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Aviation History
1920
1920 - 0959.PDF
SEPTEMBER 2, 1920 AEROPLANE PERFORMANCES AS INFLUENCED BY THEUSE OF A SUPERCHARGED ENGINE* - By GEORGE DE BOTHEZAT, Aerodynamical Expert, National Advisory Committee for Aeronautics THE question of the influence of the use of a superchargedengine on aeroplane performance will be treated here in a first approximation, but one which gives an exact idea of theadvantage of supercharging. The method used may be directly extended to treat this problem without any of thesimplifying assumptions made. These assumptions are made exclusively to allow an easier survey of the problem. Let us consider an aeroplane which climbs first with anordinary engine, not supercharged (called in the following case I), and afterwards climbs with a supercharged engine(case II), and let us find the difference of the ceilings reached in the two cases. We will asume in both cases the power am(j of the motorat sea level to be the same and the efficiency t\ of the pro- peller to be maintained constant all the time. This is quitepossible, to a certain extent for a propeller with an adjustable pitch, a conclusion reached by theory and experimentallyverified. In case I, we can consider in a first approximation thepower a m of the motor to be proportional to the density,that is, to be expressed in the form . . - < -• • . .„. (l) am=mSwhere 5 is the air density at a given height, H. m a constant coefficient characteristic for the motor con-sidered, assuming the number of revolutions of the motor to be kept nearly constant; $.._• Technical Notes: U.S. National Advisory Committee for Aeronautics. O.0C92 -At sea level we have (2) •ml,.where 8 0 is the corresponding air density.The power expended for horizontal flight at any altitude is equal to (3) an = r\am = tfmS = QV •:•:;••-.where r, is the propeller efficiency, Q the propeller thrust, V the flying speed. On the other hand, the equations of thehorizontal steady motion are of the form •- (4) " P = Ry = ky 5AV2 J \ - "^ (5) Q =? R. = ft. »AV» where P is the total weight of the aeroplane, kK and ky the drag and lift coefficients (functions of the angle of attack only), A the wing area. Comparing (3) and (5), we find (6) " QV = nmS = kx 8AV3 and following (7) -••-:• : --*'• ^=A*V3 -••-«: -•.,--:::. :. ' -. • an equation that fixes the relation between the angle of attack i and the speed V for horizontal flight at any altitude in case I. I call climbing curve or C curve) the curve of V plotted against i according to equation (7). Let us now plot on a system of (V, i) axes the system of curves see equation (4)). (8) S = **V2 ' ; '"'••• ' ' "~ for different values of 8. I call the last curves velocity curves see figure). As the height H reached by an aeroplane is a direct function of 8 (depending upon atmospheric conditions) for the curve (8), we can use H as parameter instead of 8. If we plot on the same (V, i) axes the C curve (7), each point of intersection of a velocity curve with the C curve gives for the height H corresponding to the velocity curve considered, the velocity V and the angle of attack i of the horizontal flight at the height H of the aeroplane considered. That velocity curve which is tangent to the C curve gives the value-H, (case I) of the ceiling and the values of V and i corresponding to this ceiling. The last value of the ceiling can also be found directly as follows : Eliminating V from (7) and (8), we find At 7,! ««o' I fty* that is, the density 8, (for case I) in function of the angle of attack i. The minimum value of 8, given by the last equation will correspond to the maximum of the height H, that is, to the ceiling. Thus the value iu of the angle of attack corresponding to the ceiling in case I will be found from the relation % . ._•• ~ _ .. vj ^=Dor57(^ ) = D 8, di and (10) '' °W« Practically, the best way is to plot the curve (9) and find its minimum graphically, because kx and. ky are empirical functions. It is easy to see that_ the angle of attach *„, for which 8j is minimum, is the same angle for which the power o« expended for flight at sea level is minimum. In fact we have . •>. (11) a,, = QV = kx So AV and replacing in the last equation V by its value taken from (8) we get ' ft,* The minimum of a* takes place for an angle of attack given by di that is di D which thus is the same angle iu.On the annexed figure are represented the velocity curves and the C, curve for a good actual aeroplane, as well as the8! curve for case I, which curves fully illustrate all the foregoing. The ceiling is reached at an angle of attackof 130, at a speed of I2oft./sec, and has a value of 25,000 ft. In case II we will have the power a,^ maintained constantby the supercharger, up to a certain altitude, say 20,000 ft., for example. Afterwards the power of the.motor will again 961
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