question_answer

A solid cylinder of radius R made of a material of thermal conductivity \[{{K}_{1}}\] is surrounded by a cylindrical shell of inner radius R and outer radius 1R made of a material of thermal conductivity\[{{K}_{2}}\]. The two ends of the combined system are maintained at two different temperatures. Then there is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is

A)  \[{{K}_{1}}+{{K}_{2}}\]          

B)  \[\frac{{{K}_{1}}{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\]

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

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

Correct Answer: D

Solution :

: Area of cross-section of inner cylinder \[=\pi {{R}^{2}}\] Area of cross-section of outer shell \[=\pi {{(2R)}^{2}}-\pi {{R}^{2}}=3\pi {{R}^{2}}\] Rate of heat flow in inner cylinder \[{{H}_{1}}=\frac{{{K}_{1}}\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}\] Rate of heat flow in outer shell \[{{H}_{2}}=\frac{{{K}_{2}}3\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}\] Rate of heat flow in the combined system \[H=\frac{K4\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}\] At steady state, \[H={{H}_{1}}+{{H}_{2}}\] \[\therefore \] \[\frac{K4\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}=\frac{{{K}_{1}}\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}\] \[+\frac{{{K}_{2}}3\pi {{R}^{2}}({{T}_{1}}-{{T}_{2}})}{L}\] \[4K={{K}_{1}}+3{{K}_{2}}\]  or   \[K=\frac{{{K}_{1}}+3{{K}_{2}}}{4}\]