Question

Reconsider Prob. 7-67E. Using EES (or other) software, investigate the effects of air temperature and wind velocity on the rate of heat loss from the arm. Let the air temperature vary from \(20^{\circ} \mathrm{F}\) to \(80^{\circ} \mathrm{F}\) and the wind velocity from \(10 \mathrm{mph}\) to \(40 \mathrm{mph}\). Plot the rate of heat loss as a function of air temperature and of wind velocity, and discuss the results.

Short answer

Answer: The rate of heat loss from an arm increases as air temperature decreases and wind velocity increases. This is because the temperature difference between the arm's surface and the surrounding air and the convective heat transfer coefficient both increase, leading to a higher rate of heat transfer.

Step-by-step solution

Understand the problem context

In this problem, we are trying to find the rate of heat loss from an arm. The heat loss will depend on air temperature and wind velocity. The goal is to generate a plot displaying the relationship between these variables and the heat loss rate.

Choose an appropriate heat transfer model

We can use the convective heat transfer model to estimate the rate of heat loss from the arm to the surrounding air. The convective heat transfer equation is given by: \(Q = hA(T_{s} - T_{a})\) where \(Q\) is the heat transfer rate, \(h\) is the convective heat transfer coefficient, \(A\) is the surface area of the arm, \(T_{s}\) is the surface temperature of the arm, and \(T_{a}\) is the air temperature.

Account for the effect of wind velocity

Wind velocity affects the convective heat transfer coefficient (\(h\)). To include the effect of wind velocity, we can use the following expression for the heat transfer coefficient: \(h = f(V)\) where \(f\) is a function that relates wind velocity (\(V\)) to the heat transfer coefficient.

Generate the necessary plots

With the appropriate equations and functions in place, we can use software (e.g., EES, MATLAB, or Python) to generate the required plots. In this step, we will evaluate the heat transfer rate and create two plots: 1. Rate of heat loss as a function of air temperature (varying air temperature from \(20^{\circ}\mathrm{F}\) to \(80^{\circ}\mathrm{F}\) while keeping wind velocity constant) 2. Rate of heat loss as a function of wind velocity (varying wind velocity from \(10\mathrm{mph}\) to \(40\mathrm{mph}\) while keeping air temperature constant)

Discuss the results

Analyze the graphs generated in Step 4. Typically, the following observations can be made: 1. The rate of heat loss increases as the air temperature decreases. This is because the temperature difference between the arm's surface and the surrounding air increases, resulting in a higher rate of heat transfer. 2. The rate of heat loss increases as the wind velocity increases. This is because the convective heat transfer coefficient increases with wind velocity, leading to a higher rate of heat transfer. In conclusion, air temperature and wind velocity significantly influence the rate of heat loss from an arm. These findings can be used to inform activities and clothing choices in different environmental conditions as well as designing heating systems for outdoor spaces.

Key concepts

Rate of Heat Loss

The rate of heat loss is a measure of how quickly heat energy is transferred from one body to another. Specifically, in our exercise, we're looking at how heat moves away from an arm into the surrounding air. This process is heavily influenced by various factors, most notably through convection, which is the transfer of heat by the movement of fluids such as air or water. When investigating this rate, we use the equation: \[ Q = hA(T_{s} - T_{a}) \] where

• \( Q \) represents the heat transfer rate in BTU/hr or watts,
• \( h \) is the convective heat transfer coefficient, indicating how well heat is transferred,
• \( A \) stands for the surface area of the arm being analyzed,
• \( T_{s} \) denotes the surface temperature of the arm,
• \( T_{a} \) is the ambient air temperature.

The larger the difference between the arm temperature and the air temperature, \( T_{s} - T_{a} \), the greater the potential for heat loss. As the convective heat transfer coefficient \( h \) increases, typically influenced by wind, the rate of heat loss \( Q \) also increases. This understanding is vital in many practical scenarios, such as designing clothing or building heating systems to minimize energy loss.

Air Temperature

Air temperature plays a key role in determining the rate of heat loss. In simple terms, the colder the air surrounding the arm, the greater the heat loss. This is due to the enhanced temperature difference between the arm's surface temperature and the ambient air temperature \( T_{a} \). As outlined in the heat transfer equation, the rate of heat loss \( Q \) depends on this difference:\[ Q = hA(T_{s} - T_{a}) \]When the air is significantly colder than the body part in question, the disparity \( T_{s} - T_{a} \) becomes larger, accelerating heat loss to the environment. This concept is essential for understanding how protective clothing is designed to keep people warm in varying temperatures.

• For a range from \(20^{\circ}\mathrm{F}\) to \(80^{\circ}\mathrm{F}\), as the temperature increases, heat loss generally diminishes.
• This trend is why environments with frigid temperatures pose more of a challenge for retaining body heat.
• Consequently, this understanding allows for informed clothing decisions or heating adjustments relevant to outdoor activities or occupations.

Wind Velocity

Wind velocity impacts the rate of heat loss predominantly by influencing the convective heat transfer coefficient \( h \). Essentially, wind increases air movement, enhancing the convective heat transfer and thus accelerating the rate at which heat is carried away from the body. This effect is often felt very quickly as a more pronounced chill on a windy day. The relationship between wind velocity and the heat transfer coefficient can be expressed by a functional relationship \( h = f(V) \) where \( V \) is the wind speed. Generally, the main effects of increasing wind velocity include:

• Increased air circulation over surfaces, which enhances convective heat transfer.
• A rise in \( h \), leading to greater overall heat loss at a faster pace.
• Even at constant temperatures, wind amplifies the sensation of cold, emphasizing the need for wind-resistant clothing.

In practical terms, understanding wind's role in heat loss ensures that outdoor clothing, structures, and activities are designed with adequate protection to moderate or counteract the increased heat dispersion caused by high wind speeds.