditions transfer cases
(V)
and
phenomena of
the
the
Eq.(2.2.12),
ultimately
p
Particular by
?
W/, )1
,
jk
and
exchange. obtained
2 LT
x ) r
n
(2.2.2(f)
(2.2.23)
x of
+
B, 2
Fo / g,(u)du
A2a
Fo f g,(u)du 0
x
functions
* g2(Fo)
auxiliary give
the
determine
the
r (VI)
reflect
including
boundary
a large
class
radiation
heat
conditions
can
assigning values of t h e parameters A. and
be B..
45 R E F E R E N C E S
1.
Crank,
3 . , The
Mathematics
of
Diffusion,
(Clarendon
Press,
1965) 2.
Luikov, A.V., and Mikhailov Transfer
3.
Yu., A Theory of Energy and Mass
(Pergamon P r e s s , Oxford,
Smirnov,
V.I.,
Equations
Integral
A
Course
1965).
Equations of
Shukla,
K.N.,
Tripathi, Transfer
6.
Tripathi,
G.,
Series
Differential on
Higher
Ph.D. Thesis,
Banaras
1964).
Heat and Mass Diffusion,
Hindu University ( V a r a n a s i , India 5.
Partial
International
Mathematics vol.IV (Addison Wesley, 4.
and
1973).
Shukla K.N. and Pandey R.N.,
Int.3.
Heat Mass
16 (1973) 985-90. G,
Shukla K.N. and
Technology 13 (1975) 281-83.
Pandey
R.N.,
Indian Journal of
46 Chapter 3 HEAT AND MASS TRANSFER
WITH
CHEMICAL TRANSFORMATIONS
In this chapter, general expressions for transfer of heat and moisture in a sphere in presence of phase and chemical transformations are obtained. Further the expressions for mean values of the temperature and moisture content and their approximate solutions applicable for small values of the generalised time are derived. Numerical values of the characteristic roots and v. are presented in the form of tables and figures. Tire field of moisture content inside the sphere is also studied graphically. There
exist
undergo thermal
a
number
destruction
of
processes
in t h e i r
in
which
substances
dry mass along with the
phase
change of the associated matter when they are treated t h e r m a l l y . The thermal
effects
become more apparent
when the majority h y d r o x i d e are
are
being
pressure, the or
of
heat
affected
temperature,
effect
by
(as
the
and the
that action
of moisture are
inter-related
of
factors
the
various
like
concentration e t c . At low temperature
heating,
being negligible,
evolved
of p r e s s u r e
absorbed
range
magnesium
heated.
The transfer and
in a certain temperature
of mineral substances like b i c a r b o n a t e s ,
case
may
be)
the amount of acts
as
a
heat
heat
source
motivates the process of heating. The equations of heat and
and
moisture
transfer in t h i s case become
Vo
8T
dlv( X
q Srad
^
T)
+
" E — 3T
^
+
(IX)
l-\
q
q 0
and
C
m^0 I f
Equation
dlV
U
m 6rad
u
(IX) is modified
+X
m6m S r a d
from
Eq.
T)
(I)
v
E
lf
of
' the
(X)
previous
47
chapter by including a term
Q/c
yQ
which
is
due to
evolution
absorption of heat in the chemical reaction in the body.
The
or term
E
8u/ 8t occurring in equation (X) is due to molecular transfer and
is
of
very
small
order
and
therefore,
it
may
be
neglected
in
engineering calculations according to Lebedev's assumption . In
the
chemical
reaction,
there
is
also
the
formation
of
gaseous products. The transfer phenomenon of the gaseous products of decomposition is described by the equation 3u, c
3u
Y
div
d o "57
(x
d §
rad
+ x
%
Luikov and Mikhailov
2
d
6
d 8
rad
T)
(XI)
T7
have solved these equations under the
convective type of boundary conditions. They have further
simplified
the problem by considering the uniform potential distributions and a 3 order. Ralko investigated experimentally
chemical reaction of f i r s t
the phenomenon of simultaneous heat and mass transfer in the process of kilning the kaolin and the magnesium hydroxide. He found that the chemical body
conversions
and
then
begin
spread
at
T r i p a t h i , Shukla and Pandey ' mass transfer
the
gradually
the
the capillary
interior
of
porous
the
body.
investigated the simultaneous heat and
in the presence of
for an infinite
surface of into
phase and chemical
transformation
plate and sphere. In these investigations, the system
is subjected to the generalised boundary conditions and the transfer potentials are assumed arbitrary functions of the space-coordinates. The
present
chapter
simultaneous
transfer
of
and
chemical
deals with
a theoretical
study
of
heat and moisture in a sphere with
transformations
under
the
generalised
the
phase
boundary
conditions. The analysis is further simplified for a chemical reaction of f i r s t order and for uniform i n i t i a l distributions. 3.1
Statement of the Problem The
internal
heat
and
moisture
transfer
for
unidimensional
48
bodies in the form of a s p h e r e may be d e s c r i b e d
by the system of
differential equations . 3T
, ^T
dt
q M
3u 37" 3t
2
a
2 3T .
ep
3u
Q
r
c
3t
c
3r
or
q
2 , 3 u a ™ ( —5" m .2 3r
+
2 3u > T -^ I r d r
8u
d
s
, , , ,,
3 t
'
q
2 , , 3 T 2 3T > a 6 ™ ( —y * 7 T7 ' m m m -.2 r or 3r
+
/ ^ i ->i (3.1.2)
and 3u
d
, ^Ud
a
TT
{
d
3u
2
+
T^ F
d ,
) + a
"17
6
.
d d
(
dr 3u - g - p , 0 < r < R,
where
T
T(r,t)
and
t
u
distributions respectively.
chemical
the
thermal
reaction
2
F
3T ,
3? >
> 0;
(3.1.3)
u(r,t)
are
the
heat
and
moisture
The thermophysical p r o p e r t i e s are assumed
constant in writing the E q s . (3.1.1 In
, 32T TT + or
3.1.3).
decomposition
depends
upon
the
of
the
body,
concentrations
of
the
rate
of
the
reacting
components and the products of decomposition. The r a t e of reaction to a
first
approximation
is
a
function
of
the
concentration
of
the
reactants and thus 3u
s
~gf
kf< (U s ) ,
where f (u ) is a given The transformed
system
of
(3....)
function. differential
by using dimentionless
equations variables.
(3.1.1)
(3.1.**)
is
49 a t
_ £
v
and
R '
"
u
T
q
rFo„ _
_L
fl
2 ' °1 R
x0
Q
'
°2 "
the
field
T
_y_
0 ' u
B
D
s
3
0 u s
u. 64 = -Sg- , U
d
and s i m i l a r i t y (i)
numbers:
the
Luikov
number
of
p r o d u c t s of decomposition i n r e l a t i o n to
Lu
a m = — and m a q
(ii) p r o d u c t s of
the
Lu .
Posnov
d
m
temperature
matter
and
the
field
number
for
bound
matter
and
the
gaseous
decomposition
=
(iii)
6.T0 and
— Q u
the
Pn
d
— 0 ud
Kossovich
number
gaseous p r o d u c t s of
„ Ko
bound
a d — a q
6mT° Pn
of
for
the
bound
matter
and
the
(3.1.*)
now
decomposition
0
m
p u . „ = — — « - and Ko . d T0 c T q
(iv)
t h e Hess number
Q, u° d
c
s ^
T0
q T
and
-
Ge
kR2
Ia — d
The becomes
system
of
,
(u
0\n'-1
sS )
differential
equations
(3.1.1)
50
32(x9,)
3(x9,) -, r-
3 Fo
3(x0 ac
3(x62)
,
+ e Ko
. 2 3x ) Lu
3Fo
m
3(x64) ',, r3 Fo
Lu, d
m
m
32(x6^) =; _ 2
3x
K o . -5—p
3 Fo
32(x9,) r—— + Lu ., 2 3x
3(x63)
-, c
,
d 3 Fo
(3.1.5)
32(x9 ) =—— ,
Pn
m
.
+ Lu, Pn, d d „
3x
(3.1.6)
2
32(x6]) ^ 2
3x
, 31x6^ rrr- - rW„ 3 Fo
(3.1.7)
Fo > 0.
(3.1.8)
0
and
3 6, r-p^
Lu d Ge f ( 6 3 ) ,
The
boundary
equations ( 3 . 1 . 5 )
8.
O.Fo)
T
0 < x < 1
conditions
for
and
the
system
of
differentiai
( 3 . 1 . 8 ) can be described as
A, 9 , ( 1 , F o )
+ B, 6 2 ( 1 , F o )
92
x
(l,Fo)
+ A2 9,
x(1,Fo)
94
x(1,Fo)
+ A3 9,
x
v(0,Fo)
0 ,
(',Fo)
*,(Fo),
(3.1.9)
f B2 9 2 ( 1 , F o )
$2
(3.1.10)
t B 3 8 ^ ( 1 , Fo)
<> t 3(Fo)
(3.1.11)
and
6.
(3.1.12)
1 ,x where A. and B coefficients
(i = 1,2,3) are aggregates of known
in t h e dimensioniess
forms
and
$. (Fo) (i
thermophysical 1,2,3)
p r e s c r i b e d fluxes which a r e to be determined by the e x p e r i m e n t s .
are
51 Subscript
, x stands f o r the p a r t i a l
To transfer
make
a
potentials
complete
ei(x,o) where
3.2
f.(x)
statement
are a r b i t r a r y
i n i t i a l moment of t i m e
differentiation we
functions
of
with
shall
r e s p e c t to
suppose
that
space-coordinate
the the
i.e.
r(x),
(3.1.13)
are some known functions in dimensionless
form.
Solution of the Problem Applying
equations
the Laplace t r a n s f o r m a t i o n
(3.1.5
3.1.7), d
1
differential
A T
l —^,2 dx
- xf.(x) 1
to the set of
we o b t a i n 2
A s f,
+ eKo
m
(st
xf.(x)) 2
2
+ L u , Ge K o . x ? ( 6 , ) , d d 3 d^f, * \
at
x.
-
x
f
2
U )
Lu
d'f +
m - ^ 2
(3.2.1)
Lu
m
Pn
(3
m ^ 2
'2-2)
and
s*L - x f (x)
° % — 2 + Lu d Pn d
Lu
d
dx
dx
Lu . Ge
+ —S
w
A
x
f(8,),
o
value
of
Eq.
¥_
and
(3.2.1) its
homogeneous d i f f e r e n t i a l
(3.2.3)
*
w h e r e we have i n t r o d u c e d another From
*j —j
we
function
solve
derivative
in
for Eq.
equation of o r d e r
f
/*.
f_
x and
(3.2.2), four:
6.
.
on s u b s t i t u t i n g we
obtain
the
a non-
52
—j-1. t dx
(1+ e K o
Pn + r- 1 —)s — = - L + f — * i = R ( x , s ) , m Lu . 2 Lu 1 m dx m
m
(3.2.*) where
R(x,s)
j - ^ - (x f j ( x ) m
t K o d L u d Ge x f ( 6 3 ) )
x f"(x)
-2f,(x)+eKo x f " ( x ) + 2eKo f ' ( x ) - L u , Ge K o , x i m I m 2 d d
f"(9,) 3
2Lu . Ge Ko , f ( 6 J . d d 3
The f.(x,s)
solution
(3.2.5)
of E q . ( 3 . 2 . 4 )
can be w r i t t e n as
C. e x p ( v . / s x ) + C ? e x p ( - v . / s x ) + C ,
+ C^ e x p ( - v 2
/sx)
*
=
(vf
{—'— sinh(x-£)v|(/s v. /s
w h e r e C. a r e a r b i t r a r y
2
exp(v?/sx)
[ i
- v 2 )s
R(CJS)
o
— ' — s i n h ( x - £ ; ) v 2 / s } d£ , v2/s
constants and v . ( j
(3.2.6)
1,2) a r e d e t e r m i n e d
by
the equation
v.2
j '
= 2| { l( 1 + E K O
m
Pn
m
+ —!—) + (_1)J
Lu
m
/[(UeKo
m
Pn
m
+
r^—)2
Lu
rLu* - ] J}-
m
(3.2.7)
m
The
expression
for
the
mass
transfer
potential
under
the
53 transform
will
derivative
in E q . ( 3 . 2 . 1 ) .
^ 1 f2(x,s) =
be
found
by
substituting
T h e r e f o r e , we
f. (x,s)
its
second
have
2
2 [O-v,
Ko
and
) Cj
exptvj/sx)
+ (l-v2
)C2
exp(-v.Zsx)
m 2 + ( 1 - v 2 ) Cj
+
-
2 exp(v2 / s x ) + ( l - v 2
I
x
2 2— ( V j - v 2 )s
f I
1-v ^
R
(C> )
l
/sx)
2
y
sinh(x-£ ) v . / s
1
sinh(x-g)v2/s}
y
v
' "
S
) C^ e x p ( - v 2
d£ + j
F(x,s)],
(3.2.8)
where F(x,s)
-xf,(x)+ I
The conditions
Ko
constants and
the
m
C.
xf_,(x) 2
L u . Ge K o . x f ( 0 - > ) . d d 3
are
to
be
determined
condition
of
symmetry.
by
The
the
latter,
boundary on
the
a p p l i c a t i o n of Laplace t r a n s f o r m , now becomes
f,(0,s)
The c o n d i t i o n s two
(3.2.9)
(3.2.9)
reduce t h e four a r b i t r a r y
constants
to
i.e. C.
and
0
thus
the
- - C 7 and C ,
expressions
s i m p l i f c a t i o n reduce to
in
Eq.(3.2.6)
-C.
and
Eq.
(3.2.8),
after
54 A
|
iMx.s)
C sinh v. / s x
|-7-smh(x-C)v/s V - r
O
^ D sinh v ? / s x
+
—K— s i n h ( x - £ ) v , / s ]
1
V —V O
X
r
- — (V| - v 2 )s
/•R(Cs) i
d£
(3.2.10)
£
and * —p zKo
A
¥_(x,s) I
[(l-v, m
-> )C s i n h 1 .
X +
(v,
' - v 2 )s
1 -v? -|
where
2
I — V
? R(c,s) { — ^ £ 1
sinh
o ( l - v . )D s i n h v , / s x 2 I
v,/sx + 1
(x-C)v2/s}
sinh
tx-C>v/s
. d? + J F ( x , s )
,
C and D a r e new constants t o be d e t e r m i n e d
boundary
(3.2.11)
by t h e f i r s t
two
conditions. The
expressions
in
Eq.
(3.2.10)
and E q .
(3.2.11)
can be
w r i t t e n as
A r 6. ( x , s ) = — s i n h
/ D v . / s x + — sinh
{ —J-r- s i n h ( x - £ ) v . / s
1 v./sx +
=
~
(
R{£ , s )
( v , - v 2 ) s x >0 7 - sinh(x-£) v /s } d £ (3.2.12)
and
0_(x,s)
—JT— [ ( 1 - v . m 1 +
—~2 2— ( v | "v2 )SX
) — sinh
x
/ 0
v . / s x + ( l - v ? ) — sinh
. 2 '"vl R(C,s) { — ^ '
sinh(x-c)v]/s
v? ^ix
55
'v
V
2
2
J
s i n h ( x - C ) v 2 / s } d £ + -± F ( x , s ) ]
(3.2.13)
The boundary conditions ( 3 . 1 . 9 ) and (3.1.10) are transformed as
e,
x(i,s)+A,e!(t,s)+B,e2(i,s)
62
x(1,s)
*,(s)
(3.2.1^)
and + A2e)
x(1,s)+B282(1,s)
*2(s).
Substituting the vaiues of A6 . ( x , s )
and i t s first d e r i v a t i v e at
x
1 in Eq. ( 3 . 2 . H ) and Eq. ( 3 . 2 . 1 5 ) , we obtain
C
v JVP7Q Q 1 2 1 v 27
[
7( v~. ^ - v 2~)s 2
(Q
2
R
Q
1
2R2
(3.2.15)
P
2S1
+
P
2S2)
a m
Q 2 B, F ( 1 , s )
P2B2 F ( 1 , s ) ) ]
and
°' w ^ [z^i
(P S| PS2 Q|Ri
'
'
•
w
-(P,J 2 (.|-Q,{,(s)). - g J - j (P, i H i ^ i - Q . B . F d . s ) + P,B2 F ( 1 , s ) ) ] , where
56
P. J
B 2 1 (-1+A.+(1-v ) —r;—)sinh ' J eKom
1-v.2 Q. - ( A . + - p — ! — ) v . / s j 2 Kom j
R J
v . / s + v . / s cosh v . / s , j j j
cosh v . / s + ( ( B - - 1 ) j 2
B 7 i i (_1+A,+(1-V; ) - ^ - ) - V 1 J eKom v./s
' /
R(C,s)sinh(1-5)v./s j
0
1 / R(C,s)cosh(1-C)v / s J 0
+
(1-v.2) —p—J -A.,) s i n h v . / s , e Kom 2 j
d?
d£
and
S.
2
((B2-!)(1-v
) ^ ~
2
1-v + ( g ^
1
. | 0
+A2)
m
Thus,
the
-A2) — ^ 7 - /
R(£,s)sinh(1-Ov ./sd£
R ( C , s ) c o s h ( l - C ) v VsdC '
solutions
for
transfer
•
potentials
for
heat
and
matter under t h e t r a n s f o r m can be w r i t t e n as A
8.(x,s)
= t ( Q i sinh
2
/sx-?2
-(P.
sinh
v _ / s x - Q_ sinh
sinh
(Q,
sinh
A
$.(s)
(s)}/x(Q.P2-P
sinh v 2 / s x - P 2 sinh
v 2 / s x - Q 2 sinh
V)/sx)}
v./sx)-(R.-R_)x
/ { (v| -v
)(QIP2-P,Q2)xs}
2
2 2
* -
Q )
v
{(5.-S2)(P|
♦
v./sx) *
v./sx)
2 * 2 , ( v . - v - )sx
I 0
RCS.sH^Wsinhtx-Uv/s 1
57
—y— s i n h ( x - 5 ) v _ / s ) d ^
- P 2 sinh
+ ( P . sinh v . / s x
v , / s x ) ( 3 F 8 ( ^ s ) + B2
+ B. F ( l , s ) ( Q 2 sinh
(eKo m (Q ] P 2
v./sx
F(l,s))/(EKom(Q|P2-P|Q2)xs)
Q. s i n h
v?/sx)/
P 1 Q2) x s )
(3.2.16)
and ^ 2(x,s)
[ ( Q . ( 1 - v _ 2) s i n h
v V s x - Q ( 1 - v 2) s i n h
A
v . / s x ) $ (s)
6 (P)(l-v2 )sinh
v2/sx
P2(l-V) )sinh
2
2
x f Q ^-
m
( e Ko + [(S,-S2)
v ( /sx) $2(s) ] /
P,Q2))
{ Pj(l-v2 )sinh
v2/sx
P2( 1 - v ] ) s i n h
2 2)
- (Rj-R /((v,2
(Q](1-v2 )
2
sinh
v2/sx
Q2( 1 - V j 2 ) s i n h
Vj/sx}
2
v22)(Q,P2
P,Q2) eKom xs)
1 f
v,/sx}
'
( v .1 - v2.JTZ )eKom x s —2
I0
v
' " R(?
1 ' S ) * V^
|
2
sinhfx-Ov/s
1 -v 2
r— s i n h ( x - C ) v . / s } d £ + ( (21 - v _
v_ *s
2
2
) P , sinh v . / s x
1
2
]
58
P2(l-V|2)sinh
(e2Ko2
v,/sx)
(Q,P2
xs
* B2 F ( ! , s ) ) /
+ B. F( 1 , s ) (Q 2 ( 1 - v , 2 ) s i n h
e2Ko2
v2/sx)/(
F(x,s)
eKo
dF { ,S) dl
P]Q2)xs)
Q](1-v22)sinh
+ -p-J
(
v,/sx
(Q|P2-P]Q2)xs)
.
(3.2.17)
'
m The product equation
expression
under
(3.2.3)
condition
f o r t h e transfer
t h e transform with
potential by s o l v i n g
t h e help of t h e m o d i f i e d
of t h e gaseous the differential
form of t h e boundary
( 3 . 1 . 1 1 ) , and t h u s :
s i n h v<s/Lu .)
A 6
i s obtained
4
( x
'
s )
=
x [ ^s/Lud)cosh
B.,
A
[Ms)
, ,
3
f i
*
^/Lud)+
(Bj-Dsinh
'
vfs/Lu d >]
9 8.
; / { £ f , ( 0 + Lu. Pn.(
is Lu ,)
Q
d
**
d
3 8.
^- + 2 ^2
?(6 ) } sinh /(s/Lu^ «-1)dt -A- f f 1 o *
Lu
(
d%
{ ^/L|U
2 A
^N
3 6.
3 6.
T ^+ 3x
2
_
3-x-'x=C s + f
f(6
o
+ —zri
x
3 6.
^
3) 1
yv
36.
: / (£f„U)+Lu, P n . ( — y 1 * 2 ~ )
x ^ Lud)
J
Q
it
d
d
3 x
2
~)
+ jJ- J d0
s i n h / ( s / L u d ) ( 5 - D - c o s h / ( s / L u d ) ( C - l ) } d£ ? A
.
8 x
3x x = ?
x= £
UfJO
59 Lu . Ge A + — | Cf03)) 0
sinh
4s/Lud) (C-x)dC
,
(3.2.18)
„2-
A
39,
are,
w h e r e t h e values of —5— and — = — are determined dx
» z. d X
by
(3.2.16).
A
The e x p r e s s i o n s
for
the
functions
9 (x,s)
<J>.(s) and f ( 9 , ) ,
context.
the t r u e nature of w h i c h
Therefore,
to
e x p r e s s i o n s , we s h a l l (i)
the
determine
i s not
the
inversion
the
terms
yet defined
inverted
form
in
the
of
these
analysis,
where
apply: theorem
of
the
complex
the e x p r e s s i o n s contain a l l the w e l l defined (ii)
contain t h e
1
A
convolution
theorem
for
the
terms, terms
of
A
$.(s)
and
f(9j).
To a p p l y expression
in
the
the
inversion
denominator
is always greater
theorem, has only
we s h a l l simple
suppose
roots
that
and i t s
the
degree
than t h a t of the e x p r e s s i o n s of the numerator.
Now
to f i n d the r o o t s of denominator, we equate i t to zero and thus
fQ(s)
This
s(Q,
P2
P, Q 2 )
0.
gives
(i)
s = sn
(ii)
s - sn
where s
,
s a t i s f i e s the equation
VSn> or more
0 (a zero r o o t )
Q
1
P
2
P
,
Q
2
°
clearly
Q , P , v nl n2
P. nl
Q n 7 - 0. n2
(3.2.19)
60 where
the
hyperbolic
sines
cosine by s u b s t i t u t i n g s characteristic
equation
P .= u v . nj n j
= -y
and cosines 2 , u
(3.2.19).
cos y *n
are
changed
being
into
sine
and
root
of
the
and Q . are
given
the
The values of P
2 1 v. + (-l+A, + ( 1 - v . ) —r;—)sin y j 1 j eKom' >n
v. j
(3.2.20)
and i 2 . 2 1 -v . I-v. ( A , + —r;—^—)y v . cos y v . + ( ( B _ - 1 ) —r;—*— 2 eKo n J n j 2 eKo m ' ' m
Q . v n iJ
A.Jsiny v. 2 n J) (3.2.21)
For
determination
of
d e r i v a t i v e of t h e denominator
<<Sn>
where
^
* %
v,
*n
Qn2 A n , +
the
r e s i d u e , we need the 2 at s = - y . This gives
value
,
of
the
(3-2.22)
v ^ B ^
v ^ A ^
v , ? ^ ,
(3.2.23)
The q u a n t i t i e s A . and B . are nj nj A . nj
B 2 1 ( A , + ( 1 - v . ) —7}—) 1 j cKo m
cos y n
v. j
y v . sin n j
y v. n j
(3.2.24)
and B 2 ) —Q j eKo
2
B . ni
(1-v
The
1-v.2 cos y v - ( A , + —Q—>—) y v . s i n n j 2 eKo n l
inverted
expressions
of
the
transfer
y v. n I
(3.2.25)
potentials
6 (x,Fo)
can be w r i t t e n as
e,(x,Fo)
-
oo p Fo I f / t(Qn, n=l n o
sin y n v 2
x
- ( P , siny v . x - P _ siny v,x)4>_(u)] nl n 2 nz n 1 2
Qn2 sin y v
e x rp ( - y
n
x)«
Fo-u)du
(u)
61
a 1
f
2
m
; 2 2 2, ^ i H T ( v . - v - ) x n=t K n n
-(RnrRn2)(Qn1
+
s i n
m
1
, exp(-y^
2
(P
pH V
n=1
n1
n n
Fo)](B2 F,(l,s)
B
s i n
Qn2
°° I.
[
nr5n1)(Pn1
x
V 2
B A7-BT " \
EKo-
[(S
+
sinp
V l
x )
Sin
V
nv2
]ex
x
-
P
P("p n
2X"Pn2
n2
s i n
Fo)
sin
v
l
x)
3F.(l,s) ^ )
°°
+ —77—[- -7— eKo A. m 1
— Ly — „ ; — (Q _, sinu v . x - vQ . siny v-.x) v x , y f n2 n I n1 n 2 n=l n n
.
Lu . Ge Ko . FO
e x p ( - i £ Fo)] F . ( l , s ) n '
+ —~ v,
=-* / v2 6
nv2
P
B.
H, (u^-v-^1 2
GO
+
x
I -TTT~ n=1 n n
Lu.
Ge Ko r f
+
v. - v 2
(P
nl
FO Jf 0
sinU
H
x
I (u)[- r -
+
n2
2 -
1
Q . s i n y v_ x )
FO
B
i0H 3 ( u , [ A T I7 1 2 exp(-yn
exp(-y
Fo-u)]
sin
^ n v | x > e x P ( - t^Fo^u)]
°° 1 I T p r (Qn2 siny v n=1 n n
x
Lu . Ge Ko , du + —-rjrm
»
x n=l 2 Kyn4n- ( P n l
sin
V 2 X"Pn2
sin
Lu . Ge Ko , FO . F o - u ) ] d u + —^ B, / H^(u)[- j - + m 0 I
Vlx)x 2
-
du
Vl
x )
62 00
I —y— (Q n 2 s i n u n v ( n=l n n
x
Qn] sinyn v2x) e x p ( - p n
Fo-u)]du, (3.2.26)
OO
9
2(x'Fo)
d<^-^ I 7V(Qn1(,-v22)sinun m n=1 n n -Qn2(l-V|2)sinun
v
x
2
(Pn|(l-v22)sinun
v]x)$|(u)
v2x
2 2 -P - , 0 - v , )sin M v.x)4>_(u)] e x p ( - U Fo-u)du n2 1 n 1 2 n
+ riT— [ f . ( 1 ) - f , ( 0 ) + eKo m f , ( l ) £Ko
I
m
m
2
e Ko f,(G)- J m 2
tS-i 2
ff(l) I
a 2
m (v.
(Pn|(l-v22)sinynv2x-
- v _ )x
n=l
n n
Pp2( I - v, 2 ) s i n ^ v , x)-(R*n)-R"n2)
2 2 2 ( Q n | ( l - v 2 )sin u n v 2 x - Q n 2 ( l-Vj ) s i n y n VjX)] e x p ( - u n Fo) OO
+
I. I 7 V [ ( , - V 2 2 ) P n 1 n=l n n
~^2— e Ko x m -
SlnlJ
nV2X-('-Vl2)Pn2
3F ( 1 , s )
exp(-u n 2 F o ) [ B 2 F , ( , , s ) ,
'
d
2
2)
V
lx]
» -
I
- ^ n n
2 2 sin un v ) x - Q n ] ( 1 - v 2 ) s i n p n v 2 x ) e x p ( - u n F o ) F . ( 1 , s )
Lu . Ge Ko .
♦
n
2B.
]
e Ko x n=l m
2 ( O - V j )Q n
SlnW
'
( v . - v . , )eKo x 1 2 m
eKo x
»
I H,(Fo^,)[-B;=L.
Fo
2 I
0
2
n=l
,
^ ( P n n
n
,
63
2 ( 1 - v , )sinu v , x c n 2
2 2 P , ( l - v . )sinu v . x ) e x pr ( - u u)]du n2 1 n 1 n
2 L u . Ge Ko , 2 T (v, - v A K o x
♦
I
2
m
2
Qn2(1-v, Fo j
Fo °° 1 / H (Fo-u) I —^0 ' n=l V n
2 [Q , ( 1 - v - )siny v x ni z n z
*"Ud ^ e
2
) s i n u n v , x ] e x p ( - p n u)du
^°rl
- ^ ^ m
x Lu . Ge Ko , Fo f H s ( C , u ) ( x - ? ) d u d£ + -=S— ° J H (Fo-u)
0
0
E
Ko x m
eKo x [—^T
i
0
B
2
CO
2
P
I. U V ^ n= 1 n n
n1(,-V22)
n
V
iX
u)]du ♦
m
[Q 2 ^ ' " v i
+
£-|
m
+
eKo
m
2
x
'
( ,
"v1
2 ) s i n
E-Ko-
m
[f
l(x)-
eK
%
V X
! J
°° J n=l
2 v _ x ] e x p ( - u u)du
V x
F o )
x / iAO 0
n2
0
2 i - Q i ^ ' " v 2 )si n U
)sinU
V
P
2Lu , Ge K o , B. Fo ^ ^ - ^ j H (Fo-u)
9
exp(-u
slnM
°mf2(x)- /
f
0
,(^d^
d£.
(3.2.27)
Equation ( 3 . 1 . 8 ) can d i r e c t l y be integrated as 93(x,Fo) To obtain
f 3 (x)
Lu d Ge
the inversion
obtain the roots of the equation
Fo J f( Sjjdu
expression
of
.
%Ax.,s),
(3.2.28) we have to
64 <£/Lu.) cosh
«6/Lu.> + ( B - - l ) s i n h /(s/Lu d ) = 0 .
(3.2.29)
This will give an infinite set of simple roots s = s _ , where s_ ° E q . ( 3 . 2 . 2 9 ) . For more prr e c i s e value of s , m m s a t i s f i e s the changing the
hyperbolic
sine
and
cosine
into ordinary
sine and
cosine,
we
obtain s
2 -v , m '
m
where v
are the roots of the c h a r a c t e r i s t i c equation V
V
V
7 7 ^ cos -rr^- * ( B , - 1 ) s i n y-j-™- - 0 /Lu, /Lu. 3 /Lu. d d d and thus the inverted expression for the transfer
(3.2.30)
potential
A
9.(x,s)
becomes I
-2(Lud)?
fy(x.Fo)
Fo <= f I 0 m=l
(B
3
(
v s i n ( v / / L u . )x / A u . ) - ( v m / / L u . ) s i n ( v m / / L u .)) m d m d m d
e x p ( - v m F o - u ) $5, ( u ) d u 2
»
1 ( 1 - B 3 ) s i n ( ( v m / / L u d ) ( 5 - 1 ) ) - ( v m / / L u d ) c o s ( ( v m / / L u d ) ( 5 - l ))
x *■ I x ( B , cos (v / /Lu , ) - ( v m=l 0 3 m d m
/ / L u ,)sin( v / / L u ,)) d m d
f^(C)exp(-v^old^
-2Lu d Pn d
°° Fo 1 I J I m=1 0 0
(C-1»]/[x(B3
[(l-B3)sin((Vm//Lud)(C-l))-(vm//Lud)cosl(vm//Lud)
cos(vm//Lud)-(Um//Lud)sin(vm//Lud))]
65
•y
sin( v / A u , ) x m d
e x rp ( - v
3
0i
o0i
Fo-u)(—^— + 2 - — ) _df 2 3x x=£ s
m
du
» FO I IT I I I [ ( ' - B , ) s i n l ( v m / / L u , ) ( £ - ! H v m / / L u . ) c o s « v m / / L u .) W ' - , i i J m d ^ ' m d m d o m=l 0 0
U - U ) ] / [ x ( B , cos(v //Lu , ) - ( v m / A u . ) s i n ( v / / L u .))] 5 m d m d m d e x p ( - v^Fo^Tl) f ( 0 3 ) d u d ? + ^
J G(x,£,Fo)d£,
(3.2.31)
where
(-UA, + ( . - v 2 ) ^ i - ) J R ' ( C l - p 2 ) s i n { , . 5 ) U n V
R*nj
1 , , + f R (C,-U„)cos(1-C)unv.d£; J 0
S
;
=((B2-.)(.-v2)
^
-A2) ^
,
j
R^C-U^sind-O^vdC
J 1 , * A2) j R ( C . - l i n ) c o s ( l - O ^ v . d e 0
1-v,2 +( ^ ~ m
d £
,
2 R ( C
u
'" n
)
L^m
X
f
i(x)"x
f' 1 '(x)-2f' 1 '(x)-2f 1 '(x)
it
+ eKo
m
x f-(x) 2
i
+ 2 eKo
H,(Fo)
= L"l[g|(s)-g2(s)],
H3(Fo)
. 3f(0,) L~' [ 3 x -
H^(Fo) - l'h-(HSj))x=]l, -2f'( 83)],
m 2
fAx),
H2(Fo)
L"'[h,(s)-h2(s)],
(UB2)f(83)]
H5(Fo)
L"'[LU-
xf(03)-xf"(03)
66 H,(Fo) = - L u . Ge Ko, L~' b d a
[ - f s
G(C,Fo)
, L-'[(s/Lud)-
g (,)
( ( B 2 - i ) ( . - v 2 ) j J L - _A 2 ) - J ^ | (U(e 3 ) ^ -
J
P n
(6,)], i
J
d
( - ^
♦
m
-2f'(93)sinh(1-5)v./s •'
^ >
j
d?
+(A2
< j * f P 3> I 1
x = ?
0
+£
«"(e3)
m
1-v.2 1 (Cf(63) j - J KQ J ) f m 0 m
-Cf"(63)-2f'(e3))cosh(l-C)v Vs d 5
and
h
J
(-, + A, + (i-v 23 ^L-)
(S)
'
^ J m
j
-2f'(e3))sinh(1-£)v Vs
dC
-2f'( 03))cosh(1-C)v./s
d£.
3.3
-xf.(x) 1
T eKo
m
Cf"(6
)
xf-,(x). 2
Analysis of the Result For a k - t h
order" chemical r e a c t i o n equation ( 3 . 1 . 8 )
ae,(x,Fo) ~ ^ The s o l u t i o n under
c f"(e 3 )
m
+ / ( j ^ - C £( 6 ) 0 m
J
F,(x,s) I
Uf(e 3 ) ^ 0
becomes
. - L u d Ge[ 6 3 ( x , F o ) ] K
expressed
by
the
Eq.
, k > 0
(3.2.28)
is
(3.3.1) thus
modified
as
67
93(x,Fo)
= f3(x)[1-(k-1)Lud
1
Ge Fo f ^ '
'(x)]-1
/ ( k _ l ]
,
k? 1
(3.3.2) and
6 3 (x,Fo)
f3(x)+exp(-Lud
Generally greater reaction
than of
2
it
order
common
interest
(3.2.27)
and
is
are
found
rare I
to
(3.2.31)
that
and,
takes
Ge F o ) , k
the
in
the
chemical
various
place
reduce
1
(3.3.3)
reactions
power
frequently.
complicated
of
plants,
Therefore
it
expressions
9(x,0)
fo.
Therefore
the Eqs.
p o t e n t i a l s are uniform
is
Further,
initially:
(constant)
(3.3.<0
(3.2.26),
(3.2.27) and
(3.2.31) reduce
to
6 (x,Fo)
- -
» |i FO I if- j [(Pn2 sinun n=1 n 0
-(Q - s i n u v . x - Q n j
v)X-pnl
sin Un v 2 x ) *
f
sinynv2x)
(Fo-u) ]
2 exp(-unu)du
00
* I -(A,
I ]lV [ ( P n2 B 2- B 1
n=l
Qn)
Q
n n
for(B,
n2)f02-Qn2 A. *°, > s l n l V. x
Qn,-B2 Pnl)fo2)sinynv2x]
Q * , s i n / L ud, „ < - < » ' -Kod[,+A1
exp(-^
Ge v - , x - ( / _ , s i n / L u , Ge 22 d x(Q2
P'J-P'J
v.x 1
Q") CD
e x p ( - L u . Ge Fo)
of
(3.2.26),
f o r the f i r s t o r d e r chemical r e a c t i o n .
we s h a l l suppose t h a t the t r a n s f e r
order
chemical
+2A. L u . Ge K o d
\ n=1
x n n
Fo)
68 Q„0 n2
s i n y v . x - vQ . n I n
siny 2
n
v_x 2
-> , 2n„ > exp(-u F o);
/ , -, s •. (3.3.5)
(Lu . G e - u ) x d 'n
6
2(x'Fo)
lKS~H m
[(P
I. T~ n=1 n
n2(,"vl
)si^nv,x
2 -Pnl(l-"2
-Q i ( ' - v 2
+
?
2 )
s i n
)sinun v ( x
2 U. v 2 x ) $ j ( F o - u ) ]
°°
exp(-u
u)du
1
2
I , l l V [ ( ( P n 2 V B , Qn2)fVQn2 A1 f o , ) ( , - V ! » S l n V l X n=l Ti n
^ H m
-(A,
2 )sinun v2 x)$2(Fo-u)-(Qn2(1-v]
Qp] f o r ( B ,
Q.(l-v,
Qn)-B2 Pnl)fo2)(1-v22)sinUnv2x]exp(-p2
)sin/Lu.
" A 1 K °d
Ge v - x - Q _ ( 1 - v .
( Q ^
tf,
e x pr ( - L u , Ge F o ) + 2A, Ge Ko , d 1 d
e^(x,Fo)
B , f o,, 3 k x
» I m=l
FO i Am 0
x
°° 1 —-— L) . u f n=1 Mn n v,x 2
_ , 2 .- . exp(-pn Fo);
e x p ( - L u d Ge Fo)
, x
Ge v .
P 2 )x
2 2 Q _ ( l - v . ) s i n yM v . x -vQ , ( 1 - v _ , ) s i n uM n2 1 n 1 nl 2 n 2 ( L u , Ge - uH )x d n
63(x,Fo)-fo3
)sin/Lu,
Fo)
, , , .. (3.3.6)
(3.3.7)
2 s i n
^v
/ / L u ,) x ) e x p ( - v
°° A v o v m m , 2 ._ > ) —= s i n -r, x e x vp ( - v Fo) L. 2 /Lu , m m= 1 v d m
Fo-u)$,(u)du
69
+
Lu , Ge d ~ ^o ~
7
+
°°
B, W o
s i n ' G e x e x p ( - L u . Ge Fo) x(/Ge c o s ^ e + ( B 7 - 1 ) s i n / G e ) i °° A r m 1 ^v mm=l m
n3
v S1R
^
, W~ o
ex
. m X 1 r (Lu . „ , r Ge-v . h ) d d m
P < - ^ d G e Fo)
CXP(
_ 2,- > ^
, "
FO
y
1 M , Y1 n=l n
H(pn. 0
D
P
n2
n2
D
nl)$2(u)
^
D
n2
Qn2
D
nl>
u u Vn n n 2 * , ( u ) ] s i n T L — x e x p ( - u n F O - U X T J — COS 7 5 — d d o
U
.
+ (B,-1)sin -rr^-V 3 /Lu. d
,
du + x
co LT
. m=l
FO
A I m i 0
[(P . D , - P ,D ,),j>,(u) ml m2 m2 m l * 2
v j -1 -(Q , D _-Q _,D , ) * , (u)]sin -TT^- X exp(-v Fo-u)(P ,Q ,-Q ,P , ) v K m1 m2 ^m2 ml 1 /Lu. m ml^m2 ^ml m2 d 00
du
|
-A,
fo (
I -irV[(B2Pn2-BlQn2)f02 n=l n n
, Qn2
exp^Fo)
fo
+
2 D m2- A 1
D
+
nrQn1Dn2)](7Cu7
I
cos
D
nl+ (QnlBrPnlB2)fo2
TTu^ + < V 1 ) s i n
^
n2
)
J -m_ [ ( B 2 P m 2 - B l Q m 2 ) f o 2 D m] + ( Q ^ B , - ? ^ ) m=1 v m
fo (
l Qm2DmrQmlDm2)] ^ m . ^ " ^ ! ^
eX
P("vm
2Lu . Ge Ko , A. °° , y n ~ 1 —3T- (Q |D ,-Q ,D ,)sin j y ^ - x x nl n2 n2 n / L u n=1 WJn ' d
exp(-u*Fo)
D
( ^ d
cos7T^-+(B3-l)s1n o
^ J o
Fo)
70
Lu , Ge Ko . A. x
sin ^ -
oo A l t -2 m=1 v m
, (
x exp(-^Fo)
Qm2
D
m l-Qmi
- \ ^ -
D
m2
)(P
(
P
)
ml 3m2-Qml m2 "
(Q°2 D'-Q' D^MP', Q^-QT, P* 2 )"'
d
sin /Ge x ( / G e cos / G e + ( B ^ - l ) s i n / G e ) "
e x p ( - L u . Ge Fo)
2 <*>
+ 2Pn
I
d
d
U FO
Y1 1 i< p n l ~ 2 ni
n=l *n 0
2
v?
sin y
nv2 n
v2^-1/Lud
x
P
v.
n2 ~~1—]
l
ni
v^-1/Lud
2 V
sin
v
U
x ) « (u)
(Qnl
2 V
2 siny v
2
x-Q
v2 - 1 / L u d
sin
1
v, - 1 / L u d
u v, x)
+ 2Pn d
I ^ [ { (B2Pn2-B Q n=l n
)*o 2 -A, fo, Q n 2 } v
/ ' -1/Lud
sin^v,
2 +
f
(B
,QnrB2Pn1)fVA|
f
°1 Q nl
}
2^,, v2 -1/Lud
s i n
V
x ] 2
2
e x p ( - u 2 Fo) + 2A, Lu d Ge K0(J Pn d
oo
V
J - y - (Q n 2 n=l n n v.
' -1/Lu.
1 V
sinM
n
V
exp(-u
1X
2
n
Q
nl
d
2
2 2 — v 2 -1/Lu d
Fo)+A, P n , Ko, I a o
°°
sinu
n
v
2x)(Lud
v
Ge
2-1 " Un>
2 l
) (Q* —= s i n / L u , Ge)v, x *• , 2 2 , d I n=l v. - 1 //TL u ,
71
v
2 2
-1
-Q°l — j sin^Lu. Ge^v2x)(P. v2 -1/Lud
-
+ f
Pn
eKom
d
fo 2
Qo-QiP?)
e x p ( - L u , Ge Fo)
[Lud(v,2-1/Lud)(v22-I/Lud)]
,
(3.3.S)
where V
Am m
V
V
-1
2 v / L u . ( B , cos -7H7- - y , - ™ - s i n - r r ^ - ) m d 3 /Lu. /Lu, /Lu. d d d
(3.3.9)
and
n
2 v. = P n . ( B , sinu v.+ u v . c o s u v . ) — = — ' d 3 n n > n J J v.2-l/Lu,
. "J
+ A , ( u v. cosu v -sinu v . ) . 3 n j n j n j
(3.3.10)
P ., vQ ., D and P*, vQ"., D° are w r i t t e n i n t h e same form mj' mj' mj j ' j ' j P , Q , D .; the c h a r a c t e r i s t i c root being r e p i a c e d by v and r nj ,nj nj ° ' m ( L u , Ge)1, respectively.
as
Since the
system
matter also
and
the
is
quantity
small
the
transfer
inconsiderable,
thermophysical for
Eq. (3.3.7)
remain
products,
Eq.(3.3.9)
a(x,Fo)
the
. -
gaseous
of
matter
the
term
coefficient
expression
of
in comparison
heat
A,
have
and
unchanged takes the
products
to the through Pn .
the
the
decomposition of
the
gaseous
and
the
no i m p o r t a n c e .
moisture
and
of
quantity
transfer,
expression
In
in
associated
products
is
corresponding this
case,
Eq. (3.3.6) for
the
the and
gaseous
form
» FO v _ I I A m sin T J - ™ - x e x p ( - v Fo-u)4> ( u ) d u m=1 0 d
72 B Q fo,,
°°
A
—5— m=li. iv
Sin
m
+
Lu , Ge F^B3 o
« A I -2 m=l v m
v
vnr: d
X e x p ( v 0 Fo)
-m
v TEST d
sin
9 x(Lud
Ge
-I
V
. s i n / G e x e x p ( - L u . Ge F o ) + T—- [ B , / >- ^ — 7 5 — , , . /r .— W 3 x ( / G e cos/Ge+(B,-1 )sin/Ge) o 5
7
e x p ( - v m Fo)
1] e x pK ( - L u , Ge Fo) d (3.3.11)
The equations for 6 . ( x , F o ) ,
i
1,2,3,4 contain a convergent
s e r i e s so that they a r e suitabie for p r a c t i c a l the generalised
the exponential function certain
purposes.
F u r t h e r , as
time Fo i n c r e a s e s , t h e terms of t h e s e r i e s of t h e c r i t e r i o n Fo diminish
rapidly.
value of Fo > F o . , t h e nature of the transfer
mostly determined by t h e p r e s c r i b e d fluxes For t h e constant p r e s c r i b e d = Ki and $,(Fo) m 5 exchange of heat,
containing For a
potentials
is
^.(Fo).
fluxes i . e . * . (Fo)
Ki ,
K i . , where Ki a r e t h e Kirpichev c r i t e r i a for t h e d matter and gaseous products r e s p e c t i v e l y , t h e
e x p r e s s i o n s ( 3 . 3 . 5 ) , ( 3 . 3 . 6 ) and (3.3.11) reduce to
6.(x,Fo)
=
B 2 Ki - B . Ki -^-5
1
-
A, B 2
Q*. s i n K
°d
[I+A
» I
2 I
E . -~L sinu v
n=]
.=1
>4LUJ Ge) v - x - Q *
1
x ( Q 2 P,
x
2
x exp(-u
n j
sin^Lu .
Ge) v . x
P ^ )
e x p ( - L u d Ge F o ) ] Ki m 9 (x,Fo) = H - ^
2
B2
2
I n=|
I
.^
E
n 2 — J - (1-v )sinu v x
Fo) n
,
n j
(3.3.12)
2 x exp(-M n Fo) n
73
2 2 Q*(l-v_ ) sin*{Lu. Ge) v . x - Q ? ( l - v . )sinv(Lu ,Ge) v. x
A Ko
i d
x(cr2 p*
py^p
exp (-Lu d Ge Fo); 6 3 (x,Fo)
fo 3
(3.3.13)
e x p ( - L u d Ge Fo)
(3.3.14)
and Ki , 6 (x.Fo) = - g S _ b * 3
+
. 1
<=° A J _™ [ K i d m=1 v^
x
B
fo
i
4
Lu , Ge B , — d i-jW ( L u . Ge-v^)
m Sln
m TUT d
X
ex
, 2,- s P<-vmF°> o
o
I n n vT [ ' - B 3
d
]
m
sin/Ge x x ( ^ e cos/Ge+(B,-l)sWGe) 5
e x p ( - L u d Ge F o ) ;
i
]
(3.3.15)
where E
[P
nl = F T " n n
n2(KVB2
fo
2)+Qn2(A,
f
VB1
f
VKiq>
Lu . Ge Ko . A
, / , 2 ' L u . Ge-u d n
^
and E
n2
j f V tP nl (W m -B 2 to i> + «nl (A l f0 1 +,i 1 & 2- Ki q>- A 1 J T ^ T n n The
Q
nl ]
Lu d Ge-U n mean
value
of
the
transfer
potentials
in
sphere
are
obtained from the relation 1
< 6 (Fo) 1
>
3 / 0
x
2
e.(x,Fo)dx. *
(3.3.16)
74
Thus
the
expressions
for
mean
values
of
the
transfer
potentials are obtained from E q s . ( 3 . 3 . 1 2 ) , ( 3 . 3 . 1 3 ) and ( 3 . 3 . 1 5 ) as B , Ki - B , Ki <6](x,Fo) >
2
q
1
I
m +3
2
°°
2
y
y ^ ( ^ v
n=1 j = 1
E .
c
°*Vj
1
n j
-
- s i n g v.) e x p ( - u Fo)+3Ko , [ - ■=■ + A. { (^Lii. Ge) v_
Q 1 sin/(Lu , Ge) v ? ) — y -(/(Lu . Ge) v. v2
cosy(Lu . Ge) v 7
Q
"1
2 , 1 G e ) v . ) — j ] j——g— (Q 2
a coSf{Lud Gei Vj-sin/tLu d
V,
d
Pj-P^)
d
Ge F o ) ,
(3.3.17)
exp(-Lu
a Ki2
<6
2 (x,Fo)>
exp(-u
g-S 2
OD I n=
+
3
I
2 i=1
En .
Y -SI A
Fo) + 3 :
1-v v. .
(
2L)(unvj
'
'
'
cosw v - s i n u n v . )
) 1Kod =— (/j-u, Ge) v_ cos»{Lu, Ge) V ? d
-sin L u . Ge v_) —=- -(/(Lu , Ge) v, cos/iLu. G e ) v . d l l d I d V 2 Q
°2 . "' sin /(Lu d Ge) v , ) — j } (Q2P"-P*2Q* ) exp(-Lu V l
Ge Fo) (3.3.18)
and <eltU,Fo)>
Ki . -^~ 3
♦ 3
°= A Lu , J - n y - * [Ki m=l
v m
B fo
75
Lu , Ge B-.
v 7-JlTrj Wo(LuH G e - v " ) ^ d m 2
v cos -p. /Lu
[1-
WQ
/Lu
d
] * d
3B3(/Ge cos / G e - s i n / G e ) cos/Ge+(B3-l)sin/Ge)]
(
e x p ( - v m Fo)-
v sin -p.
eGe x p (Fo) -Lud 3.4
x
Ge(/Ge
(3.3.19)
Analysis of the Solution The
graphical
characteristic
roots
method.
this
For
u (n
1 , 2 , . . . , °° )are
purpose,
the
obtained
trigonometrical
by
equation
(3.2.19) i s written in the form
u
N
A,-l
(3 1 1) U.*.U
'
where M = Q n 2 usin uvj
Qn|usinyv2
and N
cosuv
Q j(uv2
Q
2
+
( u v , cos iiv,
2 C-v2 '
+(l-v,
2
e
B 1 K0
m
sinuv2)
B l ) -^p-
sinuv,). m
The M/N
and
a b c i s s a e of the
characteristic to determine 0.3,
Ko
the
points
straight
line
roots u
at different
the
= 1.2,
values of u e
m
slope of the line Y located in the f i r s t
0.5,
B,
of
interaction
Y = u /(A.-I) at 1.8,
give
of the
the curves Y values
of
the
A.. Figure 3.1 has been plotted different B,
A.
(Lu
10.0 and A,
0.3,
Fe
0.5).
The
i
u / ( A . - 1 ) is (Aj-1)~ . For A. > 1, the line is quadrant and for A. < 1, it is located in the
(a) 0.05 < A.
< 5.0
(b)
51.0
r o o t s of E q . ( 3 . 4 . 1 ) 6.0 < A, <
F i g u r e 3.1 - Determination of c h a r a c t e r i s t i c
77 fourth
quadrant.
unity,
then the s t r a i g h t
Further,
if
the
thermophysical
line Y = u / ( A . - 1 )
coefficient
A.
is
becomes parallel to the
ordinate a x i s and we find u, It
can
1.20, also
u2
be
2.30,
seen
u3
that
3.5,
there
is
an
infinite
number
of
the
of
the
c h a r a c t e r i s t i c roots occurring in the ascending order P, < u 2 <. u 3 In most characteristic
of
the
roots
cases, are
the
first
sufficient
to
or
first
study
two values
the
various
transfer
processes. The determined
values from
of
the
other
the Eq.
characteristic
(3.2.30)
in a similar
values of the c h a r a c t e r i s t i c roots thermophysical coefficients Figure 3.2(a) and
v_ to
B,(0 of
Lu , for
roots
v
can also be m way. The first two
at different
v
values of Lu . and
m
d
A, are given in table I .
shows the relation of the c h a r a c t e r i s t i c roots v . small
values of the thermophysical
coefficients
< B , < I) and Figure 3 . 2 ( b ) shows the same for large
B,(1.0
<
characteristic
B, < root
51.0). v.
From the c u r v e s ,
varies
significantly
coefficient
B 3 in the range (0
values of
v. are almost uniform.
we see that the
with
< B 3 < 5.0)
values
the
first
thermophysical
and for B 3 > 5.0,
The second c h a r a c t e r i s t i c
the
root
v_
shows a very small variation in the range (0 < B , < 5.0) but after t h a t , it also becomes uniform in the interval (11.0 The quantities
expressions v. which
for
depend
the
transfer
v. = v . ( F e j
potentials
on the s i m i l a r i t y
Luikov for mass
j
, Lu ) .
m'
m
< B, <
51.0).
contain
number of Fedrov
other and
2.8628 2.9930 3.0406
2.4566 2.5435 2.6164 2.6770 2.8803 2.9960 3.0415 3.0655 3.0803
1.0144 1.1444 1.1278 1.2852 1.4314 1.4965 1.5203 1.5327 1.5400-
0.5087
0.5232
0.2028
0.2288
0.2455
0.2570
0.2862
0.2993
0.3040
0.3065
0.3080
2.00
3.00
4.00
5.00
11.00
21.00
31.00
41.00
51.00
0.6160
0.6131
0.6083
0.5992
0.5760
0.5354
0.4913
2.3562
0.7854
0.4712
0.1570
3.0801
3.0651
2.5704
2.4557
2.2889
2.0288
1.5708
1.5044
1.3525
1.1656
0.9208
1.00
2.3021
0.5828
0.4604
2.3455
2.2800
0.4604
0.4560 2.3239
0.1165
0.50
0.5423
0.4860
0.4217
0.6762
0.0920
0.30
2.2558
2.2555
2.2530
0.2710
0.4511
0.4511
0.2430
0.2105
0.3450
0.2445
0.7522
0.0542
0.10
0.4506
2.2510
2.2490
1 0.0000
V
0.4648
0.0486
0.08
0.1222 0.1725
2 2.2467
V
0.4691
0.0421
0.06
0.4502
0.4498
0.0000
1
0.25
0.1352
0.0345
0.04
2
V
0.1504
0.0244
0.02
V
0.4493
0 .01
0.90
0.0000
1
0.0
3
V V
2
1
6.1606
6.1311
9.2403
9.1953
9.1218
8.9790 6.0831
8.5884 5.9921
7.7112 5.7606
5.3540
7.3671
5.2829
6.0864
4.9132
6.8667
4.7124
4.7124
5.0870
4.5132
4.0575
3.4968
2.7624
1.6269
1.4580
1.2651
1.0350
0.7335
0.0000
V V
2
18.4818
18.3933
18.2493
17.9763
17.2818
16.0620
15.6987
15.2610
14.7396
14.1372
14.0733
13.9437
13.8126
13.6803
13.5351
13.5336
13.5204
13.5069
13.4937
13.4802
9.0 1
15.4005
15.3255
15.2030
14.9650
14.3140
12.8520
12.2785
11.4445
10.1440
7.8540
7.5220
6.7625
5.8280
4.6040
2.7115
2.4300
2.1085
1.7250
1.2225
0.0000
V
FOR DIFFERENT Lu ,
4.6911
4.6479
4.6042
4.5601
4.5117
4.5112
4.5068
4.5023
4.4979
4.4934
1 .0
VALUES OF THE CHARACTERISTIC ROOTS v
0.70
B
TABLE 1 :
V
2
30.8030
30.6555
30.4155
29.9605
28.8030
26.7700
26.1645
25.4350
24.5660
23.5620
23.4555
23.2395
23.0210
22.8005
22.5585
22.5560
22.5340
22.5115
22.4895
22.4670
25.0
£
79
Figure 3.2 - Relation of t h e c h a r a c t e r i s t i c roots v (a)
0 < B3 < I
(b)
m
to Lu
1.0 < B 3 < 51.0
80
The
various values of v. for
different
Lu
and Fe
j
m
in t a b l e 2. From the t a b l e , vary' significantly with Lu m o J increase little
of
each
with
of
the
of
the
affect
gaseous
products
criteria.
The
quantity
are
roots
v
only affect
0.(x,Fo)
potentials of
formed
Fe
v_
increases
very
but it
decreases
with
m
products
the transfer
6
we see that the quantity v. does not and Fe m and it diminishes with the
the increase of the c r i t e r i o n
the increse of Lu . m The c h a r a c t e r i s t i c
are given m
only
due
and
the
the transfer characteristic
heat and m a t t e r . to chemical
potentials roots
Since the
reaction,
the
u
gaseous
quantities
Ki . , v , B , and Wo can be associated with the process of chemical a m i reaction. The quantities Lu , and Ge are coming together in the expressions
of
the
transfer
potentials
of
heat
and
matter
and
they
govern the r a t e of chemical reaction so they a r e r e s p o n s i b l e for transfer
of
Ko ,
the
is
heat
and
mass due to thermal
Kossovich
criterion
for
destruction.
the
gaseous
The
products
and
defines the nature of the chemical r e a c t i o n . For Ko . > 0, i t that
the
reaction
is
of
endothermic
type
and
proceeds
the
quantity it
signifies with
the
absorption of heat and for Ko . < 0, it signifies that the reaction i s of exothermic type and proceeds with the evolution of h e a t . Now potentials
we
consider
applicable
for
approximate
solutions
small
of
values
for
these
the generalised
transfer time,
Fo.
For small values of Fo, the values of s is large and for large s we have v./s sinh v . / s
i?
cosh v
/ s *?
? e
]
Under t h e s e approximations and r e s t r i c t i n g to the terms of o r d e r s only, the E q s . (3.2.16) e,(x,Fo)-fo. I
I
(3.2.19)
give
= 77-[(N._ fo--A.N,_ f o . ) / F o i e r f c ( v . Mx
- ( N . . fo 2 -N 2 ]
1L
I
\
Aj f o , ) / F o
11
I
i erfc(v2
1
f^x/Fo)
Hoc/2/Fo)
3/2
81
w\ —
r-v (^
N O 0 ON iO
-3-^-1 ON m
JS
N
00
-3-
00
(N
OO
—
0 - 3 -
o r ^ i
O
CN
O
CN
O
CN
O
CN
—
CN
O ^ J -
N
O
CN
^3-00 O
—
ON
tf>
N N as
.
N
•
_
O
CN
OO
j - o
CN.
O
— N
u"\ N
0 0 0
O
I A C N
o
C
N
O
C
N
O
f
N
O
C
N
CN
N ON
o ^ N w\ -3- (N — ON C N N O ' J S O O v D v O — ON N ON .joo M o o —
m
> u_
o
■*
3
< > CN ID -I to
O
C
N
r^i\D N .3CN, ^
-3-30> OO
O \£ f\. N
\£> O N ON o o o
vD M u"\ oo
-3" (M oO
N
00 u^ vr\ _ — .3OS .3-
ON ON
ON
0 - 3 -
O
CN
O
CN
O
CN
O
CN
O
O
—
— CM
W\
O
r*N
O
ON
tr\
\C
c»N in
^ CN, CN -3- -3.3- N 0 0 " " \ \ 0 0 0 - 3 - C N . C N — A ON CN ON N ON 3-
ON
O
-3-
O
O
N
W
CN
-3" CN 00 TN ^ \ D N
O
-3"
N
O
C
N
CN.
0 - 3 -
O
vO ON O 0 ON
U"\ 0 0 S f ON
ON 00 — ""N
^ ON O O
O
fN
M
fv.CN «"\ — O
S
0 0 0
ON
CN \D
G
V
D
o — ^
N
00 CT>
O t N
O C N
O C N
O f N
O
v O — — CN i r \ —
c*N — 3- CN n O N
(■** \D —
0 0 0 0 0 — ON CN
0*^1 — OO oo —
O
N
N
O
O t N
CN,
N
H
O
N
CN
O
N
v£) — OO —
—
O O O O O O N
O f N
O C N
O
—
-3" — ^
O C N
O C N
O t N
O C N
O
—
ON
0O ^N vD O v f l N VO
ON
-3- 00 3- (N i A * CN.
rr\ \T\ — ■& ^ N <js _
N 00 N N (MvO
C\ rr\ —
ON
O as cjOO
O t N
O t N
O —
O
—
c> W\
N ©o - a - j "A g^ ON o
CN. 3-
— CN r«-\
CN —
ON
OO
Of-vj
O —
O
—
0 - 3 -
O
CN.
O C N
tN
rs. CN 00 ON
(n .* — CN
N
O
O
CN
O C N
-3-
0 - 3 "
oo OO -3" vD
-3N J\
N^>
ON
o N — f«N
O C N
OV
N
°*N
ON
vo 3r«N ON
CN
•— ON 00 — sD OOtN ON ON CN 00 N CN \ 0 v O O OO.* O O N CN^O N 0 0 C N r * % . O N v O \ D v O 0 N u ~ \ N » . — ON\D ONtN O N O ON 00 O N N
Or^\
O C N
O C N
O t N
O —
O
—
m o o .3-r-K (*>\o oooo ON S *r\,oo r ^ . o T^.fn ■& a\ — ■& r ^ t n m o i o v m &■ & O N o o o N N O N O o o v o o o f ^ N v o r v m ON-* O N — O N O N O N t N O N O O N O O O N r ^
•— •
O
*
O — O C N O N 0 - 3 — ■—,
—
>
>>
CN
—
>>
OC'N.
O C N
O t N
O C N
O —
O O O O
O O N O — O 00 Ou'N
O O O \D O f N . O C N
O O O O O O O O
O N 0 " A O C N O O O
—
—
—
C N C N v O —
—
tN
—
>>
(N
CN
—
>>
CN
CN
—
>>
fN
CN
—
CN
> >
—
O
O c - \ O O O O N O v O
—
—
—
—
tN
—
P U-
o 0
«
" O
—
\
O O
—
u
O
N C
O w N f N O
O
^ O C * N O
.
«
—
> >CN
> >
r -1
O
CN — o
CN oo
\ 0 \ D 00 OO u*N ON OS M
r>.vr\ O .3- vs. ON 0 N O N O 0 ON^JON
tN
3
vo (N
Y * N - 3 - C N . 3 -
— w\ 3- .=»• — OO — (N r ^ J 0 0 0 N O N O CN oo ON — -3-oo oo CN (N o ^D-=f o o < ^ r ^ . - 3 - > n O N - * \ o ( N i A — O O N O O ON "N. ON CN o\ *c o> m as — as o oo oo
"^
<
HN
o -3-
\ 0 — ON u™\ N N0 OS us.
O
1/1 UJ UJ
-
OO CN. lAVD 0O<*>
\D
>
3
tr\ —
ON
^
ON
rt
-
r<\ U"A
u"\
CN
( N O N O N O O N O O
o o —
vO N v~\
O
O0
ON
f
^ ^ O
15.00
5.00
*.00
3.00
2.00
1.00
0.80
0.60
0.40
1.0835
0.9999
V
1.2096 0.2166
1.1627
1.1128
1.0591
1.1471
1.1007
1.0518
2
0.2250
0.2345
0.2454
V
v
1.1915
0.3696
0.3846
0.4018
0.4222
0.9999
2
1.2170
1.1692
1.1180
1.0622
l 0.2581
V
0.4108
0.4276
0.4472
0.4706
0.9999
I 0.4072 0.9999
2
|
v
1.2300
0.4693
1.2340
1.2750
0.2025
1.2965 0.2092
0.3449 1.2541
1.3050
0.3831
1.3198
0.4374
1.3513
0.5232
1.4596
0.6850
1.5193
0.7358
1.6229
0.7954
1.8316
0.8632
0.3565
1.2621
0.3961
1.2761
0.4524
1.3065
0.5411
0.5617 1.2587
1.4142
0.7071
1.4749
0.7580
1.5811
0.8164
1.7957
0.8804
1.3650
0.7325
1.4269
0.7835
1.5364
0.8402
1.7582
0.8992
0.5000
1.1807
1.1274
1.0682
2
0.4889
0.5120
0.5404
V
v
1.0000
1.2069
0.5858
1.3106
0.7629
1.3740
0.8137
1.4878
0.8676
1.7185
0.9200
1.1496
0.6150
1.2483
0.8010
1.3136
0.8511
1.4338
0.9003
1.6764
0.9431
l 0.5773
2
l
V
v
2
0.6525
V
0.7071
1.2395
1.1705
0.9019
1.0000
1.1180
0.8543
1.3712
1.0003
2
0.9414
1.2909
1.6310
1.5811
1.0000
0.9694
1.0000
1 0.9996
v
v
2
|
l
v
v
V
v
2
l
v
v
Table 2 (cont'd)
1.3145
0.1964
1.3372
0.0044
1.3461
0.3714
1.3614
0.4240
1.3937
0.5073
1.5023
0.6656
1.5611
0.7161
1.6624
0.7765
1.8660
0.8473
1.3528
0.1903
1.3763
0.3249
1.3855
0.3608
1.4012
0.4120
1.4341
0.4930
1.5426
0.6482
1.6006
0.6984
1.6999
0.7594
1.8991
0.8325
1 .3900
0.1857
1.4142
0.3162
1.4236
0.3512
1.4396
0.4010
1.4729
0.4800
1.5811
0.6324
1.6383
0.6823
1.7359
0.7436
1.9312
0.8187
1.4261
0.1810
1.4508
0.3082
1.4604
0.3423
1.4766
0.3909
1.5102
0.4682
1.6180
0.6180
1.6745
0.6676
1.7706
0.7291
1.9623
0.8057
8
55.00
45.00
35.00
25.00
1.0505
0.9999
1.0500
1.0497
2
1.0496
0.9999
v
0.1284
1 0.1348
v
2
0.1420
0.9999
v
l 0.1490
v
2
0.1609
0.9999
V
l 0.1690
v
2
0.1903
l 0.2000
V
v
Table 2 (cont'd)
1.0968
0.1229
1.0971
0.1358
1.0976
0.1539
1.0985
0.1820
1.1420
0.1180
1.1424
0.1304
1.1431
0.1478
1.1443
0.1747
1.1854
0.1137
1.1859
0.1257
1.1867
0.1424
1.1881
0.1683
1.2272
0.1098
1.2277
0.1214
1.2286
0.1375
1.2302
0.1625
1.2676
0.1063
1.2682
0.1754
1.2691
0.1331
1.2709
0.1573
0.1002 1.3446
1.3067
1.3453
1.3815
0.0976
1.3822
0.1078
1.3833 0.1108
0.1221
1.3464
1.3853
0.1443
0.1255
1.3483
0.1483
0.1031
1.3073
0.1402
1.3083
0.1291
1.3102
0.1526
1.4174
0.0951
1.4181
0.1051
1.4192
0.1190
1.4213
0.1407
8
84 + (N,., K i 31 m
N , . Ki ) / F o 21 q
i erfc(v. 2
( N , - Ki - N - - Ki )/Fo i e r f c ( v . 32 m 22 q 1
Nx/2/Fo)
fo
+ K o d ( l - e x p ( - L u d Ge Fo))
V
x
'
F o M o
2
(Nn
£
-K^-M^[(N12 m
fc^-A,
+ ( N , , Ki
31 -
N2|
f
(3.4.2)
V
l
N22)d-v,VFo
fo])(l-v22)/Fo
N_. K i ) ( l - v
m
A
21
q
2
e r f c ( ^ ^ )
i erfc(v2
fo
)/Fo i erfc(v-
1~/2/Fo)
2
2
N,_ K i ) ( l - v . 2 ) / F o 22 q 1
(N,~ Ki 32 m
i
i erfc(v. foc/2/Fo)] 1
(3.f.3)
and
94(x,Fo)
= f(Kid
B3 fo4)v(Lud F o U erfc( f ^ x / 2 / L U j F o ) ,
(3.4.
where N J
= (-])J((B2+A2B1)
v
♦ B
,-v.2 L_ m
v),
] r R r
l-v.2 N
2j =
(A
%
2 ^ n ^ Y
and (v. H
-v., '
2 g
eKo
m
)/Lu 5
.
(
-|)J
V
j
85
The e x p r e s s i o n s approximating 3.2.19)
sinh
for
v/s
9(x,Fo) x
v.
near centre can be obtained /s
x,
thus
the
Eqs.
by
(3.2.16
become
6.(0,Fo) - fo. -, ' [ { ( 1 - v 2 + e K o A,)Ki - eKo Ki } x 1 1 , 2 2 , 1 m 2 q m m ' (v 2 - v , ) v ? erfc ^j%2/-Fo
+ (eKo
m
l{
? ( l - v , +eKo A-)Ki -eKo Ki J } erfc 2 m 2 q m m
v./2/Fo 1
B_ f o - + ( A . f o . - B, fo,,) (1 - v 2 +eKo A.)) x 2 2 1 1 1 2 2 m 2
erfc v. /2>4 : o
B^fo0+( A, f o . - B . f o . ) (I - v, 2 +eKo A.,))
(eKo
1
m
2
2
1
1
1
2
1
m 2
erfc v 2 /2vFo]+ K o d ( 1 - e x p ( - L u d Ge F o ) ) ,
¥°'F0)-f02
, 2 2 , , [ ( l - v 2 2 ) ( , - v , 2 + e K o m A2>Ki - K i J ( v 2 - v , )eKo m
erfc v . / 2 / F o 2
(1 -v , 2 ) ((1 -v 2 + eKo A,)Ki 1 2 m 2 q
erfc v , / 2 / F o +(B. fo, + (A, f o . - B , I
(l-v
2
2
(3.4.5)
2
2
1
I
I
Ki ) m
fo,)( 1 -v ,P_ +eKo 2
I Z
m
x
A_)) 2
x
) e r f c v - ^ v f o - t B , fo- + A. f o . - B . fo,) (1 -v , 2 + e Ko A,)) 2 2 2 1 1 1 2 2 m 2
( l - v , 2 ) erfc v , / 2 / F o ]
x
(3.4.6)
and 9(0,Fo)
= 2Lu d (Ki d
B ? fo^) erfc l / 2 / L u d F o .
(3.4.7)
86
The
Eqs.
(3.4.2
3.4.5)
functions which are sufficiently the
influence
temperature thermal almost
of
these e r r o r
over
the
destruction linearly
for
can be seen that proportional
contain
integral
of
the
error
small. From Eq. ( 3 . 4 . 2 ) , neglecting functions,
initial
distribution
(chemical
reaction)
small
the
values of
the formation
we find
that the excess of
in the of
time.
the
process body
From
is
and
due
it
to
varies
the Eq. ( 3 . 4 . 4 ) ,
of the gaseous products
to the square root of the generalised
time,
is
it
directly
Fo at
the
surface of the body. Figure 3.3 shows the relation between (8-.-fo-J/Ki and Fo for ° I 2 m the surface and centre of the s p h e r e (Ki Ki ) under the simple v
m
Y
q
boundary conditions of second kind (A, B. = B,, 0, E ' I l i = 1.2 and A- - P 0 . 5 ) . The transfer potentials for with
time. In small range of the generalised time, the matter
transferred
from
the
reaction
surface
speedily
for in
small
values
comparison
of
m is
unaffected is
chemical
matter
generalised transfer
the
0 . 5 , Ko
the
with
the
of moisture from the c e n t r e . Figures 3.4 and 3.5 show
the
distribution of moisture and i t s gradient inside the s p h e r e . From the c u r v e s , we observe that the moisture transfer
from the layer
nearer
to the surface occurs at a fast r a t e and the rate slows down as the layer goes farther
to the surface.
The moisture is transferred
from
the centre towards the surface. The e x p r e s s i o n s small
values
for
mean
values of
of Fo are obtained
6.(x,Fo),
by using the Eq.
i
(3.3.16).
we have < (e , ( x , F o ) - f o 1 ) > = ^
(Nn
fo 2
TT(V,,FO)
2
[ ( N 1 2 i°2~A\
N 2 ] A] f 0 ] ) n ( v 2 ,
(N„
N 2 2 fo,) 3i ( v ] ; F o )
Fo) + (N 3 ] K i m - N 2 ] Ki )
„ Ki ) 7i ( v . , F o ) ] 32 Ki m- N 22 q 1
1,2,3
for
Thus,
87
Figure 3.3 - ( 0- - f o _ ) / K i
versus Fo for sphere
88
Figure 3A
Distribution of moisture in a sphere (Ki m
Figure 3.5
Ki ) q
Distribution of moisture gradient in a sphere
89 + K o d ( l - e x p ( - L u d Ge Fo))
(3.4.7)
< ( 6 2 ( x , F o ) - f o 2 ) >= ^ [ ( N , 2 fo 2 -A, N 2 2 fo, )(I - v , 2 ) it ( v ( , F o )
-(Nn
+
(N
N 2) A, f 0 ) ) ( 1 - v 2 2 ) IT ( v 2 , F o )
fo 2
3I
Kl
N
m
Kiq)(.-v22)ir(v2,Fo)
2I
N 2 2 Ki ) ( ' - v , 2 ) T ( v ) f
(N 3 2 Ki m
Fo)]
(3.4.8)
and <e^(x,Fo)>
3(Ki d
B 3 f0i()TT ( L u d , F o ) ,
(3.4.9)
where 3/2 - ^ 3v. A J
Tt ( v . , F o ) = — ^ - ( 4 F o ) 3 / 2 i 3 erfc ( v . / 2 v F o ) - F o / v . J J J v. J J - ' ,2,3; Equations
(3.4.7
3.4.9)
,
l/Lud •
v3
contain the term
i
erfc
(v / 2 / F o )
which is very small for small values of Fo. Neglecting the influence of
this
depending process.
term, upon
we the
see
that
these
generalised
transfer
time
Fo
processes in
the
are
beginning
linearly of
the
90 REFERENCES 1.
Lebedev, P . D . , Int.3.Heat Mass Transfer
2.
Luikov,
A.V. and
Mikhailov,
Y.A.
Transfer" (Pergamon P r e s s , Oxford,
"Theory
Ralko, A.V., I n t . 3 . Heat Mass Transfer
k.
Shukla,
5.
1973).
Tripathi,
and
G.,
Shukla
and mass transfer Chemical
transformation
International (Belgrade, 6.
Tripathi, Transfer
K.N.
in an infinite under
Mass
I (1961) 273-279.
"Heat and Mass Diffusion",
Hindu University ( V a r a n a s i ,
of Energy and
1965).
3.
K.N.,
1 (1961) 294-301.
Pandey
P h . D . Thesis
R.N.,
Banaras
simultaneous
heat
plate in presence of phase and boundary
conditions,
Seminar on Recent Development on Heat
generalised
Exchangers,
1972). G.,
Shukla,
K.N. and Pandey,
18 (1975) 351-362.
R.N.,
Int. J.Heat
Mass
91
Chapter *f HEAT AND MASS TRANSFER DURING INTENSIVE DRYING
An analytical approach has been made to determine the temperature, moisture and pressure distributions in the drying of an infinite plate. The expressions for mean values of these distributions over the plate thickness have also been obtained. The variations in these distributions and their gradients with respect to space and time are presented graphically. Analytical result indicates that the process is intensified by the filtrational drying. In the process of drying, the moisture is transferred the
material
which
evaporates
from
the
surface
of
inside
material
to
surrounding medium. In general, the rate of drying depends upon the intensity
of moisture from within the material towards its surface.
3 7 a The experimental researches of Lebedev , Maximov proved that intensive
the
phenomena of the exchange of
drying
hydrodynamical
are
influenced
forces.
Luikov
by
the
and others have
heat and matter
action
has shown that
of for
the
in
various
non-isothermal
conditions the total flow of mass in this case is equal to the sum of the mass flow through the process of diffusion, thermodiffusion and filtration. The system of differential equations of the exchange of heat and moisture with
the molecular
and filtrational transfer of energy
and matter can be presented as: C
q Y0 | T =
div( X
q Srand
I i
c
m yQ | f =
div(X
T
>
+
Ci(qm
m Srad
u
+ X
£ P cm ^ 0 f f grad T)
m
&
Srad
(XII)
T
*
X
p 8 r a d P>
92 and
C
where
Y
p
3p 0 I t
the
first
the change i n denotes the
the
tionai)
denotes
on the
for
(XII)
Eqs.
moisture
porous
the
convective
convective a number
type
drying of
Tien
basic
Toei
In (XII
the XIV)
hand
due
content
and
Eq.
bodies
arbitrary
functions
of
like
V I
„ ,
the second
term
transformation
and
heat
by
are
currents
the
t e r m of r i g h t may
infinite
be
plate
conditions 2
have
intensive
chapter,
The of
denotes
of
differential (filtra-
hand s i d e
of
neglected
in
and
and
with
proposed
a
sphere uniform
new
porous b o d i e s .
the
solved transfer
space
system
under
of
the
potentials
coordinates
at
are the
for
the
in
body
laws.
differential most
of
initial
However,
h e a t i n g , the surface of
in
under
model
the c o n v e c t i v e t y p e of i n t e r a c t i o n
been
conditions.
(XII)
,
hydrodynamicai
(XIV)
mechanism of c a p i l l a r y
boundary
the
Eq.
heat,
phase of
( X I V )
i n v e s t i g a t e d the condensation process g Mikhailov has solved the system
Okazaki
present
of
XIV)
the t h i r d
boundary
have
of
to
3T 3T
BIT
^
.
of
processes
side
transfer
(XIII
and
does not a l w a y s f o l l o w
3u 3T
m ^0
terms.
insultation,
for
distributions.
and
£ C
diffusion
convective
and the l a s t term of
equations the
due to
the
comparison to t h e o t h e r
the
right
a zonal c a l c u l a t i o n ,
Ogniewiez
"
temperature
the
the
. P)
in
motion r e s p e c t i v e l y For
Eq.
, Srad
temperature
Similarly
equations
,, p
term
change
third
matter.
,.
d l v U
=
equations
general
type
of
to
be
moment
of
supposed initial
time. 4.1
Statement of the Problem The system of
transfer 2R for
of energy
differential
and matter
equations
in an i n f i n i t e
with
molecular
p l a t e of f i n i t e 9 10 the zonal c a l c u l a t i o n s may be d e s c r i b e d as '
and
molar
thickness
93
3T
C
,
q Y0 "37 =
C
3 T
q ^7
Yn I T
*
m ' 0 3t
_
+
e P C
Y m
^ ^
3u
i,
A J ^ * J
m _ 2 3x
(
0 a"'
^ |
m . 2 3x
, ,\
*-K,)
(4.1.2)
P ^ 2 r dx
and c
3p , 3 p Yn j f = ^ — o p 0 3t P g 2
3u ecYn"5-m ' 0 3t
,, , , . (4. . 3)
where the thermophysical p r o p e r t i e s a r e assumed constant. For s i m p l i c i t y ,
we s h a l l
transform
equations
(4.1.1
in the dimensionless form by defining the non-dimensional x
r / R , Fo
a t / R 2 , 6!
T / T ° , &2 = u/u° and Q^
and Luikov number for the field to temperature
a — , a
Lu
q Kossovich
number pu° T
Posnov
number
Pn
—Q- ,
u and Bulygin number Bu
=
=
a —2- ; a q
c
c pPP -E-5c T q
p/p°
of matter and filtration in relation
field Lu
4.1.3)
variations:
,
94 where
the
characteristic
entity
denotes
the
respective
potential
drop. The E q s . (4.1.1 32 9 —r1 -, 2 3x
36 -SET3Fo
4.1.3)
become,
39 + eKo -s=^ , 3Fo
(4.1.4)
32 9 326 Lu Bu Lu — ^ T Lu Pn —=-^ + g
39 ~
-2 3x
dro
, 2 3x
Ko
329, — ^
(4.1.5)
,.2 dx
and 3 0, 3
3 2 6, 3
,
„ Ko
3 9, 2
,.
-
T~E~ - Lu —=— eH "5c- • 3 Fo p _ 2 Bu 3Fo The boundary conditions for equations ( 4 . 1 . 4 )
the
system
( 4 . 1 . 6 ) may be p r e s c r i b e d
9,
x(',Fo)
62
x(l,Fo)-A2
+ A, e i ( 1 , F o ) + B ] 6 2 ( l , F o )
6,
x(l,Fo)+B2
, ,>
(4.1.6) of
differential
as = *t(Fo),
92(l,Fo)+C|
9j
(4.1.7)
x(1,Fo)
$2(Fo)
(4.1.8)
and 93 ( l , F o ) where
A.,
B,
(1
1,2)
physicai coefficients
and
which may be determined In order that
the
therefore
system
and
C.
are
(4.1.9)
aggregate
of
known
,(Fo) are p r e s c r i b e d fluxes at the
thermosurface
by the experiment.
to simplify is
$ 3 (Fo) ;
the
present
problem,
symmetrical, thermally
and
we s h a l l
suppose
geometrically
and
95 8
(0,Fo)
0
(4.1.10)
1, x
For the
the complete
potential
statement
distributions
at
the
of the
problem
initial
moment of
cribed functions of the space v a r i a b l e S^x.O) 4.2
we shall
specify
time as
pres
i.e.
f^x).
(4.1.11)
Solution of the Problem The solution
4 . 1 . 6 ) is obtained
of the system of differential
equations
by the application of Laplace transform.
the Laplace transform
to E q s .
(4.1.4
4.1.6)
— ~ - + eKo(s6 2 dx
f2(x)),
(4.1.4 Applying
and using the
initial
conditions ( 4 . 1 . 1 1 ) , we obtain sG,
f,(x)
7 A
s6_
f.(x)
d 8 Lu — ~ , L dx
£
f ( x )
L
Z
I
(4.2.1)
■) A
7 A
d^8 + Lu Pn — ~ , I dx
d^8 + Lu ^M —-J. n l\0 , L r dx
(4.2.2)
and s
2 •*• 3_£K^ dx
K
Eliminating help
of
Eq.
d
(4.2.1),
from
Eq.
we
find
A
differential equations in 8. and 8.
( s
-
h
(4.2.2) a
set
and of
Eq.
two
(4.2.3)
homogeneous
with
A
/sx)+L(x,s),
(4.2.4)
where 2 V
Ux s)
-
the
partial
8 , which gives
I C* e x p ( - v . y s x ) + I D. e x p ( v J J j=1 J j=1 J
1
u > 2 3 )
-—2—2T7-2—277^—27^( V ] - v 2 ) ( v 2 - v 3 ) ( v 3 - v ] )s
[
2
x
- V
f
VT7sJ-/R(x's) * '
o
96
v
sinh
vJs(x-x')dx'*
2 3
2
"vl
*-
2
2
v
+
-V
— v
x
I
v?ys(x-x ' )dx'
x /
-2— I R ( x ' , s ) Js J o
J
R(x',s)sinh
0
sinh
v, 7 s ( x - x ' ) d x ' ] . 3
and
R(x s)
uTT^f,(x)
'
+ {
ur
+
l f ) s fi' (x)
p -ffv(x) The determined for
g
2
by
3 I,
the
C
j
(
C , D a r e t h e a r b i t r a r y c o n s t a n t s to be J ) b o u n d a r y and s y m m e t r y c o n d i t i o n s . T h e e x p r e s s i o n
by s u b s t i t u t i n g t h e v a l u e s of into E q .
'-VJ2)
f2(x) +
EKO j ^ - £ * V ( x ) . U P
coefficients
(4.2.4)
d b
3
p
♦ eKo f * v ( x )
6L i s o b t a i n e d
from E q .
esf (x)
^
( 4 . 2 . 1 ) . This
eXp(
"Vj
'SX)
,
Ffe
+
d
3 I , ■>*(1-v j
,
f (x) E-KoT l
— s —
f
9. and
9./dx
gives
)exp(Vjysx)
,
MCo"
L ( x
'
s )
e-Ko^L"(x's)(4.2.5)
On ( s 67 4.2.5),
%
substituting
f?(x))
from
the
Eq.
value
(4.2.3)
of
d 6,/dx
and
making
from use
of
Eq.
(4.2.2)and
Eqs.
(4.2.4
we o b t a i n
r k
C
I,
{
+
3
j
°j e x p ( - v . ; s x )
( x )
1-e
s
* —
E
^
I 5 Bu
f
[L(x,s)- i
. iv, , ~ ^x>s>
] ,
1
( x )
s
+
_i-
^
D*0j
exp(v.ysx)
Lu Ko , " , v I ,", . , 5 ( f , ( x ) - —— f . ( x ) ) Bu 2 I e Ko
L"(x,s)]
e
-|^
[(I*
1 -=2
£KoPn)L"(x,s)
(4.2.6)
97 The s y m b o l s o . and v are defined as J J (l-e)(l-v.2)-Lu
o.
v.2(l-v.2)-eKoPnLu
v.2
(4.2.7)
and Vj
where
7(yj
y
* | « ) ;
j
1,2,3;
(4.2.8)
are the r o o t s of the c u b i c
equation
3 y
11
+
Tt,y + 1T2
1
a
'
a2 1 "
+ B
( ,
-e)
+
0 ;
a
'
2 3 27 °
*2
+
U7
+ £ Ko
L ^
1 o 3aB
+
Pn
Y
'
P 6
(I-E)
j^-t Lu
(I
t EKo Pn ♦
)
T^-
Lu
r^— Lu P
and 1 Lu Lu
Y
' P
Equations hyperbolic A
3
6.
I
ft, 2
(4.2.4
4.2.6)
can
also
be
expressed
in
the
form 3
C. cosh
- i eKo
J >,
sinh
v. y s x
+
C ( l - v 2 ) cosh j j
v. y s x
I
D. sinh
v. / s x j
L(x,s) ♦ -pj^—
v. / s x
+ -4eKo
L"(x,s) EKos
| >,
t L(x,s),
D.()-v.2) j )
f l(xJe~l
f +
(4.2.9)
x
2(x - s
) -
(4.2.10)
98 and 3 i —5— [ eBu A
A
9, 3
Co J
J
cosh
, 3 + —=— Y EBU / ,
v. / s x j *
Do sinh j J
v. J
/sx
a s
e
Bu
s
+ — H1- L L ( x , s ) e bu ,iv, , ( x s > t ' j . s
2 2 Bu s
e Ko
n
- L"(X,S)] s
- ^ ebus
I
[(UeKoPn)L"(x,s)
(it.2.11)
The c o n d i t i o n s of symmetry
under t h e t r a n s f o r m
become
3 6.(0,s)
3x which
reduce ' I
A
6.(x,s)
the
C
J
j=' (1 - e) }
-g- sinh
cosh
/
J
sinh
* {a 1 v 3 " ° 3 v i
v^sx
+
v^/sx
2 2 2 2 v_ - o 2 v 3 " ' v 2 ~ v 3 )
l°2vi
"°|v2 " " " £ ^
"^v3 ~v]
v
1
ys sinh
I —-j—
r t- L ' ( 0 , s ) L { o ,
v /sx
1 v
° >
sinh
v3/s
v
xj- -
/sx
v,/sx 2
~v2 M
2
L{a,
+(0,(0,-0.)
+{a.(o, 1 I I
2
( o 2 - o 3 )+6 , ( v 2 - v 3 ) } *■
1 7- sinh v^/s
|
Hl-e)}
-o,) 2
+ ^i'v3
? 2 ~vi M *
2 2 t 6,(v. -v, ) } * 1 1 2 '
x
99
\
7 sinh v . V s x ] v./s 3
+ (v, -v.
j
)
2
+ —=-[(v_ 2 2
2
2
7—sinh v - / s x
1
v, /s
\
v, ) 7—sinh 3 v . / s
+ ( v , -v., )
1
I
v./sx 1
7—sinh
v,/ s
z
i
* L(x,s),
82(x,s)
(f.2.12)
I C (1-v 2 ) j=1 ' '
~
cosh v V s x '
+ - ^
e
v
v
£
v
v
" ° 2 3 - C - ) ( 2 " 3 )J
y s sinh
-('-
H
3
+ {c^Vj - o , v 2 - 0 - e ) ( v .
[
i
+ { a,(o3
" ,
)}
-v2 )}
v
/s
s i n n
I-V32 - ^s
2 2 '~v2 - a j ) + B , ( v 3 - V j ) } - ys I-V32 ) } - /
+ Bt C v1 -v2
v2/sx
sinh
(«,(V 0 3 )+ 6 1 ( V 2 2 - V 3 2 ) J 7 7 s -
+ {a)(a]-o2)
Vj/sx
2
,_v <■ ( 0 ^ 3 - ^ v ,
l'(0,s)
2
,.v V
[{0^2
v,/sx]
v3/sx]
Slnh V S X
/
sinh
v2/sx
sinh
v3/sxj
2 2
. —- -T[(v tKo S
2 2
-v
2 3
2 1-v, 7 v,/s
)
- 7 ! 1
♦ (v32-v,2) x
, 2 1-v3 + ( v , -v.. ) 7— s i n h I 2 v3/s 2
s i n h v., / s x 2
sinh v / s x
2
v, / s x ] 3
a s
GKOS
EKO
eKos
100 and 9
3(X'S)
C
ifc \
( l - e ) ( v 2 -v3 )}
-
( l - e ) ( v 3 -Vj
(,
fcosh
j°,
- ys
)}-
ys
sinh
sinh
-E)(v.2-v22)} ^ T
S
+
/ v2/sx
sinh
2
"I
2
Tl
3/SX]"
"°|v2
-i^al
2,
2" [ ( v 2 " v 3
)
°1
. .
V^FSinh
V
( a
2-°3
,
, 2
/SX
+ (V
°3 7~^T
2,
2
1- £ f 1 ( x ^ — 5 —
)
LuKo - — 2 Bu s
sinh
V 3
°2
/5XJ
„
3 "V1 » V 7 ^ *
1
2
)
+ { af (a - ^ )+6 ( ( v 3 -v ] ) }
2 2 + { a, ( a , - o 3 ) + 6 , < v , " v 2 ' }
2
°3
f3U)
v_./sx + ( v . - v 0 ) 7-sinh 2 I 2 v 3 /s
[L(x,s)
"°3V|
+ {c>|V 3
+ {o2vi
v2/sx
V
lkL'(°'s)[{a3V22-°2v32
-
sinh v^/sx
cBu s
sinh
/sx
v(/sx
s i n h
+ Bj(v2 -v 3 ) } - ys
°2 -j—gr
v
,,", > 2 (X)
(f
v,/sx] 3
1 TOS
, ", f
l
+ s
.,
( X ) ) +
U"(x,s)]
- ^ - [(1+EKoPn)
s
e DUS
1-e. EBG
L"(X,S)
t
Lii*!] s (t.2.1f)
where
101
a, = j J - tfj(0)
e Ko f^O)
B, = jyp [ eBu f j ( 0 )
L'"
+ L'"(0,s)]
(l-E)fj(O)
( L U ( 1 + E KoPn) + ( 1-e))<
(0,S)],
6,
tu
[eKo f ^ O )
Lv(0,s)]
f'j'(O)
and
°rv2
I
"v3 '
*
a
2(v3 " v |
'
+ a
3
( v
| "v2
'"
A p p l i c a t i o n of Laplace t r a n s f o r m to the boundary U.I.7)
to ( 4 . 1.9)
9,
x
('.s)
e2;Xd,s)
conditions
leads to
+ A, 6 , ( 1 , s )
+
t B,82(1,s)
+ *,(s),
(4.2.15a)
A 2 e1;X(i,s) + B 2 e2(i,s)+c, e3>x(i,s)
$2(s) (4.2.15b)
and
63(l,s)
$3(s).
(4.2.15c)
S u b s t i t u t i n g t h e values of Q. and - g — from E q s . 4.2.14)
into E q .
(4.2.15),
we
I C|P|+L'(0,s)[{o3v22-a2 j
+
find v32-(1-e)(v22-v32)}
{ a , v 3 -CT 3 V, - ( 1 - e ) ( v 3 - v ,
)}
R2+fo2v,
R,
-a^2
(4.2.12
102 -(1-e)(Vl2-v22)
} R3]
[{a](o2-o3)+B1(v22-v32)}R|
1
♦ {a^Oj-o^+BjCvj2^,2)}
* ~f s
R2+{B1(v]2-v22)
+
at(aro2)}R3]
[(v22-v32)R1+(V32-v12)R2+(v]2-v22)R3J+L'(1,s)
a f,(1)
f.(D
L(l,s)
, ",,
,
(f.2.16)
3 X C Q. - L ' ( 0 , s ) [ { o 3 v 2 2 - a 2 v 3 2 - ( l - e ) ( v 2 2 - v 3 2 ) } S ) j=l + t ° | v 3 -°3vi -(1-e)(v3 -v(
- ( l - e ) ( V ] 2 - v 2 2 ) } S3J
)} S 2 + {a 2 V!
-^Vj
^ { a , ( o 2 - o3)+B,(v22-v32)}s,
+ {a,( O J - O J J + BJCVJ - v (
) } S2+
{oj( o ] - o 2 ) + B](v]
-v2
)}
53]
2 2 2 2 2 2 2 + - y [ ( v 2 " v 3 ) 5 | + ( v 3 -Vj )S 2 +(v ] - v 2 )S 3 ] s
+(
+
V ik B2
„
?il c i >L * (1 ' s) -iik L " (, ' s)+ ^ L ( , - s ) .
C
„
f '(l)
fj(l)
a
103
L1V( 1 , s ) „ i - « — C
Lu
LuKo r , '" , , , — T C , ( f 2 (1) Bus
a ■ — f'," ( 1 ) ) - 3 2 ( S ) = 0
a and I C j=l
(4.2.17)
X + L'(0,s)[ {a3v22-o2v32- (1-e)(v22
v^)
} T,
a + { a , v - c^Vj - ( 1 - e ) ( v - v )} T + { a ^ a,v 3
3
-(1-^(v,2-v22)}T3
+
{a)(o3-a])
+6)(v3
2
e
]
2
2
- ^ F [ { a , ( a 2 - a 3 ) + e,(v22-V32)}
-v]2)}
T,
l y { a, (° ,-
T3
V [(v22-v32)T1+(v32-v,2)T2+(v)2-v22)T3]
«■
E Bus +
f,
, e^J
f,(D -V-
£
t^(f20)
, ilf
+
ffefjd))
, „ [UI,S)-IL(I,S)]
|^[(U e KoPn)L"(, > S )
Bus (1
t
'
s )
]
3 (s)
s
0 ,
(4.2.18)
J
where P. j
vVssinfi J
1-v Q
j
-FKt
v . / s + (A. + J 1
2
1-v.2 J B.) E Ko 1
1-v B
2
C0Sh
V
j
/ s+
{
^Kt
cosh v . A J
2
,
a c, +
A
2
+
Tlu")
V
j
/ SSinh
V
/
S
104
a.
o.
X. j
J cosh v . / s , T . eBu J ' J
R. j
cosh v . / s j
—4—sinh v-/s
v /s, j
2 B. I + ( A . + ( 1 - v . ) —Tp— ) 7— sinh v . / s 1 j e Ko VVs j
and I-v. S. = (—7^— j £ Ko
The
+ A_ + C. o ) cosh v . / s 2 I j ]
constants
1 -v . + —7}—*— — 7 — B EKO v./s 2
C. a r e determined
by E q s .
sinh
(4.2.16
v./s.
J*
4.2.18)
and a f t e r some manipulations the t r a n s f e r p o t e n t i a l s are arranged as
SjU.s)
- y v"-y [ t ( Q 2 - j - Q 5 ) cosh V j / s x
( Q j - ^ - Q3)
cosh
♦(P^X^-Q,
x
1
Q 2 )cosh
A i
v3/sx}
<J>( ( s )
A—
+ ( P . Q _ - Q . P _ ) cosh v , / s x
T
3 I j=l
{(P2- ^
P3)
cosh
A-
( P , - j T - P , ) c o s h v / s x f {TT— P.
+ t r - { (P 2 Q 3 -P 3 Q 2 )cosh v V s x
v?/sx
T7— P 2 ) c o s h v 3 / s x } $ ( s )
(P.Q.-Q.P.Jcosh
}l'3(s)
3 + I
v/sx
L. „, / - r cosh
M. 3 N . ! ojTlVr c o s n v./sx + J, -= — cosh SY s; J j=1 s^ 0 (s) n^
v./sx J
v/sx
v,/sx
105
+ L'(0,s)[{a3v2
- c ^ v ^ - ( l - e ) ( v 2 - v , ) } —-y- sinh v . A x
+ la|v3 "°3vi
+ {o2v
|
0-e)(v,
o,)+
2
♦ {a,(o3
0|)
—-T—
- (1-e)(v! -v2 )}'y
a,v2
+ -L{a(o
v. ) }
f
v
8. (v_
B|(v3
3^}
Vj ) }
j
2 2 I ♦ {of I ( a , - a 2 ) + B1 ( v j ' - v 2 ) } - ^ s sinh
♦ -|[(v22-v32)
+ (v|2-v22)
^ s i n h
J-j.
v,/sx
sinh
sinh
v
i,/sx
v2/sx
v3/sx]
♦ (v32-v,
v,/sx
sinh v 3 A x ]
, sinh
v—~ 1/s
-
sinh v 2 ^sx
2
) ^
Sinh
v2ysx
* L(x,s);
U.2.19)
X
° 2 <*' s)
9
( l - v ? )cosh v , / s x
(P2
[
EKO^S)
Y~
p
+
7
1
cosh
cosh v
9
X~ ^ 2 ^ ' " v 3 ' c o s h
' Y ~ Qi
3>('-v|2»
X
i < V xf Q 3 ) ( , - v , 2 )
v
,
/ s x
"(Pi
)T
p
. ' s * - < V xj Q3)
v
3>
/
^
sx} $ . ( s )
3)('-v22) cosh v - »4x
106
+(
)T p i" T~ p 2 ) ( , - v 3 2 ) c o s h
v
3/
sx
1 Vs)
v]v*x -(P]Q3-P3Q|)(l-v22)
+ jf- { ( P j Q j - P j Q j H l - v ^ J c o s h
cosh v 2 / s x + ( P 1 Q 2 - P 2 Q ) ) ( 1 - v 3 2 ) c o s h
3
L.
* £,
eKofo(s)
(,
v
)(cosh
" j
3 COSh
V./sX
+
J
y
N.
j
/ s x +
(1-v
B —7—4-
j=1 V
v
s )
2
Vj/sx}
*,(s)
3
M.
x
_
i=1 e K o s ^ U )
(1
V
" j
)
*
)
=*
COSh
€Ko s 2
V
J
/ SX
.
+
L1(Q,s) eKo
2 2 ,. 2 2. w {o v 3 2 "°2 V 3 - < ' - e > < v 2 - v 3 )
1-v + {a,v3 -ojv, - ( 1 - e ) ( v w -v, )} - ^
+
sinh
,
* FK^
[
2 v 2) } (v
l«,^2-°3 ) + 6 1
2
2" 3
2 1
1-v
2
+ {oi|( a , - o 2 ) + B , ( V | - v 2 B
2
2
2,
U
v
r =rLv_ -v, ) — ; , , 2 2 3 v,/s cKos !
V7s~
2
. .
vVsx
Slnh
v
/sx
2
2
1-v 3 2 )} - ys sinh
1
v2/sx
2 1
* { a* ( c u - o . ) + B, ( v . - v . )} — 7 — sinh 2
/ v/sx
2
,.v 2 {c^v. - a v _ - ( 1 - E ) ( v . - V _ ) } — y — sinh
I
+
2 I . . ^ 7 ^ - sinh
r
/
v./sx
v3/sx] .
2
2,
U v
2
2
sinh v . / s x + v , - v . ) 7—x I 3 I v0/s 2
107
2
' "v3 sinh v.. *£x + ( v , - v 0 ) 7— sinh v , / s x l 2 1 2 v,/s 3 2
f (S)
L
'
<*> s >
klU^sl
£ Ko
c Kos
+
eKos
2
f
+
2^x* s
if./.^u;
and
X
° 3 ( x ' s ) = e -fc [
{ (Q
2
cosh v V s x + ( v - Qi X { ( P 2 - yX +(
X
x j Q 3 )0 1
COsh
V s x " ( V Xj Q3)°2
x~ Q 2 ) °3
cosn
v ^ / s x } * (s)
X P 3 ) o , cosh Vj/sx - ( P , - x 1
p 3
)a2
cosn
v >/sx 2
X
X7 3
P
, " X7 P 2 ) 0 3 3
COSh
V
3/SX
}
*2(s)
+
XT { J
(P
2VP3Q2)01
cosh v. /sx - ( P . Q - - P , Q . ) o2 cosh v 2 / s x + ( P . Q ? - P ? Q . ) a , cosh
3 I
*,(s)]+
3
3 +
L a . —±— ..I / s cosh v /sx+
.f. £Bu *Q(S)
N.
J
o.
3 M. ) —B^— U
t|
S
a. ...S f ,
cosh
V '
,
vJ / s x
2
2
.1 e i l y ' s T C ° S h
V /SX+
(1- e)(v2 - v 3 ) }
sinh v} /sx* {°, v 3 - OJVJ - ( l - e ) ( v . - V . ) }
0-> —7—sinh vVs
y i/s
J
7EU L ' ( 0 ' S ) [ | ° 3 V 2
v./sx}
"°2 v 3
2 7 2 2 ^ 3 v_,/sx + {a,v - a. v - d - e ) ( v . -v_ ) } j— sinh 2 2 I 1 2 I 2 ' v,/s
v,/sx] 3
108
+
lEk[(V02-°3)+VV22-v32)}77TSinh
V /SX
'
a + {ai(03-a, ) + M
+
v
"vi
3
) '
e,(v,2-v22M^sinh
y~
v/sx]
sinh
aj — sinh v. / s
v,/sx 1
a, —Sr- s i n h / s x ] r v,/s
*/sx
+ {
- V K V / ' V / )
e Bus
2 2 ^2 * (v, -v, ) 7-sinh 3 I v ? /s
f ,(x) ♦- ^ s
. f.(x) - ^ — e Bu s
Lu i / l „ n , . „ , - 5 7 — r[(1+eKo Pn)L"(x,s) eBus
tf
2
+
3
v
2 2 v_,/sx + ( v . - v 0 ) 2 1 2
, . + 7 ^ - [ L C x . s ) - -1 eBu s
1 ,iv, , - L ( x , s ) ]n S
L"(x,s)]
LuKo =- x BUS
2
2(x)" l i b V x ) ] '
Cf.2.21)
where
VS)
L]
(P
, ^ P 3 ) ( Q2
= L'(0,s)
L(
XjV^I
^Q3)(P2
XJ P 3 >.
^v22-o2v32-(l-e)(v22-v32))H1+(a|v32-o3v)2
- ( 1 - e ) ( v 3 2 - v ] 2 ) ) H 2 + ( a 2 v | 2 - o 1 v 2 2 - ( l - e ) ( v ] 2 - v 2 2 ) ) H}]
+
[L'd.sMA,-
^
)L(!,s)]
( Q 2 - -^ Q 3 ) . [ { A 2 + -
^
109
* H^T
v(p
(
c
i
) L
'
( ,
'
s ) +
O^" L(l,s)-(l-eKoPn) j
. -2. p ) + rJ_Li L ( i s ) . C - e K o P n ) 2 X 3 3 ; LeBu L U ' S ' eBu
^ L"'(1,s)] x
L (,,s>i
.,x
'
X j V xf
2 2 2 2 2 2 L 2 = L ' ( o , s ) [ ( a 3 v 2 - o 2 v 3 - U - e ) ( v 2 - v 3 JjGj + t O j V , - ° 3 V ) -(1-e)(v32-v12))G2+(a2v)2-o|v22-(1-e)(v)2-v22))G3] n
y
♦ L ' ( , , s ) + (A ) + ^ ) L ( 1 , s ) B L'(1,s) + ^
(Q,- ^ Q
)
(
L(1,s)-(l-eKoPn) ^M_ L "'( 1 ,s) ] ( P ,
^ iiS L(l - s) - ^ f l ^ 1 1 L " ( , ' s ) ] x; -L 3
3
V
- ^
f
i^c,)
X ^ - Pj)
L ' ( 0 , s ) [ ( o 3 v 2 - o 2 v 3 ~ { 1 - e ) ( v 2 - v 3 JJI^+tojV, -OJVJ
- ( 1 - e ) ( v 3 - v ( ) ) I 2 + ( o 2 v ) - a , v 2 - ( l - e ) ( v ) - v 2 )I 3 J Q
X
[L'(l,.MArt-fc)L(l,.)]
[ (
V
Tib
+
FB^f
C
1
) L
'
( ,
'
s ) +
(Q2^
X
^Q,)
ijcf Ul.sMl-eKoPn),,
a
110
^
CP 2 Q,-P,Q 2 ),
[a,(o2-o3)+6,(v22-v32)]H)
M,
♦ [a1(aro2)
+B 1
+ [a ] ( o 3 - a ) ) + B , ( v 3 2 - v | 2 ) j H 2
(v12-v22)]H3+B1[f2(1)-Er^
y
^ L " ( l , s ) j
II
f^.)-
[-
i f e L',(,'s)-
f,(l)+B2f2(0-
^f,(
FK!
, ) + C
L
1
ris
"(,'S)-
f3
C
.L"(,-S)
(I)
+ f
a a -^»3(,)-^Ii(,,-iSL"(,»»>+i5rLlv
(, s)]
-
(P2Q3-P3Q2),
M2
2 2 2 2 [ a,( o 2 - o 3 ) + B, ( v 2 - v , ) ] G 1 + [ a , ( o 3 - o , ) + g | ( v 3 - v ( ) ] G 2
+[ a](oro2)
+
B](v12-v22)JG3+B1[f2(l)-
y
(
^
L"(1,s)-
^
f,(1)]
Q
[
L
(, s)
V Xj V " - r f c " ' - e - f e
L
"(,'s)-
te
SL"(,'S)
+
f;(,)-f;(,)+B2f2(D
e -fe* t (i) + c,i;(o-
+
rtuc,
.-^p3)+tf3(,)-FBff,(,)
LiV(1 s)](p
-
^vja)
111
^ L - d . s K ^ L - O . s ) ]
^(P1Q3-P3Q|)
and I a](o2-a3)+6,(v22-v32)]
-M 3
+[a((o,-o2)+B,(v,
(
2
I]+[ 0,(03-0,)
-v22)]I3+B,[f2(«)-
*7 V ^7 Q2>- [" i h L"(,-s) f2(.)-f;(l)+B2f2(l)-
+
G
S
j
X
p
R
^
Cjfjd)
T
Q )+
j 2- xf 3 )- j
j
bL"(,'s)]x
^(P2Q,-P,Q2).
X
s (p
? 1
! i p 2 M i 3 ( , ) . I g J £,(„
l^fL"(l,s)+^_L-(1,s)]
H
^ f , d )
s^fe f i ( l ) - n £ C,L"(I-S)
- | j f,(l)+C,f3U)-
t " _ C , L iv (,,s)] ( ^ P ,
+
+8,(v32-v,2)]I2
'
/
P
r
X
XT
'
P
)
3 "
R (Q
3
Q
." X7 3
j
T )+
X^
3
(P
3Q.-P1Q3)
3
and !
j
S
J(^
p
r *7
)p
2"Rj(^ V
The coefficients a, = 0 and
much
+
xj fp2Q|-ptQ2»-
N. are the first t h r e e terms of
M.
where
6, is replaced by B 2 -
The are
X7
Eqs.
(4.2.19
complicated
and
4.2.21) contain
for
the
A
6,(x,s)
terms
under the
.(s)
which
transform are to
be
112 determined $.(x,s)
by
we s h a l l (i)
$.(s)
the
experiment.
To o b t a i n
the
Laplace
inversion
for
follow:
the
method
of
convolution
for
the
undefined
terms
like
and (ii)
t h e method of r e s i d u e theorem of complex
To
determine
expression
in
degree
always
is
numerator.
the
the
residue
denominator greater
Now to f i n d
we
has
than
shall
only that
first
analysis.
assume
non-repeated of
t h e roots of t h e
the
roots
that and
its
in
the
expressions
denominator
the
we set i t
equal
to z e r o . T h r e e cases thus a r i s e : (i) gives s
(f
t h e denominator
contains only
the e x p r e s s i o n
s , an i n f i n i t e set of s i m p l e r o o t s w h e r e s
V ^ P 3 ) ( ( V *7
make
the
^ Q 3 ) ( P 2 " sj P 3>
values
of
s
more
precise
T.fs),
are given
this by
°" we
change
the
h y p e r b o l i c sine and cosine to o r d i n a r y sine and c o s i n e , t h i s g i v e s 2 s = - u , w h e r e u are t h e r o o t s of t h e c h a r a c t e r i s t i c equation n n n ^
< P nl
^TPn, n3
) ( (
"(Qnl nl
3n2
i T X , n3
%3)
^ 7 n3
1
^ n3
(P
n? nz
iT^ X _ n3
P
«^ n3
°
(4.2.22)
where l-v P
v
nj
Q . nj
% j
-(
Sln
I -v. —sr— e Ko
V j ^ V
2
TKOT"
a C. + A_ + —£ 2 eBu
, '
B
1)COS
Vj'
u v. sin u v. M n j n j
l-v + —TZ—'—B_ cos u v . eKo 2 n j
113 a. X„ ■
TTa
nj
eBu
COS U
(ii) s
V.
,
j
'
n
t h e denominator
0, a zero r o o t i n a d d i t i o n to the (iii)
t h e denominator
a double r o o t at s To t (s) s
has the e x p r e s s i o n s f n ( s ) ,
n
at
get
(i)
the
s = 0 and
inversion, its
we
shall
derivative
with
gives
and
has the e x p r e s s i o n
0 in a d d i t i o n to the
this
2 s * n ( s ) , this
gives
(i). first
obtain
respect
the
to s ,
values
"^(s)
of
at s
and 0 as
FRo\Ca,(v22-v32)+CT2(v32-V,2)+a3(vl2-V22)]>
V°>
V S n>
It • n f
[v.
n
X X A ,- ^ ( v , A ,+b , P , ) ] [ Q , - ~ Q ,] n X , 3 n3 n n3 n2 X -. ^n3 n3 n3
X - [ v , B _- ~ ( v , B ,+b - Q , ) ] [ P . 2 n2 X , 3 n3 n2 n3 nl n3
X xr^- P , ] X , n3 n3
X + [V, B .- ^ 1 nl X , n3
Pn,] nJ
-lv. A,2 n2
X (V, B ,+ b , ) ] [ P ,- y ^ 3 n3 n3 n2 X . n3
X X ^ ( v , A , + b - P „ J J L Q n 1 - xr^-QnJ, X ., 3 n3 n2 n3 nl X n3 n3 n3
An. -
M n V j cos P n v j + ( A ] +
B . nj
(l-v. j
2
B 2 ) - = — sin eKo
!-v.2 ^ - B, + 1) s i n u ^ . ,
(4.2.24)
f K
'"V,2 y v.+ „ J n j eKo
(u v . cos p v .+sin|i v . ) ,
a C i l +A 0 + - £ — ) 2 £ Bu
(4.2.23)
x
(4.2.25)
114 b
v . tan p n v - v 3 tan u n v 3
Cf.2.26)
and 2
V0) =[ V
B
«ET-
a2C +A
2
1-v.
o2
2+ I B I T }
v
1-v
a
+ B-[ v ' 2 eKo
(
3
(A +
i"^ 1-v
2
^
o
.
2
1-v
T1
2
o2C +
^Ko~
SBn-}"* ^
V
1-v
1-v ][v/(1+|(A,+ „ 2 1 1 eKo
2
'"v1 B-[ „ ' 2 eKo
B,)] 1
2
v
2
[v/(1+{(A.* 2 1 l-v
B.)I
B
(
2
I-v
-rv T^~ r ^ V w . +A
2
+
°2C2, TB^)-
a
l V
^
2,'-v32 3 (TKO~
+A
2 2 i - i v/(A.+ a, <: I
2
B )][v 2(
.
-
B
2]
,
— a.
v, 3
2 ]
2
1
a
°2 2 -£• v * a, 3
2
a.
°l 1_v3 — „ a , eKo
2 ^
3
i K o
B.))I
2
l-v, - ~ e Ko
'-v
2
B-
(,+
o2
— „ 0, eKo
°1 2 '~v3 -{2 — v , Z ( A , + - ~ a. I 1 e Ko
1-v
' ~v3 3
B.)] 1
e Ko
2
1-v
. ^fo + iB 2 )
. ° 2 C K , ° 1 + 2 eTu^'2 ~
v
|
2 i " V -TK^
B
,. , 2]
°J A
The
inverted
expressions
of
6.(x,s)
distributions
6.(x,0)
can be w r i t t e n as
f,(x)
f.
(constant)
for
uniform
initial
115
6 (x,Fo)
Fo f 0
2
oo V n=l
3 y j=l
u
— F (u)cos T n '
2
exp(-yn
Fo-u)du
t 2
» J n=1
v.x '
u
A
3 , £ — E . cos U v.x J j=l n '
x
2 exp(-yn Fo),
9
2(X'F0)
= i f e
f 0
(4.2.27)
1 Y1 C - v ^ F j= I n '
I, n= 1
co
x ex P (-u^ F5ru) du
+
_|_ j;
3
'
(u)cos
pnv J x
E .
£ _ a i (i-v 2 )
n=1 j = l
J
n
2 * cosu v. x e x p ( - u Fo) n j n
(4.2.28)
and
6,(x,Fo) 3 '
_ —5— eBu
Fo <^ 3 y [ — F ( u ) a . cos y v . x L) L) > , .. y njJ 1 n 1 0 n=1 j = l n ' '
2
exp(-yn
- exp(-y^
2
Fo-u)du+ ^
» J n=l
3 En . V j - 1 - creasy j=l n '
vx '
Fo);
(f.2.29)
where X
F ,(u) nl
= (Q , - ^ Q n2 X ,
X
-)«,(u)-(P-n3 I n2
T T ^ P J1> ( u ) X , n3 2
♦ xV (Pn? ^J- P n3 < W V u ) ■
116
a "■*
n3
' JTJ IPn20„3-P„3Q„2»»3<">I.
a n;
'"
+
E
n, =
(A
>
n3
n3
r7 (P n,Qn2-Q n , P n2 )$ 3 (u) '
.VB,V^n2-^Qn3^B2f2^'Pn2-^fPn3) "J
n3
f,(1 ) + v^ (P ., Q ,-P , Q ,) , X , n2 n3 n3 n2 n3 En2
X x^Q n 3 )-B 2 f 2 (U(P n , n3
-[(A^B^XQ^
+f
3
( , )
xT( n3
P
n1
Q
n3"Pn3
Q
M
X /J-Pn3) n3
) ]
and E , n3
X (A,f,+B,f,)(71^ Q . I I 12 X , nl n3
-
B
2f2
( , ) (
rf n3
P
nr
The Eqs. (4.2.27
X y-^Q,,,) X n2 n3
Xn3 ^TPn2>+
V 3" ^ " ' ^
2
* ^
2
^ '
}
4.2.29) represent the transfer
of heat, moisture and f i l t r a t i o n respectively
'
processes
and contain convergent
series. As Fo increases, the terms of the exponential order diminish
117 rapidly Thus
and
for
for
Fo
determined
>
Fo > a c e r t a i n Fo.,
the
value,
nature
of
Fo. ,
the
by t h e f l u x e s p r e s c r i b e d
they
transfer
at the
ultimately potentials
- Ki m
and * , ( F o ) i
fully
surface.
Now keeping the surface f l u x e s as constant
4>.(Fo) = K i , 4> (Fo) I q I
vanish. are
i.e.
Ki
(4.2.30) p
we o b t a i n
6.(x,Fo) 1 '
- -T^H— ( B 0 K i - K i B , ) - 2 A.B, 2 q m l 1 *• ^
cos y
9
°° Y L , n=l
3 f —L— (F*.-E ) * > . 11T nj nj j=l n n ' '
2 v x exp(-u Fo), n j n
2< X ' Fo) V
i f e I.
2
cos u n
I, Vh<} -hi*"'****
n=1
v
x exp(-y j
r
(4.2.31)
j=l 2
n n
'
'
'
Fo)
(4.2.32)
n
and
a 2 cos u v . x e x p ( - u Fo), n j n
w h e r e t h e values of F (4.2.30).
are
obtained
(4.2.33)
by
replacing $ .(Fo)
from
Eq.
118 4.3
Analysis of the Result The e x p r e s s i o n s for the transfer
and filtration heat
and
surface effect with
contain a large number of dimensionless
moisture fluxes.
moisture,
not
transfer,
Like
the
all
these
of
thermophysical dimensionless
parameters of
coefficients
parameters
thermophysical
moisture and
of
coefficients
the
heat
and
have
equal
on the t r a n s p o r t p r o c e s s . Some of these coefficients a r e linked the
transfer
others w i t h analysis
the
of
moisture, of
Luikov
dimensionless dependence
of
transfer
thermophysical
some with
filtration.
and
are
transfer
of
heat
and
On the comparison
with
the
we
find
dependent
that
of
the
these various
parameters of heat and mass. Let us now examine of
each
of
the
thermophysical
for
mass
heat from
coefficients
under
the the
conditions.
The thermophysical coefficients number
the
Mikhaiiov ,
coefficients
convective type of boundary
Biot
potentials of heat,
and
mass
and
the
surface.
heat
and
with
the increase of the coefficients
values of these coefficients,
A. and B,, correspond
they The
influence
transfer
A
the
to the
transfer
process
of
intensifies
and B_ and for the smaller
the moisture is transferred
at the slow
rate from within the material towards the surface and t h i s causes a slow rate d r y i n g . The
coefficient
characterises potential
the
A_ correspond
relative
difference
of
to the
drop
in
mass
and
it
is
heat
Posnov
content
number
produced
responsible
for
the
which by
the
internal
transfer of heat and m a t t e r . The coefficients
B. and C. are the complex coefficients
depend
on the number of
is
product
the
moisture
is
coefficient the
moisture
dimensionless
(1-e),
transferred
Lu,
in
numbers.
The coefficient
form,
it
and Bi . At e _ 1, the total m vapour mform, the influence of the
the
has
characterises, within
the
the the
material
B.
Ko
vanishes and at e= 0, the total moisture is t r a n s f e r r e d
liquid
coefficient
of
and
maximum combined
and
its
influence. effect
of
evaporation
Further, the
from
transfer the
in this of
surface.
119
The coefficient difference transfer
C. is the r e l a t i v e
in mass content
Prescribing
Bi
q
*.(Fo)
, B, I
Bi
I
E
q l
the
mainly
values
(1 = 1,2),
Ko Bi
q
by the
responsible
for
Bi
of
the
known
potential the
mass
Pn, B_ - B i , ' 2 m'
K , 9 ^ , 4>_(Fo)=Bi
q
thermo-physical
C. and the surface fluxes
K,, A0 1 ' 2
6^ + e Ko Bi
( 1 -e) and
is
caused
due to f i l t r a t i o n .
coefficients A , B
A, I
and
filtration
I 2
C.
$.(Fo) as
5 ^ -,—E Ko Lu
1
s S , *,(Fo)
2
m
obtain
the
2
,
'
- 1
3
Lu
_
K]
q
in
Eqs.
transfer
(4.2.31
4.2.33),
we
expressions
for
the
potentials under the convective type of boundary conditions: oo
6 (x,Fo)
QC.
2
3
I n=l
I —-^— E . c o s U R V X e x p ( - u j=l n n ' '
Fo), (4.3.1)
oo
6 (x.Fo)
6^
- j ^ \ n=l
3
I -l_(l-v )E*JCOSMVX j=l n n ' ' '
expC^Fo), (4.3.2)
and 6
3(x'Fo)
'
°°
3
ifc J ,
A,
IjV
%
E
nj
C0S U
nVjX
e X
P(^n
Fo)
'
(4.3.3)
where E*,
U K o
V
l )
Biq(Qn2
K ^ Q
n 3
)
(Pp2
X x^Pn3
) B i
m'
120
B i (Q . q xn1 M
(eKo K . - l ) I
E*, n2
^ Q ,)-(P , X , vn3 nl n3
~ P„,) X , n3 n3
Bi m
and
(EKoK
<3
r1)Biq(r:^ ^
r r ^
n3
n3
r>3 x
i
irX1 ,
P
Bi
7> n2
m m
•
n3 These expressions are equivalent to Mikhailov . Transfer phenomena are studied g r a p h i c a l l y Lu
e
0.3, '
0.7, Bi - 20, Ko = 9 . 0 , Pn = 0 . 2 5 , ' m ' '
10~ 3 and Bi
Bu
for Lu
500
P
10.
q
The first value of the c h a r a c t e r i s t i c root is obtained Uj
0.81
The analysis shows that Eqs.
(*.3.l
as
4.3.3)
the d i s t r i b u t i o n s e x p r e s s e d
contain a s e r i e s which
converges
by
very
with the increase of the generalised time, Fo and for Fo > first term of the s e r i e s is sufficient Figures temperature time,
and
4.1
and
4.2
moisture
represent
inside
Fo. From the figure,
to give the accurate
the
body
the at
the
fastly 0.7 the
result.
distributions
different
of
generalised
we see that the moisture content of the
surfaces of the material are smaller than those of the internal ones and
the centre has the maximum value of the moisture content.
reverse body. internal centre
takes The
place
surface
in has
case the
ones.
This
causes
towards
the
surface.
of
temperature
greater a
rapid
distributions
temperature
than
transfer
moisture
The evaporation
of from
the
those
The
of
the
of
the
from
surface
the also
Figure 4.1 - Temperature distribution inside the piate
Figure
5
122
takes
very
rapidly.
This
intensifies
the
process
of
drying
of
material. Figures k.3 heat,
matter
different
and k.k
and
show the variation of the gradients of
filtration
in
different
generalised times. The curves for
parts
of
the
body
at
the gradients of matter
and f i l t r a t i o n are symmetrical and also these gradients have negative signs.
The
negative
sign
of
these
gradients
indicate
a lack
of
agreement between the direction of the moisture flow vector and the direction moisture
of
the
gradients
from the central
filtrational
gradient
and
corresponds
layers
favour
the
to
the
towards the surface. molar
motion
of
the
transfer
of
Further
the
vapour
from
centre to the surface. The heat gradient has the positive sign which shows
that
the
flow
centre. Further rapidly
with
of
heat occurs from the surface towards
the
the fluxes of the layers nearer to the surface
fall
the generalised
time
and as i t
increases,
an almost
steady state is arrived at. These results follow the same pattern as those of Lebedev . Figure
4.5
gives
a
comparative
study
of
the
moisture
transfer from the surface and average moisture transfer of the body. This
shows
moisture is
that
during
transferred
from the surface.
the
period
very
rapidly
of
drying
the
average
than the transfer of
of
the
moisture
Figure <*.3(a)
- Thermal gradient in the plate
Figure 4 . 3 ( b )
- Moisture gradient
in the
plate
CO
124
Figure 4 . 3 ( c )
- F i l t r a t i o n a l gradient in t h e
plate
Figure 4.4(a)
-
V a r i a t i o n of t h e r m a l w i t h Fo
gradient
F i g u r e
-
V a r i a t i o n of m o i s t u r e w i t h Fo
gradient
8
Figure 4 . 4 ( c )
- Variation of f i l t r a t i o n a i gradient w i t h Fo
Figure 4.5 - A comparison of surface moisture t r a n s f e r w i t h average moisture transfer
127 REFERENCES 1.
Ogniewicz,
Y and Tien, C.L,
Int 3 Heat Mass Transfer 2« (1981)
421-30 2.
Toei, R and Okazaki, M . J . , Engg. Phys 19 (1970)
3.
Lebedev,
P,
Drying
with
Infra-red
Rays,
1123-1131 (Goseuergoizdat,
Moscow 1955) 4.
Lebedev, P . D . , Int 3 Heat Mass Transfer
5.
Luikov, A.V., Gosud
Yavleniye perenosa V Kapillarno-Poristykh
izdad
Transporter
Takhnik
Scheinungen
oteoretisches in
Luikov,
A.V.
and
Mikhailov,
Yu.
Maksimov,
G.,
Proceedings
of
9.
the
All
Mikhailov, Y., Int 3 Heat Mass Transfer Shukla,
K.N.
"Heat
Hindu University, 10.
Tripathi
G. ,
and
Mass
(Varanasi,
Shukla,
K.N.
Transfer 20 (1977) 451-458.
iiteratury,
Korpern,
edited
in by
of
Energy
and
Mass
1965)
Drying Section of Energetics (Profizdat, 8.
lelkakh
1954)
"Theory
Transfer!! (Pergamon P r e s s , Oxford, 7.
Koli
Kapillarporosen
Luiko, A.V. (Academic P r e s s , Berlin, 6.
1 (1961) 294-301
Union
Conferences
on
1958) 1 (1961) 37-45
Diffusion",
Ph.D.Thesis,
Banaras
1973) and
Pandey
R.N.,
Int.
3 Heat Mass
128 Chapter 5 HEAT AND MASS TRANSFER DURING FREEZE DRYING The governing equations for heat and mass transfer are described for a capillary porous body in a hot gas stream. Expressions for temperature and moisture distributions are derived. Freeze drying is a process where the substances a r e by
drying
and then is
in the placed
supplied
state.
The material
in a vacuum chamber
to the
sublimates surface
frozen
and
material
vapour
through
under
whereupon
passes
out
reduced
the
from
the c a p i l l a r i e s of the
preserved
to be dried frozen
inner
body.
is
frozen
pressure. water
region
Heat
component
towards
The vapour
the
evaporates
from the body surface in a thin layer into t h e chamber. The flow of vapour
takes
problem
place
can
processes
be
in
in
the
visualised
capillary
capillaries as
a
porous
combined
body
reported King of
in
theoretical
and
literature
on the
performed freeze
phase
heat
change.
and
with
mass the
The
transfer
process
of
surface.
experimental
foods.
investigations have been 257 drying a n a l y s i s ' ' . Gunn and
freeze
an experimental
dried
a
together
evaporation in a thin layer at the body Some
with
study
Clark
and
for
the
King
mass
characteristics
devised
an
improved
convective drying technique on the basis of fundamental mechanism of heat
and
mass
and
Sunderland
the
continuum
studies of
phenomena in freeze
presented transition
consider
sublimation
liquid
transfer it
merely without
a study and
the
free
taking
into
account
food
drying
molecule
heat conduction
to vapour in the c a p i l l a r i e s .
dried
on freeze
flow the
stuffs.
the
process
change
Lebedev
the phase change in a study of the ice sublimation
All
in the
phase
Hill
considering
regime.
problem
Luikov and
by
from
considered
process.
The present c h a p t e r devotes on the freeze drying application of the diffusion
process in c a p i l l a r y
porous body.
129 5.1
Mathematical Model Consider
capillary with
the
porous
ice,
the
material
the
supercooled
dehydration, of
frozen
body,
liquid
ice
and
sublimes
core
of
thin
boundary
compared
to
the
as a r e s u l t
of
them
incompressible
of
be
to v a p o u r .
sound,
The
boundary
v is the o v e r a l l Eq.
(5.1.1)
Eq.
(5.1.1)
velocity
2 ox— 2 a
TT
that
and
the
we
layer
sublimation
liquid
undergoes
flows
from
the
is
variation
occurring
variation
can
for
small
in
assume
equation
in a
velocity
3v 2 t ~ ) . 3y 2 3 T2 Re, v . VT << a =3yZ
can be of o r d e r
of t h e f l u i d
density
it
heat
as
an
transfer
(5.1.1)
and the sec-
vp(3v/3y)
2
,
stream in t h e boundary
~ -. ,3v,2 ( )
v +
7T
where layer.
,, . 0. (5lK2)
— I7
3y
p
define
VT10 3
T -T ' *20 10
3
where
(T9n-T.n)
stream
and
stream.
fluid
pressure
a
can be w r i t t e n as
3T — 2
Let us
process of
3v + — - (-a-* 2c 3 z
s m a l l Reynolds number, RHS of
is
filled
may be w r i t t e n as
3T 32T -5-s. <■ v . V T - = a 43 t 2 - 2
ond t e r m of
It
which are
The vapour
the the
neglected
i n an i n c o m p r e s s i b l e f l u i d
For v e r y
form.
of
supercooled
motion a r e so s m a l l may
fluid.
tubular
In the
the
Figure 5 . 1 . If
velocity
the
a
to the surface from where i t evaporates
layer,
the
caused
by
body
gas.
and
a phase change from l i q u i d
inner
in
pores and c a p i l l a r i e s
The
the
is
v
F
'
a a
m
the
body
other
v
v
_y_
and
terms
initial v are
2
m
(T ^ f 5 q v 20 ' l 0 ;
temperature is
the
defined
drop
average in
the
between
velocity
of
nomenclature.
the
fluid
the
fluid
On
using
130
Figure 5.1 - Laminar boundary l a y e r flow in freeze
drawing
131 these variables, the Eq.(5.1.2) becomes 2 3 6, _, ■) 3 8v ^ (5 .,. 3) .2 X 3y 3y ' a 3 8, 3 3Fo
where
the
asterisk
is
dropped
in
Eq.(5.l.3)
and
the
subsequent
expressions. The equations
for heat and moisture t r a n s f e r in the c a p i l l a r y 7 8 porous body can be w r i t t e n in dimensionless form as usual ' . 39 -^3Fo
i.
7TF—
3Fo
36 ~ *T < x ^— L > x 3x 3x
Lu
—
x
3
^—
3x
,
(x
39 W 1 3Fo
eKo
2,
Lu P r
—~—)
3x
x
(5.1.*)
3
,
K— ( x
3x
0 < x
I .
,,
—,—)
. _,
(5.1.5)
3x
< 1 , Fo > 1
The boundary c o n d i t i o n s are d e s c r i b e d as 39 ( 1 , F o ) '-^ 3x 36,(1,Fo) ^ 3x
36 (o.Fo) ( F o ) = K . ^ 1 A 3y
38 ( l , F o ) + A, — 4 2 3x
t B, 6.(1,Fo) 2 1 36
ejd.Fo)
=
63(o,Fo)
M
(5.1.6)
*,(Fo) 2
0
(1,Fo) -j^
(5.1.7)
(5.1.8)_
and
83(l,Fo)
The
vparameter
conductivity
of
K, ( = X vapour
=
0
X/ X ) q to the
r e s i s t a n c e f r o m the body
characterises
body
surface.
(5.1.9)
and
M(=
the
relative
X /hR)
i s the
thermal thermal
132
We
shall
further
assume
that
the
temperature
and
d i s t r i b u t i o n s , at the i n i t i a l moment of time a r e p r e s c r i b e d
e
B^x.O)
e3(y,o) 5.2
=
,0(x)'
e
( x 2
'0)
6
=
as
20(x)'
e3(y)
Solution of the Problem In order to obtain the solution of Eq.
the
moisture
flow
is
expressed
as
v
fully
developed
and
the
( 5 . 1 . 3 ) , we assume that
velocity
distribution,
y2
I
v
is
(5.2.1)
On substituting for 5— from the Eq.( 5 . 2 . 1 ) , and after taking Laplace transform,
the solutio for
6,(y,s)
-
6, is obtained as
C. cosh f(s/K,)y + C ? s i n h *fs/IC)y 2
* " K X Pr f [V
2K +
~T]
+ L
3(y>s)
(5.2.2)
s
Equations procedure.
(5.1.4
5.1.5)
Using the fact
are
also
that t h e s e transfer
solved
by
the
potentials are
usual bounded
at the c e n t r e , we have 6,(x,s)
2 I D IQ(v. / s x ) j=l '
+ L,(x,s)
(5.2.3)
and
6 (x,s)
2 I :_ 1
D (1 1
Vj) I . ( v J
u
/sx)
+ L,(x,s)
J
*■
where L
l(x's)
I D/s
x ^
9
10(x1)[Vvl
/sx
>I0(v2
v sx
'
|)
(5.2.f)
133 IQ(v2
x
I
L7(x,s) '
=
/sx)IQ(v]
= !-= (l-v*)(l-v|)D
IQ(v2 / s X j )
/s
(1-v2)
| 0
ZsXjJJdXj,
?
6,n(x.)[(l-v;)In(v1 20 ' 1 0 1
/sx)
IQ(v2 /sx)IQ(v2 ^sx, )]dx,,
y
a 3(y,s) = a L
and v2
| [ 1 + e K o P n + l / L u + ( - 1 ) J "M (l+e K o P n + 1 / L u ) 2 - 4 / L u } ]
=
After
using t h e boundary c o n d i t i o n sinh
9
3
( y
'
s )
C
sinh
1
/(s/K,)y
s^h^s/^) The
arbitrary
boundary
C,
+
conditions
/s
L|(1,s)-*)(s)
Eq.(5.2.2)
[ — - j -
2
(5.1.6
C.
and
+
L3(l,s)]
D, I | ( v , / s ) + v 2
L3
(j=l,2)
are
^- 2 -5) determined
as
/s D2I2(v2 / s )
(1,s)]
+ ( 8 K 2 P r E / s 3 + L 3 ( 1 , s ) ) cosh / ( s / K x ) / s i n h / ( s / K x )
D,
(Q2F,(s)
P 2 F 2 ( s ) ) / ( P , Q 2 - P2Q,)
-(Q,Ft(s)
P,F2(s))/(P]Q2-P2Q1)
and D2
x
2K,
C.
5.1.8)
8K. P r E —^2 s
becomes
8K PrE
)
"KxPr f ^ + - ^ ^ 3 c y . « )
constants
/(Kx/s)[V|
+
/(s/Kx)(1-y)
cosh As/t^
(5.1.9), 2
by
the
134 where
F
^
s
)
[ - L . ( 1 , s )
3(0,s)-ML]'(1,s)]x
cosh
// s ) [ L( j ( l , ss) - $ (/s ) ] s K J + inh / (s/K )
v
]
x
/
(
K
a -[8K PrE/s +LJ(1,s)] / ( K / s ) s i n h / ( s / K ) 2
x
x
+8K2PrE/s3
F2(s)
P. J
a
The
L2(l,s)-B2L2(l,s)-A2L]'(1,s)
[In(v./s)+Mv. J
/sl.(v . /s)]cosh
J
'
/(s/K,) A
)
v. / K sinh / ( s / K ) I . ( v /s) J A A I J
^ Qj
* L3(1,s)
*2(s)
u
x
(A2
general
determined.
+
1-v2 -^l) £
/sI,(Vj
Vj
transformed We
shall
/s)
solution
simplify
l-v2 ♦ ^ 1 B2I0(Vj
for this
the
present
analysis
by
temperature and moisture d i s t r i b u t i o n s are i n i t i a l l y
ejQ(x)
%(y)
0 , j
=
systems
are
now
assuming
that
the
constant
1,2
i
and the p r e s c r i b e d *.(Fo) I
/s)
fluxes
Ki q
are
, $.,(Fo) z
Ki m
Hence F,(s)
(8Kx2PrE/s3+l/s)(l+cosh / ( s / K , )) / ( K . / s ) L 8 K , P r E / s 2 + Ki / s Jsinh A A q
F,(s) z
Ki
/s m
/(s/K,) A
i.e.
135
Thus the temperature and moisture distributions in the body, (5.2.3
Eqs.
5.2.4) become 2 D.I ( v . I > H- J s n s ) j=1
e.(x,s) I
/sx) +
8K x PrE -_ (Ucosh( / s / K . ) s *(s) *
/(s/Kx)sinh / ( s / l ^ ))(Q2I0(V]
e (x s 2 ' >
,
2
e -fe
.1
D (1-v2)I.(v
J
/ s x ) - Q , IQ( v 2 / s x ) )
/sx)
8K^PrE
sWH
(,+cosh ( S / K
r^7-r
j=1
(5.2.6)
x>
eKos "Hs)
/(s/K
) 2
0
x) s i n h / ( s / K x ) ) ( Q 2 (I v Q , ( l -/v s x ) ) ) a 2
where
the
J Ki
D,
(1+cosh / ( s / K , )
D2
-(1+cosh / ( s / K x )-Ki
1
A
c o e f f i c i e n t s / ( s / K , )sinh / (s/K, ))Q,
q
A
A
Ki P. , m
L
/(s/Kx)sinh /(s/Kx))Q1
D.
+
L
KimP,
and f(s)
=
P,Q2
P2Q,
Also, the temperature distribution in the fluid stream becomes 03(y,s)
i + [Vj / s
-Kiq/s
D 1 I l ( v ) / s ) + v2 / s D 2 I , ( v 2 / s )
8K x PrE/s ]
♦ ( l / s + 8K x PrE/s )
/
s i n h / ( s / K x )( 1-y) ^(s)cosn 7<JlK^)
( s / K
sinh / ( s / K x )(1-y) /(s/Kx)SInh/(s/Kx)
are
136 , sinh / ( s / K . ) y (l/s+8ICPrE/s ) sinh ■ , // ,(s/K. ,v ) r— X
8KjPrE + —^3 s
(1+cosh / ( s / K x ) - / ( s / K x ) s i n h / (s/K^ ))
{v, /sI^Vj
/ s ) Q 2 -v 2 / s l 2 ( v 2 / s j Q , }
P 9 2Ki +<*K, Pr \ (y + — - ) X 2 ' s
sinh / ( s / K , )(1-y) 7-7—n^i r—77-7i?—S / ( s / K . )cosh / ( s / K . ) For inversion,
x
let us consider
the expression
(5.2.8)
(5.2.6)
for
6.(x,s).
The roots of the denominator (1) sT (s) = 0 are (a)
s
0, a zero root
(b)
s
s , where s satisfies the Eq.(5.2.9) n n
Also
*(s ) = 0 n
(5.2.9)
the
denominator
( i i )
s
a ( c)
e
in
I.(v.
s o a t tr z0 ei rrpo ol
na do d ie t ti ot sh r o o t s the
n
modified
y
Let
us replace
/s)
by the usual Bessel functions 3 (uv ) and J . ( u v . )
hyperbolic functions cosh / ( s / K metrical functions cos u / / K .
Bessel
gb i v e) sE q . ( 5 . 2 . 9
functions
) and sinh / ( s / K
and sin u / / K .
I n (v;
/ s)
and
and the
) by the trigono
by substituting u for
/ s , then we have 2 n'
n where y
P
n
is obtained by the Eq.(5.2.10)
n,Qn2
P
n2 Q nl
°
(5
-2-,0)
137 Here P
nj = " o ' V j ' - H ' j W j +v-/K
sin(pn/
x
and
1 1
"
8
^n//KX
/Kx)3,(ynvj)
(5.2.11)
9
1-v^ - ( A - + -rr-L 2 eKo
Q v nj
The
inversion
is
9
l-v ) u V.lAv v . ) + -rr—1- B 0 J n ( u v ) n j 1 *n j eKo 2 0 Hn j
carried
out
with
the
integration of the complex a n a l y s i s . This gives (a) Laplace inversion at s = 0
lim s-0
eK
"
D.IQ(v . / s x ) [—1—urTil K ' i 2 1-v
° [ < 2 - K V TKO"
FQ eS
B
°]
. 2 l-v
2 - K i m - ( 2 - K i q ) FTtT B2
* Klm]/(v2-v2)B2 -
(2-Kiq)
(b) Laplace inversion at s = lim s +sn
s ,
D . I_( v . / s x ) j 0 j ' s Fco , [ 2 —< <-, e J
ro
(s
iH r D n j' W j n n
x )
ex
P<^n Fo) '
where f
n
= P ' vQ - - P ' vQ ,+P . 0 ' nl n 2 n2 n t nl ^n2
P , Q j n2 v n l
help
(5.2.12)
of
contour
138
D . nl
=
[ U c o s T-P- + K i /K, q M X
s i n ( u / / K , ) ]Q 0 - K i P _, n A n2 m n2
u n
(5.2.13) U
A.
D , - -[1+cos - * £ - + Ki —n2 /K, q u M A n
p\ nj
Mu
v2 J „ ( u j O n
n
sin(U / / K , ) ] Q .+Ki P . n A ^nl m nl
v )cosU / / K , + [ ( 1 - K ^ v 2 ) u J n (U v . ) j n A A j ' n 0 n j
-(MIL-K,)V Ji(U v.)](■::—J 2 A j i n j li » K , , Q . nj
(c)
1 - v. (A_ + -i?— L ) 2 cKo
sinh / ( s / K x ) )
Thus
the become
6,(x,Fo)
(2-Ki
for
A
/sx))esFo]
temperature
and
)+8K2PrE[Fo+(Nn-1'1)(Fo/B2)
2B 2 oo
*
,,
*
2
I.
j=1
V
1
S ^7 n=1
VMnvjx)
J. ^T%
n= 1
u 4* n n
n n ,,
cos
'
ex
P<^nFo)
' n
/KT A
f
n
7K" A
(5.2.16)
/(s/K. )
:T2-t N <2 B 2- 2B 2V N ,iV f ?> ] 2
(5.2.15)
root.
(Q2In(v, /sx)-Q)I()(v2
expressions
the body
+
zero
v B J . ( U v.) J 2 1 n j
2 2 8K P r E - - r - [-qTTVT- C + c o s h ^ ( s / K , )f ( s ) X ds2
lirrf s-0
2!
) sin P / / K , n A
1 -v 7 u v 3AV v . ) + - ^ n j 0 n j e Ko
Laplace i n v e r s i o n at a t r i p l e
yr
(5.2.14)
Sin
n .
7K7' X
x
moisture
content
in
139
(
Qn2VVlx)AlVV2x)eXP(-PnFo)
(5 2 ,7)
- -
and 62(x,Fo)
- Klm/B2+8KX2
2
+ —jrr K °
°° 7 n=l
2 ( y —s— D (l-v.)3n(vj j=1 ^ n "J J °
, 2.- > exp(-UnFo)
+
7KT
Sin
^ [ N ^ F o ^ - N ^ f , ]
+
16 - ^
r I
1
n=1
y ¥ n n
7iT)(Qn2 °"Vf)
A
n
v.x) J
x
,, n ( H c o s ^ X
V^V1x)
A
nI( 1 - V 2 ) 3 0( Unv 2x ) ) -Q a where a N
n
N|2
= [0.25(x2-2/K
)B2-eKoA2 + (l+0.5B2)
x (E K o P r u l / L u ) ]
= -[(x2+2/Kx)B2/8Kv + {eKoA2-(H0.5B2)
x
(eKoPn+1/Lu)}/2Kx-{(l+0.5B2)x2/Lu
+ | ( e K o A 2 + 1 ) ( 1 + eKoPn+1/Lu) -(1+eKoPn+l/Lu)2+l/Lu}]
2)
=
N
1
eK o ( A 2
a N
22
=
[C/K
)(£KoA2-(l+0.625B2)eKoPn) + (l+0.5B2)eKoPnx2
140 T, -
f
=
2
[(M + l/K x -1/8)B 2 -A 2+ eKoPn+1/Lu]
I
+
^HT^
(M-Y 2 )(1+eKoPn))B 2 -l/Lu
-(1 + e KoPn + l/Lu)(eKo(Pn-A 2 )+1/Lu)
-4 eKoA^I^+4(1+0.5 B 2 ){(M+1/K X
5.3
1 IH )/Lu+e KoPn/lcJ ]
Determination of the Characteristic Roots Eq.(5.2.10)
determining
the
characteristic
roots
is
a
transcendental equation which may be solved
by graphical method as
discussed
method.
method
in
is
subroutines
Chapter based
3
on
available
or
the with
the
numerical
initial
guess.
the computer
roots of the transcendental Equations. an interval
(a,b)
Also
library
The
there
numerical
are
which
several
calculate
The routine attempts
the
to locate
containing a simple zero of the function by
binary
search starting from the i n i t i a l guess point. Table 3 gives the f i r s t seven roots of the Eq.(5.2.10) 0.2, Pn
0.3)
Table 3,
it
the
values
two
variation
is
for a set of parameters ( e= 0.5, Ko
for various combinations of Lu, K.and M. From the
is observed of
Lu
observed
that the first i.e.
Lu
3.0
root
u . does not change for
and
5.0.
in the second root
u
for
Also
a
the two
significant values of
Lu. From consideration there exists an infinite
of
Eq.(5.2.10)
and Table 3 it
number of roots u
follows
that
of increasing order.
The
infinite series in Eqs(5.2.15- 5.2.16) converge rapidly 2 term exp (-u Fo)
because of the
and
a n i f i c at n n
i c a lg c u l aet i n th s value f
o ( ,x
6 .
F o )
henc
M
0.1
0.1
Lu
5.0
3.0
0.04970
0.11135
0.15787
0.32061
0.001
0.005
0.010
0.040
0.32061
0.040
0.03513
0.15787
0.010
0.07854
0.11135
0.005
0.0005
0.04970
0.001
0.0001
0.03513
1
0.07854
M
0.0005
X
0.0001
K
2
0.97078
1.10782
1.00331
1.04390
0.94870
1.02109
1.06819
1.04111
1.05009
0.94404
1.08970
0.9896
U
3
3.92895 2.77765
1.64827
2.99471
3.02925
3.05542
2.96872
2.77954
2.97291
2.99339
3.02902
3.05531
2.96869
^4
2.97575
1.91152
2.05136
1.93454
2.09402
1.66746
2.09377
2.14559
2.04107
1.93341
2.02675
P
5
4.10102
5.14510
4.11187
4.02382
4.03944
3.97414
4.10014
3.92867
4.11146
4.02377
4.03941
3.97414
W
Table 3 : Characteristic roots of Eq. (5.2.10)
5.23615
6.11962
5.20465
5.00592
4.46139
5.07307
5.23414
5.23327
5.01013
5.01942
5.30822
5.07396
^6
7
5.93602
6.43645
6.10566
6.00967
5.93555
6.07891
5.93581
6.11512
6.10332
6.00903
5.93513
6.07886
y
5
142
5.k
Asymptotic Solution The s e r i e s e x p r e s s i o n s of E q s . ( 5 . 2 . 1 5
rapidly
W
the
5.2.16) converges quite
because of the increasing order of the c h a r a c t e r i s t i c roots l <^2
exponential
<■
U
<%'
3
function, exp
(- u
2
Fo)
decreases
increase in u . Therefore for sufficiently
rapidly
with
an
large Fo, the E q s . ( 5 . 2 . 1 5
5.2.16) simplifies to e.(x,Fo)
(2-Ki )B,+8K?PrE[Fo+ ^ -
(N-.-f,)Fo
N
i2 v (2B 2V N n*i> +f f
* i
]
(5
-*-,)
2B 2
and Ki - ~
e2(x,Fo)
+
, N-. N 8K^PrE[-^-!-Fo + l
i
-N9I1\ ^ ' '],
Fo > Fo. The asymptotic expression
for
8,(0,
Fo) can be obtained
(5.4.2) by
taking
the limiting values of s as 6 , ( 0 , Fo)
lim s -+0
L"' 8 ( 0 , s )
which gives
a l"v2
2
8K^PrE +(
—r~ s
1
+
i°
S
j
/(S/K.)
8K^PrE +
A
— 3 — s
(5
-*- 3)
143 Hence e 3 (0,Fo)
^Ko
1-
[Ki q
(Vj-v2)B2
+2
+ 8K v PrEFo]
/(Fo/Kx)(^y-K^PrEFo+lMK^PrEFo
(5 A A)
The estimate of Fo. depends upon the various parameters involved in the p r o c e s s , e . g . for a set of parameters ( e= 0.5, Ko - u Fo 5.0, M = 0 . 1 , K 0 . 0 * ) , p . = 0.32, e 0.077 at Fo it
is
reasonable
to assume
Fo.
=25
as the limiting
0.2,
Lu =
25. Hence
value for
the
asymptotic calculation. For estimate of the heat transfer from the body surface, let us define a dimensionless number S as a r a t i o of the specific heat from the body surface to the enthalpy
heat transferred
by the
flux fluid
stream, 39, ( I , F o ) / 3 x S(F0)
=
PC v m ( T , ( 1 , t ) - T 2 ( 0 , t ) )
or in dimensionless form 36, (I , F o ) / 3 x S(Fo)
From Eq. ( 5 . 4 . 1 ) , 3 6. ( 1 , F o ) —^ The
(5
= PeCe^l.FcO-e^O.Fo)]
8.(l,Fo)/3x
, 4K£PrE[Fo
(^Kx
value of PeS(Fo)
-"-5)
can be obtained as + 2/Lu t V j / B j ) ]
versus Fo is plotted
(5.4.6) in Figure 5.2in
the
assymptotic cases for the two s e t s of the parameters (M - 0 . 1 , K, = 0 . 1 ; M = 1.0, 0.2,
Lu
transferred the fluid state.
K.
5.0,
= 0.01). EPr
uniformly
The other parameters are ( E = 0 . 5 , K0 =
1). with
The
respect
c u r v e s indicate to the enthalpy
that heat
the
heat
is
transfer
in
stream and for a very large Fo, the body attains saturated
144
Figure 5.2 - Pe S(Fo) v e r s u s Fo
145 REFERENCES 1.
Clark,
3.P.
and
King,
CD.,
Chem Engg Progr.
Symposium
Ser.
no.100(1971) 67 2.
Dyer,
D . F . and Sunderland 3 . E . ,
Trans Am. Soc Mech. Engrs 90c
3.
Gunn,
i*.
Hill
5.
Kessler, H.G., Chem-Ing Tech 34 (1962) 163
6.
Luikov
A.V.
(1973)
1087-96
(1960) 379-84 R.D.
and
King,
C.3.,
Chem Engg Progr.
Symposium
Ser.
no. 100 (1971) 67 I.E.
and
Sunderland
3.E.,
Int.
3.
Heat Mass Transfer
14
(1971) 625-38
7.
Luikov, Transfer,
8.
Ortega, Nonlinear
9.
A.V.
and
Lebedev
D.P.,
Yu.A
Mihailov,
and
(Pergamon P r e s s , Oxford, 3.M.
and
Equations
London,
1970)
Shukla,
K.N.,
Heat
Int.
3.
Heat Mass Transfer
Theory
of
W.C.,
in
Variables,
and
Mass
Hindu University ( Varanasai
Diffusion,
1973)
and
Mass
1965)
Rheinboldt, Several
Energy
16
Iterative
solution
(Academic
Ph.D.
thesis,
of
Press, Banaras
146 Appendix-1 THE INTEGRAL TRANSFORMS The solutions texts.
for
method
integral
simultaneous
Laplace,
transform
of
heat
Fourier
and
Al
and
mass
Hankel
is used to transform
Hankel transforms for s p a t i a l
transforms
has
been
transfer
transforms
in are
used the
to
find
preceding
used.
Laplace
the time v a r i a b l e while Fourier and
variables.
Laplace Transform Let 9
9(Fo)
be a piecewise continuous function By
piecewise
single
valued
specified
of the independent
variable,
function,
we mean
function
a
number
continuous and
interval 2 as
has
(0,Fo).
finite
The Laplace
that of
the
is
discontinuities
transform
of
9
Fo.
in
a an
G (Fo)
is
thus defined
oo
6 (s)
j 6 ( F o ) e " s F o dFo, 0
(A.I) A
where s
£ + in
is some complex number.
The function
called the Laplace transform of 9 ( F o ) . It should the
transform
6 (s)
exists
if
the
integral
6(s)
is
be noted here that
(A.I)
converges.
Let us
assume that the assymptotic value of the function 9 ( F o ) is expressed as | 9 ( F o ) | < Me0 F o
for
( a > 0 , M > 0)
>-sFoi „ . . - ( s - o ) F o 0 , c 19(Fo)e | < Me M > 0 and
a 2. 0 and for all Fo >
0, the integral
(At)
147
If the problem is solved in terms of the Laplace then t h e inversion is performed
with the help of contour
transform, integration
involving the solution of complex integral -1
6(Fo) -
L
A
[ 6(s)
. 2iT
=
where the integration along t h e s t r a i g h t
]
00
a+i e s e i < > a-i°°
F
ds,
is c a r r i e d
line
out in a complex plane s
(A2)
£ + in
s = o ( a constant) p a r a l l e l to the imaginary
axis. For i l l u s t r a t i o n ,
consider
6(Fo) = C ( Fo + c 2 , where C. and C_ are constants. The Laplace transform is obtained as below: GO
J 6 ( F o ) e s F o dFo 0
6(s) oo
/ (C.Fo + C - ) e s F o dFo 0
C, I Fo e s F o dFo 1 0
+
C,1 [-Fo isF° " 0
+
C, I e s F o dFo 0 1S / esFodFo + C7 ] e sFo dFo] 0 ^ 0 00
(C./s + C,)[-isF°/s] £ 1 0 C,/s2 + C2/s
148 For inversion of 6 ( s ) . consider the integral . 9(Fo)
To evaluate
=
a+i°°
2?T / a-i°°
the
integral
C. (
Cf
^~ s
(A.<0,
P )e
s
ds
we
{M)
follow
the
residue
theorem
of
complex a n a l y s i s which s t a t e s t h a t A
If bounded
the by
function the
singularities
lying
8(s)
closed
is
analytic
contour the
contour,
for
in the
finite
then
the
I
s r o , -i
region
number
values
of
of
the
Here s = 0 is a double and simple roots of the two i n t e g r a l s ,
the
integral (A.2)
inside
everywhere
except
is
2ni x sum of the r e s i d u e s . The integral (A-4) is rewritten as 1
Tr,
I
1
2 i T [ C l o-ioo I . s~2e
sro ,
ds
„
/
+ C
residues of which are determined
2
ia-i°= . 7
e
as
, 2 sFo sFo C, iim -j— (—=) * C- lim ( — ) 1 _ ds l l „ s s-K) s s-»0 C. lim (Fo.e
) + C-
C.Fo + CHence 9(Fo)
2TT i x sum of t h e r e s i d u e s 2TI i x iz—— (C.Fo + C-.) Z1T 1
= C.Fo + C ?
I
2
ds]
149 A1.I used
Laplace Transform Properties: The frequently,
(i) LIC, e, (Fo) + c 2 e 2 (Fo)]
following
properties
c, S (s) + c 2 e 2 (s)
are
(A.5)
J 9 (Fo)e"sFodFo
(ii) Le'(Fo)
0
sa § ( s ) 6 a
a L9 n (Fo)
s n 9(s)
sn_l
sn"29'(0)
9(0)
n
9 "'(0) where
6r(Fo) i s the nth d e r i v a t i v e of Fo ( i i i ) L[ I 9(Fo)dFo]
=
j
(iv)
=
|
L[ 9 ( a F o ) ]
(A.8)
9(Fo).
8(s)
(A.9)
9 (s/a)
(A.10)
□
a T he
ei
(v)
L[ 6 ( F o / a ) ] = a 9 (as) n v e e Lr as p mlt ar s ca n os f oia rl, s l i. n e a ri
L
"
'
A
[C, 8 , ( s )
+ C2 9 , ( s ) ]
C,
9,(Fo)
.
(A.11) e
* C292(Fo)
(A.12)
( v i ) Convolution
properties;
If L 9,(Fo)
= 6.(s)
-1 L
6,(s) 1
9,(s)
and L 8 2 (Fo) = § 2 (s) Fo j e.(Fo-u) 0
9,(u)du
150
Fo J
=
The solution
of
coefficients of
the
function
the
the
homogenous
advantages
9(°°)
lim s-» °°
s9(s)
9(0)
the l i m i t
exists.
the
appendix,
t r a n s f o r m s for the function A2
is
generally
differential
terms this
applicable
equations
i n 9 are independent
procedure (eg.
series
rapidly
for
is
that
the
expansion) small
Fo.
where of
for the
F o . One
subsidiary to
yield
Also
if
a the
then
s9(s)
of
of
converges
lim s-»0
end
method
be manipulated
inverse
|6(Fo)|<M,
provided At
of
can often
whose
transform
62(Fo-u)du
9)(Fo)*92(Fo)
nonhomogeneous l i n e a r
unique
equation form
Laplace
e,(u)
a table
is
given
for
the
appropriate
9(Fo)
Finite Fourier Sine Transform Let Ti
9(p)
-
/ 9(x) 0
s i n px d x ,
where p i s a p o s i t i v e i n t e g e r . 3 thus defined as
(A2.I)
The i n v e r s i o n f o r m u l a f o r
9 (p)
is
1,
by
CD
6 (x)
If
the
|
function
i n t r o d u c i n g the y
we h a v e ,
TI x ,
I 9(p) p=1
9(x)
changes
variable dy
= Tidx
sin
px,
in
the
range
0 <
x
<
then
151
1
.
/ 6(x) sin(pTix) dx 0
Tt
— f 8(y/Tr) sin py dy * 0 (l/TT) 9 ( y / n )
For example, f d29 , .. d8 , , l' ) — 5 - sin (Tipx)dx = — sin (iTpx) 0
a x
dx^
0
} d9 Tip
) -7— COS T! p x
dx
0 .1
1
a - Tip[0j 0
7T p J e s i n pTTx d x ] 0
= ( T p ) 2 e (p) TI P L e , ( i ) a A3 e l FH ai nnk m ei t T r a n s f o r
e(o)]
a T he e f i en i tH a nmtk ri s a n s fd o i ryd e fe i n e tb nh r e l a t i o
a 6(p)
t = / x6(x) 0
Jn(px)
dx,
(A3.1)
where p i s the positive root of the equation 3n
(p)
= 0,
(A3.2)
J (p) is the Bessel function of the first kind of order n. n The inversion formula for 9(p) may be e x p r e s s e d as 6 (x)
I ap 3 n ( p x ) P
where a
[ P
^ J„,(p)
]
/ x8(x)J 0
(px)dx
(A3.3)
n
n+1
a a a
a
152
differential differential
equation has an advantage in reducing i t to an ordinary equation
and
thus
simpler form than the original.
transforming
the
problem
into
a
153 Table-1A : LAPLACE TRANSFORMS OF SOME FUNCTIONS
Function, 6(Fo)
Transform, Q (s)
1
l/s
Fo
1/s 2
Fon-'/(n-l)|
t^
1
-
2
'
3
*-
>
1/s
l//(nFo)
I//S
2(FO/TT)*
i3/2
0n
r- n - j 2 .Fo [1.3.5 (2n-1)]/n Fom-\
, , ., , 4 Cn=l,2,3, — . )
5(lH*)
r(m)/sm
m>0
. ±aFo e
1/(s*a)
aFo c Fo e
1/(s-a)2
[1/(n-l)j]Fon"'
eaFo,
(n=l,2,3,-.)
l/(s-a)n
cos aFo
s / ( s +a )
s i n aFo
a //r( s 2+a2),
cosh aFo
s /it( s 2- a2).
s i n h aFo
a /ii( s 2-a2,)
e
bFo
- e
aFo
2 /fir Fo 3 )
/(s-a)
-/(s-b)
154 , , a )> 27F^
erfc(
,, , /.2r. . exp(ba)exp(b Fo)
bexp(-a/s) s(bVs)
x erf (b/Fo + a/2/Fo) exp(ab) exp(b2Fo) erfc(b/Fo+a/2/Fo) b a2 7T7Fo~)exp(- Wj~
exp(-a/s)/[/s( b+/s) ] 1 1 + /(s/b)
bex P (a/b + bFo) x
ex
b> 0
P("a/s)
erfc { ( 2 ^ ) + /(bFo) } I(i^exp(-4a2Fo)-[-^]erfc b
^
' (/s+b)
s
exp(-Vs)
+ —j exp(ab+b Fo)erfc{ jvp— +b/Fo }
(-b)n
exp(ab + b Fo)erfc(0 /■-
2 / F o
—!— Y ' (-2b/Fo) m n (-b) m = o
+ b/Fo)
i s ^ i (/s + b)
exp(-a/s)
i m erfc(=-|-} 2 ^°
^exp(bFo) {exp(- J^)eric(j^
/bFo)
_J_exp(-a/s)
+ e x p ( ^ ) e r f c [ ^ - - V(bFo)] n+2
2F° 2HT1
„ /1 n ( l +
n,n + 2 n + 2 , , a , 1 erfc( 2> 27F7 )
exp(-a/b) erfc( 2
a
11(40) , , . 4nTl- exp(-a/s) /s
M
-/bFo)
+ exp(a/b) ertc(~^
T /bFo)
772b lexp(-a/2b) e r f c ( ^
I exp (-a/s+b)
/>bFo)
-exp (a/2b) e r f c ( - ^ - + /2bFo) ]
^
^
^
exp[-a(s + 2b)* ]
155
a 2
i
P
a a
a /(TTFO)
* 7T
e x p
(
"
a
F o )
x
/s/(s+a )
a/Fo _ J exp(x )dx 0 1 2 — e x p ( - a Fo)erf a / F o
1//s(s-a )
a / 2 2 —j— e x p ( - a Fo) /
2 2 l / / s ( s +a )
a 2 a/iT
F o
2 exp(x )dx
Q
2 exp (a Fo)erfc a / F o
l//s(/s+a)
/(b-a)
1/(s+a)/(s+b)
exp(-aFo)erf[(b-a)Fo]'
■—5—exp(- a / 0
exp(-a/s)
2/(TTFOJ)
I
erf ( a / 2 / F o ) CO
j e"x a/2/Fo 1
= erfc ( a / 2 / F o )
^ exp(-a/s)
y
dx, a > 0
a2
^UFoTexp(- « ^ ' Fo 7—L-r f -r->(Fo (n-1) I '0 Ait x)
1 a
* °
71
x)n"'exp(-a/x)dx, a» 0
ex
1 , g n+j
P(-a/s>
exp(-2/(as))
156 3 n (aFo) °
I
=—=/(s2+a2)
0(aF°)
v f 2*2> / ( s -a )
exp[-i(a+b)Fo]Io(i(a-b)Fo)
/{(s+a)Lb)}
u
s
I 7(FFoTcos
0 y-E" 2 / a F o
I 7s
1 •(TIFO)
-, /•pr 2,4Fo
1 -a/s "372-e s
sin
-a/s e
1 2 —rr e x p ( - F o /<*a), a > 0
2 2 e x p ( a s )erfc as
erf 2~,
- e x p ( a s )erfas
-
a > 0
^Fo(Fo+a)
e a s erf
a >0
1 a2 exp( 2Fo~ ~ W^
/as
Ko(a/s)
* The e r r o r function erfx and the integration of erfex a r e defined x 2 erfx = ( 2 / / T T ) \ e x p ( - x )dx 0 erfcx
(2//TT)
f e x p ( - x )dx x
CO
i erfcx
C erfc Z, d £ = v r e x p ( - x )-x x
.n , I erfcx
I r -n-2 , ^— Li erfcx
erfcx
.n-1 , -> 2x I erfcxJ
as:
157 REFERENCES
1.
Abramowitz,
M and
Stegun,
I.A.,
"Handbook
of
Mathematical
Functions" Dorer Publications Inc, (New York 1965) 2.
Luikov, A.V.,
"Analytical Heat Diffusion Theory", Academic P r e s s
(New York 1968) 3.
Sneddon,
I.A.,
"The use of Integral
book Company (New York 1972)
Transforms",
Mc Graw
Hill
158 NOMENCLATURE
a a
2 diffusivity [m / s e c ] 2 Coefficient of mass diffusivity [m / s e c ] 2 Coefficient of thermal diffusivity [m / s e c ]
Coefficient of fluid-thermal m
a
q
c
Specific mass capacity [kg/kg mole]
c
Specific heat capacity
E
Eckert number
Fe
Fedorov number, (= eKoPn)
Fo
Fourier number
h
Enthalpy
I, (x)
Modified Bessel function of order k
[ k c a l / k g K]
[kcal]
j
Flux of ith
J.lx) k
Bessel function of order k
phase
K
Coefficient of icing
Ko
Kossovich number
K.
Relative thermal conductivity of vapour to the body
A
Lu
Luikov number
M
Dimensionless thermal r e s i s t a n c e 2
P
P a r t i a l vapour p r e s s u r e [N/m ]
Pn
Posnov number
Pr
Prandtl number
q
Heat flux
Q
Heat source [W/m]
r
Position vector
R
C h a r a c t e r i s t i c length
Re
Reynolds number
a
a [W/m 2 ]
[m]
[m]
s
Laplace parameter
T
Temperature distribution
VT
Temperature gradient
T.
Temperature distribution of the frozen body [K]
T.n
Initial
T?
Temperature distribution of the fluid
T._
Initial temperature [K]
t
Time [ s e c ]
u
Mass content [kg/kg mole]
v
m
x,y a a_ ,a-, i '
[K]
[K/m]
temperature of the frozen body [K]
Mean fluid velocity
[K]
[M/sec]
Dimensionless s p a t i a l coordinates 2 Heat transfer coefficient [W/m K] Dimensionless thermophysicai
coefficients 2
6
Mass transfer coefficient
[kg/m
g ,@_
Dimensionless thermophysicai
6
Thermogradient coefficient
sec]
coefficients
[l/K]
a a yQ
Density [kg/m ]
T
Form factor
6
Dimensionless transfer
e
Phase change c r i t e r i o n
(plate-0, cylinder-l,
sphere-2)
potential 2
y
Dynamic viscosity
[kg/m
X
Coefficient of mass conductivity
X
Coefficient of thermal conductivity
9
Laplace transform of 6
6
Fourier cosine transform of 6
Hankel transform of 6
sec]
[Kg/m sec moles [W/m K]
