question_answer

A composite hollow sphere with steady internal heating is made of 2 layers of materials of equal thickness with thermal conductivities in the ratio of' 1:2 for inner to outer layers. Ratio of inside to outside diameter is 0.8. What is ratio of temperature drop across the inner and outer layers?

A) 0.4                               

B) 1.6

C) 2 in 0.8            

D) 2.5

Correct Answer: D

Solution :

\[{{t}_{1}}-{{t}_{2}}=Q\,\left[ \frac{{{r}_{2}}-{{r}_{1}}}{4\pi {{k}_{1}}{{r}_{1}}{{r}_{2}}} \right]\] \[{{t}_{1}}-{{t}_{3}}=\left[ \frac{{{r}_{3}}-{{r}_{2}}}{4\pi {{k}_{2}}{{r}_{2}}{{r}_{3}}} \right]\] \[{{r}_{2}}-{{r}_{1}}={{r}_{3}}-{{r}_{2}}\] \[\frac{{{t}_{1}}-{{t}_{2}}}{{{t}_{2}}-{{t}_{3}}}=\left( \frac{{{k}_{2}}}{{{k}_{1}}} \right)\,.\,\left( \frac{{{r}_{3}}}{{{r}_{1}}} \right)=\frac{2}{0.8}=2.5\]