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Thermal Resistance

A composite s...

Question

A composite slab has two layers of different materials, having thermal conductivities $k_{1}$ and $k_{2}$ . If each layer has same thickness, then what is the equivalent thermal conductivity of the slab?

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is A $\frac{2 k_{1} k_{2}}{(k_{1} + k_{2})}$
Let $t$ be the thickness of each slab and
$A$ be the cross sectional area.

From thermal resistance analogy,
$\Rightarrow \frac{t}{k_{1} A} + \frac{t}{k_{2} A} = \frac{2 t}{k_{e q} A}$
$\Rightarrow \frac{t A (k_{1} + k_{2})}{k_{1} k_{2} A^{2}} = \frac{2 t}{k_{e q} A}$
$\Rightarrow k_{e q} = \frac{2 k_{1} k_{2}}{k_{1} + k_{2}}$

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