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Exact analytical and closed-form solutions to the transient diffusion in bi-layer composites with external mass transfer resistance are reported. Expressions for the concentrations and the mass permeated are derived in both the Laplace and time domains through the use of the Laplace transform Inversion Theorem. The lead and lag times, which are often of importance in the characterization of membranes and arise from the analysis of the asymptotic behavior of the mass permeated through the bi-layer composite, were also derived. The presented solutions are also compared to previously derived limiting cases of the diffusion in a bi-layer with an impermeable wall and constant concentrations at the upstream and downstream boundaries. Analysis of the time lag shows that this membrane property is independent of the direction of flow. Finally, an outline is provided of how these transient solutions in response to a step function increase in concentration can be used to derive more complex input conditions. The importance of adequately handling boundary layer effects has a wide array of applications such as the study of bi-layers undergoing phenomena of heat convection, gas film resistance, and absorption/desorption.

Transient Diffusion in Bi-Layer Composites With Mass Transfer Resistance: Exact Solution and Time Lag Analysis