Total air temperature

The difference between TAT and SAT is called ram rise (RR) and is caused by compressibility and friction of the air at high velocities.

(2) :${\displaystyle RR_{\mathrm {total} }=TAT-SAT\,}$

In practice the ram rise is negligible for aircraft flying at (true) airspeeds under Mach 0.2

For airspeeds (TAS) over Mach 0.2, as airspeed increases the temperature exceeds that of still air. This is caused by a combination of kinetic (friction) heating and adiabatic compression

• Kinetic heating. As the airspeed increases, more and more molecules of air per second hit the aircraft. This causes a temperature rise in the Direct Reading thermometer probe of the aircraft due to friction. Because the airflow is thought to be compressible and isentropic, which, by definition, is adiabatic and reversible, the equations used in this article do not take account of friction heating. This is why the calculation of static air temperature requires the use of the recovery factor, ${\displaystyle {e}}$. Kinetic heating for modern passenger jets is almost negligible.
• Adiabatic compression. As described above, this is caused by a conversion of energy and not by direct application of heat. At airspeeds over Mach 0.2, in the Remote Reading temperature probe (TAT-probe), the outside airflow, which may be several hundred knots, is brought virtually to rest very rapidly. The energy (Specific Kinetic Energy) of the moving air is then released (converted) in the form of a temperature rise (Specific Enthalpy). Energy cannot be destroyed but only transformed; this means that according to the first law of thermodynamics, the total energy of an isolated system must remain constant.

The total of kinetic heating and adiabatic temperature change (caused by adiabatic compression) is the Total Ram Rise.

Combining equations (1) & (2), we get:

${\displaystyle RR_{\mathrm {total} }={T_{s}{\frac {\gamma -1}{2}}eM_{a}^{2}}}$

If we use the Mach number equation for dry air:

${\displaystyle M_{a}={\frac {V}{a}}}$

where ${\displaystyle a={\sqrt {\gamma R_{sp}T_{s}}}}$

we get

(3) :${\displaystyle RR_{\mathrm {total} }={eV^{2}{\frac {\gamma -1}{\gamma 2R_{sp}}}}}$

Which can be simplified to:

${\displaystyle RR_{total}={\frac {V^{2}}{2C_{p}}}e}$

by using ${\displaystyle R_{sp}={C_{p}-C_{v}}}$

and

${\displaystyle \gamma ={\frac {C_{p}}{C_{v}}}}$

${\displaystyle a=}$ local speed of sound.

${\displaystyle \gamma =}$ adiabatic index (ratio of heat capacities) and is assumed for aviation purposes to be 7/5 = 1.400.

${\displaystyle R_{sp}=}$ specific gas constant. The approximate value of ${\displaystyle R_{sp}}$ for dry air is 286.9 J·kg−1·K−1.

${\displaystyle C_{p}=}$ heat capacity constant for constant pressure.

${\displaystyle C_{v}=}$ heat capacity constant for constant volume.

${\displaystyle T_{s}=}$ static air temperature, SAT, measured in kelvins.

${\displaystyle V=}$ true airspeed of the aircraft, TAS.

${\displaystyle e=}$ recovery factor, which has an approximate value of 0.98, typical for a modern TAT-probe.

By solving (3) for the above values with TAS in knots, a simple accurate formula for ram rise is then:

${\displaystyle RR_{\mathrm {total} }={\frac {V^{2}}{87^{2}}}}$

• In-Flight Temperature Measurements
• Measurement of Temperature on Aircraft
• TAT Sensor Operation and Equations
• TAT Sensor Heater Error Effect
• High speed flight - Viscous Interaction