Wind axis rotors with high aerodynamic efficiency.

Svrcek, Daniel ; Behulova, Maria

1. INTRODUCTION

Aerodynamic design of a blade for wind-mill rotor with high efficiency in a chosen design point requires correct definition of input conditions (Svrcek, 2009). Precise calculation depends on the supposed operation of the designed wind-mill rotor. The adjustment of the number of a rotor blades and a blade depth is possible using the circulation along the propeller blade. Circulation is connected with geometrical characteristics of the propeller blade defining the specific propeller of a wind-mill motor for a given designed mode.

2. POWER OF WIND-MILL ROTOR

The rotor-propeller thrust T can be evaluated as the product of the propeller area S and pressure difference in front and behind of propeller [DELTA]p dependent on the velocities [v.sub.0] and [v.sub.1] (Fig. 1)

T = S [DELTA]p = 0.5[pi] [R.sup.2] [rho]([v.sup.2.sub.0] - [v.sup.2.sub.1]). (1)

The total power [W.sub.c] which is possible to attain from the air flow with the velocity of [v.sub.s] is given by the relationship

[W.sub.c] = T [v.sub.s] = 0.5[pi] [R.sup.2] [rho]([v.sup.2.sub.0] - [v.sup.2.sub.1]) [v.sub.s]. (2)

Considering the axial (inlet) interfacial factor a given by the relationship [v.sub.0] - [v.sub.s] = a.[v.sub.0] which influences the air flow in the rotation axis, the power convertible to the needful power of the generator drive (alternator) W can be calculated as

W = 2[pi] [R.sup.2] [rho] [v.sup.3.sub.0]a[(1 - a).sup.2]. (3)

The expression for the maximal power which is possible to withdraw from the energy flowing through the rotor area transforms to the task to find out the maximum of the term

W/[W.sub.c] = 4.a[(1 - a).sup.2] (4)

which is for a = 1/3 and the power W = 0.5925 [W.sub.c]. (Svrcek, 2009).

[FIGURE 1 OMITTED]

Taking into account mechanical losses and the efficiency of mechanical energy transformation to electric energy through alternator, it is possible to consider the rotor with the efficiency of 45% as a very good. Then the wind-mill power is

W = 0.225 [pi] [R.sup.2] [rho] [v.sup.3.sub.0]. (5)

The wind velocity [v.sub.0] depends on the weather conditions of the wind-mill motor localization. Applying the relationship (5), the rotor perimeter can be calculated for the supposed power W

R = [[W/(0.866 [rho] [v.sup.3.sub.0)].sup.0.5]. (6)

To reduce the noise level, the tip speed of a propeller blade should be not higher than [u.sub.0] = 60 m.[s.sup.-1] (Filakovsky, 2007) from which it follows for the propeller revolutions n

[u.sub.0] = [omega]R = 2[pi] nR [??] n = [u.sub.0]/(2[pi]R). (7)

3. ROTOR CALCULATION IN A DESIGN POINT

For the supposed power W of a designed wind-mill rotor, the input parameters [v.sub.0]--the swell wind velocity, D--the propeller diameter, n--the wind motor revolutions and z--the number of propeller blades must be defined. For the initial design, the number of propeller blades is obviously chosen z = 3. The task is to find out the geometry of a propeller blade with the maximal efficiency [eta]. In the propeller practice, the term velocity ratio is introduced by the relationship (Broz & Slavik, 1979)

[lambda] = [v.sub.0]/(nD). (8)

The thrust T, drag Q and power P of propeller are defined by equations (Svrcek, 2009)

T = [rho][n.sup.2] [D.sup.4] [c.sub.T], Q = [rho][n.sup.2][D.sup.4][c.sub.Q], P = [rho][n.sup.3][D.sup.5][c.sub.P] (9)

in which [c.sub.T], [c.sub.Q], and [c.sub.P] are the thrust, drag and power coefficients, respectively. For the maximal efficiency, the trust must be maximize and/or the absorbed power minimize what leads to the following requirement

[eta] = T[v.sub.0]/P [right arrow] max, [eta] = [lambda][c.sub.T]/[c.sub.P] [right arrow] max. (10)

In another words, it is necessary to find such distribution of lifting forces--circulations along the propeller blade <0, R> which will assure the maximal efficiency. Using circulation T defined by the equation (11) the lift on the propeller blade L and in a discrete area of propeller crossection dL are given by the relationships

[GAMMA] = 0.5 [w.sub.1] [c.sub.L] b, (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [c.sub.L] is the lift coefficient and b is the blade depth. To simplify the calculations, it is useful to introduce non-dimensional quantities

[bar.[GAMMA]] = [GAMMA]/4[pi][R.sup.2][omega], [bar.b] = zb/4[pi]R, [bar.r] = r/R, [[bar.v].sub.1] = [v.sub.1]/R[omega], [[bar.u].sub.1] = [u.sub.1]/R[omega]. (13)

Applying Fig. 1, the real influent velocities in the rotation plane of a propeller disc can expressed as

[[bar.w].sub.1] = [square root of ([bar.u].sup.2.sub.1] + [[bar.v].sup.2.sub.1])], [[bar.u].sub.1] = [[bar.u].sub.0] + [[bar.u].sub.i], [[bar.v].sub.1] = [[bar.v].sub.0] + [[bar.v].sub.i]. (14)

Substituting from the relationship (8), the velocities [[bar.u].sub.1] and [[bar.v].sub.1] are

[[bar.u].sub.1] = [bar.r] + [[bar.u].sub.i], [[bar.v].sub.1] = 0.5 [lambda]/[pi] + [[bar.v].sub.i]. (15)

Based on the Zhukovsky whirl scheme of concentric semi-infinite cylinders (Broz&Slavik, 1979), the relationships between induced velocities and circulation can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Zhukovsky theory is valid for the propeller with infinite number of blades. With decreasing number of blades the deviation from the really measured values enhances. In this reason, the error is corrected using correction coefficient C dependent on the number of blades z and velocity ratio [lambda] (Broz & Slavik, 1979)

C = 1 + 1.803 [z.sup.-1.16] [[lambda].sup.2] - 0.459[z.sup.-1.062] [[lambda].sup.3] + 0.0243[z.sup.-0.835] [[lambda].sup.4] (17)

Then the relationships (16) can be rewritten to the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

from which it is possible to determine geometric relations and constructional parameters of a propeller--the angle of a blade twist [phi], the blade depth b and the blade thickness t. From Fig. 1, the following geometric relations are apparent

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Finally, it can be concluded that non-dimensional circulation [bar.[GAMMA]] represents the single unknown for expression of all constructional parameters of a propeller.

4. DISTRIBUTION OF OPTIMAL CIRCULATION

The calculation of optimal circulation is based on a iterative method using which it is possible to approximate the values of circulation in single propeller sections along its perimeter. The criterium of accuracy appears to be sufficient when the two subsequent values of circulation differ in absolute value less than [absolute value of [bar.[GAMMA]] - [[bar.[GAMMA]].sub.(-1)]] [less than or equal to] [1.10.sup.-4]. Using iteration constants [h.sub.1], [h.sub.2], [sigma], k and the drag to lift ratio [mu], the optimal circulation [bar.[GAMMA]] can be calculated from the relationship (Broz & Slavik, 1979, Svrcek, 2009)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

By the back substitution, resulting integral quantities can be calculated. Applying circumferential force in the plane of rotation N, the power of wind-mill motor W is (Svrcek, 2009)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

5. CONCLUSION

The described mathematical methodology together with applied iteration method enables to find the optimal distribution of circulation along the blade span. Therefore, for the chosen design point of the wind-mill motor, the constructional parameters of a propeller with regard to the maximal power characteristics can be defined. For more accurate calculation, the correction coefficient taking into account the number of blades and the velocity ratio was introduced and applied as well.

6. ACKNOWLEDGMENTS

The research has been supported by the project VEGA MS and SAV of the Slovak Republic No. 1/0837/08 and 1/0256/09.

7. REFERENCES

Broz, V. & Slavik, S. (1979). Optimal distribution of circulation along propeller blade. Research Report of V-ZLU No. V 1348/79, Praha

Filakovsky, K. (2007). Design of a propeller blade for wind motor. Transfer, VZLU, Vol. 4, pp. 16-22, ISSN 1801-9315.

Svrcek, D. (2009). Aerodynamic Design of Propeller Blade with Optimum Circulation Layout. Proceedings of the XXIII. MicroCAD Int. Scientific Conference, pp. 43-50, ISBN 978-963-661-878-0, Miskolc, March 2009, University of Miskolc

Svrcek, D. (2009). Rotor of wind-mill motor. Acta Metallurgica Slovaca SI, 15, pp. 314-322, ISSN 1335-1532