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Aviation History
1943
1943 - 0647.PDF
MARCH IITH-, 1943 FLIGHT 261 Relation of Height to Pressure Hoiv Corrections are Applied to Indicated Altitudes By CHARLES WILLIAMS THE height of an aircraft is measured in practice byobserving the difference between the atmosphericpressure at the position of the aircraft and that at the ground level. This method of calculating height depends on the principle that, at a given height, the decrease in pressure per square centimetre, measured in gravitational units, is equal to the weight of a column of air one square centimetre in cross section extending to that height. The weight of such a column of air depends on a number of factors, all of which are variable. The temperature at ground level, the rate of decrease of temperature with height and the humidity of the air all affect its weight to a considerable extent. Also, since pressures are now measured in terms of the millibar, which is a pressure of one thousand dy»es per square centimetre and is inde- pendent of gravity, the relation between pressure in gravitational units and pressure in millibars depends on the value of g, the acceleration of gravity, which differs slightly in different latitudes. Given a knowledge of these factors at a particular time and place, it is not difficult to make an accurate calculation of height, but obviously such a calculation cannot be made in an aircraft every time its height is required. The alterna- tive is to assume standard values for the variable factors. A suitable pressure gauge can then be calibrated directly in terms of height, and methods of correction can be devised which can be applied to the indicated height when one or more of the factors is known to differ from the standard value adopted for calibration. Altimeter Calibration The values adopted for calibration of altimeters according to the I.C.A.N. Law are these : The pressure at mean sea level is to be 1013.2 millibars. The temperature at this level is to be 15- degrees Centigrade, which is equal to 288 degrees absolute. The weight of one etibic centimetre of air at this pressure and temperature is to be 0.0012256 gramme. The lapse rate, th'at is, the decrease of temperature with height, is to be 6.5 degrees Centigrade per kilometre, which is 1.98 degrees Centigrade per thfusand feet. The value of g is that at 45 degrees latitude, which is 9S0.66 centimetres per second per second. These are the data necessary for the calculation of height in terms of pressure. Let W be the weight in grammes of a cubic centimetre of air at sea level, at absolute temperature T and pressure P. Then, since density varies inversely as absolute tem- perature and directly as pressure, the height of a cubic centimetre of air at height /; at some other temperature / and pressure p will be : ; _ W.p.T _ W.p.T P.t ~P(T~~ Lh) where L is the lapse rate. The weight of a column of air one kilometre high having a cross section of one square centimetre at temperature t and pressure p would therefore be 100,000 W.p.T. grammes P(T - Lh) and the weight of a very short column of air of height dh (measured as a fraction of a kilometre) would be 100,000 W.p.T, ,, P(T - Lh) dh- where dh stands for difference in /;. This weight is equal to the difference in pressure in gravitational units between the pressure at h and the pressure at h + dh. Remembering that a gramme weight is equivalent to a force of g dynes, and that one millibar is a pressure of 1.000 dynes per square centimetre, and denoting the difference in pressure between heights h and h -f- dh by — dp, we have dh — IO° W-8-P-T- 11 ~ P P T - Lh) which can be put in the form — dp _ TOO W.g.T . dLh •~p PIT (T - Lh) I Readers who are acquainted with the differential calcurus will recognise this as a familiar type of differential equation, but for the benefit of those who are not, it may be explained that dpIp represents the rate of change of the logarithm of p. To make the matter clearer by an example, if p is represented on a slide rule, which is of course a logarithmic scale, and when p is 3 the distance between the divisions 3 and 3.1 is fourteen hundredths of an inch, then when p is 6 the distance between the divisions 6 and 6.1 will be only seven hundredths of an inch. When p is doubled the length of a division is halved. The equation (1) therefore asserts that the rate of change of the logarithm of p is proportional to the rate of change of the logarithm of T — Lh). Or to put it in mathematical form, if P is the, pressure corresponding to height O and p . that for height h then inn W P T log P - logp = pl' ' (log T - log (T - Lh) .. U) Substituting for W,g,T.P.L. the standard sea levei values given in paragraph four, we get for the standard pressure p -at height /; log 1013.2 — logp = 5.2559 (log 288 — log (288 — 6.5/;)) if h is in kilometres. Or if h is in thousands of feet log 1013.2 - logp = 5.2559 (log 288 - log (288 - 1.98A)) (3) From equation (3) the pressure at any required height under standard conditions can be calculated. Specimen results are :— Height in feet Pressure in millibars o .. .. .. 1013.2 5,000 .. .... 843.0 10,000 .. .. .. 696.6 15,000 .. .. .. 57x-6 20,000 .. .. .. 4&5-4 25,000 . . .. . . 375-8 30,000 . . . . . . 300.7 If an altimeter, is calibrated in accordance with these figures the result will be that whenever a pressure of, say, 465.4 millibars is experienced, the altimeter will read 20,000 1eet This will be the correct height only if all the factors which enter into the pressure-height equation have at the time and place in question their standard values, which is most unlikely. Making the Corrections The methods and limits ot correction which are possible can best be studied by considering equation (2) and varying each factor in turn. The equation may be put in the form Wtog P — log p = -f (log T — log (T — Lh)) X a constant where T and P are now the temperature and pressure at sea level, not necessarily the standard temperature and pressure. Also, since it is generally the temperature at the aircraft which is known, and not the sea level temperature, it will be convenient to make use of the relation between the aircraft and sea level temperatures, t •= T — Lh to put the equation hi the form W / Lh\log P — log p — y log I 1 H—-I x a constant .. (4; Since we arc considering the correction to be applied to a
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