Page 134 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

5

Steady State Heat Conduction

in Multi-dimensions

5.1 Introduction

As seen in the previous chapters, a one-dimensional approximation is easy to implement and
is also economical. However, the majority of heat transfer problems are multi-dimensional
in nature (Bejan 1993; Holman 1989; Incropera and Dewitt 1990; Ozisik 1968). For such
problems, the accuracy of the solution can be improved using either a two- or a three-
dimensional approximation. For instance, conduction heat transfer in an infinitely long
hollow rectangular tube, which is exposed to different boundary conditions inside and out-
side the tube (Figure 5.1(a)), and heat conduction in a thin plate, which has negligible
heat transfer in the direction of the thickness may be approximated as two-dimensional
problems.
In certain situations, it is often difficult to simplify the problem to two dimensions
without sacrificing accuracy. Most complex industrial heat transfer problems are three-
dimensional in nature because of the complicated geometries involved. Heat transfer in
aircraft structures and heat shields used in space vehicles are examples of such problems. It
is, however, important to note that even geometries that are simple but which have complex
boundary conditions become three-dimensional in nature. For example, the same hollow,
rectangular tube mentioned previously, but in this case having non-uniform conditions
along the length, is a three-dimensional problem. Also, if the hollow rectangular tube is
finite, again it may be necessary to treat it as a three-dimensional problem (Figure 5.1).
One typical and simple example of three-dimensional heat conduction is that of a solid
cube subjected to different boundary conditions on all six faces as shown in Figure 5.1(b).
Another approximation, commonly employed in heat conduction studies, is the axisym-
metric formulation. This type of problem is often considered as a two-and-a-half-dimensional
case as it has the features of both a two- and a three-dimensional approximation. If a geom-
◦
etry is generated by revolving a surface through 360 with reference to its axis then it
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)