Introduction

A wide variety of practical and interesting phenomena are governed by the 1D heat conduction equation. Heat transfer through a composite slab, radial heat transfer through a cylinder, and heat loss from a long and thin fin are typical examples. By 1D, we mean that the temperature is a function of only one space coordinate (say x or r). This indeed is the case in steady-state problems. However, in unsteady state, the temperature is also a function of time. Thus, although there are two relevant independent variables (or dimensions), by convention, we refer to such problems as 1D unsteady-state problems. The extension dimensional thus always refers to the number of relevant space coordinates.

The 1D heat conduction equation derived in the next section is equally applicable to some of the problems arising in convective heat transfer, in diffusion mass transfer, and in fluid mechanics, if the dependent and independent variables of the equation are appropriately interpreted. In the last section of this chapter, therefore, problems from these neighbouring fields will be introduced. Our overall objective in this chapter is to develop a single computer program that is applicable to a wide variety of 1D problems.

1D Conduction Equation

Consider the 1D domain shown in Figure 2.1, in which the temperature varies only in the x direction although cross-sectional area A may vary with x.