Introduction to Finite Element Analysis and Design, 2nd Edition

Chapter 4Finite Elements for Heat Transfer Problems

4.1 INTRODUCTION

In this chapter, we will demonstrate the use of finite element analysis (FEA) in heat transfer problems, especially heat conduction in a solid. This is different from the thermal stress problem discussed in chapter 1. For simplicity, we will derive the equations for one‐dimensional heat transfer. Although, on the surface, the heat transfer problem looks different from the structural mechanics problem, there are a number of similarities between the two. The thermal conductivity is the material property that plays the role of Young’s modulus, and the temperature gradient is analogous to strain. Similarly, heat flow across the boundary of the solid is analogous to the surface traction in structural analysis, and internally generated heat is similar to the body force. In heat transfer problems, we solve for the temperature field instead of the displacement field. Table 4.1 compares the terms that are used in structural mechanics and their counterparts in heat conduction. Thus, as far as the finite element method is concerned, the two problems are similar, if these terms are interpreted appropriately.

Table 4.1 Analogy between structural and heat conduction problems

Structural Mechanics	Heat Transfer
Displacement (vector)	Temperature (scalar)
Stress (tensor)	Heat flux (vector)
Displacement boundary conditions	Temperature boundary conditions
Traction boundary conditions	Surface heat input boundary ...