Gases at Sonic Velocity - Choked Flow

[engineer-1]

Greetings to all in this fine community,

If you are familiar with Crane Tech Paper 410, there's an equation to calculate gas flow rate in SCFH. It takes into account pipe losses, pipe diameter, and inlet and outlet pressures.

q'h = 40,700*Y*(d^2)*((DP*P'1)/(K*T1*Sg))^0.5

q'h = SCFH
d = pipe diameter in inches
T1=absolute temp in degrees Rankine
K= total resistance coefficient in the pipeline
Sg=Specific gravity of gas
Y=net expansion factor (tabulated and given)

The flow is going from a larger diameter pipe upstream to a smaller diameter downstream.

Question is, this equation and model is used for short pipe length (around 40 ft and smaller). What happens to the model if your pipe length is longer by say a factor 15?

The inlet pressure is 200 psig. The outlet pressure is atmosphere. The flow has to be choked since k=1.4, and the ratio of P1 to P2 is much higher than 1.9

Would the flow reach sonic velocity inside the pipe or at discharge?

Regards,

 

[SteamKing]

If the flow inside the pipe is choked, then at some point in the pipe, the flow has reached a speed of Mach 1.

 

[engineer-1]

Thanks for the input. Now.
We know the flow will be choked based on [ 2/(k + 1) ]-k/(k − 1) which for say air (k=1.4) yields 1.89. This is the minimum pressure ratio (e.g. upstream P to P discharge) when we model using nozzle theory. In this case this pressure ratio is 12. Now typically sonic velocity occurs at pipe discharge. But is this true when we start to increase the length of the pipe? It seems to me that according to what you mention, sonic flow could occur inside the pipe and remain till discharge. Excellent point!

 

[engineer-1]

... bear in mind that the term -k/(k − 1).. in the expression [ 2/(k + 1) ]-k/(k − 1) means "to the power of". It just didn't paste correctly.