Question

Air whose density is \(0.078 \mathrm{lbm} / \mathrm{ft}^{3}\) enters the duct of an air-conditioning system at a volume flow rate of \(450 \mathrm{ft}^{3} / \mathrm{min}\) If the diameter of the duct is 10 in, determine the velocity of the air at the duct inlet and the mass flow rate of air.

Short answer

Answer: The velocity of the air at the duct inlet is approximately 825.69 ft/min, and the mass flow rate of air is approximately 35.10 lbm/min.

Step-by-step solution

Calculate the cross-sectional area of the duct

The first thing we need to do is to calculate the cross-sectional area of the duct. The duct is circular, and its area can be calculated using the formula: \(A = \pi(\frac{d}{2})^2\) where A is the cross-sectional area, d is the diameter of the duct. Given the diameter of the duct as 10 inches, we have: \(A = \pi(\frac{10}{2})^2\) Convert inches to feet by multiplying by 0.08333 (1 inch = 0.08333 ft): \(A = \pi(\frac{10 * 0.08333}{2})^2\) Now, we can calculate the value of A: \(A \approx 0.545 \mathrm{ft}^{2}\)

Calculate the air velocity

Now that we have the cross-sectional area of the duct, we can use the volume flow rate formula to find the velocity. Volume flow rate Q can be written as: \(Q = A * v\) where Q is the volume flow rate, A is the cross-sectional area, and v is the velocity. Given the volume flow rate Q as 450 ft^3/min, we can solve for the velocity v: \(v = \frac{Q}{A} = \frac{450\ \mathrm{ft}^{3} / \mathrm{min}}{0.545\ \mathrm{ft}^{2}}\) Calculate the value of v: \(v \approx 825.69 \ \mathrm{ft} / \mathrm{min}\) The velocity of the air at the duct inlet is approximately 825.69 ft/min.

Calculate the mass flow rate

With the velocity and density given, we can now calculate the mass flow rate (m_dot) using the formula: \( \dot{m} = \rho * Q \) where \(\dot{m}\) is the mass flow rate, \(\rho\) is the density of air, and Q is the volume flow rate. Given the density \(\rho = 0.078 \ \mathrm{lbm} / \mathrm{ft}^{3}\) and the volume flow rate Q = 450 ft^3/min, we can calculate the mass flow rate: \(\dot{m} = 0.078\ \mathrm{lbm} / \mathrm{ft}^{3} * 450\ \mathrm{ft}^{3} / \mathrm{min}\) Calculate the value of m_dot: \(\dot{m} \approx 35.10\ \mathrm{lbm} / \mathrm{min}\) The mass flow rate of air is approximately 35.10 lbm/min.