Hi

Ok, I have a pressure vessel (in this case, the tank of a compressor)
maintaining a pressure of 2.5 bar.  The air leaves the vessel into a
tube of 8mm ID. This tube is 1m long and constructed of pvc. The tube
connects to a manifold. The input to the manifold is 6mm in diameter.
There are 8 exit branches, each of 4mm diameter. Connected to these
are 8 further tubes of 1m length 6mm ID. These let the air flow to
atmospheric pressure.  The whole system is at STP (25oC and 1 bar) and
there is no height change in the tubes. How do I work out the air flow
velocity and reynolds number through the system
so I can calculate losses due to friction factors? If I wanted the
pressure in the final tube to be 2.5 bar, what would the pressure have
to be in the compressor to provide this to all 8 tubes? Its driving me
crazy, I have no idea how to apply Bernoulli etc. Thank you very much
for your time.

If no friction then Berloulli's equation says energy is conserved and
consists of terms for Pressure, K.E. and gravitational P.E.   There is
stated to be no height difference in this device so reducing it to two
terms (and we cancel a factor of 2).   Assuming no K.E. in the initial
tank we have:
    Pressure_difference = density * speed ** 2
so 1.5 bar (or 2.5 depending in the presure measurement)
suggests sqrt(1.5e5/1.3) =  339 m/s  IF NO FRICTION.   This is a high
speed suggesting that the ideal of no friction will be very far wrong.

I think the next thing to do is estimate the pressure at the end of
the first pipe, in the style of a potential divider and assuming
laminar flow everywhere.    If the reynolds numbers from that indicate
turbulence then that's a bad assumption too.