Conductive mass transport from a semi-infinite lattice of particles

Abstract

A method is presented for computing the instantaneous rate of conductive mass transport from a semi-infinite lattice of particles with arbitrary shape, in the limit as the particle size becomes small compared to the particle separation. A fixed rate of transport is imposed above the lattice, and the concentration is assumed to be constant and uniform over the particles' surface. Transport due to forced or natural convection is neglected, and the particle dissolution or evaporation is assumed to be limited by diffusion. In the mathematical formulation, the concentration field induced by each particle layer is expressed in terms of the doubly-periodic Green’s function of Laplace’s equation in three dimensions, which is evaluated using either a Fourier series or an alternative representation involving rapidly converging Ewald sums. The rate of transport from each layer is found using the method of matched asymptotic expansions resulting in a system of linear algebraic equations. Numerical results are presented for lattices with different configurations at various particle volume fractions, showing exponential decay of the rate of transport with distance from the top layer, in agreement with theoretical predictions. Having obtained general expressions for the rate of transport, a system of differential equations governing the evolution of the radii of dissolving spherical particles is derived. Numerical solutions illustrate the distribution of the particle radii after a periodic state has been established.