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

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Answer & Solution

Correct Answer

(D) \(\frac{{{K_1} + {3K_2}}}{4}\)

Detailed explanation

Equivalent thermal resisitance,

\(\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

\(\frac{{k\pi {{\left( {2R} \right)}^2}}}{L} = \frac{{{k_1}\pi {R^2}}}{L} + \frac{{{k_2}\pi \left[ {{{\left( {2R} \right)}^2} - {R^2}} \right]}}{L}\)

\( \Rightarrow 4k = {k_1} + 3{k_2}\)

\( \Rightarrow k = \frac{{{k_1} + 3{k_2}}}{4}\)

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