The heat transferred to and from a vertical surface, such as...

The heat transferred to and from a vertical surface, such as a window pane, by convection in the surrounding air has been found to be equal to $0.4 \times 10^{-4}(\Delta t)^{5 / 4} \mathrm{cal} / \mathrm{sec} \cdot \mathrm{cm}^2$, where $\Delta t$ is the temperature difference between the surface and the air. If the air temperature is $25^{\circ} \mathrm{C}$ on the inside of a room and $-15^{\circ} \mathrm{C}$ on the outside, what is the temperature of the inner surface of a window pane in the room? The window pane has a thickness of 2 mm and a thermal conductivity of $2 \times 10^{-3} \mathrm{cal} / \mathrm{sec} \cdot \mathrm{cm} \cdot{ }^{\circ} \mathrm{C}$. Heat transfer by radiation can be neglected.

Key Concepts

Thermal Equilibrium (Heat Flux Balance)

Thermal equilibrium is achieved when the rate of heat entering a system equals the rate of heat leaving it, resulting in no net temperature change over time. In problems involving multiple modes of heat transfer, setting the convective and conductive heat fluxes equal at an interface allows one to solve for unknown temperature distributions.

Thermal Conductivity

Thermal conductivity is a material-specific property that measures the ability of a material to conduct heat. It plays a critical role in determining how quickly a material can transfer heat via conduction and is a key factor in analyzing thermal systems, particularly when balancing heat transfer through different mechanisms.

Conductive Heat Transfer

Conductive heat transfer involves the transmission of thermal energy through a material due to a temperature gradient within the material. This process is guided by Fourier's law of heat conduction, which relates the heat flux to the thermal conductivity of the material and the temperature gradient across it.

Convective Heat Transfer

Convective heat transfer describes the process where heat is transferred between a surface and a moving fluid (such as air), often governed by the temperature difference between the surface and the fluid. The heat transfer rate in convection can depend non-linearly on the temperature difference and is typically characterized by an empirical or theoretical law relating the two.

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