# Blade-Element Theory

The practical application of propellers is obtained through blade-element theory, as described herein. A propeller-blade cross-sectional profile has the same functions as that of a wing aerofoil – that is, to operate at the best L/D.

Figure 10.30 shows that a blade-element section, dr, at radius r, is valid for any number of blades at any radius, r. Because blades are rotating elements, their properties vary along the radius.

Figure 10.30 is a velocity diagram showing that an aircraft with a flight speed of V with the propeller rotating at n rps makes the blade element advance in a helical manner. VR is the relative velocity to the blade with an angle of attack a. Here, в is the propeller pitch angle, as defined previously. Strictly speaking, each blade rotates in the wake (i. e., downwash) of the previous blade, but the current treatment ignores this effect and uses propeller charts without appreciable error.

Figure 10.30 is the force diagram of the blade element in terms of lift, L, and drag, D, that is normal and parallel, respectively, to VR. Then, the thrust, AT, and force, AF (producing torque), on the blade element can be obtained easily by decomposing lift and drag in the direction of flight and in the plane of the propeller rotation, respectively. Integrating this over the entire blade length (i. e., nondimensionalized as r/R – an advantage applicable to different sizes) gives the thrust, T, and torque – producing force, F, of the blade. The root of the hub (with or without spinner) does not produce thrust, and integration is typically carried out from 0.2 to the tip, 1.0, in terms of r/R. When multiplied by the number of blades, N, this gives the propeller performance.

Therefore, propeller thrust:

1.0

T = Nx ATd(r / R) (10.37)

0.2

and force that produces torque:

1.0

F = Nx AFd(r / R) (10.38)

0.2

By definition, advance ratio: J = V/(nD)

It has been found that from 0.7r (i. e., tapered propeller) to 0.75r (i. e., square propeller), the blades provide the aerodynamic average value that can be applied uniformly over the entire radius to obtain the propeller performance.

It also can be shown that the thrust-to-power ratio is best when the blade element works at the highest lift-to-drag ratio (L/Dmax). It is clear that a fixed-pitch blade works best at a particular aircraft speed for the given power rating (i. e., rpm) – typically, the climb condition is matched for the compromise. For this reason, constant-speed, variable-pitch propellers have better performance over a wider speed range. It is convenient to express thrust and torque in nondimensional form, as follows. From the dimensional analysis (note that the denominator omits the 1/2): Nondimensional thrust,

Tc = Thrust/(p V2 D2)

Thrust coefficient,

Ct = Tc x J = Thrust x [V/(nD)]2/(pV2D2) = Thrust/(pn2D4) (10.39)

In FPS system:

where a = ambient density ratio for altitude performance Nondimensional force (for torque), TF = F/(p V2D2) Force coefficient:

Cf = Tf x J = F x [V/(nD)]2/(pV2D2) = F/(pn2D4) (10.41)

Therefore, torque:

Q = force x distance = Fr = Cf x (pn2 D4) x D/2

Figure 10.34. Static performance: three bladed propeller performance chart – AF100 (for a piston engine)

or torque coefficient:

(10.44)

The wider the blade, the higher the power absorbed to a point when any further increase would offer diminishing returns in increasing thrust. A nondimensional number, defined as the total activity factor (TAF) = N x (105/16) /o’0 (r/R)3(b/D)d(r/R), expresses the integrated capacity of the blade element to absorb power. This indicates that an increase in the outward blade width is more effective than at the hub direction.

A piston engine or a gas turbine drives the propeller. Propulsive efficiency np can be computed by using Equations 10.35,10.39, and 10.44.

Propulsive efficiency,

np = (TV)/[BHP or ESHP]

= [CT x (pn2D4) x V]/[CP x (pn3D5)]

= (Ct/Cp) x [V/(nD)] = (Ct/Cp) x J (10.45)

The theory determines that geometrically similar propellers can be represented in a single nondimensional chart (i. e., propeller graph) combining the nondimensional parameters, as shown in Figures 10.34 and 10.35 (for three-bladed propellers) and Figures 10.36 and 10.37 (for four-bladed propellers). Considerable amount of

Figure 10.35. Three-bladed propeller performance chart – AF100 (for a piston engine)

coursework can be conducted using these graphs. These graphs and the procedures to estimate propeller performances are from [16], a courtesy of Hamilton Standard. These graphs are replotted retaining maximum fidelity. The reference provides the full range of graphs for other types of propellers and charts for propellers with a higher activity factor (AF).

Static computation is problematic when V is zero; then np = 0. Different sets of graphs are required to obtain the values of (Єт/Cp) to compute the takeoff thrust, as shown in Figures 10.34 and 10.36. Finally, Figure 10.38 is intended for selecting the design CL for the propeller to avoid compressibility loss. Thrust for takeoff performance can be obtained from the following equations in FPS:

In flight, thrust:

T = (550 x BHP x np)/V, where V is in ft/s = (375 x BHP x np)/V, where V is in mph (10.46)

For static performance (takeoff):

Tto = [(Ct/Cp) x (550 x BHP)]/(nD) (10.47)

Figure 10.36. Four-bladed propeller performance chart – AF180 (for a high – performance turboprop)

Figure 10.37. Four-bladed propeller – performance chart – AF180 (for a high – performance turboprop)

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