Gas-Flow Calculations: Don’t Choke
AFT Arrow™ Technical Paper
Authors: Brian A.Winters, P.E., Orbital Sciences Corporation and Trey Walters, Applied Flow Technology
From the NASA Proceedings of the Eighth Annual Thermal and Fluid Analysis Workshop, July 1997
ABSTRACT
Flow of gases in pipe systems is commonplace in chemical-process plants. Unfortunately, the design and analysis of gas-flow systems are considerably more complicated than for liquid (incompressible) flow, due mainly to pressure-induced variations in the gas-stream density and velocity. Here, we review practical principles and present some key equations governing gas flow, and assess several assumptions and rules of thumb that engineers sometimes apply in order to simplify gas-flow analysis and calculations.
In a broad sense, the appropriate term for gas flow is compressible flow. In a stricter sense, however, such flow can be categorized as either incompressible or compressible, depending on the amount of pressure change the gas undergoes, as well as on other conditions.
Accurately calculating truly compressible flow in pipe systems, especially in branching networks, is a formidable task. Accordingly, engineers often apply rules of thumb to a given design situation involving gas flow, to decide whether the use of (simpler) incompressible-flow calculations can be justified. Such rules of thumb are helpful, but they can lead one astray when used without a full understanding ofthe underlying assumptions.
CONCLUSION
The methods discussed in this article can help the engineer assess endpoint sonic choking, but restriction and expansion choking are somewhat more complicated. Accordingly, the estimation methods in this article may not be applied to all choking situations.
For new designs that require a lot of pipe, the engineer should consider the potential cost savings if smaller pipe sizes can be used. If significant cost savings prove to be possible, it may be prudent to invest in developing a detailed model that can more accurately determine the system’s capability over a range of pipe sizes. A detailed model may also help assess the wisdom of making modifications proposed for an existing system
Compressors, blowers and fans raise the system pressure and density. These changes in properties inside the gasflow system further limit the applicability of incompressible methods, beyond the cautions already discussed. Take special care in applying the incompressible-flow methods and estimation equations in this article to such systems.
Below is an excerpt. Use the links above to view the full paper.
Sonic Chocking
Sonic choking In almost all instances of gas flow in pipes, the gas accelerates along the length of the pipe. This behavior can be understood from Equations (2), (3) and (5). In Equation (3), the pressure falls off, due to friction. As the pressure drops, the gas density will also drop (Equation [5]). According to Equation (2), the dropping density must be balanced by an increase in velocity to maintain mass balance.
It is not surprising, then, that gas flow in pipelines commonly takes place at velocities far greater than those for liquid flow — indeed, gases often approach sonic velocity, the local speed of sound. A typical sonic velocity for air is 1,000 ft/s (305 m/s).
When a flowing gas at some location in the pipeline experiences a local velocity equal to the sonic velocity of the gas at that temperature, sonic choking occurs and a shock wave forms. Such choking can occur in various pipe configurations (Figure 1).
The first case, which can be called endpoint choking, occurs at the end of a pipe as it exits into a large vessel or the atmosphere. In this situation, the gas pressure cannot drop to match that at the discharge without the gas accelerating to sonic velocity. A shock wave forms at the end of the pipe, resulting in a pressure discontinuity.
The second case, which might be called expansion choking, crops up when the cross-section area of the pipe is increased rapidly; for example, if the system expands from a 2-in. pipe to one of 3-in. pipe. This can also happen when a pipe enters a flow splitter such that the sum of the pipe areas on the splitting side exceeds the area of the supply pipe. A shock wave forms at the end of the supply pipe, and a pressure discontinuity is established.
The third case, which may be called restriction choking, occurs when the gas flows through a restriction in the pipe, such as an orifice or valve. In such a case, the flow area of the gas is reduced, causing a local increase in velocity, which may reach the sonic velocity. A shock wave forms at the restriction, with a pressure discontinuity similar to the first two cases.
Figure 2 shows stagnation-pressure and Mach-number profiles for expansion choking and restriction choking; both involve supply air at 100 psia and 1,000°R discharging to 30 psia. Endpoint-choking behavior appears in Figure 7, discussed later. For a given process situation, the choked flowrate can be determined from Equation (10a), by inserting a Mach number of 1 into Equation (10b).
These equations can be derived from the continuity equation [4, p.97]. In practice, it is difficult to apply these equations to choked conditions, because the local conditions, P0 and T0, are not known at the point of choking. For instance, to apply the equations to endpoint choking, one must calculate the stagnation pressure and temperature at the end of the pipe, upstream of the shock wave — but these two variables depend on the flowrate, which is not yet known.
The only way to solve such a problem accurately is by trial and error: first, assume a flowrate and march down the pipe; if M reaches 1 before the end of the pipe, repeat the procedure with a lower assumed flowrate; repeat until M reaches 1 right at the pipe endpoint. Obviously, this calculation sequence is not practical without a computer.