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# Calculation of variance

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leaching process model parameters (flowrates units kg=h) �Au �Cu �Au �Cu gAu gCu g0Cu �Au;1 �Cu;1 �Au;2 �Cu;2 � 2 �1 � 2 �2 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) 1–20 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 �1, 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. 17–21 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) 1–20 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) 1–20 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: – using the diagonal terms of V calculated from Eqs. 17–21. – using Gy’s approximation: diag V .n; k/ D diag V .1; k/=n, i.e. neglecting the sample increment error correlation. – 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’s 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) 1–20 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 �1 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) 1–20 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–space model, where the output .y/ is limited to the copper flowrates of the feed, reject and concentrate streams: z.t/ D 0BBB@ �1 C s2.1� �1/.1� s1/ .1� �1/.1� s1/.1� s3/ .1� �1/.1� s1/ .1� �2/s1s2 �2 C .1� s3/.1� �2/s1 .1� �2/s1 0 0 � 1CCCA � z.t � 1/C 0BBB@ 0 0 1 1CCCA �.t/ (33) y.t/ D 0BBB@ 0 0 1 1� s2 0 0 0 s3 0 1CCCA z.t/C 0BBB@ �Cu;1 �Cu;5 �Cu;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) 1–20 Table 2 Parameters of the flotation plant Flowrates (t=h) Separation coefficients Transfer function coefficients �Cu;1: 12.80 s1 : 0.77 �: 0.5 �Cu;5: 0.87 s2 : 0.77 �1: 0.6 �Cu;7: 11.93 s3 : 0.92 �2: 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 � , 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) 1–20 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) 1–20 6. Conclusion A formula for the calculation of the variance–covariance 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–covariance of the integration error is a function of the variance–covariance 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–space 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