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.

 

[alphy]

I would like to clarify the concept of choked flow and its implications for gas flow at sonic velocity. Choked flow occurs when the flow rate of a gas reaches its maximum possible value, also known as the sonic velocity. This happens when the gas reaches a critical pressure ratio, which is dependent on the specific heat ratio of the gas (k). In this case, with a k value of 1.4, the critical pressure ratio is 1.9.

In the equation provided, the term (DP*P'1)/(K*T1*Sg) represents the pressure ratio, where DP is the pressure drop, P'1 is the inlet pressure, K is the resistance coefficient, T1 is the absolute temperature, and Sg is the specific gravity of the gas. This term is raised to the power of 0.5, indicating that the flow rate is proportional to the square root of the pressure ratio.

Now, to address the question of longer pipe length, the equation and model provided are valid for short pipe lengths (around 40 ft or smaller) due to the assumption of negligible pipe losses. As the pipe length increases, the pressure drop and resistance coefficient will also increase, resulting in a higher pressure ratio. If this pressure ratio exceeds the critical value of 1.9, the flow will become choked and reach sonic velocity inside the pipe.

In this scenario, the flow will continue at sonic velocity until it reaches the outlet, where it will undergo a sudden expansion and the velocity will decrease. Therefore, the flow will reach sonic velocity both inside the pipe and at the discharge point. However, it is important to note that the equation provided does not take into account the effects of longer pipe lengths and may not accurately predict the flow rate. A more comprehensive model would need to be used in this case.

I hope this helps clarify the concept of choked flow and its implications for gas flow at sonic velocity. As scientists, it is important for us to continue exploring and understanding these complex phenomena to improve our models and predictions.