Conduction with Heat Sources

Abstract

In this chapter we study problems of conduction combined with internal heat generation, that is the conversion of mechanical energy into internal energy. Here we list some examples of heat generation, denoting the energy “dissipated” per unit volume and unit time by $\dot{q}$.

• Dissipation of electrical energy $\dot{q} = {I^{2} } {\text{/}} {k_{e}}$, where I is the current density, in amps cm⁻², while k_e is the electric conductivity, in ohm⁻¹ cm⁻¹.

• Dissipation of mechanical energy (viscous dissipation), $\dot{q} = \mu \left| {\nabla v} \right|^{2}$, where μ is viscosity and $\nabla v$ the velocity gradient.

• Dissipation of chemical or nuclear energy.

First, in Sect. 10.1, we consider the general solution of this problem in plane, cylindrical and spherical geometries in the case of a uniform heat production. Then, in Sect. 10.2, we examine the case of a linear source term, using either regular (Sect. 10.3) or singular (Sect. 10.4) perturbation expansion methods to solve the limit cases corresponding to small heat generation or small conduction, respectively.