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Home / Issues / № 2, 2014


Ainakul K. Yershina
Many farms of the republic are very interested in the use of independent sources of electrical energy, and primarily the wind energy available in abundance in almost every area of our country. This is due to the remoteness of some farms from the electrical power supply system, the expeditionary nature of the work of individual consumers, finally, just a shortage of electricity in the republic and therefore the desire to get away from the conjunctural-market vagaries of the allocation of fuel and energy resources. Interest to the problem of individual wind use is called by continuous rise in energy carriers prices. Wind power has become competitive with traditional methods of electricity generation.


Wind turbine torque

Consider the scheme of counterclockwise rotation of one of the working blades of the wind turbine with a certain angular velocity"" in Figure 1. Each of the positions shown in Figure 1 corresponds to the rotation of the blade at different angles "". Lifting power appears if wind flux leaks to the rotating turbine at a certain rate "". In fact, during the rotation of the turbine the air environment leaks to the blade at a speed equal to the linear speed of rotation of the latter (). By constructing a vector parallelogram from velocities and  it is easy to find the resultant vector  - speed of attack and angle of attack ""


                                =,                           (1)

where  - specific speed of the turbine [1,2].

To determine the angle of attack the law of tangents should be used and inserting we will receive

                                                                          .                                            (2)

The (1) and (2) formulas results the very important connections

                                              .                                                             (3)

The (1) - (3) formulas allow come to the following conclusions:

а) as the lifting power  is always directed normally to the  attack speed, it is only in the positions "A" and "C" (see Fig. 3) the lifting power is absent because sinq=0 and tga=0 at all other q angles the "pulling" force, equal to  is applied to the blade;

b) the greatest value  occurs when the blades pass the points "B" and “D”;

c) a pair of blades arranged symmetrically about the axis of rotation, lead to occurrence of a force couple (+) and (-), creating a torque which increases when the blade is moved from the point "A" to point "B", and decreases from "B" to "C" (subsequently on the opposite side of the blade the moment of force increases from point "C" to point “D”, and then decreases to point "A").

Lifting force of the  airfoil is directed perpendicular to the  attack velocity vector and is determined with the formula [3]

                                      ,                                                                 (4)


Figure 1. The scheme of one of the working wind turbine blades rotating counterclockwise.


where L –length of blade of a symmetric profile with a chord equal to b, - lift force coefficient.

The rotational moment is created by the tangential component of the lifting force, which is designed on the section chord line, or what is the same, on a tangent to the circumference of the movement of the blade:

                                                  .                                                                        (5)

A part of this force compensates for the resistance force of air to the movement of the blade, which can be written as the following dependence

                                               ,                                                   (6)

where - resistance coefficient of the airfoil. Thus, the joint moment that acts on the blade is equal to

or                                            .                                  (8)

where r0 - distance from the axis of rotation to the blade substantially equal to the length of strides.

Due to the fact that symmetrical working blades were used in our study and the strokes connecting them to the rotation shaft are made in the form of NASA 0021 symmetrical airfoils, the  and coefficients of these profiles in the range of  are well described by dependencies



Substituting the last expression to (8), we obtain

Factorizing  from the square brackets, taking into account (1) and (3), and rewrite as follows:


If we take into account that the preferred values of the coefficient of wind energy (x) of the Darrieus turbine are in the range of  (see. Figure 4). In this case, at the nominal work of the Darrieus wind turbine the value of << 1. The latter allows expanding the elements with radical expression in powers in small parameter and limiting with first two elements:



Thus, in the case of two-bladed Darrieus turbine with straight working blades the moment generated by the pair of forces will be expressed by the equation


Following evaluation, it must be concluded that the second element in square brackets is the share of interest from the other two elements, and can be neglected. As a result, the moment of couple of forces depending on the rotation angle of the turbine ()for  will be of the form

.                                    (14)

Full torque of two-bladed Darrieus wind turbine per one turn will be found by integrating relation (14) from 0° to 2


So, in the final form we will obtain

                                .                                                (15)

These data indicate that the torque that is developing by the lifting force is much higher than resistance force to the air environment to the movement of the blades.


       Determination of wind power and energy efficiency

Capacity of the wind turbine will be determined by multiplying  by its angular velocity

                             =.                                   (16)

On the other hand it is known that the power of the turbine can be determined, if the value of the wind power efficiency (see Fig. 2) and disk area F are given. In the case the two-bladed Darrieus turbine under consideration F=2pr0L. Hence we have [1]


Equating the last two relations to each other, we will find


Typically, the ratio . Hence


Equating the first derivative  on to zero, we will define the maximum value and the value at which is achieved,


or , then .

As can be seen from the values obtained,  and  at which the maximum value of the wind power efficiency, they are not much different from their experimental values (see. x(c) influence curve for the Darrieus wind turbine in Figure 2).

The deviation of the theoretical values of x parameter from the experimental ones, apparently is due to the fact that the additional resistance associated with dragging of bearings, the presence of two strokes etc., as well as the decrease in the flow rate at the treatment of the blade in the downwind area (СDА semicircle (see Fig. 2) did not take into account while the calculation

Dependence of the wind power efficiency x for different types and designs of wind turbines on their degree of rapidity c [4].



Figure 2. Dependencies of wind power utilization factor x for different types and constructions of wind turbine on the degree of their specific speed c.



1. Shefter Ya.I. Windpower generating units. M., 1972. 287 p. (in Russian).

2. Yershina A.K., Yershin Sh.A., Zhapbasbayev U.K. Principles of a Darrie wind turbine theory. Almaty, – 104 p, 2001. (in Russian).

3. Loitsyanskii L.G. Mechanics of fluids. M., 1970. 904 p. (in Russian).

4. Turyan CJ., J. Strickland, H., Berg D.E. The power of wind-electric units with a vertical-axis rotation. Aerospace techniques. №8.- pp. 105-121. 1988. (in Russian).

Bibliographic reference

Ainakul K. Yershina ON AERODYNAMICS OF DARIUS WIND TURBINE WITH STRAIGHT BLADES. International Journal Of Applied And Fundamental Research. – 2014. – № 2 –
URL: www.science-sd.com/457-24632 (12.12.2019).