Question

It is well-known that at the same outdoor air temperature a person is cooled at a faster rate under windy conditions than under calm conditions due to the higher convection heat transfer coefficients associated with windy air. The phrase wind chill is used to relate the rate of heat loss from people under windy conditions to an equivalent air temperature for calm conditions (considered to be a wind or walking speed of \(3 \mathrm{mph}\) or \(5 \mathrm{~km} / \mathrm{h})\). The hypothetical wind chill temperature (WCT), called the wind chill temperature index (WCTI), is an equivalent air temperature equal to the air temperature needed to produce the same cooling effect under calm conditions. A 2003 report on wind chill temperature by the U.S. National Weather Service gives the WCTI in metric units as WCTI \(\left({ }^{\circ} \mathrm{C}\right)=13.12+0.6215 T-11.37 V^{0.16}+0.3965 T V^{0.16}\) where \(T\) is the air temperature in \({ }^{\circ} \mathrm{C}\) and \(V\) the wind speed in \(\mathrm{km} / \mathrm{h}\) at \(10 \mathrm{~m}\) elevation. Show that this relation can be expressed in English units as WCTI \(\left({ }^{\circ} \mathrm{F}\right)=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}\) where \(T\) is the air temperature in \({ }^{\circ} \mathrm{F}\) and \(V\) the wind speed in \(\mathrm{mph}\) at \(33 \mathrm{ft}\) elevation. Also, prepare a table for WCTI for air temperatures ranging from 10 to \(-60^{\circ} \mathrm{C}\) and wind speeds ranging from 10 to \(80 \mathrm{~km} / \mathrm{h}\). Comment on the magnitude of the cooling effect of the wind and the danger of frostbite.

Short answer

Question: Describe the relationship between air temperature, wind speed, and the risk of frostbite according to the Wind Chill Temperature Index (WCTI). Answer: The Wind Chill Temperature Index (WCTI) indicates that as air temperature decreases and wind speed increases, there is a more significant cooling effect on the human body, increasing the risk of frostbite. It is essential to take precautions such as wearing layered clothing, covering exposed skin, and minimizing time spent outdoors in cold and windy conditions to mitigate this risk.

Step-by-step solution

Convert the WCTI equation from metric units to English units

First, we need to convert the temperature. Fahrenheit to Celsius conversion formula is: \(T_c = \frac{5}{9}(T_f - 32)\) Next, we have to convert the wind speed from \(\mathrm{km/h}\) to \(\mathrm{mph}\). The conversion factor is: \(1 \; \mathrm{mph} = 1.609 \; \mathrm{km/h}\) So, \(V_{\mathrm{mph}} = \frac{V_{\mathrm{km/h}}}{1.609}\) Now, we have to substitute the new expressions for \(T_c\) and \(V_{\mathrm{mph}}\) into the original WCTI equation to derive the equivalent equation in English units.

Prepare a table for WCTI

We will now prepare a table for WCTI for air temperatures ranging from \(10^{\circ} \mathrm{C}\) to \(-60^{\circ} \mathrm{C}\) and wind speeds ranging from \(10 \mathrm{~km} / \mathrm{h}\) to \(80 \mathrm{~km} / \mathrm{h}\) using the WCTI equation in English units: WCTI \(\left({ }^{\circ} \mathrm{F}\right)=35.74+0.6215 T-35.75 V^{0.16}+0.4275 T V^{0.16}\)

Comment on the cooling effect and the danger of frostbite

The table generated in step 2 shows how the wind chill temperature index (WCTI) decreases as the air temperature decreases and the wind speed increases. This means that colder air temperatures combined with higher wind speeds result in a more significant cooling effect on the human body. High wind speeds and low temperatures increase the risk of frostbite, especially on exposed skin. It is essential to take appropriate precautions, such as wearing layered clothing, covering exposed skin, and minimizing time spent outdoors in such conditions.

Key concepts

Convection Heat Transfer

Convection heat transfer plays a key role in understanding the wind chill temperature index. This concept refers to how heat is transferred between a surface, such as human skin, and a fluid flowing past it, like air. As wind speed increases, the convection heat transfer coefficient goes up. This means that more heat is drawn away from the body, creating a quicker, more intense cooling effect.
In calm conditions, the air moving over skin is minimal, leading to slower heat loss. But when the wind picks up, it works like a fan, whisking away the warmth faster. Therefore, the wind chill index helps quantify this increased rate of heat loss due to higher convection in windy weather.

Temperature Conversion

Temperature conversion is crucial when expressing the wind chill temperature index (WCTI) in different units. For example, converting temperatures from Celsius to Fahrenheit. The conversion formula is:
\[ T_c = \frac{5}{9}(T_f - 32) \]
where \(T_c\) is the temperature in Celsius and \(T_f\) is the temperature in Fahrenheit.
Another conversion necessary is adjusting wind speed from kilometers per hour (km/h) to miles per hour (mph). This is achieved by the formula:
\[ V_{\text{mph}} = \frac{V_{\text{km/h}}}{1.609} \]
Using these conversions allows for the wind chill formula to be accurately expressed and applied in different measurement systems, bridging the gap between metric and English units seamlessly.

Wind Speed Effect

The effect of wind speed on perceived temperatures is profound. As wind speed increases, the wind chill temperature index decreases, making it feel much colder than the actual air temperature. This is because high winds expedite the removal of heat from the body.
With a higher wind chill value, a person feels as though they are in a colder environment. For instance, if the actual air temperature reads 30°F with no wind, it may feel like 0°F if there's a strong breeze.
This demonstrates how crucial wind speed is in determining the wind chill factor and preparing us for safer outdoor activities during cold, windy days.

Frostbite Risk

Understanding frostbite risk is essential in cold environments, especially when wind chill factors are significant. Frostbite occurs when skin and tissues freeze, typically affecting extremities like fingers, toes, and the nose. The combination of low temperatures and high wind speeds greatly increases this risk.
With higher wind speeds, the body loses heat rapidly, and exposed skin can freeze in minutes if not protected. The wind chill temperature index serves as a crucial tool for assessing these risks.

• When wind chill values fall below certain thresholds, the likelihood of frostbite rises.
• Proper clothing and limited exposure to cold and windy conditions are vital to prevent frostbite.

Staying informed on wind chill values helps take proactive steps to protect against this cold weather hazard.

Cooling Effect

The cooling effect of wind is amplified through the wind chill temperature index. While the actual air temperature remains constant, strong winds enhance the cooling effect, making it feel colder. This sensation arises because wind increases the rate of heat loss from the body.
By stripping away the warm layer of air next to the skin, wind forces the body to lose more heat to the surrounding environment. Even if the temperature is mild, a significant wind can create a strong cooling sensation, leading to underestimating the need for warmer clothing.
The WCTI is a crucial measure; it helps predict how different combinations of temperature and wind speed can affect thermal comfort and safety. By understanding this index, we can better prepare and dress appropriately, ensuring we stay warm and safe outdoors.