I would like to determine the massflow rate of (a) steam (b) air flowing through a cylindrical pipe knowing that p_inlet = 5 bars p_outlet = 1 bar. Internal diameter is 0.5m. Length is 30m.

Surface roughness is assumed equal to 5E-05 m. Only steady state situation is necessary. Temperature at the inlet is assumed in both cases to be equal to 150°C. For derivation, one can assume adiabatic flow.

What would be the steps to determine the massflow rate ? I assume one must first check whether the flow is compressible or not ? Then whether the flow is subsonic, sonic, supersonic ? Then also if laminar or turbulent ? And then eventually, what would be the equation to be used to determine the massflow rate ?

  • $\begingroup$ Your first step would be to know the internal diameter and length :) $\endgroup$
    – Solar Mike
    May 21, 2017 at 9:56
  • $\begingroup$ Added. Thanks for spotting this missing info $\endgroup$ May 21, 2017 at 11:20
  • $\begingroup$ I am not sure but, do we apply reynolds transportation theorem? $\endgroup$
    – Fennekin
    May 23, 2017 at 16:03

1 Answer 1


In pipe flow, we don't typically talk about subsonic / supersonic / transonic flow. We instead differentiate between choked and non-choked flow. Choking occurs when the upstream pressure is greater than the downstream pressure by some factor (dictated by the gas or fluid). That factor is usually ~1.5, depending (roughly) on the ratio of specific heats. You are clearly dealing with choked flow in your case. Choking means the flow speed at the exit is the speed of sound. Any excess pressure past the point of choking does not increase the flow speed. It will merely increase the density of the fluid.

In this case, it does not matter whether your fluid is compressible or gas or liquid phase, because this is a relatively uncomplicated problem. In others, where you may have a restrictor plate or valve, it would.

The equation for the condition of choked flow:

p* / p0 = (2 / (gamma + 1)) ^ (gamma / (gamma - 1))

where p* is the downstream pressure, p0 is the upstream pressure, and gamma is the ratio of specific heats. If your downstream pressure is lower than p*, then your flow is choked.

In a straight cylindrical pipe we can (almost) always assume flow is laminar, but there are equations to check that for us. The Reynolds number for a pipe flow can be found by:

Re = rho V  D / mu

with rho being density (kg / m3), V being flow speed (m/s), D being the pipe diameter (m), and mu being the dynamic viscosity (N s / m^2).

For choked flow, we can simply use a mass flow rate equation based on pressure differential. If you were looking for mass flow rate without already knowing the downstream pressure, or looking for the pressure drop caused by the friction, we'd have to take account the surface roughness. Similarly more work is required with non-choked flow.

The mass flow rate equation for choked flow:

mdot = Cd A sqrt(gamma rho0 p0 (2 / (gamma +1)) ^ (gamma + 1) / (gamma - 1))


Cd - the coefficient of discharge, defined next

A - pipe exit area

gamma - ratio of specific heats

rho0 - upstream density (which you should compute based on T, p0)

p0 - upstream pressure

Cd = mdot / (A sqrt(2 rho deltaP))

with deltaP being the pressure drop across the exit (here, 5 bars - 1 bar = 4 bars). You'll need to convert units here as appropriate (i.e. to Pascals).

The coefficient of discharge basically is a refinement to the mass flow rate; it takes into account that the cross section of the flow in the orifice at its maximum speed is not the same size as the total area of the orifice. If you're bored and want to learn more about that phenomena, look up vena contracta.

There's a wide array of pipe flow, valve, orifice, etc. problems out there. If you're looking for more information, Crane's TP-410 publication is the de facto reference for these sort of standard pipe flow problems. I also really like Robert Blevin's Applied Fluid Dynamics Handbook, as it has a host of information not available anywhere on the internet (for estimating pressure losses due to all sorts of pipe configurations).

  • $\begingroup$ Thank you for your detailed and clear answer and the appropriate references! So even with a gas you can't go supersonic in a straight pipe ? is there an explanation to that ? $\endgroup$ May 23, 2017 at 20:30
  • $\begingroup$ Pressure perturbations travel at the speed of sound. If the downstream pressure decreases, the flow upstream cannot know about it - because the pressure perturbation travels upstream at the same rate as the sonic flow travels downstream! Therefore, the downstream conditions that cause flow changes can never propagate upstream past the point where the flow reaches M = 1. en.wikipedia.org/wiki/Isentropic_nozzle_flow#Supersonic_flow You require a positive area change to go supersonic when you already have sonic flow. (See Zucrow & Hoffman, a great great book, or Hill & Peterson) $\endgroup$ May 23, 2017 at 20:47

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