Environmental Engineering Reference
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be observed (for 10 cm electrode separation), a small error considering the froth-
depths usually used in industry (1 m), they would be propagated to the actuator, e.g. ,
a pinch valve, when a tight level control is used, decreasing the lifetime of the actu-
ator and increasing the variance of the control loop. To correct the above mentioned
problems, a new algorithm was developed by Maldonado et al. [24]. While keeping
the main features of Gregoire's algorithm, it substantially improves its smoothness
when the interface is near an electrode or moving from one conductivity cell to
another, by considering the weighted average between the two possible values of
froth depth determined by the largest slope method. The steps of the algorithm are
the following: (a) calculation of the conductivity profile; (b) determination of the
possible cells containing the interface, i.e. , where the largest conductivity changes
occur; (c) estimation of the X value for these two possible cases using Gregoire's
algorithm; (d) calculation of the corresponding two possible froth depth values; and
(e) calculation of a weighted average froth depth estimate. That is to say, the froth
depth is determined as a linear combination of the two possible froth depth values,
where the weighing parameters are a function of the conductivities above and below
the cell containing the interface. The experimental results obtained have shown that
the proposed method gives very good froth depth estimates, improving accuracy and
smoothness, which makes this sensor very suitable for control and optimization pur-
poses. The availability of a conductivity profile in the upper part of the column also
opens the door to the evaluation of the bias rate as will be shown in the following
section.
6.3.2 Bias Rate Sensor
Two different approaches have been proposed so far for estimating bias rate. The
first one considers some sort of water mass balance on either zone of the column,
above or below the interface, and makes use of measurements of external flow rates
to estimate an internal variable, the bias rate. The second approach uses column
internal variables, measured near the interface, for estimating the bias rate. The ra-
tionale behind the methods proposed in this category is the change in some internal
variable with variation of the water content in the upper part of the column (above
the interface), resulting from a change in the bias rate.
A method based on the first approach estimates bias rate ( J b )byawatermass
balance in the column section below the feed port assuming a steady-state column
operation, as shown in Equation 6.8 where J wt and J wf are the superficial velocity
(volumetric flow rate divided by column cross-sectional area) of the liquid compo-
nent of tails and feed streams, respectively:
=
.
(6.8)
To implement this measurement method, four relatively expensive instruments
are required: two magnetic flow meters and two gamma-ray density meters, one of
J b
J wt
J wf
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