Performance Optimization of A Low-Speed Multi-Blade Wind Turbine For Pumping: Analysis Of Power And Torque Coefficient

General assumptions :

The following assumptions are necessary to validate the theories and fundamentals of wind turbines, particularly in the case of low-speed, multi-bladed turbines. These simplifying assumptions are commonly used in the Blade Elementary Momentum (BEM) method.

• The fluid is incompressible downstream of the wind turbine. Upstream, we only have the axial wind speed.
• The wind flow is steady.
• The rotor is treated as an actuator disc in the theories of Kutta-Joukowski, Froude and Betz
• The wind turbine under consideration is a low-speed turbine with a Tip Speed Ratio (TSR) between 0.6 and 2
• The low-speed wind turbine has several blades, and the analysis considers a blade count between 4 and 35.
• The blades are formed from rigid sheet metal plates with a NACA 0012 profile
• The analysis is two-dimensional
• The aerodynamic analysis does not consider any external elements interfering with the flow and movement of the rotor.
• The drag force is negligible in the low-speed wind turbine.

Rankine–Froude theory and the Betz limit :

Both theories can only be considered subject to certain conditions and assumptions that ensure the validity of the study.

Fig. 2: Wind parameters upstream and downstream (Source: Author)

 

The law of continuity and the conservation of momentum give the axial force: 

T=ρSdV(V1-V2)

                                              (1)

 

Bernoulli’s theorem, as applied in this situation, does not take into account changes in altitude. By applying Bernoulli’s theorem upstream and downstream, we obtain the axial force given by:

T=12ρSd(V12-V22)

                                       (2)

 

Equations 1 and 2 determine the relationship between the wind upstream, downstream and at the rotor disc, by transitivity of the equality of equations 1 and 2:

V- V2=12(V1-V2)

                                        (3)

 

Thus, the power equation is given by

P=14ρSd(V12-V22)(V1+V2)

                         (4)

 

Using this formula, we can analyse the maximum power as a function of the downwind wind speed.

Kutta–Joukowski vortex theory :

Previously, the flow was assumed to be strictly axial with no rotation. In reality, however, it is necessary to take rotational motion into account in order to extract the useful torque. This approach yields results that are closer to reality. ^{[13][14] [15]}

This study is carried out in accordance with the given conditions and assumptions. There is no interference between adjacent blade elements. The airflow around a blade element is considered to be two-dimensional.

The proof will be carried out in the same way as that for Froude’s theory, taking into account the element radius dr. The continuity equation, which in this case applies to a surface element dS tangent to the disc’s axis, and the element mass flow rate, yields the tangential force and a relationship between the induced wind speeds upstream and downstream:

dFt=ρdSdVr(ω1'+ω2')

                                (5)

 

ω2= 2 r r22ω

                                                (6)

 

Identification of sources of interference :

In aerodynamic calculations, a value a – known as the aerodynamic interference factor – is used to express the induced velocity V as follows:

a=V1-VV1

                                                           (7)

 

V2=(1-2a)V1

                                               (8)

 

As with the axial interference factor, we shall introduce a coefficient a’ which is the tangential interference factor. Since the induced angular velocity ω arises from the rotation of the rotor at speed Ω, it is appropriate to express it in terms of the latter:

ω=a'Ω

                                                                 (9)

 

Elementary forces and torque :

The elementary axial force is obtained by differentiating equation 2 over a ring dr and inserting the axial interference given by equation 10. Since

Sd=π×rayons2⟹ dSd=2πrdr

                (10)

 

dT=4πrρaV12(1-a)dr

                                (11)

 

The elementary tangential moment arises from the force driven by a tangential rotational motion from equation 7 on an elementary ring. By inserting the tangential and axial interference factors:

dFt=4ρπV1Ωr2(1-a)a' dr

                           (12)

 

dMt=4ρπV1Ωr31-aa'dr

                           (13)

 

Aerodynamic forces :

In this section of the study, we will focus on a blade element. The aerodynamic force is the force generated by the action of the wind on the wind turbine blade. It consists of lift and drags. Its formula is:

R= 12ρCaW2S

                                                 (14)

 

With C_a the aerodynamics coefficient.

Ca2=CL2+CD2

                                               (15)

α : pitch angle, which is the angle between the chord line and the plane of rotation

ϕ : inflow angle, which is the angle between the relative wind speed and the axis of rotation

β : angle of attack, the angle between the relative wind speed and the chord line

W being the relative wind speed defined as the combination of the true wind and the wind created by the blade’s movement, i.e. the sum of the induced axial wind speed, the induced tangential wind speed and the propeller’s rotational speed:

W= V1.(1-a)+Ωr(1+a')                           (16)

Regarding its two components, the drag D follows the direction of the relative wind and the lift L is perpendicular to the drag.

R2=L2+D2                                                    (17)

As we are interested in the force generating the rotation of the blade parallel to the plane of rotation, and the thrust force perpendicular to this plane, we must project the components of the aerodynamic force onto these two axes. The lift and drag forces are represented over a distance dr from the axis of rotation, which has a chord length c . By projecting onto the axis of rotation, we obtain the thrust force of a blade Fx  and the rotational force of a blade Fy . We then obtain the values of the axial and tangential forces and the flow angle for the n number of blades.

tanϕ=Ωr(1+a')V1(1-a)                                                (18)

dFt=ndFy=n(dLcosϕ+dDsinϕ)                 (19)

dT=ndFx=n(dLsinϕ+dDcosϕ)                  (20)

dMt=ndFyr                                                        (21)

Correlation between interference and angles using the Glauert method

• Characteristic coefficients in wind turbines

The solidity coefficient, also known as local solidity, is the ratio of the blade area to the rotor disc area:

σ=cn2rπ                                                                  (22)

The specific speed λ, technically known as the Tip Speed Ratio or TSR, is the ratio of the tip speed of the blades to the axial wind speed. It forms the basis for classifying wind turbines by speed. A slow-speed wind turbine has a specific speed λ ≤ 3, whilst a high-speed wind turbine has a specific speed λ > 3. Let R denote the blade length; the specific speed λ and the relative specific speed λr are given by:

λ=ΩRV1                                                               (23)

λr=λrR                                                            (24)

• Axial and tangential forces

In Glauert’s calculation, the drag coefficient C_D becomes negligible, which is consistent with rigid, flat blades. Equations 13 and 22 both represent the axial force, and equations 14 and 21 represent the tangential force. By substituting the two equations and expressing the relative velocity geometrically in terms of V1  and Ωr

a(r)=σCLsinϕ4cos2ϕ+σCLsinϕ                                       (25)

a'(r)=σCLcosϕ4sinϕcosϕ_-σCLcosϕ                                  (26)

Optimal disc area :

This step allows the dimensional parameters of the multi-blade wind turbine to be determined. Some equations are only valid for slow-speed wind turbines with a low specific speed. In the case of a pumped-storage wind turbine, as with most low-speed wind turbines, the thin plate option is the most common. Indeed, for practical and cost-effective reasons, manufacturers have replaced the traditional wooden blades with steel plates, generally rigid sheets, to eliminate deflection. This assembly gives us the following figure :

Fig. 5: Perspective view of the wind turbine (Source : Author)

 

 

Fig. 6: Front view of the wind turbine (Source : Author)

We can see from this diagram that, when viewed from the front, the wind turbine resembles n isosceles trapeziums, where n is the number of blades. The following diagram focuses on a single blade :

Fig. 7: Blade of wind turbine (Source : Author)

 

Let cmax  be the chord lenght at the tip of the blade and cmin  the chord length at the root of the blade. Since a section is required to support the blade, Rmin  is the distance between the axis of rotation and the root of the blade, and R is the distance between the tip of the blade and the axis of rotation; the area of the blade is therefore :

Spâle=12cmax+cminR- Rmincosα           (27)

Sd=12ncmax+cminR- Rmincosα            (28)

According to Thales’theorem :

cmincmax=RminR=q                                                 (29)

Sd=12ncmaxRcosα1-q2                              (30)

To maximise the area, we must differentiate this equation with respect to q which gives a maximum when q=0, This is the equation of a triangle. However, space is required for the wind turbine hub and its assembly. Therefore, we apply a margin of 0.1:

cmin=0,1cmax;Rmin=0,1R                       (31)

As the disc area is given by these two equations, we obtain a relationship between the maximum chord and the radius

Sd=RminR2πrdr                                              (32)

Sd=πR2-Rmin2                                         (33)

cmaxR=2πncosα                                                      (34)

This relationship allows us to determine a precise ratio between the width of a blade and the blade length. However, the blade width corresponds to the chord length. Thus, the dimensional parameters can be calculated.

The chord is symmetrical and increases linearly. The relationship 90 allows us to express the chord length at a point of distance r from the axis of rotation. The symmetry about the axis of rotation and the chord length at the tip of the blade imply that:

cr=2πncosαr                                                   (35)

The formula for the strength determined in equation 24 remains constant regardless of the radius considered: σ(r)=1cosα                                                 (36)

Détermination of the optima angles :

The system comprises the pitch angle, the inflow angle and the angle of incidence. These three angles must be determined to obtain the exact dimensions. The basic equation for the solution is derived from the analysis of Figure 24

α=π2-β-ϕ                                                   (37)

• Pitch angle

As a low-speed wind turbine, optimisation primarily concerns the rotational speed, which determines the system’s flow rate. Storm (1937) and Shepherd (1954) proposed a semi-empirical model for multi-bladed wind turbines intended for pumping, based on Prandtl’s relation ^[16][17][18]. This theory is based on assumptions that are suitable for our study. Firstly, the specific speed must be low and less than 1.5. Secondly, it is only valid for blades with a thin profile, such as the metal sheets in our case. Finally, the calculation coincides with torque optimisation ^[16][18]. This method is derived from the Prandtl correction in a pumping wind turbine ^[17].

Pmax=12ηCpρSdV13                                          (38)

η=1- 1,39nsinα2                                         (39)

Pmax=14ρV13ncmaxx21- 1,39nsinα21-q2cosα )                                                (40)

1- 1,39nsinα2cosα=maxsinα=max                          (41)

By finding the optimal consensus value of the two equations.

fn,α= 1- 1,39nsinα2cosα- sinα=0    (42)

∂fn,α∂α≠0 sur  α∈0;π2et n∈4;+∞           (43)

This means that equation 114 has one and only one solution over its domain of definition. For graphical resolution, we will consider only n ∈ [3; 35] for practical and economic reasons.
 

• Angle of attack

The aerodynamic context is based on the NACA profile under study. In the case of a multi-blade wind turbine, preference is given to flat plates with a NACA 0012 profile, which does not optimise power output but rather torque. This profile is non-curved, hence the ‘00’ prefix. Its thickness is 12%, and its shape is symmetrical and linear. According to empirical studies of the NACA profile, this type of profile corresponds to a linear increase in lift ^[33][34];

CL≈2πβ                                                          (44)

CD≈0,01                                                         (45)

CD  is not a limiting factor in this type of wind turbine. Optimisation involves maximising torque based on the CL  coefficient to achieve high torque at a low specific speed (λ<2). In this approach, the best solution would be to set the pitch angle close to the stall angle. However, a compromise must be made to keep drag negligible and to provide a safety margin to prevent stalling. In practice, the optimal value is determined in advance to be between 7° and 10°. This value is significantly higher than that used for high-speed wind turbines. ^{[16][17][18][19]}

βopt=7°~10°                                                   (46)

By selecting an optimal pitch angle value based on the formulas and empirical results, the optimal angle of attack is simply given by formula 109

βopt=π2-α-ϕ                                              (47)

• Optimal inflow angle

Iterative method

The flow angle or inflow angle is defined by equation 20. By integrating the specific velocity equation (25 and 26), which will be a given assumption in the study:

tanϕ=λr(1+a')(1-a)=λr(1+a')R(1-a)                              (48)

ϕ(r)=arctan⁡λr(1+a')R(1-a)                                 (49)

However, the interference values also depend on the angle ϕ(r), as shown in Equations 82 and 87. Solving these equations requires the use of an iterative method. The algorithmic process is suitable for solving a system of equations and for optimisation purposes. The first step is to formulate the assumptions and determine the various values to be imposed. In our case, the specific speed varies between 0.6 and 2 for a low-speed wind turbine, and the number of blades is greater than 4. The pitch angle is determined graphically from equation 114. This value is used to determine the chord length (equation 107) and the strength value (108). The next step involves initialising the interference values a and a^' to 0. This reduces equation 124 to :

ϕ0r=arctanλr=arctan⁡(λrR)                      (50)

Using this data, we perform a change of variables : μ=rR . Thus, the variations are transformed into μ over the interval [0, 1]. Using this value of ϕ(μ), we calculate the optimal angle of incidence βopt(μ)  from equation 122 and the lift coefficient (equation 119), the exact values of the interferences and the value of the angle ϕ1 . The algorithm involves a loop to determine ϕi+1(μ)  until:

ϕi+1-ϕi<ϵ                                                   (51)

ϵ  is known as a convergence test. In the case of a low-speed wind turbine, ϵ=10-4rad

Glauert correction

The Glauert correction involves modifying the interference values when the blade is heavily loaded. This is the case with a multi-blade pumped storage wind turbine. The solidity is greater than 1, which is suitable for blades that may overlap one another. Using the blade element method, Hermann Glauert observed that the axial induction factor and the tangential induction factor given by equations 82 and 87 take on physically impossible values. Glauert’s correction is necessary in the case of high solidity and multiple blades for a wind turbine, when combined with the results from the blade element method. The problem lies in the wake, which creates instabilities; the axial interference exceeds the maximum of 0.5 and the tangential interference diverges to take on negative values. The BEM is not incorrect, but the simplified axial moment assumption becomes invalid. Glauert proposed empirical formulas that have been fitted to experimental data.^[11][13][17].

amaxr=12(1+1-σCL)

                            (52)

 

This maximum value will be the value of a

 if a>amax

 

a'r=12(2+K-(2+K)2-4)

                  (53)

 

K=σCL2sinϕ

                                                         (54)

 

It’s the value a'

 if a'<0

 

These equations ensure a physically valid solution by avoiding divergences and remaining consistent with experiments and real-world cases.
 

Power coefficients and rotor torque:

• Rotor torque and the torque coefficient

The torque generated by the rotor is based on formula 13 studied for the blade element. By making the variable μ=rR  and inserting λr=ΩrV1=λμ . The moment therefore relates to the variation of μ  between 0,1 and 1 according to equation 33.

Mt=4ρπV12λR30,11μ41-aa'dμ                  (55)

The torque coefficient is determined by

Ct=Mt12ρSdV12R                                                     (56)

With Sd=2πR  which gives us:

Ct=8λ0,11μ41-aa'dμ                                (57)

• Power and power coefficient

The power established at the rotor level for our study concerns only the power associated with the tangential forces that generate the torque. Therefore, the fundamental relationship:

P=ΩM                                                            (58)

Cp=ΩRV1M12ρSdV12R                                                (59)

Cp=λCt                                                           (60)

• Solution method

The trapezoidal method is best suited for the solution. The integral is defined as the area between the curve drawn by f(μ)  and the x'Ox . The trapezium method approximates this area as the sum of several trapeziums with bases f(μi)  an f(μi+1)  and height μi+1-μi=∆μ=0,1  in our case.

I≈i=0,11fμi+f(μi+1)2∆μ                                 (61)

Ct=4λ∆μi=0,11μi41-aia'i+μi+141-ai+1a'i+1                                                      (62)

For torque and power, it will suffice to transpose the values obtained by iteration and adapt them to equations 70 and 67.

RÉSULTS AND DISCUSSION

Résults:

The results for the power coefficient and torque coefficient are shown in the following graphs, which illustrate the variation in the tip speed ratio TSR between 0,6 and 2 and the number of blades between 4 and 35

• Betz limit

The following figure shows the Betz limit derived from equation 4.

Fig. 9: Power coefficient derived to Betz limit (Source : Author)
 

Fig. 9 shows the power coefficient according to the Rankine-Froude-Betz theory. The curve rises slightly, starting from 0 at zero downstream velocity, to reach 0.5926 when the downstream velocity is one-third of the upstream velocity. It then falls sharply to negative values. It therefore has a maximum at (1/3; 0.5926)

• Torque coefficient

The following figure shows the torque coefficient Ct  derived to λ  and n

Fig. 11 shows the variation of the power coefficient Cp as a function of the specific speed ratio λ for different numbers of blades. For rotors with few blades, an increase in the specific speed ratio leads to a gradual increase in the power coefficient. In this range, the trend remains relatively steady and close to a quasi-linear behaviour. As the number of blades increases, the growth in the power coefficient accelerates and the curves take on a more pronounced shape. However, for high-aspect-ratio rotors, the power coefficient reaches a maximum at lower values of λ. Beyond this point, the power decreases as the TSR continues to increase. In the region of high λ values, some local irregularities are also observed on the curves, linked to the limitations of the aerodynamic model used.

Fig. 11 also shows the variation of the power coefficient as a function of the number of blades for different values of λ. For low TSRs, increasing the number of blades leads to a gradual improvement in the power coefficient. This variation generally follows an upward trend with a parabolic shape. However, as the TSR increases, the power coefficient reaches a maximum for an intermediate number of blades. Beyond this point, a further increase in the number of blades leads to a decrease in the power coefficient. It is also noted that an increase in the TSR gradually shifts this maximum towards lower numbers of blades. In the region close to these maxima, some curves exhibit slight local irregularities.

Discussion :

• Effect of the Tip Speed Ratio λ

The results obtained show that the aerodynamic coefficients Cp  (power coefficient) and Ct  (torque coefficient) are highly dependent on the specific speed ratio λ. For low values of λ (approximately 0.6–1), a gradual increase in Cp and Ct is observed. This trend can be explained by the fact that at low specific speeds, the tangential component of the relative velocity on the blades increases gradually. This improves aerodynamic lift and promotes torque generation on the rotor. As λ increases to values between 0.8 and 1.4, the results show the emergence of a peak in aerodynamic performance. In this range, the maximum values obtained are approximately :

Ct≈34%-42,9%                                           (63)

CP. ≈30%-40,5%                                          (64)

These results are close to the values reported in the literature for multi-blade pumped-storage wind turbines. For example, Burton et al. (2011) generally report power coefficients between 0.30 and 0.38 for this type of rotor [6]. The values obtained in this study reach approximately 0.405, representing a difference of around +2 to +3%. This slight improvement can be explained by the inclusion of Prandtl and Glauert aerodynamic corrections, which allow for better consideration of tip effects and interference in high-aspect-ratio rotors.

Furthermore, Manwell et al. (2010) indicate that the optimal TSR for multi-blade pumped storage wind turbines generally lies around the following values ^[13]:

λopt≈1-1,5                                                   (65)

The results obtained in this study confirm this trend, as maximum performance is observed within this same range of values. This consistency confirms the physical validity of the model used. Unlike modern wind turbines designed for electricity generation, which generally operate with λ values between 6 and 10, pumped-storage wind turbines are optimised to operate at low specific speeds to produce high torque. ^[4]

• Effect of the numbers of blades n

The results also show that increasing the number of blades initially leads to an improvement in the power coefficient and torque coefficient. This trend can be explained by the increased rotor stiffness (Equation 24), which allows a greater proportion of the wind’s kinetic energy to be captured. As the number of blades increases, the active surface area of the rotor also increases, which improves the transmission of aerodynamic forces to the motor torque.

However, as the TSR increases, too many blades can reduce the aerodynamic efficiency of the rotor. In this case, the high aspect ratio acts as a partial obstruction to the wind flow by increasing axial interference, which reduces power (Equation 27). The airflow is slowed upstream of the rotor and axial interference becomes more significant. This reduces the effective wind speed passing through the rotor and progressively decreases energy efficiency. These observations are consistent with the experimental results reported by Hansen (2015), which show that an excessive increase in aspect ratio and the number of blades improves static torque but leads to a decrease in overall energy efficiency. ^[7]

• Non-linear behaviour for high numbers of blades and high TSR

The resulting flow fields also exhibit irregularities for high values of the number of blades and the aspect ratio, particularly for n > 25 and λ > 1.6. In this region, the coefficients Cp and Ct  show slight fluctuations. These irregularities can be explained by several aerodynamic phenomena. When the rotor’s solidity becomes very high, the wake behind the wind turbine becomes more complex. Interactions between the vortices generated by each blade can cause local disturbances in the distribution of wind speeds. These wake and aerodynamic interference effects can then produce local variations in the aerodynamic coefficients. ^[9][11] Furthermore, the Blade Element Momentum (BEM) model used in this study has certain limitations when the rotor aspect ratio is very high. Under these conditions, certain simplifying assumptions of the model, notably the quasi-axisymmetric flow and the independence of the blade elements, become less valid. Glauert’s empirical corrections improve the stability of the calculation, but they may also introduce small variations in the results during the iterative process. ^[31][32][33]

• Comparison with the typical performance of multi-blade pumped-storage wind turbines

The results obtained are generally consistent with the performance data reported in the literature. Typical values observed for pumped-storage wind turbines are given by

λopt≈0,8-2                                                   (66)

Cpmaximum≈30%-45%                             (67)

Ctmaximum≈30%-50%                             (68)

The maximum Cp values observed in the figure fall within this range, confirming the physical consistency of the model used. These performance figures remain below the theoretical Betz limit Cp,max =0.593. Furthermore, this is to be expected for high-stiffness multi-blade rotors, where the primary objective is torque production rather than maximum energy efficiency.

The maximum values obtained in this study (Cp≈40,5%  et Ct≈42,9% ) fall within these ranges. This confirms the physical consistency of the simulation model used. However, these performance figures remain below the theoretical Betz limit :

Cpmax≤0,593                                                (69)

This difference is to be expected for high-stiffness multi-blade rotors. In this type of wind turbine, the primary objective is not to maximise energy efficiency, but to produce high torque at low rotational speeds, which is necessary to drive mechanical pumping systems efficiently.

CONCLUSION

This study analysed the aerodynamic behaviour of a low-speed, multi-bladed horizontal-axis wind turbine used for water pumping. The analysis focused on the torque coefficient and the power coefficient. The modelling was based on the fundamental Rankine–Froude and Kutta–Joukowski theories, combined with the blade element method (BEM). The Prandtl and Glauert corrections were also applied to account for the high solidity characteristic of multi-bladed rotors. The iterative method used in this study enabled the determination of the rotor’s key aerodynamic parameters, including the inflow angle, the axial and tangential interference factors, and the power and torque coefficients. The parametric analysis carried out for specific speed ratios ranging from 0.6 to 2 and for a number of blades varying between 4 and 35 shows the existence of an optimal operating range for pumped storage wind turbines. The results indicate that the best performance is achieved for a specific speed ratio between 0.8 and 1.4. Within this operating range, the power coefficient can reach approximately 0.40 and the torque coefficient can exceed 0.40 depending on the rotor configuration. These results are consistent with the performance generally observed for multi-blade wind turbines used for pumping.

The study also identifies several useful guidelines for the design of this type of wind turbine. The rotor design should prioritise a specific speed ratio close to 0.8 to 1.2 to achieve a good balance between power output and the torque required for mechanical pumping. Increasing the number of blades improves torque when the specific speed ratio is low, which facilitates rotor start-up and allows operation even at moderate wind speeds. However, when the number of blades becomes too high, the rotor’s mass increases significantly and may reduce aerodynamic efficiency as the specific speed ratio rises. For pumping applications in rural areas, an intermediate number of blades, generally between twelve and twenty, appears to offer a good balance between starting torque, rotor stability and energy efficiency. The use of simple airfoils such as the NACA 0012 or thin plates is also well-suited to this type of wind turbine, as these profiles allow for simple manufacturing, good mechanical robustness and sufficient torque to drive a pumping system. These results can serve as a basis for the design of pumped-storage wind turbines intended for rural areas, particularly in contexts where ease of manufacture, robustness and low cost are important criteria. However, certain avenues for further work could be explored to improve and complement this study. An experimental validation of the model would be useful to compare the results obtained with the actual performance of a prototype. The study could also be extended to consider other aerodynamic effects, such as drag, three-dimensional effects and wake interactions between the blades. Finally, it would be interesting to analyse the overall operation of the system in greater detail by coupling the wind turbine’s aerodynamic model with the mechanical pumping system and the pump’s hydraulic characteristics. Thus, the proposed modelling constitutes a useful tool for the analysis and optimisation of pumped-storage wind turbines, particularly for water supply applications in rural areas.

REFERENCES

1. Le secteur de l’énergie à Madagascar, Ministère de l’Energie 2014
2. Francis Meunier : Energie renouvelable, collection Idées Reçues, Edition Le Cavalier Bleu
3. UNESCO/IRC : Windpump handbook, International Reference Centre for Community Water Supply, 1982.
4. Wilson et Lissaman: Applied Aerodynamics of Wind Power Machines1974
5. Mathew S. : Wind Energy: Fundamentals, Resources Analysis and Economics2006
6. Manwell et Al : Wind Energy Explained2010
7. Fraenkel P. : Water pumping devices - A handbook for users1986
8. RATSIMBAZAFY : Détermination des grandeurs caractéristiques et des relations entre elles pour le mécanisme de transmission et de rotation d’une éolienne lente destinée au pompage d’eau, 2011
9. Gharib-Yosry A. et Al:  Experimental Analysis of Wind Driven water pumping system, 2021
10. Istam et Al:  Aerodynamic models for Darrieus type VAWT2008
11. Louis-Charles FORCIER : Conception d’une pale d’éolienne de grande envergure à l’aide de techniques d’optimisation structurale, 2010
12. Anika Chatterjee : Design Optimization of Wind Turbine Blades Using Finite Element Analysis, 2021
13. Burton et Al : Wind Energy Handbook, 2011
14. BETZ : Wind Energy
15. Jacques Dehard : Actions du vent sur les structures, 2007
16.  Désiré LE GOURIERES : Energie éolienne, théorie, conception et calcul pratique des installations, Edition Eyrolles 1982
17. Shepherd, D.G. : Aerodynamics of Pumping Windmills. U.S. Department of Agriculture, Agricultural Research Service, 1954
18. Wood, D.:  Small Wind Turbines: Analysis, Design, and Application. Springer, 1991
19. Storm, G.F. : Performance of Multi-bladed Windmills. U.S. Department of Agriculture Bulletin, 1937.