# 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

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 $$\rho_{MSL} = 1.225\ kg/m^3$$

5.3: Exercises is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.