ings

Powered sailplanes, like this SF-25C-S-Falke, obtain good performance on low power by virtue of high aspect ratio.

By L. D. Sunderland (EAA 54771 5 Griffin Dr. Apalachm, NY 13732 and

Bill Johnson (EAA 94689) 14536 SE 51 St. Bellevue, WA 98006

HEAR THAT rate of climb is a function of span loading or weight per unit of wing span and that low powered airplanes should have long wings? This is one aerodynamic fact that is not necessarily obvious. Neither is it a simple matter to dig it out of the textbooks. For those who would like to see why this is true, we'll show how the equation for calculating rate of climb is derived. We'll also give some simple examples which you can solve on your pocket calculator to illustrate the effects of weight and span on airplane climb performance. Rate of climb is a function of the amount of power

available (P a > from the powerplant and propeller less the power required (P r ) for level flight. To express this in equation form we begin by stating that power available for climb Pc is equal to excess power, or Pc = Pa - Pr. The power available of course depends upon the powerplant and propeller selected. In a climb, propeller

efficiency may be about 70%, so in this case the power available is .7 times the brake horsepower of the engine at a specified ,rpm. Pa = .7 bhp. 46 FEBRUARY 1979

Power required is found by the equation P r = DV 375 where D is drag in pounds and V is airplane velocity in mph. Now, the big problem remaining is to derive the equation for drag. As you know, drag is composed of two parts, parasite drag (Dp), or the drag on the airplane when it is generating no lift and induced drag (Dj) or drag due to lift generation. By definition, we then have D = Dp + Dj. The dotted lines in Figure 1 show the relationship of both induced drag and parasite drag with aircraft velocity for a typical large airplane with stall speed around 80 mph. The curves can be interpreted as pounds of drag or the thrust required to overcome drag. Notice how parasite drag increases exponentially (at an increasing rate) with velocity while induced drag decreases exponentially with velocity. The lower solid line represents total airplane drag which is simply the sum of the other (dotted) curves above the stall speed. Total drag is thus at a minimum approximately at the speed where induced drag equals parasite drag. This is about the speed for maximum lift to drag ratio L/D. In Figure 1, the total drag curve is also the power required curve. The vertical separation between the power required curve and the power available curve is a measure of the power available for climb. At the speed

where these two curves cross, all available power is used just overcoming drag and none is available for climbing.

Losses Causing Induced Drag

Figure 2. — Lift Distribution on Rectangular Wing.

thing which causes a reduction in lift like twist, tip loss, interference, etc. We can now substitute the coefficients in the equa-

100

200

300

400

V (mph)

500

600

Figure 1. — Thrust required versus speed.

It will be remembered that drag can be calculated by multiplying a coefficient Crj times the product of a reference area S and dynamic pressure p, thus: D = CrjSp.

The coefficients for the components of total drag add to form a total drag coefficient so Crj = Crj

+ Cj).. We

won't derive the equation for induced drag coefficient C

ratio — . b is span and c is average chord, e is called airc plane efficiency factor or Oswald factor. (Oswald was an aerodynamicist who did the performance calculations for

the DC-3). The factor e was used by early aerodynamicists as a sort of catch-all to adjust the geometric aspect ratio to account for losses due to wing tip shape, twist, taper ratio and fuselage interference effects. Today more sophisticated techniques are available for determining these

effects. However, they are too involved for consideration here so we shall use the factor e. The equation for induced drag is really an infinite series, that means it has many 2

CL terms, and —— is only the first term although it does TrAe have the predominant effect. Some examples of airplane efficiency factors are given in Table 1.

P-38J P-40F P-47D P-51B P-63A

e

0.76 0.70 1.02 0.86 0.86

Airplane

F-80 B-17 B-24D B-25D B-29A

CLqS, so CL =

D = [C D L

°P W2

D = CUD Sq + P TrqSAe S = cb and A = — sq

L2

here, but it is given as Cr>. = —i— where A is aspect u i nAe

Airplane

C 2 / L \ tion for total drag, D = I CD + -=,- JSq.Nextsub\ ^p nAe/ stitute for CL. In level unaccelerated flight, L = W = W

e

0.82 0.85 0.78 0.78 0.94

Table 1

An infinite aspect ratio wing (too long to measure and no tips) has zero induced drag. Anything that is done to degrade the lift on this ideal wing causes induced drag. To illustrate, assume that the solid upper line in Figure 2 shows the lift distribution over a segment of an infinitely long wing. It is naturally constant. The dotted line shows the lift distribution of this same wing segment if it were removed from the very long wing. The shaded area represents induced drag. It is due to any-

D = C

L W S q CL, b ~

= = = = = = =

W2 7rqb2e

lift airplane weight wing area dynamic pressure wing lift coefficient wing span mean chord

Crj S is referred to as equivalent parasitic area, designated by2 the symbol f. Dynamic pressure q= l/2pv2 or V q = -OQ-J- for sea level conditions when V is velocity in mph. The total drag equation is thus: fV 2 39 1W2 + 39T 7rV2b2e We can now write the final power available for climb equation for sea level conditions: P c = (Pa - P r)

375

——\ Substituting for total drag:

V 375

\391 r P

c = P *

,625

W2 3.01 Vb 2 e) SPORT AVIATION 47

This equation includes the term

(v)

— I which is the

weight per unit of span quantity squared. This tells us

that power available for climb is directly proportional to span squared and indirectly proportional to weight squared. A 10% increase in span squared will give a 10% increase in power available for climb and, conversely, a 10% increase in weight will reduce power available for climb by 10% . To convert power to rate of climb in ft./min., something we can more readily appreciate, multiply by

been measured at 2.4 to 2.8 square feet. Assume an average of 2.6. Span is 20.8 feet. Efficiency factor e was calculated by Henderson to be .76 for a T-18 (which has a rectangular wing). Flight tests show that 90 mph is best climb speed. My gross weight is 1450 pounds. RC =

[

H7

,,

87 5

'

903 / 2.6 i.