E1 Steady-State Heat Conduction

1 Introduction

Conduction heat transfer is the transfer of energy caused by interaction between adjacent molecules of a stagnant substance, gaseous, liquid, or solid, subject to a temperature difference between some solid boundaries, which means a difference of molecular kinetic energy. Unless they are confined to small spaces, gases and liquids tend to move when heated and transfer heat by the regime of “natural convection.” Thus, heat conduction mostly occurs in solid bodies. The basic equation for the heat flow $\dot Q$ (rate of heat transferred) is the Fourier law of heat conduction equation (1):

$$\dot Q = - \lambda { A}{{{\rm d}T } \over {{\rm d}x}}.$$

((1))

With $\lambda$ being the thermal conductivity of the material, A the area of the body normal to the heat flow in the direction of x (one dimensional flow), and ${\rm {d}}T /{\rm {d}}x$ the temperature gradient. If there is no dependence on time, there is a steady flow i.e., steady conduction.

In the following, the...