Question

Air enters a 7-cm-diameter and 4-m-long tube at \(65^{\circ} \mathrm{C}\) and leaves at \(15^{\circ} \mathrm{C}\). The tube is observed to be nearly isothermal at \(5^{\circ} \mathrm{C}\). If the average convection heat transfer coefficient is \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\), the rate of heat transfer from the air is (a) \(491 \mathrm{~W}\) (b) \(616 \mathrm{~W}\) (c) \(810 \mathrm{~W}\) (d) \(907 \mathrm{~W}\) (e) \(975 \mathrm{~W}\)

Short answer

Answer: The rate of heat transfer from the air in the tube is approximately 616 W.

Step-by-step solution

Find the surface area of the tube

To find the surface area (A) of the tube, we need its diameter (D) and length (L). We have D=7 cm and L=4 m. First, we need to convert the diameter to meters: D=0.07 m. The surface area of the tube is given by the formula A = πDL.

Calculate the temperature difference

Given that air enters at 65°C and leaves at 15°C, the average air temperature inside the tube is \(T_{avg} = (65+15)/2 = 40^{\circ}\mathrm{C}\). The temperature of the tube's surface is nearly isothermal at 5°C. So, the temperature difference between the air and the surface of the tube is \(\Delta T = T_{avg} - T_s = 40^{\circ}\mathrm{C} - 5^{\circ}\mathrm{C} = 35^{\circ}\mathrm{C}\).

Apply the heat transfer formula

The rate of heat transfer (Q) from the air in the tube can be found using the formula Q = hAΔT, where h is the average convection heat transfer coefficient, A is the surface area of the tube, and ΔT is the temperature difference. We have h = 20 W/m²°C, A from step 1, and ΔT from step 2.

Calculate the rate of heat transfer

Now, we will plug in the values from the previous steps into the formula Q = hAΔT and calculate the rate of heat transfer: Q = 20 W/m²°C × π × 0.07 m × 4 m × 35°C. After calculating, we get Q ≈ 616 W. Hence, the rate of heat transfer from the air is 616 W (Answer option 'b').

Key concepts

Convection Heat Transfer

Convection heat transfer is the process of heat exchange between a surface and a fluid in motion. The movement of the fluid enhances the transfer of heat, making it a dynamic and rather effective process. Imagine a scenario where air flows through a tube, like in our exercise. As the air comes into contact with the tube's surface, heat is transferred between the air and the tube.

Convection can be further categorized into:

• Natural Convection: Where fluid motion is caused by buoyancy forces that result from density changes due to temperature variations in the fluid.
• Forced Convection: Occurs when external means, like a fan or pump, induces the fluid flow.

The convection heat transfer coefficient (

Heat Transfer Coefficient

opics is crucial because it quantifies how effectively a fluid carries heat away from the surface. In our example, this coefficient is given as 20 W/m²°C. It generally depends on factors like fluid velocity, fluid properties, and the type of flow.

Isothermal Process

An isothermal process is one in which the system's temperature remains constant. This typically means heat must be exchanged with the surroundings to counterbalance any incoming or outgoing energy. In our given exercise, the tube's surface is described as nearly isothermal, maintaining a consistent temperature of 5°C throughout the process.

Maintaining an isothermal condition in an object or environment assists in simplifying the calculation of heat transfer, since the temperature of the tube does not fluctuate with time. This constant temperature makes it easier to predict and calculate the rate at which heat is transferred into or out of the process, enhancing stability and consistency in measurements.

Temperature Difference

Temperature difference, represented as ΔT, is a vital factor when assessing heat transfer. It is the difference in temperature between two environments or mediums involved in a thermal process. In the exercise at hand, it refers to the disparity between the average air temperature in the tube and the tube's surface temperature.

To compute ΔT, an average air temperature is first determined, by taking the mean of the entering and leaving temperatures:

Average Temperature of Air

This average is \[ (T_{in} + T_{out}) / 2 = (65^ ext{°C} + 15^ ext{°C}) / 2 = 40^ ext{°C} \]. Then, the temperature difference is discerned by subtracting the tube's isothermal temperature from this average:\[ΔT = T_{avg} - T_s = 40^ ext{°C} - 5^ ext{°C} = 35^ ext{°C}\] This ΔT plays a crucial role in the heat transfer equation, impacting how much heat is transferred over time.

Heat Transfer Coefficient

The heat transfer coefficient, denoted as "h," is an essential factor in analyzing heat exchange efficiency between a surface and a fluid in contact. It typically varies depending on the nature of the fluid and its flow characteristics. In the context of our problem, this physic-based coefficient is given as 20 W/m²°C.

A higher heat transfer coefficient signifies that the fluid can efficiently carry away energy from the surface. Factors such as fluid velocity, surface roughness, and the types of materials involved can influence the value of 'h'.
The heat transfer equation used in our exercise—\[ Q = hAΔT \] relies heavily on the coefficient to estimate how much heat a fluid transfers per unit area per degree of temperature difference. Consistently large coefficients are essential in applications like heat exchangers, HVAC systems, and electronic cooling, optimizing the systems' cooling or heating efficiency.