Process Observers and Data Reconciliation Using Mass and Energy Balance Equations - Advanced Control and Supervision of Mineral Processing Plants - page 23

• The process is operating in a stationary regime while the frequency of the varia-
tions around the steady-state is of the same range as the process natural dynamics
and of significant amplitude. The steady-state method could be applied to aver-
aged measurements in a time window sufficiently large to significantly dampen
the dynamics of the variations around the steady-state values. The time window
must be at least two times larger than the process time constants. Measurement
variances should be augmented to take account of the residual dynamics of the
variations around the steady-sate values.
Obviously, stationary methods could be applied to instantaneous measurements
when the process is stationary, i.e. , when the process is not operating during sig-
nificant changes of the underlying steady-state values. In other words, the process
must be sampled at a time sufficiently far from a deterministic change of the average
operating conditions. When the process is clearly in a transient regime or cyclic or
batch, a dynamic reconciliation method should be used to take account of the lags
occurring between the various process states.

2.3 Models and Constraints

The component stream networks, where the material components (or enthalpy) are
flowing through the industrial unit, are described as oriented graphs, made of p
branches representing streams, and n n nodes representing accumulation equipments.
In the following, emphasis is placed on mass balance to alleviate the presentation.
Energy balance equations have the same structure as mass balance equations, en-
thalpy (or heat) being added to the list of the n 1 components (including total

mass) that must be conserved. Since reconciliation techniques are quite similar
when energy balances are considered, there is no need to repeat the expression
“mass and/or energy balance” throughout the text. The selected components used
for writing balance equations can be either phases, or species as minerals, metals,
atoms, molecules, ions, or classes of physical properties (size, density), and en-
thalpy. Each component may have its own network characterized by an incidence
matrix M i , whose entries 1,

+

1, and 0 represent either a node input, or output, or a
not connected stream. The process states used to write the mass conservation con-
straints are usually mass flowrates and mass fractions, therefore leading to bilinear
equations. When considering component flowrates - instead of total flowrates and
component mass fractions - the state equations can be kept linear. This case will be
considered first.

−

2.3.1 The Dynamic Linear Mass Balance Equation

The dynamic mass conservation equations for any i component (including phases,
species, properties and the total material flowing in the various streams) are