Kinetic limits of wind propulsion while facing the wind - Sycomoreen
motion between the 4 previous entities. One will suppose that the ground (1) is immobile and that the speeds are positive by convention relatively to Rg :.
Kinetic limits of wind propulsion while facing the wind Precisions on http://sycomoreen.free.fr calculated notably for the Pascal HA PHAM’s ROBIPLAN turbine
Setting of the context One considers a mobile which moves facing the wind with the only power of the wind via a turbine. There is 4 entities in relation : 3 V 1. The ground which supports the motion. vent 2. The mobile which translates itself 4 3. The turbine mounted on the mobile 4. The wind crossing the turbine. Vmob 2 ex All these elements get speeds relatively to a fixed O referential Rg which is supposed galilean. In this 1 survey, the reference ( O, ex ) is the only axis of motion between the 4 previous entities. One will suppose that the ground (1) is immobile and that the speeds are positive by convention relatively to Rg : - Vmob absolute speed of the mobile (2) - Vwind absolute speed of the wind (4)
Calculation of the limits A mobile moves against the wind with the only power of it extracted from a wind-turbine. The drag of the turbine is taken in account as the only strentgh which works against the motion and one supposes too that the recovered power from the turbine is totally and perfectly transmitted to the mobile. 1 3 The kinetic incidental power on the turbine is : Pkin = ρ S (Vmob + Vwind ) 2 The mechanical recovered power is : Pmot = η Pkin =
η
ρ S (Vmob + Vwind ) where η is the wind-output 3
2 2 The dragging strentgh coming from the slowing of the wind is : Ft = ρ S (Vmob + Vwind ) (1 − k )
where k is the slowing factor, while the wind is crossing the turbine, its speed collapse relatively to the mobile.
The braking power due to the drag of the turbine is : Pt = FV t mob = ρ SVmob (Vmob + Vwind ) (1 − k ) 2
The top speed limit is attempt while the mechanical power is the same as the braking power :
η
2
ρ S (Vmob + Vwind ) = ρ SVmob (Vmob + Vwind ) (1 − k ) 3
ROBIPLAN’s limits : η is optimum when the wind is slowed 2 times (k = 1/2) 1 2 2 4 For MONOROBI : η = and Vmob = Vwind For BIROBI : η = and Vmob = Vwind 4 3 10 10 9 48 is optimum when the wind is slowed 3 times (k = 1/3) Vmob = Vwind 16 100 4 16 Ultimate limit : η = maximum of Betz when the wind is slowed 3 times (k = 1/3) Vmob = Vwind 5 27
Limit of tri-blade turbine : η =
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