(534b) A Critical Comparison of Numerical Methods for Solving Coupled Multicomponent Fluxes for Complex Mixtures across Asymmetric Membranes

Membrane processes are becoming an increasingly attractive option for many industrial chemical separations. Relevant applications include water purification, carbon capture, hydrogen separation, olefin/paraffin separation, and benzene derivative concentration. Additionally, many membrane materials are being used, including highly selective zeolites, carbon nanotubes, ionic polymers, MOF mixed-matrix polymers, graphene-oxides, and other specialty polymers. Despite recent advances in membrane processes, widespread industrial adoption of membranes is still hindered by (i) the lack of reliable predictive models for membrane performance when confronted with complex multicomponent streams, (ii) the lack of methods and databases for parameterizing such models using pure component data and limited multicomponent measurements, and (iii) the lack of efficient and reliable numerical solution methods that are compatible with standard process simulation environments and enable technoeconomic comparison of membrane and non-membrane processes on a consistent basis.

In this presentation, we discuss and compare several numerical methods for computing the fluxes of complex multicomponent mixtures across asymmetric membranes. To capture the thermodynamic non-idealities and coupled diffusion processes occurring in such mixtures, these fluxes must be modeled using a Maxwell-Stefan (MS) framework that includes rigorous chemical potential driving forces and the ability to incorporate complex thermodynamic sorption models, strong cross-diffusional coupling, and variable permeant-membrane diffusivity models. In general, this leads to a challenging two-point boundary value problem (BVP) in differential-algebraic equations (DAEs).

There are a variety of numerical approaches that can be used to solve this problem in principle, but all of them are fairly advanced. To date, the membrane literature has overwhelmingly either avoided solving these equations using various approximations, or solved the equations using basic full discretization methods such as finite differences. These equations can also be solved using more sophisticated discretization techniques or using shooting approaches. In the 2020 Spring AIChE Conference, we presented a novel single-shooting algorithm and, more recently, have proposed a multiple-shooting approach. In general, it is known that discretization and shooting methods have various advantages and disadvantages relative to one another, and the best choice depends on mathematical properties of the system. For example, shooting methods are more appropriate for stiff systems, where full discretization would require a prohibitive number of discretization points. On the other hand, full discretization may be preferred when the permeants have nearly linear transmembrane concentration profiles, or when single-shooting would lead to an unstable initial value problem. However, there have been no comprehensive studies to date investigating which algorithm is most appropriate for complex membrane simulations, and how the physical features of the permeant-membrane system under investigation impact the mathematical properties that dictate solution strategy. From our extensive work with experimental collaborators over the past year, we have identified certain physical features, such as the presence of highly nonideal sorption thermodynamics, strong cross-diffusional coupling, and/or components with very low membrane compositions, that have a very significant impact on the required solution strategy. In this presentation, we will review several numerical solutions approaches, from the simplest to the most complex, and provide both numerical results and physical insights indicating when and why each solution algorithm is most efficient and reliable for the simulation of asymmetric membrane transport within process simulation environments.

This presentation will also include a showcase of our recently developed tool, asyMemLocal, released through the RAPID Software Toolbox, that allows the end user to directly use the presented membrane transport model and numerical methods for any industrial or academic applications. This tool includes scripts to (i) fit parameters for the sorption and diffusion models based solely on single component data, (ii) run single complex mixture permeation simulations, (iii) loop over various sorption models and experimental conditions to see how well each model predicts single component permeation, and (iv) run any number of mixture simulations for all included diffusion models, sorption models, and thermodynamic assumptions to test which model choice performs best for the given polymer membrane material.