Environmental Engineering Reference
In-Depth Information
sorption and solvent extraction processes when a metal transfers from a solid to a
liquid phase or from a liquid phase to another one.
When the same species is analyzed at two different levels, for instance in ore and
particle size classes, the species content in the ore must be simultaneously consistent
with particle size distribution and species assays in size classes. The example of
Section 2.3.1 shows a case study for gold, while Hodouin and Vaz Coelho [53]
show another example for processing of a uranium ore. In the example of Section
2.3.4 there are many such additional constraints.
Also, when chemical equilibrium between phases is assumed, the thermodynamic
equilibrium conditions must be satisfied [29]. Finally, inequality constraints may
also be in force. For instance, species mass fractions must have values between 0
and 1, and flowrates must be positive.
2.3.6
Summary of Stationary Conservation Equations
Equations 2.11 or 2.16 and 2.17 coupled to specific additional constraints can be
gathered into the following form:
f
(
X
)=
ε
,
(2.20)
where
X
is a vector which gathers all the
n
X
state variables and
f
the
q
conservation
equations. In the linear case all the equations are gathered into
MX
=
ε
,
(2.21)
where
M
is a matrix of coefficients containing blocks such as
M
0
and
M
i
and pos-
sibly other terms depending upon the additional constraints. It may happen that
constant terms are also present in the constraints; Equation 2.21 would then become
MX
=
K
+
ε
.
(2.22)
To simplify the presentation, this case will not be covered in the following devel-
opments. If this is required, the modification of the formulae to take account of
K
would not be complex.
2.4 Sensors, Measurement Errors and Observation Equations
Measurement errors play a central role in data reconciliation, since one important
feature of the technique is to correct data that is contaminated by measurement in-
accuracies. This section proposes a description of the statistical properties of mea-
surement errors, with some emphasis on particulate material sampling and analysis.
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