A spherical shell has inner radius 3.00 cm and outer

Chapter 20, Problem 82

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A spherical shell has inner radius 3.00 cm and outer radius 7.00 cm. It is made of material with thermal conductivity k 5 0.800 W/m ? C. The interior is maintained at temperature 5C and the exterior at 40C. After an interval of time, the shell reaches a steady state with the temperature at each point within it remaining constant in time. (a) Explain why the rate of energy transfer P must be the same through each spherical surface, of radius r, within the shell and must satisfy dT dr 5 P 4pkr 2 (b) Next, prove that 3 40 5 dT 5 P 4pk 3 0.07 0.03 r 22 dr where T is in degrees Celsius and r is in meters. (c) Find the rate of energy transfer through the shell. (d) Prove that 3 T 5 dT 5 1.84 3 r 0.03 r 22 dr where T is in degrees Celsius and r is in meters. (e) Find the temperature within the shell as a function of radius. (f) Find the temperature at r 5 5.00 cm, halfway through the shell.