Validation of a closed-form solution to Fick’s diffusion laws for non-steady state sorption by a plane sheet

Fick's second law is a partial differential equation that describes the time-dependent concentration distribution of a diffusing species. It is fundamental to fields like environmental and materials engineering. Traditional solutions rely on infinite trigonometric series, which necessitate truncation and can be computationally demanding. In contrast, the authors previously introduced a new closed-form analytical solution (based on error functions) for sorption into and out of a plane sheet of thickness l. This study validates the proposed closed-form solution by directly comparing it with the classical trigonometric series solution. A comparison of total uptake predictions, as a function of the dimensionless ratio z=l/4Dt (where D is the diffusion coefficient and t is time), reveals: • For z>3, the closed-form solution perfectly matches the trigonometric series solution • For z<3, maximum deviations of 8.8% were observed between both solutions. However, a simple correction term f(z) was derived, reducing the deviations to a negligible maximum of 0.22%The results confirm that the proposed closed-form solution offers a simple, reliable and efficient alternative for calculating and modelling sorption into and out of a plane sheet.