ss, 9

ispua

heical techniques for adaptive grid renement and for treatment of grid cells that do not t the irregularboundaries are implemented in order to achieve a fully automated grid construction for any impeller

uide a retion. H

ties in correlating the design parameters with the blade geometry(the rst two) or convergence problems (the latter). A quasi-3Dmethod is recently proposed [4], which performs a blade-to-bladesolution and saves computer time by using only one representativehub-to-shroud surface. All the above models are based on theinviscid simplied assumption. The application of fully 3-dimen-sional turbulent ow analysis for the impeller/runner design is in-

immersed boundary methods [14,15].In the present work a cell-cut sharp-interfacemethod developed

for the automated generation of Cartesian grids in irregular geom-etries is incorporated in a computer algorithm for the simulation ofa centrifugal pump impeller operation. The shape optimization pro-cess with the use of Cartesian grid has been successfully applied forlaminar ows in various geometries (eg. [16]). In order to accelerateconvergence the computational domain contains only the pumpimpeller in a 2-dimensional approach, while special modelling isimplemented to reproduce the characteristic performance curves

* Tel.: +30 2107721080; fax: +30 2107721057.

Computers & Fluids 38 (2009) 284289

Contents lists availab

rs

lseE-mail address: j.anagno@uid.mech.ntua.grcontinuous increase in computing power, the inverse designnumerical optimization is still a laborious task, because it needsa large number of ow eld evaluations. Such computations maybe very costly, especially when the entire 3D domain in both theimpeller/runner and the casing of the machine are simulated,and many design variables are incorporated. For this reason, fewreal 3D inverse design methods have been developed, as theinverse time marching method [1], the pseudo-stream functionmethod [2], and the Fourier expansion singularity method [3].These methods are very time-consuming and exhibit some difcul-

complex geometries. Some recent numerical works are directed to-wards that advanced approach [12,13].

The computer time needed by these models depends stronglyon the generation cost of the body-tted grid, as well as on the gridquality. An alternative practice in complex domains is the use ofCartesian grids that require a much reduced construction effort.The main drawback of those is the inability of their lines to followa non-orthogonal or irregular boundary. Several numerical tech-niques to improve the accuracy in such regions have been pub-lished; most of them they can be classied as cell-cut or1. Introduction

The numerical simulation of thehydraulic turbomachinery has becomincrease efciency and reduce cavita0045-7930/$ - see front matter 2008 Elsevier Ltd. Adoi:10.1016/j.compuid.2008.02.010design, as well as to produce results of adequate precision and accuracy. After estimating the additionalhydraulic losses in the casing and the inlet and outlet sections of the pump, the performance of the pumpcan be predicted using the numerical results from the impeller section only. The regulation of variousenergy loss coefcients involved in the model is carried out for a commercial pump, for which thereare available measurements. The predicted overall efciency curve of the pump was found to agree verywell with the corresponding experimental data. Finally, a numerical optimization algorithm based on theunconstrained gradient approach is developed and combined with the evaluation software in order tond the impeller geometry that maximizes the pump efciency, using as free design variables the bladeangles at the leading and the trailing edge. The results veried that the optimization process can convergevery fast and to reasonable optimal values.

2008 Elsevier Ltd. All rights reserved.

ow for the design inquisite tool in order toowever, in spite of the

creased in the past few years [511], usually with the aid ofcommercial CFD software. However, the use of NavierStokes val-idation in inverse design optimization methods is still not a com-mon practice, since in addition to the time-consumingcalculations there is a need for automated mesh generation inAvailable online 9 July 2008A fast numerical method for ow analysipump impellers

John S. Anagnostopoulos *

School of Mechanical Engineering/Fluids Section, National Technical University of Athen

a r t i c l e i n f o

Article history:Received 23 January 2007Received in revised form 28 February 2008Accepted 28 February 2008

a b s t r a c t

A numerical methodologypump impeller and to comdomain is discretized withequations are solved with t

Compute

journal homepage: www.ell rights reserved.and blade design in centrifugal

Heroon Polytechniou Avenue, Zografou, 15780 Athens, Greece

developed to simulate the turbulent ow in a 2-dimensional centrifugalte the characteristic performance curves of the entire pump. The owpolar, Cartesian mesh and the Reynolds-averaged NavierStokes (RANS)control volume approach and the ke turbulence model. Advanced numer-

le at ScienceDirect

& Fluids

vier .com/locate /compfluid

of the entire pump. With this methodology the computer cost perevaluation is much reduced compared to a fully 3D simulation ofthe pump, giving thus a quick and reliable estimation of the opti-mum values of the free design parameters. The method is expectedto be more successful for low specic speed impellers, where theow is mainly 2-dimensional.

2. The numerical methodology

For the simulation of the ow in a 2-dimensional pump impel-

savings in computer cost than each of the local grid renement orthe higher-order discretization methods alone [17].

2.2. Geometry representation

A cell-cut, sharp-interface grid construction method, developedand tested with success in various applied studies in the past (eg.[16,18]), is modied and further improved in order to increase itsaccuracy near the irregular boundaries. With the new method nocell-merging is performed, but all the grid cells that are totally or

J.S. Anagnostopoulos / Computers & Fluids 38 (2009) 284289 285ler the incompressible NavierStokes equations are expressed inpolar coordinates and a rotating with the impeller system. Usingcircular inner and outer boundaries, the governing equations fora horizontal space (no gravity) become:

Continuity :r~w 0 1Momentum : ~w r~w2~x~w ~x~x~rrp

q 1qr~s 2

where ~w is the uid velocity in the rotating system (relative uidvelocity), x is the angular rotation speed of the impeller, p and qare the uid pressure and density, respectively, and ~s is the stresstensor that includes both the viscous and the turbulence viscosityterms. The standard ke turbulence model is adopted, which is suit-able for high Re number ows, as the examined here. The system ofthe averaged form of the above equations is numerically solvedwith the nite volume approach and a collocated grid arrangement,using a preconditioned bi-conjugate gradient (Bi-CG) solver. Settingcyclic boundary conditions the solution can be restricted to the 1/zpart of the impeller, where z is the number of blades.

2.1. Numerical grid

As stated in the Introduction, the use of Cartesian grids fornumerical design optimization provides the signicant advantageof a fast and automated grid generation process. Other desirablefeatures include the easiness in the construction and control oflocally or adaptively rened regions, as well as the capability touse discretization schemes of higher, in general, accuracy, com-pared to other grid types. All the above features are developedand incorporated in the computation algorithm used for the pres-ent study. The numerical technique, more details of which can befound in Anagnostopoulos [17], introduces a multiple stencil thatallows the application of second-order discretization schemes toany grid cell, regardless of its renement ratio or local grid topol-ogy, therefore it is applicable not only to rened but also to com-pletely unstructured Cartesian grids. Moreover, in spite of theincreased accuracy the resulting expressions remain simple and ro-bust [17]. Finally, this method can be easily extended to 3-dimen-sions. An indicative picture of such a computational grid adaptivelyrened in two consecutive layers around the blades of a centrifugalimpeller is shown in Fig. 1. For a given accuracy of the results, itwas found that the above technique achieves considerably greaterFig. 1. (a) Indicative structure of the Cartesian grid, and (bpartly lled with the uid (Fig. 2) are solved using the same gen-eral equation of the following linearized form:

AP cVSP UP Xi

ciAiUi cVSU ;

AP Xi

ciAi; i E;W;N; S;U;D 3

where Ai are the coefcients linking the dependent variableUP withits neighbours on the adjacent grid volumes, and SU, SP are addi-tional source terms. The geometric coefcients ci and cv representthe free portion (not blocked by the solid boundary) of the cell-facesand volume, respectively.

The uid variables (velocities, pressures, etc) are computed atthe centroid of the Cartesian cells, which for a partly lled cell doesnot coincide with its geometric centre, as shown in Fig. 2. For thisreason, special stencils are introduced to compute the cell-face val-ues and the gradients of the ow variables, making a compromisebetween simplicity and accuracy, which results in cost-effectiverelations of almost second-order accuracy. The additional termsare included in the coefcients of the general equation (3), whereasall the needed geometric quantities are computed by a pre-pro-cessing algorithm. As a result, after dening the geometry of thecomputational domain, the grid construction process can be per-formed in a fast and fully automated way. Wall boundary condi-tions are also set automatically to every boundary cell (e.g. cellsP1P4 in Fig. 2).

The above partly-lled cells (PFC) method preserves the accu-racy of the boundary representation and retains the conservationproperty. Moreover, it does not affect the stability of the solutionalgorithm, while its simplicity makes it easily applicable to both2D and 3D complex geometries. Only the solvable cells are storedand take part in the computations, as shown in the example ofFig. 1.

2.3. Impeller head and power calculations

The net energy added to a unit mass of uid by the impeller canbe calculated after computing the total energy of the uid at theimpeller inlet and outlet (Fig. 3). Hence the uid head H12 isobtained from the ux-weighted relation:

H12 H2 H1 1QuZE

p2 p1q g

c22 c212g

dq 4) detailed view of the grid at the blade leading edge.

ersA

Fig. 2. Treatment of partly lled grid cells.

r 1 c 1 w 1

c 2

w 2

r 2

p r e s

s u r e

s i d e

s u c t i o

n s i d e

w r

w u

w

n

w

1 2

r 2

r 1

r 1

2

Fig. 3. Sketch of a centrifugal pump impeller.P1

P2

B

C

P3

P4D

E

solid

fluid free cellpartly filledblocked

286 J.S. Anagnostopoulos / Computwhere c is the absolute velocity of the uid, Qu the volume ow ratethrough the impeller and g the gravity acceleration, while thesubscripts 1 and 2 denote conditions at the impeller inlet and exitradius, respectively (Fig. 3). The integration is approximated by asummation over the radial ow rates dq at all grid cells facing theinlet or the outlet periphery of the impeller with cumulative areaE. On the other hand, the power absorbed from the impeller, Nu,can be calculated from the torque Mu developed on the blades:

Nu x Mu x Z r2r1

~r ~n p ~r ~sw cot b b dr 5

where ~n is the unit vector normal to the blade surface, ~sw the wallshear stress, b the blade angle and b the impeller width, whereasthe integration covers both the pressure side and the suction sideof the blade (Fig. 3). The impeller width b is a function of radius r,according to the real impeller geometry. Finally, the theoretical (Eu-ler) head of the impeller can be obtained as:

Hu Nuq g Qu6

2.4. Pump characteristic curves

Although the simulation is restricted to the impeller section, thenumerical results can be used as a basis to estimate the perfor-mance of the entire pump. To achieve this, the additional hydrauliclosses are properly expressed and abstracted from the head results,according to the following analysis (Fig. 3). The effect of the impel-ler shroud and hub surfaces can be computed using Darcys law,from the relation:

dhf 1QuZE

Z r2r1

kdrDh

w2r w2u2g

dr dq 7where k is the friction coefcient, wr and wu are the ow radial and(relative) tangential velocity components, and Dh 2b is thehydraulic diameter, with b(r) the impeller width.

At the exit of the real pump impeller the ow enters into thespiral volute and decelerates. The sudden expansion losses are pro-portional to the absolute exit velocity of the uid, namely:

dh1 k1QuZEc22 dq 8

Away from the design ow rate conditions additional losses ap-pear at the suction side, due to the impeller incidence and the inletpipe recirculation, as well as at the volute tongue. All these lossescan be included in an approximate expression of the type:

dh2 k2 1 QQ0

29

where Q0 is the design ow rate. The rest hydraulic losses in the in-let and outlet sections of the pump, as well as in the spiral volute,can be taken proportional to the square of the pump ow rate Q:

dh3 k3 Q2 10Using the above relations, the pump head H can be estimated by

abstracting from the computed uid head in the impeller H12 (Eq.(4)) all the above losses:

H H12 dhf dh1 dh2 dh3 11The mechanical losses at the shaft bearings, along with the disk

friction power, are taken into account introducing the mechanicalefciency factor, gm, hence the pump power N is:

N Nu=gm 12Also, the volumetric losses due to the leakage ow are expressed bythe volumetric efciency factor:

gq Q=Qu 13

and the overall pump efciency can be nally computed from therelation:

g q g H QN

HHu

gq gm 14

In the above expressions the values of the coefcients k1k3 andof the efciency factors gq and gm were regulated with the aid ofavailable experimental data of the examined impeller, as will bedescribed in chapter 3.2 below. These coefcients need to be cali-brated for every new impeller, and this constitutes a drawback ofthe methodology. However, there are several alternative ways toperform this calibration even without measurements, like by theuse of statistical data or of theoretical and empirical expressionsfor centrifugal pumps [19]. Also, a set of numerical results for theow in the actual impeller geometry, obtained by applying a moreaccurate but costly 3D solver, can be used instead of experiments.

2.5. Numerical optimization

In order to nd the combination of the impeller design variablesthat maximizes the objective value, an optimization algorithm isdeveloped based on the unconstrained gradient approach. Thismethod is selected after some preliminary numerical tests, whichshowed that the cost function (here the pump efciency) doesnot exhibit local maxima outside a global maximum region. How-ever, the pump efciency is not analytic function hence a problemof non-continuity and scattering arises. The algorithm is specially

& Fluids 38 (2009) 284289designed to operate even for such discrete data, using an adjust-able with trial-and-error step size along the gradient direction.Also, the gradients are computed using forward nite differences

at the beginning, and central differences when the cost functionapproaches maximum, and with a variable step size. The algorithmconverges very fast within the region where the cost function max-imizes, although due to scattering it cannot always nd the abso-lute maximum. However, the fast performance allows repeatingthe calculations from different starting values, in order to verifythe close approximation of the optimum.

3. Results

3.1. Accuracy and precision checks

The accuracy of the representation of the blade geometry withthe PFC method was tested at rst, along with the grid-dependency

width b2 = 9 mm, and inlet and exit angle b1 = 26, b2 = 49. Theblade shape is a simple circular arc of constant thickness 5 mm,and with both ends rounded (Fig. 4).

The measured operation characteristic curve H Q for thispump is depicted in Fig. 6a, along with the numerical results for

50 52 54 56H12 (m)

10.2

10.4

10.6

Nu (K

W

Coarse gridRefined grid

Fig. 5. Predicted uid head and corresponding impeller power values.

0 20 40 60 80 100 120 140Q (m3/h)

20

40

60

80

100

Hea

d (m

)

predictions - Hu predictions - H12 measured head H fitted head H

0 20 40 60 80 100 120Q (m3/h)

20

40

60

80

100

Pum

p eff

icien

cy,

(%

)

measurements predictions

Fig. 6. Measured and computed characteristic curves: (a) Fluid, impeller and pumphead; (b) pump overall efciency.

J.S. Anagnostopoulos / Computersof the ow eld results. These numerical test were performedusing the base-case geometry of Fig. 4, which is a periodically sym-metric section of the centrifugal impeller (z = 9 blades). Then, theimpeller is rotated step-by-step at small fractions (1/20) of the tan-gential resolution of the grid, producing a number of different gridtopologies around the blade boundary line. Consequently, the scat-tering in the corresponding results is due to the numerical errorintroduced by the PFC method.

The computed ow led drawn in Fig. 4 at the best efciencypoint (BEP) of the pump is similar for all the examined cases, how-ever the values for the uid head and the impeller power (Eqs. (4)and (5)) are not exactly the same. The latter are concentrated inFig. 5, for two grids: a coarse (8000 nodes) and a ne (27,000nodes) that has two rened layers around the blade, as in theexample of Fig. 1. The precision of the rened grid results is satis-factory, since the mean and maximum deviation from the meanvalues are of the order of 0.5% and 1.5%, respectively, for boththe Nu and the H12. These deviations are roughly one-fourth ofthe corresponding ones with the coarse grid (Fig. 5), conrmingthat the PFC method preserves the accuracy of the discretizationscheme.

On the other hand, the differences in the mean values betweenthe two grids represent the grid-dependency of the results, andthey are again of the order of 1%, which is an adequate accuracy.Consequently, the rened grid is selected to be used for the restcalculations.

3.2. Regulation of the model

The adjustable coefcients ki involved in the model Eqs. (7)(14)are regulated using the characteristic curves of a commercial cen-trifugal pump operating with a new impeller, constructed in theLaboratory. The impeller has nine two-dimensional (non-twisted)blades with inlet and exit diameter D1 = 70 mm, D2 = 190 mm, exitFig. 4. Pressure contours and ow streamlines in the standard impeller (b1 = 26,b2 = 49).10.8

11

)

& Fluids 38 (2009) 284289 287the uid head H12 and the impeller head Hu, obtained by Eqs. (4)and (6), respectively. The head H12 that the water acquires in theimpeller is, as expected, higher than the measured pump head H,

because the former does not include the additional losses in therest pump sections. On the other hand, it is less than the theoret-ical head Hu, due to the mechanical losses of the ow in theimpeller.

Next, the efciency coefcients gq and gm were determinedfrom statistical data, whereas using least-squares regression anal-ysis the values of the adjustable coefcients k1, k2 and k3 were reg-ulated so as the pump head H computed from Eq. (11)approximates well the corresponding experimental curve, asshown in Fig. 6a.

Finally, the overall efciency of the pump is calculated from Eq.(14) and the resulting characteristic curve is plotted in Fig. 6b.Although that curve is not produced by tting, the agreement withthe corresponding gQ measurements is very good, and this veri-

70.5% to about 73.5%). However, the BEP is shifted to smaller owrates compared to the standard design (52 m3/h compared to62 m3/h) because b1 is smaller, and the same is valid for the max-imum head. The latter is now reasonably higher because of the lar-ger exit angle b2 = 54 (standard b2 = 49).

60

64

68

72

76

Pum

p ef

ficie

ncy

(%

)

Starting values1 (deg), 2 (deg), Q (m3/h)

35, 45, 45 35, 60, 60

288 J.S. Anagnostopoulos / Computers & Fluids 38 (2009) 284289es the consistency of the followed modelling strategy and thevalidity of the adjusted coefcients for this particular pump. More-over some 3D ow effects, which become even stronger at off-de-sign operation of the pump, are also included in the values of theabove coefcients.

3.3. Optimal blade design

The objective here is to maximize the best efciency value ofthe pump, using as free design variables the inlet and the exit bladeangles. However, the exact location of the best efciency point(BEP) of the pump depends on the blade design therefore it hasto be determined for each examined set of blade angles. This wouldneed the construction of the gQ characteristic curve of the pump,by computing several points on it after corresponding runs of theevaluation algorithm. In order to accelerate the computations analternative and much faster technique is implemented, accordingto which the unknown volume ow rate at the BEP is treated asan additional free design variable, and its value is obtained afterthe convergence of the optimization procedure.

During the optimization the evaluation algorithm is capable togenerate the grid and solve the ow equations for a wide rangeof different blade congurations without user interference. Twoextreme geometry examples are plotted in Fig. 7, along with thecorresponding ow eld results. The use of large blade angles inFig. 7a results in a short and almost straight blade, which howevercannot lead properly the ow and a large recirculation zone isformed at the pressure side in agreement with the theory, sincethe volume ow rate is much smaller than the optimal forb1 = 35. On the other hand, the blade with small angles inFig. 7b seems to performmore smoothly, however the ow passageis much longer and narrower and consequently the mechanicallosses are increased.

The convergence rate of the optimization algorithm describedpreviously in chapter 2.5 is shown in Fig. 8, for two different setsFig. 7. Computed ow eld for different blade shapof starting values. Although the latter are selected far from theoptimal region, the algorithm reaches there in less than 20 evalu-ations, whereas nal convergence occurs in about 6080 evalua-tions. The whole procedure takes about 1520 h in a P4 PC.

The corresponding variation of the blade angles during optimi-zation is depicted in Fig. 9, where contour lines of constant ef-ciency are also plotted, computed for the optimal ow rates. Thetwo paths converge to the same region, although not exactly tothe same point, due to non-continuity effects. The optimal inletblade angle is about 21, which is more consistent with the presentrotation speed (3000 rpm) and the impeller inlet diameter (the reallaboratory blade had been constructed with an inlet angle of 26, inorder to operate the pump effectively as turbine too). On the otherhand, the exit blade angle exhibits a wider optimal region, rangingbetween 50 and 58, which is in agreement with theoretical andstatistical data. For example, the classic blade number selectionrelation of Peiderer [20]:

z 6:5 D2 D1D2 D1 sin

b1 b22

15

for b2 = 54 gives z = 8.6? 9 blades.The calculated characteristic curves of the pump with the opti-

mized impeller are drawn in Fig. 10 along with the correspondinginitial blade curves (shown also in Fig. 6). The maximum efciencyof the pump with the optimal blade shape is about 3% higher (from

0 20 40 60 80Evaluations

Fig. 8. Convergence history of the optimization algorithm.es: (a) b1 = 35, b2 = 72; (b) b1 = 21, b2 = 22.

and thus to accelerate the numerical design process. Although thesuitability of the method for high specic speed pumps needs to beevaluated, its accuracy can be enhanced by introducing additionaladjustable coefcients to account for the 3D effects.

60de

g)A

J.S. Anagnostopoulos / Computers & Fluids 38 (2009) 284289 28915 20 25 30 35Inlet blade angle (deg)

40

45

50

55

Exit

blad

e ang

le (

B

Fig. 9. Variation of the design variables during optimization.

60

70

80

(%

)

4. Conclusions

A numerical methodology for the calculation of the ow eld ina centrifugal pump impeller and the prediction of the pump perfor-mance curves is developed, regulated, and tested against experi-mental and statistical data, with encouraging results. Theproposed ow simulation algorithm is suitable for hydrodynamicdesign in hydraulic pumps and turbines, thanks to the fast andfully automated generation of the Cartesian grid and the increasedprecision in representing irregular boundaries. The main advantagevalidated in the present study is that the methodology provides theability to localize the optimal range of the free design variables atlow computer cost.

A more elaborate regulation and identication of the optimaldesign would require the simulation of the ow in the 3D impel-ler/runner geometry as well as in the inlet section and the casingof the machine, by performing costly numerical solutions of Na-vierStokes equations in three dimensions. Consequently, themuch faster modelling approach proposed here can be used as astarting optimization tool in order to locate the region of maxima,

design optimization of a Tesla-type valve for micropumps. IASME Trans

0 20 40 60 80 100 120Q (m3/h)

20

30

40

50

H (m

),

Initial blade design Optimal blade design

H

Fig. 10. Pump characteristic curves with the standard and the optimal impeller.2005;2(6):184652.[17] Anagnostopoulos J. Discretization of transport equations on 2D Cartesian

unstructured grids using data from remote cells for the convection terms. Intl JNum Meth Fluids 2003;42:297321.

[18] Anagnostopoulos J, Mathioulakis D. A ow study around a time-dependent3-D asymmetric constriction. J Fluids Struct 2004;19:4962.

[19] Neumann B. The interaction between geometry and performance of acentrifugal pump. London: Mechanical Engineering Publications; 1991.

[20] Peiderer C. Die Kreiselpumpen. 5th ed. Berlin: Springer-Verlag; 1961.Acknowledgements

The project is co-funded by the European Social Fund (75%) andNational Resources (25%) Operational Program for Educationaland Vocational Training II and particularly the ProgramPythagoras.

References

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[2] Xu JZ, Gu CW. A numerical procedure of three-dimensional design problem inturbomachinery. ASME Trans J Turbomach 1992;114:54882.

[3] Borges JE. A three-dimensional inverse method for turbomachinery: Part-1Theory. ASME Trans J Turbomach 1994;112:34654.

[4] Peng G, Cao S, Ishizuka M, Hayama S. Design optimization of axial owhydraulic turbine runner: Part I an improved Q3D inverse method. Intl J NumMeth Fluids 2002;39:51731.

[5] Benra FK. Economic development of efcient centrifugal pump impellers bynumerical methods. World Pumps 2001(May):4853.

[6] Yulin W, Jianming Y, Shuliang C. Advanced design for large hydraulic turbinerunner. In: ASME/JSME FEDSM 99, San Francisco, California, Paper S-287, July1823, 1999.

[7] Garrison LA, Richard KF, Sale MJ, Cada G. Application of biological designcriteria and computational uid dynamics to investigate sh survival in Kaplanturbines. In: HydroVision 2002, Portland, Oregon, July 29August 2, 2002.

[8] Cao S, Peng G, Yu Z. Hydrodynamic design of rotodynamic pump impeller formultiphase pumping by combined approach of inverse design and CFDanalysis. ASME Trans J Fluids Engin 2005;127:3308.

[9] Asuaje M, Bakir F, Kouidri S, Rey R. Inverse design method for centrifugalimpellers and comparison with numerical simulation tools. Int J Comput FluidDynamics 2004;18(2):10110.

[10] Gehrer A, Schmidl R, Sadnik D. Kaplan turbine runner optimization bynumerical ow simulation (CFD) and an evolutionary algorithm. In: 23rdIAHR symposium on hydraulic machinery and systems, Yokohama, Japan.Paper F125; 2006.

[11] Miyauchi S, Kasai N, Fukutomi J. Optimization of meridional shape design ofpump impeller. In: 23rd IAHR symposium on hydraulic machinery andsystems, Yokohama, Japan. Paper F310; 2006.

[12] Mazzouji F, Couston M, Ferrando L, Garcia F, Debeissat F. Multicriteriaoptimization: viscous uid analysis mechanical analysis. In: 22nd IAHRsymposium on hydraulic machinery and systems, Stockholm, Sweden, June29 July 2. Paper A04-1; 2004.

[13] Kueny JL, Lestriez R, Helali A, Dmeulenaere A, Hirsch C. Optimal design of asmall hydraulic turbine. In: 22nd IAHR symposium on hydraulic machineryand systems, Stockholm, Sweden, Paper A02-2, 2004.

[14] Ye T, Mittal R, Udaykumar HS, Shyy W. An accurate Cartesian grid method forviscous incompressible ows with complex immersed boundaries. J ComputPhys 1999;156:20940.

[15] Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary nite-difference methods for three-dimensional complex owsimulations. J Comput Phys 2000;161:3560.

[16] Anagnostopoulos J, Mathioulakis D. Numerical simulation and hydrodynamic

A fast numerical method for flow analysis and blade design in centrifugal pump impellersIntroductionThe numerical methodologyNumerical gridGeometry representationImpeller head and power calculationsPump characteristic curvesNumerical optimization

ResultsAccuracy and precision checksRegulation of the modelOptimal blade design

ConclusionsAcknowledgementAcknowledgementsReferences