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Ind. Eng. Chem. Res. 2005, 44, 1358-1369
Concentration of Inorganic Salts in the Permeate during Nano- or Ultrafiltration Promoted by Water-Soluble Polyelectrolytes in the Feed Solution P. Prokopovich, V. Starov,* and R. G. Holdich Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K.
Nano- or ultrafiltration of inorganic salts in the presence of a polyelectrolyte in the feed solution is investigated in this work. The membranes are completely impermeable to the polyelectrolyte. At low concentrations of polyelectrolyte, a gel layer on the membrane surface is not formed. At such polyelectrolyte concentrations, the concentration of inorganic salt in the permeate stream can be higher than that in the feed solution. This salt concentration effect is the reverse of what is obtained with conventional membrane processes, where the permeate salt concentration is lower than, or equal to, the salt concentration in the feed solution. It is shown that, in the nanoor ultrafiltration of inorganic salts in the presence of a polyelectrolyte, the ratio of the salt concentration in the permeate to that in the feed is improved when the initial salt concentration in the feed solution is low. Concentration polarization has a negative impact on this concentrating effect. A theory that elucidates the observed phenomenon is presented, together with experimental data for potassium chloride solutions and N,N-dimethyl-N-2-propenyl-2-propen-1aminium chloride homopolymer filtered on an ultrafilter with a pore size of 5 nm. Introduction The phenomenon of salt concentration in the filtrate1 is further investigated in this work. An analysis of the literature available on this subject showed that similar effects for reverse osmosis were first reported in 1974 by Lonsdale and Push.2 In subsequent papers, the process was mathematically modeled on the basis of the Nernst-Planck equations for ion fluxes with the use of some phenomenological parameters.3-6 In this paper, the mathematical elucidation of the effect is based on a description of the salt concentrations on both sides of the membrane, determined by the Donnan equilibrium, and the process is shown to be effective with ultra- and nanofiltration membranes. Other factors influence the rejection ability of membranes with respect to various components,7-9 whereby the flow of an electrically neutral solution through a porous solid matrix gives rise to an electrical potential accelerating the motion of one ionic species while retarding that of the others. Bryk and Tsapyuk8 believed that electrostatic forces mediate a component occurring in solution in the dissociated state. This was substantiated by examples showing the effect of charges of solution components on the rejection of a low-molecularweight electrolyte. This influence can be also explained by the Donnan effect.9 Perry and Linder10 interpreted rejection of a negative salt in terms of the theory of reverse osmosis, developed by Spiegler and Kedem,11 with allowance for the Donnan exclusion correction in the driving force component responsible for salt transport. In the present paper, the basis of the previous theory12,13 is developed to provide a mathematical model * To whom correspondence should be addressed. Tel.: +44 (0)1509 222508. Fax: +44 (0) 1509 223923. E-mail:
[email protected].
Figure 1. Schematic representation of nano- or ultrafiltration process: 0, feed solution; 1, concentration polarization zone of thickness δ; 2, active layer of the membrane of thickness h; 3, permeate.
describing the transfer of a low-molecular-weight electrolyte through an ultrafiltration membrane in the presence of a polyelectrolyte that is completely rejected by that membrane. Let us consider the nano- or ultrafiltration of an inorganic salt, KCl, that is assumed to be a strong electrolyte and a water-soluble polyelectrolyte in the feed solution. The membrane is assumed completely impermeable to the polyelectrolyte molecules. The salt completely dissociates according to KCl f K++ Cl-. The concentrations of the salt and water-soluble polymer in the feed solution are c0+ and c0, respectively. A schematic representation of the process is given in Figure 1. The process can be rationalized as follows: (1) During ultrafiltration, potassium and chloride ions pass through the membrane. (2) As the chloride concentration decreases, the polymer chloride will dissociate to release more chloride ions. (3) Chloride ions passing through the membrane, originating from the polymer chloride complex, will be accompanied by potassium ions, to maintain electroneutrality, thus concentrating the potassium ions within the permeate.
10.1021/ie049696l CCC: $30.25 © 2005 American Chemical Society Published on Web 01/12/2005
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1359
The above scheme provides a simple mechanism by which the potassium will concentrate in the permeate and, in practice, all three of the above steps will occur simultaneously. Theory In the dissociation of a water-soluble polymer, superscript 0 is used below for the feed solution. Dissociation in the Feed Solution (Region 0 in Figure 1). It is assumed that each molecule of the polyelectrolyte has N chains (N ) 5 in Figure 2). Let i be the number of dissociated sites; then ei is the charge on this molecule, where e is the electron charge (i ) 2 in Figure 2). Let c0i (i ) 0, 1, ..., N) be the concentration of polyelectrolyte molecules with 0, 1, ..., N, dissociated sites. Let c0 be the total concentration of polyelectrolyte in the feed solution. Hence, applying conservation, one obtains
c00
+
c01
+ ‚‚‚ +
c0N
)c
0
Figure 2. Water-soluble polymer with N ) 5 sites capable of dissociation. Number of dissociated sites is i ) 3 in this figure.
Figure 3. Dissociation/association of an individual site.
Rearrangement of system 4 gives
q0 ) 0 q1 ) 0
(1)
-qN-1 ) 0
(5)
The reaction rates for the dissociation and association are
Introducing the dimensionless constant R ) kp/Kpc0-, we conclude from eqs 5 and 2
q0 ) -kpNc0 + Kpc0- c01
c01 ) NRc00 ) c00RC1N
q1 ) -kp(N - 1)c01 + Kpc0-2c02
c02 )
l 0 qi ) -kp(N - i)c0i + Kpc0-(i + 1)ci+1
(2)
where c0- is the unknown concentration of chloride ions in the feed solution (to be determined) and kp and Kp are the dissociation and association reaction constants, respectively (Figure 3). The concentration of chloride ions, c0-, is unknown because part of the ions come from the dissociation of KCl and the other unknown part comes from dissociation of the polyelectrolyte. Using eq 2, the unsteady-state association/dissociation reactions of the polymer can be described by the following system of equations
dc0 ) q0 dt
or
c0i ) c00RiCiN, i ) 1, 2, 3, ..., N where
CiN )
N(N - 1)‚‚‚(N - i + 1) i!
That is, all unknown concentrations c0i (i ) 1, 2, ..., N) are expressed via only one unknown concentration, c00, of undissociated polyelectrolyte molecules. To determine the unknown concentration c00, we substitute all expressions 6 into the conservation law (eq 1), which results in
or
l
0 c00(1 + RC1N + R2C2N + ‚‚‚ + RNCN N) ) c
(3)
Under steady-state conditions, all time derivatives vanish in eqs 3, resulting in the following system of algebraic equations
q)0 -q0 + q1 ) 0
or
c00(1 + R)N ) c0 That is, the concentration of undissociated polyelectrolyte molecules, c00, can be expressed via the total electrolyte concentration, c0, as
c00 )
l -qN-1 ) 0
(6)
0 c00 + c00RC1N + c00R2C2N + ‚‚‚ + c00RANCN N ) c
dc1 ) -q0 + q1 dt dcN ) -qN-1 dt
N(N - 1) N-1 NRc01 ) c00R2 ) c00R2C2N 2 2
(4)
c0 (1 + R)N
(7)
According to eqs 6 and 7, all other concentrations can
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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005
be expressed as
RiCiN c0i ) c0 , i ) 0, 1, ..., N (1 + R)N
(8)
Let us introduce two dimensionless values R0 ) k/KC0 and λ ) c0+/c0. Using these values, eq 14 can be rewritten as
RN R0 ) -λ 1+R R
Let us introduce an average value of dissociated sites z0 of the polyelectrolyte molecules in the feed solution as N
z0 )
ic0i ∑ i)1
Equation 15 is a quadratic equation for the determination of the unknown value of R, which can be easily solved. However, before doing so, we slightly transform eq 15. If we divide eq 12 by c0 and use eq 9, we arrive at the following equation
(9)
N
(15)
0
z -
c0i ∑ i)1
c0c0
+
c0+ c0
)0
or Substitution of expressions 7 and 8 into eq 9 gives N
z0 + λ )
N
iRiCiN ∑iRiCiN ∑ i)1 i)1
c0 z0 )
)
c0i (1 + R)N
(1 + R)N
(10)
From the latter expression, we can determine the unknown value of R as
ic0i + c0+ - c0- ) 0 ∑ i)1
(11)
Note that the concentration of chlorides, c0-, is to be determined. Substitution of eqs 7 and 8 into eq 11 results in
c0
R0
R)
The concentration of the electrolyte, KCl, is fixed in the feed solution and is equal to c0+. According to the electroneutrality condition N
R0 R
(16)
z0 + λ where z0 is also an unknown value. Substitution of eq 16 into eq 15 results in an equation for the determination of the unknown averaged charge, z0
z02 + z0(λ + R0) - NR0 ) 0
(17)
The solution to this equation, which should be positive, is
N
(1 + R)
N
iRiCiN ) c0- - c0+ ∑ i)1
(12)
z0 ) N
Let us calculate the sum
iRiCiN ∑ i)1
in the latter equation. According to the binomial law, (1 + R)N ) 1 + RC1N + 2 R C2N + ‚‚‚ + RNCN N. Differentiation of both sides of the latter expression with respect to R gives N-1 N(1 + R)N-1 ) C1N + 2C2NR + ‚‚‚ + NCN NR
or, multiplying by R N RN(1 + R)N-1 ) RC1N + 2C2NR2 + ‚‚‚ + NCN NR ) I
Hence
I ) RN(1 + R)N-1 )
RN (1 + R)N 1+R
(13)
Using eqs 12 and 13, we conclude
c0RN ) c0- - c0+ 1+R
λ 1+ 0+ R
x(
)
λ 1+ 0 R
2
4N + 0 R
0 zj + 1 D D(50)
c+(x) )
[
] ( )
j+ j+ β h exp νx + c+(0+) β hν D β hν
(59)
where Equation 50 shows that presence of polyelectrolyte ions in the concentration polarization region results in the enhancement of the convective transport of cations. Solution of eq 50 subject to the boundary condition
c+(-δ) ) c0+
(51)
results in the following expression for the cation con-
β h β h)D h D Note that, because of the concentration jump on the membrane surface,13 c1+(0-) * c+(0+). This jump is calculated below using the equality of chemical potentials at the membrane surface (see below).
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The two unknown values, c+(0+) and j+, are determined below. Equality of the chemical potentials at the membrane surface reads13
c1+(0-) ) c+(0+)γ+ exp(+∆φ1)
(60)
c1-(0-) ) c-(0+)γ- exp(-∆φ1)
(61)
where γ( ) exp(Φ h () represents the distribution coefficients and Φ h ( represents the dimensionless (in kT units) potentials of specific interactions of ions with the membrane material; ∆φ0 is the electrical potential jump across the membrane surface, x ) 0. The latter conditions should be combined with the electroneutrality condition
c+(0+) ) c-(0+)
(62)
After some rearrangement, we can conclude from eqs 60-62
[c1+(0-) c1-(0-)]0.5 c+(0+) ) γ
(63)
Using eq 59
c+(h-) )
1 γ
x
zjc0 exp
νδ Def
j+
ν(1 + ω)
[
+ c0+ -
j+ ν(1 + ω)
] [ exp
]
ν(1 + ω) δ D+
(64)
Equality of chemical potentials and flow conditions at the second membrane surface (x ) h) provides
c+(h-)γ+ exp(z+∆φ2) ) c3+
(65)
c-(h-)γ- exp(-∆φ2) ) c3-
(66)
where c3+, c3-, and ∆φ2 are the concentrations of cations and anions in the permeate and the electrical potential jump across the membrane surface, x ) h, respectively. The flow condition gives the following expressions for the concentrations in the permeate solution
c(3 )
j( ν
c+(h-) - c-(h-) ) 0
where γ j ) γ/R j and the concentration c+(0+) is given by eq 64, which should be substituted into eq 71. This gives the required equation for the determination of the unknown flux j+ j+ ν
)
( )x
x
νδ Def
zjc0 exp
Using eqs 65, 66, and 68 and rearranging gives
c+(h-) ) 0.5 where γ ) γ0.5 - γ+ .
j+ νγ
[
j+
+ c0+ -
ν(1 + ω)
1 + (γ j -1)(1 - e
(69)
j+ ν(1 + ω) (R j /D)νh
] [ exp
ν(1 + ω) δ D+
]
)
(72) Dividing both sides of this equation by c0+ and introducing a new unknown dimensionless cation flux, Λ ) j+/νc0+, eq 72 can be rewritten as
Λ2 )
[
] [
ν(1 + ω) Λ Λ exp δ + 1D+ (1 + ω) (1 + ω)
[
c0+
( )]
exp -
]
νδ [1 + (γ j -1)(1 - e(Rj/D)νh)]2 Def
The positive solution of the latter equation is Λ) 2(1 + ω)
x[1 - exp(-Pe)]2 + 4A exp(-Pe)(1 + ω)2 + 1 - exp(-Pe) A)
[
c0+
zjc
0
( )]
exp -
νδ [1 + (γ j -1)(1 - e(Rj/D)νh)]2 Def Pe )
(73)
ν(1 + ω) δ D+
where Pe is the outer Peclet number. Note that the rejection coefficient is R ) 1 - Λ. We refer below to Λ as the degree of concentration in the permeate solution or simply as degree of concentration. If the velocity, ν, tends to zero, that is, Pe , 1, then according to eq 73
Λ|νf0 )
(68)
(71)
1 + (γ j -1)(1 - e(Rj/D)νh)
(67)
Note that the superscript 3 denotres the permeate solution. The electroneutlality condition reads
(70)
νc+(0+)γ
j+ )
zjc0
( )x
]
From eqs 69 and 70, we find
0.5 where γ ) γ0.5 - γ+ . Using the expressions for the concentrations on the membrane surface from the feed solution side (eqs 49 and 52), eq 63 can be rewritten as
c+(0+) )
[
j+ j+ (Rj/D)νh c+(0+) e R jν R jν
zjc0 c0+
(74)
i.e., Λ is independent of the membrane rejection properties; increases with the degree of dissociation of the polymer, zj, and the polymer concentration in the feed solution, c0; and decreases with the concentration of cations in the feed solution. According to the assumptions, Λ in eq 74 is much greater then 1 (a negative rejection). If velocity ν f ∞, that is, Pe . 1, then we conclude using eq 73
Λ|νf∞ ) 1 + ω
(75)
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1365
According to the definition of ω
ω)
(
)
D+ 1 1 >0 zj + 1 D D-
To deduce the dependency of Λ on the filtration velocity, ν, the expression for A in eq 73 can be rewritten as follows
A) a)
[
c0+
zjc0
]
exp(-aPe) {1 + (γ j - 1)[1 - exp(-bPe)]}2 D+
(1 + ω)Def
, b)
ν(1 + ω) R j D+h , Pe ) δ D (1 + ω)δ D+
If γ j ) 1, that is, the membrane does not reject cations at all (but still completely rejects polyelectrolyte), then Λ according to eq 73 decreases from the maximum value given by eq 74, which is much greater than 1, to the minimum value given by eq 75. Filtration Velocity and Applied Pressure Difference. In this section, filtration through a membrane in the presence of a polyelectrolyte solution (without added KCl) at different concentrations is considered. The critical concentration of the polyelectrolyte, below which a gel layer on the membrane does not form, is deduced. The concentration of the polyelectrolyte on the membrane surface is given by eq 48, which can be rewritten as
c1m ) c0 exp(Pem)
(76)
where
Pem )
V ) Km[∆p - cm1RT] ) Km∆p - Kmc0RTePe where ∆p is the applied pressure difference and Km is the permeability of the active layer of the membrane. Thus 0 Vδ Km∆pδ Kmc RTδ ) exp(Pem) Def Def Def
or
Pem ) Pe0 - χ exp(Pem)
Figure 5. Inverse dependency as compared with Figure 4.
Equation 77 is a nonlinear equation for the determination of the dimensionless filtration velocity Pem. The latter equation can be rewritten as
Pe0 ) Pem + χ exp(Pem)
Vδ Def
The filtration velocity, taking into account the osmotic pressure, is
Pem )
Figure 4. Dimensionless pressure as a function of dimensionless filtration velocity according to eq 78.
(77)
that is, as a reverse function of the dimensionless pressure, Pe0, on the dimensionless filtration velocity, Pem. The latter dependency is shown in Figure 4. Rotation of this figure gives the required dependency of the dimensionless filtration velocity, Pem, on the dimensionless pressure, Pe0 (Figure 5). Note that the filtration velocity can be easily obtained using Figure 5 as
Def ν ) Pem δ The filtration process proceeds in the way described above until the concentration on the membrane surface (eq 76) reaches the concentration of gel-layer formation, cg. Figure 6 shows the procedure for the determination of the critical pressure
c0 exp(Pecr) ) cq
where
Pe0 )
Km∆pδ Def
is the dimensionless filtration velocity in the absence of concentration polarization and
Kmc RTδ Def
(79)
or
Pecr ) ln
cq c0
Using eq 79, the latter equation takes the form
0
χ)
(78)
Pecr + χePecr )
Kδ ∆p Def cr
(80)
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rejection to dissolved ions, which must be avoided for the concentration of solute within the permeate process to work. Experimental Section
Figure 6. Determination of the critical pressure difference.
Figure 7. Filtration velocity as a function of applied pressure difference.
or after substitution of eq 80
(
∆pcr ) ln
)
cq cq Def +χ c0 c0 Kδ
(81)
which determines the critical applied pressure. If ∆p > ∆pcr, then a gel layer forms
c1(-l) ) cg where l is an unknown thickness of the gel layer. Using eq 48, the equation becomes
cg ) c0eV(δ-l)/Def or
ln
cg V(δ - l) ) c0 Def
(82)
If the thickness of the gel layer is much less than the thickness of the concentration polarization layer, l , δ, then according to eq 82, the filtration velocity remains constant
νlim )
Def cg ln 0 δ c
(83)
That is, the dependency of the filtration velocity, ν, on the applied pressure has the form presented in Figure 7. In the following analysis, only pressures ∆p < ∆pcr are considered, because the cake provides an additional
All experiments were carried out in a dead-end filter cell with a volume of 350 mL, equipped with a magnetic stirrer. Filtration was performed using UFM-50 acetylcellulose membranes (Polimersintez Research Corporation, Vladimir, Russia) that were circular and had an active diameter of 68 mm. The stirrer was operated at a rate of 500 rpm that was maintained constant throughout the experiments. The UFM-50 membranes were preshrunk by filtering distilled water at pressures of 0.5-4.0 bar. The pretreatment was performed until a linear dependence of the filtration rate on the pressure drop across the membrane was obtained. Filtered compressed air was used in the filter cell to provide the pressure for filtration. Analytical-grade potassium chloride was used as a low-molecular-weight inorganic solute. A commercial synthetic water-soluble polymer, N,N-dimethyl-N-2propenyl-2-propen-1-aminium chloride homopolymer (9CI) [trademark ZETAG 7125, CIBA SC (WT) Bradford, West Yorkshire, U.K.), was used as the polyelectrolyte. The working solutions were prepared using a 450 g/L polymer solution and a 0.1 mol/L potassium chloride solution. The metal ion content in the experiments, c0+, varied from 0.4 × 10-3 to 3.0 × 10-3 mol/L. The polymer concentrations used were 0.1, 0.5, 1.0, and 5.0 g/L. The molecular weights of the polymer and the monomer unit were 138000 and 161, respectively. That is, the number of dissociable units was N ) 138000/161 ≈ 857. The corresponding molar concentrations of the polymer solutions, c0, were 7.25 × 10-7, 3.6 × 10-6, 7.25 × 10-6, and 3.6 × 10-5 mol/L. The metal contents in the solutions were determined by atomic absorption spectrometry (Varian). The concentration of polymer in solution was determined by the dry residue method with a drying temperature of 90 °C. Results and Discussion Theoretical predictions based on eq 73 were compared with four sets of experimental data. The first three sets for comparison were taken from ref 1, where experimental details (similar to those used in this paper) are presented. The molecular weights of the polymer and the monomer unit in ref 1 were 35000 and 161, respectively. That is, the number of dissociable units was N ) 35000/161 ≈ 217, which is substantially smaller than in our experiments. The degrees of concentration in the permeate solution of the three chlorides KCl, NaCl, and RbCl were reported in ref 1. Predictions obtained by applying the theory derived in this paper, i.e., eq 73, are compared with the experimental data in Figures 8-10. In all cases under consideration, inorganic salts were not rejected by the membrane (the average membrane pore size was more than 15 nm), so that γ j was set to zero in eq 73 and so was ω, in accordance with this observation. As can be seen from these figures, the agreement between the experimental data and the modeling presented above is good, for the condition when the membrane does not offer any rejection of the cation from the permeate.
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1367
Figure 8. Concentration of KCl solutions in the permeate. Experimental data from ref 1; theoretical curves according to eq 73.
Figure 9. Concentration of NaCl solutions in the permeate. Experimental data from ref 1; theoretical curves according to eq 73.
In the later set of tests, where the finer-pore membrane did provide some rejection of the cation, the first set of tests investigated how the permeate rate varied with filtration pressure in the presence of various concentrations of polymer solution. The results are reported in Figure 11. The solution temperature was maintained between 19 and 20 °C. From this figure, it is evident that, in all cases, the permeate rate increased with increasing pressure until a transmembrane pressure of about 2.5 bar. After this pressure was reached, the curves with polymer present tended to plateau, indicating the possibility of formation of a gel layer, or cake, on the membrane surface. As the effect of the concentration of solute in the permeate is dependent on the avoidance of a gel layer, subsequent experimental tests were performed at pressures less than 2.5 bar. A key variable in the modeling, eq 73, is the diffusion coefficient of the polymer, which was measured by an electrical conductivity technique to be 48.3 × 10-8 cm2/s for the experiments with significant cation rejection by the membrane and 200 × 10-8 cm2/s for the tests with
Figure 10. Concentration of RbCl solutions in the permeate. Experimental data from ref 1; theoretical curves according to eq 73.
Figure 11. Plots of the linear filtration velocity ν versus the transmembrane pressure drop for the ultrafiltration of ultrapure water and aqueous solutions of a mixture containing KCl and N,Ndimethyl-N-2-propenyl-2-propen-1-aminium chloride homopolymer (9CI). The initial polymer concentrations were c0p ) 0.1, 0.5, 1.0, and 5.0 g/L.
no significant membrane rejection. The diffusivity of cations was deduced from the literature. The negative rejection coefficients can be seen from the data presented in Figures 12 and 13 for the membranes that inherently reject the cations used in this study. The rejection, or concentration, is plotted as a function of the superficial velocity of the convective flow toward the membrane, which varies as a consequence of the operating pressure. The rejection of pure (polymer-free) KCl solutions on the initial membrane is shown in Figure 14, which illustrates that the membrane exhibits some significant salt rejection capability. For the numerical analysis, γ j is provided by
γ j)
1 1 - Rmax
where Rmax is the maximum rejection taken from Figure 14 for each experiment. In Figure 15, the measured data and the results predicted according to eq 73 are compared. The main
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Figure 12. Retention coefficient versus linear filtration velocity in aqueous solutions of a mixture containing KCl and N,Ndimethyl-N-2-propenyl-2-propen-1-aminium chloride homopolymer (9CI) ultrafiltration. The initial polymer concentrations were c0p ) 0.1, 0.5, 1.0, and 5.0 g/L.
Figure 13. Retention coefficient versus linear filtration velocity in aqueous solutions of a mixture containing KCl and N,Ndimethyl-N-2-propenyl-2-propen-1-aminium chloride homopolymer (9CI) ultrafiltration. The initial polymer concentration was c0p ) 0.5 g/L, and the initial concentrations of KCI were c0KCl ) 2.5 × 10-3 and 3.3 × 10-4 mol/L.
Figure 14. Retention coefficient versus linear filtration velocity in aqueous solutions containing different concentrations of KCl.
difference between the data and the model presented here and in Figures 8-10 is that, in this case, the membrane inherently rejects the cations, whereas in the earlier figures, the membrane was neutral toward the cations. The agreement between the model and experimental values is less good for the membrane demonstrating cation rejection, but it is still acceptable. Hence, the phenomenon of the concentration of salts in the permeate during ultra- and nanofiltration appears to be successfully modeled by the foregoing analysis, which can accommodate membranes that demonstrate partial cation rejection and zero cation rejection. However, the process is dependent upon the complete rejection of the
Figure 15. Comparison of experimental data and theoretical predictions made according to eq 73 with membranes showing an inherent rejection ability for potassium.
dissociating polymer species and the absence of a gel layer, or cake, on the membrane surface. Possible practical applications of the metal concentration phenomenon are quite wide, ranging from the separation of low-molecular-weight inorganic components to the partition of multicomponent mixtures of low-molecular-weight electrolytes. In all cases, it should be possible to recycle the dissociating polymer species using the ultra- or nanofiltration membrane. Thus, the technique provides another means for metal, or cation complexation, separation, and concentration. The separation, or concentration, efficiency is particularly favorable when the cation concentration is low, which suggests that it is best applied to the recovery or decontamination of streams containing low concentrations of metal ions. Conclusions It has been demonstrated that it is possible to concentrate cations within the permeate stream from an ultra- or nanofilter by using a partially dissociating polymer in aqueous solution. The degree of concentration in the permeate, compared to the remaining concentration in the feed, is several 100 percentsoften as high as 400%. The polymer remains retained by the membrane filter and can be reused within the process. A new modeling analysis has been presented and validated against previously published experimental work. In this case, the membrane was sufficiently open that it had no inherent rejection capability for the cations under investigation. The agreement between the model and the data was very good. In subsequent experimental work, performed with a finer-pore membrane that had a significant ability to reject cations, the concentration of cations in the permeate was still observed, despite the inherent ability of the membrane to reject the transfer. The model was suitably adapted to account for the inherent rejection of the membrane, and the agreement between data and model was less good but still provided a reasonable representation of the process. Thus, it appears that the important parameters influencing this process have been identified and quantified for single-component transfer studies. Further work will include multicomponent studies, both as experiments and modeling, to investigate the process as a means of separation as well as s simple technique for the concentration of dilute species.
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Acknowledgment P.P.’s research is supported by a Development Fund award from Loughborough University. The authors also acknowledge a NATO CLG grant award that contributed to part of the work reported here. We are grateful to Dr. S. Kosvintsev for his advice and fruitful discussions, which have helped us considerably during this project. Literature Cited (1) Starov, V. M.; Filippov, A. N.; Volchek, K. A.; Gagarinskaja, I. L.; Tsezura, E. S.; Kaj, H. S.. Concentration of Inorganic Low Molecular Weight Components of Solutions in Filtrate in Ultrafiltration Process at the Presence of Polyelectrolytes. Chem. Technol. Water 1991, 13 (2), 116. (2) Lonsdale, N. K.; Push, W. Donnan-membrane effects in hyperfiltration of ternary systems. J. Chem. Soc., Faraday Trans. 1 1975, 71, 501-514. (3) Vonk, M. W.; Smith, J. A. M. Thermodynamics of ternary systems in reverse osmosis. Desalination 1983, 48, 105. (4) Vonk, M. W.; Smith, J. A. M. Positive and negative ion retention curves of mixed electrolytes in reverse osmosis with a cellulose acetate membrane. An analysis on the basis of the generalized Nernst-Planck equation. J. Colloid Interface Sci. 1983, 96 (1), 121. (5) Gorskii, V. G.; Starov, V. M.; Churaev, N. V. Reverse Osmosis Separation of Ternary Electrolyte Solutions. Colloid J. 1988, 50 (3), 379 (English Translation).
(6) Fridrikhsberg, D. A. Kurs kolloidnoi khimii (Texbook of Colloid Chemistry); Khimya: Leningrad, U.S.S.R., 1984. (7) Helfferich, F. Ionenaustauscher. Band I. Grundlagen Struktur-Herstellung-Theorie; Verlag Chemie GmbH: Weinheim, Germany, 1959. (8) Bryk, M. T.; Tsapyuk, E. A. Ultrafiltration; Naukova Dumka: Kiev, 1989. (9) Migalatii, E. V.; Tarasov, A. N.; Pushkarev, V. V. Effect of Membrane Charge on the Semipermeability Properties. In Abstracts of Papers. The 4th AllsUnion Conference on the Methods of Membrane Separation of Mixtures; Moscow, 1987; Part 1, p 46. (10) Perry, M.; Linder, C. Intermediate reverse osmosis ultrafiltration (RO UF) membranes for concentration and desalting of low molecular weight organic solutes. Desalination 1989, 71, 233. (11) Spiegler, K. S.; Kedem, O. Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes.Desalination 1966, 1, 311-326. (12) Dorokhov, V. M.; Martynov, G. A.; Starov, V. M.; Churaev, N. V. Substantiation of the Selection of a Calculation Model of a Reverse Osmosis Membrane. Colloid J. 1984, 46 (2), 202 (English Translation). (13) Starov, V.; Churaev, N. Separation of Electrolyte Solutions by Reverse Osmosis. Adv. Colloid Interface Sci. 1993, 43, 145.
Received for review April 14, 2004 Revised manuscript received December 7, 2004 Accepted December 10, 2004 IE049696L
