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Influence of the empirical coefficients of cavitation model on predicting cavitating flow in the centrifugal pump
• • ABSTRACT
Influence of the empirical coefficients of cavitation model on predicting cavitating flow in the centrifugal pump
KEYWORD
Cavitation flow , Centrifugal pump , Experiment , Numerical simulation
• ### INTRODUCTION

The occurrence of unsteady cavitation in pump is nearly inevitable, where the local pressure drops below the saturated vapor pressure, especially for those applied on vessels and offshore platforms, since the particles contained in seawater can increase the probability of cavitation generation. Cavitation may cause various problems, like vibration, noise and material erosion, which would deteriorate the pump performance and cause damage to the pump (Brennen et al., 1995; Ding et al., 2011; Bruno and Frank, 2009). In the recent years, owing to the continuous improvement of Computational Fluid Dynamics (CFD) technologies and computational capabilities, the prediction of pump cavitation performance based on CFD method has been beneficial for preliminary pump design (Liu et al., 2010; 2012; Wang et al., 2011; Dijkers et al., 2005). Thus, it makes the cavitation model play a significant role in numerical simulation progress. During the last decades, great efforts have been made in the development of cavitation models (Athavale et al., 2002; Coutier-Delgosha et al., 2003). These models can be put into two categories, namely interface tracking methods (Senocak and Shyy, 2004; Hirschi et al., 1998) and homogeneous equilibrium flow models (Zwart et al., 2004; Kunz et al., 2000; Singhal et al., 2002; Schnerr and Sauer, 2001; Merkle et al., 1998; Delannoy and Kueny, 1990). The former assumes that the cavity region has a constant pressure equal to the vapor pressure of the corresponding liquid and the computations are calculated only for the liquid phase. However, these methods are limited to 2-D planar or axisymmetric flows because of the difficulties dealing with complicated 3-D models. In the second category, the homogeneous equilibrium flow models assume the flow to be homogenous and isothermal, applying either a barotropic equation of state or a transport equation for both phases. The barotropic equation links the density to the local static pressure (Delannoy and Kueny, 1990). A recent experimental study implied that the vorticity production is an important aspect of cavitating flows, especially in the cavity closure region (Gopalan and Katz, 2000). But in the barotropic law, the gradients of density and pressure are always parallel, which leads to zero baroclinic torque. Therefore, the barotropic cavitation models cannot capture the dynamics of cavitating flows, particularly for cases with unsteady cavitation flows (Senocak and Shyy, 2002). Furthermore, this method is prone to instability because of high pressure-density dependence, which makes it difficult to reach the convergence levels of noncavitating flow simulations (Marina, 2008). Conversely, these limitations can be avoided by applying the transport equation models (TEM). In this approach, volume or mass fraction of the two phases are solved by an additional transport equation with different source terms. Besides, there is another apparent advantage of this method, which could predict the impact of inertial forces on cavities like elongation, detachment and drift of bubbles. In the past years, a great number of transport equation models are proposed (Zwart et al., 2004; Kunz et al., 2000; Singhal et al., 2002; Schnerr and Sauer, 2001; Merkle et al., 1998). These models apply different condensation and evaporation empirical coefficients to regulate the mass and momentum exchange. However, most of these empirical coefficients are calibrated on simple hydraulic machinery, such as hydrofoil or blunt body. When these models are employed in pumps, the accuracy of numerical simulation is strongly dependent on users’ experience to choose proper coefficients. Among this kind of TEM models, because of its effectively and stability, the Zwart-Gerber-Belamri model (hereafter ZGB model) was widely used for different cases (Zwart et al., 2004; Hagar et al., 2012; Liu et al., 2012).

In this study, the influence of the empirical coefficients on predicting the cavitation performance of a centrifugal pump was investigated. To this aim, the ZGB model was considered. Moreover, the experiments were carried out to validate the numerical simulations.

### EXPERIMENTAL SETUP AND TEST PUMP

The experiments were performed on a closed platform in the Research Center of Fluid Machinery Engineering and Technical of Jiangsu University. Fig. 1 shows the centrifugal pump closed test rig. Two pressure transducers, JYB-KO-HAGL- 1, are installed in the upper and down steam, with a measurement accuracy of ±0.5％FS (The FS is interpreted as the full scale of the pressure transducer, which is ±100kPa). To avoid the disturbance from the pump, a turbine flowmeter LWGY-80A is mounted far away from the pump, with an accuracy of ±0.5％ for the flow-rate value between 16-100m3/h. Measurement uncertainties were estimated to be less than ±2％ and ±1.5％ respectively, contributing to an uncertainty of ±2.8％ of the pump head and ±2％ of the cavitation number, according to Eqs. (9) and (11). All of the measured data are acquisitioned synchronously and then processed by a pump test system, TPA, developed by our research center. The ambient temperature is about 22° and after 3 times tests, the system is ceased for hours until the water temperature cools down. The basic parameters of the test pump are listed as follows: the volume flow rate Q = 0.014m3/s, the rotation speed n = 2,900r/min, the impeller diameter D2 = 168 and the blade number Z = 5. The impeller test model is given in Fig. 2(a), which is manufactured by the rapid prototyping technology. Its meridional shape is also plotted in Fig. 2(b). ### >  Governing equations

The set of governing equation consists of the mass continuity (1) and momentum Eq. (2) plus a transport Eq. (3) to define vapor generation:

The mixture density is defined by the vapor volume fraction, expressed as:

where p is the pressure, ρm is the mixture density, ui is the velocity, μ and μt stand for the laminar viscosity and turbulent viscosity, α is the volume fraction, and represent the source terms for evaporation and condensation. The subscripts m, l, v indicate the mixture, liquid and vapor, respectively.

### >  Turbulence model

The RNG k-ε model was adopted for solving the transport equations of the turbulent kinetic energy and its dissipation rate, which is based on the renormalization group analysis of the Navier-Stokes equations. The RNG k-ε model has been proved to give good predictions in cavitating flows (Zhou and Wang, 2008; Chang and Wang, 2012; Yang et al., 2012). To improve the numerical simulations, a modification of the turbulent viscosity was taken into account to reduce the turbulent viscosity of the mixture, proposed by Coutier-Delgosha et al., (2003). The turbulent viscosity μt is defined as

where the exponent n is a constant and Fig. 3 shows the relationship between different n values with f(ρm). It can be noted that, with increasing n, the turbulence viscosity would reduce obviously. As recommended by Coutier-Delgosha et al., (2003), n = 10 was employed in the present study. ### >  Cavitation model

All the simulations were conducted by using the ANSYS-CFX commercial software and the ZGB model was considered in this paper, which is deduced from the Rayleigh-Plesset equation:

where Fvap and Fcond are the empirical calibration coefficients of evaporation and condensation, respectively. And rnuc is the nucleation site volume fraction, RB stands for the nucleation site radius (hereafter NSR). Vaporization is initiated at nucleation sites, which can be regarding as the non-condensible gases. pv represents the water vaporization pressure. The recommended values of these coefficients are: Fvap = 50, Fcond = 0.01, rnuc = 5 × 10−4, RB = 2 × 10−6m and pv = 3574Pa. In this paper, the coefficients, Fvap, Fcond and RB, were studied to find out the influence on predicting the cavitating flows in centrifugal pump.

### >  Meshing

To get a good accuracy computing results, the structured hexahedral grids were generated by GridPro commercial software. Fig. 4(a) shows the computational fluid domain of the centrifugal pump. The grids near the blade surface region layer were refined, which is locally zoomed up in Fig. 4(b). The Y plus on the blade surface is ranged from 0.2 to 35 (Fig. 5). To get a relatively stable upper and down stream flow, two prolongations, whose lengths are four times of the pipe diameter, were assembled on the impeller and volute.  A mesh independence test was performed based on the pump head H under non-cavitation condition, calculated by Eq. (9).

where ptin and Pptout represent the total pressure of the inlet and outlet, respectively.

Table 1 gives the simulation results with five different kinds of grid density. It is obviously that when the cell number is over two million, the discrepancy of the pump head is within 1％. Ultimately, considering the simulation efficiency, the total cell number of all the domains are set as 1.20×106. Pump head with different cell numbers.

### >  Boundary condition

In the simulation process, since the pump impeller is a rotating part, whereas the other parts, the prolongations and volute casing, are stators, the Multiple Reference Frame (MRF) approach was employed, which allows the analysis of situations involving rotator/stator fluid domains and has been demonstrated that it has good accuracy (Ding et al., 2011; Tan et al., 2012). The interfaces were imposed between the impeller and inlet prolongation and volute. The pressure and mass flow rate boundary conditions were fixed at the inlet and outlet, respectively. Moreover, no slip boundary condition was applied on the solid surface of the pump. All the calculations were firstly carried out under non-cavitation condition to obtain a steady solution. Then, the pressure loaded on the inlet was decreased progressively until the desired cavitation number was reached.

### RESULTS AND DISCUSSIONS

In the convenience of comparing the results, two dimensionless parameters are defined as:

where u2 is the circumferential velocity at the impeller outlet, pin represents the static pressure of the inlet.

### >  Influence of the nucleation site radius

In Fig. 6, the pump head drop characteristic curves are shown, calculated by different NSR, while the other empirical coefficients are set as default. To distinguish the experimental results from the numerical simulations, the experimental results are plotted as dash line, whereas the straight lines with symbols stand for the computed results. Apparently, the smaller NSR, such as RB = 2 × 10−8m, the closer the results to the experiment. Under non-cavitation conditions, compared with the experiment, a slight overestimation of the pump head can be observed in simulations. Here we define a scaleless parameter-the critical cavitation number σc as the σ value when the pump head drops by 3％. The σc values of each conditions are 0.027, 0.048 and 0.059, successively from RB = 2 × 10−4m to 2 × 10−8m, whereas the value in the experiment is 0.138, almost three times larger than RB = 2 × 10−6m (default).   ### >  Influence of the evaporation and condensation coefficients

Since the evaporation and condensation coefficients have much more influence on the calculation, more schemes are chosen. The results are given in Fig. 9. As seeing, the smaller Fvap value, the greater discrepancy between the numerical simulation and the experiment. It can be observed that when Fvap＜50(default value), the pump head starts to drop at a very low cavitation number. For Fvap = 5, the critical cavitation number σc = 0.027 and for Fvap = 0.5, this value drops to 0.013. It is implied that reducing the Fvap value would greatly impact the computed result. In contrast, when increasing the evaporation coefficient, a slightly improvement of the head drop curve can be noticed. But compared with the experiment result, σc = 0.138, the critical cavitation number only goes up to 0.054 and 0.06, respectively for Fvap = 500 and Fvap = 5000. Figs. 10 and 11 present the vapor volume fraction distribution and blade loading distribution with various evaporation coefficients under the same conditions as Figs. 7 and 8. From Fig. 10, it can be find out that both of the cavity size and length are getting smaller and shorter as the evaporation coefficients declining, leading to diminishing the low blade loading region, as can be seen in Fig. 11. In addition, it is obviously that when the Fvap value increases, the cavity region with high vapor volume fraction become larger.  Fig. 12 shows the head drop curves with different condensation coefficients. Similarly, five values are selected to investigate. We can see when increasing Fcond from 1e-2 to 1e-1 and 1, the head drop curve nearly has no change. The critical cavitation number σc are 0.047 and 0.045, respectively for Fcond = 1e-1 and 1, compared with σc = 0.048 as Fcond = 1e-2 (default value). While Fcond drops to 1e-3, a better prediction result is obtained. The pump head starts to decline when σ is around 0.07, contributing to a higher critical cavitation number σc = 0.059. But when the condensation coefficient Fcond reduces to 1e-4, a tremendous improvement can be noticed in the figure. The head drop curve has a good agreement the experiment, with a critical number σc of 0.116, compared with 0.138 in the experiment. Since the cavitation number in the case of Fcond = 1e-4 is higher than the others, the operating condition of σ = 0.1 is chosen to study the vapor volume fraction distribution, which is shown in Fig. 13. It is noticed that with the condensation coefficient dropping, the cavity length is getting longer. As Fcond = 1e-4, the bubbles nearly cover all over the suction side of the blade, which produces a more accuracy prediction results. The reason lies in that, with lower condensation coefficient, the mass of vapor transferred into liquid decreased, making the cavity length much longer. Furthermore, we can find that the cavity of high volume fraction is almost unchanged with decreasing Fcond. Also, the blade loading with various condensation coefficients are plotted in Fig. 14. No remarkable difference can be observed, that is to say, the most affective factor for blade loading is the cavity region of high volume fraction, but not the cavity length.  ### CONCLUSIONS

To investigate the influence of the empirical coefficients of cavitation model on predicting cavitating flow in centrifugal pump, numerical simulation and experiment are presented in this paper. The widely used Zwart-Gerber-Belamri cavitation model is considered. Within this model, three coefficients are analyzed, namely the nucleation site radius RB, evaporation and condensation coefficients, Fvap and Fcond. During the simulation process, when one of these coefficients is studied, the others are set as default. The conclusions could be arrived at as follows:

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• [ ]  Fig. 1 Experimental setup. • [ Fig. 2 ]  Test impeller model (a) and impeller meridional shape (b). • [ ] • [ ] • [ ] • [ ] • [ ] • [ ] • [ Fig. 3 ]  Relationship between function f(ρm) and mixture density ρm with different n values. • [ ] • [ ] • [ Fig. 4 ]  Pump computational grids (a) and grid refinement on blade (b). • [ Fig. 5 ]  Yplus on the blade surface. • [ ] • [ Table 1 ]  Pump head with different cell numbers. • [ ] • [ Fig. 6 ]  Influence of the NSR on pump head drop curve. • [ Fig. 7 ]  Vapor volume fraction distribution with various NSR when σ = 0.07 at Span = 0.5. • [ Fig. 8 ]  Blade loading distribution with various NSR on the middle streamline at Span = 0.5. • [ Fig. 9 ]  Influence of the evaporation coefficient on pump head drop curve. • [ Fig. 10 ]  Vapor volume fraction distribution with various evaporation coefficients when σ = 0.07 at Span = 0.5. • [ Fig. 11 ]  Blade loading distribution with various evaporation coefficients on the middle streamline at Span=0.5. • [ Fig. 12 ]  Influence of the condensation coefficient on pump head drop curve. • [ Fig. 13 ]  Vapor volume fraction distribution with various condensation coefficients when σ = 0.1 at Span = 0.5. • [ Fig. 14 ]  Blade loading distribution with various condensation coefficients on the middle streamline at Span = 0.5. (우)06579 서울시 서초구 반포대로 201(반포동)
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