Question 5.2: The jet plane in Fig. 5–12 is equipped with a piezometer tha......
The jet plane in Fig. 5–12 is equipped with a piezometer that indicates an absolute pressure of 47.2 kPa, and a pitot tube B that reads an absolute pressure of 49.6 kPa. Determine the altitude of the plane and its speed.
The piezometer measures the static pressure in the air, and so the altitude of the plane can be determined from the table in Appendix A.
For an absolute pressure of 47.2 kPa, the altitude is approximately
h = 6 km
Fluid Description. We will assume that the speed of the plane is slow enough so that the air can be considered incompressible and inviscid—an ideal fluid. Doing so, we can apply the Bernoulli equation, provided we observe steady flow. This can be realized if we view the motion from the plane.* Thus, the air at A, which is actually at rest, will have the same speed as the plane when observed from the plane, V_A = V_p. The air at B, a stagnation point, will appear to be at rest when observed from the plane, V_B = 0. From Appendix A, for air at an elevation of 6 km, \rho _a = 0.6601 kg/m³.
Bernoulli Equation. Applying the Bernoulli equation at points A and B on the horizontal streamline, we have
\frac{P_A}{\gamma} + \frac{V_A^2}{2g} + z_A = \frac{P_B}{\gamma} + \frac{V_B^2}{2g} + z_B\frac{47.2 (10^3) N/m^2}{(0.6601 kg/m^3) (9.81 m/s^2)} + \frac{V_p^2}{2(9.81 m/s^2)} + 0 = \frac{49.6 (10^3) N/m^2}{(0.6601 kg/m^3)(9.81 m/s^2)}+ 0 + 0
V_p = 85.3 m/s
NOTE: We will show in Chapter 13 that this speed is about 25% the speed of sound in air (343 m/s), and since 25% < 30%, our assumption of air being incompressible is valid. Most planes are equipped with either a
piezometer and a pitot tube, or a combination pitot-static tube. The pressure readings are directly converted to altitude and air speed and are shown on the instrument panel. If more accuracy is needed, corrections are made by factoring in a reduced density of the air at high altitudes. Although this device has served the aviation industry well for many years, it is important that for proper operation, the opening of any pitot tube must always be free of debris such as caused by nesting insects or ice formation.
*If the flow is observed from the ground, then the flow is unsteady since the velocity of the air particles will change with time as the plane flies past these particles. Remember that the Bernoulli equation does not apply for unsteady flow.