question_answer

A cylinder of radius R of material of coefficient of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of a material of coefficient of thermal conductivity K2. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in the steady state. The effective coefficient of thermal conductivity of the system is

A)                  \[{{k}_{1}}+{{k}_{2}}\]                                 

B)                  \[\frac{{{k}_{1}}+3{{k}_{2}}}{4}\]

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

D)                  \[\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]

Correct Answer: B

Solution :

                 Both the cylinders are in parallel for the heat flow from one end as shown                                 Then      \[{{K}_{eq}}=\frac{{{K}_{1}}{{A}_{1}}+{{K}_{2}}{{A}_{2}}}{{{A}_{1}}+{{A}_{2}}}\]                 Where \[{{A}_{1}}=\] Area of cross-section of inner cylinder                 \[{{A}_{1}}=\pi {{r}^{2}}\]                 \[{{A}_{2}}=\]Area of cross section of cylindrical shell                 \[=\pi [(2{{R}^{2}}-{{R}^{2}})]\]                 \[=3\pi {{R}^{2}}\]                 \[{{K}_{eq}}=\frac{{{K}_{1}}{{(\pi R)}^{2}}+{{K}_{2}}{{(3\pi R)}^{2}}}{\pi {{R}^{2}}+3\pi {{R}^{2}}}\]                 \[=\frac{{{K}_{1}}+3{{k}_{2}}}{4}\]