question_answer

Three bars of equal lengths x and equal area of cross-section \[A\] are connected in series. Their thermal conductivities are in the ratio of \[2:4:3.\] If the open ends of the first and the last bars are at temperature \[200{}^\circ C\] and \[18{}^\circ C\],  respectively in the steady state, calculate the temperatures of Both the junctions.

A) \[116{}^\circ C,74{}^\circ C\]              

B) \[120{}^\circ C,180{}^\circ C\]

C) \[125{}^\circ C,50{}^\circ C~\]                       

D) \[130{}^\circ C,40{}^\circ C\]

Correct Answer: A

Solution :

Suppose \[{{\theta }_{1}}\]and \[{{\theta }_{2}}\]be the temperature of junctions \[B\]and\[C\], respectively.

In the steady state, the rate of flow of heat through each bar will be same.

\[\frac{Q}{t}=\frac{2K\times A\left( 200-{{\theta }_{1}} \right)}{X}\]\[=\frac{4K\times A({{\theta }_{1}}-{{\theta }_{2}})}{x}\]

\[=\frac{3K\times A\left( {{\theta }_{2}}-18 \right)}{X}\]   \[2\left( 200-{{\theta }_{1}} \right)=4\left( {{\theta }_{1}}-{{\theta }_{2}} \right)=3\left( {{\theta }_{2}}-18 \right)\] \[200-{{\theta }_{1}}=2{{\theta }_{1}}-2{{\theta }_{2}}\operatorname{and}4{{\theta }_{1}}-4{{\theta }_{2}}=3{{\theta }_{2}}-54\]

\[\Rightarrow 3{{\theta }_{1}}-2{{\theta }_{2}}=200\,\operatorname{and}4{{\theta }_{1}}-7{{\theta }_{2}}=-54\]\[\Rightarrow \left( -8+21 \right){{\theta }_{2}}=\left( 800+162 \right)\]\[\Rightarrow {{\theta }_{2}}=\frac{962}{13}=74{}^\circ \operatorname{C}\]

\[{{\theta }_{1}}=\frac{200+2\times 74}{3}=116{}^\circ \operatorname{C}\]