Piet Lammertse on propeller theory

INTRODUCTION

3 Thrust, power and efficiency

3.1 The thrust coefficient

This text will use the thrust from the propeller ( not yhe torque ) as the starting point of the design.

We will use two definitions of the non-dimensional thrust coefficient out of the many used in the literature. In analogy with the lift coefficient C_L of a wing, we define the non-dimensional thrust coefficient C_T. This is especially useful when applying the propeller in an aircraft :

C_{T} \equiv \frac{T}{½ ρ V^{2} . π R^{2}}

(3.1)

This definition is based on V instead of on the more usual Ω R. This reflects an emphasis on the requirements of the aircraft, rather than on the available torque or RPM of the powerplant.

The thrust coefficient C_T of an aircraft propeller in cruise is typically in the same ballpark as the lift coefficient C_L of the wing. Not by coincidence probably, the frontal area of a propeller is typically smaller than the wing area by roughly the same ratio as the aircraft's lift-to-drag ratio.

Inside classical propeller theory, it will be more convenient to divide by ρV ² instead of by ½ ρV ². We denote this by a prime. This "other" non-dimensional thrust coefficient is called T ' :

T' \equiv \frac{T}{ρ V^{2} . π R^{2}}

(3.2)

It follows that :

T' = ½ C_{T}

(3.3)

Mechanical power has the dimension of thrust times velocity. We will therefore make power non-dimensional dividing by V ³ instead of by V ² :

P' \equiv \frac{T}{ρ V^{3} . π R^{2}}

(3.4)

3.2 The net power

Mechanical power equals force times velocity. It also equals energy per second, and torque times rate of rotation ( with the rotation expressed in radians per second ).

For an airplane, the net output power that a propeller exerts on it is the thrust multiplied by the flying speed :

P = T . V

(3.5)

We can write (3.2) as a mechanical power instead of a force :

T' = \frac{T . V}{ρ V^{2} . π R^{3}}

(3.6)

We can therefore interpret T ′ not only as the non-dimensional thrust, but also as the non-dimensional net power T.V exerted by the propeller on the aircraft, by the definition of (3.4).

3.3 Power loss and efficiency

Efficiency is defined as net power output divided by gross power input. Intuitively, loss of efficiency is associated with a loss of output for a given input. However, losses in a propeller are more easily calculated as increases of the power input needed for a given power output.

The efficiency is still the same ratio : output over input. But the losses are now written as a fraction of the input, instead of as a fraction of the output. The two ratios are not the same, and we need to exercise some care.

The extra input still goes into energy lost to the wake. This is why we still call these extra inputs "losses".

3.4 Loss ratios

The momentum loss and the viscous loss as we define them are actually increases on the input power needed for a given output power T . V. We will call these increases P_m and P_v . The propeller's efficiency then becomes :

η \equiv \frac{P_{out}}{P_{in}} = \frac{T . V}{T . V + P_{m} + P_{v}}

(3.7)

We can divide top and bottom by the net output power T.V :

η = \frac{1}{1 + \frac{P_{m}}{T . V} + \frac{P_{v}}{T . V}}

(3.8)

We will call the two fractions in the denominator, the momentum loss ratio and the viscous loss ratio.

These power "losses" are really power increases on the net power. They are non-dimensional, and they can be expressed as ratios to the non-dimensional thrust and output power. By (3.4) and (3.6) we have :

\frac{P}{T . V} = \frac{P^{′}}{T^{'}}

(3.9)

η = \frac{1}{1 + \frac{{P'}_{m}}{T'} + \frac{{P'}_{v}}{T'}}

(3.10)