(422e) On the Numerical Solution of Rigorous Membrane Module Models Involving Nonideal Multicomponent Mixtures

Dylan Weber¹, Chau-Chyun Chen², Joseph Scott^1†

1 Department of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

2 Department of Chemical Engineering, Texas Tech University, Lubbock, TX, 79409, USA.

Corresponding author; Email: joseph.scott@chbe.gatech.edu

Abstract

In this presentation, we consider the problem of efficiently and reliably solving rigorous models of membrane processes involving nonideal multicomponent mixtures. Numerous applications of membranes involve nonideal multicomponent mixtures, including water purification, carbon capture, hydrogen separation, olefin/paraffin separation, and benzene derivative concentration. Moreover, these applications use many different membrane materials that can interact with such mixtures to produce complex coupled transport phenomena that cannot be captured with simple constant permeability models (e.g., zeolites, carbon nanotubes, ionic polymers, mixed-matrix MOF, graphene-oxide, and specialty polymers).

Membrane process models can be viewed as having two distinct parts: (i) a local flux model, which relates the driving force at an arbitrary point on the membrane surface to the transmembrane flux of each component at that point, and (ii) a global module model, which governs the driving force at every point on the membrane surface as a function of flowrates, module geometry, flow configuration, etc. For nonideal multicomponent mixtures with coupled transport, evaluating the local flux model at a single point on the membrane already requires the solution of nonlinear two-point boundary-value problem in differential-algebraic equations (DAEs). We presented an efficient and reliable shooting algorithm for solving this problem based on a general Maxwell-Stefan (MS) coupled transport model in a recent contribution to the 2020 AIChE Spring Meeting. However, embedding such a local flux model within a global model of an entire membrane module leads to a considerably more difficult problem. Industrially, membrane process modules can take the form of hollow fiber, spiral wound, flat plate, tubular, or plate-and-frame units. Moreover, each of these module types can be operated with various flow configurations, including co-current, counter-current, or crossflow patterns. Finally, there is also significant interest in membrane modules with chemical reactions occurring on the feed side or within the membrane itself. As a result, accurate models of complete membrane modules often take the form of a multidimensional boundary value problems in nonlinear partial differential algebraic equations (PDAEs), often with stiffness along multiple dimensions.

This problem of solving membrane module models numerically has been researched extensively over the past half century. Existing methods include full discretization approaches that solve all of the governing equation simultaneously at a set of nodal points, as well as a variety of shooting-type algorithms applicable under various simplifying assumptions. However, nearly all existing methods for simulating full membrane modules with nontrivial flow configurations assume a very simple form for the embedded local flux model. Typically, the local flux is computed using a constant permeability linear driving force model, with slightly more complex nonlinear algebraic expressions used in some cases. On the other hand, prior works using detailed local flux models that can capture coupled multicomponent transport using the Maxwell-Stefan framework have done so only under the assumption of well-mixed volumes on the feed and permeate sides. This reduces the system to a single local flux calculation, but can significantly over-estimate performance metrics compared to practical industrial units. To date, an effective numerical solution procedure for models including both detailed MS local flux models and realistic module configurations has not been proposed in the literature.

In this talk, we will present our recent progress towards an efficient and reliable numerical method for solving full membrane module models for nonideal multicomponent mixtures with detailed MS-based local flux calculations and realistic flow configurations. Numerical experiments comparing our tailored membrane algorithms to general purpose full discretization and shooting methods will be presented. Finally, we will discuss the applicability of key simplifying assumptions that can significantly reduce computational complexity in certain cases.