fe_136_01_014501

Multiobjective Optimization Designof a PumpTurbine Impeller Basedon an Inverse Design Using aCombination Optimization Strategy

Wei Yange-mail: [email protected]

Ruofu Xiaoe-mail: [email protected]

College of Water Resources and Civil Engineering,

China Agricultural University,

Beijing 100083, China

This paper presents an automatic multiobjective hydrodynamicoptimization strategy for pumpturbine impellers. In the strategy,the blade shape is parameterized based on the blade loading dis-tribution using an inverse design method. An efficient responsesurface model relating the design parameters and the objectivefunctions is obtained. Then, a multiobjective evolutionary algo-rithm is applied to the response surface functions to find a Paretofront for the final trade-off selection. The optimization strategywas used to redesign a scaled pumpturbine. Model tests wereconducted to validate the final design and confirm the validity ofthe design strategy. [DOI: 10.1115/1.4025454]

Keywords: pumpturbine, inverse design method, CFD,optimization

1 Introduction

Pumping storage plants play an important role in power grids.When the grid has high load, the plant can generate power as aturbine. When the load is low, the plant can use the redundant gridpower to pump water up to a reservoir. Then, the stored water canbe used to generate power when needed. Such systems needpumpturbines to work reliably in a range of operating conditions.Fast changes of the discharge rate and flow direction requireadjusting the variable diffuser/guide vanes, which leads to com-plex three-dimensional (3D) flows [13] and fluidstructure inter-actions [46] in the pumpturbine system. As a result, the designprocess must take into account both the pump and turbine per-formance. The pump efficiency and the turbine efficiency bothhave to be improved. In addition, stability limits in both operatingmodes have to be shifted so that the overall operating range canbe extended with reasonable cavitation performance. This is a realchallenge for designers because the design targets for the twooperations influence each other and are sometimes contradictory.

Most pumpturbine designs deal with the impeller geometricparameters [7]. The impeller geometry is changed to improve theperformance. Appropriate impeller geometry is based on the rela-tionships between the geometric parameters and the impeller per-formance. These relationships can be found using existing designknow-how or using computational fluid dynamics (CFD) tools toevaluate the performance after changes in the geometry, which isa time-consuming job especially for pumpturbines. Automaticdesign optimization based on geometric parameterization of theblade shape has been used in turbomachinery designs by couplingan optimization method, CAD based blade generators, and a CFD

code [8]. However, the method is less practical for multiobjectiveand multipoint tasks such as pumpturbine designs [9], whichrequire a very large number of simulations. The simulations arerelated to the large number of geometric parameters necessary toaccurately represent the blade geometry. Also there is no directrelationship between the geometric design parameters and thehydrodynamic performance.

Design experience plays an important role in current pumptur-bine designs. Accumulated design experience is used to reducethe number of simulations and make the time for the whole opti-mization process compatible with industrial standards. However,the major drawbacks of this design strategy are that the designresult depends on talented designers with rich design experienceand this method does not easily produce better pumpturbine con-figurations than existing designs. These drawbacks are related tothe parametric description of the blade, which is conventionallyperformed using only geometric parameters.

A good solution to this problem is to use a blade parameteriza-tion based on an inverse design method [1013]. Inverse designmethods have been widely used for the design of various kinds ofturbomachines [1416], proving that it is a valuable alternative tothe iterative use of direct methods. One main design parameter inthe inverse design approach is the blade loading on both the huband the shroud along the meridional direction. The blade loadingdistributions have a more direct relationship to the hydrodynamicperformance because they influence the hydrodynamic flow fieldin a more straightforward way. Fewer design parameters are thenrequired to describe the blade shape than a purely geometricexpression of the blade. Therefore, an optimization design methodusing the inverse method to parameterize the blade geometry canreduce the overall optimization time. The optimization design pro-cess then gives the optimal blade loading distributions, instead ofthe optimal combination of the geometric parameters. This is amore general result which can be applied to similar design prob-lems without repeating the optimization process. A good examplewas given by Bonaiuti and Zangeneh [10]. They applied the opti-mization design strategy to the design of a centrifugal compressorstage and a single stage axial compressor and validated thestrategy.

However, direct application of this design strategy to pumptur-bines may still result in high computational costs since the CFDcalculations are necessary for both the pump and turbine opera-tions. Thus, this analysis used inverse method to parameterize theblade and generated the impeller database for CFD analyses. Adesign-of-experiment (DOE) method was used to determine thetest sample points. Based on CFD results a response surface relat-ing design parameters and objective functions was built for finalmultiobjective optimization.

2 PumpTurbine Design Strategy

2.1 Optimization Process. An optimization pumpturbinedesign strategy was developed using a three-dimensional (3D)inverse design method, CFD analyses, and a multiobjectivegenetic algorithm (MGA) [17]. The 3D inverse design methodwas used for the blade parameterization. The CFD analyses wereused to evaluate the pumpturbine performance. A response sur-face methodology (RSM) [18] then coupled with an orthogonalDOE technique was used to generate a function relating the objec-tive functions and the design parameters. Second-order polyno-mials were used for the RSM models:

yj aj0 Xni0

ajixi Xni 6k

aji;kxixk Xnik

aji;kxixk (1)

The orthogonal DOE technique was used to determine a tableof design configurations for the RSM model. The MGA algorithmwas applied to the approximated response functions to determinethe optimal set of design configurations (Pareto front). Then, the

Contributed by the Fluids Engineering Division of ASME for publication in theJOURNAL OF FLUIDS ENGINEERING. Manuscript received May 13, 2013; final manuscriptreceived September 3, 2013; published online October 15, 2013. Assoc. Editor:Frank C. Visser.

Journal of Fluids Engineering JANUARY 2014, Vol. 136 / 014501-1CopyrightVC 2014 by ASME

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design solution was found as a trade-off between the varioushydrodynamic performance parameters.

After trade-off selection a CFD calculation was performed tovalidate the optimized design based on the RSM model. If theCFD agreed with the RSM model then a final optimal configura-tion was achieved. Otherwise the CFD results were added to thedatabase to update the RSM model until the final design perform-ances estimated from the RSM model agreed with the CFDresults. The pumpturbine design strategy process is illustrated inFig. 1.

For the same design head and revolution speed the pump diam-eter is larger than the turbines based on the pump and turbinedesign theory and experience. For both pump and turbine it is eas-ier to meet power and efficiency demands with larger dimension.So the pumpturbine design process usually starts from the pumpmode. In this way the impeller will be larger than normal for theturbine mode and it is easier to meet turbine design requirements.Here the primary dimensional parameters of the pumpturbineimpeller were first defined based on the pump. The meridionalshape of the impeller was then determined using a one-dimensional (1D) analysis with the shape kept fixed during theoptimization process. The primary impeller geometry was calcu-lated using a 3D inverse design method for the pump operation.

2.2 Multiobjective Genetic Algorithm. For pumpturbineoptimization finding a set of optimal trade-offs called Pareto frontbetween various hydraulic performances is concerning. So a mul-tiobjective genetic algorithm was used here. A general multiobjec-tive optimization problem can be described as a vector function fthat maps a tuple of parameters (decision variables) to a tuple of nobjectives. Formally,

min=max y f x f1x; f2x; :::; fnx

subject tox x1; x2; :::; xm 2 Xy y1; y2; :::; yn 2 Y

Here X is blade loading parameter space and Y represents the hy-draulic efficiencies of the pumpturbine impeller.

The set of solutions of a multiobjective optimization problemconsists of all decision vectors for which the corresponding objec-tive vectors cannot be improved in any dimension without degra-dation in another. These objective vectors are known as Paretooptimal. The niched Pareto genetic algorithm combines tourna-ment selection and the concept of Pareto dominance. Two com-peting individuals and a comparison set of other individuals arepicked at random from the population. If one of the competingindividuals is dominated by any member of the set and the other isnot, then the latter is chosen as winner of the tournament. If bothindividuals are dominated (or not dominated), the result of thetournament is decided by sharing: The individual that has the leastindividuals in its niche is selected for reproduction.

2.3 Blade Parameterization. The RSM model reliabilitydepends on the number of parameters and the physical relation-ships between the objective functions and the design parameters.The RSM model is less reliable when dealing with too manydesign parameters such as geometric parameters which may haveno direct influence on the performance. The problem can besolved for the blade by using an inverse design method to parame-terize the blade geometry. During the inverse design process, theblade geometry can be represented by the blade loading distribu-tion (meridional derivative of the circulation), which has less pa-rameters and more direct influence on the impeller performance.

An incompressible 3D inverse design method based on Borges[16] was used. The input design parameters required by themethod are as follows:

Fluid properties and design specifications. Fluid density,revolution speed, number of blades, and discharge rate.

Meridional channel shape. The hub, shroud, trailing edge,and leading edge contours of the impeller.

Normal blade thickness distribution. Spanwise distribution of the circulation at the inlet and out-

let. The circulation difference between the inlet and the out-

let determine the Euler work of the impeller. Blade loading distributions at the hub and the shroud. Stacking condition imposed on the high pressure side of the

impeller. This was at the leading edge for turbine operation

and the trailing edge for pump operation.

The blade loading distribution and the stacking condition wereused to parameterize the blade geometry. The other input parame-ters were held constant during the optimization design process.The trailing edge leaning angle was used as the stacking conditionfor the pump operation since it plays an important role in thesuppression of secondary flows in centrifugal and mixed flowimpellers [19].

2.4 CFD Analyses. A three-dimensional, turbulent, andsteady flow simulation was used for CFD analyses. In the pumpturbine optimization process, as shown in Fig. 1, CFD plays twoFig. 1 Pumpturbine design process

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roles. One role is to calculate the objective functions which arehydraulic efficiencies here. The other role is to validate the opti-mized results and to add points to the database for updating theRSM model. The objective function calculations used a simplifiedsingle passage model with periodic boundary conditions on theimpeller to reduce the calculation time. The validation simulationsused a full passage model including the case, guide vane, impeller,and draft tube to evaluate the overall performance.

The commercial software ANSYS was used to conduct theCFD analysis. ANSYS BladeGen was used for the 3D flow pas-sage generation with the flow passage then imported into theANSYS ICEM CFD for grid generation. The mesh was thenimported into ANSYS CFX 13.0 for the flow solution. In ANSYS13.0, all these steps can be executed automatically. All the calcu-lation was conducted on a workstation with two CPUs of IntelXeon E5606 with 2.13 GHz, 32GB memory and 1TB hard drive.

The space discretization was based on a cell-centered finite vol-ume scheme with the system of governing equations advanced intime using the explicit second order scheme. The shear stresstransport (SST) k-x model, which has been widely validated fornumerical analysis of pumpturbines [2022], was used for theturbulence closure. A frozen rotor model was used for the presentdomain including both stationary and rotation parts.

3 Design Example of a Scaled PumpTurbine

A 1:9 scaled pumpturbine was redesigned as a test case. Thepump and turbine design specifications are shown in Table 1. Themeridional channel shape was designed based on the pump operat-ing modes according to the design conditions based on a 1D flowanalysis commonly used in the conventional design process. Thefinal design configurations, which remained unchanged during theoptimization loop, were an impeller high pressure side diameterD1 515.4 mm, impeller high pressure side exit widthb 57.2 mm, impeller low pressure side shroud diameterD2 300 mm, and impeller low pressure side hub diameterDh 156.8 mm as given in Fig. 2.

3.1 Design Parameters. The meridional channel was notchanged during the optimization process. Then, the impeller wasparameterized through the blade parameterization. In the 3Dinverse design method, the blade shape is determined according tothe prescribed blade loading distribution, which is proportional tothe meridional derivative of the circulation. For incompressible

flow, the meridional derivative of the circulation is related to theblade loading as

p p 2pZqWbl

@rVh@m

(2)

Here m 0 means at the leading edge and m 1 means at the trail-ing edge. The blade pressure loading can be modified by adjustingthe meridional derivative of the circulation @rVh=@m. The bladeloading distribution parameters were used as the design parame-ters. A typical three-segment blade loading distribution for boththe hub and shroud is shown in Fig. 3, where em m=mtotal is thenormalized meridional distance and @rh=@ em is the normalizedblade loading with eVh Vh=U. There are eight parameters (hh,em1h, kh, em2h, hs, em1s ks, em2s) for the blade loading distributions onboth the hub and the shroud.

In Fig. 3 the first parabolic curve is the meridional derivative of

r eVh a1 em5 b1 em4 c1 em3 d1 em2 e1 em f1 (3)subject to

em 0; rfVh Ci; @rfVh=@ em h; @2rfVh=@ em2 0em em1; rfVh C1; @rfVh=@ em B1; @2rfVh=@ em2 k

where a1, b1, c1, d1, e1, f1 are the undetermined coefficients.The second linear curve is the meridional derivative of

r eVh a2 em2 b2 em c2 (4)subject to

em em1; rfVh C1; @rfVh=@ em B1; @2rfVh=@ em2 kwhere a2, b2, c2 are the undetermined coefficients.

The third parabolic curve is the meridional derivative of

r eVh a3 em4 b3 em3 c3 em2 d3 em e3 (5)subject to

em em2; rfVh C2; @rfVh=@ em B2; @2rfVh=@ em2 kem 1; rfVh 0; @2rfVh=@ em2 0

where a3, b3, c3, d3, e3 are the undetermined coefficients.

Table 1 Design parameters of pumpturbine

Parameters N (r/min) Q (m3/s) H (m) ns gd D1 (mm) D2 (mm) Dh (mm) B (mm)

Pump 1200 0.402 55.8 136 90.6% 515.4 300 156.8 57.2Turbine 1200 0.456 63.9 91.7% 515.4 300 156.8 57.2

Fig. 2 Sketch of the impeller meridional channel shape Fig. 3 Typical blade loading parameterization

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For both the hub and shroud:

h was varied from 0 to 1. em1was varied from 0.05 to 0.45. em2was varied from 0.55 to 0.95. k was varied from 4 to 4.

Zangeneh [19] pointed out that the stacking condition can beused to control secondary flows in the impeller region. In addition

to the eight blade loading parameters, the stacking condition onthe trailing edge at DD1 in pump mode was also selected as theninth design parameter as shown in Fig. 4, where b is the rakeangle and here c is our design parameter called the lean angle.The relationship between b and c can be determined asc 2b tan b=D1 by geometric consideration shown in Fig. 4. Alinear stacking was imposed on the high pressure edge of theblades. The slope c was varied from 10 to 10 deg (hub preced-ing the shroud in the rotational direction means positive stacking).

All design variables are subject to being exchanged independ-ently at a probability of 50%. The mutation operator produces ran-dom disturbances to the design variable in the amount of 60.05for parameter em1 and em2,6 0.5 for parameter k,6 0.1 for parame-ter h, and 61 for parameter c. The probability of mutation is ini-tially 15% and it decreases linearly to 1% over 150 generations.

3.2 Objective Functions. A pumpturbine has to work asboth a pump and a turbine, which makes the design job a multiob-jective and multipoint task. This study used four performance pa-rameters for the pump turbine system for both pump and turbineoperations as the objective functions:

Fig. 4 Definition of lean angle c

Fig. 5 Pumpturbine optimization work flow chart

Fig. 6 Single impeller passage model for CFD calculations



1. Pump mode hydraulic efficiency gp for the pump designmass flow rate.

2. Pump mode hydraulic efficiency gp80 for the 80% pumpdesign mass flow rate to influence the hump performance forthe pump mode.

3. Turbine mode hydraulic efficiency gt for the turbine designmass flow rate.

4. Turbine mode hydraulic efficiency gt80 for 80% of the tur-bine design mass flow rate.

3.3 Optimization Procedure. The optimization procedurewas carried out in an automatic way by integrating all the codes

and software into the Isight platform together. The optimizationwork flow was shown in Fig. 5. The work flow is a softwareimplementation of the design process shown in Fig. 1. In the workflow: Isight software was used for DOE database generation,MGA searching, and platform establishment. MATLAB codes wereused for the inverse design method and the RSM model genera-tion. ANSYS products were used for geometry generation, gridgeneration, and CFD analyses.

Fig. 7 Whole machine passage model for CFD calculations

Fig. 8 Pareto front for the optimization results

Fig. 9 Comparison of blade loading distributions for the base-line and optimized designs

Fig. 10 Pump hydraulic efficiency simulation results for thebaseline and optimized designs

Fig. 11 Turbine hydraulic efficiency simulation results at thedesign head Hd for the baseline and optimized designs



Since four energy performance parameters were used as theobjective functions, four CFD calculations were used for eachdesign configuration. The calculations were reduced by using anorthogonal DOE model to determine a table of design configura-tions. There was a total of nine parameters with two levels usedfor each parameter which gave an orthogonal DOE table L64(29)with 64 tests in total.

The inverse design method was then used to generate the 64different impeller geometries for the CFD evaluations. If theinverse design computation did not converge, an actuator duct(AD) design that assumes axisymmetric flow (infinite number ofblades) was used instead. At least 256 times CFD calculationswere needed for the whole optimization process. The calculationtimes were reduced by simplifying the simulation domain to con-tain only the impeller and a single blade passage with periodicboundary conditions as shown in Fig. 6. All the configurationswere analyzed using an H-type grid with 131 grid points in thestreamwise direction, 38 in the pitchwise direction, and 40 in thespanwise direction. O-type grid clustering was imposed close tothe blade/walls to have a Y 1. An example of the computa-tional grid is shown in Fig. 6. After optimization, a whole machinepassage CFD model, as shown in Fig. 7, was used to validate thefinal design configuration.

For single domain simulation the boundaries are shown in Fig.6. For whole machine passage simulation, as shown in Fig. 7, theinlet is located on the spiral case flange and the outlet is locatedon the draft tube flange. For both the turbine and pump operatingmodes, the boundary conditions were imposed on the solid walls,on the periodic boundaries, at the inlet, and at the outlet of thecomputational domain as shown in Figs. 6 and 7. In single domain

simulation: for pump inlet the mass flow rate of 401 kg/s wasgiven and the flow angle was normal to the inlet boundary. Forturbine inlet three velocity components in the cylindrical coordi-nate were given. The axial velocity was assumed to be zero. Theradial velocity was calculated from the flow rate and was given as4.9 m/s. The circumferential velocity was determined from theEuler equation by assuming zero velocity circulation at the outletand was given as 16.4 m/s. In whole machine passage simula-tion: the boundary conditions for pump mode were same as thesingle domain simulation. For turbine mode total pressure of622,000 Pa computed from the working head was imposed at thespiral case inlet. A static pressure of 1 atm was imposed at thedraft tube outlet.

3.4 Optimization Results. The CFD simulations were usedto generate four RSM functions relating the objective functionsand the design parameters. They all had high values of R [2](above 99%) and R2a (above 96%); thus, confirming the validity ofthe RSM models. The RSM hydraulic efficiency curve for thedesign mass flow rate coincided with the curve for the 80% designmass flow rate for both the pump and turbine operations, whichmeans the two kinds of efficiencies are positive correlated. Theefficiencies for the pump and the turbine, however, were competingobjective functions here. The efficiencies for the 80% mass flowrate gp80 and gt80 were then used as the objectives for postprocess-ing. Therefore, the resulting Pareto front consisted of configurationsmaximizing gp80 and gt80 or a compromise between the two.

Different choices on the Pareto front would have led to differ-ent optimized configurations. Figure 8 shows the Pareto front of

Fig. 12 3D velocity streamlines in the blade passage for the pump mode at thedesign point

Fig. 13 Velocity vectors at the 50% spanwise view for pump mode at the designpoint



the multiobjective optimization for a single impeller passagemodel based on the final RSM models. The axes of Fig. 8 repre-sent the percent variation of the 80% mass flow rate efficiencieswith respect to a baseline configuration. The baseline configura-tion was also designed by the inverse method with an initial bladeloading distribution as shown in Fig. 9 and zero stacking condi-tion. The efficiency of pump mode at design condition is 90.2%,which did not meet the design efficiency 90.6%. And the effi-ciency of turbine mode at the design condition is 92.5%. The

main problems of the baseline design are the hump performancecurve as shown in Fig. 17 and a lower efficiency at the pumpmode. Any configurations on the Pareto front can be used for dif-ferent design objectives. The analysis considered impeller effi-ciencies for both pump and turbine mode with the chosenconfiguration indicated in Fig. 8. The chosen configuration had a0.24% higher turbine impeller efficiency and a 1.18% higherpump impeller efficiency based on the final RSM functions whichwere validated by CFD calculations.

The optimized values of the nine parameters (hh, em1h, kh, em2h,hs, em1s, ks, em2s, c) equal (0.11, 0.16, 0.56, 0.81, 0.18, 0.26, 0.22,0.64, 3.49 deg). The blade loading distributions for the baselineand final designs are compared in Fig. 9. The optimized bladeloading distributions are after-loaded on both the hub and theshroud. The maximum loading difference between the hub and theshroud, however, are still on the fore part of the impeller which isgood for controlling the secondary flow [19]. The optimizationresults were confirmed for the whole passage flow simulation ofthe pumpturbine for both the baseline and the optimized configu-rations. The simulation hydraulic efficiency results of both pumpand turbine modes for the baseline and final designs are shown inFigs. 10 and 11. After optimization, the pumpturbine efficienciesfor both the pump and turbine modes were improved for all thesimulation points and verified the optimization method.

The final design has better flow in the impeller than the baselineconfiguration especially for the pump mode. The 3D velocitystreamlines in the baseline blade passage shown in Fig. 12 indi-cate an obvious secondary flow near the shroud corner, which wascompletely eliminated in the final design. There was cross flowfrom the suction side to the pressure side near the pressure side inthe baseline design as shown in Fig. 13, which was also

Fig. 14 Three-dimensional velocity streamlines in the bladepassage for turbine mode at the design point

Fig. 15 Velocity vectors at the 50% spanwise view for turbinemode at the design point

Fig. 16 Model test rig for pumpturbine

Fig. 17 Measured Q-H and Q-g curves of the pump mode forthe baseline and optimized designs



eliminated in the final configuration. The averaged blade angle atthe leading edge increased by 2.38 deg from baseline to final geom-etry. There were no obvious flow improvements in the final designfor the turbine operating mode. No secondary flow or cross flowwere found in the impeller passage as shown in Figs. 14 and 15.

4 Model Test Results

The model tests were performed on a stand hydraulic machin-ery test rig which has about 60:2% of composition error for effi-ciency measurement. Model tests of both the baselineconfiguration and the final design, as shown in Fig. 16, were usedto verify the optimization results. The measured pump efficiencycurves shown in Fig. 17 for various flow rates indicate that theoptimized impeller has higher efficiencies at all the operatingpoints. These results are consistent with the CFD results. Themeasured head curves in Fig. 17 show that the final design hasbetter performance for small flow rates and the unstable humpperformance curve was improved after optimization.

The turbine efficiency curves at the design head Hd 63.9 m inFig. 18 show that the optimized design had higher efficiencies.This is consistent with the optimization objectives gt (turbine effi-ciency at the design point) and gt80 (turbine efficiency at 80%design mass flow rate). For pumpturbine design the best efficien-cies for pump operation are usually at lower heads than for turbine

operation. And the pump mode operates at higher heads than theturbine mode. Combining these two aspects, the turbine operationof a pumpturbine system is usually operated far from its effi-ciency optimum. Figure 19 shows the turbine efficiency curves forthe rated head for turbine operation Hr 83.2 m, the optimizeddesign has a better performance at low flow rate and an almostsame performance at high flow rate. Thus, both the CFD and thetest show that the turbine performance at the design point and lowflow rate of the rated point is improved.

5 Conclusions

A multiobjective optimization strategy based on a 3D inversedesign method and CFD, RSM, DOE, and MGA methods wasdeveloped for pumpturbine designs. The design started from thepump operation. During the process the meridional geometry wasfixed and the design focused on the blade loading distributionsand the blade trailing edge lean angle (stacking condition) atpump mode, which more directly influence the impeller perform-ance. The hydraulic efficiencies for both pump and turbine opera-tions at design and off-design points were chosen as the designobjectives. An orthogonal DOE technique was used to decidewhich design parameter combinations were investigated. CFDtools were used to evaluate the objective values for differentdesign configurations with a single impeller passage model usedfor the optimization process and a full machine passage modelused for validation after the optimization. The CFD results wereused to generate RSM functions relating the design parametersand the objectives. Then, a Pareto front was achieved to choose aconfiguration as the final design for the various design demands.Both CFD simulations and the model test were conducted to con-firm the final design and validate the optimization results.

The use of the inverse-based blade parameterization with fewerdesign parameters than conventional optimization strategiesreduces the complexity of the RSM correlations. Use of the DOEtechnique to define the CFD models to limit the number of config-urations substantially reduced the computational costs. Only fourobjective functions were used here but more pumpturbine param-eters could be used in this optimization strategy by selecting moreobjectives from the CFD computations.

Although the case presented in this paper only analyzes thehydrodynamic performance, the method is suitable for multidisci-plinary optimizations, where stress and vibration analyses can becoupled with the hydrodynamic analyses. The strengths of the 3Dinverse design method coupled with CFD analyses can be usednot only with pumpturbine designs but also for optimization ofall kinds of turbomachinery.

Acknowledgment

This work was supported by the National Science Foundationof China (No. 51209206).

Nomenclature

b impeller blade exit width for pump modeB1 blade loading at meridional position em1B2 blade loading at meridional position em2C1 velocity circulation at meridional position em1C2 velocity circulation at meridional position em2Ci inlet velocity circulationD1 impeller blade trailing edge diameter for pump modeD2 impeller blade leading edge diameter at shroud for pump

modeDh impeller blade leading edge diameter at hub for pump

modeH water head

Hd design head for turbine modeHr rated head for turbine mode

k slope of the linear segment

Fig. 18 Measured Q-g curves of the turbine mode for the base-line and optimized designs at the design head Hd

Fig. 19 Measured Q-g curves of the turbine mode for the base-line and optimized designs at the rated head Hr



m meridional distanceem normalized meridional distanceem1 intersection between the first parabolic section and the

linear segmentem2 intersection between the linear segment and the second

parabolic sectionmtotal total length of meridional streamline

n impeller rotational speed in rpmns specific speed

p pressure on pressure side of bladesp pressure on suction side of bladesQ volumetric flow rater radial coordinate of impeller

R2 ratio of the regression sum of squares to the total sum ofsquares

R2a R2 adjusted to the number of parameters in the RSMmodel

U peripheral speed at r 0.5D1Vh circumferential average of tangential velocityeVh normalized circumferential average of tangential velocity

Wbl relative velocity at blade surfacex decision vectorxi design parametersX parameter spacey objective vectoryj performance parametersY objective spaceZ number of blades

Greek Letters

aji polynomial parameters of RSM modelb rake anglec lean angle of trailing edge at pump modeh blade loading at leading edgeg efficiencygd target efficiency at design conditiongp hydraulic efficiency of pump modegt hydraulic efficiency of turbine modeq water densityx impeller rotational speed in rad/s

Subscripts

80 for 80% design mass flow rateh for hubs for shroud

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