school form 9 purdue university 8/89)132.235.17.4/paper-gu/phd_ori.pdf · 2006-05-15 · graduate...
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Graduate School Form 9 (Revised 8/89)
PURDUE UNIVERSITY GRADUATE SCHOOL
Thesis Acceptance
This is to certify that the thesis prepared
Entitled Inclusion Chromatography Using Cyclodextrin-Containing Resins and Studies of Nonlinear Chromatographic Theories
Complies with University regulations and meets the standards of the Graduate School for originality and quality
For the degree of Doctor of Philosophy
Signed by the final examining committee:
Approved by:
0 is This thesis is not to be regarded as confidential
Major Professor
INCLUSION CHROMATOGRAPHY
USING CYCLODEXTRIN-CONTAINING RESINS AND
STUDIES OF NONLINEAR CHROMATOGRAPHIC THEORIES
A Thesis
Submitted to the Faculty
of
Purdue University
by
Tingyue Gu
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
August 1990
ACKNOWLEDGEMENTS
The invaluable guidance and support provided by my Major Professor, Dr. G. T.
Tsao, throughout my stay at Purdue University is gratefully appreciated. My thanks also
go to Professors R. P. Andres, K. C. Chao and M. R. Ladisch for serving on my
advisory committee.
I must thank my group leader, Dr. G.-J. Tsai, for his valuable help during the course
of this work. The help from Dr. Z. Chen in the experimental work is also
acknowledged. The administrative and clerical support from Norma Leuck and
Carolyn Wasson, and the friendship of my colleagues in the Laboratory of Renewable
Resources Engineering (LORRE) are appreciated.
This work was financially supported by a National Science Foundation grant, and a
grant and a fellowship provided by the Corn Refiner's Association. Without their
generous support this work would not have been possible.
TABLE OF CONTENTS
Page
... LIST OF TABLES ......................................................................................................... VIII
.......................................................................................................... LIST OF FIGURES ix
LIST OF SYMBOLS ....................................................................................................... xv
... ................................................................................................................. ABSTRACT xvm
.................................................................................... CHAPTER 1 . INTRODUCTION 1
................................................................................... 1.1 Inclusion Chromatography 1 ................................................ 1.2 Studies of Nonlinear Chromatographic Theories 3
........................................................................ CHAPTER 2 . LITERATURE REVIEW -8
........................................................................................................ 2.1 Cyclodextrins 8 ........................................................... 2.1.1 Historical Background (Szejtli. 1982) -8
2.1.2 The Structures of CDs ................................................................................... -9 .............................................................. 2.1.3 Some Important Properties of CDs 10
.............................................. 2.1.4 Types of Guest Molecules in CD Complexes 1 ......................................... 2.1.5 Binding Forces of CD Complexes .................. .. 15
............................................. 2.2 CDs as Catalysts (Bender and Korniyama. 1978) -15 ........................................................................................ 2.3 CD-Containing Resins 16
............................................................................... 2.3.1 With Inorganic Support 1 6 .................................................................................. 2.3.2 With Organic Support 18
.................... 2.4 Applications of CDs. CD Derivatives and CD-Containing Resins 19 .................................................................................... 2.4.1 Analytical Chemistry 19
2.4.2 Applications in the Agricultural Industry .................................................... -20 2.4.3 Applications in Food and Tobacco Indusmes .............................................. 20 2.4.4 Applications in the Pharmaceutical Industry ............................................... 21 2.4.5 Applications in the Chemical Industry ......................................................... 21 2.4.6 Other Applications ...................................................................................... -22
2.5 Dynamics of Nonlinear Multicomponent Chromatography in Fixed-Beds ....... 22 2.5.1 Equilibrium Theory ..................................................................................... -22 2.5.2 Plate Models ................................................................................................. 23 2.5.3 Rate Models ................................................................................................. -24
Page
................... 2.5.3.1 Rate Expressions and Particle Phase Governing Equations 25 ............................... 2.5.3.2 Adsorption Kinetics and Affinity Chromatography 26
............................................ 2.5.3.3 Governing Equations for Bulk Fluid Phase 26 2.5.4 General Multicomponent Rate Models ........................................................ 27 2.5.5 Solution to the General Multicomponent Rate Models ................................ 27
2.5.5.1 Discretization of Particle Phase Equations ............................................. 28 ................................................. 2.5.5.2 Discretization of Bulk Phase Equations 28
2.5.5.3 Solution to the ODE System .............................. .,,.,. + .....,,,,. . .............,,,,. 29
CHAPTER 3 . MATERIALS AND EXPERIMENTAL PROCEDURES ..................... 31
.................................................................. 3.1 Syntheses of CD- Containing Resins 31 ....................................................................................................... 3.1.1 Materials 31
3.1.2 MMA Method ............................................................................................... 33 3.1.3 BA Method ................................................................................................... 34
........................................................................................ 3.2 Adsorption Isotherms -35 3.2.1 Chromatographic Methods ........................................................................... 35 3.2.2 Batch Adsorption Method ............................................................................ 36
3.3 Inclusion Clrro~na~ography Using Cyclodexnin-Containing Resins .................. 37 .......................................................................................... 3.3.1 Column Packing -37
..................................................................................... 3.3.2 Nongradient Elution 37 ............ 3.3.3 Stepwise Displacement Experiments and Evaluation of Displacers 40
...................................................... 3.3.4 Experiments with Real Sample Systems 40 ................................................................................... 3.3.4.1 Corn Steep Water 40
3.3.4.2 Phenylalanine-Containing Industrial Waste Samples ............................ 41
CHAPTER 4 . EXPERIMENTAL RESULTS AND DISCUSSIONS ........................... 43
4.1 Properties of Synthesized CD-Containing Resins ............................................. -43 4.2 Reaction Mechanisms ........................................................................................ 47
......................................................................................... 4.3 Adsorption Isotherms 47 4.4 Separation of a Binary Sample by Nongradient Elution .................................... 52 4.5 Displacement Experiments and Evaluation of Displacers ................................. 54 4.6 Application to Real Systems .............................................................................. 63
4.6.1 Corn Steep Water ......................................................................................... 63 4.6.2 Phenylalanine-Containing Industrial Waste Samples .................................. 63
CHAPTER 5 . THEORY AND MATHEMATICAL MODELING ............................... 66
5.1 A General Multicomponent Rate Model for Axial Flow Chromatography ....... 66 5.1.1 Model Assumptions ...................................................................................... 66 5.1.2 Model Formulation ...................................................................................... -68
5.2 Numerical Solution to the Model ....................................................................... 70
Page
. . ................................................................................................ 5 .2.1 Discretmttlon 70 ....................... ....................................... 5.2.2 Solution to the ODE System ... 72
.................................................................................... 5.2.3 Isotherm Expressions 73 .................................... 5.3 Efficiency and Robustness of the Numerical Procedure 75
5.4 Extension of the Rate Model .............................................................................. 79 .............................................................. 5.4.1 Addition of Second Order Kinetics 80
.......................................................................................... 5.4.2 Solution Strategy 81 .............................................................. 5.4.3 Addition of Size Exclusion Effects 82
5.4.4 Solution Strategy .......................................................................................... 84 ....... 5.5 Multicomponent Adsorption Systems with Uneven Saturation Capacities 85
5.5.1 Systems with Physically Induced Uneven Saturation Capacities ................ 86 ....................................................................... 5.5.2 Kinetic and Isotherm Models 86
5.5.3 Systems with Chemically Induced Uneven Saturation Capacities .............. 88 .................................................................................... 5.5.4 Isotherm Cross-Over 89
5.6 Other Extensions of the Rate Model .................................................................. 98 5.7 The Question of Column Boundary Conditions ................................................. 98
CHAPTER 6 . OPTIMIZATION OF STEPWISE DISPLACEMENT ........................ 108
............................................................. 6.1 Introduction 108 6.2 Results and Discussion (Gu et al.. 1990c) ........................................................ 112
6.2.1 Effect of Feed Concentration of Displacer (CO2) ....................................... 112 6.2.2 Effect of Adsorption Equilibrium Constant of Displacer (bz) ................... 115
....................................................................................... 6.2.3 Industrial Practice 119 6.3 Summary .......................................................................................................... 120
CHAPTER 7 . MASS TRANSFER EFFECTS IN ............................. MULTICOMPONENT CHROMATOGRAPHY 124
7.1 Effects of Parameters PeLi . Bii and qi ............................................................. 124 .......................................................................................... 7.2 Effect of Flow Rate 128
7.3 Effect of Mass Transfer in a Case with Unfavorable Isotherm ........................ 133
CHAPTER 8 . DISPLACEMENT EFFECTS IN MULTICOMPONENT CHROMATOGRAPHY ................................. 137
8.1 Introduction ..................................................................................................... -137 ....................................................... 8.2 Results and Discllssion (Gu et al.. 1990b) 138
8.2.1 Displacement Operation ........................................................................... 139 8.2.2 Frontal Adsorption ...................................................................................... 142 8.2.3 Elution ....................................................................................................... -147
8.2.3.1 Effect of the Adsorption Equilibrium Constants .................................. 149 8.2.3.2 Low Adsorption Saturation Capacity ................................................... 152
Page
............. 8.2.3.3 High Sample Feed Concentrations (Concentration Overload) 152 ............................................... 8.2.3.4 Large Sample Size (Volume Overload) 152
.............................................................................. 8.2.3.5 More Component(s) 155 8.3 Summary ........................................................................................................... 155
................... CHAPTER 9 . SYSTEM PEAKS IN MULTICOMPONENT ELUTION 160
..................................................................................................... 9.1 Introduction -160 ........................................................ 9.2 Boundary Conditions for the Rate Model 162 ........................................................ 9.3 Results and Discussion (Gu et al., 1990e) 163
..................... 9.3.1 Modifier Affinity Is Weaker Than Those of Sample Solutes 163 ............................. 9.3.2 Modifier Affinity Is Between Those of Sample Solutes 167
.................... 9.3.3 Modifier Affinity Is Stronger Than Those of Sample Solutes 169 ....................... 9.3.4 Effect of Modifier Concentration on System Peak Patterns 173
................................. .................. 9.3.5 Effect of Modifier on Sample Solutes .. 177 ........................................................................... 9.3.6 Effect of Type of Sample 182
................................................. 9.3.7 Effect of Sample Solutes on the Modifier -184 ............................................................ 9.3.8 Summary of System Peak Patterns -187
........... 9.3.9 Binary Elu~iorl with Two Different Modifiers in the Mobile Phase 190 ........................................................................................ 9.4 Concluding Remarks 190
.................................................. CHAPTER 10 . AFFINlTY CHROMATOGRAPHY 196
.................................................................................................... 10.1 Introduction 196 ........................................................................... 10.2 Effect of Reaction Kinetics -198
................................................................................. 10.3 Effect of Size Exclusion 206 10.4 Interaction Between Soluble Ligand and Macromolecule ............................. 210
................................................. 10.4.1 Modeling of Reaction in the Fluid Phase 210 ...................................................................................... 10.4.2 Solution Strategy 212
.......................... 10.5 Modeling of the Three S rages in Affinity Chromatography 213 ......................................................... 10.6 Inhibition in Affinity Chromatography 217
10.7 Summary ......................................................................................................... 217
CHAPTER 1 1 . MULTICOMPONENT RADIAL FLOW CHROMATOGRAPHY ..................................................................... -221
1 1.1 Introduction ................................................................................................... 221 ............................................. 11.2 General Multicmponent Rate Model For RFC 225
..................................................................................... 11.3 Numerical Solution ....229 11.3.1 Using v value at V = 0.5 for averaging Dbi and ki ................................... 232 11.3.2 Using v value at X = (XI + Xo)/2 for averaging Dbi and ki .................... 232
11.4 Results and Discussion (Gu et al., 1990d) ................................................. 233 1 1.4.1 Simulations of Different Chromatographic Operations ........................... 233
vii
Page
1 1.4.2 Effect of Vo ........................................................................................... -238 .. ................................................ 1 1.4.3 Effects of Pei. q; and Bii on elution ..... -241
1 1.4.4 Effect of treating Dbi and ki as variables ................................................. 241 1 1.4.5 Comparison of RFC and AFC ................................................................. -249
........................................................... 11.5 Extensions of the General RFC Model 252 ......................................................................................................... 11.6 Summary 252
............................. . CHAPTER 12 CONCLUSIONS AND RECOMMENDATIONS 256
12.1 Syntheses of Cyclodextrin-Containing Resins ............................................... 256 ........................................... 12.2 Studies of Nonlinear Chromatographic Theories 257
BIBLIOGRAPHY ......................................................................................................... 260
VITA .............................................................................................................................. 275
... Vll l
LIST OF TABLES
Table Page
............................................. 2.1 Some Important Physical Properties of Cyclodextrins 10
.............................................. 5.1 Parameter Values Used for Simulation in Chapter 5 107
.............................................. 6.1 Parameter Values Used for Simulation in Chapter 6 123
.............................................. 7.1 Parameter Values Used for Simulation in Chapter 7 136
.......................... 8.1 Parameter Values Used for Simulation in Chapter 8 .. ............... 158
8.2 Summary of Nonlinear Multicomponent Elution ........................................................... (Compared with Pure Component Elutions) 159
............................................. 9.1 Parameter Values Used for Simulation in Chapter 9 194
9.2 Possible System Peak Combinations in a Binary Elution with a Competing Modifier in the Mobile Phase ..................................................... 195
................... ................... 10.1 Parameter Values Used for Simulation in Chapter 10 .., 220
1 1.1 Comparison of Dimensionless Variables and Parameters ................................... 227
.......................................... 11.2 Parameter Values Used for Simulation in Chapter 11 255
LIST OF FIGURES
Figure Page
.................................................................. 1.1 Structures of c yclodextrins (Szej tli. 1982) 2
2.1 Schematic representation of the formation of a ........................................................ cyclodexmn inclusion complex (Szej tli. 1982) -12
3.1 Slurry packing ........................................................................................................... 38
.................................................................................................... 3.2 Experimental setup 39
4.1 IR spectrum of a P-CD resin ..................................................................................... 45
................................................................................... 4.2 IR spectrum of an a-CD resin 46
4.3 Schematic representation of the structure of a CD resin ........................................... 48
............................................................ 4.4 Adsorption isotherm of Phe on a P-CD resin 49
.................................. 4.5 Adsorption isotherms of Trp and Aspartame on a P-CD resin 50
.......................................................... 4.6 Adsorption isotherm of Phe on an a-CD resin 51
4.7 Multicomponent elution using a P-CD column ......................................................... 53
................................................... 4.8 Typical response curve of a two-stage operation 55
4.9 Displacement of Phe with 100% Methanol ............................................................... 58
4.10 Displacement of Phe with 100% Ethanol ................................................................ 59
.......................................................... 4.1 1 Displacement of Phe with 100% n-Propanol 60
.............................................. 4.12 Adsorption and displacement of an inert component 61
4.13 Effect of displacer concentration ............................................................................. 62
.................................................... 4.14 Recovery of Phe from a cell steep water sample 65
Figure Page
5.1 Anatomy of a chromatographic column .................................................................... 67
5.2 Solution strategy for the general multicomponent rate model .................................. 71
5.3 Effect of the number of interior collocation points ........................................................................ in the simulation of frontal adsorption 76
...... 5.4 Effect of the number of interior collocation points in the simulation of elution 77
........ 5.5 Convergence of the concentration profiles of a stepwise displacement system 78
5.6 Concentration cross-over point ................................................................................. 91
...................................... 5.7 Peak reversal due to increased component 1 concentration 93
............................................ 5.8 Peak reversal in binary elution without size exclusion 95
.............................................................................. 5.9 Peak reversal with size exclusion 96
5.10 Cross-over of breakthrough curves ......................................................................... 97
5.1 1 Single-component breakthrough curves with different ................................ boundary conditions at the column exit (Peclet number = 50) 101
........................................ 5.12 Breakthrough concentration profiles inside the column 103
5.13 Two-component elution with different boundary conditions at the column exit .................................................................. 104
5.14 Single component breakthrough curves with different boundary conditions at the column exit (Peclet number = 300) ............................. 105
................................................................................... 6.1 Displacement chromatogram 110
.................................................................. 6.2 Single Component Langmuir Isotherms 113
6.3 Effect of displacer concentration on displacement ................... .. ......................... 114
6.4 Same conditions as Figure 6.3, except that the concentration of thc displacer is lower ............................................................. 116
6.5 Effect of displacer concentration on displacement for a case in which b2 < bl ....................................................................................................... 117
Figure Page
............................................................ 6.6 Effect of on displacement performance.. 118
..................... 6.7 Frontal operation switched to forward flow displacement operation. 121
6.8 Same conditions as Figure 6.7, except that flow direction was ...................................................................... reversed for the displacement process 122
7.1 Effect of Peclet numbers on two-component frontal adsorption ........................... 125
............................................. 7.2 Effect of Peclet numbers on two-component elution 126
..................... 7.3 Effect of Peclet numbers on two-component stepwise displacement 127
7.4 Effect of qi and Bii on two-component frontal adsorption ..................................... 129
...................................................... 7.5 Effect of qi and Bii on two-component elution 130
7.6 Effect of qi and Bii on two-component stepwise displacement ............................. 131
........................................ 7.7 Effect of interstitial velocity on two-component elution 132
.......... 7.8 Effect of mass transfer on an elution system with an unfavorable isotherm 135
.................................................... 8.1 Two-component stepwise displacement process 140
................................................................. 8.2 Three-component displacement system -141
......................................................... 8.3 Binary frontal adsorption with a roll-up peak 143
.................................................. 8.4 Ternary frontal adsorption with two roll-up peaks 145
...................................................... 8.5 Binary frontal adsorption with no roll-up peak 146
........................................................................................................ 8.6 Ternary elution 148
................................................. 8.7 Binary elution showing slight displacement effect 150
...................................................... 8.8 Binary elution with increased a1 and bl values 151
................................................. 8.9 Ternary elution with increased saturation capacity 153
8.10 Binary elution with large sample size ................................................................... 154
xii
Figure Page
............................................................. 8.1 1 Effect of added component in the sample 156
9.1 Binary elution with a weak modifier (Type I sample) ............................................ 164
9.2 Binary elution with a weak modifier (Type I1 sample) .......................................... 166
9.3 Modifier affinity is between those of sample solutes (Type I sample) .................... 168
.................. 9.4 Modifier affinity is between those of sample solutes (Type I1 sample) 170
9.5 Binary elution with a strong modifier (Type I sample) ........................................... 171
9.6 Binary elution with a strong modifier (Type I1 sample) .......................................... 172
.......... 9.7 Same conditions as Figure 9.5. except that the modifier affinity is stronger 174
.......... 9.8 Same conditions as Figure 9.7. except that the modifier affinity is stronger 175
9.9 Same conditions as Figure 9.3, .................................................... except higher modifier concentration (CO3 = 1.0) 176
9.10 Same conditions as Figure 9.4, ................................................... except higher modifier concentration (CO3 = 1.0) 178
9.1 1 Binary elution without modifier ............................................................................ 179
9.12 Effect of added modifier (Type I sample) ............................................................. 180
.......................................................... 9.13 Effect of added modifier (Type I1 sample) 181
........................................................ 9.14 Effect of type of sample at large sample size 183
9.15 Same conditions as Figure 9.2, except that Component 2 has a weaker affinity ....................................................... 185
9.16 Same conditions as Figure 9.8, ....................................................... except that Component 1 has a weaker affinity 186
9.17 Binary elution showing only one system peak (Type I1 sample) .......................... 188
9.18 Binary elution showing one positive. and one negative peak. respectively (Type I1 sample) .......................................... 189
xiii
Figure Page
9.19 Binary elution with two modifiers (Type II sample) ............................................. 191
9.20 Binary elution with two modifiers (Type I sample) .............................................. 192
10.1 Effect of reaction rates in frontal analysis ............................................................. 199
. . . 10.2 Fast reaction rates vs . equilibrium ......................................................................... 200
............................................................... 10.3 Effect of reaction rates in zonal analysis 202
10.4 Slow kinetics as rate limiting step in frontal analysis ......................................... 203
.......................................... 10.5 Mass transfer as rate limiting step in frontal analysis 204
10.6 Comparison of slow kinetics with slow mass transfer .......................................... 205
................................ 10.7 Effect of slow intraparticle diffusion and film mass transfer 207
10.8 Size exclusion effect in presence of adsorption .................................................. 208
10.9 Size exclusion effect in absence of adsorption ............................. A 9
10.10 Frontal adsorption stage combined with wash stage ................................................ ............................... in affinity chromatography .. 214
10.11 Effect of soluble ligand in the elution stage ......................................................... .................... of affinity chromatography .. 215
10.12 Effect of soluble ligand concentration in elution ................................................ 216
........................................ 10.13 Effect of soluble ligand inhibitor in zonal elution 218
10.14 Effect of competing binding inhibitor in zonal elution ..................................... 219
....................................................... 1 1.1 Structure of a cylindrical radial flow column -222
.... 11.2 Comparison of inward and outward flow RFC and AFC in frontal adsorption 234
11.3 Comparison of inward and outward flow RFC and AFC in displacement ............ 235
11.4 Comparison of inward and outward flow RFC and AFC in elution ...................... 236
..................................................................... 11.5 Binary elution in ion-exchange RFC 237
Figure Page
11.6 Effect of RFC flow direction in reverse flow displacement .................................. 239
......................................................... 11.7 Effect of Vo on elution in inward flow RFC 240
......................................................... 11.8 Effect of Pei on elution in inward flow RFC 242
......................................................... 11.9 Effect of q i on elution in inward flow RFC 243
....................................................... 11.10 Effect of Bii on elution in inward flow RFC 244
...................................................... 11.11 Effect of Dpi on elution in inward flow RFC 245
........................... 11.12 Effect of treating Dbi and ki as variables in inward flow RFC 246
......................... 11.13 Effect of treating Dbi and ki as variables in outward flow RFC 247
............................................................................ 11.14 Affinity RFC with inward flow 253
LIST OF SYMBOLS
constant in Langmuir isotherm for component i, biCr
adsorption equilibrium constant for component i, k&/kd,
Biot number of mass transfer for component i, kiRp/(%Dpi) or kiRp/($iDpi )
averaged Bii
bulk phase concentration of component i
feed concentration profile of component i, a time dependent variable
concentration used for nondimensionalization, max ( Cfi (t) }
concentration of component i in the stagnant fluid phase inside particle macropores
concentration of component i in the solid phase of particle (mole adsorbate / unit volume of particle skeleton)
critical concentration for concentration cross-over in a binary isotherm
critical concentration for selectivity cross-over defined in Eq. (5-50)
adsorption saturation capacity for component i (mole adsorbate / unit volume of particle skeleton)
adsorption saturation capacity based on the unit volume of the bed
concentration of component i in the stationary phase based on the unit volume of the bed
concentration of component i in the fluid phase based on the unit volume of the bed
=Cbi/coi
=%i/Coi
= C p / q i
= Cr/Coi
dimensionless column hold-up capacity for component i
xvi
axial or radial dispersion coefficient of component i
averaged Dbi
effective diffusivity of component i, porosity not included
Damk8her number for adsorption, W a i Coi 1
in AFC, Vb Eb (kai Coi 1 in
v Q - RFC
DamkCYher number for desorption, Lk& in AFC, Vbebh/Q in RFC
size exclusion factor for component i,
axial bed length of the radial flow column
film mass transfer coefficient of component i
adsorption rate constant for component i
desorption rate constant for component i
column length for AFC
number of interior collocation points
number of quadratic elements
number of components
Peclet number of axial dispersion for component i, vL/Dbi
Peclet number of radial dispersion for component i, v(X1-Xo)
Dbi
volumetric flow rate of the mobile phase
radial coordinate for particle
particle radius
= R/R,
time
interstitial velocity
bed volume for RFC column, nh(Xf-xg)
7~11x8 dimensionless constant for RFC, - or xi?
Vb x T - x ~ rrh(x2-x;) x2-x;
dimensionless volumetric coordinate, or Vb x Y - x ~ E [0,11
radial coordinate for RFC column
outer radius of RFC column
xvii
inner radius of RFC column
axial coordinate for AFC
= Z/L
Greek Letters
a =24v+vo ( 41+Vo-6 ) for RFC
Eb bed void volume fraction
% particle porosity
&;i accessible particle porosity of cornponcnt i
dimensionless constant, %DPiL %Dpi V b ~ b
in AFC, - - in RFC R;V R$ Q
Si dimensionless constant for component i, 3Biiqi (l-eb)/eb vt
dimensionless time, - in AFC, - L
Qt in RFC V b ~ b
Timp dimensionless time duration for a rectangular pulse of the sample
0 Lagrangian interpolation function
P liquid density
P liquid viscosity
6ij discount factors for extended multicomponent Langmuir isotherm
xviii
ABSTRACT
Gu, Tingyue. Ph.D., Purdue University, August 1990. Inclusion Chromatography Using Cyclodextrin-Containing Resins and Studies of Nonlinear Chromatographic Theories. Major Professor: George T. Tsao.
Cylcodextrins are a family of cyclic-oligosaccharides, containing usually six to
eight glucopyranose units. Because of their unique donut-shaped steric structure they
are able to form inclusion complexes selectively with some guest molecules.
In this work, a novel method of synthesizing cyclodextrin-containing resins for
inclusion chromatography has been developed. Resins synthesized with this method
possess a very high content of cyclodextrins and good physical properties. Experiments
on the adsorption isotherms and stepwise adsorptioIs/desorption column operations have
been carried out to separate some biomolecules. Efforts have been made to recover
phenylalanine from industrial waste samples using a p-Cyclodextrin column. Some
water soluble aliphatic alcohols have been found to be good displacers.
A robust and efficient numerical procedure has been developed to solve a general
multicomponent rate model which considers axial dispersion, external film mass
transfer, intraparticle diffusion and nonlinear isotherms. The method uses finite element
and orthogonal collocation methods to discretize the bulk phase and particle phase
partial differential governing equations, respectively. The resulting ordinary differential
equation system is then solved by Gear's stiff method. The model has also been
xix
extended to include second order kinetics, size exclusion, and the reaction between
soluble ligands and macromolecules in the fluid phase for the study of affinity
chromatography.
Computer simulations of stepwise displacement qualitatively proved some of the
experimental observations. A nlethoclology has been developed to derive kinetic and
isotherm models, which have successfully demonstrated isotherm cross-over and peak
reversal phenomena for multicomponent systems with uneven saturation capacities. A
unified approach based on a displacement effect has been proposed to explain the
dominating interference effect in multicomponent chromatography. System peak
patterns in binary elution have been summarized. Various aspects, including operational
stages, the rate-limiting step and two types of inhibition in affinity chromatography
have been analyzed.
Multicomponent rate models for radial flow chromatography (RFC) have also been
solved using an accurate numerical treatment in which the radial disperion and the
external film mass transfer coefficients are treated as variables. A comparison of, and
mathematical analogy between, RFC and conventional axial flow chromatography have
been made.
CHAPTER 1 - INTRODUCTION
This work consists of two parts. The first deals with the various aspects of inclusion
chromatography using cyclodextrin-containing polymeric resins. The second part is a
collection of studies of nonlinear chromatographic theories, ranging from the adsorp-
tion kinetics and isotherms to the analysis and optimization of column dynamics.
1.1 Inclusion Chromatography
Cyclodextrins (CDs) are a family of cyclic oligosaccharides consisting of usually 6,
7, or 8 a-(1,4)-linked D-glucopyranose units, which are called a-CD, P-CD and y-CD,
respectively (Figure 1.1). Among them, P-CD is the most abundant and most important.
The structure of CDs, which can be described as a truncated cone or a donut, gives
them a very attractive and useful feature: the ability to form inclusion complexes selec-
tively with a variety of compounds, such as aromatic amino acids, and many other
important biomolecules (Szejtli, 1982). Their capability for selective inclusion has
received much attention in many fields such as separations, drug delivery, artificial
enzymes.
Bccause CDs are soluble in water to some extent, they have to be immobilized
before they can be used as stationary phases for inclusion chromatography, which is
defined as a kind of chromatography that utilizes selective inclusions to provide
Figure 1.1 Structures of c yclodextrins (Szej tli, 1982).
selectivities (Zsadon et al., 1979; Szejtli et al., 1987). In this work a novel and success-
ful method of synthesizing CD-containing resins is presented. In this method a cross-
linking agent is used to link the CD molecules directly via chemical bonds. The syn-
thesis can be carried out under very moderate reaction conditions. Resins synthesized
using this method possess a very high content of CDs, and good physical properties in
terns of insolubility, hardness and wettability.
Experiments on the adsorption isotherms of some biomolecules such as aromatic
amino acids on CD-containing polymeric resins have been performed. Some water
soluble aliphatic alcohols have been found to be good displacers in inclusion chroma-
tography using CD-containing resins. Experiments have been performed to show the
effect of the displacer concentration on the stepwise displacement operation. Experi-
ments have also been carried out to recover phenylalanine from some industrial waste
samples such as cell steep water, using inclusion chromatography. It has been con-
cluded that such a recovery process is possible though it may not be economically feasi-
ble.
1.2 Studies of Nonlinear Chromatographic Theories
With the rapid growth of preparative and large scale chromatography in separations,
many chromatographic operations are in the nonlinear concentration range with
significant mass transfer or even kinetic resistances. The study of nonlinear chromatog-
raphy becomes more and more demanding. Much work has been done in the past two
decades, but many topics of practical importance still remain unsolved.
In this work, a robust and efficient numerical procedure has been developed to solve
a general nonlinear multicomponent rate model which considers axial dispersion, exter-
nal film mass transfer, intraparticle diffusion and complicated nonlinear isotherms. It
uses quadratic finite elements for the discretization of the bulk phase partial differential
equations and orthogonal collocation for the particle phase equations in the model,
respectively. The resulting ordinary differential equation system is solved by Gear's
stiff method (Gear, 1972). The model has been extended to include second order kinet-
ics and size exclusion factor. An alternative boundary condition at the column exit to
the Danckwerts boundary condition is also analyzed.
A methodology is presented in Chapter 5 for the development of kinetic and isoth-
erm models for multicomponent adsorption systems with uneven saturation capacities
for different components which are either physically induced or chemically induced.
The extended multicornponent Langmuir isotherm derived with this methodology,
which is therrnodynamic;ally consistent, has been very successfully used to explain
isotherm cross-over and to demonstrate peak reversal phenomenon under column over-
load conditions.
Computer simulations of stepwise chromatography based on the general rate model
with the multicomponent Langmuir isotherm have been carried out in Chapter 6. It has
been shown that a desirable displacer often should have a suitable affinity which does
not have to be higher than that of the displaced component if the objective of the opera-
tion is to displace the presaturated component from the column efficiently such as in an
integrated fermentation-column separation system. If the displacer concentration is
high enough roll-up in stepwise displacement occurs. Reverse flow displacement may
be beneficial if the previous frontal adsorption allows only low level column loading.
The study qualitatively proved some of the experimental observations such as the effect
of the concentration of the displacer on the displacement efficiency in inclusion
chromatography of aromatic amino acids using CD-containing resins.
Theoretical study of mass transfer effects in Chapter 7 points out the effects of mass
transfer on the resolution and performance of multicomponent chromatographic separa-
tions. The influence of the mass transfer related dimensionless parameters in the rate
model is analyzed. It is found that an elution peak with unfavorable isotherm will not
show the expected anti-Langmuir peak asymmetry if mass transfer effects are
significant.
In Chapter 8 a unified approach to a better understanding of multicomponent
interference effects under mass transfer conditions is proposed. It has been shown that
a displacement effect can be used to explain the dominating interference effects arising
from the competition for binding sites among different components in multicomponent
chromatography. It has been concluded that the conccntration profile of a component
usually becomes sharper due to the displacement effect from another component, while
the concentration front of the displacer is usually diffused as a consequence. Five fac-
tors stemming from equilibrium isotherms which tend to escalate the displacement
effect in multicomponent elution have been investigated. They have important implica-
tions for interference effects in multicomponent elution under column overload condi-
tions.
In multicomponent elution, competing modifiers sometimes are added to the mobile
phase to compete with sample solutes for binding sites in order to reduce the retention
times of strongly retained sample solutes (Snyder et al., 1988a). Peaks in the chromato-
gram corresponding to the modifiers are called system peaks (Levin and Grushka,
1986). Studies of the behavior of system peaks may provide useful information on the
effects of modifiers on the sample solutes and interpretation of complicated chromato-
grams. In Chapter 9, system peaks are studied systematically by the general multicom-
ponent rate model. Systems peak patterns have been summarized for binary elutions
with one competing modifier in the mobile phase for samples which are either prepared
in the mobile phase or in an inert solution. Binary elutions with two competing
modifiers have also been investigated briefly.
In Chapter 10, the kinetic and mass transfer effects are discussed. The rate limiting
step in chromatography is investigated numerically. The general model has been
modified to account for the reaction in the fluid phase between macromolecules and
soluble ligands for the study of affinity chromatography. The adsorption, wash and elu-
tion stages in affinity chromatography are simulated and analyzcd. Two kinds of inhi-
bition, one using the competing binding inhibitor, another the soluble ligand, in affinity
chromatography are investigated.
A general multicomponent rate model for radial flow chromatography (RFC) has
been solved in Chapter 11 with the same methodology as that for the conventional axial
flow chromatography (AFC). The radial dispersion and the external film mass transfer
coefficients are treated as variables in the model. Mass transfer effects and the
difference between inward flow and outward flow in radial flow chromatography have
been investigated. The comparison of, and mathematical analogy between, RFC and
AFC have been carried out. The question of whether it is necessary to treat dispersion
and film mass transfer coefficients as variable-dependent on the linear flow velocity is
addressed. The RFC rate model has also been extended to include second order kinet-
ics, size exclusion effects and liquid phase reactions for the study of affinity RFC.
In the last chapter, recommendations for future work regarding the experimental
verifications of some of the theories developed in this work and further expansion of
current models and theories are discussed.
CHAPTER 2 - LITERATURE REVIEW
This chapter is a general review of the literature on cyclodextrins (CDs) and the
dynamics of nonlinear chromatography. Reviews are also given as introductions in
some self-contained chapters in this work in order to maintain their integrity.
2.1 Cyclodextrins
2.1.1 Historical Background (Szejtli, 1982)
According to Szejtli (1982) the first publication about CDs was dated 1891 by Vil-
liers who isolated a small amount of a crystalline substance from a culture medium of
Bacillus amylobacter grown on a starch-containing medium. His data on its physical
properties were later found in good agreement with Schardinger's study on P-CD pub-
lished in 1903. Scardinger isolated a bacillus, which he named Bacillus macerans. This
micro-organism is still commonly used for CD production today.
Pure CDs were first prepared in the mid-thirties by Freudenberg and Jacobi (1935).
They also discovered y-CD. Higher homologues were discovered by French (1957) and
also by Thomas and Stewart (1965).
Many reviews on CDs are available in the literature. The following relatively recent
works are the most helpful:
reviews up to 1978 by Bender and Korniyama (1978),
up to 1981 by Hinze (1981),
up to 1982 by Szejtli (1982), and
up to 1988 by Szejtli (1988).
The last two works in the list are monographs which contain comprehensive reviews on
CDs. Between 1981 and 1986, there were four international symposia on CDs (Szejtli,
1982b; Atwoocl et al., 1984; Atwood and Davies, 1987).
2.1.2 The Structures of CDs
CDs are cyclic oligosaccharides consisting of, usually 6, 7, or 8 a-(1,4)-linked D-
glucopyranose units and they are called a-, P- and y-CD, respectively. Higher homolo-
gues have also been found, but they have negligible ability to form inclusion com-
plexes, and thus, are not important. Sundarajan and Rao (1970) showed that lower
homologues can not be formed due to steric reasons.
Figure 1.1 shows the structure of P-CD and spatial dimensions of a-, P- and y-CD.
The structures of CDs trace back to the structure of arnylose since CDs are produced
from starch by enzymatic reactions. Amylose has linear, nonbranched molecules with
several hundred or thousand glucopyranose units linked together by a-1,4 glycosidic
bonds. When cut by CD-producing enzymes, a variety of CDs are formed with different
numbers of ring units. among which P-CD is the most abundant and the cheapest pro-
duct (Szejtli, 1982). Figure 1.1 also shows that the primary hydroxyl groups of a CD
molecule are all on the outside of its structure and all secondary hydroxyl groups are on
the two edges of the ring. The inside of the cavity has no hydroxyl groups, and thus is
basically hydrophobic.
2.1.3 Some Important Properties of CDs
CDs possess no reducing end-groups. They generally show positive results in tests
characteristic of non-reducing carbohydrate, e.g. they are stable in alkaline solutions.
Their densities are in the range of 1.42 - 1.45 g/cm3 (Szejtli, 1982). Due to the hydroxyl
groups on the outside of their structures, CDs are soluble in water to a certain extcnt
and their solubilities increase greatly at high temperatures. The experimental correla-
tions for the aqueous solubilities of a-, P-, and y-CDs were reported by Lammers and
Wiedenhof (1967), Jozwiakowski and Connors (1985) and Szejtli (1988).
Solubilities of CDs generally decrease in the presence of organic solvents due to
complex formation, but in the case of ethanol and propanol a maximum point exists on
the organic solvent concentration vs. CD solubility curve (Szejtli, 1982). Some impor-
tant properties of CDs are listed in Table 2.1 (data Erom Szejtli, 1982).
Table 2.1 Some Important Physical Properties of Cyclodextrins
Property Molecular weight
0
Diameter of cavity 0
Volume of cavity A Solubility in water g/100 ml, 25OC
a-CD 972
4.7-6
176 14.5
P-CD 1135
8
346 1.85
y-CD 1297
10
510 23.2
CDs, though not sensitive to alkalis, can be hydrolyzed by acids partially or com-
pletely. The complete hydrolysis of CDs yields glucose, which can be used for quantita-
tive analysis of CDs (Szejtli, 1982). Cyclodextrins can also be analyzed with other
methods, such as chromatographic and photometric methods (Szej tli, 198 8). Extensive
studies on the toxicology of CDs have been carried out. CDs are now generally con-
sidered nontoxic, and in Japan, P-CD has been approved as a food additive (Szejtli,
1982).
The most important property of CDs is their ability to form inclusion complexes
with some guest molecules selectively. Inclusion complexes are molecular compounds
in which a guest molecule is trapped in the cavity of a host molecule. Other names used
in the literature for an inclusion complex include adduct, clatherate, complex, cryptate,
and molecular compound.
2.1.4 Types of Guest Molecules in CD Complexes
The cavities of CD molecules can accommodate a variety of guest molecules.
Because of the hydrophobicity of the cavity, CDs form inclusion complexes preferen-
tially with guest molecules which have bulky hydrophobic functional groups, such as
aromatic rings (Figure 2.1). Below is a list of types of guest molecules:
(1) Halogen and Hydrogen
Alpha-CD can form inclusion complexes with C12, Br2, while P-CD with Br2 and
12. y-CD can form an inilusion complex with Iz. Szejtli and Budai (1977) prepared
s o m e CD complexes with hydrogen halides (i.e., HCl, HBr and HI). The molecular
Figure 2.1 Schematic representation of the formation of a cyclodextrin inclusion complex (Szejtli, 1982).
ratio in the HCl-p-CD complex is 1.8:l. It is interesting that when the complex is
exposed to air for several months, this ratio will drop to 1:l and thereafter remain con-
stant (Szejtli, 1982).
(2) Gases
Alpha-CD can form complexes with with some common gases, such as C4, Xe, 02,
C02, ethylene, methane, propane, butane, etc. with different host/guest molar ratio
ranging from below one to above one (Cramer and Henglein, 1956, 1957). Such crystal-
line complexes formed by simply exposing a-CD solution to gases at elevated pressure
are stable even after storage for 18 months at room temperature. Upon dissolving in
water, they release gas bubbles (Szejtli, 1982).
(3) Fatty acids and their esters
Fatty acids, their methyl and ethyl esters, monoglycerides and their esters with sac-
charose are excellent guest compounds with CDs. Szejtli et al. (1977) reported that CDs
preferably form complexes with unsaturated acids. Pauli and Lach (1965) showed that
phenyl-substituted saturated fatty acids complex with P-CD much more easily than the
corresponding unsaturated fatty acids.
(4) Nucleic acids
Differential UV-spectroscopic studies (Hoffrnann and Bock, 1970) showed that
AMP and IMP form inclusion complexes with P-CD.
(5) Aromatic Compounds
Benzene and its derivatives generally are excellent guest molecules for inclusion
complexion with CDs because they possess bulky hydrophobic aromatic ring(s) in their
structure. Naphthalene forms inclusion complexes with P- and y-CDs but fails with a-
CD because its molecular size is too big for a-CD. Similarly, anthracene can form an
inclusion complex only with y-CD according to Szejtli (1982).
(6) Aliphatic alcohols
Methanol, ethanol, propanols and butanols are all capable of forming inclusion
complexes with CDs. Their use is very important in chromatography using CD-
containing resins (Szej tli, 1982).
(7) Biomolecules and Drugs
Many biomolecules possess bulky hydrophobic functional groups in their structure.
They can be included or partially included into cavities of CDs. The three aromatic
amino acids fall into this case. Other examples include some hormones, vitamins and
proteins and other pharmaceutically important molecules (Szejtli, 1982).
(8) Inorganic ions (Szejtli, 1982)
Some inorganic salts promote the solubility of CDs in water due to the formation of
inclusion complexes. Such inorganic ions include Cli, I-, SCN-, Br-, NOT and C1-.
Trommsdorf (1974) patented a CD silver complex. The preparation of a P-CD-copper
complex was reported by Matsui et ul. (1972).
Some molecules are not able to form stable inclusion complexes with CDs, but they
are able to form stable ternary complexes together with another guest molecule.
2.1.5 Binding Forces of CD Complexes
Calorimetry studies of the formation of a variety of CD complexes showed a favor-
able enthalpy change and an unfavorable (or slightly favorable) entropy change. This
indicates that the interaction force for the formation of a CD complex cannot be the
classical apolar binding such as that in the formation of enzyme-substrate complexes
because apolar binding is characterized by a very favorable entropy change (Bender and
Komiyama, 1978). The nature of the binding forces still remains controversial. Several
proposals were made to explain the binding forces involved in the formation of CD
complexes (Bender and Komiyama, 1978):
(1) Van der Wads interactions between guest and host,
(2) hydrogen bonding between the guest and the hydroxyl groups of CD,
(3) release of high energy water molecules in complex formation, and
(4) release of strain energy in the molecular ring of CD.
2.2 CDs as Catalysts (Bender and Komiyama, 1978)
It has been known that CDs can accelerate many different kinds of reactions such as
cleavage of ester and amide bonds, decarboxylation, oxidation, intramolecular acyl
migration, etc. Using CDs and their derivatives as artificial enzymes is a very active
research area. A good review on this can be found in a monograph by Bender and
Komiyama (1978).
2.3 CD-Containing Resins
Because of their ability to form inclusion complexes, CDs have been used in
chromatography in both the stationary phase and the mobile phase (Hinze, 1981; Love
and Arunyanart, 1986; Szejtli, 1987) when used in the stationary phase for chromatog-
raphy, CDs can retain those components that can be included in their cavities. Due to
their solubility in water, CDs must first be immobilized in order to be used as a station-
ary phase for chromatography. CD-containing resins can be classified into two
categories based on the support used:
2.3.1 With Inorganic Support
So far, only silica gel has been successfully used as an inorganic support. Fujimura
et al. (1983) flrst synthesized chemically bonded CD silica stationary phase for liquid
chromatography (LC) and tested the retention behavior of some aromatic compounds.
To prepare the resin, they used 10% solution of [3-[(2-aminoethyl) amino ] propyl 1
trimethoxysilane in dry toluene to react with silica gels to form diarnine-modified silica
gels. Then they used p-toluenesulfonyl chloride in dry pyridine to tosylate the primary
hydroxyl groups of a- or P-CD. The resulting CD tosylate, after being dried, was then
reacted with diamine-modified silica gels to form diarnino type silica gels with chemi-
cally bonded CDs. Feitsrna et al. (1985) used Fujimura's method to synthesize P-CD-
containing stationary phase, and carried out a study of enantiomer separation of some
aromatic carboxylic acids on HPLC. Kawaguchi et al. (1983) also immobilized CDs on
silica gel by condensation of carboxylated silica gel with CDs having amino groups by
using l-ethyl-3-[3-(dimethyl-amino) propyl] carbodiimide. They separated 0-, m-, and
p- isomers of several disubstituted benzene derivatives effectively on HPLC with such
resins. They also claimed that amino acids, aromatic alcohols, aromatic carboxylic
acids, or naphthalene derivatives were successfully separated using the P-CD-
containing resin. Shirashi et al. (1986) also immobilized CDs on silica gel in a simpler
method which did not involve the synthesis of modified CDs. In his method silica gel
was first chlorinated with thionyl chloride or silicon tetrachloride, and then was reacted
with P-CD.
The commercial P-CD-boned phase column by Advanced Separation Technology,
Inc. (Whippany, NJ) is packed with 5 pm silica covalently bonded with P-CD
molecules (Hinze et al., 1985; Chang et al., 1986). The P-CD molecules are chemically
bonded to silica gel via a stable spacer 6 to 10 atoms in length (Hinze, 1985). Alpha-
and y-CD bonded phase HPLC columns are also available from the same company.
Some researchers have used this kind of column to separate substituted phenolic com-
pounds in reversed phase HPLC (Chang et al., 1986) and to separate diastereomers and
structural isomers (Armstrong et al., 1985; Hinze et al., 1985; Marziani and Sisco,
1989).
Silica-based CD-containing resins are excellent for HPLC packing because of the
known advantages of silica used as a support. Unfortunately, such resins are not useful
when used as a separation medium to carry out preparative scale separations. The syn-
thesis of silica-based resins inherently limited the loading of CD on silica gel to only a
few weight percentages, typically 3%. Such resins are also very expensive.
2.3.2 With Organic Support
One of the first reported and very commonly used CD stationary phases is a polym-
eric 0-epichlorohydrin resin, abbreviated ECP m n z e et al., 1981). It has been used to
separate various natural products (vitamins, amino acids ), perfumes, aromatic amino
acids, o- or p-nitrophenol substituted chlorobenzoic acids derivatives and the diastereo-
mers of Co(NH3)4-glucose-6-phosphate ATP, but the swelling of such resins (up to
500% in water) can be quite undesirable in large-scale use.
CD polymer gels, obtained from their polymerization in solution with poly (vinyl
alcohol) using ethylene glycol-bis(epoxypy1) ether as cross-linking agent (abbreviated
a-, p-, y-CDP), have also been employed as stationary phases for inclusion chromatog-
raphy (Hinze et al., 1981). Aromatic acids can be easily separated from non-aromatic
ones and, under appropriate conditions, even from each other using either the a-, P-, or
y-CDP gels.
Two chemically bonded P-CD gels, P-en-Bio-Gel and P-en-agarose, were prepared
by Tanaka et al. (1981) by coupling mono-(6-~-aminoethylamino-6-deoxy)-~D to
either succinylhydrazide Bio-Gel P-2 or 1,4-butanediol diglycidyl ether. Both have
been utilized as stationary phases in the LC separations of the o-, m-, and p-isomers of
nitroaniline and dinitrobenzoic acid in experiments.
Several a- and P-CD-containing polyurethane resins (CDPU), cross-linked with dif-
ferent diisocyanates have been used with some success in both LC and GC (Case and
Case, 1970; Hayakawa et al., 1979; Mizobuchi et al., 1980, 1981). Mizobuchi et al.
(1981) reported that P-CD polyurethane resins contain high P-CD contents (ca. 65
wt%) and are stable in organic solvents. But they become unstable in acidic solutions.
The a-CD polyurethane resin also contains a high a-CD content, but it is unstable in
organic solvents. The LC separation of the aromatic amino acids on polyurethane resins
was also reported by Mizobuchi et al. (1981).
Unlike silica-based resins, it is possible to synthesize CD-containing resins which
contain a large weight percentage of CD. A large content in the composition of resins is
essential for such resins as effective adsorption media for large scale use, since CD has
a large molecular weight and one such big molecule usually can only include at most
one guest molecule. Other physical properties, such as hardness, swelling rate, insolu-
bility and wettability are also very important in the practical use of CD-containing
resins. When the CD content in the resins is high, the resins tend to be weak in hardness
and less insoluble. Note that pure CDs are starch-like substances with little physical
strength.
2.4 Applications of CDs, CD Derivatives and CD-Containing Resins
CDs, CD derivatives and CD-containing resins have found amazingly diversified
uses in various fields mainly because of their ability to form inclusion complexes.
2.4.1 Analytical Chemistry
Commercial CD-containing HPLC columns provide a very useful means for the
separation and analysis of enantiomeric, positional and structural isomers and many
biomolecules, especially drugs. Many researchers have used commercial CD columns
for the analysis of a variety of compounds ( Hinze et al., 1985; Armstrong et al., 1985;
Chang et al., 1986; Feitsma et al., 1987; Ahmed and El-Gizawy, 1987; Marziani and
Sisico, 1989). CDs can also be used in HPLC mobile phase ( Debowski et al., 1982;
Zukowski et al., 1985; Fujimura et al., 1986; Gazdag et al., 1988; Bazaant et al., 1988;
Shimada et al., 1988, 1989; Mohseni and Hurtubise, 1990; Italia et al., 1990) to adjust
retention behavior of those components which interact with them. Selective precipita-
tion using CDs is also useful for chemical analysis.
2.4.2 Applications in the Agricultural Industry
Many synthetic pesticides, herbicides, insecticides and fungicides form inclusion
complexes with CDs. Several advantages may be obtained by this, including stabiliza-
tion, enhanced wettability, solubility and bioavailability, micronizing effect, reduced
contact effect and Auxin-like effect (Pagington, 1987).
2.4.3 Applications in Food and Tobacco Industries
CDs can be used to protect food aroma and to eliminate unpleasant tastes. Reiners
and Birkhang (1970) showed a process for the removal of free fatty acids from veget-
able oils based on the formation of inclusion complexes. Wagner et al. (1988) discussed
a fruit juice debittering process using CD polymers cross-linked by epichlorohydrin. Yu
(1988) reported a decaffeination process using the same type of resin. CD-containing
tobacco filter can be used to retain a large portion of nicotine and tar (Miwa and
Tanaka, 1975, 1976).
2.4.4 Applications in the Pharmaceutical Industry
Many pharmaceutical chemicals are capable of forming inclusion complexes with
CDs. These include antiseptics, chemotherapeutic agents, insecticides, anthelmintics,
pesticides, vitamins, steroids and non-steroidal anti-inflammatory agents (Szejtli, 1982).
For such uses, CDs are usually modified to achieve better results. For example, since
P-CD has a small solubility in water, in order to increase the solubility of a drug via
inclusion complex, it is often modified to increase its solubility.
CDs can be used to promote the bioavailability of some drugs (Szejtli, 1982;
Thuaud et al., 1990). Several patents by the Ono Company (Hayashi et al., 1972, 1974)
described the preparation of CD complexes of different prostaglandin derivatives to
reduce the high instability of prostaglandins (especially in the E series).
CD-containing resins may also be used to separate some antibiotics from their fer-
mentation broth (Szejtli, 1982).
Some insoluble cross-linked CD polymers were found effective to accelerate heal-
ing of oozing wounds like burns and ulcers (Fenyvesi, 1988).
2.4.5 Applications in Chemical Industry
Several cases of applications of CDs in the chemical industry were reviewed in
detail by Szejtli (1982). The following examples are worth mentioning. Buckler et al.
(1969) mentioned that CD polymers can separate branched and straight-chain olefins,
xylene isomers for the production of detergents and plastics, and also upgrade jet-fuels
and lubricating oils by selective removal of straight-chain hydrocarbons.
Gerhold and Broughton (1969) developed a continuous process for the separation of
p-xylene from other C8 aromatic hydrocarbons including m-xylene and ethylbenzene.
An aqueous solution of a-CD was used in the adsorption column. Such separations
cannot be easily achieved by distillation since their boiling points are very close.
2.4.6 Other Applications
CDs and their derivatives are used in cosmetics. There are many dozens of patents
in this area (Szejtli, 1982). CDs can also be used as catalysts (Bender and Korniyama,
1978). Potentially they may also find use in environmental engineering since they may
be able to form inclusion complexes with guest compounds which are environmental
hazards.
2.5 Dynamics of Nonlinear Multicomponent Chromatography in Fixed-Beds
A very comprehensive review on the dynamics and mathematical modeling of
adsorption and chromatography was given by Ruthven (1984). Models in this area are
generally classified into three categories (Ruthven, 1984): equilibrium theory, plate
models and rate models. The emphasis of the review below is on the rate models since
the theoretical studies in this work are largely based on such models.
2.5.1 Equilibrium Theory
Glueckauf (1949) is considered the first person who developed the equilibrium
theory of multicomponent isothermal adsorption (Ruthven, 1984). The theory was
further developed into the interference theory by Helfferich and Klein (1970) which is
mainly aimed at stoichiometric ion-exchange systems with constant separation factors,
and a mathematically parallel treatise for systems with multicomponent Langmuir
isotherms by Rhee and coworkers (Rhee et al., 1970; Rhee and Amundson 1982).
Equilibrium theory assumes direct local equilibrium between the mobile phase and
the stationary phase, neglecting axial dispersion and mass transfer resistances. The
theory gives good interpretations of experimental results for chromatographic columns
with fast mass transfer rates such as many analytical and some preparative columns. It
can provide general locations of the concentration profiles of a chromatographic system
but fails to provide accurate details if mass transfer effects in the system are significant
(Lee et al., 1989). Equilibrium theory has been widely used for the study of multicom-
ponent interference effects (Helfferich and Klein, 1970) and ideal displacement
development (Rhee and Amundson, 1982). Many cases of practical application have
been reported (Glueckauf, 1947; Helfferich and Klein, 1970; Helfferich and James,
1970; Bailly and Tondeur, 1981; Frenz and Horvath, 1985; Yu et al., 1987; Frenz and
H o ~ a t h , 1988)
2.5.2 Plate Models
Plate models may also be called staged models or staged theories (Wankat, 1986).
Generally speaking, there are two kinds of plate models. The h t kind is directly
analogous to the "tanks in series" model for nonideal flow systems (Ruthven, 1984). In
such a model, the column is divided into a series of small artificial elements. Inside
each element the content is assumed to be completely mixed. This gives a set of first
order ordinary differential equations (ODE'S) that describe the adsorption and
interfacial mass transfer jrocesses. Many researchers have contributed to this kind of
plate model (Martin and Synge, 1941; Yang, 1980; Villermaux, 1981; Ruthven, 1982,
1984). However, plate models of this kind generally are not suitable for multicom-
ponent chromatography since the equilibrium stages may not be assumed equal for dif-
ferent components.
The other kind of plate model is formulated based on the distribution factors which
determine the equilibrium of each component in each artificial stages, and the model
solution involves recursi~~e iterations, rather than solving for ODE systems. The most
popular one of this kind should be the Craig distribution models. By considering the
blockage effect the Craig models are applicable to multicomponent systems. Descrip-
tions of Craig models were given by Eble et al. (1987b), Seshadri and Derning (1984),
and Solms et al., (1971). In recent years, Craig models have been extensively used for
the study of column-overload problem (Eble et al., 1987a, 19876).
2.5.3 Rate Models
Rate models are also called rate equation models. The word "rate" refers to the rate
expression or rate equation for the mass transfer between the mobile phase and the sta-
tionary phase. A rate model usually consists of two sets of differential mass balance
equations, one for the bulk fluid phase, another for the particle phase. Different rate
models have different complexity. A comprehensive review of rate models was given
by Ruthven (1984). Reviews were also given by Mansour et al. (1982), Yu and Wang
(1989) and Lee (1989).
2.5.3.1 Rate Expressions and Particle Phase Governing Equations
The solid film resistance hypothesis was first proposed by Glueckauf and Coates
(1947). It assumes a linear driving force between the equilibrium concentrations in the
stationary phase (determined from the isotherm) and the average fictitious concentra-
tions in the stationary phase. This simple rate expression has been used by many
researchers (Rhee and Amundson, 1974; Bradley and Sweed, 1975; Ruthevn, 1984;
Golshan-Shirazi and Guiochon, 1988a, 1989a; Farooq and Ruthven, 1990) because of
its simplicity, but this model cannot provide the details of the mass transfer processes.
The fluid film resistance mechanism which also assumes a linear driving force is
widely used (Ruthven, 1984). It is often called external mass transfer resistance. If the
concentration gradient inside the particle phase is ignored, the model then becomes the
lumped particle model, which has been used by some researchers (Zwiebel et al., 1972;
Santacesaria et al., 1982a; 1982b) If the mass transfer Biot number, which reflect the
ratio of the characteristic rate of film mass transfer over that of intraparticle diffusion, is
much larger than 1, the external film mass transfer resistance can be neglected with
respect to pore diffusion.
In many cases both external mass transfer and intraparticle diffusion must be con-
sidered. A local equilibrium is often assumed between the concentration in the stagnant
fluid phase inside macropores and the solid phase of the particle. Such a rate mechan-
ism is adequate to describe the adsorption and mass transfer between the bulk fluid
phase and the particle phase, and inside the particle phase in most chromatographic
processes. The local equilibrium assumption here is different from that made for the
equilibrium model which assumes a direct equilibrium of concentrations in the solid
and the liquid phase without any kind of mass transfer resistances.
2.5.3.2 Adsorption Kinetics and Affinity Chromatography
In some cases, the adsorption and desorption rates may not be high enough and the
assumption of the local equilibrium between the concentration in the stagnant fluid
phase inside macropores and the solid phase of the particle is no longer valid. Kinetic
models must be used. Some kinetic models were reviewed by Ruthven (1984) and Lee
(1988, 1989). The second order kinetics has been widely used in affinity chromatogra-
phy (Chase, 1984a,b; Arnold et al., 1985a,b, 1986a,b; Arne and Liapis, 1987; 1988a). If
the saturation capacities for all the solutes are the same, the second order kinetics
reduces to common Langmuir isotherm when equilibrium is assumed.
2.5.3.3 Governing Equations for Bulk Fluid Phase
The partial differential equations for the bulk fluid phase can be easily obtained
with differential mass balances. They usually contain the following terms: axial disper-
sion, convection, transient and the interfacial flux. Such equations themselves are gen-
erally linear if physical parameters are not concentration dependent. They become non-
linear when coupled with nonlinear rate expressions.
Analytical solutions may be obtained using Laplace transform (Ruthven, 1984; Lee,
1989) for many isothermal, single component systems with linear isotherms. The linear
operator method (Ramlaishna and Amundson, 1985) can also be used to solve problems
in linear chromatography. For more complex systems, especially those involving
nonlinear isotherms, analytical solutions generally cannot be derived (Ruthven, 1984).
With the rapid growth of the availability of fast and powerful computers and develop-
ment of efficient numerical methods, it is now possible to obtain numerical solutions to
complex rate models that consider various forms of mass transfer mechanisms (Gu et
al., 1990a). Complex rate models are now becoming more and more popular especially
in the study of preparative and large scale chromatography.
2.5.4 General Multicomponent Rate Models
A rate model which considers axial dispersion, external mass transfer, intraparticle
diffusion and nonlinear isotherms is considered a general multicomponent rate model.
Such a general model is adequate in most cases to describe the adsorption and mass
transfer processes in multicomponent chromatography. In some cases surface adsorp-
tion and size exclusion, adsorption kinetics, etc., may have to be included to give an
adequate account for a particular system.
Several groups of researchers have proposed and solved various general multicom-
ponent rate models using different numerical approaches (Liapis and Rippin, 1978; Yu
ang Wang, 1989; Mansour, 1989; Gu et al., 1990a).
2.5.5 Solution to the General Multicomponent Rate Models
A general multicomponent rate model consists of a coupled PDE system with two
sets of mass balance equations in the bulk fluid- and particle phases for each com-
ponent, respectively. The transient PDE system becomes nonlinear if any nonlinear
isotherms or nonlinear kinetics are involved in the system.
The fmite difference method is a very simple numerical procedure and can be
directly applied for the solution to the model (Mansour et al., 1982; Mansour, 1989),
but this procedure often requires a huge amount of memory space, and its efficiency and
accuracy are not competitive compared with other advanced numerical methods, such
as orthogonal collocation (OC), finite element and OCFE.
The general strategy for solving a nonlinear transient PDE system numerically using
the advanced numerical methods is to discretize the spatial axes in the model equations
first, and then solve the resulting ODE system using an ODE solver.
2.5.5.1 Discretization of Particle Phase Equations
The OC method is a very accurate, efficient and simple method for discretization. It
has been widely used for particle problems (Villadsen and Michelsen, 1978; Finlayson,
1980) and is obviously the best choice for the particle phase governing cquations of
general multicomponent rate models (Liapis and Rippin, 1978; Yu ang Wang, 1989; Gu
et al., 1990a).
2.5.5.2 Discretization of Bulk Phase Equations
Concentration gradients in the bulk phase can be very stiff thus the OC method is no
longer suitable, since global splines using high order polynomials are very expensive
(Finlayson, 1980) and sometimes not stable. The method of OCFE uses linear elements
for global spline and collocation points inside each element. No numerical integration
for element matrices is needed because of the use of linear elements. This discrctiza-
tion method can be used for systems with stiff gradients (Finalyson, 1980).
The finite element method with higher order of interpolation functions (typically
quadratic, or occasionally cubic) is a very powerful method for stiff systems. Its highly
streamlined structure provides unsurpassed convenience and versatility. This method is
especially useful for systems with variablc physical pararnctcrs, such as in radial flow
chromatography and nonisothermal adsorption with or without chemical reaction.
Chromatography of some biopolymers also involves variable axial dispersion
coefficient (Antia and Horvath, 1989).
2.5.5.3 Solution to the ODE System
If the finite element is used for the discretization of bulk phase equations and OC
for the particle phase equations, an ODE system then results. The ODE system with
initial values can be readily solved using an ODE solver such as subroutine IVPAG of
the IMSL (1987), which uses the powerful Gear's stiff method (Gu et al., 1990a).
If the discretization of the bulk phase equations is carried out using the OCFE, an
ODE system coupled with some algebraic equations which come from the continuity
of boundary fluxes (Finlayson, 1980; Yu and Wang, 1989) result. The system can be
solved using an available differential algebraic equation solver.
Such a system can also be conveniently solved with an ODE solver if one manipu-
lates the user-supplied function subroutine which evaluates the concentration deriva-
tives for the ODE solver to eliminate those algebraic equations in an in situ fashion.
This is possible since the trial concentrations are given as arguments for the subroutine.
This approach helps reduce the total number of equations in the final system. It was
apparently adopted hy Gardini et al. (1985) for a multi-phase reaction engineering
problem.
CHAPTER 3 - MATERIALS AND EXPERIMENTAL PROCEDURES
This chapter describes the syntheses of cyclodextrin (CD)-containing polymeric
resins and the experimental procedures of inclusion chromatography using such resins.
Due to considerations for technical secrecy the names of some of the compounds used
for polymer synthesis are withheld and in such cases coded names are assigned for
them.
3.1 Syntheses of CD-Containing Resins
3.1.1 Materials
The materials used in all the experiments and their sources are listed below. Some
substance names are withheld because part of the experimental methud is under con-
sideration for an application of a patent.
(1) CDs
Alpha-, P-CDs were purchased from the Ensuiko Sugar Company, Japan through its
U.S. representative, the Hereld Organization (Gaithersburg, MD 20879). Prior to syn-
thesis, CDs were purified to get rid of the water that was trapped in CD cavities. The
purification method is described below.
First, wash CD with spectrum grade acetone (Fisher Scientific) three times by shak-
ing the mixture of CD and acetone for 2 hours in a shaker. After each wash, use centri-
fugation to separate CD and acetone. After the treatment with acetone, treat CD with
cyclohexane (99%, in bulk volume from Purdue Chemistry Stores) in the same way
also for three times. Finally dry the CD in a vacuum oven at 200°F.
(2) Cross-Linking Agent BA (Compound Name Withheld)
Purchased from Aldrich. Prior to use it was dried in an oven at 100°C overnight.
(2) Cross-linking Agent SM (Compound Name Withheld)
The 75:25 (mole) copolymer with an average molecular weight of 1,900 was pur-
chased from Aldrich. Used as received.
(4) Methyl Methacrylate (MMA Monomer)
Purity 99%. Purchased from Aldrich. Inhibitor was removed by distillation before
use.
(5) Divinylbenzene
Tech. grade, 55% pure, and mixture of isomers. Remainder mainly 3- and 4-
ethylvinylbenzene. Purchased from Aldrich. Used as received.
(6) Catalysts
All catalysts were purchased from Aldrich and used as received.
(7) N,N-Dimethyl Formamide (DMF)
Certified ACS grade. Purchased from Fisher Scientific. It was mixed with CaO and
refluxed overnight and then was distillated to get rid of trace amount of H20 before use.
(8) Amino Acids
All amino acids used in the experiments were purchased from Aldrich and used as
received. All are of L form, unless otherwise stated as of D form.
(9) L-Aspartyl-L-Phenylalanine Methyl Ester (Aspartame)
Purity 96%. Purchased from Aldrich and used as received.
(10) Corn Steep LiquorIWater
Purchased from Sigma. 50% solids. Rich in vitamins, amino acids, and protein.
Phenylalanine content 1.5 wt% (dry base).
(1 1) Phenylalanine-Containing Industrial Waste Samples
The mother liquor samples and cell steep water samples containing phenylalanine
were obtained from an industrial source. The mother liquor samples contain 2.5-3.2
wt% phenylalanine, 5.4% inorganic salts (e.g. sod2-, ca2', m-).
3.1.2 MMA Method
Beta-CD molecules are first modified to chemically bond the easily pol ymerizable
functional group (- C = C -) to the structure of the CDs. This is done by reacting maleic
anhydride with P-CD in DMF with an initiator at room temperature (Shen et al., 1987)
Since one P-CD molecule contains 21 hydroxyl groups, the maximum number of
hydroxyl groups that can be modified is 21. The mono-carboxyl vinylenic P-CD which
is a P-CD derivative with only one hydroxyl group being modified is used. The CD
derivative is copolymerized with MMA monomer and a cross-linking agent divinylben-
zene.
3.1.3 BA Method
In this method no prior modification of CD is needed. BA is used as a cross-linking
agent and DMF as a solvent. CD molecules are directly cross-linked by BA under the
catalysis of a very potent catalytic system. The polymerization is carried out in one
step under room temperature. Typical experimental procedure is as follows:
In a 1000 rnl round bottom flask, 40 g of purified P-CD is dissolved in 600 rnl dried
DMF, then 10 g of BA is added and dissolved in the solvent by vigorous stirring. After
this, catalysts are added. The flask is then sealed and the inside solution is stirred
rigorously. The reaction liquid becomes thicker and thicker during the course of the
reaction and after half an hour the liquid becomes a kind of sticky opaque gel indicating
the formaaon of a highly cross-linked polymer
The post treatment of reaction product begins with the evaporation of DMF in the
gel by using a rotary vacuum evaporator at 85OC. After this, the product is put into a
vacuum oven at 200°F to get rid of remaining DMF from the product. The dried pro-
duct is further treated with methanol in a Soxhlet extraction apparatus for several days
to wash away the remaining catalysts. Finally the product is dried in a vacuum oven at
200°F to give the final product which was lumps of particles with a light yellow color.
The product typically weighs around 48 g. The combined weight of P-CD and BA in
the feed at the beginning of the reaction was 50 g.
Samples of the final product are sent to the Micro-analysis Laboratory in the Chem-
istry Department of Purdue University for element analysis to determine the contents of
C, H, 0, N.
Alpha-CD-containing resins can also be synthesized by the same method. It was
also found that a compound named SM in this work can be used to replace BA as the
cross-linking agent.
3.2 Adsorption Isotherms
In order to find out the adsorption behavior and capacity of some compounds on the
resins, adsorption isotherm must be obtained by experiments. isotherms data are also
essential in the mathematical modeling of chromatography. Generally speaking, there
are two ways to carry out this study.
3.2.1 Chromatographic Methods
Such methods have become popular these days. They can be used to dynamically
measure isotherms. There are two principal methods (Jacobson, 1984): Frontal Analysis
(FA) and Frontal Analysis by Characteristic Points (FACP). The FA method is not
efficient because each frontal analysis gives only a single equilibrium data point. FACP
assumes that each point at the rear front curve is at equilibrium. This assumption may
cause considerable error if the mass transfer effect is very prominent. Chromatographic
methods avoid the waste of resins, but the amount of sample solutes used may be large.
3.2.2 Batch Adsorption Method
Batch adsorption is a traditional method and is still very popular today. Typical pro-
cedure used in this work is described as follows. One gram of resin is placed into a 25
ml test tube with a half-inch magnetic stirrer. Then 10 ml adsorption solution with con-
centration C, is added. The test tube is sealed and placed in a thermal bath with a con-
stant temperature. The solution is stirred for 24 hrs to ensure that the adsorption equili-
brium is reached. After that, the supernatant volume (V,) is obtained by filtration. The
supernatant is further filtrated through a 0.5 pm membrane filter and then analyzed on
an HPLC for equilibrium concentration (C,).
Experimental isotherm data are expressed by plotting equilibrium bulk concentra-
tion (C,) vs. concentration in resins (C,), where C, = (CoVo - CeVe)/(Vo - V,) in
which V, is the initial volume of the liquid and C, the initial concentration of the sam-
ple solute. This kind of isotherm curve shows how many times the solute is concen-
trated at each equilibrium bulk concentration (C,) point. In order to show the adsorp-
tion capacity an isotherm can also be expressed by plotting equilibrium bulk concentra-
tion vs g solute adsorbed/ g resins [ = (C, - C,)V,/W, where W is total resin weight].
The unit for the concentration in the solid phase can be converted to the unit based on
the unit volume of the solid volume of the resin particle, excluding pores, which is used
for the general rate models described in Chapter 3, provided that the porosity and den-
sity of the resin are known.
3.3 Inclusion Chromatography Using Cyclodextrin-Containing Resins
3.3.1 Column Packing
Cyclodextrin resins to be used were first ground and sieved, and particles with a cer-
tain mesh range (e.g. 120-170 mesh) were selected for column packing. The slurry-
packing method was used (Figure 3.1). The resin was washed with water for several
times to get rid of fine powders on the surface of resins caused by grinding in order to
prevent clogging at the column frit. After this, the resin was soaked in ethanoVwater
(v:v=l: 1) solution for a couple of hours or overnight to let resins swell. The resins were
then packed into a 250 x 7.1 mm stainless steel column. This step is not a necessity
since swelling is not a problem with CD-containing resins synthesized by the BA
method.
It was observed during experiments that a well-packed column gave only a very
small presure drop. The flow of the mobile phase through the column can be induced by
gravity alone.
3.3.2 Nongradient Elution
A 250 x 7.1 mm column was packed with 120-170 mesh resins containing 63 wt%
P-CD. A typical chromatographic system was constructed with a column packed with
CD-containing resin, a Waters HPLC pump, a Waters U6K sample injector and an
ISCO v4 absorbance detector (Figure 3.2). The following operational conditions were
used.
Stainless Steel Container
Beaker
HPLC Pump
Figure 3.1 Slurry packing.
Pump
Three-Way Valve
Mobile Phase Solutions
Column
UV Detector
Fraction Collector
Figure 3.2 Experimental setup.
Mobile phase 30 % (v) EtOH in water Flow rate 0.6 mVmin Sample volume 0.05 rnl Sample composition 0.02 M Phe and 0.02 M Trp in water Detector UV @ 254 nm Paper speed 15 mdhr
3.3.3 Stepwise Displacement Experiments and Evaluation of Displacers
A 250 x 7.1 mm stainless steel column packed with 10.9 g of resins which had a P-
CD content of 38 wt% was used. The experimental procedure is described as follows.
Prior to adsorption the column is eluted with distilled water until it is equilibrated with
water. At one point, the mobile phase is switched to a solution containing a sample
solute. After the breakthrough curve levels off, the mobile phase is switched to a
displacer solution (e.g., ethanoywater solution) to begin the displacement of the
adsorbed component till the desorption curve levels off. The column is regenerated for
a next experiment by eluting pure water through the column.
3.3.4 Experiments with Real Sample Systems
So far two real systems have been tested to recover L-phenylalanine which is the
major raw material used in the synthesis of aspartame (NutraSweet), a very popular
artificial sweetener. Samples were supplied by an industrial source.
3.3.4.1 Corn Steep Water
The original corn steep water was a very thick and sticky paste-like substance full
of solid particles. It was diluted ten times with distilled water and passed through a filter
paper. After this pretreatment the sample was then eluted through a 250~7.1 mm
column packed with 100-120 mesh resins which contained 63 wt% of P-CD in a frontal
adsorption fashion. The subsequent displacement operation was carried out using 100%
ethanol.
3.3.4.2 Phenylalanine-Containing Industrial Waste Samples
A preparative-scale column was used in the adsorption and displacement experi-
ments. A 1.5 x 30 cm column packed with 80 wt% 60-80 mesh and 20 wt% 80-100
mesh resins which contains 63 wt% P-CD was used.
(1) Mother Liquor Samples
Mother Liquor samples from the production of phenylalanine are often saturated
with phenylalanine. They also contained a lot of impurities including various salts, pro-
teins and oligopeptides. Due to the viscousity of such samples they had to be diluted 10
times before frontal adsorption. The stepwise displacement column operation was car-
ried out using 100% ethanol as the dispacer.
(2) Cell Steep Water
A fourth-wash cell steep water was used for the recovery of phenylalanine. The
phenylalanine content was determined by IPLC to be 0.022 M. Before the frontal
adsorption, the sample was filtrated through a 5p membrane filter and its pH was
adjusted to 7 with concentrated NaOH solution. The displacement operation was carried
out using 100% ethanol. A fraction collector was used to collect the effluent for the
analysis on HPLC. Operational conditions of the experiment are listed below:
Phe Concentration in the Sample Column Resin Flow Rate of Ad/Desorption Displacer Fraction Collector Detector Temperature
0.022 M (pH = 7) 1.5 x 30 cm 63 wt% P-CD 1.65 mVmin 100% EtOH 1 tube14 min UV @ 254 nm 25' C
Usually a phenylalanine sample is analyzed by using reversed-phase gradient
HPLC. The speed of such a method is very low, say 40 min per sample. Since the
effluent samples contained only a few components the following nongradient (or quasi-
isocratic) HPLC method was found to be workable.
Column Waters 1-1 BONDAPAK C18 P/N 27324(1986) Mobile phase 0.1 % TFA in water. Flow rate 1.5 ml/min Detector UV @ 254 nm
CHAPTER 4 - EXPERIMENTAL RESULTS AND DISCUSSIONS
4.1 Properties of Synthesized CD-Containing Resins
With the MMA method described in Section 4.1.2 a copolymer was produced which
contained only a very small amount of p-CD. Such a polymer is also very brittle, thus
not suitable for chromatographic used, and the method was abandoned.
With the method described in section 4.1.3, which uses compound BA as the cross-
linking agent, P-CD-containing resins were produced. The P-CD contents can reach up
to 70 wt% (determined from element analysis) by using the BA method. The physical
properties of such resins are also quite good in terms of hardness, insolubility in water
organic solvents, low swelling and good wettability. The density of such resins is
around 1.3 g/cm3. The resins have a slight yellow coloration and look like beach sand.
It was found that the resins are quite acid resistant, but they decompose in alkaline solu-
tions.
P-CD-containing resins with higher (than 70 wt%) P-CD content can also be syn-
thesized using the BA method, but such resins have less desirable physical properties
because when the CD content is too high there is not sufficient cross-linking. The phy-
sical properties of a cyclodextrin-containing resin largely depend on the degree of
cross-linking (Billmeyer, 1984). The higher the degree of cross-linking, the harder and
more insoluble the resin becomes. Note that a fair amount of the cross-linking agent BA
content is required to achieved a sufficiently high degree of cross-linking.
Figure 4.1 is the IR spectrum of a P-CD-containing resin, which shows a big -OH
absorbance peak, a clear indication of the existence of a large content of P-CD in the
copolymer. The analysis of the steep water of the product after copolymerization of P-
CD and BA prior to the methanol treatment gave no trace of dissolved P-CD. This
proves that all 0-CD was copolymerized during the reaction. The amount of BA reacted
was easily determined by the common acid-base titration of the steep water and was
found to be very close to 100%.
It is also possible that some polymers were washed away during the extraction with
methanol in the post-treatment stage because their degree of polymerization was not
high enough to make them absolutely insoluble in organic solvents. It was also found
that some catalyst residues remained in the resin, which is typical of the catalytic sys-
tem used in this method and experiments were carried out to try to reduce the content of
the residues.
The BA method was also used to synthesize a-CD-containing polymeric resins.
Similar results were obtained. Figure 4.2 is the IR spectrum of a a-CD resin syn-
thesized with the BA method, which contains 63 wt% of a-CD. The a-CD resins syn-
thesized using BA method are insoluble in organic solvent, unlike a-W polyurethane
resins which are unstable in organic solvents (Mizobuchi et al., 1981).
It was found that cross-linking agent SM can be used to replace BA for the syn-
thesis of CD-containing resins. Equally desired results were obtained as far as CD con-
tents and physical properties are concerned.
4.2 Reaction Mechanisms
Figure 4.3 is a schematic illustrating the assumed structure of the resins produced
by directly cross-linking with a strong cross-linking agent, such as BA and SM. The
structure is a random copolymer of the CD and the the cross-linking agent. The pro-
posed reaction mechanism, which is characteristic of the catalyst system used for the
synthesis, was reported elsewhere (Gu et al., 1987).
4.3 Adsorption Isotherms
All isotherms listed in Figures 4.4-4.6 show Langmuir-like behavior in the testcd
concentration ranges. Unlike in gas adsorption the leveling off cannot be achieved in
the systems in this work, because of the solubility limitation.
Comparing Figure 4.4 with Figure 4.5 it is easy to find that at the same equilibrium
bulk fluid concentration the resins which contains 63.3 wt% of P-CD adsorbs more
tryptophan than phenylalanine in terms of weight amount. This is in part because
tryptophan's ability to form inclusion complexes with P-CD is stronger than that of
phenylalanine and also because the molecular weight of tryptophan is larger than that of
phenylalanine.
Figure 4.4 shows that increased temperature reduces the adsorption of phenylalan-
ine on P-CD resins. This is also a general behavior because the formation of inclusion
complexes usually involves a favorable enthalpy change. In phenylalanine case the tem-
perature effect is not strong. Usually when temperature is increased to 70 - 90°C most
complexes will break up. We failed to carry out phenylalanine isotherm experiments
Figure 4.3 Schematic representation of the structure of a CD resin.
Phenylalanine Adsorption on Beta-CD Resins
Phenylalanine Adsorption on Beta-CD Resins
0.00 0.05 0.1 0
Equlibrium Phe Concentration (M)
Figure 4.4 Adsorption isotherm of Phe on a P-CD resin.
Adsorption of Trp on Beta-CD Resins (24°C)
0.00 0.01 0.02 0.03
Equilibrium Trp Concentration (M)
Aspartame Adsorption on Beta-CD Resins (21.5"C)
0.00 0.01 0.02
Equilibrium Aspartame Concentration (M)
Figure 4.5 Adsorption isotherms of Trp and Aspartame on a j3-CD rcsin.
Adsorption of Phenylalanine on Alpha-CD Resins (21.5 "C)
0.00 0.05 0.1 0 0.1 5
Equilibrium Phe Concentration (M)
Figure 4.6 Adsorption isotherm of Phe on an a-CD resin.
filtrated through 0.5pm membrane filter for HPLC analysis. It is likely that the superna-
tant is a suspension containing low molecular weight polymers containing cyclodext-
rins.
4.4 Separation of a Binary Sample by Nongradient Elution
Figure 4.7 is the chromatogram of a nongradient elution with conditions stated in
Section 3.3.2. It shows that at the penetrable volume time a sharp peak appears. This
peak was caused by the water content in the sample. Similar experiments showed that if
the sample was dissolved in an Na2HP04 / C6H8O7 buffer, a peak representing the
buffer components would also appear near the penetrable volume time because the
buffer components do not adsorb on the P-CD-containing resin. After the first unre-
tained peak, there are two peaks which correspond to Yhe and Trp, respectively. The
two amino acid peaks partially overlap under c m n t operational conditions. A base-
line separation can be achieved if finer resin particles and a longer column are used
under optimized operational conditions. It can also be achieved by reducing the sample
size. Some researchers (Hinze, 1982), when testing their own synthesized P-CD resins,
have used very long columns (up to 100 cm) and very small flow rate (as low as 1
ml/hr) to achieve a base-line separation of some aromatic compounds.
The chromatogram shown in Figure 4.7 shows that it is very easy to separate
aromatic amino acids from compounds that do not form inclusion complexes with P-
CD, but the separation between the two aromatic amino acids (Phe and Trp) is achiev-
able, but not as easily.
cm- COO-
H
Phenylalanine Tryprop han
Mobile Phase 30% (v) EtOH Flow Rate 0.6 Wmin Sample 0.02 M Phe & 0.02 M Trp in Watcr Sample Size 0.05 ml W Detector 254 nrn
C o i n S izt ZQx7.l rnm Resins 63% (wt) Beta-CD
Figure 4.7 Multicomponent elution using a 8-CD column.
In Figure 4.7,30 (v/v)% EtOH was used as the mobile phase. The use of an organic
modifier in the mobile phase for a CD column helps reduce retention times for some
strongly retained components. Without the organic modifier these components have
excessive retention times and their peaks tend to be very broadened. Other water solu-
ble aliphatic alcohols and acetonitrile can also be used as organic modifiers since they
are able to form inclusion complexes with cyclodextrin (Szejtli, 1982). The higher the
concentration of an organic modifier in the mobile phase the shorter the retention times
of sample components.
It is worthwhile pointing out that unlike many other aliphatic alcohols, tert-butyl
alcohol may form a ternary inclusion complex with the cyclodextrin and the sample
solute. In such a case, the addition of this organic modifier in the mobile phase tends to
increase the retention time of the sample solute (Tan et al., 1988) rather than reduce it.
Thus, tert-butyl alcohol is no longer a competing modifier. If increased retention times
are desired then ten-butyl alcohol may be used.
4.5 Displacement Experiments and Evaluation of Displacers
A scrics of displacement chromatography experiments, including the adsorption of
phenylalanine and its displacement by alcohol and other organic solvents, were carried
out to evaluate the displacement efficiency. A typical response curve is shown in Fig-
ure 4.8. The column is first equilibrated with pure water. At time A, a sample solution
which contains an amino acid is used as the mobile phase for frontal adsorption. After
the UV absorbance curve levels off at time C, the mobile phase is shifted to the
displacer. At point D the displacer begins to elute out of the column and brings the
UV Absorbance
Displacer I
Time
Figure 4.8 Typical response curve of a two-stage operation.
desorbed sample solute with it. The concentration of the desorbed sample solute in the
displacer goes up at point D to a maximum (point E) very quickly and then comes down
until no solute is eluted out when the absorbance curve levels off at point F. This kind
of stepwise displacement response with a big roll-up peak is an indication that the
displacer used is quite good.
Experiments showed that aliphatic alcohols (such as methanol, ethanol, propanol
and butyl alcohol) and some other organic solvents like tetrahydrofuran, and acetoni-
trile gave excellent displacement results in displacing phenylalanine and tryptophan.
Among these displacers alcohols are favored. Ethanol is especially attractive as a
displacer. The reasons are as follows: (1) It is quite safe to use in food-related indus-
tries; (2) Its low boiling point makes the removal of it from the product energy efficient
and it causes no contamination in the final product; and (3) It is a10 economical to use
because it is cheap and recyclable.
As a matter of fact, phenylalanine and tryptophan dissolve only slightly in ethanol,
isopropanol and n-propanol, etc. Why then such alcohols still displace them from the
resins efficiently? A very interesting phenomenon was observed in the experiments. It
was found found that at the peak maximum area (point E in Figure 4.8) the displacer
was actually oversaturated with the sample solute. This oversaturation was so excessive
that the eluted effluent appeared to be quite milky.
Figures 4.9-4.11 show several experimental chromatograms These and other
chromatograms presented show that the order of the time required for displacement by
alcoholic displacers is methanol > ethanol > (iso, n)-propanol > butyl alcohol, i.e., the
larger the molecular size of the displacer the quicker the displacement process.
Experiments on the adsorption and desorption of L-analine, which does not adsorb
on P-CD resin, were carried out for reference. Figure 4.12 shows that the breakthrough
and displacement curves are both quite stiff since alanine does not retain on the resin.
The effect of dsplacer concentration on the efficiency of the stepwise displacement
operation was also studied experimentally. For a column presaturated with 0.01 M
phenylalanine, an EtOH water solution with different volumetric ratios was introduced
as the displacer. Figure 4.13 shows that the higher the EtOH concentration the higher
the displacement efficiency. It was found that if the EtOH concentration was not high,
oversaturation of the sample solute (Phe) in the effluent did not occur since the solubil-
ity of the sample solute increased when the water content in the displacer stream was
increased. It should be pointed out that a displacer may have to be diluted if it is harm-
ful to the sample solute. For example, some proteins may denature or lose their activi-
ties when contacted with organic solvent at certain concentration.
The following operational conditions were used for the study of concentration
effects.
Column Size 250 x 7.1 mrn Resin 63 wt% P-CD Flow Rate 1.0 ml/min Detector UV @ 254 nm
The finding that somc organic solvents can be used as effective displacers in step-
wise displacement chromatography using P-CD-containing resins gives encouraging
support to the feasibility of the practical use of cyclodextrin-containing resins.
Cclumn Size: 250~7.1 mm Resins: 38%(wt) Beta-CD Flow Rate: 0.7 mllmin
I b
Time (hr)
Absorbance (254nrn)
Shift to 100% MeOH
t ,
Figure 4.9 Displacement of Phe with 100% Methanol.
1
,K Brea'kthrough Curve of 0.01 M Phe
t Absorbance (254nrn)
Step Change to 0.01M Phe '
Column Size: 250~7.1 mrn Resins: 380h(wt) Beta-CD Flow Rate: 0.7 mllmin
Shift to 100°/o €.OH
Figure 4.10
I
Displacement of Phe with 100% Ethanol.
- Time (hr)
Breakthmugh Curve of 0.01 M Phe
Column Size: 250~7.1 mm Resins: 38%(wt) Beta-CD Flow Rate: 0.7 rnllrnin
I
Time (hr)
Figure 4.1 1 Displacement of Phe with 100% n-Propanol.
Absorbance
Shift to Water
Time (hr)
Figure 4.12 Adsorption and displacement of an inert component.
I I Column Size: 250~7.1 mm Resins : 63%(wt) B e a t 4 0
tOO%(v) EtOH Detectgr : UV @' 254 nm Desorption Conditons: Row Rate: 1.0 mllmin. Column Was At Equiiibriurn With 0.01 M Phe Before Desorption.
O.01M L-Phe , / Breakthrough Curve
10% EtOH .
Figure 4.13 Effect of displacer concentration.
4.6 Application to Real Systems
Several experiments were carried out to apply CD-containing resins to recover
phenylalanine from some industrial sample systems.
4.6.1 Corn Steep Water
The raw corn steep water is actually not a liquid but rather a paste-like mixture
which contains many components other than phenyalanine (Reiners et al., 1973). It
was found that before adsorption the corn steep water had to be diluted at least 10 times
and filtrated, and at this time its phenylalanine content was less than 0.3 wt% which is
no longer attractive as a source for the recovery of phenylalanine compared with other
industrial waste samples.
4.6.2 Phenylalanine-Containing Industrial Waste Samples
Experiments were carried out to recover Phenylalanine from several industrial
waste samples.
(1) Mother Liquors From Commercial Phenylalanine Production Process
Mother liquors are much better sources than corn steep water for the recovery of
phenylalanine, but a protein assay showed that they contain massive amount of impuri-
ties such as proteins and oligopeptides, which tend to compete with phenylalanine for
inclusion with PCD cavities. Experiments showed that displacement effluent samples
still contain at least eight impurities and their amount is on the same order as that of
phenylalanine.
Efforts were made to use a pre-column to get rid of impurities before adsorption on
a fKD column. Several adsorbents were tried including polyvinylpyridine, lignine,
molecular sieves, and different activated carbons. It was found that the Darco G-60
activated carbon showed some effects, but it also adsorbed phenylalanine to some
extent and the regeneration of such carbon column was very difficult. Such activated
carbon is also expensive. It is unlikely that such a recovery process can be commer-
cially feasible.
(2) Cell Steep Water
Analysis of the sample shows that it contains at least four major components. Figure
4.14 shows the adsorption and desorption chromatogram with the operational condi-
tions stated in Section 3.3.4.1. The desorption curve shows a maximum point. The
curve levels off around the point when 165ml ethanol is eluted through the column. The
total amount of phenylalanine recovered in each run was determined to be 0.4 g, which
is around 2% of the dry weight of the resin inside the column.
It seems that it is possible to recover phenylalanine from cell steep water by using
the P-CD resin, but such process may not be commercially viable since phenylalanine is
not a highly priced product.
The ability of EtOH to form inclusion complexes with P-CD is considerably weaker
than that of Phe (Szejtli, 1982), but EtOH can still efficiently displace Phe from a P-CD
column if EtOH concentration is high enough, The theoretical explanation and
verification of this phenomenon are presented in the Chapter 6.
Adsorptionldesorption of Cell Steep A
E Water on Beta-CD Column at 2 5 ' ~
Adsorption Sample pH=7.0 Concentration=0.022 M Flow Rate 1.65 rnllmin Column Size 250x1 6 mm Displacer: 100% EtOH
Desorption
I "
TIME (min)
Figure 4.14 Recovery of Phe from a cell steep water sample.
CHAPTER 5 - THEORY AND MATHEMATICAL MODELING
In this work, various theoretical studies were carried out for axial flow chromatogra-
phy using computer simulations based on a general nonlinear multicomponent rate
model and its extensions. Models have been further extended in Chapter 10 for the
study of inhibition effects in affinity chromatography. Modeling and theoretical studies
on radial flow chromatography will be discussed in Chapter 11.
5.1 A General Multicomponent Rate Model for Axial Flow Chromatography
5.1.1 Model Assumptions
Figure 5.1 shows the anatomy of a chromatographic column with axial flow. The
following basic assumptions are needed for the formulation of the general rate model.
[l] The multicomponent fixed-bed process is isothermal.
121 The bed is packed with porous adsorbents which are spherical and uniform in
size.
[3] The concentration gradients in the radial direction of the bed are negligible.
[4] Local equilibrium exists for each component between the pore surface and the
stagnant fluid phase in the macropores.
Bulk Phase
Column
cg Particle Phase
Figure 5.1 Anatomy of a chromatographic column.
[5 ] The diffusional and mass transfer coefficients are constant and independent of
the mixing effects of the components.
5.1.2 Model Formulation
Based on these basic assumptions, the following governing equations can be formu-
lated from the differential mass balances for each component in the bulk fluid- and the
particle phases.
with the initial and boundary conditions
Eqs. (5-1) and (5-2) are coupled via Cpi,R=Rp which is the concentration of component
i at the surface of a particle. In Eq. (5-2), Ch is the concentration of component i in the
solid phase of the adsorbents based on the unit volume of the solid, excluding pores. It
is directly linked to the multicomponent isotherms which couple the PDE system based
on assumption [4]. Concentrations Chi and Cp are based on the unit volume of mobile
phase fluid. The effective diffusivities Dpi in this work do not lump in the particle
porosity.
By introducing the following dimensionless terms
the PDE system can be transformed into the following dimensionless forms.
For frontal adsorption Cfi(z)/Coi = 1
For elution 1 O<zLz,,
0 else
After the sample introduction (in the form of frontal adsorption):
if component i is displaced, Cfi (z)/Ca = 0
if component i is a displacer, Cfi (z)/Co; = 1
Note that all the dimensionless concentrations are based on Coi which is equal to the
maximum of the feed profile CG-(z). For example, in an elution, if component i is a
sample solute in the sample which is injected as a rectangular pulse, the profile of Cfi(z)
is then of a rectangular shape, and its upper boundary value is the value of Coi.
5.2 Numerical Solution to the Model
An efficient and robust numerical procedure has been developed by Gu et al.
(1990a) for the solution to the above PDE system. It involves two parts. First, the spa-
tial axes, z and r, are discretized. And then the resulting ODE system (with initial
values) is solved with an ODE solver (integrator). An overview of the general strategy
for the solution is shown in Figure 5.2.
5.2.1 Discretization
Equations (5-9) and (5-10) can be discretized into a set of ODE'S by the finite ele-
ment and the OC method, respectively. Using the Galerkin approximation and the first
weak form (Reddy, 1984), Eq. (5-9) becomes
[DBi] [c&] + [AKBi] [chi] = [PBi] + [AFBi] (5- 17)
where (DBi%,n = I b b d z (5- 18)
in which m, n E (1, 2, 31, and the superscript e indicates that the finite element
matrices and vectors are evaluated over each individual element before global assem-
bly. Four point Gauss-Lcgcndre quadraturcs (Reddy, 1984) are used for integrations.
The superscript prime in this work indicates a first order time derivative. The bold face
Coupled PDE System
Bulk Phase PDEs Ns
t Finite Element Discretization
Ns(2Ne+ 1)
Coupled ODE System Ns(2Ne+ l)(N+ 1)
Gear's Stiff Method
1
Concentration Profiles Inside Column
I c . ( z , z ) VS. Z I bl
Figure 5.2 Solution strategy for the general multicomponent rate model.
variables indicates matrices or vectors. The natural boundary condition
(PBi) 1 ,+ = - chi + Cfi(z)/Coi will be applied to [ A D i ] and [AFBi] at z=0. (PBi) = 0
anywhere else.
Using the same symmetric polynomials as defined by Finlayson (1980), Eq. (5-10)
is transformed to the following equation by the OC method.
in which gi = (1 - %) cii + ~p cpi. Note that gi for each component i contains the non-
linear multicomponent isotherms. The value of ( c ~ ~ ) ~ + ~ (i.e. c,~,,~) can be obtained
from the boundary condition at r = 1, which gives
where the matrices A and B are the same as defined by Finlayson (1980).
5.2.2 Solution to the ODE System
If Ne quadratic elements [i.e., (2 Ne + 1) nodes] are used for the z-axis in bulk
phase equations and N interior OC points are used for r-axis in particle phase equations,
the above discretization procedure gives Ns (2 Ne + 1) (N + 1) ODE'S which are then
solved simultaneously by Gem's stiff method (IMSL, 1987). A function subroutine
must be supplied to the ODE solver to evaluate concentration derivatives at each ele-
ment node and OC point with given trial concentration values. The concentration
derivatives at each element node (&) are easily determined from Eq. (5-17). The con-
centration derivatives at each OC point (ck) are coupled because of the complexity of
the isotherms which are related to gi via cii. At each interior OC point, Eq. (21) can be
rewritten in the following mamx form.
a g i GPij = - , cPj = - dcp' , RHi = right hand side of Eq. (5-21). a% dz
Since the rnamces [GP] and [RH] are known with given ma1 concentrations at each
interior OC point, the vector [cb] can be easily determined from Eq. (5-24). Using this
approach, we can deal with complex nonlinear isotherms here without any iteration.
5.2.3 Isotherm Expressions
The numerical procedure discussed above can accommodate any type of nonlinear
isotherms as long as they do not cause mathematical singularities. The following two
common types of isotherms are used in this work.
(1) Langmuir Isotherm
where ai =C"bi. Note that bjCoj can be treated as a dimensionless group for each
component. With this the entire model system can then be treated with dimensionless
parameters alone. This helps reduce the total number of parameters involved in discus-
sions.
(2) Stoichiometric Isotherm with Constant Separation Factors
where cq, = l/aji = %akj, and gi = 1. Ci is the concentration of ion component i in the
stagnant fluid inside particles. C is the saturation capacity and is considered equal for
all components. ci is the concentration of ion component i in the solid of the particles.
This type of isotherm is widely used in ion exchange and all the concentrations are
based on the unit volume of the column rather than of the respective phases as in the
case of Langmuir isotherms (Helfferich and Klein, 1970).
The stoichiometric isotherm can be converted into an isotherm shown below, which
is the same in algebraic expression as the Langmuir isotherm except that the "I+" in the
denominator in the Langmuir isotherm expression is dropped.
j=l j= 1
The following relationships are needed for the conversion.
bi = q,~, and ai = biCm = a, NS (1 - ~ b X 1 - q (i = 1,2, ..., Ns)
where ion component Ns is assigned as the basis of the separation factors. Note that the
units of ai and bi in the Langrnuir isotherm and the converted stoichiometric isotherm
are not the same. In the stoichiometric isotherm, the concentrations of components can-
not all be zero at the same time, which means that the column is never "empty."
5.3 Efficiency and Robustness of the Numerical Procedure
The solution to the rate model provides the effluent history and the moving concen-
tration profiles inside column for each component. The concentration profile of each
component inside the stagnant fluid phase and the solid phase of the particle can also be
obtained, but they are rarely used for discussions.
Generally speaking, one interior collocation point (N = 1) is sufficient for some
cases, while more often N = 2 is needed, especially when Dpi values are small, which in
turn give large Bii and small qi values. N = 3 is rarely needed. The number of elements
Ne = 5-10 is usually sufficient for systems with non-stiff or slightly stiff concentration
profiles. For very stiff cases, Ne = 20-30 is often enough.
Insufficient N tends to give diffused concentration profiles as shown in Figures 5.3
to 5.5. Using N = 1 instead of N = 2 in Figures 5.3 to 5.5 (dashed lines) save about 60%
CPU time on a SUN 41280 computer, but the concentration profiles diff to some extent
from the converged ones (solid lines). In Figure 5.5, the dotted lines are obtained by
using three quadratic finite elements (Ne = 3) and one interior collocation point (N = 1)
with a CPU time of only 13.2 seconds. Though the dotted lines show a certain degree
of oscillation, they still provide the general shapes of the converged concenmtion
Ne=lO, N=2; 12.1 min
. - - - - Ne=lO, N=l; 4.0 min
Dimensionless Time
Figure 5.3 Effect of the number of interior collocation points in the simulation of frontal adsorption.
Ne=7, N=2; 14.8 min
- - - - - - - Ne=7, N=l; 5.4 min
10 20 30
Dimensionless Time
Figure 5.4 Effect of the number of interior collocation points in the simulation of elution.
Ne=12, N=2; 6.9 min
Ne=12, N=l; 2.7 min ........... Ne=3, N=l; 13.2 sec
..... ..... 1 2 (Displacer)
... . . ...... ..... . . . . . . a _ . . - - ...
2 4 6
Dimensionless Time
Figure 5.5 Convergence of the concentration profiles of a stepwise displacement system.
profiles, which take 6.9 minutes of CPU time. This means that one may use small Ne
and N values to get the rough concentration profiles very quickly and then decide what
to do next.
The efficiency and robustness of the numerical procedure are further demonstrated
by many simulate effluent histories for the discussions in the following chapters, includ-
ing cases involving very stiff concentration profiles.
The FORTRAN code based on the numerical procedure discussed above is capable
of simulating many kinds of multicomponent chromatographic processes, including
frontal analysis, displacement development, simple nongradient elution, nonlinear gra-
dient elution, and some multistage operations. Each mode of simulation is designated
with a process index in the code, which is included in the date input. Simulated exarn-
ples of these processes will appear in discussions in the coming chapters.
The input data for the FORTRAN code contains the number of components, ele-
ments and interior collocation points, process index, time control data, dimensionless
parameters, isotherm type and parameters. Note that the code is based on the dimen-
sionless PDE systems and Coi can be combined with bi to form a dimensionless group
biCo;. The initial conditions are reflected in the process index, or entered in the data
file.
5.4 Extension of the Rate Model
The assumption that a local equilibrium exists for each component between the pore
surface and the stagnant fluid phase in the macropores (Section 5.1.1) may not be
satisfied if the adsorption and desorption rates are not high, or the mass transfer rates
are relatively much faster. In such cases, isotherm expressions cannot be inserted into
Eq. (5-2) to replace CE. Instead, a kinetic expression is often used. The so-called
second order kinetics has been widely used to account for reaction kinetics in the study
of affinity chromatography (Chase, 1984a,b; Arnold et al., 1985a, 1986a, 1986b; Arve
and Liapis, 1987b). A general rate model with second order kinetics has been applied to
affinity chromatography by Arve and Liapis (1987b).
5.4.1 Addition of Second Order Kinetics
The second order kinetics assumes the following reversible binding and disassocia-
tion reaction.
where Pi is component i (macromolecule) and L represents the immobilized ligand. In
this elementary reaction, the binding kinetics is of second order and the disassociation
first order, as shown by the rate expression below.
where k~ and kd, are the adsorption and desorption rate constants for component i,
respectively. The rate constant kai has a unit of concentration over time while the rate
constant kdi has a unit of inverse time.
If the reaction rates are relatively large compared to mass transfer rates, then instant
adsorptionfdesorption equilibrium can be assumed such that both sides of Eq. (5-28)
can be set to zero, which consequently gives the Langmuir isotherms with the equili-
brium constant bi = ki/kdi for each component.
Introducing dimensionless groups Daa = L(k,COi)/v and = Lkdi/v which are
defined as the Damkblher numbers (Froment and Bischoff, 1979) for adsorption and
desorption, respectively, Eqs. (5-28) can be nondimensionalized as follows.
If the saturation capacities are the same for all the components, at equilibrium, Eq. (5-
d 29) gives biCoi = ~ a f / D a t and ai = CYbi = cmDaf/Dai for the resultant multicom-
ponent Langmuir isotherm. The Damkblher numbers reflect the characteristic reaction
times to that of the stoichiomenic time. The bi = kG/kdi values for affinity chromatogra-
phy are often very large (Chase, 1984b), but it is erroneous to jump to the conclusion
depending on this alone that the desorption rate must be much smaller that that of
adsorption, since the two processes have different reaction orders, and the concentration
Cpi is often very small at the adsorption side, Eq. (5-28). It is obvious that the dimen-
sionless Damklllher numbers provide a better comparison in this regard.
5.4.2 Solution Strategy
Adding the second order kinetics to the general rate model does not complicate the
numerical procedure for its solution since the discretization process is untouched. One
only has to add Eq. (5-29) in the final ODE system.
The following equation should be used to replace Eq. (5-21)
The final ODE system consists of Eqs. (5-17), (5-29) and (5-30). With the trial values
of Chi, Cpi and C$ in the function subroutine in the FORTRAN code, their derivatives
can be easily evaluated from the three ODE expressions.
If Ne elements and N interior collocation points are used for the discretization of the
Eqs. (5-1) and (5-2), there will be Ns(2Ne + 1)(2N + 1) ODE's in the final ODE's sys-
tem, which are Ns(2Ne + l)N more than in the equilibrium case. These extra ODE's
come from Eq. (5-29) at each element node and each interior collocation point for each
components.
The relationship among the kinetic effects, reaction equilibrium and mass transfer
rates will be discussed in Chapter 10 which deals with various aspects of affinity
chromatography.
5.4.3 Addition of Size Exclusion Effects
In some chromatographic systems, large solute molecules have considerable size
exclusion effects, which means that such large molecules either cannot access part of
the small macropores in the particles or the entire particle at all. This is especially true
in affinity chromatography in which large macromolecules are often present, and some-
times even larger complexes can be formed between the macromolecules with the solu-
ble ligands (see Chapter 10).
In recent years, there have been three ACS Symposium Series on size exclusion
chromatography (Provder, 1980, 1984, 1987). Several mathematical models have been
proposed for size exclusion chromatography (Yau et al., 1979; Kim and Johnson, 1984;
Koo and Wankat, 1988) among which the model proposed by Kim and Johnson is par-
ticularly helpful for this work. Their model is similar to the general rate model
described in Section 5.1.2 of this work, except that their model considers size exclusion
single component systems involving no adsorption. They introduced an "accessible
pore volume fraction" to account for the size exclusion effect.
In this work, the symbol E ; ~ is used to denote the accessible porosity (i.e., accessible
macropore volume fraction) for component i. It implies that for small molecules with
no size exclusion effect, E:~ = t+, and for large molecules that are completely excluded
fiom the particles = 0. For any medium-sized molecules 0 < e;i < $. It is con-
venient to define a si-~e eexclusivn fdctor 0 I Ffx ?" 1 such that e; = FfXep. F$ is a func-
tion of the distribution coefficient of component i. It is also a function of the particle
size distribution if the particle sizes can not be assumed to be equal (Kim and Johnson,
1984). To include the size exclusion effect, Eq. (5-2) should be modified as follows.
acs, where the first term (1-E$- should be dropped or set to zero, if a component does
at
not bind with the stationary phase. It should be pointed out again that in the equation
above Cii values are based on the unit volume of the solids of the particles excluding
the pores measured by the particle porosiry t+. For a component which is completely
excluded from the particles (i-e. E;; = O) , adsorbing only on the outer surface of the par-
ticles, Eq. (5-31) degenerates into the following interfacial mass balance relationship.
This equation can be combined with the bulk phase governing equation [Eq. (5-I)] to
give the following equation which is similar to a lumped particle model.
where Chi either follows the multicomponent isotherms or the expression for reversible
binding, Eq. (5-28). If component i does not bind with the stationary phase, C$ = 0 and
the fourth term in Eq. (5-33) is dropped for that component. A reminder again that the
solid phase concentration of component i, Cii, is based on the unit volume of the solid
part of the particle excluding pores, i.e., the unit volume of the solid skeleton. The
dimensional form of Eq. (5-33) is listed below.
5.4.4 Solution Strategy
If no component is totally excluded, the addition of the size exclusion effect in the
rate models is very simple. One only has to use E : ~ D ~ ~ to replace %Dpi in the expres-
sion of Rii and qi, and the €2 in in Eq. (5- 10).
Mathematically, a singularity occurs in the model equation system when a com-
ponent (say, component i) is totally excluded from the particles (i.e., e& = 0) if one
does not use Eq. (5-34) to replace Eqs. (5-9) and (5-10). It turns out that for numerical
calculation, there is no need to worry about this singularity, if E;~ is given a very small
value below that of the tolerance of the ODE solver, which is set to throughout
this work. It is found that this treatment gives the results which haw the samc valucs
for the first five significant digits as those obtained by using Eq. (5-34).
One should be aware that the size exclusion of a component affects its saturation
capacity in the isotherm. It also affects the effective diffusivity of the component since
the tortuosity is related to accessible porosity. It is clear that using size exclusion in a
multicomponent model often leads to the use of uneven saturation capacities for a com-
ponent with significant size exclusion and a component without size exclusion. This
may cause problems when the multicomponent Langmuir isotherm is used in terms of
thermodynamic inconsistency.
5.5 Multicomponent Adsorption Systems with Uneven Saturation Capacities
Many multicomponent adsorption sys tems have different saturation capacities for
different components. In such cases, the multicomponent Langmuir isotherm is thermo-
dynamically inconsistent (Ruthven, 1984). Although it can be considered only as an
experimental expression used for correlation, it may not be used for extrapolation over
a wider concentration range (Ruthven, 1984). In this work, the saturation capacities are
based on the unit of molecular counts, such as mole, not the weight of solutes per unit
volume of particle skeleton. The differences in saturation capacities can be caused by
physical or chemical reasons.
5.5.1 Systems with Physically Induced Uneven Saturation Capacities
In size exclusion chromatography, adsorption is considered as a side effect which
should be avoided. With the rapid growth in the separation of large molecules such as
proteins by using various chromatographic methods such as affinity chromatography
and ion-exchange, the involvement of the size exclusion effect becomes often unavoid-
able. This is a very important issue which deserves special attention. For a multicom-
ponent system with size exclusions, the adsorption saturation capacities cannot be con-
sidered equal for components with widely different degrees of size exclusion. The com-
ponent with a larger degree of size exclusion tends to have a smaller saturation capa-
city, since some binding sites on the surfaces of macropores are blocked due to size
exclusion.
5.5.2 Kinetic and Tsothenn Models
In this section a novel mathematical treatment is presented for systems with uneven
saturation capacities for components with different degrees of size exclusion, which fol-
low the second order kinetics for binding reactions. It is assumed that one molecule can
occupy only one binding site and its binding or size exclusion does not block the acces-
sibility of other vacant binding sites. That one molecule can only take one binding sitc
is a reasonable assumption for affinity chromatography involving low density immobil-
ized ligands.
Based on these basic assumptions Eq. (5-28) can be modified to give the following
kinetic expression.
Xii NS -- - ki Cpi (Cq - 6ij C;j) - kdi Cii (5-35)
at j= 1
where constants 0 < liij 5 1 are named 'discount factors' in this work, which are used to
discount the values of C:j which belong to the components that have a lower degree of
size exclusion, i.e.,
Recalling the earlier definition Fy = E;&, it is clear that for a component with a
higher degree of size exclusion its Fy value is smaller, and so is its saturation capacity
Cr. One may reasonably assume that Gij = Ci/CT for those 6, values that are
apparently not equal to unity. II hose hij values are obtained from experimental corre-
lation, the model then becomes a semi-theoretical one. If adsorption equilibrium is
assumed, Eq. (5-35) becomes
Rearrangement gives
The equation above can be rewritten in the following matrix form.
[A1 -rarc;1- rc;1= 0
which gives the following extended multicomponent Langmuir isotherm.
where
1 i=j ~ ; = b ; q ; C ? , Bij=biCpiGij, and I . .=
For a binary system in which component 1 has a higher degree of size exclusion
than component 2, one obtains = 622 = 621 = 1 and 8i2 < 1. The extended binary
Langmuir isotherm becomes
It is obvious that the above two isotherm expressions reduce to the common Langmuir
isotherm expressions if Cy = CF and 612 = 1. The extended binary Langmuir isotherm
has only one extra constant 6i2 apart from Cy = CT than the common Langmuir isoth-
erm, and 612 may often be reasonably set to CT/CT.
The extended multicomponent isotherm can also be applied to some adsorption sys-
tems with uneven saturation capacities which are not induced by size exclusion.
5.5.3 Systems with Chemically Induced Uneven Saturation Capacities
In some multicomponent systems, uneven saturation capacities do not arise from
different degrees of size exclusion, but they are induced by an adsorption mechanism at
the molecular level. For example, for a binary system, suppose the binding sites (or
ligands) are a racemic mixture and they make no difference to component 2, but only
part of them are active and usable for component 1; thus component 1 has a lower
saturation capacity than component 2. The mathematical treatment for such a system is
the same as that for systems with uneven saturation capacities which are induced by
size exclusion which has been discussed in the previous section. The discount factor
6i2 may also be reasonably set to the ratio of CT/CF. For systems with more than two
components, the determination of 6ij values may not be that simple.
5.5.4 Isotherm Cross-Over
With uneven saturation capacities, isotherm "cross-over" may occur. This happens
when a component which has a smaller saturation capacity has a larger adsorption
equilibrium concentration. In this work the isotherm concentration cross-over point CE
is defined as the concentration in the stagnant fluid inside macropores for a pair of com-
ponents, at which their concentrations in the solid phase (CE) are the same.
The concentration cross-over point for the binary isotherms Eqs. (5-41) and (5-42)
can be derived by subtracting the two isotherm expressions and setting = q2 = C;.
(blCr - b2Cr)CE + blb2[2Cy - (1 + 612)Cr] (CJ2 c;, - q2 = (5-43) 1 + bl Ci + b2Ci + (1 - 812)blCzb2Cz
Letting the left hand side of the equation above be zero, one obtains
(bl C r - b2Cr)C; + bl b2[2CT - (1 + 612)CT] (C;)? = 0 (5-44)
which gives a nontrivial solution
the denominator of the equation above is positive since CT > CT. Thus the binary
isotherm has a cross-over point if and only if the cross-over concentration has a positive
value, which requires
Figure 5.6 shows the concentration cross-over point in a 3-D space.
Isotherm concentration cross-over often signals a selectivity change. The selec-
tivity cross-over point in this work is defined as the critical concentration C i = CP1 and
Cp2 with which the relative selectivity of the two components,
a~;~/ac,, Relative Selectivity =
a ~ ; ~ / a c ~ ~
is unity. The selectivity cross-over point can be found by following an approach similar
to that for the concentration cross-over point. Setting the left hand side of Eq. (5-47) to
unity yields
from which the following critical selectivity cross-over concentration can be easily
obtained with = Cp2 = Cg and 612 = CTICT.
Figure 5.6 Concentration cross-over point.
In the equation above for C i > 0, the following equation must be satisfied.
which readily reduces to Eq. (5-46). Thus, both concentration cross-over and selectivity
cross-over require the fulfillment of Eq. (5-46).
It has been known that selectivity depends on the concentration range, and selec-
tivity reversal may occur in the operational concentration range (Antia and Horvath,
1989). Selectivity reversal may cause the reversal of the sequence of elution peaks.
since the migration speed of a component is primarily determined by its aC:i/aC+i
value (Helfferich and Klein, 1970). Peak reversal in elution usually happens in volume
overload cases if the feed concentration is very high.
Figure 5.7 shows a binary elution case in which component 1 has a smaller satura-
tion capacity and a higher adsorption equilibrium constant than component 2. Parameter
values used for simulation are listed in Table 5.1 at the end of this Chapter. The dashed
lines show that component 2 has a smaller retention time than component 1 when the
feed concentration of component 1 is low, while the solid lines show that component 2
has a higher retention time when the concentration of component 1 is increased ten
fold. It is interesting to note that in Figure 5.7 (solid lines) the tail end of the component
1 peak is behind that of the component 2 peak. Apparently at low concentrations
\ 1 (R;;ntion Time = 3.18)
1
, P o Size ~xclusionl
4 6 8
DIMENSIONLESS TIME
Figure 5.7 Peak reversal due to increased component 1 concentration.
component 1 has a higher affinity than component 2. All figures in this chapter, which
use the extended binary Langmuir isotherm, have 612 =Cy/CF.
Figure 5.8 has the same conditions as Figure 5.7, except that in the solid line case
the concentrations of components 1 and 2 are both 2.0 in Figure 5.8. The peak reversal
phenomenon is also present in Figure 5.8. If the uneven saturation capacities are not
chemically induced, but by size exclusion, peak reversal still can be present. Figure 5.9
clearly shows such a case in which component 1 has a size exclusion factor of
Fy = 0.5.
The selectivity reversal is also very interesting in frontal adsorption. The solid lines
in Figure 5.10 show that the breakthrough curves cross over each other when the feed
concentrations are high.
A detailed treatment of peak reversal due to isotherm selectivity cross-over is con-
siderably more difficult, involving cornqdiuated arguments and is recommended as a
future study. Peak reversal is not necessarily the consequence of selectivity reversal,
although selectivity reversal facilitates the peak reversal phenomenon. In this section
the extended binary Langmuir isotherm successfully demonstrated the peak reversal
phenomenon without extreme differences between the feed concentrations, and adsorp-
tion equilibrium values of the two components The extended isotherm also serves as a
valuable isotherm model for experimental correlation of isotherm data showing uneven
saturation capacities. A similar approach can be readily applied to common
stoichiometric ion-exchange systems (Gu et al., 1990g).
DIMENSIONLESS CONCENTRATION
1 (Retention Time = 2.15)
2 4 6 8 10
DIMENSIONLESS TIME
Figure 5.9 Peak reversal with size exclusion.
No Size Exclusion
Figure 5.10 Cross-over of breakthrough curves.
5.6 Other Extensions of the Rate Model
The general rate model can also be modified to account for the interaction between
adsorbates and soluble ligands as in affinity chromatography. This extension is consid-
crably more conlplicated. Details will be shown in Chapter 10.
A set of general rate models has also been formulated and solved for radial flow
chromatography to which Chapter 11 has been devoted. Radial flow chromatography
has a notable characteristics of having variable radial dispersion and external film mass
transfer coefficients. The numerical procedure presented in this chapter is a superb
scheme for such a case.
5.7 The Question of Column Boundary Conditions
In this work, the Danckwerts boundary conditions (Eqs. (55,611 are used for the
two column ends in axial flow chromatography. The validity of the Danckwerts boun-
dary conditions ( Danckwerts, 1953) in transient axial dispersion models has been dis-
cussed for many years by many researchers. Comprehensive reviews were given by
Pmlekar (19831, and Parulekar and Ramkrishna (1984). For axial dispersions in some
linear systems, Parulekar and Ramkrishna (1984) provided some physically more rea-
sonable alternatives to the Danckwerts boundary condition for transient systems based
on analytical analysis. Unfortunately, for nonlinear systems, such an approach is gen-
erally not possible. Recently. Lee and coworkers (Lee et al., 1988b; Lee. 1989) dis-
cussed the use of alternative boundary conditions for both column inlet and exit in some
rate models.
In nonlinear chromatography, the Danckwerts boundary conditions are generally
accepted. However, for the column inlet some researchers (Brian et al., 1987; Lin et al.,
1989) implied that it is better to use finite values for the concentration flux instead of
zero as in Danckwerts boundary conditions. This is equivalent to assuming that the
column is semi-infinitely long, and the effluent history is detected at z=1. This alterna-
tive boundary condition is hardly appropriate, since it tends to destroy the mass balance
of the model system.
In the effluent history of a fi-ontal analysis, each breakthrough curve can be
integrated to find out whether it matches the dimensionless column hold-up capacity for
the corresponding component which is expressed by the following expression.
CAi consists of three parts, the amount of component i adsorbed onto the solid, that in
the stagnant fluid inside particles, and that in the bulk fluid. It is actually equal to the
first moment of the breakthrough curve. It is equivalent to the expression of the first
moment for a single component sys tem with Danckwerts boundary conditions and
Langmuir isothenn derived by Lee et al., (1990) from differential mass balance cqua-
tions. The theoretical hold-up capacity should also be equal to the area integrated from
the following expression.
where 2, is a time value at which the breakthrough curve has already leveled off. Since
the hold-up capacity reflects the steady state of the column, mass transfer and disper-
sion effects should not affect its value. The above two equations are very helpful in
checking the mass balance of an effluent history in frontal analysis and stepwise dis-
placement.
The use of the Danckwerts boundary condition at the column exit needs no effort,
since in the finite element method, the zero flux as a natural boundary condition is a
default. The implementation of the aforementioned alternative boundary condition at
column exit can also be easily accommodated in the existing code with Danckwerts
boundary conditions.
In the function subroutine of the code which evaluates concentration derivatives, the
trial concentration values are given; thus acbi/az at z=1 can be obtained by using the
concentration values at the three element node for derivative calculations. In the actual
code, adding six lines and a simple subroutine for the derivative calculations will
suffice for the modification needed. Note that the natural boundary condition at z=1 is
and it should be added to [AFBi] at z=1.
Figure 5.11 shows single component breakthrough curves with the Danckwerts
boundary condition and the alternative boundary condition, respectively. Parameter
values uscd for simulation are listed in Table 5.1. It is obvious that the use of the alter-
native boundary condition results in a later breakthrough, and thus a larger capacity
area which is not equal to the theoretical value, unlike in the case of Danckwerts
1
- E Danckwerts Boundary Condition at z=1 0 . II .--- u 0.8 - Alternative Boundary Condition at z=1 ert
Peclet Number = 50
Theoretical dimensionless column hold-up capacity = 4.6
Area = 4.600 - - - - - Area = 4.692
2 3 4 5
Dimensionless Time
Figure 5.11 Single-component breakthrough curves with different boundary condi- tions at the column exit (Peclet number = 50).
boundary condition. This violation of a basic mass balance is clearly undesirable. In
fact, any attempt to change the Danckwerts boundary conditions at one column end
while leaving the other end intact may lead to such a violation since the complete
Danckwerts boundary conditions are mass balanced.
Figure 5.12 shows the concentrations profiles inside the column at different times. It
shows that when the Danckwerts boundary condition is used at z=1, the concentration
curves bend up trying to approximate the zero flux requirement. Note that this boun-
dary condition may not be completely satisfied with limited element nodes, but the chi
values at z=1 are easily converged. Figure 5.12 also shows that the concentration
profiles are quite smooth at the column exit when the alternative boundary condition is
used. This is due to the fact that the alternative boundary condition assumes that the
column has no discontinuity at the column exit. In Figure 5.12, when z is not very
small, the concentration patterns are very similar. This is the so-called "constant pat-
tern" phenomenon (Ruthven, 1984).
The mass balance violation may not occur or be noticeable in elution, as is shown
by Figure 5.13, in which the areas for each component for cases with Danckwerts boun-
dary condition and the alternative boundary conditions at z=1, are found to be 0.2000
which is the value of sample size zhp.
It is clearly shown in Eq. (5-53) that the (PBi) 1 ,I can be set to zero if PeL; values
art: large. Figure 5.14 has the same conditions as Figure 5.11, except that the Peclet
number in Figure 5.14 is 200 which is much larger than that in Figure 5.1 1. Figure 5.14
shows that the differences are quite small from using the Danckwerts boundary
Danckwerts Boundary Condition at z=1 ................. Alternative Boundary Condition at z=1
\
0 0.2 0.4 0.6 0.8 I
Dimensionless z
Figure 5.12 Breakthrough concentration profiles inside the column.
Danckwerts Boundary Condition at z=1 - - - - - - . Alternative Boundary Condition at z=l
0 5 10 15 20
Dimensionless Time
Figure 5.13 Two-component elution with different boundary conditions at the column exit.
- 0 1 2 3 4 5 6 7
Dimensionless Time
Figure 5.14 Single component breakthrough curves with different boundary condi- tions at the column exit (Peclet number = 300).
condition or the alternative boundary condition at the column exit when the Peclet
number is not small. This is in agreement with Brian et al. (1987).
As a matter of fact, in a common axial flow chromatograph, the Peclet numbers for
axial dispersion often run into the hundreds and even thousands or higher; the differ-
ence from using alternative boundary conditions at the column exit is very much negli-
gible. Thus seeking an alternative boundary condition for the column exit seems mean-
ingless.
* Table 5.1 Parameter Values Used for Simulation in Chapter 5
* In all cases, q, = 0.4. For Figures 5.3 to 5.5, ~p = 0.5, and for all other figures, ep = 0.4. For elution cases, sample sizes are: Figure 5.4, zhp = 0.1; Figure 5.7 to 5.9, zi, = 1.0; Figure 5.13, zhp = 0.2. The error tolerance of the ODE solver is tul=l0-'. Double precision is used in the Fortran code. CPU times on SUN 4,280 computer are: Figure 5.13,3,18 min and Figure 5.14,0.37 min.
CHAPTER 6 - OPTIMIZATION OF STEPWISE DISPLACEMENT
6.1 Introduction
Stepwise displacement chromatography has received considcrable attention recently
in microbiological processes for in situ removal of toxic fermentation product(s) (Yang,
1988; Yang et al., 1988, 1989). Lee et al. (1988a) used a PVP resin for the in situ
removal of lactic acid during fermentation. The P-CD resin described in this work
potentially may be used for in situ separation of tryptophan during fermentation. This
kind of in situ separation reduces product inhibition, and thus enhances productivity. In
such adsorption-combined fermentation processes, chromatographic columns are cou-
pled with the fermenter to remove the product(s) simultaneously via preferential
adsorption and the adsorbate(s) is(are) then recovered through a displacement opera-
tion. This kind of stepwise displacement is also widely used to recover biomolecules
from a dilute solution after they are adsorbed onto a column. In both cases, frontal
adsorption proceeds the displacement process which often concentrates the adsorbate(s)
by using a suitable displacer. This kind of stepwise displacement operation is somewhat
different from the classical displacement chromatography or displacement development
first classified by Tiselius (Tiselius, 1943) and extensively reviewed by Horvath and
coworkers (Horvath et a/., 1981; Horvath, 1985; Frenz and Horvath, 1988; Antia and
Horvath, 1989).
Classical displacement chromatography was described by many researchers as a
process in which a column packed with solid adsorbent is equilibrated with the mobile
phase that has no or weak affinity to the adsorbent. A sample of mixtures is then intro-
duced to the column. The sample usually takes up a fraction of the column volume in
the inlet section. Subsequently, a development agent (called displacer) is pumped into
the column. The displacer must have a higher affinity to the stationary phase than any of
the components in the sample, i.e., its adsorption isotherm overlies those of the feed
components (Horvath, 1985; Ruthvcn, 1984). Provided that the column is sufficiently
long, and isotherm curves are all favorably shaped, sample components will eventually
migrate inside the column with the same speeds to form individual product zones. The
series of such zones is usually called a displacement train ( Horvath et al., 1981; Antia
and Horvath, 1989; Phillips et al., 1988; Golshan-Shirazi er ul., 1989; Katti .and
Guiochon, 1988). Figure 6.1 shows a displacement chromatogram with two sample
solutes and one displacer. Parameter values used in simulation are listed in Table 6.1 at
the end of this Chapter. Compared with elution chromatography, the displacement
dcvclopmcnt has two distinct advantages: (1) the displacement effect reduces tailing
(Figure 6.1); (2) sample loading can be higher (Frenz and Horvath, 1988). These
features make the displacement development operation a very attractive alternative to
elution in preparative scale chromatography (Horvath, 1985).
The main difference between the displacement development and the stepwise dis-
placement studied in this chapter is often the operation purpose itself. The former
desires the products to be separated into a displacement train containing individual pro-
duct zones in the effluent stream, while the latter requires the efficient displacement of
m . . . , . . . , . . .
3 (Displacer)
1 I
4 6
Dimensionless Time
Fie- 6.1 Displacement chromatogram.
the adsorbates. In other words, the desorption chromatography does not require a well-
defined displacement train in the effluent; rather it requires the displacement of
adsorbed component(s) with a minimum amount of displacer in a minimum length of
time in order to obtain concentrated product(s). The product(s) in the effluent after dis-
placement may be furthe] purified if necessary after the stepwise displacement. A typi-
cal use of the stepwise displacement, as we have already mentioned, is the in situ
separation during fermentation (Yang, 1988; Yang et al., 1988, 1989; Lee et al.,
198 8,). Another important difference is that the displacement development takes a
sample which usually occupies only a fraction of the column inlet section while the
stepwise displacement has no such limitation. The strong affinity of the displacer which
is required in the displacement development should not be mistaken as a requirement
for the stepwise displacement.
In this chapter, the effects of adsorption characteristics of a displacer on the dis-
placement efficiency of stepwise displacement will be discussed. It will be shown that
the selection of displacer in stepwise displacement will be somewhat different from that
of the displacement development, As a matter of fact, a displacer with a stronger
affinity is often not an ideal choice for the stepwise displacement. The analysis is based
on computer simulation using the general multicomponent rate model discussed in
Chapter 6 instead of the equilibrium theory which has been widely used in the theoreti-
cal studies of the displacement development ( Frenz and Horvath, 1988; Antia and Hor-
vath, 1989; Ruthven, 1984; Helfferich and Klein, 1970; Hclfferich and James, 1970;
Rhee and Amundson, 1982; Yu et a/., 1987). The use of the general model gives a
more realistic picture of the displacement process since it accounts for diffusional and
mass transfer effects which are often quite important in preparative and large scale
chromatography. The displacer is treated as one of the components in the model equa-
tions and is included in the multicomponent isotherms.
6.2 Results and Discussion (Gu et al., 1990c)
The multicomponent Langmuir isotherms (Eq. (5-25)) with uniform adsorption
capacities will be used in this study. Figure 2 shows the Langmuir isotherm curves for
single component cases. Such isotherm curves can be used to obtain the isotachic con-
ditions in the study of ideal displacement development (Tiselius, 1943; Horvath, 1985).
Figure 6.2 shows the Langmuir isotherm curves for single component cases. Such
isotherm curves can be used to obtain the isotachic conditions in the study of ideal dis-
placement development (Tiselius, 1943; Horvclth et a1 ., 198 1; Horvath, 1985; Frenz and
Horvath, 1988; Antia and Horvath, 1989). In this study Figure 6.2 helps illustrate the
meaning for each physical parameter in the Langmuir isotherm. For simplicity, only
the two component displacement chromatographic processes will be discussed, in
which component 1 is the component to be displclced and cornponcnt 2 the displacer.
6.2.1 Effect of Feed Concentration of Displacer (Cf2)
Figure 6.3 shows that the higher the displacer concentration in the mobile phase, the
higher the roll-up peak on the concentration profile of component 1 (see Table 6.1 for
parameter value). This is due to the fact that a higher displacer concentration in the feed
gives a faster migration rate for the concentration front of the displacer inside the
column (Eq. (6-111, a larger b 2 q 2 value (Eq. (6-2)) and, hence, a better displacement
Figure 6.2 Single Component Langmuir Isotherms.
2 4 6
Dimensionless Time
Figure 6.3 Effect of displacer concentration on displacement.
efficiency. Figure 6.4 shows a case in which the displacer does not give much help in
the displacement of component 1 from the column because the concentration of the
displacer is too low. This kind of situation was mentioned by some researchers (Rhee
and Amundson, 1982; Morbidelli, 1985). In Figure 6.5, the affinity of the chsplacer is
lower than that of the adsorbate. It shows that if the concentration of the displacer is
sufficiently high a desirable displacement of the adsorbate can also be achieved. The
study of the effect of ethanol concentration on the efficiency of the displacement of
phenylalanine from a column packed with P-CD-containing resin in Chapter 4 qualita-
tively proved this argument.
6.2.2 Effect of Adsorption Equilibrium Constant of Displacer (b2)
The effect of the b2 value on displacement performance is shown in Figure 6.6. It
can be seen that an increase in b2 delays the appearance of the roll-up peak, gives a
sharper displacer front, and reduces the tailing of the displaced component. The max-
imum roll-up peak occurs somewhere in the middle range of the b2 value. If the pri-
mary goal of displacement is to obtain a large fraction of pure component 1, a larger b2
is obviously favorable. However, if the mixing of displacer in the product is not a set-
back, such as in the case when the displacer is a volatile organic solvent and the product
is readily recovered by evaporation after the displacement, a larger b2 is not always
favorable. As a matter of fact, if the displacement is terminated when the major portion
of the product has been recovered, then a small q may be a better choice because the
roll-up peak appears earlier.
With Displacer (2) ....... . . Without Displacer
1 2
* I * I - _ - - I
2 4 6
Dimensionless Time
Figure 6.4 Same conditions as Figure 6.3, except that the concentration of the displacer is lower.
2 4 6
Dimensionless Time
Figure 6.5 Effect of displacer concentration on displacement for a case in which b2 < b l .
2 4 6
Dimensionless Time
Figure 6.6 Effect of on displacement performance.
Compared with the displacement development, the conclusion for stepwise dis-
placement is somewhat different. The displacement development requires a displacer
which has an affinity higher than any other component in the sample. However, this is
hardly true for the stepwise displacement as we have already discussed in the cases of
Figures 6.5 and 6.6.
6.2.3 Industrial Practice
In industrial application, column operations often include both adsorption and
desorption. The adsorption stage uses the frontal operation until the concentration of the
key adsorbed species in the effluent reaches a certain fraction, say lo%, of its influent
value. Then, the process switches to the desorption operation in the second stage.
Switching before the complete frontal adsorption operation is due to the concern of loss
of valuable products or release of environmentally hazardous adsorbates in the column
exit or overall process efficiency during the operation. Also, the displacement operation
can be carried out in either flow direction, forward or reverse. The reverse flow dis-
placement operation may reduce the fouling of the column due to clogging of irregular
or fractured adsorbent particles. In some cases, especially when the frontal adsorption
period is brief, reverse flow displacement may give better displacement efficiency.
Reverse flow regeneration in gas adsorption is a standard practice and forward flow is
used only in special situations which preclude the use of reverse flow (1 1). Reverse
flow was also reported by some researchers in the chromatographic separation of
biomolecules (Chase, 1985; Arve and Liapis, 1988).
Figures 6.7 and 6.8 show chromatograms for forward and reverse flow operations,
respectively. In both cases, the frontal adsorption operation is switched to the displace-
ment operation at z = 2.0. The reverse flow case, as shown in Figure 6.8, gives two
peaks for the displaced component. The first peak is due tu the sudden change in flow
direction when the displacer is introduced and the second peak is due to the displace-
ment effect from the displacer. Compared with Figure 6.7, Figure 6.8 shows that the
reverse flow displacement gives a better displacement efficiency as far as the rninimiza-
tion of displacer amount is concerned. Note that in the reverse flow displacement
mode, a fast moving displacer is usually desired because of the high adsorbate concen-
tration on the other end of the column at the beginning of displacement.
Simulations shown in Figures 6.7 and 6.8 reveal some interesting features in the
practical application of adsorption and desorption processes. However, optimization of
such a combined adsorption/desorption process requires more detailed information
about the entire process.
6.3 Summary
This chapter presents some interesting effects of isotherm characteristics of the
displacer on the optimization of stepwise displacement and some considerations in
industrial practice. It is concluded that a displacer with a high feed concentration, and a
suitable adsorption equilibrium constant is often a desirable choice when the purpose of
the displacement operation is to displace and to concentrate the adsorbed species and to
minimize the amount of displacer employed. Reversed flow direction for the stepwise
displacement is advantageous under certain operational conditions.
+Frontal +b- Displacement - 1
2 4 6
Dimensionless Time
Figure 6.7 Frontal operation switched to f o m d flow displacement operation.
*Frontal -1- Displacement - REVERSE FLOW
0 2 4 6 8
Dimensionless Time
Figure 6.8 Same conditions as Figure 6.7, except that flow direction was reversed for the displacement process.
* Table 6.1 Parameter Values Used for Simulation in Chapter 6
Species
Physical Parameters
Numerical Parameters
* In all cases, q., = 0.4, ep = 0.5. The error tolerance of the ODE solver is tol=10-~. Double precision is used in the Fortran code. CPU times on SUN 41280 comput- ers are (solid lines): Figure 6.1,99.6 min; Figure 6.3,5.0 min: Figure 6.7,2.1 min and Figure 6.8,2.3 min.
CHAPTER 7 - MASS TRANSFER EFFECTS IN
MULTICOMPONENT CHROMATOGRAPHY
For analytical and some preparative columns, mass transfer resistances are usually
negligible and equilibrium theory often suffices (Helfferich and Klein, 1970). But for
preparative columns with smaller plate numbers and large scale columns mass transfer
effects are often significant and usually cannot be neglected.
The study of mass transfer effects for single component systems has been carried
out by many researchers (Ruthven, 1984). Recently, Lee et al. (1989) studied the mass
transfer effects in nonlinear multicomponent elution in ion exchange chromatography.
They compared the difference between the general rate model and the equilibrium
theory under various mass transfer conditions.
7.1 Effects of Parameters PeLi, Bii and qi
The Peclet number (PeLi) reflects the ratio of the convection rate over the dispersion
rate while the Biot number (Bii) reflects the ratio of the external film mass transfer rate
over the intraparticle diffusion rate.
Figures 7.1 to 7.3 show that the increase of Pei values (while fixing other parame-
ters) sharpens the concentration profiles in the effluent history in cases of frontal
adsorption, elution and stepwise displacement. This well-known result has been
Figure 7.1 Effect of Peclet numbers on two-component frontal adsorption.
2
.5
1
.5
0
1 Pe,, Pe L2 400,400 - - - - - - 200, zoo
... ...... 80, 80
-
-
I . .L ." I
0 1 2 3 4 5 6
Dimensionless Time
0 5 10 15 20 25
Dimensionless Time
Figure 7.2 Effect of Peclet numbers on two-component elution.
Dimensionless Time
Figure 7.3 Effect of Peclet numbers on two-component stepwise displacement.
reported by numerous researchers and summarized by Ruthven (1984). Parameter
values used for simulation in this chapter are listed in Table 7.1 at the end of the
chapter. Figures 7.4 to 7.6 show that increasing q; or Bi; (while fixing other parame-
ters) gives sharper concentration profiles. They also show that increasing the &men-
sional parameter ki or Dpi also has the same effect. It is clear that mass transfer effects
tend to diffuse concentration profiles.
7.2 Effect of Flow Rate
The flow rate in an axial flow chromatographic column is directly proportional to
the interstitial velocity v. This velocity affects Dbi and ki values. Meanwhile the intra-
particle diffusivities, Dpi can be regarded as independent of v (Gu er al., 1990d).
Dispersion is caused by molecular diffusion and turbulent mixing or eddy diffusion
(Ruthven, 1984). A simple linear approximation for a single component system may be
represented by (Ruthven, 1984; Jonsson, 1987)
where yl and y2 are constants which normally have values of about 0.7 and 0.5, respec-
tively. The molecular diffusion (yl Dm) of a liquid is negligible compared to eddy diffu-
sion, even at low Reynolds numbers (Ruthven, 1984). Thus in Eq. (7-1) the second
term, eddy diffusivity, is the dominant term in liquid chromatography, especially when
the flow velocity is not low, thus Db oc V. This relationship has been acknowledged by
some researchers (Weber and Can, 1989; Lee, 1989). For simplicity in discussions the
multicomponent mixing effects on Dbi, Dpi and ki for multicomponent systems are
2 3 4
Dimensionless Time
Figure 7.4 Effect of Ti and Bii on two-component frontal adsorption.
0 5 10 15 20 25 30
Dimensionless Time
Figure 7.5 Effect of qi and Bii on two-component elution.
2 3 4
Dimensionless Time
Figure 7.6 Effect of qi and Bii on two-component stepwise displacement.
10 15 20
DimensionIess Time
Figure 7.7 Effect of interstitial velocity on two-component elution.
ignored. Thus Dbi oc V, and PeLi = vL/Dbi is independent of v for each component.
The relationship between ki and v can be simply expressed as ki = vlB (Lee et al.,
1990). It is in agreement with the two different experimental correlations reported by
Pfeffer and Happel (1964) and Wilson and Geankoplis (1966) (Ruthven, 1984) for
liquid systems at low Reynold's (Re = 2Rpvp/p) numbers which are the range for liquid
chromatography (Kaizuma el al., 1970; Homath and Lin, 1976). From the relationship
ki = v'", one obtains Bi; = k; = v'".
Figure 7.7 clearly shows a case in which the sharpness and resolution of the elution
peaks decrease when v is doubled (dashed lines). The values of qi and Bii for both
cases are listed in the legend of the figure. Note that, q; .c l/v and the comparison is
based on the dimensionless time.
The effect of increasing v is somewhat similar to that of decreasing Dpi, since both
result in the increase of Bii and the decrease of Tli. But the increasing v reduces the
sharpness and resolution of the peaks more severely because the increase of Bii values
are smaller than that in the case of decreasing Dpi.
7.3 Effect of Mass Transfer in a Case with Unfavorable Isotherm
For a common elution peak, the peak front is often sharper than its rear boundary if
the isotherm is in the nonlinear range and the isotherm is of favorable type, i.e., con-
cave downward or a2c;/ac; < 0, for which the Langmuir isotherm is a typical exam-
ple. This is due to the well-known self-sharpening effect of favorable isotherms (Gid-
dings, 1965; Antia and Horvath, 1989).
Some adsorption systems, namely cooperative adsorption systems, have unfavor-
able isotherms (Antia and Horvath). It was found that in elution when the isotherm is
of unfavorable type, the peak front tends to get diffused and its rear boundary sharpened
(Giddings, 1965; Antia and Horvath, 1989). This type of peak shape may be termed as
having anti-Langmuirian asymmetry (Golshan-Shirazi and Guiochon, 1988a). Such a
phenomenon has also been observed in experiments and is a well-known fact in non-
linear chromatography. This is generally true for systems with fast mass transfer rates.
For systems with slow mass transfer rates this may not be the case. Figure 7.8 shows
that a single component elution with an inert mobile phase gives a peak of anti-
Langmuirian asymmetry, and if ki or Dpi is decreased to some extent the peak sym-
metry will be reversed to that of common Langmuir type. In Figure 7.8, a parabolically
shaped unfavorable isotherm is used for the simulation.
This phenomenon of peak shape reversal due to mass transfer effects may be attri-
buted to the reason that the diffusive effect of slow mass transfer affects the peak front
more severely than its rear boundary. An experimental proof should be very helpful.
15 I Isotherm /
Dimensionless Time
Figure 7.8 Effect of mass transfer on an elution system with an unfavorable isoth- erm.
* Table 7.1 Parameter Values Used for Simulation in Chapter 7
-.- In all cases, q, = $ = 0.45. For elution cases sample sizes are z~~ = 0.6. The error tolerance of the ODE solver is tol=10-~. Double precision is used in the Fortran code.
CHAPTER 8 - DISPLACEMENT EFFECTS IN
MULTICOMPONENT CHROMATOGRAPHY
8.1 Introduction
Chromatography for analytical purposes usually involves small and dilute samples.
Thus, interference effects among different species are often negligible. With the rapid
growth of chromatography in bioseparation, preparative and large-scale chromatogra-
phy become more and more important. High feed concentrations and large sample
volumes are often used. In such cases, competition among different species can no
longer be ignored, and the study of interference effects becomes important in the under-
standing and optimization of multicomponent chromatography.
In the past, systematic studies of interference effects were based mostly on the
equilibrium theory (Helfferich and Klein, 1970; Rhee et al., 1970, 1982). Both assume
direct local equilibrium between the liquid phase and the stationary phase, and neglect
mass transfer effects which can be quite important in preparative and large-scale
chromatography.
In this chapter, the general multicomponent rate model described in Chapter 5 is
used to simulate various chromatographic processes for the study of multicomponent
interference. The model is able to describe some important phenomena such as roll-up
in all the three major modcs of chromatography, frontal, displacement and elution under
mass transfer conditions. The use of the general rate equation model involving various
mass transfer mechanisms gives a more accurate account, and thus, helps the visualiza-
tion of the dynamics of the chromatographic processes.
Tiselius (1940) was the first to use the phrase "displacement effect" to describe the
competition for binding sites in ml~lticomponent elution. It will be shown in this work
that the displacement effect is, in fact, the dominating factor in multicomponent
interference interactions which are directly attributed to the competition for binding
sites among different components, and this effect exists in all the three major opera-
tional modes. Many observed multicomponent interactions due to competitive adsorp-
tion can be satisfactorily explained by using this simple concept. Although a few sys-
tems exist with synergistic (cooperative) isotherms (Helfferich and Klein, 1970) where
the presence of other solutes enhances adsorption, the competitive isotherms are the
most common type in practical opcration (Wankat, 1986). In this chapter thc Langxnuir
isotherm is chosen. The conclusions in most cases can be readily extended to mul-
ticomponent systems with other types of competitive isotherms.
8.2 Results and Discussion (Gu et al., 1990b)
For comparison and simplicity, we ignore the component mixing effect on some
physical properties, such as diffusion and mass transfer coefficients. All the computer
simulations have been carried out on a SUN 41280 computer. Parameter values used for
simulations are listed in Table 8.1 at the end of the chapter, or mentioned during discus-
sion.
8.2.1 Displacement Operation
The displacement effect is most noticeable and also relatively well-known in dis-
placement chromatography. Figure 8.1 (solid lines) shows a simulated chromatogram
(effluent history) of a stepwise displacement process in which component 2 (displacer)
is introduced at z = 0 to a column pre-saturated with co~x~ponen~ 1 via a step change. A
roll-up peak appears in the concentration profile of component 1, which is a clear indi-
cation of the displacement effect. The concentration profile of component 1 is shar-
pened compared with the dashed line which represents the corresponding desorption
operation when only an inert mobile phase is used to "wash out" component 1 from the
column, In other words, the use of the displacer reduces the tailing and thus concen-
trates component 1. This is also evident in Figure 8.2 which shows a simulated
chromatogram of a binary displacement system, in which components 1 and 2 are intro-
duced to the column via frontal adsorption lasting z,p = 4.0 before component 3
(displacer) is pumped into the column. In this volume overload case, component 2 has
two peaks, in which the roll-up peak is due to the displacement effect from the displacer
(component 3). Such a concentrating effect was proven by experiments carried out by
Helfferich (1962). In Figure 8.2 there is also a roll-up peak for component 1 which is
the result of the displacement effect from component 2. The first smaller componcnt 2
peak should not be mistaken as a displacement band of a separate component.
Competition can be viewed as a mutual interaction. The displaced component,
while being displaced, in return also exerts some influence on the displacer. Figure 8.1
illustrated that the concentration hont of the displacer is diffused by component 1 in the
Binary Displacement System
Single Component Breakthrough Curve of Component 2
1 2
Displacement of Component 1 ' with Inert Mobile Phase
\
, ='* I
0 2 4 6 8
Dimensionless Time
Figure 8.1 Two-component stepwise displacement process.
Three-Component Displacement Train . - --- Binary Elution Without Dispiacer
3 (Displacer)
Dimensionless Time
Figure 8.2 Thrcc-component displacement system.
displacement process. The concentration profile of component 2 is actually its break-
through curve under the interference of component 1. As compared with the break-
through curve of pure component 2 (dotted line in Figure 8.1), the concentration front
of component 2 becomes diffused due to component 1.
The concentrating effect and roll-up phenomenon in displacement chromatography
with negligible mass transfer effects were predicted by the ideal theories including the
interference theory (Tiselius, 1943; Glueckauf, 1949; Helfferich and Klein, 1970; Rhee
and Amundson, 1982; Ruthven, 1984; Frenz and Horvath, 1985; Antia and Horvath,
1989). The general model presented in this work describes the roll-up phenomenon
under mass transfer conditions.
8.2.2 Frontal Adsorption
Figure 8.3 shows a simulated chromatogram of a binary frontal adsorption process
in which component 1 has a weaker affinity than component 2. The concentration
profile of component 1 reaches a maximum which is larger than its feed concentration
before leveling off. This roll-up phenomenon is the result of a displacement effect. The
concentration front of component 1, which has a weaker affinity, migrates faster than
the concentration front of component 2 inside the column. Component 1 takes advan-
tage of the relative absence of component 2 and initially occupies a disproportionate
share of binding sites. When the concentration front of component 2 catches up, it dis-
places some portions of component 1 such that the concentration of component 1 may
exceed its feed value causing the roll-up. The column finally reaches adsorption equili-
brium when each component occupies its share of binding sites according to the
Binary System .----- Single Component Systems I
Dimensionless Time
Figure 8.3 Binary frontal adsorption with a roll-up peak.
governing multicomponent isotherms. Experimental observations and simulations for
this kind of roll-up in frontal adsorption with mass transfer effects have been reported
by many researchers (Thomas and Lombardi, 1971; Rhee and Amundson, 1974; Hsieh
et al., 1977; Liapis and Rippin, 1978; Liapis and Litchfield, 1980; Santacesaria et al.,
1982; Kapoor and Yang, 1987).
A comparison of the breakthrough curves of the binary system and their correspond-
ing pure component breakthrough curves in Figure 8.3 indicates that earlier break-
throughs result for each component in the binary system. This reflects the lower dimen-
sionless hold-up capacity of each component in the column compared to the
corresponding pure component case.
Figure 8.4 shows a ternary system in which a third component, which has a stronger
affinity than the other two, is added to the binary system shown in Figure 8.3. Two
roll-up peaks appear and, by the same token, they can be explained by the displacement
effect. Note that the last component, which has the strongest affinity, does not roll-up in
any isothermal frontal adsorption case. Also, note that the component in the middle
(component 2) displaces component 1, while it is displaced by component 3 itself.
Roll-up peaks do not always exist or are noticeable in frontal adsorption, especially
when the saturation capacities of the components are low, or components have very
similar physical properties. Figure 8.5 has the same conditions as Figure 8.3, except that
its bi values are 10 times of those for Figure 8.3 (Table 8.1), thus the saturation capacity
(Ci = ai/bi) values for Figure 8.5 are 1/10 of those for Figure 8.3. In Figure 8.5, the
roll-up phenomenon is not noticeable, but the displacement effect is still evident.
I Ternary System I .----. Single Component Systems
2 4 6
Dimensionless Time
Figure 8.4 Ternary frontal adsorption with two roll-up peaks
Binary System
- - - - - - Single Component Systems
1 2 3
Dimensionless Time
Figure 8.5 Binary frontal adsorption with no roll-up peak
8.2.3 Elution
Multicomponent elution with an inert mobile phase results in a shortened retention
time for each component pigure 8.6). The retention times here are based on the first
moments rather than the positions of peak maxima. The peak height of component 1,
which has a weaker affinity than component 2, is increased, indicating that less band
spreading occurs as compared with the corresponding single component case, Its front
is diffused and tailing reduced for the component 2 peak. Conversely, the peak height
of component 3 is significantly decreased and its front is severely diffused.
These observations again can be explained by a displacement effect. When the three
components are migrating inside the column with different speeds depending primarily
on their adsorption affinities, they separate from each other. Since component 2 has a
higher affinity than component 1, it travels behind and displaces and concentrates com-
ponent 1, thus reducing the tailing of the component 1 peak. This results in a slightly
shorter retention time, a larger peak height, and less band spreading for component 1.
The displacement effect for such a case was also mentioned by other researchers
(Tiselius, 1940; Antia and Horvath, 1989).
Mutual displacement causes the portion of component 2 that is in the mixing zone
with component 1 to migrate faster than for the pure component case inside the column,
while the unmixed portion of component 2 migrates with the same speed as the pure
component case. This causes the diffusion of the front of the component 2 peak. In
comparison, the displacement effect of component 3 on component 2 reduces the tailing
ofthe component 2 peak. In return, component 2 diffuses the front of the component 3
Ternary System - - - - - - Single Component Systems
4 6
Dimensionless Time
Figure 8.6 Ternary elution.
peak. Since component 3 elutes last, thus, the tail end-pint of its peak hardly changes as
compared to its pure component case. The effect of surrounding components is further
illustrated by component 2 (Figure 8.6) where the diffusion effect of component 1
reduces the peak height of component 2, while the displacement effect of component 3
tends to do the opposite. Therefore, the net effect of these two influences will deter-
mine the relative peak height of component 2.
The influence of the displacement effect on nonlinear multicomponent elution is
summarized in Table 8.2, with the understanding that the effects listed in the table may
not always be noticeable depending on the severity of the displacement effect. The
severity of the displacement effect depends on the level of competition among all the
components and the nonlinearity of the system. In multicomponent elution five factors
have impacts on the displacement effect.
8.2.3.1 Effect of the Adsorption Equilibrium Constants
As the values of bi increase, the nonlinearity of the isotherm and the competition for
binding sites also increase. This escalates the displacement effect. If the values of bi in
a binary elution system are similar, the contact time between the two components is
maximized as they migrate through the column and separate from each other. This
increases the displacement effects. Figure 8.8 has the same conditions as Figure 8.7,
except that in Figure 8.8, the affinity of component 1 is larger, thus closer to that of
component 2 (see Table 8.1). Compared with Figure 8.7, the displacement effect in
Figure 8.8 is obviously more pronounced,
.----- Single Component Systems
2 4 6
Dimensionless Time
Figure 8.7 Binary elution showing slight displaccmcnt effect.
Binary System . - - - - - a Single Component Systems
al =2, bl = 4
Dimensionless Time
Figure 8.8 Binary elution with increased a1 and bl values.
8.2.3.2 Low Adsorption Saturation Capacity
Lower saturation capacity means fewer binding sites, and often increased competi-
tion for binding sites, especially in a system with large bi values. In Figure 8.6, a sys-
tem with small saturation capacity (Cy) was used in order to show a case with pro-
nounced displacement effects. Figure 8.9 is obtained from Figure 8.6 by increasing the
value of ai for the three components by 4-fold. The displacement effect is more pro-
nounced in Figure 8.6 than in Figure 8.9.
8.2.3.3 High Sample Feed Concentrations (Concentration Overload)
Increasing Qi is equivalent to increasing bi and reducing Cr proportionally, as is
shown by the isotherm expression, Eq. (5-25). Thus, the displacement effect escalates
when the feed concentrations of the sample are increased.
8.2.3.4 Large Sample Size (Volume Overload)
When a larger sample size is used, the contact time between the components will
increase, thus making the displacement effect more noticeable. Figures 8.7 and 8.10
have the same conditions except that in Figure 8.10, the sample size = 2.5) is
much larger than that in Figure 8.7 (zimP = 0.1). The first half of the effluent history in
Figure 8.10 actually represents the concentration profiles of the frontal adsorption
curves, due to severe volume overload, with a roll-up peak.
The use of large sample size in elution is not rare. In order to promote throughput,
the column is often overloaded in terms of either sample size or sample concentration
0 10 20 30 40
Dimensionless Time
Figure 8.9 Ternary elution with increased saturation capacity.
4 6 8
Dimensionless Time
Figm 8.10 Binary elution with large sample size.
(Bauman et al., 1956; Giddings, 1965; Knox and Pyper, 1986; Eble et al., 1987). Over-
load often increases the nonlinearity of the system and thus the displacement effect.
8.2.3.5 More Component(s)
Adding more component(s) in the sample will increase the competition for binding
sites among components. It also increases the nonlinearity of the isotherms (Eq. (9)),
thus escalating the displacement effect. The increased displacement effect in Figure
8.1 1 is obtained by adding one more component to the case presented in Figure 8.7.
When an additional component is present as a competing modifier in the mobile
phase, the displacement effect becomes rather complicated. The peaks corresponding to
the concentration profile of the modifier in a chromatogram are often referred to as sys-
tem peaks (Levin and Grushka, 1987), which will be discussed in detail in the next
chapter.
8.3 Summary
For multicomponent chromatography involving competitive isotherms, the dominat-
ing interference effect can be attributed to the displacement effect, which occurs not
only in the displacement mode but also in the other two major modes of chromatogra-
phy, frontal and elution. Five factors that may escalate the displacement effect in elu-
tion chromatography were investigated. In short, these five factors either promote com-
petition for binding sites among components or prolong such competition, or both.
From a mathematical point of view, these factors can be interpreted as being able to
either increase or prolong the nonlinearity of the isotherms. It has also been shown that
Ternary System - - - - - - Single Component Systems
4 6 8
Dimensionless Time
Figure 8.11 Effect of added component in the sample.
roll-up exists not only in frontal adsorption and displacement, but also in elution. Tt is
concluded that the displacement effect tends to reduce the tailing of the displaced
species, while the concentration front of the displacer is diffused by the displaced
species in all the three major modes of chromatography.
The use of a general nonlinear multicomponent rate eyuativn model provides a sys-
tematic study of interference effects in multicomponent chromatography. The model
accounts for various diffusional and mass transfer effects. The graphical representation
of the results aids the visualization of multicomponent interactions, and thereby pro-
motes a better understanding of the primary causes of the interference effects. The dis-
cussion presented in this work should be useful in the optimization of the chromato-
graphic separation processes.
Figures
8.1
* Table 8.1 Parameter Values Used for Simulation in Chapter 8
* In all cases, q, = 0.4 and ~p = 0.5. For elution cases, .rhP = 0.1, or otherwise as mentioned. The error tolerance of the ODE solver is tol=10". Double precision is used in the Fortran code.
Species
1
2 1 2 3 1 2 3 1
2 1 2 3 1
2 1
2 1 2 3 1 2 3
Physical
PeL;
300
300 200 200 200 300 300 300 300
300 300 400 500 300
350 300
350 300 4 0 500 300 350 350
Numerical Parameters
Ne
12
25
8
8
18
12
11
20
14
q;
10
15 10 10 10 1
1 1
1 30 40 90 40
50 40
50 30 40 90 40 50 50
N
2
1
2
2
1
1
1
1
1
Bii
6
8 10 10 10 20
1 2 0 20 20
20 8 7 6
10
9 10
9 8 7
6 10 9 9
Parameters ai
3
2 2
30 80
1 10 20
1
10 1 4 9 0.4
4 2
4 4
26
36 0.4 4
10
bi x Coi
6 x0.l
4 x 0.4 2 x 0.2
30 x 0.2 80 x 0.24 2xO.l
20 x 0.1 40 x 0.1 20 x 0.1
200xO.l l o x 1 40 x 0.1 90 xO.1 0.8 x 0.1
8 x0.l 4x 0.1
8 x 0.1 l o x 1 40 x 0.1
90 x 0.1 0.8 x 0.1
8 xO.1 20 x 0.1
Table 8.2 Summary of Nonlinear Multicomponent Elution (Compared with Pure Component Elutions)
Peak Position in Chromatogram
first peak
* The tail end-point does not change much while the tail may become flatter (see Figures 8.6 and 8.8).
Retention Time (First Moment)
middle peak(s)
last peak
decreases
Peak Height
decreases
decre'eases
increases
increases
Front Flank
or decreases
decreases
Tailing
sharpens decreases
diffuses
diffuses
decreases
*
CHAPTER 9 - SYSTEM PEAKS IN MULTICOMPONENT ELUTION
9.1 Introduction
In elution chromatography, a modifier (or modifiers) is sometimes added to the
mobile phase in order to compete with sample solutes for binding sites (Snyder et al.,
1988a). This often helps reduce the retention time and band spreading of sample
solutes.
Mobile phase modifiers are used not only in analytical chromatography, but also in
preparative and large scale chromatography. Other kinds of modifiers are also used in
chromatography. For example, some modifiers reduce the affinities of the sample
solutes with the adsorbent by binding with the sample solutes rather than with the
adsorbent. This chapter deals with the kind of modifier that competes with the sample
solutes for binding sites and its concentration in the feed is constant unlike in gradient
elution.
Peaks attributed to the modifier in an elution chromatogram are called system peaks
(Dreux et al., 1982; Cassidy and Fraser, 1984; Levin and Grushka, 1987). A positive
system peak, which is above the base-line value of the modifier concentration, is also
called a displacement peak (Solms et al., 1971; McCorrnick and Karger, 1980). A
negative one, which is below the base-line value, is called a vacancy peak (McCormick
and Karger, 1980).
Comprehensive reviews on system peaks were given by Levin and Grushka (1986)
and more recently by Golshan-Shirazi and Guiochon (1988a,b, 1989a). Solrns et al.
(1971) used a plate model and simulated three cases of single component elution with a
mobile phase containing a competing modifier. Much more detailed simulations,
including binary elution, were carried out by Golshan-Shirazi and Guiochon (1988a,
1989a) using a semi-ideal model with nonlinear multicomponent Langmuir isotherms.
They also performed experiments which qualitatively proved some of their model pred-
ictions (Golshan-Shirazi and Guiochon, 1988b, 1989b).
There are two different types of samples for elution chromatography with the
mobile phase containing a modifier. The first type, named Type I sample in this work,
consists of those samples which are prepared by dissolving sample solutes in a solution
which has the same composition as the mobile phase, thus the feen stream to the
column contains the competing modifier with a constant concentration. This kind of
system is a strictly isocratic elution process if the modifier concentration in the feed is
constant. Modeling of system peaks with Type I samples was carried out by Solrns et
al. (1971), and Golshan-Shirazi and Guiochon (1988a,b, 1989a,b). The second type of
sample, Type I1 sample, is those samples which are prepared based on an inert (blank)
solution, i.e., the samples contain no modifier. In such cases, system peaks have dif-
ferent patterns from those with Type I samples because of the deficit of modifier intro-
duced during the sample injection. Experiments with both types of samples were car-
ried out by Levin and Grushka (1987). They also investigated elution systems contain-
ing more than one modifier in the mobile phase.
This chapter extends the previous theoretical studies in the literature on elution
chromatography with a competing modifier which has a constant concentration in the
mobile phase, based on the computer simulations using the general multicomponent
rate model described in Chapter 5. The effect of modifier on the elution performance of
binary-solute systems with Type I or Type I1 sample will be extensively studied and
system peak patterns will be summarized for both cases. Binary elution with two
modifiers in the mobile phase will also be briefly discussed.
9.2 Boundary Conditions for the Rate Model
The general multicomponent rate model described in Chapter 5 will be used for the
study. The modifier (or modifiers) is treated as one of the components in the governing
equations of the model. Langmuir isotherms is used, in which the modifier is considered
as one of the competing components.
The following boundary conditions are needed for the modifier(s). For molfier(s)
in systems with Type I sample,
For modifier(s) in systems with Type I1 sample,
Coi 1 1 else
In this chapter, parameter values used for the simulation are listed in Table 9.1, or
mentioned during discussions. In all runs, E,, = 0.4 and E,, = 0.4, and the rectangular
sample size was zimP = 0.1, unless otherwise specified. The ODE solver's (IMSL's
IVPAG subroutine) tolerance is tol=10-~. Double precision is used in the Fortran code.
CPU times for some cases are listed in Table 9.1,
9.3 Results and Discussion (Gu et al.. 19%)
The competing modifier is treated as one of the components in the governing equa-
tions of the model, and it is also considered as a competing component in the isotherms
shown above. Hence, a binary elution with a competing modifier in the mobile phase
constitutes a three-component system. Based on theoretical simulation, the discussion
of the system peaks and the interrelationship between the modifier(s) and the sample
solutes will be divided into several sections. Binary elutions with single competing
modifier are discussed first.
9.3.1 Modifier Affinity Is Weaker Than Those of Sample Solutes
Figure 9.1 (solid lines) shows a simulated effluent history (chromatogram) of a
binary elution with Type I sample and a competing modifier (component 3) in the
mobile phase. Components 1 and 2 are the two sample solutes. The affinity of the
modifier is smaller than those of the sample components. Parameter values used for
simulation are listed in Table 9.1 at the end of this chapter. Note that the scale for the
modifier concentration shown in Figure 9.1 (as well as in all other figures) is Cb3 - 1.
The actual base-line value for the modifier concentration is cb3 = 1. By transforming the
base-line value to zero (is. cb3 - 1 = O), the effluent history becomes more presentable.
It is obvious that a negative system peak does not indicate negative concentrations, but
rather concentration values that are below the base-line value.
Dimensionless Time
Figure 9.1 Binary elution with a weak modifier (Type I sample).
The case shown in Figure 9.1 gives one positive system peak and two negative sys-
tem peaks which are due to the displacement effect of two sample solutes on the
modifier. This kind of system peak pattern (with Type I sample) was also reported by
Golshan-Shirazi and Guiochon (1989a) using a semi-ideal ~llodel. A mass balarice of
each species has been checked to evaluate the accuracy of the numerical solution. For
the modifier, the numerical integration (using the subroutine QDAGS from IMSL,
1987) of the concentration profile of the modifier (cb3 - 1.0) in Figure 9.1, which con-
sists of 400 data points, from z = 1 to z = 15 gives a value of 0.0000, which is in agree-
ment with its theoretical value zero. For the sample solutes, mass balances are also
held.
Figure 9.1 (dashed lines) also shows the binary elution case in the absence of the
modifier. It is evident that the use of modifier results in the decrease of the retention
time and the spreading of the band and the increase of peak height of each sample
solute.
Figure 9.2 shows a chromatogram with a Type I1 sample. Other conditions for Fig-
ure 9.2 are the same as in Figure 9.1. It can be seen that in Figure 9.2 there are three
negative system peaks and no positive system peak. The numerical integration of the
concentration profile for the modifier (component 3) of the three system peaks was
found to be -0,1000. This negative value indicates the deficit of modifier introduced
during sample injection. It is equal to the sample size, zhp = 0.1. In Figure 9.2 the peak
at the front is a negative system peak, instead of a positive system peak as shown in
Figure 9.1, because the large negative system peak due to the deficit of the modifier
I I I I I ' Sample Contains Components 1 and 2 I I I ' - - - - - I I
Sample is Blank . I
Dimensionless Time
Figure 9.2 Binary elution with a weak modifier (Type I1 sample).
during sample injection negated the positive system peak. This can be easily verified
by checking the concentration profile of the modifier when a blank sample, which con-
tains only an inert carrier liquid, is employed. This case is also shown in Figure 9.2
(dashed line). It gives only a single large negative peak, and the peak area has been
found to be equal to the injection pulse size, TimP = 0.1.
It should be pointed out that positive system peak(s) does occur in some cases
involving Type I1 sample, if the positive system peak overcomes the negative one due
to sample introduction, as we will encounter shortly. The number and direction
(positivelnegative, i.e., upwardldownward) of system peaks for the modifier are deter-
mined primarily by sample type and their relative affinity to those of the sample solutes,
and of course, the number of sample solutes.
9.3.2 Modifier Affinity Is Between Those of Sample Solutes
Figure 9.3 gives the chromatogram for the case shown in Figure 9.1 except that the
affinity of the modifier is between those of the two sample solutes (Table 9,1), Figure
9.3 shows one positive system peak and two negative system peaks, which are similar
to Figure 9.1. However, the retention time of the positive system peak is prolonged and
the peak height is increased. Both changes are due to the increase of affinity of the
modifier. Because of the increase of modifier affinity, there are more modifier
molecules adsorbed onto the stationary phase that can be dislodged by the sample
solutes. It should not be surprising if one considers the case in which the modifier has
no mnity to the column packing, resulting in a flat concentration profile for the
modifier. On the other hand, if the affinity of the modifier further increases when its
Dimensionless Time
Figure 9.3 Modifier affinity is between those of sample solutes (Type I sample).
affinity is already not far from the leveling off range of the Langmuir isotherm, the
increase of modifier's loading in the stationary phase can be overshadowed by the
affinity increase that could make it too difficult to be dislodged by the sample solutes.
In such cases, the increase of modifier affinity may result in a reduced positive system
peak at the front.
Figure 9.4 has the same conditions as Figure 9.3, except that Type II sample is used
in the case of Figure 9.4. The effluent history shown in Figure 9.4 gives one positive
and two negative system peaks, quite different from Figure 9.2 which has the same con-
ditions as Figure 9.4, except that the affinity of the modifier in Figure 9.4 is stronger.
This is due to the fact that the displacement effects from components 1 and 2 cause a
larger positive system peak, and this positive system peak overcomes the negative sys-
tem peak which is caused by the lack of modifier in the sample. The fact that the posi-
tive system peak in Figure 9.4 is smaller than that in Figure 9.3 is in agreement with
this argument.
9.3.3 Modifier Affinity Is Stronger Than Those of Sample Solutes
Figure 9.5 shows the case in which the affinity of the modifier is stronger than both
sample solutes. In this case, the system peaks include two positive ones and one nega-
tive one. The first positive system peak partially overlaps with the component 1 peak,
and it departs from the component 1 peak when the component 2 peak starts to take off.
The corresponding case with the Type I1 sample is shown in Figure 9.6 which gives a
similar system peak pattern.
I Type I1 Sample 1
4 6 8
Dimensionless Time
Figure 9.4 Modifier affinity is between those of sample solutes (Type I1 sample).
2 4 6
Dimensionless Time
Figure 9.5 Binary elution with a strong modifier (Type I sample).
4 6
Dimensionless Time
Figure 9.6 Binary elution with a strong modifier (Type I1 sample).
In Figure 9.5, if the affinity of the modifier is further increased, the first positive sys-
tem peak will no longer overlap with component 1 peak, as is shown clearly in Figure
9.7. Figure 9.8 shows the degenerated case which is obtained by simply increasing the
affinity of the modifier shown in Figure 9.7. It is obvious that the merger of two positive
system peaks in Figure 9.8 is due to the partial overlapping of the component 1 peak
with the component 2 peak.
9.3.4 Effect of Modifier Concentration on System Peak Patterns
The increase of modifier concentration obviously will reduce the retention times of
the sample solutes because the modifier competes with sample solutes for adsorption
sites. This effect has been extensively investigated by Golshan-Shirazi and Guiochon
(1 988a,b) both theoretically and experimentally.
Increasing modifier concentration also affects system peaks. Figure 9.9 shows the
case in which the modifier concentration is ten times higher than that in Figure 9.3.
Comparing Figure 9.3 with Figure 9.9 (both with Type I sample), it can be seen that the
system peaks in Figure 9.9 are much smaller than those in Figure 9.3. This means that
the disturbance caused by the sample solutes to the concentration profile of the modifier
becomes smaller if the concentration of the modifier increases. It also implies that if the
modifier concentration is sufficiently large, its concentration in the model system can be
considered as a constant. This simplifies the simulation. Note that in all figures the con-
centration scale is dimensionless concentration, Thus a smaller peak does not neces-
sarily mean a smaller dimensional concentration.
4 6
Dimensionless Time
Figure 9.7 Same conditions as Figun 9.5, except that the modifier affinity is stronger.
2 3 4 5
Dimensionless Time
Figure 9.8 Same conditions as Figure 9.7, except that the modifier affinity is stronger.
Dimensionless Time
Figure 9.9 Same conditions as Figrue 9.3, except higher modifier concentration (Cm = 1.0).
Figure 9.10 shows the case of the Type 11 sample, in which the concentration of
modifier is ten times higher than in Figure 9.4. The increase of modifier concentration
changes the first system peak from a positive one (shown in Figure 9.4) to a negative
one (shown in Figure 9.10). The reversal of the peak direction occurs because when the
modifier concentration is increased the negative system peak which is caused by the
deficit of the modifier during sample injection overcomes the positive system peak
caused by the displacement effect from the sample solutes on the modifier.
9.3.5 Effect of Modifier on Sample Solutes
Figure 9.1 1 shows a binary elution without a modifier. The retention time and the
resolution of the two solute peaks are both unnecessarily high for the complete separa-
tion of the two components. In such an elution system, adding a proper modifier may
reduce the process duration while still achieve a base-line separation. Figure 9.12
shows the effect of the added modifier in the aforementioned binary elution system. It
is clear that the retention times and the resolution of the two sample solutes are drasti-
cally reduced. The base-line separation of the two sample solutes is still achieved while
the elution duration is cut to one-fifth of that in Figure 9.11. The concentrations of the
product peaks are much higher and the band spreadings of these peaks are largely
reduced when the modifier is used. This is because of the displacement effect from the
modifier. Figure 9.13 has the same conditions as Figure 9.12, except that in Figure 9.13
Type I sample is employed. It is clear that the result shown in Figure 9.13 is also desir-
able.
0 1 2 3 4 5 6
Dimensionless Time
Figure 9.10 Same conditions as Figure 9.4, except higher modifier concentration (Cm = 1-0).
10 15 20
Dimensionless Time
Figure 9.1 1 Binary elution without modifier.
Type I1 Sample
Dimensionless Time
Figure 9.12 Effect of added modifier (Type I sample).
Dimensionless Time
F i p 9.13 Effect of added modifier (Type II sample).
Golshan-S hirazi and Guiochon (1988a,b, 1989a,b) reported that when the modifier
concentration was high the common Langmuir shape elution peaks might reverse their
asymmetry and become anti-Langmuir, i.e., the tailing of a peak becomes smaller than
the diffused front flank of the peak. Figures 9.12 and 9.13 show that at low modifier
concentration levels, the phenomenon of peak shape reversal also occurs if the adsorp-
tion equilibrium constant of the modifier (b3) is high enough (see Table 9.1 for parame-
ter values) and the adsorption capacity is low. Interesting, in Figures 9.12 and 9.13,
component 1 peak still retains its Langmuir type peak shape while the peak shape of
component 2 becomes anti-Langmuir type.
More discussion of the effect of modifier in the binary elution with type I sample is
given by Golshan-Shirazi and Guiochon (1 989a,b).
9.3.6 Effect of Type of Sample
The difference in the system peak pattern due to sample type is quite obvious. It is
clear that the sample type may affect the direction, size and location of system peaks.
These effects have been shown during the previous discussion.
On the other hand, the sample type also affects the elution pattern of sample solutes.
By comparing some of the figures shown in this chapter, one may quickly find that the
influence of sample type on sample solutes is usually quite small. This situation may be
changed if the sample size is large. The cases shown in Figures 9.3 and 9.4 have the
same conditions except the type of sample. The sample size in both case is .r*p = 0.1
Their corresponding cases with ~h~ = 1.0 are shown in Figure 9.14. It is clear that the
4 6 8
Dimensionless Time
Figure 9.14 Effect of type of sample at large sample size.
difference in the concentration profiles of the two sample solutes is not small when a
large sample size is used.
9.3.7 Effect of Sample Solutes on the Modifier
In the discussion above, it has already been pointed out that system peaks are the
result of the displacement effects of the sample solutes on the modifier arising from
competition for binding sites. This was also revealed by many other researchers (Solms
et al., 1971; Golshan-Shirazi and Guiochon, 1988a). In the case of the Type I1 sample,
the deficit of modifier concentration during the introduction of the sample also plays a
role which may cause a negative system peak at the front (Figure 9.2) or reduce the size
of the positive system peak at the front (Figure 9.14). It may even negate the positive
system peak (Figure 9.15).
The relative affinities of the sample solutes also affect the system peaks as shown in
Figure 9.15. Figure 9.15 has the same conditions as Figure 9.2, except that in Figure
9.15 the affinity of component 2 is smaller, thus closer to that of component 1 (Table
9.1). Figure 9.15 (with Type II sample) shows that when the component 1 and com-
ponent 2 peaks overlap to some degree, the two corresponding negative system peaks
will degenerate into a single one. A similar system peak pattern in the case of the Type
I sample was observed by Golshan-Shirazi and Guiochon (1989a) using a semi-ideal
model. The comparison of Figure 9.8 with Figure 9.16 proves that the partial overlap-
ping of the peaks for sample solutes may also cause the merger of the positive system
peaks.
Dimensionless Time
Figure 9.15 Same conditions as Figure 9.2, except that Component 2 has a weaker affinity.
I Type I Sample I
2 3 4 5
Dimensionless Time
Figure 9.16 Same conditions as Figure 9.8, except that Component 1 has a weaker affinity.
9.3.8 Summary of System Peak Patterns
Table 9.2 summarizes all possible combinations of system peak patterns for binary
elution with one competing modifier. There are twice as many combinations for cases
with the Type 11 sample than those with the Type I sample. This table also gives indi-
cations for system peak combinations in single component elution since degenerated
cases are included in the table. It is interesting to point out that Figure 9.17 gives a
severely degenerated case in which the overlapping of component 1 and component 2
peaks causes the degeneration of their corresponding negative system peaks. The posi-
tive displacement peak and the peak that would be caused due to the deficit of modifier
during sample introduction negated each other. Figure 9.18 has the same conditions as
Figure 9.17, except that the modifier concentration is 0.1 which is lower than that in
Figure 9.17. Because of the decrease of the modifier concentration the previously
degenerated peak (in Figure 9.17) becomes very prominent in Figure 9.18.
In general, both sample types, I and 11, could have a maximum of only three system
peaks for binary elution with one competing modifier. For binary elutions with Type I
sample, the minimum number of system peaks should be two because the existence of a
positive system peak necessitates the existence of a negative system peak in order to
meet the mass balance, which requires that the sum of peak areas of positive system
peaks be equal to the sum of peaks areas of negative system peaks. On the other hand,
this requirement does not apply to those cases with Type II sample. In such cases, the
minimum number of system peaks is one as shown in Figure 9.17.
2 3 4
Dimensionless Time
Figure 9.17 Binary elution showing only one system peak (Type 11 sample).
Dimensionless Time
Figure 9.18 Binary elution showing one positive, and one negative peak, respectively (Type II sample).
9.3.9 Binary Elution with Two Different Modifiers in the Mobile Phase
As the discussion above indicates, system peak behavior can be very complex and
elusive. This fact was also realized by previous researchers (Levin and Grushka, 1987;
Golshan-S hirazi and Guiochon, 1988a,b, 1989a,b). The situation can be further compli-
cated if there is more than one modifier in the mobile phase. In practice, the multiple
modifier cases are not rare. Experiments by Levin and Grushka (1987) showed that dif-
ferent modifiers gave different sets of system peaks.
Figure 9.19 shows a case involving two sample solutes (components 1 and 2) and
two different modifiers (components 3 and 4). The first modifier (component 3) has a
weaker affinity than the second modifier (component 4). Figure 9.20 has the same con-
ditions as Figure 9.19, except that a Type I sample is employed. It is interesting to note
that there is a positive system peak at the tail of the concentration profile of the first
modifier (component 3) in both figures. This kind of tail has never been observed in
simulations for single modifier systems. Its presence is likely due to the involvement of
a second modifier in the system.
9.4 Concluding Remarks
The interrelationship between sample solutes and the modifier(s) in elution chroma-
tography has been investigated by using extensive computer simulations based on a
general rate equation model. It has been concluded that for binary elution with one
competing modifier in the mobile phase, there are three system peak patterns if Type I
samples are used, and six if Type I1 samples are used. In addition, the binary elution
0 2 4 6 8 10
Dimensionless Time
Figure 9.19 Binary elution with two modifiers (Type I1 sample).
4 :. . . . . . . . . . . 0 . . . . .
Lines 1 and 2 - Sample Solutes Lines 3 and 4 - Modifiers
.......-- I ....
0 2 4 6 8 10
Dimensionless Time
Figure 9.20 Binary elution with two modifiers (Type I sample).
system with two different competing modifiers has been also briefly discussed.
This study shows that system peaks in some cases can be very complex and may not
be fully explained by qualitative arguments, although the ultimate cause behind system
peaks described in this work may be simply attributed to the displacement effect due to
the competitive nature of the isotherms involving all the components in the system
including the modifier, and the deficit of modifier during sample introduction if Type I1
samples are used.
In gradient elution the modifier concentration is continuously changed during the
elution process. The situation in gradient elution is more complicated since the
modifiers used in gradient elution usually may not be considered as only competing
with sample solutes for binding sites. The mutual interaction is quite complex, which is
why nonlinear gradient is often needed for the optimization of the elution. The com-
puter implementation for the accommodation of gradient elution is remarkably easy;
one only has to modify thc Cfi(~)/Coi term in Eq. (5-13) for the modifier component. A
detailed study of multicomponent nonlinear gradient chromatography using experimen-
tal correlations will be reported elsewhere.
* Table 9.1 Parameter Values Used for Simulation in Chapter 9
CPU times on SUN 41280 computer for some cases (solid lines): Figure 9.1, 10.1 min; Figure 9.4, 10.8 rnin; Figure 9.5,7.3 min; Figure 9.15,9.4 rnin; Figure 9.20, 18.6 min.
Numerical Parameters
Ne - -
- -
- -
- -
- -
A
Figures Species Physical
Parameters bi x Coi PeLi Bii qi ai
Table 9.2 Possible System Peak Combinations in a Binary Elution with a Competing Modifier in the Mobile Phase
Sample
Type I
Type I1
System Peak Combinations (positive peak(s)/negative peak(s)) I
112 (Fig. 9.1)
013 (Fig. 9.2)
11
111 (Fig. 9.8)
012 (Fig. 9.15)
I11
211 (Fig. 9.5)
011 (Fig. 9.17)
IV
111 (Fig. 9.18)
V
211 (Fig. 9.6)
VI
112 (Fig. 9.4)
CHAPTER 10 - AFFINITY CHROMATOGRAPHY
10.1 Introduction
Affinity chromatography has seen rapid growth in recent years. It is widely con-
sidered as the most powerful means of separating and purifying enzymes, antibodies,
antigens, and many other proteins and macromolecules that are of important use in
scientific research and development of novel pharmaceuticals. Affinity chromatography
not only purifies a product, but also concentrates the product to a considerable extent
(Chase, 1984a). Over the years, this subject has been reviewed by many people, most
recently by Chase (1984a), and Liapis (1989). Affinity chromatography is also often
referred to as biospecific adsorption, since it utilizes the biospecific binding between the
solute molecules and the immobilized ligands. The monovalent binding between a
ligand and a solute macromolecule is often considered as a second order kinetics as
shown in Eqs. (5-27) and (5-28).
The class of monoclonal antibody often used in affinity chromatography is irnmuno-
globin G, which has two identical antigen binding sites. If the binding of one antigen
does not interfere with the binding of another antigen onto the other binding site of the
same antibody, then bindings can be considered as two monovalent bindmgs. If the
antigen has more than one binding site that can be recognized by the antibody, mul-
tivalent bindings are possible (Chase, 1984a).
There are two kinds of bindings in affinity chromatography, specific and non-
specific. The specific binding usually involves only the target macromolecule and the
ligand. Non-specific binding is undesirable, but often unavoidable. It can be caused by
unintended ion-exchange effects, hydrophobic effects, etc.
The operational stages of affinity chromatography often include adsorption, wash-
ing and elution. The column is regenerated after each cycle. The adsorption stage is
carried out in the form of frontal adsorption. In order to obtain a sharp concentration
front for the target macromolecule, small flow is often used (Chase, 1984a). The wash-
ing stage right after the adsorption stage is aimed at removing the impurities in the bulk
fluid and in the stagnant fluid inside particles, and impurities bonded to the support via
non-specific binding (Chase, 1984a).
The elution stage removes the bonded target macromolecules from the ligands. Elu-
tion can be carried out by using a soluble ligand which is often the same as that immo-
bilized in the support, if the soluble ligand is present in a higher concentration and is
relatively inexpensive. The other method is often called non-specific desorption, which
uses a variety of eluting agents, such as pH, protein denaturants, chaotropic agents,
polarity reducing agents, temperature (Chase, 1984a) to destabilize the binding between
the macromolecules and the ligands.
Elution in affinity chromatography carries a different meaning from that used in
other forms of chromatography, such as adsorption, ion-exchange, in which elution
means impulse analysis. To avoid confusion, impulse analysis in affinity chromatogra-
phy is referred as zonal analysis (Arnold et al., 1986a; Lee, 1989; Lee et al., 1990).
The Langmuir isotherm for biospecific binding, which comes from the second order
kinetics is characterized by a very large equilibrium constant and a very small satura-
tion capacity that indicates that the ligand density of an affinity matrix is often quite
low. Because of the large equilibrium constant, the isotherm is oftcn nonlinear even
though the concentrations are often very low. A universal function has been developed
by Lee et al. (1990) to measure the effects of isotherm nonlinearity in zonal analysis.
General rate models have been developed by h e and Liapis (1987b, 1988b) for
affinity chromatography. Their models consider various mass transfer mechanisms and
a second order kinetics between the immobilized ligands and the macromolecules, and
between the soluble ligands and the macromolecules during elution.
10.2 El'fect of Reaction Kinetics
A general rate model with a second order kinetics has already been described in
Chapter 5. The three breakthrough curves in Figure 10.1 show the effect of reaction
rates. Parameter values used for its simulation are listed in Table 10.1 at the end of the
Chapter. The solid curve shows that when the Darnklilher numbers for binding and dis-
sociatior1 are low the breakthrough curve takes off sharply at a earlier time and levels
off later on quite slowly. This indicates that when the reaction rates are slow while the
mass transfer rates are not, may solute molecules do not have chances to adsorb onto
the solid phase, and those that do are released slowly.
Figure 10.2 shows that when the reaction rates increase to some extent the break-
through curves will be very close to that of the equilibrium case. In fact the equilibrium
6 8 10
DIMENSIONLESS TIME
Figure 10.1 Effect of reaction rates in frontal analysis.
Equilibrium
2 3 4
DIMENSIONLESS TIME
Figure 10.2 Fast reaction rates vs. equilibrium.
case is the asymptotical limit of the breakthrough curves calculated from the kinetic
model.
Figure 10.3 shows the effect of reaction rates in a single component zonal elution
case. The solid line shows that the elution peak appears early with a very sharp front,
however it has a very long tailing. This indicates that when the reaction rates are very
low, a large portion of the solute molecules do not have a chance to bind with the
ligands and they are eluted out quite quickly. On the other hand, those molecules that
do bind with the ligands are dissociated very slowly, causing a long tail. This is par-
tially reflected by the breakthrough curve shown as the solid line in Figure 10.1 since
the two modes are interrelated. Figure 10.3 also shows that the peak front appears later,
and the peak height reduces when the reaction rates increase. When the reaction rates
further increase the appearance of the peak front is further delayed, and the peak height
increases, since the diffusion of the peak front due to slow reaction rates is reduced.
The increase of reaction rates reduces the tailing effect (Figure 10.3).
In Figure 10.4 the solid curve is the same as that in Figure 10.1. Figure 10.4 shows
that the slow reaction rates are the rate-limiting step in this case. On the contrary, Fig-
ure 10.5 shows a case in which the mass transfer rates are rate-limiting, since the reac-
tion rates in this figure are relatively much higher than the mass transfer rates.
A breakthrough curve of an adsorption system with slow mass transfer rates is sirni-
lar to that of slow reaction rates. They both take off sharply at initial time and level off
later slowly. In Figure 10.6, one may find that the two cases differ in a revealing way.
For the slow mass transfer case, the breakthrough curve takes off earlier (at T < 1) than
DIMENSIONLESS CONCENTRATION
( 4 Curves Overlap )
5 10 15
DIMENSIONLESS TIME
Figure 10.4 Slow kinetics as rate limiting step in frontal analysis.
DIMENSIONLESS CONCENTRATION
'1 = 0.1, Bi = 5; Slow Mass Transfer
- - - - - - '1 = 10, Bi = 10; Slow Reaction
DIMENSIONLESS TIME
Figure 10.6 Comparison of slow kinetics with slow mass transfer.
in the case with slow reaction rates, since in the former case, many solutes do not enter
the particles, while in the latter case they do. The slow mass transfer here means that
both external film mass transfer and the intraparticle diffusion rates are low. The solid
line in Figure 10.7 shows that the take-off of the breakthrough curve is very sharp if
film mass transfer coefficient is small even through the intraparticle diffusion
coefficient is not small, since many solutes do not have a chance to penetrate the liquid
film into the particles.
10.3 Effect of Size Exchsion
The effect of size exclusion on the adsorption saturation capacity has already been
discussed in Chapter 5. The reduction of the column hold-up capacity due to the effect
of size exclusion is clearly shown in Figure 10.8. The two single-component break-
through curves have the same conditions except that the dashed line case has a size
exclusion effect and half of thc particlc porosity is unaccessible, i.e., FeX = = 0.5.
The capacity in the size exclusion case has been taken to be half of that with no size
exclusion (solid line). Figure 10.8 shows that in the case of size exclusion, the break-
through curve tends to be sharper. This is also true for systems with no adsorption as
shown in Figure 10.9. In both Figures 10.8 and 10.9, the column hold-up capacity area
can be checked using Eq. (5-51). The dotted curve in Figure 10.9 shows a case with
complete size exclusion. The area above the curve is found to be unity. One may expect
this kind of case when using a totally unpenetrable and nonadsorbing substance such as
blue dextrin to measure the bed voidage.
DIMENSIONLESS TIME
Figure 10.7 Effect of slow intraparticle diffusion and film mass transfer.
4 6 8 10 12
DIMENSIONLESS TIME
Figure 10.8 Size exclusion effect in presence of adsorption.
0 1 2 3 4
DIMENSIONLESS TIME
Figure 10.9 Size exclusion effect in absence of adsorption.
By comparing the retention times of component 1 peaks in Figures 5.5 and 5.6 in
Chapter 5, one easily finds out that size exclusion effect reduces retention times in elu-
tion chromatography.
10.4 Interaction Between Soluble Ligand and Macromolecule
Soluble ligands are often used to elute the adsorbed macromolecules in the elution
stage, if the ligands are not expensive and can be easily separated from the macro-
molecules after elution (Chase, 1984a). A rate model involving soluble ligand used for
the elution of a single adsorbate has been reported by Awe and Liapis (1987b, 1988a)
for finite baths and fixed-beds.
10.4.1 Modeling of Reaction in the Fluid Phase
The extended rate model described in Section 5.4.1 can be further extended to
include a binding reaction in the bulk fluid and the stagnant fluid in the particles
between macromolecule P (component 1) and soluble ligand I (component 2). The
complex formed from the binding of P and I is PI (component 3).
The binding between the macromolecule and the immobilized ligand L forms PL.
It is assumed that each macromolecule can bind with only one ligand, I or L, and there
is no interaction between the two different ligands, I and L.
Size exclusion effects in such a system which involves large molecules such as P,
PL and possibly L, may not be negligible, thus should be included in the model. The
rate models described in Sections 5.4.1 and 5.4.3 can be readily extended. For simpli-
city, only a three-component system is going to be discussed, since a generalized sys-
tem in this case is too complicated.
(1) Bulk Fluid Phase Governing Equation
where f(i) = -1 for components 1 and 2 (i=1,2), and f(i) = 1 for component 3 (i=3).
(2) Particle Phase Governing Equations
in which g(i) = 1 for i = 1, and g(i) = 0 for i = 2,3, since only component 1 binds with
the immobilized ligand. The use of sign changers, f(i) and g(i), is purely for the com-
pactness of the model system in its written form. Note that Cil represents the concen-
tration of the macromolecule in the solid phase (i.e., [PL]) based on the unit volume of
the particle skeleton.
The model system is more general than the similar one presented by Arve and
Liapis (1987b) since size exclusion is included. The model system is actually more like
that for a fixed-bed reactor than for chromatography, since there is a new component
(PI) forming and leaving the column.
Defining the following dimensionless constants
the PDE system c m be expressed in dirnensionlcss forms as follows.
Since Q3 is not known before simulation, it is replaced by G1 for the nondimensional-
ization of the concentrations of component 3 such that Cb3 = Cb3 CO1 and Cp3 = cp3 CO1.
10.4.2 Solution Strategy
The existing model can be modified to implement the fluid phase reaction. For the
bulk fluid phase, the finite element vector (AFBi)h should now include the last term of
Eq. (10-6) and the following expression should be used to replace Eq. (5-20).
The modification of the particle phase governing equation is quite straightforward.
Details are omitted here.
10.5 Modeling of the Three Stages in Affinity Chromatography
Figure 10.10 shows an affinity chromatographic process with a wash stage after the
frontal adsorption stage. The non-specifically bound impurities are not included in the
simulation. Their effluent histories can be simulated in a separate run and then superim-
posed onto the current figure, since they do not interact with the macromolecule.
Because no soluble ligand or other active eluting agent is used for the elution, the
chromatogram shows a very long tail which indicates that the recovery of the macro-
molecule is difficult and not efficient for this system.
Figure 10.1 1 has the same condition as Figure 10.10, except that soluble ligands are
used for elution at z = 15 after the wash stage which started at z = 14. Compared with
Figure 10.10, it is clear that elution using soluble ligands helps reduce tailing and the
time needed for the recovery of the product. If a higher concentration of soluble ligand
is used for elution, the elution stage will be shorter, and the recovered product will have
a higher concentration as Figure 10.12 shows, in which the soluble ligand concentration
in the feed is five times that in Figure 10.1 1.
-~rontal-l k- Elution with Soluble Ligand - -
\Wash (14 - 15 )
Complex
0 5 10 15 20 25 30 35 40 45 50 55 60
DIMENSIONLESS TIME
Figure 10.1 1 Effect of soluble ligand in the elution stage of affinity chromatography.
10.6 Inhibition in Affinity Chromatography
There are two types of inhibitors that are commonly used in affinity chromatogra-
phy (Lee, 1989), namely, (I) soluble ligand inhibitor, and (11) competing binding inhibi-
tor,
Figrue 10.13 shows that soluble ligand inhibitor reduces the retention time and tail-
ing of the macromolecule peak in zonal elution since a large amount of macromolecules
bind with the soluble ligands instead of the immobilized ligand.
The competing binding inhibitor effect is in essence the same as that of the compet-
ing modifier discussed in Chapter 9. It also helps reduce the retention time and tailing
of macromolecule peak as shown in Figure 10.14. In Figures 10.13 and 10.14, the base
line of the concentration profile of the inhibitor has been moved from the actual cb2 = 1
to 0 in order to present the effluent history better. Both cases are strict isocratic elution.
10.7 Summary
In this Chapter, various aspects of affinity chromatography, including the effects of
reaction kinetics, mass transfer, size exclusion have been discussed. The general rate
model has been further modified to account for the reaction between macromolecules
and soluble ligands in the bulk fluid and in the stagnant fluid inside particle macropores.
The role of soluble ligand in the elution stage in affinity chromatography has been
investigated. The effects of both soluble ligand inhibitor and competing binding inhibi-
tor in zonal elution have also been discussed briefly.
%
'1 : ; Zonal Elution without Inhibitor I I ' I I \ I I I I
Soluble Ligand As Inhibitor I I
0 2 4 6 8 10 12 14
DIMENSIONLESS TIME
Figure 10.13 Effect of soluble ligand inhibitor in zonal elution.
I d
0 2 (Inhibitor) d 0
DIMENSIONLESS TIME
Figure 10.14 Effect of competing binding inhibitor in zonal elution.
Fig ure(s)
10.1 10.2 10.3 10.4 10.5 10.6 10.6 (dash) 10.7 10.8 10.9 10.10
10.1 1
10.12
10.13
10.14
* Table 10.1 Parameter Values Used for Simulation in Chapter 10
Species PeLi
1 400 1 400 1 400 1 400 1 400 1 400 1 400
1 400 1 400 1 200 1 300 1 300 2 300 3 300 1 300 2 300 3 300 1 300 2 300 3 300 1 300
2 300
Physical Numerical Parameters I Parameters
* Parameters are for solid line curves in figures unless indicated otherwise. In all cases, eb = = 0.4. The sample size for zonal analysis are: Figure 10.3, zimp = 0.5; Figure 10.13, zimP = 1.0; Figure 10.4, zhp = 0.8. For the solid line case in Figure 10.6, the parameters for Langmuir isotherm is a = 5, b x Co = 5. For Fig- ure 10.7, a = 5 in the Langmuir isotherm. In Figures 10.10 to 10.12, the size exclusion factor for all components is FfX = 0.8. The error tolerance of the ODE solver is t o l = l ~ - ~ . Double precision is used in the Fortran code. CPU times on SUN 41280 computer are: Figure 10.2 (solid line), 9.9 sec; Figure 10.3 (solide line), 3.8 rnin; Figure 10.12,24.7 min.
CHAPTER 11 - MULTICOMPONENT RADIAL FLOW CHROMATOGRAPHY
11.1 Introduction
Chromatography has long been established as an effective means of separation. It
becomes more and more popular in the age of rapid development of biotechnology.
The demand for efficient preparative and large scale liquid chromatographic separation
processes is ever increasing. Radial flow chromatography (RFC), since its introduction
in the commercial markrt i11 the 11licl 1980s (McCornlick, 1988), has proved to be a
promising alternative to conventional axial flow chromatography (AFC). Compared to
AFC, the RFC geometry (Figure 11.1) provides a relatively large flow area and short
flow path. These factors enable a larger volumetric flow rate and shorter shift time in
liquid chromatographic separations. If soft gels or affinity matrix materials are used as
separation media, the low pressure drop of RFC helps prevent bed compression (Saxena
et al., 1987; Ernst, 1987). A full range of sizes from SO rnL to 200 L in bed volume of
RFC columns both prepacked and unpacked can be obtained from commercial com-
panies. Separation of various biological products has been reported (Chen and Hou,
1985; Saxena et al., 1987; Saxena and Wed, 1987; Huang et al., 1988; Plaigin et al.,
1989; Lee et al., 1990). An experimental case study of the comparison of RFC and
AFC was carried out by Saxena and Weil (1987) for the separation of ascites using
QAE cellulose pacldngs. They reported that by using a higher flow rate, the separation
time for RFC was one-fourth that needed for an longer AFC column with the same bed
volume. It was claimed that by using RFC instead of AFC, separation productivity can
be improved quite significantly (McCormick, 1988). RFC is especially suitable for
affinity chromatography in which solutes are usually strongly retained. This permits the
use of high flow rate and short flow path for fast treatment of a large volume of Sam-
ples. RFC is advantageous for separation processes in which the effluent from the
column is being recycled and the adsorbate will not be lost in the effluent stream
(Liapis, 1989), such as in the case of in situ separation during fermentation (Yang et al.,
1989).
Radial flow packed-bed reactors have been used for a variety of industrial applica-
tions such as ammonia and methanol syntheses, catalytic reforming, and vapor-phase
desulfurization (Balakotaiah and Luss, 198 1; Strauss and Budde, 1978). Unlike the
radial flow reactor RFC is used primarily for liquid phase chromatographic separation.
There are quite a few papers that are related to the theoretical and experimental studies
of radial flow reactors. A brief review was given by Hlavacek and Votruba (1977).
More recent papers include Strauss and Budde, 1978; Balakotaiah and Luss, 198 1;
Chang et al., 1983; Lopez de Ramos and Pironti, 1987; and Tharakan and Chau, 1987.
The study of RFC may provide some useful information for the understanding of mass
transfer processes in radial flow reactors.
Because of the special flow geometry in RFC some complications may arise in
mathematical modeling. Since the linear flow velocity (v) in the RFC column changes
constantly along the radial coordinate of the column (Figure 11. I), unlike in most cases
of AFC, the radial dispersion and external mass transfer coefficients are no longer con-
stants. This important feature was rarely considered in mathematical modelings of RFC
in the literature in the past. Extensive theoretical studies have been reported for single
component ideal RFC, which neglects radial dispersion, intraparticle diffusion, and
external mass transfer resistance. In such studies a local equilibrium and a linear isoth-
erm were often assumed. The earliest theoretical treatment of RFC was done by
Lapidus and Amundson (1950). A similar study was carried out by Rachinskii (1968).
Later Inchin and Rachinskii (1973) included bulk fluid phase molecular diffusion in
their modeling. Lee et al, (1988) proposed a unified approach for moments in chroma-
tography, both AFC and RFC. They used several single component rate models for the
comparison of statistical moments for RFC and AFC. Their models included radial
dispersion, intraparticle diffusion, and external mass transfer effects. Kalinichev and
Zolotarev (1977) also carried out an analytical study on moments for single component
RFC in which they treated the radial dispersion coefficient as a variable is considered.
A rate model for nonlinear single component RFC was solved numerically by Lee
(1989) by using the finite difference and orthogonal collocation. His model considered
radial dispersion, intraparticle diffusion, external mass transfer, and nonlinear isoth-
erms. His model used averaged radial dispersion and mass transfer coefficients instead
of treating them as variables. A nonlinear model of this kind of complexity has no
analytical solution and must be solved numerically.
Rhee et al. (1970) discussed the extension of their multicomponent chromatography
theory for ideal AFC with Langmuir isotherms, which is a parallel treatise to the
interference theory developed by Helfferich and Klein (1970), to RFC. Apart from this,
so far no other detailed theoretical treatment of nonlinear multicomponent RFC is avail-
able in the literature. With the development of powerful computers and efficient
numerical methods, more complicated treatment uf multicompvnent RFC now becomes
possible. A general model for multicomponent RFC can provide some very valuable
information.
In this chapter a numerical procedure is presented for solution to a general rate
model for multicomponent RFC. The model is solved by using the same basic
approached presented in Chapter 5 for AFC models. The solution of the model enables
the discussion of several important issues concerning the characteristics and perfor-
mance of RFC and its differences from AFC. And also the question of whether one
should treat dispersion and mass transfer coefficients as variables.
11.2 General Multicomponent Rate Model For RFC
Consider a fixed bed with cylindrical radial flow geometry (Figure 11.1) which is
filled with uniform spherical porous solid adsorbents. Suppose the process is isother-
mal and there is no concentration gradient in the axial direction of the column.
Although it can be a problem in some real cases, the possible maldistribution of flow
streams is ignored in this theoretical study. Also, local equilibrium is assumed for each
component between the pore surface and the liquid phase in the macropores in particles.
Based on these basic assumptions, the following governing equations for component i
in the bulk fluid and particle phases via mass balances in the two phases can be fomu-
lated.
where in Eq. (11-1) +v is for outward flow and -v for inward flow. Note that in Eq.
(1 1-1) Dbi and ki are variables which are dependent on v.
The initial and boundary conditions are
ac, R=O, - i - ki
aR =O R=Rp,-- -(cbi-qi, R = R ~
aR qDpi
For outward flow
and for inward flow
These equations can be expressed in the following dimensionless forms,
where for bulk phase equation (Eq. (11-9)) local volume averaging method (Slattery,
1981; Lee, 1989) has been used for its nondimensionalization. A comparison of the
definitions of the dimensionless variables and parameters used in AFC and RFC is
listed in Table 11.1.
Table 1 1.1 Comparison of Dimensionless Variables and Parameters
In Eq. (1 1-9) the radial flow Peclet number is defined as Pei = v(X1-XO)/DW. The
introduction of radial flow Peclet number and the use of local volume averaging method
in the transformation, streamlines the analogy and comparison between the RFC model
and the AFC model (Gu et al., 1990a). One may find that the dimensionless RFC rate
model looks very similar to the dimensionless AFC rate model shown in Section 5.1.2,
except that in the RFC model there is an extra variable a and 5 is not a constant. Also,
in RFC, there are two different flow directions, which are reflected by the f sign in Eq.
(1 1-91.
Initial conditions:
7 = O, Cbi=Cbi (0, V), Cpi = Cpi (0, l, V)
Boundary Conditions for outward flow:
For frontal adsorption, Cfi (z)/C& = 1
For elution, Cfi (z)/Coi
After the sample has been introduced:
(in the form of frontal adsorption)
if component i is displaced,
if component i is a displacer,
= i 1 O ~ Z S Z ~ ~ ~
0 else
For inward flow one only needs to swap V = 0 in Eq. (1 1-13) with V = 1 in Eq. (1 1-14).
The general solution strategy for the coupled partial differential equation (PDE)
system (Eqs. (1 1-9) and (1 1-10)) is the same as that for the case of AFC discussed in
Chapter 5. Compared to AFC, the solution for RFC seems to be more complicated
because of variations in some physical properties of the system as mentioned earlier.
The finite element method is apparently an ideal approach for this kind of system.
11.3 Numerical Solution
Eqs. (11-9) and (11-10) are transformed to a set of ODES by the finite element
method and the orthogonal collocation (OC) method (Villadsen and Michelsen, 1978;
Finlay son, 1980), respectively. Using Ihe Galerkin approximation (Reddy, 19841, Eq.
(1 1-9) becomes
where (DBi)Ln = j + m + n d ~ (11-18)
in which m, n = 1, 2, 3, and the superscript e indicates that the finite element matrices
and vectors are evaluated over each individual element before global assembly. In Eq.
(1 1-19), +$,, is for outward flow and 4, for inward flow. The natural boundary con-
dition (PB;) = - chi + Cfi(%)/Qi is applied to [AKB;] and [AFBi] at V=O for outward
flow or (PBi) = chi - Cfi(%)/COi at V=l for inward flow. (PBi) = 0 elsewhere. Note that
in Eq. (1 1-19) a is a function of V.
The particle phase equation can be discretized with N interior collocation points.
The ODE system resulting from the numerical discretization can then be solved by
Gear's stiff method. The procedure is the same as that for AFC which has been shown
in Chapter 5.
In this numerical procedure Dbi and ki values are mated as variables which are
dependent on the variations of v along the radial coordinate V. Meanwhile intraparticle
diffusivities (Dpi) are regarded as independent of the variations of v. In Chapter 7, it
has been shown that for axial dispersion, Db = V. This also applies to RFC. Thus the
Pei can be treated as constants that are independent of v. It has also been shown in
Chapter 7 that
Since ki = vln and
one has
Bii = ki = (l/X)lB = (v+v~)-"~
If Bii,v,l (i.e., Bii,x=x, ) values are given as input values, then
For ti at any V position one has
Eq. (1 1-25) is used for the evaluation of Eq. (1 1-19). In Eq. (1 1-19), due to the special
geometry of radial flow chromatography, there are two space coordinates (V) dependent
variables, a, and ti. The finite element integral in Eq. (11-19) is evaluated for each
local element and a, ti can be dealt with routinely without any trouble, since in this
chapter finite element integrals are evaluated using four point Gauss-Legendre quadra-
tures (Reddy, 1984). The ability to deal with variable physical properties with ease is
one of the well-known advantages of the finite element method. Accuracy is another
notable advantage of the method. The accommodation of variable Bii in Eq. (1 1-22) for
the particle phase is also very easy. Since particle phase equations must be solved at
each finite element node (with given nodal position, V) in the function subroutine, BiiVv
values can be readily obtained from Eq. (1 1-24) and inserted in Eq. (1 1-22).
It is very helpful to study the effects of treating Dbi and ki as variables compared to
treating them as constants, as is the case in most of the existing papers in the literature.
There are two easy ways to take averaged Dbi and ki values for the modeling. In both
cases Pei = v(x1-&))/Dbi will no longer be constant. In all the RFC cases in this
chapter in the input data for simulation Bii,v,l values are given.
The discussion below shows how to modify the algorithm to accommodate cases
using the averaged Dbi and ki values in order to make comparisons.
1 1.3.1 Using v value at V = 0.5 for averaging Dbi and ki
Since Dbi v, from Eq. (1 1-22) it is easy to obtain
Eq. (1 1-24) gives
11.3.2 Using v value at X = (X1+Xo)/2 for averaging Dbi and k;
- pei Pei Dbi
For Bii one has
which gives
If using averaged values of Dbi and ki instead of treating them as v dependent vari-
ables, one needs to use a/& in Eq. (1 1-27) or Eq. (1 1-29) to replace W e i in Eq. (1 1-
19), and use Zi to replace Bi; in Eq. (1 1-23). Bii should also be used to calculate ti.
11.4 Results and Discussion (Gu et al., 1990d)
The multicomponent Langmuir isotherm and the stoichiomemc isotherm are used
for the simulations. Parameter values used in simulations are listed in Table 11.2, or
mentioned during discussions.
1 1.4.1 Simulations of Different Chromatographic Operations
Figure 11.2 shows the simulated breakthrough curves for two components in inward
and outward flow RFC. The corresponding breakthrough curves in AFC are shown for
comparison in Figure 11.2. They were obtained by using the same dimensionless
parameters and isotherms in the RFC case, except that Bil = Biz = 1.1 15 were used.
These two Bi values for AFC are the ones its corresponding RFC possesses at V=0.5.
Figure 11.2 clearly shows that in RFC, inward flow provides sharper concentration
profiles than outward flow. This is in agreement with the results obtained by Lee (1989)
for single component RFC. For a comparison similar to that shown in Figure 11.2, the
simulated effluent history for a step change displacement process is presented in Figure
11.3. In this case the column is presaturated with component 1. Component 2 is intro-
duced via a step change as a displacer to displace the adsorbed molecules of component
1. Again, inward flow RFC offers sharper concentration profiles, which are favorable
for separation. The same conclusion also holds for the two component elution case
shown in Figure 1 1.4.
Figure 11.5 shows the chromatogram for isocratic elution of a binary sample (corn-
ponent 1 and component 2) with a small ion (component 3) as the eluant (component 3)
- - - - - - RFC with Outward Flow ------
I I I I I
0 1 2 3 4 5 6 7
Dimensionless Time
Figure 11.2 Comparison of inward and outward flow RFC and AFC in frontal adsorption.
in inward flow RFC. The separation factors are constants in this case (al3 = 0.6,
~ 1 2 3 = 3.0). This is similar to the binary elutions with a competing modifier in the
mobile phase with Type I sample discussed in Chapter 9 for the case of AFC. The
roll-up of the component 1 peak in Figure 10.5 is due to the volumc overload
(7imP = lo).
Multi-stage operations can also be simulated with our code. Figure 11.6 shows the
effluent history of a reverse flow displacement process, in which the displacer (com-
ponent 2) is introduced with a reversed flow direction after an incomplete period of
frontal adsorption of component 1, which lasted z = 3. Such reverse flow displacement
operation is actually very common in industrial AFC practice (Ruthven, 1984) aimed at
improving process efficiency and reducing column clogging. It has been used in the elu-
tion stage of affinity chromatography by Chase (1985). In Figure 11.6, the combination
of outward flow adsorption and inward flow displacement gives better results, since its
process time is slightly shorter.
11 A.2 Effect of Vo
Vo represents the ratio of the central cavity volume of an RFC column to the bed
volume. The effect of its value on elution process is shown in Figure 11.7, in which the
RFC column's XI and h are fixed while changing the Xo value. In Figure 11.7, the solid
lines are the same as those in Figure 11.4. The way Vo values are changed from 0.04 to
0.1 requires that the Pei values for Vo =0.1 be reduced to 86.88% of those for
Vo = 0.04 case, and for qi values the percentage is 94.54%. These two percentage
values can easily be obtained by checking the changes in (XI-&) and Vb and their
Frontal -- ( Displacement C
Outward Displacement Outward Adsorption,
2
I
0 2 4 6 8 10
Dimensionless Time
Figure 11.6 Effect of RFC flow direction in reverse flow displacement.
relationship with Pei and q;. Figure 11.7 shows that the peak heights are reduced and so
is the peak resolution when Vo is increased from 0.04 to 0.1. This has the same effect as
reducing column length in AFC.
1 1.4.3 Effects of Pei, qi and Bii on elution
Figures 1 1.8 to 11.10 clearly show that the increase of Pei, qi, or Bii value increases
the peak heights and the peak resolution. In these t h e figures the solid line curves are
the same as those in Figure 11.4. In Figure 11.8 Pel = Pe2 = - case is plotted in order
to show the errors that one may encounter if radial dispersion is neglected. In the case
shown in Figure 11.8 such errors are quite large. Figure 11.9 also indicates that the
peaks are sharper if film mass transfer coefficients for the two components are larger
since the dimensionless parameter qi is proportional to ki. Figure 11.1 1 shows that the
resolution of the elution peaks is decreased if the intraparticle diffusion coefficients are
reduced as reflected in the dimensionless parameters, qi and Bii.
11.4.4 Effect of treating Dbi and ki as variables
The two component frontal adsorption system shown in Figure 11.2 is chosen as a
case study. Figure 1 1.12 shows three sets of inward flow breakthrough curves obtained
from using the variable Dbi and ki average values evaluated at V=0.5, or (Xl+Xo)/2,
respectively. These three sets of curves show some differences in the sharpness and
height of the "roll-up" peak. The corresponding outward flow case is given in Figure
1 1.13. In Figures 1 1.12 and 1 1.13, it is obvious that there are some differences among
the three sets of curves in each figure caused by the way Dbi and ki are treated. The
average Dbi and ki evaluated at (X1+Xo)/2 are higher than those evaluated at V=0.5
since (X1+Xo)/2 is closer to the center of the radial flow column than V=0.5, and v is
higher at (X1+Xo)/2. Higher Dbi values give lower Ei values and the concentration
profiles tend to be mort: diffused, while higher ki values give higher Bii values and the
concentration profiles tend to be sharper. Between these two compromising factors, the
Dbi factor is more dominant than the ki factor, since the dependence of ki on the coordi-
nate X or V is much weaker.
It is well known in mass transfer studies that both Peclet and Biot numbers show
some asymptotic behavior. When Pei values become larger, the system becomes less
sensitive to the changes in Pei values. When Bii values are in the range of above 1, the
higher the Bii values the less sensitive the system to the increase of Bii values. When
Bii values are sufficiently small, the internal concentration gradient inside a particle can
be neglected. On the other hand, when the Bii values are sufficiently large, the external
interface mass transfer resistances become negligible. These arguments are very help-
ful in determining the parameter ranges in which the treatment of Dbi and ki as vari-
ables become important. Generally speaking, when the Pei values are large, the errors
caused by using averaged Dbi values are small. If Bii values are not close to 1, then the
treatment of ki values as variables will have little effect. In most cases such averaging
treatment causes some error, but they are not terribly severe. If an averaging treatment
is necessary, such as in some analytical treatments in which simplification may be
essential, one should not be inhibited from doing so. A good averaging method obvi-
ously can reduce error. It is difficult to provide exact general rules for averaging Dbi
and ki values, since the system is quite complex. From Figures 1 1.12 and 1 1.13, and
other extensive simulations, including different RFC operation and different flow direc-
tions, it is found that the averaging point should be at an X position farther away from
Xo than (X1+Xo)/2. V = 0.5 sometimes proves to be too close to XI as an averaging
point, while at other times it becomes too far away, but generally speaking it is a better
choice than (X1+Xo)/2. For numerical solutions it is desirable to treat Dbi and ki as
variable, since it is not only possible but also quite convenient if a suitable numerical
procedure, such as the one presented in this chapter, is used.
1 1.4.5 Comparison of RFC and AFC
If the radial dispersion tern in Eq. (1 1-9) is neglected, and ki values are treated as
constants independent of the variation of v, i.e., ti and Bii are treated as constants, then
the RFC's dimensionless PDE system (Eqs. (1 1-9) and (1 1-10)) become the same as the
corresponding AFC's dimensionless PDE system presented by Gu et al. (1990a).
Inward and outward flow difference will also disappear. This can be easily verified by
using a coordinate transformation with V* = 1 - V. This conclusion is, of course, valid
for more simplified cases, such as the ideal RFC which neglects radial dispersion, exter-
nal film mass transfer, and intraparticle diffusion. Rhee et al. (1970) pointed out that
system equations for ideal radial flow chromatography can be transformed to the system
equations for ideal AFC with Langmuir isotherms. A similar conclusion was also
reached by Rice (1982) and Huang et al. (1988).
The effects of radial dispersion on elution has already been shown in Figure 11.9. It
is often very important to account for radial dispersion in the modeling of RFC. The a
values in the radial dispersion term (Eq. (11-9)) are in the neighborhood of 1. For
vo = 0.04, one has 0.328 5 a 2 1.672. Comparing the definition of radial flow Peclet
number in RFC with the axial flow Peclet number in AFC, it is reasonable to say that
their ratio is close to (XI-&)/L. Obviously, Peclet numbers for radial flow in RFC are
much smaller than those for axial flow in AFC. A ratio between 1:20 to 1:5 should not
be uncommon. Furthermore, RFC is designed primarily for preparative and production
scale separation; thus Peclet numbers for radial flow tend to be even smaller. Typical
values may often be below or not very far above 100. This is in agreement with the esti-
mation by Tharakan and Chau (1986) in their study of a radial flow bio-reactor for
mammalian cell culture. In preparative and large scale AFC Peclet numbers often
reach hundreds or higher and neglecting axial dispersion (i.e., assuming Peclet numbers
equal to infinity) often does not give large errors. For RFC, one may not be able to
make such assumptions without risking substantial errors. Thus, one should be very
cautious when assuming negligible radial dispersion in RFC. In the discussion above it
has been mentioned that, in RFC, radial dispersion coefficients are reversely propor-
tional to the radial coordinate R. This is actually a very important identity of RFC aris-
ing from its special flow geometry, which also differentiates RFC from AFC in terms of
dimensionless mathematical expressions. Neglecting radial disperson in RFC means
that its identity in mathematical modeling is partially lost.
As already shown in Figures 11.3 to 11.5, if one sets the corresponding dimension-
less parameters and isotherm expressions the same for both RFC and AFC, their differ-
ences in simulated effluent histories are similar. In order to have similar dimensionless
constants for an AFC and an RFC, the AFC column should be a short one. Note that
the dimensionless parameter Vo is unique in RFC, and for the corresponding AFC one
needs to pick the Bii values from the variable Bii values of RFC. Because of this kind
of close analogy, some data obtained from a short AFC column may be used for refer-
ence in RFC. This also helps in transforming an existing AFC setup to RFC. It is also
obvious that many studies for AFC, such as multicomponent interference, can be quali-
tatively applied to RFC.
The difference between RFC and AFC may arise from the differences in the dimen-
sionless parameters, especially Pei and qi. In reality, Pe; and qi values for RFC
columns are usually several times smaller than in a longer AFC column. Notice that vi
values are proportional to the "dead volume time" of the bed and this time value for
RFC is often several times smaller than in AFC because of the shorter flow path in
RFC. Figures 11.9 and 11.10 show that both Pei and qi values are very important to the
sharpness of the concentration profiles and peak resolutions. The higher the Pei and qi
values the better resolution. Thus, generally speaking, RFC provides lower resolution
than does AFC. This is why RFC is not intended for analytical purposes. The effect
arising from the difference of Bii values in RFC and AFC is not discussed here because
of some uncertainties. Bii values for RFC can be either higher or lower than those in
AFC depending on valuc ranges of v in RFC and AFC. RFC usually has a higher
volumetric flow rate, but it does not necessarily have larger linear flow velocities since
its cross-sectional flow area is much larger than that of AFC's.
The most important advantage of RFC over AFC with longer column is, again, that
in RFC the cross-sectional area perpendicular to flow direction is very large and the
flow path is relatively short. These two factors help reduce the pressure drop in the bed
and permit a much higher flow rate, and thus promote productivity.
11.5 Extensions of the General RFC Model
All the extensions to the basic general multicomponent rate model for AFC in Sec-
tion 5.1.2 have also beer? applied to the RFC model. Such extensions include second
order kinetics, size exclusion effect and reaction in the liquid phase for modeling of
biospecific elution using soluble ligand. These extensions have been carried out with
ease. Details are omitted here, since the necessary modifications for adding second
order kinetics involving only the particle phase governing equation, in which the AFC
and the RFC model do not differ except that the ki values in RFC are variables. The
adding of reaction terms for macromolecule and soluble ligand interaction involves the
bulk fluid phase, but it does not touch the characteristic terms of RFC model. It has
also been easily implemented. Figure 11.14 shows a simulated chromatogram of an
affinity separation process with frontal adsorption, wash, and elution stages. A
manuscript (Gu et al., 1990h) currently in preparation uses the extended RFC model for
a study of affinity chromatography.
The methodology presented in this chapter has also been used to solve the general
model for spherical flow radial flow chromatography (Gu et al., 1990f).
11.6 Summary
A general nonlinear multicomponent rate model for RFC has been developed.
Radial dispersion and mass transfer coefficients are treated as variables in the model.
The model is solved numerically by using finite element and orthogonal collocation
*~rontal+l 1- Elution with Soluble Ligand
Complex
. Wash .. . - (14 - 15) . . - . . . . . . . . . . . . - . - C I _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - - - - - - - - - - - - - - . - . . - . . C C - - C . . . . )
e Soluble Ligand . ' * . . : # f . # : r : I : I : r : I : I :I I I I I I I I
-. *. .. .. ... . . ...... '"" I - - ~ . . . . . . . ~ . . . . , , ~ , , J , ~ ~ . ~ ~ ~ . ~ J . . ~ ~ I I I
I Inward Flow RFC (
20 25 30 35 40
DIMENSIONLESS TIME
Figure 1 1.14 Affinity RFC with inward flow.
methods for the discretizations of bulk fluid and particle phase PDEs, respectively.
Various chromatographic operations have been simulated. The diffusional and mass
transfer effects on RFC elution process have been studied. It has been found that for
simple chromatographic operations inward flow is generally better than outward flow in
RFC, since inward flow generally provides sharper concentration profiles. The treat-
ment of radial dispersion and mass transfer coefficients by using averaged values
instead treating them as variables causes some errors and such error may be reduced by
properly taking the averages.
In numerical calculations treating the radial dispersion and mass transfer
coefficients as variables adds very little complexity if the finite element method is used
for the discretization of the bulk phase governing equation. It has been found that,
unlike in AFC, dispersion in the flow direction is often very important in RFC. The
dynamic RFC behavior is similar to that of AFC with a shorter column, which has low
Pei and Ti in non-dimensional analysis values. The theoretical treatment of RFC with
comparison to AFC provides some useful information and it is helpful for the scale-up
of RFC, either from a smaller scale RFC or from AFC to RFC. The general model has
also been extended to include second order kinetics, size exclusion effect and reaction
in the liquid phase for modeling of biospecific elution using soluble ligand. The numer-
ical procedure presented in this chapter also serves as an example of how to deal with
fixed bed problems involving variable physical properties.
* Table 11.2 Parameter Values used for Simulation in Chapter 11
In all runs, ep = 0.4 and &b = 0.4, and Vo = 0.04 or otherwise as specified. For elution cases, sample size is Tig = 0.2. For Figure 11.5, a13 = 0.6, az = 3, Col = CO2 = 0.1CO3 = 0.04N and C = 1.8 meq/ml bed. The cm, Gi/Cal, Daa, ad and FIX values for Figure 11.14 are the same as those for Figure 10.12 in Chapter 10, which are listed in Table 10.1. The ODE solver's (IMSL's IVPAG subrou- tine) tolerance is t o l = l ~ - ~ . Double precision is used in dle Fumn code. CPU times on SUB 41280 computer for some simulations are: Figure 11.2 (solid lines), 4.7 min; Figure 11.3 (solid lines), 0.9 min; Figure 11.4 (solid lines), 4.5 min; Fig- ure 11.14, 14.4 rnin.
CHAPTER 12 - CONCLUSIONS AND RECOMMENDATIONS
12.1 Syntheses of Cyclodextrin-Containing Resins
A novel method has been developed to synthesize cyclodextrin-containing polym-
eric resins. This method uses a cross-linking agent to copolymerize directly with cyclo-
dextrin molecules. The resins synthesized with this method contain a very high content
of cyclodextrins, and possess quite good physical properties in terms of insolubility,
hardness and wettability. The P-cyclodextrin resin synthesized with the novel method
has a density of about 1.3 jz/cm3 and a very small swelling rate in water and organic
solvent. Although the resins are non-sp herical ground particles, their hydrodynamic
performance is still quite good.
This method can be further improved by finding a better way for the post-treatment
to get rid of the remaining impurities. Suspension synthesis, if possible, can produce
spherical resin particles. Unfortunately, the yield of suspension synthesis may be con-
siderably lower than that of the established method.
It seems that cyclodextrin-containing polymeric resins have a good prospect for
applications in separations, especially those which are used in areas such as p h m d -
ceutical and biochemical industries. The experiments in this work show that such resins
may be applied to separation systems which do not contain large amounts of contarn-
inants and the target adsorbate does not have a very strong tendency to form inclusion
complexs with cyclodexmns on the resin. Otherwise, contamination and competition
for adsorption will make the separation process unfeasible.
12.2 Studies of Nonlinear Chromatographic Theories
In this work, a general multicomponent rate model has been solved numerically
with an efficient and robust procedure. Various studies in many different areas of
modem nonlinear chromatographic theories have been carried out.
In Chapter 5, an extended Langmuir isotherm is derived for adsorption systems with
uneven saturation capacities which are either physically induced or chemically induccd.
Using an extended binary Langmuir isotherm, the isotherm cross-over and the peak
reversal phenomenon has been successfully demonstrated under conditions that do not
involve large differences in concentrations or adsorption equilibrium constants of the
two components. The extended isotherm may serve as a working model for correla-
tions of experimental isotherm data for systems with uneven saturation capacities.
Experimental verification of the isotherm model is strongly recommended.
The discussion on stepwise displacement processes in Chapter 6 has shown that the
displacer's affinity does not have to be stronger than that of the displaced component
unlike the conventiunal displacement development.
The effects of mass transfer coefficients and flow rates on the performance of vari-
ous chromatographic operations have been demonstrated via computer simulation in
Chapter 7. The influence of the dimensionless parameters used in the general multicom-
ponent rate model has also been investigated. Section 7.3 points out that the elution
peak will show the common Langmuir asymmetry for a single component elution with
an unfavorable isotherm if the intraparticle diffusion rate is low or the flow rate is
sufficiently high. This interesting theoretical finding needs actual experimental
verification, which can be easily carried out.
The approach presented in Chapter 8 for the interpretation of the multicomponent
interference effect based on the concept of a displacement effect is simple and easy to
understand. The computer simulation in the Chapter helps the visualization of the
interference effect under mass transfer conditions.
The interaction between the competing modifier in the mobile phase and the sample
solutes has been studied based on the general rate model in Chapter 9. The patterns of
system peaks summarized in Table 9.2 helps the interpretation of peak sources in
chromatograms involving system peaks.
The general multicomponent rate model has been modified to account for the reac-
tion between soluble ligands and macromolecules in affinity chromatography. Various
aspects of affinity chromatography have been discussed in Chapter 10 including the
simulation of frontal adsorption, wash, and elution stages. Two types of inhibition have
been discussed.
A general rate model for multicomponent radial flow chromatography (RFC) has
also been solved in Chapter 11 based on an accurate numerical procedure in which the
radal dispersion and film mass transfer coefficients are treated as variables which
depend on the radial coordinate of the RFC column. RFC is often used for affinity
c~mmtography. Affinity RFC has only been briefly discussed in Chapter 11. A
study of affinity RFC is recommended as future work for which a manuscript is
currently in preparation.
Quite a few Fortran codes have been developed based on the numerical procedures
described in this work. Inquiries regarding the availability of the computer codes
should be addressed directly to the candidate's academic advisor, Prof. George T. Tsao.
BIBLIOGRAPHY
BIBLIOGRAPHY
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VITA
VITA
Tingyue Gu was born on February 19, 1963 in Zhenjiang, Jiangsu Province, China.
He obtained his B.S. degree in Chemical Engineering from Zhejiang University, Hang-
zhou, China in June 1985. He then attended the Beijing Language Institute in Beijing
between August 1985 and May 1986.
In August 1986, he joined Purdue University as a graduate student in the School of
Chemical Engineering under the guidance of Professor George T. Tsao. He bypassed
his M.S. degree in December 1988, and is expected to receive his Ph.D. degree in
Chemical Engineering in August 1990. During his stay at Purdue University, he served
as a teaching assistant and a graduate research assistant in the School of Chemical
Engineering and the Laboratory of Renewable Resources Engineering (LORRE).
He is a member of AIChE and ACS. His research interests include bioseparations,
mass transfer processes, numerical computation for large scale partial differential equa-
tion systems and computer simulation.