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 \[K=\frac{{{K}_{1}}+3{{K}_{2}}}{4}\]You need to login to perform this action.
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