APCE Propeller Ct and Cp Coefficient ExpansionW B Garner, July 2017

IntroductionThere is a need for a way to estimate in-flight performance of model aircraft under a variety of flight conditions. To do so requires a means to estimate propeller performance as a function of airspeed and applied power and rpm. While static thrust testing is useful, it does not provide the necessary information to account for forward airspeed and the effects of that airspeed on propeller performance.

This document presents a method for obtaining the desired information from a combination of static testing and a computer model validated against wind tunnel tested propellers. The available wind tunnel data is limited so the method includes a means for extending the combination of testing and computation to propellers of the same family, albeit with potentially greater inaccuracy.

In particular the results obtained are for APC Thin Electric propellers and based on the wind tunnel testing done at the University Of Indiana Urbana Campus Aeronautics Department (UIUC) and the Wichita State University Aerospace Engineering Department (WSU).

ResultsThe results are provided in two tables, one for the Ct coefficients, and the other for the Cp coefficients. The value of Ct or Cp is given in the form of a polynomial with the advance ratio, J, as the variable.

Ct =tcof3*J^3 + tcof2*J^2 + tcof1* J + tcof0

Cp=pcof5*J^5 + pcof4*j^4 + pcof3*J^3 + pcof2*J^2 + pcof1*J + pcof0

The table headings list the diameter, the pitch, each of the coefficients, the rpm at which they were obtained and the source. UIUC stands for the University of Illinois Urbana Campus, WSU for the Wichita State University and WBG are the author’s initials, showing he was the source.

The diameters range from 8 inches to 13 inches. 7 inch diameter propellers were to have been included but measuring pitch turned out to be problematical because of the small dimensions involved. The computations were too unreliable to be included. Larger propellers may be included in the future.

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Table 1 Cp Coefficientsdiameter pitch pcof5 pcof4 pcof3 pcof2 pcof1 pcof0 test rpm source

8 4-

0.0821 0.0123 0.0411 7000 uiuc8 6 -0.115 0.0628 -0.0198 0.0749 6700 uiuc8 8 -0.1029 0.1016 -0.0556 0.116 6400 uiuc9 4.5 0.3639 -0.4287 0.038 0.0074 0.0398 6900 uiuc9 6 0.4545 -0.7492 0.2672 -0.0189 0.0519 6700 uiuc9 7.5 0.303 -0.6682 0.3753 -0.0746 0.083 6000 uiuc9 9 -0.1195 0.1043 -0.0435 0.103 5900 uiuc

10 5 0.322 -0.4558 0.096 -0.0009 0.0399 6900 uiuc

10 6 0.0394-

0.1155 0.0131 0.0465 6000 wbg10 7 0.4487 -0.7668 0.3035 -0.0319 0.0482 6700 uiuc

10 8 1.149-

2.5054 1.716-

0.4508 0.0384 0.0558 7050 wbg10 10 -0.1385 0.1114 -0.0202 0.0943 6740 wbg11 5.5 0.303 -0.4192 0.0811 0.0003 0.0319 6900 uiuc11 7 0.4283 -0.708 0.2605 -0.0242 0.0481 6700 uiuc11 8 0.2069 -0.487 0.2417 -0.0321 0.0524 6000 uiuc

11 10 1.8429 -4.517 3.6638-

1.1316 0.0741 0.0832 5900 uiuc12 6 0.0569 -0.1296 0.0164 0.032 7000 wbg12 8 -0.0477 -0.0367 0.0067 0.0459 7000 wbg12 10 -0.0779 0.0101 0.0009 0.062 7000 wbg12 12 -.2742 .04632 -.267 .047 .076 ~6500 wsu13 4 0.0389 -0.1038 0.012 0.0171 7000 wbg13 6.5 -0.0128 -0.0603 0.0101 0.0281 7000 wbg13 8 -0.156 0.0783 -0.012 0.0346 7000 wbg13 10 -0.0842 0.0208 0.0037 0.0486 7000 wbg

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Table2 Ct Coefficients

diameter pitch tcof4 tcof3 Tcof2 tcof1 tcof0 test rpm source8 4 -0.0774 -0.1004 0.1025 7000 uiuc8 6 -0.1382 -0.0016 0.1211 6700 uiuc8 8 -0.1363 0.0476 0.1274 6400 uiuc9 4.5 0.4482 -0.5602 0.0218 0.1003 6900 uiuc9 6 0.234 -0.4231 0.0474 0.1125 6700 uiuc9 7.5 -0.1559 0.027 0.1229 6000 uiuc9 9 -0.1383 0.0265 0.1362 5900 uiuc10 5 0.4457 -0.601 0.0572 0.098 6700 uiuc10 6 -0.1628 -0.0429 0.1028 6500 wbg10 7 -0.1665 -0.0068 0.1132 6500 uiuc10 8 -0.1552 0.0033 0.1175 7050 wbg10 10 -0.1822 0.1135 -0.0112 0.1106 6740 wbg11 5.5 -0.181 -0.0436 0.087 6000 uiuc11 7 0.5594 0.06813 0.0339 -0.0087 0.1053 6000 uiuc11 8 -0.1242 -0.0225 0.113 6000 uiuc11 10 -0.1601 0.0523 0.11 5500 uiuc12 6 -0.1399 -0.0629 0.0826 6000 wbg12 8 -0.1385 -0.0257 0.0978 6000 wbg12 10 0.068 -0.2269 0.0405 0.1049 6000 wbg12 12 -.0965 .0413 -.0395 .1203 ~ 6500 wsu13 4 -0.0869 -0.1138 0.0557 5500 wbg13 6.5 -0.1318 -0.0652 0.0814 5500 wbg13 8 -0.1312 -0.0482 0.0902 5500 wbg13 10 0.244 -0.3675 -0.011 0.0199 0.0905 5500 wbg14 12 -0.1473 0.0262 0.0998 3500 uiuc

CommentaryObtaining good matches between calculated and wind tunnel results is difficult and subject to substantial error. The degree of match is best for low pitch/diameter ratio propellers as these propellers operate mostly in the linear region where math models work reasonably well. As the p/D ratio increases, the fraction of the propeller subject to stall conditions increases, making theoretical mathematical models unreliable. Empirical models are useful in covering these high p/D conditions.

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Note that the results are generated for specific RPMs. Increasing rpms beyond those listed improves performance slightly while decreasing it can lead to substantial reductions in performance. As the rpm decreases, the Reynolds Number effects become dominate near the hub and near the tip of a blade.

Use of these results is at the discretion of the user; the author makes no guarantee as to its utility or application.

Example Comparison of UIUC data and computed data for CtPolynomial representations of all of the UIUC measured propellers where extracted and used for references to compare against the computer generated versions. The UIUC data at the maximum RPM displayed in the charts was used as they show the least influence of really low RNs. They are also more representative of conditions likely to be found during normal sport flight. One set of data for the 12x12 propeller is from WSU.

Curves labeled “Ct UIUC” are derived from the UIUC data base. Those labeled “Ct calc” are from the computer program. The curves labeled “adjusted” are the computed curves adjusted for measured values at J = 0.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.0200.0000.0200.0400.0600.0800.1000.1200.1400.160

apc thin e 8x6

Ct uiuc

J

Ct

4

5

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Generally the matches are close with some exceptions. For instance, the 8x4 and 8x6 comparisons show that the match is good at low J but diverges at high J. The reason for this divergence is not known.

Example Comparison of Wind Tunnel Data and Computed Data for CpThe computed drag coefficients match reasonably well at low pitch to diameter ratios but can deviate significantly at high ratios, especially at low advance ratios. Hence computed values for Cp for low advance ratios are unreliable and measured static values substituted in the final results. The graphs show how this substitution was done. The measured Cpo was inserted and the same value used until the resulting line met the computed line. There is some judgement required in doing so. Some of the calculated graphs show considerable deviation from the wind tunnel data (8x6, 11x7 for example), so using the results for propellers not in the wind tunnel data base should be used with the understanding that considerable errors may be present.

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MethodologyThe basic methodology is to measure static thrust and absorbed power of a specific propeller and to determine the associated thrust and power coefficients, Cto and Cpo, respectively. Using measured physical properties of the propeller (pitch & chord distributions), computer generated estimates of Ct and Cp with advance ratio, J, are obtained in the linear region of the associated curves. The static values are then used to anchor the computed curves and to extrapolate the coefficient values over the value of J near zero range where the computations have the most error.

Computer ProgramThe computer program is based on the QPROP theory developed by Dr. Mark Drella of MIT. The method requires detailed knowledge of the physical and aerodynamic properties of the propeller. These properties include the distribution of pitch and chord length with position along a blade. The UIUC data base contains these measurements for some of the tested props, but not all. Those not tested were measured in the shop or were estimated based on the UIUC data for similar size props.

The theory also requires knowledge of the airfoil characteristics of lift and drag including the effects of Reynolds Number. This requirement is the greatest source of uncertainty as the coefficients are not well known or are not actually knowable under stalled conditions. Never the less it is possible to approximate the required information with the aid of comparison to wind tunnel data and airfoil type stated by the manufacturer.

UIUC Propeller Data Base ObservationsThe methods for direct measurement of propeller coefficients at UIUC are well documented with the data base and were confirmed by comparison to measurements from other institutions. Hence there is high confidence in the accuracy of the results. The same cannot be said for the accompanying geometry

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measurements of the chord and pitch distributions. No information is provided on how these measurements were made or their accuracy. They are therefore suspect in terms of their accuracy & need to be verified to use them in the computer program. These measurements seem to be incidental to the main research question related to low Reynolds Number effects, so accuracy may have not been of concern.

Chord DistributionsThe UIUC data base lists chord distributions for some, but not all, of the propellers. The shape of the distributions is nearly the same for all propellers. The values of c/R (ratio of chord length to blade length) are a function of blade length, decreasing as diameter increases. Figure 1 is a plot of c/R for several propellers showing how it varies with diameter. Note that there are some deviants among the shown set that may be correct or may be due to measurement errors.

Figure 1 Sample c/R Distributions

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Measurements of chord distributions for several size props using a caliper compare closely to those in the UIUC data base. Hence there is confidence that measurements on props not in the data base are accurate. The computer program assumes that the chord distributions are the same for props of the same size and takes into account the changes with that size.

Pitch DistributionsThe normalized pitch distributions with blade station are nearly the same for all propellers as indicated in Figure 2. The plot is of the ratio of actual pitch to name plate pitch. The actual pitch is generally greater than the name plate pitch over most of the blade but is reduced near the tip and near the hub. There are exceptions such as the pitch distribution for the 10x5 propeller. It was found that using the actual pitch distributions rather than the name plate distributions provides a better match between measured and computed prop coefficients.

Figure 2 Normalized Pitch distributions.

Attempts to replicate the pitch distributions in the UIUC data base were only partially successful. Several methods were used to try to replicate those results, none were satisfactory. The pitch distributions from the UIUC data base were incorporated into the program using an average value for all. It is assumed that the propellers not in the data base have the same distributions. This seems to be the case from examining the shop measured pitch results for all propellers tested. The shop data is not considered accurate enough to be used in the program but is good enough to confirm the right shape. However, where WBG data is significantly different the WBG data was used.

Appendix 0ne discusses the various pitch measurement methods tried.

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Airfoil CharacteristicsAPC says that the airfoil characteristics near the hub are similar to the Eppler E63 airfoil, chosen because it has good low Reynolds Number characteristics. The outer portion of the blade is similar to the Clark Y airfoil. The two types transition over the blade length. How that transition is done is not known.

Another factor is the influences of low Reynolds Number on lift and drag along the blade length.

Figure 3A is an example of the Reynolds Number to be expected along the length of a 5 inch blade under zero airspeed, using the chord distribution for a 10 inch diameter propeller. The RN is very low near the hub, increases around the middle, and then decreases near the tip. Note that it is a function of rpm that varies from near zero at the hub to maximum at the tip.

Figure 3A Example Reynolds Number along the Blade of a 10 inch Diameter Propeller, Zero Airspeed

Figure 3B is an example of Reynolds Number for a 10x7 propeller operating at an air speed of 30 mph. The introduction of airspeed increases the Reynolds Number at low r/R but barely changes the high end number. The low end is dominated by the airspeed while the high end is dominated by the propeller speed.

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Figure 3B Reynolds Number at 30 mph

Figure 4 is a plot of the assumed drag coefficient as a function of Angle of Attack (AOA) for different Reynolds Numbers, derived from a Profili program for generating airfoil coefficients. The values of Cd were modified (increased) to better match observed results near J = 0 as real world drag is always greater than theoretical drag. The program calculates the Reynolds Number for each blade station and uses the value of Cd closest to it from the discrete value set.

Figure 4 Airfoil Cd Plots

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The original attempt at a Cl polar plot was to use a Clark Y airfoil and generate a series of lift coefficients as a function of Reynolds Number. However, this turned out to be too complicated and did not interact properly with an internal convergence algorithm required to solve the QPROP equations. A single profile was substituted instead. While this introduces some potential errors, they are tolerable. The plot was derived from a GOE 693 plot, modified by reducing the slope somewhat. It was established by trial and error, observing the effect on several of the plots simultaneously. The function is truncated at the low and high ends where the blade goes into reverse or stall conditions.

It was found that the results are sensitive to the value of the zero crossing angle of the lift coefficient. Ct is approximately proportional to Cl and Cl is proportional to Angle of Attack, so shifts in the Cl function zero crossing directly affects the resulting Ct values. It was found that the computed and UIUC measured Ct values could be closely matched in most cases by adjusting the zero crossing angle and pitch angle, but the adjustments were different for different propellers. Hence a single compromise Cl zero crossing value was assumed for the non-UIUC propellers.

Figure 5 Airfoil Lift Coefficient Plot with Reynolds Number

Static Test Measurement SystemThe static test system was composed of a motor/propeller test mount, an ESC, a 72 MHz receiver and transmitter to drive the ESC, a regulated power supply, a data logger at the input to the ESC that recorded voltage current and rpm, and a calibrated scale for measuring thrust force.

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Any measurement system is subject to errors and generally it is desired to reduce those errors to an “acceptable” level. The definition of “acceptable” depends on the end objectives and how difficult (or easy) it is to achieve that objective. In the present case there are multiple sources of potential errors, multiplicative in nature. It is desirable to minimize the potential errors within the capabilities of the available measuring instruments. Calibration against standards is highly desirable where it can be done, and averaging of readings used where calibration may not be available.

First, some definitions are presented.

Accuracy is how close a measured value is to the actual (true) value.

Precision is how close repeated measured values are to each other.

Standard Deviation is a measure of the dispersion of measured results around the average or mean.

Measuring Instruments Used

Voltage, Current and RPMAn Eagletree Systems Elogger V4 was used to record samples of voltage, current and RPM. The sampled data is available for off-line reduction and analysis using Excel.

The corrected value of voltage is estimated to be within +/- 1% of the true value. The voltage is not used in determining Cto or Cpo.

Current accuracy is not specified so it will be assumed that it is 1.5%.

There are no specifications for the RPM sensor. The Elogger has a crystal controlled time base used to count revolutions. Assuming a cheap crystal, the accuracy would be better than +/- 50 ppm or 0.05%.

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The logger samples the rpm with a resulting jitter around the mean. By averaging many samples the rpm can be estimated to within 0.1% with high confidence.

Force Measurement

Force (thrust) was measured using either an American Weigh Scales model SC-2KG unit or a USPS postal scale. Both units come calibrated from the factory but may drift with time and battery voltage. The units were calibrated using a set of calibrated weights good to 0.2 grams. Estimated accuracy is within +/- 0.2 gram for the AWS scale and 0.1 ounce for the USPS scale.

The test set consists of a vertical shaft with the motor mounted at its top and a horizontal shaft pressing on the scale at its end. The shafts are attached to an axle at their junction. The lever lengths are slightly different so that the thrust force is 0.95 times the scale measured force. The remaining error is negligible.

The motor-prop combination does produce vibration, showing up as jitter in the scale readings. A one pound weight is attached to the horizontal arm to reduce the jitter but some occurs, especially at high force levels. The scale is read by eye so there is some error introduced due to this jitter. An overall accuracy of 0.5% is assigned to the force measurements.

Air Density MeasurementAir density is calculated from air pressure, air temperature and dew point temperature. These parameters are measured using an ennoLogic model eA980R anemometer. The anemometer was compared to a local airport pressure reading that showed the anemometer read about 1% low. The nominal accuracy is about +/- 0.25%.

Accuracy SummaryVoltage +/- 0.9%Current +/- 1.5%RPM +/- 0.1%Force +/- 0.5%Air density +/- 0.25%, 1% bias

Thrust Coefficient Accuracy

The thrust coefficient is defined as:

Cto= Tρ∗N 2∗D4

T is thrust, lbfRho is air density, slugs/ft^3N is revolutions per secondD is prop diameter, feet

The estimated standard deviation is about +/- 0.6%

It is likely that the real result will be within twice that range or about +/- 1.2%.

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Power Coefficient CpoThe thrust coefficient made use of direct measurement of thrust and rpm. However, the prop shaft power is available indirectly through measurements of current, rpm and motor calibration functions. Hence it is subject to considerably more uncertainty.

Cpo = Pshaft

rho∗rpm3∗D5

Pshaft is a function of current, rpm, and Kv and residual nonlinearity losses.

Pshaft= rpmKv

( Itotal−Inl)

Where:Rpm is the revolution rate per minuteKv is the motor voltage constant, rpm/voltItotal is the total current into the motor, measured externallyInl is the no load current, including the effects of load on bearings.1

Rho and rpm are the same as for the thrust measurementsPshaft errors are conservatively estimated as +/-5%.Rho error: 0.25%Rpm error: 0.1%The estimated standard deviation is about +/- 5%.The approximate maximum range of error is twice or about +/- 10%.

1 Normally no load testing is done without any load on the shaft. Adding weight increases the bearing load and the no load current increases as well. Washers were added to simulate the weight of the prop & adaptor, resulting in an increase in no load current.

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This graph shows how closely the calculated shaft power compares to the wind tunnel calculated shaft power. All used measurements were made at full throttle & adjusted to match wind tunnel derived power.

The next graph shows the match for an APCE 9x7.5 propeller

The match is not as good as for the 10x7 propeller, especially at low rpm.

While this kind of match is good enough for some applications, it was decided that all subsequent measurements would be made at full throttle where it is possible to calibrate out most of the error by comparison to wind tunnel data.

Appendix One: Pitch Measurement Several methods for measuring pitch angle were tried, none totally acceptable. The first method was to make a HoopteeToo propeller pitch gauge enlarged and modified to read pitch angle directly, Figure 5. The gauge is satisfactory over a mid-range of angles but is subject to variability at small and large angles. The problem is that it is hard to judge when the gauge is parallel to the bottom of the prop when the chord is short, especially near the tip. It also suffers because it measures the bottom angle, not the chord line angle, introducing an error of about 1.5 degrees assuming a Clark Y airfoil. This method was abandoned.

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Figure 5 HoopteeToo Pitch Gauge

The second method tried was to modify a digital helicopter blade pitch gauge to better match the leading edge of a prop. The gauge is made for symmetrical airfoils while prop blades are asymmetric. The result is that the leading edge does not line up properly in the gauge. The solution was to fill in one side of the gauge with a hardwood plug that has a shallow groove in the middle to match the prop leading edge. Figure 6 shows the gauge modification while Figure 7 shows how the gauge is used.

Figure 6 Digital Pitch Gauge Modification

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Figure 7 Digital Pitch Gauge in Use

The gauge works well at moderate angles of attack but becomes subject to variation as the angle decreases and the chord becomes shorter. It is also difficult to measure large angles near the hub where the twist causes the gauge to slip easily and slight radius changes cause large pitch angle changes. Note that this gauge hangs off the blade, causing some twisting & reduction in angle.

A second gauge was obtained that was balanced around the mid-point, reducing any twisting. Measurements were made with the gauge upright and then inverted below the blade. The gauge was zeroed out and operated in the relative mode.

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To improve the results measurements are made three times above and three times below and the average of all adopted. It has been found that the angle precision (repeatability) is better than +/- 0.7 degrees except near the hub.

The third approach tried was to use MH-Aerotools.de prop analysis program. The technique requires taking photos of the top and side of a prop and running them through the program. This method did not work as the program requires the prop diameter to match exactly between the two photos, not possible with the photo software available. A similar method uses a scanner to take the two pictures but even though the length match error was with less than 0.5%, the program would not accept the photos.

Another method was to make vertical traces on flexible cardboard of the leading and trailing edges relative to a plane, and to trace the plan form on to another piece of paper. The pitch angle was the computed from these measurements. The method works OK where the chord is long and the difference in heights between the leading and trailing edges are large, the method suffers as the chord length decreases and the height difference becomes small. This method was abandoned as it gave considerable variation in results when repeated.

The final method attempted was to use a caliper to make the same measurements as the paper tracing method. This method proved to be fairly repeatable, although the variation was greatest near the prop tips. (Figure 6)

Figure 6 Digital Calipers Used to Measure Pitch Angle

Figure 7 plots measured pitch angle as a function of normalized blade station for three measurement methods that are compared to the UIUC data.

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Figure 8 Pitch Angle Comparisons

The degree of match to the UIUC data varies with method. The digital gauge method matches the UIUC data well at large angles but diverges at low angles. The results of this method are repeatable. The Hooptee method angles are everywhere lower than the UIUC data and show the same divergence at small angles as does the digital gauge method, but repeatability is poor. The direct method (using the calipers) tracks almost the same as the digital gauge method. All three methods diverge from the UIUC data at low angles, raising the possibility that the UIUC data is in error.

The 11x7 prop used for all of these presented measurements was then sliced in sections using a band saw. Figure 9 is a photo of one of the slices.

Figure 9 11x7 propeller slice

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These photos were then copied to paper and the pitch angle measured using trigonometry. Figure 10 shows how the sliced results compare to all of the other methods tried.

Figure 10 Angular Measurement Comparisons

The sliced method shows the same low angle characteristic relative to the UIUC data, i.e. lower. The conclusion is that the UIUC data is in error or that the pitch distribution has changed since the UIUC data was taken, over estimating the pitch angle on the outer 50% of the blade. The sliced method is the most accurate available. However, it is laborious to implement and destroys the evidence.

It was decided to use the digital pitch gauge #2 for all measurements. These measurements were used to verify that the propeller under test conformed reasonably well to the UIUC data. The UIUC average pitch distribution was incorporated in to the computer program and adjusted only where there was a major difference between the UIUC data and the measured data. There were a few cases where this occurred, notably the 9x6 propeller measured a narrower chord distribution but had a greater pitch distribution that more or less negated each other.

An experiment was run where the pitch distribution was assumed to be uniform (ideal) over the full blade length. It made virtually no difference in the computed results. Hence the distribution is not critical, only the actual pitch.

References:

http://m-selig.ae.illinois.edu/props/propDB.html

http://soar.wichita.edu/bitstream/handle/10057/773/t05031.pdf

“QPROP Formulation”, Mark Drela, MIT Aero & Astro, 2006 web.mit.edu/drela/Public/web/qprop

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