Multiscale Simulation Method for Flow and Mass-Transfer

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Multiscale Simulation Method for Flow and Mass Transfer Characteristics in a Reverse Osmosis Membrane Module Toru Ishigami, and Hideto Matsuyama Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03087 • Publication Date (Web): 02 Nov 2015 Downloaded from http://pubs.acs.org on November 3, 2015

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Industrial & Engineering Chemistry Research

1

Multiscale Simulation Method for Flow and Mass

2

Transfer Characteristics in a Reverse Osmosis

3

Membrane Module

4

Toru Ishigami, †,‡ Hideto Matsuyama*,†

5

†

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Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan

7

‡

8

University, 1866 Kameino, Fujisawa, Kanagawa 252-8510, Japan

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*Corresponding author. E-mail: [email protected]. Tel.: +81-78-803-6180. Fax: +81-78-

Center for Membrane and Film Technology, Department of Chemical Science and Engineering,

Department of Food Bioscience and Biotechnology, College of Bioresource Sciences, Nihon

10

803-6180.

11

KEYWORDS: Computational fluid dynamics, Membrane modules, Porous media modeling,

12

Reverse osmosis membrane

13

ACS Paragon Plus Environment

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ABSTRACT

2

This paper presents a numerical simulation method for calculating flow and mass transfer

3

characteristics in an entire membrane sheet module comprised of feed and permeate channels.

4

The channels, including the spacers, were coarse-grained as the porous media for the simulation

5

of the entire membrane sheet. This is because the previous unit cell concept, which directly

6

calculates the flow and solute concentration fields around the spacers, cannot be extended to the

7

meter- sized computational domain owing to computational load limitations. We first carried out

8

the unit cell simulation of the spacer-filled channel to obtain the flow and mass transfer

9

characteristics. The obtained flow and mass transfer characteristics were then used for modeling

10

porous media. The flow and solute mass fraction fields were then calculated for a membrane

11

sheet, and the effect of spacer arrangement on the membrane sheet performance was investigated.

12

From the relationship between the pressure drop and water permeation rate, significantly

13

different correlations were found for two parameters: the distance between spacers and the angle

14

between spacers.

15


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1

1. Introduction

2

Spiral-would membrane (SWM) modules with reverse osmosis (RO) membranes are widely

3

used in desalination processes. The use of RO membranes in desalination plants is expected to

4

continue growing 1,2. In order for freshwater production from seawater to be economical, the

5

optimal design of all parts that comprise the module, as well as operating conditions, is required.

6

Computational fluid dynamics (CFD) approaches have been applied to study the flow and mass

7

transfer characteristics in such a module. A number of researchers calculated the flow and solute

8

concentration fields around the spacers, and systematically investigated the effects of the spacer

9

geometry 3-5 and arrangement 3,6-8 on the flow and mass transfer characteristics inside the feed

10

channel of the module. Recently, a simulation model that considered the local water and salt

11

permeation fluxes across the RO membrane was constructed 9. This study showed that the

12

concentration polarization, local water flux, and salt concentration distribution of the RO

13

membrane surface were successfully described using the simulation model. Many CFD studies,

14

including the previous study, clarified that the spacer geometry and arrangement significantly

15

affected mass transfer, water permeation rate, and energy loss (pressure drop in the feed channel).

16

Interestingly, the spacer design influences the ratio of mass transfer enhancement to energy loss.

17

These studies only focused on the millimeter-size periodic unit cell of the feed channel, which

18

was specified using spacer arrangement. On the other hand, only a few studies dealt with the

19

numerical flow and mass transfer characteristics through membrane sheets or envelopes 10–13.

20

The unit cell simulation (direct numerical simulation) could not be extended to the entire

21

membrane sheet, or envelope, because of computational load limitations. Despite its small-scale

22

computational domain, three-dimensional unit cell simulation requires a certain number of grids

23

because of the presence of closely separated mesh spacers. However, if the spacers are


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collectively or individually modeled in a simulation of the entire membrane sheet 14–16, the

2

effects of the spacer geometry and arrangement, which are important in the module design,

3

cannot be investigated in detail. Kostoglou et al. constructed a simulation method that can

4

calculate the hydrodynamic solute concentration and the water permeation flux through an entire

5

membrane sheet 10,12,13. In addition to the unit cell simulation, a simulation of the entire

6

membrane sheet is far more effective for optimal design of SWM modules by qualitatively

7

evaluating the effect of the spacer geometry and arrangement on the module performance e.g.

8

water production rate, salt rejection and pressure drop.

9

In this study, a numerical simulation technique was constructed in order to calculate the flow

10

and solute concentration fields through a membrane sheet comprised of a pair of feed and

11

permeate channels. The unit cell simulation was used to derive equations on flow and mass

12

transfer characteristics that were used in the membrane sheet simulation. The equations could be

13

derived in experimental, analytical approaches, etc. However, the experimental approach can not

14

consider the local but overall or mean flow and mass transfer characteristics and the analytical

15

approach can not deal with the complicated geometry such as the spacer-filled channel. A

16

numerical simulation method for the entire membrane sheet and a unit cell simulation

17

coordination method were developed. First, the flow and mass transfer characteristics in the feed

18

channel were obtained using the unit cell simulation method proposed in the previous study 9.

19

Then, this new technique was used to investigate the effect of the spacer arrangement on pressure

20

drop (energy loss) and permeation rate through the membrane sheet.

21 22

2. Numerical Methods


4

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1

In this study, a multiscale simulation method that coordinates the millimeter-scaled unit cell

2

simulation with meter-scaled membrane sheet simulation was proposed. First, the flow and salt

3

concentration fields in a spacer-filled channel were directly calculated. Then, the correlation

4

between the flow and mass transfer characteristics, which were the resistance factors and

5

Sherwood number, were obtained for the membrane sheet simulation. A porous media model

6

17,18

7

visualize the spacer-filled channel as porous media based on Darcy’s law. The flow and solute

8

concentration fields were calculated using the porous media model with the correlation obtained

9

from the unit cell simulation of the spacer-filled channel.

was applied for the membrane sheet simulation. This method is generally used to roughly

10

2.1. Unit cell simulation of flow and solute concentration fields in spacer-filled channel

11

To calculate the fluid velocity, pressure, and solute mass fraction fields, continuity, momentum,

12

and solute mass transport equations were numerically solved using a commercial CFD software

13

package, Fluent 13.0.0, which adopts the finite volume method with an unstructured grid system.

14

The computational domain of the spacer-filled channel is shown in Figure 1. This computational

15

domain consists of a series of three units with periodic spacer geometries. As in the previous

16

study, the top and bottom walls are assumed to be the impermeable wall and the RO membrane

17

(permeable wall), respectively. It should be noted that the membrane boundary conditions of the

18

permeable wall do not influence the flow and mass transfer characteristics described in Section

19

2.2, because the permeation flux is much lower than cross-flow velocity. Although we just

20

applied the unit cell simulation method that we developed in the previous paper 9, the permeable

21

wall is no need to derive the equations on flow and mass transfer characteristics. The geometrical

22

characteristics of spacers include the ratio of the distance between the spacers (W/d) and the

23

angle between the spacers (θ). The computational results of the central unit were applied in order


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1

to analyze the flow and mass transfer characteristics of the spacer-filled channel because the

2

other units were affected by the inlet and outlet boundary conditions. Although the previous

3

studies 3,7,8 have used one periodic cell, the periodic boundary condition in streamwise direction

4

can not be applied in the present simulation model because of the changes of the flow rate and

5

the salt mass fraction with permeation. Details were given in the previous paper 9.

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Figure 1. Computational domain and boundary conditions for the unit cell simulation of a

8

spacer-filled channel 9.

9 10

2.2. Correlation specification for flow and mass transfer characteristics of the porous media model


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1

A porous media model was utilized for the calculation of flow and solute mass fraction fields

2

in the membrane sheet. The momentum relationship for porous media is defined in the following

3

equation.

4

1 µ  ∇p = − u + C ρ u u  2 α 

5

Here, p [Pa] is the pressure, µ [Pa s] is the viscosity, 1/α is the viscous resistance factor (α [m2]

6

is the permeability), u [m s–1] is the superficial velocity (mean cross-sectional velocity) vector, C

7

[m–1] is the inertial resistance factor, and ρ [kg m–3] is the fluid density. The first and second

8

terms on the right hand side of Eq. (1) are viscous and inertial loss, according to Darcy’s law. It

9

should be noted that the present simulation method can not be applied to the case of high

(1)

10

Reynolds number, because this model is based on the Darcy’s law. The simulation of the spacer-

11

filled channel with various superficial velocities was carried out, and the correlation between the

12

pressure gradient and the superficial velocity was then plotted. The viscous and inertial

13

resistance factors were obtained and used as fitting parameters for a second-order polynomial

14

equation, which corresponds to the correlation between the pressure gradient and the superficial

15

velocity in Eq (1).

16 17

18

The empirical equation for the mass transfer characteristics in the feed water channel is as follows 19,20:

Sh = b ⋅ Re a ⋅ Sc 0.25

(2)

19

The detailed calculation procedures of Sherwood, Reynolds and Schmidt numbers and the mass

20

transfer coefficient in the unit cell are shown in our previous paper 9. As for the case of the


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resistance factors, the correlation between the Sherwood and Reynolds numbers was plotted

2

using the simulation results from the spacer-filled channel. The coefficients, a and b, were

3

obtained as fitting parameters of the exponential equation, Eq. (2), which presents the correlation

4

between the Sherwood and Reynolds numbers. The exponent of the Schmidt number Sc in Eq.

5

(2) was set at 0.25 in this study, based on many previous studies 19-22, although a recent paper

6

reported by Koutsou proposed the exponent 0.4. In the near future, we will investigate the

7

Schmidt number dependence on the mass transfer characteristics.

8 9 10

2.3. Simulation of flow and solute concentration fields in a reverse osmosis membrane module based on porous media modeling The governing equations for the porous media model are two-dimensional continuity,

11

momentum, and mass transport equations in the direction of the flat sheet. In the simulation, the

12

flow and mass transfer in the height direction were not solved numerically, but were dealt with

13

theoretically, as explained in Section 2.4. Numerical computations were employed in both the

14

feed and permeate sides.

15

∇ ⋅ ( ρu ) = S v

(3)

16

∇ ⋅ ( ρ uu ) = −∇ p + ∇ ⋅ ( µ ∇ u ) + S p

(4)

17

∇ ⋅ ( ρum) = ∇ ⋅ ( ρD∇m) + Ss

(5)

18

The coarse-graining procedure proposed in this study added the source term into the

19

momentum equation Eq (4). The source term in Eq (4), Sp [kg m–2 s–2], represents the momentum

20

change corresponding to the energy loss between the fluid and spacers. When substituting Eq (1)

21

into Eq (4), the resulting equation is represented by Eq (6).


8

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1

1 µ  S p = − u + C ρ u u  2 α 

2

The source terms in Eq (3), Sv [kg m–3 s–1], and Eq. (5), Ss [s–1], are the fluid and solute mass

(6)

3

changes resulting from permeation, respectively.

4

Sv = −

J v ρA V

(7)

5

Ss = −

J s ρs A V

(8)

6

Jv [m3 m–2 s–1] and Js [m3 m–2 s–1] are the water and solute permeation fluxes, respectively. The

7

calculation procedure is explained in next section. V [m3 m–1] is the cell area adjacent to the RO

8

membrane.

9 10

2.4. Permeation model The present permeation model is essentially the same as in the previous study 9. It is the

11

numerical simulation of a spacer-filled channel. Based on non-equilibrium thermodynamics, the

12

water flux, Jv, and solute permeation flux, Js, are given by the following equations 23,24 :

13

J v = Lp {p f − p p − (π w − π p )}

(9)

14

J s = P ( m w − m p ) − (1 − σ ) m s J v

(10)

15

where Lp [m Pa–1 s–1] is the water permeability coefficient, P [m s–1] is the sodium permeability

16

coefficient, π [Pa] is the osmotic pressure, σ [–] is the sodium reflection coefficient, and ms is the

17

average solute mass fraction between the feed and permeate channel. The subscripts f and p

18

denote the feed and permeate sides, respectively, and w denotes the membrane surface on the


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1

feed side. In order to obtain Jv and Js in Eqs. (9) and (10), the solute mass fraction of the

2

membrane surface on the feed side, mw, is needed.

3 4 5

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The solute mass balance that considers permeation, advection, and diffusion across the RO membrane surfaces is described by the following equation 25. J v mf − D ∇ mf = J v mp

(11)

6

It should be noted that this equation was established under the assumption that seawater density

7

was constant 26. The present simulation considered the local seawater density, as explained in

8

Section 2.5. Therefore, these are mismatched. However, the difference may hardly affect the

9

mass transfer characteristics because the calculated change in the seawater density was quite

10

small. The following equation was obtained by integrating Eq. (11) along the boundary layer

11

thickness.

J vδ D

12

mw = mp + (mb − mp ) exp

13

Here, δ [m] is the boundary layer thickness, and is given as follows:

14

δ=

15

16

(12)

D k

(13)

The mass transfer coefficient, k, is given by the following equation.

k=

ShD dh

(14)

17

Here, dh [m] is the hydraulic diameter ( dh = 2H ) 19, . By substituting Eqs. (13) and (14) into Eq.

18

(12), the solute mass fraction on the membrane surface of the feed side, mw, can be obtained.


10

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J vdh ShD

1

m w = m p + ( m b − m p ) exp

2

The Sherwood number, Sh, in Eq (15) is calculated using Eq. (2), which describes the mass

3

transfer characteristics obtained from the results of the unit cell simulation of the spacer-filled

4

channel.

(15)

5

2.5. Numerical method

6

Figure 2 shows the computational domain of the membrane sheet simulation. Two

7

computational domains were used for the calculation of the feed and permeate channels as shown

8

in Figure 2(b). The fluid flow and solute mass transfer inside the RO membrane were not

9

numerically calculated, but were theoretically calculated, using the permeation model as

10

explained in the previous section. Therefore, the computational domain did not include the RO

11

membrane. Both the feed and permeate channels were two-dimensional flat sheets. The channel

12

heights were used only when algebraic equations, such as Eq. (14), were calculated. The channel

13

heights of the feed and permeate sides were 530 µm and 270 µm, respectively 26. The dimensions

14

of the domain were Lx = 1.0 m and Ly = 1.0 m. The domain was equally divided into 333 × 256

15

meshes, and the mesh size was almost equivalent to the size of the computational domain of the

16

space-filled channel. It should be noted that the mesh used in this study would be superfluous

17

and can be optimized by confirming the mesh size dependency on the numerical results. For the

18

inlet boundary condition of the feed channel (y = 0), the uniform velocity and solute mass

19

fraction were set to 4.22 × 10–2 m s–1 25 and 3.5 × 10–2, respectively. Just like the membrane

20

envelope of a commercial SWM, a wall boundary condition was applied at the x = 0 and x = Lx

21

boundaries. For the permeate channel, the wall boundary condition was applied at the x = Lx, y =

22

0 and y = Ly boundaries 10. The transmembrane pressure at the inlet boundary of the feed side (y


11


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Page 12 of 36

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= 0) was set to 5.0 MPa. The physical properties of the fluid were based on those of seawater.

2

The Miyake formula was used to calculate the osmotic pressure 27. The equations for density,

3

viscosity, and mass diffusivity were taken from previous studies 28. For the flow characteristics

4

of the permeate channel, an experimental formula by Schock and Miquel 19 , which utilized a

5

Toray PEC1000 membrane, was applied.

6

1 2 1 ρu 2 dh −0.8 λ = 13 Re

∇p = λ

(16)

7

Here λ [–] denotes the friction coefficient. The friction coefficients of the permeate channel in

8

the x- and y-directions were equal under the assumption of homogeneous structure in the planar

9

direction. The geometries of the permeate channels of current commercial RO membranes have

10

not been well documented and there have been no CFD studies on their flow characteristics.

11

Thus, the above equation was used in this study. The other parameters of the RO membrane were

12

taken from the previous study 9, as shown in Table 1.

13

Table 1. Physical properties of the RO membrane used in this study 9. Water permeability coefficient

1.08 × 10–11 m Pa–1 s–1

Sodium reflection coefficient

1.00

Sodium permeability coefficient

6.90 × 10–8 m s–1

14 15

Numerical computations were carried out using the commercial CFD software package, Fluent

16

13.0.0. The user-defined function (UDF) was incorporated in order to implement the permeation

17

model described in Section 2.4. The UDF is an optional function in the Fluent software package,


12

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1

programmed by users, which can be dynamically linked to the solver. The numerical procedure

2

for the membrane sheet simulation with the permeation model is as follows:

3

(1) The flow and solute mass fraction fields are updated by numerically solving Eqs. (7)–(9).

4

(2) The local mass transfer coefficient is calculated by substituting the velocity obtained in

5 6 7 8

step 1 into Eqs. (2) and (18). (3) The solute mass fraction for the membrane surface of the feed side is theoretically calculated using Eq. (19). (4) The water and solute permeation fluxes are calculated using Eqs. (13) and (14),

9

respectively. In this case, except for the solute mass fraction calculated above, the pressure

10

and osmotic pressure for the cells of both the feed and permeate sides are required in order

11

to solve these equations. In this calculation, variables with the same position vector

12

between the feed and permeate sides are used.

13 14 15

(5) The source terms in the continuity and mass transport equations are calculated by substituting the water and solute permeation fluxes into Eqs. (11) and (12), respectively. (6) The source term in the momentum equation is calculated using Eq. (10).


13


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Page 14 of 36

1 2

Figure 2. (a) Schematic of the membrane sheet simulated in this study (b) Two-dimensional

3

computational domain and boundary conditions for the membrane sheet simulation.

4 5

3. Results and Discussion

6

To demonstrate the numerical simulation method that was proposed in this study, the effects of

7

the Reynolds number (Re) defined at the inlet boundary of the feed side, the spacer spacing, and

8

the angle between spacers (θ) on the flow and mass transfer characteristics throughout the entire

9

membrane sheet were investigated. The Reynolds number was varied in four steps, and the

10

values used were 4.63, 23.4, 35.3, and 47.0. The normalized distance between spacers, W/d, was

11

also varied in four steps, and the values used were 4.76, 9.52, 14.3, and 19.0. It should be noted


14

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1

that the Reynolds number is much smaller than those in previous studies. This Reynolds number

2

is set under the assumption that the permeate water and recovery rate are 2.3 m3 d–1 and 50% of

3

each membrane leaf, respectively, which are the usual values in current RO systems using SWM

4

modules. The spacer angle was varied in three steps, and the values used were 0.417π, 0.5π, and

5

0.590π. These numerical conditions were the same as those from the previous study 9 that

6

presented a unit cell simulation of a spacer-filled channel.

7

3.1 Resistance factors and mass transfer coefficient

8

Figure 3 shows the fitting curves of resistance factors and Sherwood number obtained using

9

the results of the unit cell simulation of the spacer-filled channel for W/d = 4.76 and θ = 0.417π.

10

∆p/∆y is the pressure drop per unit length in the streamwise direction. The pressure gradient

11

shown in Figure 3(a) is comparable with recent papers reported by Kerabelas et al. 29, 30, although

12

the spacer designs were different. In addition, R2 values (the correlation coefficients) in both

13

figures are greater than 0.999, and this explains the good agreement with the theoretical equation.

14

This implies that the simulation results and analytical method were reasonable. The viscous and

15

inertial resistance factors obtained from fitting were 1.95 × 108 m2 and 1.82 × 103 m–1,

16

respectively. Additionally, the resulting correlation between the Sherwood and Reynolds

17

numbers was as follows:

18

Sh = 1.685 Re 0.406 ⋅ Sc 0.25

(17)

19 20


15


8000

(a)

6000

Δp/Δy [Pa m–1]

4000 Simulated

2000

Fitted

0 0.00

0.01

0.02

0.03

0.04

Superficial velocity [m s–1] 10

(b)

Sh/Sc0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 36

Simulated Fitted

1 1

1

10

100

Re

2

Figure 3. Fittings for the results of the unit cell simulation of the spacer-filled channel (a)

3

Relationship between superficial velocity and pressure drop (b) Relationship between Reynolds

4

and Sherwood numbers (W/d = 4.76 and θ = 0.417π).

5


16

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1


The resistance factors and the mass transfer characteristics of the feed channel for each

2

condition are summarized in Table 2. R2 values for all cases were greater than 0.999, showing

3

that the theoretical equation correlated well with the simulation results. From this, it can be seen

4

that the resistance factors increased with decreasing distance between spacers (cases 1–4) and

5

increasing angle between spacers (cases 1, 5, and 6). These tendencies were consistent with the

6

results for the pressure drop simulated in the previous study 9. The relationship between the

7

Sherwood and Reynolds numbers, and the pressure drop and velocity in the streamwise direction

8

is provided in Supporting Information (Figure S1), which is essentially the same as Table 2.

9 10

Table 2. Viscous and inertial resistance factors and the equations of mass transfer characteristics

11

for the feed channel obtained using fitting of results for each case of the unit cell simulation. Resistance factors in y-direction Distance Angle between between spacers spacers W/d

θ

Viscous resistance factor

Inertial resistance factor

1/α

C2

Equation of mass transfer characteristics

Case 1

4.76

0.417π

1.95×108

1.82×103

Sh = 1.685Re0.406Sc0.25

Case 2

9.52

0.417π

1.02×108

5.91×102

Sh = 1.392Re0.406Sc0.25

Case 3

14.29

0.417π

7.99×107

3.37×102

Sh = 1.253Re0.399Sc0.25

Case 4

19.05

0.417π

7.03×107

2.14×102

Sh = 1.124Re0.404Sc0.25

Case 5

4.76

0.500π

2.64×108

4.32×103

Sh = 1.487Re0.408Sc0.25

Case 6

4.76

0.590π

4.23×108

1.20×104

Sh = 1.226Re0.389Sc0.25

12


17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

3.2 Distribution of flow, solute concentration, and water flux in the membrane module

2

Figure 4 shows the pressure distribution in the feed channel (Figure 4(a)), the water

Page 18 of 36

3

permeation flux across the RO membrane (Figure 4(b)), the solute mass fraction in the feed

4

channel (Figure 4(c)), and the velocity vector and pressure distribution in the permeate channel

5

(Figure 4(d)). The results of case 1 were used as an example because the distribution profiles

6

were quite similar for the other cases, although the absolute values were different. The results of

7

other cases were provided in the Supporting Information (Figure S2).

8

As seen in the pressure distribution for the feed channel (Figure 4(a)), the pressure decreased

9

in the streamwise direction. This is mainly because of the shear stress caused by the presence of

10

the spacers 9. Furthermore, the spacing between the contour lines also increased in the

11

streamwise direction. This is because the flow rate in the feed channel decreased in the

12

streamwise direction because of permeation. The tendensies were observed in recent papers. 29, 30.

13

The water flux across the RO membrane was higher and lower at the upstream and downstream,

14

respectively (Figure 4(b)). This is because the transmembrane pressure, which is the driving

15

force for water permeation, decreases due to the pressure drop in the feed channel, as shown in

16

Figure 4(a). As shown in Figure 4(c), the solute mass fraction in the feed channel increased in

17

the streamwise direction because the solute was concentrated by the permeation process. The

18

spacing between contour lines became larger in the streamwise direction. This was because the

19

concentration rate decreased in the streamwise direction due to the water flux decrease, as seen

20

in Figure 4(b). From the velocity vector in the permeate channel (Figure 4(d)), the velocity was

21

higher near the exit of the permeate channel and upstream from the feed channel. This was

22

because the flow rate in the permeate channel increased towards the exit of the channel, with the

23

permeate and permeation flux increasing upstream from the feed channel, as shown in Figure


18

Page 19 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

4(b). On the other hand, the pressure in the permeate channel showed the highest value at x = Lx,

2

y = 0. It should be noted that the pressure in the permeate channel was much smaller than the

3

present transmembrane pressure of 5.0 MPa at the inlet of the feed channel. Thus, the pressure

4

distribution hardly affected the simulation results under the present condition.

5

6 7

Figure 4. Distributions for simulated channels (a) Pressure in the feed channel (b) Water flux

8

across the membrane surface (c) Solute mass fraction in the feed channel (d) Velocity vector and

9

pressure in the permeate channel.


19


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

3.3 Effect of spacer arrangement on performance

2

The effect of spacer arrangement on performance was examined. Figure 5 shows the

Page 20 of 36

3

relationship between the pressure drop between the inlet and outlet of the feed channel, and the

4

water permeation rate. It is well known that the relationship between pressure drop and mass

5

transfer enhancement is usually a trade-off 19. However, the spacer arrangement and geometric

6

design may affect their ratio. The water permeation rate was obtained by integration of the water

7

flux through the surface area of the membrane. The spacer performance was improved when the

8

plots were located in the left and upper regions of the figure, representing a lower pressure drop

9

and higher water permeation rate.

10

As can be seen in this figure, there are two scenarios that show correlations. Increasing the

11

distance between spacers brings about a decrease in pressure drop and water permeation rate,

12

illustrating the usual trade-off correlation. On the other hand, increasing the angle between

13

spacers decreases the water permeation rate, and therefore, the pressure drop becomes large.

14

Generally, the mass transfer coefficient increases when the pressure drop increases. These

15

tendencies were consistent with the relationship between Sherwood and Reynolds numbers, and

16

pressure drop and streamwise velocity shown in Figure S1, indicating that the results of the

17

membrane sheet simulation were significantly affected by the flow and mass transfer

18

characteristics derived from the unit cell simulation. The water flux was mainly affected by the

19

osmotic pressure difference under the present condition. Comparing the distribution of the solute

20

mass fraction (Figures 4(b) and S2) in feed channel (bulk), the significant changes can not be

21

observed. However, the solute mass fraction on the membrane surface differed, because the

22

Sherwood numbers (mass transfer characteristics) were varied as shown in Table 2 and Figure S1.

23

The change in the Sherwood number brings about the change in the solute mass fraction on the


20

Page 21 of 36

1

membrane surface as shown in Eq.15. Consequently, the water flux was affected as described in

2

Eq. 9.

3 Water permeation rate [m3 s–1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


9.0E-06–6 9.0×10 Case 1

8.8E-06–6 8.8×10

Case 5

8.6E-06–6 8.6×10 Case 2

8.4E-06–6 8.4×10

8.2×10 8.2E-06–6

Case 3

8.0×10 8.0E-06–6

Case 4

7.8×10 7.8E-06–6 0

5

10

Case 6

15

20

25

Pressure drop [kPa]

4 5

Figure 5. Relationship between pressure drop in the feed channel and water permeation rate.

6

Black circle, case 1 (control). Blue diamonds, effect of distance between spacers. Green triangles,

7

effect of angle between spacers. Details of each case are shown in Table 2.

8 9 10

Conclusions In this study, we constructed a new simulation method for calculating the flow and solute mass

11

transfer fields in a membrane sheet comprised of a pair of feed and permeate channels. The

12

spacer-filled channel was coarse-grained as two-dimensional porous media, because the

13

simulation method for the spacer-filled channel cannot be directly extended owing to its

14

computational load limitations. The flow and mass transfer characteristics obtained from the unit

15

cell simulation of the spacer-filled channel were coordinated with the results from the membrane


21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 36

1

sheet simulation. The simulation method can quantitatively predict the effect of the millimeter-

2

scale spacer arrangement on the flow and solute mass fraction for the meter-scale channels and

3

performance. In addition, the method could be applied to various conditions, e.g. low reflection

4

coefficient, small transmembrne pressure, which were not shown in this study. Although the

5

simulation results were not validated, they should be compared with the experimental results of

6

future work. This simulation method will be a useful tool for optimizing the design of a SWM

7

module.

8 9

FIGURE CAPTIONS

10

Figure 1. Computational domain and boundary conditions for the unit cell simulation of a

11

spacer-filled channel 9.

12

Figure 2. (a) Schematic of the membrane sheet simulated in this study (b) Two-dimensional

13

computational domain and boundary conditions for the membrane sheet simulation.

14

Figure 3. Fittings for the results of the unit cell simulation of the spacer-filled channel (a)

15

Relationship between superficial velocity and pressure drop (b) Relationship between Reynolds

16

and Sherwood numbers (W/d = 4.76 and θ = 0.417π).

17

Figure 4. Distributions for simulated channels (a) Pressure in the feed channel (b) Water flux

18

across the membrane surface (c) Solute mass fraction in the feed channel (d) Velocity vector and

19

pressure in the permeate channel.


22

Page 23 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

Figure 5. Relationship between pressure drop in the feed channel and water permeation rate.

2

Black circle, case 1 (control). Blue diamonds, effect of distance between spacers. Green triangles,

3

effect of angle between spacers. Details of each case are shown in Table 2.

4

Table 1. Physical properties of the RO membrane used in this study 9. Water permeability coefficient

1.08 × 10–11 m Pa–1 s–1

Sodium reflection coefficient

1.00

Sodium permeability coefficient

6.90 × 10–8 m s–1

5 6

Table 2. Viscous and inertial resistance factors and the equations of mass transfer characteristics

7

for the feed channel obtained using fitting of results for each case of the unit cell simulation. Resistance factors Distance Angle between between spacers spacers W/d

θ

Viscous resistance factor

Inertial resistance factor

1/α

C2

Equation of mass transfer characteristics

Case 1

4.76

0.417π

1.95×108

1.82×103

Sh = 1.685Re0.406Sc0.25

Case 2

9.52

0.417π

1.02×108

5.91×102

Sh = 1.392Re0.406Sc0.25

Case 3

14.29

0.417π

7.99×107

3.37×102

Sh = 1.253Re0.399Sc0.25

Case 4

19.05

0.417π

7.03×107

2.14×102

Sh = 1.124Re0.404Sc0.25

Case 5

4.76

0.500π

2.64×108

4.32×103

Sh = 1.487Re0.408Sc0.25

Case 6

4.76

0.590π

4.23×108

1.20×104

Sh = 1.226Re0.389Sc0.25

8 9 10

ACKNOWLEDGEMENTS


23


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Page 24 of 36

1

This research was funded by the Japan Society for the Promotion of Science (JSPS) through the

2

Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST

3

Program), initiated by the Council for Science and Technology Policy (CSTP). Furthermore, the

4

authors greatly appreciate the support of the Mega-ton Water System Project led by Dr. Kurihara,

5

Toray Industries, Inc.

6 7

NOMENCLATURE

8

C = inertial viscous factor, m–1

9

d = spacer diameter, m

10

D = mass diffusivity, m2 s–1

11

dh = hydraulic diameter, m

12

H = channel height, m

13

J = permeation flux, m s–1

14

k = mass transfer coefficient, m s–1

15

km = mean mass transfer coefficient, m s–1

16

L = channel length in the y-direction, m

17

Lx = length of the computational domain of the membrane sheet simulation in the x-direction, m

18

Ly = length of the computational domain of the membrane sheet simulation in the y-direction, m

19

Lp = water permeability coefficient, m Pa–1 s–1

20

m = solute mass fraction

21

p = pressure, Pa

22

P = sodium permeability coefficient, m s–1

23

Re = Reynolds number


24

Page 25 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

Sv = source term for the continuity equation, kg m–3 s–1

2

Ss = source term for the transport equation, s–1

3

Sc = Schmidt number

4

Sh = Sherwood number

5

t = time, s

6

T = temperature, ºC

7

u = fluid velocity vector, m s–1

8

V = cell volume, m3 m–1

9

W = channel length in the x-direction, m

10

x = x-coordinate, m

11

y = y-coordinate, m

12

z = z-coordinate, m

13 14

Greek letters

15

α = viscous resistance factor, m2

16

δ = boundary layer thickness, m

17

π = osmotic pressure, Pa

18

∆p/∆y = pressure difference per unit length in the streamwise direction, Pa m–1

19

λ = friction coefficient

20

µ = fluid viscosity, Pa s

21

ρ = fluid density, kg m–3

22

σ = sodium reflection coefficient

23


25


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Subscripts

2

f = feed side

3

p = permeate side

4

s = solute

5

v = fluid

6

b = bulk

7

w = membrane surface of the feed side

Page 26 of 36

8 9 10

REFERENCES 1.

11 12

desalination: water sources, technology, and today's challenges. Water Res. 2009, 43, 2317. 2.

13 14

Greenlee, L.F.; Lawler, D.F.; Freeman, B.D.; Marrot, B.; Moulin, P. Reverse osmosis

Khawaji, A.D.; Kutubkhanah, I.K.; Wie, J.M. Advances in seawater desalination technologies. Desalination 2008, 221, 47.

3.

Shakaib, M.; Hasani, S.M.F.; Mahmood, M. Study on the effects of spacer geometry in

15

membrane feed channels using three- dimensional computational flow modeling. J. Membr.

16

Sci. 2007, 297, 74.

17

4.

18 19

Li, F.; Meindersma, W.; de Haan, A.B.; Reith, T. Novel spacers for mass transfer enhancement in membrane separations. J. Membr. Sci. 2005, 253, 1.

5.

Karode, S.K.; Kumar, A. Flow visualization through spacer filled channels by

20

computational fluid dynamics. I. Pressure drop and shear rate calculations for flat sheet

21

geometry. J. Membr. Sci. 2001, 193, 69.


26

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1


6.

Santos, J.L.C.; Geraldes, V.M.; Velizarov, S.; Crespo, J.G. Investigation of flow patterns

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and mass transfer in membrane module channels filled with flow aligned spacers using

3

computational fluid dynamics (CFD). J. Membr. Sci. 2007, 305, 103.

4

7.

Koutsou, C.P.; Yiantsios, S.G.; Karabelas, A.J. A numerical and experimental study of mass

5

transfer in spacer-filled channels: Effects of spacer geometrical characteristics and Schmidt

6

number. J. Membr. Sci. 2009, 326, 234.

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8.

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Fimbres-Weihs, G.A.; Wiley, D.E. Numerical study of mass transfer in three-dimensional spacer-filled narrow channels with steady flow. J. Membr. Sci. 2007, 306, 228.

9.

Ishigami, T.; Matsuyama, H. Numerical modeling of concentration polarization in spacer-

10

filled channel with permeation across reverse osmosis membrane. Ind. Eng. Chem. Res.

11

2015, 54, 1665.

12 13 14

10. Kostoglou, M.; Karabelas, A.J. Mathematical analysis of the meso-scale flow field in spiralwound membrane modules. Ind. Eng. Chem. Res. 2011, 50, 4653. 11. Kostoglou, M.; Karabelas, A.J. A mathematical study of the evolution of fouling and

15

operating parameters throughout membrane sheets comprising spiral wound modules. Chem.

16

Eng. J. 2012, 187, 222.

17 18

19 20

12. Kostoglou, M.; Karabelas, A.J. Comprehensive simulation of flat-sheet membrane element performance in steady state desalination. Desalination 2013, 316, 91. 13. Kostoglou, M.; Karabelas, A.J. Modeling scale formation in flat‐sheet membrane modules during water desalination. AIChE J. 2013, 59, 2917.


27


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14. Marcovecchio, M.G.; Aguirre, P.A.; Scenna, N.J. Global optimal design of reverse osmosis

2

networks for seawater desalination: modeling and algorithm. Desalination 2005, 184, 259.

3

15. Avlonitis, S.A.; Pappas, M.; Moutesidis, K. A unified model for the detailed investigation

4 5 6 7

of membrane modules and RO plants performance. Desalination 2007, 203, 218. 16. Oh, H.J.; Hwang, T.M.; Lee, S. A simplified simulation model of RO systems for seawater desalination. Desalination 2009, 238, 128. 17. Hayes, A.M.; Khan, J.A.; Shaaban, A.H.; Spearing, I.G. The thermal modeling of a matrix

8

heat exchanger using a porous medium and the thermal nonequilibrium model. Int. J. Therm.

9

Sci. 2008, 47, 1306.

10

18. Wang, Y.; Brannock, M.; Cox, S.; Leslie, G. CFD simulations of membrane filtration zone

11

in a submerged hollow fibre membrane bioreactor using a porous media approach. J.

12

Membr. Sci. 2010, 363, 57.

13 14

19. Schock, G.; Miguel, A. Mass transfer and pressure loss in spiral wound modules. Desalination 1987, 64, 339.

15

20. Ikeda, K.; Kimura, S.; Ueyama, K. Characterization of a nanofiltration membrane used for

16

demineralization of underground brackish water by application of transport equations.

17

Membrane 1998, 23, 266.

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21. Taniguchi, M. Establishment of analysis method for a seawater desalination plant adopting a brine conversion two-stage RO process. Membrane 2002, 27, 180.


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


22. Derzansky, L.J.; Gill, W.N. Mechanisms of brine-side mass transfer in a horizontal reverse osmosis tubular membrane. AIChE J. 1974, 20, 751. 23. Katchalsky, A.; Curran, P. Nonequilibrium Thermodynamics in Biophysics; Harward University Press, Cambridge, MA, 1965. 24. Spiegler, K.S.; Kedem, O. Thermodynamics of hyperfiltration (reverse osmosis): Criteria for efficient membranes. Desalination 1966, 1, 311. 25. Kimura, S.; Sourirajan, S. Analysis of data in reverse osmosis with porous cellulose acetate membranes used. AIChE J. 1967, 13, 497. 26. Minegishi, S.; Kihara, M.; Nakanishi, T. Spiral type separation membrane element. Japanese Patent JP2000000437, 2000. 27. Miyake, Y. Freezing point, osmotic pressure, boiling point and vapor pressure of sea water. Bull. Chem. Soc. Japan 1939, 14, 58. 28. Sekino, M. Performance data analysis for hollow-fiber reverse-osmosis modules in seawater desalination plants. Kagaku Kougaku Ronbunsyu 1994, 20, 574. 29. Karabelas, A.J.; Koutsou, C.P.; Kostoglou, M. The effect of spiral wound membrane

16

element design characteristics on its performance in steady state desalination — A

17

parametric study. Desalination 2014, 332, 76.

18

30. Koutsou, C.P.; Karabelas, A.J.; Kostoglou, M. Membrane desalination under constant water

19

recovery – The effect of module design parameters on system performance. Sep. Purif.

20

Technol. 2015, 147, 90.


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Page 30 of 36

1 2

Table of Contents

3


30

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Figure 1 90x126mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2 140x104mm (300 x 300 DPI)


Page 32 of 36

Page 33 of 36

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Figure 3 90x163mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4 140x124mm (300 x 300 DPI)


Page 34 of 36

Page 35 of 36

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Figure 5 90x75mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table of Contents 85x26mm (300 x 300 DPI)


Page 36 of 36

Multiscale Simulation Method for Flow and Mass-Transfer

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