# 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 K1 and that of the outer cylinder is K2. Assuming no loss of heat , the effective thermal conductivity of the system for heat flowing along the length of the cylinder is :Option 1) $\frac{K_{1} + 3 K_{2}}{4}$Option 2) $\frac{K_{1} + K_{2}}{2}$Option 3) $K_{1} + K_{2}$Option 4)$\frac{2K_{1} + 3 K_{2}}{5}$

Equivalent Thermal Conductivity in Parallel -

$\frac{K_{1}A_{1}+K_{2}A_{2}+..........K_{n}A_{n}}{A_{1}+A_{2}+..............A_{n}}$

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Both are in parallel combination

$k_{ea}=\frac{k_{1}A_{1}+k_{2}A_{2}}{A_{1}+A_{2}}$

$A_{1}=\pi R^{2}$

$A_{2}=\pi \left (4 R^{2} \right )-\pi R^{2}=3\pi R^{2}$

$k_{ea}=\frac{k_{1}\times \pi R^{2}+k_{2}\left ( 3\pi R^{2} \right )}{4\pi R^{2}}$

$k_{ea}=\frac{k_{1}+3k_{2}}{4}$

Option 1)

$\frac{K_{1} + 3 K_{2}}{4}$

Option 2)

$\frac{K_{1} + K_{2}}{2}$

Option 3)

$K_{1} + K_{2}$

Option 4)

$\frac{2K_{1} + 3 K_{2}}{5}$

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