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