APC Sport Propeller Performance ProgramW. B. Garner, November 2012

IntroductionUnderstanding how model airplane propellers work and how to predict their performance in

actual usage has been a goal for a long time. Starting in 2009 I undertook research on the subject anddeveloped an Excel based computer program for that purpose. The model was based on the SimpleBlade Element Theory, known to have deficiencies at low Advance Ratios. At the time there was nogood measured data available to compare the results against. It was clear from what little data wasavailable, mostly from static tests, that the computer model did not match very well, especially at lowadvance ratios. And so the ability to predict performance within reasonable error bounds remainedunresolved.

In 2011 a paper by John Brandt & Michael Sellig of the University of Illinois at Urbana-1Champaign (UIUC) was published at an AIAA conference titled “Propeller Performance Data at Low

Reynolds Numbers”. This paper describes wind tunnel test findings on about eighty model airplanepropellers from several different manufacturers. Of particular value was a link to the test data itself.This data, when compared to the Excel program-produced data, showed marked differences in manycases so substantial that the computed results were worthless.

Examination of the measured data revealed that there was considerable variation in Ct and Cpcoefficients for the same diameter and pitch propellers between manufacturers and even between typeswithin a single manufacturer's line of products. This led to the conclusion that a single genericcomputer program would not produce good results across all of the propeller types. Instead, a basicprogram could be developed with variants adapted to each type of propeller.

The other consideration was to find a better theoretical model on which to base the program. A2

search over the Internet found a theory, called QPROP , that was implementable and corrected a majordefect in the Simple Blade Element Theory, that of no direct accounting for vortex effects.

The computer program described in this document focuses on the APC Sport line of propellers.There are a sufficient number of them in the UIUC data base to allow a reasonable comparison ofcomputed and measured results over a range of pitches and diameters. The author uses APC props inmost models so they have a direct relevance to practical applications.

QPROP ModelThe computer program is based upon the QPROP theoretical model. This theory includes the

effects of the vortexes caused by the rotation of a blade in a fluid, in this case air. The rotation of theblades causes the air to swirl around behind the propeller, resulting in changes to the lift and drag alongthe length of the blade compared to, say, a wing. The theory does not require the use of the concept ofaspect ratio or of induced drag, both required by the Simple Blade Element Theory.

The theory does include non-linear equations that must be solved to derive a specific parameter,in this case an angle. Newton's Method is used to obtain the solution. Appendix A contains the formulas

1

2

Brandt, John B. & Sellig, Michael S., “Propeller Performance Data at Low Reynolds Numbers”, 49 AIAAAerospaceth

Sciences Meeting, 4-7 January 2011, Orlando FL, AIAA 2011-1255Drela, Mark, “QPROP Formulation”, MIT Aero & Astro, June 2006.

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out of the QPROP paper.

The model selects a radius, r, computes the incremental thrust and torque at that radius,increments the radius while repeating the incremental computation, then sums the increments to obtainthe total thrust, T, and torque, Q. The associated power, P, is computed from the torque.

The measured blade physical properties vary from prop type to prop type. For example, theblade angle, beta, does not necessarily result in a constant pitch over the radius, nor are the chordwidths and distributions the same. The blade width profiles are also different, making the blade lift anddrag coefficient profiles different. The result of these differences are differences in the resulting Ct andCp coefficient variations with advance ratio, J.

APC Sport Propeller Characteristics

The UIUC paper describing the results of the wind tunnel tests contains some selectedinformation about the propellers tested. In particular, the three types of APC props are brieflypresented. That information is extracted and presented here.

Figure 1. APC Sport, Thin Electric and Slow Flier Propellers

Figure 1 shows photos of the three types of APC propellers tested. The plan-forms, widthdistributions and thicknesses are different, each applicable to different airplane model types. The Sportmodel is for gas-fueled engines that develop maximum torque and power at high rpm. The ThinElectric model is for electric motors that develop maximum torque and power at low rpm but need tooperate at loads somewhat less than do the Sport propellers at moderate rpm. The Slow Flier model isintended for low rpm electric motor applications. The remainder of this document is devoted to theSport propellers.

Pitch Characteristics

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The measured data provide the pitch information along the radius in the form of the blade angle,beta. This data can be converted to pitch data and compared to other propellers. Figure 2. plots localpitch normalized to the name-plate pitch as a function of normalized radius for several propellers.

Normalized p/pname vs r/R, APC Sport Props

1.500

1.400

1.300

1.20010x610x81.10011x4

1.000

0.900

0.800

0.700

0.600

0.500

11x511x6p/pavgPoly. (p/pavg)

0 0.2 0.4 0.6 0.8 1 1.2

r/R y = 2.38E+01x5 - 8.83E+01x4 + 1.22E+02x3 - 7.93E+01x2 + 2.44E+01x - 1.59E+00

Figure 2. APC Sport Propeller Normalized Pitch as a Function of Normalized Radius

There are several items of note. The curve shapes are similar, with maximum pitch at a radius ofabout 0.3, gradually decreasing to a radius of about 0.8 and then sharply decreasing to the tip radius.The pitch matches the name plate value at a radius of about 0.9 to 0.95, considerably different from theconventional location of 0.75. In fact, most of the blade operates at pitches significantly higher than thename-plate value. Therefore assuming a constant pitch with radius in a model is not valid. The modelused in this document is shown by the curve labeled 'Poly' which is curve fitted to the average pitch forthis group of propellers. With the exception of the 11x4 results ( yellow curve), the error is relativelysmall.

Chord Width Characteristics

Figure 3. shows the normalized chord width as a function of normalized radius for a series of11 inch propellers. They are essentially identical. Figure 4 shows the same information for 10 inchpropellers. The 11 and 10 inch curves are essentially the same over the region of interest (r/R >.3).

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c/R for D = 11 inchesy = 2.2114x 5 - 6.7435x 4 + 6.8679x 3 - 2.9915x 2 +0.5354x + 0.1368

0.18

0.16

0.14

0.12

0.1

11 x 411 x 511 x 611 x 7

0.08

0.06

0.04

0.02

0

11 x 811 x 9Poly. (11 x 6)

0 0.5 1 1.5r/R

Figure 3. 11 Inch Propeller Normalized Chord Width

c/R for D = 10 inchesy = -0.1656x 4 - 0.7895x 3 + 1.3224x 2 - 0.5809x

+ 0.23040.18

0.16

0.14

0.12

0.1 10 x 610 x 8

0.08

0.06

0.04

0.02

0

Poly. (10 x 6)

0 0.5 1 1.5r/R

Figure 4. 10 Inch Propeller Normalized Chord Width

The chord width curve fitted to the 11 inch data was adopted for this model.

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Lift and Drag Coefficient Profiles

The blade lift and drag coefficient profiles, Cl(alpha) and Cd(alpha) are not known. They mayactually vary along the radius. Nevertheless it is necessary to have such profiles to implement ananalytical program. What is known is the approximate physical profile of a blade cross-section. Theblades are flat bottomed with rounded leading edges and sharp trailing edges. The cord line appears tobe slightly angled relative to the flat bottom, on the order of one to two degrees. The maximumthickness is about 13 % at the 35% of chord location. Camber is unknown, but fairly small.

Given these characteristics a search was conducted using the Profili wing section program tofind candidates for the model. There are fairly large numbers of sectional profiles with similarcharacteristics so one was picked as being representative. The profile chosen is that of the GOE 693section, plotted in Figure 5. The shape and slope of the lift coefficient profile does not vary very muchbetween similar physical sections, so it is fairly representative of its class. What does vary betweensimilar sections is the drag coefficient profiles. They are especially sensitive to Reynolds Number (Re),as seen in the Figure. The curves for an Re of 150,000 was picked as the reference – more on thissubject when the results are discussed.

Figure 5. GOE 693 Cl and Cd Profiles

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Figure 6. is a graph showing the profile as implemented in the program. The upper ends of theprofiles have been extended about four degrees to make them operate over a larger range. This

GOE 693 @ Re=150000y = -1.1913E-04x 3 - 2.1824E-03x 2 + 1.1983E-01x + 4.1540E-01

1.5 0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

1

0.5

0

CLCDPoly. (CL)Poly. (CD)

-10 -5 0 5 10 15 20

-0.5

-1

y = -2.2133E-08x5 + 1.9884E-06x 4 - 2.0802E-05x 3 + 1.8687E-04x 2 -Alpha, degrees5.2905E-04x + 1.3870E-02

Figure 6. Cl & Cd Profiles Used in the Programextension was based upon the results of various adjustments used to match measured results. Notshown but included in the detailed model is making Cl constant at angles beyond some value, in thiscase, 17 degrees.

The shape and magnitude of the Cl and Cd profiles have an effect on the propeller Ct and Cpprofiles. Refer to Figure 7 showing the geometrical relationships among blade physical andperformance variables. Wt is the effective rotational velocity while Wa is the free stream velocity, witha resultant velocity of W at angle phi relative to the horizontal. The ratio of Wa to Wt is essentially theadvance ratio, J. When this ratio is large, corresponding to large phi, the angle of attack is small & thedrag component is also small compared to the lift component. The result on Ct is to make itproportional to the lift coefficient, Cl, which is nearly linear at small alpha. (large phi). The powercoefficient Cp is also small. As phi decreases (or J decreases), the blade angle of attack increasesresulting in an increase in thrust & Ct and in the power coefficient Cp. However, at some value thepeak of the lift curve is met or exceeded so that lift no longer increases but may actually decrease. At

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the same time the drag increases until the blade is essentially stalled, at which point the drag becomesunpredictable.

Figure 7.In summary, the low advance ratio region is controlled primarily by blade stalled conditions,

marked by limited Ct and flattened Cp. Drag dominates the net effect. At high advance ratio the bladeis operating in the linear lift, low drag, region so Ct is primarily controlled by the Cl characteristic.Hence adjustments to the model to match measured Ct and Cp are primarily to Ct in the high J regionand primarily to Cd in the low J region.

Comparing measured versus calculated results over a range of pitch to diameter ratios (p/D)showed that to match Ct it was necessary to adjust the Cl zero crossing as a function of p/D. MatchingCp required an adjustment to the magnitude of the drag coefficient, Cp. These adjustments are builtinto the program.

Results

Figure 8 plots generic Ct values for a ten inch propeller with pitches ranging from 4 to 8. Thecorresponding p/D ratios vary from 0.4 to 0.8. Figure 9 plots Cp for the same set of propeller sizes.

Thrust Coefficient for Various p/D ratiosAPC Sport 10 inch Diameter Propellers

0.140

0.120

0.100

0.0800.40.5

0.060

0.040

0.020

0.000

-0.020

0.60.70.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

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Figure 8.

Power Coefficient for Various p/D RatiosAPC Sport 10 inch Diameter Propellers

0.0800.070

0.060

0.050

0.040

0.030

0.020

0.010

0.40.50.60.70.8

0.000-0.010 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

-0.020

J

Figure 9.

Measure d & Calculated Ct for APC 10x6 Prop

0.12

0.1

0.08

0.06

0.04

0.02

0

measured

calc

-0.02 0 0.2 0.4 0.6 0.8 1

J

Figure 10

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Measure d & Calculate d Ct for APC 11x8 Prop

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

measuredcalc

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.02

Figure 11

Measured & Calculated Ct for APC 9x7 Prop

0.12

0.10.08

0.06

0.04

0.02

0

measuredcalc

-0.02 0 0.2 0.4 0.6 0.8 1

J

Figure 12

Figures 10 -11 compare measured and calculated Ct for three prop sizes. Figures 10 and 11show close agreement over the whole range of J while figure 12 shows substantial disagreement at highJ. Figure 12 is atypical, however, indicating that not all props follow the same design rules even in thesame type. If the assumed pitch is reduced a little the curves begin to match, indicating that thegeneralized pitch function is off a little bit for this diameter.

Figures 13 – 15 are representative plots of Cp for three sizes of propellers. The 10x6 results(Figure 13) are very close, a result of the 10 inch props being used as the reference. The calculatedresults for the 11x8 propeller (Figure 14) are slightly lower than the measured results. The calculatedresults for the 9x5 propeller (Figure 15) are higher than the measured results. Experimentationindicates that the primary source of error is the variation in actual pitch from that assumed in theanalysis.

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Measured & Calculated Cp, APC 10x6 Sport

0.06

0.05

0.04

0.03

0.02

0.01

0

MeasuredCalculated

0 0.2 0.4 0.6 0.8 1-0.01

J

Figure 13. Cp for 10x6 Sport Prop

Measured & Calculated Cp, 11x8 APC Sport

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

Meas uredCalculated

0 0.2 0.4 0.6 0.8 1

J

Figure 14. Cp for 11x8 Sport prop

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Measured & Calculated Cp, 9x5 APC Sport

0.05

0.045

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

Meas uredCalculated

0 0.2 0.4 0.6 0.8 1

J

Figure 15. Cp for 9x5 Sport Prop

Summary

The computer program does a decent job of matching Ct & Cp to measured values with someexceptions. There are several reasons for the differences.

The effects of Reynolds number are not included. The calculations tend to overestimateperformance for low rpm.

The actual pitch distribution varies somewhat with size from the assumed model. It may bepossible to include a diameter adjustment to partially compensate.

The actual lift and drag coefficient profiles may be different, even varying along blade radius.

The results are good for diameters of 9 to 11 inches. Match goodness outside this range isunknown.

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