Question

A car mechanic is working in a shop whose interior space is not heated. Comfort for the mechanic is provided by two radiant heaters that radiate heat at a total rate of \(4 \mathrm{~kJ} / \mathrm{s}\). About 5 percent of this heat strikes the mechanic directly. The shop and its surfaces can be assumed to be at the ambient temperature, and the emissivity and absorptivity of the mechanic can be taken to be \(0.95\) and the surface area to be \(1.8 \mathrm{~m}^{2}\). The mechanic is generating heat at a rate of \(350 \mathrm{~W}\), half of which is latent, and is wearing medium clothing with a thermal resistance of \(0.7 \mathrm{clo}\). Determine the lowest ambient temperature in which the mechanic can work comfortably.

Short answer

Answer: The lowest ambient temperature at which the mechanic can work comfortably is approximately \(273~\mathrm{K}\), or \(0^\circ \mathrm{C}\).

Step-by-step solution

Calculate the heat rate received from the radiant heaters

To determine the heat rate received by the mechanic directly from the radiant heaters, we can multiply the total heat rate of the heaters by the percentage of the heat that strikes the mechanic directly. Heat rate received from the radiant heaters: \(Q_{r} = 0.05 \times 4000~\mathrm{J/s} = 200~\mathrm{W}\)

Calculate the heat rate lost by the mechanic

The mechanic loses heat through the following ways: 1. By radiation (assuming the surroundings are at ambient temperature). 2. Through clothing (assuming the surroundings are at ambient temperature). 3. Half of the generated heat is latent and not used by the mechanic. First, we calculate the heat loss through radiation using the equation: \(Q_{rad} = \sigma \times \epsilon \times A \times (T_{m}^{4} - T_{a}^{4})\) Where \(\sigma \approx 5.67 \times 10^{-8}~\mathrm{W/m^2\cdot K^4}\) is the Stefan-Boltzmann constant, \(\epsilon = 0.95\) is the emissivity, \(A = 1.8~\mathrm{m^2}\) is the surface area of the mechanic, \(T_{m}\) is the mechanic's effective temperature, and \(T_{a}\) is the ambient temperature. Second, we calculate the heat loss through clothing using the equation: \(Q_{clo} = \frac{T_{m} - T_{a}}{R}\) Where \(R = 0.155\times0.7~\mathrm{K\cdot m^2/W}\) is the thermal resistance of the clothing. The total heat loss \(Q_{loss} = Q_{rad} + Q_{clo} + 350/2~\mathrm{W}\)

Set up an equation balancing the heat rates and solve for the ambient temperature

To find the lowest ambient temperature at which the mechanic can work comfortably, we need to balance the heat rates: \(Q_{r} = Q_{loss}\) \(200~\mathrm{W} = 5.67 \times 10^{-8}\times 0.95 \times 1.8 \times (T_{m}^4 - T_{a}^4) + \frac{T_{m} - T_{a}}{0.155\times0.7} + 175~\mathrm{W}\) Assuming that the comfortable work environment is when the mechanic's effective temperature is at \(37^\circ \mathrm{C}\), i.e. the normal body temperature, we have \(T_{m}= 310~\mathrm{K}\). Using this assumption, we can solve for the ambient temperature: \(200 - 175 = 5.67 \times 10^{-8}\times 0.95 \times 1.8 \times (310^4 - T_{a}^4) + \frac{310 - T_{a}}{0.155\times0.7}\) Solving for \(T_{a}\), we find: \(T_{a} \approx 273~\mathrm{K}\) Thus, the lowest ambient temperature in which the mechanic can work comfortably is approximately \(273~\mathrm{K}\), or \(0^\circ \mathrm{C}\).

Key concepts

Radiant Heating

Radiant heating is a method of transferring heat using electromagnetic waves, mainly in the infrared spectrum.
This form of heating does not rely on heating the air directly, but rather emits heat that is absorbed by objects and surfaces in its path.
In the context of our mechanic's shop, radiant heaters emit energy, part of which gets absorbed by the mechanic and his surrounding environment.
Some key points about radiant heating:

• It provides instant warmth since it's not dependent on warm air circulation.
• Efficiency is high as heat is delivered directly to the subject.
• It minimizes energy loss in large open spaces.

This heating method is crucial for maintaining comfort in spaces that lack conventional heating systems.

Thermal Comfort

Thermal comfort is the state of mind that expresses satisfaction with the surrounding environment's thermal conditions.
It involves maintaining a temperature where the body feels neither too hot nor too cold.
In determining thermal comfort:

• Personal factors like clothing and activity level matter significantly.
• Environmental factors include air temperature, radiant temperature, humidity, and airflow.

For the mechanic in the exercise, comfort is impacted by both his clothing's thermal resistance and the ambient conditions created by the radiant heaters.
Achieving balance between heat gained from the environment and heat lost is key to maintaining comfort.

Emissivity

Emissivity is a measure of an object's ability to emit infrared energy, relative to a black body at the same temperature.
It scales from 0 to 1, where 1 indicates a perfect emitter.
Important aspects of emissivity:

• Materials with higher emissivity are better radiators of heat.
• It affects how efficiently a body loses heat through radiation.
• In the exercise, the mechanic's emissivity is given as 0.95, indicating he emits radiation effectively.

Understanding emissivity helps in calculating heat radiation losses, which is an important part of thermal balance in this scenario.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is fundamental in understanding radiative heat transfer.
It describes the power radiated from a black body in terms of its temperature:
\[ Q = \sigma imes \epsilon imes A imes T^4 \]Where:

• \( \sigma \) is the Stefan-Boltzmann constant \( \simeq 5.67 \times 10^{-8} \ \mathrm{W/m^2\cdot K^4} \)
• \( \epsilon \) is the emissivity of the object
• \( A \) is the surface area
• \( T \) is the absolute temperature in Kelvin

In the exercise, this law is applied to calculate the heat loss by radiation from the mechanic.
It shows how important it is to account for temperature and surface area when determining heat transfer through radiation.