Problem 23
A \(1.0 \mathrm{~m} \times 1.5 \mathrm{~m}\) double-pane window consists of two 4-mm-thick layers of glass $(k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( that are separated by a \)5-\mathrm{mm}\( air gap \)\left(k_{\text {air }}=0.025 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$. The heat flow through the air gap is assumed to be by conduction. The inside and outside air temperatures are \(20^{\circ} \mathrm{C}\) and \(-20^{\circ} \mathrm{C}\), respectively, and the inside and outside heat transfer coefficients are 40 and \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine \((a)\) the daily rate of heat loss through the window in steady operation and \((b)\) the temperature difference across the largest thermal resistance.
Problem 24
Consider a \(1.5-\mathrm{m}\)-high and \(2.4\)-m-wide glass window whose thickness is \(6 \mathrm{~mm}\) and thermal conductivity is $k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Determine the steady rate of heat transfer through this glass window and the temperature of its inner surface for a day during which the room is maintained at \(24^{\circ} \mathrm{C}\) while the temperature of the outdoors is \(-5^{\circ} \mathrm{C}\). Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be \(h_{1}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and $h_{2}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and disregard any heat transfer by radiation.
Problem 25
Consider a \(1.5\)-m-high and 2.4-m-wide doublepane window consisting of two \(3-\mathrm{mm}\)-thick layers of glass $(k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( separated by a \)12-\mathrm{mm}\(-wide stagnant airspace \)(k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. Determine the steady rate of heat transfer through this double-pane window and the temperature of its inner surface for a day during which the room is maintained at $21^{\circ} \mathrm{C}\( while the temperature of the outdoors is \)-5^{\circ} \mathrm{C}$. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be \(h_{1}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(h_{2}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and disregard any heat transfer by radiation.
Problem 28
A wall is constructed of two layers of \(0.6\)-in-thick sheetrock $\left(k=0.10 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)$, which is a plasterboard made of two layers of heavy paper separated by a layer of gypsum, placed 7 in apart. The space between the sheetrocks is filled with fiberglass insulation $\left(k=0.020 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)$. Determine (a) the thermal resistance of the wall and (b) its \(R\)-value of insulation in English units.
Problem 29
To defog the rear window of an automobile, a very thin transparent heating element is attached to the inner surface of the window. A uniform heat flux of \(1300 \mathrm{~W} / \mathrm{m}^{2}\) is provided to the heating element for defogging a rear window with thickness of \(5 \mathrm{~mm}\). The interior temperature of the automobile is \(22^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The outside ambient temperature is \)-5^{\circ} \mathrm{C}$, and the convection heat transfer coefficient is $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. If the thermal conductivity of the window is \)1.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the inner surface temperature of the window.
Problem 30
A transparent film is to be bonded onto the top surface of a solid plate inside a heated chamber. For the bond to cure properly, a temperature of \(70^{\circ} \mathrm{C}\) is to be maintained at the bond, between the film and the solid plate. The transparent film has a thickness of \(1 \mathrm{~mm}\) and thermal conductivity of \(0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), while the solid plate is \(13 \mathrm{~mm}\) thick and has a thermal conductivity of \(1.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Inside the heated chamber, the convection heat transfer coefficient is $70 \mathrm{~W} / \mathrm{m}^{2}\(. \)\mathrm{K}$. If the bottom surface of the solid plate is maintained at \(52^{\circ} \mathrm{C}\), determine the temperature inside the heated chamber and the surface temperature of the transparent film. Assume thermal contact resistance is negligible.
Problem 31
To defrost ice accumulated on the outer surface of an automobile windshield, warm air is blown over the inner surface of the windshield. Consider an automobile windshield with thickness of \(5 \mathrm{~mm}\) and thermal conductivity of \(1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The outside ambient temperature is \(-10^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the ambient temperature inside the automobile is $25^{\circ} \mathrm{C}$. Determine the value of the convection heat transfer coefficient for the warm air blowing over the inner surface of the windshield necessary to cause the accumulated ice to begin melting.
Problem 32
The roof of a house consists of a 15 -cm-thick concrete slab $(k=2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( that is \)15 \mathrm{~m}\( wide and \)20 \mathrm{~m}$ long. The convection heat transfer coefficients on the inner and outer surfaces of the roof are 5 and \(12 \mathrm{~W} / \mathrm{m}^{2}\). , respectively. On a clear winter night, the ambient air is reported to be at \(10^{\circ} \mathrm{C}\), while the night sky temperature is \(100 \mathrm{~K}\). The house and the interior surfaces of the wall are maintained at a constant temperature of \(20^{\circ} \mathrm{C}\). The emissivity of both surfaces of the concrete roof is \(0.9\). Considering both radiation and convection heat transfers, determine the rate of heat transfer through the roof, and the inner surface temperature of the roof. If the house is heated by a furnace burning natural gas with an efficiency of 80 percent, and the price of natural gas is \(\$ 1.20 /\) therm ( 1 therm \(=105,500 \mathrm{~kJ}\) of energy content), determine the money lost through the roof that night during a 14-h period.
Problem 36
Heat is to be conducted along a circuit board that has a copper layer on one side. The circuit board is \(15 \mathrm{~cm}\) long and \(15 \mathrm{~cm}\) wide, and the thicknesses of the copper and epoxy layers are \(0.1 \mathrm{~mm}\) and \(1.2 \mathrm{~mm}\), respectively. Disregarding heat transfer from side surfaces, determine the percentages of heat conduction along the copper \((k=386 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) and epoxy $(k=0.26 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ layers. Also determine the effective thermal conductivity of the board.
Problem 38
Consider a house that has a \(10-\mathrm{m} \times 20-\mathrm{m}\) base and a 4 -m-high wall. All four walls of the house have an \(R\)-value of $2.31 \mathrm{~m}^{2}{ }^{\circ} \mathrm{C} / \mathrm{W}\(. The two \)10-\mathrm{m} \times 4-\mathrm{m}$ walls have no windows. The third wall has five windows made of \(0.5-\mathrm{cm}\)-thick glass $(k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}), 1.2 \mathrm{~m} \times 1.8 \mathrm{~m}$ in size. The fourth wall has the same size and number of windows, but they are doublepaned with a \(1.5-\mathrm{cm}\)-thick stagnant airspace $(k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( enclosed between two \)0.5$-cm-thick glass layers. The thermostat in the house is set at \(24^{\circ} \mathrm{C}\), and the average temperature outside at that location is \(8^{\circ} \mathrm{C}\) during the seven-month-long heating season. Disregarding any direct radiation gain or loss through the windows and taking the heat transfer coefficients at the inner and outer surfaces of the house to be 7 and $18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively, determine the average rate of heat transfer through each wall. If the house is electrically heated and the price of electricity is $\$ 0.08 / \mathrm{kWh}$, determine the amount of money this household will save per heating season by converting the single-pane windows to double-pane windows.
