JPEE  Vol.4 No.1 , January 2016
On the Maximum of Wind Power Efficiency

In our paper we demonstrate that the filtration equation used by Gorban’ et al. for determining the maximum efficiency of plane propellers of about 30 percent for free fluids plays no role in describing the flows in the atmospheric boundary layer (ABL) because the ABL is mainly governed by turbulent motions. We also demonstrate that the stream tube model customarily applied to derive the Rankine-Froude theorem must be corrected in the sense of Glauert to provide an appropriate value for the axial velocity at the rotor area. Including this correction leads to the Betz-Joukowsky limit, the maximum efficiency of 59.3 percent. Thus, Gorban’ et al.’s 30% value may be valid in water, but it has to be discarded for the atmosphere. We also show that Joukowsky’s constant circulation model leads to values of the maximum efficiency which are higher than the Betz-Jow-kowsky limit if the tip speed ratio is very low. Some of these values, however, have to be rejected for physical reasons. Based on Glauert’s optimum actuator disk, and the results of the blade-element analysis by Okulov and S&oslashrensen we also illustrate that the maximum efficiency of propeller-type wind turbines depends on tip-speed ratio and the number of blades.

Cite this paper: Kramm, G. , Sellhorst, G. , Ross, H. , Cooney, J. , Dlugi, R. and Mölders, N. (2016) On the Maximum of Wind Power Efficiency. Journal of Power and Energy Engineering, 4, 1-39. doi: 10.4236/jpee.2016.41001.

[1]   Gorban’, A.N., Gorlov, A.M. and Silantyev, V.M. (2001) Limits of the Turbine Efficiency for Free Fluid Flow. Journal of Energy Resources Technology, 123, 311-317.

[2]   Okulov, V.L. and van Kuik, G.A.M. (2011) The Betz-Joukowsky Limit: On the Contribution to Rotor Aerodynamics by the British, German and Russian Scientific Schools. Wind Energy, 15, 355-344.

[3]   Betz, A. (1920) Das Maximum der theoretisch möglichen Ausnützung des Windes durch Windmotoren. Zeitschrift für das gesamte Turbinenwesen, 26, 307-309. (In German)

[4]   Joukowsky, N.E. (1920) Windmill of the NEJ Type. Transactions of the Central Institute for Aero-Hydrodynamics of Moscow. (In Russian) (as cited by [5])

[5]   van Kuik, G.A.M. (2007) The Lanchester-Betz-Joukowsky Limit. Wind Energy, 10, 289-291.

[6]   Sørensen, J.N. (2011) Aerodynamic Aspects of Wind Energy Conversion. Annual Review of Fluid Mechanics, 43, 427-448.

[7]   Hartwanger, D. and Horvat, A. (2008) 3D Modelling of Wind Turbine Using CFD. NAFEMS UK Conference 2008 “Engineering Simulation: Effective Use and Best Practice”, Cheltenham, 10-11 June 2008, 14 p.

[8]   Blackledge, J., Coyle, E. and Kearney, D. (2011) A Stochastic Model for Wind Turbine Power Quality Using a Levy Index Analysis of Wind Velocity Data. The Third International Conference on Resource Intensive Applications and Services, Venice, 22-27 May 2011.

[9]   van Kuik, G.A.M., Sørensen, J.N. and Okulov, V.L. (2015) Rotor Theories by Professor Joukowsky: Momentum Theories. Progress in Aerospace Sciences, 73, 1-18.

[10]   Glauert, H. (1935) Airplane Propellers. In: Durand, W.F., Ed., Aerodynamic Theory, Vol. IV, Division L, Springer, New York, 169-360.

[11]   Okulov, V.L. and Sørensen, J.N. (2008) Refined Betz Limit for Rotors with a Finite Number of Blades. Wind Energy, 11, 415-426.

[12]   Okulov, V.L. and Sørensen, J.N. (2010) Maximum Efficiency of Wind Turbine Rotors Using Joukowsky and Betz Approaches. Journal of Fluid Mechanics, 649, 497-508.

[13]   de Groot, S.R. and Mazur, P. (1969) Non-Equilibrium Thermodynamics. North-Holland Publishing Comp., Amsterdam/ London.

[14]   Landau, L.D. and Lifschitz, E.M. (1981) Lehrbuch der theoretischen Physik-Hydrodynamik. Akademie-Verlag Berlin. (In German)

[15]   Budó, A. (1990) Theoretische Mechanik. In: Rompe, R. and Schmutzer, E., Eds., Hochschulbücher für Physik, VEB Deutscher Verlag der Wissenschaften, Berlin, 615. (In German)

[16]   Mölders, N. and Kramm, G. (2014) Lectures in Meteorology. Springer International Publishing.

[17]   Prandtl, L. (1905) über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III. Internationalen Mathematiker Kongresses, Heidelberg, 8-13 August 1904, B. G. Teubner, Leipzig, 485-491. (In German)

[18]   Rotta, J.C. (1972) Turbulente Strömungen. B. G. Teubner, Stuttgart. (In German)

[19]   Sanderse, B., van der Pijl, S.P. and Koren, B. (2011) Review of Computational Fluid Dynamics for Wind Turbine Wake Aerodynamics. Wind Energy, 14, 799-819.

[20]   Reynolds, O. (1895) On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Philosophical Transactions of the Royal Society of London, 186, 123-164.

[21]   Prandtl, L. (1918) Tragflügeltheorie—I. Mitteilung. Nachrichten der K. Gesellschaft zu Göttingen, Mathematisch-Physikalische Klasse, 451-477.

[22]   Kramm, G. and Meixner, F.X. (2000) On the Dispersion of Trace Species in the Atmospheric Boundary Layer: A Re-formulation of the Governing Equations for the Turbulent Flow of the Compressible Atmosphere. Tellus, 52A, 500-522.

[23]   Montgomery, R.B. (1954) Convection of Heat. Archiv für Meteorologie,Geophysik und Bioklimatologie, A7, 125-132.

[24]   Fortak, H. (1969) Zur Energetik der planetarischen Grenzschicht. Annalen der Meteorologie (NF) 4, 157-162. (In German)

[25]   van Mieghem, J. (1949) Les equations générales de la mécanique et de l'énergétique des milieux turbulents en vue des applications à la météorologie. Inst. R. Météor. Belgique., Mém., XXXIV, 60. (In French)

[26]   van Mieghem, J. (1973)Atmospheric Energetics. Clarendon Press, Oxford, UK.

[27]   Herbert, F. (1975) Irreversible Prozesse der Atmosphäre—3. Teil (Phänomenologische Theorie mikroturbulenter Systeme). Beiträge zur Physik der Atmosphäre, 48, 1-29. (In German)

[28]   Libby, P.A. and Williams, F.A., Eds. (1980) Turbulent Reacting Flows. Springer-Verlag, Berlin.

[29]   Pichler, H. (1984) Dynamic der Atmosphäre. Bibliographisches Institut, Zürich. (In German)

[30]   Cox, G. (1995) Basic considerations. In: Cox, G., Ed., Combustion Fundamentals of Fire, Academic Press, London, San Diego, New York, 3-30.

[31]   Kramm, G., Dlugi, R. and Lenschow, D.H. (1995) A Re-evaluation of the Webb-Correction Using Density-Weighted Averages. Journal of Hydrology, 166, 283-292.

[32]   Thomson, D. (1995) The Parameterization of the Vertical Dispersion of a Scalar in the Atmospheric Boundary Layer. Atmospheric Environment, 29, 1343.

[33]   Venkatram, A. (1998) Response. Atmospheric Environment, 32, 259.

[34]   Kowalski, A.S. (2012) Exact Averaging of Atmospheric State and Flow Variables. Journal of the Atmospheric Sciences, 69, 1750-1757.

[35]   Hesselberg, T. (1926) Die Gesetze der ausgeglichenen atmosphärischen Bewegungen. Beiträge zur Physik der freien Atmosphäre, 12, 141-160. (In German)

[36]   Lumley, J.L. and Yaglom, A.M. (2001) A Century of Turbulence. Flow, Turbulence and Combustion, 66, 241-286.

[37]   Herbert, F. (1995) A Re-evaluation of the Webb Correction Using Density-Weighted Averages—Comment. Journal of Hydrology, 173, 343-344.

[38]   Eliassen, A. and Kleinschmidt Jr., E. (1957) Dynamic Meteorology. In: Flügge, S., Ed., Handbuch der Physik, Bd. XLVIII. Springer-Verlag Berlin/Heidelberg/New York, 1-154.

[39]   Dutton, J.A. (1995) Dynamics of Atmospheric Motion. Dover, New York.

[40]   de Vries, O. (1979) Fluid Dynamic Aspects of Wind Energy Conversion. AGARDograph 243, AGARD, Brussels.

[41]   Türk, M. and Emeis, S. (2010) The Dependence of Offshore Turbulence Intensity on Wind Speed. Journal of Wind Engineering and Industrial Aerodynamics, 98, 466-471.

[42]   Barthelmie, R.J., Frandsen, S.T., Nielsen, M.N., Pryor, S.C., Rethore, P.-E. and Jørgensen, H.E. (2007) Modelling and Measurements of Power Losses and Turbulence Intensity in Wind Turbine Wakes at Middelgrunden Offshore Wind Farm. Wind Energy, 10, 517-528.

[43]   Barthelmie, R.J., Churchfield, M.J., Moriarty, P.J., Lundquist, J.K., Oxley, G.S., Hahn, S. and Pryor, S.C. (2015) The Role of Atmospheric Stability/Turbulence on Wakes at the Egmond aan Zee Offshore Wind Farm. Journal of Physics: Conference Series, 625, conference 1.

[44]   Stull, R.B. (1988) An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Dordrecht, Boston, London.

[45]   Garratt, J.R. (1994) The Atmospheric Boundary Layer. Cambridge University Press, Cambridge, New York, Melbourne.

[46]   Mellor, G.L. and Yamada, T. (1974) A Hierarchy of Turbulence Closure Models for Planetary Boundary Layers. Journal of the Atmospheric Sciences, 31, 1791-1806.<1791:AHOTCM>2.0.CO;2

[47]   Mellor, G.L. and Yamada, T. (1982) Development of a Turbulence Closure Model for Geophysical Fluid Problems. Review of Geophysics and Space Physics, 20, 851-875.

[48]   Janjic, Z.I. (2001) Nonsingular Implementation of the Mellor-Yamada Level 2.5 Scheme in the NCEP Meso Model. National Centers for Environmental Prediction (NCEP), Office Note #437.

[49]   Lorenz, E.N. (1967) The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization, WMO-No. 218.Tp.115, Geneva.

[50]   Bernhardt, K. and Lauter, E.A. (1977) Globale physikalische Prozesse und Umwelt. Zeitschrift für Meteorologie, 27, 1-20. (In German)

[51]   Holton, J.R. (1979) An Introduction to Dynamic Meteorology. Academic Press, New York, San Francisco, London.

[52]   Peixoto, J.P. and Oort, A.H. (1992) Physics of Climate. Springer-Verlag, New York, Berlin, Heidelberg.

[53]   Hilbert, D. and Cohn-Vossen, S. (1952) Geometry and the Imagination. Chelsea Publishing Company, New York.

[54]   Lass, H. (1950) Vector and Tensor Analysis. McGraw-Hill, New York, Toronto, London.

[55]   Alekseenko, S.V., Kuibin, P.A. and Okulov, V.L. (2007) Theory of Concentrated Vortices. Springer, Berlin, Heidelberg, New York.

[56]   Ross, H.K., Cooney, J., Hinzman, M., Smock, S., Sellhorst, G., Dlugi, R, Mölders, N. and Kramm, G. (2014) Wind Power Potential in Interior Alaska from a Micrometeorological Perspective. Atmospheric and Climate Sciences, 4, 100-121.

[57]   Goorjian, P.M. (1972) An Invalid Equation in the General Momentum Theory of the Actuator Disc. AIAA Journal, 10, 543-544.

[58]   Betz, A. (1926) Wind-Energie und Ihre Ausnutzung durch Windmühlen. Vandenhoeck & Ruprecht, Göttingen, Germany. (In German)

[59]   Rankine, W.J.M. (1865) On the Mechanical Principles of the Action of Propellers. Transaction of the Institute of Naval Architects, 6, 13-39.

[60]   Froude, W. (1878) On the Elementary Relation between Pitch, Slip and Propulsive Efficiency. Transaction of the Institute of Naval Architects, 19, 22-33.

[61]   Froude, R.E. (1889) On the Part Played in Propulsion by Difference in Pressure. Transaction of the Institute of Naval Architects, 30, 390-405.

[62]   Wilson, R.E. and Lissaman, P.B.S. (1974) Applied Aerodynamic Performance of Wind Power Machines. Oregon State University, Corvallis.

[63]   Snel, H. (1998) Review of the Present Status of Rotor Aerodynamics. Wind Energy, 1, 46-69.<46::AID-WE3>3.3.CO;2-0

[64]   Mathew, S. (2006) Wind Energy: Fundamentals, Resource Analysis, and Economics. Springer.

[65]   Sharpe, D. (2004) A General Momentum Theory Applied to an Energy-Extracting Actuator Disc. Wind Energy, 7, 177-188.

[66]   Trefftz, E. (1921) Zur Prandtlschen Tragflächentheorie. Math. Ann., 82, 306-319.

[67]   Joukowsky, N.E. (1912) Vortex Theory of Screw Propeller, I. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 16 (1), 1-31 (In Russian). French translation in: Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929) 1-47 (as cited by [12]).

[68]   Joukowsky, N.E. (1914) Vortex Theory of Screw Propeller, II. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 17 (1), 1-33 (In Russian). French translation in: Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929) 48-93 (as cited by [12]).

[69]   Joukowsky, N.E. (1915) Vortex Theory of Screw Propeller, III. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 17 (2), 1-23 (In Russian). French translation in: Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929) 94-122 (as cited by [12]).

[70]   Joukowsky, N.E. (1918) Vortex Theory of Screw Propeller, IV. Trudy Avia Raschetno-Ispytatelnogo Byuro, no 3, 1-97 (In Russian). French translation in: Théorie tourbillonnaire de l’hélice propulsive (Gauthier-Villars, Paris, 1929) 123-198 (as cited by [12]).

[71]   Betz, A. (1919) Schraubenpropeller mit geringstem Energieverlust—Mit einem Zusatz von L. Prandtl. Nachrichten d. K. Gesellschaft d. Wissenschaften, Göttingen, Math.-phys. Klasse, 193-217. (In German)

[72]   Goldstein, S. (1929) On the Vortex Theory of Screw Propellers. Proceedings of the Royal Society of London. Series A, 123, 440-465.

[73]   Okulov, V.L. (2004) On the Stability of Multiple Helical Vortices. Journal of Fluid Mechanics, 521, 319-342.

[74]   Haltiner, G.J. and Martin, F.L. (1957) Dynamical and Physical Meteorology. McGraw-Hill, New York, Toronto, London.

[75]   Thomas, T.Y. (1961) Concepts from Tensor Analysis and Differential Geometry. Academic Press, New York, London.

[76]   Sokolnikoff, I.S. (1964) Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. 2nd Edition, John Wiley & Sons, New York, London, Sydney.

[77]   Eisenreich, G. (1971) Vorlesungen über Vektorund Tensorrechnung. BSB B.G. Teubner Verlagsgesellschaft, Leipzig. (In German)

[78]   Teichmann, H. (1973) Physikalische Anwendungen der Vektor-und Tensorrechnung. Bibliographisches Institut Mannheim, Wien, Zürich. (In German)

[79]   Fortak, H. (1967) Vorlesungen über theoretische Meteorologie-Kinematik der Atmosphäre. Freie Universität Berlin, Institut für Theoretische Meteorologie. (In German)