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Efficient clusters and patterned farms for Darrieus wind turbines
Sustainable Energy Technologies and Assessments 19 (2017) 125–135
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Sustainable Energy Technologies and Assessments journal homepage: www.elsevier.com/locate/seta
Original article
Efficient clusters and patterned farms for Darrieus wind turbines Mohammed Shaheen ⇑, Shaaban Abdallah Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, United States
a r t i c l e
i n f o
Article history: Received 6 March 2016 Revised 8 November 2016 Accepted 13 January 2017
Keywords: Renewable energy Wind power Vertical axis wind turbines (VAWTs) Darrieus wind turbines Wind farm power density
a b s t r a c t Recent studies show that interactions between Darrieus vertical axis wind turbines (VAWTs) arranged in close proximity mutually enhance the output power of individual turbines. In this paper, multi-Darrieus turbine clusters are studied for the development of efficient patterned vertical axis wind turbine farms. Numerical solutions are performed for a single Darrieus turbine, clusters of two Co- and counter-rotating turbines in parallel and in oblique configurations, and triangular three turbine clusters. The commercial CFD software FLUENT 14.5 is used for the numerical simulation. The single Darrieus turbine results are validated against experimental data. The results of the simulation of the multi-turbine clusters are used to develop an efficient triangular shaped three turbine cluster having an average power coefficient up to 30% higher than an isolated turbine. Efficient Darrieus wind turbine farms are developed using the enhanced three turbine cluster. The farms consist of multiple clusters with scaled geometrical ratios of the three turbine cluster keeping the same topology of the cluster. This resulted in patterned farms that have the same power enhancement ratio of the three turbine cluster. Numerical simulation of a farm that consist of nine turbines confirm the enhancement and the pattern progression for larger farms. Ó 2017 Elsevier Ltd. All rights reserved.
Introduction Horizontal axis wind turbines (HAWTs) has the highest efficiency of all types of wind turbines and this made most of the wind farms consist of HAWTs. Horizontal axis wind turbines (HAWTs) dominate the majority of the wind industry for the last 30 years. 74% of the small wind turbines manufacturers invested in the HAWT, however 18% adopted VAWTs [1]. Mostly wind turbine farms consist of HAWTs because of the higher efficiency of an isolated HAWT compared to a VAWT of the same size. Land-based HAWT farms have successfully been implemented, a modern HAWT farm consists of an array of 30 to 150 turbines [2]. A large amount of land is required to separate HAWTs from the adjacent turbines’ wakes. The available power extracted from a HAWT farm is limited by the requirement of separating each turbine from adjacent turbines wakes. To maintain 90% of the performance of isolated HAWTs, the turbines in a HAWT farm must be spaced 3–5 turbine diameters apart in the cross-wind direction and 6–10 diameters apart in the downwind direction [3]. A Typical HAWT farm has a power densities, defined as the power extracted divided by the area of its foot print, between 2 and 3 W/m2 [3]. Researchers at CalTech found that the power density of VAWT farms can be increased up to 30 W/m2 by optimizing the placement of the tur⇑ Corresponding author. E-mail address: [email protected] (M. Shaheen). http://dx.doi.org/10.1016/j.seta.2017.01.007 2213-1388/Ó 2017 Elsevier Ltd. All rights reserved.
bines, this enables them to extract energy from adjacent turbines’ wakes [4]. The efficiency of two counter rotating H-rotor VAWTs set at a distance of 1.5D to 2.0D was tested experimentally, an increase of 11% over the efficiency obtained by their stand-alone counterpart was achieved [5]. A geometric arrangement based on the configuration of shed vortices in the wake of schooling fish showed an increase in the performance of an array of 16 16 wind turbines by one order of magnitude [6]. A Study on of the effect of the relative angular position on aerodynamic interference between two curved blade Darrieus VAWTs showed a power reduction for the downstream turbine at small oblique angular positions, as the tip speed ratio of the upwind turbine increases, the blade circulation increases accompanied by an increase in the strength of the shed vortices, resulting in higher induced velocities, and consequently lower angles of attack at the downstream turbines, and correspondingly lower torque and power output [7]. Numerical analysis on two dimensional clusters of Savonius VAWTs showed that the efficiency of turbines operating completely in the wake of other turbines can be improved by choosing their angular position such that they see a higher flow velocity due to the presence of other turbines and associated stream-tube contraction [8]. This study is interested in development of efficient Darrieus VAWT farms. A single Darrieus turbine is solved numerically to validate the numerical model by comparing the results with experimental data in reference [9]. Performance of clusters of two coand counter-rotating turbines in parallel and in oblique configura-
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Nomenclature turbine swept area As = DH, m2 torque coefficient, T/(0.5 qV21 RAS) static torque coefficient, Ts/(0.5 qV21 RAS) power coefficient, TX/(qV31 RH) rotor diameter, m bucket diameter, m overlap ratio of the rotor buckets, e = S/d rotor height, m rotor radius, m gap distance between turbines, m cluster height, m
AS Cm Cms Cp D d e H R S S1
T Ts V1
q
X k h
a
u
tions is studied to be used in the development of an efficient triangular three turbine cluster. The developed cluster is simulated at different tip speed ratios in order to investigate the enhancement in its turbines’ power coefficients compared to isolated turbines. The cluster is used as a building unit for constructing efficient triangular Darrieus VAWT farms. The developed farms consist of multiple clusters with scaled geometrical ratios of the three turbine cluster keeping the same topology of the cluster. This resulted in patterned farms that have the same power enhancement ratio of the three turbine cluster. Numerical simulation of a farm that consist of nine turbines confirm the enhancement and the pattern progression for larger farms. Numerical simulation model Turbine geometry and aerodynamics The geometry used in this study is a three blade Darrieus VAWT with Straight-blades founded in reference [9]. The geometric features are listed in the Table 1. The spokes that connect the turbine blades with the rotating shaft are connected to each blade at the center of pressure of the airfoil (0.25C). Fig. 1 shows the velocity triangles at different azimuth positions of the turbine rotor blade and the force analysis on the blade. The aerodynamics of VAWTs involve high local angles of attack and the wake from the upwind blades and from the axis [10]. The tangential force FT and the normal force FN are originally the transformation of the components of lift FL and drag forces FD . The expression of FT and FN can be determined as follows:
F T ¼ F L sin£ F D cos£
ð1Þ
FN ¼ FL cos£ FD sin£
ð2Þ
The amount of torque and power torque acting on one blade can be obtained respectively from the following expressions:
T¼
R ð2pÞ
Z 2p
F T ðhÞdh
torque, N/m static torque, N/m free stream velocity, m/s density, kg/m3 rotation speed, rad/s tip speed ratio, XR=V1 azimuth angle of the turbine oblique angle between rotors angle of resultant flow velocity
ð3Þ
0
Table 1 Turbine Geometrical Features. Turbine Category
Vertical Axis Wind Turbine
Turbine Type Blade Airfoil Blade Shape Number of Blades Turbine Rotor Diameter (mm) Turbine Height (mm) Blade Chord Length (mm)
Darrieus Type NACA 0021 Straight Blades 3 1030 1414 85.8
R P ¼ 2p x
Z 2p
F T ðhÞdh
ð4Þ
0
From the power and torque equation, the enhancement of the power of VAWTs is affected mainly by the tangential force which increases when the lift force increases on the blades in addition to the value of the resultant flow angle. Computational domain Different domains are tested by solving the single Darrieus turbine at different speeds in order to obtain the appropriate size of the computational domain. The best compromise is found in Fig. 2, where the upstream, upper and lower boundaries of the domain are at a distance 15D (D is the turbine diameter) from the turbine center, while the downstream boundary is at 30D. The domain is divided into 3 sub-domains: A rotating region (1) with a torus shape includes the turbine blades, a fixed region (2) extends from the outer interface of the rotating region to the boundaries of the domain and a fixed region (3) extends from the inner interface of the rotating region to the shaft of the turbine. The outer diameter of the rotating region is 1.2D and its inner diameter is 0.8D . The upstream boundary of the domain is defined by velocity inlet boundary condition (BC) (x-component = 9 m/s), the upper and lower boundaries are defined by symmetry BC and the downstream boundary is defined by pressure outlet BC [11– 13]. Grid generation The sliding mesh model is used to capture the unsteady flow parameters generated due to the rotation of the turbine blades. A non conformal mesh with unstructured triangular cells is created using ANSYS meshing. An inflation with 10 levels of quadrilateral cells is imposed on the airfoil’s surfaces to account for the boundary layer with a maximum thickness of 1 mm and growth rate of 1.1. Five grid refinements levels ranging from 47,591 up to 450,705 cells are tested at all wind speeds and rotational speeds in order to obtain y+ < 1 as required by the transition SST turbulence model [14,15]. A mesh of 173,768 cells is found to be adequate as the relative variation in the output quantities are below 1 103 as shown in Table 2. The grid structure is shown in Fig. 3(a) and a close view of the grid in the vicinity of the rotor blade leading edge is shown in Fig. 3(b). Solver settings and turbulence model Fluent code is used to solve the unsteady Reynolds averaged Navier-Stokes equations using finite volume method [15–18]. SIM-
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Fig. 1. Darrieus VAWT Rotor Blades, Velocity Triangles and Force Analysis [9].
Fig. 2. Computational Domain.
@ðqcÞ qðU jc Þ @ ¼ Pc1 Ec1 þ P c2 Ec2 þ þ @t @xj @xj
Table 2 Grid Refinement Levels for Single Turbine Simulation. Grid level
Number of cells
Cm
Cp
1 2 3 4 5
47,591 81,034 173,768 235,451 492,605
0.063561 0.0989 0.136 0.137 0.1365
0.166529 0.259118 0.35632 0.35894 0.35763
PLE algorithm (Semi-Implicit Method) with a second order spatial discretization for all pressure, momentum, and turbulence equations and a Least Squares Cell Based algorithm is used for gradients is used [14]. In order to capture the laminar and transitional boundary layers correctly, the transition SST turbulence model introduced by Menter [19] is adopted. This leads to major improvements in the prediction of adverse pressure gradient flows [20]. It is based on SST k-x transport equations coupled with two additional transport equations, one for intermittency (c), and the other for the momentum-thickness Reynolds number Reht as follows [21]:
@ðqReht Þ @ðqU j Reht Þ @ þ ¼ Pht þ @t @xj @xj
rht l þ lt
lþ
lt @ c rc @xj
@Reht @xj
ð5Þ
ð6Þ
The transition model interacts with the SST turbulence model by modification of the K-equation. The definitions for all terms in Eqs. (5)–(7) are discussed in details in the user guide of fluent and Ref. [19].
@ @ @ @k ~ k Y k þ Sk þG ðqkÞ þ ðqkui Þ ¼ Ck @t @xj @xj @xj
ð7Þ
A Steady flow simulations is run for few time steps obtained initially to set up the flow domain to reasonable parametric values, then unsteady flow simulations are obtained using the sliding mesh model (SMM) [21]. Different time step sizes are tested and a time step size corresponding to a turbine rotation of 1.0 degree is used for the time dependent solution [15,22,23]. A drop of at least 3 orders of magnitude in each time step as well as a minimum of 20 iterations per time step is considered as convergence criteria
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Fig. 3. Grid Structure.
for the time dependent solution. The torque (T) is calculated about the turbine center from the resultant pressure and viscous forces on the turbine blades. The torque coefficient (Cm) is calculated by Eq. (8) at each time step and averaged after periodic convergence [10], then the power coefficient (Cp) is calculated from Eq. (9):
C ms ¼
Ts
ð8Þ
0:5qV 21 RAs
Cp ¼ Cm k
ð9Þ
Results and discussion Single Darrieus turbine numerical solution results and validation The average power coefficient of the single Darrieus turbine is calculated at different tip speed ratios (k) using time dependent solution for the rotating turbine. Fig. 4 shows that the fully turbulent k-omega SST model under predicts the values of the power coefficient and that the transition SST turbulence model has the closest results to the experimental data, similar findings are reported in reference [24]. The maximum average power coefficient for the single turbine is found to be 0.345 corresponding to k equal to 2.6. Fig. 5 shows the velocity contours around the Darrieus turbine blades, two regions of high velocity magnitude are induced on both sides downstream the turbine. The kinetic energy
Power Coefficient Cp
0.4
Experimental Data K-w SST Transion SST
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1
1.5
2
2.5
3
3.5
Tips Speed Rao Fig. 4. Cp at Different tip speed Ratios, Numerical Results Vs Experimental data [5].
Fig. 5. Velocity Contours around the Darrieus Turbine Blades.
in these two regions can be used to enhance the efficiency of turbines placed in the near vicinity. Steady and unsteady Loading of the wind turbine Operation of an H-type Darrieus vertical axis wind turbine at low blade speed ratios resulted in complex flow-blade interaction mechanisms. These include dynamic stall resulting in large scale vortex production, vortex impingement on the source blade, and significant flow momentum extraction. The presentation of such analysis of dynamic stall for the flow around the VAWT blade aerofoil is complicated because the relative velocity vector to the blade aerofoil changes in both direction and magnitude around the azimuth as shown in Fig. 6. To study the instantaneous lift coefficient against incidence angle, the instantaneous velocity magnitude should be used in the lift and drag coefficient. Quasi – steady force analysis A quasi-steady two dimensional study is performed for a single blade, the instantaneous relative velocity and angle of attack are calculated graphically at different azimuth angles. The absolute values of the angles of attack are similar at the corresponding posi-
129
Li Coefficient (Cl)
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-30
-20
-10
1 0.8 0.6 0.4 0.2 0 -0.2 0 -0.4 -0.6 -0.8 -1
10
20
30
Cl xfoil Cl Fluent
Angle of Aack ( Fig. 8. Lift Curve, Numerical Results of Fluent Vs Xfoil.
Unsteady force analysis The unsteady flow simulations is obtained using the sliding mesh model. The torque (T) is calculated about the turbine center from the resultant pressure and viscous forces on the turbine blades. The torque of a single blade is plotted at different azimuth angles as shown in Fig. 11, the results shows a great discrepancy between the unsteady and the quasi-unsteady results shown in Fig. 10. The average torque generated by a single blade performed by the unsteady simulation is found to be higher than that of the value calculated by the quasi-steady solution. The discrepancy in the first half of rotation can be explained as the result of the change
0.25 0.2 0.15 0.1
CD Xfoil CD Fluent
0.05 0
-30
-20
-10
0
10
20
30
Angle of Aack ( ) Fig. 9. Drag Curve, Numerical Results of Fluent Vs Xfoil.
Torque (N.m)
tions in the first and second half of the turbine rotation as shown in Fig. 7. The lift and drag coefficients for the blade airfoil are computed at different Reynolds numbers due to the change in the relative velocity magnitude at different azimuth angles. Plots of lift and drag coefficients at different angles of attack are shown in Figs. 8and 9 using two solvers Fluent 14.5 and Xfoil. The lift and drag forces are calculated at different positions and the torque generated from both forces is calculated. The torque of the turbine is a resultant of the positive torque generated by the lift forces and the negative torque generated by the drag force. Fig. 10 shows the quasi-unsteady torque generated by a single turbine blade in one complete cycle, the results are symmetric for the first and second half cycle with two peaks values for the torque along the azimuth, and a positive average generated torque.
6 5 4 3 2 1 0 -1 0 -2
30
60
90
120 150 180 210 240 270 300 330 360
Azimuth Angle ( ) Fig. 10. Torque at Different Azimuth Angles (Quasi-steady Simulation).
40
Torque (N.m)
Fig. 6. Change in the Relative Velocity Vector Due to the Blade Aerofoil Changes Position around the Azimuth.
Drag Coeffciient (Cd)
0.3
20 0 0 -20
30
60
90
120 150 180 210 240 270 300 330 360
Azimuth Angle
Fig. 11. Unsteady Torque Generated by a Single Blade of the Darrieus Turbine.
Angle of Aack ( )
30 20 10 0 -10
0
30
60
90 120 150 180 210 240 270 300 330 360
in the axial velocity due to stream lines expansion and the discrepancy in the second half is due to the same reason in addition to the interference with the vortex shed from the upstream blades. Another reason is that the quasi-steady solution assumes that the flow velocities normal to the free-stream direction are zero which is not true.
-20 -30
Two Darrieus turbine clusters solution results
Azimuth Angle ( )
Fig. 7. Angle of Attack of the Turbine Blade Airfoil at Different Azimuth Angles.
The performance of multi-turbine co-rotating and counterrotating Darrieus turbine clusters is studied. All possible configura-
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tions of parallel and oblique configurations are simulated at different gap distances.
Two parallel turbines Two parallel co- and counter-rotating turbines are simulated at tip speed ratio (k = 2.6) and different gap distances using the same numerical method for the single Darrieus turbine. For the corotating turbines, rotors (1) and (2) are both rotating counter clockwise as shown in Fig. 12(a). For the counter rotating turbines, two cases (A) and (B) are studied. In case (A): Rotor (1) rotates counter clockwise and rotor (2) rotates clockwise where their inward blades move in the same direction of the wind as shown in Fig. 12(b), In case (B): Rotor (1) rotates clockwise and rotor (2) rotates counter clockwise where their inward blades move opposite to the wind direction shows the velocity contours for the two parallel co-rotating turbines as shown in Fig. 12(c). The numerical results in Fig. 13 show an enhancement in the average power coefficient of the two parallel turbine clusters for the three cases compared to isolated turbines (represented by the single turbine). The comparison of the results for the three cases shows that the most efficient configuration for two parallel Darrieus turbines is the counter-rotating configuration case (A). The maximum average power coefficient for case (A) is found to be 0.42 achieved at a gap distance equal to 0.5D as shown in Fig. 13. The enhancement in
Fig. 13. Avg. Power Coefficient for Three Cases of Two Parallel Darrieus Turbines.
the average power coefficient decreases as the gap distance increases. The results show that the counter-rotating turbines enhance the performance of each other, this enhancement vanishes as the gap distances increases between the turbines. The comparison shows that case (A) where the inward blades moves with the wind direction is more efficient than case (B), an average power coefficient of 0.42 is achieved at a gap distance 0.5D in case (CA), this represents 16% higher than the isolated turbines.
1
(a) Velocity Contours around Two Parallel Co-Rotating Turbines
Wind Direcon
2
1
1
Wind Direcon
Wind Direcon
2
2
(b) Case (A)
(c) Case (B)
Fig. 12. Velocity Contours around Two Parallel Darrieus Turbines.
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2
1
Wind Direcon
1
2
(a) Case (C)
(b) Case (D)
Fig. 14. Velocity Contours around Two Parallel Counter-Rotating Darrieus Turbines.
Two oblique turbines Two oblique co- and counter-rotating Darrieus turbines are simulated. The performance is studied at a tip speed ratio (k = 2.6) and different gap distances using the same numerical method for the single Darrieus turbine.
Fig. 15. Avg. Power Coefficient for Two Cases of Two Co-Rotating Oblique Darrieus Turbines at different Gap Distances
Two Co-Rotating oblique turbines. For the co-rotating configuration, two cases (C) and (D) are studied. Rotor (1) is upstream and rotor (2) is downstream. In case (C): rotor (2) is set at an oblique angle +ve 60o and in case (D): rotor (2) is set at an oblique angle ve 60° as shown in Fig. 14(a) and (b). In case (C) turbine (1) has an isolated turbine performance, an enhancement occurs in the power coefficients of turbine (2) due to the Magnus effect of turbine (1). In case (D) both turbines are enhanced in performance due to the Magnus effect of the other turbine (2). The inward blade of turbine (2) faces a high velocity region downstream turbine (1) as shown in Fig. 14(b). The comparison of the average power coefficients for both cases with the isolated turbine performance in Fig. 14 show that case (D) has higher efficiency than case (C). The maximum average power coefficient for case (D) is found to be 0.37 achieved at a gap distance equal to 0.5D as shown in Fig. 15. The enhancement in the average power coefficient in the two cases decreases as the gap distance increases. Two counter-rotating oblique turbines. Two cases (E) and (F) of two oblique counter-rotating Darrieus turbine clusters are numerically studied. Rotor (1) is upstream and rotor (2) is downstream. In case (E): rotor (2) is set at an oblique angle +ve 60o to rotor (1) and their inward blades move in the same direction of the wind as shown in
2 1
Wind Direcon
Wind Direcon
1
2
(a) Case (E)
(b) Case (F)
Fig. 16. Velocity Contours around Two Oblique Counter-Rotating Darrieus Turbines.
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ficient. The results show that two counter-rotating turbines in close proximity enhance the performance of each other, this enhancement decreases as the gap distances increases between the turbines. The comparison shows that case (E) where the inward blades moves with the wind direction is more efficient than case (F) specially at small gap distances. The maximum average power coefficient for case (E) is found to be 0.41 achieved at a gap distance equal to 0.5D. The enhancement in the average power coefficient in the two cases decreases as the gap distance increases. Three turbine clusters simulation Fig. 17. Avg. Power Coefficient for Two Cases of Two Counter-Rotating Oblique Darrieus Turbines.
1
2 Wind Direcon
3
Fig. 18. Velocity Contours around Triangular Three Turbine Darrieus Cluster at k = 2.6.
Fig. 16(a). In case (F): rotor (2) is set at an oblique angle -ve 60o to rotor (1) and their inward blades move opposite to the wind direction Fig. 16(b). The performance of the two oblique turbines in both cases are studied at different gap distances (S). Fig. 17 shows a comparison between the average power coefficients for both cases (E) and (F) and the isolated rotor power coef-
The numerical results for the performance of two co- and counter-rotating parallel and oblique turbines show that case (A) is the most efficient configuration for two parallel turbines, and that case (E) is the most efficient configuration for two oblique turbines. This finding is used to develop an efficient triangular three turbine cluster. The base of the triangle is perpendicular to the wind direction and consists of two counter rotating turbines (1) and (3) as in case (A). The center of the two turbines (1) and (2) are set at a distance of two turbine diameters from each other. Turbine (3) is set downstream in an oblique position and counterrotating to turbine (1) as in case (E) as shown in Fig. 18. The simulation of the triangular three turbine cluster is performed for different triangle heights, where turbine (2) is set at different distances from the line joining the centers of the two upstream turbines (1) and (3). The computational domain for the three turbine cluster simulation shown in Fig. 19 is the similar to the domain used in the single turbine simulation. A grid sensitivity study is performed, the mesh is refined five times from 133,175 to 2,257,340 cells until there is no significant change observed in the torque coefficient of the three turbines as shown in Table 3. A grid with 400,487elements is found to be adequate.
Table 3 Grid levels for Three Turbine Cluster Simulation. Grid Level
No of Cells
Cm1
Cm2
Cm3
1 2 3 4 5
133175 219245 400487 993007 2257340
0.0013 0.0002 0.0004 0.0004 0.0004
0.1409 0.1394 0.1374 0.1374 0.1372
0.0373 0.0380 0.0378 0.0378 0.0378
Fig. 19. Computational Domain for the Three Turbine Cluster Simulation.
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0.5 0.4 0.3 0.2 0.1
Cp1
Cp2
Cp3
0 1
1.5
2
2.5
3
3.5
Tips Speed Rao Fig. 20. Cp of the Three Turbines of the Cluster at Different Tip Speed Ratios.
Table 4 Power Coefficients for the Three Turbine Cluster at Different Tip Speed Ratios. k
Cp1
Cp2
Cp3
Average
1.8 2.2 2.6 3.0
0.11 0.26 0.40 0.42
0.11 0.23 0.43 0.48
0.12 0.25 0.41 0.45
0.11 0.25 0.42 0.45
Power Coefficient Cp
The performance of the three turbine cluster is studied at different tip speed ratios (k) as shown in Fig. 20. The computed power coefficients of the three turbines are shown in Table 4. The results show enhancement in the power coefficients of the three turbines compared to their isolated counter-parts. At k > 2.6 a ratio of 1 : 1.2 : 1.1 is found between the power coefficients of the three rotors (1), (2) and (3) respectively. Fig. 21 shows the average power coefficient of the three turbines at different tip speed ratios compared to the isolated turbine, the maximum average power coefficient of the cluster is 0.47 at k = 2.6 and this represents about 3.3 times the power coefficient of an isolated turbine at k > 2.6. At k = 2.6 an aligned array of three isolated turbines generates 702 W and occu-
0.6
Single Rotor Average
0.5 0.4
pies an area of 11 square meters, resulting in a power density of 63 W/m2. The developed three turbines cluster (one meter diameter turbines) generates 917 W and occupies an area of 3.7 square meters, resulting in a power density of 248 W/m2. The efficiency of the developed three turbine cluster is 30% higher than the isolated three turbines, also the power density of the developed three turbine cluster is 4 times the isolated three turbines. In order to confirm that the separation distance for turbine (2) is the most efficient, the cluster is tested at different distances. Fig. 22 shows that the power coefficient of the three turbines is enhanced compared to a single isolated turbine at different distances for turbine (2) defined as cluster height. Turbine (2) has the highest power coefficient due to the incoming higher velocity it is subjected to coming out of the stream tube created by the two downstream turbines (1) and (3). Fig. 23 shows the average power coefficient of the three turbines of the cluster compared to isolated turbines, the maximum average power coefficient for the three turbines is 0.47 at 0.6D cluster height, this represents an enhancement of 30% higher than that of the isolated turbines and the enhancement decreases at smaller cluster heights. The ratio between the power coefficients of the three rotors (1), (2) and (3) is approximately 1:1.2:1.1 respectively. This cluster is used as the building unit of an efficient Darrieus VAWT farm.
Patterned nine Darrieus wind turbine farm The triangular three turbine cluster is used as a building unit to construct efficient VAWT farms having the same topology as the cluster, the triangular farm has identical geometrical ratios of the
Avg Power Coeffcient Cp
Power Coefficient Cp
0.6
0.5
Avergae
0.45 0.4 0.35 0.3 0
0.5
1
1.5
2
2.5
3
Cluster Height (Percentage of Rotor Diameter)
0.3 0.2
Fig. 23. Avg. Power Coefficient for Three Turbine Cluster (II) Vs Single Turbine(for k = 2.6).
0.1 0 1
1.5
2
2.5
3
3.5
1
Tips Speed Rao Fig. 21. Avg. Cp of Three Turbines at Different Tip Speed Ratios Vs Single Turbine.
Cluster (A)
2
3
0.54
Power Coefficient Cp
Single Turbine
Cp1
Cp2
Cp3
Single Turbine
4
0.49
Wind Direcon
Cluster (B)
5
0.44 6 7
0.39 0.34 0
0.5
1
1.5
2
2.5
Cluster Height (% of Rotor Diameter)
3
Fig. 22. Avg. Power Coefficient for Three Turbine Cluster at Different Distances (for k = 2.6).
Cluster (C)
8
9
Fig. 24. Velocity Contours around Nine Savonius Wind Turbine Farm.
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Table 5 Results of the Nine Darrieus Wind Turbine Farm. Cluster A
Isolated Rotor
Rotor (1)
Rotor (2)
Rotor (3)
Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (1) Cluster B Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (4) Cluster C Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (7)
0.36
0.39 8% 1.0 Rotor (4) 0.50 38% 1.0 Rotor (7) 0.39 8% 1.0
0.45 25% 1.15 Rotor (5) 0.55 52% 1.1 Rotor (8) 0.46 28% 1.18
0.41 13% 1.05 Rotor (6) 0.51 41% 1.02 Rotor (9) 0.42 17% 1.07
Isolated Rotor 0.36
Isolated Rotor 0.36
Table 6 Total Power Generated by the three clusters in Nine Turbines Wind Farm.
Power Coefficient Total Power (watt) Ratio Compared to Cluster (A)
Cluster (A)
Cluster (B)
Cluster (C)
Average
0.42 858 1.0
0.52 1014 1.18
0.42 839 1
0.453
three turbine cluster. A nine turbine triangular farm is developed using two three turbine clusters in the first row and one similar cluster in the second row as shown in Fig. 24 which is consistent with cluster topology. Rotors (1), (2) and (3) represent cluster (A), rotors (3), (4) and (5) represent cluster (B) and rotors (6), (7) and (8) represent cluster (C). Numerical simulation of a farm is performed to confirm the pattern and the enhancement ratio of the three turbine cluster. The results of the power coefficient of the
Total Power (Wa)
3030
Developed Farm
2530
Isolated Turbine Farm
2030 1530 1030 530 30 0
3
6
9
12
Number of Turbines Fig. 25. Total Power Generated by the developed farms Vs the Power Generated by the Same Number of Isolated Turbines.
nine turbines summarized in Table 5 confirm the pattern, within each three turbine cluster (A, B and C) the ratio between the power coefficients of individual turbines is similar to the ratio achieved for the individual turbines in a single cluster. The numerical results in Table 6 show that the ratio between the total power generated by clusters (A), (B) and (C) is about 1.0:1.2:1.0. The enhancement in the performance of the patterned nine turbine farm is similar to that of a single three turbine cluster. An aligned array consisting of nine isolated Darrieus turbines generates 2017 W/m and occupies an area of 245 square meters, resulting in a power density of 8 W/m2. On the other hand, this developed nine Darrieus wind turbines farm generates 2711 W/m and occupies an area of 25 square meters, resulting in a power density of 108 W/m2. The efficiency of the developed three turbine cluster is 28% higher than the isolated three turbines, also the power density of the developed three turbine cluster is 13 times the isolated three turbines. Fig. 25 shows the total power achieved by the developed cluster and farms compared to the output from the same number of isolated turbines, the total power generated by the nine turbine farm 28% higher than that of the isolated turbines. Fig. 26 shows the power density of the developed farms compared to the power density of the same number of isolated turbines (5 rotor diameters apart cross field and 10 rotor diameters apart downstream), the power density achieved by the developed nine turbine farm is 13 times than the isolated nine turbine farm. Conclusion
Power Density (W/m2)
300
Developed farm
250
Isolated Turbine Farm
200 150 100 50 0 0
3
6
9
Number of Turbines Fig. 26. Power Density Vs the Number of Turbines in a Farm.
12
The performance of a single Darrieus VAWT, clusters of two coand counter- rotating parallel and oblique turbines, triangular three-and nine-turbine clusters are numerically studied. The numerical results for the single turbine are used for the validation of the numerical method, the results compare very well with the experimental data. The results for the two turbine clusters show an enhanced performance compared to isolated turbines. An efficient triangular three turbine cluster is developed based on the numerical results of the two turbine clusters. The developed three turbine cluster has an average power coefficient up to 30% higher than that of an isolated turbine. The cluster generates 4 times the power generated by an isolated turbine with a power ratio 1.0:1.2:1.0 between its individual turbines. Darrieus VAWT farms are developed using the efficient triangular three turbine cluster as the building unit. The farms have the same geometric topology
M. Shaheen, S. Abdallah / Sustainable Energy Technologies and Assessments 19 (2017) 125–135
of the three turbine cluster. The developed farms are patterned and have the same enhancement ratios in the power coefficients as the three turbine cluster. The power scaling factor, the pattern and ratio of the enhanced power coefficient are confirmed by numerical solution of a nine turbine farm. The power density of the nine turbine farm is about 13 times a nine isolated turbine farm. The scaling factor of 3.2 can be used to predict the performance of larger farms with the same topology to save processing time and man power. The enhancement in the power generated by the VAWTs leads to an increase in the centrifugal force affecting the blade and turbine mounting which has to be accounted for when performing a complete design for the turbine clusters.
References [1] World Wind Energy Association. 2014 half year report; August 2014. [2] U.S. Department of Energy. 2013 Market report on wind technologies in distributed applications energy. Effic Renewable Energy; 2014. [3] MacKay DJC. Sustainable energy—without the hot air. Cambridge, UK: UIT Cambridge Ltd.; 2009. [4] Dabiri JO. Potential order-of-magnitude enhancement of wind farm power density via counter-rotating vertical-axis wind turbine arrays. J Renew Sustainable Energy 2011;3:043104. [5] Shyu Lih-Shyng. A pilot study of vertical-axis turbine wind farm layout planning. Adv Mater Res 2014;953–954:395–9. [6] Whittlesey RW, Liska S, Dabiri JO. Fish schooling as a basis for vertical axis wind turbine farm design. Bioinspiration Biomimetics 2010;5:035005. [7] Schatzle PR, Klimas PC, Spahr HR. Aerodynamic interference between two darrieus wind turbines. J Energy 1981;5:84–8. [8] Shaheen Mohammed, Abdallah Shaaban. Development of efficient vertical axis wind turbine clustered farms. Renew Sustain Energy Rev 2016;63:237–44. [9] Abu-El-Yazied Taher G, Doghiem Hossam N, Ali Ahmad M, Hassan Islam M. Investigation of the aerodynamic performance of darrieus vertical axis wind turbine. IOSR J Eng 2014;4:18–29.
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Contents lists available at ScienceDirect
Sustainable Energy Technologies and Assessments journal homepage: www.elsevier.com/locate/seta
Original article
Efficient clusters and patterned farms for Darrieus wind turbines Mohammed Shaheen ⇑, Shaaban Abdallah Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, United States
a r t i c l e
i n f o
Article history: Received 6 March 2016 Revised 8 November 2016 Accepted 13 January 2017
Keywords: Renewable energy Wind power Vertical axis wind turbines (VAWTs) Darrieus wind turbines Wind farm power density
a b s t r a c t Recent studies show that interactions between Darrieus vertical axis wind turbines (VAWTs) arranged in close proximity mutually enhance the output power of individual turbines. In this paper, multi-Darrieus turbine clusters are studied for the development of efficient patterned vertical axis wind turbine farms. Numerical solutions are performed for a single Darrieus turbine, clusters of two Co- and counter-rotating turbines in parallel and in oblique configurations, and triangular three turbine clusters. The commercial CFD software FLUENT 14.5 is used for the numerical simulation. The single Darrieus turbine results are validated against experimental data. The results of the simulation of the multi-turbine clusters are used to develop an efficient triangular shaped three turbine cluster having an average power coefficient up to 30% higher than an isolated turbine. Efficient Darrieus wind turbine farms are developed using the enhanced three turbine cluster. The farms consist of multiple clusters with scaled geometrical ratios of the three turbine cluster keeping the same topology of the cluster. This resulted in patterned farms that have the same power enhancement ratio of the three turbine cluster. Numerical simulation of a farm that consist of nine turbines confirm the enhancement and the pattern progression for larger farms. Ó 2017 Elsevier Ltd. All rights reserved.
Introduction Horizontal axis wind turbines (HAWTs) has the highest efficiency of all types of wind turbines and this made most of the wind farms consist of HAWTs. Horizontal axis wind turbines (HAWTs) dominate the majority of the wind industry for the last 30 years. 74% of the small wind turbines manufacturers invested in the HAWT, however 18% adopted VAWTs [1]. Mostly wind turbine farms consist of HAWTs because of the higher efficiency of an isolated HAWT compared to a VAWT of the same size. Land-based HAWT farms have successfully been implemented, a modern HAWT farm consists of an array of 30 to 150 turbines [2]. A large amount of land is required to separate HAWTs from the adjacent turbines’ wakes. The available power extracted from a HAWT farm is limited by the requirement of separating each turbine from adjacent turbines wakes. To maintain 90% of the performance of isolated HAWTs, the turbines in a HAWT farm must be spaced 3–5 turbine diameters apart in the cross-wind direction and 6–10 diameters apart in the downwind direction [3]. A Typical HAWT farm has a power densities, defined as the power extracted divided by the area of its foot print, between 2 and 3 W/m2 [3]. Researchers at CalTech found that the power density of VAWT farms can be increased up to 30 W/m2 by optimizing the placement of the tur⇑ Corresponding author. E-mail address: [email protected] (M. Shaheen). http://dx.doi.org/10.1016/j.seta.2017.01.007 2213-1388/Ó 2017 Elsevier Ltd. All rights reserved.
bines, this enables them to extract energy from adjacent turbines’ wakes [4]. The efficiency of two counter rotating H-rotor VAWTs set at a distance of 1.5D to 2.0D was tested experimentally, an increase of 11% over the efficiency obtained by their stand-alone counterpart was achieved [5]. A geometric arrangement based on the configuration of shed vortices in the wake of schooling fish showed an increase in the performance of an array of 16 16 wind turbines by one order of magnitude [6]. A Study on of the effect of the relative angular position on aerodynamic interference between two curved blade Darrieus VAWTs showed a power reduction for the downstream turbine at small oblique angular positions, as the tip speed ratio of the upwind turbine increases, the blade circulation increases accompanied by an increase in the strength of the shed vortices, resulting in higher induced velocities, and consequently lower angles of attack at the downstream turbines, and correspondingly lower torque and power output [7]. Numerical analysis on two dimensional clusters of Savonius VAWTs showed that the efficiency of turbines operating completely in the wake of other turbines can be improved by choosing their angular position such that they see a higher flow velocity due to the presence of other turbines and associated stream-tube contraction [8]. This study is interested in development of efficient Darrieus VAWT farms. A single Darrieus turbine is solved numerically to validate the numerical model by comparing the results with experimental data in reference [9]. Performance of clusters of two coand counter-rotating turbines in parallel and in oblique configura-
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Nomenclature turbine swept area As = DH, m2 torque coefficient, T/(0.5 qV21 RAS) static torque coefficient, Ts/(0.5 qV21 RAS) power coefficient, TX/(qV31 RH) rotor diameter, m bucket diameter, m overlap ratio of the rotor buckets, e = S/d rotor height, m rotor radius, m gap distance between turbines, m cluster height, m
AS Cm Cms Cp D d e H R S S1
T Ts V1
q
X k h
a
u
tions is studied to be used in the development of an efficient triangular three turbine cluster. The developed cluster is simulated at different tip speed ratios in order to investigate the enhancement in its turbines’ power coefficients compared to isolated turbines. The cluster is used as a building unit for constructing efficient triangular Darrieus VAWT farms. The developed farms consist of multiple clusters with scaled geometrical ratios of the three turbine cluster keeping the same topology of the cluster. This resulted in patterned farms that have the same power enhancement ratio of the three turbine cluster. Numerical simulation of a farm that consist of nine turbines confirm the enhancement and the pattern progression for larger farms. Numerical simulation model Turbine geometry and aerodynamics The geometry used in this study is a three blade Darrieus VAWT with Straight-blades founded in reference [9]. The geometric features are listed in the Table 1. The spokes that connect the turbine blades with the rotating shaft are connected to each blade at the center of pressure of the airfoil (0.25C). Fig. 1 shows the velocity triangles at different azimuth positions of the turbine rotor blade and the force analysis on the blade. The aerodynamics of VAWTs involve high local angles of attack and the wake from the upwind blades and from the axis [10]. The tangential force FT and the normal force FN are originally the transformation of the components of lift FL and drag forces FD . The expression of FT and FN can be determined as follows:
F T ¼ F L sin£ F D cos£
ð1Þ
FN ¼ FL cos£ FD sin£
ð2Þ
The amount of torque and power torque acting on one blade can be obtained respectively from the following expressions:
T¼
R ð2pÞ
Z 2p
F T ðhÞdh
torque, N/m static torque, N/m free stream velocity, m/s density, kg/m3 rotation speed, rad/s tip speed ratio, XR=V1 azimuth angle of the turbine oblique angle between rotors angle of resultant flow velocity
ð3Þ
0
Table 1 Turbine Geometrical Features. Turbine Category
Vertical Axis Wind Turbine
Turbine Type Blade Airfoil Blade Shape Number of Blades Turbine Rotor Diameter (mm) Turbine Height (mm) Blade Chord Length (mm)
Darrieus Type NACA 0021 Straight Blades 3 1030 1414 85.8
R P ¼ 2p x
Z 2p
F T ðhÞdh
ð4Þ
0
From the power and torque equation, the enhancement of the power of VAWTs is affected mainly by the tangential force which increases when the lift force increases on the blades in addition to the value of the resultant flow angle. Computational domain Different domains are tested by solving the single Darrieus turbine at different speeds in order to obtain the appropriate size of the computational domain. The best compromise is found in Fig. 2, where the upstream, upper and lower boundaries of the domain are at a distance 15D (D is the turbine diameter) from the turbine center, while the downstream boundary is at 30D. The domain is divided into 3 sub-domains: A rotating region (1) with a torus shape includes the turbine blades, a fixed region (2) extends from the outer interface of the rotating region to the boundaries of the domain and a fixed region (3) extends from the inner interface of the rotating region to the shaft of the turbine. The outer diameter of the rotating region is 1.2D and its inner diameter is 0.8D . The upstream boundary of the domain is defined by velocity inlet boundary condition (BC) (x-component = 9 m/s), the upper and lower boundaries are defined by symmetry BC and the downstream boundary is defined by pressure outlet BC [11– 13]. Grid generation The sliding mesh model is used to capture the unsteady flow parameters generated due to the rotation of the turbine blades. A non conformal mesh with unstructured triangular cells is created using ANSYS meshing. An inflation with 10 levels of quadrilateral cells is imposed on the airfoil’s surfaces to account for the boundary layer with a maximum thickness of 1 mm and growth rate of 1.1. Five grid refinements levels ranging from 47,591 up to 450,705 cells are tested at all wind speeds and rotational speeds in order to obtain y+ < 1 as required by the transition SST turbulence model [14,15]. A mesh of 173,768 cells is found to be adequate as the relative variation in the output quantities are below 1 103 as shown in Table 2. The grid structure is shown in Fig. 3(a) and a close view of the grid in the vicinity of the rotor blade leading edge is shown in Fig. 3(b). Solver settings and turbulence model Fluent code is used to solve the unsteady Reynolds averaged Navier-Stokes equations using finite volume method [15–18]. SIM-
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Fig. 1. Darrieus VAWT Rotor Blades, Velocity Triangles and Force Analysis [9].
Fig. 2. Computational Domain.
@ðqcÞ qðU jc Þ @ ¼ Pc1 Ec1 þ P c2 Ec2 þ þ @t @xj @xj
Table 2 Grid Refinement Levels for Single Turbine Simulation. Grid level
Number of cells
Cm
Cp
1 2 3 4 5
47,591 81,034 173,768 235,451 492,605
0.063561 0.0989 0.136 0.137 0.1365
0.166529 0.259118 0.35632 0.35894 0.35763
PLE algorithm (Semi-Implicit Method) with a second order spatial discretization for all pressure, momentum, and turbulence equations and a Least Squares Cell Based algorithm is used for gradients is used [14]. In order to capture the laminar and transitional boundary layers correctly, the transition SST turbulence model introduced by Menter [19] is adopted. This leads to major improvements in the prediction of adverse pressure gradient flows [20]. It is based on SST k-x transport equations coupled with two additional transport equations, one for intermittency (c), and the other for the momentum-thickness Reynolds number Reht as follows [21]:
@ðqReht Þ @ðqU j Reht Þ @ þ ¼ Pht þ @t @xj @xj
rht l þ lt
lþ
lt @ c rc @xj
@Reht @xj
ð5Þ
ð6Þ
The transition model interacts with the SST turbulence model by modification of the K-equation. The definitions for all terms in Eqs. (5)–(7) are discussed in details in the user guide of fluent and Ref. [19].
@ @ @ @k ~ k Y k þ Sk þG ðqkÞ þ ðqkui Þ ¼ Ck @t @xj @xj @xj
ð7Þ
A Steady flow simulations is run for few time steps obtained initially to set up the flow domain to reasonable parametric values, then unsteady flow simulations are obtained using the sliding mesh model (SMM) [21]. Different time step sizes are tested and a time step size corresponding to a turbine rotation of 1.0 degree is used for the time dependent solution [15,22,23]. A drop of at least 3 orders of magnitude in each time step as well as a minimum of 20 iterations per time step is considered as convergence criteria
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Fig. 3. Grid Structure.
for the time dependent solution. The torque (T) is calculated about the turbine center from the resultant pressure and viscous forces on the turbine blades. The torque coefficient (Cm) is calculated by Eq. (8) at each time step and averaged after periodic convergence [10], then the power coefficient (Cp) is calculated from Eq. (9):
C ms ¼
Ts
ð8Þ
0:5qV 21 RAs
Cp ¼ Cm k
ð9Þ
Results and discussion Single Darrieus turbine numerical solution results and validation The average power coefficient of the single Darrieus turbine is calculated at different tip speed ratios (k) using time dependent solution for the rotating turbine. Fig. 4 shows that the fully turbulent k-omega SST model under predicts the values of the power coefficient and that the transition SST turbulence model has the closest results to the experimental data, similar findings are reported in reference [24]. The maximum average power coefficient for the single turbine is found to be 0.345 corresponding to k equal to 2.6. Fig. 5 shows the velocity contours around the Darrieus turbine blades, two regions of high velocity magnitude are induced on both sides downstream the turbine. The kinetic energy
Power Coefficient Cp
0.4
Experimental Data K-w SST Transion SST
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1
1.5
2
2.5
3
3.5
Tips Speed Rao Fig. 4. Cp at Different tip speed Ratios, Numerical Results Vs Experimental data [5].
Fig. 5. Velocity Contours around the Darrieus Turbine Blades.
in these two regions can be used to enhance the efficiency of turbines placed in the near vicinity. Steady and unsteady Loading of the wind turbine Operation of an H-type Darrieus vertical axis wind turbine at low blade speed ratios resulted in complex flow-blade interaction mechanisms. These include dynamic stall resulting in large scale vortex production, vortex impingement on the source blade, and significant flow momentum extraction. The presentation of such analysis of dynamic stall for the flow around the VAWT blade aerofoil is complicated because the relative velocity vector to the blade aerofoil changes in both direction and magnitude around the azimuth as shown in Fig. 6. To study the instantaneous lift coefficient against incidence angle, the instantaneous velocity magnitude should be used in the lift and drag coefficient. Quasi – steady force analysis A quasi-steady two dimensional study is performed for a single blade, the instantaneous relative velocity and angle of attack are calculated graphically at different azimuth angles. The absolute values of the angles of attack are similar at the corresponding posi-
129
Li Coefficient (Cl)
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-30
-20
-10
1 0.8 0.6 0.4 0.2 0 -0.2 0 -0.4 -0.6 -0.8 -1
10
20
30
Cl xfoil Cl Fluent
Angle of Aack ( Fig. 8. Lift Curve, Numerical Results of Fluent Vs Xfoil.
Unsteady force analysis The unsteady flow simulations is obtained using the sliding mesh model. The torque (T) is calculated about the turbine center from the resultant pressure and viscous forces on the turbine blades. The torque of a single blade is plotted at different azimuth angles as shown in Fig. 11, the results shows a great discrepancy between the unsteady and the quasi-unsteady results shown in Fig. 10. The average torque generated by a single blade performed by the unsteady simulation is found to be higher than that of the value calculated by the quasi-steady solution. The discrepancy in the first half of rotation can be explained as the result of the change
0.25 0.2 0.15 0.1
CD Xfoil CD Fluent
0.05 0
-30
-20
-10
0
10
20
30
Angle of Aack ( ) Fig. 9. Drag Curve, Numerical Results of Fluent Vs Xfoil.
Torque (N.m)
tions in the first and second half of the turbine rotation as shown in Fig. 7. The lift and drag coefficients for the blade airfoil are computed at different Reynolds numbers due to the change in the relative velocity magnitude at different azimuth angles. Plots of lift and drag coefficients at different angles of attack are shown in Figs. 8and 9 using two solvers Fluent 14.5 and Xfoil. The lift and drag forces are calculated at different positions and the torque generated from both forces is calculated. The torque of the turbine is a resultant of the positive torque generated by the lift forces and the negative torque generated by the drag force. Fig. 10 shows the quasi-unsteady torque generated by a single turbine blade in one complete cycle, the results are symmetric for the first and second half cycle with two peaks values for the torque along the azimuth, and a positive average generated torque.
6 5 4 3 2 1 0 -1 0 -2
30
60
90
120 150 180 210 240 270 300 330 360
Azimuth Angle ( ) Fig. 10. Torque at Different Azimuth Angles (Quasi-steady Simulation).
40
Torque (N.m)
Fig. 6. Change in the Relative Velocity Vector Due to the Blade Aerofoil Changes Position around the Azimuth.
Drag Coeffciient (Cd)
0.3
20 0 0 -20
30
60
90
120 150 180 210 240 270 300 330 360
Azimuth Angle
Fig. 11. Unsteady Torque Generated by a Single Blade of the Darrieus Turbine.
Angle of Aack ( )
30 20 10 0 -10
0
30
60
90 120 150 180 210 240 270 300 330 360
in the axial velocity due to stream lines expansion and the discrepancy in the second half is due to the same reason in addition to the interference with the vortex shed from the upstream blades. Another reason is that the quasi-steady solution assumes that the flow velocities normal to the free-stream direction are zero which is not true.
-20 -30
Two Darrieus turbine clusters solution results
Azimuth Angle ( )
Fig. 7. Angle of Attack of the Turbine Blade Airfoil at Different Azimuth Angles.
The performance of multi-turbine co-rotating and counterrotating Darrieus turbine clusters is studied. All possible configura-
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tions of parallel and oblique configurations are simulated at different gap distances.
Two parallel turbines Two parallel co- and counter-rotating turbines are simulated at tip speed ratio (k = 2.6) and different gap distances using the same numerical method for the single Darrieus turbine. For the corotating turbines, rotors (1) and (2) are both rotating counter clockwise as shown in Fig. 12(a). For the counter rotating turbines, two cases (A) and (B) are studied. In case (A): Rotor (1) rotates counter clockwise and rotor (2) rotates clockwise where their inward blades move in the same direction of the wind as shown in Fig. 12(b), In case (B): Rotor (1) rotates clockwise and rotor (2) rotates counter clockwise where their inward blades move opposite to the wind direction shows the velocity contours for the two parallel co-rotating turbines as shown in Fig. 12(c). The numerical results in Fig. 13 show an enhancement in the average power coefficient of the two parallel turbine clusters for the three cases compared to isolated turbines (represented by the single turbine). The comparison of the results for the three cases shows that the most efficient configuration for two parallel Darrieus turbines is the counter-rotating configuration case (A). The maximum average power coefficient for case (A) is found to be 0.42 achieved at a gap distance equal to 0.5D as shown in Fig. 13. The enhancement in
Fig. 13. Avg. Power Coefficient for Three Cases of Two Parallel Darrieus Turbines.
the average power coefficient decreases as the gap distance increases. The results show that the counter-rotating turbines enhance the performance of each other, this enhancement vanishes as the gap distances increases between the turbines. The comparison shows that case (A) where the inward blades moves with the wind direction is more efficient than case (B), an average power coefficient of 0.42 is achieved at a gap distance 0.5D in case (CA), this represents 16% higher than the isolated turbines.
1
(a) Velocity Contours around Two Parallel Co-Rotating Turbines
Wind Direcon
2
1
1
Wind Direcon
Wind Direcon
2
2
(b) Case (A)
(c) Case (B)
Fig. 12. Velocity Contours around Two Parallel Darrieus Turbines.
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2
1
Wind Direcon
1
2
(a) Case (C)
(b) Case (D)
Fig. 14. Velocity Contours around Two Parallel Counter-Rotating Darrieus Turbines.
Two oblique turbines Two oblique co- and counter-rotating Darrieus turbines are simulated. The performance is studied at a tip speed ratio (k = 2.6) and different gap distances using the same numerical method for the single Darrieus turbine.
Fig. 15. Avg. Power Coefficient for Two Cases of Two Co-Rotating Oblique Darrieus Turbines at different Gap Distances
Two Co-Rotating oblique turbines. For the co-rotating configuration, two cases (C) and (D) are studied. Rotor (1) is upstream and rotor (2) is downstream. In case (C): rotor (2) is set at an oblique angle +ve 60o and in case (D): rotor (2) is set at an oblique angle ve 60° as shown in Fig. 14(a) and (b). In case (C) turbine (1) has an isolated turbine performance, an enhancement occurs in the power coefficients of turbine (2) due to the Magnus effect of turbine (1). In case (D) both turbines are enhanced in performance due to the Magnus effect of the other turbine (2). The inward blade of turbine (2) faces a high velocity region downstream turbine (1) as shown in Fig. 14(b). The comparison of the average power coefficients for both cases with the isolated turbine performance in Fig. 14 show that case (D) has higher efficiency than case (C). The maximum average power coefficient for case (D) is found to be 0.37 achieved at a gap distance equal to 0.5D as shown in Fig. 15. The enhancement in the average power coefficient in the two cases decreases as the gap distance increases. Two counter-rotating oblique turbines. Two cases (E) and (F) of two oblique counter-rotating Darrieus turbine clusters are numerically studied. Rotor (1) is upstream and rotor (2) is downstream. In case (E): rotor (2) is set at an oblique angle +ve 60o to rotor (1) and their inward blades move in the same direction of the wind as shown in
2 1
Wind Direcon
Wind Direcon
1
2
(a) Case (E)
(b) Case (F)
Fig. 16. Velocity Contours around Two Oblique Counter-Rotating Darrieus Turbines.
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ficient. The results show that two counter-rotating turbines in close proximity enhance the performance of each other, this enhancement decreases as the gap distances increases between the turbines. The comparison shows that case (E) where the inward blades moves with the wind direction is more efficient than case (F) specially at small gap distances. The maximum average power coefficient for case (E) is found to be 0.41 achieved at a gap distance equal to 0.5D. The enhancement in the average power coefficient in the two cases decreases as the gap distance increases. Three turbine clusters simulation Fig. 17. Avg. Power Coefficient for Two Cases of Two Counter-Rotating Oblique Darrieus Turbines.
1
2 Wind Direcon
3
Fig. 18. Velocity Contours around Triangular Three Turbine Darrieus Cluster at k = 2.6.
Fig. 16(a). In case (F): rotor (2) is set at an oblique angle -ve 60o to rotor (1) and their inward blades move opposite to the wind direction Fig. 16(b). The performance of the two oblique turbines in both cases are studied at different gap distances (S). Fig. 17 shows a comparison between the average power coefficients for both cases (E) and (F) and the isolated rotor power coef-
The numerical results for the performance of two co- and counter-rotating parallel and oblique turbines show that case (A) is the most efficient configuration for two parallel turbines, and that case (E) is the most efficient configuration for two oblique turbines. This finding is used to develop an efficient triangular three turbine cluster. The base of the triangle is perpendicular to the wind direction and consists of two counter rotating turbines (1) and (3) as in case (A). The center of the two turbines (1) and (2) are set at a distance of two turbine diameters from each other. Turbine (3) is set downstream in an oblique position and counterrotating to turbine (1) as in case (E) as shown in Fig. 18. The simulation of the triangular three turbine cluster is performed for different triangle heights, where turbine (2) is set at different distances from the line joining the centers of the two upstream turbines (1) and (3). The computational domain for the three turbine cluster simulation shown in Fig. 19 is the similar to the domain used in the single turbine simulation. A grid sensitivity study is performed, the mesh is refined five times from 133,175 to 2,257,340 cells until there is no significant change observed in the torque coefficient of the three turbines as shown in Table 3. A grid with 400,487elements is found to be adequate.
Table 3 Grid levels for Three Turbine Cluster Simulation. Grid Level
No of Cells
Cm1
Cm2
Cm3
1 2 3 4 5
133175 219245 400487 993007 2257340
0.0013 0.0002 0.0004 0.0004 0.0004
0.1409 0.1394 0.1374 0.1374 0.1372
0.0373 0.0380 0.0378 0.0378 0.0378
Fig. 19. Computational Domain for the Three Turbine Cluster Simulation.
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0.5 0.4 0.3 0.2 0.1
Cp1
Cp2
Cp3
0 1
1.5
2
2.5
3
3.5
Tips Speed Rao Fig. 20. Cp of the Three Turbines of the Cluster at Different Tip Speed Ratios.
Table 4 Power Coefficients for the Three Turbine Cluster at Different Tip Speed Ratios. k
Cp1
Cp2
Cp3
Average
1.8 2.2 2.6 3.0
0.11 0.26 0.40 0.42
0.11 0.23 0.43 0.48
0.12 0.25 0.41 0.45
0.11 0.25 0.42 0.45
Power Coefficient Cp
The performance of the three turbine cluster is studied at different tip speed ratios (k) as shown in Fig. 20. The computed power coefficients of the three turbines are shown in Table 4. The results show enhancement in the power coefficients of the three turbines compared to their isolated counter-parts. At k > 2.6 a ratio of 1 : 1.2 : 1.1 is found between the power coefficients of the three rotors (1), (2) and (3) respectively. Fig. 21 shows the average power coefficient of the three turbines at different tip speed ratios compared to the isolated turbine, the maximum average power coefficient of the cluster is 0.47 at k = 2.6 and this represents about 3.3 times the power coefficient of an isolated turbine at k > 2.6. At k = 2.6 an aligned array of three isolated turbines generates 702 W and occu-
0.6
Single Rotor Average
0.5 0.4
pies an area of 11 square meters, resulting in a power density of 63 W/m2. The developed three turbines cluster (one meter diameter turbines) generates 917 W and occupies an area of 3.7 square meters, resulting in a power density of 248 W/m2. The efficiency of the developed three turbine cluster is 30% higher than the isolated three turbines, also the power density of the developed three turbine cluster is 4 times the isolated three turbines. In order to confirm that the separation distance for turbine (2) is the most efficient, the cluster is tested at different distances. Fig. 22 shows that the power coefficient of the three turbines is enhanced compared to a single isolated turbine at different distances for turbine (2) defined as cluster height. Turbine (2) has the highest power coefficient due to the incoming higher velocity it is subjected to coming out of the stream tube created by the two downstream turbines (1) and (3). Fig. 23 shows the average power coefficient of the three turbines of the cluster compared to isolated turbines, the maximum average power coefficient for the three turbines is 0.47 at 0.6D cluster height, this represents an enhancement of 30% higher than that of the isolated turbines and the enhancement decreases at smaller cluster heights. The ratio between the power coefficients of the three rotors (1), (2) and (3) is approximately 1:1.2:1.1 respectively. This cluster is used as the building unit of an efficient Darrieus VAWT farm.
Patterned nine Darrieus wind turbine farm The triangular three turbine cluster is used as a building unit to construct efficient VAWT farms having the same topology as the cluster, the triangular farm has identical geometrical ratios of the
Avg Power Coeffcient Cp
Power Coefficient Cp
0.6
0.5
Avergae
0.45 0.4 0.35 0.3 0
0.5
1
1.5
2
2.5
3
Cluster Height (Percentage of Rotor Diameter)
0.3 0.2
Fig. 23. Avg. Power Coefficient for Three Turbine Cluster (II) Vs Single Turbine(for k = 2.6).
0.1 0 1
1.5
2
2.5
3
3.5
1
Tips Speed Rao Fig. 21. Avg. Cp of Three Turbines at Different Tip Speed Ratios Vs Single Turbine.
Cluster (A)
2
3
0.54
Power Coefficient Cp
Single Turbine
Cp1
Cp2
Cp3
Single Turbine
4
0.49
Wind Direcon
Cluster (B)
5
0.44 6 7
0.39 0.34 0
0.5
1
1.5
2
2.5
Cluster Height (% of Rotor Diameter)
3
Fig. 22. Avg. Power Coefficient for Three Turbine Cluster at Different Distances (for k = 2.6).
Cluster (C)
8
9
Fig. 24. Velocity Contours around Nine Savonius Wind Turbine Farm.
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Table 5 Results of the Nine Darrieus Wind Turbine Farm. Cluster A
Isolated Rotor
Rotor (1)
Rotor (2)
Rotor (3)
Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (1) Cluster B Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (4) Cluster C Power Coefficient (Cp) Enhancement % Compared to Isolated Rotor Ratio Compared to Rotor (7)
0.36
0.39 8% 1.0 Rotor (4) 0.50 38% 1.0 Rotor (7) 0.39 8% 1.0
0.45 25% 1.15 Rotor (5) 0.55 52% 1.1 Rotor (8) 0.46 28% 1.18
0.41 13% 1.05 Rotor (6) 0.51 41% 1.02 Rotor (9) 0.42 17% 1.07
Isolated Rotor 0.36
Isolated Rotor 0.36
Table 6 Total Power Generated by the three clusters in Nine Turbines Wind Farm.
Power Coefficient Total Power (watt) Ratio Compared to Cluster (A)
Cluster (A)
Cluster (B)
Cluster (C)
Average
0.42 858 1.0
0.52 1014 1.18
0.42 839 1
0.453
three turbine cluster. A nine turbine triangular farm is developed using two three turbine clusters in the first row and one similar cluster in the second row as shown in Fig. 24 which is consistent with cluster topology. Rotors (1), (2) and (3) represent cluster (A), rotors (3), (4) and (5) represent cluster (B) and rotors (6), (7) and (8) represent cluster (C). Numerical simulation of a farm is performed to confirm the pattern and the enhancement ratio of the three turbine cluster. The results of the power coefficient of the
Total Power (Wa)
3030
Developed Farm
2530
Isolated Turbine Farm
2030 1530 1030 530 30 0
3
6
9
12
Number of Turbines Fig. 25. Total Power Generated by the developed farms Vs the Power Generated by the Same Number of Isolated Turbines.
nine turbines summarized in Table 5 confirm the pattern, within each three turbine cluster (A, B and C) the ratio between the power coefficients of individual turbines is similar to the ratio achieved for the individual turbines in a single cluster. The numerical results in Table 6 show that the ratio between the total power generated by clusters (A), (B) and (C) is about 1.0:1.2:1.0. The enhancement in the performance of the patterned nine turbine farm is similar to that of a single three turbine cluster. An aligned array consisting of nine isolated Darrieus turbines generates 2017 W/m and occupies an area of 245 square meters, resulting in a power density of 8 W/m2. On the other hand, this developed nine Darrieus wind turbines farm generates 2711 W/m and occupies an area of 25 square meters, resulting in a power density of 108 W/m2. The efficiency of the developed three turbine cluster is 28% higher than the isolated three turbines, also the power density of the developed three turbine cluster is 13 times the isolated three turbines. Fig. 25 shows the total power achieved by the developed cluster and farms compared to the output from the same number of isolated turbines, the total power generated by the nine turbine farm 28% higher than that of the isolated turbines. Fig. 26 shows the power density of the developed farms compared to the power density of the same number of isolated turbines (5 rotor diameters apart cross field and 10 rotor diameters apart downstream), the power density achieved by the developed nine turbine farm is 13 times than the isolated nine turbine farm. Conclusion
Power Density (W/m2)
300
Developed farm
250
Isolated Turbine Farm
200 150 100 50 0 0
3
6
9
Number of Turbines Fig. 26. Power Density Vs the Number of Turbines in a Farm.
12
The performance of a single Darrieus VAWT, clusters of two coand counter- rotating parallel and oblique turbines, triangular three-and nine-turbine clusters are numerically studied. The numerical results for the single turbine are used for the validation of the numerical method, the results compare very well with the experimental data. The results for the two turbine clusters show an enhanced performance compared to isolated turbines. An efficient triangular three turbine cluster is developed based on the numerical results of the two turbine clusters. The developed three turbine cluster has an average power coefficient up to 30% higher than that of an isolated turbine. The cluster generates 4 times the power generated by an isolated turbine with a power ratio 1.0:1.2:1.0 between its individual turbines. Darrieus VAWT farms are developed using the efficient triangular three turbine cluster as the building unit. The farms have the same geometric topology
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of the three turbine cluster. The developed farms are patterned and have the same enhancement ratios in the power coefficients as the three turbine cluster. The power scaling factor, the pattern and ratio of the enhanced power coefficient are confirmed by numerical solution of a nine turbine farm. The power density of the nine turbine farm is about 13 times a nine isolated turbine farm. The scaling factor of 3.2 can be used to predict the performance of larger farms with the same topology to save processing time and man power. The enhancement in the power generated by the VAWTs leads to an increase in the centrifugal force affecting the blade and turbine mounting which has to be accounted for when performing a complete design for the turbine clusters.
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