Problem 4 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

On a hot day, a fan is blowing warm air over a person. The air is at a temperature of \(45^{\circ} \mathrm{C}\) and the person can be assumed to be represented as a flat plate with dimensions of \(1 \mathrm{~m} \times 0.2 \mathrm{~m}\) having a surface temperature of \(28^{\circ} \mathrm{C}\). If the convection heat transfer coefficient is found to be \(100 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\), determine the rate of heat transfer from the hot air to the person.

Step-by-Step Solution

Answer

The rate of heat transfer is 340 W.

Step 1: Identify Known Values

First, let's identify and list all the known values from the problem. The air temperature \(T_{\infty}\) is \(45^{\circ} \mathrm{C}\), the surface temperature \(T_s\) is \(28^{\circ} \mathrm{C}\), and the convection heat transfer coefficient \(h\) is \(100 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). The surface area of the person (flat plate) is \(1 \mathrm{~m} \times 0.2 \mathrm{~m} = 0.2 \mathrm{~m}^2\).

Step 2: Determine the Temperature Difference

The temperature difference between the hot air and the surface of the person is the driving force for heat transfer. We calculate this as \(\Delta T = T_{\infty} - T_s = 45^{\circ} \mathrm{C} - 28^{\circ} \mathrm{C} = 17^{\circ} \mathrm{C}\).

Step 3: Calculate the Rate of Heat Transfer

The rate of heat transfer \(\dot{Q}\) can be calculated using the formula for convective heat transfer: \[ \dot{Q} = h \cdot A \cdot \Delta T \] Substituting the given values, we get:\[ \dot{Q} = 100 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K} \times 0.2 \mathrm{~m}^2 \times 17^{\circ} \mathrm{C} = 340 \mathrm{~W} \]

Key Concepts

Heat Transfer Calculation

Heat transfer calculation is essential to understanding how thermal energy moves from one object to another. When it comes to convection heat transfer, there's a particular equation that governs the process. This equation is \[ \dot{Q} = h \cdot A \cdot \Delta T \]where \(\dot{Q}\) is the rate of heat transfer, \(h\) is the heat transfer coefficient, \(A\) is the area through which heat is being transferred, and \(\Delta T\) is the temperature difference across that area.

• To perform these calculations, start by identifying the known values such as the heat transfer coefficient, the surface area, and the temperature difference.
• Substitute these into the equation and solve for the rate of heat transfer.

Breaking it down, the equation tells us how effectively heat is transferring between the surroundings and an object. So, whether you're looking to find out how many watts of power are being transferred to, or from an object, this is the go-to formula. In our specific example, using the convection heat transfer formula, we found the rate of heat transfer to be 340 W, illustrating the process efficiently.

Temperature Difference

Temperature difference (\(\Delta T\)) is a key player in heat transfer scenarios. It represents the thermal gradient driving the transfer of heat. The greater the difference in temperature between two objects or environments, the more heat will flow between them naturally. It's important to remember:

• Temperature difference is calculated simply by subtracting the surface temperature from the temperature of the surrounding air or fluid.
• This difference is often expressed in Celsius or Kelvin.

In the exercise provided, the temperature difference was fixed at \(17^{\circ} C\), calculated as \(45^{\circ} C - 28^{\circ} C\). This means that there's a 17-degree capacity for heat to move from the air to the cooler surface. Understanding how temperature difference impacts heat transfer is crucial for effectively managing heating and cooling processes. It is this difference that propels the movement of heat, directly affecting how energy efficient and effective our systems can be.

Heat Transfer Coefficient

The heat transfer coefficient ( \(h\) ) is a measure of the convective heat transfer ability. Essentially, it signifies how readily heat transfers between an object and a fluid or air flowing around it.

• The larger the heat transfer coefficient, the more efficient the heat transfer will be.
• This coefficient is dependent on the properties of the fluid (such as viscosity and thermal conductivity) and the nature of the surface interaction.

In practical applications, engineers use the heat transfer coefficient to optimize designs for heating and cooling processes. In our original problem, \(h\) was found to be \(100 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\). This value represents a fairly active transfer scenario, where heat easily flows between the air and the surface. Recognizing the heat transfer coefficient as an intermediate -- connecting the rate of heat transfer with the temperature difference and area -- allows for precise adjustments in systems to enhance efficiency and performance. Understanding and calculating the heat transfer coefficient is paramount for those involved in energy management and thermal regulation challenges.