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Aerodynamic performance prediction of Darrieus-type wind turbines.


The power generation technologies aim at increasing the use of unconventional and clean energy. Therefor, activities related to research and development, manufacturing and operation of wind plants are stimulated.

Considering that the wind potential in Romania is small as extent and intensity, INCAS aims to build wind turbines suitable for the wind potential available in Romania. A combined aerorotor was designed, consisting of a Savonius rotor and a Darrieus turbine which ensure its operation at low wind speeds of 4-5 m/s. The Savonius rotor built from lightweight materials and having small dimensions decouples by means of a centrifugal clutch when the Darrieus turbine reaches the nominal working regime, allowing a substantially increased couple [2].

A five-bladed Darrieus turbine will be tested in the subsonic wind tunnel. Testing will be done for different construction and arrangement of blades [1].


2.1 The global method

For turbines with straight vertical blades, the rotation axis is perpendicular to the wind direction. . Using the global method, inspired by Betz model, results the maximum power coefficient estimation, in which a tube of curent is equal with turbine rotor's dimensions. [3].

Let's consider the turbine area divided in two: S 'and S". Passing through the two surfaces, [5] the current modifies its speed twice. The force exerted by the fluid on S', and S" respectively, is

F' = Q([V.sub.[infinity]] - [V.sub.2]) = [rho]S'[V.sub.2]([V.sub.[infinity]] - [V.sub.2]) (1)

F" = Q([V.sub.2] - [V.sub.3]) = [rho]S" [V.sub.2] ([V.sub.2] - [V.sub.3]) (2)

Using the linear nature of the velocity transformation law, [V.sub.2] = k[V.sub.[infinity]] and [V.sub.3] = k[V.sub.2] = k2 [V.sub.[infinity]], the average power supplied by S' si S"

[P.sub.m] = F' [V'.sub.m] + F" [V".sub.m] (3)

where [V'.sub.m] and [V".sub.m]are the average speeds across the respective surfaces:

[V'.sub.m] = ([V.sub.[infinity]] + [V.sub.2] = 0.5 [V.sub.[infinity]] (1 + k) (4)

[V".sub.m] = 0.5([V.sub.3] + [V.sub.2] = 0.5. k [V.sub.[infinity]] (1 + k) (5)

After the necessary replacing the provided average power is obtained:

For a [k.sub.max] = 1 / 4[square root of (5)] (determined experimentally):

[P.sub.max] = 0,5[rho]S"[V.sup.3.sub.[infinity]] 4 4[square root of (125)] / 25 (6)

in this case the power coefficient is

[C.sub.p] = 4 4[square root of (125) / 25 = 0.53 (7)

This method of calculation overestimates the performances.

2.2 The method of the gust factor and momentum

This method involves an analysis of the blade as being composed of distinct elements, which don't influence each other from the aerodynamic point of view. The velocity induced on each element is determined using the momentum equation; the aerodynamic forces on the element are calculated using the lift and drag coefficients of the considered profile section [3-5].

For a gust element we have:--the element aerodynamic centre, R--the distance from F to the Darrieus rotor axis, [theta]--the angle between the Ox axis and the R radius, z--the F height and [delta]--the angle of the normal element in F to the blade element and the horizontal Oxy plane (for cylindrical rotors [delta] = 0 ). Position of F with respect to the Oxyz axis system is determined by the R, [theta], z coordinates.

Let V be the absolute wind speed, W its relative speed against the considered gust element and U the appropriate transport velocity U, [omega]--turbine rotor angular velocity.

U = [omega]r (8)

W = V - U = V - [omega]r (9)

The relative velocity components along the specified axes are

[W.sub.r] = Vsin[theta], [W.sub.t] = [omega]r + Vcos[theta], [W.sub.v] = 0 (10)

The director cosines of the normal to blade element in the considered system of axes are: cos[delta], 0, sin[delta].

To calculate the aerodynamic forces acting on the blade element the [W.sup.u] component of the relative wind speed shall be considered

[W.sup.2.sub.u] = [W.sup.2.sub.t] + [W.sup.2.sub.n] (11)

[W.sup.2.sub.u] = [([omega]r + Vcos[theta]).sup.2] + [(V sin[theta]cos[delta]).sup.2] (12)

For ([alpha]) local incident angle, defined by the relation

tga = [W.sub.n] / [W.sub.t] = Vsin[theta]cos[delta] / [omega]r+Vcos[theta] (13)

[C.sub.n]--normal aerodynamic coefficient, [C.sub.t]--tangential aerodynamic coefficient,

[C.sub.z]--lift coefficient

[C.sub.x]--rag coefficient the following can be written:

[C.sub.n] = [C.sub.z] cos [alpha] + [C.sub.x]sin[alpha] (14)

[C.sub.t] = [C.sub.z] sin [alpha] - [C.sub.x]cos[alpha] (15)

The [alpha] incident angle varies over a complete rotation of the considered blade element, being dependent on the angle [theta].

The elementary normal force (dN) and the tangential force (dT) acting on blade element depends on the coefficients and [C.sub.t]

dN = q[C.sub.n]c ds (16) dT = q[C.sub.t]c ds (17) q = 1/2 [rho]p [W.sup.2.sub.u] (18)

q--represents the dynamic pressure, c--length of profile chord, ds--width of considered blade element, dz--the height of considered blade element, dF--the elementary aerodynamic force on wind direction,

dz = ds cos [delta] (19)

dN = q [C.sub.n] cdz/cos [delta] (20)

dT = q [C.sub.t] cdz/cos [delta] (21)

dF = dN cos [delta] sin[theta]-dT cos[theta] (22)

The dF elementary aerodynamic force varies over a blade complete rotation around the axis Darrieus rotor. If the chord c is constant along the length of the blade, the aerodynamic force exerted on the Darrieus rotor on wind direction is calculated with:


[n.sub.p]--number of wind turbine blades.

For power calculation and for momentum

P = M[omega] (24)

dM = r dT = qc[C.sub.t] / cos[delta] r dz (25)

The momentum given by the dN component is null.

Over the whole Darrieus rotor we have the momentum expression:


The relation for power calculation becomes:


[lambda] = [omega]R / [V.sub.1] (28)

[lambda]--the end speeds ratio, [V.sub.1]--wind speed.

The power coefficient [C.sub.p] is given by the relation



[C.sub.m]--momentum coefficient, S--surface described by the rotor, perpendicular to the wind direction. To calculate the momentum coefficient we consider the following relation existing between it and the power coefficient:

[C.sub.m] = [C.sub.p] / [lambda] (31)

It follows that knowing one of the two coefficients we can immediately determine the other.

The dependence [C.sub.p]([lambda]) is significantly influenced by the solidity ratio [sigma] which is an important design parameter generally defined as the ratio of blades total surface to the area described by them in their rotational motion.

[sigma] = n/s [[integral].sup.b.sub.0] c(y) dy (32)

b--blade span, c(y)--blade chord in section y



The prediction based on the calculating method for the gust factor and momentum gives closer results to the obtained experimental data in comparison with other calculation methods. The INCAS concerns have highlighted the possibility of obtaining experimental data for the verical shaft rotor in the subsonic wind tunnel. Combined results for a combined Darrieus- Savonious rotor allowed to design an installation for water extracting.

The calculation method for the gust factor and momentum was utilized to conceive a multi-blades Darrieus rotor intended to produce electricity.


The prediction of the aerodynamic performances of a Darrieus turbine enables the wind installations manufacturer to estimate the structural characteristics of the rotor appropriate to the wind potential of the operation area. Applying the lifting surface analysis to the quasi-steady and nonsteady movement conditions the essential characteristics of a Darrieus turbines are highlighted.

The calculation method for the gust factor and momentum ensures getting a range of types and sizes depending on the average wind speeds, without requiring a laborious numerical calculation.

Based on the designed Darrieus turbine performances the required generator can be established.The forces calculated by the gust factor and moment method allow sizing the turbine base support.


The work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/6/1.5/S/16.


Atkins, R. (1989) Measurements of surface pressure on an operating vertical-axis wind turbine, Sandia National Laboratories, Report SAND89-7051, 1989

Dumitrescu, H., Cardos, V., Frunzulica, F. & Dumitrache, A. (2007), Aerodinamica nestationara, aeroelasticitate si aeroacustica pentru turbine de vant, Ed. Academiei Romane, Bucuresti 2007

Lapin, E. (1975) Theoretical performance of vertical-axis wind machines, American Society of Mechanical Engineers Paper 75-WA/ENER-1, Huston, Texas, 1975

Paraschivoiu, 981) Double-multiple streamtube model for Darrieus wind turbines, second DOE-NASA Wind Turbines Dynamics Workshop, NASA-CP-2185, 1981

Tomescu, N, (2006) Analiza aerodinamica a convertorului eolian, raport tehnic, proiect instalatie eoliana, INCAS
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Author:Bogateanu, Radu
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EXRO
Date:Jan 1, 2010
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