# Inexplicable Results in a Vacuum Experiment

We've been doing the following experiment in our Physics Lab course and are coming up with some strange results which we're at a loss to explain.

The purpose of the experiment is to determine the conductance of various narrow pipes.

The outline of the experiment is as follows:

• Seal off a chamber from the atmosphere by closing tap 2 (See diagram below).
• Depressurise the chamber with a rotary vane pump.
• Seal off the chamber from the rotary vane pump by closing tap 1 and detach from the rotary pump.
• Open tap 2 and record how the pressure in the chamber varies as a function of time.

According to the theory we've been given, the flow through the narrow pipe will be viscous and thus the throughput will be given by:

$$Q = -V\frac{dP}{dt} = A(P+P_a)(P-P_a)$$

where $V$ is the volume of the chamber, $P$ is the pressure of the chamber, $P_a$ is the atmospheric pressure and $A$ is a constant which depends on the dimensions of the narrow pipe.

Solving this equation, one would expect that:

$$P = P_a\mbox{Tanh}\left(\frac{AP_a}{V}t + C\right)$$

The following plot shows one of our actual results:

You won't be surprised to learn that fitting the above equation to this graph produces a terrible fit. What is really baffling us however is the way the pressure actually noticeably decreases a little after reaching the peak and then slowly starts to increase again. We've repeated the experiment with different equipment and the phenomena isn't going away so it doesn't appear to be a case of equipment failure. We've also noticed that the pressure decrease is more pronounced when the narrow pipe has a higher conductivity (i.e. wider or shorter pipes).

So what is causing this inexplicable pressure drop? Once the chamber has filled with air, why would its pressure drop to below atmospheric and then start to rise towards it again?

One possibility we've thought of is that our experiment involved turbulent flow as opposed to laminar flow (we've done the calculations and this is certain). The theory we've managed to find describes the difference but is ambiguous as to whether the equations for pipe conductivity are applicable to only laminar flow or also turbulent flow. Is this relevant or are we barking up the wrong tree? If it is relevant, how would this explain the pressure drop? You should bear in mind that we haven't taken any classes on fluid dynamics yet so we have very little knowledge to work with.

• Excuse me, but I don't think your formula $Q=-V\frac{dP}{dt}$ is good. Its RHS unit is watts. Shouldn't it be m³/s? – André Chalella Nov 13 '14 at 17:01
• Your expression for a viscous pipe flow -- what speed regime do you think that is valid to use? And what speed do you think the air entering the pipe will be at given your pressure ratios? – tpg2114 Nov 13 '14 at 17:43
• Also, "pressure drop" relative to what? What is your final expected pressure? Does your pressure peak overshoot the expected pressure or not? – tpg2114 Nov 13 '14 at 17:45
• Quite frankly, wasn't it supposed to fill up much more slowly than that? Seems too fast. Intuitively, I don't think such a fast flow would be laminar (air isn't very viscous right?). – André Chalella Nov 13 '14 at 19:28
• @AndréNeves Since this is a homework question, let's be careful about how much we guide the answer. Not that we've crossed that line yet, but I have to keep reminding myself not to answer the question (and your previous comment as the answer to one is sort of the answer to the other). – tpg2114 Nov 13 '14 at 20:35

I'm not a fluid mechanics expert, but my mechanical systems knowledge suggests it might be simply a natural oscillatory behavior, which is always present but in this case is more noticeable due to the aggressive initial response (i.e fast influx of air) your chamber experiences.

So what is causing this inexplicable pressure drop? Once the chamber has filled with air, why would its pressure drop to below atmospheric and then start to rise towards it again?

It works like this, in a nutshell: initially, your chamber is in a vacuum. Air loves vacuum, so lots of it enter your chamber very quickly. Now your chamber has more pressure than the outside environment, so the dynamic is reversed: air leaves, until the inside pressure is lower than the outside. That happens a couple times until the system stabilizes. The graph below shows this behavior in a general manner.

I must say, however, that your graph isn't of a very typical oscillatory behavior. But it might be that.