Blade Element Theories

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In order to design a propeller or to predict the performance of an existing propeller more accurately, it is necessary to examine the aerodynamics of the blade in detail. Figure 6.9a presents the front view of a three-bladed propeller that is rotating with an angular velocity of <0 rad/s and advancing through the air with a velocity of V. Two cylindrical surfaces concentric with the axis of rotation and a differential distance of dr apart cut the propeller blade at a radius of r from the axis. The blade element thus defined is illustrated in Figure 6.9b. Here we are looking in along the blade. The section is moving to the right (due to rotation) and toward the top of the page as the propeller advances into the air. The velocities influencing the element are shown relative to the element.

For the following analysis, the pitch angle, /8, of the section is defined relative to the zero lift line of the airfoil section. In this regard, however, one must be careful, since propeller pitch angles are frequently tabulated with respect to the chord line or to a flat lower surface.

The qpitcl^ of a propeller has reference to the corresponding quantity for the ordinary screw. In fact, the early literature refers to propellers as “airscrews.” If the propeller “screws” itself through the air without slipping, the distance it would move forward in one revolution is the pitch, p. From Figure 6.9b,

Figure 6.9 Velocities and forces acting on a propeller blade, (a) front view of a three-bladed propeller, (b) Blade element as seen looking in along blade.

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Propellers are sometimes categorized by their pitch-diameter ratios. Thus,

P

— = 7tx tan /3

where x = r/R, the relative radius of the blade section.

A constant pitch propeller is one whose pitch does not vary with radius.

For such a propeller,

/3 = tan-‘^ (6.24)

At the tip, x equals unity so that,

f}(x = l) = tan-1^^

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The terms constant pitch, fixed pitch, and variable pitch are somewhat confusing. “Constant pitch” refers to the propeller geometry as just defined. “Fixed” or “variable” pitch refers to whether or not the whole blade can rotate about an axis along the blade (feathering axis) in order to vary the pitch angles of the blade sections all along the blade. Some propellers are equipped with governors to maintain a constant rpm as the engine throttle is varied. This is done by increasing the blade pitch angles as the propeller rpm tends to increase due to increased power or, vice versa, by decreasing the pitch for reduced power. Such a propeller is called a “constant speed” propeller.

Referring to Figure 6.9b, the contribution of one-blade element to the thrust, T, and torque, Q, will be,

dT = dL cos (ф + at) – dD sin (ф + a,) (6.25a)

dQ = r[dL sin (ф + a,) + dD cos (ф + a,)] (6.25b)

dL and dD are the differential lift and drag forces, respectively. Similar to finite wing theory, a, is an induced angle of attack resulting from the induced velocity, w. dL and dD can be calculated by

dL = {pV? cQdr (6.26a)

dD = pVe2cCd dr (6.26 b)

The chord, c, is usually a function of the local radius, r. The section Q is primarily a function of the section Q. It can also depend on the local Reynolds and Mach numbers. Q can be found from

С, = а(р-ф-сч) (6.27)

We are now at somewhat of a dilemma. We need a„ which is a function of w, in order to get the blade loading. But w depends in turn on the blade loading.