XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 957
attach to the extractant and transfer into the organic phase
thus transferring the H+ into the aqueous phase.
A general equilibrium constant can be defined from the
basic chemical reaction:
k
Ln eq 3 3
3
3
=
+
+6LnRH
6 6RH @eq
@eq6H
@eq
@eq
The equation and be redefined specifically for solvent
extraction:
k x
yeq
rh n
h
eq
eq
i
N
e
eq
n
0
e
Dx
=
-
+
/
d nf p
where yeq and xeq are the molar concentration of the element
in the organic and aqueous phases at equilibrium, h+eq is
the molar concentration of the hydrogen ion at equilibrium
(derived from pH), rh0 is the initial molar concentration of
the extractant, N indicates the total number of elements in
the system, ne is the valence number of the element i, and
Δx is the difference in the aqueous molar concentration
before and after the test for element i. The denominator
on the right side of the formula denotes the remaining free
extractant in the system, which is calculated based on the
transfer of elements from the aqueous to the organic phase.
The equation above shows how the equilibrium con-
stant can be readily determined for any element in any
extractant system through laboratory tests that reach equi-
librium conditions. Many of the terms, including, xeq, h+eq,
rho0, ne, and Δx are either known from the test conditions
or measured after the test is complete. Conversely, yeq can
be readily calculated through a mass balance. Turgeon et
al. (2016), recommend a procedure whereby lab tests are
conducted at multiple organic to aqueous (O/A) ratios
to obtain the data necessary to estimate the equilibrium
constants for each element. Multiple O/As are used to
ensure robust calculations as the equilibrium constant is
not affected by the O/A ratio. Moreover, Turgeon recom-
mends a reconciliation process whereby the mass balance
and charge balance are reconciled through aqueous assays
and final pH measurements.
We have expanded this reconciliation process to also
include direct measurement of the loaded organic con-
centration, yeq. This is accomplished through microwave
digestion of the organic solution followed by ICP-MS. Our
initial results suggest that this improves the reconciliation
process by adding a separate and independent confirmation
of the loaded organic concentration.
Using the chemical equilibrium framework, Turgeon
et al. (2023) expanded the formula to model a solvent
extraction system for the design case (i.e., the case where
the equilibrium constant and initial conditions are known,
but the final concentrations must be calculated). The equi-
librium formula was altered to include the aqueous and
organic volume flows in the system as well as incorporating
a mass transfer term, Δe, that represents the molar transfer
of metal ions from the aqueous to the organic phase. The
result equation is presented as (note that the Turgeon et al.,
(2023) paper has a sign error in the first numerator, which
has been corrected here):
Keq
Q
Q x
Q
Q y
Q
Q rh
Q
Q h
aq
aq
org
org RM
org
org e
aq
aq eq e
0
0
0 #K
/n
/n
De
De
De
De
=-
+
-
++0 J
L
K
K
K
K
Nne
P
O
O
O
O
O
where Keq is the equilibrium constant of the element in
question, Q is the volumetric flow for either the organic
(org) or the aqueous (aq), y, x, rh, h+ are the concentrations
of the organic, aqueous, extractant, and h-plus ions respec-
tively at equilibrium (eq) or initial (0), ne is the valence
number of the element, and Δe is the molar transfer rate
of an element.
As described above, for the design case, all initial
concentrations, flow rates, and equilibrium constants are
known, and this equation can thus be solved for Δe, which
can in turn be used to calculate the final concentrations for
each mixer settler. For each mixer settler, this equation must
be solved simultaneously for all elements under consider-
ation in the system. Note that the equation is highly non-
linear with respect to Δe due to the exponential term on the
right-hand side. This system of highly nonlinear equations
thus necessitates a sophisticated solution approach, which
must then be iterated when solving for multiple units with
complex recycle loops.
Linear Circuit Analysis
Linear circuit analysis (LCA) is a framework for solving
and analyzing separation circuits using mathematical rela-
tions. Originally developed by Meloy (1983), the approach
was largely used to optimize separation efficiency in par-
ticulate separation circuits however, Williams et al. (1986)
also showed how the process could be used for multi-feed
circuits, including those used in gas adsorption and distilla-
tion. The LCA methodology was later generalized by Noble
and Luttrell (2014) who developed a matrix approach to
the LCA solutions. This technique can be widely used to
analyze and assess mass flows in complex circuits using
simple transfer and connection functions, and the general
methodology is not limited to simple particulate systems.
Recently, Noble et al. (2019) authored a review paper that
attach to the extractant and transfer into the organic phase
thus transferring the H+ into the aqueous phase.
A general equilibrium constant can be defined from the
basic chemical reaction:
k
Ln eq 3 3
3
3
=
+
+6LnRH
6 6RH @eq
@eq6H
@eq
@eq
The equation and be redefined specifically for solvent
extraction:
k x
yeq
rh n
h
eq
eq
i
N
e
eq
n
0
e
Dx
=
-
+
/
d nf p
where yeq and xeq are the molar concentration of the element
in the organic and aqueous phases at equilibrium, h+eq is
the molar concentration of the hydrogen ion at equilibrium
(derived from pH), rh0 is the initial molar concentration of
the extractant, N indicates the total number of elements in
the system, ne is the valence number of the element i, and
Δx is the difference in the aqueous molar concentration
before and after the test for element i. The denominator
on the right side of the formula denotes the remaining free
extractant in the system, which is calculated based on the
transfer of elements from the aqueous to the organic phase.
The equation above shows how the equilibrium con-
stant can be readily determined for any element in any
extractant system through laboratory tests that reach equi-
librium conditions. Many of the terms, including, xeq, h+eq,
rho0, ne, and Δx are either known from the test conditions
or measured after the test is complete. Conversely, yeq can
be readily calculated through a mass balance. Turgeon et
al. (2016), recommend a procedure whereby lab tests are
conducted at multiple organic to aqueous (O/A) ratios
to obtain the data necessary to estimate the equilibrium
constants for each element. Multiple O/As are used to
ensure robust calculations as the equilibrium constant is
not affected by the O/A ratio. Moreover, Turgeon recom-
mends a reconciliation process whereby the mass balance
and charge balance are reconciled through aqueous assays
and final pH measurements.
We have expanded this reconciliation process to also
include direct measurement of the loaded organic con-
centration, yeq. This is accomplished through microwave
digestion of the organic solution followed by ICP-MS. Our
initial results suggest that this improves the reconciliation
process by adding a separate and independent confirmation
of the loaded organic concentration.
Using the chemical equilibrium framework, Turgeon
et al. (2023) expanded the formula to model a solvent
extraction system for the design case (i.e., the case where
the equilibrium constant and initial conditions are known,
but the final concentrations must be calculated). The equi-
librium formula was altered to include the aqueous and
organic volume flows in the system as well as incorporating
a mass transfer term, Δe, that represents the molar transfer
of metal ions from the aqueous to the organic phase. The
result equation is presented as (note that the Turgeon et al.,
(2023) paper has a sign error in the first numerator, which
has been corrected here):
Keq
Q
Q x
Q
Q y
Q
Q rh
Q
Q h
aq
aq
org
org RM
org
org e
aq
aq eq e
0
0
0 #K
/n
/n
De
De
De
De
=-
+
-
++0 J
L
K
K
K
K
Nne
P
O
O
O
O
O
where Keq is the equilibrium constant of the element in
question, Q is the volumetric flow for either the organic
(org) or the aqueous (aq), y, x, rh, h+ are the concentrations
of the organic, aqueous, extractant, and h-plus ions respec-
tively at equilibrium (eq) or initial (0), ne is the valence
number of the element, and Δe is the molar transfer rate
of an element.
As described above, for the design case, all initial
concentrations, flow rates, and equilibrium constants are
known, and this equation can thus be solved for Δe, which
can in turn be used to calculate the final concentrations for
each mixer settler. For each mixer settler, this equation must
be solved simultaneously for all elements under consider-
ation in the system. Note that the equation is highly non-
linear with respect to Δe due to the exponential term on the
right-hand side. This system of highly nonlinear equations
thus necessitates a sophisticated solution approach, which
must then be iterated when solving for multiple units with
complex recycle loops.
Linear Circuit Analysis
Linear circuit analysis (LCA) is a framework for solving
and analyzing separation circuits using mathematical rela-
tions. Originally developed by Meloy (1983), the approach
was largely used to optimize separation efficiency in par-
ticulate separation circuits however, Williams et al. (1986)
also showed how the process could be used for multi-feed
circuits, including those used in gas adsorption and distilla-
tion. The LCA methodology was later generalized by Noble
and Luttrell (2014) who developed a matrix approach to
the LCA solutions. This technique can be widely used to
analyze and assess mass flows in complex circuits using
simple transfer and connection functions, and the general
methodology is not limited to simple particulate systems.
Recently, Noble et al. (2019) authored a review paper that