Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Committee Member

Dr. Lea Jenkins, Committee Chair

Committee Member

Dr. Chris Cox

Committee Member

Dr. Vince Ervin

Committee Member

Dr. Scott Husson


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 reaction. 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. To gain insight into the advection-diffusion-reaction problem, we begin by 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 SUPG formulation, and a fully implicit SUPG formulation. For the implicit adsorption rela-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 relationship is evaluated at the previous time step. As expected, the solvability and stability 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, considering both constant and transient adsorption rates. Additionally, we numerically investigate the effects of varying velocity profiles by comparing results from simulations in-volving five different profiles.