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Kitano MajidiTechnische Universitaet Berlin,

Carnotstr. 1A,D-10587, Germany

e-mail: kitano.majidi@tu-berlin.de

Numerical Study of UnsteadyFlow in a Centrifugal PumpComputational fluid dynamics (CFD) analysis has been used to solve the unsteady threedimensional viscous flow in the entire impeller and volute casing of a centrifugal pump.The results of the calculations are used to predict the impeller/volute interaction and toobtain the unsteady pressure distribution in the impeller and volute casing. The calculatedunsteady pressure distribution is used to determine the unsteady blade loading. The caculations at the design point and at two off-design points are carried out with a multipleframe of reference and a sliding mesh technique is applied to consider the impeller/voluteinteraction. The results obtained show that the flow in the impeller and volute casing isperiodically unsteady and confirm the circumferential distortion of the pressure distribu-tion at the impeller outlet and in the volute casing. Due to the interaction betweenimpeller blades and the tongue of the volute casing the flow is characterized by pressurfluctuations, which are strong at the impeller outlet and in the vicinity of the tongue. Thesepressure fluctuations are died away in the casing as the advancement angle increaseThese reduced pressure fluctuations are spread to the discharge nozzle; the pressufluctuations are also reflected to the impeller inlet and they affect the mass flow ratethrough the blade passages.@DOI: 10.1115/1.1776587#

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IntroductionSingle-stage centrifugal turbo-machines are mostly desig

with a spiral volute casing. The asymmetric shape of the spvolute and tongue results in a circumferential distortion offlow conditions at the outlet of the impeller. This distortionespecially pronounced at off-design points. The circumferennon-uniformity of the pressure field causes unbalanced raforces that must be considered by designing rotor-beasystems.

Due to the relative movement between impeller and volute cing the flow at the outlet of the impeller is strongly interactiwith the volute flow. The unsteady interaction between these cponents generates pressure fluctuations, which are responsibunsteady dynamic forces. These unsteady dynamic forces giveto vibration of the pump components and generate hydranoises. The physics of the hydrodynamic forces and the reasothe vibration and noise generation are extensively reported in@1#.

Considerable attention has already been focused to studyunsteady interaction in centrifugal turbomachines. Both expmental and numerical approaches have contributed to the unstanding of the complex flow fluctuations due to the unsteinteraction. There are numerous examples of the experimentavestigations of which Arndt et al.@2#, Kaupert and Staubli@3#, andHagelstein et al.@4# are a representative sample. In addition, sonumerical studies have been undertaken to capture the unsinteraction and to predict the pressure fluctuations. Some ofstudies, e.g., Hillewaert and Van den Braembussche@5#, considerthe flow as inviscid and some authors, e.g., Longatte and Ku@6#, use a two-dimensional model. In recent years, improved cputational algorithms as well as hardware development havetributed to enhance CFD capability. It is now feasible to use Ccodes for a realistic prediction of the complex three-dimensioturbulent flow in the entire pump and perform unsteady calcutions see, for example, Zhang et al.@7# and Gonza´lez et al.@8,9#.However, the knowledge about the unsteady pressure fluctuaand the unsteady blade loading is still not satisfying. Furtherm

Contributed by the International Gas Turbine Institute~IGTI! of THE AMERICANSOCIETY OF MECHANICAL ENGINEERSfor publication in the ASME JOURNAL OFTURBOMACHINERY. Paper presented at the International Gas TurbineAeroengine Congress and Exhibition, Vienna, Austria, June 13–17, 2004, Pape2004-GT-54099. Manuscript received by IGTI, October 1, 2003; final revisMarch 1, 2004. IGTI Review Chair: A. J. Strazisar.

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the design of the centrifugal pumps has already reached a levelthat only through a detailed understanding of the internal flow anincrease of the overall performance can be achieved. Due to thecurved passages inside the impeller and the volute the flow is tobe considered as three-dimensional. Additionally, since the flowfollowing blade passages as well as the volute casing interactswith viscous boundary layers, secondary flows are generated.Therefore, a correct simulation of the impeller/volute interactionrequires the simultaneous solution of the three-dimensional un-steady Navier-Stokes equations in both the impeller and volute.

Centrifugal Pump and Test ConditionsThe impeller considered in this study is a commercial one. It is

shrouded and has five backswept blades. The blade profile variesbetween the hub and the shroud. The blade angle at the inlet variesfrom 18.5 deg~from tangential! at the shroud to 30.0 deg at thehub. The blade angle at the outlet is 23.5 deg. The single volutecasing is unvaned. The shape of the single volute casing is de-signed according to the theory of a constant average velocity forall sections of the volute~Stepanoff,@10#!. The main dimensionsand characteristic of the investigated pump and the test conditionsfor this study are presented inTable 1.

Numerical Model and Computational MethodsThe numerical simulations have been carried out using the

commercial code CFX-TASCflow. The flow solver of the codeemploys for incompressible turbulent flow the continuity equationand the three-dimensional time-averaged Navier-Stokes equationsIn this study, the eddy-viscosity assumption is used to model theReynolds stresses. The eddy viscosity is determined by means ofthe Standardk-« turbulence model. The walls are modeled usinga log-law wall function. The transport equations are discretizedusing an element based conservative finite volume method.

The numerical calculations are carried out with a multipleframe of reference approach, whereby the impeller flow field issolved in a rotating frame and the casing in a fixed one. The gridfor these two frames of reference should be generated separatelyThe employed code requires provision of structured or block-structured grids.Figure 1 shows the computational grids used tomodel the impeller and the volute. Both of the grids are blockstructured. The grid of the impeller models all impeller blades andpassages. In order to enhance a fully developed flow before enter-

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ing the impeller, some part of the suction nozzle is also modeTherefore, the inlet surface of the impeller was extended upstrof the physical inlet region of the blades. The grid of the impelis generated in 16 blocks, i.e., one block for the suction nozfive blocks for the inlet region, one block for each of the fivimpeller passages and each of the regions connecting bladesages with the inlet surface of the volute. All of these sub gridspinched H-grids. The grid for the impeller has 476,568 nodes. Tblades are defined as solid bodies, therefore only 76% ofnodes are active nodes. The grid of the casing is generated inblocks, one for the volute casing and one for the discharge nozand has 356,664 nodes.

As already mentioned the numerical calculations are carriedwith a multiple frame of reference. The two frames of referenare connected in such a way that for steady state calculationsrelative position of the impeller and casing does not chanthrough the calculations, i.e., that the grids of the impeller andcasing are connected by means of a frozen-rotor interface.unsteady calculations the grids are connected by means of a r

Fig. 1 Grids of the computational domain

Table 1 Main characteristics of the investigated pump

Impeller:d25508 mm Impeller outlet diameterb2572.5 mm Impeller outlet passage width

Volute Casing:d35523 mm Base circle diameterb3594.3 mm Volute width at the base circle

Design Point:mopt5730.0 kg/s Mass flow rateH546.68 m Total headn51482 rpm Rotational speedNs568 min21 Specific speed

Test Conditions:~1! Mass flow rate5730.0 kg/s (m/mopt51)~2! Mass flow rate5541.04 kg/s (m/mopt50.74)~3! Mass flow rate5978.2 kg/s (m/mopt51.34)

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stator interface, i.e., that they change their relative positionthrough the calculation according to the angular velocity of theimpeller. The reference change occurs as the flow crosses the interfaces; the appropriate transformation occurs across the interfacwithout any interface averaging. The rotor/stator approach ac-counts for the interaction between two frames.

For the steady-state calculations the following boundary condi-tions are assumed: At the inlet of the computational domain themass flow rate, the turbulence intensity, the eddy length scale anda reference pressure at one grid point are specified. It is assumedthat the absolute velocity vectors at the inlet are perpendicular tothe inlet grid surface and point into axial direction. The turbulentintensity was assumed to be 5%. Furthermore, the eddy lengthscale was assumed to be 10% of the diameter at the inlet plane. Athe outlet for all variables~with exception of pressure! a zero-gradient condition was assumed. Both in the fixed frame and inthe rotating frame the solid walls, i.e., the impeller blades, huband shroud, the casing walls and the walls of the suction nozzlesare modeled using a no-slip boundary condition.

For unsteady calculations in a previous study, the same boundary conditions as for the steady state calculations were assumedThe results are published in@11#. Some authors, e.g., Gonzalezet al.@9# and Longatte and Kueny@6#, suggested that the assump-tion of a fixed mass flow rate at the inlet of the computationaldomain is physically unsuitable for unsteady calculations and inparticular for considering the rotor/stator interaction. Therefore, inthe present study for unsteady calculations the following boundaryconditions are used: instead of a fixed mass flow rate at the inletthe total absolute pressure, and at the outlet the static pressure ia single grid face are specified.

The time step of the unsteady calculations has been set to2.0243* 1024 seconds. This time step is related to the rotationalspeed of the impeller and is chosen in such a way that one complete impeller revolution is performed after each 200 time steps.The chosen time step is small enough to get the necessary timresolution. The number of iterations in each time step has been seto 4. This number of iterations is in most cases sufficient to reducethe maximum residuals by three orders of magnitude. The averagevalues of residuals~rms values! reduce by four orders of magni-tude. All parameter settings, for example the time step and thenumber of iterations in each time step, have been retained un-changed for all test conditions. The calculations are carried out forfour and half impeller revolutions, i.e., for 900 time steps, at thedesign point—test condition 1—as well as for five impeller revo-lutions, i.e., for 1000 time steps, at both off-design points—testconditions 2 and 3. At each test condition at first a steady-statecalculation is carried out and the result is used to initialize theunsteady calculation at this test point.

Results and DiscussionsThe results of the unsteady calculations are discussed for dif-

ferent nodal points which are shown inFig. 2 ~the nodal points areshown exaggeratedly big in order to distinguish them!. The nodalpoints are selected in one blade passage~passage 3! and in thevolute casing and in the discharge nozzle of the casing. In theblade passage the points 1PM, 2PM, 1SM, and 2SM are located amidspan of the passage, whereby the points 1PM and 2PM arelocated at the blade pressure side at the inlet and outlet of theblade passage respectively. The points 1SM and 2SM are locateat the suction side (P stands for pIressure side,S for sIuction side,andM for mI idspan). The nodal points 1VM and 2VM are locatedin vIolute mI idspan at different angular advancementsw, measuredfrom the volute tongue. The nodal points 1TM and 2TM are lo-cated directly at the tIongue at mI idspan of the volute casing,whereby the point 1TM is located at the impeller side of thetongue~at w50) and the point 2TM is located at the dischargenozzle side of the tongue. The points 1DM and 2DM are locatedrespectively at begin and end of the dI ischarge nozzle at mI idspan

Transactions of the ASME

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of the casing. Furthermore,Fig. 2 shows the relative position ofthe impeller and casing for steady state calculations as well atime50 of unsteady calculations.

Unsteadiness Inside the Impeller. In Fig. 3, the time histo-ries of the pressure coefficientCp at the investigated nodal pointsinside the blade passage are shown, where the time-average ocomputed mass flow rate closely captures the design massrate. By the set of boundary conditions used in the presentproach, i.e., total pressure at the inlet of the computational domand the static pressure at a single grid face at the outlet, the demass flow rate can only roughly be adjusted. For the results psented here, a time-average value of 734.1 kg/s (m/mopt51.005)

Fig. 2 Location of the investigated nodal points, as well as therelative position of the impeller and volute casing at time Ä0„starting the unsteady calculations, initiated from steady-statesolutions …

Fig. 3 Unsteady pressure distribution at midspan of the bladepassage at m Õm optÉ1.0

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has been computed. The static pressure is normalized using a dynamic pressure based on the impeller outlet tip velocity:

Cp5p2pref

0.5ru22 . (1)

Fig. 4 Unsteady pressure distribution at midspan of the bladepassage at m Õm optÉ0.74

Fig. 5 Unsteady pressure distribution at midspan of the bladepassage at m Õm optÉ1.34

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As it can be seen inFig. 3, the final periodic unsteady solutiois achieved after one impeller revolution, whereby each imperevolution needs 200 time steps~40,486 ms!. At the design point~Fig. 3! the calculations are carried out for four and half impelrevolutions, i.e., for 900 time steps. And at both off-design poinFigs. 4 and 5, for five impeller revolutions, i.e., for 1000 timesteps.

According toFig. 3 negativeCp values at the inlet of the bladepassage can be observed. A negativeCp refers to a pressure that ilower than the pressure at the reference point, which is selectethe impeller eye. Furthermore, at the inlet of the blade passagthe immediate vicinity of the leading edge, a pressure surplus

Fig. 6 Unsteady mass flow rate through each blade passageat m Õm optÉ0.74

Fig. 7 Unsteady mass flow rate through each blade passageat m Õm optÉ1.3

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the suction side~nodal point 1SM! compared to the correspondingvalues on pressure side~nodal point 1PM! can be observed. Thisresult indicates a negative blade incidence, which is actually typi-cal for mass flow rates in excess of the design mass flow rate.Indeed, as already mentioned, the mass flow rate exceeds margin-ally the design mass flow rate.

Fig. 8 Unsteady pressure distribution in the volute casing atm Õm optÉ1.0

Fig. 9 Unsteady pressure distribution in the volute casing atm Õm optÉ0.74

Transactions of the ASME

Figures 4 and 5show the time histories of the pressure coficients in the blade passage at off-design pointsm/mopt'0.74 andm/mopt'1.34, respectively. The phenomenon inFig. 3—a nega-tive blade incidence—can also be observed inFig. 5, but muchmore intensely. According toFig. 4, representing a part load operation point, at the inlet of the blade passage the pressure anodal point 1PM is higher than at the suction side nodal po1SM.

ConsideringFigs. 3–5, it can be seen that the amplitude of tpressure fluctuations within the impeller passage grows in ma

Fig. 10 Unsteady pressure distribution in the volute casing atm Õm optÉ1.34

Fig. 11 Unsteady pressure distribution at the tongue of the

casing at m Õm optÉ1.0

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tude as the trailing edge of the blade is approached. The Compari-son ofFig. 3 with Figs. 4 and 5, shows that the amplitudes of thepressure fluctuations at off-design points are considerably largerthan the amplitudes at the design point. However, the amplitude ofthe pressure fluctuations at a mass flow rate higher than the design

Fig. 12 Unsteady pressure distribution at the tongue of thecasing at m Õm optÉ0.74

Fig. 13 Unsteady pressure distribution at the tongue of thecasing at m Õm optÉ1.34

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Fig. 14 Pressure distribution at midspan of the pump „a…, pressure distribution „b…, and secondary flow „c… in the cross-sectional plane of the volute casing at angular advancement wÄ60 deg at three different time steps at off-design point„m Õm optÉ0.74…, †13‡

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point is obviously more critical. Comparing the pressure flucttions at the pressure side~nodal points 1PM and 2PM! to suctionside ~nodal points 1SMand 2SM! reveals that in all operatingpoints the amplitude of the pressure fluctuations on the presside of the blade passage is larger than on the suction side.

Since the flow is incompressible, strong pressure fluctuationthe impeller outlet caused by the interaction between the blaand tongue of the volute casing, are reflected upstream toimpeller inlet~see the pressure distribution for nodal points 1P1SM!. Thus, pressure fluctuations at the impeller inlet influenthe mass flow rate through the blade passages, as it can befrom Figs. 6 and 7. According to these Figs. the mass flow ra

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through each blade passage varies with time according to the retive position of the blade passage to the tongue of the volucasing. The periodic pressure distribution at the impeller-inlet anoutlet leads to a periodic flow, which results in a cyclic acceleration and deceleration of the fluid flow inside each blade passagAt the upper part ofFig. 7, the relative position of the passage 1to the tongue of the volute casing at time step zero as well aexemplary at the begin and at the end of one deceleration and oacceleration-phase are shown; they are linked to the corresponing points of the graph for passage 1. Deviations of the exarepeatability inFigs. 6 and 7are partly numerical. However, alsoin a real pump an exact repeatability of mass flow rates throug

Transactions of the ASME

Fig. 15 Pressure distribution „a… and secondary flow „b… in the cross-sectional plane of the volute casing at angular advance-ment wÄ260 deg at three different time steps at off-design point „m Õm optÉ0.74…

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each blade passage at every impeller revolution can notachieved, since the initial values at each impeller revolution vslightly.

Unsteadiness Inside the Volute-Casing. The time historiesof the pressure coefficientCp at the investigated nodal points ithe volute casing and at the tongue of the casing are showFigs. 8–13, where both design and off-design points are takinto account. By ConsideringFigs. 8–10 it is evident, that pres-sure fluctuations assume most pronounced intensities in polocated closest to the tongue, e.g., points 1VM, 1 TM. The amtude of these pressure fluctuations die away with advancemew ~2VM!, but is still significant at the discharge nozzle~1DM!. Atthe outlet of the discharge nozzle~2DM! the pressure fluctuationshave almost vanished.

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According toFigs. 11–13, the pressure fluctuations at the im-peller side of the tongue, nodal point 1TM, are strong and havelarge amplitudes. At the discharge side of the tongue at nodalpoint 2TM these pressure fluctuations have died away and thusthe amplitudes of the fluctuations are considerably smaller.

Contour Plots at Various Instances. The unsteady calcula-tions capture the interaction between the tongue and the bladesFigure 14~a! shows the contour plots of the pressure coefficient atthe midspan of the impeller at three different time steps. Accord-ing to this figure the tongue and the blades are interactingstrongly. As each blade approaches the tongue, very strong pressure fluctuations in the vicinity of the tongue can be observed.These pressure fluctuations are propagated in the volute and arspread to the discharge nozzle. The fluctuations are also reflecte

APRIL 2005, Vol. 127 Õ 369

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in the impeller and are spread to the inlet region. Part (b) of thisfigure shows the distribution of the pressure coefficient atcross-sectional plane of the volute casing at the advancemanglew560 deg at the same time steps as in part (a). In order todistinguish the pressure fluctuations, the minimum and maximvalues of the color scale of contours remain in all the time stethe same. It means contour level scales are consistent betweetime steps. It is to distinguish that at different time steps the prsure fluctuates strongly~colors vary strongly!, but the pressuregradients in the cross-sectional plane remain in all time stepsmost the same, at all the time steps there are two pronounminimum value regions in the pressure distribution of the crosectional plane. These pressure gradients are arising from thesage curvature~centrifugal forces! and they are in combinationwith boundary layers responsible for the secondary flows involute casing. Since these pressure gradients do not vary wtime, the secondary flow, shown in part (c) of Fig. 14, does notvary too. The secondary vectors are the projections of the calated velocity vectors onto the investigated cross-sectional plaThe centers of the vortices of the secondary flow in part (c) cor-respond to the two pronounced minimum value regions in part (b)of the figure. The high velocity core in the cross section is drivby the impeller.

In Fig. 15, the pressure coefficient as well as secondary flovelocity vectors are depicted in a cross-sectional plane of the cing at an angular advancement ofw5260 deg. By consideringFig. 15, it is obvious that pressure fluctuations die away wiincreasing angular advancementw ~the colors remain in all timesteps almost unchanged!. In this cross-sectional plane only ondeficit region can be observed and correspondingly the seconflow has only one vortex. As already mentioned, the secondflows are caused by pressure gradients perpendicular to thedirection, whereby the pressure gradients depend on the pascurvature; they vary according to the shape of the cross-sectioplane and according to the local radius of the spiral volute~see,for example@12#!.

Blade Loading Results. Using the calculated pressure distrbution in the impeller, the magnitude of blade loading canestimated. To calculate the blade loading the static pressureintegrated over the entire surface of the blades. The unsteady psure distribution results in an unsteady blade loading, whichshown inFig. 16. The blade loading is normalized using the folowing equation:

Fig. 16 Unsteady blade loading at different test conditions

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Fnormalized5F radial

0.5ru22pd2b2

, (2)

wherer means the density of the fluid,d2 is the impeller diameter,andb2 is the width of the impeller at the outlet. Applying Eq.~2!to different test conditions, blade loading appears to assume—aexpected—much larger values at off-design conditions~1.3 and0.74 mass flow ratios!. According to Fig. 16, at off designpoints—specially at 1.3 mass flow ratio—the amplitudes of thefluctuations due to flow unsteadiness are equal to the average nblade loading values. This unsteady blade loading gives rise todynamic effects, which are one of the most important reasons ovibration and hydraulic noises. In order to appraise the virtue ofthe present unsteady computational approach with semi-empiricacorrelation given by Stepanoff@10# may be used. Stepanoff hassuggested that the radial thrust due to due to the nonuniformity othe circumferential pressure distribution is a function of totalhead,H, impeller diameter,d2 , impeller width,b2 , and an ex-perimental coefficientKr

F radial–Stepanoff5KrrgHd2b2 . (3)

The coefficientKr depends on the operating point and reachesits maximum at shutoff:

Kr5~0.36••••••0.6!F12S m

moptD 2G . (4)

Comparing normalized time-averaged loading values~Fig. 16!,numerical calculations appear to be less responsive to off-desigconditions. On the other hand, the coefficient in Eq.~4! is more orless uncertain in the setup considered here.

ConclusionThe results obtained show that the flow field in the impeller and

volute casing of centrifugal pumps is periodically unsteady. It wasconfirmed that due to the interaction between impeller and volutecasing the flow is characterized with pressure fluctuations, whichare strong at impeller outlet and at the vicinity of the tongue. Thelarge amplitudes of the pressure fluctuations at impeller outlet andin tongue region die away in the casing as the advancement anglincreases. These reduced pressure fluctuations are spread to tdischarge nozzle and are reflected to the impeller inlet. Using theunsteady pressure distribution inside the impeller, unsteady bladloading, which is one of the most important reasons of vibration,was calculated. The comparison of the numerical results obtainedin this study with the published experimental results show a quali-tative good agreement of the results considering the behavior othe unsteady pressure fluctuations.

Nomenclature

b2 5 outlet passage width of the impellerCp 5 pressure coefficientd2 5 outlet diameter of the impeller

F radial 5 radial force~blade loading!p 5 pressure

u2 5 impeller outlet tip velocityr 5 density

References@1# Gulich, J. F., 1999,Kreiselpumpen, Ein Handbuch fu¨r Entwicklung, Anlagen-

planung und Betrieb, Springer, Berlin.@2# Arndt, N., Acosta, A. J., Brennen, C. E., and Caghey, T. K., 1990, ‘‘Experi-

mental Investigation of Rotor-Stator Interaction in a Centrifugal Pump WithSeveral Vaned Diffusers,’’ ASME J. Turbomach.,112, pp. 98–107.

@3# Kaupert, K. A., and Staubli, T., 1999, ‘‘The Unsteady Pressure Field in a High

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Specific Speed Centrifugal Pump Impeller—Part I: Influence of the Volute,ASME J. Fluids Eng.,121, pp. 621–629.

@4# Hagelstein, D., Hillewaert, K., Van den Braembussche, R. A., Engeda, AKeiper, R., and Rautenberg, M., 2000, ‘‘Experimental and Numerical Investgation of the Flow in a Centrifugal Compressor Volute,’’ ASME J. Turbom-ach.,122, pp. 22–31.

@5# Hillewaert, K., and Van den Braembussche, R. A., 1999, ‘‘Numerical Simulation of Impeller-Volute Interaction in Centrifugal Compressors,’’ASME J. Tur-bomach.,121, pp. 603–608.

@6# Longatte, F., and Kueny, J. L., 1999, ‘‘Analysis of Rotor-Stator-Circuit Inter-actions in a Centrifugal Pump,’’ ASME Paper FEDSM99-6866.

@7# Zhang, M., Wang, H., and Tsukamoto, H., 2002, ‘‘Numerical Analysis of Unsteady Hydrodynamic forces on a Diffuser Pump Impeller due to Rotor-StatInteraction,’’ ASME Paper FEDSM2002-31181.

@8# Gonzalez, J., Ferna´ndez, J., Blanco, E., and Santolaria, C., 2002, ‘‘NumericaSimulation of the Dynamic Effects Due to Impeller-Volute Interaction in a

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Centrifugal Pump,’’ ASME J. Fluids Eng.,124, pp. 348–355.@9# Gonzalez, J., Santolaria, C., Blanco, E., and Ferna´ndez, J., 2002, ‘‘Unsteady

Flow Structure on a Centrifugal Pump: Experimental and Numerical Ap-proaches,’’ ASME Paper FEDSM2002-31182.

@10# Stepanoff, A. J., 1957,Centrifugal and Axial Flow Pumps, Krieger PublishingCompany, Malabar, FL.

@11# Majidi, K., 2003, ‘‘Numerical Calculation of Impeller/Volute Interaction in aCentrifugal Pump,’’Proceedings of the 5th European Conference on Turbo-machinery Fluid Dynamics and Thermodynamics, pp. 597–606.

@12# Majidi, K., and Siekmann, H. E., 2000, ‘‘Numerical Calculation of SecondaryFlow in Pump Volute and Circular Casing Using 3D Viscous Flow Tech-niques,’’ Int. J. Rotating Machinery,6~4!, pp. 245–252.

@13# Majidi, K., 2004, ‘‘Unsteady Radial Thrust of a Centrifugal Pump due to theImpeller/Volute Interaction,’’Proceedings of the 10th International Symposiumon Transport Phenomena and Dynamics of Rotating Machinery, Pacific Centerof Thermal-Fluids Engineering, Paper ISROMAC10-2004-032.

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