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Aviation History
1920
1920 - 1212.PDF
NOVEMBER .$, 1920 A STUDY OF AEROPLANE RANGES AND USEFUL LOADS BY J. G. COFFIN (Concluded from page 1201) PART III. Effect of Wind on Range Calculations Best Flight Speed. a retarding or a helping wind it will be shownIF there below that the conditions for maximum range must be changed. The following is a proof of an important method for finding the proper attitude of flight with or without winds. Let curve I be the required thrust-speed curve and II the required power-speed curve for the machine. Consider a machine of constant weight W which is flying with air speed V against a wind speed w. The ground speed is then V — w. In order to fly a ground distance ds it will take a time ds V—• w' If the thrust is T and " a " is the rate of gas consumption per delivered power the gas consumed in flying this distance is ds , v aPi( = aP,, . (30) V — w As " a " is assumed constant and ds is fixed, for this expres- sion to be a minimum we must have d_ P (V -w) P' - P dVV — w~°~ (V^a/)8 and since. V — w eannnt be infinite the condition is : P ' _ corresponding weight of the machine, and the values thus obtained are the D/L's corresponding to- economical flight under the assumed conditions- This means that the subtangent to the power-speed curve is (V — w) and the equation is fulfilled by the following construc- tion : Lay off w, figure 9, on the V-axis, to the right of the origin, if a contrary wind, or to the left if a following wind, and draw a tangent from this point to the P — V curve ; it isp seen that the slope or tangent P' is equal to vv • -. For calm air the tangent is drawn from the origin. As the slope of a line drawn from the origin to any point on the P — V curve is p VT J -*** « • - - - always-y = -^r = T, it follows that the thrust varies'as the slope of such a line, and as the tangent from the origin to the P — V curve has evidently the minimum slope, this shows that in calm air the machine must fly at minimum thrust, as is otherwise evident. Thus the minimum points of the T — V curves lie directly over the points of tangency of lines from the origin to the P — V curves. If there is a head wind this condition of minimum thrust no longer holds, and more power is required for most economic flight, which corresponds, of course, to a greater thrust. As the power curve is limited to the right by the maximum output of the power plant, it is seen that for economical flight there is a limiting head wind corresponding to the distance OH, where H is the intersection of the tangent to the P — V curve at its limit with the V-axis. It is, of course, possible to make headway against stronger winds, but the condition for economical flight in such a case is no longer fulfilled. When KMS a helping wind the tangent is drawn from a point on the left of O, and it is evident that as the following wind increases in speed it pays to use less and less power, the limit for an infinite wind being minimum power. In other words, it pays to let the wind carry the machine along with the least use of the power plant. Curiously enough, this corresponds to a thrust greater than the minimum which is proper in calm air. While for economic flight in calm air the machine must fly at minimum thrust, and hence at maximum L/D* for the machine for all loads, this simplicity does not obtain for econo- mic flight in the wind. Not only does the L/D change for a given load with varying winds, but also for a constant wind it varies with the load. Fortunately these variations are small for any reasonable head winds and for a change of load equal to the weight of the machine empty. Referring to figure 9, the proper L/D's for the machine, and hence the proper angles of incidence, may be determined by the method demon- strated above. Assuming a head wind of w miles per hour, draw tangents to the required horsepower curves from abscissa-f-w. Read off on the thrust curves the thrusts corresponding to the points of tangency on the power curves, divide these thrusts by the • In the following D represents the total drag on the machine and corresponds to (D -j- R) in the preceding pages. Single Curve Method. A much simpler method will now be described to accomplish the same result requiring the drawing of but a single curve for the whole procedure. The method is based upon the following considerations : The equations for horizontal flight may be written W = LAV2 T = DAV P = TV From these we obtain V = N/LA W '2. w. L (31) (32) (33) These equations show that as the load changes the corres- ponding speeds for any given angle of incidence vary as W1 , the thrusts as W and the powers as W3 2. Consider now any required power-speed curve. Fig. (9). The required power curves for any other weight W, can be calculated from this given curve by multiplying the speeds by WA1/2 ^ /W.\8/a ' =A* '2 and the corresponding powers by and plotting these values on the same sheet. If required the PAS 2 P . ,thrust curves can be obtained by plotting y—, 2 = y A against V1 2. T T*t Wtrt« ° 1 Heiul Wir.a T A 1 H fW Fig. 9 Consider what effect a change in loading has upon the equa- tion of condition for economical flight. dP P dV ~ V - w becomes for a new loading W] 8/* PA.3/2 VA1/2 — W where This reduces to A == W w A1/*which indicates that instead of plotting P—V curves for various loads and drawing tangents from the abscissa w it is sufficient to plot but one curve, and as the load increases draw tangents from abscissas j,z as the load changes. The point of tan- gency determines values of P and V which correspond to a required power PA3/2 and a flying speed VA1/2 for the new condition. As the main interest here is to find the variations in L/D,or, what is the same thing, in D/L, we continue as follows : pA3 /a p^ = v is the new corresponding thrust, andSince ^-, ,2 = 1214
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