eFunda: Theory of Pitot Static Tubes




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Pitot Static Tubes: Theory


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Introduction






The Pitot tube (named after Henri Pitot in 1732) measures a fluid
velocity by converting the kinetic energy of the flow into potential
energy. The conversion takes place at the stagnation point, located at the Pitot
tube entrance (see the schematic below). A pressure higher than the
free-stream (i.e. dynamic) pressure results from the kinematic to
potential conversion. This "static" pressure is measured by
comparing it to the flow's dynamic pressure with a differential
manometer.



Cross-section of a Typical Pitot Static
Tube
Converting the resulting differential pressure measurement into a
fluid velocity depends on the particular fluid flow regime the Pitot
tube is measuring. Specifically, one must determine whether the
fluid regime is incompressible,
subsonic
compressible, or supersonic.





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Incompressible Flow






A flow can be considered incompressible if its velocity is less
than 30% of its sonic velocity. For such a fluid, the Bernoulli
equation describes the relationship between the velocity and
pressure along a streamline,


Evaluated at two different points along a streamline, the
Bernoulli equation yields,


If z1 = z2 and point 2
is a stagnation point, i.e., v2 = 0, the above
equation reduces to,


The velocity of the flow can hence be obtained,


or more specifically,






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Subsonic Compressible
Flow






For flow velocities greater than 30% of the sonic velocity, the
fluid must be treated as compressible. In compressible flow theory,
one must work with the Mach number
M, defined as the ratio of the flow velocity v to the
sonic velocity c,


When a Pitot tube is exposed to a subsonic compressible flow (0.3
< M < 1), fluid traveling along the streamline that
ends on the Pitot tube's stagnation point is continuously
compressed.


If we assume that the flow decelerated and compressed from the
free-stream state isentropically, the velocity-pressure relationship
for the Pitot tube is,


where g is the ratio of specific heat at
constant pressure to the specific heat at constant volume,


If the free-stream density rstatic is not available, then one can
solve for the Mach number of the flow instead,


where is the speed of sound (i.e. sonic
velocity), R is the gas constant, and T is the
free-stream static temperature.




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Supersonic Compressible
Flow






For supersonic flow (M > 1), the streamline terminating
at the Pitot tube's stagnation point crosses the bow shock in front
of the Pitot tube. Fluid traveling along this streamline is first
decelerated nonisentropically to a subsonic speed and then
decelerated isentropically to zero velocity at the stagnation point.



The flow velocity is an implicit function of the Pitot tube
pressures,


Note that this formula is valid only for Reynolds
numbers R > 400 (using the probe diameter as the
characteristic length). Below that limit, the isentropic assumption
breaks down.
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