Any wind energy specialist will say that an open-rotor wind turbine with an efficiency greater than the Betz limit is technological equivalent of a bargain price for this bridge.
Perpetual motion machines, time travel, and open-rotor wind turbines with an efficiency greater than 59.3% cannot exist.
Turbine Power Output = ๐๐๐๐.๐๐ ๐พ๐พ
๐ต๐ต๐ต๐ต๐ต๐ต๐ต๐ต ๐๐๐๐๐๐๐ต๐ต๐๐ ๐๐๐๐๐ต๐ต๐๐๐๐๐ต๐ต = 59.3%12๐๐๐๐03๐ด๐ด = ๐๐๐๐.๐๐๐๐๐พ๐พ
How is it possible that our turbine broke the Betz limit?
In the 1920โs, three researchers (Lanchester,
Betz, and Joukowski) independently derived
actuator disc momentum theory to express the
maximum efficiency of an open-rotor turbine.
Albert Betz
This theory sets the maximum efficiency of any open-rotor wind turbine as ๏ฟฝฬ๏ฟฝ๐พ = ๐ช๐ช๐๐,๐ถ๐ถ๐ถ๐ถ โ
๐๐๐๐๐๐๐ผ๐ผ๐๐
๐๐, ๐๐๐ค๐ต๐ต๐๐๐ต๐ต ๐ช๐ช๐๐,๐ถ๐ถ๐ถ๐ถ < ๐๐๐๐.๐๐๐
From the time that the Betz limit was published, reaching it has become the greatest challenge for inventors, theoreticians, and developers of the wind industry.
augmented turbine
Ideal Cp (Betz Limit, 59.3%)
It is now an accepted fact that well-optimized augmented wind turbines can achieve efficiencies above that of a Betz wind turbine of comparable size.
The reason that wind augmenters can achieve a greater efficiency is that the combination of a nozzle (aka concentrator) and diffuser accelerates the airflow through the throat and controls the expansion of the flow to a larger exit area than the stream tube of a Betz turbine.
Wind augmentation is making a comeback since its appearance in the 1970โs especially because of the development of more accurate analytical and computational models enabling optimization of the augmenter geometry.
โข The work of Igra [8],[9] and a team of researchers at Grumman Aerospace [6],[7] in the late 1970โs laid the groundwork for the majority of subsequent research.
โข The analytical approach developed by De Vries [10] introduced the base pressure coefficient, a very important parameter in the evaluation of augmenter performance.
โข Schaffarczyk and Phillips [11] analyzed the performance of the augmenter in terms of the loading applied to the turbine rotor.
โข Lawn [12] evaluated sets of diffuser configurations in terms of the resistance coefficient and base pressure coefficient.
โข Jamieson [13] derived a generalized version of Betzโs momentum theory using the axial induction factor ๐๐๐๐๐๐ to show that the maximum efficiency of augmented turbines is 88.8% energy conversion.
Modeling of wind augmentation that would also include airborne systems remains a relatively unexplored topic, as illustrated by the scarcity of related literature outside patent documents.
โข Since the fluid flow does not perform any work in the concentrator and diffuser sections of the wind augmenter, the flow can be modeled with Bernoulliโs equation with a correction factor to account for the pressure loss due to gradual contraction and expansion of the flow created by the wind augmenter.
โข The two empirical correction factors are defined as the ratio of the difference in static pressure and difference in dynamic pressure between the two locations in the augmenter:
๐๐๐๐ =๐๐0 โ ๐๐1
12๐๐๐๐1
2 โ 12๐๐๐๐0
2, ๐๐๐ท๐ท =
๐๐3 โ ๐๐212 ๐๐๐๐2
2 โ 12๐๐๐๐3
2
โข The measures the obstruction to flow caused by the turbine rotor [12].
โข It is defined as the ratio of the drop in static pressure across the turbine and dynamic pressure in the throat:
๐พ๐พ =โฌ๐๐ ๐ฅ๐ฅ1, ๐๐,๐๐ ๐๐๐ด๐ด โโฌ๐๐ ๐ฅ๐ฅ2, ๐๐,๐๐ ๐๐๐ด๐ด
12๐๐โฌ (๐๐2 ๐ฅ๐ฅ1, ๐๐,๐๐ ๐๐๐ด๐ด
=๐๐1 โ ๐๐212๐๐๐๐1
2
โข The is defined as the ratio of the difference in static pressure between the augmenter intake and exhaust pressures and the free stream dynamic pressure [10]:
๐ถ๐ถ๐๐๐๐ =โฌ๐๐ ๐ฅ๐ฅ0, ๐๐,๐๐ ๐๐๐ด๐ด โโฌ๐๐ ๐ฅ๐ฅ3, ๐๐,๐๐ ๐๐๐ด๐ด
12๐๐โฌ (๐๐2 ๐ฅ๐ฅ0, ๐๐,๐๐ ๐๐๐ด๐ด
=๐๐0 โ ๐๐312๐๐๐๐0
2
โข Summing the pressure drops and gains across each section of the augmenter,
๐๐0 โ ๐๐1 + ๐๐1 โ ๐๐2 + ๐๐2 โ ๐๐3 + ๐๐3 โ ๐๐0 = 0. โข Dividing by half of the air density 1
2โ ๐๐ and solving for ๐๐1,
๐๐1๐๐0
=๐๐๐๐ + ๐๐๐ท๐ท
๐ด๐ด0๐๐2
๐ด๐ด12โ ๐ด๐ด0๐๐2
๐ด๐ด32+ ๐ถ๐ถ๐๐๐๐
๐๐๐๐ + ๐พ๐พ
โข Many critics of wind augmentation point out that augmented turbines are often compared to the same turbines without the augmenter, making the reported performance increase meaningless.
โข A conservative approach calls for comparison between turbines of the same intake area.
(a) (b)
โข The coefficient of performance is defined as the ratio of the
to that of an of the same diameter as the
augmenter intake, operating at the Betz theoretical maximum efficiency.
๐ถ๐ถ๐๐ =๏ฟฝฬ๏ฟฝ๐๐ด๐ด๐ด๐ด๐ด๐ด
๏ฟฝฬ๏ฟฝ๐๐ต๐ต๐ต๐ต๐ต๐ต๐ต๐ต=
12 ๐๐ ๐๐๐ต๐ต๐พ๐พโฌ๐๐3 ๐ฅ๐ฅ1, ๐๐,๐๐ ๐๐๐ด๐ด
1627
12 ๐๐ ๐๐03๐ด๐ด0
=2716
๐พ๐พ๐๐๐ต๐ต๐๐13 โ ๐ด๐ด1๐๐03 โ ๐ด๐ด0
=2716
๐พ๐พ๐๐๐ต๐ต โ๐ด๐ด1๐ด๐ด0
โ๐๐๐๐ + ๐๐๐ท๐ท
๐ด๐ด0๐๐2๐ด๐ด12
โ ๐ด๐ด0๐๐2๐ด๐ด32
+ ๐ถ๐ถ๐๐๐๐
๐๐๐๐ + ๐พ๐พ
1.5
โข Any expression for the coefficient of performance of the augmented turbine should reduce to the Betz limit when the effects of the wind concentrator and diffuser are removed.
โข Setting ๐๐๐๐ = ๐๐๐ท๐ท = 1, the base pressure coefficient to ๐ถ๐ถ๐๐๐๐ = 0, and ๐ด๐ด1 = ๐ด๐ด0,
๐ถ๐ถ๐๐,๐๐๐๐ =๏ฟฝฬ๏ฟฝ๐๐๐๐๐
๏ฟฝฬ๏ฟฝ๐๐ต๐ต๐ต๐ต๐ต๐ต๐ต๐ต=
2716
๐พ๐พ๐๐๐ต๐ต โ1 + ๐ด๐ด0๐๐2
๐ด๐ด12โ ๐ด๐ด0๐๐2๐ด๐ด32
1 + ๐พ๐พ
1.5
โข Since ๐ถ๐ถ๐๐๐๐ = 0, the resistance coefficient becomes
๐พ๐พ =๐ด๐ด12
๐ด๐ด0๐๐2โ๐ด๐ด12
๐ด๐ด32
โข The coefficient of performance becomes
๐ถ๐ถ๐๐,๐๐๐๐ =2716
๐๐๐ต๐ต๐ด๐ด12
๐ด๐ด0๐๐2โ๐ด๐ด12
๐ด๐ด32
1 + ๐ด๐ด0๐๐2๐ด๐ด12
โ ๐ด๐ด0๐๐2๐ด๐ด32
1 + ๐ด๐ด12๐ด๐ด0๐๐2
โ ๐ด๐ด12๐ด๐ด32
1.5
โข Introducing Jamiesonโs axial induction factor
๐๐๐๐๐๐ =๐๐0 โ ๐๐1๐๐0
,
โข The coefficient of performance reduces to
๐ถ๐ถ๐๐,๐๐๐๐ =2716
๐๐๐ต๐ต๐ด๐ด0๐๐๐ด๐ด1
1 โ๐ด๐ด0๐๐2
๐ด๐ด32=
2716
๐๐๐ต๐ต๐๐1 ๐๐02 โ ๐๐32
๐๐03
=2716
๐๐๐ต๐ต 4๐๐๐๐๐๐ 1 โ ๐๐๐๐๐๐ 2
Looks Familiar?
โข The maximum of the expression on the previous slide occurs for ๐๐๐๐๐๐ = 1 and ๐๐๐๐๐๐ = 1
3.
โข Substituting,
๐ถ๐ถ๐๐,๐๐๐๐ =2716
๐๐๐ต๐ต (413
1 โ13
2
= ๐๐๐ต๐ต
The open-rotor turbine is at the Betz limit!
Extraneous Solution (Violates continuity)
00.5
11.5
22.5
33.5
44.5
5 00.1
0.20.3
0.40.5
0.60.7
0.80.9
10.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
y
Velo
city
Rat
io, (
U 1/U0)
Throat Velocity Ratio vs. Resistance and Base Pressure Coefficients
00.5
11.5
22.5
33.5
44.5
5
00.1
0.20.3
0.40.5
0.60.7
0.80.9
10
0.5
1
1.5
Cp
Coefficient of Performance vs. Resistance and Base Pressure Coefficients
For this reason, simulations [18] attempting to model the augmenter as a stationary volume of revolution (no rotating turbine) are insufficient to create an accurate estimation of the power output of the wind augmenter.
Other simulations [6],[9] attempted to model the behavior of augmenters as an internal flow problem, in which the turbine was placed inside a duct with a set of boundary conditions at the entrance and exit of the duct.
However, it has been recently demonstrated by Werle and Presz that these assumptions are inaccurate since the system must be treated as an external flow problem, in which the flow around the wind augmenter is equally important as the flow through it.
With the new analytical model of the turbine and these computational considerations, it is possible to perform high resolution CFD simulations to predict the performance of augmented turbines.
The simulations solved the incompressible Reynolds Averaged Navier Stokes equation with a two-equation linear-eddy viscosity model, supplemented with an algorithm to calculate the four dimensionless pressure coefficients and ๐ถ๐ถ๐๐
โข The flow volume extends 2 chord lengths upstream of the wind augmenter, 1.5 chord lengths laterally, and 3 chord lengths downstream.
โข The boundary conditions of the simulations are the free stream velocity passing normally through the farthest upstream plane and a static pressure of 0 Pa at the farthest downstream plane.
โข To verify the computational model,
performance predictions were compared to the actual performance of
A 0.4 kW augmented
airborne wind turbine.
A 60 W wind tunnel test model
MARLECยฎ 504E turbine: โข 60 W pancake generator โข 3-phase AC output
rectified to 12 VDC โข 6-bladed rotor with
safety ring
๐ถ๐ถ๐๐ =๏ฟฝฬ๏ฟฝ๐๐๐๐ด๐ด๐ด๐ด๐๐๐ต๐ต๐๐๐ต๐ต๐ต๐ต๐๐
๏ฟฝฬ๏ฟฝ๐๐๐๐ต๐ต๐ต๐ต๐ต๐ต=๐๐๐๐.๐๐ ๐พ๐พ11.05 ๐๐
= 1.104.
Dimensionless Parameters Power Wind Speed ฮพ N ฮพ D K CPE ๐๐๐ต๐ต CP ๏ฟฝฬ๏ฟฝ๐
3.5 m/s 1.932 2.121 1.264 1.702 80% 1.133 12.5 W
๐๐๐ต๐ต๐๐๐๐๐ต๐ต๐๐๐ต๐ต ๐ธ๐ธ๐๐๐๐๐๐๐๐ = 2.5%
๏ฟฝฬ๏ฟฝ๐๐๐๐ต๐ต๐ต๐ต๐ต๐ต = 59.3% 12๐๐๐๐3๐ด๐ด =11.05 W
โข The true advantages of wind augmentation are seen by comparing the power curves of the augmented turbine to an equivalent conventional turbine in low wind conditions.
โข The nearest comparable wind turbine is the 0.4 kW AIR 30 Turbine manufactured by Primus Windpower.
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10 12 14 16
Pow
er O
utpu
t (W
)
Freestream Wind Velocity (mph)
Power Output vs. Wind Speed: AIR 40 vs. Augmented MARLEC
โขThe same performance prediction technique was applied to the โEnergy Sharkโ augmented airborne wind electric generation system.
โข A-PEGASUS (Airborne Portable Electric Generation and Storage Universal System) is an innovative, patent-pending airborne electric generation technology.
โข The system comprises a tethered aerostat with an augmented horizontal axis wind turbine, a set of control systems to regulate the internal pressure and altitude of the tethered airship, and a hydrogen generation, recovery, and storage system.
โข By integrating airborne generation with wind augmentation, the technology has great potential to create a new market for portable, cost-effective, self-sustaining distributed generation systems in geographic regions previously deemed unprofitable for development of renewable energy.
โข The system can carry additional payloads
enabling it to perform other missions including: Meteorological observation Reconnaissance Aerial surveillance Radio telecommunications
Tethered Aerostat Specifications: โข Length: 7 m (21 ft) โข Volume: 20 m3 (1.75 tanks) โข Payload: 10 kg (each) โข PVC Thickness: 0.45 mm (18 mil)
Wind Augmenter: โข Intake-to-throat area ratio: ๐ด๐ด0
๐ด๐ด1= 2.0
โข Intake diameter: 1.7 m
Primus AIR-30 Wind Turbine โข Rated Power: 0.4 kW โข 3-phase AC output rectified into 48VDC โข Cut-in speed (un-augmented): 3.6 m/s
System Schematic
Turbine Wiring Diagram
Dimensionless Parameters Wind Speed ฮพ N ฮพ D K CPE ๐๐๐ต๐ต CP
5.0 m/s 0.983 1.198 0.321 0.410 75% 0.605 7.5 m/s 1.114 1.206 0.277 0.386 75% 0.506 10 m/s 1.029 1.096 0.235 0.392 75% 0.416
Sample Dimensionless Parameters Wind Speed Power Output (W) Predicted CP Actual CP
2.5 m/s 0 (Cut-in speed) ------------- -------
5.5 m/s 62 0.605 0.620
8.0 m/s 179 0.506 0.528
10 m/s 325 0.416 0.404
0
50
100
150
200
250
300
350
400
0 2.5 5 7.5 10
Aver
age
Pow
er O
utpu
t
Wind Speed (m/s)
Energy Shark Airborne Turbine Power Curve
Measured Power Output CFD Predictions Cubic Fit (Experimental) Cubic Fit (CFD)
0 2 4 6 8 10 12 140
100
200
300
400
500
600
700
Wind Speed (m/s)
Pow
er O
utpu
t (W
)
CFD PredictionsEnergy Shark1.17m diameter Betz turbineAIR-30 Turbine
Comparison of CFD Predictions and Power Curves of Airborne Turbine, AIR-30, and Betz Turbine
โข The power curve of the Energy Shark prototype was compared to that of four other conventional wind turbines of similar size that are currently on the market:
02
46
810
1214 1000
1500 20002500
30003500 4000
45005000
5500 6000
0
100
200
300
400
500
600
700
800
900
Purchase Cost (USD)
Wind Speed (m/s)
Pow
er O
utpu
t (W
)
Marlec 910AIR 30Energy SharkWhisper 100Ampair 600
โข The evaluation of the cost-competitiveness of the technology was performed by comparing the monthly energy production of the five turbines for two situations where the wind speed distributions are given by Rayleigh distributions centered at average wind speeds of 2.5 m/s and 5.0 m/s.
MODEL RATED
POWER OUTPUT
PURCHASE COST
MONTHLY ENERGY
PRODUCTION (2.5 m/s AVG)
MONTHLY ENERGY
PRODUCTION (5.0 m/s AVG)
IMAGE
Energy Shark (50m Altitude) 400 W
$849 (turbine) $2157 (2 airships) $421 (augmenter) $673 (helium)
$4100
35.0 kWh 89.9 kWh
Marlec 910 200 W
$1,280 (turbine) $155 (controller) $472 (15 m tower)
$1907
4.6 kWh 17.4 kWh
Primus Windpower Air 30 400 W
$849 (turbine) $404 (15 m tower)
$1253 4.2 kWh 30.6 kWh
Southwest Whisper 100 900 W
$2,875 (turbine) $1,225 (17 m tower)
$4100 21.4 kWh 1077 kWh
Ampair 600 600 W
$3280 (turbine) $1138 (controller) $1,225 (17 m tower)
$5643
37.1 kWh 1574 kWh
1. The costs presented in the table do not include the installation costs of the turbines, which can vary significantly depending on terrain, soil quality, and necessary wire gauge for the transmission distance. Since the Energy Shark turbine requires minimal ground work, the decrease in installation costs further enhances its cost-competitiveness.
2. The prototype Energy Shark demonstrator uses helium for its lighter-than-air gas, while any future production models will utilize hydrogen, thereby significantly reducing the operation and maintenance costs of the system.
3. An AIR-30 turbine was used to simplify the initial development process of the system. However, a permanent magnet alternator (for DC applications) or induction generator (grid-connected applications) can be substituted, resulting in substantial cost and weight savings.
4. Future airborne systems can also take advantage of economies of scale, making the technology even more cost-effective.
Follow-up research and development is recommended to advance this technology in the following areas: Replace the current aerostats with models designed to
be compatible with hydrogen and rated for extreme winds up to 20 m/s;
Develop and implement the hydrogen generation, storage, and recovery system for use in the aerostats;
Develop a set of ground controls that would enable fully autonomous operation of the system;
Substitute a grid-connected induction generator in place of the current permanent magnet alternator;
Implement Grade-A aircraft Dacron fabric for the wind augmenter (used in an earlier 60W wind tunnel model), rather than the polyester film which had a strong tensile strength but poor puncture resistance.
As always, more research is needed. It will be focused on evaluating the modelsโ ability to predict the performance of ducted hydrokinetic turbines.
The operational test model will go to a community in Jikawa Province, Papua New Guinea for whom we have just finished designing a rainwater harvesting and distribution system after preliminary survivability tests.
[l] Y. Ohya and T. Karasudani, "A Shrouded Wind Turbine Generating High Output Power with Wind-lens Technology," Energies, vol. 3, no. 4, pp. 634-639, 2010.
[2] B. William, "Lighter than air wind energy conversion system". United States of America Patent 4350897, 21 September 1982.
[3] L. Potter, "Funneled Wind Turbine Aircraft". United States of America Patent 7786610, 22 May 2007. [4] B. Glass, "Power-Augmenting Shroud for Energy-Producing Turbines". United States of America Patent 8253265, 28
August 2012. [5] A. Anderson, "Portable Self-Inflating Airborne Wind Turbine System". United States of America Patent 13/926073, 25
June 2013. [6] K. Foreman, R. Oman and B. Gilbert, "Fluid Dynamics of DAWT's," Journal of Energy, vol. 2, pp. 368-374, 1978. [7] K. Foreman, R. Oman and B. Gilbert, "A Progress Report on the Diffuser Augmented Wind Turbine," in 3rd Biennial
Conference an Workshop on Wind Energy Conversion Systems, Washington DC, USA, 1975. [8] O. Igra, "Shrouds for Aerogenerators," AIAA Journal, vol. 14, no. 10, pp. 1481-1483, 1976. [9] O. Igra, "Research and Development for Shrouded Wind Turbines," Energy conservation and Management, vol. 21, pp.
13-48, 1980. [10] O. De Vries, "Fluid Dynamic Aspects of Wind Energy Conversion," AGARD-AG-243, 1979. [11] A. Schaffarczyk and D. Phillips, "DESIGN PRINCIPLES FOR A DIFFUSOR AUGMENTED WIND-TURBINE BLADE," in EWEC,
2001. [12] C. Lawn, "Optimization of the Power Output from Ducted Turbines," Proceedings of the Institution of Mechanical
Engineers, Part A: Journal of Power and Energy, vol. 217, no. 1, pp. 107-117, 2003. [13] P. Jamieson, "Beating Betz: Energy Extraction Limits in a Constrained Flow Field," Journal of Solar Energy Engineering,
vol. 131, no. 3, 2009.
[14] D. Phillips, P. Richards, G. Mallinson and R. Flay, "Computational Modelling of Diffuser Designs for a Diffuser Augmented Wind Turbine," in 13th Australasian Fluid Mecahnics Conference, Melbourne, Australia, 1998.
[15] R. Ghajar and E. Badr, "An experimental study of a collector and diffuser system on a small demonstration wind turbine," International Journal of Mechanical Engineering Education , vol. 36, no. 1, pp. 58-68, 2008.
[16] M. Werle and W. Presz, "Ducted Wind/Water Turbines and Propellers Revisited," Journal of propulsion and Power, vol. 24, no. 5, pp. 1146-1140, 2008.
[17] A. Aranake, V. Lakshminarayan and K. Duraisamy, "Computational Analysis of Shrouded Wind Turbine Configurations," in 51st AIAA Aerospace Sciences Meeting, Grapevine, TX, 2013.
[18] T. Matsushima, S. Takagi and S. Muroyama, "Characteristics of a highly efficient propeller type," Renewable Energy, vol. 31, no. 9, pp. 1343-1354, 2006.
[19] M. Mashud and M. Ali, "HIGH-PERFORMANCE WIND TURBINE: A NEW APPROACH," in Proceedings of the ASME 2011 5th International Conference on Energy Sustainability, Washington DC, 2011.
[20] G. Van Bussel, "The science of making more torque from wind: Diffuser experiments and theory revisited.," Journal of Physics: Conference Series, vol. 75, no. 1, 2007.
[21] E. Hau, Wind Turbines: Fundamentals, Technologies, Applications, Economics, Berlin: Springer Heidelberg, 2006.
[22] B. Launder and B. Sharma, "Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc," Letters in Heat and Mass Transfer, vol. 1, no. 2, pp. 131-138, 1974.
Slide No. Image Source
2,3 Martin St-Amant
5 V.L. Okulov
6 Unkwnown
7 Eric Hau [21]
10 (CW from top left) Patrick Charpiat; โHot Cake Syrupโ (Kyushu University); Paul Gipe; Paul Gipe
40 Primus Windpower
43, 44 (CW from top left) Altaeros Energies; Pierre Rivard (Magenn); Makani Power
60 Dr. Larry Hull/ Centralia Rotary
All other images are the individual work of the researcher.
The research team would like to thank
for helping us to beat the Betz limit.