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Aviation History
1929
1929 - 0860.PDF
TO FLIGHT 32 THE AIRCRAFT ENGINEER APBIL 26, 192* 3. The Influence of the Decreasing Weight of Fuel During the Flight. During a long-distance flight the weight of fuel decreases. In order to fly always in the position of minimum drag the pilot must reduce the speed of the plane, so that the induced drag is the same as the sum of the profile and parasite drag : J'w-7^i-£-<W«-A.V (16) If we suppose the flight to be at a constant altitude, then W — = const. After the time T the weight of fuel consumed will be Wf = w • W. The total weight at thig time is W, = W(l — w) (17) and the speed from equation (16) is V, = Vv/ 1 - w (18) The total drag is T>f = D(1 — w) (19) and the power B, = B (1 - ivf* (20) When these conditions are fulfilled at all periods we can reach the maximum cruising radius, and we have the minimum consumption of fuel for all distances, or dWf = ce • Bt • dt. Substituting for W/ and Bt we obtain 75 • 6 • W , .. -») The maximum cruising radius will be 1 i*T> __ V, • dt (21) (22)1,000.1 0 Substituting for Yt and dt from equations (18) and (21), we find by integration that 1 75 • € • WR = • log, (1 — w) 1,000 c, • D Se v ; assuming a constant efficiency of propeller. In Section 2 we have already determined the cruising radius at a constant speed as 1 75 • f • W • w whence R coa" 1,000 log, (1 — w) D w (23) By integration of equation (21) between the limits 1 to (1 — w) we obtain the duration of the flight for the distance R, or c, • D • V 1 -w The ratio of this duration to that of Section 2 is 1( w 1 - (24) These relations are shown in Fig. 5, as they vary with the value w. The curve R shows the increase of the maximum cruising radius in percentage as compared with the cruising radius at a constant speed, and also the curves of the duration of flight T and of the speed V. The curve W shows the decrease of the total weight. For a flight with these conditions the efficiency of the propeller does not change: because the efficiency of the propeller remains constant if its angle of attack (its pitch) does not change; the revolutions of the propeller change with the cube root of power; from equations (18) and (20) the speed changes also in this relation, and so the ratio V - is constant. The propeller must be adapted for the flight m *» at the angle of minimum drag in order to obtain the best result. By flying at a constant altitude we can attain the maximum cruising radius only by reducing the speed. From this a loss results on the cruising radius, due to the greater specific fuel consumption when throttling the engine. Flying at the higher altitudes, it will be possible to maintain constant speed and the best efficiency. 4. Variation of Long-Distance-Flight Conditions with Altitude. From the equations of the induced, profile and parasite drag, and from equation (1), it is seen that the condition of flying at the angle of minimum drag obtains also for flight at higher altitudes. We shall assume that the altitude correction of the carburettor works faultlessly and that the fuel consumption remains the same as at sea-level. This is true, because the increase in the consumption for the altitude will be equivalent to the increase caused by the throttling of the engine at sea-level. The fundamental equation of the chart of Fig. 2 will change as follows: where Pk • V»« = pB • Vo* = const (25) The speed must increase with the altitude inversely as the square root of the air density. When we substitute for V in equation (8) we obtain ~ l K A / -L2 _ V v. 3.6 c. (26) p0 3-0 .LBO W From equations (10) and (11) we obtain the power loading in flight at the altitude h : •"HA — ... (27 or, at a constant propeller efficiency, Lm • Vo » I™ • Vh = const (28) The power loading is inversely proportional to the velocity and therefore directly proportional to the square root of the air density. The power is directly proportional to the velocity. From equation (26) we find that the cruising radius remains constant for the same specific fuel con- sumption, and therefore altitude flying shortens the duration of the flight. The propeller efficiency remains constant if the relation V- is constant. This is due to the fact that the total resistance N from equations (11) and (25) remains constant, and also the propeller thrust: Ci is the coefficient of the type of propeller and D is the diameter. 338A
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