Modeling, Analysis, and Simulation of Adsorption in Functionalized Memebranes

239 Pages·2016·2.74 MB·English

by  Anastasia Bridner Wilson

Clemson University
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Modeling, Analysis, and Simulation of Adsorption
in Functionalized Memebranes
Anastasia Bridner Wilson
Clemson University
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Modeling, Analysis, and Simulation of Adsorption in
Functionalized Membranes
A Dissertation
Presented to
the Graduate School of
Clemson University
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Mathematical Sciences
by
Anastasia Bridner Wilson
August 2016
Accepted by:
Dr. Lea Jenkins, Committee Chair
Dr. Chris Cox
Dr. Vince Ervin
Dr. Scott Husson
Abstract
The emergence of biopharmaceuticals, and particularly therapeutic proteins, as a
leading way to manage chronic diseases in humans has created a need for technologies that
deliver purified products efficiently and quickly. Towards this end, there has been a signif-
icant amount of research on development of porous membranes used in chromatographic
bioseparations. In this work, we focus on high-capacity multimodal membranes developed
by Husson and colleagues in the Department of Chemical and Biomolecular Engineering at
Clemson University.
Chromatographic performance of such membranes, particularly the adsorption ca-
pabilities of the membranes, depends of a large number of variables making it unrealistic to
scan the available options and determine the conditions resulting in the best performance
experimentally. Consequently, the goal of this work is to develop a modeling framework ca-
pable of describing the process under continuous flow conditions and software tools capable
of simulating the protein chromatography process under the effect of complex adsorption
relationships.
In this work, we consider the reactive transport, or advection-diffusion-reaction,
problem to model the chromatography process. We focus on the case of highly advective
flows as one of the advantages of using membranes in chromatography is the capacity to
maintain high protein binding capacity at high flow rates. Toward this end, we utilize a
streamline upwind Petrov-Galerkin (SUPG) finite element method to numerically solve the
advection-dominated advection-diffusion-reaction equation for porous media.
The complicating feature of the problem arises from modeling the adsorption reac-
ii
tion. The most accurate, thermodynamically consistent model, or isotherm, for multimodal
adsorption, recently developed by Nfor and colleagues, is highly nonlinear and implicitly
defined. Even the next best model, Langmuir’s isotherm, while not implicitly defined is still
nonlinear. As such we develop and analyze discretization methods incorporating nonlinear,
potentially implicit, adsorption isotherm models.
Togaininsightintothe advection-diffusion-reactionproblem, webeginby analyzing
the SUPG formulation for the steady state case of the advection-diffusion equation. We
also analyze the time-dependent linear cases incorporating constant and linear adsorption
models. Although the constant and linear adsorption models do not represent realistic
adsorption relationships, the linear analysis serves as a template for the nonlinear analysis.
When incorporating nonlinear adsorption, we consider two cases: adsorption with
an explicit representation as in Langmuir’s isotherm and adsorption with an implicit equa-
tion as in Nfor’s isotherm. In the case of an explicit adsorption relationship, three different
formulations are analyzed: a time-integrated mixed methods formulation, a time-integrated
SUPGformulation,andafullyimplicitSUPGformulation. Fortheimplicitadsorptionrela-
tionship, a simple formulation is proposed which not only deals with the implicit definition
of the isotherm but also deals with the nonlinearity: the right hand side of the isotherm
relationshipisevaluatedattheprevioustimestep. Asexpected,thesolvabilityandstability
for this relationship are all shown to have a requirement on the time step size.
We provide numerical validation for each of the a priori error estimates. We also
compare results of our algorithm with data obtained from laboratory experiments. To im-
prove the accuracy of the numerical simulations, we incorporate non-instantaneous adsorp-
tion,consideringbothconstantandtransientadsorptionrates. Additionally,wenumerically
investigate the effects of varying velocity profiles by comparing results from simulations in-
volving five different profiles.
iii
Dedication
To my husband, Dustin, without whose continuous support over the years this work
would never have been finished.
iv
Acknowledgments
I would like to thank Dr. Lea Jenkins, my Ph.D. dissertation advisor, for the
extensive amount of time she has spent with me over the years helping me learn about
modeling and fluids in porous media, making me a better programmer, teaching me how to
write papers and present my research, thoroughly editing everything I wrote with her, and
advising me when I was looking for employment after graduation. Additionally I would like
to thank Dr. Scott Husson and Juan Wang for providing numerous experimental data sets
for comparison over the years and for answering my seemingly unending questions about
separations processes and membrane chromatography. I would also like to thank Dr. Vince
Ervin for all the help he has provided in checking the analysis portions of my research and
Dr. Chris Cox for serving on my committee with Drs. Husson and Ervin and providing
ideas for future collaborative research projects.
v
Table of Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Review of Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 The Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 The Physics Involved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 The Reactive Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . 40
3 SUPG Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 SUPG Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Finite Element Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Finite Element Approximation & Assumptions . . . . . . . . . . . . . . . . 55
4.2 Boundedness and Coercivity of Bilinear Form . . . . . . . . . . . . . . . . . 56
4.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Time-Dependent Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Case 1: Constant Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Case 2: Linear Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Analysis for Nonlinear, Explicit Adsorption . . . . . . . . . . . . . . . . . 123
6.1 Time-Integrated Mixed Method Formulation . . . . . . . . . . . . . . . . . 123
vi
6.2 Time-Integrated, SUPG Formulation . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Fully Implicit SUPG Formulation . . . . . . . . . . . . . . . . . . . . . . . . 154
7 Nonlinear Analysis with Implicit Adsorption . . . . . . . . . . . . . . . . 168
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2 Solvability and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.1 Steady-State Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 Linear Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.3 Nonlinear Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
vii
List of Tables
8.1 Approximation errors and experimental convergence rates for the approxi-
mation to the steady-state problem. As ∆t and h are cut in half, the H1
error is reduced the same amount which is consistent with the theoretical
convergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2 Approximationerrorsandexperimentalconvergenceratesforthefully-explicit
approximation with a linear adsorption model. As ∆t and h are cut in half,
the H1 error is reduced the same amount which is consistent with the theo-
retical convergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3 Approximationerrorsandexperimentalconvergenceratesforthefully-implicit
approximation with a linear adsorption model. As ∆t and h are cut in half,
the H2 error is reduced the same amount which is consistent with the theo-
retical convergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.4 Time-integrated approximation errors and convergence rates for nonlinear
adsorption with Langmuir adsorption model. As ∆t and h are cut in half,
the time-integrated error is reduced the same amount which is consistent
with the theoretical convergence rates. . . . . . . . . . . . . . . . . . . . . . 186
8.5 Approximation errors and convergence rates for nonlinear adsorption with
Langmuir adsorption model. As ∆t and h are cut in half, the H1 error is
reducedthesameamountwhichisconsistentwiththetheoreticalconvergence
rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.6 Approximation errors and convergence rates for nonlinear adsorption with
Nfor adsorption model. As ∆t and h are cut in half, the H1 error for C and
the L2 error for q is reduced the same amount which is consistent with the
theoretical convergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 189
viii
List of Figures
1.1 Theproteinchromatographyprocess: thecolumnmaterialsseparatethepro-
teins as the solution is pushed through the column. . . . . . . . . . . . . . . 2
1.2 An electron microscopy image of a membrane used in membrane chromatog-
raphy. Note the porous structure of the membrane that provides many ben-
efits for chromatography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
8.1 Comparison of numerical and experimental breakthrough curves using in-
stantaneousadsorption. NotethetypicalS-shapecharacteristicandtheclose
match of breakthrough for both curves. . . . . . . . . . . . . . . . . . . . . 194
8.2 Effect of varying the mass transfer coefficient k on breakthrough curve.
m
Notice as k increases, the breakthrough curves asymptotically approach
m
the results with instantaneous adsorption as shown in Figure 8.1. . . . . . . 197
8.3 Comparison of transient adsorption rate with constant adsorption rates. . . 198
8.4 Comparisonofnumericalandexperimentalbreakthroughcurvesusinginstan-
taneous adsorption and non-instantaneous adsorption. Note the numerical
result obtained using a transient adsorption rate are significantly closer to
the experimental results than those obtained using instantaneous adsorption. 199
8.5 The2Devolutionoftheconcentrationinthemembraneassumingaspatially-
constant velocity profile. Notice that the membrane is essentially saturated
with protein by t = 20 as shown in the last image. . . . . . . . . . . . . . . 204
8.6 The 2D evolution of the concentration in the membrane assuming a single
parabola velocity profile. Saturation of the membrane does not occur in this
case until after the final image shown; specifically, it occurs at approximately
t = 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.7 The 2D evolution of the concentration in the membrane assuming a double
parabola velocity profile. Saturation of the membrane does not occur in this
case until after the final image shown; specifically, it occurs at approximately
t = 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.8 The 3D evolution of the concentration in the membrane assuming a single
parabola velocity profile. Saturation of the membrane does not occur in this
case until after the final image shown; specifically, it occurs at approximately
t = 72.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.9 The 3D evolution of the concentration in the membrane assuming a velocity
profile involving five parabolas. Saturation of the membrane does not oc-
cur in this case until after the final image shown; specifically, it occurs at
approximately t = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
ix