1.3.1 Introduction

The generation of thermal energy within a conductive medium may occur through ohmic dissipation, chemical or nuclear reactions, or absorption of radiation. According to the first law of thermodynamics, energy cannot be generated (excluding nuclear reactions); however, it can be converted from other forms (e.g., electrical energy) to thermal energy. The energy balance that we use to solve these problems is then strictly a thermal energy conservation equation. The addition of thermal energy generation to the 1-D steadystate solutions considered in Section 1.2 is straightforward and the steps required to obtain an analytical solution are essentially the same.

1.3.2 Uniform Thermal Energy Generation in a Plane Wall

Consider a plane wall with temperatures fixed at either edge that experiences a volumetric generation of thermal energy, as shown in Figure 1-7.

Figure 1-7: Plane wall with thermal energy generation and fixed temperature boundary conditions.

The problem is assumed to be 1-D in the x-direction and therefore an appropriate differential control volume has width dx (see Figure 1-7). Notice the additional energy term in the control volume that is related to the generation of thermal energy. A steadystate energy balance includes conduction into the left-side of the control volume ( _x, at position x), generation within the control volume ( ?), and conduction out of the rightside of the control voume ( , at position x + dx):

or, after expanding the right side:

The rate of thermal...