Critical, or choked, flow is not only an interesting academic problem but is also important in many practical applications, such as in power generation and in chemical process industries where, without a precise knowledge of critical flow behavior, safety or performance of a system may be compromised. Experimental data may be available for a specific application but reliance often has to be placed upon theoretical models, which, in turn, must be validated against well-qualified data.
Single-phase choking is well understood [see, for example, Roy (1988)] but, with the introduction of a phase change, the behavior of fluid becomes far more complex [for a detailed review of two-phase critical flow models and data, see Elias and Lellouche (1994)]. The occurrence of choking in a system is conventionally defined as the maximum mass flow rate as a function of downstream pressure. To illustrate this, consider Figure 1, where a large reservoir is connected by a flow path of arbitrary geometry to another reservoir whose thermodynamic state can be controlled precisely.
Reducing P_{1} promotes the flow of fluid from A to B at a rate which increases with pressure drop until the velocity at some point in the connector achieves the local sonic velocity.
A choked plane forms at this location, and further reductions in downstream pressure have no effect on conditions upstream as the rarefaction waves travel at the local sound speed and are stalled at the choked plane. However, further reductions in P_{1} will increase pressure drop across the choked plane—where the pressure gradient is now mathematically indeterminate—although physically the pressure drop occurs over a finite distance, resulting in a large pressure gradient. The ratio of the critical pressure P_{c} at the choked plane to the inlet pressure P_{0} is known as the critical pressure ratio (P_{c}/ P_{0}). The geometry of the flow path has a direct bearing on the flow. For a converging/diverging nozzle, the choked plane forms at the minimum flow area. For example, in an isentropic flow in a De Laval nozzle, the critical pressure ratio is given by:
For air, with γ = 1.4, the critical pressure ratio is 0.528 and for steam (γ = 1.3), 0.546. The maximum flow rate is given by:
where A_{c} is the area of the choked plane, which in our example is the nozzle throat (the minimum flow area), and ρ_{0} is the fluid density, corresponding to P_{0}, T_{0}.
For steady, 1-D horizontal flow, the Euler equation reduces to:
and introducing the sonic velocity c^{2} = dP/dp gives,
where M is the Mach number, defined by the ratio of flow velocity to local sonic velocity. Invoking continuity of mass then gives:
Thus, for subsonic velocities, a decrease in flow area results in an increase in flow velocity, whereas for supersonic velocities, the converse is true. De Laval nozzles are often used in steam turbines, where a condensation shock may form downstream of the throat if the static pressure falls sufficiently for thermodynamic conditions to cross the Wilson line (a line approximately parallel to the saturation line and 115 kJ/kg below it) on the steam/water h-s diagram.
Therefore, repeating the experiment with the two reservoirs now connected by a De Laval nozzle will give the characteristic family of axial pressure profiles as in Figure 2.
Generalizing to two-phase critical flow—where it is now assumed that reservoir A contains liquid at or near saturation conditions—it can be seen that for a sufficiently long flow path, the static pressure of the fluid accelerating through the connector will eventually fall to a level where flashing to vapor begins. Bubble nucleation and growth rely on heat transfer at the vapor/liquid interface, which introduces a time delay (typically ~ 1ms) in the development of voids, similar in many respects to the case of a condensation shock described above. The degree of sustainable superheat prior to void formation depends upon the availability of nucleation sites on the walls and in the bulk of the fluid. Shin and Jones (1993) have considered this problem and have introduced a critical flow model which included the effect of wall nucleation and bubble growth.
Once flashing to vapor occurs, the presence of vapor bubbles reduces the average density of the fluid, and hence the mass flow rate. It also has an impact on local sound speed, which exhibits a dependence upon both pressure and frequency as well as the character of the pressure disturbance itself, i.e., a pressure pulse or a continuous wave [see Chen et al. (1983) for a detailed discussion].
A theoretical criterion for determining critical flow in two-phase systems can be derived from the method of characteristics and the previously-mentioned mathematical indeterminacy at the choked plane. For a detailed discussion, see Giot (1981). Various models for critical flow have been developed over many decades; early attempts include the Homogenous Equilibrium Model (HEM) through to more sophisticated models, including a number of full two-fluid six equation models. However, none of the currently-available critical flow models, to this author's knowledge, have been able to account for the full measure of critical flow parameters.
In the absence of a sufficiently accurate critical flow model, trends in the data may be identified to allow extrapolation from available experiments. Holmes and Allen (1995) have identified a number of data trends in two-phase critical flow for the purposes of Pressurized Water Reactor safety studies. Amongst the most important are that increasing inlet stagnation pressure leads to generally increasing choked flow rates, although evidence existed that this increase is not always monotonic and—for a given inlet stagnation pressure—the variation in fluid density and sound speed with quality combines to produce a monotonically decreasing mass flow rate with increasing quality. Also, sharp-edged inlet geometries generate vena contracta, reducing the area of the choked plane and hence mass flow rates, whereas well-rounded or gradually converging inlets maximize flow rates. The effects of inlet geometry are more marked for saturated inlet conditions and shorter flow paths. Exit geometry may also affect flow rates for short flow paths, where delayed flashing can shift the choked plane downstream of the minimum flow area in a diverging nozzle, resulting in a choked plane with a larger surface area. For short flow paths, where the effects of delayed flashing dominate mass flow, the correlating parameter is flow path length, whereas longer flow paths allow equilibration, and wall friction dominates so that the L/D ratio is the appropriate correlating parameter. Choking in complex flow paths, such as safety valves, may differ from design values owing to flow separations and the presence of noncondensable gases and particulates which reduce flow rates by promoting earlier flashing to vapor than would be experienced in a pure liquid.
REFERENCES
Chen, L-Y., Drew, D. A., and Lahey, Jr. R. T. (1983) An Analysis of Wave Dispersion, Sonic Velocity and Critical Flow in Two-Phase Mixtures. NUREG/CR-3372, July.
Elias, E. and Lellouche, G. S. (1994) Two-Phase Critical Flow. Int. J. of Multiphase Flow, Annual Reviews in Multiphase Flow 1994, edited by G. Hetsroni. DOI: 10.1016/0301-9322(94)90071-X
Giot, M. (1981) Critical Flow. Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. Hemisphere Publishing Corporation.
Holmes, B. J. and Allen, E. J. (1995) A Review of Critical Flow Data for Pressurized Water Reactor Safety Studies. Critical Reviews of Multi-Phase Science and Technology, to be published.
Roy, D. N. (1988) Applied Fluid Mechanics. Ellis Horwood Ltd.
Shin, T. S. and Jones, O. C. (1993) Nucleation and Flashing in Nozzles-1 A Distributed Nucleation Model. Int. J. Multiphase Flow, 19, 6, 943-964. DOI: 10.1016/0301-9322(93)90071-2
References
- Chen, L-Y., Drew, D. A., and Lahey, Jr. R. T. (1983) An Analysis of Wave Dispersion, Sonic Velocity and Critical Flow in Two-Phase Mixtures. NUREG/CR-3372, July.
- Elias, E. and Lellouche, G. S. (1994) Two-Phase Critical Flow. Int. J. of Multiphase Flow, Annual Reviews in Multiphase Flow 1994, edited by G. Hetsroni. DOI: 10.1016/0301-9322(94)90071-X
- Giot, M. (1981) Critical Flow. Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. Hemisphere Publishing Corporation.
- Holmes, B. J. and Allen, E. J. (1995) A Review of Critical Flow Data for Pressurized Water Reactor Safety Studies. Critical Reviews of Multi-Phase Science and Technology, to be published.
- Roy, D. N. (1988) Applied Fluid Mechanics. Ellis Horwood Ltd.
- Shin, T. S. and Jones, O. C. (1993) Nucleation and Flashing in Nozzles-1 A Distributed Nucleation Model. Int. J. Multiphase Flow, 19, 6, 943-964. DOI: 10.1016/0301-9322(93)90071-2