5.3: Exercises

Exercise \(\PageIndex{1}\) Pitot tube

Figure 5.19L Pitot Tube.

Aircraft use pitot tubes to measure airspeed. They consists of a tube pointing directly into the fluid flow, such that the moving fluid is brought to rest (stagnation pressure of the air, \(p_T\)). Typically, pitot tubes include also a static port to measure the static pressure of the air (\(p_{\infty}\)). See Figure 5.19 as illustration considering the air as a compressible flow.

Consider the following measurements on board the aircraft:

• The Pitot tube measures a stagnation pressure \(p_T = 36975\ Pa\).
• The static part measures a static pressure of \(p_{\infty} = 22500\ Pa\).

Assume also that:

• the air can be considered an ideal gas.
• the air should be considered a compressible fluid. For compressible flow, one has that

\[P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} M^2 \right )^{\tfrac{\gamma}{\gamma - 1}},\]

with \(\gamma = 1.4\) the adiabatic coefficient of air, and \(M\) the Mach number:

\[M = \sqrt{V_{TAS}}{\sqrt{\gamma RT}}\]

Calculate the calibrated airspeed of the aircraft (CAS).⁸

Answer

Notice that one can apply Bernoulli’s equation to fluid’s stream line within the Pitot tube. Assuming compressible flow, one has:

\[P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} M^2 \right )^{\tfrac{\gamma}{\gamma - 1}}.\]

Assuming also the air can be considered an ideal gas, one has:

\[P = \rho \cdot R \cdot T.\]

In addition, one has the following relation:

\[M = \dfrac{V_{TAS}}{\sqrt{\gamma RT}}.\]

All in all, elaborating with this three equations, one has:

\[P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} \dfrac{\rho_{\infty} \cdot V_{TAS}^2}{\gamma \cdot P_{\infty}} \right )^{\tfrac{\gamma}{\gamma - 1}}.\]

Considering \(P_T - P_{\infty} = \Delta P\) and isolating \(V_{TAS}^2\):

\[V_{TAS}^2 = \dfrac{2\gamma}{\gamma - 1} \cdot \dfrac{P_{\infty}}{\rho_{\infty}} \left (\left (\dfrac{\Delta P}{P_{\infty}} + 1 \right )^{\tfrac{\gamma - 1}{\gamma}} - 1 \right ).\]

However, one should notice that only with the pitot tube and the static port, neither temperature nor density can be measured. Thus, \(V_{TAS}\) can not be directly calculated (we would need additional instruments/sensors). This is the reason behind the Calibrated Airspeed (CAS) concept: the true airspeed an aircraft would have if flying with standard mean sea level conditions. Thus, CAS is defined as follows:

\[V_{CAS}^2 = \dfrac{2\gamma}{\gamma - 1} \cdot \dfrac{P_{MSL}}{\rho_{MSL}} \left (\left (\dfrac{\Delta P}{P_{MSL}} + 1 \right )^{\tfrac{\gamma - 1}{\gamma}} - 1 \right ).\label{eq5.3.8}\]

Indeed, the airspeed that is displayed in the cockpit to the pilot (referred to as Indicated Airspeed) is the CAS speed corrected with instrument errors.

Now, entering in Eq. (\(\ref{eq5.3.8}\)) with the values given in the statement, one has the solution to the problem:

\[V_{CAS} = 150\ m/s.\nonumber\]

---

8. assume mean sea level conditions are the standard ones according to ISA, \(P_{MSL} = 101325\ Pa\) and \(\rho_{MSL} = 1.225\ kg/m^3\)