Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 26

where it is clear that these equations contain flowrates and mass fractions cross-
products, thus giving a bilinear to the equation set.
Example. For the same example as in Section 2.3.1, f 0 would be the ore flowrate, and
c i would correspond to copper, lead, zinc, gold and the 10 size class mass fractions.
To keep the bilinear structure, it is necessary, in this particular case, to ignore the
data levels corresponding to slurry and gold content in size classes.

2.3.4 Multi-linear Constraints

Process measurements usually concern flowrates and material compositions. In that
case, process states are defined with two levels of properties as depicted in the bi-
linear case above. Unfortunately, when the performances of a mineral or metallurgi-
cal processing plant must be deeply assessed, more than two levels of material prop-
erties need to be handled [53]. Material streams may contain various phases (ore,
carbon, aqueous, organic, and gas phases) which are characterized by many prop-
erties such as flowrates, particle density and size distributions, as well as mineral
and chemical compositions of each phase and particle class. The mass conservation
equations complexity can rapidly increase with the level of detail needed for process
analysis.
Example. Gold ore processing plants involve different phases (slurry, water, carbon
and ore) that are characterized by various properties (size distributions, along with
chemical and mineral compositions), as well as mineral and chemical compositions
of ore and carbon size classes. Only a few studies on data reconciliation in the
gold ore processing industry are available [25-27],[54]. Figure 2.4 shows a possible
multi-level representation of the stream materials. First, the slurry phase is divided
into liquid, ore and carbon phases. Then, each phase is subdivided into its specific
components:

the liquid phase into chemical reagents (CN − ,O 2

• ) and leached species (Au, Ag,

Cu);

• the ore phase into populations of particles (such as coarse and fine) which are

subsequently split into classes of particles (such as − 38μm, + 38 / − 53μm,

+ 75μm), each particle size class being characterized by its mineral (na-

tive gold, pyrite, hematite) content, and, subsequently, the mineral metal contents
(Au, Ag, Cu);

53 / −

• the carbon phase into size classes, each class being characterized by its metal

content.

In such a complex system with six different levels of information, the mass con-
servation system becomes 6-linear. In addition, various components are simultane-
ously considered at various levels, thus creating a complex set of additional con-
straints that are necessary to ensure gold conservation is consistent between the
different levels. This case study is discussed in [28], and the next section gives some
more information on the additional constraints that are usual in MMP plants.