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The performance and annual energy output have to be predicted to maximize the economic benefits from a wind turbine. Mathematically predicting the performance of Darrieus type lift based turbines are challenging due to the inconsistent angle of attack, blade wake interaction and local induced velocities giving rise to complex flow physics. A reliable and validated mathematical model is therefore essential to optimize the various design parameters prior to manufacture. The objective of the current study is to evaluate widely employed aerodynamic models based on their prediction accuracy, limitations, and computational requirements. Double multiple stream tube models have been discussed in detail and the predictions are experimentally validated through the wind tunnel test of three-bladed H-Darrieus rotor in terms of torque and power coefficient. The possible sources for the deviation between the predicted and measured values have been discussed and concluded with potential solutions.

The wind power continued to be the major source of renewable energy and a potential alternative for the fossil fuels from the ecological perspective. Decentralized power generation has seen significant growth from the smart grid context, and the small wind turbines become an indispensable part of it. Of several wind turbines designs, Darrieus type lift based turbines hold a prominent place for its unmatched advantages such as simple design, low noise and Omni directionality. Extensive research has been carried in the past in an effort to optimize these turbines for the urban environment and highly turbulent flows [

For any analytical model, the procedure to compute the performance is described as

1) Calculating the local angle of attack and the relative velocity corresponding to the blade orbital position

2) Calculating the induced velocity accounting the blade wake interaction

3) Choosing the suitable mathematical model along with the mathematical expressions

4) Calculation of blade tangential forces and the normal forces

5) Pre-Stall airfoil characteristics at different Reynolds numbers

6) Aspect Ratio consideration to account for the blade tip loss effects

7) Dynamic Stall Model to account for the unsteady effects

8) Flow Curvature Model to account for the circular blade motion.

The cascade model was prescribed by Hirsch and Mandal [

v u v 0 = ( v 1 v 0 ) k i (1)

v d v 0 = ( v 2 v 1 ) k i v 1 v 0 (2)

k i = ( 0.425 + 0.332 ( N c / R ) ) (3)

Q = ρ h R 2 ∫ 0 2 π ( W o 2 sin α 0 cos α 0 − W i 2 sin α i cos α i ) d θ (4)

With h, as the height of the turbine and R, as the radius of the turbine and c is the airfoil chord. The overall torque Q generated by the rotor can be deduced from Equation (4). The cascade model is able to predict the performance of the rotors with solidities and also at high tip speed ratio quite accurately compared to other mathematical models. Other geometric characteristics such as blade pitch, flow curvature effect and aspect ratio effects can be readily integrated into this model. To improve the predicting capability, the dynamic stall model was accounted along with the flow curvature effects [

The vortex model is based on the influence of vorticity in the wake of the blade in which a bound vortex filament replaces each blade element. From the Helmholtz theorem [

d v = Γ d l × r 4 π І r 3 І (5)

The undisturbed freestream velocity added with the induced velocity by the entire vortex filaments comprises the fluid velocity at any point in the flow field. The relationship between the lift L per unit span on a blade segment and the bound vortex strength Γ B is given by the Kutta-Joukowski law as in Equation (6).

L ′ = ρ v Γ (6)

The relation between the bound vortex strength and the lift coefficient C l , and U r , the local relative fluid velocity is given as

Γ B = 1 2 C l C U R (7)

The vortex model was improved in recent years by incorporating dynamic stall effects [

The kinetic energy in the wind invariably creates thrust in the incoming direction. The change in kinetic energy between the upstream wind v 0 and downstream wind v 2 is the useful power P extracted from the wind. Wind rotors can be conveniently considered as a permeable disk as shown in _{D} and the torque by the blades. In the single streamtube model as shown in

F D = m x ( v o − v 2 ) (8)

v 1 = ( 1 − a ) v 0 (9)

v 2 = ( 1 − 2 a ) v 0 (10)

v 1 = v 0 + v 2 2 (11)

P = 2 ρ v 0 3 a ( 1 − a ) 2 (12)

Since the upstream and downstream wind velocity varies considerably, double actuator disk model was introduced as shown in

v 1 = ( 1 − a ) u 0 (13)

v 2 = ( 1 − 2 a ) u 0 (14)

v 3 = ( 1 − 2 a ) ( 1 − a ′ ) u 0 (15)

v 4 = ( 1 − 2 a ′ ) ( 1 − 2 a ) u 0 (16)

The value of the axial inductions factors determines the operating state of the rotor. For a < 0, the rotor is in are in the propeller state where the streamtube is contracting, and the thrust force is directed upstream and acts as a propulsion device as energy has to be expended to the fluid if a < 0. If the value of induction factor lies between 0 and 0.4, the rotor will be in windmill state and the momentum theory can be applied as the streamtube is expanding and the thrust force is directed downstream acting as a drag force. In this case, energy can be extracted from the free stream.

An improved version of the single streamtube model was introduced by Lissaman [

blade element and the moment theory are applied for each streamtube. Rotor power, torque is obtained by averaging the values from each streamtube. The model proposed by [

The MST was extended by Paraschivoiu [

C Q = N C H 2 π S ∫ − π / 2 π / 2 ∫ − 1 1 C T ( η cos δ ) ( W V ∞ ) 2 d θ d ς (17)

C ′ Q = N C H 2 π S ∫ π / 2 3 π / 2 ∫ − 1 1 C ′ T ( η cos δ ) ( W ′ V ∞ ) 2 d θ d ς (18)

C P = ( C Q + C ′ Q ) X E Q (19)

The power coefficient C P of the turbine is obtained by Equation (19). The current DMST model accounts for the secondary effects including streamtube expansion, blade geometry, rotating tower and the effect of struts and spoilers. Boeing-Vertoll stall model was the recent addition. A good agreement with the experimental data has been observed for different solidity and high-tip speed ratio. By far the DMST model predicts more accurately compared to other

available mathematical models. Hence the DMST model is suitable to analyze the performance of current three bladed H-rotor under investigation.

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 the bottom end and spherical bearings on the top end as shown in

The power coefficient is derived from the torque measured and plotted against TSR as shown in

Design Parameters | Values |
---|---|

Rotor Diameter (mm) | 400 |

Rotor Height (mm) | 300 |

Solidity | 0.43 |

Center Shaft Diameter (mm) | 20 |

Operating Reynolds Number | 20,000 - 100,000 |

Airfoil Chord (mm) | 100 |

Airfoil | NACA 0018 |

Number of Blades | 3 |

End plate diameter (mm) | 420 |

End plate thickness (mm) | 8 |

Support | Both ends support |

The operating Reynolds (Re) number range of a straight bladed Darrieus turbine of 0.4 m diameter and 0.1 m chord length is from 20,000 to 80,000 for the wind speed of 3 - 12 m/s. From Equation (20), the Re can be calculated for the given chord length and the wind speed.

Re = ρ C V ∝ γ (20)

where ρ is the density of the air ( kg / ( m ^ 3 ) ) , C is the chord length in (m), V ∝ is the freestream wind speed ( m / s ) . γ is the kinematic viscosity ( m 2 / s ) . The Re of this range falls in the low Re category and the airfoil characteristics are highly complex to predict, unless it is experimentally measured. Airfoil operating in this range stage a complex phenomenon on the upper surface of the

airfoil, and the flow behavior is markedly different for a static and dynamic airfoil. Due to low Re, the boundary layer lacks energy to reattach as a turbulent flow after the laminar separation bubble burst. As the Re increase beyond 2 × 10^{5}, the probability of reattachment is likely to occur. From the previous studies [^{5}. The Re range under investigation is below this range and Xfoil it is not a surprising factor that Cp is predicted much higher than the experimental value by DMST due to high lift/drag ratio computed by Xfoil. It is widely accepted that low Re prediction of Xfoil varies significantly with the wind tunnel measured data.

Another source of error is from extrapolation the initial data obtained from Xfoil as the airfoil characteristics are required for 360˚. Two methods are widely used to extrapolate are Viterna [

The 3D printed airfoils are polished before testing, yet the surface roughness sensitivity is more pronounced for a small blade. The surface roughness decreases the airfoil lift coefficient and increases skin friction drag which is not accounted in Xfoil prediction. Hence these are all the notable sources of error contributed by the Xfoil to the DMST to over predict the Cp to such a high value as shown in

No experiments are perfect and so the current experiment, and it is inevitable to conduct an experiment free of errors, but can be kept minimal. The size of the model under investigation is comparatively smaller, and any error in the experiments can significantly influence the torque output. Deep groove ball bearings support the center shaft on the bottom and spherical bearings on top to accommodate shaft misalignment. Even a minor misalignment in the shaft will resist the shaft rotation and affect the rotor torque substantially. The bearing friction also contributes their part in reducing the shaft torque. The bearing friction can be minimized by periodic lubrication. Flexible couplings connecting the rotor shaft to the torque sensor also consumes some of the rotor torque. Collectively, these resistive torques results in recording a lower torque value, though the generated torque is higher. In this case, the Cp displayed by the experiment should be higher and a step closer to the DMST predictions.

Though DMST model predicts the aerodynamic performance of Darrieus rotor more accurately compared to other mathematical models, the inaccuracies do exist for certain design and operating specifications. The discrepancies between the predicted value and the experimental value are proportional to the solidity of the turbines operating at low Tip Speed Ratio (TSR). The solidity σ can be calculated from Equation (21) as 1.5, with r as the rotor radius.

σ = N c r (21)

From the past validation studies, the DMST model over predicts the performance for such high solidity turbines. Further increase in Cp by DMST model is due to dynamic stall effects, which is cumbersome to predict due to rapidly changing AOA and the turbulence generated. The airfoil characteristics are sensitive to turbulence with loss of lift. For the current study, the blade tip losses are not included due to the presence of end plates in the test model, to prevent the flow around the blades. The blade tip losses are not eliminated with end plates, but are minimized. The DMST model predicts negative Cp if the tip loss factor is included and a possible reason is low aspect ratio. The models under investigation have the aspect ratio of 1.2, whereas the aspect ratios of larger turbines are in the order of 5 to 15. Hence the increase in Cp by DMST prediction, is due to the fact the tip loss cannot be included for small aspect ratio wind tunnel models. Flow curvature effects are more pronounced for small rotor diameter turbines when it operated at higher RPM. The flow curvature effects are induced due to the orbital motion of the blades in Darrieus turbine. In reality, the blades are subjected to curvilinear flow rather than rectilinear flow as assumed. The airfoil lift is significantly reduced due to centrifugal effects on the boundary layer proportional to the TSR. The study by [

The objective of the current study is to evaluate the aerodynamic models widely used to predict the performance of Darrieus turbine in order to choose a suitable model to obtain aerodynamic characteristics for 10 kW H-rotor. Albeit, these models are well proven and validated against large turbines, the small VAWTs presents different scenarios due to low Re, flow curvature and low aspect ratio, dynamic stall effects. Hence the chosen mathematical model is experimentally validated against wind tunnel test of a scaled rotor. It is imperative to identify the sources of error in prediction to propose potential solutions. Xfoil induces substantial error in computing the lift and drag characteristics of airfoil due to low Re. The error is compounded by 360˚ interpolations, which relies on the initial input. Following are the conclusions that can be drawn for validating the DMST model prediction with wind tunnel test of a three-straight bladed H-rotor.

The DMST model over predicts the Cp by 23.1% for Re 1.4 × 10^{5}, 20.5% for Re 1.7 × 10^{5}, 29.2% for Re 1.9 × 10^{5}, 20.2% for Re 2.0 × 10^{5}, 24.6% for Re 2.3 × 10^{5} and 21.8% for Re 2.6 × 10^{5}. As Re increases the predicted value is closer to the experimental value. Xfoil induces substantial error in estimating the airfoil characteristics, especially for low Re. Hence 360˚ polar obtained through the experiments will allow the DMST model to predict with higher accuracy. It is advisable to eliminate the tip loss in an experimental setup with sufficient end plate diameter, rather than including the tip loss factor in the DMST model, which predicts negative Cp due to low aspect ratio.

The errors in the experiment tend to reduce the power coefficient value of the experimental results, resulting in a further deviation from DMST prediction. The DMST has its part of contribution to the error such as low Re performance of the airfoil, flow curvature effect, inaccurate dynamic stall model. Hence the final statement to conclude this study is that the DMST model prediction for small VAWT is far from acceptable accuracy compared to experimental results.

This research is 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., Rashmitha, S.R., Srikanth, N. and Lim, T.-C. (2017) Wind Tunnel Validation of Double Multiple Streamtube Model for Vertical Axis Wind Turbine. Smart Grid and Renewable Energy, 8, 412-424. https://doi.org/10.4236/sgre.2017.812027