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A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$. The thermal conductivity of the material of the inner cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is
$\frac{{{K_1} + {K_2}}}{2}$
$K_1 + K_2$
$\frac{{2{K_1} + {3K_2}}}{5}$
$\frac{{{K_1} + {3K_2}}}{4}$
Solution
Equivalent thermal resisitance,
$\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}$
$\frac{{k\pi {{\left( {2R} \right)}^2}}}{L} = \frac{{{k_1}\pi {R^2}}}{L} + \frac{{{k_2}\pi \left[ {{{\left( {2R} \right)}^2} – {R^2}} \right]}}{L}$
$ \Rightarrow 4k = {k_1} + 3{k_2}$
$ \Rightarrow k = \frac{{{k_1} + 3{k_2}}}{4}$
