Constructal Optimization of Internal Flow Geometry in Convection

Abstract

In this paper “constructal theory,” is used to predict the formation of geometric shape and structure in finite-size fluid systems subjected to heating from below. Two classes of system are considered as tests: (i) single-phase fluid layers, and (ii) porous layers saturated with single-phase fluids. It is shown that the minimization of thermal resistance across the layer can be used to account for the appearance of organized macroscopic motion (streams) on the background of disorganized motion (diffusion). By optimizing the shape of the flow, it is possible to predict analytically the main structural and heat transfer characteristics of the system, e.g., the onset of convection, the relation between Nusselt number and Rayleigh number, the geometric shape of the rolls, and the decreasing exponent of RaH as RaH increases. The convective flow structure emerges as the result of a process of geometric optimization of heat flow path, in which diffusion is assigned to length scales smaller than the smallest macroscopic flow element (elemental system). The implications of this test of constructal theory are discussed in the context of the wider search for a physics law of geometric form generation in natural flow systems.