Question

A steady-flow compressor is used to compress helium from 15 psia and \(70^{\circ} \mathrm{F}\) at the inlet to 200 psia and \(600^{\circ} \mathrm{F}\) at the outlet. The outlet area and velocity are \(0.01 \mathrm{ft}^{2}\) and \(100 \mathrm{ft} / \mathrm{s}\) respectively, and the inlet velocity is \(50 \mathrm{ft} / \mathrm{s}\). Determine the mass flow rate and the inlet area.

Short answer

Question: Determine the mass flow rate and the inlet area of a steady-flow compressor handling the compression of helium. Given parameters are the pressure, temperature, and area at both the inlet and outlet, and the velocities at both points. Answer: To solve this problem, follow these steps: 1. Convert the given parameters to SI units. 2. Compute the density of helium at the inlet and outlet using the ideal gas law. 3. Determine the mass flow rate using the formula m_dot = rho * v * A, where rho is the density, v is the velocity, and A is the area at the outlet. 4. Determine the inlet area using the same mass flow rate and the formula A = m_dot / (rho * v) with the velocity and density at the inlet. With the given parameters in SI units, compute the density, mass flow rate, and inlet area.

Step-by-step solution

Convert the given parameters to SI units

Working in SI units (International System of Units) is convenient for calculations in physics. Convert the temperatures from Fahrenheit to Kelvin using the formula: \(K = (°F + 459.67) × 5/9\), the pressures from psia to Pa by using the fact that 1 psi = 6894.76 Pa, the velocities from ft/s to m/s (1 ft/s = 0.3048 m/s), and the areas from ft^2 to m^2 (1 ft^2 = 0.092903 m^2).

Compute the Density of Helium at the Inlet and Outlet

Use the ideal gas law, namely \(PV = nRT\), where P is pressure, n is the amount of substance, V is volume, R is the ideal gas constant, and T is temperature. Here, we are interested in calculating the density which is the mass (m) per unit volume (V), so we can rearrange the equation in terms of density: \(\rho = P / RT\). Here, we have to use the specific gas constant for helium (R = 2077 J/(kg.K)). Plugging in the values for temperature and pressure at the inlet and outlet, we get density at the inlet and outlet.

Determine the Mass Flow Rate

The mass flow rate (m_dot) can be calculated using the formula: \(m_dot = rho * v * A\), where rho is the density, v is velocity, and A is area. As we have the density, the velocity, and the area at the outlet, substitute these values into the formula to get the mass flow rate.

Determine the Inlet Area

Since the flow is steady, the mass flow rate at the inlet and outlet are the same. With the mass flow rate known, rearrange the formula from step 3 to solve for the area, A = m_dot / (rho * v). Substitute in the mass flow rate and the velocity and density at the inlet to find the inlet area.