Piet Lammertse on propeller theory

ROTATION

9 Rotational momentum loss

9.1 Local loss and local efficiency

Compared to the axial actuator disk, the pitch angle of the blade elements causes an extra induction u, at right angles to the axial inducton v. This leaves extra kinetic energy in the wake which shows up as an extra momentum loss.

Using the far wake values w and w ′ in figure 7.2, we have :

w ′ = \frac{w}{cos φ}

(9.1)

With the definition w ≡ b . V from (4.1), the kinetic energy dP_m left behind in the far wake of a ring becomes :

d P_{m} = ½ d m^{•} . {(w ′)}^{2} = ½ d m^{•} . V^{2} . \frac{b^{2}}{{cos}^{2} φ}

(9.2)

Dividing by the local thrust (7.5) gives us the local momentum loss ratio :

\frac{d P_{m}}{d T . V} = \frac{½ d m^{•} . V^{2} . \frac{b^{2}}{{cos}^{2} φ}}{d m^{•} . V . b . V} = ½ . \frac{b (x)}{{cos}^{2} φ}

(9.3)

This local loss depends on the local induction ratio, but not on the mass flow through the ring, since dm^● drops out of (9.3). Compared to the axial loss ratio b of (4.16), the rotation increases the momentum loss by a factor of 1 ⁄ cos² φ.

We can reach the same result by the non-dimensional route. Applying the steps from (7.15) tot (7.17) to the ring momentum loss (7.2) gives :

d P_{m} ′ = \frac{2 x . d x .}{X^{2}} . (1 + ½ b) . \frac{½ b^{2}}{{cos}^{2} φ}

(9.4)

Dividing by ( 7.14 ) for the ring thrust, first off the 2x . dx / X² terms cancel, leaving :

\frac{d P_{m} ′}{d T ′} = \frac{1}{b} . \frac{(1 + ½ b)}{(1 + ½ b)} . \frac{½ b^{2}}{{cos}^{2} φ}

(9.5)

Like in the dimensional (9.3), one term of b and both the heavy loading terms in dm ′ cancel, leaving only :

\frac{d P_{m} ′}{d T ′} = \frac{½ b}{{cos}^{2} φ}

(9.6)

Once again, this ratio is independent of the local mass flow. By (3.9) and (3.10), the ring efficiency becomes :

d η_{m} = \frac{1}{1 + \frac{d P_{m} ′}{d T ′}} = \frac{1}{1 + ½ . \frac{b (x)}{{cos}^{2} φ}}

(9.7)

The symbol dη_m, with a "d" in front, is used here loosely for the local momentum efficiency, but is not an infinitesimal. Local ratios and efficiencies cannot be integrated over the rings like the overall thrust and power can.