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So I was thinking of using a barometric sensor to measure the compression of air in a tube in order to measure water level in a tank (Basic diagram below).

Air pressure in water tube

So based on the equation $ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $, I assumed that temperature remains constant (realistically it doesn't, but I will account for it), and given that the face area of the tube also remains constant, you can figure out the pressure via this equation: $ \frac{P_i h_i}{h_f} = P_f $.

So lets assume that the tube is halfway in the water, so $ \frac{h_i}{h_f} = 2 $, which means that $ P_f = 2P_i $. Does this mean that the water applies ~200kpa (2x mean sea level atmospheric pressure) of pressure on the air? And obviously we can't compress the air to infinity or even to extreme levels (otherwise expensive air compressors would never have been needed) just by forcing the tube below the water - so whats the limit on how much force (or pressure) water can apply on air in such an environment?

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There is a better way to measure water level. Run a tube to the bottom of the water tank that you want to measure the level of. Put a pressure sensor at the top of the tube. Run a slow flow rate of air into the tube. The pressure required to force air out the bottom of the tube is directly related to the depth of water that the air is forced to go through, and that measured pressure follows the equation $P=\rho g h$. The density of fresh water is 1000 $kg/m^3$, so the only unknown in the pressure equation is $h$, which is the height of the water that the air is bubbling through.

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  • $\begingroup$ You are using ideal gas equation for which you have to careful since it has limitations david's answer is best here $\endgroup$ – Prateek Mourya Sep 3 at 1:56
  • $\begingroup$ But how could I measure whether air bubbles are escaping? By seeing whether the pressure changes in the tube - and if there's no change in pressure that means that I can use the measured pressure to find the depth of the water? $\endgroup$ – explodingfilms101 Sep 3 at 17:26
  • $\begingroup$ @explodingfilms101, you can look for air bubbles rising up from the bottom of the tube. Also note - if you establish a small and continuous air flow rate in the tube, you are guaranteed that air is bubbling out the bottom of the tube via conservation of mass, unless there is a hole somewhere in the tube. If you are "fancy" enough to put some kind of high frequency pressure recorder on your pressure sensor, you should also see "noise in the record", indicating that the instrument is functioning,. $\endgroup$ – David White Sep 3 at 19:32
  • $\begingroup$ @sheltonBenjamin, thanks for the vote of confidence. Note that this is not my original idea ... this is actual industry practice for processes that are difficult to measure via normal means (e.g., the depth of liquid sulfur in a sulfur tank). Hopefully, the "pure" physicists in this forum will not get too annoyed that I posted a real world application for an answer (these usually show up on the engineering exchange). $\endgroup$ – David White Sep 3 at 19:35
  • $\begingroup$ Right but I'm trying to find a way to automate it. If the air bubbles even if the airflow is small - how would I be able to figure out the depth? So assuming that the tube is 2m long and half way in the water, that would mean that the water pressure at the bottom of the tube is ~10kPa. Would the air pump need to add an additional 10kPa above equilibrium conditions in order for the air to bubble out? If that is true - would you be able to see the current equilibrium pressure by measuring the air pressure in the tube? $\endgroup$ – explodingfilms101 Sep 3 at 20:02

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