Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 52

The air that passes through the disc undergoes an overall change in velocity, U_∞ − U_W, and a rate of change of momentum equal to the overall change of velocity times the mass flow rate:

(3.3)

The force causing this change in momentum comes entirely from the pressure difference across the actuator disc and the axial component of the pressure acting on the curved surface of the streamtube. This latter pressure is usually assumed to be ambient and therefore to give zero contribution, without further explanation. In fact this pressure is different from ambient due to the axial variation of velocity along the streamtube, but the integral of its axial component from far upstream to far downstream can be shown to be exactly equal to zero, the streamwise contribution upstream of the actuator disc exactly balancing the downstream contribution that opposes the stream (Jamieson 2018).

Therefore,

(3.4)

To obtain the pressure difference (p_D⁺ − p_D⁻), Bernoulli's equation is applied separately to the upstream and downstream sections of the streamtube: separate equations are necessary because the total energy is different upstream and downstream. Bernoulli's equation states that, under steady conditions, the total energy in the flow, comprising kinetic energy, static pressure energy, and gravitational potential energy, remains constant provided no work is done on or by the fluid. Thus, for a unit volume of air,

(3.5a)

Upstream, therefore, we have

(3.5b)

Assuming the flow speed to be at low Mach number M (typically M < 0.3 is sufficient), it may be treated as incompressible (ρ_∞ = ρ_D) and to be independent of buoyancy effects (ρgh_∞ = ρgh_D) then,

(3.5c)

Similarly, downstream,

(3.5d)

Subtracting these equations, we obtain

(3.6)

Equation (3.4) then gives

(3.7)

and so,

(3.8)

That is, half the axial speed loss in the streamtube takes place upstream of the actuator disc and half downstream.