Environmental Engineering Reference
In-Depth Information
and sampling execution allows elimination of this type of error, which is usually
almost impossible to quantify.
Preparation errors.
They are related to the operations that condition the final
sample to be analyzed by the measuring equipment (X-ray fluorescence device for
instance). The preparation involves secondary sampling, and thus induces new fun-
damental and integration errors. Drying, grinding are also operations that may in-
duce contamination or transformation of the phases, by oxidation for example.
Analysis error.
Finally, as the analytical method is based either on mechanical
methods (sieving for instance) or chemical principles (titration) or spectral analy-
sis (X-ray fluorescence, atomic adsorption), the analyzing device itself necessarily
induces errors. Usually, these errors can be quantified but cannot be avoided.
Total measurement error.
The variance of the resulting measurement error is ob-
tained by summing the variances of all the contributing errors.
2.5
Observation Equations
The general form of a process observation equation is
Z
=
g
(
X
),
(2.28)
Y
=
g
(
X
)+
e
,
where
X
is the vector of state variables,
Z
the measured process variables and
Y
the
measurement value of
Z
. There are two different ways to define the measurements.
Logically, the measured variables should be selected as the process variables that are
directly measured by on-line sensors or laboratory analytical instruments. However,
there is frequently some raw measurement preliminary processing. For instance the
particle masses retained on sieves are converted to mass fractions [58], or slurry
volume flowrates and densities converted to ore flowrates, thus assuming that the
ore specific mass is a known parameter. The propagation of the measurement errors
through the preliminary processing must be evaluated properly, since the calculation
process may not only increase the variance, but also create covariance terms in the
V
matrix. These covariance terms structure the reconciliation results and should not
be ignored, as shown for instance by Hodouin
et al.
[103]and Bazin and Hodouin
[55] for particle size distributions.
Moreover, when the measured variables are not state variables, one may find it
easier to combine measured variables in such a way to obtain state variable measure-
ments even if those have not been directly measured. An advantage of this procedure
is that the observation equation becomes linear and can be written as
Z
=
CX
,
(2.29)
Y
=
CX
+
e
,
where
C
is a matrix of coefficients with values pointing at measured state variables.
A possible drawback of the method is that the covariance of the pseudo-measured
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