Page 238 - Introduction to Naval Architecture

PROPULSION 223

The total torque is obtained by integration from the hub to the tip of
the blade.
The thrust power of the propeller will be proportional to TV a and the
shaft power to 'ZnNQ. So the propeller efficiency will be TV^/^jnNQ.
Correspondingly there is an efficiency associated with the blade
element in the ratio of the thrust to torque on the element. This is:

But from Figure 9.12,

This gives a blade element efficiency:

This shows that the efficiency of the blade element is governed by the
'momentum factor' and the blade section characteristics in the form of
the angles <p and /?, the latter representing the ratio of the drag to lift
coefficients. If /? were zero the blade efficiency reduces to the ideal
efficiency deduced from the momentum theory. Thus the drag on the
blade leads to an additional loss of efficiency.
The simple analysis ignores many factors which have to be taken into
account in more comprehensive theories. These include:

(1) the finite number of blades and the variation in the axial and
rotational inflow factors;
(2) interference effects between blades;
(3) the flow around the tip from face to back of the blade which
produces a tip vortex modifying the lift and drag for that region
of the blade.

It is not possible to cover adequately the more advanced propeller
theories in a book of this nature. For those the reader should refer to
2
a more specialist treatise. Theory has developed greatly in recent
years, much of the development being possible because of the
increasing power of modern computers. So that the reader is familiar
with the terminology mention can be made of:

(1) Lifting line models. In these the aerofoil blade element is replaced
with a single bound vortex at the radius concerned. The strength