Question

The heat transfer by conduction through a thick cylinder of inner radius r₁, outer radius r₂, Higher temperature T₁, lower temperature T₂, length of cylinder l and thermal conductivity k is given by
1. \(\frac{{2\pi \;lk\;\left( {{T_1} - {T_2}} \right)}}{{2.3\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}\)
2. \(\frac{{2.3\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}{{2\pi \;lk\left( {{T_1} - {T_2}} \right)}}v\)
3. \(\frac{{2\pi \left( {{T_1} - {T_2}} \right)}}{{2.3\;lk\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)\;}}\)
4. \(\frac{{2\pi \;lk}}{{2.3\;\left( {{T_1} - {T_2}} \right)\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}\)

Answer 1

Correct Answer - Option 1 : \(\frac{{2\pi \;lk\;\left( {{T_1} - {T_2}} \right)}}{{2.3\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}\)

Explanation:

For hollow cylinder:

\(Q = \frac{{{T_1} - {T_2}}}{{\frac{{\ln \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}{{2\pi lk}}}} = \frac{{2\pi lk\;\left( {{T_1} - {T_2}} \right)}}{{\ln \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}} = \frac{{2\pi lk\;\left( {{T_1} - {T_2}} \right)}}{{2.3\log \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}\)

The radial heat transfer rate through hollow cylinder increases as the ratio of outer radius to inner radius decreases.

Heat conduction through the plane wall	\(Q = \frac{{{T_1} - {T_2}}}{{\frac{l}{{kA}}}}\)
Heat conduction through a hollow cylinder	\(Q = \frac{{{T_1} - {T_2}}}{{\frac{{\ln \left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}{{2\pi\; lk}}}}\)
Heat conduction through the hollow sphere	\(Q = \frac{{{T_1} - {T_2}}}{{\frac{{{r_2} - {r_1}}}{{4\pi k{r_2}{r_1}}}}}\)