A cylinder of radius R made of a material of thermal conductivity K...

A cylinder of radius

R

made of a material of thermal conductivity

{K}_{1}

is surrounded by a cylindrical shell of inner radius

R

and outer radius

2R

made of a material of thermal conductivity

{K}_{2}

. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is–

Options

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

Correct answer

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

Step-by-step solution

Parallel combination of cross - section area

\piR^{2}

and

\pi\left[\left(2R\right)2 - R^{2}\right] = 3\piR^{2}

\dfrac{1}{R\text{eq}} = \dfrac{1}{R1} + \dfrac{1}{R2}

with

R = \dfrac{L}{KA}

\dfrac{{K}_{eq}\text{.}4\piR^{2}}{L} = \dfrac{{K}_{1}\piR^{2}}{L} + \dfrac{{K}_{2}\left(3\piR^{2}\right)}{L}

⇒ {K}_{eq} = \dfrac{{K}_{1} + 3{K}_{2}}{4}

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