I want to measure the density in different layers of a suspension. To do this I want to place pressure sensors at different heights. Let's assume that the sensors are not by orders of magnitude more exact than neccessary (because I want to detect small differences in density, like 1% over heights of 1m or smaller, and maybe I'm on a budget for the sonsors). The idea would be to place the sensors on a vertical pole with a known distance between two. The pole will be placed in the liquid and not be moved vertically during the measurements. Possibly I will use two sensors at each height.

Now, I know that - labeling the sensors downward, 0 at the top - that $(p_1-p_0)+(p_2-p_1)=p_2-p_0$ Now my questions are, how do I use this fact to combine the readings into a coherent, exact picture and how do I calculate the actual error of my result?

Here's my approach:

  • Set up linear equations for the "true" pressures that follow the relation given above.
  • find - numerically, with some equation solver - the tupel of "true" pressures that satisfy the linear equations and have the least mean square deviation from the measured pressures

Is this approach basically correct? How would I calculate the error of the true pressure from the error of my sensors?

  • 1
    $\begingroup$ Are we to assume that in $(p_1-p_0)+(p_2-p_1)=p_2-p_0$ you have three separate sensor clusters or is this some kind of arrangement where you measure two values then move the package down so that the top sensor now occupies the position previously occupied by the lower one and measure again? These kinds of details matter because the go to the degree to which the uncertainties are or are not correlated. A diagram (or several) would really help here. $\endgroup$ – dmckee Mar 5 '13 at 17:05
  • $\begingroup$ BTW--I talk a bit about errors being correlated or uncorrelated in the above comment and my answer. Do you understand what I mean by this and how to work with correlated uncertainties? If not, reading on that subject is certainly the first thing to do. $\endgroup$ – dmckee Mar 5 '13 at 17:31

In the case you have indicated, the two instances of $p_1$ represent the same measurement, so their uncertainties are 100% correlated and since you have one $+p_1$ and one $-p_1$ both the measurement and the uncertainties cancel out. No help there.

Now, you have suggested have more than one sensor at each measurement location. That will help some, though you should be aware that in general each measurement will be of the form $$ p_{i,j} \pm \Delta_\text{cal} p_{i,j} \pm \Delta_\text{noise} p_{i,j} $$ where $p_{i,j}$ represents the $j$th sensor at the $i$th height; \Delta_\text{cal} and $\Delta_\text{noise}$ represent the uncertainty due to miscalibration and measurement noise respectively. The miscalibration effect will be constant from measurement to measurement (i.e. 100% correlated with other measurement from the same sensor), which the noise uncertainty should be uncorrelated.

You can get a bit of a win by averaging many measurements in time so that the mean noise goes to zero with a very small uncertainty-of-the mean, but unless your sensors are pretty noisy in the first place this may not be a big deal.

If the liquid is question is uniform in density over the whole column (incompressible and uniform in temperature) then you a measure of the non-linearity of the data: $$ A = \left( p_2 - p_1\right) - \left( p_1 - p_0\right) = \frac{p_2 - 2p_1 + p_0}{2} \,.$$ Naively the uncertainty due to noise on this is of order $\frac{1}{2}\sqrt{4\Delta_\text{noise}} = \Delta_\text{noise}$ (assuming the same performance parameters for all the measurement location).

This doesn't actually help you correct any given measurement but it provides a figure of merit for detecting a disagreement between the sensors (or a failure of the uniform density assumption). Sometimes this is useful in and of itself as it provides a hint that you no longer understand your machine and re-calibration may be in order.

  • $\begingroup$ thanks so far - but I assume non uniform density over the column (since I want to measure the density differences between different layers of a suspension) $\endgroup$ – mart Mar 6 '13 at 8:16

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