Title Page
 Table of Contents
 List of Tables
 List of Illustrations
 Previous work
 Numerical analysis
 Results of calculations
 Comparisons between experimental...
 List of symbols
 Literature cited
 Biographical sketch

Title: Application of numerical methods in analysis of fixed bed adsorption fractionation.
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Permanent Link: http://ufdc.ufl.edu/UF00089981/00001
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Title: Application of numerical methods in analysis of fixed bed adsorption fractionation.
Physical Description: Book
Language: English
Creator: Johnson, Adrain Earl Jr.
Publisher: Adrain Earl Johnson, Jr.
Place of Publication: Gainesville, Fla.
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Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
        Page vi
    List of Illustrations
        Page vii
        Page viii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Previous work
        Page 6
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        Page 8
        Page 9
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        Page 27
        Page 28
    Numerical analysis
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
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        Page 38
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    Results of calculations
        Page 40
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    Comparisons between experimental and calculated results
        Page 62
        Page 63
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    List of symbols
        Page 131
        Page 132
    Literature cited
        Page 133
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    Biographical sketch
        Page 173
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        Copyright 1
        Page i
Full Text






February, 1958


The author wishes to express his deep appreciation

and sincere thanks to Professor R. D. Walker, Jr., for his

encouragement, interest, and many suggestions during the

course of this investigation; to Mr. Carlis Taylor of the

University of Florida Statistical Laboratory for his very

valuable help in the preparation of the computer program and

in the obtaining of the computer solutions; to Dr. H. A.

Meyer for authorizing the use of the facilities of the

Statistical Laboratory and Computing Center for this work;

to Dr. Mack Tyner, Dr. T. M. Reed, Dr. E. E. Muschlitz, and

Dr. R. .W. Cowan, of the graduate committee, for their help-

ful suggestions and criticisms; to the faculty and graduate

students of the Department of Chemical Engineering for

their cooperation and interest; and to his wife for the

assistance and unwavering support which she has given.



ACKNOWLEDGEMENTS.............................. .. ....... ii

LIST OF TABLES........................................... v

LIST OF ILLUSTRATIONS .................................... vii

I. INTRODUCTION .................................... 1

II. BACKGROUND...................................... 3

III. PREVIOUS WORK.................................... 6

IV. THEORY.......................................... 16

A. The Fixed Bed Binary Liquid Adsorption
Process ................................ .... 16
B. Derivation of Equations.................... 21
C. The Dimensionless Parameters H and T....... 24
D. Boundary Conditions for the Liquid Phase
Fixed Bed Process.......................... 26

V. NUMERICAL ANALYSIS ............................... 29

A. Numerical Methods........................... 29
B. Description of Integration Procedure........ 32
C. Computer Program............................ 37

VI. RESULTS OF CALCULATIONS........................ 40

A. Problem Solutions.......................... 40
B. The Asymptotic or Ultimate Adsorption Wave. 43
C. The Shape of the Asymptotic Wave............ 48
D. Computation of HETS From Fixed Bed Data.... 52

VII. EXPERIMENTAL................................... 56

A. Adsorbent.................................... 56
B. Adsorbates.................................. 56
C. Experimental Procedures.................... 57




CALCULATED RESULTS.............................. 62

A. Adsorption Fractionation Experiments of
Lombardo.................................... 63
B. Toluene-Methylcyclohexane Fractionation
on Silica Gel............................... 64
C. Toluene-Methylcyclohexane Adsorption on
Activated Alumina......................... 67
D. Use of Constant-Alpha Type Equilibrium
Diagram .................................... 68
E. HETS of Column Packing..................... 69
F. Discussion of Calculations................. 71
G. Discussion of Intraparticle Diffusional
Resistance................................. 72

IX. CONCLUSIONS.................................... 75

X. LIST OF SYMBOLS................................. 131

XI. LITERATURE CITED............................... 133


BIOGRAPHICAL SKETCH...................................... 173


Table Page

1. Numerical Integration Formulae..................... 77

2. Summary of Adsorption Fractionation Calculations.. 78

3. Determination of Specific Pore Volumes.......... 80

4. Adsorption Equilibrium Data for Toluene-Methyl-
cyclohexane on Davison 6-12 Mesh Silica Gel....... 81

5. Adsorption Equilibrium Data for Toluene-Methyl-
cyclohexane on Alcoa 8-14 Mesh Activated Alumina.. 82

6. Adsorption Equilibrium Data for Benzene-N-
Hexane on Davison "Thru 200" Mesh Silica Gel....... 83

7. Summary of Fractionation Experiments.............. 84

8. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel...................... 86

9. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel..................... 87

10. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel...................... 88

11. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel..................... 89

12. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel...................... 90

13. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel..................... 91

14. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel..................... 92

15. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel...................... 93


Table Page

16. Toluene-Methylcyclohexane Fractionation on
Davison 6-12 Mesh Silica Gel.................... 94

17. Toluene-Methylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina................ 95

18. Toluene-Methylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina............... 96

19. Toluene-Methylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina............... 97

20. Toluene-Methylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina.............. 98

21. Toluene-Mechylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina............... 99

22. Toluene-Methylcyclohexane Fractionation on
Alcoa 8-14 Mesh Activated Alumina............... 100

23. Benzene-N-Hexane Fractionation on Davison
"Thru 200" Mesh Silica Gel...................... 101

24. Benzene-N-Hexane Fractionation on Davison
"Thru 200" Mesh Silica Gel...................... 102

25. Benzene-N-Hexane Fractionation on Davison
"Thru 200" Mesh Silica Gel ................... 103

26. Calibration of Refractometer for MCH-Toluene
Solutions at 30C................................. 104


Figure Page

1. Flow Diagram of Computer Program................ 105

2. Liquid Phase Composition History, Computer
Solution to Problem 1........................... 106

3. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 1........................... 107

4. Liquid Phase Composition History, Computer
Solution to Problem 9.......................... 108

5. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 9 ........................... 109

6. Liquid Phase Composition History, Computer
Solution to Problem 51 .......................... 110

7. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 51........................... 111

8. Liquid Phase Composition History, Computer
Solution to Problem 52 .......................... 112

9. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 52 .......................... 113

10. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 52.......................... 114

11. Liquid Phase Composition History, Computer
Solution to Problem 99.......................... 115

12. Ultimate Adsorption Wave Shapes, Computer
Solution to Problem 99.......................... 116

13. Adsorption Equilibrium Diagram for MCH-Toluene
on Davison 6-12 Mesh Silica Gel.................. 117

14. Adsorption Equilibrium Diagram for MCH-Toluene
on Alcoa 8-14 Mesh Activated Alumina............ 118



Figure Page

15. Adsorption Equilibrium Diagram for Benzene-
Hexane on Davison "Thru 200" Mesh Silica Gel.... 119

16. Benzene-Hexane Fractionation With Silica Gel.... 120

17. Benzene-Hexane Fractionation With Silica Gel.... 121

18. Benzene-Hexane Fractionation With Silica Gel.... 122

19. MCH-Toluene Fractionation With Silica Gel....... 123

20. MCH-Toluene Fractionation With Silica Gel....... 124

21. MCH-Toluene Fractionation With Silica Gel....... 125

22. MCH-Toluene Fractionation With Alumina.......... 126

23. MCH-Toluene Fractionation With Alumina........... 127

24. Effect of Liquid Velocity on Overall Mass
Transfer Coefficient............................. 128

25. Effect of Liquid Velocity on HETS............... 129

26. Calibration of Refractometer for MCH-Toluene
Solutions....................................... 130



This dissertation describes the results of a study

made on the process of adsorption fractionation of binary

liquid solutions. Based on a theoretical analysis of the

factors controlling the process, mathematical partial dif-

ferential equations expressing column operation were derived

and solved by numerical integration with the aid of an IBM

650 digital computer. Particular emphasis was placed upon

the statement of the boundary conditions for the liquid

adsorption process, as it is believed that the proper bound-

ary conditions have not been used in previous work.

Computed solutions to column operation were com-

pared with experimental data taken in this study and with

other published data. It was found that good agreement be-

tween calculated and experimental data may be obtained in

systems in which the external particle film resistance to

diffusion apparently controls. Agreement in cases where

intraparticle diffusion contributes to the total diffusional

resistance is not as good, but is considered useful. The

success with the external film controlling case indicates

that when a suitable theory on intraparticle resistance is

derived, numerical integration by means of a computer will

prove the best means of obtaining satisfactory solutions,



because of the apparent impossibility of obtaining analytical

solutions to the equations.

It was found that, through a fortuitous circumstance,

computer solutions based on constant-alpha type equilibrium

diagrams fit the data for the case of intraparticle diffu-

sion contributing more closely than solutions using the

actual equilibrium diagram of the system.

In the course of this work equations were developed

for determining the rate of advance and the shape of the

ultimate adsorption wave, which is formed in columns of

sufficient length. In addition, a method was developed for

evaluating the height of an equilibrium stage (HETS) in an

adsorbent bed from data of fixed bed adsorption fractiona-

tion experiments. The determination of HETS is of importance

in the design of continuous countercurrent adsorption


This work also included the development of a com-

puter program for solving the partial differential equations.

The resulting program and a brief description of the

numerical methods used are presented.


Historically, the most frequently encountered prob-

lem in the chemical and related industries has been the

necessity of separating relatively pure materials from

mixtures of. two or more components, thereby producing either

finished products for sale or intermediate products to be

further processed. One portion of chemical engineering

science, the unit operations, is devoted entirely to the study

of the various methods for separating materials.

Research in the unit operations is usually aimed

either at the development of new, more economical, or more

exacting separation methods, or at the development of more

precise theories and formulae for expressing the phenomena

of the known methods so that they may be put to better use.

In the past decade, a new tool has been made available which

can help the scientist and engineer to investigate mathe-

matical theories and methods in a manner undreamed of twenty

years ago. This tool is the high speed electronic computer,

digital or analog. Such machines have many capabilities,

but one of the most important to technical research is their

ability to solve complicated mathematical equations, both

algebraic and differential, which are otherwise insoluble.

In the past, many a theorist has been forced to

abandon a set of equations which he believed might express a



phenomenon because the solution to the equations could not

be provided by the most expert mathematician; instead, the

theorist resolved the difficulty by making restricting

assumptions about the process which simplified the equations

and permitted a solution. Such solutions are quite useful

in the design of a process, but they are always only approxi-

mations. Sometimes their use leads to serious and costly

mistakes, not only in the design of industrial processes,

but in the interpretation of the phenomenon being investigated.

In June, 1957, an electronic digital computer, IBM

type 650, was installed at the University of Florida Sta-

tistical Laboratory. This machine, with its auxiliary

equipment, represents the beginnings of a computing center,

which will be available to the University and the State on a

basis similar to that of the other facilities of the Labora-


In anticipation of the installation of this computer,

the subject research was initiated in the field of chemical

engineering unit operations with the view of utilizing the

computer for providing a solution to equations which promise

to express a theory more precisely than previous treatments.

The unit operation chosen for a study was adsorption, which

is a relatively new entrant to the commercial field of large-

scale separation process. The "Arosorb" (1) and "Hyper-

sorption" (2, 60) processes for the separation of petroleum

hydrocarbons are examples of commercial applications of



The analysis of fixed bed liquid phase adsorption

fractionation is complicated by the fact that it is

inherently unsteady-state, or transient; partial differen-

tial equations are required to express the process behavior

mathematically. The fixed bed adsorption fractionation

process, being mathematically complex and hence in an early

stage of treatment, was chosen for study with the dual

purpose in mind of advancing the frontier of knowledge in

this field and of demonstrating what the computer can mean

to research.


In this section, the progress in adsorption research

is traced from the turn of the century to the present. In

general only those publications which deal with multicom-

ponent adsorption equilibria or rate of adsorption are

discussed. However, any paper of unusual interest is also



The early investigators concerned themselves with

the nature of adsorption and with the equilibrium relation-

ships of various systems of adsorbate and adsorbent.

Freundlich (3) proposed his now famous isotherm for correlat-

ing the adsorption data of many systems. He was an early

exponent of the theory that adsorption is a surface pheno-

menon (4), (5), which was not altogether accepted by the

scientists of his day. Travers (6) suggested that since

adsorption depends upon temperature it should be considered

a "solid solution" phenomenon; this was refuted by Wohlers

(7), who concluded that chemical bonds must account for the

process because the adsorbed material usually does not react

normally. Michaelis and Rona (8) suggested that adsorption

is caused by a lowering of the surface tension of the sol-

vent by the adsorbent. Reychler (9) demonstrated that the



Freundlich isotherm was compatible with his own chemical

reaction theory; Duclaux (10) theorized that adsorption is

a result of differences in temperature which exist in minute

cavities of the solid, causing liquifaction. Many investi-

gators, Geddes (11), Schmidt (12), Katz (13), Langmuir (14),

Polanyi (15), Williams (16), proposed equations different

from that of Freundlich. Some of these proposals were merely

the result of curve fitting, but others, such as those made

by Langmuir and Polanyi, were based on theories which ade-

quately explain certain features of adsorption. By 1920,

when Polanyi introduced his equation, which utilized one

"characteristic" curve to account for the adsorption of a

vapor or gas under all conditions of temperature and pressure

of a given system, it was generally recognized that adsorp-

tion may be explained by more than one theory, depending

upon the system, and may involve physical forces, chemical

forces, or a combination of both.

Theoretical analyses based on thermodynamic considera-

tions became prevalent towards the last of this period;

speculations concerning the heat of adsorption were made.

Polanyi (17) discussed adsorption from the standpoint of the

3rd law, Langmuir (14) suggested that unbalanced crystal

forces account for physical adsorption, Williams (18) derived

an adsorption isostere equation from thermodynamic reasoning,

Lamb and Coolidge (19) concluded that the total heat of ad-


sorption equals the heat of condensation plus the work of

compression. Very little was done with liquid adsorbates;

interest in vapor phase adsorption predominated. Gurvich

(20), however, noted that, on the same adsorbent and at

their own vapor pressure, approximately equal volumes of

various liquids were adsorbed.

One of the earliest investigations of the rate of

adsorption was performed by Berzter in 1912 (21). As with

most of the early studies, Berzter used a gas-air mixture,

from which he adsorbed the gas onto activated charcoal. He

fitted an empirical equation to his rate data without

determining the significance of the factors. Other inves-

tigations of similar systems were made by Rakovskii (22),

Gurvich (23), who discovered that subdivision of the adsorbent

particles affected the rate of adsorption but not the ad-

sorption equilibrium, Freundlich (24), Dietl (25), and Hernad

(26). Little was concluded from these studies except that

the velocity of batch-type adsorption decreased as the ad-

sorption progressed, and that diffusion might play a role in

determining the rate.


The role of adsorption in catalysis was foreseen by

Polanyi, who in 1921 showed in a theoretical paper (27) that

adsorbents should by their nature accelerate chemical reac-

tions, because of the reduction in the required activation


energy upon adsorption. However, Kruyt (28) disagreed; he

believed that adsorption should decrease the rate of reac-

tions because of the immobility of the adsorbed molecules.

An important concept was developed by Mathews (29)

who, in 1921, pointed out that the term adsorption should

properly be used to describe a phenomenon in which the con-

centration of a substance tends to be different at the

interface between two phases from the concentration in the

main body of either phase, thus broadening the scope of


A typical early paper on kinetics was published by

Ilin (30), who proposed that the rate of adsorption of a

constituent from a gas in a batch process is proportional

to e-kt. Such a relation, although it may fit a set of data,

sheds little light on the factors which determine the

instantaneous rate of adsorption.

During the late 1920's interest was aroused in the

equilibrium relationships for a system in which the adsorbate

contains more than one component attracted by the adsorbent.

Levy (31), and Klosky (32) submitted equations for correlat-

ing such data. These studies probably suggested the concept

that adsorption could be used to fractionate binary mixtures.


Additional equations for correlating the kinetics of

batch adsorption were proposed by Tolloizko (33), Constable


(34), Kondrashon (35), Ilin (36), Rogenskei (37), and

Crespi (38, 39), most of which were still rather empirical.

The idea that the rate depends upon the approach to the

equilibrium condition was appearing in various mathematical

forms in most of this work. A few workers began to consider

the kinetics as similar to those of chemical reactions,

requiring activation energies, etc. Taylor (40) approached

the subject in this manner; but Nizovkin (41) decided that

diffusion of the gases controls the entire process. Crespi

(42) derived a rate equation from Langmuir and Freundlich

isotherms which was also of the chemical kinetics type.

Later, Damkohler (43) showed that the Langmuir derived equa-

tion applies only where establishment of the equilibrium

takes 10-5 seconds, otherwise diffusion of the material to

the adsorption site is controlling.

Brunauer, Emmett, and Teller published their impor-

tant paper which dealt with the derivation of adsorption

isotherms on the assumption that condensation forces are

responsible for multimolecular layer adsorption (44). Sta-

tistical mechanical approaches to the explanation of adsorp-

tion equilibrium were presented by Wilkins (45) and Kimball

(46). Experimental studies of adsorption from binary liquid

solutions were performed by Ruff (47), Jones, et. al. (48),

and Kane and Jatkar (49).



In 1940 Brunauer, Deming, and Teller (50) combined

the recognized five types of vapor isotherms into one equa-


One of the first papers dealing with the kinetics of

adsorption in a column was that of Wilson (51) who developed

equations assuming instantaneous equilibrium, no void space

between particles, and a single adsorbed component. This

paper showed mathematically the existence of an adsorption

band which moves through the adsorbent column, and thus

qualititatively agreed with known facts. Martin and Synge

(52) pointed out the analogy between a moving bed adsorption

column and distillation. Mathematical equations were developed

for the steady state case to compute the number of equilibrium

stages required for a given separation.

DeVault (53) extended the work of Wilson by develop-

ing differential equations and their solutions for single

solute adsorption which considered the void space between

particles. Differential equations for multiple solutes were

derived but not solved. There was reasonable agreement with

selected previously published data.

Thomas (54) proposed a kinetic theory which leads to

a Langmuir type isotherm at equilibrium. The adsorption

step was assumed to control with no diffusional resistance.

Solutions for the case of multiple solutes were impossible.


S Amundsen in his first paper on the mathematics of

bed adsorption (55) developed differential equations based

on the assumptions of irreversible adsorption and a rate

proportional to the concentration of the adsorbate in the

gas stream and to the approach to equilibrium on the ad-

sorbent. In a later paper he took into account the desorp-

tion pressure exerted by the adsorbate.

In 1947 Hougen and Marshall (56) developed methods

for calculating relations between time, position, tempera-

ture, and concentration, in both gas and solid phase in a

fixed bed, with the restriction that the adsorption isotherms

be linear. Analytical solutions of the partial differential

equations were obtained and plots of the solutions were made.

The interest in multicomponent adsorption equilibria

grew rapidly in the late 1940's. Many papers were published

for both gases and liquids showing isotherms for various

experimentally investigated systems, and various modifica-

tions of the Brunauer, Emmett, and Teller isotherms were

proposed. Such papers were authored by Wieke (57), Mair (58),

Arnold (59), Spengler and Kaenker (61), Lewis and Gilliland

(62), and Eagle and Scott (63). Industrial applications

were described by Berg (60), who explained the Hypersorption

process for separation of light hydrocarbons, and by Weiss



1950 to Present

Since 1950 the mathematics of adsorption kinetics

have been even more intensively investigated. Amundsen and

Kasten (65, 66) have continued to approach the problem

analytically, mainly in the field of ion exchange, which is

closely related, but also in the field of adsorption. They

developed very complicated analytical solutions to the ad-

sorption of gases in fluidized beds, assuming intraparticle

diffusion controlling and/or the adsorption process itself

controlling, but still restricting the equilibrium rela-

tions to straight lines.

Eagle and Scott (63) presented extensive data for

equilibria of binary liquid systems and some batch kinetic

data, which permitted the evaluation of apparent diffusi-

vities within the particles of adsorbent.

Mair (67) treated fixed bed adsorption fractiona-

tion as a distillation process, and developed design equa-

tions utilizing a theoretical stage concept. Experimental

results were given for separation factors and for HETS.

Kasten and Amundsen (68) showed that in liquid

systems the rate is most often controlled by the diffusional

process instead of the adsorption process. They also

developed equations based on mass transfer and intraparticle

diffusion for a gas-solid moving bed adsorber, for the case

of one adsorbate, or multiple adsorbates assuming individual

isotherms for each. The effects of fluid phase resistance,


intraparticle diffusion, and adsorption resistance, on the

solution of the bed height required were shown.

Hiester (69) considered the performance of ion

exchange and adsorption columns mathematically. Approxi-

mate solutions of mass transfer differential equations were

given which can be used to predict column behavior.

J. B. Rosen (70) published a solution of the general

problem of transient behavior of a linear fixed-bed system

when the rate is determined by liquid film and particle


Gilliland and Baddour (71) considered the kinetics

of ion exchange, wherein an overall coefficient representing

all resistances to transfer was used successfully, and pre-

sented a solution to the partial differential equations

previously derived by Thomas. This is an isolated instance

where the equilibrium equation used was not restricted to

a straight line. Experimental data correlated very well,

so that use of experimentally determined rate constants pre-

dicted the elution curves of other experiments satisfac-


Lombardo (72) considered the problem of binary

liquid adsorption fractionation from the pseudo-theoretical

stage standpoint, and obtained solutions to the stepwise

equations which he proposed by means of a card programmed



Hirschler and Mertes (1) performed experiments batch-

wise, similar to those of Eagle and Scott for liquid phase

binary adsorption. Internal diffusivities were computed

from the data.

Lapidus and Rosen (73), considering ion exchange,

developed equations similar to adsorption fractionation

equations, using a lumped resistance, and were able.to show

that an asymptotic solution usually exists. Solutions to

the asymptotic equation were obtained with a Langmuir type



It can be seen from the foregoing literature survey

that there has been some very creditable work done towards

the mathematical treatment of adsorption and ion exchange

kinetics, especially in recent years. Nevertheless, it

appears that there are enough variations in the different

phenomena of vapor phase adsorption, ion exchange, and liquid

phase adsorption to warrant a treatment based specifically

on the system being considered. The electronic computer is

best suited for individual treatment of a difficult problem,

since the results obtained by computer analysis are in the

form of numerical answers to the specific problem with par-

ticular boundary conditions. To obtain general answers

comparable to an analytical solution, it is necessary to run

the problem repeatedly on the computer, varying the para-

meters and boundary conditions each time, until enough

answers are computed to permit the drawing of graphs and

curves which present the desired coverage of the variables.

A. The Fixed Bed Binary Liquid Adsorption Process

The basic assumptions made to define the fixed bed

fractionation of a binary liquid are described below. These

are the conditions on which the calculations made in this

study were based. The following discussion points out the


conditions which are peculiar to the liquid phase process.

1. A constant composition feed liquid consisting only of the

two completely miscible components A and B, is fed at a con-

stant rate into a column of solid adsorbent. The selectivity

of the adsorbent results in a gradual removal of A from the

liquid as it travels through the bed.

2. The velocity profile of the liquid flowing through the

column is assumed to be rod-like. [To promote this condition

experimentally, especially at the liquid front during the

filling of the bed, the liquid was fed through the adsorbent

column from bottom to top.

3. The adsorbent is initially free of adsorbate liquid.

4. Equal volumes of pure A, pure B, or any mixture of the

two are adsorbed per unit of adsorbent; i.e. the pore volume

of the adsorbent is the same for both A and B.

5. Mixtures of A and B are volumetrically additive.

6. The activation energy of the adsorption process at the

surface of the adsorbent is considered small enough to per-

mit the diffusion process to control the rate of adsorption.

7. The combined resistance to diffusion, consisting of both

the external surface film and an intraparticle resistance,

is considered together as one diffusional resistance, express-

ible in the usual manner of the mass transfer "film" theory.

Items 1 and 2 above are standard with the fixed bed

adsorption process. The development of the theory has not


progressed to the point that a better account of the velo-

city profile of a fluid in a fixed bed can be made. This is

an item which could conceivably be included in a computer

analysis when a suitable fluid flow theory is available.

Item 3 is a deceivingly simple statement which needs

further discussion. In the gas phase process, which is used

as a model by almost every published paper on adsorption

fractionation, the bed is initially free of adsorbate, but

contains inert gas or carrier vapor. Since the bed is

initially already full of fluid, there is little effect of

the adsorption process upon the quantity of fluid flowing

while the bed is filling. The mathematical boundary con-

ditions used to express this case state that when the feed

fluid reaches a given position in the bed the quantity of

adsorbate on the adsorbent is zero. In addition, the time

required for the fluid to reach a bed point is that required

to push the original gas out of the void space between the

adsorbent particles. Neither of these conditions is typical

of binary liquid adsorption. First, the time required to

fill the bed to a given point is that required to fill the

void space and the pore volume of the adsorbent bed to that

point. Secondly, the composition of the liquid on the ad-

sorbent at the foremost point of liquid penetration into the

bed continually changes as the filling "front" progresses.

In a sufficiently long bed, the composition eventually


becomes zero with respect to component A, the more strongly

adsorbed component, but the length of bed required to arrive

at this condition is of importance. To the author's know-

ledge, no treatment of the liquid phase process to date has

considered either of these two points. As will be seen later,

the development of the boundary conditions accurately

describing the binary liquid adsorption case is involved.

Item 4 has been shown to be true of liquid phase

adsorption in a great many instances, and is usually assumed.

It is particularly true of members of homologous series and

otherwise chemically related compounds. As a consequence

of item 4, it is generally convenient to assume item 5 also,

thus permitting compositions to be expressed in volume frac-

tions rather than mole fractions.

Although some investigators continue to use a chemical

kinetics type of rate equation, it is generally accepted that,

as stated in item 6, diffusion controls liquid phase adsorp-

tion processes.

The manner in which the diffusion is taken into

account is a topic of considerable interest. As stated in

item 7 the subject treatment assumes that the external and

intraparticle resistance can be lumped together as one over-

all resistance. This admittedly is not as precise a theory

as desired; however, other investigators have not devised a

theory of intraparticle diffusion for liquid adsorption which

is acceptable to the author, nor has the author. One con-


trolling factor in this study was the limitation of the

storage capacity of the IBM 650 computer. It was found

that over 60 per cent of the machine capacity was required

to store the "program," the sequence of instructions which

the machine follows to solve the problem. The remaining

storage was not sufficient to permit the addition of a third

independent variable, particle radius, to the other two,

time and bed depth. It would have been necessary to include

particle radius if intraparticle diffusion were treated as

a separate item. The required storage is available on larger

computers, however. Based on the results of the computations

of this study, it now appears that particle radius might

have been handled with the IBM 650, if the ranges covered by

the other two variables, time and bed depth, were suitably


All analytical solutions which have been published

to date have of necessity each been based on a particular

form of the adsorption equilibrium relationship, which

expresses the relation between x, the composition of the

unadsorbed liquid phase, and y, the composition of the ad-

sorbent-free adsorbed phase. This diagram is similar in

appearance to the usual vapor-liquid equilibrium diagram.

Because of the extreme difficulty in solving the equations,

most have assumed a straight line relationship, which is

strictly applicable only to dilute gas phase adsorption.


The Langmuir equation, (y = x/a+bx) has been used for an

approximate solution, assuming chemical kinetics to be the

controlling rate. Neither of these forms expresses satis-

factorily the equilibrium of liquid phase adsorption over a

very wide range. In fact, usually no one algebraic expression

fits adsorption equilibria over the complete diagram. It is

quite often necessary to fit two or more algebraic expressions

to liquid phase adsorption equilibrium data. Because of this

an analytical solution cannot be generally applicable to

different systems. Moreover, an analytical solution is very

complex, even when based on the simplest straight line

equilibrium relation. The computation of the infinite series

which usually result in analytical solutions could easily

require a computer. It is of importance that a computer

solution can be obtained no matter how complex the equilibrium

relationship, thus "tailoring" the solution to the particular

system under study, and thereby removing a basis for conjec-

ture when comparing the calculated solution with the experi-

mental results.

B. Derivation of Equations

A material balance (using volume fraction composi-

tions) for component A, the more strongly adsorbed component,

can be made over a differential section of the adsorption bed.

Equating the loss from the fluid stream to the gain by the


adsorbed and unadsorbed phases,

[ .e dLde = + AdLde [fv(x/60)L + PbVp(y/6e)LJ
rearranging gives:

(ox/AL)e + (Afv/Q)(ox/oe)L = (PbVpA/Q)(oy/be)L (1)

which is the equation of continuity written in volume


The classical mass transfer rate equation for dif-

fusion of component A between phase 1 and 2 across a film

whose area per unit volume of bed is unknown is,

rA = KLa (CA1 CA2)

for equimolar countercurrent diffusion of components A and B.

It is assumed that the conditions of equimolar

countercurrent diffusion are approximated closely enough by

the adsorption process, in which component B is displaced by

component A, to permit this form of mass transfer equation

to be used. The coefficient, KLa, is assumed to remain con-

stant as CA varies. Thermodynamically, it is possible that

the coefficient, KLa,would be more constant if based upon

activities instead of molar concentrations; use of the above

equation may assume ideal solutions. If such is true, a

further refinement would be to include the activity coef-

ficient in the above relation.

Writing the mass transfer rate equation for a

differential section of an adsorption bed,


KLa (CA CA*) (AdLde) = (pbVpA/Vm)(6y/6e)L(dLde)

Here, CA represents the composition of the unad-

sorbed bulk liquid phase, and CA* is the composition of the

liquid phase which is in equilibrium with the adsorbed phase.

Note the assumption that the resistance of the adsorption

process itself is negligible, so that CA* may be used in

the above equation.
Rearrangement gives:

CA CA* (pbVp/KLaVm) ()L

but by definition CA x/Vm

substitution for CA gives:
x x* = (PbVp/KLa)(6y/6e)L (2)

Equations (1) and (2), with the equilibrium x-y

relation for the system under consideration, represent the

mathematical problem to be solved, given suitable boundary

Before attempting a solution, it is desirable to

transform equations (1) and (2) into a dimensionless form so

that a solution using a particular equilibrium relation will

be as general as possible, thereby permitting evaluation of

the solution without prior knowledge of such parameters as

fv, Pb, Vp, Q, A, and KLa. To effect such a transformation,
two new independent variables are chosen:


Let T = (KLa/pbVp) [e (Afy/Q)(L)l (3)

and H (KLaA/Q)(L) (4)

The resulting transformation equations are,

(Oy/O )L (6y/ T)H(KLa/pbVp)

(Ox/80)L = (6x/6T)H(KLa/pbVp)

(Ox/DL)e = -(AfvKLa/QpbVp) (x/aT)H + (KLaA/Q) (x/aH)T

Substitution of these relations into equations (1) and (2)


(bx/WH)T = -(6y/OT)H (5)
x x* = (3y/6T)H (6)

Equations (5) and (6) together with the x-y equilibrium

relationship, express, in the desired dimensionless form, the

mathematical relations which the adsorption process obeys,

according to the assumptions listed originally. A solution

of these equations, based on the proper boundary conditions,
should be correlatable with experimental data. It should be

noted that all of the physical properties involved in defin-

ing H and T are readily measured experimentally except KLa.

The evaluation of KLa must hinge upon the matching of the

calculated solution with experimental data.

C. The Dimensionless Parameters H and T

It is important to the statement of the boundary

conditions and to the understanding of the results of the

calculation that the physical significance of the dimension-

less parameters, H and T, be understood. Multiplication of


both sides of equations (3) and (4) by (pbVp/KLa) gives,

(PbV /KLa)(T) 0 (Afv/Q)(L) (7)

(PbVp/KLa)(H) = (APbVp/Q)(L) (8)
The net dimension of both sides of equations (7) and

(8) is time. A study of the right hand side of these equa-

tions will reveal the following interpretation of T and H.

The parameter T is proportional to the actual time

elapsed since introducing feed liquid into the adsorption

bed in excess of that which is required to fill the void

volume of the bed to point L by the feed flow rate Q.

The parameter H is proportional to the time that

would be required to fill the adsorbed phase volume of the

bed to point L by the feed flow rate Q. The proportionality

constant is the same as the one for T.

An alternate way of expressing the above would be to

state that T is proportional to the volume of liquid which

has entered the bed in excess of that required to fill the

void volume to point L, and H is proportional to the volume

of liquid which is required to fill the adsorbed phase (pore)

volume of the bed to point L.

Some reflection will show that for a given bed depth,

L, if H = T, then the liquid front has just reached point L

and both the void and pore volumes of the bed are filled to

the point L.


D. Boundary Conditions for the Liquid
Phase Fixed Bed Process

Inspection of equations (5) and (6) shows that there

are two dependent variables, x and y, and two independent

variables H and T. Only first order partial derivatives are

present, suggesting that only two boundary conditions, one

fixing x and one fixing y along two different axes, will

suffice. Physically, it can be seen that for a given bed

and given flow rate, the inlet feed composition and the

initial condition of the bed determine the resulting column

operation. Since the properties of the bed, the flow rate,

etc., are contained within H and T, they need not be con-

sidered mathematically.

The condition of constant inlet feed composition

corresponds to the condition that at L = O

x xF, for all 0 > O

This is easily converted to the dimensionless system by

the condition

at H = O

x = XF, for all T > 0

In other papers, the second boundary condition has

been met by considering that at T = O

y = 0, for all H > 0

which is equivalent to the physical case of a bed saturated

at 0 0 with a liquid of composition pure B. This would

insure that y = O initially, and that when the feed liquid


front arrives at point L by pushing out the liquid in the

void volume of the bed, the composition, y, of the liquid on

the adsorbent is zero. This, however, is not the case for

an initially dry bed.

For the initially dry bed, it was seen above that the

instant of filling corresponds to T = H. It then will suffice

to state a boundary condition for y along the boundary T = H,

if possible. Considering the physical problem, as a given

adsorbent particle fills, there are two extreme cases which

may occur. The liquid phase and the adsorbed phase at the

moment of filling may be in complete equilibrium, indicating

that diffusion of components A and B happened more rapidly

than the filling; or, the other extreme, the liquid phase and

the adsorbed phase may be of the same composition at the

moment of filling, indicating that the diffusion process is

very slow compared to the rate of filling. In actual fact,

it is of course probable that the physical process which

occurs is somewhere between the two extremes, depending upon

the filling rate. However, for lack of a better criterion,

it is certainly more probable that in the majority of cases

the diffusion rate.is quite slow compared to the filling rate.

It has been shown (73) that each individual particle takes

something on the order of one minute or more to come to

equilibrium in batch experiments, and the filling process,

even in the smallest columns, is completed at the rate of

many, many particles per minute.


The boundary condition chosen in this study, based

on the above observations, is

for T = H, all T and all H_ O

x y

which expresses mathematically that as each particle in the

bed fills, the rate of diffusion of components A and B is

negligible compared to the filling rate. Note that such a

boundary condition is not easily applied when attempting

an analytical solution to a set of equations, but, as will

be seen in the description of the numerical method, it pre-

sented no insurmountable problem in computer analysis.


A. Numerical Methods

The general procedure for solving differential equa-

tions by means of numerical techniques is covered by many


To solve a partial differential equation or equa-

tions, it is necessary to substitute, in effect, a set of

simultaneous differential equations, which are integrated

numerically and simultaneously by standard numerical tech-

niques. The voluminous number of computations required and

the quantity of numbers to keep track of during the integra-

tion make it imperative that the modern high speed computer

be used when dealing with partial differential equations.

The adsorption problem can be demonstrated graph-

ically in the following manner.
x y

sF y = xF
x x(T,H) y y(T,H)

T/ \ H-T T H-T



The two sketches portray the three dimensional pic-

ture of the desired relationships. The surface, x x(T,H)

and the surface y = y(T,H) represent the functions which

satisfy the partial differential equation and its boundary

conditions. Along the boundary H = O, x is shown to be con-

stant, xF, the feed composition. Also along this boundary,

y increases from xF, its initial value as the first drop fills

the first section of the column, to YF*, the value in equi-

librium with the feed. Along the boundary H = T, both the

x and y surfaces follow the same curve, as prescribed by

the second boundary condition. The general shape of the

curve is known before hand, but the actual boundary condi-

tion is merely that x = y. The values of the two function

between these two boundaries make up the surfaces represent-

ing the solution to the problem.

A rectangular grid has been superposed at the base

of the figures. This grid represents the finite values of

H and T at which the numerical solution provides values of

x and y. As the grid is made smaller the resulting numeri-

cal solution will approach the true solution very closely,

but also many more points must be computed. In this problem,

capacity was available in the computer to compute values for

a grid composed of 200 T and 200 H points. From the sketch

one can see that this would involve the computation of x and

y for a total of 20,000 grid points each time the problem is

worked. As the computer required about four seconds to


compute each point, the computer time required would have

been prohibitive, except that it was found unnecessary to

compute all of the points. Since the physical problem is

such that an adsorption "wave" is formed in both the liquid

and adsorbed phases, and that this "wave" moves through the

column, there are a great many points before and behind the

wave whose composition is fixed. In front of the wave is a

section of the column containing pure B, where both x and y

are zero; behind the wave is a section of a column in which

the liquid composition is xF and the adsorbed phase compo-

sition is YF the value in equilibrium with the feed liquid

composition. In both of these sections no mass transfer

takes place, and it is not necessary to compute changes in

the values of x and y. This fact was incorporated into the

computer "program," and the computer did not bother to com-

pute values for x and y outside of the wave itself. The

criteria chosen for the wave boundaries were that a point

was considered inside the wave whenever the compositions of

the point immediately adjacent to it were such that either

y > 0.00005 or xF-x > 0.00005. A check computation made with-

out these restrictions revealed that the results of the

solution were not affected within four significant figures,

which was considered sufficient precision. This maneuver

cut the computation time down to 5-10 hours per solution.


B. Description of Integration Procedure
M C) r_- r-4 (q m
I t + ++

--- i-1 H
-) 1


The numerical integration procedure can be described

as follows:

Given the value of (x,y)ij for a particular grid

point, (i,j), (see sketch above) within the desired H and T

boundary, x x* at this point may be computed from the

equilibrium x y relationship. From equations (5) and (6),

p. 24, the partial derivatives (6x/5H)T and (6y/6T)H at

(i,j) should equal -(x x*) and ( x x*) respectively.

The value of x at the neighboring grid point (i+l,j) may be

estimated by a suitable formula for numerical integration.

The simplest formula is that used by Schmidt in heat trans-

fer calculations, which consists of assuming that (bx/cH)T

is constant between the point (i,j) and the point (i+l,j).

To put it more elegantly, a straight line may be fit over

the AH increment from (i,j) to (i+l,j) utilizing the value

of x and the slope, (6x/oH)T, both evaluated at (i,j).

Similarly, the value of y at the grid point (i,j+l) may be


computed by fitting a straight line over the AT increment

from (i,j) to (i,j+l) utilizing the slope Oy/ST)H and value

of y at the point (i,j). However, this is the crudest of the

numerical integration formulae. For the resulting solution

to be even approximately close to the true solution, it is

necessary to use very small AH and AT increments. If inte-

gration formulae be used which fit higher degree polynomials

to the curve in the neighborhood of the point (i,j), the

precision of the integration process is vastly improved, and

much larger AH and AT increment sizes can be used.

It was decided, by trying alternate integration

formulae on the computer, that, to obtain the degree of pre-

cision required and yet cover a large range of the H and T

variables with the 200 increments allotted, it would be

necessary to use integration formulae which fit at least

second degree polynomials to each integration step. The

formulae used are listed in Table 1.

Formula number 1, which fits a second degree poly-

nomial over two increments, was used to compute values of y

at points corresponding to (i,j+l) in the sketch. This equa-

tion requires no trial and error. Formula number 2, which

fits a third degree polynomial over two increments (thus

requiring a trial and error solution) was used to compute

values of x at points corresponding to (i+l,j) in the sketch.

Two different formulae were used simply because it was


impractical to fit a third degree equation in both direc-

tions, as a double trial and error procedure would have

been required. Use of such a double trial and error proce-

dure would have increased the computing time by a factor of

about 20. It was, therefore, necessary to compute in one

direction without a trial and error procedure, and the T

direction was arbitrarily chosen.

The above discussion holds for the computation of

all "normal" interior points; however, for points near the

boundaries T 0 and T H different formulae were required

to maintain at least second degree precision for all cal-


It is of interest to describe in detail the first

few steps in the computation of a solution, so that an

accurate picture of the manner in which the boundary condi-

tions were applied may be seen. The procedure followed in

starting a numerical integration is outlined below:
0 H

1 2

6 7 9

10, 4



1. Refer to the above sketch of the H and T axis with the

superposed grid. At T 0 and H O, both x and y were set

equal to xF, the feed liquid composition. This corresponds

to the condition that the first differential layer of parti-

cles in the column is filled with feed liquid in both void

and pore volume.

2. The value of x for points 1, 3, 6, 10, etc., was fixed

equal to xF. This meets the boundary condition that x is

always xF at the column inlet.

3. The value of y at grid number 1 was computed first, using

integration formula number 3, Table 1. This formula fits

by trial and error a second degree curve over one increment

to the desired relationship that dy/dT x x* at constant

H. The equilibrium relationship must of course be used to

compute x* from values of y.

4. The value of both x and y at grid point 2 was computed

next using integration formula number 4, since x y along

the H T axis. This computation is also trial and error,

fitting a second degree equation over one increment to the

desired relationship that dx/dH -(x x*) at constant T.

5. The value of y at point 3 was next computed using inte-

gration formula 5 which fits a third degree equation over

two increments. All subsequent values of y along the H O

axis were computed by this formula.

6. The values of x and y at point 4 were next computed by

trial and error simultaneously using formulae 4 and 6, which


fit second degree equations over one time increment. Since

it was desired to fit at least second degree equations in

every integration step, the simultaneous calculation of

x and y for this point was required.

7. The value of x and y (equal) for grid point 5 was com-

puted using formula 2, which fits a third degree equation

by trial and error over two time increments. All subsequent

points along the H T diagonal were calculated using this


8. The value of y at point 7 was computed by formula 1,

which fits a second degree equation over two time increments

without trial and error. This is the first instance in which

formula number 1, for a "normal" point, was used.

9. The value of y at point number 6 was computed by formula


10. The value of x at point 7 was computed by formula 4,

which fits a second degree equation by trial and error over

one increment. All subsequent values of x along the H AH

axis were computed by formula 4.

11. Values of x and y at point 8 were computed simultaneously

in order to use at least second degree equation accuracy.

Formulae 2 and 3 were used, involving a double trial and

error. All subsequent values of x and y along the diagonal

neighboring the H T diagonal were computed with these

formulae. This is the only instance of double trial and

error involved.in this procedure.


12. The value of x and y (equal) for grid point 9 was

computed by formula 2.

13. Subsequent calculations proceeded, using formula 1 to

compute values of y, and using formula 2 to compute values

of x for all normal interior points. Points on and neighbor-

ing to the boundaries H 0 and H T were calculated as

noted in steps 5, 7, 10, and 11.

C. Computer Program

The development of a computer program to perform

the computations described above was a tedious, drawn-out

process rife with rewrites and changes in procedure. The

IBM programming procedures first had to be learned, largely

from the manuals available for this purpose, but with a good

deal of help from the University of Florida Statistical

Laboratory personnel. The specific programming method chosen

was the IBM SOAP II method, developed for the Type 650

machine. The program itself is much too long and complicated

to be discussed here, although a complete copy of the SOAP II

program is included in the Appendix. The SOAP II procedure

has the advantage that the program is listed in a symbolic

code as well as the numerical machine code. The printed

program also includes comments inserted specifically to help

orient the operator as to the calculations being performed

in each particular section of the program. In Figure 1 a

"Flow Diagram" of the program is presented. It must suffice


to point out here that if the program as listed in the

Appendix be punched into standard IBM cards according to

the SOAP II format, and if the instructions accompanying

the program be followed, any competent 650 operator could

utilize this program to solve a binary liquid phase adsorp-

tion fractionation problem, limited, of course, to the basic

assumption as to the mechanism involved on which the work

was based. The program listed in the Appendix uses the alpha-

type equilibrium diagram familiar to distillation processes.

To work a problem, it is necessary only to read the program

into the Type 650, then to read in one "problem" card, which

provides the information as to the feed composition, magni-

tude of alpha, size of the AT and AH increment, and frequency

desired in the punching of the answer cards. The computed

answers are punched by the machine at pre-determined incre-

ments of AT, chosen arbitrarily for each problem on the

problem card.

If it is desired to perform a calculation using the

equilibrium diagram of a specific system, as was done in this

work, a sub-routine for computing x* from y must be added to

the program listed here in such a way that it replaces the

equilibrium diagram calculation of the listed program. Again,

this is a fairly simple task for an experienced IBM programmer.

The author will be glad to furnish additional information

other than that given herein to enable any interested party

to make use of the program. Although the development of the


program represents some four to six months of intensive

effort, it, like any other computer program, is now

available for future use at any time.


A. Problem Solutions

The numerical solution to the binary liquid adsorp-

tion fractionation problem was run twenty-three times on

the IBM 650 computer. Each time the problem was solved,

there were two parameters which were subject to change.

These were the composition of the feed liquid, xF, and the

shape of the x-y equilibrium diagram relating the composi-

tion of the adsorbed and unadsorbed phases at equilibrium.

The latter parameter actually consists of one or more equa-

tions which express the x-y relationship over the range

x = O to x = 1. These equations were included as a sub-

routine of the computer program, and to make a change it was

merely necessary to place in the deck of program cards the

proper subroutine deck for the x-y relationship desired.

The problems which were computed are summarized as

to the parameters used in each solution in Table 2. Four

of the solutions were run for comparison with experimental

data of Lombardo (73) (Benzene-Hexane fractionation on

Silica gel) and with experimental data of this study (Toluene-

Methylcyclohexane on Silica gel). The remaining solutions

were run to provide a set of curves for use in evaluating the

effect of the parameters on column operation. This latter



group of nineteen solutions was based on a constant-alpha.

type of equilibrium diagram as is used in correlating vapor-

liquid equilibria of distillation systems. In adsorption,

alpha is defined exactly as in distillation: (y/l-y) -

(1-x/x) -a Although adsorption equilibria seldom cor-

relate perfectly with a constant alpha, this is one of the

few algebraic formulae which approximate the general shape

of the adsorption x-y diagram over the complete range. It

was believed that a set of solutions based on the constant-

alpha equation should prove useful in interpreting the effect

on column performance of varying the two parameters, and it

was hoped that solutions obtained with constant-alpha dia-

grams which approximate the experimental equilibrium data

might be used successfully to predict column performance.

The answers to the solutions were punched by the

computer onto standard IBM cards as they were calculated.

Each card contained six answers plus identifying informa-

tion. The six answers were actually three sets of x and y

values corresponding to three bed grid points (H) at a given

time increment (T). To provide enough answers to draw smooth

curves through the points, it was not necessary to punch out

x and y values for every one of the grid point intersections.

In fact, this was not desirable at all, because the relative

change in the values of x and y between adjacent grid points

was quite small (in numerical integration this is a necessary

prerequisite for accuracy). Usually the choice was to punch


out answers for every tenth dimensionless time (T) incre-

ment. The entire adsorption wave was punched out at this

time increment, but, as explained before, the constant

composition sections in front of and following the wave were

not punched.

The information from the cards was then printed in

list form by means of an IBM 403 tabulating machine. From

these lists of calculated data points, graphs of the solu-

tion were prepared. It was found that there were three

graphs required to portray the information from each solu-

tion. On one, values of x, the liquid phase composition,

were plotted against H, the dimensionless bed depth para-

meter, along lines of constant T, the dimensionless time

parameter. A second plot was required to give the same

information about y, the adsorbed phase composition. A

third plot was made of the ultimate, or asymptotic, wave

shapes which are reached by the adsorption wave as it travels

down the bed. Typical graphs of problem solutions are shown

in Figures 2-11. Only those solutions referred to in this

dissertation are shown. The tabulated data from which the

graphs were computed were much too voluminous to include here.

It is planned to compile the data and graphs for all of the

twenty-three solutions under a separate cover for ready


It was found that in every problem solution an ulti-

mate wave shape was formed provided sufficient distance along


the bed depth parameter H was covered. Several authors have

discussed the existence of the adsorption wave, and some have

speculated upon the conditions or requirements that an ulti-

mate or invariant shape be formed. The discovery that an

invariant wave shape was formed in these problem solutions

prompted a further analysis of the conditions necessary for

its formation.

B. The Asymptotic or Ultimate Adsorption Wave

It is an experimental fact that if an adsorption

column is long enough (and if there is no adsorption azeo-

trope) eventually there will be set up three distinct zones

which travel through the column. Refer to the following


< zone 1 zone 2 zone 3

0 y

0 D
Bed Depth, L


In zone 1, the adsorbent has preferentially adsorbed

component A from the liquid phase passing over it until the

composition of the adsorbed phase has reached YF the compo-

sition in equilibrium with the feed. When this occurs, there

is no tendency for further exchange of material between the

two phases, and XF is also constant in zone 1. In zone 2,

mass transfer is taking place, and the composition of both

y and x vary with bed depth, L, and with time, 0 Zone 2

is the adsorption wave. However, the continual removal of

component A from the liquid phase as it travels through the

bed eventually becomes complete, and the composition of the

liquid at the head of the wave becomes zero with respect to

component A. Zone 3 represents the portion of the bed over

which pure component B is passing. As in zone 1, there is

no tendency for mass exchange between the phases and compo-

sitions are constant with e in this section. Remember, how-

ever, that all three zones are traveling through the column.

Although these three zones will be formed in any

adsorption fractionation experiment (with the exception of

azeotropes), the questions of importance are (1) How soon

will the three zones be formed? (2) What is the width of

zone 2? (3) Does zone 2 reach an ultimate non-changing shape?

(4) If so, at what rate does zone 2 travel through the



The numerical solutions obtained with the IBM 650

in this work provided the answers to these questions in each

case investigated, but did not shed light upon other cases,

e.g., equilibrium diagrams of different shape from those

studied here. This, admittedly, is one of the main draw-

backs to numerical solutions.

If one starts with the assumption that a zone 2 of

non-changing shape is formed, its velocity may be calculated.

Since zone 3 is continually building up in length as the

adsorption progresses, zone 2 must move through the column

at a rate slower than the rate that the liquid passes through.

Therefore, if we imagine a column in which the adsorbent is

made to move in the opposite direction from the liquid, there

is a certain rate of adsorbent movement which will cause

zone 2 to remain stationary. There is, then, a counter-

current moving bed which is exactly analogous to the fixed

bed operation. See the following diagram.
Countercurrent Case Fixed Bed Case
Zone 2 Stationary Zone 2 Moves

y0 p x-0 x-O

Zone Kone
2 2

W i
y-y* x-xF Q
X Xp


If Q is the volumetric flow rate of the liquid

through the stationary bed, Q/Af is the velocity of the

liquid through the bed void volume. This velocity would

have to be reduced by an amount equal to the velocity of

travel of the adsorbent in the countercurrent case, in order

to maintain the same relative velocity of fluid through the

bed in the two cases. If W is the mass rate of flow of ad-

sorbent required to maintain zone 2 stationary, W/pbA is the

velocity of the adsorbent through the bed. Therefore, the

countercurrent liquid feed velocity may be related to the

fixed bed velocity.
Q'/Afv = Q/Afy W/pbA (9)

where Q' represents the volumetric liquid
feed rate in the countercurrent case.

and Q represents the volumetric liquid
feed rate in the fixed bed case.

A volumetric material balance on component A about zone 2 for

the countercurrent case gives,

(WVp)yF* (Q')(xF) (10)

Substituting for Q' from equation (9),

(WVp)yF* (Q fvW/Pb)(xF) (11)

A little study will show that the velocity of the wave, Vw,

when the liquid feed rate is Q, is equal to the velocity of

the adsorbent bed required to maintain zone 2 stationary when

the liquid feed rate is Q'. Solving equation (11) for W/pbA,

the adsorbent bed velocity, gives,


Vw W xFfv (Q/Afy) (12)
PbA (XFfv + VppbYF*)

Therefore, equation (12) above gives the velocity of the

wave traveling through a bed, based on the assumption that

a wave of invariant shape is formed.

It is desirable to transform this velocity into a

velocity in terms of the dimensionless parameters H and T.

V, may be considered as the ratio of AL/AO which is required

to maintain a given x or y composition in the wave constant.

Similarly, the adsorption wave velocity in dimensionless

parameters would be the value of the ratio of AH/AT corre-

sponding to Vw. This transformation may be obtained by

substituting for AH/AT using equations (3) and (4), which

define T and H in terms of 0 and L.

AT (KLa/pbV )(A0) (KLaAfv/QPbVp)(AL) (13)

AH (KLaA/Q)(AL) (14)

From (13) and (14),

AT/AH (Q/ApbVp)(LO/AL) (fv/PbVp) (15)

Therefore, designating the wave velocity in terms of the

dimensionless parameters as Vwd,

1/Vwd = (Q/ApbVl)(l/Vw) (f/PbVp) (16)

Substitution for Vw from equation (12) above, yields the

simple relation,

Vwd = XF/YF* (17)

Equation (17) points out that the velocity at

which the adsorption wave moves through the column in terms


of the dimensionless parameters is merely the ratio of the

feed liquid composition to the adsorbed phase composition in

equilibrium with the feed. Note that the physical properties

of the bed do not enter into the relation. This relation can

be verified readily by inspection of the calculated solu-

tions (Figures 2-10) to the adsorption fractionation problem.

In every case, after sufficient bed depth H was reached, the

wave reached an ultimate shape and a velocity, AH/AT, which

equalled xF/YF*.

C. The Shape of the Asymptotic Wave

The concept of a stationary wave maintained by a

movement of the adsorbent bed countercurrent to the liquid

flow can also be utilized in computing the ultimate wave

shape. Consider again a diagram of the liquid and adsorbed

phase composition plotted vs. bed depth, L, for the counter-

current bed, and assume that the wave is being maintained

stationary by appropriate flow of the liquid and adsorbent.

o y


k 0
0 L

Bed Depth, L


A volumetric balance for component A over section dL yields,

(dy/dL)WVp)(dL) = (dx/dL)(Q'dL) (18)

Note that total differentials may be used since the wave is

assumed to be stationary. Rearrangement and integration

between limits gives,

yF* xF

/ *dy = Q'/WVp dx
0 0

This integration is easily performed, so that,

yF*/xF = Q'/WVp (19)
Equation (19) relates the flow rates required for main-

tenance of the stationary bed to the feed liquid composition

and equilibrium adsorbed phase composition. Equation (19)

is equivalent to equation (12); note that Q' is used in the

former and Q in the latter. If the upper limit of the above

integration be made indefinite, there results,

y (Q'/WVp)x (yF*/yF)x (20)
Equation (20) points out the relation that must hold between

y and x at a given point in the adsorption wave, if the

wave is to become invariant, as was assumed. This, then,

places a restriction upon the shape of the equilibrium x-y
diagram which will permit an adsorption wave of ultimate

or invariant shape to become established. Consider the

following diagram.





The straight line OA, which connects the origin with the

equilibrium curve at the point representing the feed condi-

tion, can be thought of as the operating line for this process.

Everywhere along the invariant adsorption wave, whether the

wave is stationary or moving down the column, x and y for a

given bed point at a given instant must fall on the line

OA, that is obey equation (20). This relation may also be

verified by referring to any of the calculated curves for the

ultimate wave shapes (Figures 2-10).

It is apparent that if the equilibrium curve were to

cross the line OA, then the liquid and adsorbed phase compo-

sitions could not possibly follow line OA. Hence, a require-

ment for the formation of an invariant adsorption wave is

that the equilibrium diagram may not cross the "operating"

line connecting (xF,yF*) with the origin.


Further information about the invariant wave may

be derived by equating the rate of mass transfer between the

two phases using the proposed mass transfer rate equation.

Again considering section dL in the countercurrent bed,

(dy/dL)(WVp)(dL) = KLa(x x*)(A dL) = (dx/dL)(Q')(dL)

Thus, rearranging and integrating,

2LA (L2 ) (21)
Sdx/(x-x*) = (KLaA/Q') dL = KL. (L2 Ll) (21)

xI L1

The integration is indicated between two arbitrary composi-

tions because, theoretically, an infinite length of bed,

based on the assumed rate mechanism, is required for the

entire wave; this is because at the two ends of the wave the

driving force for mass transfer is zero. However, by inte-

grating between two compositions other than the extreme ends

of the wave, the wave shape as a function of depth, L, can

be obtained. The left hand integral can be evaluated because

x is related to y from the equilibrium diagram, and y is

related to x by equation (20). A convenient lower limit

for the integration is xF/2, so that by integrating in both

directions to various values of x, the ultimate wave shape

may be obtained and plotted. Since the right hand side of

equation (21) before integration equals dB, the wave shape

equation may be written in terms of the dimensionless para-




Sdx/(x x*) = Hx H (22)


In most cases the left-hand integral must be ob-

tained by numerical means because of the difficulty in

integrating the expression analytically. In effect, the

computer solutions which were run in this work performed

this integration, as evidenced by the ultimate wave shapes

which were obtained. It is important to remember that the

computer solutions also provided the relations for column

operation before the ultimate wave shape was formed. An

inspection of the graphs of the solutions shows that in

general, 90 per cent of the wave (excluding the ends) becomes

invariant by the time the composition of the liquid at the

filling front drops to less than 5 per cent of the feed

composition. Therefore, with a knowledge of the rate of

advance of the adsorption wave (Vwd = xF/yF*), the computer

solution for the ultimate wave shape, and the computer solu-

tion during filling of the first portion of the bed, the

composition of both phases may be quickly calculated for

any H and T thereafter.

D. Computation of HETS From Fixed Bed Data

Because continuous countercurrent moving bed ad-

sorbers are readily analyzed by an equilibrium stage concept,

in which the number of theoretical stages in the column


necessary to give a given separation may be readily deter-

mined, the experimental determination of the height equiva-

lent to a theoretical stage (HETS) has always been of

interest. It is apparent that an experimental apparatus

utilizing the countercurrent principle could be built and

the determination of HETS made by suitable experiments.

However, it is not easy to construct true countercurrent

apparatus in the laboratory. It would be more desirable to
devise a means of predicting the HETS of a moving bed from

a simple fixed bed experiment.

The analysis of the adsorption process made in the

previous sections affords a way of doing this. It has been

pointed out how the establishment of an invariant wave shape

is possibly subject to one restriction concerning the shape

of the equilibrium diagram, a restriction which is almost

always met. It was also shown that the movement of the

ultimate wave through the column is equivalent to a counter-

current experiment in which the adsorbent and liquid feed

rates are adjusted to maintain the same velocity of feed

liquid through the bed and to maintain the adsorption wave

stationary. It was further shown that.the flow rates between

the two cases can easily be related.

This leads to the conclusion that every fixed bed

experiment in which the column is long enough for the ad-

sorption wave to be established is exactly equivalent to a

continuous countercurrent experiment. The one difficulty


is that the operating line for the continuous countercurrent

experiment is such that the adsorbent at both ends of the

column is in equilibrium with the liquid.

If the number of plates required for this separation

were to be stepped off, there would, of course, result an

infinity of plates because of the two pinched sections.

However, it is suggested that the HETS may nevertheless be

obtained from the fixed bed experiment.

Since the experimental effluent volume vs. composition

curve for the adsorption wave can be readily obtained, it may

be transformed into a liquid composition vs. bed length quite

readily, assuming the void fraction of the bed has been

measured. Then, instead of determining the number of stages

required for the complete separation, it is suggested that

the number of stages be stepped off between the equilibrium

and operating line for some arbitrary separation, say from

0.9xF to O.lxF. See the following diagram.

0.9xp r Y* eA


SI N stages
x Y

O. lxF 0-
L1 L2 0.1 0.9


The bed depth required for the liquid composition to change

from 0.9xF to O.lxF can be determined from the wave shape

which was computed from the experimental effluent curve,

and a simple division by the number of theoretical stages

stepped off will give the HETS. Whether or not this HETS

will be constant for any pair of compositions is subject to

conjecture. Nevertheless, the procedure described above

affords a method of determining HETS from fixed bed experi-

ments which should, if correlatable, be exactly analogous to

the HETS required in the design of a continuous counter-

current bed.


A. Adsorbent

Commercial Davison silica gel (6-12 mesh) and Alcoa

alumina (8-14 mesh) were used as adsorbents. A large parti-

cle size was chosen as there were already available in.the

literature both equilibrium data and kinetic data on systems

using small particle size adsorbents. It was planned to

secure data in this work with large particle sizes, which

together with the previously published data of Lombardo (73)

for 200 mesh silica gel would provide a good basis for com-

parison with calculated results. Both the silica gel and

the alumina were heated to 2000C. and stored in airtight

desiccators prior to use. This insured that their pore

volumes were free of volatile contaminants, thereby promot-

ing the reproducibility of experimental data.

B. Adsorbates

Methylcyclohexane and toluene were used as adsorb-

ates. These compounds have similar molecular weights, but

are of different chemical configuration. There is a definite

selective adsorption exhibited by both the silica gel and

the alumina for toluene when binary solutions of these two

liquids are adsorbed onto the adsorbents. Toluene is, there-

fore, component "A" for these systems.


C. Experimental Procedures

1. Specific Pore Volume, Vp

A weighing bottle containing a weighed quantity of

adsorbent was exposed in a closed desiccator, maintained at

normal room temperature, to the vapors of the pure adsorbate

(contained in a beaker also placed in the desiccator) for a

period of two weeks. At the end of this time, which had

previously been shown to be adequate for equilibrium to be

established, the adsorbent was re-weighed to determine the

weight of adsorbate taken up by the adsorbent. From these

data the weight of adsorbate adsorbed per gram of adsorbent

and Vp, the specific pore volume of the adsorbent, milli-

liters per gram, were calculated. The results of these

experiments are presented in Table 3.

2. Determination of x-y Equilibria

Approximately 20 grams of adsorbent was measured

into a 50 ml. flask, to which 20 ml. of a particular mixture

of toluene and methylcyclohexane was added. The flasks were

closed and left at room temperature for a minimum of four

hours. A sample of the liquid phase was removed and its

refractive index determined. From a refractive index cali-

bration curve (Figure 26, Table 26, Appendix) previously

obtained for toluene-MCH solutions, the composition of the

liquid phase, x, was determined. Values of y the composi-

tion of the adsorbed phase in equilibrium with the liquid


phase, were calculated from a material balance of the system.

This method of equilibrium determination has been used pre-

viously by Lombardo (73), Eagle and Scott (63), and Perez

(75). It has proven to be quite accurate over the largest

portion of the x-y diagram, assuming that the specific pore

volumes of the two adsorbents are very nearly the same. The

equilibrium data and diagrams for the two systems inves-

tigated here are presented in Tables 4 and 5 and in Figures

13 and 14. The equilibrium data and diagram of Lombardo (73)

for the Benzene-Hexane silica gel system are shown in Table 6

and Figure 15. In order to make the computations for ad-

sorption fractionation, suitable empirical equations had

to be fitted to the x-y diagrams. The equations which were

used for each system are given in each table.

3. Adsorption Fractionation Experiments

The experimental apparatus used was quite simple,

consisting of three pyrex glass adsorption columns, each

2.43 cm. in diameter, of varying lengths. The lengths were

approximately six inches, twelve inches, and twenty-four

inches. Each column was equipped with a side arm near the

top for removal of the effluent, since the liquid was fed

through the columns from bottom to top. A metal charging

bomb of approximately 400 ml. capacity was connected to a

nitrogen cylinder. The bomb was equipped with a filling

connection and valve which could be closed after charging

the bomb with feed liquid. During a run the feed liquid was


forced by nitrogen pressure from the bomb through poly-

ethylene tubing through a capillary tube flowmeter into the

inlet at the bottom of a column. A pressure regulating valve

on the nitrogen cylinder permitted very precise control of

the flow rate, as indicated by a manometer attached to the

capillary. It was thus possible to make a set of three runs

(one each through the three columns) in which the flow rate

and feed composition were maintained constant.

The columns were packed with adsorbent prior to a

run by carefully pouring the adsorbent into the column while

tapping continuously with a rubber mallet. The tapping was

continued and adsorbent was added until the top of the ad-

sorbent was level with the exit side arm, and the surface

of the adsorbent ceased to settle. By weighing the columns

before and after packing, the quantity of adsorbent added

was ascertained.

A run was started by opening the stopcock at the

bottom of the column and adjusting the nitrogen pressure

to give the desired manometer reading. The small capillary

orifices used in the flowmeter produced pressure drops of

about ten inches of mercury, so that only minor adjustments

of the nitrogen regulating valve were required during a run

to compensate for the rise in liquid level as the column


The effluent liquid was collected in graduated

cylinders, and samples of five drops (1/4 ml.) were collected


at regular intervals. The large diameter column was chosen

so that samples of five drops could be taken at about 5-10

ml. intervals, thereby giving instantaneous compositions

rather than average compositions, which would have resulted

if a very smallcolumn diameter were used.

An electric stopcloclk was started at the moment the

liquid reached the first particle of adsorbent, and record-

ings of the time vs. volume of effluent liquid collected

were made. The average flow rate during the run was ascer-

tained from this time and volume record. The refractive

indices of the samples collected were measured after com-

pletion of a run, and tables and charts of effluent liquid

composition vs. quantity of liquid collected were prepared.

The column experiments and the data obtained during

the course of these experiments are shown in Tables 8-22.

The experiments shown are only those which are referred to

in this dissertation. In addition, the data of Lombardo

for the benzene-hexane-silica gel system are presented in

Tables 23-25. Table 7 summarizes these data as to the

nature of the run and certain other factors.

4. Determination of Fraction Voids and Bed Density

In order to compare experimental data for the ad-

sorption fractionation runs with the calculated results

obtained with the computer, it was necessary to evaluate

the bed density and the fraction voids in each adsorbent bed.

This was done by taking various sizes of graduated cylinders,


50, 100, and 200 ml., and filling them carefully with ad-

sorbent. Bed densities were calculated from the weights

before and after filling and the cylinder volumes. By

tapping the cylinders with rubber mallets during the filling,

as was done when packing the adsorption columns, it was

possible to obtain reproducible bed densities. The bed

density, Pb, used in the equations of this dissertation, is

the grams of dry adsorbent per total volume of dry bed. It

was assumed that since the bed densities were reproducible,

the void volumes would also be reproducible. Consequently,

after weighing the cylinders filled with dry adsorbent, the

cylinders were then filled with pure toluene or methylcyclo-

hexane. Time was allowed for the pore volumes to fill, and

additional liquid was added until the liquid remained level

with the top of the adsorbent. A third weighing ascertained

how much liquid was added, and the void space in the bed

was then computed as the difference between the volume of

liquid added and the volume of liquid known to have been

adsorbed into the adsorbent pore volumes. The void fractions

and bed densities obtained in this manner for the adsorbent

beds are listed at the top of Tables 8-25, which present

the results of the adsorption fractionation runs.



The only method of comparing the results of the

computer calculations with the experimental data obtained

in this study and in the work of Lombardo is to test whether

the effluent composition curves of the adsorption fractiona-

tion experiments can be satisfactorily correlated by the

computed solutions.

It has been explained that there is one unmeasured

property of the system, KLa, which is contained in both of

the dimensionless parameters, H and T, used in the calcula-

tions. The success of the calculations depends on whether

for a given experiment a value of KLa can be found which

results in a good agreement between the experimental and the

calculated effluent curves, and whether the values of KLa so

obtained correlate with the flow rate of liquid through the


In fitting the calculated results to the experimental

data, there are two criteria which are considered. First,

the general shape of the adsorption wave should be approxi-

mated, and second, the wave should be at the proper location

in the bed at the proper time. It has been pointed out that

in a long enough column, the wave will eventually come to an

ultimate shape and an ultimate velocity. In the experiments



performed by Lombardo, the columns were sufficiently long

for this to occur. Since Lombardo did not make duplicate

runs at different column lengths, there was only one check

point for each run.

In those cases where the length of the column is

large compared to the length of the adsorption wave, there

is very little interest (other than academic) in an exact

solution to the problem of wave shape. A rough estimate of

the wave length in such a case, combined with the assumption

that the wave reaches the ultimate velocity within a few

wave lengths into the column (which it usually does) will

suffice to predict with good accuracy the quantity of pure

B which can be produced with a given column.

It is those cases in which the wave length is a sub-

stantial fraction of the column length that a more accurate

knowledge of the adsorption wave shape and position is re-

quired. It is precisely this case that cannot be handled

by the ultimate wave velocity and shape, but which requires

the complete solution, which was provided by the computer.

,The experiments performed in this work were aimed at creat-

ing conditionsiof column operation which would require the

computer solution.

A. Adsorption Fractionation Experiments of Lombardo

The effluent volume vs. composition curves for three

fractionation experiments on the benzene-hexane-silica gel


system are presented in Figures 16, 17, and 18. The experi-

mental data for each curve are listed in Tables 23, 24, and

25. These data which were published by Lombardo (72) were

the results of a Ph.D. thesis on adsorption fractionation.

Since the columns were long enough for the establishment of

the ultimate wave shape, the data were fitted to the calcula-

tions by means of the ultimate wave shape. The calculated

curve at H constant of Figure 12 was compared with the

experimental curves, and the value of KLa which best fit

each was chosen. The computed points using the chosen values

of KLa are also plotted in Figures 16, 17, and 18, and

curves are drawn through both the computed and the experi-

mental data. It can be seen that there is good agreement

between the shapes of the computed and the experimental

curves. The values of KLa used are plotted vs. the super-

ficial fluid velocity, Q/A, in Figure 24, curve A. It can

be seen that there was a good correlation between KLa and

the liquid velocity for the three runs.

B. Toluene-Methylcyclohexane Fractionation on Silica Gel

The effluent volume vs. composition curves for three

sets of fractionation experiments with the toluene-MCH-

silica gel system are shown in Figures 19, 20, and 21. The

experimental data for each of the nine runs are listed in

Tables 8, 16. A summary of all adsorption fractionation runs

is given in Table 7.


Each set consists of three separate fractionation

runs made under identical conditions except for the quantity

of gel used. It was desired to perform duplicate experi-

ments with different column heights so that the value of KLa

would be subject to three separate checks. These experiments

were run at rates which insured that the invariant or asump-

totic wave front was not established. Two computer solu-

tions, one at xF of 0.5 and one at xF at 0.1, both based on

the equilibrium diagram for this system, are shown in Figures


In order to fit the calculated solutions to the experi-

mental data it was necessary, as with the Lombardo data, to

find the value of KLa which best fit the curve shapes and

positions in the bed. Here, however, there were three curves

to be checked by the same KLa. The computed solutions are

plotted on the graphs of the experimental data for comparison.

It can be seen that the agreement between wave shapes

was not as good as resulted with Lombardo's data, although

the rate of movement of the waves through the column corre-

lated well. In each set only one value of KLa was needed to

correlate all three runs. The values of KLa used are plotted

vs. superficial liquid velocity, Q/A, in Figure 24, curve B.

The calculated curves are steeper and show.the charac-

teristic "s" shape more definitely than the experimental

curves. Previous investigators (76), have pointed out that

in ion exchange two factors may affect the steepness of the


curves. These factors are the relative adsorbability of the

adsorbent, and the relative contribution of intraparticle

diffusion to the total diffusional resistance. Two calcu-

lated curves using the same xF and KLa but using equilibrium

diagrams exhibiting considerable differences in relative

adsorbability will have different shapes. If the adsorbent

selectivity is low, the wave will be less steep, for a given

mass transfer coefficient, than if the selectivity is quite

high. This is because the magnitude of x-x*, the driving

force for mass transfer, is much lower for the column of low

selectivity. A similar effect will occur if the intraparticle

diffusion resistance contributes appreciably to the total

diffusional resistance. The concentration gradients set up

inside the particle tend to increase the quantity of compo-

nent A near the external film above the value of x*, which is

computed from the average adsorbed phase composition. This

causes the adsorption wave to have a shape which cannot be

duplicated exactly by adjusting KLa in the assumed rate


It may be concluded that intraparticle diffusion is

a definite contributor to the diffusional resistance in the

large particle size gel used in these experiments. This is

in qualitative agreement with theory, since the average length

of the internal diffusion paths per unit of surface area in-

creases with particle size. On the other hand, the Davison


"thru 200" mesh silica gel used by Lombardo was apparently

of a particle size small enough to permit the external film

resistance to control.

The values of KLa which best fit the computed curves

to the experimental data for the large particle size gel did

correlate with superficial liquid velocity, however, as seen

in Figure 24. It was gratifying to find that such a correla-

tion was possible even though feed compositions of 0.1 volume

fraction toluene and 0.5 volume fraction toluene were used.

Apparently the intraparticle diffusional resistance was not

altogether controlling at these flow rates, since there was

an increase in effective KLa when the fluid velocity through

the bed was raised.

C. Toluene-Methylcyclohexane Adsorption on
Activated Alumina

Data for the adsorption fractionation of toluene and

MCH mixtures on activated alumina are plotted in Figures 21

and 22. There were no computer calculations made using the

equilibrium diagram of this system, so that there is no com-

parison presented here between experiment and calculations.

It can be seen, however, that for the same feed compositions

and range of liquid flow rates as was used in the silica gel,

the sharpness of the fractionation, as measured by the shape

of the effluent curves, was better than that of the silica



D. Use of Constant-Alpha Type Equilibrium Diagrams

In the previous discussion of the numerical integra-

tion process, it was mentioned that a number of computer

solutions were obtained to the adsorption fractionation

problem using equilibrium curves of the constant-alpha type

in anticipation of the possibility of using them for approxi-

mate solutions to specific cases, whenever the true equilib-

rium curve of the system could be approximated by a constant

alpha curve. The results of such an approximation would

certainly be more valuable than the application of solutions

based on straight line equilibrium diagrams, which can approxi-

mate only a very small portion of an equilibrium curve.

It was hoped that a constant-alpha curve based on an

average alpha over the range of the fractionation experiment

would approximate the solution closely enough to be used in

many systems. It was discovered, however, that it is quite

important to use an equilibrium diagram which exhibits the

exact relative adsorbability, alpha, for the system at the

fedd composition. That this is necessary was shown in the

previous discussion on ultimate velocity of the adsorption

wave. For the wave to come to the proper ultimate velocity

(and, presumably, approach it in the proper manner) the value

of xF/YF* used in a calculation must be exact. This means

that, not an average alpha, but the alpha of the feed compo-

sition must be used. In adsorption systems, alpha is very


high at low values of x, and decreases with an increase in x.

This is demonstrated in Figures 13, 14, and 15.

An example of the results when a constant-alpha type

equilibrium diagram is substituted for the true diagram can

be seen in Figure 19. The computer solution for an alpha

of 3.0 (Figures 4 and 5) was fitted to the experimental data

of a run by choosing an appropriate value of KLa, as before,

which best fit the data. The resulting curves are shown in

Figure 19 on the same plot with the experimental data and the

curves obtained from the computer solution. It can be seen

that the approximate x-y diagram fit the experimental wave

shape better than the true x-y diagram. This anomaly is ex-

plained by reference to the previous discussion concerning

the effect of intraparticle diffusion on the adsorption wave

shape. Internal diffusion broadens the wave in a manner

similar to a low adsorbent selectivity. Since the alpha of

3.0 was lower average selectivity than the true equilibrium

curve, yet was the proper value at the feed composition, the

effect of intraparticle diffusion caused this solution to fit

the experimental data more closely.

E. HETS of Column Packing

A method was derived in the Results of Calculations

section for determining the HETS of column packing from fixed

bed experiments. It was suggested that the effluent curves

from fixed bed runs, when known to be of the ultimate or

asymptotic shape, can be transformed into column length units;

and the number of stages required for a given change in x for

a countercurrent column equivalent to the fixed bed experi-

ment may be determined by a graphical procedure. Dividing

the column length by the number of stages required for the

composition change results in a value for HETS which may then

be used in the design of countercurrent adsorption columns

operating with the same relative velocity of liquid through

the adsorbent as was maintained in the fixed bed experiment.

This procedure was applied to the experimental runs

of Lombardo, since the effluent composition curves were

thought to be invariant. The calculated values of HETS for

the three runs are plotted vs. Q/A, the superficial liquid

velocity in Figure 25, curve A. A definite correlation is

noted, with a strong dependence of HETS on the liquid velocity.

The number of stages required for a change in x from 0.05 to

0.45 was graphically determined in Figure 15. Three equi-

librium stages were stepped off.

The suggested procedure could not be applied to the

runs made with the toluene-MCH-silica gel system, as the

asymptotic adsorption waves were obviously not established.

However, an estimate was made based on the calculated ulti-

mate wave shapes and the relation between H and L which had

been established by choosing KLa values. In this case about

3.2 stages were required for the separation from an x of 0.05

to 0.45, and 3.0 stages were required in run F-4 for a


separation from an x of 0.02 to 0.09. The values of HETS for

the toluene-MCH-silica gel system are also plotted in Figure

25 against Q/A.

It is significant that in both cases a trend is

established. There is a marked increase in HETS with the

velocity of the liquid through the bed. It is apparent that

more data of this type are required to establish whether HETS

is actually as strongly dependent upon liquid velocity as is

indicated here. If, however, the trends indicated here are

true, it would be very important in the design of a counter-

current adsorber to size the column diameter for a given

service properly.

F. Discussion of Calculations

The comparison of the computed curves for adsorption

fractionation with two systems has shown that a very good

agreement with experimental adsorption fractionation results

when the adsorbent particle size and liquid flow rates are

such that the external film is the major resistance to mass

transfer between the adsorbed and liquid phases. In these

cases the apparent or effective overall coefficient corre-

lates well with liquid velocity through the adsorbent bed.

It was found that a fair approximation of the column opera-

tion is obtained when the intraparticle diffusion contributes

to the diffusional resistance. However, the wave shape is

definitely not duplicated by the calculated curves. Through


a fortuitous circumstance, namely, that increased intra-

particle resistance affects the adsorption wave shape

similarly to a decreased adsorbent selectivity, it was seen

that when intraparticle resistance contributes to the diffu-

sional resistance, computer solutions based on constant-

alpha equilibrium diagrams may correlate better than solution

using the true equilibrium diagram, if care is taken to use a

constant-alpha solution which is exactly equal to the value

of alpha at the feed composition. The latter restriction was

found to be required in order for the velocity of the calcu-

lated ultimate adsorption wave to be correct. It is recog-

nized that the use of constant-alpha diagrams in cases where

the external film resistance controls would probably result

in an incorrect wave shape.

The method proposed in this dissertation for evalua-

tion of HETS was used on the data presented here with some

success. A correlation of HETS with liquid velocity through

the bed was obtained, but the indicated dependence of HETS

upon liquid velocity seemed high.

G. Discussion of Intraparticle Diffusional Resistance

Since the computed solutions of this work do not yield

an exact fit with data of large particle size adsorbent, the

next logical improvement in the method of analysis which was

used here would be to include in the basic equations a mathe-

matical expression for the intraparticle resistance.


The most important new consideration in such an

analysis would be that the adsorbed liquid phase would no

longer have just one composition, y, at a given L and 0,

but its composition would also be a function of r, the radius

of the particle.

It is very difficult to propose a mathematical model

to explain the adsorption forces when intraparticle diffu-

sion is considered. One possible procedure would be to

assume an external film resistance, characterized by KLa,

with the bulk unadsorbed liquid phase composition, x, on one

side and a pseudo-liquid phase composition, xR*, on the other,

where xR is the liquid phase composition in equilibrium with

yR, the composition of the adsorbed phase liquid at the ex-

ternal particle radius r = R. Diffusion within the particle

in the adsorbed phase could be assumed to follow Fick's law

for diffusion within a sphere, using an effective diffusivity,

D, for the diffusion inside the particle.

By equating the rates of mass transfer across the ex-

ternal film to the Fick's law expression for the diffusion

rate at r R, the intraparticle and external diffusion may

be related. Numerical integration of the resulting equations,

applying the proper boundary conditions, should provide a


One important limitation which would be encountered

is that both KLa and D, the effective internal diffusivity,

would be unknown parameters. Experiments would have to be


designed to evaluate D when KLa was negligible, and then to

add the effect of KLa in cases where D had previously been


The addition of an extra unknown parameter, D, and

an additional independent variable, r, makes the problem a

much more difficult one than was solved in this work. It

is believed, however, that the techniques demonstrated here

will be applied in the future, using faster and larger

capacity computers if necessary, to approach more closely

the exact solution to adsorption fractionation problems.


1. The application of the proposed equations for adsorption

fractionation was demonstrated for systems with small

adsorbent particle size and low flow rates, in which the

external film resistance presumably controls.

2. The boundary conditions of the liquid phase adsorption

fractionation process were properly defined and applied

in a numerical solution.

3. A complete IBM 650 program for solving the proposed

equations has been developed and presented.

4. The basic thesis, that a numerical approach can provide

useful solutions to problems otherwise insoluble, has

been proved.

5. The use of a solution based on a constant-alpha type

equilibrium curve which approximates the true equilibrium

curve was found to give qualitative accuracy. The shape

of the adsorption wave is distorted, but its rate of

travel down the column is closely approximated. The

results of this distortion can be used to advantage in

systems in which intraparticle diffusion contributes to

the total diffusional resistance.

6. Differential equations for evaluating the ultimate wave

shape were derived, and the velocity of the ultimate wave

was found to. be dependent upon xF, the feed liquid compo-



sition, and YF the adsorbed phase composition in

equilibrium with the feed liquid.

7. A method for determining from fixed bed experiments the

height equivalent to a theoretical stage (HETS) and of

an adsorbent bed was proposed and demonstrated.



Yi+l,j Yi,j + [AT] [(3/2)(Oy/TT),j (/2)y/T) (1

This formula fits a second degree polynomial over
two AT increments.

Xi, Xi (2)
j+ j +[AH] [(5/12)(Ox/H)i,j+i + (2)
(2/3)(x/oH)i,j (l/12)(ox/H)i, j-l]

This formula fits a third degree polynomial over
two AH increments. Trial and error is required.

Yi+l,j Yi,j + [AT][(1/2)(6y/ST)i, + (l/2)(bY/ T)i+l,j- (3)

This formula fits a second degree polynomial over
one AT increment. Trial and error is required.

ij+l = xij + AH [(1l/2)(6x/H)ij, + (l/2)(Ox/6H)i,j+1] (4)

This formula fits a second degree polynomial over
one AH increment. Trial and error is required.

yi+l,j J yi,j + [AT] [(5/12)( y/6T)ilj +
(2/3)(6y/bT)ij (1/12)(/y/6T)i_l,j1

This formula fits a third degree polynomial over
one AT increment. Trial and error is required.




Calculation Vol. Frac. Comp.
Number A in Feed

51 0.5

52 0.1

98 0.1

99 0.5

2 0.5

3 0.3

4 0.1

5 0.7

6 0.9

7 0.9

8 0.7

9 0.5

10 0.3

11 0.1

12 0.1

13 0.3

14 0.5

15 0.1

16 0.3

x-y Equilibria

Toluene-MCH-Silica Gel**

Toluene-MCH-Silica Gel**

Benzene-Hexane-Silica Gel*

Benzene-Hexane-Silica Gel*

a = 2.0

a = 2.0

a = 2.0

a = 2.0

a 2.0

a = 3.0

a = 3.0

a = 3.0

a = 3.0

a 3.0

a = 5.0

a 5.0

a 5.0

a 7.0

a = 7.0


Table 2 (Continued)

Calculation Vol. Frac. Comp.
Number A in Feed

17 0.5

18 0.1

19 0.3

20 0.5

x-y Equilibria

a 7.0

a = 9.0

a = 9.0

a = 9.0

* Data of Lombardo (73)

** Data of this work



Run No. Adsorbent Adsorbate

1 6-12 Mesh Toluene
Silica Gel
2 "


8-14 Mesh Methyl-
Act. Alumina cyclohexan
it i


Wt. Ad-
sorbent, g.









Wt. ad-









g. Adsorbate
g. Adsorbent


















For 6-12 Mesh Silica Gel., Average Vp .402

For 8-14 Mesh Activated Alumina Average Vp .188













(cf. Figure 13)

Volume Fraction
Toluene in Liquid


Volume Fraction
Toluene in
Adsorbed Phase






x = .203y

(x/l-x) = .1725(y/l-y)1412 ;

x/l-x = .459(y/l-y) .589;

0 y 0.15

0.15 y .776

.776 < y < 1

_ ___

-- F ,



(cf. Figure 14)

Volume Fraction
Toluene in Liquid

Volume Fraction
Toluene in
Adsorbed Phase


.0178 .127 8.03
.0475 .214 5.46
.233 .480 3.04
.417 .685 3.04
.648 .832 2.69
.874 .960 3.46



(Data of Lombardo)
(cf. Figure 15)

Volume Fraction
Benzene in Liquid

Volume Fraction
Benzene in
Adsorbed Phase


0.045 0.298 9.009
0.115 0.485 7.247
0.209 0.615 6.046
0.319 0.723 5.572
0.428 0.771 4.500
0.546 0.841 4.398
0.653 0.875 3.72
0.771 0.922 3.51
0.882 0.966 ---

Empirical Equations

y = x/(.9398x + .1475)

; .226 < x ( .500

y = x/(1.1354x + .1032) ;

0 < x < .226



Run No. System Adsorbent

Toluene-MCH 6-12 Mesh
Silica Gel
It it

Column Diam.,
















Wt. Adsorbent,














Inverse Rate,














Feed Comp.














8-14 Mesh

TABLE 7 (Continued)

Run No. System Adsorbent

F-6b "

F-6c "

B-2(Lom- Benzene-
bardo) N-Hexane

B-3 "


"Thru 200"
Mesh Silica

Column Diam.,


Wt. Adsorbent,




Inverse Rate













- -



Run No. F la (cf. Figure 19)

= 2.47 cm.

= 195.0 g.

= .679 g./cc.

S0.500 Vol. fi

Time. sec.





Ave. Inverse Rate = 12.7 sec./cc.

= .402 cc./g.

- .293

r. Sample Size

Total Vol.
Effluent, cc.


= 7 drops

Vol. Fraction


Col. Diam.

Wt. Gel.

Sample No.


.... N. ..




Run No. F lb (cf. Figure 19)

Col. Diam.

Wt. Gel.

- 2.47 cm.

- 95.20 g.

Ave. Inverse Rate 12.7 sec./cc.

= .402 cc./g.

= .679 g./cc.

- 0.500 Vol. fr.

Sample Size

Sample No.

Time. sec.



Total Vol.
Effluent, cc.


Vol. Fraction


- .293

= 7 drops



Run No. F ic (cf. Figure 19)

Col. Diam. = 2.47 cm.

Ave. Inverse Rate = 12.73 sec./cc.

Sample No.

- 45.35g.

= .679 g./cc.

= 0.500 Vol. fr. Sample Size

Total Vol.

Time, sec.

(1 ml.


Effluent, cc.


= .402 cc./g.

= .293

= 7 drops
(except as

Vol. Fraction


Wt. Gel.

) ---





Run No. F 2a (cf. Figure 20)

Col. Diam. = 2.47 cm.

Ave. Inverse Rate = 20 sec./cc.

Wt. Gel.


= 191.20 g.

- .679 g./cc.

= 0.500 Vol. fr.

= .402 cc./g.

= .293

Sample Size

= 5 drops

Sample No.

Time, sec.



Total Vol.
Effluent, cc.


Vol. Fraction





Run No. F 2b (cf. Figure 20)

Col. Diam. = 2.47 cm.

Ave. Inverse Rate = 20 sec./cc.

Wt. Gel.

- 95.15 g.

= .402 cc./g.

- .679 g./cc.

Vol. fr.

Sample Size

- 5 drops

Samn e Non

Time. sec.



Total Vol.

Vol. Fraction




- .293


. No Ti s e



Run No. F 2c (cf. Figure 20)

Col. Diam.

Wt. Gel.

= 2.47 cm.

= 47.55 g.

Ave. Inverse Rate = 20 sec./cc.

= .402 cc./g.

= .679 g./cc.

= 0.500 Vol. fr. Sample Size

Sample No.

Time, sec.



Total Vol.
Effluent, cc.


Vol. Fraction


= .293

= 5 drops




Run No. F 4a (cf. Figure 21)

Col. Diam. 2.47 cm.

Ave. Inverse Rate = 5.75 sec./cc.

- 195.35 g.

= .679 g./cc.

= 0.100 Vol. fr.

= .402 cc./g.

- .293

Sample Size

= 5 drops

Sample No.

Time, sec.






Total Vol.
Effluent, cc.


Vol. Fraction


Wt. Gel.


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