Question

Air \(\left(c_{p}=1000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters a 20 -cm-diameter and 19-m-long underwater duct at \(50^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) at an average velocity of \(7 \mathrm{~m} / \mathrm{s}\) and is cooled by the water outside. If the average heat transfer coefficient is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the tube temperature is nearly equal to the water temperature of \(5{ }^{\circ} \mathrm{C}\), the exit temperature of air is (a) \(8^{\circ} \mathrm{C}\) (b) \(13^{\circ} \mathrm{C}\) (c) \(18^{\circ} \mathrm{C}\) (d) \(28^{\circ} \mathrm{C}\) (e) \(37^{\circ} \mathrm{C}\)

Short answer

The air enters the duct at a temperature of \(50^{\circ} \mathrm{C}\) and a pressure of 1 atm. Assume the average air velocity in the duct is 7 m/s and the heat transfer coefficient between the duct and cooling water is 35 W/\(\mathrm{m^2\cdot K}\). The duct's diameter is 20 cm. a) \(80^{\circ} \mathrm{C}\) b) \(60^{\circ} \mathrm{C}\) c) \(50^{\circ} \mathrm{C}\) d) \(40^{\circ} \mathrm{C}\) e) \(37^{\circ} \mathrm{C}\) Answer: (e) \(37^{\circ} \mathrm{C}\)

Step-by-step solution

Calculate the mass flow rate of air

To find the mass flow rate, first, we need to calculate the area of the duct. Use the diameter of the duct (20 cm) to calculate its area: Area = \(\pi\frac{D^2}{4}=\pi\frac{0.2^2}{4}=0.0314\ \mathrm{m^{2}}\) Now, we find the mass flow rate using the formula mass_flow_rate = density × velocity × area. The density of air at 1 atm and \(50^{\circ} \mathrm{C}\) is approximately 1.164 \(\mathrm{kg/m^3}\). After that, use the given average velocity, mass_flow_rate = 1.164 \(\mathrm{kg/m^3} × 7\ \mathrm{m/s} \times 0.0314\ \mathrm{m^{2}}\) mass_flow_rate = 0.256 \(\mathrm{kg/s}\)

Determine the rate of heat transmission

Using the conservation of energy principle, the rate of heat transmission (Q) can be determined as follows: Q = mass_flow_rate × cp × (T_exit - T_inlet) where cp = 1000 \(\mathrm{J/(kg\cdot K)}\), T_inlet = \(50^{\circ} \mathrm{C}\), and T_exit is the exit temperature we want to find.

Calculate the heat transfer rate

With the tube's heat transfer coefficient (35 \(\mathrm{W/m^2\cdot K}\)) and its temperature (5 \(^{\circ} \mathrm{C}\)), we can determine the heat transfer rate caused by the cooling water outside. First, calculate the surface area of the tube: Surface_area = \(\pi D\times L =\pi\times 0.2\times 19 = 11.969\ \mathrm{m^2}\) Now, we can write out the heat transfer formula (Q) as follows: Q = heat_transfer_coefficient × Surface_area × (T_tube - T_exit) Plugging in the values we have: 0.256 \(\mathrm{kg/s}\) × 1000 \(\mathrm{J/(kg\cdot K)}\) × (T_exit - 50) = 35 \(\mathrm{W/m^2\cdot K}\) × 11.969 \(\mathrm{m^2}\) × (5 - T_exit)

Solve for exit temperature

Simplify the equation from Step 3 and solve for T_exit: 256 × (T_exit - 50) = 418.45 × (5 - T_exit) 256T_exit - 12800 = 2092.25 - 418.45T_exit 674.45T_exit = 22892.25 T_exit = 33.96 \(^{\circ} \mathrm{C}\) Here, we see that the exact T_exit is not equal to any of the given options in the question. However, since it is closer to \(37^{\circ} \mathrm{C}\), we select option (e) as the closest match. The exit temperature of the air is approximately (e) \(37^{\circ} \mathrm{C}\).

Key concepts

Mass Flow Rate

The mass flow rate is a fundamental concept in fluid mechanics and heat transfer, especially in problems involving the cooling or heating of a fluid as it travels through a system. It's the measure of the mass of fluid passing a given point per unit time, which is crucial for calculating the rate of heat transfer in a system.

In the exercise, the mass flow rate of air is determined by multiplying the air's density by its velocity and the cross-sectional area of the duct. A higher mass flow rate often means more thermal energy is carried by the fluid, affecting the system's overall capacity to transfer heat. This calculation is the first step in analyzing heat transfer in the given duct, setting the stage for subsequent calculations involving heat exchange. It's vital to use the correct density corresponding to the air's temperature and pressure conditions. A calculation error here would lead to an inaccurate assessment of the system's thermal energy handling.

Heat Transfer Coefficient

In heat transfer problems, the heat transfer coefficient is a measure of the heat transfer rate per unit area per unit temperature difference. It's essentially a measure of a material's ability to allow thermal energy to pass through it. Factors like material properties and fluid velocity can influence the heat transfer coefficient.

In the context of the exercise, the heat transfer coefficient relates to how effectively the water outside the duct can absorb thermal energy from the air inside. The higher the coefficient, the more efficient the heat transfer. For precise calculations, it's essential to know the accurate value of this coefficient, as it directly impacts the rate at which heat is removed from the air in the duct. This coefficient is a crucial figure in the formula that determines the rate of heat transfer from the air to the surrounding water.

Thermal Energy Conservation

Thermal energy conservation is a principle stating that the total rate of heat loss from the fluid must equal the rate of heat gain or loss through the system boundaries. It's rooted in the first law of thermodynamics which deals with energy conservation.

Applied in the exercise, we calculate the rate at which the air is losing thermal energy. By setting up an energy balance equation using the mass flow rate and the heat transfer coefficient, we can solve for the air's exit temperature. The essential idea is that the thermal energy removed from the air within the duct must be equal to the thermal energy absorbed by the surrounding water. The law of thermal energy conservation serves as a check to ensure the energy entering and exiting the system is accounted for. Understanding this conservation principle is essential, as it is the key to solving many thermal systems' problems and ensuring the correct application of heat transfer equations.