Heat and Mass Transfer in Packed Beds
Topics in Chemical Engineering A scrin o( monogr3phs :�nd texts edited by R. Hughes.
Ullit·tnil)• of So/fonl. UK.
Volumt I
ltEAT AND MASS TRANSFER IN PAfKED BEDS N. W:ak:ao :�nd S. Kaguci
YoktJitamll NatimJDI U11ia·�nity. lDfNZII
Volume 2
THREE-PHASE CATALYTIC REACTORS P. A. Ramachandran and R. V. Ch:audhari Nalimuzl Oltmic'a/IAbonzto,•. Poo1111. lnJill
Additim�al •·ohmrn ;, prrparatio11
ISSN: 0211·5883 Thki boot b pan of 1 S�tdcs. 'fh( rubUskrs •ill K«Pl C'Onllnu.alton orckr"J whktl ma)' � can«lkd sl m)' lirM :and •·hidl pro,·ldc for aulomalk biUI"J and sNppb� of e-xhlllle ln the trrln upon publk:al6on. Pln�r •·rite lor dcralh.
HEAT AND MASS TRANSFER IN PACKED BEDS N. Wakao and S. Kaguei Yokollama Natimml Utriversity, Japan
GORDON AND BREACII SCIENCE I,UBUSI-IERS New York
London
Paris.
Copyright © 1982 by Gordon and Breach, Science Publishers, Inc. Gordon and Breach, Science Publishers, Inc. One Park Avenue New York, NY 10016 Gordon and Breach Science Publishers Ltd. 42 William IV Street London, WC2N 4DE Gordon & Breach 58, rue Lhomond 75005 Paris
Library of Congress Cataloging in Publication Data Wakao, Noriaki, 1930Heat and mass transfer in packed beds. (Topics in chemical engineering, ISSN 0277-5883; v. 1 ) Includes bibliographies and indexes. 1 . Fluidization. 2. Heat- Transmission. 3. Mass transfer. I. Kaguei, Seiichiro. II. Title. III. Series. TP156.F65W34 660.2'842 81-1 3203 AACR2 ISBN 0-677-05860-8 ISBN 0-677-05860-8, ISSN 0277-5883. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without prior permission in writing from the publishers. Printed in Great Britain.
Contents Introduction to the Series
ix
Preface
xi
Notation
xv
1
PARAMETER ESTIMATION FROM TRACER RESPONSE MEASUREMENTS
1.1 1 .2 1 .3
Dispersed plug flow of an inert system Adsorption chromatography Effect of dead volume associated with signal detecting elements 1.4 Assumption of an infinite bed References 2
I
31 57 62 71
2. I 2.2 2.3
Effect of dispersion on conversion Fluid dispersion coefficients in a reacting system Fluid dispersion coefficients in adsorption beds References
72 72 79 87 92
3
DIFFUSION AND REACTION IN A POROUS CATALYST
94
4
PARTICLE-TO-FLUID MASS TRANSFER COEFFICIENTS
FLUID DISPERSION COEFFICIENTS
3.1
Assumption of a concentric concentration profile in a spherical catalyst pellet 95 3. 2 Effectiveness factors for first-order irreversible reactions under isothermal conditions 97 3.3 Pore diffusion of gases 1 08 1 27 3 . 4 Juttner modulus for first-order reversible reactions References 1 36
4.1 4.2 4.3
Review o f the published gas phase data Review of the published liquid phase data Re-evaluation of the mass transfer data References v
138 1 40 149 151 1 58
vi 5
CONTENTS
STEADY-STATE HEAT TRANSFER
5 . 1 Steady-state bed temperature 5 . 2 Effective radial thermal conductivities 5 . 3 Wall heat transfer coefficients 5 .4 Overall heat transfer coefficients 5 . 5 Effective axial thermal conductivities References 6
THERMAL RESPONSE MEASUREMENTS
206 208 210 230
Frequency response measurements of Gunn and De Souza Parameter estimation from one-shot thermal input Fluid thermal dispersion coefficients Transient effective thermal conductivities of quiescent beds 6 . 5 Assumption o f an infinite bed References
232 237 242
UNSTEADY-STATE HEAT TRANSFER MODELS
243
6.1 6. 2 6.3 6.4
7
161 1 62 175 1 97 201 202 204
Step and frequency responses for the Schumann, C - S and D-C models 7 . 2 Assumption of a concentric temperature profile in a solid sphere in the D-C model 7 . 3 Effect of fluid thermal dispersion coefficients on the Nusselt numbers of the D-C model 7 . 4 The C-S model 7 . 5 The Schumann model References 7. 1
8
PARTICLE-TO-FLUID HEAT TRANSFER COEFFICIENTS
A review and correction of the data obtained from steadystate measurements 8 . 2 A review and correction of the data obtained from unsteady-state measurements 8 . 3 Correlation of Nusselt numbers References
244 248 250 253 261 263 264
8.1
272 285 292 294
APPENDICES A PHYSICAL PROPERTIES
A. I Some fundamental physical constants in SI units A . 2 Conversion factors A.3 Physical properties of the elements and some inorganic and organic compounds
296 296 297 301
CONTENTS
A.4 A.5 A.6 A. 7 A.8 A.9 A.IO B
Physical properties of some gases Physical properties of some liquids Physical properties of plastics Thermal conductivities of miscellaneous solids Prediction of diffusion coefficients in binary gas systems Data of diffusion coefficients in binary gas systems Diffusion coefficients of gases in water
vii 304 307 310 315 316 317 322
COMPUTER PROGRAMS (FORTRAN 77)
323
Prediction of response s�gnal by the method of Section 1.1.6. 2 � calculation of root-mean-square-errors for construction of two-dimensional error map B . 2 Data input
323 345
B. 1
C DERIVATIONS OF MOMENT-EQUATIONS IN DIGITAL COMPUTER
C.l Derivations of Eqs. {1.63) and {1.64) C.2 Derivation of first moment of the system discussed in Section I 3 .
346 346 348
Author Index
353
Subject Index
359
Introduction to the Series Chemical engineering covers a very wide spectrum of learning and the number of subject areas encompassed in both undergraduate and graduate courses is inevitably increasing every year. This wide variety of subjects makes it difficult to cover the whole subject matter of chemical engineer ing in a single book. The present series is therefore planned as a number of books covering areas of chemical engineering which, although important, are not treated at any length in graduate and postgraduate standard texts. Additionally, the series will incorporate recent research material which has reached the stage where an overall survey is appropriate, and where sufficient information is available to merit publication in book form for the benefit of the profession as a whole. Inevitably, with a series such as this, constant revision is necessary if the value of the texts for both teaching and research purposes is to be maintained. I would therefore be indebted to individuals for criticisms and for suggestions for future editions. R. HUGHES
ix
Preface Nori Wakao commenced his research in heat and mass transfer in packed bed reactors about two and a half decades ago when he was a graduate student. He was first interested in steady-state heat transfer in packed beds, particularly in axial and radial effective thermal conductivities. He was surprised to find from his experimental results that the axial effective thermal conductivities were larger than the radial conductivities, but soon learned that Wilhelm had also found, from tracer dispersion measurements, that the axial gas dispersion coefficient was larger than that in the radial direction. A packed bed is a heterogeneous system composed of solid particles and fluid flowing in the interstitial space among the particles. Because o f this heterogeneity and complexity, the packed bed has not lent itself to the application of exact hydrodynamic theory. However, instead of an exact theory, a rather conventional or statistical approach has often been made. One of the typical examples is the assumption made by Wilhelm that a packed bed may be regarded as a series of mixing cells, each con taining a single particle. The development of the digital computer has made application of this idea possible for the analysis and design of packed bed reactors. Ranz's model on fluid m ixing in a sphere-lattice has also made us visualize where the lateral mixing comes from in a packed bed. His model was later extended by Yagi and Kunii for the interpretation and correla tion o f the radial effective thermal conductivities. In packed bed heat transfer, what Gunn found from frequency thermal response measurements is of great significance. As far as the authors know, he was the first to observe the large axial fluid thermal diffusivities from the conventional model based on the assumption of the in traparticle tem perature having radial symmetry. The interpretation of his finding was attempted separately by Vortmeyer and Wakao. One of the long-lasting subjects of discussion in the past four decades has been the anomalous decrease in particle-to-fluid heat and mass transfer coefficients with decreasing flow rate at low Reynolds number. In fact, the anomaly had been experimentally observed by a number of investigators. The authors have shown from theory that fluid dispersion coefficients for Xl
xii
P REFACE
mass depend upon the type of system: inert bed, reacting bed, or bed with only mass transfer taking place between particle surface and fluid . They have shown that the particle-to-fluid transfer coefficients never continue to decrease beyond a certain Reynolds number, if proper values of fluid dispersion coefficients are employed. With regard to gaseous diffusion in porous media, the Wicke-Kallenbach type of apparatus is widely used for the determination of effective dif fusivity. In the early work, there was some confusion in interpreting the effective diffusivity data in a binary gaseous system. It had been recog nized that the inverse relation between the ratio of diffusion flux and the square root of the molecular weight of the gases in a binary gaseous system applied .only to the Knudsen diffusion region. Hoogschagen, however, found experimentally that the same relation applied to the bulk diffusion as well. This important observation has led to the succeeding theoretical development made by Evans, Watson and Mason, Scott and Dullien, and Rothfeld on the gas diffusion in a capillary tube, covering the whole range from the bulk through the transitional stage to the Knudsen region. Pollard and Present made a theoretical computation of the gaseous self diffusion coefficient so that they had to conduct the elaborate computa tion manually. But it is amazing that their results are in agreement with those predicted from a simple formula derived intuitively by Bosanquet. The importance of pore diffusion in catalysis was believed to have been first pointed out separately by Damkohler, Thiele and Zeldowitsch at about the same time in the late 1 930s, but, in fact, a similar work was reported by J ti ttner as early as in 1 909. The work by Wheeler on his parallel pore model has contributed significantly to the theoretical develop ment of diffusion in catalysis. Regretfully, despite all of these and since the pioneer work of Juttner, research progress in the transport phenomena in porous catalysts has been rather slow. The recently developed chromatography measurement techniques have been successfully applied for the determination of some o f the rate para meters of interest in packed bed systems. Contributions toward the development of the various estimation techniques are made by Ostergaard and Michelsen, Anderssen and White, Smith, Silveston and Hudgins, and others. In this preface, only the names of some researchers and their contribu tions are mentioned; however, the valuable contributions made by many other investigators should also be credited.
PREFACE
xiii
Lastly, the authors hope that this book will help graduate students and researchers in chemical engineering understand the phenomena of heat and mass transfer in packed bed reactors. We wish to acknowledge the helpful assistance of Dr W. K. Teo and Dr R. Hughes in offering many constructive comments on the entire manuscript. N. WAKAO S. KAGUEI
Notation
a
b*n b�
c c' ci en c* c*a c*s '
'
c *(II)
area (m 2 ) = A � /A � , amplitude ratio for frequency w amplitudes for the harmonic components with frequency w of the input and response signals, respectively: in Example 1 . 1 (s); in Chapters 6 and 7 (K) particle surface area per unit volume of packed bed; a = 3(1 - Eb)/R for spherical particles (m-1) capillary tube radius (m) mean pore radius ( m) coefficient defined by Eq. ( 1 .34a) (s- 1 ) ; defined by Eq. (1.4 l a) (mol m -3); defined by Eq. (6 . 1 7a) (K) n-th root of Eqs. (5 .4c) or (8.6) coefficient defined by Eq. ( 1 .46a) (mol m -3) coefficient defined by Eq. ( 1.42a) (mol m -3) ; defined in Eq. (6 . 1 6) (K) cosine components with frequency w of input and response signals, respectively: in Section 1 . 1.5 (mol s m-3); in Example 1 . 1 (s) coefficient defined by Eq. ( 1 .34b) (s-1); defined by Eq. ( 1 .4 1 b) (mol m- 3); defined in Eq. (6. 1 7b) (K) coefficient defined by Eq. ( 1 .46b) (mol m -3) coefficient defined by Eq. ( 1 .42b) (mol m-3); defined in Eq . (6. 1 6) (K) sine components with frequency w of input and response signals, respectively: in Section 1 . 1 .5 (mol s m -3); in Example 1 . 1 (s) concentration in the bulk fluid (mol m -3) concentration in empty/inert bed sections (mol m -3) input and response concentrations, respectively (mol m - 3) fluid concentration in a unit cell (mol m -3) antisymmetric and symmetric components of C*, respec tively (mol m-· 3 ) fluid concentration in two cells in contact (mol m-�) XV
xvi
HEAT AND MASS TRANSFER IN PACKED BEDS
Cexit CF Cin Cps' Cs Cs c
c *(II) Ca Cad * Cad Cj D' Da
(Dax)mixing De D1·
concentrations of reacting species A and B, respectively (mol m-3) concentrations at the pellet surface of reacting species A and B, respectively (mol m -3) exit concentration (mol m-3) specific heat of the fluid (J kg-1 K-1) inlet concentration (mol m-3) concentrations at the particle surface (mol m-3) specific heat of the solid particle (J kg-1 K - 1 ) gaseous tracer concentration in the intraparticle pore volume (mol m-3) intraparticle gaseous concentration in a unit cell (mol m-3) antisymmetric and symmetric components of c*, respec tively (mol m-3) intraparticle gaseous concentration in two cells in contact (mol m-3) gaseous concentration in the macropores (mol m-3) amount adsorbed in a particle (mol kg- 1 ) amount adsorbed in a unit cell (mol kg-1) gaseous concentration in microporous particle (mol m-3) dispersion coefficient in empty bed sections (m2 s-1) gaseous effective diffusivity in the macropores, defined per unit cross-sectional area of the pellet (m2 s-1) axial fluid dispersion coefficient based on unit void area (m2 s-1) axial fluid dispersion coefficient based on unit void area in bed of non-porous inert particles (m 2 s-1) turbulent contribution to Dax (m2 s-1) intraparticle effective diffusivity (m 2 s-1) gaseous effective diffusivity in the micropores, defined per unit cross-sectional area of the microporous particle (m 2 s- 1 ) Knudsen diffusivity (m 2 s-1) diffusivity of m-th component through external film on catalyst pellet (m 2 s-1) self-diffusion coefficient for species m (m 2 s- 1 ) binary molecular diffusion coefficient for species m and n (m2 s- 1 ) particle diameter (m)
N O TA TION
xvii
radial fluid dispersion coefficient based on unit void area (m2 s-1) radial fluid dispersion coefficient based on unit void area in bed of non-porous inert particles (m2 s-1) turbulent contribution to Dr (m2 s-1) effective diffusivity of adsorbate in the micropores, defined per unit cross-sectional area of the microporous particle (m2 s- 1 ) column diameter (m) molecular diffusion coefficient (m2 s-1) activation energy of intrinsic chemical reaction (J mol-1) activation energy of overall reaction (J mol- 1 ) effective diffusion coefficient in a cylindrical unit cell (m2 s- 1 ) effective diffusion coefficient in a bed of non-porous (E0)inert inert particles (m2 s- 1 ) effective diffusion coefficient m two cells in contact (m2 s-1) axial fluid dispersion coefficient based on column cross section (m2 s-1) catalyst e ffectiveness factor Ef transfer function F(s) transfer function (C -S model) [F(s)lc-s transfer function (D-C model) [F(s)lo-c [F(s)]schumann transfer function (Schumann model) plf .. view factor from surface i to surface j overall view factor from surface ito surface j plf.. volumetric flow rate (m3 s-1) Fv radiation correction factor defined by Eq. (5 . 59) f in versed transfer function, i . e . impulse response signal (s- 1 ) f(t) upF, fluid mass velocity per unit area of bed cross G section (kg m-2 s-1) particle-to-fluid heat transfer coefficient (W m-2 K-1) particle-to-fluid heat transfer coefficient evaluated with aax = 0 (W m-2 K-1 ) radiant heat transfer coefficient based on the unit area of the particle surface (W m-2 K-1) h' radiant heat transfer coefficient based on cross-sectional area (W m-2 K-1 ) =
r
xviii HEAT AND MASS TRANSFER IN PACKED BEDS radiant heat transfer coefficient with radiating gray gas (W m-2 K-1) wall heat transfer coefficient (W m-2 K-1) imaginary part of F(inrr/r) modified Bessel function of the first kind and n-th order imaginary part of F(iw) (-1 )1/2 i diffusion flux in a capillary tube (mol m-2 s-1) J diffusion flux in porous media (mol m-2 s-1) Nu/(Pr113Re ) J factor for heat transfer = Sh/(Sc113Re), J factor for mass transfer Bessel function of the first kind and n-th order overall rate constant (s-1) effective thermal conductivity of cylindrical unit cell with stagnant fluid (W m -1 K -1) adsorption equilibrium constant (first-order) (m3 kg-1) chemical reaction equilibrium constant modified Bessel function of the second kind and n-th order adsorption rate constant (first-order) (m3 kg-1 s-1) effective thermal conductivity of a quiescent bed (W m-1 K-1) conduction contribution to k� (W m-1 K-1) (k�)coND (k�)RAD-COND combined radiation and conduction contribution to k� (W m-1 K-1) (k� )coNTACT contact contribution to k� (W m-1 K-1) effective axial thermal conductivity (W m-1 K -1) keax effective fluid phase thermal conductivity, defined per keF unit area of bed cross-section (C-S model) (W m-1 K-1) effective radial thermal conductivity (W m-1 K-1) effective solid phase thermal conductivity, defined per unit area of bed cross-section (C-S model) (W m-1 K-1) fluid thermal conductivity (W m-1 K-1) particle-to-fluid mass transfer coefficient (m s-1) particle-to-fluid mass transfer coefficient evaluated with Dax = 0 (m s-1) solid thermal conductivity (W m-1 K-1) chemical reaction rate constant (first-order) (s-1) distance/height (m) h* r
=
=
,
NOTATION
N No NH Nu Nu' Nu" Nu"' Nur nd nx p
Peax (Pea x) mixing Per (Per ) mixing P� ( )
Pr Po
p
Pg Q Qp q
Qlatera l Qy Qx
xix
half length of dead volume section (m) actual length of pore (m) distance between a response measuring point and bed exit (m) molecular weight molar mass (kg mol-1) n-th moments of input and response signals, respectively (s") = m �;m ; (s) n-th weighted moments of input and response signals, respectively (s") diffusion rate (mol s-1) = Dax f(L U), mass dispersion number = Ci.ax f(LU), thermal dispersion number = h p D p lkF, Nusselt number (modified D-C model) Nusselt number (original D-C model) Nusselt number (C-S model) Nusselt number (Schumann model) = hrDp /ks, radiant Nusselt number molar flux from gas to pellet (mol m-2 s-1) diffusion rate passing axially through a cross-sectional area of a unit cell (mol s-1) total pressure (Pa) D p U/Dax' axial Peclet number = D p U/(Dax) mixing, turbulent contribution to Peax DP U/Dn radial Peclet number = D p U/(Dr)mixing , turbulent contribution to Per associated Legendre function of the first kind and n-th order = CFJ..LfkF, Prandtl number atmospheric pressure (Pa) emissivity of gray surface emissivity of gray gas radiant heat transfer rate between two hemispheres (W) heat transfer rate from fluid to particle (W) average axial heat conduction rate in a cell (W) lateral heat flow rate per unit area (W m -2) rate of heat generation per unit volume of solid (W m -3) axial heat conduction rate in a cell (W) =
=
XX
HEAT AND MASS TRANSFER IN PACKED BEDS
R R' Re Rg Rn Rp RT Rv Rw
particle radius ( m) cylindrical cell radius (m) = Dp UPF I JJ.. = DpG/JJ.., Reynolds number gas constant (J K-1 mol-1) real part of F(inrr/r) total reaction rate in a single pellet (mol s-1) column radius (m) reaction rate based on stoichiometry (mol m-3 s-1) real part of F(iw) radial distance variable from the center of a particle (m) r radial distance variable in a cylindrical packed bed (m) r radial distance variable from central axis of a unit cell (m) radial distance variable in microporous particle (m) radius of microporous particle (m) chemical reaction rate per unit volume of catalyst pellet (mol m-3 s-1) reaction rate per unit volume of reactor (mol m-3 s-1) 'x Sc = JJ.. f(pFD v), Sclunidt number = krDp/D , Sherwood number Sh v = klDp/Dv7 Sherwood number evaluated with Dax 0 sht pore surface area per unit mass of porous so1id (m2 kg-1) Sg particle surface area (m2 ) SP Laplace operator (s-1) s temperature (K) T temperature at central axis (K) Tc exit fluid temperature (K) Texit fluid temperature (K) TF fluid temperature in a unit cell (K) Tt input and response temperature signals, respectively (K) T},TV response temperature signal (C -S model) (K) (TP)c-s response temperature signal (D-C model) (K) ( Tp)D-c response temperature signal (original D-C model) (K) CTU)o-c (TV)schumann response temperature signal (Schumann model) (K) inlet fluid temperature (K) Tin bed exit temperature (K) TL mixed mean temperature (K) Tm temperature at particle surface (K) Tps solid temperature (K) Ts solid temperature in a unit cell (K) r; =
Tw To
t
u Uo
u
NOTATION
Xxi
wall temperature (K) temperature at bed inlet (K) time variable (s) interstitial fluid velocity (m s-1) overall heat transfer coefficient (W m-2 K-1) superficial fluid velocity (m s-1) fluid velocity in empty section (m s-1) reactor volume (m3) pore volume per unit mass of porous solid (m3 kg-1) particle volume (m3) mean molecular velocity (m s-1) conversion axial distance variable (m) Bessel function of the second kind and n-th order mole fraction of species m
Greek symbols
a:F O:r as 'Y o 'Y 1
A
8 8 8H
Do 81
€ €a €b , " €f, €f, €f, €f*
= 1 + 12 /1 1 accommodation coefficient fraction defined by Eq. (5 .25) axial fluid thermal dispersion coefficient (m2 s-1) axial fluid thermal dispersion coefficient defined by Eq. (6.3) {m2 s-1) = kFf(CF PF), fluid thermal diffusivity (m2 s-1) radial fluid thermal dispersion coefficient (m 2 s-1) = ks/(Cs Ps), solid thermal diffusivity (m2 s-1) defined by Eq. {6. 1 9a) defined by Eq. {6. 1 9b) (s) difference defined by Eq. { 1 .5 5) (s) diffusibility temperature jump coefficient in Eq. (5 . 52) (m) coefficient defined by Eq. (6. 2 1 ) defined by Eq. { 1 .63a) defined by Eq. { 1 .64a) (s) root-mean-square error macropore void fraction of a pellet bed void fraction root-mean-square-errors (frequency response)
XXii
HEAT AND MASS TRANSFER IN PACKED BEDS
Ei Eip Ep
€5 Es 8 1\ Ao A ad A.
/\H A.o
J1
,.... n ,,.... n
,I
,II
Jln 'Jln
*I *II
a a� a� r,r
*
f
r
<}>
¢
w
,t,.l ,t,.II '+'w' '+'w w
micropore void fraction of pellet volume fraction of a microporous particle in pellet intraparticle void fraction volume fraction of solid root-mean-square-error (step response) angle variable (rad) fractional contact area coefficient defined by Eq. ( 1 .86a) coefficient defined by Eq. { 1 .90a) coefficient defined by Eq. (6.38b) mean free path (m) mean free path at P0 (m) fluid viscosity (Pa s) n-th central moments of input and response signals, respectively (sn ) n-th weighted central moments of input and response signals, respectively (sn ) fluid density (kg m-3) particle density {kg m-3) Stefan-Boltzmann constant (W m-2 K-4) variance defined by Eq. (6.1 9) (s2) variance defined by Eq. ( 1.64) ( s2) half period (s) tortuosity factor = L/U, mean residence time (s) angle variable in spherical coordinates (rad) = R(kx fDe)112 Jtittner modulus for first-order irreversible chemical reaction Jtittner modulus for first-order reversible chemical reac tion, defined by Eq. (3.88) = ¢� - ¢� , phase shift (rad) phases for harmonic components with frequency w of input and response signals, respectively (rad) pressure parameter defined by Eq. (5 . 5 7) frequency (rad s-1)
1
IN
Parameter Estimation from Tracer Response Measurements
chapter, the techniques o f parameter estimation from the measurements of tracer input and response signals are discussed. A moment method was first proposed for the estimation of the packed bed parameters described by a dispersed plug flow model. The moment method is interesting in theory, but, in practice, its shortcomings are that tailing and the frontal portions of the signal are overly weighted in the evaluation of the moments. Response signals usually have long tails, and the experimental errors in the tailing portion, as well as truncation of the tailing portion give serious errors in the moment analysis. To overcome this disadvantage, modified methods have been p roposed. These include: a weighted moment method and transfer function fitting b y Q.>stergaard and Michelsen [ I ]; Fourier analysis b y Gangwal et al. [2]; curve fitting by Clements [3] and others. Clements suggested that accurate parameter determination should be made by curve fitting in the time or Laplace domain. Anderssen and White (4, 5] showed that if an optimum weighting factor was chosen, the weighted moment method was almost as good as the analysis in the time domain. A similar conclusion was also reached by Wolff et al. [6]. In general, the moment, weighted moment, transfer function fitting and Fourier analysis all deal with the measured signals multiplied by a time function called a weighting factor. It is easily understood that the best weighting factor is unity, i.e. the parameters are best determined by real-time analysis.
1.1
THIS
Dispersed Plug Flow of an Inert System
Consider an inert tracer imposed on a stream of fluid in a packed bed of non-porous particles. The concentration input signal, C�xpt(t), and the I
2
HEAT AND MASS TRANSFER IN PACKED BEDS
concentration response signal, C��pt(t), are measured at two downstream points, at a distance, L, apart in the bed. If the concentration, C, is uniform in the radial direction, the fundamental equation according to the dispersed plug flow model is
ac a2c -=Dax ax2 at
--
u
ac ax
( 1.1)
-
where D ax is the axial fluid dispersion coefficient, U is the interstitial fluid velocity, and x is the axial distance. Assuming an infinite packed bed ( for the criterion , see Section 1.4), and if the initial tracer concentration is zero throughout the bed, the transfer function of�the bed within !� a distance L is expressed in terms of the measured signals, C xpt(t) and c pt(t), by
I c!�p. 00
exp (-st) dt
0 F(s) = ------
(1.2)
00
I c!xpt
exp (-st) dt
0
= exp
(-2N0 1-
[I- (I + 4Nofs)112]
)
( 1 .3)
where f and N0 are the mean residence time and mass dispersion number, respectively, defined as follows:
L
f=
u
Dax N0 = -·
LU
1.1.1
.
( 1 3a)
.
( 1 3 b)
Moment Method for Impulse Response
If the input signal is a delta function, the denominator of Eq. (I .2) � !� becomes fooo C xp t dt, which is identical to fooo C pt dt. The transfer
PARAMETER ESTIM ATION
3
function is then 00
F(s) =
J c!�pt
exp (-st) dt
0
(1.4)
00
Differentiating with respect to s, it gives 00
dnF(s) = (-It ds n
--
J c!�p1t"
exp ( st) dt -
o
oo
------
J c!�P' dt
( 1 .5)
0
Therefore, ( 1 .6) where 00
( 1 .7)
��
and is called the n-th moment of C pt.
4
HEAT AND MASS TRANSFER IN PACKED BEDS
The second central moment or variance is defined as: 00
00
( 1 .8)
( 1 .9) For the transfer function of Eq. ( 1 .3), it is shown that ( 1 . 1 0) (1.11) The mean residence time, f, and the mass dispersion number, N0, are determined, therefore, from the first moment and the variance of the impulse response, respectively. 1.1.2
Moment Method for One-Shot Input
Mathematically, delta input is the only possibility. Even if a tracer is injected instantaneously, the tracer is usually being imposed on a fluid flowing in a column, which results in some d iffusion from the very beginning. If tracer concentration-time curves are measured at two downstream points, the tracer can be introduced in any type of one-shot input. It is advantageous not to have to be concerned about the shape of the concen tration curve on introducing the tracer. Some mathematical manipulations give : ( 1 . 1 2) ( 1 .13) where M: and J..L� are the first and second central moments of input signal, respectively, and are defined in a similar way to the response signal
PARAMETER ESTIMATION
5
moments in Eqs. (1.7) and (1.8). When the input signal is of delta func tion, M� = 0 and J.L! = 0, so that Eqs. (1.12) and (1.13) reduce, corre spondingly, to Eqs. (1.10) and (1.11). However, the moment method has the shortcoming that the weighting factor, tn, puts a large weight on the tailing portion of the signal. Errors in the tailing portion are magnified in the evaluation of moments, particu larly o f the higher moments. Also the errors in the frontal portion are magnified in evaluating the central moments. 1.1.3
Weighted Moment Method
To overcome the shortcomings of the moment method, �stergaard and Michelsen [ 1 ] proposed using the form of the right-hand side of Eq. (1.5) as the basis of their analysis. This modified technique is called the weighted moment method. The weighting factor, tn exp (-st), which is zero at both t = 0 and at longer times, may obviously avoid the disad vantages of the moment method. The parameters involved in the transfer function are then determined from the modified moments as defined below:t Zeroth weighted moment 00
*m 0-
I
Cexp (-st) d t
0
00
------
I
(1.14)
Cdt
0
n-th weighted moment 00
m* =
n
I
Ctn exp (-st) dt
0
------
00
I
0
(1.15)
Cdt
t Note that (/)stergaard and Michelsen [ 1 ] defined m�fm� as the n-th weighted moment.
6
HEAT AND MASS TRANSFER IN PACKED BEDS
n-th weighted central moment is defined in terms of Mi= m'f fm6 as 00
f.l� =
J
C(t - Mt t exp (-st) dt
0
-----v.J
f
( 1.16)
Cdt
0
The weigh ted moments are related to the transfer function as follows:
m6n - =F(s) m61
( 1 . 1 7)
F'(s) mi11 mi1 ----=--m611 m61 F(s)
( 1 . 1 8)
pf11
pf1
m6II
m61
----
=
d [F'(s)]
-
--
ds F(s)
( 1 . 1 9)
( 1 .20)
( 1 .2 1 ) For the transfer function given by Eq. ( 1 .3), it is shown that
F'(s)=-r-(1 + 4N0rs - ) -112 F(s)
( 1 .22)
(2n)1 dn F'(s) - [--] = --· (- ft+1 N JS(l + 4N0rsrn-112• dsn F(s) n!
( 1 .23)
The two parameters,
f
and N0, can hence be determined from any set
PARAMETER ESTIMATION 7 of two moment equations, for example, from Eqs. (1.17) and ( 1 . 1 8), or from Eqs. ( 1 .18) and ( 1 . 1 9). However, the problem is what optimal values of s to use in the para meter estimation; the weighting factors, exp (- st) for zeroth moment, t exp ( st) for first moment, (t- Mi)2 exp (- st) for second central moment etc. all give weight on different portions of the signal curve. In other words, the optimal value of s depends on the order of the moment. Anderssen and White [ 5] have suggested the following equation for the estimation of the optimal s values: -
s=
nhighest ------
{max-input +l max-response -/:).to
(1.24)
where nhighest is the highest order of moment used for the parameter estimation, Llto is the difference in time delay between the input and response signals, and tmax is the time when a signal reaches the maximum point. 1.1.4
Transfer Function Fitting
The transfer function of Eq. ( 1 .3) may be rewritten as: - [In F(s)r1 = fs(ln F(s)r2- N0 or
[ ]
F'(s) -2 4No -- =-s+(f)-2. f F(s)
( 1 .25)
( 1 .26)
Q>stergaard and Michelsen [ 1 ] recommended that the transfer function, evaluated from the measured input and response signals, should be plotted as -[In F(s)r1 versus s[ln F(s)r2, or [F'(s)/F(s)r2 versus s. The parameters f and N0 are thus obtained from the slope and intercept of a straight line, according to Eq. ( 1 .25) or Eq. ( 1 .26). Again, the problem is the selection of s values for the plot. I f s is too large, the large weight is given to the frontal portion of the signal, while Eq. (I .3) shows that if s is too small, the transfer function itself has little dispersion e ffect. Hopkins eta!. (7) recommended that transfer function fitting should be made within the restricted range 2 �sf �5 .
8
HEAT AND MASS TRANSFER IN PACKED BEDS
1.1.5
Fourier Analysis
Gangwal et al. [2] applied Fourier analysis to the estimation of parameters in adsorption chromatography. Input and response signals may be considered to be composed of numerous harmonic components. Fourier analysis evaluates decay in amplitude and the phase shift for the harmonic components between the input and response signals. Substitution of s =iw (where w is frequency) into a t ransfer function gives
F(iw) = Rw + ilw.
( 1 .27)
For F(s) according to Eq. ( 1 .3), Rw and lw are as follows: Rw
=
Iw
=
exp (y) cos z
- ex p (y) sin z
( 1 .28)
where
I
y=---a cosb 2 ND
( 1 .28a)
z =a sinb
( l .28b)
a= and
[C�J +(::H,.
( 1 . 28c)
( 1 .28d) From the input and response signals measured, F(iw) is evaluated from 00
F(iw) =
J c�!pt
exp (-iwt) dt
0
00
-------
J c!xpt
0
exp (-iwt) dl
( 1 .29)
PARAMETER ESTIMATION
a� -ib� a 1w - ib 1w
9
( 1 .30)
where 00
a� =
f C:xvt
cos wt dt
( 1 .30a)
sin WI dt
( 1 .3Gb)
cos wt dt
( 1 .30c)
. sm wt dt.
( 1 .30d)
0
00
b�
=
f C!xpt
0
00
a� =
f c!�pt
0
00
buw -
J
0
en expt
By equating Eqs. ( 1 .27) and ( 1 .30), the parameters involved in the real and imaginary parts of Eq. ( 1 .28) are determined in terms of the Fourier coefficients evaluated from the signals measured. 1.1.6
Cuave Fitting in the Time Domain
This is a method in which the response signals measured are compared in the time domain with those predicted based on assumed parameter values. I f the two signal curves agree well, the parameter values used for the prediction may be regarded as correct. The following methods may be applied. 1 . 1 .6. 1
Prediction of the signal in response to one-shot input by a convolution integral
Equation ( 1 .2) indicates that the response signal is calculated by a
10
H EAT AND MASS TRANSFER IN PACKED BEDS
convolution integral as: t
cgtc(t)
=I c!,p,(n/(t - n
d�
( 1 .3 1 )
0
where f(t), the impulse response of a delta input, is the Laplace inversion of the transfer function defined as:
f(t) or
=
£-1 [F(s)]
( 1 .32)
00
I f(t)
( 1 .32a)
exp (-st) dt = F(s).
0
In the case of dispersed plug flow of an inert fluid, f(t) is easily found from Eq. ( 1 .3) to be
1
f(t) =
[ G}]
3 112 exp
2f rrNo
( 1 .33 )
t
4No f
However, in many cases, the transfer functions are much more compli cated so that the inversion cannot be made easily by conventional methods. Under such conditions, the inversion has to be performed in terms of a Fourier series. Over the period, 0 to 2 7, where 2 7 is a period of time sufficiently long enough for the tailing portion of the response signal to vanish, f(t) is expressed as: a0
oo
n=l
f(t) = - + I 2
(
an
n1Tt
n1rt
7
7
cos-+ bn sin-
)
( 1 .34)
PARAMETER ESTIMATION
where 1 an = -
7
f f(t) 2-r
7
2-r
0
1
( 1 .34a)
cos- dt
0
f bn = - f(t) I
nrrt 7
]
nrrt
sin- dt. 7
( 1 .34b)
On the other hand, substitution of s = inrr/7 into Eq. ( 1 .32a) gives
f f(t) n;t dt 00
cos
0
and
=Rn
f f(t) 00
-
[ ( n,.")]
= Real F i
0
nrrt
( 1 .35)
[
(
nrr \-
)]
sin ---; dt = Imag F i ;
( 1 .36)
Note that in the present case of dispersed plug flow of an inert fluid, F(O) = 1 , so that
Ro =I.
( 1 .37)
The signal, in response to an imperfect pulse input, is zero at t = 27, so that the response to a perfect delta input, o r the impulse response f(t), must become zero at a time shorter than 27. The terms on the left hand sides of Eqs. ( 1 .35) and ( 1 .36) may then be integrated from 0 to 27 instead of 0 to and Eq. ( 1 .34) is rewritten in terms of Rn and In evaluated' from the transfer function as: 00 ,
nrrt nrrt') 1 1 oo f(t) = - + - L Rn cos- - In sin2 7 7 n=l 7 7
(
1
·
( 1 .38)
12
HEA.T AND MASS TRANSFER IN PACKED BEDS
Equation (1.38) can also be derived more directly from the following inversion integral: 00
f(t) =
-1 J F(iw)
2rr
( 1 .39)
exp (iwt) dw.
- 00
The integration is rewritten as:
i::lw "" f(t) = - L F(ini::lw ) exp (ini::lwt) 2rr n =oo
�: { F(O) + .t (F(inAw)
exp (inAwt)
+ F(- inAw) exp (-inAwt)l =
i::lw 2rr
J F(O) + 2 I [Rn cos (ni::lwt) \ n=l
)
}
- In sin (nAwt)J .
( 1 .40)
By considering F(O) = 1 and writing i::lw = rr/r, it is easily shown that Eq. ( 1 .40) reduces to Eq. ( 1 .38). The response curve, C��tc(t), is then computed from Eqs. (1.31) and (1.38) and may be compared with the experimental curve, C��pt(t). The comparison is made over the entire region or any arbitrary time interval.
1 . 1 .6.2 Prediction of the signal in response to one-shot input by Fourier series The input signal measured is expressed by a Fourier series as: 1
00
(
a0 nrrt . nrrt Cexpt(t) =- + L an cos-+ bn sm T T 2 n=l
)
with the Fourier coefficients evaluated by the following expressions:
( 1 .4 1 )
PARAMETER ESTIM ATION
an-
- I Cexpt
_
and
bn
2r
I
I
r
nrrt
cos- dt
= I Cexpt . 2T
1
1
-
r
(1.4 I a)
r
0
nnt sm- dt
( 1 .41 b)
r
0
13
where 2r is again a period of time long enough to let the tail of the response signal vanish. The response signal is also predicted by a Fourier series of the form:
at
C�lic(t) =-2
+
(
"" nrrt nrrt L a� cos-+ bJ sin-
n=J
T
T
)
( 1 .42)
where the Fourier coefficients, expressed in terms of C��c(t), are
�I
t- r
a nand
t
2T II cos Ccalc
0
2T
nrrt r
dt
. II sm bn =- Ccalc - dt.
1
r
nrrt
J
r
0
� 1 .42a)
( 1 .42b)
The transfer function can be written as: 00
J C�c ------I dxpt exp (-sr) dt
F(s) =
0
00
exp (-st) dt
0
(1 .43)
14
HEAT A N D MASS TR ANSFER IN PACKED BEDS
Substitution of s = inrr/r and consideration of the response signal being zero at t � 2r give 2r
(
Fi
_rr n r
)
=
I c2.lc
e xp (-imrt/T) dt
r C!xpt
exp (-inrri/T) dt
___ _ o
( 1 .44)
2r
'
0
From Eqs. ( 1 .41a). ( 1.4 1 b), ( l .42a), ( 1 .42b) and ( 1 .44), it is shown that
at- ib t = (a 11
n
( 1111)
n - ib n ) F i
•
( 1.45)
r
The Fourier coefficients in Eq. ( 1 4 2 ) are evaluated, therefore, with the measured input signal and the t ransfer function. Similarly, the response signal measured is also expressed as: .
1 nrrt . rr n t a6 C��pt(t) = -2 + L \a � cos- + b� smoo
n= I
T
.
T
)
( 1.46)
with
I
I
an* -_
T
2r
nrrt
C..,11 expt cos- dt T
0
( 1 .46a)
and * _
1
bn -T
2r
I 0
u
.
rr n t
Cexpt sm -dt. r
( 1.46b)
The root-mean-square-error between C��pt and C��tc over the entire region is then evaluated by Eq. ( 1 .47) or Eq. (I .48).
PARA M E T E R ESTIMATION
2r
€=
I (C!�pt -c��Jc)2
1/2 dt
0
( 1 .47)
2r
I (C!�pt)2
15
dt
0
( 1.48)
Note that as far as the tracer imposed is not dispersed in the column (inert or reversible adsorption system), a6 = ai; . In the evaluation of the confidence range by curve fitting in the time domain, the following general criteria are often adopted: fitting is good if € < 0.05 ; fitting is poor if € � 0.05. The one-shot input method, discussed in Sections 1 . 1 .6 . 1 and 1 . 1 .6.2, requires that the tracer be imposed on a stream of an inert carrier fluid under the condition of C = 0 at t � 0. However, even if the measured input signal is any arbitrary function of time, without the imposed restric tion, the signal in response to it can also be predicted by using a convolu tion integral. Let us assume that the impulse response, f(t), becomes ' zero at any time greater than r . Equation ( 1.3 1 ) is then rewritten as:
C��c(t) =
t
I c!xpt(�)f(t -�)
d�.
( 1 .49)
' t-r
Therefore, if the input signal is measured over a time interval from t 1 to t 2 ' (with the restriction that t 2 - t 1 > r ) , the response signal in the range ' from t 1 + r to t 2 can be predicted from Eq. ( 1 .49). The computed response signal, C��c(t), is then compared with the measured signal, c��pt(t), in the time domain.
16
HEAT A N D MASS TRANSFER IN PACKED BEDS
Example 1 . 1
Table 1 . 1 lists the input and response signal readings for nitrogen, an inert tracer imposed on a laminar flow of hydrogen in a packed bed
TABLE 1 . 1 Measured input and response signals for nitrogen. t(S)
Readings
Readings Input
Response
0.0 0.5 1 .0 1 .5 2.0 2.5 3.0 3.5 4.0 4.5
0.0 9.6 40.6 84.5 1 16.7 1 3 1 .6 134.1 129.0 1 19.5 106.6
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5
95.3 82.2 70.1 61.3 5 2.6 45.0 39.2 33.8 28.5 25.0
0.0 2.3 1 1.2 21.0 35.0 48.6 6 1 .5 70.7 79.0 82.5
10.0 10.5 1 1 .0 1 1.5 12.0 12.5 13.0 13.5 1-4.0 14.5
21.9 19.0 16.5 14.5 1 2.3 1 1 .0 9.5 8.4 7.3 6.6
83.0 83.0 80.5 76.0 70.0 62.8 55.8 49.7 43.5 38.6
15.0 15.5 16.0
5.8 5.2 4.5
33.7 29.6 25.5
t(s)
Input
Response
16.5 17.0 17.5 1 8.0 18.5 19.0 19.5
4.0 3.5 3.3 3.0 2.7 2.3 2.1
22.5 19.5 17.0 14.5 1 2.8 1 1.0 9.8
20.0 20.5 2 1 .0 2 1 .5 22.0 22.5 23.0 23.5 24.0 24.5
1 .8 1.7 1.5 1.4 1.3 1 .2 1.1 1.0 0.9 0.8
8.5 7.5 6.5 5.5 4.5 4.0 3.5 3.1 2.7 2.4
25.0 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.0
2.0 1.7 1 .4 1.3 1.2 1.1 1.0 0.9 0.8 0.7
30.0 30.5 3 1 .0 3 1 .5 32.0 32.5 33.0
0.6 0.5 0.4 0.4 0.2 0.1 0.0
17
PARAMETER ESTIMATION
(L = 1 0.7 em and Eb = 0.4) of glass beads (at 20°C and atmospheric pressure). The transfer function is expressed by Eq. ( 1 .3). Find f and N0.
SOLUTION Moment method The input and response signals normalized by Eq. ( 1 .50) are shown m Figure 1 . 1 (a) (note that C1 and en are dimensionless).
0 . 2 .--- ------� l
inpu t
c
c 0
el
0 ....
c
"u
l
r�sponse
-
0.1
c 0 v
10
(a )
t
r s1
30
I . S r----�
r�spons�
,......, Vl .......
1.0
-
u
(b)
0.5
10 t
[sJ
20
30
18
HEAT AND MASS TRANSFER IN PACKED B E D S 2 �------� i n put r�sponse
N -
�
-I
u
(c)
[5)
t
1 . 1 Curves for moment calculation for Example 1 . 1 : (a) normalized input and response signals; (b) Ct for first moment: (c) C(t -M1) 2 for variance.
FIGURE
00
I
00
C1 dt
0
=
I
0
en dl
=
1 s.
( 1 . 50)
The first moments are the areas under the curves, Ct versus t, in Figure 1 . 1 {b). The areas give Mt = 5.4 s and Ml1 = 1 2.0 s, and consequently, f = 6.6 s. Similarly, the variances are the areas under the curves, C(t - M 1)2 versus t , in Figure 1 . 1 ( c). The zig-zag curves obtained from the slightly scattered data at long periods of time are apparently the result of small errors in the tailing portions of the signals. This shows that the variance values them selves may have considerable errors. In any case, the difference between these two variances gives N0 0.0087. =
Weighted moment method The weighting factors for the zeroth, first , second and third weighted moments are shown in Figures 1.2(a)-(d). Figures 1.3(a)-{d) show Cexp (-st), Ct exp (-st), C(t -M()2exp (-st) and C(t - M()3 exp (- st) versus t curves. The zeroth, first and second weighted central moments are the areas under the curves of Figures 1.3(a), (b) and (c), respectively. In Figure 1 .3{d), the curve is negative for t < Mi
PARAMETER ESTIMATION
19
and becomes positive for t > Mi. The third central moment is then the difference between the area of the positive portion of the cutve (t > M i) and that of the negative portion (t < M{). In Figure 1 .3( c), the tailing portions are again zig-zag when s is small� but apparently become smooth with an increase in s. The same trend is seen in Figure 1 .3( d). This is due to the fact that a large value of s makes the weight shift from the tailing portion to the frontal portion. The small errors associated with the frontal portions are then magnified when the moments are evaluated with large s values.
Ill I
"
0 0
10 t
(a)
[ S]
20
30
10
I I I
c'
,-,
I
V)
Ill
I
5
I I I I I
I
.,
-
(b)
I I
I
...__,
0.05
0 0
10
20 t
l S]
30
20
HEAT AND MASS TRANSFER IN PACKED BEDS
1 00 c'
,...
I \
I
...-.
N
�
I
. .,
\
\
I
50
VI
\
I
V)
-
\
I I I
I I
I I I
N
-
0.1
\
\ \
\ \ \ \
I I
I
0
10
0
(c)
t
1 00 0
...... V)
I I I
�
VI
I
500
..,
I I I I
M
I
-
0
' '
',
[51
20
30
c ' ....
I ' ' I \
I
I I
M
'
\
\\
\
\\ 11
I
( Sl FIGURE 1.2 Weighting factors versus t for weighted moment calculation: (a) exp (- st) for zeroth weighted moment; (b) t exp (- st) for first weighted moment; (c) t2 exp (-st) for second weighted moment; (d) t3 exp (-st) for third
(d)
0
10
20
30
weighted moment.
Table 1.2 lists the parameter values obtained in the range s = 0.01 to 2 s-1 from: (i) Eqs. ( 1 . 1 7 ) and ( 1 . 1 8), (ii) Eqs. ( 1 . 1 8 ) and ( 1 . 1 9), and (iii) Eqs. ( 1 . 1 9) and ( 1 .20). Compared with the values of f and N0 obtained in (i) and (ii), the data determined in (iii) are entirely inconsistent and erroneous. Some of the parameter values are found to be negative or even
PARAMETER F.STIMATION 2 1 imaginary. This discrepancy is obviously due to the fact that the third moments themselves, as seen in Figure 1 .3( d), have large errors. As shown in Figure 1 .4 , the values of f obtained in (i) and (ii) in the range s < 0.4 s- 1 are slightly s-dependent� and they decrease at higher s values. The N0 values obtained in (i) and (ii) are, as shown in Figure 1 . 5 , highly s-depend ent. First, they increase with an increase in s, and then decrease after reaching maximum values.
0 . 2 .-------�
til
IQI
u
s = 0
O. J
\
0.05 .
�
..
10
(a)
.
I. 5
t -
.
.�
,
J. o til
IQl
-
u
(b)
..
·.•. s = 0 [ s - 1 ) .
.
r··
I
[ SJ
. •
.
•
•
s
= 0
.
.
0.5
t
[ S)
20
30
22
HEAT AND MASS T R A N S F E R IN PACKED BEDS 2
.------
,...., N V)
s = 0
��
.......
•
Ill I a, N * �
�
•
0.05 ·.•
I
-
s = 0 s = 0
u
.. . . .
10
(c)
,...., M V) ......
[ S]
20
- - .. .
..
..
.. .
30
' . .' ' ' ' 0.05 ..
4
'
•
,
.. .,/
0
- 8 ' ------�----�--�
(d) t [ S) FIGURE 1.3 Curves for weighted moment calculation for Example 1 . 1 (solid and dashed lines are for input and response signals, respectively) : (a) C exp (-st) for zeroth weighted moment; (b) Ct exp (-st) for first weighted moment; (c) C(t -M';)2 exp (-st) for second weighted central moment; (d) C(t - M1* )3 exp (-st) for third weighted central moment. 0
10
20
30
From Figure 1 .2, the values of s suited for both input and response curves seem to be s � 0.1 s-1 s =
0 . 1 to 0.2 s-1 s = 0.2 to 0.3 s-1
for zeroth weighted moment for first weighted moment for second weighted moment.
PARAMETER F.STIM ATION
23
TABLE 1 . 2 The values o f r and .fv'o obtained from the '"·eightcd moment method. -----
- ---·----
(i)
- -- - - ----( iii)
4- · --------------
(ii)
Oth and 1st moments
s ( s- • )
r (S)
No
0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.6 0.8 1 .0 2.0
6.6 6.6 6.6 6.6 6.6 6.6 6.7 6.7 6.7 6.6 6.6 6.5 6.1
0.0 1 1 0.0 1 2 0.0 1 5 0.0 1 7 0.019 0.020 0.024 0.024 0.024 0.022 0.020 0.0 1 7 0.009
a Imaginary values obtained.
Oth, 2nd and 3rd moments
Oth, 1st and 2nd moments -----r
/Vo
{S)
6.6 6.6 6.6 6.6 6.7 6.7 6.7 6.7 6.6 6.4 6.3 6.1 5.4
0.0 1 1 0.014 0.017 0.020 0.022 0.023 0.025 0.024 0.022 0.018 0.0 1 4 0.0 1 1 0.003
-0.13 -0.22 -0.49 - 0.98 - 1 .9 - 3.9 6.1 4.0 3.6 3.2 3.5 a a
(s)
T
,vo 24.4 9.7 2.3 0.68 0.2 1 0.055 0.031 0.085 0.12 0.22 0.42 a
a
-----·
7 �------� - - -------
, ....
( .)
I
-��
OUJ
ana
lsr
6 1-
I -I -I --------� --� s � L--� � l -0.2
0
0.4
S
FIGURE
1 .4
r
I S- I I
0 .6
0.8
1 .0
versus s, obtained from weighted moment for Example 1.1.
24
HEAT A N D MASS TRANSFER IN PACKED BEDS 0 . 03 ,.-------,
0 . 02 0 z:
0.01
0 �--�---�---� 1 .0 0.6 0.8 0.4 0.2 0 S
FIGURE
(S-I
l
1 .5 No versus s, obtained from weighted moment for Example 1.1.
If we roughly assume that s = 0 . 1 s- 1 is good for the analysis (i) based on the zeroth and first weighted moments, and s = 0 . 1 5 s- 1 for the analysis (ii) using the zeroth, first and second weighted moments, the parameter values are determined as: (i) from Eqs. ( 1. I 7) and ( 1 . 1 8 ) f = 6.6 s and No = 0.020 (ii) from Eqs. ( 1 . 1 8) and ( 1 . 1 9) f = 6.7 s
and
N0 = 0.024.
Transfer function fitting With the measured signals, the transfer function, F(s), is evaluated from Eq. ( 1 .2). Then, as suggested by Eq. ( 1 .25), F(s) is plotted as -(ln F(s)r1 versus s (ln F(s)r2 in Figure 1 .6. I f we examine the graph closely, we will find that the points with large s values are crowded together toward 'the origin of the graph and the so-called straight line actually crosses the y-axis at a very small negative value. The value of the intercept is so small that the determination of N0 is seriously affected by small errors in the points near the origin or by a slight change in the slope of the straight line. If Eq. ( 1 .25) is rewritten as:
-
In F(s) s
= - No
[In F(s)F s
+r
( 1 .5 I )
PARAMETER ESTIMATION
25
1. 5 !-.. -
-
"'
--
I. 0
u.. c -
.......
1
0.5
/
0
/
0.1
0
-2
[s-1 ]
s ( l n F (s))
FIGURE
1 .6
0.2
1 2 - [ In F(s)r versus s [ ln F(s)r for Exam·ple
1.1.
the data, replotted as -[In F(s)]/s versus [In F(s)] 2/s in Figure 1 .7, show that a straight line cannot be drawn over the entire region of s from 0.0 1 to 1 s-1. But a relatively good straight line exists in the range s = 0.2 to 0.4 s- 1 , from which the parameters are determined as: f 6.7 s and N0 = 0.024. =
6 . 8 .------. 6. 6
Ill
.,.
..... -
6.4
"'
� 6.2 c
.......
I 6 .0 5. 8
1.0
'------J. __ __ _...__ . __ __._ __ __ _._ __ __ _.__ __ __ ...._ ___, __
10
0
FIGURE
1.7
2
20
{ I n F (s)) /s
30 rsJ
-[In F(s)]/s versus [ln F(s) J2/s for Example 1 . 1 .
26 HEAT AND MASS TRANSFER IN PACKED BEDS Figure 1 .8 also indicates that a plot based on Eq. ( 1 .26), [F'(s)/F(s)r2 versus s. again does not give a straight line over the range s == 0 to 1 s-1 . But, a linear portion in the range s 0. 1 to 0.3 s-1 yields f = 6. 7 s and N0 = 0.025. =
....
,......
I
Ill
.......
0.03
N , ,......_, Ill
�
-
-
Ill
-
0 . 025
.. �
-
0.02 0 fiGURE
1 .8
s
0.4
0.2
[F'(s)/F(s)r 2
Fourier analysis The amplitude, A�, and phase, input signal are
¢�,
[ s-1 l
0.6
versus
0.8
s for Example
J.O
1.1.
for the harmonic component of the
and
( 1 .5 2)
respectively. Similarly, those of the response signal are
and
( 1 . 53)
PARAMETER ESTIMATION
27
The amplitude ratio Aw and the phase shift w between the two signals are then
( 1 .54)
and r�,. '+' W
r�,. II _ r�,. I = '+' W '+'W •
The amplitudes, A� and A�, and the ratio, A w , are plotted versus w in Figure 1.9, and the phases, ¢� and ¢�, and the phase shift, ¢w, are plotted versus w in Figure 1 . 1 0. (Note that -¢w is often called a phase lag.) Fourier analysis should be made in a frequency range where the ampli tudes A � and A� are not very small, and the amplitude ratio, Aw, is appreciably away from unity (or the phase shift ¢w is away from zero). Therefore, both low and high frequencies are found inadequate for Fourier analysis. The parameter values obtained at various frequencies are listed in Table 1 .3. Figure 1 . 1 1 is a sensitivity test of the parameter values obtained. The y-axis is the difference �' defined by 00
00
I d!1c
A=
exp (-iwt) dt -
( 1 .5 5)
exp (-iwt) dt
0
0
0
I C�1c
1.0
-
·-
0 cr
..
.. "0 :J -
0.5
0.
E
�
O L_
L_ __ __ __ L_ __ __ __ L---� --�
__ __ __
0
0.2
0 .4 w
FIGURE 1.9
0.6
0.8
1.0
[rad · s1 ]
Amplitudes and amplitude ratio versus w for Example 1.1.
28
HEAT AND MASS TRANSFER IN PACKED BEDS r-'\
"'0
0 ...
1t
I I I
"' I '
w
.l: II)
I I
-
Ql II) 0 .l: a.
I
0
' '
'
' '
ca Ql II) 0 .I:. a.
'
'
'
'
'
......
•w ' ' - •w '
-n
0.2
0
0.4 lA)
FIGURE 1 . 1 0
[rad s-1 ]
0.6
......
1.0
0.�
·
Phases and phase shift versus w for Example 1.1.
TABLE 1 .3 The values of :; and No obtained from Fourier analysis. w
(rad s- 1) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-
j
No
6.7 6.7 6.7 6.7 6.7 6.6 6.6 6.6
0.0 1 3 0.021 0.028 0.029 0.030 0.029 0.029 0.030
(s)
where C��lc is a response signal predicted using the values of f and N0 listed in Table 1 .3 , and C��lc is that calculated using either 1 . 1 f and N0, or f and 1 . 1 N0 values. Figure 1 . 1 1 demonstrates explicitly that the d iffer ence, D., is much larger for a I0% increase in f values than the corre sponding increase in N0 values. This indicates that f has a greater effect upon the shape of the response signal than N 0 , and consequently, f may be determined more accurately than N0. The f curve reaches a maximum value at w = 0.32 rad s- 1 , and this indicates that f is determined most accurately at this w value, or in the
PARAMETER ESTIMATION
29
0 . 1 5 �-------.
-T-curvt> (colc_ula
� 0. 1
tf'd w i t h
l . 1 � and N0
<3 ., u c " ... "
)
- 0 . 05 0
(cal cui
)
a ted w i t h � and 1 . l N 0
0
0
FIGURE 1 . 1 1
0.4
0. 2
0.8
0.6
1.0
w [ rad · s-1 ] Sensitivity test in :r and N o values obtained for Example 1.1.
range, for example, w = 0.3 to 0.35 rad s- 1 . Similarly, N0 should be esti mated in the range w = 0.5 to 0.7 rad s- 1 from the No curve. The para meter values are then found to be f = 6.7 s and No = 0.029.
Curve fitting in the time domain Curve fitting between the response signal measured and that predicted is first tested over the central time intervals. Table 1 .4 lists the parameter values determined from the curve fitting and the root-mean-square-errors between the two signal curves, as defined according to Eq. ( 1 .56).
I
t2
E=
(c!�p. - c�lc)2 d t
tl
I (c!�ptl2 dt t2
tl
Figure 1 . 1 2 is a plot of N0 versus
f
1/2 ( 1 .56)
on a map of the root-mean-square-
30
HEAT AND MASS TRANSFER IN PACKED BF.OS
The values of i and
TABLE 1 .4
Nn obtained from curve-fitting and one-point-fitting in the time domain.
Time interval from 1 1 (s)
Cun·e-fitting
(S)
No
€
6.7 6.7 6.7 6.7
0.030 0.032 0.031 0.030
0.02 0.01 0.02 0.02
6.6 6. 9
0.030 0.030
0.02 0.01
r
to /2 (s)
�·-----
7.5 9.0
6.5
5.0 One-point-fitting at
t (S)
14. 0 1 1 .0 1 9.0 33.0
7.0 1 0. 03
a Time when C�� t U ) is at the highest peak. p
0 .04
0
z
0 .0 2
M
0 �----�----L---� 1.0 6.6 6.2 'T
[S]
FIGURE 1 . 1 2 Error map in the plot of No versus :r obtained from curve-fitting in the time domain, for Example l.l (least-error corresponds to point labeled + ) ; line W shows the values obtained by the weighted moment method (using zeroth and first moments); line W ' shows the weighted moment method (zeroth, first and second moments); line F shows the fourier analysis; point M shows the moment method; and point T shows transfer function fitting.
PARAMETER ESTIMATION
31
error, when the two curves C��pt and C��lc are compared over the time interval t 1 5 to t 2 3 3 s. From the least error point (labeled +), the parameters are found to be f = 6.7 s and N0 0.030. Parameter estimation by curve fitting can also be made at any arbitrary single time within the domain. This is tested at t = I 0 s when the response curve measured reaches a maximum, and at t 7 s in the frontal portion of the response signal. In both cases, the parameter values are, as listed in Table 1 .4, in good agreement with those obtained from curve fitting over the central time intervals. Figure 1 . 1 2 compares the values of f and N0 estimated u sing the moment method, weighted moment method, transfer function fitting, Fourier analysis and real-time methods of analysis. As the contour map reveals, the parameter values determined using the time domain analysis is, by far, the most accurate. As depicted, the parameter values predicted by Fourier analysis are close to those from the time domain analysis, while the moment method yields less accurate values. On the other hand, the transfer function fitting and weighted moment methods are found, even with their proper values of s, to give data which are not as good as the Fourier analysis. In Figure 1 . 1 3, the response signal measured is compared with those predicted using parameter values obtained by the different methods of analysis. As shown, Curve A (moment method) deviates significantly from Curve E (time domain) which matches well with the measured response signal. Curve D (Fourier analysis) and Curve E almost overlap with each other, while Curve B (weighted moment) and Curve C (transfer function fitting) are between Curve A and Curves D and E. =
=
=
=
(End of Example)
1.2
Adsorption Chromatography
When mass transfer takes place from the fluid surrounding a particle into the particle or vice versa, it has usually been assumed that the intraparticle concentration o f the mass transferring species is concentric (the same as radial symmetry or center symmetry). According to the Dispersion -Concentric model (D- C model: fluid in dispersed plug flow and concentric intraparticle concentration), p hysical adsorption in a packed bed with an imposed signal is described by the
32
HEAT AND MASS TRANSFER IN PACKED BEDS 0 . 2 �------� •
�xpt>rimt>ntat
c 0 a � c: " v c: 0 u
0. I
O L_--��---L----�--L_-=�-� 10
0
20
30
t rsI FIGURE 1 . 1 3 Comparison of response signal measure-d and those predicted with the following i' and N D values, for Example 1 . 1 : Curve
7-(s)
No
€
A
6.62 6.64
0.0087 0.020
0.156 0.063
6.66 6.73 6.68
0.024 0.029 0.030
0.042 0.029 0.025
B c 0
E
Method of analysis Moment Weighted moment (from Oth and 1 st) Transfer function Fourier analysis Time domain
following equations: ( 1 . 57 )
( 1 .5 8) and at
r=R
( 1 .59 )
PARAMETER ESTIMATION with
C = C = Cad = 0 oc -= 0 or
at
t=0
at
r
33
=0
where
a=
particle surface area per unit volume of packed bed ; a = 3 ( 1 - Eb)/R for spherical particles C = tracer concentration in the bulk fluid c = tracer concentration in the intraparticle pore volume Cad = amount adsorbed in the particle Dax axial dispersion coefficient of the adsorbing species De = in traparticle effective diffusivity kr = particle-to-fluid mass transfer coefficient R = particle radius r = radial distance variable U = interstitial fluid velocity Eb = bed void fraction Ep = intraparticle void fraction Pp = particle density. =
When c is small, the physical adsorption rate is assumed to be first-order ( 1 .60) where
KA = adsorption equilibrium constant ka
=
adsorption rate constant.
Under the assumption of an infinite bed, the solution in the Laplace domain is ( 1 .6 1 )
34
HEAT AND MASS TRANSFER IN PACKED BEDS
where
<7a = Dax [ LV
I+
- 11 2
4D
]
U:x (s + q )
( 1 .6 I a)
Dea 1 ---------krR ¢ coth ¢ - I
q = EbR De -+
a
( I .6 I b )
a
( 1 . 6 I c) Also, note that F(inrr/r), which is needed for the prediction of the response signal, is expressed as: ( 1 .62) where
Rn
=
exp
[ C�ax - ) ] [ C�ax - ) ]
( l .62a)
811 L cos (8L)
In = - exp
811 L sin (8L)
( 1 .62b)
( l .62c)
( l .62d)
(3
'Y =
(U) a = --
2
2Dax
( 1 .62e)
Q
+
kra
EoDax
(
I-
--
X cos M sinh N + Y sin M cosh N X 2 + Y2
)
( 1 .62f)
{3
=
X=
Y=
w Dax
De
krR
De
krR
_
kra EhD ax
(X
PARAMETER ESTIMATION 35 sin M cosh N - Y cos M sinh N
X 2 + Y2
(
(N cos M cosh N - M sin M sinh N) + 1 -
(
(N sin M sinh N + M cos M cosh N) + I -
)
( 1 .62g)
De krR De krR
)
)
cos M sinh N ( 1 .62h) sin M cosh N ( I .62i) ( 1 .62j)
( 1 .62k)
( 1 .621)
----
w2 Ki w = - P pka De K't.w2 + ki 1'l 1T
w =-· i
1.2.1
( 1 .62m)
( 1 .62n)
Parameter Estimation from First and Second Central Moments
The five parameters, Dax , D e , kr, k a and KA, are involved in first-order reversible adsorption. Kubin [8] and Kucera [9] showed that all five para meters were determinable, in principle, from five moments generated from a single measurement. However, in practice, higher moments have magni fied errors and it is not feasible to determine so many parameters from a single measurement. This is demonstrated in Example 1 . 2 . Schneider and Smith [ I 0], instead, proposed a method for estimating the parameter values from the first and second central moments based on the response signals from a series of measurements with varied flow rates and particle sizes.
36
HEAT AND MASS TRANSFER IN PACKED BEDS The difference in first moment between the response and input signals
is
MP - Mf
=-
L
u
( I + 8 o)
( 1 .63)
where ( 1 .63a) Similarly, the difference in second central moment or variance, ait, is a2 ,II M - ,... 2 _
_ 111 ,... 2 ( 1 .64)
where ( I .64a) The method of Schneider and Smith [ 1 01 begins by plotting Mf1 - M: versus L/ U and a'JJ/ (2L/ U) versus u-2 for a series o f measurements at various flow rates. The KA value is estimated from the slope of the straight line passing through the origin of the graph of the first moment. A straight line is also drawn for the plot of the variances, and then 8 1 and Dax can be obtained from the intercept and slope of the straight line, respectively. The analysis is based on the assumption that both Dax and kr are not influenced by U. This assumption is satisfied if the measurements are made at low flow rates. The values of 8 1 obtained for various particle sizes are plotted against R 2• If krR is constant, independent of R , a stratght line is drawn. I f the value of krR is known, the values of k a and De can be determined from the intercept and slope of the straight line. The assumption of a constant krR value is, as seen from Eq. (4. 1 1 ), valid for measurements made at low Reynolds numbers. The method of analysis is demonstrated in Example 1 . 2 .
1.2.2
PARAMETER ESTIMATION Parameter Estimation
by Curve Fitting in the
37
Time Domain
The techniques outlined in Section 1 . 2 . 1 are far more superior than the moment method p roposed by Kubin [8] and Kucera [9), which employs higher moments in the estimation of adsorption parameters. As demon strated in Example 1 . 1 , however, the method of analysis in the time domain is, among all the analytical techniques, the most reliable. The method will be employed, therefore, for parameter estimation in adsorp tion chromatography. The response signal may be predicted, u sing the measured input signal, C�x pt(l), and F(in1rjr) given by Eq. ( 1 .62), from either Eq. ( 1 . 3 1 ) or Eq. ( 1 .42). The predicted signal, C��1c(t), is then compared with the measured signal, c:� pt(!), in the time domain. Of the five parameters, Dax • De , kr, k a and KA , mentioned in Section 1 . 2 . 1 , kr can be estimated using Eq. (4. 1 1 ), k3 for physical adsorption is usually very large [2, 1 0] and may be assumed to equal infinity [ 1 1 ] , thus, the parameters remaining to be determined are D3x , De and KA . Note that when k3 Eqs. ( 1 .58) and ( 1 .60) become = oo,
(E p + p pKA)
( )
l oc = De - !__ r 2 at r2 or or
oc
( 1 .65)
and consequently Eq. ( 1 .6 1 c) reduces to ( 1 .65a) Also, both M and N defined by Eqs. ( 1 .62j) and ( 1 .62k) become
[
w
]
M = N = R - ( EP + pP KA ) 2 De
1/2
( 1 .65b)
Based on the above equations, Wakao et al. [ 1 2 ] have shown that the value of KA and a relationship between Dax and D e are the only accurate results obtainable from the time domain analysis of a single measurement . They have also shown that i f measurements are made with various flow rates, values of Dax and De could be determined from a series of Dax-De relationships obtained for different flow rates. This is also demonstrated in Example 1.2. (The Fortran programs for the computations of cg.�c(t) from Eqs. ( 1 .42), ( 1 .45) and ( 1 .6 1 ), and E from Eq. ( 1 .48) are listed in Appendix B.)
38 HEAT AND MASS TRANSFER IN PACKED BEDS
txample 1 . 2
Adsorption chromatography measurements were made at 20°C and atmo spheric p ressure by imposing nitrogen on a stream of carrier gas of hydrogen in a packed bed (L = 20.4 em and Eb = 0.38) of spherical activated carbon particles (0.2 em in size and Ep = 0.59). Table 1 .5 lists the measured input and response signals of the four runs at Re = 0.05 L 0.1 1 , 0.30 and 0.4 7. For illustration� the data for Run 3 are plotted in Figure 1 . 1 4 in normalized form, as defined by Eq. ( 1 .50). The transfer function is given by Eq. ( 1 .6 1 ) . Find what parameters are determinable. The molecular diffusion coefficient, Dv, for the nitrogen - hydrogen system at 20°C and atmospheric pressure is 0.76 X I o-4 m 2 s-1• SOLUTION
Moment method
The first moments, M P and M L are calculated from the measured input and response signals, respectively. Figure 1 . 1 5 is a plot of M P -M � versus L/U for all four runs. From the straight line drawn in the graph, PpKA is found to be 5 .30. The second central moments are also evaluated and plotted, in Figure 1 . 1 6 , as alt/(2L/U) versus u- 2 . I f we assume that the straight line drawn in the graph represents the data points, we obtain, with the aid of a value of krR predicted from Eq. (4. 1 1 ) and the assumption that ka = 00, that De = 0.53 X I o-6 m 2 s-1 and Dax = 0.46 X 1 o-4 m 2 s- 1 or EbDax/Dv 0.23. It should be noted, however, that the four points in Figure 1 . 1 6 lie on a somewhat convex curve. Strictly speaking, it is not correct to represent them by a straight line. =
Assuming that ka = oo and using Eq. ( 4. 1 1 ) for kr, we consider first the effect of Dax on KA . Since De is not yet known, calculations are made based on the assumed values of De = 0.4 x 1 0-6 and I x 1 0-6 m2 s- 1 . The error map, in Figure 1 . 1 7 for Run 3 , shows that for either value of De , and for any value of Dax in the range EbDax !Dv = 0 to 0.6, the least-error contour corresponds to PpKA = 5 .29. Figure 1 . 1 8 is a similar error map displaying the effect of PpKA on De for EbDaxiDv = 0.2 and 0.4. Again the least error contour corresponds to PpKA = 5 .29. Therefore, we conclude that a single run determines an accurate value of adsorption equilibrium constant.
Curve fitting in the time domain
TABLE 1 .5 Adsorption gas chromatography data. Input and response readings. Time
nlit (s)
Run 1 = 0.051 0.70 2 cm s- •
Run 2
Re
U
=
Run 3 = 0.30 U = 4.20 cm s· •
Re = O. l l U
=
Run 4 = 0 47 U = 6.54 e m s· •
Re
1 .55 em s-•
Re
- -
11
Input
Response
Input
Response
Input
- -·-·-·
-
0 1 2 3 4 5 6 7 8 9
0.0 2 1 .9 9 1 .4 1 7 1 .9 208.4 306.8 335.3 346.8 345.3 331.3
10 11 12 13 14 15 16 17 18 19
3 1 3. 2 289.7 267.2 244.7 220.6 1 99. 1 1 80 . 1 160.6 144.5 1 29.5
20 21 22 23
1 1 4.5 103.5 92.5 82.9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.0
0.0 2.0 3 5 .5 1 1 0.0 1 8 7.9 249.9 283.9 300.9 297.9 280.9 254.4 227.4 200.3 1 74.3 1 5 1 .8 1 30.3 1 1 1.8 94.3 80.8 69.3 58.8 49.7 42.7 35.7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.0 6.5 30.5 8 1 .5 1 3 3.0 1 84.0 225.5 25 1 .0 270.5 2 8 1 .0
282.5 273.5 257.0 238.0 207.0 186.0 158.5 1 3 8.5 1 1 2 .0 99.0 83.5 73.5 60.0 52.5
Input
Response
---- -··- -- ·-·- - -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.0 0.6 0.8
l.l 1.3
..
-- -
·
-
---
Response
-· _.,______ _____ +_ 00 3.3 41.3 1 1 9. 3 193.3 239.3 262.8 265.8 239.8 2 1 6.8 1 73.3 1 30.7 89.2 70.2 47.7 37.7 27.7 22.2 18 . 2 1 5 .2 1 3.2 1 1 .7 1 0.2 9.2
-
-
-
-
-
-
-
-
-
-
-
-
"'C
>
:::e
> � 7 -l 7 ;:c
.
U> -l .
.
� > -l 0 z -
-
w ..0
� 0
TABLE 1 .5 (Continued) nl:l.t
Time (s)
·--·
Run 1 Re = 0.05 1 U = 0.702 em s-•
Run 2 Re = 0. 1 1 U = 1.55 em s-•
Run 3 Re = 0.30 U = 4.20 em s-•
Run 4 Re = 0.47 V = 6 54 em s-• Input
Input
Response
0.0
45.0 40.0 34.5 3 1 .5 27.5 24.0
1 .7 2.2 3.2 4.7 8.3 1 2.3
8.7 8.2 7.7 7.2 6.7 6.2
0.0 0.5 1 .0 1 .0
1 1 .1 9.1 7.6 6.6 6.1 5.1 4.6 4.0 3.5 3.0
1.1 1 .6 2.1 2.6 4.1 5.1 6.2 8.2 1 1 .2 14.2
22.0 1 9.5 1 7 .5 1 6.0 ] 4.5 1 3.0 1 2.0 1 1 .0 1 0.5 9.5
17.3 23.3 30.9 40.4 50.4 6 1 .0 72.5 83.0 93.0 100.1
5.7 5.2 4.7 4.7 4.2 4.2 3.7 3.6 3.1 3.1
1 .6 2.6 3.6 5.1 7.6 ] 0.1 1 3.1 17.1 2 1 .2 26.2
2.5 2.5 2.0 2.0 1 .5 1 .4 1 .4 1 .4
19.2 24.3 30.8 37.3 45.3 53.8 62.8 7 1 .4
9.0 8.0 7.5 7.0 6.5 6.0 5.5 5.5
107.1 111.1 1 14.1 1 1 5.2 1 1 4.2 1 1 1 .2 106.7 101.3
2.6 2.6 2.1 2.1 2.1 2.1 1 .6 1 .6
3 1 .2 37.7 43.2 49.2 55.2 62.3 67.3 73.3
n
Input
Response
Input
24 25 26 27 28 29
74.4 66.4 59.4 52.8 47 . 3 42.3
0.5 0.4 0.9 0.9 1 .4 1 .5
30.7 25.2 2 1 .7 1 8.2 1 5 .7 1 2.6
-
30 31 32 33 34 35 36 37 38 39
38.3 34.8 30.2 27.7 25.2 22.2 20.1 18.6 1 7. 1 1 5 .6
3.3 4.3 6.2 8.2 1 0.7 1 3 .2 16.1 21.1 25.6 30.0
40 41 42 43 44 45 46 47
14.0 1 2.5 1 1 .5 10.5 9.5 8.4 7.4 7.1
36.5 4 1 .0 47.0 53.9 60.9 68.9 75.8 83.3
Response -
-
Response -
:t � >
....,
>
z 0 $;
> (/) (/)
..., :;c
>
z (/)
., {T1 :;c -
z
"t:
>
(')
� {T1
0 til {T1
0 (/)
48 49
6.9 n.s
89.8 95.8
1 .4 1 .1
79.9 89.4
5.0 5.0
94.3 87.8
1 .6 1.6
78.3 83.8
50 51 52 53 54 55 56 57 58 59
6.3 5.8 5.3 5.3 5.2 4.7 4.2 4.2 4.1 3.8
102.7 108.7 1 15.2 1 19.7 124.1 1 29.1 1 3 1 .6 133.5 1 36.0 137.0
0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.8 0.8 1 .0
97.4 105.9 1 1 3.4 1 20.5 1 25.5 1 30.5 1 34.0 136.0 137.1 1 37.6
4.5 4.5 4.0 4.0 3.5 3.5 3.0 3.0 2.5 2.5
80.4 72.4 64.4 56.9 50.5 43.0 37.5 32.0 27.1 22.6
1 .6 1 .6 1 .6 1 .6 1.1 1.1 1.1 1.1 1.1 1.1
87.3 90.3 92.3 93.9 94.4 94.4 93.9 92.4 89.9 86.9
60 61 62 63 64 65 66 67 68 69
3.6 3.9 4.0 3.7 3.5 3.5 3.5 3.1 2.9 2.9
1 38.5 1 39.4 1 37.9 1 37.4 1 36.8 1 33.3 1 3 1 .8 1 29.8 127.2 123.7
1.3 1 .2 1.2 0.9 0.7 0.7 0.7 0.7 0.7 0.6
1 35 . 1 1 32 . 1 128.6 125.6 120.2 1 14.7 109.2 103.7 97.2 91.8
2.0 2.0 1 .5 1 .5 1 .0 1 .0 1.0 1.0 0.5 0.5
19.1 16.2 1 3.7 1 1 .2 8.7 6.8 5.3 4.3 3.3 2.4
0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
83.9 79.5 75.0 7 1 .0 67.0 63.0 58.5 54.0 50.6 45.6
70 71 72 73 74 75 76 77 78 79
2.9 2.5 2.3 2.3 2.3 2.0 1.7 1 .4 1 .2 0.9
1 20.7 1 16.6 1 12 . 1 109.1 104.1 100.0 95.5 90.5 86.4 82.9
0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.4 0.3 0.2
85.3 80.3 74.3 68.8 62.3 56.9 52.9 47.4 42.9 38.9
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
1 .4 0.9 0.4 0.0
0.5 0.4 0.2 0.1 0.0
42.1 38.6 34.6 3 1 .6 28.1 25.6 23.2 20.2 1 7. 7 15.7
-
-
-
-
-
-
-
-
-
-
-
"t: >::0
>s: ::r1
-l tTl ::0
tTl (./J
-l s: >-
-l
0 z
-
�
-
TABLE 1.5 (Continued) ··--·--- ----·--·----
Time n tlt
(s) n
Run 1 Re 0.051 U 0.702 em s-1 =
=
Input
Response
Run 2 Re = 0. 1 1 U = 1 .55 em s-1 Input
- -·-- ---
80 81 82 83 84 85 86 87 88 89
0.6 0.6 0.6 0.8 0.5 0.3 0.0 -
-
-
79.4 74.9 69.8 66.8 62.8 59.7 57.2 52.7 49.7 46.6
0.0 -
-
-
-
-
-
-
Response -
34.9 3 1 .0 28.0 25.0 23.0 20.5 18.6 16.1 14.1 13.1
Run 4 Re == 0.47 U 6.54 em s · 1
Run 3 Re :.:: 0.30 U = 4.20 em s-1 Input
Response
·--·-- ·
0.5 0.5 0.5 0.0 -
-
-
-
-
-
=
-
-
-
-
-
-
-
-
--
Input - - · ----
-
-
-
-
-
-
-
-
Response
_ _ L
___ __ -
14.2 1 2.7 1 1 .2 9.8 8.8 7.3 6.3 5.3 4.8 4.3
� N :X: � > -l > z 0
s: >
{/) r.f)
-l ;;c
> z
r.f) � tr. :::0 -
z
�
>
()
� tr. -
90 91 92 93 94 95 96 97 98 99 100 101 102
-
-
-
-
-
-
-
-
-
-
43.6 40.1 37.5 35.5 33.5 30.5 28.4 26.4 24.4 22.3 20.8 19.8 1 8.3
-
-
-
-
-
-
-
-
-
-
-
1 1 .6 10.1 9.2 7.7 6.7 5.7 5.2 4.8 4.3 3.8 3.8 3.3 2.3
-
-
-
-
-
-
-
__ ,
-
-
-
-
-
-
-
-
-
-
-
-
-
-
v -
-
-
-
-
-
-
-
-
3.8 3.4 2.9 2.4 1.9 1 .4 0.9 0.9 0.4 0.5 0.5 0.0
0' tTl
0 r.f)
103 104 105 106 107 108 109 110 111 112 113 1 14 115 116 117 118 119 120 121 1 22 123 124 1 25 126 1 27 1 28
t.. t (s)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1 6. 2 1 4.2 1 3.2 1 2.7 1 1 .] 9.6 8.6 7.0 6.5 5.5 5.0 4.9 4.4 3.4 3.3 3.3 3.8 3.3 2.7 2.2 1 .7 2.1 0.6 0.6 0.6 0.0
-
-
-
1.9 1.9 1 .4 1 .4 0.9 0.9 1 .0
-
0.5 0.0
-
-
-·
-
-
-
-
·
-
-
--
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-·
-
-
--
-·
-
-·
-
-·
-
-
·-
--
-
'-:::
> :;c :> 3::
� -
t'T1 :;c
tTl (/) ...., ....
3:: :>
...., -
Time interval 2.5
5.0
1 .25
2.5
0 . 3 1 25
J .25
0.3 1 25
0.625
0 z
� w
44
HEAT AND MASS TRANSFER IN PACKED BEDS t1 . ·�s
A
0.2
c
0
.) , ).,
0
8.1
'-
� 1);\:: i. resr.onsc
. - 1\
-
c Q.l u c 0 u
5 0 . 02
! s;
exnt I
10
t
[S]
FIGURE 1 . 1 4 Input and response signals measured from adsorption chromato graphy; response signals predicted for Run 3 in Example 1.2: Response signal predicted with Curve A B
c
P
pKA
5.29 5 . 37 5. l l
De
€ tJDax /Dv
(m 2 s-•)
€
0.24 0.24 0.24
0.63 X 10-" 0.63 X 10-" 0.63 x 1 0" 6
0.016 0.05 0.10
·----·---
·----
Using the value of P pKA obtained, the error map for various values of ebDax!D v and De is shown in Figure 1 . 1 9 . For example, the values of these two parameters within the shaded part of the figure show that C��Ic(t) differs from C��pt(t) by a root-mean-square-error, e, defined by Eq. ( 1 .4 7), of less than 0.025. The region with the least error may be visualized as a valley in a three-dimensional error map. The valley expands with an increase in De. The graph indicates that simultaneous determination of the two parameters De and Dax is not feasible. However, if we know one of them, the value of the other can be determined from the contour for € �0.025.
PARAMETER ESTIMATION 45 300
V)
,..._
.._.
-
200
-£ �
--
100
0 "------.J__--'---__....J 0 10 20 30 FIGURE 1 . 1 5
L/U
[ SJ
M!1 - M! versus L/U for Example 1.2.
100 (/)
,.....
....._.
_Jf::> -...:...-
N
N
tf
50
O L-----.J__-----L--�---��--� 0 �5 1 15 2 25 FIGURE 1 . 16
u-2 [s2 ·m-2J
alvJ/(2L/U) versus u - 2 for Example 1.2.
x 104
46
HEAT A N D MASS T R A N S F E R IN PACKED BEDS 6 r-------�
< � c. a.
5
'·
0.2
FIGURE 1 . 1 7
/ _/
- - ,: = 0 . 05 -
"'--- ;; = 0 . 1
e:b0ox10v
0.6
0.4
Error map in the plot of Pp/(A versus Example 1.2.
ebDax iDv
for Run 3 in
6 �-------,
--E
=
0.1
-·
4 �------��--�0 0.5 I .5 x!o-6 D e tm2 s- 1 1 ·
FIGURE 1.18
Error map in the plot of PpKA versus D e for Run 3 in Example 1.2.
Figure 1.20 shows the error map for the four runs with varied flow rates. The contours for E � 0.025 are steep with respect to De at high flow rates and nearly flat at low flow rates. Thus, De has a larger effect on the response curve at high flow rates, but is not important at low flow rates. In the laminar flow range, Dax is constant , not depending on flow rate, so
PARAMETER ESTIMATION
47
0.4
> 0 ....... X 0 Cl .Cl 1,.)
0.2
(
<
0 . 025
O L------L��U---�L---� 1 .5 0 0.5
De
FIGURE 1 . 19
xlo-G
1 rm2 · s- J
Error map in the plot of ebDax/Dv versus De for Run 3 in Example 1.2.
Re
=
0 . 47
-- -_____.-; --
--··
0.4
> Cl ....... X 0 Cl w .Cl
( = 0 . 025
/
/
/
/
/
/
/
/
...::............... ....... ....... ... ....... ..............
0 . 30
0' I I
················ ······ · · · · · · · · ·
0.2
0
0
FIGURE 1.20
0.5
De
1.5 rm2 · s - 1 J
x l o -6
Error map in the plot of EbDax/Dv versus De for various flow rates, for Example 1.2.
that the best values of Dax and De correspond to the basin where all four valleys overlap. This is shown more clearly in Figure 1 .2 1, which is a map of the arithmetic mean error for all the runs. The least-error point ( + in the graph) corresponds to EbDax /Dv = 0.24 and De = 0.63 x 1 0-6 m 2 s-1.
48
HEAT AND M ASS TRANSFER IN PACKED BEDS
0.4
> 0 ' X 0
0
w .o
•
0.2
<'"'-
£ =
£ = 0 . 05
""--- £ =
0
0
0.5
De
0 . 025
lm2 · s - I J
0.1
1.5
xJo-6
FIGURE 1 . 2 1 Map of the mean -error i n the plot of EbDax iDv versus D e for Example 1.2 (least - error corresponds to point labeled + ) .
With these parameter values determined, the predicted response curve, C��tc(t), is that labeled A in Figure 1 . 1 4 . It agrees well with the experi mental curve; the root-mean-square-error is E = 0.01 6. Curves B and C illustrate the large effect of P pKA on the response. A 1 .5% increase in P pKA from 5 .29 to 5.37 increases the error, E, from 0.0 1 6 t o 0.05. Similarly, a 3 .4% drop in this parameter, from 5.29 to 5. 1 1 , increases the error to 0. 1 0. In adsorption chromatography, the adsorption equilibrium constant has the largest effect on the response curve. As illustrated, an accurate value of KA can be determined from a single measurement. In order to establish accurate values of the axial dispersion coefficient and the effective diffusivity in the particle, input-response curves need to be measured over a range of fluid velocities. These conclusions apply only when the particle-to-fluid mass t ransfer process and the adsorption rate have little effect on the response curve. Such conditions are usually met for physical adsorption of gas in beds of small adsorbent particles. It should be noted also that the intraparticle diffusivity cannot be accurately determined solely from response curves measured at low flow rates. This is evident from Figure 1 .20 where the least error valley extends horizontally. Error maps such as Figures 1 .20 and 1 . 2 1 are particularly useful for determining the range of operating conditions best suited for the evaluation of accurate parameter values.
PARAMETER ESTIMATION 49 J a r---.----.--�--�
f.
<
0 . 025
l = 0 . 025 r
=
0 . 05
� = 0.1
ka.
FIGURE 1.22
1 0-
1
1m3 · kg - I
·
s-
IJ
1
Error map in the plot of Sh versus k3 for Run 3 in Example 1.2.
The measured response signal may be used to evaluate the influence of the adsorption rate constant. Calculated response curves for Run 3 are obtained using ebDax!Dv = 0.24, De = 0 . 63 X 1 0-6 m 2 s-1 and with various assumed values of Sherwood number and k3• Figure 1 .22 is the resulting error map. The curves show that, for Sh > 2, for example, any value for k a greater than about 0.09 m3 kg- 1 s- 1 will have a small ·(e < 0.025) effect on the response signal. (End of Example)
Figure 1 .20 shows that the ebDax!Dv-De contour becomes flat as the flow rate decreases. The con tours are regarded as the valleys on a three dimensional error map. If we take a look at a moment of the system, the moment value must be constant everywhere along the least-error (or deepest valley) line in the middle of the contour. For example, according to Eq. ( 1 .64), the second central moments or variances should be constant along the deepest valley. In other words, the following condition prevails: ( 1 .66)
50 HEAT AND M ASS TRANSFER IN PACKED BEDS Therefore, the slope of the line in a graph of Dax versus De is dDax --
dDe
=
1
---
[
Oo
] - (Sc)(Re) Dv
2
( 1 .67)
20EbaR ( 1 + Oo) De
Simultaneous determination of Dax and De needs at least two curves crossing each other on a graph of Dax versus De, one horizontal or nearly horizontal and the other steep. The former with dDax/dDe 0 is obviously obtained at low Reynolds numbers. The curve with high dDax/dDe is obtained, on the contrary, at high flow rates. If, however, o0 or ( E p + P pKA) is small, the slope dDax/dDe is low even at high flow rates. The value of P p KA is accurately determined from a single chromato graphy measurement. In most cases, De/Dv is less than 0 . 1 . Therefore, if the value of (E p + P p KA) is large, probably larger than about unity, both Dax and De may be determined from the time domain analysis of the measurements made in a laminar flow range. If the value of (E p + P p KA ) is small, however, intraparticle diffusion makes little contribution to the overall transport in adsorption chromatography. �
1 .2.3
Relationship Between Overall Gaseous Effective Diffusivity and Macropore and Micropore Diffusivities
When a porous pellet has a bidisperse macropore/micropore structure, the simplest model is, of course, the one in which the bidisperse pores are not distinguished, and the concept of overall effective diffusivity is introduced. The analysis in the preceding sections is based on such a model and is called Model I in this section. Molecular sieves have a typical b idisperse pore structure. In macropores, gaseous diffusion is considered to take place, but the micropores have been assumed to be in : (i) gas phase by Antonson and Dranoff [ 1 3 ] , Ruckenstein et al. [ 1 4] , Hashimoto and Smith [ 1 5 ] , and Kawazoe and Takeuchi [ 1 6 ] ; (ii) adsorbate phase by Kawazoe and Takeuchi [ 1 6] , Schneider and Smith [ 1 7] , Kawazoe et a!. [ 1 8], and Lee and Ruthven [ 1 9 ] . Let us call them Models II and IlL respectively.
Model II Model III
Macropores
Micropores
Gas phase Gas phase
Gas phase Adsorbate phase
PARAMETER ESTIMATION
In
51
the models, the spherical pellet is assumed to be composed of a n assemblage of fine spherical m icroporous particles. The fundamental equations for Models 1-111 are listed together with the first and second central moments for impulse response in Table 1 .6. Besides the notation listed below Eq. ( 1 .59), the following symbols are used:
Ca = gaseous concentration in the macropores Cact, c�d, c�d = amount adsorbed per unit mass of pellet, used in Models I , I I and I l l , respectively ci = gaseous concentration in microporous particle Da = gaseous effective d iffusivity in the macropores, based on cross-sectional area of the pellet Di = gaseous effective diffusivity in the micropores, based on cross-sectional area of microporous particl e _ Dsi = effective diffusivity of adsorbate in the micropores, based on cross-sectional area of microporous particle KA , K/... , KA = adsorption equilibrium constants, used in Models I , II and I I I , respectively ka, k�, k� = adsorption rate constants, used in Models I , I I and I I I , respectively n d , nd , n� = molar fluxes from gas to pellet, used in Models I , I I and I I I , respectively ri = radial distance variable in microporous particle r0 = radius of microporous particle Ea = macropore void fraction of pellet Ei = micropore void fraction of pellet, accessible for adsorbing Ei P = gas species E p = volume fraction of microporous particles in pellet Ea + Ei · A comparison of the first moments of the three models indicates that
Model I
Model II
Model III
( 1 . 68 )
Therefore, ( 1 .69)
VI tv
:r: (Tl > -i > z 0 s: >en en -i ::c > z en o-r: !Tl �
TABLE 1 .6
-
Models for gas adsorption in a packed bed.
z
Model l II
Model I I
Model l
'"0
> ()
Macro-micropore model
Single gaseous diffusion model Particle
€
�� = D� a (r2
p ar
r2 ar
?.C:) ar
_
acad Pp at
Macropores: Microporous particle:
Gas
2c ac a -- = Dax a- U ac -- nd 2 ax ax €h ar nd = D� (
:�t = kr(C-(c)R
I
Gas diffusion in macropores.
Gas diffusion in macropores.
Gas diffusion in micropores
Adsorbate diffusion in microporcs
€a aca = D� a (,2 �E!! Jt r ar ar
aci - a ( ·2 aci )- -ac�u EipD-i €· r p at ff arj arj p at _
1
Ca = (Cj),n
2 ac ac a n. - = Da a c- u --d 2 ax €h x ax at -
a nd = Da ( ;a ,
)- ��ii!roPi(��ari )
}R = kr[C- (ca)R)
I
-
i
r 0
�
0
�..!i) - �PpDsj_ ('?car;.Q)
Ea ?.�!! D� � (r2 at r ar ar =
ac�� Dsi � ( � �c:��) r = � at , a,. a,. I
I
I
'o
I
a2c ac a ac = Dax ax·2 -- U ax €h "u at ac ., nu = Da (a,a) =kriC - (ca>Rl R "
i
rI}
to tT: 0 (/)
Adsorption rate
First moment of impulse response
(}cad = ot
response
_ Cad
l !.L [ �
1
U
aR
e 3 h
P
Ai1J\
L!..-1:. .�0�] u
2
0
=
aR
3 Eh -
3En
-
x
[
aR (e
- I+
U
'2L u
3Eh
·-
[e
·
� ]
1
� + o�) u
2
2
[
A
(
I
SOc
ka
+
p
) ]
R2 !_ --
krR __
+
·
3
PpKA)2
U
r,
�
�t
K " A
- ( E . + p pKA ) .1 ., .l Eh 11
..
where
o�= aR- - (E a + PpKA)
I
[
aR p (KA)
-
C
+
ori
= kll
. (I + 2L il lj2 lj o I + Dax
I
I
oa
I ' +
2
ro
k rR
I 5EipDi
Je h -
2 o' = ' )2 �-�--- + ( Ep + Pp /\A ka 3En I x
Sl
/, --- I +
p + pp KA )
ul + Dax
aR
Ejp
K�
o� = - (Ep + PrKA ) Je h
(Ep + PpK )
aR Pp(KA)2 + (e -
I -
L
•
· (ac;d) lc. - (c�d)rJ, ] [ aR [ �0�]
Pr D
where
where
00
,,
or
(e + P PK
01 + Dax
U
ac�� = k� (c· - c�d)
(c KA )
Mil = - - l +
11 _ Second central J.1 2 mom ent of impulse
. ka
)R
2 -
3
(Ei + PpK�
)2 ]
-
[
11
K ll )2 �.l!iP_ll_ �- --+ (e., + P KA) 2 r � t JEipka JE h l' 2 2 Yo Ppl'o. A : I . --- + +
r:ll _ aR_ O
x
,
( 5o;·, krR )
R
3
II
1 5 D�i
]
�
> � > s: M -l tTl � tTl (f) -l 3:: > -l 0 z -
-
V\ w
54
H E A T AND M A S S TRANSFER I N PACKED BEDS
Similarly: from a comparison of adsorption terms of the second central moments, the following expression is obtained:
Model Ill
Model II Pp(KA.)2
ro(PpK:\)2
k�
3€jp k�
Consequently,
, ka = ka
3 Ej p a/1 k
= ____ :... _; ..___
Y oPp
(1
-
Ej
)" 2 .
PpKA
( 1 .70)
( 1 .7 1 )
The effective diffusivity terms of the models are related through Eq. ( 1 .72).
Model l 1
Mode/ III ( 1 .72)
Equations ( 1 .69), ( 1 .7 1 ) and ( 1 .72) show the relationships between the parameters of the different models. Equation ( 1 .68) also indicates that, as far as the first moments of the models are concerned, the controlling parameters are intraparticle void fraction for gaseous diffusion plus the adsorption equilibrium constant. It is clear from Eq. ( 1 .72) that the De value of Model I is close to the Da value of Model I I (because of small r0/R) and also between the Da and Dsi values of Model I I I (because Dsi is also small). The parameters used in the models are reciprocally convertible, but unless chromatography measurements are made with varied tempera tures, no answer is given to the question about whether the micropores are in the gas phase (Model II) or in the adsorbate phase (Model III).
1.2.4
PARAMETER ESTIMATION 55 Assumption of Concentric Intraparticle Concentration
Take a look at an adsorbent particle in a packed bed. When an adsorbing species, imposed as a step-function or a one-shot input on a carrier stream, passes over the particle, the intraparticle concentration of the adsorbing species cannot have radial symmetry. If the tracer species starts to pass over the particle from left to right, for example, it is apparent that more of the species penetrate into the particle from the left hand side. However, Eq. ( 1 .58) is based on the assumption that the intraparticle concentration is radially symmetric. The purpose of this section is to examine whether the assumption causes any errors in the parameter estimation from adsorption chromatography under isothermal conditions. The intraparticle concentration of the adsorbing species is expressed in the spherical coordinates shown in Figure 1 .23. Writing the gaseous con centration and adsorbate concentrations as c* and c:d, respectively� the mass balance equation is ac*
- =
€P at where
(
ac:d -at
D e\72c* - pp
) [
(
( 1 .73)
·)
]
1 a a 1 a2 1 a a 1 \7 2 = - - r2 - + - - - sin e - + -- -- · ( 1 .73a) r2 Sin 8 ae ae I Sin 2 8 a 2 r2 ar . ar
FIGURE 1 .23
Spherical coordinates.
56 HEAT AND MASS TRANSFER IN PACKED BEDS When the adsorption is first-order and reversible, the adsorption rate is
( 1 .74) Integrating Eq. ( 1 .73) over a spherical surface with radius r and dividing this by 41Tr2, it follows that:
1 _
41Tr2
271'
f
1
d
0
f
J
-1
( 1 . 7 5) If the order of differentiation and integration is reversed, and considering (3c */3<1>) = o = (3c */3<1>) = 2rr, Eq. ( 1 .75) becomes
( 1 .76) where 1 271' X = .,. 4
1
c* dcos 0
( 1 . 76a)
c:d dcos 0.
( 1 .76b)
I I d
-1
0
and 21T
Y=
X
:I I 1
d
.,.
0
-1
It is clear that and Y are the average concentrations of c* and c:d, respectively, over a spherical area 47Tr2 inside the spherical pellet. Equation
PARAMETER ESTIM ATION 57 ( 1 .76) i s identical to Eq. ( 1 .58) which i s derived under the assumption that the intrap article concentration has radial symmetry. Similarly, integrating Eq. ( 1 .74) over a spherical area with radius r and dividing by 41Tr2 , we obtain Eq. ( 1 .60). Therefore, it is concluded that Eqs. ( 1 .5 7)-( 1 .60), derived assuming that the concentration has radial symmetry, are valid as long as the adsorp tion rate is first-order.
1 .3
Effect of Dead Volutne Associated with Signal Detecting Eletnents
In chromatography measurements, the input and response signals are measured using detecting elements inserted into the packed bed. The elements are usually placed in a shallow empty section installed in the column. Kaguei et al. (20] examined the effect of dead volume upon parameter estimation. 1 .3 . 1
Packed Beds of Glass Beads
Suppose the detecting elements are inserted into the middle of the dead volume between the packed beds of glass beads, as shown in Figure 1 .24. The material balance equations for a tracer injected into the column give:
In the dead volumes (concentration C'), Section 1 (0 < x < L0) and Section 3 (L + L0 < x < L + 3L0)
ac' a2c' ac' = D' - u' at ax2 ax
-
--
( 1 . 77)
where D ' is the dispersion coefficient and u' is the fluid velocity in the dead volume section.
In the packed beds (concentration C), Section 2 (Lo < x < L + Lo) and Section 4 (x > L + 3L0)
ac at
- = Dax
a2c ac - U ax2 ax
-
( 1 .78)
58
HEAT AND MASS TRANSFER IN PACKED BEDS Carr ie r gas
1
Injector
sect i o n
Calming
Section 1
Detector I
Section 2 Section 3
L
Detectorll
2 L0
Section 4
FIGURE 1.24
A column consisted of packed beds and dead volumes associated with concentration detecting elements.
with
c' = C = o C=O
at
t= 0
at
x=
00•
Conditions at the boundary between the packed bed and dead volume section are also needed. The mass balance equation at the boundary gives I
I
u C -D
I
acl ( ax
= Eb UC - Dax
-)
ac ax
·
( 1 .79)
In the column shown in Figure 1 .24, there cannot be a concentration discontinuity at the boundary. Obviously, U 1 = EbU, and consequently, the dispersion fluxes are also equal at the boundary.
PARAMETER ESTIMATION 59 C' = C ac ac' D' - = EbDax ax ax If we simply write transfer function is 00
I
ell
0
at x
= L o , L + L0 and L + 3L o .
C'(x = 0) = C1 and C'(x = L + 2L0) = C11, the
e xp (-st) dt
F(s) = -----00
I c•
e xp (-st) dt
0
a(Lo/L) a oa s(8 + 'Y exp (- ao)] exp (As + 2Ao) {[8 + -y e x p (- ao)F [8- -y e xp (- ao)] - 8 -y ( l - ex p (- 2an)][-y + 8 exp (- a o )] e x p (-as)} ( 1 .80) where
( 1 .80a)
( 1 .80b)
( 1 .80c)
( 1 .80d)
( 1 .80e)
60
HEAT A N D
MASS T R A N S F E R IN PACKED BEDS a8 =
a
D
=
- (1 + U /,U ,
Dax
4Dax
)1/2
-s 2
( I .80f)
(I + 4D ' ) s
( t . 80g)
ti2
Lou' --
u '2
.
D'
With the F(s) and the input signal measured� the response signal may be predi<.:led hy the te hni q u es mentioned in Section 1 . 1 .6. The parameter values are then determined from a comparison of c��Jc(t) and c��pt(l) in the tim e domain. Kaguei et a!. f20l found. from the inpu t-response signal measurements made in a c lu m n (L 20.4 em and L n 0.4 to 1 .0 em) at Re = 0.0 1 to 0 . 1 7 : that the tlow rates obtained under the consideration of the dead volume (Veale.> agreed well with the experimental data (Uex pt); the values obtained by ignoring the dead volumes ( U!atc) were, however, I 0-20% lower than Uexpt or Veale· The difference between Uexpt and Udale may be evaluated by equating the two first moments, obtained separately from Eqs. ( 1 .80) and ( 1 .3). in wh i h the flow rates are distinguished u sing Ucxp t for Eq. ( 1 .80) (dead volume considered ) and Udale for Eq. ( 1 .3) (dead volu e ignored) The rcsul t is
c
o
=
=
c
m
uexp� t Veale
-
_
] +
.
( 1 .8 l )
�
where
v
= �-- exp sinh (3
[- (2 + �) o:L L
( 1 . 8 l a)
l
(3
( l . 8 I b)
( 1 .8 1 c)
The dead volume effect may be ignored if � � 1 . Otherwise, Eq. ( 1 .80) should be used as the transfer function of the system. Large � values result
PA RAMETER ESTIMATION
61
from a column with L 0/L being not very small. As a matter of fact, in such a bed the input signal is closely followed by the response signal. 1 .3 .2
Adsorption Packed Beds
In a similar way to the preceding section for an inert packed bed, the effect of dead volume in adsorption packed beds can be tested by com paring the adsorption equilibrium constants, KA (dead volume considered) and K1 (dead volume ignored). Again, this is done by equating the two first moments based on Eq. ( 1 .6 1 ) (dead volume ignored) and Eq. ( 1 .80) (dead volume considered). Note that in applying Eq. ( 1 .80) in the calcula tion of the first moment of the adsorption system, as of Eq. ( I . 80f) should be replaced by a s of Eq. ( I .6 1 a). The ratio of K2 to KA is found to be ( 1 .82) where ( 1 .82a) Equation ( 1 .82) suggests that if the measurement is made in a short column of adsorbent particles with a low adsorption equilibrium constant, then K1 is appreciably different from KA . In most adsorption columns, however, response signals come far behind the input ones� such that the dead volume may safely be ignored. Example 1.3
The chromatography data listed in Table 1 .5 (see Example 1 .2) arc those measured using detecting elements (tungsten filaments) installed in the middle of empty sections (L0 = 0.4 em) of the column. Find P pKA for Run 3 , taking into account the dead volume associated with the detecting clements. Assume that ka = oo and kr is estimated from Eq. (4. 1 1 ), as in Example 1 .2 .
62
HEAT AND MASS TRANSFER IN PACKED BEDS
SOLUTION
1
At the low flow rate (Re = 0.30), D is equal to the molecular diffusion ° coefficient, Dv, for the nitrogen-hydrogen system at 20 C and atmospheric pressure. Using the transfer function of Eq. ( 1 .80) (with the modification for adsorption), the response signals are predicted with various values of Pp KA , Dax and De. Figure 1 .25 is the error map of Pp KA versus ebDax!Dv with De as a parameter, which corresponds to Figure 1 . 1 7 when the dead volume is ignored or when L D is assumed to be zero. 6
:::.:0. ::: a.
5
�------�
"""" '\.
-··
""'-
4
FIGURE 1.25
e:
=
e:
=
0 . 05 0. I
-
�--------------�----------------J � --------------� 0 0.2 0.4 0.6 e:bDax1Dv
PpKA
Error map in the plot of versus EbDaxiDv for Example 1.3, when dead volume is considered.
The value of P pKA with the dead volume considered is then found to be 5.23. In the solution of Example 1 .2, the value of P v K1 = 5. 29 is obtained without considering the dead volume. P pK1 is about 1 % higher than the P pKA . Also, Eq. ( 1 .82a) indicates that �ad = 0.0 1 2 or 1 .2% for Run 3. In any case, the difference between KA and Kl in adsorption chromato graphy is small and the dead volume may be ignored in the estimation of parameter values. (End of Example)
1.4
Assumption of an Infinite Bed
As mentioned in Section 1 . 1 , a packed bed in which the input and response signals are measured is assumed to be infinite. It is necessary,
PARAMETER ESTIMATION 63 however, to know the validity of this assumption which has simplified the solution to Eq. ( 1 . 1 ) considerably. Suppose a response signal is measured in a packed bed at a distance , /, away from the bed exit. It is considered, in general, that if l is short, the shape of the response signal is influenced by the length l, but that the effect diminishes with increasing length. Therefore, a critical bed length exists beyond which the shape of the response signal is independent of bed length. The critical length is affected by flow rate, and, in an adsorption column, also by the magnitude of the adsorption equilibrium constant.
Packed bed
Empty col umn
- 00
x=O inPut
FIGURE 1 .26
1 .4 . 1
x= L
x= L+£
response
A column used for examination in an infinite bed.
Packed Beds of Glass Beads
Figure 1 .26 shows a packed bed of glass beads connected to an infinitely long empty column. Unsteady-state mass balance equations are as follows:
In the packed bed (0 < x < L + l; concentration C)
a2c ac - = Dax 2 ax at -
-
v
ac ax
-
( 1 .83)
In the empty column (x > L + l; concentration C')
ac' a2c' ac' - u' - = D' at ax ax2 --
( 1 .84)
64
HEAT A N D M ASS TRANSFER IN PACKED BEDS
with
c = c' = o C= C
'
and
EbDax
ac ' ' -=D -
ac
ax
ax
c' = O
at
t=0
at
x=L +1
at
x
= 00•
The two boundary conditions at the bed exit listed above are the same as those employed in Section I .3 . 1 . The transfer function is found to be
�(s) =
I - A exp [- a8(l/L)] I - A exp {- a 8 [ 1 + (l/L)]}
exp ('As)
( 1 .85)
where 'A'
'A s A = ---' 'A s + a s - 'A
[ (
( 1 .85a)
)]
4D 's 11 2 'A = 1- I+U '2 2EbD ax ,
Lu'
·
( 1 .85b)
'A8 and a8 are the same as those defined in Section 1 .3 . 1 and u' = EbU. The first moment of an impulse response, for example, is then L
M II1 = - ( I + 1\o) u
where
(
)
D' - I exp (- (l/L)/No] ( I - exp (- 1 /ND)] 1\o = No Eb2 Dax
( 1 .86)
( 1 .86a)
is a measure of deviation of M ? from that of an infinite packed bed. The measure of deviation becomes large when l/L decreases and/or ND increases.
PARAMETER ESTIMATION 65 At intermediate and high flow rates, N0 is low, but, with the decrease in flow rate, N0 becomes relatively large and this makes A0 large. The mass dispersion number is rewritten as No =
ax Dp €b D -- -- . (Sc)(Re) Dv L
( 1 .87)
At low flow rates, the dispersion coefficient, D', in the empty column is identical to the molecular diffusion coefficient, Dv. The axial dispersion coefficient, Dax, in such a packed bed of glass beads is (0.6 "' 0.8) x Dv (Eq. 2.29). With the assumptions that D' Dv, Dax = 0.7 Dv and €h = 0.4, a measure of A0, the deviation from an infinite packed bed, is shown in Figure 1 .27 as a function of N0 and l/L . The packed bed may be assumed to be infinite if the measure of devia tion, A0, is less than 0.0 1 . It is then found that a packed bed with No < 0.03 can be assumed to be infinite if the response measurement is made at 1/L > 0. 1 . If N0 < 0.006, the criterion for an infinite packed bed is met at 1/L > 0.0 1 . =
No
FIGURE 1.27
Effects of 1/L and No on Ao for laminar flow in inert beds.
66 HEAT AND M ASS TRANSFER IN PACKED BEDS According to Eq . ( 1 .87), the dispersion number for a laminar gas flow (Re � 1 ) with Sc 1 in a bed of Eb 0.4, for example, is �
=
Nn � - -
0.3 Dp
( 1 . 88)
Re L
=
If L/Dp 100, the condition of N0 < 0.03 is satisfied at Re > 0 . 1 . Simi larly, if L/Dp 200, N0 becomes less than 0.03 at Re > 0.05 . With the data given in Example 1 . 1 , the effect of l on the response signal is examined. As listed in Table 1 .4, we have found from the time domain analysis of the data under the assumption of an infinite bed that f = 6.7 s and N0 = 0.030. The molecular diffusion coefficient, Dv, is 0.76 x 1 0-4 m2 s-1• With these values, the response curves at various locations are com puted. As illustrated in Figure 1 .28, if l/L < 0.0 1 , the response signals predicted are significantly different from those measured. However, if l/L > 0.1, the signals computed do not differ appreciably from the signal measured. Therefore, we find that, as far as the packed bed of Example 1 . 1 is concerned, the bed of glass beads may be assumed to be infinite, if the response signal is measured at l/L > 0.1.
=
o.2
c 0
�0-1 c ell u c
.Q
0
c
u
-
c o. 1 QJ
t
0 ....
u
LIL
c 0 u
1 - co •
(s)
20
exptal
00�--�-40-�1�0--�--�20���35�30���--� FIGURE 1 .28
t [S]
Effect of l/L on response curves predicted.
1 .4.2
PARAMETER ESTIMATION
67
Adsorption Packed Beds
A similar computation is made for an adsorption bed. The bed end is again assumed to be connected to an infinitely long empty column. The system is then described by Eqs. ( 1 .57)-( I .60) and ( 1 .84) with the same con ditions as listed in the preceding section. The transfer function is F(s) =
1 - A exp [-o8(l/L)] �
I -A cxp {- a8 [ 1 + (1/L)]}
exp
(AB) A
( 1 .89)
where (1 .89a)
�B = �
(
LU
2 Dax
)
- as
( 1 .89b)
and A' is defined by Eq. ( 1 .85b) and o8 by Eq. ( 1 .61 a). The first moment of the impulse response is then ( 1 .90) where Aad
= No r
exp [- (l/L)/No ] [ l - exp (- 1/Nn)]
( 1 .90a)
and ( 1 .90b) and o0 is defined by Eq. ( 1 .63a). If (Ep + Pp KA ) = 0, Aad becomes J\0 for inert bed . This Aad is a measure of deviation from the infinite adsorption column. The Aad is relatively large at low flow rates. Assuming that D ' = Dv, Dax = 0.7 Dv and
HEAT AND MASS TRANSFER IN PACKED BEDS
68
No
(a )
- So = 100 - - - So = (X)
No
(b) FIGURE 1.29
Effects of 1/L, beds: (a) 60
No and o0 on I A adl for laminar flow in adsorption 1 a nd 1 0 ; (b) o0 == 100 and ==
oo.
0.4, similar to the preceding section, Aad for an adsorption column at low flow rates is evaluated as a function of N0, 80 and l/L . Figures 1.29(a) and (b) show the relationships between 1 /\ad l , 80 and N0 at l/L 0, 0.0 1 and 0 . 1 . Again, the criterion for an infinite bed may be given by I Aad I = 0.0 1 .
Eb
=
=
PARAMETER ESTIMATION
o.ol
ill = 0
Q.l
69
IAoo l = o.o1
10
FIGURE 1.30
No
Relationships between No, 50 and 1/L at I J\ad l flow in adsorption beds.
=
0.01 for laminar
Figure 1 .30 shows the relationship between N0, 80 and 1/L , which satisfies l i\.ad l = 0 .0 1 . When No and o0 are known, we can easily see where in the bed the response signals should be measured in order to satisfy the assump tion of an infinite bed length. The impulse responses at Re 0.05, for example, are computed with the following data (the same as Example 1 .2): =
Eb = 0.38 E p = 0.59 D p = 0.2 em De = 0.63 X 1 o-6 m 2 s-1 Dv = 0.76 X 1 o-4 m 2 s- 1 Eb DaxfDv = 0.24
L = 20.4 em
70 HEAT AND MASS TRANSFER IN PACKED BEDS 0 . 0 2,...----. L/L 1 - co Q.Q1
0
c 0
-
0
L. -
c Ql u c 0 u
0.01
0
(a)
100
t
x1o-5
200
[ S)
300
400
4
c 0
-
0
L. -
c Ql
g
0 u
2
0.5
0
(b) FIGURE 1 . 3 1
t [ S)
PpKA on impulse responses 0.05: (a) P pKA 1 ; (b) P pKA
Effect of
==
=
for E x ample 1 .2 with 1000.
Re
==
P AR AM ETER ESTIMATION
71
kr from Eq. (4 . 1 1 )
ka = 00 .
As shown in Figure 1 .3 1 , the impulse response at l/L = 1 to differs considerably from those at l/L = 0 and 0.01, when Pp KA is small. I t is interesting to note that the peak of the impulse response with l/L = 1 to is higher than those with l/L 0 and 0.01 as long as P p KA is small. When P pKA is high, however, the peak o f the impulse response with l/L = 0 becomes the highest. This comes from the fact that with an increase in 50 or P p KA, the value of r turns from positive to negative. The first moment of the impulse response then decreases and the response peak gets high. The moment equations, such as Eqs. ( 1 .63) and ( 1 .64), may be derived from the transfer function in a digital computer. The computer programs for the derivations are listed in Appendix C. oo
oo
=
REfERENCES
[3] [4] [5] (6] [7]
K. (,l)stergaard and M . L. Michelsen, Can. J. Chern. Eng. 47, 107 ( 1 969). S. K. Gangwal, R. R. Hudgins, A. W. Bryson and P. L. Silveston, Can. J. Chern. Eng. 49, 1 1 3 ( 1 9 7 1 ). W. C. Clements, Chern. Eng. Sci. 24, 957 (1969). A . S. Anderssen and E . T . White, Chern. Eng. Sci. 25, 1015 (1970). A. S. Anderssen and E. T. White, Chem. Eng. Sci. 26, 1203 ( 1 9 7 1 ) . H . J . Wolff, K . H. Radeke and D. Gelbin, Chern. Eng. Sci. 34, 101 (1979). M. J . Hopkins, A. J . Sheppard and P. Eisenklam, Chern. Eng. Sci. 24, 1 1 3 1
[8] (9] [ 10 j [11J (12j [ 13]
M . Kubin, Co/lee. Czech. Chern. Commun. 30, 1 104; 2900 (1965). E . Kucera, J. Chromatography 1 9 , 237 (1965). P. Schneider and J. M . Smith, A/ChE J. 14, 762 (1968). N . Wakao, K. Tanaka and H . Nagai, Chern. Eng. Sci. 3 1 , 1 109 (1976). N . Wakao, S. Kaguei and J . M . Smith, f. Chern. Eng. Japan 12, 481 ( 1 979). C. R. Antonson and J. S. Dranoff, Chern. Eng. Prog. Symp. Ser. 65 (No. 96),
[14]
E. Ruckenstein, A. S. Vaidyanathan and G. R. Youngquist,
[1) [2)
(1969).
27 (1969).
26, 1305 ( 1971).
[15] [ 1 6] [17] [18] [ 1 9] (20]
Chern.
Eng.
Sci.
N . Hashimoto and J . M . Smith, !nd. Hng. Chern. Fund. 12, 353 (1973). K. Kawazoe and Y . Takeuchi f. Ch ern. Eng. Japan 1, 431 ( 1 974). P . Schneider and J . M . Smith, A!ChE J. 14, 886 ( 1 968). K . Kawazoe, M . Suzuki and K . Chihara , J. Chern. Eng. Japan 7, 1 5 1 ( 1 974). L. K. Lee and D. M. Ruthven, Can. J. Chern. J::ng. 57, 65 ( 1 979). S. Kaguei, K. Matsumoto and N . Wakao, Chern. Eng. Sci. 35, 1809 (1 980). ,
2
Fluid Dispersion Coefficients
IT IS well recognized that conversion in a chemical reactor depends largely on the degree of fluid dispersion in the reactor. Axial fluid dispersion coefficients have been obtained mainly from tracer injection measure ments, as discussed in Section 1 . 1 . In packed beds of non-porous particles such as glass beads, no tracer species penetrates into the particles, and it is considered that axial dispersion of the tracer, while flowing in the bed, is described by Eq. ( 1 . 1 ). However, in the case of mass transfer taking place inside the particle, the intraparticle concentration is, as mentioned in Section 1 .2 for an adsorption bed, usually assumed to have radial symmetry. This conven tion has been introduced to simplify the solution to the problem. Dis persion itself is a hydrodynamic phenomenon which occurs while the fluid is flowing in the interstitial space of a packed bed. Fluid dispersion should, therefore, be independent of what is occurring inside the particle. However, the question is whether the assumption of concentric intra particle concentration can describe the mass transfer phenomenon sufficiently enough or not . If not, this may superficially alter the value of the dispersion coefficient which appears in the fundamental equations derived under the assumption of radially symmetric concentration. In this chapter, the theoretical treatment of the effect of dispersion on chemical conversion, the significance of fluid dispersion coefficients and their evaluation in reactive, non-reactive and adsorption packed bed systems are discussed .
2.1
Effect of Dispersion on Conversion
In a continuous flow reactor in which a chemical reaction is taking place under steady-state conditions, the mass balance equation for a reacting 72
FLUID
DISPERSION COEFFICIENTS 73
species is:
(2. 1 ) where rx is the reaction rate per unit volume of reactor and defined as the production rate of the species under consideration. For a reactant disappearing in the reactor, the reaction rate is negative . Equation (2 . 1 ) is analytically soluble only when the reaction rate is zeroth-order or first-order with respect to the concentration of the reactant. The reaction rate, for example, with first-order kinetics is: rX
= -KC
(2 .2)
where K is the reaction rate constant. When the reaction proceeds only in a fluid phase, K is the intrinsic rate constant of the homogeneous chemical reaction. If, however, the reaction takes place in a porous catalyst particle, K is then an overall rate constant, which depends, in general, not only upon the intrinsic chemical reaction rate, but also on the mass transfer rates both at the particle surface and inside the particle. If the reaction is not zeroth-order or first-order, Eq. (2.1) should be solved numerically. In any case, solving Eq. (2 . 1 ) requires two axial boundary conditions. The Danckwerts boundary conditions [ 1 ] are widely used where:
dC Uq n = UC - Dax dx
dC =0 dx
at x = 0 (inlet)
(2.3)
at x = L (exit)
(2 .4)
where Cin is the concentration of the reactant in the fluid flowing into the reactor. The inlet condition, Eq. (2. 3), has been derived under the assumption that no fluid dispersion occurs before the reactor, in which the fluid is flowing in the dispersed plug flow mode. The exit condition is established intuitively. With the reaction rate given by Eq. (2.2) and the Danckwerts boundary
74
HEAT A N D MASS T R A N S F E R IN PACKED BEDS
conditions, the solution to Eq. ( 2 . 1 ) under isothermal conditions is:
where
A=
B)2 (_!J_ ) B = ( Kf)t/2 (1 +
exp
2N0
2
B
(
- ( l - )2 exp -
�) 2 N0 ,
(2 .5a)
(2 .5b)
1 + 4N0 -
Eb
Da x
(2.5c)
Nn = LU f =- ·
L
(2 .5d)
u
The Danckwerts boundary conditions were first verified by Wehner and Wilhelm [ 2 ] , under the condition that Cin was the concentration at x � O. In their verification, they considered a column consisting of three sections, as shown in Figure 2 . 1 . Sections 1 and 3 are inert zones in which no reaction occurs and are assumed to be semi-infinite in length. Between them is a reaction zone (Section 2), where a first-order reaction is pro ceeding. Under steady-state conditions, mass balance equations for the reactant in the three separate sections arc: In Section
1 (x < 0;
concentration
u
-
dC' dx
- Eax '
C')
dx2 =
d2 C ' --·-
0
(2.6)
F L U I D DISPERSION COEFFICIENTS
S.ction
x :z - oo
3
2
+---
�
x:L
x:O
75
X : oo
� ��eac�� � E"
Dispersion
ax
fiGURE
A column composed of three sections.
2.1
In Section 2 (0 < x < L; concentration C) u
dC dx
- Eax
d 2C dx 2
+ KC = 0
(2.7)
In Section 3 (x > L; concentration C") U
dx
dC"
, - E,a x
d 2 C" dx 2
= 0.
(2.8)
The fluid velocity, u, and the axial fluid dispersion coefficients, E�x , Eax and E�'x , are a]] based on the cross-section of the column. The boundary conditions are: (a) C' = Cin (b) uC ' - E�x
dC ' dx
= uC - Eax
dx
dC
(c) c' = c dC dC" (d) uC - Eax - = uC" - E�� dx dx (e) C = C " (f) c " = finite
} }
at x =
-
oo
at x = 0
at x
-
atx
=L = oo
.
Conditions (b) and (d) result from the conservation of reactant at the bed inlet and outlet. Conditions (c) and (e) come from the intuitive argu ment that the concentrations should be the same at the intersections.
76
HEAT AND MASS TRANSFER I N PACKED BEDS
The solutions are :
Cin - C ' Ci n - C(O)
=
exp
(
x
LN[)
)
for x � 0
(:2 .9)
where (2 .9a)
Eax ' No' - Lu
(2 .9b)
_
and
c
- = Eq. (2.5) Cin ell
for 0 :s:;;; x :s:;;; L
(1
Cexit - = -- = 2AB exp 2 No Cin Cin
)
for x � L .
( 2 . 1 0)
The axial fluid dispersion coefficient, Dax , based on unit void area, is equal to Eax/Eb, and the interstitial fluid velocity, U, is equal to u/Eh· Therefore, the mass dispersion number for Section 2 , for example, is
Eax Dax No = - = - . Lu L U
(2 .1 1 )
Wehner and Wilhelm [2] found that the concentration profile in Section 2 was identical to that predicted from the solution obtained under the Danckwerts boundary conditions. It is also found that N � has no e ffect on the concentration profiles in Sections 2 and 3 . As depicted in Figure 2 .2, if N0 is zero, the concentration profile at x = 0- approaches that of a step function. With increasing N0, the concentration gradient decreases. I n any case, the concentration profile in Section 2 depends only upon N0. Also, the concentration gradient is zero at x = L . The fluid dispersion in Section 3 has no effect on concentration profile in any section.
FLUID DISPERSION COEFFICI ENTS
Sec t i on I
2
77
3
Concentration profiles in Sections 1 , 2 and 3 .
FIGURE 2.2
Equation (2 . I 0) shows that the exit concentrations, Cexit • in the follow ing two extremes are:
For a plug-flow reactor (with N0 = 0)
( )
--
Kf Cexit = exp - Cin Eb
(2 . 1 2) For a continuous stirred tank reactor (with N0
-- --Cexi t
-
)
= oo
1
1 KV 1+Fv
(2 . 1 3)
where f is the mean residence time, V is the reactor volume and Fv is the volumetric flow rate; f Eb V/Fv. =
78
HEAT A N D MASS TRANSFER I N PACKED BEDS
....� ... X
c: ·X Q) u
I
i
l
-f.-
rfi'� ! I
i I
-- -
���
\\\1\,l'\. �
\\[\ r-...\\.�"'-�04' 1\\1\1\
\\ \i\ �
�
l� \�\�0� 0� .... �\l�1\ �� o o
�
KV
· �
10
Fv
FIGURE 2.3
Effect of dispersion on conversion for a first-order reaction.
Figure 2 . 3 is a graphical presentation of Eq. X, is defined by
C ·
X= 1-�
qn
(2 .I 0).
The conversion,
(2 . 1 4)
Obviously, a plug flow reactor requires the least reactor volume to attain a given conversion. With an increase in N0 , the reactor volume becomes larger, and the maximum is that of a single continuous stirred tank reactor. Since the first confirmation by Wehner and Wilhelm [2 ), the Danek werts boundary conditions have been successively re-examined and employed for different reaction schemes, both isothermal and non isothermal, by van der Laan [ 3 ] , Fan and Bailie [4), Bischoff [S], Bischoff and Levenspiel [6), Fan and Ahn [7], Carberry and Wendel [8], Liu and Amundson [9], Hofmann and Astheimer [ 1 0], Schmeal and Amundson [ 1 1 ] , Mears [ 1 2 ) , Wen and Fan [ 1 3 ) and Chang et al. [ 1 4]. The criteria, in applying the Danckwerts boundary conditions, are given by Gunn [ 1 5 ] as: the Danckwerts conditions are realized only when the tracer imposed moves to some extent against the main direction of flow in the reactor, i.e. when diffusional dispersion is controlling. For dynamic study,
FLUID DISPERSION COEFFICIENTS
79
X = R
X = -R R'
�
I
Mass f low
FIGURE 2.4
Single cell and coordinates.
however, appropriate boundary conditions other than those of Danckwerts have often been employed. More detailed information is given by Carbonell [ 1 6 ] , for example.
2.2
Fluid Dispersion Coefficients in a Reacting Systent
The fluid dispersion coefficient needed for the design o f a packed bed reactor is that for the reacting species in the fluid. It had long been assumed that the dispersion coefficient for a reacting species was the same as that in an inert system. However, Wakao et al. [ 1 7] have studied the dispersion coefficient for a reacting species, and have found it to be considerably different from that under inert conditions. First, their study of the dispersion coefficient at zero flow rate is outlined below. Suppose a short cylinder as shown in Figure 2 .4, consisting of a spherical catalyst particle and a stagnant fluid, is the smallest element or
80
HEAT AND MASS TRANSFER I N PACKED BEDS
unit cell of a multiparticle system. The height of the cylinder is assumed to equal the sphere diameter 2R, and the radius R ' of the cylinder is 1.05R, the void volume fraction in the cell is then 0.4. Also, no mass transfer is assumed to occur across the side of the cylinder. When an isothermal, first-order, irreversible, catalytic reaction proceeds in the sphere under steady-state conditions, mass balance equations for the reactant give n... v2 c*
De \12 c * where
=o -
k xc * = 0
( ) l
for r > R
(2 . 1 5 )
for O < r < R
(2 . 1 6)
(
)
]
1 a2 a 1 1 a a a V2 = -1 -- (2 . 1 6a) r 2 - + - -- - sin O - + r2 or ar r2 sin e ae ae. sin2 e o 2 with
ac* oc* c* = c* and Dv - = De ar ar ac* ar'
-=0
at r
I =
--
at r = R
R'
where
C*
=
concentration in stagnant fluid c* = concentration in a particle De = intraparticle effective diffusivity Dv = molecular diffusion coefficient kx = first-order reaction rate constant r = radial distance in spherical coordinates r' = radial distance in cylindrical coordinates.
'The complete solution to Eqs. (2 . 1 5 ) and ( 2 . 1 6) is analytically difficult. Therefore , instead, let us assume that what happens in the unit cell (Cell A) may be considered to take place in two separate unit cells (Cell B and Cell C) as shown in Figure 2 . 5 . Suppose the following boundary con ditions are imposed:
F L U I D DISPERSION COEFFICIENTS
Ct . c2 2
Ct . c2 2
c,
N, � x = -R
FIGURE 2.5
Ns 1
c2
� N2
__.....
X =
Ct - C 2 2
x= R
Ns 2
�
-R
x= R
No, ___.
81
Ct - C z 2
-+
Na 2
x = -R x=R Equivalence o f cell models: (a) Case A; (b) Case B ; (c) Case C.
Cell A (Figure 2.5a, C* and c* are the concentrations in the stag
nant fluid and particle, respectively). at x = -R atx = R
)
(2 . 1 7)
Cell B (Figure 2 . S b , C*5 and c*5 are the concentrations in the
stagnant fluid and particle, respectively). c* s =
cl ---
+ c2 2
at x = -R (2 .1 8) at x = R
Cell C (Figure 2. Sc, C*3 and c*3 are the concentrations in the
stagnant fluid and particle, respectively). c*a =
cl -c2 2
c*a = -
c1 - c2
2
a t x = -R (2 . 1 9) atx = R.
82
HEAT A N D MASS TRANSFER I N PACKED BEDS
Similar to Cell A, it is assumed i n Cells B and C that no mass t ransfer occ urs across the si des of the cylinders. The solutions in Cell A are then r ela ted t o those of Cells B and C by the fol lowing eq uations: (2 .20 ) (2 .2 1 ) In
Cell
B: the boundary conditions given b y Eq. ( 2 . 1 8) indicate that the c oncentrations. C*5 and c*5, are even functions of x or symmetric with respect to x; hence the diffusion rates at both ends of the cell are eq ual but in opposite directions, i.e. N� N� . In other words, ther e is no net diffusion fl ux of the species passing across the cell. In Cell C: to the contrary, the boundary conditions, Eq . (2 . 1 9), show that the concentrations, C*a and c*a, are odd functions of x or anti symmetric with respect to x ; the diffusion rates are, therefore, eq ual and in the same direction, i. e . N� = N�. Thus, the number of moles of reacting species depleted due to chemical reaction in the left hemisphere of the solid particle is compensated by the appearance of an eq ual amount prod uced in the right hemisphere. As a result, there is no net change in the total number of moles of reacting species in the particle. The boundary concentrations and the diffusion rates of the proposed cell models are summarized in Table 2 .I. The above analysis shows that there is no net change in the number of moles of reacting species in the solid particle in Cell C, whereas, in Cell B, the net diffusion flux is zero. Therefore, the changes in the number of
=
-
TABLE 2 . 1 Characteristics o f cell models. -·----·-
Concentrations
Boundary concentrations
Diffusion rates across end-faces
Cell
fluid
Solid
x=-R
x=R
x=-R
x=R
A
c*
c*
c2
Nl
N2
B
s c*
s
cl
c*
C1 +C2
C1 +C2
c
c*
a
c*
a
C1-C2
2
---
2
2
C1-C 2
2
s
N1-N2 NI=
2
a N1 +N 2 NI = 2
:
N =-
N1-N2
2
a N1 +N2 N2= 2
· ·--
FLUID DISPERSION COEFFICIENTS
83
moles of reacting species in the par ticle and the diffusion rate in C ell A must correspond to those in C ell B and Ce ll C , respective ly. Hence, the diffusion rate across C ell A can be determined from that of C ell C and vice versa. The diffu sion coe fficie n t of t he re acting specie s in Ce ll A c an, therefore, be assume d to be the same as that in C ell C. The effective diffusion coeffi cient, £0, oft he reacting specie s in Ce ll C is define d as: N 1 + N2
=
2
(nR '2) Eo
( c1 - c2) 2R
(2 . 2 2 )
.
Using a grid ne twork for C ell A , steady-state conce ntrations a t nodal points are computed for DefDv 10 - 3 to 10 , and the Ji.i ttner modulus The pre dicted concentrat ion (see Section 3 . 2) R(kx/De)112 = 0 to profile s are illustrated for the case s where 0 . 5 and 5 in Figures 2 .6(a)-(c). The catalyst effectiveness factor (see C hapte r 3) is close to unity at = 0 . 5 , and is 0 .48 at 5. In either case, Figure 2.6 reveals that the intraparticle concentration is not radially symmetric, especially, at lower values. The diffusion rates, N 1 and N2 , in C ell A are evaluate d with the calcu lated concentration gradie nts and the grid conductances at both e nds of
=
=
00•
=
=
...,
\ � I � ? I
I / 1/
I
N
u II
I
J 1/
I I 1/
I
II
I I
�
....
II �IT
r-.....
-
l I 1/
I I II,/
Nr--. ,,..._ I I' r-... !\ � \ " I
' [\ I \ I
\ \
I
1.9
(a )
1.8
1.6
,... �
Center 1.4 of sphere
'
I
1.2
1.1
i I
i
II N
u
84
HEAT AND MASS TRANSFER I N PACKED BEDS
I
I1-Iv �� v-v
.....
1\v /
N II
cS
17 I 1/I J I rr
1/ I 1/
1/
I
I
n
1.5 1.0 0.7 0.5
v
/
� v '/ 1/ l7 7 v I 7 17 Ill I I II II v j 7
1---"
1/
I
/ v v
0.1
(b)
�v
17v r7 II II' II 'J
i N
II
v II IJ
I
1/
17 17
1/
I
1.98
II
I
I
1.95
I
I If
I
1.90
l----
v
I
�
"
I
v
I
v
I If
I
�
Center of sphere
\
I
1\
!
I
I
'
T I
I f\T :v
II
N
u
\ 1\ \
0.2
l
I
!
I
0.5 0.7
;
1.895
I I
[7 r---.loo..J ......
I'--l'-U �
['..
f-
I
1\
I
0.1
I1
\ \.\\ I
1\ \
"'
-
:
II
r\ 1\'I· ' 1\ \
I
1\
v
/
""'
!'..... � "" 1'. " t\1\. I '\ t\. '\
t-...
_;;:........ . .
!I i' T: ·
\
0.07 Center of sphere
[/ r7
1.87
I
I'�
!"':
�...... rT I� -v
f.-
II
-
!
-
I ,{
I
-�
[\
)
0.2
I T ';?
1-"
I
7
I
'\
I':
i\
rT I
1.85
1\.. t\.
'\
\
'
1\ \
1\'
1\
\
\ '
II
s
1.87
(C) FIGURE 2.6 Calculated concentration profiles of reacting species in the cell with De = 0.08 X 10-4m2 s-1 and Dv = 0.7 X 10-4 ffi2 S-1: (a) C1 = 2, C2 = 1 and = 0 .5; (b) cl = 2' c2 = 1 and = 5; (c) cl = 2, c2 = 1. 9 and = 0. 5.
FLUID DISPERSION COEFFICIENTS 85
the cylinder. With these diffusion rates, and given C 1 and C2 , the effective diffusion coefficients are evaluated fr om Eq. (2. 22). The coefficient values obtained are then plotted in Figure 2.7 as £0/Dv versus De!Dv with ¢ as a parameter. As depicted in Figure 2 .7, the £0 values are surprisingly large, and also, they depend on the Jii ttner modulus, ¢, or catalyst effectiveness factor, as well as on the flui d and intra partic le diffusivities. I t should be noted that concentration profiles depend upon the assumed values of C 1 and C2 (refer to Figures 2 .6a and c), but t he values of £0 defined by Eq. (2.22) are independent of the boundary concentration values as long as the reaction is first-order.
10 > 0
-
"
0
w
v
,/
v'Lv,.,�
101
FIGURE 2.7
10
De/Dv
Effective diffusion coefficients i n a quiescent bed under first-order reaction conditions.
In Figure 2.7, the curve for ¢ = 0 corresponds to the diffusi onal dis persion when no chemical reaction takes place in a bed of porous particles. The curve for ¢ 0 approaches (£0)inertfDv = 0.23 when chemical reaction De!Dv goes to zero (non-porous particles). When ¢ = is fast and proceeds only at the surface, so that the reactants are at zero concentrations (for irreversible reaction) or at equilibrium concentrations =
00,
86
HEAT AND MASS TRANSFER IN PACKED BEDS
(for a reversible reaction) on the catalyst parti cle surface as well as withi n the particle. Under such conditions, the contribution due to di ffusional d is persion is large as shown in Fi gure 2.7: £ 0/Dv = 20. The fluid dispersi on coefficient i s consi dered to consi st of diffusi onal and turbulent contributions. The turbulent contribution is well recogni zed and, i n terms of the Peclet number, is given as: (Per)mixing = DpU/(Dr)mixing (Peax)mixing
�
= DpU/(Dax)mixing
10
(2.23)
2.
(2.24)
�
The diffusional contribution corresponds to the effective diffusi on coeffi ci ent at z ero fl ow rate and is i sotropic. In the range of lami nar flow, Re < 1 , the d si persi on coeffici ent only consists of a diffusi onal contri bution. At Reynolds number greater than 5 , fl ow is turbulent. Hence, the radial dispers ion coefficients are gi ven by the following equations : £0 D= r €b £0 =- + O.lDPU €b
for Re <
1
(2 .25) for Re > 5 .
S imilarly, the axial di spersi on coefficients are Eo =Dax €b £0 =-+0.5DpU €b
for Re < 1
(2.26)
for Re > 5.
The diffusional contri bution, obtained by Edwards and Ri chardson [ 1 8], Evans and Kenney [ 1 9], S uzuki and S mith [20] and Wakao eta!. [2 1 ] from di spersi on measurements of non-adsor bing speci es in packed beds of non-porous particles , has been expressed as: · (£0 )mert ., Dv
= (0.6 0.8) €b . "'
(2.27)
FLUID DISPERSION COEFFICIENTS
87
Ther efore, if a particle (both extern al and internal surfaces) is involved in neither a reaction nor in a mass transfer pr ocess, the radial and axial dispersion coef ficients in such an inert bed are for Re < 1
(Dr )inert = (0. 6..,_, 0.8)Dv
for Re > 5
= ( 0.6 ...... 0.8)Dv + O.lDP U (Dax )inert = (0.6'""" 0.8)Dv =
for Re < 1 for Re > 5 .
(0.6"' 0.8 )Dv + 0 . 5DP U
(2.28)
(2 .29)
l n the past the radial and axial dispersion coef ficients for reacting species have been assumed simply to be identical to the corresponding dispersion coeff icients in an inert bed. However, as long as the turbulent contributions are not dominant in Eqs. (2 .25 ) and (2.26) it has been shown that the dispersion coefficients of r eacting species a re significantly different from those under inert conditions. I t should be noted that if a chemical reaction occurs homogeneously only in the fl uid phase in a packed bed, the dispersion coefficients for the reacting species are given by Eqs. (2 .28) and (2 .29). 2. 3
Fluid Dispersion Coefficients in Adsorption Beds
L et us examine again the dispersion coeff icient for an adsorbing gaseous species at zero flow rate b ased on the cel l models (Figure 2 .4). The con centration of the adsorbing species in the gas phase, C*, is governed by the following equation: ac* - =Dv V2C* ar
for r > R
(2.30)
where V 2 is given by Eq . (2.1 6a). The intraparticle concentration, c*, is expressed by Eq. (1 .73). In physical adsorption, the adsorption rate constant is so large that it may be assumed that ka = oo (ref er to Section 1 . 2.2). Equation ( 1 .74) then gives c;d KAc*. Substituting this into Eq. ( 1 .73), we have
=
forr < R .
(2 . 3 1 )
88 HEAT AND MASS TRANSFER IN PACKED BEDS Equations (2 .30) and (2 . 3 1 ) are numerically solved under the following conditions: C* =c* = 0
at t= 0
C* =0
a t x =R
a t x = -R
C* = 1
C* =c*
and Dv
ac*
ac* =De ar ar
atr = R
-
.
c*
are functions of three variables, x , r1 = r sin 8, and t. The diffusion rate, nx, passing axially through a cross-sectional area of the unit cell is defined as: C* and
nx
=
-211
R'
I
0
D
ae -
ax
I
r dr
I
(2. 32)
where D = De' E> = c*
for OR.
The average diffusion rate, N, in the unit cell is expressed in terms of the transient effective diffusion coefficient, E0(t), as: R
N=
�I
2
-R
nx
dx
(2.33) where �E> = Ci=-R- Ci=R· From Eqs. (2 . 32 ) and (2. 33), we have
FLUID DISPERSION COEFFICIENTS
' R
R
--
.., J J..
R'2�e
dx
-R
0
ae , , D-r d r . ax
De/Dv
0.8 >
c.: ......
0 l I
=-
=
89
(2.34)
0.1
0.01
.). 6 0,/1 0.2
o ��--�����--�����--�����--��� 10-4
FIGURE 2.8
10-3
10-2
t
I::: I
10-l
1
Transient effective diffusion coefficients under adsorption conditions.
As an illustration, the calculations made with the following data are shown in Figure 2.8: R = 0.1 em R' = l.OSR
Dv = 0.8 X Io-4 m2 s-1 €p = 0.5 PpKA = 1, 10 and 100
De/Dv = 0.01 and 0.1. As shown, the effective diffusion coefficient s, E0 (t), are high at t = 0, but dec rease r apidl y with increasing time until steady-state values are
90 HEAT AND MASS TRANSFER IN PACKED BEDS reached. The decrease becomes more gradual with an increase in Pp KA, but the time to reach the steady-state values is always very short. More over, the steady-state values of £0/Dv depend only upon De!Dv· The s teady-state values of £0/Dv, attained by the adsorbing species, are f ound to be the same as those under inert conditions.
FIGURE 2 . 9
Two unit cells in contact.
For two cells in contact [wr iting C*(II) and c*(II)] as shown in Figure 2.9; if the initial and boundary conditions are chosen as: c*(II) = 0
at t = 0
=
1
at x
=
-2R
=
0
atx
=
2R.
C*(II)
=
C*(II) C*(II)
The solutions C*(II) and c*(II) are expressed, similar to Eqs. (2. 20) and (2. 2 1 ) , as follows: C*(II) = C*(II)5 + C*(II)a
(2.35)
and c*(II)
=
c*(II)5 + c*(II)a
(2.36)
where C*(II)5 and c*(II)5 are symmetric with respect to x and are the
FLUID DISPERSION COEFFICIENTS
91
sol u tions obtained under the conditions: C*(II)5 = c*(II)5 = 0
at t = 0
C*(II)5
a t x =-2R
1 /2
=
C*(II)5 = 1 /2
a t x = 2R
and C*(II)3 and c*(IIl are an tisymmetric with resp ect to x and are the solutions under the conditions: C*(II)3 = c*(II)3
=
at t = 0
0
C*(II)3 = 1 /2 C*(II)3
=
at x - 2R =
-1 /2
at x
=
2R
or C*(II)a = 0
at x = 0.
Similar to Eq. (2.34), the effective diffusion coefficient, E0(II), for the two cells in con tact is
Eo(ll) =-
2
R'2Ae(II)
2R
R'
I I dx
-2R
D
0
a[e(n)s + e(II)a] r ' dr ' ax
(2 . 37)
where A8(11) = C*(II)X=-2R- C*(II)X=2R = 1
8(11)5 = c*(II)5, 8(11)3 = c*(II)3, D = De
8(11)5 = C*(II)5, 8(11)3
=
C*(II)3, D
=
Dv
for O < r < R for r >R .
Since ae(II)5jax is antisymmetr ic and ae(II)ajax is symmetric with resp ect t o x , Eq. (2.37) becomes R'
D -2R
0
ae(II)3 ax
r ' dr ' .
(2.38)
92
HEAT AND MASS TRANSFER IN PACKED BEDS
As mentioned already, the concentrations at x = - 2R and x = 0 are 8(11)3 = 1/2 and 0, respectively, and consequently �e for a single cell is 1/2 . Substitution of this into Eq. (2 . 34) gives Eq. (2.38) for the effective diffusion coe ff icient of the two cells. Therefore, the e ffe ctive diffusion coeffi cient of the single cel l is found to be identical to that of the two cells in contact . Similarly, the effective diffusion coefficients for the number of cell s� corre sponding to 22, 23 2 n in contact are the same as that of the singl e unit cell. The effective diffusion coeff icient or dif fusional dispersion term is thus expected to be the same as that of the unit cell. The transie nt time which is proportional to the square of the particle siz e wil l be longer for l arger particles; but whe n fluid is flowing, the e ffe ctive diffusion coe f icient is considered to attain a steady-state value in a much shorter time . Anyway, the transie nt time is usually so short compared to the mean residence time that the dispersion coefficient for an adsorbing species may be assume d to be constant over the entire adsorption period. More over, the dispersion coefficient for an adsorbing species is generally considered to be equal to that under inert conditions. In fact, Kaguei et al. [22] me asured, in packed beds of activated carbon particle s , the dispersion coefficients for adsorbing species, nitrogen, imposed on a carrier stream of hydrogen. They found that the dispersion coeffi cients under adsorption conditions were practically the same as those for an inert bed . .
•
•
REFERENCES ,Chem . Eng. Sci.2, l ( 1 5 9 3). [ l ) P. V. Danckwerts , 9( 15 9 6). [ 2 ) J . F. Wehner and R. H. Wilhelm, Chem. Eng. Sci. 6 8 [ 3 ) E. T. van der Laan, Chern. Eng. Sci. 7 , 81 7 ( 1 5 9 8 ). [ 4 ] L. T. Fan and R. C. Bailie, Chern. Eng. Sci. 1 3 , 63 ( 1 960). 5( ] K . B. Bischoff, Chern. Eng. Sci. 1 6 , 1 3 1 ( 1 9 6 1 ) . 7 2 4 5 ( I 962) . [ 6 J K . B. Bischoff and 0 . Levenspiel, Chern. Eng. Sci. 1 , 1 190 (196 2 ). [ 7 ) L. T . Fan and Y. K . Ahn,lnd. Eng. Chern. Process De s . Dev. , , 1 2 9 ( 1 963). 8[ ) J. J. Carberry and M. M. Wendei,A/ChE J. 9 [ 9 ] S . L. Liu and N . R. Amundson,Ind. Eng. Chern. Fund.2,81 3 ( 1 963). [ 1 0) H . Hofmann and H. J . Astheimer, Chern. Eng. Sci. 1 8 , 643 ( 1 963). [ 1 1 ) W. R. Schmeal and N. R. Amundson, A /ChE J. 21 , 1 202 1( 966). [ 21 ) D . E. Mears, Chern. Eng. Sci.2 6, 1 36 1 ( 1 97 1 ). C. Y. Wen and L. T. Fan, Models for How Systems and Chemical Reactors, 3 [1] Marcel Dekker, New York ( 1 9 5 7 ). K. Chang, K. Bergevin and E. W . Godsavc,J. Cltem. Hng. Japan 1 5 , 1 2 6 S. [14 1 (19 8 2 ). 8 ). D. 1. Gunn, Trans. In st. ('hem. Eng. 4 6 , CE5 1 3 ( 1 96 [ 5] 1 , 1 0 3 1 ( 1 979). [ 1 6 1 R . G . Carbonell, Chem. Eng. Sci. 34
FLUID DISPERSION COEFFICIENTS 93 [17] [18] [ 1 9] {20) (21) [22]
N. Wakao, S. Kaguei and H. Nagai, Chern. Eng. Sci. 33, 1 8 3 ( 1978). M . F. Edwards and J. F. Richardson, Chern. Eng. Sci. 23, 109 ( 1 968). E. V. Evans and C. N . Kenney, Trans.Jnst. Chern. Eng. 44, T 1 89 ( 1 966). M. Suzuki and J. M. Smith, Chern. Eng. J. 3, 256 ( 1 972). N . Wakao, Y. lida and S . Tanisho, J. Chern. Eng. Japan 7 , 438 ( 1 974). S. Kaguei, D . I . Lee and N. Wakao, Kagaku Kogaku Ronbunshu 6, 397 ( 1 980).
3 Diffusion and Reaction in a Porous Catalyst
PoROUS SOLID catalysts used for gas catalytic reactions have specific surface areas of tens to hundreds of square meters per gram. This enormous amount of surface area results mainly from the fine interconnecting pores in the catalyst pellet.t If a chemical reaction is very fast, it proceeds only at the external surface of the pellet. If, however, the reaction is very slow, the reactant gas may diffuse deep into the pores of pellet, even to the center of the pellet, and the chemical reaction takes place everywhere uniformly in the pellet. In the laboratory, reaction rate determined directly by measurements using a differential reactor, for example, is the overall rate. The overall rate constant does not necessarily mean the intrinsic chemical reaction rate constant. For the design of industrial packed bed reactors, one needs to know the overall reaction rate, not the intrinsic chemical reaction rate. The overall rate is governed not only by the chemical reaction, but also by the diffusion rate through the pores inside the catalyst pellet as well as at the pellet's external surface. If we simply measure activation energy from the overall reaction rate constants, the activation energy may differ from that of the in trinsic chemical reaction . The importance of diffusion is often underestimated by some catalyst chemists. Wheeler [ 1 ,2] made an extensive study on the role of pore diffusion in catalysis. Also, Dullien [3), Jackson [4), Petersen [5], Satterfield [6] and Smith [7] have reviewed the subject of pore diffusion associated with chemical reaction well. The review article of Youngquist [8) will also help readers understand the basic principles of diffusion and reaction in a porous catalyst. In this chapter, the validity of the assumption of in trapellet concen tra tion being radially symmetric is examined first. The importance of the t A catalyst pellet is often made by compressing fine particles. In this chapter the word "pellet" is used to distinguish it from the fine particles. 94
DIFFUSION AND REACTION 95
catalyst effectiveness factor and the Jiittner modulus in the catalytic reaction system, their relationships and evaluation for various systems are then discussed. I n the section on pore diffusion of gases, the mechanisms of pore diffusion and their interpretations are elucidated; moreover, the measurement of effective diffusivities and their predictions using various proposed models are reviewed. 3.1
Assumption of a Concentric Concentration Profile in a Spherical Catalyst Pellet
The objective of this section is to examine whether the steady-state con centration profile in a catalyst pellet may be assumed to be radially symmetric when evaluating the reaction rate in a pellet. Let us illustrate this by considering an isothermal, first-order, irrever sible reaction proceeding in a spherical catalyst pellet of radius, R, under steady-state conditions. The mass balance equation for a reactant is given by Eq. (2 . 1 6) in Section 2. 2 . Integration of Eq. (2.16) over a surface area with radius r and dividing this by 41Tr 2 gives (3 . 1 )
Following the same procedure as given in Section 1 . 2 4 Eq. (3 . 1 ) becomes, under isothermal conditions: .
1 d D e -r 2 dr
(r2 de-) - kxc=O dr
,
(3.2)
where 1
J
c* dcos8.
(3.2a)
-1
This c represents an average concentration over a surface with radius r in the pellet; therefore, the assumption of radially symmetric intrapellet
96 HEAT AND MASS TRANSFER IN PACKED BEDS
concentration may be used in evaluating the chemical reaction rate if the reaction is first-order with respect to the reactant concentration. This assumption can also be verified by considering the fact that the solution to Eq . (2 .16)may be expressed by a series of Legendre functions: c*(r , 8, ) =
oo
n
L
L
n=Om=-n
f:f (r) P:f(cos8) exp (im)
(3. 3 )
where P:f(cos8), with m defined in the range -n to n, is an associate Legendre function of the first kind. The coefficient,[� (r ), is given by
r;:'(r) = (
27T
1
-I)m c:: J J ')
0
d
X exp (-im) dcos8
-1
c*(r, 8, ) P;;m(cosO ) (3.4)
which should satisfy Eq . (3.5).
De[�
�( 2 df �)- n(n+I) t�J -kx � 0. r f = r 2 dr dr r2
(3.5)
The overall reaction rate ,RP, throughout a pellet is: R p =-kXJc* dv
= -kx
R
J
0
r2dr
21r
1
J J
0
d
-1
c*(r,8, )dcos8.
(3.6 )
Substitution of Eq. (3.3 )into (3.6)and consideration of the orthogonality of the Legendre function give 27T
l
J J
o
d
-I
P';: (cos8)exp (im) dcos8= 4rr
=0
when n = 0
when n =I= 0 .
(3.7 )
DIF FUSION AND REACTION 97 The overall reaction rate is then R
R p = -4rrk,
,
J
0
j'g(r) r2 dr.
(3.8)
Since Pg(cos())= 1 Eq. (3 .4) shows that f�(r)=
21T
1
�J J
4
0
d
-1
c*(r,O,
(3.9)
Also, Eq. (3. 5) reduces to
- ( -)
1 d d[0 2 0 - kx [0 r De o dr r2 d r
=
0.
(3.10)
According to Eq. (3.8), the overall reaction rate in a pellet may be evaluated only in terms of [g(r). As Eq. (3. 9) shows, this is an average concentration of c* over a surface area with radius r. Therefore, although c* was originally a function of r, ()and , Eq. (3.10) indicates that .f8(r) has center symmetry.
3.2
Effectiveness Factors for First-order Irreversible Reactions under Isothermal Conditions
A catalyst effectiveness factor is an indication of how much internal surface area is being utilized in a given reaction . The effectiveness factor depends not only upon the intrinsic chemical reaction rate, but also on the rates of the diffusion processes. If the reaction proceeds only at the external surface, the catalyst effectiveness factor is low, whereas if the internal pores are being used effectively for the chemical reaction, the factor is large. The concept of a catalyst effectiveness factor was first introduced by the German scientist Jli ttner [9] as early as 1909. The work of Thiele [ 1 0] on the catalyst effectiveness factor, published in 1939 and written in English, is popular among researchers in catalysis. Also in the Iate 1930's,
98 HEAT AND MASS TRANSFER IN PACKED BEDS a similar theoretical work was reported in a German paper by Damkohler [ 11) and in a Russian paper by Zeldowitsch [ 12]. The parameter indicating the reaction-to-diffusion rate ratio is widely known as the Thiele modulus. The importance of this parameter was, in fact, first recognized by Hittner. We shall, therefore, call it the Hittner modulus in this book. 3. 2 .I
Effectiveness Factors for a Spherical Catalyst and Overall Reaction Rate Constants
Let us consider again a spherical catalyst pellet of radius R , as shown in Figure 3.1. The steady-state mass balance equation for a reacting species is 1 d
( -) 2
de
(3.11)
r + Yv=O De-dr r 2 dr
where rv is the chemical reaction rate, defined as the production rate per unit volume of catalyst pellet. For a reactant, the rate is negative. In the preceding section it was shown that, for a first-order reaction, the intrapellet concentration may be assumed to be of radial symmetry. When the reaction is first-order and irreversible with respect to the reactant concentration, e, Eq. (3.11) becomes Eq. (3. 2). Based on the following boundary conditions: de
-= 0 dr
atr
=
(3.1la)
0 �--
c
Pore FIGURE 3.1
Concentration profile in a catalyst pellet.
DIFFUSION ANO REACTION 99 and
c
=
at r = R
cs
(J.l l b)
where C5 is the reactant concentration at the surface of pellet, the solu tion to Eq. (3 .2) under isothermal conditions is:
R
c=Cs r
sinh
[r(;J'J [ ( )112]
k sm h R -x .
(3 . 1 2 )
.
De
At steady-state conditions, the overall reaction rate throughout a single catalyst pellet is
2 e( ) -
R P = 41TR D or
de
dr R
(3 . 1 3 )
cr2 d r.
(3. 1 4)
R
R p = -41Tkx
f
0
Substitution of Eq. (3 . 1 2) into either Eq. (3. 1 3) or (3 . 1 4) gives
(3 . 1 5 ) In terms of the Jiittner modulus,¢, which is defined as:
2 )1 1 ( De'
kx = R Eq. (3 . 1 5) becomes
Rp= -
47TR 3 --
3
�
(3.16)
)
3 1 <j> k xCscoth
·
( 3 . 1 7)
100
HEAT AND MASS TRANSFER IN PACKED BEDS
As a matter of fact , (4rrR3/3) kx Cs is the overall reaction rate in a pellet when the intrapellet concentration is Cs everywhere, in other words, there is no in trapellet diffusion resistance. Equation (3.17) is expressed as:
( 3 . 1 8) where
(3.19) This Er is the catalyst effectiveness factor, or sometimes called the catalyst utilization factor. The variation of Er with¢, according to Eq. {3.19), for a first-order irreversible reaction is shown in Figure 3. 2 over a wide range of¢. As shown, Er is approximately given as:
In a chemical reaction controlling region for¢< 0.6
Er = 1
--
�
�"
( 3.20)
'
11-
-Chemical
reoc t ion
...
� "" �"'
-f',.."\ diffusion f',.
Mixed c on t ro l I ing ---H-Pore
controlling
controlling
t'-.
10
FIGURE 3.2
Catalyst effectiveness factor versus Hittner modulus, for a first-order reaction in a spherical catalyst pellet.
DIFFUSION AND REACTION 1 0 1
In a pore d�ffusion controlling region 3 Er = -
for¢> 10.
¢
(3. 2 1 )
A small Hittner modulus means that the catalyst pellet is small, the
chemical reaction is slow and/or there is a high intrapellet diffusion rate. Under these conditions, chemical reaction takes place uniformly through out the pellet and thus the catalyst effectiveness factor is unity. On the other hand, if the catalyst pellet is large and chemical reaction is fast and/or intrapellet diffusion is slow, then, the Jii ttner modulus is large, and consequently, the catalyst effectiveness factor becomes low. Under steady-state conditions, the overall reaction rate is equal to the rate at which the reactants are supplied to the pellet surface from the bulk fluid. Therefore, Eq. (3 . 1 8) is related to the rate of mass transfer of the reacting species as follows:
-Rp = -
47TR3 --
3
kx C5Er
3
47TR2kr(C-C5)
C
47TR3 -
=
--
1
R + kx Er 3kr
(3.22)
--
where Cis the reactant concentration in the bulk fluid and kr is the mass transfer coefficient at the pellet surface. The reaction rate, rx, defined on the basis of the unit volume of a packed bed catalytic reactor, is
rx =
1- €b 41TR 3
R p.
(3.23)
3 The overall rate constant,K, of Eq. (2.2) is then
K=
1
-
Eb
I R --+ kxEf 3kr
----
(3 .2 4)
102 HEAT AND MASS TRANSFER IN PACKED BEDS
i t
i FIGURE 3 . 3
t
Same catalyst volume
t
Same flow rate
Reaction rate measurements with different sized columns.
Suppose, as illustrated in Figure 3.3, reaction rate measurements are to be made with several different sized cylindrical columns of packed bed reactors under the same reaction conditions (same volumetric flow rates, catalyst volume, temperature and pressure ). Everything is the same except for the column size or the fluid velocity. In Eq. (3.24), the intrapellet term, kxEf, is not affected by flow rate, but the mass transfer coefficient, kr, depends upon the fluid velocity. The mass transfer coefficient is low at low fluid velocity, but increases with an increase in fluid velocity (see Chapter 4). Therefore, the overall rate constants and the conversion are low at low fluid velocities in large columns, and high at high fluid velocities in small columns. However, if the size of the column is sufficiently small, or the fluid velocity is high enough, such that, the external diffusional resistance is negligible, then the overall rate constant becomes (3 .25)
Under such conditions, the overall rate constant and the conversion do not depend on the fluid velocity any more. This is the region in which the intrapellet diffusion/reaction is rate controlling. This may be further classified into chemical reaction control and/or pore diffusion control, as governed by the following equations:
In a chemical reaction controlling region for< 0.6
(3.26)
DIFFUSION AND REACTION 103
In a pore dzffusion controlling region K=
3(1- €b ) R
(D ekx ) 11
for¢>10.
2
(3. 27)
The region, where the overall rate constant depends upon the fluid velocity, is extrapellet diffusion controlling . The overall rate constant is:
In an extrapellet diffusion controlling region (3.28) It is also interesting to note that the overall rate constants are inversely proportional to the catalyst pellet size when diffusion, either externally or internally, is the rate controlling step. The intrinsic chemical reaction rate constant, kx, is expressed in terms of the activation energy, E, as: kX
=
( _!_)
k0 exp -
(3 .29 )
Rg T
where R g is the gas constant and Tis temperature. The overall rate constant, K, is also expressed as: (3.30) where E' is the activation energy based on the overall rate constant. There fore, from Eqs. (3.25), (3.29) and (3.30), we obtain
E'
E
RgT
RgT
ln K0--= ln (1- €b) + In k0-- + In Er. Differentiation of this with respect to - 1 / (RgT ) gives dln Er E'=E+ ----
(
1
d --
RgT
)
(3. 30a)
) 04 HEAT AND MASS TRANSFER IN PACKED BEDS
=£+
--
dinEr din -din dln kx
---( ) din k x
d-
With Eqs. (3 . 1 6) and (3. 29), we obtain
(
1 dinE r E' = E 1 +2 din
)
1
(3. 3 1 )
_ _
R gT
·
(3 . 32)
From Eqs. (3. 20) and (3 . 2 1 ), Eq. (3.32) is simplified to:
E' = E E
E ' =2
for< 0.6 for > 10.
(3. 32a)
If the overall rate constants are measured in a pore diffusion con trolling region, an Arrhenius plot of the rate constants is expected to give half the activation energy of the intrinsic chemical reaction. This is illustrated in Figure 3 .4. At high temperatures, k xEr � 3kr/R and the reaction rate is extrapellet diffusion controlling. In this region, an Arrhenius plot of K will yield a small activation energy, of the order of about 1 0 kJ mol-1, corresponding to the value of kr. 3.2.2
Effectiveness Factors for Other Geometries
Effectiveness factors for non-spherical pellets in which a first-order chemical reaction is occurring under isothermal and steady-state condi tions are listed below: 3 . 2 .2 .1
Flat plate (of height L) with one side and edges sealed
I f the plate is contacted with reactants on one side only, then: Er=
---
tanh L L
(3.33)
DIFFUSION AND REACTION Extrapellet
Pore
diffusion
diffusion
control! ing
] 05
Chemical
controlling
react ion
control! inn
liT FIGURE
3.4
Arrhenius plot of overall rate constants.
3 .2.2.2 F1at plate (of height 2L) with edges sealed Jf the plate is contacted with reactants on both sides, then: Ef
3 .2.2.3
=
Eq . (3.33).
Infinitely long cylinder (of radius R) or finite length cylinder (of radius R) with ends sealed
I f the cylinder is contacted with reactants on the cylinder surface [ 1 3 ] , then: Ef �
-
2/ 1 (¢) ¢_1_o_(¢ -)
(3.34 )
where ¢ R(kxfDe)112; /0 and 11 are the modified Bessel functions of the first kind, respectively, of zeroth and first-order. =
106 HEAT AND MASS TRANSFER IN PACKED BEDS 3.2.2.4
Finite length cylind�r (of radius R and height 2L)
If the cylinder is contacted with reactants on all surfaces [ 1 4), then: (3 . 3 5 )
where U n- l)rr 2
p -
¢
R L 1
k )J/2
=R I .2.
\De
and im i s an m-th root of the following Bessel function of the first kind and zeroth-order:
3 .2.2.5
Ring (of inside radius Ri, outside radius R and height 2L)
If the ring is contacted with reactants on all surfaces ( 1 4] , then:
E= r
I-
8
1 [ I j:nk� � n=l �
1 -�2 m l
lo(�im) lo� ( im)
-loUn,)][ +
where
kn
=
(2n - 1 ) rr ----
2
R
p=L
loi ( m)
¢2
] ¢2 i�z (knP)2 +
+
(3.36)
D IFFUSION AND REACTION
)t/2 k ( x ¢=R -
1 07
De .
R· {3 = R I
and jm is an m-th root of
where Y0 is the Bessel function of the second kind and zeroth-order. For a ring (with sealed ends) contacted with reactants on the inside and outside surfaces [ 1 4], then: 2
+ E = r ( l - {32 ) ""-+.
{ [Ko(f3¢) - Ko(¢))
[It(¢) - f3/t(f3¢)) + [fo(¢) - /o(f3¢)) [{3K t(f3¢) - K t(¢) ] } (3.3 7 ) Ko(f3¢) /o(¢) - fo(f3¢) Ko(¢)
------
where K 0 and K 1 are modified Bessel functions of the second kind of zeroth and first-order, respectively. For a ring (with sealed ends and inside surface) contacted with reactants on the outside surface [ 14], then:
Er =
2
[ It(¢) K t(f3¢) - ft(f3¢) K t(¢) ] .
( 1 - {32) ¢ fo(¢) Kt(f3¢) + /t(f3¢) Ko(¢)
(3.38)
For a ring (with sealed ends and outside surface) contacted with reactants on the inside surface [ 14], then:
Er =
[
]
2 {3 It(¢) Kt(f3¢) - ft(f3¢) K t (¢) . ( l - {32) ¢ ft(¢) K0({3¢) + 1o(f3¢) K 1 (¢)
(3 . 39 )
3 . 2 . 2 . 6 Sphere (of radius R ) with the catalyst coated on an inert solid sphere (of radius R i) [ 14]
Er =
[
coth [( 1
- {3) ¢] + {3¢ I ( I - (P) ¢ 1 + {3¢ coth [(I - {3) ¢) ¢ 3
]
where ¢ and {3 are the same as those defined in Section 3 . 2 . 2 . 5 .
(3.40)
1 08 HEAT AND MASS TRANSFER IN PACKED BEDS
The effectiveness factors for extruded catalysts with cross-sections, such as a letter L, a dumb-bell, a trilobe or a quadrulobe, have also been numerically computed by Suzuki and Uchida [ 1 5 ) . Low effectiveness factor means that the reaction takes place in the vicinity of surfaces exposed to the reactants. Aris [ 13] has shown that Eq. (3 . 33) holds for any shape of catalyst pellet at high values of ¢, if L is taken as the ratio of the pellet volume to the area of surfaces exposed to the reactants. 3. 3
Pore Diffusion of Gases
In porous media, diffusion of non-adsorbing inert gas occurs through the pore volumes. There are two types of pore volume diffusion. I f the pore is large, normal diffusion due to molecule-molecule collisions takes place. I f the pore diameter is smaller than the length of the mean free path of the gas molecules, diffusion proceeds by molecule-wall collisions, and is called Knudsen diffusion [ 16 ] . If the pores are filled with liquid, bulk liquid phase diffusion is the only transport process. For physical adsorbing species, diffusion takes place not only through the pore volumes, but also along the pore surfaces. The physically adsorbed molecules migrate on to the pore walls. The amount adsorbed is in equilibrium with its gas phase concentration. and consequently, pore volume diffusion and surface diffusion both proceed in the direction of decreasing gas phase concentration in the pore. At low temperatures, the mechanism of adsorption is predominantly physical. Under such conditions, overall intrapellet diffusion is largely due to surface diffusion. For catalytic reaction at temperatures higher than the boiling points of the reacting species, surface diffusion is generally con sidered to be of little importance [ 1 , 2]. 3.3.1
3 .3 . 1 . 1
Diffusion in a Capillary Tube
Knudsen diffusion
Consider the steady-state countercurrent diffusion of two inert gases, 1 and 2, in a capillary tube (radius a and length L ) in which the pressures at both ends are kept constant . I f the diameter of the capillary tube is smaller than the length of the mean free path of the species, diffusion takes place only by molecule-wall collisions. There is little chance of
DIFFUSION AND REACTION 1 09
molecule-molecule collisions occurring in the capillary tube, and molecules collide only with the capillary tube wall. I f the concentration of the species under consideration is higher at one end, say the inlet end (x = 0), than the other, outlet end (x = L), molecules collide with the tube wall most frequently at the inlet end. The frequency of collision decreases with increasing distance of x , and then the species diffuse in the capillary tube from the inlet toward the outlet. In the Knudsen region, the diffusion flux is proportional to the differ ence in concentration between the ends of the capillary tube, but inversely proportional to the tube length. The molar flux for species 1, 1 1 , across a distance, L , in the direction of increasing x is (3.4 1 )
where .Ll C1 = (C1)x = O - (C1 )x = L and DK1 is the Knudsen diffusivity of species 1 . In general , Knudsen diffusivity, DK, is related to the mean molecular velocity, v, and the capillary tube radius, a , by the following equation: 2va DK = - · 3
(3 .42)
For an ideal gas of molar mass, M, the velocity is (3.43)
Thus (3 .44)
In SI units, the above equation is given as:
( T )1n DK = 3 .068a ,M
where D K i s i n m2 s- 1 , a i s i n m, T is in K and M is in k g mol- 1 •
(3.44a)
1 10
HEAT AND MASS TRANSFER IN PACKED BEDS
Similarly. the flux of species 2 in the Knudsen diffusion region is
(3.45) Equations (3 .4 1 ) and (3 .45) are always valid even if the total pressures at either end of the capillary tube are not equal. However. if the ends are at the same p ressure. the diffusion flux ratio becomes
(3 .46) In the Knudsen diffusion region, the molecules collide only with the capillary tube wall, so that there is no interaction between J 1 and 1 2 . However, if the capillary tube is large , the situation is completely different. Molecules of species 1 collide with molecules of species 2 as well as with its own kind. This is the normal or bulk diffusion.
3.3 .1 .2
Normal diffusion
In the normal diffusion region, the flux is expressed by
(3.47) where c 1 is the concentration of species 1 in a capillary tube, y 1 is the mole fraction of species 1, and D 12 is the binary diffusion coefficient for species 1 and 2 . The first term on the right hand side of Eq. (3.47) is Fick's first law of diffusion. In the second term, 1 1 + 1 2 is the total molar flux or diffusive flow of mixture of species 1 and 2 in the direction of increasing x, and y 1 (J1 + 12) is the transport rate of species 1 by the flow. The diffusion flux of species I , relative to a fixed coordinate system , is the sum of the Fick's diffusion flux and the flux carried by the diffusive flow . Similarly, the diffusion flux for species 2 is
(3.48)
DIFFUSION AND REACTION ) 1 1
Note that (3 .49) 3 .3 . 1 . 3
Combined Knudsen diffusion and normal diffusion
If both Knudsen and normal diffusion occur simultaneously then the diffusion flux may be expressed by the following form: 1
de
1
1
D 12
DK1
-+ -
Y t ( J 1 + J2)
-1 + ----
dx
D12 1 +DKl
(3. 50)
or 1
de 1
1-ay1
1
D12
DK 1
--- + --
dx
(3. 50a)
where (3.50b) Equation (3. 50) reduces to Eq. (3.47) for normal diffusion when the capillary tube is large, or DK 1 ;p D 12. When DK 1 � D 1 2 , or the capillary tube is small and diffusion is of the Knudsen type, Eq. (3. 50) becomes (3 . 5 1 )
Equation (3 .4 1 ) is an integrated form of Eq. (3 . 5 1 ) over a distance L . Equation (3 . 50) was derived in 1 96 1 by Evans et a!. [ 1 7 ] t o describe the diffusion processes from normal to Knudsen diffusion through a transition region. Jn the diffusion equations based on Chapman-Enskog kinetic theory for a multicomponent system, Evans et al. assumed that one of the components consisted of giant gas molecules which were
1 12
HEAT AND MASS TRANSFER IN PACKED BEDS
uniformly distributed and fixed in space. Thus, from the dusty gas model, they obtained Eq. (3 . 50) for a binary gas system. Scott and Dullien [ 1 8 ] obtained Eq. (3 .50) from momentum transfer arguments. Rothfeld [ 1 9] also derived, from similar momentum transfer discussions, a diffusion equation for gas in porous media, which reduces to Eq. (3 . 5 0) in the case of diffusion in a single capillary tube. Rearranging Eq. (3.50) for species 1 gives (3 . 52) Similarly, for species 2 (3 . 5 3 ) Since D 12 = D21 , addition o f Eqs. (3.52) and (3.53), at constant total pressure, gives (3 . 54) which further reduces to Eq. (3 .46). The ratio of constant pressure dif fusion fluxes is, thus, always governed by the relationship in Eq. (3.46), irrespective of the modes of diffusion: Knudsen, transition or normal.
3.3 . 1 .4
Selfdiffusion in a capillary tube
Self-diffusion occurs in a single species system. The flux of self-diffusion is usually measured by the countercurrent diffusion of isotopes technique. I f J1 is the flux of the isotopes or labeled molecules, and J2 is the flux of the unlabeled molecules; since M 1 M2 , Eq. (3 .46) shows that J 1 + J2 0 . The flux of self-diffusion in a capillary tube i s then: =
=
(3 . 5 5 )
DIFFUSION AND REACTION 1 1 3
where 1 1 1 -=-+-· D 1 D 1 1 D Kl
(3 .55 a)
As a matter of fact, Bosanquet [20] obtained Eq. ( 3 . 5 5a) by assuming that the resistance to transport is the sum of the resistances due to both molecule-wall collisions and self-diffusion. In Eq. (3.55a), D 1 1 is the self diffusion coefficient defined as: (3. 56 ) where A 1 is the length of the mean free path of the molecules of species 1 . Wheeler [ 1 , 2 ] developed an exponential combining formula for D 1 : (3.57) where, from Eqs. ( 3 .42) and (3 . 56), i t is shown that (3 . 57a) Pollard and Present [2 1 ] carried out elaborate calculations for D1 from molecular theory; the result is
( --
D 1 = D1 1 I
-- )
3 A1 6 A1 -+ Q rr a 8 a
'
(3.58)
where
in which
'Y
is Euler's constan t . As shown in Figure 3 . 5 , the Bosanquet [20]
1 14
HEAT AND MASS TRANSFER IN PACKED BEDS \
\
\
.....-1
\ ,
\/Wheeler
�
�t + -P 1 o r dloI:------ se-nr e� O. 5 f---��.-+-P�
'
c.
2 FIGURE 3 . 5
Q/). 1
3
---
5
Comparison o f self-diffusion coefficient formulae.
formula obtained by the additive resistance law is in surprisingly good agreement with the rigorous expression of Pollard and Present. 3.3.2
Diffusion in a Porous Solid
Effective diffusivity, D e � equation:
m
a porous solid is defined by the following
b. C N = AD e L
(3 . 59)
where N is the diffusion rate, A is the cross-sectional area perpendicular to the direction of diffusion and b.C is the difference in concentration across the distance L .
3.3.2.1
Measurement of effective dzffusivity in a porous solid
The Wicke-Kallenbach [22, 2 3 ] type of apparatus shown in Figure 3 .6 is widely used for effective diffusivity measurements in a porous solid. When gas 1 and gas 2 sweep across the top and bottom of the cylinder of a porous solid under constant total pressure, countercurrent diffusion takes place through the pores of the cylinder. From measurements of concentra tion and flow rate of exit streams under steady-state conditions, the steady countercurrent diffusion rate is determined. The effective diffusivity defined by Eq. (3. 59) is then evaluated.
DIFFUSION AND REACTION Porous solid
Gas 1
Gas 2
Flow
1 15
meter
measurement
'--- Mano�ter
fiGURE
3.6
Wicke-Kallenbach type of apparatus for steady countercurrent diffusion measurements.
For a cylindrical pellet, the cross-sectional area is a constan t , and the concentration profiles of the diffusing species in the pellet are linear and parallel under steady-state conditions. For spherical particles, however, the cross-sectional area varies, and as the calculations made by Kaguei et al. (24] show: the steady-state concentration profiles are nonlinear, as shown in Figure 3.7. This makes it rather difficult to define A in Eq. (3 . 59). Kaguei et al. (24] devised a method for the evaluation of the effective diffusivity in spherical pellets. I n their method, they suggested that the pellets should be glued into the holes made through a plate with a thickness less than the diameter of the pellet. The ends of the pellets sticking out of the plate should then be shaved off, as shown in Figure 3 . 8 . They recommended that the average cross-sectional area, which should be taken as A in Eq. (3. 59), should be found from Figure 3 . 9 . In the graph A 1 and A 2 are the cross-sectional areas o f the pellet exposed on either side of the plate. Also, A0 is the cross-sectional area of the spherical pellet a t the equator, or A0 rrR 2 , where R is the radius of the pellet. In the range where both A 1 /A0 and A 2 /A0 are greater than 0 . 3 , the curves shown in Figure 3 . 9 are expressed within an error of I % by the use of log-mean areas, A � and A�, defined as follows: =
1 1 6 HEAT AND MASS TRANSFER IN PACKED BEDS c
FIGURE
3.7
=
,
Steady-state concentration profiles in spherical pellet with sides shaved off.
A=
L L t L2 -+A; A;
(3.60)
where
A; =
A0 -At In
A; =
(��)
A o - A2 ln
(3 . 60a)
A) L:
and L 1 and L 2 are the distances between the center of the sphere and the plate surfaces (refer to Figure 3 . 8 ) ; L = L1 + L 2 . With the mean area, A , the effective diffusivity, . D e , for a spherical pellet is determined from measurements of the countercurrent diffusion rate, N, and the difference in concentration, D.C, across the distance L .
DIFFUSION AND REACTION A
l
Porous so l i d
--..----,·············· ········· ............ ·········· ..---
/
L
FIGURE 3.8
0.2
A 2
Sphere with sides shaved off.
----4---+--4---�
�Q2 FIGURE 3. 9
0.4
A1 /Ao
0.6
0.8
Mean diffusion area of a spherical pellet.
1.0
1 17
1 18
HEAT AND MASS TRANSFER IN PACKED BEDS
3.3. 2 . 2
Prediction of effective diffusivity from the proposed models
Wheeler's [ 1 , 2] parallel pore model assumes that the porous structure can be represented by a number of parallel capillaries of the same size. If there are n such capillary tubes of radius, ii, and length, Le, in a unit mass of porous solid of height, L , the total surface area of the tubes is Sg =n21TiiLe, and the total tube volume is Vg = n1r(a? Le. The model further assumes that Sg is equal to the internal surface area usually measured in a BET apparatus, and Vg is equal to the pore volume of the solid. The mean pore radius is then calculated by a = _
2 Vg -· Sg
(3 . 6 1 )
The diffusion flux in a pore of mean radius, a, is given by Eq. (3. 62 ). 1
dc 1
1 - ay 1 I --- + --
dx
D12
where -
DKl
=
J5K I
2iJt ii -- ·
3
(3.62 )
(3.62a)
The diffusion rate per unit solid area, le, is related to the diffusion flux , J , through a straight capillary tube of radius ti and length L by: (3 .63) = oJ
(3.63a)
where ; and o are the tortuosity factor and diffusibility, respectively. The tortuosity factor and diffusibility have been studied by many investigators. Wheeler [2] suggests that the diffusion rate per unit solid area is (3.64)
DIFFUSION AND REACTION 1 1 9
where m is the number of pores per unit solid area. The porosity of the solid, Ep, is equal to m1T(ii)2 L e/L , so that Eq. (3. 64) becomes (3.65) Moreover, Wheeler assumes that the pores intersect any plane at an angle of 7T/4 rad, or Le/L = 2 1 12 . The tortuosity factor of E q . (3.63) is then 2 . On the other hand, the random pore model o f Wakao and Smith [ 2 5 ] , which will be described later, predicts 0 =
€� .
(3.66)
The diffusion flux is expressed in terms of effective diffusivity as: (3.67) where Det = o
1 1 - ay1
1
-----
---
Dn
(3.67a)
+ -=-DK1
The diffusion flux measured in a Wicke-Kallenbach apparatus corresponds to the one obtained from the integration of Eq. (3.67):
(3.68)
where P is the total pressure. The effective diffusivity is then D 12 1 + -=-- - a(yl )out DKl
(3.69 )
1 2 0 HEAT AND MASS TRANSFER IN PACKED BEDS Similar to Eq. (3.46), the diffusion fluxes through a porous solid at uniform total pressure are related by Eq. (3 .70). (3.70) In the earlier work, Eq. (3 .70) was considered to hold only in the Knudsen region, whereas in the normal diffusion region the assumption of equimolar countercurrent diffusion according to Eq. (3.7 1 ) had often been used.
lel + le2 = 0 .
(3.7 1 )
This invalid assumption could have arisen due to confusion with diffusion in a closed vessel. Hoogschagen [26], Henry et al. [27], Evans et al. [28] and Scott and Cox [29] measured both fluxes in steady countercurrent diffusion. They all observed that the measured fluxes were inversely proportional to the square roots of the molecular weights. Hoogschagen showed that the pore sizes of the tablet used in their experiments were much larger than the length of the mean free path of the gas molecules. Evans et al. and Scott and Cox also showed that their data were obtained within the normal diffusion region . Based on the assumption that normal diffusion would be equimolar, Henry et al. regarded their experimental data as proof of Knudsen diffusion. For porous solids having monodisperse pores, the diffusion flux may be estimated by Eq. (3 .63). However, pelleted type materials prepared by compressing particles of catalyst powder have a bidisperse pore structure. The powder particles themselves are microporous, but the spaces between the powder particles are macropores. Usually micropores are defined as pores of radius less than about 1 0 nm and macropores are larger than this. In most cases the pore size distribution of micropores is measured by low temperature nitrogen adsorption [30] and that of macropores by mercury porosimetry [3 1 ] . In both methods, the pores are assumed to be interconnected cylindrical capillaries. In 1 962 Wakao and Smith [25] presented a model, which is known as the random pore model, for the estimation of effective diffusivity of a bidisperse pore structure. Figure 3 . 1 0 is a diagrammatic representation of the system used to describe the model. The dotted squares represent
DIFFUSION AND REACTION 1 2 1
powder particles having micropores and the spaces between the squares represent the macropores. Writing the volume fractions of the macropores, micropores and solid as Ea , Ei and E5, respectively, gives Ea + Ej
FIGURE 3.10
+ €5 = l .
(3.72)
Random pore model of Wakao and Smith [ 2 5 ] for a bidisperse porous solid.
Suppose the sample is cut at a plane and the two surfaces are then rejoined. If the void area fraction on each of the two surfaces is E, the void area fraction on the plane rejoined at random will be the possibility of two successive events, or E2. Therefore, the diffusion through a unit area of the rejoined plane can be divided into three additive parts, parallel mechanisms:
Mechanism 1 Diffusion through the macropores with an area of Ei and average pore radius a a. Mechanism 2 Diffusion through the microporous particles having an area of ( l - Ea)2 and an average pore radius ai. Mechanism 3 Diffusion through the macropores and micropores in series. The area for this contribution is 2Ea( l - Ea).
1 22
HEAT AND MASS TRANSFER IN PACKED BEDS
The diffusion rate per unit cross-sectional area of the porous media is then
Mechanism 1 Je = - e 2n a'-'a
de -
dx
Mechanism 2 - ( 1 - €a )2D·1
de -
dx
Mechanism 3 - 2 ea ( 1 - €a )
(3.73)
The diffusivities , Da and Di , for the macropores and micropores, are given as an example for species 1 by 1
D t a = ---- --l - ay
Dl2
�+
---
1 - ay 1
D 12
1
--
.f5Kt , a
(3.73a)
1 +
1 f5Kl , i
where .i5K1 , a and i5K 1 , i are the Knudsen diffusivities for species 1 in macro pores and micropores, respectively: .i5 K 1 , a = 2tJ 1 0.3/3 and i5 K 1 , i = 2tJ 1 ad3. By evaluating aa and ai from the pore size distribution, Wakao and Smith showed that the diffusion fluxes, experimentally measured using alumina pellets in a He-N2 system, were explained well by Eq. (3.73). The random pore model of Wakao and Smith was later extended by Cunningham and Geankoplis [32] to a more complicated tridisperse pore structure. Foster and Butt [33] proposed a model for computing effective dif fusivity from pore size distribution. The model considers the void volume in porous media to be composed of two major arrays of conical ducts, shown in Figure 3 . 1 1, which are made up of straight cylindrical capillary segments. One of the ducts is narrowest at the center of the solid and the other widest at the center. These two ducts, centrally convergent and
DIFFUSION AND R EACTION
1 23
converging Array
Diverging Array
FIGURE 3.1 1
Convergent-divergent pore array model of foster and Butt [ 33 J.
centrally divergent, are the inverse of each other. The shape of these ducts is rletermined from the pore size distribution. Diffusion flux through the ducts is estimated by trial and error calcula tion. By using an assumed value for the flux, the change in concentration across each capillary segment may be estimated from Eq. (3. 50a) or its integration. For the i-th segment of length L;, it is shown that
(3.74) Starting with the concentration at one end of the solid sample, the con centration change is computed for each capillary segment. Mixing between the two ducts is assumed to take place at various points. The computation is carried out along the total length of the two ducts. The correct flux is then determined by comparing calculated and given concentrations at the end. Johnson and Stewart [34] also presented a method for predicting the rate of diffusion through a porous solid. The rate of diffusion in each pore is calculated based on the dusty gas model of Evans et al. [ 1 7 ] and the total diffusion rate is then evaluated by integration over the entire range of pore size distribution. However, they stated that, because of possible anisotropy and other related effects, a diffusion or Knudsen permeability measurement was needed for accurate predictions.
1 24
HEAT AND MASS TRANSFER IN PACKED BEDS
3.3.2.3
Effective diffusivity and surface diffusion
For adsorbent particles, the contribution of pore volume diffusion may be estimated from pore size distribution. However, the effective diffusivity for an adsorption system consists of not only pore volume diffusion, but also pore surface migration. For a strong adsorbent, surface diffusion makes a larger contribution to the total transport. A considerable amount of work has been reported on the measurement of surface diffusion in porous solids. I n the earlier work of Babbit [35], and Gilliland et a/. [36], a two-dimensional spreading pressure working on the adsorbed layer was regarded as the d riving force for surface diffusion. However, in recent work, a model based on the random hopping of adsorbed molecules between adjacent sites has been employed more fre quently, for example, in the work of Higashi et al. [37], Smith and Metzner (38], Weaver and Metzner [39], Gilliland et al. [40] and Ponzi et a/. (4 1 ] . I t was also pointed out by Thakur et a/. [42] that the effect of collisions between gas molecules and mobile molecules in the adsorbed phase should be taken into account in evaluating surface diffusion flux. However, it seems that at present a satisfactory prediction of surface diffusion for every gas-solid system is still far too difficult. Surface diffusion should be measured in a constant pressure Wicke-Kallenbach apparatus with both non-adsorbing (molecular weight M , diffusion flux N) and adsorbing (molecular weight Ma , diffusion flux Na) gases. The pore volume diffusion flux measured for the non-adsorbing gas is then corrected for the molecular weight of the adsorbing species. Subtraction of this from the total diffusion flux measured for the adsorbing species gives the surface diffusion contribution of the adsorbing species: Surface diffusion = N3 - N
)' 1 2 · ( M
Ma
I f the permeability measurements are made under Knudsen diffusion conditions with an inert gas and an adsorbing gas separately, the surface diffusion of the adsorbing species may also be evaluated by the correction for the molecular weight ratio. The permeability, or forced-flow measure ment, should be made completely in Knudsen diffusion region. Therefore, this method cannot be applied to porous solids with macropores. How ever, this restriction is not imposed on a constant pressure countercurrent diffusion measurement . Effective diffusivities comprising of pore volume and surface diffusion may be determined directly from a chromatography measurement as well. As a matter of fact, effective d iffusivities of adsorb-
DIFFUSION AND REACTION 1 25
ents, particularly of strong adsorbents, are easily determined from adsorption chromatography measurements at various flow rates. 3.3.2.4
Effective diffusivity in multicomponent systems
Rothfeld [ 1 9] has shown that Eq. (3 .50a) can be extended to a multi component system as: d cm Jm = -Dm dx
(3 .75)
where 1
Dm = ------- Yn Ym I m -- + Dmn D Km r n
-(�•)
(3.75a)
in which Dmn is the binary molecular diffusion coefficient for species m and n . I f a chemical reaction with the stoichiometry (3.76) where
am > 0 for reactants am < 0 for products proceeds in a porous catalyst, the reaction rates, rv , or production rates of the reacting species are related as follows:
- (rv) I
ai
= .. .=
- (rv)m
am
= . . . = R v.
(3.77)
Rv is often referred to as the reaction rate based on stoichiometry. The diffusion fluxes are related to the stoichiometric coefficients of the reaction, e .g. -
=
- ·
(3.78)
1 26
HEAT AND MASS TRANSFER IN PACKED BEDS
The diffusion flux in a porous solid having a monodispersc pore struc ture, for example, is then
d cm = lem -Dem- dx
(3.79)
where
Dem
=8
1
(3.79a)
---
Also, it should be noted that the diffusivity of m-th component through the so-called external film on a catalyst pellet, in which the chemical reaction of Eq. (3.76) takes place, is given by Eq. (3.75a) in conjunction with Eq. (3.78) and = • Thus
DKm
(Dm )ext
film
=
00
1 -------
' Yn _ (·aamn ) m n L
(3.80)
y
There is a difference in the flux ratio between the two systems under reactive and inert conditions: the ratio is governed by Eq. (3.46) in con stant pressure countercurrent diffusion and by Eq. (3.78) in diffusion with a chemical reaction. A question arises, i.e. are the two effective dif fusivities under reactive and inert conditions the same, even if y 1 , for example in a binary gas system, is so small that the flux ratio effect is of no significance. The answer is that they will be the same provided that the mean pore radius under the reaction conditions is identical to that under inert conditions. Balder and Petersen [43] measured the reactive effective diffusivities from experiments on the hydrogenolysis of cyclopropane on a platinum/ alumina catalyst and found them to be almost the same as the non-reactive diffusivities measured for the same gas system. A similar conclusion was also reached by Toci et [ 44], who studied hydrogenation of ethylene on a nickel/diatomaceous earth catalyst.
al.
DIFFUSION AND REACTION 1 2 7
Ryan et al. [45] developed a theory for the calculation of components of an effective diffusivity tensor under reaction conditions in a spatially periodic porous media. They showed that the effective diffusivities should be independent of the rate of chemical reaction.
3.4
Jiittner Modulus for First-order Reversible Reactions
Suppose a reversible chemical reaction,
aA � bB
(3 . 8 1 )
with first-order kinetics, (3.82) where Keq is the equilibrium constant, takes place in the presence of inert component, I , in a spherical solid catalyst under isothermal con ditions. Writing Eq. (3 . 1 1 ) for each species as follows:
( c)
DeA d ' 2 d A - - r - + (rv)A = O dr r2 d r
(3 .83)
des Des d - - r2 dr r2 dr
(3.84)
( -)
+
(rv)n = 0.
Also, from Eq. (3 .77) (rv )A
- --
a
=
(rv)s
--
b
(3.85)
therefore, (3 . 85a)
1 28
HEAT AND MASS TRANSFER IN PACKED BEDS
I f cA ' c8 � c1 , the effective diffusivities, D eA and D e8, are approxi mately independent of cA and c 8 . With these assumptions, Eq. (3 . 85a) is integrated to give (3.86) where CAs and Css are the concentrations of A and B , respectively, at the pellet surface. The reaction rate is then rewritten as: (3.87) where
) ( + ------bDeA
Css cAe -
aDe B
Keq
+
CAs
aDeB
The Hi ttner modulus is, therefore , modified as: ¢R
(3.87a)
bD �
= R [kx ( -+ 1
DeA
b
)] 112
aDesKeq
·
(3. 88)
The catalyst effectiveness factor for a first-order reversible reaction is evaluated from Eq. (3 . 1 9) with the modified modulus. In the case of a = b and DeA � De s , the modulus reduces to ¢R
k ( 1 ] 1/2 [ x =R _ 1 + - ) . DeA
Keq
(3.89)
For catalyst pellets with a bidisperse macropore/micropore structure, Carberry [46] introduced a concept of macropore and micropore effective ness factors, and developed a theory for an overall pellet effectiveness factor expressed in terms of the macropore and micropore effectiveness factors.
DIFFUSION AND REACTION 1 29
Example 3.1 Ethylene was hydrogenated at 1 30°C and 0 . 1 MPa in a 1 . 5 em diameter differential reactor with fifty 0.26 em spherical pellets of nickel/silica catalyst. The feed was a mixture of 1 0 cm3 s- 1 C2 H4 , 90 cm3 s- 1 H2 and 1 00 cm3 s- 1 N2 at 20°C and 0 . 1 MPa. The reaction product, ethane, in the exit stream was found to be 0 . 5%. The catalyst pellets are believed to have monodisperse pores of average diameter 50 nm. The intraparticle void fraction is 0.46. Estimate the catalyst effectiveness factor and intrinsic chemical reaction rate constant. SOLUTION
Let us use the following notation:
Molar flow rate Reactor inlet
Component
Reactor outlet
------ -· - - -
C2 H4 : A H2 : B C2 H 6 : s N2 : I
total
FA J Fs J Fs t = 0 Fu
FA2 Fs2 Fs2 F12 = F1 1
FI
F2
Fv1 and Fv2 are the volumetric flow rates at reactor inlet and outlet,
respectively. The reaction stoichiometry is
A + B = S. The changes in the number of moles arc
and then
(3.90)
1 30 HEAT AND MASS TRANSFER IN PACKED BEDS (i) Reactor inlet The inlet molar flow rates arc FA I = F Bl = FI
I
=
( 1 0) (273) (22 ,400) (293) (90) ( -? 73) (22 ,400) (293) ( 1 00) (273) (22 ,400) (293)
=
4 . 1 6 x 1 0 -4 mo l s- 1
= 37.4 x 1 0-4 mol s- 1 =
41.6
x
1 0-4 mol s- 1 .
The total rate is F1
=
(4 . 1 6 + 3 7.4 + 4 1 . 6) x 1 0-4
=
8 3. 2
x
I 0-4 mol s - 1 .
The mole fractions are YA l = 0.05 YB1 = 0.45 y 11
= 0.50.
The volumetric flow rate at 1 30 °C and 0 . 1 MPa is
The ethylene concentration is CA l =
(ii)
4.16
X
10-4
2 .75
X
10-
4 =
1 . 5 1 mol m-3.
Reactor outlet F � F2 F2
=
0 . 005
=
F1 - Fs2 = 83 . 2
x
1 0-4 - Fs2
DIFFUSION AND REACTION 1 3 1
therefore , Fs2 = 0 .4 1 4
10-4 mol s- 1 1 F2 = 8 2 . 8 x I 0-4 mol sx
and FA2 = 3 .75 F82
=
x
1 0-4 mol s-1
37.0 x 1 0-4 mol s- 1 .
The mole fractions are FA2 YA2 = - = 0 . 045 F2 Fs2 YB2 = - = 0 .447 F2 Ys2 = 0.005 Fu y12 = - = 0 . 5 02 . F2
For the differential reactor it is clear that
therefore,
1 32
HEAT AND MASS TRANSFER IN PACKED BEDS
The concentration of reactant A is
FA2 3.75 X I 0-4 = l.37mol m- 3. CA2 = -= 2.74 x 1 o-4 Fv2
(iii ) Calculation of (R p)A The reaction rate throughout a single catalyst pellet is
(R )A = P
FA2 -FA I
number of pellets 0.41 x 1 o-4 50
---- =
-8 .2 x 1 0- 7 mol s-1.
The average concentration of reactant A in the differential reactor is
The volume of a single pellet is
therefore, from Eq. (3 .22) , we find 1 R -- + kxEf 3kr
(
4rrR 3 ) (CA )av 3
- (R p)A
= ------
=
(9 .2 X 1 0-9 )( 1. 44) (8 .2 x 10-7)
= 0.016s.
DIFFUSION AND REACTION 1 33
(iv) Estimation of k r From Eq. (3 .80) the diffusivity of ethylene in the external film of catalyst pellet for the reaction of Eq. (3. 90) is (3. 9 1 )
The binary gas diffusivities a t 1 30°C and 0 . 1 MPa are estimated t o be
2 2 2
DA B = 0 . 9 1 6 X 1 0-4 m s - 1 DAs = 0. 2 1 6 x 1 0 -4 m s- 1 DAI = 0.284
X
1 0 -4 m s- 1 .
The composition varies to some extent across the external film of a catalyst pellet. But , if we assume that the y values in Eq. (3.9 1 ) can be equated with the average mole fractions of the bulk gas in the reactor, then, the average mole fraction of gas A is YA =
YA I + YA2 = 0 . 048 . 2
Similarly YB
= 0.449
Ys = 0.0025
and YI
=
0.501
therefore,
(DA) ex t film =
1 0. 449 - 0 . 04 8 0.0 02 5 + 0 . 048 0.501 ----- + + ----0.9 1 6 X ] 0-4 0.2 1 6 X 1 0-4 0.284 X 1 0-4
1 34 HEAT AND MASS TRANSFER IN PACKED BEDS 1
1
------ - ------
=
2 . 436 X 104
(0.438 + 0 .234 + 1.764) X 104
1 4. 1 1 x t o - s m2 s- .
For calculating viscosity and density of the bulk gas in the reactor, the gas is assumed to be a mixture of H2 and N2 in the volume ratio 90:100. The viscosities of H2 and N2 at 130°C are fJH 2 = 1 .10 x 10-5 Pa s and fJN 2 = 2 .24 x 1o-s Pa s. From the chart of Bromley and Wilke [4 7], the viscosity of the gas mixture is then estimated to be fJ 2 .15 x t o-s Pa s . The densities of H2 and N2 at 0°C and 0. 1 MPa are PH 2 = 0.0898 kg m-3 and PN2 = 1.251 kg m-3, so that the density of the gas mixture at 1 30°C is =
PF =
2 73 (0 .0898) + (.!OO) ( l .251) 403 190 190
[(� )
]
= 0.475kg m-3.
The average volumetric flow rate is: (Fv 1 + Fv2)/2 = 2.745x I o-4 m3 s- 1 • The superficial gas velocity is then 2 .745 10-4 -1 u = 7T(0.75 2 2 = I .55m s x 1o - ) X
therefore,
Re =
2Rup F
fJ
=
and
Sc = From Eq. (4.1I )
fJ
5) (0 . 4 75) (2.6 X 10-3) (1.5 - = 89.0 (2 . 15x l 0-5)
PF(DA)ext film
- -
(2 .15x 10-5)
(0 .475) (4.11 X 10-5 )
= 1.10.
DIFFUSION AND REACTION 1 35
therefore, kr = (v)
( 1 8.8) (4. 1 1 x l 0- 5 ) --·-·----
(2 . 6 X 1 0-3)
= 0 . 297 m s 1 • _
Estimation of Er and kx ( 1 . 3 X 1 0 -3) = 0 . 0 1 6 - - = 0.0 1 6 - ---3kr kxEf (3) (0.297)
I
R
--
= 0 . 0 1 6 - 0. 00 1 46 = 0 . 0 1 454 s
therefore,
From Eq. (3 .79a), the effective diffusivity of ethylene in the catalyst pellet is DeA = o -----1 YI YB -YA Ys + YA ++ -- + DAs DAI DAB jjKA
(3 . 92)
The Knudsen diffusivity of ethylene at 130°C in pores of radius a= 25 nm is 15KA = 3 . 068a
(
T )112 MA
= (3 .068) (25 X 1 0-9)
(
403 )J/2 0.028
= 9 . 20 X 1 0-6 m 2 s-1 .
Average mole fractions in the catalyst pellet should be used for the y values in Eq. (3.92). However, if in place of y we simply take the average mole fractions of the bulk gas in the reactor calculated in (iv), and i f we assume o= E�, DeA = (0.46)2 ----- = 1 . 59 1 ---- + 2 . 436 X 1 04 9 . 2 0 X 1 0-6
X
1 0-6 m 2 s- 1
1 36 HEAT AND MASS TRANSFER IN PACKED BEDS
therefore, (kxEr)
R2
-
DeA
=
(68 .8)
(1.3
X
1 . 59
1 0-3? X
1 0 _6
=
73 . 1 .
The left hand side is ErcJ}. I f ¢ > 1 0 , Er = 3/¢, and Er¢2
73.1
= - =
3
=
3¢. Therefore,
24 .4 .
In fact, this shows that the condition ¢ > 1 0 is fulfilled. We can then see that the catalyst effectiveness factor, Er, is 3/24.4 0 . 1 2 3 , and the intrinsic chemical reaction rate constant, kx, is 560 s- 1 . =
REFERENCES [1] [2) [3] [ 4J [5] [6J [7] [8) [9) [ 1 0] [1 1] [12] (13] [ 14] [15] [16] [ 17] {18] [ 1 9] [ 20] [211 [ 22] [ 23 ]
A . Wheeler, Advances in Catalysis, Vol. 3 , Academic Press, New York (195 1 ). A . Wheeler, in Catalysis, edited by P. H . Emmett, Vol. 2 , Reinhold, New York ( 1 955). F. A. L . Dullien, Porous Media: Fluid Transport and Pore Structure, Academic Press, New York ( 1 979). R. Jackson, Transport in Porous Catalysis, Elsevier, New York ( 1 977). E . E . Petersen, Chemical Reaction Analysis, Prentice-Hall, New Jersey ( 1 965). C. N . Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, Massachusetts ( 1 970). J . M . Smith , Chemical Engineering Kinetics, 2nd edn., McGraw-Hill, New York ( 1 970). G . R . Youngquist, /nd. Eng. Chern. 62 (No. 8), 5 2 ( 1 970). F . Hittner, Z. Phys. Chemie 65 , 595 (1909). E . W. Thiel e , /nd. Eng. Chern. 3 1 , 9 1 6 ( 1 939). G . Damkohler, Der Chemieingenieur, Vol. 3, Akadem. Verlag., Leipzig, p. 430 ( 1 937) . J . B . Zeldowitsch, Acta Physicochim. URSS 1 0 , 583 (1939). R . Aris , Chem. Eng. Sci. 6 , 262 ( 1 957). S . Kasaoka and Y . Sakata, Kagaku Kogaku 3 1 , 164 ( 1 967). T. Suzuki and T. Uchida,J. Chern. Eng. Japan 1 2 , 425 ( 1 979). M . Knudsen, Ann. Phys. 28 , 75 ( 1 909). R . B . Evans, G . M . Watson and E. A . Mason, ]. Chern. Phys. 35, 2076 ( 1 96 1 ). D . S . Scott and F. A. L. Du llien , AIChE J. 8 , 1 1 3 ( 1 962). L. B . Rothfeld, A /ChE J. 9 , 1 9 ( 1 963). C. H . Bosanquct, British TA Rept. BR-507, September 27 ( 1 944), quoted in [ 2 1 ]. W. G . Pollard and R . D. Present, Phys. Rev. 7 3 , 762 ( 1 948). E . Wicke and R. Kallenbach, Kolloid Z. 97, 1 35 ( 1 94 1 ) . P. B . Weisz,Z. Phys. Chern. 1 1 , 1 (1957).
DIFFUSION AND R EACTION (2 4 ] (2 5] [2 6 j [2 ] 7 [ 28 1 [2 91 [ 30 1 [3 1 ] [ 32] [ 3] 3 ] [ 34 (35 ] [ 36 ] [ 37 ] [8 3] [ 39 ] [40] [4 1 ] [4 2 ] [43 ] [44 ] [45 ] ] [46 [471
1 37
S. Kaguei, K . Matsumoto and N. Wakao, Kagaku Kogaku Ronbunshu 6 , 2 06 ( 19 8 0). N . Wakao and J . M . Smith, Chem. Eng. Sci. 1 , 7 8 2 5 ( 1 96 2 ). J . Hoogschagen, Ind. Eng. Chem. 47, 906 ( 1 955). J . P. Henry, B . Channakesavan and J . M . Smith, AIChE J. 7 , 1 0 ( 1 96 1 ) . R. B. Evans, J . Truitt and G . M . Watson , ]. Chem. Eng. Data 6 , 5 2 2 ( 1 9 6 1 ). D . S. Scott and K . E. Cox, Can. J. Chem. Eng. 3 , 2 01 ( 1 960). R . P . Barrett, L . G . Joyner and P. P. Halenda, J. A mer. Chem. Soc. 7 3 , 3 7 3 ( 1 951). H . L . Ritter and R . C. Drake, Ind. t:ng. Chem., Anal. Ed. 1 7 , 8 7 7 ( 1 945). R. S. Cunningham and C. J . Geankoplis, lnd. Eng. Chem. Fund. , 7 5 3 5 ( 1 96 8 ). 1 0 ( 1 966). R . N. Foster and J. B. But t , A ICht: J. 1 2 , 8 M . F . L. Johnson and W. E. Stewart, J. Cata!. 4 , 2 4 8 ( 1 965). 449 A, 2 8 Res. J. an. C J . D . Babbit, ( 1 950). E . R . Gilliland, R. F . Baddour and J . L. Russcl , A IChEJ. 4 , 90 ( 195 8 ). K . Higashi, H . Ito and J . Oishi, f. Japan Atom. Energy Soc. 5 8 , 46 ( 1 963). R . K. Smith and A . B. Mctzner, J. Phys. Chem. 8 6 , 2 74 1 ( 1 964). J . A . Weaver and A. B. Metzner, AIChE J. 1 , 2 655 ( 1 966). E . R. Gilliland, R. F. Baddour, G . P. Perkinson and K. L. Sladek, Ind. Eng. 3 95 ( 1 974). Chern. Fund. 1 , M . Ponzi, J . Papa, J . B. P. Rivarola and G. Zgrablich, A IChE J. 23 , 347 ( 1 977). S . C. Thakur, C. F . Brown and G . L. HalJer, A!ChE J. 26 , 355 ( 1 9 8 0). 8 ). J . R. Balder and E. E. Petersen, ]. Catal. 1 1 , 195 ( 1 96 R . Toei, M . Okazaki, K . Nakanishi, Y . Kondo, M. Hayashi and Y. Shiozaki, 6 50 ( 1 973). J. Chem. Eng. Japan , D. Ryan, R. G. Carbonell and S . Whitaker, Chem. Eng. Sci. 35 , 10 ( 19 8 0). J. J . Carberry,A/ChE J.8 , 557 ( 1 962 ). L . A . Bromley and C. R . Wilke, Ind. Eng. Chern. 43, 1 64 1 ( 1 95 1 ).
4 Particle-to-Fluid Mass Transfer Coefficients
ONE OF the important parameters needed in the design of packed bed
systems is the particle-to-fluid mass transfer coefficient. In the past four decades, a substantial amount of work has been devoted to the study of this parameter. Particle-to-fluid mass transfer studies were first carried out by Gamson
eta/. [I]
and Hurt
[2],
both in
1943.
They obtained mass transfer coeffi
cients from measurements of the rates of evaporation of water from wet porous particles. Hurt
[2]
also reported mass transfer coefficients derived
from the measurement of rates of naphthalene sublimation. Since their pioneering work, a large number of experimental studies have been carried out on mass transfer coefficients in packed bed systems.
[3], and Pfeffer and [4] applied a free surface cell model to the creeping flow region. Le Clair and Hamielec [5-7] proposed a zero vorticity cell model, and El Kaissy and Homsy [8] applied the free surface cell model, zero vorticity Theoretical work has also been in progress. Pfeffer
Happel
cell model and distorted cell model to a multiparticle system at low Reynolds numbers. Nishimura and Ishii
[9]
also applied the free surface
cell model to the study of mass transfer at high Reynolds numbers. These models which are based on different assumptions. generally give different and inconsistent values of particle-to-fluid mass transfer coefficients. Therefore, theoretical prediction of transfer coefficients is far from satisfactory. When mass transfer occurs between a flowing fluid in a packed bed and the particle surface on which the concentration of the transferring species is constant, the resistance to mass transfer is considered to reside on the fluid side. In such a system, the unsteady mass balance equation of the transferring species, according to the dispersed plug flow model, may
138
MASS TRANSFER COEFFICIENTS
139
be expressed as: a2c
ac
ac
a
(4. 1)
-=D.ax --U---kr(C-C) ps 2 at aX aX €b where a=particle surface area per unit volume of packed bed
C= concentration of transferring species in the bulk fluid Cps =concentration of transferring species at the particle surface Dax =axial fluid dispersion coefficient kr =particle-to-fluid mass transfer coefficient U= interstitial fluid velocity Eb =bed void fraction.
In the experimental measurements of mass transfer coefficients, most investigators have chosen to ignore the dispersion effect. For instance, in the experiments conducted by Satterfield and Resnick
[ 10]
on the
catalytic decomposition of hydrogen peroxide in a packed bed of metal spheres, they obtained, for such a fast reaction, the mass transfer coeffi cients under the assumption of ideal plug flow, or no fluid dispersion in
2.2 (Figure 2.7), the stagnant E0/Dv, of the dispersion coefficient for a fast reaction (c/> =oo) is as large as 20. Under such conditions the axial fluid dispersion coefficient is given by Eq. (4.2): the bed. However, as discussed in Section
term,
EbD
� =20 +
Dv
As can be seen from Eq.
(4.) 1,
0.5(Sc)(Re)
for
Re > 5.
(4.2)
the values of the mass transfer coefficients
obtained under the assumption of ideal plug flow
(Dax =0),
therefore,
will be significantly different from those obtained using the large disper sion coefficients given by Eq.
(4.2).
Evaporation, sublimation and dissolution follow the same sequence of steps as a fast chemical reaction on a particle surface. Mass transfer takes place between the bulk fluid and the particle surface where the mass trans ferring species is at a constant concentration, and no intraparticle diffusion is involved in the overall mass transfer. The bed in which this type of mass transfer proceeds should therefore have a dispersion coefficient as
1 40 HEAT AND MASS TRANSFER IN PACKED BEDS
given by Eq. (4.2). Wakao and Funazkri [ 1 1 ] corrected the literature data for the axial fluid dispersion coefficient of Eq. (4.2) and obtained an empirical correlation for particle-to-fluid mass transfer coe fficients. A critical review of the published mass transfer coefficient data and their correction for axial dispersion effect are made in the folJowing sections.
4. 1
Review of the Published Gas Phase Data
The numerous packed bed mass transfer coefficients, reported in the literature, were obtained using various experimental methods under different conditions. For the purposes of data correlation, the following criteria have been adopted in the selection of the data: a)
The particles in the bed are all active. Distended and diluted bed data are not considered.
b ) The number of particle layers in a mass transferring bed are greater than two. Table 4 . 1 (for the gas phase) and Table 4.2 (for the liquid phase) list selected experimental work together with methods and operating conditions. 4. 1 . 1
Evaporation o f Water into an Air Strea m : Steady-state Measurements
Since Gam son et a!. [ 1 ] and Hurt [2] reported their results on the evapora tion of water into air in I 943, the same system has been repeatedly studied by many investigators. Mass transfer coefficients were determined from the rate measurements during constant rate evaporation. In the work of Gam son et al. [ I ] (Re = 1 00 to 4000), and Wilke and Hougen [ 1 2 ] (Re = 45 to 250), the particle surfaces were assumed to be at wet-bulb temperatures. Hurt (2] was the first to measure particle surface temperatures (for the two runs at Re = I 50 and 370). Galloway et al. [ 1 3 ] (Re = 1 50 t o I 200) found that the differences between the measured temperatures and the wet-bulb temperatures were less than 0 . 3°C. Bradshaw and Myers [ 1 4] (Re = 200 to 4000) observed, in some of their experi mental runs, that the surface temperatures were at wet-bulb values. However, De Acetis and Thodos [ 1 5 ] (Re = 60 to 2 1 00) pointed out that
MASS TRANSFER COEFFICIENTS 1 4 1
considerable temperature differences existed between the measured surface temperatures and the wet-bulb values when flow rates were low. Since the experimental findings of De Acetis and Thodos, subsequent studies carried out by Thodos and coworkers [ 1 6-2 1 ] on the determina tion of transfer coefficients were all based on the measured surface temperatures. In the work of Hougen et al. [ 1 , 1 2], in which they assumed wet-bulb surface temperatures, there is some reservation about the reliability of their results, particularly those obtained at lower Reynolds numbers, Re = 45 to 1 50. Nevertheless, their data are not substantially d ifferent from those of Thodos et al., which were determined based on experi mentally measured surface temperatures. Therefore, all the data obtained from studies of water evaporation are included in the correction and correlation section later in this chapter, provided that the information on bed height and void fraction, required for correction for axial fluid disper sion, is given. 4.1.2
Evaporation of Organic Solvents: Steady-state Measurements
The determination of mass transfer coefficients from the rates of evapora tion of organic solvents from particle surfaces into a stream of inert gases has been the subject of extensive investigation by Thodos and coworkers. The systems employed by Hobson and Thodos [ 1 6] are n-butanol, toluene, n- octane and n-dodecane in air, nitrogen, carbon dioxide and hydrogen� those of Petrovic and Thodos [ 1 9] are n-octane, n-decane, n- dodecane and n-tetradecane in air; and that of Wilkins and Thodos ·[22] is n-decane in air. In these studies, the mass transfer coefficients were determined based on the measured temperatures of the particle surfaces under steady-state conditions. 4 . 1 .3
Sublimation of Naphthalene: Steady-state Measurements
The rates of sublimation of naphthalene were measured by Hurt [2], Resnick and White [23], Chu et a!. (24], and Bradshaw and Bennett [25] in their determinations of mass t ransfer coefficients. Compared to the liquid-gas system, the naphthalene-gas system has the advantage that the adiabatic temperature drop is small. But, the disadvantage of the system is that the vapor pressure of naphthalene has not been thoroughly investigated.
TABLE 4 . 1 Gas phase mass transfer experimental data3•
......
t0
�
Year 1943
1943
1945
Steady or unsteady Experimental state Investigator method conditions Gamson et al. Evaporation of water [1 ]
Hurt [ 2 ]
Wilke and Hougen [ 1 2 ]
Steady
Evaporation Steady of water Sublimation Steady of naphthalene
Evaporation of water
Steady
1949
Resnick and White [ 2 3 ] b
Sublimation Steady of naphthalene
1 95 1
Hobson and Thodos [ 1 6 ]
Evaporation of water and organic solvents
Steady
Particle Material Celite
Naphthalene
Celite
Shape
Size (mm)
Sc
Fluid
Re
l:luid dispersion considered
2 . 3 , 3.0, 5.6, Air 8.4, 1 1 .6 Cylinder 4 . 1 X4.8, 6.8 x 8.5, 9 .8 X l 1 .7 , 1 4. 0 X 1 2.5, 1 8.8 X 1 6.9
0.6 1 -0.62
Cylinder 9.5 x 9.5
0.6 1
1 5 0 & 370
Air Cylinder 4 . 8 X 4.8, 9.5 X 9.5 H2 Flake 2.0, 2.8, 4. 1 ' 5 .6
2.5 4.0
7-670
Cylinder 3.1 X 3.1 , 4 . 8 X 4.3, 6.6 X 7 .2 , 9.7 x 8.6, 1 3.4 x 1 2.8, 1 5 . 1 x 16.3, 1 8 . 2 X l 6 .9
0.6
Sphere
Air
Air
No
:rl )> ....j )> z 0 s: )> (/l (/l ....j
45-250
No
:::0 )> z (/l
"Tl �
:::0 z '"0
)> n "" M
No
0
CD t'T1 0
(/l
Naphthalene
Granule 0.5, 0.8, 1.0, Air (ground) 1 . 1 C02 H2
Porous packing
Sphere
9.4
1004000
:t
Air, N2, C02, H2
2.39 1 .4 7 4.02
0.83-25
No
0.6 1 -5 . 1
8.6-330
No
Glass beads Lead shot Celite
1954
Satterfield and Resnick [ 10]
Decomposition of H202
1957
Galloway etal. [ 1 3 j De Acetis and Thodos ( 1 5 j Bradshaw and Bennett [ 25] McConnachie and Thodos r 1 81 Bradshaw and Myers [ 1 4 J
Evaporation Steady of water Evaporation Steady of water Sublimation Steady of naphthalene Evaporation Steady of water
Celite
Sphere 0.7 Air 2.57 Sphere 0.7, 1 .3, 2.0 Cylinder 5 . 3 , 5.5, 8.5, 1 3 .7 , 14.1 2.0 Vapor Sphere 5 . 1 0.7-0.9 mixture of H202 and water Air Sphere 1 7 . 1 0.6 1
Celite
Sphere
Air
0.61
60-2100
No
Naphthalene
2.57
440-9900
Yes
Celite
Air Sphere 9.5 Cylinder 6.4, 9.5, 1 2.7 Sphere 15.9 Air
0.61
1 00-25001:
No
Evaporation of water
Kaoline AMT Kaosorb Celite
Sphere Sphere Cylinder Cylinder
0.6
400-65 00c No
Celite
Sphere
1953
1960 1961 1963 1963
1963 1 964 1967 1968
Chu et a!. [24 j Sublimation Steady of naphthalene coated on particles
Sen Gupta and Thodos [20] Sen Gupta and Thodos 121] Mailing and Thodos [ 1 7 ] Petrovic and Thodos [ 1 9 ]
Steady
Steady
Rape seed Polished catalytic metal
1 5 .9
4.7 8.8 4.0 X 4 . 1 4.2X 4.2, 6.2 X 4.9 1 5 .9
Air
Steady
Evaporation of water
Steady
Evaporation of water
Steady
Celite
Sphere
1 5 .7-1 5.9
Evaporation Steady o f water and heavy hydrocarbons
Celitc
Sphere
1 .8, 2.2, 2.6, Air 3 . 1 ' 9.4
Sphere
1 5.9
No
1 5-160
No
150-1 200
No
$: >(/) (/) -l :;x:l
>z
Evaporation of water
('elite
20-2000
Air Air Air
0.61 0.61
800-2000
No
2000-6000 No
0.61
300-8500
Yes
0.6-5.45
3-230
Yes
(/) � rt1 :::c (")
0 tr'l � "11 (") rr.
z -l (/)
� w
-
TABLE 4 . 1 (Continued)
Year
Investigator
Steady or unsteady state Experimental conditions method
Particle Material
Shape
2.6, 3 . 1
Wilkins and Thodos [22]
Evaporation of n-decane
Steady
Celite
1974
Wakao and Tanisho [27]
Pulse response, nonadsorption
Unsteady
Vanadium Cylinder 3 . 1 X 4.7 diatomaceous Granule 1 . 1 earth (ground)
Miyauchi et al. [ 28]
Pulse chromatography, chemical reaction
Unsteady
Pulse chromatography, adsorption
Unsteady
Gangwal e t al. Pulse chromato[30] graphy, adsorption
Unsteady
1 976
Wakao ct al.
[29]
1 977
Porous packing
Sphere
Sphere
Sc
Size (mm) Fluid
1 969
1976
� �
0. 7, 1 .0, 1 . 2, 1 .4
1.1,
Re
rtuid dispersion considered
Air
3.72
1 5 0- 1 80
Yes
H2
1 .5
0.06-1 .8
No
C.1H8, N2, 0.5-2.0 H2, He
1 - 1 60
Yes
:r: tTl >-"'"l >z 0
s: > (/) (/)
-"'"l ::c
> z (/) �
(Tl :;c -
z
.,
>
Activated carbon
Silica gel
Sphere
2.2
H2
1.5
0. 1 - 1 .0
Yes
(") r:
tTl 0 t:l'
tTl 0
0.2
a Diluted beds, distended beds and data with a single particle layer are not included. b Criticized by Bar-Ilan and Resnick [26 ]. c Re = DpG/JJ. except for Refs. [ 1 4 ] and [ 1 8] where R e = DpG/[JJ-0 -eb) ] .
He
-
(/)
0.05-0.3
Yes
TABLE 4.2 Liquid phase mass transfer experimental data3.
Year
Investigator
Steady or unsteady state Experimental method conditions
Particle Material
Shape Sphere
Size (mm)
Fluid Water
Sc
Rc
Fluid dispersion considered
1 949
Hobson and Extraction of Unsteady Thodos [ 3 l ] b iso-butanol and methyl ethyl ketone
Celite
1949
M cCune and Willielm [ 3 2]
Dissolution of Steady 2-naphthol
1950
Gaffney and Drew [ 3 3 ]
Dissolution of Steady organic acids
3.2, 4.8, 6.4 Water 1 1 9 0-1 5 1 0 14-1 7 7 0 1 . 3, 2 . 1 6.4, 9.5, 1 2.4 Acetone 160-180 0.8-1480c Succinic acid Sphere Cylinder 6 . 3 n-bu tanol 1 0 1 001 3 300 Salicylic acid Sphere Benzene 340-430 9.6, 1 2 .9 Cylinder 6.3 Benzoic acid Cylinder 3.9,4.0, 4 . 1 , Water 1 1 70-1 6 1 0 1-1 80 4.2, 4.3, 5 .5, 6.2, 6.8 Benzoic acid Granule 0.6, 0.8, 1.4, Water 990-1 100 l-60 2-naphthol
lshino eta/.
[34]
Dissolution of Steady benzoic acid
1953
Evans and Gerald [35 )
Dissolution of Steady benzoic acid
1953
Dryden et al.
Dissolution of Steady 2-naphthol and benzoic acid
2-naphthol Benzoic acid
Dissolution of Steady lead
Lead
1 95 1
1956
[ 36 ] d
Dunn et a!.
[ 37 )
9.4, 1 6 . 1
780-870
3-35
Sphere Flake
(ground) 2 . I
Cylinder 6.3 Cylinder 6.3 Sphere
Water
2.0, 2 . 1 , 2.2, Mercury
4.4
8 1 0-1 150 0 . 0 1 3-7.2c 1 20--140
32-1500�.-
No
No No
s: > (/) (/) >-1
No
No No
No
� > z (/) '"rj tT1 ::c
n 0
rr. '"rj '"rj () tT1
z
...;
(/)
...... .+;:.. Vt
� 0\ :r: r:-: > -1 � z v
TABLE 4.2 (Continued)
Year
1 956
Investigator Selke et al.
[ 38 ]
Steady or unsteady state Experimental method conditions Ion exchan�e
Unsteady
Particle Material
Shape
Size (mm)
Amberlite
Sphere
0.4, 0.5 , 0.6, Copper 0.9 sulfate
IR- 1 20
Fluid
Sc
520 & 1 1 30
Re
Fluid dispersion considered
2.7-1 20 c
No
solution
1958
Wakao et a/.
1963
2-naphthol
Cylinder 8.0. 8.1, 8.5
Water
1460-1760 0.4-3000
No
Benzoic acid
Sphere
6 . 1 , 6.3
Water
940- 1 140 0.04-·53
et a/. [401
Dissolution of Steady benzoic acid
No
1966
Wilson and Geankoplis
Dissolution of Steady benzoic acid
Benzoic acid
Sphere
6.4
Water
Williamson
1969
1 975
Kasaoka and Nitta [ 42]
860-1 1 00 0.00 1 6- 1 1 5 2 300propylene 70 600 6CVir.-
[4 1 ]
No
glycol Dissolution of Steady benzoic acid coated o n particles
Upadhyay and Dissolution of Steady Tripathi [43] benzoic acid
Steel
Sphere
Benzoic acid
Cylinder 6.0, 7.7, 8 . 1 , Water
2.8, 4.1' 6.4 Water,
benzoic acid aqueous solution
8.6, 9.0, 1 1.2
.. -
:;o -
z
Dissolution of Steady 2-naphthol
[ 39)
� > (/) (/) -1 ;:t: > z (/)
350-2850 l-100
No
720-1 350 2-24 1 oc
No
-;
(') " rr: v
to � 0 (/)
1975
Miyauchi
et a/. [ 44 I
Pulse chromatography, chemical reaction
Unsteady
1976
Appel and Cathodic Steady Newman [ 45) reduction of ferricyanidc to ferrocyanide
1977
Kumar et a!.
[ 46 I
Dissolution of Steady benzoic acid
Sulfonic acid- Sphere ion exchange resin
0.9, 1.5
Water
Stainless steel Sphere
4.0
Mixture 1 3901450 of aqueous solution of ferrocyanide, fcrricyanidc and potassium nitrate 770Water,
Benzoic acid
Cylinder 5.5 X 2.5,
8.8 X 3.4, 8.8 X 4.5, 9.6 x 2.8, l 2.8 X 3. l , 1 2.8 X 3.8, 1 2.8X 4.9
5 ) 0-640
60%-
Yes
0.008-0. I 7
No
0.0 1 -600
No
42 400
propylene l!lycol
s: > (/) (/) ....,
------- ------ -- ·
a Diluted bed data are not included. b Criticized by Gaffney and Drew [ 33 ] , and Williamson et a/. l40j. c Re = DpG/JJ except for Refs. [ 3 3 ) and f 36-38J where Re =- lJpG/(J.Lq,) , and Ref. l43 1 where Rc d Natural convection at Re < 5E b ·
0.01-5
-- · ·----------
· =-
DpG/IJJ.( 1 - ch) ) .
� > z c.r: "!1 tTl � t; 0 !'T1 -r: "T1
(=)
tr1 z ...., -
c.r:
-+:>. -....)
1 48 HEAT AND MASS TRANSFER IN PACKED BEDS TABLE 4.3 Vapor pressures of naphthalene reported in various sublimation studies. Reference
Vapor pressure a t 25°C (Pa)
Hurt [ 2 1 Resnick et a!. fJ 3 1 Chu et a!. [241 Bradshaw et a/. [25 ] Andrews [ 4 7 ] Handbook of Chemistry and Physics [ 4 8 1
15.33 11.1 11.7 Not mentioned 9.64 1 1.7
a Value estimated by extrapolation, according to Resnick et a!. [231. b Data recommended in the International Critical Tables [ 491.
As compared in Table 4.3, different vapor pressures have been measured or assumed in different studies. The disagreement in the measured vapor pressures seems to have resulted in the disagreement in the obtained mass transfer coefficients. I n fact, small errors in the vapor pressure value as well as in the measured outlet pressure are greatly magnified in the calcula tion of mass transfer coefficients, particularly when the difference between the two p ressure values is small. This is very critical in the experimental determination of mass transfer coefficients, especially at low flow rates. Table 4.3 indicates that quite high vapor pressures were assumed by Hurt [2]. This is probably the reason why he obtained relatively low t ransfer coefficients. Resnick and White [23] also obtained low t ransfer coefficients, which, according t o Bar-Ilan and Resnick [26] , are attributable to the improper experimental techniques employed in their measurements. Thus, except for the transfer coefficients reported by Hurt and Resnick et al., all the other data mentioned above are included in the data correc tion and correlation. 4 . 1 .4
Diffusion Controlled Catalytic Reaction on Particle Surfaces: Steady-state Measurements
Satterfield and Resnick [ 1 0] conducted experiments on the catalytic decomposition of hydrogen peroxide at a metal surface. The reaction is so fast that mass transfer between the particle surface and the bulk fluid is the rate controlling step. The mass transfer coefficients can be easily evaluated from measurements of the overall rates.
MASS TRANSFER COEFFICIENTS 1 49 4.1.5
Pulse Gas Chromatography : Unsteady-state Measurements
Mass transfer coefficients in non-adsorption and adsorption systems were determined by Wakao and Tanisho [27] and Wakao etal. [29], respectively. The data were obtained from unsteady-state, pulse chromatography measurements. They obtained anomalously low transfer coefficients, but Wakao [50] has shown that the assumption of concentric intraparticle con centration inherent to the original Dispersion-Concentric model (in which solid phase mass diffusion in the axial direction is not considered) is responsible for the low coefficient values. Their original data, therefore, are not included in the data correlation. Gangwal et al. (30] also made adsorption chromatography measure ments and found the limiting particle-to-fluid mass transfer coefficient in terms of the Sherwood number being not less than unity. Miyauchi et al. [28] made chromatography measurements of a chemical reaction between potassium hydroxide, presoaked on to particles, and carbon dioxide imposed as a pulse on a carrier gas flowing in a packed bed. They measured the overall mass t ransfer resistance, comprised of gas phase and solid phase diffusion resistances. They mentioned that the limiting Sherwood number was 1 2.5 at a bed void fraction of 0.5. 4.2
Review o f the Published Liquid Phase Data
In some published data, the coefficients were determined based on measurements conducted at very low Reynolds numbers. The problem at low flow rates is that there is interference by natural convection. The con vection effect becomes increasingly important as the Reynolds number decreases beyond a certain critical value. This c ritical Reynolds number, above which natural convection may be ignored is generally not clear. According to Dryden et al. [36], the critical Reynolds number for a bed packed with 6.3 mm particles is related to the bed void fraction, Eb, by Re � 5E b . In order to avoid the possible natural convection effect, the liquid phase data for Re < 3 are not considered in the data correlation. 4.2.1
Dissolution of a Solid into a Liquid Stream : Steady-state Measurements
Mass transfer coefficients determined from the rates of dissolution of
1 50
HEAT AND MASS
TRANSFER
IN
PACKED BEDS
solids into liquid streams are numerous. The data that will be considered for the correlation are obtained from the following work. The rates of dissolution of spherical or cylindrical particles of benzoic acid into water flowing through the beds were measured by Ishino et al. [34 L Dryden et al. [ 36], Williamson et al. [ 40], Wilson and Geankoplis [ 4 1 ] , Kasaoka and Nitta [ 42], Upadhyay and Tripathi [ 43], Kumar et a!. [ 46]. Evans and Gerald [35] used finely ground granular particles of benzoic acid in their measurements of rate of dissolution. Dissolution of benzoic acid into propylene glycol was studied by Wilson and Geankoplis [ 4 1 ], and Kumar et al. [46]. Dissolution of 2-naphthol into water was investigated by McCune and Wilhelm [32], Dryden et al. [36], and Wakao et al. [39] . Dissolution of succinic acid and salicylic acid into acetone, n-butanol and benzene was studied by Gaffney and Drew [ 33 ]. Dissolution of lead into mercury was investigated by Dunn et al. [37]. 4.2.2
Electrochemical Reaction: Steady-state Measurements
Appel and Newman [ 45] applied a limiting current method to obtain the mass t ransfer coefficients at low flow rates: Re .::::;: 0 . 1 7 . 4.2.3
Extraction of Liquids: Unsteady-state Measurements
Hobson and Thodos [3 1 ] conducted experiments on the extraction of iso butanol and methyl ethyl ketone, presoaked on to porous Celite particles, into water flowing through the bed. They measured the variation with time of the effluent concentration and evaluated the initial extraction rate from the extrapolation of the curve, having a rapidly changing slope, to "zero time". However, the work has been criticized by Gaffney and Drew (33], and Williamson et al. (40] for the uncertainty involved in the evalua tion of initial rates by the method of extrapolation. Therefore, the data obtained by Hobson and Thodos are not used in the data correlation. 4.2.4
Ion Exchange : Unsteady-state Measurements
Ion exchange in Amberlite particles was studied by Selke et a!. [38]. Their mass transfer coefficients, determined in the Reynolds number range 1 to 40, are considerably larger than those obtained by many other investiga tors. The graphical method they applied for the determination of transfer
MASS TRANSFER COEFFICIENTS 1 5 1
coefficients does not seem to give accurate coefficient values. Their data will not be included in the correlation. 4.2.5
Pulse
Liquid Chromatography :
Unsteady-state Measurements
Miyauchi et al. [ 44] carried out chromatography measurements of the reaction between sodium hydroxide, imposed as a pulse on a stream of water, and sulfonic acid in ion exchange resin particles. They determined the overall mass transfer resistance, employing similar techniques to those used in the gas chromatography measurements [28]. They reported that the limiting Sherwood number was 1 6.7 at a bed void fraction of 0.4. 4.3
Re-evaluation of the Mass Transfer Data
As a result of the analysis in the preceding sections, the data which have satisfied the requirements for the correlation are as follows: a)
Evaporation of water [ 1 , 2, 1 2-2 1 ] ;
b)
Evaporation of organic solvents [ 1 6, 1 9 , 22] ;
c)
Sublimation of naphthalene [24, 2 5 ] ;
d) Diffusion controlled reaction on particle surfaces [ 1 0]; e) Dissolution of solids [32-37, 39-43, 46]. The dissolution data are available for a very wide range of Reynolds number, 0.00 1 6 to 3000. But, to avoid any possible natural convection effect, as mentioned already, the data [36] at Reynolds number less than about three are not included in the data correlation. Incidentally, all the data selected are those obtained under steady-state conditions, with solid particles having a constant concentration of mass transferring species at the surface. The data obtained from u nsteady-state measurements have not passed the criteria. I n general, two rate parameters are involved in the analysis of steady-state measurements: the particle-to fluid mass transfer coefficient and the fluid dispersion coefficient. In the analysis of unsteady-state mass transfer by pulse chromatography and ion exchange, additional parameters, such as intrapartic)e diffusivity and intra particle void fraction, are involved. It is conceivable that this makes the determination of the transfer coefficients in unsteady-state processes more complicated.
1 5 2 HEAT AND M ASS TRANSFER IN PACKED BEDS Under steady-state conditions Eq. (4. 1 ) can be rewritten as: d 2C
dC
U - - D ax dx dx2
a
+ - kr(C - Cps) = €b
O.
(4.3)
The following two types of packed beds have been used, so far, for the mass transfer measurements: a)
Empty column before the mass transferring packed bed (0 < x < L )
b) Inert packed bed (-l < x < O ; concentration C ' ) before the mass transferring packed bed (0 < x < L ; concentration C). In type (a), the Danckwerts boundary conditions are used U(C - Cm) = D
ax
dC dx
dC
at x = 0 (4.4) atx = L.
-= 0 dx
I n type (b), same-sized particles are usually packed in both beds, but the fluid dispersion coefficient, (D ax)inert ' in the inert bed may be differ ent from Dax in the mass transferring bed. The system is described by the following conditions: U(C ' - Cin) = (Dax) inert
dC ' dx
dC ' d 2C ' · U - - (Dax)me rt - = 0 dx2
dx
c' = c (D ax) inert
dC
-=0 dx
at x = - I for -/ < x < 0 at x = 0
dC' dx
= D ax
dC dx
(4.5)
at x = 0 at x = L .
Following the discussion of Wehner and Wilhelm (5 1 ], outlined in Section 2 . 1 , it is easily shown that both types (a) and (b) give the same
M ASS TRANSFER COEFFICIENTS 1 53
effluent concentration: 4A exp
Cps - Cexit Cps - Cm
(1
LU ) + A)2 exp (A 2D0
-
( �Uax ) 2
( 4.6) -, L U ) ( 1 - A)2 exp (-A - Du
where
)112 --A = (. 1 + 4kraDax €bU2 If
Dax
�
0, Eq. (4.6) reduces to
Sht aL] ----= exp [ (Sc)(Re) Cps - Cexit Cps - Cin
where
(4.6a)
Sh
t
-
(4.7)
is a Sherwood number evaluated under the assumption of
Dax = 0. Our main concern is about the mass transfer coefficients re-evaluated with Dax given by either Eq. (2.26), in its general form, or by Eq. (4.2) for the mass transfer systems under consideration. The Sherwood number based on the re-evaluated mass t ransfer coefficient is denoted by Sh. From the Sh t data reported in the literature, it is feasible to calculate Sh values by equating Eqs. (4.6) and (4.7) when information on the bed
height (or number of particJe layers) and bed void fraction is given. Mass transfer coefficients have been obtained by some researchers [ 17, 1 9 , 22, 25] by assuming that the axial Peclet number equals two. These data are also easily converted values. The values are re-evaluated for all the steady-state measurements listed in the preceding section except for those given by Gamson [ 1 ], Hurt [2], and Bradshaw and Myers ( 1 4) who gave no detailed data about and/or €b. The values recalculated from the data of Satterfield and Resnick [ 1 0] , and Petrovic and Thodos [ 1 9] are shown together with their data in Figure 4 1 A considerable difference can be seen between the two Sherwood values. The difference increases with a decrease in Reynolds number.
into Sh
Sht
eta!.
L
Sh
..
Sht
1 54 HEAT AND MASS TRANSFER IN PACKED BEDS
+_c (./)
<>a-.3��
102 10
u c d
£. (./)
o<x>� <>._ � •.o•._,. . o woo. o� • . <>
<>.
.
0
10
r-
•
•
.. � . � . . . ..... .q • •• • .
-
• .
•
-
•
Satterfield & Resn i c k I
10
I
Re
Sh
Sh
0
•
<>
Pet rovic & T h o d o s
FIGURE 4 . 1
0
� .
102
•
103
Sht o f Satterfield and Resnick [ 1 0 ] and Petrovic and Thodos [ 1 9 ] and Sh values corrected for fluid dispersion.
Figure 4.2 illustrates the Sh values re-evaluated from the evaporation data with a Schmidt number, Sc, of 0.6. For mass (and heat) transfer of a single sphere, the following Ranz and Marshall [52] equation is popularly recognized: (4.8) Figure 4.2 shows that the Sh values for packed beds are generally higher than those predicted from Eq. (4.8) for single spheres (dashed line), but the difference diminishes as the Reynolds number is lowered. It seems that with the decrease in Reynolds number, the Sherwood number for packed beds reduces and approaches the same limiting value of two for single spheres. Accordingly, the following empirical equation may be assumed:
Sh
=
2 + et.Sc113RefJ.
(4. 9 )
MASS TRANSFER COEFFICIENTS I S S 1 o3
a
� +
a
I>
1 o2
v 6
0
.c. If)
( 194 5) (1951) Galloway et ol ( 1 957) De Acetis and Thodos ( 1960) McConnachie and Thodos ( 1963) Sen Gupta and Thodos ( 1 9 6 3 , 64) Molting and Thodos ( 1967) Petrovic and Thodos ( 1 968) Wilke and Hougen
Hobson and Thodos
Sc:
10
_ _____..
..---
____
0
0.6
o---
__
---· -----+
;......
+
--
--
-�>-"
,
,
... ,
S1ngte sphere
----
10 FIGURE 4.2
----
-�-� < � -------
"'
0
Re
Sh versus Re, for water evaporation (solid line showing Eq. 4.1 1 ).
It should be noted that Petrovic and Thodos [ 19) have obtained an equation for the gas phase data corrected for an axial Peclet number of two : for 3 < Re < 900 .
(4 . 1 0)
At high Reynolds numbers, the data shown in Figure 4.2 are satis factorily represented by the Petrovic-Thodos relationship as well. This is quite natural since, at high Reynolds numbers, the second term on the right-hand side of Eq. (4.2) is dominant: axial Peclet number being two. However, at lower Reynolds numbers, the re-evaluated data are higher than those according to Eq. ( 4 . 1 0). As mentioned already, the correction for larger axial dispersion coefficients gives higher mass transfer coeffi cients, particularly at low flow rates. In a liquid phase system, Sh values are relatively large so that the liquid phase data are good to use for the determination of a and {3 values. Figure 4.3 is a plot of the liquid phase Sh data as (Sh - 2)/Sc113 against Re . In the
1 56 HEAT AND MASS TRANSFER IN PACKED BEDS 1 03 •
'Y
•
....... M
-u lf) .......
l 02
McCune and Wilhelm Gaffney and
Drew
lshino et al
..
Evans and Gerald
•
Dunn e t al
...
Williamson et a l
�
Wakao et al
*
{ 1 949) ( 1 9 50) (1951) ( 1 95 3) ( 1 956) { 1958) ( 1963)
•
.A *
W i l s o n a n d Geankoplis Kasaoka and Upadhyay and Kumar e t al
Nitta Tripathi
( 1 966) ( 1969) ( 1975) ( 1 977)
....... N
l
.c. lf)
-
10
( S h - 2 ) / S c 1 13
10
FIGURE 4 . 3
=
1 . 1 R e 0 ·6
1 02 Re
(Sh - 2)/Sc113 versus Re, for liquid phase data.
wide range of Reynolds number from about 3 to 3000, the Sherwood data fit well a single straight line corresponding to the following correlation:
Sh = 2 + 1 . 1Sc 1 13Re 0· 6.
(4. 1 1 )
Equation (4.6) reduces to Eq. (4.7) when A � 1 , i.e. Dax � 0 or k rf(eb U) = Sh/(Sc Re) is low. This indicates that liquid phase (high Sc) coefficients are affected little by the axial fluid dispersion unless Re is very low, whereas gas phase (low Sc) coefficients are considerably affected by the fluid dispersion, particularly at low Reynolds numbers. In Figure 4.4, both liquid phase and gas phase Sh data are plotted against (Sc113 Re0 •6 )2. The reason the square of Sc 1 13 Re0· 6 is taken is in order to enlarge the plot in the x-axis direction. It is seen that the data are well correlated by Eq. (4.1 1 ) , represented by the solid line. In Figure 4.5, the range of the recalculated Sh values are compared with
MASS TRANSFER COEFFICIENTS 1 57 LIQUID-PHASE
GAS-PHASE
• McCune and Wilhelm ( 1 945) ( 1 949) Wilke and Hougen (1951) • Gaffney and Drew ( 1 950) Hobson and Thodos (1953) • Ishino et al . ( 1 95 1 ) ., Chu et a/. ( 1 954) o Satterfield and Resnick • Evans and Gerald ( 1 953) ( 1 957) + Galloway et al. • Dunn et al. ( 1 956) ( 1 960) a De Acetis and Thodos � Wakao et al. ( 1 958) (1961) " Bradshaw and Bennett ... Williamson eta/. ( 1 963) ( 1 963) 1> McConnachie and Thodos ( 1 966) * Wilson and Geankoplis ( 1 963, 64) v Sen Gupta and Thodos ( 1 969) � Kasaoka and Nitta ( 1 967) A Mailing and Thodos (1 975 ) ... Upadhyay and Tripathi ( 1 968) o Petrovic and Thodos • Kumar et a/. (1 97 7) ( 1 969) 1t Wilkins and Thodos 1 � �� e
CP
����r-�
..c. Ul
10
10
FIGURE 4.4
( sPJReo.G )2
Correlation of Sherwood numbers, Sh, for gas and liquid phase data.
the corresponding Sht data. When Sc113Re0·6 is large, the Sh values are only slightly higher than the Sht data. As Sc113Re0· 6 decreases, however, the difference between the two Sherwood values becomes more prominent and then significant at low Sc 1 13Re0•6. In many studies, the mass transfer coefficients determined, assuming Dax = 0, have been correlated in terms of the ]M ass factor (see Dwivedi and Upadhyay (53], for instance). The fact that the Sht values decrease with decreasing Reynolds number appears as if a ]M ass factor correlation were valid even at very low flow rates. However, this is not exactly correct because recalculated Sh values are found to approach a limiting value.
1 58
HEAT AND MASS TRANSFER IN PACKED BEDS
+ .c
(/)
10
'U c 0
.c (/)
1 01 ��--���--�--�����--��� 3 10 102 1 10 2 1 ( Sc /3 Re0 .6 )
fiGURE 4.5
Comparison of Sh and Sh t , for gas and liquid phase data.
There is somewhat of an uncertainty about the limiting Sherwood value of two. The limiting value may be higher or lower than two. However, it is not likely that particle-to-fluid mass transfer is the rate controlling step at low flow rates. It seems that a Sherwood number greater than a certain value, say Sh <: 1 , will explain mass transfer phenomena in packed beds at low flow rates equally well. The important thing is that the Sherwood number does not keep decreasing with a decrease in Reynolds number. In fact, if the Sherwood number keeps decreasing, no mass t ransfer will occur between a particle and its surrounding fluid at very low or zero flow rates. This is certainly against the nature of mass t ransfer in packed beds.
REFERENCES
[ l l B . W . Gamson, G. Thodos and 0. A . Hougen, [2) [3 ]
39, 1 ( 1 943).
Trans. A mer. Jnst. Chern. Eng.
D . M . Hurt, Ind. f:n�. Ch em 35, 5 2 2 ( 1 943). R. Pfeffcr, Jm.J. f:ng. Chem. Fund. 3, 380 ( 1 964). .
MASS TRANSFER COEFFICIENTS 1 59 [4j [5) [6J [7] [8) [9] [10) [11] [ 12] [ 13] [ 14] [15] [ 16 ] [ 17 ) [ 18] [ 19] ( 20 ] [21] [ 22 ] [23] [24] [251 [ 26 ) [27] [28] [ 29] [30] (31] [32] [33] [34 ] [35) [361 [ 37] [38] ( 39] [ 40] (4 1 ] [42) [43) [44] [ 45 ] (46 )
R. Pfeffer and J . Happel, A /ChH J. 1 0, 605 ( 1 964 ). B . P. Le Clair and A . E . Hamielec, Ind. Eng. Chem. Fund. 7, 542 ( 1 968). B. P. Le Clair and A . E. Hamielec, Ind. Eng. Ch em. Fund. 9, 608 ( 1 970). B. P. Le Clair and A . E. Hamielec, Can. J. Ch en1. Eng. 49, 7 1 3 ( 1 9 7 1 ) . M . M. El-Kaissy and G. M. Homsy, Jnd. Eng. Chem. Fund. 1 2, 82 ( 1 973). Y . Nishimura and T . Ishii, Chem. Eng. Sci. 35, 1 205 ( 1 980). C . N . Satterfield and H . Resnick, Chem. Eng. Prog. 50, 504 ( 1 954). N . Wakao and T. Funazkri, Chern Eng. Sci. 33, 1 375 ( 1 978). C . R . Wilke and 0 . A . Hougen, Trans. Amer. lnst. Ch ern. Eng. 4 1 , 445 ( 1 945). L . R . Galloway, W . Komarnicky and N . Epstein, Can. J. Chem. Eng. 35, 1 39 ( 1 957). R . D . Bradshaw and J . E . Myers, A IChE J. 9, 590 ( 1 963). J . De Acetis and G. Thodos, lnd. Eng. Chern. 5 2, 1003 ( 1 960). M . Hobson and G. Thodos, Chem. Eng. Prog. 47, 370 ( 1 95 1 ). G . F. Mailing and G . Thodos, Int. J. Heat Mass Trans. 1 0, 489 ( 1 967). J . T . L. McConnachie and G . Thodos, AIChEJ. 9, 60 ( 1 963). L . J. Petrovic and G . Thodos, Ind. Eng. Chem. Fund. 7 , 274 ( 1 968). A . Sen Gupta and G . Thodos, AIChE J. 9, 7 5 1 ( 1 963). A . Sen Gupta and G. Thodos, Ind. Eng. Chern. Fund. 3, 2 1 8 (1 964). G . S . Wilkins and G . Thodos, AIChE J. 1 5 , 47 ( 1 969). W. Resnick and R. R. White, Chern. Eng. Prog. 45, 377 ( 1 949). J . C. Chu, J . Kalil and W. A. Wetteroth, Chern. Eng. Prog. 49, 1 4 1 ( 1 953). R. D . Bradshaw and C . 0 . Bennett, AICht: J. 7 , 48 ( 1 9 6 1 ) . M . Bar-llan and W . Resnick, Ind. Eng. Ch ern. 49, 3 1 3 ( 1 957). N . Wakao and S . Tanisho, Chem. Eng. Sci. 29, 1 99 1 ( 1 974). T. Miyauchi, H . Kataoka and T . Kikuchi, Ch em. t:ng. Sci. 3 1 , 9 ( 1 976). N. Wakao, K . Tanaka and H. Nagai, Chern. t"ng. Sci. 3 1 , 1 109 ( 1 976). S. K . Gangwal, R . R. Hudgins and P. L . Silveston, The Limiting Sherwood Number for Mass Transfer in Packed Beds, Proceedings of the European Congress on Transfer Processes in Particle Systems, Nurnberg, March ( 1 977). M . Hobson and G. Thodos, Chern. Eng. Prog. 45, 5 1 7 ( 1 949). L . K . McCune and R. H. Wilhelm, Ind. Eng. Chern. 4 1 , 1 1 24 ( 1 949). B. J . Gaffney and T . B. Drew, Ind. Eng. Chem. 42, 1 1 20 ( 1 950). T. lshino, T. Otake and T. Okada, Kagaku Kogaku IS, 255 ( 1 9 5 1 ). G . C . Evans and C. F . Gerald, Chern. Eng. Prog. 49, 1 35 ( 1 953). C. E. Dryden, D. A. Strang and A. E. Withrow, Chem. Eng. Prog. 49, 1 9 1 ( 1 953). W. E. Dunn, C. F. Bonilla, C. Ferstenberg and B. Gross, A /ChH J. 2 , 1 84 ( 1 956). W . A. Selke, Y . Bard, A . D. Pasternak and S. K. Aditya, AICht: J. 2 , 468 ( 1 956). N . Wakao, T. Oshima and S. Yagi, Kagaku Kogaku 2 2, 780 ( 1 958). J . E. Williamson, K. E. Bazaire and C. 1 . Geankoplis, Ind. Eng. Chenz. Fund. 2 , 1 26 ( 1 963). E . J. Wilson and C. J. Geankoplis, Ind. Eng. Chem. Fund. 5, 9 ( 1 966). S. Kasaoka and K . Nitta, Kagaku Kogaku 3 3 , 1 23 1 ( 1969). S. N. Upadhyay and G. Tripathi, J. Chem. Eng. Data 20, 2 0 ( 1 975). T. Miyauchi, K. Matsumoto and T. Yoshida, J. Chern. t:ng. Japan 8, 228 ( 1 975). P . W. Appel and J. Newman, A /ChE J. 22, 979 ( 1 976). S. Kumar, S. N . Upadhyay and V. K . Mathur, Ind. Hng. Chem. Process Des. Dev. 1 6 , 1 ( 1977).
1 60
HEAT AND MASS TRANSFER IN PACKED BEDS
[47] ( 48)
M . R . Andrews, J. Phys. Chern. 30, 1 497 (1 926). Handbook of Chemistry and Physics, edited by R. C. Weast, 53rd edn., Chemical Rubber Company, Ohio, D-164 (1 972-73). International Critical Tables, Vol. 3 , McGraw-Hill, New York, p. 208 ( 1928). N . Wakao, Chem. Eng. Sci. 31, 1 1 1 5 ( 1 976). J. F. Wehner and R . H . Wilhelm, Chern. Eng. Sci. 6, 89 (1956). W. E . Ranz and W . R . Marshall, Chern. Eng. Prog. 48, 1 7 3 ( 1 952). P. N. Dwivedi and S. N . Upadhyay, Ind. Eng. Chem. Process Des. Dev. 1 6 , 1 5 7 ( 1 977 ).
[49] (50] [5 1 ) [52) [53]
5
Steady-State Heat Transfer
IN THIS chapter, a packed bed used as a heat exchanger will be considered. Fluid may be heated or cooled from the column wall while flowing through a packed bed. In such a packed bed operated under steady-state conditions, a difference in local temperature between the fluid and the particle may exist, but the overall solid and fluid temperature profiles are considered to be identical to each other, as sketched in Figure 5 . I . In estimating the overall steady-state temperature profiles, the heterogeneous packed bed may be assumed to be a homogeneous single phase. The temperature profiles in the bed are then predicted in terms of effective thermal conductivities and wall heat transfer coefficients.
Sol i d phase temp ( Ts) Fluid phase temp (TF) Local
temperature
FIGURE 5 . 1
Steady-state temperature profiles in a packed bed (of heat exchanger type).
]61
162
HEAT AND MASS TRANSFER IN PACKED BEDS
In the following sections, analytical solutions of the steady-state temperature profiles under different conditions will be shown. In addition, the various empirical formulae proposed for the estimation of e ffective thermal conductivities as well as wall heat transfer coefficients are discussed. 5. 1
Steady-state Bed Ten1perature
For a cylindrical packed bed operated as a steady-state heat exchanger, the heat balance equation gives cc.;·
( )
a 2T aT � � r aT + keax = ker ax ax r ar ar 2
(5.1)
where =
specific heat of fluid at constant pressure G fluid mass velocity per unit area of bed cross-section k eax = effective axial thermal conductivity ker = effective radial thermal conductivity T bed temperature. Cr:
=
=
A t intermediate and high flow rates, the axial second derivative, a2Tjax2, is very small compared to the other terms. Equation (5. 1 ) then reduces to GCF
( )
aT � r aT ker� · = ax r ar ar
(5.2)
Equation (5 . 2) was first solved by Hatta and Maeda [ 1 ] in 1 948, and a few years later, by Coberly and Marshall [2], both based on the following boundary conditions:
T = T0 aT -=0 ar aT = hw(Tw - T) ar
ker -
at x
=
0
at
r=0
at
r = RT
(5.3)
STEADY-STATE H EA T T RANSFER
1 63
where hw = wall heat transfer coefficient RT = column radius T0 = bed inlet temperature Tw = wall temperature.
The solution to Eq. (5 .2) with the conditions given by Eq. (5.3) is: Tw - T =
Tw - To
; lo(anr/R T) exp (- a�y)
?
- � 2 n = l an [ I + (an/B) ] tCan) 1 J
(5 .4)
where (5.4a)
(5.4b ) and an is an n-th root of the following equation of Bessel functions (J0 is a Bessel function of first kind and zeroth-order, and /1 is that of first kind and first-order): (5 .4c) 5 1 1 .
.
Solution Deep in a Bed
When y, as defined by Eq. (5 .4b ), is greater than about 0.2, the series in Eq. (5 .4) converges so rapidly such that only the first term of the series is significant. Therefore, Tw - T
2Jo(a l r/R T) exp (-aty)
Tw - To
a d I + (a1/B)2] 11(a 1)
(5.5)
where (5.5a)
1 64
HEAT AND MASS T RANSFER IN PACKED BEDS
Equation (5.5) gives the temperature profile deep in the bed. At r = 0 (T Tc ), Eq. (5.5) reduces to =
2 exp (-ary)
Tw - Tc
a1 [1
Tw - To
+
(adB)2] J1(a1)
(5.6)
which gives the temperature profile· along the central axis of the bed under the conditions specified .
5.1.2
Mixed Mean Temperature of the Fluid
The mixed mean temperature, Tm, is the average radial temperature defined as:
Tm =
RT
�J
Rr
T(r) r dr.
(5.7 )
0
Hence, Eq. (5 .4), when expressed in terms of Tm, becomes (5.8) For y > 0.2, the above equation converges to 4 exp (-ary) ai[l + (adB)2]
(5.9)
In terms of the radial mixed mean temperature the steady-state heat balance equation of the heat exchanger is GCF
dTm
-
dx
=
2 Uo
-
Rr
(T.w
-
T.m )
( 5 . 1 0)
where U0 is the overall heat transfer coefficient between the wall and the
STEADY-STATE HEAT TRANSFER 1 6 5 bed. The solution to Eq. (5 . 1 0) is
( 2 Uox ) = exp Tw - To GCF R T
Tw - T
m
·
(5 . 1 1 )
At sufficiently large values of x, comparison of Eqs. (5.9) and ( 5 . 1 1 ) gives (5 . 1 2) where DT is the column diameter or 2Rr, and k F is the thermal con ductivity of the fluid flowing in the bed. 5 .1.3
Solution when the Bed Inlet Temperature is a Function of Radial Distance
With a radial temperature profile at the bed inlet, i.e. T = T0(r)
at x = 0
the solution to Eq. (5 .2), when axial heat conduction is ignored, is (5 . 1 3) where
(5 . 1 3a)
5 . 1 .4
Solution when Axial Heat Conduction is Considered
When flow rates are low, the axial heat conduction term cannot be ignored. Taking into account the axial heat conduction, Eq. (5. 1 ) is solved for a semi-infinite bed. Under the boundary conditions expressed by
} 66
HEAT AND MASS TRANSFER IN PACKED BEDS
Eq. (5.3), the solution is
Tw- T = 2 I lo(an rfRT) ex� (-a�z) Tw - To n=l an[l + (an/B) ] JI(an)
(5. 1 4)
2y --z = -112 2 ( an ) kerkeax] 1 + [1 + 4 GCFR T
(5. 1 4a)
where
and y is defined in Eq. ( 5 .4b). When the inlet bed temperature is a function of r, and when axial heat conduction is taken into consideration, the solution of Eq. (5. 1 ) for a semi-in finite bed is:
(5. 1 5)
where the coefficient is defined in Eq. (5 . 1 3a). At very low flow rates, Eq. ( 5 . 1 4a) reduces to
en
-z = -a_ n (k k ) 1/2 GCFRT y
er
_ _
(5. 1 6)
eax
As will be shown in Sections 5 . 2 and 5 . 5 , the effective radial and axial thermal conductivities at low flow rates are equal to the effective thermal conductivity of a quiescent bed. Therefore, with Eq. ( 5 .4b), Eq. (5 . 1 6) becomes X
(5. 1 6a)
At sufficiently large values of x, only the first term of the series in Eqs.
S T EADY-STATE HEAT TRANSFER
16
7
(5 . 1 4) and (5 . 1 5 ) is dominant. Hence (- a1xfRT) Tw - T 210(a 1 rfRT) ------2 /B a , [ l + (a j 11(a1) Tw - To 1 ) exp
=
Tw - T Tw - To(O)
2c 110(a 1 rfRT) exp (-a1 x/Rr) [ I + (a 1/B)2] Jf(a I )
(5 . 1 7) (5 . 1 8)
Equations (5 . 1 7) and (5 . 18) show that the bed temperature depends upon B , which is defined as hwRT/ker , but not on the individual values of ker and hw . 5 .I.5
Determination of Effective Radial Thermal Conductivities and Wall Heat Transfer Coefficients
The two heat transfer parameters, ker and hw, are easily determined from measurements of axial temperature profiles. The condition required is that the measurements should be made at high flow rates under which axial heat conduction may be ignored. The temperature profiles at any radial position will do, but the temperature measurements along the central axis of the bed, where radial temperatures level off, are most preferable. If the measurements are made near the wall, where radial temperature profiles are usually steep, small errors in radial location of the thermocouples will result in considerable errors in the measured temperatures. I f the measured temperatures at the center of the bed, Tc, are plotted as In [(Tw - Tc)/(Tw - T0)] versus x, a straight line will be produced at suffi ciently large values of x. This straight line means that this is the region where Eq. (5 .6) holds. The slope and intercept of the straight line are slope = - ai
(
ker
)2
( 5 . 1 9)
GCE-R T
and intercept = In Also, Eqs.
2
2
a I [ I + (a tfB) ] l1 (a 1 )
·
(5.20)
(5 .6) and (5 . 9) show that, when x is large, the mixed mean
1 68 HEAT AND M ASS TRANSFER IN PACKED BEDS temperature of the effluent fluid, Tm , is related to Tc by Tw - Tm
2 1t (a t )
Tw - Tc
a1
(5 . 2 1 )
Therefore, the parameters, ker and hw, can be determined b y either (a) Eqs. (5 . 1 9) and (5 .2 I ), or (b) Eqs. (5 . 1 9) and (5 .20), both in conjunc tion with Eqs. (5.4a) and (5 .5a). In method (b), the value o f a 1 1 based on Eqs. (5 .20) and (5.5a), is easily effected by a slight change in the value of the intercept obtained from the extrapolation of the linear relationship between In ((Tw - Tc)f(Tw - T0)] and x . On the other hand, a 1 is more safely determined from Eq. (5. 2 1 ). Therefore, ker and hw may, in general, be more accurately evaluated from method (a) than method (b). Coberly and Marshall (2], however, evaluated the values of aTjax, aTjar and a2Tjar2 by graphical differentiation of the measured tempera ture profiles and then found ker directly from Eq. (5.2). The local values of ker were then averaged to give a mean value to be used for the entire bed. With this mean ker , the value of a 1 was obtained from the slope of a straight line portion in a plot of In [(Tw - T)/('l.'w - T0)] versus x . And finally hw was calculated using Eqs. (5.4a) and (5.5a). De Wasch and Froment [3] compared the measured temperature profiles a t the bed exit and those computed from Eq. (5.4) and obtained ker and hw by adjusting them to give the best fit. The various methods of determination of ker and hw from measured temperature profiles are illustrated in Examples 5 . 1 and 5 .2 . txample 5. 1 Table 5 .1 lists the experimental conditions and the temperatures measured along the central axis of a cylindrical packed bed. Temperatures a t the bed inlet were found to be almost constant in the radial direction. Find kec and hw. SOLUTION (i) Estimation from Eqs. (5. 1 9) and (5. 21) The data are plotted as In [(Tw - Tc)/(Tw - T0)] versusx in Figure 5 . 2 . The straight line portion has a slope of -0.064 cm- 1 . With Tm and Tc at the
bed exit , a 1 is found from Eq. (5.2 1 ) to be 1.63. Based on the slope and a l l
STEADY-STATE HEAT TRANSFER 1 69 TABLE 5 . 1 Experimental conditions and axial temperatures measured along a bed central axis. Bed diameter: 3.6 em Bed height: 20 em Glass bead: Dp 6.0 mm Air mass velocity : G = 0.761 kg m-2 s-• Specific heat of air: Cr: = 1000 J kg- • K-• Inlet air temperature: T0 = 1 6 .5°C Wall temperature: Tw = l 00°C =
Height, x (em)
Temperature at central axis, Tc (C)
0 4 8 12 16 20
16.9 21.9 35 . 3 49.3 61.1 69.8
Mixed mean temperature of effluent air:
78.8°C
2 . 0 ------1 .0 . 0.8 Uj 0 1- , 1-
0.6
'3:i. 1-13: 0 . 4
1-
0.2
a
exPe r i mental
_ __, _ _ ...._ _ _ ,__ _ ---l "---'--' 0 . 1 '-4 0 8 16 20 12
FIGURE 5 .2
x
Ccml ln [ (Tw - Tc)/(Tw - T0)J versus x for Example 5 . 1.
ker is determined from Eq. (5. 1 9) to be 0.59 W m-1 K -1 , and then, accord ing to Eqs. (5 .4a) and (5.5a), hw = 70 W m-2 K- 1 .
(ii) Estimation from Eqs. (5. 1 9) and (5.20) In Figure 5.2, the extrapolation of the straight line portion to the y-axis gives an intercept equal to In 1.24. From Eqs. (5.5a) and (5.20),
1 70
HEAT AND MASS TRANSFER IN PACKED BEDS
1 14 = -�
2 ---
a 1 { 1 + [Jo(a t)/lt(a t)]2} lt (a t )
and it is found that a 1 = 1 . 3 5 . Also, from Eq. (5 . 5a), B = 1 . 2 1 . From both the slope of the straight line, - 0 .064 cm- 1 , and a b ker can be deter mined from Eq. (5. 1 9) to be 0.87 W m-1 K - 1 . Hence, from Eq. ( 5 .4a) hw = 58 W m-2 K-1. ,
(iii) Hstimation of ker and hw from curve fitting Temperatures along the central axis of the bed are computed from Eq. (5 .4) with various assumed values of ker and hw, and then compared with the measured temperatures. The errors between the computed and measured temperatures, defined by Eq. (5. 22),
(5. 22)
are shown on an error map in Figure 5 . 3 . (N is the number of points o f comparison; in this case N = 5 , corresponding to x = 4 to 2 0 em.) From the least-error point on the map (labeled + ) , the following data are obtained: ker = 0.7 1 W m-1 K-1 and hw = 63 W m-2 K-1. Also, the data obtained in Solutions (i) and (ii) are indicated as 0 and x on the map, respectively. As shown, the results obtained from Solution (i) are more accurate (with less error) than those of Solution (ii). For reference, the axial temperature profiles predicted with the data of ker and hw obtained in Solution (iii) are shown in Figure 5 .4. It is shown that the first term in the series of Eq. (5 .4) becomes dominant only when x > 7 em. Example 5.2
The experimental conditions and the measured temperatures listed in Table 5.2 are those of Bunnell e t a/. [ 4]. Find ker and hw. Also, examine the effect of k eax · SOLUTION
With an assumed value of keax = 0.7 W m- 1 K- 1 , bed temperatures are computed from Eq. ( 5 . 1 5). The predicted temperatures are then com pared with the measured data. Figure 5 .5(a) shows the resulting error map;
STEADY-STATE HEAT TRANS FER 1 7 1
1.2 1.0 I
� I
E
-
::3:
'-'
� <1.> .:.<:
X
0.8 0.6
/c
0.4
0.1
0.2 0
100
50
0
hw
150
[ W · m- 2 . K- 1 J
200
250
FIGURE 5 . 3 Error map in the plot of ker versus hw for Example 5 . 1 (Solution iii): + indicates the least-error point; 0 represents the data ob tained in Solution (i); x shows those in Solution (ii). 1 . 5 r-------�
1 .0 u 0 ff1 I > > ff-
0.5 " experimental 0
FIGURE 5 .4
0
4
8 x
12
[em)
16
20
Comparison of measured and calculated temperature profiles for Example 5 . l.
HEAT AND MASS TRANSFER IN PACKED BEDS
1 72
TABLE 5 . 2 Experimental conditions and data measured b y Bunnell et a/. [ 4 ]. Bed diameter: 5 .08 em (2 in) Cylindrical alumina pellet: D = 0.32 em Ck in) Air mass velocity: G = 0.199 g m-2s-1 ( 1 4 7 lb ft-2 h-1) Specific heat of air: CF = 1050 J kg-1 K -1 Wall temperature: Tw = 100°C
lk
Temperature (°C)
em
X
(in)
0 5.1 1 0.2 1 5. 2 20 .3
r/R r =
(0) (2) (4) (6) (8)
0
0.1
0.2
4003 3983 3963 303 3 1 83 229 2343 180 1 823 154 1 5 73
0.3
0.4
0.5
0.6
0.7
3923
3883 280 213 171 146
3793
3673 241 1 86 152 1 34
3493
0.8
0.9
3203 2253 191 150 1 29 1 20
a Read from their Figures 4-6. €
being given by Eq. ( 5 . 22) with N = 20, corresponding to x = 5 . I to 20.3 em. The least-error point (labeled +) indicates that ker = 0.3 1 W m-1 K-1 and hw = 1 1 3 W m-2 K-1. Figures 5 . 5(b) and (c) show the effect of keax: Figure 5 .5(b ) ker versus keax with hw = 1 1 3 W m-2 K-1; Figure 5 .5( c) hw versus keax with ker = 0.3 1 W m-1 K-1. 0 . 6 �------� keax
�
I :>.::
r-1 I E .
•
-1 -1 0. 7 W·m ·K
0.4
£ = £ =
3: .
.... .:.£Q)
0
(a)
E
0.2
0
50
hw
100
[W·m-2·K-1 J
150
=
0 . 05 0 . 02 0 , 05
200
STEADY-STATE HEAT TRANSFER 1 7 3 0 . 6 r-------�
I �
0.4
...-i
r
=
0 . 05
.
...-1 I
E :i L
(1) �
0.2
0
'-
(
10-
=
0 . 05
e:
2
0 . 02
10 -1
( W · m- 1 · K- 1 J
keox
(b)
150
<
0 . 02 - 0 . 05
e:
=
E
:
0 . 02
E
>
0 . 05
rl I � .
N I E .
�
100
3: .I:;.
-
0 , 05
50
0
(c)
� -� -----� ------� ----� 2 1 010-1 1 keo x
[ W - m- 1 · K- 1 J
FIGURE 5 . 5 Error maps for the data o f Bunnell et a/_ ( 4 ] : (a) ker versus hw ( + indicates the least-error point); (b) ker versus ke a x ; (c) hw versus keax .
1 74
HEAT AND M A S S TRANSFER IN PACKED BEDS
Bunnell et al. [4] found that their data of ker were correlated by k er
-
k ..-
= 5 0 + 0.061 Re.
( 5 . 23)
.
From Eqs. (5 . 23) and (5 .69). keax at this flow rate (Re . 30, according to Bunnell et al.) is estimated to be roughly 0.7 W m- 1 K - 1 . However, Figures 5 . 5(b) and (c) indicate that, if keax � 1 W m-1 K-1, axial heat conduction makes little contribution to the bed temperatures. ·
experimental
•
1 . o,_
X =
0 - - - - - - - - - - - ----:.:-..:;: C4= .::
.......
x =
--
--
5.1
em
... ....
..... ..... .... ........
,_ ;. c;
� c . ..... I �: f- i � .....
...
'
...
'
0.4
' ' ' ' ' ' ' ' '
'
0.2 O . ll FIGURE 5 . 6
0.6
' '
0.8
' ' ' '
1.0
r/RT
Measured and calculated radial temperature profiles (solid curves calculated from Eq. 5 . 1 5 ; dashed curves from Eq. 5 . 14).
Figure 5 .6 shows the temperatures (solid lines) predicted from Eq. (5 . 1 5) with the measured T0(r), and those (dashed lines) computed from Eq. (5 . 1 4) assuming that the bed inlet temperature is uniform at T0(r = 0) = 400°C. The temperatures measured at x = 5 . 1 , 1 0.2 and 1 5 .2 em show better agreement with the solid lines than with the dashed lines. It is clear, in this example, that the radially varied temperatures at the bed inlet should be taken into consideration. (End of Example)
STEADY-STATE HEAT TRANSFER
5.2
1 75
Effective Radial Thennal Conductivities
Hatta and Maeda [ 1 , 5 ] correlated their data of e ffective radial thermal conductivities into the form: kerfkF = a 'Reb ' and obtained a' and b ' as functions of the Reynolds number. Coberly and Marshall [ � ] , however� found that their data were best fitted using a linear function of the Reynolds number as follows: ker
- = a + bRe kF
( 5 . 24 )
where Dp = particle diameter k F = fluid thermal conductivity Re = Dp UPFIJ1, Reynolds number u = superficial fluid velocity, based on an empty column J1 = fluid viscosity PF = fluid density. In 1 95 2, Ranz [6] proposed a model for radial fluid mixing. His model is primarily based on the assumption that the bed is composed of inter connected cells and that lateral mixing occurs from a single cell to the neighboring cells through randomly connected channels. The model further assumes that the average lateral fluid velocity is equal to a certain fraction of the apparent axial fluid velocity, u , i.e. a u . The lateral heat flow rate per unit area, through N interconnecting cells from temperatures T0 to TN in the bed , is then qlateral
= aup F CF(To - Tt) =
Ci.UpFCF(Tt - T2)
=
Ci.UpFCF(TN - 1 - TN)
=
cx. upFCF (To - TN ) . N
(5 .25)
1 76
HEAT AND
MASS TRANSFER IN PACKED BEDS
In terms of the effective radial thermal conductivity, Eq. (5 .25) is re written as:
( 5 26) .
where l is the distance between the centers of two adjacent cells or the width of a single cell. Therefore, ( 5 .27) From a theoretical consideration of a high flow rate in a rhombohedral packing of spheres: o: is found to be 0 . 1 7 9 . In addition, Ranz assumes that l = Dp/ 2 . Consequently,
(ker)mixing kF
=
0.0895 (Pr)(R e)
(5 .28)
where Pr is the Prandtl number, CF J..Lik F . Yagi and Kunii [7, 8] proposed the following equation for the predic tion of ker :
ker kF
- =
k� af3(Pr)(Re) kF -
( 5 . 29)
+
where (3 comes from their assumption that l (3D p/ 2 . They found o: (3 = 0.1 from the data reported by Hatta and Maeda [ 1 , 5 J for cylindrical particles in the range of particle-to-column diameter ratio of 0.036 to 0.24. Also, they proposed a formula for estimating the values of k�, the effective thermal conductivity of a quiescent bed, which is equivalent to a in Eq. (5 . 24). Actually, the value of o:(3 in Eq. (5.29) can be obtained based on a simple mass and heat transfer analogy. Turbulent fluid d ispersion in the radial direction has already been found to be (Per)mixing 1 0 or eb(Dr)mixing 0 . 1 Dpu. Since the turbulent radial fluid dispersion coeffi cient, eb(Dr )mixing, and the effective radial thermal conductivity, (ker)mixing, are both based on the unit cross-sectional area of a packed bed, and result from the lateral fluid mixing in the bed ; then, (ker)mixing =
�
�
�
STEADY-STATE HEAT TRANSFER
ker
-=
kF
k�
-+
kF
0. 1 (Pr)(Re).
1 77
(5.30)
Similar to Eq. (2.25), it may be shown that
k er = k� = k� + 0. 1DpuCFPF
for laminar flow (Re < 1) for turbulent flow (Re > 5).
(5 .30a)
However, the term, 0. 1DpuCFP F, is usually small compared to k� at low flow rates, such that Eq. (5.30) may be applied over the entire range of flow from laminar to turbulent.
5.2.1
Effective Thermal Conductivities of Quiescent Beds
Many theoretical studies have been carried out on the estimation of effec tive thermal conductivities of quiescent beds. As listed in Table 5.3, the studies fall into two main groups: one assuming unidirectional heat flow, and the other considering two-dimensional heat flow. Unidirectional heat conduction was assumed by Schumann and Voss [9] for a hyperbolic solid; by Krischer and Kroll [ 1 2] for slab-array; and by Schliinder [ 1 3 ] for spheres. Also, the unidirectional heat flow of combined conduction and radiation was assumed by Argo and Smith [ 1 0], and Schotte [ 1 1 ] for spheres; by Yagi and Kunii [7, 8) for slab-array; and by Zehner and Schliinder [ 1 4] for spheroid solids. The two-dimensional heat flow model is obviously more realistic than the unidirectional model. Deissler and Boegli [ 1 5 ] numerically solved the Laplace conduction equation by the relaxation method for a unit cell of spheres in a cubical array. The computed isotherms, in the cell with solid to-fluid thermal conductivity ratios of ks/kF = 3 and 30, are shown in Figure 5 .7. The isotherms come closer together in the vicinity of the point of contact as the ratio ks/k F increases from 3 to 30. It is clear that the heat flow is not unidirectional. By considering that a packed bed may consist of a bundle of long cylinders as shown in Figure 5 .8 , Krupiczka [ 1 6 , 1 7] solved numerically
.
-------
Heat
----,---- ·
··
Year flow ---+
Unidirectional
--·
.
TABLE 5 03 Theoretical studies of k�o (Partly adapted from Zehner and Schliinder [ 1 4 jo)
....
·-
Mod e l
Investigator
-
--·--
Sch u m an n anJ
-o-o
1934
-
·
Solid
•e•c: V
Voss (9)
l lyperbolic
xy
A
= nz(m + I )
Result
I leal t ransfer Con d uc t i o n
/. K ().:
o- --
��. F
1+ = q,
���
.l ] - fh 0
C ) l 1\ I o
0
-
·- ' s ·
-
-J 00
lo 1\ I·•
••• • -·-·
+
ks 0
+ m )) m ( l + m ) ( � '·: _ 1) 1 (�t-C � k S· k sm II+k kF m - - 1 ) + .!: II
X
(
1...
-
1 960
Argo and Smith l J O] Schotte I l l
1 954 Vagi an d K unii [7)
I
I
I
X
I
!IiTITIJF:
I
.
-
•··-··
__ . I
0::
I
Sphere
radiation
Sphere
Condu�.:tion
Slab
Conduction
and radiation and radiation
ks
ks
·
·
( + -m1 ) - m 0
1
k�
-
kF
= Et> (I + 0 .8 9 5 +
-
hrs _
h
rv
00895( I
hrvDp) · -
kr
-
- eh)
1/¢ + hr.;Dp /k10
(k s
k�
1
-=- - t p
)
(-I !_ )
3ks
00
€tJ ooo
-
- ·•oo•·-
0-
2( t - eh) ·-·
w m-2 K 1
.l
002268 0 l
o o o• -
J+
·-
·
�-_2���
= .
-
------2k r 1 + - .. -
¢= fun <.:
� > (/) (/)
t'!j :;c
Equation for k�
·
> z 0
....; :;c > z (/)
Equation for k�
Conduction and
· · - -
.
E11 = m( l + m ) ln 1 953
-·-
::r: r:: > ....;
_
p
· ·-
--
P
( T_ )
3
1 00
·
w m-2 K- 1
z ""t: > ("') " tr. 0 ttJ !":"l 0 (/)
1956
E-- _, t-� · -- =t
Krischer and
Kroll [ 1 2 }
t t-
-
Slab
Conduction
- -=-1 ··=t
kF
_ ,.
k 11
•
-
kF
mffi ffiS t
� = En + ( I - €h)
k..-
Sch!Under [ 1 3 }
I· t
I
� � �I
I I
I I
Sphere
Conduction
k,.
-
Zehnerand
Schliinder [ 1 4 J
Two-dimensional 1 1 958
-
-
Deissler and
Boegli
[15]
1 966
Krupiczka
1 969
Wakao and
I
[ 1 6]
Kato ( 1 8)
1971
Wakao and
Yortmeyer
[ I 9}
_
2 -···z2 r +-
lB - (B - I ) z }2
=1
Conduction
and rad iat ion
--
--··.___·__L_ -··-
-
--
--
·-
-------
�-- ----
·- -
I + - -·k.fkr knfkF --
= --�
[
In Lk /k_}j � k ..k ·F-
1 --
Graph
1 -- -
for k�
Conduction
Sphere
Graph for
k�
I Cylinder
Conduction
Eq. (5.32) fo r k�
Sphere
Conduction
Figure 5.9
.......
and
Sphere
__
· - --
radiation
C'on duct ion
and radiation
··----
_
1
]
ks
C/) ....,
� )>0 -<
(/; ...., I
for fine
.
I
1 -/
B = I, sphere B = 00, cylinder
particles and/or
�--- ·
-
-
ks
-
1 970
k�
k ..
I - €0 €h + k s/k..-
k� -=
-- -
1 966
1
= --- --
ks
at ----
low
pressure
for k�
)> ...., !T!
:c tT! > ...., ...., :::0 ;l> z C/) ..:.; l'#
-
Graphs
for k�
:::0
-..] \D
-
1 80
HEAT AND MASS TRANSFER IN PACKED BEDS Heat flow
_____ + _ ___
__
r----£ �---, (O)
T = 1.0
Heat flow
* ---, -----
r--
T = 1 .0
0.8
0.4
0.2 0
(b)
Isotherms in a cubic lattice of spheres, after Deissler and Boegli [ 1 5 ) .
FIGURE 5.7
the following heat b alance equations: for a solid
O
for a fluid
r> R
with the boundary conditions temperature of ad = 1 temperature of be = 0 ab and de are adiabatic Ts = TF
at r = R .
o Ts o TF ks - = k F or
or
From the computed temperatures, the effective thermal conductivities of the quiescent cylinder beds (Eb = 0.2 1 5) are found to be expressed by
(
)
k� ks ks log 10 - = 0.785 - 0.057 log 1 0 - logto- · kF
kF
kF
(5.32)
STEADY-STATE HEAT TRANSFER 1 8 1 Heat f l o w
(O)
FIGURE 5.8
(b)
R
Model composed of cylinders, after Krupiczka [ 16, 1 7 ] : (a) model composed of cylinders, Eb = 0 . 2 1 5 ; (b) cross-section.
Extending the results to a spherical lattice, Krupiczka proposed the following equation for effective thermal conductivities of quiescent beds of spherical particles: (5.33) with
When the fluid in the bed is gas, heat transfer by radiation becomes significant at high temperatures. Under such conditions, combined conduc tion and radiation should then be considered as the mechanism of heat transfer from the surface of a particle to surfaces of neighboring particles. Wakao and Kato [ 18] calculated steady-state temperatures in a unit cell of spheres arranged in an orthorhombic lattice with € b 0.395, taking radiant heat transfer into account. The temperature computations were made by the relaxation method for a network of grids. The e ffective thermal conductivities, k�, were then evaluated from the computed temperature profiles. Figure 5 .9 shows the values of k� in a plot of k�fkF versus ks/k F . Moreover, Wakao and Kato [ 1 8 ] found that the values of k� were approximately expressed by Eq. (5 .34) =
k�
=
(k�)COND + (k�) RAD-COND·
(5.34)
......
104
10
lL .:::L ...... 0 (]) .:::L
10
3
. r:....
I
-
O r tho rhomb i c l a t t i c e
-
I
£ b = 0 . 395
� . I
:
r
eqn ( 5 . 48 )
.
.
.. · - ·
!
iI
. . ! I . . ! ' •
.
.
'
.
. .
I
.
'
.
'
'I
·'·····
: : i : : ::::
. . . ... .
:
1
..
··-·· ·
��
i
:
Ii
· ·
Zl '
:I: I:Tl ;> ...., ;> z 0 s: > (/) (/) ...., :;o ;> z (/) 'T! tTl :;o
. .
•
I
.: .: :.. t ... . : i1 . ' . .. .
k Nu r= h rDp/ s h
2
, ! --
I
00 N
.
··
. !
. . ..
.
z
10
'"Cl
1
..
- ·-··-·
• • •• • ·-··-·- --
10- 1
1
10-
I
3
jI
I •,
10-
0 • •
,..__ -·· ······- .-•
' ,1 I:
' I : ; :::-. .
.
2
10-
1
:1
• ••••
'
. ,1
.
1
I ..
•
1,
i
i
.
i ! ·j •
10
I
:
.
-
· ··
r
.
--
10
2
.
;
1
> (') � tTl 0 Ill tT1 0 (/)
. .. .. ..
. . . .. . .
•
. . . .
.1
10
3
1
,
ksl kF FIGURE 5.9
Static effective thermal conductivity for an orthorhombic lattice of spheres.
1
10
4
STEADY-STATE HEAT TRANSFER
183
The first term is the e ffective thermal conductivity when heat transfer occurs only by conduction, which corresponds to the curve with a radiant Nusselt number, Nur , equal to zero in Figure 5 .9. The second term shows the increase in e ffective thermal conductivity due to combined radiation and conduction at high temperatures. The values of (k �) RAD -COND are approximately correlated by (k�)RAD-CON D kF
=
( )
t k 0 .7 0 7Nu �·96 ks L l for 20 < s < 1 000 and Nu r < 0.3 kF kF
(5 .35)
where Nur is defined as:
(5.35a) in which, hr is the radiant heat transfer coefficient defined by Eq. (5.48) in the following section. 5.2.2
Radiant Heat
Transfer Coefficients
Radiant heat flux between two large gray surfaces at temperatures T1 and T2 is given by: q
'
=
_
a_
2 -- 1 p
(Ti - T� )
(5 .36)
where a, the Stefan-Boltzmann constant, is 5.67 x I 0-8 W rn-2 K -4, and p is the emissivity o f the gray surfaces. The radiant heat flux may be written approximately in terms of an average temperature, T, as: qI
4aT3
--
2
-- ) p
( T - T2)· t
(5.37 )
Equation (5.37) is often expressed in the following form: q
'
=
h�(Tt - T2).
(5 .38)
184
HEAT A N D MASS T R ANSFER IN PACKED BEDS
The radiant heat transfer coefficient,�' is then (5.39)
This formula, first derived by Damkohler [20], was applied to heat transfer in packed beds by Argo and Smith [10], Yagi and Kunii [7,8], and Kunii and Smith [21]. It should be mentioned that in packed beds the radiant heat flux, q1, and the heat transfer coefficient, h�, are both defined per unit area of bed cross-section. As well as Eq. (5.39), other formulae have also been used by different investigators. The one employed by Schotte [II] is (5.40)
Chen and Churchill [22] derived Eq. (5.41) by applying the radiant two flux model proposed by Hamaker [23]: hr I
=
4a
2
(a + 2b)Dp
T3
(5.41)
where DP is the particle diameter, a = absorption cross-section, and b =back scattering cross-section, both, defined per unit solid volume. Vortmeyer [24] also applied the two-flux model and obtained the following formula: hr
2B + p(l-B)
I
=
(2- p)(I- B)
4aT3
(5 .42)
where B is a geometrical bed constant. The value of B was found to be 0.1 by Vortmeyer and Borner (25]. Wakao and Kato [ 18 J proposed a formula, which includes an overall view factor, for the evaluation of the radiant heat transfer coefficient based on the unit area of a particle surface. They assumed that every two hemispheres in contact were circumscribed with a diffusively reflective cylindrical wall, Rw, as shown in Figure 5.1 0. The overall view factor, F12, between the two hemispheres is defined as:
STEADY-STATE HEAT TRANSFER
185
Heat f l ow
Hemisphere
surface 1
Hemisphere surface 2
R : d i ffusively refl ective wa l l w
FIGURE 5.10
Two hemispheres in contact.
(5.43)
where F12
=view factor from surface I to surface 2 F lR = view factor from surface 1 to wall Rw FR2 = view factor from wall Rw to surface 2 FRR = view factor of the wall Rw by itself. For the cylindrical cell under consideration, the following relationships prevail: Fl2 +FIR=
I
FRR + F R1 + FR2 = I
Moreover, the two view factors,
F;i
(5.44)
and Fj;, are related reciprocally to
J 86
HEAT AND MASS TRANSFER IN PACKED BEDS
their respective areas, Ai and Ai, by the following equation:
Flj··
AJ·
(5.45)
The view factor between the two hemispheres in contact, F12, has been computed to be 0 . 1 52. Hence, from Eqs. (5.43) to (5.45) the overall view factor, F12, is shown to be 0.576. The radiant heat transfer rate, Q, between the two hemispheres at temperatures T1 and T2, respectively, is Q=
Aa
2(�- 1) P
+
(T1- T i)
l
(5.46)
F12
where A is the surface area of the hemisphere. In terms of the radiant heat transfer coefficient, hr, based on the unit area of a particle surface, the radiant heat transfer rate equation is written as: (5.47)
( )
4a
h
= r
2
2
l-1
p
0.2268
- - 0.264
p
+-l
T3
F12
( )
T 3
1 00
(W m-2 K-1).
(5.48)
I t should be noted that the definitions of hr and h� are based on different unit areas. In the cell shown in Figure 5 . 1 0, if the cross-sectional area of the cell is denoted by A', the two heat transfer coefficients are related in the following manner: (5.4 9) The area of the hemisphere is roughly twice the cross-sectional area of the
STEADY-STATE HEAT T R A N S F E R
cell, i.e. A
�
1 87
2A', therefore, h'
hr =-· 2 r
(5.50)
These formulae are compared in terms of h;/( 4aT3) in Table 5 .4. It is found that the values of radiant heat transfer coefficients predicted from Eq. (5 .42) by Vortmeyer [24] and Eq. (5 .48) by Wakao and Kato [ 1 8] are in fairly good agreement. TABLE 5.4
Comparison of radiant heat transfer coefficients.
Emissivity, p 0.2 0.5 0.8 1
5 .2.3
Damkohler [20] Eq. (5.39) 0.11 0.33 0.67 1
Schotte [ 11] Eq. (5.40) 0.2 0.5 0.8 1
V ortmeyer [ 24] Eq. (5.42) with 8 = 0.1 0.23 0.48 0.85 1.2
Wakao and Kate [18] Eq. (5.48) 0.21 0.54 0.89 1.2
Effect of Gas Radiation
In the case of the void space in the cell of Figure 5 . 1 0 being filled with a radiating gray gas, Wakao (26] has shown that the radiant heat transfer coefficient given by Eq. (5 .48) can be corrected by the introduction of emissivity of the gray gas, Pg, as follows: (5.5 1 )
The gas emissivity is a function of gas temperature, size o f the void and the partial pressures of the radiating gases in the void. In packed beds, the size of the void is usually small so that the gas emissivity is very low. For example, the emissivity of pure C02 at a temperature of as high as 900°C
1 88
HEAT AND MASS TRANSF ER I N PACKED BEDS
in a large void of 1 em in an average beam length is found [27] to be only 0.05. Because of the low gas emissivities, the effect of radiating gas on the radiant heat transfer coefficient in a packed bed is usually insignificant. In other words, gas radiation has little effect on the effective thermal conductivity of packed bed. Equation (5 .48) can be applied, therefore, even for radiant heat transfer in packed beds containing radiating gases. 5.2.4
Effective Thermal Conductivities of Quiescent Beds at Low Pressures
The effective thermal conductivities of quiescent beds shown in Figure 5.9 are those computed assuming that fluid thermal conductivity is kept constant and does not depend on pressure. This is true only under normal pressure conditions where the length of the mean free path of the gas molecule is considerably less than the characteristic lengths of the inter stitial volumes. However, there are always some regions near the contact points of the particles where this condition is not fulfille d. Under normal pressure conditions heat transfer in these regions can be neglected since the area of these regions is very small compared with the total heat trans fer area. The effective thermal conductivities of quiescent beds for particles larger than about 1 mm, under normal pressure conditions, can be predicted from Figure 5 . 9. In beds with particle diameters of a few millimeters, the effective thermal conductivity is constant at normal pressure, but decreases with decreasing pressure when the pressure is sufficiently low. This decrease in effective thermal conductivity is due to a temperature jump at the particle surface. For fine powder particles, however, this effect of decreasing effective thermal conductivity usually occurs at atmospheric or even higher pressures. The temperature jump effect is well known from the kinetic theory of gases. One of the first investigations was made by Smoluchowsky [28] who proposed an equation, which was later modified by Hengst [29], of the following form:
28
1 +-
Dp � In
-
kF
l -ks
28
1 +Dp
STEADY-STATE HEAT T R ANSFER
1 89
where ky is the gas thermal conductivity under normal pressure, 8 is the temperature jump coefficient defined as: 8 = cjP ( c is an experimental constant and P is the pressure), a is the effective thermal conductivity in a vacuum, and b is the correction factor for beds compared with a cubic lattice of spheres. Kling [30] measured the effective thermal conductivities for steel shot hydrogen/air systems. He applied the above formula to the measured data and by choosing suitable values for the parameters b and c was able to obtain good agreement between the predicted and theoretical results. In addition to the above, measurements of effective thermal conduc tivities in various types of beds have been reported: Fulk [3 1] for beds of perlite particles at air pressure from 0 . 1 MPa to 0 . 1 Pa; Kaganer and Glebova [32] for silica gels at air pressure from 0 . 1 MPa to 1 Pa; Masamune and Smith [33] for beds of glass beads and steel shot at air pressure from 0. 1 MPa to 1 Pa; Swift [34] for fine particles of uranium, zirconium and oxidized uranium in helium, nitrogen, argon and methane at pressures from 0.1 MPa to 1 or 1 x 1 o-3 Pa; Wakao et al. [35] for glass beads in helium, hydrogen, nitrogen, acetylene and ethylene and lead shot in hydrogen in the pressure range 0 . 1 MPa to 0 . 1 Pa; and Luikov et al. [36) for quartz sand and powdered plexiglass at air pressure from 0 . 1 MPa to 0 . 1 Pa. The results obtained by these investigators all show the same characteristic S-shaped curves when the effeGtive thermal conductivities are plotted against pressure using logarithmic coordinates. 5.2.4 . 1
Heat transfer between two parallel plates at low pressure
As mentioned already, the kinetic gas theory predicts a temperature jump at the surface. The temperature jump leads to a decrease in the gas conductivity, and consequently, to a decrease in the effective thermal conductivity of packed beds. Following some simple arguments by Chapman and Cowling [37], the temperature jump, in the case of two parallel plates under the condition that the length of the mean free path, A, is less than the distance between the two plates, l, may be described by �T=
( 2 - a ) A( dT) ·.
fr
d x inner
(5.53)
where a is the accommodation coefficient, and (dT/dx)inner is the
1 90
HEAT AND MASS TRANSFER IN PACKED BEDS
temperature gradient in the range where the gas can still be considered as a continuum. With the further assumption that the temperature jump occurs along one mean free path length, the thermal conductivity of gas is given by Eq. (5.54). k*-
F-
k r
----
2(2-a):\ I+ l a k ...
(5.54)
---
I+
2(2 - a) :\0Po
--
a
lP
where P0 is the atmospheric pressure and :\0 is the length of the mean free path at P0. The values of :\0 at 0 and 20°C are listed for some gases in Table 5.5. When the pressure is further decreased to a range where :\ ""'l , the assumptions leading to Eq. (5 .53) are no longer valid. Now the molecules just hit the surfaces without colliding with each other. From the kinetic theory of gases, the rate of heat transfer in this range should be exactly proportional to the pressure. Since Eq. ( 5.54) also shows this linear TABLE 5.5 Mean free paths of several gases at atmospheric pressure. Mean free path (nm) Gas
0°C
20°C
88.6 83.5 54.9 157.9 249.2 93.8 83.9 58.4 89.3 55.3
97.5 91.1 60.7 172.1 270.9
--
Ar
co
C02
H2 He Kr N2
NH3
02
Xe
91.7 65.1 98.0
The Boltzmann mean free paths evaluated from the data at 0.100 MPa in the Handbook of Chemistry and Physics [ 38].
STEADY-STATE HEAT TRANSFER
191
pressure dependency, the equation may be used to predict kt for very low pressure ranges. In fact, the error associated with the application of Eq. (5 .54) in the case of/.. .---- l, is not as serious as one might think; since in packed beds within this pressure range, solid-solid conduction and radia tion contributions are dominant. 5 .2.4.2
Calculation of the effective thermal conductivities of quiescent beds
Wakao and Vortmeyer [ 1 9] proposed a method for the computation of effective thermal conductivities of quiescent beds of fine powder, taking the effect of contact conductivity into account. The calculation proce dure is identical to that employed by Wakao and Kato [ 18] except that kf. of Eq. (5.54) is used as the distance dependent gas thermal conductivity. The conductivity of a grid in the direction of heat flow or the x direction is
k*Fx-
kF
(5.55)
--
1+
D __!_
lx\}1
and that in the direction perpendicular to the heat flow or they-direction is (5.56)
where lx and ly are the distances (refer to Figure 5. l 0) between two surface elements on the x andy grids, respectively. The pressure parameter, \[!, is defined as: Dp a 'l'=A 2(2-a) ---
Dp
=-
a
/...0P0 2(2 -a)
(5.57) P.
1 92 HEAT AND MASS TRANSFER IN PACKED BEDS The steady-state temperatures at nodal points are computed by the relaxation method, and consequently, the effective thermal conduc tivities can be evaluated. Based on the calculated results, Wakao and Yortmeyer [ 1 9] suggested the following expression for the model: k� = (k�)cOND + (k�)RAD-COND + (k�)cONTACT·
(5.58)
The conduction term, (k�)coNo, is shown as (k�)coNolkF versus ksfkF in Figure 5 . 1 1 . It should again be noted that kF is the gas thermal conductivity under normal pressure. The combined radiation and conduction term, (k�)RAD-CON o, is approximately expressed as: (k�)RAD-COND ks
-
l.8Nur ----
1 +
0.7Nur
[
for Nu r < 0.5.
(5.59)
The term 1.8Nurf(I + 0.7Nur) corresponds to (k�)RAD-CO No/ks in a vacuum or at 'l! = 0. Figure 5 . 1 2 shows the correction factor, [, as a function of ks/kF and \f!.
�
=
10,000
5,000
2,000 1,000
u.. .:X.
500 200
10
100
......
0 z: 0 u
50
20
�
0 (l) �
10
�
5
1
2
1
0.5
'---'-----L--'-....I-!..-.I.--l..-...J..-I..l.--L.--'--JL-1-1...--1..--L--L..-'-'
1 0- 1
1
10
J.'IGURE 5.11
102 kslkF
103
(k�)coNo/kF versus ksfkr·
10
.2 0.1
4
STEADY-STATE HEAT TRANSFER 1 0.8 0.6 '-o z 0.4
.....
0.2 0.1 0.08
,_��� ��P v���� o:s
���t::;��
... �5 ���� �� zo'� 'i"'V f-
10
5 0'
OO H ... zoo'1V 00
10
1
1
I
�
QQ) �
w-
•
rr
(k�)CONTACT1ks
l w- - for
A <
4 10-
'· =
18A
....
.... �
,."'
3x!o-4
/'
/
2
3 w-
kslkF
Correction factor f versus ks/ kF.
FIGURE 5.12
V) � ...... .... u <( .... ;z 0 u
193
v
/
� t-
[7
/
v I
I
I 11
II
1
A
FIGURE 5.13
versus A. (k�)CONTACT/ks
The solid-solid contact term
(k�)coNTACT
is very small under normal
pressure conditions, but becomes increasingly significant with lowering pressure. Figure
5.13 is a plot of (k�)coN TACTIks versus A, in which the
HEAT AND MASS TRANSFER IN PACKED BEDS
1 94
{A
Theoretical �
=
1
=
5 lxlo6 5xlo-
2xlo-6
Swift
U<80ilm)-N2 25°(
0 0Q) �
10-
3
1------'--
FIGURE 5.14
(Pal
P
k� versus pressure, for a uranium powder-nitrogen system.
fractional contact area, /\, is the ratio of the solid - solid contact area to the projected area of a particle , rrD�/4. When 1\ is small, (k�)coNTACT is expressed as (k�)CONTACT =
ks
for 1\ < 3 X I 0-4.
IBI\
(5.60)
Among the published data of effective thermal conductivities of quiescent beds, Swift's [34) graphs are regarded as the most comprehensive. Thus, some of his data on fine particles of uranium and zirconium are compared with the model proposed by Wakao and Vortmeyer (19] as follows: Figure 5 . 14 demonstrates how the contact areas between two particles affect the asymptotic value of k� at low pressure. The data in a vacuum consist of solid - solid contact and radiation conductivities. Because of the fine powder and low temperature, the radiation conductivity at P < 1 Pa is small: [(k�)R AD-CONol 'l!=o = 5 x 1 o-4 W m-1 K-1• In calcu lating the theoretical curves, the accommodation coefficient is assumed to be unity. Therefore, (k�)coNTACT 2.22 x 1 o-3 W m-1 K -1 and conse quently the contact area fraction for uranium powder (ks 2 4 .7 W m-1 K-1) is found to be 1\ = 5 X ] 0 -6. The data, given in Figures 5. 15(a)-( c), are for beds of zirconium powder (ks 26 W m- 1 K -1) filled with argon, nitrogen and helium, respectively. At a pressure of 1.3 Pa, the measured data are equal to each other. This consists of (k�)coNTACT and [(k�)R AD-CONDl'll=o· The contact area fraction, 1\, is eventually found to be 7 X 1 o-6. The graphs demonstrate =
=
=
STEADY-STATE HEAT TRAN SFER
195
{'
Theore' ical r. •
7xlo-6
Swift
Zr< 19Q;.ml -Ar 25°C
P
(Q)
Theoretical :.
7xF;-:, �
=
r.
i
\.
[PcJ
= 1. 0 0.3
Swift
Zr 09Ql.m) -N 2 25oc
[Po)
(b)
...... ..... I
:..:
..... I
E
.
{
Theoretical
1 1 10-
A
•
7xJo-6
3: ...
ocv .:.£
Swift
10-2
Zr0901lml-He 25"C
10-3 1 10(C)
FIGURE 5.15
1
10
p
k� versus pressure: (a) zirconium powder-argon system; (b) zirconium powder-nitrogen system; (c) zirconium powder-helium system.
1 96 HEAT AND MASS TRANSFER IN PACKED BEDS how the effective thermal conductivities are influenced by the values of the accommodation coefficient. It is found that the experimental data are best fitted by assuming a= I for atgon and nitrogen, and a= 0. 3 for helium. The low accommodation coefficients for lighter gases such as helium and hydrogen are in good agreement with the reported literature values [39-4 1 ]. It should be pointed out that the effective thermal con ductivities for such fine particles are still influenced by the accommoda tion coefficient even at normal pressure. The small contact area fraction, A= (5 "' 7) x 1 0-6, as found from Figures 5 . 1 4 and 5 . 1 5 , is due to the fact that a rather small apparatus (cylindrical bed: 5 .08 em (2 in) long and 1 . 9 em (0.75 in) diameter) was employed in the measurements. Kling [30] measured the effective thermal conductivities of beds of steel shot-hydrogen/air, and determined the values of the parameters b and c in Hengst's equation, Eq. (5.52), for the systems studied. The contri butions of solid-solid contact and radiation to the effective thermal con ductivities were not stated in their paper, therefore, only the gas-solid conduction contribution (the first and second terms on the right hand side of Eq. 5 . 52) is compared with the (k�)coN o term of Eq. (5 . 58). Note that, because of the high thermal conductivity of steel shot (ks fkF = 200 10
.......
rl I ::.<: rl I E
Kling: Steel sllot<3.18rm1l-H , air 2 33·c l
-=::o.::s.:..---1 Ai r
....:3:...
0 z: C> u C>QJ .:.<:.
10-1
Kling's data and equation
(k�lCOND
�
<with aH = 0.3, aair= ll 2
.... _ _ .. --... .....__ .. _.._ .L-.I L....L."-L... _ - .I... --L.L.--L.. ...J -.JL.....J... L.--1 .J-J 10 -2 ..._ 3 10 10
P
FIGURE 5.16
[Po)
(k�)COND for steel shot.
STEADY-STATE HEAT TRAN SFE R 1 97
for hydrogen and ks/kF = 1500 for air), the correction factor, /, for radiation conductivity is almost unity. Therefore, (k�)RAD-CON D is independent of pressure and (k�)RAD-COND + (k�)coNTACT corresponds to a in Eq. (5 . 5 2). Figure 5.16 shows good agreement between the Hengst Kling equation and the model proposed by Wakao and Vortmeyer [ 19 ]. The contact conductivity is considered to depend on such physical properties as hardness, plasticity and thermal expansion of the particles, roughness of the particle surfaces, bed weight and external mechanical load. If the particle is covered with an oxide film, for instance, an addi tional resistance should be accounted for in (k�)coNTACT· At present, there is no way of predicting the value of (k�)coNTACT, and this should be evaluated only from effective thermal conductivity measurements in a vacuum. 5.3
Wall Heat Transfer Coefficients
Wall heat transfer coefficients, hw, have also been measured by many research workers. From an examination of the published data in the literature Li and Finlayson [42] found that a large number of the data of hw had a bed length effect or entrance effect: some of the hw data
Re
FIGURE 5.17
hwD pfkF versus
Re
for a spherical particle-air system, after Li and finlayson [ 42].
198
HEAT AND MASS TRANSFER IN PACKED BEDS
were obtained from Eqs. (5 19) to (5.2 1 ) even though plots of the ln ((Tw- T)/(Tw- T0)l-x relationships did not quite form a straight line. .
5 .3 .I
Spherical Particle-air System
Based on the published hw data of Yagi and Wakao (43] and Kunii et al. [44], which are considered to be free from the length effect, Li and Finlayson [42] plotted hwDp/kF versus Re as shown in Figure 5.17, and from which they obtained the following correlation: h D -� k..-
=
(5.61)
0.17Re0·79
for 20�Re �7600 and 0.05 �Dp/Dr �0.3, where Dr is the column diameter. 5.3.2
Cylindrical Particle-air System
Similarly, Li and Finlayson [42] correlated the selected hw data obtained by Hatta and Maeda [1, 5], Coberly and Marshall (2], Felix (45], Phillips et al. [46], and Hashimoto et al. (47] as shown in Figure 5.18. They suggested the following empirical relationship between hwDp/kF and Re: hwDP kF
=
0.16Re0·93
(5.62)
for 20 � Re � 800, 0.03 � Dp/Dr � 0.2 and Dp = 6 Vp/Sp, where Sp and Vp are the surface area and volume of a particle, respectively. Yagi and Kunii [ 48] correlated their data of hw into the following form: hwD P kF
=a' + 0.054(Pr)(Re)
(5.63)
for 20 �Re � 2000, where a' depends on the shape of the solid particle, Dp and Dp/Dr. The a' values for glass beads and Celite cylinders were found to be: a
I
1 .2 3 5
D P (mm)
Dp/Dr
0.8 ...__ 0.9 1.8 "'3 .2 4.3 "'6.4
0.02 "'0.04 0.04"'0.07 0.08"'0.1 7
STEADY -STATE HEAT TRANSFER
1 99
l.L -""' ....... 0. Cl
-� 10
1 10 Re
FIGURE 5.18
hwDpfkF
versus Re for a cylindrical particle-air system, after Li and Finlayson [42 ].
However, Gunn and Khalid [49] found that their experimental data with metal and glass beads, for Re = 2 to 400, were larger than the values estimated from Eq. (5.61 ) . The heat balance equation, Eq. (5. 1 ), is generally derived based o n the assumption that the fluid mass velocity per unit area of bed cross-section is constant. However, Schwartz and Smith [50], in their work on the measurement of radial fluid velocities, found that the radial velocity profile, as shown in Figure 5.19, reached a maximum value at about one particle diameter away from the column wall. Their experimental results subsequently led to studies of radial voidage variations by several groups of researchers. In the experimental work carried out by Kimura eta!. [5 1] and Roblee et al. [52], the void space in the bed was filled with molten wax, which was then allowed to solidify. A slab of the bed composed of the particles and wax was mounted in a turning lathe chuck. Annular layers were shaved off and the void fraction of each layer was found from the volume of the melted wax collected for the layer. Kimura et al. measured the radial void age variation for beds of broken pieces of limestone, and Roblee et a!. did this for beds of cork spheres, wood cylinders, carbon Raschig rings and carbon Berl saddles. In the case of spheres, the voidage variation,
200 HEAT AND MASS TRANSFER IN PACKED BEDS 1.4�----� 2RT = 4 in 1.2 1.0
l/8-in
cylinder
> ....,
g
·-
0.8
<1> > <1> Ol 0 ' <1> > 0 ....... >
n.G
....,
u 0 <1> > ..., c: 0 c....
1.2
1.0
0.8
0.6
0
0.2
Dis tance
0.4 from
0.6
wall,
0.8
1.0
1 - r/RT
FIGURE 5.19 Radial velocity profiles measured by Schwar tz and S mith [50], for tin (0.32 e m) and tin (0.64 e m) diameter cylinders in a 4 in (10 e m) diameter column.
shown in Figure 5.20, indicates that the voidage reaches a minimum at one particle radius from the wall, within alternate maximums and minimums occurring at successive particle radii. The amplitude of the cycling decreases as the distance from the wall increases. Pillai [53], instead, counted the number of particles in the vicinity of the wall in a two-dimensional vessel. The results show, similar to those of Roblee et al. [52], that the voidage variation follows a heavily damped oscillatory curve with high voidage at about one particle diameter away from the wall. This explains well the experimental results of Schwartz and Smith [50] that velocity is at a maximum at about one particle diameter from the wall. The large voidage variation near the wall makes the velocity profile and effective thermal conductivity near the wall differ from those in the bed core. In solving Eqs. (5 . 1) and (5 .2), however, both G and ker are assumed to be constant. Hence, the wall heat transfer coefficient not
STEADY-STATE HEAT TRANSFER
201
1.0 c 0 ·�
u 0
'...... "0 0 >
0.8 0.6
0.4 0. 2 0 0
1
2
3
4
Distance from wall in particle diameter
FIGURE 5 .20 Radial voidage variation measured by Roblee et al. [52], for 0.76 in (1. 9 em) diameter spheres in a 6.7 in (17 em) diameter column. only stands for the intrinsic heat transfer at the wall, but also includes the effects resulting from radial variations in the fluid velocity and ker in the vicinity of the wall.
5.4
Overall Heat Transfer Coefficients
The overall heat transfe£ coefficients, defmed according to Eq.
(5.1 0),
have been determined experimentally for various solid-fluid systems. Empirical correlations obtained by Li and Finlayson
[42]
are presented as
follows:
5.4.1
Spherical Particle-air System
UoDr = 2.03Re 0·8 exp kF
--
for
6DP
__
Dr
(5.64)
20 �Re � 7600 and 0.05 �Dv/Dr � 0.3.
5.4.2
Cylindrical Particle-air System
UoDr --
kF
for
(- )
=
1.26Re0•95 exp
20 �Re � 800 and 0.03 �Dp/Dr � 0.2.
( ) 6DP Dr
- __
(5.65)
202
HEAT A N D MASS TRANSFER I N PACKED BEDS
5.5
Effective Axial Thennal Conductivities
Yagi et al. (54) were the first to obtain the effective axial thermal conductivities of packed beds. As shown in Figure 5.2 I, their axial steady state heat transfer measurements were made with the bed packed in an adiabatic column. The packed bed was heated from the top by an infra red lamp so that heat penetrated downwards into the bed, while air flowed countercurrently upwards through the bed from the bottom. From the following heat balance equation: (5.66) where x is the distance in the bed measured from the exit, the axial temperature is found to be T- T0a exp
(-
G _ C_ ..-x_
keax
)
(5.67)
Infrared lamP
Heat
Adiabatic column
t
Air
FIGURE 5.21
Apparatus used for effective axial thermal conductivity measurements by Yagi eta!. [54].
STEADY-STATE HEAT TRANSFER
203
where T0 is the temperature of air flowing into the packed bed. Therefore, the values of keax were obtained from measurements of axial temperature profiles. The data were found to be correlated by (5.68) with o = 0.8 for glass beads and limestone broken pieces, and o = 0.7 for metal spheres at low air flow rates. By the time Yagi et al. had finished their work on the measurements of keax. no research had been reported on effective axial thermal conductivity. Since they thought that keax should be lower than ker· they were rather puzzled to find that the coefficient, o, of the second term on the right hand side of Eq. (5.68) was larger than that of Eq. (5.30) for effective radial thermal conductivity, and hesitated to publish their results. However, in the March 1957 issue of A!ChE J., McHenry and Wilhelm (55] published the results of their work on mass dispersion in packed beds in which they had made frequency response measurements with the binary gas systems, HrN2 and C2H4- N 2 , in beds of non-porous particles. They found that Peax = 1.88 ± 0.15 in the range of Reynolds number, 10 to 400. Also, McHenry and Wilhelm came to realize, from a theoretical considera tion, that an axial Peclet number of two should be expected if the frequency response measurements were made with a series of n perfect mixers, where n was the number of particles traversed between the bed inlet and outlet. Soon after this, in the June 1957 issue of A!ChE J., Aris and Amundson [56] published a theoretical work in which they reached the same conclusion as that of McHenry and Wilhelm. As far as beds of Raschig rings with flowing water are concerned, axial fluid dispersion coefficients were measured as early as 1953 by Danckwerts (57] and Kramers and Alberda (58]. Danckwerts obtained Peax = 1/1.8 at Re = 24, and Kramers and Alberda found Peax 1/(1.1 ± 0.1) for Re 75 to 150. Their Peax results are lower than those of McHenry and Wilhelm [55]. In any case, the work of McHenry and Wilhelm made Yagi et al. (54] submit their results on the effective axial thermal conductivity for publica tion, and it appeared in A!ChE J., three years after the completion of the work. The axial fluid mixing coefficient per unit cross-sectional area of a packed bed has been shown to be 0.5DP u. According to a heat- mass analogy, the contribution of turbulent flow to the effective axial thermal =
=
204
HEAT AND MASS TRANSFER IN PACKED BEDS
conductivity is considered to be 0.5DPuCrPF· Therefore, Eq. (5.69) is recommended for the evaluation of keax in a wide range of Reynolds number. keax kr
=
k0 ___:_
kr
+ 0.5(Pr)(Re).
(5.69)
REFERENCES S. Hatta and S. Maeda, Kagaku Kogaku 12,56 (1948). C. A. Coberly and W. R. Marshall, Clzem. Eng. Prog. 47, 1 4 1 (1951). A. P. De Wasch and G. f. Fromcnt, Chenz. Eng. Sci. 27,567 (1972). D. G. Butu1ell, H. B. Irvin, R. W. Olson and J. M. Smith, IHC 41, 1977 (1949). S. Hatta and S. Maeda, Kagaku Kogaku 13,79 (1949). W. E. Ranz, Chem. Eng. Prog. 48, 247 ( 1952). S. Yagi and D. Kunii, Kagaku Kogaku 18, 576 ( 1 954). S. Yagi and D. Kunii, A!ChE J. 3, 373 (1 957). T. E. W. Schumann and V. Voss, Fuel 13, 249 (1934). W. B. Argo and J. M. Smith, Chern. Eng. Prog. 49, 443 (1953). W. Schotte, AIChEJ. 6, 63 (1960). 0. Krischer and K. Kroll, Die wissenschaftlischen Grundlagen der Trocknungstechnik., Bd. 1, Berlin-Gottingen-Heiderberg (1956). [ 13] E. U. SchlUnder, Chern. Ing. Tech. 38,967 (1966). [ 14] P. Zehner and E . U. SchlUnder, Chern. Ing. Tech. 42, 933 (1970). [ 15] R. G. Deissler andJ. S. Boegli, Trans. ASME 80, 1 4 1 7 (1958). [ 16] R. Krupiczka, Chemia Stosowana 2B, 183 (1966). [17] R. Krupiczka,Int. Chem. Eng. 7, 122 (1967). [18] N. Wakao and K. Kato,J Chern. Eng. Japan 2,24 (1969). [ 19] N. Wakao and D. Vortmeyer, Chern. Eng. Sci. 26, 1753 (1971). (20] G. Damkohler, Der Chemie Ingenieur, Eucken-Jacob, Vol. 3, Akadem. Verlag. Leipzig, p.445 (1 937). [21] D. Kunii andJ. M. Smith,A!ChEJ. 6,7 1 (1960). [22] J. C. Chen and S. W. Churchill, A!ChE J 9,35 (1963). [ 23] H. C. Hamaker, Philips Res. Reports 2,55, 103, 112, 420 ( 1 947). [24] D. Vortmeyer, Fortschr. Ber. VDI-Z Reihe 3, Nr. 9, VDI-Verlag., DUsseldorf (1966). [25 j D. Vortmeyer and C.J. Borner, Chern. Ing. Tech. 38,1077 (1966) . (26] N.Wakao, Chern. Eng. Sci. 28, 1117 (1973). [27] H. C. Hottel, Heat Transmission, edited by W. C. McAdams, 3rd edn., McGraw-Hill, New Yark, Ch. 2 (1954). [28] M. Smoluchowsky, Wiener Akad. 107, 304 ( 1 898). [29] G. Hengst,Dissertation Technische Universitat MUnchen (1934). [30] G. Kling, VDI-Forschung 9,28 (1938). (31] M. M. fulk, Progress in Cryogenics 1, 63 (1959). [32] M.G. Kaganer and L. I. Glebova, Kislorod 1, 13 (1959). [33) S. Masamune and J. M. Smith, Ind. J::ng. Chern. Fund. 2, 136 (1963). [34] D. L. Swift, Int. J. Heat Mass Transfer 9, 1061 ( 1 966). [35) N. Wakao, S.Omura and M. Fukuda, Kagaku Kogaku 30, 1 119 (1966). [36] A. V. Luikov, A. G. Shashkov, L. L. Vasiliev and Y. E. fraiman, Int. J. //eat Mass Transfer 11, 1 1 7 (1968). [1] (2] [3] [4] (5] [6 ] [7] [8] [9] [10] [ 11] [12]
STEADY-STATE HEAT TRANSFER
(37] [38] [39] [40] [41] [42] [ 43] [44] (45] [46] [47] [48] [49] (50] (51] (52) [53] [54] (55) (56] [57] (58]
205
S. Chapman and T. G. Cowling, The Mathematical Theory of Non· Uniform Gases, Cambridge University Press, p. 104 (1960). Handbook of Chemistry and Physics, edited by R. C. Weast, 53rd edn., Chemical Rubber Company, Ohio, F-174 (1972-73). M.Knudsen,Ann. Phys. 34,593 (1911). J. H.Wachmann,PhD thesis, University of Missouri (1957). M. L. Wiedmann and P. R . Trumpler, Trans. ASME 68, 57 (1946). C. H.Li and B. A.Finlayson, Chenz. Eng. Sci. 32, 1055 (1977). S. Yagi and N . Wakao, AICIIE J. 5, 79 (1959). D. Kunii, M. Suzuki and N. Ono, J. Chem. Eng. Japan l , 21 (1968). T. R. felix, PhD thesis, University of Wisconsin (1951). B. D. Phillips, F. W. Leavitt and C. Y. Yoon, Chern. Eng. Prog. Synzp. Ser. 56 (No. 30), 219 (1960). K. Hashimoto, N. Suzuki, M. Teramoto and S. Nagata, Kagaku Kogaku 4, 68 (1966). S. Yagi and D.Kunii, A!Ch£ J. 6, 97 (1960). D. J. Gunn and M. Khalid, Chern. Eng. Sci. 30, 261 (1975). C. E . Schwartz and J. M. Smith,/nd. Eng. Chem. 45,1209 (1953). M. Kimura, K. Nono and T. Kaneda, Kagaku Kogaku 19,397 (1955). L. H. S. Roblee, R. M. Baird and J. W. Tierney, A!ChE J. 4,460 (1958). K. K. Pillai, Chern. Eng. Sci. 32, 59 (1977). S. Yagi, D. Kunii and N. Wakao, A!ChE J. 6, 543 (1960). K. W. McHenry and R. H. Wilhelm, A!ChE J. 3,83 (1957). R. Aris and N. R. Amundson, A!ChE J. 3,280 (1957). P. V. Danckwerts, Chern. Eng. Sci. 2, 1 (1953). H. Kramers and G. Alberda, Chern. Eng. Sci. 2, 173 (1953).
6 Thermal Response Measurements
THE TECHNIQUES
of parameter estimation from tracer input-response signals, as discussed in Chapter 1 , may be applied to the estimation of heat transfer parameters from thermal responses. The Dispersion-Concentric model (D-C model), based on the assumption that fluid is in the dispersed plug flow mode and that the intraparticle temperature/concentration profile has radial symmetry or is concentric, has been used widely for the analysis of unsteady-state heat transfer in packed beds as well as for adsorption and catalytic reaction systems. For a bed of inert spherical particles, the unsteady-state heat balance equations based on the D- C model are (6. 1 ) (6.2) with atr =R where particle surface area per unit volume of packed bed specific heat of fluid CF Cs = specific heat of solid particle hp = particle-to-fluid heat transfer coefficient ks thermal conductivity of solid particle 206 a =
=
=
(6.2a)
THERMAL RESPONSE MEASUREMENTS
207
particle radius, Dp / 2 T..- =temperature o f fluid Ts = temperature of solid particle U interstitial fluid velocity
=
=
=
=
=
The rate parameters involved in unsteady- state heat transfer are usually
=
(0.6"' 0.8)
=(0.6"' 0.8) aF + O.SDp U
for Re 5
(6.3)
where O'f is the thermal diffusivity of fluid. However, in 1974, Gunn and De Souza [ 1 ] found, from frequency response measurements, that the values of
208
HEAT A N D MASS TRANSFER IN PACKED BEDS
In the following sections we shall outline the techniques for parameter measurement by thermal response. The prediction of fluid thermal disper sion coefficients from axial effective thermal conductivity by model comparison will also be illustrated. 6.1
Frequency Response Measurements of Gunn and De Souza
In applying the technique of parameter measurement by thermal frequency response, Eqs. (6. J) and (6.2) based on the D- C model are solved under the following conditions: at x = 0 (within a bed)
T�(t) =A� cos wt
at x
Tr= 0
= oo
where A� is the amplitude of the input signal, T}(t), with frequency The stationary-state solution of the response signal, T¥(t), at x = L, is
TV(t) =A� cos (wt
--
8L)
w.
(6.4)
where A� is the amplitude of the response signal, which, for a bed of inert particles, is defined as:
A� = A:U exp
[C,�x - ) L] 81)
(6.4a)
and
(6.4c)
(6.4d)
T H E R M A L RESPONSE M EASUREMENTS
p =
Bw {12 [cosh (2'w) + cos (2'w) ] - (H - 1 ) [cosh (2'w)- cos (2 'w)] + 'w(H- 2) [sinh (2'w) + sin (2'w)] }
q
=
209
HBw'w [sinh (2'w ) - sin (2¢'w)]
(6 .4e) (6.4f)
ksaH REb Bw 2 = {(H -1) ( cosh(2¢'w) - cos(1'w)] + 2¢ 2 [cosh(2¢'w) + cos(2'w)] +
w -R I
2'w(H - 1) [sinh (2¢'w) - sin (2'w)]}
(we )
p 11 s s 2 2ks
(6 .4g) ( 6.4h) (6.4i)
Hence, the amplitude ratio, Aw, of rV(t) to T�{t) is then
=
exp
[(�-o�) L]
2aax When 2'w< I, Eqs. (6.4e) and (6.4f) become
(6.5)
(6.6)
q
aR
= wCsPs -.
3€b
(6.7)
Gunn and De Souza [ I ] made thermal frequency response measure ments in packed beds of glass and metal spheres over the range of Reynolds number from 0.05 to 330 for air flow. In their analysis, first, they computed theoretical amplitude ratios from Eq. (6.5) with various assumed values of aa x, h p and ks, and then compared these with measured
210
HEAT A N D MASS TRANSfER IN PACKF.D BEDS
values. At low Reynolds numbers. say Re < 1 , the amplitude ratio was found to be sensitive only to aax· In fact, at low flow rates and inter mediate frequencies, "(2 becomes much greater than one. Using this and Eqs. (6.6) and (6.7), Eq. (6.5) simplifies to In Aw
LU =
- -
2aax
(
l
+
aR Cs P s -
3Eb CFPF
)1/2 (--w )1/2 lw /2 2 aax
L.
(6.8)
According to Eq. (6.8), a graph of ln Aw versus is linear, and the value of aax may be found from the intercept or the slope of the straight line. Equation (6.8) also shows that the amplitude ratio depends only upon aax, and is independent of hp and ks. In fact, the imposition of a relatively slow sinusoidal change, at low flow rates, allows a particle and its surrounding envelope of fluid to reach thermal equilibrium, so that the attenuation of the temperature wave is governed by the axial thermal dispersion alone. At higher Reynolds number, thermal response is found to be equally sensitive to both aax and hp. The sensitivity of the response to ks is generally low, particularly for high thermal conductivity particles. Values of aax, obtained by Gunn and De Souza [ 1 ] , are plotted in terms of Dp U/aax in Figure 6. 1 . Included in the graph are the mass dispersion data reported by Edwards and Richardson [2] and Gunn and Pryce [3]. These mass dispersion data, which were obtained for argon-air systems (with a Schmidt number of 0.77) in packed beds with Eb 0.36 to 0.38, are in good agreement with the solid line representing Eq. (6.3) based on Pr 0.77 and Eb = 0.37. In any case, the aax values Gunn and De Souza determined from thermal responses are much greater than the a�x values estimated according to mass dispersion analogy, except those at high Reynolds numbers. =
=
6.2
Parameter Estimation from One-shot Thermal Input
Parameter estimation may be made with a one-shot heat input provided that the temperature-time curves are measured at two points downstream. If the temperature signal measured at the first downstream point is Tl.-(t), the response signal, T�!(t), at a distance L from the first measuring point
T H E R M A L RESPONSE MEASU RF.MENTS
211
data Mass Heat
(.�]
[3] 1.1]
(81
!
0 •
argon-a i r i n g l a s s beads
+ 0 X • o 6 l ai r i n
glass beads
&l
10 eon ( 6 . 3 >
-g 0 => I X c. o 0
'I
10- 1
ecm < 6 . 28 >
1
FIGURE 6.1
10
Re
Com parison of Dp U/aax and Dp U/a�x·
is obtained by solving Eqs. (6. 1 ) and (6.2) under the following conditions:
TF = Ts = O r ... = T�·(t) TF 0 =
at t at x
=
0
=
0 (within a bed)
at x = 00•
There are two mathematical techniques for the solution of the response signal (refer to Section 1 . 1 .6). 6.2.1
Prediction of Response Signal Using a Convolution Integral
By using a convolution integral, the response signaL T�! caic(t), is
212
HEAT AND MASS TRANSFER IN PACKED BEDS
f T�t
rP calc{t)
=
(6.9)
exptm f(t - n d�
0
where f(t) is the Laplace inversion of the transfer function. The transfer function, F(s), of an inert bed within the distance L is (6. 10)
where 1
- ----ks
cf>s = R
hpR
2 1 )1 ( --,;; CsPs
+
1
(6. 1 Oa)
cf>s coth cf>s - 1
(6. 1 0b)
s
The inversion is expressed, in terms of a Fourier series, over a period of 2r, as: f(t)
1
=
1
oo
(
nrrt
nrrt
r
r
- + - L Rn cos - - In sin -
2r
r n=l
)
(6. 1 1 )
where 2r is a period of time longer than the time required for the tailing portion of the measured response signal to vanish� Rn and In are the real and imaginary parts of F(inrr/r), respectively, i.e. (6. 1 2)
and R n = l\ exp
[(�-of/) ] 2aax
L cos (8L)
)
w = mrj-r
(6. 1 3)
T H E R M A L RESPONSE MEASUREMENTS
In
= -
( [( _!!__ - ) L] (OL)) exp
2aax
e11
sin
w = mrjr
213
(6. 1 4)
where e and 77 are defined by Eqs. (6.4b) and (6.4c). With these Rn and In values the response signal is then calculated from Eqs. (6.9) and (6. 11):
TF calc(t ) II
-
I
2T
t
I 0
TF ex pt(n d� I
oo [ +- L 1 T
n =I
t
Rn
I 0
T�- expt(�)
COS
-n1r T
(t - �) d �
(6.15)
6.2.2
Prediction of Response Signal by Fourier Series
The response signal, expressed by a Fourier series over the period 2r, is a0t
(
)
t n1rt , ant cos n1rt TF calc(t) = + L - + b n sin - . 2 n =I T T 11
00
(6.16)
Similar to the derivation in Section 1.1.6. 2, the Fourier coefficients are determined by (6.1 7) where an - -1 T _
2T
I 0
n1rt TFI expt cos - dt T
(6.17a)
214
HEAT AND MASS TRANSFER IN PACKED BEDS
1 bn - _
T
2T
J
1 nrrt TF expt sm - dt. .
(6. 1 7b)
T
0
Following the method discussed in Section 1. 1.6, the predicted response signal, TtJ calc(t), is then compared with the measured signal T P expt (t ). Parameters are determined by minimizing the difference between the two response signals, for example, using the root-mean-square-error, €, defined by Eq. (6. 1 8). (The Fortran programs listed in Appendix B can easily be modified for use in the computations of the thermal response signal and the root-mean-square-error.)
J ( TfJexpt - T�! 2T
1/2
catc) 2 dt
0 f=
f (TP expt)'
(6. 18)
2T
dl
0
6.2.3
Detennination of Particle-to-fluid Heat Transfer Coefficients and Axial Fluid Thennal Dispersion Coefficients
The determination of the heat transfer parameters, particle-to-fluid heat transfer coefficient and the axial fluid thermal dispersion coefficient, is demonstrated in Example 6 . 1 . Example 6.1 Thermal response measurements were made using a bed of glass beads. Table 6.1 lists the experimental conditions and the measured input and response signals (data for Runs 2 and 3 are from Shen et al. (4] ). Find the heat transfer parameters. SOLUTION
For illustration purposes, the normalized input and response signals (similar to Eq. 1.50) for Run 2 with Re 1 7 .6, are shown in Figure 6.2. The response signals computed with various assumed values of the three =
215
THERMAL RESPONSE M EASU KFMENTS
TABLE 6 . 1 Experimental conditions and temperature signals recorded in a bed of glass beads. Glass beads: Dp = 0.5, 1.3 and 2.7 mm; Cs = 670 J kg-1 K-• ; P S = 2500 kg m-3; = 0.88 W m-• K-•. Fluid: air a t 0.1 MPa. Average temperature of the air in the bed = 20°(; kF = 0.026 W m-1 K-1• The response signals were measured in the packed beds at 4 em from the bed exit. Input and response signals measured (before normalized ; the figures are ten times the actual temperature increase (K) above the room temperature).
ks
Time
Run 1
Dp 11�!
(s)
=
Eb = L = Re =
11
0 1 2 3 4 5 6 7 8 9
0.0 12.5 33.0 52.5 60.0 63.2 65.2 64.9 64.0 62.2
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Run 2
0.5 mm 0.39 1 . 1 em 0.54
TFi l
Run 3
Dp =
1 .3 mm Eb = 0.39 L = 1 . 3 em Re = 1 7 .6
TFil
Dp = 2.7 mm Eb L Re
0.40 2.4 em = 229 =
=
TFi l
0.0 3.3 9.5 1 5 .2 21.0 26.9 30.1 33.2
0.0 0.9 0.9 0.9 1 .0 1.8 3.1 5.9 9.8 1 3.9
0.0 26.7 67.2 95.0 122.0 129.5 127.9 122.0 1 1 2.9 103.9
0.0 0.7 3.0 4.6 8.0 1 2.0 16.4 20.8 25 . 3 29.9
60.6 58.0 55.4 52.8 50.4 47.8 45.4 43.2 4 1 .0 38 .8
36.6 39.0 41.4 42.4 42.9 43.0 43.0 42.9 42 .8 42.1
19.8 25.9 32.9 39.8 46.3 52.8 58.8 64.3 68.7 72.5
0.0 0.1 0.2 0.5 0.8 1.0 1.3 1.9
94.6 85.4 76.9 68.9 6 1 .4 54.4 48.7 42.8 37.9 32.9
34.1 38.1 4 1 .9 45.3 47.9 50 .2 52.4 53.6 54.6 55.2
36.9 35.0 33.1 31.5 29.8 27.7 25.9 24.9 23.9 22.9
4 1 .0 40.1 39.0 38.3 37.3 36.2 35 . 1 33.7 32.7 31.6
75.0 77.1 78.0 78.6 78.6 77.7 76.5 74.7 72.7 70.2
2.8 3.8 4.4 5.8 6.9 8.7 10.1 1 2.1 14.3 16.7
29.3 25.9 22.4 19.8 17.2 15.1 1 3. 3 1 1 .6 10.2 9.0
55.6 55.6 55.6 54.8 53.8 52.6 5 1 .3 49.1 47. 5 45.6
216
H E A T AND MASS TRANSFER IN PACKED BEDS
TABLE 6.1 (Continued) Time
Run
Run 2
1
Run 3
Dp = 0.5 mm €b = 0.39 L = 1 . 1 em Re = 0.54
Dp
1 .3 em Re = 17.6
Re
r}
r}I
r}
rV
T}
rV
30 31 32 33 34 35 36 37 38 39
21.6 20.4 19.5 18.4 17.5 16.2 15.3 14.5 13.7 13.1
30.5 29.3 28.2 27.0 25.9 24.8 23.7 22.7 21.6 20.7
67.7 64.9 62.1 59.3 56.1 53.4 50.5 47.6 44.8 42.5
18.9 20.9 22.9 25.3 27.7 29.8 31.9 34.3 36.3 37.9
7.9 6.9 5.9 5.3 4.8 4.3 3.7 3.2 2.7 2.5
43.6 4 1 .7 39.6 37.7 35.6 33.5 31.5 29.5 27.5 25.7
40 41 42 43 44 45 46 47 48 49
12.6 12.1 11.6 10.8 10.3 9.6 9.3 9.0 8.7 8.4
20.2 19.6 18.8 18.1 17.5 16.6 16.0 15.4 14.6 14.2
39 .6 37.4 34.5 32.5 30.4 28.5 26.5 24.5 22.9 21.5
39.9 41.8 42.8 44.3 45.5 46.5 47.4 47.7 48.1 48.7
2.2 2.0 1.7 1.6 1.4 1.2 1 .1 1.0 0.9 0.8
24.3 22.5 20.7 19.3 17.8 16.6 15.5 14.4 13.3 12.2
50 51 52 53 54 55 56 57 58 59
8.0 7.7 7.4 7.0 6.7 6.3 6.0 5.7 5.4 5.2
13.6 12.9 12.4 12.0 11.5 11.1 10.6 10.2 9.8 9.3
19.8 18.4 17.3 16.2 15.0 13.8 12.6 11.5 10.4 9.9
48.7 48.7 48.7 48.7 48.3 47.7 47.1 46.4 45.1 44.3
0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5
11.3 10.4 9.5 8.6 7.7 7.1 6.5 6.0 5.5 5.0
60 61 62 63 64 65 66 67 68 69
5.0 4.7 4.5 4.4 4.3 4.0 3.8 3.7 3.5 3.3
8.9 8.5 8.1 7.8 7.5 7.2 7.0 6.7 6.4 6.2
9.3 8.8 8.2 7.5 6.8 6.3 5 .8 5.5 5.2 4.8
43.4 42.4 41.3 40.3 39.1 37.9 36.7 35.5 34.3 33.1
0.5 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.1 0.1
4.5 4.3 3.9 3.5 3.2 3.0 2.7 2.4 2.3 2.1
nflt
(s) n
€b L
=
1.3 mm
= 0.39
=
Dp = 2.7 mm €b L
=
= =
0.40 2.4 em 229
217
T H E R M A L RESPONSE M E A S U R E M ENTS
TABLE 6.1 (Continued) Run 1
Run 2
Dp = 0.5 mm fb = 0.39
Dp = 1.3 mm fb = 0.39 L = 1.3 em Re = 17.6
Dp
r}
rV
T}
rP
0.1 0.0
2.0 1.9 1.8 1.7 1.5 1.4 1.3 1.2 1.1 1.1
Time nt:l.t (s)
L
Re
= =
1.1 em 0.54
Run 3 fb
L Re
=
2.7 mm
= 0.40 = =
2.4 em 229
11
TFI
TilF
70 71 72 73 74 75 76 77 78 79
3.2 2.9 2.7 2.5 2.4 2.1 2.0 1.9 1.8 1.7
6.0 5.7 5.5 5.2 5.0 4.7 4.4 4. 1 3.9 3.7
4.4 4.2 4.0 3.7 3.4 3.2 2.9 2.7 2.5 2.3
31.9 30.6 29.4 28.2 27.0 25.8 24.6 23.4 22.3 21.3
80 81 82 83 84 85 86 87 88 89
1.5 1.4 1.2 1.2 1.1 0.9 0.9 0.9 0.8 0.7
3.5 3.3 3.2 3.1 3.0 2.9 2.7 2.6 2.5 2.4
2.1 2.0 1.8 1.6 1.5 1.4 1.3 1.1 1.0 0.9
20.3 19.3 18.3 17.4 16.4 15.5 14.6 13.9 13.2 12.6
1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.5 0.5
90 91 92 93 94 95 96 97 98 99
0.6 0.6 0.5 0.4 0.4 0.2 0.1 0.1 0.1 0.0
2.2 2.0 1.9 1.8 1.6 1.5 1.4 1.3 1.2 1.1
0.8 0.7 0.6 0.5 0.4 0.4 0.3 0.3 0.3 0.3
1 1 .9 11.3 10.6 10.0 9.4 8.9 8.4 7.8 7.3 6.8
0.4 0.4 0.3 0.2 0.2 0.1 0.1 0.0
1.0 0.9 0.8 0.6 0.4 0.3 0.2 0.0
0.2 0.2 0.2 0. 1 0. 1 0.1 0.1 0.1 0.0
6.5 6.2 5.9 5.6 5.2 4.9 4.6 4.3 4. 1 3.9
100 101 102 103 104 105 106 107 108 109
218
HEAT AND MASS TRANSFER IN PACKED BEDS
TABLE 6 . 1 (Continued) Time
Run 1
Run 2
Dp = n t:l. r (S)
0.5 mm €b = 0.39 L = 1 . 1 em Re = 0.54 il TF
n
Dp = Eb =
L
=
Re = 1
Tr
1 . 3 mm 0.39 1 . 3 em 17.6
Dp
rV
TF
1 10 111 112 113 114 1 15 1 16 117 1 18 119
3.7 3.5 3.4 3.2 3.0 2.8 2.6 2.4 2.3 2.1
120 121 122 1 23 124 1 25 1 26 127 128 129
2.0 1.9 1.8 1 .7 1.6 1.6 1.5 1.4 1.3 1.2
130 131 1 32 1 33 1 34 135 136 137 138 1 39
1.1 1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
140 141 142
0.2 0.1 0.0
Time interval t:l.t (s)
60
60
2
Run 3
2
=
€b = L = Re = I
1
2 . 7 mm 0.40 2.4 em 229 ll TF
1
219
T H E R M A L RESPONSE MEASUREM ENTS
x 1 o-2
2 r-------. Re
=
17 . 6 •
experimen ta l
c a l c u l ated w i t h
Nu
-
-LL 1-
-g 1 0
20
0 . 03
5
0 . 06
10
- LL I-
E
0 . 01
o bL���----�--��=-�-��0
100
200
t
FIGURE 6.2
300
(sJ
Normalized input and response signals measured and response signals predicted, for Run 2 in Example 6 . 1 .
parameters, O::a x• hP (in terms of Nu = hpDp /kF) and k5, are compared with the measured response signal. The difference between the two curves is then evaluated using Eq. (6.1 8). Figures 6.3(a)-(c) show the relationships between the three parameters, O::ax• Nu and ks at Re = 0.54, 1 7.6 and 229, respectively. As shown, particle thermal conductivity has little effect on the thermal responses, when the conductivity is high and/or the flow rate is low. In this example, the values of ks, Cs and Ps for the glass beads are provided. The unknown parameters are, therefore, O::ax and hp. At the average bed temperature of 20°C, the thermal conductivity, kf', of air is 0.026 W m-1 K - 1 • The thermal conductivity of the glass particles is ks 0.88 W m-1 K - 1 or kslkF = 34; therefore, according to Figure 5.9, k�fkF = 6.4 for the glass-air system. The values of O::ax estimated from =
220
HEAT AND MASS TRANSFER IN PACKED BEDS
Eq. (6.28) are compared with a�x predicted from Eq. (6.3) below: Re 0.54 1 7.6 229
xlo-
1 O!ax (m 2 s- )
a�x (m 2 s- 1 )
3.5 x 1 o-4 6.7 x 1 0-4 47 x 1 o-4
(0.23 "' 0.27) X 10-4 (3 .4 "' 3 .5) x 1 o-4 43 x l 0-4
4
10 Re
8
.
**
6
N E
4 X 0 Cl
/ I
2
/
/
E:
I
.; M
�**
0 . 03
* **
o.os
0 . 08 0.1
-M
o � �
10-2
/
/
0 . 54
ks � o . os w . m- 1 · K - 1
.--.
..--1 I (/)
=
� -- --
-----
--
_. ---
----
10 - 1
-�
----
10
1
Nu (Q)
xl0 -
4 12 �------�--��Re
..--1 I
=
17 . 6
8 k
(/)
N E '-'
X 0 Cl
s
[W·m-1·K- 1 J
CD (i)
4
*
)( )(
®
0
0
10
20 Nu
(b)
E:
0.2
0 . 02
0.88
0 . 02
0 . 88
0 . 06
0.88
0.1
50
0 . 02
30
40
THERMAL RESPONSE MEASUREMENTS
221
x o-4 l
120 �-------, Re
,......, rl I (/) . N E
X 0 �
=
229
80 ks
40
G) ©
[W·m-l.K-l]
(:
0.2
0 . 02
5
0 . 02
0 . 83
0 . 02
50
0 . 02
0 �------����--�80 60 20 40 0 (C)
FIGURE 6. 3
Nu
Relationships between aa x • Nu and kg, for glass beads, for Example 6.1 : (a) Re = 0.54; (b) Re = 1 7 .6; (c) Re = 229.
From the contours with the least root-mean-square-errors (or valley in the three-dimensional error map) drawn in Figures 6.3(a)-(c), the particle to-gas heat transfer coefficients can be estimated using the above o:ax values as: Re
Nu
€
0.54 1 7.6 229
0.1 "' 00 8 "-' 1 2 26 "' 30
0.03 0.02 0.02
At Re = 0.54, any Nusselt number, in the range 0.1 to gives good agreement between the predicted response curves and the measured signal. According to Eq. (8.20), however, the Nusselt number at this Reynolds number is expected to be 2 .7 . The Nusselt numbers determined at Re 1 7.6 and 229 concur well with those predicted from Eq. (8.20) , Nu 8 at Re = 1 7.6 and Nu 28 at Re = 229. For Run 2 with Re = 1 7.6, the e ffect of Nu on the shape of the predicted response curves is illustrated in Figure 6.2. It can be clearly seen that the response signal, computed with Nu 1 0, matches the measured signal fairly well; while the curves 00 ,
=
=
=
=
222
HEAT AND M ASS T R A N S F E R I N PACKED BEDS
predicted, with Nu = 20 or 5 , differ considerably from the measured signal. Figure 6.3(a) reveals that, at Re 0.54, a3x is almost independent of Nu if Nu is greater than about 0 . 1 . Within a confidence range of E = 0.03, the value of aax falls in the range (2 .8 "' 3 .6) x 1 o -4 m2 s-1, which agrees well with the value of 3 . 5 x 1 0-4 m2 s- 1 estimated from Eq. (6.28), but differs significantly from the a�x value, (0.23 "' 0.27) x 1 0-4 m2 s- I, pre dicted using Eq. (6.3). =
(End ofExample)
As illustrated in the above example, definite values of the Nusselt number can not be determined from thermal response measurements at low flow rates. This arises from the fact that, when the flow rate is low, thermal equilibrium is nearly attained between the particle and its surrounding envelope of fluid, such that particle-to-fluid heat transfer makes little contribution to the overall heat transfer. Figures 6. 3(a) and (b) show that if the a�x values from Eq. (6.3) are assumed, the root-mean-square-error, €, is large, i.e. the agreement between the measured and predicted response curves is not good. In any case if the a �x values are assumed, the Nusselt numbers estimated from the contours labeled * and ** are much smaller than those obtained using the large values of aax from Eq. (6.28). At high flow rates, as seen from Figure 6.3(c), the Nusselt number is little affected by the axial fluid thermal dispersion coefficient. Also, the a3x value predicted from Eq. (6.28) is close to the a� x value estimated from Eq. (6.3). 6.2.4
Determination of Particle Thermal Conductivities
If particle-to-fluid heat transfer coefficients are known, the particle thermal conductivity can be determined from thermal response measure ments. The difference in the second central moment or variance, OA, between the response and input signals is (6. 1 9) where (6. l 9a)
THERMAL RESPONSE MEASUREM ENTS
223
(6. 1 9b) As discussed in Section 1.2 .2, the least-error line in the middle of the contour corresponds to a'A = constant or d(a 'A ) = 0. The slope of the line in a graph of aax versus ks is then daax
- =
dks
1
[
'Y o
kF
]2
- (Pr)(Re) .
20EbaRCFPF (I + 'Yo) ks
( 6 .20)
According to Eq. (6.20), the least-error line is nearly horizontal, when the flow rate is sufficiently low and/or when the solid thermal conduc tivity is high. With an increase in flow rate and/or a reduction in solid thermal conductivity, the slope of the least-error line is increased and eventually the line becomes vertical above a certain flow rate for low thermal conductivity solids. The particle thermal conductivity is determined, therefore from the bottom of the overlapping valleys of the contour maps generated from response measurements made at both high and low flow rates. If the particle thermal conductivity is expected to be high, then, response measurement at extremely high flow rates is required in order to obtain a steep valley in an error map. The following example illustrates the determination of particle thermal conductivity from one-shot thermal input. txample 6.2 Thermal response measurements were carried out by Shen et al. [5] with a bed of spherical polystyrene foam particles. Table 6.2 lists the experi mental conditions and the input and response signals measured. Find the thermal conductivity of polystyrene foam particles. SOLUTION
Equation (6.28) may be rewritten as: (6.21)
224
HEAT AND MASS TRANSFER IN PACKED BEDS
TABLE 6.2 Experimental conditions and temperature signals recorded in a bed of polystyrene foam particles. Spherical polystyrene foam: D p = Bed void fraction: Eb
=
3.0
mm; Cs =
0.42.
1 260 J kg-1
K - 1 ; PS =
Distance between the input and response signal measuring points: L =
Fluid : air at
0.1
MPa.
Average temperature of the air in the bed =
20°C;
60 kg m-3•
3.0 em.
kF = 0.026 W m-1 K - 1 •
Input and response measured (before normalized; the figures are ten times the actual temperature increase (K) above the room temperature).
Time
Run
1
Run
nllt
(s) n
Re
TFI
=
11
rV
Re =
T}
2 26
rV
Run
Re I TF
=
3 107
TilF
Run
Re =
T}
4
416
rV
0.0 0.3 0.7 1.6 3.2
0.0 4.0 20.0 49.0 79.0 103.0 117.1 120.8 1 1 5.0 104.0
0.0 0.4 3.0 9.1 2 1 .0 38.0 57.0 73.0 87.9
0.0 1.2 6.8 18.0 35.0 56.2 78.3 98.0 1 1 4.4 124.7
0.0 0.1 2.9 7.2 1 5 .4 25.6 38.5 53.8 69.5
0.0 0.3 6.0 19.0 37.9 60.0 85.0 108.0 130.1 150.0
0.0 0.7 4.8 14.0 28.0 45.5 64.0 83.2 103.5
55.9 6 1 .2 63.7 64.8 64.9 63.5 61.1 57.8 54.1 50.4
5.4 8.4 12.3 16.5 21.7 26.8 32.1 37.4 42.1 46.2
89.0 75.0 6 1 .9 50.9 41.8 34.0 29.0 24.5 20.8 17.6
98.1 100.7 99.8 93.7 83.9 73.9 63.5 53.7 44.9 37.9
129.2 1 30.1 127.1 120.5 1 1 2.4 102.8 92.2 83.8 75.0 66.7
82.0 96.0 104.3 1 1 1.2 115.1 1 16 . 2 1 15.4 1 1 1.9 106.4 100.1
166.6 182.8 193.0 195.0 190.0 180.0 166.1 1 5 1 .4 138.2 123.0
122.1 140.3 157.6 170.5 178.0 180.2 177.1 170.4 162.0 1 5 1 .8
46.9 43.1 39.7 36.6 33.4 30.7 27.9 25.8 23.8 21.8
49.2 51.7 53.2 54.2 54.3 54.5 53.4 5 1 .8 49.3 47.0
15.0 1 2.6 1 1.0 9.8 8.7 7.9 7.0 6.1 5.8 5.2
31.1 25.8 20.9 18.2 15.7 13.8 11.7 10.4 9.0 7.9
58.4 52.0 45.9 40.3 35.6 31.2 28.3 25.0 22.4 20.0
92.5 85.0 77.7 70.5 63.3 56.1 50.4 44.9 39.3 34.9
108.0 95.0 83.9 73.6 64.7 57.0 50.1 44.0 38.8 33.6
140.0 1 27.8 1 16.2 104.7 94.0 84. 1 74.6 66.2 58.6 5 1 .6
0 1 2 3 4 5 6 7 8 9
0.0 0.2 1.5 4.2 8.6 15.2 23.4 32.3 41.3 49.2
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
T H E R M A L RESPONSE MEASU REMENTS
2 25
TABLE 6.2 (Continued) Run 1
Time nfl.t
(s)
Re
=
11
Run 2 Re
=
26
Run 3 Re
==
107
Run 4 Re
=
416
T�
r}I
r}
r}I
TFI
rV
r}
rV
30 31 32 33 34 35 36 37 38 39
20.3 18.8 17.6 16.4 15.1 14.0 13.1 12.4 1 1.8 10.8
44.5 42.0 39.2 36.6 34.2 31.4 29.4 27.2 25.3 23.7
4.9 4.4 4.1 3.9 3.6 3.4 3.2 3.0 2.7 2.5
7.2 6.3 5.8 5.5 4.9 4.6 4.3 3.9 3.7 3.6
18.0 16.0 14.3 13.0 11.8 10.8 9.9 9.0 8.2 7.5
30.9 27.1 23.9 21.4 19.2 17.0 15.5 1 3.8 12.4 1 1 .4
29.9 26.0 23.0 20.7 18.2 16.2 14.3 13.1 1 1 .8 10.6
45.3 39.9 35.1 31.0 27.3 24.2 2 1 .4 1 8.9 16.7 14.8
40 41 42 43 44 45 46 47 48 49
10.2 9.8 9.0 8.6 8.2 7.7 7.3 6.9 6.5 6.1
2 1 .8 20.3 19.2 17.8 16.6 1 5 .4 14.5 1 3.6 12.9 12.2
2.2 2.0 2.0 2.0 1.9 1.8 1.7 1 .6 1.5 1.5
3.1 2.9 2.8 2.6 2.4 2.2 2.1 1.9 1.8 1.8
7.0 6.5 6.0 5.6 5.1 4.9 4.7 4.5 4.2 4.0
10.4 9.5 8.8 8.0 7.4 7.0 6.3 5.8 5.5 5.2
9.7 8.8 7.9 7.2 6.9 6.1 5.6 5.1 4.8 4.3
1 3.2 1 1.7 10.5 9.4 8.4 7.5 6.9 6.3 5.7 5.1
50 51 52 53 54 55 56 57 58 59
5.7 5.5 5.2 4.9 4.7 4.5 4.3 4.1 3.9 3.7
1 1 .4 10.6 10.0 9.4 8.8 8.6 8.0 7.7 7.4 6.9
1.4 1 .3 1.2 1.1 1.0 1.0 1.0 1.0 1.0 0.9
1.7 1.7 1.6 1.6 1.5 1.5 1 .4 1.3 1.2 1.2
3.9 3.7 3.6 3.4 3.3 3.1 3.0 2.9 2.7 2.6
4.8 4.6 4.4 4.2 4.0 3.8 3.6 3.5 3.3 3.1
4.1 3.9 3.7 3.5 3.2 3.1 2.9 2.8 2.6 2.5
4.8 4.6 4.3 3.9 3.6 3.4 3.3 3.1 2.9 2.8
60 61 62 63 64 65 66 67 68 69
3.5 3.4 3.3 3. 1 3.0 2.9 2.8 2.6 2.5 2.4
6.7 6.4 5.9 5.6 5.4 5.2 4.9 4.8 4.4 4.3
0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6
1.1 1.1 1.0 1 .0 1.0 0.9 0.9 0.9 0.8 0.8
2.5 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.1 2.0
3.0 2.8 2.7 2.6 2.6 2.5 2.4 2.3 2.2 2.2
2.4 2.3 2.2 2.2 2.1 2.0 1.9 1.8 1.7 1 .6
2.7 2.5 2.4 2.3 2.2
n
2.1
2.1 2.1 2.0 2.0
226
HEAT AND MASS T R A N S F E R I N PACKED BEDS
TABLE 6.2 (Continued) Run I
Time nAt
(s)
Re
=
11
Run 3
Run 2 Re
I
=
26
Re
I
=
107
Run 4 Re
=
416
11
TF
il TF
TF
il TF
TF
il TF
T
}
il TF
70 71 72 73 74 75 76 77 78 79
2.4 2.3 2.2 2.1 2.0 1.9 1.8 1 .7 1.6 1.6
4.1 3.9 3.7 3.6 3.4 3.3 3.2 3.1 3.0 2.9
0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0. 3
0.8 0.7 0.7 0.7 0.6 0.6 0.5 0.5 0.5 0.4
1.9 1.9 1.8 1 .8 1.7 1.7 1.6 1 .6 1.5 1.5
2.1 2.1 2.0 2.0 1 .9 1.8 1.8 1.8 1.7 1.7
1.5 1 .5 1 .4 1.4 1 .4 1. 3 1.3 1.3 1 .3 1.2
2.0 1 .9 1. 9 1 .9 1 .9 1 .8 1 .8 1 .8 1 .7 1 .7
80 81 82 83 84 85 86 87 88 89
1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1.0
2.8 2.7 2.5 2.5 2.4 2.3 2.1 2.0 1.9 1.8
0.2 0.2 0.2 0.1 0.1 0.1 0.0
0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1
1.4 1.4 1.3 1.3 1.3 1.2 1.1 1.1 1 .0 1 .0
1 .6 1.6 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2
1.2 1.2 1.1 1.1 1.1 1.0 1.0 1.0 1 .0 0.9
1.7 1.7 1.6 1 .6 1.6 1 .5 1.5 1.5 1.5 1.4
90 91 92 93 94 95 96 97 98 99
0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.4
1.7 1.6 1.5 1.4 1. 3 1.2 1.1 1.0 0.9 0.8
0. 1 0.0
1.0 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7
1.1 1.1 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7
0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6
1.4 1.4 1 .3 1.3 1.3 1.2 1.2 1.1 1.1 1.1
100 101 102 103 104 105 106 107 108 109
0.4 0.3 0.3 0.2 0.2 0.1 0.0
0.7 0.6 0.6 0.5 0.5 0.4 0.3 0.2 0.2 0.1
0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5
0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5
0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3
1 .0 1 .0 1 .0 0.9 0.9 0.9 0.8 0.8 0.8 0.7
I
227
T H E R M A L RESPONSE MEASUREMENTS
TABLE 6.2 Run 1
Time 11 D.t
(s) II
Re
ll
= --
I
TF
1 10 111 112 113 1 14 115 1 16 117 118 1 19
TFII
Run
Re
T}
==
(Continued)
2 26
T Fil
0.1 0.0
120 121 1 22 123 124 125 126 127 128 129
Run
Re
3 107
= ---
I TF
TFil
0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2
0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4
0.2 0.2 0.2 0.1 0.1 0.1 0.1. 0.0
0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3
1 30 131 1 32 1 33 1 34 1 35 1 36 137 138 139
0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1
140 141 142 143 144 145 146
0.1 0.1 0.1 0.1 0.1 0.1 0.0
Time interval D.t (s)
2.0
2.0
2.0
2.0
0.5
0.5
Run
Re
I
Tr:
0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0
=
4
416
TFII
0.7 0.7 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.0
0.2
0.2
228 HEAT AND MASS TRANSFER IN PACKED BEDS where a
�
k (e)
--
=
(6.2 I a)
EbCFPF
The particle-to-gas heat transfer coefficients, hp, are estimated from Eq. (8.20). Response signals are then predicted by assuming various values of a The predicted signals are compared with the measured signal, ax and ks. by evaluating the error, e of Eq. (6.1 8), between the two signals. Figure 6.4(a) is an error map of a plot of 8 H versus ks. The values of 8H and ks are then determined from the area where all four valleys overlap. This is clearly depicted in Figure 6.4(b ), which is an arithmetic mean of the four error maps. From the least-error point, labeled +, the value of 8H is found to be 0.55 and ks to be 0.048 W m- 1 K -1. Similar calculations are made with Nusselt numbers 0.5 and 1.5 times the values estimated from Eq. (8.20). The average error maps shown in Figure 6.4(c ) indicates that, in the response measurements with the extremely low thermal conductivity particles at high flow rates, the particle-to-gas heat transfer is a significant parameter, having a large 1.5
£
=
Re
0 . 015
11
26
107
1.0
416
;A':/r ,'1:
0.5
0
I .I I I II I : . • I I
0 (Q)
0.0 5
I I
:
0.]
0 . 15
TH ERMAL RESPON SE MEASUREMENTS
229
1 . 5 r-------,---.---� P o l y s t y rene foam 0art i c les - a i r 1.0 c
=
0 . 02
0.5
0
0
0 . 05
ks
(b)
J.5
0 . 15
rNu
Nu
1.0
c
=
=
=
I .5
0.5
x x
----�--�--�
ean < 8 . 20 > ean < 8 . 20 >
0 . 02
0.5 - --
c = 0 . 02 ... ... --- - - - --- - -- - -
0.015
-----� ._ --� --� --= � � � ---0� 0
0.05
0.15
(C)
FIGURE 6.4 Error maps in the plot of cS H versus ks, for polystyrene foam particles, for Example 6.2: (a) effect of Re on cSH-ks relationship; (b) average error map for four runs; (c) effect of Nu on cSH-ks relationship: least-error point labeled + result ing from h p of Eq. (8.20); • from h p of 0.5 times Eq. (8.20); o from h p of 1.5 times Eq. (8.20).
230
H EAT AND MASS TRANSFER I N PACKED BED S
influence on the values of o H and ks. The least-error point moves up and left along the locus with an increase in the Nusselt value, and down and right with decreasing Nusselt number. Equation (8.20) is the empirical correlation obtained from numerous experimental data reported in the literature. Suppose the data scattering is roughly ±50%. A 50% increase in Nusselt value, or 1.5 times the Nusselt values of Eq. (8.20), gives, as seen in Figure 6.4(c): o H 0.69 and ks 0.042 W m- 1 K -1. A 50% decrease in Nusselt values results in a greater effect, i.e. oH 0. 1 1 and ks 0.095 W m-1 K -1 . However, the value of o H can be neither as small as 0. 1 1 nor as large as 0.69. Most previous work in the literature indicates that 81-1 0.5 (or o 0.45 to 0.55), so that the thermal conductivity of the polystyrene H foam particles is found, from the measurements, to be ks = 0.05 W m-1 K -1 (or ks = 0.054 to 0.048 W m-1 K -1 , corresponding to o H 0.45 to 0.55). (End of Example) =
=
=
=
=
=
=
6.3
Fluid Thermal Dispersion Coefficients
Gunn and De Souza [ I ] pointed out that the contribution of solid phase conduction to thermal dispersion causes the axial fluid dispersion coeffi cient for heat to be larger than that for mass. Quantitative interpretations of the fluid thermal dispersion and solid heat conduction relationship were subsequently furnished by Vortmeyer [6] and Wakao [7]. Vortmeyer based his analysis on a comparison of the Continuous-Solid Phase model (C-S model, see Discussion in Section 7.4) and the Single Phase model (the heterogeneous packed bed is assumed to be a homogeneous single phase). He obtained an expression which relates the axial fluid thermal dispersion coefficient to the effective solid phase conductivity. On the other hand, from a comparison of the D-C model and the Single Phase model, Wakao derived a formula relating the two heat transfer parameters. Since the C-S model, employed by Vortmeyer, is considered to be inadequate for describing the phenomenon of heat transfer in packed beds, only Wakao's interpretation will be discussed in this section. Consider a packed bed with constant heat generation in the particles. Based on the D-C model, the steady-state heat balance equations are given as: (6.22)
THERMAL RESPONSE M EASUREMENTS
23 1
and (6.23) where qv is the rate of heat generation per unit volume of solid particle. Solving Eq. (6.23) and substituting this into Eq. (6.22) , it follows that: dTF d 2 TF U ax a -2 - -+ dx
dx
aR 3 EbCFPF
qv =
0.
(6.24)
Based on the Single Phase model, on the other hand, the heat balance equation of the packed bed may be simply described in terms of the axial effective thermal conductivity, keax, as: (6.25) Under steady-state conditions, the bed temperature T is equal to the fluid temperature TF so that a comparison of Eqs. (6.24) and (6.25) yields keax aax = ---
(6.26)
EbCFPF
or (6.27) The effective axial thermal conductivity, discussed in Section 5.5, is given by Eq. (5.69). Substitution of Eq. (5.69) into Eq. (6.27) gives aax
-
aF
=
I -
[
k�
-
Eb kF
.
]
+ 0.5 (Pr)(Re) .
(6.28)
Equation {6.28) , obtained by Wakao [7], is widely employed for estimating the axial thermal dispersion coefficient, aax, in packed bed heat transfer. Values of the dispersion coefficient calculated from Eqs.
232
HEAT AND MASS TRANSFER IN PACKED BEDS
(6.28) and (6.3) agree reasonably well at high Reynolds numbers, but are significantly different from each other when the flow rates are low. In general, aax values predicted using Eq. (6.28) agree fairly well with reported experimental data. As an illustration, the experimental aax data obtained by Gunn and De Souza [ 1 ] are compared with those predicted according to Eq. (6.28). The data of Gunn and De Souza, shown in Figure 6. 1 , are those for the glass-air system with ks 0.79 W m- 1 K- 1 and kF 0.026 W m-1 K -1 (at an assumed temperature of 25°C). At ksfk F = 30, the effective thermal conductivity of the quiescent bed is found, in terms of k�fkF, from Figure 5.9, to be 6 . 1 . With this value of k�/kF and Pr 0.7, the Dp U/aax-Re relationship predicted from Eq. (6.28) is shown as the lower solid line in Figure 6 . 1 . As depicted, the experi mental data match the theoretical values well, but differ considerably from those predicted according to Eq. (6.3), based on the mass dispersion analogy, over a range of Reynolds number from about 0 . 1 to 1 00. In corporated in Figure 6 . 1 are some aax data determined from one-shot thermal response measurements by Gunn et al. [8]. These data, which were obtained from an analysis in the Laplace domain, also fit the theoretical line well. Wakao et al. [7, 9] also determined, from a time domain analysis of one-shot thermal response measurements, values of aax for air flow in beds of polystyrene foam particles, glass beads and lead shot. They reported that their results were correlated well by Eq. (6.28). Corresponding to Eq. (6.27), the radial fluid thermal dispersion coeffi cient is given as: =
=
=
-
=
--
(6.29)
where ke r is the effective radial thermal conductivity. 6.4
Transient Effective Thennal Conductivities of Quiescent Beds
The axial thermal dispersion coefficient, aax, obtained from thermal response measurements is related, through Eq. (6.26), to the effective axial thermal conductivity, keax, which is usually determined under steady state conditions. The purpose of this section is to discuss the effective thermal conductivity under unsteady-state conditions; a subject which has been examined by Kaguei et al. [ 1 0].
THERMAL RESPONSE MEASUREMENTS
233
The conduction of heat in a cylindrical cell (radius R', length 2R) shown in Figure 6.5 is considered. The system consists of a solid sphere (radius R , temperature Ts) and a stagnant fluid envelope (temperature T�·). For convenience, either end of the cylindrical cell is assumed to be at constant temperature. The temperatures, Ts and T';:-, are then sym metric around the central axis of the cylinder. The system is described by the following equations (in both cylindrical (r', x) and spherical (r, 8 ) coordinates): for 0 < r' < R' and -R < x < R (6.30) where V
2
=
1
a
( ) 2 a
- r or r2 or -
-
+
T * = Ts and a = as
1 r
2
sin (}
a
-
o(}
(
·
a
sm (} o(}
for 0 < r < R
X =
R
X =
-R
t
fiGURE 6.5
Heat flow
Heat transfer cell and coordinates.
)
234 HEAT AND MASS TRANSFER IN PACKED BEDS and T* Tt- and for r > R with atx =-R T� =TI Tt T2 atx R and - T* T*FS at r R . oTs aT{ kF=ksar or aF
a:=
=
=
=
=
The rate of axial heat conduction through a cross-section (at x) of the cell is q
x
R'
, =- 2rr J k oT* ax -
0
,
r dr
(6.31)
where k ks and T* =
=
Ts
for 0< r< R
and k kF and T* Tt for r>R. Therefore, the average axial heat conduction rate in the cell is =
=
R
q
=2� I qx dx -R
(6.32)
THERMAL RESPONSE MEASUREMENTS
235
The effective thermal conductivity, K0(t), may be defined as: (6.33) or (6.34) Eb
Suppose the cylinder (R' l.OSR and 0.4) shown in Figure 6.5 is heated under the conditions: T* 0 at t 0; T* 1 at -R(T1 1) and T* -1 at R (or T2 -1); aT*/ar' = 0 at r' R'. The tempera ture profiles in the lower half (-R< < 0) of the cylinder are computed with a grid network: the temperatures at the nodal points are calculated at sequences of time. Figures 6.6(a) and (b) illustrate the increase in temperature in the cylinder, with a 1 mm glass sphere in it, at 0.001 and 0.09 s, respectively. The transient effective thermal conductivities are evaluated by the follow ing rewritten form of Eq. (6.34): 1 (6.35) K0(t) � Ki,j [Ttj(t)- Tt+I,j(t)] �Aj =
x =
=
=
=
=
=
rrR'2
=
x =
=
=
=
x
t,j
where, referring to Figure 6.6(a), Ki.i is the rod conductance between nodes (i, j) and (i 1, j), and �Ai is the area represented by the rod at j. The transient effective thermal conductivity, K0(t), of the cylinder con taining the 1 mm glass sphere is 0.2k�, at t 0.001 s (Figure 6.6a); and 0. 78k�, at t 0.09 s (Figure 6.6b ), where k� is the steady-state effective thermal conductivity of a quiescent bed. Figure 6.7 shows the conductivity-time curves for the cylinder contain ing a 1 mm glass/lead sphere and stagnant air. It is shown that the conduc tivities increase rapidly up to the steady-state values. The rise in conduc tivity for the lead-air system, is almost instantaneous, reaching its steady-state value in about 0.01 s. For the glass-air system, the increase is more gradual; but the transient time is still very short. Since a packed bed may be visualized as composed of many unit cells connected in series, as discussed in Section 2.3, the conductivity of a +
=
=
236
HEAT AND MASS TRANSFER IN PACKED BEDS Heat
r·
"'
0.4 0.2 o. 1
--
;-.... r--
��� -v .......- �� -;;;oo � � � , V"
o.8
0.6
--
�
flow
I
-
��..../ ...-
/...-::: / , / 7./• / /
7 / I 7 /i I /; I
_j_....... .-' .I
,....
.......-
,
.
0.2 0.1
i
( Lj j ffij
1i-r-r--
t
=
0. 001 s
1/0( t) � 0
"e R
=
=
0.2
0.05 em
:
(Q)
T* = 0
Heat flow
'it
0.6 ---
0.4
---
0.1
(b)
'l
/� vlA
v /, // /v J I 1/ I I -
0.2
FIGURE 6.6
��---1--:� : ooc::: ........ ... ... / ,... ......., -.,...,... . . �", v "-..., ..../ . h7 -
T* = 0.8
......
"' 1
I
I
l
T*
�
"�r---.
p = 1
- 0.6
--
--
1--f-.
t = 0.09 s
�(Jl ke R
=
= 0.78
0.05 em
0.8
0.4
0.2
ff-.ffff-
'---
0.1
T"
=
0
Grid network and computed temperature profiles, for glass (1 mm diameter)-air system: (a) t = 0.001 s; (b) t = 0.09 s.
THERMAL RESPONSE MEASUREMENTS
1.0
0.8 OQJ �
:::::
'-'
::..:
Lead - air
237
-=--=--:=.---:=;--:.:::.;:--.=.;:..:---:.=--..-----�
,--------------- -----------------
I I I .
0.6
0
O.LI'
Particle diameter
=
0.1 em
0.2
0 �------�--��0
0.1
0.2
t
FIGURE 6.7
0.3
O.LI
[sJ
Transient effective thermal conductivities for glass (1 mm diameter) air and lead (1 mm diameter)-air systems.
quiescent bed is, thus, considered to be the same as that of the single cell illustrated. When fluid is flowing in the bed, the conductivity is con sidered to attain its steady-state value in a much shorter time. In fact, the transient time is generally very small compared to either the period of frequency response (see Littman et al. [ 11] and Gunn and De S ouza [ 1]), or the residence time in an input-response measurement (Wakao [7]). Hence, the effective thermal conductivity, defined by Eq. (6.33), may be satisfactorily assumed to be constant and equal to its steady-state value, in an overall unsteady-state heat transfer process.
6.5
Assumption of an Infinite Bed
The heat transfer parameters involved in Eqs. (6.1), (6.2) and (6.2a) based on the D-C model have been determined by Gunn and De S ouza [I] from frequency response measurements, and by Gunn et al. [8] and Shen et al. (4, 5] from one-shot response measurements. In the analysis, the packed beds were assumed to be of infinite length. This assumption simplifies the solution to Eqs. (6.1 ) -(6.2).
238
HEAT AND MASS TRANSFER IN PACKED BEDS
As in Section 1.4, it is necessary to examine where the response signal should be measured in a bed o f finite length in order to satisfy the infinite bed assumption. Let us assume that a finite packed bed is connected to an infinitely long empty column, as shown in Figure 1.26. Consider that a temperature change is imposed on a fluid flowing in the bed at x < 0, and the fluid temperature-time curves are monitored at x = 0 and x = L, which is at a distance,/, away from the bed exit. According to the D-C model� the unsteady-state heat balance equa tions for the packed bed (x < L + /) are given by Eqs. (6.1) and (6.2), and for the empty column (x >L + l � fluid temperature Tf-� fluid velocity u ) by
ar�k}.- a2Tf.· = at CrPF ax2
-
-----
u
arf.-
(6.36)
-·
ax
The initial and boundary conditions are
TF=Ts=Tf:=O
at t =O
arF arf. TF = TF and keax - ke .... ax ax Ti· 0 I
=
=
I
at X
=
at x
= oo
L+l
where k�- is the axial fluid thermal conductivity in flowing system, and keax is the effective axial thermal conductivity of the packed bed under unsteady-state conditions. As discussed in the preceding section, the unsteady-state conductivity is the same as that under steady-state conditions. The transfer function of the system between the gas temperatures at x 0 and x = L is then =
F(s)
=
l- AH exp (-aH(/jL)]
1-AH exp{-aH(l + (ljL)]}
exp (A.H)
(6.37)
where
(6.37a)
THERMAL RESPONSE MEASUREMENTS 239 (6.37b) LU
OH =-(1 + B) tt2 aax
(6.37c)
,AH LU [ - (.1 + 4krs )1'2] CFPF and is defined by Eq. (6.1 Oa). The first moment, M}l, of the impulse response is MP =-F'(O) =-Lu (I + �H)( 1 - AH) where 2aax
=--
1
u
2
(6.37d)
-
B
(6.38) (6.38a) (6.38b) (6.38c)
and rH= I -
ki.---ehkeax (1 �H)
(6.38d) For an infinite bed (1/L ) Eq. (6.38b) shows that AH 0; there fore, AH is a measure of the deviation of the first moment from that of an infinite bed. In gas-solid systems, �H � 1 and keaxlkr > 1 and hence, Eq. (6.38d) shows that fH 1 and Eq. (6.38b) reduces to (6.39) +
= 00 ,
�
=
240 HEAT AND MASS TRANSFER IN PACKED BEDS Based on Eq. (6.39), the relationships between /\H and NH at values of l/L between 0 and 10 are presented in Figure 6.8. The criterion for an infinite bed may be given by /\H 0.0 Thus, a packed bed with NH < 0.3 may be assumed to be infinite if the response signal is measured at ljL > 1. If NH I, however, the response signal should be measured at 1/L 4. =
I.
>
=
1
:c <
-.L... --L. .J.-. .L-J... . ....I.. __.L .... .... .;._ � .J...-'.1..__.. .... .... .. ..., 10-3 L..-
10-2
10-1
10
1
NH
NH A.H. With Eq. (6.28), the thermal dispersion number given by Eq. (6.38c) is rewritten as (6.40) For most solid-gas packed bed systems, the ratio, k�fkF, is in the range 5 t.o 15, and Pr'"" 1. The condition NH < 0.3 at l/L > 1 is met, therefore, at Re > 2 to 6 (corresponding to k�/kF 5 to 15) for beds with L/Dp 10. If L/Dp 20, the condition is satisfied with Re > 1 to 3. Shen et al. [4] applied the technique of a one-shot thermal input to the determination of the Ci.ax-Nu relation for the flow of air through a finite packed bed of glass beads (Example 6.1). Their measured input and FIGURE 6.8
=
Effects of 1/L and
=
on
=
THERMAL RESPONSE MEASUREMENTS
241
1
0
L....-_....__..IL...J..L..�-.L..--.1.1..-..L-.l.I..-�-.L...-...L..J 2 10 -
1 10-
10
1 Re
FIGURE 6.9
x1o-2
Effects of l/L and Re on AH, for Run 2 in Example 6.1.
2 �-------, Re
=
17.6 ·experimental
i/L
_,..--- 0
c
0.1 0.04 0.01
-u.. �
g
0
1-
-u.. �
200
100 t
FIGURE 6.10
Is I
300
Effect of 1/L on computed response signal, for Run 2 in Example 6.1.
242 response signals at Re 17.6 are illustrated in Figure 6.2. Using their data (L/Dp I0; Pr 0. 7; k�/kr: 6.4 ), the 1\H-Re relationships at various 1/L values are computed from Eqs. (6.39) and (6.40), and presented in Figure 6.9. The graph shows that at Re 17.6, the condition, i\H < 0.01, is met, when 1/L is greater than 0.3. In the experiments of Shen et al., the response signals were measured at 1/L � 3. The measured response signals are, therefore, free from the bed-end effect, and the assumption of an infinite bed is valid. The response signals, T��(t), at 1/L 0,- 0.1 and- are computed using the heat transfer parameters (aax 7 1o 4m2 s 1 and 10, which are the same as those employed in Figure 6.2), and compared, in Figure 6.10, with the response signal measured at 1/L 3. As depicted, the deviation of the computed response curves from the measured signal increases with decreasing ratio of IfL (in the graph the deviation is indicated in terms of the root-mean-square-error, defined by Eq. 6.18 ). Noticeable deviation is seen, when l/L is 0.1 or less. Therefore, it may be concluded that the assumption of infinite bed is valid in beds of finite length, if response signals are measured at proper locations in the bed. Figure 6.8 demonstrates the locations where response signals should be measured in order to satisfy the criterion expressed in terms of /\H. HEAT AND MASS TRANSFER IN PACKED BEDS =
=
=
=
=
=
=
oo
X
Nu
=
�
€,
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] ( 11]
D. J. Gunn and J. F. C. De So u za Clzem. Eng. Sci. 29, 1363 (1974). M. F. Edwards and J. F. Richardson, Chem. Hlzg. Sci. 23, 109 (1968). D. J. Gunn and C. Pryce, Trans.lnst. Chern. Eng. 47,T341 (1969).
,
J.
Shcn, S. Kaguei and N.
Wakao, Clrem. Eng. Sci.
36, 1283 (1981).
J. Shen, S. Kaguci and N. Wakao,J. Chern. Eng Japan 14,413 (1981). D. Vortmeyer, Chem. Eng. Sci. 30,999 (1975). N. Wakao, Chem. Eng. Sci. 31, IllS (1976). D. J. Gunn, P. V. N arayanan and A. P. W ardle, Sixth Int. Heat Transfer Conf
Toronto, V ol. 4, p. 19 (1978). N. Wakao, S. Tanisho and B. Shiozawa, Kagaku Kogaku Ronhunshu 2, 422 (1976). S. Kaguei, B. Shioz awa and N. Wakao, Chern. t:ng. Sci. 32, 507 (1977). H. Littman, R. G. Barile and A. H. Pulsifer, Ind. Eng. Chern. Fund. 1, 554 (1968).
7 Unsteady-State Heat Transfer Models
models proposed to describe the phenomena of heat transfer in packed beds under unsteady-state conditions are the Schumann model, the Continuous- Solid Phase model (C-S model) and the Dispersion Concentric model (D-C model). The Schumann model, the least complex of the three, is simply based on the assumption of ideal plug flow of the fluid and considers no heat conduction resistance in the solid particle. The C-S model takes the solid phase heat conduction effect into con sideration by assuming that the solid is in a continuous phase. Moreover, fluid thermal dispersion is also considered in the C-S model. The D-C model is, by far, the most widely practised model in the solution of problems of unsteady-state heat transfer in packed beds. The model, which has already been discussed in Chapter 6, is based on the fluid having dispersed plug flow and the intraparticle temperature having radial symmetry. However, the assumption that the intraparticle temperature profile is radially symmetric does not portray the real temperature profile in the solid particle. Similar to the discussion in Section 1. 2.4, if the temperature profile in particle was concentric, no heat conduction would take place across the particle. To compensate for this shortcoming, solid phase heat conduction has to be superficially included in the fluid thermal dispersion term. Solid phase conduction is important in unsteady-state heat transfer, particularly at low flow rates. In this chapter, the unsteady-state heat transfer models are discussed and compared with respect to their response signals. The Nusselt numbers, Nu, predicted from the corresponding response signals, are assessed to determine the validity of the models. The underlying assumption of an intraparticle concentric temperature profile in the D-C model is also verified. 243 THE THREE
244
HEAT AND MASS TRANSFER IN PACKED BEDS
7.1
Step and Frequency Responses for the Schumann, C-S and D-C Models
Equations (7.1 )-(7 .7) listed in Table 7.1 are the fundamental equations of the S chumann, C-S and D-C models for unsteady-state heat transfer in inert packed beds. Besides the notation employed in C hapter 6, the follow ing symbols are used (in the C-S model): keF
=
effective fluid phase thermal conductivity
kes = effective solid phase thermal conductivity 7 .I. I.
Transfer Functions for Response Signals at Bed Exit
Transfer functions governing the thermal input signal, T�(t), imposed on a fluid entering the bed, and the response signal, T�(t), of a fluid leaving a bed of length, L, are shown below: 7 .1.1.1
The Schumann mode I
The boundary condition for the Schumann model (3] is at x = 0 (inlet) .
TF = T}(t)
(7.8)
The transfer function, relating the input and response signals, is
(F(s)]schumann
=
where kl=
and k2=
7 . 1.1 . 2
The C-S model
exp
[- sL ( 1 u
+
k1
s +k2
)]
hpa
---
€bCFPF
hpa
----
(1- Eb) CsPs
The boundary conditions employed by Littman eta!. [ 1, 2] are
(7.9)
(7.9a)
(7.9b)
TABLE 7.1 Heat transfer models and the fundamental equations.
,.
Schumann model
Cf:J20 C-S model
�
-----,.
D-(' model
Fundamental equations
Assumptions
Model
�
original
-----modi fled
Fluid in plug flow
oTF
-----;;( ==No temperature gradient within the particle Fluid in dispersed plug flow
(1 - <'b)
-
ilTF ill
Axial heat conduction in the solid phase Fluid in dispersed plug flow Dispersion coefficient of Eq. (6.3) for the original D-C model; that of Eq. (6.28) for the modified D-(' model Particle temperature with radial symmetry
.. ------·-··---
=
ilT.,.
at
il l
_
hpa
t:hC FPF
o2T1.-
<-'hCFPF
ilx2
ill
::-: Ocax
=--
kes
· �-
-U
il2Ts
CsPS ilx2
i'lr
��
S r2 ilr
llTs
ilTF ilx 1-
-
f:bCt.-PF
CsPS
.
<7 F-Ts)
(T""-Ts>
hpa il2TF ilT..-U I TF- (Ts)t.d 2 oX i'lx f. h C1:P 1: "
--
, (r2 ) (-)
Ill
(7.2)
CsPs
-- - lipa - -- -,h-p a ilTs
(7 .1)
(TF- Ts)
hpa (TF- Ts>
=---
keF
i'l Ts ==a
ks
ilx
C>Ts
o- eb)
--
ilTF
U
· .
.
a Ts
(7.4) (7 .5)
(7.6)
i'lr
lipCT.,--Ts>
(7.3)
at
r
(7.7)
R
--··------
-----
---
In the C'-S model, keF= effective fluid phase thermal conductivity and kcs -·effective solid ph<Jsc thermal conductivity, both basL·d on the bed cross-section. The same symbols are used by Littman et al. 11, 21, but kc F ;;: cb(k I! ,,.)Littman a nd kcs .:. ( 1- c-h) Ckes)Littman·
::t M ? o-:1 o-:1 ;;z:: ? z C/) "'11 rT1 ::0 s: 0 0 rT1 [""'
C/)
t.J � VI
246
HEAT AND MASS TRANSFER IN PACKED BEDS
(7. 1 0)
atx
=
L (exit).
The transfer function is
[F(s)lc-s =
4
L P; ex p (m;)
i= 1
(7 .11)
where m; is an i-th root of the following equation:
(7 .12)
and P; is an i-th root of
4
L
i=I
(. haL2) P; = 0 l-w; V; +
p
HEAT T RANSFER MODELS 4
L mi exp (mi)Pi
i= 1 4
L
i= 1
in which
7.1 .1.3
(
vi+
hpaL2
1
+w·
)
l
=
247
0 (7.13)
exp (mi)Pi = 0
The D-C model
Assuming that the _Danckwerts boundary conditions [ 4] may be applied to the fundamental heat balance equations. Then, at x = 0 (inlet) (7. 1 4) at x = L (exit). The transfer function is given as:
[F(s)] D-C=
(
4A exp
)
( ) . LU
-
2aax
LU 2 Aexp - ( l- A) 2exp (I +A) 2aax
(
-A
LU
-
)
(7.15)
2aax
where
A= (I+B) 112 and B is defined in Eq. (6.1 0a). Note that as = ks/(CsPs)-
(7.1 5a)
248
HEAT AND MASS TRANSFER IN PACKED BEDS
7 .1.2
Prediction of Step and Frequency Responses
7 .1.2.1
Step response
With the following conditions at t = 0
TF= Ts=O Tf·(t) =Tin
at t > 0
(7 .16)
the response, TP(t), is obtained as: 1 2 1 TU(t) -- =-+-I Imag(F(s) exp (st)Js=i(2n-l)7T/r* (7.17) Tin 2 1T n 1 2n - 1 00
=
where ;* is a time sufficiently long enough to allow the step response to attain a steady-state value. 7 .1.2 .2
Frequency response
The response, TP(t), under stationary conditions, is obtained using
T}(t) =A� cos wt
(7 . 18)
as _.,..1- = Real(F(s)
TU(t) Aw
exp (st)ls=iw
(7 .19)
where A� is the amplitude of the harmonic component with frequency w of the input signal. When the responses are measured at the bed exit, the transfer functions, F(s) in Eqs. (7.17) and (7.19) are substituted by Eqs. (7.9), (7.11) and (7 .15) for the Schumann, C -S and D -C models, accordingly.
7.2
Assumption of a Concentric Temperature Profile in a Solid Sphere in the D-C Model
The fundamental equations, Eqs. (7 .5)-(7. 7), based on the assumptions of dispersed plug flow and concentric intraparticle temperature profiles, are
HEAT TRANSFER MODELS
249
rather simple and can easily be solved. The question is whether the assumption of intraparticle temperature being radially symmetric can be verified in the calculation of the heat transfer rate. The purpose of this section is to examine the validity of this assumption. For a particle in a packed bed, the unsteady-state heat balance equation is given in terms of temperature Ts as: (7.20) where V2
) -- a ] a2
= - a ( -a ) [ -- aea ( 1
- r2 2 r ar ar
1
a 1 - sin () - + ae sin2 ()
1
+-
r2 sin ()
2
--
·
(7. 20a)
Following the same procedure given in Section 1 . 2.4, Eq. (7.20) is replaced by l
2rr
:2f J
4
r
d
0
-1
a:s
ar�
2
-r2dcos()=at 4trr2
rr
0
l
2 T s 2 dcos e . (7. J J d
V
-1
r
21 )
Changing the order of differentiation and integration and then taking (aT;/a)=O = (aT;/a)=2rr into consideration, it results in the follow ing expression: (7.22) where
X=
1 4"
2rr
1
f f r;
0
d
-1
dcos 8.
(7. 22a)
This X is nothing but the average value of r; on a spherical surface with radius r. The actual temperatures, r;, in a particle are not radially
250
HEAT AND MASS T RANSFER IN PACKED BEDS
symmetric, but the intraparticle temperature profile, similar to the intra particle concentration profile argument, can be expressed in terms of average temperatures which are radially symmetric. The rate of heat transfer from a tluid to a particle is then
Qp(t)
= ks(f J a;,� R' d
0
= ksR2
-1
2rr
(�f ar 0
d
1
dcos
8)R )
r r; dcos
�
-
1
8
'R
·
(7.23)
S ubstitution of Eq. (7.22a) into Eq. (7.23) gives (7.24)
Therefore, we may conclude that the heat transfer rate, Qp(t), may be evaluated in terms of the intraparticle surface mean temperature.
7.3
Effect of Fluid Thermal Dispersion Coefficients on the Nusselt Numbers of the D-C model
The axial solid phase heat conduction contribution is not taken into con sideration in the model as originally proposed. To overcome this problem, Wakao [5] proposed including it in the fluid dispersion term. He has shown that the axial fluid thermal dispersion coefficient, a3x, is related to the effective axial thermal conductivity, keax, according to Eq. (6.26) or Eq. (6.28). The model, which takes into account axial solid phase heat conduction, is henceforth referred to as the modified model in order to distinguish it from the original model in which thermal dispersion is simply based on a mass dispersion analogy. The effect of fluid thermal dispersion coefficients on the particle-to fluid heat transfer coefficients based on the two models will be examined using the following example:
D-C
D-C
D-C
D-C
HEAT T RANSFER MODELS
Example 7.1
251
Let us consider that the thermal frequency response signals are measured over a distance, L 1.5 em, in an infinitely long column packed with glass beads (1 mm) and bed void fraction, Eb 0.4. Air is flowing in the bed at Re < 2. Additional data are: thermal conductivities, kF = 0 . 0 27 and 0 .88 W m-1 K- 1; specific heats, CF 1000 and Cs = 670 J kg-1 K-1; ks densities, PF = 1.2 and Ps = 2500 kg m-J. Find Nusselt numbers based on the original D-C model. =
=
=
=
SOLUTION
The frequency response signals, based on the D-C models, should be computed using Eqs. (6.10) and (7. 19) from the above information together with axial thermal dispersion coefficients and particle-to-fluid heat transfer coefficients. The frequency response signal based on the modified D-C model is denoted by (TV)o-c· This is computed using aax from Eq. (6.28) and values of Nusselt number, Nu, ranging from 1 to (Note that (TU)o-c is not influenced by Nu in the range 1 to ) S imilarly, the frequency response signal, based on the original D-C model, is calculated using a�x from Eq. ( 6.3) and various assumed values of Nu'. This signal is denoted by (T�)o-c. The correct values of Nu' are then determined by minimizing the difference between the two calculated response signals using the root mean-square-error, Er, defined as: oo.
oo.
2n
T
€f
=
0
[
J [(T�')o-cl2
112 (7.25)
dt
0
Figure 7.1 compares the two frequency response signals, (T�)o-c and (T¥)b-c, computed at w = 0.00047T rad s- 1 and Re 2. We can say that the two signals are in: =
a) b) c)
Very good agreement, if Er < 0.01.
Relatively good agreement, ifO.Ol < Ef < 0. 05.
Poor agreement, if Ef � 0.05.
252 HEAT AND MASS TRANSFER IN PACKED BEDS Glass - air
w
,.1
Re
-�6
Nu'
0.5
- u.. 1-
-g
0 u I Q
-�
-u.. 1-
=
--··--
1.0 u
=
0.0004n rad·s-1
2
II'
aa/°F
'f
0.7
0.01
0.7
0.05
0.7
0.1
0.7
0.3
0
-0.5
�
-l.O L-----�--�--�
1
0
FIGURE 7.1
2
Comp arison of
t
[S)
3
4
5
(TV)D-C and (TV)o-c for Exa mple 7.1.
is the error map of Nu' versus Re at a frequency, w = 0.00017T Figure rad s-1. Based on the criteria, (a)-(c) above, the following results may be drawn:
7. 2
any value of Nu' greater than about 0.03 gives very a') A t Re = good agreement (Ef< 0 .0 I), therefore, no definite Nu' value can be determined. b) At Re 0.9 with Er 0.01 (Point A), Nu' is found to be 0.015;
2,
=
=
similarly, at Re = 0.5 with Ef = 0.05 (Point B), Nu' is shown to be 0 .008.
c') For Re < 0.5, the fit is always poor (Er � 0.05) with any value of Nu'.
Therefore, it can be concluded that, within an acceptable confidence range, curve AB is the only Nu'-Re relationship to be found from thermal response at this frequency. S imilar Nu-Re relationships for a number of different systems with varied response frequencies are estimated and presented in Figure 7 .3. As
HEAT TRANSFER MODELS
Glass - air
w
_ ...._ 10-4 ..__ 0.05 0.1
=
0.0001n rad·S-
1
__._ _ __._ ----L --'---JJ
_
FIGURE 7. 2
253
Re
0.5
1
2
Error map in the plot of Nu' versus Re for Example 7 1 .
.
shown, each response frequency will predict only a certain range of the Nu'-Re relationship. This frequency dependency had been experimentally observed by Turner and Otten [6] and Gunn and De Souza [7). If, in Figure 7.3, the Nu'-Re relationships for the various systems are interpreted as Nu'-PrRe, then Figure 7.4 shows that all the data fall almost within the same range bounded by the two curves. At high Reynolds number, the mixing terms in Eqs. (6.3) and (6.28) predominate, so that the difference between a�x and aax is small; hence, the two Nusselt values predicted from the original and modified D-C models should be close. At low Reynolds number, as shown in Figure 7.4 (Re< 2), the original D-C model which uses a�x from Eq. (6.3) is found to give entirely different and anomalously low Nusselt values. (End of Example)
7.4
The C-S Model
Based on the assumption that the bed of solid particles is in a continuous phase, the C-S model, proposed by Littman and Barile [1 ], takes into
Lead - air
Glass - air L
D
=
P
=
0.01
w
=
L
1.5 em
0.03cm
Ef
<
<
Dp
0.0008•
'/
�
0.00\ 1
.I I ;;.:_
0 0002n ·
=
=
0.01
0,05
rad·s -1 ;
0.002n
1
w
=
--�,_.....J
__ _ _
Re (a)
1
1 2 10-
Re
(b)
1
10
0.1 em
£f
<
<
L
w
0.05 10-
1
1 0.0004n rad·s0.0002n 0.0001•
:r: trl )> >-! )> z 0 3:: )>
Glass - oil
1.5 em
1
=
D P
=
=
0.1�
1.5 em
0.1 em
1 rad·s-
,,"' 0.0411\-:-:�"/
0.08n�
U> U> ....,
:;o )> z U> "T1 � ,
..·· ..
"'
�/
I
1
rO.OOOl• 10- 3.__ 1 10-
N VI �
0.01 10-
2
10-
1
�l� ,"-. -- �
�
<
0.05
-
z
�0,02TT 0.01n
�
'"tl
0.006�
__, "--'='"__.._ ____,
____
(C)
cf
'----- 0.00211
L-... -.,----i.---' 10-3 10-2 .._... 2 10 - 1 1 2 10-4
Re
<
10-3
Re
(d)
FIGURE 7.3 Nu' versus Re for Example 7.1: (a) and (b) glass-air systems with Pr = 0.7; (c) lead-air system with Pr = 0.7; (d) glass-oil system with Pr = 10 500.
_
2
5x1o-3
)> (') " trl 0 0" m 0 U>
HEAT T RANSFER MODELS
255
10 Re < 2
#
1
/
v
//v
�1 IL /,I
Iv
v
I Iv I
I/I I Iv FIGURE 7.4
10
1
50
PrRe
Nu' versus (Pr)(Re) for Example 7.1.
consideration the solid phase heat conduction in addition to the fluid phase thermal dispersion. In this section, the C-S model will be examined by assessing the heat transfer parameters obtained from the model with respect to the equivalent parameters according to the modified D-C model. The transfer functions and the frequency responses, given in Section 7. 1, will be used in the analysis. The Nusselt numbers of these two models are denoted as Nu and Nu" for the modified D-C model and the C-S model, respectively. 74 1 .
.
From the C-S Model to the Modified D-C Model
Based on the C-S model and thermal frequency response measurements, Littman et al. [ 2] obtained particle-to-fluid heat transfer coefficients in terms of Nusselt numbers, Nu ", and effective thermal conductivities of the solid phase, kes, for several solid-gas systems. Based on their results for
256
HEAT AND MASS TRANSFER I N PACKED BEDS
kes and Nu", the heat transfer parameters, aax and Nu, can be predicted
from the D-C model. The method of converting rate parameters of one model in to the corresponding parameters of another model and vice versa is given by Kaguei et al. [8]. Using the data given by Littman et al. (2] and by assuming several frequency values, response signals, (TP.)c-s, based on the C-S model may be evaluated from Eqs. (7.11) and (7.19). As an illustration, let us con sider their data for Run No. 28 (copper-air, Dp 0.72 mm, Re = 2 .74 , L = 1.32 em, Eb = 0.46), in which the Nusselt number, Nu", and the effec tive solid phase thermal conductivity, kes, are reported to be 1.0 and 0.45 W m-I K -�, respectively. Using the given data, the amplitude ratio frequency and the phase lag-frequency relationships at a Reynolds number of 2.74 are estimated as shown in Figure 7.5. The response signals, (T�)c-s, are then calculated over the amplitude ratio 0.04 to 0.5 or the frequency range 0.00217T to O.Ol57T rad s-1. On the other hand, thermal response signals based on the D-C model, (T�)0_c, over the same fre quency range and at the same Reynolds number can also be computed from Eqs. (7.15) and (7.19) with various assumed values of aax and Nu. =
1
......
'0 0 .....
0
+J 0 .....
�
-
� (/) 0 .s= 0.
a.
10-2� _,__ -
O.Oln
w
FIGURE 7.5
__. 10-J.
____. ___ ..__ _ �_
_
O.OOln
Amplitude ratio versus frequency, and phase lag versus frequency computed for Run No. 28 (Re 2.74) of Littman e t a!. [2]. =
HEAT TRANSFER MODELS
x1o-4
257
18 �------�--� Run No.28 of Littman et
ol -1 w = 0.0021n rod•S
Re = 2.74 15
Nu"= 1.0
cf
=
0.1
...... I (/)
"!....
c; .......
10
>< 0 �
5
0.015n
1
FIGURE 7.6
10 Nu
cxax-Nu relationship for the D-C model evaluated using data of Run No. 28 (Re = 2.74) of Littman et al. [2].
From the two sets of response signals, (TV)c-s and (TV)o-c, the aax-Nu relationship of the D-C model is determined by curve fitting. Figure 7.6 depicts such a relationship with frequency as a parameter at a Reynolds number of 2.74. The contour map is constructed with a root mean-square-error, Er, defined by Eq. (7.26), of less than 0. 1 .
hT [(TPk-s- (TP)o-cF €f I
=
0
2rrfw
J [(T}1)c_5]' 0
'" dt (7.26)
dt
258
HEAT AND MASS TRANSFER IN PACKED BEDS
From the contour map with w = 0.0157T rad s-1, frax is determined to be approximately 11 X 10-4m2 s-1, which agrees well with the value, 11.2 x 1 o-4m2 s-1, predicted from Eq. (6.28). However, the contour map indicates that the Nusselt value cannot possibly be determined according to the D-C model with aax from Eq. (6.28), i.e. the modified D-C model. At this low Reynolds number, the large and frequency-dependent con fidence range of Nusselt number according to the modified D-C model indicates that the Nusselt number, Nu, may be regarded even as large as infinity, or there is little resistance to particle-to-fluid heat transfer. However, according to the C-S model, the particle-to-fluid heat transfer coefficient is given as Nu" 1. =
7 .4.2
From the Modified D-C
Model to the C-S Model
The preceding section has demonstrated that the exact value of Nusselt number based on the modified D-C model cannot be ascertained from heat transfer parameters obtained according to the C-S model. The purpose of this section is to reverse the procedure and to see what the C-S model Nusselt numbers, Nu", are generated from those based on the modi fied D-C model, Nu. The data for the glass-air system given in Example 7.1 are used in this illustration as follows: The frequency response signals based on the models, (TP)o-c and (TlJ)c-s will be predicted at Reynolds numbers of 0.1 and I. Using Nu � 1 and frax estimated from Eq. (6.28), (TP)o-c values can be evaluated from Eqs. (7. 1 5) and (7.19) over the frequency range 0.00027T to 0.0027T rad s-1 for Re 0. 1 , and 0.00277' to 0.0177' rad s-1 for Re 1. Note that any Nusselt number greater than unity will do; it makes no difference to the calculated response signals, (Tll)n-c- (TP)c-s values can be computed using Eqs. (7. 1 1 ) and (7.19) with various assumed values of kes and Nu" at the same Reynolds numbers. In the computation, the values of keF in Eqs. (7.12) and (7.13) are estimated using the following relationships: =
=
=
0.5(Pr) (Re) kF
at Re < 0.8
(7 .27)
at Re > 0.8.
(7.28)
A Reynolds number of 0.8 is tentatively chosen as the borderline in the application of Eqs. (7. 27) and (7. 28) with Eb 0.4 and Pr = 0.7. . 11.{ From the fitting of (Ty:II )c-s and (TFII )o-c, the kes-1vu re1 atwnsh"1p according to the C-S model can be determined. Figures 7 . 7(a) and (b) =
II
260
H EAT AND M ASS TRANSFER IN PACKED BEDS
show the relationship with a root-mean-square-error, Er, defined by Eq. (7.29), of 0.05.
J [(T�!)o-c - (T�!)c_s)2
112
2rr/w
€f
II
=
0
dt (7 .29)
2rr/w
r
..
0
[C TV)o-cP dt
Littman et al. [2] observed experimentally that the kes values predicted according to the C-S model depended upon the frequency values applied. However, kes cannot be a function of frequency. As illustrated in Figures 7.7(a) and (b), it is only the confidence range of the kes-Nu" relationship that depends on the frequency value.
10
o Li ttmon et a!
�---o·---r-- --�� 0
0
0 0 0 0 0
1
:: :J :z
1
10
Re
FIGURE 7.8
Nu" versus Re relationship for the
C-S model.
HEAT TRANSFER MODELS
261
Figure 7.8 showsNu" predicted in the amplitude ratio range 0.04 to 0.5 for the glass-air and copper-air systems over a range of Reynolds numbers from 0.01 to 5. The numerousNu" data (within the shaded area) obtained over this range of Reynolds numbers are compared with the experimental data determined by Littman et al. As illustrated, the predicted values agree reasonably well with the experimental data in the upper Reynolds number range. At low flow rates, the estimated Nusselt numbers, Nu", are signifi cantly lower than the corresponding values determined according to Eq. (8.20); moreover, the continuous decrease in Nu" with decreasing flow rate predicted by the C-S model is, as discussed in Chapter 8 , illogical and contradictory. The anomaly appears to indicate that the C-S model does not depict satisfactorily the phenomena of heat transfer in packed beds.
7.5
The Schumann Model
The Schumann model [3] is based on the assumptions o f fluid plug flow with no dispersion and no temperature gradient existing in the solid particle. In this section, the relationship between the Nusselt number, Nu, of the modified D-C model and Nu"' of the Schumann model is examined. From step response measurements Handley and Heggs [9] obtained heat transfer coefficients based on the Schumann model. Using the information obtained from their work on the solid-gas system, the response curve, (T�) schumann, is predicted from Eqs. (7.9) and (7.17) based on the Schumann model. The (T¥)o-c curves based on the modified D-C model are also computed from Eqs. (7.15) and (7 .I 7) with various assumed values of the heat transfer coefficient. Figure 7.9 shows a com parison of the (T¥)schumann and (T¥)o-c curves. The error, €5, is evaluated from
(7.30)
where t 1 and t 2 are chosen for
262
HEAT AND MASS TRANSFER IN PACKED BEDS
and
respectively. From Figure 7.9 it appears that the agreement between the two curves is good when E s 0.02. In Figure 7.1 0, the original Nusselt numbers, Nu"', of Handley and Heggs (9] are compared with those re-evaluated according to the modified D-C model. Some of the recalculated data are plotted with a confidence range indicating that Es = 0.02. It is seen that the data re-evaluated according to the modified D-C model are generally higher than the corresponding values based on the Schumann model. The differ ence between the two Nusselt numbers, Nu and Nu'", appears to widen with a lowering of the Reynolds number. From the analyses, it may be concluded that the Schumann model, which suffers from its over-simplified assumptions, is the least reliable for predicting heat transfer parameters. Both the C-S model and the original D-C model, although applicable at high Reynolds number, are found to give erroneously low and anomalous heat transfer coefficients at low Reynolds number. The modified D-C model, which also takes into con sideration the effect of axial heat conduction in solid particle and uses Eq. (6.28) in the prediction of the axial fluid thermal dispersion coeffi cient, describes the phenomena of heat transfer in packed beds more closely.
<
1
c: c:
�
::J .c: u V)
Num
c:
-u.. f-f--
c: 0
u
I 0
�
Nu -- 16
0.5
"0
=
9.
c: -
-u.. f-f--
10,
� 1 >schumann
� 0.005 0 . 05
'
0.02
12 23
0.02
44
0 . 05
}
II
o-c
0 50
0
Dp
FIG U RE 7.9 eb
=
0.36,
150
100
t
200
[S)
Comparison of step response curves, lead-air system; Re�lOO, = 3 mm, L = 3 . 3 em (these data are from Handley and Heggs [9 )) .
HEAT TRANSFER MODELS
•
Num of Handley and Heggs
I confidence range o Nu
102 f-
10 -
1
FIGURE 7.10
Nu"'
10
263
�� 0o�j�r c5
=
0.02.
f B.·,�rjj •
L
••
-
.•
-
I
I
Re
data of Handley and Heggs [9} and Nu re-evaluated based on the modified D-C model.
REFERENCES [ 1] H. Littman and R. G. Barile, Chern. Eng. Prog. Syrnp. Ser. 62 (No. 67), 10 (1966). [2] H. Littman, R. G. Barile and A. H. Pulsifer, Ind. Eng. Chern. Fund. 7, 554 (1968). [3} T. E. W. Schumann, J. Franklin Jnst. 208, 405 (1929). [ 4] P. V. Danckwerts, Chern. Eng. Sci. 2, l (1953). [5] N. Wakao, Chern. Eng. Sci. 31, 1115 (1976). [ 6 ] G. A. Turner and L. Otten, Ind. Eng. Chern. Process Des. Dev. 12,417 (1973). [7) D. J. Gunn and J. F. C. De Souza, Chern. Eng. Sci. 29, 1363 (1974). [ 8} S. Kaguei, B. Shiozawa and N. Wakao, Chern. Eng. Sci. 32, 507 (1977). [ 9 } D. Handley and P. J. Heggs, Trans. Jnst. Chern. Eng. 46, T251 (1968).
8 Particle-to-Fluid Heat Transfer
Coefficients
design and analysis of packed bed catalytic reactors, it is neces sary to know the temperatures of the fluid and the catalyst particles in which the chemical reactions are taking place. In general, fluid tempera ture is measured with little difficulty, but the measurement of the solid surface temperature is not easy. This is particularly true of packed bed reactors. The particle temperature or temperature drop at the particle surface then has to be estimated in terms of the heat transfer coefficient between the particle and the fluid. Because of the importance of the particle-to-fluid heat transfer coeffi cient in packed bed reactor, a considerable effort has been made to evaluate this parameter. Experimental determinations of heat transfer coefficients for a wide variety of systems have been made using various experimental techniques, under either steady-state or unsteady-state conditions. Table 8.1 summarizes, in chronological order, the heat transfer work reported in the literature together with their corresponding experimental methods and operating conditions. Figure 8.1 compares the numerous experimental data on heat transfer coefficients published in the literature over a wide range of Reynolds numbers. The transfer coefficients are expressed in terms of Nusselt numbers, Nu, in the graph. As shown, the reported data at high Reynolds numbers are consistent; however, at low flow rates, the data are quite incompatible and indicate a continuous decrease in Nusselt number with decreasing Reynolds number. This anomalous decrease in Nusselt number at low Reynolds numbers has been the subject of long dispute. There are two opposing views: one supports the unlimited decline in Nusselt number based on experimental observations, whereas the other argues that a limiting Nusselt number should exist at zero flow rate. Gunn and De Souza [23], for example, have shown from frequency response measure ments, that the limiting Nusselt number is about 10. FoR THE
264
HEAT TRANSFER COEFFICIENTS
265
Some theoretical studies have also been carried out to explain the observed experimental results. There is, of course, no exact theory which describes satisfactorily the transport phenomena in packed beds. The proposed models which are based on different assumptions often predict different and contradicting Nusselt numbers. Two completely divergent conclusions have been drawn based on different assumptions. From an analogy between electrostatics and heat, Cornish [31 1 explained that the heat transfer coefficient in a dense system of particles was con siderably less than that for a single sphere in an infinite medium. Kunii and Suzuki [321 pointed out that fluid flowing in channels in a bed was the reason for this anomaly. Nelson and Galloway [33 1 claimed that the anomaly could be explained by a renewal of the fluid element surrounding each particle. Schltinder [34 1 showed that if a packed bed was a bundle of parallel capillaries the transfer coefficient should continuously decrease with a decrease in the flow rate at lower Reynolds numbers. Martin [35] pointed out that non-uniform packing of the particles in a bed caused a decrease in Nusselt number at low flow rates. On the other hand, by applying a free surface model, Pfeffer and Happel [36] obtained a Nu-Re relationship with a limiting Nusselt number of about 1 3, as the Reynolds number dropped to zero in a bed with a void fraction of 0.4. From an analysis of steady-state heat transfer in a con centric hollow sphere with a stagnant fluid, Miyauchi [37] showed a limiting Nusselt number of about 18, at a bed void fraction of 0.4. S0rensen and Stewart [38] studied creep flow through a cubic array of spheres and found a limiting Nusselt number of about 3.9. Schllinder [39] p roposed that the Nusselt numbers for particles in packed beds are greater, by a factor of 1 + 1.5( 1 -Eb), than those for flow over single spheres; assuming this, he obtained a limiting Nusselt number of 3.8 for a packed bed with Eb = 0.4. Based on a stochastic model o f extraparticle void space, Gunn [40) derived a theoretical equation for the particle-to-fluid transfer coefficient, which yields a limiting Nusselt number of 4 at zero flow rate. Wakao et al. [29, 30, 41, 42], however, have shown that it is a defect in the fundamental equations which is mainly responsible for the anomalous decrease in Nusselt number at lower Reynolds numbers. In the following sections a critical review is made of the published particle-to-fluid heat transfer coefficient data. Selected reliable data will be revised and correlated to give an empirical formula which may be used for the accurate prediction of particle-to-fluid heat transfer coefficients.
266
HEAT AND MASS T RANSFER IN PACKED BEDS
TABLE 8.1 Heat transfer experimental dataa.
Year
Investigator
Experimental method
Steady or unsteady state conditions Material
1943 Gamson eta!. Evaporation of water [ 1]
Steady
1943 Hurt [2]
Evaporation of water
Steady
1945 Wilke and Hougen [3]
Evaporation of water
Steady
1952 Eichhorn High frequency Steady and White [4] dielectric heating particles 1954 Satterfield Decomposition Steady and Resnick of H202 [ 5] 1957 Galloway Evaporation Steady eta/. [6] of water 1958 Glaser and Thodos [7]
1958 oaumeistcr and Bennett (8] 1960 De Acetis and Thodos [9J 1 9 6 1 Kunii and Smith [ 10]
1963 M cConnachie Evaporation of and Thodos water [ 1 1]
Shape
Size (mm)
Sphere
2.3, 3.0, 5.6, 8.4, 1 1 .6 Cylinder 4. 1X4.8, 6.8 X 8.5, 9.8X1 1.7, 14.0X 1 2.5, 1 8 . 8 X 16.9 Cylinder 9.5 X9.5
Celite
Cylinder 3.1X 3.1, 4.8 X4.3, 6.6x7.2, 9.7x8.6, 1 3.4 X 12.8, 1 5 . 1 X 16.3, 18.2X 16.9
Dowex-50 Sphere
0.1, 0 . 3 , 0.4, 0.5, 0.7
Polished catalytic metal Celite
Sphere
5.1
Sphere
17.1
Monel
Sphere
4.8
Brass Steel Steel
Sphere Cylinder Cube Sphere
6.4, 6.4, 6.4, 3.9,
Celite
Sphere
15.9
Steady
Glass Sand
Sphere
0.1, 0.4, 0.6, 1.0 0.1, 0.2
Steady
Celite
Sphere
15.9
Heating metallic Steady particles by passing electric current through the beds High frequency Steady induction heating particles Evaporation of Steady water Axial heat conduction in beds
Celite
Particle
7.9 9.5 9.5 6.3, 9.5
HEAT TRANSFER COEFFICIENTS
267
TABLE 8.1 (Continued) Determination of the heat transfer coefficients
Fluid
Pr
Particle temperature
Re
No
Measured
No
Surface assumed to be at wet-bulb temperature
No
Heat transfer coefficients were not determined, but obtainable from their data
1-18 Measured
No
The measurements were criticized by Littman et al. [ 18]
15160
Measured
No
1501200
Measured
No
100b 9200
Measured
No
20010 400
Measured
No
322100
Measured
No
0.001-1
Continuous solid phase with axial heat conduction assumed Measured
Yes
0.720.75
1004000
Air
0.76
Air
0.73
72950 45250
Air col
0.7 0.8
Air
0.71
Hl C02
0.72 0.67
Air
0.7
Air
0.72
He, Air, C02 , Water
Air
0.72
Remarks
Surface assumed to be at wet-bulb temperature
Air
Vapor 1.0 mixture of H202 and H20 Air 0.72
Fluid dispersion considered
110b 2500
No
The measurements were criticized by Jeffreson [28]
The anomalously low data were criticized by Littman et al. [18] and Gunn et al. [23]
268
HEAT AND MASS TRANSFER IN PACKED BEDS
TABLE 8.1 (Continued) ---
Year
..
..
·---·--·
lnvestiga tor
Steady or unsteady state Experimental conditions method
Particle Material
Shape
Size (mm)
1963 Bradshaw and Evaporation of water Myers [ 12]
Steady
K aoline AMT Kaosorb Celite
Sphere Sphere Cylinder Cylinder
4.7 8.8 4.0X4.1 4.2 X4.2, 6.2 X4.9
1963 Sen Gupta and Thodos [ 13] 1964 Sen Gupta and Thodos [ 14] 1967 Mailing and Thodos [15] 1967 Lindauer [ 16]
Evaporation of water
Steady
Celite
Sphere
15.9
Evaporation of water
Steady
Celite
Sphere
15.9
Evaporation of water Frequency response
Steady
Celite
Sphere
15.7-15.9
Sphere Sphere
1.0, 1.8, 3 . 2 0.5
1968 Handley and Heggs [ 17]
Step response
Unsteady Steel
1968 Littman et a!. [ 18]
frequency response
1970 Bradshaw et a!. [ 19]
Step response
Unsteady Alumina Steel Hematite
1 9 7 1 Goss and Turner [20]
Frequency response
Unsteady Soda-lime Sphere glass Borosilicate Sphere glass Methyl Sphere methacrylate
Unsteady Steel Tungsten
3.2, 6.4, 9.5 Sphere Cylinder 4.8X4.8, 6.4X6.4, 6.4 X 12.7 Sphere 3.0, 6.1, 9 . 1 Lead Sphere 9.5 Bronze 6 . 1, 9.1 Soda glass Sphere 3.0 Lead glass Sphere Alumina- Sphere 3.2 silica Unsteady Copper Sphere 0.5, 0.6, 0.7, 1 . 1 Glass Sphere 0.5 Lead Sphere 2.0 Sphere Sphere Sphere
1 3.2, 25.4 25.2 1 1. 1 4.0 5.0 4.8
HEAT TRANSFE R COEFFICIENTS
269
TABLE 8.1 (Continued) --·------ ----
Determination of the heat transfer coefficients
Fluid
Pr
Re
Particle temperature
Fluid dispersion considered
Air
0.7
40065ooh
Measured
No
Air
0.72
8002000
Measured
No
Air
0.72
20006000
Measured
No
Air
0.71
Measured
No
Air
0.7
1858500 2318 200
No
Air
0.7
804000
No temperature gradient assumed in the particle No temperature gradient assumed in the particle
Air
0.7
2100
Air, N 2
0.74
150b 600
Air
0.7
16003000
Continuous solid phase with axial heat conduction assumed Center-symmetric temperature profile assumed in the particle Center-symmetric temperature proflle assumed in the particle
Remarks
No
The data were corrected by Jeffreson [ 28] for fluid dispersion
Yes
The method of determining Nu data was criticized by Kaguei et a!. [29]
Yes
Yes
270
HEAT AND MASS T R ANSFER IN PACKED BEDS
TABLE 8.1 (Continued)
Year
Investigator
19 73 Turner and Otten [21]
1974 Balakrishnan and Pei [22]
Steady or unsteady Experimental state method conditions Material Frequency response
Microwave heating particles
Unsteady Soda-lime glass Ceramic Sintered glass Fertilizer Iron ore Epoxy Steady Iron oxide Nickel oxide Vanadium pentoxide Nickelmolybdenum oxide Cobaltmolybdenum Unsteady Glass
Particle Shape Sphere
Size (mm) 4.0
Spheroid 7.4 4.6
Sphere
3.1 4.8 3.5
Sphere Sphere
6.4 6.4, 12.7
4.8 Sphere Cylinder 5.6 X5.6, 5.6 X8.3 Cylinder 3.2X6.4
Cylinder 3.2 X6.4
197 4 Gunn and De Souza [23]
Frequency response
1975 Bhattacharyya and Pei {24] 1975 Cybulski et .1!. [25]
Microwave heating particles Radial heat conduction in beds
1976 Wakao et al. [ 261
Shot response
Unsteady PolystyreneSphcre Sphere Glass Lead Sphere
0.8 0.3, 1.1' 1.6 0.9
1981 Shen et al. [27]
Shot response
Unsteady Glass
1.3, 2.7
Steady Steady
Steel Lead Ferric oxide Siliconcopper
0.3, 0.5, 1.2, 2.2, 3.0, 6.0 Sphere 3.2, 6.3 Sphere 0.8 3.2, 7.6 Sphere Cylinder 5.1X5.1 Irregular 0.1 Sphere
Sphere
HEAT TRANSFER COEFFICIE NTS
271
TABLE 8.1 (Continued) Determination of the heat transfer coefficients Fluid dispersion considered
Particle temperature
Pr
Re
Air
0.7
12004600
Center-symmetric temperature profile assumed in the particle
Yes
Air
0.7
3404400b
Measured
No
Air
0.7
0.05330
Center-symmetric temperature profile assumed in the particle
Yes
Measured
No
Continuous solid phase with radial heat conduction assumed
No
Center-symmetric temperature profile assumed in the particle
Yes
No definite Nu data were obtained, but it was shown that Nu cannot be smaller than 0.1
Center-symm�tric temperature profile assumed in the particle
Yes
The results are shown in Figure 8.10
Fluid
Air
0.7
Air
0.7
Air
0.7
Air
0.7
110830 0.240.64
0.2-6
5.1229
Remarks
Nusselt numbers at
Re < 1 were not determined
The anomalously low Nu data were criti-
cized by Wakao et al.
(30]
a Diluted beds, distended beds and data with a single particle layer are not included. b Re = DpG/J.L except for Refs. [71 where Re Sp2G/[J.L0-eb) 'It], in which Sp is the
particle surface area and 'It is the shape factor; Refs. [ 1 1 , 12, 22] where Re = DpGI [J.L ( l - eb)] and Ref. [ 19] where Re = DpG/[6J.L(l-eb) J. =
272
HEAT AND MASS TRANSFER IN PACKED BEDS
�} Kunii and Smith
C Cybulski e r a!. D Eichhorn and White E Littman eta!. F Gunn and De Souza G Balakrishnan and Pei H Bhattacharyya and Pei I Glaser and Thodos J Satterfield and Resnick
K Bradshaw eta!. L Handley and Hcggs Hougen etal. Hurt M , Galloway eta!. Thodos eta!. Bradshaw and Myers N Lindauer 0 Baumeister and Bennett P Turner eta!.
1 1 \ t
( 1 961) (1975)
(1 952) (1968) (1974) (1 974) (1975) (1958) (1954)
(1970) (1968) (1 943, 45) (1943) (1957) (1 960-67) ( 1 963) (1967) (1958) (1971, 73)
A: Liquid-Solid; B-P: Gas-Solid 103 1 02
10 1 10-1
?
::I z:
10-2 1o-3
�--�--�-----�--�
10-6 1 0- 3
10 -2
10-1
1
10 Re
FIGURE 8 . 1
8.1
Heat transfer coefficients published in the literature.
A Review and Correction of the Data Obtained from Steady-state Measurements
Similar to the discussion in Chapter 4, the data selection is confined to the heat transfer measurements which satisfy the following conditions:
273
HEAT TRANSFER COEFFICIENTS
a)
The particles in the bed are all active. Distended and diluted bed data not included.
b) The number o f particle layers in a heat transfer bed are greater than two. 8.1.1
Simultaneous Heat and Mass Transfer Studies: Evaporation of Water and Diffusion-controlled Chemical Reactions at Particle Surfaces
Mass transfer data obtained from simultaneous heat and mass transfer studies have been used for the data correlation in Chapter 4. The corre sponding heat transfer data which were determined without considering the effect of thermal dispersion are re-assessed in this section. The data include those obtained from measurements of the rates of evaporation of water by Gamson et a/. [ 1] , Hurt [2], Wilke and Hougen [3], Galloway et a/. [ 6], Thodos et a/. [9, 11, 13-15] and Bradshaw and Myers [ 12], as well as those determined from the catalytic decomposition of hydrogen peroxide on metal spheres by Satterfield and Resnick (5]. These measure ments were made using solid particles with a constant surface temperature, Tps' throughout the bed. In packed beds with a constant particle surface temperature, the resist ance to heat transfer resides only on the fluid side. If the contribution due to axial fluid thermal dispersion is neglected, the system may be described by the following heat balance equation under steady-state conditions (with the notation used in Chapter 6): d TF
�
U- + dx
�
h a
€bCFPF
(TF- Tps)
=
0
for
0
<x < L
(8. 1 )
where h is the heat transfer coefficient with the axial fluid thermal dispersion coefficient, a3x, equal to zero. When a fluid , at a temperature Tin is flowing into a bed of length the exit temperature, Texit, is
L,
(8.2) I f , however, axial fluid thermal dispersion is taken into consideration, then, the heat balance equation becomes
274
HEAT AND MASS TRANSFER IN PACK ED BEDS
(8.3) With the following Danckwerts boundary conditions: U( TF
d TF
at x
-Tm)=aax dx
-
and
dTF dx =0
at
-
=
0 (inlet)
(8.3a)
x = L (exit)
(8.3b)
the exit fluid temperature is then
Tps- Texit Tps- Tin where
4A exp
(
(l+A)2exp A
LU 2 aax
)
(!!!_) 2aax
(
-(l-A)2exp -A
LU 2aax
)
(8.4)
(8.4a)
h�,
By applying Eqs. (8.2) and (8.4), the heat transfer coefficients, reviewed in this section, can be converted into hp, which takes into account the axial thermal dispersion effect. The values of the axial fluid thermal dispersion coefficient, a3x, used in Eq. (8.4) are estimated from Eq. (6.28). All the published data, except those from articles by Gamson et al. [ 1 ], Hurt [2], and Bradshaw and Myers [ 1 2], in which no detailed information on bed height and/or void fraction is available, are thus corrected. "8.1.2
Temperature Measurements in Beds with
no
Heat Generating
Particles
Kunii and Smith [10] and Cybulski et al. [25] determined heat transfer coefficients based on the C-S model from steady-state heat transfer measurements.
HEAT T RANSFER COEFFICIENTS
275
Heat transfer coefficients determined from axial heat transfer measure ments by Kunii and Smith [ 1 O] were subjected to criticism by Littman et al. [ 18] and Gunn and De Souza (23) . They pointed out that the anoma lously low heat transfer coefficients obtained by Kunii et al. were due to incorrect interpretation of an algebraic relationship between the heat transfer coefficient and the effective thermal conductivity of the bed. Gunn and De Souza (23] further elucidated that, if Kunii et al. had inter preted the algebraic relationship correctly, infinitely large heat transfer coefficients would have been obtained. Cybulski et al. [25] obtained anomalously low heat transfer coefficients by fitting the steady-state radial gas temperatures measured at the bed exit and the corresponding gas temperatures estimated based on the C-S model. Wakao et a!. (30] have shown that the gas and solid temperature profiles at the bed exit predicted based on the C-S model do not agree with each other at all. As a matter of fact, the solid and gas .Phase tempera tures, under steady-state conditions and in a bed with no heat generating particles, are considered to be identical with each other, as illustrated in Figure 5 . 1 . Thus, the C-S model cannot be applied to steady-state heat transfer measurements unless there is a heat source or sink existing in the solid particles. This is demonstrated in the example as follows.
txample 8.1 Suppose the steady-state radial heat transfer measurements are made with the following solid-gas system: Fluid : air Solid: glass beads of 0.5 mm diameter k F 0.030 W m-1 K-1 ks 0.88 W m-1 K -1 Cs = 670 J kg-1 K-1 Ps = 2500 kg m-3 Wall temperature, Tw 1 00°C Fluid temperature at bed inlet, T0 50°C Radius of bed, Rr 2 em Length of bed, L 2 em Void fraction, Eb 0.4 Re = 0 . 1 . =
=
=
=
=
= =
276
HEAT AND MASS T R ANSFER IN PACKED BEDS
Predict the radial temperature profile at the bed exit. Also, examine whether the C-S model may be applied to steady-state heat transfer in a packed bed. Assume no heat transfer resistance at the column wall (this is the same assumption made by Cybulski et al. (25]). SOLUTION
(i) Temperature at the bed exit Bed temperatures, under steady-state conditions, may be estimated from Eq. (5.1) based on the Single Phase model. At such a low flow rate (Re =0.1) axial heat conduction should be taken into consideration. Using the same boundary conditions employed by Cybulski et al. (25]:
T= T0 ar -=0 ar T= Tw
atx=O (inlet) at r=0 at r=Rr
together with the Danckwerts condition at the bed exit:
aT -=0 ax
atx
=L
the solution to Eq. (5.l) for a packed bed of finite length, L, gives
(8.5) where Fn
=
an -�n
--- --
an exp (-�n)- �n exp (-an)
an=A(l+Bn) �n=A(l-Bn)
A=
GC FL
2keax
--
(8.5a) (8 .5b) (8.5c) (8.5d)
B
n =
[
]
HEAT TRANSFER COEFFICIENTS
1 +4
( GCFRT ,
a
n
)2
k k er
112
277
(8.5e)
eax
and an is an n-th root of
(8.6) The bed exit temperature, TL, calculated from Eq. (8.5), is represented by the solid lines in Figures 8.2(a)-(d)
.
(ii) Solid and gas phase temperatures from the C-S model Under steady-state conditions the fundamental equations based on the C-S model are: GCr
[ � � (r arF)
a T F h a(TF arF- k + -Ts) eF p ax r ar ar ax2 2 1 a a rs a Ts] , kes r- +- + hpa�Tr-Ts)=O. r ar ar ax2
[ ( )
'Y
2
]
+
=
--
0 (8.7) (8.8)
At this low flow rate (Re 0.1 ) the effective radial and axial fluid phase 'Y thermal conductivities are considered to be the same, i.e. 1 . If axial fluid phase thermal conduction is ignored, then, 'Y is set equal to zero. The C-S model was employed by Littman et al. [ 1 8] and Vortmeyer and Schaefer [43] in their studies of axial unsteady-state heat transfer in packed beds. Let us assume that their boundary conditions may be used for steady-state heat transfer as well. The axial boundary conditions of Cybulski et al. [25], Littman et al. [I 8], and Vortmeyer and Schaefer [43) are designated as Case A , C and D , respectively, and are shown in Table 8.2. Case B which has the boundary conditions somewhat similar to those of Case A , except that axial fluid phase heat conduction is considered, is also included in Table 8.2. Note that, in solving the fundamental equations, Littman et al. con sidered the axial fluid phase conduction term, while Vortmeyer and Schaefer ignored it. Cybulski et al. also neglected the axial fluid phase heat conduction term in their application of the C-S model for radial steady-state heat transfer measurements. In addition to the axial boundary conditions given in Table 8.2, the following radial conditions are needed for solving Eqs. (8.7) and (8.8): =
,
=
11\---- - ----·;-- ·:�--� s-- [6 -666i>
100 �;.--
--
-
-
koO�l---
s
---
TFL(0.1-"') TsL<0.1-oo)
\
\
� 0 ' <1)
:J ...,
\
u 0
I
80
�
<1) t-
\
\
\
i
I \ TFL <0.0001) \
I ! i
I
70
\'
\
(O)
\
<1)
5 80
0 ' <1) a
+-'
::---....- I
r/R
\. \. \
I
�
t-
70
TFL<20> Ts L<20> TFL(") TsL <"") TL =
=
\. TFL<0.0001) \
I
"'
60 �------�--._--� 1.0 0.5 0 0.5 1.0
00
(0
I
I
,-, u 0
t.J -..)
I
=
90
-.. \� - - --�� � ���s_t.l \\ s:<0�0008> ,\ T \. ,. 90 TF � . 0008) \
100 ----
\'
\
I
I
! I
""
60L-------�� ------�� -----�� ------� 1.0 0.5 0.5 1.0 0 (b)
r/R
:r: !'T1 > -l > z 0 s: > (/) v: -l ::c > z (/) -r: tTl ;l:l
-
z "0 > ()
� � ti o::l � 0 (/)
1 oo
�:: �:-:-:-Js:-::-:=--.---- \';: = : ��-,-_-_ �-i oooi-) \
\
:
,......
\ ·,� \
Q)
\
901-
100
s�- l - 0-.
\
...., 0 L Q)
a c:: Q) I-
TFL(4)
TFL (oo)
.
\
TL \
=
=
� 80
0 L Q) a E Q) I-
:::J ....,
\'TFL(0.0001) \ \ \
70
\
(C)
TL
=
T sL<40 �)
\TF L(0.0001 ) \' \'
:t M > >-1 >-1 ;:c > z (/) .,...
""-
:"'1 ::0 (')
60 �------�--� 0 0.5 1.0 0.5 1.0
F.r.�------�---L--�
0.5
� I . #
TFL(40-�>
0
"' -
1.0
,-'i
,...... u
TsL<4> TsL (oo)
\
70
-
90
.
3 80
SL-( Q.-
T sL<0.0005)
" ooaJ ;�:<-O.O
.
u 0 ......
----- - - ----. . _-- ----- - -.- - , __T 0001> --- -
0
r/R
0.5
1.0
r/R
fo'JGURE 8.2 Comparison of bed exit temperatures calculated basl�d on thl' Single Phase modd ('/'1.) and lhl: C-S 111odd ('/'H. and TsL), for glass (0.5 mm diametcr)-air system at Rc = 0.1. For cxampk, Tn.(0.0005) and '�'sL(O.OOO.S) indit:i..lll� lhl· gas and solid temperatures, respectively, at the
bed exit calculalcd with Nu
=
0.0005: (a) Case A; (b) Case B; (c) Casl'
C;
(d) Case D.
0 M -
.. � (') -
� z >-1 (/)
tv -.J \0
N 00 0
TABLE 8.2 Comparison of axial boundary conditions for C-S models.
Case A
In fundamental Eqs. (8. 7) and (8.8) 'Y
TF
0
To
aTs =0 ax aTs =0 ax
Ts
1
To
c
1
aTs T11 -kes- = (1- c=b)hp(TF - Ts) ax
0
Ts<x = 0), TF(X = L) and Ts(x =L) arc
x=L
x=O
B
D
Conditions for
Axial boundary conditions
aT s T0 -kes- = (1-c=b)hp(TF-Ts) ax
Ts
TF
aTs =0 ax
Cybulski et al. {25] for steady-state analysis
aTs =0 ax
aTF =O ax aTF =O ax
those assumed by
aTs kes- = (1- c=b) hp(Tf- Ts) ax
Littman et al. [ 181 for unsteady-state analysis
Vortmeyer and aTs kes - = (1- c=b)hp(3(Tr-Ts) Schaefer [431 for ax
(3=
1
-------
1 + (1- c=b)hp/(GCF)
unsteady-state analysis
:I: � ;p. � ;p. z 0 :s:: ;p. (/) (/) � " ;p. z
{f) .,...
rr1 " -
z ., ;p. () :::-:: !'1'1 0 t%' tTl 0
{/)
HEAT TRANS FER COE FFICIENTS
arF ars -=-= 0 ar ar
28 1
at r= 0
TF= Ts = Tw The temperatures at the bed exit, TFL and TsL are, hence,
(8.9)
(8.1 0) where an is a root of Eq. (8.6), and m; is a root of the follo:wing equation:
(8.11)
Pn; and Qn; are obtained from
3+-y
� L pnz·=
i=l
1
Lhp'A' Pni l-kes i=I
3+-y r
L
(8.12)
-- (
Lhp'A'
m; +
ke s
) ]
Qn; exp (m;)= 0
282
HEAT AND MASS TRANSFER IN PACK ED BEDS
and
3+-y
L m; exp (m;) Pni
i=I
=
0
(8. 1 3)
where -y, A. and A.' for Cases A-D are tabulated as follows: Case
'Y
A.
A.'
A B
0
0 0
0 0
c
1 0
1 - €b 1 - €b
1 - €b (I - €b) �
D
See Table 8.2 for {3.
In the case of /' 0, Pni and Qn; can only be determined from the relationships given by Eq. (8 . 1 2). The effective fluid phase thermal conductivity, keF• is given by Eq. (7.27). The effective solid phase thermal conductivity, kes, is assumed to be related to the effective axial thermal conductivity, keax• and keF by: =
keax =keF+ kes·
(8. 1 4)
At Re 0. 1, keax k�, where k� is the effective thermal conductivity of a quiescent bed. Hence, =
�
kes = k�-keF·
(8.15)
The solid temperature (TsL) and gas temperature (TFL) profiles at the bed exit are then computed. In Figures 8 .2(a)-(d), the predicted profiles are compared with the temperature (TL) profile estimated in Solution (i) based on the Single Phase model. In Cases B-0, it is found that
TL is in good agreement with TFL predicted with certain small Nusselt numbers, but TFL is different from the corresponding TsL value. b) TL concurs well with both TFL and TsL computed with infinitely a)
large Nusselt numbers.
HEAT TRANSFER COEFFICIENTS
283
Therefore, if a measured temperature profile is compared with the com puted TFL values, as done by Cybulski et al. [25), it is expected that two different Nusselt values, one very small and the other infinitely large, will be found. The C-S model reduces to the Single Phase model when Ts TF is infinitely large. or the particle-to-fluid heat transfer coefficient, Since the packed bed temperature under steady-state conditions is described by the Single Phase model, the application of the C-S model to steady-state heat transfer is expected to yield, superficially, infinite values of Nusselt number. However, in Case A of Cybulski et a/. (25], infinitely large Nusselt numbers are not obtained; TFL and TsL estimated with large Nusselt numbers are considerably different from TL. This is due to the assumptions inherent in Case A: axial fluid phase conduction is neglected, and the solid phase is assumed to be adiabatic at both ends. From a comparison of TL and TFL' small Nusselt numbers are obtained. The Nusselt numbers determined by Cybulski et al. are of this magnitude. However, attention should be paid to the fact that in Cases A-D, when Nusselt numbers are small, the TsL value is entirely different from the corresponding TFL value. As mentioned already, TFL and TsL should be the same under steady-state conditions. The small Nusselt numbers, obtained from the fitting of TL and TFL do not represent the intrinsic heat transfer coefficients. Therefore, it is concluded that the C-S model cannot be applied to steady-state heat transfer analysis. For reference, the Nusselt numbers erroneously obtained from the fitting of TL and TFL are shown in Figure 8.3. (End of Example)
hp,
�
'
8.1.3
Heat Transfer Between Heat Generating Particles and
Fluid
In their heat transfer measurements, Eichhorn and White [ 4] , Baumeister and Bennett [8] and Pei et al. [22, 24] employed high frequency heating to generate heat in solid particles. According to Eichhorn and White, solid temperature profiles in the bed are a linear function of the axial distance; thus, the solid temperature at the bed exit can be estimated by linear extrapolation. They assumed that the temperature difference between the solid and gas phases through out the bed was the same as the difference between the extrapolated solid temperature and the measured gas temperature at the bed exit. With these assumptions they determined the heat transfer coefficients. Their results
284
HEAT AND MASS TRANSFER IN PACKED BEDS
Case A, D
10 -2
axial fluid
z
dispersion ignored
:J
(/) :J 0
Case
c 0 L L w
axial fluid dis�ersion
10-3
considered
Re FIGURE 8.3
B, c
1
Erroneous Nusselt numbers obtained by fitting TL and TFL·
indicate that the temperature difference between the two phases is rela tively small (0.5 to 2.1°C). Considering the errors (±0.6°C) associated with the measured solid temperature profiles, the heat transfer coefficients determined are, therefore, not reliable enough. Moreover, it is doubtful whether the solid temperature at the bed exit is predicted by simple extrapolation. To verify this, the temperature profile in the bed is analysed based on the D-C model as follows: When heat is generated at a constant rate in particles, the steady-state fluid and solid temperatures in a bed of finite length a re described by Eqs. (6.22) and (6.23) according to the D-C model. With the Danckwerts boundary conditions, Eqs. {8.3a) and (8.3b), the solutions to Eqs. (6.22) and (6.23) are expressed as:
{8.16) and (8.17)
HEAT TRANSFER COEFFICIENTS
285
where Qv is the rate of heat generation per unit volume of solid. From Eq. (8.1 7), the average (volume mean) particle temperature is (8. 1 8) Equation (8. 1 6) shows that the fluid temperature increases almost linearly in the axial direction but levels off before it reaches the bed exit. Accordingly, Eq. (8. 1 8) indicates that the temperature gradient of fs should be approximately constant within the bed and becomes zero at the bed exit. The bed exit effect, however, was ignored by Eichhorn and White [4] in their extrapolation of the solid temperatures to the bed exit value. Hence, the temperature difference between the gas and solid predicted by Eichhorn and White is expected to be larger than the actual difference in the bed. In their investigation Baumeister and Bennett [8] found significant temperature gradients in both the radial and axial directions in the bed. Their results on the large radial temperature differences were subjected to criticism by Jeffreson (28]. Pei et al. [22, 24], on the other hand, found that the solid temperature was uniform throughout the bed; based on this, they evaluated the heat transfer coefficients. Their observed uniform solid temperature in the bed contradicts the findings of previous investigators, e.g. Eichhorn and White (4], and Baumeister and Bennett [8]. Glaser and Thodos (7] applied an electric current directly through a bed of metal spheres to generate heat in the particles. However, in apply ing direct electric heating there is a chance that heat transfer may occur only at, or near, the solid-solid contact points where most of the heat is generated. If this is the case, then the measured heat transfer coefficients would be different from those we are concerned with. Therefore, based on the above considerations, the data reviewed in this section are not to be included in the data correlation.
8.2
A Review and Correction of the Data Obtained from
U nsteady-state Measurements
Heat transfer coefficients in unsteady-state packed bed systems are usually obtained using step, frequency and shot response techniques. The deter mination of transfer coefficients from the response measurements is based
286
HEAT AND MASS TRANSFER IN PACKED BEDS
on the unsteady-state models: the Schumann model, the C-S model and the D-C model, discussed in Chapter 7. The Schumann model [ 44] , which assumes no temperature gradient in solid particles and no dispersion in fluid phases, is not realistic and is inadequate for describing unsteady-state heat transfer. Both the C-S model and the original D-C model (with fr�x from Eq. 6.3) have been shown to be incapable of predicting accurate transfer coefficients at low Reynolds numbers. So far, the modified D-C model (with frax from Eq. 6.28) is the most successful model proposed. Owing to the varying assumptions underlying the different models, the values of the heat transfer coefficients based on these models are generally incompatible, especially at low Reynolds numbers; thus, they cannot be compared simply on an equal basis. Therefore, the published unsteady-state heat transfer data are treated according to the modified D-C model and then correlated together with the re-assessed steady-state heat transfer data. 8.2.1
Heat Transfer Data obtained from Step Response Measurements
Step response measurements were made by Handley and Heggs [ 1 7], Furnas [45], Saunders and Ford [46], Lof and Hawley [47], and Coppage and London [48], in the determination of heat transfer coefficients. They all determined heat transfer coefficients based on the Schumann model. Bradshaw et a/. [ 1 9 ] also determined heat transfer coefficients based on the Schumann model, and then converted the data into those according to the original D-C model. In the earlier studies [45-48], the heat transfer coefficients determined by graphical methods are not reliable. Therefore, only the data of the two relatively recent measurements by Handley and Heggs [ 17], and Bradshaw et a/. [ 1 9 ] are considered in the modified D-C model. The data of Handley and Heggs have already been re-evaluated in Section 7.5. The heat transfer coefficient data of Bradshaw et a/. [ 1 9 ] are revised as follows: Assuming that a step temperature change is imposed on the fluid entering the packed bed, the response signals are predicted with the data presented in their paper. The predicted signals are denoted by ��(T}.I)Schumann· The response signals based on the modified D-C model, (T )n-c, are also computed with the frax values from Eq. (6.28) and various assumed Nusselt values. By fitting (T�� )schuman n and (TV) n-c in the time domain, the correct Nusselt values of the modified D-C model,
HEAT TRANSFER COEFFICIENTS o
Nu'
287
I of Bradshaw et o l
-
10
1 � 2-� --� � � ��-� -� � � � 103
10
4 10
Re
FIGURE 8.4
et a/. [ 19] and Nu re-evaluated based on the modified 0-C model.
Nu' of Bradshaw
Nu, are determined. The re-evaluated values of Nu are plotted together with their corresponding data, Nu 1, in Figure 8.4. For some of the data,
the confidence ranges are indicated in terms of the root-mean-square-error, €5 , equal to 0.02, as defined by Eq. (7.30). As illustrated, the agreement between Nu and Nu 1 is fairly good. This is not unexpected since the measurements were made at high Reynolds numbers (Re 490 to 2200), thereby, Eqs. (6.3) and (6.28) give almost the same axial fluid thermal dispersion coefficient values. =
8.2.2
Heat Transfer Data obtained from Frequency Response Measurements
Heat transfer coefficients were obtained from frequency response measure ments by Lindauer [ 1 6], Littman et al. [ 1 8], Goss and Turner [20], Turner and Otten [21], Gunn and De Souza [23], and Littman and Sliva [49]. Lindauer [ 16] employed the fundamental equations based on the Schumann model to determine heat transfer coefficients, but no information on the frequency range is given in his paper. Therefore, the heat transfer data cannot be converted into those based on the modified D-C model. The data of Littman et al. [ 1 8 ] , obtained on the C-S model, are examined and revised in Section 7 .4, according to the modified D-C model. Goss and Turner [20], Turner and Otten [21], and Gunn and De Souza [23], all reported heat transfer coefficients and axial fluid thermal dispersion coefficients based on the D-C model. The following tests are carried out to assess the reliability o f these data:
288
HEAT AND MASS TRANSFER I N PACKED BEDS
8.2.2. 1
Confidence test on the data of Gunn and De Souza [23]
Using the reported aax and Nu data of Gunn and De Souza [23], frequency response signals, (TV)Ref, are first predicted from Eqs. (6. 1 0) and (7.19). For the same system, the response signals, (TV)o-c, are also computed from Eqs. ( 6. 1 0) and (7 19) with various assumed values of aax and Nu. The signals, (TV)o-c and (Tl.J)Ref , are then compared in terms of the following root-mean-square-error: .
21TjW *
€f
=
f 0
[(TP )Rer - (Tt')o-cl' dt 21Tfw
f 0
1/2
(8.19)
[ (TP ) Rerl' dt
The resulting error maps are constructed as shown in Figures 8.5(a) and (b). In Figure 8.5(a) (Re ::::::: 1 1), the horizontal contour with Ef 0.05 indicates that the precise value of Nusselt number cannot be determined. Although Gunn and De Souza obtained Nu ::::::: 50, the contour shows that the Nusselt number can be any value greater than about I 0. In Figure 8.5(b) (Re ::::::: 33), the contour with Ef 0.05 reveals that the Nusselt value should fall in the range 20 to 90. The confidence ranges o f Nusselt numbers, thus estimated, are shown in Figure 8.6. As illustrated, the ranges are large at low Reynolds numbers ; moreover, their original data show considerable scattering. Therefore, their data will not be considered in the correlation in Section 8 . 3. =
=
8.2.2.2
Confidence test on the data of Turner et al. [20, 21]
No detailed information on the frequencies employed in the measurements is given in the papers of Goss and Turner [20], and Turner and Otten [21]. There is, however, a simulated example presented in the paper o f Goss and Turner. This example (Re 950) is subjected, therefore, to the sensi tivity test. As shown in Figure 8.7, the contour with an error, Ef 0.05, is steep. As far as this example is concerned, the Nusselt number determined is considered to be reliable enough. In fact, the measurements made by Turner et al. are at Reynolds numbers as high as 1200 to 4600. At such :::::::
=
HEAT TRANSFER COEFFICIENTS
289
Gunn and De Souza 10
Gloss - air Re
8
D P
11
-
0 . 115 em
=
-+
6 X 0 (j
Ef
4
0 . 05 0.1
2
O L---�--����--�---� 10
1
Nu
10
2
10
3
(a)
x1o-
4 14
Gunn and De Souza 12
G l ass - air Re
10
D
P
-
=
33 0 . 22 em
......
...-1 I V>
N E
8
'-'
X 0 (j
6 4 2
0
10
(b)
Nu
FIGURE 8 . 5 Error maps in the plot o f ()ax versus Nu, for data of Gunn and De Souza [ 2 3 ] (+ shows the data obtained by them): (a) Re :=, 1 1 , Eb = 0.4, D p = 1 . 1 5 mm, L = 3 em and the amplitude ratio = 0. 3 ; (b) Re ::=, 33, Eb = 0.4, = 2.2 mm, L = 3 em and the amplitude ratio = 0.3.
Dp
290
HEAT AND MASS TRANSFER IN PACKED BEDS
103 o Nu of Gunn and
Souza
De
I confi dence range
E:f
102
=
0 . 05
I
::J z:
10
1
?
f } ? 9 �� 9
10 Re
FIGURE 8.6
Nu
e;
data of Gunn and De Souza [ 23] with the confidence range indicating = 0.05.
Goss and Turner 0 . 12
Re D
0 . 10 ,...., rl I (/) N E
P
w
•·
=
=
950 0.2
em
0 . 5 rad · S -1
0 ' 08
L..J
0 . 06 X 0 (:l
0 . 04 0 . 02 0
10 Nu
FIGURE 8.7 Error map for a simulated example of Goss and Turner [ 20 ] ( + shows the data obtained by them); Re � 950, w = 0.5 rad s · 1 , other data listed in their Table 1 .
HEAT TRANSFER COEFFICIENTS
291
high flow rates, the effect of axial fluid thermal dispersion on the overall heat transfer is small. Hence, heat transfer coefficients obtained under such conditions are usually reliable and quite consistent. The data of Turner et al. will be included in the data correlation. Heat Transfer Data obtained from Shot Response Measurements
8.2 . 3
From the analysis of shot response measurements, Wakao et al. [26, 41] examined heat transfer coefficients based on the modified D-C model with aax from Eq. (6.28). No definite Nusselt numbers could be obtained, over the but it was found that they fall within the range 0.1 to Reynolds number range 0.2 to 6. Based on the modified D-C model, Shen et al. [27] determined heat transfer parameters from curve fitting in the time domain using the one shot input technique. Their results are shown in Figure 8. 8 for Re = 5 .I. As depicted, the axial fluid thermal dispersion coefficient, aax' and the particle-to-fluid heat transfer coefficient, h p , cannot be determined simul taneously from a single measu rement. It is also revealed that aax is almost independent of the Nusselt number at a Nusselt number greater than about 3 , according to the contour with a root-mean-square-error, e, defined by Eq. (6. 1 8), o f 0.03. oo
xlo-4 12 �--�--���r---�-���--�-
Re
10 .-t .-. I
ks
=
>
5.1
0.2
W·m- 1 . K - 1
8
0 . 06 0 . 03
2
FIGURE 8.8 Re
Nu
Error map in the plot of aax versus Nu, from Shen et a!. [ 27 ]; 5 . 1 , eb = 0.39, D p= 1 . 3 mm (glass beads) and L = 1.3 em.
=
HEAT AND MASS T RANSFER IN PACKED BEDS
292
8.3
Correlation of N usselt Numbers
Figure 8 . 1 reveals a mixture of heat transfer data obtained from the different models. As mentioned already, some of the data are less reliable and some have been criticized for the improper methods employed in the analysis ; these data will not be considered in the correlation. From the review given in the preceding sections, the reported heat transfer measure ments which have satisfied the predetermined criteria are as follows: a)
Steady-state measurements (3, 5 , 6, 9, 1 1, 13-15];
b)
Unsteady-state measurements ( 1 7, 19-21].
The heat transfer coefficients re-evaluated according to the modified D-C model are plotted in Figure 8.9. As depicted, the re-evaluated particle to-fluid heat transfer coefficients, expressed in terms of Nusselt numbers, are quite consistent and compatible. The values are considerably higher than their corresponding original values, in particular, at low Reynolds number. More importantly, the recalculated values show no tendency to decrease further with decreasing Reynolds number at low Reynolds number, and as the trend predicts, a limiting Nusselt number is approached at zero flow rate. Steady-state o
e
+ t>
o
v A
Wilke and Hougen
Satterfield
and Resnick
Galloway et al
DeAcetis and Thodos
McConnachie and Sen Gupta and
Mailing
and
Thodos
Thodos
Thodos
( 1945) ( 1954) ( 1957 ) ( 1960 ) ( 1963 ) ( 1963,64) ( 1967)
•
0
10
UnsteOdY-StOte • • Y •
1
1
Handley and
Heggs
Bradshaw et al
Goss and Turner
Turner and Otten
10 1 2 < P r /3Re0 . 6 >
FIGURE 8.9
Correlation of re-evaluated Nusselt numbers.
( 1968) ( 1970) ( 1971 ) ( 1973)
HEAT T RANSFER COEFFICIENTS
Based on an analogy with Eq. (4. 1 1 ) for mass transfer, Wakao [50] proposed the following correlation (solid line in Figure 8.9):
Nu
=
2 + 1 . 1 Pr113 Re 0·6.
293 et
al.
(8.20)
At lower Reynolds numbers, Figure 8.9 shows that the fitting is not as good as in the case of the mass transfer coefficient expressed in terms o f Sherwood number, shown i n Figure 4.4. This is not unusual in view o f the fact that heat transfer measurements and determination of heat transfer cofficients are often more difficult than mass t ransfer measurements. For instance, the ratio, lH eat!JM ass, has been found to be 1.37 and 1.5 1 by Satterfield and Resnick [ 5 ] and De Ace tis and Thodos (9], respectively; on the other hand, McConnachie and Thodos [ 1 1 ] , Sen Gupta and Thodos [ 1 3 , 1 4] and Mailing and Thodos [ 1 5 ] found the ratio to be approximately 1.0. Considering all these, we may say that the heat transfer data shown in Figure 8.9 are well represented by Eq. (8.20). The question of the limiting Nusselt number at zero flow rate has been the subject of much controversy. Different limiting Nusselt numbers have been estimated based on different models. Gunn and De Souza [23] obtained a limiting Nusselt value of 1 0 from frequency response measure ment. But, Wakao et al. [26, 27, 4 1 ] have demonstrated, from one-shot measurements, and using curve fitting in the time domain, that no definite Nusselt values can be obtained at low Reynolds numbers. Figure 8.8 shows that, at Re 5 . 1 and with € = 0.03, the Nusselt number varies from about The fact that any Nu value within this range will yield approxi 3 to mately the same value of aax• suggests that, at this low Reynolds number, particle-to-fluid heat transfer makes little contribution to the overall heat transfer in the system. This is further demonstrated by the Nu- Re relation ship, given in Figure 8 . 1 0 obtained by Shen et al. [27]. It appears that the limiting Nusselt value may be somewhat higher than that predicted accord ing to Eq. (8.20). However, as indicated, the confidence range increases significantly at lower Reynolds number. The high uncertainty in Nusselt values, at low flow rates, again implies the insignificant role o f particle-to fluid heat transfer in the overall heat transfer process. This deduction is not unreasonable considering the fact that, at low flow rates, a particle and its surrounding envelope of fluid are likely to be in thermal equi librium. The authors feel that although there should be a limit to the decrease in Nusselt number with lowering Reynolds number, the particular limiting value is not practically important. For this reason, the relationship =
00•
Appendix A . Physical Properties Sources of the data: Chemical Engineers Handbook, 4 edn., Maruzen, Tokyo (1978).
A.l
Some Fundamental Physical Constants in SI Units Value in SI units3
Quantity
Remarks
Avogadro number
NA =
6.022 045(31) X 1 02 3 mol-1
Boltzmann constant
k
1.380 622(44) X 10 - 2 3 J K - 1
k
Gas constant
Rg
=
8.314 41(26) J K - 1 mol- 1
Planck's constant
h
=
1.987 19 cal K - 1 mol-1 = 6.236 32 X 104 cm 3 mmHg K-1 mol-1 = 82.056 8 cm 3 atm K - 1 mol - 1 = 1 0. 7 31 4 ft 3 lb in-2 °F-1 lb-mol- 1
Standard volume of ideal gas
vo
=
a
=
c
=
Stefan-Boltzmann constant Velocity of light in a vacuum a
=
=
R gfNA
=
6.626 176(36) x 1 o-34 J s 22.413 83(70) X 1 0 - 3 m3 mol-' 5.670 32(71) X l 0- 8 W m- 2 K - 4
a =
21T5k4/(l5 h3 c2)
2.997 924 58(1) X 1 0 8 m s- 1
The numbers in parentheses are the uncertainties in the last digits of the quoted value.
SI prefixes prefix 10 - 1 10 -2 10-3 10 - 6 10-9 10 - 1 2 10 - I S 1 0 - 18
deci centi milli micro nano pico femto at to
d c m
J.J.
n p f a
prefix 10 1 02 103 10 6 109 1012 5 101 1 0 18
296
deca hecto kilo mega giga tera peta exa
da h k M G T p
E
APPENDIX A
A. 2
297
Conversion Factors
Sl units are shown in the first column. The digit is on FORTRAN E-format (for example, E + 2 = 102). 1)
Length (L) m
em
in
ft
yd
1 1.000 00 E-2 2.54000 £-2 3.048 00 £-1 9.144 00 E-1
1.000 00 E + 2 1 2.540 00 E + O 3.04800 E + 1 9.144 00 E+ 1
3.937 0 1 E + 1 3.937 01 E-1 1 1.200 00 £ + 1 3.60000 E + 1
3.280 84 E + O 3.280 84 E - 2 8.333 33 E-2 1 3.000 00 E + O
1.093 61 E + O 1.093 6 1 E-2 2.777 7 8 E-2 3.333 33 E--1 1
1 A = 10-8 em, I ,u (micron) = 1 0 - 3 mm = 10-4 em, 1 mile = 5280 ft = 1609.3 m. 2)
Mass (M) kg
g
oz
lb
1 1.000 00 E-3 2.834 95 E-2 4.535 92 E-1
1.000 00 E + 3 1 2.834 95 E + I 4.535 9 2 E + 2
3.527 40 £ + 1 3.527 40 E-2 1 1.600 00 E + 1
2.204 6 2 E + O 2.204 6 2 E-3 6.25000 E-2 1
1 tonne (metric) = 0.9842 long ton (British) = 1.102 short ton (USA); 1 long ton (British) = 2 240 lb = 1.0 I 6 05 tonne (metric); 1 short ton (USA) = 2000 lb = 0.907 1 8 tonne (metric). 3)
Specific volume (L 3 M -•) m 3 kg-•
cm 3 g-1
I kg-1
in3lb-1
ft3 lb-1
1 1.000 00 E--3 3.6 1 2 73 £-5 6.242 80 E--2
1.000 00 E + 3 I 3.612 73 E - 2 6.242 80 E + I
1.000 00 E + 3 1 3.612 7 3 E-2 6.242 80 E + 1
2.767 99 E + 4 2.767 99 E + 1 1 1.728 0 0 E + 3
1 .601 85 E + 1 1.601 85 E-2 5.787 04 E - 4 1
kg 1- 1
lb in-3
Ib n-)
3.6 1 2 73 £-5 3.6 1 2 7 3 E - 2
6.242 80 E - 2 6.242 8 0 E + 1 1.728 00 £ + 3 1
4)
Density (ML -3) kg m - 3
g cm- 3 --·-
1 1.000 00 E + 3 2.767 99 E + 4 1.601 85 E + 1
--·-----·
1.000 00 E-3 1 2.767 99 E + 1 1.601 85 E-2
1.000 00 E-3 1
2.767 99 E + 1 1.601 85 E-2
1
5.787 04 E-4
298 5)
HEAT AND MASS TRANSFER IN PACKED BEDS
Surface tension (MT-2)
- ·-·-· ---- --
··-· -----· � · .
N m- 1
=
dyn cm-1 = erg cm- 2
J m- 2
· --- ---- . ··--· -·-·
1 1 .000 00 E-3 9.806 6 5 £ + 0 1 .75 1 2 7 E + 2 6)
kgf m-1 ---- ·-···
1 .000 00 E + 3 1 9.806 65 £ + 3 1.75 1 27 £ + 5
-·-··-
·
-"
·
lbf in-1 -· --·· - - - · ·
1 .0 1 9 7 2 E - 1 1 .0 1 9 7 2 E-4 1 1 .785 80 E + 1
--·
5 . 7 1 0 1 5 E-3 5 .7 1 0 1 5 E-6 5.599 74 E-2 1
Force (ML T -2) dyn
kgf
poundal
1
1 .000 00 E + 5
1 .0 1 9 72 E - 1
7.233 0 1 E + O
2.248 09 E - - 1
9.806 6 5 E + 0
1
7.093 1 6 E + 1
2.204 6 2 E + O
1 .3 8 2 5 5 E - 1
9.806 65 E + 5 1 .382 5 5 E + 4
1 .409 8 1 E-2
1
3 . 1 08 1 0 E--2
4.448 2 2 E + 0
4.448 22 E + 5
4.535 9 2 E - 1
3 .2 1 7 40 £ + 1
1
N
7)
lbf
Pressure (ML - • T-2) Pa
bar
atm
kgf cm- 2
lbf in-2 (psi)
l 1 .000 00 E + 5
1 .000 00 E--5 1
9.869 2 3 E-6
1 .0 1 9 7 2 E-5
1 .450 38 E-4
9.869 23 E - 1
1 .0 1 9 7 2 E + O
1 .450 38 E + 1
1 .0 1 3 2 5 £ + 5
1 .0 1 3 2 5 E + 0
1
1 .033 2 3 E + O
1 .469 60 £ + 1
9.806 65 E + 4
9.80665 E - 1
9.678 4 1 E - 1
6.894 76 E-2
6.804 60 E-2
1
1 .422 34 E + 1
6.894 7 6 E + 3
7.03069 E-2
1
Pa
dyn cm- 2
mmHg (torr)
in Hg
lbf n- 2
7 . 5 0 0 6 2 E-3 7 . 5 0 0 6 2 E-4
2.953 00 E-4
l
2.953 00 E-5
2.088 5 3 E-2 2.088 53 E-3
1 1.000 00 E - 1
10
1 .333 2 2 E + 2 3.386 39 E + 3
1.333 2 2 £ + 3
1
3.937 0 1 E-2
2.784 5 0 £ + 0
3.386 39 £ + 4
1
4.788 03 E + 1
4.788 03 E + 2
2.540 00 E + 1 3.591 3 1 E - 1
1 .4 1 3 90 E-2
7.072 6 2 E + 1 1
erg
calth
Btuth
kgf m
1 .000 0 0 E + 7 1
2.390 06 E - 1 2.390 06 E-8
9.484 5 2 E-4
4.1 84 00 E + 7 1.054 35 E + 1 0 9.806 65 E + 7
1 2.5 1 9 96 E + 2 2.343 85 £ + 0
1 .0 1 9 72 E - 1 1 .0 1 9 7 2 E-8 4.266 49 E - 1 1 .075 1 4 £ + 2 1
8)
Work, heat, energy (ML 2T - 2) J
1 1 .0 0 0 0 0 £ - 7 4 . 1 84 00 £ + 0 1 .054 35 E + 3 9.806 65 E + O
J
calIT
Btu IT
1
2.388 46 E - - 1 1 2 . 5 1 9 97 E + 2 8.59845 E + 5 6.41 1 8 7 £ + 5
9.478 1 3 E-4 3.968 30 E-3 1 3.4 1 2 1 3 £ + 3 2.544 4 2 E + 3
4 . 1 86 80 E + O 1 .055 0 6 E + 3 3.600 00 E + 6 2.684 5 2 E + 6
9 .484 5 2 E - l l 3.968 32 E - 3 1 9.301 1 3 E-3
HP h
kW h 2 . 7 7 7 78 E 7 1 . 1 6 3 00 E-6 2.930 72 E--4 1 7.45 7 00 £--1 -·
3.725 06 E--7 1 .559 6 1 E-6 3.930 16 E-4 1 .341 02 E + O 1
APPENDIX A
9)
299
Specific enthalpy (L 2T-2)
J kg- •
calth g-•
caln g-•
Btuth lb- 1
Btun lb-•
1 4 . 1 84 00 E + 3 4 . 1 86 80 E + 3 2.324 44 E + 3 2.326 01 E + 3
2.390 06 E - 4 1 1 .000 67 E+O 5.555 55 E-1 5.559 29 E-1
2.388 46 E - 4 9.993 3 1 E--1 1 5.551 84 E - 1 5.555 5 8 E-1
4.302 10 E-4 1 .800 00 E +O 1 .801 20 E+O 1 1 .000 67 E +O
4.299 21 E-4 1 .798 79 E + 0 1.800 00 E +O 9.993 3 1 E-1 1
calrr g-l Oc-l
Btu th lb-1 o r - '
Btu IT lb-• o r- - •
2.388 46 E-4 9.993 3 1 E-1 1 9.993 3 1 £-1 1.00000 E +O
2.390 06 E-4 1 .000 00 E +O 1 .000 67 E +O 1 1 .000 67 E +O
2.388 46 E-4 9.993 3 1 E-1 1 .000 00 E +O 9.993 3 1 E-1 1
kgf m s-•
lbf ft s-•
HP
PS
1 .0 1 9 7 2 E-1
7.375 62 £-1 7.233 02 £ + 0 1 5.500 00 E + 2 5 .424 76 E + 2
1 .341 02 E-3 1 . 3 1 5 09 E-2 1 .8 1 8 1 8 E-3 1 9.863 20 E-1
1.359 62 E-3 1 .333 3 3 E-2 1 .843 40 E-3 1 .0 1 3 87 E +O 1
10)
Specific heat (L 2T- 2 o-•) J kg-'
4 . 1 84 4.186 4 . 1 84 4 . 1 86
1 1)
K-I
2.390 06 E-4 1 1 00 E + 3 80 E + 3 1 .000 67 E+O 00 E + 3 1 .000 00 E +O 80 E + 3 1 .000 67 E+O
Power (ML 2T-3) w
1 9.806 65 E + 0 1 .355 82 E + 0 7.45 7 0 0 £ + 2 7.354 99 £ + 2
12)
calth g-1 Oc-1
1
1 .382 55 E--·1 7.604 02 E + 1 7.50000 £ + 1
Viscosity (ML _ , r - ' ) Pa s
poise
kgf s m- 2
kgf h m- 2
lb h-I n - •
1 1.00000 E - 1 9.806 65 E+ 0 3.530 39 E+ 4 4.133 7 9 E - 4
1 .000 00 E + 1 1 9.806 65 E + 1 3.530 39 £ + 5 4 . 1 3 3 79 E-3
1.019 7 2 E-1 1 .0 1 9 7 2 E-2 1 3.600 00 E + 3 4.2 1 5 28 £-5
2.832 55 E-5 2.832 55 E-6 2.777 78 E-4 1 1 .1 7 0 9 1 £-8
2.419 0 9 E + 3 2.4 1 9 09 £+ 2 2.372 32 E + 4 8.540 3 8 E + 7
Pa s
lbf s in- 2
lbf s ft - 2
lbf h in- 2
tbf h n - 2
1 6.894 76 E + 3 4.788 03 E + 1 2.482 1 1 E + 7 1 .723 69 E + 5
1 .450 3 8 E-4
2.088 54 E-2 1 .440 00 E + 2 1 5 . 1 84 00 E + S 3.600 00 E + 3
4.028 8 3 E-8 2.777 78 E-4 1 .929 01 E-6 1 6.944 44 E-3
5.801 5 1 E-6 4.00000£-2 2.777 78 E-4 1 .440 00 E + 2 1
1
6.944 44 E-3 3 .600 00 E + 3 2.50000 E + 1
1
300
HEAT AND MASS TRANSFER IN PACKED BEDS
Thermal conductivity (ML T- 38 - 1)
13)
W m- 1 K 1 -
1 4 . 1 84 00 E + 2 1.162 22 E+O 1 . 7 29 5 8 £ + 0 1 .441 3 1 E - 1
14)
calth S-1 cm-1 oc - 1
kcalth h- 1 m- 1 oc - 1
Btuth h - 1 ft- 1 o r: - •
Btu th in h - 1 n - : or-- 1
2.390 06 E-3 1 2.777 7 8 E-3
8.604 2 1 E - 1 3.600 00 E + 2
5.78 1 76 E·-1 2.4 1 9 09 E + 2
6.938 1 1 £ + 0 2.90 2 9 1 E + 3
1 1 .488 1 7 £ + 0
3.444 82 E-4
1 .240 14 E--1
6.7 1 9 6 8 £ - 1 1 8.333 33 E-2
8.063 6 2 £ + 0
4 . 1 3 3 7 9 E-3
1 . 200 00 E + 1 1
Diffusivity (L 2 T - 1) m 2 s- 1
stokes (cm 2 s - 1 )
m 2 h- 1
in 2 s-•
[t2 h- 1
1
1 .000 00 E + 4 1
3.600 00 E + 3
1 .550 00 E + 3
3.875 0 1 E + 4
1.000 00 E-4
1 .550 00 E - 1
3.875 0 1 E + O
2.777 7 8 E-4
2.777 78 E + O
3.600 00 E - 1 1
6.45 1 60 £-4
6.451 60 E + O 2.580 6 4 £ - 1
2.322 5 8 £ + 0 9.290 3 1 E-2
4.305 56 E - 1 1
2.500 00 E + 1
4.000 00 E-2
1
2 . 5 8 0 6 4 E-5
15)
Heat flux (MT -3) W m- 2
calt h cm- 2 S- 1
kcalth m- 2 h - 1
Btu t h ft- 2 h - 1
1
2.39006 E-5 1
8.604 2 1 E - 1
3.172 1 1 E-1 1 . 327 2 1 E + 4
4 . 1 84 0 0 E + 4 1 .1 6 2 2 2 E + O 3 . 1 5 2 48 £ + 0
16)
2.777 7 8 E-5
3.600 00 E + 4 1
7.534 6 1 E-5
2 .7 1 2 46 E + O
3.686 69 E - 1 1
Heat transfer coefficient (M T - 3 e-1)
W m-2 K - 1
calth cm-2 s- 1 oc - 1
kcalth m-2 h- 1 oc - 1
Btuth ft- 2 h- 1 O F - I
1
2.390 06 E-5
8.604 2 1 £ - 1
4 . 1 84 00 E + 4 1 .162 22 E + O 5.67446 £ + 0
1
3.600 00 E + 4 1 4.882 4 1 E + O
1 .762 2 8 E -- 1 7.373 4 1 £ + 3 2.048 1 7 £ - 1 1
17)
2.777 7 8 E-5 1 .356 2 3 E-4
Temperature (8) T (K)
=
T (OC) =
273.15 +
T (°C)
T (°F) - 3 2 1.8
1 .076 39 E + 1
A.3
Physical Properties of the Elements and Some Inorganic and Organic Compounds TABLE A . 3 (a) Elements and inorganic compounds
Name Air Argon Boron bromide Boron chloride Boron fluoride Bromine Cyanogen Carbon monoxide Carbon dioxide Phosgene Carbon oxysulfide Carbon disulfide Chlorine Deuterium Heavy water Fluorine Germanium tetrachloride Hydrogen Hydrogen bromide Hydrogen cyanide Hydrogen chloride Hydrogen fluoride Hydrogen iodide Water
Formula -
Ar BBr3 BC13 Bf' 3 Br 2 C2 N2
co
C0 2 C0Cl2
cos
CS 2 Cl 2 02 0 0 2 F2 GeC14 H2 HBr HCN HCl HF HI H 20
Molecular weight 28.97 39.94 250.57 1 1 7.19 67.82 1 59.83 52.02 28.01 44.01 98.92 60.07 76. 1 3 70.91 4.02 20.03 38.00 2 1 4.43 2 .0 1 6 80.92 27.03 3 6 47 20.01 1 27.93 1 8 .02
Specific gravitya 1 .2928 1 .7828 -
( 1 .434)0 3.065 ( 3 . 1 1 9 ) 20 2.3348 1 .2 5 01 1 .9768 ( 1 .434 ) 0 2.7149 ( 1 .2927) 0 3.2204 -
( 1 . 1 07 1 4) 2 5 1 .6354 ( 1 .8443) 30 0.0898 3.6445 (0.6876) 2 0 1 .6394 (0.987) 1 5 5.7245 s (0.99708) 2
Melting point o ( C) -
- 1 89.2 -46 - 1 07 - 1 27 -7.2 -34.4 -207 -56.6 (530 kPa) - 1 04 - 1 38.2 - 1 08.6 - 1 0 1 .6 -254.4 3.82 -2 2 3 -49.5 -259.1 -88.5 -14 -Ill -83 -50.8 0.0
Boiling point o ( C)
Critical temperature (K)
Critical pressure ( X 106 Pa)
- 1 94 - 1 85.7 96 1 2 .5 - 1 00.4 58.78 -20.5 -192 -78.5 (sublimation) 8.3 -50.2 46.3 -3 4.6 -249.6 1 0 1 .4 2 -187 84.0 -252.7 -67.0 26 -85 1 9 .4 -35.4 100.0
1 32.5 151 573 452.0 260.9 584 401 1 34.2 304.3
3.77 4.86
455.2 378.2 546.2 4 1 7.2 38.8 644.7 1 1 8.2 550.2 33.3 363.2 456.7 324.6 503.4 424.1 647.4
-
3.87 4.99 10.3 5.98 3.55 7.40 5.67 6.18 7.7 0 7.71 1 .76 22.15 2.5 3 3.85 1 .30 8.51 5.39 8.27 .. 8.3 1 22.1 3
Critical density ( X 103 kg m- 3 ) 0.35 0.531 0.90
0.848 0.3 1 1 0.460 0.52 0.441 0.573
-
-
0.0310 -
0.20 0.42 -
-
0.323
"t!
>
t'T1 "t!
z 0 X
-
>
w 0 .....
TABLE A. 3(a) - contd
Name Hydrogen peroxide Hydrogen suUide Hydrogen selenide Helium Mercury Iodine Krypton Nitrogen Ammonia H ydrazine Nitric oxide Nitrous oxide Nitrogen peroxide Neon Oxygen Ozone Phosphine Radon Sulfur Sulfur dio x ide Sulfur trioxide Silicon chloride Silicon fluoride Silane Tin chloride Xenon
Formula H 20 2 H 2S H 2 Se He Hg 12 Kr N2 NH3 N 2 H4 NO N20 N 04 2 Ne 02 03 PH 3 Rn
s
S0 2 S03 SiC14 SiF4 SiH4 SnCl4 Xe
Molecular weight
Specific gravity3
34.02 34.08 8 1 .2 2 4.00 200.61 2 5 3 .84 83.70 28.02 1 7.03 32.05 30.01 44.02 92.02 20.18 32.00 48.00 34.00 2 2 2 .0 32.06 64.06 80 .06 1 6 9 .89 1 04.06 32.09 260.53 1 3 1 .30
( 1 .438) 2 0 1 .5392 0 . 1 769 ( 1 3.546)20 (4.93 r o 3.6431 1 .2507 0.7708 ( 1 .0 1 1 ) I S 1 .3401 1 .9 7 8 1 ( 1 .448) 2 0 0.87 1 3 1 .4289 2.1415 1 .5293 9.73 -
2.9268 2 ( 1 .97) 0 ( 1 .50) 2 0 -·
1 .44 (2.23) 5.7 168
Melting point (oC)
Boiling point
-0.89 --82.9 . - 64 <-272.2 -38.87 1 1 3.5 - 1 69 -209.86 -77.7 1 .4 -161 -- 102.3 -9.3 -248.67 - 2 1 8.4 - 1 92.5 -- 1 3 2.5 -71 1 20 -77.5 16.83 -70 -95.7 -- 1 8 5 -30.2 - 140
1 5 1 .4 -59.6 -42 -268.9 356.9 1 84.35 - 1 5 1 .8 - 1 95.8 -33.4 1 1 3.5 -151 -90.7 2 1 .3 - 245.9 -183 -· 1 1 1 .9 -85 --62 444.6 - 1 0.0 44.6 57.6
tc>
-
-112 1 14.1 - 1 09.1
Critical temperature (K) .. 373.6 4 1 1 .2 5.3 < 1 82 3 826.2 209.4 126.1 405.6 653.2 1 7 9.2 309.7 4 3 1 .2 44.5 1 54.4 26 1 . 1 324.2 377.2 1313 430.4 49 1 .5 506 2 7 1 .7 269.7 5 9 1 .9 289.8
Critical pressure ( X 1 06 Pa) ·-
9.01 8.92 0.229 > 20 -
5.50 3.39 1 1 . 30 14.69 6.59 7.27 10 2.62 5 .04 5 .5 3 6.48 6.28 1 1 .8 7.87 8.47 5.07 4.86 3.75 5.90
Critical density ( X 103 kg m-3) 2.86 -
0.0693 4-5 1 .10 0.3 1 1 0 0.235 ... 0.52 0.45 1 .785 0.484 0.430 0.326 0.30 0.52 0.630
1 . 1 55
a The figures in parentheses are the specific gravities of the liquid at the temperature (0C) indicated in the superscript; others are densities (kg m-3) of the gas at atmospheric pressure and 0°C.
'..J..J 0 I'...) :I: :-r:
>-
"""
>z 0 > �
(/) (/) """
:;tJ >z (/)
.,
� :;tJ -
z "'C
(')
>
:::'\ �
0
tt � v (/)
TABLE A. 3(b)
Organic compounds
Name Methane Ethane Propane n-butane /so butane n-pentane /so pentane Neopentane n-hexane n-heptane n-octane n-nonane n-decane Cyclopentane Cyclohexane Ethylene Propylene Butadiene- ( 1 , 3) Acetylene
a The
Formula CH4 C H6 2 C3H11 C4 H t o (CH3)2CHCH3 Cs H t 2 (CH3)2CHC2 H 5 (CH3)4C C6 H I 4 C H I6 7 C8 H I S C9 H 2o C , o Hn (CH 2 ) s (CH 2 ) 6 CH,=CH2 CH2=CHCH3 CH2=CHCH=CH 2 CH=CH
Molecular weight 1 6 .04 30.07 44 . 1 0 58.1 2 58.12 72 . 1 5 72.15 72.15 86.18
100.21 1 1 4.23
1 2 8 .25 142.28 70.13 84.16 �8.05 42.08 54.09 26.04
Specific gravity3
Melting point (oC)
Boiling point (C)
Critical temperature (K)
Critical pressure (X 1 06 Pa)
0.7 1 6 7 1 . 3567
- 1 8 2.5 ..-· 1 83 . 3
- 1 6 1 .5 --88.6 -· ·4 2 . 1
191.1 305.6
4.641
-- · 1 59.6 - 1 29.7 -·- 160.0 - 1 6 .6 -95.3
- 1 1 .7 36.1 28.0 9.5
425.3 408.2 469.8 46 1 .0 433.8
3.797 3.648
2.0200 2.5985
co.5 9 8 3 r ' 3•6 (0.6262)2() 20 (0.6201 )
(0.6 1 3)0 (0.6594)20 (0.6838)20 (0.7025)20
(0.7 1 8)20 (0.730)20 0 (0.745)2 2 (0.779) 0 1 .2644 co .647 r 79 -
1 . 1 708
- 1 87.7 - 1 38.4
-90.7 -56.8 -53.7
-29.7 --93.3 6.5
- 1 69 - 1 85
- 1 08.9 - 8 1 .5 ( 1 1 9 kJ>a)
-0.5
68.7
98.4 1 25.7 1 5 0.5 1 74.0 49 80
-- 1 03.9 -47.0 --4.5 -84
370.0
507.9 540.2 569.4
594.6 603.6
5 1 1 .8 554.2 282.9 365.5 425 309.2
4.894 4.257
3.375 3.33 3 . 1 99 3.034 2.736 2.497
2.31 2.15 4.52 4.09
5.12 4.5 6
4.33
6.28
Critical density
(X 103
kg m·3) 0.162 0.203 0.220 0.228 0.221 0.232 0.234 0.338 0.234 0.235 0.235
0.236 0.270
0.22 0.233 0.245 0.231
figures in parentheses are the specific gravities of the liquid at the tt.•mpcrature (0(') indic::.�tcd in the superscript : oth
;t> '"C
'"C
tr1 z c >< -
;t>
w 0 w
304
HEAT AND MASS TRANSFER IN PACKED BEDS
A.4
Physical Properties of Some Gases
4A � �...r;. l/ :j..-,j< --+-,..<: Ll . O - -+-+ --+ 1---r-- -+--1--+--1--+· v f-// 3 . 6 +� --++-1--4-- f---1-· +-,..o<:tV ..,.-1i� /V ..q.::: v -¥-J : V -A · -+-+-+-+ �-1---1-.. - -- ·..--1--1-1--4-- -1--+---1--"'4 ...,. .£1- L+,.. .4 4 -lh4· -+ -+-4+---+. ---f-/� -r- 1--· -· .. -- -H- - "' ---.? ;;ol<:"::;;-i:> 7 1--+--r-r---+t' -- --r -· --1-[- . � .. "'1.--4£-+--,4-+ -1-1-1 ..?''./ 7 /17 / -·. --f----i- - . !:>'b-k --1--4---1 --�� 2 . 8 I-- - -17 .' 0-A-+. .. . t;>-- 1- 4 �--l��."'-+-4+--1� t· - + .. 1--f� -+-++-+ ·� !;./. 1 -f-0 -· 1/ + 1-. t-+ �r-· - - .. : v .. . -t· 7 � � f.- -+--+-+-+-1 2 . 4 1---, -+-1- !--1---1- -4-._ --l-- 1---1--+-1 . . . t 1-1--�17 I/ .A' . ..t·�"-.:'.L--+----4-/ f-- ... . - -.± l v /Vi---17v ' / ./ ' +.,£--1.� � / :..+, -+-4�1-1 ! ' -.� 2.0 1---1-· f-- .. i-· V..... � � 1 '1.--t;' H----- ·7 /� ·-t· I i I/ i � [7 , I 1.8 1---.. - I; 0;f , 7-17/ :7:_I-
-
·-
- --
III
.......
I 0
....
>-
(/) 0 u (/) >
-
1 '
·
-
-
-
-
•
- -
-
"
1.4 1 .2
. .
-
- --•-
-
.I
� ��J:t t-r-� �':_ 'l: �- --� vv . v � � _..-0V : -!I t?'t7 Vl!;:( · l Jt Jt -· [7-� �'1-r-t-�-rt . t-i · rr-� · .
1.6
r.::: r·-7" <:§)
+
-
1---1--t-- �r.t?'
-
.
"'!"
-
·
.
· -
--·
.-.
- ·- I-·
-
.•
A
.
/ -· . . t�-+- ,L..-- - · &
.L i /
--- -r /
.:
-·
'
·r
.
-
.
I
I
·
·� �;i · -- : -i [7 .. • -t·-+ · ·r · I
-
,L!- � : j +- , -
+
!
V� l ! / -i! ;
- V--
1 . 0 -� , - -.,..<1- - -+I -�
/[{.+�+
I 1--i.
t·
1 I
I
•
.
,
-
.
:
lI
j
- -- �
I
f--I- r-
1 - · --�---+--t-1I I '
t -�
:
•
!
.
t
:
I
;
�:_t "
!· : - ---+- +· 4 ' .
I
-r.
t: !
-·
- � � r+ - .... _____--� · ... - t -
1- + i .. . t-. - 4-+ . ! - + -! • f.- ..
. . -1
---��---�
-
� - --
-
;
-
· H--- · · + ++-1--t- + ; v- Vvk': -- ---..r--t- i -i · -+j +r-7�t I : -t+- �· + � v -1-- - �- --r- t 1 t -r- -, �-----· �- -1· --- -r-1v-"-----1.---'T-+t- .... t...L t-4�--.I-I ---1.-4-... --t-�---+---+--+0 6 � ... ---'-+--...I.-'- .L... ..L. � ..L. -4-1- -'-+J. ....L. ...+ ..I --�--� o.8
1
.
-
1-
'
-
.
- 100
. .
.
1·
•
..
+-
•
•
-
-
J I
.
:
I
! ;
�
Temperature
FIGURE A.4(a)
I
·
100
0
•
200
- - -·
r oc )
-
-
300
Viscosities of gases.
-
400 500 600 700
APPENDIX A
I
j
I i : I I
...--.
...... I �
.
...... I Cl �
I
I
I
!
"" 0 ......
._,
2
0 (l) .c.
I I
i
I
!
I
I !
I
I
I
,
I
u
·-
1
l
f- Q 2
--
---.--:CO?.
.
I !
I 1 mO. _ __otm) ste o
l---
:
H < read ing x 10> f-2
r .L/
-�co
f N2
1-
u (l) c. (/)
I
---�
I
.._
I
!
i
I
. ....,
....,
!
I ! I ! V1 y l ll I l I !/ \.,�� � I !
/1 !/'f , Vi I v _)7' -� i -r I
3
!
\ '\.
'L
\ai r
r- H e < read ing x 1 0 > �
-
_
0 -100
0
100
200
300
Temperature
FIGURE A.4(b)
4 00[
oC
500
J
600
Specific heats of gases.
700
305
306
HEAT AND MASS TRANSFER IN PACKED BE DS
�� c...;
0.1 0 . 09
i
·
0 . 08 I ::,..:: -I
E
3:
>0-J ·>
u
0 . 07 0 . 06
l
.. t .
I -t I
·· t . .!
�
..
j -1 -r I I
-· r !
'
.
I I
I
I
. ·t
.
I
·- �
I
I
i I
IJ
j
I
t
I
!
I
.! I
f j !. I
I
I
j
..
.
' I
t I
I
I
t. . I
-· ·
I
.. _ _
� ..1. .
-
r - � �� .
+ -
I
...
�
�
"'
c.,'(,;
!
0 . 05
_,
0 u
-g
a E
L Q) .r:: f-
0 . 04 0 . 03 0 . 02
+- ·
0.01 0
- 100
0
1 00
200
300
Temperature
FIGURE A.4(c)
400
("CJ
500
600
Thermal conductivities of gases.
700
APPENDIX A
A. 5
Physical Properties of So1ne Liquids
;
,
glycerine
+-r-
��-4-+-+-L-LJI eth ler�e olvrnl
I
l
-
!
!
i
H o
L 1--1----1-+--+-+--+- b'i> nz
·
I
I
I !
I
! I
ene
I
! I -�
i
I
I
i
o l uene ethono
400
���+-��-+�4-�� ��00
0
10
20
30
40
50
Temperature FIGURE A.S(a)
60
70
80
[°C]
Densities o f liquids.
90
1
307
308
--�--�--��--r-,--r�����-r�
HEAT AND MASS TRANSFER IN PACKED BEDS 10
-2
I -+-t--+ --+ -t1 +--1--+-t--t--1-i-1 ,9 / !t + 4--1-+--1-++-1 ! f----- k(;-�· +--+-----lf-·-·� ·1-•----f- • • t -+ . (."),.._ -+-+--t-i-!-+ -+ +-f I i -?. ./ + i� , l I
-+ +--+-
8 6
.
..
.
•
·
..
-
,
: ��: �--i--i � -+ : +-t -+-+ --+ 1 1 -+ -0:--ro � /�ne ! -�nj tro/ 1--+!-+-_1�---+1\ -9 ' -+-t 1--r-..! '\'<"�I ! i !; --+ ---+-
�-�--+-�r � --+-_,-+--+-� ��� -�-� �=: 1= = =� � � �
4
-·
•
I
3
-
I j
·-
0 0..
-
. . ··-
+
i
2
I
i
I
V) 0 u V)
>
10
0
20
30
40
Tempe rature
FIGURE A.S(b)
50
[°C)
60
70
80
Viscosities of liquids.
4
.
rl I �
...,
rl I 0> .::.!.
3
.
r<"\ 0 rl
2 .._,
0
.....
u ·u
�F--
r+H9 ( read i ng -H -+ +-t -+l-+ +-+ 4-1 ' ..,. -
L I
'
'
ri . 1 ' -r+ -r x u
-�--- cs� 1-4--4-+---1 I-+-l--t -+ l1-+ . +-i--+ - -+-.f-+--1 i! +-+··-r-t-+-+-1-+-+-+-t-++--lr-+-t-t-t-t-1 1-+ -+-r-+ -+'
i
1-+-+-1-+ 0 -100
-50
0
50
Temperature
[°CJ
FIGURE A.S(c)
100
Specific heats of liquids.
150
90 100
APPENDIX A
0.4
.--.
rl I ::..::
.
0.3
�
rl I E :X
1-.
>
...,
..., >
0.2
t--1-.
(1) .c; t-
1-1-
1-t---t-
.._
0.1
carbon
t-t--t-.._ t-
-100
b
-
�-� '--+ac t lorh · ��
-50
1-
0
. n9 \\g{ reocn _t!::!:
1-
H o< reading x 1 0 > 2
50
Tempe rature
FIGURE A.S(d)
methanol
e thanol
te t. r
-
0
e t h y l ene g ! y c o l
1.._1-t- t'-re n z e e r-t.._......_ .._ n r-,-r.... r- .._1-n tiJtj�'- -t-t-.._r-t--1--t-.._1-- I I I--de�---��1- ��--t::���-r-n-
u :::l -o c= 0 u
� L..
.._g l y c e r ine
--
100
150
[°C)
Thermal conductivities of liquids.
200
309
310
HEAT AND MASS TRANSFER IN PACKED BEDS
A. 6
Physical Properties of Plastics
A.6.1
Polystyrene and Polyvinylchloride3
3
I
103
L
8
6 4 I
E:
""
0\ �
....,
>-
Vl c:: Q) 0
4
I - _I_
2 -
10 8 6
--
kg·m-3 )-
Pol y ( v i nyl chloride) ( O e n s i t y : l ,400 -
Polystyrene(Oens i t y : l ,050
V
2 2 10 8 6
f-
r
2
kg·m- 3 ) -
· Po1 ystyrene exranded �1i th a i r (Oensity:60 k g · m - 3 ,
7with a i r ( Oensity:60 kg·m-3) l l 1-tI I' I1 II ' · c hlor1de · ) expanded ILpo1y( v1ny with a i r ( Oensity:20 kg.m-3) .,. 1 t 4 I Poly(vinyl chloride) ex�anded
-
L
4
-
-
-
-
--
--
Polystyrene expanded �1i th a i r _ (Oens i ty : l O
kg · m-3)
2 1 -200
FIGURE
A.6(a)
-100
0
Temperature
100
[•C)
200
300
Densities of polystyrene and polyvinylchloride.
APPENDIX A
::..::
2
I
I
0• �
�
• ••
-�--
··
.
....
t'"l 0
,
e �
'
._.: I� il' " *<.,� �-- ·
Polystyrene expanded with a i r ' (Oenslty : 1 0 kg·m-3)
--,
·
_
.
1
-Po 1 ys tyrene expanded w i t h a i r ( Oensity:60
kg.m· 3 )
_..,
�Po1y(vinyl . chloride) · (Oensi t y : l ,400 kg·m-3)
· · · - --· ·
0
....
--
u
0 -200
---
- I 00
:
. ·· -
I
I
0
y� w i t h
Po1y(vinyl c h l o r i d e ) expanded a i r{Oensi ty : 60 kg·m·3)
' P o l y ( v inyl c h 1 o r i de)expanded w i t h a i r (Oensity:20 kg · m · 3 ) ·-··-!- . · t--·· ·--··-
:-.. t ---+- -
-' · · ·-· --- ·-
"u
,
311
r
-+-�>--�
.
-
t-
:
[ "CJ
Temperature
400
300
200
100
Specific heats of polystyrene and polyvinylchloride.
FIGURE A.6(b)
0.4 I
.....
::..::
,.....
rl I E 3
>... . >
....
u ::J '0 c
0 u 0 E
.s:: I-
l..
0.2
0.1
0.04 0 . 02 0.01 -200
FIGURE A.6(c)
...
.. .
-
0.06
\
I
-;:::::;;;
0 . 08
�-Po1y ( v i n yl
. .. ....
-
·-·
\r
/
-100
. r-· ...
Polystyrene{Ocns i t y : 1 , 050 k g . m
-
--
. . ... ...
-
i
-
0
1
Po I y (
-
hI
1
-3
)
-·
/+ i � , ,, ) XP:>::____ : � kg.m-3) l
'\
kg.m- 3 )
---1--
Polystyrene expanded w i t h a i r ( Ocns i ty : l O
r\ /� K' \ ·%�v 1\ v /
c h 1 o r i d c ) ( Oens i ty : 1 ,400
;
) "'
a i r (Ocns i ty : ?O
,_
-�
kq ·nl - 3 )
"'''
1•I th
1
I r-- j )
Po I ys tyrene expa ndrd w i t h <> i r
kg.m" 3 )
(Oensity:60
-3 I I
P o 1 y ( v i n y l chloridc)expanded w i t h
I
I
I
a i r ( Oe n s i t y : 6 0 k g . m
I
100
Temoerature
200
( °C )
300
400
Thermal conductivities of polystyrene and polyvinylchloride.
312
HEAT AND MASS TRANSFER IN PACKED BEDS
.... ,
�
-�--t- I I I Polystyrene expande
4
.l wit h
a i r( Oensi ty : lO kg · m - 3 )
.,.,.
2
,-_
I I
L 1 .
_i I
I 1.
I ,I
I I
-
�
� P o l y ( v l nyl c h lor1de ) expand
.. ....
-
-
with a i r(Oensi t y : 20 kg·m-3)
10I
I I J l l I I� Poly(vinyl chloride) expanded
6
8
,...-, .-I
N E
V)
6 -
......
4
-
--
>... '
> V) ::l ...... ......
2
'0
� \..
10-
.....
7
8
v .c. .....
6
4
'
\
--
\
1-
--
Polystyrene expanded with a i r
I
(Oensity : 60 kg·m- 3 )
l::��� � t-.. i/ 1' ,
I .I I
�
�Polys t ren (D en i ty : 1 ,050 kg·m-3)
�r--
J
I
with a i r ( Oensity:60 kg·m-3)----
1
---
Poly(vinyl chloride) � (Oens i t y : l ,400 kg·m- 3 )
2
8 10-200
-
100
0
100
Temperature
FIGURE A.6(d)
200
c•cJ
300
Thermal diffusivities of polystyrene and polyvinylchloride.
3 Source: C. Y. Ho, P. D. Desai, K. Y. Wu, T. N. Havill and T. Y. Lee, Proceedings of the 7th Symposium on Thermophysical Properties, sponsored by ASME and NBS, Maryland, p. 1 9 8 (1977).
APPENDIX A
A.6 . 2
Polyethylene, Polypropylene and Polytetrafluoroethylene3 0.4
I
:::.::
E
>......
r----.. -l....:-:-... 1--I-. .. --� ::: � �---- ::::b-... ��
r-----
......
::!:
�
1 1':::::r-:: ....--... 1--r--1-........ .. 4 .. ---------.....
......
I
r--r----r-0.3
>
u ......
'0 c: 0 u
-
-
:::;
� ......
OJ .r:::. I-
313
2
-
r----r--
--
.....__
!':::::::
r--.....__
3
7.
0.2
20
40
60 Temperature
FIGURE A.6(e)
(°C)
80
100
Thermal conductivities of polyethylene, polypropylene and poly tetrafluoroethylene.
a Source: K. W. Jackson and W. Z. Black, Proceedings of the 7th Symposium on Thermophysical Properties, sponsored by ASME and NBS, Maryland, p. 1 4 1 (1977).
Sample
Material
Specifications-Use
Density (kg m- 3 )
Supplier Hercules Inc. ® Hifax 1900 Polyethylene
1
Ultrahigh molecular weight polyethylene
Used in high abrasion applications and cases where lubricity is important
2
Glass reinforced polypropylene with carbon black
40% continuous glass fiber, 59.5% polypropylene, 0.5% carbon black
1 088
GRTL ® Azdel Laminate
3
Glass reinforced polypropylene without carbon black
40% continuous glass fiber, 60% polypropylene
1 170
GRTL Azdel® Laminate
4
Crosslinked Polyethylene EVA base, CaCO 3 filler
Ethylvinylacetate: 10% vinyl acetate filler; 28.5% by weight CaC03 filler
1 112
Union Carbide
949
3 14
HEAT A N D MASS TRANSFER IN PACKED BEDS
5
Low density polyethylene
Unfilled crosslinked polyethylene
6
Crosslinked polyethylene EVA base, Si02 filler
7
Polytetrafluoroethylene
Ethylvinylacetate : 10% vinyl acetate filler; 28.5% by weight Si0 2 (silane treated) filler ® Standard Teflon thermoplastic
972
Union Carbide
1061
Union Carbide
2 1 54
Dupont
APPENDIX A
A.7
Thermal Conductivities of Miscellaneous Solids
Aa
Ag Cu
I--
IJI'rg
.....
-
.
3t
:11
Brass Ni
Ff>
st O l n less c;teel 18Cr- 8Nl ·
� I r7 �t:--I� � �\
10
heat storag
.....
�2�
>
0 E L.. Q) .r::. 1-
I
I
- Pb
>-
..... u ::J "0 c 0 u
AI
�1--:---
I ::.c I E
315
f-2'
·� po;;;:
L .
10- I
POCk
- ---
.......
J, tl
' � \
'
i reclay b r i c k burnt L450° c , '-- - I 330° c
Oi I ShO)P
bsbe�tos ock 1ool
___... .-.
( \aspestqs,40 85% moan s i a
�
mognes i tP
t=
as estos , J ocseJy
- ."...
wrought i ron 0 . 5�C
=
.
rO ' l Sh::Jif' [D
:: gloss'.;.._ -=:::1::
f- ))b- 6
-
fel t ,20 J (minot ions/ i n
laminations/in
2 0
200
400
600
800
Temperature
(°C)
Heat sto rage material 1 . LiOH, Dens i tY L404 kg . m
-3
2. LIOH, Dens i t Y ! , 421 3 . L!N03 , Dens ity 2 , 1 39 4 . L!N03 , Densi t y 2 , 2 1 5
5. 63-Li OH/37-L l C I , Dens i t ies ! , 635 & 1 , 648
6.
Na2 B4o7,
Dens i t ies 2 , 322 & 2,354
FIGURE A.7
1 000
1200
1400
o i I shale [l) 6 . 8 �wilton
�J 29 . 6 GJ 4 7 . 9
glass
I > boros i l icate
2>
w i ndow
Thermal conductivities of solids.
Permission has been granted to use material from the following sources: R . P. Tye, A. 0. Desjarlais and J . G. Bourne, Proceedings of the 7th Symposium on Thermophysical Properties, sponsored by ASME and NBS, Maryland ( 1 9 77) p. 1 89 for heat storage materials. R. Nottenburg, K. Rajeshwar, J . Dubow and R. Rosenvold, ibid. p. 396 for oil shales. Others are from A. I. Brown and S. M . Marco, Introduction to Heat Transfer, 3rd edn., McGraw-Hill, New York ( 1 958); used with permission o f the Publisher.
316
HEAT AND MASS TRANSFER IN PACKED BEDS
A.8
Prediction of Diffusion Coefficients in Binary Gas Systems
The following equations are recommended for predicting diffusion coefficients in binary gas systems: D 1 2 = diffusion coefficient (m 2 s- 1 ) between species 1 and 2
M 1 , M2 = molecular weights P = total pressure (Pa) T = temperature (K). Hirschfelder-Bird-Spotz 's formula: J. 0. Hirschfelder, Trans. Amer. Soc. Mech. Eng. 71, 921 (1949).
where
B = 1.08 X 10-4 a
12 =
a ., a 2
=
a1 + a2
2
[
1 - 0. 2 3
(
1
M.
+
1
R. B. Bird and E. L . Spotz,
)] 1'2
M2
(nm)
collision diameters of species 1 and 2, respectively (nm).
[( )( )]
nn = collision integral for diffusion, function of kT/E 1 2
€ 1 2 = energy o f molecular interaction:
k a,
=
€ 12
k
=
€1
€2
k k
11 2
Boltzmann constant.
no, and k/£ 1 2 see R. B. Bird, W.
E. Stewart and E. N. Lightfoot, Trans port Phenomena, John Wiley, New York (1960). For E/k,
Andrussow's formula: L. Andrussow, Z. Elektrochem. 54,566 (1950); 55 , 51 (1951). D .2 =
2 8.025 X 10 ' P ·'s [ l + (M, + M2) 11 ] P(V:'3 + v�n) 2 (M.M2 ) 1' 2 -
where V, and V are the liquid molar volumes at normal boiling points (m 3 mol -1 ) of 2 species 1 and 2, respectively.
FUjita's formula: S. Fujita, Kagaku Kogaku 28 , 2 5 1 (1 964 ). D ., =
( p[(::r (:: T 6.70 x 10- 8T'· 8 3 I
+
1
M
+
,
)
1 1'2 M
,
where Tc and Pc are the critical temperature (K) and critical pressure (Pa), respec tively; see Table A . 3.
A.9
'·
Data of Diffusion Coefficients in Binary Gas Systems
The figures times 1 0 - 4 give the diffusion coefficients (m 2 s- 1 ) at the temperature (K) indicated in parentheses.
Air
air
Ar
Ar
co
0 . 1 880 (295.7)
C02
0.1420 0 . 1 6 5 2 0 . 1 5 2 0 (27 6.2 ) ( 3 1 7.2 ) (2 96 .1 ) d d d
02
0.5650 0.5750 0.5490 0.4740 (296.8) (296.8) (295.7) (295.7) d d d d
H2
0.6 1 1 0.8280 0.7430 0.6650 1 .2400 (273.2) (287 .9) (295.6) (298.2) (288.2) a d d d d
d
co C02
02
>
H2
't!
't!
H20
0.2770 ( 3 1 2.6 ) d
He
0.6242 0.8090 0.7020 0.6780 1 .2500 1 . 1 320 1 .4 1 40 (2 76.2) (323.2) (29 5.6 ) (32 3.2 ) (29 5 . 1 ) (2 98 .2) (4 98 .2) d d d d d d d
0.2 1 1 0 (32 8.6 ) d
trl z 0 >< >
1 .0200 (307 .3) d
-
H20
He
---- I
w .......)
-
Kr
N2
w .....
0 . 1 190 (2 73.0)
d
0.1940 0.2120 0 . 1 7 3 0 0.5420 0.674 0.3030 0.7050 (293.0) (295.8) (300.0) (296.8) (273.0) (328.6) (293.0)
:
d
d
0.2 7 1 0 (273.0)
Ne
d
N H3
0.2470 0.2320 0.2400 (295 . 1 ) (295 . 1 ) (295 . 1 )
d
d
00
Kr
d
I
I
d
a
d
I
b
d
en en
Ne
--1 it' > z r.n "T) � it'
d
0.6300 0.7454 (296.8) (27 3.0)
d
� >
0.2230 (273.0)
(303.9)
> z 0
N2
d
1 . 1 50
::X: � > --1
0.8310 (296.6)
d
d
0.2480 (2 95 . 1 )
NH
d
3
-
.,
z
0.0531
02
d
0.1 7 8 (273.2)
a
a
d Ethane
0.3 1 80 0.8090 (273.2) (32 9.0 ) (323.2)
a
d
d
0.181 (273.2)
d
a
0 . 1 090 (296.6)
d
d
0.7260 0 .3 3 1 0 1 .0050 (298.0) (328.8) (373.0)
(29 8.2 )
0.5370 (298.0)
0.1480 (298.0)
d
d
02
a
0.3060 0.3960 (295.7) (286.2)
0.220 0.1 5 3 (298.2) (273.2)
c
0.697
a
d
0.2190 (289.0)
N 20
a
0 . 1 85 0.139 (273.2) (27 3.2 )
0.0887 (296.8)
SF6
Methane
0.535 (273.2)
(194.8)
N 20
d
d
0.216
c
d
I
I
II
I II
> (') � trl 0 t:C trl 0
en
0.486 0.2330 (273.2) (328.5) d a
0.1 1 6 (273.2) a
Ethylene
0.1630 (291.2) d
1 o.0863 1(298.0) I d
Propane
I
0.0860 (298.0) d
n-butane
0.36 1 0 (287 .9) d
0.0960 (298.0) d
isobutane
0.277 (273.2) a
0.0905 (298.0) d
n-hexane
0.0663 (288.6) d
0.2900 (288.7) d
2,3dimethyl butane
0.0657 (288.9) d
Cyclohexane Methyl cyclopentane n-heptane
0.5740 (4 1 7 .0) d
0.0757 (288.6) d
0.0753 (288.6) d
0.30 1 0 (288.8) d
0.075 1 (288.7) d
0.0753 (288.4) d
0.0719 (288.9) d
0.3190 (288.5) d
0.0760 (288.6) d
0.0744 (288.6) d
0.0731 (287 . 1 ) d
0.3180 (288.5) d
0.0760 (288.6) d
0.0742 (287 . 1 ) d
0.0658 (303.2) d
0.2 18 0 0.2830 (303.2) (303.2) d d
0.2650 (303.2) d
"t:
>
"t: trl z 0 ......
>< >
w
0.0740 (303.2) d
\D -
--
2,4-
n-octane
d
0.0505 0.0587 (273.2) (303.2)
2,2,4-
d
d
0.0599 (303.2)
trimethyl pentane
d
0.2080 0.2770 (303.2) (303.2)
d
0.2 1 20 0.2920 (303.2) (303.2)
d
d
d 0.3060 ( 3 64. 1 )
n-decane
d Benzene
0.0962 (298.2)
d
0.0744 (303 . 1 )
0.2630 (303.2)
(303.2) (303.3)
d
a
'
I0.2240 0.2970
0.0655 (303.2)
dimethyl pentane
d
0.2480 (303.2)
1 0.2530 d
I
I
(303.2)
d
l
I
I
w N
0
d
0.0726 (3 0 3 . 1)
0.0705 (303.1 )
d
d
0.0713 (303.3)
0.0705 (303.0)
d
d
0.0841 (363.6)
l
d
0.0528 (273.2)
0.4036 ( 3 1 1 .3 )
a
0.3840 (298.2)
d
d i
0 . 10 2 2 ( 3 1 1 .3)
o.1 0 1 1 (3 1 1 .3)
d
d
'
Toluene
0.0920 0.07 1 ( 3 1 2 .6) (273.2)
d Methanol
0.0879 (273.2)
a Ethanol
0.1 350 (298.2)
d
Propanol
0.0850 (273.2)
a
0.506 (273.2)
1 .0320 (423.2)
a
d
0.0685 (273.2)
0.375 (273.2)
0.4940 (298.2)
a
a
d
0.0577 (273.2)
0.315 (273.2)
0.6760 (423.2)
a I
a
a
I 1
i
i' '
a
0. 1 3 2 (273.2)
i
d
I I
i
I
I I
I
I
X tTl > >-! > z 0 � > en en >-! ::0 > z en '"'1 tTl ::0 -
z "0
> (') � tTl 0 ttl cr. 0 en
0.6770 (423.2) d
0.0990 2-propanol (299.1) d
Butanol
0.0870 (299 . 1 ) d
0.0476 (273.2) a
0.2716 (273.2) a
0.5870 (423.2) d 0.0914 (298.0) d
0.0918 (298.0) d
E thylene oxide
Pyridine
0.4370 ( 3 1 7 .9 ) d
0 . 1 068 (31 7.9) d
0 . 1 050 (318.3) d
Piperidine
0.4030 (314.7) d
0.0953 (314.9) d
0.0953 (315.0) d
Thiophene
0.4000 (302.2) d
0.0992 (302.1) d
0.0975 (302. 1 ) d
Nitrobenzene
0.0855 (298.2) d
I
0.3720 (298.2) d
The data are reproduced from (a) International Critical Tables, Vol. 5, McGraw-Hill, New York (1929), p. 6 2 . (With permission of National Academy Press, Washington, D .C.). (b) R . Paul and I . B. Srivastava, J. Chern. Phys. 35, 1621 (1961). (c) C. R. Mueller and R . W. Cahill, J. Chern. Phys. 40, 651 (1964). (d) E. N. Fuller, P. D . Schettler and J. C. Giddings, Ind. Eng. Chern. 5 8 (No. 5 ) , 19 ( 1 966).
"'e .,
>
t':j z 0 -
>< > w N
-
322 A.l 0
Diffusion Coefficients of Gases in Water
xlo-9
10 8
.-i I (/)
......
N E
....
c: (1)
·-
....
u
6
l.j
3
2 1
0.8-
.....
xlo-9
0
l.j 3
(1) 0 u
c:
(/)
.... ::J
....
Cl
,......
6 ,.....
2
1
0.8
r(.)'2. __... ..../ ....- �'2. :...-. ... -� __...v ._e--/v C.\'2. vr-
�
0
-
;_'L\-\
I
10
20
30
Temperature
FIGURE A . l 0
,--..--,__.
l.jQ
so
60
70
[0()
D iffusion coefficients of gases in water.
Appendix B . Computer Programs (Fortran 7 7 )
B. I
Prediction of Response Signal by the Method of Section 1.1.6.2; Calculation of Root -mean-square-errors for Construction of Two -dimensional Error Map
········································�················· ·�···········�
••••••••••••••••••••••••••• • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • •• ••• •• • • • • • • • •• •• •• ••
----------
MAIN PHOGRAM
•• ••
Curve-fitting in time domain by fourier a n a l y s i s . .. prediction of responsEc' signal from measured input s i g na l ; calculation o f root-mean-square-errors between measurEd ana • •
•• ••
error map .
•• ••
** •• ••
**
predicted signals for the construction o f two-dimensional ••
----- �ummary of Flow Proc . 1 . normalization of measured input & response signals 1 . 1 . read t i t l e of e x perimental run 1 . 2 . reac �easured input signal
••
read measured response signal
•• ••
1 . 3 . normaliz� measured inp�t signal normalize measurca response signal 1 . 4 . print input & response signals
••
••
II
•• II
** •• •• II
•• •• II
•• •• •• II
••
II
•• ••
�·l A l N
II :
SUb.
NORSlG
- -
- - - - - -- - - - - --
su.e . RIJSJGL
�Lib .
... •• •• ••
RD::>H;L
It 'll
.SUb.
NUhl"iLZ
II 'II
SUB.
NORt-JLZ
••
.SUb.
PHIN"l
.. .
.S U B . r'Ol.Jt:XP Proc . 2 . fourier expansions o f measurea signals - -- - ----------2 . 1 . set e x pansion condition .SlJb. fo'OUCOt:: 2 . 2 . c a l c Fourier coer o f i n p u t signal SUB. fo'OUCO£ calc Fourier coer of response signl .SUb . kDPAHM Proc . 3 . reading of packea-bed parameter values ::i i.J b . PH I:: C UR Proc . 4 . prediction of response curve .SUE . CLCCOt:. 4 . 1 . c a l c F. coef o f predictfd curve at various frequencies S U B . TRAN.Sr 4 . 1 . 1 . c a l c transfer !'unction -------4 . 1 . 2 . c a l c Fourier coef ficients of predicted signal SUb. CLCCUR 4 . 2 . c a l c response curve � print -------4 . 3 . c a l c root-mean-square-error & p r i n t SUb. H!viSEhli Pro c . s . calculation o f root-mean-squart-Prrors for construction of error m � p sue . kDVAt
323
**
•• ••
It t II
.. .
••
l li *"
· - .. .
•• 'lt ti ·� . .. l it
. .. ••
324
HEAT AND MASS TRANSFER IN PACKED BEDS
••
kepE a t c a l c by varying vertical
param
••
••
R e p e a t c a l c by varying norizontal param
••
••
5.4.
calc F .
5.5.
c a l c root-mean-squar�-err·or
5.6.
p r i n t errors
coef of predicted curve a t
: SU b .
various frequencies ( same a s 4 . 1 )
it lt ••
CLCC0�
1 RA I�.Sf
SUb . �
- - - - - 1 -------
1* u tt l ••
•• ••
--------
**
**
i ttl
••
••
••
- - - - - V a r i a b l e s i n COMMON Blocks - - - -
••
COMMON
data on input & rE sponse s i g n 3 l s
/�lGNAL/
s t a r t i ng
times o f sign3ls
.. .
TOIN , TOHES
rea l ;
••
D'l l N , DTIU:S
real ;
timf i n t e r v a l s
**
NIN ,
integer;
number o f data
re.al array ;
norm : l i z e d sign31s
•• ••
NRES
CIN , CUMMON
CHES /l•OATA
fAU
••
I
data on fourier e x pansion hal f period number of terms zerotn fourier c oe f f i c i e n t s Fourier cosinE c o e f f i c i e n t s F o u r i e r sine coefficients
NH:RM
integer;
••
AOIN , AOHES
••
AIN ,
ARES
••
BIN ,
I:lRES
rea l ; r�al array; real a r r a y ;
••
COMMON
••
PM
( ---IN
•• •• ••
rea l ;
••
••
••
i n pu t , ---RE�
:
r€sponse
p�cked-bed
/ PBPARM/
re&l a r r a y ;
•• ••
)
•• •• ••
parameters
••
packed-bed parameter values ( sE E SUB.
••
kOPAHM)
•• .. . ••
• • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • it lt i tt tt tttt lt lt l tt tt l tt tt tt tt tt tt lt ji f tt ll tt tt tt l l lt l tt tt l it ti lt lt lt l tt a tt lt tt tt l tt lt lt l lt lt lt tt tt tt tt l lt lt l lt w a l lt A I It it l
p�·J( 2 0 )
CONMON
/PI::S P A Rt-J/
CALL
NORSIG
CALL
FOUBXP
CALL
fWPARM( Pl"l )
CALL
PR£CUR
CALL
HMSERR
•
•
•
•
Proc . l .
signal norma l i z a t i o n
Proc . 2 .
Fourier expansion
Proc . j .
parter reading
Proc . 4 .
curve prcd ic tion
Proc . 5 . r . m . s
•
STOP EN D
. c .
c a 1 c u l
APPENDIX B
325
,..•............,, ................. ..•...•.•......••..•..•.....•.......• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • •• •• • �;UnSlG •• .SUbHUU'! l t\ 1:. • •• •• • • t ti t l E and measured signals are read from file ' 1 ' ) • • • • ' source main program • routines used : SUB . H031GL , NORM LZ & P H 1 N 1 • • none d 3 t a us�a • rE:turned d a t a : COI'1MON /.SlGf'jAL/ • working a r e2 s : liTL� , COI� , CD H �S , A l N , ARES • • t i t l e of experimental run ll1L� character; • • CDl � , CORES: real array; measured input & response signals • AIN , ARt:.S : real ; area of curves • .. ( ---IN : input , ---R�S r�spons€ ) • fOrmat Of' input data from fi l e I 1 I : • • TllLE : FOHtviAT( MO ) • • It • • • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •
Pro� .
1 .
norma l i z a t i o n o f measured input & response
N O H.SI IJ
SUbHOUTlt-11:: COMI"'Oti
/SIGNAL/ TOI N ,
OT IN ,
�IN ,
signals
CIN ( 2 00 ) ,
TOHES , O T R E:S , NR E S 1 C R ES ( 200 )
$
Ollvii:: NSlU�
C O I N ( 2 0 0 ) 1 CDRES ( 20 0 )
CHARACTI::R
TITLE*80
REA D ( 1 1 1 0 0 )
TITLE
..
� H l 1� ( b , 600 )
Proc . 1 . 1 .
read
title
Proc . 1 . 2 .
read signals
Proc . 1 . 3 .
norm a l i z e
TlTL�
W R IT £ ( 6 1 6 1 0 ) CALL
RD.SlGL ( TO l N 1
CALL
kOSlGL ( T O R �S 1 0TRES , NR E S , CORES)
DT l N ,
NIN1
CDI N )
NI� ,
CALL
tiuHMLZ ( OT I N 1
CALL
NORMLZ(OTRI::S , NR�S, CORES,
CALL
PR1Nt ( T OI N 1
DT1N 1
COI N ,
NIN ,
AlN,
CI N )
A R E S , CRES) Proc . 1 . 4 .
COIN ,
AlN ,
print
CIN,
T O HES I OT � f ES t N R ES I CDR ES t A RES I CRES)
* * * * * * * * * * * ** * * * * * * * 1 0 0 fOkHAT(
A80
)
F 0 R M A T
*************************************
f Oki"'AT( 1 t1 1 I 1 H O , 8 2 ( 1 li + ) I 11i I 1 H + I 80X , 1 H + I 1 H � 1 H + I H i I 1 H + , o o x l 1 H + I 1 H t 82 ( 1 H+ ) I ) 6 1 0 F'Oki'1AT( I 1 11 0 1 , , ..,. . . MEASURE;D SIGNALS H u tt • ) 600
I
lH+,
A80 ,
326
HEAT AND MASS TRANSFER IN PACKED BEDS
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •
•
• •
SUBROUTINE
•
•
Proc . 1 . 2 .
•
source
SUb . NORSIG
*
•
routine used
none
*
data
none
•
returned
It
I
•
HDSIGL ( TO , DT , N , CD ) read measured signal
used data :
from f i l e
•
•
•
TO DT
rea l ;
time
•
N CD
integer;
number o f data
real
measurea signal
• •
working area
I
•
format o f input d a t a from
array;
starting time of signal
• • •
interval points
•
*
CWORK
• •
CwORK: real array; data in one block
• •
•
TO , DT , N , CD rea l ;
•
•
'1'
file
TO , DT : FORMAT( 2F 1 0 . 0 ) CwORK: FORMAT( 10F8 . 0 )
'1'
• •
it
•
*
•
Negative value of CWOR K ( )
•
indicates e n d of d a t a .
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
SUBROUTINE •
DIMENSION
RDSIGL ( TO , DT , N , C D ) CD(200 ) , CwORK ( 1 0 )
REA D ( 1 , 1 0 0 )
TO , DT
+++++
read & check
+++++
clear counter
+++++
read one b l o c k
+++++
block division
+++++
memory o v e r
wRlT£ ( 6 , 60 0 ) TO , DT WRITE( 6 , 6 1 0 ) I F ( TO . LT . O . O WRITE ( 6 , 62 0 ) IF( DT . LE . O . O •
• •
N : 0
REAL>( 1 , 1 1 0 , END:90) ( CwORK, .. ) , 1 : 1 , 1 0 ) ( CWORK( I ) , 1: 1 , 1 0 ) wRITE ( 6 , 6 3 0 )
DO 1 0 1 = 1 , 1 0 IF(
•
CWORK ( l ) . LT . O . O + 1
RETUkN
� = N
GO TO 9 1 I F ( N . G1' . 200 ) CD( N ) : CWORK ( l ) ti
?
1 0 CONTINUE GO TO 1 p r i n t error messages
90 WRITE ( 6 , 64 0 ) STOP 9 1 WRITE ( 6 , 650) STOP
APPENDIX B ************•*******
F 0 R M A T
*************************************
1 0 0 FORMAT( 2 F 1 0 . 0 ) 1 1 0 FORMAT( 600 FORMAT(
10Fd . O ) 1 HO , 5 X ,
610 620 630 640 650
1HO, l HO , lH , 1 HO , 1 HO ,
FORMAT( FORMAT( FORMAT( FORMAT( FORMAT( END
327
' TO = ' , F 1 0 . 5 , 5X ,
' DT = ' , F 1 0 . 5 I
' STARTING TIME ( T O ) IS NEGATIVE . ' I ) ' TIME INTERVAL ( D T ) IS NOT POSITIVE ' I 1 0 F8 . 2 ) ' END OF DATA CANNOT BE FOUN D . ( SU B . RDSIGL ) ' I ' NUMBER OF DATA EXCEEDS 2 0 0 . (SUB . RDSIGL ) ' I ) .
328
HEAT A N D MASS TRANSFER IN PACKED BEDS
•• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • •
•
Proc . 1 . 3 .
•
�OHMLZ ( DT , N , C D , AREA , CN )
.SUbHOUTl�l:.
tt
norma l i z e
measured
* •
signal
*
• I
source
.SUB. NURSlG
routine useo
none
•
DT , N , CD returned d a t a : AR�A , CN AH�A rea l ; real array; CN oata
• • II
usco
method o f
II
•
s e e SU6.
•
RDSlGL
• •
ar�a o f curve normalized signal integration : trapezoidal rule
• • •
..
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • •• • • • • • • • • • • • • • • • • • • • •
.SUBROUTINE
�ORMLZ ( DT , N , CD , AREA , CN )
lHMI:;NSlON
CD(200 ) , C N ( 2 0 0 )
•
+++++
calc AREA
+++++
check AREA
+++++
normalize
A.kt;A = 0 . 0 i)(J 1 0 1 = 1 . �
AReA = AREA + CD( l ) 1 0 CUNTlNUc AHt::A = AREA * OT •
lf(
AR�A . �Q . O . O )
w k 11'� ( 6 ' 600 )
THEN
DG 20 1 = 1 , N
20
CN ( l ) = C D ( l ) CONTINUe Ri::T UHN 8L.S� l F ( AREA . LT . O . O WR1Tt: ( 6 , 6 1 0 ) E.ND lF
THEN
DO 30 1= 1 , N C � ( l ) = CD ( l ) /AHEA 30 CuNTlNUE. ri i::TlJkN 11**1*1****�***1****1*****1*11*1***** F 0 R M A T ' A HEA O F CURV£ lS ZERO ( RETUHNED SIGNAL I S NOT NORMAL
* * * * * * * * * • • • ** * • * * * *
600 FORMAT( 1 H O , ;fll.i::D ) . ' I ) 6 1 0 FURMAT( t:ND
1�0,
' A REA OF CUHVt: IS Ni::G ATlVE . '
I )
APPENDIX B
329
·················�·············�···························�············ •
• * 1
SUBROUTINE
PRINT (TOIN , DTIN, N I N , COIN, A I N , CIN , TORES , DTRES , NRES, CDRES , ARES,CRES)
$
• 1
Proc . 1 . 4 .
I •
print input & response signals ( measured
&
I 1 I
normalized) * •
source routine used data used
SUB. NORSIG none TOIN , DTIN, N I N , COIN, AlN , CIN , TORES , DTRES , NRES, CDRES , A RES, CRES returned data : none
* I • •
• • •
see SUB. NORSIG
1 •
•
•
··································�·····································
SUBROUTINE $ DIMENSION WRITe(6 , 60 0 ) wRITE ( 6 , 6 1 0 ) wRIT £ ( 6 , 620 )
PRINT(TOIN , DTlN, N l N , COIN , A I N , CIN , TORES , DTRES , NRES, CDRES , ARES,CRES) CDIN(200 ) ,CIN ( 200 ) , CDRES(200 ) , CRES(200) +++++ heading TOIN , DTIN , NIN , AIN TORES , DTRES,NRES, ARcS print input
I
wRH£ ( 6 , 63 0 ) •
M =
&
response
+++ set number of aata
MAX(NlN , NHgS) DO 10 I= 1 , M +++++ set times TIN : TOlN + FLOAT ( l - l ) *DTIN IRES= TORES+ FLOAT ( l- 1 ) 1DTRES I
+++++ print signals IF( l . GT . NI N ) THEN �RI1e ( 6 , 640 ) I , TRES, CDRES ( I ) ,CRES ( I ) ELSE IF( I . GT . NRES ) THEN WRIT8 ( 6 ,650 ) I , TlN , CDI N ( I ) , C lN ( I ) ELSE Wk1TE ( 6 , 66 0 ) I , TIN , CDlN ( l ) , C IN ( I ) , TRES , CDRES ( I ) ,CkES ( l ) £ND I F 10 CONTINUE RETURN
11111**111*1*1111111
F 0 R M A T *1*1111*1*111*1111111111111111*111111 600 FORMAT( II 1 H0 , ' 1 1 1 1 1 INPUT AND RESPONSE SIGNALS 1 1 1 1 1 ' I l H O , lOX $ , 6X , ' STARTING TIME' , 5X , ' TIME INTERVA L ' , SX , ' NO OF DATA' , 5X , ' AREA' ) 6 1 0 FORMAT( 1 H O , SX , ' INPUT ' , 4 X , f 1 0 . 4 , 7X , F 1 0 . 4 , 8X , I 5 , 5X , E 1 3 . 5 ) 620 FORMAT( 1 H O , 5X , ' RESPONSE ' , 4X , F 1 0 . 4 , 7X , F 1 0 . 4 , 8X , l 5 , 5X , £ 1 3 . 5 ) 6 3 0 FORMAT( 1 H O , 1 5X , ' = = = = = = = = INPUT = = = = = = = = ' , 1 5X , ' = = = = = = = RESPONSE $ ::::: : : 1 I 1 H , 4X , ' N ' , 2 ( 7 X , ' T1ME READING (NORMALIZED ) ' ) ) 640 FORMAT( 1 H , 1 5 , 38X , 3X , F 1 0 . 4 , F 1 0 . 1 , 3X , F 1 0 . 5 ) 650 FORMAT( 1 H , 1 5 , 3X , F 1 0 . 4 , F 1 0 . 1 , 3X , f' 1 0 . 5 ) 660 FORMAT( 1 H , 1 5 , 2 ( 3X , F 1 0 . 4 , F 1 0 . 1 , 3X , F 1 0 . 5 , 2X) END
330
HEAT AND MASS TRANSFER IN PACKED BEDS
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •• •• •• FOUEXP SUBROUTINE •• •• •• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • Proc . 2 . Fourier expansions o f measured input & response signals* • ( ha l f period & number of terms are read from file ' 2 ' ) • * • • main program • source • SUB. FOUCOE • routine used • COMMON /SIGNAL/ data used • • see main program returned d a t a : COMMON /FDATA I • see main program • format o f input data from file ' 2 ' : • • • TAU , NTERM: FORMAT( F 1 0 . 0 , 1 1 0 ) • TAU real; • • half period • NTERM integer; number of terms • • • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• SUBROUTINE COMMON $
COMMON
$
FOUEXP /SIGNAL/ TOIN , DTIN, NlN , CIN ( 200 ) , TORES , DTRES , NR�S,CRES(200) /FDATA I TA U , NTERM, AOl N , A I N ( 2 0 0 ) , BIN ( 200 ) , AORES , ARES(200) ,BRES(200) Proc . 2 . 1 .
• TA U , NTERM READ ( 2 , 20 0 ) �RlT� ( 6 , 60 0 ) TA U , NTERM
•
•
Stt
paramters
+++++ check parameters lF( TA U . LT . ( TORES+DTRES*FLOA T ( N R ES ) ) /2 . 0 ) THEN TAU = (TORES + DTRES 1 FLOAT ( N R ES ) ) /2 . 0 1 1 . 5 �RIT� ( 6 , 6 1 0 ) TAU END IF THEN IF( NTERM . LT . 1 . OR . NTERM . G T . IFIX(TAU/DTRES+0 . 5 ) NT£RM = I F I X ( TAU/DTRES + 0 . 5 ) WRIT£ ( 6 , 620 ) NTERM END IF IF( NTERM . G T . 200 THEN NTERM = 200 WRIT£ ( 6 , 620 ) NTERM END IF Proc . 2 . 2 . calc F . coef WRITE ( 6 , 63 0 ) CALL FOUCOE ( TOIN , DTIN, N l N , CIN , TAU , NTERM , A O I N , AIN , BIN ) WR1TE ( 6 , 64 0 ) CALL FOUCOE(TORES, DTRES , NRES , CRES , TA U , NTERM , AORES , ARES, BRES) RETURN
APPENDIX B
331
11***••••••1****1*1***1*111*1111***** F 0 R M A T 200 FORMAT( F 1 0 . 0 , I 1 0 ) 600 FORMAT( // 1 H O , ' * * * * * FOURIER EXPANSION * * 1 * 1 ' I 1 H O , 5X , ' HALF $PERIOD = ' , E 1 3 . 5 , 5X , ' NO OF TERMS = ' , I5 ) 6 1 0 FORMAT( 1 H O , ' HALF PERIOD ( T A U ) IS REPLACED BY ' , E 1 5 . 7 )
••••••••••••••••••••
620 FORMAT( 630 FORMAT( 640 FORMAT( END
1 HO , ' NUMBER OF TERMS (NTERM) IS REPLACED BY ' , I5 1 HO , 5X , ' +++++ CONVERSION CHECK (INPUT) +++++' ) 1 H O , 5X , '+++++ CONVERSION CHECK ( RESPONSE) +++++'
)
332
HEAT AND MASS TRANSFER IN PACKED BEDS
······�····························································I····
•
I
I
SUBROUTIN�
• * *
Proc . 2 . 2 .
F0UCOE(TO , OT , NT , CN , TA U , NTERM,
AO , A , B )
calculate Fourier coefficients of signals (eqns( 1 . 4 1 a ) * 1 response signal) •
& ( 1 . 4 1 b ) for input ; eqns ( 1 . 46 a ) & ( 1 . 46b) for
•
•
source
SUB.
•
data used CN ( N ) returned data : AO A(N)
T O , OT , N T , CN , TAU , �TERM;
• • •
* • • I
B(N)
real; AO , A , E real ; real; real ; real ;
PI
• li
I
* • •
see SUB. RDSIGL & FOUEXP signal value a t n-th point
• •
I
• • •
zeroth fourier coefficient n-th Fourier cosine coefficient
•
n-th Fourier sine coefficient
• •
circle circumference-to-diameter ratio
1
COTEST : r e a l ; real; �Q SOTEST: r e a l ;
calculated signal at t=O by Fourier series•
RAT
ratio of SOTEST to SO
• It
FOUEXP
working areas: Pl , COTEST , SO , SOTEST , RAT
I
• • •
real;
integral of signal squared i n tegral of signal squared , predicted in terms of F'ourier coefficients
( COTEST & SOTEST are used to check convergence of Fourier series . ) integration formula : trapezoidal rule
1
• * • * • 1
•
l f l l l l l l l l l l l l l tt l l l l l l · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
SUBROUTlNt; DlMENSlON
FOUCOE ( TO , DT , NT , CN , TA U , NTERM, CN(200 ) , A ( 200 ) , B(200 )
DATA
PI I 3 . 1 4 1593 I
AO = 0 . 0 so = 0 . 0 DO 1 0 1 = 1 , NT AO = AO
SQ = SQ 10 CONTINUr:
AO , A , B )
+++++
+
CN ( I )
+
CN ( I )11£
calc AO & SO
AO = A01DT/TAU SQ = SQ*DT •
•
w.kiTE(6 , 6oo l
COTEST = A012 . 0
SOTEST : 2 . 01 ( AOI2 . 0 ) 1 1 2
1
+++++
heading
+++++
initial set
TAU +++++
DO 20 N= 1 , NTERM X : 0.0
y = 0.0
+++++
SV : 1'LOA T ( N ) 1PIITAU DO 30 I= 1 , NT Z : SV * ( TO + FLOA T ( I - 1 ) * DT ) X = X + CN ( I ) * C O S ( Z ) Y : Y + CN ( l ) 1SlN ( Z)
ca l c A ( N ) & B ( N )
save SV
APPENDIX B
30
CONTINUE A ( N ) : X1DT/TAU B ( N ) : Y1DT/TAU
•
•
333
+++
COTEST = COTEST SQTEST : SQTEST
+ +
A(N) ( A ( N )•12
+
clc COTEST
& SQTEST
B ( N ) 1 1 2 ) 1TAU +++++
print check data
IF( MOD ( N , 1 0 ) . E Q . O ) THEN RAT = SQTEST/SQ WRIT� ( 6 , 6 1 0 ) N , RA T , COTEST END IF 20 CONTINUC RETURN
11111*11*11111111111 600 FORMAT( l H O , 6 1 0 FORMAT ( 1 H I END
F 0 R M A T
1*1*11*111111111111111***************
l O X , ' NO O F TERMS' I 8X , ' RATIO' I 7 X , 1 ox ' I 5 ' 1 ox ' F 1 0 . 5 5X F 10 5 ) ' ' •
' VALUE AT T:O '
)
334
HEAT AND MASS TRANSFER IN PACKED BEDS
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • RDPARM(PM) SUBROUTINE • •
• Proc. 3 .
• • • •
reading of packed-bed parameter values from file ' 3 '
source routine used data used returned data: PM
• • •
PM( 1 ) PM ( 2 ) PM ( 3 ) PM( 4 )
•
• •
main program none none PM real array; packed-bed parameters length o f bed 'L
• • •
I
•
'U 'SA ' 'EB '
interstitial flow rate surface area per unit volume bed void fraction
PM( 6 )
'R 'EP '
particle radius intraparticle void fraction particle density molecular d i ffusivity axial dispersion coefficient
• "
PM( 1 1 ) :
'DAX ' ; 'DE I , 'SH ' ;
•
PM( 1 2 ) :
SKF , 'SKA ' ;
• • • • • •
PM( 5 ) PM( 1 ) PM( 8 ) PM( 9 ) PM( 1 0 ) :
• • •
•
, 'RHOP ' ; 'DV ' ;
• • • • • • • •
• intraparticle effective d i ffusivity • Sherwood number ( = 2*SKF*R/DV) • particle-to-fluid transfer coefficient adsorption rate constant; •
set SKA=O when SKA is infinitely large • adsorption equil ibrium cunstant • • • number o f parameters • ( I P = 1 3 in this program) • PNAME ( I ) : character; parameter symbol for PM ( I ) • format o f input data from file ' 3 ' : • PM( 1 ) to PM( 1 3 ) : FORMAT( 8 F 1 0 . 0 )
• • •
PM( 1 3 ) : ' K A '; working areas : I P , PNAME integer; IP
I I
• •
• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
I
SUBROUTINE
RDPARM( Pl-1)
DIMENSION CHARACTER*4
PM(20) PNAME ( 2 0 ) PNAMI:: I ' L I RHOP 1 , I DV
DATA
$
IU t I DAX
I t I
I SA t 'DE
I t I
t
I EB
t ' I SH '
I 'IR ' , ' SKA
' EP ' , 'KA
I t
t I
I
lP = 1 3 • READ ( 3 , 30 0 ) • WRITt-:(6 , 600 ) WRIT E ( 6 , 6 1 0 )
+++++
read param values
+++++
print
( P M( l J , l= 1 , 1 P )
( I , PNAME ( l ) ,PM( I ) , I = 1 , I P )
RETURN ********************
F 0 R M A T
1***1***1*************11*************
300 FORMAT( 8F 1 0 . 0 ) 600 FORMAT( II 1 H O , ' ** * * * PACKED-BED PARAMETERS * 1 1 * 1 1 6 1 0 FORMAT ( ( 1 H , 4 ( 2 X , 1 2 , ' ) ' , A4 , ' = ' , E9 . 3 ) ) ) END
I )
APPENDIX B
335
····································································�··· 1111111111111111111111111111�··········································· II
••
PRECUR
SUBROUTINE
II
II
••
**
lllllllllllllllllt••········································llllllllllll I
Proc . 4 .
* I
prediction of response curve & root-mean-squa re-error
I I
I
source
•
routines used : SUB.
I
data used
•
returned data : none
1
I
working areas: AOCLC , ACLC, BCLC , NT , ERR AOCLC , A C L C , BCLC , N T see SUB. CLCCOE ERR : real; root-mean-square-error between
1
I •
main program
I
CLCCOE & CLCCUR
I
COMMON /SIGN A L / , /FDATA I & /PBPARM/
I
method o f calculation
I
I
•
*
1
measured & predicted curve s , eqn( 1 . 4 8 ) 1 1 ( 1 . 45 ) & ( 1 . 48 )
eqns( 1 . 42 ) ,
I
I
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l • l l l l l l l l l l l l l ii l l l l t t • t t l l l l l
SUBROUTINE COMMON
PRECUR /SIGNAL/ TOIN , DTIN, NIN , CIN(200 ) , TORES , DTRES , NRES, CRES(200 )
COMMON
/FDATA I TAU , NTERM, A O I N , AlN(200 ) , ElN( 200 ) , AORES , A RES ( 2 00 ) , BRES(200) /PBPARM/ PM(20) ACLC(200 ) , BCLC(200) +++++ heading
$ $ COMMON DIMENSION •
WRITE ( 6 , 60 0 ) CALL
Proc . 4 . 1 . c a l c F . coeff CLCCOE ( P M , TA U , NTERM , NT , AOIN , AI N , hl N , AOCLC , ACLC, BCLC)
CALL
CLCCUR (TAU , NT , AOCLC , ACLC , BCLC , 0 . 0 , 2 . 01TAU, DTRES)
•
Proc . 4 . 2 . calc response
I
I
Proc . 4 . 3 . calc r . m . s . e . X : 2 . 01 ( AORES/2 . 0 - AOCLC/2 . 0 ) 11 2 Y : 2 . 0 1 ( AORES/2 . 0 ) 112 DO 40 I : l , NT X : X + ( ARES ( I )-ACLC ( l ) ) 1 1 2 + ( BRES ( l ) -BCLC( l ) ) * 1 2 Y = Y + ARES ( I ) 1 1 2 + BRES ( I ) 1 • 2 4 0 CONTINUE ERR : SQRT (XIY ) WRITE ( 6 , 6 1 0 ) ERR RETURN
ltltttttlllltttltltl
6 0 0 FORMAT( // 1 H O , 6 1 o FORMAT< 1 H o , 5X , END
F 0 R M A T
111*1111111111*1111111111111**1*11111
CALCULATION O f RESPONSE CURVE ' HOOT-MEA N-souARE-ERRCF< = ' , Fo . 4 >
' *1111
11111'
)
336
HEAT AND MASS TRANSFER IN PACKED BEDS
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •
•
•
CLCCOE ( PM , TA U , NTERM , NT , AO l N , A I N , BIN ,
SUBROUTJNE
•
AOCLC , ACLC, BCLC)
$
•
I
• •
Proc . 4 . 1 .
..
•
calculate Fourier coefficients of predicted curve at various frequencies
• • •
•
source routine used data used returned data :
•
SUB. PRECUR (Proc . 4 ) , or SUB. RMSERR (Proc . 5 )
SUB. TRANSF PM,TAU , NTERM , AOIN , A I N , BIN see SUB. PRECUR AOCLC , ACLC , BCLC real; zeroth coefficient AOCLC n-th cosine coefficient ACLC ( N ) : real; n-th sine coefficient hCLC ( N ) : real;
• • • it *
•
II
integer;
NT
I
�orking areas: PI , W real ; PI
li
•
I
minimum number of terms needed for convergence of Fourier series for predicted signal
I
• • • • • • • • •
circle c ircumference-to-diameter ratio*
real ; frequency w method of calculation: eqn( 1 . 45 )
•
• •
•
1
• • • • • • • • • • • • • • • • • • • • • l l l l l l l l l lt t t t t t l t t l l l l l l l l l l l l l l l l l l l l l t l l l l l l l l l l l
•
CLCCOE ( PM , TA U , NTERM , N T , AOIN , AIN , BIN , AOCLC , ACLC , BCLC)
SUBROUTINE
$ DIMENSION DATA •
CALL AOCLC
=
•
PM(20 ) , A l N ( 2 00 ) , BI N ( 2 00 ) , ACLC(200 ) , BCLC ( 20 0 ) P I I 3 . 1 4 1 593 I +++++ zero frequency TRANSF ( O , O . O , P M , A N , BN ) AOIN*AN Repeat clc varying freq
DO 1 0 N : l , NTERM W : FLOAT ( N ) 11Pl/TAU Proc . 4 . 1 . 1 . calc transf
•
TRANSF( l , W , PM , A N , B N )
CALL
Proc . 4 . 1 . 2 . calc coeffi
•
ACLC ( N )
=
AIN ( N ) * AN
BCLC ( N )
=
BIN ( N )1AN - AIN ( N ) * BN
+
B IN ( N ) 1BN +++++
IF( AN112+BN112 . LT . 1 .0E-8 ) NT = N RETURN END I F 1 0 CONTINUE NT
=
Rt:TURN
r:rw
NTERM
THEN
check conversion
337
APPENDIX B
·················�···········�··�···············�·······�··············· •
•
I
•
SUBROUTINE
I
•
Proc . 4 . 1 . 1 .
•
source data used
• I
TnANS F ( I C T L , W , PM ,
ICTL
*
PM returned data
•
AN bN
•
•
I
•
•
• I
•
ir 1:
• I
• I • •
•
•
calculate transfer function for adsorption system
I *
•
A N , BN )
•
A N , BN real ; real ;
=
•
• SUB. CLCCOE • ICTL , w ,PM integer; = 0 , initial call for parameter s e t 1 * nonzero , otherwise real array; packed-bed parameters, see SUB. ROPARM* •
real part of transfer function imaginary part o f transfer function
I
•
save variables: 0 1 , 02 , 03 , 04 , 0 5 , 0 6 , 07 , D8 , D9 0 1 = L*U/( 2 . 010A X ) ; 0 2 = 4 . 01DAX/U1*2; D3
I
SA1DE/ ( EB1R ) ; D4
=
OE/(R*SKF ) ;
•
05
06 = RHOP1KA; 07 = KA/SKA ; 08 working areas: C S , CPHl , COTH , CQ , CF complex1 1 6 ; Laplace operator CS CPHl : complex1 1 6 ; phi-a in eqn( 1 . 6 1 c ) CEX : complex* 1 6 ; e x p { - 2 * phi-a) COT H : complex1 1 6 ; cotn(phi-a)
=
1 /DE ;
=
R;
•
D9
=
EP
•
• I
• •
CQ complex* 1 6 ; q in eqn( 1 . 6 1 b ) CF complex* 1 6 ; transfer function, eqn( 1 . 6 1 ) method of calculation: eqn ( 1 . 61 ) For other forms of transfer func tion, only IP and PNAM� ( in Proc . 3 ) and Proc . 4 . 1 . 1 need to be alterd .
* * • it
* *
I
•
· · · · · · · · · · · · · � · · · · · · · · · · · · · · · l l l l l l il li l l l l l l li l l l l l l l l l l l l l l l l l l l l l l l l l l l l
SUBROU'flNE
TRANSF ( lCTL, W , P M , A N , B N )
DIMENSION
PM(20) 0 1 , D2 , 0j , 04 , D5 , D6 , D7 , D8,D9
REAL*d SAVE •
Cul"'PLEX* 1 6
0 1 , D2 , 0j , D 4 , 05 , D6 , 07 , D3 , D9 CS , CPHI , CEX , COTH , CQ , CF +++++
IF'( lCTL . EQ . 0
THEN
*
+++++
initial c�ll ? set save variables
D1 = P�{ 1 ) 1PM( 2 ) / { 2 . 01PM ( � ) ) D2 : 4 . 0 1P M ( 9 ) / P M ( 2 ) 1 * 2 Dj : PM( 3 ) �PM( 1 0 ) / ( PM { 4 ) *PM ( 5 ) ) SKF = PM( 1 1 )1PM ( 8 ) / ( 2 . 01PM{ 5 ) ) D4 = PM( 1 0 ) / { PM ( 5 ) *SKf) 05 = 1 . 0/PM( l O ) 06 = PM ( 7 ) 1PM( 1 3 ) l F ( PM( 1 2 ) . EQ. O . O THEN 07 = 0 . 0 ELSE 07 = PM( 1 3 )/PM( 1 2 ) END IF 08 = PM{ 5 ) 09 = PM{ 6 ) END IF •
+++++
zero frequency ?
338
HEAT AND MASS TRANSFER IN PACKED BEDS IF( w . EQ . O . O ) THEN A� = 1 0 •
BN = 0 . 0 RETUHN t:NO IF •
+++++ calc transf�r func CS = CMPLX ( O . O , W ) CPHl= o�• CDSQR T ( CS*05*(09 + 06/ ( 07*CS+1 . 0 ) ) ) CEX : COEXP (- 2 . 0*CPHI) COTH= ( 1 . 0+CEA ) / ( 1 . 0-CEX) CQ = 0 3 / ( 0 4 + 1 . 0/ ( CPHI*COTH- 1 . 0 ) ) CF : COEXP ( 0 1 1 ( 1 . 0 - COSQRT ( 1 . 0+D2*( CS+CQ ) ) ) ) AN = CF BN = OHIAG ( C F ) RETURN END
APPENDIX B
to It • ..
•
*
�UjjHOUTI�E Proc . 4 . 2 .
• •
CLCC U R ( TA U , NT , AO , A , B , T 1 , T2 , DT ) calculate response curve to T 2 )
I
( in
time range from T l
source
SUB.
*
routine usea
none
f
data used
.. li *
•
*
rea l ;
zeroth Fourier coefficient
•
r ea l ;
n-th Fourier cosine coefficient
*
rea l ;
n-th Fourier sine coefficient
real;
time range
•
DT
real ;
t i m e int€rval
see main program
returned data :
none
It
working area s :
PI , DTW , T , C , IN , ER R
PI DT�
HN) C{N)
ti ..
IN
It
• *
*
PRECUR
TAU , NT : AO A(N) .b( N ) T l , T2
to
•
* * *
PRECUR
� A U , N1 , AO , A , B , T 1 , T2 , DT & SUB.
•
• •
w i t h Fourier coefficients and print
I
I
339
• •
*
•
real;
c i r c l e c ircumference- to-diameter ratio
real;
time interval ,
real;
time a t n-th point
real;
s i g n a l v a l u e a t n - t h point
integer;
number of data
methoa of c a l c u l a t i o n :
working area o f DT
• •
• • •
points
•
eqn ( 1 . 4 2 )
•
•••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••
.:>UbR0UTINE DltviEN.SION DATA
CLCC U R ( T A U , N T , AO , A , b , T 1 , T2 , DT ) A ( 200 ) , B { 2 0 0 ) , C ( 50 0 ) , T ( 500 ) Pl I 3 . 1 4 1 595 I +++
wRlT£ ( 6 , 6oo ) lF(
DT . BQ . O ) IN = 1 DTW: 01
se t number of points
T l , T 2 , DT THEN
eLSE lN = IF1X l ( T2-T 1 ) 1Dt I F ( IN . GE . O ) THEN
+
0.5)
I� = IN + 1 DTw: DT eLSe IN = -IN + 1 DTW: -DT eND 11-' I F ( 1 N . GT . 500 THEN I N = 500 WHlT� l 6 , 6 1 0 ) lN eND IF END IF •
+++++
initial
set
DO 1 0 1 : 1 , I N T ( l ) = T 1 + FLOA T ( l- l ) * DTW C ( l ) = AOI2 . 0 1 0 C0NTlNUE +++
calc response curve
340
HEAT AND MASS TRANSFER IN PACKED BEDS DG 20 N : , I �T
•
+++++ save SV
SV : FLOAT ( N ) 1 P I /1 A U
D O 3 0 1 : 1 1 1N
Z : SV * T ( I )
jO
it
C ( l ) : C ( I ) + ( A ( N ) * COS ( Z )
+
CONTlNUt.
B ( N ) *SIN ( Z ) )
20 Cu!�TlNl.Jt:: +++++ print wRlTE ( 6 1 620 ) wRl T d 6 1 63 0 )
( T { l ) 1 C( l ) 1
KE'!U.KN **** �1**11***1**1*** 600 f0kMAT( $ =' 1
1 H O I 5X I
F7 . � )
l : 1 1 IN )
F U R M A T
* * * * * * * * * * 1 1 • * * 1 1 1 * 1 ** 1 * * * * * * * * * * * * • *
' TIME RANGE : ' I F7 . 2 1
I
TO
I
I
F7 . 2 ,
6 1 0 fORMAT( 1 H O I ' N UMEER OF DATA ( I N ) 1s RcPLACED .h Y ' I 620 FOHMAT( 1 H 0 1 2X 1 5 ( ' 11ME H t::S PONSE ' ) ) 6 3 0 FOHNA'f ( ( H i I 5 ( F7 2 I F9 5 ) ) ) .
l:.ND
.
5X I
I5 1 )
' INTERVAL
APPENDIX B
341
l l l � l l it a it t t l l � l t a � l w l l a l l • • • � l l l a l f l l l l l l l l l & l l l l ll l l l l l l l l l l l l l l l l l l l l l ····· ····� ········· ···�······················llllllllltlltlllllllfllllll
•• ••
••
.SUbHOUll�l::
k�JSEHR
•• .., . . . ........ 1···�·····················································
• Proc .
*
5.
•
I
• calculation of root-mean-square- errors between measured • • and predicted signals for construction of error map
I
'
•
I
source
•
data used returned d a t a : none �orking areas: NP1 , N l , VS l , D P l , NP 2 , N 2 , VS2 , DP2
main program
routine usea
• ir
•
S U B . RDVARI & CLCCOE COMMON /FDATA I & /PBPARM/
• see main program see SUB.
RDVARI
* * •
PMw , PM l , ER R , ASQ,IQ
•
• • • • *
• •
• •
PMw
real array;
Pi'1 1 eRk ASQ 1Q
real array;
parameter set with varying horizontal and vertical parameters horizontal parameter values
real array; real ; charac te r ;
root-mean-square-errors for PM1 denominator of eqn( 1 . 48 ) character for heading
values
method of calculation : eqn( 1 . 48 )
• • • • • •
• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
�
SUBROUTINE:; COMMON
COMMON DIMENSION DIMENSION CHARACTER DATA
RMSEHR /FDATA I TA U , NTERM, AOIN, A I N ( 2 00 ) , B I N ( 2 00 ) , AORES , ARES ( 200 ) ,BRES(200) /PBPARMI PM(20 ) PMW ( 20 ) , P M 1 ( 1 1 ) , ERR ( 1 1 ) ACLC ( 200 ) , BCLC ( 200) IQ1 1 0 IC I I = = = = = = = = = ' I
wHlT£ ( 6 , 60 0 ) •
CALL
Proc .. 5 . 1 . set two param RDVAR1 ( N P 1 , N 1 , VS 1 , DP 1 , NP2 , N 2 , VS 2 , DP2) Proc . 5 . 2 .
calc ASQ
ASQ = 2 . 0* ( AOHESI2 . 0 ) * 1 2 l)() 1 0 t-1: 1 , NTEHM ASQ : ASQ + ARES ( N ) 1 1 2 + BRES ( N ) * 1 2 1 0 CON'i1NUE ASCi = SQRT ( ASQ) •
00 20 1 : 1 ' 20 PMW ( I ) = PM ( 1 ) 20 CONTINUE
++++ initial set o f PMW
Proc . 5 . 3 .
set hor param
DO 30 1 1 = 1 , N 1 + 1 PM 1 ( 1 1 ) = V S 1 + DP 1 *FLOA l ( I 1 - 1 ) 30 CONTINUE ..
+++++
heading
342
HEAT AND MASS TRANSFER IN PACKED BEDS ( PM 1
w rl 1 Tt. ( 6 , 6 1 0 ) lt..kiT£( 6 , 62 0 ) •
(1�,
(l 1 ) , 1 1 = 1 , N 1 +1 ) I 1=1 ,N1+1 )
Repeat calc verticl prm
DC 40 1 2 = 1 , N 2+1 PM� ( N P 2 )
+++++ set ver prm value
= V � 2 + DP2*FLOA T ( I 2- 1 )
Repeat calc horizon prm
oi
lJO
50 I 1 = 1
, N 1 +1
+++++ set hor prm value PM� ( hP 1 )
= PM1 l i 1 )
CALL
Proc . 5 . 4 . calc F . coeff CLCCO�( PMW , TA U , NTERM , NT , AOIN , A I N , BIN , AOCLC , ACLC, BCLC)
*
DO
60
X X
2 . 0* ( AORES/ 2 . 0
=
60 I = 1 , N1' =
X +
- A OC LC / 2 . 0 ) * * 2
Proc . 5 . 5 . calc r . m . s . e .
(ARES ( l )-ACLC ( I ) J 1 * 2 + (BRES ( l ) -BCLC( I ) ) * * 2
CONTlNUc
*
+++++ store r . m . s . e .
eRR ( I l ) 50
SQ.k T ( X ) / ASQ
=
CuNTlNUE Proc . 5 . 6 . print errors
•
lt.RITi ( 6 , 6� 0 )
PMW ( N P 2 ) , ( ERR ( I 1 ) ,
1 1 = 1 , N 1+ 1 )
40 CUNTlNut:
Rt::TiJkN F 0 R M A
1
*****11***1111** 1****1********�****** 60J FOHMA1' ( / / 1 H0 , ' *1 * 1 * CALCULATION O F ROOT-MEAN-SQUARE-ERRORS * * * * * ' ) 6 1 0 FORi•iAT( 1 H O , 1 2 X , 1 1 8 1 0 . 3 ) t>20 FOHI'lAT( 1 H , 1 2X , 1 1 A 1 0 )
k * •� • • • � � t • • • • • • • • • •
6 j 0 FC,Ki•iA1 (
ElJD
1H
I
£:: 1 1 . 4 ,
1 1F10.5
APPENDIX B
343 , .....
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
•
• •
RDVARI( N P 1 , N l , VS 1 , DP 1 ,
SUBROUTINE
N P 2 , N 2 , VS 2 , DP 2 )
•
* •
Proc . 5 . 1 .
s e t two variable parameters
•
( data on horizontal
vertical parameters are read
from f i l e
'4'
)
&
•
• • • I
•
source
SUB.
*
routine used
none
RMSERR
•
•
data used
none
•
•
returned d a t a :
N P 1 , N 1 , V S 1 , DP 1 , NP 2 , N2 , VS 2 , DP 2
•
•
It
N P 1 , NP 2
integer;
parameter numbers
•
•
N1,
r ea l ;
number o f d i v isions
•
It
V S 1 , VS2
r ea l ;
initial
•
v g 1 , V�2
real;
final values
N2 :
(---1
It
horizontal
param e t e r ,
'4'
•
values
---2 :
:
•
vertical
paramet e r )
•
I
format o f i n p u t data
I
N P 1 , N 1 , VS 1 , VB 1 :
FORMA1 ( 2I 5 , F 1 5 . 0 , F 1 0 . 0
•
•
NP 2 , N 2 , VS 2 , V E 2 :
FORMAT(
•
from
file
I
2I5 , F 1 5 . 0 , F 1 0 . 0
I 111111**••·······················1······································
SU8ROUTIN£
RDVAR1(N P 1 , N 1 , V 3 1 , 0P 1 ,
READ( 4 , 400 )
N P 1 , N l , VS 1 , vg 1
NP2 , N 2 , VS 2 , DP 2 ) +++++
•
read
two parametr
WRl'fE ( 6 , 600 ) N P 1 , N 1 , V S 1 , VE 1 ht::A D( 4 , 400 )
NP2 , N2 , V$ 2 , V E 2
wRlTd 6 , 6 1 0 ) N P2 , N 2 , VS 2 , V g 2 •
. OIL
I F ( N P 1 . LT . 1 l F ( NP2 . LT . 1 I F ( N 1 . LT . O N1
. OR . )
�1
GO TO 90
N P 2 . GT . 20
GO TO 90
)
THEN
: 0
� R I TE ( 6 , 62 0 )
�LSh
+++++ check parm values N P 1 . GT . 20 )
If(
�1
� l . Gl . 1 0 )
THEN
10
=
WRlT� ( 6 , 62 0 )
N1
eND I F
I F ( � 2 . LT . O J N2
THBN
0
=
WR1TE ( 6 , 63 0 )
N2
�LS� I F ( N 2 . GT . 1 0 N2
)
THEN
10
WRITE ( 6 , 63 0 ) =
N2
i:.ND I F
+++++ calc incremEnts IF(
N 1 . EQ . O DP1 = 0
TH�N
ELSt: DP1
=
( V E 1 -VS 1 ) /FLOAT ( N 1 )
eND I F
IF( N 2 . EQ . O DP2
=
THEN
0
ELSE DP2 END u·
=
(VE2-VS2) /FLOA T ( N 2 )
344
HEAT AND MASS TRANSFER IN PACKED BEDS R.E��T UHN print error message
li
90 WHITt:. ( 6 , 64 0 ) STOP F 0 R M A T • • • * * * * * * * * * * * * * • * * * * * ** * • • • • • • • • • • • • 400 FOriMAT( 2 I 5 , F 1 5 . 0 , F l O . O ) 600 FORMAT( l H O , 5X , ' HORIZONTAL PARAM. NO . ' , 1 2 , 5X , ' NO OF DIVI�ION $= ' , 1 3 , 3 X , ' ( ' , e 1 0 . 3 , ' TO ' , E 1 0 . 3 , ' ) ' ) PARAM. NO . ' , 1 2 , 5X , ' NO OF DIVIS ION 6 1 0 FOHMAT( 1 H O , 5X , ' VERTICAL $= ' , 1 3 , 3X , ' ( ' , E 1 0 . 3 , I TO ' , E 1 0 . 3 , 1 ) 1 ) 620 FOHMAT( 1 HO , ' NUMBt;R OF DlVlSlON ( HORIZONTAL PARAMETER ) IS REPLACED $ BY I I 15 I ) 630 FORMAT( 1 HO , ' NUMBER OF DlVlSION (V£RT1CAL PARAMETER) IS REPLACED $BY I I 15 I ) 640 FORMAT( 1 H O , ' PARAMETER NUMBER IS NOT ADEQUATE. ( SU B . RDVA R I ) ' I )
••••••••••*•••••••••
END
APPENDIX
B .2
B
345
Data Input
The measured input and response signals are stored in me '1 ' ; the half period and number of terms for Fourier expansion are in me '2'; the packed-bed parameters are in me '3 ' ; control parameters for construction of two-dimensional error map are in file '4'. The following illustration is based on the given data of Run 3 in Example 1 . 2 :
file '1 ': measured input and response signals
ADSORPTION GAS CHROMATOGRAPHY 0.0 0.3125 133.0 184.0 6.5 30.5 0.0 8 1 .5 282.5 273.5 257.0 238.0 207.0 186.0 40.0 73.5 45.0 8 3 .5 60.0 52.5 1 3.0 1 9 .5 14.5 1 7.5 22.0 1 6 .0 7.0 7.5 6.0 9.0 6.5 8.0 4.5 4.0 4.5 4.0 3.5 3.5 1 .5 2.0 2.0 1 .0 1.5 1 .0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.0 -1.0 1 .25 23.75 1 .7 0.6 0.8 1.1 1 .3 0.0 23.3 50.4 40.4 17.3 30.9 12.3 107.1 1 1 1 .1 100.1 1 14.1 1 15.2 1 14.2 87.3 50.5 80.4 64.4 72.4 56.9 1 9.1 1 3.7 22.6 16.2 1 1 .2 8.7 0.9 -1.0 2.4 1 .4 0.0 0.4
DATA RUN 3 (RE = 0.30) 225.5 158.5 34.5 12.0 5.5 3.0 1 .0 0.5
25 1 .0 138.5 3 1 .5 1 1 .0 5.5 3.0 1 .0 0.5
270.5 1 1 2.0 27.5 10.5 5.0 2.5 0.5 0.5
2 8 1 .0 99.0 24.0 9.5 5.0 2.5 0.5 0.5
2.2 6 1 .0 1 1 1.2 43.0 6.8
3.2 72.5 106.7 37.5 5.3
4.7 83.0 1 0 1 .3 32.0 4.3
8.3 93.0 94.3 27.1 3.3
1 .0
0.76E-4
file '2': parameters for Fourier expansion
50
70.0
file '3': packed-bed parameters
1860. 0.042 0.204 0.48E-4 0.63E-6 2.6
0.38 0
0.001 5.29
0.59
file '4': vertical and horizontal parameters for construction of error map
9 13
5 10
0.5E-4 0.4E-4 5.0 5.5
Appendix C. Derivations of Moment-equations in digital computer C.l
Derivations of Eqs.
( 1.63) and (1.64)
Using the relations given by Eqs. ( 1 . 18) and ( 1 . 1 9) , Eqs. ( 1 .63) and (1.64) are derived from the transfer function of Eq. (1.6 1 ) . A REDUCE 2t program for the derivations together with the obtained moment-equations are listed below.
C OMMENT
************************************************************* • •
•
MOMENT EQUATIONS FOR I N F I N I T E ADSORPT I O N BEDS
• • •
* * • • • •
* • • • •
• •
DAX DE EB EP I< A L R RHOP: s
SA SI
u
a x i a l d i spe r s i o n coe f f i c i en t i n t r a p a r t i c l e e f f ec t i v e d i f f us i v i t y bed v o i d f r a c t i o n intraparticle void fraction a d s o r p t i o n e qu i l i b r i um constant d i s tance between m e a s u r i n g p o i n t s pa r t i c 1 e r a d i u s pa r t i c l e d e n s i t y Laplace o p e r a t o r s u r f a c e a r e a p e r un i t v o l um e o f p a c k e d bed adsorption rate constant pa r t i c l e-to- f l u i d t r a n s f e r coe f f i c i e n t i n t e r st i t i a l f l o w r a t e
* • • • • • • •
* • • • •
••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••• ,
C OMMENT
************************************************************* * • LAMB ( SQ) : l og a r i thm o f t r a n s f e r f u n c t i o n * *
SQ = S +Q ( P ( S ) ) , P ( S ) = (PHI-A) * * 2 ,
Q(P) PHI-A
: :
e qn ( l . 6 l b ) e qn ( l . 6 l c )
• •
••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••• ,
PROC EDURE LAMB ( S Q ) ; ( L * U/ ( 2 * DAX ) ) * ( l - ( 1 +4 * DAX * SQ/U * * 2 ) * * ( l / 2 ) ) ; C OMMENT
* * * * * * * * * * * * * * ****************************************** ***** * EXP ( X , N ) : p o l y n o m i a l e x p r e s s i o n o f E* * X * * N : h i g he s t o r d e r •
,
••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••
PROC EDURE EXP ( X , N ) ; 1 + ( FOR I : = l : N S UM X * * I / ( FOR J : = l : I C OMM E NT
PRODUC T J ) ) ;
************* ************************************************ * * Q ( P , N ) : pol ynom i a l e x p r e s s i o n o f Q ( P ) * Q ( P ) : e qn ( l . 6 l b ) , P (PHI-A) * * 2 , N : highest order* ****************** * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * ** * * · =
I
t A . C . Hearn, UCP-19, Univ. of Utah, March 1973.
346
APPENDIX C
347
PROCEDURE Q ( P ,N) ; BEGIN INTEGER M ; M : = 2 *N $ LET X* * 2 = P ; COTH : = ( EX P ( X , M) + EXP ( - X , M) ) / ( E XP ( X , M+ l ) - E X P ( - X , M+ l ) ) $ RETURN ( SA * DE/ ( R* EB) ) / ( DE/ ( R * S K F ) + l / ( X * COTH- 1 ) ) ; END; COMM ENT
************************************************************* • P ( S ) : PHI-A s q u a r e d * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * · •
I
PROCEDURE P ( S) ; R* * 2 * { S/ D E ) * { E P+RHOP * S K A* KA/ { KA * S+SKA) ) ; COMM ENT
************************************************************* * • d e r i v a t io n of f i r s t moment
• * f i r s t moment : F l = - L l * { l +R l ) , Rl = Q l * P l * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * ** * * · I
Ll Ql Pl Rl
: = SUB { S Q= O � D F { LAMB { S Q ) , SQ ) ) $ : = S U B { Y = 0 1 DF { Q { Y , l ) , Y ) ) $ : = SUB { S=O , DF { P { S ) , S ) ) $ : = Ql *P l $
OFF E X P ; F l : = - L l * { l +R l ) $ ON D I V ; W R I T E " FIRST MOMENT = : OMMENT
" ,
Fl;
************************************************************* *
d e r i v a t i o n o f second second
•
*
central
c e n t r a l moment : F2 = L 2 * { l +R l ) * * 2 + L l * { Q 2 * P l * * /. + Q l * P /. )
moment
* •
*
••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••
I
ON E X P ; L 2 : = SUB { SQ=O , DF { LAMB { SQ ) , S Q , 2 ) ) $ Q 2 : = S UB { Y=O , DF { Q { Y , 2 ) , Y , 2 ) ) $ P 2 : = SUB ( S=O , DF ( P ( S ) , S , 2 ) ) $ OFF EXP;
F 2 : = L 2* ( l + R l ) * * 2 + L l * ( Q 2* P l * * 2 + Q l * P 2 ) $ WRITE " SECOND CENTRAL MOMENT = " , F 2 ;
END;
Computer output: FIRST MOMENT = I/3*EB( - l h L*U(- lh (3*EB + EP*R*SA + KA*R*RHOP*SA) SECOND CENTRAL MOMENT = 2/9*DAX*EB( - 2h L*u ( - 3h (3*EB + EP*R*SA + KA*R*RHOP*SA)2
- 1 /45*DE( - l h EB( - l h L*R( - 2h sKA( - 1 ) 3 *SKF( - lh u( - Ih (-30*DE*KA2 *R *RHOP
*SA*SKF
+
SKA*(-IO*DE*SA - 2*R*SA *SKF)*(EP*R2 + KA*R2 * RHOP)2 )
348
C.2
HEAT AND MASS TRANSFER IN PACKED BEDS
Derivation of First Moment of the System Discussed in Section
1.3
Laplace transformations of Eqs. (1.77) and (1.78) yield (also refer to Figure 1.24) (Cl ) where
0 < x < LD
L + LD < x < L
+
for Section i = 1
3L0
for Section i = 3
and (C2) where
Lo < x < L + Lo
[
x >L +
for Section i = 2
3 L o , and B4 = 0
for Section i = 4
Consider the following two-component vectors: C\ =
and
C1 ·
D' �i -
l
for i = 1 and 3
(C3)
for i = 2 and 4
(C4)
and L + 3L D
(CS)
The boundary conditions are then expressed as at x = L0, L
+
Lo
Also, from Eqs. (Cl)-(C4) the following equations are derived : (C6)
(C7) (C8) (C9)
APPENDIX C 349 and (C10)
where M(a, {3, -y)
=
-
1 {3 a
[
{3 - a exp [a
-
=
(C3):=L + 2 LD (Cth=o
(a
a{3-y(l - exp [a -{3])
Transfer function of the system between x F(s)
-� -
{3]
(
( 1 O ] M ;\0, A0 + ao,
=
-
{3 exp[a- {3])
]
0 and x = L + 2Lo is then
[ ]
) eb D1
![_ Lo
( 1 -exp [a -{3])
axA B L
exp ( ;\ 8 + 2;\o]
(C 1 1)
A REDECE 2 program for the derivation of fust moment according to Eq. (1.18) is listed together with the obtained moment-equation below. (Note that in the program the matrix elements are expressed as linear function of s. This is good enough for the derivation of fust moment.)
COMMENT
************************************************************* * • * MOMENT EQUAT ION FOR ADSORPTION/DEAD VOLUME SYS T EM * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * ,· •
C OMMENT
************************************************************* * M ( I , J , A L , BE , GAM) : m a t r i x ' s ( I , J ) -component expressed * * * as l i n e a r f u n c t i o n o f S * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,· PROC EDURE B EG I N ; I F I=l I F I=l I F I=2
M ( I , J , AL , B E ,GAM) ;
( BE-AL* E ** ( AL-BE) ) / ( BE-AL) $ : =- ( 1/GAM) * ( l - E * * ( AL-BE) ) / ( BE-AL) $ : = A L* B E* GAM* ( 1 -E * * ( AL-BE) ) / ( BE - A L ) $ * * THEN MX : = ( AL-BE I F I = 2 AND J=2 E * ( AL-BE) ) / ( BE-A L ) $ SUB ( S=O , M X ) + S U B ( S=O , DF ( M X , S ) ) * S ; RETURN END; AND J = l THEN AND J=2 THEN AND J=l THEN
MX MX MX
:=
350
HEAT AND MASS TRANSFER IN PACKED BEDS
COMMENT
*************************************************************
d e r i v a t i o n o f f i r s t moment a x i a l d i spe r s i o n c o e f f i c i e n t DAX d i s pe r s i o n c o e f f i c i e n t i n d e a d v o l um e s e c t i o n DD bed v o i d f r a c t i o n EB packed bed l e n g t h L ha l f l en g t h o f dead v o l ume sec t i on LD Laplace o pe r a t o r S U i n te r s t i t i a l f l u i d v e l o c i t y f l ui d v e l o c i t y i n d e a d v o l ume s e c t i o n UD
* * * * * * * * *
* * * * * * * * *
,
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * ·
C OMME NT
p a r a m e t e r s a s l i n e a r f un c t i o n o f s LAMB : eqn ( 1 . 8 0 d } , LAMD : e qn ( l . 8 0 e } S I G B : e qn ( l . RO f } , S I G D : e qn ( l . 8 0 g } AL = EB * DAX/DD, BE = EB * U * LD/DD Q l = l + D E L O , DELO : e qn { 1 . 6 3 a }
************************************************************* * * * * •
• * * • *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * · ,
UD = EB * U , EB * DAX = A L* D D , EB * U * LD = B E * D D ; LAMB : = - { L/ U } * Q l * S $ LAMD : = - { LD/UD) * S S S IGB : = U * L/DAX + 2 * ( L/ U } * Q 1 * S$ S IGD : = UD * LD/DD + 2 * { LD/UD) * S $ LET
C OMMENT
*************************************************************
ma t r i c e s and
*
vectors
i n e qn ( C 1 1 )
*
* ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * · ,
MATRIX M l ( 2 , 2 } , M 2 ( 2 , 2 ) , M 3 ( 2 , 2 } , V l ( 1 , 2 ) , V2 ( 2 , 1 ) ; V 1 : = MAT ( { 1 , 0) } $ V2 : = M A T ( ( l ) , ( EB * DAX * LAMB/ L ) } $ FOR 1 : = 1 : 2 DO B EG I N ; FOR J : = l : 2 D O B EG I N ; M l ( I , J ) : = M ( I , J , LAMD, LAMD+ S I G D , DD/LD) S M 2 ( I , J } : = M ( I , J , LA M B , LAMB+S I G B , EB* DAX/L ) $ M 3 ( I , J ) : = M ( I , J , 2 * LAMD , 2 * ( LAMD+S IGD) ,DD/ ( 2 * LD) ) $ END; END; � OMMENT
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * *
transfer
*
function
:
( FNUM/FDEN) * E * * FEXP
•
• • ••••••••••••••••••••••••••••••••••••••••• •••••••••••••••• • • • ,
FNUM FDEN FEXP C OMMENT
:= V l * M l *V 2 $ : = V l * M l * M 2 * M 3 *V 2 S : = LAMB+2 * LAMD$
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * *
d e r i vation o f
*
f i r st momen t ,
Fl
:
f i r st moment
*
••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••• • ,
Fl
: =
ON D I V ; WRI T E END;
- S UB ( S= O , DF ( FNUM , S ) } / S U B ( S = O , FNUM) + S UB ( S=O , DF ( FDEN , S ) ) / S UB ( S=O , FDEN) - S UB ( S= O , DF ( FEXP , S ) ) $
" FIRS T MOMENT =
•
Fl;
APPENDIX
C
351
Computer output: * FIRST MOMENT = £(( - BE* DAX - L U)/DAXh AL*DD*EB( - l hQ l *u(-2 ) E(( - 3 * BE * DAX - L *U) /D AX h AL *DD *EB( - l h Q l * u( -2 ) + 2 * BE*DD *EB( -2 h u(-2 ) - E(( - BE*DA X -L*U) /DAX) _
+
E(( - 3 * BE*DA X - L * U)/DA X) *DD*EB ( -2 h u( 2 ) + L* Q l *U( - 1 ) * DD*E B ( -2 h u( - 2 ) -
Author Index
Aditya, S. K. 146, 1 4 7 , 1 50 Ahn, Y . K. 7 8 Alberda, G. 203 Amundson, N. R. 78, 203 Anderssen, A. S. 1 , 7 Andrews, M. R . 1 48 Andrussow, L . 3 1 6 Antonson, C . R . 50 Appel, P. W. 1 47 , 1 5 0 Argo, W . B. 1 7 7 , 1 7 8 , 1 8 4 Aris, R . 1 0 5 , 1 0 8 , 203 Astheimer, H. J. 78 Babbit, J . D. 1 24 Baddour, R. F. 1 2 4 Bailie, R . C . 7 8 Baird, R . M. 199, 200, 2 0 1 Balakrishnan, A. R. 270, 2 8 3, 285 Balder, 1. R. 1 26 Bard, Y . 146, 147, 1 5 0 Bar-Ilan, M . 1 4 4 , 1 4 8 Barile, R. G. 237, 244, 245 , 25 3, 255 , 25 6, 257 , 260, 267, 268 , 275 , 277, 280, 287 Barret, R. P. 1 20 Baumeister, E. B. 266, 283, 285 Bazaire, K. E. 146, 147, 150, 1 5 1 Bennett, C . 0. 1 4 1 , 1 4 3 , 1 4 8 , 1 5 1 , 1 5 3 , 266, 283, 285 Bergevin, K. 78 Bhattacharyya, D. 270, 283, 285 Bird, R. B. 3 1 6 Bischoff, K . B . 78 Black, W. Z. 3 1 3 Boegli, 1 . S. 1 7 7 , 1 79 , 1 80 Bonilla, C. F. 1 4 5 , 1 4 7 , 1 5 0 , 1 5 1 Borner, C. 1 . 1 84 Bosanquet, C . H. 1 1 3 Bourne, 1. G . 3 1 5
Bradshaw, A. V. Bradshaw, R. D.
268, 286, 287, 292 140, 1 4 1 , 1 4 3 , 1 44, 1 4 8 , 1 5 1 , 1 5 3 , 268 , 273, 274 Bromley, L. A. 1 3 4 Brown, A . I . 3 1 5 Brown, C . F . 1 24 Bryson, A. W. 1 , 8 , 37. Bunnell, D. G. 1 70, 1 7 2 , 1 7 3 , 174 Butt, J. B . 1 22 , 1 2 3
Cahill, R . W. 3 2 1 Carberry, J. 1. 7 8 , 1 28 Carbonell, R. G. 79, 12 7 Chang, K . S. 7 8 Channakesavan, B . 1 20 Chapman, S. 189 Chen, 1. C. 1 84 Chihara, K. 5 0 Chiu, Y. T. 268, 286, 287, 292 Chu, 1. C. 1 4 1 , 1 43, 148 , 1 5 1 Churchill, S. W. 1 8 4 Clements, W . C. 1 Coberly, C. A. 1 6 2 , 1 6 8 , 1 7 5 , 1 98 Coppage, 1. E. 286 Cornish, A. R. H. 265 Cowling, T. G. 1 8 9 Cox, K. E. 12 0 Cunningham, R . S. 1 2 2 Cybulski, A. 270, 274, 275 , 276, 277, 2 80, 283
Damkohler, G. 9 8 , 184, 1 8 7 Danckwerts, P . V . 7 3 , 203, 247 De Acetis, 1 . 140, 143 , 1 5 1 , 266 , 273, 292 , 293
Deissler, R. G. 1 77 , 1 79 , 1 80 Desai, P. D. 3 1 2 Desjarlais, A. 0 . 3 1 5
353
354
HEAT AN D MASS TRANSFER IN PACKED BEDS
De Souza, J. F. C . 207, 209, 210, 2 1 1 , 230, 232, 237, 253, 264, 267, 270, 275 , 287, 288, 289, 290, 293 De Wasch, A. P. 168 Drake, R. C. 120 Dranoff, J. S. 50 Drew, T. B. 145 , 14 7 , 150, 1 5 1 Dryden, C . E. 145 , 147, 149, 150, 1 5 1 Dubow , J . 3 1 5 Dullien, F . A . L. 94, 1 1 2 Dunn, W. E. 145 , 147, 150, 1 5 1 Dwivedi, P . N . 157 Edwards, M. F. 86, 210, 2 1 1 Eichhorn, J . 266, 283, 285 Eisenklam, P. 7 El-Kaissy, M. M. 1 3 8 Epstein, N. 140, 1 4 3 , 1 5 1 , 266, 273, 292 Evans, E. V. 86 Evans, G. C . 1 4 5 , 150, 1 5 1 Evans, R . B . 1 1 1 , 120, 1 2 3 Fan, L . T. 78 Felix, T. R. 198 Ferstenberg, C. 1 4 5 , 147, 150, 15 1 Finlayson, B . A. 197, 198, 199, 201 Ford, H. 286 Foster, R . N. 1 2 2 , 1 2 3 Fraiman, Y . E. 189 Froment, G. F. 168 Fujita, S . 3 1 6 Fukuda, M . 189 Fulk, M. M. 189 Fuller, E. N. 321 Funazkri, T. 140, 293 Furnas, C. C. 286 Gaffney, B. J. 1 4 5 , 147, 150, 1 5 1 Galloway, L. R . 140, 1 4 3 , 1 5 1 , 266, 273, 292 Galloway, T. R. 265 Gamson, B. W. 1 38 , 140, 1 4 1 , 142, 1 5 1 , 1 5 3, 266, 273, 274 Gangwal, S. K. 1 , 8, 37, 144, 149 Geankoplis, C. J. 122, 146, 147, 150, 151 Gelbin, D. 1 Gerald, C. F. 145 , 150, 1 5 1 Giddings, J . C . 321 Gilliland, E. R. 124 Glaser, M. B. 266, 285
Glebova, L. I. 189 Godsave, E. W. 78 Goss, M. J. 268 , 28 7 , 288, 290, 292 Gross, B. 145, 147, 150, 1 5 1 Gunn, D . J. 78, 199, 207, 209 , 210, 2 1 1 , 230, 232, 237, 253, 264, 265, 267 , 270, 275 , 287, 288, 289, 290, 293 Halenda, P. P. 120 Haller, G. L . 124 Hamaker, H. C. 184 Hamielec, A. E. 138 Handley, D. 2 6 1 , 262, 268, 286, 292 Happel, J. 1 38 , 265 Hashimoto, K . 198 Hashimoto, N. 50 Hatta, S. 162, 1 7 5 , 176, 1 9 8 Havill, T . N. 3 12 Hawley, R. W. 286 Hayashi, M . 126 Heggs, P. J . 26 1 , 262, 268, 286, 292 Hengst, G. 188 Henry, J. P 120 Higashi, K. 1 24 Hirschfelder, J. 0 . 3 1 6 Ho, C . Y . 3 1 2 Hobson, M . 1 4 1 , 142, 145 , 150, 1 5 1 Hofmann, H . 78 Homsy, G. M. 1 3 8 Hoogschagen, J. 120 Hopkins, M. J. 7 Hottel, H . C. 188 Hougen, O. A. 1 38 , 140, 1 4 1 , 142, 1 5 1 , 1 5 3 , 266 , 27 3 , 274, 292 Hudgins, R . R. 1 , 8 , 37, 144, 149 Hurt, D. M. 1 38 , 140, 1 4 1 , 142, 148, 1 5 1 , 1 5 3, 266 , 27 3 , 274 .
Iida, Y. 86 Irvin, H . B . 170, 172, 1 7 3 , 174 Ishii, T. 1 38 Ishino, T. 145, 150, 15 1 Ito, H . 124 Jackson, K . W. 3 1 3 Jackson, R . 94 Jeffreson, C. P. 267, 269, 285 Johnson, A. 268 , 286, 287, 292 Johnson, M. F. L. 1 2 3 Joyner, L . G. 120 JUttner, F. 97
AUTHOR INDEX
Kaganer, M. G 189 Kaguei, S. 37, 5 7 , 6 0 , 79 , 9 2 , 1 1 5 , .
2 1 4 , 223, 2 32 , 2 37, 240, 256, 265, 269, 270, 27 1 , 275 , 29 1 , 293, 294 Kalil, J. 1 4 1 , 1 4 3, 1 48 , 1 5 1 Kallenbach, R . 1 1 4 Kaneda, T. 1 9 9 Kasaoka, S . 1 06 , 1 0 7 , 146, 1 5 0 , 1 5 1 Kataoka, H . 144, 1 4 9 , 1 5 1 Kato, K . 1 7 9 , 1 8 1 , 1 84, 1 87, 1 9 1 Kawazoe, K . 5 0 Kenney, C . N . 8 6 Khalid, M. 1 9 9 Kikuchi, T. 144, 1 49 , 1 5 1 Kimura, M . 1 9 9 Kling, G. 1 89 , 1 9 6 Knudsen, M . 1 0 8 , 196 Komarnicky, W. 140, 143, 1 5 1 , 266, 27 3 , 292 Kondo, Y. 126 Kramers, H. 20 3 Krischer, 0. 1 7 7 , 1 7 9 Kroll, K . 1 7 7 , 1 79 Krupiczka, R. 1 7 7 , 1 7 9 , 1 8 1 Kubin, M . 35 , 37 Kucera, E. 3 5 , 37 Kumar, S. 147, 150, 1 5 1 Kunii, D. 1 76 , 1 7 7 , 1 7 8 , 1 8 4, 1 9 8 , 202, 203, 265, 266, 274, 275
Leavitt, F. W. 1 9 8 L e C lair, B . P . 1 38 Lee, D. I. 9 2 Lee, L. K. 5 0 Lee, T. Y. 3 1 2 Levenspiel, 0. 78 Li, C. H. 1 97 , 1 9 8 , 1 9 9 , 2 0 1 Lightfoot, E . N. 3 1 6 Lindauer, G . C . 2 6 8 , 287 Littman, H. 2 37, 244, 245 , 2 53, 255 , 256, 257, 260 , 267 , 268 , 2 75, 2 7 7 , 280, 287 Liu, S. L. 7 8 LOf, G. 0. G . 286 London, A. L. 286 Luikov, A. V. 189
Maeda, S. 1 6 2 , 1 7 5 , 1 7 6 , 1 9 8 Mailing, G. F . 1 4 1 , 1 4 3 , 1 5 1 , 1 5 3 , 268 , 273, 292, 293 315
Marco, S. M.
Marshall, W. R.
3 55
1 5 4 , 1 6 2 , 168, 17 5 ,
198
Martin, H. 265 Masamune, S. 1 89 Mason, E . A. 1 1 1 , 1 2 3 Mathur, V . K. 14 7 , 1 5 0, 1 5 1 Matsumoto, K. 5 7 , 60, 1 15 , 1 4 7 , 1 5 1 McConnachie, J . T. L . 1 4 1 , 143, 144, 1 5 1 , 266, 273, 292, 293
McCune, L. K. 145, 150, 1 5 1 McHenry, K . W. 203 McLachlan, N. H . 268 , 2 86 , 287 , 292 Mears, D. E. 7 8 Metzner, A . B . 1 24 Michelsen, M . L. 1 , 5, 7 Miyauchi, T. 144, 1 47 , 149, 1 5 1 , 265 Mueller, C. R. 3 2 1 Myers, J . E. 140, 1 4 3 , 1 4 4 , 1 5 1 , 1 5 3 , 268 , 273, 274
Nagai, H.
3 7 , 7 9, 144, 149, 265, 2 7 1 ,
275
Nagata, S. 198 Nakanishi, K. 1 26 Narayanan, P. V . 2 1 1 , 232, 237 Nelson, P . A. 265 Newman, J. 147, 1 5 0 Nishimura, Y , 1 38 Nitta, K. 146, 150, 1 5 1 Nono, K . 1 99 Nottenburg, R. 3 1 5 .
Oishi, J . 124 Okada, T. 1 45 , 1 5 0 , 1 5 1 Okazaki, M . 126 Olson, R. W. 170, 1 7 2 , 1 7 3 , 174 Omura, S . 189 Ono, N. 198 Oshima, T . 146, 1 5 0 , 1 5 1 Q>stergaard, K. 1 , 5 , 7 Otake, T. 1 45 , 1 5 0 , 1 5 1 Otten, L . 2 5 3 , 270, 287, 288 , 292 Papa, J. 124 Pasternak, A. D. 146, 147 , 1 5 0 Paul, R. 3 2 1 Pei, D . C . T. 270, 283, 285 Perkinson, G. P. 124 Petersen, E. E. 94, 126 Petrovic, L. J . 1 4 1 , 1 4 3 , 1 5 1 , 15 3 , 154, 15 5
Pfeffer, R.
1 38 , 265
356
HEAT AN D MASS TRANSFER IN PACKE D BEDS
Phillips, B. D. 1 9 8 Pillai, K. K. 200 Pollard, W. G. 1 1 3 Ponzi, M . 124 Present, R. D. 1 1 3 Pryce, C. 210, 2 1 1 Pulsifer, A. H . 237, 244, 245, 255, 256, 257, 260, 267, 268, 275, 277, 280 , 287 Radeke, K. H . 1 Rajeshwar, K. 3 1 5 Ranz, W . E. 154, 175 Resnick, H . 1 39 , 143, 148, 1 5 1 , 1 5 3 , 154, 266, 273, 292, 293 Resnick , W. 1 4 1 , 142, 144, 148 Richardson, J. F . 86, 210, 2 1 1 Ritter, H. L. 120 Rivarola, J . B . P. 124 Roblee, L. H . S. 199, 200, 201 Rosenvold, R. 3 1 5 Rothfeld, L. B . 1 12 , 125 Ruckenstein, E. 50 Russel, J. L. 124 Ruthven, D. M. 50 Ryan, D. 127 Sakata, Y. 106, 107 Saunders, 0 . A. 286 Satterfield, C. N. 94, 139, 143, 148, 1 5 1 , 1 5 3, 154, 266, 273, 292, 293 Schaefer, R. J. 277, 280 Schettler, P . D. 321 Schliinder, E. U . 177, 178, 179, 265 Schmeal, W. R . 78 Schneider, P . 35, 36, 37, 50 Schotte, W. 177 , 1 7 8 , 184, 187 Schumann, T. E. W. 1 77, 178, 244, 2 6 1 , 286 Schwartz, C. E. 199, 200 Scott, D. S. 1 1 2, 120 Selke, W. A. 146, 147, 150 Sen Gupta, A. 1 4 1 , 143, 1 5 1 , 268, 273, 292, 293 Shashkov, A. G. 189 Shen, J . 214, 223, 237, 240, 270, 291 , 293, 294 Sheppard, A. J . 7 Shiozaki, Y . 126 Shiozawa, B. 232, 256, 265, 269, 270, 2 9 1 , 293
Silveston, P. L. 1 , 8, 37, 144, 149 Sladek, K. L. 124 Sliva, D. E. 287 Smith, J. M. 35, 36, 37, 50, 86, 94, 1 1 9, 120, 1 2 1 , 170, 172, 173, 1 74, 177' 178, 184, 189' 199' 200, 266, 274, 275 Smith, R . K. 124 Smoluchowsky, M. 188 S¢rensen, J . P . 265 Spotz, E. L. 3 1 6 Srivastava, I. B . 3 2 1 Stewart, W. E. 1 2 3 , 265, 3 1 6 Strang, D . A. 145, 147, 149, 150, 1 5 1 Suzuki, M . 50, 86, 198, 265 Suzuki, N . 198 Suzuki, T. 108 Swift, D. L. 189, 194 Takeuchi, Y. 50 Tanaka, K. 3 7 , 144, 149 Tan�ho , S. 86, 144, 149, 232, 270, 2 9 1 , 293 Teramoto, M. 198 Thakur, S. C. 124 Thiele, E. W. 97 Thodos, G. 1 3 8 , 140, 1 4 1 , 142, 143, 144, 145, 150, 1 5 1 , 153, 154, 155, 266, 268, 273, 274, 285 , 292, 293 Tierney, J. W. 199, 200, 201 Toei, R. 126 Tripathi, G. 146, 147, 150, 1 5 1 Truitt, J . 120 Trumpler, P. R . 196 Turner, G . A. 253, 268, 270, 287, 288, 290, 292 Tye, R. P. 3 1 5 Uchida, T. 108 Upadhyay, S. N . 157
146, 147, 150, 1 5 1 ,
Vaidyanathan, A. S. 50 Van Dalen, M. J. 270, 274, 275, 276, 277, 280, 283 Van Den Berg, P. J. 270, 274, 275, 276, 277, 280, 283 van der Laan, E. T. 78 Vasiliev, L. L . 189 Verkerk, J. W. 270, 274, 275, 276, 277, 280, 283
AUTHOR IN DEX
Vortmeyer, D. 179, 184, 187, 19 1 , 192, 194, 197, 230, 277, 280 Voss, V. 177, 178 Wachmann, J. H. 196 Wakao, N. 37, 57, 60, 79, 86, 92, 1 15 , 1 19 , 120, 1 2 1 , 1 40, 144, 146, 149, 150, 1 5 1 , 179, 1 8 1 , 184, 187, 1 89 , 1 9 1 , 192, 194, 197, 198, 202, 203, 214, 223, 230, 2 3 1 , 232, 237, 240, 250, 256 , 265 , 269, 270, 27 1 , 275 , 29 1 , 293, 294 Wardle, A. P. 2 1 1, 232, 237 Watson, G. M. 1 1 1 , 120, 123 Weaver, J. A. 124 Wehner, J. F. 74, 76, 78, 152 Weisz, P. B . 1 14 Wen, C . Y. 78 Wendel, M. M. 78 Wetteroth, W. A. 1 4 1 , 143, 148, 1 5 1 Wheeler, A. 94, 108, 1 1 3, 1 1 8 Whitaker, S. 127 White, E. T. 1 , 7 White, R. R. 1 4 1 , 142, 148, 266, 283, 285
357
Wicke, E . 1 14 Wiedmann, M. L. 196 Wilhelm, R. H . 74, 76, 78, 145 , 150, 1 5 1 , 152, 203 Wilke, C. R. 134, 140, 1 4 1 , 142, 15 1 , 266, 273, 292 Wilkins, G. S. 1 4 1 , 144, 1 5 1 , 153 Williamson, J . E. 146, 147, 150, 151 Wilson, E. J. 146, 150, 151 Withrow, A . E . 145, 147, 149, 150, 151 Wolff, H . J . 1 Wu, K. Y. 3 1 2
Yagi, S. 146, 150, 1 5 1 , 1 7 6 , 1 7 7 , 178, 1 84 , 198, 202, 203 Yoon, C . Y. 198 Yoshida, T. 147, 1 5 1 Youngquist, G. R. 50, 94
Zehner, P. 177, 178, 179 Zeldowitsch, J . B. 98 Zgrablich , G. 124
Subject Index
Accommodation coefficients 189, 194-196 Activation energies 103 based on intrinsic chemical reaction rate constants 1 0 3 based o n overall rate constants 1 0 3 effect o f effectiveness factor 103105 Adsorption chromatographies 3 1 -50, 149 adsorption equilibrium constants 3 3 , 5 1-54, 6 1-62 adsorption rate constants 3 3 , 3 7 , 49, 54 effect of dead volume 61-62 fundamental equations 32-33, 52-53 parameter estimations 35-50 curve-fitting in time domain 3 7 , 38 moment method 35-36, 38 Analogy between heat and mass transfer 176, 203
Bidisperse pore structure 50, 120 effective diffusivity 5 1 -54, 1 20122 Bulk diffusion (normal diffusion) 1 10 Catalyst effectiveness factors 97-108, 1 36 Catalytic decomposition of hydrogen peroxide 148 Center-symmetry (See also Concentra tion/temperature proftles in particle) 3 1 , 55-56, 95-97, 248-250
Chemical reaction rate 9 8 , 1 0 1 , 102, 1 35-136 Computer programs 323-351 Concentration profiles in particle in adsorption system 5 5 -56 in reaction system 8 3-84, 95-99 Contact area, effect on effective thermal conductivity 193-196 Continuous Solid-Phase (C-S) model assumptions 245 boundary conditions 246, 280 comparison with D-C model 253261 fundamental equations 245, 277 relation with Single Phase model in steady-state heat transfer 2302 3 1 , 275-283 transfer functions 246 Convergent-divergent pore array model 122-123 Conversion factor of units 297-300 Conversion in chemical reactor 7 8 Correction factor, radiation 192 Curve fitting 9-15, 29-3 1 , 37 , 2 14, 25 1-252, 257, 261-262, 278279, 288 criterion for fitting 1 5 , 251-252, 262 Danckwerts boundary conditions 7 3 , 247 discussion 74-76 steady-state heat transfer 274 steady-state mass transfer 7 3 , 152 Dead volume effect, associated with detecting elements in parameter estimation 57-62 adsorption chromatographies 61-62 tracer dispersion in inert beds 57-61
359
360
HEAT AN D MASS TRANSFER IN PACKED BEDS
Densities of gases 301-303 of liquids 301-303, 307 of plastics 3 1 0 Diffusion in a capillary tube combined Knudsen and normal l l l- 1 1 2 in binary gas system 1 10 in multicomponent system 125 Knudsen 108-110 normal (bulk) 1 1 0 self-diffusion 1 12-114 Diffusion in external film 126 Diffusion in porous solid diffusibility 1 1 8-1 1 9 diffusion flux 1 1 8-120 measurements by adsorption chromatography 3 8 , 47-48 measurements in Wicke-Kallenbach type apparatus 1 14-1 1 7 , 124 parallel pore model 1 18 surface diffusion (surface migration) 124 tortuosity factor 1 1 8 with bidisperse pore structure 1 20123 with monodisperse pores 120 Diffusion coefficients in binary gas systems 316-321 data 3 17-321 prediction 3 1 6 of gases in water 322 Diffusion models for bidisperse pore structure 50-54, 120-123 Diffusivities, effective in bidisperse pore structure 5 1 -54 in porous solid of adsorbing species 3 3 , 46-48 of reacting species 8 3-85 in quiescent beds 79-85 Dispersed plug flow 1 , 1 3 8 Dispersion, effect o n conversion 72-78 Dispersion coefficients adsorption system 33, 44-48, 8792 inert system 2, 8 7 reaction system 79-86, 1 39 Dispersion-Concentric (D-C) model for adsorption chromatographies 3 1 , 149 fundamental equations 32 intraparticle concentration
profiles 55-56 transfer functions 33-35 for heat transfer 206 fundamental equations 206, 245 intraparticle temperature profiles 248-250 transfer functions 212, 247 for reaction system intraparticle concentration profiles 95-99 D-C model, modified for heat transfer in packed beds comparison with C-S model 253-261 comparison with original D-C model 251-253 comparison with Schumann model 261-262 D-C model, original 245, 251-25 3 Dispersion number (See Mass/thermal dispersion number)
Effective thermal conductivities 162 ax�l 1 62 , 1 7 3 , 202-204, 2 3 1 , 282 correlation of 204 rad�l 162, 167, 170-1 77, 232 correlation of 177 Effective thermal conductivities of quiescent beds 177-183 at low pressures 188-197 combined conduction and radiation contribution 182-183, 192 conduction contribution 192, 196 estimation of 178-179, 1 8 1 , 1 8 2 radiation contribution 1 8 3 , 192 solid-solid contact 192-194 transient effective thermal conductivities 232-237 definition of 235 Effectiveness factors activation energy, relationship with 103-105 definition 100 evaluation, example 1 35-136 for finite length cylinder 106 for flat plate 104-105 for infinitely long cylinder 105 for ring 106-107 for sphere with catalyst coated on inert solid sphere 107
SUBJECT INDEX Effectiveness factors continued for spherical catalyst pellet 100-
104, 128 in chemical reaction controlling region 100 in pore diffusion controlling region
101
Juttner modulus, relationship with
99, 1 28
Equilibrium constants of adsorption 3 3 , 38, 54, 6 1 o f chemical reaction 127 Error, root-mean-square- 1 5 , 29, 170,
214, 2 5 1 , 257: 260, 2 6 1 , 288 Error map 30, 46-49, 62, 1 7 1-173, 220-2 2 1 , 228-229 , 253, 289-291 Evaporation of water into air 140-1 4 1 , 2 7 3-274 Fourier analysis of one shot response
8-9 , 26-29 amplitude, amplitude ratio 26-27 phase, phase shift 26-28 sensitivity test 27-29 Frequency responses (See also Thermal responses, Input/response signals) 208-2 1 0 , 248 , 25 1 -260,
287-291 Gas constant
296
Heat conduction (See also Heat flow) gas, at low pressures 189-191 grid network , relaxation method
177, 182, 1 9 1 , 235-236 Heat flow, unidirectional 177, 178-179 Heat flux, radiant 183-187 between two large gray surfaces
183 in packed beds 184-187 Heat transfer, unsteady-state (See Thermal responses or Response signals) Heat transfer coefficients (See Particle to-fluid heat transfer coefficients) Impulse response (See Response signals) Infinite packed bed adsorption packed beds 67-71 thermal dispersion in inert beds
237-242 tracer dispersion in inert beds
62-66
361
Input signals data for N2-H1 in adsorption system
39-43 data for N2-H1 in inert system 1 6 delta input 2 Fourier coefficients of measured signals 1 2 frequency 208 , 248, 287 normalized 17 one-shot input, heat 2 1 0-214, 2 9 1 one-shot input, tracer 4, 9 , 1 2 step input 248 J Hea t factor 293 JMass factor 1 5 5 , 293 JUttner modulus 83, 97-99 definition 99, 1 28 for first order irreversible reactio n
83-85, 99 for fust order reversible reaction
127-128 relationship with effectiveness factor 100 Knudsen diffusion (See Diffusion in a capillary tube) Laplace transform convolution integral 9-10, 1 5 , 2 1 2 inversion integral 1 2 inversion o f transfer function b y Fourier series 10-1 1 , 2 12-21 3 Lateral fluid mixing 175-176 Limiting Nusselt number (See Nusselt number) · Limiting Sherwood number (See Sherwood number) Log-mean area 1 15 -1 16 Mass dispersion number 2 , 65-69, 76 Mass transfer coefficients (See Particleto-fluid mass transfer coefficients) Mean free paths 1 1 3 , 189-190 Mean pore radius 1 1 8 Mean residence time 2 Mixed mean temperature 164, 168 Moments 2-5 , 1 7 -1 8 , 35-36, 45 , 64,
67, 222-223
central moments 4, 36, 49-50 definition 3-4 of delta input, impulse response 2-4 of one-shot input 4-5 variance 4, 3 6 , 49, 222
362
HEAT AND MASS TRANSFER I N PACKED BEDS
Monodisperse pore structure
120
Naphthalene, vapor pressures 148 Natural convtction 149 Normal diffusion 1 10 Nusselt number, particle-to-fluid (See also Particle-to-fluid heat transfer coefficients) anomalous decrease of 25 3, 255, 2 6 1 , 264-265 , 284 correlation of 293 determination by thermal response measurements 2 1 9-222, 253, 257-262, 286-291 effect of fluid thermal dispersion coefficient 2 1 9-222, 257, 289, 290-291 limiting value of 265, 293-294 of C-S model 260 of modified D-C model 289-290, 294 of original D-C model 255, 287 of Schumann model 263 Nusselt number, radiant 1 8 3 Overall heat transfer coefficients 164, 201 correlation 201 relationship with radial effective thermal conductivity 165 Overall rate constants (See Rate constants) Packed bed models cubic lattice of spheres 1 7 7 , 180 rhombohedral packing of spheres 182 Parallel pore model 1 18 Parameter estimations 1-71 from steady-state heat transfer determination of effective thermal conductivities axial 202 radial 167-173 determination of wall heat transfer coefficients 167-173 determination of Nusselt numbers 273-285 from tracer response measurements, by one-shot techniques 4, 9 curve fitting in time domain 9-15, 29-3 1 , 37
Fourier analysis 8-9, 26-29 moment method 4-5, 17-1 8, 36, 38 transfer function fitting 7 , 24-26 weighted moment method 5-7 18-24 from thermal response measurements frequency techniques 209-2 10, 25 1-260, 287-290 one-shot techniques 2 1 0-214, 29 1 step response techniques 2 6 1 262, 286-287 Particle-to-fluid heat transfer coefficients anomalous decrease of 25 3, 255, 26 1 , 264-265 , 284 based on C-S model 260, 275 based on D-C model 2 2 1 , 228230, 287, 290 based on Schumann model 263 ' 286 correlation of 293 limiting value of 265, 293-294 Particle-to-fluid mass transfer coefficients (See also Sherwood number) 49 correlation of 154-157 limiting value of 1 5 8 re-evaluation o f 15 1-158 Peclet number 86, 176, 203 Physical adsorption 33, 37 Physical constants 296 Physical properties (in table) of gases 301-303, 317-321 of inorganic compounds 301-302, 3 1 7-318 oforganic compounds 303, 3 1 8-321 '
Radial symmetry 3 1 , 55-56, 95-97, 248-250 Radial variations in fluid velocity (See Velocity profile) Radial voidage variations (See Voidage) Radiation conductivity of bed of fine particles a t low pressures 192 correction factor 192-193 effect of gas radiation 187-188 heat transfer coefficients 183-187 Random pore model 1 20-122 Rate constants, overall (See also Reaction rate) 73, 101-104
S U BJECT IN DEX
Rate constants, overall - continued in catalyst packed beds 7 3 , 1 0 1 104 in chemical reaction controlling region 102 in extrapellet diffusion controlling region 103 in intrapellet diffusion-reaction controlling region 102-103 in pore diffusion controlling region 103 Reaction rate effect of column size 102 overall reaction rate throughout a pellet 8 2 , 96-97, 99-100 Reactors continuous stirred tank reactor 77, 7 8 plug flow reactor 7 7 , 78 Response signals data for N2-H2 in activated carbon system 39-43 data for N2-H2 in inert system 1 6 Fourier coefficients of 14 frequency response 208-210, 248, 252, 287 impulse response 10 normalized 1 8 one-shot response 2 1 4 , 223, 291 prediction by convolution integral 9-12, 2 1 1 -2 1 3 prediction by Fourier series 12-15 , 21 3-214 step response 248, 262, 286 Root-mean-square-error (See Error) Schumann model assumptions of 245 comparison with D-C model 261262 fundamental equations 245 transfer function 244 Sherwood number, particle-to-fluid 49 correlation of 156-158 limiting value of 1 5 8 o f single sphere 154 Single Phase model 230, 276 SI units 296-300 Specific heats of gases 305 of liquids 308 of plastics 3 1 1
363
Standard volume of ideal gas 296 Stefan-Boltzmann constant 1 83, 296 Step responses (See Response signals) Sublimation 1 4 1 , 148 Surface diffusion (See Diffusion in porous solid) Techniques of parameter estimation (See Parameter estimations) Temperature profiles, steady-state in packed beds along central axis 164, 169, 1 7 1 deep in a bed 1 6 3 determination of heat transfer coefficients from axial temperature profiles 167-170 from radial temperature profiles 168, 170 for bed inlet temperature being a function of radial distance 165, 166 radial profiles 174, 276-279 under consideration of axial heat conduction 165-167 Temperature profiles in particle assumption in D-C model 245, 248-250 average (volume-mean) 285 Thermal conductivities of gases 306 of liquids 309 of plastics 3 1 1 , 3 1 3 of solids 3 J 5 Thermal diffusivities of plastics 3 1 2 Thermal dispersion coefficients, axial fluid 207, 2 1 0-21 1 , 2 19-220, 230-232 determination by frequency response 209-21 0 determination by one-shot response 2 1 4-222 effect on Nusselt numbers 229230, 257, 289-291 radial 232 Thermal dispersion number 240 Thermal responses (See also Response signals) criterion for infinite packed bed 2 37-242 for finite packed beds 244, 246247
364
HEAT AN D MASS TRANSFER
IN
Thiele modulus (See Jiittner modulus) Tortuosity factor 1 18 Transfer functio ns 2 according to C-S model 246 according to D-C model 2 1 2, 247 according to Schumann model 244 for adsorption chromatography 33-35 bed length effect 6 7 dead volume effect 6 1 for tracer dispersion in inert beds 2 bed length effect 64 dead volume effect 59-60 relationship with weighted moments 6 Transfer function fitting 7, 24-26 Two hemispheres in contact 184-186 Unidirectional heat flow model 178-179
177,
PACKED BEDS
Vapor pressures, of naphthalene 148 Velocity profile, radial variations 199-200 View factors 184-186 Viscosities of gases 304 of liquids 308 Voidage, radial variations 199-201 Wall heat transfer coefficients 162, 197-199 correlation of 197-199 determination of 167-173 Weighted moments 5-7, 18-24 central moments 6, 22 Weighting factors 1 Wet-bulb temperature 140-14 1 Wicke-Kallenbach type apparatus 1 14-1 1 5
