Calculation of variance
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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