Thermal Resistance 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 k1 and two layers of foam with thermal conductivity k2 for insulation. The left surface of the wall is at temperature T1 and exposed to air

Question

Thermal Resistance

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 [latex]k_{1}[/latex] and two layers of foam with thermal conductivity [latex]k_{2}[/latex] for insulation. The left surface of the wall is at temperature [latex]T_{1}[/latex] and exposed to air with heat transfer coefficient [latex]h_{1}[/latex]. The right surface of the wall is at temperature [latex]T_{2}[/latex] and exposed to air with heat transfer coefficient [latex]h_{2}[/latex]. Assume that [latex]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}=[/latex] [latex]38^{\circ} \mathrm{C}[/latex], and [latex]T_{2}=20^{\circ} \mathrm{C}[/latex]. The thickness of the brick layer is [latex]0.1 \mathrm{~m}[/latex], the thickness of each foam layer is [latex]0.03 \mathrm{~m}[/latex], and the cross-sectional area of the wall is [latex]16 \mathrm{~m}^{2}[/latex].

a. Determine the heat flow rate through the wall.

b. Determine the temperature distribution through the wall.

Accepted Answer

a. The heat transfer through the insulated wall can be represented using a thermal circuit with five thermal resistances connected in series as shown in Figure 7.25. Two modes of heat transfer, conduction and convection, are involved. The corresponding thermal resistances are

[latex] \begin{aligned} & R_{1}=\frac{1}{h_{1} A}=\frac{1}{10 imes 16}=6.25 imes 10^{-3\circ} \mathrm{C} \cdot \mathrm{s} / \mathrm{J} \ & R_{2}=R_{4}=\frac{L_{2}}{k_{2} A}=\frac{0.03}{0.17 imes 16}=1.10 imes 10^{-2 \circ} \mathrm{C} \cdot \mathrm{s} / \mathrm{J} \ & R_{3}=\frac{L_{1}}{k_{1} A}=\frac{0.1}{0.5 imes 16}=1.25 imes 10^{-2\circ} \mathrm{C} \cdot \mathrm{s} / \mathrm{J} \ & R_{5}=\frac{1}{h_{2} A}=\frac{1}{10 imes 16}=6.25 imes 10^{-3\circ} \mathrm{C} \cdot \mathrm{s} / \mathrm{J} \end{aligned} [/latex]

The total thermal resistance is

[latex]R_{\mathrm{eq}}=\sum\limits_{i=1}^{5} R_{i}=4.70 imes 10^{-2 \circ} \mathrm{C} \cdot \mathrm{s} / \mathrm{J} .[/latex]

Thus, the heat flow rate through the insulated wall is

[latex]q_{\mathrm{h}}=\frac{\Delta T}{R_{\mathrm{eq}}}=\frac{38-20}{4.70 imes 10^{-2}}=382.98 \mathrm{~W}.[/latex]

b. Note that the heat flow rate stays the same through the insulated wall. Thus, from left to right, the heat flow rate through each layer is

[latex] \begin{array}{ll} ext { Air: } & q_{\mathrm{h}}=R_{1}\left(T_{1}-T_{3} ight) \ ext { Foam: } & q_{\mathrm{h}}=R_{2}\left(T_{3}-T_{4} ight) \ ext { Brick: } & q_{\mathrm{h}}=R_{3}\left(T_{4}-T_{5} ight) \ ext { Foam: } & q_{\mathrm{h}}=R_{4}\left(T_{5}-T_{6} ight) \ ext { Air: } & q_{\mathrm{h}}=R_{5}\left(T_{6}-T_{2} ight) \end{array} [/latex]

With the given temperatures [latex]T_{1}[/latex] and [latex]T_{2}[/latex], we have [latex]T_{3}=35.61^{\circ} \mathrm{C}, T_{4}=31.40^{\circ} \mathrm{C}, T_{5}=26.61^{\circ} \mathrm{C}[/latex], and [latex]T_{6}=22.40^{\circ} \mathrm{C}[/latex]. Figure 7.26 shows the temperature distribution through the wall. Note that the temperatures shown in Figure 7.26 are the values when the heat transfer process reaches steady state.

Question 7.6: Thermal Resistance Consider heat transfer through an insulat...

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