question_answer

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)  \[\frac{1}{4}({{K}_{1}}+3{{K}_{2}})\]

Correct Answer: D

Solution :

                 Both the cylinders are in parallel, for the heat flow from one end as shown Hence,\[{{K}_{eq}}=\frac{{{K}_{1}}{{A}_{1}}+{{K}_{2}}{{A}_{2}}}{{{A}_{1}}+{{A}_{2}}};\]where\[{{A}_{1}}=\]area of cross-section of inner cylinder\[=\pi {{R}^{2}}\]and \[{{A}_{2}}=\]area of cross-section of cylindrical shell \[=\pi \{{{(2R)}^{2}}-{{(R)}^{2}}\}=3\pi {{R}^{2}}\] \[\Rightarrow \]               \[{{K}_{eq}}=\frac{{{K}_{1}}(\pi {{R}^{2}})+{{K}_{2}}(3\pi {{R}^{2}})}{\pi {{R}^{2}}+3\pi {{R}^{2}}}\]                 \[=\frac{{{K}_{1}}+3{{K}_{2}}}{4}\]