ILASS Americas, 24 th Annual Conference on Liquid Atomization and Spray Systems, San Antonio, TX, May 2012 Evaluation of Cavitation in a Liquid-Liquid Ejector Carsten Mehring * Central Engineering, Parker Aerospace 16666 Von Karman Avenue Irvine, CA 92606-4917 USA Abstract This paper summarizes the numerical analysis of a liquid-liquid ejector pump with focus on the accuracy of the employed cavitation model and in view of its potential application as a dual- or multi-liquid mixing and injection system. While the use of gas-phase or gas/liquid-phase ejection systems has found wide application in gas/hybrid burners and scrubbers for air pollution control; the use of liquid-phase ejectors for the mixing and preparation of different liquids and additives prior to atomization or other processing steps is not common place. As a metered-delivery and mixing device, the ejector system has to be properly designed in order to omit onset of cavitation within the employed working fluids and across the system operating range. Cavitation would not only impact ejector performance, it could also affect steady-state flow conditions and result in poor mixing of the liquid phases which can affect the quality of the subsequent processing steps. In order to utilize CFD analysis as part of the design process for the prescribed system, it is important that the employed cavitation model accurately predicts onset and extend of cavitation within the system. As a first step, the present work focuses on the numerical analysis of a liquid fuel ejector pump over a range of operating conditions and including onset of cavitation. The analysis is carried out by employing ANSYS/CFX v13.0 with its implementation of the Rayleigh-Plesset cavitation model. Comparison with empirical data shows that, the numerical analysis accurately tracks ejector performance and the employed cavitation model accurately predicts the onset of cavitation within the ejector. The relevance of dissolved gases and viscous stresses on the cavitating ejector flow is discussed. * Corresponding author: Carsten.Mehring@Parker.com
ILASS Americas, 24 th Annual Conference on Liquid Atomization and Spray Systems, San Antonio, TX, May 2012 Introduction Ejector or jet pumps are commonly used to pump fuel within an aircraft fuel system. One application includes scavenging fuel from remote corners or the bottom of fuel tanks and discharge that fuel at the inlet to the main fuel feed pump(s). See Figs. 1 and 2. In an ejector or jet pump, a driving fluid is expanded through a nozzle (motive flow), converting its pressure energy into kinetic energy thereby reducing its static pressure according to Bernoulli s Principle. The low pressure high-speed fluid zone downstream of the motive nozzle draws in and entrains the surrounding suction fluid fed from a separate inlet. Motive and suction fluids mix as the static pressure of the mixture increases further downstream when kinetic head is transformed back into pressure head. The suction fluid can be of the same type as the motive flow or it can be a different fluid. The fact that an ejector pump does not have any moving parts allows the pumping of suction fluids that cannot be delivered by other pumps, e.g., due to impurities such as particle loading, for example. Figure 1. Schematic of aircraft fuel system including ejector pump for fuel scavenging [1]. In the fuel scavenging application described above, the fuel scavenged from the bottom of fuel tanks is often contaminated with considerable amounts of water (originally dissolved within the fuel or entering the fuel tank via condensation through vent lines); consequently, an important function of ejector pumps in this application is to disperse any water present within the scavenged liquid volume (i.e., suction fluid) into fine drops, so that the fuel with its finely dispersed water droplets can be safely consumed by the engine without any performance impact. Since traditional ejector pumps do not have any moving parts, they are highly reliable if used within their operating range, latter being limited by the onset of cavitation. Here, two types of cavitation are to be distinguished; i.e., gaseous cavitation and vapor cavitation both of which are very different in nature. The formation and collapse of gaseous cavitation bubbles is a gas diffusion process taking place across the bubble interface. This is in contrast to vapor cavitation, where bubble formation and collapse is the result of a phase change of the liquid from liquid-phase to gas-phase, i.e., a very rapid process taking place in microseconds. This rapid phase-change causes the release of considerable amounts of energy when vapor bubbles collapse, which can result in cavitation damage if the bubble collapse occurs close to walls. Gaseous cavitation, on the other hand, will not cause any material damage; however, just like vapor cavitation, it can be the performance limiting factor in ejector pumps. The present paper investigates a fuel ejector pump whose performance limit is determined by the onset of vapor cavitation. Accordingly, comparison of the subsequently presented CFD results with actual test or empirical data provides a means to evaluate the performance of the cavitation sub-model employed within the analysis. Problem Set-Up and Operating Conditions Figure 3 shows a cut-away of the fluid inverse generated from the ejector pump geometry considered for this study. The suction flow is fed from a side inlet into a plenum chamber where the suction fluid interacts with the motive flow which enters the cylindrical plenum axially. Geometric dimensions of motive nozzle, plenum, mixing section and diffuser section are fixed. Various operating points of the ejector pump are analyzed governed by motive and suction flow static inlet pressures and mixture flow static pressure at the diffuser outlet. Figure 2. Two typical ejector pumps out of the Parker product line.
The analysis was carried out by employing ANSYS CFX v13.0 using constant fluid properties and the SST (Shear Stress Transport) turbulence model of Menter. Cavitation was modeled by employing the homogeneous multiphase model together with the Rayleigh- Plesset Cavitation Model assuming a nucleation site diameter of 2μm. No changes have been made to the other cavitation model parameters, i.e., cavitation condensation/vaporization coefficients = 0.01 and 50, respectively; and nuclei volume fraction of 0.0005. Planar symmetry conditions were employed, i.e., only one half of the ejector was analyzed (see Fig. 3). Total mesh count was 3.1 million and care was taken in order to properly resolve wall shear layers and free shear layers of the discharging motive jet (see Figure 4). Figure 3. Fluid inverse of ejector pump including inlet and exit flow boundaries. Table 1 summarizes the analyzed operating points, identified by flow ratio Φ = W i / W m with W i and W m representing the induced/suction flow and motive flow, respectively; and the non-dimensional total pressure ratio formed by the differences of total downstream and suction pressures (P t,d - P t,s ), and total motive and downstream pressures (P t,m - P t,d ), respectively. Case Flow Ratio Pressure Ratio Φ= W i /W m (P t,d P t,s )/(P t,m P t,d ) 1 8.4 0.0094 2 11.6 0.0086 3 13.6 0.0079 4 13.7 0.0075 Table 1. Analyzed ejector pump operating conditions.. Each analysis was carried out by specifying motive, suction and downstream static pressures at the respective boundaries. Motive and suction static pressures were held constant. The outlet static pressure was adjusted producing the total pressure ratios and flow rates summarized in Table 1. Reference pressure was 2.5924 psia corresponding to an altitude of 41,000 feet. Pumping fluid was Jet A with a vapor pressure of 0.294 psia corresponding to a fuel temperature of at 140 F. Figure 4: Hexahedral mesh (3.1 million nodes). Top: mesh in symmetry plane. Bottom: Mesh near motive nozzle in symmetry plane. Results Figures 5 and 6 illustrate the flowfield within the prescribed fuel ejector pump under design operating conditions at flow ratio Φ = 8.4 and pressure ratio (P t,d - P t,s )/(P t,m - P t,d ) = 0.0094 (Case 1 in Table 1). As the suction flow is entrained by the discharging motive jet, its static pressure drops while being accelerated through the throat of the pump (i.e., the interface between plenum and mixing section in Fig. 3). Note that, suction flow directly entrained from the suction side inlet and not entering the inner plenum volume accelerates around the tight bend connecting suction and mixing
tubes resulting in lower static pressure in the throat region adjacent to the suction inlet tube. As suction and motive flow mix downstream of the throat and as motive flow kinetic energy is converted back into pressure head, the overall static pressure of the mixture increases in the mixing and diffuser section. As seen in Fig. 5, pressure recovery occurs early in the mixing section and is only enhance further downstream in the short diffusing section. The prescribed flow characteristics do not change as the ratio of induced flow to suction flow Φ = W i / W m increases with decreasing diffuser exit static pressure. As the exit static pressure decreases, the depressed suction pressure in the throat also decreases until it drops below the fuel vapor pressure leading to the onset of vapor cavitation. CFD analysis indicates that the operating condition at which this first occurs in the present ejector pump is at flow ratio Φ = 13.6 and pressure ratio (P t,d - P t,s )/(P t,m - P t,d ) = 0.0079, i.e., Case 3 in Table 1. Figure 5: Normalized static pressure for Case 1 (see Table 1) in symmetry plane (top) and in various axial cross-sections (bottom). Scale does not apply to motive nozzle; normalized motive pressure P s,m /P s,d = 55.58). Figure 6: Top: Suction flow streamlines for Case 1 (see Table 1) colored by normalized velocity magnitude (range 0-6.59). Bottom: Velocity contour lines in ejector symmetry plane (range 0 24.31). Normalization w.r.t. average exit flow velocity. The suction flow streamline plot shown in Fig. 6 illustrates that, as static pressure recovers after the throat, the adverse pressure gradient causes local flow separation on both sides of the entrance to the mixing section. Figure 7: Suction flow streamlines for Case 3 (Table 1) colored by normalized static pressure. Also shown cavitation bubble (i.e., iso-surface of vapor-phase volume fraction 0.5. Pressure normalization w.r.t. absolute static exit pressure P s,d. Figure 7 illustrates suction streamlines for this case colored by normalized static pressure. The cavitation region is identified by and iso-surface of vapor-phase volume fraction with value 0.5. Cavitation is pronounced on the lower section (suction-inlet side) of the mixing tube, with no cavitation on the opposed side and only limited cavitation on the side walls in the entry section of the mixing tube. In addition, there is a very small toroidal cavitation pocket located at the wall of the motive nozzle in the nozzle exit plane. While the overall flow characteristics have not changed, the flow recirculation regions described for Case 1 (see Fig. 6) have considerably decreased in size and shifted downstream. Note that, the flow recirculation zone on the cavitating side of the mixing tube is limited to the cavitation zone located farthest downstream in the mixing tube. The extend of the cavitation region for Case 3 is also illustrated in Fig. 9. As the downstream static pressure is further reduced (Case 4 in Table 1), the cavitation region significantly expands downstream into the mixing section (see Fig. 8) resulting in no further increase in flow ratio (see Fig. 9). The ejector pump has reached its performance limit
and in the present case, the limiting factor is fuel vapor cavitation. Figure 8: Contour lines of vapor-phase volume fraction in various axial cross sections of the ejector pump for Case 4 of Table 1. Figure 9 identifies ejector pump performance based on pressure ratio (P t,d - P t,s )/(P t,m - P t,d ) delivered at a given flow ratio Φ = W i / W m as predicted by the prescribed CFD analyses (red dots). Also shown is the performance curve for the investigated pump based on empirical data. As indicated by the 2D-plot, the employed CFD analysis accurately tracks pump performance within its operating range. In addition, the employed Rayleigh-Plesset Cavitation Model only slightly overpredicts the onset of vapor cavitation and hence the performance limit for this pump, i.e., (Φ CFD - Φ emp )/Φ emp = 0.07. Note that, at the performance limit of the investigated pump, i.e., where the performance curve ends in the 2D-plot of Fig. 9, test data shows that the pressure ratio drops sharply to zero. The same behavior is observed when comparing the CFD results for Case 3 and Case 4 described previously and illustrated in Figs. 7 through 9. At the performance limit, any decrease in pressure ratio leads to a rapid expansion of the cavitation region from the throat of the pump downstream into the mixing region. While onset of cavitation is found near the pump throat on the suction-inlet side, full blockage of the pump by fuel vapor will take place near the midsection of the mixing region. Additional Considerations The analysis results presented above were produced using ANSYS/CFX v13.0 with cavitation modeled based on the Rayleigh-Plesset equation for bubble dynamics [2,3]. Input parameters for this cavitation model are: Liquid saturation pressure, mean diameter of nucleation sites, nuclei volume fraction, two empirical coefficients (Cavitation Condensation/Vaporization Coeff.), and two computational model parameters (max. density ratio, cavitation rate under-relaxation factor). The prescribed cavitation model and the underling equation describing bubble dynamics do not include the effect of viscous stress on cavitation as described in Refs. [4-6] and as reviewed in Ref. [7] nor the effect of gases dissolved within the liquid phase and diffusing into the bubble [8-10]. Effect of Dissolved Gases Figure 9: Top: Comparison of empirical ejector pump performance data including performance limit and calculated performance for Cases 1-4 summarized in Table 1. Flow Ratio Φ = W i / W m, Pressure Ratio = (P t,d - P t,s )/(P t,m - P t,d ). Also shown, iso-surfaces of vapor-phase volume fraction (= 0.5) for cases with fluid cavitation, i.e., Case 3 (middle) and Case 4 (bottom). The influence of the gas content within a nucleus on cavitation inception has been described in Refs. [2,8]. For larger gas-contents, as a consequence of larger amounts of dissolved gases within the fluid, cavitation inception occurs at larger nuclei sizes and at larger fluid pressures. Also, bubbles or nuclei with higher gas content can become visible at pressures above the vapor pressure without leading to vapor cavitation. Their growth or decay is dominated by gas diffusion through the bubble interface and is governed by the amount of gas which can be kept in solution within the surround-
ing liquid. This phenomenon is referred to as gaseous cavitation [9]. It is worthwhile to note that, the performance of ejector pumps can be limited by both, vapor cavitation (as in the present case) or by gaseous cavitation. However, cavitation wear or damage of material walls as a consequence of bubble collapse near walls is solely a result of vapor cavitation, i.e., the rapid collapse of vapor bubbles and the associated energy release as a consequence of a rapid phase-change process. Gaseous cavitation, i.e., the growth and dissolution of gas bubbles within a liquid (driven by a slow diffusion process) does not cause material damage. Moreover, the energy released during vapor bubble collapse or vapor cavitation (measured in terms of implosion pressure and temperature) is significantly reduced with increasing gasphase content within the vapor bubble [10]. In context with the influence of dissolved gases on vapor cavitation, an improved cavitation model, has been proposed by Singhal et al. [3]. Figure 10: Contour lines of turbulent eddy viscosity for Case 3 (Table 1) in ejector symmetry plane. Effect of Viscous Stress Vapor cavitation occurs when the liquid pressure drops below the critical pressure or critical stress also identified as the breaking strength of the liquid. In an idealized case this critical pressure is the vapor pressure of the liquid at the given temperature [6]. Within the present analysis the prescribed criterion for cavitation onset has been employed. However, as proposed and discussed by Joseph [4,5], viscous stresses might have to be considered when analyzing cavitation. In consideration of the prescribed work, the critical pressure formulation of Martynov et al. [7] has been reviewed. According to Ref. [7], cavitation occurs when the fluid pressure drops below the critical pressure defined by 2 1 where p v denotes the liquid vapor pressure, μ the dynamic viscosity of the liquid, μ t turbulent/eddy viscosity and S ij max the maximum principal component of the strain-rate tensor. max Within the present CFD analysis, S ij is obtained by a user-defined expression transforming the local strainrate tensor into principal coordinates. For the considered operating conditions, the fuel vapor pressure is p v = 0.294 psia and the dynamic viscosity μ = 8.23 10-4 Pa s. Figures 10 and 11 show contour lines of turbulent eddy viscosity and maximum principal strain rate in the symmetry plane of the ejector under Case 3 operating conditions (see Table 1). Figure 11: Contour lines of maximum principal strain rate for Case 3 (Table 1) in ejector symmetry plane. While maximum principal strain rates are highest in the motive nozzle, they will not impact cavitation due to high motive operating pressures. However, consideration of viscous stress does impact the extend of the cavitation region present under Case 3 operating conditions. Considering an average value of (μ S ij max ) av = 1646 Pa as laminar stress contribution in the vicinity of the cavitation region in Fig. 7, the critical pressure for cavitation to occur increases from the vapor pressure value of 0.294 psia to 0.7716 psia. Figure 12 shows the corresponding pressure iso-surface for Case 3, illustrating a significant increase in the cavitation region under consideration of the laminar viscous stress term in Eqn. (1). Note that, with eddy viscosities averaging 0.2 Pa s in the cavitation region currently predicted for Case 3 (see Figs. 7 and 10), consideration of the turbulent viscous stress term in Eqn. (1) would result in unrealistically high critical pressures and cavitation onset not consistent with the experimentally observed performance limit of the investigated ejector pump.
8. Baur, T., et al., Third International Symp. On Cavitation, Grenoble, France, April 1998. 9. Holl, J.W., J. Basic Eng., Trans. ASME 82: 941-946. 10. Cavitation in Control Valves, Samson Technical Information Report, Samson AG, V74/Training, Frankfurt, Germany. 11. Singhal, A.K., Althavale, M.M., Li, H., Jiang, Y., J. Fluids Engineering 124: 617-624 (2002). Figure 12: Iso-surface of vapor-phase volume fraction 0.5 (blue) and static pressure iso-surface p s = p v +μ (S ij max ) av = 0.7716 psi (orange) for Case 3 (Table 1). Summary A liquid fuel ejector pump has been analyzed at various operating conditions including operation at its performance limit induced by vapor cavitation. The analysis was carried out by using ANSYS/CFX v13.0 and its implementation of the Rayleigh-Plesset cavitation model. Performance predictions for the pump agreed well with empirical data. Onset of cavitation was shown to be accurately predicted by the employed cavitation model and without considering the effect of dissolved gases within the liquid fuel. Consideration of laminar viscous stress was demonstrated to expand the cavitation region predicted with the current model, indicating an improvement in the prediction of cavitation onset and ejector pump performance limit. Consideration of turbulent viscous stress based on eddy-viscosity resulted in excessive critical pressure predictions and cavitation onset not in line with empirical ejector pump data. References 1. Langton, R., Clark C., Hewitt, M., and Richards, L., Aircraft Fuel Systems, Wiley & Sons, 2009. 2. Brennen, C.E., Cavitation and Bubble Dynamics, Oxford Univ. Press, 1995. 3. Leighton, T.G., Derivation of the Rayleigh-Plesset Equation in Terms of Volume, ISVR Technical Report No 308, Institute for Sound and Vibration Research, University of Southampton, 2007. 4. Joseph, D.D., Phys. Review, E., 51 (3): 1649-1650 (1995). 5. Joseph, D.D., J. Fluid Mech., 366:367-378 (1998) 6. Dabiri, S., Sirigniano, W.A., and Joseph, D.D., Phys. Fluids 19: 072112 (2007). 7. Martynov, S.B., Mason, D.J., and Heikal, M.R., Effect of Viscous Stress on Cavitation Flow in Nozzles, Engineering Research Center, School of Engineering, Univ. Brighton, UK, 2006.