1025a anderson beating betz

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 = 𝟏𝟏𝟏𝟏.𝟏𝟏 𝑾𝑾

𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑃𝑃𝑃𝑃𝑃𝑃𝐵𝐵𝑃𝑃 𝑂𝑂𝑂𝑂𝐵𝐵𝑂𝑂𝑂𝑂𝐵𝐵 = 59.3%12𝜌𝜌𝑈𝑈03𝐴𝐴 = 𝟏𝟏𝟏𝟏.𝟎𝟎𝟎𝟎𝑾𝑾

How is it possible that our turbine broke the Betz limit?

In the 1920’s, 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 �̇�𝑾 = 𝑪𝑪𝒑𝒑,𝑶𝑶𝑶𝑶 ∙

𝟏𝟏𝟏𝟏𝝆𝝆𝑼𝑼𝒐𝒐

𝟑𝟑, 𝑃𝑃𝑤𝐵𝐵𝑃𝑃𝐵𝐵 𝑪𝑪𝒑𝒑,𝑶𝑶𝑶𝑶 < 𝟎𝟎𝟓𝟓.𝟑𝟑𝟑

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 1970’s especially because of the development of more accurate analytical and computational models enabling optimization of the augmenter geometry.

• The work of Igra [8],[9] and a team of researchers at Grumman Aerospace [6],[7] in the late 1970’s laid the groundwork for the majority of subsequent research.

• The analytical approach developed by De Vries [10] introduced the base pressure coefficient, a very important parameter in the evaluation of augmenter performance.

• Schaffarczyk and Phillips [11] analyzed the performance of the augmenter in terms of the loading applied to the turbine rotor.

• Lawn [12] evaluated sets of diffuser configurations in terms of the resistance coefficient and base pressure coefficient.

• Jamieson [13] derived a generalized version of Betz’s momentum theory using the axial induction factor 𝑎𝑎𝑂𝑂𝑂𝑂 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.

• 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 Bernoulli’s 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.

• 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:

𝜉𝜉𝑁𝑁 =𝑃𝑃0 − 𝑃𝑃1

12𝜌𝜌𝑈𝑈1

2 − 12𝜌𝜌𝑈𝑈0

2, 𝜉𝜉𝐷𝐷 =

𝑃𝑃3 − 𝑃𝑃212 𝜌𝜌𝑈𝑈2

2 − 12𝜌𝜌𝑈𝑈3

2

• The measures the obstruction to flow caused by the turbine rotor [12].

• It is defined as the ratio of the drop in static pressure across the turbine and dynamic pressure in the throat:

𝐾𝐾 =∬𝑃𝑃 𝑥𝑥1, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴 −∬𝑃𝑃 𝑥𝑥2, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴

12𝜌𝜌∬ (𝑈𝑈2 𝑥𝑥1, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴

=𝑃𝑃1 − 𝑃𝑃212𝜌𝜌𝑈𝑈1

2

• 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]:

𝐶𝐶𝑃𝑃𝑃𝑃 =∬𝑃𝑃 𝑥𝑥0, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴 −∬𝑃𝑃 𝑥𝑥3, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴

12𝜌𝜌∬ (𝑈𝑈2 𝑥𝑥0, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴

=𝑃𝑃0 − 𝑃𝑃312𝜌𝜌𝑈𝑈0

2

• Summing the pressure drops and gains across each section of the augmenter,

𝑃𝑃0 − 𝑃𝑃1 + 𝑃𝑃1 − 𝑃𝑃2 + 𝑃𝑃2 − 𝑃𝑃3 + 𝑃𝑃3 − 𝑃𝑃0 = 0. • Dividing by half of the air density 1

2⁄ 𝜌𝜌 and solving for 𝑈𝑈1,

𝑈𝑈1𝑈𝑈0

=𝜉𝜉𝑁𝑁 + 𝜉𝜉𝐷𝐷

𝐴𝐴0𝑃𝑃2

𝐴𝐴12− 𝐴𝐴0𝑃𝑃2

𝐴𝐴32+ 𝐶𝐶𝑃𝑃𝑃𝑃

𝜉𝜉𝑁𝑁 + 𝐾𝐾

• 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.

• A conservative approach calls for comparison between turbines of the same intake area.

(a) (b)

• 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.

𝐶𝐶𝑝𝑝 =�̇�𝑊𝐴𝐴𝐴𝐴𝐴𝐴

�̇�𝑊𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵=

12 𝜌𝜌 𝜂𝜂𝐵𝐵𝐾𝐾∬𝑈𝑈3 𝑥𝑥1, 𝑃𝑃,𝜃𝜃 𝑑𝑑𝐴𝐴

1627

12 𝜌𝜌 𝑈𝑈03𝐴𝐴0

=2716

𝐾𝐾𝜂𝜂𝐵𝐵𝑈𝑈13 ∙ 𝐴𝐴1𝑈𝑈03 ∙ 𝐴𝐴0

=2716

𝐾𝐾𝜂𝜂𝐵𝐵 ∙𝐴𝐴1𝐴𝐴0

∙𝜉𝜉𝑁𝑁 + 𝜉𝜉𝐷𝐷

𝐴𝐴0𝑃𝑃2𝐴𝐴12

− 𝐴𝐴0𝑃𝑃2𝐴𝐴32

+ 𝐶𝐶𝑃𝑃𝑃𝑃

𝜉𝜉𝑁𝑁 + 𝐾𝐾

1.5

• 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.

• Setting 𝜉𝜉𝑁𝑁 = 𝜉𝜉𝐷𝐷 = 1, the base pressure coefficient to 𝐶𝐶𝑃𝑃𝑃𝑃 = 0, and 𝐴𝐴1 = 𝐴𝐴0,

𝐶𝐶𝑝𝑝,𝑂𝑂𝑂𝑂 =�̇�𝑊𝑂𝑂𝑂𝑂

�̇�𝑊𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵=

2716

𝐾𝐾𝜂𝜂𝐵𝐵 ∙1 + 𝐴𝐴0𝑃𝑃2

𝐴𝐴12− 𝐴𝐴0𝑃𝑃2𝐴𝐴32

1 + 𝐾𝐾

1.5

• Since 𝐶𝐶𝑃𝑃𝑃𝑃 = 0, the resistance coefficient becomes

𝐾𝐾 =𝐴𝐴12

𝐴𝐴0𝑃𝑃2−𝐴𝐴12

𝐴𝐴32

• The coefficient of performance becomes

𝐶𝐶𝑝𝑝,𝑂𝑂𝑂𝑂 =2716

𝜂𝜂𝐵𝐵𝐴𝐴12

𝐴𝐴0𝑃𝑃2−𝐴𝐴12

𝐴𝐴32

1 + 𝐴𝐴0𝑃𝑃2𝐴𝐴12

− 𝐴𝐴0𝑃𝑃2𝐴𝐴32

1 + 𝐴𝐴12𝐴𝐴0𝑃𝑃2

− 𝐴𝐴12𝐴𝐴32

1.5

• Introducing Jamieson’s axial induction factor

𝑎𝑎𝑂𝑂𝑂𝑂 =𝑈𝑈0 − 𝑈𝑈1𝑈𝑈0

,

• The coefficient of performance reduces to

𝐶𝐶𝑝𝑝,𝑂𝑂𝑂𝑂 =2716

𝜂𝜂𝐵𝐵𝐴𝐴0𝑃𝑃𝐴𝐴1

1 −𝐴𝐴0𝑃𝑃2

𝐴𝐴32=

2716

𝜂𝜂𝐵𝐵𝑈𝑈1 𝑈𝑈02 − 𝑈𝑈32

𝑈𝑈03

=2716

𝜂𝜂𝐵𝐵 4𝑎𝑎𝑂𝑂𝑂𝑂 1 − 𝑎𝑎𝑂𝑂𝑂𝑂 2

Looks Familiar?

• The maximum of the expression on the previous slide occurs for 𝑎𝑎𝑂𝑂𝑂𝑂 = 1 and 𝑎𝑎𝑂𝑂𝑂𝑂 = 1

3.

• Substituting,

𝐶𝐶𝑝𝑝,𝑂𝑂𝑂𝑂 =2716

𝜂𝜂𝐵𝐵 (413

1 −13

2

= 𝜂𝜂𝐵𝐵

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 𝐶𝐶𝑝𝑝

• The flow volume extends 2 chord lengths upstream of the wind augmenter, 1.5 chord lengths laterally, and 3 chord lengths downstream.

• 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.

• 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

MARLEC® 504E turbine: • 60 W pancake generator • 3-phase AC output

rectified to 12 VDC • 6-bladed rotor with

safety ring

𝐶𝐶𝑝𝑝 =�̇�𝑊𝑎𝑎𝐴𝐴𝐴𝐴𝑎𝑎𝐵𝐵𝑎𝑎𝐵𝐵𝐵𝐵𝑎𝑎

�̇�𝑊𝑏𝑏𝐵𝐵𝐵𝐵𝐵𝐵=𝟏𝟏𝟏𝟏.𝟏𝟏 𝑾𝑾11.05 𝑊𝑊

= 1.104.

Dimensionless Parameters Power Wind Speed ξ N ξ D K CPE 𝜂𝜂𝐵𝐵 CP �̇�𝑊

3.5 m/s 1.932 2.121 1.264 1.702 80% 1.133 12.5 W

𝑃𝑃𝐵𝐵𝑃𝑃𝑃𝑃𝐵𝐵𝑃𝑃𝐵𝐵 𝐸𝐸𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2.5%

�̇�𝑊𝑏𝑏𝐵𝐵𝐵𝐵𝐵𝐵 = 59.3% 12𝜌𝜌𝑉𝑉3𝐴𝐴 =11.05 W

• 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.

• The nearest comparable wind turbine is the 0.4 kW AIR 30 Turbine manufactured by Primus Windpower.

0

10

20

30

40

50

60

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90

100

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)

Freestream Wind Velocity (mph)

Power Output vs. Wind Speed: AIR 40 vs. Augmented MARLEC

•The same performance prediction technique was applied to the “Energy Shark” augmented airborne wind electric generation system.

• A-PEGASUS (Airborne Portable Electric Generation and Storage Universal System) is an innovative, patent-pending airborne electric generation technology.

• 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.

• 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.

• The system can carry additional payloads

enabling it to perform other missions including: Meteorological observation Reconnaissance Aerial surveillance Radio telecommunications

Tethered Aerostat Specifications: • Length: 7 m (21 ft) • Volume: 20 m3 (1.75 tanks) • Payload: 10 kg (each) • PVC Thickness: 0.45 mm (18 mil)

Wind Augmenter: • Intake-to-throat area ratio: 𝐴𝐴0

𝐴𝐴1= 2.0

• Intake diameter: 1.7 m

Primus AIR-30 Wind Turbine • Rated Power: 0.4 kW • 3-phase AC output rectified into 48VDC • Cut-in speed (un-augmented): 3.6 m/s

System Schematic

Turbine Wiring Diagram

Dimensionless Parameters Wind Speed ξ N ξ D K CPE 𝜂𝜂𝐵𝐵 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

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400

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Aver

age

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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

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700

Wind Speed (m/s)

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)

CFD PredictionsEnergy Shark1.17m diameter Betz turbineAIR-30 Turbine

Comparison of CFD Predictions and Power Curves of Airborne Turbine, AIR-30, and Betz Turbine

• 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

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Purchase Cost (USD)

Wind Speed (m/s)

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Marlec 910AIR 30Energy SharkWhisper 100Ampair 600

• 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 models’ 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; “Hot Cake Syrup” (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.