Calculation of variance
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Calculation of variance


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leaching process model parameters (flowrates units kg=h)
\ufffdAu \ufffdCu \ufffdAu \ufffdCu gAu gCu g0Cu \ufffdAu;1 \ufffdCu;1 \ufffdAu;2 \ufffdCu;2 \ufffd
2
\ufffd1 \ufffd
2
\ufffd2 
0.9 0.8 0.5 0.75 0.05 0.19 0.004 1 100 0.5 95 2510-4 0.25 40
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 11
Fig. 3. Autocorrelograms of the four species flowrates.
Fig. 3 shows the autocorrelograms of the feed and discharge species flowrates and
Fig. 4 the six cross-correlograms between the four species flowrates as calculated from
Vyy.l/ after normalization by the y variances. The autocorrelations in the feeds are due
to the autoregressive nature of the disturbance generator. The larger extent of correlation
in the discharges are explained by the additional correlation introduced by the process.
The cross-correlation between the feeds is caused by the common random event \ufffd1, and
the cross-correlation between the discharges, and between the feeds and discharges are
complex mixtures of the process dynamics and the feed disturbance correlations.
Figs. 5 and 6 show the integration error variances calculated using the expression
of V of Eqs. 17\u201321 normalized by the species flowrate variances. Two different time
intervals [0; N T ] are considered (Fig. 5, N D 10; Fig. 6, N D 100) and in each case the
number of sample increments is varied from 1 to N . As it should be, the measurement
accuracy increases as the number of increments increases. Also it is logical that for
n D 1 the integration errors are lower than the flowrate variances, since the sampling
Fig. 4. Crosscorrelograms of the four species flowrates.
12 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
Fig. 5. Variance of the integration error as a function of the number of increments for a 10-h time interval.
Fig. 6. Variance of the integration error as a function of the number of increments for a 100-h time interval.
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 13
Fig. 7. A comparison of three different evaluation methods of the integration error variance.
interval (10 h or 100 h) is smaller than the duration of the autocorrelation of the flowrate,
specially at the process discharge.
Fig. 7 shows a comparison of the integration error variance calculated with the
following three methods:
\u2013 using the diagonal terms of V calculated from Eqs. 17\u201321.
\u2013 using Gy\u2019s approximation: diag V .n; k/ D diag V .1; k/=n, i.e. neglecting the
sample increment error correlation.
\u2013 neglecting the autocorrelation of the flowrates, i.e. calculating the integration error
variance as .diag Vyy.0//=n, the usual formula for random sampling of an uncorrelated
population.
Fig. 7 shows that Gy\u2019s method gives a good approximation and that neglecting the
flowrate autocorrelation produces a drastic overestimation of the error variance.
In addition to the integration error variance, the matrix V gives also the covariance
between the integration errors for the different species and=or the different streams. This
information, contained in the off-diagonal terms of V , is useful when the measured
species flowrates are to be reconciled before being used for subsequent calculations,
a procedure which is strongly recommended to improve the accuracy of process
performance evaluation (Hodouin et al., 1984, 1998). Fig. 8 gives the correlation
14 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
Fig. 8. Correlation coefficient of the four species.
coefficients of the four species integration error as a function of n for N D 100.
The correlation in the feed and the discharge are positive as they should be since
the random variations of Cu and Au flowrates are strongly coupled. The correlation
between the feed and the discharge are positive or negative depending on the number of
samples. This is related to the phase lag introduced by the process itself between its feed
and discharge. It is logical that the correlation tends to C1 or \ufffd1 when the number of
samples increases because the integration error tends simultaneously toward zero.
5.2. Flotation process
A second illustration of the method is now considered for the flotation circuit of
Fig. 1. The dynamic model structure for a flotation single unit is depicted in Fig. 9. G1,
G2, G3 are three unitary transfer functions and s is the separation coefficient evaluated
with respect to the variations around the nominal values. H is the ARMA generator of
the feed disturbance. If a decoupled model is considered for the two metals (Cu and
Zn), one can write independently the model for copper and the model for zinc. For
illustrating purposes it is here sufficient to consider only one metal (copper for instance).
Assembling the various parts of the model and with a rough calibration from data
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 15
Fig. 9. Basic empirical linear model of a separation unit.
obtained through a phenomenological flotation simulator (Makni, 1996), one obtains the
model of Fig. 10, and the following equivalent state\u2013space model, where the output .y/
is limited to the copper flowrates of the feed, reject and concentrate streams:
z.t/ D
0BBB@
\ufffd1 C s2.1\ufffd \ufffd1/.1\ufffd s1/ .1\ufffd \ufffd1/.1\ufffd s1/.1\ufffd s3/ .1\ufffd \ufffd1/.1\ufffd s1/
.1\ufffd \ufffd2/s1s2 \ufffd2 C .1\ufffd s3/.1\ufffd \ufffd2/s1 .1\ufffd \ufffd2/s1
0 0 \ufffd
1CCCA
\ufffd z.t \ufffd 1/C
0BBB@
0
0
1
1CCCA \ufffd.t/ (33)
y.t/ D
0BBB@
0 0 1
1\ufffd s2 0 0
0 s3 0
1CCCA z.t/C
0BBB@
\ufffdCu;1
\ufffdCu;5
\ufffdCu;7
1CCCA (34)
Fig. 10. Model of the flotation circuit of Fig. 1 (only the dynamic variations around the steady-state
operation conditions are presented).
16 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
Table 2
Parameters of the flotation plant
Flowrates (t=h) Separation coefficients Transfer function coefficients
\ufffdCu;1: 12.80 s1 : 0.77 \ufffd: 0.5
\ufffdCu;5: 0.87 s2 : 0.77 \ufffd1: 0.6
\ufffdCu;7: 11.93 s3 : 0.92 \ufffd2: 0.4
With the parameter numerical values listed in Table 2 for a discretization period
T D 0:1 h and a variance of 0.41 for the white noise \ufffd , one obtains the autocorrelograms
and cross-correlograms of Fig. 11, the integration error variances of Fig. 12 and the
correlation of the integration error for the three main streams (feed, concentrate, reject)
as a function of n presented in Fig. 13. The dynamic model presented here is quite
simple (only six parameters), but at the same time it is realistic and can be calibrated
from the measurements of the natural disturbances of an industrial flotation circuit,
using the same strategy as the one presented for the gold leaching tank.
Fig. 11. Autocorrelograms and crosscorrelograms of the species flowrate for the three main streams of the
flotation circuit of Fig. 1.
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 17
Fig. 12. Variance of the integration error for the three main streams.
Fig. 13. Integration error correlation.
18 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
6. Conclusion
A formula for the calculation of the variance\u2013covariance matrix of the sampling
integration error of the streams of a complex mineral processing unit has been proposed
in this paper. It assumes that the streams are synchronously sampled for measuring
the mass flowrates of the various species of interest in the processed material. The
variance\u2013covariance of the integration error is a function of the variance\u2013covariance
and autocovariance matrices of the flowrates involved in the flowsheet, a function of the
number of increment samples and finally a function of the sampling period.
The covariance and autocovariance matrices of the flowrates are estimated through a
state\u2013space dynamic model of the process complex unit. The application of the method
to two simple processes, a leaching tank and a flotation circuit, shows how the method
works and discusses the correlation which appears between the measurement errors of
the different species on the various streams. This information is useful when weighting
the reconciliation criterion for data filtering using material balance constraints around
a complex