Page 72 - Fluid Power Engineering

50 Chapter Four

According to the actuator disk theory from Chapter 2, Eq. (2-32), the
power extracted by the turbine in an annulus of area δA r at distance
r from the center is:

1 3 2
δP = ρδA r v 0 4a(1 − a) (4-14)
2
Equating (4-10) and (4-14) gives:

2
4 (1 − a) λ b = 4a(1 − a) 2
r
a(1 − a) (4-15)
2
λ =
r
b
λ r is a function of angular speed, radius, and upstream wind speed,
so it is not a function of a and b. Differentiating by b gives:
da λ 2 r a(1 − a)
= = (4-16)
db 1 − 2a b(1 − 2a)

From Eqs. (4-12) and (4-16):

1
a =
3

This is the same axial flow induction factor that was derived from the
actuator disk theory in Chapter 2.
Substituting Eq. (4-15) in (4-13) yields:

8ra(1 − a) 2
δC P
=
δr R 2
(4-17)
R
8ra(1 − a) 2 16
2
C P = dr = 4a(1 − a) =
R 2 27
0
The maximum power coefficient as computed using the rotor disk the-
ory is the same as the actuator disk theory. The rotor disk theory does
not alter the axial component of velocity, but introduces a tangential
component in the wake (after the wind passes through the turbine).
Question: How can the maximum power coefficient be the same
eventhoughadditionalkineticenergy(intheformoftangentialspeed)
has been imparted to air at the rotor?
The explanation lies in the observation that the tangential speed
causes additional pressure drops, which is in addition to the pressure