The rear window of an automobile is defogged by passing warm...

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 $T_{\infty, \rho}=-10^{\circ} \mathrm{C}$ and the associated convection coefficient is $h_{o}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ ? (b) In practice $T_{\infty, \rho}$ 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 $-30 \leq T_{\infty, 0} \leq 0^{\circ} \mathrm{C}$.

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 $T_{\infty, \rho}=-10^{\circ} \mathrm{C}$ and the associated convection coefficient is $h_{o}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ ?
(b) In practice $T_{\infty, \rho}$ 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 $-30 \leq T_{\infty, 0} \leq 0^{\circ} \mathrm{C}$.

Key Concepts

Thermal Resistance Concept

Thermal resistance models the opposition to heat flow, analogous to electrical resistance in circuits. It provides a systematic way to account for the resistances imposed by conduction through a solid and convection at the surfaces. By summing these resistances, one can relate the temperature differences across layers to the heat flux, enabling easier analysis of composite systems.

Steady State Energy Balance

In steady state conditions, the temperatures at different points of a system do not change over time, meaning the rate of heat input equals the rate of heat output. This concept is fundamental in setting up energy balance equations for calculating temperature profiles, as it indicates that the net heat flow through each component of the system remains constant.

Conduction Heat Transfer

This concept involves the transfer of heat within a material or between materials in direct contact due to a temperature gradient. In scenarios such as heat flowing through a pane of glass, Fourier’s law is used to describe how heat diffuses from the warmer side to the cooler side due to the thermal conductivity of the material.

Convection Heat Transfer

Convection refers to the heat exchange between a surface and a surrounding fluid that is in motion. The rate of convective heat transfer is governed by the convection heat transfer coefficient, which reflects how efficiently heat is carried away from (or supplied to) the surface by the fluid flow, and it is crucial for calculating the boundary conditions in heat transfer problems.

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