Question 6.P.17: Explain the principle of operation of the pitot tube and ind...
Explain the principle of operation of the pitot tube and indicate how it can be used in order to measure the total flowrate of fluid in a duct. If a pitot tube is inserted in a circular cross-section pipe in which a fluid is in streamline flow, calculate at what point in the cross-section it should be situated so as to give a direct reading representative of the mean velocity of flow of the fluid.
The principle of operation of a pitot tube is discussed in Section 6.3.1. It should be emphasised that the pitot tube measures the point velocity of a flowing fluid and not the average velocity so that in order to find the average velocity, a traverse across the duct is necessary. Treatment of typical results is illustrated in Problem 6.16. The point velocity is given by u=\sqrt{(2 g h)} where h is the difference of head expressed in terms of the flowing fluid.
For streamline flow, the velocity distribution is discussed in Section 3.3.4 and:
u_s / u_{ CL }=1-\left(s^2 / r^2\right) (equation 3.32)
where u_s and u_{CL} are the point velocities at a distance s from the wall and at the axis respectively and r is the radius of the pipe. The average velocity is:
u_{ av }=u_{\max } / 2 (equation 3.36)
When u_s=u_{ av }=u_{\max } / 2,
u_s / u_{\max }=\left(u_{\max } / 2\right) / u_{\max }=1-\left(s^2 / r^2\right)
and: 0.5=s^2 / r^2 from which s=\underline{\underline{0.707 r}}