To calculate the available flow rate of instrument air to a control valve.  The tubing is 3/8" SS and the pressure is 80 psig minimum.  I do not know the velocity of the air or the pressure drop in the line so I'm having a hard time applying Bernoulli's. The flow rate (and velocity) is varying over a really wide range.  When the system first calls for air you have the maximum dP and very high flow rate for a few miliseconds.  As the actuator fills, the dP goes down and the velocity with it.  As pressure in the actuator approaches spring pressure the dP (and velocity) reaches a steady value for a few seconds.  Finally, with the actuator at full travel the dP (and velocity) again becomes continuously variable until it reaches supply pressure (which probably isn't 80 psig, generally actuators expect 15-30 psig and people use a regulator to cut system pressure to those  control values).
If you calculate the initial velocity with max dP, you'll find a huge friction loss (which reduces the dP so you have to iterate on this step a few times).  When you get a stable value, determine how much air got into the actuator in 50 mS at that velocity and recalculate the pressure in the actuator.  With the new dP you can start over and calculate a new velocity.  It is interesting to put a 100 Hz datalogger on the actuator to see the pressure traverse, it is impressive to see the real data match the theory.
A rigorous solution to this problem has eluded engineers throughout all time.  You can get an answer that will work with the above process, but it is kind of a pain.  I've done it when I was trying to determine if I had a problem with a slow-acting valve or if the latency really was that long.  Once when I was doing this arithmetic I found that the valve really was taking far too long to operate and discovered that it had the wrong main spring in it.  Replaced the spring and the valve worked as advertised.

MORE NEWS