Consider heat transfer through an insulated wall, as shown in Figure 7.24. The wall is made of a layer of brick with thermal conductivity \(k_{1}\) and two layers of foam with thermal conductivity \(k_{2}\) for insulation. The left surface of the wall is at temperature \(T_{1}\) and exposed to air, with heat transfer coefficient \(h_{1}\). The right surface of the wall is at temperature \(T_{2}\) and exposed to air, with heat transfer coefficient \(h_{2}\). Assume that \(k_{1}=0.5 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right), k_{2}=0.17 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right), h_{1}=h_{2}=10 \mathrm{~W} /\left(\mathrm{m}^{2 .}{ }^{\circ} \mathrm{C}\right), T_{1}=38^{\circ} \mathrm{C}\), and \(T_{2}=20^{\circ} \mathrm{C}\). The thickness of the brick layer is \(0.1 \mathrm{~m}\), the thickness of each foam layer is \(0.03 \mathrm{~m}\), and the cross-sectional area of the wall is \(16 \mathrm{~m}^{2}\).

a. Determine the heat flow rate through the wall.

b. Determine the temperature distribution through the wall.

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T Air FIGURE 7.24 Heat transfer through an insulated wall. L +42 Air T