A particular house wall consists of three layers and has a

Chapter , Problem 7.48

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A particular house wall consists of three layers and has a surface area of \(30 \mathrm{~m}^{2}\). The inside layer is 10 mm thick and made of plaster board with a thermal conductivity of \(k=0.2 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The middle layer is made of fiberglass insulation with \(k=0.04 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The outside layer is 20 mm thick and made of wood siding with \(k=0.1 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The inside temperature is \(20^{\circ} \mathrm{C}\), and the convection coefficient for the inside wall surface is \(h_{i}=40 \ \mathrm{W} /\left(\mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\right)\). The convection coefficient for the outside wall surface is \(h_{o}=70 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\right)\). How thick must the insulation layer be so that the heat loss is no greater than 400 W if the outside temperature is \(-20^{\circ} \mathrm{C}\)?