Conduction with Internal Heat Generation

CONDUCTION WITH INTERNAL HEAT GENERATION:

Applications: current carrying conductor, chemically reacting systems, nuclear reactors. Energy generated per unit volume is given by V Eq.

Plane wall with heat source: Assumptions: 1D, steady state, constant k, uniform

Consider one-dimensional, steady-state conduction in a plane wall of constant k, with uniform generation, and asymmetric surface conditions: Heat diffusion equation.

A medium through which heat is conducted may involve the conversion ofmechanical, electrical, nuclear, or chemical energy into heat (or thermal energy).In heat conduction analysis, such conversion processes are characterizedas heat generation.

For example, the temperature of a resistance wire rises rapidly when electriccurrent passes through it as a result of the electrical energy being convertedto heat at a rate of I2R, where I is the current and R is the electricalresistance of the wire The safe and effective removal of this heataway from the sites of heat generation (the electronic circuits) is the subjectof electronics cooling, which is one of the modern application areas of heat transfer.

Likewise, a large amount of heat is generated in the fuel elements of nuclearreactors as a result of nuclear fission that serves as the heat source for the nuclearpower plants. The natural disintegration of radioactive elements in nuclearwaste or other radioactive material also results in the generation of heat throughout the body. The heat generated in the sun as a result of the fusion ofhydrogen into helium makes the sun a large nuclear reactor that supplies heatto the earth.

Another source of heat generation in a medium is exothermic chemical reactionsthat may occur throughout the medium. The chemical reaction in thiscase serves as a heat source for the medium. In the case of endothermic reactions,however, heat is absorbed instead of being released during reaction, and thus the chemical reaction serves as a heat sink. The heat generation term becomes

a negative quantity in this case.

Often it is also convenient to model the absorption of radiation such as solarenergy or gamma rays as heat generation when these rays penetrate deepinto the body while being absorbed gradually. For example, the absorption ofsolar energy in large bodies of water can be treated as heat generation throughout the water at a rate equal to the rate of absorption, which varies withdepth. But the absorption of solar energy by an opaque bodyoccurs within a few microns of the surface, and the solar energy that penetratesinto the medium in this case can be treated as specified heat flux on the surface.

Note that heat generation is a volumetric phenomenon. That is, it occursthroughout the body of a medium. Therefore, the rate of heat generation in amedium is usually specified per unit volume.

The rate of heat generation in a medium may vary with time as well as positionwithin the medium. When the variation of heat generation with positionis known, the total rate of heat generation in a medium of volume V can be determinedfrom In the special case of uniform heat generation, as in the case of electricresistance heating throughout a homogeneous material, the relation in

reduces to E ·gen _ e · genV, where Egen is the constant rate of heat generation per unit volume.