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Aviation History
1910
1910 - 0631.PDF
article, that v = V tan §.; v* = V2tan20, from which we derive the final expression for energy in the form Energy (per sq. ft. per sec.) ^ V3 tan28 g (where V = flight-speed in miles per hour; 8 = the angle of deflection ; p = density ; g = gravity). If we assume that this energy is transmitted without loss from some product of thrust and speed of flight we may write TV = Jv»tan*/B whence T = ± V2 tan 28 = ^L tan28 (where T = thrust for aerodynamic resistance alone, V = flight speed m.p.h. ; p = density ; g = gravity ;.$= angle of deflection). Here, therefore, we have a very simple expression for the aero dynamic resistance alone, which we may equate to the formula for skin friction in order to obtain the condition of minimum total resistance. V2 Thus, — tan 2 (3 = -ooooi8 V2 400 . •. tan 2 8 = -0072 . •. tan 8 = '085 . •• 8 = 40 51' = 50 approx. Having established the angle ot the plane, the next step is obviously to write down an expression for the total thrust, which will be equal to twice the aerodynamic resistance, in the form T = -T tan 2/3 200 From this the ratio of thrust to lift may be evolved thus :— Thus, for skin friction, tan 8 tan 8 —i-, and for aero-tan 0 __ \200 / tan " 28 V- tan8 which shows that the gliding angle is equal to the angle of deflec tion for the condition of minimum resistance, in other words the minimum gliding angle for air is 5°, and the minimum coefficient of horizontal flight is I in 12 or -085. For any other condition than that of minimum resistance the above equations do not hold good, because the aerodynamic resistance will no longer be equal to the skin friction, and, therefore, the total thrust will not be adequately represented by twice the aerodynamic resistance. A fairly simple formula for thrust may, however, be derived by first establishing an independent ratio between the lift and each resistance separately. dynamic resistance, p = V* ~ = "—, whence the total thrust tan 8 zoo is obtained by adding together the above expressions, T _ Aan 8 •QQ36\ _ tan 28 + -0072 P ~ \ 2 + tan 8/ ~ 2tan8 The results obtained by this formula are shown graphically in the accompanying chart of thrust per unit-load for different angles. The characteristic shape of the graph therein represented is very interest ing and should be borne m mind by all those interested in the design of aeroplanes. It shows the condition of minimum resist ance coinciding with the angle of 50 and illustrates how inefficient it would be to use planes flatter than this value. That part of the graph to the left of the angle of 5° quite evidently marks a sort of danger zone that should be avoided, and the mere fact that the angle of 5" itself is such a critical point suggests that it would be far safer to keep on the high side. Thus we should suggest an angle of say 8° as the lowest safe value for preliminary experiments until there is more knowledge of the subject. With an angle of 8° there is still a little latitude before entering the danger zone, and the thrust factor is not seriously increased. If we investigate by the aid of this chart the conditions prevailing with larger angles we find that the coefficient of thrust for an angle of 210 is *2 ; that is to say, the gliding angle is I in 5, or about ilj°. Once more it is necessary to emphasise the fact that the chart represents the thrust required for the propulsion of the plane alone, and must be augmented by that required to overcome the body resistance and the skin friction of the supplementary surfaces in order to obtain an estimate of the total thrust required for a practical machine. In conclusion we will take two numerical examples, the first representing a machine with 5° planes for minimum resistance, and the second representing a machine with planes having an angle of deflection of 200. We will suppose that the load carried in both cases is 1,000 lbs., and that each machine has an extra resistance represented by 200 sq. ft. of skin friction. We will further assume that both machines carry the load with the same loading, which, for f>R •Sfi •54 •i? •fl •48 -46 •44 4? •4 •^8 26 24 18 16 12 08 06 04 02 THRUS T ne r Uni t Lo« H \ . 1 1 1 - 1 1 THRUST per Unit Load for Different Angles calculated from the formula T_ tan^+oora 2 tan^s [ planes alone ) V A •5C •52 •5 48 42 •54 3Z 28 or, >\t Ifi 14 •12 •1 -nft •Of. •04 02 "Tfi 4Ti 8 10 Q 1416 18 20 2224 26 28303234383840 42 44/3 629
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