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. $\frac{7}{48\pi \text{KR}}$
• D. $\frac{6}{49\pi \text{KR}}$

Answer 1

Answer: (C)

${\mathrm{R}}_{\mathrm{eff}}={\int }_{\mathrm{R}}^{2\mathrm{R}}\frac{\mathrm{dx}}{\mathrm{K}\left(4\pi {\mathrm{x}}^2\right)}+{\int }_{2\mathrm{R}}^{3\mathrm{R}}\frac{\mathrm{dx}}{2\mathrm{K}\left(4\pi {\mathrm{x}}^2\right)}$
$=\frac{1}{4\pi \text{K}}{\left[-\frac{1}{\text{x}}\right]}_{\text{R}}^{2\text{R}}+\frac{1}{8\pi \text{K}}{\left[-\frac{1}{\text{x}}\right]}_{2\text{R}}^{3\text{R}}$
$\Rightarrow {\text{R}}_{\text{eff}}=\frac{7}{48\pi \text{KR}}$