Question 102 of 126

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:

A$\dfrac{2{K}_{1}+3{K}_{2}}{5}$B$\dfrac{{K}_{1}+{K}_{2}}{2}$C${K}_{1}+{K}_{2}$D$\dfrac{{K}_{1}+3{K}_{2}}{4}$

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Answer: (D) $\dfrac{{K}_{1}+3{K}_{2}}{4}$

Let ${q}_{1}$ and ${q}_{2}$ be the rate of flow of heat through inner part from outer part respecting net flow of heat


$q={q}_{1}+{q}_{2}$

$=\dfrac{{K}_{1}\pi {R}^{2}}{l}.\Delta T+\dfrac{{K}_{2}.\pi \left[{\left(2R\right)}^{2}-{R}^{2}\right]}{l}.\Delta T$

$=\dfrac{\pi {R}^{2}}{l}\Delta T\left({K}_{1}+3{K}_{2}\right)$

If the cylinder is replaced by a single material of thermal conductivity $K$ then

$q=\dfrac{K.{\left(2R\right)}^{2}}{l}.\Delta T$

On comparison,

$K=\dfrac{{K}_{1}+3{K}_{2}}{4}$.