Two-phase Flow in Jet Pumps for Different Liquids

Abstract

This research was performed to determine the performance of a jet pump when the viscosity and density of the primary (or power) liquid are different from the viscosity and density of the secondary (or pumped) liquid. Engineers face this problem in several engineering applications (12). Using one-dimensional assumptions, a theory has been developed (34) and validated (56) for jet pumps for water–water flow. The one-dimensional theory has been used (789) for two-phase flow in jet pumps. However, little research has investigated the use of two different liquids (10). The second objective of this research is to study the mixing process analytically and experimentally. The one-dimensional theory, was altered to account for difference of the viscosities and densities of the two liquids. Experimental verification of the theoretical results was carried out by tests to measure the performance of the jet pump and the wall static pressure variation along the mixing chamber.The one-dimensional flow assumptions are 1Streams are one-dimensional at the entrance and exit of the mixing chamber.2Mixing is completed in the mixing chamber area.3Spacing between the nozzle exit and inlet of mixing chamber is zero, which is attainable due to the small nozzle diameter, relative to the large diameter of the mixing chamber.Momentum and continuity equations (3) were applied to different sections of the jet pump, namely the suction, inlet, and outlet of the mixing chamber and the outlet of the diffuser. From this, the relationships between M, N, and η and the performance of the jet pump were determined.N=numerator∕denominator, where1numerator=A+B−C2denominator=D−A−EWhereA=[1−Kdiff−(At∕Adiff)2]B=2[(1+M)−1{R−1(1+ΓM)−1−1}+M(1+M)−1{ΓM(1−R)−1(1+ΓM)−1−1}]−KtC=ΓM2[(1+M)−1(1−R)−2(1+ΓM)−1Cds−2]D=(1+ΓM)(1+M)−1[R−2{Cdn(1+M)}−2]E=2[(1+M)−1{R−1(1+ΓM)−1−1}+M(1+M)−1{ΓM(1−R)−1(1+ΓM)−1−1}]−KtThe coefficient of loss in the mixing chamber, Kt, is set as a function of Reynolds number in the form3Kt=ζ∕Reβwhere Re=(ρmVtDt)∕μm, Vt=(Qp+Qs)∕At, ρm=(ρpQp+ρsQs)∕Qd, μm=αμo+(1−α)μw, and (ζ and β) are constants, derived from the experimental data.Calculation of M versus N and η were carried out for the cases of water–water flow and water–oil flow, assuming Cdn=0.96, Cds=0.9, and Kt=377.5∕Re0.63 to estimate the impact of the velocity and density on performance.Figure 1 shows the changes in M, N, and η, under differing density ratios, γ holding viscosity ratio, Rv constant. Figure 2 shows that, at the highest efficiency, both M and η are increased by increasing γ, but the pressure ratio, N, does not change appreciably. Further, at constant (γ) and differing Rv, Figs. 34 show that, at the highest efficiency points, both M and η decrease appreciably as Rv increases, while N does not change much.A diagrammatic sketch of the test loop is shown in Fig. 5. It consists of a transparent Perspex jet pump and a water pump driven by a 1.5 horsepower electric motor. The pump has a duty of 1.33×10−3m3∕s at 33.5m head of water and is connected to a lower sheet-metal water tank. A 235l sheet-metal suction tank for the secondary liquid is elevated 1m above ground level, and a 290l “accumulating and separator” liquid reservoir is used. Two orifice meters were used, one to measure the flow rate of the power fluid and the other to measure the flow rate of the secondary fluid from the suction tank. Mercury and water differential manometers are used to measure the differential heads across the orifice meters. The pump dimensions were calculated according to the best ratios developed by previous research (3) for R=0.34 in a similar system. Calibrated pressure gauges measured pressure heads in the suction chamber, Pn, and at the outlet point of the jet pump, Pd. Static pressures across the mixing chamber and diffuser section were measured by a pressure gauge at four points, equally spaced along the mixing chamber and at the midpoint of the diffuser. The power liquid was always water, while three types of liquids were used as suction fluid: (1) water; (2) light crude oil (Sp.Gr.=0.85 and ν=29.4 centistokes at 25°C); and (3) very viscous crude oil (Sp.Gr.=0.98 and ν=357 centistokes at 25°C). The jet pump efficiency curves are plotted in Fig. 6, for the water/water flow, water/light oil flow, and water/heavy oil flow. The theoretical results plotted against experimental results in Fig. 7, for the cases of water/water flow and water/light oil flow. Uncertainty analysis was carried out using the method developed by Kline and McClintock (11) and gave the results shown in Table 1.A computer program was used to solve the Navier–Stokes equations for the axially symmetric two-phase turbulent flow (12) based on the finite-difference method, together with a two-equation turbulence model. The flow is bubbly, so the coefficient of drag (F) is taken as4F=−CD(ρwd′∕μ)∣v−u∣where CD is the drag coefficient, calculated by an empirical correlation (13), ρw is the density of the continuous phase, and d′ is the droplet diameter.Turbulent correlations appearing in the equations of the mean motion of both phases and the kinetic energy of turbulence and its dissipation rate for the carrier fluid are modeled through turbulent exchange coefficients and mean gradients.The interaction between the phases was accounted for by including interfacial drag in the corresponding momentum equations. The parameters to be determined were the pressure, velocity component of each phase, the void fraction of each phase, and the turbulence energy and its dissipation.The solution domain is subdivided into a number of finite cells, each of which encloses an imaginary grid node, at which all of the dependent variables are calculated with the exception of the velocities, which are caused to lie on the cell boundaries where they are used for mass flux computations. Algebraic finite difference counterparts of the differential equations were derived for each cell by a combination of formal integration and approximation.Static pressure was measured along the mixing chamber and diffuser, and in the outlet of the diffuser. The measured pressure distributions are plotted in Fig. 8 for the three secondary liquids. The pressure distribution is negative in the inlet of the mixing chamber due to the acceleration of velocity in the mixing zone, and begins to rise near the middle of the mixing chamber due to the decrease of total momentum. The mixing process is begun in the mixing chamber and is completed in the diffuser. Kt is experimentally derived by applying following equations: 1The actual pressure rise in the mixing chamber:5(ΔPms)=(Pd−Ps)−0.5ρmηdiff(Vt−Vdiff)2The theoretical pressure rise in the mixing chamber (ΔPth):6ΔPth=[(1∕At){mpVn+msVs−Vt(mp+ms)}]and the coefficient of pressure losses:7Kt=(ΔPth−ΔPms)∕(0.5ρmVt2)3Reynolds number (Re) plotted versus Kt in Fig. 9.The validity of the theoretical analysis is proved by comparing the experimental performance curves with the theoretical curves (Fig. 7). At a constant viscosity ratio and variable density ratio, it is observed that there is no appreciable change for N with increasing density. At best efficiency, both M and η increase gradually with increasing density ratio. This is probably related to the fact that, as the density ratio increases, Reynolds number increases. However, the coefficient of pressure loss in the mixing chamber (Kt) decreases slightly at high Reynolds number and consequently has only a small influence on the pressure ratio N. At constant density and varying viscosity ratio, N decreases slightly with an increase in the viscosity ratio. At best efficiency, both M and η decrease sharply with an increasing viscosity ratio. The reason for this is probably the same as for the effect of the density ratio, but the change is limited, especially at higher Reynolds numbers. Consequently, N changes only slightly.A one-dimensional theoretical equation for the performance of the jet pump is developed when the viscosity and density of the primary liquid are different from those of the secondary flow. Comparison between the results of the theoretical equations and experimental results shows close agreement.The empirical formula: Kt=379∕Re0.63, is obtained experimentally by relating the pressure loss coefficient in the mixing chamber (Kt), to the Reynolds number calculated from the density, viscosity, and velocity of the mixture inside the mixing chamber (Kt−Re). To pump an extremely viscous crude oil, a longer mixing chamber is needed to get a good mixing between this oil and the powered water flow.The authors wish to express their gratitude to all who provided help and guidance in various ways at the different stages of this research. Our deepest appreciation is conveyed to cosupervisor, Professor Mohsen Abou Elleial, for his guidance in overcoming problems occurring during this research. Thanks also to Dr. Mostafa El-Sallak, for his valuable help in supplying some references.SymbolMeaningUnitAaream2Cddischarge coefficient—DdiametermmKpressure loss coefficient—mmass flow rateKgm/sMmass ratio=ms∕mp=(ρsQs)∕(ρpQp)—Npressure ratio=(Pd-Ps)∕(Pn-Pd)—PpressurePaQflow rateL∕s=10−3m3∕sRarea ratio=An∕At—ReReynolds number=ρmVtVt∕μm—Rvviscosity ratio=μs∕μp—Sp.Gr.specific gravityTdimensionless pressure=[2(P−Ps)∕ρmVt2]—uvelocity of the dispersedphase (light oil or heavy oil)m/svvelocity of the carrier phase (water)m/sVvelocitym/sW-HOwater–heavy oil—W-LOwater–light oil—W-Wwater–water—Xdistance from inlet to mixing chambermmρdensity of fluidKgm∕m3γdensity ratio=ρs∕ρp—Γ=ρp∕ρs—ηefficiency=100[Qs(Pd-Ps)∕Qp(Pn-Pd)]=100M×N%αvoid fraction=Qs∕(Qs+Qp)—μdynamic viscosity of the fluidpa sνviscosity of the fluidcentistokesSubscriptsddeliveryDdragmmixturemsmeasurednnozzle or normalooilssuctiontmixing chamberdiffdiffuserththeoretical