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Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN
j o u r n a l of MEMBRANE SCIENCE ELSEVIER
Joumal of MembraneScience 119 (1996) 183-190
Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN v
Z. Palat2~, A. ZS.kova Department of Chemical Engineering, University of Pardubice, s. Cs. legi1565, 532 10 Pardubice, Czech Republic
Received 12 February 1996; revised 27 March 1996; accepted10 April 1996
Abstract This paper deals with the determination of the membrane mass transfer coefficient for sulfuric acid in an anion-exchange membrane NEOSEPTA-AFN. This quantity has been determined on the basis of experiments carried out in a batch dialysis cell using the method of numerical integration of the basic differential equation describing the time dependence of sulfuric acid concentration and subsequent optimization procedure. The experiments carried out made it possible to calculate the membrane mass transfer coefficient for sulfuric acid over the concentration range from 0.1 to 1.9 kmol m -3 in the external solution. Keywords: Diffusion dialysis; Membrane mass transfer coefficient; Diffusivity of sulfuric acid; Anion-exchange membrane; Membrane/solution equilibrium
1. Introduction The Japanese firm Tokuyama Soda Co. is one of the important producers of ion-exchange membranes and their products find broad application in diffusion dialysis and electrodialysis. However, the physicochemical data of these membranes published so far are not adequate to their industrial application. Application of and research into basic properties of NEOSEPTA membranes have been dealt with in several papers [1-10], the main attention being focused upon the equilibria between solution and membrane [1,4-6,9] and the working conditions or separation factors [3,10]. Since the NEOSEPTA-AFN membrane is often used in the separation of sulfuric acid from aqueous solutions containing beside sulfuric acid also its salts, and since the equilibrium HzSO 4 solution-membrane has not been studied yet, this paper is focused
upon the said equilibrium and experimental determination of the membrane mass transfer coefficient for sulfuric acid. Although the HSO 4 and SO 2- ions, which result from dissociation of the acid, predominantly penetrate through the membrane (the transport of dissociated forms of sulfuric acid through the membrane was dealt with in [11]), the present paper, for simplicity, considers the transport of acid without regard to this fact. The experimental determination of the membrane mass transfer coefficient can adopt a two compartment batch cell with mixing where unsteady state mass transport takes place. With the volume changes taken into account, the unsteady state transport of component A is described by the following differential equations
dCIA dT --
0376-7388/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0376-7388(96)00122-6
kI A V I (cI--cli)
dW I
W I dT
(1)
184
Z. Palate, A. Z.fkovf / Journal of Membrane Science 119 (1996) 183-190
dc I d--T- = -
kMA -----7-(cIM-c~M) V
dCXA 07"-
k~ a W I (CIAIi--C~)
ClA dV l V' d'c
(2)
c~ dV I W I aT
(3)
Eqs. (1) and (3) describe the transport of component A through the liquid films at both sides of the membrane, whereas Eq. (2) concerns the transport of component A through the membrane. The concentrations of component A at the interface and in the membrane are interrelated as follows:
C~M
=
~3CJAi j = I, II
(4)
Modification of Eqs. (1)-(3) using Eq. (4) leads to the differential Eq. (5) which describes the time dependence of concentration of component A in compartment I of the cell dc~ aT
Z
1/,¢AICI- ,r. II '/-'AIICA
VI 1/¢I 1 --+--+--
1t¢II
CI
dV I
VI aT
(5)
If experimental data are available concerning the time dependence of A concentration in liquid volume of compartment I of the cell and data enabling an estimate of mass transfer coefficients in liquid, then - knowing the partition coefficients ~ ( j = I, II) one can adopt Eq. (5) to determine the membrane mass transfer coefficient k M. The procedure is based on numerical integration of Eq. (5) complemented by a suitable optimizing procedure.
in 25 ml distilled water. Due to ion exchange between the membrane and H 2SO 4 solution, the original solutions and - partially - the extracts, also, contained beside sulfuric acid hydrochloric acid. Concentrations of the two acids were determined in the following way. The first portion of sample was titrated with standard NaOH solution under argon to determine the overall concentration of the acids. In the second portion, sulfate ions were precipitated by addition of Ba(NO3) 2 and the HC1 content was determined by argentometry. The equivalence points were determined potentiometrically (using the universal titrator OH-407, Radelkis, Hungary). In the titrations of original solutions and extracts we used 0.1 and 0.05 M NaOH solutions, respectively. The concentrations of AgNO 3 solutions were 0.01 and 0.005 M. The fact that at the beginning of experiments the membrane is in the C1- form leads to a complex analytical procedure but, on the other hand, it is possible to determine the concentration of the fixed charges in the membrane, also.
2.2. Dialysis cell The dialysis of sulfuric acid was monitored with the help of experimental arrangement shown in Fig. 1. The main part is a batch cell with two compartments equipped with stirrers. The cell is described in detail elsewhere [11].
2. Experimental 2.1. Concentration of H 2504 in membrane The concentration of sulfuric acid in the membrane was determined by the method based on saturation of the membrane with the acid and subsequent extraction of acid into water. The membrane of 25-35 cm 2 surface area was cut into 4 - 6 pieces, rid of salt residues by thorough washing in distilled water, and shaken with 50 ml sulfuric acid of given concentration for 18 h. Then the membrane was rinsed with distilled water for a short time to remove the acid remaining at the surface of the membrane whereupon it was repeatedly shaken for 2 h ( 3 - 4 x )
8
[-I
711 13 II t
I
161
12
Fig. 1. Scheme of experimental arrangement: 1, dialysis cell; 2, partition; 3, membrane; 4, stirrers; 5, conductivity probes; 6, conductometers; 7, thermostat; 8, thermostat reservoir; I and If, compartments of cell.
Z Palate, A. Ztkovd/Journal of Membrane Science 119 (1996) 183-190
In all the experiments, the anion-exchange N E O S E P T A - A F N membranes of the Japanese firm Tokuyama Soda Co. Ltd. were adopted. The surface area of each membrane was 62.2 cm 2, thickness 0.17 mm (in C1- form). In the experiments, the compartments I and II of the cell were filled with sulfuric acid of a given concentration and with distilled water, respectively. The initial concentration of sulfuric acid was varied within the limits from ca. 0.1 to 1.9 kmol m -3, rotational speed of the stirrers was in the range from 1.17 to 13.67 s-1. At the beginning of each measurement the volume of liquid in both compartments of the cell was 1 1 and the height of levels was adjusted at the same value by addition of glass beads. In a given experiment using the same rotational speed of the two stirrers (n I = nil), we determined the HzSO 4 concentrations and level heights to find the volume changes in the two compartments with time. The measurement was finished after attaining equal concentrations in the two compartments of the cell. At the lowest initial H 2 SO 4 concentration (i.e. 0.1 kmol m -3) the conductimetry was adopted for determination of the acid concentration, whereas in the other cases the H2SO 4 concentration was determined titrimetrically with NaOH and potentiometric determination of equivalence point. The changes in level heights were measured by means of a modified micrometer screw with a needle attached at the centre of the emerging piston. In all the experiments, temperature was kept at a constant value of 20 + 0.5°C.
185
3.0
? E E
~alk~
2.0
! o
1.0
0.0~
i
0.0
i
0.5
i0
i
i
1.
i
1.5
cA
,
i
J
2,0
kmol.m
2.5
-3
Fig. 2. Dependence of H 2 S O 4 concentration in membrane upon
HzSO4 concentration in external solution: A, experimental values; ---, values calculated from Eq. (6).
its value was 4.7 kmol m -3 (the concentration is referred to the volume of H2SO 4 in the membrane). The dependence of sulfuric acid concentration in membrane upon concentration of H2SO 4 in external solution is presented in Fig. 2 where from it can be seen that the a n i o n - e x c h a n g e membrane N E O S E P T A - A F N shows a considerable affinity to sulfuric acid. The H 2 SO 4 concentration in membrane is always higher than that in the external solution, the differences being particularly marked for the n 2 s o 4 concentrations in the external solution up to ca. 0.5 kmol m -3. Eq. (6) was suggested to correlate the dependence of H2SO 4 concentration in the membrane vs. that in external solution. CA
3. Results and discussion 3.1. H 2 S O 4 concentration in m e m b r a n e
The concentration of sulfuric acid dissolved in the membrane was calculated on the basis of H2SO 4 concentrations in the individual extracts and the solution volume in the membrane which was determined from density of HzSO 4, the weight of the membrane saturated with acid and that of dried membrane in HSO 4 form. The membranes were dried in vacuum at 60°C. The concentration of fixed charges was calculated from the amount of C1- ions released -
CAM = P l + p 2 C A
+P3 c2
(6)
where the Pi constants (i = 1, 2, 3) were determined by nonlinear regression (p~ = 3.88 × 10 -2 _+ 5.6 × 10 -6, P 2 = 0 . 4 9 5 _ 2 . 1 × 10 -4 m 3 kmol-~,p3 = - 8 . 7 6 × 10 -2 _+ 3.4 X 10 -4 m 6 kmol-2). 3.2. M e m b r a n e mass transfer coefficient
The membrane mass transfer coefficient k M was determined from Eq. (5) in which the mass transfer coefficients in liquid, k L, were determined from the criterion equation Sh = C R e l / 2 S c l / 3
(7)
186
Z Palate, A. ZZtkovf / Journal of Membrane Science 119 (1996) 183-190
Modification of Eq. (7) taking into account the definition of R e number in stirring leads to the relation k L = C p - 1/602/3n1/2 (8) The C constant depends on geometry of the dialysis cell. In the case considered, the constant was determined from a series of 10 measurements realized at constant initial concentration of sulfuric acid in compartment I, c~0 = 0.1 kmol m - 3 , and at rotational speed of stirrers varying from 1.17 to 13.67 s - l . The procedure of determination of C, which is based on numerical integration of differential Eq. (5) with subsequent optimizing, can be summarized in the following steps. 1. The calculation of derivative ( d V I ) / ( d r ) from the second order polynomial approximating the time dependence of volume of liquid in compartment I of the cell. 2. Initial estimate of C and k M (0.5 and 5.5 × 10 -7 m s - l ) . 3. Integration of differential Eq. (5) by the Runge-Kutta 4th order method with integration step h = 1800 s. In this step one obtains the values of calculated concentrations in the same time intervals as those of the experimental values. In each integration step it is necessary to calculate the concentrations at the liquid-membrane interface and diffusivity and kinematic viscosity of H2SO 4. Since the diffusivity and viscosity of acid depend on its concentration, it was necessary to determine these quantities in both liquid films by an iteration procedure using literature data [12-14]. (a) The estimate of DAJ, v~ ( j = I, II), for r = 0 these values were determined for the concentration CA = CAO,
(b) Calculation of the mass transfer coefficients k£ ( j = I, II) from Eq. (8). (c) Calculation of H2SO 4 concentrations at the liquid-membrane interface, i.e. c,{i ( j = I, II), by solving the equation set (1)-(3) completed by Eq. (6) for j = I, II. This set of equations was treated by the Newton-Raphson method. (d) Correction of diffusivities and kinematic viscosities for the concentrations
+ cL cJ
2
j = I, II
(9)
(e) Procedures (b)-(d) were repeated until the relative changes of physical quantities DAj, v~ ( j = I,
II) were below 0.2%. The DAJ, VA i (j-----I, II) values fulfilling the criterion for finishing the iteration were used as initial estimates in the next integration step. The values of partition coefficients ~ J ( j = I, II) in Eq. (5) were calculated from the concentrations c~M and c~i ( j = I, II) - see Eq. (4). 4. Calculation of H2SO 4 concentration in compartment II of dialysis cell from the mass balance 1
, l , I _ _ _l,l II,/,calc = V T , / ~ C[ A I0, l ,V0 CAk CAk,calc TlI,l~ "/k )
(10)
5. Calculation of the objective function
1=1 k = l -- [
I1,1
-r tCAk,exp --
^II,l 2/ CAk ,calc ) /
(11)
where n is the number of measurements (n = 10) and m the number of concentration-time pairs obtained in one measurement 6. Realization of one step of the optimizing procedure. We used the simplex method - the algorithm by Nelder and Mead. This step provides the corrected values of the C and k M parameters. 7. Procedures 2 - 6 were repeated until reaching the minimum of the objective function (11). In the way given we obtained the values of C = 0.714 + 5.3 × 10 -3 and k M = 4.50 × 10 -7 + 1.6 × 10 -9 m s -1 . The suitability of the procedure suggested can be judged from Fig. 3 giving the plot of time dependence of calculated and experimental H2SO a concentrations in both compartments of the cell. From the picture it is obvious that there is a good agreement between model (5) and experiment. Fig. 4 shows contours of the objective function (11) it is evident that in the range of the parameters C and k M investigated there is only one minimum of the objective function. For the value C = 0.714 found we then determined the values of membrane mass transfer coefficients in the individual measurements using a procedure similar to that of determining C and k M for the concentration c[0 = 0.1 kmol m -3. The difference between the two procedures concerned the points 2 (the initial estimate for k M only), 5 (in the definition of objective function (I 1) the summation over I was dropped out), and 6 (the optimizing procedure only
Z. Palate, A. Zdkovd / Journal of Membrane Science 119 (1996) 183-190
187
8.0-
0.12
E
.
6.0
0.09
,i. m
0.06
~-.
4.0
ii; 0.03
2.0
0.00( 0
,
,
,
I
I
I
4
6
12
16
0.3
T , h
0.1
Fig. 3. Time dependence of H 2 SO 4 concentration in compartment I ( O ) and in compartment II ( O ) for n I = n II = 9.33 s l, c~ ° = 0.099 kmol m -3, t = 20°C: - - , values calculated from model (5).
0.5
1.0
1.5
2.0
C Fig. 4. Contours of objective function (11): ---, level step 1 X 10 . 2 kmol 2 m - 6 ; - - , level step 1 × 10 .4 kmol 2 m - 6 ; + , minimum.
Table 1 Membrane mass transfer coefficients and diffusion coefficients for H 2 S O 4 in N E O S E P T A - A F N membrane cI0 (kmol m - 3)
n (s - 1)
k N X 107 (ms - 1)
s X 108 (m s 1)
O A X l 0 II (m 2 s - l )
0.099
1.17 1.17 2.08 2.08 5.50 5.50 9.33 9.33 13.67 13.67 1.17 13.67 1.17 2.08 5.50 9.33 13.67 1.17 1.17 13.67 1.17 2.08 5.50 9.33 13.67
4.75 4.48 4.51 4.08 4.66 4.47 4.24 4.21 4.49 4.49 9.90 9.49 12.50 12.46 12.12 13.46 12.67 13.95 14.21 14.14 12.82 13.24 12.14 13.49 13.91
1.5 1.8 1.0 1.6 1.5 1.3 0.9 1.1 0.6 0.9 2.7 2.6 6.6 7.0 1.7 5.8 2.3 2.8 3.4 3.5 6.6 3.6 2.9 2.5 7.4
7.84 7.39 7.44 6.74 7.69 7.38 7.01 6.95 7.42 7.41 16.34 15.66 20.63 20.57 19.99 22.22 20.92 23.01 23.46 23.33 21.15 21.85 20.02 22.27 22.95
0.595 1.019
1.480
1.849
lc M X 107 (m s 1)
DA X l 0 II (m 2 s - 1)
4.44 -I- 0.13
7.33
9.70 + 0.27
16.00
12.64 + 0.52
20.87
14.10 _+ 0.32
23.27
13.12 -t- 0.50
21.65
Z. Palate, A. Z,dkovd / Journal of Membrane Science 119 (1996) 183-190
188
corrected the k M values). Table 1 gives the calculated k M values for the individual measurements. From Table 1 it is clear that the membrane mass transfer coefficients are independent of rotational speed of the stirrers, but they are affected by the initial HzSO 4 concentration in compartment I of the cell. The dependence of membrane mass transfer coefficient upon the initial H2SO 4 concentration is presented in Fig. 5, where from it follows that increasing the initial H2SO 4 concentration results at first in an increase in the membrane mass transfer coefficient k M and then (above a concentration of ca. 1.4 kmol m 3) in its mild decrease. Table 1 presents besides the k M values also the values of diffusion coefficients in the membrane for sulfuric acid calculated from k M ( D A = 8kM). Experimentally it was found that the membrane thickness in HzSO 4 solution was independent of the H2SO 4 concentration in the range from 0.1 to 2.0 kmol m -3. The calculation of D A considers the constant value of 6 (6 = 0.165 × 10 -3 m). As compared with the diffusion coefficients for HzSO 4 in water in the same concentration range, those in the membrane are distinctly lower.
3.3. Mass transfer resistances in liquid films and in membrane In the context with determination of the membrane mass transfer coefficient or diffusivity of a 20
7 (n E
~5
% __,£= lo
5
o00
0;
i
lo
,;
210
25
cA , k m o l . m -3 Fig. 5. D e p e n d e n c e o f m e m b r a n e m a s s transfer coefficient for H 2 S O 4 u p o n the initial H 2 S O 4 c o n c e n t r a t i o n in c o m p a r t m e n t I o f cell.
10
7 E d
?
8
6
12) c2
4
ml~lNIItll~ -D - - - -O -tl 0 i J i J J i
0
20
i
I
40
i
-r
i
,
i
610
r
h
Fig. 6. T i m e d e p e n d e n c e o f m a s s transfer resistances in m e m b r a n e ( a , • ) a n d in liquid films (in c o m p a r t m e n t I, 0 ; in c o m p a r t m e n t II, O ) f o r n l = n u = 1 . 1 7 s - 1 ( - ) a n d n ~=n u=13.67s-l(---) for c I 0 = 1.019 k m o l m 3.
component in membrane it is important to mention also the contribution of membrane resistance to the overall mass transfer resistance which includes the resistance in membrane and resistances in liquid films. These resistances can be characterized by the individual members in the denominator at the righthand side of differential Eq. (5). Fig. 6 represents time dependences of the individual mass transfer resistances for the lowest and the highest rotational speed of the stirrers for the initial concentration of H 2 8 0 4 in compartment I of the cell equal to 1.019 kmol m -3. From these dependences it can be seen that, due to H 2 SO 4 concentration changes in both compartments of the dialysis cell, the mass transfer resistance in both liquid films is changed. In compartment I of the cell the resistance is slightly increased (the resistance-time dependence can be approximated by a straight line), whereas the resistance in compartment II distinctly decreases. From Fig. 6 it is obvious that, even at the highest stirrer rotational speeds used, the mass transfer resistance in any of the liquid films cannot be considered negligible. As compared with the resistance in the individual films, the resistance in membrane assumes a relatively high value which remains constant even if the stirrer rotational speed is distinctly increased. Figs. 7 and 8 present the dependences of average relative resistances in the liquid films and in the membrane upon the stirrer rotational speed, and they
Z. Palate, A. Zfkovf / Journal of Membrane Science 119 (1996) 183-190 1 O0
80
60 r~
40
2o
o
0
i 0
i
r
i 4
i
i
i
E ~
i
i
8
i
i
12 n
,
i -1
i 16
s
Fig. 7. Dependence of relative mass transfer resistances in membrane and in liquid films upon the stirrer rotational speed for c~0 = 0.099 kmol m 3: A, resistance in membrane; O, resistance in film in compartment I; ©, resistance in film in compartment II. concern the initial H 2 SO 4 concentrations cI0 = 0.099 and 1.019 kmol m -3, respectively. Both figures show that the predominant part of resistance is concentrated in the membrane, the proportion of the membrane resistance being increased with increasing stirrer rotational speed. At the conditions given it is possible to attain a state when the resistance in the membrane makes ca. 70% of total resistance at the highest stirrer rotational speed, i.e. n ~= n I I = 13.67 S -1 "
1 O0
80
60 rY
40
Sometimes the resistance in both liquid films at both sides of the membrane is neglected in calculations of membrane mass transfer coefficient k M. However, such an approach can lead to considerably inaccurate k M values. The k~ values determined under the presumption of negligible resistance in the liquid films are always lower than the k M values obtained with the help of the nonsimplified Eq. (5). In the present work it has been found that differences between the two coefficients are increased with decreasing stirrer rotational speed and of the initial H2SO 4 concentration in compartment I of the cell. Even at the highest stirrer rotational speed and the highest initial HzSO 4 concentration, the k~ values reach about 8 5 - 8 8 % of the k M value, whereas for the lowest stirrer rotational speed and the lowest initial HzSO 4 concentration the k~ values make 49% of the k M value.
4. Conclusion The dialysis of H2SO 4 has been followed in the H 2 S O 4 - H 2 0 system in a batch dialysis cell with anion-exchange membrane NEOSEPTA-AFN. A model has been suggested for description of transport of H2SO 4 through the membrane: the model also considers the mass transfer resistance in both the liquid films at both sides of the membrane. The quantification of the process adopts the membrane mass transfer coefficient. The membrane mass transfer coefficient has been determined from experimental time dependence of H2SO 4 concentration in both compartments of the cell using the method of numerical integration of the basic differential equation with subsequent optimization. The treatment of the model suggested also necessitated determination of equilibrium of H2 8 0 4 solution-membrane.
5. List of symbols 0
4
8
-1 n
,
16
s
F i g . 8. D e p e n d e n c e o f r e l a t i v e m a s s t r a n s f e r r e s i s t a n c e s in m e m b r a n e a n d in liquid f i l m s u p o n t h e stirrer rotational s p e e d f o r C~o = 1.019 k m o l m - 3 : A , r e s i s t a n c e in m e m b r a n e ; 0 , r e s i s t a n c e in f i l m in c o m p a r t m e n t I; © , r e s i s t a n c e in f i l m in c o m p a r t m e n t II.
189
A C C
D D d
surface area of membrane (m) constant in Eq. (7) ( - ) molar concentration (kmol m - 3 ) diffusivity (m 2 s - 1) mean diffusivity (m 2 s - l ) diameter of stirrer (m)
Z. Palate,A. Z~fkov6/ Journal of MembraneScience 119 (1996) 183-190
190 h kL kM kM
integration step (s) mass transfer coefficient in liquid (m S-1) m e m b r a n e mass transfer coefficient (m s - 1 ) m e a n m e m b r a n e mass transfer coefficient (m S -1 )
kM
n Pl P2 P3 R
Re R% s
Sc Sh V
m e m b r a n e mass transfer coefficient calculated with p r e s u m p t i o n o f negligible resistance o f liquid film ( m s - 1) stirrer rotational speed ( s - l ) constant in Eq. (6) ( - ) constant in Eq. (6) (m 3 k m o l - l) constant in Eq. (6) (m 6 k m o 1 - 2 ) mass transfer resistance (s m -1 ) Reynolds number (= nd2/u) relative mass transfer resistance (%) standard deviation (m s - l ) Schmidt number (= v/D) Sherwood number (= k L d / D ) v o l u m e ( m 3)
5.1. Greek symbols
/.,
m e m b r a n e thickness (m) k i n e m a t i c viscosity (m 2 s - 1 )
gr
time (s) partition coefficient ( - )
6
5.2. Indexes A calc exp i M 0 I II
related to c o m p o n e n t A calculated value experimental v a l u e related to interface related to m e m b r a n e initial related to c o m p a r t m e n t I o f cell related to c o m p a r t m e n t II o f celt
Acknowledgements This w o r k was financially supported by the Grant Agency o f the C z e c h R e p u b l i c , g r a n t no. 104/93/2159.
References [1] A. Narebska and A. Warszawski, Diffusion dialysis. Effect of membrane composition on acid/salt separation, Sep. Sci. Technol., 27 (1992) 703-715. [2] J.D. Edwards and M.M. Benjamin, Diffusion dialysis for recovery of acids from concentrated process solutions: The importance of chemical speciation, Environ. Sci. Technol., 24 (1990) 880-885. [3] Y. Kobuchi, H. Motomura, Y. Noma and F. Hanada, Application of ion exchange membranes to the recovery of acids by diffusion dialysis, J. Membrane Sci., 27 (1986) 173-179. [4] I. Tugas, G. Pourcelly and C. Gavach, Electrotransport of protons and chloride ions in anion exchange membranes for the recovery of acids. Part I. Equilibrium properties, J. Membrane Sci., 85 (1993) 183-194. [5] G. Pourcelly, I. Tugas and C. Gavach, Electrotransport of HCI in anion exchange membranes for the recovery of acids. Part II. Kinetics of ion transfer at the membrane-solution interface, J. Membrane Sci., 85 (1993) 195-204. [6] H. Miyoshi, M. Yamagami, M. Chubachi and T. Kataoka, Characteristic coefficients of cation-exchange membranes for bivalent cations in equilibrium between the membrane and solution, J. Chem. Eng. Data, 39 (1994) 595-585. [7] L.J. AndrOs, F.A. Riera, R. Alvarez and R. Audinos, Separation of strong acids by electrodialysis with membranes selective to monovalent ions. An approach to modelling the process, Can. J. Chem. Eng., 72 (1994) 848-853. [8] K. Hanaoka, R. Kiyono and M. Tasaka, Thermal membrane potential across anion/exchange membranes in KC1 and KIO 3 solutions and the transported entropy of ions, J. Membrane Sci., 82 (1993) 255-263. [9] H. Miyoshi, M. Chubachi, M. Yamagami and T. Kataoka, Characteristic coefficients for equilibrium between solution and Neosepta or Selemion cation exchange membranes, J. Chem. Eng. Data, 37 (1992) 120-124. [10] P. Sridhar and G. Subramaniam, Recovery of acid from cation exchange resin regeneration waste by diffusion dialysis, J. Membrane. Sci., 45 (1989) 273-280. [11] Z. Palat3~ and A. Zfikovfi, Diffusion dialysis of sulfuric acid in a batch cell, Collect. Czech. Chem. Commun., 59 (1994) 1971-1982. [12] S. Bretsznajder, Wiasno~ci gaz6w i cieczy, Wydawnictwa naukovo-techniczne, Warszawa, 1966, p. 490. [13] P. Pitter, Hydrochemick6 tabulky, SNTL Praha, 1987, p. 119. [14] International Critical Tables, Vol. 5, McGraw-Hill, New York, 1926-1933, p. 12.
Joumal of MembraneScience 119 (1996) 183-190
Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN v
Z. Palat2~, A. ZS.kova Department of Chemical Engineering, University of Pardubice, s. Cs. legi1565, 532 10 Pardubice, Czech Republic
Received 12 February 1996; revised 27 March 1996; accepted10 April 1996
Abstract This paper deals with the determination of the membrane mass transfer coefficient for sulfuric acid in an anion-exchange membrane NEOSEPTA-AFN. This quantity has been determined on the basis of experiments carried out in a batch dialysis cell using the method of numerical integration of the basic differential equation describing the time dependence of sulfuric acid concentration and subsequent optimization procedure. The experiments carried out made it possible to calculate the membrane mass transfer coefficient for sulfuric acid over the concentration range from 0.1 to 1.9 kmol m -3 in the external solution. Keywords: Diffusion dialysis; Membrane mass transfer coefficient; Diffusivity of sulfuric acid; Anion-exchange membrane; Membrane/solution equilibrium
1. Introduction The Japanese firm Tokuyama Soda Co. is one of the important producers of ion-exchange membranes and their products find broad application in diffusion dialysis and electrodialysis. However, the physicochemical data of these membranes published so far are not adequate to their industrial application. Application of and research into basic properties of NEOSEPTA membranes have been dealt with in several papers [1-10], the main attention being focused upon the equilibria between solution and membrane [1,4-6,9] and the working conditions or separation factors [3,10]. Since the NEOSEPTA-AFN membrane is often used in the separation of sulfuric acid from aqueous solutions containing beside sulfuric acid also its salts, and since the equilibrium HzSO 4 solution-membrane has not been studied yet, this paper is focused
upon the said equilibrium and experimental determination of the membrane mass transfer coefficient for sulfuric acid. Although the HSO 4 and SO 2- ions, which result from dissociation of the acid, predominantly penetrate through the membrane (the transport of dissociated forms of sulfuric acid through the membrane was dealt with in [11]), the present paper, for simplicity, considers the transport of acid without regard to this fact. The experimental determination of the membrane mass transfer coefficient can adopt a two compartment batch cell with mixing where unsteady state mass transport takes place. With the volume changes taken into account, the unsteady state transport of component A is described by the following differential equations
dCIA dT --
0376-7388/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0376-7388(96)00122-6
kI A V I (cI--cli)
dW I
W I dT
(1)
184
Z. Palate, A. Z.fkovf / Journal of Membrane Science 119 (1996) 183-190
dc I d--T- = -
kMA -----7-(cIM-c~M) V
dCXA 07"-
k~ a W I (CIAIi--C~)
ClA dV l V' d'c
(2)
c~ dV I W I aT
(3)
Eqs. (1) and (3) describe the transport of component A through the liquid films at both sides of the membrane, whereas Eq. (2) concerns the transport of component A through the membrane. The concentrations of component A at the interface and in the membrane are interrelated as follows:
C~M
=
~3CJAi j = I, II
(4)
Modification of Eqs. (1)-(3) using Eq. (4) leads to the differential Eq. (5) which describes the time dependence of concentration of component A in compartment I of the cell dc~ aT
Z
1/,¢AICI- ,r. II '/-'AIICA
VI 1/¢I 1 --+--+--
1t¢II
CI
dV I
VI aT
(5)
If experimental data are available concerning the time dependence of A concentration in liquid volume of compartment I of the cell and data enabling an estimate of mass transfer coefficients in liquid, then - knowing the partition coefficients ~ ( j = I, II) one can adopt Eq. (5) to determine the membrane mass transfer coefficient k M. The procedure is based on numerical integration of Eq. (5) complemented by a suitable optimizing procedure.
in 25 ml distilled water. Due to ion exchange between the membrane and H 2SO 4 solution, the original solutions and - partially - the extracts, also, contained beside sulfuric acid hydrochloric acid. Concentrations of the two acids were determined in the following way. The first portion of sample was titrated with standard NaOH solution under argon to determine the overall concentration of the acids. In the second portion, sulfate ions were precipitated by addition of Ba(NO3) 2 and the HC1 content was determined by argentometry. The equivalence points were determined potentiometrically (using the universal titrator OH-407, Radelkis, Hungary). In the titrations of original solutions and extracts we used 0.1 and 0.05 M NaOH solutions, respectively. The concentrations of AgNO 3 solutions were 0.01 and 0.005 M. The fact that at the beginning of experiments the membrane is in the C1- form leads to a complex analytical procedure but, on the other hand, it is possible to determine the concentration of the fixed charges in the membrane, also.
2.2. Dialysis cell The dialysis of sulfuric acid was monitored with the help of experimental arrangement shown in Fig. 1. The main part is a batch cell with two compartments equipped with stirrers. The cell is described in detail elsewhere [11].
2. Experimental 2.1. Concentration of H 2504 in membrane The concentration of sulfuric acid in the membrane was determined by the method based on saturation of the membrane with the acid and subsequent extraction of acid into water. The membrane of 25-35 cm 2 surface area was cut into 4 - 6 pieces, rid of salt residues by thorough washing in distilled water, and shaken with 50 ml sulfuric acid of given concentration for 18 h. Then the membrane was rinsed with distilled water for a short time to remove the acid remaining at the surface of the membrane whereupon it was repeatedly shaken for 2 h ( 3 - 4 x )
8
[-I
711 13 II t
I
161
12
Fig. 1. Scheme of experimental arrangement: 1, dialysis cell; 2, partition; 3, membrane; 4, stirrers; 5, conductivity probes; 6, conductometers; 7, thermostat; 8, thermostat reservoir; I and If, compartments of cell.
Z Palate, A. Ztkovd/Journal of Membrane Science 119 (1996) 183-190
In all the experiments, the anion-exchange N E O S E P T A - A F N membranes of the Japanese firm Tokuyama Soda Co. Ltd. were adopted. The surface area of each membrane was 62.2 cm 2, thickness 0.17 mm (in C1- form). In the experiments, the compartments I and II of the cell were filled with sulfuric acid of a given concentration and with distilled water, respectively. The initial concentration of sulfuric acid was varied within the limits from ca. 0.1 to 1.9 kmol m -3, rotational speed of the stirrers was in the range from 1.17 to 13.67 s-1. At the beginning of each measurement the volume of liquid in both compartments of the cell was 1 1 and the height of levels was adjusted at the same value by addition of glass beads. In a given experiment using the same rotational speed of the two stirrers (n I = nil), we determined the HzSO 4 concentrations and level heights to find the volume changes in the two compartments with time. The measurement was finished after attaining equal concentrations in the two compartments of the cell. At the lowest initial H 2 SO 4 concentration (i.e. 0.1 kmol m -3) the conductimetry was adopted for determination of the acid concentration, whereas in the other cases the H2SO 4 concentration was determined titrimetrically with NaOH and potentiometric determination of equivalence point. The changes in level heights were measured by means of a modified micrometer screw with a needle attached at the centre of the emerging piston. In all the experiments, temperature was kept at a constant value of 20 + 0.5°C.
185
3.0
? E E
~alk~
2.0
! o
1.0
0.0~
i
0.0
i
0.5
i0
i
i
1.
i
1.5
cA
,
i
J
2,0
kmol.m
2.5
-3
Fig. 2. Dependence of H 2 S O 4 concentration in membrane upon
HzSO4 concentration in external solution: A, experimental values; ---, values calculated from Eq. (6).
its value was 4.7 kmol m -3 (the concentration is referred to the volume of H2SO 4 in the membrane). The dependence of sulfuric acid concentration in membrane upon concentration of H2SO 4 in external solution is presented in Fig. 2 where from it can be seen that the a n i o n - e x c h a n g e membrane N E O S E P T A - A F N shows a considerable affinity to sulfuric acid. The H 2 SO 4 concentration in membrane is always higher than that in the external solution, the differences being particularly marked for the n 2 s o 4 concentrations in the external solution up to ca. 0.5 kmol m -3. Eq. (6) was suggested to correlate the dependence of H2SO 4 concentration in the membrane vs. that in external solution. CA
3. Results and discussion 3.1. H 2 S O 4 concentration in m e m b r a n e
The concentration of sulfuric acid dissolved in the membrane was calculated on the basis of H2SO 4 concentrations in the individual extracts and the solution volume in the membrane which was determined from density of HzSO 4, the weight of the membrane saturated with acid and that of dried membrane in HSO 4 form. The membranes were dried in vacuum at 60°C. The concentration of fixed charges was calculated from the amount of C1- ions released -
CAM = P l + p 2 C A
+P3 c2
(6)
where the Pi constants (i = 1, 2, 3) were determined by nonlinear regression (p~ = 3.88 × 10 -2 _+ 5.6 × 10 -6, P 2 = 0 . 4 9 5 _ 2 . 1 × 10 -4 m 3 kmol-~,p3 = - 8 . 7 6 × 10 -2 _+ 3.4 X 10 -4 m 6 kmol-2). 3.2. M e m b r a n e mass transfer coefficient
The membrane mass transfer coefficient k M was determined from Eq. (5) in which the mass transfer coefficients in liquid, k L, were determined from the criterion equation Sh = C R e l / 2 S c l / 3
(7)
186
Z Palate, A. ZZtkovf / Journal of Membrane Science 119 (1996) 183-190
Modification of Eq. (7) taking into account the definition of R e number in stirring leads to the relation k L = C p - 1/602/3n1/2 (8) The C constant depends on geometry of the dialysis cell. In the case considered, the constant was determined from a series of 10 measurements realized at constant initial concentration of sulfuric acid in compartment I, c~0 = 0.1 kmol m - 3 , and at rotational speed of stirrers varying from 1.17 to 13.67 s - l . The procedure of determination of C, which is based on numerical integration of differential Eq. (5) with subsequent optimizing, can be summarized in the following steps. 1. The calculation of derivative ( d V I ) / ( d r ) from the second order polynomial approximating the time dependence of volume of liquid in compartment I of the cell. 2. Initial estimate of C and k M (0.5 and 5.5 × 10 -7 m s - l ) . 3. Integration of differential Eq. (5) by the Runge-Kutta 4th order method with integration step h = 1800 s. In this step one obtains the values of calculated concentrations in the same time intervals as those of the experimental values. In each integration step it is necessary to calculate the concentrations at the liquid-membrane interface and diffusivity and kinematic viscosity of H2SO 4. Since the diffusivity and viscosity of acid depend on its concentration, it was necessary to determine these quantities in both liquid films by an iteration procedure using literature data [12-14]. (a) The estimate of DAJ, v~ ( j = I, II), for r = 0 these values were determined for the concentration CA = CAO,
(b) Calculation of the mass transfer coefficients k£ ( j = I, II) from Eq. (8). (c) Calculation of H2SO 4 concentrations at the liquid-membrane interface, i.e. c,{i ( j = I, II), by solving the equation set (1)-(3) completed by Eq. (6) for j = I, II. This set of equations was treated by the Newton-Raphson method. (d) Correction of diffusivities and kinematic viscosities for the concentrations
+ cL cJ
2
j = I, II
(9)
(e) Procedures (b)-(d) were repeated until the relative changes of physical quantities DAj, v~ ( j = I,
II) were below 0.2%. The DAJ, VA i (j-----I, II) values fulfilling the criterion for finishing the iteration were used as initial estimates in the next integration step. The values of partition coefficients ~ J ( j = I, II) in Eq. (5) were calculated from the concentrations c~M and c~i ( j = I, II) - see Eq. (4). 4. Calculation of H2SO 4 concentration in compartment II of dialysis cell from the mass balance 1
, l , I _ _ _l,l II,/,calc = V T , / ~ C[ A I0, l ,V0 CAk CAk,calc TlI,l~ "/k )
(10)
5. Calculation of the objective function
1=1 k = l -- [
I1,1
-r tCAk,exp --
^II,l 2/ CAk ,calc ) /
(11)
where n is the number of measurements (n = 10) and m the number of concentration-time pairs obtained in one measurement 6. Realization of one step of the optimizing procedure. We used the simplex method - the algorithm by Nelder and Mead. This step provides the corrected values of the C and k M parameters. 7. Procedures 2 - 6 were repeated until reaching the minimum of the objective function (11). In the way given we obtained the values of C = 0.714 + 5.3 × 10 -3 and k M = 4.50 × 10 -7 + 1.6 × 10 -9 m s -1 . The suitability of the procedure suggested can be judged from Fig. 3 giving the plot of time dependence of calculated and experimental H2SO a concentrations in both compartments of the cell. From the picture it is obvious that there is a good agreement between model (5) and experiment. Fig. 4 shows contours of the objective function (11) it is evident that in the range of the parameters C and k M investigated there is only one minimum of the objective function. For the value C = 0.714 found we then determined the values of membrane mass transfer coefficients in the individual measurements using a procedure similar to that of determining C and k M for the concentration c[0 = 0.1 kmol m -3. The difference between the two procedures concerned the points 2 (the initial estimate for k M only), 5 (in the definition of objective function (I 1) the summation over I was dropped out), and 6 (the optimizing procedure only
Z. Palate, A. Zdkovd / Journal of Membrane Science 119 (1996) 183-190
187
8.0-
0.12
E
.
6.0
0.09
,i. m
0.06
~-.
4.0
ii; 0.03
2.0
0.00( 0
,
,
,
I
I
I
4
6
12
16
0.3
T , h
0.1
Fig. 3. Time dependence of H 2 SO 4 concentration in compartment I ( O ) and in compartment II ( O ) for n I = n II = 9.33 s l, c~ ° = 0.099 kmol m -3, t = 20°C: - - , values calculated from model (5).
0.5
1.0
1.5
2.0
C Fig. 4. Contours of objective function (11): ---, level step 1 X 10 . 2 kmol 2 m - 6 ; - - , level step 1 × 10 .4 kmol 2 m - 6 ; + , minimum.
Table 1 Membrane mass transfer coefficients and diffusion coefficients for H 2 S O 4 in N E O S E P T A - A F N membrane cI0 (kmol m - 3)
n (s - 1)
k N X 107 (ms - 1)
s X 108 (m s 1)
O A X l 0 II (m 2 s - l )
0.099
1.17 1.17 2.08 2.08 5.50 5.50 9.33 9.33 13.67 13.67 1.17 13.67 1.17 2.08 5.50 9.33 13.67 1.17 1.17 13.67 1.17 2.08 5.50 9.33 13.67
4.75 4.48 4.51 4.08 4.66 4.47 4.24 4.21 4.49 4.49 9.90 9.49 12.50 12.46 12.12 13.46 12.67 13.95 14.21 14.14 12.82 13.24 12.14 13.49 13.91
1.5 1.8 1.0 1.6 1.5 1.3 0.9 1.1 0.6 0.9 2.7 2.6 6.6 7.0 1.7 5.8 2.3 2.8 3.4 3.5 6.6 3.6 2.9 2.5 7.4
7.84 7.39 7.44 6.74 7.69 7.38 7.01 6.95 7.42 7.41 16.34 15.66 20.63 20.57 19.99 22.22 20.92 23.01 23.46 23.33 21.15 21.85 20.02 22.27 22.95
0.595 1.019
1.480
1.849
lc M X 107 (m s 1)
DA X l 0 II (m 2 s - 1)
4.44 -I- 0.13
7.33
9.70 + 0.27
16.00
12.64 + 0.52
20.87
14.10 _+ 0.32
23.27
13.12 -t- 0.50
21.65
Z. Palate, A. Z,dkovd / Journal of Membrane Science 119 (1996) 183-190
188
corrected the k M values). Table 1 gives the calculated k M values for the individual measurements. From Table 1 it is clear that the membrane mass transfer coefficients are independent of rotational speed of the stirrers, but they are affected by the initial HzSO 4 concentration in compartment I of the cell. The dependence of membrane mass transfer coefficient upon the initial H2SO 4 concentration is presented in Fig. 5, where from it follows that increasing the initial H2SO 4 concentration results at first in an increase in the membrane mass transfer coefficient k M and then (above a concentration of ca. 1.4 kmol m 3) in its mild decrease. Table 1 presents besides the k M values also the values of diffusion coefficients in the membrane for sulfuric acid calculated from k M ( D A = 8kM). Experimentally it was found that the membrane thickness in HzSO 4 solution was independent of the H2SO 4 concentration in the range from 0.1 to 2.0 kmol m -3. The calculation of D A considers the constant value of 6 (6 = 0.165 × 10 -3 m). As compared with the diffusion coefficients for HzSO 4 in water in the same concentration range, those in the membrane are distinctly lower.
3.3. Mass transfer resistances in liquid films and in membrane In the context with determination of the membrane mass transfer coefficient or diffusivity of a 20
7 (n E
~5
% __,£= lo
5
o00
0;
i
lo
,;
210
25
cA , k m o l . m -3 Fig. 5. D e p e n d e n c e o f m e m b r a n e m a s s transfer coefficient for H 2 S O 4 u p o n the initial H 2 S O 4 c o n c e n t r a t i o n in c o m p a r t m e n t I o f cell.
10
7 E d
?
8
6
12) c2
4
ml~lNIItll~ -D - - - -O -tl 0 i J i J J i
0
20
i
I
40
i
-r
i
,
i
610
r
h
Fig. 6. T i m e d e p e n d e n c e o f m a s s transfer resistances in m e m b r a n e ( a , • ) a n d in liquid films (in c o m p a r t m e n t I, 0 ; in c o m p a r t m e n t II, O ) f o r n l = n u = 1 . 1 7 s - 1 ( - ) a n d n ~=n u=13.67s-l(---) for c I 0 = 1.019 k m o l m 3.
component in membrane it is important to mention also the contribution of membrane resistance to the overall mass transfer resistance which includes the resistance in membrane and resistances in liquid films. These resistances can be characterized by the individual members in the denominator at the righthand side of differential Eq. (5). Fig. 6 represents time dependences of the individual mass transfer resistances for the lowest and the highest rotational speed of the stirrers for the initial concentration of H 2 8 0 4 in compartment I of the cell equal to 1.019 kmol m -3. From these dependences it can be seen that, due to H 2 SO 4 concentration changes in both compartments of the dialysis cell, the mass transfer resistance in both liquid films is changed. In compartment I of the cell the resistance is slightly increased (the resistance-time dependence can be approximated by a straight line), whereas the resistance in compartment II distinctly decreases. From Fig. 6 it is obvious that, even at the highest stirrer rotational speeds used, the mass transfer resistance in any of the liquid films cannot be considered negligible. As compared with the resistance in the individual films, the resistance in membrane assumes a relatively high value which remains constant even if the stirrer rotational speed is distinctly increased. Figs. 7 and 8 present the dependences of average relative resistances in the liquid films and in the membrane upon the stirrer rotational speed, and they
Z. Palate, A. Zfkovf / Journal of Membrane Science 119 (1996) 183-190 1 O0
80
60 r~
40
2o
o
0
i 0
i
r
i 4
i
i
i
E ~
i
i
8
i
i
12 n
,
i -1
i 16
s
Fig. 7. Dependence of relative mass transfer resistances in membrane and in liquid films upon the stirrer rotational speed for c~0 = 0.099 kmol m 3: A, resistance in membrane; O, resistance in film in compartment I; ©, resistance in film in compartment II. concern the initial H 2 SO 4 concentrations cI0 = 0.099 and 1.019 kmol m -3, respectively. Both figures show that the predominant part of resistance is concentrated in the membrane, the proportion of the membrane resistance being increased with increasing stirrer rotational speed. At the conditions given it is possible to attain a state when the resistance in the membrane makes ca. 70% of total resistance at the highest stirrer rotational speed, i.e. n ~= n I I = 13.67 S -1 "
1 O0
80
60 rY
40
Sometimes the resistance in both liquid films at both sides of the membrane is neglected in calculations of membrane mass transfer coefficient k M. However, such an approach can lead to considerably inaccurate k M values. The k~ values determined under the presumption of negligible resistance in the liquid films are always lower than the k M values obtained with the help of the nonsimplified Eq. (5). In the present work it has been found that differences between the two coefficients are increased with decreasing stirrer rotational speed and of the initial H2SO 4 concentration in compartment I of the cell. Even at the highest stirrer rotational speed and the highest initial HzSO 4 concentration, the k~ values reach about 8 5 - 8 8 % of the k M value, whereas for the lowest stirrer rotational speed and the lowest initial HzSO 4 concentration the k~ values make 49% of the k M value.
4. Conclusion The dialysis of H2SO 4 has been followed in the H 2 S O 4 - H 2 0 system in a batch dialysis cell with anion-exchange membrane NEOSEPTA-AFN. A model has been suggested for description of transport of H2SO 4 through the membrane: the model also considers the mass transfer resistance in both the liquid films at both sides of the membrane. The quantification of the process adopts the membrane mass transfer coefficient. The membrane mass transfer coefficient has been determined from experimental time dependence of H2SO 4 concentration in both compartments of the cell using the method of numerical integration of the basic differential equation with subsequent optimization. The treatment of the model suggested also necessitated determination of equilibrium of H2 8 0 4 solution-membrane.
5. List of symbols 0
4
8
-1 n
,
16
s
F i g . 8. D e p e n d e n c e o f r e l a t i v e m a s s t r a n s f e r r e s i s t a n c e s in m e m b r a n e a n d in liquid f i l m s u p o n t h e stirrer rotational s p e e d f o r C~o = 1.019 k m o l m - 3 : A , r e s i s t a n c e in m e m b r a n e ; 0 , r e s i s t a n c e in f i l m in c o m p a r t m e n t I; © , r e s i s t a n c e in f i l m in c o m p a r t m e n t II.
189
A C C
D D d
surface area of membrane (m) constant in Eq. (7) ( - ) molar concentration (kmol m - 3 ) diffusivity (m 2 s - 1) mean diffusivity (m 2 s - l ) diameter of stirrer (m)
Z. Palate,A. Z~fkov6/ Journal of MembraneScience 119 (1996) 183-190
190 h kL kM kM
integration step (s) mass transfer coefficient in liquid (m S-1) m e m b r a n e mass transfer coefficient (m s - 1 ) m e a n m e m b r a n e mass transfer coefficient (m S -1 )
kM
n Pl P2 P3 R
Re R% s
Sc Sh V
m e m b r a n e mass transfer coefficient calculated with p r e s u m p t i o n o f negligible resistance o f liquid film ( m s - 1) stirrer rotational speed ( s - l ) constant in Eq. (6) ( - ) constant in Eq. (6) (m 3 k m o l - l) constant in Eq. (6) (m 6 k m o 1 - 2 ) mass transfer resistance (s m -1 ) Reynolds number (= nd2/u) relative mass transfer resistance (%) standard deviation (m s - l ) Schmidt number (= v/D) Sherwood number (= k L d / D ) v o l u m e ( m 3)
5.1. Greek symbols
/.,
m e m b r a n e thickness (m) k i n e m a t i c viscosity (m 2 s - 1 )
gr
time (s) partition coefficient ( - )
6
5.2. Indexes A calc exp i M 0 I II
related to c o m p o n e n t A calculated value experimental v a l u e related to interface related to m e m b r a n e initial related to c o m p a r t m e n t I o f cell related to c o m p a r t m e n t II o f celt
Acknowledgements This w o r k was financially supported by the Grant Agency o f the C z e c h R e p u b l i c , g r a n t no. 104/93/2159.
References [1] A. Narebska and A. Warszawski, Diffusion dialysis. Effect of membrane composition on acid/salt separation, Sep. Sci. Technol., 27 (1992) 703-715. [2] J.D. Edwards and M.M. Benjamin, Diffusion dialysis for recovery of acids from concentrated process solutions: The importance of chemical speciation, Environ. Sci. Technol., 24 (1990) 880-885. [3] Y. Kobuchi, H. Motomura, Y. Noma and F. Hanada, Application of ion exchange membranes to the recovery of acids by diffusion dialysis, J. Membrane Sci., 27 (1986) 173-179. [4] I. Tugas, G. Pourcelly and C. Gavach, Electrotransport of protons and chloride ions in anion exchange membranes for the recovery of acids. Part I. Equilibrium properties, J. Membrane Sci., 85 (1993) 183-194. [5] G. Pourcelly, I. Tugas and C. Gavach, Electrotransport of HCI in anion exchange membranes for the recovery of acids. Part II. Kinetics of ion transfer at the membrane-solution interface, J. Membrane Sci., 85 (1993) 195-204. [6] H. Miyoshi, M. Yamagami, M. Chubachi and T. Kataoka, Characteristic coefficients of cation-exchange membranes for bivalent cations in equilibrium between the membrane and solution, J. Chem. Eng. Data, 39 (1994) 595-585. [7] L.J. AndrOs, F.A. Riera, R. Alvarez and R. Audinos, Separation of strong acids by electrodialysis with membranes selective to monovalent ions. An approach to modelling the process, Can. J. Chem. Eng., 72 (1994) 848-853. [8] K. Hanaoka, R. Kiyono and M. Tasaka, Thermal membrane potential across anion/exchange membranes in KC1 and KIO 3 solutions and the transported entropy of ions, J. Membrane Sci., 82 (1993) 255-263. [9] H. Miyoshi, M. Chubachi, M. Yamagami and T. Kataoka, Characteristic coefficients for equilibrium between solution and Neosepta or Selemion cation exchange membranes, J. Chem. Eng. Data, 37 (1992) 120-124. [10] P. Sridhar and G. Subramaniam, Recovery of acid from cation exchange resin regeneration waste by diffusion dialysis, J. Membrane. Sci., 45 (1989) 273-280. [11] Z. Palat3~ and A. Zfikovfi, Diffusion dialysis of sulfuric acid in a batch cell, Collect. Czech. Chem. Commun., 59 (1994) 1971-1982. [12] S. Bretsznajder, Wiasno~ci gaz6w i cieczy, Wydawnictwa naukovo-techniczne, Warszawa, 1966, p. 490. [13] P. Pitter, Hydrochemick6 tabulky, SNTL Praha, 1987, p. 119. [14] International Critical Tables, Vol. 5, McGraw-Hill, New York, 1926-1933, p. 12.