Question

Consider an air compressor operating steadily. How would you compare the volume flow rates of the air at the compressor inlet and exit?

Short answer

Answer: In a steadily operating air compressor, the volume flow rate of the air at the compressor inlet is equal to the volume flow rate of the air at the compressor exit.

Step-by-step solution

Understand the continuity equation for compressible fluids

For compressible fluids like air, the continuity equation can be expressed as follows: \[\rho_1 A_1 v_1 = \rho_2 A_2 v_2\] where - \(\rho_1\) and \(\rho_2\) are the air densities at the inlet and exit, respectively (kg/m³) - \(A_1\) and \(A_2\) are the cross-sectional areas at the inlet and exit, respectively (m²) - \(v_1\) and \(v_2\) are the air velocities at the inlet and exit, respectively (m/s) This equation states that the mass flow rate at any two points in a flow system must be constant if the flow is steady and there are no mass sources or sinks.

Define volume flow rate

Volume flow rate (\(Q\)) is defined as the volume of fluid passing through a particular point in the system per unit time, mathematically represented as: \[Q = Av\] where - \(Q\) is the volume flow rate (m³/s) - \(A\) is the cross-sectional area (m²) - \(v\) is the fluid velocity (m/s) Using the volume flow rate definition, we can rewrite the continuity equation in terms of volume flow rates at the inlet and exit: \[Q_1 = Q_2\]

Compare the volume flow rates at the inlet and exit

Following the conclusion drawn from the continuity equation, the volume flow rates at the inlet (\(Q_1\)) and exit (\(Q_2\)) of the air compressor are equal if the compressor is operating in steady-state condition. Therefore, in a steadily operating air compressor, the volume flow rate of the air at the compressor inlet is equal to the volume flow rate of the air at the compressor exit.