This is the first post
of a mini-series that treats altitude measurement, this series will
cover from the basic theory to practical application and further to
GPS sensor aiding.

The most used method
for altitude measurement in airborne vehicles is by mean of static pressure
readings, this measure is referred as barometric altitude.

Working principle is
simple and can be visualized recalling Stevin's law for hydrostatic pressure
calculation, as introductory example I will use the calculation of a
bathtub water level.

Refer to the below figure for a section view of a bathtub. The Bathtub is filled with water at a temperature \(T_{W}\) of 40 degrees Celsius and current atmospheric pressure \(P_{atm}\) is 101325 Pa, our sensor is reading the pressure on the bathtub floor \(P_{F}\) that is 105000 Pa.

*Figure 14.1 Section view of a bathtub*

Water is considered still over all the tub volume, as the water pressure and themperature have a limited range of variation hence the density is considered constant and then \(\rho=f(T_{W})=992 kg/m^3\).

To calculate the pressure at the bathtub floor the definition of pressure is used,

*Equation 14.1, definition of pressure*

$$Pressure=\frac {Force over one square meter }{(m^2)}$$

Consider the force
acting on the tub floor, the water layer close to the floor should
withstand to the weight of all the water and air that lies on it.

The force on the floor ot \(FF\) is then is simply the force exerted by the weight of the water and air column. Pressure of air \(P_{atm}\) at the interface with water is known and equal to 101325 Pa hence using eq.14.1 \(P_{atm}=\frac {Force over one square meter }{(m^2)}\)

Regarding the water part of the column, given \(g\) as the gravity acceleration constant, \(h\) as the water column height and given \(S=one square meter\) as the column section hence

$$FF=P_{atm}S +(Weight of water for a column of one square meter of section S)g$$

$$FF=P_{atm}S + h\rho gS$$

therefore

$$PF=\frac {P_{atm} S+ h\rho gS}{S}= P_{atm} +\rho gh$$

So it is practical to write \(PF-P_{atm}=\rho gh\)

With the example data \(PF-P_{atm}=9929,81h=3675 \)

\(h=0,37 m\)

More appetible result is that the pressure \(P(z)\) at a determinated deep \(z\) is then

E

*quation 14.2*
$$P(z)=P_{atm}+\rho gz $$

Calculate the
atmospheric pressure at a given z altitude is exactly the same thing
that calculate the pressure in a bathtub, it's necessary to calculate
the force exerted by the over head air column weight.

To define how change
the air density with altitude and have the chance to take this into
account an explicit formulation for air density is needed.

To calculate the air
density a widely used model are the
model and the US 1962/72. For altitudes included in the troposphere,
less than 11000m, they are equivalent, a huge number of altimeters
use this models formulas to convert pressure readings in altitude
indications.

To calculte the air density two common models are the ISA model and the US 1962/72. For altitudes included in the troposphere, less than 11000m, they are equivalent, a huge number of altimeters use this models formulas to convert pressure readings in altitude indications.

To calculte the air density two common models are the ISA model and the US 1962/72. For altitudes included in the troposphere, less than 11000m, they are equivalent, a huge number of altimeters use this models formulas to convert pressure readings in altitude indications.

Here below a short list
of model main assumptions, reference altitude is the mean sea level altitude.

a- latitude 45°

b-No wind or other forms of eddies,

c-Thermal gradient constant across the trophosphere of 6,5 °C each 1000 meter
d-Dry air, perfect gas \(Cp/Cv=1,4\)

Molecular weight, from the standard air composition is 28,9644 kg/kmol

R=8,314462175 J/°K/mol

e-Standard sea level conditions, reference conditions

\(T_{0}=288,15°K\)

\(P_{0}=101325 Pa\)

All the
altimeters installed on aircrafts have the possibility to compensate
for local pressure conditions, many are capable to more sophisticated
compensations for temperature and humidity.

The
atmosphere is by far idealized so
no local weather conditions are taken into
account, all the coefficients are given to fit at best 45° latitude
standard conditions, humidity impact directly into the altitude
reading values. Practically
speaking the barometric altitude reading can be amazingly off when
the aircraft is flying through clouds in a really warm day, tens
meters of error should not surprise us cause the instrument is
operating with a wrong density profile.

There
are techniques to compensate for a,d and e assumptions.

All
the altimeters installed on aircrafts have the possibility to
compensate for local pressure conditions, many are capable to more
sophisticated compensations for temperature and humidity.

Leaving to this reference the derivation of barometric formula is possible to introduce a numeric Scilab example, in this example two different calculation methods are employed; Initially the barometric, or pressure, altitude is calculated using a user specified external pressure and formulas from the Goodrich 4081 Air data handbook equation 4.1 then the calculated altitude is used in the ISA/US1962/72 pressure formula. If the implementation of the direct formula is the same of the inverse formula then the input pressure must be equal to ISA pressure.

You can run the
example, and examine the calculation details, here is the scilab
barometricr1.sce file,
and verify that results confirm the equivalence between the two
methods. Just test what is the Scilab output with a 105000 Pa input.

The atmospheric model
calculates the following data about air

-\(\rho=f(h)\)

-\(T=f(h)\)

-\(p=f(h)\)
As example calculation result have a look
to the following figure that shows density vs altitude, you can
visualize the graph setting graph=1 into the example Scilab file and
executing it.

*Figure 14.2 Atmosphere density vs altitue ISA or US1962/1972 model.*

The
model is completed with the viscosity calculation, carried out with
the Sutherland's formula, as the other values you get the viscosity
using the supplied example Scilab.

With
this data about the atmosphere you have everything for calculate, in
a practical closed form, all the performances parameters of a fixed
wing aircraft. It is so easy to have an initial evaluation of the
performance of a particular airplane configuration, take off
distance, climbing characteristics and in general every performance
parameter inside the flight envelope.

Atmospheric models are
a coarse approximation of reality, anyway real time information on
pressure, temperature and altitude can be used to adapt the model to
current climate conditions.Into the next post a practical approach to
solving the issue will be introduced.

2007 David G.Hull, Fundamentals of Airplane Flight Mechanics, Springer Chapter 3