Question

Consider two identical people each generating \(60 \mathrm{~W}\) of metabolic heat steadily while doing sedentary work, and dissipating it by convection and perspiration. The first person is wearing clothes made of 1 -mm-thick leather \((k=\) \(0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) that covers half of the body while the second one is wearing clothes made of 1 -mm-thick synthetic fabric \((k=0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) that covers the body completely. The ambient air is at \(30^{\circ} \mathrm{C}\), the heat transfer coefficient at the outer surface is \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the inner surface temperature of the clothes can be taken to be \(32^{\circ} \mathrm{C}\). Treating the body of each person as a 25 -cm-diameter, \(1.7-\mathrm{m}\)-long cylinder, determine the fractions of heat lost from each person by perspiration.

Short answer

To determine the fraction of heat loss due to perspiration for two individuals wearing different clothing materials, we first calculated the heat transfer by convection using the clothing thickness, thermal properties, and heat transfer coefficient. The calculations provided us with the values of \(q_{1}\) and \(q_{2}\) for person 1 (leather) and person 2 (synthetic fabric) respectively. Then, we calculated the fraction of heat loss due to perspiration for each person by comparing the total metabolic heat generated (60 W) to the amount dissipated by convection. The resulting values were \(F_{1}\) for person 1 and \(F_{2}\) for person 2, which can be expressed as percentages to indicate the fractions of heat loss due to perspiration for each individual.

Step-by-step solution

Calculate the heat dissipation by convection for each person

We can use the formula for heat conduction through a cylindrical wall: \(q = \dfrac{2 \pi \mathrm{L} k \Delta T}{\ln (r_{2}/r_{1})}\) where: - \(q\) - heat transfer by convection (W) - \(L\) - length of the cylinder (1.7 m) - \(k\) - thermal conductivity of the material (W/m⋅K) - \(\Delta T\) - temperature difference between the inner and outer surfaces (K) - \(\ln (r_{2}/r_{1})\) - logarithmic mean radius of the cylindrical wall The temperature difference can be calculated as: \(\Delta T = T_{\text{inner}} - T_{\text{outer}} = 32 - 30 = 2 \mathrm{~K}\) Now we can calculate the heat dissipation by convection for each person using their respective material properties. Person 1 (Leather): Using the given thickness and thermal conductivity for leather: \(k_{1} = 0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) \(r_{1} = 0.125 \mathrm{~m}\) (radius of the body) \(r_{2} = r_{1} + 0.1/2 = 0.125 + 0.001 = 0.126 \mathrm{~m}\) (radius of the body plus leather) We can now calculate the heat transfer \(q_{1}\) by substituting these values into the formula: \(q_{1} = \dfrac{2 \pi (1.7) (0.159)(2)}{\ln (0.126/0.125)}\) Person 2 (Synthetic Fabric): Using the given thickness and thermal conductivity for synthetic fabric: \(k_{2} = 0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) \(r_{1} = 0.125 \mathrm{~m}\) (radius of the body) \(r_{2} = r_{1} + 0.1/2 = 0.125 + 0.001 = 0.126 \mathrm{~m}\) (radius of the body plus synthetic fabric) We can now calculate the heat transfer \(q_{2}\) by substituting these values into the formula: \(q_{2} = \dfrac{2 \pi (1.7) (0.13)(2)}{\ln (0.126/0.125)}\) Don't forget, Person 1 is only covered by the clothing around half of their body, so the heat loss from convection needs to be multiplied by 0.5. Finally, calculate \(q_{1}\) and \(q_{2}\).

Calculate the fraction of heat loss due to perspiration

We will now calculate the fraction of heat lost due to perspiration for each person by comparing the total metabolic heat generated (60 W) to the amount dissipated by convection (\(q_{1}\), \(q_{2}\)). Fraction of heat loss due to perspiration = \(\dfrac{\text{Metabolic Heat - Heat Dissipated by Convection}}{\text{Metabolic Heat}}\) Person 1 (Leather) Fraction of heat loss due to perspiration: \(F_{1} = \dfrac{60 - 0.5q_{1}}{60}\) Person 2 (Synthetic Fabric) Fraction of heat loss due to perspiration: \(F_{2} = \dfrac{60 - q_{2}}{60}\) Calculate \(F_{1}\) and \(F_{2}\) and interpret them as percentages to find the fractions of heat loss due to perspiration for each person.

Key concepts

Convection

Convection is a method of heat transfer that occurs through the movement of fluids like air or water. It plays a crucial role in heat dissipation from the human body. The convective heat transfer depends on the temperature difference between the body's surface and the surrounding air. The rate of heat transfer through convection can be described by Newton’s Law of Cooling:

• The greater the temperature difference, the higher the heat dissipation via convection.
• The heat transfer coefficient, which depends on the properties of the fluid and the nature of the surface, also affects this process.

In the given exercise, the ambient air temperature is set at 30°C, and the body's covered surface is at 32°C. The heat transfer coefficient is 15 W/m²K, indicating efficient heat transfer from the body. Understanding convection helps in realizing how our bodies naturally manage heat under varying ambient conditions.

Thermal Conductivity

Thermal conductivity is a material property that describes its ability to conduct heat. Materials with high thermal conductivity can transfer heat more efficiently. In our exercise, we have two different clothing materials for comparison:

• Leather with thermal conductivity of 0.159 W/m·K.
• Synthetic fabric with thermal conductivity of 0.13 W/m·K.

The leather allows better heat conduction compared to the synthetic fabric. This property determines how effectively each material can dissipate heat away from the body into the environment. Effective thermal conductivity is crucial for designing clothing that can regulate temperature under different conditions.

Perspiration

Perspiration, or sweating, is the body's natural mechanism to cool down. It involves the secretion of sweat, which evaporates and carries heat away from the body. This process is vital for maintaining body temperature, especially in warm environments or during physical activity. Here, perspiration complements convection, aiding in the removal of metabolic heat produced by the body. The fraction of heat loss due to perspiration is calculated by subtracting the fraction dissipated by convection from the total metabolic heat generated. Understanding perspiration helps explain human thermoregulation, particularly how the body prioritizes heat loss modes to maintain optimal body temperature under varying circumstances.

Metabolic Heat

Metabolic heat is the heat produced by the body's basic life processes and physical activities. Humans generate metabolic heat steadily, which needs to be dissipated to maintain a stable core body temperature. In our scenario, each individual creates 60 W of metabolic heat while engaging in sedentary work. The body's heat dissipation process involves:

• Convection, which transfers heat to the surrounding air.
• Perspiration, which releases heat through sweat evaporation.

Managing this metabolic heat efficiently ensures that the body maintains homeostasis, avoiding overheating.

Cylindrical Heat Conduction

The concept of cylindrical heat conduction applies to the body model used in this problem. The human body is approximated as a cylinder with a given diameter and length. This simplifies calculations by using formulas specific to cylindrical shapes.Heat transfer through a cylindrical wall is calculated using:\[q = \frac{2 \pi L k \Delta T}{\ln (r_{2}/r_{1})}\]Where:

• \(L\) is the length.
• \(r_{1}\) and \(r_{2}\) are the inner and outer radii, respectively.
• \(k\) is the thermal conductivity, and \(\Delta T\) is the temperature difference.

Using this setup, we calculate heat dissipation specific to the cylindrical geometry of the body.

Temperature Gradient

A temperature gradient is the difference in temperature between two points divided by the distance over which the change occurs. It is a driving factor for heat transfer. In our problem, the temperature gradient between the inner clothing surface and the ambient air is 2 K (32°C - 30°C). This gradient influences how much heat is transferred from the body through the clothing to the air.

• A higher gradient means faster heat transfer.
• A significant gradient ensures that the body can efficiently dissipate excess metabolic heat.

Understanding temperature gradients is essential in assessing how effectively heat transfer occurs in different materials and conditions.