960 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
A similar structure to the LCA system above can be
defined for the generic solvent extraction circuit here,
including the recycle flows for each mixer settler. The chal-
lenge, though, is that the separation in a mixer-settler is not
readily defined as a simple mass transfer percent (i.e., a P
matrix), and as such, the LCA methodology must be modi-
fied accordingly. As discussed above, a mixer settler can,
however, be modelled by solving for the mass transfer term
(e) when the equilibrium constant is known. As defined
in the Turgeon framework, the mass transfer term defines
the molar flow rate from the aqueous phase to the organic
or vice versa. When solved independently, Δe can define
the fixed difference between aqueous or organic streams
around a mixer settler. For example, using the circuit above,
X2 can be defined as the difference between X4 and ΔeN2,
adjusting to ensure the sign convention is properly applied.
The same applies for Y2 and Y4, X6 and X8, etc.
Noting that Δe represents a fixed mathematical rela-
tionship between streams within the superstructure, the
Noble and Luttrell (2014) LCA equation can be redefined
as:
I P F IC #!6E@ -=l 6 6X@ 6 @
where the new term, E, is a nx1 vector (note: it can be
expanded to be an nxk matrix for multicomponent sys-
tems). In the vector E, a value of ΔeMC is given if stream
n is a product of a mixer settler, and all other values are 0.
An example set of the matrices for the aqueous flows
(x) in the example system shown in Figure 2 are:
⃗ =
⎣0
⎢0
⎢0
⎢0
⎢0
⎢0
⎢0
⎢
⎢
⎢
⎢
⎢
⎡
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0⎥
0 0 0 0 1 0 0 0⎥
0 0 0 0 0 1 0 0⎥
0 0 0 0 0 0 1 0⎥
0 0 0 0 0 1 0 0⎥
0 0 0 0 0 0 0 1⎥
0 0 0 0 0 0 0 0⎦
⎥
⎥
⎥
⎥
⎥
⎤
⃗ =
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
(1 – r)
(1 – r)
(1 – r)
r
r
A similar structure to the LCA system above can be
defined for the generic solvent extraction circuit here,
including the recycle flows for each mixer settler. The chal-
lenge, though, is that the separation in a mixer-settler is not
readily defined as a simple mass transfer percent (i.e., a P
matrix), and as such, the LCA methodology must be modi-
fied accordingly. As discussed above, a mixer settler can,
however, be modelled by solving for the mass transfer term
(e) when the equilibrium constant is known. As defined
in the Turgeon framework, the mass transfer term defines
the molar flow rate from the aqueous phase to the organic
or vice versa. When solved independently, Δe can define
the fixed difference between aqueous or organic streams
around a mixer settler. For example, using the circuit above,
X2 can be defined as the difference between X4 and ΔeN2,
adjusting to ensure the sign convention is properly applied.
The same applies for Y2 and Y4, X6 and X8, etc.
Noting that Δe represents a fixed mathematical rela-
tionship between streams within the superstructure, the
Noble and Luttrell (2014) LCA equation can be redefined
as:
I P F IC #!6E@ -=l 6 6X@ 6 @
where the new term, E, is a nx1 vector (note: it can be
expanded to be an nxk matrix for multicomponent sys-
tems). In the vector E, a value of ΔeMC is given if stream
n is a product of a mixer settler, and all other values are 0.
An example set of the matrices for the aqueous flows
(x) in the example system shown in Figure 2 are:
⃗ =
⎣0
⎢0
⎢0
⎢0
⎢0
⎢0
⎢0
⎢
⎢
⎢
⎢
⎢
⎡
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0⎥
0 0 0 0 1 0 0 0⎥
0 0 0 0 0 1 0 0⎥
0 0 0 0 0 0 1 0⎥
0 0 0 0 0 1 0 0⎥
0 0 0 0 0 0 0 1⎥
0 0 0 0 0 0 0 0⎦
⎥
⎥
⎥
⎥
⎥
⎤
⃗ =
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
(1 – r)
(1 – r)
(1 – r)
r
r