Hydrodynamic design of axial hydraulic turbines

DANIEL BALINTEftimie Murgu University of ResitaFaculty of Engineering - CCHAPT

P-ta Traian Vuia 1-4, RO-320085 ResitaROMANIA

daniel.balint@gmail.com

VIOREL CAMPIANEftimie Murgu University of ResitaFaculty of Engineering - CCHAPT

P-ta Traian Vuia 1-4, RO-320085 ResitaROMANIA

v.campian@uem.ro

Abstract:This paper presents a complete methodology of the hydrodynamic design for the runner of axial hydraulicturbines (Kaplan) using the finite element method. The procedure starts with the parametric design of the meridianchannel. Next, the stream traces are being computed in the meridian channel using the finite element method. Thefinite element method is implemented numericaly in an original software calledQTurbo3D. The last stage is todesign the runner blade usingQ3D techniques.

Key–Words:design, axial hydraulic turbines, meridian channel, runner blade,QTurbo3D

1 Introduction

The hydrodynamic design of the hydraulic turbinesshould be ourdays fast enough to obtain a blade pat-tern for the optimization. Even if a reliable optimiza-tion procedure will adjust the runner blade surface, thetime spent to converge to a suitable blade depends onthe first guess of the blade made in the design.

The hydrodynamic design of the runner blade isperformed usualy withQ3D techniques and next theblade hydrodynamics is computed with3D codes aspresented in [1] and [2]. The parametrization of therunner blade surface is directly linked to the speed ofthe optimization loops and also to its numerical con-vergence as presented in [3].

A procedure of the hydrodynamic design and op-timization of hydraulic turbines is presented in [4]where an original software calledTurboCADoptimisdeveloped. The softwareQTurbo3D is based on thedesign ideas ofTurboCADoptim, but other power-ful numerical techniques are being employed. Alsospecial programming tools likenumpyandpsycoofPythonprogramming language are used for having afaster computing time. For having a friendly acces tothe code, an original graphical user interface is devel-oped inQt4 library and next it is translated toPyhtonprogramming viaPyQt4tool.

The first part of the hydrodynamic design of therunner of axial turbines is to design the hydrodynamicmeridian channel. For this, a fast and reliable tech-nique of parametrization is used as presented in [5].Next the stream traces are computed in the 2D axi-symmetric meridian channel of the turbine. For thispart, a finite element method presented in [6] is em-

ployed. The last part is to design the runner bladesurface and to assure a smooth three dimensional linkbetween the designed foil cascades.

2 Meridian channel designThe hydrodynamic design of the meridian channel isbased on the design guidelines presented by Wu in [5],chapter 12. A special graphical user interface has beendeveloped inQTurbo3Das shown in Fig. 1.

The design starts from the headH and powerPof the turbine. With these, the specific speed is:

ns =21447√H + 10

− 924 (1)

and next the speed of the runner becomes:

n =nsH

5/4

√P

(2)

The speed of the runner should be adjusted to asynchronous one linked to a particular electrical polepairspe as:

n =3000

pe(3)

The discharge of the turbine is computed as:

Q =P

ρgH(4)

whereρ is the density of the water andg is the gravity.The peripheral velocity coefficient is:

kuR = 0.90163+ 0.001414ns (5)

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 104

Figure 1: Parametric design of Kaplan hydraulic turbines

Figure 2: Building the meridian channel of the hydraulic turbine

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 105

and then the diameter of the runner becomes:

D = kuR

√2gH

nπ(6)

The relative hub diameter is:

ν = 0.2718 +83.2338

ns(7)

The number of the blades in the runner equals to8 for heads greater than35m, 4 blades if the head ofthe turbine is less than18m and in the rest of the headrange the runner contains6 blades. The distributorheight is usualy 40% of the runner diameter while therelative runner depth is 21% of the runner diameter.

The hub curve is as follows (relatively to the run-ner diameter): height of the sphere over the runner shaft is 5%.

elbow radius is 50%.

the ogive radius is 60%.

the ogive length is 90%.

the ogive lowest diameter is 10%.The shroud curve is as follows (also relatively to

the runner diameter): the elbow 1st center and radius is 8.1% and 5%.

the elbow 2nd center and radius is 8.7% and 10%.

the cone 1st and 2nd radii equal to 15.4% and5.52%.

the cone starts with a diameter of 98.1% and has anangle of 8.3 degrees.

The output of the meridian channel of the turbineis plot in real-time when scrolling the bar as shownin Fig. 1. Multiple choices are available to be set asparameter and modified in the design algorithm.

The next stage is to split the meridian channel inthree parts: distributor, runner and cone. For this, theinterfaces between the three parts should be deliveredby slope and interface as presented in Fig. 2.

3 Computation of the stream tracesusing the finite element method

The stream traces in the meridian channel of the hy-draulic turbine are computed using the finite elementmethod presented in [6]. First of all, the meridianchannel of the turbine should be meshed as presentedin Fig. 3. The mesh element is a quasi-linear quadri-lateral one. The flow is assumed to be inviscid andincompressible. Both plane and axi-symmetric solverare available inQTurbo3D. The stream traces com-puted are quickly visualized and interpreted as shown

in Fig. 4. The region of the stream traces in the runnerpart is cut off and parameterized in order to have thestream tubes for the design of the runner blade. Theyare transposed in coordinates and fitted with a 5th or-der polinomial function.

The two-dimensional axi-symmetrical flow is as-sumed incompressible, irrotational and inviscid. Theproblem formulation uses the stream functionψwhere the velocities are obtained as:

vx =∂ψ

∂y(8)

vy = −∂ψ∂x

(9)

The stream function formulationψ satisfies the2D Laplace equation:

∇2ψ = 0 (10)

The computational domain is shown in Fig. 3.The stream functionψ is constant on the hub andshroud curves, thereforeψ = 0 for the hub andψ = 1for the shroud respectively.

The elemental equation is:

n∑

j=1

ψj

∫

Ωe

∇Ni · ∇Nj dΩ =

∫

Se

Ni∂ψ

∂nds (11)

wheren = 4 is the number of quadrilateral elementnodes.

The matrix form becomes:

[Ee] ψe = Ψe (12)

where the element matrix and the right-hand-sideterms are as follows:

Eij =

∫

Ωe

(

∂Ni

∂x

∂Nj

∂x+∂Ni

∂y

∂Nj

∂y

)

dΩ (13)

Ψi =

∫

Se

Ni∂ψ

∂nds (14)

4 Hydrodynamic design of the tur-bine runner blade

For starting the design of the Kaplan runner blade, fewparameters should be available from plant geometry orfrom the previous meridian channel design: the headof the turbine, the power, the runner speed and diam-eter together with the relative hub diameter, but alsothe number of blades.

The inlet and outlet velocity profiles should be de-livered related to the flow produced by the distributor

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 106

Figure 3: The mesh of the meridian channel with quadrilateral elements

Figure 4: Computing the streamlines using the finite element method

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 107

Figure 5: Designing the runner of the turbine using theQ3D technique

Figure 6: Vizualization of the designed 3D runner blade

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 108

of the one designed for the draft tube inlet. The dif-ference between them shows the loading of the cas-cades, therefore a law of the radial loading distribu-tion should be imposed. Usualy, the loading at huband shroud cascades should be decreased in order toreduce the unsteady effects due to the clearance be-tween the hub/shroud curves and runner blade. It isknown that this clearance is usualy 0.1% of the runnerdiameter and it generates vortex ropes at off-designoperating regimes.

Next, the relative chord length of the foils shouldbe delivered (the values in the middle span section andat the hub) or they will be imposed as:

(l/t)med = zbl78

360(15)

(l/t)hub = 1.1(l/t)med (16)

wherezbl is the number of blades.The thickness function of the foils should be de-

livered toQTurbo3Dby its maximum thicknessdmax

and the position of the maximum thicknessxdmax(the

values are relative to the chord length of the foil):

yd =dmax

2

m

√x− x

m

√xdmax

− xdmax

(17)

whereyd is the thickness value for a foil with unitchord length and placed orizontaly with the leadingedge in the origin of coordinate system. The coeffi-cientm of the root in the above formula is computedin the range of20÷60% of the relative position of themaximum thickness as:

m(xdmax) =

4∑

i=0

uixidmax

(18)

where the coefficients have the valuesu0 = 10.985,u1 = −69.7486, u2 = 187.632, u3 = −237.949 andu4 = 115.706.

Next, the relative position of the maximum load-ing from leading to trailing edge of the foils shouldbe delivered. Usualy, a value of 35% is used, but itshould be related also to the position of the blade axisin order to avoid torsion moments of the blade.

For this part of the hydrodynamic design of therunner blade, a graphical user interface is built inQTurbo3Das shown in Fig. 5. Then, the three di-mensional blade is plot using theGnuplot library aspresented in Fig 6.

5 ConclusionThis paper shows a method to employ the finite ele-ment method in the hydrodynamic design of the run-ner of axial turbines. The finite element formulationpresented in [6] is implemented in an original soft-wareQTurbo3Ddeveloped by the authors.

The meridian channel of the turbine is meshedby quasi-linear quadrilateral elements directly inQTurbo3Dand the stream traces are being computedusing the finite element method (the solver is a part ofQTurbo3D).

In conclusion, the design of the runner becamefaster and more easily to adapt in parametrical opti-mization procedures.

Acknowledgements: The present work has beenco-funded by the Sectoral Operational ProgrammeHuman Resources Development 2007-2013 of theRomanian Ministry of Labour, Family and SocialProtection through the Financial Agreement POS-DRU/89/1.5/S/62557.

References:

[1] High Performance Computing Center,COVISE,http://www.hlrs.de/organization/vis/covise/.

[2] Advanced Design Technology,TURBOdesign,http://www.adtechnology.co.uk/products/turbo-design1/.

[3] L. Ferrando,Surface Parametrization and Op-timum Design Methodology for Hydraulic Tur-bines (PhD Thesis), Swiss Federal Institute ofTechnology, Lausanne 2006.

[4] D. Balint, Numerical Computing Methods for3D Flows in the Distributor and the Runnerof Kaplan Turbines (PhD Thesis), Politehnica,Timisoara 2008.

[5] R. Krishna,Hydraulic Design of Hydraulic Ma-chinery, Aldershot, Avebury 1997.

[6] R. Resiga, S. Muntean, Periodic boundaryconditions implementation for the finite ele-ment analysis of the cascade flows,Sc. Bul.of Politehnica University of Timisoara, 1999,pp. 151–160.

Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements

ISBN: 978-960-474-298-1 109