1170 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
Manis, A.A., Soldenhoff, K.H., Ho, E.M., and Macintosh,
P.D., 2021. Nanofiltration in Hydrometallurgy. in
Nanofiltration: Principles, Applications, and New
Materials 2th ed. Edited by Wiley.
Muniruzzaman, M., and Rolle, M., 2016. Modeling mul-
ticomponent ionic transport in groundwater with
IPhreeqc coupling: Electrostatic interactions and geo-
chemical reactions in homogeneous and heterogeneous
domains. Advances in Water Resources 98:1–15.
Szymczyk, A., and Fievet, P., 2005. Investigating transport
properties of nanofiltration membranes by means of a
steric, electric and dielectric exclusion model. Journal of
Membrane Science 252 :77–88.
Xuan, W., 2022. Développement d’un procédé hydrométal-
lurgique pour le recyclage des électrodes positives de type
NMC contenues dans les batteries lithium-ion usagées.
PhD thesis, Université de Lorraine.
Zante, G., 2020. Développement de procédés hydrométallur-
giques pour l’extraction du lithium et le recyclage des bat-
teries lithium-ion. PhD thesis, Université de Strasbourg.
Yaroshchuk, A., 2013. Solution-diffusion–electro-migra-
tion model and its uses for analysis of nanofiltration,
pressure-retarded osmosis and forward osmosis in
multi-ionic solutions. Journal of Membrane Science
447:463–476.
SUPPLEMENTARY MATERIAL
Transport equations used in the SEDE model
Extended Nernst-Planck equation
j Di, dx
dc
RT
z FK c D
dx
d{ K civ
,c i
i i i,d i i, =-Ki,d -+(1)
with,
K K
6r
,d
,t
=(2)
K
K
2K
2
,c
,t
i i,s z
=
-^h (3)
K a
a
4
9 2 1
,t i n n i
n n i
n
2 2
5
1
2
0
4
3
r m m
m
=-+-
+
-
=
=+
/
/
^^1 h hnD :1
(4)
K b
b
4
9 2 1
,s i n n i
n n i
n
2 2
5
1
2
0
4
3
r m m
m
=-+-
+
-
=
=+
/
/
^^1 h hnD :1
(5)
with, a1 =–73/60 a2 =77.293/50.400 a3 =–22.5083 a4
=–5.6117 a5 =–0.3363 a6 =–1.216 a7 =1.647 b1 =
7/60 b2 =–2.227/50.400 b3 =4.0180 b4 =–3.9788 b5
=–1.9215 b6 =4.392 b7 =5.006
/r r ª
i i,Stokes p =(6)
r
k T
6 ,Stokes i
i,
B r∑D =(7)
1
i i
2 m =-^h (8)
dx
dci
K D A
J
c c x RT
z Fci
dx
d{
,,
,c
d i k
V
i i
i =-+_K ^hi- (9)
Electrical potential gradient inside pore
dx
d{
RT
F z c K D
z K D dx
dci
A
JV z K c
,d i i i i
i i i,d i
K i i i,c i
2
=
-+
/
//
(10)
Relation between permeate volume flow J
V and ionic
molar flow j
i
ji A
J ci x
k
V =
+^h
(11)
Partitioning equation at the membrane/solution interface
exp(- c
c
k T
z eD{
||
i
i
B
i D_0 0 0 DWi,_0
=
+
-
+
+-i +-i ^0
^0
h
h (12)
exp(-
c
c
k T
z eD
|,_x |
i
i
B
i D_x x i x {DW
=
+
+
-
-+-+^x
^x
h
h
i i (13)
with,
ln
ln
W T W
k T
,||i B i i,image_0
B
0 0 0 z
ci
ci
=-k +
+
-
+
+-+-
^0
^0
_
h
h
i i
(14)
ln
ln
W T W
k T
,||i x x B i i,image_Dx
B
i
z
c
ci
=-k +
+
D Dx D
+
-
-+-+
^x
^x
_
h
h
i i
(15)
W I
K
dkw
2
,image
i
o
w w
w w
0 1
0 1
0 1
0 1 ?
r =
+b^k
-b^k 8
^vhK
^kwhK
^kwh
^vh
^vhK
^vhK
^k
^k hK
hI h
h (16)
k Tr
z eh2
8rf i
p B p
i
0
?f =
^(17)
()"d v k int
w
2 n =+(18)
expK
Frp RTf0fb
z c k T
z e
,image int
int
int
i i i B
i
D 2
n
zi
{D
=
-DW
/
J
L
K-
K
N
P
O
O
O
(19)
Manis, A.A., Soldenhoff, K.H., Ho, E.M., and Macintosh,
P.D., 2021. Nanofiltration in Hydrometallurgy. in
Nanofiltration: Principles, Applications, and New
Materials 2th ed. Edited by Wiley.
Muniruzzaman, M., and Rolle, M., 2016. Modeling mul-
ticomponent ionic transport in groundwater with
IPhreeqc coupling: Electrostatic interactions and geo-
chemical reactions in homogeneous and heterogeneous
domains. Advances in Water Resources 98:1–15.
Szymczyk, A., and Fievet, P., 2005. Investigating transport
properties of nanofiltration membranes by means of a
steric, electric and dielectric exclusion model. Journal of
Membrane Science 252 :77–88.
Xuan, W., 2022. Développement d’un procédé hydrométal-
lurgique pour le recyclage des électrodes positives de type
NMC contenues dans les batteries lithium-ion usagées.
PhD thesis, Université de Lorraine.
Zante, G., 2020. Développement de procédés hydrométallur-
giques pour l’extraction du lithium et le recyclage des bat-
teries lithium-ion. PhD thesis, Université de Strasbourg.
Yaroshchuk, A., 2013. Solution-diffusion–electro-migra-
tion model and its uses for analysis of nanofiltration,
pressure-retarded osmosis and forward osmosis in
multi-ionic solutions. Journal of Membrane Science
447:463–476.
SUPPLEMENTARY MATERIAL
Transport equations used in the SEDE model
Extended Nernst-Planck equation
j Di, dx
dc
RT
z FK c D
dx
d{ K civ
,c i
i i i,d i i, =-Ki,d -+(1)
with,
K K
6r
,d
,t
=(2)
K
K
2K
2
,c
,t
i i,s z
=
-^h (3)
K a
a
4
9 2 1
,t i n n i
n n i
n
2 2
5
1
2
0
4
3
r m m
m
=-+-
+
-
=
=+
/
/
^^1 h hnD :1
(4)
K b
b
4
9 2 1
,s i n n i
n n i
n
2 2
5
1
2
0
4
3
r m m
m
=-+-
+
-
=
=+
/
/
^^1 h hnD :1
(5)
with, a1 =–73/60 a2 =77.293/50.400 a3 =–22.5083 a4
=–5.6117 a5 =–0.3363 a6 =–1.216 a7 =1.647 b1 =
7/60 b2 =–2.227/50.400 b3 =4.0180 b4 =–3.9788 b5
=–1.9215 b6 =4.392 b7 =5.006
/r r ª
i i,Stokes p =(6)
r
k T
6 ,Stokes i
i,
B r∑D =(7)
1
i i
2 m =-^h (8)
dx
dci
K D A
J
c c x RT
z Fci
dx
d{
,,
,c
d i k
V
i i
i =-+_K ^hi- (9)
Electrical potential gradient inside pore
dx
d{
RT
F z c K D
z K D dx
dci
A
JV z K c
,d i i i i
i i i,d i
K i i i,c i
2
=
-+
/
//
(10)
Relation between permeate volume flow J
V and ionic
molar flow j
i
ji A
J ci x
k
V =
+^h
(11)
Partitioning equation at the membrane/solution interface
exp(- c
c
k T
z eD{
||
i
i
B
i D_0 0 0 DWi,_0
=
+
-
+
+-i +-i ^0
^0
h
h (12)
exp(-
c
c
k T
z eD
|,_x |
i
i
B
i D_x x i x {DW
=
+
+
-
-+-+^x
^x
h
h
i i (13)
with,
ln
ln
W T W
k T
,||i B i i,image_0
B
0 0 0 z
ci
ci
=-k +
+
-
+
+-+-
^0
^0
_
h
h
i i
(14)
ln
ln
W T W
k T
,||i x x B i i,image_Dx
B
i
z
c
ci
=-k +
+
D Dx D
+
-
-+-+
^x
^x
_
h
h
i i
(15)
W I
K
dkw
2
,image
i
o
w w
w w
0 1
0 1
0 1
0 1 ?
r =
+b^k
-b^k 8
^vhK
^kwhK
^kwh
^vh
^vhK
^vhK
^k
^k hK
hI h
h (16)
k Tr
z eh2
8rf i
p B p
i
0
?f =
^(17)
()"d v k int
w
2 n =+(18)
expK
Frp RTf0fb
z c k T
z e
,image int
int
int
i i i B
i
D 2
n
zi
{D
=
-DW
/
J
L
K-
K
N
P
O
O
O
(19)