A spherical shell of thermal conductivity has inside

Chapter , Problem 39

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A spherical shell of thermal conductivity has inside radius and outside radius (Figure 20-19). The inside of the shell is held at a temperature and the outside of the shell is held at temperature with In this problem, you are to show that the thermal current through the shell is given by where is positive if heat is transferred in the direction. Here is a suggested procedure for obtaining this result: (1) obtain an expression for the thermal current through a thin spherical shell of radius and thickness when there is a temperature difference across the thickness of the shell; (2) explain why the thermal current is the same through each such thin shell; (3) express the thermal current through such a shell element in terms of the area the thickness and the temperature difference across the element; and (4) separate variables (solve for in terms of r and dr) and integrate.

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