Derive the expression for thermal resistance for the composite plane wall shown in Figures given below. Assume area of cross-section to be 'A' for the Fig. a.
Consider a composite plane wall consisting of three different materials arranged in series, each having thickness (L_1, L_2, L_3), thermal conductivities (k_1, k_2, k_3), and a common cross-sectional area (A). Let the Temperature at the left and right faces be (T_1) and (T_4), respectively. Heat flows steadily in one direction, and under steady-state conditions, the heat flow rate (q) remains constant through all layers.
For one-dimensional heat conduction through a plane wall, Fourier’s law gives:
[
q = \frac{kA (T_{\text{hot}} - T_{\text{cold}})}{L}
]
This can be rearranged in terms of thermal resistance:
[
R = \frac{L}{kA}
]
Thus, each layer offers a resistance to heat flow:
For material 1:
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