Darrieus wind turbines are experiencing a renewed interest for their application in decentralized power generation and urban installation. Much attention and research efforts have been dedicated in the past to develop as an efficient standalone Darrieus turbine. Despite these efforts, these vertical axis turbines are still low in efficiency compared to the horizontal axis counterparts. The current architecture of the turbine and their inherent characteristics limit their application in low wind speed areas as confirmed experimentally and computationally by past research. To enable and extend their operation for weak wind flows, a novel design of Adaptive Darrieus Wind Turbine (ADWT) is proposed. The hybrid Darrieus Savonius rotor with dynamically varying Savonius rotor diameter based on the wind speed enables the turbine to start, efficiently operate and stop the turbine at high winds. As the wake of Savonius rotor has a profound impact on the power performance of the combined rotor, the wake of two buckets Savonius rotor in open and closed configuration is reviewed. The current study aims to develop an analytical model to predict the power coefficient and the influence of other design parameters on the proposed design. The formulated analytical model is coded in python, and the results are obtained for the 10 kW rotor. Parametric analysis on the chord length and the diameter of the closed Savonius rotor is performed in search of an optimized diameter to maximize the annual energy output. Blade torque and the rotor torque are evaluated with respect to azimuthal angle and compared with conventional Darrieus rotor. The computed results show that peak power coefficient of ADWT is 13% lower than the conventional Darrieus rotor at the rated wind speed of 10 m/s.
Renewable energy sources are increasingly popular for their emission-free power generation. The tremendous increase in the wind turbine installation is expected to continue in the future with the current worldwide installation of 468 GW [
The past attempts contribute significantly for the enhancement of startup characteristics, yet one design has not been singled out as an implementable solution for both startup and over speed regulation without affecting the performance of the Darrieus turbine at higher Tip Speed Ratio (TSR). Rotors with cambered airfoils are found to generate higher starting torque at low wind speed compared to symmetric airfoils [
Of several solutions that are discussed above, hybrid Darrieus and Savonius turbine is a potential candidate that be redesigned to improve the low wind speed performance and over speed regulation. The conventional hybrid Darrieus turbine integrates a Savonius rotor to a common shaft with the Darrieus rotor. The strategy is that the high torque generated by the Savonius rotor accelerates the rotor to higher TSR. Similar to other concepts, the hybrid Darrieus-Savonius also suffers from poor performance when the Darrieus rotor accelerates beyond 1. The optimum TSR for a two-bladed Darrieus rotor lies between 3 to 5 [
rotor and decelerate the rotor when it rotates beyond the rated rpm. The construction and the mechanical arrangement are less complex making this concept commercially implementable.
The analytical model of the ADWT in open configuration is similar to the conventional hybrid Darrieus-Savonius rotor. In order to simplify the development of analytical model, the Savonius buckets are arranged in line with the blades of Darrieus rotor. The simplified configuration for ADWT is with two Savonius buckets without an overlap mounted on the same axis with the Darrieus rotor with two blades. The velocity diagram of the ADWT is shown in the
Before proceeding with the analytical solution it is indispensable to predict the wake flow pattern of a two-bladed Savonius rotor. Countless studies focused on the flow over the Savonius rotor rather than on the downstream wake. A detailed flow investigation and adequate knowledge is a must to generalize the wake structure on the downstream. It is possible to deduce the wake flow pattern on the downstream within the flight path of the Darrieus blades by examining the flow leaving the Savonius buckets. Typical flow of a two bucket Savonius rotor is presented in the Figures 1(d)-(f). The flow is characterized by a sequence in which flow is first attached to the convex side of advancing blade. As the rotation advances, the flow transfers from the advancing blade’s convex surface to returning blade’s concave side. The flow simultaneously enters in-between center shaft space, followed on the upstream flow on the convex side of the returning blade. The vortex shedding from the returning and advancing blade tips contributes to increased wake width. The wake pattern is highly turbulent due to alternating suction and pressure zone occurrence on the concave and convex side of the blade. The flow on a single stage Savonius rotor is different from the double stage rotor [
V ∞ = u V ∞ i (1)
And the equilibrium induced velocity is
V e = ( 2 u − 1 ) V ∞ i (2)
With V e as the input velocity for the downstream half-cycle of the rotor the induced velocity at the end of the streamtube is
V ′ = u ′ ( 2 u − 1 ) V ∞ i (3)
The relative velocity for the for the upstream half-cycle of the rotor, − π / 2 < θ < π / 2 , is given by the expression
W 2 = V 2 [ ( X − sin θ ) 2 + cos 2 θ cos 2 δ ] (4)
where X = ω r / V represents the local tip speed ratio. The general expression for the angle of attack is
α = sin − 1 [ cos θ cos δ ( X − sin θ ) 2 + cos 2 θ cos 2 δ ] (5)
By equating the blade element theory and the momentum equation for each stream-tube
f u p ( V V ∞ ) 2 = π η ( V V ∞ ) [ ( V ∞ i V ∞ ) − ( V V ∞ ) ] (6)
f u p u = π η ( 1 − u ) (7)
η = r R D (8)
where f u p is the function that characterizes the upwind conditions
f u p = N c 8 π R D ∫ − π / 2 π / 2 ( C N cos θ | cos θ | − C T sin θ | cos θ | cos δ ) ( W V ) 2 d θ (9)
C N = C L cos α + C D sin α (10)
C T = C L sin α − C D cos α (11)
Airfoil coefficients C L and C are obtained from the wind tunnel test or from literature and interpolating for local Reynolds number and the local angle of attack.
Defining the blades local Reynolds number as R e b for local conditions given by
R e b = W c / V ∞ (12)
The turbine Reynolds number will be
R e b = ( R e t n / X ) ( X − sin θ ) 2 + cos 2 θ cos 2 δ (13)
For each blade in the upstream position, the non-dimensional force coefficients as functions of the azimuthal angle θ are given by
F N ( θ ) = c H S ∫ − 1 1 C N ( W V ∞ ) 2 ( η cos δ ) d ζ (14)
F T ( θ ) = c H S ∫ − 1 1 C T ( W V ∞ ) 2 ( η cos δ ) d ζ (15)
By integrating for the entire blade
T u p ( θ ) = 1 2 ρ ∞ c R D H ∫ − 1 1 C T W 2 ( η cos δ ) d ζ (16)
The average half cycle of the rotor torque produced by N/2 of the N blades is given by:
T u p ¯ = N / 2 π ∫ − π / 2 π / 2 T u p ( θ ) d θ (17)
The average torque coefficient will be:
C Q 1 ¯ = N c H 2 π S ∫ − π / 2 π / 2 ∫ − 1 + 1 C T ( η c o s δ ) ( W W ∞ ) 2 d θ d ζ (18)
Thus, the power coefficient for the upstream half can be written as
C P 1 = ω R D V ∞ C Q 1 ¯ (19)
Similarly for the downstream half cycle. The relative velocity for the for the downstream half-cycle of the rotor, π / 2 < θ < 3π / 2 , is given by the expression
W ′ 2 = V ′ 2 ( X ′ − sin θ ) 2 + cos 2 θ cos 2 δ (20)
where X ′ = ω r / V ′ represents the local tip speed ratio. The general expression for the angle of attack is
α ′ = sin − 1 [ cos θ cos δ ( X ′ − sin θ ) 2 + cos 2 θ cos 2 δ ] (21)
By equating the blade element theory and the momentum equation for each stream-tube
f d w ( V ′ V ∞ ) 2 = π η ( V ′ V ∞ ) [ ( V ∞ i V ∞ ) − ( V ′ V ∞ ) ] (22)
f d w u ′ = π η ( 1 − u ′ ) (23)
f d w = N c 8 π R ∫ π / 2 3π / 2 ( C ′ N cos θ | cos θ | − C ′ T sin θ | cos θ | cos δ ) ( W ′ V ′ ) 2 d θ (24)
F ′ N ( θ ) = c H S ∫ − 1 1 C ′ N ( W ′ V ∞ ) 2 ( η cos δ ) d ζ (25)
F ′ T ( θ ) = c H S ∫ − 1 1 C ′ T ( W ′ V ∞ ) 2 ( η cos δ ) d ζ (26)
T d w ( θ ) = 1 2 ρ ∞ c R H ∫ − 1 1 C ′ T W ′ 2 ( η cos δ ) d ζ (27)
T d w ¯ = N 2 π ∫ π / 2 3π / 2 T d w ( θ ) d θ (28)
The downstream torque coefficient is given by
C Q 2 = T d w ¯ 1 / 2 ρ ∞ V ∞ 2 S R D (29)
C Q 2 = N c H 2 π S ∫ π / 2 3π / 2 ∫ − 1 + 1 C ′ T ( η c o s δ ) ( W W ∞ ) 2 d θ d ζ (30)
C P 2 = ( ω R V ∞ ) C Q 2 ¯ (31)
C P − D = C P 1 + C P 2 (32)
For the Savonius rotor as shown in the
Suppose pressure difference on retreating side is Δ P r
δ Q p = Δ P r sin θ ⋅ 2 r cos θ ∗ Area (33)
Assuming advancing side contributes to negative torque through drag
δ Q = ( Δ P r sin θ ⋅ 2 r cos θ − δ Q p − Δ P a sin θ ⋅ 2 r cos θ ) 4 r h (34)
Q = ∫ 0 π / 2 Δ ( P r − Δ P a ) sin θ cos θ ∗ 4 r 2 h d θ (35)
Q = 2 r s 2 H s ∫ 0 π / 2 Δ ( P r − Δ P a ) sin 2 θ (36)
Average power p is obtained by integrating torque from 0 to π
P s = ω ⋅ Q = ω π ∫ 0 π Q d α (37)
Normalized power coefficient can be given as
C p − s = P 0.5 ρ V e 3 ( 4 r s 2 H s ) (38)
Let’s assume
Δ P r 0.5 ρ v e − s 2 = S 1 (39)
V e − s is equivalent velocity or relative velocity, v i s is absolute velocity.
V e − s = v e s − v i s (40)
Consider retreating side: Δ P p value is known. Solving the integral,
Q p = 2 r 2 H s 1 2 ρ s 1 ( v e s 2 ) (41)
Q p = r 2 H s ρ s 1 ( v ∞ − v i s ) 2 (42)
Q M = r 2 H s ρ s 1 ( v ∞ 2 + 2 r 2 w 2 − π 2 v ∞ r ω cos α − 2 v ∞ r ω cos α − 2 v ∞ r ω sin α ) (43))
Similarly, for the advancing side:
Δ P a / 0.5 ρ v ∞ 2 = S 2 (44)
Again, resolving the component will yield the equation of the torque for advancing side.
Q D = 2 ρ r 2 α s 2 H s S 2 (45)
Combining all the parts together the final equation for the C p is given as
C P − S = [ S 1 φ 4 − S 1 φ 2 2 π + S 1 − S 2 S 1 ( φ 3 4 ) ] (46)
The power coefficient of ADWT rotor in open configuration
C P = C P − D + C P − S (47)
The power coefficient under the influence of Savonius rotor in closed condition can be derived by treating it as a nominal cylinder placed in a steady and homogenous wind flow. Even this simplified assumption gives rise to complex flow wake structures. The objective of the model is to predict the wake width and the velocity deficit due to the presence of cylinder. The stream tubes that are influenced by the wake width are identified and the input velocity is modified with the velocity deficit calculated during the iteration. However there will be axisymmetric flow acceleration on either side of the wake which is not accounted. The cylinder can be considered as non-rotating as the cylinder TSR is low, and the flow field displayed by both rotating and non-rotating flow for low rpm of 80 ~ 100 is similar. The current approaches and the assumptions will lead to the development of an analytical model that can be well integrated into the existing subroutine coded in python.
A wake boundary occurs between two fluids carrying different momentum along with their flow. The momentum deficit in the particular region of unidirectional flow is highly unstable and gives rise to the zone of turbulence mixing layer downstream at a point where the two streams meet for the first time. Though at far downstream the static pressure tends to equalize within the flow and wake, the velocity or the momentum deficit continues to travel along with the flow.
Detailed studies by the previous researchers have emitted a very important conclusion which relates the cylinder wake, its growth to the ratio of rotational to rectilinear speed ratio. The experimental studies by Prandtl [
∂ U ∂ X + V ∂ U ∂ Y = − 1 ρ ∂ P ∂ X + γ ∂ 2 U ∂ Y 2 (48)
The above equation is the governing partial differential equation
− ∂ P ∂ y = 0 (49)
∂ U ∂ X + ∂ V ∂ Y = 0 (50)
For cylinder wake which is relatively thick we cannot consider
∂ P ∂ X = 0 (51)
Assuming, X is considerably large
U V α = 1 (52)
V V α = 0 (53)
U d the difference between the velocities ( V α − U ) should be an even function (symmetric wake)
V must be an odd function (asymmetric) in order to get the solution for governing differential equation.
Drag prediction from Boundary layer assumptions
D = ρ ∝ U ∫ − ∝ + ∝ ( V ∝ − U ) d y (54)
Ud2 can be neglected from
D = ρ ∞ V ∫ − ∞ ∞ ( V ∞ − U ) d y (55)
D = ρ ∞ V ∫ − ∞ ∞ U d d y (56)
Comparing it with conventional drag equation
1 2 ρ ∞ V ∞ ⋅ d ⋅ C d = ρ ∞ V ∞ ∫ − ∞ ∞ U d d y (57)
∫ − ∞ ∞ U d d y = 1 2 V ∞ ⋅ d ⋅ C d (58)
U d V ∞ ∝ C D d 2 b (59)
From Prandtl’s mixing length theory we know that
v ′ = u ′ ⋅ const = const ⋅ l ⋅ d u / d y (60)
v ′ + u ′ are turbulent velocity components. Also rate of increase of width ‘b’ of mixing zone is proportional to transverse velocity v ′
D b D t = const ⋅ l b u d = const ⋅ U max (61)
D b D t ∝ v ′ = D b D t ∝ l ∂ u ∂ Y (62)
Average value of ∂ u ∂ Y i.e.; velocity deficit along Y direction is considered to be proportional to transverse velocity U max b
D b D t = const ⋅ l b u max (63)
Let
l b = Δ = D b D t = const ⋅ Δ ⋅ U max (64)
For two dimensional
D b D t = const ⋅ l b u d = const ⋅ Δ ⋅ U d (65)
Equating above expressions
D b D t = const d b d x (66)
d b d x ∝ Δ u V ∞ (67)
Introducing equation of wake width variation into drag equation
We derived previously
2 b d b d x ∝ Δ C D d (68)
Using variable differential form
b ∝ ( Δ x C D d ) 1 / 2 (69)
We know from 2-D incompressible flow equations
∂ U ∂ t + U ∂ U ∂ x + V ∂ U ∂ Y = 1 ρ ∂ C ∂ Y (70)
From Prandtl mixing length theory
− V ∞ ∂ V d ∂ x = 2 l 2 ∂ U d ∂ Y ∂ 2 U d ∂ Y 2 (71)
We introduce K = Y b as independent variable. (m=constant)
b = m ⋅ Δ ( C D d x ) 1 / 2 (72)
U d = V ∞ ( x C D d ) 1 2 f ( k ) (73)
Put this function again in differential equation to get value of f ( x )
f ( k ) = m Δ 18 Δ 2 ( 1 − k 3 / 2 ) 2 (74)
And from momentum equation of experimental data the constant m Δ yields the value equivalent to 10 Δ
b = 10 Δ ( x C D D ) 1 / 2 (75)
U d V ∞ = 10 18 Δ ( x C D d ) 1 / 2 ( 1 − ( y b ) 3 / 2 ) 2 (76)
According to experiments by H. Reichardt
b ′ 2 = 1 4 ( x C D d ) 1 / 2 (77)
Substituting y = 0 in the U d equation then the velocity deficit is at central region which practically explains the maximum deficit velocity since the distribution is close to Gaussian distribution.
From Boundary layer theory and momentum equation after neglecting small terms we get
C D T 4 = Δ V m V e ⋅ L D T (78)
( V e / Δ V m ) 2 = ( x + x 0 ) / ( b D T ) (79)
For small Reynolds number of 104, the following result holds true
( L / D T ) 2 = c ( x + x 1 ) / D T (80)
Δ V e V e = Δ V m ⋅ 2 L V e N T R T f θ 0 (81)
The developed mathematical model is applied to 10 kW ADWT and conventional straight bladed Darrieus turbine. The dimensional details of the configurations are listed in
Description | Value | Unit |
---|---|---|
Rated power | 10 | kW |
Rated wind speed | 10 | m/s |
Starting wind speed | 3 | m/s |
Rotor height | 7 | m |
Rotor diameter | 6 | m |
Number of blades | 2 | - |
Blade airfoil | NACA 0018 | - |
Blade chord length | 70 | mm |
Number of struts | 6 | - |
Strut airfoil | NACA 0018 | - |
Strut chord length | 0.1 | m |
Parametric chord length | 20, 50, 70 | mm |
Parametric closed Savonius rotor diamter | 600, 750, 1200, 2000 | mm |
Parametric height of Savonius rotor | 3.5, 7 | m |
Darrieus blade on the downstream half when it enters the wake zone of the cylinder, yet the advantage that can be reaped in the high TSR is attractive to incite an investigation. Blade torque and the rotor torque values are evaluated for the above-said architectures and compared with the conventional Darrieus turbine.
A parametric analysis is conducted to evaluate the effect of solidity on the power and torque coefficient on a conventional Darrieus rotor. The solidity is varied by changing the chord length of the blades. The chosen chord for the study is 20 mm, 40 mm and 70 mm. The results are plotted as a function of TSR as shown in
the starting torque will be generated by Savonius rotor if the ADWT is in open condition.
The normal and the tangential forces are plotted as a function of azimuthal angles shown in the
The power coefficients are evaluated for various diameters and the conventional Darrieus rotor. The diameters are investigated for the full length and half-length Savonius rotor including the blade tip loss as shown in
To validate the derived analytical model, the computed results are compared against the experimental results. An analytical model of the closed condition can be predicted close to the reality, experimentation has been carried out with dif-
ferent diameter cylinders that will represent the closed Savonius rotor integrated with Darrieus rotor. The performance of the Darrieus rotor with 80 mm, 115 mm, 130 mm, 150 mm is compared with the conventional Darrieus rotor. The rotors are investigated for the Reynolds number of 268,737 corresponding to the wind speed of 9 m/s, as unsteady and flow separation behavior at low Re is more pronounced at low speeds. The Darrieus rotor consists of two blades of NACA 0018. The cylinders under investigation are made in two halves, so that the cylinders can be interchanged without disturbing the Darrieus rotor arrangement. The Darrieus rotor diameter is 400 mm and height is 300 mm. The various diameters employed in the wind tunnel test along with the Darrieus rotor are shown in the
Subsonic open circuit wind tunnel used for this study has a square cross-section of 700 mm × 700 mm. The wind tunnel is powered by 900 mm diameter axial flow fan (Multi-Wing) with the power capacity of 6 kW. A variable frequency drive regulates the wind speed from 0 - 15 m/s. The settling chamber with flow straightener streamlines the air flow. The flow velocity in the test section is uniform within 0.1%. The airfoils can be interchanged in the test setup without disturbing the center shaft or end plates to minimize the errors induced by shaft misalignment. The test turbine is mounted on the aluminum shaft with deep groove ball bearings on bottom end and spherical bearings on top end as shown in
amperage to the magnetic particle brake. The difference between the rotor torque and the braking torque was measured by the torque sensor (LORENZ Transducers). The output data from the above sensors are logged by DEWE 43V data acquisition system. The test rotor has a frontal swept area of 400 mm × 300 mm and occupies nearly 39% of the test section area, which is more than the allowable blockage. The total blockage including solid and wake blockage for the non -standard shapes are accounted. The experimental values are blockage corrected before comparison with the analytical outcome.
The analytical computation and the experimental results are compared as shown in
An analytical model has been developed for the proposed novel Adaptive Darrieus Wind Turbine and the results are obtained for 10 kW rotor. The mathematical model has been developed for ADWT in open configuration and closed configuration. For simplification, the Savonius rotor in the open configuration is assumed to be a bluff body with the wake width equal to the diameter of the
bluff body, whereas in the closed configuration, it is treated as a nominal cylinder. The open configuration of ADWT is highly complex to model unless both the rotors are assumed to be aerodynamically independent which is not in reality. In the open configuration, the Savonius rotor dominates during the starting and low TSR with negligible torque from Darrieus rotor. Hence the current study emphasizes on the performance of the ADWT in closed configuration. Since the rpm of the Darrieus rotor is low in closed configuration, the cylinder is treated as non-rotating. The velocity deficit has been calculated for various diameter cylinder and the stream tubes that is encompassed by the wake width are modified with deficit velocity. The power coefficient is predicted against various TSR and the blades, rotor torque are calculated as a function of azimuthal angle. All the parameters that are investigated are compared for full and half-length Savonius rotor. The results indicate that the power loss between full and half-length Savonius is negligible. On various rotor diameter studies, 12.8% loss is reported for 2000 mm Savonius rotor. Hence an optimum configuration can be with 70 mm chord and full-length Savonius rotor of diameter 1200 mm from aerodynamic and structural perspective. The future work continues with scaling the proposed ADWT to 10 kW to monitor the performance in the field conditions. The dynamic variation of Savonius rotor will highlight the structural issues arising out, which has not been considered so far in the study. The 10 kW system is expected to shed light on the required actuation systems and the cost.
This research was supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Energy Innovation Research Programme (EIRP Award No. NRF2013EWT-EIRP003-032: Efficient Low Flow Wind Turbine).
Kumar, P.M., Ajit, K.R., Srikanth, N. and Lim, T.-C. (2017) On the Mathematical Modelling of Adaptive Darrieus Wind Turbine. Journal of Power and Energy Engineering, 5, 133-158. https://doi.org/10.4236/jpee.2017.512015
c Blade chord, m
C D Drag coefficient
C F N , C F ′ N Upwind and downwind elemental blade normal force coefficients
C L Lift coefficient
C N Normal force coefficient
C P Darrieus rotor power coefficient
C P − D ADWT rotor power coefficient
C F T , C F ′ T Upwind and downwind elemental blade tangential force coefficients
CQ Rotor torque coefficient
C T Tangential force coefficient
f Free-vortex frequency
f d w , f u p Downwind and upwind functions
F N , F ′ N Upwind and downwind blade normal force coefficients
F T , F ′ T Upwind and downwind blade tangential force coefficients
H Half-height of the rotor, m
l Blade length, m
N Number of blades
R , R D Darrieus rotor radius at the equator, m
R e t Turbine Reynolds number
R e b Blade Reynolds number
S Rotor swept area, m2
s Rotor solidity
u , u ′ Upwind and downwind interference factors
V , V ′ Upwind and downwind induced velocities, m/s
V e Induced equilibrium velocity, m/s
V ∞ Wind velocity at the equator level, m/s
W , W ′ Upwind and downwind relative inflow velocity, m/s
X , X ′ Upwind and downwind local tip-speed ratio
X E O Tip-speed ratio at the equator
z Local turbine height, m
α , α ′ Upwind and downwind local angle of attack, degree
β Rotor maximum diameter/height ratio
δ Angle between the blade normal and the equatorial plane, degree
θ Azimuthal angle, degree
ρ ∞ Freestream density, kg/m3
ω Turbine rotational speed, s-1
η r/R, Non-dimensional Cartesian coordinate
ζ z/H, Non-dimensional Cartesian coordinate
A s Savonius turbine swept area, m2
C P − S Savonius power coefficient
r ′ Savonius bucket radius, m
H s Savonius rotor height, m
N s Number of Savonius buckets
Q Savonius turbine torque, N・m
φ Savonius Turbine tip-speed ratio
P r Pressure on the returning Savonius bucket, Pa
P a Pressure on the advancing Savonius bucket, Pa
v e − s Equivalent velocity of Savonius rotor, m/s
v i s Absolute velocity of Savonius rotor, m/s
v ∞ Freestream wind speed, m/s