Separation of Multicomponent Electrolyte Solution - American

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Ind. Eng. Chem. Res. 2009, 48, 9268–9277

Simulation and Optimal Design of Electrodeionization Process: Separation of Multicomponent Electrolyte Solution Anjushri S. Kurup, Thang Ho, and Jamie A. Hestekin† Ralph E. Martin Department of Chemical Engineering, UniVersity of Arkansas, FayetteVille, Arkansas 72701

Electrodeionization (EDI) technology combines the operation principle of both ion exchange and electrodialysis in one single unit and overcomes the disadvantages of either unit. This technology has been widely used in the production of ultrapure water due to its better performance and economical operation at low feed concentrations. As a result, there have been several studies that deal with application of this technology for the removal of ions from water. However, except for a few studies, most of the reported works deal with the experimental study of EDI. This study deals with the development of a mathematical model for deionization of electrolyte solutions containing more than one ion and ions that are multivalent in nature. The model validity is verified by comparing the results with a few experimental observations. This is followed by a sensitivity study of the EDI unit to explore the effect of various operating and system parameters on the ion removing ability of the EDI unit. Consequently, a systematic optimization exercise that can simultaneously handle more than one objective for superior system performance of the experimental unit is demonstrated. Due to the optimization study, it is shown that many times EDI is not operated at optimized conditions and decreasing the flow rate, among other variables, might have a significant effect on system performance. 1. Introduction Over the years, scarcity of fresh water supplies, deteriorating quality of available water sources, and stringent water quality requirements in many industries has forced the development of new processes that can eliminate most of the unwanted ions and impurities from water. Ion-exchange resins have been used for decades in such water purification processes. In a typical ion-exchange process, water containing the ions that need to be removed is contacted with ion-exchange resin particles leading to an exchange between the ions in solution and the preferred ions on the resin, resulting in deionized water. This process can be operated as batch or continuous mode. Both processes, however, necessitate regeneration of ion-exchange resins resulting in effluents causing further environmental concerns. Another promising technology used for the removal of ions from water is electrodialysis (ED). ED utilizes electric potential to migrate ions across ion-selective membranes into appropriate collection chambers.1-4 Although very efficient in operation, this process suffers from the limitation of high power consumption at low ion concentrations. This difficulty led to the development of a promising substitute process, electrodeionization (EDI), that couples both ion-exchange resins and electrodialysis in one unit.5-7 EDI can be operated in both continuous and batch modes and overcomes the need to regenerate the resins as a separate step. Further, EDI can be operated at low concentrations with lower power consumption, thus overcoming the major disadvantage of ED. EDI is therefore considered very important in the production of ultrapure water.6,8 A typical EDI unit is similar to conventional ED and consists of two chambers: the feed or diluate chamber and the concentrate chamber. These chambers are separated by ion-selective cationor anion-exchange membranes (Figure 1). However, in the case of EDI, the diluate chamber is filled with ion-exchange resins and the electrolyte solution is percolated through the ionexchange bed. In the production of ultrapure water, the ion† To whom correspondence should be addressed. Tel.: (479) 5753492. Fax: (479) 575-7926. E-mail: [email protected].

exchange bed generally consists of a mixture of both cationand anion-exchange resins,7,9 although an alternative configuration comprising just cation-exchange resin separated by cationexchange membranes has also been reported.10-12 Ion-exchange resin present in the diluate chamber results in high ionic conductivity13 compared to high resistance in the diluate chamber of ED. H+ and OH- generated due to the hydrolysis of water results in the regeneration of the bed within the unit, and hence chemical regeneration as those used in typical ionexchange operations is not required. Thus EDI combines the advantages of both ED and ion exchange while overcoming the disadvantages of both processes. The EDI unit as described earlier suffers from a few limitations especially due to the presence of loose resin in the diluate chamber. One of these is uneven flow distribution causing a decrease in separation efficiency.14 Another limitation that significantly affects the separation efficiency of the unit is leakage of ions between the compartments. These limitations can be overcome to a considerable extent by immobilizing the resins. An effort in this direction led to the development of wafer enhanced electrodeionization (WE-EDI).15 WE-EDI consists of a wafer instead of loose resin sandwiched between the two ionexchange membranes, and the wafer is synthesized by binding the ion-exchange resins using a polymer as binding agent. The experimental setup used in this study is a WE-EDI unit as described by Arora et al.15 The EDI unit, as described earlier, employs various phenomena such as ion exchange, transport of ions through the resin bed, and membranes that are strongly coupled to each other. Modeling and simulation are valuable tools to study and design any unit, especially the scaling up of units, which makes it imperative to be able to predict its dynamic behavior and optimize the operating conditions. Computer simulations help accomplish such goals by reducing the number of indispensable experiments. Most studies in the literature concerning EDI are based on experimental investigations carried out with a given stack design, feed solution composition, and required product water quality and do not deal much with relating the different

10.1021/ie801906d CCC: $40.75  2009 American Chemical Society Published on Web 09/23/2009

Ind. Eng. Chem. Res., Vol. 48, No. 20, 2009

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Experimental validation is followed by a sensitivity study in order to obtain the important design parameters that influence the separation power of the EDI unit. This is further followed by rigorous optimization of the current EDI unit that aims at extracting the maximum performance of the unit at minimum process costs. Two representative cases are shown in this paper that demonstrate the advantages of performing such systematic optimization studies on the EDI unit. A detailed optimization study such as this should give new insights into the proper operating parameters for the EDI unit. 2. Mathematical Model

Figure 1. Schematic representation of an EDI setup.

experimental parameters to the overall process performance. In one of the first reported studies, Glueckauf5 presents a comprehensive analysis of the mass transfer in the EDI unit with an ion-exchange resin packed in the feed compartment. The paper describes the development of a steady state model, and analytical solutions are derived for electrolyte solutions that contain a monovalent ion. The model solutions are also compared to experimental results. The paper also discusses the optimal design of the EDI unit for the removal of ions from water by maximizing the capacity of the unit while minimizing the power consumption. In a later reported study, Verbeek et al.16 discuss the development of a two-dimensional model of an EDI unit with separate cation and anion exchange resin compartments. The model considers several ions generally important in the production of ultrapure water along with removal of carbon dioxide and silicate. A few simulation results showing the concentration profiles of a few ions in the liquid phase and the solid phase are reported in the paper. The work of Danielsson et al.17 demonstrates the development of a steady state model to describe the EDI unit that uses an ion-exchange textile instead of loose resin in the feed compartment. The model is derived by performing volume averages of conservation equations at microscopic levels. The paper shows the effect of model parameters through simulation studies of nondimensionalized equations. Spoor et al.18 use a one-dimensional transport model to describe the migration of ions through the solid phase and discuss the comparison of experimental and simulation results. An investigation on the mass transfer characteristics of a typical EDI unit is also reported.19 Thus, in the literature, a steady state model and a transient model have been described. In this work we propose to use a steady state model to describe the WE-EDI unit as described earlier. The mathematical model used in this work is an extension of the steady state model described by Glueckauf.5 In his work, Glueckauf5 has given a mathematical representation of an EDI setup for the deionization of NaCl solution. In the present work, we have extended this model to aid deionization of water containing multiple ions such as sodium and potassium. The model also accounts for the presence of multivalent ions such as calcium. In our study, the validity of the mathematical model is obtained by comparing the model predictions with experimental observations for different operating conditions.

In this work, the wafer is modeled as a packed bed between two membranes. Similar to the work described by Glueckauf, it is assumed that the transfer of ions through the diluate compartment filled with ion-exchange resin is a two-stage process. In the first stage the ions are transferred or diffused from the bulk solution to the liquid film around the resin particles. When the ions reach the solid-liquid interface, they get exchanged with the mobile (counter)ions of the resin. The cations get exchanged with the H+ ions of the cation-exchange resin, while the anions get exchanged with the OH- ions of the anion-exchange resin. In the later stage these ions present in the resin particles are carried through by electric current flowing through the exchanger bed and reach the opposite membrane wall, where they are collected in the concentrate compartment. The current flowing through the diluate compartment is complex since it is composed of two phases (solid or resin phase and liquid phase) with different properties. The current can follow one or even a combination of the following paths: purely through the resin phase, purely through the interstitial liquid phase, and partly through the liquid and partly through the resin phase. The mobility of ions in the liquid phase is generally higher than in the resin phase. However, it is well-known that the conductivity of an ion-exchange resin is considerably higher than that of liquid especially when the ionic concentration of the solution is quite low.13 Since EDI units are typically operated at quite low ionic concentrations, it is reasonable to assume that the ion-conduction path is predominantly through the resin phase, which reduces the complexity of the model considerably. For dilute solutions, generally used in EDI systems, the first stage is diffusion-controlled in the aqueous phase since the ions have to diffuse through a thin film of unstirred solution to the surface of the exchanger particles.13 In the bulk phase, axial dispersion is neglected and the transport is primarily convection dominated. Exchange kinetics of ion transport from water to resin is totally controlled by a film-diffusion process. When an electrolyte solution contains n components including the counterions on the resin, then there are n - 1 unknown solute concentrations (Ci) and an unknown electrostatic potential (φ). Conservation equations need to be written for each of the n 1 solutes, and the equation for electroneutrality of the solution completes the set of equations that need to be solved. These equations are necessary for both the solid phase (ion-exchange resin) and the liquid phase, which are linked together by ionexchange equilibrium relations. It is assumed that the width of the diluate chamber/resin wafer is narrow enough to avoid any bulk concentration difference along the width of the chamber. The mass transport in a dilute electrolyte solution involves ion transport due to convection, diffusion, and migration (due to the electric field). Due to the two-stage process as described earlier, two mass balance equations involving ion fluxes in two dimensions are needed to model the EDI setup. A diluate chamber of width W (which is

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equal to the wafer thickness/width) and total area A packed with a known ratio of cation- and anion-exchange resins and in contact with an electrolyte solution flowing at a flow rate f in the direction of the y-axis is considered. A potential gradient applied across the cell causes the ions to migrate through the resin phase in the direction of the x-axis. At steady state, combining the equations for species conservation in the liquid and solid phases, the following two balance equations can be written for each ionic species i in the solution to describe the transport processes occurring in the cell:

f

From eqs 1, 2, 5, 6, and 8 we have finally ∂ηi F 3Rβγ (Cb - Cio) ) Di ∂w J r 2(1 - R) i p

(10)

(11)

) Jelec,i | W dA

(2)

as )

Cbi

∆

Coi

(3)

3Rβγ rp

3Rβγ (Cbi - Coi ) rp2(1 - R)

(12)

b ex Cbi ) Cin i at A ) 0 and Ci ) Ci at A ) A

(13)

Equation 12 is based on the assumption that the membrane at w ) 0 is totally impermeable to the counterions. For example, an anion-exchange membrane present at w ) 0 does not allow any cations and allows only anions to pass through. An instantaneous equilibrium between the liquid film around the resin particles and the solid or resin phase is assumed in this model. The liquid film concentrations and solid phase concentration are linked together by the ion-exchange equilibrium relations j i (CHo)zi C

Ki )

(5)

The amount of ions being transported due to a potential gradient depends on the valence, mobility, and the concentration of the ion and is expressed as j i ∂φ Jelec,i ) ziujiC ∂w

Cio ) 0 at w ) 0

(4)

The film thickness can be assumed5 as ∆ ) rp(1 - R) and eq 3 can now be written as

(6)

A is defined as the effective membrane cross-sectional area, which can be easily measured. It is assumed that half the current is to remove cations while the other half is to remove anions. Since in most cases cations generally have lower mobilities than anions, in this case the condition that cation transport is limiting is assumed. Thus, J ) Itot/2A is the current density expressed as

(14)

j H)zi Cio (C

Additionally, the following conditions are necessary in order to describe the problem adequately. (i) There is no net current in the liquid film surrounding the resin particles: n

m

∑ (z J

i diff,i)

+

i)1

∑ (z J

j diff,j)

)0

(15)

j)1

where i ) 1, n is the counterion species, and j ) 1, m is the co-ion in the solution. If the co-ions do not participate in any chemical reactions and since they do not adsorb on the cation exchange resin, we can assume Jdiff,j ) 0. (ii) There exists electroneutrality condition at every position of the solution: n

m

∑ (z C ) + ∑ (z C ) ) 0 i i

i)1

j j

(16)

j)1

In the solid phase, we can write this as

n

i elec,i,

(9)

The following boundary conditions are required to solve the above two equations:

Next, as is defined as the total particle surface per volume of the bed. For a packed bed with packing ratio R, defined as the volume of solids to the total volume and particle radius rp, as is given5 as 3R /rp. Assuming the fraction of accessible particle surface area for ion exchange as β and the ratio of cation to anion exchange resin in the bed as γ, we have5

∑zJ

where 1 ) Na+ and 2 ) H+

(1)

Jdiff,i ) Di

J)F

ηiJ J ) , j2 F j 1 + uj2C F uj1C

ηW J dCbi ) dA Ff

where the flux due to diffusion across a narrow film of thickness ∆ can be written as

asJdiff,i ) Di

j1 uj1C

∂Jelec,i ) ∂w

asJdiff,i dCbi

Jelec,1 )

n ) all counterion species in solution

i)1

(7) From eqs 6 and 7 we have J

∂φ ) ∂w

n

F

∑

(8)

ji zi2ujiC

i)1

For example, for a 1:1 electrolyte (such as NaCl) and with ion-exchange resin with H+ ions as counterion, we have

n

∑ z Cj

i i

j fixed ) -ωC

(17)

i

where ω is the sign of fixed charges in the resin. As seen from the equations, the electric potential is dependent on concentrations of the ions present in the solution and hence needs to be determined. Also, the ionic concentration is also dependent on the electric potential. This interdependence causes ion fluxes to be coupled with each other and they need to be solved simultaneously even for dilute solutions. The nonlinearity and the coupling of the governing set of equations make the


models difficult to solve analytically without making numerous simplifications and necessitate the development of computational algorithms as will be done in this work. For 1:1 electrolyte such as NaCl, analytical solutions exist for the above equations as shown by Glueckauf.5 However, when there are more than one counterions present in the solution and especially if they are multivalent ions, an analytical solution of these equations is very difficult to obtain and the equations have to be solved by numerical techniques. In this work, eqs 10 and 11 were discretized using finite difference method and the resulting coupled nonlinear algebraic equations along with the other conditions were solved simultaneously using the subroutine DNEQNF from the IMSL library in FORTRAN 90 in order to describe the entire EDI setup. The subroutine DNEQNF uses the modified Powell’s hybrid algorithm to solve the nonlinear equations. In all of the computational results reported in this paper, 50 plates each are used to discretize the differential equations along each dimension. 3. Validity of the Computational Model and the Approach

Figure 2. Comparison of model solution from numerical approach described in this work with the analytical solution mentioned in Glueckauf.5 Table 1. System Parameters As Used in Glueckauf5

The EDI setup can be described by the mathematical model as explained in the above section. In order to confirm the validity of the model and the numerical method proposed in section 2, it is worthwhile to compare it with the case where analytical solutions are possible. As mentioned earlier, for a 1:1 electrolyte solution such as NaCl the above differential equations can be solved with the boundary conditions and are reported in the work of Glueckauf5 and are given as below: -a ln(1 - ηW) - (a - 1)ηW )

J b bC W ) Z F

1 - ηW,in ηW,in + ln 1 - ηW,ex ηW,ex

bWA/f ) -a ln

(18)

ηW )

(C )W aC - (a - 1)(C )W b

o

,

a)

ujH , ujNaK

b)

parameter

value

T DNa-H rp ujH/ujNa K R β W A B

20 °C 2.1 × 10-5 cm2/s 0.04 cm 6.9 1.4 0.6 0.8 0.31 cm 4.9 2.3 × 10-2

Table 2. Comparison of Exit Concentration Values As Predicted by the Present Model with Those Reported by Glueckauf5 equiv/106 cm3 (Na)

(19)

where o

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D 3Rβγ rp2 (1 - R)

Equations 18 and 19 are solved simultaneously to describe the EDI unit used for the removal of Na ions from water. The nonlinear algebraic equations are solved using the bisection method in FORTRAN 90. The analytical solutions obtained were compared with the results reported in Glueckauf,5 and a perfect match between the two was observed. In order to validate the model and the computational technique, eqs 10 and 11 described in section 2 were solved numerically along with the boundary condition to obtain ηW for different values of the parameter Z ) JbCbW/F and the results were compared to the analytical solution of eqs 18 and 19. Figure 2 shows the comparison between the numerical and the analytical solutions for three different values of a, and it can be clearly seen that the numerical predictions are in excellent agreement with the analytical solutions for all the a values. The system parameters needed to solve these equations are given in Table 1. To further validate the model, we compared the model predictions to the experimental data reported in the paper by Glueckauf.5 Table 2 shows the exit concentration values predicted by the model described in this work for given areas, current densities, and flow rates in comparison to the experimentally measured outlet concentrations reported.5 It should be noted that the reported experimental operating parameters such as area and flow rates

expt no.

current density, 2J (mA/cm2)

f (cm3/s)

A (cm2)

Cinb

Cexb

reported Cexb

1 2 3 4 9 11

3.30 3.25 3.10 3.00 3.20 5.20

41 41 57 50 50 50

21 700 21 200 27 400 27 500 25 100 19 200

1.83 1.28 1.26 1.22 5.00 3.70

0.139 0.078 0.101 0.060 1.317 0.823

0.143 0.096 0.117 0.074 1.350 0.825

in Glueckauf5 are rounded off. With these rounded-off operating values and initial concentrations as input parameters, it can be observed that the model predicted results are in good agreement with those reported, demonstrating the validity of the numerical approach used in this work. Usage of the approximate operating values has obviously caused the slight discrepancies in the values observed. 4. Experimental Details The experimental WE-EDI unit20 consists of an ion-exchangeresin wafer sandwiched between two membranes. The wafer is synthesized by using predetermined amounts of cationic and anionic exchange resins along with sucrose and polyethylene. In this study the cationic and anionic ion exchange resins used were Amberlite IRA-400 chloride and Amberlite IR-120 plus sodium forms, respectively. A Carver Model 3851 Pneumatic Press was used for synthesis of the wafer. The ED stack consists of the Micro Flow Cell (Electrocell, Amherst, NY) which has a PTFE frame, and stainless steel electrodes, has a filtration area of approximately 0.001 m2 per cell pair. Cationic 5B membrane and anionic ACS membrane from Tokuyama, Japan,

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Table 3. Operating Conditions Used in the Experimental Setup Described in This Work parameter

value

rp V Itotal F area

0.04 cm 150 mL 0.04 A 2.5 mL/s 10 cm2

Table 4. System Parameters Used for the Computational Predictions Described in This Work value parameter β R γ D × 105 (cm2/s) uj × 105 (cm2/V/s) K j fix (mequiv/mL) C

K

1.957 11.43 2.9/1.3

Ca 0.8 0.6 0.5 0.792 4.626 5.2/1.3 2.0 × 10-3

H

9.312 54.396 -

were used to assemble the experimental setup. A one cell pair setup system was used for all of the experimentation. The WEEDI system was operated in batch mode with flow rates in all chambers set to 150 mL/min (2.5 mL/s). The concentration of the rinse solution was 0.3 M Na2SO4 for single component (a standard rinse concentration in industry) and 0.3 M NaCl for multicomponent experiments in order to prevent calcium precipitation. Solutions with the same starting concentration were used in diluate and concentrate chambers for each experimental setup. For single ion removal, a solution with a concentration of ∼2000 mg/L K+ (as KCl) was used. For twocomponent experiments, a solution with a concentration of ∼1500 mg/L K+ (as KCl) and ∼1500 mg/L Ca2+ (as CaCl2) was used. The volumes of the diluate, concentrate, and rinses were 150 mL each. The experiments were operated at a constant current of 0.04 A for 8 h, while the voltage started at 4 V and increased to 8 V as the solution became more dilute and the membrane fouled. Two separate tanks are used to continuously recirculate solution that needed to be deionized through the diluate and the concentrate compartments. Samples (2 mL) were drawn every hour for the first two hours and every two hours thereafter. Experimental results were corrected for the change in total volume as samples were taken. The system was run until there was a considerable drop in the solution concentration. Samples were analyzed using inductively coupled mass spectrometry (ICP-MS). 5. Results and Discussion 5.a. Experimental Validation. Any modeling and computational exercise becomes more valuable if we can compare the predictions with experimental observations. Hence, in this study we have attempted to compare the observed computational predictions with a few experimental observations. The above model was used for a WE-EDI unit for the removal of monovalent electrolyte solutions with only one salt present such as NaCl or KCl. The comparisons between the experimental and computational predictions are reported in work of Ho.20 In this paper, a few comparisons between experimental results of a WE-EDI unit used to deionize electrolyte solutions containing more than one salt including polyvalent ions such as calcium are reported. The experimental operating parameters, design, and system parameters used in the computational predictions are shown in Tables 3 and 4. The experimental and the computational predictions for deionization of solution with a feed concentration of 1.39 ×

Figure 3. Comparison of experimental and model predicted concentration profile for the deionization of calcium ions from solution with time.

10-5 mol/mL calcium chloride are shown in Figure 3. For these conditions, the current efficiencies for these experiments were between 0.82 and 0.88 for the transport of both cations and anions. The liquid phase diffusion coefficients used in the model are taken from Newman.21 The solid phase mobility of the ions is linked with the solid phase diffusion coefficients using the following relation. uji )

j iF D RT

(20)

j i ) 0.15Di17 is used to relate the An approximate relation D solid phase diffusion coefficient to the liquid phase diffusion coefficient. The ion-exchange equilibrium selectivity coefficients are taken from the selectivity scales reported in Schweitzer.22 For a given feed concentration, the governing equations described in an earlier section are solved to obtain the ionic concentration at the exit of the WE-EDI unit. This concentration value is then used to obtain the concentration in the tank by solving equations governing the dynamics of a continuously stirred tank, and the entire process is repeated. Figure 3 shows that there is a good match between the experimental and computational predictions. This clearly shows that the present model can be successfully applied to predict separation of divalent ions using WE-EDI. Similar experiments were performed to study removal of more than one ion from solution. Deionization of potassium and calcium ions was studied. Experimental and corresponding computational predictions are shown in Figure 4. Two different sets of experimental data for different thicknesses/widths of wafer, viz., 1.78 and 1.27 mm, are used to check the validity of the current model. The concentrations of potassium and calcium in the feed solutions were 1.87 × 10-5 and 1.11 × 10-5 mol/mL for a wafer of 1.78 mm thickness and 1.98 × 10-5 and 1.2 × 10-5 mol/mL for a wafer of 1.27 mm, respectively. Both runs were performed with wafers synthesized with a 50:50 ratio of cation- and anion-exchange resins. Another experiment with a 25:75 ratio of cation- to anion-exchange resin was also performed, and the comparison of experimental observations with computational predictions is shown in Figure 5. The wafer used had a width of 2 mm. These results clearly demonstrate that the computational model prediction matches considerably well with the experimental observations even for a system that consists of multiple electrolytes with multivalent ions. Thus, this demonstrates that this model can be used for variations in thickness, ion type, and resin selectivity. 5.b. Sensitivity Analysis. The results presented in the above section prove the validity of the model used in this study to


IR )

Cin - Cex Cin

or %IR )

Cin - Cex × 100 Cin

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(21)

Capacity (CAP). CAP is the amount of ion that can be separated by the EDI unit of a given contact area. CAP )

f (C - Cex) A in

(22)

Power Consumption (Π). The power consumption of a typical EDI unit can be considered proportional to the square of the operating current flowing through the unit.5 If Ω is the resistance per unit area of the cell, then the power consumption can be defined as Π ) I2ΩA

(23)

In this study, since current and area are the operating variables, we have chosen to use the power consumption prefactor defined as PC ) I2A

Figure 4. Comparison of model predicted and experimental concentration profiles for deionization of both potassium and calcium ions from solution with EDI unit consisting of wafers: (a) width ) 1.78 mm and (b) width ) 1.27 mm.

Figure 5. Comparison of model predicted and experimental concentration profiles for deionization of both potassium and calcium ions from solution with EDI unit consisting of wafers with cation and anion exchange ratio as 25:75, width ) 2.0 mm.

reasonably predict the experimental observations of an EDI or WE-EDI unit. Hence, it was proposed to perform a sensitivity analysis of the various operating parameters on the unit performance which will help us identify the most important parameters that can influence the EDI separation performance. This sensitivity exercise will further aid in optimal design of EDI as will be described in later sections. These studies are based on a single pass through the EDI unit. In this work, a few performance criteria were identified in order to quantify the separation capacity of the EDI unit and are as described below. Ion Removal Rate (IR). IR is the amount of each ionic species removed in one pass through the EDI unit.

(24)

The operating conditions described earlier (Tables 3 and 4) were used as the base conditions for the sensitivity simulations. The design parameters such as flow rate, current, contact area, width of wafer, and the radius of the particle were varied around the base conditions and the ion removal rate obtained was noted. The other system parameters such as the selectivity coefficient, mobilities, voidage, etc. were kept constant at the base values. The chosen variables can be easily varied at the unit operation facility and hence were chosen in this study. Figure 6 shows the results obtained from these simulations. The results show the expected trend that the ion removal rate increases with an increase in current and a decrease in flow rate. Also, as the area and width of the wafer increase, the ion removal rate also increases and then remains constant. From the figures, it can be noted that only the parameters current and flow rate have a drastic effect on the ion removal rate of the unit. For example, the ion removal rate can reach as high as 5.6% (potassium) and 6.9% (calcium) for a flow rate of 0.05 cm3/s and 1.0% (potassium) and 0.6% (calcium) when the unit is operated at a constant current of 0.36 A. A higher operating current aids in the increase in IR for both potassium and calcium, with potassium removal being the more sensitive. This is expected since the potassium ion being a monovalent ion has a higher solid phase mobility than the divalent calcium ion13 and is transported more easily with an increase in current. This selectivity could be changed if a resin was used that had an even higher selectivity of calcium to potassium. An increase in the operating flow rate causes a decrease in IR, which means that the unit is being overloaded with ionic solution and hence the separation power of the unit decreases, while a decrease in the flow rate drastically increases the IR especially for calcium ion. The other parameters, viz., area, width, and particle radius, do have an effect although not as pronounced as the above parameters. Overall, it can be observed that when the unit is operated in stringent conditions such as less area, low width, and high operating flow rates, removal of calcium deteriorates more than that of potassium, while when the unit is operated in surplus conditions such as high area, more width, and low flow rates, the calcium removal is much better than that of potassium. This is understandable since removal of divalent calcium has been reported to be more difficult than removal of monovalent potassium ion.23 However, it should be noted that, in this study, the sensitivity analysis was performed by changing only one

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Figure 6. Sensitivity study demonstrating change in IR with respect to the operating and design parameters governing the performance of EDI: % IR vs (a) f for K ion and (b) f for Ca ion and (c) J, (d) area, (e) W, and (f) rp for both K and Ca ions.

parameter at a time while keeping all others constant. This can mask any presence of interactions between the operating parameters. Nevertheless, the study gives an understanding of how and to what extent the operating parameters influence the ion removal rate of the EDI unit. 5.c. Multiobjective Optimization for Design of EDI Unit. Problem Formulation. In order to obtain the best possible performance from a system, optimization of the unit is crucial. However, for an industrial process unit it is sometimes required to fulfill more than one criterion or objective simultaneously in order to obtain an optimal economical operation. These objectives can sometimes be of a conflicting nature. In such scenarios optimization of a unit involves using single (scalar) objective functions that incorporate several objectives with arbitrary weightage factors. However, this methodology has limitations and may possibly lead to inferior optimal solutions.24,25 One of the limitations is that the optimal results are sensitive to the values of the weighting factors used, which are difficult to assign on an a priori basis. The benefits of performing multiobjective optimization in which more than one objective can be considered simultaneously over single objective optimization studies have been reported, and the improvements in process performance obtained by performing such studies have been well reported in the literature.26,27 In a typical multicriterion optimization with conflicting objectives, instead of trying to find the unique and best (global) optimum design solution, the goal is to find a set of equally good solutions, which are known as Pareto optimal solutions.24,25 A Pareto set is such that, when one moves from any one point to another within the set, at least one objective function improves while at least one other deteriorates. In this work, a few two-objective optimization problems have been solved. As a demonstration of the improvement in performance, an optimization exercise is performed only on the

WE-EDI unit used to remove just potassium ion. The operating conditions and the system parameters are same as those given in Tables 3 and 4. A potassium ion feed concentration of 2.96 × 10-5 mol/cm3 is used in the study. The experimental and computational predictions of the concentration profile showing a drop in potassium ion with time in this case is shown in the work of Ho.20 In this work, an optimization technique based on a genetic algorithm, a nondominated sorting genetic algorithm with jumping genes (NGSA-II-JG),28 is utilized. NSGA-II-JG is a type of evolutionary algorithm and allows handling of complex optimization problems. The use of NSGA-II-JG in solving complex optimization problems has been well demonstrated.26,27 Problem 1: Maximization of Ion Removal Rate with Minimization of Power Consumption Prefactor of the WE-EDI Unit. Since the WE-EDI unit utilizes current to transfer ions, one of the obvious factors that affect the economics of this unit is power consumption. The power requirement might possibly increase with an increase in the ion removal requirement of the unit. Hence, in our study the first multiobjective optimization problem solved is maximization of the ion removal rate (IR) with simultaneous minimization of the power consumption prefactor (PC). From the sensitivity analysis described in an earlier section, it is clear that the parameters flow rate and operating current are the most influential parameters on the separation performance of EDI. Hence in our optimization study we chose these two parameters as decision variables. The parameters area and the width of the cell although not that influential were added to check if the optimization algorithm chooses a different value than what is used in the experimental design. All other parameters, other than the above four, were fixed at the experimental design point. The formulation can be mathematically represented as maximize J1 ) %IR[I, f, A, W]

(21a)

minimize J2 ) PC[I, f, A, W]

(21b)

decision variables: 0.01 e I e 0.05 A 3

(21c)

0.5 e f e 3 cm /s

(21d)

8.0 e A e 12.0 cm2

(21e)

0.1 e W e 0.2 cm

(21f)

The boundaries for the decision variables were chosen based on sensitivity analyses of the decision variables. The decision variables were held within a range so that they were in the practical operating range of the WE-EDI system. All other parameters were fixed in the optimization excercise, and the values are given in Tables 3 and 4. The optimization problem was solved using an adaptation of a genetic algorithm, the elitist nondominated sorting genetic algorithm with jumping genes (NSGA-II-JG) discussed above. The results obtained from the optimization exercise are shown in Figure 7. It can be clearly seen that as the ion removal rate increases the power consumption prefactor of the WE-EDI unit also increases, as expected. For example, for an ion removal rate of 0.6% the power consumption prefactor is as low as 0.0008 A2 · cm2 while for an ion removal rate of 2.5% the power consumption prefactor increases to 0.02 A2 · cm2. The plot of capacity as the ion removal rate increases is also shown in Figure 7a. Figure 7b shows the change in the four decision variables as the ion removal rate increases. It can be seen that as the ion removal rate increases the current required also increases,


Figure 7. (a) Pareto optimal solutions obtained for the simultaneous minimizing of both power consumption prefactor and ion removal rate and comparison with the experimental operating point. (b) Values of decision variables (operating parameters) obtained from solution of optimization problem of simultaneous minimizing both power consumption prefactor and ion removal rate.

reaching a value of 0.0435 A for an IR value of 2.55%. The value of the flow rate chosen is the lowest bound defined for the optimization problem. This means that at lower flow rates the unit can remove more ions, which is evident from the sensitivity analyses. The results also show that the maximum value of the width of 0.2 cm is chosen by the algorithm in order to maximize the ion removal rate while the area of the cell varied between 11 and 8 cm2. Figure 7 also shows the comparison between the ion removal rate and the power consumption prefactor when the unit is operated at the experimental conditions with the optimal results. It can be clearly seen that the ion removal rate obtained from the optimization is quite higher compared to that obtained from the original experimental conditions for the same power consumption prefactor. For example, for a power consumption prefactor of 0.004 A2 · cm2 the unit when operated at the given experimental condition gives an ion removal rate of only 0.27%, while the unit when operated at the optimized conditions gives an ion removal rate of 1.41%, an increase of 1.14%. This clearly shows the improvement obtained by performing such a systematic optimization exercise as above. Although this is a single cell-pair system, this study could certainly be extended to an industrial EDI or WE-EDI system with multiple cells. Problem 2: Simultaneous Maximization of Ion Removal Rate and Capacity of WE-EDI Unit. In the above section, we observed the improvement in performance of the WE-EDI unit when it was aimed to maximize the ion removal rate while minimizing the power consumption prefactor. It was therefore further aimed to analyze the change in unit performance when both ion removal rate and capacity of the unit are maximized. Once again the same set of decision variables as mentioned in the earlier problem were used with the same bounds. Similar

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Figure 8. (a) Pareto optimal solutions obtained by solving problem 2: simultaneous maximization of both capacity and ion removal rate. (b) Values of operating parameters of EDI obtained from solution of problem 2.

to the earlier problem, the remaining parameters were fixed at values as described earlier. The optimization problem was formulated and the mathematical representation is as given below. The decision variables are also given below. maximize J1 ) %IR[I, f, A, W]

(22a)

maximize J2 ) CAP[I, f, A, W]

(22b)

decision variables: 0.01 e I e 0.05 A

(22c)

3

0.5 e f e 3 cm /s

(22d) 2

8.0 e A e 12.0 cm (22e) 0.1 e W e 0.2 cm (22f) Figure 8a shows the Pareto optimal solutions obtained by solving the optimization problem described above. An interesting trend is observed between the ion removal rate and the capacity of the WE-EDI unit. We can see that below a typical ion removal rate (2.55%) the unit can be operated at a high capacity of ∼0.047 mol/cm2 with simultaneous increase in the ion removal rate. However, after this point, in order to obtain a high ion removal rate the capacity decreases quite sharply, which means the unit can handle only considerably lower flow rates of feed solutions. It should be noted that this relation would have been difficult to make with a single parameter study. As seen in the earlier problem, again the power consumption prefactor increases with an increase in the ion removal rate. Figure 8b shows values for the four decision variables used in the problem. The optimization algorithm chooses the maximum value of operating current in order to simultaneously maximize both the ion removal rate and the capacity of the unit. The maximum value of the width of the cell is again chosen in this problem. The flow rate chosen decreases from a value of 1.1

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operated at quite high capacities compared to those used in the experimental operating point while still achieving a high ion removal rate, however at the cost of a high power consumption prefactor (plot of power consumption prefactor vs ion removal rate). Figure 9 also shows the comparison between the results obtained by solving problems 1 and 2. It can be seen that in both cases the ion removal rate achievable is high and is comparable while in order to reach such high ion removal rates while achieving the other optimization goal one of the performance criteria has to be compromised. In problem 1, in order to obtain a low power consumption prefactor the values of capacity obtained are lower than what we can obtain from problem 2. The reverse trend is observed in the case of problem 2. Figure 9b also shows the trend of variation of the variables operating current and flow rate for the respective problems. Thus it can be observed that, at the same power consumption, improvement in the ion removal rate over the experiments is obtained when the unit is operated at the optimized conditions. 6. Conclusions

Figure 9. (a) Comparison between the solutions obtained by solving problem 1, problem 2, and the experimental operating point. (b) Comparison of values of key operating parameters obtained from solution of problem 1, problem 2, and the experimental operating point.

cm3/s for an ion removal rate of 1.1% to a value of 0.5 cm3/s for an ion removal rate of 2.9%. The area chosen is constant until an ion removal rate of 2.54% and then increases sharply in order to increase the ion removal rate. Figure 9 shows the comparison between the optimization results obtained and the performance obtained when the unit is operated at original experimental conditions. From the plot of capacity vs ion removal rate it can be clearly seen that with systematic optimization, as seen in this case, the unit can be

This paper deals with the development of a mathematical model for the prediction of deionization of electrolyte solutions containing multivalent ions such as calcium. The model is validated by comparing the computational predictions with observed experimental trends. It is shown that the mathematical model is able to predict the experimental observations reasonably well. Sensitivity studies reveal that the ion removal rate of a WE-EDI unit depends strongly on operating variables such as the flow rate and operating current and weakly on the contact area and width of the wafer. As the flow rate decreases and the operating current increases, the ion removal rate also increases. The paper also demonstrates the improvement in separation power of a typical WE-EDI unit used for deionization of solution containing monovalent electrolyte such as potassium chloride. It can be seen that with systematic and rigorous optimization exercise, the performance of the current experimental unit in terms of higher ion removal rate, lower power consumption prefactor, and higher unit capacities can be obtained. Hence, this study demonstrates the importance of developing relatively simple mathematical models which when coupled with systematic optimization studies further aid in the efficient design and economic operation of industrial process units such as the EDI. These optimization studies are first of their kind for an EDI or WE-EDI system. The modeling and the multiobjective optimization of the current experimental unit has enabled identifying optimal values of key operating variables for superior performance at minimal operating costs. Also, the current modeling and optimization framework can be readily adapted to optimize other EDI systems. However, it is worth noting that there are other relevant variables that have not been considered in the present study for the optimization of the EDI unit such as resin properties and size, mass transfer boundary layers, etc. Nevertheless, the importance of the analysis of the effect of such variables is recognized and is being considered for future studies. Acknowledgment The authors would like to recognize the University of Arkansas and the Jim L. Turpin Professorship in Chemical and Biochemical Separations for partial support of this work. Nomenclature A ) membrane cross-sectional area, cm2 Cib ) bulk concentration of species i, mol/cm3

Ind. Eng. Chem. Res., Vol. 48, No. 20, 2009 ) film concentration at the interface between liquid and resin/ solid particle of species i, mol/cm3 j i ) solid phase concentration of counterion species i, mol/cm3 C CAP ) capacity of EDI unit, mol/cm2 Cf ) feed concentration, mol/cm3 D ) diffusion coefficient of ion in water, cm2/s F ) Faraday constant, C/mol f ) flow rate, cm3/s I ) current carried by cationic species, A IR ) ion removal rate of EDI unit Itot ) total current through the unit, A J ) current density, A/cm2 Jdiff,i ) diffusion flux of species i, mol/cm2/s Jelec,i ) flux of species i due to electric potential, mol/cm2/s K ) ion-exchange equilibrium selectivity coefficient n ) number of counterion species PC ) power consumption prefactor of EDI unit, A2 · cm2 rp ) radius of resin particle, cm uji ) mobility of counterion species i in solid phase, cm2/V/s V ) volume of tank solution, mL W ) width of the diluate chamber, thickness/width of wafer, cm zi ) valence of species i Cio

Greek Symbols R ) packing ratio β ) fraction of total ion-exchange resin surface area accessible for exchange of ions γ ) ratio of cation- to anion-exchange resin η ) local current utilization at W Π ) power consumption of the EDI unit, W φ ) electric potential across the EDI cell, V Ω ) resistance of unit area of cell, ohm/cm2 ω ) sign of fixed charges in resin Subscript H ) hydrogen ion Superscripts in ) input point of the cell ex ) exit point of the cell

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ReceiVed for reView December 10, 2008 ReVised manuscript receiVed July 10, 2009 Accepted August 24, 2009 IE801906D

Separation of Multicomponent Electrolyte Solution - American

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