Removal of heavy metal ions by polymer enhanced ultrafiltration

Removal of heavy metal ions by polymer enhanced ultrafiltration

Desalination 286 (2012) 193–199 Contents lists available at SciVerse ScienceDirect Desalination journal homepage: www.elsevier.com/locate/desal Rem...

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Desalination 286 (2012) 193–199

Contents lists available at SciVerse ScienceDirect

Desalination journal homepage: www.elsevier.com/locate/desal

Removal of heavy metal ions by polymer enhanced ultrafiltration Batch process modeling and thermodynamics of complexation reactions Rafael Camarillo b,⁎, Ángel Pérez a, Pablo Cañizares a, Antonio de Lucas a a b

Department of Chemical Engineering. University of Castilla-La Mancha, Faculty of Chemical Sciences. Av. Camilo José Cela, 12. 13005 Ciudad Real (Spain) Department of Chemical Engineering. University of Castilla-La Mancha, Faculty of Environmental Sciences and Biochemistry. Av. Carlos III, s/n. 45071 Toledo (Spain)

a r t i c l e

i n f o

Article history: Received 22 July 2010 Received in revised form 5 November 2011 Accepted 7 November 2011 Available online 29 November 2011 Keywords: Polymer enhanced ultrafiltration Copper removal Poly(ethylenimine) Modeling Thermodynamics

a b s t r a c t A batch process of polymer enhanced ultrafiltration (PEUF) for the removal of Cu(II) ions from aqueous effluents has been developed in an installation equipped with mineral membranes at laboratory scale. It has been proposed a model based on mass conservation equations and kinetics of complex formation reactions of macromolecular species comprising a metallic ion and a water-soluble polymer. The model predicts the temporal evolutions of metal concentration in permeate and rejected streams. Moreover, it enables to establish the operating pH values to tackle the stages of metal retention and polymer regeneration, and also to calculate conditional formation constants of the macromolecular complex as a function of pH and temperature. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Membrane filtration processes have been successfully applied for the separation and/or enrichment of several species from different aqueous streams [1–3]. Micellar enhanced ultrafiltration (MEUF) is a physico chemical membrane separation technique aimed at improving the performance of ultrafiltration membranes by capturing the small size pollutants into larger structures called micelles [4]. More effective retention of certain inorganic ions can be achieved by complexing previously these ions with water-soluble polymers by means of polymer enhanced ultrafiltration (PEUF) [5]. At the end of this process, a diluted permeate stream (discharged as a less pollutant waste or employed for specific purposes) but also a retentate stream with a high concentration of metallic ions bound to the polymer are produced. In order to make feasible both technical and economical viability of the process, these ions in concentrated stream should be released from polymer [6,7]. This regeneration of polymer can be chemically achieved in three successive steps by adding firstly an acid (with protons that compete with metal ions for binding sites of polymer), a second ultrafiltration process, and finally, the return of regenerated polymer into process after neutralization.

⁎ Corresponding author. Tel.: + 34 925 26 88 00x5414; fax: + 34 925 26 88 40. E-mail address: [email protected] (R. Camarillo). 0011-9164/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.desal.2011.11.021

PEUF processes have been modeled in batch [8–9], semi-continuous [10–12], and continuous [13] modes. Models help to obtain a deeper understanding of the whole process and predict results under untested conditions [14]. These models allow predicting rejection coefficients [15] or to measure equilibrium constants for complexation of metal ions by synthetic and natural ligands [16]. In this paper, a well-known couple of metal ion (Cu2+) and watersoluble polymer (PEI) has been selected. They can be considered as standard substances to develop a PEUF process. In the case of copper, because it is a heavy metal that has been recycled for hundreds of years. For poly(ethylenimine) (PEI), because it is a chemical chameleon that can be evaluated as a selective binder or even used like polymeric vector for gene delivery [17–18]. Although in bibliography there are plenty of papers related to the modeling of PEUF processes, few of them are based on the kinetic theory of reactions applied to partially open systems. On the one hand, the aim of this paper is to develop a kinetic model simpler than those previously proposed that allows: 1. To successfully predict the copper ion concentrations in permeate and rejected streams; 2. To calculate conditional formation constants of the macromolecular complex. On the other hand, operating conditions to develop both metal retention and polymer regeneration stages have been optimized. Moreover, thanks to the calculation of the formation enthalpy of PEI–Cu complex, a thermochemical regeneration of polymer is proposed, since the increase in permeate flux compensates the energy needs, working at milder pH values and reducing the consumption of strong inorganic acids.

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R. Camarillo et al. / Desalination 286 (2012) 193–199

2. Theory 2.1. The reaction of complex formation and the influence of pH value The reaction of complex formation between Cu(II) ions and amine groups of macromolecular ligand, considering the latter as a polyfunctional ligand, can be written in a simplified form as: M þ mL ↔MLm

ð1Þ

where M symbolizes the free metal ions and L the repeating unit for polymer. Charges have been intentionally omitted. The selected polymer, poly(ethylenimine) (PEI), is obtained from monomer ethylenimine. The monomer consists of a three-membered ring. Two corners of the molecule consist of –CH2– linkages. The third corner is a secondary amine group, –NH–. In the presence of a catalyst this monomer is converted into a highly branched aliphatic polyamine with about 25% primary amine groups, 50% secondary amine groups, and 25% tertiary amine groups. In Fig. 1, the structure for this polymer proposed by the supplier (Aldrich) is depicted. In order to simplify the polymer structure it will be considered as a macromolecule composed of repeating structural units denoted L (ethylenimine). The functional groups show an acidic character, which means the amine groups can experience a competitive reaction of protonation into LH. L þ H ↔LH

ð2Þ

In absence of metallic ions, the amounts of LH and L in solution are related with the pH value by a Hasselbach type equation: pH ¼ pK a −n ⋅

logð1−α Þ α

½L  ½L 0

being [L]0 the initial concentration of PEI in solution. From Eqs. (3) and (4) the concentration of [LH] and [L] as function of pH can be obtained:

  pK a −pH ½LH ¼ ½L0 1−10 n

ð5Þ

ð7Þ

At a fixed pH value in solution and considering that [L] is the concentration of free ligand (neither protonated nor taking part in metallic complex MLm), a conditional complex formation constant can be defined as: ½MLm  ½M ½Lm



Kf ¼

ð8Þ

The value of this complex formation constant will depend on the pH value of solution. 2.2. Modeling of batch process The batch mode is the simplest one and requiring the minimum filtration area (A) to reach a determined separation per unit of time. In concentration mode, a feed with initial volume V is filtered at a constant pressure, removing permeate stream, so the system volume diminishes with time, while retained species are concentrated in feed stream. Equations that describe the change of volume and total mass of a determined solute in a batch operation in absence of chemical reaction [20] are: −

ð4Þ

  pK a −pH −1 ½L ¼ ½L0 1 þ 10 n

½L0 ¼ ½L þ ½LH þ m½MLm 

ð3Þ

where pKa is the dissociation constant of LH, n is a constant depending on each polyelectrolyte which accounts for nearest neighboring interaction, and α is the protonation degree of PEI expressed as: α¼

with pH of the relative concentration of both PEI forms in solution, L and LH, with respect to the initial concentration in solution can be observed in [19]. As can be seen, the PEI becomes less charged as the pH value increases. In the presence of metallic ions, taking into account Eq. (1), the initial concentration of ligand in solution [L]o will be:

dV ¼ JV ⋅ A dt f

ð9Þ f

t

p

V 0 ⋅C S0 ¼ V t ⋅ C St þ A ⋅ ∫ Jvt ⋅ C st ⋅ dt 0

In these equations, V is system volume, Jv is permeate flux, and Cs is molar concentration of solute at each time t. Subindexes 0 and t refer to initial conditions and conditions at an arbitrary time t, respectively. Superindexes f and p refer to feed and permeate streams, respectively. If we consider a system where a complex MLm is formed from a transition metal ion (M) and the representative constitutional unit of a polymer (L), this case can be described in kinetic terms by the reversible chemical reaction (charges are omitted): k

1 MLm M þ mL ↔

ð6Þ

Von Zelewsky et al. [17] found that this kind of polymer exhibits a pKa of 7.69 and n equal to 7 for protonation degree values higher than 0.3. On the contrary, when the protonation degree is lower than 0.3, pKa and n can reach 8.39 and 2, respectively. The change

ð10Þ

k2

ð11Þ

Conditional formation constant will be the ratio between kinetic constants of complex formation (k1) and dissociation (k2). In systems with metal concentration small enough with regard to polymer concentration, the rate of complexation reaction will be: m



k1 ⋅½M⋅½L ¼ k1 ⋅½M 

ð12Þ

while the rate of dissociation can be written as follows: k2 ⋅½MLm 

Fig. 1. Poly(ethylenimine) (PEI) structure (Aldrich).

ð13Þ

Evolution of metal concentration in retentate and permeate streams can be described with a model based on the theory of kinetics of reactions of complexation-dissociation and the irreversible transfer of M across the membrane [21–22]. Designate the concentrations of retained metal form MLm as C2 and of free metal form M as C1.

R. Camarillo et al. / Desalination 286 (2012) 193–199

From the point of view of formal kinetics, such a model can be written as a system of differential equations:

3. Experimental

  d V ⋅C f1

3.1. Materials

dt   d V ⋅C f2 dt



f

f

p

ð14Þ

f

p

ð15Þ

¼ k2 ⋅V  C 2 −k1 ⋅ V ⋅ C 1 −A⋅ J V ⋅ C1



f

¼ k1 ⋅V  C 1 −k2 ⋅ V ⋅ C 2 −A⋅ J V ⋅ C2

The sum of these two equations expresses the variation of total mass of metal in system:

V⋅

  dC f f p ¼ A ⋅ J v ⋅ C −C dt

ð16Þ

In all these equations, A refers to membrane area (m 2), Jv is permeate flux (L h − 1 m − 2 or LHM), and V is volume (L). This system of differential equations can be solved by a numerical method of finite growths, which allows to calculate free metal and complexed metal concentrations in retentate and permeate streams (C1f, C2f, C1p, C2p), kinetic constants (k1⁎, k2) and conditional formation constant (Kf*):  113 0 0 A ⋅ J v ðtÞ ⋅ C f1ðtÞ −C p1ðtÞ f f  f 4 @ @ AA5 ¼ C 1 ðtÞ þ Δt ⋅ k2 ⋅ C 2 ðtÞ  k1 ⋅ C 1 ðtÞ þ V ðt Þ 2

f C 1 ðtþ1Þ

ð17Þ 2 f C 2 ðtþ1Þ

0

0

f f  f ¼ 4C 2 ðtÞ þ Δt ⋅ @−k2 ⋅ C 2 ðtÞ þ k1 ⋅ C 1 ðtÞ þ @

 113 A ⋅ J v ðtÞ ⋅ C f2ðtÞ −C p2 ðtÞ AA5 V ðtÞ

ð18Þ

195

Feed solutions were prepared with branched poly(ethylenimine) (Mw = 25,000 Da) 99 wt.% in water and copper (II) nitrate hydrate, both of analytical grade and supplied by Aldrich. Sodium hydroxide and nitric acid of analytical grade supplied by Aldrich were used to control pH in solution. Sodium nitrate of analytical grade supplied by Aldrich was used to control ionic strength in solution. Copper concentrations in feed and permeate streams were measured by means of AA spectrophotometer Varian SpectrAA 220, and polymer concentrations with a Shimadzu 5050A Total Organic Carbon analyzer. 3.2. Apparatus Laboratory-scale experiments were performed using an ultrafiltration plant consisting of a 2 L stirred reactor with a jacket (1) connected to a thermostatized bath (2), a feed gear pump supplied by Liquiflo (400 L h − 1at 4 bar) (3), a flow meter to know the feed flow rate (4), and a tubular Micro Carbosep 20 UF module with axial outlets with M2 Rhodia-Orelis ceramic membranes (MWCO 15 kDa) with effective membrane area of 0.004 m 2 (5). These membranes have a support layer of graphite and an active layer of ZrO2 that withstands temperatures 1–95 °C, feed flow rates 50–300 L h − 1 and transmembrane pressures 0–10 bar. A detailed scheme of the laboratory-scale installation is shown in Fig. 2. 3.3. Operation procedures i) Total recirculation experiments. In the first place, feed tank is filled with working solution. The thermostatized bath is adjusted at a certain temperature in order to heat reactor jacket. The system is turned on, and feed flows around installation without

5

4

3

1

Fig. 2. Experimental set-up.

2

R. Camarillo et al. / Desalination 286 (2012) 193–199

transmembrane pressure until stationary conditions are reached. With feed stream, concentrated stream and recirculation valves, feed flow rate and transmembrane pressure are controlled. Feed flow rate is measured with a variable area flow meter, and permeate flux is measured by weighting the liquid leaving the system within a determined period of time. In this mode, the permeate stream is returned to feed tank. Samples of permeate stream and reactor are taken in order to measure polymer and metal ion concentration. In this mode, influence of some operating variables (polymer concentration, transmembrane pressure and feed flow rate) on design parameters (permeate flux and rejection coefficients) is studied. ii) Batch mode experiments. The procedure is similar to the previous one, except that in this case before sending permeate stream to a separate reservoir, we let 10 minutes pass until stationary regime is attained and to let possible fouling phenomenon start. Polymer and metal ion concentrations, feed flow rate, transmembrane pressure, ionic strength and temperature were previously established with total recirculation experiments. In batch mode experiments the only variables are pH value (for enrichment and regeneration modes) and operation time. iii) Membrane cleaning procedure. This procedure consisting in successive alkali and acid cleanings is explained in previous papers [23]. 4. Results and discussion 4.1. Previous experiments in total recirculation mode In order to study the effect of transmembrane pressure, polymer concentration and feed flow rate on permeate flux, a series of previous experiments in total recirculation mode were accomplished. Unlike previous research [15,22], far higher polymer and metal concentrations (1–3 wt.% and 250–1500 mg L − 1) were essayed. The aim is to check the technical viability of the proposed process under experimental conditions closer to reality at industrial scale operation. It must be highlighted that all experiments have been performed with aqueous solutions containing only polymer or metal ions at their natural pH values (≥11 for polymer solutions and b5 for metal solutions). Fig. 3 shows the effect of polymer concentration, feed flow rate and transmembrane pressure on permeate flux. The asymptotic trend in curves implies the appearance of a concentration polarization phenomenon when transmembrane pressure exceeds 3 bar, polymer concentration is 3 wt.% or feed flow rate is 100 L h − 1. In

1.0

Cu 250 ppm Cu 500 ppm Cu 750 ppm Cu 1500 ppm 0.1

0.0 1

2

3

4

PTM (bar) Fig. 4. Influence of transmembrane pressure, polymer concentration and metal concentration on metal and polymer and metal ion rejection coefficients at 25 °C. Tangential velocity v = 1.25 m s− 1.

Fig. 4 the influence of transmembrane pressure and polymer concentration on polymer and metal ion rejection coefficients is depicted. The analysis of both figures calls for caution to not exceed 3 bar in transmembrane pressure. Otherwise polymer rejection coefficient would be endangered and concentration polarization would be enhanced. Moreover, henceforth solutions with initial polymer concentration 1 wt.% and feed flow rate 125 L h − 1 (tangential velocity 1.25 m s − 1) will be used. In Fig. 4 it can be also observed that copper rejection coefficient diminishes as metal concentration increases. To ensure that free metal ion rejection coefficient is almost zero, the ionic strength of the essayed solutions will be controlled by adding an inert electrolyte (NaNO3) until 0.15 M concentration. 4.2. Experiments in batch mode Batch mode has been selected because it is the most suitable method when treating industrial effluents with feed flow rates lower than 5 m 3 d − 1 [24]. In the first place, two experiments were accomplished to analyze the effect of ionic strength on permeate flux and polymer rejection coefficients. Fig. 5 describes the effect of ionic strength on temporal evolution of these latter two parameters. As it was to be expected, given that PEI is a weak polyelectrolyte, the increase of ionic strength in the medium has effect on the hydrodynamic radius of gyration of polymer molecule. Park and Choi [25]

0

50

2

4

6

8

100 PEI 1 %; 100 l/h PEI 1 %; 125 l/h PEI 3 %; 100 l/h PEI 3 %; 125 l/h

80

30

20

10

0

1

2

3

4

PTM (bar) Fig. 3. Influence of transmembrane pressure, polymer concentration and feed flow rate on permeate flux at 25 °C.

0.8

R PEI 1 % R PEI 1 % + 0.15 M (NaNO3)

60

0.6

40

0.4

Jv PEI 1 % Jv PEI 1 % + 0.15 M (NaNO3)

20

0

10 1.0

R

Jv (l h-1 m-2)

40

Jv (l h-1 m-2)

PEI 1 % PEI 3 %

0.9

R

196

0.2

0 0

2

4

6

8

0.0 10

Time (h) Fig. 5. Temporal evolution of permeate flux and polymer rejection coefficients as a function of ionic strength. ΔPTM = 3 bar; v = 1.25 m s− 1; T = 25 °C.

R. Camarillo et al. / Desalination 286 (2012) 193–199 1.0

100

Jv (l h-1 m-2)

R pH 10.2 R pH 4.8 R pH 4.0 R pH 3.2 R pH 2.0 R pH 1.0

60

0.8

R (PEI)

0.9 80

0.7 Jv pH 10.2 Jv pH 4.8 Jv pH 4.0 Jv pH 3.2 Jv pH 2.0 Jv pH 1.0

40

0.6

20

0.5 0

2

4

6

8

10

Time (h) Fig. 6. Temporal evolution of permeate flux and polymer rejection coefficient as a function of pH value. ΔPTM =3 bar; v=1.25 m s− 1; [PEI]0 =0.2325 M; [Cu2+]0 =0.0075 M; T=25 °C.

197

In Fig. 6, one can check that polymer rejection coefficients are always higher than 0.9 during the 10.5 h of operation no matter which pH value is used. However, Fig. 7 indicates that metal ion rejection coefficients significantly vary with pH. When pH value interval is 3.2–10.2, they are in the order of 0.9, which means that macromolecular complex is stable enough at these pH values. The decrease in rejection coefficients for lower pH values can be explained in terms of a loss in complex stability as protons from acidic medium begin competing with copper ions for active sites of macromolecular ligand. At pH 1, this complex is so dissociated that metal rejection coefficients are only slightly higher than 0.1. Finally, the experiment at pH 4 was repeated at 50 °C. Results appear in Fig. 8. The increase in temperature leads to an increase in permeate flux that is related to the decrease in feed solution viscosity. Polymer and metal rejection coefficients do not significantly decreased and keep higher than 0.9 when temperature is raised from 25 to 50 °C. 4.3. Numerical solution of PEUF model in batch mode

verified that the intrinsic viscosity of an aqueous solution comprising PEI and pure water decreases from 32.0 to 23.0 mL mg − 1 in presence of NaCl 1 M. For this reason, in Fig. 5 both a slight increase in permeate flux and a decrease in polymer rejection coefficients are observed when NaNO3 is added. In any case, the increase of ionic strength does not provoke significant changes in permeate flux, and polymer rejection coefficients keep a high value. Next, six ultrafiltration experiments with aqueous solutions of macromolecular complex PEI–Cu were developed. The value of pH of these solutions was varied from 1 to 10.2. The initial concentrations of polymer and copper were 1 wt.% ([L] = 0.2325 M) and 475 ppm (0.0075 M), respectively. In Figs. 6 and 7, temporal evolutions of permeate flux and polymer and metal rejection coefficients as functions of pH value are shown. Regarding permeate flux, Fig. 6 shows that, at pH 4, this parameter diminishes from 78.3 to 44.8 (L h − 1 m − 2) as polymer concentration in feed reservoir rises from 1 to 1.7 wt.% after 10.5 h. This trend is almost linear and the flux decline is inversely proportional to the increase in the viscosity of feed solution. Experiments at pH 4.8, 3.2 and 2.0 follow a similar evolution. Experiments at pH 1 and 10.2 also show a similar trend except for initial flux. It is lower (≅ 60 L h − 1 m − 2) due to the membrane suffers from a serious fouling phenomenon at extreme pH values. This fact has been more deeply explained in a previous paper [23]. At pH 1 the dominant fouling mechanism is cake formation, whereas at pH 10.2 is pore blocking.

Modeling of PEUF process allows estimating the design parameters to dimension the pilot or industrial scale plants where proposed treatment may be carried out. Nowadays, references about modeling of PEUF processes are more and more numerous, but rather diverse at the same time [10,14,21,26], being the main differences the operation mode and the aim of the modeling: 1. Prediction of rejection coefficients, or 2. Calculation of equilibrium constants. As explained in Theory, the solution of the system of differential Eqs. (14) and (15) allows to calculate the concentrations of free metal (C1) and complexed metal (C2) in feed (f) and permeate (p) streams, kinetic constants (k1* and k2), as well as conditional formation constant (Kf*). Given that analytical solution of these equations would be too complicated, they were expressed in discrete form, being substituted by the Eqs. (17) and (18). The new system of equations can be solved by the method of finite growths. To calculate the initial values (at time t = 0) of C1f, C2f, C1p, C2p, k1⁎, k2, Jv(t) and V(t), the following conditional relationships and simplifying hypotheses were established: • The total amount of metal in feed and permeate streams can be measured at zero time (and at each Δt) and their values can be expressed in terms of concentrations of free and complexed metal as: f

f

f

C ðt¼0Þ ¼ C 1ðt¼0Þ þ C 2ðt¼0Þ

ð19Þ

1.0

180

1.0

160 0.8

Jv (l h-1 m-2)

R (Cu)

0.6 pH 10.2 pH 4.8 pH 4.0 pH 3.2 pH 2.0 pH 1.0

0.4

0.2

PEI 25 oC PEI 50 oC Cu 25 oC Cu 50 oC

120 100

0.8

R

0.9

140

o

Jv 50 C Jv 25 o C

80

0.7

60 0.6

40

0.0 0

2

4

6

8

10

Time (h) Fig. 7. Temporal evolution of metal rejection coefficients as a function of pH value. ΔPTM = 3 bar; v = 1.25 m s− 1; [PEI]0 = 0.2325 M; [Cu2+]0 = 0.0075 M; T = 25 °C.

0.0

1.5

3.0

4.5

6.0

7.5

9.0

Time (h) Fig. 8. Temporal evolution of permeate flux and polymer rejection coefficients with temperature at pH 4. ΔPTM = 3 bar; v= 1.25 m s− 1; [PEI]0 = 0.2325 M; [Cu2+]0 = 0.0075 M.

R. Camarillo et al. / Desalination 286 (2012) 193–199 p

p

p

C ðt¼0Þ ¼ C 1ðt¼0Þ þ C 2ðt¼0Þ

ð20Þ

• The concentration of ligand in feed and permeate streams can be measured at zero time (and at each Δt), which allows to calculate the ligand rejection coefficient. If we suppose that metallic complex (RMLm) and ligand rejection coefficients (RL) are the same, it is fulfilled that:   p f C 2ðt¼0Þ ¼ C 2ðt¼0Þ ⋅ 1−RLðt¼0Þ

ð21Þ

• If before starting the experiment in batch mode, system is kept in total recirculation mode the time necessary to reach stationary conditions, Eqs. (14) and (15) at zero time turn into: f



f

p

f

p

k2ðt¼0Þ ⋅V ðt¼0Þ ⋅C 2ðt¼0Þ −k1ðt¼0Þ ⋅V ðt¼0Þ ⋅C 1ðt¼0Þ −A ⋅ J vðt¼0Þ ⋅C 1ðt¼0Þ ¼ 0 ð22Þ 

f

k1ðt¼0Þ ⋅V ðt¼0Þ ⋅C 1ðt¼0Þ −k2ðt¼0Þ ⋅V ðt¼0Þ ⋅C 2ðt¼0Þ −A ⋅ J vðt¼0Þ ⋅C 2ðt¼0Þ ¼ 0 ð23Þ • The membrane area (A) is a known parameter. If initial volume of system is also known and permeate flux at t = 0, t = t + 1, etc. is experimentally measured, we can determine the temporal variation in volume of system V(t) in terms of Jv(t) by means of Eq. (9). • If the residence time of feed solution in reactor is long enough to enable the reaction to attain the equilibrium state [16,22], Eqs. (12) and (13) will be equal and an additional hypothesis could be established: 

f

f

k1 ⋅C 1 ¼ k2 ⋅C 2

ð24Þ

Once initial values (t = 0) for C1f, C2f, C1p, C2p, k1⁎, k2, Jv(t) and V(t) are established, an iterative procedure is developed to obtain temporal evolution of free metal and complexed metal concentrations in process streams by optimization of initial values of k1* and k2 with the following sequence of calculations: 1. Solving the system of discrete equations until the ending time of each experiment. 2. Determination of the values of C f (t) (calculated). 3. Calculation of the sum of squares of errors.  2 f f SSE ¼ ∑ C ð expÞ−C ðcalÞ

ð25Þ

4. Development of a procedure of non-linear optimization implemented in the spreadsheet-application Microsoft Excel (Solver) to minimize the value of SSE. This model requires a series of restrictions which provide physical meaning to results obtained (positive concentrations and kinetic constants, C1f/C f and C1p/C p ratios comprised between 0 and 1, etc.). As example, in Table 1 experimental results and values from balances of copper by numerical method of finite growths are depicted, when a pH value equal to 2 is imposed. In Fig. 9, the experimental and calculated values of C f for experiments at pH 4 (25 and 50 °C) have been depicted. Generally speaking, the fitting is rather good, which allows to state that the model simulates satisfactorily a batch process of ultrafiltration of a metallic complex.

Table 1 Solution of copper balance for experiment at pH 2, 25 °C, [PEI]0 = 1 wt.%, [Cu2+]0 = 475 ppm. t (h)

Jv (l/h·m2)

Cf (ppm)

Cp (ppm)

C1f (ppm)

C2f (ppm)

Cf (calc.) (ppm)

(Cfexp–Cfcalc)2 (error)

0 1.5 3 4.5 6 7.5 9 10.5

76.143 69.214 66.130 62.162 57.345 52.719 48.312 43.396

477.3 491.8 506.5 536.4 553.8 591.1 614.2 651.1

239.0 230.6 230.6 241.0 246.2 253.0 266.8 267.8

0.844 0.422 0.464 0.496 0.352 0.278 0.191 0.088

458.5 479.4 502.4 528.8 558.0 590.3 626.2 664.8

459.4 479.8 502.9 529.3 558.4 590.5 626.4 664.9

320.1 144.3 13.1 49.7 20.7 0.3 149.2 192.1 SSE = 889.4

Taking into account Eqs. (3)–(8) and solving the mathematical model previously described, conditional formation constants for complex Cu–PEI are calculated for experiments performed at 25 °C and pH values between 1 and 5. Since Cu 2+ cation is very acidic, at pH values ≥5 it can also react with hydroxyl ions in the medium. The precipitation of hydroxides affects both the permeate flux and the copper rejection coefficient (Figs. 6–7). For this reason, this model has not been applied to experiment at pH 10.2. In the development of the model we have not considered this side reaction that can alter remarkably the concentration of free metal. At pH values higher than 3, conditional formation constant of complex is high enough (6.83 × 10 5 and 1.08 × 10 7 at pH 3 and 4.8, respectively) to provide copper rejection coefficients above 92%. At pH 1, the value of conditional formation constant is so small (1.32 × 10 − 3) that allows to tackle satisfactorily the polymer regeneration stage. In Fig. 10, conditional formation constants of complex Cu–PEI obtained by both spectrophotometric methods [29] and solving PEUF model versus pH value are depicted. We can see that both sets of results can be fitted rather well with the same plot. This observed trend confirms the relationship between the formation constant of complex Cu–PEI and the pH value of the medium, justifying the selection of determined operation pH values for copper recovery stage (pH 4–5) and polymer regeneration stage (pH 1). These conditional formation constants for Cu–PEI complex have been also measured at two different temperatures, keeping a constant pH value of 4. In this way, constant values of 3.23 × 10 6 and 8.34 × 10 4 for 25 and 50 °C, respectively, have been obtained. With the aim of obtaining a magnitude order for reaction enthalpy of formation of complex Cu–PEI, Van't Hoff's equation was applied taking into account the two values of conditional constant calculated at 25 and

1400

Experimental 25 oC Experimental 50 oC Calculated pH 4

1200

Cf (ppm Cu)

198

1000

800

600

4.4. Determination of conditional constants for Cu–PEI complex 400

Although the measurement of complex formation constants can be accomplished by potentiometric titrations [27] or spectrophotometric analyses [28–29], in this section conditional formation constants will be calculated from ultrafiltration experiments in batch mode [16,26].

0

2

4

6

8

10

Time (h) Fig. 9. Temporal evolution of total concentration of Cu in feed stream for experiments at pH 4 and different temperatures. ΔPTM = 3 bar; v = 1.25 m s− 1; [PEI]0 = 0.2325 M; [Cu2+]0 = 0.0075 M.

R. Camarillo et al. / Desalination 286 (2012) 193–199 10

An increase in temperature during polymer regeneration process could make possible working at less extreme pH values together with cost savings by strong inorganic acids reduction.

8

log Kf*

6

References

4 2

PEUF MODEL SPECTROPHOTOMETRIC SPLINE

0 -2 -4 1

2

3

4

5

6

pH Fig. 10. Variation of conditional formation constant of complex Cu–PEI with pH value. T = 25 °C.

50 °C. If we suppose that reaction enthalpy keeps constant within the temperature range essayed, Van't Hoff's equation has the expression: ln

K f 2 K f 1

! ¼

199

  ΔH 1 1 − R T1 T2

ð26Þ

Entering the values of K⁎f for two different temperatures in this equation, we can calculate an approximated value (from only two temperatures) of reaction enthalpy change in the formation of metallic complex: 4

DH ¼ −11:7012  10 J=mol ¼ −117:012 kJ=mol This reaction is exothermic, in which formation of metallic complex is hindered at high temperatures. Although it would be more correct to calculate the heat released during the neutralization of each amine functional group with a proton, if we compare the value of enthalpy of formation of complex with the enthalpy of neutralization of a proton (ΔHneutr. = − 56.02 kJ/mol), the first one is clearly higher. On the other hand, using reaction enthalpy calculated, and supposing negligible the change within the temperature range 25–100 °C, we can estimate the conditional formation constant of complex at different temperatures. In this way, at a temperature of 75 °C, the conditional formation constant at pH 4 acquires a value similar to that one calculated at pH 2 and 25 °C (1.98·10 3). This circumstance allows to state that combining the two effects (temperature and pH), thermochemical regeneration of polymer can be achieved at milder pH values than those at 25 °C. Moreover, the increase of operation pH value during polymer regeneration stage would diminish the reagents consumption and minimize the appearance of determined types of corrosion. 5. Conclusions A new kinetic model has been successfully applied to predict the temporal evolution of metal concentration in both permeate and rejected streams of a batch polymer enhanced ultrafiltration process. Specifically, the removal of copper ions by poly (ethyleneimine) has been selected to verify its practical application. This model allows us as well determining not only the most suitable operating pH values for metal retention (pH = 4–5) and polymer regeneration processes (pH = 1) at 25 °C, but also calculating pH- and temperature-depending conditional complex formation constants, whose values are significantly similar to those obtained in previous works by means of potentiometric and spectrophotometric methods.

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