The rear window of an automobile is defogged by attaching a...

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 $h_{o}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} .$ (b) In practice $T_{\infty \rho \rho}$ and $h_{o}$ vary according to weather conditions and car speed. For values of $h_{o}=2,20$, 65 , and $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine and plot the electrical power requirement as a function of $T_{\infty \rho}$ for $-30 \leq T_{\infty, o} \leq 0^{\circ} \mathrm{C}$. From your results, what can you conclude about the need for heater operation at low values of $h_{o}$ ? How is this conclusion affected by the value of $T_{\infty, 0}$ ? If $h \propto V^{n}$, where $V$ is the vehicle speed and $n$ is a positive exponent, how does the vehicle speed affect the need for heater operation?

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 $h_{o}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} .$
(b) In practice $T_{\infty \rho \rho}$ and $h_{o}$ vary according to weather conditions and car speed. For values of $h_{o}=2,20$, 65 , and $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine and plot the electrical power requirement as a function of $T_{\infty \rho}$ for $-30 \leq T_{\infty, o} \leq 0^{\circ} \mathrm{C}$. From your results, what can you conclude about the need for heater operation at low values of $h_{o}$ ? How is this conclusion affected by the value of $T_{\infty, 0}$ ? If $h \propto V^{n}$, where $V$ is the vehicle speed and $n$ is a positive exponent, how does the vehicle speed affect the need for heater operation?

Key Concepts

Dependence of Convective Coefficient on Vehicle Speed

Often, the convective heat transfer coefficient is not a constant but depends on the speed of the vehicle. With the relationship h ? V^n, an increase in speed enhances convective heat transfer, thereby affecting the heat loss from the system. This has direct implications for the power required to maintain a desired temperature, linking mechanical motion with thermal performance.

Parameter Sensitivity and Environmental Effects

This concept deals with how variations in environmental conditions, such as ambient temperature and convective heat transfer coefficient, impact the overall heat transfer and required heating power. It underscores the importance of analyzing different scenarios to understand the system's performance under various operational conditions, leading to robust design decisions.

Steady-State Energy Balance

A steady-state energy balance involves setting the net energy change to zero so that the power input equals the power output. In the context of maintaining a specific surface temperature, this balance encompasses all modes of heat transfer—conduction through the glass and convection at both surfaces—allowing for the determination of the required electrical power input.

Conduction Heat Transfer

This concept involves the transfer of heat through a solid material by means of thermal conduction. It is governed by Fourier’s law, which relates the temperature gradient within the material to the heat flux, and is crucial in determining how temperature differences across a medium (like a glass pane) drive heat transfer.

Convective Heat Transfer

Convective heat transfer occurs at the boundaries of a surface where a fluid (such as air) is in contact with the solid. It is characterized by a heat transfer coefficient that depends on the fluid properties and flow conditions. Understanding convection is essential when evaluating heat loss or gain at the surfaces, especially from ambient air and interior air.

Thermal Resistance Network Analysis

In many heat transfer problems, different modes of heat transfer (such as conduction through a solid and convection at surfaces) can be modeled as resistances in series. This network approach enables the calculation of overall heat flux based on the sum of individual thermal resistances, providing a systematic way to design and analyze systems with layered or multiple interfaces.

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