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Parametric design of a Francis turbine runner by means of a three-dimensional inverse design

method

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Parametric design of a Francis turbine runner by meansof a three-dimensional inverse design method

K Daneshkah1and M Zangeneh

2

1Advanced Design Technology, London WC1E 7JN, UK

2Department of Mechanical Engineering, University College London, London WC1E

7JE, UK

E-mail: k.daneshkhah@adtechnology.co.uk

Abstract. The present paper describes the parametric design of a Francis turbine runner. Therunner geometry is parameterized by means of a 3D inverse design method, while CFD

analyses were performed to assess the hydrodymanic and suction performance of differentdesign configurations that were investigated. An initial runner design was first generated andused as baseline for parametric study. The effects of several design parameter, namely stackingcondition and blade loading was then investigated in order to determine their effect on thesuction performance. The use of blade parameterization using the inverse method lead to amajor advantage for design of Francis turbine runners, as the three-dimensional blade shape isdescribe by parameters that closely related to the flow field namely blade loading and stacking

condition that have a direct impact on the hydrodynamics of the flow field. On the basis of thisstudy, an optimum configuration was designed which results in a cavitation free flow in the

runner, while maintaining a high level of hydraulic efficiency. The paper highlights designguidelines for application of inverse design method to Francis turbine runners. The design

guidelines have a general validity and can be used for similar design applications since they arebased on flow field analyses and on hydrodynamic design parameters.

1.Introduction

The hydraulic design of Francis turbine runners requires accomplishment of several targets and constraints. A

high level of efficiency and a cavitation-free flow in the runner is usually desirable. The flow in Francis turbinerunners is highly rotational and three-dimensional and therefore only three-dimensional methods will provideeffective solution for a Francis runner. A considerable improvement in the design of Francis turbines have beenobtained by the use of Computational Fluid Dynamics (CFD). CFD results provide a better understanding of the flowphysics and they are now commonly used in industry, ref [1-4]. Although these methods are very useful for analysisin different design configurations, they cannot be directly used as a design tool as they do not provide any directinformation on how to change the runner shape. So the designer needs to rely on trial and error to improve the runner

geometry. Such an approach, with its reliance on empiricism, may restrict the part of design space that is being usedin the design as the designer tends to stay within the bounds of successful previous designs.

A major improvement in the design of Francis runners can be achieved by the application of 3D inverse designmethod for the design of the runner shapes. Unlike conventional direct design methods, where the blade geometry isdescribed by geometrical parameters, inverse design uses hydrodynamic parameters like the blade loading, tocompute the blade shape, offering a major advantage in the design process. Such an approach allows designers todirectly relate their understanding of flow physics in the design process and hence access a larger part of the designspace. The application of 3D inverse design method has already resulted in important design breakthroughs such assuppression of secondary flows in radial and mixed flow impeller impellers [5-6], improvement of suctionperformance and efficiency of water jet pumps [7], suppression of corner separation in pump diffusers [8] andimprovement of cavitation in a Francis turbine runner [9].

In this present paper, a parametric design study of a Francis turbine runner is carried out where an inverse designmethod is used to parametrically describe the runner geometry and CFD analyses are performed to evaluate the

hydrodynamic and suction performance of different configurations. First, a baseline design was created using thebasic design specifications of the Francis turbine runner. Next, the impact of stacking condition on the runner

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performance was assessed. The aim of this study was to understand the effect of stacking condition of on the runnerefficiency and its suction performance. Then, the effect of blade loading was studied for an optimum stackingconfiguration obtained in the previous step so that a cavitation-free flow in the runner is achieved, while maintaininghigh level of hydraulic efficiency.

2.Inverse Design Method

The commercial 3D inverse design code TURBOdesign-1 was used as the design methodology in this study.Turbodesign-1 [10] is a three-dimensional inviscid inverse design method, where the distribution of thecircumferentially averaged swirl velocity rV is prescribed on the blade meridional channel and the corresponding

blade shape is computed iteratively.The circulation distribution is specified by imposing the spanwise rV distribution at blade leading and trailing

edge and the meridional derivative of the circulation drV/dm (blade loading) inside the blade channel. The pressureloading (the pressure difference across the blade) is directly related to the meridional derivative of rV throughmomentum equation of an incompressible flow in the blade passage in pitch-wise direction, which is given below:

(1)

Where p+ and p correspond to the static pressure on pressure and suction side of the blade, B is the blade number, is the density and Wm is the pitch-wise averaged meridional velocity.

The input design parameters required by the program are as follows:

Meridional channel shape in terms of crown, band, leading and trailing edge contours.

Normal thickness distribution at two or more spanwise sections.

Fluid properties and design specifications.

Number of blades.

Inlet flow conditions in terms of spanwise distributions of total pressure and velocity components.

Inlet and exit rV spanwise distribution. By controlling its value, the runner head is controlled

Blade loading distribution (drV/dm) at two or more spanwise sections. The code then automatically

interpolates the blade loading in spanwise direction to obtain two-dimensional distribution of the loadingover the whole meridional channel.

Stacking condition. The stacking condition must be imposed at a chord-wise location between leading andtrailing edge. Everywhere else the blade is free to adjust itself according to the loading specifications.

One unique feature of TURBOdesign1 is that it allows designers to vary one parameter (e.g stacking or blade

loading) while fixing the other parameters. The program then automatically arrives at the blade shape that satisfiesthe necessary specific work at the correct flow rate and specified blade loading or stacking. It is this feature of thecode that is used in this paper for parametric study.

In order to verify the different configurations that were designed, CFD calculations were performed using thecommercial software ANSYS CFX 12.1. The computational domain was discretized by means of a hybrid H-C-Otype structured mesh with approximately 375K nodes per blade passage. The Reynolds Averaged Navier-Stokesequations were solved using a finite-volume approach and k- model with standard wall function implementationwas used for the turbulence closure. The average value of total pressure, which occurs at the runner inlet was

imposed as a boundary condition at the inlet of the computational domain. For cavitation analysis, a two phaseRayleigh-Plesset model is used. The interphase transfer is governed by a mixture model where the interfacelength scale is 1 mm. Flow is assumed to be homogeneous and isothermal at 293.15 K. The saturation pressure is

3619 Pa and the mean nucleation site diameter is 2m.

3.Design of Baseline Configuration

A Francis turbine runner with specific speed of vs=0.35 was selected for this study, where the specific speedis defined by

( )(2 / )m

rVp p B Wm

+ =

1/2

1/2 3/4

(2 )

Qvs

gH

= (2)

2

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The runner meridional geometry is presented in Fig.1. The runner maximum diameter is 157.5 mm and itsaxial length is 140 mm. The runner meridional shape is usually fixed by design constraints and therefore it wasnot changed during the design process. The runner has 13 blades with a maximum profile thickness of 7 mm atthe crown and 4 mm maximum thickness at the band. The runner operating conditions are listed in Table1.

Before proceeding with the parametric study, a baseline design was created using TURBOdesign-1. Thedesign specifications and inlet condition were imposed according to their values at the operating condition. A

free-vortex flow distribution (uniform spanwise rV ) was assumed at the runner inlet. The value of rVwaschosen to produce the available head at runner inlet. A zero stacking was imposed at runner LE.

Table 1. Francis Runner Design Specifications

Rotational speed 1350

Runner Head 42 m

Design flow rate 0.45 m^3 min-1

Inlet total pressure 415 kPa

Guide vane opening 73 degRequired Shaft Power 165 kW

Figure 1 Francis runner meridional contour

Streamwise Distance

BladeLoading

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

Crown

Band

Figure 2 Blade Loading distribution

Figures 2 represents the normalized loading distribution of the baseline runner design. The loading is definedat two sections (band and crown) and it is then interpolated over the meridional channel. Each loadingdistribution is plotted against the normalized streamwise distance from leading edge (streamwise distance=0) totrailing edge (streamwise distance=1). Both sections are mid-loaded with a constant loading from 25% to 75%of blade chord. The value of blade loading at the leading edge controls the flow incidence at design point (seeequation [1]). The baseline design runner geometry obtained by the inverse code is presented in Fig.3.

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Figure 3 Baseline Design 3D geometry

Figure 4 3D View of computational mesh

4.CFD Analysis of Baseline Configuration

CFD analysis is performed for the baseline design in order to investigate detailed flow field at design and off-design conditions using a single-phase flow model. The flow is assumed to be steady-state and axi-symmetric,therefore only one flow passage in the runner is modeled. Figure 4 shows the computational mesh at runner mid-span for the baseline runner. In order to ensure of the accuracy of CFD results, a mesh dependency study wasperformed for the baseline runner. Three mesh sizes with the same mesh topology were investigated; the coarse

mesh has a mesh size of 90K nodes per passage with an average value of Y+ at midspan of about 120, themedium mesh has a total mesh size of about 375K nodes and an average Y+ at midpan of about 20, the fine mesh

has a total mesh size of about 700K nodes and an average Y+

at midspan of 10.The runner performance characteristics at design flow corresponding to a guide vane opening of about 18 deg.

is presented in Fig. 5. for the three different mesh. The results confirm that a mesh independent solution isreached for the medium size mesh This mesh size is used for all computation in the present work hereafter. Theperformance characteristics also show that runner achieves the required power output with a good efficiency andperforms well at off-design condition. In this figure, the runner head, power and hydraulic efficiency are plottedagainst non-dimensional blade velocity given by:

The hydraulic efficiency is given by:

Figure 6 shows the velocity vectors on the suction and pressure surfaces on the runner. The flow is roughlyaligned with the streamwise direction on the suction side of the blade, whereas near the pressure side inside theboundary layer the flow is forced towards the band, which indicates its strong three-dimensional character andthe distinct secondary flows in Francis runner. Figure 7 shows the runner pressure distribution at threespanwise sections, i.e., crown , midspan and band. The low pressure region on the band suctions side indicatesthat this area is prone to severe cavitation. This is further confirmed by a two-phase flow cavitation analysis, as it

can be seen by contours of water vapor volume fraction in Fig.8, confirming strong cavitation on the shroud nearthe trailing edge region.

/ 2uK U gH=

T

QH

=

(3)

(4)

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Ku

Head[m]

0.5 0.6 0.7 0.8 0.932

34

36

38

40

42

Coarse

Medium

Fine

(a)

Ku

Power

[kW]

0.5 0.6 0.7 0.8 0.9140

145

150

155

160

165

170

175

180

185

Coarse

Medium

Fine

(b)

Ku

0.5 0.6 0.7 0.8 0.90.92

0.94

0.96

0.98

1

Coarse

Medium

Fine

(c)

Figure 5 Runner performance characteristics at design flow rate, Runner Head (a), Shaft Power (b), RunnerEfficiency (c)

(a)(b)

Figure 6 Baseline design: Velocity vector on the blade suction surface (a) and pressure surface (b) at design

point

Streamwise Distance

StaticPressure[kPa]

0 0.2 0.4 0.6 0.8 1-100

-50

0

50

100

150

200

250

300

Crown

Midspan

Band

Figure 7 Baseline design: blade pressure

distribution at design point

Figure 8 Baseline design: contours of watervapour volume fraction at design point

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StreamwiseDistance

StaticPressure[kPa]

0 0.2 0.4 0.6 0.8 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

StaticPressure[kPa]

0 0.2 0.4 0.6 0.8 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

StaticPressure[kPa]

0 0.2 0.4 0.6 0.8 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

5.Parametric Study of the Runner Stacking Condition

The stacking condition has a significant effect on spanwise work distribution and three-dimensional flow

structure in the Francis runner. Three stacking configurations were investigated using the inverse design code byvarying the stacking to -15, -30 and -45 degrees. The negative sign indicates the direction of stacking in such away that the pressure loading is reduced at the band and increased at the crown. This is done in order to reduce

the low pressure region on the band suction surfaces and associated cavitation region. All the other runner designparameters were kept unaltered.

Figure 9 shows the 3D geometries of the runner for different stacking conditions and Fig. 10 presents thecorresponding blade pressure distributions at design condition obtained from a single-phase flow analysis foreach case. As it can be seen from these plots by increasing the stacking to -15 degrees, the loading at the band isreduced and increased at the crown, however there is a still a low pressure region at about 20% chord followedby another low pressure region from 70-95% chord on the band suction surface where cavitation can occur.Increasing of stacking to -30 degrees, results in a roughly uniform spanwise pressure loading where the lowpressure region is significantly reduced and is limited to a small region between 75%-90% chord from midspanto band on the suction surface. Further increase of stacking to -45 degrees, results in a very low pressure region

on the crown suction section from 40% chord onward which extend up to midspan. The results of cavitationanalysis, presented in Fig.11 in form of water vapour volume fraction contours on the blade surfaces confirmsthe observations obtained from single-phase flow analysis.

(a)(b) (c)

Figure 9 3D blade geometries at -15 deg (a), -30 deg (b) and -45 deg (c) stacking

(a) (b)(c)

Fig. 10 Blade pressure distributions for -15 deg (a), -30 deg (b) and -45 deg (c) stacking design configuration

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Figure 14Design S30_MF: Contours of water vapor volume fraction

(a) (b)

Figure 15Design S30_MF: Velocity vector on the blade suction surface (a) and pressure surface (b)

XY

Z

BaselineDesignS30_MF

(a)

X

Z

Baseline

DesignS30_MF

(b)

XY

Z

Baseline

DesignS30_MF

(c)

Figure 16 Comparison of baseline and DesignS30_MF blade section geometries at crown (a), midspan (b)

and band (c)

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Ku

Head[m]

0.5 0.6 0.7 0.8 0.932

34

36

38

40

42

Baseline

DesignS30_MF

(a)

Ku

Power[kW]

0.5 0.6 0.7 0.8 0.9140

145

150

155

160

165

170

175

180

185

Baseline

DesignS30_MF

(b)

Ku

0.5 0.6 0.7 0.8 0.90.965

0.97

0.975

0.98

0.985

0.99

Baseline

DesignS30_MF

(c)

Figure 17 Comparison of the baseline and DesignS30_MF runner performance characteristics at design flowrate, Runner Head (a), Shaft Power (b), Runner Efficiency (c)

7.Conclusion

In this paper, a 3D inverse design method was applied to a Francis turbine design. Effect of inverse designparameter (stacking condition and blade loading) on the flow field inside the runner was studied in a parametricway. The aim of design was to obtain a cavitation free runner with high hydraulic efficiency. The flow field andsuction performance obtained by CFD with single-phase and two-phase flow models were compared between

different designs.The effects of stacking condition on the spanwise work distribution and the associated pressure field was

studied in details. By a combination of stacking condition and blade loading parameters, the static pressure fieldinside the runner was optimized so that the low pressure region on the blade suction side was eliminated and acavitation free runner was realized.

It was shown that parameterization of blade geometry using the inverse design flow related parameters can

provide the designer with control over the pressure field inside the runner, which can be used effectively tosuppress cavitation phenomena without deteriorating the hydraulic efficiency. The design guidelines presented inthis paper can be applied easily to the optimization of other Francis turbine runners. The 3D inverse method is an

extremely powerful and practical design tool for designing hydraulic turbine runners.

Nomenclature

BHLE

Ku

m

PQrT

Number of bladesRunner head [m]Leading edge

Non-dimensional blade velocityMerdional distance

Static pressure [Pa]Flow rate [m

3/s]

Radius [m]Torque [N.m]

TE

UV

vsW

Trailing edgeBlade velocity[m/s]Absolute velocity[m/s]

Specific SpeedRelative velocity[m/s]

Circumferential directionDensity [ kg/m

3]

Rotational Speed [rad/s]

References

[1] Drinta P, Sallaberger M 1999 Hydraulic Turbines- Basic Principle and State-of-the-Art Computation FluidDynamics ApplicationProc. Institute of Mechanical Eng.vol 213 (Part C) pp 85-102

[2] Sallaberger M 1996 Quasi-Three-Dimensional and Three-Dimensional Flow Calculation in a FrancisTurbineIGTI(Birmingham) p 96-GT-38

[3] Keck H, Goede E and Pestalozzi J 1990 Experience with 3D Euler Flow Analysis as a Practical Design

ToolIn Proc. of 16

th

IAHR Symp.(Sao Paolo, Brazil)[4] Nagafuji T, Uchida K, Tezuka K and Sugama K 1999 Navier Sokes Prediction on Performance of a

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Francis Turbine with High Specific SpeedASME Fluids Eng. (FEDSM99-7815)[5] Zangeneh M, Goto A and Takemura T 1996 Suppression of Secondary Flows in a Mixed Flow Pump

Impeller by Application of Three-Dimensional Inverse Method ASME J. of Turbomachinery118 536-561

[6] Zangeneh M, Goto A and Harada H 1998 On the Design Criteria for Suppression of Secondary Flows inCentrifugal and Mixed Flow Impellers ASME J. of Turbomachinery120 723-35

[7] Bonaiuti D, Zangeneh M, Aartojarvi R and Eriksson J 2010 A Parametric Design of a Waterjet Pump byMeans of Inverse Design, CFD Calculations and Experimental Analyses ASME J. of Fluids Eng.132031104

[8] Goto A, Zangeneh M 2002 Hydrodynamic Design of Pump Diffuser Using Invese Design Method andCFDASME J. of Fluids Eng.124 319- 328

[9] Okomoto H, Goto A 2002 Suppression of Cavitation in a Francis Turbine Runner by Application of 3DInverse Design MethodASME Fluids Eng. (FEDSM2002-31192)

[10] Zangeneh M 1991 A 3D Design Method for Radial and Mixed Flow Turbomachinery Blades Int. J. of

Numerical Methods in Fluids13 599-624

10