# Actuator Disc Theory

Perhaps the simplest approach to the aerodynamic modelling of the wind turbine is the actuator disc theory which ignores the flow rotational effects. In this case, one may integrate the increment of the axial forces dFx, across the cross-sectional area of the turbine disc, and obtain

Fx = 1 p(V1 – V4)(V1 + V4)pR2. (4.3.7)

For steady-state flow, the mass flow rate across the wind turbine disc can be obtained using the equation,

m = pAV2. (4.3.8)

Applying the conservation of linear momentum equation on both sides of the actuator disc and using the fact that the thrust is the product of the pressure drop and the area gives,

Fx = pAV2(V1 – V4) = ADp = 2p(V1 – Vt)(V1 + V4)A. (4.3.9)

Hence,

V2 = 2 (V1 + V4) (4.3.10)

and

V4 = (2V2 – V1) = V1(1 – 2a). (4.3.11)

Hence, in the case of an annular element of the stream tube,

dFx = (P2 – P3)dA = 2p(V1 – Vt)(V1 + V4)2nrdr

The expression simplifies to,

dFx = 2pVj24a (1 — a)2nrdr. (4.3.13)

Thus, for the case of the actuator disc model,

Fx = 1 pVj24a(1 — a)nR2. (4.3.14)

The power absorbed by the wind turbine is

P = FxV2 = 1 pV134a(1 — a)2 pR2. (4.3.15)

The power in the wind is

Pwind = 2 pV/pR2. (4.3.16)

The power coefficient is

P2

CP = ——— = 4a(1 — a)2. (4.3.17)

Pwind

The axial force or thrust coefficient is

Ct = 4a(1 — a). (4.3.18)

Updated: October 27, 2015 — 12:09 pm