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A metal block A of square cross-section of area a, length d and thermal conductivity $k_{1}$ is in contact with two blocks B and C each of area of cross section a/2, length d/2 and thermal conductivity $k_{2}$ and $k_{3}$ respectively. If the left side A is temperature $T_{2}$ and the right side B and C at temperature $T_{1}$ $(T_{1} < T_{2}) .$ The effective thermal conductivity is

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$\frac{k_{1} k_{2} k_{3}}{k_{1} + k_{2} + k_{3}}$

$\frac{3}{2} {(\frac{1}{k_{1}} + \frac{1}{k_{2} + k_{3}})}^{- 1}$

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

$\frac{2}{3} (k_{1} + k_{2} + k_{3})$

answer is D.

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Detailed Solution

$R = R_{1} + {(\frac{1}{R_{2}} + \frac{1}{R_{3}})}^{- 1}$ ;

$\frac{3}{2} \frac{d}{k_{e f f} A} = \frac{d}{k_{1} A} + \frac{d / 2}{(\frac{k_{2} + k_{3}}{2}) A}$

$k_{e f f} = \frac{3}{2} {(\frac{1}{k_{1}} + \frac{1}{k_{2} + k_{3}})}^{- 1}$

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