A composite slab has two layers of different materials with thermal conductivities k₁ and k₂. If each layer has the same thickness, then the equivalent thermal conductivity of the slab will be

A composite slab has two layers of different materials with thermal conductivities k and k. If each layer has the same thickness, then the equivalent thermal conductivity of the slab will be

Right Answer is:

2 k₁ k₂/ (k₁ + k₂)

SOLUTION

​For a solid plate, thermal resistance is given by:

Rthermal = ΔT/Q

Since the slabs are in contact with each other side by side. Therefore, the thermal resistances will be in series.

The equivalent thermal resistance to the flow of heat is given by,

$\begin{array}{l} {R_{eq}} = {R_1} + {R_2}\\ \\ \dfrac{{{L_{eq}}}}{{{k_{eq}}A}} = \dfrac{{{L_1}}}{{{k_1}A}} + \dfrac{{{L_2}}}{{{k_2}A}} \end{array}$

Since L1 = L2 = L

$\begin{array}{l} \dfrac{{2L}}{{{k_{eq}}}} = \dfrac{L}{{{k_1}}} + \dfrac{L}{{{k_2}}}\\ \\ \dfrac{{{k_{eq}}}}{2} = \dfrac{{{k_1}{k_2}}}{{{k_1} + {k_2}}}\\ \\ {k_{eq}} = \dfrac{{2{k_1}{k_2}}}{{{k_1} + {k_2}}} \end{array}$

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