Modelling the transport of Cl− ions through the anion-exchange membrane NEOSEPTA-AFN

Modelling the transport of Cl− ions through the anion-exchange membrane NEOSEPTA-AFN

Journal of Membrane Science 165 (2000) 237–249 Modelling the transport of Cl− ions through the anion-exchange membrane NEOSEPTA-AFN Systems HCl/membr...

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Journal of Membrane Science 165 (2000) 237–249

Modelling the transport of Cl− ions through the anion-exchange membrane NEOSEPTA-AFN Systems HCl/membrane/H2 O and HCl–FeCl3 /membrane/H2 O Zdenˇek Pa1atý ∗ , Alena Žáková, Petr Doleˇcek ˇ legi´ı 565, 532 10 Pardubice, Czech Republic Department of Chemical Engineering, University of Pardubice, Cs. Received 4 September 1998; received in revised form 12 July 1999; accepted 15 July 1999

Abstract The paper deals with the diffusion dialysis of hydrochloric acid in a mixed batch cell with an anion-exchange membrane NEOSEPTA-AFN. The systems HCI/membrane/H2 O and HCl–FeCl3 /membrane/H2 O were investigated. The overall dialysis coefficient and the membrane mass transfer coefficient were used for a quantitative evaluation of the mass transfer rate of Cl− ions in the membrane. Both these quantities were calculated by numerical integration of differential equations, which describe the unsteady state transport of Cl− ions, with subsequent optimization procedure. The model suggested presumes the zero flux of Fe3+ ions, existing equilibria and mass transfer resistances on both sides of the membrane. It was found that, in the range of HCl concentration from 0.27 to 3.48 kmol m−3 , and of Fe3+ ions from 0.17 to 1.66 kmol m−3 , the rate of transport of Cl− ions decreases with increasing concentration of Fe3+ ions in the solution dialyzed. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Diffusion dialysis; Anion-exchange membrane; HCl–FeCl3 solutions; Solution/membrane equilibrium

1. Introduction For many years, separation processes based on the different rate of transport of components through ion-exchange membranes have been used frequently in industrial practice. Diffusion dialysis and electrodialysis belong among them. The main advantage of diffusion dialysis is the low consumption of energy during the process. It is preferably used for the separation of inorganic acids from acid waste waters containing, besides acids, their salts [1–3]. Although these processes are used on a large scale, the phys∗ Corresponding author. Tel.: +42-40-60-37-503; fax: +42-40-6037-068. E-mail address: [email protected] (Z. Pa1at´y).

ical chemical data of the membranes used are not published to such an extent so as to enable the optimization of the process. A permeability coefficient, a membrane mass transfer coefficient or a diffusion coefficient of the component in the membrane are the basic characteristics of the ion-exchange membranes. In order to determine them, a simple experimental apparatus — a two-compartment batch cell — is often used. In the course of the experiment, the time dependencies of the concentration of components in both compartments are recorded. A correct data treatment demands not only information on equilibria, which exist in the system given, but also further information concerning mass transfer resistances in the diffusion films on both sides of the membrane.

0376-7388/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 9 9 ) 0 0 2 3 9 - 2

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The aim of this paper is to give a better understanding of the transport of Cl− ions in the presence of Fe3+ ions through the anion-exchange membrane if the transport of Cl− ions in the diffusion films is taken into account. The systems HCl/H2 O and HCl–FeCl3 /H2 O, which can be encountered in practice, were chosen as model systems.

2. Theory Generally, the transport of electrolytes through the membrane is described by the Nernst–Planck equation. If ionic equilibria between many species must be taken into account, its application can be considerably complicated. Moreover, other physical properties such as membrane conductivity or membrane potential at various compositions of the solution must be known. In order to describe the membrane transport, the mass transfer coefficient, which can easily be determined from only the concentration data, is used here. It is true that it does not fully reflect the actual condition in the membrane, but from the chemical engineering point of view, it can be considered as an adequate parameter. In the systems HCl/membrane/H2 O and HCl–FeCl3 / membrane/H2 O, Cl− ions are transported through the anion-exchange membrane. Taking into account the volume changes, the unsteady state transport of Cl− ions is described by the following differential equations: VI

d[Cl− ]I dV I +[Cl− ]I =−kLI A{[Cl− ]I −[Cl− ]Ii } (1) dτ dτ

of Cl− ions depends on the total concentration of HC1 and FeCl3 and on the equilibria which exist in this system. The following ionic equilibria can be considered [4]: Fe3+ + Cl− FeCl2+

(4)

Fe3+ + 2Cl− FeCl+ 2

(5)

Fe3+ + 3Cl− FeCl3

(6)

Fe3+ + 4Cl− FeCl− 4

(7)

Using the substitution I d[Cl− ]I dcCl− d[Cl− ]I · = I dτ dτ dcCl −

Eqs. (1)–(3) can be rewritten to give the forms I dcCl −



V

VI

− I ]



− I dV

+ [Cl ]

dτ − [Cl− ]II M}

I

=

(2)

d[Cl− ]I dV I − II +[Cl− ]I = −kLII A{[Cl− ]II i − [Cl ] } dτ dτ (3)

Eqs. (1) and (3) describe the transport of Cl− ions through the diffusion films on both sides of the membrane, whereas Eq. (2) concerns the transport of Cl− ions through the membrane. The actual concentration

kLI A{[Cl− ]I − [Cl− ]Ii } + [Cl− ]I (dV I /dτ ) I ) V I (d[Cl− ]I /dcCl −

I dcCl −

dτ =−

− I I kM,Cl− A{[Cl− ]IM − [Cl− ]II M } + [Cl ] (dV /dτ ) I ) V I (d[Cl− ]I /dcCl −

(10) I dcCl −

=−

−kM,Cl− A{[Cl− ]IM

=−

(9)

dτ I d[Cl

(8)

− II − I I kLII A{[Cl− ]II i − [Cl ] } + [Cl ] (dV /dτ ) I ) V I (d[Cl− ]I /dcCl −

(11)

Eqs. (9)–(11) express (in the differential form) the dependence of the total concentration of chloride ions I cCl − in compartment I on time. The concentration of Cl− ions at the interface and in the membrane are interrelated as follows: j

j

[Cl− ]M = 9 j [Cl− ]i

j = I, II

(12)

Modification of Eqs. (9)–(11) using the relations (12) leads to the equation

Z. Pa1at´y et al. / Journal of Membrane Science 165 (2000) 237–249 I dcCl −

dτ =−

KCl− A{[Cl− ]I − (9 II /9 I )[Cl− ]II } +[Cl− ]I (dV I /dτ ) I ) V I (d[Cl− ]I /dcCl −

(13)

where KCl− =

1 (1/kLI ) + (1/(9 I kM,Cl− )) + (9 II /9 I kLII )

(14)

is the overall dialysis coefficient, whose reciprocal value is a sum of mass transfer resistances in the diffusion films and the membrane. I If the experimental data cCl − = f (τ ) are available, it is possible, using the set of the differential Eqs. (9)–(11), to determine the membrane mass transfer coefficient for Cl− ions. The procedure, which demands further information enabling an estimate of the mass j transfer coefficients kL (j = I, II) and the calculation of the partition coefficients 9 j (j = I, II), is based on a numerical integration of the differential Eq. (10) with subsequent optimization. In a similar way, it is possible to use Eq. (13) to determine the overall dialysis coefficient KCl− . In the case of the system HCl/membrane/H2 O, Eqs j (9)–(11) can be simplified with cCl− = [Cl− ]j (j =I, I II) and the derivative d[Cl− ]I /dcCl − equal to unity.

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repeatedly for 2 h (three to four times) in 25 ml of distilled water. In the case of the measurement of equilibria using the solutions containing FeCl3 , the membrane still contained the rest of Fe3+ ions which were extracted into 0.5 M HCl. The concentration of HCl in the extracts without Fe3+ was determined conductometrically. The total concentration of Cl− ions in the extracts with Fe3+ ions was determined titrimetrically with 0.02 M AgNO3 and potentiometric determination of equivalence point. The concentration of Fe3+ ions in these extracts was determined spectrophotometrically — the reaction between Fe3+ ions and rhodanide being used. The concentration of HCl in the solution was determined titrimetrically with 0.1 M NaOH, the total concentration of Cl− ions in the solutions containing, besides HCl, FeCl3 also, was determined titrimetrically with AgNO3 . In both cases, the equivalence points were found potentiometrically. The ferric ions concentration was determined chelatometrically. 3.2. Dialysis experiments Dialysis of hydrochloric acid was investigated in a two-compartment cell with stirrers. The cell is described in detail elsewhere [5]. The experimental set up is shown in Fig. 1. In all the experiments, the anion-exchange membranes NEOSEPTA-AFN manufactured by TOKUYAMA SODA Co. Ltd. were used. The basic

3. Experimental 3.1. Concentrations of Cl− and Fe3+ ions in the membrane The concentrations of Cl− and Fe3+ ions in the membrane which is equilibrated with the solution of a given composition were determined by the method based on the saturation of the membrane with hydrochloric acid and/or with a mixture of hydrochloric acid and ferric chloride followed by the extraction of the components into water. The membrane of 25–40 cm2 surface area, which was kept in a 0.5 M NaCl, was rid of salt by thorough washing in distilled water and shaken with the solution of a given composition for 18 h. Then, the membrane was wiped quickly with blotting paper to remove the solution adhering the surface of the membrane whereupon it was shaken

Fig. 1. Experimental set up: 1 batch cell; 2 partition; 3 membrane; 4 stirrers; 5 thermostat; 6 thermostat reservoir; I, II compartments of cell.

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physical properties of the membrane used were as follows: thickness 165 ␮m, water content 0.418 g g−1 of dry membrane in Cl− form and concentration of fixed charges (considered as monovalent) 4.7 kmol m−3 . The area of each membrane was 62.2 cm2 . At the beginning of each experiment, the compartments I and II were filled with hydrochloric acid and/or with a mixture of hydrochloric acid and ferric chloride and distilled water, respectively. The initial hydrochloric acid concentration at cFe3+ = 0 was changed in the range from 0.28 to 3.64 kmol m3 , rotational speed of the stirrers (nI = nII ) was varied within the limits from 1.07 to 13.67 s−1 . In the experiments concerning dialysis of hydrochloric acid in the system HCl–FeCl3 /membrane/H2 O, the initial concentration of HCl was in the range from 0.27 to 3.48 kmol m3 and that of ferric ions was varied in the limits from 0.17 to 1.66 kmol m3 . Rotational speed of the stirrers was kept constant (nI = nII = 9.33 s−1 ). The initial volumes of liquid in both the compartments were always 1 l. In the course of each measurement, the concentration of HCl and/or the concentration of Cl− and Fe3+ and the height of liquid levels (for the determination of the volume changes in both the compartments) were determined. The concentrations of both components in solution were determined in the same way as in the case of the measurement of solution/membrane equilibria. The changes in the liquid levels were measured by means of a modified micrometer screw with a needle attached in the center of the emerging piston. In all the experiments, temperature was kept at the constant value of 20 ± 0.5◦ C. The duration of the experiment was from 45 to 96 h. 4. Treatment of data and discussion 4.1. Concentrations of Cl− and Fe3+ ions in the membrane NEOSEPTA-AFN The concentrations of Cl− and Fe3+ ions in the membrane were calculated from the concentrations of these components in the individual extracts and the solution volume in the membrane which was determined from density of the HCl–FeCl3 solution, the weight of the membrane saturated with the solution, and that of dried membrane in Cl− form. The membranes were dried in vacuum at 60◦ C.

Fig. 2 presents the dependencies of the total concentration of Cl− ions in the membrane (i.e. the total concentration of Cl− ions sorbed in the membrane without taking into account the concentration of Cl− ions which are in equilibrium with fixed charges) upon the total concentration of Cl− ions in the solution — the concentration of Fe3+ in the solution is a parameter of the curves. Curve 1 is valid for the zero concentration of Fe3+ ions and curve 2 was found for the concentrations of Fe3+ ions in the limits from 0.10 to 1.75 kmol m3 . From Fig. 2, it can be seen that the total concentration of Cl− ions in the membrane depends not only upon the total concentration of Cl− ions in the solution but also upon the concentration of Fe3+ ions in the solution. The total concentration of Cl− ions in the membrane increases with increasing concentration of both the ions in the solution. For further utilization of the data obtained concerning the solution/membrane equilibrium, the dependencies mentioned above were approximated by the following empirical equations: 2 cCl− ,M = p11 cCl− + p21 cCl −

cFe3+ = 0

(15)

cCl− ,M = p02 {1 + p12 exp [−p22 (cCl− − p32 )]}−1/p12 0.10 < cFe3+ < 1.75 kmol m−3

(16)

The pij constants (i = 0, 1, 2, 3; j = 1, 2) in Eqs (15) and (16) were determined by non-linear regression: ip21 = 7.39 × 10−2 m3 kmol−1 p11 = 8.45 × 10−1 −3 p02 = 17.51 kmol m jp12 = 1.66; p22 = 1.15 m3 −1 kmol ; p32 = 4.71 kmol m−3 . The dependencies of the total concentration of Fe3+ ions in the membrane upon the total concentration of Cl− ions in the solutions are given in Fig. 3. Curve 1 is valid for the total concentration of Fe3+ ions in the solution equal to 0.1 kmol m−3 , while curve 2 was obtained for the concentrations of Fe3+ ions in the solution in the range from 0.50 to 1.75 kmol m−3 . From the dependencies presented, it is rather surprising that the anion-exchange membrane, which in fact transports negligible amounts of Fe3+ ions, has such a high affinity for these ions. The concentrations of Fe3+ ions are very high as compared to those in the solution — they reach the value of about 5.5 kmol m−3 . The high content of Fe3+ ions was proved qualitatively by the visual observation that an intensive yellow-brown

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241

Fig. 2. Dependence of the total concentration of Cl− ions in the membrane upon the total concentration of Cl− ions in solution: 1 – cFe3+ = 0; 2 – cFe3+ ∈ h0.10; 1.75i kmol m−3 (cFe3+ 䊊 – 0.10 kmol m−3 ; N – 0.50 kmol m−3 ; +– 1.00 kmol m−3 ; 䊐 – 1.50 kmol m−3 ; 䊉 – 1.75 kmol m−3 ).

colouring of the membrane equilibrated with the solution was found. With respect to further utilization of the equilibria, the dependencies obtained were approximated by the following empirical equations: 2 cFe3+ ,M = p03 + p13 cCl− + p23 cCl −

cFe3+ = 0.1 kmol m−3

(17)

cFe3+ ,M = p04 {1 + p14 exp [−p24 (cCl− − p34 )]}−1/p14 0.50 < cFe3+ < 1.75 kmol m−3

(18)

The constants p03 , p13 , p23 , p04 , p14 , p24 , p34 of the empirical Eqs. (17) and (18) were determined by non-linear regression: p03 = 6.02 × 10−2 kmol m−3 ; p13 = −8.66 × 10−2 ; p23 = 6.20 × 10−2 m3 kmol−1 ;

Fig. 3. Dependence of the total concentration of Fe3+ ions in membrane upon the total concentration of Cl− ions in solution: 1 – cFe3+ = 0.10 kmol m−3 ; 2 – cFe3+ ∈ h0.50; 1.75i kmol m−3 (cFe3+ : N – 0.50 kmol m−3 ; +– 1.00 kmol m−3 ; 䊐 – 1.50 kmol m−3 ; 䊉 – 1.75 kmol m−3 ).

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p04 = 5.52 kmol m−3 ; p14 = 6.21 × 10−1 kmol m−3 ; p24 = 8.82 × 10−1 m3 kmol−1 ; p34 = 4.32 kmol m−3 .

4.2.1. Overall dialysis coefficient In order to calculate the overall dialysis coefficient for Cl− ions, the procedure which is based on a numerical integration of the differential Eq. (13) with subsequent optimization was used. In Eq. (13), the ratio 9 II /9 I was supposed to be equal to unity. All of the procedure, which furthermore presumes that the membrane does not transfer Fe3+ ions at all, can be summarized into the following steps: 1. A numerical calculation of the derivative (dVI /dτ ). 2. The initial estimate of the overall dialysis coefficient KCl− . 3. The integration of the differential Eq. (13) by the Runge–Kutta fourth order method with an integration step of 900 s. In this step, we obtain the calculated values of the total concentrations of Cl− I,i ions in compartment I of the cell cCl − ,calc . − The actual concentrations of Cl ions in compartment I of the cell [Cl− ]I at a given time were calculated from the following set of relations describing the equilibria of ions:

I [FeCl+ 2]

(19)

− β2I = 0

(20)

[FeCl3 ]I − β3I = 0 [Fe3+ ]I ([Cl− ]I )3

(21)

[Fe3+ ]I ([Cl− ]I )2

I [FeCl− 4]

[Fe3+ ]I ([Cl− ]I )4

− β4I = 0

(22)

balance equations − I + I I I cFe 3+ − [FeCl4 ] − [FeCl3 ] − [FeCl2 ]

− [FeCl2+ ]I − [Fe3+ ]I = 0

(23)

− I − I I cCl − − [Cl ] − 4[FeCl4 ] − 3[FeCl3 ] I 2+ I − 2[FeCl+ 2 ] − [FeCl ] = 0

I [H+ ]I + 3[Fe3+ ]I + 2[FeCl2+ ]I + [FeCl+ 2] I − [Cl− ]I − [FeCl− 4] =0

4.2. Dialysis experiments

[FeCl2+ ]I − β1I = 0 [Fe3+ ]I [Cl− ]I

and the electroneutrality condition

(24)

(25)

where the negligible concentration of OH− ions is presumed. The set of Eq. (19) through Eq. (25) was solved by the Newton–Raphson method. The stability constants βiI (i = 1, 2, 3, 4) were taken from the literature [4]. In the calculation of these constants, ionic strength was corrected iteratively. I The derivative d[Cl− ]I /dcCl − was determined numerically. The actual concentration of Cl− ions in compartment II of the cell [Cl− ]II was supposed to be equal to that of HCl in this compartment, i.e. the complete dissociation of acid was presumed. 4. The calculation of the total concentration of Cl− II,i ions in compartment II of the cell cCl − ,calc from mass balance. 5. The calculation of the objective function n  2  X I,i I,i cCl F KCl− = − ,exp − cCl− ,calc i=1

+



II,i cCl − ,exp

II,i − cCl − ,calc

2  (26)

6. The realization of one step of the optimization procedure. The golden section search was used. This step provides the corrected value of the overall dialysis coefficient. 7. Repetition of the procedure from 3–6 until the minimum of the objective function (26) was reached. In Fig. 4, the values of the total concentrations of chloride ions in both the compartments corresponding to the minimum of the objective function (26) are compared with those found experimentally. On the basis of small deviations between the calculated and measured values of the total concentrations of chloride ions, it can be concluded that a good agreement between the model and the experiment exists. The model suggested, which is represented by Eqs. (13) and (19)–(25), presumes that the anion-exchange membrane does not transport Fe3+ ions at all. However, it was found experimentally that the membrane used transports Fe3+ ions partially. The amount of Fe3+ ions transported increases with increasing concentration of these ions and the duration of the

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243

Fig. 4. Comparison of experimental (䊉, 䊊) and calculated values (––—) of the total concentrations of Cl− ions in both compartments of I −3 and cI −3 the cell (䊉 – compartment I, 䊊 – compartment II) at cFe 3+ ,0 = 0.90 kmol m HCl,0 = 3.48 kmol m .

experiment — the maximum fraction of transported I −3 , Fe3+ ions was about 8% (cFe 3+ ,0 = 1.66 kmol m τ = 94 h). Fig. 4 shows a typical course of dialysis in the system HCl–FeCl3 /membrane/H2 O in the batch cell. From Fig. 4, it is evident that, during the experiment, the total concentration of Cl− ions in compartment I decreases, while that in compartment II increases. However, the rates of the changes in the total concentration of Cl− ions in the individual compartments

of the cell decrease until the constant non-zero difference between the total concentrations of Cl− ions in both compartments is reached. This concentration difference increases with increasing concentration of Fe3+ ions in compartment I of the cell. In Fig. 5, the time dependence of HCl concentraI = tion in both the compartments is given for cHCl,0 I −3 −3 3.48 kmol m and cFe3+ ,0 = 0.90 kmol m . From Fig. 5, it can be seen that, after a sufficiently long duration of experiment, the concentration of hydrochloric

Fig. 5. Time dependencies of HCl concentration in both compartments of the cell (䊉 – compartment I, 䊊 – compartment II) at I −3 and cI −3 cFe 3+ ,0 = 0.90 kmol m HCl,0 = 3.48 kmol m .

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acid in compartment II reaches higher values than that in compartment I, this increase being more significant at higher concentrations of Fe3+ ions in compartment I. For that reason, a good efficiency of the continuous dialyser, which separates HCl from the mixture HCl–FeCl3 , can be expected. Here, it is suitable to mention some results in the paper [6] in which it is shown that in the description of the transport of an acid through the anion-exchange membrane in multicomponent systems, it is sufficient to consider the total concentrations of the acid only. However, this conclusion is in contradiction with the conclusion of our work, from which it is evident that the driving force in the dialysis of a multicomponent system can be expressed on the basis of the actual concentrations of ions transported through the membrane. If the driving force were considered as a difference between the concentrations of hydrochloric acid in compartments I and II, after a certain time, its sign would be changed and the process would take place in an opposite direction. In the dialysis of multicomponent systems, it is then sufficient to respect the existing physico-chemical phenomena (i.e. ionic equilibria) without creating a new theory as mentioned in [1] in connection with the dialysis of HNO3 in the HNO3 –HF system containing Fe3+ ions. Fig. 6 shows the dependence of the coefficient KCl− upon the concentration of hydrochloric acid — the

total concentration of Fe3+ ions is a parameter of the curves. The dependence of the coefficient KCl− upon the concentration of Fe3+ ions is presented in Fig. 7 — in this case, the concentration of hydrochloric acid is a parameter of the curves. From both figures, it is evident that the overall dialysis coefficient for Cl− ions, which was calculated under the presumption 9 II /9 I = 1, is affected considerably by the presence of Fe3+ ions in the solution dialyzed. At low concentrations of Fe3+ ions (cFe3+ ,0 = 0.21 kmol m−3 ) the overall dialysis coefficient increases markedly with increasing HCl concentration, while at high concentrations of Fe3+ ions, the increasing HCl concentration does not lead to a significant change in KCl− . In this case, a weak maximum can be observed on the depenI ) and the values of the overdence KCl− = f (cHCl,0 all dialysis coefficient are much lower than those for cFe3+ ,0 = 0.21 kmol m−3 . At the constant initial HCl concentration in compartment I, the overall dialysis coefficient for Cl− ions decreases if the concentration of Fe3+ ions is increasing — this decline is affected by the initial concentration of HCl in compartment I. 4.3. Membrane mass transfer coefficient kM,Cl − The membrane mass transfer coefficient was determined in a similar way as the overall dialysis coefficient KCl− . The procedure, however, was more

Fig. 6. Dependence of the overall dialysis coefficient KCl− upon the initial HCl concentration in compartment I of the cell: 䊉 – I −3 ; 䊊 – cI cFe = 1.66 kmol m−3 . 3+ ,0 = 0.21 kmol m Fe3+ ,0

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245

Fig. 7. Dependence of the overall dialysis coefficient KCl− upon the initial concentration of Fe3+ ions in compartment I of the cell: 䊉 – I I = 0.32 kmol m−3 ; 䊊 – cHCl,0 = 3.43 kmol m−3 . cHCl,0

complex than that used above and it demanded data on the solution/membrane equilibrium. Similarly, as in the case of the overall dialysis coefficient KCl− , the presumption of the zero flux of Fe3+ ions through the membrane was considered — the supposed concentration profiles of Cl− and Fe3+ ions are shown in Fig. 8. All of the procedure, which is based on a numerical integration of the differential Eq. (10), can be summarized into the following steps: 1. A numerical calculation of the derivatives (dV/dτ ) I ). and (d[Cl− ]I /dcCl − 2. The initial estimate of the membrane mass transfer coefficient.The integration of the differential Eq. (10) by the Runge–Kutta fourth order method with the integration step h = 900 s. In this step, one obtains the calculated values of the total concenI trations of Cl− ions in compartment I cCl − ,calc in the same time intervals as those of the experimental values. In order to calculate these values, it was necessary in each integration step to determine the concentrations of Cl− ions in the membrane, i.e. [Cl− ]IM and [Cl− ]II M . The procedure used is as follows: (a) The estimate of the concentrations [Cl− ]Ii and [Cl− ]II i . (b) The calculation of the total concentration of Cl− ions at the interface in compartment I by solving the set of Eqs. (19)–(25). For a specified

actual concentration of Cl− ions and the total concentration of Fe3+ ions, the original set of Eqs. (19)–(25) is transferred to a set of linear equations, which can easily be solved by the Gauss elimination method. In this step, the condition of electroneutrality is supposed to be valid at the solution/membrane interface also — a similar presumption was used in [7]. (c) The calculation of the total concentrations of Cl− and Fe3+ ions in the membrane. For the I I , Eqs. (16)–(18) calculation of cCl − ,M and c 3+ Fe ,M were used (a linear interpolation was adopted for the concentration range of Fe3+ ions from 0.10 to 0.50 kmol m−3 ), the total concentration of Cl−

Fig. 8. Concentration profiles of Cl− and Fe3+ ions in membrane and diffusion films.

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ions in compartment II (which is equal to the acid concentration) was determined from the sorption isotherm valid for cFe3+ = 0 (Eq. (15)). (d) The calculation of the actual concentration of Cl− ions in the membrane using the set of Eqs. (19)–(25) with a slightly modified Eq. (25) — the + was added to the left-hand side of Eq. term cM (25) to respect fixed charges in the membrane. (e) The calculation of the mean concentration of Cl− ions in the diffusion film in compartment II of the cell: [Cl− ]II m =

− II [Cl− ]II i + [Cl ] 2

(27)

(f) The calculation of the mass transfer coefficients from the equation j

kL = Bn1/2 D 2/3 ν −1/6 ,

j = I, II

(28)

where the constant B is equal to 2.39. Diffusivity and kinematic viscosity of HCI were taken from the literature [8–11]. The mass transfer coefficient was calculated only for free HCl. (g) The calculation of the corrected concentrations of Cl− ions at the interface from Eqs. (1)–(3). (h) The procedure from (3a)–(3g) was repeated until the relative changes in the actual concentraj tions Cl− ions [Cl− ]i (j = I, II) were below 0.5%. 3. The calculation of the total concentration of Cl− in compartment II using the mass balance. 4. The calculation of the objective function (26) (as a function of kM,Cl− ). 5. The realisation of one step of the optimising procedure — the golden section search was used. In this step, the corrected value of the coefficient kM,Cl− is obtained. 6. The procedure from 3–6 was repeated until the minimum of the objective function (26) was reached. In the case of the system HCl/membrane/H2 O, the membrane mass transfer coefficient for Cl− ions can be calculated using a slightly modified procedure I given above as it is d[Cl− ]/dcCl − = 1, moreover, the steps (3b) and (3d) need not be considered. The constant B in Eq. (28) was calculated using all experimental data concerning diffusion dialysis in the system HCl/membrane/H2 O. The procedure used was based on the following idea: the membrane mass

transfer coefficient is a parameter which is particularly affected by membrane properties. Its value would not be dependent on hydrodynamic conditions, i.e. on the rotational speed of stirrers. From this point of view, the constant B was calculated in such a way so as to achieve the minimum of standard deviations of membrane mass transfer coefficients for a given initial acid concentration in compartment I. Simultaneously, the achievement of the best coincidence between calculated and experimental values of the hydrochloric acid concentration in both compartments was sought too. As mentioned in the calculation of diffusivity of Cl− ions, the relations taken from the literature [8–11] concerning the aqueous solutions of hydrochloric acid were used. In reality, the rate of diffusion of each ion in a mixture of electrolytes is affected by the concentration of all other ions in the mixture. For the case of concentrated mixtures of electrolytes, the reliable theory or empirical correlations in order to calculate diffusivity are not at hand. In order to judge the effect of the change in diffusivity on the determination of the membrane mass transfer coefficient kM,Cl− , three verifying calculations were carried out, diffusivity of Cl− calculated for the HCl solution, its half and double values being used. The membrane mass transfer coefficients calculated did not differ significantly. On the basis of the results obtained from the basic data measured at the different rotational speeds of the stirrers in the system HCl/membrane/H2 O, it was found that the membrane mass transfer coefficient for Cl− ions (equal to that for HCl) is a function of the initial concentration of HCl in compartment I — see Fig. 9. From Fig. 9, it is evident that increasing the initial concentration of HCl results at first in an increase in the membrane mass transfer coefficient and then (above a concentration about 2.6 kmol m−3 ) in its mild decrease. A similar course of the dependence of the membrane mass transfer coefficient upon the initial concentration of the component dialyzed in compartment I was observed in the case of dialysis of sulfuric acid also using an identical membrane [12]. However, the values of the membrane mass transfer coefficient for Cl− ions are higher than those for sulfuric acid. The results concerning the system HCl–FeCl3 membrane H2 O are presented in Figs. 9 and 10. Fig. 9 shows the dependence of the membrane mass transfer coefficient upon the concentration of HCl; Fig. 10

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Fig. 9. Dependence of the membrane mass transfer coefficient kM,Cl− for Cl− ions upon the initial HCl concentration in compartment I I I −3 ; 䊉 – cI −3 . = 0.21 kmol m−3 ; 䊊 – cFe of cell: N – cFe 3+ ,0 = 0 kmol m 3+ ,0 = 1.66 kmol m Fe3+ ,0

concerns the dependence kM,Cl− = f (cFe3+ ,0 ). The initial concentration of Fe3+ ions (in Fig. 9) and HCl (in Fig. 10) in compartment I are parameters of the curves. From these graphical presentations, it is evident that the dependencies of the membrane mass transfer coefficient upon the concentration of HCl are very similar to those for the overall dialysis coefficient. Furthermore, the values of the membrane mass transfer coefficient in the presence of Fe3+ ions are always lower than those for pure hydrochloric acid.

The highest reduction of the membrane mass transfer coefficient can be observed at the highest concentration of Fe3+ ions in the solution dialyzed. From Fig. 10, it can be seen that, at the HCl concentration of 0.32 kmol m−3 , at first, the membrane mass transfer coefficient remains practically constant — up to cFe+3 ,0 ≈ 0.9 kmol m−3 — then, it decreases considerably, while at the highest HCI concentration, a weak minimum on the dependence kM,Cl− = f (cFe3+ ,0 ) can be identified.

Fig. 10. Dependence of the membrane mass transfer coefficient kM,Cl− for Cl− ions upon the initial concentration of Fe3+ ions in I I compartment I of the cell: 䊉 – cHCl,0 = 0.32 kmol m−3 ; 䊊: – cHCl,0 = 3.43 kmol m−3 .

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(i = 1, 2, 3, 4) constants (–) (i = 1, 2, 3, 4) constants (m3 kmol−1 ) (i = 2, 3) constants (kmol m−3 ) volume (m3 ) actual concentration (kmol m−3 , in Eqs. (19)–(22) dimensionless)

From the comparison of the membrane mass transfer coefficients obtained for dialysis in the systems HCl/H2 O and HCl–FeCl3 /H2 O, it is evident that the presence of Fe3+ ions brings down considerably the value of the membrane mass transfer coefficient for Cl− ions.

p1i p2i p3i V []

5. Conclusion

6.1. Greek symbols

A mathematical model, which on the basis of experimental data on the transport of Cl− ions through the membrane in the batch cell enables the calculation of the membrane mass transfer coefficient for Cl− ions in the systems HCl/membrane/H2 O and HCl–FeCl3 /membrane/H2 O, was developed. This model represented by the ordinary differential equation describes the time dependence of the total concentration of Cl− ions in the compartment with a higher concentration of Cl− ions. It presumes a zero flux of Fe3+ ions, considers ionic equilibria in the solution and membrane and takes into account mass transfer resistances in the liquid films and the solution/membrane equilibrium. The overall dialysis coefficient and also the membrane mass transfer coefficient were determined by the procedure based on a numerical integration of the basic differential equation followed by the optimization procedure. It was found that both the overall dialysis coefficient and the membrane mass transfer coefficient for Cl− ions decrease with increasing content of Fe3+ ions.

βi ν 9 τ

(i = 1, 2, 3, 4) complexity constants (–) kinematic viscosity (m2 s−1 ) partition coefficient (–) time (s)

6.2. Indexes calc Cl− exp Fe3+ HCl I II i M m

calculated value related to Cl− ions experimental value related to Fe3+ ions related to hydrochloric acid related to compartment I of cell related to compartment II of cell related to interface related to membrane mean value

Acknowledgements 6. Symbols A B c + cM D F KCl− kL kM,Cl− n p0i

membrane area (m2 ) constant in Eq. (28) (–) total molar concentration (kmol m−3 ) concentration of fixed charges (kmol m−3 ) diffusivity (m2 s−1 ) objective function (kmol m−3 ) overall dialysis coefficient (m s−1 ) liquid mass transfer coefficient (m s−1 ) membrane mass transfer coefficient for Cl− ions (m s−1 ) rotational speed of stirrers (s−1 ) (i = 2, 3, 4) constants (kmol m−3 )

This work was financially supported by the Grant Agency of the Czech republic, Grant number 104/93/2159. References [1] Y. Kobuchi, H. Motomura, Y. Noma, F. Hanada, Application of ion exchange membranes to the recovery of acids by diffusion dialysis, J. Membr. Sci. 27 (1986) 173–179. [2] Y. Kobuchi, H. Motomura, Y. Noma, F. Hanada, Application of the ion-exchange membranes acids recovery by diffusion dialysis, in: Proceedings of the Europe–Japan Congress on Membrane and Membrane Processes, Stresa, Italy, 1984. [3] Y. Kobuchi, Developing Membrane Techniques and its Application — Ion-exchange Membrane, Tokuyama Soda Co., Ltd., Tokyo, 1984.

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[8] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, 2nd ed., Butterworths, London, 1959. [9] S. Bretsznajder, Wlasno´sci gazów i cieczy, Wydawnictwa naukovo-techniczne, Warsaw, 1966. [10] International Critical Tables, vol. 5, McGraw-Hill, New York, 1926–1933. [11] Landolt-Börnstein, 6. Auflage II. band, Teil 5, Bandteil a, Transport phänomene (Viskozität und diffusion), Springer, Berlin, 1969, p. 305. [12] Z. Palatý, A. Žáková, Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN, J. Membr. Sci. 119 (1996) 183–190.