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I want to calculate each step like IAS -> CAS -> EAS -> TAS as a chain. Each chain step should depend on the previous chain step.

IAS is clear. This is function of pressures. With Bernoulli equation, IAS = f(p_total - p_static)

CAS is not clear. But I know that static error is taken into account, CAS = f(IAS,static error)

EAS is not clear. I know that EAS takes compressiblity effect into account.

TAS is again clear, dependend on density it is easy to calculate with TAS = f(EAS, rho, rho_0)

How to calculate EAS from CAS immediate with EAS = f(CAS)?

I know that one can calculate EAS with EAS = f(Mach, a_0), also by taking in account the compressibility of air.

Thank you very much!

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Airspeed is measured with a pitot tube. A pitot tube has two pressure measurement ports. One that measures the total pressure $p_t$. This port is facing the incoming airflow. The other measures the static pressure $p$ and is placed perpendicular to the airflow. The difference between the two pressures is called impact pressure (pressure rise do to the airflow impacting the pitot tube) and is denoted $q_c$.

The impact pressure is related to the speed of the airflow the pitot tube is exposed to. If the flow is considered incompressible (which is an acceptable approximation for speeds up to 200 knots) the impact pressure can be derived from Bernouilli's equation.

$q_c = \frac{1}{2}\rho V^2$

  • $q_c$ is the impact pressure in Pa
  • $\rho$ is the density in kg/m3
  • $V$ is the true airspeed in m/s

Equivalent airspeed

The airspeed indicator is calibrated for standard sea level conditions, where $\rho$ is 1.225 kg/m3. In reality the aircraft will fly at altitude and therefor the actual density of the air is lower. Therefor the airspeed as measured by impact pressure will be lower as well. For example if an aircraft flies 75 m/s (about 146 knots) at 6000 ft the density will be 1.02393 kg/m3.

$q_c = \frac{1}{2} 1.02393 \cdot 75 ^2 = 2879.8 \textrm{ Pa}$

The equivalent airspeed at sea level for the same $q_c$ is:

$V_{EAS} = \sqrt{\frac{2 q_c}{\rho_0}} = \sqrt{\frac{2 \cdot 2879.8}{1.225}} = 68.6 \textrm{ m/s}$

Your airspeed indicator (assuming no errors) will read only 68.6 m/s (133 knots) despite the fact that you are moving with 75 m/s (146 knots) with respect to the air.

Conversion of true airspeed to the equivalent airspeed can be done directly by:

$V_{EAS} =V\cdot \sqrt{\frac{\rho}{\rho_0}}$

  • $V_{EAS}$ equivalent airspeed (m/s)
  • $V$ true airspeed (m/s)
  • $\rho$ actual air density (kg/m3).
  • $\rho_0$ density at standard sea level conditions (1.225 kg/m3)

Calibrated airspeed

The effects of the lower density on your aispeed indicator become more pronounced the higher you go. Once you go faster than about 100 m/s true airspeed the effects of compressibility can no longer be ignored and the above no longer applies. Airspeed indicators are corrected for effects of compressibility and therefore don't use the equivalent airspeed but instead use calibrated airspeed for calibration.

$V_{CAS}=a_{0}\sqrt{5\left[\left(\frac{q_c}{p_{0}}+1\right)^\frac{2}{7}-1\right]}$

  • $V_{CAS}$ is calibrated airspeed
  • $a_{0}$ is the speed of sound under standard sea level conditions (340.3 m/s)
  • $p_0$ is the static air pressure under standard sea level conditions (101325 Pa)
  • ${q_c}$ is the impact pressure

The impact pressure is also a bit more complex for compressible flow:

$\;q_c = p\left[\left(1+0.2 M^2 \right)^\tfrac{7}{2}-1\right]$

  • $p$ the static pressure
  • $M$ the Mach number

Indicated airspeed

The airspeed that is actually indicate on the airspeed indicator deviates from the calibrated airspeed because of several error factors:

  • instrument error
  • position error
  • installation error

Instrument error are errors within the airspeed indicator when converting the static pressure and the total pressure to a speed indication. In mechanical instruments these are often more pronounced that in digital systems.

Position error are errors in the position of the static port (not measuring exactly static pressure, but also some effects of moving air) and position of the total pressure port (not exactly measuring the full ram rise).

Finally there are installation errors, which include for example leaking tubes between the instrument and the pitot ports.


Given the above, we can now derive the relation between Calibrated Airspeed and Equivalent airspeed

Calibrated airspeed is dependent on the impact pressure, which is in turn depending on the Mach number.

The Mach number is the ratio between the true air speed and the speed of sound $a = \sqrt{\gamma R T}$. We can now express the Mach number as a function of the equivalent airspeed:

$M = \frac{V}{\sqrt{\gamma R T}} = \frac{V_{EAS}\sqrt{\frac{\rho_0}{\rho}}}{\sqrt{\gamma R T}}$

From the ideal gas law it follows that $p = \rho R T$ and so we can simplify the Mach number to :

$ M = V_{EAS}\sqrt{\frac{\rho_0}{\gamma p}} $

From this it follows that impact pressure for compressible flow is:

$\;q_c = p\left[\left(1+0.2 M^2 \right)^\tfrac{7}{2}-1\right] = p\left[\left(1+ \frac{\rho_0}{5\gamma p} V_{EAS}^2 \right)^\tfrac{7}{2}-1\right] $

That brings the relation between CAS and EAS to :

$V_{CAS}=a_{0}\sqrt{5\left[\left(\frac{p}{p_{0}}\left[\left(1+ \frac{\rho_0}{5\gamma p} V_{EAS}^2 \right)^\tfrac{7}{2}-1\right]+1\right)^\frac{2}{7}-1\right]}$

Solving it for EAS = f(CAS) is left to the reader.


Parts of this answer are taken from this answer.

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  • $\begingroup$ All what is explained here is information that you find in most books, "the easy way", but Im looking for more mathematical explanation. Is there some authors who goes deeper? $\endgroup$ – ptiza_v_nebe May 25 at 11:21
  • $\begingroup$ Hello @ptiza_v_nebe, I don't understand what you are looking for. All the mathematical relations between the speeds are in the answer, except for the relation between IAS and CAS because that relation is specific to each aircraft. Can you explain what you are missing? $\endgroup$ – DeltaLima May 25 at 16:32
  • $\begingroup$ My question is still EAS = f(CAS). How to accomplish this? How does ADIRU calculate all the speeds if it has only pressures from pitot tube and Temperature? Other thing, what you have mentioned is only summarize of all knowledge about air characteristics and using pitot tube. What I need for example is, where does the formula for CAS come from? What are the roots with all physical relations? Is there no author who can explain things about air data holistically? $\endgroup$ – ptiza_v_nebe May 26 at 9:09
  • $\begingroup$ Hello @ptiza_v_nebe The ADIRU follows these equations and may have a lookup table to correct the for pitot port position errors. For the fundamental physics behind the compressible flow, I would start with the Euler equations. You can integrate these to get to Bernoulli's equation (assume $\rho$ is constant) or to the compressible variant (assuming adiabitic compression), taking into account the appropriate thermodynamic equations. It is a bit too much to cover that here in a single answer. $\endgroup$ – DeltaLima May 27 at 11:20
  • $\begingroup$ DeltaLima! You are the best! Thank you! $\endgroup$ – ptiza_v_nebe May 27 at 15:50

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