APPLICATION OF NUMERICAL METHODS
IN ANALYSIS OF FIXED
BED ADSORPTION FRACTIONATION
By
ADRAIN EARL JOHNSON, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
February, 1958
ACKNOWLEDGEMENTS
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.
TABLE OF CONTENTS
Page
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
iii
TABLE OF CONTENTS (Continued)
Page
VIII. COMPARISONS BETWEEN EXPERIMENTAL AND
CALCULATED RESULTS.............................. 62
A. Adsorption Fractionation Experiments of
Lombardo.................................... 63
B. TolueneMethylcyclohexane Fractionation
on Silica Gel............................... 64
C. TolueneMethylcyclohexane Adsorption on
Activated Alumina......................... 67
D. Use of ConstantAlpha 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
APPENDIX. IBM 650 COMPUTER PROGRAM FOR SOLVING
ADSORPTION FRACTIONATION EQUATIONS................. 139
BIOGRAPHICAL SKETCH...................................... 173
LIST OF TABLES
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 TolueneMethyl
cyclohexane on Davison 612 Mesh Silica Gel....... 81
5. Adsorption Equilibrium Data for TolueneMethyl
cyclohexane on Alcoa 814 Mesh Activated Alumina.. 82
6. Adsorption Equilibrium Data for BenzeneN
Hexane on Davison "Thru 200" Mesh Silica Gel....... 83
7. Summary of Fractionation Experiments.............. 84
8. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel...................... 86
9. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel..................... 87
10. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel...................... 88
11. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel..................... 89
12. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel...................... 90
13. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel..................... 91
14. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel..................... 92
15. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel...................... 93
LIST OF TABLES (Continued
Table Page
16. TolueneMethylcyclohexane Fractionation on
Davison 612 Mesh Silica Gel.................... 94
17. TolueneMethylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina................ 95
18. TolueneMethylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina............... 96
19. TolueneMethylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina............... 97
20. TolueneMethylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina.............. 98
21. TolueneMechylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina............... 99
22. TolueneMethylcyclohexane Fractionation on
Alcoa 814 Mesh Activated Alumina............... 100
23. BenzeneNHexane Fractionation on Davison
"Thru 200" Mesh Silica Gel...................... 101
24. BenzeneNHexane Fractionation on Davison
"Thru 200" Mesh Silica Gel...................... 102
25. BenzeneNHexane Fractionation on Davison
"Thru 200" Mesh Silica Gel ................... 103
26. Calibration of Refractometer for MCHToluene
Solutions at 30C................................. 104
LIST OF ILLUSTRATIONS
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 MCHToluene
on Davison 612 Mesh Silica Gel.................. 117
14. Adsorption Equilibrium Diagram for MCHToluene
on Alcoa 814 Mesh Activated Alumina............ 118
vii
LIST OF ILLUSTRATIONS (Continued)
Figure Page
15. Adsorption Equilibrium Diagram for Benzene
Hexane on Davison "Thru 200" Mesh Silica Gel.... 119
16. BenzeneHexane Fractionation With Silica Gel.... 120
17. BenzeneHexane Fractionation With Silica Gel.... 121
18. BenzeneHexane Fractionation With Silica Gel.... 122
19. MCHToluene Fractionation With Silica Gel....... 123
20. MCHToluene Fractionation With Silica Gel....... 124
21. MCHToluene Fractionation With Silica Gel....... 125
22. MCHToluene Fractionation With Alumina.......... 126
23. MCHToluene 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 MCHToluene
Solutions....................................... 130
viii
I. INTRODUCTION
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,
1
2
because of the apparent impossibility of obtaining analytical
solutions to the equations.
It was found that, through a fortuitous circumstance,
computer solutions based on constantalpha 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
columns.
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.
II. BACKGROUND
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
3
4
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
tory.
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
5
adsorption.
The analysis of fixed bed liquid phase adsorption
fractionation is complicated by the fact that it is
inherently unsteadystate, 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.
III. PREVIOUS WORK
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
mentioned.
19001920
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
6
7
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
8
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 gasair 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 batchtype adsorption decreased as the ad
sorption progressed, and that diffusion might play a role in
determining the rate.
19201930
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
9
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
adsorption.
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 ekt. 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.
19301940
Additional equations for correlating the kinetics of
batch adsorption were proposed by Tolloizko (33), Constable
10
(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 105 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).
11
19401950
In 1940 Brunauer, Deming, and Teller (50) combined
the recognized five types of vapor isotherms into one equa
tion.
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.
12
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
(64).
13
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 gassolid moving bed adsorber, for the case
of one adsorbate, or multiple adsorbates assuming individual
isotherms for each. The effects of fluid phase resistance,
14
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 fixedbed system
when the rate is determined by liquid film and particle
diffusion.
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
torily.
Lombardo (72) considered the problem of binary
liquid adsorption fractionation from the pseudotheoretical
stage standpoint, and obtained solutions to the stepwise
equations which he proposed by means of a card programmed
calculator.
15
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
isotherm.
IV. THEORY
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
16
17
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 rodlike. [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
18
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
19
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
20
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
restricted.
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
sorbentfree adsorbed phase. This diagram is similar in
appearance to the usual vaporliquid 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.
21
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
22
adsorbed and unadsorbed phases,
!(Qx)
[ .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
fractions.
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,
23
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 xy
relation for the system under consideration, represent the
mathematical problem to be solved, given suitable boundary
conditions.
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:
24
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)
gives,
(bx/WH)T = (6y/OT)H (5)
x x* = (3y/6T)H (6)
Equations (5) and (6) together with the xy 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
25
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.
26
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
27
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.
28
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.
V. NUMERICAL ANALYSIS
A. Numerical Methods
The general procedure for solving differential equa
tions by means of numerical techniques is covered by many
texts.
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)
H H
T/ \ HT T HT
29
30
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
31
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 xFx > 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 510 hours per solution.
32
B. Description of Integration Procedure
M C) r_ r4 (q m
I t + ++
12
 i1 H
) 1
i+1
1+2
T
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
33
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
34
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
culations.
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
3
6 7 9
10, 4
H T
35
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
36
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
formula.
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
5.
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.
37
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, drawnout
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
38
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 predetermined 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 subroutine 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
39
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.
VI. RESULTS OF CALCULATIONS
A. Problem Solutions
The numerical solution to the binary liquid adsorp
tion fractionation problem was run twentythree 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 xy 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 xy 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 xy 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) (BenzeneHexane 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
40
41
group of nineteen solutions was based on a constantalpha.
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/ly) 
(1x/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 xy 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 constantalpha 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
42
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 211. 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
twentythree solutions under a separate cover for ready
reference.
It was found that in every problem solution an ulti
mate wave shape was formed provided sufficient distance along
43
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
diagram.
< zone 1 zone 2 zone 3
0 y
44
0 D
Bed Depth, L
44
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 nonchanging shape?
(4) If so, at what rate does zone 2 travel through the
column?
45
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
nonchanging 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 x0 xO
Zone Kone
2 2
W i
yy* xxF Q
X Xp
46
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,
47
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
48
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 210) 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.
xF
o
o y
0d
k 0
0 L
Bed Depth, L
49
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 xy
diagram which will permit an adsorption wave of ultimate
or invariant shape to become established. Consider the
following diagram.
50
YF
y
0
XF
x
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 210).
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.
51
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/(xx*) = (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
meter:
52
x
Sdx/(x x*) = Hx H (22)
xF/2
In most cases the lefthand 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
53
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
f
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
54
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
HETS N/L2L1
SI N stages
x Y
O. lxF 0
L1 L2 0.1 0.9
xF XF
55
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.
VII. EXPERIMENTAL
A. Adsorbent
Commercial Davison silica gel (612 mesh) and Alcoa
alumina (814 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.
56
57
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 reweighed 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 xy 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 tolueneMCH 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
58
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 xy 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 BenzeneHexane 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 xy 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 twentyfour
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
59
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
filled.
The effluent liquid was collected in graduated
cylinders, and samples of five drops (1/4 ml.) were collected
60
at regular intervals. The large diameter column was chosen
so that samples of five drops could be taken at about 510
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 822.
The experiments shown are only those which are referred to
in this dissertation. In addition, the data of Lombardo
for the benzenehexanesilica gel system are presented in
Tables 2325. 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,
61
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 825, which present
the results of the adsorption fractionation runs.
VIII. COMPARISONS BETWEEN EXPERIMENTAL
AND CALCULATED RESULTS
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
bed.
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
62
63
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 benzenehexanesilica gel
64
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. TolueneMethylcyclohexane Fractionation on Silica Gel
The effluent volume vs. composition curves for three
sets of fractionation experiments with the tolueneMCH
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.
65
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
610.
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
k.
66
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 xx*, 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
relation.
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
67
"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. TolueneMethylcyclohexane 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
gel.
68
D. Use of ConstantAlpha 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 constantalpha 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 constantalpha 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
69
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 constantalpha 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 xy diagram fit the experimental wave
shape better than the true xy 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
70
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 tolueneMCHsilica 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 F4 for a
71
separation from an x of 0.02 to 0.09. The values of HETS for
the tolueneMCHsilica 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
72
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
constantalpha 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 constantalpha 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.
73
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 pseudoliquid 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
solution.
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
74
designed to evaluate D when KLa was negligible, and then to
add the effect of KLa in cases where D had previously been
evaluated.
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.
IX. CONCLUSIONS
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 constantalpha 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
75
76
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.
77
TABLE 1
NUMERICAL INTEGRATION FORMULAE
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, jl]
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 +
(5)
(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.
78
TABLE 2
SUMMARY OF
ADSORPTION FRACTIONATION CALCULATIONS
XF,
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
xy Equilibria
TolueneMCHSilica Gel**
TolueneMCHSilica Gel**
BenzeneHexaneSilica Gel*
BenzeneHexaneSilica 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
79
Table 2 (Continued)
XF
Calculation Vol. Frac. Comp.
Number A in Feed
17 0.5
18 0.1
19 0.3
20 0.5
xy Equilibria
a 7.0
a = 9.0
a = 9.0
a = 9.0
* Data of Lombardo (73)
** Data of this work
TABLE 3
DETERMINATION OF SPECIFIC PORE
Run No. Adsorbent Adsorbate
1 612 Mesh Toluene
Silica Gel
2 "
Methyl
cyclohexane
it
814 Mesh Methyl
Act. Alumina cyclohexan
it i
Toluene
11
Wt. Ad
sorbent, g.
29.56
21.28
33.44
21.17
43.73
23.68
21.30
36.48
Wt. ad
sorbate,
10.77
7.85
9.98
6.26
6.36
3.473
3.563
5.636
g. Adsorbate
g. Adsorbent
.366
.369
.299
.296
.1455
.1467
.1673
.1616
Adsorbate
Density,
g./cc.
.872
.872
.774
.774
.774
.774
.872
.872
For 612 Mesh Silica Gel., Average Vp .402
For 814 Mesh Activated Alumina Average Vp .188
VOLUMES
Vp
cc./g.
.420
.424
.387
.382
.188
.189
.192
.185
81
TABLE 4
ADSORPTION EQUILIBRIUM DATA FOR
TOLUENEMETHYLCYCLOHEXANE ON
DAVISON 612 MESH SILICA GEL
(cf. Figure 13)
x
Volume Fraction
Toluene in Liquid
Phase
.0350
.0372
.0647
.0847
o124
.128
.149
.182
.210
.132
.243
.304
.344
.411
.489
.526
.578
.641
.704
.741
.796
.871
.933
y
Volume Fraction
Toluene in
Adsorbed Phase
.289
.269
.344
.420
.462
.476
.499
.540
.570
.467
.605
.656
.687
.726
.768
.787
.817
.839
.866
.877
.906
.943
.949
a
Relative
Adsorbability
(y/ly)(lx/x)
11.211
9.526
7.581
7.825
6.064
6.188
5.688
5.276
4.987
5.761
4.770
4.366
4.186
3.798
3.459
3.330
3.259
2.919
2.717
2.492
2.470
2.450
1.336
Empirical
Equations
x = .203y
(x/lx) = .1725(y/ly)1412 ;
x/lx = .459(y/ly) .589;
0 y 0.15
0.15 y .776
.776 < y < 1
_ ___
 F ,
82
TABLE 5
ADSORPTION EQUILIBRIUM DATA FOR
TOLUENEMETHYLCYCLOHEXANE ON
ALCOA 814 MESH ACTIVATED ALUMINA
(cf. Figure 14)
x
Volume Fraction
Toluene in Liquid
Phase
y
Volume Fraction
Toluene in
Adsorbed Phase
a
Relative
Adsorbability
(y/ly)(lx/x)
.0178 .127 8.03
.0475 .214 5.46
.233 .480 3.04
.417 .685 3.04
.648 .832 2.69
.874 .960 3.46
83
TABLE 6
ADSORPTION EQUILIBRIUM DATA
BENZENENHEXANE ON
DAVISON "THRU 200" MESH SILICA GEL
(Data of Lombardo)
(cf. Figure 15)
x
Volume Fraction
Benzene in Liquid
Phase
y
Volume Fraction
Benzene in
Adsorbed Phase
a
Relative
Adsorbability
(y/ly)(1x/x)
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
TABLE 7
SUMMARY OF FRACTIONATION EXPERIMENTS
Run No. System Adsorbent
TolueneMCH 612 Mesh
Silica Gel
It it
Column Diam.,
cm.
2.47
it
"n
Fla
Flb
Flc
F2a
F2b
F2c
F4a
F4b
F4c
F5a
F5b
F5c
F6a
Wt. Adsorbent,
g.
195
95.2
45.35
191.2
95.2
47.55
195.35
96.6
45.3
255.2
121.5
59.7
62.8
Inverse Rate,
sec./cc.
12.73
12.7
12.7
20
20
20
5.75
5.75
5.75
8.13
8.13
8.13
16
XF
Feed Comp.
0.5
0.5
0.5
0.5
0.5
0.5
0.1
0.1
0.1
0.1
0.1
0.1
0.5
814 Mesh
Alumina
I
TABLE 7 (Continued)
Run No. System Adsorbent
F6b "
F6c "
B2(Lom Benzene
bardo) NHexane
B3 "
B4
"Thru 200"
Mesh Silica
Gel
It
Column Diam.,
cm.
0.8
Wt. Adsorbent,
g.
129.9
267.9
20
Inverse Rate
sec./cc.
16
16
880
650
246
1.9
FeedComp.
0.5
0.5
0.5
0.5
0.5
 
86
TABLE 8
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
Run No. F la (cf. Figure 19)
= 2.47 cm.
= 195.0 g.
= .679 g./cc.
S0.500 Vol. fi
Toluene
Time. sec.
2155
2305
2380
2450
2512
2590
2790
3135
3195
3270
3335
3395
Ave. Inverse Rate = 12.7 sec./cc.
= .402 cc./g.
 .293
r. Sample Size
Total Vol.
Effluent, cc.
.15
5.50
10.85
16.20
21.55
26.90
32.25
37.60
42.85
48.30
59.00
64.35
69.70
75.05
80.40
85.75
91.10
96.45
111.80
= 7 drops
x
Vol. Fraction
Toluene
.02
.055
.102
.150
.199
.240
.279
.308
.328
.356
.380
.395
.408
.412
.428
.439
.441
.453
.466
Col. Diam.
Wt. Gel.
Sample No.

.... N. ..

87
TABLE 9
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
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.
Toluene
Sample Size
Sample No.
Time. sec.
1487
1539
1604
1666
1797
1869
1933
Total Vol.
Effluent, cc.
.15
2.50
4.85
7.20
9.55
11.90
14.15
16.50
18.85
21.10
23.45
28.80
34.15
39.50
44.85
50.20
55.55
60.90
66.25
71.60
x
Vol. Fraction
Toluene
.125
.208
.240
.268
.290
.321
.327
.343
.359
.370
.370
.395
.408
.420
.428
.435
.437
.453
.463
.465
 .293
= 7 drops
88
TABLE 10
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
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
Toluene
Total Vol.
Time, sec.
(1 ml.
it
Effluent, cc.
.15
0.70
1.70
2.70
3.70
4.70
5.70
6.70
7.70
8.70
30.70
36.05
41.40
= .402 cc./g.
= .293
= 7 drops
(except as
noted)
x
Vol. Fraction
Toluene
.196
.230
.255
.282
.300
.310
.330
.342
.356
.367
.438
.448
.450
Wt. Gel.
"'
) 


.
89
TABLE 11
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
Run No. F 2a (cf. Figure 20)
Col. Diam. = 2.47 cm.
Ave. Inverse Rate = 20 sec./cc.
Wt. Gel.
Pb
= 191.20 g.
 .679 g./cc.
= 0.500 Vol. fr.
Toluene
= .402 cc./g.
= .293
Sample Size
= 5 drops
Sample No.
Time, sec.
3644
3747
3853
3957
4065
4180
4275
4515
4606
4710
4825
4924
5020
5121
5216
Total Vol.
Effluent, cc.
.10
5.35
10.60
15.85
21.10
26.35
31.60
36.85
42.10
47.35
52.60
57.85
63.10
68.35
73.60
78.85
84.10
89.35
Vol. Fraction
Toluene
.004
.004
.050
.0Q50
.112
.172
.230
.278
.305
.339
.368
.385
.396
.408
.418
.425
.430
.438
__
90
TABLE 12
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
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.
Toluene
Sample Size
 5 drops
Samn e Non
Time. sec.
1650
1725
1762
1809
1847
1890
1930
1975
2022
2066
2183
2298
2398
2499
2611
2713
2282
3037
3037
Total Vol.
Effluent.
x
Vol. Fraction
Toluene
cc.
.10
4.35
6.60
8.85
11.10
13.35
15.60
17.85
20.10
22.35
27.60
32.85
38.10
43.35
48.60
53.85
59.10
64,35
69.60
.035
.090
.136
.100
.060
.205
.180
.278
.328
.341
.366
.390
.405
.406
.436
.440
.444
.452
.460
 .293

. No Ti s e
91
TABLE 13
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
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
Toluene
Sample No.
Time, sec.
835
868
935
975
1020
1050
1118
1156
1198
1283
1394
1491
Total Vol.
Effluent, cc.
.10
2.35
4.60
6.85
9.10
11.35
13.60
15.85
18.10
20.35
22.60
27.85
33.10
48.85
x
Vol. Fraction
Toluene
.122
.210
.252
.312
.326
.340
.370
.395
.405
.416
.425
.440
.450
.468
= .293
= 5 drops
_r
92
TABLE 14
TOLUENEMETHYLCYCLOHEXANE FRACTIONATION ON
DAVISON 612 MESH SILICA GEL
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.
Toluene
= .402 cc./g.
 .293
Sample Size
= 5 drops
Sample No.
Time, sec.
983
1041
1163
1222
1272
1337
1393
1445
1560
1619
1730
1906
1969
Total Vol.
Effluent, cc.
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
170
180
x
Vol. Fraction
Toluene
.0
.0
.0
.0
.0
.0
.0
,0
.0
.0
.006
.010
.016
.019
.023
.032
.035
Wt. Gel.
_____
