Page 88 - Fluid Power Engineering

66 Chapter Five

If the pressure drop term above containing b is ignored, then:

2
4 sin β
a = 1 + 1 (5-15)
σC x
Torque is equal to the rate of change of angular momentum, therefore:
1 2
δQ = ρv Nc(C L sin β − C D cos β)rδr = δ ˙m(2ωrb)r
res
2
(5-16)
= ρv 0 (1 − a)2πrδr(2ωrb)r (5-17)
Using Eqs. (5-9) and (5-12), the above equation becomes:
v 2
res σC y = 4b(1 − a) (5-18)
v 0 ωr
(1 − a)(1 + b)
σC y = 4b(1 − a) (5-19)
sin β cos β

4 sin β cos β
b = 1 − 1 (5-20)
σC y
Equations (5-15) and (5-20) for a and b require an explanation:
In the two equations, β is a function of a and b, as indicated

in Eq. (5-2).
σ, blade solidity is a function of radius r

C x and C y are functions of β, v res
The values of a and b are computed iteratively; an algorithm for
2
solving this is presented by Hansen. Since pitch changes along the
length of the blade, the solution algorithm involves solving the above
equations for smaller slices along the length of the blade. Two correc-
tions are required for the equations that solve for a and b, because of
the assumptions made thus far:

1. Prandtl’s Tip Loss Factor, which corrects for the assumption
of infinite number of blades
2. Glauert correction when a is high. The momentum theory
used above does not apply when a ≥ 0.5, as discussed in Eq.
1
(2-25). In fact, correction factors are applied for a > .
3
2
The corrections are described by Hansen. Having computed a and
b, expressions for torque, power and the power coefficient may be
derived.
1 2 3 1 3
Torque Q = ρv π R λ 8b (1 − a) μ dμ (5-21)
0
2
0