Three plane slabs of thicknesses

PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter

Problem

Three plane slabs of thicknesses d1d_1, d2d_2 and d3d_3 and thermal conductivities k1k_1, k2k_2 and k3k_3 respectively are fixed together to form a single composite wall of total thickness d1+d2+d3d_1+d_2+d_3. The slabs are in perfect thermal contact and their faces are normal to the direction of heat flow. (i) Derive an expression for the effective thermal conductivity keffk_{\text{eff}} of the composite wall in terms of the individual thicknesses and conductivities. (ii) Using your result, find keffk_{\text{eff}} when d1=2 cmd_1=2\,\text{cm}, d2=3 cmd_2=3\,\text{cm}, d3=5 cmd_3=5\,\text{cm} and k1=0.50 W m−1 K−1k_1=0.50\,\text{W m}^{-1}\,\text{K}^{-1}, k2=0.20 W m−1 K−1k_2=0.20\,\text{W m}^{-1}\,\text{K}^{-1}, k3=0.10 W m−1 K−1k_3=0.10\,\text{W m}^{-1}\,\text{K}^{-1}. (iii) If slab 2 is replaced by another material of conductivity k2′=0.15 W m−1 K−1k_2'=0.15\,\text{W m}^{-1}\,\text{K}^{-1}, determine the thickness d2′d_2' that must be used so that the effective thermal conductivity of the wall remains unchanged.

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