Page 110 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

4

Steady State Heat Conduction

in One Dimension

4.1 Introduction

A one-dimensional approximation of the heat conduction equation is feasible for many
physical problems, for example, plane walls, fins, and so on (Bejan 1993; Holman 1989;
Incropera and Dewitt 1990; Ozisik 1968). In these problems, any major temperature vari-
ation is in one direction only and the variation in all other directions can be ignored.
Other examples of one-dimensional heat transfer occur in cylindrical and spherical solids
in which the temperature variation occurs only in the radial direction. In this chapter, such
one-dimensional problems are considered for steady state conditions, in which the temper-
ature does not depend on time. Time-dependent and multi-dimensional problems will be
discussed in later chapters.

4.2 Plane Walls

4.2.1 Homogeneous wall

The differential equations that govern the heat conduction through plane walls have already
been discussed in Chapter 1. The steady state heat conduction equation for a plane wall,
shown in Figure 4.1, is
2
d T
kA = 0 (4.1)
dx 2
where k is the thermal conductivity and A is the cross-sectional area perpendicular to
the direction of heat flow. The problem is complete with the following description of the

Fundamentals of the Finite Element Method for Heat and Fluid Flow R. W. Lewis, P. Nithiarasu and K. N. Seetharamu
 2004 John Wiley & Sons, Ltd ISBNs: 0-470-84788-3 (HB); 0-470-84789-1 (PB)