SIMPLIFIED PERFORMANCE CALCULATIONS FOR LIGHT AIRPLANES
Go to Table II for Pr at other speeds. You say this is great, but my airplane three view does not show Cdo or e, or a way to calculate them. Sorry 'bout that, but you have to estimate them from the following information. TABLE I DEFINITION OF TERMS
Cl
= Lift Coefficient
C f = Airplane Min. Drag Coef. Based on Wetted Area Cdo = Airplane Min. Drag Coef. Cl 2 Cdi = Airplane Induced Drag Coef. = —j-=r— = Wing Aspect Ratio —=— o
Frisco, TX 75034
LAVE YOU EVER questioned quoted performance or wanted a way to estimate what your planned bomb should do? These calculations are generally a lot of work, but since you and I don't have to guarantee the numbers for the sales department I will give you an old lazy engineer's method of performance calculation. There are
some possible errors in this method at maximum and minimum speeds. You can use the system for about ±5$ accuracy. Better yet, figure -5f7t or +(F/e — if we weren't optimists we would be doing something else. Only sea level standard day performance is covered because this is the standard reference condition. Approximate density altitude corrections for rate of climb are given later. Actually, all performance calculations involve determining power required and power available. From this information the remaining items are determined. For light airplanes, I stop at rate of climb because this gives enough to evaluate a design. Let's go through the required calculations and estimations without worrying about where they came from. You will need a scale three view drawing of the airplane, an engine power curve and a pocket calculator with a square root function. There is an established relationship in the power required at L/Dmax and any other speed. This is shown in Table II. Fortunately, there is also a simple direct method of calculating L/Dmax, speed at L/Dmax, and power required at L/Dmax. Here is how it is done. Ae 4 Cdo
L/Dmax
Cl
At
Cdi = Cdo
L/Dma x
80
"Ae
(L/Dmax)
And
L = W
But
q = 1/2 p
Then
V
L/Dmax
S
= Wing Area — Square Feet
B
= Wingspan — Feet = Span Efficiency Factor
e W T Pr Pa BMP l> V
N D
= Design Weight — Pounds
= Thrust = D = Drag - Pounds = Air Horsepower Required = Air Horsepower Available = Brake Horsepower
= = = =
.002378 (Sea Level, Std. Day) Flight Velocity — FPS or MPH as Noted Prop RPM Prop Diameter
= Prop Efficiency TABLE
V/V L/D Max .6 .75 .8 1.0 1.2 1.5 2.0 2.5
PR/PR L/D Max .95 .88 .885 1.0
1.3 2.0 4.2 8.0
TABLE III SOME STANDARD AIRPLANE Cf VALUES AIRPLANE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cf
Piper J-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .014 Cessna 170-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .008 Cessna 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .009 Bellanca 14-13-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .007 Anderson Greenwood AG-14 (Twin Boom Pusher) . . . . . . . . . . . . . . . . . . . .010 Stinson L-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .015 Stearman (PT-13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .011 P-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0058 P-63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .005 P-51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0041 P-47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0052
™ • ^T-./-V m n- i Cf /~TUby — Wetted (Surface) Area To ™ Obtain CDO Multiply
Cl Sq V
Wing Area
2
TABLE IV 2W
in F.P.S .
Cl
'L/Dmax
Wing Location
High Wing
Estimating Span Efficiency Factor
.9-.95 well filleted, no acute angles or sharp breaks
V FPS
Mid Wing Low Wing
1 . 446
Now
T =• D
Finally
Pr
L/Dmax
.9 No fillets, no acute angles to .8, no fillets and
acute angles
I think in MPH -
So
TrAE
B2
By Al Backstrom. EAA 1162 Rt. I
W L/Dmax
T L/Dmax T L/Dmax V MPH L/Dma» 375
Go to Table II to calculate Pr at other airspeeds
.85 filleted .S-.85 filleted,
.8 No fillets
.75 no fillets, no acute angles .6-.7 no fillets with acute angles Note: A leading edge with a pie section at the side of the fuselage can be considered as filleted.
no acute angles
SPORT AVIATION 59
Estimating Airplane Drag There are several ways to estimate Cdo, but about as easy and accurate a way as any is to use information from flight tested airplanes. By comparing the airplane cleanliness being studied to similar airplanes and using measured values, a reasonable estimate can be made. Table III lists skin friction coefficients for a number of airplanes. When using the high speed airplane values for moderate speed designs, they should be increased by 15% to 30% to account for scale factor. A reasonable
four speed points, but no more than six, as you only need enough to draw the curve. The calculations for a constant speed propeller requires only the calculation of ——at a series of speeds. Read propeller efficiency from Figure 1 and multiply by engine rated power for power available. This is also shown in the example. To calculate the rate of climb; plot the power required and power available curves together, the difference in these is the excess horsepower. Rate of climb (FPM) = ExcessHPx_33,000
90
W
'< BEST CASE *
80 U
1/
Z
UJ
5
70
01
z 60 UJ u cc
s^ 4
100
^ TYPICAL
- •
90 PERCENT OF MAXIMUM 1 EFFICIENCY
,'
t
1'
70 60
f/
Ul Q.
80
50
f
50 •2
.3
.4
.5
.6
.7
.8
DESIGN
.9
1.0
1.1 1.2
.4
1.3 1.4
Calculate Design
ND
From Figure 1 read the maximum propeller efficiency.
.8
.9
1.0 1.1 1.2 1.3
PERCENT NORMAL R.P.M.
100 90 80
1.0
Estimating Wing Span Efficiency
In estimating the span efficiency the primary consideration is wing location and wing fuselage intersection details. Table IV will serve as a guide in these estimations. The span efficiency variations are also related to the wing-fuselage size relationships, i.e. a large fuselage on a small wing will generally have a lower span efficiency than vice versa. Now for the calculation of power available. From your power required curve find V at your intended engine rated brake horsepower x .7. Estimate prop diameter or calculate by the procedure in Lu Sunderland's article in November 1973 S/>ORT AVIATION.
.7
FIGURE 2. VARIATION OF EFFICIENCY WITH V/ND
ND
approximation of wetted area is: Wing area x 2.05 + tail area x 2.02 + fuselage (side area x 2 + top area x 2) -2 x wing area in fuselage. If the fuselage is round multiply the fuselage area by .8. If landing gear areas are small they can be neglected. Do not count the wingfuselage mutual areas twice as noted above, but don't worry with the small mutual areas around the tail. Do include large gear fairings, nacelles, etc.
.6
PERCENT DESIGN MIND
V
MAXIMUM EFFICIENCY VERSUS DESIGN V/(ND) FIGURE 1. MAXIMUM EFFICIENCY VERSUS DESIGN V/ND.
.5
1.1
1.2 1.3 1.4
1.5
PERCENT DESIGN AIRSPEED VARIATION OF REVOLUTIONS PER MINUTE WITH AIRSPEED FIGURE 3. VARIATION OF R.P.M. WITH AIRSPEEC
Using Figure 7 you can make an approximation of rate of climb at varying density altitudes. As noted at 1500 feet pressure altitude and 80°F the rate of climb is 84V( of the sea level standard day performance. So you understand all that good stuff; I'm glad, but just to help put it all together let's go through an example. Figure 4 shows our airplane and Figure 5 is our engine power curve. As the engine is liquid cooled and the airplane is quite clean, I assumed the skin friction coefficient for the P-51 with a 20% increase because of lower
speeds. The span efficiency is estimated as .85 since the
As propeller rpm at full throttle will vary with airspeed, prepare a table of c/< design V and actual V. From this
airplane is most nearly a filleted low wing design. The
calculate c/< design rpm and actual rpm using Figure 3.
wetted area was calculated to be 342 square feet; so Cdo 342 is: —^f- x .0048 = .013, and our power required is cal-
Calculate =-=-_=— at each point and the ratio of this to
culated as follows
design -rj=-
Plot this data on graph paper as shown in Figure 6. As .7 x 85 equals 59.5 the closest easy number to this on our power required curve is 175 mph. This will be our
With this ratio go to Figure 2 and find
% maximum efficiency.
From this information and your power curve the power available is calculated. The tabular presentation in the
example shows the step by step procedure. Use at least 60 SEPTEMBER 1979
design speed point for
. Construct a table as shown
SMOOTHIE II CONCEPT LAYOUT 2 PLACE FOR STAGGERED SEATING ENGINE 85 HP MERCURY O.B. DRIVE SYSTEM FROM SPRATT CONTROLWING WING — SPAN 25' — AREA - 130 SQ. FT. WEIGHT GUESS: EMPTY 575, GROSS 1050
5500
NOTE: THIS CURVE FOR ILLUSTRATION NOT A MFC'S CURVE
ESTIMATER (NOT ON MFC'S INFO)
5000
RPM
4000
3000 60
70
BRAKE HORSEPOWER
FIGURE 5 SPORT AVIATION 61
below and calculate the power available. Don't let the table confuse you, it is all simple repetitious (translatedboring) punching the calculator. Figure 1 shows a best expected and service type propeller efficiency. I have used the best expected curve due to the very low propeller interference provided by this design. Plot this data with your power required curves as shown in Figure 6 and calculate the rate of climb as shown below. Well, there we are with our basic performance laid out. Now with our example we can see that the propeller selected was pitched too high. By using the constant speed propeller curve we can go to best case condition to select another propeller to check. In this machine a propeller designed for around 135 mph would raise the rate of climb to an acceptable level. My own measure of adequate climb performance is will it be comfortable to fly in the high plains on a summer afternoon? This means looking at climb performance for 3500 feet pressure altitude and 100°F. From Figure 7 you can see that this is only 64% of the sea level standard day performance. Have fun in checking out the quoted performance max numbers or in finding out what your dream machine should do.
./
"I x 4.8 x .85
Cl < L / D m a x ) -
4.8
- r-i
-
i 1 1*/ unax j
•
1
T V 375
prIL/Dnax) 1
From Table 1
66.8 X 9 1 "375
2
3
(T)x 90.1
V/V(L/Dmax)
From Table 1
V
Pr/
Pr(L/Dmax)
3 % DESIGN RPH
4 ENGINE RPH
5 B.H .P.
6 PROP RPK
.95
15.3
67.6
.88
14.2
.8
72.1
.BBS
1.0
90.1
1.0
16.1
1.2
108.1
1.3
21.0
1.5
135.1
2.0
32.2
3.0
180.2
4.2
67.7
2.5
225.3
8.0
128.8
8
7 V ND
* DESIGN V/ND
1.08
1.03
5350
90
2572
1.28
1.05
1.0
1.0
5200
81
2500
1.21
1.0
1.08
11
10
PA
1
.98
.81
72.9
1.0
.83
67.2
.89
.97
.80
58.4
.78
.90
.75
42.5
2200
.79
.65
. 77
.65
40.3
2175
.60
.49
.59
.49
29.4
.96
4992
73
2400
.714
.90
4680
66
2250
100
.571
.88
4576
63
.428
.87
4524
60
75
9 % DESIGN
.94
.857
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