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Example RC Plane Design for Long Flight Times

Basic Theory, Example Application and Variations

William Garner

12/1/2019

Achieving long flight times in fixed wing RC aircraft is a subject that comes up in online forums involving FPV. This document presents the theory on flight time and design, explains each component of the theory and provides a series of examples illustrating how it would actually work in practice. The keys to long flights are: 1) High aspect ratio wings with low Reynolds Number airfoils, 2) a high pitch/diameter propeller, low resistance and low no-load current motor, matched to each other so that both operate at maximum efficiency at the nominal cruise airspeed, 3) Lithium Ion batteries where the weight of the battery is approximately the same as the rest of the plane, 4) emphasis on light weight construction designed for moderate aerodynamic forces and 5) light as practical wing loading. Following these guidelines it is possible to design a plane that can fly for many hours with no payload. A payload can be added by trading battery weight for payload weight with a corresponding reduction in flight time.

1

Contents Purpose ......................................................................................................................................................... 2

Theory ........................................................................................................................................................... 2

Equation .................................................................................................................................................... 2

Interpretation ........................................................................................................................................... 3

Example Application ..................................................................................................................................... 6

Example #1 Battery Size as Variable ....................................................................................................... 10

Example # 2 Varying Non-Battery Weight .............................................................................................. 10

Example # 3 Varying Wing Span and Chord ............................................................................................ 11

Example # 4 Adding a Payload ................................................................................................................ 12

Example #5 Different Airfoil .................................................................................................................... 12

Example # 6 Varying No-Load Current .................................................................................................... 14

Propeller Efficiency, Motor Efficiency and Reynolds Number .................................................................... 14

References .................................................................................................................................................. 15

Appendix 1 Plane Component Weight Estimates ....................................................................................... 16

Appendix2 Matching a Motor to a Propeller .............................................................................................. 19

Part 1 Maximum Efficiency Cruise .......................................................................................................... 19

Part 2 Full Power Operation.................................................................................................................... 21

Summary ..................................................................................................................................................... 24

Appendix 3 Formulas and Excel Functions .................................................................................................. 26

2

Purpose The use of RC planes for FPV use is now wide spread. One of the frequent questions that arise is how to

obtain long flight durations or long flight ranges. This document explores the first of these applications

though the second is closely related.

The purpose of this document is two-fold. The first purpose is to present the theory of flight duration

time with explanations of how the various components affect the results. The second purpose is to

present specific examples as a way to illustrate the application of the theory.

Theory The derivation of the theory is provided in a companion document (Reference 1) so will not be repeated

here. The basic equation developed in that reference is presented with discussions on each of the

components of the theory.

Equation The duration equation is as follows:

( )

√

∑

Definitions

Kp: Propeller efficiency defined as the ratio of power used for thrust to power input to the propeller

Km: Motor efficiency defined as the power out of the motor into the propeller divided by the input

power to the motor. This includes external losses such as battery or cable resistive losses.

Kb: The battery specific energy density defined as the ratio of Watt-Hours to per unit battery weight

Wb: The weight of the battery

Wa: The weight of the plane without the battery.

ρ: The density of air, weight per unit volume

g; The gravitational constant

Sw: The effective wing area

Cl: The wing lift coefficient at the operating point angle of attack (AOA)

∑ : The sum of the drag coefficients including wing profile and induced drag, fuselage drag and tail

drag at the AOA operating point.

3

Interpretation The equation consists of three major parts. One part is the power system efficiency which is a function

of the propeller’s reaction to air speed and rpm and the motor’s reaction to load and rpm.

Another part consists of the physical fixed parts such as weight and wing area.

The third part incorporates the aerodynamic properties of the plane primarily that of the wing lift and

drag. These properties are a function of airspeed and Reynolds number.

Each of the terms in the equation will be described in more detail with the objective of showing how

they affect the overall results.

Power System Efficiency

Power system efficiency is the measure of how well power out of the battery transfers power to the

thrust airstream. It is the product of the propeller efficiency, Kp, and the motor efficiency, Km.

Kp is actually a function of the propeller pitch, diameter, revolution rate and airspeed. Figure 1-1 is a

generic plot of Kp as a function of the advance ratio, J, and the ratio of pitch to diameter. J is defined as:

Where V is the airspeed, n is the revolution rate and D is the diameter.

Figure 1-1 Generic Propeller Efficiency Properties

There are two observations about these curves related to duration flying. The first is that the greater the

ratio of pitch to diameter the better the peak efficiency; hence ‘best’ efficiency favors the highest

pitch/diameter ratio.

4

The second observation is that there is an optimum value of J that makes the efficiency the greatest.

Note that, for a given diameter, it depends on the ratio of V/ n. V tends to be determined by the

aerodynamic properties of the plane (sufficient to fly at the optimum point), leaving n as the primary

adjustable variable determined by the power system properties. This aspect of the issue will be

illustrated in the example section.

Motor efficiency is determined by a combination of characteristics and the operating conditions. Figure

2-2 is a typical plot of motor performance as a function of input current. Note that the efficiency is low

at low current levels. It rises quickly as current increases, reaches a maximum and then slowly declines.

However, the greater the current the greater the battery current drain and the less the flight duration.

Therefore a goal is to select a motor that produces low current at high efficiency at the required power

output and rpm demanded by the propeller.

Figure 2-2 Typical Motor Performance Characteristics

Battery Power Density

Maximum flight time occurs when the value of Kb is maximized. Table 1-1 lists candidate battery types

with their associated energy densities.

5

Table 1-1 Battery Energy Densities

Type Energy Density, W-Hrs/Kg Notes

Lithium Polymer, 30C 140

Lithium Polymer, 10C 175

Lithium Ion 200 20 Amp/Cell Discharge

Lithium Ion 250 10 Amp/Cell Discharge

Lithium Ion batteries are the clear type choice.

Plane Weight Components

The ratio of battery weight to non-battery weight, Wb/Wa, is key factor in time duration. In particular

the value of K (Wb,Wa) is of interest (Figure 1-2):

( )

(

)

Figure 1-2 Weight Ratio Results

This result indicates that the ratio of battery to non-battery weights tends to a maximum for a ratio of

about 1:1. Beyond that point the time gain is small while the plane becomes heavier and less

aerodynamic. 1:1 is a good goal for maximum flight duration.

Wing Loading

Time increases as wing loading decreases.

Air Density

Air density is more or less constant for a given altitude and is not something a pilot can change other

than by relocating.

6

Lift and Drag Coefficients

Lift and drag coefficients are a function of air speed (AOA ) and Reynolds number. Since the longest

flight times are achieved at low air speeds where the Reynolds number tends to be small, selecting an

air foil for low Reynolds number qualities is essential.

Example Application As stated the formula does not lend itself to observing the actual effects of flight on time duration since

some of the parameters are air speed dependent. Therefore a hypothetical design will be presented and

its performance with airspeed estimated. Variations will be explored to show how the changes modify

the results.

Plane Dimensions

The reference plane has the dimensions given in Table 3-1.

Table 3-1 Reference Plane Dimensions

Item Dimension Notes

Bw, Wing Span 120 (3.048) Inches (meters)

Cw, Wing Chord 10 (254) Inches (mm)

Sw, Wing Area 1200 (0.774) In2 (m2)

St, Tail Area 240 In2 (20% of wing area)

Sf, Fuselage Cross Section 12 In2 (Sized to battery dimensions)

Wa, Weight, no Battery 35 Ounces

Plane Weight Model

One of the objectives is to estimate the changes in flight time due to changes in weight. The following

formula provides an estimate based on chord and span dimensions. It was derived from sample builds

and known equipment weights (motor, prop, etc.).

( ) , Ounces

Power System Model

The power system consists of the propeller, motor and battery.

The battery is assumed to be composed of three 3AH, 3.6 Volts cells each in series, 10.8 Volts total with

P series sets in parallel. Each set is rated 32.4 Watt-Hours and weighs 5 ounces.

Total weight is P*5 ounces and capacity is P*32.4 Watt-Hours.

7

The propeller is an APCE 12x 12, selected for its pitch/diameter ratio of 1 and because there is wind

tunnel data available for its thrust and power coefficients as a function of the ratio of J, the advance

ratio. Figure 3-2 plots the coefficients as a function of advance ratio. Note that the maximum efficiency

is about 70% at J = 0.7; the goal for the optimum operating point in this case.

Figure 3-2 Propeller Coefficients

The primary motor losses are resistive loss and iron loss. The motor wires have resistance and the loss is

given by:

, Watts

I is the current, Amps Rm is the resistance Ohms. Minimizing Rm minimizes resistance loss. Iron loss is caused by motor friction and circulating currents in the metal parts on which the wire is

wound. It takes a minimum amount of current to start and keep the motor running. This current is

known as the no-load or idle current and is proportional to rpm. It is measured at a fixed input voltage,

Vo and is designated Io. The loss is given by:

Watts

Kv is a motor constant.

The lower the value of Io is the lower the losses. Therefore a motor with low resistance, Rm, and low Io

is desirable.

8

The propeller and motor are coupled together and interact. Changing the applied voltage changes the

RPM which changes the propeller speed which in turn changes the current drawn from the motor which

in turn modifies the RPM. There is an equilibrium condition for any voltage setting (throttle setting), but

it is not readily calculated. Note that the resistive and iron losses are also changed as the current is

changed.

The motor specifications are for a T-Motor model MN3110 – Kv700.

Kv = 700 rpm/volt Rm = 0.095 Ohms Io = 0.3 Amps Vo = 10.0 Volts Weight = 80 Grams

Airfoil Reference Model

Figure 3-3 is a plot of the relationship between lift coefficient and power coefficient for an S3010 airfoil,

showing how Cdp is affected by Reynolds number. This air foil was chosen because it has relatively good

(stable) characteristics as Reynolds number decreases. At the relatively slow speeds near stall where

best time duration is found the Reynolds number can be below 100,000 where drag increases

significantly. In order to use this data the plots were transformed into a data set and interpolation

employed to find values not explicitly in the plots.

Figure 3-3 S3010 Airfoil Coefficient Plots

9

Figure 3-4 is the same data shown as a function of AOA, as well as a drawing of the airfoil. It lends itself

to straight forward construction.

Figure 3-4 S3010 Polar Plots

The remaining drag coefficients are for the wing induced drag, the fuselage drag and the tail drag.

Induced Drag Coefficient:

Fuselage drag coefficient:

This value assumes a circular cross section with a smooth

ovoid shape along its length.

Tail drag coefficient:

This value assumes a symmetrical true airfoil.

Note that both Cdp and Cdi are functions of Cl.

10

Example #1 Battery Size as Variable Calculating the value of time given all of these variables is complicated. The details of how to do them

are contained in Appendix 2.

The first example provides estimates of time duration as a function of air speed with the battery sizes as

a variable.

Example 1 Hours as a Function of Airspeed with the Number of Parallel Battery Sets as a Variable

This graph illustrates two characteristics of this design. First, the time starts off at a minimum, climbs

steeply, reaches a peak and then declines more slowly. The peak occurs at airspeed approximately 1.3

times the stall speed.

Second, the time increases rapidly from p=1 to about p=4, then the increase slows and nearly flattens

out beyond P=7. At P= 7 the battery weight equals the non-battery weight.

The graph also shows that it is possible to achieve very long flight times, on the order of at least 10

hours or more.

Example # 2 Varying Non-Battery Weight Example # 2 fixes P=6 and varies the non-battery weight by plus and minus 10%. Increasing the weight

by 10% decreases the peak time by about 5 % while decreasing the weight by 10% increases the peak

time by about 8%.

11

Example #2 Effect of Varying the Non-Motor Weight on Time Duration

Example # 3 Varying Wing Span and Chord This example shows the effects of varying wing span and chord.

Example # 3 Variation with Changes in Wing Span and Chord Dimensions

Increasing the span from 120 inches to 144 inches increases the peak time for either chord length. The

narrower chord has a slight advantage over the 10 inch chord version. Reducing the span to 96” with a

chord length of 10 inches greatly reduces the peak time. This shows the value of long wing span as it

minimizes the induced drag present at these low end speeds where the AOA is high and so is the lift

coefficient.

12

Example # 4 Adding a Payload The previous examples showed various effects for the cases where there was no weight allocated to a

payload. Those are good examples where the objective is to achieve maximum flight time only. Of some

interest is what happens when a payload is added.

This example assumes that a payload of 1 pound (16 ounces) is added when the battery P = 3. Total

weight is then 66 ounces. The reference is taken for P=3 and a weight of 50 Ounces, equal to the weight

of the battery and the bare plane.

The no-payload maximum time is about 13 hours. It is reduced to about 9 hours when the payload is

added. This result indicates that it is possible to design a plane carrying substantial payload (in this case,

24% of the total weight) and cruise for several hours.

Example #4 Times With and Without Separate Payload

Example #5 Different Airfoil The reference airfoil, an S3010, was selected because it has good low Reynolds Number properties and

is straight forward to build. The reference design flies relatively slowly, so would a slower speed design

outperform the S3101? To answer that question a very high lift coefficient airfoil, an S1223 was analyzed

and compared to the reference design. The airfoil plots show that the peak Cl can exceed 2, but that the

drag coefficient rises rapidly from its smallest value and is sensitive to Reynolds Number.

13

The comparison to the reference is shown in graph Example #5 Airfoil Comparison.

Example #5 Comparison of Different Airfoils

The S1223 airfoil operates at lower airspeed than the S3010 but is not as efficient. It also has a narrower

range of airspeeds in which to fly efficiently. It will also be difficult to build accurately.

14

Example # 6 Varying No-Load Current No-load current is required to start and keep the motor running. The greater the current required, the

more power is wasted in the motor. This example shows the effect of increasing the value of Io from 0.3

Amps to 1.0 Amps, a value fairly typical of many RC motors.

The effect of increasing Io is dramatic, showing how critical it is to achieving long flight duration.

Propeller Efficiency, Motor Efficiency and Reynolds Number A goal of any such design is to match the motor and propeller such that both achieve near maximum

efficiency. Figure 4-1 shows the results for the reference design. Both the propeller and motor are

operating near maximum efficiency across the air speed range. The Reynolds Number is greater than

100,000 so that the wing drag is stable and well behaved.

Figure 4-1 Reference Model Power Efficiency and Reynolds Number with Air Speed

15

Table 4-1 is a printout of detailed calculations for the reference design. It shows how the makeup of the

total drag coefficient changes as the air speed and therefore the lift coefficient, changes. At low speeds

it is dominated by induced drag that decreases as the speed increases while the parasitic term also

decreases but relatively slowly. This trend points to decreasing the induced drag by increasing the wing

span while keeping the wing area relatively constant. Example # 3 illustrates this point.

TABLE 4-1 Example Detailed Calculations

References #1 this document develops the theory for flight duration.

https://rcaeronotes.wordpress.com/design-and-analysis-notes/ , Designing for Flight Duration Time

#2 this document contains tables of APCE propeller Ct and Cp coefficients as a function of advance ratio

for diameters from 8 to 19 inches. It is in Excel format.

https://rcaeronotes.wordpress.com/home/propellers-2/, APCE Propeller Thrust and Absorbed Power

Excel Program

#3 this site contains information on a huge collection of airfoils and their associated polar plots.

http://airfoiltools.com/ (not secure)

vmph Cl RE Cdp Cdi Cdfus+tal Cdsum Drag, oz Pdrag, W RPM Pprop, W Pbattery hours Motor-Eff Prop-Eff

12.5 1.22 97611 0.028 0.0395 0.0067 0.074 3.9 6.1 2146 9.5 11.7 16.7 0.82 0.64

13.5 1.05 105411 0.020 0.0290 0.0067 0.055 3.4 5.8 2061 8.4 10.3 18.9 0.81 0.69

14.5 0.91 113211 0.017 0.0218 0.0067 0.046 3.3 5.9 2057 8.3 10.2 19.1 0.81 0.71

15.5 0.79 121011 0.016 0.0167 0.0067 0.039 3.2 6.2 2085 8.6 10.5 18.5 0.81 0.73

16.5 0.70 128811 0.015 0.0130 0.0067 0.035 3.2 6.6 2125 9.0 11.1 17.6 0.81 0.73

17.5 0.62 136611 0.014 0.0103 0.0067 0.031 3.2 7.0 2171 9.5 11.7 16.6 0.82 0.74

18.5 0.56 144411 0.013 0.0082 0.0067 0.028 3.2 7.5 2219 10.1 12.3 15.7 0.82 0.74

19.5 0.50 152211 0.012 0.0067 0.0067 0.025 3.3 7.9 2271 10.7 13.1 14.9 0.82 0.74

20.5 0.45 160011 0.011 0.0055 0.0067 0.023 3.3 8.4 2331 11.5 14.0 13.9 0.82 0.73

21.5 0.41 167811 0.010 0.0045 0.0067 0.021 3.4 9.0 2397 12.4 15.0 12.9 0.82 0.73

22.5 0.38 175611 0.010 0.0038 0.0067 0.020 3.5 9.7 2468 13.4 16.2 12.0 0.83 0.73

23.5 0.35 183411 0.009 0.0032 0.0067 0.019 3.6 10.5 2544 14.5 17.6 11.1 0.83 0.72

16

Appendix 1 Plane Component Weight Estimates The calculations in this document are based on measured and estimated values for component weights.

Table 1 lists the components used in this document except for the wing and tail boom.

Table 1 Component Weights

Weight Ounces Source

prop 1 APCE

spinner 1 Estimate

esc 1.3 YEP 40A

rx 0.9 measured

tlm log 0.9 measured

tlm tx 0.9 measured

servos (2) 0.9 measured

tail 2 Estimate

fus 4.5 measured*

sub tot 13.4

misc 2.6

total 16

The weights of items marked with an asterisk are based on measured values for prototypes build for the

purpose here. Photo A1-1 shows the boom, made of balsa. It weighs 1.5 ounces and is 24 inches long.

The actual boom would be on the order of 36 inches and weigh more. The boom length is dependent

upon the wing span so its weight is included in that of the wing.

Photo A1-1 Balsa Boom

Photo A1-2 shows the fuselage. It was designed around a P=6 battery pack, located on the C.G.

However, the weights should be comparable. The fuselage is more or less rectangular while an actual

fuselage would have a more rounded and streamlined shape.

17

Photo A1-2 Prototype Fuselage

The wing weights were generated from several sources. One was from the one-piece wing shown in

Photo A1-3. It is 9 feet in length with an average chord on 8 inches, constructed of balsa, some ply with

carbon fiber main spar and basswood outer spars. It weighs 15 ounces.

Photo A1-3 Nine Foot Wing

18

In addition, a prototype wing panel was laid out, its part weights estimated and then constructed. Photo

A1-4 shows the section. The estimated weight was 2.54 ounces while the measured weight was 2.68

ounces.

Photo A1-4 Prototype Wing Panel

This panel is all balsa with balsa spars and shear webs. It is 24 inches in length, 10 inches in chord and

uses an S3101 airfoil. It is suitable for outer wing panels but the spars are inadequate to handle the

stresses at wing center. The solution is to replace the balsa spars with carbon fiber spars but keep the

rest of the design.

The wing weight is composed of two types. One type is the spars, leading edge and trailing edge which

are proportional to length. The other type is proportional to area, assuming the thickness is not

changed. These parts include the ribs, sheeting, braces, shear webs and covering.

Between all of these inputs a formula was developed for estimating wing weight, including the boom, as

a function of chord and span.

Wing = (0.0108*chord+0.0266)*span + 3, ounces

19

Appendix2 Matching a Motor to a Propeller There are two parts to matching a motor to a propeller. The first part is to match them at the nominal

cruising speed and seek maximum efficiency for both. The second part is to determine how the model is

required to fly with full power for climbing and maneuvering and select a motor that also meets that

requirement.

Part 1 Maximum Efficiency Cruise The first objective is to select the advance ratio J that maximizes propeller efficiency while also providing

the required thrust at the desired cruise air speed. The second objective is to find motor Kv and other

parameters that meet the propeller requirements of RPM and power while also achieving maximum

motor efficiency. This is an iterative process as there is no simple way of making the match, nor is there

a single answer.

Objective 1

Givens:

Vmph Cruise airspeed D Propeller diameter A set of values for propeller Ct and Cp for specific values of J around the peak propeller

efficiency (Figure3-2)

Find: The value of rpm where rpmj = rpmt

( ) V in mph, D in inches

√

(

)

T in Oz, D in inches, rhog = .002378*sigma

V = 16.5 mph

D = 12 inches

T = 3.5 ounces

Table A2-1

J Ct Cp rpmj rpmt

.6 .090 .079 2479 1918

.7 .079 .077 2125 2047

.8 .065 .073 1859 2257

20

Plot rpmj and rpmt versus J

The two values of rpm are equal when J =0.71 and rpm = 2050. These values are then used to estimate a suitable value of motor Kv. The value of Cp is about 0.76.

Objective 2 Finding Motor Kv

This step requires some guesses about the value of motor Io, Vo and Rm. The battery voltage is already established.

Table A2-2 Motor Givens

Item Value Units

Ebattery 10.8 Volts

Rm .1 Ohms

Io .03 Amps

Vo 10 Volts

Cp 0.76

Rpm 2050 Rev/minute

The next step is to determine the motor output power required by the prop at the given rpm.

(

) (

) Watts

Pprop = 10.5 Watts

21

Using the Excel function pinesc and a list of possible values for Kv the required battery power is found as is the associated motor efficiency. Table A1-3 lists the results. Note that there is a wide range of Kv values to choose from with the smaller values having the best efficiency. These results are not sufficient for making a decision. For that the full power operation needs to be considered.

Table A2-3

Part 2 Full Power Operation Full power operation occurs when the throttle is raised to maximum and the full battery voltage is

applied to the motor. The propeller responds, increasing in rpm and power draw until an equilibrium

state is met. There is no simple equation to find this equilibrium state so some form of iteration is

necessary. It is possible to write an Excel function that does the iteration. What follows uses a manual

method.

The objective is to find an rpm where the motor output power and the propeller input power is the

same.

( )

( )

(

)

(

)

The motor equation does not include the no-load current as it is small relative to the resulting full power current and adds little to the results. For this calculation use the static (J=0) values for Cp = 0.082 & Ct = 0.12 as the greatest power consumption occurs then. The motor values are those from Table A2-2. Conduct a series of calculations for each value of Kv for the motor and the propeller as a function of current, I. Also calculate the associated propeller thrust. The next three tables show the results for Kv values of 500, 600 and 700. For each there is a current and rpm where the motor and propeller are in equilibrium. The graphs show the cross-overs and the extracted matches are shown in the last graph and table.

Kv Pbattery efficiency

500 12.7 0.83

600 12.8 0.82

700 13.0 0.81

800 13.3 0.79

900 13.7 0.77

22

Table A2-4 Kv=500 Trades

kv 500

I rpm Pmotor Pprop Tprop

10 4900 98 144 30

15 4650 139.5 123 27

20 4400 176 104 25

25 4150 207.5 87 22

30 3900 234 73 19

TableA2-5 Kv = 600 Trades

kv 600

I rpm Pmotor Pprop Tprop

10 5880 98 249 44

15 5580 139.5 213 39

20 5280 176 180 35

25 4980 207.5 151 31

30 4680 234 125 28

23

Table A2-6 Kv = 700 Trades

Table A2-7 Matched Values

Figure A2-1 Full Power Matches

kv 700

I rpm Pmotor Pprop Tprop

10 6860 98 395 60

15 6510 139.5 338 54

20 6160 176 286 48

25 5810 207.5 240 43

30 5460 234 199 38

Kv Imatch Pmatch Tmatch

500 13 130 29

600 20 180 35

700 27 220 42

24

Figure A2-1 can be used to choose a value for Kv based on full power thrust and power demands. There

are some rules of thumb sometimes employed for this purpose. One is to determine the ratio of thrust

to weight; the greater this ratio the greater the aerobatic capability. One set says that at a ratio of 0.5:1

the plane can do mild climbs and maneuvers while at 1:1 can do about anything except hover. In the

present case there is only a requirement for mild climb and maneuvering capability so 0.5:1 is a choice.

Since the weight is 65 ounces the thrust should be at least 33 ounces; from the chart that corresponds

to a Kv =600. The maximum current is 20 Amps, the output power is about 170 Watts and the battery

power is Pbattery = 20*10.8 = 216 Watts.

There is another rule of thumb that says a motor can safely handle input power at the rate of 3 Watts

per gram. Therefore the motor should weigh about 72 grams or 2.5 ounces.

Summarizing the derived motor characteristics:

Table A2-8 Derived Motor Characteristics

Parameter Value Units & Notes

Kv 600 Rpm/volt

Rm <= 0.1 Ohms

Io 0.3 Amps

Vo 10 Volts

I max >=20 Amps

P max 220 Watts

Weight >2.5 ounces

Finding a motor that meets all of these requirements may be a challenge. As long as the values for Rm

and Io are respected the value of Kv can vary, keeping in mind the effects on full power performance.

Summary Long flight times may be achieved by using the following guidelines.

1 Employ a high aspect ratio wing with a low Reynolds Number airfoil

2. Choose a wing chord that keeps the Reynolds number above about 100,000 if possible at the cruise

airspeed.

3. Employ a high pitch/diameter ratio propeller and a motor with low resistance and no-load current.

4. Match the propeller and motor both for maximum efficiency at the desired cruise airspeed.

5. Employ Lithium Ion batteries

6. Select the ratio of battery weight to non-battery weight to fit the mission. For flights with no payload

make the batteries weight approximately half of the total weight.

25

7. For flights with a payload trade battery weight for payload weight and flight time.

8. Construct for low as possible wing loading while maintaining adequate protection for modest

maneuvering.

Following these guidelines will result in the potential for multi-hour flight times.

As indicated by the examples there are many ways that the actual flight times may be less than

predictions. A few are battery capacity less than advertised, power system efficiency less than predicted,

overall drag greater than estimated, power used for other things like servos, climb and other maneuvers

and how well the pilot can maintain the desired cruise speed. Despite these potential losses it is possible

to achieve multi-hour flights with a plane specifically designed for it.

26

Appendix 3 Formulas and Excel Functions

The process of calculating a single time value is as follows:

Select an airspeed value greater than stall: √

Calculate Cl for level flight:

Calculate the value of sigma: (the ratio of air density at some altitude relative to sea level)

Calculate Reynolds number: chord in inches, V in mph Calculate the values of the drag coefficients and sum:

Cdp from the Excel function cdsx3010(cl,re)

Induced Drag Coefficient:

Fuselage drag coefficient:

Tail drag coefficient:

Calculate the drag: Set the propeller thrust equal to the drag and find the RPM required.

Use the Excel function: rpmmatchy(Trequired,vmph,diameter,tcof4,tcof3,tcof2,tcof1,tcof0,sigma)

Calculate the prop absorbed power meeting the RPM: Use the Excel function:

( ) Calculate the input power to the motor using the Excel function:

( )

Eb is battery voltage, rbattery is battery resistance, resc is ESC resistance Calculate the time by dividing the battery watt-hours by the motor input Watts.

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Function cdsx3010(cl, re) ‘Calculates the profile drag coefficient for an S3010 air foil ‘ cl is lift coefficient, re is Reynolds number rx = re / 1000 Select Case rx Case Is < 60 cdsx3010 = 0.0502 * cl ^ 4 - 0.0973 * cl ^ 3 + 0.0588 * cl ^ 2 - 0.0041 * cl + 0.0205 Case 60 To 80 lower = 0.0502 * cl ^ 4 - 0.0973 * cl ^ 3 + 0.0588 * cl ^ 2 - 0.0041 * cl + 0.0205 upper = 0.1023 * cl ^ 4 - 0.2608 * cl ^ 3 + 0.2339 * cl ^ 2 - 0.0786 * cl + 0.0252 cdsx3010 = (rx - 60) / 20 * (upper - lower) + lower Case 80.00001 To 100 lower = 0.1023 * cl ^ 4 - 0.2608 * cl ^ 3 + 0.2339 * cl ^ 2 - 0.0786 * cl + 0.0252 upper = 0.1009 * cl ^ 4 - 0.2631 * cl ^ 3 + 0.244 * cl ^ 2 - 0.0868 * cl + 0.0241 cdsx3010 = (rx - 80) / 20 * (upper - lower) + lower Case 100.00001 To 150 lower = 0.1009 * cl ^ 4 - 0.2631 * cl ^ 3 + 0.244 * cl ^ 2 - 0.0868 * cl + 0.0241 upper = 0.1204 * cl ^ 4 - 0.3099 * cl ^ 3 + 0.2777 * cl ^ 2 - 0.0934 * cl + 0.0203 cdsx3010 = (rx - 100) / 50 * (upper - lower) + lower Case 150.00001 To 200 lower = 0.1204 * cl ^ 4 - 0.3099 * cl ^ 3 + 0.2777 * cl ^ 2 - 0.0934 * cl + 0.0203 upper = 0.0763 * cl ^ 4 - 0.1935 * cl ^ 3 + 0.1739 * cl ^ 2 - 0.0581 * cl + 0.015 cdsx3010 = (rx - 150) / 50 * (upper - lower) + lower Case Is > 200 cdsx3010 = 0.0763 * cl ^ 4 - 0.1935 * cl ^ 3 + 0.1739 * cl ^ 2 - 0.0581 * cl + 0.015 End Select End Function Function thrust(vmph, rpm, diameter, tcof4, tcof3, tcof2, tcof1, tcof0, sigma)

‘Calculates thrust in ounces. Parameters tcofx are the propeller Ct polynomial coefficents

j = 1056 * vmph / rpm / diameter n = rpm / 60 d = diameter / 12 ct = tcof4 * j ^ 4 + tcof3 * j ^ 3 + tcof2 * j ^ 2 + tcof1 * j + tcof0 thrust = ct * 0.002397 * sigma * n ^ 2 * d ^ 4 * 16 End Function

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Function rpmmatchy(Trequired, vmph, diameter, tcof4, tcof3, tcof2, tcof1, tcof0, sigma)

‘Finds the rpm required to meet a given thrust (in ounces). The tcofx are CT polynomial coefficients

guess = 10000 For i = 1 To 5 rpm1 = guess rpm2 = 0.8 * guess rpm3 = 1.2 * guess delrpm = rpm3 - rpm2 r1 = thrust(vmph, rpm1, diameter, tcof4, tcof3, tcof2, tcof1, tcof0, sigma) - Trequired r2 = thrust(vmph, rpm2, diameter, tcof4, tcof3, tcof2, tcof1, tcof0, sigma) - Trequired r3 = thrust(vmph, rpm3, diameter, tcof4, tcof3, tcof2, tcof1, tcof0, sigma) - Trequired delr = r3 - r2 delrpm = -r1 * delrpm / delr guess = guess + delrpm Next i rpmmatchy = guess End Function

Function pprop(vmph, rpm, diameter, pcof5, pcof4, pcof3, pcof2, pcof1, pcof0, sigma)

‘Finds the propeller input power for the given airspeed, vmph, and rpm.

‘The pcofx are the propeller power polynomial coefficients

j = 1056 * vmph / rpm / diameter n = rpm / 60 d = diameter / 12 cp = pcof5 * j ^ 5 + pcof4 * j ^ 4 + pcof3 * j ^ 3 + pcof2 * j ^ 2 + pcof1 * j + pcof0 pprop = 1.345 * cp * 0.002376 * sigma * n ^ 3 * d ^ 5 End Function

Function pinesc(wout, rpm, io, vo, rm, kv, Eb, Rbattery, resc)

'Computes power into ESC given output power and rpm Emf = rpm / kv Im = wout / Emf Inl = io * Eb / vo iout = Im + Inl Emot = iout * (rm + resc) + Emf A = iout * Rbattery b = Eb c = Emot d = (b - (b ^ 2 - 4 * A * c) ^ 0.5) / 2 / A pinesc = iout * Eb * d End Function