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Model Aircraft Performance Calculator Guidebook

        Manual
        developed by:      Design Services Printware

                            Contents
                           a) General instructions
                           b) Technical information
                                     -Engine power
                                     -Airspeed
                                     -Thrust
                                     -Wing loading
                                     -Stalling speed
                                     -Density Altitude
                           c) Coefficient of Lift data
                           d) References

Copyright C 1997 Design Services

Instructions

The slide rule is constructed of two circular disk’s, connected at the center so as to allow rotation of the inner disk relative to the outer disk.

To use the slide rule, grasp the outer disk in the hands and rotate the inner disk with the thumbs.

To calculate a result, the inner disc is rotated to align one number on one disc, with another number on the other disc, the answer is read by looking at the range of numbers visible through the window of the inner disk, the answer is indicated by the arrowhead pointing at the window, or from the numbers adjacent to the window.

To see how this works, lets calculate the stalling speed of a an airplane. The plane has a 50 inch span with a 10 inch average chord, the airfoil is symmetrical. The plane weighs 4 pounds. The altitude of the flying site is 3000 Ft. The air temperature is 80 Degrees F.

First calculate the wing area by multiplying the wingspan by the average wing chord
50" x 10" = 500 Sq. Inches.

Locate the weight section on the outer disk, hold the disk with the weight section uppermost, rotate the inner disk until the wing area in square inches is uppermost, align the wing area of 500 Sq. Inches with the weight, 4 Lb.

Locate the Ounces Sq. Ft window in the inner disk, read the calculated wing loading from where the arrow head is pointing. Ex. 18.5 Ounces Sq. Ft.

The stalling speed is related to the coefficient of lift of the wing. For a symmetrical airfoil the maximum coefficient of lift is approximately 0.8 at a 10 degree angle of attack.

Locate the stalling speed window. Locate the coefficient of lift mark of 0.8, the stalling speed at sea level is aligned with this mark, Ex. 24 MPH.

Rotate the outer disk until the pressure altitude section is uppermost, rotate the inner disk until the density altitude section is uppermost. Align the arrowhead to the left of the density altitude window with the pressure altitude ,Ex. 3(3000Ft).

Locate the temperature band at the top of the density altitude window , by eye, align the 80 temperature to the % stall speed the from the lowest row of numbers, Ex. 117%.

The calculated stalling speed is 1.17 x 24 = 28.1 MPH.

Now lets calculate the approximate HP of an engine.

The engine is rotating a 10 inch diameter, 6 inch pitch Propellor at 12,000 RPM. The altitude of the flying site is 3000 Ft. The air temperature is 80 Degrees F.

Locate the RPM section on the outer disk, hold the disk with the RPM section uppermost, rotate the inner disk until the Propellor diameter in inches is uppermost, align the Propellor diameter of 10 inches with the RPM 12(12000).

Locate the Pitch-Inches window in the inner disk, read the calculated horsepower at sea level from the Propellor pitch of 6 inches.
Ex. 0.8 H.P.

Locate the temperature band at the top of the density altitude window , by eye, align the 80 temperature to the % engine power from the middle row of numbers, Ex. 86%.

The calculated horsepower is .86 x 0.8 = .69 H.P.
To calculate the approximate engine capacity to generate 0.8 H.P. at this altitude and temperature, divide the engine capacity by 0.86.

Ex. .40 Cu In/0.86=0.46 Cu In

Now calculate the speed and thrust capability of a Propellor at sea level. A 10 inch diameter Propellor with a six inch pitch is rotating at 12,000 RPM.

Locate the RPM section on the outer disk, hold the disk with the RPM section uppermost, rotate the inner disk until the Propellor diameter in inches is uppermost, align the arrow pointing to the airspeed and thrust windows in the inner disk with the RPM 12(12000).
Locate the Airspeed window in the inner disk, locate the Propellor pitch in inches and read the airspeed capability.

Ex 68 MPH.

Locate the thrust window in the inner disk and read the thrust in ounces from the Propellor diameter.

Ex. 58 Ounces=3.6 Lb.

NOTE: THAT ALL THE CALCULATIONS CAN BE REVERSED,
BY STARTING WITH THE REQUIIRED RESULT AND CALCULATING THE NEEDED INPUTS

Technical Section

Horsepower Calculation and Propellor selection
A practical method of determining a Power Coefficient , using test results, is the following;

Pc = H.P./(p x RPM^3 x Diameter^5)

An empirical equation for the power absorbed by a model Propellor, uses pitch to substitute for the theoretical blade loading, and a Propellor factor for model engine usage derived from published test data.

H.P. =(RPM^3 x Diameter^4 x Pitch)/Propellor factor 1.4x10^17

The horse power formula used, is fairly accurate for calculating the horse power generated by an engine, but it is more useful as a method for selecting the appropriate propeller for a given application.

Mathematically this is the measured RPM raised to the third power, multiplied by the propeller diameter in inches raised to the forth power, multiplied by the propeller pitch, all divided by a factor for model propellers. The Propellor factor has been determined by comparing the calculated results against published data on engine horsepower and Propellor sizes.

The RPM and diameter are easy to verify but the actual pitch of a propeller is not so easy. Pitch is traditionally determined by measuring the angle of the back surface of the propeller blade, at 75% of the propeller radius. A pitch measuring gages can be purchased or constructed. Note that marked pitch sizes may not represent the actual Propellor pitch.

If, for example, a engine is rotating a 10" x 6" propeller at 12,000 RPM. The H.P. can be determined from the slide rule by aligning the propeller diameter, 10, on the outer circumference of the inner disc, with the measured RPM on the outer disc, 12. The engine H.P. can be read in the pitch window from the Propellor pitch, 6, located on the inner disc,
Ex =0.8 H.P.

It is important to note that the RPM measured may not be the RPM at which the maximum BHP of the engine is generated or the RPM at which the maximum torque is generated. To maximize the performance of the engine, the RPM at which the maximum BHP and Torque needs to be determined from the manufacturer’s data or published reports in the media.

In the previous example, assume that the maximum BHP is generated at 15000 RPM, and that the max. BHP is +10%, to select an appropriate propeller, chose a pitch, Ex 10", align the pitch value with the H.P. calculated previously plus 10%, 0.9 in the pitch window on the inner disk, (note that this value will approximate the actual maximum H.P. capability, so some adjustments may be required). From the RPM selected ,15000, the propeller diameter can be calculated, 7.7" in this example. Note that if an overly large value of pitch is selected, the calculated propeller diameter can be too small to generate enough thrust for good performance, for the size and weight of the model.

Ex, to maximize thrust, a larger diameter is required, 12". align the Propellor diameter with the desired RPM, 12000, and note that for the available horsepower, .8, a practical pitch is not available, so assume that a 4" pitch is acceptable, align the HP, .8 with the pitch, and the new RPM will be, 10600.
This RPM also could be compatible with the maximum torque RPM.

The same procedure can be used for Propellor selection, for example a four bladed Propellor of the same pitch, 6, and RPM, is required for the above example
Divide the horsepower by two, = .4
Align the new H.P. in the H.P. window with the pitch value, 6. and read the new Propellor diameter, 8.5, from the required RPM.
Home

Airspeed Capability

The airspeed capability of the Propellor is determine by the pitch.

Airspeed = RPM x 60 x Pitch/(12 x 5280)

For the above example, 6" pitch, 12000 RPM;

Align the Airspeed and Thrust arrow line with the measured RPM, 12000, the airspeed capability of the Propellor is determined from the airspeed window, by reading the airspeed from the value of the pitch, 6, approximately 68 MPH.

Ex = 68 MPH

This assumes a -10% factor for slip and a +10% airborne RPM increase from the static RPM.

Note that the speed capability of the Propellor may not be achieved in practice if the model is to big and /or heavy for the size of the motor used . Too big and the model drag is too high, too heavy and the drag due to the angle of attack required to generate the required lift, is too high.
Home
Thrust Capability
The theoretical maximum thrust available from a Propellor is defined as;

T = (H.P.)^2/3 x (2pA)^1/3

A is the area of the Propellor.
p = Air density

this is modified by the losses associated with the Propellor and the efficiency, which is a function of the advance ratio J.
The advance ratio is a function of the forward velocity of the Propellor and the RPM.
A fixed Propellor, at a fixed RPM, then has its maximum efficiency at one airspeed, so for model purposes the pitch selection is critical to performance.
Especially for high pitch values, the Propellor can be stalled at low velocities (take off and landing) , e.g. ducted fan jets, racers.

A more practical method, using test results, is to determine the Thrust coefficient using the following.

Ct = Thrust/ RPM^2 x Diameter^4 x Air Density

Thrust =RPM^2 x Diameter^4 x Air Density x Thrust Coefficient(Ct)

Thrust Coefficient is a combination factor that combines the design of the Propellor, (its shape), the Advance Ratio J, and the Reynolds number of the advancing blade. As these factors are not easily determined for model propellers, a coefficient is determine from measured static thrust figures.

The advance ratio J is of interest , as, as the speed of the aircraft increases the thrust (blade lift) will decrease due to the reducing angle of attack of the advancing blades. This will limit the maximum speed of the model in horizontal flight.

Align the Speed and Thrust arrow with the measured RPM, 12000, and read the approximate thrust

58 Oz (3.6Lb)

from the Propellor diameter ,10.

How much thrust is enough? From practical experience if the static thrust is equal to the model weight the model will have excellent performance.
To be able to hover vertically and for better vertical climbing ability 1.5 to 2 times the model weight is required.

A electric glider using a 05 motor, is a low performance model and would have a thrust to weight ratio of about 0.5. (8 x 3 at 10,000 RPM), and a speed capability of 29 MPH.
Home
Wing loading

The wing loading in Ounces per square foot is calculated by dividing the projected or measured wingspan which would include the area loss due to dihedral, by the average chord. The average wing chord is calculated by averaging the root chord and the tip chord.
(Root chord + Tip chord)/2.

The effect of wing loading varies with size of the model, as the chord increases in width, the effective Reynolds number increases, which will increase the "efficiency" of the airfoils used for model purposes. That is; Larger models can fly with higher relative wing loading.

From the wing loading the stalling speed of the wing can be found.
The stalling speed is a function of the coefficient of lift of the airfoil.
The coefficient of lift is a function of the airfoil shape and its angle of attack.
for example a symmetrical airfoil has its maximum coefficient of lift at an angle of 10 degrees and is approximately equal to 0.8
Home
The stalling speed is calculated from;

Stalling speed = .68 x (( Wing loading/16)/(.00119 x Cl))^.5

To calculate the speed, align the wing area in Sq. Inches with the weight of the model in pounds.
Read the wing loading in Ounces/Sq. Ft from the wing loading window.

Ex; 500 Sq. In Area, model weight 4.5Lb
Wing loading = 21 Oz/SqFt.

The stalling speed at the calculated loading for a particular coefficient of lift can be read from the stalling speed window.

Ex; At 21 Oz/Sq. Ft wing loading and a coefficient of lift of 0.8 (symmetrical airfoil at 10 Degrees incidence), the stalling speed is 26 MPH

Note that as the coefficient of lift of the airfoil increases, the stalling speed decreases. For example flaps increase the lift coefficient and allow a lower landing speed, flaps also increase drag, so more power is needed.
The stalling speed depend on the density of the air, so at elevated altitudes and /or temperatures the stalling speed will increase. This can be calculated in the Density Altitude window.
Home
Density Altitude

As altitude above sea level increases the air density decreases.
If air density decreases, the performance of the engine decreases and the stalling speed of the wing increases, and for equal lift the model speed must increase.
As the air temperature increases the air density decreases, and causes the same effect as above.
To calculate the density altitude relative to temperature and the local pressure altitude;
Align the temperature arrow on the inner disc with the local pressure altitude in thousands, on the outer disc, and read the density altitude in the window.

Ex Pressure altitude = 3.3 (3300 Ft), temperature 80 Deg F,
Density altitude = 5.2 (5200 Ft)

Also the approximate engine horsepower is 85% of original, and the increase in stalling speed, 118%, can be read in the window

Home
Coefficients of Lift

Airfoil Cl Angle of attack

FLAT PLATE                0.7                   15
SYMETRICALL            0.8                   10
CLARK-Y                      1.2                   10
N60                                 1.25                 10
SD7032A                        1.25                 12
SELIG 2091                    1.35                11
FX-63                              1.6                  11

Home
   References:

Radio Control SCALE AIRCRAFT
GORDON WHITEHEAD Publisher: RM Books Ltd

MODEL AIRCRAFT AERODYNAMICS
MARTIN SIMONS Publisher: Argus Press

Aerodynamics Aeronautics and Flight Mechanics
BARNES W. McCORMICK Publisher: Wiley

THEORETICAL AERODYNAMICS
L.M. MILNE-THOMPSON Publisher: Dover Publications, Inc

AIRFOILS AT LOW SPEEDS
Selig, Donovan, Fraser Publisher H. A. Stokely

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