A) \[{{K}_{1}}+{{K}_{2}}\]
B) \[\frac{{{K}_{1}}+3{{K}_{2}}}{4}\]
C) \[\frac{{{K}_{1}}.{{K}_{2}}}{{{K}_{1}}+{{K}_{2}}}\]
D) \[\frac{3{{K}_{1}}+{{K}_{2}}}{4}\]
Correct Answer: B
Solution :
| Rate of flow of heat in the combined system =rate of flow of heat through cross-section of inner cylinder +rate of flow of heat through cross- section of outer shell\[=\frac{KA\left( {{\theta }_{1}}-{{\theta }_{2}} \right)}{l}\]\[=\frac{{{K}_{1}}{{A}_{1}}\left( {{\theta }_{1}}-{{\theta }_{2}} \right)}{l}+\frac{{{K}_{2}}{{A}_{2}}\left( {{\theta }_{1}}-{{\theta }_{2}} \right)}{l}\] |
| \[\Rightarrow KA={{K}_{1}}{{A}_{1}}+{{K}_{2}}{{A}_{2}}\]\[\Rightarrow K\pi {{\left( 2R \right)}^{2}}={{K}_{1}}\left( \pi {{R}^{2}} \right)+{{K}_{2}}\pi \left[ {{\left( 2R \right)}^{2}}-{{R}^{2}} \right]\]\[\Rightarrow \pi {{R}^{2}}\left( K\times 4 \right)=\pi {{R}^{2}}\left( {{K}_{1}}+3{{K}_{2}} \right)\]\[\therefore K=\frac{{{K}_{1}}+3{{K}_{2}}}{4}\] |
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