Desalination 329 (2013) 29–34
Contents lists available at ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Desalination using capacitive deionization at constant current Y.A.C. Jande a,b, W.S. Kim a,⁎ a b
Department of Mechanical Engineering, Hanyang University, 55 Hanyangdaehak-ro, Sangnok-gu, Ansan Kyeonggi-do 426-791, Republic of Korea Department of Materials Science and Engineering, Nelson Mandela African Institution of Science and Technology, Arusha, Tanzania
H I G H L I G H T S • • • • •
The The The The The
phenomenon of ion adsorption–desorption when charging CDI cell at constant current is described by simple equations. equation representing the lowest concentration has been derived. lowest concentration is the steady state condition of the CDI cell charged at constant current. steady state condition depends on the dead volume, applied current, ﬂow rate, feed concentration, and capacitance. average puriﬁcation at a given time interval can be estimated by using the derived equation.
a r t i c l e
i n f o
Article history: Received 31 July 2013 Received in revised form 24 August 2013 Accepted 31 August 2013 Available online 25 September 2013 Keywords: Capacitive deionization Constant current charging Lowest concentration
a b s t r a c t Capacitive deionization (CDI) is an emerging technology of desalinating brackish/seawater to attain freshwater. The process involves polarization of the two electrodes electrically using direct current; thus the cations and anions are attracted towards the oppositely charged electrode. So far most of the experiments/models involve the charging of the CDI cell at constant voltage. However, charging at constant voltage leads to having a shorter time in a given CDI cell cycle when the system has reached its lowest efﬂuent concentration. This is undesired phenomena. To overcome this problem desalination process is preferred to be performed at constant current. The dynamic response model to describe the variation of the efﬂuent concentration with time under constant current charging has been derived and validated. Also, the effect of processing parameters such as applied current, ﬂow rate, CDI cell dead volume, and capacitance on the lowest efﬂuent concentration is analyzed. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The world population is increasing at a tremendous rate; in which currently there are around seven billion people on earth . According to the 2004 United Nations report, by 2050 the population is expected to reach around ten billion people . Water plays an important role in fulﬁlling the human needs in all aspects of life. Agricultural sector requires around 40% of total useful withdrawn water, 20% for domestic use, 10% for industrial use, and 30% for environmental . Qualitatively the water demand is expected to increase as the population increases. The direct source of useful freshwater is limited in the world and it is estimated to be less than 0.01% . The World Health Organization has set less than 500 ppm total dissolved solids (TDS) to be an acceptable value for the water to be useful for human consumption . Therefore, there is a need of ﬁnding new methods of acquiring water from all other sources (i.e., seawater, brackish water, and groundwater) through desalination to attain the acceptable TDS value. There are different processes ranging from multiphase to single phase used in desalting brackish/seawater. Membrane distillation processes ⁎ Corresponding author. Tel.: +82 31 400 5248; fax: +82 31 418 0153. E-mail address: [email protected]
(W.S. Kim). 0011-9164/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2013.08.023
(direct contact membrane distillation, vacuum membrane distillation, air gap membrane distillation, sweeping gas membrane distillation, and thermostatic sweeping gas membrane distillation) involve heating and the freshwater is achieved with the aid of hydrophobic membrane . Pure water can also be achieved through freeze desalination from brackish water through fractional crystallization in which during the formation of ice crystals impurities can be separated . Reverse osmosis, on the other hand, is a single phase process involving high pressure (greater than atmospheric pressure) application . These desalination processes consume more energy for a given amount of fresh water produced . There are other types of desalination processes that remove ions from the stream of saline water at atmospheric pressure. Electrodialysis (ED) process removes ions from the feed solution stream upon the application of direct current (DC). Another type of ion removal related desalination process is capacitive deionization (CDI), which also uses DC power. However, the operation principle of ED and CDI is not the same. ED normally removes ions from one solution to the other solution through ionexchange membranes. Furthermore, ED consumes much more energy than CDI for a given unit of water produced [9,10]. In this study desalination via CDI is the subject of concern. When DC power is applied to the CDI electrodes one of the electrodes becomes positively charged and the other one is negatively charged. In a
Y.A.C. Jande, W.S. Kim / Desalination 329 (2013) 29–34
traditional electrochemical cell (batteries) cathode and anode terminals are deﬁned during discharging, whereas when CDI is used to desalinate saline water the cathode and anode are deﬁned during charging in which ions are adsorbed . Cations are attracted towards the cathode and anions towards the anode during charging. During CDI discharging, in which the system is either short circuited or power is supplied in a reversed direction, ions are desorbed from the electrodes. The charging and discharging processes completes the full cycle of a given CDI cell. Fig. 1 shows the operation of the CDI cell during water puriﬁcation stage. During the CDI operation the electrical double layer develops on the electrodes, in which the Gouy–Chapman–Stern model can be used to describe it [11,12]. A traditional CDI which only contains two porous, experiences poor performance due to the mixing in the pore volume solution. The Coulombic efﬁciency (coulombs of ions per coulombs of electronic current) can be improved by using ion-exchange membranes (good charge barrier membranes); and this type of CDI cell is normally known as membrane capacitive deionization (MCDI) [14–17]. The Coulombic efﬁciency can approach unity when good charge barrier membranes are used. Also, the Coulombic efﬁciency can be improved by using polarized electrodes; that is by proper electrostatic redesign of the electrodes . The author in Ref.  used ﬂowing electrodes to desalinate seawater. This type of CDI does not involve regeneration process; hence it can be the bright future in desalination industry. The capacitance of the CDI cell is improved by using good material with better ion adsorption property. Many researches on electrode materials are underway such as the use of graphene, carbon nanotubes, and other nanocomposite materials [20,21]. Currently most of the CDI cell works under constant voltage during adsorption period, and during desorption the system are short circuited or under reversed constant voltage (Fig. 2a). The lowest concentration under this operating condition persists for a very short period of time. This is undesirable phenomena, since water recovery (amount of pure water divided by the total waste water fed) becomes very low. To avoid this problem, the CDI cell is charged at constant current. There are two types of constant current CDI operation depending on how the CDI cell is being discharged [11,22]. During discharging, a CDI cell is either short circuited (zero voltage application) as shown in Fig. 2b or a reversed constant current is applied as shown in Fig. 2c. During discharging at constant current the CDI cell is targeted to have zero voltage after sometime; that is complete discharge. Modeling of the ion adsorption and desorption in CDI cell serves a lot in terms of design simplicity and economy. The author in Ref.  used pore simulation tool, which rely on Poisson's equation and Nernst–Planck equations, to predict the efﬂuent concentration of a CDI cell charged at constant voltage. In Ref. , the models describing the variation of voltage and concentration with total capacity are presented. The ﬁrst order differential equations are presented in Ref.  to describe the adsorption and desorption phenomena for the CDI cell under constant voltage operation. The parameters related to the models were found using Levenberg–Marquardt method. A
Fig. 2. Theoretical variation of efﬂuent concentration with time when the CDI cell (a) charged at constant voltage, (b) charged at constant current, but discharged at zero voltage, and (c) charged and discharged at constant current.
Fig. 1. Operation of the CDI during half cycle .
Y.A.C. Jande, W.S. Kim / Desalination 329 (2013) 29–34
differential equation representing adsorption behavior of the CDI cell is available in Refs. [26,27], but there is no analytical solution to describe the desorption process when the CDI cell is charged at constant voltage. The author in Ref. , has presented equations to describe both adsorption and desorption dynamic responses under constant voltage based on the concept described in Ref. . So far there are few models describing the dynamic response of the CDI cell charged at constant current. Refs. [22,28,29] analyzed the response of CDI under constant current by discretizing the spacer channel and electrodes into ﬁnite number of ideally-stirred volumes (subcells) placed in series. The two-porosity model that considers the two types of porosity in the electrodes, i.e., macropores in between the particles and micropores inside the particles, is used to describe the ion adsorption in the porous electrode during charging/discharging basing on modiﬁed-Donnan model to describe the electric double layer. In this study, the equations to describe the adsorption and desorption are presented, speciﬁcally the dynamic response of the system when it is under constant current operation. The derivation is based on the concept described in Refs. [13,15] which is for constant voltage. Furthermore, the equation for predicting a lowest efﬂuent concentration will be derived so as to analyze the behavior of the CDI cell when applied current, ﬂow rate, capacitance, dead volume and feed concentration are varied.
where Vc is the dead volume of the CDI cell (the spacer volume plus all other ﬂow channels inside the CDI cell [15,31])—this dead volume is different from the one described in Refs. [22,28,33,34], which represent the volume after the cell where some more mixing takes place. Thus, under constant current the efﬂuent concentration is:
2. Derivation of the adsorption and desorption equations
where C is the capacitance of the CDI cell. Before the CDI cell starts the next cycle, it should be fully discharged. But, if the desorption process do not fully discharge a cell, Eq. (8) have to be modiﬁed to be as follows:
2.1. Adsorption cycle In Ref.  the model for the CDI cell being charged and discharged at constant current is presented. Generally the efﬂuent concentration is a result of mixing between the feed concentration and the puriﬁed water within the spacer volume. Therefore, at constant current the rate of molar puriﬁcation is given by: Mp ¼
where Vs is the spacer volume and ϕ is the ﬂow rate. Under constant current the amount of moles adsorbed by the CDI electrodes is: M¼
Examining Eq. (6) when duration of charging the CDI cell becomes ϕ large, the term e−V c t approaches zero. Therefore, for any CDI cell under constant current operation the lowest efﬂuent concentration is C alowest ¼ C f −
λI : zFϕ
The lowest efﬂuent concentration is thus affected by the three CDI operating parameters namely feed concentration, applied constant current, and ﬂow rate. The CDI charging (desorption) time under constant current depends on the desired target voltage, Vt, thus it can be found using the following formula: ta ¼
CV t I
C ðV t −V ia Þ I
where Via is the initial potential within the cell before adsorption process begins for the new cycle.
λI −ϕt 1−e V c : zFϕ
2.2. Desorption cycle
where λ is the differential charge efﬁciency (the ratio of the salt adsorbed to the amount of charge) [11,26,27,30], I is the applied constant current, z is the average of the partial molar ionic valences of the feed solution, and F is the Faraday's constant. The time period that the saline water is in the spacer channel is termed the spacer residence time and it can be computed from the following equation: T¼
C ef ðt Þ ¼ C f ð1−f Þ þ C s f ¼ C f −
Mp dt ¼
where t is the time. The concentration at any given time t inside the spacer is given as: Μ C s ðt Þ ¼ C f − Vs
−Vϕ t c
C ðt Þ ¼ C f þ
λI −ϕt 1−e V c : zF
Also, the desorption time depends on how much voltage the CDI cell has to discharge; but usually complete discharge is the desired one in order to have good performance in salt removal efﬁciency of the CDI cell. Therefore, the time needed to fully discharge a CDI cell is given by: td ¼
CV id I
where Via is the potential across the CDI cell at the start of desorption cycle. If the discharge is done partially, the time to discharge a CDI cell to a potential Vtd is given by: td ¼
where Cf is the feed concentration. As it has been mentioned earlier, the exit concentration depends on the mixing phenomena happening within the spacer volume—according to Andelman the mixing phenomena is like that of continuous stirred tank reactor [31,32]. Andelman  reported the fraction of the puriﬁed water to ﬂow cell dead volume as: f ¼ 1−e
There are two ways of discharging a CDI cell . In discharging the cell at constant current the efﬂuent concentration equation is like that of adsorption, but for this case the current direction has been reversed. Therefore, the equation is given as:
C ðV id −V td Þ : I
Alternatively, the CDI cell can be discharged by short circuiting; that is supplying zero voltage to it in which its equation is found in Ref. . 2.3. Purity of pooled puriﬁed water Generally puriﬁed water is accumulated at the certain location, such as tank and bucket, before use. The puriﬁcation process in CDI cell is a transient phenomenon; implying that the concentration of water exited from the CDI cell depends on time. Therefore, the concentration of
Y.A.C. Jande, W.S. Kim / Desalination 329 (2013) 29–34
pooled puriﬁed water, Cpw is the average of solute concentration at a given time interval; the time of pooling water into a given device such as tank. If pooling starts at time t1 and ends at time t2, then Cpw can be found as follows:
C pw ¼ ¼
1 t 2 −t 1 þ 1
Zt 2 C ef ðt Þdt t1
1 V −ϕt −ϕt ð13Þ C f ðt 2 −t 1 Þ−σ ðt 2 −t 1 Þ þ c e V c 2 −e V c 1 t 2 −t 1 þ 1 ϕ
λI . Time t2 is the time the CDI cell takes to be charged to the where σ ¼ zFϕ required voltage. To obtain concentration in ppm, the Cpw is multiplied by the molecular weight of a given salt.
3. Results and discussion The model has been validated using experimental data found in Ref. , in which the CDI cell has been charged at constant current and then discharged at constant current (Fig. 3). The efﬂuent concentration dropped to a minimum and then prolonged at that value for sometimes before desorption period begins. The experimental data indicates 8.5 mM as the minimum concentration, while the model shows 9.64 mM as the minimum concentration. The CDI cell has been charged until it reaches 1.6 V and then it is fully discharged (until the voltage across the CDI cell is zero). Generally, the model agrees well with the experimental data. Fig. 4a shows the variation of applied current with the expected lowest concentration given by Eq. (7), while other parameters (feed concentration and ﬂow rate) are kept constant. As in constant voltage charging, constant current also behave in the similar way. As current increases the lowest efﬂuent concentration decreases, implying that for a given feed concentration highly pure water can be attained by utilizing higher current. However, care has to be taken by considering the maximum voltage a given CDI cell can hold and also the possibility of water to undergo electrolysis process. As it has been mentioned earlier, charging the CDI cell at constant current prolongs the lowest efﬂuent concentration period, which is the desired phenomena. The variation of ﬂow rate with the lowest efﬂuent concentration is shown in Fig. 4b, while other parameters are kept constant. If the ﬂow rate becomes too high the resident time within spacer becomes small
Effluent concentration (mM)
25 20 15 10 5 0 0
Time (s) Fig. 3. Validation of the model equation using experimental data from Ref. . During adsorption applied current is 1 A and during desorption — 1 A. The ﬂow rate is 60 mL/min, feed concentration is 20 mM NaCl, spacer volume is 6.758 mL, dead volume is 6.758 mL, and charge efﬁciency is assumed to be unity. If good charge barrier membranes  or through the electrostatic redesign of the electrode itself , charge efﬁciency can approach unity.
Fig. 4. The effects of (a) applied current when Cf = 20 mM NaCl and ϕ = 60 mL/min (b) ﬂow rate when Cf = 20 mM NaCl and I = 1 A, and (c) feed concentration when I = 1 A and ϕ = 60 mL/min, on the lowest concentration of the CDI cell.
and hence the lowest efﬂuent concentration becomes high. The danger of not performing any desalination process can occur when the chosen ﬂow rate is high. Fig. 4b shows that the ﬂow rate above 600 mL/min results in poor desalination, and it can be interpreted that at this highest ﬂow rate the feed concentration is equal to the efﬂuent concentration.
Y.A.C. Jande, W.S. Kim / Desalination 329 (2013) 29–34
In Fig. 4c, the variation of the lowest efﬂuent concentration with the feed concentration is shown. The given CDI parameter set (applied current and ﬂow rate), desalinates well a certain feed concentration. If the feed concentration increases while the CDI parameters are kept constant, the lowest efﬂuent concentration increases. Therefore, in the design stage proper CDI operating parameters should be chosen to meet the existing feed concentration. By using genetic algorithm the desired lowest efﬂuent concentration can be achieved. The procedure on how to ﬁnd out the appropriate CDI cell operating parameters to achieve the desired lowest efﬂuent concentration by using this method is well described in Ref. . In determining the desired lowest efﬂuent concentration, Eqs. (6) to (12) have to be thoroughly used. The design has to consider simultaneously the desired lowest efﬂuent concentration and the target CDI voltage during charging and discharging. The target CDI voltage depends on charging time and discharging time, in which all of them depend on other CDI operating parameters (feed concentration, capacitance, ﬂow rate, current, and dead volume). The dead volume affects the purity of puriﬁed water negatively. If all other CDI parameters are kept constant, the increase in dead volume increases the lowest efﬂuent concentration. The equilibrium position (steady state condition) is achieved after longer time if larger dead volume is used. Fig. 5 shows that at time = 624 s, the efﬂuent concentration is 5.7 ppm (0.098 mM NaCl) when Vc = 6 mL, is 231.8 ppm (3.963 mM NaCl) when Vc = 13 mL, and is 1298 ppm (22.184 mM NaCl) when Vc = 20 mL. The target voltage during charging and discharging is 2 V and 0 V, respectively. In Ref.  the CDI cell having a capacitance of 1746 F and spacer volume of 13 mL is operating at constant voltage; the lowest efﬂuent concentration of 23,953 mg/L was achieved when the feed concentration is 32704 mg/L. Fig. 6 shows how the capacitance affects the prolongation time in CDI cell operated at constant current. The charging time varies with capacitance for the same target voltage (required CDI voltage during charging). In this case, only CDI capacitance is varied while current is 5.6 A, ﬂow rate is 6.216 mL/min, and dead volume is 6 mL. A CDI cell with capacitance 1746 F takes 624 s to be charged to a potential of 2 V, with capacitance 3492 F takes 1247 s to be charged to a potential of 2 V, and with capacitance 5238 F takes 1872 s to be charged to 2 V. If the capacitance increases from 1746 F to 3492 F, and from 1746 F to 5238 F the prolongation time increases by 623 s and 1248 s, respectively. The prolongation time relates with steady state duration of the CDI cell charged at constant current, implying that the lowest efﬂuent concentration persists for longer time during puriﬁcation period.
Fig. 5. The effect of CDI cell dead volume on the lowest efﬂuent concentration; the CDI operating parameters are: capacitance = 1746 F, ﬂow rate = 6.216 mL/min, and current = 5.6 A.
Fig. 6. The effect of CDI cell capacitance on the lowest efﬂuent concentration prolongation time; other CDI operating parameters are: ﬂow rate = 6.216 mL/min, current = 5.6 A, and dead volume = 6 mL.
Fig. 7 shows the variation of the concentration of pooled puriﬁed water with the pooling starting time. The ending time is held constant; and it is the time CDI cell takes to be charged until the voltage reaches 2 V. The CDI cell takes around 624 s to be charged to a 2 V. For this particular case, water below 500 ppm is obtained when the pooling starting time is above 117 s. According to WHO, the drinkable water should have TDS below 500 ppm . Therefore, any desalination activity via CDI cell, the required puriﬁcation level has to be predetermined. In charging the CDI cell at constant current, the applied constant electron current translates into an equally large ionic current in the CDI cell contributing to the ionic ﬂux of anions and cations . During adsorption the efﬂuent concentration decreases to a minimum value and remains at this state for a longer time until desorption stage is initiated. Hence, the lowest concentration condition of the desalted stream is prolonged resulting in higher probability of acquiring ultrapure water. On the other hand, charging the CDI cell at constant voltage has the disadvantage of all-time-variation of the efﬂuent concentration during CDI operation. At the initial stage of ion adsorption the electrical double layers (EDLs) do not contain charges; therefore across the spacer channel the driving force is at maximum resulting in large ion ﬂux towards
Fig. 7. Numerical averaged puriﬁcation during a particular length of puriﬁcation time; the CDI operating parameters used are: C = 1746 F, ϕ = 6.216 mL/min, Vc = 6 mL, I = 5.6 A, and Cf = 0.56 mol/L NaCl.
Y.A.C. Jande, W.S. Kim / Desalination 329 (2013) 29–34
the electrodes. As the ion adsorption progresses the potential in the EDLs increases resulting in the steady decrease in voltage across the spacer channel. Consequently, initially the efﬂuent concentration will decrease to the minimum and then starts to increase until the efﬂuent concentration becomes equal to the feed concentration . 4. Conclusion Charging the CDI cell at constant current prolongs the steady state condition and also the salt concentration of the desalted water can be adjusted accurately by varying electrical current to a certain set point . The prolonged steady state condition may increase the chance of obtaining very high pure water compared to charging the CDI cell at constant voltage. While the increase in current decreases the lowest efﬂuent concentration (in the sense that the puriﬁed water become purer), the increase in ﬂow rate increases the lowest efﬂuent concentration. The CDI cell dead volume inﬂuences the time it takes the CDI cell to reach the steady state which in turn affects the purity level of the puriﬁed water, and when it is smaller the system reaches the equilibrium state very fast. Also, if the capacitance of the CDI cell is big enough, the steady state condition prolongs for a larger time until the target voltage is reached. Also, the end user in desalination via CDI cell gets the average puriﬁcation for a particular puriﬁcation time. This study has proposed a model to predict the possible purity level of pooled puriﬁed water during CDI desalination process. These kinds of study will help to produce the desired CDI cell for the targeted application and hence minimize the cost. References  U.S. and World Population Clock, U.S. Census Bureau, Department of Commerce, United States, 2013.  World Population to 2300, Department of Economic and Social Affairs, Population Division, New York, United Nations, 2004.  D. Seckler, U. Amarasinghe, D. Molden, R. d. Silva, R. Barker, World Water Demand and Supply, 1990 to 2025: Scenarios and Issues, International Water Management Institute, Colombo, Sri Lanka, 1998.  R. Fujioka, L.P. Wang, G. Dodbiba, T. Fujita, Application of progressive freezeconcentration for desalination, Desalination 319 (2013) 33–37.  WHO, Guidelines for Drinking-water Quality, Fourth edition World Health Organization, Malta, 2011.  M. Khayet, T. Matsuura, Membrane Distillation: Principles and Applications, Elsevier, Amsterdam, 2011.  M. Abbasi, M.R. Sebzari, S.R. Hossein Abadi, T. Mohammadi, Integrated membrane pilot plant for reﬁnery wastewater treatment in order to produce boiler feedwater, Desalin. Water Treat. 51 (2013) 2543–2553.  T.J. Welgemoed, Capacitive Deionization Technology: Development and Evaluation of an Industrial Prototype, University of Pretoria, Pretoria, 2005.  T.J. Welgemoed, C.F. Schutte, Capacitive Deionization Technology™: an alternative desalination solution, Desalination 183 (2005) 327–340.  A. Al-Karaghouli, L.L. Kazmerski, Energy consumption and water production cost of conventional and renewable-energy-powered desalination processes, Renew. Sustain. Energy Rev. 24 (2013) 343–356.
 S. Porada, R. Zhao, A. van der Wal, V. Presser, P.M. Biesheuvel, Review on the science and technology of water desalination by capacitive deionization, Prog. Mater. Sci. (2013), (in press).  P.M. Biesheuvel, M.Z. Bazant, Nonlinear dynamics of capacitive charging and desalination by porous electrodes, Phys. Rev. E 81 (2010).  Y.A.C. Jande, W.S. Kim, Predicting the lowest efﬂuent concentration in capacitive deionization, Sep. Purif. Technol. 115 (2013) 224–230.  M.D. Andelman, G.S. Walker, Charge Barrier Flow-through Capacitor, in: U.S. Patent (Ed.), Biosource, Inc., Worcester, MA (US), United States of America, 2004, pp. 1–28.  M. Andelman, Flow through capacitor basics, Sep. Purif. Technol. 80 (2011) 262–269.  J.-S. Kim, J.-H. Choi, Fabrication and characterization of a carbon electrode coated with cation-exchange polymer for the membrane capacitive deionization applications, J. Membr. Sci. 355 (2010) 85–90.  Y.J. Kim, J.H. Choi, Improvement of desalination efﬁciency in capacitive deionization using a carbon electrode coated with an ion-exchange polymer, Water Res. 44 (2010) 990–996.  M.D. Andelman, Polarized Electrode for Flow-through Capacitive Deionization, in: W.I.P.O.-I. Bureau (Ed.), Patent Cooperation Treaty (PCT), 2012, pp. 1–70.  S.-i. Jeon, H.-r. Park, J.-g. Yeo, S. Yang, C.H. Cho, M.H. Han, D.K. Kim, Desalination via a new membrane capacitive deionization process utilizing ﬂow-electrodes, Energy Environ. Sci. 6 (2013) 1471.  J. Yang, L. Zou, N.R. Choudhury, Ion-selective carbon nanotube electrodes in capacitive deionisation, Electrochim. Acta 91 (2013) 11–19.  B. Jia, L. Zou, Graphene nanosheets reduced by a multi-step process as highperformance electrode material for capacitive deionisation, Carbon 50 (2012) 2315–2321.  R. Zhao, P.M. Biesheuvel, A. van der Wal, Energy consumption and constant current operation in membrane capacitive deionization, Energy Environ. Sci. 5 (2012) 9520.  B.G. Jeon, H.C. No, Development of a two-dimensional coupled-implicit numerical tool for analysis of the CDI operation, Desalination 288 (2012) 66–71.  T.Y. Ying, K.L. Yang, S. Yiacoumi, C. Tsouris, Electrosorption of ions from aqueous solutions by nanostructured carbon aerogel, J. Colloid Interface Sci. 250 (2002) 18–27.  J.-H. Ryu, T.-J. Kim, T.-Y. Lee, I.-B. Lee, A study on modeling and simulation of capacitive deionization process for waste water treatment, Taiwan Inst. Chem. Eng. 41 (2010) 506–511.  P.M. Biesheuvel, B. van Limpt, A. van der Wal, Dynamic adsorption/desorption process model for capacitive deionization, J. Phys. Chem. 113 (2009) 5636–5640.  R. Zhao, P.M. Biesheuvel, H. Miedema, H. Bruning, A. van der Wal, Charge efﬁciency: a functional tool to probe the double-layer structure inside of porous electrodes and application in the modeling of capacitive deionization, J. Phys. Chem. Lett. 1 (2009) 205–210.  R. Zhao, O. Satpradit, H.H. Rijnaarts, P.M. Biesheuvel, A. van der Wal, Optimization of salt adsorption rate in membrane capacitive deionization, Water Res. 47 (2013) 1941–1952.  P.M. Biesheuvel, R. Zhao, S. Porada, A. van der Wal, Theory of membrane capacitive deionization including the effect of the electrode pore space, J. Colloid Interface Sci. 360 (2011) 239–248.  I. Cohen, E. Avraham, A. Soffer, D. Aurbach, Water desalination by capacitive deionization — advantages limitations and modiﬁcation, ECS Trans. 45 (2013) 43–59.  M. Andelman, Personal Communication, Mespilus Inc., Worcester, 2012.  D.E. Seborg, T.F. Edgar, D.A. Mellichamp, F.J. Doyle III, Process Dynamics and Control, Third ed. John Wiley & Sons, Inc., United States, 2011.  S. Porada, M. Bryjak, A. van der Wal, P.M. Biesheuvel, Effect of electrode thickness variation on operation of capacitive deionization, Electrochim. Acta 75 (2012) 148–156.  S. Porada, L. Borchardt, M. Oschatz, M. Bryjak, J. Atchison, K.J. Keesman, S. Kaskel, M. Biesheuvel, V. Presser, Direct prediction of the desalination performance of porous carbon electrodes for capacitive deionization, Energy Environ. Sci. (2013), (in press).