Clemson University TigerPrints All Dissertations Dissertations 8-2016 Modeling, Analysis, and Simulation of Adsorption in Functionalized Memebranes Anastasia Bridner Wilson Clemson University Follow this and additional works at: hps://tigerprints.clemson.edu/all_dissertations is Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Recommended Citation Wilson, Anastasia Bridner, "Modeling, Analysis, and Simulation of Adsorption in Functionalized Memebranes" (2016). All Dissertations. 1687. hps://tigerprints.clemson.edu/all_dissertations/1687
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Clemson UniversityTigerPrints
All Dissertations Dissertations
8-2016
Modeling, Analysis, and Simulation of Adsorptionin Functionalized MemebranesAnastasia Bridner WilsonClemson University
Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations
This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations byan authorized administrator of TigerPrints. For more information, please contact [email protected].
Recommended CitationWilson, Anastasia Bridner, "Modeling, Analysis, and Simulation of Adsorption in Functionalized Memebranes" (2016). AllDissertations. 1687.https://tigerprints.clemson.edu/all_dissertations/1687
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 theoreticalconvergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2 Approximation errors and experimental convergence rates for the fully-explicitapproximation 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 Approximation errors and experimental convergence rates for the fully-implicitapproximation 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 nonlinearadsorption with Langmuir adsorption model. As ∆t and h are cut in half,the time-integrated error is reduced the same amount which is consistentwith the theoretical convergence rates. . . . . . . . . . . . . . . . . . . . . . 186
8.5 Approximation errors and convergence rates for nonlinear adsorption withLangmuir adsorption model. As ∆t and h are cut in half, the H1 error isreduced the same amount which is consistent with the theoretical convergencerates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.6 Approximation errors and convergence rates for nonlinear adsorption withNfor adsorption model. As ∆t and h are cut in half, the H1 error for C andthe L2 error for q is reduced the same amount which is consistent with thetheoretical convergence rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 189
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List of Figures
1.1 The protein chromatography process: the column materials separate the pro-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-stantaneous adsorption. Note the typical S-shape characteristic and the closematch of breakthrough for both curves. . . . . . . . . . . . . . . . . . . . . 194
8.2 Effect of varying the mass transfer coefficient km on breakthrough curve.Notice as km increases, the breakthrough curves asymptotically approachthe results with instantaneous adsorption as shown in Figure 8.1. . . . . . . 197
8.3 Comparison of transient adsorption rate with constant adsorption rates. . . 1988.4 Comparison of numerical and experimental breakthrough curves using instan-
taneous adsorption and non-instantaneous adsorption. Note the numericalresult obtained using a transient adsorption rate are significantly closer tothe experimental results than those obtained using instantaneous adsorption. 199
8.5 The 2D evolution of the concentration in the membrane assuming a spatially-constant velocity profile. Notice that the membrane is essentially saturatedwith protein by t = 20 as shown in the last image. . . . . . . . . . . . . . . 204
8.6 The 2D evolution of the concentration in the membrane assuming a singleparabola velocity profile. Saturation of the membrane does not occur in thiscase until after the final image shown; specifically, it occurs at approximatelyt = 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.7 The 2D evolution of the concentration in the membrane assuming a doubleparabola velocity profile. Saturation of the membrane does not occur in thiscase until after the final image shown; specifically, it occurs at approximatelyt = 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.8 The 3D evolution of the concentration in the membrane assuming a singleparabola velocity profile. Saturation of the membrane does not occur in thiscase until after the final image shown; specifically, it occurs at approximatelyt = 72.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.9 The 3D evolution of the concentration in the membrane assuming a velocityprofile involving five parabolas. Saturation of the membrane does not oc-cur in this case until after the final image shown; specifically, it occurs atapproximately t = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
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8.10 Comparison of breakthrough curves for five different velocity profiles. Al-though some of the velocity profiles resulted in remarkably similar break-through curves (e.g. single and double parabola cases in 2D), there is enoughvariation in the curves to indicate further investigation into the velocity ifwarranted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
x
Chapter 1
Introduction
Biopharmaceuticals, and particularly therapeutic proteins and cells, are in high
demand for treating many severe and chronic illnesses such as diabetes, cancer, hemophilia,
anemia, and infectious diseases [101]. Currently, the U.S. Food and Drug Administration
has approved about 130 therapeutic proteins for use, and over 500 products are being
developed, 200 of which are cell therapeutics for transplant medicine, cancer treatment,
and aids therapy [1]. The global market for biophameceuticals is projected to exceed $142
billion in 2017 while the protein therapeutics market specifically is expected to grow 15%
annually through the next decade [72]. By the end of 2016, 50% of the top 100 drugs will
most likely be biologics, that is, drugs derived from biotechnology [44].
While the increase in product development is immensely beneficial to society, there
are significant challenges that must be addressed to ensure the supply meets the needs of
the population. In particular, manufacturers are striving to develop new production meth-
ods to increase production capacity while also minimizing the cost of production. Although
building new biomanufacturing facilities is an option for integrating new production meth-
ods, the significant financial risk of building and validating the new facilities often outweigh
the benefits of the new method. Consequently, technologies that could be integrated into
current biomanufacturing facilities would have the most economic benefit.
A typical biomanufacturing facility comprises upstream and downstream processes.
1
In the upstream process, biopharmaceuticals are produced by cell culture technologies us-
ing genetically engineered cells that excrete the product of interest into a solution. The
downstream process is designed to recover and purify the product from solution.
Downstream purification, normally including chromatography, often limits the pro-
duction capacity. In capture-step protein chromatography, the protein serum is passed
through a column containing materials that separate the target proteins (see Figure 1.1).
Once the materials in the column are saturated with protein, breakthrough occurs. Break-
through is evidenced by the outflow from the column containing a “significant” amount of
protein, for example 10% of the value of the feed concentration appears in the effluent of
the column for 10% breakthrough. After breakthrough occurs, an elution solution is pushed
through to remove the protein from the column.
Figure 1.1: The protein chromatography process: the column materials separate the proteinsas the solution is pushed through the column.
Current practice utilizes resin chromatography in several steps. In resin-based chro-
matography, small resin beads are used to isolate and purify the product or to remove
process impurities from the product solution usually through ion-exchange and diffusion-
based processes. Although reliable, it has low throughput, characterized by long processing
times that cause product degradation, with losses as high as 50% of the product manufac-
tured [139]. Its low efficiency is considered a “must be addressed” factor to avoid production
capacity constraints and decrease production cost in biopharmaceutical industries [99].
2
To address these challenges, membrane chromatography has been studied as an
alternative to resin chromatography [12, 13, 14, 33, 53, 107, 148, 149, 154, 158, 175, 176],
particularly now that strategies have been developed by Husson and colleagues and others
to increase the binding capacities of membrane adsorbers to values that meet or exceed
Figure 1.2: An electron microscopy image of a membrane used in membrane chromatog-raphy. Note the porous structure of the membrane that provides many benefits for chro-matography.
Protein binding capacity is a key performance metric in membrane chromatography
as it determines the volume of membrane needed for purification. Currently, commercial
membranes are limited to products in which protein capture is accomplished primarily by
one adsorption mechanism, e.g., ion exchange, hydrophobic interactions. Most ion-exchange
membranes lose protein binding capacity at relatively low conductivity and do a poor job of
removing unwanted protein aggregates. Thus, additional processing steps often are required
to reach high purity causing lower yield and higher costs [89, 170]. Recent research efforts
by Husson and colleagues have focused on multimodal membrane-based chromatography, in
which protein capture and aggregate removal are enhanced by coupling orthogonal mech-
anisms for adsorption [166]. Initial work produced multimodal membranes characterized
by high protein binding capacities at ionic strengths and used a thermodynamic model to
3
provide insights on the nature of protein-membrane interactions under different test condi-
tions.
Chromatographic performance by multimodal membranes depends on membrane
properties, protein properties, and operating conditions, e.g., protein concentration, pH,
ionic strength, salt type, flow rate. The number of variables makes it unrealistic to scan the
available options experimentally and discover the conditions that would result in the best
performance. Thus, to limit the number of experiments needed for process development,
it is imperative to develop a modeling framework capable of describing the process under
continuous flow conditions.
To model this process and thereby create a simulation tool, we consider the reac-
tive transport problem (advection-diffusion-reaction problem) given by the coupled set of
equations [154, 50, 5, 156]
ω∂C
∂t+ ρs (1− ω)
∂q
∂t+∇ · (uC)−∇ · (D∇C) = f
q = g (q, C)
where C is the liquid phase concentration of the material of interest, q is the solid phase
concentration of the material of interest, ω is the porosity of the membrane (0 ≤ ω ≤ 1),
ρs is the density of the membrane, u is the Darcy velocity, and D represents the diffusivity
of the fluid through the membrane. A more detailed description of the modeling equations
and the involved physics is given in Chapter 2.
The solid phase concentration is the concentration of material adsorbed onto the
membrane. The equation q = g (q, C) describes the (usually explicit and nonlinear) func-
tional relationship between the adsorptive properties of the membrane and the liquid phase
concentration C. Notable example relationships include the Langmuir and Freundlich
isotherm functions. However, recent developments in the understanding of these adsorp-
tion processes has led to newly defined, implicit representations of q as a function of C
[114, 116, 120, 167]. (See section 2.1.4 for more details on the different adsorption models.)
4
Etzel and colleagues were some of the first to mathematically model membrane
chromatography [154, 175]. In [154], Suen and Etzel first analyzed protein separation in
membranes due to affinity using a Langmuir model for adsorption. Affinity separations rely
on a very specific chemical reaction that causes the protein to bind to the membranes, or
more specifically to the ligands in the coating on the membranes. Etzel and colleagues were
able to use analytical solutions to their model to determine the effects of flow velocity and
membrane thickness and porosity on the performance of the membrane.
As affinity membranes use just one type of reaction to separate proteins, Yang, Bitzer
and Etzel continued analyzing membranes in the context of ion-exchange chromatography in
[175]. The authors investigated the performance of ion-exchange membranes by comparing
numerical results from different isotherm models to experimental results from literature.
Fitting the numerical solution to experimental results provided a means of determining
the phenomena that most affected the mass transfer rate and predicting the operating and
membrane-design parameters needed for sharp breakthrough curves.
The two most common isotherm models used in the literature are Langmuir’s and
Freundlich’s isotherms which were both proposed in the early nineteenth century. However,
both models have their disadvantages especially when it comes to predicting multimodal
adsorption. Additionally, neither model is able to accurately predict adsorption relation-
ships for varying salt concentrations: if the salt concentration of the solution changes, then
the fitted isotherm parameter values also change.
Consequently, much research and experimentation has been conducted in the past
decade to develop isotherm models capable of accurately predicting multimodal adsorption
under varying salt concentrations. In [67], Ghose et al. explored protein separations based on
hydrophobic tendencies of proteins in a liquid. Based on experimental results, the authors
developed an isotherm model to better predict the relationship between the amount of
protein in the liquid and the amount of protein bound to the membrane; the isotherm
model that was developed is an exponentially modified version of Langmuir’s isotherm.
While this isotherm accurately simulated adsorption in a range of circumstances, it was
5
developed in the context of one mode of adsorption.
In [65], Gao et al. investigated mixed-mode protein adsorption focusing on the effect
of salt concentration and pH levels. Based on experimental data, they proposed an empir-
ical modification to Langmuir’s isotherm which was similar to the model proposed in [67].
Although accurate in its predictions, the proposed isotherm was not thermodynamically
consistent.
Understanding the importance of having an accurate, thermodynamically-consistent
mixed-mode adsorption model, Mollerup and colleagues developed an adsorption model
based on thermodynamic principles [112, 113, 114, 115, 116]. In [112], Mollerup laid the
foundation of thermodynamic principles in the context of biopharmaceutical development,
specifically the area of applied thermodynamics of chromatography. Mollerup later discussed
thermodynamic principles of ligand binding in order to develop models for multi-component
adsorption due to ion-exchange and hydrophobic chromatography [113]. In [115], Mollerup
et al. focused specifically on modeling adsorption due to ion-exchange, describing a strategy
for estimating the parameters in the ion-exchange adsorption model. Mollerup and asso-
ciates summarized the development of their multimodal adsorption model in [114, 116] in
order to support an in-depth understanding of their model, both the development and im-
plementation. The authors emphasized that the parameters in their model are not only easy
to determine from experimental data but also have useful, well-defined physical meaning.
In [120], Nfor et al. applied Mollerup’s thermodynamic framework for adsorption to
the case hydrophobic interaction and ion-exchange chromatography. The resulting adsorp-
tion model gave extremely good fits with various sets of data found in literature and thus
can be used with any mixed mode adsorbent having hydrophobic and ion-exchange inter-
actions. Additionally, the adsorption model developed by Nfor et al. is applicable under
varying salt concentrations. As one of the known parameter values is the salt concentra-
tion of the solution, then changing the salt concentration of the solution changes only that
known value while the fitted parameters remain unchanged.
While the isotherm model developed by Nfor was very accurate for a wide variety of
6
data sets, it is by definition highly nonlinear and implicit in its definition making it difficult
to incorporate into numerical simulation. As a result, it has yet to be incorporated into a
full-model simulations of protein chromatography.
The goal of this work, therefore, is to develop software tools capable of simulating
the protein chromatography process under the effect of complex, implicit adsorption rela-
tionships in the presence of highly advective flows. Toward this end, we choose to utilize
a streamline upwind Petrov-Galerkin (SUPG) finite element method to numerically solve
the advection-dominated advection-diffusion-reaction equation for porous media. In this
work, we develop and analyze SUPG discretizations schemes for the cases of linear, non-
linear explicit, and nonlinear implicit adsorption relationships. We first review previous
mathematical analysis and numerical methods applied to the advection-diffusion-reaction
equation.
1.1 Review of Mathematical Methods
Extensive research has been conducted over the past decades mathematically study-
ing the general advection-diffusion equation. Despite this extensive research, designing, an-
alyzing, and implementing a scheme to solve the problem is still challenging especially in
the cases of advection dominated flow and nonlinear adsorption. Many times mixed meth-
ods are used in conjunction with the advection-diffusion equation because of the inherent
mass conservation properties of the mixed method. However, in the case of flow that is
dominated by advection, it is well known that numerical instabilities occur when standard
numerical methods are used (even mixed methods) because of sharp layers in the solutions
that are not resolved correctly (unless the spatial discretization parameter is taken to be
extremely small) [8, 17, 42, 70, 82]. While many numerical solutions have been proposed
to resolve these instabilities, two major families of methods have emerged: the streamline
upwinding Petrov Galerkin finite element method (SUPG) and the discontinuous Galerkin
finite element method (DG) [74]. These techniques often complicate the analysis of the
7
scheme so much of the analysis of the advection-diffusion-reaction problem is completed
without any stabilizing element involved or in a different context such as a mixed finite
element method.
The presence of a reaction terms, especially a nonlinear one, generally complicates
the analysis and numerical solution. The analysis is complicated even further by the consid-
eration of equilibrium adsorption instead of non-equilibrium (see [134] for a specific compar-
ison). The one case of reaction which actually simplifies the analysis is when the adsorption
is accounted for by a function of the unknown being added to the diffusive and reactive
terms as in
∂C
∂t−∇ · (D∇C) +∇ · (uC) +RC = f.
Since the reaction term RC does not have a temporal derivative associated with it, the
analysis is somewhat simpler. Also many times the reaction is taken to be linear, i.e. R is
not a function of C, to simplify the analysis and implementation even further.
Below we present a review of previous literature dealing with the advection-diffusion-
reaction equation and/or transport in porous media, specifically that which studies advection-
dominated flow. Since DG is one of the most common techniques for dealing with advection-
dominated flow, a brief summary of literature using DG is included; however since we have
chosen to employ the SUPG method, most of the review pertains to the SUPG method or
techniques inspired by and/or combined with upwinding.
1.1.1 Finite Element Analysis of Advection-Diffusion Equation
Extensive analysis is available for DG, SUPG, and mixed methods when applied
to the advection-diffusion-reaction equation. However, much of the analysis is conducted
under simplifying assumptions such as diffusion dominance and linear or no adsorption.
8
1.1.1.1 Discontinuous Galerkin Method
Just recently, Miller et al. provided a succinct summary of DG methods as applied
to water resource problems, but more extensive recent summaries are available in [40, 71,
143]. Specifically in [143], many different situations of elliptic and parabolic equations were
presented and analyzed. The linear parabolic problem given by the time-dependent diffusion
equation was analyzed for stability and error bound in the semi-discrete and fully-discrete
cases. For the transport equation, stability and error bounds were presented and proven
for the semi-discrete case while the fully-discrete case was studied numerically with a few
examples.
The fully discrete case was analyzed by Cockburn and Shu in [41]. The DG method
applied specifically to the transport problems in porous media was studied by Riviere and
Wheeler in [144]. In [82], Johnson presented the DG method for the advection-diffusion
equation. Some simple error estimates for the purely-diffusive problem with linear adsorp-
tion were presented.
In [36], Cockburn, et al. analyzed a new formulation for the stationary advection-
diffusion-reaction problem combining both discontinuous Galerkin and mixed method tech-
niques which they referred to as the local discontinuous Galerkin-hybridizable (LDG-H)
method. This unified method was first introduced in [38] in the context of a purely diffu-
sive problem (no advection or reaction terms) and then analyzed in this same situation in
[36] and [39]. The LDG-H methods cannot be applied to the purely hyperbolic case (when
there is no diffusive term) so [37] was restricted to the diffusion-dominated case. Although
convergence properties for this hybrid DG method were proven in [36] and [39] in the con-
text of the purely diffusive problem, [38] presented only numerical tests of the convergence
properties of the scheme applied to the advection-diffusion-reaction problem. Egger and
Schobel completed an analysis using this method for the advection-diffusion problem a year
later in [52]. Nonlinearity in the advection-diffusion equation was considered in [173] with
error estimates of the semi-discrete DG method stated and proven.
9
1.1.1.2 Upwinding
Summaries and comparisons of different upwinding schemes for advection-dominated
problems can be found in [8] and [42]. In [8], Bank, Burgler, Fichtner, and Smith summa-
rized upwinding schemes for the advection-diffusion problem. All methods were compared
and contrasted to the standard FE Galerkin discretization. In [42], Ramon Codina presented
an overview of the several finite element methods for solving the advection-diffusion-reaction
equation, particularly the advection-dominated case, including the SUPG method.
Initially the idea of upwinding was applied in centered finite difference methods to
avoid numerical oscillations in the case of small diffusion [163]. As the initial idea added too
much artificial diffusion, an improvement was made by introducing this artificial diffusion
only along the streamlines [75, 76, 92]. One more improvement was added to this method
by viewing the streamline diffusion method in the context of a weighted residual method
[111]; this streamline upwinding Petrov-Galerkin weighted residual method is sometimes
referred to as the “standard” SUPG method [42, 77].
As a weighted residual method, the upwinded test function should be applied to all
the terms (including the temporal derivative terms). However as this contributes a skew-
symmetric component to the mass matrix, many authors suggested and implemented a
modification to the classical SUPG method in which only the stationary terms are upwinded
[34, 54, 82, 85]. The idea of upwinding was more recently used in conjunction with mixed
method schemes to provide stability to methods which are inherently mass-conservative (see
Upwind-Mixed & Hybrid Mixed Methods paragraph in section 1.1.1.3).
Some analysis of the SUPG method has been completed over the years, but much
of the time simplifying assumptions were made. In [20], Brooks and Hughes showed that
when implemented as a Petrov-Galerkin weighted residual method, the SUPG method does
not add the same amount of artificial diffusion as classical upwind methods. They reviewed
information known about the phase accuracy and damping for the algorithm from work
completed by Tezduyar in [157]. They also showed the accuracy of the SUPG method on
several numerical experiments of the linear advection-diffusion equation.
10
Hughes and Mallet analyzed the time-dependent advection-diffusion problem with
linear approximation elements in [78]. This case is significantly simpler than that considered
in this dissertation since the second-order term arising from upwinding the diffusive term
drops out with linear elements. Also, research has been conducted to determine the best
choice for the upwinding parameters in both the case of linear elements [43] and quadratic
elements [78].
In [81, 82, 84, 85], Johnson considered the general case of elements but presented
theoretical analysis (stability and error bounds) of the stationary situation of the linear hy-
perbolic equation given by the advection-diffusion-reaction equation with linear adsorption.
The linear adsorption, in this case given by the term C, simplified the analysis somewhat
by adding stability and enabling easier writing of the analysis in terms of the H1 norm. In
[119, 118], the fully discrete case of the advection-diffusion-reaction equation with the same
adsorption term as in [82, 85, 84] was analyzed. The case of variable coefficients was also
analyzed in [118].
Franca, Frey, and Hughes compared different stabilizing techniques for approxi-
mating the stationary advective-diffusive model including SUPG [62]. Stability and er-
ror bounds were proven assuming scalar diffusion, using a slightly modified test function,
v + δ(u · ∇v + d∆v), and were written in terms of a seminorm involving ‖∇C‖0 and
‖δ1/2u · ∇C‖0. The proven error bounds did not explicitly show the dependence on the
upwinding parameter, and no reaction term was considered.
In [106], Melenk and Schwab investigated the hp-stability and error bounds for
the stationary 1D advection-dominated problem with linear adsorption. In [74], Houston,
Schwab and Suli extended this analysis to the case of two dimensions for the purely advective
problem with simple linear adsorption. The 2D advection dominated diffusion problem with
linear adsorption was then analyzed by Gerdes et al. in [66]. All analysis was completed
based on a positivity assumption on the difference between the reaction and advection
parameters.
A new stabilization method for the linear scalar advection-diffusion problem with
11
no adsorption was proposed by Dutra do Carmo and Alvarez [51]. The new formulation was
based on the classical SUPG formulation, and depending on the regularity of the approxi-
mate solution, the new method degenerated into either the SUPG method or the Consistent
Approximate Upwind (CAU) method. The accuracy and stability of the new formulation
was demonstrated numerically. In [64], the authors analyzed the Consistent Approximate
Upwind (CAU) Petrov-Galerkin method for the steady-state advection dominated reaction-
diffusion problem and compared it to the SUPG method. A simple linear adsorption term
written as RC was included in the analysis. Although no new analysis was completed on the
SUPG method, stability and continuity of the bilinear form associated with the steady-state
equation was recalled from previously completed analysis.
In [15], Bochev, Gunzburger, and Shadid analyzed the transient advection-diffusion
problem. A classical SUPG spatial discretization and an implicit time integration tech-
nique were analyzed for stability over a range of parameters values showing that the SUPG
method provided stability for any parameter value even when combined with an implicit
time integration method.
In [34], Chrispell, Ervin and Jenkins analyzed a fractional step θ-method for the
scalar advection-diffusion problem with a linear reaction term using SUPG to stabilize.
The solvability of the system was proven along with O(δ) error bounds. Also, work was
done to compute the optimal θ for the method used, and numerical computations verifying
the theoretical results were presented.
In [23], Burman analyzed the stability and convergence of the time-dependent ad-
vection equation (no diffusion) using the “classical” Streamline Upwind Petrov-Galerkin
method spatial discretization and a few different A -stable finite difference discretization in
time. Note that since the classical SUPG method was used, even the temporal term was
upwinded. Stability and convergence bounds were proven for both the backward Euler and
the Crank-Nicholson temporal discretizations. Because the temporal terms was upwinded,
the stability and convergence rates were O(δ) (which is to be expected), but the bound
controlled only the discrete material derivative∥∥δ1/2 (∂t(C − Ch) + u · ∇(C − Ch))
∥∥, not
12
the expected weighted streamline derivative∥∥δ1/2u · ∇(C − Ch)
∥∥. Similar results for BDF2
were stated without proofs. Numerical results supporting the analysis were shown. No
reaction term was considered in any of the analysis or numerical experiments.
Most recently, John and Novo considered the evolutionary advection-diffusion-reaction
equation [80], analyzing the “classical” SUPG method. As with much of the previous anal-
ysis involving SUPG (e.g. [34, 51]), the reaction was a linear term written as RC which
simplified the analysis somewhat allowing more easily for terms to be hidden in the reaction
term.
1.1.1.3 Mixed Methods
Mixed methods for second order problems such as the transport equation have been
studied extensively throughout the years (see for example [87, 108, 132, 133, 137, 138, 140,
146, 162]). In [87], Douglas and Roberts obtained global error estimates for the two- and
three-dimensional mixed finite element method for the Dirichlet problem of the general
second order elliptic equation. Although a reaction term was considered, the coefficient
function only depended on space and therefore was linear in the unknown. In [162], Vohralık
proved a posteriori error estimates for the advection-diffusion-reaction equations using the
lowest-order mixed finite element discretization. The error estimates were proven assuming
the reaction term was linear on the triangulation.
Sacco and Saleri proposed a way to stabilize the advection-diffusion equation using
a new family of mixed finite volume methods in [146]. Their formulation of the advection-
diffusion equation was stationary but allowed for a nonlinear reaction term written simply
as RC. The reaction coefficient function R was permitted to be a function of C, but
it was assumed to be nonnegative and sufficiently regular. Also positivity assumptions
were made on the interaction of the reaction coefficient function with the velocity function.
The stabilization employed the lowest-order Raviart-Thomas finite element spaces plus a
suitable quadrature formula for the mass matrix. The quadrature formula was employed as
a lumping technique to deal with the fullness of the discrete matrix instead of the alternative
13
technique of hybridization of the mixed formulation.
In [108], Micheletti, Sacco and Saleri studied the reaction-diffusion problem. They
analyzed a stabilization method similar to the method in [146] applied to the advection-
diffusion problem, i.e. they used lowest order Raviart-Thomas finite element spaces with a
suitable quadrature formula for the mass matrix. Here they assumed simply the adsorption
coefficient function R was constant and nonnegative. Radu and Wang analyzed a nonlinear
parabolic equation (specifically Richards’ equation) in [138] and obtained an optimal order
of convergence which was supported by numerical experiments.
In [132], Radu and Pop applied a mixed finite element method to the nonlinear
reactive transport problem taking into account both advection and diffusion. Newton’s
method was used to resolve the nonlinearity due to the adsorption isotherm. The method
was first analyzed in the context of nonlinear equilibrium adsorption when the isotherm was
nondecreasing and the derivative of the isotherm was Lipschitz continuous (as in Langmuir’s
isotherm). Then the case of non-Lipschitz adsorption (as in Freundlich’s isotherm) was
analyzed. For the case of non-Lipschitz adsorption, they first applied a regularization to
the isotherm; this regularized isotherm, which approached the Freundlich isotherm as the
regularization parameter went to zero, was then nondecreasing with a Lipschitz continuous
derivative. In both cases, they give a sufficient condition for the quadratic convergence
of Newton’s method to be preserved. Numerical examples were included to verify the
theoretical results.
Radu and Pop continued their analysis of the nonlinear reactive transport problem
in [133]. They again used a mixed finite element method with a Newton linearization but
instead considered non-equilibrium adsorption. The authors considered only the case of a
Freundlich-type adsorption. The same regularization step as in [132] was made to allow for
the convergence analysis to be completed. A sufficient condition for quadratic convergence
with Newton’s method was given and proven, and numerical verification of the theoretical
results was provided.
14
Time-Integrated Mixed Methods When dealing with nonlinear adsorption in the
transport equation, it is helpful to look at the analysis of Richards’ equation since it can
be written in a form that has the same mixed collection of derivatives as the advection-
diffusion-reaction equation. One technique that has been especially helpful in the analysis
of Richards’ equation is a time-integrated technique first introduced by Nochetto and Verdi
in [121]. The method first integrates in time to eliminate any temporal derivatives which
allows for less regularity on the solution. Nochetto and Verdi proved stability and error
estimates for this method applied to the degenerate parabolic problem focusing specifically
on the two-phase Stefan problem.
An extension of this time-integrated method was made by Arbogast, Wheeler and
Zhang in [6]: they analyzed this method applied to a nonlinear, possibly degenerate,
advection-diffusion equation. However, no adsorption was considered. They assumed Lip-
schitz continuity on all the nonlinear functions involved, and they obtained nearly optimal
bounds in the sense that all bounding terms reduced to approximation error except one
which involved the difference of two discrete projection (an extremely small quantity when
using RT spaces).
Woodward and Dawson analyzed Richards’ equation using this time-integrated
method in [171]. They analyzed the method over two ranges of saturation corresponding to
the possibilities of nondegeneracy and degeneracy. Their analysis considered a nonlinearity
in the volumetric water content θ which they assumed to be Lipschitz continuous. They
also assumed the absolute permeability tensor K was Lipschitz continuous in the case of
partially saturated to saturated flow. They obtained optimal error results when considering
the lowest order RT spaces.
In [129] and [130], Radu, Pop and Knabner extend the analysis of [6] and [171] on
Richards’ equation to the situation of less regularity for the coefficient functions. Specif-
ically, the function k for the conductivity of the medium was considered to be bounded,
but not necessary Lipschitz continuous. For simplicity of the analysis, they considered ho-
mogeneous Dirichlet boundary conditions and analyzed only the situation of variably- to
15
fully-saturated regimes. Most importantly, they showed equivalence of the time-integrated
mixed method and the original mixed method approached for Richards’ equation.
Based on the theoretical results in [129], Pop, Radu and Knabner proposed an it-
erative scheme to solve nonlinear elliptic problems (including Richards’ equation). They
proved convergence of the scheme based on a contraction argument and showed numeri-
cal support of the theoretical results. In [147], Schneid, Knabner, and Radu extended the
work done in [129] to consider the possibly degenerate case of Richards’ equation. Also,
the homogeneous Dirichlet boundary conditions were relaxed somewhat to consider possi-
bly non-homogeneous Dirichlet conditions as long as the Dirichlet function was sufficiently
smooth. Pop, Radu, and Knabner continued the analysis of this method in [136] by consid-
ering less regularity on the diffusion function; instead of Lipschitz continuity, they assumed
only Holder continuity. This low level of regularity allowed for modeling the cases of both
fast diffusion and slow diffusion.
This time-integrated method was also applied to transport of reactive solutes in
porous media although not necessarily in the context of mixed methods. Barrett and Kn-
abner looked at both the case of non-equilibrium adsorption [9] and the case of equilibrium
adsorption [10] in porous media. For simplicity in the analysis, they assumed that the pro-
cess was not advection-dominated and therefore dropped out the advection term from the
equation. Also, they focused on a formulation for the equilibrium term of the adsorption
similar to the Freundlich isotherm as this isotherm degenerates at C = 0, i.e. the func-
tion is only Holder continuous, not Lipschitz continuous at C = 0. The time-integrated
method was used so that the lower regularity of the adsorption function would not hinder
the analysis.
In [134] and [135], Radu, et al. presented and analyzed the Euler implicit-mixed
finite element scheme based on the lowest order RT elements for the reactive transport
equation in porous media coupled with Richards’ equation to describe the flux. The time-
integration strategy was employed in this case again to allow for less regularity on the
function representing water content. They proved convergence for the case of equilibrium
16
adsorption where they assumed Holder continuity of the adsorption equation. No upwinding
or stabilization technique was considered for the case of advection-dominate flow.
Upwind-Mixed & Hybrid Mixed Methods More recently, methods combining both
the stabilizing effects of upwinding techniques and mass-conserving effects of mixed methods
have been introduced and analyzed for the advection-diffusion equation.
Following an idea initially suggested in [31], Dawson developed and analyzed a time-
splitting method for solving an advection-dominated parabolic equation in which a Godunov
procedure was used to approximate the advection and a mixed finite element discretization
was used to approximate the diffusion [46, 47]. In [46], error estimates for the 1D case
were presented along with numerical results for the original algorithm and a few variants of
the scheme. In [47], Dawson extended the ideas to multiple dimensions and derived error
estimates.
In [4], Arbogast and Wheeler developed a new finite element method which they
called the “characteristics-mixed method” to approximate the solution to the advection
dominated transport equation. The test functions were piecewise constant in space and
approximately follow the characteristics of advection in time. They assumed the diffusion
tensor was positive definite (precluding the case of degeneracy) and considered no adsorp-
tion. They proved stability and error bounds on the approximation for the restricted case
of a rectangular domain with periodic boundary conditions.
Dawson later analyzed a scheme similar to the one presented in [4] applied to the
nonlinear advection-diffusion-reaction equation in [48]. Only the 1D situation was consid-
ered, and adsorption was assumed to be instantaneous (equilibrium). Error estimates for
the semi-discrete formulation were stated and proven, and convergence of the fully discrete
formulation was illustrated with numerical results.
In [49], Dawson and Aizinger continued analyzing the scheme used in [48] with more
relaxed assumptions on the data. The diffusion tensor was no longer assumed to be positive
definite (they considered the possibility of zero diffusion), and more realistic boundary con-
17
ditions than homogeneous Dirichlet were allowed. However, the reaction term was simplified
significantly in that it was merely a linear term given by qC with q a nonpositive constant.
Stability and error bounds were derived, and numerical experiments were presented that
supported the theoretical bounds.
In [21], Brunner, Radu, and Knabner analyzed an upwind-mixed hybrid finite ele-
ment method for the transport problem that differed from the one developed and analyzed
by Dawson and Aizinger in [49]. This upwind-mixed scheme was designed to improve the
stability in the case of strongly advection-dominated problems while maintaining the effi-
ciency of the classical mixed method previously analyzed [7, 19, 86, 87]. It was based on an
Euler-implicit mixed hybrid finite element discretization using RT elements of lowest order.
This method was introduced and studied numerically in the context of reactive transport
simulation in [137]. They considered the linear parabolic advection-diffusion-reaction prob-
lem where the reaction term was of the form RC with R constant and derived optimal order
convergence in time and space for the fully discrete formulation.
Radu, et al. studied the problem of multicomponent reactive transport in porous
media using a mixed hybrid finite element discretization scheme [131]. The advection-
diffusion-reaction problem was coupled with Richards’ equation for flow in possibly un-
saturated mediums and mass balance equations for each species. Although no theoretical
analysis was presented, a detailed description of the discretization and solution process was
given along with numerical results showing good convergence; Newton’s method was used
ultimately to resolve the nonlinearities in the system due to the adsorption function.
Radu and Brunner continued their work with multicomponent reactive transport in
porous media in [22]. They presented a mass conservative modified mixed finite element
scheme using BDM1 elements; the scheme was modified based on the hybrid form for
the mixed finite element scheme. Numerical results were presented that showed second
order convergence for the flux variable which was an improvement over the classical BDM1
elements which are known to be only first order accurate. The new scheme was also shown to
be more robust for high Peclet numbers using classical BDM1 elements. For the strongly
18
advective problem, Radu and Brunner combined the modified BDM1 approach with an
upwinding technique, the same used in [137].
1.1.2 Numerical Methods for Advection-Diffusion Equation
In addition to the numerical schemes analyzed in the literature discussed in section
1.1.1, other numerical schemes used for transport in porous media are considered with little
or no theoretical analysis. However, much numerical experimentation is included confirming
the expected behavior for the schemes. A few papers summarizing numerical techniques for
transport in porous media are available [24, 109, 137]. In [24], Celia focused on a finite dif-
ference approximation to the advection-diffusion-reaction equation and only briefly touched
on finite element methods and a few modifications of finite elements such as upwinding tech-
niques and the method of characteristics. In [137], Radu, et al. summarized the Galerkin
finite element method, finite volumes, and mixed hybrid finite elements as applied to the
contaminant transport problem. Miller et al. [109] provided a summary of many different
models pertaining to water resources systems, including the reactive transport problem,
along with numerical simulation methods pertaining to each model.
As already mentioned, the mixed-derivative form of Richards’ equation is very sim-
ilar in form to the advection-diffusion-reaction equation so solution methods for Richards’
equation can many times be applied to the transport problem. Summary papers concern-
ing numerical methods for Richards’ equation are available as well [26, 88]. In [26], Celia
and Binning presented summaries of numerical methods based on a few different forms
of Richards’ equation including both finite element and finite difference methods. Ju and
Kung [88] restricted their attention to finite element methods for Richards’ equation.
1.1.2.1 Operator Splitting
Operator splitting is very likely the most common numerical solution method for
the advection-diffusion-reaction problem [25, 56, 63, 68, 90, 91, 94, 95, 100, 103, 104, 123,
126, 128, 142, 161, 168, 174]. In the case of advection-dominated flow with no reaction,
19
time-splitting methods can deal with the advection dominance by splitting the advection
and diffusion operators; then the advection step is solved using a higher-order explicit
formulation and small time step while the diffusive step is solved with a lower order implicit
method and large time step [168]. If a reaction term is present, an additional step can be
taken at each time step to solve the reaction term as in [90, 91, 94, 95, 141, 142]. If prior
knowledge is known of the inherent physics in the problem, decoupling techniques can be
developed to accurately approximate the fully-coupled system [56].
One of the benefits of splitting methods is that different spatial discretizations can
be employed for each sub-step, e.g. finite volume method for diffusion term and analytical
solution for advection and reaction as in [90, 141, 142]. Another benefit is that nonlinearities
are much easier to resolve. If nonlinearity in the reaction term or diffusion/dispersion tensor
is allowed, then some linearization scheme must be implemented; however, each nonlinearity
is solved separately in a simpler problem. One of the downfalls of splitting methods is that,
depending on the interaction of the physics in the problem, an exceedingly small step size
may be necessary to ensure stability and accuracy of the method [73, 93, 145, 169]. The
theory of classical operator splitting algorithms has been well known for some time (see
[68, 103, 104, 123, 128, 161, 174]), and new splitting techniques have been proposed and
Table 8.1: Approximation errors and experimental convergence rates for the approximationto the steady-state problem. As ∆t and h are cut in half, the H1 error is reduced the sameamount which is consistent with the theoretical convergence rates.
8.2 Linear Convergence Rates
The numerical validation of the fully explicit and fully implicit a-priori estimates
were obtained using a linear isotherm model with K1 = K2 = 1 so that
q(C) = 1 + C.
For both cases, we take u =< 1, 1 >, D = I, Ω = [0, 1] × [0, 1], and Vh to be the space
of continuous, piecewise linear functions. The right-hand side f , along with the boundary
182
conditions and initial conditions, are determined by the true solution
C(x, y, t) = t(x− x2
)sin(πy).
8.2.1 Fully Explicit Discretization
Recall from sections 5.1.2 and 5.2.2 that the predicted convergence rate for the fully
explicit formulation using linear approximating functions is
‖C − Ch‖0,1 ≈ O(h+ ∆t)
with the time step restriction
∆t ≤ K(u,D, δ)h2.
In this case, the upwinding parameter δ does not affect the convergence rate although it
does affect the constraint on the time step. The numerical results presented in Table 8.2
are consistent with the error estimates shown previously. The numerical results presented
in Table 8.1 are consistent with this estimate.
8.2.2 Fully Implicit Discretization
Recall from sections 5.1.3 and 5.2.3 that the predicted convergence rates for the
fully implicit formulations using linear approximating functions are
‖C − Ch‖0,1 ≈ O(h(1 + δ) + ∆t).
In this case, the convergence depends on δ and just as in the steady-state case if we take
δ ≈ O(h), then we preserve the convergence rate in h. The numerical results presented in
Table 8.2: Approximation errors and experimental convergence rates for the fully-explicitapproximation with a linear adsorption model. As ∆t and h are cut in half, the H1 erroris reduced the same amount which is consistent with the theoretical convergence rates.
8.3 Nonlinear Convergence Rates
The numerical validation of the nonlinear a-priori error estimates were obtained
using a standard nonlinear isotherm model. For all three cases, we take u =< 1, 1 >,
D = I, Ω = [0, 1] × [0, 1], and Vh (and Wh if appropriate) to be the space of continuous,
piecewise linear functions. The right-hand side f , along with the boundary conditions and
initial conditions, are determined by the true solution
Table 8.3: Approximation errors and experimental convergence rates for the fully-implicitapproximation with a linear adsorption model. As ∆t and h are cut in half, the H2 erroris reduced the same amount which is consistent with the theoretical convergence rates.
8.3.1 Time-Integrated Discretization
For the nonlinear time-integrated a-priori error rates, we use Langmuir’s isotherm
with qmax = Keq = 1 where
q (C) =qmaxKeqC
1 +KeqC=
C
1 + C.
185
We simplify the problem formulation to a single (nonlinear) transport equation in
one unknown Cn using
∂q
∂t=
dq
dC· ∂C∂t
=1
(1 + C)2 ·∂C
∂t.
We compute solutions by lagging the nonlinearity q′(C) as [125]
1
(1 + Cn)2 ·Cn − Cn−1
∆t≈ 1
(1 + Cn−1)2 ·Cn − Cn−1
∆t.
Although the error analysis of the time-integrated discretization has yet to be com-
pleted, we expect from other time-integrated analysis that the convergence rates using linear
elements to be ∥∥∥∥∥∫ T
0C dt−
N∑n=0
Cnh∆t
∥∥∥∥∥0
≈ O ((1 + δ) (h+ ∆t)) .
We present numerical results in Table 8.4 that are consistent with this expectation.
Table 8.4: Time-integrated approximation errors and convergence rates for nonlinear ad-sorption with Langmuir adsorption model. As ∆t and h are cut in half, the time-integratederror is reduced the same amount which is consistent with the theoretical convergence rates.
8.3.2 Backward Euler Discretization
For the nonlinear a-priori convergence rates of the backward Euler discretization, we
again use Langmuir’s isotherm with qmax = Keq = 1 and we compute solutions by lagging
the nonlinearity q′(C) as [125].
Although the error analysis of this discretization has yet to be completed, we expect
186
the convergence rates using linear elements to be
‖C − Ch‖0,1 ≈ O ((1 + δ)(h+ ∆t)) .
We present numerical results in Table 8.5 that are consistent with this expectation.
Table 8.5: Approximation errors and convergence rates for nonlinear adsorption with Lang-muir adsorption model. As ∆t and h are cut in half, the H1 error is reduced the sameamount which is consistent with the theoretical convergence rates.
187
8.3.3 Two Unknowns Discretization
To verify the nonlinear a-priori convergence rates for the two unknowns discretiza-
tion, we use Nor’s isotherm with
Keq = Λ = z = cs = cm = qmax = 1, ν = n = 0.5, and Ks = Kp = −1
so that
q = (1− q) exp−1− C
and consequently
q =C exp−1− C
1 + C exp−1− C.
Although the error analysis of the discretization with two unknowns has yet to be
completed, we expect the convergence rates using linear elements to be
‖C − Ch‖0,1 ≈ O ((1 + δ)(h+ ∆t))
and
‖q − qh‖0,0 ≈ O ((1 + δ)(h+ ∆t)) .
We present numerical results in Table 8.6 that are consistent with this expectation.
8.4 Numerical Simulations
For the comparison with data generated in the laboratory, we choose parameter
values to mimic the associated laboratory conditions. The data was obtained by Juan
Wang in the Department of Chemical and Biomolecular Engineering at Clemson University
as part of Scott Husson’s Bioseparations and Advanced Separation Materials research group.
The multimodal membrane (MMM) is modified from commercially available regener-
ated cellulose membrane. The regenerated cellulose membranes purchased from Whatman,
Table 8.6: Approximation errors and convergence rates for nonlinear adsorption with Nforadsorption model. As ∆t and h are cut in half, the H1 error for C and the L2 error for qis reduced the same amount which is consistent with the theoretical convergence rates.
189
Inc. have an average effective pore size of 1 µm, 0.7 mm thickness and 47 mm diameter. A
stack of 10 MMMs is placed in a Mustang Coin® module (Pall Corporation, Port Washing-
ton, NY) with one piece of 25 µm nominal pore diameter filter paper (Whatman 5) placed
on each side of the stack. The resulting height of the module is approximately 0.7 cm.
The effective membrane diameter within the module is 1.4 cm; however, a diameter of 1.6
cm is used for calculation of the membrane bed volume and consequently in computational
simulations because radial distribution of the adsorbing species within the membrane stack
is likely to happen. The module is installed in an AKTA purifier. The membrane bed
porosity is directly measured to be 0.84.
A protein-rich serum is pushed vertically through the membrane using a constant
pressure differential providing a constant flow rate of 0.1 mL/min. The vertical velocity is
thus chosen to have a parabolic velocity profile with zero velocity at the sides of the module
and maximum velocity chosen to ensure the correct flow rate. We assume a 0 cm/min
horizontal velocity.
Values for the diffusion of protein molecules in porous membranes range from 6 ×
10−4 cm2/min (for small proteins such as glycine) to 2.28×10−5 cm2/min (for large proteins
such as immunoglobulin G) [154]. We choose a diffusion value of 2.28 × 10−5 cm2/min as
immunoglobulin G was used in the experiment.
Since the experimental velocity is larger than this diffusion, we use a more general
model for the dispersion tensor, D, than simply diffusion, d0. We choose a model for low
Peclet numbers [50] to account for the spreading effect due to both diffusion and velocity:
DL = ωd0 + αL |u| , DT = ωd0 + αT |u|
where ω is the porosity, d0 is the coefficient of diffusion, |u| is the mean microscopic ve-
locity, and αL and αT are the longitudinal and transverse dispersiveness respectively. The
190
dispersion tensor is then expressed in its principal direction of anisotropy giving
D =
DL 0
0 DT
with DL the longitudinal dispersion coefficient (in the direction of the flow) and DT the
transverse dispersion coefficient (in the direction at a right angle to the velocity). In our
case the velocity is purely vertical so that the longitudinal direction is vertical and the
transverse direction is horizontal.
The values for the porosity and the diffusion coefficient have been discussed above.
The value for the mean microscopic velocity |u| is approximated by means of the constant
flow rate. As the experimental flow rate is 0.1 mL/min, then
|u| ≈ flowrate
πr2= 0.0497 cm/min.
The values for the longitudinal and transverse dispersivities, αL and αT respectively,
are known to vary depending on the physical properties of the membrane, but they cannot
be obtained experimentally. However, much is known about their behavior from previous
experimentation [50, 2, 172]. Experimental values for αL on a small scale range from
around a few millimeters to nearly a hundred centimeters depending on the porosity [50, 2],
and αL decreases as the porosity increases [172]. Consequently for our purposes with a
porosity value closer to 1, αL should be on the lower end of this range on the order of a few
millimeters. The specific value of αL is chosen to match up the experimental and numerical
points of breakthrough.
Additionally, αT is generally found to be between 15 and 1
100 of αL [50]. Through
comparison of the experimental and numerical breakthrough curves, larger values of αL give
more accurate numerical results. We therefore take αT = 15αL.
We consider two isotherm models for the numerical simulation results shown below.
191
First we consider Nfor’s model
q = KeqΛν+n
(1
zcs
)ν ( 1
cm
)n(1− q
qmax
)ν+n
C exp Kscs +KpC
to capture the multimodal adsorption properties of the MMMs. The parameter values were
obtained by fitting the model to known data and are as follows:
Keq = 502, Λ = 0.34009 M, ν = 0.1, n = 1.0, z = 1, cs = 0.37 M,
cm = 55.5 M, qmax = 216 mg/mL, Ks = −0.1 M−1, and Kp = −0.1 mL/mg.
where M denotes a Molar unit (=moles/L). Note that an absence of units implies the
quantity is unitless. For the simulations involving Nfor’s isotherm, we take h and ∆t to
be small enough to ensure the error is not significantly affecting the results; specifically we
choose h = 0.0125 and ∆t = 0.0078125.
As a comparison, we also consider Langmuir’s model
q (C) =qmaxKeqC
1 +KeqC
where qmax and Keq were obtained by fitting the model to known data; specifically,
qmax = 150 mg/mL and Keq = 2.06 mL/mg.
For the simulations involving Langmuir’s isotherm, we again choose h and ∆t to be small
enough to ensure the error is not significantly affecting the results; specifically we choose
h = 0.0125 and ∆t = 0.03125.
The metric of productivity is important in evaluating bind-and-elute chromatog-
raphy processes. Previously [33], Husson and coworkers defined productivity as the mass
of protein that can bind per volume of membrane per time, and provided a convenient
192
expression to calculate it:
Productivity =Bdynamic
tbreak
where Bdynamic is the dynamic binding capacity and tbreak is the time it takes to reach the
point of 10% breakthrough, i.e., the time at which 10% of the value of the feed concentration
appears in the effluent from the column. From a productivity standpoint, differences in
dynamic binding capacity do not impact productivity nearly as significantly as differences
in residence times [33]. Therefore, high productivities can be achieved when the time to
breakthrough is short; that is, the membrane quickly reaches its capacity to bind proteins.
The information on protein breakthrough is visualized using a breakthrough curve,
which is a plot of the concentration at the effluent as a function of accumulated effluent
volume. We choose to assess the validity of our approach by comparing our simulated break-
through curve with the experimental curve generated from laboratory data. Breakthrough
curves are greatly affected by diffusion and dispersion values. Consequently, having accu-
rate diffusion and dispersion values is important when analyzing breakthrough curves. Since
upwinding adds some artificial diffusion along the velocity streamlines, we take the upwind-
ing parameter δ to be 0 to ensure there is no artificial diffusion affecting the breakthrough
results.
Figure 8.1 shows experimental results along with results generated using the disper-
sion model described above. The initial baseline absorbance reading indicates that protein
is binding to the column. As the membrane bed capacity is reached under flow conditions,
the unbound protein begins to break through the column, illustrated by the initial increase
in concentration. Both experimental and model breakthrough curves exhibit the typical
S-shape characteristic of favorable binding within a packed bed. Experimentally, break-
through occurs after approximately 1.366 mL of fluid has passed through the column which
is matched very closely by the model.
The model shows complete bed saturation, i.e., the effluent concentration reaches the
feed value, after approximately 3 mL of fluid has passed through the column; whereas, the
193
experimental breakthrough curve does not reach saturation during the measurement period.
Such behavior for the experimental data is typical for the membranes used in this study.
At high degrees of loading, the adsorption rate decelerates due to surface exclusion from
previously adsorbed molecules [155]. Since the model does not account for the kinetic rate
of adsorption, it is not surprising that it does not capture this long-term tailing behavior.
Figure 8.1: Comparison of numerical and experimental breakthrough curves using instan-taneous adsorption. Note the typical S-shape characteristic and the close match of break-through for both curves.
Although it is standard practice to stop flow at 10% breakthrough, Nadarajah and
Mehta [117] proposed an overload and elute purification strategy for purification of pro-
tein antibodies to be able to utilize bed capacity more fully. Also, recently Husson and
coworkers found that the kinetic rate of IgG binding with multimodal ligands becomes the
rate limiting step at high process flow rates for our first-generation multimodal membranes
[167]. Therefore, extension of the model to incorporate transient kinetic adsorption rates is
of interest.
Given the accuracy of the numerical results obtained with Langmuir’s model in
194
this case, one might question why Nfor’s model should be considered given that Langmuir’s
model is simpler. Recall from the discussion of Nfor’s model in Chapter 1 that the constants
in the Nfor model are independent of salt concentration, which is not the case for the
Langmuir model. Thus, we could predict breakthrough at different salt concentrations
with the same model parameters for Nfor; whereas we would need to reevaluate model
parameters for Langmuir anytime salt concentration is varied. Put simply, Nfor is predictive
and Langmuir is not.
8.4.1 Non-Instantaneous Adsorption
To incorporate non-instantaneous, or non-equilibrium, adsorption, we consider a
system of three equations:
ω∂C
∂t+ (1− ω)ρs
∂q
∂t+∇ · (uC)−∇ · (D∇C) = f (8.4.1)
∂q
∂t= km(q∗ − q) (8.4.2)
q = g(q, C) (8.4.3)
where km is the adsorption reaction rate and q∗ is an equilibrium concentration, i.e. the
concentration at the interface between the concentration in the mobile phase and the con-
centration in the adsorbed phase. The same initial and boundary conditions are applied to
C and q with an additional initial condition on q∗, namely q∗ = 0.
Applying the finite element method and linearizing (8.4.3) similar to the system
with two unknowns by evaluating g at the previous time step gives the following variational
195
formulation: For n = 1, 2, ..., N , find (C, q∗, q) ∈ V ×W ×W such that
(ωCn − Cn−1
∆t, v
)+
((1− ω)ρs
qn − qn−1
∆t, v
)+ (∇ · (uCn), v)− (∇ · (D∇Cn), v) = (f, v) ,(
qn − qn−1
∆t, w
)= (km((q∗)n − qn), w) ,
(qn, z) =(g(qn−1, Cn−1), z
).
for all v ∈ V , w ∈W , z ∈W .
Figure 8.2 shows results using non-instantaneous adsorption with h and ∆t chosen
to be 0.0125 and 0.0078125 respectively; as with the simulations involving instantaneous
adsorption, the upwinding parameter was chosen to be δ = 0. Breakthrough curves resulting
from various constant reaction rates are compared. Although km = 0 corresponds to no
adsorption, the results for that case are provided as a point of comparison. Recall that
as km → ∞, the behavior of the system (8.4.1)-(8.4.3) approaches that of instantaneous
adsorption. The results in Figure 8.2 support this as there is no distinguishable difference
between the breakthrough curves for km ≥ 10. In fact, most of the variation in the results
occurs between km = 0 and km = 1. For lower values of km, the shape of the breakthrough
curves changes in addition to the 10% breakthrough point changing quite drastically. For
0 ≤ km ≤ 1, the 10% breakthrough point ranges from 0.915 to 1.299. The effects of a
finite reaction rate, as shown in Figure 8.2, support the supposition that incorporating
non-instantaneous adsorption would provide more accurate numerical results.
Additionally, the influence of reaction rate becomes important at higher flow rates,
i.e. shorter residence times. This can be related to the first Damkohler number, which is
a dimensionless quantity representing the ratio of the reaction rate to the rate of convec-
tive mass transport through the membrane bed. Husson and coworkers discuss the first
Damkohler number and the influence of reaction rate with varying flow rates in [32].
It would stand to reason that the rate of adsorption would not stay constant through
196
Figure 8.2: Effect of varying the mass transfer coefficient km on breakthrough curve. No-tice as km increases, the breakthrough curves asymptotically approach the results withinstantaneous adsorption as shown in Figure 8.1.
the entire process. In fact, the reaction rate most likely decreases with time as the mem-
branes become saturated with protein. It may therefore be beneficial to consider a varying
adsorption rate. Having a reaction rate dependent on the amount of concentration in the
adsorbed phase would be most accurate, but for simplicity in the algorithm, considering
a reaction rate that is solely a function of time way be sufficient. Figure 8.3 shows the
breakthrough curve resulting from one such function as compared to the results shown in
Figure 8.2.
The reaction rate function for the results in Figure 8.3 was chosen to have a large
value for small t that decreases with time in order to give results closer to life; specifically,
we have
km(t) =K
(t− 1)p
where K and p are chosen to match up the 10% breakthrough point and the shape of the
breakthrough curve. In this case, with all other parameters chosen as in the Numerical
197
Figure 8.3: Comparison of transient adsorption rate with constant adsorption rates.
Simulations section above, we pick
K = 10(13.66)p and p = 10.
The values in the numerator, 10 and 13.66, were chosen to match up 10% breakthrough
point, while the power of 10 was chosen to decrease the adsorption faster to better match
up the breakthrough curve. A comparison of the numerical and experimental results, as
shown in Figure 8.4, confirms that the results with a transient adsorption rate are in fact
closer to the experimental results.
8.4.2 Varying Velocity Profiles
In order to investigate the effects of varying velocity profiles on the breakthrough
curves, we ran two- and three-dimensional simulations involving five different velocity pro-
files. With the exception of the simulation involving spatially-constant velocity, the velocity
profiles were parabolic in nature with between 1 and 5 vertices of maximum magnitude.
198
Figure 8.4: Comparison of numerical and experimental breakthrough curves using instan-taneous adsorption and non-instantaneous adsorption. Note the numerical result obtainedusing a transient adsorption rate are significantly closer to the experimental results thanthose obtained using instantaneous adsorption.
More detailed descriptions of the five cases are below.
In 2D, there were three velocity profiles considered:
1. Spatially-constant velocity with a value chosen to ensure a flow rate of 0.1 mL/min.
2. Parabolic velocity with the vertex at the center of the inflow boundary, zero values
on side boundaries, and the value at the vertex chosen to ensure a flow rate of 0.1
mL/min.
3. Velocity profile with two parabolas having zero value at the center of the inflow bound-
ary, vertices at ±√
22 r, and values at the vertices chosen to ensure a flow rate of 0.1
mL/min.
In 3D, there were two velocity profiles considered:
1. Paraboloid velocity profile with the vertex at the center of the inflow boundary, zero
199
value on the entire circular side boundary, and a vertex value chosen to ensure a flow
rate of 0.1 mL/min.
2. Velocity profile with five paraboloids having the vertices placed in an X design (1 at
the center of the inflow boundary and the other 4 all 23r distance away from the center
at 45, 135, 225, and 315), all zero values on and outside circles of radius 13r from
the vertices, and values at all five vertices chosen as the same value to ensure a flow
rate of 0.1 mL/min.
All the results in this section were obtained using deal.II with temporal discretization
value being ∆t = 0.03125 and a final simulation time of T = 80. The spatial discretization
value varied somewhat depending on the velocity profile and consequently is described in
each subsection below. The upwinding parameter δ was chosen to be 0 in each case to
ensure there is no artificial diffusion affecting the breakthrough results.
The results for all five cases assumed instantaneous adsorption using a Langmuir
isotherm model with qmax = 150 mg/mL and Keq = 2.06 mL/mg. Also, a low Peclet
dispersion model was used with d0 = 2.28 × 10−5 cm2/min, |u| = 0.0497 cm/min, αL =
0.001, and αT = 1/10 ∗ αL = 0.0001. The porosity and dimensions of the membrane were
taken to be the same as simulation values described in previous sections: width= 0.7 cm,
height= 1.6 cm, ω = 0.84. The inflow concentration was also the same: Cin = 3 mg/mL.
8.4.2.1 2D Constant Velocity Results
For the case of a two-dimensional, spatially-constant velocity profile, six initial spa-
tial refinements in deal.II were used resulting in a spatial discretization parameter of ap-
proximately 0.02, i.e. h ≈ 0.02. This value is much smaller than what is necessary to
get accurate breakthrough curve results considering the values chosen for the dispersion;
however, h was chosen to be this small to ensure that the concentration front as shown in
Figure 8.5 were fully resolved.
The evolution of the concentration front in the membrane is shown in Figure 8.5.
200
The shape of the concentration front follows the shape of the velocity profile (spatially
constant over the width of the membrane), and the front progressively moves through the
height of the membrane as time elapses. The membrane is very nearly saturated by t = 20
as shown in the last image in Figure 8.5.
8.4.2.2 2D Single Parabola Velocity Results
For the case of a two-dimensional, parabolic velocity profile, six initial spatial re-
finements were used again to give h ≈ 0.02. The evolution of the concentration front in
the membrane is shown in Figure 8.6. The shape of the concentration front follows the
shape of the velocity profile (parabolic over the width of the membrane), and the front
progressively moves through the height of the membrane as time elapses. The saturation
of the membrane occurs at a later time when compared to the case of a constant velocity
profile; in this case, saturation occurs at approximately t = 70 which is later than the last
image shown in Figure 8.6.
8.4.2.3 2D Double Parabola Velocity Results
For the case of a two-dimensional velocity profile involving two parabolas, we again
chose h ≈ 0.02. The evolution of the concentration front in the membrane is shown in
Figure 8.7. The shape of the concentration front follows the shape of the velocity profile
(two parabolas over the width of the membrane), and the front progressively moves through
the height of the membrane as time elapses. The saturation of the membrane occurs later
than in the case of constant velocity but earlier than the case of a single parabolic velocity
profile. In this case, saturation occurs at approximately t = 65 which is later than the last
image shown in Figure 8.7.
8.4.2.4 3D Single Parabola Velocity Results
For the case of a three-dimensional, single parabolic velocity profile, four initial
spatial refinements in deal.II were used resulting in a spatial discretization parameter of
201
approximately 0.08, i.e. h ≈ 0.08. Although this value is larger than that chosen for the
two-dimensional cases, it is still small enough to give accurate breakthrough curves results
and resolve the concentration profiles well enough for visual inspection. A smaller value was
not used because of time and computational constraints (i.e. a value of h ≈ 0.08 resulted in
a run time of about 9 days on a Dell Precision M4500 with an Intel Core i7 X940 @2.13GHz
and 8 GB of RAM running in Ubuntu 14.04 LTS).
The evolution of the concentration front in the membrane is shown in Figure 8.8. As
with the two-dimensional cases, the shape of the concentration front follows the shape of the
velocity profile, and the front progressively moves through the height of the membrane as
time elapses. The saturation of the membrane occurs slightly later than the two-dimensional
case with a single parabolic velocity profile; specifically, saturation occurs at approximately
t = 72.5 which is later than the last image shown in Figure 8.8.
8.4.2.5 3D Quintuple Parabola Velocity Results
For the case of a three-dimensional velocity profile involving five parabolas, three
initial spatial refinements in deal.II were used resulting in a spatial discretization parameter
of approximately 0.17, i.e. h ≈ 0.17. Although this value is larger than that chosen for the
other three-dimensional cases, it is still small enough to give accurate breakthrough curves
results. The concentration profiles could be resolved somewhat better especially on the
outflow boundary, but they are resolved enough to see the general behavior of the profile.
The larger value of h was chosen to reduce the computational time of the simulation; in this
case the run time was about 25 hours on the same machine (Dell Precision M4500 with an
Intel Core i7 X940 @2.13GHz and 8 GB of RAM running in Ubuntu 14.04 LTS).
The evolution of the concentration front in the membrane is shown in Figure 8.9.
As with all previous cases, the shape of the concentration front follows the shape of the
velocity profile, and the front progressively moves through the height of the membrane as
time elapses. The saturation of the membrane occurs later than in any of the other cases.
In this case, saturation occurs at approximately t = 80 which is at the end of the simulation
202
and occurs later than the last image shown in Figure 8.9.
8.4.2.6 Comparison of Breakthrough Curves
As the purpose of running these five simulations was to see the effects of varying
velocity profiles on breakthrough, a comparison of the breakthrough curves was generated
and is shown in Figure 8.10. The cases of a single parabolic velocity profile and a double
parabolic velocity profile resulted in very similar breakthrough curves while the other three
cases are distinctly different. Even with two of the two-dimensional cases being similar, there
is enough variation in the breakthrough curve results to indicate that the velocity profile
can drastically affect the breakthrough results, and consequently a further investigation into
velocity profiles, perhaps incorporating variable velocity, should be conducted.
We note here the difference between the 2D and 3D single parabolic velocity profile
results. This difference can most likely be contributed to how the outflow concentration is
computed in the two-dimensional case. For both the 2D and 3D cases, the concentration is
averaged over the outflow; this results in the average being taken over width of the outflow
in the 2D case instead of the area of the outflow in the 3D case. Another reason for the
differences in the 2D and 3D single parabolic cases could be the fact that the 2D simulations
are taken as a slice through the center of the membrane which may not accurately take into
account the cylindrical nature of the membrane in the chromatography column.
203
(a) t = 0.03125 (b) t = 6.5
(c) t = 13 (d) t = 20
Figure 8.5: The 2D evolution of the concentration in the membrane assuming a spatially-constant velocity profile. Notice that the membrane is essentially saturated with protein byt = 20 as shown in the last image.
204
(a) t = 0.03125 (b) t = 7.5
(c) t = 15 (d) t = 23
Figure 8.6: The 2D evolution of the concentration in the membrane assuming a singleparabola velocity profile. Saturation of the membrane does not occur in this case until afterthe final image shown; specifically, it occurs at approximately t = 70.
205
(a) t = 0.03125 (b) t = 7.5
(c) t = 15 (d) t = 23
Figure 8.7: The 2D evolution of the concentration in the membrane assuming a doubleparabola velocity profile. Saturation of the membrane does not occur in this case until afterthe final image shown; specifically, it occurs at approximately t = 65.
206
(a) t = 0.03125 (b) t = 7.5
(c) t = 15 (d) t = 23
Figure 8.8: The 3D evolution of the concentration in the membrane assuming a singleparabola velocity profile. Saturation of the membrane does not occur in this case until afterthe final image shown; specifically, it occurs at approximately t = 72.5.
207
(a) t = 0.125 (b) t = 1.875
(c) t = 3.75 (d) t = 5.75
Figure 8.9: The 3D evolution of the concentration in the membrane assuming a velocityprofile involving five parabolas. Saturation of the membrane does not occur in this caseuntil after the final image shown; specifically, it occurs at approximately t = 80.
208
Figure 8.10: Comparison of breakthrough curves for five different velocity profiles. Althoughsome of the velocity profiles resulted in remarkably similar breakthrough curves (e.g. singleand double parabola cases in 2D), there is enough variation in the curves to indicate furtherinvestigation into the velocity if warranted.
209
Chapter 9
Summary and Future Work
We conclude with a brief summary of work completed and possible future extensions.
9.1 Summary
In this work, we considered the time-dependent, multi-dimensional, advection-
diffusion-reaction equation used to model reactive transport in porous media. Specifically,
we focused on modeling the mechanism of protein chromatography as part of a protein sep-
arations process with the goal of developing software tools capable of simulating the protein
chromatography process under the effect of complex, implicit adsorption relationships in
the presence of highly advective flows.
With this goal in mind, we developed and analyzed various streamline upwind
Petrov-Galerkin (SUPG) finite element discretizations of the reactive transport problem.
Beginning with the steady-state equation of the SUPG formulation, we showed the bilinear
form was coercive and bounded in the H1-norm; in addition, we derived H1 error bounds
for the steady-state equation. The error bound was derived with an explicit dependence on
the upwinding parameter δ.
Continuing with the transient problem, we analyzed the cases of constant and lin-
ear adsorption. We showed solvability of the fully discrete (both Forward and Backward
210
Euler) SUPG formulations and derived L∞(L2) stability and L2(H1) error bounds for the
semi-discrete and fully-discrete problems. All the stability and error bounds for the linear
formulations were shown with an explicit dependence on the upwinding parameter δ.
As the implicit adsorption relationship was nonlinear, we also developed and an-
alyzed finite element formulations involving a nonlinear adsorption term. In the case of
an explicit adsorption relationship, three different formulations were analyzed: a time-
integrated mixed methods formulation, a time-integrated SUPG formulation, and a fully
implicit SUPG formulation.
For the time-integrated mixed methods formulation, error estimates for the semi-
discrete and fully discrete cases were proven in terms of a “time-integrated” error:
∥∥∥∥∫ T
0(C − Ch) dt
∥∥∥∥2
0
+
∥∥∥∥∫ T
0(Q−Qh) dt
∥∥∥∥2
0
.
Solvability of the time-integrated SUPG formulation was also shown. For the fully implicit
SUPG formulation, the solvability was proven with the use of the Newton-Kantarovich
theorem to deal with the nonlinearity arising from the fully implicit discretization; the
stability was proven in terms of the L2 spatial norm evaluated at the final time. The stability
bounds for both SUPG formulations were written in terms of the upwinding parameter δ.
For the implicit adsorption relationship, a simple formulation was proposed which
not only dealt with the implicit definition of the isotherm but also dealt with the nonlinear-
ity: the right hand side of the isotherm relationship was evaluated at the previous time step.
As expected, the solvability and stability bound for this relationship were shown to have a
requirement on the time step size based on the Lipschitz continuity of the right hand side
of the isotherm. While this time step requirement is unfortunate, it is not an unreasonable
restriction in real world simulations. As with the previous nonlinear stability bounds, the
stability bound for the case of implicit adsorption was explicitly written in terms of the
upwinding parameter δ.
We also showed results from various numerical experiments. The first three ex-
211
perimental results supported the steady-state and fully-discrete linear convergence rates
while the next three experiments showed the expected, although yet unproven, nonlinear
convergence rates including one using the implicit adsorption model proposed by Nfor.
Results from simulations were also provided. The comparison of numerical results
using instantaneous adsorption with experimental results (Figure 8.1) showed fairly good
breakthrough curve predictions from both Langmuir’s adsorption model and Nfor’s adsorp-
tion model; the good predictions were most likely due to the addition of the low Peclet
dispersion model which allowed some freedom in the choice of dispersivities. However,
while the 10% breakthrough point was matched by appropriate choices of the dispersivities,
the shape of the breakthrough curve did not match up as well later on.
Consequently, we also investigated the affects of non-instantaneous adsorption in the
hopes of obtaining better matching results. Incorporating non-instantaneous adsorption,
particularly that involving a transient adsorption rate, greatly improved the shape of the
numerical breakthrough curve in the case of Nfor’s adsorption model significantly.
Last, we investigated the effects of varying velocity profiles on the breakthrough
curves by running two-dimensional and three-dimensional simulations involving five different
velocity profiles. Images showing the evolution of the concentration fronts for each case
were included. A comparison of the breakthrough curves for the five simulations showed
significant differences caused by the different velocity profiles.
9.2 Future Work
The analysis presented in this work considered many different finite element dis-
cretizations of the reactive transport equation; however, the error analysis of most of the
nonlinear cases has yet to be completed and therefore should be the first step in further anal-
ysis. Later analysis should include the three-equation system involving non-instantaneous
adsorption. Considering a constant adsorption rate, or similarly one dependent on time,
would result in an additional linear equation in the system being analyzed. Since the ad-
212
sorption rate in real world simulations most likely depends on the adsorbed concentration,
considering an adsorption rate function dependent on q in the analysis would be another
extension of the current analysis.
Numerically, more work involving non-instantaneous adsorption needs to be com-
pleted. While the numerical results incorporating a transient adsorption rate are indeed
much closer to the experimental results, improvements can still be made by varying other
parameters such as the dispersivities, αL and αT . In addition, other functions can be used
for the transient adsorption rate, e.g., a logarithmic function of time, or the adsorption rate
can be made to be a function of the adsorbed concentration, q.
Due to the initial results comparing different velocity profiles, further numerical
study should be conducted into the effects of velocity on the breakthrough curves. As part
of this study, a variable velocity should be incorporated into the model to determine if the
assumption of temporally-constant velocity is accurate.
Since there are now a number of parameters which can be varied to increase the ac-
curacy of the numerical results, finding the best set of parameters could be greatly hastened
by creating an optimization algorithm. The algorithm, posed over the set of parameters
involved in the dispersion model and the transient reaction rate function, would minimize
the difference between the experimental and numerical breakthrough curves.
The minimization problem could also be adapted to verify the measured values of
the parameters used in the governing equations. It can be posed over any subset of the
parameters, giving additional insight into their specific effects on the separation process.
The effect of individual parameters became apparent recently when a change was made to
the porosity value used in simulations. Initially the manufacturer’s porosity value was being
used, but a change was made after a different value was obtained experimentally by Juan
Wang in the Department of Chemical and Biomolecular Engineering at Clemson University.
The experimentally determine porosity value resulted in a numerical breakthrough curve
which was much closer to the experimental breakthrough curve than the results using the
manufacturer’s porosity.
213
Last, a numerical analysis of the different algorithms developed for the reactive
transport problem would be very informative. All the different algorithms we developed,
including a couple of discretizations not described in this work, worked successfully for
the parameter values used in the comparison experiments. However, if we wish to apply
algorithms described in this work to other applications, it would be beneficial to know which
parameter sets are feasible for each algorithm. Consequently, future work should include
an in-depth comparison of the different algorithms, including a sensitivity analysis of the
algorithms to parameter values.
214
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