The wind chill, which is experienced on a cold, windy day,...

The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is $3 \mathrm{~mm}$ thick and whose interior surface is maintained at a temperature of $36^{\circ} \mathrm{C}$. On a calm day the convection heat transfer coefficient at the outer surface is $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, but with $30 \mathrm{~km} / \mathrm{h}$ winds it reaches $65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. In both cases the ambient air temperature is $-15^{\circ} \mathrm{C}$. (a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day? (b) What will be the skin outer surface temperature for the calm day? For the windy day? (c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at $-15^{\circ} \mathrm{C}$ on the windy day?

The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is $3 \mathrm{~mm}$ thick and whose interior surface is maintained at a temperature of $36^{\circ} \mathrm{C}$. On a calm day the convection heat transfer coefficient at the outer surface is $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, but with $30 \mathrm{~km} / \mathrm{h}$ winds it reaches $65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. In both cases the ambient air temperature is $-15^{\circ} \mathrm{C}$.
(a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day?
(b) What will be the skin outer surface temperature for the calm day? For the windy day?
(c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at $-15^{\circ} \mathrm{C}$ on the windy day?

Key Concepts

Interface Temperature and Energy Balance

Determining the temperature at the interface between different media, such as the skin or the outer surface of a tissue layer, requires balancing the conductive heat flow from within with the convective heat loss to the environment. This balance is key to analyzing how changes in convective conditions (like wind speed) affect the actual surface temperature.

Wind Chill Effect

The wind chill effect describes how increased air movement enhances the convective heat transfer from a surface, leading to more rapid cooling than under calm conditions. It is an important concept when assessing how environmental conditions impact heat loss, especially on exposed surfaces like human skin.

Thermal Resistance

Thermal resistance is a way to model the opposition to heat flow through a material or across an interface. In layered systems, such as a tissue layer and its surrounding environment, resistances from conduction within the material and convection at the surface combine in series to determine the overall heat transfer rate.

Convective Heat Transfer

This concept involves the transfer of heat between a surface and a moving fluid, characterized by the convective heat transfer coefficient, which quantifies the effectiveness of the fluid in removing or supplying heat. It is central to problems involving different wind speeds or flow conditions where the medium’s motion alters the rate of energy exchange at the surface.

Steady-State Heat Transfer

This concept assumes that the temperatures at various points in a system do not change over time, meaning the heat entering any part of the system is equal to the heat leaving it. It allows the use of equilibrium equations to find unknown temperatures or heat fluxes in scenarios like skin temperature and environmental cooling.

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