Understanding Choked Flow in Compressible Pipes in PIPE-FLO Advantage 17
As a compressible gas flows down a pipeline, friction acts to reduce the static pressure of the gas. As the pressure decreases, the static temperature and density of the gas decreases, causing the velocity of the gas to increase in order for the conservation of mass to hold true for a steady state condition. The gas velocity is limited to the speed of sound at the flowing conditions. When the gas reaches the sonic velocity, it is considered "choked flow" or "critical flow." The Mach Number (the ratio of the gas velocity to the static speed of sound) plays an important role in determining how the gas properties vary with flow.
Choked flow in real-world applications can result in inadequate over-pressure protection for safety relief systems, bottle-necked production rates, damage to equipment, or environmental excursions and costly fines.
Choked Flow in Compressible Pipes
The adiabatic Fanno flow relationships used for Compressible Pipes show that when friction is the only influence on the fluid properties, the Mach Number will always move toward sonic conditions with Ma = 1.0. There are certain configurations that can propel the gas velocity into the supersonic region (Mach Numbers greater than 1.0) but friction will act to slow the gas velocity and reduce Mach Number back to unity, often achieved through a shock wave. An increase in flow area can result in supersonic flow if it is choked at the pipe inlet to the expansion. Heat transferred into a gas can also propel a gas into supersonic flow conditions.
Using PIPE-FLO Advantage to Evaluate Choked Flow
Consider the classic explanation of achieving choked flow in a pipeline: while holding the inlet pressure constant, as the downstream pressure decrease, the mass flow rate increases until choked (or sonic) flow is reached. For the system modeled in Figure 1, there are 5 pipe segments of 100 mm pipe each 25 m long.
With a system inlet total pressure of 250,000 Pa a and outlet total pressure of 110,000 Pa a, the gas flows at 7237 kg/h (or 5919 sm3/h) as shown below. The Mach Number steadily increases from 0.2523 at the inlet boundary to 0.7771 at the outlet boundary. Note the conditions that change from inlet to outlet of each pipe: Volumetric Flows, Velocities, Mach Numbers, Total Pressures, Static Pressures, Static Temperatures, and Static Densities. Also note that the calculated Mass Flow is less than the Choked Mass Flow. The calculation for Choked Mass Flow is based on the calculated Inlet Total Pressure. As Total Pressure decreases, so does the Choked Mass Flow as seen by comparing each pipe's value.
Figure 1. Demonstrating choked flow: moderately high mach numbers but not choked.
Figure 2 shows the Outlet Pressure is reduced to 107,000 Pa a and results in a slight increase of Mass Flow to 7258 kg/h, Pipe 5 Outlet Mach of 0.864, a Choked Mass Flow of 7310 kg/h.
Figure 2: Demonstrating choked flow: reduced outlet pressure results in higher mass flow and Mach Numbers.
Figure 3 shows a decrease of the outlet pressure to 105,369 Pa a and results in an Outlet Mach Number of 0.9972, slightly increased Mass Flow to 7268 kg/h which is almost equal to the Choked Mass Flow of Pipe 5.
Figure 3: Demonstrating choked flow: reduced outlet pressure very close to choked conditions with Outlet Mach almost at 1.0.
In Figure 4, the outlet pressure is reduced by 1 Pa a and results in the exact choked flow conditions and accurate results throughout the system.
Figure 4: Demonstrating choked flow: reduced outlet pressure by 1 Pa a results in choked conditions with Outlet Mach equal to 1.0.
Modeling Beyond Choked Flow Conditions Results in Inaccuracies
If the outlet pressure is reduced slightly below the value that results in choked flow, a physically impossible condition in the real world, PIPE-FLO Advantage attempts to solve the system by limiting a pipe's outlet Mach Number to 1.0. The calculated results can be evaluated to see how inaccuracies begin to appear. In Figure 5, the outlet pressure is reduced to 105,000 Pa a, a value slightly below the choked total pressure. Note that all Mach Number upstream increase, but inaccuracies show up in the calculated total pressure at the boundary (105,421 Pa a vs. a setting of 105,000 Pa a) and the conservation of mass is not conserved between Pipe 4 and Pipe 5 (7270 kg/h vs. 7271 kg/h). Also note that the calculated Mass Flow is greater than the Choked Mass Flow, another indication that the system is set up to model conditions that are physically impossible to achieve.
Figure 5: Demonstrating choked flow: reduced outlet pressure below the choked outlet pressure results in Outlet Mach equal to 1.0 but inaccuracies begin to appear.
Now consider reducing the outlet pressure even further. In Figure 6, the outlet pressure is reduced to 100,000 Pa a. The calculated pressure at the boundary is 102,235 Pa a, the upstream Mach Numbers are increasing, the mass flow is much higher than the choked mass flow at Pipe 5, and the disparity between the mass flows of PIpe 4 and 5 are even greater.
Figure 6: Demonstrating choked flow: reduced outlet pressure below the choked outlet pressure results in Outlet Mach equal to 1.0, upstream Mach Numbers increase, and even greater inaccuracies result.
One more condition to evaluate is one that may show a pipe with both the inlet and outlet Mach Numbers equal to 1.0, again a physical impossibility but one that may be encountered. In Figure 7, the outlet pressure is reduced to 75,000 Pa a. The outlet Mach Number of Pipe 5 is still limited to 1.0 and the upstream Mach Numbers attempt to compensate to the point that the inlet Mach Number of Pipe 5 is at 1.0 as well. The inaccuracies can be seen in the calculated pressure at the Outlet compared to the setting, as well as the different mass flow rates between Pipe 3, 4, and 5.
Figure 7: Demonstrating choked flow: reduced outlet pressure below the choked outlet pressure results in Inlet and Outlet Mach equal to 1.0.
PIPE-FLO Advantage provides accurate calculated results when the modeled conditions result in sub-sonic flow with all Mach Numbers < 1.0. When conditions are modeled that are close to the choked flow conditions, results may be accurate but they must be closely evaluated. When discrepancies such as the mass flow rate of two consecutive pipes are encountered, the results cannot be relied upon and the modeled conditions changed to sub-sonic flow conditions. Discrepancies between the setting of pressure boundaries and the calculated pressure, and comparing a pipe's Mass Flow to its Choked Mass Flow, are also indications that the results cannot be relied upon and the modeled conditions changed.