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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.
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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
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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
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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
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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
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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
014501-8 / Vol. 136, JANUARY 2014 Transactions of the ASME
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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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