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A lumped parameter model to predict hydrochloric acid recovery in diffusion dialysis
Journal of Membrane Science 188 (2001) 61–70
A lumped parameter model to predict hydrochloric acid recovery in diffusion dialysis Moon-Sung Kang, Kye-Sang Yoo, Suk-Jung Oh, Seung-Hyeon Moon∗ Department of Environmental Engineering, Kwang-Ju Institute of Science & Technology, 1 Oryong-dong, Puk-gu, Kwangju, South Korea Received 1 June 2000; received in revised form 9 November 2000; accepted 31 January 2001
Abstract A lumped-parameter model has been developed to predict the performances of diffusion dialysis using anion-exchange membrane (NEOSEPTA® -AFN) for hydrochloric acid recovery in terms of various operating parameters. The operating parameters investigated in this study included the feed concentration, the retention time, and the ratio of feed to water (recipient) flow rate. The mass transfer coefficients of liquid films in continuous dialyzer were estimated from the Sherwood number (Sh) which was correlated to the Schmidt number (Sc) and the Reynolds number (Re). The results obtained from diffusion dialysis using continuous dialyzer showed that the model gave reasonably good agreement with the experimental results in predicting the recovery yield and the recovered acid concentration in the range of 0.05–3 mol dm−3 HCl. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Diffusion dialysis; Anion-exchange membrane; Hydrochloric acid; Continuous dialyzer; Acid recovery
1. Introduction Diffusion dialysis using ion exchange membranes has been employed for many years to recover acids from various waste solutions [1,2]. Yet diffusion dialysis is one of the membrane separation techniques under examination as far as the effectiveness in separation is concerned [3]. The operation of diffusion dialysis utilizes the difference in concentration of permeable species in solutions partitioned by an ion exchange membrane, and diffusion dialysis requires no external forces to promote separation. Operation of a diffusion dialyzer needs only electrical energy to pump the liquids into the compartments [1–5].
∗ Corresponding author. Tel.: +82-62-970-2435; fax: +82-62-970-2434. E-mail address: [email protected] (S.-H. Moon).
Despite wide use of diffusion dialysis to recover acid from high-concentration acidic wastewater, recovery from low-concentration solutions has not been attempted. As the wastewater treatment cost has been increasing recently, recovery of acid from even low-concentration acidic wastewater is now of interest. In addition, the experiments to examine the recovery yield and the recovered acid concentration under various conditions are laborious. In this regard, a model for the prediction of the acid recovery in diffusion dialysis of low-concentration feed solutions can be an important tool. The objective of this study was to develop a model to estimate the diffusion dialysis performance for practical use. A lumped-parameter model was used in this study to predict the performance of diffusion dialysis for HCl recovery in terms of the operating parameters. The operating parameters investigated in this study includes the feed concentration, the retention time, and
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Nomenclature a A b c C C d dH D E h hM j k kL km M Ntu Nu Pr Q Re Sc Sh t u U V x
constant in Eq. (4) membrane effective area (m2 ) constant in Eq. (5) constant in Eq. (5) molar concentration (mol dm−3 ) thermal fluid capacity rate (cal s−1 ◦ C−1 ) constant in Eq. (5) hydraulic diameter (m) diffusion coefficient (m2 s−1 ) recovery yield/efficiency heat transfer coefficient (W m−2 ◦ C−1 ) mass transfer coefficient (m s−1 ) molar flux (mol m−2 s−1 ) thermal conductivity (W m−1 ◦ C−1 ) mass transfer coefficient in liquid film mass transfer coefficient in membrane mass transfer rate of acid (mol s−1 ) number of heat or mass transfer units Nusselt number (=hx/k) Prandtl number (=cp µ/k) flow rate (dm3 s−1 ) Reynolds number (=dH ρu/µ) Schmidt number (=µ/ρD) Sherwood number (=kL dH /D) time (s) linear velocity (m s−1 ) overall mass/heat transport coefficient (m s−1 ) volume (dm−3 ) distance between both membranes/length (m)
Greek symbols δ membrane thickness (m) ε effectiveness φ mass-fraction field µ viscosity (kg m−1 s−1 ) θ temperature field ρ density (kg m−3 ) ψ partition coefficient Ψ function Subscripts H high concentration i related to input/interface
lm L m o 0
log mean low concentration related to membrane related to output initial
the ratio of the feed to recipient flow rate, where fresh water was used as the receiving liquid in the permeate compartment. The overall mass transfer coefficient in continuous dialyzer was estimated from the Sherwood number determined from continuous dialysis experiments and the results of cell experiments. The experimental results obtained in continuous diffusion dialysis were compared with the performance predicted by the model equations. Since volume change is not considered in the mass transfer correlation, the model was applied to diffusion dialysis of lowconcentration acid feed, where volume change is negligible.
2. Theory 2.1. Diffusion coefficient in ion exchange membrane When acid solution and water are contacted by an anion-exchange membrane, diffusion of the acid through the membrane takes place. The molar flux j of the acid through the membrane can be described by Fick’s first law. Integration between the membrane surfaces contacting each compartment gives j = Dm
CHm − CLm δ
(1)
where CHm and CLm are the concentration of the acid at the membrane surfaces contacting the feed and permeate solutions, respectively. Dm is the diffusion coefficient of the acid in the membrane and δ the membrane thickness. The flux is correlated with the change of the acid concentration in the permeate side according to the acid mass balance j=
VL dCL A dt
(2)
where VL and CL are the volume and the acid concentration of the permeate side, respectively and A
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the effective membrane area. Combination of Eqs. (1) and (2) gives Eq. (3). dCL Dm A = (CHm − CLm ) dt δVL
(3)
Since CHm and CLm are not easily measured experimentally, an equilibrium relationship between the concentration of the acid in the membrane and that in the bulk solution is required. The acid concentration in the membrane was linearly correlated with the bulk concentration in this study Cm = aC
(4)
where Cm and C are the acid concentration in membrane and that in the bulk solution, respectively. The constants a was determined from the equilibrated acid uptake by the membrane. Combining Eqs. (3) and (4), the diffusion coefficient can be calculated from the time course of the acid transfer in diffusion dialysis [6]. 2.2. Mass transfer coefficient in diffusion dialysis In view of the complexity of mass transfer in diffusion dialysis, fundamental equations of mass transfer are rarely available. Instead empirical methods, guided by dimensional analysis and semi-empirical analogies, are relied on to give useful relation, Eq. (5) [7,8]. This analogy is valid with an assumption that the volume change in the streams is negligible. Therefore, we can obtain the mass transfer coefficients of the liquid film from the equation k L dH dH ρu c µ d =b D µ ρD
(5)
where, kL is the mass transfer coefficient in the liquid boundary layer. The group kL dH /D is the Sherwood number, dH ρu/µ the Reynolds number, µ/ρD the Schmidt number. The hydraulic diameter, dH can be replaced with 2x for the flow between two membranes. Here, x represents the distance between the membranes. In this equation, the constants c and d were chosen as 0.50 and 0.33, respectively, from the analogy of heat transfer in a flat plate [4,5,9], and the constant b was determined experimentally. The overall heat transfer coefficient in a heat transfer process
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represents the combined heat transfer rate of convection in the hot medium, conduction, and convection in the cold medium. Similarly the overall mass transfer coefficient in a diffusion dialyzer consists of the mass transfer resistance in the feed, diffusion through a membrane, and the mass transfer resistance in the permeate. Therefore, we can determine the overall mass transfer coefficients in continuous diffusion dialyzer from the following equation [4,5] U=
1 (1/kLH ) + 1/(ψkm ) + (1/kLL )
(6)
where ψ is the partition coefficient and km the mass transfer coefficient in membrane determined from the diffusion coefficient divided by the membrane thickness. 2.3. Performance equations of diffusion dialysis The performance of diffusion dialysis is usually characterized by the recovery yield and the recovered acid concentration. In a counter-current diffusion dialysis operation, typical operating parameters affecting the performance are the mass transfer coefficient, the effective area, the feed concentrations, and the ratio of the feed to water flow rate. These variables can be determined from the dimension of the system and the operating conditions with an exception of the overall mass transfer coefficient. The mass transfer coefficient is a complex variable depending on the various variables as correlated in Eqs. (5) and (6). The mass transfer rate in diffusion dialysis can be defined as Eq. (7) [2,10,11]. M = UA Clm
(7)
where M is the mass transfer rate of acid and U the overall mass transfer coefficient in continuous dialyzer, Clm the log mean concentration difference between the both solutions and A is the membrane effective area. Since the diffusion dialyzer is a countercurrent flow system, the log mean concentration difference is denoted as the driving force. The mass transfer coefficients were calculated using Eq. (7) experimentally and then we determined the constant b in Eq. (5) using a nonlinear regression method.
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When fresh water is fed in the permeate compartment, the recovery yield can be defined as follows: E=
QL CLo QH CHi
The number of mass transfer unit UA/QH is a dimensionless expression of the mass transfer characteristic of diffusion dialysis. QH /QL is the ratio of the inlet acid flow rate to the inlet water flow rate. To determine unknown recovered concentration (CLo ), an algebraic relation between recovery yield and two operating parameters was derived from the formula used for heat transfer operation [12]. 1 − exp[−(UA/QH )(1 − (QH /QL ))] 1 − (QH /QL ) exp[−(UA/QH )(1 − (QH /QL ))] (10a)
In the limiting case, i.e. QH /QL → 1, Eq. (10a) can be rewritten to the form E=
UA/QH 1 + UA/QH
QH QL 1 − exp[−(UA/QH )(1 − (QH /QL ))] × 1 − (QH /QL ) exp[−(UA/QH )(1 − (QH /QL ))]
(8)
Dimensionless operating parameters are very useful for estimating the acid recovery performances. The dimensionless parameters for depicting the operating conditions are (i) the number of mass transfer unit (UA/QH ) and (ii) the ratio of the feed to water flow rate (QH /QL ). Therefore, Eq. (8) can be expressed as a function of these two dimensionless operating parameters as follows: UA QH E=Ψ (9) , QH Q L
E=
CLo = CHi
(10b)
Then the recovered acid concentration CLo is derived from the combination of Eqs. (8) and (10) as follows:
(11)
The summary of analogous quantities for heat and mass transfer is given in Table 1 [8,9,12]. 3. Experimental An anion-exchange membrane, NEOSEPTA® -AFN (Tokuyama Co., Japan), was used in all the experiments. The membrane (in Cl− form) thickness was measured as 0.17 mm. The feed solutions were prepared by diluting 32% HCl solution (Merck, Germany). All experiments were performed at room temperature. 3.1. Water contents In order to determine the acid concentration in the membrane, the membrane water content was measured. Small pieces of the membrane (area = 9 cm2 ) were soaked in HCl solution of various concentrations for 24 h. Then the weights of the wet membranes were measured immediately. The membrane samples were then dried at 60◦ C under a vacuum condition until a constant weight was obtained. The water contents were determined from the weight difference between the wet and dry membranes. 3.2. HCl uptake by the membrane The saturated concentration of HCl in the membrane was determined by soaking the membrane in
Table 1 Analogies between heat and mass transfer at low mass transfer rates Heat transfer quantities ∂θ hL = −L Nu = k ∂y w √ √ hL 3 Nu = 0.332 Re Pr (for Pr > 0.6) k AU Ntu = C min ε=
C min /C C max ) 1 − e−Ntu (1−C (Cmin /Cmax : capacity rate ratio) C min /C C max ) −N (1−C tu C min /C C max ) e 1 − (C
Mass transfer quantities hM L ∂φ Sh = = −L D ∂y w √ √ hM L 3 Sh = 0.332 Re Sc (for Sc > 0.6) D AU Ntu = CH E=
1 − e−Ntu (1−QH /QL ) (QH /QL : flow ratio) 1 − (QH /QL ) e−Ntu (1−QH /QL )
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HCl solution and extracting the absorbed acid. At the beginning of the tests, membrane pieces (area = 25 cm2 , Cl− form) were washed with distilled water until no salt residues could be detected in the water. Then the sheets were soaked in 100 ml of HCl solution (0.125–3.00 mol m−3 ) and then the solution was stirred for 24 h. The membrane sheets were rinsed with distilled water for a short time to remove the acid remaining at the surface of the membrane. The surface-rinsed membranes were immersed in distilled water for 24 h and this extraction step was repeated three times. Then the amount of the acid extracted from the membrane was measured by microtitration (Metrohm 702 SM Titrino, Swiss) with 0.05 mol dm−3 NaOH standard solution. 3.3. Dialysis cell tests The diffusion coefficients of the acid were determined from batch dialysis cell tests. Fig. 1(a) shows the dialysis cell used in this study. The two-compartment cell, separated by the AFN membrane, was made of plexiglass. The effective area of the test membrane was 16 cm2 . At the beginning of each experiment, the feed side was filled with HCl solution and the permeate side with distilled water. In order to ensure a homogeneous concentration in the cell, each compartment was agitated using a stirrer (model 57022 clamp, Caframo) at 400 rpm. The initial volume of solution in each side was 200 ml and the acid concentrations in both sides were measured every 30 min by acid–base titration. The diffusion coefficients were determined from initial concentration gradients obtained from these data.
Fig. 1. Schematic diagram for diffusion dialyzers: (a) batch diffusion dialysis cell; (b) continuous diffusion dialyzer. The dotted line represents the anion-exchange membrane through which an acid permeates.
3.4. Diffusion dialysis of hydrochloric acid using continuous dialyzer The dialysis coefficients and performance data according to various operating conditions were obtained from continuous dialysis experiments. The detailed experimental conditions are listed in Table 2. The continuous diffusion dialyzer, consisting of 10 sheets of AFN membranes with an effective area of 200 cm2 each, were placed 1 mm apart from each other using a plastic net spacer and rubber gaskets. The dialyzer used for the experiments was TSD-2 diffusion dialyzer (Tokuyama Co., Japan). As shown in Fig. 1(b), the acid solution and water were contacted
Table 2 Various operating conditions for acid recovery in continuous operation Acid concentration (mol dm−3 )
Specific retention time (h cm−1 )
Flow rate ratio
0.06 0.22 0.60 1.05 1.58 1.95 2.60 3.20
0.045 0.55 1.06 1.62 2.53 4.74 6.87 8.22
0.20 0.30 0.47 0.77 1.11 1.65 3.08
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counter-currently. Since the linear flow velocity in a stack was low, air bubbles entrained in the feed solution could be retained in the dialyzer. The trapped air bubbles would disturb the uniform distribution of fluid and reduce the mass transfer area resulting in a significant deterioration of dialysis performance. In order to avoid this problem distilled water was heated and cooled down to room temperature to remove dissolved air, and the water flow was reversed for a short period prior to running each experiment to expel bubbles within the compartments. An experiment was completed when the recovered acid concentration reached a steady state. At the steady state, the concentrations and the flow rates of both streams were measured. The acid concentrations were determined by acid–base titration using NaOH standard solution. 4. Results and discussion 4.1. Water content and HCl uptake by the membrane The water contents were calculated as weight of water per unit mass dry membrane. The water contents were not affected by the acid concentrations investigated in this study, up to 3 mol dm−3 HCl. The average water content was calculated as 0.365 (g-water/g-dry membrane). The dependence of the absorbed HCl in the membrane on the concentration of bulk solution is presented in Fig. 2, where the HCl concentration in the membrane was expressed as mole of the acid absorbed per solution volume in the membrane. It was observed that the concentration in the membrane was nearly proportional to the bulk concentration in the considered concentration range studied. The constant in Eq. (4) was determined as a = 0.9037 from Fig. 2.
Fig. 2. Dependence of HCl concentration in membrane on HCl concentration in the bulk solution: (䉱) experimental data; (- - -) calculated from Eq. (4).
where CH0 is the initial feed concentration. Eq. (12) was solved using a computer program to calculate the diffusion coefficient from the initial transport rate of HCl in the dialysis cell. Dependence of the diffusion coefficient in the AFN membrane on the concentration in the feed solution is shown in Fig. 3. The diffusion coefficients of hydrochloric acid in the membrane were compared with that of hydrochloric acid at infinitely dilute aqueous solution (ca. 3.30E−5 cm2 s−1 at 25◦ C) calculated from the diffusion coefficients of H+ and Cl− [7] and the diffusion coefficients in the membrane were one order of magnitude lower values than that in the bulk solution.
4.2. Diffusion coefficient in membrane Batch cell experiments were performed as described in Section 3. The diffusion coefficient according to the feed concentrations was calculated from Eqs. (3) and (4) and the mass balance. Eq. (3) substituted by Eq. (4) and HCl balance (C H0 V H = C H V H + C L V L ) gave a first-order differential equation as following [4]: dCL Dm A a(CH0 VH − CL VL ) = dt δVL VH − aCL
(12)
Fig. 3. Dependence of the diffusion coefficients in the membrane on the HCl concentration in the feed solution: ( ) experimental data; (- - -) calculated from Eq. (12).
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4.4. Recovery with various operating conditions in continuous operation
Fig. 4. Dependence of the mass transfer coefficients in continuous diffusion dialyzer on the HCl concentration in the feed solution: (䉱) experimental data; (- - -) calculated from Eq. (6).
4.3. Mass transfer coefficient in continuous dialyzer The mass transfer coefficients in continuous dialyzer were calculated from the experimental data using Eq. (7). The relation between the overall mass transfer coefficient and the acid concentration in the feed solution is presented in Fig. 4. The mass transfer coefficients in liquid films were determined using correlation (Eq. (5)) of Sherwood number and the mass transfer coefficient in membrane was calculated from the diffusion coefficient (k m = D m /δ) as a function of log-mean acid concentration. From the experimental results, the constant b in Eq. (5) can be determined using the physical properties. The equation for viscosity of the acid solution, as a function of the concentration, was obtained from the literature [13]: (µ − µo )µo = 0.0030 + 0.0620C 0.5 + 0.0008C (13) where µo is the viscosity of water. The constant b in Eq. (5) was determined as 0.95 by nonlinear regression with the mass transfer coefficients and other physical properties. The partition coefficient ψ in Eq. (6) was determined as 0.9737 from the correlation (Eq. (14)) between the acid concentrations at the interface and in the membrane [5], assuming that the interface concentration was the mean value of the acid concentrations in bulk and membrane phases. Cm = ψCi
(14)
Recovery was defined as the ratio of moles of acid in the recovered stream to the feed expressed in Eq. (8). Plugging the experimental data into the equation gave the recovery in diffusion dialysis. As described in Table 2, continuous dialysis experiments were performed under various operating conditions, i.e. the acid concentration in the feed solution, the retention time of the feed solution, and the ratio of the feed to water flow rates. In these experiments, we found that the flow rate difference between the inlet and outlet was insignificant. It was observed that the volume change was less than 2% in all experiments. It implies that the water transfer through the membrane was negligible unlike diffusion dialysis of high-concentration acid. Therefore, the model developed in this study from the analogy to heat transfer can be reasonably applied to diffusion dialysis of low-concentration acid. In the first set of experiments, effects of the feed concentration were examined from 0.05 to 3 mol dm−3 HCl, while the other two variables were fixed. Considering the relation of retention time to the membrane area, we defined the specific retention time as an effective membrane area divided by the flow rate. The specific retention time of the feed stream was 8.92 h cm−1 and the ratio of the feed and permeate flow rates was unity in each experiment. The effect of acid concentrations in the feed solution on acid recovery by diffusion dialysis is presented in Fig. 5. Since the volume change in the solutions was negligible during the diffusion dialysis, the recovery was proportional to the recovered acid concentration. As the feed concentration increased, the acid uptake increased (Fig. 2), and the diffusion coefficient and the mass transfer coefficient increased (Figs. 3 and 4). Consequently, the acid recovery increased with the feed concentration, the increasing rate being retarded due to the limited concentration difference at higher concentrations. In this case, the resistance in membrane affects the acid recovery yield significantly. The second operating parameter investigated was the flow rate to change the retention time in the stack. The specific retention time of both feed and permeate streams was changed from 0.42 to 8.34 h cm−1 . The feed concentration was fixed at 1.27 mol dm−3 HCl, and the ratio of the feed to water flow rate was unity.
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Fig. 5. Dependence of the acid recovery in the continuous diffusion dialysis on the HCl concentration in the feed solution (specific retention time = 8.92 h cm−1 ). Symbols: experimental data; dotted lines: calculated from Eq. (10).
The acid recovery as a function of the specific retention time of the feed is presented in Fig. 6. As expected the recovery increased with the increasing retention time, but not linearly at a greater retention time due to decreased concentration difference. When the specific retention time was increased, the resistances in liquid films and membrane also increased simultaneously. The third operating parameter investigated was the ratio of the feed to water flow rates. The ratio was varied from 0.1 to 3.0. During the experiment, QH was fixed and QL was varied. The feed solution used in this experiment was 1.22 mol dm−3 . The dependence
Fig. 6. Dependence of the acid recovery in the continuous diffusion dialysis on the specific retention time of the feed solution (feed concentration = 1.27 mol dm−3 ). Symbols: experimental data; dotted lines: calculated from Eq. (10).
Fig. 7. Dependence of the acid recovery in the continuous diffusion dialysis on the ratio of feed to water flow rate (feed concentration = 1.22 mol dm−3 ). Symbols: experimental data; dotted lines: calculated from Eq. (10)].
of the acid recovery in diffusion dialysis on the ratio of the feed to permeate flow rates is shown in Fig. 7. This result shows that the recovery yield decreased with the increasing flow ratio, due to the decreased concentration difference across the membrane and decrease in the diffusion coefficient at a higher flow ratio. Consequently, the resistances in the permeate side liquid film and membrane increased with the increasing the flow ratio. 4.5. Recovery yield and recovered acid concentration predicted by the model In order to estimate acid recovery with various operating conditions, the physicochemical parameters of the diffusion dialysis system, such as water content, membrane thickness, equilibrium concentration and diffusion coefficients obtained in this study, were substituted in the model. Since the log-mean concentration in diffusion dialysis was unknown initially, the overall mass transfer coefficients were determined by the iteration procedure shown in Fig. 8. In this iteration not only the overall mass transfer coefficient, but also the recovery yield and the recovered acid concentration were determined eventually. The numerical simulation of diffusion dialysis was repeated for the range of conditions depicted in Figs. 5–7. Each simulation result is compared with the corresponding experimental results. The dotted lines of Figs. 5–7 represent the recovery yields and
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Fig. 9. Acid recovery yields and recovered concentrations predicted by the model developed in this study for 1 mol dm−3 HCl feed. Solid line: recovery yield; dotted line: recovered concentration (mol dm−3 ). Fig. 8. Algorithm for estimating mass transfer coefficient and acid recovery in diffusion dialysis.
recovered acid concentrations predicted with variation of the feed concentration, the retention time, and the ratio of the feed to water flow rates, respectively. The comparison between experimental results and the predicted values show good agreement. Under the experimental conditions investigated in this study, all the mass transfer resistances in the feed side, the membrane, and the permeate side were found to be significant in determining the overall mass transfer coefficient. Considering the simulation was performed with physicochemical properties for HCl in the membrane and one experimentally determined constant b in Eq. (5), which is subject to a diffusion dialysis system, the model has been developed with minimal empirical factors. The model developed in this study is capable of simulating the variation of the three major operating variables of diffusion dialysis. As an example of the applications, 1 mol dm−3 HCl feed solution was considered for acid recovery by diffusion dialysis. The concentration of the recovered acid is an important criterion for successful implementation of diffusion dialysis in light of recycling the
recovered acid. In the case that the feed concentration is fixed, the retention time and the ratio of the feed to permeate flow rate are the major operating variables. The estimated acid recovery yields and recovered concentrations for 1 mol dm−3 HCl solution are shown in Fig. 9 in terms of the two operating variables. The results show requirement of the two operating variables, the retention time and the ratio of the feed to water flow rate, to obtain a selected acid recovery yield and recovered concentration. It is obvious in this example that the recovery yield may be sacrificed unless the system is operated at a high retention time and a low ratio of the feed to water flow rate, as shown in Figs. 6 and 7.
5. Conclusions In order to predict the performance of a diffusion dialysis process, a mathematical model was developed. For the model equations, the key parameters including water content, membrane thickness and equilibrium acid concentration in the membrane were measured experimentally. The diffusion coefficients of acid in the membrane were calculated from the
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acid transport rate in a dialysis cell. The overall mass transfer coefficient in continuous diffusion dialyzer was determined from the correlation for Sherwood number and the membrane diffusion coefficient. The physicochemical properties were used to predict the performance of diffusion dialysis of HCl in the range of 0.05–3.0 mol dm−3 HCl. It is expected that the model developed in this study can be further applied to diffusion dialysis of other inorganic acids such as sulfuric acid, nitric acid, fluoric acid. Acknowledgements This work was supported in part by Brain Korea 21 from the Ministry of Education through the Graduate Program for Chemical & Environmental Engineering at Kwangju Institute of Science & Technology. 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. [2] S.J. Oh, S.-H. Moon, T. Davis, Effects of metal ions on diffusion dialysis of inorganic acids, J. Membr. Sci. 169 (2000) 95.
[3] A. Narebska, A. Warszaski, Diffusion dialysis transport phenomena by irreversible thermodynamics, J. Membr. Sci. 88 (1994) 167. [4] Z. Palatý, A. Žáková, Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN, J. Membr. Sci. 119 (1996) 183. [5] Z. Palatý, A. Žáková, P. Dolecek, Modeling the transport of Cl− ions through the anion exchange membrane NEOSEPTAAFN; systems HCl/membrane/H2 O and HCl–FeCl3 /membrane/H2 O, J. Membr. Sci. 165 (2000) 237. [6] A. Heintz, C. Illenberger, Diffusion coefficients of Br2 in cation exchange membranes, J. Membr. Sci. 113 (1996) 175. [7] E.L. Cussler, Diffusion, Cambridge University Press, Cambridge, 1984, p. 150, 226. [8] R.B. Bird, W.E. Stewart, E.N. Lighfoot, Transport Phenomena, Wiley, New York, 1960, p. 640. [9] E.R.G. Eckert, R.M. Drake, Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972, p. 731. [10] Y. Kobuchi, H. Motomura, Y. Noma, F. Hanada, Application of the ion-exchange membranes-acids recovery by diffusion dialysis, in: Proceedings of the 1986 Europe–Japan Congress on Membranes and Membrane Processes, Plenum Press, New York, 1986. [11] P. Sridhar, G. Subramaniam, Recovery of acid from cation exchange resin regeneration waste by diffusion dialysis, J. Membr. Sci. 45 (1989) 273. [12] W.M. Kays, A.L. London, Compact Heat Exchangers, 2nd Edition, McGraw-Hill, New York, 1964, p. 17. [13] Encyclopedia of Chemical Technology, Vol. 12, 3rd Edition, Wiley, New York, 1984, p. 989.
A lumped parameter model to predict hydrochloric acid recovery in diffusion dialysis Moon-Sung Kang, Kye-Sang Yoo, Suk-Jung Oh, Seung-Hyeon Moon∗ Department of Environmental Engineering, Kwang-Ju Institute of Science & Technology, 1 Oryong-dong, Puk-gu, Kwangju, South Korea Received 1 June 2000; received in revised form 9 November 2000; accepted 31 January 2001
Abstract A lumped-parameter model has been developed to predict the performances of diffusion dialysis using anion-exchange membrane (NEOSEPTA® -AFN) for hydrochloric acid recovery in terms of various operating parameters. The operating parameters investigated in this study included the feed concentration, the retention time, and the ratio of feed to water (recipient) flow rate. The mass transfer coefficients of liquid films in continuous dialyzer were estimated from the Sherwood number (Sh) which was correlated to the Schmidt number (Sc) and the Reynolds number (Re). The results obtained from diffusion dialysis using continuous dialyzer showed that the model gave reasonably good agreement with the experimental results in predicting the recovery yield and the recovered acid concentration in the range of 0.05–3 mol dm−3 HCl. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Diffusion dialysis; Anion-exchange membrane; Hydrochloric acid; Continuous dialyzer; Acid recovery
1. Introduction Diffusion dialysis using ion exchange membranes has been employed for many years to recover acids from various waste solutions [1,2]. Yet diffusion dialysis is one of the membrane separation techniques under examination as far as the effectiveness in separation is concerned [3]. The operation of diffusion dialysis utilizes the difference in concentration of permeable species in solutions partitioned by an ion exchange membrane, and diffusion dialysis requires no external forces to promote separation. Operation of a diffusion dialyzer needs only electrical energy to pump the liquids into the compartments [1–5].
∗ Corresponding author. Tel.: +82-62-970-2435; fax: +82-62-970-2434. E-mail address: [email protected] (S.-H. Moon).
Despite wide use of diffusion dialysis to recover acid from high-concentration acidic wastewater, recovery from low-concentration solutions has not been attempted. As the wastewater treatment cost has been increasing recently, recovery of acid from even low-concentration acidic wastewater is now of interest. In addition, the experiments to examine the recovery yield and the recovered acid concentration under various conditions are laborious. In this regard, a model for the prediction of the acid recovery in diffusion dialysis of low-concentration feed solutions can be an important tool. The objective of this study was to develop a model to estimate the diffusion dialysis performance for practical use. A lumped-parameter model was used in this study to predict the performance of diffusion dialysis for HCl recovery in terms of the operating parameters. The operating parameters investigated in this study includes the feed concentration, the retention time, and
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Nomenclature a A b c C C d dH D E h hM j k kL km M Ntu Nu Pr Q Re Sc Sh t u U V x
constant in Eq. (4) membrane effective area (m2 ) constant in Eq. (5) constant in Eq. (5) molar concentration (mol dm−3 ) thermal fluid capacity rate (cal s−1 ◦ C−1 ) constant in Eq. (5) hydraulic diameter (m) diffusion coefficient (m2 s−1 ) recovery yield/efficiency heat transfer coefficient (W m−2 ◦ C−1 ) mass transfer coefficient (m s−1 ) molar flux (mol m−2 s−1 ) thermal conductivity (W m−1 ◦ C−1 ) mass transfer coefficient in liquid film mass transfer coefficient in membrane mass transfer rate of acid (mol s−1 ) number of heat or mass transfer units Nusselt number (=hx/k) Prandtl number (=cp µ/k) flow rate (dm3 s−1 ) Reynolds number (=dH ρu/µ) Schmidt number (=µ/ρD) Sherwood number (=kL dH /D) time (s) linear velocity (m s−1 ) overall mass/heat transport coefficient (m s−1 ) volume (dm−3 ) distance between both membranes/length (m)
Greek symbols δ membrane thickness (m) ε effectiveness φ mass-fraction field µ viscosity (kg m−1 s−1 ) θ temperature field ρ density (kg m−3 ) ψ partition coefficient Ψ function Subscripts H high concentration i related to input/interface
lm L m o 0
log mean low concentration related to membrane related to output initial
the ratio of the feed to recipient flow rate, where fresh water was used as the receiving liquid in the permeate compartment. The overall mass transfer coefficient in continuous dialyzer was estimated from the Sherwood number determined from continuous dialysis experiments and the results of cell experiments. The experimental results obtained in continuous diffusion dialysis were compared with the performance predicted by the model equations. Since volume change is not considered in the mass transfer correlation, the model was applied to diffusion dialysis of lowconcentration acid feed, where volume change is negligible.
2. Theory 2.1. Diffusion coefficient in ion exchange membrane When acid solution and water are contacted by an anion-exchange membrane, diffusion of the acid through the membrane takes place. The molar flux j of the acid through the membrane can be described by Fick’s first law. Integration between the membrane surfaces contacting each compartment gives j = Dm
CHm − CLm δ
(1)
where CHm and CLm are the concentration of the acid at the membrane surfaces contacting the feed and permeate solutions, respectively. Dm is the diffusion coefficient of the acid in the membrane and δ the membrane thickness. The flux is correlated with the change of the acid concentration in the permeate side according to the acid mass balance j=
VL dCL A dt
(2)
where VL and CL are the volume and the acid concentration of the permeate side, respectively and A
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the effective membrane area. Combination of Eqs. (1) and (2) gives Eq. (3). dCL Dm A = (CHm − CLm ) dt δVL
(3)
Since CHm and CLm are not easily measured experimentally, an equilibrium relationship between the concentration of the acid in the membrane and that in the bulk solution is required. The acid concentration in the membrane was linearly correlated with the bulk concentration in this study Cm = aC
(4)
where Cm and C are the acid concentration in membrane and that in the bulk solution, respectively. The constants a was determined from the equilibrated acid uptake by the membrane. Combining Eqs. (3) and (4), the diffusion coefficient can be calculated from the time course of the acid transfer in diffusion dialysis [6]. 2.2. Mass transfer coefficient in diffusion dialysis In view of the complexity of mass transfer in diffusion dialysis, fundamental equations of mass transfer are rarely available. Instead empirical methods, guided by dimensional analysis and semi-empirical analogies, are relied on to give useful relation, Eq. (5) [7,8]. This analogy is valid with an assumption that the volume change in the streams is negligible. Therefore, we can obtain the mass transfer coefficients of the liquid film from the equation k L dH dH ρu c µ d =b D µ ρD
(5)
where, kL is the mass transfer coefficient in the liquid boundary layer. The group kL dH /D is the Sherwood number, dH ρu/µ the Reynolds number, µ/ρD the Schmidt number. The hydraulic diameter, dH can be replaced with 2x for the flow between two membranes. Here, x represents the distance between the membranes. In this equation, the constants c and d were chosen as 0.50 and 0.33, respectively, from the analogy of heat transfer in a flat plate [4,5,9], and the constant b was determined experimentally. The overall heat transfer coefficient in a heat transfer process
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represents the combined heat transfer rate of convection in the hot medium, conduction, and convection in the cold medium. Similarly the overall mass transfer coefficient in a diffusion dialyzer consists of the mass transfer resistance in the feed, diffusion through a membrane, and the mass transfer resistance in the permeate. Therefore, we can determine the overall mass transfer coefficients in continuous diffusion dialyzer from the following equation [4,5] U=
1 (1/kLH ) + 1/(ψkm ) + (1/kLL )
(6)
where ψ is the partition coefficient and km the mass transfer coefficient in membrane determined from the diffusion coefficient divided by the membrane thickness. 2.3. Performance equations of diffusion dialysis The performance of diffusion dialysis is usually characterized by the recovery yield and the recovered acid concentration. In a counter-current diffusion dialysis operation, typical operating parameters affecting the performance are the mass transfer coefficient, the effective area, the feed concentrations, and the ratio of the feed to water flow rate. These variables can be determined from the dimension of the system and the operating conditions with an exception of the overall mass transfer coefficient. The mass transfer coefficient is a complex variable depending on the various variables as correlated in Eqs. (5) and (6). The mass transfer rate in diffusion dialysis can be defined as Eq. (7) [2,10,11]. M = UA Clm
(7)
where M is the mass transfer rate of acid and U the overall mass transfer coefficient in continuous dialyzer, Clm the log mean concentration difference between the both solutions and A is the membrane effective area. Since the diffusion dialyzer is a countercurrent flow system, the log mean concentration difference is denoted as the driving force. The mass transfer coefficients were calculated using Eq. (7) experimentally and then we determined the constant b in Eq. (5) using a nonlinear regression method.
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When fresh water is fed in the permeate compartment, the recovery yield can be defined as follows: E=
QL CLo QH CHi
The number of mass transfer unit UA/QH is a dimensionless expression of the mass transfer characteristic of diffusion dialysis. QH /QL is the ratio of the inlet acid flow rate to the inlet water flow rate. To determine unknown recovered concentration (CLo ), an algebraic relation between recovery yield and two operating parameters was derived from the formula used for heat transfer operation [12]. 1 − exp[−(UA/QH )(1 − (QH /QL ))] 1 − (QH /QL ) exp[−(UA/QH )(1 − (QH /QL ))] (10a)
In the limiting case, i.e. QH /QL → 1, Eq. (10a) can be rewritten to the form E=
UA/QH 1 + UA/QH
QH QL 1 − exp[−(UA/QH )(1 − (QH /QL ))] × 1 − (QH /QL ) exp[−(UA/QH )(1 − (QH /QL ))]
(8)
Dimensionless operating parameters are very useful for estimating the acid recovery performances. The dimensionless parameters for depicting the operating conditions are (i) the number of mass transfer unit (UA/QH ) and (ii) the ratio of the feed to water flow rate (QH /QL ). Therefore, Eq. (8) can be expressed as a function of these two dimensionless operating parameters as follows: UA QH E=Ψ (9) , QH Q L
E=
CLo = CHi
(10b)
Then the recovered acid concentration CLo is derived from the combination of Eqs. (8) and (10) as follows:
(11)
The summary of analogous quantities for heat and mass transfer is given in Table 1 [8,9,12]. 3. Experimental An anion-exchange membrane, NEOSEPTA® -AFN (Tokuyama Co., Japan), was used in all the experiments. The membrane (in Cl− form) thickness was measured as 0.17 mm. The feed solutions were prepared by diluting 32% HCl solution (Merck, Germany). All experiments were performed at room temperature. 3.1. Water contents In order to determine the acid concentration in the membrane, the membrane water content was measured. Small pieces of the membrane (area = 9 cm2 ) were soaked in HCl solution of various concentrations for 24 h. Then the weights of the wet membranes were measured immediately. The membrane samples were then dried at 60◦ C under a vacuum condition until a constant weight was obtained. The water contents were determined from the weight difference between the wet and dry membranes. 3.2. HCl uptake by the membrane The saturated concentration of HCl in the membrane was determined by soaking the membrane in
Table 1 Analogies between heat and mass transfer at low mass transfer rates Heat transfer quantities ∂θ hL = −L Nu = k ∂y w √ √ hL 3 Nu = 0.332 Re Pr (for Pr > 0.6) k AU Ntu = C min ε=
C min /C C max ) 1 − e−Ntu (1−C (Cmin /Cmax : capacity rate ratio) C min /C C max ) −N (1−C tu C min /C C max ) e 1 − (C
Mass transfer quantities hM L ∂φ Sh = = −L D ∂y w √ √ hM L 3 Sh = 0.332 Re Sc (for Sc > 0.6) D AU Ntu = CH E=
1 − e−Ntu (1−QH /QL ) (QH /QL : flow ratio) 1 − (QH /QL ) e−Ntu (1−QH /QL )
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HCl solution and extracting the absorbed acid. At the beginning of the tests, membrane pieces (area = 25 cm2 , Cl− form) were washed with distilled water until no salt residues could be detected in the water. Then the sheets were soaked in 100 ml of HCl solution (0.125–3.00 mol m−3 ) and then the solution was stirred for 24 h. The membrane sheets were rinsed with distilled water for a short time to remove the acid remaining at the surface of the membrane. The surface-rinsed membranes were immersed in distilled water for 24 h and this extraction step was repeated three times. Then the amount of the acid extracted from the membrane was measured by microtitration (Metrohm 702 SM Titrino, Swiss) with 0.05 mol dm−3 NaOH standard solution. 3.3. Dialysis cell tests The diffusion coefficients of the acid were determined from batch dialysis cell tests. Fig. 1(a) shows the dialysis cell used in this study. The two-compartment cell, separated by the AFN membrane, was made of plexiglass. The effective area of the test membrane was 16 cm2 . At the beginning of each experiment, the feed side was filled with HCl solution and the permeate side with distilled water. In order to ensure a homogeneous concentration in the cell, each compartment was agitated using a stirrer (model 57022 clamp, Caframo) at 400 rpm. The initial volume of solution in each side was 200 ml and the acid concentrations in both sides were measured every 30 min by acid–base titration. The diffusion coefficients were determined from initial concentration gradients obtained from these data.
Fig. 1. Schematic diagram for diffusion dialyzers: (a) batch diffusion dialysis cell; (b) continuous diffusion dialyzer. The dotted line represents the anion-exchange membrane through which an acid permeates.
3.4. Diffusion dialysis of hydrochloric acid using continuous dialyzer The dialysis coefficients and performance data according to various operating conditions were obtained from continuous dialysis experiments. The detailed experimental conditions are listed in Table 2. The continuous diffusion dialyzer, consisting of 10 sheets of AFN membranes with an effective area of 200 cm2 each, were placed 1 mm apart from each other using a plastic net spacer and rubber gaskets. The dialyzer used for the experiments was TSD-2 diffusion dialyzer (Tokuyama Co., Japan). As shown in Fig. 1(b), the acid solution and water were contacted
Table 2 Various operating conditions for acid recovery in continuous operation Acid concentration (mol dm−3 )
Specific retention time (h cm−1 )
Flow rate ratio
0.06 0.22 0.60 1.05 1.58 1.95 2.60 3.20
0.045 0.55 1.06 1.62 2.53 4.74 6.87 8.22
0.20 0.30 0.47 0.77 1.11 1.65 3.08
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counter-currently. Since the linear flow velocity in a stack was low, air bubbles entrained in the feed solution could be retained in the dialyzer. The trapped air bubbles would disturb the uniform distribution of fluid and reduce the mass transfer area resulting in a significant deterioration of dialysis performance. In order to avoid this problem distilled water was heated and cooled down to room temperature to remove dissolved air, and the water flow was reversed for a short period prior to running each experiment to expel bubbles within the compartments. An experiment was completed when the recovered acid concentration reached a steady state. At the steady state, the concentrations and the flow rates of both streams were measured. The acid concentrations were determined by acid–base titration using NaOH standard solution. 4. Results and discussion 4.1. Water content and HCl uptake by the membrane The water contents were calculated as weight of water per unit mass dry membrane. The water contents were not affected by the acid concentrations investigated in this study, up to 3 mol dm−3 HCl. The average water content was calculated as 0.365 (g-water/g-dry membrane). The dependence of the absorbed HCl in the membrane on the concentration of bulk solution is presented in Fig. 2, where the HCl concentration in the membrane was expressed as mole of the acid absorbed per solution volume in the membrane. It was observed that the concentration in the membrane was nearly proportional to the bulk concentration in the considered concentration range studied. The constant in Eq. (4) was determined as a = 0.9037 from Fig. 2.
Fig. 2. Dependence of HCl concentration in membrane on HCl concentration in the bulk solution: (䉱) experimental data; (- - -) calculated from Eq. (4).
where CH0 is the initial feed concentration. Eq. (12) was solved using a computer program to calculate the diffusion coefficient from the initial transport rate of HCl in the dialysis cell. Dependence of the diffusion coefficient in the AFN membrane on the concentration in the feed solution is shown in Fig. 3. The diffusion coefficients of hydrochloric acid in the membrane were compared with that of hydrochloric acid at infinitely dilute aqueous solution (ca. 3.30E−5 cm2 s−1 at 25◦ C) calculated from the diffusion coefficients of H+ and Cl− [7] and the diffusion coefficients in the membrane were one order of magnitude lower values than that in the bulk solution.
4.2. Diffusion coefficient in membrane Batch cell experiments were performed as described in Section 3. The diffusion coefficient according to the feed concentrations was calculated from Eqs. (3) and (4) and the mass balance. Eq. (3) substituted by Eq. (4) and HCl balance (C H0 V H = C H V H + C L V L ) gave a first-order differential equation as following [4]: dCL Dm A a(CH0 VH − CL VL ) = dt δVL VH − aCL
(12)
Fig. 3. Dependence of the diffusion coefficients in the membrane on the HCl concentration in the feed solution: ( ) experimental data; (- - -) calculated from Eq. (12).
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4.4. Recovery with various operating conditions in continuous operation
Fig. 4. Dependence of the mass transfer coefficients in continuous diffusion dialyzer on the HCl concentration in the feed solution: (䉱) experimental data; (- - -) calculated from Eq. (6).
4.3. Mass transfer coefficient in continuous dialyzer The mass transfer coefficients in continuous dialyzer were calculated from the experimental data using Eq. (7). The relation between the overall mass transfer coefficient and the acid concentration in the feed solution is presented in Fig. 4. The mass transfer coefficients in liquid films were determined using correlation (Eq. (5)) of Sherwood number and the mass transfer coefficient in membrane was calculated from the diffusion coefficient (k m = D m /δ) as a function of log-mean acid concentration. From the experimental results, the constant b in Eq. (5) can be determined using the physical properties. The equation for viscosity of the acid solution, as a function of the concentration, was obtained from the literature [13]: (µ − µo )µo = 0.0030 + 0.0620C 0.5 + 0.0008C (13) where µo is the viscosity of water. The constant b in Eq. (5) was determined as 0.95 by nonlinear regression with the mass transfer coefficients and other physical properties. The partition coefficient ψ in Eq. (6) was determined as 0.9737 from the correlation (Eq. (14)) between the acid concentrations at the interface and in the membrane [5], assuming that the interface concentration was the mean value of the acid concentrations in bulk and membrane phases. Cm = ψCi
(14)
Recovery was defined as the ratio of moles of acid in the recovered stream to the feed expressed in Eq. (8). Plugging the experimental data into the equation gave the recovery in diffusion dialysis. As described in Table 2, continuous dialysis experiments were performed under various operating conditions, i.e. the acid concentration in the feed solution, the retention time of the feed solution, and the ratio of the feed to water flow rates. In these experiments, we found that the flow rate difference between the inlet and outlet was insignificant. It was observed that the volume change was less than 2% in all experiments. It implies that the water transfer through the membrane was negligible unlike diffusion dialysis of high-concentration acid. Therefore, the model developed in this study from the analogy to heat transfer can be reasonably applied to diffusion dialysis of low-concentration acid. In the first set of experiments, effects of the feed concentration were examined from 0.05 to 3 mol dm−3 HCl, while the other two variables were fixed. Considering the relation of retention time to the membrane area, we defined the specific retention time as an effective membrane area divided by the flow rate. The specific retention time of the feed stream was 8.92 h cm−1 and the ratio of the feed and permeate flow rates was unity in each experiment. The effect of acid concentrations in the feed solution on acid recovery by diffusion dialysis is presented in Fig. 5. Since the volume change in the solutions was negligible during the diffusion dialysis, the recovery was proportional to the recovered acid concentration. As the feed concentration increased, the acid uptake increased (Fig. 2), and the diffusion coefficient and the mass transfer coefficient increased (Figs. 3 and 4). Consequently, the acid recovery increased with the feed concentration, the increasing rate being retarded due to the limited concentration difference at higher concentrations. In this case, the resistance in membrane affects the acid recovery yield significantly. The second operating parameter investigated was the flow rate to change the retention time in the stack. The specific retention time of both feed and permeate streams was changed from 0.42 to 8.34 h cm−1 . The feed concentration was fixed at 1.27 mol dm−3 HCl, and the ratio of the feed to water flow rate was unity.
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Fig. 5. Dependence of the acid recovery in the continuous diffusion dialysis on the HCl concentration in the feed solution (specific retention time = 8.92 h cm−1 ). Symbols: experimental data; dotted lines: calculated from Eq. (10).
The acid recovery as a function of the specific retention time of the feed is presented in Fig. 6. As expected the recovery increased with the increasing retention time, but not linearly at a greater retention time due to decreased concentration difference. When the specific retention time was increased, the resistances in liquid films and membrane also increased simultaneously. The third operating parameter investigated was the ratio of the feed to water flow rates. The ratio was varied from 0.1 to 3.0. During the experiment, QH was fixed and QL was varied. The feed solution used in this experiment was 1.22 mol dm−3 . The dependence
Fig. 6. Dependence of the acid recovery in the continuous diffusion dialysis on the specific retention time of the feed solution (feed concentration = 1.27 mol dm−3 ). Symbols: experimental data; dotted lines: calculated from Eq. (10).
Fig. 7. Dependence of the acid recovery in the continuous diffusion dialysis on the ratio of feed to water flow rate (feed concentration = 1.22 mol dm−3 ). Symbols: experimental data; dotted lines: calculated from Eq. (10)].
of the acid recovery in diffusion dialysis on the ratio of the feed to permeate flow rates is shown in Fig. 7. This result shows that the recovery yield decreased with the increasing flow ratio, due to the decreased concentration difference across the membrane and decrease in the diffusion coefficient at a higher flow ratio. Consequently, the resistances in the permeate side liquid film and membrane increased with the increasing the flow ratio. 4.5. Recovery yield and recovered acid concentration predicted by the model In order to estimate acid recovery with various operating conditions, the physicochemical parameters of the diffusion dialysis system, such as water content, membrane thickness, equilibrium concentration and diffusion coefficients obtained in this study, were substituted in the model. Since the log-mean concentration in diffusion dialysis was unknown initially, the overall mass transfer coefficients were determined by the iteration procedure shown in Fig. 8. In this iteration not only the overall mass transfer coefficient, but also the recovery yield and the recovered acid concentration were determined eventually. The numerical simulation of diffusion dialysis was repeated for the range of conditions depicted in Figs. 5–7. Each simulation result is compared with the corresponding experimental results. The dotted lines of Figs. 5–7 represent the recovery yields and
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Fig. 9. Acid recovery yields and recovered concentrations predicted by the model developed in this study for 1 mol dm−3 HCl feed. Solid line: recovery yield; dotted line: recovered concentration (mol dm−3 ). Fig. 8. Algorithm for estimating mass transfer coefficient and acid recovery in diffusion dialysis.
recovered acid concentrations predicted with variation of the feed concentration, the retention time, and the ratio of the feed to water flow rates, respectively. The comparison between experimental results and the predicted values show good agreement. Under the experimental conditions investigated in this study, all the mass transfer resistances in the feed side, the membrane, and the permeate side were found to be significant in determining the overall mass transfer coefficient. Considering the simulation was performed with physicochemical properties for HCl in the membrane and one experimentally determined constant b in Eq. (5), which is subject to a diffusion dialysis system, the model has been developed with minimal empirical factors. The model developed in this study is capable of simulating the variation of the three major operating variables of diffusion dialysis. As an example of the applications, 1 mol dm−3 HCl feed solution was considered for acid recovery by diffusion dialysis. The concentration of the recovered acid is an important criterion for successful implementation of diffusion dialysis in light of recycling the
recovered acid. In the case that the feed concentration is fixed, the retention time and the ratio of the feed to permeate flow rate are the major operating variables. The estimated acid recovery yields and recovered concentrations for 1 mol dm−3 HCl solution are shown in Fig. 9 in terms of the two operating variables. The results show requirement of the two operating variables, the retention time and the ratio of the feed to water flow rate, to obtain a selected acid recovery yield and recovered concentration. It is obvious in this example that the recovery yield may be sacrificed unless the system is operated at a high retention time and a low ratio of the feed to water flow rate, as shown in Figs. 6 and 7.
5. Conclusions In order to predict the performance of a diffusion dialysis process, a mathematical model was developed. For the model equations, the key parameters including water content, membrane thickness and equilibrium acid concentration in the membrane were measured experimentally. The diffusion coefficients of acid in the membrane were calculated from the
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acid transport rate in a dialysis cell. The overall mass transfer coefficient in continuous diffusion dialyzer was determined from the correlation for Sherwood number and the membrane diffusion coefficient. The physicochemical properties were used to predict the performance of diffusion dialysis of HCl in the range of 0.05–3.0 mol dm−3 HCl. It is expected that the model developed in this study can be further applied to diffusion dialysis of other inorganic acids such as sulfuric acid, nitric acid, fluoric acid. Acknowledgements This work was supported in part by Brain Korea 21 from the Ministry of Education through the Graduate Program for Chemical & Environmental Engineering at Kwangju Institute of Science & Technology. 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. [2] S.J. Oh, S.-H. Moon, T. Davis, Effects of metal ions on diffusion dialysis of inorganic acids, J. Membr. Sci. 169 (2000) 95.
[3] A. Narebska, A. Warszaski, Diffusion dialysis transport phenomena by irreversible thermodynamics, J. Membr. Sci. 88 (1994) 167. [4] Z. Palatý, A. Žáková, Transport of sulfuric acid through anion-exchange membrane NEOSEPTA-AFN, J. Membr. Sci. 119 (1996) 183. [5] Z. Palatý, A. Žáková, P. Dolecek, Modeling the transport of Cl− ions through the anion exchange membrane NEOSEPTAAFN; systems HCl/membrane/H2 O and HCl–FeCl3 /membrane/H2 O, J. Membr. Sci. 165 (2000) 237. [6] A. Heintz, C. Illenberger, Diffusion coefficients of Br2 in cation exchange membranes, J. Membr. Sci. 113 (1996) 175. [7] E.L. Cussler, Diffusion, Cambridge University Press, Cambridge, 1984, p. 150, 226. [8] R.B. Bird, W.E. Stewart, E.N. Lighfoot, Transport Phenomena, Wiley, New York, 1960, p. 640. [9] E.R.G. Eckert, R.M. Drake, Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972, p. 731. [10] Y. Kobuchi, H. Motomura, Y. Noma, F. Hanada, Application of the ion-exchange membranes-acids recovery by diffusion dialysis, in: Proceedings of the 1986 Europe–Japan Congress on Membranes and Membrane Processes, Plenum Press, New York, 1986. [11] P. Sridhar, G. Subramaniam, Recovery of acid from cation exchange resin regeneration waste by diffusion dialysis, J. Membr. Sci. 45 (1989) 273. [12] W.M. Kays, A.L. London, Compact Heat Exchangers, 2nd Edition, McGraw-Hill, New York, 1964, p. 17. [13] Encyclopedia of Chemical Technology, Vol. 12, 3rd Edition, Wiley, New York, 1984, p. 989.