A Solution Procedure for the Equations that Govern Three-dimensional Free Convection in Bulk Stored Grains

Abstract

This article presents a mathematical formulation of the physical laws that govern the behavior of free convection and diffusion processes that occur in grain bulks. The analysis applies to the transient behavior of three-dimensional systems, and this represents a significant improvement on previous analyses. The momentum equation is based on Darcys law, and a one concentration equation is proffered to describe the convection and diffusion of fumigants through the interstitial air and grain kernels. The momentum equation is expressed in terms of a vector potential that ensures mass is conserved. The governing equations are discretised on both uniform and non-uniform grids and solved using an alternating direction implicit method. During each real-time step, the components of the vector potential are evaluated by solving a parabolic equation by means of a false transient method. The walls and roof of the grain store have been taken as being isothermal, while the floor is assumed to be adiabatic. Numerical experiments have been performed to investigate the effects of the false transient parameter, the grid size, and non-uniformity. It is shown that a 313131 non-uniform grid yields accurate solutions to the partial differential equations. Graphical results are presented for the temperature, vector potential, grain moisture content, dry matter loss, and pesticide decay and fumigant concentration at selected cross-sections of a grain store with a simple geometry.