125

Scientific Bulletin of the Politehnica University of Timisoara

Transactions on Mechanics Special issue

The 6th International Conference on Hydraulic Machinery and Hydrodynamics Timisoara, Romania, October 21 - 22, 2004

CAD TECHNIQUE USED TO OPTIMIZE THE FRANCIS RUNNER DESIGN

Teodor MILOŞ, Assoc. Prof * Department of Hydraulic Machinery “Politehnica” University of Timişoara

Mircea BĂRGLĂZAN, Prof. Department of Hydraulic Machinery “Politehnica” University of Timişoara

*Corresponding author: Bv Mihai Viteazul 1, 300222, Timişoara, România Tel.: (+40) 256 403683, Email: milost@home.ro

ABSTRACT The article introduces step by step a computer aided design technique for the radial-axial turbomachinery runner. The technique is applied for hydraulic turbines Francis type. The design method used for the turbine’s runner is based on conformal mapping. The result shows the advantages of CAD for optimizing the shape of the runner especially in respect of flexibility and computing-time.

KEYWORDS Hydraulic turbine Francis type, conformal mapping design method, computer aided design technique.

NOMENCLATURE

4/5T

SSCP

H

Pnn = [-] specific speed

( ) 43

2 TgH

Qπωυ = [-] nondimensional specific speed

π =3.14159 SP stereomechanical power

TH head Q rate of flow n speed of rotation ω angular speed g gravity acceleration

1. INTRODUCTION The theory and practice, accumulated till now, in the field of radial-axial hydraulic turbines, Francis type, runners offers a great amount of information referring to the options and methods of solving the encountered problems. Namely, in this article, the adopted solutions are focussed on the size and shape of the runner’s blades.

The classic design through grapho-analytical methods is cumbersome and didn’t put in evidence the usual positive and negative aspects of the designing options. So it was obvious that it is imposed as a necessity to realize a professional soft for a runner design together with an adequate graphic interface operating in real time. So any option chose during design calculus in a specific step of designing may be verified in ensemble and detail from all points of view. The final graphic post-processing would be realised only for the optimal solutions, through a 3D repre-sentation of the blades separately and the runner as a whole for the final approval of the geometric shape. Nowadays in Romania a lot of our hydropower plants are in the process of modernization. In this article it was investigated the refurbishment of the hydraulic turbine’s runner, medium head Francis type, with the following value for the most important parameter: the rated specific speed 238=SCPn , re-spectively the nondimensional specific speed 413.0=υ . The soft used is completely original and is in a continuous process of upgrading.

2. FLOW DOMAIN CONFIGURATION IN THE RUNNER ZONE AND FEM ANALYSIS OF THE AXIS-SYMMETRIC MERIDIONAL FLOW

Usually the initial design data establishes the operation parameters of the hydraulic turbine for the most frequently hydroenergetic conditions. The hydraulic turbine’s runner needs to show suitable energetic and cavitation characteristics in a large interval (extension) of the main parameters values. The calculus method for the important dimensions of the runner are given in literature [1], [2], [3] and [4]. For our design it was chose a modified Bovet method [3] and [4].

126

The reason consists in the believable experimental results published, till now, for this subject. Also the above mentioned method fits for the soft construc-tion. In Table 1 are presented the nondimensional (rated) results obtained.

Table 1 ns nq ν r1e r1i 237.90 65.17 0.4132 1.071 0.837 r2e r2i le li b0 1.0 0.492 1.154 4.933 0.433 x2e y2e rmax re ri 0.437 0.131 1.404 1.131 1.004

The Bovet relations give the geometry of the hub and shroud (the generating curves) in the runner’s blade zone (Fig. 1)

Figure 1. FEM analysis domain together with the characteristic dimensions

All the runner’s dimensions are rated in respect with the runner hydraulic outlet radius 1r e2 = . The real value of the outlet radius is calculated with the formula:

3/1

e2e2 )/Q(R

ω⋅φΠ

= (1)

where )(fe2 ν=φ with the numerical values between 0.24 and 0.28. In the investigated case 25.0e2 =φ . FEM analysis of the flow was extended with about 20% before the leading edge of the runner’s blades and also with the same quantity after the trailing edge of the blades.

This extension assures an uniform set of conditions for the liquid flow through the runner together with the limit conditions on the boundary. The discretization of the analysed domain through FEM was realized with quadrilateral finite iso-parametric elements with a total amount of 1111 grid knots (Fig. 2)

Figure 2. The discretization of the analysed domain with FEM

Figure 3 Streamline hydrodynamic spectra

Using specific techniques for smoothing and inte-rpolation with spline - functions it is obtained the meridional image of the hydrodynamic flow spectra (Fig.3) and the meridional velocity field variation along the streamlines (Fig.4).

127

Figure 4 The liquid velocity variation along the streamlines.

3. LEADING AND TRAILING EDGE SHAPE AND KINEMATICS OF THE FLOW (VELOCITIES VALUES AND ANGLES)

Analytical modality of the leading and trailing edge shape was treated in previous articles of the authors [5]. In this issue it was used the parabola arcs method, to match a definite point R1i at the leading edge and R2e at the trailing edge. The minimum of these curves is established only after the variation of the meridional velocities of the flow in the neighbourhood of the edges, is known, fig. 5.

Figure 5. Leading and trailing edge shape and their position.

Once there are established the leading and trailing edges position the blade domain is cut up from the whole domain of the hydrodynamic field. Further for

increasing the calculus precision of the bladed zone it are recalculated through interpolation, the domain of every streamline in 100 equal intervals, respectively 101 knots. The kinematics and angular elements of the blades are calculated in the first and in the last point of every truncated streamline which may be defined between the entrance and exit of the runner’s blade. Well known relations are used for the inlet and outlet of the flow between two blades namely in the blades channel. At them it was added the necessary corrections for the entrance and exit kinematics and flow angles by the obstruction of the flow through the blades of variable thickness and also the circulation of the flow in the neighbourhood of the leading and trailing edges. In tables 2 and 3 there are given the results of the calculus for kinematics and angles of the flow at the blade inlet and outlet. Table 2. Kinematics and angles of the flow at blade inlet

SL 0 2 4 6 8 10 U1 34.55 36.67 38.65 40.54 42.39 44.27W1 9.04 12.89 17.00 21.35 26.53 32.17V1 30.97 29.88 29.44 29.73 31.28 33.86α1 14.58 19.06 24.46 30.75 38.64 46.30β1 59.59 49.18 45.81 45.38 47.41 49.55

Vm1 7.80 9.76 12.19 15.20 19.53 24.48ρ1 0.88 0.87 0.87 0.87 0.88 0.89

Table 3. Kinematics and angles of the flow at blade outlet

SL 0 2 4 6 8 10 U2 20.31 24.91 29.10 33.37 37.57 41.29W2 21.82 27.38 32.26 36.64 40.74 44.49V2 7.98 11.38 13.92 15.14 15.74 16.57β2 21.45 24.56 25.56 24.41 22.73 21.87

Vm2 7.98 11.38 13.92 15.14 15.74 16.57ρ2 0.71 0.76 0.78 0.78 0.78 0.78

4. CAMBER SURFACE OF THE BLADE DETERMINED THROUGH CONFORMAL MAPPING METHOD

Any method of designing the camber surface of the blade uses as initial elements the blade angles at en-trance 1β and exit 2β . Usually the values of the blades angles between entrance and exit obtain values between extreme limits with a continuous variation between them. Technically the conformal mapping method reduces to straighten out the liquid particles trajectories

128

a) meridional plan

b) image-plan of the conformal mapping

c) horizontal plan

Figure 6. The correspondence between the geometric elements from the meridional plane with them from transversal projection of the blade through the image-plane of the conformal mapping.

on the blades to multiple cylindrical coaxial surfaces. Mathematically there are established bi-univoque correspondence between the camber curve of the blade lying on the streamsurface and the plane-image of the conformal mapping. This analytical transformation is possible if it are observed the conditions which need to be fulfilled from the function A(x) in the image plane (surface) Fig. 6 These conditions expressed mathematically are: • To intersect the point M:

A(Lin.) = AM. • To intersect the point O:

A(O) = 0. • The angle of the tangent to the curve A(x) in M

to be β1

1M tgdx

)x(dAβ=

• The angle of the tangent to the curve A(x) in O to be β2

2O tgdx

)x(dAβ=

The mathematical function which is most suitable for usual grapho-analytical solution is a third degree polynom, which has four unknown coefficients exactly equal with the before mentioned four conditions. In fig. 7 there are drawn ( ) ( )xfxA = for all the eleven streamlines chose and presented in meridional plane.

Figure 7. The function ( ) ( )A x f x= for the 11 stream-lines

In accord with the fig. 6, the wrapping angle iϕ∆ of the blade, in transversal projection resulted from the relation:

21

1

+

+

+−

=∆

=∆ii

ii

medi

ii RR

AAR

Aϕ (2)

129

Using a special algorithm it may be obtained the same total wrapping angle for every streamline, also the chose maximum angle value so that the function A (x) gave a curve without inflexions and the profile grid, used in the next steps of the design method, to be with simple curvature for the camber. In fig. 8 there are given the image of the camber surface of the blade obtained for a total (maximum) wrapping angle of 45o.

Figure 8. Camber surface of the blade in meridional plane and in transversal projection perpendicular on the axis of rotation.

5. CONSTRUCTION OF THE REAL BLADES USING NACA PROFILES

Once the camber surface is determined and verified in space – especially to not have distorsion – it follows the embarkment of this surface with profiles of adequate thickness, which will confer the necessary mechanical strength, the maximum energy transfer and the minimum drag by the liquid flow. For the designed runner with known specific speed it was chosen NACA profiles with 6 numbers (items) from the series “65” for whom the thickness relative func-tion is given by the formula:

⎥⎥⎦

⎤⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

⎢⎣

⎡+−

−

−=

32

8385.24478.2

2758.00675.117.01

1

lx

lx

lx

lx

lx

lx

ld

lyd

(3)

Abscissa x in this case is associated to the profile’s chord which in image plane Fig. 9 is the straight line OO`. The NACA profile which wrap the camber curve is initially referring to the coordinate system x`o`y`. Later the necessity to have A(x) = F(x) for the pressure and suction side of the profile develops a calculus and transposition of the profile contour points in the new coordinate system (A(x),ox).

Figure 9. Transposition of thickness on the camber line from image plan.

The transposition of the profiles for all the 11 streamlines produce the shape from fig. 10. It is necessary to observe that for the sake of clarity and nonsuperposition the drawing was so made that one profile is translated from another with 0.12 units.

Figure 10. Profiles transpositions on the camber curve from the image plane (the translation between profiles is with 0.12 units)

Finally, for a 3D image of the whole runner it was used an Auto Lisp program under AutoCAD environ-ment through which there are sketched the pressure and suction surfaces of the blades and the closing surfaces of the blades at the hub and the shroud fig. 12.

130

a) suction side

b) pressure side

Figure 11. Profiles contour curves in meridional plane and in transversal projection

Figure 12. 3D image of the runner in an isometric perspective view.

6. CONCLUSIONS 1. The erected soft presented in this article is a program

with remarkable flexibility. So it permits: • the optimal choosing of the leading and trailing

edge. • the rectification of the flow velocities. At the

inlet in the runner to match the wicket gates and by the outlet of the runner for the swirl and pressure oscillations (surge) in the draft-tube.

• the recursive calculus of the kinematics of the flow, the entrance and exit angles of the runner and the liquid particle trajectory from inlet to outlet of the runner zone, through exactly tacking into account the thickness function on every streamline.

2. The computing time for obtaining an optimal variant of radial-axial hydraulic turbine, Francis type, is smaller in comparison with the necessary length for a classic design

REFERENCES 1. Anton I., (1979), Turbine hidraulice, Ed. Facla,

Timişoara. 2. Bărglăzan M., (2001), Turbine hidraulice şi trans-

misii hidrodinamice, Ed. Politehnica, Timişoara. 3. Wislicenus G.F., (1965), Fluid mechanics of Turbo-

machinery, Dover Publ., New York 4. Ed. Krishna R., (1997), Hydraulic Design of

Hydraulic Machinery, Avebury, Aldershot. 5. Bărglăzan M. Miloş T., About Meridional Plan

View of a Francis Turbine Runner. Proc of the 5th Internat Conf. Hydraulic Machinery and Hydro-dynamics. Timişoara pp. 43-51.

6. Anton L.E., Miloş T., Pompe centrifuge cu impulsor, Editura „Orizonturi Universitare”, Timişoara, 1998.