958 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
explains the historical development and theoretical under-
pinnings of LCA in addition to industrial case studies high-
lighting its utility.
The matrix LCA approach defines a circuit superstruc-
ture using a series of four matrices, namely P, F, X, and IC.
Table 1 provides a brief description of the matrix dimen-
sions and characteristics.
According to the Noble and Luttrell (2014) approach,
the four matrices are related by the following equation:
where I is the m × m identify matrix, and the other terms
are as defined in Table 1. A typical use for this equation is to
define P, F, and IC based on the circuit structure and opera-
tional parameters and then solve for X. Notwithstanding,
the equation could, in principle, be solved for any unknown
if given the other three variables, though this may yield
infeasible or nonsensical solutions. Moreover, while the
nomenclature used here as well as in the Noble and Luttrell
(2014) publication presumes 2D dimensionality for the
matrices (i.e., a single component analysis), the methodol-
ogy can be further generalized to multi-component analysis
by adding a dimension representing the species and defin-
ing the overall dimensionality accordingly.
To demonstrate the methodology, consider a two-unit
separation circuit with the stream connections as shown in
Figure 1.
The system nodes are defined as follows: N1 is where
the feed and the recycle combine. N2 and N3 denote unit
separators with one feed stream and two output streams.
Likewise, the streams are also defined: X1 is the circuit feed,
X2 is the feed to the first unit separator which is comprised
of X1 and X6. X3 is the concentrated stream out of the first
unit separator while X4 is the waste material from the sep-
arator. Likewise, X5 is the concentrated stream from the
second unit separator, and X6 is the waste material that is
Table 1. Definition of the matrices used in the linear circuit analysis process
Symbol Name Size Function Value
P Products
Matrix
Rows (m): #of streams
Column (n): #of nodes
Defines which streams are
the products of which nodes
p if stream m is a product of node n,
where p is the proportion of the feed
exiting in that stream.
0 if stream m is not a product of node
n
F Feeds Matrix Rows (m): #of streams
Column (n): #of nodes
Defines which streams feed
which nodes
1 if stream m feeds node n
0 if stream m does not feed node m
X Mass Flow
Vector
Rows (m): #of streams Defines the steady-state mass
flow for each stream
x, where x is the mass flow of stream m
IC Initial
Condition
Vector
Rows #(m): of streams Defines the circuit feed or
other initial conditions
ic, where ic is the feed mass flow or
initial condition for stream m
Figure 1. Flow diagram of an example rougher-cleaner separator system
explains the historical development and theoretical under-
pinnings of LCA in addition to industrial case studies high-
lighting its utility.
The matrix LCA approach defines a circuit superstruc-
ture using a series of four matrices, namely P, F, X, and IC.
Table 1 provides a brief description of the matrix dimen-
sions and characteristics.
According to the Noble and Luttrell (2014) approach,
the four matrices are related by the following equation:
where I is the m × m identify matrix, and the other terms
are as defined in Table 1. A typical use for this equation is to
define P, F, and IC based on the circuit structure and opera-
tional parameters and then solve for X. Notwithstanding,
the equation could, in principle, be solved for any unknown
if given the other three variables, though this may yield
infeasible or nonsensical solutions. Moreover, while the
nomenclature used here as well as in the Noble and Luttrell
(2014) publication presumes 2D dimensionality for the
matrices (i.e., a single component analysis), the methodol-
ogy can be further generalized to multi-component analysis
by adding a dimension representing the species and defin-
ing the overall dimensionality accordingly.
To demonstrate the methodology, consider a two-unit
separation circuit with the stream connections as shown in
Figure 1.
The system nodes are defined as follows: N1 is where
the feed and the recycle combine. N2 and N3 denote unit
separators with one feed stream and two output streams.
Likewise, the streams are also defined: X1 is the circuit feed,
X2 is the feed to the first unit separator which is comprised
of X1 and X6. X3 is the concentrated stream out of the first
unit separator while X4 is the waste material from the sep-
arator. Likewise, X5 is the concentrated stream from the
second unit separator, and X6 is the waste material that is
Table 1. Definition of the matrices used in the linear circuit analysis process
Symbol Name Size Function Value
P Products
Matrix
Rows (m): #of streams
Column (n): #of nodes
Defines which streams are
the products of which nodes
p if stream m is a product of node n,
where p is the proportion of the feed
exiting in that stream.
0 if stream m is not a product of node
n
F Feeds Matrix Rows (m): #of streams
Column (n): #of nodes
Defines which streams feed
which nodes
1 if stream m feeds node n
0 if stream m does not feed node m
X Mass Flow
Vector
Rows (m): #of streams Defines the steady-state mass
flow for each stream
x, where x is the mass flow of stream m
IC Initial
Condition
Vector
Rows #(m): of streams Defines the circuit feed or
other initial conditions
ic, where ic is the feed mass flow or
initial condition for stream m
Figure 1. Flow diagram of an example rougher-cleaner separator system