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In Situ Aircraft Temperature
Measurement
MJ Mahoney
Initial Release: September 16, 2002
Last Revision: October 10, 2007
Introduction
While analyzing some data from a recent field campaign (CAMEX-4),
I noticed that I was not getting very good agreement between the ER-2
Nav Data Recorder outside air temperature (OAT, or static air
temperature) and the measured MTP outside air temperature. Then I
remembered that the Nav Data
Recorder pressure altitudes were in error because of a leak in the
static air line, which caused the
actual altitude to be underestimated by 600 meters or more. So the
question
was: could this explain the differences in the temperatures as well?
Keeping things simple to start off, I recalled that the relationship ( a derivation is provided below ) between
static air temperature (SAT or Ts), and
total air temperature (TAT or Tt), was given by:
(1) 
where 
is the ratio of the specific heats at
constant pressure and volume, and 
is the Mach Number,
which is the ratio of the aircraft's true air speed (TAS or va)
, and the local velocity of sound vs . Now the velocity of
sound (to first order) depends only on the static air temperature ( 
),
where all the quantities have been previously defined except
the Gas Constant, R , which for dry air is 287.053 J kg-1 K
-1 .
I didn't know off hand how Equation (1) was derived; I had simply been
told it years ago when I asked someone how SAT and TAT where related.
So this got me thinking because I wanted to know to determine how large
the SAT error would be if the pressure altitude of an airplane was
incorrect by 600 meters.
First off, I though there was a catch-22 in Equation (1). If T
t was measured (see below), then Ts could be
determined if the Mach Number was known, but Mach Number also depends
on the velocity of sound, which in turn depends on the thing that you
want to know, T s , and the true air speed. It turns
out that Mach Number can be determined independently if the total and
static
pressure are known by using the equation.
(2) 
Then a measurement of the static air temperature can be determined from:
(3) .JPG)
As it turns out, Tt cannot be measured because that would require 100%
conversion of the kinetic energy of the adiabatic flow into heat in the
probe, and this can never be achieved. The measured temperature (Tm
) is instead corrected to approximate Tt very accurately by
applying a correction that depends on aircraft attitude and Mach
Number. So there IS a catch-22, but since the corrections are small,
the Mach Number effect on the correction is second order.
Derivation of Equation (1)
It all begins with the First Law of Thermodynamics -- the conservation
of energy. A closed system has three forms of energy: internal,
kinetic, and potential, and what the First Law of Thermodynamics states
is that any changes in the total energy (
) of the system
-- that is, the sum of changes in the internal energy (
), the kinetic energy (
), and the potential energy (
) -- during a thermodynamic state change must equal the heat added to
the system (Q) minus the work done by the system (W). That is,
(4) 
or equivalently,
(5) 
The subscripts "1" and "2" just refer to the two different
thermodynamic states, before and after the state change. Now for our
purposes, and without loss of generality, we will assume that there is
no change in potential energy, and note that the mechanical work done
during the state change is just:
(6) 
where p and V are the pressure and volume. With this substitution
we can introduce the initial and final enthalpies, or energy content,
defined by: 
and 
. Equation (5) can then be written as:
(6) 
.
While some state variables, such a temperature, do not depend on the
amount of material present, others, such as the internal energy and
enthalpy, do. Because of this it is, convenient to define the latter
state variable per unit mass, and give them names like specific
internal energy and specific enthalpy, and denote them by lower case
letters such as h1 and h2 for the specific
enthalpy.
If a thermodynamic state change happens fast enough (or if the
apparatus is well insulated), negligible heat is added or taken
away (that is, the process is adiabatic) and Q = 0. On rearranging
Equation (6) and dividing through by the mass (m) to use specific
enthalpy, we obtain the energy equation for invicid flow of an ideal
gas:
(7) 
Now consider what happens when air is flowing by a total air
temperature probe on an aircraft. If the aircraft is effectively
stationary, the probe will measure the static air temperature Ts
due to the average thermal motion of the air molecules hitting it. If
the aircraft has a true air speed va, the temperature
measured by the probe will clearly increase because of the extra
translational energy of the air molecules hitting it. Thus, for the
initial state (1) we have v1 ~ 0,
and for the final state (2), v2 = va, so that
Equation (7) becomes:
(8) 
Now from introductory physics classes, we also know that the the
specific heat at constant pressure is simply the derivative of specific
enthalpy with respect to temperature, or 
, or using
infinitesimal notation, 
. Substituting for the specific
enthalpy difference in Equation (8), letting T1 = Tt
and T2 = Ts, and rearranging, we have:
(9) 
Now the velocity of sound (a) at any temperature (T) is given by 
,
so on substituting for Ts and Tf, we can
write:
(10) 
On dividing through by vs, noting that vs =
a, the speed of sound at temperature Ts, and that the Mach Number is , we
have
(11) 
Finally, using the expression ( ) for the speed of sound again,
we see that the ratio of the
velocities squared on the left hand side of Equation (11) is simply the
ratio of the temperatures, or:
(12)
And we are done with the derivation of Equation (1). Of course, there
are a lot of simplifying assumptions, such as adiabatic, invicid
flow, that go in to deriving this expression, and these need to
be accounted
for in order to obtain very accurate temperature measurements.
Derivation of Equation (2)
The derivation of an expression for the Mach Number in terms of
the total pressure and static pressure is actually a little involved,
and again involves some thermodynamics. During a quasi-static adiabatic
process, the change in internal energy is given by the work done, or: 
. But the change in internal energy is also given by: 
. Equating these two expressions, and using the Ideal Gas
Law to eliminate the pressure, we can write:
(13) 
Noting that 
and rearranging terms, Equation (13) becomes:
(14) 
where we have also used the fact that 
and .
This equation can also be written as: .JPG)
, which on
integrating from the static temperature to the total temperature, and
from the initial volume(Vs) to the final volume (Vt
), gives:
(15) 
,
or, noting that the volume and density are inversely related,
(16) 
This establishes the fact that for a quasi-static adiabatic process 
, a result which we will now
use to relate total and static temperatures to total and static
pressures. Using the Ideal Gas Law to eliminate temperature in terms of
pressure and density, the total to static temperature ratio can be
written:
(17) 
Substituting this result into the left hand side of Equation (12)
and rearranging to solve for the Mach Number results in Equation (2),
which
is repeated here:
(18)
Since 
the coefficient inside the radical is 5 and the exponent of the
pressure
ratio is 2/7.
Measuring the Total Air Temperature
Having derived Equation (1), it is necessary
to point out some of the potential error sources. The whole point for
using this equation is that
it is very difficult to measure the static air temperature accurately;
it is easier to calculate it based on other measurements. However, it
is
also difficult to measure the total air temperature accurately. Before
proceeding, we need to be clear about what is meant by the various
temperatures:
- Static Air Temperature (SAT or Ts): This is the
physical temperature of the air which the aircraft is flying through.
It
is also know as the outside air temperature (OAT); it is the
temperature
that we need to determine.
- Total Air Temperature (TAT or Tt): This is the
temperature that would be measured by a probe if all of the kinetic
energy of the air resulting from the aircraft's motion was absorbed.
Because this is impossible, it can never be measured!
- Recovery Temperature (Tr): This is the
temperature that the total air temperature is approximated by because
of the incomplete recovery of the kinetic energy of the air by the
temperature probe.
- Measured Temperature (Tm): As the name implies,
this is the temperature that is actually measured by the aircraft's
temperature probe. It differs from the recovery temperature, Tr,
because of parasitic heating or cooling of the temperature sensor.
The figure to the right shows the relationship between these
temperatures. As might be expected, Tr is always less than Tt,
and both these approach Ts as the Mach Number approaches
zero. All
of these temperatures can be be related to the flight speed, but T
m is more involved. As shown by the cyan and white vertical box, T
m may be higher or lower than Tt and Tr due
to sensor design, location, and parasitics for any particular flight
condition (that is, speed and attitude). In extreme case such as for
vortex-producing probe
designs or for severe weather conditions, Tm may actually be
less
than T s. For a well-designed temperature probe, however, it
is
possible to achieve Tm > 0.995 Tt.
The parameter that relates the Tr to Tt and T
s is called the recovery factor ( r ), which is defined by:
(19) 
Using this definition in Equation (1) we obtain:
(20) 
For most total temperature sensors, the recovery factor varies with
Mach Number. It is now more common to use the recovery correction (
)
defined by:
(21) 
Typically, the recovery correction varies with Mach Number below M = 1,
but is constant for higher Mach Numbers. The recovery factor is related
to the recovery correction by:
(22) 
Based on wind tunnel measurements for a specific total air temperature
probe, a chart for the dependence of the recovery correction on
Mach
Number can be created. This in turn can be used in Equation (22) to
obtain
the recovery factor (r), which in turn, by assuming that Tm
very
closely equals Tr , allows Equation (20) to be solved for T
s. How well all this can be done depends on how well the
total
temperature probe is characterized for a number of other important
effects:
- Time Constant: How fast the probe responds depends on the
heat capacity of the sensor parts
- Thermal Conduction: If the aircraft body is much warmer
than the probe (for example, in the sunlight on the ground), heat may
be transferred to the probe.
- Thermal Radiation: If the probe is very warm as it might
be on the ground, it may radiate some of it's energy away, thus
appearing cooler than it actually is.
- Airflow Direction: Although TAT probes are designed to be
insensitive to the direction of the air stream entering the probe,
there
nevertheless remains some dependence, especially for large attitude
excursions.
- Self-Heating: Some probes require a small constant
current to flow through them in order to generate a voltage
proportional to resistance (which depends on temperature). This current
can increase the temperature of the sensing element.
- De-icing: Some probes have de-icing elements which when
activated, can cause heat to flow toward the sensing element.
Generally the probe manufacturer will provide tables or equations which
can be used in the Air Data Computer (ADC) to correct for these effects.
Since we basically have all we need to calculate the true air speed
(TAS), we finish off with that. Since TAS is simply Mach Number times
the speed
of sound at the static air temperature, we can use Equation (21) to
solve
for Tt in terms of Tr and , substitute this
value of Tt into Equation (3) for Ts , and then
use this value of Ts in the expression for the speed of sound ( ), to
obtain:
(23) 
.
The Bottom Line
Having gone through all of this, we still need to answer the question
that started it all: "Could the pressure altitude error explain the
fact
that based on radiosonde comparisons the Nav Data Recorder static air
temperature
was 1.5 K too warm. The answer is yes. The ER-2 flies at Mach = 0.715
and
the flight data indicates this Mach Number. The Air Data Computer (ADC)
calculates
Mach number using static and dynamic pressure, therefore, it appears
that
the dynamic pressure measurement must be correct, and that the reported
total
pressure is in error. Based on comparisons with radionsondes, I
concluded
that the pressure altitude was in error by up to 700 meters. If I
change
the static pressure to reflect this, but use the same dynamic pressure,
the
effect is to increase the static temperature by 1.9 K. However, I was
also
told by DFRC personnel that the ADC used a recovery factor of 1
in
deriving the static air temperature from Equation (20). If the correct
recovery
factor for Mach 0.715 is used (r = 0.98), the effect of using a
recovery
factor of 1 is to reduce the static air temperature by 0.4 K. The net
result
of these two effects is that the static air temperature is 1.5 K too
warm,
in perfect agreement with our comparisons against radiosondes.
The End
Just for the Fun of it!
I started off writing this page by saying that to understand in situ
temperature measurements you had to understand how a refrigerator
works, but this made the discussion more complicated than it needed to
be. But since this was written, here it is anyway! What we are
specifically interested in to understand refrigerators is called a
throttling process (or porous plug experiment) that leads to a state
change. We start off with a fluid or gas at high pressure and squeeze
it through a porous plug or needle valve in a manner that keeps the
pressure constant on both sides of the porous plug and allows no heat
to enter or escape during the process. Eventually all the fluid or gas
is
transferred from one side of the plug to the other. When that is done,
the
volume on the high pressure side (denoted by subscript "t") will go
from V
t to zero, and on the low pressure side (denoted by subscript
"s")
will go from 0 to Vs. The net work done is: W = Vt
p t - V s ps .
If the process happens fast enough or is well insulated, no
heat is added or taken away (that is, the process is adiabatic) and Q
= 0. Then First Law of Thermodynamics can be caste in the form.
(A1) 
, or 
where we have introduced the thermodynamic quantity, H, which called
the enthalpy of the process -- a conserved quantity. If we write
Equation (A1) per unit mass (m) of air, and note that 
, where
is
the density, then we obtain:
(A2) 
, or
ht = hs
where the lower case is used indicate the specific internal energy and
enthalpy. Using the Ideal Gas Law (
), we can
also write:
(A3) 
And this is how a refrigerator works. The change in internal energy
results in a temperature change of the gas, which is needed for cooling.
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