Effects of interface model on performance of a vortex pump in CFD simulations

A vortex pump (VP) is a rotodynamic pump that sucks fluid into a semi-open impeller from a vaneless side chamber and then discharges the fluid, thereby transmitting the mechanical energy of the impeller to the fluid. Because the fluid moves rotationally in the chamber, any solid matter in the fluid is carried away from the pump without blocking the impeller. This unique blockage-free feature gives VPs extensive applications for waste-water treatment and in the chemical, petro-chemical, food, and metallurgical industries.

Numerous studies have investigated the hydraulic performance and fluid mechanics of VPs experimentally. For example, how the impeller, volute, and suction pipe affected the hydraulic performance of a VP was examined in Refs. 1 and 2 with water as the flowing fluid. In Ref. 3, the hydraulic performance of a VP was measured with air as the flowing fluid; the hydraulic loss was analyzed, and a flow model was proposed that comprised a through flow and a circulating flow in the VP. In Ref. 4, the hydraulic losses in a VP were modelled based on measurements of the fluid field between the impeller and side chamber. That model was verified experimentally in Refs. 5 and 6 and analytically in Ref. 7. In Ref. 8, the hydraulic performance of a VP was measured for various particle sizes and concentrations, and the patterns of particle erosion in the pump were visualized. In Ref. 9, measurements were made of the improved performance of a VP when the impeller was moved into the side chamber. In Ref. 10, the hydraulic and suction performances of a VP were measured when pumping rapeseeds, wheat grains, and soya beans at 6% volumetric concentration. In Refs. 11 and 12 the air–water two-phase flow in a VP was mapped by using probes. Furthermore, dilute salt crystal–water two-phase flow fields in the impeller^13–15 and side chamber¹⁶ of a VP were measured by using particle image velocimetry. In Ref. 17, the hydraulic performance of a VP with different impeller diameter, number of blades, and blade discharge angle was measured under non-woven textile–water two-phase flow conditions.

Numerical simulations have become increasingly vital for predicting the hydraulic performance of VPs and the details of the fluid flow therein. Initially, the 3D potential flow in the impeller of a VP was simulated by using the streamline curvature method¹⁸ and the boundary element method.¹⁹ Later, 3D, steady, incompressible, and turbulent single-phase flows in VPs were simulated using computational fluid dynamics (CFD).^20–34 Also, numerical simulations have been performed of 3D steady turbulent solid–water^35–38 fiber–water³⁹ and air–water^40,41 two-phase flows. Furthermore, unsteady non-cavitating and cavitating water flows in a VP were simulated using Fluent in Ref. 42 and ANSYS CFX in Ref. 43. In Ref. 44, the vortex pattern in the side chamber of a VP was predicted using ANSYS CFX.

The present author has also conducted numerical simulations of the hydraulic performance of a VP and the flow therein^45,46 and of a VP as a turbine.^47–49 A critical issue that arose in those studies was that the predicted head of the VP was higher than the experimental value measured for water flow. Clearly, it is disappointing that the predicted head disagrees with the experimental measurement, and it suggests that the modelling procedure involved some improper treatment. Therefore, reported herein is a systematic investigation conducted using the CFD software ANSYS CFX 2019 R2 to resolve the issue, which it turns out is due to the choice of impeller geometrical model and interface model.

The hydraulic performance of the motor-connected VP (model 32WB8-12) studied herein has been measured under water single-phase flow conditions²⁴ and solid–water¹⁰ and air–water¹² two-phase flow conditions. This VP was used in CFD simulations^45–49 to investigate how the liquid viscosity affects its hydraulic performance and fluid flow. The pump is shown schematically in cross section in Fig. 1(a), and its specifications and main geometrical dimensions are given in Table I.

As in the CFD simulations reported in Refs. 20–25 the axial symmetry of the suction pipe and impeller is ignored, and a complete fluid domain is generated as shown in Fig. 1(b). This comprises three individual fluid domains, i.e., the suction pipe, the impeller, and the volute, leading to a fluid–fluid interface at the boundary between the suction pipe and the impeller and another one at the boundary between the impeller and the volute. When these interfaces are crossed, the reference frame for the governing equations changes.

The finite volume method is used to discretize Eqs. (1)–(4). Pressure–velocity coupling is accomplished using the SIMPLE algorithm; the pressure and velocity are assigned in a staggered mesh, which is the PRESTO scheme in Fluent 6.3. The second-order upwind scheme is selected for the advective terms in Eqs. (2)–(4), but the central difference scheme is selected for the dissipation terms in those equations.

At the entrance of the suction pipe, the velocity-inlet boundary condition is imposed with an axial velocity, 5% turbulence intensity, and 32-mm hydraulic diameter. The imposed axial velocity is calculated from the specified flow rate and the cross-sectional area of the pipe. At the volute nozzle outlet, a relative static pressure is specified along with 5% turbulence intensity and 23-mm hydraulic diameter. For the walls along which the flow passes, the no-slip boundary condition is imposed with zero roughness.

The under-relaxation coefficients for the pressure–velocity coupling equation, the momentum equation, and the equations for the turbulent kinetic energy and its dissipation are 0.3, 0.5, 0.8, and 0.8, respectively. The convergence criterion is 1 × 10⁻⁵ for the residuals of those equations.

The working liquids are water and five different machine oils, and their densities and kinematic viscosities are given in Table II along with the corresponding impeller Reynolds number. Oils 1–5 are 10#, 22#, 32#, 68#, and 100# mineral oils, respectively; the number before the “#.” is the kinematic viscosity at 40 °C in centistokes (1 × 10⁻⁶ m²/s or 1 mm²/s) based on ISO 3448. These viscous oils are used only to study how viscosity affects the hydraulic performance of the VP.

The meshing software Gambit is used to generate three meshes, referred to as meshes 1–3. The mesh in the suction-pipe fluid domain is hexahedral, whereas those in the impeller and volute fluid domains are hybrid (tetrahedral near walls and interfaces but cubic in the core flow). The mesh information is summarized in Table III along with the mean dimensionless wall distance of mesh 2 at the best-efficiency point (BEP) (Q = 8 m³/h) and ν = 1 cSt. Fig. 2 shows the predicted hydraulic performance curves of the VP with the three meshes at ν = 1 cSt in the MRF system in comparison with the experimental data from Ref. 24.

In Fig. 2, the efficiency shows some agreement between prediction and experiment, but the predicted head and shaft power are higher than the experimental data. Specifically, the errors in the head and shaft power are 6.92%–12.4% and 8.39%–14.8%, respectively, depending on the operational conditions, while the error in the efficiency is only between −1.01% and 1.57%, see Table IV.

This phenomenon was observed in Refs. 45–49 and why the predicted head is higher than the experimental data is clarified in Sec. III.

From Table III, the mean y⁺ values are close to or higher than 100 at the BEP, especially for the suction-pipe wall. Because Gambit cannot create an inflation boundary-layer mesh for a hybrid mesh, the complete fluid domain is read into ANSYS CFX and re-meshed using the ANSYS meshing module to generate an unstructured mesh with an inflation boundary-layer mesh near the wall to reduce the mean y⁺ values. Figure 2 shows the hydraulic performance curves of the pump predicted using the frozen-rotor interface model (identical to MRF in Fluent) with two meshes, i.e., mesh A (3 430 427 cells) and mesh B (5 283 082 cells); see Table V for full details about these meshes. For mesh A, the mean y⁺ values are 13.46–27.73 on the blades, 53.35–65.81 on the volute wall, 55.35–52.38 on the chamber wall, 24.37–35.79 on the impeller inside and outside surfaces, and 7.19–29.08 on the suction-pipe wall. For mesh B, even smaller mean y⁺ values are observed. These values are much smaller than the counterparts of the meshes in Fluent 6.3 listed in Table III. The smaller mean y⁺ values do reduce the head to make it closer to the experimental data, but they are not effective at the over-load point. The efficiency predicted with the smaller mean y⁺ values is significantly lower than that with the larger mean y⁺ values. These facts imply that the y⁺ value is not fully responsible for the predicted head being higher than the experimental data.

In ANSYS CFX, the computational domains can comprise fluid, solid, and porous domains depending on the problem to be solved. There is a domain interface between any two domains contacting each other, and there are six interface types, i.e., fluid–fluid, fluid–porous, fluid–solid, porous–porous, solid–porous, and solid–solid. However, for isothermal fluid flows in hydro-turbomachinery, the domain interface type is fluid–fluid.⁵³ The interface model specifies how the solver models physical processes across the interface. ANSYS CFX has two types of interface model, i.e., periodical connection and general grid connection. Periodical connection involves translational periodicity and rotational periodicity, while general grid connection involves grid connection and reference-frame change among the frozen rotor model (FRM), stage model (SM), and transient rotor model (TRM).⁵³ A detailed interface-model tree is shown in Fig. 4.

Periodical connection is a grid connection between a pair of symmetric or axisymmetric edges/planes/surfaces in 2D or 3Dspace. When fluid crosses a pair of symmetric or axisymmetric geometrical elements,the fluid velocity or heat flux remains unchanged. General grid connection includes grid connection either with or without reference-frame change; in the formercase, a stationary reference frame changes into a rotational reference frameor vice vs, or from one rotational reference frame to another with different rotationalangular speeds.

In the FRM, even though the reference frame changes across the interface, the relative orientation of the two fluid domains remains fixed across the interface. If the pitch of the two fluid domains varies across the interface, then the fluid flow parameters are scaled by the pitch change. The FRM is equivalent to the MRF model in Fluent, and it produces a steady solution by taking account of the interaction between rotor and stator to some degree with minimal computational time. However, the mixing loss between rotor and stator is ignored.

In the SM, the relative position between rotor and stator is not fixed, and circumferential averaging is performed to ensure that the fluxes of fluid variables on both sides of an interface are conserved; as a result, steady solutions are produced in each reference frame. This circumferential averaging at the interface induces a mixing loss that is equivalent to real fluid mixing taking place at the interface to some degree. The SM accounts for time-averaged interaction effects between rotor and stator but ignores transient interaction effects. Furthermore, the SM is not suitable for a rotodynamic pump with a volute in off-design conditions where the fluid variables vary markedly in the circumferential direction. In the circumferential averaging, a constraint can be applied to the total pressure or velocity.

In the TRM, the motion of the rotor relative to the stator varies with each timestep, and the transient interaction of the flow between the stator and the rotor is predicted. The TRM uses the flow field predicted by the FRM or SM as the initial flow field and is quite time-consuming in comparison with those two models. The TRM in ANSYS CFX is the same as the sliding mesh model in Fluent.

To clarify how the choice of interface model affects the performance of the VP, periodical connection and the FRM, SM, and TRM are adopted here. Because the impeller has eight repeated flow passages, just one is chosen in the impeller fluid domain, and thus only 1/8 sectors of the casing chamber and suction pipe are included in the fluid domain as shown in Fig. 5. The pitches of the interface between the suction pipe and the impeller are equal, but those of the interface between the volute and the impeller are different with an 8:1 (volute to impeller) pitch ratio. There is a periodical connection between the two periodical surfaces in the impeller and suction-pipe fluid domains.

First, the mesh size independence is checked with the FRM, and details of the four unstructured meshes used for this purpose, i.e., meshes I–IV, in which six-node wedge elements are dominant, are listed in Table VI. Inflation mesh layers are arranged on the walls of the suction pipe, volute, blade, hub, and chamber to handle the fluid flow in the boundary layers over them. Except for meshes I and II, the dimensionless wall distance y⁺ is smaller than 13, and the scalable wall function in Eq. (9) is basically satisfied.

In Fig. 6, the head H_BEP, shaft power P_BEP, and overall efficiency η_BEP at the BEP (Q_BEP = 8 m³/h) are plotted against the number of mesh elements under water flow conditions. As can be seen, η_BEP remains almost constant with the number of mesh elements from mesh III to mesh IV, so mesh size independence is achieved with meshes III and IV.

Second, the performance curves of the VP are predicted using ANSYS CFX based on the fluid domains shown in Fig. 5 and the FRM, and these are compared in Fig. 7 with the experimental data from Ref. 24. The head curves predicted with the four meshes exhibit much better agreement with the experimental data than do those in Fig. 2 predicted with the FRM and the fluid domains shown in Fig. 1. For example, the mean errors in the head in Fig. 7 are 2.79% and 0.53% compared with the errors of 7.99% and 11.09% in the head in Fig. 2. against the measured head data at the part-load point (Q = 4 m³/h) and the BEP (Q_BEP = 8 m³/h), respectively.

The shaft power and efficiency curves predicted with the fluid domains in Fig. 5 and the FRM are slightly poorer in agreement with the experimental data than are those in Fig. 2. The mean errors in the shaft power are 18.17% and 6.66% in Fig. 7 against the measured data in comparison with the errors of 9.35% and 12.30% in Fig. 2 at the part-load point (Q = 4 m³/h) and the BEP (Q_BEP = 8 m³/h), respectively. Regarding the overall efficiency, the errors are −4.25% and −1.39% in Fig. 7 compared with −0.09% and 0.25% in Fig. 2. at the two operative points, respectively. Note that the shaft power in Ref. 24 was calculated from the electrical power consumed by the electrical motor and its efficiency. However, the rated efficiency was obtained from the product catalogue of the motor manufacturer and so was not sufficiently accurate, causing the experimental shaft power as well as the efficiency calculated in Ref. 24 to have large errors when compared with the experimental head and flow rate.

Third, how the choice of interface model affects the hydraulic performance curves of the VP is clarified. The head, shaft power, and efficiency predicted with the fluid domains in Fig. 5 and the FRM, SM, and TRM are shown in Fig. 8 for mesh III and water. Compared with the FRM, the SM leads to a significantly low head and small shaft power as well as poor efficiency against the corresponding experimental data. The TRM results in a slightly lower head and shaft power but a higher efficiency. The head predicted using the TRM agrees well with that estimated using the FRM except at Q = 2 and 4 m³/h. However, the shaft power predicted with the former is considerably lower than that given by the latter and is closer to the experimental data.

Based on the head predicted by using the FRM, SM, and TRM in the fluid domains with periodical connection, the FRM and TRM are more suitable for predicting the hydraulic performance of the VP than is the SM. However, the TRM requires much more computational resources than does the FRM and so is limited in applications. Therefore, unless unsteady turbulent flow is happening in the VP, the FRM is recommended as the best interface model for predicting the pump performance.

Figure 9 shows the hydraulic performance curves of the VP for kinematic viscosity ν = 1, 24, 48, 60, 90, and 120 cSt and the corresponding densities in Table II as predicted using the FRM and the fluid domains with periodical connection shown in Fig. 5. Even though the disc friction power is not a parameter of the hydraulic performance, it can affect the shaft power greatly at high viscosity and so was extracted from the ANSYS CFX result files and is also shown in Fig. 9.

Generally, the head, efficiency, and disc friction power rise with increasing kinematic viscosity but are insensitive to density at a given flow rate. The shaft power depends on kinematic viscosity, density, and flow rate. At zero flow rate, the shaft power depends on kinematic viscosity. With increasing flow rate, the shaft power is increasingly affected by density but is decreasingly influenced by kinematic viscosity; i.e., the denser and thicker the fluid, the greater the shaft power.

As shown in Fig. 10, k_Q is identical at 0.6BEP, BEP, and 1.2BEP and increases slightly with decreasing Re, whereas k_H and k_η decrease with decreasing Re. For given Re, we have k_Q ≥ k_H ≥ k_η, and the operational condition influences k_H more substantially than it influences k_η.

Figure. 11, k_Q, k_H, and k_η of the VP are compared with the correction factors of centrifugal pumps with a closed impeller in Ref. 54 (n_s = 92), Refs. 55 and 56 (n_s = 47), and Ref. 57 (n_s = 107) and with a semi-open impeller in Ref. 58 (n_s = 121, 1-mm blade side-tip gap) at BEP based on the measured performance curves and the correction factors predicted by using the empirical correlations of KSB (a notable pump manufacturer) and the American Hydraulic Institute presented in Ref. 59. As can be seen, the correction factors of the VP differ significantly from the other factors.

Basically, k_Q and k_η of the VP are larger but k_H is smaller than the counterparts of the centrifugal pumps based on the experimental data at given Re. The same is true for k_Q and k_η of the VP in comparison with those estimated by the empirical correlations for centrifugal pumps, but k_H is comparable between the two types of pump. Therefore, the head of the VP is influenced more strongly by viscosity than is that of centrifugal pumps. The efficiency of the VP is reduced less by viscosity than is that of centrifugal pumps, while the flow rate of the VP is least affected by viscosity compared with centrifugal pumps.

In Fig. 12(a), the efficiency of the VP at BEP is compared with those of the centrifugal pumps reported in Refs. 55–61. The impellers of the centrifugal pumps in Refs. 55–57, 60 were closed, but those in Refs. 58 and 61 (n_s = 85) were semi-open. The efficiency of the centrifugal pumps with a closed impeller suffers from substantial incremental loss with increasing viscosity (decreasing Re) compared with the pumps with a semi-open impeller. The efficiency of the VP decreases less rapidly with increasing viscosity than does that of the centrifugal pumps with a semi-open impeller, and this is due to the smaller disc friction loss power in the VP.

The P_d/P curves calculated from Eqs. (14) and (15) at BEP are shown in Fig. 12(b). For the centrifugal pump with a closed impeller, the disc friction loss power ratio is 15% in water flow conditions, increasing to 55% with increasing viscosity. However, for the VP the ratio is less than 5%, and this suggests that the impeller disc friction loss power is a minor factor in reducing the efficiency of the VP.

In the fluid domains shown in Fig. 5(a), the pitch of the interface in the impeller fluid domain is 1/8 of that in the volute fluid domain. For the FRM in ANSYS CFX, the flow-variable profiles on the interface of the impeller fluid domain are scaled up to those on the interface of the volute fluid domain in the pitch-wise (circumferential) direction. The continuity, momentum, and turbulence model equations, among others, are scaled accordingly based on the pitch change.⁵³ This scaling process is akin to averaging or artificial mixing so that the flow variables are distributed more uniformly in the pitch-wise direction. The static pressure and absolute velocity of water in the mid-span plane of the vaneless chamber and the static pressure and relative velocity in the mid-span plane of the impeller at 0.5BEP (Q = 4 m³/h) and BEP (Q = 8 m³/h) are shown in Figs. 13 and 14, respectively. The inner part (the diameter of which is equal to the inner diameter of the suction piper) and outer part (the diameter of which is in between those of the suction pipe and the volute) of the mid-span plane are upstream and downstream of the impeller, respectively.

As seen in Fig. 14, the pressure and flow in the mid-span plane of the impeller are axisymmetric in the fluid domains with periodical connection, but those in the domains with the whole impeller are not axisymmetric as shown in Fig. 13. Obviously, the pressure and velocity are more uniform in the fluid domains with pitch change or periodical connection than in those with the whole impeller. As a result, the fluid flow experiences an artificial mixing process to reduce the head rise, and the predicted head is closer to the experimental measurement.

That the head curve predicted by CFD simulation is above the experimental head curve is common for VPs^{40,42,63–69} but the underlying reasons remain unclear. In the present study, this issue has been clarified in detail by using the CFD software ANSYS CFX 2019 R2, and the literature to date contains no such investigation into this phenomenon. Because the main focus in the present study was how the choice of interface model affects the hydraulic performance of a VP, other issues such as the vortex in the chamber, efficiency decomposition, slip factor, and hydraulic losses in the impeller and volute were not examined, and the interested reader should refer to Ref. 45 for those aspects.

Three interface models were applied to the fluid domains with the whole impeller shown in Fig. 1 to identify the differences in their effects on the hydraulic performance curve from the corresponding effects on the performance curve based on the fluid domains with periodical connection. The predicted hydraulic performance curves based on the whole-impeller model by using the three interface models with mesh A are shown in Fig. 15. As can be seen, the FRM results in a significantly higher head and larger shaft power than the experimental data at over-load points (Q ≥ 8 m³/h). The SM leads to considerably lower head and shaft power than the experimental values and so is unsuitable for the VP. The TRM produces a head that agrees with the experimental observation, but it also produces a smaller shaft power and thus a noticeably higher efficiency than the experimental efficiency at over-load points (Q ≥ 8 m³/h). Overall, the effects of the SM and TRM on the performance in the fluid domain with the whole impeller resemble those in the domain with periodical connection.

Figure 16 shows the pressure and absolute velocity in the mid-span plane of the chamber and the pressure and relative velocity in the mid-span plane of the impeller as predicted by using the TRM in the fluid domains with the whole impeller for water. Compared with the pressure and absolute velocity in the chamber and the relative velocity in the impeller predicted by means of the FRM in Figs. 13 and 14, respectively, the pressure and flow pattern predicted with the TRM vary more uniformly in the circumferential direction, particularly at the part-load point.

The CFD simulations of turbulent flow in the VP reported herein were based on the standard k–ɛ two-equation model only, and wall roughness was not considered. Therefore, future work should consider other turbulence models and rough walls in the continued study of how the choice of interface model affects the hydraulic performance of the VP.

In the study reported herein, how the choice of interface model among the FRM, SM, and TRM affected the hydraulic performance of a VP with a specific speed of 76 and smooth walls in CFD simulations was investigated by using fluid domains with 1/8-impeller and whole-impeller geometrical models. Three-dimensional, steady, turbulent flows of water and viscous oils in the VP were simulated by using the Reynolds-averaged Navier–Stokes equations and the standard two-equation k–ɛ model as well as a scalable wall function in the CFD software ANSYS CFX 2019 R2. The influences of wall roughness and the whole-impeller geometrical model on the hydraulic performance were discussed.

The results showed that the head overestimation of the VP is related to using the FRM and the fluid domains with the whole-impeller geometrical model. If instead the 1/8-impeller model (periodical connection) and FRM are used, the predicted head of the pump is likely to agree well with the experimental observation. The TRM can predict a head that agrees well with the experimental head, but it requires huge computational resources and also overestimates the pump efficiency.

The flow patterns in the chamber and impeller predicted with the 1/8-impeller geometrical model were distributed more uniformly along the circumferential direction than were those predicted with the whole-impeller geometrical model by using the FRM, and the flow patterns predicted with the whole-impeller geometrical model by using the TRM were in between.

The hydraulic performance of the VP was predicted when pumping liquids with various kinematic viscosity in the 1/8-impeller geometrical model using the FRM, and the flow rate, head, and efficiency correction factors were obtained and correlated with the impeller Reynolds number.

The author thanks the School of Engineering at the University of Glasgow, where the CFD simulations reported in this paper were conducted under a license for ANSYS CFX.

The authors have no conflicts to disclose.

Wenguang Li: Conceptualization (lead); Data curation (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.