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Parametric design of a Francis turbine runner bymeans of a three-dimensional inverse designmethodTo cite this article K Daneshkah and M Zangeneh 2010 IOP Conf Ser Earth Environ Sci 12012058

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

K Daneshkah1 and M Zangeneh2 1Advanced Design Technology London WC1E 7JN UK 2Department of Mechanical Engineering University College London London WC1E 7JE UK

E-mail kdaneshkhahadtechnologycouk

Abstract The present paper describes the parametric design of a Francis turbine runner The runner geometry is parameterized by means of a 3D inverse design method while CFD analyses were performed to assess the hydrodymanic and suction performance of different design configurations that were investigated An initial runner design was first generated and used as baseline for parametric study The effects of several design parameter namely stacking condition and blade loading was then investigated in order to determine their effect on the suction performance The use of blade parameterization using the inverse method lead to a major advantage for design of Francis turbine runners as the three-dimensional blade shape is describe 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 this study 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 design guidelines 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 are based 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 turbine runners is highly rotational and three-dimensional and therefore only three-dimensional methods will provide effective solution for a Francis runner A considerable improvement in the design of Francis turbines have been obtained by the use of Computational Fluid Dynamics (CFD) CFD results provide a better understanding of the flow physics and they are now commonly used in industry ref [1-4] Although these methods are very useful for analysis in different design configurations they cannot be directly used as a design tool as they do not provide any direct information 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 used in 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 design method for the design of the runner shapes Unlike conventional direct design methods where the blade geometry is described by geometrical parameters inverse design uses hydrodynamic parameters like the blade loading to compute the blade shape offering a major advantage in the design process Such an approach allows designers to directly relate their understanding of flow physics in the design process and hence access a larger part of the design space The application of 3D inverse design method has already resulted in important design breakthroughs such as suppression of secondary flows in radial and mixed flow impeller impellers [5-6] improvement of suction performance and efficiency of water jet pumps [7] suppression of corner separation in pump diffusers [8] and improvement 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 design method 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 the basic design specifications of the Francis turbine runner Next the impact of stacking condition on the runner

25th IAHR Symposium on Hydraulic Machinery and Systems IOP PublishingIOP Conf Series Earth and Environmental Science 12 (2010) 012058 doi1010881755-1315121012058

ccopy 2010 IOP Publishing Ltd 1

performance was assessed The aim of this study was to understand the effect of stacking condition of on the runner efficiency and its suction performance Then the effect of blade loading was studied for an optimum stacking configuration obtained in the previous step so that a cavitation-free flow in the runner is achieved while maintaining high 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 the circumferentially averaged swirl velocity rVθ is prescribed on the blade meridional channel and the corresponding blade shape is computed iteratively

The circulation distribution is specified by imposing the spanwise rVθ distribution at blade leading and trailing edge and the meridional derivative of the circulation drVθdm (blade loading) inside the blade channel The pressure loading (the pressure difference across the blade) is directly related to the meridional derivative of rVθ through momentum equation of an incompressible flow in the blade passage in pitch-wise direction which is given below

(1)

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

The input design parameters required by the program are as follows

bull Meridional channel shape in terms of crown band leading and trailing edge contours bull Normal thickness distribution at two or more spanwise sections bull Fluid properties and design specifications bull Number of blades bull Inlet flow conditions in terms of spanwise distributions of total pressure and velocity components bull Inlet and exit rVθ spanwise distribution By controlling its value the runner head is controlled bull Blade loading distribution (drVθdm) at two or more spanwise sections The code then automatically

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

bull Stacking condition The stacking condition must be imposed at a chord-wise location between leading and trailing 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 (eg stacking or blade

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

In order to verify the different configurations that were designed CFD calculations were performed using the commercial software ANSYS CFX 121 The computational domain was discretized by means of a hybrid H-C-O type structured mesh with approximately 375K nodes per blade passage The Reynolds Averaged Navier-Stokes equations were solved using a finite-volume approach and k-ε model with standard wall function implementation was 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 phase Rayleigh-Plesset model is used The interphase transfer is governed by a mixture model where the interface length scale is 1 mm Flow is assumed to be homogeneous and isothermal at 29315 K The saturation pressure is 3619 Pa and the mean nucleation site diameter is 2μm

3 Design of Baseline Configuration A Francis turbine runner with specific speed of vs=035 was selected for this study where the specific speed

is defined by

( )(2 ) mrVp p B Wm

θπ ρ+ minus partminus =

part

12

12 34(2 )QvsgH

ωπ

= (2)


2

The runner meridional geometry is presented in Fig1 The runner maximum diameter is 1575 mm and its axial length is 140 mm The runner meridional shape is usually fixed by design constraints and therefore it was not changed during the design process The runner has 13 blades with a maximum profile thickness of 7 mm at the 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 The design specifications and inlet condition were imposed according to their values at the operating condition A free-vortex flow distribution (uniform spanwise rVθ ) was assumed at the runner inlet The value of rVθ was chosen 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 045 m^3 min-1 Inlet total pressure 415 kPa Guide vane opening 73 deg Required Shaft Power 165 kW

Figure 1 Francis runner meridional contour

Streamwise Distance

Bla

deLo

adin

g

0 02 04 06 08 1-4

-3

-2

-1

0

CrownBand

Figure 2 Blade Loading distribution

Figures 2 represents the normalized loading distribution of the baseline runner design The loading is defined

at two sections (band and crown) and it is then interpolated over the meridional channel Each loading distribution is plotted against the normalized streamwise distance from leading edge (streamwise distance=0) to trailing 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 (see equation [1]) The baseline design runner geometry obtained by the inverse code is presented in Fig3


3

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 was performed 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 the medium 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 is reached for the medium size mesh This mesh size is used for all computation in the present work hereafter The performance characteristics also show that runner achieves the required power output with a good efficiency and performs well at off-design condition In this figure the runner head power and hydraulic efficiency are plotted against 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 roughly aligned with the streamwise direction on the suction side of the blade whereas near the pressure side inside the boundary layer the flow is forced towards the band which indicates its strong three-dimensional character and the distinct secondary flows in Francis runner Figure 7 shows the runner pressure distribution at three spanwise sections ie crown midspan and band The low pressure region on the band suctions side indicates that 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 Fig8 confirming strong cavitation on the shroud near the trailing edge region

2uK U gH=

TgQH

ωηρ

=

(3)

(4)


4

Ku

Hea

d[m

]

05 06 07 08 0932

34

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CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

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1

CoarseMediumFine

(c)

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

(a)

(b)

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

point

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Figure 7 Baseline design blade pressure

distribution at design point

Figure 8 Baseline design contours of water vapour volume fraction at design point


5

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

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CrownMidspanBand

Streamwise Distance

Sta

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ress

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[kP

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0 02 04 06 08 1-100

-50

0

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CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

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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 by varying the stacking to -15 -30 and -45 degrees The negative sign indicates the direction of stacking in such a way 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 design parameters were kept unaltered

Figure 9 shows the 3D geometries of the runner for different stacking conditions and Fig 10 presents the corresponding blade pressure distributions at design condition obtained from a single-phase flow analysis for each case As it can be seen from these plots by increasing the stacking to -15 degrees the loading at the band is reduced and increased at the crown however there is a still a low pressure region at about 20 chord followed by 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 low pressure region is significantly reduced and is limited to a small region between 75-90 chord from midspan to 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 cavitation analysis presented in Fig11 in form of water vapour volume fraction contours on the blade surfaces confirms the 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


6

(a)

(b) (c)

Figure 11 Contours of water vapour volume fraction at -15 deg (a) -30 deg (b) and -45 deg (c) stacking design configuration

6 Parametric Study of Blade Loading The design with 30 degrees stacking which has a mid-loaded distribution both at the crown and the band

(Design S30_MM) is selected for further investigation of the blade loading distribution Since cavitation occurs toward in the blade aft part from midspan to band a fore-loaded distribution is specified at the band while the crown loading remains unaltered as shown in Fig12 All the other design parameters of the runner are unaltered

Figure 13 shows the blade pressure distribution at design point obtained from single-phase flow analysis for this design (Design S30_MF) The pressure distribution plots show that that the low pressure region is raised above the water vapour pressure at all sections and hence a cavitation-free design can be expected at design operating condition This is further confirmed by a two-phase flow cavitation analysis as shown in Fig14 in terms of water vapour volume fraction contours on the blade surfaces where no region of cavitation can be observed at least the design conditions

Figure 15 shows the velocity vectors on the suction and pressure surfaces on the runner Comparing to secondary flow structure of the baseline design with no stacking secondary flow on the pressure surface is reduced close to the crown but is increased towards the band This is due to a different spanwise work distribution in the runner caused by the stacking condition which increases the blade loading towards the crown and decreases it towards the band

Figure 16 shows a comparison of blade sections between the baseline design and Design S30_MF at crown midspan and band The effect of the prescribed stacking condition and loading distribution on the resulting blade geometry obtained from the inverse design method can be clearly seen in these figures The overall flow turning of the baseline design is 204 236 and 323 degrees and for Design S30_MF is 255 217 and 214 degrees at crown midspan and band respectively This agrees with the increase of the blade loading at crown and its reduction at the hub for Design S30_MF due to the prescribed stacking condition

Finally Fig17 shows a comparison of the baseline runner performance characteristic with that of Design S30_MF The results show similar head and power and efficiency characteristics for both designs

S tre a m w ise D ista nce

Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd

Figure 12 Design S30_MF Blade loading distribution

S tream wise D istance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand

Figure 13 Design S30_MF Blade pressure distributions


7

Figure 14 Design S30_MF Contours of water vapor volume fraction

(a)

(b)

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

XY

Z

BaselineDesignS30_MF

(a)

X

Z


(b)

XY

Z


(c)

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


8

Ku

Hea

d[m

]

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(b)

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(c)

Figure 17 Comparison of the baseline and DesignS30_MF runner performance characteristics at design flow rate 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 design parameter (stacking condition and blade loading) on the flow field inside the runner was studied in a parametric way The aim of design was to obtain a cavitation free runner with high hydraulic efficiency The flow field and suction 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 field inside the runner was optimized so that the low pressure region on the blade suction side was eliminated and a cavitation 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 to suppress cavitation phenomena without deteriorating the hydraulic efficiency The design guidelines presented in this 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

B H LE Ku m P Q r T

Number of blades Runner head [m] Leading edge Non-dimensional blade velocity Merdional distance Static pressure [Pa] Flow rate [m3s] Radius [m] Torque [Nm]

TE U V vs W θ ρ ω

Trailing edge Blade velocity[ms] Absolute velocity[ms] Specific Speed Relative velocity[ms] Circumferential direction Density [ kgm3] Rotational Speed [rads]

References [1] Drinta P Sallaberger M 1999 Hydraulic Turbines- Basic Principle and State-of-the-Art Computation Fluid

Dynamics Application Proc Institute of Mechanical Engvol 213 (Part C) pp 85-102 [2] Sallaberger M 1996 Quasi-Three-Dimensional and Three-Dimensional Flow Calculation in a Francis

Turbine IGTI (Birmingham) p 96-GT-38 [3] Keck H Goede E and Pestalozzi J 1990 Experience with 3D Euler Flow Analysis as a Practical Design

Tool In Proc of 16th IAHR Symp(Sao Paolo Brazil) [4] Nagafuji T Uchida K Tezuka K and Sugama K 1999 Navier Sokes Prediction on Performance of a


9

Francis Turbine with High Specific Speed ASME 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 Turbomachinery 118 536-561

[6] Zangeneh M Goto A and Harada H 1998 On the Design Criteria for Suppression of Secondary Flows in Centrifugal and Mixed Flow Impellers ASME J of Turbomachinery 120 723-35

[7] Bonaiuti D Zangeneh M Aartojarvi R and Eriksson J 2010 A Parametric Design of a Waterjet Pump by Means of Inverse Design CFD Calculations and Experimental Analyses ASME J of Fluids Eng132 031104

[8] Goto A Zangeneh M 2002 Hydrodynamic Design of Pump Diffuser Using Invese Design Method and CFD ASME J of Fluids Eng 124 319- 328

[9] Okomoto H Goto A 2002 Suppression of Cavitation in a Francis Turbine Runner by Application of 3D Inverse Design Method ASME 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 Fluids 13 599-624


10

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

K Daneshkah1 and M Zangeneh2 1Advanced Design Technology London WC1E 7JN UK 2Department of Mechanical Engineering University College London London WC1E 7JE UK

E-mail kdaneshkhahadtechnologycouk

Abstract The present paper describes the parametric design of a Francis turbine runner The runner geometry is parameterized by means of a 3D inverse design method while CFD analyses were performed to assess the hydrodymanic and suction performance of different design configurations that were investigated An initial runner design was first generated and used as baseline for parametric study The effects of several design parameter namely stacking condition and blade loading was then investigated in order to determine their effect on the suction performance The use of blade parameterization using the inverse method lead to a major advantage for design of Francis turbine runners as the three-dimensional blade shape is describe 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 this study 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 design guidelines 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 are based 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 turbine runners is highly rotational and three-dimensional and therefore only three-dimensional methods will provide effective solution for a Francis runner A considerable improvement in the design of Francis turbines have been obtained by the use of Computational Fluid Dynamics (CFD) CFD results provide a better understanding of the flow physics and they are now commonly used in industry ref [1-4] Although these methods are very useful for analysis in different design configurations they cannot be directly used as a design tool as they do not provide any direct information 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 used in 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 design method for the design of the runner shapes Unlike conventional direct design methods where the blade geometry is described by geometrical parameters inverse design uses hydrodynamic parameters like the blade loading to compute the blade shape offering a major advantage in the design process Such an approach allows designers to directly relate their understanding of flow physics in the design process and hence access a larger part of the design space The application of 3D inverse design method has already resulted in important design breakthroughs such as suppression of secondary flows in radial and mixed flow impeller impellers [5-6] improvement of suction performance and efficiency of water jet pumps [7] suppression of corner separation in pump diffusers [8] and improvement 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 design method 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 the basic design specifications of the Francis turbine runner Next the impact of stacking condition on the runner


ccopy 2010 IOP Publishing Ltd 1





(1)










is defined by

( )(2 ) mrVp p B Wm


part

12

12 34(2 )QvsgH

ωπ

= (2)


2






Streamwise Distance

Bla

deLo

adin

g

0 02 04 06 08 1-4

-3

-2

-1

0

CrownBand





3








2uK U gH=

TgQH

ωηρ

=

(3)

(4)


4

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42

CoarseMediumFine

(a)

Ku

Po

we

r[

kW

]

05 06 07 08 09140

145

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185

CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

098

1

CoarseMediumFine

(c)


(a)

(b)


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Streamwise Distance

Sta

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ress

ure

[kP

a]

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5

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(a) (b) (c)


(a)

(b)

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6

(a)

(b) (c)









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(a)

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Nomenclature

B H LE Ku m P Q r T









9









10





(1)










is defined by

( )(2 ) mrVp p B Wm


part

12

12 34(2 )QvsgH

ωπ

= (2)


2






Streamwise Distance

Bla

deLo

adin

g

0 02 04 06 08 1-4

-3

-2

-1

0

CrownBand





3








2uK U gH=

TgQH

ωηρ

=

(3)

(4)


4

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42

CoarseMediumFine

(a)

Ku

Po

we

r[

kW

]

05 06 07 08 09140

145

150

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CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

098

1

CoarseMediumFine

(c)


(a)

(b)


point

Streamwise Distance

Sta

ticP

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[kP

a]

0 02 04 06 08 1-100

-50

0

50

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CrownMidspanBand





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Streamwise Distance

Sta

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Sta

ticP

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[kP

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0

50

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CrownMidspanBand




(a) (b) (c)


(a)

(b)

(c)



6

(a)

(b) (c)









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deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

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C row nB a nd



Sta

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[kP

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7


(a)

(b)


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(a)

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(b)

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Z


(c)



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Ku

Hea

d[m

]

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Ku

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W]

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155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10






Streamwise Distance

Bla

deLo

adin

g

0 02 04 06 08 1-4

-3

-2

-1

0

CrownBand





3








2uK U gH=

TgQH

ωηρ

=

(3)

(4)


4

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42

CoarseMediumFine

(a)

Ku

Po

we

r[

kW

]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185

CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

098

1

CoarseMediumFine

(c)


(a)

(b)


point

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand





5

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand




(a) (b) (c)


(a)

(b)

(c)



6

(a)

(b) (c)









Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd



Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand



7


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10








2uK U gH=

TgQH

ωηρ

=

(3)

(4)


4

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42

CoarseMediumFine

(a)

Ku

Po

we

r[

kW

]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185

CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

098

1

CoarseMediumFine

(c)


(a)

(b)


point

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand





5

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand




(a) (b) (c)


(a)

(b)

(c)



6

(a)

(b) (c)









Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd



Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand



7


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42

CoarseMediumFine

(a)

Ku

Po

we

r[

kW

]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185

CoarseMediumFine

(b)

Ku

η

05 06 07 08 09092

094

096

098

1

CoarseMediumFine

(c)


(a)

(b)


point

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand





5

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand




(a) (b) (c)


(a)

(b)

(c)



6

(a)

(b) (c)









Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd



Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand



7


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand

Streamwise Distance

Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

CrownMidspanBand




(a) (b) (c)


(a)

(b)

(c)



6

(a)

(b) (c)









Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd



Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand



7


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10

(a)

(b) (c)









Bla

deLo

adin

g

0 0 2 0 4 0 6 0 8 1-4

-3

-2

-1

0

C row nB a nd



Sta

ticP

ress

ure

[kP

a]

0 02 04 06 08 1-100

-50

0

50

100

150

200

250

300

C rownM idspanBand



7


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10


(a)

(b)


XY

Z


(a)

X

Z


(b)

XY

Z


(c)



8

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10

Ku

Hea

d[m

]

05 06 07 08 0932

34

36

38

40

42


(a)

Ku

Pow

er[k

W]

05 06 07 08 09140

145

150

155

160

165

170

175

180

185


(b)

Ku

η

05 06 07 08 090965

097

0975

098

0985

099


(c)




Nomenclature

B H LE Ku m P Q r T









9









10









10

Parametric design of a Francis turbine runner by - IOPscience

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Transcript of Parametric design of a Francis turbine runner by - IOPscience