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Aviation History
1936
1936 - 1409.PDF
SlPFLEMENT TO FLIGHT 5766 32 THE AIRCRAFT ENGINEER MAY 28, 1936 be neglected, but in some cases it is important and must be considered, as, for instance, in that of a heavily loaded machine required for flight over mountainous districts in the tropics. From the basic equations Lift = SV2 . k, . pa and Drag = SV2Ad . pa it will be seen that for any given altitude of the machine the speed and thrust horse-power required vary inversely as the square root of a, the relative density of the air. The torque horsepower absorbed by the airscrew is also affected by the density, and varies directly as a, as is shown by the equation Q = K2D5A QPO In this equation the term Q includes a figure for the value of the brake horsepower given out by the engine, since it must be equal to that absorbed by the airscrew. The B.H.P. of the engine is itself affected by atmospheric conditions ; the law governing the variation of the power is incapable of exact definition, but seems to depend upon pressure, density, and at times temperature. The Power Factor In standard atmosphere, in which the characteristics vary only with height, it is reasonable to assume that the power factor for altitude, p„ is constant at each individual altitude. In a non-standard atmosphere, however, this is no longer the case, and it will be necessary to derive a set of values of pt for the particular atmosphere under consideration. In the absence of more accurate informa tion the only course appears to be that of using the same combinations of relative pressure (p), relative density (a), and relative absolute temperature (T) as those which give close approximations to the power factors for altitude in general use for estimates of performance in standard atmos phere. These are as follows :— P11 P. = P*v and P. Normally aspirated engines Supercharged engines In using these factors it should be remembered that the corrections must be applied to the B.H.P. at full throttle at zero altitude in standard atmosphere, and not to the ground level conditions in the new non-standard atmos phere under consideration, and, in the same way, the values of p, a, and T are the ratios of the pressure, density, and absolute temperature at any given height in any atmosphere to the corresponding values at zero height in standard atmosphere. Before making any attempt to estimate the performance of a machine in a non-standard atmosphere it is first necessary to determine the conditions prevailing at a series of tape-line heights in that atmosphere. The method of doing so is shown in the following example. Suppose that the worst atmospheric conditions likely to be encountered by an air line operating in the tropics occur when the barometric pressure at mean sea level is 730 mm., and the corresponding temperature is 45° C, the only further information being that the temperature gradient may be taken as being the same as that in standard atmosphere. It is stated also that the engines used are normally aspirated. We will proceed to find the values at tape-line heights of o, 5,000, 10,000, and 15,000 feet in the new atmosphere of p, a, and T relative to the corresponding values at zero height in standard atmosphere. (1) At o feet Absolute temperature = 45 + 273° C. 45 + 273 = 73^ 760 m .961 0.961 whence o0 = 1.104 = 0.87 For normally aspirated engines 0.961 IJ Pt = 1.104* = 0.9109 (2) At 5,000 feet. The pressure per unit area at any height is equal to the pressure at H = o less the weight of the column of air, of unit cross sectional area, extending from H = o to the height in question. If P is the absolute pressure and p the absolute density, the difference in pressure is given by dP=-pg.dH On the assumption that air is a perfect gas, we have P PS = where 6 is the absolute temperature, and R is the gas constant for air. Combining the two equations we get — = dK p Re whence, by integration between the appropriate limits log,PH - logeP0 = —-^ (HH - H.) or los H "where H is the height interval. • T = J5 + 273 BS I.IO4 p R0 If H is in feet and 6 in degrees centigrade, the value of the gas constant, R, is 96.04, and the equation becomes 1 * H For rigid accuracy, particularly when the height interval is considerable, the value of 6 used in this equation should be a harmonic mean between the values of 0 at the upper and lower limits of the height interval under consideration, its value being given by 0.00198 H 6 mean = log,— where the suffixes o and H denote the lower and upper limits respectively. For our purposes it will be sufficiently accurate to take the arithmetic mean, which gives a value of 313.05° C. instead of 313.70 C, the harmonic mean as given by the above equation. Relative Pressure The value of p, the relative pressure, obtained by the use of the above equations is the ratio of the pressures at the upper and lower limits of the height interval in the atmosphere under consideration, and does not of necessity indicate the pressure relative to that at zero height in standard atmosphere. Denoting the relative pressure for the height interval by p' and employing the appro priate values of H and 8 we get 1 5,000 log -= 2 p 221.14 x 3T3-05 and p'= 0.8468. We have found already that the value of p at mean sea level is 0.961, so at 5,000 feet p = 0.8468 X 0.961 = 0.8138 The remaining constants are found as before : • Absolute temperature = 318 — 0.00198 x 5,000 = 308.i°C. whence T = 1.0698 _ 0.8138 1.0698
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