Kaplan turbine blade design with Octavian Popa method.

Manea, Adriana Sida ; Barglazan, Mircea ; Stroita, Daniel Catalin 等

1. INTRODUCTION

For the projection of axial hydraulic turbines the method of conformal mapping permits the determination of runner blade airfoil on the base of some computations which permits the theoretical estimation of the airfoil energetical and cavitational behaviour. Although the calculus method suppose the consideration of the potential flow, the results obtained theoretically, permit the analysis of the possible performances of the airfoil in operation in real flow.

The method, of conformal mapping used at the design of the axial turbines runners, have the base on the O. POPA method (Popa, 2004). This method permits to obtain the aerodynamic characteristics of the airfoils obtained through arbitrary parameters which enter in the conformal mapping of the chosen function. Through the modification of the parameters we obtain different airfoils which appertain to the same family.

In the case of a conformal mapping the velocity circulation doesn't change. The transformation must be made as the trailing edge to correspond to a stagnation point on the generative circle.

If we know the dynamics on the external unit disc domain, through a conformal mapping function we can obtain the movement in the physical plane on the airfoil external domain.

The method of conformal representation can be applied for airfoils networks, accomplishing the conformal mapping of the external of a circle network on the external domain of a airfoils network (Stepanov, 1962).

At the design of an axial turbo machine, it intersects the runner with co-axial cylinders. The evolute of a cylindrical section on the tangent plane forms a linear airfoils network. The movement trough this airfoils network is planar.

2. O. POPA METHOD

The method, of Professor Octavian Popa for airfoils networks, suppose to consider the airfoils boundary appertaining to the network represent the image through the conformal mapping function.

Conformal to O. Popa method, the parameter m used in the design is the solution of the transdescendent equation:

l/t = 2/[pi] {(cos [lambda]) Arth(2m/M cos [lambda]) + (sin [lambda])arctg (2m/M sin [lambda])} (1)

[lambda]--install angle of network

M = [square root of 1 + [m.sup.2] + 2[m.sup.2] cos(2[lambda])] (2)

The network is obtained through parametric presentation:

X = 1/2 + [X.sub.L]([psi]) (3)

Y = [a.sub.0]/2 + [N.summation over (n=1)] [[a.sub.n] cos(n[psi]) + [b.sub.n] sin(n[psi])] (4)

The coefficients of the network depend on the coefficients [a.sub.n] and [b.sub.n] of the isolated airfoil, on the network [lambda] and on network pitch to chord ratio t/l.

A plane airfoils network is described through geometrical parameters of the airfoil from the network: relative arrow f/l, relative thickness d/l and through parameters which define the emplacement of the airfoil in the network: install angle [lambda] and pitch to chord ratio t/l, where the pitch to chord ratio is obtained by the number of runner blades (Anton, 1979). The emplacement angle of the airfoil in the network is [beta] = [pi]/2 - [lambda], and the smooth of the network is the ratio l/t.

For the construction of the runner blade, we start from a generative airfoil with the known coordinates x, y and we obtain the coefficients of the trigonometrical polynomial approximation using the formulas:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. THE CHOOSING OF THE GENERATIVE AIRFOIL

In order to construct the runner blade, we chose as a generative airfoil a theoretical airfoil, for which were obtained the coefficients [a.sub.n] and [b.sub.n] of the trigonometrical polynomial. The pressure distribution calculated with O. Popa method is presented in fig. 1.

[FIGURE 1 OMITTED]

4. THE DESIGN OF RUNNER BLADE

For a turbine, with the characteristics presented in table we choose as a generative airfoil a theoretical airfoil and the thickness and the arrow were calculated for the characteristic dimensions.

The blade was calculated with O. Popa method (Popa, 2007) in 11 sections having the following elements:

Tab. 2. Calculated blade

Radius Chord f/l d/l [[beta].sub.s]
 [m] length [m] [[degrees]]

0.5 0.53915 0.052 0.13 36.56811
0.575 0.60762 0.0375 0.12 29.41753
0.65 0.67285 0.024 0.11 24.98855
0.725 0.73486 0.0185 0.1 22.0816
0.8 0.79362 0.015 0.09 19.85673
0.875 0.84916 0.0123 0.08 18.03378
0.95 0.90145 0.011 0.07 16.6774
1.025 0.95051 0.0096 0.06 15.43556
1.1 0.99634 0.008 0.05 14.50612
1.175 1.03893 0.007 0.04 13.41805
1.25 1.07829 0.0063 0.03 12.42049

In fig.2.--4. are presented the variation diagrams of kinematic and geometric parameters (Barglazan, 1999) for theoretical airfoil used in the design of the turbine runner:

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The runner obtained for turbine characteristics, using the O. Popa method is presented in fig.5.

5. CONCLUSIONS

The O. Popa blade design method proposed for hydraulic machinery permits the using of the same generative airfoil for different blade sections. The airfoil is approximated analytically and it's characteristics also through the coefficients of the trigonometrical polynomial approximation of the contour.

6. REFERENCES

Anton, loan, (1979). Hydrodynamic Turbine, Ed. Facla, Timisoara,

Barglazan, Mircea, (1999). Hydraulic Turbine and Hydrodynamics Transmisions, Ed. Politehnica, Timisoara, ISBN 973-9389-39-2.

Popa, Octavian, (2007). Fluid Mechanics, vol I, II, Tempus Timisoara, ISBN 978-973-88147-2-1, Vol I, ISBN 978973-88147-3-8, Vol II, ISBN 978-973-88147-4-5.

Popa, Octavian, (2004). Fluid Mechanics, Chosen scientific papars, vol I, II, Tempus Timisoara, vol.I, ISBN 97397842-8-3, 973-97842-9-1, vol.II, ISBN 973-97842-8-3, 973-87105-0-2.

Stepanov, G. I., (1962). Hydrodynamics Airfoil Network, Gosudarstvenoe, Izdatelstvo fiziko-matematiceskoi literaturi, Moskva.

Tab. 1. Turbine caracteristics

Calculus net head [H.sub.c]=18 m
Maximum flow [Q.sub.c]=35 [m.sup.3]/s
Nominal power P=5662,5 kW
Runner diameter D=2500 mm
Speed n=230,8 rot/min