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1944-3994/1944-3986 © 2017 Desalination Publications. All rights reserved.
Desalination and Water Treatment
www.deswater.com
doi: 10.5004/dwt.2017.20169
63 (2017) 113–123
February
Numerical modelling and simulation of membrane-based extraction
of copper (II) using hollow ber contactors
Amir Muhammada, Mohammad Younasa,*, Stéphanie Druon-Bocquetb, Julio Romeroc,
José Sanchez-Marcanob
aDepartment of Chemical Engineering, University of Engineering and Technology, Peshawar, P.O. Box 814, University Campus,
Peshawar 25120, Pakistan, Tel. +92919218180; emails: m.younas@uetpeshawar.edu.pk (M. Younas),
engr.amir@uetpeshawar.edu.pk (A. Muhammad)
bInstitut Européen des Membranes, UMR 5635, CNRS, ENSCM, UMII, Université de Montpellier II, CC 047, 2 Place Eugène
Bataillon, 34095 Montpellier cedex 5, France, Tel. +33 4 67 14 91 65; email: Stephanie.Druon-Bocquet@univ-montp2.fr
(S. Druon-Bocquet), Tel. +33 4 67 14 91 49; email: Jose.Sanchez-Marcano@univ-montp2.fr (J. Sanchez-Marcano)
cLaboratory of Membrane Separation Processes (LabProSeM) Department of Chemical Engineering, University of Santiago de Chile,
Av. Lib. Bdo. O’Higgins 3363, Estación Central, Santiago, Chile, Tel. +56 2 718 18 21; email: julio.romero@usach.cl
Received 20 May 2016; Accepted 14 August 2016
abstract
This work describes the mathematical modelling and numerical simulation of a hollow fiber mem-
brane contactor for copper (II) extraction from aqueous solutions with 1,1,1-trifluoro-2,4-pentanedione
commonly known as TFA diluted in n-decanol. Model was developed for convection–diffusion mass
and momentum transfer using continuity and Navier–Stokes equations. Model equations were solved
using a computational fluid dynamics code, and results were validated with experimental data. After
validation of model, simulation was performed to check the effects of hydrodynamics conditions on
contactor performance. 49% of copper (II) was extracted from aqueous solution for feed flow rate of
8.3 × 10–7 m3.s–1, and for partition coefficient equals to 1. However, simulation results indicated that
extraction could be greatly improved by decreasing feed flow and increasing partition coefficient.
Simulation was also run to study the distribution profiles of copper (II) concentration, flux and veloc-
ity in 2-D.
Keywords: Copper (II); Dispersion free extraction; Hollow fiber membrane contactor; Computational
fluid dynamics; TFA
1. Introduction
Copper (II) recovery from aqueous streams has always
been a relevant research area due to its importance from envi-
ronmental perspective or in the context of hydrometallurgical
process [1–3]. Membrane-based solvent extraction represent-
ing a non-dispersive phase contact operation, in which the
interface between feed and extractant streams is supported
by a macroporous membrane wall, is a useful emerging tech-
nique for copper (II) extraction. In this process, mixing of
phases is avoided, and problems of flooding, entraining or
downstream phase separation do not occur [4]. Moreover, on
the contrary to classical extraction columns, there is no need
for density differences between the feed and extractant, lead-
ing to a greater choice of extractants. Hollow fiber membrane
contactors (HFMCs) offer several advantages such as: com-
pact process design, modularity, controlled interfacial area
and higher volumetric efficiency due to large specific contact
area [5–7].
Extensive research work has been reported on extraction
processes based on HFMCs by several research teams like
Prasad and Sirkar [8], Yang and Cussler [9] as well as exten-
sive review by Gabelman and Hwang [10] and Pabby and
A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123114
Sastre [11]. HFMCs have been used in extraction processes
for the last 2 decades [12]. Extraction of copper (II) using che-
lating extractants like LIX 84-1 and LIX 622N has been widely
studied [13–15]. Literature study reviewed that experimen-
tal investigation covered the extraction kinetics, distribution
equilibria and dispersion-free extraction of copper (II) with
extractants. Nonetheless, not enough literature on theoretical
study of copper (II) with extractants using HFMC is avail-
able. Furthermore, 1,1,1-trifluoro-2,4-pentanedione (TFA)
has not been used as extractant for copper (II) extraction,
although TFA has been found to be an efficient extractant for
copper (II) extraction [16,17].
Mass transfer in membrane contactors has been modelled
by several research teams with resistance-in-series approach
[10,12,18,19]. This approach is based on three mass trans-
fer resistances in series as solute moves from feed to solvent
across membrane. Models developed with this approach were
analytically solved and based on negligible axial diffusion.
Mass transfer coefficients were estimated from empirical cor-
relations, which are not very accurate. Meanwhile, numerical
models and their solutions offer a description of axial diffusion
approximation. These models are based on mass and momen-
tum balance equations and do not require estimation of mass
transfer coefficients. Numerical analysis and simulation also
known as computational fluid dynamics (CFD) of HFMC offer
a detailed description of solute transfer across the membrane
surface [20–23]. In recent years, several researchers used CFD
to describe mass and momentum transport through HFMC
[24–26]. Although comprehensive 2-D model was developed
and simulated for gas absorption [27–30], simulation of mass
and momentum transport of liquid–liquid extraction, in gen-
eral, and of copper (II) extraction, in particular, through HFMC
is lacking in literature. Marjani and Shirazian [31], Rezakazemi
et al. [32] and Nosratinia et al. [33] studied the numerical sim-
ulation of ammonia removal from wastewaters using mem-
brane contactors. Ghadiri and Shirazian [34] investigated
computational simulation of mass transfer in extraction of
alkali metals through membrane contactors. Fadaei et al. [35]
simulated the membrane contactor for copper (II) extraction
with di-(2-ethylhexyl)phosphoric acid (D2EHPA) through
CFD techniques. Effect of partition coefficient and profile
study of hydrodynamics have not yet been studied for mem-
brane-based solvent extraction of copper (II).
This work proposes a steady-state numerical model of
an HFMC to study the behavior of copper (II) extraction
from an aqueous phase using an organic phase containing
TFA as extractant agent. The objective of the current work is
the identification and description of the copper (II) transfer
mechanism in liquid–liquid HFMC through CFD. Numerical
model was developed on a hypothesized symmetrical unit of
HFMC. Concentration profile for copper (II) extraction with
TFA through convective and diffusive mass transport in a
hollow fiber contactor was developed. The model was then
integrated in a more global model taking also into account
feed and solvent phase recycling. The integrated model of the
extraction process was validated with experimental results.
Simulations were done in order to assess the effect of the
operating variables on the concentration and flux profiles
of copper (II) as well as on the velocity profiles in the con-
tactor. Partition coefficient and fluid flow rates effects upon
extraction efficiency were also investigated.
2. Materials and methods
2.1. Reagents and solutions
1-Decanol and TFA were obtained from Sigma-Aldrich®
(France) and used without further purification. Their char-
acteristics such as density and viscosity were found in
Rigglo et al. [36], Faria et al. [37] and Mutalik et al. [38]
or provided by the manufacturer. The feed solution is an
aqueous solution of copper (II) prepared by dissolving ana-
lytical reagent grade CuSO4.5H2O (purchased from Fisher
Chemical®, France) in deionized water. A master solution
of 15.75 mol m−3 of copper (II) was prepared and was then
diluted to 7.87 and 3.15 mol m−3 or less as per requirement.
Organic phase was prepared by dissolving the known vol-
ume of TFA in the desired diluent.
2.2. Membrane process configuration and its properties
The experimental work was based on dispersion-free
extraction of copper (II) from the feed with TFA through the
HFMC. The hollow fiber contactor module was provided by
Liqui-celTM (USA). Copper (II) solution was prepared by dis-
solving CuSO4 in deionized water to achieve the desired cop-
per (II) concentration. TFA was used as extractant agent and
was diluted in n-decanol. The runs were carried out at room
temperature (298 ± 5 K). The schematic setup of the plant is
shown in Fig. 1. The aqueous and organic phase volumes
were taken as 1.0 × 10–3 m3 in two storage tanks and were
agitated constantly by magnetic stirrers.
Aqueous feed stream with known copper (II) concentra-
tion (3.15 mol m–3) was fed inside the fibers. Organic phase
was flown countercurrently in the shell side. The concen-
tration of TFA was taken as 160 mol.m–3. This achieves the
partition coefficient value of 1 [16,17]. As the membrane
material was hydrophobic, the organic phase penetrated
through the pores of the membrane and immobilized up to
the pores mouth at the aqueous phase-membrane interface.
A pressure of 180 kPa was exerted on aqueous phase side
of the fiber preventing the organic phase flow to enter into
the lumen side. Nonetheless, the exerted pressure was less
than the critical penetration pressure in order to prevent
the mixing of both phases [39]. Aqueous phase samples,
H
F
M
C
Feed
tank
Solvent
tank
Fig. 1. Experimental set up: recycled copper (II) extraction with
TFA in HFMC.
115A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123
at different pre-assumed time intervals, were taken from
aqueous phase storage tank during the experiment, and
the concentration of copper (II) was quantified by atomic
absorption spectrometry. Prior to experiments, deionized
water and organic phase solution were introduced into the
membrane module for at least 30 min or till the stabiliza-
tion of operating parameters like temperature, flow rates
and pressure drop. Time delays could also occur in achiev-
ing steady state; however, these effects were neglected in
model formulation. The deionized water phase was then
replaced with an aqueous phase solution containing cop-
per (II). Experiments lasted for about 70–90 min. Physico-
chemical and transport properties of both phases and
geometrical parameters of the membrane contactor are
summarized in Table 1.
3. Theoretical description of the mass transfer
3.1. Model approach and its considerations
Fibers of the contactor were assumed as regularly packed
axially in the shell of the HFMC module. Each fiber had its
hypothesized flow area. This original configuration, named
as “flow-cell”, resembled a concentric circular tube where
the thickness of the inner tube denoted the thickness of the
porous membrane [40]. The module was an assembly of iden-
tical “flow-cells” parallel to each other. The model behavior
of single “flow-cell” was extrapolated to the whole con-
tactor. Three sections of a single “flow-cell”, i.e., tube side,
membrane and shell side are shown schematically in Fig. 2.
The model domain and its boundaries are also shown. R1,
R2 and R3 denote the inner radius of fiber, outer radius of
fiber and the “flow-cell” radius, respectively. The radial posi-
tion of a single “flow-cell” section, R = 0, shows the center of
fiber. C1, C2 and C3 are concentrations of copper (II) in aque-
ous phase, membrane pores and organic phase, respectively.
These concentrations depend on axial (z) and radial (r) posi-
tions of the module.
Copper (II) is considered as complexed with TFA at the
interface, and this process was considered fast enough to
achieve equilibrium at the interface on aqueous phase bound-
ary layer [16,17]. TFA–Cu2+ complex formation at interface
could be taken as a hypothetical simple extraction. Apparent
partition coefficient between both phases: copper (II) in aque-
ous phase and TFA–Cu2+ in organic phase was considered in
model development.
The description of copper (II) transfer involved the devel-
opment of a mathematical model based on mass and momen-
tum conservation equations, which have been applied in
three sections at proximities of the membrane contactor.
Following assumptions have been made while deriving mass
and momentum conservation equations:
• Both liquids were assumed to be incompressible and
Newtonian fluids.
• Steady-state conditions were considered.
• The membrane was exclusively wetted by organic phase.
• No mixing of aqueous and organic phases.
• No convective mass transfer through membrane’s pores.
• Flow was laminar in both sides of the contactor.
• The flow in lumen side was fully developed.
• Equilibrium was established at interface of aqueous
phase boundary layer.
• The system was isothermal.
• Partition coefficient was assumed to remain constant
throughout contactor.
Table 1
Geometrical and simulation parameters of the hollow fiber mem-
brane contactor module (X-50) considered for simulations [16,40]
Parameters Symbols Values
Fiber inside radius R11.1 × 10–4 m
Fiber outside radius R21.5 × 10–4 m
Flow-cell (shell side) radius R32.47 × 10–4 m
Length of the module L121.8 × 10–3 m
Porosity ε0.4
Tortuosity τ1/ε
Packing fraction Pf0.37
Number of fibers n7,400
Diffusion coefficient of Cu(II)
in aqueous side (tube side)
Daq 2.79 × 10–9 m2.s–1
Diffusion coefficient of Cu-TFA
complex in organic side
(shell side)
Dorg 1.23 × 10–10 m2.s–1
Diffusion coefficient of Cu-TFA
complex compound in mem-
brane
Dmem Dorg (ε/τ) m2.s–1
Initial concentration of copper
(II) in aqueous phase
Cini 3.15 mol.m–3
Initial concentration of TFA in
organic phase
Corg 160 mol.m–3
Partition coefficient P1
Velocity of aqueous phase Vaq 0.0033 m.s–1
Velocity of organic phase Vorg 0.0017 m.s–1
(a) (b)
r
z
Organic
phase inlet
Aqueous
feed inlet
R=0
C1C2C3
Symmetry
Symmetry
Fig. 2. (a) Hypothetical single fiber flow-cell HFMC module
and (b) single axial–radial flow-cell with model domain and its
boundaries.
A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123116
3.2. Governing equations
Transport of copper (II) in fibers of the module occurs due
to convection–diffusion mass transfer. The transfer of copper
(II) with chemical reaction and for unsteady-state conditions
could be described by the following continuity equation [41]:
∂
∂=−
∇+
Ciaq
ii
t
NR
,.
′
′ (1)
where Ci
′ is the dimensionless concentration given as:
CCC
ii
′=/0
Ci is the concentration of solute at any point in membrane
module, while C0 is the inlet concentration.
In Eq. (1), Ni is the combined flux (diffusive + convective);
Ri
′ denotes the reaction rate of the specie i and subscript “aq”
is written for aqueous phase.
∇ is gradient vector given as:
∇= ∂
∂
∂
∂
RZ
or
where R and Z are given as follows:
R = r/R3, Z = z/L
where r and z are radial and axial axis, respectively, while L
denotes total length of module.
According to Fick’s law, combined flux could be
estimated by:
NDCCu
i=− ∇+
i,aq i, aq i,aq z,aq
′′
(2)
where Di,aq is the diffusion coefficient of specie i in aqueous
phase, and uz,aq is axial velocity of aqueous phase along the
length of module. The two terms on right-hand side of Eq. (2)
show diffusive and convective fluxes, respectively.
Combining Eqs. (1) and (2) results in Eq. (3):
∂
∂=∇
−+
C
t
DC Cu Ri
i,aq
i,aq i, aq i,aq z,aq
′
′′ ′ (3)
As mentioned earlier, the reaction between solute and
organic extractant could be considered as instantaneous at
aqueous phase interface, and the term Ri
′ in Eq. (3) vanishes.
Thus, it can be described by the partition coefficient at the inter-
face of aqueous phase boundary layer with the membrane. At
steady-state condition, accumulation term becomes null, so
Eq. (3) takes the following form in cylindrical coordinates:
DC
Rr
C
R
C
ZuC
iaq
iaqiaq iaq
Zaq
i
,
,,,
,
,
∂
∂+∂
∂+∂
∂
=∂
2
2
2
2
1
′′′
aaq
Z
′
∂ (4)
As the flow inside the fiber was assumed to be laminar
and fully developed, the velocity distribution of aqueous
phase could be given by [41]:
uur
R
zaq,=−
21
1
2
(5)
where u is the mean velocity of aqueous phase inside the fiber.
Transfer of solute across the hydrophobic membrane
occurs through diffusion and is given by following steady-
state continuity equation:
−′
DC
ii,,memmem
∇=
0 (6)
This equation in cylindrical coordinates is expressed by
Eq. (7):
−
′′′
DC
RR
C
R
C
Z
i
iii
,
,,,
mem
memmem mem
∂
∂+∂
∂+∂
∂
=
2
2
2
2
10 (7)
The solute transfer in the shell of the module occurs
through diffusion and convection, and is described by fol-
lowing steady-state continuity equation:
DC
RR
C
R
C
Zu
i
iii
z,
,,,
,org
orgorg org
or
∂
∂+∂
∂+∂
∂
=
2
2
2
2
1
′′′
gg
org
∂
∂
C
z
i,
′
(8)
where Ci,org
′ is the concentration of solute in the shell (flow-
cell) of the module.
The velocity distribution in the shell side was calculated
by using Navier–Stokes equation and was then coupled with
continuity equation to characterize the flow in the shell side
and to find the concentration profile of solute in this region.
Navier–Stokes equation for incompressible flow under
steady condition is given as [41]:
ρµuu uu pF u
T
..
(,
∇−∇∇+∇
()
+∇
=∇
=0 (9)
where ρ denotes density, u the velocity vector, µ the dynamic
viscosity, p the pressure and F the body force term such
as gravity. Gravity forces are ignored as the module was
installed horizontally during experimentation. The bound-
ary conditions for these equations are summarized in Table 2.
3.3. Numerical simulations
Model equations summarized above were solved using
CFD techniques in COMSOL MultiphysicsTM software.
COMSOL software uses numerical finite element method
(FEM) to solve mathematical equations. The nite element
analysis was combined with adaptive meshing and error con-
trol using numerical solver of UMFPACK. This solver was
well suited for solving stiff and non-stiff, non-linear bound-
ary value problems [42]. A scale factor of 200 was applied
in axial direction due to the large difference between length
and radius of the module. “Scaling” avoided the excessive
number of meshes and nodes thus minimized the calculation
time. Adaptive mesh refinement was used to generate small
meshes across the membrane and near the mass transfer
boundaries. This was important near the interfaces especially
where the fluid dynamics was the most sensitive to the con-
ditions and the most influential on the overall mass transfer
[43]. The COMSOL mesh generator thus generated around
13,000 triangular elements of varying sizes. Geometrical and
simulation parameters of HFMC module are provided in
Table 1. The numerical scheme adopted for solving the equa-
tions is shown in Fig. 3. Axial–radial magnified mesh of mod-
ule geometry is shown in Fig. 4. The numerical and dynamic
models were coupled and integrated in Matlab.
117A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123
4. Results and discussion
4.1. Model validation
4.1.1. Steady-state once-through mode
The developed model was validated for both once-
through (steady state) and recycled (dynamic) modes. For
once through, the aqueous solution and spent solvent were
not recycled and collected in different tanks after passing
through the contactor. Concentration of copper (II) was mea-
sured at the outlet of contactor for different partition coeffi-
cients. Initial copper (II) concentration in aqueous feed was
taken as 3.15 mol.m–3, while initial concentration of TFA in
1-decanol was varied from 83 to 1,230 mol.m–3 in order to
achieve the required partition coefficients. These values are
listed in Table 3. Experiments were performed at 298 K, and
pH was maintained at 4.96 (±0.2). These parameters were
continuously monitored and controlled as they could alter
the partition coefficient [16]. Experiments were run to inves-
tigate the extraction of solute at these four selected partition
coefficients. Results obtained from experimental setup were
Table 2
Boundary conditions of copper (II) transport in HFMC
Boundary Fiber side Membrane Shell side Momentum transport
z = 0 Cin = C0Symmetry Convective flux –
r = 0 Axial symmetry – – –
r = R1C1 = C2/P C2 = P * C1– –
r = R2–C2 = C3C3 = C2Wall, no slip condition
r = R3––∂C3/∂r = 0
(symmetry)
Wall, no slip condition
(when R3 shows shell of module)
Axial symmetry (when R3 shows
shell of assumed flow-cell)
z = LConvective flux Symmetry Cin = 0 u = uin
Real System
(HFMC)
Feed tanks
(Aq/org phase)
Geometrical
symmetry (Unit cell)
Steady state model
(Unit cell)
Flow model
Fiber side
Diffusion mass
transfer model
membrane side
Flow model
Shell side
Finite element analysis
Meshing
Subdomain
setting/ Boundary
conditions
Solve /
integration
Total fibers /
flow-cells
Dynamic model
(Mass balances)
t+ Δt
t≤t
f
Model validation
Experimentation
Simulation
Improve theory Improve model
s
y
S
F
i
i
r
y
R
e
)
v
e
/
(
H
F
M
C
)
a
s
e
S
o
l
v
i
n
t
e
g
r
a
T
o
T
T
t
a
l
f
i
f
f
b
f
l
f
f
o
w
-
c
e
)
S
o
l
v
m
i
c
m
n
D
y
n
a
m
(
M
a
s
s
t
t
m
t
d
e
M
n
≤
f
f
f
Matlab
Comsol
s
t
a
t
e
l
Solute concentration from exit of contactor
Fig. 3. Model algorithm/numerical scheme.
r
z
Fig. 4. Axial–radial magnified mesh screens of the flow-cell.
A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123118
compared with those of simulation as given in Table 3. It can
be observed that good agreement exists between experimen-
tal data and model findings for higher partition coefficient.
Model results deviated for lower value of partition coeffi-
cient. While performing the simulation, it was assumed that
partition coefficient remained constant; however, the parti-
tion coefficient changes as the feed and solvent move along
contactor. A higher partition coefficient means greater affin-
ity of solvent for solute. Solubility of solute slightly changes
when a higher partition coefficient is used. However, in
case of lower partition coefficient, the effect on solubility is
significant.
4.1.2. Unsteady-state recycled mode
In order to validate the model for recycled mode, simu-
lations were compared with extraction experiments of cop-
per (II) from aqueous solutions using the organic extractant
phase (n-decanol) containing TFA (described in section 2).
Unsteady-state mass balance was carried out around the feed
and extractant considering an extended model in total recy-
cled mode and to obtain the copper (II) concentration in feed
tank. The developed dynamic model around the feed tank
was given as follows [16]:
Ct tt
TCtCt t
T
t
aq
zL t
aq
+∆
()
=∆
()
+
()
−∆
=1 (10)
where Taq is the mean residence time, given by Eq. (11):
TV
Q
aq
aq
aq
= (11)
where Vaq was the volume of feed tank, and Qaq was the volu-
metric flow rate. Ct(t) and Ct(t + ∆t) were the concentration of
copper (II) in the tank or at inlet of the contactor at time t and
t + Δt, respectively. Cz=L(t) was the copper (II) concentration
at the exit of the contactor entering back into the feed tank
at time t. This concentration was calculated using following
equation:
CCrds
ds
zL==
()
∫
∫ (12)
where S is the area of tube side.
Similar equation could also be developed for copper (II)
concentration in the organic phase tank.
The algebraic model Eq. (10) was coupled with the numer-
ical model (Eqs. (1)–(9)) and solved in Matlab. Simulation
was carried out according to the scheme presented in Fig. 3.
The results were compared with experimental data of copper
(II) concentration in feed tank. The operational and structural
parameters used in model simulation were the same as for
experimental data. The results were plotted as copper (II)
concentration to initial concentration in feed tank as a func-
tion of time in Fig. 5. The figure shows a good agreement
between model predictions and experimental data. A slight
deviation may be observed in the first 30 min; however, this
behavior could be done by the low value of partition coef-
ficient equal or less than 1 as the case is in current simula-
tion where partition coefficient value of 1 has been taken. If
a higher partition coefficient was used, the model would be
more accurate (as shown for steady state). The same trends
have been observed in a previous study published by Younas
et al. [16], where model results matched very well with exper-
imental data for the highest values of partition coefficient. It
can also be noted from Fig. 5 that the concentration drop is
rapid at the start of the extraction process in early 20 min
of continuous recycling. This is due to higher concentration
gradient in early extraction. However, as far as the extraction
process progress, the concentration gradient across the mem-
brane pores decreases, and thus, the rate of transfer of copper
(II) decreases.
4.2. Profiles study
4.2.1. Velocity profile
The velocity plays an important role in the design and per-
formance of HFMCs for liquid–liquid extraction. Simulation
was carried out to observe the velocity distribution of organic
phase in the “flow-cell” (shell) of HFMC based on parame-
ters reported in Table 1. Axial–radial velocity profiles in the
shell of module are plotted in Fig. 6. There are two types of
velocity profiles in the annulus of imaginary flow-cell. For
the flow-cells near the shell of module velocity as given in
Fig. 6(a), here R3 is a solid boundary (shell of module), and
Table 3
Comparison of simulation results with experimental data for
different partition coefficients: Cini = 3.15 mol.m–3, pH = 4.96,
T = 298 K
Initial concentration
of TFA, mol.m–3
P Cz=L/C0
(Sim)
Cz=L/C0
(Exp)
% Std.
Deviation
83 0.5 0.81 0.98 8.73
160 1.0 0.70 0.79 5.66
855 4.1 0.38 0.41 3.47
1,230 6.9 0.13 0.12 2.43
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0510 15 20 25 30 35 40 45 50 55 60 65 70 75
Ctank/C0
Time (mins)
Model simulation
Experimental data
Fig. 5. Comparison of model results with experimental
data: Cin (Cu2+) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3, P = 1,
vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and T = 298 K.
119A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123
hence, no slip condition is assumed. It can be clearly observed
that flow is in developing stage near the inlet of the module,
i.e., velocity profile is not parabolic in this region. This veloc-
ity field is drawn in Fig. 6(b) for velocity of organic phase in
“flow-cell” vs. radius at different positions of module length.
Velocity variation can be seen from entrance till z/L = 0.87.
At this point, the flow becomes fully developed. The max-
imum velocity of 0.0017 m.s–1 is noted in the center of the
“flow-cell”; the corresponding Reynolds number is equal to
0.05. Fig. 6(c) shows velocity profile inside module where a
symmetry has been assumed at r = R3, and hence, a maximum
velocity can be observed at this imaginary boundary.
The flow characterization in “flow-cell” is described by
means of the Navier–Stokes equation. The entry effects are
also taken into account using this approach, which increases
the accuracy of the model.
4.2.2. Flux distribution profile
Fig. 7 shows the copper (II) flux distribution inside the
fiber along the length of the module. The transport of copper
(II) takes place due to convection and diffusion. Copper (II) is
transferred from aqueous phase across the porous membrane
in r-direction because of the concentration gradient. The mass
transfer in radial direction is governed by molecular diffusion
[44]. Simulations were performed with a partition coefficient
value equal to 1.0, which is coherent with the assumption of
instantaneous transfer of copper (II) by total complexation
at the aqueous-organic interface. The Reynolds number of
aqueous and organic phases in the fiber and “flow-cell” were
equal to 0.73 and 0.05, respectively. It can be observed that
convective flux in z-direction (axial) is predominant upon
axial diffusive flux. This is due to the fact that velocity is
signicant in axial direction, which causes high convective
flux of copper (II). A maximum convective flux is observed
near the center of the fiber, as the velocity in this region is
the highest one. Thus, diffusion is dominant in radial direc-
tion or in the boundary layers adjacent to porous membrane.
Convection is dominant in axial direction, and velocity of
aqueous phase affects the transport of copper (II) along the
axis of module length. However, to increase the removal rate
of copper (II) from aqueous phase, concentration gradient
in r-direction should be enhanced. This can be achieved by
using aqueous and organic phases with higher diffusivities
of copper (II). Diffusion coefficient may be increased with
increasing temperature; however, a higher temperature may
degrade membrane material, so the factor must be kept in
mind while selecting the process temperature.
z/
L
r/R
1
(a)
(b)
r
/
R
/
/
1
Fig. 7. (a) Axial–radial convective flux field in fiber side and
(b) axial–radial diffusive flux field in fiber side. Feed concen-
tration Cin (Cu2+) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3, P = 1,
vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and T = 298 K.
z/
L
r/R
3
(a)
(b
)(
c)
Fig. 6. Velocity profile in shell: (a) 2-D field distribution where R3 shows shell of module, (b) radial profile and (c) 2-D field distribution
where R3 shows symmetry boundary: Cin (Cu2+) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3, P = 1, vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and
T = 298 K.
A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123120
It can also be observed from Fig. 7 that diffusive flux is
higher in r-direction at the feed inlet. For example, at z/L = 0.2,
diffusive flux increases from 0 to 1.2 × 10–6 mol m–2.s–1 in
r-direction from r = 0 to r = R1, i.e., from the center of tube
toward the membrane pores entrance. At z/L = 0.8, the flux
is enhanced from 0 to 8.3 × 10–7 mol m–2.s–1, which represents
69% of the earlier one. Copper (II) concentration is higher
at z/L = 0.2 as compared with that at z/L = 0.8. More copper
(II) is available for transport at earlier point, which conse-
quently increases the flux of copper (II) and vice versa at later
point. As a result, copper (II) concentration falls, and thus,
r-direction flux is reduced as the feed moves along the z-axis
of the module.
Diffusive and convective flux is calculated radially inside
the fiber in the middle of the module at z/L = 0.5. The flux
is plotted vs. r/R1 in Fig. 8. r/R1 = 0 indicates the center of
fiber while r/R1 = 1 shows the boundary of fiber wall. It can
be observed that convective flux is maximum at the cen-
ter of the fiber and is calculated to be 2.53 × 10–5 mol.m–2.s–1
while diffusive flux is found to be 0.42 × 10–7 mol.m–2.s–1 at
that point. Diffusive flux then increases while convective flux
decreases in radial direction. Diffusive flux becomes maxi-
mum in the boundary layer of aqueous phase achieving 10.44
× 10–7 mol.m–2.s–1. It can be inferred that convection is domi-
nant in the center of the fiber. Nevertheless, copper (II) trans-
port in boundary layer near the fiber wall is dominated by
diffusion. Increasing the diffusion will increase the removal
efficiency of solute. This can be achieved by lowering the
velocity or by using an effective extractant with higher parti-
tion coefficient [16,40].
4.2.3. Concentration profile of copper (II)
Assembly of single fiber and “flow-cell” as sketched in
Fig. 1 was simulated in 2-D and 3-D. The concentration of
copper (II) was calculated radially in direction of flow along
the length of module. Dimensionless concentration profile
C/C0 of copper (II) is plotted in Fig. 9 in 2-D for countercur-
rent flow, in all three compartments in which the transport
of copper (II) is involved, i.e., inside the fiber, the membrane
pores and the shell side. Feed enters at z = 0 inside fibers
with copper (II) initial concentration of 3.15 mol m–3. Organic
phase flows countercurrently in shell of the module at z = L
with TFA (II) initial concentration of 160 mol.m–3. Fig. 9 indi-
cates that organic phase fills the pores of membrane and
immobilizes in the fiber at membrane mouth. The reaction of
copper (II) with TFA is considered instantaneous and is sim-
ply considered through the partition coefficient at the inter-
face between the aqueous phase boundary layer to that of
the organic phase at the membrane’s mouth. The TFA-copper
complex molecule diffuses through the membrane pores into
shell side due to concentration gradient and is then swept out
by the incoming organic phase.
Concentration profile plotted in Fig. 9 indicates the
decrease of copper (II) concentration in the aqueous phase
flows along the module length in z-axis. On the other hand,
the concentration of TFA-copper complex in organic phase
increases along its flow path. This behavior is given by the
continuous transfer of copper (II) from aqueous feed to
organic phase. It can also be observed that the decrease in
copper (II) concentration in aqueous phase is smooth through
the module length under the stated hydrodynamics condi-
tions. Copper concentration falls to half of its initial value at
exit, i.e., at z/L = 1 for once through under given conditions.
However, this is not always the case. The fall in concentration
strongly depends upon the partition coefficient. Changing
the partition coefficient alters the concentration profile. For
a partition coefficient greater than unity, the fall in copper
(II) concentration is higher near the inlet and then decreases
slowly throughout the module. A higher partition coefficient
means greater affinity of TFA toward copper (II).
Fig. 8. Radial profiles of convective and diffusive flux at z/L = 0.5.
Feed concentration Cin (Cu2+) ) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3,
P = 1, vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and T = 298 K.
Fig. 9. 2-D concentration profile of a “flow-cell”, C_all/C0. Feed
concentration Cin (Cu2+) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3,
P = 1, vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and T = 298 K.
121A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123
4.3. Effects of process parameters on extraction efficiency
4.3.1. Effect of partition coefficient
Partition coefficient is an important parameter in liquid–
liquid extraction process. It is defined as the ratio of solute
concentration in organic phase to that of aqueous phase at
equilibrium. To study the effects of partition coefficient
using the current mathematical model, simulations were
performed for arbitrarily selected values of partition coeffi-
cient. Flow rates of aqueous and organic phases were kept
constant at 8.3 × 10–7 and 1.0 × 10–6 m3.s–1, respectively. It can be
observed that an increase in partition coefficient increases the
extraction of copper (II) across the membrane. Extraction effi-
ciency is measured by taking the percentage ratio of copper
(II) transferred from aqueous to organic phase to total copper
(II) concentration available at module inlet in aqueous phase.
Extraction efficiency of copper (II) removal from aqueous
phase in once-through mode has been measured and plotted
as function of partition coefficient in Fig. 10. Efficiency was
calculated as follows:
Efficiency =−
×
=
1 100
0
C
C
zL (13)
It was found that the extraction efficiency increases
much more rapidly at low values of partition coefficient.
On the other hand, at higher values, the partition coefficient
has a very little effect on extraction efficiency. For example,
extraction efficiency of copper (II) transport is 34% with a
partition coefficient equal to 2. Extraction efficiency increases
at the rate of 23% per unit value till partition coefficient value
of 4. Above this value, extraction efficiency increases expo-
nentially and become negligible at values greater than 8. This
means that there is no need to nd an extractant or diluent
with which the solute achieves a very high partition coeffi-
cient (that may either be toxic or expensive). Nevertheless,
too small partition coefficient, i.e., a value near the unity, will
result in low extraction efficiency. Nevertheless, it should be
kept in mind that partition coefficient is not the only param-
eter that affects the extraction efficiency; feed flow rate has
also a great effect on removal of solute. These effects are
explained in the subsequent section. However, the same
trends for partition coefficient effects on extraction can be
observed for any value of flow rate used. Partition coefficient
depends on various factors like the nature of extractant, pH,
temperature and type of diluents [16,45].
4.3.2. Effects of flow rates
Interest has also been taken to investigate the effects
of flow rates on extraction efficiency. Simulation was per-
formed to measure the extraction efficiency of copper (II)
from aqueous phase at partition coefficient equals to one.
Copper (II) concentration was calculated at the inlet and
outlet of HFMC module at various flow rates. Extraction
efficiency was calculated for different values of flow rates
of aqueous and organic phases. Flow rate of one phase
was varied at a time while that of other phase was kept
constant, and the result has been plotted in Fig. 11. It is
indicated from the Fig. 11 that the extraction efficiency
decreases with increasing aqueous phase (feed) flow rate.
This is due to the fact that the contact or residence time of
copper (II) decreases as the flow rate increases, resulting
in lower mass transfer across the membrane. However, the
increase in organic flow rate in shell side has the opposite
effect on extraction efficiency of copper (II). The extraction
efficiency increases with increasing organic flow rate as
shown in Fig. 11. In fact increase in organic phase flow rate
provides more TFA (extractant) to be in contact with cop-
per (II) in the pores of the membrane. Thus, more copper
(II) molecules are extracted from aqueous phase to organic
phase. Increasing the flow rate of organic phase in shell
side decreases the concentration of copper (II) at the mem-
brane-shell interface and thus results in higher concentra-
tion gradient, which in turn increases the mass transfer [27].
It can also be observed that this effect is significant only at
low flow rates. The effect on extraction efficiency becomes
negligible at flow rate of organic phase greater than 8.0 ×
10–8 m3.s–1. Because no more solute is available for removal
at boundary layer of organic phase in shell side at higher
flow rate. Nevertheless, Marjani and Shirazian [44] reported
that organic flow rate has negligible effects on extraction
Fig. 10. Effect of partition coefficient on extraction efficiency. Feed
concentration Cin (Cu2+) = 3.15 mol.m–3, Cin (TFA) = 160 mol.m–3,
vaq = 0.0033 m.s–1, vorg = 0.0017 m.s–1, and T = 298 K.
Fig. 11. Effects of aqueous and organic flowrates on extraction
efficiency. Cin(Cu) = 3.15 mol.m–3, Cin(TFA) = 160 mol.m–3, P = 1,
T = 298 K.
A. Muhammad et al. / Desalination and Water Treatment 63 (2017) 113–123122
efficiency. However, it is true for only high flow rate values
of organic phase. Meanwhile, these effects are significant at
lower flow rate values. Fadaei et al. [35] have also showed
increase in extraction efficiency of solute with the increase of
organic phase flow rate. However, this increase was not sig-
nificant. They found that extraction efficiency ranges from
12% to 65% when feed flow rate is varied from 1 × 10–8 to 9 ×
10–8 m3.s–1 for a constant organic flow rate of 1 × 10–8 m3.s–1. In
current study, copper (II) extraction efficiency was found to
reach 95% while flow rate was ten times greater as reported
in previous studies.
5. Conclusions
A numerical model was developed for extraction of cop-
per (II) with TFA diluted in n-decanol using HFMC. Fibers
were considered as straightened and parallel to each other.
A hypothetical “flow-cell” was assumed surrounding each
fiber. CFD simulation was performed using coupling of
mass and momentum transport equations. Numerical CFD
model was then integrated with dynamic model developed
across the feed tank. The integrated model was successfully
validated with experimental results for the extraction of
copper (II) with TFA. Extraction efficiency of 49% has been
determined in about 60 min of recycled-based extraction of
copper (II) for partition coefficient equal to 1. Extraction effi-
ciency greatly depends upon partition coefficient and feed
flow rate. It increases exponentially with partition coefficient
and decreases with feed flow. Steady-state CFD simulation
of HFMC revealed that diffusion and convection controlled
regions along the path of feed flow strongly depend on the
feed flow rate and partition coefficient. The 2-D simulation
permits the visualization of concentration, velocity and flux
distribution both axially and radially.
Symbols
C — Concentration, mol.m–3
C’ — Dimensionless concentration
D — Diffusion coefficient, m2.s–1
F — Body force term (N) such as gravity, N
N — Combined flux (diffusive + convective), mol.m–2.s–1
n — Number of fibers
P — Partition coefficient
p — Pressure, N.m–2
Pf — Packing fraction
Q — Volumetric flow rate, m3.s–1
Re — Reynolds number
R’ — Rate of reaction, s–1
R — Radius, m
r — Radial axis, m
u — Mean velocity, m.s–1
V — Volume of tank, m3
v — Velocity, m.s–1
T — Time constant, min
t — Time, min
ρ — Density, kg.m–3
µ — Dynamic viscosity, pa.s
ε — Porosity
τ — Tortuosity
∇ — Gradient vector
Subscripts
1 — Fiber side boundary layer
2 — Membrane pores
3 — Shell side boundary layer
aq — Aqueous phase
i — Specie, solute (copper (II))
in — Inlet
mem — Membrane
org — Organic phase
r — Radial axis
z — Axial axis
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