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Journal of Membrane Science 213 (2003) 181–193
Recovery of phenol from aqueous solutions
using hollow fibre contactors
M.J. González-Muñoz, S. Luque, J.R. Álvarez, J. Coca∗
Department of Chemical and Environmental Engineering, C/Julian Claveria s/n, University of Oviedo, 33071 Oviedo, Spain
Received 3 April 2002; received in revised form 6 November 2002; accepted 6 November 2002
Abstract
The purpose of this study is to characterise recovery of phenol from an aqueous solution using a hydrophobic polypropylene
membrane contactor. The effects of temperature and hydrodynamics on the overall mass transfer coefficient were determined.
Integration of the extraction and stripping stages was also carried out thereby allowing removal of more than 99% of the
original phenol, while the organic phase is simultaneously regenerated.
© 2002 Elsevier Science B.V. All rights reserved.
Keywords: Phenol; 1-Decanol; Membrane contactor; Mass transfer coefficient; Extraction-stripping
1. Introduction
Phenols are often present in wastewaters from many
industrial processes, such as refineries (6–500mg/l),
coking operations (28–3900mg/l), coal processing
(9–6800mg/l), and manufacture of petrochemicals
(2.8–1220mg/l). Phenols are also the main organic
constituents present in condensate streams in coal
gasification and liquefaction processes [1,2]. Other
sources of waste stream water containing phenols are
pharmaceutical, plastics, wood products, paint, and
pulp and paper industries (0.1–1600mg/l) [3–8].
The maximum allowed concentration of phenol in
non-chlorinated water is 0.1mg/l while that in chlo-
rinated water is 0.001–0.002mg/l. Some standards
adopted for water supplies are as low as 1–2ppb
[9–11].
∗Corresponding author. Tel.: +34-98-510-3443;
fax: +34-98-510-3434/3443.
E-mail address: jcoca@correo.uniovi.es (J. Coca).
Phenols are very difficult to degrade, because they
are toxic for micro-organisms. Furthermore, phenol re-
acts with the chlorine used in the treatment of potable
water, yielding compounds even more toxic and more
resistant to biodegradation than phenol [12].
When the phenol concentration is low (<50ppm),
biological, chemical or electrochemical oxidation
processes may be used for its removal. However, at
higher concentrations, the treatment process should
be designed to permit its recovery and reuse [8].A
comparative study of processes for the recovery of
phenol from aqueous effluents at concentrations over
50ppm shows that liquid–liquid extraction is the most
economic non-destructive process [13].
This work focused on recovery of phenol by
membrane-enhanced solvent extraction. In this tech-
nique a solid membrane acts as an interface between
aqueous and organic phases, allowing transfer of so-
lute through the pores of the membrane, but prevent-
ing mixing of the bulk phases. The use of membrane
contactors overcomes some of the disadvantages of
conventional liquid–liquid extraction, such as the
0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S0376-7388(02)00526-4
182 M.J. Gonz´alez-Muñoz et al./Journal of Membrane Science 213 (2003) 181–193
Nomenclature
a,b,cparameters for exponential
curve (Eq. (13))
Aflow area in the shell (m2)
Amsurface area of membrane (m2)
cdefined in Eq. (7)
caphenol concentration in aqueous
phase (mg/l)
cophenol concentration in organic
phase (mg/l)
cdefined in Eq. (11)
dfoutside diameter of fibres (m)
Ddistribution coefficient
DHhydraulic diameter (m)
Dshell inside diameter of shell (m)
Dtoutside diameter of centre tube
of the module (m)
Jsol solute flux (g/s)
Kaoverall mass transfer coefficient
defined for the aqueous
phase (m/s)
Lmodule length (m)
Nfnumber of fibres in the module
Qdefined in Eq. (9)
Qaaqueous phase flow rate (m3/s)
Qoorganic phase flow rate (m3/s)
Re Reynolds number
ttime (s)
vflow velocity (m/s)
Vdefined in Eq. (8)
Vavolume of aqueous phase (m3)
Vovolume of organic phase
(m3)
Vsvolume of stripping phase
(m3)
xdistance along the module (m)
Greek letters
φdefined in Eq. (12)
µviscosity (Pas)
ρdensity (kg/m3)
Subscripts
a aqueous phase
o organic phase
s stripping phase
Superscripts
in module inlet
out module outlet
∗equilibrium
0 initial conditions
formation of stable emulsions, the need for different
densities of the immiscible phases and restrictions
to a narrow range of operating conditions because
of loading or flooding constraints. Additional advan-
tages of this technique are a high surface area per unit
volume when hollow fibre modules are used, and the
possibility of changing the hydrodynamics of both
phases independently. The most important drawback,
however, is the slower mass transfer rates because of
the additional mass transfer resistance offered by the
pores of the membrane [14–19].
The solutions used contained phenol concentrations
of 3000 mg/l. The process involved both extraction and
stripping steps in order to achieve both high recovery
and, simultaneously, a high concentration of phenol in
the final NaOH stripping solution [20,21].
2. Experimental
2.1. Materials
Phenol, sodium hydroxide, sulphuric acid (96%
(w/w)), phenolphthalein solution (0.1% (w/w)), hex-
ane, toluene, methylcyclohexane, n-heptane, and met-
hylisobutylketone (MIBK) were obtained from
Panreac. Potassium hydrogen phthalate and trib-
utylphosphate were purchased from Probus. Petrosol
15/20 and Petrosol D15/20 were donated by Cepsa.
Hostarex A237 was purchased from Clariant while
1-decanol (purum, 97% (w/w)), Amberlite LA-2, tri-
octylphosphine oxide (TOPO, >97% (w/w)), trioctyl-
methylammonium chloride (TOMAC) and triocty-
lamine (TOA, 95% (w/w)) were supplied by Fluka.
All chemicals were used without further purification.
2.2. Experimental apparatus
A schematic diagram of the experimental appara-
tus is shown in Fig. 1. The equipment consists of a
polypropylene hollow fibre membrane contactor and
M.J. Gonz´alez-Muñoz et al./Journal of Membrane Science 213 (2003) 181–193 183
Fig. 1. Schematic diagram of extraction-stripping apparatus.
two jacketed vessels containing either the aqueous or
the organic phases. The hollow fibre membrane mod-
ule is a Liqui-Cel Extra-Flow phase contactor that
contains 10176 hydrophobic polypropylene Celgard®
fibres. Each fibre is 15cm length and has a nominal
internal diameter of 240m, a nominal thickness of
30m and a pore size of 0.03m. The effective sur-
face area is 1.4 m2. The membrane module is provided
with a central tube (0.0222m diameter) and a central
baffle to facilitate good distribution of the flow through
Fig. 2. Liqui-Cel Extra-Flow membrane contactor with flow of aqueous phase through the lumens of the hollow fibres.
the shell side (see Fig. 2). The internal diameter of the
casing is 0.0555m.
The aqueous phase is pumped through the lumen of
thefibres, and the organic phase flows through the shell
side of the module. Transfer of solute from the aque-
ous feed stream to the organic phase occurs through
the pores of the membrane. Since the membrane is hy-
drophobic, a slight overpressure of the aqueous phase
(0.6–0.8bar) is necessary to stabilise the interface in
the pores so as to avoid bulk mixing of the two phases.
184 M.J. Gonz´alez-Muñoz et al./Journal of Membrane Science 213 (2003) 181–193
Ethylene glycol from a constant temperature bath is
pumped through the jacket of each vessel in order to
ensure a constant temperature throughout the experi-
ment.
In order to carry out the extraction and stripping
processes simultaneously, a second membrane module
is used. In the process of interest, the organic phase
is pumped through the first module. and then, now
loaded with solute, is transferred to the shell-side of
stripping module. In this second module, the organic
phase is contacted with the aqueous stripping solution
of NaOH, thereby achieving continuous regeneration
of the organic phase.
2.3. Methods
2.3.1. Selection of the extraction system
Batch extraction experiments were carried out using
several solvents (1-decanol, hexane, toluene, methyl-
cyclohexane, n-heptane, MIBK, Petrosol 15/20 and
Petrosol D15/20).
In order to study physical extraction by the afore-
mentioned solvents, a known volume (10–20ml) of
aqueous phase, containing 3000mg/l of phenol, was
mixed with the same volume of an organic phase in
a screw-capped flask. The flasks were immersed in a
constant temperature bath (20◦C) and stirred for at
least 4h, long enough to approach equilibrium. After
phase separation, phenol concentrations were mea-
sured for both phases. For the solvents providing the
highest degrees of extraction, distribution coefficients
were measured using different ratios of the volumes
of the aqueous and organic phases (with a constant
initial concentration of phenol in the aqueous phase).
After the physical extraction experiments, the re-
active extractant was added to the solvent (diluent)
with the highest distribution coefficient at a concen-
tration of 5% (v/v). The aforementioned commercial
extractants were tested in the extraction experiments.
Sulphuric acid salts of TOA were obtained by mixing
80lofH
2SO4(96% (w/w)) with 10 ml of TOA [22].
All experiments in the reactive extractant phase of the
work were carried out at 20◦C with a volume ratio of
2 (aqueous/organic).
2.3.2. Membrane contactor experiments
The experiments were designed to study the effects
of hydrodynamics (Reynolds number and flow pattern,
i.e. co-current or counter-current operation) and tem-
perature (in the range 20–40◦C) on the overall mass
transfer coefficient. The effect of the Reynolds number
was studied by establishing a specific flow rate for one
phase while varying the flow rate of the other phase.
The Reynolds number for the shell side was calculated
using the expression provided by the manufacturer,
which takes into account the geometry of the module:
Re =DHvρ
µ(1)
where DHis the hydraulic diameter, vthe average lin-
ear velocity and ρand µare the density and viscosity
of the organic phase, respectively. DHwas calculated
as follows:
DH=4(π/4)D2
shell
Nfπdf=D2
shell
Nfdf(2)
where Dshell is the inside diameter of the shell, Nfthe
number of fibres, and dfis the outside diameter of a
single fibre. The value of DHin this case is 10−3m.
The average linear velocity is given by
v=Qo
A(3)
where Qois the volumetric flow rate, and Ais the area
of the shell through which the organic phase flows,
calculated as
A=1
4π(D2
shell −D2
t−Nfd2
f)(4)
where Dtis the outside diameter of the central (col-
lection and distribution) tube (see Fig. 2).
For the stripping experiments, the effect of the mole
ratio of NaOH to phenol on the recovery of phenol was
studied to determine the optimum value of this ratio,
i.e. the ratio that gives the highest recovery of phe-
nol at the lowest concentration of NaOH. The effect
of hydrodynamics during the stripping process has not
been studied since the same behaviour as in the aque-
ous phase during the extraction process is expected.
Both the feed and stripping solutions are diluted aque-
ous streams. Therefore, influencing parameters such
us density, viscosity and interfacial tension would not
change significantly, and the optimum operating con-
ditions should be similar in both cases.
During the membrane contactor experiments, sam-
ples of each phase were taken at specific times in order
to monitor the time course of the phenol concentration
profiles in each phase.
M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193 185
Once the optimum conditions (hydrodynamics and
flow pattern) for the extraction process were estab-
lished, experiments involving the integrated process
were carried out. The combination of the extraction
and stripping processes in the two membrane contac-
tors permits one to transfer the solute from an aque-
ous phase (feed) to a second aqueous phase (stripping)
while continuously regenerating the organic phase.
2.4. Analytical methods
Phenol solutions were analysed by HPLC, us-
ing a Hewlett Packard Hypersil MOS column. The
UV absorbances at 254 and 280nm were measured
using a diode array detector. The mobile phase
was a mixture of methanol and water (55/45 (v/v))
for the analysis of phenol in aqueous phases, and
methanol/acetonitrile/water (40/40/20 (v/v/v)) for de-
termination of phenol in the organic phase [22]. The
flow rate was 1ml/min in all cases.
The NaOH concentration in the stripping phase was
determined by titration with potassium hydrogen ph-
thalate using phenolphthalein as the indicator.
2.5. Mass transfer coefficient calculation
The transfer of phenol from the aqueous phase to
the organic phase can be described in terms of the
overall mass transfer coefficient, Ka:
Jsol =KaAm(ca−c∗
a)(5)
where Jsol is the solute flux, Amthe membrane area,
cathe concentration of phenol in the aqueous solution
at time t, and c∗
ais the concentration of phenol in the
aqueous phase in equilibrium with the organic phase
at this same time.
Through a mass balance, the flux of solute in Eq. (5)
can be related to the depletion of solute in the aqueous
phase. If one assumes a high degree of mixing in the
feed vessels (the solute concentration in the vessel
being equal to that entering the module) and a constant
distribution coefficient, D, the solute concentration in
the aqueous phase may be obtained [16,26,27]. The
equation for co-current flow is (a detailed derivation
can be found in Appendix A):
ca=Vc0
a
1+V+c0
a
1+Vexp(−ct)(6)
where
c=Qa
Va
(1+V)
(1+Q) 1−exp −AmKa
Qa(1+Q) (7)
and
V=Va
VoD(8)
Q=Qa
QoD(9)
For counter-current flow [26]:
ca=Vc0
a
1+V+c0
a
1+Vexp(−ct) (10)
where
c=Qa
Va(1+V)1−exp[φ(Q −1)]
1−Qexp[φ(Q −1)](11)
and
φ=AmKa
Qa(12)
The initial phenol concentration in the aqueous phase
is c0
a,Vaand Voare the volumes of the aqueous and or-
ganic phases, respectively. Qaand Qoare the flow rates
of the aqueous feed and organic phase, respectively.
3. Results and discussion
3.1. Selection of the extraction system
For the solvents used in the preliminary extraction
experiments the equilibrium concentration of phenol
in the organic and aqueous phases are shown in Fig. 3.
The solvents that produced the greatest degree of ex-
traction (1-decanol, toluene and MIBK) were further
studied to obtain the equilibrium isotherms (20◦C)
shown in Fig. 4. In these experiments mass balances
closed, in general, within 3%.
Two of the most important characteristics of a sol-
vent to be used to extract a particular solute, namely,
a high distribution coefficient and a low solubility in
water [6,23] are presented in Table 1 for 1-decanol,
toluene and MIBK. On the basis of these data,
1-decanol was selected for further study. This solvent
has both a high distribution coefficient and very low
solubility in water.
186 M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193
Fig. 3. Distribution of phenol between water (䊐) and several solvents (䊏). Initial phenol concentration in aqueous phase: 3000ppm.
Volume ratio (Va:Vo)=1:1.
Solutions of reactive extractants at a concentrations
of 5% (v/v) in 1-decanol were then tested to assess
their potential for use as extraction solvents for phenol
(see Fig. 5). The degree of extraction obtained using
a reactive extractant was slightly higher than that ob-
Fig. 4. Extraction isotherms at 20◦C: (䊐) methylisobutylketone; (䊉) 1-decanol; () toluene.
tained using just the diluent (1-decanol) itself. Conse-
quently, use of a reactive extraction system cannot be
justified. Hence, all membrane contactor experiments
were carried out using pure 1-decanol as the organic
solvent (extractant).
M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193 187
Table 1
Distribution coefficients and solubility in water of some solvents
[24,25]
Solvent Distribution coefficient
at 20◦CSolubility in water
(mg/l) at 25◦C
MIBK 83.2 17,000
1-Decanol 25.4 37
Toluene 1.7 526
3.2. Calculation of mass transfer coefficient
The time dependence of the concentration of phe-
nol in the aqueous phase can be expressed (see
Eqs. (6)–(12)) as the sum of two terms
ca=a+bexp(−ct)(13)
where aand bdepend only on known parameters. The
value of ccan be obtained from a plot of ln(ca−a)
against time, cbeing the slope of the straight line thus
obtained. Then, Kacan be determined from either Eq.
(7) or Eq. (11), for co-current or counter-current flow,
respectively. For a co-current flow pattern:
Ka=−Qa
(1+Q)Amln1−cVa
Qa1+Q
1+V (14)
Fig. 5. Percentage of phenol removed from aqueous solution (3000ppm) using 1-decanol together with different extractants (5% (v/v)) at
20◦C. Volume ratio (Va:Vo)=2:1.
3.3. Influence of hydrodynamics and flow pattern
The data for the extraction experiments provide
very regular and smooth solute concentration profiles
(Fig. 6). The overall mass transfer coefficient, Ka,
is depicted in Fig. 7a as a function of the Reynolds
number of the organic phase (shell side), Reo, for
the extraction experiments performed at 20◦C with
co-current flow of aqueous and organic phases.
The Reynolds number of the organic phase (Reo)
was varied in the range 0.1–0.65 (Qo=8–50l/h)
while keeping the tube side Reynolds number constant
(Rea=4; Qa=28l/h). For these experiments, Kain-
creased with Reountil an asymptotic value is reached
at about Reo=0.4. When the Reynolds number of
the aqueous phase was varied while keeping the shell
side Reynolds number constant (Reo=0.32), no fur-
ther increase in the overall mass transfer coefficient
was observed. Therefore, the system had reached
the conditions in which the controlling resistance to
mass transfer is the diffusion through the pores of the
membrane.
The same set of experiments was then repeated
with a counter-current flow pattern (Fig. 7b). The re-
sults were more erratic but the average values for Ka
were similar to those for co-current flow. This result
188 M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193
Fig. 6. Phenol concentration profiles in the (䉬) aqueous and (䊐) organic phases during the extraction of phenol from an aqueous solution
with 1-decanol at 20◦C using a membrane contactor. Rea=4 and Reo=0.47. Counter-current flow.
Fig. 7. Influence of the organic phase hydrodynamics (Reo) on the overall mass transfer coefficient defined for the aqueous phase (Ka)
during the extraction of phenol from an aqueous solution with 1-decanol at 20◦C using a membrane contactor (Rea=4). (a) Reproducible
stable values were obtained in co-current flow. (b) Relative to (䊉) co-current flow and (䊐) counter-current flow experiments yielded
similar average values, but with a significantly poorer precision.
indicates that for this system the flow pattern does not
have a significant influence on the overall mass trans-
fer coefficient, but that the system is more stable when
operated with co-current flow patterns. As mentioned
in Section 2.3.2, the effect of hydrodynamics during
the extraction and stripping processes should be the
same.
In all these experiments, the mass balances closed,
in general, within 7%.
3.4. Influence of temperature
Temperature directly affects the physico-chemical
properties of the system (i.e. density, viscosity,
M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193 189
Fig. 8. Influence of the mole ratio of NaOH to phenol on removal of phenol during stripping of phenol using a membrane contactor at
20◦C. Reo=0.32 and Res=7.
interfacial tension and mutual solubility) but not the
distribution coefficient. Thus, temperature has an indi-
rect influence on the overall mass transfer coefficient.
As the temperature changed from 20 to 40◦C, an in-
crease of 60% in the overall mass transfer coefficient
was observed.
Fig. 9. Phenol concentration ((䉫) aqueous phase; (䊏) organic phase; () stripping phase) as a function of time during a simultaneous
extraction and stripping experiment, using 1-decanol as the organic phase and 1.8M NaOH as the stripping phase at 20◦C. Va=6l,
Vo=0.7l and Vs=0.4l. Rea=7, Reo=0.32 and Res=7.
3.5. Influence of the mole ratio of NaOH to phenol
on stripping efficiency
Phenol can be easily stripped from the loaded or-
ganic phases by contacting them with an aqueous solu-
tion of sodium hydroxide, yielding the phenol sodium
190 M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193
salt in aqueous solution. Stripping experiments were
performed using a membrane contactor at 20◦C with
Reo=0.32 and Res=7. The experimental proto-
col was similar to that employed in the extraction ex-
periments described above. The variable to be opti-
mised in this case is the mole ratio of NaOH to phe-
nol. Examination of Fig. 8 shows that it is necessary
to utilise a ratio of at least 4 to remove about 97%
of the phenol from an organic solution with a phe-
nol concentration of 3g/l. This ratio is significantly
higher than the stoichiometric of the reaction of phenol
with NaOH.
3.6. Integrated extraction-stripping process
In an integrated extraction-stripping process, the
phenol extracted from the aqueous phase into the
organic phase, is simultaneously transferred to a strip-
ping solution (aqueous sodium hydroxide). In this
system, two membrane contactors are used to carry
out the extraction and stripping processes forming a
closed loop cycle. This scheme permits continuous re-
generation of the organic phase and if the volumetric
ratios are properly adjusted, may yield a more con-
centrated phenol solution. Experiments were carried
out at 20◦C using optimum hydrodynamic conditions
(co-current flow pattern in both contactors with Rea=
7, Reo=0.32, and Res=7). Typical concentration
profiles for the case where the aqueous, organic and
stripping phase volumes were 6, 0.7, and 0.4l, respec-
tively, are shown in Fig. 9. A concentration ratio of
14 was obtained. The associated percentage removal
of phenol exceeded 99%. Values of phenol recovery
and the degree of concentration reached in different
experiments are summarised in Table 2.
Table 2
Phenol recovery and concentration ratio reached in several inte-
grated extraction and stripping experiments using membrane con-
tactors at 20◦C(Vo=0.4 l of 1-decanol in all cases)
Va(l) Vs(l) Va:Vsratio Phenol
recovery (%) Concentration
ratio
0.5 0.6 8.3 99.8 8.2
0.6 0.4 15.0 99.2 14
20 0.9 22.2 90.3 16
10 0.4 25.0 97.8 21
25 0.9 27.8 93.5 21
4. Conclusions
Experiments were conducted to recover phenol
from aqueous solutions using both physical and chem-
ically extractants. Physical extraction using 1-decanol
as solvent gave the best results because of its high
distribution coefficient and low solubility in water.
When membrane contactor experiments were car-
ried out using 1-decanol as solvent, the system was
more stable when co-current flows were used. Con-
ditions in which diffusion through the membrane
becomes the controlling resistance to the mass trans-
fer were easily reached. A 60% increase in the mass
transfer coefficient is observed as the temperature
increases from 20 to 40◦C.
A concentrated aqueous sodium hydroxide solution
is an effective stripping solution. A mole ratio of 4:1
(NaOH:phenol) was needed to recover more than 99%
of the extracted phenol.
It is possible to carry out extraction and stripping
simultaneously, running the system as a closed loop
process. Concentration ratios of up to about 20-fold
with 98% recovery were obtained.
Acknowledgements
This research was funded by CICYT Project
QUI1999-0888.
Appendix A
A differential mass balance for phenol in the mod-
ule (Fig. 10) taking into account the transfer of the
component from the aqueous into the organic phase is
as follows:
Qa[ca(x, t ) −ca(x +dx, t)]=−Qadca(x, t )
=KadAm(ca(x, t ) −c∗
a(x, t )) (A.1)
where Qais the flow rate of the aqueous feed phase,
Kathe overall mass transfer coefficient with respect
to the aqueous phase, Amthe membrane area, ca(x,t)
the concentration of phenol in the aqueous solution at
the axial position xand at time t, and c∗
a(x,t)isthe
concentration of phenol in the aqueous phase in equi-
librium with that in the organic phase at the same time
M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193 191
Fig. 10. Schematic of the solute transfer in a membrane contactor system.
and location, co(x,t). The equilibrium relationship can
be expressed as
c∗
a(x, t ) =co(x, t )
D(A.2)
where Dis the distribution coefficient.
The decrease in solute molar flow in the aqueous
phase should be equal to the increase in solute molar
flow in the organic phase, as expressed in the module
overall mass balance:
Qa(cin
a(t) −cout
a(t)) =Qo(cout
o(t) −cin
o(t)) (A.3)
where Qois the flow rate of the organic phase and
the superscripts “in” and “out” refer to the module
inlet and outlet, respectively (being cin
a(t) =ca(0,t)
and cout
a(t) =ca(L, t )). A similar mass balance could
also be written at any point (x) of the module as
follows:
Qa(ca(x, t ) −cout
a(t)) =Qo(cout
o(t) −co(x, t )) (A.4)
Substituting co(x,t) from Eq. (A.2) into Eq. (A.4),
and c∗
a(x,t)inEq. (A.1), and integrating this equation
between x=0 (where and ca(x =0,t)=cin
a(t)) and
x=L(where ca(x =L, t) =cout
a(t)), the following
equation is obtained:
cout
a(t) −cout
o(t)/D
cin
a(t)(1+Q) −cout
a(t)Q −cout
o/D
=exp −KaAm
Qa(1+Q)(A.5)
where
Q=Qa
QoD(A.6)
and taking into account the overall mass balance to
the module, Eq. (A.3),Eq. (A.5) can be expressed as
cout
a(t) −cout
o(t)/D
cin
a(t) −cin
o(t)/D =exp −KaAm
Qa(1+Q)(A.7)
A simultaneous unsteady-state balance to the aqueous
phase feed tank, assuming perfect mixing, gives
−Vadcin
a(t)
dt=Qa[cin
a(t) −cout
a(t)] (A.8)
At any time the depletion of the amount of solute
present in the aqueous phase should be equal to its
increase in the organic phase:
Va(c0
a−cin
a(t)) =Vo(cin
o(t) −c0
o)(A.9)
where c0
a=cin
a(t =0)and c0
o=cin
o(t =0)are the
initial solute concentration in the aqueous and organic
phases, respectively.
Combining Eqs. (A.3) and (A.9), the concentration
of solute in the organic phase at the module outlet
can be expressed as a function of (i) the initial solute
concentration in the aforementioned phase, (ii) known
quantities, such as phase volumes and flow rates, and
(iii) the solute concentration in the aqueous phase:
cout
o(t) =Qa(cin
a(t) −cout
a(t))
Qo
+Va(c0
a−cin
a(t))
Vo+c0
o(A.10)
Combining Eqs. (A.10) and (A.7), allows to obtain
an expression for the solute concentration in the
aqueous phase at the module outlet [cout
a(t)]asa
function of the inlet concentration [cin
a(t)] and some
192 M.J. Gonz´alez-Muñoz et al./ Journal of Membrane Science 213 (2003) 181–193
process parameters:
cout
a(t) =1
(1+Q) cin
a(t) −V[c0
a−cin
a(t)]+c0
o
D
exp −KaAm
Qa(1+Q)
+1
(1+Q) Qcin
a(t) +V[c0
a−cin
a(t)]+c0
o
D
(A.11)
where
V=Va
VoD(A.12)
Substituting Eq. (A.11) into Eq. (A.8) and integrating
between t=0 and time t:
cin
a(t) =(c0
aV−c0
o/D)
(1+V) +(c0
a−c0
o/D)
(1+V)
exp −Qa
Va
(1+V)
(1+Q)
×1−exp −KaAm
Qa(1+Q)t(A.13)
Eq. (A.13) is identical to Eqs. (6) and (7) in Section
2.5, when the solute concentration in the aqueous
phase is measured in the feed tank [cin
a(t) =ca],
assuming perfect mixing, and the initial solute con-
centration in the organic phase, c0
o, is zero. Thus, Ka
can be obtained from the exponential term, as shown
in Eq. (14).
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