Violating Betz Limit?
Introduction
At American Wind, Inc we have known for some time that Betz Limit does not apply to ducted wind turbines. Our technology uses advanced ducting as well as other air flow manipulating technologies to get more energy from the air than any other wind turbine. During our research, we came across a study on wind turbine ducting that was done completely independent of American Wind, Inc’s input. The study was done by Clarkson University in New York uses an off the shelf wind turbine and doubles the power output by simply adding ducting, this violates Betz Limit… or does Betz Limit even apply to ducted wind turbines? According to the man who wrote Betz Limit, Albert Betz, It does not.
We’d like to give credit to the Authors of this paper Benjamin Kanya and Kenneth D. Viser
Experimental validation of a ducted wind turbine design strategy
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, USA
Abstract
A synergistic design strategy for ducted horizontal axis wind turbines (DWTs), utilizing the numerical solution of a ducted actuator disk system as the input condition for a modified blade element momentum method, is presented. Computational results of the ducted disk have shown that the incoming flow field for a DWT differs substantially from that of a conventional open rotor. The rotor plane velocity is increased in the ducted flow field, and, more importantly, the axial velocity component varies radially. An experimental full-scale 2.5 m rotor and duct were designed, using this numerical strategy, and tested at the University of Waterloo’s wind turbine test facility. Experimental results indicated a very good correlation of the data with the numerical predictions, namely a doubling of the power output at a given velocity, suggesting that the numerical strategy can provide a means for a scalable design methodology.
1. Introduction
Wind energy has long been acknowledged as having the potential to supplement and even displace the carbon-based fuel needs of our society. The wide adoption of small wind energy, namely that with a swept rotor area of less than 200 m2, has been hampered, however, by higher unit costs and lower efficiency than that of their large-scale counterparts. Studies on small turbines at Clarkson University have focused on improving their efficiency, particularly at lower wind speeds, with a focus on the key metric of cost per unit energy produced, namely USD kWh−1. Increasing the energy extraction for a given turbine size or reducing the manufacturing and operating costs are both options that increase the adoption of small wind by consumers. Other important factors that must also be considered include noise signature issues, sensitivity to wind directional changes, and issues of visibility and community acceptance.
The ducted wind turbine (DWT) concept has been fraught with controversy over the years yet still shows promise in improving the USD kWh−1 issue. DWTs are created by enclosing a conventional horizontal axis wind turbine with a lifting surface geometry revolved around the rotor axis. The duct captures a larger stream tube than an open rotor, as illustrated in Fig. 1. A substantial increase in velocity, exceeding even the free stream, is observed at the rotor face and the associated increase in mass flow rate increases the power output of the turbine. A properly designed DWT can improve the key areas mentioned above, leading to a much more effective small turbine design. There are, however, issues with DWTs that need to be addressed before their full potential can be realized, the foremost being the tradeoff of increased energy production against the increased use of materials, which usually results in a higher unit cost.
This paper reports on recent experimental results that validate a synergistic design strategy of the duct and the rotor. The numerical flow field of the optimized ducted actuator disk geometry is used as the input to the blade element momentum rotor design code. In this way, the influence of the duct on the flow field of the rotor is accounted for and the rotor geometry modified appropriately.
2. Background
Although studies on the potential performance gains of ducted turbines can be traced back to the 1920s, Foreman and Gilbert’s (1979), Foreman et al.’s (1978) and Gilbert et al.’s (1978) extensive testing in the 1970s proposed that this occurred because the duct reduces the pressure behind the turbine, relative to that behind a conventional wind turbine, causing more air to be drawn through. They suggested that they could have a performance efficiency of Cp=1.57, defined as
where A is the rotor area. The maximum Cp for an un-ducted open rotor is 0.593, commonly known as the Betz limit. This leads to the definition of an “augmentation ratio” of r=2.65, where
Hanson (2008) suggested that it is the lift generated by the shroud, as shown by de Vries (1979), that induces an increased mass flow through the rotor, resulting in an increase in the power coefficient proportional to the mass flow. Although one might surmise that, via this increased mass flow rate and velocity, a DWT can then exceed the Betz limit, this is incorrect because a much larger stream tube has been captured, and the assumptions applied to the open-rotor case do not apply per se. Unfortunately, such claims by inventors that they have “beaten Betz” have only served to give DWTs a bad reputation.
Many studies have investigated the feasibility and associated augmentation factors seen in DWTs in an effort to further their development (Hu and Cheng, 2008; Igra, 1976, 1984; Hansen et al., 2000; Werle and Presz Jr., 2008; van Bussel, 2007; Oman et al., 1977; Leoffler Jr. and Vanderbilt, 1978; Ohya et al., 2002, 2008; Politis and Koras, 1995; and Jamieson, 2009) with the largest prediction of 7 by Badawy and Aly (2000); however, conclusions have been quite varied. Werle and Presz Jr. (2008) used fundamental momentum principles and concluded that the possible augmentation factor could only approach 2 and that earlier studies had incorrect assumptions, leading to overly optimistic predictions. Hansen et al.’s (2000) viscous numerical results predicted ideal Cp values approaching 0.94, and an augmentation factor of 1.6. He also indicated that if the duct geometry could be made to keep the flow attached, the augmentation factor could be improved further.
A review article by van Bussel (2007) substantiates the above arguments regarding mass flow and indicates that the increase in the mass flow, and thus the augmentation ratio, is proportional to the ratio of the diffuser area to the rotor area. He concludes that the amount of energy extracted per unit volume of air with a DWT remains the same as for a bare rotor, but since the volume of air has increased, so has the total energy extracted. He also noted that Cp values above 1, corresponding to augmentation ratios on the order of 2, are achievable with diffuser-to-inlet area ratios on the order of 2.5. In addition to the experimental data, van Bussel reported on the effect of reducing the back pressure, which can also have a profitable effect on the performance.
Figure 2 Commercial attempts at large ducted turbines: (a) Vortec 7, (b) Ogin.
This potential increase in power generation has continued to drive DWT research; however, no commercial design has been able to realize these augmentation factors and no commercially viable DWT has been successful. A good example of this type of failure is seen in the Vortec 7 from New Zealand in Fig. 2a (Phillips, 2003; Windpower Monthly, 2018). A more recent example is that of the demise of Ogin (Boston Globe, 2018) in Fig. 2b. Perhaps the most promising experimental field results have been that of Ohya at Kyushu University in Japan on ducted turbines with a brim at the trailing edge (Ohya, 2014). Experimental data have been obtained on several units, including 500 W, 3 kW, 5 kW and 100 kW units, with measured power coefficients approaching a Cp=1.0. Ohya (2014) also reported no appreciable increase in the noise levels generated by the turbine while running.
Recent results of a synergistic design strategy coupling the duct flow field to the rotor design at Clarkson University have indicated two key design aspects. First, the presence of the duct modifies the axial velocity at the rotor, as shown in Fig. 3a (Jedamski and Visser, 2013) from a nominally uniform distribution to one with a radial variation. Second, moving the rotor to a location aft of the throat (Fig. 3b) provides an increased power output for a given duct geometry (Visser, 2016). Most rotor designs seek to exploit the high velocity at the throat of the duct; however, the presence of the rotor modifies the velocity where it is stationed, and more power can be extracted from the design, for a given duct, by moving the rotor aft. The optimum blade design for the rotor is not that which would be required of an open rotor but is different in planform shape and twist, due to the presence of the flow field generated by the duct. Venters et al. (2017) have also indicated Cp values, based on the duct exit area, of greater than 0.593, possibly pointing the way for a wind energy extraction device that is more efficient than a turbine of equal diameter.
Perhaps the most enticing aspect of the DWT concept is the potential for increased energy production in lower speed wind regimes, opening up many more areas to a viable distributed wind energy solution. Based on the above promising results, this investigation was undertaken to experimentally validate the synergistic design strategy.
Figure 3 Key design aspects for the Clarkson ducted turbine. (a) Non-uniform velocity distribution. (b) Aft rotor location.
3. Investigative methods
In order to make a comparison of the experimental data to existing available turbines, a rotor diameter of 2.5 m was selected for the design to compare it to a commercially available turbine, the Excel 1 by Bergey Windpower (2017). The Excel 1, illustrated in Fig. 4, is a 2.5 m diameter open rotor with a maximum output of 1.2 kW. The blades are constant chord and untwisted. The 2.5 m ducted prototype rotor was designed specifically for the ducted turbine environment. The test plan focused first on evaluating the open-rotor design against the Bergey open rotor and then examining the effect of the duct. Details of the numerical design are presented below followed by the experimental methods overview.
Figure 4 Bergey Excel 1.
Figure 5 Numerical duct results. (a) Flow Field Solution. (b) Extracted velocity profile.
3.1 Numerical approach
The numerical design strategy used a two-part scheme. First, the flow field of the duct, with an actuator disk, was determined using the Navier–Stokes solver FLUENT. The grid had a boundary layer mesh with a growth rate of 1.1 and the first mesh point was set at y+≈1. The boundary layer thickness was calculated as a function of Rec, based on duct chord, for each case and enough inflation layers were used to span the entire boundary layer to make sure the most accurate results that one can obtain from a Reynolds averaged Navier–Stokes (RANS) numerical solution were obtained. The actuator disk was covered with 200 quadrilateral elements (i.e., for the 2.5 m rotor, each element in our axisymmetric model covered 6.25 mm) to model the turbine in the 2-D axisymmetric model. There was a refined unstructured triangular grid around the duct which was surrounded by a large structured quadrilateral grid covering further the upstream and downstream of the actuator disk. The rest of the domain was meshed with unstructured quadrilateral elements. As mentioned above, the boundary layer thickness and y+ was calculated for each case. The k−ω shear stress transport (SST) turbulence model was utilized, which, among the two-equation turbulence models, gives better prediction of flow separation. Further details of the methods employed can be found in Bagheri-Sadeghi et al. (2018), and an example of the solution is shown in Fig. 5a.
From the field solution, the axial velocity field was then extracted (Fig. 5b) and used as an input for Clarkson’s in-house blade element momentum (BEM) code, mRotor. The rotor design in mRotor uses a fairly standard BEM strategy by Glauert to determine the optimum rotor shape (Kanya and Visser, 2010). Figure 6 illustrates the typical forces and velocities, including the axial induction factor “a”, the factor by which the upstream flow velocity is slowed by the time it reaches the rotor plane. For an ideal open rotor, a=1/3 to maximize the power extracted. Other local variables at radius, r, to be noted include the following: θ, the blade pitch angle; α, the angle of attack; ϕ, the angle which the velocity vector, W, makes with the rotor plane; ω, the angular velocity; dQ, the elemental torque; dT, the elemental thrust; Vo, the upstream velocity; L, lift; D, drag; and a′, the angular velocity induction factor.
Figure 6 Blade element momentum forces and velocites.
In addition to obtaining the rotor plane velocity profile from the numerical solution, a second piece of information, the thrust coefficient at the rotor, CT, rotor, was also extracted and the axial interference factor, a, was determined for input to mRotor from the following relation:
where