# Propeller Theory

The fundamentals of propeller performance start with the idealized consideration of momentum theory. Its practical application in the industry is based on the subsequent “blade-element” theory. Both are presented in this section, followed by engineering considerations appropriate to aircraft designers. Industrial practices still use a propeller that is supplied by the manufacturer and wind-tunnel-tested generic charts and tables to evaluate its performance. Of the various forms of propeller charts, two are predominant: the NACA method and the Hamilton Standard (i. e., propeller manufacturer) method. This book prefers the Hamilton Standard method used in the industry ([16]). For designing advanced propellers and propfans to operate at speeds greater than Mach 0.6, CFD is important for arriving at the best compromise, substantiated by wind-tunnel tests. CFD employs more advanced theories (e. g., vortex theory).

Momentum Theory: Actuator Disc

The classical incompressible inviscid momentum theory provides the basis for propeller performance ([21]). In this theory, the propeller is represented by a thin actuator disc of area, A, placed normal to the free-stream velocity, V0. This captures a stream tube within a CV that has a front surface sufficiently upstream represented by subscript “0” and sufficiently downstream represented by subscript “3” (Figure 10.33). It is assumed that thrust is uniformly distributed over the disc and the tip effects are ignored. Whether or not the disc is rotating is irrelevant because flow through it is taken without any rotation. The station numbers just in front and aft of the disc are designated as 1 and 2.

The impulse given by the disc (i. e., propeller) increases the velocity from the free-stream value of V0, smoothly accelerates to V2 behind the disc, and continues to accelerate to V3 (i. e., Station 3) until the static pressure equals the ambient pressure, p0. The pressure and velocity distribution along the stream tube is shown in Figure 10.33. There is a jump in static pressure across the disc (from p1 to p2), but there is no jump in velocity change.

Newton’s law states that the rate of change of momentum is the applied force; in this case, it is the thrust, T. Consider Station 2 of the stream tube immediately behind the disc that produces the thrust. It has a mass flow rate, hi = pAdiscV2, and

Figure 10.33. Control volume showing the stream tube of the actuator disc

the change of velocity is AV = (V3 – V0). This is the reactionary thrust experienced at the disc through the pressure difference multiplied by its area, A.

Thrust produced by the disc T = the rate of the change of momentum = m AV

= P Adisc x (V – Vo)xV>

= pressure across the disc x Adisc = Adisc X (P2 – Pi) (10.23)

Equation 10.23 now can be rewritten as:

P(V3 – V0) x V2 = (p2 – pi) (10.24)

The incompressible flow in Bernoulli’s equation cannot be applied through the disc imparting the energy. Instead, two equations are set up: one for conditions ahead of and the other aft of the disc. Ambient pressure, p0, is the same everywhere.

Ahead of the disc:

P0 + 1/2P V2 = P1 + 1/2P V12 (10.25)

Aft of the disc:

P0 + 1/2P V32 = P2 + 1/2P V22 (10.26)

Subtracting the front relation from the aft relation:

1/2P (V32 – V>2) = (P2 – P1) X 1/2P (V22 – V2) (10.27)

Because there is no jump in velocity across the disc, the last term is omitted. Next, substitute the value of (p2 – p1) from Equation 10.24 in Equation 10.25:

1/2(V32 – V? = (V3 – V0) X V2 or (V3 + V0) = 2V2

Note that (V3 – V0) = AV, when added to Equation 10.26, gives 2V3 = 2V2 + A V, or:

Using conservation of mass, A3V3 = AVi, Equation 10.23 becomes:

T = pAdiscV1 X (V3 – V0) = Adisc(p2 — P1) or (P2 — P1) = p V1 X (V3 — V0) (10.30)

This means that half of the added velocity, AV/2, is ahead of the disc and the remainder, AV/2, is added aft of the disc.

Using Equations 10.29 and 10.30, thrust Equation 10.23 can be rewritten as:

T = AdscPV1 x (V3 — V0) = AdscP(V0 + AV/2) x AV (10.31)

Applying this to an aircraft, V0 may be seen as the aircraft velocity, V, by dropping the subscript “0”. Then, the useful work rate (power, P) on the aircraft is:

P = TV (10.32)

For the ideal flow without the tip effects, the mechanical work produced in the system is the power, Pideai, generated to drive the propeller force (thrust, T) times velocity, V1, at the disc.

Pideai = T(V + AV/2) (the maximum possible value in an ideal situation) (10.33) Therefore, ideal efficiency:

Пі = P/ Pideai = (TV)/[T(V + AV/2)] = 1/[1 + (AV/2V)] (10.34)

The real effects have viscous, propeller tip effects and other installation effects. In other words, to produce the same thrust, the system must provide more power (for a piston engine, it is seen as the BHP, and for a turboprop, the ESHP), where ESHP is the equivalent SHP that converts the residual thrust at the exhaust nozzle to HP, dividing by an empirical factor of 2.5. The propulsive efficiency as given in Equation 10.4 can be written as:

np = (TV)/[BHP or ESHP] (10.35)

This gives:

np/Пі = {(TV)/[BHP or SHP]}/{1/[1 + (AV/2V)]}

= {(TV)[1 + (A V/2V)]/[BHP or SHP]} = 85 to86% (typically) (10.36)

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