Development of an analytical solution to diffusion equation for multidimensional solids of arbitrary shape

Abstract

Mass transfer processes between solids and fluids are ubiquitous in the food engineering field and using simple analytical models for their simulation or estimation of mass diffusivities remains widespread. Unfortunately, these solutions are available only for a limited number of simple solid geometries which might not be applicable to all food shapes; thus, this study introduces a simple method to obtain the volume average concentration in multidimensional solids of arbitrary shape (SAS). The proposed method approximates the (analytical or numerical) solution of the SAS from the analytical solution of a properly sized box by minimizing a weighted similitude index between the original and box shapes. Besides, the method was generalized to consider the isotropic shrinkage of foods (that is, size reduction while maintaining the aspect ratio). The applicability of the equivalent box approach was exemplified by solving inverse problems for the estimation of caffeine diffusivity during aqueous extraction of green coffee beans and water diffusivity during lentil drying using data available from literature. The results were compared with those obtained by the finite element solution of the 3D mass transfer model using the real product shape. Mass diffusivities for caffeine in green coffee and for water in lentils estimated with the equivalent box approach were not statistically different ($p>0.05$) to those estimated by numerically solving the 3D mass transfer model under the same assumptions. This method represents a simple and reliable way to model mass transfer in complex-shaped foods.