Prediction of Cavitation Performance of Axial Flow Pump by Using Numerical Cavitating Flow Simulation with Bubble Flow Model

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1 Ca-OS-6-1 Fith International Symposium on Caitation (CAV) Osaka, Japan, Noember 1-4, Prediction o Caitation Perormance o Axial Flow Pump by Using Numerical Caitating Flow Simulation with Bubble Flow Model Masashi Fukaya Mechanical Engineering Research aboratory Hitachi, td. 5, Kandatsu, Tsuchiura, Ibaraki -1, JAPAN ukaya@merl.hitachi.co.jp Tomoyoshi Okamura Research & Deelopment aboratory Hitachi Industries Co.,td. 6, Kandatsu, Tsuchiura, Ibaraki -1, JAPAN tomoyoshi_okamura@gm.hitachi-hic.jp Yoshiaki Tamura Department o Computational Science and Engineering Toyo Uniersity 1, Kujirai, Kawagoe, Saitama , JAPAN tamtam@eng.toyo.ac.jp Yoichiro Matsumoto Department o Mechanical Engineering Uniersity o Tokyo 7--1, Hongo, Bunkyo-ku, Tokyo , JAPAN ymats@mech.t.u-tokyo.ac.jp ABSTRACT The prediction o caitation perormance by using a numerical low simulation is signiicant to reduce the productie cost when designing a pump impeller blade. The caitation has seeral appearances in the pump such as sheet caitation, cloud caitation, and ortex caitation. Cloud caitation is composed o a lot o tiny bubbles with complex behaior. The bubble low model is a caitation model or numerical simulation, in which the bubble dynamics is treated in detail. In this study, we deelop a new numerical simulation code that includes the bubble low model. The code considers the distribution o the number density o bubbles, and the transitional and olumetric motions o bubbles. It was applied to a rotating low in an axial low pump. The predicted caitation perormance o the pump agreed qualitatiely with the experiment. The predicted caitation distribution on the impeller blade also agreed with that isualized. The code has the potential to simulate bubble behaior in caitation erosion. INTRODUCTION When a pump is downsized rom the iewpoint o cost reduction, the relatie low elocity increases near the impeller blades. As a result, the reduction o caitation perormance and the caitation erosion become signiicant problems. Recently, numerical caitating low simulations hae been applied to rotating machinery such as pumps, inducers, and water turbines [1-7]. The reports on these simulations predict the caitating low pattern and the caitation perormance. Howeer, caitation erosion has not been ocused on yet. Caitation erosion especially occurs when cloud and ortex caitations appear in the turbomachinery. Caitation erosion has a close relation with bubble behaior, thereore, bubble dynamics needs to be predicted to simulate caitation erosion. 1 The bubble low model [7,8] is a representatie model or the numerical caitating low simulation. In this model, a lot o tiny spherical bubbles are initially assumed to be in the low. The bubble olume aries with the pressure dierence between the bubbles and the liquid. The ariations in the bubble radius and bubble pressure are described with the Rayleigh-Plesset equation. A oid raction is obtained rom the bubble radius and the number density o the bubbles. In the preious study [9], we applied a numerical caitating low simulation including the bubble low model to six axial low pumps with dierent blade proiles, blade angles, and number o blades. To reduce the calculation time, a twodimensional isolated hydrooil simulation was conducted to calculate the lit and drag coeicients. The total head o the pump was calculated rom the caitation perormance o the ie hydrooils that constitute the impeller blade. The predicted ariation o the total head against NPSH qualitatiely agreed with the experimental data. In the present study, we deeloped a bubble-low-model simulation code or the turbomachinery and applied it to an axial low pump. We then ealuated the caitation perormance o the pump, the distributions o the liquid pressure and the oid raction, and also the distributions o the number density o the bubbles, the bubble radius and the bubble pressure. The results suggest that this code can possibly to simulate bubble behaior in caitation erosion. NOMENCATURE c: coeicient or pseudocompressibility C: chord length d: diameter o cylindrical surace D: diameter o blade tip

2 : olume raction Eˆ, Fˆ, ˆ : luxes in,, H: head Ĥ : source term g: acceleration due to graity : caitation length NPSH: Net Positie Suction Head Q: low rate unknown ector Qˆ : p: pressure r: bubble radius Re: Reynolds number t: time T: surace tension u,, w: elocity U, V, W contraariant elocity U t : peripheral elocity at blade tip : head coeicient : iscosity : rotating speed : density Subscripts cal: calculated exp: experimental B: bubble d: design point : gas phase i: x, y and z s j: ξ,η and ζ s : liquid phase R: three-percent drop o total head : apor NUMERICA METHOD oerning Equations The ollowing assumptions concerning the bubbles in the low are made in the simulation code. The gas phase consisting o spherical bubbles is compressible. No collision and coalescence occurs. The bubbles are illed with apor and non-condensable gas. The eects o eaporation and condensation on the bubble surace are modeled according to the pressure ariation o the non-condensable gas. Non-condensable gas pressure aries with isothermal expansion and adiabatic contraction [1]. Mass transer between the gas and the liquid phases is negligibly small compared to liquid mass. The density and momentum o the gas phase are small enough to be negligible. The goerning equations are summarized as ollows, Qˆ Eˆ Fˆ ˆ Eˆ Fˆ ˆ Hˆ t, (1) Eˆ y, Eˆ / J,, H / J, where Fˆ,Ĝ, Fˆ and Ĝ are not described. Equation (1) is composed o a) the conseration o the olumetric raction o the liquid phase, b) the conseration o momentum, c) a pressure equation based on pseudo-compressibility, which is deried orm the conseration o the olumetric ractions d) and the conseration o the number density o bubbles. In Eq. (1), the liquid and gas elocities mean the absolute elocities in a rotating coordinate system. Thereore, Coriolis orce appears in the source term Ĥ. The olumetric motion o a bubble is described by the Rayleigh-Plesset equation [11], r p u Qˆ / J w p n x yx y yy z yz / B x xx D r Dt x zx y xy y zy T 1 Dr p p 4, () r r Dt z xz z zz J U, () where both T and p are constant. To aoid diergence in the calculation, the iscorsity in Eq. () is assumed to be much larger than that o the actual low. Caitation is expressed as the increase o the oid raction, which is calculated by the ollowing equation, Dr ( ) Dt 4 r n. (4) The translational motion o a bubble is soled considering the orce balance o the bubble, F F Ai pi F is a constant o.5 or a spherical bubble, F pi is the orce o the acceleration o the surrounding luid, wu z p c U c U nu ˆ pb p 1 ( u 4 where F Ai is the added mass orce, 4 i FAi t U i t U Di F i F Ci u U U, (5) x p p u D r 4c r n Dt i u i )( u i u r u r u j r u r u j j j i i i ), (6)

3 F F Di and F i are the drag and lit orces, F F Pi Di i where 4 r 1 r 1 r u t C D C u i u U u u j 4. (1.15 Re 687 bub bub u ( u ) / Based on Eqs. (5)-(1), the relatie elocity o the bubble to the liquid phase is soled with the bubble radius, the liquid elocity and so on. The details o the goerning equations and the algorithm o the calculation are described in Re. [7]. The Reynolds number o the low in this simulation is Howeer, no turbulent model is used in the simulation code. The introduction o the turbulent model into the code is a subject or study in the near uture. Simulated Region and Boundary Conditions The deeloped simulation code is applied to an axial low pump that operates at a high speciic speed. The luid is water at a temperature o 9 K. The pump has our impeller blades. The blade proile is based on the NACA 65 series. The low rate o water at the design point is 4.1 m /min. The blade-tip diameter is 8 mm and the hub diameter is 1 mm. Figure 1 shows the numerical mesh and the boundary conditions in the simulation. A region between the pressure side and suction sides o the impeller blades is inestigated by using periodical boundaries. Cylindrical upstream and downstream channels o.5 m are connected to the region that includes the impeller blades. The tip clearance between the impeller blade and the casing is not considered in this simulation. The grid numbers are 75, 9 and 9 in the axial, radial and peripheral s. At the inlet boundary, the ollowing is assumed. To ix the low rate, the liquid elocity is uniorm constant at be 7.9 m/s. There is no elocity dierence between the liquid phase and the bubbles. The oid raction o.1 and the bubble radius o m are gien. The number density o the bubbles becomes m -. The static pressure has the Neumann condition. At the outlet boundary, the static pressure is aried to ijk i j i, (7) u CD ), (1) Re r u u Rebub, (11) r.5 C.59( ), (1) u u F C k i ( u ), (8) j u j, (9) is the orticity ector, and F Ci is the Coriolis orce, (1 ) 4 r u (1 ) u, (1) Inlet low boundary ψ Casing wall Hub Fig. 1 Simulated region and boundary conditions S.S. o the impeller blade P.S. o the impeller blade Case Case Case 4 Pressure boundary Rotating Exp. Cal. Cal. (re. [?]) [9] Case NPSH Fig. Caitation perormance o the axial pump change the NPSH condition. Other parameters such as the liquid and bubble elocities, the bubble radius, and the number density o the bubbles hae the Neumann condition at the outlet. Nonslip condition is assumed on the casing wall, and the peripheral elocity caused by the impeller rotation is added on the suraces o impeller blade and hub. For other parameters except the liquid and bubble elocities, the Neumann condition is assumed on the casing wall and the suraces o impeller blade and hub. RESUTS AND DISCUSSION Caitation Perormance Figure shows a comparison between the calculated caitation perormance o the axial low pump and the experiment. The NPSH was obtained rom the total head that was low-rate-aeraged just upstream o the region that included the impeller blades. The total head was equialent to the lowrate-aeraged total pressure dierence between just upstream and downstream o the impeller-blades-region. The NPSH and the total head were diided to be dimensionless NPSH and by U t /(g), simultaneously. In the experiment, the total head went up once beore the NPSH R o.7 when the NPSH was decreasing. The present numerical results shown by squares do.

4 Rotating Rotating Case 1 (NPSH =.99) Contour line o apor pressure Case (NPSH =.46) Contour line o apor pressure Case (NPSH =.8) Contour line o apor pressure Case 4 (NPSH =.) eading edge Caitation (Edge o caitation) Flow Hub Rotating Tip (a) Static Pressure Fig. Predicted static pressure and oid raction distributions Rotating Flow (a) Obsered (b) Predicted Fig. 4 Obsered and predicted caitating regions d/d ((a) NPSH =.7, (b) NPSH =.8 (Case )) Fig. 5 Caitation length rom the leading edge in Case 4 /C (b) Void Fraction Cal. Cal. (Re.[9] [?]) (Hub) (Tip)

5 Rotating (m- ) (m/s) (m- ) (m/s) Case 1 (NPSH =.99 Case (NPSH =.8 (a) Number Density o bubbles (b) Absolute elocity o water (c) Void raction Fig. 6 Features o predicted bubble low on a cross section near the leading edge o the impeller blade not obiously hae the local increase o the total head. The predicted caitation perormance o the pump, howeer, agrees qualitatiely with the experiment. The predicted NPSHR was.. Figure also shows the preious numerical result [9] as triangles. In the preious study, the total head was obtained rom two-dimensional calculations o ie isolated hydrooils constituting an impeller blade. When the NPSH was high and caitation did not occur, the predicted total head exceeded the experiment. This result is alid because the losses caused by the leakage low at the tip clearance, the secondary low between the impeller blades, and the boundary layer near the casing wall were not considered in the preious simulation. In the present simulation, on the other hand, the aboe secondary low and the boundary layer are considered, resulting in a total head reduction compared to the preious prediction. The present prediction o the total head was lower than the experiment. This is because no turbulent model is used in the present code and the simulated low near the wall was not suiciently exact. Furthermore, a lack o the numerical mesh number is also suspicious. In the present simulation, the eect o pressure interaction between the impeller blades is included. Although the predicted caitation perormance is not necessarily in good quantitatie agreement with the experiment, the low around the impeller blades is simulated well. with the region where the oid raction exceeds.1. In this study, thereore, the region where the oid raction is oer.1 is regarded as the caitating region. The caitating region noticeably expands near the blade tip with a decrease in the NPSH. The obsered and predicted caitating regions are compared in Fig. 4. Figure 4(a) shows an image taken by a highspeed camera rom a nearly radial through the acrylic casing o the pump. Near the surace o the impeller blade, the caitating region expands rom the hub side to the blade-tip side, while the tip caitation partially obstructs the iew. Figure 4(b) shows the oid raction distribution and the caitating region colored red. Concerning the caitating region near the impeller-blade surace, the prediction is in good qualitatie agreement with the obseration. The predicted caitation length is shown in Fig. 5. The caitation length is deined as the distance between the leading edge o the impeller blade and the downstream edge o the caitating region. The caitation length is measured on ie cylindrical suraces between the hub and the tip o the impeller blade. The caitation length is nondimensionalized by the chord length C o a hydrooil that is cut out rom the impeller blade at each cylindrical surace. In Fig. 5, the caitation lengths predicted in the preious study are also shown as triangles. The preious caitation length increases at a constant rate with an increase in the radial position. In the present study, the increasing rate o the caitation length near the blade tip is larger than that near the hub. The present result is in better agreement with the obseration shown in Fig. 4(a) than the preious one. Caitating Region Figure shows the numerical results o the static pressure and the oid raction distributions in Cases 1-4 in Fig.. In static pressure distributions, we draw the contour lines o the apor pressure o Pa. In oid raction distributions, the red regions indicate that the oid raction is higher than.1. In Cases, and 4, the region below the apor pressure corresponds well Number Density Distribution o Bubbles Figure 6 shows the numerical results o the number density 5

6 Rotating (m/s) (a) Bubble Radius Ratio (b) Bubble Pressure Fig. 7 Features o predicted bubbles in Case (c) Slip Velocity o the bubbles, the absolute elocity o the water and the oid raction in Cases 1 and. In both cases, the bubble number density becomes large in a circular band between the middle o the impeller blade and the casing wall, as shown in Fig. 6(a). The number density gradient in the radial in Case 1 is larger than that in Case. The number density distribution o the bubbles correlates with the absolute water elocity. As shown in Figs. 6(a) and (b), there is a tendency that the number density increases in regions where the water elocity is high. The water carries the bubbles and gathers them in the high elocity region. The number density distribution o the bubbles also correlates with the oid raction. Figure 6(c) shows that the caitating region spreads around the suction side near the blade tip in Case. When the bubbles go into a low-pressure region and rapidly swell keeping the continuity o oid raction, the bubble number density decreases. In Case, thereore, the bubble number density is reduced around the caitating region. In both Cases 1 and, another thin circular band o a large number density appears just close to the casing wall. This is because the centriugal orce acts on the water and the water carries the bubbles towards the outer side. Consequently, the bubbles gather near the casing wall. Bubble Behaior Figure 7 shows the bubble behaior such as the radius, the pressure and the elocity in Case. The bubble radius ratio is deined as the ratio o the local bubble radius to the initial bubble radius o m. In Fig. 7(a), the bubble radius ratio in the caitating region is larger than that in the non-caitation region. Figure 7(b) shows the bubble pressure p B obtained rom Eq. (). The bubble pressure decreases when the bubbles swell as shown in Fig. 7(a). The bubble elocity is inluenced by the swell o the bubble based on Eq. (5). Figure 7(c) shows the slip elocity, that is, the elocity dierence between the bubble and the water. Near the region where the bubble radius aries, the slip elocity is increased. As mentioned aboe, the caitation perormance o the pump, the caitating region, the number density o bubbles and the bubble behaior can be qualitatiely ealuated by the deeloped simulation code. These results indicate that the present code has the potential to simulate the bubble behaior in caitation erosion. CONCUSION We deeloped a new numerical simulation code that includes the bubble low model or turbomachinery. The simulation code was applied to an axial low pump with a high speciic speed. The predicted caitation perormance o the pump agreed qualitatiely with the experimental data. The predicted caitation distribution on the impeller blade was also in qualitatie agreement with that isualized. Using the present code it is possible to ealuate the number density o bubbles and the bubble behaior such as the radius, the pressure and the elocity. The unction o the code has the potential to simulate the low and bubble behaior in caitation erosion in the turbomachinery. REFERENCES [1] Hirschi, R., et al., 1997, Centriugal pump perormance drop due to leading edge caitation: Numerical predictions compared with model tests, ASME Fluids Eng. Di. Summer Meeting, FEDSM97-4. [] Song, C. C. S., et al., 1999, Simulation o Caitating Flows in Francis Turbine and Drat Tube, rd ASME/JSME Joint Fluids Eng. Con., FEDSM [] Mahesh, M., et al.,, Application o The Full Caitation Model to Pumps and Inducers, Proc. 8th Int. Symposium on Transport Phenomena and Dynamics o Rotating Machinery (ISROMAC-8), Honolulu. [4] Homann, M., et al., 1, Experimental and Numerical Studies on A Centriugal Pump with D-Cured Blades in Caitating Condition, Proc. 4th Int. Symposium on Caitation, (1), B7.5. [5] Visser, F. C., 1, Some User Experience Demonstrating The Use o Computational Fluid Dynamics or Caitation Analysis and Head Prediction o Centriugal Pumps, ASME Fluids Eng. Con., FEDSM [6] Meditz, R. B., et al., 1, Perormance Analysis o Caitating Flow in Centriugal Pumps Using Multiphase CFD, ASME Fluids Eng. Con., FEDSM

7 [7] Tamura, Y., et al.,, Numerical Method or Caitating Flow Simulations and its Application to Axial Flow Pumps, Proc. 9th Int. Symposium on Transport Phenomena and Dynamics o Rotating Machinery (ISROMAC-9), FD-ABS- 19. [8] Kubota, A. et al., 199, A New Modelling o Caitating Flows: A Numerical Study o Unsteady Caitation On a Hydrooil Section, J. Fluid Mech., ol. 4, pp [9] Fukaya, M., et. al.,, Prediction o Suction Speciic Speed o Axial Flow Pump by Using Numerical Simulation o Two-Dimensional Caitating Flow, Proc. o The 4 th International Conerence on Pumps and Fans (4 th ICPF), Beijing, pp [1] Takemura, F. and Matsumoto, Y., 1994, Internal Phenomena in Bubble Motion, Bubble Dynamics and Interace Phenomena, KUWER, pp [11] Plesset, M. S., 1954, On the Stability o Fluid Flows with Spherical Symmetry, J. Appl. Phys., Vol. 5, pp