Question

A composite spherical shell is made up of two materials having thermal conductivities K and 2K respectively as shown in the diagram. The temperature at the innermost surface is maintained at T whereas the temperature at the outermost surface is maintained at 10T. A, B, C and D are four points in the outer material such that AB = BC = CD.

The effective thermal resistance between the inner surface of the shell and the outer surface of the shell for the radial heat flow is

• A. $\frac{1}{8 \pi \text{KR}}$
• B. $\frac{1}{7 \pi \text{KR}}$
• C. $\frac{7}{4 8 \pi \text{KR}}$
• D. $\frac{6}{4 9 \pi \text{KR}}$

Answer 1

Answer: (C)

$\left(\text{R}\right)_{\text{eff}} = \displaystyle \int _{\text{R}}^{2 \text{R}} \frac{\text{dx}}{\text{K} \left(4 \pi \left(\text{x}\right)^{2}\right)} + \displaystyle \int _{2 \text{R}}^{3 \text{R}} \frac{\text{dx}}{2 \text{K} \left(4 \pi \left(\text{x}\right)^{2}\right)}$
$= \frac{1}{4 \pi \text{K}} \left[- \frac{1}{\text{x}}\right]_{\text{R}}^{2 \text{R}} + \frac{1}{8 \pi \text{K}} \left[- \frac{1}{\text{x}}\right]_{2 \text{R}}^{3 \text{R}}$
$⇒ \, \, \text{R}_{\text{eff}} = \frac{7}{4 8 \pi \text{KR}}$