Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 31

be representative of the average whole batch properties. Based on Gy [56] and Pitard
[57], the main error sources for particulate materials are now brieflypresented.
Fundamental error. Unless the material is perfectly homogeneous at a micro-
scale level, sampling a limited number of particles to characterize the whole prop-
erties of an ore batch inherently leads to measurement errors. According to Gy, the
errors induced in a sample by ore constitution heterogeneity will be called a fun-
damental error. It cannot be eliminated. However, it can be evaluated. It has a zero
mean and a variance that can be estimated by

E x s

2

c d lib
d

x ∗

− 1

M s

σ F fgd 3

= = ,

(2.25)

x ∗

where x s and x ∗ are the mineral contents, respectively, of the ore sample and the
ore batch, M s the sample mass, d lib the liberation particle size, d the sieve opening
retaining 25% of the particles, f the shape factor, g the size distribution factor, and
c the composition factor defined as

x ∗ )/ x ∗ )(( x ∗ ) x ∗ ρ gan c =(( 1 − 1 − ρ min + ), (2.26)

where ρdenotes the mineral or gangue density.
Integration error. This error is induced by the particle distribution heterogeneity,
either spatially distributed for a fixed batch of ore, or as a function of time for a
flowing material. Hybrid integration errors may also occur when a sample is taken
fromagivenlocationofaflowing stream, rather than from all the possible locations
in the stream (using, for instance, a cross-stream cutter). This error can be evalu-
ated if the signal heterogeneity statistics, such as mineral content autocovariance, or
geostatistical variograms of the particle batch, are known. For a time related con-
centration of a stream sampled in a time window of width T , the definition of the
integration error is simply

⎧
⎨

⎫
⎬
⎭ −

⎧
⎨

⎫
⎬
⎭ .

composition
of the incre-
ment reunion

average stream
composition over
time window T

e I = (2.27)

⎩ ⎩

The variance of e I can be calculated from the sampling strategy parameters and
the autocovariance of e I . This calculation can be extended to a composition vector
of several streams [79].
Materialization error. This arises when extracting the sample from the ore batch
to be characterized, i.e. , when executing the designed sampling scheme. For exam-
ple, when cutting a stream flow with a moving sampler, errors may occur if all the
particles do not have the same probability of entering into the sampler, either be-
cause the sampler opening is too tight or its speed too high or not uniform. Other
materialization errors may arise if the sampler is overflowing or when gathering the
various sampling increments to generate a composite sample. Spillage and contam-
ination are also materialization error sources. Careful design of sampling devices