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The first graphs produced by the Mathcad file are the plots of the raw turbidity data vs. time and raw turbidity vs. settling velocity. The settling velocity (Vs) is calculated by dividing the distance between the ball valve and the zone illuminated by the infrared LED (z) of FRETA by the time elapsed(t).

{latex}
\large
$$
V_s = {z \over t}
$$

{latex}

For experiments involving varying alum dosage, another graph was produced to obtain the residual turbidity vs. settling velocity. Residual turbidity is the turbidity resulting from the flocs that failed to reach the capture velocity, which is 0.12 mm/s for the sedimentation tanks of the AguaClara plants. From this, we can get the mean residual turbidity for each variation of alum dose. The mean residual turbidity is the mean of all the raw turbidity measured after a time of approximately 22 min till the end of the sedimentation state. It corresponds to a range of sedimentation velocities from 0.12 mm/s to 0.089 mm/s. See example of the residual turbidity graph: Analysis Alum Dose Experiment.
TThen data smoothing and normalization are performed on the raw turbidity data. The smoothing allows us to exclude outlying data points caused by really large flocs passing in front of the light sensor of FRETA, which create turbidity fluctuations. The normalization allows us to compare data sets with varying influent turbidities. The plot of normalized turbidity vs. Vs (settling velocity) can be interpreted as a cumulative distribution function (CFD) of turbidity with respect to Vs. A CFD describes the probability that a variable is less than or equal to some value.
To make the analysis more robust, the experimental data was fit to a gamma distribution. Then a derivative of the CFD of the gamma distribution gives a probability distribution of the particle population with respect to Vs. From the fitted data, the data processor retrieves the mean sedimentation velocity for each alum dose as well as the coefficient of variation of the distribution. The coefficient of variation is the standard deviation of the distribution divided by the mean. It is an indication of the width of the sedimentation velocity distribution.

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