Derivation of the permeation equation for diffusion of gases and vapors in flat membrane by using Laplace transform

Abstract

Generally, the model of forced diffusion of a penetrant through nonporous polymer membranes can be quantitatively described by a partial differential equation of parabolic type, which is known as Fick’s second law. In this article, the detailed explanation of application of the integral transform method (especially Laplace transform) for the solution of Fick’s second law at given initial and boundary conditions is presented. Obtained final expression for the concentration profile inside a flat membrane and the diffusion flux through a membrane were verified on permeability data of carbon dioxide and cyclohexane through low-density polyethylene membrane. While CO₂ permeation data can be successfully fitted by obtained model, in the case of cyclohexane vapors, when the diffusion coefficient cannot be supposed to be constant due to strong polymer–penetrant interactions (swelling), the agreement between model and experimental data is lower.

Keywords:

• Laplace transform
• Diffusion
• Gas/vapor permeation
• Flat membrane

Acknowledgements

The financial support of Czech Ministry of Education, Youth and Sports (Grant MSM No.6046137307 and grant MSM 6046137306) and Grant Agency of Czech Republic (P106/10/1194) is gratefully acknowledged.