Problem 9 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

A sunbather who loses no heat through the surface pointing away from the sun is exposed to solar radiation at a rate of \(750 \mathrm{~W} / \mathrm{m}^{2}\). The absorptivity of the person can be estimated to be \(0.55\) with an air temperature of \(27^{\circ} \mathrm{C}\). Determine the surface temperature of the person if the convection heat loss is \(0.9\) of the solar energy absorbed by the person and assume that the convection heat transfer coefficient is \(35 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). Disregard heat loss from radiation.

Step-by-Step Solution

Answer

The surface temperature of the person is approximately 310.75 K.

Step 1: Calculate Solar Energy Absorption

The rate of solar energy absorbed by the person is given by the solar radiation multiplied by the absorptivity. The absorbed energy rate, \( Q_{\text{absorbed}} \), can be calculated using:\[Q_{\text{absorbed}} = \text{Radiation} \times \text{Absorptivity} = 750 \times 0.55 = 412.5 \mathrm{~W/m^2}\]

Step 2: Determine Convective Heat Loss

Given that convective heat loss is 0.9 of the absorbed solar energy, calculate the convective heat loss, \( Q_{\text{convective}} \).\[Q_{\text{convective}} = 0.9 \times Q_{\text{absorbed}} = 0.9 \times 412.5 = 371.25 \mathrm{~W/m^2}\]

Step 3: Set Up the Heat Transfer Equation

The convection heat loss can be described by the equation:\[Q_{\text{convective}} = h \times A \times (T_s - T_{\infty})\]Where- \( h = 35 \mathrm{~W/m^2~K} \)- \( T_s \) is the surface temperature of the person (in Kelvin)- \( T_{\infty} = 27^{\circ} \mathrm{C} = 300.15 \mathrm{~K} \)- \( A \) cancels out as it appears both in calculation of heat absorbed and lost.Now we have:\[371.25 = 35 \times (T_s - 300.15)\]

Step 4: Solve for Surface Temperature

Rearrange the equation for \( T_s \):\[T_s - 300.15 = \frac{371.25}{35}\]Calculate \( T_s \):\[T_s = \frac{371.25}{35} + 300.15 = 310.75 \mathrm{~K}\]

Key Concepts

Solar Radiation

Solar radiation is the energy emitted by the sun in the form of electromagnetic waves, primarily visible and ultraviolet light. When this energy reaches the Earth, it is either absorbed, reflected, or transmitted by different surfaces. In the context of biofluid mechanics, especially when dealing with human exposure, it's crucial to consider how much of this solar radiation is absorbed by the body. The rate of solar energy absorption by a surface is determined by several factors:

• **Solar Radiation Intensity**: This is the power per unit area received from the sun, typically measured in watts per square meter (W/m²). In the given exercise, the solar intensity is 750 W/m².


• **Absorptivity**: This refers to the fraction of the radiation that is absorbed by a material. Different materials (or skin types) have varying absorptivities. For instance, an absorptivity of 0.55 means that 55% of the incoming solar energy is absorbed.

Understanding these concepts allows us to calculate the actual amount of energy absorbed, which is critical for further analysis such as estimating heat transfer through convection.

Convective Heat Loss

Convective heat loss refers to the transfer of heat from the body's surface to the surrounding fluid, which is typically air. This process occurs when the air in contact with the body is heated, becoming less dense, and then moving away, carrying the heat with it. This is an important cooling mechanism that helps regulate body temperature when exposed to external heat sources. In the exercise, we learn that convective heat loss is a certain percentage of the absorbed solar energy. Specifically:

• The convective heat loss is modeled to be 0.9 of the absorbed energy, meaning 90% of the energy absorbed from solar radiation is lost through convection.


• The efficient removal of this heat is crucial as excessive heat retention can lead to thermal discomfort or heat stress.

This concept shows how energy absorbed from the environment is not just retained in the body but dynamically interacts with the environment, facilitating thermal regulation.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, denoted by \( h \), is a measure of how effectively heat is transferred from a surface to a fluid (or vice versa) due to convection. This is an essential parameter in calculating heat transfer rates and can vary based on factors such as fluid velocity, surface properties, and temperature differences.In the provided problem, the convection heat transfer coefficient is given as 35 W/m²·K, which is a typical value for natural convection in air. It informs us about:

• **Rate of Heat Transfer**: A higher \( h \) value indicates a better heat transfer rate, which translates into more effective cooling in scenarios like the sunbather example.


• **Role in Equations**: The coefficient is used in the formula \( Q_{\text{convective}} = h \times A \times (T_s - T_\infty) \), to determine how heat moves from the body's surface \( T_s \) to the ambient temperature \( T_\infty \).

Understanding this coefficient and its application helps in designing systems for thermal comfort, making it crucial for studies in biofluid mechanics.