Question

A cylinder of radius $r$ and of thermal conductivity $K_{1}$ is surrounded by a cylindrical shell of inner radius $r$ and outer radius $2 r$ made of a material of thermal conductivity $K_{2} .$ The effective thermal conductivity of the system is

• A. $ \frac{1}{3}({{K}_{1}}+2{{K}_{2}}) $
• B. $ \frac{1}{2}(2{{K}_{1}}+3{{K}_{2}}) $
• C. $ \frac{1}{4}(3{{K}_{2}}+2{{K}_{1}}) $
• D. $ \frac{1}{4}({{K}_{1}}+3{{K}_{2}}) $

Answer 1

Answer: (D)

Both the cylinders are in parallel, for the heat flow from one end as shown

Hence, $K_{e q}=\frac{K_{1} A_{1}+K_{2} A_{2}}{A_{1}+A_{2}}$
where $A_{1}=$ area of cross-section of inner cylinder $=\pi R^{2}$ and
$A_{2}=$ area of cross-section of cylindrical shell
$=\pi\left\{(2 R)^{2}-(R)^{2}\right\}=3 \pi R^{2} $
$\Rightarrow K_{e q}=\frac{K_{1}\left(\pi R^{2}\right)+K_{2}\left(3 \pi R^{2}\right)}{\pi R^{2}+3 \pi R^{2}} $
$=\frac{K_{1}+3 K_{2}}{4}$