Development of a computer simulation program of batch ion-exchange membrane electrodialysis for saline water desalination

Desalination 320 (2013) 118–133

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Development of a computer simulation program of batch ion-exchange membrane electrodialysis for saline water desalination Yoshinobu Tanaka ⁎ IEM Research, 1-46-3 Kamiya, Ushiku-shi, Ibaraki 300-1216, Japan

H I G H L I G H T S • • • • •

Computer simulation program of a batch electrodialysis process is developed for saline water desalination. A salt solution is supplied to an electrodialyzer with an instantaneous open/shut operation. Drinking water is produced in a one-stage and a two-stage batch electrodialysis process. Multi-stage operation is economically disadvantageous compared to the single-stage operation. Limiting current density is expressed by the function including temperature.

a r t i c l e

i n f o

Article history: Received 4 February 2013 Received in revised form 18 April 2013 Accepted 19 April 2013 Available online 20 May 2013 Keywords: Ion-exchange membrane Multi-stage batch electrodialysis Computer simulation program Mass transport Drinking water production Limiting current density

a b s t r a c t A computer simulation program of batch ion-exchange membrane electrodialysis is developed for saline water desalination. For proceeding with the computation, the relationships between the salt concentration in desalting cells, cell voltage and the five electrodialysis parameters are expressed by the equations using the continuous electrodialysis program. A salt solution is supplied to the batch electrodialysis process, and the performance changes with time are computed with an instantaneous open/shut feed operation of a solution taking salt concentration and cell voltage as parameters. A salt solution is supplied to a one-stage and a two-stage batch electrodialysis process and the performance of drinking water production is calculated. From cell pair number and energy consumption, the multi-stage operation is economically disadvantageous compared to the single-stage operation. Limiting current density of an electrodialyzer is expressed by the function including temperature and computed. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Ion-exchange membrane electrodialysis is applied widely to desalination and concentration of electrolyte solutions. It is further applied to separate ionic species dissolving in a solution. Saline water desalination is the most fundamental application of ion-exchange membranes and the desalination processes are classified to a continuous (one-pass flow), a batch and a feed-and-bleed process. Among these processes, the batch process is applicable to operating a small or middle scale electrodialysis such as; seawater desalination in an island [1,2]; seawater desalination in a vessel [3]; desalination of brackish groundwater [4,5] and organic acid mixture [6]; waste water treatment [7,8]; alkaline black liquor treatment [9]; demineralization of why [10] and sugar liquor [11]; removal of boron [12] and silver [13]; etc.

⁎ Tel./fax: +81 29 874 5400. E-mail address: [email protected]. 0011-9164/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2013.04.022

The performance of a batch process has been discussed from various view point as follows; Parulekar investigated energy consumption in a constant current and constant voltage batch operation [14]. Demircioglu et al. introduced equations expressing the ionic mass balance around diluate circulation tank and discussed energy consumption [15]. Moon et al. predicted the performance of continuous and batch electrodialysis processes based on the fundamental principles of electrochemistry, transport phenomena and thermodynamics [16]. Ortiz et al. developed a mathematical model for a batch process and discussed mass balance, ohmic drop and membrane potential [17]. 2. Development of computer simulation programs and the target of the investigation The computer simulation programs which had been published in a series of articles are developed from the investigation of seawater concentration for salt manufacturing carried out in Japan. The development and the target are summarized as follows.

Y. Tanaka / Desalination 320 (2013) 118–133

119

2.1. Basic study

3. Mass transport in an electrodialyzer

Mss transport across an ion-exchange membrane pair is expressed by the overall mass transport equation [18]. Distribution of electrodialysis conditions in an electrodialyzer is discussed [19]. Limiting current density of an ion-exchange membrane and of an electrodialyzer are defined [20]. Direct electric current electric resistance of an ion-exchange membrane is calculated from alternating current electric resistance [21]. Energy consumption in electrodialysis is discussed [22]. Computer simulation program for seawater concentration is developed [23]. Reliability of the program is discussed by comparing the calculated data with experimental data [24] and industrial data [22,23].

Fig. 1, illustrates a one-stage (stage no. = 1) batch electrodialysis process. In this process, the valve V1 is opened at first and a definite volume (Q0) of a raw salt solution (salt concentration C 0) is supplied to the circulation tank. Next, V1 is closed, V2 is opened and the solution is supplied to the inlets of desalting cells (De) integrated into the stack (marked gray) in the electrodialyzer. The solution flowing out from the outlets of desalting cells is returned to the circulation tank. The raw feeding solution is also supplied to the inlets of concentrating cells (Con) for preventing scale formation in the concentrating cells. A part of the solution flowing out from the outlets of concentrating cells is further supplied to electrode cells and partition cells (Part), which are integrated for preventing the influence of electrode reactions to the performance of the electrodialyzer. Then, the solution is discharged to the outside of the process with the residual solution flowing out from the outlets of concentrating cells. Next, the electrodialysis is proceeded applying constant voltage between electrodes and circulating the salt solution between the circulation tank and desalting cells. After the salt concentration in desalting cells decreased to a targeted value, V2 is closed, V3 is opened and the desalted solution (the output of the process) is extracted to the outside of the process. In this electrodialysis process, the following parameters are defined.

2.2. Application of the program to saline water desalination The program is developed for discussing the performance of the electrodialysis process for saline water desalination [25–27]. After the publication of the program, it is revised taking account of the following phenomena or processes. (a) Influence of operating temperature [28] The phenomenon is discussed taking account of the influence of temperature on physical properties of saline water. (b) Influence of temperature on the limiting current density This phenomenon is described in Appendix A in this article. (c) Electric current screening effect of a spacer [29] The electric current screening effect is determined by the volume ratio of spacer rods in the cells. (d) Solution is supplied to concentrating cells [29] In seawater concentration, concentrated solution is the output of the process and it is extracted from concentrating cells. However in desalting process, a raw solution is supplied to concentrating cells for preventing scale precipitation. (e) Pressure drop in the electrodialyzer [29] The pressure drop is influenced by the electrodialyzer specifications, operating conditions and solution viscosity. (f) Reasonability of standard deviation of solution velocity ratio in desalting cells α [30]. The influence of α on the limiting current density is discussed.

2.3. Program development in the next stage It is necessary to improve the availability of the program at present. The electrodialysis process for saline water desalination is classified to the continuous, batch and feed-and-bleed process. The program for discussing the above processes is classified to the continuous program describing the functions of an electrodialyzer and the batch and feed-and-bleed program describing the function of a circulating tank. The continuous program is the basis of all programs and it is explained definitely by arranging equations systematically [29]. The performance of the continuous process and feed-and-bleed process can be computed with a single computation with the stand alone software. Calculation is carried out in the spread sheet with a use of common software (Excel), several pieces of papers, an ink cartridge and ordinary hardware (computer) within about 10 min. The program is scheduled to be integrated in web sites [31]. So, readers can operate the program in the web sites (Companion site). The program aims to function as a pilot plant operation. However, the batch process developed in the previous investigation [26] includes the integration equation which cannot be calculated with common software integrated in the web site. In this investigation the open/shut solution feeding operation is introduced in the program instead of the integration operation. Such an improvement makes possible to calculate the batch process in the web site.

Flow-pass thickness in a desalting cell and a concentrating cell: a (cm) Flow-pass width in a desalting cells and a concentrating cell: b (cm) Flow-pass length of a desalting cell and a concentrating cell: l (cm) Membrane area: S = bl (cm 2) Number of desalting cells: N Number of concentrating cells: N + 1 Salt concentration of a raw solution: C 0 (mg dm −3) Solution volume supplied into the circulation tank at first: Q 0 (dm 3) Salt concentration at the inlets and outlets of desalting cells: C′in, C′out (mg dm − 3) Salt concentration at the inlets and outlets of concentrating cells: C″in (= C 0), C″out (mg dm − 3) Linear velocity at the inlets and outlets of desalting cells u′in, u′out (cm s −1) Linear velocity at the inlets and outlets of concentrating cells: u″in, u″out (cm s −1) Standard deviation of solution velocity ratio ξ (= (u − ū) / ū, u; linear velocity in each desalting cell, ū; average linear velocity in desalting cells) in desalting cells in a stack: σ Temperature: T (°C) In the multi-stage process (stage no. ≧ 2), a salt solution (salt concentration C′in) produced in the circulation tank in the previous stage is supplied to the circulation tank in the next stage to operate the next batch operation. In the multi-stage operation, salt concentration supplied to the circulation tank is defined as C′in,0 instead of C 0 defined in the one-stage operation. Fig. 2 illustrates the structure of the desalting cell and concentrating cell. 4. Performance of an electrodialyzer In order to discuss the performance of the batch process, the performance of an electrodialyzer has to be defined by determining the relationship between salt concentration at the inlets of desalting cells C′in · cell voltage Vcell and the five parameters (salt concentration at the outlets of desalting cells C′out, salt concentration at the outlets of concentrating cells C″out and linear velocity at the outlets

120

Y. Tanaka / Desalination 320 (2013) 118–133 Dischaged solution

C" out C' out Q' out

K

u" out

u' out

A

K

K

Part

Con no.1

u" in

De no.1

u" out

u' out

u" out

A

K

A

K

Con no.N

u' in

u" in

De no.N

A

Con no.N+1

u'in

Part

u" in

C' in Q' in C 0 = C" in Feeding solution C 0 or C' in,0

Q" in

V1

V2

Electrodialyzer V3

C'in Circulation tank

K ; Cation-exchange membrane A ; Anion-exchange membrane + ; Anode - ; Cathode

De ; Desalting cell Con ; Concentrating cell Part ; Partition cell

Desalted solution

Fig. 1. Batch electrodialysis process.

of desalting cells u′out, current density I/S, ion flux across a membrane pair JS). The relationship is calculated from the following continuous electrodialysis program developed in the previous investigation [29]. The program is integrated into the spread sheet and computation is carried out with Excel software. 4.1. Electrodialysis program

(concentration: C 0) is supplied to the inlets of concentrating cells (Con) at the average linear velocity of u″in. By supplying an electric current I, ions and solutions are transferred from desalting cells to concentrating cells across an ion-exchange membrane pair and the flux is defined by JS and JV respectively. In desalting (concentrating) cells, salt concentration is decreased (increased) from C′in (C 0) under applied average current density I/S and reaches average salt concentration C′out (C″out) at the outlets of desalting (concentrating)

At first, the mass transport in a continuous process is illustrated in Fig. 3. A salt solution (concentration: C′in) is supplied to the inlets of desalting cells (De) at average linear velocity of u′in. For preventing scale formation in concentrating cells, a feeding (raw) solution

C' out A

C" out

K

A V out

Concentrating slot

l slot,c

u' out

u" out

De

Con i

x =l

outlet

i out

x

b slot,c

u'

b

Concentrating cell

p

C' p u" p C" p JS JV

a

a

u' in

u" in

Vp

V in

i = I/S

x = pl

x =0

inlet

i in

Desalting cell

Anode l

Cathode C' in 0 C

b slot,d

Electrodialyzer

l slot,d Desalting slot Thickness; a Fig. 2. Structure of a desalting and a concentrating cell.

De: Desalting cell, Con: Concentrating cell K: Cation exchage membrane, A: Anion exchange membrane J S, J V: Fluxes of ions and solutions across membrane pairs at x = pl C' p , C" p : Electrolyte concentration in desalting and concentrating cells at x = pl u' p , u" p : Linear velocity in desalting cells and concentrating cells at x = pl V in = V p = V out Fig. 3. Mass transport in a continuous process.

Y. Tanaka / Desalination 320 (2013) 118–133

input

T λ, µ, φ r alter

ρ

Start

121

control key 3 * I/S

input C' in , u' in , u" in a, b, l

control key 1 JS, JV, η C' out , C" out

C' p *,C" p *

decision point 1 no C' p = C' p *, C" p = C" p * yes input N, σ , ε

C' out,j , r' out,j, r memb,out,j

ζ a1, a2, a3

C' p,j

ζ

ζ

inout

inout

control key 2 p

r' p,j , r memb,p,j

C' p , C" p

inp

= ζ

decision point 2

no

inp

yes a1, a2, a3

ζ

in ,

i in , ζ

out ,

i out

V Ω ,in , V memb,in , VΩ ,out , V memb,out

V cell , E control key 3 * V cell decision point 3 V cell * = V cell

no

yes

End

Fig. 4. Computer simulation of a continuous process. Colored figure entries show key variables operated in a computer simulation.

Y. Tanaka / Desalination 320 (2013) 118–133

cells. Salt concentration change in desalting cells causes current density change along the flow-pass from iin at the inlets to iout at the outlets. The current density becomes i at x distant from the inlets of desalting cells. I/S, JS, JV, C′p, C″p, u′p and u″p are altogether the values at x = pl distant from the inlets of desalting cells. Vin, Vout and Vp are voltage difference between electrodes respectively at the inlets (x = 0), the outlets (x = l) and x = pl of desalting cells (Vin = Vp = Vout). Fig. 4 illustrates the computer simulation program of the continuous electrodialysis process in Fig. 3. The explanation of the program (Fig. 4) is described definitely and systematically in the previous article [29]. This section excludes the definite explanation of the program and remarks only the functions of the following three decision points in Fig. 4. (1) Decision point 1 Control key 1 (C′p⁎ and C″p⁎) is equalized to respectively C′p and C″p for preventing circulation reference as follows. 



C’p ¼ C’p

C”p ¼ C ’’p :

ð1Þ

3000 C'out = (1-0.9114Vcell0.9) C'in

2500

0.2 V/pair 0.3

C'out (mg/dm3)

122

2000 0.4 0.5

1500

0.6

1000

500

0

0

500

1000

1500

2000

2500

3000

3500

C'in (mg/dm3) Fig. 5. Salt concentration at the outlets of desalting cells. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3.

(2) Decision point 2 Control key 2 (p⁎) is adjusted to realize ζ inout ¼ ζ inp ¼ ζ out :

ð2Þ

Eq. (2) is equivalent to the electric current non-uniformity coefficient defined by ζout = iout / (I / S). (3) Decision point 3 Control key 3 (I/S⁎) is adjusted to realize 

V cell ¼ V cell :

ð3Þ

The computation is carried out by inputting the electrodialyzer specifications listed in Table 1.

The continuous program is developed based on the assumptions and experiments. The coefficients (α, β1, β2, γ1, γ2, γ3, δ1, δ2, ε1, ε2) that appeared in Eqs. (4)–(8) are determined using the Numerical figure processor; Cut (Win 32 Ver 7, Programmed by Ge-Xin, Evergreen company, Japan). The standard errors of the coefficients are due to the errors in the assumptions and experiments applied for developing the continuous program. Numerical figures presented in this article are created using the Cut. These coefficients are calculated under the specifications defined in Table 1. It should be added that number of cell pairs N = 300 pairs is inputted but it does not influence to the computed results. 5. Performance of a batch process

4.2. Results Five parameters (C′out, C″out, u′out, I/S, JS,) are plotted against C′in taking Vcell as a parameter and shown respectively in Figs. 5, 6, 7, 8 and 9. The five parameters are expressed by the following five functions of C′in and Vcell.

0

C”out ¼ C –β1 V cell

0:9

u’out ¼ γ1 þ γ2 V cell

ð4Þ

þ β2 V cell

1:1

–γ3 V cell

I=S ¼ −δ1 V cell þ δ2 Vcell J S ¼ −ε1 V cell þ ε2 V cell

0:9

0:9

C’in

C’in

0:9

0:9

C’in

C’in

0:8

ð5Þ

0:8

20000

ð6Þ

0:8

0:8

ð8Þ

Table 1 Electrodialyzer specifications.

C"out = C0 - 2315Vcell0.9 + 45.84Vcell0.9 C'in0.8

18000 16000

ð7Þ

The relationships between the values on the x-axis and on the y-axis in Figs. 5–9 are determined from the computer simulation using the continuous program published in the previous article [29].

a b l u′in u″in σ T

The equations are developed by incorporating the imaginary feeding cell; Fe and shutter; Sh at the outlet of the circulation tank as illustrated in Fig. 10 in which definite volume (Q0 dm 3) of a solution (salt concentration: C 0 or C′in,0 mg/dm 3) is prepared at first.

C"out - C0 (mg/dm3)

C’out


 0:9 C’in ¼ 1–αV cell

5.1. Relationship between operation time and the performance of an electrodialyzer in a one-stage batch operation; the open/shut solution feeding operation

0.6 V/pair

14000

0.5

12000

0.4

10000 0.3

8000 6000

0.2

4000 cm cm cm cm/s cm/s °C

0.05 100 100 10 1 0.1 25

2000 0

0

500

1000

1500

2000

2500

3000

3500

C'in (mg/dm3) Fig. 6. Salt concentration at the outlets of concentrating cells. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3.

Y. Tanaka / Desalination 320 (2013) 118–133

10.02

1.8 u'out = 10.000 + 7.487 × 10-3 Vcell1.1

10.01

1.6

- 1.3452 × 10-4Vcell0.9 C'in0.8

I/S (10-2 A/cm2)

9.99

u'out (cm/s)

I/S = -2.622 × 10-3 Vcell + 4.418 × 10-5 Vcell0.9 C'in0.8

0.6 V/pair

1.4

10.00

0.2 V/pair

9.98 0.3

9.97 0.4

9.96

0.5

0.5

1.2 0.4

1.0 0.3

0.8 0.6

0.2

0.4

0.6

9.95 9.94

123

0.2 0

500

1000

1500

2000

2500

3000

0.0 0

3500

500

1000

C'in (mg/dm3)

1500

2000

2500

3000

3500

C'in (mg/dm3)

Fig. 7. Linear velocity at the outlets of desalting cells. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3.

Fig. 9. Ion flux. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3.

For proceeding with the computation, the operation time t is divided at an interval of Δt minutes. The shutter repeats an instantaneous imaginary open/shut operation and supplies a definite flow rate (Q′in) of a solution from the circulation tank to desalting cells through the feeding cell at an interval time of Δt minutes.

at the inlets of the electrodialyzer (desalting cells) C′in. Table 2 gives salt concentration changes in the circulation tank (in the feeding cell) (C′in,n) and at the outlets of the electrodialyzer (C′out,,n) and solution volume in the circulation tank (Qn) in each step n. The mass balance and volume balance in the circulation tank in each step are introduced as follows.


 3 −3 Q ’in dm =Δtmin ¼ abNu’in  ð60Δt Þ  10

n¼1

ð9Þ


 3 −3 Q ’out dm =Δtmin ¼ abNu’out  ð60Δt Þ  10

0

C in;1 ¼

Before the step calculation; C′out,0 u′out,0 Q′out,0 C′out,0

Mass balance:

ð10Þ

C 0in;0 Q 0 −C 0in;0 Q 0in þ C 0out;0 Q 0out;0 : Q1

is computed from Eq. (4). is calculated from Eq. (6). is computed from Eqs. (6) and (10). is calculated from Eq. (5).

C' out,n

ð11Þ

Q' out,n

Steps during operation time are numbered as n = 1 (t = Δt min), n = 2 (t = 2Δt min), n = 3 (t = 3Δt min), ­­­n = n (t = nΔt min). The solution in the circulation tank is mixed vigorously and its concentration is the same to the concentration in the feeding cell and

De

Con

1.8 I/S = -2.622 × 10-3 Vcell + 4.418 × 10-5 Vcell0.9 C'in0.8

1.6 1.4

I/S (10-2 A/cm2)

ED

0.6 V/pair 0.5

Q' in

1.2 0.4

Qn C' in,n

1.0 0.3

0.8

C' in,n P

0.6

Circulation tank

0.2

0.4 0.2 0.0

0

500

1000

1500

2000

2500

3000

3500

C'in (mg/dm3) Fig. 8. Current density. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3.

Shut.

Fe

Shut.; Shutter Fe; Feeding cell P; Pump ED; Electrodialyzer De; Desalting cell Con; Concentrating cell Fig. 10. Instantaneous open/shut solution feed batch operation.

124

Y. Tanaka / Desalination 320 (2013) 118–133

Table 2 Changes of salt concentration and solution volume In a batch electrodialysis process. Step n

0 1 2 n n+1

Time t

Circulation tank

Feeding cell (ED inlet)

ED outlet

min

mg/dm3

dm3

mg/dm3

mg/dm3

dm3/Δt min

0 Δt 2Δt nΔt (n + 1)Δt

C′in,0 C′in,1 C′in,2 C′in,n C′in,n + 1

Q0 Q1 Q2 Qn Qn + 1

C′in,0 C′in,1 C′in,2 C′in,n C′in,n + 1

C′out,0 C′out,1 C′out,2 C′out,n C′out,n + 1

Q′out,0 Q′out,1 Q′out,2 Q′out,n Q′out,n + 1




 S cm2 NV cell ðI=SÞaverage A=cm2  10−3 3
 En kWh=m ¼ 3 Q n dm −3  60ð min=hÞ  10 t ð minÞ

ð22Þ

The computation is carried out by inputting the data in Table 1 into the spread sheet software (Excel) and the performance changes with time in the batch process are calculated. 5.2. Computation

Volume balance: Q 1 ¼ Q 0 –Q ’in þ Q ’out;0

ð12Þ

n ¼ 2: Mass balance: 0

0

C in;2 ¼

0

0

0

0

C in;1 Q 1 −C in;1 Q in þ C out;1 Q out;1 : Q2

ð13Þ

Volume balance: Q 2 ¼ Q 1 –Q ’in þ Q ’out;1 :

ð14Þ

C′in,1 and Q 1 are calculated using Eqs. (11) and (12). C′out,1 and Q′out,1 are calculated using Eqs. (4), (6), and (10). n¼n Mass balance 0

C in;n ¼

C 0in;n−1 Q n−1 −C 0in;n−1 Q 0in þ C 0out;n−1 Q 0out;n−1 : Qn

ð15Þ

Volume balance Q n ¼ Q n−1 –Q’in þ Q ’out;n−1

ð16Þ

C″out,n and I/Sn are computed substituting C′in,n and Vcell into Eqs. (5) and (7). I/Sn and JS,n are computed using Eqs. (7) and (8). (I/S)average is computed using the following equation.   I ¼ S average

n X

In order to improve the precision of computations, Δt should be as small as possible, but it consumes a lot of sheets. Figs 11–13 give the changes of C′in, I/S and E with operating time t calculated putting Δt = 1, 2, 3, 4, 6 and 10 min and C 0 = 1000 mg/dm 3, Vcell = 0.4 V/pair, N = 300 pairs and Q0 = 30 m3. C′in, I/S and E at t = 60 min are plotted against Δt and given in Fig. 14 which shows that the precision is lowered at Δt = 10 min. From the above plotting, the interval time is determined as Δt = 4 min in this investigation. Table 3 exemplifies the spread sheet of one-stage batch electrodialysis for saline water desalination computed by setting C 0 (= Cin,0) = 2000 mg/dm3, Vcell = 0.4 V/pair, N = 300 pairs and Q0 = 30 m 3. The fundamental parameters in the table are fixed in all computation in this program. The calculation is carried out similarly changing C 0 and Vcell and the performance of the batch process is plotted against operation time t. The followings are the performance of a one-stage batch operation calculated by setting Vcell and C0 (= C′in,0) as parameters and N = 300 pairs and Q0 = 30 m 3. Fig. 15 shows changes of C′in with time. C′in is equivalent to the salt concentration in the circulation tank. So it corresponds to the salt concentration of product water. It equals to Cin,0 at t = 0 and decreases toward lower concentration. The decreasing rate is increased with Vcell. Fig. 16 shows I/S, which increases with C 0 at t = 0. It is decreased with t and decreasing rate is increased with Vcell. C″out is increased with C 0 and decreased with t (Fig. 17). Solution volume in the circulation tank Q corresponds to the output of the batch process. Q is slightly decreased with the increase of t and Vcell (Fig. 18). Energy consumption E is increased with t, Vcell and C0 (Fig. 19). Current efficiency η is decreased slightly with the increase of t and Vcell and decrease of C0 (Fig. 20). Desalting ratio α increases from 0 to 1 with the increase of t, increases with Vcell and independent of C 0 (Fig. 21). Water recovery

I=Sn

n¼1

n

1200

ð17Þ 1000

Current efficiency ηn, desalting ratio αn, water recovery Ren and energy consumption En are computed as follows. J S;n

n¼1 n X

ð18Þ

I=Sn

800

C'in (mg/dm3)

F ηn ¼

n X

600

400

n¼1 0

α n ¼ 1−

Ren ¼

C in;n 0

C in;0

Qn   Q 0 þ Q }in dm3 = min  t ð minÞ


 3 −3 Q ”in dm =min ¼ abðN þ 1Þu”in  ð60s=minÞ  10

ð19Þ

ð20Þ ð21Þ

200

0

0

20

40

60

80

100

120

140

t (min) Fig. 11. Salt concentration at the inlets of desalting cells. Δt = 1 (○), 2 (Δ), 3 (□), 4 (●), 6 (▲), 10 (■) min. C0 = 1000 mg/dm3, Vcell = 0.4 V/pair, N = 300 pairs, Q0 = 30 m3.

Y. Tanaka / Desalination 320 (2013) 118–133

0.6

E (kWh/m3) I/S (A/dm2) C'in (mg/dm3)

0.5

I/S (A/dm2)

0.4

0.3

0.2

0.1

0.0

125

0

20

40

60

80

100

120

0.5

0.4

0.3

0.2 I/S E

0.1

0.0

140

C'in

0

1

2

3

4

5

6

7

8

9

10

11

12

Δ t (min)

t (min) Fig. 12. Current density. Δt = 1 (○), 2 (Δ), 3 (□), 4 (●), 6 (▲), 10 (■) min. C0 = 1000 mg/dm3, Vcell = 0.4 V/pair, N = 300 pairs, Q0 = 30 m3.

Fig. 14. Δt versus C′in, I/S and E. t = 60 min, C0 = 1000 mg/dm3, Vcell = 0.4 V/pair, N = 300 pairs, Q0 = 30 m3.

Re is decreased with the increase of t and independent of C 0 and Vcell (Fig. 22).

6.2. Two-stage batch operation

6. Drinking water production by a multi-stage batch electrodialysis process 6.1. One-stage batch operation The threshold of acceptable esthetic criteria of human drinking water is 500 mg/dm3. So, the drinking water (salt concentration 400 mg/dm3) is produced according the process described in Section 5.1. Computation is carried out by setting C 0 (= Cin,0 in the first-stage) = 1000, 2000 and 3000 mg/dm 3, Q0 = 30 m 3 and operation time t = 60 min (1 h, n = 15) and changing cell voltage as Vcell = 0.3, 0.4, 0.5 and 0.6 V/pair. Number of cell pairs N is calculated by adjusting N to realize C′in = 400 mg/dm3 at t = 60 min. Table 4 exemplifies the spread sheet computed at Δt = 4 min by setting C0 (= Cin,0) = 2000 mg/dm3 and Vcell = 0.4 V/pair. Fig. 23 shows the relationship between N and Vcell taking C0 as parameters. Fig. 24 shows the relationship between E and Vcell.

6.2.1. Average or total performances A salt solution is supplied to the first-stage in the two-stage batch electrodialysis process. Specifications of an electrodialyzer on each stage are given in Table 1. The solution is electrodialyzed applying constant cell voltage Vcell (V/pair) at Δt = 4 min in each stage to produce drinking water in the second-stage. Number of desalting cells is NI in the first-stage and NII in the second-stage. Computation is carried out according the process described in Section 5.1. Table 5 exemplifies the spread sheet of two-stage batch electrodialysis for drinking water production computed by setting Q0 = 30 m 3, Vcell = 0.4 V/pair, Cin,0 in the first-stage (= C 0) = 2000 mg/dm 3, Cin,0 in the second-stage = 1000 mg/dm 3 and operation time in each stage t = 60 min (1 h, n = 15). The fundamental parameters are the same to the values in Table 1. C′in,0 and Q0 in the second stage are replaced with C′in and Q at t = 60 min (n = 15) in the first-stage. Drinking water is produced by inputting NI and adjusting NII to realize C′in in the second-stage = 400 mg/dm3 at t = 60 min (n = 15). Average or total performances (I/Stotal,average, ηaverage, αtotal, Retotal and Etotal) in the two-stage operation are calculated as follows.

0.12

I=Stotal;average ¼

0.10

ηaverage ¼

N I ηI þ NII ηII NI þ NII

0.08

E (kWh/m3)

NI I=Saverage;I þ N II I=Saverage;II NI þ NII

α total ¼ 1−

C 0in;n;II C 0in;0;I

ð23Þ

ð24Þ

n ¼ 15

ð25Þ

0.06

Retotal ¼ 0.04


 3 Etotal kWh=m ¼ E1 þ E2

0.02

0.00

Q n;II Q 0;I þ Q}in;I t n;II þ Q }in;II t n;II

0

20

40

60

80

100

120

140

t (min) Fig. 13. Energy consumption. Δt = 1 (○), 2 (Δ), 3 (□), 4 (●), 6 (▲), 10 (■) min. C0 = 1000 mg/dm3, Vcell = 0.4 V/pair, N = 300 pairs, Q0 = 30 m3.

n ¼ 15

ð26Þ ð27Þ

6.2.2. Results and discussion Salt solution in the first stage circulation tank (C′in,I) is supplied to the second stage circulation tank (C′in,II). Here, junction salt solution supplied from stage I to stage II is defined and its concentration is termed C′junc = C′in,I. Fig. 25 exemplifies the influence of C′junc to N,

126

Y. Tanaka / Desalination 320 (2013) 118–133

Table 3 Electrodialysis program of one-stage batch electrodialysis for saline water desalination.

1. Fundamental parameters u'in

10.00

cm/s

α

0.9114

δ1

2.622E-03

u" in

1.00

cm/s

β1

2315

δ2

4.418E-05

flow-pass thickness

β2

45.84

ε1

2.481E-08

ε2

4.160E-10

a

0.05 cm

b

100 cm

flo-pass width

γ1

10.000

l

100 cm

flow-pass length

γ2

7.487E-03

γ3

1.345E-04

T

25

F

o

C

96485 As/eq

σ

standard deviation

0.10

2. Computation C

0

2000.00

C' in,0 = C

0

Q0

2000.00 30000.00

V cell

No.

0.4

C' out,0

1200.91

C" out,0

9773.80

t

Q

3

input

N

300 cell

3

301 cell

mg/dm mg/dm

input

N +1

dm

3

input

Δt

V/pair

input

Q' in

3600.00

Q" in

90.30

3

dm /min

9.9769

Q' out,0

3591.70

dm /Δ t min

Q' out

I/S

C' out 3

C" out 3

3

3

interval

3

u' out,0

C' in 3

input

dm /Δ t min

3

mg/dm

desalting cell no., cell pair no. concentrating cell no.

4 min

3

mg/dm

input

cm/s 3

η

JS 2

α

Re

2

I/S average

E 3

2

mg/dm

mg/dm

mg/dm

dm /Δ t min

A/cm

4 29991.70

1904.30

1143.45

9435.75

3592.06

7.096E-03

6.677E-08

0.9078

0.0478

0.9878

0.0189

0.7096

8 29983.76

1813.15

1088.72

9110.57

3592.40

6.782E-03

6.381E-08

0.9078

0.0934

0.9760

0.0370

0.6939

1726.34

1036.59

8797.81

3592.73

6.481E-03

6.098E-08

0.9078

0.1368

0.9644

0.0543

0.6786

29968.89

1643.65

986.94

8496.97

3593.05

6.191E-03

5.825E-08

0.9078

0.1782

0.9531

0.0709

0.6638

20

29961.93

1564.90

939.65

8207.63

3593.35

5.912E-03

5.562E-08

0.9078

0.2176

0.9420

0.0867

0.6492

24

29955.29

1489.89

894.62

7929.35

3593.65

5.644E-03

5.310E-08

0.9078

0.2551

0.9312

0.1018

0.6351

7 1680

28

29948.93

1418.47

851.73

7661.72

3593.93

5.386E-03

5.067E-08

0.9078

0.2908

0.9207

0.1162

0.6213

8 1920

32

29942.86

1350.44

810.88

7404.32

3594.20

5.138E-03

4.833E-08

0.9077

0.3248

0.9104

0.1299

0.6079 0.5948

s

m

dm

1

240

2

480

3

720

12

29976.15

4

960

16

5 1200 6 1440

9 2160

eq/cm s

kWh/m

A/dm

36

29937.07

1285.66

771.99

7156.78

3594.46

4.899E-03

4.608E-08

0.9077

0.3572

0.9003

0.1430

10

2400

40

29931.53

1223.98

734.95

6918.72

3594.72

4.670E-03

4.392E-08

0.9077

0.3880

0.8905

0.1556

0.5820

11

2640

44

29926.25

1165.23

699.67

6689.79

3594.96

4.449E-03

4.185E-08

0.9077

0.4174

0.8809

0.1675

0.5695

12

2880

48

29921.20

1109.30

666.09

6469.65

3595.19

4.237E-03

3.985E-08

0.9077

0.4454

0.8715

0.1788

0.5574

13

3120

52

29916.39

1056.04

634.10

6257.94

3595.41

4.033E-03

3.793E-08

0.9076

0.4720

0.8623

0.1896

0.5455

14

3360

56

29911.81

1005.32

603.65

6054.37

3595.63

3.837E-03

3.608E-08

0.9076

0.4973

0.8532

0.1999

0.5340

15

3600

60

29907.43

957.03

574.65

5858.61

3595.84

3.648E-03

3.430E-08

0.9076

0.5215

0.8444

0.2097

0.5227

16

3840

64

29903.27

911.05

547.05

5670.38

3596.03

3.467E-03

3.259E-08

0.9076

0.5445

0.8358

0.2190

0.5117

17

4080

68

29899.30

867.27

520.76

5489.39

3596.23

3.292E-03

3.095E-08

0.9076

0.5664

0.8273

0.2279

0.5010

18

4320

72

29895.53

825.59

495.73

5315.35

3596.41

3.125E-03

2.937E-08

0.9076

0.5872

0.8190

0.2363

0.4905

19

4560

76

29891.94

785.90

471.90

5148.01

3596.59

2.963E-03

2.785E-08

0.9075

0.6070

0.8109

0.2442

0.4803

20

4800

80

29888.52

748.12

449.21

4987.11

3596.76

2.808E-03

2.639E-08

0.9075

0.6259

0.8029

0.2518

0.4703

21

5040

84

29885.28

712.14

427.61

4832.40

3596.92

2.659E-03

2.499E-08

0.9075

0.6439

0.7951

0.2589

0.4606

22

5280

88

29882.20

677.89

407.05

4683.66

3597.08

2.516E-03

2.364E-08

0.9075

0.6611

0.7875

0.2657

0.4511

23

5520

92

29879.28

645.29

387.47

4540.64

3597.23

2.378E-03

2.234E-08

0.9074

0.6774

0.7800

0.2721

0.4418

24

5760

96

29876.50

614.24

368.83

4403.14

3597.37

2.245E-03

2.109E-08

0.9074

0.6929

0.7726

0.2781

0.4327

25

6000 100

29873.88

584.69

351.08

4270.93

3597.51

2.118E-03

1.989E-08

0.9074

0.7077

0.7654

0.2838

0.4239

26

6240 104

29871.39

556.56

334.19

4143.83

3597.65

1.995E-03

1.874E-08

0.9074

0.7217

0.7583

0.2892

0.4153

27

6480 108

29869.04

529.77

318.11

4021.62

3597.78

1.878E-03

1.763E-08

0.9074

0.7351

0.7514

0.2942

0.4068

28

6720 112

29866.81

504.28

302.80

3904.14

3597.90

1.764E-03

1.657E-08

0.9073

0.7479

0.7446

0.2990

0.3986

29

6960 116

29864.71

480.00

288.22

3791.18

3598.02

1.656E-03

1.554E-08

0.9073

0.7600

0.7379

0.3034

0.3906

30

7200 120

29862.73

456.90

274.35

3682.59

3598.13

1.551E-03

1.456E-08

0.9073

0.7716

0.7313

0.3076

0.3827

31

7440 124

29860.86

434.90

261.14

3578.19

3598.24

1.450E-03

1.361E-08

0.9073

0.7826

0.7248

0.3115

0.3751

32

7680 128

29859.11

413.96

248.57

3477.82

3598.35

1.354E-03

1.270E-08

0.9072

0.7930

0.7185

0.3151

0.3676

33

7920 132

29857.46

394.03

236.60

3381.32

3598.45

1.261E-03

1.182E-08

0.9072

0.8030

0.7123

0.3185

0.3603

34

8160 136

29855.91

375.05

225.20

3288.56

3598.55

1.171E-03

1.098E-08

0.9072

0.8125

0.7061

0.3217

0.3531

Colored table entries show key variables operated in a computer simulation.

E and I/S calculated by putting C 0 = 2000 mg/dm 3, Vcell = 0.4 V/pair and Q0 = 30 m 3. The plots at C′junc = 400 and 2000 mg/dm 3 correspond to N and E values in the one-stage operation integrated with

NI and NII alone and these values are less than the values in the two-stage operation; NI + NII, EI + EII. From cell pair number and energy consumption, the multi-stage operation is assumed to be

Y. Tanaka / Desalination 320 (2013) 118–133

3200

20000

2800

18000 0.6 16000

127

0.5

2400 14000

C"out (mg/dm3)

C'in (mg/dm3)

0.5

2000 0.6 0.4

1600 0.3

1200

10000 8000 6000

Vcell = 0.2 V/pair

800

0.4

12000

0.2 V/pair

4000 400 0

0.3

2000 0

50

100

150

200

250

300

0

350

0

50

100

150

200

250

300

350

t (min)

t (min) Fig. 15. Salt concentration at the inlets of desalting cells. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

Fig. 17. Salt concentration at the outlets of concentrating cells. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

disadvantageous compared to the single-stage operation. N and E values in the one-stage batch operation described above are computed in Fig. 23 (N = 635.4 pairs) and Fig. 24 (E = 0.2993 kW h/m 3) at C 0 (C′in,0) = 2000 mg/dm3 and Vcell = 0.4 V/pair in Section 6.1. I/Saverage,I is larger than I/Saverage.II. Desalination is advanced progressively in the first stage and purification is advanced profoundly in the second stage to produce drinking water.

the normal distribution of the solution velocity ratio σ = 0.1 which functions as a safety factor for operating the electrodialyzer stably [30]. In order to evaluate the real limiting current density of the electrodialyzer (I/S)lim,real, (I/S)lim is plotted against average current density I/S taking Vcell, C 0 and salt concentration at the inlets of desalting cells C′in as parameters (Fig. 27). (I/S)lim,real is determined from the intersections between the plots and the I/S = (I/S)lim line drawn in Fig. 27. From the intersection, the relationship between C′in and (I/S)lim,real is introduced as shown in Fig. 28. (I/S)lim,real is computed on the basis of the electrodialyzer specifications defined in Table 1 and it is expressed by the function of C′in (Fig. 28). This phenomenon suggests that the limiting current density is the performance of the electrodialyzer and it is not the performance of the batch process.

7. Limiting current density The influence of temperature T on the limiting current density (I/S)lim is discussed in Appendix A. (I/S)lim is calculated at T = 25 °C in this section. (I/S)lim and average current density I/S are plotted against C′in at C 0 = 3000 mg/dm 3 in Fig. 26. Almost the same plots are obtained at C 0 = 2000 and 1000 mg/dm 3. I/S is recognized to be less than (I/S)lim; I/S b (I/S)lim, so the electrodialysis is assumed to be operated stably. Linear velocities in desalting cells are not uniform between the cells and are expressed by the normal distribution [19]. The above (I/S)lim is computed assuming that the standard deviation of

8. Conclusion The electrodialysis program of a batch electrodialysis program is revised using the instantaneous open/shut solution feed operation and applying the continuous program. The revised program is applicable to

30100

1.6 1.4

0.5

1.2 1.0

30000 29950

0.4

Q (dm3)

I/S (A/dm2)

30050

0.6

0.8 0.3

29900

0.6

29850

0.4

29800

0.2 V/pair 0.6

0.2 0.0

29750

0.2 V/pair

0

50

100

150

200

250

300

350

t (min) Fig. 16. Current density. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

29700

0

50

100

0.4

0.5

150

0.3

200

250

300

350

t (min) Fig. 18. Solution volume in the circulation tank. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

128

Y. Tanaka / Desalination 320 (2013) 118–133

1.0

1.0

0.9

0.9

0.8

0.6 0.5

0.3

0.4

0.2 V/pair

0.8

0.6

0.7

0.7

0.6

0.6 0.4

0.5

α

E (kWh/m3)

0.5

0.4

0.4

0.3

Vcell = 0.2 V/pair

0.3

0.5

0.3

0.2

0.2

0.1

0.1

0.0

0

50

100

150

200

250

300

0.0

350

0

50

100

150

200

250

300

350

t (min)

t (min) Fig. 19. Energy consumption. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

Fig. 21. Desalting ratio. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

discuss the performance a batch electrodialysis process for saline water desalination.

p

Nomenclatures a flow-pass thickness in a desalting and a concentrating cell (cm) b flow-pass width in a desalting and a concentrating cell (cm) C salt concentration (mg dm −3) 0 salt concentration of a solution supplied into the circulation C tank at first (mg dm −3) E energy consumption (kW h m −3) F Faraday constant (As eq −1) i current density (A cm −2, A dm −2) I electric current (A) I/S average current density (A cm −2, A dm −2) ion flux across a membrane pair (eq cm −2 s −1) JS solution flux across a membrane pair (cm 3cm −2 s −1) JV l flow-pass length in a desalting and a concentrating cell (cm) n step number in an electrodialysis operation N number of desalting cells in a stack (electrodialyzer)

Q Q0

dimensionless distance from the inlet of a desalting cell at which current density equals to the average current density I/S of an electrodialyzer solution volume in the circulation tank (dm 3) solution volume in the circulation tank supplied into the circulation tank at first (dm 3) water recovery ion-exchange membrane area (cm 2) electrodialysis operation time (s) temperature (°C) linear velocity (cm s −1) voltage difference between electrodes (V) cell voltage (V pair −1) distance from the inlet of a desalting cell (cm)

Re S t T u V Vcell x

Greek letters α desalting ratio Δ interval time (min)

0.911

1.1

0.910

1.0

0.909

0.9

0.908

Re

η

0.8 0.907 0.6 0.5

0.906

0.4 0.3

0. 2

0.7

0.2 V/pair

,0 .3

0.6

,0 .4 , 0. 5

0.905

V/

0.5

0.904 0.903

,0 .6

0

50

100

150

200

250

300

350

t (min) Fig. 20. Current efficiency. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

0.4

0

50

100

150

200

250

300

pa

ir

350

400

450

t (min) Fig. 22. Water recovery. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. N = 300 pairs, Q0 = 30 m3.

Y. Tanaka / Desalination 320 (2013) 118–133

129

Table 4 Electrodialysis program of one-stage batch electrodialysis for drinking water production.

1. Fundamental parameters u' in u"

in

10.00

cm/s

α

0.9114

δ1

2.622E-03

1.00

cm/s

β1

2315

δ2

4.418E-05

flow-pass thickness

β2

45.84

ε1

2.481E-08

ε2

4.160E-10

a

0.05 cm

b

100 cm

flo-pass width

γ

1

10.000

l

100 cm

flow-pass length

γ

2

7.487E-03

T

25 oC

γ

3

1.345E-04

F

96485 As/eq

σ

standard deviation

0.10

2. Computation C

0

C' in,0 = C

0

Q0

2000.00

mg/dm

2000.00

mg/dm

30000.00

Vcell C'out,0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

N*

input

adjust N* to realize C' in = 400 mg/dm at t = 60 min.

3

mg/dm

9773.80

t

input

3

V/pair

1200.91

C" out,0

No.

dm

0.4

3

mg/dm

3

3

input

N +1

input

Δ t

636.4 cell 7624.80

3

Q" in

190.92

3

u' out,0

9.9769

Q' out,0

7607.22

C'out 3

C" out 3

concentrating cell no.

4 min

Q' in

C'in

Q

desalting cell no., cell pair no.

635.4 cell, pair

Q' out 3

3

input

interval

3

dm / Δ t min 3

dm /min cm/s 3

dm / Δ t min

η

JS

I/S 2

s

m

dm

mg/dm

mg/dm

mg/dm

dm / Δ t min

A/cm

240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600 3840 4080

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68

29982.42 29966.45 29951.96 29938.83 29926.95 29916.22 29906.54 29897.83 29890.01 29883.00 29876.74 29871.17 29866.23 29861.87 29858.04 29854.69 29851.80

1797.25 1614.93 1450.98 1303.59 1171.10 1052.01 944.98 848.80 762.38 684.73 614.97 552.30 496.00 445.43 400.00 359.20 322.56

1079.17 969.69 871.25 782.75 703.19 631.69 567.42 509.67 457.78 411.15 369.26 331.63 297.83 267.46 240.18 215.69 193.68

9053.52 8391.77 7783.88 7225.53 6712.75 6241.86 5809.48 5412.50 5048.05 4713.48 4406.37 4124.47 3865.74 3628.28 3410.35 3210.35 3026.82

7608.83 7610.31 7611.67 7612.92 7614.07 7615.12 7616.09 7616.98 7617.79 7618.54 7619.23 7619.86 7620.44 7620.97 7621.46 7621.90 7622.32

6.727E-03 6.090E-03 5.504E-03 4.966E-03 4.471E-03 4.018E-03 3.601E-03 3.218E-03 2.867E-03 2.545E-03 2.249E-03 1.977E-03 1.727E-03 1.499E-03 1.289E-03 1.096E-03 9.189E-04

α

Re

3

eq/cm s 6.330E-08 5.729E-08 5.177E-08 4.671E-08 4.205E-08 3.778E-08 3.386E-08 3.025E-08 2.695E-08 2.391E-08 2.112E-08 1.857E-08 1.622E-08 1.406E-08 1.208E-08 1.027E-08 8.604E-09

I/S average

E

2

kWh/m 0.9078 0.9078 0.9077 0.9077 0.9077 0.9076 0.9076 0.9075 0.9075 0.9074 0.9074 0.9073 0.9073 0.9072 0.9072 0.9071 0.9071

0.1014 0.1925 0.2745 0.3482 0.4145 0.4740 0.5275 0.5756 0.6188 0.6576 0.6925 0.7239 0.7520 0.7773 0.8000 0.8204 0.8387

0.9746 0.9505 0.9276 0.9057 0.8849 0.8651 0.8461 0.8280 0.8106 0.7940 0.7780 0.7627 0.7480 0.7339 0.7202 0.7071 0.6945

0.0380 0.0725 0.1036 0.1318 0.1572 0.1800 0.2004 0.2187 0.2350 0.2495 0.2623 0.2736 0.2834 0.2920 0.2993 0.3056 0.3108

2

A/dm 0.6727 0.6409 0.6107 0.5822 0.5552 0.5296 0.5054 0.4824 0.4607 0.4401 0.4205 0.4019 0.3843 0.3676 0.3516 0.3365 0.3221

Colored table entries show key variables operated in a computer simulation.

0.8

1600

0.7 30 0

0

1800

1400 30 0 0

1200

00

1000

E (kWh/m3)

20

N (pair)

0.6

800 C0

600

=1 0

400

00

mg

0.5 20

0.4

00

0.3 3

/dm

0.2

1 00 0 C =

3

mg/

dm

0

0.1

200 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Vcell (V/pair) Fig. 23. Cell pairs integrated in an electrodialyzer in a one-stage electrodialysis process. C′in (t = 60 min) = 400 mg/dm3, Q0 = 30 m3.

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Vcell (V/pair) Fig. 24. Energy consumption in a one-stage electrodialysis process. C′in (t = 60 min) = 400 mg/dm3, Q0 = 30 m3.

130

Y. Tanaka / Desalination 320 (2013) 118–133

Table 5 Electrodialysis program of two-stage batch electrodialysis for drinking water production. Total performance I/S total C

C'

2

0.3624

0 feed sol desalted sol

E

total

0.3163 kWh/m3

V cell

3

η

total

0.9070

C' in

3

α

total

0.8000

NI *

Re total

0.7147

adjust N I *

A/dm

2000.00

mg/dm

400.00

mg/dm 3

Q

0 feed sol

30.000

m

Q

desalted sol

29.858

m min

Δ t

3

4

C' in input

0.4 V/pair 282.565 pair to realize C' in

adjust 3

I

= 1000 mg/dm at t = 60 min in stage I 3

400.00 mg/dm

II

N II *

interval

input

1000.00 mg/dm3

I

369.58 pair

adjust 3

adjust N II * to realize C' in II = 400 mg/dm at t = 60 min in stage II N I + N II 652.145 1. Fundamental parameters u' in

10.00

cm/s

α

0.9114

δ1

2.622E-03

u" in

1.00

cm/s

β1

2315

δ2

4.418E-05

a

0.05

cm

flow-pass thickness

β2

45.84

ε1

2.481E-08

b

100

cm

flo-pass width

γ

1

10.000

ε2

4.160E-10

l

100

cm

flow-pass length

γ

2

7.487E-03

T

25

γ

3

1.345E-04

F

96485

σ

0.10

o

C

As/eq standard deviation

2. Computation Stage I C

0

C' in,0 = C Q

0

2000.00

mg/dm

3

input

N

II

2000.00

mg/dm

3

input

N

II

3

input

Q' in

30000.00

0

V cell

dm

0.4

Q' out,0

3382.96

mg/dm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

C' in

Q

240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600 3840 4080

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68

3

C' out 3

mg/dm 1909.87 1823.77 1741.52 1662.96 1587.92 1516.24 1447.78 1382.40 1319.95 1260.31 1203.34 1148.95 1096.99 1047.38 1000.00 954.76 911.55

C"out 3

3

dm /min cm/s 3

dm / Δ t min

Q'out 3

mg/dm 1146.79 1095.09 1045.71 998.54 953.48 910.44 869.33 830.07 792.57 756.76 722.56 689.89 658.70 628.91 600.46 573.29 547.35

3

dm / Δ t min

85.07

3

9773.80

dm 29992.18 29984.68 29977.48 29970.58 29963.96 29957.62 29951.54 29945.71 29940.13 29934.78 29929.67 29924.77 29920.08 29915.60 29911.32 29907.22 29903.31

concentrating cell no.

9.9769

C" out,0

m

desalting cell no., cell pair no.

cell

u'out,0

mg/dm

s

cell

283.565

3

1200.91

t

282.565 3390.78

Q" in

V/pair

C' out,0

No.

+1

2

dm /Δ t min 3383.28 3383.58 3383.88 3384.16 3384.44 3384.70 3384.95 3385.20 3385.43 3385.66 3385.88 3386.09 3386.30 3386.49 3386.68 3386.87 3387.04

η

JS

I/S

3

mg/dm 9455.49 9148.60 8852.73 8567.49 8292.51 8027.43 7771.89 7525.55 7288.10 7059.21 6838.58 6625.91 6420.92 6223.34 6032.91 5849.36 5672.46

A/cm 7.115E-03 6.819E-03 6.534E-03 6.259E-03 5.994E-03 5.738E-03 5.492E-03 5.255E-03 5.026E-03 4.805E-03 4.593E-03 4.388E-03 4.190E-03 4.000E-03 3.816E-03 3.639E-03 3.469E-03

α

Re

2

eq/cms 6.694E-08 6.416E-08 6.147E-08 5.889E-08 5.639E-08 5.399E-08 5.167E-08 4.943E-08 4.728E-08 4.520E-08 4.320E-08 4.127E-08 3.941E-08 3.761E-08 3.588E-08 3.422E-08 3.261E-08

I/S average

E 3

0.9078 0.9078 0.9078 0.9078 0.9078 0.9078 0.9078 0.9077 0.9077 0.9077 0.9077 0.9077 0.9077 0.9076 0.9076 0.9076 0.9076

0.0451 0.0881 0.1292 0.1685 0.2060 0.2419 0.2761 0.3088 0.3400 0.3698 0.3983 0.4255 0.4515 0.4763 0.5000 0.5226 0.5442

0.9885 0.9773 0.9664 0.9557 0.9452 0.9350 0.9249 0.9151 0.9056 0.8962 0.8870 0.8780 0.8692 0.8605 0.8521 0.8438 0.8356

kWh/m 0.0179 0.0350 0.0514 0.0672 0.0823 0.0967 0.1106 0.1238 0.1365 0.1486 0.1602 0.1713 0.1818 0.1919 0.2016 0.2108 0.2196

2

A/dm 0.7115 0.6967 0.6823 0.6682 0.6544 0.6410 0.6279 0.6151 0.6026 0.5904 0.5785 0.5668 0.5554 0.5443 0.5335 0.5229 0.5125

Stage II C

0

C'in,0 = C Q

0

1000.00

mg/dm

3

replace

N

1000.00

mg/dm

3

replace

N II +1

3

replace

29911.32

0

V cell

0.4

dm

3

Q'out,0

600.46

mg/dm

mg/dm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

s 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600 3840 4080

C'out

C'in

Q m 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68

3

dm 29905.96 29900.91 29896.17 29891.70 29887.51 29883.58 29879.89 29876.43 29873.20 29870.18 29867.36 29864.73 29862.29 29860.02 29857.92 29855.97 29854.18

3

mg/dm 940.82 885.13 832.73 783.41 737.01 693.35 652.26 613.61 577.24 543.02 510.82 480.53 452.04 425.22 400.00 376.27 353.95

concentrating cell no.

9.9879 cm/s 3

4429.60 dm / Δ t min C" out

3

mg/dm 564.92 531.48 500.02 470.41 442.54 416.33 391.66 368.45 346.61 326.06 306.73 288.54 271.43 255.33 240.18 225.93 212.53

desalting cell no., cell pair no.

370.58 cell 111.17 dm3/min

u'out,0

5032.91

369.58 cell, pair

4434.96 dm3/ Δ t min

Q' in

3

C" out,0 t

*

Q" in

V/pair

C'out,0

No.

II

Q'out 3

mg/dm 4792.48 4563.45 4345.28 4137.47 3939.51 3750.96 3571.37 3400.31 3237.39 3082.21 2934.42 2793.66 2659.61 2531.93 2410.34 2294.55 2184.27

Colored table entries show key variables operated in a computer simulation.

3

dm /Δ t min 4429.92 4430.21 4430.50 4430.77 4431.03 4431.27 4431.50 4431.73 4431.94 4432.14 4432.33 4432.52 4432.69 4432.86 4433.02 4433.17 4433.31

η

JS

I/S 2

A/cm 3.584E-03 3.364E-03 3.153E-03 2.953E-03 2.762E-03 2.581E-03 2.408E-03 2.243E-03 2.086E-03 1.936E-03 1.794E-03 1.658E-03 1.529E-03 1.406E-03 1.289E-03 1.177E-03 1.071E-03

α

Re

2

eq/cms 3.370E-08 3.162E-08 2.964E-08 2.776E-08 2.596E-08 2.425E-08 2.262E-08 2.107E-08 1.959E-08 1.818E-08 1.684E-08 1.556E-08 1.435E-08 1.319E-08 1.208E-08 1.103E-08 1.003E-08

I/S average

E 3

0.9072 0.9072 0.9071 0.9071 0.9070 0.9070 0.9069 0.9069 0.9068 0.9068 0.9067 0.9067 0.9066 0.9065 0.9065 0.9064 0.9064

0.0592 0.1149 0.1673 0.2166 0.2630 0.3067 0.3477 0.3864 0.4228 0.4570 0.4892 0.5195 0.5480 0.5748 0.6000 0.6237 0.6461

0.9852 0.9708 0.9568 0.9433 0.9301 0.9173 0.9048 0.8927 0.8809 0.8694 0.8582 0.8473 0.8367 0.8263 0.8162 0.8063 0.7967

kWh/m 0.0118 0.0229 0.0333 0.0430 0.0522 0.0607 0.0686 0.0760 0.0829 0.0893 0.0952 0.1007 0.1058 0.1104 0.1147 0.1186 0.1221

2

A/dm 0.3584 0.3474 0.3367 0.3264 0.3163 0.3066 0.2972 0.2881 0.2793 0.2707 0.2624 0.2543 0.2465 0.2390 0.2316 0.2245 0.2176

Y. Tanaka / Desalination 320 (2013) 118–133

0.8

800 700

2.0 1.8

0.7

N I + NII

131

0.2

1.6

0.5

N II

400

0.4

I/S total average E I + E II

0.3

300 200

I/S

ra a ve

I

g eI

0.2 EI

E II

100 0 400

600

800

1000

1200

1600

1800

m

)li

1.2

2000

1.0 1500

0.8 0.6

C'in =1000 mg/dm3 800

0.4

600 500 400

0.2

0.1

1400

/S

1.4

(I

I

=

e

S

ag

3000

I/

er

(I/S)lim (A/dm2)

av

I

N (pairs)

I /S

N

500

I/S (A/dm2), E (kWh/m3)

0.6

600

0.6 V/pair

0.5

0.4

0.3

0.0 0.0

0.0 2000

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

I/S (A/dm2)

C'junc = C'in I (mg/dm3)

Fig. 27. Determination of (I/S)lim,real. C0 = 1000 (○), 2000 (Δ), 3000 (□) mg/dm3. Fig. 25. Cell pair number, current density and energy consumption in a two-stage electrodialysis process. C0 = 2000 mg/dm3, Q0 = 300 m3, Vcell = 0.4 V/pair. Operating time in each stage = 1 h. Salt concentration of a desalted solution = 400 mg/dm3.

Appendix A. Influence of temperature on limiting current density A.1. Experiment

ζ η

current density non-uniformity coefficient (= iout/(I/S)) current efficiency

Subscripts A anion-exchange membrane in inlet of a desalting and a concentrating cell or a slot K cation-exchange membrane lim limiting current density n step number in a electrodialysis operation out outlet of a desalting and a concentrating cell or a slot p point x = pl distant from the inlet of a desalting cell

Limiting current density equation of an ion-exchange membrane ilim developed in the previous investigation [20] was defined at temperature T = 25 °C. In order to evaluate ilim at T (°C), k = (ilim / C)T / (ilim / C)25 was measured in the electrodialyzer reported in the previous article [20]. The electrodialyzer is incorporated with Neosepta CL-25 T/ ACH-45 T membranes and diagonal net spacers. A NaCl solution is supplied to the desalting cell changing temperature T, linear velocity u and NaCl concentration C as follows. T ¼ 10; 15; 20; 25; 30; 35; 40BC u ¼ 6; 9; 12cm=s

Superscripts ′ desalting cell ″ concentrating cell * control key # desalting cell in which solution velocity becomes the least 0 at t = 0

3

C ¼ 0:01−0:05eq=dm

Passing an electric current I and changing the current density incrementally, the limiting current density of CL-25 T cation-exchange membrane ilim was measured from the inflection of V/I versus 1/I plot.

2.0

2.0

1.8

1.8 (I/S)lim

1.6

0.6

1.4

(I/S)lim,real (A/dm2)

I/S, (I/S)lim (A/dm2)

1.6

0.5

1.2 0.4 I/S

1.0 0.3

0.8

Vcell = 0.2 V/pair

0.6

1.4 1.2 1.0 0.8 0.6

0.4

0.4

0.2

0.2

0.0

0

1000

2000

3000

4000

C'in (mg/dm3) Fig. 26. Limiting current density. Filled: (I/S)lim, Open; I/S.

0.0

0

500

1000

1500

2000

2500

3000

C'in (mg/dm3) Fig. 28. Relationship between C′in and (I/S)lim,real.

3500

132

Y. Tanaka / Desalination 320 (2013) 118–133

2.0 1.8

Thus the limiting current density equation at T (°C) is introduced from Eq. (A1) and the equation at 25 °C developed in the previous investigation [20] as follows.

k = 0.5950 + 0.2731(T/25) + 0.1310(T/25)2

k = (ilim/C'out)T/(ilim/C'out)25

1.6 1.4


 n o 2 2  0  0n þn u0 m1 þ m2 uout C out1 2 out i lim A=cm ¼ l1 þ l2 ðT=25Þ þ l3 ðT=25Þ

1.2

m1 ¼ 83:50; m2 ¼ 24:00; n1 ¼ 0:7846; n2 ¼ 8:612  10−3

1.0

Limiting current density of an electrodialyzer (I/S)lim is introduced # # (u′out is replaced with u′in) and C′out = C′out as; by putting u′in = u′in

0.8


       2 m þ m u0#
 0# 1 2 in I i lim T T 0# n1 þn2 uin þ l3 ¼ ¼ l1 þ l2 C out S lim ζ out 25 25 ζ out

0.6 0.4

ðA3Þ

0.2 0.0 0.0

ðA2Þ

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

T/25 oC Fig. A1. Temperature dependence of limiting current density. Neosepta CL-25 T.

in which the superscript # denotes the least value in desalting cells integrated in a stack. # is also given by the folThe relationship between (I/S)lim and C′out lowing equation.   
a
I 0# 0 0# ¼ uin C in −C out S lim λl

A.2. Results and discussions k = (ilim/C)T/(ilim/C)25 is plotted against T/25 and shown in Fig. A1 and Eq. (A1). 2

k ¼ l1 þ l2 ðT=25Þ þ l3 ðT=25Þ l1 ¼ 0:5950; l2 ¼ 0:2731; l3 ¼ 0:1310

ðA1Þ

ðA4Þ

Putting Eq. (A3) = Eq. (A4);
 0#     2 C 0# n1 þn2 uin out T T Z 1 ¼ l 1 þ l2 þ l3 25 25 C 0in −C 0# out

Fig. A2. Simulation of limiting current density.

ðA5Þ

Y. Tanaka / Desalination 320 (2013) 118–133

 Z2 ¼

aζ out λl

Z1 ¼ Z2



u0# in m1 þ

m2 u0# in

! ðA6Þ

[11]

ðA7Þ [12]

in which λ (= (tK + tA − 1)/F; tK and tA are transport numbers of a cation- and an anion-exchange membrane) is the overall transport number of an ion-exchange membrane pair. Before the computation of the limiting current density, the continuous program in Fig. 4 (Section 4.1) is achieved inputting control keys and iterating the calculation at the decision points 1, 2 and 3. Next, the computation of the limiting current density (Appendix A) is achieved using the program in Fig. A2 and iterating the computation at the decision point (Z1 = Z2). Both computations are arranged in series and carried out simultaneously within about 10 min. They are the trial-error-calculations using only MS Excel software and inputting main parameters given in Figs. 4 and A2. The explanation of the program is described definitely in the previous article [29].

[13] [14] [15] [16] [17]

[18] [19] [20] [21]

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