Related questions with answers

The rear window of an automobile is defogged by passing warm air over its inner surface. (a) If the warm air is at $T_{\infty, i}=40^{\circ} \mathrm{C}$ and the corresponding convection coefficient is $h_{i}=30 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ , what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is $T_{\infty, o}=-10^{\circ} \mathrm{C}$ and the associated convection coefficient is $h_{o}=65 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ ? (b) In practice $T_{\infty, o}$ and $h_{o}$ vary according to weather conditions and car speed. For values of $h_{o}=2,65$ , and $100 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ , compute and plot the inner and outer surface temperatures as a function of $T_{\infty, o}$ for $-30 \leq T_{\infty, o} \leq 0^{\circ} \mathrm{C}$ .

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 $1300 W/m^2$ is provided to the heating element for defogging a rear window with thick- ness of 5 mm. The interior temperature of the automobile is $22^\circ C$ and the convection heat transfer coefficient is $15 W/m^2 \cdot K.$ The outside ambient temperature is $−5^\circ C$ and the convection heat transfer coefficient is $100 W/m^2 \cdot K.$ If the thermal conduc- tivity of the window is $1.2 W/m \cdot K,$ determine the inner surface temperature of the window.

A cartridge electrical heater is shaped as a cylinder of length L=200 mm and outer diameter D=20 mm. Under normal operating conditions, the heater dissipates 2 kW while submerged in a water flow that is at $20^{\circ} \mathrm{C}$ and provides a convection heat transfer coefficient of $h=5000 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ . Neglecting heat transfer from the ends of the heater, determine its surface temperature $T_{s}$ . If the water flow is inadvertently terminated while the heater continues to operate, the heater surface is exposed to air that is also at $20^{\circ} \mathrm{C}$ but for which h=50 $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ . What is the corresponding surface temperature? What are the consequences of such an event?

Question

The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating element to its inner surface. By electrically heating this element, a uniform heat flux may be established at the inner surface. (a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of $15^{\circ} \mathrm{C}$ when the interior air temperature and convection coefficient are $T_{\infty, i}=25^{\circ} \mathrm{C}$ and $h_{i}=10 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ , while the exterior (ambient) air temperature and convection coefficient are $T_{\infty, o}=-10^{\circ} \mathrm{C}$ and $h_{o}=65 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ .

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$\textbf{Known:}$

window glass thickness, $L=4$ mm $= 0.004$ m
inner surface of the window temperature, $T_{s,i}=15^\circ$ C
interior air temperature and convection factor, $T_{\infty,i}=25^\circ$ C, $h_i=10\frac{\text{W}}{\text{m}^2\text{K}}$
exterior ambient air temperature and convection factor, $T_{\infty,o}=-10^\circ$ C, $h_o=65\frac{\text{W}}{\text{m}^2\text{K}}$
$\textbf{b)}$ convection factors for plotting the graphs, $h_0=2, 20, 65, 100 \frac{\text{W}}{\text{m}^2\text{K}}$

Since there are no other parameters involved, electrical power can be found as the heat flux on the plane wall. Heat flux expression can be set as energy balance on the inner surface of the window: