Transient Heat Flow in a Semi-Infinite Solid

Consider the semi-infinite solid shown in Fig. 4-3 maintained at some initial temperature T,. The surface temperature is suddenly lowered and maintained at a temperature T0, and we seek an expression for the temperature distribution in the solid as a function of time. This temperature distribution may subsequently be used to calculate heat flow at any x position in the solid as a function of time. For constant properties, the differential equation for the temperature distribution T(x, r) is 

This is a problem which may be solved by the Laplace-transform technique. The solution is given in Ref. I as 

It will be noted that in this definition rf is a dummy variable and the integral is a function of its upper limit. When the definition of the error function is inserted in Eq. (4-8), the expression for the temperature distribution becomes 

The surface heat flux is determined by evaluating the temperature gradient at x = 0 from Eq. (4-11). A plot of the temperature distribution for the semiinfinite solid is given in Fig. 4-4. Values of the error function are tabulated in Ref. 3, and an abbreviated tabulation is given in Appendix A. 

For the same uniform initial temperature distribution, we could suddenly expose the surface to a constant surface heat flux q jA. The initial and boundary conditions on Eq. (4-7) would then become 

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TRANSIENT HEAT FLOW IN A SEMI-INFINITE SOLID... 

Fig. 4-3 Nomenclature lor transient heat flow in a semi- infinite solid...

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