In this section, the problem of steady state axial heat conduction in a rod is addressed. Heat is generated in the rod by an electric current and each end of the rod is held at a different fixed temperature. Around its circular periphery, the rod is perfectly insulated. The ordinary differential equation which arises in this problem is second order.
The solution to the problem gives the axial temperature distribution, T(z), and by differentiation, dT(z)/dz. The latter quantity is easily converted into the heat flux distribution, q(z), since q(z) = -k(dT(z)/dz). One can reasonably ask why anyone should be interested in T(z) or q(z). There are, of course, many reasons but one for each will suffice.
Let's consider first an example of why one might be interested in T(z).
Suppose we have a copper wire insulated by a plastic material with a zero thermal conductivity (a perfect insulator). The wire has heated up to its steady state temperature distribution and we might want to know for safety reasons whether the current passing through the wire will cause the insulation to melt or catch fire.
Why might we be interested in q(z)?
It is apparent that if heat is being generated in the rod that it must flow out somewhere. q(z) tells us what the heat flux is at any point in the rod. Does it flow out at both ends or only at one end? The collaborate section will deal with these considerations.
Now let us ask whether there are parameters other than the magnitude of the current which are important? Surely there are. The thermal and electrical conductivity of the wire should be important since nichrome wires are used for hot plates and copper for wiring buildings. And how do the length and diameter of the wire affect the temperature distribution? In addition, we need to discuss how the relevant variables and parameters are determined from the differential equation and discuss the issue of parametric sensitivity which is important, for example, in designing a temperature control system.
Finally we will change the boundary condition at the circular periphery of a bare rod and allow heat to be lost by convection from it. We will then see how this change alters the problem mathematically and how it affects the temperature distribution and heat flux.