ABSTRACT
In this chapter, general heat conduction equation are derived, where boundary and initial conditions and solution are discussed. Solution methods are divided into two groups: classical mathematical methods and numerical methods. The chapter demonstrates classical mathematical methods for multidimensional steady conduction and for unsteady conduction. In particular, the method of separation of variables is used, which leads to the need to construct Fourier series expansions. The chapter presents the analysis of a special class of conduction problems, in which there is a moving boundary — for example, solidification of ice from water. Numerical methods used to solve the heat conduction equation include the finite-difference, the finite-element, and the boundary-element method. The chapter also demonstrates the use of the finite-difference method for steady two-dimensional conduction, unsteady one-dimensional conduction, and a moving-boundary problem. The energy conservation principle and Fourier's law of heat conduction are used to derive various forms of the differential equation governing the temperature distribution in a stationary medium.
