A theoretical model for salt ion drift due to electric field suitable to seawater desalination

Desalination 473 (2020) 114163

Contents lists available at ScienceDirect

Desalination journal homepage: www.elsevier.com/locate/desal

A theoretical model for salt ion drift due to electric field suitable to seawater desalination

T

V. Bartzisa, , I.E. Sarrisb ⁎

a b

Dept. Food Science and Technology, Faculty of Food Science, University of West Attica, Campus 1, Ag. Spyridonos 12243, Egaleo, Athens, Greece Dept. Mechanical Engineering, University of West Attica, 12244 Athens, Greece

ARTICLE INFO

ABSTRACT

Keywords: Electric field Salt ion drift Water duct flow

A seawater desalination method due to a high voltage static electric field that drifts the dissolved salt ions is proposed in this paper. The electric field creates opposite forces to the positive and negative charged ions of continuously flowing seawater and forces them to drift to the sidewalls of the duct, the center of which is less salted then. Both the transient and the steady-state ion distributions are studied analytically. The effects of various parameters, such as the electric field intensity, the width of the duct and drift time of ions inside the duct are studied. The method is found to work well as the width of the duct and the electric field intensity increase. Moreover, significant salt concentration reduction can be succeeded, even < 95%, in reasonable time, less than 10 s, for electric fields of the order 104 V/m, which makes the proposed method efficient and low cost. In a nutshell, his method can be very helpful in the problem solution of fresh water shortage.

1. Introduction As freshwater shortage is observed at last decades to many regions of the planet and foreseen that the situation will become worse the following years due to economic development and global warming, the seawater desalination methods attract significant attention. During and after the World War II several desalination technics were developed [1–11]. Until the end of 60s thermal desalination units were established to different countries, with capability of 8000 m3/day. Units were divided to the following subcategories: Multi-Stage Flash Distillation (MSF), Multiple Effect Distillation and Vapor Compression Distillation. In the beginning of 70s the new desalination factories started to use technics with special membranes. The two main subcategories are electrodialysis and reverse osmosis (RO). Nowadays, the two main methods that are used is MSF and RO that cover up to 43% of the production each. Although these two methods have the same total production, the RO method is used to the 68% of the cases with mean production 1200m3/day, whereas MSF is used for just the 9% of the cases with mean production 8800m3/day. The main disadvantages of all these methods are that they require too much energy, consumables like membranes and possibly mechanical parts in a salty environment. Significant effort is given to develop methods that reduce the energy consumption and thus the economic cost [12,13]. Additionally, in the recent years, desalination due to electric fields was

⁎

studied with emphasis in drift velocity ion mobility in aqueous electrolyte solutions and in processes such as electrodialysis, capacitive deionization, electro-deionization, electrophoresis, electroosmosis [14–30]. A very promising method for low-cost desalination due to electric field uses nanopore membranes. The most usual studied membrane is made of graphene nanosheet (GNS) [31–37] in which the nanopores are created with unsaturated carbon atoms which are passivated by chemical functional groups. Cohen–Tanugi and Grossman [34] studied the desalination efficiency of GNS for various applied pressures and Azamat [31] for various applied electric fields. In the present paper, a new low-cost method of seawater desalination is proposed which is based on drift of the positive and negative ions (mostly Na+ and Cl− ions) due to an external electric field without the use of pressure difference between the side walls and membranes. A duct with continuous flowing of seawater is considered, an area of which is under the external application of DC electric field. The proposed configuration is similar to a usual capacitor, Fig. 1. So, the ions, merely Na+ and Cl−, are attracted by the electric potential and drift to the non-conductive sidewalls of the duct. Thus, the central area of the duct will be found in a lower salt ion concentration, a region suitable to collect saltless water. Very recent molecular dynamics simulations of an analogous configuration in nanochannels [29] proved that the proposed method can successively drift salt ions. The method is similar to capacitive deionization method [31–37], but it is simpler since no porous

Corresponding author. E-mail address: [email protected] (V. Bartzis).

https://doi.org/10.1016/j.desal.2019.114163 Received 10 July 2019; Received in revised form 20 September 2019; Accepted 21 September 2019 0011-9164/ © 2019 Published by Elsevier B.V.

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

Nomenclature C μ μ0 µ G f Δy υy ay T z E L m rNa V

Greek symbols α activity (mol/L) γ activity coefficient ε electric permittivity φ potential v dynamic viscosity of water Subscripts (1 mol) per mole y along y axis eq equilibrium mean mean value Constants rNa = 183 × 10−12m NA = 6, 023 ⋅ 1023mol−1 F = 96485.34C/mol J R = 8.314 mol K ε0 = 8.85 ⋅ 10−12F/m εr ≈ 80

concentration , mol/L chemical potential , J/mol chemical potential in standard conditions, J/mol electrochemical potential ,J/mol free Gibbs energy pseudoforce displacement along y axis velocity along y axis acceleration along y axis absolute temperature number of overflow protons or electrons electric field intensity width of the duct ion mass ion Na radius voltage

electrodes are required. The present paper is organized as follows, the general theory of ion movement in the presence of electric field is summarized at Section 2, while at Section 3 the present model equations are presented together with the most important assumptions. The analytical solution procedure is described for the equilibrium state as well as for the initial transient state. The ion concentration then is obtained as a function of the ion properties, its position, electric field intensity, duct width and time across the duct and the most important results are discussed. Finally, the major conclusions of the present study are summarized in Section 4.

(2)

µ = µ + zF

where, z is the number of overflow protons or electrons and F = 96,485.34 C/mol is the Faraday constant. It is known that if a substance goes from an initial state with free Gibbs energy, Gin, to a final state, Gf, the work Wf of the pseudoforce f that creates the ion motion is given by the relation:

Gf

(3)

Gin = Wf or G = Wf If Δy is its displacement, then:

G=

f· y

The relationship between the Gibbs energy and electrochemical G potential is given by n = µ , where n is the mole number. So:

2. General theory for ion movement

µ =

Suppose that an ion of concentration C (molarity in mol/L) is dissolved in a water solution which is flowing in the streamwise direction of a duct, and a change of the ion concentration along the y-axis of the duct may mark a possible concertation gradient, C , as in the ion cony

centration profiles of Fig. 2. Moreover, the activity gradient

y

(4)

f(1mol) y

where, f(1mol) =

f n

is the pseudoforce per mole. The pseudoforce per ion f(1mol)

is given by the relation f(1ion) = N , where ΝΑ is the Avogadro conA stant. If we consider infinite displacements from the above Eq. (4) we have:

, where

α = γC is the activity and γ the activity coefficient can be defined. Thus, activity is the effective concentration in which an ideal solution has the same properties of a given real non-ideal solution. According to theory, the chemical potential is given by the relation [38,39]:

f(1 ion) =

1 µ NA y

Also, during the ion movement a friction force also exists between

(1)

µ = µ 0 + RT ln

(5)

where, μ0 is the chemical potential in standard conditions, R = 8.314 J/ (mol K) and T is the absolute temperature. If in addition to ions, an external constant electric field of intensity Ε and potential φ is applied along the y-axis, see Fig. 2, the electrochemical potential will read as:

Fig. 1. Schematic model description.

Fig. 2. Ion concentration across the duct of width L. 2

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

the moving ion and the solvent, that is opposite to the ion velocity y , which approximately is given by the relation (we consider as usual the ion shape as a small sphere with radius r that moves with a small velocity):

f fr =

6

r·

(6)

y

where, ν is the dynamic viscosity of water. From the second Newton's law we have for the ion:

Fy = mion a y

f(1ion) + f fr = mion a y ,

or, by using Eqs. (5) and (6):

1 µ NA y

b

y

= mion a y

(7)

where, b = 6πνr, and ay is the ion acceleration along y axis. Next, we shall apply this theory to our model. 3. Model description Sea water in oceans has mean salinity of approximate 3.5% (35 g/L) which means that 1 L seawater contains 35 g salt, most of it is dissolved salt (NaCl) in ion form Na+ and Cl−. As we have already mentioned our purpose is to reduce the salinity of the water by drift of ions in the sidewalls of a continuous water flow duct. Fig. 1 shows schematically the saltwater desalination reactor device studied here, a cross-section of which is also showed in Fig. 2. It is composed of two parts: 1) a pair of plate electrodes which are connected with high voltage V and create a uniform static field of high intensity E constant in space and in time between them. The intensity E has direction vertical to the plate electrodes from positive to negative. 2) an electrically isolated duct which is placed along and between the electrodes. The internal width of the duct is L. The duct is nested inside the two electrodes taking care that the distance between the plate electrode and the duct to be the smallest possible, this way assuming that E is constant and uniform in the duct. Thus, the water flow is normal to the electric field. The steady-state and the development regions of the desalination process will be analyzed below. It should be noticed that the present study is in accordance to Debye-Huckel theory [40], where water is considered as a continuous dielectric medium. Thus, water effect due to the external electric field is to change the electric permittivity of the solution to ε = εrε0, where εr ≈ 80 is the relative permittivity of the water and ε0 = 8.85 · 10−12F/ m. Thus, water changes the electric current and potential conceptualized by the salt ions.

Fig. 3. Ion concentration along y/L for various E(V/m) values and a) L = 0.001 m, and b) L = 0.1 m.

dCeq Ceq

3.1. Steady-state solution

where, = Solving the above simple differential equation and considering y = 0 at the one sidewall of the pipeline, and y = L at the opposite sidewall, and also considering an initial ion concentration, C0, before the application of the electric field or before the entrance of the seawater in the duct, we have:

Ceq =

(8)

Substituting Eqs. (1) and (2) in Eq. (8) we have:

RT

1 eq

eq

y

Since,

+ zF =

RT Ceq = zFE Ceq y

y y

(10) zFE . RT

Under permanent external conditions, after some time, ions drift procedure stops, and an equilibrium state is succeeded. To determine the equilibrium state, υy = 0 and ay = 0 conditions are applied in Eq. (7) and so it reads as:

1 µ =0 NA y

= Ady

C0 LA Ay e e AL 1

For seawater, either the Na+ and Cl− concentrations are equal and approximately C0 = 0.6mol/L. In the following, we shall study the effect of duct width and electric intensity E on the ion concentration distribution across the duct, namely as a function of the y coordinate. For convenience, the relative concentration is shown in Figs. 3 and 4 as function of y/L:

=0 and αeq = γCeq, the above equation takes the form:

Ceq

(9)

C0

As it is already demonstrated, ion concentration changes only along the y-axis in the equilibrium, so the above equation could take the form:

=

LA e Ay e AL 1

(11)

Starting from the initial uniform concentration, C0, for E → 0, and 3

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

area (near the opposite sidewall) of relative concentration lower than y 9%. Moreover, even for 200 V/m the center of the duct at L = 0.5 will be found with higher than 9% concentration. Although high E is needed in the narrow duct to drift ions, its application is easier that as the width of the duct increases. On the other hand, for the wider duct of L = 0.1 m, > 90% of the width of the duct will be in lower than 9% relative concentration when E > 25 V/m. The important parameter of time duration for the ion drift is investigated in the next section. Thus, the actual time and consequently the length of the duct for a given water flow rate will be discussed. 3.2. Time evolution of concentration The temporal evolution of ion drift in the duct is studied here by solving Eq. (7) from the initial salt entrance stage C0 to the equilibrium state C. To start, let us define as y1 the ion displacement along y-axis, so Eq. (7) with the help of Eq. (2) takes the form:

m

d 2y1 +6 dt 2

r

dy1 1 (ln ) + RT + zF dt NA y y

here E and φ are related through: E = the above equation reads as:

y

=0

(12)

, and by replacing α = γC,

Fig. 4. Ion concentration along y/L for various L values and a) E = 1(V/m), and b) E = 100(V/m).

by increasing E, an exponential distribution is found. Concentration distributions however appear to be strongly related to the width of the duct L. In the case of a narrow duct, L = 0.001 m, desalination, i.e. to Ceq 0 , needs the application of a very strong electric field succeed C 0 because ions can move harder due to smaller free space. As Fig. 3a suggests, for E ≤ 10V/m, a very weak distortion of the C profiles are y observed and even for E = 100V/m, the center of the duct at L = 0.5, is Ceq

still at high salt concentration of C ~0.5, thus only the half of the salt 0 has been drift to the sidewalls. The effect of E is more intense as L increases, as Fig. 3b is shown for L = 0.1 m. In the L = 0.1 m case, even E = 20N/C is adequate to leave most of the water almost without salt y ions at 0.1 < L < 0.9 and to drift them all near the sidewalls of the duct. Thus, in order to intensify desalination, either stronger electric fields or longer width of the duct should be applied, as Fig. 4a and b also demonstrate. Details on the confinement distance of the ion concentration profiles as E and L increases can be found in Fig. 5a and b, respectively. There, as an example, we set as fully desalinated water the one with conCeq centration C = 0.09 , Thus, solving Eq. (11), the distance y9% of the 0 width of the duct where the relative concentration is lower than 9% of the initial one is obtained. As E increases, the narrow duct with L = 0.001 m needs almost 100 V/m and above to reach at least a short

Fig. 5. Salt confinement distance from the wall: a) as E increases for L = 0.001, 0.01 and 0.1 m, and b) as L increases for E = 2, 10 and 50 V/m. 4

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

m

d 2y1 +6 dt 2

dy1 RT 1 C zFE + = dt NA C y NA

r

where, for Na+, the number of overflow protons are z = 1, and water temperature is assumed to be T = 300K. Moreover, b, P, and D can be evaluated as: N ·s b = 6 rNa = 3.06 × 10 12 m ,

(13)

Eq. (13) is very difficult to be solved analytically to this general form, however, by assuming that the ion concentration profile should be in a similar to the steady-state distribution even in the development regime, thus C is a function of y of in the form:

C=

C0 L L

e

1

y

e

P = N = 1.6 × 10 19E , and D = N = 1.6 × 10 16 (N / m) . A A where, the effective radius of the ion Na+ is rNa = 183 × 10−12m, viscosity of the water is ν = 0.00089Pa · s, and the Avogadro number is atom J C NA = 6.023 × 10 23 mol . Also, R = 8.314 mol·K , and F = 96485.34 mol . Considering that Na+ mass is mNa = 3.81 × 10−26Kg, it may be b 1, and Eq. (22) can be simplified further as: observed that s zFE

(14)

where, Λ is a time dependent factor that its value is in the range Λ = 0 to A, for the initial value at t = 0 and the final equilibrium state, respectively. Using the above Eq. (14), it can be proved that:

1 C = C y

y1 (t ) =

Now, the time dependence value of Λ(t) should be a function that depend on the ion velocity, which is progressively decreased because ion movement takes place. For this, as a first order approximation it is considered here that the time derivative of Λ is analogous to ion velocity as: (16)

y1 (t ) = Q%·y1 (

where, κ is a proportionality constant. Integrating the above equation, and imposing the less favor condition that the initial position of the ion at t = 0 is at y1 = 0, Λ is found as:

t=

Using Eqs. (15) and (17), Eq. (13) takes the simpler form:

d 2y dy m 21 + b 1 + Dy1 = P dt dt

(18)

zFE NA

y1 (t ) =

P 2Ds

1t

be 2

(

s b m

) + be 12 t ( smb ) + s e 12 t (

s b m

) + s e 12 t ( smb )

2s (19)

where, s = b2 4Dm . Indeed, for t = 0 the initial condition y1 = 0 is recovered in Eq. (19), whereas for t → ∞, the distance covered by the ion is:

y1 (

)=

P zFE = D RT

(23)

)=

Q P 100 D

19125·ln 1

Q 100

(24)

This time interval as a function of the percentage Q is shown in Fig. 6. Thus, for the desalination process to reach its equilibrium, i.e. for Q = 100%, the time required is found to be very high. This time corresponds to the worst scenario of an ion that starts moving under the electric field from one of the sidewalls to the other that covers the distance L. Which means that the actual mean time for most of the ions to drift is lower. At a first reading of Eq. (24), the time to equilibrium does not depend on the intensity of the electric field, however, since the final state is different for each value of the electric intensity, only the usage of strong electric fields can accelerate the present method. The effect of E in the time to equilibrium is discussed hereafter for the case of duct width L = 0.1 m, however, the analysis is similar for every duct. In the following, the ion concentration is expressed as a function of time along the y-axis. If we combine Eq. (17) and Eq. (23), the parameter of the model Λ may read as:

P= where, D = and b = 6πνr. The solution of the above equation, considering that the initial conditions at t = 0 are: y1(0) = 0, υ(0) = 0, is the following: RT and NA

)

1

So, from Eq. (23), the time for an ion to cover the Q percentage of its equilibrium displacement may be expressed as:

(17)

(t ) = ·y1

(

P s bt e 2m D

The above Eq. (23) can be used to determine the time interval that the separation of the ions is mostly completed. Theoretically, this time is infinite, however, if it is considered that the ion has covered most of the width of the duct and reached for example the distance s b 2mln 0.1 P y1 = 0.9y1 ( ) = 0.9 D , then e 2m t = 0.1, or t = s b 44040s . Following the same concept, a general relation of how the time interval grows as a function of percentage Q for the final state, i.e. the equilibrium one, can be expressed as:

(15)

d (t ) = · dt

RT

=

(20)

38.68· ·(e

5.228·10 5t

1) (S. I.)

(25)

which is the ion displacement at the equilibrium state, and so:

(

) = y1 (

)=

zFE =A RT

(21)

as it is expected. Eq. (19) can be written to the more convenient form as:

y1 (t ) =

P b e D s

1 bt 2 m sinh

1 s t +e 2 m

1 bt 2 m cosh

1 s t 2 m

1

(22)

+

Our analysis will focus now on Na case, however it is similarly to be applied and to other ions. The present model needs the parameter κ to be determined. This can be done only approximately in the lack of experimental data or numerical simulations. As it is observed in Fig. 4b, for E = 100 V/m and L = 0.1 m most of the ions are concentrated at the sidewalls of the duct. So approximately, we can consider that in equilibrium as t → ∞, the ion may cover a distance y1 = L at the most (for these values of E, L). So, if we put in Eq. (20) this limit, it is found that:

y1 (

)

L

Fig. 6. Time interval grow of ion displacement as a function of final state percentage Q%.

zFE = 38683 (m 2) LRT 5

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

Table 1 Calculation of Λ and the width 2d of ion-clean part of the center of the duct at different times for E = 104 V/m and L = 0.1 m. t (s)

Λ (m

5 10 15 20 25 30 35 40

101 202 303 404 505 606 707 808

−1

)

d(m)

Width 2d(m)

– 0.033 0.039 0.042 0.043 0.044 0.045 0.046

– 0.066 0.078 0.084 0.086 0.088 0.090 0.092

10 9

t=5 t = 10 t = 15 t = 20 t = 25 t = 30 t = 35 t = 40

8 7

C/C0

6 5 4

Fig. 8. Electric field intensity dependence of time t to achieve with 2d = 0.05 m and L = 0.1 m.

width 2d of this duct if we want to reach let us say this case we have:

sinh ( d ) 0.05(e = d Le

1 0 0.05

0.06

0.07

0.08

0.09

t=

And thus, the relative concentration distribution with respect to time reads as:

C C0 mean

= 0.05. In

1) (29)

L 2

e

y

(26)

C C0 mean

= 0.05 for a specific value of E

(

)

3.82 E 5.228·10 5

ln 1

(30)

4. Conclusions From the above analysis, it can be easily concluded that with the use of electric field, the present desalination method can work, and it can achieve a satisfactory percentage of desalination. More specific, from the equilibrium solutions seems that better desalination can be succeeded as the width of the duct and the electric intensity magnitude are increased. This impressive result is crucial for the possible large-scale usage of the present method and demonstrate its capacity to desalinate important quantities of seawater. Thus, for example, only an electric field of 25N/C is required for the almost total desalination (< 9% of the initial salt concentration in up to 90% of the duct cross section) in a duct of width 0.1 m or wider. Moreover, for the ion drift duration in the duct for desalination, the present analysis was conducted from two opposite starting points. On the one hand, the time to perform water desalination assumed to be between 5–40 s, and it is found that even > 95% of the initial salt concentration of the water can be succeeded with electric fields of the order of 104 V/m for a duct with width L = 0.1 m to the most of its with (Table 1). Thus, the method can work well in reasonable times and

L

Le 2 sinh ( d) (e L 1) d

( )

As it is shown in Fig. 8, the time required for succeeding the desalination level of the example is highly increased as E → 0 even for the relative narrow internal width of 2d = 0.05 m. However, as E increases, the time is asymptotically decreased to zero.

From Eq. (25) is clear that Λ is a function of t and E and for example in the case of the electric intensity of E = 10,000 V/m and for L = 0.1 m, Λ are evaluated in the Table 1 for different times. Although a high E is considered in this case, since it refers to a duct with width L = 0.1 m, an external potential difference of the order of ∆φ = 1000V at the most should be sufficient to be applied. As time increases, and more ions are drifted and confined at the sidewalls, Λ is found to increase as resistance to drift increases. The temporal evolution of the C relative concentration profiles C = f (y, t ) of Eq. (26) is shown in Fig. 7 0 for a range of different times as the one presented in Table 1. Let us suppose in the following that we want to collect water around the center of the duct after a specific time and with specific E (this means that the value of Λ is determined as constant). The mean value of C/C0 in an area with width 2d around the center of the duct (see Fig. 2) is:

mean

( )

can also be estimated by Eq. (25). Initially, the value of Λ ≈ 148m−1 for L = 0.1 m can be found from Eq. (29). If this Λ value is substituted in Eq. (25), the required time as a function of E is found as:

Fig. 7. Ion concentration profiles along y for various values of Λ and E = 10,000 V/m, L = 0.1 m.

=

L

mean relevant concentration of

0.10

y (m)

C C0

= 0.05

Solving numerically the above equation for the values of Λ that are calculated in the Table 1, we can calculate d and 2d (the width of the inner duct) as it seems in the last two columns of the Table 1. Inversely, the time that is needed to succeed for example in an inner width of 2d = 0.05 m of the duct (the half of the width of the duct) a

2

1

C C0 mean

salt concentration C0. From the above equation we can evaluate the

3

C L = C0 e L

( )

(28)

We have to put a second smaller duct in this area, as is seen in Fig. 2 in order to pump water with salt concentration C lower than the initial 6

Desalination 473 (2020) 114163

V. Bartzis and I.E. Sarris

electric fields that can be succeeded easily in engineering applications. On the other and more important approach, lower electric fields started from 10N/C were considered. Taking the same mean desalination level of 95% at the central half of the width of the duct, duration from 9200 s for Ε = 10Ν/C, to 90s for E = 1000N/C and to 7 s για Ε = 10,000Ν/C was found that are reasonable. Thus, a desalination method based in this model can produce high quantities of saltless water in an economical operational cost fashion that can be easily compared with the other well-known methods, thus, the further investigation of this method is proposed.

electric field, J. Thermodyn. 10 (1985) 131–144. [17] M. Wright, An Introduction to Aqueous Electrolyte Solutions, Wiley, New York, 2007. [18] M. Gordon, X. Huang, S. Pentoney Jr., R.N. Zare, Capillary electrophoresis, Science 242 (1988) 224–228. [19] H. Strathmann, Ion-exchange membrane separation processes, Membrane Science and Technology, 9 Elsevier, New York, 2004. [20] C. Ganzi, Y. Egozy, A. Giuffrida, A. Jha, High purity water by electro-deionization, Ultrapure Water Journal 4 (3) (1987) 43–50. [21] I. Tironi, R. Sperb, P. Smith, W. van Gunsteren, A generalized reaction field method for molecular dynamics simulations, J. Chem. Phys. 102 (13) (1995) 5451–5459. [22] H. Berendsen, J. Postma, W. van Gunsteren, B. Pullman (Ed.), Intermolecular Forces, Reidel Publishing Co, 1981, pp. 331–342. [23] G. Wulfsberg, Principles of Descriptive Inorganic Chemistry, University Science Books, South Orange NJ, 1991, pp. 23–31. [24] J. Rasaiah, R. Lynden-Bell, Computer simulation studies of the structure and dynamics of ions and non–polar solutes in water, Philos. Trans. R. Soc. London, Ser. A 359 (2001) 1545–1574. [25] A. Belch, M. Berkowitz, J. McCammon, Solvation structure of a sodium chloride ion pair in water, J. Am. Chem. Soc. 108 (1986) 1755–1761. [26] A. Das, B. Tembe, The pervasive solvent-separated sodium chloride ion pair in water-DMSO mixtures, Proc. Indian. Acad. Sci. (Chem. Sci.) 111 (2) (1999) 353–360. [27] S. Murad, The role of external electric fields in enhancing ion mobility, drift velocity, and drift–diffusion rates in aqueous electrolyte solutions, J. Chem. Phys. 134 (11) (2011) 114504 1–7. [28] M. Sheikholeslami, M.A. Sheremet, A. Shafee, Z. Li, CVFEM approach for EHD flow of nanofluid through porous medium within a wavy chamber under the impacts of radiation and moving walls, J. Therm. Anal. Calorim. (2019), https://doi.org/10. 1007/s10973-019-08235-3. [29] F. Sofos, T. Karakasidis, D. Spetsiotis, Molecular dynamics simulations of ion separation in nano-channel water flows using an electric field, Mol. Simul. 45 (17) (2019) 1395–1402. [30] M. Suss, S. Porada, X. Sun, et al., Water desalination via capacitive deionization: what is it and what can we expect from it? Energy Environ. Sci. 8 (2015) 2296–2319. [31] J. Azamat, Functionalized graphene nanosheet as a membrane for water desalination using applied electric fields: insights from molecular dynamics simulations, J. Phys. Chem. C120 (41) (2016) 23883–23891. [32] L. Drahushuk, M. Strano, Mechanisms of gas permeation through single layer graphene membranes, Langmuir 28 (48) (2012) 16671–16678. [33] C. Sun, M. Boutilier, H. Au, P. Poesio, B. Bai, R. Karnik, N. Hadjiconstantinou, Mechanisms of molecular permeation through nanoporous graphene membranes, Langmuir 30 (2) (2014) 675–682. [34] D. Cohen-Tanugi, J. Grossman, Water desalination across nanoporous graphene, Nano Lett. 12 (7) (2012) 3602–3608. [35] T. Humplik, J. Lee, S. O’Hern, B. Fellman, M. Baig, S. Hassan, M. Atieh, F. Rahman, T. Laoui, R. Karnik, E. Wang, Nanostructured materials for water desalination, Nanotechnology 22 (29) (2011) 1–19. [36] M. Suk, N. Aluru, Water transport through ultrathin graphene, J. Phys. Chem. Lett. 1 (10) (2010) 1590–1594. [37] E. Wang, R. Karnik, Water desalination: graphene cleans up water, Nat. Nanotechnol. 7 (9) (2012) 552–554. [38] P. Atkins, J. Paola, Atkin's Physical Chemistry Oxford University Press, 8th ed., vol. 5, (2006), pp. 136–169. [39] R.G. Mortimer, Physical Chemistry, 3rd edition, 8 Elsevier, 2008, pp. 351–378. [40] P. Debye, E. Hückel, The theory of electrolytes. I. Lowering of freezing point and related phenomena, Phys. Z. 24 (1923) 185–206.

References [1] S. Dallas, N. Sumiyoshi, J. Kirk, et al., Design, economic analysis and environmental considerations of mini-grid hybrid power system with reverse osmosis desalination plant for remote areas, Renew. Energy 34 (2) (2009) 374–383. [2] A. Cipollina, G. Micale, Coupling sustainable energy with membrane distillation processes for seawater desalination, Nuclear & Renewable Energy Conference (INREC) 2010 1st International, 2010, pp. 1–6. [3] L. Garcia-Rodriguez, Renewable energy applications in desalination: state of the art, Sol. Energy 75 (2003) 381–393. [4] M. Wilf, L. Awerbuch, C. Bartels, M. Mickley, G. Pearce, N. Voutchkov, The guidebook to membrane desalination technology, Reverse Osmosis, Nanofiltration and Hybrid Systems. Processes, Design, Applications, Balaban Desalination Publications, 2006. [5] R. Guoling, F. Houjun, Technical progress in seawater desalination technology at home and abroad, China Water &Wastewater 20 (2008) 86–90. [6] E. Mathioulakis, V. Belessiotis, E. Delyannis, Desalination by using alternative energy: review and state-of-the-art, Desalination 203 (2007) 346–365. [7] E. Mohamed, G. Papadakis, E. Mathioulakis, V. Belessiotis, The effect of hydraulic energy recovery in a small sea water reverse osmosis desalination system; experimental and economical evaluation, Desalination 184 (2005) 241–246. [8] E. Mohamed, G. Papadakis, E. Mathioulakis, V. Belessiotis, An experimental comparative study of the technical and economic performance of a small reverse osmosis desalination system equipped with a hydraulic energy recovery unit, Desalination 194 (2006) 239–250. [9] E. Mohamed, G. Papadakis, E. Mathioulakis, V. Belessiotis, A direct coupled photovoltaic seawater reverse osmosis desalination system toward battery based systems a technical and economical experimental comparative study, Desalination 221 (2008) 17–22. [10] N. Li, A. Fane, W.H. Winston, T. Matsuura, Advanced Membrane Technology and Applications, Wiley, 2008. [11] T. Mezher, H. Fath, Z. Abbas, A. Khaled, Techno-economic assessment and environmental impacts of desalination technologies, Desalination 266 (2011) 263–273. [12] L. Gu, X. Chen, X. Liu, W. Liu, Ion distribution in saltwater under high-voltage static electric field, Adv. Mater. Res. 361–363 (2012) 865–869. [13] L. Gu, Simulation of ion distribution on both sides of insulation film in saltwater under static electric field, Adv. Mater. Res. 538–541 (2012) 110–115. [14] K. Maerzke, J. Siepmann, Effects of an applied electric field on the vapor−liquid equilibria of water, methanol, and dimethyl ether, J. Phys. Chem. B 114 (2010) 4261–4270. [15] H. Bakker, Structural dynamics of aqueous salt solutions, Chem. Rev. 108 (4) (2008) 1456–1473. [16] A. Morro, R. Drouot, G. Maugin, Thermodynamics of polyelectrolyte solutions in an

7