Analysing power generation in

Vertical Axis Wind Turbines (VAWT)

Esmaeil Ajdehak Mechanical Engineer Bilfinger Berger Services Australia

au.com.bbsaa@ajdehak.smaeilE ABSTRACT: This research studies the mechanism of power generation in a special type

vertical axis wind turbine (VAWT) with parabolic blades called darrieus. Studying the

resulting forces at different points in each blade that are due to the blade rotation, along with

the high amplitude of power fluctuation, a method is introduced to lower this fluctuation to a

large extent by breaking the blade into different pieces of constant pitch. According to the

power loss which is obtained as the result of tip vortices generated on each blade piece (finite

wing theory), a method is introduced to keep the generated power at a reasonable level while

lowering the amplitude of power fluctuation to a large extent.

Keywords: Vertical axis wind turbine, Aerodynamic, Power fluctuation

a : interference factor AR : ratio between wing length and Airfoil cord C : airfoil cord length CD : drag coefficient cdi : drag coefficient due to the tip vortexes CL : lift ceffecient cr : wing cord at root ct : wing cord at tip Fi : force component on each element of blade N : number of blades r : radius of blade- distance from rotor U : effective flow passing through rotor V∞: upstream wind velocity Vθ : blade speed W : relative wind velocity α : attack angle ω : angular velocity αi : induced attack angle αL : attack angle (infinite wing theory) χ : tip speed ratio δ : induced drag coefficient θ : the angle location of blade σ : solidity ψi : polar parameter of blade element i

Fig-1: Main components of a vertical axis wind turbine (VAWT)

1- Introduction: In recent years Vertical Axis Wind Turbines are more on focus in terms of optimization of generated power and cost, though in 1925 and 1926 the initial research was commenced to figure out how they work but in the mid 60th, Central Canadian wind turbine research laboratory begun a broad and detailed research on VAWTs. Nevertheless HAWT (horizontal axis wind turbines) are still the favored form of turbines used for purpose of electricity generation. The characteristics of HAWTs are low solidity ratio (ratio of blade area to swept area, (σ) and high tip speed ratio (χ). The second major type of wind turbines are VAWTs. a wide variety of VAWT configurations have been proposed, dating from the Persia VAWTs in Yazd city, used for milling grains over a thousands years ago, through to Darrieus turbine which invented in 1926 by Georges Darrieus . in fact this article is concentrated on Aerodynamic phenomena of a Darrieus wind turbine.

The basic theoretical advantages of a vertical axis wind turbine: 1- The generator and gearbox can be located on the ground 2- No need to Yaw mechanism to turn the rotor against the wind. In vertical axis wind turbines with a long blade, due to the high inertia forces on the blades a parabolic frame of blade help to distribute the forces alongside the blade to avoid the fracture in the blades.

2- Power generation in VAWTs: Before starting the analysis of VAWT, some parameters which are essential in calculations are needed to be defined:

Solidity: (σ) is the ratio between blade area to swept area in a full rotation. σ =NC/2πr Eq-1

Effective wind velocity (U) and Interference factor (a):

Wind velocity passing through rotor is somewhat less than infinite upstream wind velocity (V∞) which is due to the effect of vortexes caused by blade rotation. The real velocity to be used in the calculations is effective velocity (U):

U= V∞(1-a) Eq-2

Blade speed: (Vθ)

Rotational movement of blade around the rotor can be calculated by multiply radius and angular velocity of blades:

Vθ = r.ω Eq-3

Tip speed ratio: (χ ) is equal to ratio of blade velocity to effective upstream flow, i.e

χ=Vθ /U Eq-4

To start the calculations the physical characteristics of turbine is required. In order that, calculation has been done for a darriues turbine with a parabolic blade, 17metres maximum diameter of curve in blades and 25metres height of tower. First of all a real profile of the blade needs to be transferred to mathematical language, so by using Lagrange numerical method a parabolic function is derived from about 50 Cartesian data(x-y), the result concluded as follow :

Eq-5 y² =-18.8x+155.1

regarding the symmetrical shape of blade, calculations is done for a semi part of blade. In order to finding the length of blade the mathematical arc length equation is used:

S= ∫√ (1+x'²) dy 0<y<12.5 Eq-6

To improve and simplify the work with the parabolic equation the Cartesian system is changed to polar system by using:

x = r.sinψ , y = r.cos ψ Eq-7

Fig-2) Parameters used in polar system to calculate the blade profile

r = (-18.8cos ψ+√ (-266.5.cos² ψ+620))/2sin² ψ) 10˚< ψ<90˚ Eq-8

It can be seen in the Fig-2 for 0˚< ψ<10˚ a mathematical error is appeared. To avoid divergence in calculations this section of curve is simulated by a linear function as follow:

r =155.1/(sin ψ+18.8cos ψ) 0˚≤ ψ≤10˚ Eq-9

As parameters r and ψ have been shown in Fig-2, specify the location of any particular point on the blade from rotor.

Fig-3) tangential angle γ and profile of frame

In vertical axis wind turbines, the angle between tangent line from each point and rotor which is shown as γ is an important parameter in design of turbine in terms of cost and efficiency. This angle identifies the profile of blade which shapes the frame. It is expected in design to minimize the bending stress due to the inertia forces on the blades and distribute the force as tensile force along the blade to avoid fracture in the frame. In fact, providing a good design of profile using a cheaper material which can carry the internal tensile forces would be a key factor in lowering the final cost of structure.

In Fig-3 γ= β -π/2, Eq-9-1

to calculate β,

According to Fig-3, a mathematical relation between ψ and β is defined as:

tgβ=(tg ψ.dr/dψ+r)/(dr/dψ-rtg ψ)

then β=tg¯ ¹(tg ψ.dr/dψ+r)/(dr/dψ-rtg ψ), Eq-10

Referring to Eq-8 by derivative of r with respect to ψ:

dr/dψ = (18.8 sin ψ-2r² sin ψ.cos ψ)/(2rsin² ψ+cos ψ ) Eq-11

and for Eq-9:

dr/dψ =(-155.1(cos ψ-18.8sin ψ))/(sin ψ+18.8cos ψ)² Eq-12

using dr/dψ from Eq-11 amd Eq-12 in Eq-10, β can be calculated; consequently γ is also derived from Eq-…

dr/dψ-rtg ψ)- π/2 Eq-13 /(γ= tg¯ ¹( (dr/dψ.tg ψ+r)

where dr/dψ can be replaced from Eq-11 amd Eq-12 respectively,

In order to calculate the attack angle of wind flow on the blade, regarding the diverting the blade direction in every section of circular path around the rotor, the full period rotation is divided in 4 equal quarter and vector triangle of velocity for each sector is formed as follow.

→ → → U= W+ Vθ (14) Where vector U can be calculated from Eq-2 and vector Vθ. is also already calculated as Eq-3 and W is the relative speed. In fact W is the velocity of wind that is needed to be considered in the calculation of attack angle, lift and drag forces. The angle between vector velocity of W and cord of airfoil is attack angle.

Fig-4) Vector velocity triangles for different sections of rotation. According to Fig-4 and Fig-5 and using trigonometric function in mathematics, attack angle α is calculated in relation to θ (location of blade in rotational path), a (interference factor) and χ (tip speed ratio) for first and second area as Eq-16:

Fig-5) Vector velocities in quarters (I) and (II) tgα = U . sinθ/(U . cosθ+Vθ) = sinθ/(cosθ+Vθ/U) Eq-15

χ = Vθ/(U(1-a)) → tgα = sinθ/(cosθ+χ/(1-a)) 0≤θ≤π Eq-16

and also for the third and forth sector of rotation:

0 0

2 0.21

4 0.42

6 0.63

8 0.9

10 1.08

12 1.3

14 1.41

16 1.6

y =

5.5

69x +

.1548

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20

ATTACK ANGLE

CL

Fig-6) Vector velocities in quarters (III) and (IV) tgα = -sinθ/(cosθ+ χ/(1-a)) π≤θ≤2π Eq-17

Calculation of Interference factor (a): Regarding the relative turbulence in the flow around the rotor and blades which is due to interference of blades with air an axial interference factor a is introduced to represent the fraction of longitudinal momentum lost by air and a tangential flow interference, a = σ . χ . sinθ . cosγ 0 ≤θ≤ π Eq-18 a = -σ . χ . sinθ . cosγ π ≤θ≤ 2π Eq-19 where σ :solidity (Eq-1), χ: tip speed ratio (Eq-4), θ: location of blade, γ: blade profile identifier (Eq-13) using Eq-18 and Eq-19 in Eq-16 and Eq-17, attack angle α is calculated. Airfoil NACA0012 is selected for the VAWT, following graph is indicating the lift and drag coefficients in relation to attack angle α,

Fig-7: Lift and drag coefficients for NACA0012 To make the data shown on the graphs applicable in a computer code program, numerical method is employed to translate the data base to a mathematical function,

18; 1.05

20; 0.8622; 0.82

y =

53.5

x2 -

40.2

5x

+8.5

3

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30

ATTACK ANGLE

CL

Fig-8: numerical simulation for lift coefficient of NACA0012 According to the results:

CL = 5.569 α + 0.1548 α <16˚ CL = 53.5 α² - 40.25 α + 8.53 16˚ < α <20˚ Eq-20 CL = -1.654α² + 2.068 α + 0.4578 α > 20˚

And for Drag coefficient:

CD = 0.4963 α³ + 0.4681α² - 0.0092 α + 0.0058 Eq-21

Lift and drag force components are in perpendicular and parallel relation to relative speed W (Fig-9), therefore for an element section of blade the amount of generated force is calculated as:

Li =(1/2)CLi . ρ . W² . Si Eq-22 Di =(1/2)CDi . ρ . W² . Si Eq-23

Where in the above equations Si is the element area of the blade, Consequently the final force exerted on the blade according to Fig-9 is calculated as:

Fi = Li . sinα – Di . cosα Eq-24

Fig-9: Arrangement of aerodynamic forces in relation to the triangle velocity for a section of Airfoil

Regarding the fact that attack angle is a function of θ(location of blade in rotation around rotor) therefore according to Eq-15 and Eq-17 lift and drag forces and consequently the result force F are the function of rotation angle θ, i.e,

α = f (θ) CL ,CD = f (θ)

L , D = f (θ) F = f (θ)

After calculating the aerodynamic result force (Fi) for a particular element (i), which is located in a distant (ri) form shaft (refer to Fig-2 and Fig-3), the momentum exerted on the shaft for the element of blade is:

Ti =ri x Fi Eq-25

Where ri = ri / cosψ Eq-26

The average torque is given by the following formula:

T=∫T.dθ/2n [0,θ] Eq-27

In the above equation 2n is the number of segments which torque is calculated. However the number of elements on blade increases, the more calculations is needed to be done, therefore 72 element for each blade is selected, and according to the mathematical average function for torque introduced as follow:

T= ∫Td(θ/5) =(2π-0)/3{T[0] + Σ(4T[2i-1] + Σ2T[2i]-T[2n]}/2n 1<i<36 Eq-28

Using Eq-28 in Eq-27:

T= {T[0] + Σ(4T[2i-1] + Σ2T[2i]-T[2n]}/ 3(2n) 1<i<36 Eq-29

A simple equation relates torque to power:

P= T x ω, where ω is angular velocity of mechanism.

Regarding the complexity of calculations, a computer program is prepared in Pascal to include all the required equations in this article which it takes the inputs as the constant quantities e.g. wind velocity, physical dimension of turbine- tower height and angular velocity. The output is a graph of output power in every location of blades in the plan. The final results shown on the following graph reveals large amplitude in output power for these turbines.

(a) (b)

Fig-10: a) Shows the variable quantities for the blade speed,

b) Shows the output power in VAWT,

The results shown on the graph indicate that the minimum output occurs when the airfoil cord is in parallel to the direction of flow, whereas in perpendicular position of cord against the flow maximum output power is generated. However this is also shown a big fluctuation in power generation which in continuation of this research some solutions are made to decreases the amplitude of generated power in this type of turbines.

How to reduce the fluctuations:

With a precise look at Fig-10it is clear that the location of blade in the plan of rotation is a key to find out the reason for drift in torque and consequently power generation. It is expected that filling the gap between the spaces between two current blades would be helpful. The ideal situation is to have a spiral blade (as shown on the Fig-11 (a)) which in every spot of rotation, one element of blade is located. In this case the less lost of energy due to the vortexes at the ends of blade is expected and in fact this kind of arrangement can be considered as an infinite wing. But the problem with this could be counted as complexity in fabrication and consequently a higher cost in the final price of turbine. Moreover the spiral shape of blade would need a more complex set of bracing to reinforce the blade around the rotor Fig-11 (a). But still some other simple solutions can be replaced with this complex design.

(a) (b)

Fig-11: a) configuration of a spiral blade around the shaft,

b) Reinforcing the blade to shaft,

A practical solution to the problem is to increase the number of blades in different pitches. Fig-12 shows three type arrangements of blades by pitches (a) 45º, (b) 90º and (c) 180º.

(a) 45º (b) 90º (c) 180º

Fig-12: a) arrangement of blades in different pitches 45º, 90º and 180º

As it can be seen in the output results, in the ideal situation in absence of drag lost, a smoother output occurs compare to previous output data. The best output is seen for the arrangement of 45º (as expected). However there is an obstacle to increase the number of blades and more pitches which is due to the downwash vortexes at the end of blades. According to finite wing theory, a downwash induce flow which occurs due to the vortexes at the edge of finite blade, it causes the lowering the effective attack angle (αeff) (Fig-13), and also by making turbulence at the edge of blades cause the less effective area on the blade to make lift force, consequently by increasing the number of blades the lost of energy increases and would result in dropping the overall output power.

Fig-13: reducing effective attack angle-αeff due to induced downwash flow (v)

Therefore, in calculation of aerodynamic forces for the new arrangement of blades, attack angle αeff. should be used.

αeff = αL – αi Eq-30

Cdeff= cd+cdi Eq-31

where

αi = CL/(πAR) Eq-32

AR is the ratio between blade length and Airfoil cord,

Moreover the drag coefficient for the finite wing is calculated as:

cdi = CL²/( πeAR) Eq-33

where, e = (1+δ)¯¹ Eq-34

δ is called induced drag coefficient which according to Fig-14 is specified by (AR) and (Ct/Cr). Ct is the length of cord at tip and Cr is the length of cord at root. The goal to achieve is to minimize the drag coefficient δ. As shown on the graph the minimum occurs for Ct/Cr~0.3, therefore if this ratio applied for the turbine blades, a better efficiency of generation power is expected.

Fig-14: Induced drag coefficient δ in relation to Ct/Cr

Conclusion:

The final results have shown on the Fig-15 represent for three arrangements of 45º, 90º, 180º pitches how the fluctuations in amplitude of output could be declined. Moreover, according to the above explanation, it has been shown that however the amplitude is smoother, the bigger drop in output power is appeared. Thus, a compromise between output amplitude and power generation needs to be considered.

WASTED POWER

0100002000030000400005000060000700008000090000

100000

IDE

AL

ct/c

r=0

.3

ct/c

r=1

Pitch 180 Pitch 90 Pitch 45AMPLITUDE CHANGE

0

100000

200000

300000

400000

IDE

AL

ct/c

r=0

.3

ct/c

r=1

Pitch 180 Pitch 90 Pitch 45

infinite wing

ct/cr~0.3

ct/cr~1

(a) (b) (c)

Fig-15: Program output for different arrangement of blade pitches

For a full period rotation of blades as it is shown on (a), not only a high variation in amplitude of output is seen but also in some location of blade orientation a minus torque due to the overcoming the drag force pushed the mechanism in a reverse direction. Dividing the blades in (b) 90º and finally (c) 45º helps to avoid such big fluctuations and reverse forces but as it is shown a decline in the output power in due to the vortices an obstacle to break the blades in shorter pieces and increase the number of pitches.

0

(a) (b)

Fig-16: Variation of amplitude and wasted power in different pitches

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