Question

Air is flowing through a smooth thin-walled 4-indiameter copper tube that is submerged in water. The water maintains a constant temperature of \(60^{\circ} \mathrm{F}\) and a convection heat transfer coefficient of \(176 \mathrm{Btu} / \mathrm{h}-\mathrm{ft}^{2} \cdot \mathrm{R}\). If air (1 atm) enters the copper tube at a mean temperature of \(90^{\circ} \mathrm{F}\) with an average velocity of \(8 \mathrm{ft} / \mathrm{s}\), determine the necessary copper tube length so that the outlet mean temperature of the air is \(70^{\circ} \mathrm{F}\).

Short answer

Answer: To find the required length of the copper tube, use the following formula: \(L = \frac{\dot{m} \cdot C_p \cdot (T_{out} - T_{in})}{h_c \cdot (π \cdot d) \cdot (T_w - T_{av})}\) Substitute the known values and calculate the required length of the copper tube.

Step-by-step solution

Identifying Known Parameters

We are given the information about the air and the tube: 1. \(d\) (tube diameter) = \(4"\) 2. Water temperature, \(T_w\) = \(60 ^{\circ} \mathrm{F}\) 3. Convection heat transfer coefficient, \(h_c\) = \(176 \mathrm{Btu} / \mathrm{h}-\mathrm{ft}^{2} \cdot \mathrm{R}\) 4. Inlet air temperature, \(T_{in}\) = \(90 ^{\circ} \mathrm{F}\) 5. Air velocity, \(v_a\) = \(8 \mathrm{ft} / \mathrm{s}\) 6. Outlet air temperature, \(T_{out}\) = \(70 ^{\circ} \mathrm{F}\)

Calculate the Surface Area of the Tube

First, we need to find the surface area of the tube, assuming that the tube length is L. Surface area of the tube, \(A_t=π \cdot d \cdot L\)

Calculate the Heat Loss through Convection

We use the formula for convection heat loss, which is given by: \(Q = h_c \cdot A_t \cdot (T_w - T_{av})\) "Q" is the heat loss, and "T_{av}" is the average temperature of the air in the tube, which can be found by taking the average of the inlet and outlet temperatures: \(T_{av} = \frac{T_{in} + T_{out}}{2}\)

Calculate Mass Flow Rate of Air

Using the given air velocity, we can find the mass flow rate of the air using the density of air, which we assume to be constant for this problem at 1 atm pressure: \(\rho = 0.075 \mathrm{lb_m} / \mathrm{ft}^3\) The mass flow rate of the air, \(\dot{m}\), can be calculated as follows: \(\dot{m} = \rho \cdot v_a \cdot A_{in}\) The cross-sectional area of the tube, \(A_{in}\), can be calculated using the diameter: \(A_{in} = \frac{π \cdot d^2}{4}\)

Apply the Energy Balance Equation

Now, we can apply the energy balance equation to find the length of the tube: \(Q = \dot{m} \cdot C_p \cdot (T_{out} - T_{in})\) Here, \(C_p\) is specific heat capacity of air, approximately \(0.24 \mathrm{Btu} / (\mathrm{lb_m}\cdot \mathrm{R})\). Now, we can substitute \(Q\) from step 3, and \(\dot{m}\) from step 4 and solve for the length L: \(Q = h_c \cdot (π \cdot d \cdot L) \cdot (T_w - T_{av})\) \(\dot{m} \cdot C_p \cdot (T_{out} - T_{in}) = h_c \cdot (π \cdot d \cdot L) \cdot (T_w - T_{av})\)

Solve for the Length of the Copper Tube

Rearranging and solving for L: \(L = \frac{\dot{m} \cdot C_p \cdot (T_{out} - T_{in})}{h_c \cdot (π \cdot d) \cdot (T_w - T_{av})}\) Now substitute the known values and perform the calculation to find the required length of the copper tube.

Key concepts

Heat Loss Calculation

Understanding how heat loss is calculated is critical when analyzing thermal systems, especially in scenarios like air flowing through a copper tube submerged in water. Heat loss occurs due to the difference in temperature between the air inside the tube and the surrounding water. The key formula to remember for convection heat transfer is:

\[Q = h_c \times A_t \times (T_w - T_{av})\]
Where:

• \(Q\) represents the heat loss.
• \(h_c\) is the convection heat transfer coefficient.
• \(A_t\) is the surface area of the tube.
• \(T_w\) is the water temperature.
• \(T_{av}\) is the average air temperature within the tube.

This equation reflects the direct relationship between the temperature difference and the heat loss rate; as the temperature difference increases, so does the heat lost through convection. The aim in our exercise is to find the right length of copper tube so that the air cools to the desired temperature, thus achieving a specific amount of heat loss.

Mass Flow Rate of Air

The mass flow rate of air is a measure of how much air is passing through a given cross-sectional area per unit time. This parameter is essential in our calculations. We use the formula:

\[\r\dot{m} = \rho \times v_a \times A_{in}\]
Where:

• \(\r\dot{m}\) is the mass flow rate of air.
• \(\rho\) is the density of air.
• \(v_a\) is the velocity of air.
• \(A_{in}\) is the cross-sectional area of the tube.

In this scenario, we assume air is incompressible at a constant atmospheric pressure—which simplifies the calculation, aiding in determining the rate at which air transfers its energy to the surrounding water. Analyzing this flow rate is vital, as it will be directly used in the energy balance formula to determine the required tube length for the desired change in air temperature.

Energy Balance Equation

The energy balance equation is a representation of the conservation of energy principle. In the context of our exercise, it allows us to equate the heat loss from the air as it travels through the tube to the change in internal energy of the air due to temperature difference. Mathematically, this is expressed by the equation:

\[Q = \r\dot{m} \times C_p \times (T_{out} - T_{in})\]
Here:

• \(Q\) is the heat loss through the tube's surface.
• \(\r\dot{m}\) is the mass flow rate of air.
• \(C_p\) is the specific heat capacity of air, reflecting how much energy is required to change the air's temperature.
• \(T_{out}\) and \(T_{in}\) are the outlet and inlet air temperatures, respectively.

Using this equation, we can solve for the unknown length of the copper tube by rearranging the terms. Properly applying the energy balance ensures that all heat transfer is accounted for, leading to a reliable prediction of system behavior and accurate design criteria for the copper tube in our hypothetical application.