The Pitot tube was invented in the 18th century by French engineer Henri Pitot to measure the speed of water flows. Thanks to its simplicity and economy, this instrument is still used today in various industrial applications. Mainly to measure the volumetric flow rate in pipes and to determine the cruise speed of aircraft. There are also more advanced applications. In fact, this versatile tool is being used in the aerodynamic optimisation of Formula 1 cars, as shown in this video.
Conceptually, this simple sensor consists of a tube with a hole at the front to align with the direction of motion and a hole at the side of the tube. By measuring the difference in pressure between the two holes it is possible, as we shall see, to know the speed at which a body is moving in a fluid.
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Bernoulli’s equation
Bernoulli’s equation is one of the cornerstones of fluid mechanics and probably one of the most widely used mathematical relations in engineering. This equation describes nothing more than the conservation of energy for an ideal fluid, i.e. one with zero viscosity and incompressible along a flow line. More simply, what Bernoulli’s equation tells us in mathematical terms is that the energy of a fluid is constant along its path and is composed of three parts: gravitational potential energy, kinetic energy and pressure energy.
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In the equation the terms to keep in mind are g, which represents the gravitational acceleration (9.81 m/s2), the density ρ, the altitude z, the pressure p and the velocity v relative to the fluid. The term H is called the head and represents the maximum height to which you can raise the column of fluid by transforming all the pressure and kinetic energy, contained in the fluid itself, into potential energy. If you are unfamiliar with the subject, you may be confused by the fact that we are talking about energy, but as units we are using metres and not joules for the various terms in the equation.
In fact, Bernoulli’s equation can also be rewritten in terms of specific energy, i.e. per unit mass, simply by multiplying everything by g. However, in hydraulic applications it is preferred to use the Head concept. For example, when you buy a hydraulic pump you are always given, or at least should be given, the pump head value, which is just another name for H. With this parameter, you can immediately and intuitively know how high you can pump water from downstream to upstream, because this height corresponds exactly to the pump head value.
Bernoulli’s equation applied to the Pitot Tube
Before moving on to the mathematics, we need to try to understand how a fluid behaves near a fixed obstacle such as a Pitot tube. In general, an ideal fluid follows Bernoulli’s equation and so as it flows it has three types of energy as seen (point 1 in the image). At a certain point the fluid, flowing more or less horizontally along the current lines (z2=z1), reaches the cavity of the tube where it comes to a sudden stop at what is called the stagnation point (point 2 in the picture). The sudden stop of the fluid (v2=0) results in a transformation of kinetic energy into pressure energy. This causes the water column in the vertical portion of the pipe to rise.
The same event can be expressed in mathematical terms by using Bernoulli’s equation to derive the velocity of the fluid at point 1. This is done by calculating Bernoulli’s equation at point 1 and imposing it equal to it at the point of stagnation (point 2) due to conservation of energy along the current line.
With a bit of algebraic manipulation, we come to the conclusion that speed can be measured as the difference in pressure between the static and dynamic grips. Nowadays, Pitot tubes work with pressure sensors that replace the rudimentary graduated tubes originally designed. However, as electronics did not yet exist in the 18th century, it is interesting to understand how the French engineer was able to calculate speed using only graduated tubes.
Supposing that the tubes are subjected to atmospheric pressure (in terms of relative pressure), the difference in pressure between point 1 and 2 can be calculated simply by using the equation of fluid statics. Another pillar of fluid mechanics, where i is the pressure exerted by the water column at a point i within the fluid, and is the height of the water column from the free surface of the tubes to the point i.
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From this last relation we can see that the speed at which the fluid moves is simply given by the difference in height of the free surface in the two tubes. The rise of the free surface in the static outlet is given by the static pressure of the fluid only, i.e. the normal (thermodynamic) pressure we are all used to thinking about. On the other hand, the rise of the free surface in the dynamic inlet is given by the sum of the static pressure of the fluid, the same as measured by the static inlet, and the so-called dynamic pressure, which represents the local increase in pressure due to the sudden stopping of the fluid.
In a more intuitive way you can think of when you suddenly turn off the tap in your house, in fact, if you pay attention you will feel a strong vibration due to the water hammering against the valve you have just closed. The quicker you close the tap, the more violent this blow will be, as the kinetic energy of the water is rapidly transformed into pressure energy.
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