Theoretical calculation of the power of wind turbine or tidal turbine Pierre Lecanu, Joel Breard, Dominique Mouazé To cite this version: Pierre Lecanu, Joel Breard, Dominique Mouazé. Theoretical calculation of the power of wind turbine or tidal turbine. 09. <hal-09856v> HAL Id: hal-09856 https://hal.archives-ouvertes.fr/hal-09856v Submitted on 5 Jan 09 (v), last revised Jan 09 (v) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Theoretical calculation of the power of wind turbine or tidal turbine Pierre Lecanu, AjcInnov 7 chemin du Mont Desert 4400 Esquay sur Seulles Joel Breard, LOMC CNRS-UNILEHAVRE-Normandie Univ Dominique Mouaze, MC CNRS-UNICAEN-UNIROUEN-Normandie Univ January 5, 09 Inventor of the active lift turbine concept normandajc@laposte.net
Abstract The subject of this article is to define the power of a wind turbine or marine current turbine. The German mathematician A. Betz [] demonstrates that the power P of such turbine will reach the followinf law (cf. annex ): P = F V = [ C p a ρ S V ] V = C p ρ S V 3 with P turbine the turbine power, C p the power coefficient, ρ the density, V the velocity. According to the work of Betz, a kinetic energy approach shows that the maximum power coefficient can not exceed a maximum of 6 7 Cp maxi = Cp Betz = 6 7 Many people consider that Betz limit is a stated law. This article will attempt to demonstrate that this theoretical power coefficient can be reached and exceeded. The theory of the Betz limit is correct, it is based on the calculation of the kinetic energy. Transforming potential energy into kinetic energy provides additional motor energy. Power calculation. Stress in turbine blade : The blades are subjected to stresses due to a thrust force of wind F x = C x ρ S V The power of this thrust force is equal to P x = V F x with V = a V and a ( cf. annex ) P x = a C x ρ S V 3 with V the velocity at the position of the turbine. The total power is equal to P T = P + P x = (C p + a C x ) ρ S V 3
with (C p + a C x ) C p = 4 a ( a) C x = a ( C p) 0.5 a C x C p 0 0.5 The power P should be superior to the thrust power F x in order that additional energy to be produced. P P x C p a C x Previous equation is resolved by this way : C p = a C x 4 [4a ( a) ] = 0 4 [4a ( a) ] = (x )(x a )(x a 3 ) a = a = + (0)0.5 a 3 = (0)0.5 P P x if a (0)0.5 (= 0.8)
The calculations of the C p maximum (Betz coefficient) and the corresponding C x are following dc p da = d(a C x) da = 0 a ( 3 a) = 0 as a 0.5 a = 3 a = 3 C p = Cp Betz = 6 7 60% C x = 6 ( 3 7 ) 60%+ In the case of horizontal wind turbines (HAWT, fast wind turbine type), the stresses in the blades for a defined wind speed, are constant. dσ dβ = 0 σ Stress in turbine blade β rotation angle of the blades. In fact, some variations of the stresses are existing due to gravitational forces and the differencial velocity within the boundary layer depending on the elevation. In the case of vertical axis turbines (VAWT, Darrieus type) the blade and arm stresses are depending on the rotation angle of the blades (for a given wind speed). dσ dβ 0 π σdβ = ɛ (ɛ small) π 0 During a half-turn, the arms are submitted to compression stresses whereas extending stresses are dominant during the next half-turn. In the case of a HAWT, it is not possible to convert these stresses into additional energy as the stresses will be kept constant for a given wind speed. Alternative stresses, encountered in a vertical axis wind turbine will allow the extraction of additionnal energy.
. HAWT-VAWT comparison Following the work of E.Hau, the power coefficient of different turbines is compared (A performance of 0.7 is applied for the supplementary energy recovery system.) W ith a = 3 C p 60% a C x 40% case Coef. HAWT VAWT(Darrieus) VAWT(with conversion) perfect C p 60% 60% 60% perfect a C x 0% 0% 40% perfect C p + a C x 60% 60% 00% in practice C p 0.8 60% 48% 0.7 60% 4% 0.7 60% 4% in practice a C x 0% 0% 0.7 0.7 40% 0% in practice C p + a C x 48% 4% 6% gain / HAWT + 0% % + 9% W ith a 0.8 C p 50% a C x 50% case Coef. HAWT VAWT(Darrieus) VAWT(with conversion) in practice C p 0.8 50% 40% 0.7 50% 35% 0.7 50% 35% in practice a C x 0% 0% 0.7 0.7 50% % in practice C p + a C x 40% 35% 5% gain / HAWT + 0% % + 30%
3 Conclusion : For a Horizontal Axis Turbine HAWT the power is equal to P HAW T = C p ρ S V 3 with 0.5 a For a vertical axis turbine VAWT the power is equal to P V AW T = (C p + a C x ) ρ S V 3 with 0.5 a (0)0.5 ( 0.8) a = V a C x C p C p 6 V 7 C x a with C p = 4 a ( a) C x = a [ 4 a ( a)] Compared to a HAWT turbine, the gain of a VAWT Turbine with an energy recovery system is in practice from 0% to 30%. The stress conversion into an additionnal recovered energy consists of a potential energy transformed into a kinetic energy a C x = 0 if the turbine is not equipped with an additional energy recovery system (Darrieus turbine) a C x = λ R b R for a Active lift turbine [] [3] λ Tip speed ratio R Turbine radius R b Connecting rod radius
References [] Betz A. Das maximum der theoretisch moglichen ausnutzung des windesdurch windmotoren. Zeitschrift fur das gesamte Turbinenwesen, (6):307 309, 90. [] Pierre Lecanu, Joel Breard, and Dominique Mouazé. Simplified theory of an active lift turbine with controlled displacement. working paper or preprint, April 06. [3] Pierre Lecanu, Joel Breard, and Dominique Mouazé. Operating principle of an active lift turbine with controlled displacement. working paper or preprint, July 08.
Annexe Maximum wind power (from kinetic energy) de c E c = m V dm V + m dv = de c = ṁ V = ρ S V 3 V Wind speed at the turbine level Force applied by the wind on the rotor dv = 0 F = m dv = ṁ V = ρsv (V V wake ) V wake streamwise velocity in the far wake P = E t = mv mv wake t From theses equalities P = F V = ρsv (V V wake ) = ṁ(v Vwake) = ρsv (V V wake) V = V = V + V wake F = ρsv (V V wake ) = ρs(v V wake) defining a = V V P = F V = ρsv (V V wake ) V wake = V ( a ) as V wake 0 a defining power coefficient C p = P = 4 a ( a) ρsv 3 P ρsv 3 Search of maximum power coefficient = 4 a ( a) dc p da = 0 a ( 3 a) = 0 a = 0 or a = 3 a = C p = 6 3 7 = 0.593
The maximum power coefficient C pmaxi is defined by Betz C pmaxi = C pbetz = 6 7 60% The maximum power of the is P = ρs V 3 S = S V V = a S The power of the turbine is P = C p a P = C p ρs a The maximum power of the turbine is V 3 = C p ρsv 3 P max = C p Betz 3 P = 8 9 P = C p Betz ρsv 3 = 6 7 ( ρsv 3 ) P max = C p Betz ρsv 3 ()
Additional recovery power (from potential energy) The creates stresses in the blade. They are due to a surface strength. The energy of this force is E p = V olume F s S = m ρ F s S F s surface strength For HAWT horizontal wind turbines (fast wind turbine type), the surface strength F s are constant. de p = 0 For a VAWT, the surface strength the force depends on the time or the rotation angle F s (t) or F s (β) β = ω t ω = d β angular frequency de p 0 and π π 0 E p (β)dβ = ɛ (ɛ small) As F s = = C x ρ V E p = m C x S V V speed at the level turbine The power is P p = de p P p = de p = dm C x S V + m C x S V dv = 0 P p = a C x ρ S V 3 with a = V V () π π 0 E p (β)dβ = ɛ (ɛ small) E p max E p min the power depends on a potential energy difference P p = E p t T = π ω P p E p max E p min T P p E p max π ω
for a half-turn in particular As de p E p (β) R dβ = de p π R E p max = de p dβ π = de p = a C x ρ S V 3 and P p E p max π P p a C x ρ S V 3 π ω ω