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theoretical model with dimensionless numbers to predict the performance of electrodialysis systems on the basis of operating conditions
Water Research 98 (2016) 270e279
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An empirical/theoretical model with dimensionless numbers to predict the performance of electrodialysis systems on the basis of operating conditions Leila Karimi*, Abbas Ghassemi Institute for Energy & the Environment (IEE), 1060 Frenger Mall, ECIII, Suite 300, New Mexico State University, Las Cruces, NM, 88003, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 January 2016 Received in revised form 4 April 2016 Accepted 8 April 2016 Available online 11 April 2016
Among the different technologies developed for desalination, the electrodialysis/electrodialysis reversal (ED/EDR) process is one of the most promising for treating brackish water with low salinity when there is high risk of scaling. Multiple researchers have investigated ED/EDR to optimize the process, determine the effects of operating parameters, and develop theoretical/empirical models. Previously published empirical/theoretical models have evaluated the effect of the hydraulic conditions of the ED/EDR on the limiting current density using dimensionless numbers. The reason for previous studies' emphasis on limiting current density is twofold: 1) to maximize ion removal, most ED/EDR systems are operated close to limiting current conditions if there is not a scaling potential in the concentrate chamber due to a high concentration of less-soluble salts; and 2) for modeling the ED/EDR system with dimensionless numbers, it is more accurate and convenient to use limiting current density, where the boundary layer's characteristics are known at constant electrical conditions. To improve knowledge of ED/EDR systems, ED/EDR models should be also developed for the Ohmic region, where operation reduces energy consumption, facilitates targeted ion removal, and prolongs membrane life compared to limiting current conditions. In this paper, theoretical/empirical models were developed for ED/EDR performance in a wide range of operating conditions. The presented ion removal and selectivity models were developed for the removal of monovalent ions and divalent ions utilizing the dominant dimensionless numbers obtained from laboratory scale electrodialysis experiments. At any system scale, these models can predict ED/EDR performance in terms of monovalent and divalent ion removal. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Electrodialysis Predictive model Ion removal Selectivity Stanton number
1. Introduction In 2014, the installed capacity of electrodialysis/electrodialysis reversal (ED/EDR) systems around the world was 2.59 million gallons per day (IDA, 2015). Because of the extent to which ED/EDR is utilized, small improvements in efficiency or quality can bring significant practical benefits, so there has been ongoing interest in studying the ED/EDR process, modeling it, and optimizing its performance. The performance of ED/EDR systems e in terms of the cost and the quality of the treated water e is strongly influenced by operating conditions (Karimi and Ghassemi, 2015a; Karimi et al., 2015),
* Corresponding author. E-mail addresses: [email protected] (A. Ghassemi).
(L.
Karimi),
http://dx.doi.org/10.1016/j.watres.2016.04.014 0043-1354/© 2016 Elsevier Ltd. All rights reserved.
[email protected]
so multiple experimental and theoretical investigations have modeled ED/EDR performance on the basis of operating parameters. The vast majority of these investigations have utilized numerical solutions to model the EDR process, whose complexity has rendered analytical solutions prohibitively difficult (Ghorbani et al., 2016). Several researchers have therefore pursued the alternative approach of developing empirical/theoretical models, and the most applicable of these models have been developed based on dimensionless numbers, which allow the models to be applied at different scales. However, previously published empirical/theoretical models have evaluated the effect of the hydraulic conditions of the ED/EDR stack on the limiting current density using dominant dimensionless numbers at constant electrical conditions. To expand dimensionless empirical/theoretical models of the ED/EDR process into a wider range of operation, the purpose of the present study is to develop a dimensionless empirical/theoretical model of the ED/EDR process that can predict performance under different operating conditions
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while accommodating variations in applied voltage. The reason for previous studies' emphasis on limiting current conditions is twofold: 1) most EDR systems are operated close to limiting current conditions, and 2) it is more accurate and convenient to model the ED/EDR system using dimensionless numbers at limiting current density, where the boundary layer's characteristics are known at constant electrical conditions. At limiting current point, it is possible to find analytical solutions for the governing equations in the system, and at limiting current density, an experimental model can validate a theoretical model. Section 1.1 summarizes previously published ED/EDR models that have been developed for limiting current densities. 1.1. ED/EDR models for limiting current densities All previous dimensionless models of the ED/EDR process have focused on predicting limiting current density at different hydraulic conditions. In the previous studies, the prediction of ion removal in the EDR process under the hydraulic and electrical conditions were not discussed. A handful of studies e for example, the work of Probstein et al. (1972) e have considered both limiting current conditions and lower salt removal conditions, but such studies have not constituted comprehensive investigations of ED systems across a wide range of applied voltage. A chronological review of previously conducted studies follows. Since existing models have been developed using different systems, different types of membranes and spacers, and different feed solution characteristics, there are few commonalities in the studies apart from limiting current points. The late 1960s and the 1970s saw the development and refinement of dimensionless models for ED/EDR. In 1968, Sonin and Probstein developed an empirical equation for dimensionless current density under a wide range of operating conditions, based on assumptions of a laminar flow regime and the existence of two regions e developing and fully developed e for concentration diffusion layers (Sonin and Probstein, 1968). In 1971, Kitamoto and Takashima presented a dimensionless model that correlates the current density to the solution flow hydrodynamics and applied voltage through a modified version of Peclet and Stanton numbers. Their results also showed that, although the limiting current density depends on the operating conditions, it has a weak relationship with the ion exchange membrane properties (Kitamoto and Takashima, 1971). This result is rational, because in the electrodialysis process, the controlling step in ionic mass transfer is diffusion in the boundary layer, especially at limiting current condition. Then, in 1972, Probstein et al. conducted a study in two primary regions, zero polarization and full polarization. Although Probstein et al. presented a correlation between the limiting current density and the mean current density, their correlation was most accurate for very small levels of salt removal (Probstein et al., 1972). Huang (1977) also obtained different empirical models for ionic mass transfer at limiting current for two laminar and turbulent flow regimes (Huang, 1977). Kuroda et al. (1983) found that Sh in the ED process is largely a function of Re0.5, regardless of the type of the spacer (Kuroda et al., 1983). Kraaijeveld et al. (1995) presented an empirical model for limiting current measurement in the separation of amino acid using the electrodialysis process. They used the same form of model presented by Sonin and Isaacson (1974), but based their experiments on different spacers (Kraaijeveld et al., 1995). Later, Fidaleo and Moresi (2005) compared their experimental data with the models presented by Kuroda et al. (1983) and Sonin and Isaacson (1974), and found that Sh has a functionality of Re0.5 rather than Re0.13, which was presented Kuroda et al. (1983) and Sonin and Isaacson (1974).
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More recently, Shaposhnik and Grigorchuk (2010) also presented equations for Sh, representing the mass transfer rate, to compare the rate of ionic mass transfer in spacer-free channels and channels having inert spacers in the ED process. The generalized equations developed for Sh were used to predict the limiting current density (Shaposhnik and Grigorchuk, 2010). Fidaleo and Moresi (2005, 2006, and 2010) found a similar form of power relationship between Sh as dependent variable and two independent variables of Re and Sc, using experimental data collected for the electrodialysis with different feed solution compositions (Fidaleo and Moresi, 2010, 2006, 2005). Mitko and Turek (2014) utilized the Graetz-Leveque equation for laminar flow in a channel (Noble and Stem, 1995) to find the mass transfer coefficient in order to model the concentration polarization along the ED with segmented electrodes (Mitko and Turek, 2014). Tadimeti et al. (2015) experimentally developed another equation for Sh to predict the ionic transfer coefficients of Ca2þ and Cl from a sugar solution in a batch ED process (Tadimeti et al., 2015). Despite the wide range of conditions and design characteristics represented in these studies, each of the previously developed models has focused almost exclusively on limiting current density. Although Sonin and Probstein (1968) and Probstein et al. (1972) presented some theoretical models for partial concentration polarization and lower levels of current, the experimental results show that the presented models are valid mostly for very small levels of salt removal, reducing the applicability of the models. To improve our knowledge of ED/EDR systems and optimize the process, ED/EDR performance should be investigated in a wide range of operating conditions, including both the Ohmic region and the non-Ohmic region. Each region can be used for different purposes. Operation in the Ohmic region reduces energy consumption, facilitates targeted ion removal, and prolongs membrane life compared to limiting current conditions; however, if there are no scaling restrictions due to a high content of CaSO4 or other less soluble salts, industrial ED/EDR systems are often operated in higher levels of voltage (conditions close to or higher than limiting current conditions) to meet the maximum ion removal without damaging the membranes. However, if there is a potential for scaling and membrane damage, ED/EDR should operate under mild electrical conditions: when a current greater than limiting current is applied, the ions migrate through the ion exchange membranes faster than they move in the solution, so ions such as Ca2þ and SO2 4 can accumulate in the concentration boundary layers and precipitate (Scarazzato et al., 2015). When there is such a risk, operations at lower voltages (and lower salt removal) will be more rational. To model ED/EDR performance in a wider range of operation including the Ohmic region, the present study presents empirical models for the removal of monovalent ions (Naþ and Cl) and divalent ions (Ca2þ and SO2 4 ) in a wide range of operating condition through dimensionless numbers that were determined based on the theoretical studies presented in Section 1.2. 1.2. Theory of predictive models for ion removal in the ED/EDR process based on the dominant dimensionless numbers The quality of the desalted water in the ED/EDR process can be predicted based on the dominant dimensionless numbers in the process. The dominant dimensionless numbers in the solution phase are important because ion transport in the boundary layer, strongly influenced by concentration polarization, can be considered the controlling step in the ion transport process from dilute to concentrate stream due to high rate of electro-migration in the membrane phase (Grigorchuk et al., 2005; Huang, 1977; Tadimeti et al., 2015). The following theoretical equations show that a predictive
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model for ion removal in the ED/EDR process can be developed based on two dominant dimensionless numbers: Peclet number, which represents the hydrodynamics of the process, and dimensionless applied electrical potential, which represents the main driving force of the process. The general form of the equation for the above-mentioned dimensionless numbers can be generated theoretically as described below. The current density in the electrodialysis system is derived from the Nernst-Planck equation, which represents the mass transfer mechanisms in the ED/EDR process, and which can be written as shown in Equation (1):
ii ¼ zi F Di
dCi zi F dj DC þ dx RT i i dx
0
(2)
where Q is the total flow rate of dilute stream, DC is the difference between the concentrations of the ion in the inlet and outlet of dilute cells that occurs due to ion transport, V is the dilute solution velocity in a cell, and Ncp is the number of cell pairs in the stack. Equation (2) can be rewritten by multiplying it by h=Cr;i h , allowing the variables to be reduced to the dimensionless variables, as in Equation (3):
Q DCi ANcp VCr;i
! 0D Ci 10 zi F j 1 i Cr;i v RT D A@ A ¼ i @ Aþ@ Vh Vh v hx v hx 0v
Ci Cr;i
! 1
(4)
ji ¼
zi F j RT
(5)
x ¼
x h
(6)
Cr;i ¼
CF;i CD;i ! Ln CF;i C D;i
(7)
where St is Stanton number, j* is dimensionless electrical potential per desalting cell, x* is dimensionless direction toward the membrane, CF is feed concentration (inlet of dilute chamber), and CD is dilute concentration (outlet of dilute chamber). By applying Equations (4)e(7) in Equation (3), Equation (3) can be finalized to Equation (8).
Sti ¼
dC C dji 1 i þ i Pei dx Pei dx
(8)
From the definition of Stanton number shown in Equation (4) and Equation (8), the Stanton number (or modified Sherwood number), which is the ratio of ion flux toward the membrane to the ion transfer in the flow direction, can be defined as a function of Peclet number and dimensionless electrical potential, as follows in Equation (9).
Sti ¼ f Pei ; j*i
(9)
In this research, the model was developed experimentally due to the lack of an analytical solution for Equation (8). The empirical models, which are based on the theory, not only can be used easily, but also show a high degree of accuracy in real applications. As shown in Equation (9), the Stanton number can represent the removal of ions as a function of Peclet number and dimensionless electrical potential.
(3)
where A is the membrane active area, h is the dilute cell thickness, and Cr is the logarithmic average of ion concentration in dilute cells. Equation (3) is simplified to the form of Equation (8) using the dimensionless parameters, which are defined as shown in Equations (4)e(7):
Fig. 1. Schematic of once cell pair of the modeled electrodialysis system.
Q DCi ANcp VCr;i
(1)
where ii is the amount of current density that is carried by ion type i, z is electrical charge, F is Faraday constant, D is the diffusion coefficient, C is ion concentration, R is a gas constant, T is the temperature of the dilute stream, j is applied voltage per desalting cell, and x is the direction perpendicular to the membrane. Subscript i is the type of ion. The flow and ion movement directions are shown in the schematic in Fig. 1. By dividing both sides of Equation (1) by the DCi term ziFV and replacing ii with zi FQ ANcp , the equation then takes the form of Equation (2):
1 iFj v zRT Q DCi Di vCi Di Ci @ A ¼ þ ANcp V V vx V vx
Sti ¼
2. Materials and methods 2.1. Experimental set-up In this research, a complete laboratory-scale ED/EDR setup was designed and built by the authors to conduct the laboratory-scale experiments. The purpose of utilizing the laboratory scale set-up was to run the experiments under highly controlled operating conditions, where all the parameters were recorded every second using a data acquisition system. A complete data acquisition system, shown in Fig. 2, was designed in order to continuously and accurately monitor and record the operating conditions in the experiment. To monitor and record the operating conditions in the process, five sensors e for pH, temperature, conductivity, pressure, and flow e were installed in the six main stream manifolds (three inlet and three outlet streams). The three inlet streams to the stack were Dilute In, Concentrate In, and Electrode In (electrode rinse solution entering cathode and anode chambers). The above-mentioned sensors were installed in each inlet stream to record the operating conditions. Additionally, three outlets streams, leaving the stack, were Dilute Out (product), Concentrate Out (brine), and Electrode Out (electrode rinse solution leaving cathode and anode
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Fig. 2. Schematic of designed ED set-up equipped with a full data acquisition.
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L. Karimi, A. Ghassemi / Water Research 98 (2016) 270e279 Table 1 Investigated factors and their levels. Factor
Levels
Feed Composition (ppm)
Mono-Salt: NaCl: 200 and 1000 CaSO4: 200 and 1000 Binary-Salt Mixture: NaCl 200 þ CaSO4 200 NaCl 1000 þ CaSO4 1000 15, 35 1.6, 2.1, 2.6, 3.7, 4.2 5, 10, 15, 20, 25
Temperature (ºC) Solution Linear Velocity (cm/s) Total Applied Voltage (V) on the Membrane Pack
chambers). The outlet streams, similar to inlet streams, were also equipped with five inline sensors. For data acquisition, the system employed Canary Enterprise Historian, a data acquisition software program designed by Canary Labs. The connection between the programmable logic controller and the Historian software was established using Kepware software. The ED cell unit used in this set-up is PCCell ED 64002, a laboratory scale electrodialyzer with the nominal capacity of 4e8 L/h per dilute and concrete cell, and which is composed of 15 cell pairs. The utilized cation and anion exchange membranes were GE CR67HMR and GE AR908, respectively, with the active area of 64 cm2. The mesh spacer used in the stack had a thickness of 0.4 mm and a direct flow path. 2.2. Design of experiments The experiments were performed with a full factorial design, using 6 levels of feed compositions, 2 levels of temperature, 5 levels of solution linear velocity, and 5 levels of total applied voltage. The different levels for each factor are given in Table 1.
During the experiments, the flow rate of the streams was set through controllable micro-gear pumps and measured using the inline flow sensors. The linear velocity of the streams in the dilute cell was calculated using Equation (10).
V¼
Q whNcp ð1 4Þ
(10)
where w is the width of the dilute chamber and 4 is the fraction of the cross section of the desalting chamber filled with the fibers of the spacer. The 4-value of the spacer used in this research is 20.3%. Additionally, the overall voltage applied to the stack was recorded at two points. The first recorded voltage was the overall voltage applied to the whole ED stack, including the electrode chambers. The recorded value of the overall voltage was greater than the effective voltage applied for the desalination process due to potential drop in electrode chambers (Karimi, 2015). The second recorded voltage was the amount of applied electrical potential in the membrane pack, DjMP , excluding the electrode chambers. The amount of applied electrical potential per dilute cell could be calculated using Equation (11), given the assumption that the voltage is uniformly distributed among cells:
2.3. Experimental procedure The three inlet solutions e Dilute In, Concentrate In, and Electrode In e were prepared before starting the experiment according to the design of experiment. The compositions of the Dilute In and Concentrate In solutions were the same within experimental runs; however, trace amounts of HYPERSPERSE MDC706 antiscalant, (Hanrahan et al., 2015), were added to the Concentrate In solution to prevent scaling problems when feed water containing CaSO4 was used. The Dilute In and Concentrate In solutions were prepared according to the experimental design (shown in Table 1) by adding either NaCl, CaSO4, or both to pure water with a salinity of less than 5 ppm, provided by New Mexico State University's semi-pilot-scale RO setup. Next, the heating or cooling was done according to the required temperature of the experiment. The heating or cooling process was done using a heat exchanger with three branches of stainless steel coils that were placed in the isolated containers. The composition of the electrode rinse solution, shown as Electrode In, was different from the compositions of the other solutions to prevent the formation of Cl2 and CaSO4 scales in the electrode chambers. The electrode solutions used in the experiments were buffer solutions of CH3COONa and CH3COOH with a pH of 5.2. The required amount of NaNO3 was added to the Electrode In to make its electrical conductivity equal the conductivity of the Dilute In and Concentrate In solutions. Additionally, the temperature of the electrode rinse solution was kept at the same temperature as the other solutions. During the experiments, the set points were set according to the designed set points. Water samples were collected while the multi-component solutions were used in the experiments. Before sample collection, steady state condition was checked and confirmed.
DjDil:Cell ¼
DjMP 2Ncp
(11)
where DjDil:Cell and DjMP are, respectively, applied electrical potential for one dilute cell, and applied electrical potential in the membrane pack.
2.4. Sample analysis When single-salt solutions were used in the experiments, the measurements for ion removal and the concentration of ions in the solutions could be done using the conductivity-TDS calibration curves. These calibration curves were generated for the NaCl and CaSO4 solutions in a wide range of salinities and different levels of temperature, and were used to convert the experimentally measured conductivity values into the desired salinity values. During the experiments, the conductivity and temperature values of the streams were recorded by inline temperature and conductivity sensors. Then, the conductivity values were converted to salinity values using the models obtained for each salt, which could give the concentration of ions in the streams. Conductivity-TDS models for NaCl and CaSO4, as shown in Equations (13) and (14). The conductivity-TDS calibration curve for NaCl and the conductivity-TDS model validation for CaSO4 are shown in the supplementary information, Figs. S1 and S2, respectively. For NaCl, solution conductivity was linearly dependent on temperature at a 2% slope in the examined range of concentration (0e2000 ppm), as shown in Equation 12.
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kTð CÞ ¼ k25 C *ð1 þ ðT 25ÞÞ*0:02
(12)
TDSNaCl ¼ k25 C 0:517 21:8
(13)
where TDS and k are the concentration of NaCl and the conductivity of the solution at 25 C. For CaSO4, although the temperature dependency of CaSO4 was linear, the slope was highly dependent on the salt concentration in the investigated range of 0e1600 ppm. Therefore, a non-linear equation was fitted for the concentration of CaSO4, using the MATLAB curve fitting tool with two independent variables for conductivity and temperature, as shown in Equation 14.
TDSCaSO4 ¼ 0:7952 k1:228 T 0:498
(14)
where k and T are the conductivity of the solution measured by conductivity sensor and the temperature of the solution in degrees Celsius, respectively. The R2 value of the obtained model is 0.9981. In cases where binary mixtures of salts were used, water samples were collected during the experiments and analyzed for cations and anions using the Dionex ICS-5000 Dual Channel Ion Chromatography System. 3. Results and discussion As shown before in Equation (9), the Stanton number can represent the removal of ions as a function of Peclet number and dimensionless electrical potential. Based on the proposed theoretical approach, the collected experimental data were used to develop the empirical model for predicting ion removal in the ED process. The general form of the empirical model is shown in Equation (15):
Sti ¼ aPeb jc i
(15)
where St, Pe, and j*are Stanton number, Peclet number, and dimensionless electrical potential per dilute cell. The terms a, b, and c are model coefficients, and the subscript i represents a typical ion. It is worth mentioning that the dimensionless number Pe was calculated based on the effective diffusion coefficient of the ions available in the solutions. When the feed is a multi-ionic solution, each ion is assigned a unique effective diffusion coefficient, whose value can be calculated through the method developed by Geraldes and Afonso (2010), an approach that was used in this research. The coefficients and powers of the variables were obtained through two approaches e MATLAB and linear regression analysis e using Minitab software, after linearizing the nonlinear equation (Equation (15)) to a linear one. The model coefficients were different at Table 2 Power of variables of dimensionless ion removal model in EDR under different conditions, shown as Equation (15). Salt
Component
C0 (ppm)
a
b
c
R2
NaCl NaCl þ CaSO4 NaCl þ CaSO4 NaCl NaCl þ CaSO4 NaCl þ CaSO4 CaSO4 NaCl þ CaSO4 NaCl þ CaSO4 CaSO4 NaCl þ CaSO4 NaCl þ CaSO4
Naþ, Cl Naþ Cl Naþ, Cl Naþ Cl Ca2þ, SO2 4 Ca2þ SO2 4 Ca2þ, SO2 4 2þ Ca 2 SO4
1000 1000, 1000 1000, 1000 200 200, 200 200, 200 1000 1000, 1000 1000, 1000 200 200, 200 200, 200
0.18 0.58 1.08 0.17 0.58 0.70 0.41 0.53 0.19 0.32 1.69 0.55
0.69 1.13 1.08 0.67 0.93 0.96 0.84 0.83 0.80 0.75 0.81 0.75
0.71 1.57 1.28 0.74 1.12 1.16 0.81 0.67 0.83 0.72 0.44 0.54
0.98 0.97 0.96 0.97 0.94 0.93 0.97 0.95 0.93 0.96 0.93 0.92
275
different feed solution compositions, as shown in Table 2. The results from statistical analysis conducted by Minitab confirmed that the two considered dimensionless numbers based on the theoretical discussion, Pe and j*, were significant parameters in the developed models for ion transport in the ED process at the significance level of a ¼ 0.05. The p-values for the constant “a” in Equation (15) and the variables “b” and “c” in Equation (15) were much less than 0.05, which means these coefficients were significantly different from zero. Additionally, the obtained R2 values for all models varied between 92% and 98%, confirming that experimental data had a good fit with the presented empirical form of the models. The values obtained for the model coefficients (a, b, and c) are shown in Table 2. In the application stage, by applying the obtained values, the Stanton number for each ion can be calculated at different operating conditions. Then, based on the equation provided for Stanton number in form of Equation (4), the concentration of the ion in the product stream (Dilute Out) can be calculated through a simple mathematical equation, Equation (7), that gives the ratio of ion removal to the logarithmic average of ion concentration. Additionally, the pH values of the product during the experiments were in the range of 5.6e6.9, while the pH of the feed solution was 6.1e7.3. The variations in pH had no discernible effect on the conductivity values of the dilute stream. The presented equations showed a good goodness of fit for predicting the Stanton number of ions (see the supplementary information, Figs. S3 to S8). 3.1. Generalized form of the model In order to present a general form of the model, models were developed for the whole range of concentration, from 200 to 1000 ppm for mono-salt solutions and 400e2000 ppm for binary salt mixtures. Although the R2 value of the generalized models was slightly reduced compared to the models developed for the limited level of the concentration, the generalized models can be used much more extensively because their applications are less limited by the concentration range. The model parameters are shown in Table 3. Additionally, the comparisons of models with experimental data are shown in Figs. 3 and 4. It is proposed that the effect of initial concentration becomes larger when limiting current approaches. Therefore, for some of the experiments that were done above the Ohmic region, the effect of concentration became more significant. That is why the generalized models show less accuracy compared to the models developed for each level of concentration. Surface graphs (Figs. 5 and 6) also were plotted to show the variation of each ion's Stanton number versus Peclet number and variations in dimensionless applied voltage. Each surface plot indicates that the Stanton number follows the form of the developed models. The results demonstrate that the generalized models can predict the amount of monovalent and divalent ion removal with a good level of accuracy. Therefore, in the brackish water electrodialysis process, researchers and engineers can utilize this kind of master plots to visually predict the removal
Table 3 Power of variables in the generalized form of the dimensionless ion removal model in EDR under different conditions, shown as Equation (15). Salt
Component
a
b
c
R2
NaCl CaSO4 NaCl þ CaSO4 NaCl þ CaSO4
Naþ, Cl Ca2þ, SO2 4 Naþ, Cl 2þ Ca , SO2 4
0.29 0.37 0.82 1.17
0.73 0.81 1.05 0.86
0.71 0.80 1.32 0.57
86% 84% 86% 81%
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Fig. 3. (a) Comparison of predicted Stanton number of monovalent ions (Naþ and Cl) with the experimental values when the feed solution consists of NaCl with the concentration of 200e1000 ppm, and (b) Comparison of predicted Stanton number of divalent ions (Ca2þ and SO2 4 ) with the experimental values when the feed solution consists of CaSO4 with the concentration of 200e1000 ppm.
of each ion based on the master curve. These models are likely to offer a good fit for the removal of other monovalent and divalent ions with hydrated ion sizes similar to the ions studied in this research. According to the characteristics of the dimensionless equations and the scale-up rules, the presented models can be applied in large-scale ED/EDR processes at the same values of Peclet number and dimensionless applied voltage. According to the definition of Stanton number, it can be rewritten as Equation (16).
Sti ¼
DCi C r;i L=h
¼ aPeb j*c i
(16)
Therefore, at the same values of Pe and j*i , the ratio of L (length of the dilute channel) to h (thickness of the dilute channel) can be applied to predict the amount of ion removal in large-scale ED/EDR processes, as follows in Equation (17):
DCi C r;i DCi C r;i
! !F ¼ L
LF hL LL hF
(17)
where subscripts F and L are full scale and laboratory scale, respectively. Additionally, by utilizing the obtained model coefficients, it is possible to predict the removal of ions under a wide range of operating factors. Furthermore, these models have promising potential to predict the removal of other monovalent and divalent ions with hydrated ion sizes similar to those of the ions studied in this research.
3.2. Selectivity model in ED/EDR process based on the dominant dimensionless numbers One of the important parameters in the performance of the ED/ EDR process is the selectivity, which shows the removal ratios of ions in the process. Due to different compositions of water resources and the application fields of treated water, it is sometimes necessary to run the process under specific conditions to remove a target ion or ions more than the others. One of the most effective ways to do this is utilizing selective ion exchange membranes with a higher affinity toward the target ions (Karimi and Ghassemi, 2015b). However, the operating factors, such as solution flow rate (velocity) and especially the applied voltage can have a noticeable role in the selective removal of ions. Based on the provided model for selectivity as shown in
Fig. 4. Goodness of fit of the model when feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm: (a) Comparison of the predicted Stanton number of monovalent ions (Naþ and Cl) with the experimental values; (b) Comparison of the predicted Stanton number of divalent ions (Ca2þ and SO2 4 ) with the experimental values.
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277
Fig. 5. (a) Variation of Stanton number of monovalent ions (Naþ and Cl) when the feed solution consists of NaCl with the concentration of 200 ppm and 1000 ppm; (b) Variation of Stanton number of divalent ions (Ca2þ and SO2 4 ) when the feed solution consists of CaSO4 with the concentration of 200 and 1000 ppm.
Equation (18), the selectivity term can be written for any pair of ions with the same type of electrical charge. However, it is more rational to assume a reference cation and anion and compare the removal of the ions with respect to those reference ions. The reference ions can be chosen based on the composition of the solution. In this study, two monovalent ions, Naþ and Cl, were chosen as reference cation and anion, respectively.
Sij
ti t j ¼ Ci C j
(18)
where t is the ion transport number and subscripts i and j are two different counter ions. In Equation (18), by replacing ti and tj with i Ji , it is possible to rewrite the selectivity their equation ti ¼ PFzFz J i i
of the ED/EDR process as follows in Equation (19):
. Sij ¼ Fzi Ji=F P z J P J Fz i i j j F zi Ji Cr;i C r;j
(19)
DC by the term Q ANcp ;
where the ion flux, j, can be replaced therefore, the new form of selectivity parameter can be written as Equation (20):
Sij ¼
1 !0 DC @ i Cr;i A zi z DCj C j r;j
(20)
By multiplying the second parenthetical expression in Equation ! (20) by
L= h L= h
, according to Equation (16), the second parenthetical
expression in Equation (20) is the ratio of the Stanton numbers of
two ions, i and j. Therefore, the selectivity of the ED/EDR process can be defined as the product of the ratio of the electrical charge of ions and the Stanton numbers. The empirical equations developed for the Stanton number of ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. Additionally, by replacing the provided equations for Stanton numbers, the selectivity of ED/EDR process can be written as Equations (21)e(24). For feed water composed of binary salt mixtures (NaCl at 1000 ppm and CaSO4 at 1000 ppm), Equations (21) and (22) are obtained utilizing the Stanton model parameters for each ion reported in Table 2: 0:3 *0:9 SCa jNa Na ¼ 2:89Pe
(21)
4 ¼ 0:63Pe0:28 j*0:45 SSO Cl Cl
(22)
For feed water composed of binary salt mixtures (NaCl at 200 ppm and CaSO4 at 200 ppm), the application equations are Equations (23) and (24): 0:12 *0:69 SCa jNa Na ¼ 7:95Pe
(23)
4 ¼ 2:23Pe0:2 j*0:62 SSO Cl Cl
(24)
These equations depict that increases in applied voltage decrease the selectivity ratio of divalent ions against monovalent ions. Moreover, pilot-scale experiments have found that, as higher levels of voltage are applied, the percentage removal of ions approaches 100%, which means the selectivity values approach one (Karimi and Ghassemi, 2015a). The comparisons of experimental values for selectivity and the calculated values for selectivity are presented in the supplementary
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Fig. 6. Variation of Stanton number with variations of Peclet number and dimensionless applied voltage when the feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm for (a) monovalent ions (Naþ and Cl), and (b) divalent ions (Ca2þ and SO2 4 ).
information, Figs. S9 and S10.
4. Conclusions
3.3. Generalized model for selectivity in ED/EDR process based on the dominant dimensionless numbers
The main hypothesis of this research was that an empirical predictive model could be developed based on the dominant dimensionless numbers in the ED/EDR process, giving the ability to predict the removal of ions in the ED/EDR process regardless the scale of the process. Based on the theoretical approach, the collected experimental data were utilized to develop the empirical model based on the dimensionless analysis to predict ion removal in the ED/EDR process. The empirical model was developed for Stanton number (representing the ion removal) as a function of two dominant dimensionless numbers, Peclet number and dimensionless applied voltage. The regression analysis confirmed the significant roles of the modeling parameters, and showing a good fit of the model with provided constants. The model can be utilized in large-scale applications due to the nature of dimensionless numbers. Additionally, through the utilization of the obtained model coefficients, ion removal can be predicted under a wide range of operating factors. These correlations have promising potential to predict the removal of other monovalent and divalent ions with hydrated ion sizes similar to those of the ions studied in this research. Furthermore, the selectivity of the ED/EDR process was defined in a new form as the product of the ratio of electrical charge of ions and the ratio of their Stanton numbers. The developed empirical correlations for the Stanton numbers of monovalent ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. The developed equations and selectivity values depict that the ED/EDR process has more selective behavior for divalent ions compared to monovalent ions. However, the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values and with decreases in the Peclet number values. Moreover, at higher levels of voltage, the
The generalized models were developed for selective ion removal in the ED/EDR process based on the generalized models for Stanton numbers, which can be written as in Equation (25):
Sij ¼ 4:25Pe0:19 j*0:75 j
(25)
where i and j are couples of Ca2þ and Naþ or SO2 4 and Cl . Similar to the generalized form of models for Stanton number, Equation (25) has much a more extensive application than Equation 21 through 24. Equation (25) depicts that the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values, while it increases with increases in Pe values. Decreases in Pe can occur due to increases in superficial velocity, greater thickness of the dilute chamber, or a decrease in the ratio of divalent ions to monovalent ions in the feed composition. The comparison between the presented model and the experimental values is shown in Fig. 7. Furthermore, the selectivity of the ED/EDR process was defined in a new form as the product of the ratio of the electrical charge of ions and the ratio of their Stanton numbers. The developed empirical equations for the Stanton numbers of ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. The developed equations and selectivity values shown in Fig. 7 depict that the ED/EDR process has more selective behavior for divalent ions compared to monovalent ions. However, the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values. Moreover, at higher levels of voltage, the percentage removal of all ions approaches 100%, which means the selectivity values approach one.
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279
Fig. 7. Comparison of the ED process's predicted selectivity with experimental values for the removal of divalent ions against monovalent ions ((a) Ca2þ against Naþ, and (b) SO2 4 against Cl) when the feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm.
percentage removal of all ions approaches 100%, which means the selectivity values approach one. For further investigations, it would be worthwhile to study and verify the model at a higher range of concentration, other types of monovalent and divalent ions, and larger scales. Acknowledgment The authors would like to thank the Office of Naval Research (N000141010711) for funding this research. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.watres.2016.04.014. References Fidaleo, M., Moresi, M., 2005. Optimal strategy to model the electrodialytic recovery of a strong electrolyte. J. Memb. Sci. 260, 90e111. Fidaleo, M., Moresi, M., 2006. Assessment of the main engineering parameters controlling the electrodialytic recovery of sodium propionate from aqueous solutions. J. Food Eng. 76, 218e231. http://dx.doi.org/10.1016/ j.jfoodeng.2005.05.010. Fidaleo, M., Moresi, M., 2010. Application of the Nernst-Planck approach to model the electrodialytic recovery of disodium itaconate. J. Memb. Sci. 349, 393e404. http://dx.doi.org/10.1016/j.memsci.2009.12.002. Geraldes, V., Afonso, M.D., 2010. Limiting current density in the electrodialysis of multi-ionic solutions. J. Memb. Sci. 360, 499e508. http://dx.doi.org/10.1016/ j.memsci.2010.05.054. Ghorbani, A., Ghassemi, A., Andersen, P.K., Foudazi, R., 2016. A prediction model of mass transfer through an electrodialysis cell. Desalin. Water Treat. http:// dx.doi.org/10.1080/19443994.2015.1123195 (in press). Grigorchuk, O.V., Vasil’eva, V.I., Shaposhnik, V.A., 2005. Local characteristics of mass transfer under electrodialysis demineralization. Desalination 184, 431e438. http://dx.doi.org/10.1016/j.desal.2005.03.048. Hanrahan, C., Karimi, L., Ghassemi, A., Sharbat, A., 2015. High-recovery electrodialysis reversal for the desalination of Inland brackish waters. Desalin. Water Treat. 1e11. http://dx.doi.org/10.1080/19443994.2015.1041162.
Huang, T., 1977. Correlation of ionic mass transfer rate in ion exchange membrane electrodialysis. J. Chem. Eng. Data 22, 422e426. IDA, 2015. IDA Desalination Yearbook 2015e2016. Karimi, L., 2015. Theoretical, Experimental, and Predictive Model for Ion Removal in Electrodialysis and Electrodialysis Reversal. New Mexico State University. Karimi, L., Abkar, L., Aghajani, M., Ghassemi, A., 2015. Technical feasibility comparison of off-grid PV-EDR and PV-RO desalination systems via their energy consumption. Sep. Purif. Technol. 151, 82e94. http://dx.doi.org/10.1016/ j.seppur.2015.07.023. Karimi, L., Ghassemi, A., 2015a. Effects of operating conditions on ion removal from brackish water using a pilot-scale electrodialysis reversal system. Desalin. Water Treat. 1e13. http://dx.doi.org/10.1080/19443994.2015.1024748. Karimi, L., Ghassemi, A., 2015b. How operational parameters and membrane characteristics affect the performance of electrodialysis reversal desalination systems: the state of the art. J. Membr. Sci. Res. Articles in Press Current Issue. Kitamoto, A., Takashima, Y., 1971. Transfer rates in electrodialysis with ion exchange membranes. Desalination. http://dx.doi.org/10.1016/S0011-9164(00)80129-0. Kraaijeveld, G., Sumberova, V., Kuindersma, S., Wesselingh, H., 1995. Modelling electrodialysis using the Maxwell-Stefan description. Chem. Eng. J. Biochem. Eng. J. http://dx.doi.org/10.1016/0923-0467(94) 02940-7. Kuroda, O., Takahashi, S., Nomura, M., 1983. Characteristics of flow and mass transfer rate in an electrodialyzer compartment including spacer. Desalination 46, 225e232. http://dx.doi.org/10.1016/0011-9164(83)87159-8. Mitko, K., Turek, M., 2014. Concentration distribution along the electrodialyzer. Desalination 341, 94e100. http://dx.doi.org/10.1016/j.desal.2014.02.040. Noble, R.D., Stem, S.A., 1995. Membrane Separations Technology Principles and Applications. Elsevier, Amsterdam. Probstein, R.F., Sonin, A.A., Gur-Arie, E., 1972. A turbulent flow theory of electrodialysis. Desalination 11, 165e187. http://dx.doi.org/10.1016/S0011-9164(00) 80066-1. Scarazzato, T., Buzzi, D.C., Bernardes, A.M., Espinosa, D.C.R., 2015. Current-voltage curves for treating effluent containing Hedp: determination of the limiting current. Braz. J. Chem. Eng. 32, 831e836. Shaposhnik, V.A., Grigorchuk, O.V., 2010. Mathematical model of electrodialysis with ion-exchange membranes and inert spacers. Russ. J. Electrochem. 46, 1182e1188. http://dx.doi.org/10.1134/S1023193510100149. Sonin, A.A., Isaacson, M.S., 1974. Optimization of flow design in forced flow electrochemical systems, with special application to electrodialysis. Ind. Eng. Chem. Process Des. Dev. 13, 241e248. http://dx.doi.org/10.1021/i260051a009. Sonin, A.A., Probstein, R.F., 1968. A hydrodynamic theory of desalination by electrodialysis. Desalination 5, 293e329. Tadimeti, J.G.D., Chandra, A., Chattopadhyay, S., 2015. Optimum concentrate stream concentration in CaCl2 removal from sugar solution using electrodialysis. Chem. Eng. Process Technol. 6, 1e13. http://dx.doi.org/10.4172/2167-7048.1000216.
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An empirical/theoretical model with dimensionless numbers to predict the performance of electrodialysis systems on the basis of operating conditions Leila Karimi*, Abbas Ghassemi Institute for Energy & the Environment (IEE), 1060 Frenger Mall, ECIII, Suite 300, New Mexico State University, Las Cruces, NM, 88003, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 January 2016 Received in revised form 4 April 2016 Accepted 8 April 2016 Available online 11 April 2016
Among the different technologies developed for desalination, the electrodialysis/electrodialysis reversal (ED/EDR) process is one of the most promising for treating brackish water with low salinity when there is high risk of scaling. Multiple researchers have investigated ED/EDR to optimize the process, determine the effects of operating parameters, and develop theoretical/empirical models. Previously published empirical/theoretical models have evaluated the effect of the hydraulic conditions of the ED/EDR on the limiting current density using dimensionless numbers. The reason for previous studies' emphasis on limiting current density is twofold: 1) to maximize ion removal, most ED/EDR systems are operated close to limiting current conditions if there is not a scaling potential in the concentrate chamber due to a high concentration of less-soluble salts; and 2) for modeling the ED/EDR system with dimensionless numbers, it is more accurate and convenient to use limiting current density, where the boundary layer's characteristics are known at constant electrical conditions. To improve knowledge of ED/EDR systems, ED/EDR models should be also developed for the Ohmic region, where operation reduces energy consumption, facilitates targeted ion removal, and prolongs membrane life compared to limiting current conditions. In this paper, theoretical/empirical models were developed for ED/EDR performance in a wide range of operating conditions. The presented ion removal and selectivity models were developed for the removal of monovalent ions and divalent ions utilizing the dominant dimensionless numbers obtained from laboratory scale electrodialysis experiments. At any system scale, these models can predict ED/EDR performance in terms of monovalent and divalent ion removal. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Electrodialysis Predictive model Ion removal Selectivity Stanton number
1. Introduction In 2014, the installed capacity of electrodialysis/electrodialysis reversal (ED/EDR) systems around the world was 2.59 million gallons per day (IDA, 2015). Because of the extent to which ED/EDR is utilized, small improvements in efficiency or quality can bring significant practical benefits, so there has been ongoing interest in studying the ED/EDR process, modeling it, and optimizing its performance. The performance of ED/EDR systems e in terms of the cost and the quality of the treated water e is strongly influenced by operating conditions (Karimi and Ghassemi, 2015a; Karimi et al., 2015),
* Corresponding author. E-mail addresses: [email protected] (A. Ghassemi).
(L.
Karimi),
http://dx.doi.org/10.1016/j.watres.2016.04.014 0043-1354/© 2016 Elsevier Ltd. All rights reserved.
[email protected]
so multiple experimental and theoretical investigations have modeled ED/EDR performance on the basis of operating parameters. The vast majority of these investigations have utilized numerical solutions to model the EDR process, whose complexity has rendered analytical solutions prohibitively difficult (Ghorbani et al., 2016). Several researchers have therefore pursued the alternative approach of developing empirical/theoretical models, and the most applicable of these models have been developed based on dimensionless numbers, which allow the models to be applied at different scales. However, previously published empirical/theoretical models have evaluated the effect of the hydraulic conditions of the ED/EDR stack on the limiting current density using dominant dimensionless numbers at constant electrical conditions. To expand dimensionless empirical/theoretical models of the ED/EDR process into a wider range of operation, the purpose of the present study is to develop a dimensionless empirical/theoretical model of the ED/EDR process that can predict performance under different operating conditions
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while accommodating variations in applied voltage. The reason for previous studies' emphasis on limiting current conditions is twofold: 1) most EDR systems are operated close to limiting current conditions, and 2) it is more accurate and convenient to model the ED/EDR system using dimensionless numbers at limiting current density, where the boundary layer's characteristics are known at constant electrical conditions. At limiting current point, it is possible to find analytical solutions for the governing equations in the system, and at limiting current density, an experimental model can validate a theoretical model. Section 1.1 summarizes previously published ED/EDR models that have been developed for limiting current densities. 1.1. ED/EDR models for limiting current densities All previous dimensionless models of the ED/EDR process have focused on predicting limiting current density at different hydraulic conditions. In the previous studies, the prediction of ion removal in the EDR process under the hydraulic and electrical conditions were not discussed. A handful of studies e for example, the work of Probstein et al. (1972) e have considered both limiting current conditions and lower salt removal conditions, but such studies have not constituted comprehensive investigations of ED systems across a wide range of applied voltage. A chronological review of previously conducted studies follows. Since existing models have been developed using different systems, different types of membranes and spacers, and different feed solution characteristics, there are few commonalities in the studies apart from limiting current points. The late 1960s and the 1970s saw the development and refinement of dimensionless models for ED/EDR. In 1968, Sonin and Probstein developed an empirical equation for dimensionless current density under a wide range of operating conditions, based on assumptions of a laminar flow regime and the existence of two regions e developing and fully developed e for concentration diffusion layers (Sonin and Probstein, 1968). In 1971, Kitamoto and Takashima presented a dimensionless model that correlates the current density to the solution flow hydrodynamics and applied voltage through a modified version of Peclet and Stanton numbers. Their results also showed that, although the limiting current density depends on the operating conditions, it has a weak relationship with the ion exchange membrane properties (Kitamoto and Takashima, 1971). This result is rational, because in the electrodialysis process, the controlling step in ionic mass transfer is diffusion in the boundary layer, especially at limiting current condition. Then, in 1972, Probstein et al. conducted a study in two primary regions, zero polarization and full polarization. Although Probstein et al. presented a correlation between the limiting current density and the mean current density, their correlation was most accurate for very small levels of salt removal (Probstein et al., 1972). Huang (1977) also obtained different empirical models for ionic mass transfer at limiting current for two laminar and turbulent flow regimes (Huang, 1977). Kuroda et al. (1983) found that Sh in the ED process is largely a function of Re0.5, regardless of the type of the spacer (Kuroda et al., 1983). Kraaijeveld et al. (1995) presented an empirical model for limiting current measurement in the separation of amino acid using the electrodialysis process. They used the same form of model presented by Sonin and Isaacson (1974), but based their experiments on different spacers (Kraaijeveld et al., 1995). Later, Fidaleo and Moresi (2005) compared their experimental data with the models presented by Kuroda et al. (1983) and Sonin and Isaacson (1974), and found that Sh has a functionality of Re0.5 rather than Re0.13, which was presented Kuroda et al. (1983) and Sonin and Isaacson (1974).
271
More recently, Shaposhnik and Grigorchuk (2010) also presented equations for Sh, representing the mass transfer rate, to compare the rate of ionic mass transfer in spacer-free channels and channels having inert spacers in the ED process. The generalized equations developed for Sh were used to predict the limiting current density (Shaposhnik and Grigorchuk, 2010). Fidaleo and Moresi (2005, 2006, and 2010) found a similar form of power relationship between Sh as dependent variable and two independent variables of Re and Sc, using experimental data collected for the electrodialysis with different feed solution compositions (Fidaleo and Moresi, 2010, 2006, 2005). Mitko and Turek (2014) utilized the Graetz-Leveque equation for laminar flow in a channel (Noble and Stem, 1995) to find the mass transfer coefficient in order to model the concentration polarization along the ED with segmented electrodes (Mitko and Turek, 2014). Tadimeti et al. (2015) experimentally developed another equation for Sh to predict the ionic transfer coefficients of Ca2þ and Cl from a sugar solution in a batch ED process (Tadimeti et al., 2015). Despite the wide range of conditions and design characteristics represented in these studies, each of the previously developed models has focused almost exclusively on limiting current density. Although Sonin and Probstein (1968) and Probstein et al. (1972) presented some theoretical models for partial concentration polarization and lower levels of current, the experimental results show that the presented models are valid mostly for very small levels of salt removal, reducing the applicability of the models. To improve our knowledge of ED/EDR systems and optimize the process, ED/EDR performance should be investigated in a wide range of operating conditions, including both the Ohmic region and the non-Ohmic region. Each region can be used for different purposes. Operation in the Ohmic region reduces energy consumption, facilitates targeted ion removal, and prolongs membrane life compared to limiting current conditions; however, if there are no scaling restrictions due to a high content of CaSO4 or other less soluble salts, industrial ED/EDR systems are often operated in higher levels of voltage (conditions close to or higher than limiting current conditions) to meet the maximum ion removal without damaging the membranes. However, if there is a potential for scaling and membrane damage, ED/EDR should operate under mild electrical conditions: when a current greater than limiting current is applied, the ions migrate through the ion exchange membranes faster than they move in the solution, so ions such as Ca2þ and SO2 4 can accumulate in the concentration boundary layers and precipitate (Scarazzato et al., 2015). When there is such a risk, operations at lower voltages (and lower salt removal) will be more rational. To model ED/EDR performance in a wider range of operation including the Ohmic region, the present study presents empirical models for the removal of monovalent ions (Naþ and Cl) and divalent ions (Ca2þ and SO2 4 ) in a wide range of operating condition through dimensionless numbers that were determined based on the theoretical studies presented in Section 1.2. 1.2. Theory of predictive models for ion removal in the ED/EDR process based on the dominant dimensionless numbers The quality of the desalted water in the ED/EDR process can be predicted based on the dominant dimensionless numbers in the process. The dominant dimensionless numbers in the solution phase are important because ion transport in the boundary layer, strongly influenced by concentration polarization, can be considered the controlling step in the ion transport process from dilute to concentrate stream due to high rate of electro-migration in the membrane phase (Grigorchuk et al., 2005; Huang, 1977; Tadimeti et al., 2015). The following theoretical equations show that a predictive
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model for ion removal in the ED/EDR process can be developed based on two dominant dimensionless numbers: Peclet number, which represents the hydrodynamics of the process, and dimensionless applied electrical potential, which represents the main driving force of the process. The general form of the equation for the above-mentioned dimensionless numbers can be generated theoretically as described below. The current density in the electrodialysis system is derived from the Nernst-Planck equation, which represents the mass transfer mechanisms in the ED/EDR process, and which can be written as shown in Equation (1):
ii ¼ zi F Di
dCi zi F dj DC þ dx RT i i dx
0
(2)
where Q is the total flow rate of dilute stream, DC is the difference between the concentrations of the ion in the inlet and outlet of dilute cells that occurs due to ion transport, V is the dilute solution velocity in a cell, and Ncp is the number of cell pairs in the stack. Equation (2) can be rewritten by multiplying it by h=Cr;i h , allowing the variables to be reduced to the dimensionless variables, as in Equation (3):
Q DCi ANcp VCr;i
! 0D Ci 10 zi F j 1 i Cr;i v RT D A@ A ¼ i @ Aþ@ Vh Vh v hx v hx 0v
Ci Cr;i
! 1
(4)
ji ¼
zi F j RT
(5)
x ¼
x h
(6)
Cr;i ¼
CF;i CD;i ! Ln CF;i C D;i
(7)
where St is Stanton number, j* is dimensionless electrical potential per desalting cell, x* is dimensionless direction toward the membrane, CF is feed concentration (inlet of dilute chamber), and CD is dilute concentration (outlet of dilute chamber). By applying Equations (4)e(7) in Equation (3), Equation (3) can be finalized to Equation (8).
Sti ¼
dC C dji 1 i þ i Pei dx Pei dx
(8)
From the definition of Stanton number shown in Equation (4) and Equation (8), the Stanton number (or modified Sherwood number), which is the ratio of ion flux toward the membrane to the ion transfer in the flow direction, can be defined as a function of Peclet number and dimensionless electrical potential, as follows in Equation (9).
Sti ¼ f Pei ; j*i
(9)
In this research, the model was developed experimentally due to the lack of an analytical solution for Equation (8). The empirical models, which are based on the theory, not only can be used easily, but also show a high degree of accuracy in real applications. As shown in Equation (9), the Stanton number can represent the removal of ions as a function of Peclet number and dimensionless electrical potential.
(3)
where A is the membrane active area, h is the dilute cell thickness, and Cr is the logarithmic average of ion concentration in dilute cells. Equation (3) is simplified to the form of Equation (8) using the dimensionless parameters, which are defined as shown in Equations (4)e(7):
Fig. 1. Schematic of once cell pair of the modeled electrodialysis system.
Q DCi ANcp VCr;i
(1)
where ii is the amount of current density that is carried by ion type i, z is electrical charge, F is Faraday constant, D is the diffusion coefficient, C is ion concentration, R is a gas constant, T is the temperature of the dilute stream, j is applied voltage per desalting cell, and x is the direction perpendicular to the membrane. Subscript i is the type of ion. The flow and ion movement directions are shown in the schematic in Fig. 1. By dividing both sides of Equation (1) by the DCi term ziFV and replacing ii with zi FQ ANcp , the equation then takes the form of Equation (2):
1 iFj v zRT Q DCi Di vCi Di Ci @ A ¼ þ ANcp V V vx V vx
Sti ¼
2. Materials and methods 2.1. Experimental set-up In this research, a complete laboratory-scale ED/EDR setup was designed and built by the authors to conduct the laboratory-scale experiments. The purpose of utilizing the laboratory scale set-up was to run the experiments under highly controlled operating conditions, where all the parameters were recorded every second using a data acquisition system. A complete data acquisition system, shown in Fig. 2, was designed in order to continuously and accurately monitor and record the operating conditions in the experiment. To monitor and record the operating conditions in the process, five sensors e for pH, temperature, conductivity, pressure, and flow e were installed in the six main stream manifolds (three inlet and three outlet streams). The three inlet streams to the stack were Dilute In, Concentrate In, and Electrode In (electrode rinse solution entering cathode and anode chambers). The above-mentioned sensors were installed in each inlet stream to record the operating conditions. Additionally, three outlets streams, leaving the stack, were Dilute Out (product), Concentrate Out (brine), and Electrode Out (electrode rinse solution leaving cathode and anode
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Fig. 2. Schematic of designed ED set-up equipped with a full data acquisition.
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L. Karimi, A. Ghassemi / Water Research 98 (2016) 270e279 Table 1 Investigated factors and their levels. Factor
Levels
Feed Composition (ppm)
Mono-Salt: NaCl: 200 and 1000 CaSO4: 200 and 1000 Binary-Salt Mixture: NaCl 200 þ CaSO4 200 NaCl 1000 þ CaSO4 1000 15, 35 1.6, 2.1, 2.6, 3.7, 4.2 5, 10, 15, 20, 25
Temperature (ºC) Solution Linear Velocity (cm/s) Total Applied Voltage (V) on the Membrane Pack
chambers). The outlet streams, similar to inlet streams, were also equipped with five inline sensors. For data acquisition, the system employed Canary Enterprise Historian, a data acquisition software program designed by Canary Labs. The connection between the programmable logic controller and the Historian software was established using Kepware software. The ED cell unit used in this set-up is PCCell ED 64002, a laboratory scale electrodialyzer with the nominal capacity of 4e8 L/h per dilute and concrete cell, and which is composed of 15 cell pairs. The utilized cation and anion exchange membranes were GE CR67HMR and GE AR908, respectively, with the active area of 64 cm2. The mesh spacer used in the stack had a thickness of 0.4 mm and a direct flow path. 2.2. Design of experiments The experiments were performed with a full factorial design, using 6 levels of feed compositions, 2 levels of temperature, 5 levels of solution linear velocity, and 5 levels of total applied voltage. The different levels for each factor are given in Table 1.
During the experiments, the flow rate of the streams was set through controllable micro-gear pumps and measured using the inline flow sensors. The linear velocity of the streams in the dilute cell was calculated using Equation (10).
V¼
Q whNcp ð1 4Þ
(10)
where w is the width of the dilute chamber and 4 is the fraction of the cross section of the desalting chamber filled with the fibers of the spacer. The 4-value of the spacer used in this research is 20.3%. Additionally, the overall voltage applied to the stack was recorded at two points. The first recorded voltage was the overall voltage applied to the whole ED stack, including the electrode chambers. The recorded value of the overall voltage was greater than the effective voltage applied for the desalination process due to potential drop in electrode chambers (Karimi, 2015). The second recorded voltage was the amount of applied electrical potential in the membrane pack, DjMP , excluding the electrode chambers. The amount of applied electrical potential per dilute cell could be calculated using Equation (11), given the assumption that the voltage is uniformly distributed among cells:
2.3. Experimental procedure The three inlet solutions e Dilute In, Concentrate In, and Electrode In e were prepared before starting the experiment according to the design of experiment. The compositions of the Dilute In and Concentrate In solutions were the same within experimental runs; however, trace amounts of HYPERSPERSE MDC706 antiscalant, (Hanrahan et al., 2015), were added to the Concentrate In solution to prevent scaling problems when feed water containing CaSO4 was used. The Dilute In and Concentrate In solutions were prepared according to the experimental design (shown in Table 1) by adding either NaCl, CaSO4, or both to pure water with a salinity of less than 5 ppm, provided by New Mexico State University's semi-pilot-scale RO setup. Next, the heating or cooling was done according to the required temperature of the experiment. The heating or cooling process was done using a heat exchanger with three branches of stainless steel coils that were placed in the isolated containers. The composition of the electrode rinse solution, shown as Electrode In, was different from the compositions of the other solutions to prevent the formation of Cl2 and CaSO4 scales in the electrode chambers. The electrode solutions used in the experiments were buffer solutions of CH3COONa and CH3COOH with a pH of 5.2. The required amount of NaNO3 was added to the Electrode In to make its electrical conductivity equal the conductivity of the Dilute In and Concentrate In solutions. Additionally, the temperature of the electrode rinse solution was kept at the same temperature as the other solutions. During the experiments, the set points were set according to the designed set points. Water samples were collected while the multi-component solutions were used in the experiments. Before sample collection, steady state condition was checked and confirmed.
DjDil:Cell ¼
DjMP 2Ncp
(11)
where DjDil:Cell and DjMP are, respectively, applied electrical potential for one dilute cell, and applied electrical potential in the membrane pack.
2.4. Sample analysis When single-salt solutions were used in the experiments, the measurements for ion removal and the concentration of ions in the solutions could be done using the conductivity-TDS calibration curves. These calibration curves were generated for the NaCl and CaSO4 solutions in a wide range of salinities and different levels of temperature, and were used to convert the experimentally measured conductivity values into the desired salinity values. During the experiments, the conductivity and temperature values of the streams were recorded by inline temperature and conductivity sensors. Then, the conductivity values were converted to salinity values using the models obtained for each salt, which could give the concentration of ions in the streams. Conductivity-TDS models for NaCl and CaSO4, as shown in Equations (13) and (14). The conductivity-TDS calibration curve for NaCl and the conductivity-TDS model validation for CaSO4 are shown in the supplementary information, Figs. S1 and S2, respectively. For NaCl, solution conductivity was linearly dependent on temperature at a 2% slope in the examined range of concentration (0e2000 ppm), as shown in Equation 12.
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kTð CÞ ¼ k25 C *ð1 þ ðT 25ÞÞ*0:02
(12)
TDSNaCl ¼ k25 C 0:517 21:8
(13)
where TDS and k are the concentration of NaCl and the conductivity of the solution at 25 C. For CaSO4, although the temperature dependency of CaSO4 was linear, the slope was highly dependent on the salt concentration in the investigated range of 0e1600 ppm. Therefore, a non-linear equation was fitted for the concentration of CaSO4, using the MATLAB curve fitting tool with two independent variables for conductivity and temperature, as shown in Equation 14.
TDSCaSO4 ¼ 0:7952 k1:228 T 0:498
(14)
where k and T are the conductivity of the solution measured by conductivity sensor and the temperature of the solution in degrees Celsius, respectively. The R2 value of the obtained model is 0.9981. In cases where binary mixtures of salts were used, water samples were collected during the experiments and analyzed for cations and anions using the Dionex ICS-5000 Dual Channel Ion Chromatography System. 3. Results and discussion As shown before in Equation (9), the Stanton number can represent the removal of ions as a function of Peclet number and dimensionless electrical potential. Based on the proposed theoretical approach, the collected experimental data were used to develop the empirical model for predicting ion removal in the ED process. The general form of the empirical model is shown in Equation (15):
Sti ¼ aPeb jc i
(15)
where St, Pe, and j*are Stanton number, Peclet number, and dimensionless electrical potential per dilute cell. The terms a, b, and c are model coefficients, and the subscript i represents a typical ion. It is worth mentioning that the dimensionless number Pe was calculated based on the effective diffusion coefficient of the ions available in the solutions. When the feed is a multi-ionic solution, each ion is assigned a unique effective diffusion coefficient, whose value can be calculated through the method developed by Geraldes and Afonso (2010), an approach that was used in this research. The coefficients and powers of the variables were obtained through two approaches e MATLAB and linear regression analysis e using Minitab software, after linearizing the nonlinear equation (Equation (15)) to a linear one. The model coefficients were different at Table 2 Power of variables of dimensionless ion removal model in EDR under different conditions, shown as Equation (15). Salt
Component
C0 (ppm)
a
b
c
R2
NaCl NaCl þ CaSO4 NaCl þ CaSO4 NaCl NaCl þ CaSO4 NaCl þ CaSO4 CaSO4 NaCl þ CaSO4 NaCl þ CaSO4 CaSO4 NaCl þ CaSO4 NaCl þ CaSO4
Naþ, Cl Naþ Cl Naþ, Cl Naþ Cl Ca2þ, SO2 4 Ca2þ SO2 4 Ca2þ, SO2 4 2þ Ca 2 SO4
1000 1000, 1000 1000, 1000 200 200, 200 200, 200 1000 1000, 1000 1000, 1000 200 200, 200 200, 200
0.18 0.58 1.08 0.17 0.58 0.70 0.41 0.53 0.19 0.32 1.69 0.55
0.69 1.13 1.08 0.67 0.93 0.96 0.84 0.83 0.80 0.75 0.81 0.75
0.71 1.57 1.28 0.74 1.12 1.16 0.81 0.67 0.83 0.72 0.44 0.54
0.98 0.97 0.96 0.97 0.94 0.93 0.97 0.95 0.93 0.96 0.93 0.92
275
different feed solution compositions, as shown in Table 2. The results from statistical analysis conducted by Minitab confirmed that the two considered dimensionless numbers based on the theoretical discussion, Pe and j*, were significant parameters in the developed models for ion transport in the ED process at the significance level of a ¼ 0.05. The p-values for the constant “a” in Equation (15) and the variables “b” and “c” in Equation (15) were much less than 0.05, which means these coefficients were significantly different from zero. Additionally, the obtained R2 values for all models varied between 92% and 98%, confirming that experimental data had a good fit with the presented empirical form of the models. The values obtained for the model coefficients (a, b, and c) are shown in Table 2. In the application stage, by applying the obtained values, the Stanton number for each ion can be calculated at different operating conditions. Then, based on the equation provided for Stanton number in form of Equation (4), the concentration of the ion in the product stream (Dilute Out) can be calculated through a simple mathematical equation, Equation (7), that gives the ratio of ion removal to the logarithmic average of ion concentration. Additionally, the pH values of the product during the experiments were in the range of 5.6e6.9, while the pH of the feed solution was 6.1e7.3. The variations in pH had no discernible effect on the conductivity values of the dilute stream. The presented equations showed a good goodness of fit for predicting the Stanton number of ions (see the supplementary information, Figs. S3 to S8). 3.1. Generalized form of the model In order to present a general form of the model, models were developed for the whole range of concentration, from 200 to 1000 ppm for mono-salt solutions and 400e2000 ppm for binary salt mixtures. Although the R2 value of the generalized models was slightly reduced compared to the models developed for the limited level of the concentration, the generalized models can be used much more extensively because their applications are less limited by the concentration range. The model parameters are shown in Table 3. Additionally, the comparisons of models with experimental data are shown in Figs. 3 and 4. It is proposed that the effect of initial concentration becomes larger when limiting current approaches. Therefore, for some of the experiments that were done above the Ohmic region, the effect of concentration became more significant. That is why the generalized models show less accuracy compared to the models developed for each level of concentration. Surface graphs (Figs. 5 and 6) also were plotted to show the variation of each ion's Stanton number versus Peclet number and variations in dimensionless applied voltage. Each surface plot indicates that the Stanton number follows the form of the developed models. The results demonstrate that the generalized models can predict the amount of monovalent and divalent ion removal with a good level of accuracy. Therefore, in the brackish water electrodialysis process, researchers and engineers can utilize this kind of master plots to visually predict the removal
Table 3 Power of variables in the generalized form of the dimensionless ion removal model in EDR under different conditions, shown as Equation (15). Salt
Component
a
b
c
R2
NaCl CaSO4 NaCl þ CaSO4 NaCl þ CaSO4
Naþ, Cl Ca2þ, SO2 4 Naþ, Cl 2þ Ca , SO2 4
0.29 0.37 0.82 1.17
0.73 0.81 1.05 0.86
0.71 0.80 1.32 0.57
86% 84% 86% 81%
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Fig. 3. (a) Comparison of predicted Stanton number of monovalent ions (Naþ and Cl) with the experimental values when the feed solution consists of NaCl with the concentration of 200e1000 ppm, and (b) Comparison of predicted Stanton number of divalent ions (Ca2þ and SO2 4 ) with the experimental values when the feed solution consists of CaSO4 with the concentration of 200e1000 ppm.
of each ion based on the master curve. These models are likely to offer a good fit for the removal of other monovalent and divalent ions with hydrated ion sizes similar to the ions studied in this research. According to the characteristics of the dimensionless equations and the scale-up rules, the presented models can be applied in large-scale ED/EDR processes at the same values of Peclet number and dimensionless applied voltage. According to the definition of Stanton number, it can be rewritten as Equation (16).
Sti ¼
DCi C r;i L=h
¼ aPeb j*c i
(16)
Therefore, at the same values of Pe and j*i , the ratio of L (length of the dilute channel) to h (thickness of the dilute channel) can be applied to predict the amount of ion removal in large-scale ED/EDR processes, as follows in Equation (17):
DCi C r;i DCi C r;i
! !F ¼ L
LF hL LL hF
(17)
where subscripts F and L are full scale and laboratory scale, respectively. Additionally, by utilizing the obtained model coefficients, it is possible to predict the removal of ions under a wide range of operating factors. Furthermore, these models have promising potential to predict the removal of other monovalent and divalent ions with hydrated ion sizes similar to those of the ions studied in this research.
3.2. Selectivity model in ED/EDR process based on the dominant dimensionless numbers One of the important parameters in the performance of the ED/ EDR process is the selectivity, which shows the removal ratios of ions in the process. Due to different compositions of water resources and the application fields of treated water, it is sometimes necessary to run the process under specific conditions to remove a target ion or ions more than the others. One of the most effective ways to do this is utilizing selective ion exchange membranes with a higher affinity toward the target ions (Karimi and Ghassemi, 2015b). However, the operating factors, such as solution flow rate (velocity) and especially the applied voltage can have a noticeable role in the selective removal of ions. Based on the provided model for selectivity as shown in
Fig. 4. Goodness of fit of the model when feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm: (a) Comparison of the predicted Stanton number of monovalent ions (Naþ and Cl) with the experimental values; (b) Comparison of the predicted Stanton number of divalent ions (Ca2þ and SO2 4 ) with the experimental values.
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Fig. 5. (a) Variation of Stanton number of monovalent ions (Naþ and Cl) when the feed solution consists of NaCl with the concentration of 200 ppm and 1000 ppm; (b) Variation of Stanton number of divalent ions (Ca2þ and SO2 4 ) when the feed solution consists of CaSO4 with the concentration of 200 and 1000 ppm.
Equation (18), the selectivity term can be written for any pair of ions with the same type of electrical charge. However, it is more rational to assume a reference cation and anion and compare the removal of the ions with respect to those reference ions. The reference ions can be chosen based on the composition of the solution. In this study, two monovalent ions, Naþ and Cl, were chosen as reference cation and anion, respectively.
Sij
ti t j ¼ Ci C j
(18)
where t is the ion transport number and subscripts i and j are two different counter ions. In Equation (18), by replacing ti and tj with i Ji , it is possible to rewrite the selectivity their equation ti ¼ PFzFz J i i
of the ED/EDR process as follows in Equation (19):
. Sij ¼ Fzi Ji=F P z J P J Fz i i j j F zi Ji Cr;i C r;j
(19)
DC by the term Q ANcp ;
where the ion flux, j, can be replaced therefore, the new form of selectivity parameter can be written as Equation (20):
Sij ¼
1 !0 DC @ i Cr;i A zi z DCj C j r;j
(20)
By multiplying the second parenthetical expression in Equation ! (20) by
L= h L= h
, according to Equation (16), the second parenthetical
expression in Equation (20) is the ratio of the Stanton numbers of
two ions, i and j. Therefore, the selectivity of the ED/EDR process can be defined as the product of the ratio of the electrical charge of ions and the Stanton numbers. The empirical equations developed for the Stanton number of ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. Additionally, by replacing the provided equations for Stanton numbers, the selectivity of ED/EDR process can be written as Equations (21)e(24). For feed water composed of binary salt mixtures (NaCl at 1000 ppm and CaSO4 at 1000 ppm), Equations (21) and (22) are obtained utilizing the Stanton model parameters for each ion reported in Table 2: 0:3 *0:9 SCa jNa Na ¼ 2:89Pe
(21)
4 ¼ 0:63Pe0:28 j*0:45 SSO Cl Cl
(22)
For feed water composed of binary salt mixtures (NaCl at 200 ppm and CaSO4 at 200 ppm), the application equations are Equations (23) and (24): 0:12 *0:69 SCa jNa Na ¼ 7:95Pe
(23)
4 ¼ 2:23Pe0:2 j*0:62 SSO Cl Cl
(24)
These equations depict that increases in applied voltage decrease the selectivity ratio of divalent ions against monovalent ions. Moreover, pilot-scale experiments have found that, as higher levels of voltage are applied, the percentage removal of ions approaches 100%, which means the selectivity values approach one (Karimi and Ghassemi, 2015a). The comparisons of experimental values for selectivity and the calculated values for selectivity are presented in the supplementary
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Fig. 6. Variation of Stanton number with variations of Peclet number and dimensionless applied voltage when the feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm for (a) monovalent ions (Naþ and Cl), and (b) divalent ions (Ca2þ and SO2 4 ).
information, Figs. S9 and S10.
4. Conclusions
3.3. Generalized model for selectivity in ED/EDR process based on the dominant dimensionless numbers
The main hypothesis of this research was that an empirical predictive model could be developed based on the dominant dimensionless numbers in the ED/EDR process, giving the ability to predict the removal of ions in the ED/EDR process regardless the scale of the process. Based on the theoretical approach, the collected experimental data were utilized to develop the empirical model based on the dimensionless analysis to predict ion removal in the ED/EDR process. The empirical model was developed for Stanton number (representing the ion removal) as a function of two dominant dimensionless numbers, Peclet number and dimensionless applied voltage. The regression analysis confirmed the significant roles of the modeling parameters, and showing a good fit of the model with provided constants. The model can be utilized in large-scale applications due to the nature of dimensionless numbers. Additionally, through the utilization of the obtained model coefficients, ion removal can be predicted under a wide range of operating factors. These correlations have promising potential to predict the removal of other monovalent and divalent ions with hydrated ion sizes similar to those of the ions studied in this research. Furthermore, the selectivity of the ED/EDR process was defined in a new form as the product of the ratio of electrical charge of ions and the ratio of their Stanton numbers. The developed empirical correlations for the Stanton numbers of monovalent ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. The developed equations and selectivity values depict that the ED/EDR process has more selective behavior for divalent ions compared to monovalent ions. However, the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values and with decreases in the Peclet number values. Moreover, at higher levels of voltage, the
The generalized models were developed for selective ion removal in the ED/EDR process based on the generalized models for Stanton numbers, which can be written as in Equation (25):
Sij ¼ 4:25Pe0:19 j*0:75 j
(25)
where i and j are couples of Ca2þ and Naþ or SO2 4 and Cl . Similar to the generalized form of models for Stanton number, Equation (25) has much a more extensive application than Equation 21 through 24. Equation (25) depicts that the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values, while it increases with increases in Pe values. Decreases in Pe can occur due to increases in superficial velocity, greater thickness of the dilute chamber, or a decrease in the ratio of divalent ions to monovalent ions in the feed composition. The comparison between the presented model and the experimental values is shown in Fig. 7. Furthermore, the selectivity of the ED/EDR process was defined in a new form as the product of the ratio of the electrical charge of ions and the ratio of their Stanton numbers. The developed empirical equations for the Stanton numbers of ions can be used to calculate the selectivity of the process toward the monovalent and divalent ions. The developed equations and selectivity values shown in Fig. 7 depict that the ED/EDR process has more selective behavior for divalent ions compared to monovalent ions. However, the selectivity ratio of divalent ions against monovalent ions decreases with increases in the applied voltage values. Moreover, at higher levels of voltage, the percentage removal of all ions approaches 100%, which means the selectivity values approach one.
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Fig. 7. Comparison of the ED process's predicted selectivity with experimental values for the removal of divalent ions against monovalent ions ((a) Ca2þ against Naþ, and (b) SO2 4 against Cl) when the feed solution consists of binary salt mixtures with the concentration of 400e2000 ppm.
percentage removal of all ions approaches 100%, which means the selectivity values approach one. For further investigations, it would be worthwhile to study and verify the model at a higher range of concentration, other types of monovalent and divalent ions, and larger scales. Acknowledgment The authors would like to thank the Office of Naval Research (N000141010711) for funding this research. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.watres.2016.04.014. References Fidaleo, M., Moresi, M., 2005. Optimal strategy to model the electrodialytic recovery of a strong electrolyte. J. Memb. Sci. 260, 90e111. Fidaleo, M., Moresi, M., 2006. Assessment of the main engineering parameters controlling the electrodialytic recovery of sodium propionate from aqueous solutions. J. Food Eng. 76, 218e231. http://dx.doi.org/10.1016/ j.jfoodeng.2005.05.010. Fidaleo, M., Moresi, M., 2010. Application of the Nernst-Planck approach to model the electrodialytic recovery of disodium itaconate. J. Memb. Sci. 349, 393e404. http://dx.doi.org/10.1016/j.memsci.2009.12.002. Geraldes, V., Afonso, M.D., 2010. Limiting current density in the electrodialysis of multi-ionic solutions. J. Memb. Sci. 360, 499e508. http://dx.doi.org/10.1016/ j.memsci.2010.05.054. Ghorbani, A., Ghassemi, A., Andersen, P.K., Foudazi, R., 2016. A prediction model of mass transfer through an electrodialysis cell. Desalin. Water Treat. http:// dx.doi.org/10.1080/19443994.2015.1123195 (in press). Grigorchuk, O.V., Vasil’eva, V.I., Shaposhnik, V.A., 2005. Local characteristics of mass transfer under electrodialysis demineralization. Desalination 184, 431e438. http://dx.doi.org/10.1016/j.desal.2005.03.048. Hanrahan, C., Karimi, L., Ghassemi, A., Sharbat, A., 2015. High-recovery electrodialysis reversal for the desalination of Inland brackish waters. Desalin. Water Treat. 1e11. http://dx.doi.org/10.1080/19443994.2015.1041162.
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