Multicomponent Mixtures, Mass Transfer, concentration gradient, pressure gradients, multicomponent mass transfer theory, Chapter, driving forces, flux, binary mixtures, VSSD Leeghwaterstraat, Separation Processes, Difference equation, Delft University, J.A. Wesselingh R. Krishna, driving force, Diffusivities, Ion Exchange Equilibria, Exercises, Ion Exchange, Transport equations, Further Reading, Equipment Engineering Department of Shell Research, James Clerk Maxwell, text, friction coefficients, viscous flow, electrical fields, chapters, friction coefficient, ideal mixture, difference equations, Josef Stefan, multicomponent mass transfer, Shell Research, binary mixture, carbon dioxide, velocities, positive direction, Hans Wesselingh Groningen
Content:
Mass Transfer in Multicomponent Mixtures J.A. Wesselingh R. Krishna
Mass Transfer in Multicomponent Mixtures
Mass Transfer in Multicomponent Mixtures J.A. Wesselingh R. Krishna VSSD
© VSSD First edition 20002006 Published by: VSSD Leeghwaterstraat 42, 2628 CA Delft, The Netherlands tel. +31 15 2782124, telefax +31 15 2787585, email:
[email protected] internet: http://www.vssd.nl/hlf URL on this book and for downloading auxiliary files: http://www.vssd.nl/hlf/d004.htm All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the
prior written permission of the publisher. NUR 952 Printed version ISBN: 9789071301582 Ebook: ISBN: 9789065622242 Keywords: mass transfer, multicomponent mixtures
5 Foreword It was twenty years ago. A little before that, I had left the Equipment Engineering Department of Shell Research in Amsterdam for a less hectic job at Delft University. At least, so I thought at that moment. In my former section at Shell we had worked on catalytic crackers, on polymerisation reactors, on cleaning of oil tankers and other exciting developments, but I had found managing this a bit too much. There I was, with a lower salary, teaching separation processes to second year students, and running the undergraduate laboratory with one hundredth of my Shell budget. I had written a little book on Separation Processes, and sent it to friends in Amsterdam. One of the pieces of equipment that we had in Shell Research was (what was then) the largest distillation test column in the world. It was two and a half metres in diameter and some twenty metres high. The column was so big that we could only run it in the summer: the reboiler used the complete capacity of our boiler house. The operating pressure could be varied between vacuum and fifteen atmospheres. We had a beautiful time trying out all kinds of trays and packings. In time we started to get interested in trying to understand not only distillation of binary mixtures, but also of mixtures with more components. We started to gather measurements and to try to understand them. However, much of what we saw was baffling, to say the least. Only gradually did we realise that our binary mass transfer tools were not adequate; that we needed to try something different. That something was a young graduate from Manchester who had picked up wild ideas on mass transfer doing his PhD. His name was Krishna. I left Shell just after he arrived. One day, Krishna came along at home to visit me. He had read my book and told me politely that my approach to mass transfer was not all that good. I was a little vexed because I was professor, he was not, and besides, I had copied my ideas from wellknown handbooks. Even so, I did try to listen, and three weeks later went back to him for more explanation. It was all about multicomponent mass transfer, it had connections with thermodynamics and was quite different from anything I knew.. I had difficulties in following what he was telling me, and I can remember: `Hans, if you really want to understand this, we should give a course together.' That is where this book started. We are now twenty years, fifteen
PhD students, twentythree courses and some thousand participants further. The course has evolved and so has the book. It now has examples from membrane technology, reaction engineering, sorption processes, biotechnology; from mixtures of gases or liquids, but also porous media and
polymers. The basics have not changed: they are still almost the same as presented by James Clerk Maxwell in 1866 and (more clearly) by Josef Stefan in 1872. (Maxwell is the one of the theory of electrical and magnetic fields, and Stefan the one associated with the name of Boltzmann.) I am a little ashamed when I look in editions of the Encyclopaedia Britannica from around 1900 and see how well diffusion was then understood. It feels as if it has taken me more than a century to pick up the brilliant but simple ideas of these two longdead scientists. Krish and I hope this book will help you to do that a bit more quickly. Hans Wesselingh Groningen, May 2000
7 Contents
FOREWORD
5
1 BEGINNING...
13
1.1 Who should read this Book?
13
1.2 What this Book covers
14
1.3 Structure of the Book
15
1.4 Guidelines to the Reader
16
1.5 Guidelines to the Teacher
17
1.6 Symbols
18
1.7 Conventions
21
2 IS SOMETHING WRONG?
24
2.1 The Starting Point
24
2.2 Three Gases
26
2.3 Two Cations
28
2.4 Two Gases and a Porous Plug
28
2.5 Summary
30
2.6
Further reading30
2.7 Exercises
30
PART 1. MASS TRANSFER IN GASES AND LIQUIDS
33
3
driving forces34
3.1 Potentials, Forces and Momentum
34
3.2 Momentum (Force) Balance of a Species
36
3.3 The Driving Force: a Potential Gradient
38
3.4 The MaxwellStefan Equation
39
3.5 Simplifying the Mathematics
40
3.6 The Film Model
41
3.7 Difference Form of the Driving Force
42
3.8 Summary
44
3.9 Further Reading
44
3.10 Exercises
45
4 FRICTION
46
4.1 Friction Coefficients and Diffusivities
46
4.2 Velocities and the Bootstrap
48
4.3 Velocities and Fluxes
50
4.4 The Difference Equation
50
4.5
mass transfer coefficients
52
4.6 Summary
53
4.7 Further Reading
53
4.8 Exercises
53
5 BINARY EXAMPLES
55
5.1 Stripping
55
5.2 Polarisation
56
8
Mass Transfer in Multicomponent Mixtures
5.3 Vaporisation
56
5.4 Gasification of a Carbon Particle
58
5.5 Binary Distillation
59
5.6 Summary
60
5.7 Further Reading
60
5.8 Exercises
61
6 TERNARY EXAMPLES
63
6.1 From Binary to Ternary
63
6.2 A Condenser
64
6.3 A Ternary Distillation
65
6.4 A Ternary Reaction
68
6.5 Binary Approximation of a Ternary
68
6.6 Summary
70
6.7 Further Reading
70
6.8 Exercises
71
7 MASS AND
heat transfer74
7.1
temperature gradients74
7.2 Enthalpy
75
7.3 Mass Transfer Relation
76
7.4
energy transfer Relation
77
7.5 Condensation in Presence of an Inert Gas
79
7.6 Heterogeneous Reacting Systems
80
7.7 An Ammonia Absorber
80
7.8 Summary
81
7.9 Further Reading
81
7.10 Exercises
82
8 NONIDEALITIES
84
8.1 Chemical Potential and Activity
84
8.2 Nonideal Binary Distillation
84
8.3 A Simple Model of Nonidealities
85
8.4 Large NonIdealities: Demixing
87
8.5 MaxwellStefan versus Fick
88
8.6 When can we neglect Nonideality?
89
8.7 Mass Transfer in LiquidLiquid Extraction
90
8.8 Summary
92
8.9 Further Reading
92
8.10 Exercises
92
9
diffusion coefficientS
95
9.1 Diffusivities in Gases
95
9.2 Diffusivities in Liquids
97
9.3 How do you measure diffusivities?
100
9.4 Summary
102
9.5 Further Reading
102
9.6 Exercises
103
10 TRANSFER COEFFICIENTS
106
10.1 Introduction
106
10.2 Dimensionless Groups
106
Contents
9
10.3 Tubes and Packed Beds
108
10.4 Packed GasLiquid Columns
110
10.5 Single Particles, Bubbles and Drops
111
10.6 Using `Single' Coefficients for Swarms
117
10.7 Using Binary Coefficients in Multicomponent Calculations
118
10.8 Summary
118
10.9 Further Reading
119
10.10 Exercises
120
11 ELECTRICAL FORCES AND ELECTROLYTES
122
11.1 Electrolytes
122
11.2 The Electroneutrality Relation
123
11.3 Electrical Forces
124
11.4 Transport Relations
124
11.5 Diffusion of
hydrochloric acid125
11.6 Plus a Trace of Sodium Chloride
126
11.7 Diffusion of Proteins
126
11.8 Conduction and Friction between Ions
128
11.9 Diffusivities in Electrolyte Solutions
130
11.10 Summary
132
11.11 Further Reading
133
11.12 Exercises
134
12 CENTRIFUGAL AND PRESSURE FORCES
137
12.1 Volumetric Properties
137
12.2 The Pressure Gradient
138
12.3 Gravitational Force
138
12.4 Centrifuges
139
12.5 Gas and Protein Centrifugation
140
12.6 Difference Equation for the Pressure Force
141
12.7 The MaxwellStefan Equations (again)
142
12.8 Summary
143
12.9 Further Reading
143
12.10 Exercises
143
13 WHY WE USE THE MSEQUATIONS
146
13.1 Three Ways
146
13.2 A Mixture of Three Gases
147
13.3 The Fick Description
148
13.4 Thermodynamics of Irreversible Processes
150
13.5 The MaxwellStefan Description
150
13.6 Units
152
13.7 Further Reading
152
13.8 Exercises
153
PART 2. MASS TRANSFER THROUGH A SOLID MATRIX
155
14 SOLID MATRICES
156
14.1 The Applications
156
14.2 Membrane Processes
156
14.3 Adsorption and Chromatography
158
14.4 Heterogeneous Catalysis
160
10
Mass Transfer in Multicomponent Mixtures
14.5 Structured and Nonstructured Matrices
161
14.6 Effects of a Matrix on Mass Transfer
162
14.7 Compositions with a Matrix
163
14.8 How Further?
164
14.9 Further Reading
165
14.10 Exercises
165
15 PROPERTIES OF POLYMERS
167
15.1 A Few Words on Polymers
167
15.2 Thermodynamics of Mixtures in a Polymer
171
15.3 Summary
177
15.4 Further Reading
177
15.5 Exercises
177
16 DIFFUSION IN POLYMERS
180
16.1 Behaviour of Diffusivities
180
16.2 The Free Volume Theory
182
16.3 Summary
188
16.4 Further Reading
189
16.5 Exercises
189
17 DIALYSIS AND GAS SEPARATION
191
17.1 Dialysis
191
17.2 Gas Separation
194
17.3 Summary
197
17.4 Further Reading
197
17.5 Exercises
198
18 PERVAPORATION AND REVERSE OSMOSIS
200
18.1 Pervaporation
200
18.2 Reverse Osmosis
204
18.3 Summary
208
18.4 Further Reading
208
18.5 Exercises
209
19 ELECTROLYSIS AND ELECTRODIALYSIS
211
19.1 Introduction
211
19.2 Polarisation in Electrolysis
212
19.3 Electrodialysis
214
19.4 Summary
220
19.5 Further Reading
220
19.6 Exercises
221
20 ION EXCHANGE
223
20.1 FixedBed Processes
223
20.2 Ion Exchange Equilibria
224
20.3 Linear Driving Force Model
225
20.4 Ion Exchange (Film Limited)
227
20.5 Ion Exchange (Particle Limited)
228
20.6 Summary
230
20.7 Further Reading
230
20.8 Exercises
231
Contents
11
21 GAS PERMEATION
233
21.1 Transport in Cylindrical Pores
233
21.2 The Diffusion Coefficients
233
21.3 Looking Back
235
21.4 Transport in a Bed of Spheres
236
21.5 The Dusty Gas Model
238
21.6 Summary
240
21.7 Further Reading
240
21.8 Exercises
241
22 IN POROUS CATALYSTS
242
22.1 Introduction
242
22.2 Pressure gradients inside a particle
243
22.3 Separate Transport Equations
245
22.4 Singlevariable Pressure and Rate Expressions
245
22.5 Solution for a Slab
247
22.6 Summary
249
22.7 Further Reading
249
22.8 Exercises
250
23 IN ADSORBENTS
252
23.1 Adsorption
252
23.2 Equilibria
Langmuir isotherm253
23.3 MaxwellStefan and Fick Diffusivities
254
23.4 Macropore Diffusion
257
23.5 Transport Equations
258
23.6 Transient Adsorption of a Binary
259
23.7 Membrane Applications
260
23.8 Summary
261
23.9 Further Reading
261
23.10 Exercises
262
24 ULTRAFILTRATION
264
24.1 The Module
264
24.2 Membrane and Permeants
265
24.3 Osmotic Pressure (No Ions, no Charge)
266
24.4 Size Exclusion
266
24.5 Polarisation
267
24.6 Transport Equations
269
24.7 Inside the Membrane
272
24.8 Electrical Effects
273
24.9 Summary
275
24.10 Further Reading
276
24.11 Exercises
277
25 .........ENDING
279
25.1 Looking Back
279
25.2 Thermodynamic Models the Potentials
279
25.3 Driving Forces
281
25.4 Friction Terms
281
25.5 Friction Coefficients
282
12
Mass Transfer in Multicomponent Mixtures
25.6 Additional Relations (Bootstraps)
283
25.7 The Many Variants
285
25.8 Goodbye
286
25.9 Further Reading
287
25.10 Last Exercise
287
THANKS
289
APPENDIX 1. READING MATHCAD
293
APPENDIX 2. UNITS
296
A21 Molar basis
296
A22 Mass basis
297
A23 Volume basis
297
A24 Molar and mass diffusivities
298
A25 Molar and volume diffusivities
299
A26 Molar and mass driving force
300
A27 Molar and volume driving forces
300
A28 Difference equation for a `film'
301
A29 Difference equation, molar basis
301
A210 Difference equation, mass basis
302
A211 Difference equation, volume basis
302
A212 Summary
303
APPENDIX 3. PORING OVER PORES
304
A31 Introduction
304
A32 The System
305
A33 Forces and Velocities
305
A34 The Two Transport equations
308
A35 Comparing the Two Models
310
A36 Summary
313
A37 Further Reading
314
A38 Exercises
314
ANSWERS
317
INDEX
326
AUXILIARY FILES FOR `MASS TRANSFER IN MULTICOMPONENT MIXTURES' 329
13 1 Beginning... 1.1 Who should read this Book? This book is about the diffusion and mass transfer processes that are really important, but which are neglected in most textbooks: · those with three or more species, the `multicomponent' mixtures, · those with more than one driving force, including electrical or pressure gradients, and · those with a solid matrix such as a polymer or a porous medium. If you want to know more about these subjects, but find existing texts too difficult, then this is the book for you. Also, if you already understand the intricacies of multicomponent mass transfer, you may find it enjoyable to see how far you can get with simple means. We are assuming that you are interested in processes or products. This may be in an academic or industrial setting: in chemicals,
water treatment, food, biotechnology, pharmaceuticals ... you name it. The book assumes that you have a working knowledge of: · thermodynamics and phase equilibria: chemical potentials, enthalpies, activity coefficients, partial molar volumes and distribution coefficients, · transport phenomena: simple mass balances, binary diffusion and mass transfer coefficients, and · fluid flow, especially around particles and in porous media. If you are not too sure, do not despair. We will repeat all important concepts in a leisurely manner. However, this is not a book for a complete beginner in mass transfer; you must have heard of the above concepts. Because there are many new ideas to get used to, we have tried to avoid mathematical complexity. For the greater part of the text you do not need more than the ability to solve three linear equations with three unknowns. You can even do a fair bit with pencil, paper and a calculator. Of course you will need a computer for larger problems, but not to obtain a first understanding.
14
Mass Transfer in Multicomponent Mixtures
1.2 What this Book covers This book takes motion in a mixture to be governed by forces on the individual species. There are two kinds of forces: · driving forces, which stem from the potential gradient of a species, and · friction forces between the species, which arise from their velocity differences. Maxwell and Stefan already used this method more than a century ago. This mechanical viewpoint is much more general than Fick's law, which is usually taken as the basis of diffusion theory. It has not caught on, probably because the mathematics is thought to be difficult. This is not really a problem however: · There are simple approximations to the solutions of the equations. · The computer and numeri
Cal Techniques now make `exact' calculations much easier. Using potential gradients allows the incorporation of different driving forces: · composition gradients (or more precisely: activity gradients), · electrical potential gradients, · pressure gradients, · centrifugal fields and others. The friction approach to interaction between the species allows a consistent handling of any number of components. Working with force balances makes it easy to link the subject of mass transfer to other parts of science. Thermodynamics and transport processes become neighbours: equilibrium is simply the situation where driving forces have disappeared. The use of forces on the species in a mixture fits in the way of thinking of engineers: it is a logical extension of mechanics of a single species. For friction coefficients we can make use of the many relations for hydrodynamics of particles or porous media. These show that flow and diffusion are two sides of the same coin. With our starting points we can describe almost any mass transfer process. Examples in this book cover: · multicomponent distillation, absorption and extraction, · multicomponent evaporation and condensation, · sedimentation and ultracentrifugation, · dialysis and gas separations, · pervaporation and reverse osmosis, · electrolysis and electrodialysis, · ion exchange and adsorption · heterogeneous catalysis and · ultrafiltration.
1. Beginning ...
15
The examples treat diffusion in gases, in liquids, in electrolyte solutions, in swollen polymers and in porous media. The book includes methods for estimating multicomponent diffusivities and mass transfer coefficients. A major limitation of the book is that it mainly covers examples with a single transfer resistance, not complete pieces of equipment. Such a resistance will be a
building block for the simulation of separation columns, membrane modules or chemical reactors. The reader must be prepared to incorporate the equations into his own simulations. The approximations used should be sufficiently accurate for most engineering applications. With this book, we hope to make you feel at home in the equations of multicomponent mass transfer. However, we do not derive these equations. If you are inquisitive and have some perseverance, you will be able to retrace the fundamentals in the references that we give. 1.3 Structure of the Book This book has twentyfive chapters, covering a range of subjects. You may feel that it is a jumble of facts and problems, but there is an underlying structure. The theme is the development of the MaxwellStefan equations.
There are two main parts: on transfer in gases and liquids (Chapters 3... 13), and on transfer through a solid matrix (Chapters 14...24). Chapters 3 and 4 introduce the two sides of the MaxwellStefan equations: the driving forces for mass transfer and the frictional forces between moving species. Chapters 5 and 6 apply the equations to simple binary and ternary examples. Chapters 8, 11 and 12 complete the description of the driving forces by including the effects of nonideality in a mixture (8), electrical forces (11) and centrifugal and pressure forces (12). Chapters 9 and 10 consider parameters in the friction terms: diffusivities or mass transfer coefficients. Chapter 13 discusses the relation between the MSequations and other ways of describing mass transfer. You may wonder what has happened to Chapter 7. It introduces the effects of a temperature gradient, and does not quite fit into the structure. We could have put it almost anywhere. The second part of the book considers mass transfer through solid matrices. Chapter 14 gives a preview of the subject and discusses the two types of matrix: · polymer matrices (Chapters 15...20) and · structures with defined pores (Chapters 21...24). Chapters 15 and 16 give a
brief description of polymers and of the behaviour of diffusion coefficients in polymers. These chapters are a sideline, introducing concepts that we need further on. We continue with a series of examples with
16
Mass Transfer in Multicomponent Mixtures
different driving forces: composition gradients (Chapter 17), pressure gradients (Chapters 17 and 18) and electrical gradients (Chapters 19 and 20). In the chapters on porous media, we mainly focus on the friction side of the MSequations. Chapter 21 covers transport of nonadsorbing gases and introduces the effects of viscous flow. Chapter 22 shows how the MaxwellStefan equations are applied when
chemical reactions are taking place. Chapter 23 considers diffusion of species which do adsorb, such as in microporous adsorbents. In Chapter 24 we consider an example where viscous flow is very important: ultrafiltration. We finish by looking back at the many different aspects of the MSequations in Chapter 25. 1.4 Guidelines to the Reader The text was written to accompany overhead transparencies in a full week's course on multicomponent mass transfer. Most transparencies have found their way into the figures: they are important, not just illustrations. The figures contain all formulae and much of the other information. Not all chapters are equally important. As a minimum, we recommend that you work through Chapters 3, 4, 5, 6, 14, 17 and 21. Together, these will give you a working idea of multicomponent mass transfer theory for about two day's work. Other comments: · If you are convinced that you know all about mass transfer (as we used to be!) you should read Chapter 2. It may contain a few surprises. · Chapters 7, 8 and 12 cover subjects which, although important, can be omitted on first reading. · The Chapters 9, 10, 11 (second half), 15 and 16 are on the estimation of properties and model parameters such as diffusivities and mass transfer coefficients. You can skip these on first reading. · If you are not interested in ions, electrolytes and electrical fields you can skip Chapters 11, 19, 20 and parts of 24. (However, do note that electrical fields are much more prevalent than thought by most chemical engineers!) · If you never encounter polymers, you will not need Chapters 15...20. · When porous media play no role in your life, you can omit Chapters 21...24. It is all up to you. Chapters 2...6 contain a number of questions and small sums in the text. We recommend that you try these. The answers are buried in the text or figures. Behind each chapter is a series of exercises. These are to help you to go through the material more thoroughly than you will with a single reading. There are short questions, discussions and additions to the material of the main text. The answers are given at the end of the book.
1. Beginning ...
17
Beginning with Chapter 5, there are assignments in Mathcad a fairly accessible
programming language. There are two kinds: short ones, which you are to program yourself and longer files, which are demonstrations of more complicated problems. Our students very much favour the first type. We have marked them with and we hope that you will try a fair number of these. The second kind of files is for you to use for your own problems, to look at, to play with, to modify or to criticise. We leave it to you. Many of these files work out examples given in the text. The Mathcad assignments can be downloaded from the website http://www.vssd.nl/hlf/d004.htm. Appendix 1 contains an introduction to Mathcad: enough to allow you to read the files. At the website you will find files with only the questions, and those with complete answers. They are in three forms: for Mathcad 8 and later, for Mathcad 13 and later and for Acrobat Reader 7 and later. The pdf files in the Acrobat directory are readonly: you cannot play with the files as you can in Mathcad. The regular text of the book continues in Chapter 2. However, you should glance through the list of symbols, and the list of conventions at the end of this chapter. You may not understand all details on first reading, but you should know where they are so you can look back later. 1.5 Guidelines to the Teacher This book has evolved in a series of twentythree courses that we have given at different universities since 1982. The participants were mainly PhD and Masters students, but we have also had many participants from industry and a fair number of our colleagues: together about nine hundred of them. Most have been from
Chemical Engineering, but we have also had mathematicians, chemists, physicists, mechanical engineers, and the occasional pharmacy or biology student. Because we always have an audience coming from many different places, our courses have mostly been in five days consecutively. In such a course we have about 36 hours for lectures and computer assignments. We divide these into (roughly) 16 hours of lecturing and 20 hours of computer assignments in changing groups of two. Except for Chapters 1 and 25 (which require no lecturing time) and 2 and 14 (which take less than half an hour), all chapters need about an hour. This means that you will have to make a choice of about 15 chapters from the 19 others that we provide. The `Guidelines to the Reader' above should help you in making a choice. As a minimum for a course we recommend Chapters 3, 4, 5, 6, 14, 17 and 21. Together, these will give participants a working idea of multicomponent mass transfer theory for about two day's work. You can download auxiliary files from the website http://www.vssd.nl/hlf/d004.htm. These also contain a complete set of PowerPoint 2003 files of the colour slides that
18
Mass Transfer in Multicomponent Mixtures
we use in our course. You can use and edit these freely for your own teaching, but you are not allowed to use them for
commercial purposes. They are our property! On assessing the knowledge of students. You can of course do that in the traditional way. We also have good experience with giving each student some article on a mass transfer problem and asking him or her to construct a new Mathcad example. Our students find this difficult but instructive. Success! 1.6 Symbols Symbols used only at one or two points, are defined there and not listed here. A Avogadro constant # mol1 A nonideality parameter a activity c molar concentration mol m3 cp molar
heat capacity J mol1 D Fick diffusivity m2 s1 d diameter m D MaxwellStefan diffusivity m2 s1 E energy flux J m2 s1 E activation energy J mol1 F force per mole N mol1 F Faraday constant C mol1 g acceleration of gravity m s2 H enthalpy J mol1 h heat transfer coefficient W m2 K1 I electrical current density A m2 J flux with respect to the mixture mol m2 s1 k mass transfer coefficient m s1 K equilibrium constant M molar mass kg mol1 m mass kg N flux with respect to an interface mol m2 s1 n number of moles mol p pressure N m2 R gas constant J mol1 K1 R retention r radius m
1. Beginning ...
19
r reaction rate mol m3 s1 T temperature K t time s Tg glass transition temperature K u diffusive velocity m s1 v viscous velocity m s1 V molar volume m3 mol1 w whole (overall) velocity m s1 x mole fraction y mole fraction z distance (position) m z charge number 
Greek symbols viscous selectivity increase of... volume (void) fraction electrical potential V activity coefficient viscosity Pa s thermal conductivity J m1 K1 µ chemical potential J mol1 stoichiometric coefficient osmotic pressure N m2 density kg m3 interfacial tension N m1 stress N m2 tortuosity diffusion volume m3 mol1 angular speed rad s1 friction coefficient, structured N s mol1 m1 potential of a species J mol1 friction coefficient, unstructured N s mol1 m1
20
Mass Transfer in Multicomponent Mixtures
Superscripts
x
average of x in a film
p1
* pressure of pure `1' (vapour pressure)
V1
* volume of pure `1'
D10, 2
0 in free space
D
d 1, 2
in the dispersed phase
p1b D 1x,12=1
boiling pressure D 1,2 at x1 = 1
p1eq
pressure of `1' if a reaction were to go to equilibrium
Tref
reference
vT
thermal velocity
u1
velocity of `1' at the average composition
x, x x in two different phases
Subscripts
cw dp D1,eff
concentration of water diameter of particle or pore effective Fick diffusivity of `1'
Vf 1,M VC x1 , x1 Tref
free volume friction coefficient between `1' and the matrix molar volume of a chain element species `1' at different positions , , , reference temperature
u1, u2 u1· ui uj
species `1' and `2' tracer with properties equal to species `1' species i under consideration species j other than that under consideration
V
viscous friction coefficient
Abbreviations
ED
electrodialysis
FH
FloryHuggins
MS
MaxwellStefan
RO
Reverse Osmosis
UF
Ultrafiltration
1. Beginning ...
21
1.7 Conventions Below are a few notes on the conventions used in this book. You may want to look back at this list occasionally while you are reading the book; do not expect to understand every detail on first reading.
(1) In the drawings and sketches, the positive direction is from left to right. Velocities and fluxes in that direction are also positive.
(2) A force is directed down a potential gradient. Examples are:
F1
=

dµ1 dz
or
Fi
=
Fzi
z
This convention holds both for differential and for difference equations.
(3) When computing a difference, we begin with the value at the most positive position (the right hand value) and subtract the other: x2 = x2  x2 (4) Examples may consider compositions in many positions; these are denoted by Greek subscripts. If the problem considers different phases, these are distinguished by accents (Figure 1.1).
Fig. 1.1 Three different phases (5) A `property' can mean several different things in a mixture. We illustrate this for the molar volumes in a ternary mixture. (Figure 1.2)
22
Mass Transfer in Multicomponent Mixtures
Fig.1.2 Component volumes: pure and in a mixture In an ideal mixture, the species volumes are the same as those of the pure species, but only then. Note that a property of the mixture has no separate subscript or superscript. (6) A mixture moving through a solid matrix can be described in several ways. As an example, we consider the movement of two permeants `1' and `2' through a membrane. We can regard the membrane material as a third component `3', or as something separate the matrix `M' which is not part of the mixture (Figure 1.3). The friction forces exerted on component `1' in the two notations are:
Fig.1.3 Two notations: (a) the matrix is part of the mixture, (b) the matrix is separate Each of the terms separately has the same size in both notations. Also the velocities are identical (including u3 = uM). However, the
numerical values of both the mole fractions and the friction coefficients differ. (7) We will be describing porous media in two ways. In the first, we consider the effects of the structure of the medium: effects of pore or
particle sizes, or the effect of the void fraction. In this model it is also common to distinguish between two species velocities: diffusive and viscous (or convective) velocities. Viscous flow is governed by hydrodynamics. The sum of the two velocities is the whole (or overall velocity) . Note how the symbols for the velocities are contained in the letters of their names (Figure 1.4).
1. Beginning ...
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Fig. 1.4 Notation in the structured and nonstructured models In the nonstructured model, the effects of the structure are built into the friction coefficients. This kind of model only considers the whole velocities. The two models give the same result when applied properly. However, the friction coefficients can behave quite differently. The relations between the two sets of coefficients are complicated.
24 2 Is Something Wrong? In this chapter, we look back at how we have learned mass transfer. We see that Fick's law is incomplete and that it leads to wrong predictions, even in simple multicomponent problems. 2.1 The Starting Point We expect your working knowledge of mass transfer to be something like that summarised in Figure 2.1. Once these were also the only tools we had. So let us have a look at them. The upper part of the figure shows two `laws' which govern motion of a species i in a mixture, with respect to that mixture: · the flux of a species is proportional to its diffusivity and concentration gradient, and · the flux is proportional to a concentration difference times a mass transfer coefficient. The flux is defined relative to some `frame of reference', for example one that moves with the average molar velocity of the mixture. Fig. 2.1 Mass transfer as you have learned it The first law (Fick's law) tells us that a species should move down its concentration gradient. You may regard the second `law' as a difference form of the first, which is applicable to a
thin film. A film is a rough model for a mass transfer resistance. The two laws are handy little formulae for describing simple mass transfer problems. In passing we note that the fluxes and concentrations used here are in molar units
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(mol m2 s1 and mol m3). This implies that the diffusivity has units of m2 s1 and the mass transfer coefficient of m s1 (a velocity).
We are usually interested not in the fluxes with respect to the mixture, but in the fluxes with respect to some boundary or interface. We give these the symbol Ni (Figure 2.2). Only if the flux N of the mixture (as a whole) is zero, are the fluxes Ji and Ni equal. Otherwise, we must add a drift flux to the diffusion flux. Also here, we can write the flux relation as either a
differential equation or a difference equation. The last is simpler, but approximate. We can force the difference equation into a simple looking form by including a Stefan or drift correction. Unfortunately, this correction can have any value and sign; the result is not as useful as it looks.
Fig. 2.2 The flux with respect to an interface In most of the examples that we consider in this chapter, drift is not important. So you should expect both fluxes Ji and Ni to be equal, and directed down the concentration gradient of i. The only exception will be in the very last example, where drift is important. Conventional mass transfer in a binary mixture of gases (Figure 2.3) is especially simple. If pressure and temperature in the gas are assumed constant, the total molar concentration is also constant. By definition the sum of the fluxes with respect to the mixture is zero. If you then write down the flux equations for the two components it is immediately clear that there is only one binary diffusion coefficient. Also experiment (and the kinetic theory of gases) tells us that this diffusivity is a constant (that is: independent of composition, not of pressure and temperature). There is more to be said about conventional mass transfer theory, but this should be enough for the moment.
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Mass Transfer in Multicomponent Mixtures
Fig. 2.3 A binary gas is simple 2.2 Three Gases Look at the experiment in Figure 2.4. There are two equal glass bulbs filled with mixtures of ideal gases. The left bulb consists of hydrogen and nitrogen, and the right bulb of
carbon dioxide and nitrogen. The amounts of nitrogen in the two bulbs only differ slightly. Both bulbs are at the same pressure and temperature. At a certain moment we connect them by a capillary; this is fairly narrow, say with a diameter of one millimetre, but otherwise nothing special. Gases start diffusing from one bulb to the other. Before you read on we would like to ask you to think a moment about the questions given in the figure, and to make your own decision on which answer you choose.
Fig. 2.4 A `simple' experiment, with some questions Have you answered the questions?
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The results of the experiment are shown in Figure 2.5. The behaviours of hydrogen and carbon dioxide (bottom part of the figure) are as expected. Their compositions change monotonically in such a way that after a few days the amounts in the two bulbs will have become equal. Hydrogen (which is the smaller molecule) moves more rapidly than carbon dioxide. Nitrogen (note the difference in the composition scale) behaves quite differently. Initially it diffuses from the high concentration (bulb B) to the low concentration, and the two concentrations become equal after about one hour. However, nitrogen keeps on diffusing in the same direction, now against its concentration gradient. The gradient keeps increasing up to about eight hours after the start of the experiment. Only after that, do the two bulbs gradually go back to equal compositions.
Fig. 2.5 The strange behaviour of nitrogen With the conventional mass transfer theory of the first two figures in mind you will probably find it difficult to understand what is happening. So just try to forget these for a moment and try a different viewpoint. It is fairly obvious why hydrogen is going from left to right. There is far more hydrogen in the left bulb than in the right one; the
random thermal motions of the molecules will on average cause them to move to the right. The same mechanism causes carbon dioxide to move to the left. Now it looks as if nitrogen is being dragged along by the carbon dioxide. This is understandable: you would expect more friction between the heavy carbon dioxide molecules and nitrogen than between nitrogen and hydrogen. At least initially, the movement of nitrogen is mainly determined by the carbon dioxide and hydrogen gradients, and not by its own gradient (which is rather small). You can extend conventional mass transfer theory to explain the results, but the result is much more complicated and less convincing than that above.
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Mass Transfer in Multicomponent Mixtures
2.3 Two Cations This experiment involves a membrane that is permeable to cations, but not to anions (Figure 2.6). We bring a dilute solution of sodium chloride in the right compartment and a much more concentrated solution of hydrochloric acid in the left compartment. You would expect the sodium ions to diffuse from the right to the left until the two concentrations have become equal. They do indeed go in that direction. However, they may go on until their concentration in the left compartment is many times higher than in the right one! The experiment has similarities with the previous one. The explanation, however, is rather different. Hydrogen ions diffuse through the membrane to the right. (Remember that the chloride ions cannot.) This causes a small positive excess charge in the right compartment. The resulting electrical gradient forces the sodium ions to the left and restricts the amount of hydrogen that can be transferred. There is no such mechanism in conventional mass transfer.
Fig. 2.6 Sodium moves against its gradient 2.4 Two Gases and a Porous Plug As a last example (Figure 2.7) we consider a porous plug. It is a plug with fine openings (you might think of compressed cotton wool), but otherwise inert. On one side there is helium, on the other argon, both at exactly the same pressure and temperature. Such a situation can be maintained by having flows of pure helium and pure argon past the ends of the plug, passing out into the open. The concentration gradients of the two gases are equal. A careless application of Figure 2.3 might let you expect that the fluxes should also be equal.
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In reality, experiment shows that the helium flux is about three times higher1 than the argon flux! If you want to understand this you should realise that Figure 2.3 only tells you something about the movement of helium and argon with respect to each other. Figure 2.3 tells you nothing about the movement of the mixture with respect to the plug. This movement is balanced by friction between the two gases and the plug. To let the two friction terms cancel, helium must have the higher velocity. There is no interaction with any plug in the conventional analysis of mass transfer; clearly you should take the plug as a component of the mass transfer system.
Fig. 2.7 The fluxes need not be equalbutopposite By applying a pressure difference (Figure 2.8) you can equalise the two fluxes. The viscous flow due to the pressure gradient increases the argon flux and decreases the helium flux. This is analogous to the effect of the electrical gradient in the previous example. The pressure difference required depends on the structure of the plug: the finer the pores, the larger the pressure difference. Fig. 2.8 Equal fluxes with a small pressure difference 1 The ratio is equal to the square root of the ratio of the molar masses.
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Mass Transfer in Multicomponent Mixtures
In the threegases problem we saw earlier, the capillary is much wider. Even so, rapid diffusion of hydrogen does cause a minute pressure difference. The transport of nitrogen against its gradient is in small part due to the resulting viscous flow. 2.5 Summary In this second chapter we have briefly reviewed conventional mass transfer. We have then applied it to a few examples and seen that it does not work well. The examples suggest that a better approach to mass transfer would have to take the following phenomena into account: · friction between each pair of components, including any solid matrix, · the occurrence of other driving forces than only composition gradients. You can think of electrical and pressure gradients, and · viscous flow in heterogeneous media (solid matrices). We will be working these ideas out in the rest of this book. 2.6 Further Reading Cussler, E. L. (1997) Diffusion, Mass Transfer in Fluid Systems,
2nd Edition,
Cambridge University Press, Cambridge, 1997. A lively book and a good primer. If you decide not to proceed into the subject of multicomponent mass transfer, Cussler will give you a few excuses. Duncan, J.B. and Toor, H.L. (1962) An
Experimental Study of three component gas diffusion. A.I.Ch.E.J. 8, 3841. This describes the twobulb diffusion experiment. 2.7 Exercises 2.1 The concentrations of oxygen and nitrogen in the atmosphere change with altitude. At the top of the Mount Everest their values are about one half of those at sea level. According to Fick's law, air should be diffusing away into outer space; rough estimates show that the halflife of the atmosphere should be about ten thousand years. This is incorrect; the atmosphere is several hundred times older. Which force counteracts the diffusion and is neglected by Fick's law?
2.2 Consider a glass of water. The concentration of water in the liquid is 55 000 mol m3; that in the surrounding air 1 mol m3. So a large concentration difference exists across the water/air interface, which is thought to be a few molecules thick. It may not seem to be surprising that water is vaporising. Now we put a lid on the glass. The air in the glass becomes saturated; the water concentration might go up to 2 mol m3. We still have the concentration gradient in the interface, yet vaporisation stops. (This is obviously at variance with Fick's law, although few people seem to notice.)
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In the next chapter we will take the driving force for mass transfer to be the chemical potential gradient. What does this theory tell when we put the lid on?
2.3 We fill the two compartments of the cell in Figure 2.6 with the same
aqueous solution of NaCl. There are no concentration differences, so there is no driving force for either water or for the ions. If we now apply an electrical potential difference across the membrane, sodium will be forced through. This is not surprising: there is a driving force on the sodium, be it one that is not incorporated in Fick's law. What is surprising, is that the electrical force leads to a considerable flux of water. Water is not charged, and there is no obvious driving force. How can this be?
2.4 Flow is quite important in gas and water pipelines, in the airing of clothes, in the wetting of toilet paper... in just about everything in daily life. However, transport in media with extremely small pores, such as cellophane, seems to be dominated by diffusion. Why would this be?
2.5 Study the introduction to the review of Krishna and Wesselingh (1997), `The MaxwellStefan approach to mass transfer' Chem.Eng.Sci. 73, 861911. Here the limitations of conventional approaches to mass transfer are shown using several other examples.
JA Wesselingh, R Krishna