Monday, May 6, 2013

Pneumatic lag issues

Have a look to this pitot simulation block

Some time ago at the airfield playing around with a pressure sensor  with sample rate of 50 Hz I noticed a really poor unexpected performance of my carbon fiber Pitot. You know, the measure system is quite simple, one digital pressure sensor connected to an Arduino board, two pressure lines , the static pressure port and the total pressure port. I ended the post fly analysis with a sole sure fact, I haven't a math model for the pneumatic transmission line. I needed such a model to calculate the relationship between the amplitude of the port pressure and the pressure at sensor at different frequencies, syntetically I needed the transfer function. With a transfer function it is straightforward to determinate the overall measurement system performance and even compensate for undesired transmission line behavior. Refer to the below figure 1 for the schematics of a pneumatic line.

Figure 1 Pneumatic line layout

Input pressure cannot be present at the very same time at the pressure port and at the sensor port, it take some time for the pressure wave to propagate trough the transmission line.
At last I found a text where the authors integrated the Navier-Stokes equations to achieve a closed form solution, from now on I will proceed according to reference [7] that you find at this link.
Is very important to considerate the model assumptions, here below  the list from [7] pag 7

Length/Radius >> 1
Variable nominal values variations are small
Laminar flow inside the tubing

To get a numeric example let's pretend we have the following, possible,  parameters values

Internal radius R of line [1 ; 2 ; 3] mm
Length of pressure line L 0,6 m
Volume of the sensor Vu 50e-9 mm^3

Let's consider now three different cases from the same base design, only variable is the line internal radius R, results plotted in the decimal scaled graphic below.

Figure 2 Frequency response of pneumatic line

You can download the scilab source here

By inspection of the figure it's correct to assert that if the radius increase the frequency of resonance is increased too, with a wider transmission line volume the pressure transmission and the resonance have a higher magnitude. Without correcting our sensor readings we can use only frequencies that have a gain equal to unity, so it's clear that transmission lines can have a heavy impact on the performance, according to figure 2 we can use safetly frequencies under 15 Hz.
During the use of an instrument in non steady conditions the frequency of pressure variation at pressure ports should be considerate, if not you probably will waste resources using better sensors of pressure and have poor results. Operating with a wider bandwidth may have some drawback, the first of all is some degree of increased vulnerability with respect to unwanted pressure oscillations due to buffeting or eddies.
In figure 1 schematics the transmission line have a constant diameter, by experience, that it's more common to have at last one variation in the line diameter, this case is well covered by papers and maybe worth a future post.
Reference [7] have a plethora of figures that illustrate different cases, also the conclusion section is quite clear and concise, of course it worth the time to read through it.
So at the end the problems that I've got in the past can be explained, and some times avoided, taking into account the dynamic of the pneumatic line.


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