Inexplicable Results in a Vacuum Experiment

Question

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.

Thanks in advance.

Answer 1

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.

More info I might use:

• What is your atmospheric pressure? Use your altitude to find it, if you don't have it. We need it to find if the chamber pressure has indeed exceeded atmospheric pressure. That is called overshoot (see picture above) and will confirm our hypothesis.
• Post a new graph zooming in on the period where the chamber is filling. The 200-225 seconds window should be good.
• What were your expectations about the experiment? How long should it take to fill the chamber? Was filling up in less than ten seconds expected?

Answer 2

One environmental variable that is being ignored, but which is significant at the 5L volume you are dealing with is Temperature!

As you evacuate that much air, you’re going to have a measurable cooling effect on the air and the whole container.

As the air rushes in, it changes this temperature, but not fast enough that you can simply hand waive it’s existence. The effect of your container equilibrating to room temp could take minutes.

I see this all the time with compressor tanks where it’s the opposite effect wrt Temp. (Heating). What I observe is that one can fully discharge a tank leaving valve 100% open (let’s say half inch valve), and come back 30 to 60 seconds after there is no audible or tangible flow, close off the valve, and in a few minutes there is enough pressure built up in the system to discharge another blast of air. Sometimes 2 or 3 times hence.

The answers above about damped oscillation after exponential rise are correct, and I offer consideration of internal temp to refine this idealistic solution.