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# Calculation of variance

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the same scheme: E � yN yTN � D 1 N 2 " N Vyy.0/C N�1X iD1 .N � i/Vyy.i/C N�1X iD1 .N � i/V Tyy.i/ # (19) The third term is: E � yn yTN � D 1 nN 24n�1X jD0 NX iD1 Vyy.k0 C jk � i/ 35 (20) and the fourth one is the transpose of the third one: E � yN yTn � D �E �yn yTN� T (21) Finally V is a symmetric positive matrix which can be calculated using expressions (17) to (21), if the matrices Vyy.l/ are known. There are two techniques to evaluate Vyy.l/. The first approach consists of measuring flowrates and compositions of all the streams at the prescribed discretization period. The second one is based on a dynamic model of the plant, which can be calibrated using a limited amount of measurements. The first approach requires the implementation of the suitable instrumentation for a few days. It is a very expensive approach. The second approach is certainly less demanding, if a simple model is to be developed. If the model is linear or linearized, Vyy.l/ can be directly calculated from it as will be shown in the next section. If it is not, it must be used to simulate the plant dynamics and then estimate Vyy.l/ from simulated flowrates and compositions. 4. Linear dynamic models of mineral processing plants One way to estimate Vyy.l/ is to prepare a mathematical dynamic model of the plant where the measurements are performed. Variances can be easily calculated for systems involving linear relationships between the process variables. This is why linear models only are considered in this study. There are two types of process dynamics modelling approaches. One can use empirical linear transfer functions, calibrated from dynamic experiment results, or phenomenological models based on an analysis of the physico-chemical mechanisms prevailing in the process. Most of the time the latter approach leads to non-linear models which have to be linearized before calculating the variance matrix Vyy.l/, or directly used to generate simulated data from which the statistical properties can be evaluated. 8 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20 In the empirical approach, a complex production unit is described by a network consisting of nodes representing flowrates combination or flowrates separation, and of branches connected to the nodes and containing transfer functions which represent stream and process dynamics. The gains of these transfer functions correspond to the steady-state degree of transformation or separation of the materials involved in chemically or physically reacting systems. In the phenomenological approach, processes are described using models for trans- formation rates, mixing properties, and mass and energy transfer between phases or pieces of equipment. The linearization is then performed around nominal values of the process variables. In addition to these elements of an empirical or phenomenological model, dis- turbances must be modelled. There are two different kinds of disturbances, those corresponding to the random species flowrate variations of the feed streams (external disturbances), and those corresponding to the disturbances generated within the process units (internal disturbances). The disturbances can be modelled by AutoRegressive Moving Average (ARMA) processes driven by white noises (Box and Jenkins, 1994). The model can be decoupled for the various species or coupled, i.e. taking into account the correlations between the flowrate variations for different species. Examples will be given of uncoupled or coupled process models. Finally, any linear model of a mineral processing plant can be written under the following state–space form (the state–space equations are presented in many text books on dynamic system or automatic control, see for instance A¨ stro¨m and Wittenmark, 1990):8<:z.t C 1/ D Az.t/C B�.t/y.t/ D Cz.t/C � (22) where A, B and C are matrices of coefficients, �.t/ is a vector of random inputs (the zero mean white noises driving the external or internal disturbances), z.t/ the state vector, and � and y.t/, respectively, the steady-state values and the dynamic variations of the species flowrates. From this dynamic model, one can calculate the variance–covariance matrix of y.t/. Since z.t/ is a zero mean variable, and �.t/ and z.t/ are not correlated, it follows from Eq. 22 that: E[z.t C 1/z.t C 1/] D AE �z.t/z.t/T� AT C BV� BT (23) where V� is the variance–covariance matrix of �.t/. Since the plant is stationary: Vzz.0/ D AVzz.0/AT C BV� BT (24) equation which has a unique solution Vzz.0/. By repetitive application of Eq. 22 one obtains also: Vzz.l/ D Al Vzz.0/ (25) and then from the second part of Eq. 22: Vyy.l/ D C Al Vyy.0/CT (26) A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20 9 Eqs. 24 and 26 allow the complete evaluation of Vyy.l/ and consequently the calculation of the variance–covariance of the integration error V . 5. Examples of the method 5.1. Gold leaching process First, a single unit process with a simple chemical transformation is considered for illustrating the method. It is a perfectly mixed leaching tank where gold and copper are dissolved from the ore and transferred to the liquid phase (Fig. 2). This is for instance a cyanidation process designed for gold extraction, copper leaching being an undesirable side reaction which consumes cyanide, thus decreasing the gold dissolution kinetics. The model is linearized around an operating point and, assuming that the cyanide, ore and water feedrates are constant, the following simple equations are obtained for copper and gold flowrates in the solid phase: xAu;2.t/� �Au;2 D gAuG.�Au/[xAu;1.t � 1/� �Au;1]C g0CuG.�Cu/ � [xCu;1.t � 1/� �Cu;1] (27) xCu;2.t/� �Cu;2 D gCuG.�Cu/[xCu;1.t � 1/� �Cu;1] (28) In Eqs. 27 and 28 � stands for the nominal value of the flowrates and G.�/ are the discrete-time first-order transfer functions z�1=.1 � �z�1/ where z�1 is the backshift operator .z�1x.t/ D x.t � 1// and � a coefficient characterizing the process time Fig. 2. Model of a leaching tank. 10 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20 constant. gAu, gCu and g0Cu are model coefficients for adjusting the process gains. The first equation expresses that the marginal gold dissolution yield is constant at constant copper content but decreases when the ore copper content increases, due to cyanide consumption. The second equation simply says that the steady-state marginal copper dissolution is constant. Furthermore it is assumed that Cu and Au variations in the feed are correlated, i.e. that common random events �1.t/ drive both feed variations, while the second type of random event �2.t/ affects only copper variations, thus allowing an imperfect correlation between both metal contents. The feed composition variations are modelled by first-order autoregressive models: xAu;1.t/� �Au;1 D G.�Au/�1.t/ (29) xCu;1.t/� �Cu;1 D G.�Cu/�1.t/C G.�Cu/�2.t/ (30) where the coefficient � characterizes the disturbance dynamics and the coefficient adjusts the degree of correlation of the feed gold and copper contents. The model is graphically shown in Fig. 2, and its state–space formulation is: z.t/ D 0BBBBB@ �Au 0 0 0 0 �Cu 0 0 gAu 0 �Au 0 0 1 0 �Cu 1CCCCCA z.t � 1/C 0BBBBB@ 1 0 1 0 0 0 0 1CCCCCA �.t/ (31) y.t/ D 0BBBBB@ 1 0 0 0 0 1 0 0 0 0 1 g0Cu 0 0 0 gCu 1CCCCCA z.t/C 0BBBBB@ �Au;1 �Cu;1 �Au;2 �Cu;2 1CCCCCA (32) In a real plant, the values of the ten coefficients of the model could be obtained by a least-squares procedure applied to the measurements of the natural variations of the gold and copper contents at the inlet and outlet of the process. Data should be acquired during a period of at least five times the mean residence time, with a sampling frequency around one tenth of the mean residence time. The numerical values of model parameters as well as those of the random input variances selected for the present illustration are given in Table 1, for a discretization period T D 1 h. Table 1 Values of the