Numerical analysis of an annular water–air jet pump with self-induced oscillation mixing chamber

Geometrical and operational parameters

A schematic diagram illustrating both conventional and improved AJP designs is shown in Figure 1, where water is used as the motive fluid and connected with the annular nozzle on the periphery of the inner suction tube, while the air is used as the entrained fluid and connected with the inner nozzle. The conventional AJP geometrical parameters were referred from Park and Yang. ¹⁶ As mentioned earlier, the main difference between the conventional AJP and the improved AJP design is the self-induced oscillation chamber, which is an enlarged cylindrical cavity with a cone-shaped impinge downstream sidewall, and its geometrical configuration was referred from Liao et al. ¹⁷ All basic pump parameters are listed in Table 1. OPEN IN VIEWER

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Table 1.

Geometric dimensions of the studied AJPs.

Dimension	D i	D s	D t	D c	D d	D o1	D o2	Lt	α	β	θ
Value (mm)	50	26	26	140	36	45	35	290	59°	4°	120°

AJP: annular jet pump.

CFD modelling strategy

In the present study, the commercial CFD code ANSYS–Fluent was employed as the research platform. Both conventional and improved AJPs were assumed into 2D axisymmetric domains. ¹ Both conventional and improved AJP CAD models were meshed into quadrilateral structural elements, and mesh refinements were applied at the vicinity of gas–liquid interface.

For multiphase flow modelling, there are two commonly used approaches: the Euler–Lagrangian approach or the Euler–Euler approach. Kim et al. ¹² suggested to choose the Eulerian multiphase model to simulate the water–air multiphase flow in AJPs as dispersed-phase volume fractions of bubbly flow normally exceeds 10%. Therefore, the present work adopts the Euler–Euler method, where the different phases are treated as interpenetrating continua and the conservation equations for all phases are similar. The governing equations are as follows:

Continuity equation for phase i is

∂ ∂ t ( α i ρ i ) + ∇ · ( α i ρ i v i ) = 0

Momentum equation for the air phase yields

∂ ∂ t ( α g ρ g v g ) + ∇ · ( α g ρ g v g v g ) = - α g ∇ p + ∇ · τ ¯ g + α g ρ g g → + R → gl + ( F g + F → lift , g + F → td , g )

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The flow inside the pump was assumed to be steady and incompressible flow, controlled by the Reynolds averaging Navier–Stokes equations and the continuity equation. In this study, three widely-used k–ɛ turbulence model variations (the standard, realizable and RNG model) were selected to resolve the turbulence characteristics with the standard wall function adopted to resolve the near-wall region. Velocity boundary conditions were applied on the inlets boundaries and pressure boundary condition on the outlet boundary. The governing equations were discredited by the finite volume method and the secondary upwind scheme was adopted for spatial discretization of the convection terms. The SIMPLE algorithm was employed to solve the coupling the pressure and velocity.

Prior to performing the simulations, a mesh independence study was conducted to eliminate the influence of the numerical element size. The grid number was initially made at about 72,000 and later increased to about 421,000 to confirm that the results were grid independent. The air entrainment value was selected as grid evaluation target for all mesh configurations, and detailed numerical results are listed in Table 2. As the air entrainment predictions between the medium and fine meshes were very close, the medium mesh scale was selected for further numerical simulations.

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Table 2.

Mesh independence study.

	Elements number		Air entrainment (L/s)
AJP	Low	72,500	0.99
	Medium	234,000	1.06
	High	421,000	1.07
Improved AJP	Low	74,000	1.04
	Medium	275,100	1.1
	High	433,000	1.105

Validation of CFD simulation

To validate the accuracy of the proposed numerical modelling approach, the experimental work conducted by Park and Yang ¹⁶ was used as validation target. This referred experimental study investigated the influences of both operational parameters and geometrical parameters on overall pumping performance. In the present study, only the relevant experimental data considering the variation of geometrical parameters was selected for results comparison. In detail, the tip position (L_tn) indicating the distance between the annular nozzle exit plane and the inlet of the parallel mixing tube was adjusted, and 10 positions with an interval of 1.5 mm were studied. All tube tip positions (L_tn) along with its corresponding operating primary fluid conditions (Q_p) are listed in Table 3.

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Table 3.

Tube tip position along with its corresponding operating conditions.

Pitch	1	2	3	4	5	6	7	8	9	10
Ltn (mm)	2.5	4	5.5	7	8.5	10	11.5	13	14.5	16
Atn (mm2)	56	91	126	163	200	238	276	316	357	398
Qp (m3/h)	3.0	5.6	9.4	12.5	14.5	15.5	16.4	16.8	17.0	17.5

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Figure 2.

Numerical boundary conditions and regional mesh refinement.

Figure 3 compares the numerical simulation results with the referred experimental data based on the conventional AJP design by changing the suction tube exit tip position (L_tn). In general, the numerical predictions agree well with the experimental data, the air entrainment flow rate first increases with the increase of L_tn and peaks at 35 m³/h when L_tn = 5.5 mm. While with the further L_tn increase, the entrained air flow rate decreases rapidly, and reaches the lowest value of 6 m³/h when L_tn = 16 mm. OPEN IN VIEWER

As the internal flow regime is turbulence, the turbulence flow predicting is largely relying on the chosen of the suitable turbulence model. In the present study, three widely used turbulent models were assessed. It was demonstrated that the realizable k–ɛ turbulence model provides the closest entrainment performance prediction compared with the experimental data. However, the numerical simulation slightly under-predicted the pumping performance, especially for the prediction of the peak value when L_tn = 5.5 mm, with a discrepancy of 14%.

Despite the existence of the numerical prediction discrepancy compared with experimental data, the validity of the proposed numerical modelling approach was verified, and all further simulations were solved by the same solver configurations using the realizable k–ɛ turbulence model.

Comparison between the conventional and improved AJPs

Figure 4 compares the pumping performance of the conventional and improved AJPs in terms of the entrained air flow rate on the basis of a fixed tip position (L_tn = 5.5 mm). It was found that both designs exhibit similar air entrainment profile trend, which indicates a positive correlation between the air entrainment flow rate and the motive water flow rate. However, the improved AJP with the self-induced oscillation chamber showed a superior air entrainment performance for all considered motive water flow rates compared with the conventional AJP, and the maximum pumping improvement percentage (approximately 12%) occurs at the water flow rate of 15 m³/h. OPEN IN VIEWER

To provide a better understanding of the internal flow patterns of the studies, AJPs, the streamlines, phase fraction distribution and the turbulence kinetic energy of the gas phase based on the case with maximum performance improvement as mentioned earlier were selected for flow pattern analysis.

Figure 5 compares the streamlines between the conventional and improved AJP. For the conventional AJP, a significant flow recirculation which occupies more than half of the throat was found downstream of the mixing tube. In comparison, after integrating the self-induced oscillation chamber with the mixing tube, this flow recirculation vanished and the flow interaction between the gas phase (air) and liquid phase (water) was enhanced with the formation of a counter-clockwise vortex. OPEN IN VIEWER

Figures 6 and 7 provide a detailed phase distribution and turbulence flow characteristics comparison. For the conventional AJP (Figure 6(a)), a clear air–water phase interface was established, indicating the interaction between the motive water and the entrained air is mainly relying on the viscous fraction along the phase interface region. On the contrary, the improved AJP with self-induce oscillation chamber showed a different phase mixing pattern (Figure 6(b)). Due to the abrupt cross-sectional area expansion, the air–water phase interface was disrupted into discontinuous fragments, and a better phase mixing was reached in the self-induced oscillation chamber. Figure 7 illustrates the momentum exchange between these two phases for both AJPs. For the conventional AJP, the high turbulence kinetic energy for the entrained air occurs along the phase interface. Therefore, the momentum exchange mainly occurs along the limited region of the phase interface, which implies the energy transfer from the motive fluid to the entrained fluid is not sufficient. While for the improved AJP, larger proportion of gas phase (air) with high turbulence kinetic energy was found both in the self-induce oscillation chamber and the following downstream mixing tube. This demonstrates the self-induce oscillation chamber can considerably enhance the phase mixing and energy exchanger, which ultimately improve the AJP pumping performance. OPEN IN VIEWER

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Figure 7.

Turbulence kinetic energy comparison between (a) the conventional and (b) the improved AJPs.

Unsteady flow patterns in the improved AJP

As the energy exchange between water and air phases is highly unsteady with fluctuations, time domain analysis based on a transient flow simulation was performed to better illustrate this dynamic energy exchanging process.

Through monitoring the unsteady air volume flow rate at the outlet of the jet pump, it was found that the air volume flow rate profile shows a periodic pattern over time with its flow rate ranges between 0.01 L/s and 0.015 L/s (Figure 8), while the frequency of the volume flow rate variation remains relatively stable. To reveal this variation frequency, a Laplace transform was carried out based on the current profile, and its domain frequency was found to be 147 Hz (Figure 9). Therefore, a typical period of the energy exchanging process takes approximately 0.0068 s for the improved AJP with the given operation conditions. OPEN IN VIEWER

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Figure 9.

Time domain analysis of the transient air flow rate profile.

Figure 10 presents the phase variation of water and air over a single cycle. At the beginning phase (Figure 10(a)), a raised “air bubble” starts to protrude towards the self-induced oscillation chamber at the sloped downstream chamber edge. After that, as shown in Figure 10(b), the “air bubble” breaks away from the air–water interface, and flows into the self-induced oscillation chamber along with the counter-clockwise vortex (Figure 5(b)). Then the “air bubble” dissipates into the chamber space, where both water and air are almost sufficiently mixed (Figure 10(c)). Finally, the whole period repeats again as demonstrated in Figure 10(d), in which similar air–water interface oscillation pattern with that shown in Figure 10(a) was formed. OPEN IN VIEWER