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The other day I with my team had to measure the volumetric flow rate through a pipe only using a 2000 mm$^3$ volumetric flask and a chronometer. The end of the pipe discharged to the atmosphere. As we thought the measurement was very imprecise and we only had 3 measure to do, we wanted to make the measurement the best we could, so I told them "Hey, lets measure the time it takes to make it to 2000 mm$^3$ three times and the take the simple mean of that value, so the flow rate would be $$Q^*=\dfrac{3}{t_1+t_2+t_3} \times 2000 \text{ mm}^3/s.$$ and that's it, we will minimize manipulation errors."

But someone said "that is not the best thing to do, something better is to take 3 points $(t_1,V_1), (t_2,V_2), (t_3,V_3)$, make a linear regression and use the slope as the flow rate $Q^*$

My concerns with this method is that if we take the points too separated, then some point will be near cero and therefore the error associated with the manipulation of the flask will be big, but if we take all the points near 2000, then the linear regression will be with 3 points very near each other, and if some measurement went wrong, that would affect very much the final result.

What method do you think is more precise? Why? I suppose the statistics involved are simple but I can't solve this by myself.

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In principle you can push the whole error expression (including the correlation of the coefficients to a linear fit if you want to check that $c_0 = 0$) through symbolically ahead of time and then plug in approximate values for the flow rate and the proposed measurement volume to find an analytic answer. Obviously that is a huge amount of work for such a basic measurement, but something similar (involving Monte Carlo's rather than full symbolic expressions) is done for very expensive measurements in nuclear and high energy physics. –  dmckee Apr 21 '13 at 17:06
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up vote 2 down vote accepted

If I understood your problem correctly, assuming volumetric flow rate is constant, there must be three stages of flask filling:

  1. Process of putting flask under flow
  2. Linear filling
  3. Flask saturation

If flow rate is derivative of (Volume x Time) curve, best sampling points are in 2-d stage. Not in saturation stage.

Your advisor was right.

I've made diagram, for better understanding, but I'm lacking reputation to post it right now.

edit: Now i can post picture: enter image description here

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Yes, the flow rate is constant and we want to determine its value. You could send it via email (see my profile) as I still can't understand why it is better that way. –  Francisco Apr 21 '13 at 16:31
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Here you go, red are the one measured after filling. green ones are in linear stage, and error is smaller when you getting in linear stage. link –  Alexander Apr 21 '13 at 16:55
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