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Flight-Calc 2018 Documentation

W. B Garner, February 2018

Introduction

Background A usual way of estimating in-flight performance of electric powered model airplane is to use one of the on-line calculators, guessing about size of propellers, motors & related items, and then iterating the inputs until a ‘satisfactory’ result is obtained. While this method works it is cumbersome and doesn’t explicitly set performance goals as part of the input process. The Flight-Calc program starts with the desired flight performance criteria and builds from there.

Scope The program takes as input the characteristics of the plane to be analyzed and lets the user select a

particular flight condition for synthesis purposes. The user then selects a trial propeller size and the

program provides an estimate of the motor characteristics necessary to meet the flight criteria. This is

followed by a program that takes as input the propeller and battery size selected and a specific set of

motor specifications supplied by the user. The results are then presented in the form of a table and a

graph.

Structure The program is run in Excel and consists of five user sheets. There are other sheets that are internal to

the program, not intended for the user.

The first sheet, labeled “Notes”, provides instructions on how to run the program as well as describing

its limitations.

The second sheet, labeled “Airplane Inputs”, takes in user supplied information about the airplane’s

physical characteristics. Its output is a flight performance graph from which the user may select a

specific operating point to use in the following analysis.

The third sheet, labeled “Prop & Motor Selection”, takes as input the selected flight operating point &

user selected propeller sizes. The program then computes the required propeller rpm and input power

necessary to meet the flight conditions. Estimates of the required motor characteristics are then made

as a function of several battery voltages. The user can change the propeller size to see how the motor

characteristics must change to meet the flight criteria. The user then selects the desired battery voltage

and motor parameters for use in the fourth sheet.

The fourth sheet, labeled “Specific Prop-Motor Results”, has as input the results of sheet three and user

supplied motor specifications. The output is a Table listing the flight performance as a function of

airspeed as well as the motor and battery operating values.

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The fifth sheet, labeled “Results Summary”, is the final sheet, presenting in one place all of the

assumptions and results. The primary results are the Table from sheet four and a graph showing how

the analysis results relate to the original flight performance graph.

Airplane Input Sheet

Inputs The inputs for this sheet are wing area, wing span, weight, fuselage area, tail area and altitude. In

addition the user selects a wing type and a fuselage configuration that most closely matches the plane

from lists. The program estimates the drag as a function of airspeed, climb angle and airspeed and

estimates the thrust required to meet each flight condition.

Calculation Model Figure 1 is a diagram illustrating the forces acting on an airplane in climbing flight.

The resulting equilibrium forces are given by the following equations.

L = W*cos (Ɵ) T = D + W* sin (Ɵ) Where: L is lift W is weight T is thrust Ɵ is the angle between horizontal and the velocity vector V is the air speed The associated rate of climb, ROC, is given by:

ROC = (T*V – D*V)/W, ft/sec

And the angle of climb, AOC, is given by:

AOC=arcsine ((T-D)/W), radians

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The rate of climb or descent is zero when T =D, positive when T > D and negative when T < D. The ROC is

inversely proportional to the weight, so an increase in weight results in a decrease in climb rate. ROC

increases with a decrease in weight.

In the present case the airspeed, climb angle and weight are given and it is desired to find the required

thrust and the drag associated with the airspeed.

T = D + W*sin(AOC)

When the climb angle is zero the second term is zero and when the climb angle is 90 degrees the

required thrust is equal to the drag plus the weight. These two values of climb angle set the boundaries

of powered flight.

The program analyzes the drag as composed of four components. They are wing profile drag, wing

induced drag, tail drag and fuselage drag. The wing profile drag is modeled as a function of lift

coefficient, CL. The wing induced drag is modeled taking into account the effect of climb angle on the

resulting lift coefficient. It goes to zero when the climb angle is 90degrees, as the wing is then at zero lift

condition.

Formulas:

Cl = 𝑊

𝑟ℎ𝑜∗𝑆𝑤∗𝑉2∗cos(𝐴𝑂𝐶)

Cdprofile = F(Cl) F(Cl) is a polynomial with the variable being Cl

Profile Drag: Dprofile =𝐶𝑑𝑝𝑟𝑜𝑓𝑖𝑙𝑒 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑤 ∗ 𝑉2

Induced Drag: Dinduced =0.318∗𝑊2∗(1+𝑑𝑒𝑙𝑡𝑎)∗cos(𝐴𝑂𝐶)

𝑟ℎ𝑜∗𝑠𝑝𝑎𝑛2∗𝑉^2

Tail Drag: Dtail =𝐶𝑑𝑡𝑎𝑖𝑙 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑡 ∗ 𝑉2

Fuselage Drag: Dfus =𝐶𝑑𝑓𝑢𝑠 ∗ 𝑟ℎ𝑜 ∗ 𝑆𝑓 ∗ 𝑉2

Total Drag: Drag = Dprofile + Dinduced + Dtail + Dfuselage

Where:

rho is air density Sw is wing area V is airspeed W is weight Cl is lift coefficient span is wing span delta is a wing planform adjustment, = .05 Cdtail is tail drag coefficient (0.03) St is total tail area

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Cdfus is fuselage drag coefficient Sf is maximum fuselage cross-section area The program calculates the required equilibrium thrust for each airspeed- climb angle combination and

displays the results in the form of a graph.

Results The user selects an operating point of interest and enters the corresponding required thrust and

airspeed into an input box automatically transferred to the next sheet.

Example

This example illustrates how the program is used for selecting a flight operating point. The plane is an

electric powered stick-like one with the listed characteristics. The wing type selection number 2 is

chosen as the wing has a thick symmetrical cross section. The fuselage selection number 4 indicates that

the fuselage is shaped like a typical stick airplane with fixed landing gear.

Name Value Units

wing area 600 in^2

wing span 50 in

weight 32 oz

tail area 120 in^2

fuselage max cross section area 9 in^2

altitude 6800 feet

wing type selection number 2

fuselage type selection number 4

Drag Multiplier 1

Wing Type Selection List

Select # Description Typical Airfoil

1 Thin Symmetrical Eppler 478

2 Thick Symmetrical NACA 010

3 Thin Asymmetrical RG-14

4 Thick Asymmetrical Clark Y

5 Sail Plane S3014

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The resulting flight performance graph is shown in Figure 2.

The graph displays the required equilibrium thrust as a function of airspeed and climb angle. The top

curve is the case where the climb angle is 90 degrees (straight up) and where there is no wing lift. The

thrust must equal the plane weight plus the drag as the airspeed increases. Note that this curve

increases as the square of the airspeed, as do all of the others except near stall where the lower curves

terminate.

The lowest curve displays the thrust required at zero climb angle, or level flight. The plane stalls at about

15 MPH, the thrust decreases then increases as the airspeed increases. Other climb angle values lie

between the zero and ninety degree curves. Hence the plane operating range is bounded by these two

curves.

Fuselage Types

Select # Description Cdfus

1 War Bird, Fixed Gear 0.75

2 War Bird, Retracts 0.5

3 Trainer, Boxy, Cabin 1.5

4 Stick, fixed gear 1.2

5 Sailplane, no gear 0.25

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In this example, an operating point is selected at 40 mph and zero climb-angle. The required thrust is 2.6

pounds. Note that this thrust value should allow the plane to go vertical at about 20 mph or less.

The values of 2.6 pounds thrust and 40 mph are selected and passed to the next sheet.

Prop & Motor Selection

Summary This section is the central tradeoff part of the analysis. The user selects a propeller size, diameter and

pitch, from the list of allowed values. The program then calculates the propeller rpm and input power

required to meet the required thrust and airspeed values. Next the program estimates the required

motor current, power and Kv needed for each of four battery voltages. The user then selects the battery

voltage and corresponding motor properties for further detailed investigation.

Inputs The inputs are propeller diameter and pitch selected from a list of allowed APCE propellers ranging in

size from 8 to 14 inches. The thrust and power coefficients associated with these propellers were

derived from wind tunnel data with interpolation for sizes not included in the wind tunnel data. The

program will return an error if a size is not included in the list.

The input also includes a box for entering a Watts per gram value to help identify the weight of a

suitable motor.

Calculation Method The program then computes the value of rpm and input power required to match the required thrust for

the selected propeller.

Tcalc = 𝐶𝑡(𝐽) ∗ 𝑟ℎ𝑜 ∗ (𝑟𝑝𝑚

60)2∗ (

𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟

12) ∗ 𝑉2

J = 1056∗𝑉

𝑟𝑝𝑚∗𝐷 Where J is the advance ratio

Since rpm is not known, RPM is changed in increments until the value of Tcalc matches the value of

required T. The value of the required propeller input power is then calculated. The result is presented in

the form of a table. In this case the propeller size is 10x7.

Match Results

T Lbs Prop Watts Rpm

Pbatt Watts

Weight, grm

2.59 290 10763 341 114

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The results show that a thrust value of 2.59 pounds is achieved at a propeller input power of 290 Watts

and an rpm = 10,763. The estimated battery power is 341 Watts and the motor should weight about 114

grams.

The next part of the program takes these results and applies them to a generic electric motor model to

estimate the required Kv as a function of battery voltage.

A simple motor model is implemented, consisting of a battery in series with a battery resistance, ESC

resistance and a motor resistor and opposed by the motor EMF voltage. The total current is equal to the

battery power previously calculated divided by the battery voltage. The battery is assumed to be able to

support the full current for 6 minutes, its C rating is 35, and its voltage is a multiple of 4 Volts.

Rbattery = 0.037∗𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦

𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦 + .005

The ESC resistance is modeled as: Resc = = .281 ∗ 𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦−1.24

The motor resistance is modeled as: Rm =𝑎 ∗ 𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦−𝑏 where a = 6.9 and b = .843

Kv = 𝑟𝑝𝑚

𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦−𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦

𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦∗𝑅𝑡𝑜𝑡𝑎𝑙

The results for this example are listed in the following table.

Motor Performance Options

Ebattery Ibattery Kv

8.0 44 2176

12.0 29 1118

16.0 22 776

20.0 17 600

24.0 15 490

This table shows the tradeoff between battery voltage, current and Kv where each case matches the

thrust, prop input power and rpm requirements.

The next stage is for the user to select a battery voltage and input detailed actual motor specifications

using the selected propeller size. This example selects a voltage of 12 Volts with a corresponding Kv of

1118.

The selected values are used to estimate the thrust and other performance characteristics as a function

of airspeed as illustrated in the following table. These results are overlaid on the graph generated for

the Airplane Input section as shown in the following figure.

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Results

Vmph Tprop Iamps RPM Pout Pbattery

0 3.4 33.4 10643 289 401

10 3.4 31.9 10763 293 383

20 3.1 33.4 10643 291 401

30 2.9 33.4 10643 293 401

40 2.6 31.9 10763 290 383

50 2.2 30.5 10884 266 366

60 1.8 26.1 11246 245 314

70 1.4 21.8 11608 208 261

The graph illustrates how the thrust varies with airspeed and how it compares to the required thrust at a

given airspeed and climb angle.

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Specific Prop-Motor Results

Inputs A suitable motor will have a Kv of about 1000, have a power rating greater than 350 Watts, a current

rating greater than 30 Amps and a weight of about 114 grams. The next table lists the specifications for

an actual motor approximately meeting or exceeding these requirements. The weight is 134 grams.

Motor Parameters Kv 1000 rpm/volts

Rm 0.032 Ohms

Io 1 Amps

Vo 10 Volts

Imax 45 Amps

Pmax 670 Watts

Calculation Method There is no closed form method for matching the motor to the prop for a given set of required propeller

inputs. The method used is to increment the total current in steps from a low value to a high value,

calculate the resulting motor rpm and power. Then choose airspeed and calculate the prop input power

and thrust given the motor rpm for each current increment. Compare the motor output power to the

propeller input power until they match. At that current and rpm the solution is found. The airspeed is

then incremented to the next value and the process repeated, resulting in a table listing the results as

follows.

Formulas:

EMF = Ebattery – Itotal*Rtotal

RPM = Kv * EMF

Imotor = 𝐼𝑡𝑜𝑡𝑎𝑙 −𝐼𝑜

𝑉𝑜∗ 𝐸𝑀𝐹

Pout = 𝐼𝑚𝑜𝑡𝑜𝑟∗𝑟𝑝𝑚

𝐾𝑣

Pbattery = Itotal * Ebattery

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Results Summary The final sheet summarizes all of the inputs and assumptions used in the analysis and present the results

in the form of tables and a graph. The graph is similar to the one from the Prop and Motor Section

sheet.

The values of specific motor thrust are slightly lower than for the estimated thrust conditions, due in

part to the lower than estimated Kv.

Performance Results

Vmph Thrust, lbs I, amps RPM Pout, W Pbattery, W

0 3.2 31.9 10352 320 383

10 3.1 31.9 10352 320 383

20 3.0 31.9 10352 320 383

30 2.7 31.9 10352 320 383

40 2.5 27.6 10577 281 331

50 2.0 27.6 10577 281 331

60 1.6 23.2 10801 239 279

70 1.1 17.4 11101 181 209

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Comments The program uses the foot – pound – second units of measure. The Airplane Input section contains a

converter from metric to this system.

Drag coefficients are the parameters most subject to error in the Airplane Input section. The Airplane

Input sheet has a box labeled “Drag Multiplier”. Any number inserted into it will multiply the total

computed drag by that amount.

The motor model is a simplified version of an actual motor model. It assumes that the idle current for all

motors is equal to 1 Amp at 10 Volts. The battery, ESC and motor estimated resistances may be

different from actual devices.

Kv is the most sensitive parameter in selecting or analyzing a motor. The thrust varies as the square of

Kv. For instance, a difference of 5% will change the thrust by 1.05^2 = 1.10, or 10 %. The power is

proportional to the cube of the Kv so a 5% difference will result in a 1.05^3 =1.16, or 16% difference in

power. Actual motors tend to come in discrete Kv values, so matching the estimated Kv with a real

motor may be a challenge.

Appendices:

Airfoil Drag Coefficients

These drag coefficients were generated using the Profili XT program.

Thick Symmetrical: NACA 010

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Thin Symmetrical: Eppler 478 Thin Asymmetrical: RG-14 Thick Asymmetrical: Clark Y Sail Plane: S3014

Fuselage Drag Coefficients The fuselage drag coefficients were derived from Andy Lennon, “Basics of RC Model Aircraft Design”

Model Airplane News, 2002, Chapter 12.

The coefficient Cdfus can be estimated by the use of the values contained in the above figures, obtained

from wind tunnel tests at MIT. All of the fuselages were 43 inches long. Note that other combinations

can be derived from these results. For instance, fuselages 1 and 8 are the same except that number 8

has landing gear. The increase in the coefficient is thus due to the landing gear.

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Resistance Models The program requires values for battery, ESC and motor resistances. Models were developed for each

of these resistances based on available information.

Battery Resistance

Battery resistance is a function of the number of cells in series (Ncells), Ampere hour capacity (AH) and

Discharge rating (C ). The general form of the equation is:

Rbattery =𝑎∗𝑁𝑐𝑒𝑙𝑙𝑠

𝐴𝐻∗𝐶+ 𝑏

For new batteries the value of a is about 0.52 and b, an allowance for wire and cables, has a value of

about 0.005 ohms. A value of 35 is chosen for C. It is assumed that a full power flight time of 6 minutes

or 0.1 hours is typical. Since Time = AH/Ibattery, AH = T * Ibattery or AH = 0.1 * Ibattery. In the program

it is assumed that each cell has a voltage of 4 volts, so Ncells = Ebattery/4.

Substituting:

Rbattery = .037∗𝐸𝑏𝑎𝑡𝑡𝑒𝑟𝑦

𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦 +.005 Ohms

ESC Resistance

ESC resistance is usually very small but it is included for completeness.

Resc = . 281 ∗ 𝐼𝑏𝑎𝑡𝑡𝑒𝑟𝑦−1.24 Ohms

Motor Resistance

Rmotor = 6.9 ∗ 𝑃𝑏𝑎𝑡𝑡𝑒𝑟𝑦−0.842 +.005 Ohms

The motor resistance formula was derived by plotting the relationship between resistance and motor

power for a large number of motors. There is considerable scatter in the results so this formula may be

in error compared to a selected motor with known specifications. Nevertheless it provides a means of

estimating an approximate value for Kv. The added value of 0.005 is to account for connectors and

wires.