ISSN 1063-7842, Technical Physics, 2015, Vol. 60, No. 11, pp. 1632–1636. © Pleiades Publishing, Ltd., 2015. Original Russian Text © V.L. Okulov, R. Mikkelsen, I.V. Litvinov, I.V. Naumov, 2015, published in Zhurnal Tekhnicheskoi Fiziki, 2015, Vol. 60, No. 11, pp. 60–64.
GASES AND LIQUIDS
Efficiency of Operation of Wind Turbine Rotors Optimized by the Glauert and Betz Methods V. L. Okulov*a,b, R. Mikkelsenb, I. V. Litvinova, and I. V. Naumova a Institute
of Thermal Physics, Siberian Branch, Russian Academy of Sciences, ul. Akademika Lavrent’eva 15, Novosibirsk, 630090 Russia b Wind Power Faculty, Technical University of Denmark, Lyngby, DK-2800 Denmark *e-mail:
[email protected] Received December 22, 2014
Abstract—The models of two types of rotors with blades constructed using different optimization methods are compared experimentally. In the first case, the Glauert optimization by the pulsed method is used, which is applied independently for each individual blade cross section. This method remains the main approach in designing rotors of various duties. The construction of the other rotor is based on the Betz idea about optimization of rotors by determining a special distribution of circulation over the blade, which ensures the helical structure of the wake behind the rotor. It is established for the first time as a result of direct experimental comparison that the rotor constructed using the Betz method makes it possible to extract more kinetic energy from the homogeneous incoming flow. DOI: 10.1134/S1063784215110237
At the beginning of the 20th century (in 1912), Prof. N.E. Zhukovskii laid the basis of the vortex theory of a rotor, which is rightfully considered as a significant achievement in fluid mechanics [1]. Six years later, Albert Betz (in 1919), who was then a postgraduate student of Ludwig Prandtl, refined this theory for a new configuration of the optimal blade; the prototype of this model was the vortex model of an airfoil of a finite aspect ratio (wing of finite span) with an elliptic load distribution [2]. In a rotor with such a blade (called the Betz rotor for brevity), the distribution of the circulation along the load line replacing a rotating blade should be such that the free vortex sheet converging to it be strictly helical in shape and move uniformly in the axial direction. Analogously to the elliptic airfoil, such a field of free vortices in the wake behind the rotor must correspond to its minimal induction resistance and to the optimal operation regime [3]. Indeed, if we take into account the rotation of a blade in a uniformly incoming fluid flow, imparting the helical shape to the vortex sheet converging to the rear edge, this model appears as an obvious consequence of the theory of an elliptic airfoil, but the circulation distribution is asymmetric in this case and not elliptical any longer. The search for such a distribution turned out to be a complicated problem which could not be solved by Betz. Only in 1929, Goldstein succeeded in determining analytically the distribution of circulation along the vortex line, which replaced the optimal rotor in the Betz approximation with numbers of blades of 2, 3, and 4 [4]. However, its exact calcula-
tion turned out to be so cumbersome that Theodorsen (1945) even tried to measure it approximately using the electromagnetic analogy after replacing an idealized helical vortex wake by a folded metal foil [5]. The effective algorithm for obtaining the solution for the Betz rotor as applied to wind turbines with any speed of rotation and any number of blades has been implemented recently in [3], but the construction and testing of the rotor designed according to these calculations have not been performed as yet and are the goal of this study. Naturally, approximate methods like simulation in individual cross sections were developed simultaneously with attempts at obtaining exact solutions based on complex vortex theories of the rotor for a specific design and at constructing blades. The possibility of representing a rotor blade in terms of its individual elements cut by cylindrical sections was indicated by Drzewiecki in 1892 [1]. It should be noted that this simplified approach remains the basis for designing air propellers and water-wheels, compressors, windmills, and turbines. Using this method (i.e., assuming for simplicity that different elements of a fluid tube behave independently), Glauert calculated in 1935 the parameters of a wind turbine rotor [6]. For any fixed value of the rotor radius, he independently optimized the equations of the pulse theory in each individual thin circular section of a fluid tube, disregarding the interaction existing between them and radial changes in pressure.
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EFFICIENCY OF OPERATION OF WIND TURBINE ROTORS
(a)
(b) L U∞
D
z
Φ
Φ Ωr
γ
α φ
U∞ Vrel
Vrel
W
1 2 ub
Ωr
wU ∞
uθ
0
uz
0
Fig. 1. Velocity triangles in an arbitrary cross section of the blade: (a) for Glauert’s rotor with induced velocity W in the rotor plane; (b) for Betz’s rotor with velocity wU∞ of the helical vortex wake.
Analysis performed in this study provides the answer to the question which shape of rotors (designed in accordance with the Betz theory or optimized in sections according to Glauert) is the best for constructing wind generators and for extracting the highest kinetic energy from wind. For this purpose, we tested water models for both types of rotors in laboratory conditions. For constructing the blades of both types of rotors, two independent approaches to determining the forces acting on an element of a blade are used. The first approach is based on determining the forces acting in the blade cross section on a chosen 2D profile with detailed description of its characteristics, or tabulated in relevant handbooks, or measured independently. In our tests, we chose for correct comparison the wellstudied and approved SD7003 profile [7], which is the same for both rotors. The force acting in the axial z direction on a given element of the blade with a given profile can formally be written in the same form for both rotors (Fig. 1a):
dT = 1 ρ cN V 2 C , (1) b rel n dr 2 where c is the chord of the given blade element, Vrel is the relative velocity of the flow incoming to the profile, ρ is the density of the fluid, Nb is the number of blades, and Cn is the coefficient of the force acting on a blade element in the axial direction of the rotor and corresponding to the chosen profile. If we disregard the drag loss in view of its smallness for the chosen SD7003 profile, the axial force coefficient can be expressed in terms of the lifting force coefficient with its tabulated value CL = 0.8; i.e., C n ≈ C L cos φ, TECHNICAL PHYSICS
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where ϕ is the angle between the direction of velocity Vrel and the plane of rotation of the rotor. It should be noted that the density, the number of blades, and the lifting force coefficient in formulas (1) and (2) for the chosen profile are identical for both rotors, while other parameters (velocity and angle between it and the plane of the rotor) contain individual information on the rotors. Analysis of the velocity triangles in Fig. 1 leads to the conclusion that relative velocity Vrel of the incoming flow to the profile in any cross section of the rotor depends on wind velocity U∞, local angular velocity Ωr of the blade element, and rate W of deceleration of the flow by the rotor determined by the pulsed method in cross sections (Fig. 1a) for the Glauert rotor [6]. For the Betz rotor, the deceleration of the flow is determined in terms of the velocity induced by helical vortex wakes and equal to half the velocity of their axial motion [3]; i.e., 1 wU∞ (Fig. 1b). Further, we recall the 2 algorithm for constructing the shape of blades using the impulse theory in cross section, or boundary-element method (BEM), according to Glauert [8] and then describe the algorithm used here for designing the Betz rotor blade. Using the impulse technique applied independently in each circular element of the fluid tube, Glauert [6] determined the dependences of quantities a, a', a'X2λ2 for each local value of X = Ωr/U∞, where a and a' characterize the degree of variation in the axial and circular velocity components of the flow incoming to the profile, and λ = Ωr/U∞ is the rotation speed of the rotor of radius r. Thus, these data can be used to determine the rate of reversal flow in the rotor plane according to Glauert, W = (–aU∞, a'Ωr), which allows us to derive the following relations from the polygon of velocities (see Fig. 1a):
sin φ =
U ∞ (1 − a) Ω r (1 + a ') and cos φ = V rel V rel
or
U ∞ (1 − a) (3) . 2 sin φ Substituting relations (2) and (3) into (1), we obtain 2
2 V rel =
2
dT = ρ N bcU ∞ (1 − a) cos φ C . (4) L 2 dr 2 sin φ To determine the axial force acting in the Glauert rotor, we now apply the impulse technique to the entire flow tube containing the plane of the rotor [8]: 2
2
dT = ρ(U − u )2π ru = 4πr ρ U 2 a(1 − a), (5) ∞ R ∞ wake dr where uR = U∞(1 – a) is the value of the axial velocity in the plane of the rotor and uwake = U∞(1 – 2a) is the
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OKULOV et al.
value of the axial velocity in the far wake. After equating the right-hand sides of Eqs. (4) and (5), we obtain the following relation for determining the chord of the SD7003 profile along the blade of the Glauert rotor:
where λ = ΩR/U∞ defines the rotor tip speed ratio. Substituting relations (2) and (8) into (1) and considering that the angle between Vrel and the rotor plane is Φ in this case, we obtain
8π ra sin φ (6) . C L N b (1 − a) cos φ To perform calculations using this expression, it remains for us to determine angle ϕ, viz., the direction of the main flow in each blade cross section. In accordance with expression (3), it is defined by the formula 2
c=
U (1 − a) (1 − a) (7) ≡ tan φ = ∞ Ω r (1 + a ') X (1 + a ') with subsequent computation using tabulated data. For the Betz rotor, relative velocity Vrel = (U∞ – u z0 , Ωr + uθ0 ) is defined in terms of the axial u z0 and circular uθ0 components of the velocity induced by the wake in the plane of the rotor [3]. Using the velocity triangle (see Fig. 1b), we can find the following expressions for these components from the half-value of the axial velocity of the vortex wake relative to the velocity of the incoming flow (i.e., 1 wU∞ [9]) and angle Φ 2 between the relative velocity and the rotor plane: u z0 = 1 wU ∞ cos 2 Φ 2 and
uθ0 = 1 wU ∞ cos Φ sin Φ 2
(
)
(
)
2 ⎤ + λ r + 1 w cos Φ sin Φ ⎥ R 2 ⎦
2
12
and
V rel =
Ω r + uθ 0 , cos Φ
c=
(8)
(
)
(
2 ⎤ + λ r + 1 w cos Φ sin Φ ⎥ R 2 ⎦
)
2
(9)
1/2
(Ω r + uθ0 ).
To find now the axial force acting on all blades of the Betz rotor, we apply the vortex theory of the rotor. The distribution of the total axial load along the radial section of the rotor in this case can be determined using the Kutta–Zhukovskii theorem [3]
dT = ρ N Γ(Ω r + u ). (10) b θ0 dr In this expression, we replace the total circulation along the blades by its dimensionless value G introduced by Goldstein [4]: N bΓ = GwU ∞ h,
(11)
where h is the pitch of the helical structure of the vortex system of the wake behind the rotor and wU∞ is the constant velocity of the wake flow behind the rotor relative to the velocity of the incoming flow [9] (see above). In addition, we pass from the abstract value of the helix lead in the vortex wake to the rotor specific speed λ [9]:
(
)
(
)
(12) h = 1π 1 − 1 w . λ 2 Substituting expressions (11) and (12) into (10), we obtain
or
⎡ 2 V rel = U ∞ ⎢ 1 − 1 w cos Φ 2 ⎣
(
dT = 1 ρ cN C U ⎡ 1 − 1 w cos 2 Φ b I ∞ ⎢ dr 2 2 ⎣
(
dT = 2πρ GU w 1 − 1 w (Ω r + u ). (13) ∞ θ0 dr λ 2 Equating the right-hand sides of Eqs. (9) and (13), we obtain the expression for determining the distribution of the chord of the SD7003 profile over the blade for the Betz rotor:
)
4π w 1 − 1 w G 2
⎡ λ N bC L ⎢ 1 − 1 w cos 2 Φ 2 ⎣
) ( 2
)
2 ⎤ + λ r + 1 w cos Φ sin Φ ⎥ R 2 ⎦
To perform calculations based on this expression, it remains for us to determine angle Φ, viz., the direction of the main flow in each cross section of the blade and reduction factor w of the wake. In accordance with the velocity triangles in Fig. 1b, angle Φ is defined by the formula
tan Φ =
1/2
(14)
.
(
)
U ∞ − u z0 U ∞ ≡ 1− 1w . 2 Ω r + uθ 0 Ω r
(15)
Reduction factor w of the wake flow can be determined by solving the problem of determining the maximal energy ratio of wind (this term was introduced by TECHNICAL PHYSICS
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EFFICIENCY OF OPERATION OF WIND TURBINE ROTORS Conditions of the flow through the plane of the rotor optimized according to Glauert [6] a
a'
a'X2λ2
λX
0.25 0.27 0.29 0.31 0.33 1/3
∞ 2.375 0.812 0.292 0.031 0
0 0.0584 0.1136 0.1656 0.2144 0.2222
0 0.157 0.374 0.753 2.630 ∞
Zhukovskii) or wind wheel power coefficient (modern term). For the Betz rotor, this coefficient was determined in [3, 9]:
)(
(
)
C P = 2w 1 − 1 w I 1 − 1 wI 3 , 2 2 where I1 = 2
∫
R
0
(16)
G (r, h)rdr and I3 = 2
∫
R
0
G (r,
r 3dr are defined in terms of the Goldstein r 2 + (h/2π) 2 function determined in [3, 9]. It should be emphasized that dimensionless coefficient (16) is a function of only one universal parameter w, which is the same for all points of the wake. For this reason, it can be used for optimizing the problem. Differentiating CP with respect to w, we find that its maximal value is attained for (Table). h)
(
)
(17)
0.6
0.8
1.0
0.6
0.8
1.0
2 2 w = 2 I 1 + I 3 − I 1 − I 1I 3 + I 3 . 3I 3
c
0.2
1635
Thus, the construction of an optimal blade for the Betz rotor is determined by the solution of Eqs. (14) and (15) using parameter w calculated by formula (17).
It should be noted that for the Glauert and Betz rotors with identical diameters 2R = 0.376 m for specific tip speed ratio λ = 5, the variation of chord c of the SD7003 profile along the blade is determined using relations (6) and (14), respectively (Fig. 2a). The final angle of its arrangement in each cross section of both blades with the correction of the values of (7) and (15) for angle of attack α is shown in Fig. 2b. For all cross sections, angle α was chosen the same and equal to 4° for both rotors. The shapes of the blades in the rotors designed in this way are shown in Fig. 3. The experimental setup with insignificant modifications is identical to that described in [10–12]. The models of the rotors were placed into the test water channel. For specific tip speed ratio λ = 5 at a working temperature of 20°C, Reynolds number Re in our experiments was approximately equal to 20000. The values of the velocity of the incoming flow in the region of the rotor (U∞ = 0.65 m/s) and its oscillations during the experiment did not exceed 3%. The force characteristics of the models of the rotors were measured by strain gauges installed in the rotor attachment. We measured the torque and the arrestor force acting on the rotor for specific tip speed ratio λ = 3–9. Figure 4 shows the corresponding dependences of the power coefficient Cp and thrust coefficient CT for both wind wheels. It can be seen from the graphs that the maximal efficiency C Pmax is attained in both cases for the rated specific tip speed ratio equal to 5. At the same time, it is larger for the rotor designed using the approach proposed by Betz [2]. Indeed, in contrast to the vortex theory of the rotor, the Glauert optimization imposes more stringent constraints associated with the disregard of the variation of pressure in the radial direction and the interaction between independent cross sections. Apparently, these disregards in
0.1
0
0.2
0.4
0.2
0.4
r/R
60
γ
40 20 0
r/R Fig. 2. Distributions of chord c and angle γ of setting of the SD7003 profile for Glauert’s rotor (dashed curve) and for Betz’s rotor (solid curve). TECHNICAL PHYSICS
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Fig. 3. Photographs of blades and rotors of Glauert (left) and Betz (right).
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OKULOV et al.
energy from the uniformly incoming flow as compared to the rotor optimized according to Glauert.
0.6 Betz’s rotor Glauert’s rotor
ACKNOWLEDGMENTS This work was supported financially by the Russian Science Foundation (project no. 14-19-00487).
Cp
0.4
REFERENCES
0.2
0
2
3
4
5
1.2
6 λ
7
8
9
10
6 λ
7
8
9
10
Betz’s rotor Glauert’s rotor
1.1 1.0 CT
0.9 0.8 0.7 0.6 0.5
2
3
4
5
Fig. 4. Coefficients of power CP and thrust CT for both rotors for various values of specific speed.
optimization lead to a larger loss in the Glauert rotor efficiency. Thus, we have established for the first time as a result of direct experimental comparison that the rotor designed in accordance with the Betz optimization method makes it possible to extract higher kinetic
1. V. L. Okulov, Zh. N. Sorensen, and G. A. M. van Kuik, Development of the Optimum Rotor Theories: On the 100th Anniversary of Professor Joukowsky’s Vortex Theory of Screw Propeller (RKhD, Moskva–Izhevsk, 2013). 2. A. Betz, Gottinger Nachrichten (Gottingen, 1919), pp. 196–217. 3. V. L. Okulov and J. N. Sorensen, Wind Energy 11, 415 (2008). 4. S. Goldstein, Proc. R. Soc. London, Ser. A 123, 440 (1929). 5. T. Theodorsen, Theory of Propellers (McGraw-Hill, New York, 1948). 6. H. Glauert, Airplane Propellers, Div. L of Aerodynamic Theory IV, Ed. by W. F. Durand (Springer, Berlin, 1935), pp. 169–360. 7. M. S. Selig, J. J. Guglielmo, A. P. Broeren, and P. Giguere, Summary of Low-Speed Airfoil Data (SoarTech, Virginia Beach, 1995), Vol. 1, p. 292. 8. M. O. L. Hansen, Aerodynamics of Wind Turbines (Earthscan, London, 2008), p. 181. 9. V. l. Okulov and J. N. Sorensen, J. Fluid Mech. 649, 497 (2010). 10. V. L. Okulov, I. V. Naumov, R. F. Mikkelsen, I. K. Kabardin, and J. N. Sorensen, J. Fluid Mech. 747, 369 (2014). 11. I. V. Naumov, V. V. Rakhmanov, V. L. Okulov, K. M. Velte, K. E. Maier, and R. F. Mikkel’sen, Teplofiz. Aeromekh. 19, 268 (2012). 12. I. K. Kabardin, I. V. Naumov, R. F. Mikel’sen, V. A. Pavlov, G. V. Bakakin, and V. L. Okulov, Vestn. Novosib. Gos. Univ., Ser. Fiz. 8, 89 (2013).
Translated by N. Wadhwa
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