How to calculate the flow and velocity of air in a pipe knowing the pressure at the ends?

Question

Is there a way to calculate the air flow and velocity at the outlet of a pipe (straight) knowing that one end of the pipe is connected to a compressor working at $3 kg/cm^2$ psi and the other end is not connected to anything (atmosphere)?

One of my main concerns is regarding the density of the air, since I dont know it changes through the pipe; another problem is the fact that I'm expecting turbulent flow and I'm not sure how to handle that kind of scenario.

Also, I'm willing to assume that temperature is constant it that makes easy.

Answer 1

If you assume that the temperature is constant and the gas behaves like an ideal gas, then the analysis is straightforward. The pressure gradient along the pipe is given by $$\frac{dp}{dz}=-\frac{1}{D}\rho\frac{v^2}{2}f \tag{1}$$where f is the Darcy-Weissbach friction factor, given graphically in the literature as a function of the Reynolds number:$$Re=\frac{4\dot{m}}{\pi D \mu}\tag{2}$$where $\mu$ is the gas viscosity and the mass flow rate $\dot{m}$ is given by$$\dot{m}=\rho v A\tag{3}$$with A representing the cross sectional area of he pipe ($A=\frac{\pi D^2}{4}$). If we combine Eqns. 1 and 3, make use of the ideal gas law ($\rho=\frac{pM}{RT}$, with M representing the molar mass of the gas), and integrate between z = 0 and z = L, we obtain: $$\Delta p^2=-\frac{RTL}{MDA^2}(\dot{m})^2\tag{4}f$$Since f is a function of Re (which in turn is a function of $\dot{m}$), if you are given the inlet and outlet pressures, you will need to use trial-and-error to solve for $\dot{m}$