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86 Rotor
fV . The mass flux through the rotor disk is ˙m = ρAU = ρA∞U∞, where U and U∞ are respectively the
total velocity magnitudes at the fan and in the far wake:
U 2 =(fVxV cos α) 2 +(fVzV sin α + v) 2
U 2 ∞ =(V cos α) 2 +(V sin α + w) 2
Mass conservation (fA = A/A∞ = U∞/U) relates fA and fW . Momentum and energy conservation give
T = ˙mw = ρAU∞w/fA = ρAUfW v
P = 1
˙mw (2V sin α + w) =T V sin α +
2 w
2
With these expressions, the span of the lifting system in forward flight is assumed equal to the rotor
diameter 2R. Next it is required that the power equals the rotor induced and parasite loss:
P = Trotor(fVzV sin α + v) =TfT (fVzV sin α + v)
In axial flow, this result can be derived from Bernoulli’s equation for the pressure in the wake. In
forward flight, any induced drag on the duct is being neglected. From these two expressions for power,
Vz + fW v/2 =fT (fVzVz + v) is obtained, relating fT and fW . With no duct (fT = fVx = fVz =1),
the far-wake velocity is always w =2v, hence fW =2. With an ideal duct (fA = fVx = fVz =1), the
far-wake velocity is fW =1. In hover (with or without a duct), fW = fA =2fT and v = 2/fW vh. The
rotor ideal induced power is Pideal = Tw/2=fDTv, introducing the duct factor fD = fW /2.
For a ducted fan, the thrust CT is calculated from the total load (rotor plus duct). To define the duct
effectiveness, either the thrust ratio fT = Trotor/T or the far-wake area ratio fA = A/A∞ is specified
(and the fan velocity ratio fV ). The wake-induced velocity is obtained from the momentum theory result
for a ducted fan: λ 2 h =(fW λi/2) (fVxμ) 2 +(fVzμz + λi) 2 . If the thrust ratio fT is specified, this can
be written
fVzμz + λi =
sλ2 h /fT
(fVzμz + λi) 2 μz
+
+(fVxμ) 2 fT
In this form, λi can be determined using the free-rotor expressions given previously: replacing λ2 h , μz,
μ, λ with respectively λ2 h /fT , μz/fT , fVxμ, fVzμz + λi. Then from λi the velocity and area ratios are
obtained:
fW =2 fT − (1 − fT fVz) μz
λi
fA =
μ 2 +(μz + fW λi) 2
(fVxμ) 2 +(fVzμz + λi) 2
If instead the area ratio fA is specified, it is simplest to first solve for the far-wake velocity fW λi:
μz + fW λi =
sλ2 h2fA
(μz + fW λi) 2 + μz
+ μ2 In this form, fW λi can be determined using the free-rotor expressions given previously: replacing λ2 h , λ
with respectively λ2 h2fA, μz + fW λi. The induced velocity is
(fVzμz + λi) 2 = 1
f 2 A
μ 2 +(μz + fW λi) 2 − (fVxμ) 2