A numerical method for solving the generalized heat conduction equation

Abstract

A general shaped one-dimensional heat conducting body with side convection in a condition of steady state is formulated numerically, in dimensionless form, by dividing the body into elements and considering an energy balance for each element. The equations for each element are arranged such that the most advanced point in the grid is represented in terms of the preceding points, i.e., U[subscript i+1]= f(U[subscript i],U[subscript i-1]). In this way all of the unknown dimensionless temperatures in the body are expressed in terms of one unknown dimensionless temperature which is readily determined, thus enabling the calculation of all other temperatures. This formulation approach is expanded first to include transient states with internal energy generation and finally to include multi-dimensional bodies in both transient and steady states. All transient states are formulated implicitly to avoid stability limitations on the time increment. The data obtained include both temperature distributions and heat fluxes in dimensionless form for a body conducting heat away from a constant temperature source and convecting it to some surrounding medium. The data obtained are presented such that comparison of the one-, two-, and three-dimensional solutions are made for various values of key dimensionless parameters defining the physics of the problem.