question_answer

A cylinder of radius r and of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius r and outer radius 2r made of a material of thermal conductivity K2. The effective thermal conductivity of the system if

A) \[\frac{1}{3}\left( {{K}_{1}}+2{{K}_{2}} \right)\]                 

B) \[\frac{1}{2}\left( 2{{K}_{1}}+3{{K}_{2}} \right)\]

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

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

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 \[=\pi \{{{(2R)}^{2}}-{{(R)}^{2}}\}=3\pi {{R}^{2}}\] \[\Rightarrow \] \[{{K}_{eq}}=\frac{{{K}_{1}}(\pi {{R}^{2}})+{{K}_{2}}(3\pi {{R}^{2}})}{\pi {{R}^{2}}+3\pi {{R}^{2}}}\] \[=\frac{{{K}_{1}}+3{{K}_{2}}}{4}\]