Momentum-Blade Element Theory

The momentum-blade element theory is one means around this difficulty. If we assume a, and the drag-to-lift ratio to be small, then Ve — Vn and

Equation 6.25 can be written approximately for В blades as dT = ^ V2rca(P – ф – a,) cos ф dr

Applying momentum principles to the differential annulus and letting w ~ Vra„ we can also write, for dT,

r

Equating these two expressions for dT/dr gives the following quadratic for a,.

2 , A, <raVr o-aVr + (x+ 8?Vr/ 8?V^ ~Ф)~°

where:

Given the geometry, forward speed, and rotational speed of a propeller, Equation 6.28 can be solved for af. Equation 6.25a and 6.25b can then be numerically integrated. Using equations 6.26 and 6.27 to give the thrust and torque.

The thrust and power of a propeller are normally expressed in coefficient form. These thrust and power coefficients are defined in various ways, depending on what particular reference areas and velocities are used. Test results on propellers almost always define the thrust coefficient, CT, and power coefficient, Cp, as follows.

where n is the rotational speed in revolutions per second and D is the propeller diameter. The thrust, power, p, and D must be in consistent units. For this. convention, one might say that nD is the reference velocity and D2 is the reference area.

One would expect these dimensionless coefficients to be a function only of the flow geometry (excluding scale effects such as Mach number and Reynolds number). From Figure 6.9, the angle of the resultant flow, ф, is seen to be determined by the ratio of V to tor.

ф = tan-1 —

(ОГ

This can be written as

ф = tan ‘ —

TTX

The quantity, J, is called the advance ratio and is defined by

Thus, Ct and Cp are functions of /.

In a dimensionless form, Equations 6.25 and 6.26 can be combined and expressed as

CT = 7Г f (J2+ 7t2x2)(t[Ci cos (ф + a,) — C, j sin (ф + a,)] dx (6.32a) » Jxh

and, since P = (oQ,

Cp = f 7rx(J2 + тт2х2)а[Сі sin (ф + a,) + Q cos (ф + a,)] dx (6.32b) о JXh

xh is the hub station where the blade begins. xh is rather arbitrary, but СУ and Cp are not too sensitive to its value.

To reiterate, one would be given D, V, p, and n. Also, c and /3 would be given as a function of x. At a given station, x, a, is calculated from Equation 6.28. This is followed in order by C, and Cd and, finally, dCTldx and dCpldx. These are then integrated from jch to 1 to give Ct and Cp.

Given J and having calculated CT and Cp, one can now calculate the propeller efficiency. The useful power is defined as TV and P is, of course, the input power. Thus,

TV

т)=Ч – (6.33)

In terms of CT, Cp, and J, this becomes

(6.34)