Question

Air enters the duct of an air-conditioning system at 15 psia and \(50^{\circ} \mathrm{F}\) at a volume flow rate of \(450 \mathrm{ft}^{3} / \mathrm{min}\). The diameter of the duct is 10 in, and heat is transferred to the air in the duct from the surroundings at a rate of \(2 \mathrm{Btu} / \mathrm{s}\). Determine \((a)\) the velocity of the air at the duct inlet and \((b)\) the temperature of the air at the exit.

Short answer

Short Answer: To find the velocity of the air at the duct inlet and the temperature of the air at the exit, first calculate the area of the duct inlet using the diameter provided. Then, divide the volume flow rate by the area to determine the velocity. To find the exit temperature, first find the heat capacity of the air on the assumption that it behaves as an ideal gas. Next, compute the mass flow rate of air using the volume flow rate and the specific volume. Use the heat transfer rate, the heat capacity, and the mass flow rate to establish the temperature increase. Finally, add this temperature increase to the initial inlet temperature, resulting in the exit temperature.

Step-by-step solution

Find the area of the duct inlet

To find the area of the duct inlet, we can use the formula for the area of a circle: \(A = \pi r^2\), where \(r\) is the radius of the duct. The diameter of the duct is given as 10 inches, so the radius is half of that, which is 5 inches. Convert this to feet: \(r = 5/12\) ft. Now, we can find the area: \(A = \pi (5/12)^2\) ft\(^2\).

Determine the velocity of the air at the duct inlet

Now that we have the area of the duct inlet, we can use the volume flow rate to find the velocity of the air: \(V = Q/A\), where \(Q\) is the volume flow rate and \(A\) is the area. We are given \(Q = 450\) ft\(^3\)/min. Convert this to ft\(^3\)/s: \(Q = 450/60\) ft\(^3\)/s. Now, we can find the velocity: \(V = (450/60) / \pi (5/12)^2\) ft/s.

Find the heat capacity of the air

In order to find the temperature of the air at the exit, we need to know the heat capacity of the air. We can assume that the air is an ideal gas, and its specific heat at constant pressure is \(c_p = 0.24\) Btu/lbm\(^\circ\)F.

Calculate the mass flow rate of air

We can assume that the air is an ideal gas, and we know the specific volume of air at the given pressure and temperature is \(\nu = 13.22\) ft\(^3\)/lbm. The mass flow rate of air, \(m\), can be found from the volume flow rate and specific volume: \(m = Q/\nu\) lbm/s, where \(Q = 450/60\) ft\(^3\)/s. Now, we can find the mass flow rate: \(m = (450/60) / 13.22\) lbm/s.

Calculate the temperature increase of the air due to heat transfer

We are given that heat is transferred to the air at a rate of \(2\) Btu/s. The temperature increase, \(\Delta T\), can be found from the heat transfer rate and the heat capacity of the air: \(\Delta T = Q_{heat} \cdot c_p / (m\cdot c_p)\), where \(Q_{heat} = 2\) Btu/s, \(m\) is the mass flow rate calculated in Step 4, and \(c_p = 0.24\) Btu/lbm\(^\circ\)F. Now, we can find the temperature increase: \(\Delta T = 2/(m\cdot 0.24)\) \(^\circ\)F.

Calculate the temperature of the air at the exit

The temperature of the air at the exit, \(T_{exit}\), can be found by adding the temperature increase due to heat transfer to the inlet temperature: \(T_{exit} = T_{inlet} + \Delta T\), where \(T_{inlet} = 50^\circ\)F and \(\Delta T\) is the temperature increase calculated in Step 5. Now, we can find the temperature of the air at the exit: \(T_{exit} = 50 + \Delta T\) \(^\circ\)F.