Simulation of enhanced power generation by reverse electrodialysis stack module in serial configuration

Desalination 318 (2013) 79–87

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Simulation of enhanced power generation by reverse electrodialysis stack module in serial configuration Kwang Seok Kim a, Won Ryoo b, Myung-Suk Chun a,⁎, Gui-Yung Chung b a b

Complex Fluids Laboratory, National Agenda Research Division, Korea Institute of Science and Technology (KIST), Seongbuk-gu, Seoul 136-791, Republic of Korea Department of Chemical Engineering, Hong-Ik University, Mapo-gu, Seoul 121-791, Republic of Korea

H I G H L I G H T S • • • • •

Simulations of reverse electrodialysis (RED) stack module in serial configuration Orthogonal collocation on finite element solution to convection–diffusion problem Unique implication of Legendre polynomial and Gaussian quadrature integration Performance evaluations and validity of the module from power and energy densities Adverse ion flux can occur with condition of extremely low Peclet number.

a r t i c l e

i n f o

Article history: Received 4 February 2013 Received in revised form 25 March 2013 Accepted 26 March 2013 Available online 20 April 2013 Keywords: Reverse electrodialysis Serial stack module Convection–diffusion Power density Energy density

a b s t r a c t By extending the theoretical model previously formulated for the single unit problem, we fully analyze the performance of the reverse electrodialysis stack module aiming for electric power generation. In our prototype, each single unit is assembled in serial, where the mixing is allowed in connecting channels to exclude the ionic polarization. The numerical algorithm accompanying the orthogonal collocation on finite element method is applied with Legendre polynomial and Gaussian quadrature integrations to predict the ion concentration profile in each compartment, power, energy, and corresponding current densities. As the number of units increases from 1 to 8, the maximum power and the maximum current densities decrease from 9 to 1 mW/m2 and from 3.6 to 1.3 A/m2, respectively, but the maximum energy density increases from 1.5 to 4 mW h/m3. Pursuing the justification of the validity of our RED stack module, we determine the unique compartment thickness that makes each density maximum for the specified number of units. Power and current densities increase with increasing characteristic fluid velocity, while the energy density maintains constant (ca. 0.36 mWh/m3) prior to monotonic decreasing at Pe > 70. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Reverse electrodialysis (RED), or sometimes called electrodialysis reversal, is one of the reverse desalination processes, and recently noticed that electrical energy can be produced through the salinity gradients [1–10]. This principle can be also applied in reactivation of filtration membranes by removing fouling deposits [11]. The initial attempt to materialize the thermodynamic concept can go back to 1950s [12], and the success has been given consistent spotlights due to its potential to be a good alternative and sustainable energy source. Nowadays, the advancement of the RED process results in the hybrid system with the seawater desalination units [13], the pressure retarded osmosis [14], and the microbial power cell [15]. ⁎ Corresponding author. Tel.: +82 2 958 5363. E-mail address: [email protected] (M.-S. Chun). 0011-9164/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2013.03.023

There are many efforts to maximize the performance and efficiency of the energy harvesting in the RED by avoiding power loss [5] or by expanding module dimension [6,7]. It should be pointed out that power density is a key performance to evaluate the RED as an established process. Therefore, further studies are required to enhance the power density, inter alia, by designing more innovative stack system [9,10]. For enhancing the power generation, the effect of time-periodic pulsatile counter-current flow has been investigated by simulating the single RED unit [16]. Due to the polarization of the ions near the ion-exchange membrane surface, the voltage drop was observed before the flow gets into the steady state, which is consistent with the experimental results [2]. In this study, we develop the numerical framework capable of evaluating of performance for the RED stack module in serial configuration and with mixing between two consecutive compartments, which were not elaborated in previous studies. The module dimension is mainly

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decided by the number of units and the compartment thickness. The ion transport within compartment is analyzed by using the convection–diffusion model with the slit laminar counter-current flow. The orthogonal collocation on finite element (OCFE) method applied here to generate the mesh grid is known to be superior to the finite difference method in the aspects of the convergence, the numerical stability, and the computational load [17]. Special benefit lies on the settlement of the stability problem that is stemmed from the discontinuity of the solution. As the process parameters, we change the values of the number of units, the compartment thicknesses, and the characteristic fluid velocities to examine the behaviors of the power density, the energy density, and the current density. Although the adoption of the serial connection and the mixing should be a challenging issue, its full practical consideration is a beyond the scope of this study.

compartment. The color arrows denote connecting channels between the compartments, where mixing is accomplished for the reduction of the ionic polarization. The saline water is supplied to the bottom of the saline compartment in the unit 1, and leaves it through the top, losing sodium cations and chlorine anions to the adjacent fresh compartments. Fresh water can be supplied to the module in two ways: being introduced to the unit 1 (as illustrated in Fig. 1) and to the last unit (not shown here). In the former case, the inlet is placed at the top of the compartment to form a counter-current flow. In the latter case, however, the position of the fresh inlet depends on the number of the units N, i.e., if N is even, the inlet is at the bottom; otherwise, it is at the top. This study has mainly focused on the computations for the case where the fresh water is supplied to the unit 1. The other case is addressed briefly for comparison.

2. Theoretical considerations

2.2. Transport analysis in RED unit

2.1. The RED stack module in serial configuration

The saline and fresh streams switch their directions (upward downward) in the next compartments. To avoid any confusion, we set the Cartesian coordinates such that the origin is placed at the bottom in the middle of the unit, the x-axis is toward the fresh compartment and the z-axis toward the top (see Fig. 2), regardless of the flow directions. Consequently, the y-axis goes down the paper. According to a reasonable assumption of neglecting membrane thickness δM [16], the computational domain is defined as −δC b x b δC and 0 b z b L. Since the diffusion of anion (e.g., Cl −) takes place around 1.5 times faster than that of cation (e.g., Na+) [18], the transport of cation is the rate-controlling mechanism and can consistently represent the entire species balance. Thus, the “ion” hereafter in this paper refers to the

The stack sequence of the conventional RED module can be easily found in the literatures [5–8]. The prototype of the module studied here consists of multiple sheets of cation-exchange membranes (CEMs) and anion-exchange membranes (AEMs) with the length L and the width W, denoted in Fig. 1. They are placed in the alternating order, and both ends of the module are equipped with the anode and the cathode plates. In Fig. 2, the compartment thickness δC is much less than the length and the width, indicating a configuration of the slit channel. As shown in Fig. 1, one unit is designed to comprise a pair of compartments: one is the saline compartment and the other is the fresh

↔

Fig. 1. Schematic diagram of the serial RED stack module consisting of saline and fresh compartments, ion exchange membranes (AEMs and CEMs), connection channels, and voltage output. The M in the square indicates mixing.

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81

Fig. 2. Dashed region in Fig. 1, illustrating the transport mechanism of ions and the coordinates.

cation, otherwise specified. The convection–diffusion equation governing the ion concentration profile is provided as [19]

 j cf 

x¼0þ



 j ¼ cs 

x¼0−

þ

KD DM =δM



∂cjs ∂x

! :

ð5Þ

x¼0−

j

∂ci j j 2 j þ vi ⋅∇ci ¼ D∇ ci : ∂t

ð1Þ

Here, cij and v ij are the ion concentration and the fluid velocity vector, respectively, in the compartment i at the j-th unit (i = s for saline or i = f for fresh). The diffusivity of ion D (=1.33 × 10−5 cm2/s) is set constant. Applying the steady-state transport and the slit geometry shown in Fig. 2 to Eq. (1), the governing equation can be obtained as

D

∂2 cji ∂x

2

j

−vi

∂cji ¼0 ∂z

ð2Þ

where vij is the z component of the vector v ij. It is hypothesized in Eq. (2) that the diffusion is dominant in the lateral (x direction) ion transport within compartment and the convection is governing the streamwise (z direction) transport. There is no flux of ion at the surface of the AEM (i.e., x = − δC or δC), therefore, the Neumann condition is assigned as ∂cjs ∂x

! ¼ x¼−δC

∂cjf

!

∂x

¼ 0:

ð3Þ

x¼δC

We need to impose two boundary conditions at both sides of the CEM surface (i.e., x = 0 − for left side and 0 + for right side). Each condition can be derived by applying ion flux continuity and on the basis of the pseudo-steady transfer across the CEM, as follows ∂cjs ∂x

! ¼ x¼0−

∂cjf ∂x

! ; x¼0þ

ð4Þ

Here, DM is the diffusivity of ion through the CEM, and K is the partition coefficient of ion between water–CEM, which is assumed 0.1 [16]. Due to the mixing allowed in the connecting channel, the ion concentration at each entrance is uniform, corresponding to the mean value at the exit of the previous compartment. With these identities, we obtain each ion concentration that is addressed in the saline compartments and in the fresh compartments, respectively  j cs 

−1

z¼0;L

 j cf 

¼ δC ∫

−1

z¼0;L

δ

cs

 

z¼0;L

j−1 

¼ δC ∫ C cf 0

 

j−1 

0

−δC

z¼0;L

dx;

dx:

ð6Þ

ð7Þ

The velocity profile in the compartment can be independently obtained by solving the Navier–Stokes momentum equation for an incompressible laminar flow in the slit channel with no-slip boundary conditions. One obtains each form of the parabolic type velocity in the j-th compartments for saline and fresh water, respectively h i j j 2 vs ðxÞ ¼ ð−1Þ v0 ðx=δC Þ þ ðx=δC Þ ; j

j−1

vf ðxÞ ¼ ð−1Þ

h i 2 v0 ðx=δC Þ −ðx=δC Þ :

ð8Þ

ð9Þ

Here, the characteristic fluid velocity v0 is equivalent to (−ΔpδC 2) / (2 μL) with the pressure drop Δp between inlet and outlet of each compartment and the fluid viscosity μ.

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Eqs. (2)–(9) can be rewritten in the dimensionless form with the dimensionless variables, X = x/δC, Z = z/L, C ij = c ij/c0, and V ij = v ij/v0, where c0 is the saline feed concentration. The resulting equations are: ∂2 C ji ∂X 2

j

−PeV i

j

∂C s ∂X ∂C js ∂X  j Cf   j Cs

∂C ji ¼ 0; ∂Z

! ¼ !

∂C jf

¼  j ¼ Cs 

Z¼0;1

¼ 0;

;

¼ ∫ Cf 0

∂C s ∂X

 

Z¼0;1

 

ð21Þ

! ;

    j j  αRTδC ln cs x¼0 =cf x¼0   :     ΛF 2 DM cjs  −cjf 

 RM ¼

Z¼0;1

ð14Þ

dX;

ð15Þ

dX;

ð23Þ

x¼0

Here, the first group in the right-hand side is constant. Assuming that cf ≈ kcs and the proportionality k is less than 1 at the CEM surfaces, the logarithmic term becomes constant, and thus the membrane resistance is inversely proportional to cs, −1

RM ∝ c s :

  j j 2 V s ¼ ð−1Þ X þ X ;

ð16Þ

  j j−1 2 V f ¼ ð−1Þ X −X :

ð17Þ

Here, the Peclet number Pe = (v0/L)/(D/δC 2), and G = (KD/δC)/(DM/ δM). The Pe is an important parameter that measures the relative contribution of convection over diffusion in the z direction. 2.3. Electric power output The ion transfer from one compartment to the adjacent compartment across the membrane creates an electric circuit. The opencircuit voltage, the current and the power of the circuit depend on the concentration difference between the compartments. The equation for the open-circuit voltage ϕ induced across the membrane can be derived from the Nernst–Planck (N–P) equation with the permselectivity α, such that ϕðzÞ ¼ α

RT cj ln s x¼0 cf jx¼0 ΛF

!

kB T cj ln s x¼0 : cf jx¼0 Λe

2.4. Power density and energy density in stack module The purpose of the RED stack module in serial configuration is to maximize the power generation, which comes with the economical aspects, such as the installation, operation, and maintenance costs. The present study stresses on the physical point of view. The cost involved here, thus, is related to the overall working area (=NWL). With the above constraint, the power density PD is defined as the total amount of power ΣP j divided by the overall working area PD ¼ −

ð25Þ

Table 1 Elements of 7 × 7 matrices for the first (Aij) and the second (Bij) derivatives used in the OCFE with the 5th-order Legendre polynomial. i

ð20Þ

j 1

2

3

4

5

6

7

Aij

1 2 3 4 5 6 7

−31.0 −13.1 3.73 −1.88 1.10 −0.64 1.00

34.7 10.1 −7.63 3.37 −1.94 1.10 −1.71

5.03 3.88 1.52 −4.00 1.90 −0.99 1.50

2.13 −1.50 3.41 0.00 −3.41 1.50 −2.13

−1.50 0.99 −1.90 4.00 −1.52 −3.88 −5.03

1.71 −1.10 1.94 −3.37 7.63 −10.1 −34.7

−1.0 0.64 −1.10 1.88 −3.73 13.1 31.0

Bij

1 2 3 4 5 6 7

480 292 −21.0 −7.50 −6.30 14.4 60.0

−671 −390 59.8 −14.9 11.3 −25.0 −102

268 120 −67.0 30.0 −12.5 22.8 −89.7

−123 −36.0 35.7 −45.3 35.7 −36.0 −123

−89.7 22.8 −12.5 30.0 −67.0 120 268

−102 −25.0 11.3 −14.9 59.8 −390 −671

60.0 14.4 −6.30 −7.50 −21.0 292 480

ð19Þ

Multiplying Eq. (19) by F and the specified membrane area (WΔz) results in the electric current   FD W    IðzÞ ¼ − M Δz: cf  −cs  x¼0 x¼0 δM

      N   DM RT X L j j j j ∫ cf  −cs  ln cs  =cf  dz: x¼0 x¼0 x¼0 x¼0 NLδM j¼1 0

ð18Þ

Here, F is the Faraday constant (=96,485 C/mol), R is the universal gas constant (=8.314 J/mol K), kB is the Boltzmann constant (=1.38 × 10−23 J/K), e is the elementary charge (=1.6 × 10−19 C), T is the absolute temperature, and Λ is the valence of ion (=+1 for Na+). The permselectivity α is assumed 1 based on the literature [20,21], which means that the membrane perfectly rejects the penetrated anions. The ion flux across the CEM, JC(z), is given as   D    : J C ðzÞ ¼ − M cf  −cs  x¼0 x¼0 δM

ð24Þ

It is evident that the increasing rate of RM becomes higher at very low ion concentration. The membrane resistance was experimentally measured as a function of NaCl concentration [22], and the previous results (cf., Fig. 4 therein) presented a similar trend expected by Eq. (24).

!

¼α

ð22Þ

It is worth noticing that, in view of Ohm's law, dividing Eq. (18) by Eq. (20) with multiplication of WΔz and rearranging it lead to the resistance of the CEM (Ω cm 2)

ð13Þ

X¼0−

      DM WRT L  j  j j j ∫ cf  −cs  ln cs  =cf  dz: 0 x¼0 x¼0 x¼0 x¼0 δM

x¼0

1 j−1 

Z¼0;1

ð12Þ

X¼0þ

j−1 

−1

L

0

j

¼ ∫ Cs

j

P ≡∫ P ðzÞdz ¼ −

!

þG

X¼0−

ð11Þ

X¼1

∂X

X¼0−

0

 j Cf 

∂X

!   DM WRT   cs jx¼0  P ðzÞ ≡ ϕI ¼ − Δz: cf  −cs  ln x¼0 x¼0 cf jx¼0 δM

The total power generated in the j-th unit is computed by integrating Eq. (21) with respect to z:

j! ∂C f

X¼−1

X¼0þ

ð10Þ

If Eq. (20) is multiplied by only F, one can evaluate the current density ID (A/m 2). Then, the power P is the product of ϕ and I:

K.S. Kim et al. / Desalination 318 (2013) 79–87 Table 2 Properties of the ion exchange membranes considered in this study. Properties

Type Ionic group Exchange capacity (meq/g) Resistance, RM (Ω cm2)a Permselectivity, α Chemical stabilityb Temperature (°C) Thickness, δM (mm) a b

Neosepta® CMX

AMX

Strongly acidic, cation exchange (Na) Sulfonic acid 1.62

Strongly basic, anion exchange (Cl) Quaternary ammonium 1.25

2.9

2.4

0.99 0.91 Not attacked by 3–20% NaCl solution 40 40 0.16 0.13

N         DM RT X L j j j j ln cs  =cf  dz: ∫ cf  −cs  x¼0 x¼0 x¼0 x¼0 δC v0 δM j¼1 0

C≡Q ⋅a

ð26Þ

It is useful to understand that the energy density is the energy generation (Wh) per unit volume of fresh water. 3. Methods and input parameters for computation 3.1. Computation scheme Solving Eq. (10) subjected to Eqs. (11)–(17), the OCFE method is applied into the generation of the 2-dimensional mesh grids [17] and the approximation of the differential or integral terms in Eqs. (10)–(15). In the simulation, the use of 3 finite elements and 7 collocation points provides the sufficient convergence in the concentration profiles. The set of the collocation points is ΦC = {Φn} with Φ1 = 10 −20, Φ2 = 0.0469, Φ3 = 0.2308, Φ4 = 0.5, Φ5 = 0.7692, Φ6 = 0.9531, and Φ7 = 1.

ð27Þ

where the 7 × 7 square matrix Q = {Qnm} = {Φnm−1} and a = {an} is the row matrix of coefficients. Thus, the 1st and the 2nd derivatives of C can be given by 0

ð28Þ

00

ð29Þ

C ≡Θ⋅a

Another way to access the performance is to calculate the energy density ED. To get this, we divide the total amount of power ΣP j by the characteristic flow rate of the fresh feed (=δCWv0), such that ED ¼ −

Note that Φ2 to Φ6 are the zeros of the 5th-order Legendre polynomial [23] that is compressed by a factor 2 and then shifted to the right by 1/2. In the OCFE, the column matrix for the ion concentration C is approximated by

C ≡ Γ ⋅ a;

In 0.5 N NaCl at 25 °C. From the manufacturer's catalog.

83

where Γ = {Γ nm } = {(m −1)Φ nm−2 } and Θ = {Θ nm } = {(m−1) (m−2)Φnm−3} with 7 × 7 square matrices. Multiplying both sides of Eq. (27) by the inverse matrix Q−1 and substituting the resulting equation into Eqs. (28) and (29) lead to C′ = (Γ · Q −1) · C = A · C and C″ = (Θ · Q −1) · C = B · C. Here, the elements of the 7 × 7 matrices A and B are given in Table 1. The integration of the concentration in each unit is accomplished using the Gaussian quadrature [17], such that 1

∫ C dx ¼ ω ⋅ C 0

ð30Þ

where each element of the weighting row matrix ω (cf., ω1 = 0, ω2 = 0.118, ω3 = 0.239, ω4 = 0.284, ω5 = 0.239, ω6 = 0.118, and ω7 = 0) is multiplied with C. The algorithm for solving and evaluating the equations is programmed in Mathematica (Wolfram Research, Inc., IL), and run on the 2.8 GHz Intel Core i5 CPU. 3.2. Experimental parameters and condition For simulations in this study, the Neosepta® CMX and AMX membranes (Tokurama Soda Inc., Japan) are considered, which are widely used in electrodialysis processes and are designed to have high mechanical strength. Table 2 summarizes their detailed characteristics

Fig. 3. Ion concentration profiles in the RED stack module with N = 4, v0 = 1.0 cm/s, and δC = 0.015 cm.

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based on the previous report [22]. The membrane diffusivity of CMX is estimated as 1.94 × 10 −6 cm 2/s, from Eq. (23) and the membrane resistance. The ion transports induce the electric current via redox reactions at the electrodes. The electrodes made of titanium mesh coated with precious metal oxides are placed inside each endplate shown in Fig. 1, which are suitable as anode and cathode allowing current reversal. These electrode compartments are electrically connected to the galvanostat, and Ag/AgCl reference electrodes are used for the measurement of the stack voltage. The reference electrodes are placed within reservoirs that are connected to the electrode compartments via salt bridges. Detailed descriptions on the electrode system are available in the literature [3,8,10]. The working area of a single unit (=WL) in our prototype module is set 400 cm2 with L = W = 20 cm. We test the model cases in the range of δC from 10−3 to 0.5 cm and the condition of N = 1, 2, 4, and 8. The value of v0 is less than 5 cm/s, indicating a fact that the Reynolds number Re (=ρv0δC/μ) does not exceed about 250.

4. Results and discussion For the salinity of saline feed, we apply a 35 g/L NaCl aqueous solution, corresponding c0 = 0.599 mol/L when a complete dissociation is assumed. In fact, specifying c0 is not necessary to solve Eq. (10) because the transport properties do not depend on the ion concentration. Hence, the term “saline” is chosen to refer to the ion-donating compartment in this paper, which does not restrict the applicability of the current model within a specific range of c0 (e.g., brackish water, seawater, or brine). 4.1. Ion concentration profiles In Figs. 3 and 4, the dimensionless steady-state ion concentration in the compartments are demonstrated for N = 4 and v0 = 1.0 cm/s, where dark blue for 0 and red color for 1 represent the fresh feed and the saline feed, respectively. The white vertical dashed lines inserted between the adjacent compartments are either CEMs or AEMs, depending on their positions.

Fig. 4. Ion concentration profiles in the RED stack module with fresh water feeding into (a) the first unit and (b) the last unit, where N = 4, v0 = 1.0 cm/s, and δC = 0.15 cm.

K.S. Kim et al. / Desalination 318 (2013) 79–87

85

getting higher. Due to the mixing, the ionic polarization is not observed at each entrance. The driving force in Fig. 4a appears high in the unit 1, and decreases as the unit number increases. On the other hand, the driving force in Fig. 4b appears moderate in all the units. The membrane resistance RM along the CEM is plotted in Fig. 5, estimated from Fig. 4a. We point out that, since the flow directions in the units 1 and 3 are opposite to those in the units 2 and 4, the trend of RM of the units 1 and 3 appears opposite to that of the units 2 and 4. Due to the ionic polarization, RM becomes higher as the saline water flows through the compartment. Since the increasing resistance means a slow ionic transport, the mixing is necessary to maintain the enhanced power generation.

4.2. Performance evaluations

Fig. 3 shows the concentration profile obtained for the low Pe case (δC = 0.015 cm, Pe = 0.85 and Re = 1.5). The saline water is supplied to the bottom of the saline compartment in the unit 1, while the fresh water comes in at the top of the fresh compartment, underlying a counter-current flow. It should be noted that, due to the thin δC, the saline compartment loses almost all ions at the exit of unit 1, while the fresh compartment contains the considerable amount of ions at the exit. Hence, in the unit 2, the concentration of the fresh compartment becomes higher than that of the saline one, and the ion is transferred from the fresh to the saline compartments. This reverse transfer occurs again in the unit 4, resulting in the reduction of the net power generation. Fig. 4 shows the concentration profiles obtained for the high Pe case (δC = 0.15 cm, Pe = 85 and Re = 15). The RED stack module in Fig. 4a has the fresh feed at the top of the fresh compartment in the unit 1 (like Fig. 3), while the module in Fig. 4b has the feed at the bottom of the fresh compartment in the unit 4. In both cases, as the streams go through one unit and another, the concentration in the saline compartment is getting lower, but the concentration in the fresh water is

4.2.1. Effect of compartment thickness Based on the concentration profiles, the power density PD (Fig. 6), the energy density ED (Fig. 7), and the current density ID with the ion flux JC (Fig. 8) are plotted at various values of δC ranging from 5 × 10 −3 to 0.8 cm. For v0 = 1.0 cm/s, the Re ranges from 0.5 to 80. As the number of unit increases, both power and current densities and the ion flux decrease, but the energy density increases. These trends are related to a fact that, in the stack module of multiple units assembled with single units in serial, the ion flux in a unit gets reduced when compared to the flux in the previous one. On the other hand, since the energy density is related to the cumulative amount of the ion transfer across the membrane, an addition of units increases the energy density. According to our simulation results, the power, the energy, and the current densities show the maximum values. This feature allows us to figure out that the strong streamwise convection, i.e., high Pe, can accelerate the lateral diffusion by reducing the ionic polarization in the compartments. Fig. 6 provides also a comparison of our results and other reported data obtained in RED stack modules of N = 30 with parallel configurations and δM = 0.1 cm. Note that δC = 0.1 cm in Weinstein and Leitz [6], but δC = 0.1 cm (for fresh) and 1.0 cm (for saline) in Suda et al. [2]. Although their module geometry and several factors are not exactly same as ours, the present RED module generates similar level of power density. The actual benefit from the adoption of serial configuration is expected to get the enhancement of the energy density. However, the

Fig. 6. The power density of the RED stack module with variations of N and δC at v0 = 1.0 cm/s, where symbols denote experimental results in the literature [2,6].

Fig. 7. The energy density of the RED stack module with variations of N and δC at v0 = 1.0 cm/s.

Fig. 5. The membrane resistance along the streamwise z direction, where N = 4, v0 = 1.0 cm/s, and δC = 0.15 cm.

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Fig. 8. The current density and ion flux of the RED stack module with variations of N and δC at v0 = 1.0 cm/s.

comparison of energy density cannot be conducted due to a lack of relevant discussion in the previous studies. 4.2.2. Effects of unit number and fresh supply position In Figs. 6 and 7, δC that makes PD maximum is larger than δC that makes ED maximum. This is because the energy density is related to the amount of the fresh water consumption. Pe increases with increasing δC, which provides possibly the reduction of ionic polarization. However, in case of δC > 0.04 cm, lots of ions will stay in the saline compartment and do not participate in the power generation until they exit the module. For the same reason, as N increases, δC that makes PD maximum increases but δC that makes ED maximum decreases. As mentioned above, the RED stack module shown in Fig. 4b is more difficult to design than the module in Fig. 4a. It can be pointed out that the enhancements in PD and ED by the complicated design are not considerable. For example, the condition used in Fig. 4a results

Fig. 10. The energy density and current density of the RED stack module with variations of v0 at N = 4 and δC = 0.15 cm.

in the maximum power density for N = 4 (i.e., PD = 2.4 mW/m 2, and ED = 0.36 mWh/m 3). On the other hand, PD and ED estimated with the same condition are 2.61 mW/m 2 and 0.39 mWh/m3, respectively, which means only 10% enhancement. 4.2.3. Effect of flow rate In Figs. 9 and 10, the power, the energy, and the current densities are plotted with range of v0 from 0 to 5 cm/s. According to the literature information, v0 values used in the experimental or pilot-scale modules are almost always less than 5 cm/s. As v0 increases, power and current densities show the monotonic increase. The energy density shows a constant value for v0 less than about 1 cm/s, and then monotonically decreases. Whether to use the power density or the energy density in the device design depends on the external factors. For instance, if the membrane costs high, the module should be designed to produce the maximum power density. If the fresh water is limited, the maximum energy density becomes the key parameter. According to Fig. 6, except for the single unit, the power density displays a plateau prior to the maximum value, the width of which broadens as N increases. That means that the power density is less sensitive to the change in δC, comparing to the energy density. This behavior may be an advantage in terms of the convenient control of the power density. 5. Conclusions

Fig. 9. The power density and current density of the RED stack module with variations of v0 at N = 4 and δC = 0.15 cm.

We developed the numerical algorithm for solving the convection– diffusion model with the N–P principle to evaluate the performances of the prototype RED stack module with commercially available ion exchange membranes. Owing to the multiple units assembled by serial configuration, seawater and fresh water subsequently goes through every inherent compartment. In order to deal with the suggested multiple units, the integral of the ion concentration over the compartment thickness was provided in the discretized form. It should be recognized that, with increasing number of units N, the maximum values of the power, the current density, and the ion flux decrease, but the maximum energy density increases. While ionic polarization is reduced by the increase of compartment thickness δC, there exists δC that makes each density maximum for various values of N. With increasing characteristic fluid velocity v0 at the given compartment dimension, power and current densities increase,

K.S. Kim et al. / Desalination 318 (2013) 79–87

due to a reduction of ionic polarization. The optimum v0 can be determined to 1 cm/s because a consistent decrease in the energy density begins evidently in the range v0 > 1 cm/s. As a special feature, with a sufficiently small compartment thickness (i.e., lower Pe), the reverse transport of ion from fresh to saline compartments can occur and then reduce power and energy densities. Our numerical framework can effectively be applied to the performance evaluations and justification of the validity of the RED stack module. Nomenclature A matrix for the first derivative B matrix for the second derivative C dimensionless ion concentration c ion concentration (g/L) ion concentration in saline feed (g/L) c0 D diffusivity of ion (m 2/s) diffusivity of ion across membrane (m 2/s) DM energy density (mWh/m 3) ED e elementary charge (= 1.6 × 10 −19 C) F Faraday constant (=96,485 C/mol) G dimensionless parameter (=KDδM/DMδC) I electric current (A) current density (A/m 2) ID ion flux across membrane (g/m 2 s) JC K partition coefficient of ion between water–CEM Boltzmann constant (= 1.38 × 10 −23 J/K) kB L length of a unit (m) N number of units power density (mW/m 2) PD Pe Peclet number (= v0δC 2/LD) p pressure drop (Pa) R universal gas constant (= 8.314 J/mol K) Re Reynolds number (=ρv0δC/μ) membrane resistance (Ω cm 2) RM T absolute temperature (K) V dimensionless fluid velocity v fluid velocity (m/s) characteristic fluid velocity (m/s) v0 W width of a unit (m) X dimensionless lateral coordinate x lateral coordinate (m) Z dimensionless streamwise coordinate z streamwise coordinate (m)

Greek letters α permselectivity compartment thickness (m) δC membrane thickness (m) δM Λ valence of ion ϕ open circuit voltage (V) μ viscosity of fluid (g/cm s)

Superscripts j index for unit number m power of a number

Subscripts f fresh water M membrane

n s

87

index for collocation point saline water

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