Research Article Optimization for Cavitation Inception Performance of Pump …downloads.hindawi.com/journals/mpe/2014/234615.pdf · 2015-11-22 · Research Article Optimization for

Research ArticleOptimization for Cavitation Inception Performance ofPump-Turbine in Pump Mode Based on Genetic Algorithm

Ran Tao1 Ruofu Xiao1 Wei Yang1 Fujun Wang1 and Weichao Liu2

1 College of Water Resources and Civil Engineering China Agricultural University Beijing 100083 China2Dongfang Electric Machinery Co Ltd Deyang Sichuan 618000 China

Correspondence should be addressed to Ruofu Xiao xrfcaueducn

Received 30 June 2014 Revised 26 August 2014 Accepted 27 August 2014 Published 25 September 2014

Academic Editor Jyh-Hong Chou

Copyright copy 2014 Ran Tao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Cavitation is a negative factor of hydraulic machinery because of its undesirable effects on the operation stability and safetyFor reversible pump-turbines the improvement of cavitation inception performance in pump mode is very important due tothe strict requirements The geometry of blade leading edge is crucial for the local flow separation which affects the scale andposition of pressure drop Hence the optimization of leading edge shape is helpful for the improvement of cavitation inceptionperformance Based on the genetic algorithm optimization under multiple flow rate conditions was conducted by modifying theleading edge ellipse ratio and blade thickness on the front 20 meanline By using CFD simulation optimization was completedwith obvious improvements on the cavitation inception performance CFD results show that the pressure drop location hadmoved downstream with the increasement of the minimum pressure coefficient Experimental verifications also got an obviousenhancement of cavitation inception performanceThe stability and safety was improved by moving the cavitation inception curveout of the operating rangeThis optimization is proved applicable and effective for the engineering applications of reversible pump-turbines

1 Introduction

Cavitation is a common dangerous phenomenon in hydraulicmachinery usually causing vibration noise and damage Asa consequence it affects the stability and safety seriouslyReversible pump-turbines usually have higher speeds andheads than typical centrifugal pumps and are more likely toexperience large-scale cavitation with tremendous negativeimpacts Moreover the cavitation coefficient in pump-modeis much greater than that in turbine-mode Often criticalcavitation data are obtained by measuring the external char-acteristics However before the conditions reach the ldquocriticalcavitationrdquo conditions in the impeller the actual extent ofthe cavitation is already quite serious For the above reasonsthe cavitation inception in pump-mode is often regarded asthe crucial factor Hence improving the cavitation incep-tion performance of a reversible pump-turbine is obviouslyimportant in engineering applications

For pumps the minimum pressure point is usually loca-ted in the leading edge (LE) of the blade As a consequence

the inception cavitation often manifests as LE cavitation LEcavitation in hydraulic turbomachinery has received greatattention and been widely investigated by researchers basedon experiments and CFD simulations [1ndash5] Arakeri [6]pointed out that the cavitation inception at LE is relatedto the position of flow separation Flow separation occurswhen the boundary layer lifts off or separates from thesurface due to the geometry [7 8] In practical applicationsresearchers modified the geometry of LE to improve cavita-tion performances of pumps and inducers [9ndash11] But targetedoptimization for the cavitation inception performance ofpump-turbine is still unresolved in engineering applications

As a common optimization method the genetic algo-rithm (GA) is famous for its global optimization and parallelcomputing capabilities [12] and has been applied to thedynamics optimizations of pumps compressors and windturbines [13ndash15] In this study GA is used to search theoptimal LE geometry to improve the cavitation inceptionperformance of pump-turbine in pumpmodeThe optimizedpump-turbine model is expected to change the position of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 234615 9 pageshttpdxdoiorg1011552014234615

2 Mathematical Problems in Engineering

Flowdirection

Outlet

Inlet

Shroud

Hub

b2

D2

Ds1

Dh1

Figure 1 Meridional shape of pump-turbine impeller

Table 1 Specification of the performance and geometry parameters

Parameters ValuesPerformance

Design flow rate 119876119889 450 kgsDesign head119867119889 54mRotational speed 119899 1200 rmin

GeometryBlade number 119885 9Leading edge hub diameter 119863ℎ1 140mmLeading edge shroud diameter1198631199041 300mmTrailing edge diameter119863

2514mm

Outflow width 1198872 572mm

local separation especially under off-design conditions slowdown the dramatic pressure drop near LE and enhance thesafety and stability of pump-turbine units

2 Application Example

21 Model and Parameterization In this study a pump-turbine which has seriously bad cavitation performance wasput into optimization This pump-turbine has a 9-blademixed flow impeller whose meridional shape is shown inFigure 1 Moreover the details of performance parametersand geometry parameters are shown in Table 1

As the optimized object the LE geometry was parame-terized for controlling Figure 2 shows the parameterizationof LE shape The thickness values on the front 20 meanlineand the LE ellipsewere controlled So therewere 6 parametersincluding the thickness at 0 4 8 12 and 16meanlinelength and the LE ellipse ratio To apply the GA the originalvalues and variation range of all the 6 parameters weredetermined and shown in Table 2

Front 20 meanline

0 4 128 16 20

Leading edge ellipse ratio Thickness on the meanline

Meanline length ()

(keep constant at 20)

Figure 2 Parameterization of leading edge shape

Table 2 Original values and variation range of controlling param-eters

Parameters Original values Variation rangeEllipse ratio 3 1sim5Thickness at 0 1058mm 1058sim4058mmThickness at 4 4664mm 2664sim6664mmThickness at 8 5911mm 3911sim9911mmThickness at 12 6613mm 4613sim10613mmThickness at 16 6967mm 4967sim10967mm

22 Optimization Method After the parameterization of LEgeometry of impeller the initial generation was createdrandomly in the variation range with 10 individuals Theseindividuals were encoded with 42-digit binary code (7 digitsfor each parameter) The binary code and the parametervalues could be converted following the rule below

119909 = 119886 + 1199091015840 119887 minus 119886

2119899 minus 1 (1)

where 119909 is the decimal value of parameter 119886 and 119887 are thelower and upper limit of the variation range 119899 is the digitnumber of binary code per parameter (here 119899 = 7) and 1199091015840is the corresponding decimal value of the binary code Thebinary code had a numerical precision of 01 for the ellipseratio 0024 for the thickness at 0 and 0047 for the thicknessat other locations

As mentioned above the minimum pressure coefficient119862119901-min in the impeller was chosen as the fitness function toevaluate the cavitation inception performance To comparethe minimum pressure between different impeller geome-tries the dimensionless pressure coefficient 119862119901 was definedand is shown below

119862119901 =2 (119901 minus 119901infin)

1205881198812infin

(2)

where119901 is the pressure 119901infin and119881infin are the reference pressureand velocity and 120588 is the density of water The referencepressure and velocity values were acquired at the impellerinlet

In this study three different operating conditions includ-ing 360 kgs 450 kgs (the design condition119876119889) and 540 kgs were taken into consideration The fitness function 119865 isdefined as follows

119865 = sum

119894

119908119894119862119901119894-min (3)

Mathematical Problems in Engineering 3

Outflow

Inflow

Periodic

Periodic

Mesh elements

Figure 3 Schematic map of domain mesh and boundary conditions

where 119894 denotes the number of operating conditions whichis 3 in this study The argument 119908119894 is the weight of 119862119901119894-minunder different conditions Because of the worse cavitationperformance under off-design conditions 1199081 (the weight of119876119889 = 450 kgs) was set as 02 and 1199082 (the weight of 360 kgs)and 1199083 (the weight of 540 kgs) were set as 04

Considering the nonlinearity in this study computationalfluid dynamics (CFD) tools were used as the solver offitness function So modeling and meshing of the flowdomain were proceeded before the solving A single bladepassage was modeled as the flow domain Then structuralhexahedral elements were used in the meshing of domainA mesh independence check was conducted to guaranteethe computational accuracy The mesh schemes from 11935to 57750 nodes were checked by comparing the minimumpressure coefficient 119862119901119894-min on the blade surfaces under thedesign condition The variation of 119862119901119894-min value became lessthan 1 when the mesh node increased to 44537 Moreoverthe 119910+ value was controlled within the range of 16 to 583by setting the height of near-wall mesh elements to 3mmSo wall function could be used in the CFD solving of near-wall region Finally the mesh scheme with about 45000mesh nodes was used The Reynolds Averaged Navier-Stokesequations were solved with SST 119896-120596 turbulence model [16]In the definition of boundary conditions the impeller inflowwas set as mass flow inlet The impeller outflow was setas pressure outlet with the static pressure of 0 Pa Impellerhub shroud and blades were set as no slip wall Rotationalperiodic boundaries were given for the simplification of sim-ulation The schematic map of domain mesh and boundaryconditions are shown in Figure 3

After the CFD simulation processes the individual whohad the lowest 119865 value was eliminated and the vacancy wasfilled by the copy of the sample who had the highest 119865 valueIn the setting of genetic operations the crossover rate wasset as 065 and the mutation rate was set as 01 Optimizationwould converge when the residual of the fitness function of

the best individual became less than 1 times 10minus3 The schematicmap of the whole optimization process is shown in Figure 4

23 Optimization Monitoring and Results In the solvingof fitness function for all the 10 individuals optimizationwas running in a parallel way Every impeller individualwas simulated in a standalone CFD progress Then fitnessfunction values were transferred to a terminal machine forthe genetic operations The monitoring of fitness function119865 of all the 10 individuals is shown in Figure 5 After 20generations the optimization converged with the 119865 valuesincreasing to a higher level Individual-9 who has the highest119865 value was chosen as the final optimized impeller Thecomparisons of LE geometry between the optimized and theoriginal impeller are shown in Figure 6

3 Verifications and Analysis

31 Cavitation Inception Performance To verify the improve-ment of cavitation inception performance both the originaland optimized impellers were put into cavitation experimentsand CFD simulations The cavitation inception experimentswere conducted on the hydraulic test rig shown in Figure 7The high-speed camera was used as the measurement deviceof cavitation bubbles A ldquo3-bubblerdquo criterionwas used to iden-tify cavitation inception By lowering the ambient pressurewith the vacuum pump cavitation bubbles occurred near theblade LE as shown in Figure 7(b) When 3 bubbles appearedthe cavitation inception coefficient119862120590119894 was recorded which isdefined as

119862120590119894 =119873119875119878119867119903

119867=(119901119904120588119892 + 119881

2

1199042119892 minus 119901V120588119892)

119867 (4)

where 120588 is the density of water 119892 is the acceleration of gravity119901V is the vapor pressure 119901119904 and 119881119904 denote the pressure andvelocity at the reference position (impeller inflow) and 119867denotes the head of pump-turbine


Converged(residual of the fitness function of the

Parameterization

Original impeller

Encoding the initial generation

Geometry modeling

Meshing

Solving the fitness functions

Genetic algorithm operation

Done

Yes

New generation

Nobest individual less than 1 times 10minus3)

Figure 4 Schematic map of optimization process

0 5 10 15 20Generations

Individual 1Individual 3Individual 5Individual 7Individual 9


Fitn

ess f

unct

ionF

minus085

minus095

minus105

minus115

minus125

minus135

Figure 5 Monitoring of fitness function 119865 of all the 10 individuals

In the cavitation inception simulations mass transfer wasturned on to make the cavitation happen The saturationpressure is set to 3500 Pa under the reference pressure of1 Atm By lowering the pressure value at impeller outlet

cavitation occurred in the impeller The cavitation inceptioncriterionwas set as an average vapor volume fraction of 001in the impeller domain Figure 8 shows the experimental andnumerical 119862120590119894 values under different flow rate conditions


0

1

2

3

4

5

Eclip

se ra

tio

Leadingedge

location

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20

Thic

knes

s (m

m)

Meanline length ()

OriginalOptimized

OptimizedOriginal

0sim20 meanline

Figure 6 Comparison of LE geometry between the optimized and the original impeller

(a) Test rig

Cavitationbubbles

(b) Capturing the cavitation bubbles

Figure 7 Measurement of the cavitation inception in the test rig

As shown in Figure 8 if the 119862120590119894 curve goes acrossthe operating range cavitation would happen because thepressure may drop below the vapor pressure Experimen-tal data show that the 119862120590119894 curve went across the rangeunder off-design conditions before optimization After opti-mizing the blade LE shape even the 119862120590119894 value increasedunder the design flow rate condition (119876 = 450 kgs)

and the cavitation inception had also been improvedbecause 119862120590119894 decreased under all the off-design conditionswith the curve getting out of the range It is necessaryto study the mechanism of cavitation inception perfor-mance optimization As shown in Figure 8 the numericalsimulation had obtained a consistent variation tendency of119862120590119894 with the experimental data So it is reasonable to study


01

02

03

04

05

06

350 400 450 500 550

Operating range in EXPOptimized EXPOriginal EXP

Optimized CFDOriginal CFD

Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)

Figure 8 Cavitation coefficient 119862120590119894by experiment and numerical

simulation

and analyze the flow details through the CFD simulationresults

32 Pressure Drop at Leading Edge To study the flow detailsnear blade LE the velocity vectors and pressure contour onthe spanwise 50 surface under the design condition are plot-ted in Figure 9 Seen from the vectors fluid flowed aroundthe blade LE and separated from the side surface Adversepressure gradient generated and induced the pressure dropBefore optimization the pressure drop was very abruptbecause of the mutational geometry After optimization theimpeller got a bigger thickness around the LE Hence thepressure drop became gentle with the minimum pressurepoint moving downstream To analyze the pressure drop indetail the pressure coefficient 119862119901 distributions on the front2 meanline are plotted in Figure 10

Seen from Figure 10 pressure drop occurred on the front2 meanline on the spanwise 20 50 and 80 surfacesAs illustrated in Figure 10 the minimum pressure coefficient119862119901-min increased under all the 3 flow rate conditions excepton the spanwise 50 surface of 119876 = 450 kgs conditionThis is due to the setting of weight of 119862119901119894-min under differentconditions in the fitness function (3) In this optimizationit focused more on the cavitation inception performanceunder off-design conditions Even the cavitation coefficient119862120590119894 increased at the flow rate of 119876 = 450 kgs after optimi-zation (shown in Figure 8) this optimization was alsoglobally successful for improving the cavitation inceptionperformance under the poor-performing conditions

33 Impacts on Hydrodynamic Performances Consideringthe impacts on external characteristics after the optimization

093

minus047Cp

(a) Original

(b) Optimized

Figure 9 Flow separation and pressure drop at blade LE onspanwise 50 surface

of LE shape additional studies were carried out by analyzingthe variation of head and efficiency Figure 11 shows thecomparison curves of head 119867 and efficiency 120578 betweenoriginal impeller and optimized impeller In Figure 11 theCFD head values are the impeller head and the experimentalhead is the head of the whole passage The CFD efficiency isthe hydraulic efficiency of the impeller and the experimentalefficiency is the total efficiency of the whole passage Both119867 and 120578 decreased after the optimization of LE HoweverCFD119867 and experimental119867 changed by less than 0986 and1059 respectively Also CFD 120578 and experiment 120578 decreasedby less than 0237 and 0180 respectively There were justlittle changes on the head and efficiency It means that theoptimization was applicable and effective

4 Conclusion

With the genetic algorithm the optimization on cavitationinception performance of a pump-turbine in pump modehad been conducted The geometry of LE had been modified


Spanwise 20

0

05

1

0 1 2Meanline length ()

OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)

Figure 10 Pressure coefficient 119862119901 distributions on the front 2 meanline


40

45

50

55

60

65

350 400 450 500 550

Original CFDOptimized CFD

Original EXPOptimized EXP

Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)

(b) EfficiencyFigure 11 Comparison curves of head and efficiency

after optimization By both the CFD simulation and theverification experiment the optimization had been provedeffective with conclusions drawn as follows

As a widely used optimization method the genetic algo-rithm was used in this study to solve the nonlinear problemAfter optimization the impeller got a better cavitation perfor-mance than before The optimization was helpful to controlthe local separation With the optimized LE geometry flowseparation near leading edge was weakened and postponedto downstream The tendency of pressure drop becamegentle especially under the off-design conditions Moreoverthe minimum pressure coefficients 119862119901-min increased afteroptimization Hence this optimization was proved excellentfor improving the cavitation inception performance of pump-turbines in pump mode It is reasonable and applicable forrelevant engineering applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportgiven by the National Natural Science Foundation of China(no 51139007) and National ldquoTwelfth Five-Yearrdquo Plan forScience amp Technology Support (no 2012BAD08B03)

References

[1] C E BrennenHydrodynamics of Pumps Cambridge UniversityPress Cambridge UK 2011

[2] X Li S Yuan Z Pan J Yuan and Y Fu ldquoNumerical simulationof leading edge cavitation within the whole flow passage of acentrifugal pumprdquo Science China Technological Sciences vol 56no 9 pp 2156ndash2162 2013

[3] X J Li Y Z Pan Q D Zhang et al ldquoCentrifugal pump per-formance drop due to leading edge cavitationrdquo IOP ConferenceSeries Earth and Environmental Science vol 15 no 3 ArticleID 032058 2012

[4] D Pierrat L Gros A Couzinet et al ldquoOn the leading edge cavi-tation in a helico-centifugal pump experimental and numericalinvestigationsrdquo in Proceedings of the 3rd IAHR InternationalMeeting of the Workgroup on Cavitation and Dynamic Problemsin Hydraulic Machinery and Systems 2009

[5] T Sudsuansee U Nontakaew and Y Tiaple ldquoSimulation ofleading edge cavitation on bulb turbinerdquo Songklanakarin Jour-nal of Science and Technology vol 33 no 1 pp 51ndash60 2011

[6] V H Arakeri ldquoViscous effects on the position of cavitationseparation from smooth bodiesrdquo Journal of FluidMechanics vol68 no 4 pp 779ndash799 1975

[7] M Gad-el-Hak and D M Bushnell ldquoSeparation controlreviewrdquo Journal of Fluids Engineering Transactions of the ASMEvol 113 no 1 pp 5ndash30 1991

[8] M P Patel Z H Sowie T C Corke and C He ldquoAutonomoussensing and control of wing stall using a smart plasma slatrdquoJournal of Aircraft vol 44 no 2 pp 516ndash527 2007

[9] A Cervone G Pace L Torre et al ldquoEffects of the leading edgeshape on the performance of an axial three bladed inducer[C]rdquoin Proceedings of the 14th International Symposium on TransportPhenomena and Dynamics of Rotating Machinery 2012

[10] O Coutier-Delgosha J-L Reboud and R Fortes-PatellaldquoNumerical study of the effect of the leading edge shape oncavitation around inducer blade sectionsrdquo JSME InternationalJournal Series B Fluids andThermal Engineering vol 45 no 3pp 678ndash685 2002


[11] D Pierrat L Gros G Pintrand et al ldquoExperimental and num-erical investigations of leading edge cavitation in a helico-centrifugal pumprdquo in Proceedings of the 12th InternationalSymposium of Transport Phenomena and Dynamics on RotatingMachinery pp 17ndash22 Honolulu Hawaii USA February 2008

[12] Z Michalewicz Genetic Algorithms + Data Structures = Evolu-tion Programs Springer New York NY USA 1996

[13] R Xiao and Z Wang ldquoCentrifugal pump blade optimizationbased on a combined optimization strategyrdquo Journal of TsinghuaUniversity vol 46 no 5 pp 700ndash703 2006

[14] J Luo C Zhou and F Liu ldquoMultipoint design optimization ofa transonic compressor blade by using an adjoint methodrdquo Jou-rnal of Turbomachinery vol 136 no 5 Article ID 051005 2013

[15] L Xiong C Yan and Y Zhiquan ldquoApplication of genetic algo-rithms to HAWT rotor blades optimizationrdquo Acta EnergiaeSolaris Sinica vol 27 no 2 pp 180ndash185 2006

[16] F R Menter M Kuntz and R Langtry ldquoTen years of industrialexperience with the SST turbulence modelrdquo Turbulence Heatand Mass Transfer vol 4 pp 625ndash632 2003

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Flowdirection

Outlet

Inlet

Shroud

Hub

b2

D2

Ds1

Dh1

Figure 1 Meridional shape of pump-turbine impeller

Table 1 Specification of the performance and geometry parameters

Parameters ValuesPerformance

Design flow rate 119876119889 450 kgsDesign head119867119889 54mRotational speed 119899 1200 rmin

GeometryBlade number 119885 9Leading edge hub diameter 119863ℎ1 140mmLeading edge shroud diameter1198631199041 300mmTrailing edge diameter119863

2514mm

Outflow width 1198872 572mm

local separation especially under off-design conditions slowdown the dramatic pressure drop near LE and enhance thesafety and stability of pump-turbine units

2 Application Example

21 Model and Parameterization In this study a pump-turbine which has seriously bad cavitation performance wasput into optimization This pump-turbine has a 9-blademixed flow impeller whose meridional shape is shown inFigure 1 Moreover the details of performance parametersand geometry parameters are shown in Table 1

As the optimized object the LE geometry was parame-terized for controlling Figure 2 shows the parameterizationof LE shape The thickness values on the front 20 meanlineand the LE ellipsewere controlled So therewere 6 parametersincluding the thickness at 0 4 8 12 and 16meanlinelength and the LE ellipse ratio To apply the GA the originalvalues and variation range of all the 6 parameters weredetermined and shown in Table 2

Front 20 meanline

0 4 128 16 20

Leading edge ellipse ratio Thickness on the meanline

Meanline length ()

(keep constant at 20)

Figure 2 Parameterization of leading edge shape

Table 2 Original values and variation range of controlling param-eters

Parameters Original values Variation rangeEllipse ratio 3 1sim5Thickness at 0 1058mm 1058sim4058mmThickness at 4 4664mm 2664sim6664mmThickness at 8 5911mm 3911sim9911mmThickness at 12 6613mm 4613sim10613mmThickness at 16 6967mm 4967sim10967mm

22 Optimization Method After the parameterization of LEgeometry of impeller the initial generation was createdrandomly in the variation range with 10 individuals Theseindividuals were encoded with 42-digit binary code (7 digitsfor each parameter) The binary code and the parametervalues could be converted following the rule below

119909 = 119886 + 1199091015840 119887 minus 119886

2119899 minus 1 (1)

where 119909 is the decimal value of parameter 119886 and 119887 are thelower and upper limit of the variation range 119899 is the digitnumber of binary code per parameter (here 119899 = 7) and 1199091015840is the corresponding decimal value of the binary code Thebinary code had a numerical precision of 01 for the ellipseratio 0024 for the thickness at 0 and 0047 for the thicknessat other locations

As mentioned above the minimum pressure coefficient119862119901-min in the impeller was chosen as the fitness function toevaluate the cavitation inception performance To comparethe minimum pressure between different impeller geome-tries the dimensionless pressure coefficient 119862119901 was definedand is shown below

119862119901 =2 (119901 minus 119901infin)

1205881198812infin

(2)

where119901 is the pressure 119901infin and119881infin are the reference pressureand velocity and 120588 is the density of water The referencepressure and velocity values were acquired at the impellerinlet

In this study three different operating conditions includ-ing 360 kgs 450 kgs (the design condition119876119889) and 540 kgs were taken into consideration The fitness function 119865 isdefined as follows

119865 = sum

119894

119908119894119862119901119894-min (3)


Outflow

Inflow

Periodic

Periodic

Mesh elements









119862120590119894 =119873119875119878119867119903

119867=(119901119904120588119892 + 119881

2

1199042119892 minus 119901V120588119892)

119867 (4)




Parameterization

Original impeller


Geometry modeling

Meshing



Done

Yes

New generation






Fitn

ess f

unct

ionF

minus085

minus095

minus105

minus115

minus125

minus135





0

1

2

3

4

5

Eclip

se ra

tio

Leadingedge

location

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20

Thic

knes

s (m

m)

Meanline length ()

OriginalOptimized

OptimizedOriginal

0sim20 meanline


(a) Test rig

Cavitationbubbles






01

02

03

04

05

06

350 400 450 500 550



Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)


simulation





093

minus047Cp

(a) Original

(b) Optimized



4 Conclusion



Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Outflow

Inflow

Periodic

Periodic

Mesh elements









119862120590119894 =119873119875119878119867119903

119867=(119901119904120588119892 + 119881

2

1199042119892 minus 119901V120588119892)

119867 (4)




Parameterization

Original impeller


Geometry modeling

Meshing



Done

Yes

New generation






Fitn

ess f

unct

ionF

minus085

minus095

minus105

minus115

minus125

minus135





0

1

2

3

4

5

Eclip

se ra

tio

Leadingedge

location

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20

Thic

knes

s (m

m)

Meanline length ()

OriginalOptimized

OptimizedOriginal

0sim20 meanline


(a) Test rig

Cavitationbubbles






01

02

03

04

05

06

350 400 450 500 550



Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)


simulation





093

minus047Cp

(a) Original

(b) Optimized



4 Conclusion



Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameterization

Original impeller


Geometry modeling

Meshing



Done

Yes

New generation






Fitn

ess f

unct

ionF

minus085

minus095

minus105

minus115

minus125

minus135





0

1

2

3

4

5

Eclip

se ra

tio

Leadingedge

location

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20

Thic

knes

s (m

m)

Meanline length ()

OriginalOptimized

OptimizedOriginal

0sim20 meanline


(a) Test rig

Cavitationbubbles






01

02

03

04

05

06

350 400 450 500 550



Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)


simulation





093

minus047Cp

(a) Original

(b) Optimized



4 Conclusion



Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

Eclip

se ra

tio

Leadingedge

location

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20

Thic

knes

s (m

m)

Meanline length ()

OriginalOptimized

OptimizedOriginal

0sim20 meanline


(a) Test rig

Cavitationbubbles






01

02

03

04

05

06

350 400 450 500 550



Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)


simulation





093

minus047Cp

(a) Original

(b) Optimized



4 Conclusion



Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01

02

03

04

05

06

350 400 450 500 550



Cavi

tatio

n in

cept

ion

coeffi

cien

tC120590i

Flow rate Q (m3s)


simulation





093

minus047Cp

(a) Original

(b) Optimized



4 Conclusion



Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

Spanwise 50

0

05

1


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(a) 119876 = 360 kgs (08119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50


OptimizedOriginal

Spanwise 80


OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

05

0

1

15

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

(b) 119876 = 450 kgs (10119876119889)

Spanwise 20

0

05

1


OptimizedOriginal

Spanwise 50

0 1 2

Meanline length ()

OptimizedOriginal

Spanwise 80

0 1 2

Meanline length ()

OptimizedOriginal

Pres

sure

coeffi

cien

tCp

minus05

minus1

0

05

1

Pres

sure

coeffi

cien

tCp

minus05

minus1

minus15

minus2

0

1

2

Pres

sure

coeffi

cien

tCp

minus1

minus2

minus3

(c) 119876 = 540 kgs (12119876119889)



40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










40

45

50

55

60

65

350 400 450 500 550



Flow rate Q (m3s)

Hea

d H

(m)

(a) Head

89

90

91

92

93

94

95

96

97

Effici

ency

120578(

)

350 400 450 500 550



Flow rate Q (m3s)






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Optimization for Cavitation Inception Performance of Pump …downloads.hindawi.com/journals/mpe/2014/234615.pdf · 2015-11-22 · Research Article Optimization for