Reaching the betz limit experimentally and numerically

Document Type : Research Paper


1 College of Mechanical Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

2 Waterloo Institute for Sustainable Energy (WISE), University of Waterloo, Waterloo, ON, N2L 3G1, Canada


The Betz theory expresses that no horizontal axis wind turbine can extract more than 16/27 (59.3%) of the kinetic energy of the wind. The factor 16/27 (0.593) is known as the Betz limit. Horizontal Axis wind turbine designers try to improve the power performance to reach the Betz limit. Modern operational wind turbines achieve at peak 75% to 80% of the Betz limit. In 1919, Albert Betz used an analytical method to derive the Betz limit. He derived momentum equations of an Actuator Disc (AD) in the stream. In this research, an experimental and a numerical setup based on the Actuator Disc (AD) have been designed and tested to reach the Betz limit. A Plexiglass screen with the porosity of 0.5 mimics the wind turbine rotor. For the numerical study, a 2D flow filed is considered. The results of both experimental and numerical methods agree well with the analytical results of the Betz theory. From the experimental and numerical results, the relative errors in comparison with the Betz limit (which is 16/27) are 0.16% and 1.27%, respectively. The small amount of errors shows the possibility of reaching the Betz limit using either experimental or numerical methods. This approach can be used for modeling ideal wind turbines, ideal rotating devices or ideal wind farms either numerically or experimentally and gives the maximum possible power extractions; thus, any improvement to the performance of a system can be made by this method.


[1] Global Wind Statistics (GWEC) report 2017.
[2] Rankine, W. J., Transactions, Institute of Naval Architects, Vol. 19, P. 47, 1878.
[3] Froude, W., Transactions, Institute of Naval Architects, Vol. 6, P. 13, 1865.
[4] Froude, R. E., Transactions, institute of Naval Architects, Vol. 30, P. 390, 1889.
[5] Betz, A. (1920). Theoretical limit for best utilization of wind by wind-motors. Magazine for the Entire Turbine System, 20, 307-309.
[6] Lanchester, F. W. (1915). A contribution to the theory of propulsion and the screw propeller. Naval Engineers Journal, 27(2), 509-510.
[7] Joukowsky NE. Windmill of the NEJ type. Transactions of the Central Institute for Aero-Hydrodynamics of Moscow, 1920. Also published in Joukowsky NE. Collected Papers Vol VI. The Joukowsky Institute for AeroHydrodynamics, Moscow: vol VI, 405–409, 1937 (in Russian).
[8] Tony Burton et al., (ed), Wind Energy Handbook, John Wiley and Sons 2001 ISBN 0471489972 page 65.
[9] Jiang, H., Li, Y., & Cheng, Z. (2015). Performances of ideal wind turbine. Renewable Energy, 83, 658-662.
[10] De Lellis, M., Reginatto, R., Saraiva, R., & Trofino, A. (2018). The Betz limit applied to Airborne Wind Energy. Renewable Energy, 127, 32-40.
[11] Aubrun, S., Devinant, P., & Espana, G. (2007, May). Physical modeling of the far wake from wind turbines. Application to wind turbine interactions. In Proceedings of the European wind energy conference, Milan, Italy (pp. 7-10).
[12] Theunissen, R., Housley, P., Allen, C. B., & Carey, C. (2015). Experimental verification of computational predictions in power generation variation with layout of offshore wind farms. Wind Energy, 18(10), 1739-1757.
[13] PM Sforza, S., & Smorto, M. (1981). Three-dimensional wakes of simulated wind turbines. AIAA Journal, 19(9), 1101-1107.
[14] Dighe, V. V., Avallone, F., & van Bussel, G. J. W. (2016). Computational study of diffuser augmented wind turbine using AD force method. International Journal of Computational Methods and Experimental Measurements,4(4), 522-531.
[15] Dighe, V. V., Avallone, F., Tang, J., & van Bussel, G. (2017). Effects of Gurney Flaps on the Performance of Diffuser Augmented Wind Turbine. In 35th Wind Energy Symposium (p. 1382).
[16] Aubrun, S., Loyer, S., Hancock, P. E., & Hayden, P. (2013). Wind turbine wake properties: Comparison between a non-rotating simplified wind turbine model and a rotating model.Journal of Wind Engineering and Industrial Aerodynamics,120, 1-8.
[17] Lignarolo, L. E. M., Ragni, D., Ferreira, C. J., & Van Bussel, G. J. W. (2016). Experimental comparison of a wind-turbine and of an actuator-disc near wake. Journal of Renewable and Sustainable Energy, 8(2), 023301.
[18] Aloui, F., Kardous, M., Cheker, R., & Nasrallah, S. B. (2013). Study of the wake induced by a porous disc. In 21 st Congress Francais de Mecanique.
[19] Charmanski, K., Turner, J., & Wosnik, M. (2014, August). Physical model study of the wind turbine array boundary layer. In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting collocated with the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels (pp. V01DT39A010-V01DT39A010). American Society of Mechanical Engineers.
[20] Manwell, J. F., McGowan, J. G., & Rogers, A. L. (2010). Wind energy explained: theory, design and application. John Wiley & Sons.