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))
with:
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).
