n Ondersoek na die gedrag van die oplossing van 'n warmtegeleidingsprobleem met varierende randmediumparameters

Abstract

Governing equations and subsidiary conditions are required for a well-posed problem. Governing equations are derived from conservation laws and constitutive equations. The applicable conservation law in this dissertation, is the law for the conservation of heat energy, formulated as a partial differential equation known as the heat equation. In this equation u is the temperature and, r is the thermal diffusivity. b is a term that describes a source for heat energy and x and t denote position and time respectively. Constitutive equations are equations that attempt to formulate a mathematical model for the physical properties of a material. In this paper, subsidiary conditions consist of boundary conditions and an initial condition. Chapter l is an introduction where the most important technology is mentioned. In Chapter 2, the necessary concepts are defined and the heat equation is derived. In Chapter 3, three possible boundary conditions that apply to the thic boundary model are discussed, namely the Neumann boundary condition, the Dirichlet boundary condition and the Dynamic boundary condition. In Chapter 4 an analytical solution is obtained for each above-mentioned problems, as well as the solution of the so called of the mother problem", where two rods, each with its own physical properties, are connected. In Chapter 5 the graphs of solutions of the problems are considered. In Chapter 6 parameters (either the heat conduction coefficient K2, or the length L) of the second rod are varied in the mother problem and the solution compared with the solutions of the above-mentioned problems. Finally, Chapter 7 is a synopsis of the results of this research.