Question

Consider the conditions of Example \(3-14\) in the text except that the ambient air is at a temperature of \(30^{\circ} \mathrm{C}\). A person with skin/fat layer thickness of \(0.003 \mathrm{~m}\) is doing vigorous exercise which raises the metabolic heat generation rate from 700 to \(7000 \mathrm{~W} / \mathrm{m}^{3}\) over a period of time. Calculate the perspiration rate required in lit/s so as to maintain the skin temperature at \(34^{\circ} \mathrm{C}\). Use the perspiration properties to be the same as that of liquid water at the average surface skin temperature of \(35.5^{\circ} \mathrm{C}\).

Short answer

Answer: The perspiration rate required to maintain the skin temperature at 34°C when a person is doing vigorous exercise is approximately 1.003 L/s.

Step-by-step solution

Determine the heat removed by perspiration

When perspiration comes in contact with the skin, it evaporates and absorbs energy as heat, which is removed from the body. We can calculate the required heat removal rate by using energy balance equation for the skin and the heat generation rate due to metabolism: \(q_{removed} = q_{generation} - q_{conduction} - q_{radiation} - q_{convection}\) We are given the metabolic heat generation rate \(q_{generation} = 7000 \,\text{W}/\text{m}^3\) after exercise. As per the supplied information, the amount of heat conducted through the skin is \(q_{conduction} = 700 \,\text{W}\) (the amount before exercise), and \(q_{radiation} = 0 \,\text{W}\), since it's not mentioned in the exercise. Lastly, without losing generality, we will consider \(q_{convection} = 0 \,\text{W}\) as well. Now, we can find the removed heat: \(q_{removed} = 7000 - 700 = 6300 \,\text{W/m}^3\)

Calculate the heat absorbed per unit mass of perspiration

To calculate the heat absorbed per unit mass of perspiration, we will use the specific heat capacity (\(c_p\)) and the difference in temperatures. We are given that the average surface skin temperature is \(35.5^{\circ} \mathrm{C}\), and the skin temperature should be maintained at \(34^{\circ} \mathrm{C}\). Thus, the temperature difference (\(ΔT\)) is: \(ΔT = 35.5 - 34 = 1.5 \, \mathrm{K}\) The properties of liquid water at the average surface skin temperature of \(35.5^{\circ} \mathrm{C}\) are: \(c_p = 4186 \, \text{J/kgK}\) (specific heat capacity of water) ρ = \(1000 \, \text{kg/m}^3\) (density of water) Now let's calculate the heat absorbed per unit mass of perspiration (\(q_{absorbed}\)): \(q_{absorbed} = c_p \cdot ΔT = 4186 \cdot 1.5 = 6279 \, \text{J/kg}\)

Calculate the perspiration rate

We can now determine the perspiration rate (\(\dot{m}\)) in kg/s by dividing the heat removed by the heat absorbed per unit mass: \(\dot{m} = \frac{q_{removed}}{q_{absorbed}} = \frac{6300}{6279} = 1.003 \, \text{kg/s}\) Lastly, convert the perspiration rate from kg/s to liters per second (L/s): \(\dot{m} = \frac{1.003 \, \text{kg/s}}{1000 \, \text{kg/m}^3} = 0.001003 \, \text{m}^3/\text{s} = 1.003 \, \text{L/s}\)

Present the final answer

The perspiration rate required to maintain the skin temperature at \(34^{\circ} \mathrm{C}\) when the person is doing vigorous exercise is approximately \(1.003 \, \text{L/s}\).

Key concepts

Heat Transfer in Human Body

The human body constantly engages in heat transfer with its environment to maintain a stable internal temperature, a process vital for survival. Heat transfer in the human body occurs in multiple ways, including conduction, convection, radiation, and evaporation.

Conduction involves the transfer of heat through direct contact between the skin and other surfaces. For example, you feel warmer when sitting on a heated seat. Convection involves the heat transfer due to fluid motion, such as when a breeze moves across your skin, carrying away body heat. Radiation is the emission of thermal energy in the form of infrared rays, which can be felt as the warmth from the sun or a fire. Lastly, evaporation is the process where sweat on the skin surface absorbs heat from the body to evaporate, leading to cooling effects.

When exercising, the body produces more heat due to increased metabolic activity. To prevent overheating, the body intensifies heat dissipation primarily through evaporation - the main focus of our perspiration rate calculation. Perspiration or sweating is a powerful mechanism which exploits the high latent heat of vaporization of water. As sweat evaporates, it removes excess heat, thus cooling the body effectively. This example exemplifies how evaporation plays a critical role in thermal regulation during physical activities or in hot environments.

Metabolic Heat Generation

Metabolic heat generation refers to the heat produced within the human body by metabolic reactions. These reactions are biochemical processes that occur in cells, including the breakdown of nutrients to release energy. The body utilizes this energy to perform various functions, but not all the energy can be harnessed. A significant portion is released as heat.

In a state of rest, the body has a baseline metabolic rate that generates a certain amount of heat. When the activity level increases, such as during vigorous exercise, the metabolic rate escalates correspondingly. The body's metabolic heat generation can increase dramatically from a typical 700 watts per cubic meter to as high as 7000 watts per cubic meter, as depicted in the given exercise.

The higher metabolic heat generation necessitates efficient heat dissipation to maintain normal body temperatures. If the heat produced exceeds the body's ability to dissipate it, core temperature rises, possibly leading to hyperthermia. Understanding metabolic heat generation is crucial for calculating the perspiration rate needed to balance this heat production during increased levels of activity.

Energy Balance Equation

The energy balance equation is vital for understanding the body's thermal status and how it achieves temperature homeostasis. It balances the internal heat production against heat loss to the environment to determine if the body is gaining or losing heat overall.

In its simplest form, the equation can be expressed as:\[\begin{equation}(\text{Heat Generated}) = (\text{Heat Lost}) + (\text{Change in Body Heat Content})\end{equation}\]
In scenarios like exercise, where metabolic rate is increased, the heat generated within the body can significantly increase. If the ambient temperature is high, as in our example, heat loss via conduction, convection, and radiation may decrease, amplifying the body's reliance on evaporative cooling through perspiration.

Using the energy balance equation, we calculate the rate of heat that needs to be removed to maintain a constant body temperature. For the perspiration rate calculation, we focus on the heat generation and the heat lost through perspiration (evaporative cooling), which we've simplified and represented as the energy removed from the body. In sum, the careful application of the energy balance equation allows us to deduce the required perspiration rate to sustain skin temperature during increased metabolic activity.