Extending effective medium theory to finite size systems: Theory and simulation for permeation in mixed-matrix membranes

Abstract

We present a novel theory for estimation of the effective permeability of pure gases in flat mixed-matrix membranes (MMMs), in which effective medium theory (EMT) is extended to systems with finite filler size and membrane thickness. We introduce an inhomogeneous filler volume fraction profile, which arises due to depletion of the filler in regions adjacent to the membrane ends, into the MMM permeation model. In this way, the effective medium approach (EMA) can still be applied to systems where the dispersant size is not small in comparison to the membrane thickness, and for which a permeability profile arises in the MMM that is dependent on both filler size and membrane thickness, besides the filler-polymer equilibrium constant. It is found that increase in particle size reduces the effective membrane permeability at fixed membrane thickness, and that the effective membrane permeability increases with increase of the membrane thickness to asymptotically reach the value predicted by existing models. The present theory is validated against detailed simulations of the transport in MMMs, and theoretical predictions are found to be in agreement with those obtained from the exact calculations. Further, comparison of the exact effective permeability at different filler volume fractions is made for different packing configurations, showing variations in dispersant packing structure to have only a very weak effect on MMM performance.

Modeling Permeation through Mixed-Matrix Membranes: A Review

“…Later, Gonzo et al [145] fitted Chiew and Glandt's results for K 2 , which led to Equation (27) in Table 2. Further, the Chiew-Glandt model corresponds to the exact solution to the series truncated after K 2 -term [141,149], with the model applicable to moderate/high filler loadings (φ f ≤ 0.645) [141,145,151]. Table 2.…”

“…Recently, Monsalve-Bravo and Bhatia [151][152][153] used the Chiew-Glandt model in conjunction with the one-dimensional transport equation for the permeant to describe the permeability for various gases in flat and hollow fiber MMMs. The semi-analytical model is given by Equation (30) in Table 2, referred to as the MB-B model.…”

“…Further, the locally averaged filler volume fraction ( ) f φ in Equation (31) averages the filler volume fraction ( ) f φ over the particle volume at a given membrane location, with o r being the filler particle radius and ( , , ) R R r θ ′ the location in the particle relative to the position R in the MMM. Thus, the MB-B model incorporates effects of particle size [151], membrane geometry [152] and isotherm nonlinearity [153] in the calculation of the MMM permeability. The MB-B model reduces to the Chiew-Glandt model when 0 o r →  .…”

“…The simulation-based rigorous modeling approach (SMA) is based on numerical solution of coupled partial differential equations (PDEs) describing the transport through the MMM via the finite-element method (FEM) [20,151,169], or any other suitable method (e.g., finite volume method or boundary element method), to discretize the 3D computational system [123]. Thus, the SMA is based on the assumption that the steady-state transport through the MMM is well-described by the continuity equation [151,170], as:…”

“…In Equations (50) and (51), ∇C and D are the concentration gradient and diffusivity in a given phase, respectively. In this way, Equations (49)-(51) are solved in the 3D MMM [151], in which the resulting steady-state flux is used to calculate the MMM permeability via Equation (2) [152].…”