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EX3: Non-Constant Conductivity I

5: Non-Constant Conductivity I (Example 3)

Consider the same geometry as in : an infinite slab of material lies between the planes and , and these plane boundaries are kept at constant temperatures and respectively. The conductivity , however, is not assumed to be constant.

A more realistic assumption is that depends on the temperature ; the graphs of several k-vs-T relations, for various materials, appear in Figure 6.



Figure 6:





Specifically, let us assume that is a linear function of , which can be a good approximation over a short range of temperatures. Thus



Equation 43

where, , are given constants (). The sign of depends on the material and the temperature range as seen in figure 6.

We seek a steady state temperature distribution and the rate of heat flow at each point in the slab. As in Example 1, because the boundary surfaces and the boundary values of are independent of and , the temperature distribution within the slab will also be independent of and . Thus , and . Therefore the heat flux vector has only an x-component, .

Does actually depend on ? An argument like that in Example 1 shows that, in spite of the non-constant character of the conductivity , is constant; heat flows at the same rate through every plane between the boundary planes.

Now substituting given by (5.1) and in Fourier's law of heat conduction, we obtain the differential equation



Equation 44

 
 
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