Problem 5 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

Consider an average person standing in the middle of a room. The various surfaces are found to be at a temperature of \(15^{\circ} \mathrm{C}\) in the winter and \(26^{\circ} \mathrm{C}\) in the summer. Calculate the rates of radiation heat transfer between this person and the surfaces in both the winter and summer. Assume that the average surface temperature of the person is \(34^{\circ} \mathrm{C}\). The surface area of the person is \(1.5 \mathrm{~m}^{2}\), and the constant emissivity of the skin is \(0.92\).

Step-by-Step Solution

Answer

Use the Stefan-Boltzmann Law to calculate: winter transfer rate is higher.

Step 1: Understand the problem

We need to calculate the rate of radiation heat transfer between a person and the room surfaces during winter and summer. Given data includes the temperatures of the surfaces and the person, along with surface area and emissivity of the skin.

Step 2: Use the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the rate of radiation heat transfer is given by:\[ q = \sigma \cdot \varepsilon \cdot A \cdot (T_{p}^4 - T_{s}^4) \]where \(q\) is the heat transfer rate, \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \text{W/m}^2\text{K}^4\), \(\varepsilon\) is the emissivity of skin (\(0.92\)), \(A\) is the surface area (\(1.5 \text{m}^2\)), and \(T_p\) and \(T_s\) are the temperatures of the person and the surfaces, respectively, in Kelvin.

Step 3: Convert temperatures to Kelvin

Convert all given temperatures from Celsius to Kelvin using the formula: \(T(K) = T(°C) + 273.15\).- Winter surface temperature: \(15°C = 288.15 \text{K}\)- Summer surface temperature: \(26°C = 299.15 \text{K}\)- Person temperature: \(34°C = 307.15 \text{K}\)

Step 4: Calculate radiation heat transfer for winter

Substitute the winter surface temperature into the Stefan-Boltzmann Law:\[ q_{winter} = 5.67 \times 10^{-8} \times 0.92 \times 1.5 \times (307.15^4 - 288.15^4) \]This results in a radiation heat transfer rate for winter.

Step 5: Calculate radiation heat transfer for summer

Substitute the summer surface temperature into the Stefan-Boltzmann Law:\[ q_{summer} = 5.67 \times 10^{-8} \times 0.92 \times 1.5 \times (307.15^4 - 299.15^4) \]This results in a radiation heat transfer rate for summer.

Step 6: Compare winter and summer radiation heat transfer

The calculated values from Steps 4 and 5 show the difference in radiation heat transfer rates between the person and the surfaces in different seasons. The winter rate will be higher due to the larger temperature difference.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in physics that describes the power radiated from a perfect black body in terms of its temperature. It states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's absolute temperature. This can be mathematically represented by the formula:\[ q = \sigma \cdot \varepsilon \cdot A \cdot (T_{p}^4 - T_{s}^4) \]where:

• \( q \) is the rate of radiation heat transfer.
• \( \sigma \) is the Stefan-Boltzmann constant, \( 5.67 \times 10^{-8} \text{W/m}^2\text{K}^4 \).
• \( \varepsilon \) represents the emissivity, which is a measure of how effectively a surface emits thermal radiation.
• \( A \) is the surface area through which the heat is being transferred.
• \( T_p \) and \( T_s \) are the absolute temperatures in Kelvin of the person and the surfaces, respectively.

This formula indicates that small changes in temperature can result in large changes in the radiation heat transfer rate because of the temperature being raised to the fourth power. This is crucial in understanding how humans interact with their environment in terms of heat exchange.

emissivity

Emissivity is a crucial concept in radiation heat transfer. It is defined as the ratio of the energy radiated by a particular surface to the energy radiated by a perfect black body at the same temperature. Emissivity values range from 0 to 1.

• An emissivity of 1 means the object is a perfect emitter, like a black body.
• An emissivity of 0 indicates it emits no thermal radiation at all.

In the context of the exercise, the skin's emissivity is given as 0.92. This means the human skin is quite efficient in emitting thermal radiation. The higher the emissivity, the more effective the surface is at radiating heat.
Emissivity can be influenced by several factors such as the material's texture, composition, and temperature. Different materials have different emissivity values, making it essential to consider when calculating radiation heat transfer rates. In this exercise, the assumption is that the emissivity of the person's skin remains constant despite changing environmental conditions.

temperature conversion

Temperature conversion is the process of changing the units of temperature measurement, typically from Celsius to Kelvin, to effectively apply certain scientific formulas. Kelvin is the SI unit for temperature and is essential for calculations involving the Stefan-Boltzmann Law as it requires absolute temperatures.To convert from Celsius to Kelvin, you use the formula:\[ T(K) = T(°C) + 273.15 \]This formula shifts the Celsius value into the Kelvin scale, maintaining the relative differences between temperatures. In the exercise:

• The winter surface temperature of \(15^{\circ} \text{C}\) becomes \(288.15 \text{K}\).
• The summer surface temperature of \(26^{\circ} \text{C}\) becomes \(299.15 \text{K}\).
• The body temperature of \(34^{\circ} \text{C}\) becomes \(307.15 \text{K}\).

Converting temperatures to Kelvin ensures the calculations involve absolute zero as a reference point, which is necessary for accurate thermodynamic calculations. This universal method simplifies comparisons across different systems and conditions, as seen in the radiation heat transfer analysis of this exercise.