Practical Use of Propeller Charts

The practicing aerodynamicist will normally have available both engine and propeller operating curves as supplied by the respective manufacturers. Using these curves together with a knowledge of the airplane’s aerodynamic characteristics, one is able to estimate the airplane’s performance. In order to illustrate the procedures that are followed in using a set of propeller charts, let us again use the Cherokee 180 as an example.

An estimated curve of efficiency as a function of advance ratio for the fixed pitch propeller used on the PA-28 is presented in Figure 6.15. This curve is applicable to the aircraft pictured in Figure 3.62 with the engine operating curves of Figure 6.3. This particular propeller has a diameter of 1.88 m (6.17 ft).

As an example in the use of the engine performance charts together with the graph of propeller efficiency, assume that in steady, level flight, the pilot of a PA-28 reads a manifold pressure of 24 in., an rpm of 2400, a pressure altitude of 3000 ft, an OAT of 65 °F, and an indicated airspeed of 127 mph. From this information, together with Figure 6.15, one can estimate the drag of the airplane at this indicated airspeed and density altitude.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Advance ratio, J ~ V/nD

Figure 6.15 Estimated propeller efficiency for the Piper Cherokee PA-28.

From Figure 2.3 at 914 m (3000 ft),

£- = 0.90 Po

Thus, p = 91,163 Pa (1904 psf).

Furthermore, air obeys closely the equation of state for a perfect gas.

—= = constant = R

pT

In Equation 6.50 T is the absolute temperature.

Using standard sea level values for p0, p0, and T0, the preceding constant is seen to be

R = 287.1 (m/s)2/°R [1717 (fps)2/°K]

Thus, for this example, T = 292 °R (525 °K), so that

p = 1.087 kg/m3 (0.00211 slugs/ft3)

This corresponds to a tr of 0.888. Thus, from Figure 2.3, the density altitude is found to be 1006 m (3300 ft) and the true airspeed is calculated to be 60.4 m/s (135 mph or 198 fps).

The propeller advance ratio is defined by Equation 6.30.

J = 0.802

For this value of J, a propeller efficiency of 0.81 is read from Figure 6.15.

One can verify that the engine power for these operating conditions, from Figure 6.3, is equal to 138 bhp. Therefore, from Equation 6.33, knowing 17, P, and V, the propeller thrust can be calculated as

T-Vp T ~ V

= 310 lb (1380 N)

In steady, level flight, the propeller thrust and airplane drag must be equal. Thus, 3101b is the drag of the airplane at this particular density altitude and airspeed.

For analyzing a variable pitch propeller a set of curves for different blade pitch angles is required. These are given in Figures 6.16 and 6.17 for the propeller installed on the Piper PA-28R, the Cherokee Arrow. Here we are given both 17 and Cp as a function of J. To illustrate the use of such graphs, let us assume that they apply to the preceding example for the PA-28. Here, p = 0.00211 slugs/ft3, D = 6.17 ft, V = 198 fps, n = 40 rps, J = 0.802, and hp = 139. Thus,

P

Cp pn3D5 = 0.0633

From Figure 6.17 for the preceding cp and a J of 0.802, the blade pitch angle must be equal to 24°. Entering Figure 6.16 with this /3 and J results in an efficiency, Tj, of 0.83.

A well-designed propeller, or one carefully selected to match the engine and airplane on which it is to operate, can be expected to have a cruise efficiency of approximately 85%. At low speeds, however (e. g., during the takeoff roll), the efficiency is difficult to estimate. At zero forward speed, the efficiency of a propeller is zero by definition, even though its thrust is not zero. In fact, for the same shaft power, a variable pitch propeller will produce the most thrust at zero advance velocity (i. e., its static thrust is greater than the thrust produced in forward flight).

Figures 6.18 and 6.19 may be used to estimate the thrust attainable from a

variable pitch propeller at low forward speeds. The static thrust is first obtained from Figure 6.19 and then reduced by the factor from Figure 6.18 to give the thrust in forward flight. These curves apply only to a constant speed propeller, which will allow the engine to develop its rated power regardless of forward speed. As an example of the use of these figures, consider a propeller having a diameter of 6.2 ft, turning at 2700 rpm, and absorbing 200 hp. The power loading for this propeller is #

^ = 6.62 hp/ft2

Hence, from Figure 6.19, the static thrust to power loading should be

jr = 4-9

hp

resulting in a static thrust, T0,.for this propeller of 980 lb.

Velocity, fps Figure 6.18 Decrease of thrust with velocity for different power loadings.

From Figure 6.18, the expected thrust at a speed of, say, 50mph (22.4 m/s) can be calculated as

= (0.715X980) = 700 lb

Approximate Useful Relationships for Propellers

Figures 6.18 and 6.19 were prepared using some approximations that are fairly accurate and convenient to use. Referring to Equations 6.31 and 6.32, assume that <j, Ch and Cd are constants, so that they can be removed from under the integral sign. In addition, it is assumed that a, and xh — 0 and ас, і. With these assumptions, CT and Cp can be written as

CT = -^r-Ci f x(A2 + x2)U2dx

Cp = JCt + ^ Q Г X2(A2 + x2)1’2 dx.
o Jo

Performing the integrations, CT and Cp become

CT = ~Qf()

CP = JCT + Cdg(X)

where

/(A) = (l + A2)3/2-A3

g(A) = [(1 + A2)1,2(2 + A2) – A4log1 + ^+— ]

/(A) and g(A) are given as a function of J in Figure 6.20. C, and Cd indicate average values of these quantities as defined by Equation 6.51.

# The term JCT in the expression for C„ simply represents the useful power. The remaining term in Cp is the profile power, or the power required to overcome the profile drag of the blades. The induced power is missing, since а, was assumed to be zero. Experience shows that the induced power is typically 12% higher than the ideal value given by Equations 6.14 and 6.15. Thus, in coefficient form,

J Figure 6.20 Functions for approximating CT and CP.

Cp then becomes approximately

Cp = CTJ + CPi + ^ С^(Л) (6.54)

The average value of tr is referred to as propeller solidity and is equal to the ratio of blade area to disc area.

Propeller designs are sometimes identified by an “integrated design lift coefficient” and an activity factor. These are defined by

CLd = 3 f Cldx2 dx (6.55)

Jo

AF, mmf‘(£jx, dx (6 56)

The integrated design lift coefficient represents the average of the section design lift coefficients weighted by x2. The activity factor is simply another measure of the solidity. The higher the activity factor, the higher are the values of CT and Cp attainable by a propeller at a given integrated design CL.

Equation 6.54 represents about the best one can hope to achieve with a well-designed propeller operating at its design point. For the propeller shown in Figure 6.11, this corresponds to blade angles of around 15 to 25°. Beyond this range, the twist distribution along the blade departs too much from the optimum for these relationships to hold.