Simulation of a continuous metal separation process by polymer enhanced ultrafiltration

Journal of Membrane Science 268 (2006) 37–47

Simulation of a continuous metal separation process by polymer enhanced ultrafiltration J. Sabat´e a,∗ , M. Pujol`a a , J. Llorens b a

Departament d’Enginyeria Agroaliment`aria i Biotecnologia, Universitat Polit`ecnica de Catalunya, Edifici ESAB, Avinguda del Canal Ol´ımpic s/n, 08860-Castelldefels, Spain b Department of Chemical Engineering, Universitat de Barcelona, Mart´ı i Franqu` es 1, 08028 Barcelona, Spain Received 22 October 2004; received in revised form 14 May 2005; accepted 17 May 2005 Available online 28 July 2005

Abstract The separation of metals from aqueous streams by continuous polymer enhanced ultrafiltration (PEUF) was simulated in order to understand, evaluate and optimize the process feasibility. The model allows one to examine the influence of physico-chemical and operation variables on the metal reduction and productivity of the treated water stream. For a given metal–polymer system, the computations revealed that the most influential operation variables are the acid and base reagents expended on the process, the amount of polymer used and the recycling stream flow, which are represented by the dimensionless parameters af, pp and ro, respectively. Two of them can be free chosen while the third one is determined by a fixed treated water production and metal reduction. The selection of af and pp values should be a compromise between the costs of the reagents to regenerate the polymer and the energy spent to achieve a permeate flow. If process efficiency requirements are more exigent, higher values of af or pp are required. © 2005 Elsevier B.V. All rights reserved. Keywords: Continuous ultrafiltration; Complexation; Heavy metal; Polymer; Simulation

1. Introduction Polymer enhanced ultrafiltration (PEUF) has been proposed as a feasible method to separate a great variety of metal ions from aqueous streams [1–5]. Ultrafiltration (UF) membranes reject polymer molecules and small species bound to them. A diluted permeate that can be discharged as waste or employed for a specific purpose is obtained. Meanwhile, a retentate stream with a high concentration of metallic ions bound to the polymer is also produced. To make the process economically competitive, these ions must be released from the polyelectrolyte in order to reuse the polymer. This can be done by the addition of acid, which introduces protons that compete with the metal ions for binding sites, followed by a second ultrafiltration step which produces a retentate with a high concentration of metallic ∗

Corresponding author. Fax: +34 935521001. E-mail address: [email protected] (J. Sabat´e).

0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.05.028

ions whilst all the polymer is left protonated. The addition of a base to this polymer returns it to its original form, allowing it to bind new metal ions. In fact, this method works as a conventional ion exchange process where the diffusion limitations are drastically reduced. Several studies analyzing the entire PEUF process, reusing the polymer, have been reported. Most of them deal with the analysis of batch [6–8] or semi-continuous processes [9], but the feasibility of a continuous PEUF process applied to water softening [10] and heavy metal separation [11,12] has also been studied. The aim of the present paper is to analyze extensively the running of a continuous PEUF applied to the separation of heavy metals that join a polyelectrolyte by complexation binding. Simulations can help understand the way that the process works and determine the conditions required to achieve a given treated water composition, saving reagents and energy. As with any industrial practice, the efficiency of the process of metal separation by PEUF can be evaluated according to the productivity and quality achieved. In this

38

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

case, the productivity is determined by the ratio between the flows of the treated water stream and the raw water streams, whilst the quality can be described by the ratio of the heavy metal concentrations in the two streams. Evaluating this efficiency requires the knowledge of both the affinity between metal and polymer and the influence of the operation variables on the membrane separation steps and the compositions of the different streams. Thus, both the physico-chemical and engineering aspects of the process should be taken into account. The affinity between metal and polymer has been quantitatively evaluated measuring complex formation constants. However, the pKa of the ligand group of the polymer, and the formation of soluble metal hydroxocomplexes or insoluble hydroxides or salts should also be considered [3,13–17]. For a given metal–polymer pair, where all these equilibrium constants are fixed, the metal rejection by an UF membrane is essentially determined by the total concentrations of polymer and metal and the pH. As a rule, metal rejection increases with pH and polymer–metal concentration ratio. Further, when the polymer–metal concentration ratio is fixed, the higher the metal concentration, the higher the metal rejection achieved [15,16]. Meanwhile, the engineering parameters that determine the running of the process are the following: the acid and base reagents spent to regenerate the polymer, the amount of polyelectrolyte present in the system and the flows of the streams that connect the two membrane processes [10]. As the quantitative effects of those parameters on process efficiency are interconnected and also depend on the physico-chemical variables, an extensive arrangement of algebraic equations must be raised to describe the system. In this study, we model two situations. Firstly, we model a specific case studied by the authors in a previous work [16]: the continuous separation of cadmium using chitosan as a complexation reagent. Chitosan is a biodegradable and natural compound that contains amine groups able to complex a great variety of metal ions [18–25]. After that, we model the general case of a metal ion and a polymer with amine groups, where the values of complexation and acidity constants are varied by several orders of magnitude and their influence analyzed.

Fig. 1. Flow sheet for a continuous metal separation using polymer-assisted unites.

with an acidic stream (HCl), A, and the stream that comes from the retentate of the ultrafiltration unit of Section 1. It produces a water stream with a high metal concentration, P2, and a retentate stream that recycles the polyelectrolyte back to Section 1. Section 2 is acidic. The efficiency of the process can be evaluated by monitoring two variables: the quantity and the quality of the treated water. P1/Fo determines the productivity. To establish the quality of the treated water, metal reduction, MR, was defined as follows:
 CMP1 MR = 1 − 100 (1) CMFo when MR = 100, the treated water has zero metal; when MR = 0, the treated water has the same concentration of metal as the raw feed water. Given a feed stream, Fo, and its metal concentration, CMFo , there is a series of independent dimensionless parameters that determine the running of the process. These parameters and their meanings are summarized in Table 1. Process productivity is determined by p = P1/Fo. For a given value of ab, af is proportional to the amount of acid and base spent in order to continuously regenerate the polymer and therefore proportional to the cost of the chemicals used in the process. pp is the ratio between the concentration of the polymer concentration present in the system and the metal concentration in the feed stream. A high value of pp improves the complexation of the metal but increases the Table 1 Independent dimensionless parameters that determine the running of the continuous PEUF process Parameter

2. Continuous process Fig. 1 shows a general flow sheet for the continuous metal membrane separation process. The system has two similar sections, each consisting of an ultrafiltration apparatus with a tank. Section 1 is fed with raw water, Fo, a base stream (NaOH), B, and a polyelectrolyte stream that comes from Section 2. At the same time, Section 1 produces a treated water stream, P1, where the metal content is very low and a retentate stream that feeds Section 2. The pH in Section 1 should be high enough to ensure the polymer remains unprotonated and able to bind the main metal ions. Meanwhile, Section 2 is fed

ab af p pp r1 r2 ro

Equivalence ACA BCB ACA FoCMFo P1 Fo NP /VT CMFo PR1 R1 RR2 R2 RRP Fo

Meaning Acid–base feed ratio Acid–metal feed ratio Production of treated water Polymer–metal ratio Recycled retentate flow in Section 1 Recycled retentate flow in Section 2 Global recycling

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

viscosity of the solutions and permeates resistance to flow. ro represents the recycled polymer stream coming from Section 2 to Section 1. As ro increases, the global system becomes more mixed and the compositions in Sections 1 and 2 tend to one another. The global acid–base feed ratio, ab, must be close to 1 for a process with neutral out-streams.

3. Simulations 3.1. Cadmium–chitosan The PEUF separation of cadmium ions using chitosan as a complexing polymer has previously been studied by the authors [16]. Additional experimental data about adsorption isotherms and kinetics for cadmium(II) on chitosan can be found on the literature [18–25]. In order to understand, evaluate and optimize the feasibility of the continuous PEUF for cadmium separation, a mathematical model was developed with three sections: (1) the mass balance to determine the stream concentrations and flow rates; (2) the equilibrium of metal–polymer complexation; (3) the rules for UF membrane separation. The mass balance equations are described in Appendix A. They describe the global, metal, sodium and chlorine ion mass balances in the different parts of the system (see Fig. 1). Mass balances are completed with electroneutrality in the process streams. It is assumed that chemical equilibrium is reached in both sections. The relative quick stabilization of the pH in the ultrafiltration runs suggests fast kinetics [16]. On the other hand, it is always feasible to reach the equilibrium by providing sufficient residence time by increasing the volume in both sections of the system. In agreement with the experimental UF results, a twophase mathematical model was developed and described by Llorens et al. [16]. The model considers that a polymer solution has two differentiated regions: the polymer region and the remaining solution. The polymer region is occupied by the polymer itself, its bounded ions and some water. The formation of two complexes involving Cd2+ and the amine group of the polymer were considered: Cd(R–NH2 )k 2+ , (k = 1, 2). The estimated values of the formation constants were log β1 = 2.72 and log β2 = 3.25. According to the literature [26], four soluble cadmium hydroxide complexes exist, Cd(OH)l (l = 1–4), and their formation constants are: log βOH,1 = 3.4, log βOH,2 = 7, log βOH,3 = 9.6, log βOH,4 = 11.3. The apparent log Kps for Cd(OH)2(s) was −14.51. A titration measurement yielded a value of 6.1 for the pKa of the chitosan amine group. These values of equilibrium constants are used throughout the continuous separation simulation. The equations for the equilibrium model are summarized in Appendix B. The behaviour of the membrane separation is based on the assumption that the polymer and ions bound to the polymer are completely rejected by UF membranes while small

39

ions leak freely through the membrane. Nevertheless, the rejection of the most abundant counter-ion in the polyelectrolyte, chloride, is governed by the electroneutrality of the permeate streams. As polymers have a molecular weight distribution, a total rejection of the polymer by the UF membrane can only be achieved by a previous dialyzation in order to eliminate the shortest chains [16]. Moreover, after some running time, only the constant retainable part of the polymer will remain in the system. Therefore, at steady state, only some degraded polymer fraction could pass throughout the membranes. Equations for the membrane separations are also described in Appendix B. To run a simulation, the following data were considered: fixed values of the dimensionless parameters described in Table 1, the concentrations of the streams entering the system (CMFo , CA and CB ) and the equilibrium constants for the chemical reactions. Afterwards, the coupled system of equations for mass balances, chemical equilibriums and membrane separations was implemented in the programme “Mathematica® ” (Wolfram Research) and iteratively solved. The outputted results were the stream flow rates and compositions and therefore the final value for MR. The concentrations in the acid and base feed flows, CA and CB , were fixed at 12 and 19 mol L−1 , respectively, simulating normal commercial concentrations. Fo = 100 L time−1 was taken as a basis in all runs. In order to analyze the effects of dimensionless parameters on process efficiency, their values were realistically ranged. The recycled retentate streams in both sections, r1 and r2, were fixed at 0.99. This value seems reasonable given we want high tangential velocity to reduce concentration polarization. Trials with small changes to r1 and r2 did not produce significant differences in process efficiency. The global acid–base feed ratio value, ab, was fixed at 1. As high production is desirable, only p values over 0.9 were taken. After all these considerations, the open dimensionless parameters to be analyzed for their effect on efficiency are: ro, af and pp. 3.1.1. Simulation results A first set of simulations was run by fitting the following parameters: p = 0.9 and ab = 1. The cadmium chloride concentration in the feed stream was 10−4 mol L−1 (11.2 mg Cd L−1 ). The metal reductions calculated are shown in Fig. 2. af values are presented along the X-axis so as to more easily fix reagents expenditures. It should be remembered that chitosan cannot prevent the formation of insoluble cadmium hydroxide or carbonate at pH values beyond a point between 8.5 and 9 (the exact value depend on polymer and cadmium concentrations) [16]. Therefore, all the graphics dealing with the cadmium–chitosan system are restricted to conditions where no insoluble products evolve. This was tested by the fulfillment of Eq. (B.4). For any value of pp and ro, there is an upper limit value for af over which the pH in Section 1 is high enough to induce precipitation. Fig. 2 shows that metal reduction increases with af. This means that efficiency can be

40

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

Fig. 2. Effect of pp, ro and af on metal reduction, MR. p = 0.9, ab = 1 and CMFo = 10−4 mol L−1 . (
) pp = 250; ( ♦) pp = 125; ( ) pp = 50. Open symbols: ro = 0.2. Full symbols: ro = 0.05.

increased if more reagent is expended, as one would expect. However, if af is fixed, a decrease of ro improves the efficiency of metal separation. Obviously, efficiency depends on the extent of metal complexation and decomplexation in Sections 1 and 2. In this way, any condition that leads to a high unprotonated polymer concentration in Section 1 improves their retention and produces nearly cadmium-free water in stream P1. In contrast, a low quantity of unprotonated polymer in Section 2 helps metal ions leak through membrane 2. We can understand the running of the process by analyzing the way that ro and af affect polymer concentration, the metal concentration and the pH in Sections 1 and 2. Fig. 3 illustrates the simulated total polymer concentration in R1 and R2 over a wide range of af values and two values of ro. It can be seen that, with a low ro, the polymer tends to concentrate in Section 2, diminishing concentration in Section 1 and hindering metal separation. However, increasing the recycling parameter, ro, polymer concentrations in the two sections come closer together. Moreover, the amount of acid and base added does not affect chitosan concentration. This is because the acid and base streams, A and B, are much lower than Fo and any change in them hardly modifies the total incoming flow and polymer concentrations. The effect of ro and af on pH can be seen in Fig. 4. For a given value of ro, an increase in af produces a drop in pH2 and a rise in pH1 , which leads to the polymer being present in the appropriate forms in both sections—unprotonated in membrane 1 and protonated in membrane 2. That explains the increase in metal reduction with af for all curves in Fig. 2. Furthermore, for low values of ro, the drop in pH2 and the increase in pH1 is achieved at lower values of af. This can be understood by taking into account that when ro is aug-

Fig. 3. Effect of af and ro on retentate polymer concentration. p = 0.9, pp = 250, ab = 1 and CMFo = 10−4 mol L−1 . (
) ro = 0.05; ( ) ro = 0.2. Open symbols: Section 2. Full symbols: Section 1.

mented, not only polymer but also all the chemical species, in particular protons, are brought from Section 2 to Section 1. The total cadmium concentration in R1 and R2 is also affected by ro and af (Fig. 5). The same explanation given for polymer concentrations is applicable to cadmium concentrations: cadmium concentrations in R2 are higher than in R1 and these differences are affected by ro. These differences are also diminished by increasing af and they become

Fig. 4. Effect of af and ro on pH. p = 0.9, pp = 250, ab = 1 and CMFo = 10−4 mol L−1 . (
) ro = 0.05; ( ) ro = 0.2. Open symbols: Section 2. Full symbols: Section 1.

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

41

increasing the amount of polymer used increases the amount of acid–base addition required to return the polymer to its suitable form, thereby achieving a given value of MR. The results of these computations illustrate that values of continuous cadmium reduction near to 100%, with high productivity, cannot be expected using a reasonable amount of acid and base. It suggests that the affinity between cadmium and chitosan is not strong enough. Therefore, simulations using other magnitudes of complexation and acidity constants will be necessary to test the feasibility of the separation of a target metal by PEUF. This will be developed in the next part. 3.2. A general case

Fig. 5. Effect of af and ro on retentate metal concentration. p = 0.9, pp = 250, ab = 1 and CMFo = 10−4 mol L−1 . (
) ro = 0.05; ( ) ro = 0.2. Open symbols: Section 2. Full symbols: Section 1.

insignificant when metal rejection in membrane 2 is small (Fig. 6). In these conditions, because of the large cadmium leakage in membrane 2 and the resulting low cadmium return, CMR2 and CMR1 are kept low. pp has an influence qualitatively similar to ro: a large amount of polymer contributes to enhance the maximum MR because more complexing agent is present (Fig. 2). However, the buffering capacity of the amine groups can also be seen, as

Fig. 6. Effect of af and ro on cadmium rejection in membranes 1 and 2. p = 0.9, pp = 250, ab = 1 and CMFo = 10−4 mol L−1 . (
) ro = 0.05; ( ) ro = 0.2. Open symbols: membrane 2. Full symbols: membrane 1.

In this part, simulations are carried out in a generic system of a metal and a complexing polymer with amine ligand groups. Because of the differences between metals and their differing tendencies to interact with hydroxyl ions, the formation of precipitates or soluble hydroxocomplexes will be neglected. Thus, the polymer–metal system will be defined by the formation of complexes and the amine group acidity constants. Because soluble hydroxocomplexes compete with amine polymer groups for the complexation of the metal ions, the resulting curves will be the upper limit of MR that can be achieved with a given polymer–metal pair. For the sake of simplicity, only complexes with one amine ligand are simulated. Bearing this last restriction in mind, the two-phase model proposed by the authors and the classical one-phase model are equivalent [16] and β1 values taken from the literature can be directly used with the described program. If complexes with more ligands were to be considered, the “chemical equilibrium” equations would have to be slightly modified in order to apply the usually reported constants. 3.2.1. Effect of β1 Fig. 7 shows the effect of the complexation constant (β1 ), af and ro on the quality of the treated water. For any typical curve, several zones are differentiated. In the first zone, for low values of af, MR increases quickly with af. For high values of af, MR tends asymptotically to an upper limit value, MRmax . Obviously, a transition zone is also present. In practice, the last zone can be seen as a plateau where the acid–base added is more than that necessary to protonate quantitatively the amine group in Section 1 and quantitatively unprotonate it in Section 2. At this point, the extra addition of acid and base does not significantly improve the quality of the treated water but increases the operating costs. Two log β1 values of 3 and 5 were tested. Log β1 = 5 is high enough to reach MR values close to 100%. However, when log β1 = 3, the maximum achievable value of MR is lower than 30%. These figures clearly demonstrate that the magnitude of the complexation constant is the most important parameter in determining the feasibility of the process. The use of a polymer with a strong affinity for the relevant metal not only allows one to use lower polymer concentrations, thus making the ultrafiltration faster, but it also enhances the value

42

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

Fig. 7. Effect of β1 and ro on metal reduction, MR. p = 0.9, pp = 50, ab = 1, CMFo = 10−4 mol L−1 and pKa = 6. (
) Log β1 = 3, ro = 0.2; () log β1 = 3, ro = 0.05; () log β1 = 5, ro = 0.2; () log β1 = 5, ro = 0.05.

of the achievable MR and reduces the amounts of reagents required. 3.2.2. Effect of ro, pp and p When curves in Fig. 7 corresponding to the same β1 values but different ro values are compared, it can be seen that an augmentation of ro produces a decrease of the slope at low values of af and the value of af at which the MR plateau commences is higher. However, higher values of MR are attainable. This behavior can be explained using the description given previously for the cadmium-chitosan system. There are particular points where one value of af produces the same value of MR for both values of ro (0.2 and 0.05). Qualitatively, the effect of pp on MR is similar to that described for ro in the previous paragraph (Fig. 8). For concentrated polymer solutions, that is with high values of pp, the values of af where the MR plateau commences are higher but the maximum achievable MR is also higher. If a large amount of acid and base are added, the presence of a high concentration of ligand groups enhances the complexation of the metal in Section 1 and release in Section 2, thereby improving metal separation. Nevertheless, if a small amount of acid and base are used, the buffering capacity of the amine group results in a decrease in metal reduction with respect to that calculated with the same chemical costs but lower amounts of polymer. As usual, holding other parameters constant, an increase of productivity, p, results in a detriment of quality (Fig. 8). The curves corresponding to higher values of p provide smaller values of MR and this is more drastic when less polymer is present. It can be seen that for a pp value of 250, little difference is observed between the curves corresponding to

Fig. 8. Effect of pp and p on metal reduction, MR. ro = 0.05, ab = 1, CMFo = 10−4 mol L−1 , log β1 = 5 and pKa = 6. (
) p = 0.98, pp = 250; () p = 0.9, pp = 250; () p = 0.98, pp = 50; () p = 0.9, pp = 50.

p = 0.9 and 0.98. However, a severe drop of MR is produced by an increase of productivity when pp = 50. 3.2.3. Effect of metal concentration in the feed stream If the polymer–metal ratio, pp, is kept constant, metal reduction becomes worse when metal concentration in the feed stream is lower (Fig. 9). This is a direct consequence of

Fig. 9. Effect of metal concentration in the feed stream on metal reduction, MR. p = 0.9, ro = 0.05, pp = 50, ab = 1, log β1 = 5 and pKa = 6. (
) CMFo = 0.2 × 10−4 mol L−1 ; () CMFo = 10−4 mol L−1 ; () CMFo = 5 × 10−4 mol L−1 .

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

the mass action law for a complexation reaction (Eq. (B.1)) as already outlined in the introduction. 3.2.4. Effect of pKa and ab Altering ab allows one to modify the pH of both sections. For instance, a high ratio of acid–base lowers both pH1 and pH2 . Therefore, ab could help set suitable pHs in both sections of the system. As those appropriate pHs values depend on the amine group pKa , the two parameters will be analyzed together. The pKa of the polymer amine groups is influenced by the neighbouring functional groups and polymer chain configuration. In many cases, the pKa value is not constant but it varies with pH. As the pH goes down, the number of protonated amine groups increases and makes more difficult the binding of other protons. This produces a decrease of the observed pKa with the pH and it leads to more acid and base being needed for continuous metal separation by PEUF. Nevertheless, in this study, for the sake of simplicity, pKa will be assumed to be constant and its effect on MR is shown in Fig. 10. A polymer with a strong affinity for the metal, log β1 = 5, and a pp value that leads to a MR value close to unity has been chosen. Similar results are obtained for pKa values between 10 and 6 but the system behaves worse for a pKa value of 4. This difference cannot be attributed to the buffering capacity of the polymer, but rather to the high amount of acid required to achieve a quantitative protonation of polymer at such a low pKa . This involves a sharp difference in the rejection of metals in membrane 2. For pKa values between 6 and 10, the rejection of metal in membrane 2, RM2 , is low enough, at moderate values of af, to keep low the transport of metal to Section 1 in the RRP stream. In contrast,

Fig. 10. Effect of pKa and ab on metal reduction, MR. ro = 0.05, p = 0.9, pp = 50, log β1 = 5 and CMFo = 10−4 mol L−1 . (
) pKa = 4, ab = 1; () pKa = 6, ab = 1; () pKa = 8, ab = 1; () pKa = 10, ab = 1; () pKa = 6, ab = 0.5; (♦) pKa = 6, ab = 2.

43

at moderate values of af, when pKa = 4, RM2 is too high and much metal passes from Section 2 to Section 1, hindering the metal reduction process. A modification of the acid–base ratio was simulated in order to test its effect on system efficiency. For any pKa , taking ab = 1 as a reference value, small changes of ab cause nearly no variation of MR. However, moving far away from ab = 1 (values of ab = 0.5 or 2.0) produces a decline in MR. This effect is shown in Fig. 10 for the particular case of pKa = 6. 3.2.5. Isoefficiency curves The simulations discussed above were carried out fixing the different dimensionless parameters and calculating the obtained metal reduction. Nevertheless, the purpose of this study is to help determine the suitable parameters to achieve metal reduction for a given production. Therefore, another type of simulation was perfiormed: MR and p were fixed and the operation parameters ro, af and pp calculated. In fact, specifying two of these values fixes the third. This allows one to generate a map where the values of the operation parameters (ro, af and pp) that give the desired production and quality can be visualized usefully. A typical result of this simulation for given values of equilibrium constants (log β1 = 5, pKa = 6) and metal concentration in the feed stream (CMFo = 10−4 mol L−1 ) is drawn in Fig. 11. A set of “isoefficiency curves” are obtained, one for every value of pp, with MR = 95% and p = 0.9. Each isoefficiency curve has a ro threshold value (rot ), below which equations have no solution. That indicates that, for any pp specified, a minimum value of ro is needed to reach the desired production and quality. rot is the value of ro for which the maximum MR is the desired MR. This is

Fig. 11. Isoefficiency curves: MR = 95 and p = 0.9. pKa = 6, log β1 = 5 and CMFo = 10−4 mol L−1 , ab = 1, (
) pp = 100; () pp = 70; () pp = 50; () pp = 35.

44

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

illustrated comparing the curves corresponding to pp = 50 in Figs. 7 and 11. Fig. 7 shows that, for ro = 0.05 (open triangles), a value of MR = 95% will never be reached but it can be exceeded using ro = 0.2 (full triangles). Fig. 11 (black triangles) shows that rot = 0.063, a value in between 0.05 and 0.2. Further, the higher the polymer concentration, the lower the rot value. rot is a limiting value which can be operated because it leads to high costs. More interesting is the minimum of the isoefficiency curves (coordinates: romin , afmin ) which describes the operation conditions that optimize the consumption of acid and base. On the one hand, the rise of af in Fig. 11, when ro > romin , is motivated by the influence of ro on the slope of the first zone of the graphic MR versus af (Fig. 7). On the other hand, the sudden rise of af in Fig. 11, when ro < romin , corresponds to ro values where the desired MR is just achieved in the plateau region of the graphic MR versus af (Fig. 7). This limiting value of MR leads to an elevated value of af. The value of afmin is inversely affected by pp (Fig. 12). It should be noted that the increment of afmin is moderate for high values of pp, but it is strong when a low amount of polymer is used. Nevertheless, beyond a certain pp value, the consumption of reagents cannot be significantly diminished. It can be seen that there is a minimum af below which a desired production and quality cannot be achieved independent of pp. If production or quality demands are more stringent, isoefficiency curves move to higher values of af and ro (Fig. 13). It should be noted that the extent of the shift is moderate for high values of pp, but it is strong when a low concentration of polymer is used. This observation could already have been determined from the drop of the maximum achievable value

Fig. 13. Isoefficiency curves. Influence of production and metal reduction. pKa = 6, log β1 = 5 and CMFo = 10−4 mol L−1 , ab = 1. (
) p = 0.9, MR = 95; ( ) p = 0.9, MR = 98; ( ♦) p = 0.95, MR = 95. Open symbols: pp = 70. Full symbols: pp = 100.

of MR when pp decreases in plots of MR versus af (Fig. 8). This is corroborated by comparing afmin for different quality requirements (Fig. 12). As expected, better water quality requires higher spending on reagents. In summary, after the production and quality of treated water is fixed, a great variety of values of operations parameters (ro, af and pp) can be selected. In fact, if two of them are fixed the third one is determined. Although it would be convenient to spend little on all reagents (acid, base and polymer), the selection of af and pp values should be a compromise between these costs. Drop pp and af must be increased. Another factor to take into account is the stability of the process against small variations in operating conditions. As a result, values of ro and pp must be chosen that produce af values a little bit higher than afmin , where af is less sensitive to ro and pp.

4. Conclusions

Fig. 12. Influence of pp and MR on af minimum. pKa = 6, log β1 = 5, CMFo = 10−4 mol L−1 , p = 0.9, ab = 1. (
) MR = 98; () MR = 95; () MR = 90.

The proposed continuous metal separation process by polymer enhanced ultrafiltration can produce good quality water, at high production rates, using a convenient polymer–metal system by fixing the appropriate operating variables. Two outcome variables are important: the maximum reachable metal reduction, MRmax , and the minimum expenditure of chemicals afmin . The effects of the polymer–metal system on MRmax and afmin determined by log β1 and pKa . The most important variable is log β1 whose value directly affects MRmax . The effects of the operating variables, pp, ro, and p on MRmax and afmin are varied and interrelated. Variables pp and ro have similar

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

effects on MRmax . An increase in either allows one to reach a higher value of MRmax . The value of afmin is inversely affected by pp. As expected, variable p reduces MRmax and increases afmin . However, it should be pointed out that the intensity of the effect of any of these operating variables on MRmax and afmin depends on the values of the rest of them. As a result, the generation of isoefficiency curves is shown to be helpful in deciding the values of operation parameters that lead to the stable attainment of a treated water at a desired quality and production.

Acknowledgements The authors are grateful to the Spanish Ministerio de Ciencia y Tecnolog´ıa (Projects PPQ2002-04115-C02, 01 and 02) for funds received to carry out this study.

Nomenclature A a ab af afmin B b c ij

CA CB Cij

CPj F1 F2 Fo Ka MR MRmax NP p P1 P2

acid feed flow, (L time−1 ) dimensionless acid feed flow acid–base feed ratio acid–calcium feed ratio minimum af needed to reach a desired production and quality of out-stream base feed flow (L time−1 ) dimensionless base feed flow equilibrium molar concentrations of species i in the solution phase of volume V in the stream j (mol L−1 ) equivalent acid concentration in the acid feed flow (eq L−1 ) equivalent base concentration in the base feed flow (eq L−1 ) analytical molar concentrations of species i within the entire solution of the stream j (mol L−1 ) total (free + protonated + complexed) amine group concentration in the stream j (mol L−1 ) feed flow in the membrane of Section 1 (L time−1 ) feed flow in the membrane of Section 2 (L time−1 ) raw water feed flow (L time−1 ) polyelectrolyte acidity constant metal reduction maximum reachable metal reduction total number of polyelectrolyte equivalent units introduced in the device (eq) dimensionless production of treated water product flow of treated water (L time−1 ) product flow of calcium concentrated water (L time−1 )

pKa pp qij

r1 R1 R2 r2 Rij RM ro rot RR1 RR2 RRP V VP VT

45

−log Ka polymer–metal ratio equilibrium molar concentration of species i in the polyelectrolyte phase of volume VP in the stream j (mol L−1 ) dimensionless recycled retentate flow in Section 1 retentate flow in Section 1 (L time−1 ) retentate flow in Section 2 (L time−1 ) dimensionless recycled retentate flow in Section 2 membrane rejection of specie i in membrane j rejection of metal in a membrane dimensionless recycled polyelectrolyte minimum ro needed to reach a desired production and quality of out-stream for a fixed pp recycled retentate flow in Section 1 (L time−1 ) recycled retentate flow in Section 2 (L time−1 ) polymer recycled flow (L time−1 ) volume of the solution phase (L) volume of the polyelectrolyte phase (L) total volume of Sections 1 and 2 (L)

Greek symbols βk Stability constant of a complex formed by a metal ion and k amine groups βOHl Stability constant of a soluble complex formed by a metal ion and l hydroxyl ions

Appendix A. Mass balances The projected separation unit has been divided into five regions in order to facilitate the mass balances (Fig. 1). The global mass balances are: Fo + B + RRP + RR1 = F 1

(A.1)

F 1 = R1 + P1

(A.2)

R1 + A + RR2 = RR1 + F 2

(A.3)

F 2 = R2 + P2

(A.4)

R2 = RR2 + RRP

(A.5)

From the definitions of the dimensionless variables shown in Table 1: P1 = p × Fo

(A.6)

RR1 = r1 × R1

(A.7)

RR2 = r2 × R2

(A.8)

RRP = ro × Fo

(A.9)

A=

af × Fo × CMFo CA

(A.10)

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

46

B=

ACA abCB

(A.11)

The polymer mass balances are: RRP × CPR2 + RR1 × CPR1 = F 1 × CPF 1

(A.12)

F 1 × CPF 1 = R1 × CPR1

(A.13)

F 2 × CPF 2 = R2 × CPR2

(A.14)

From the definition of pp: NP /VT pp = CMFo

(1/2)(CPF 1 + CPF 2 ) CMFo

(A.15)

qMLk  qk cM L

βOH,l =

(B.1)

 cM(OH) l

(B.2)

 c l cM OH

The acidity constant of the amine group: (A.16) Ka =

The sodium mass balances are: B × CB + RRP × CNaR2 + RR1 × CNaR1 = F 1 × CNaF 1 (A.17) F 1 × CNaF 1 = R1 × CNaR1 + P1 × CNaP1

The model considers the feed solution to be divided into two regions. One region is occupied by the polymer, the bounded ions and some water, and its volume is Vp . The other region is the remaining solution, whose volume is V. The stability constants for the soluble complexes that the metal forms with polymer amine group and hydroxyl ion are: βk =

Np /VT is the average polymer concentration in the system. If the volume of both Sections are identical, then: pp =

Appendix B. Equilibrium model and membrane separation

(A.18)

 qL cH qLH

(B.3)

M represents a free metal ion, L a ligand amine group of the polymer (R–NH2 ) and MLk a complex formed by a metal ion and k ligands, M(R–NH2 )k . If no insoluble hydroxide is present:   Kps < cM c OH 2

R1 × CNaR1 + RR2 × CNaR2 = RR1 × CNaR1 + F 2 × CNaF 2 F 2 × CNaF 2 = R2 × CNaR2 + P2 × CNaP2

(A.19) (A.20)

The polymer and metal balance are, respectively:
  Kw qLj V  k CPj +1 = + qLj + kβk cMj qLj  Vpj Ka cOHj


Fo × CMFo + RRP × CMR2 + RR1 × CMR1

F 1 × CMF 1 = R1 × CMR1 + P1 × CMP1

(B.5)

k=1−2

The heavy metal balances are:

= F 1 × CMF 1

(B.4)

CMj (A.21)

V +1 Vpj





 cMj

= +

(A.22)

+





  l βOHl cMj cOHj

l=1−4  k βk cMj qLj

V Vpj (B.6)

k=1−2

R1 × CMR1 + RR2 × CMR2 = RR1 × CMR1 + F 2 × CMF 2 F 2 × CMF 2 = R2 × CMR2 + P2 × CMP2

(A.23) (A.24)

The chlorine mass balances are: Fo × n × CMFo + RRP × CClR2 + RR1 × CClR1 = F 1 × CClF 1 F 1 × CClF 1 = R1 × CClR1 + P1 × CClP1

(A.25) (A.26)

R1 × CClR1 + RR2 × CClR2 + A × CA = RR1 × CClR1 + F 2 × CClF 2 F 2 × CClF 2 = R2 × CClR2 + P2 × CClP2

The electroneutrality in a solution present can be formulated as follows:    V  l n+ c (n − l)βOHl c OHj + Vpj Mj l=1−4
  V + + 1 (CNaj − CClj ) + n Vpj

where polymer is V  (c − c OHj ) Vpj Hj  k k  βk cMj qLj

k=1−2

Kw qLj + =0  Ka cOHj

(B.7)

(A.27)

The definition of the rejection for any species in any membrane:

(A.28)

Rij = 1 −

As the electroneutrality equations depend on the concentrations of the different species, the are shown in Appendix B.

CiPj CiRj

(B.8)

This was formulated considering the composition of permeate and retentate streams. As r1 = r2 = 0.99, it does not

J. Sabat´e et al. / Journal of Membrane Science 268 (2006) 37–47

involve any significant error with respect to the combined average composition of the feed and retentate streams. Consequently, the equilibrium equations have only to be applied to R1 and R2 streams. Because only unbound metal leaks through the UF membranes:     l CMPj = cMj 1 + (B.9) βOHl c OHj l=1−4

The electroneutrality of permeates can be written as:     l cMj n + (n − l)βOHl c OHj l=1−4   + CNaPj − CClPj + cHj − cOHj =0

(B.10)

References [1] K.J. Sasaki, S.L. Burnett, S.D. Christian, E.E. Tucker, J.F. Scamehorn, Polyelectrolyte ultrafiltration of multivalent ions. Removal of Cu2+ by sodium poly(styrenesulfonate), Langmuir 5 (1989) 363. [2] R. Juang, C. Chiou, Feasibility of the use of polymer-assisted membrane filtration for brackish water softening, J. Membr. Sci. 187 (2001) 119. [3] M. Rumeau, F. Persin, V. Sciers, M. Persin, J. Sarrazin, Separation by coupling ultrafiltration and complexation of metallic species with industrial water soluble polymers. Application for removal or concentration of metallic cations, J. Membr. Sci. 73 (1992) 313. [4] J. Sabat´e, M. Pujol`a, J. Llorens, Two-phase model for calcium removal from aqueous solution by polymer enhanced ultrafiltration, J. Membr. Sci. 204 (2002) 139. [5] B. Rivas, E. Pereira, P. Gallegos, D. Homper, K.E. Geckeler, Metal ion binding capability of the water-soluble poly(vinyl phosphonic acid) for mono-, di-, and trivalent cations, J. Appl. Polym. Sci. 92 (2004) 2917. [6] A. Tabatabai, J.F. Scamehorn, S.D. Christian, Economic feasibility of polyelectrolyte-enhanced ultrafiltration (PEUF) for water softening, J. Membr. Sci. 100 (1995) 193. [7] P. Baticle, C. Kiefer, N. Lakhchaf, O. Leclerc, M. Persin, J. Sarrazin, Treatment of nickel containing industrial effluents with a hybrid process comprising of polymer complexation-ultrafiltration-electrolysis, Sep. Purif. Technol. 18 (2000) 195. [8] I. Korus, M. Bodzeck, K. Loska, Removal of zinc and nickel ions from aqueous solutions by means of the hybrid complexationultrafiltration process, Sep. Purif. Technol. 7 (1999) 111. [9] P. Ca˜nizares, A. P´erez, R. Camarillo, J.J. Linares, A semi-continuous laboratory-scale polymer enhanced ultrafiltration process for the recovery of cadmium and lead from aqueous, J. Membr. Sci. 240 (2004) 197.

47

[10] J. Llorens, J. Sabat´e, M. Pujol`a, Viability of the use of polymerassisted ultrafiltration for continuous water softening, Sep. Sci. Technol. 38 (2003) 295. [11] B. Schlichter, V. Mavrov, T. Erwe, H. Chmiel, Regeneration of bonding agents loaded with heavy metals by electrodialysis with bipolar membranes, J. Membr. Sci. 232 (2004) 99. [12] V. Mavrov, T. Erwe, C. Blocher, H. Chmiel, Study of new integrated processes combining adsorption, membrane separation and flotation for heavy metal removal from wastewater, Desalination 157 (2003) 97. [13] S. Kobayashi, K. Hiroishi, M. Tokunoh, T. Saegusa, Chelating properties of linear and branched poly(ethylenimines), Macromolecules 20 (1987) 1496. ´ [14] K. Ennassef, M. Persin, G. Durand, Etude par ultrafiltration de la complexation des cations argent (I) et cuivre (II) par des macroligands oligom`eres d’acide polyacrylique et mise au point de leur s´eparation, Analysis 17 (1989) 565. [15] K. Volchek, E. Krentsel, Y. Zhilin, G. Shtereva, Y. Dytnersky, Polymer binding ultrafiltration as a method for concentration and separation of metals, J. Membr. Sci. 79 (1993) 253. [16] J. Llorens, M. Pujol`a, J. Sabat´e, Separation of cadmium from aqueous streams by polymer enhanced ultrafiltration: a two-phase model for complexation binding, J. Membr. Sci. 239 (2004) 173. [17] R. Juang, M. Chen, Measurement of binding constants of poly(ethylenimine) with metal ions and metal chelates in aqueous media by ultrafiltration, Ind. Eng. Chem. Res. 35 (1996) 1935. [18] S. Taha, P. Bouvet, G. Corre, G. Dorange, Study and modelisation of some heavy metals removal by ultrafiltration in presence of soluble chitosan, Adv. Chitin Sci. 1 (1996) 389. [19] R. Juang, R. Shiau, Metal removal from aqueous solutions using chitosan-enhanced membrane filtration, J. Membr. Sci. 165 (2000) 159. [20] E. Guibal, Interactions of metal ions with chitosan-based sorbents: a review, Sep. Purif. Technol. 38 (2004) 43. [21] V.G.L. Rorrer, T. Hsien, J.D. Way, Synthesis of porous-magnetic chitosan beads for removal of cadmium ions from waste water, Ind. Eng. Chern. Res. 32 (1993) 2170. [22] R. Navarro, J. Guzman, I. Saucedo, J. Revilla, E. Guibal, Recovery of metal ions by chitosan: sorption mechanisms and influence of metal speciation, Macromol. Biosci. 3 (2003) 552. [23] V.R. Bassi, S. Prasher, B.K. Simpson, Removal of selected metal ions from aqueous solutions using chitosan flakes, Sep. Sci. Technol. 35 (2000) 547. [24] J.R. Evans, W.G. Davids, J.D. MacRae, A. Amirbahman, Kinetics of cadmium uptake by chitosan-based crab shells, Water. Res. 36 (2002) 3219. [25] M. Rhazi, J. Desbrieres, A. Tolaimate, M. Rinaudo, P. Vottero, A. Alagui, M. El Meray, Influence of the nature of the metal ions on the complexation with chitosan. Application to the treatment of liquid waste, Eur. Polym. J. 38 (2002) 1523. [26] N. Jarvis, J. Wagener, Mechanistic studies of metal ion binding to water-soluble polymers using potentiometry, Talanta 42 (1995) 219.