Mathematical model of flat sheet membrane modules for FO process: Plate-and-frame module and spiral-wound module

Journal of Membrane Science 379 (2011) 403–415

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Mathematical model of flat sheet membrane modules for FO process: Plate-and-frame module and spiral-wound module B. Gu a , D.Y. Kim a , J.H. Kim b , D.R. Yang a,∗ a b

Department of Chemical and Biological Engineering, Korea University, Seoul 136-713, Republic of Korea Department of Environmental Science and Engineering, Gwangju Institute of Science and Technology (GIST), Gwangju 500-712, Republic of Korea

a r t i c l e

i n f o

Article history: Received 23 March 2011 Received in revised form 1 June 2011 Accepted 7 June 2011 Available online 14 June 2011 Keywords: Forward osmosis Modelling Plate-and-frame module Modified spiral-wound module Concentration polarization

a b s t r a c t The forward osmosis process is considered a promising desalination method due to its low energy requirement compared to other methods. In this study, modelling and simulations for a plate-and-frame and a modified spiral-wound module are carried out for the FO process. The mathematical models consist of mass balance, a permeate flux model, and concentration polarization equations. The plate-and-frame model is formulated with consideration of flow directions, and the modified spiral-wound model is formulated with consideration of its geometric characteristics. These two sets of model equations are numerically and iteratively integrated since they are implicit and highly non-linear. The simulation for both modules was conducted by varying 4 types of operating conditions: volumetric flow rate of the feed and the draw solution, the concentration of the draw solution, flow direction, and the membrane orientation. The results for various conditions are also compared. In future research, the developed model could be applied for designing FO modules and finding optimal operating conditions. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Global water shortage has been an important issue in past decades as the population has increased drastically, and many researchers have tried to solve this problem through sea water desalination. Many desalination methods have been practiced and researched so far. Due to the high cost of membranes in the past, thermal methods such as Multi-Effect Distillation (MED) and Multi-Stage Flash (MSF) processes were widely used. However, membrane-based methods by forward osmosis (FO) and membrane distillation (MD) as well as reverse osmosis (RO) have been spotlighted in recent days since they require less energy than the thermal methods and the membranes are now readily available [1]. Of the membrane-based methods, the FO process has recently received considerable attention. This is because it does not require any external energy for separation such as hydraulic pressure in RO process and sensible heating energy in MD process. The FO process simply separates water from saline water by introducing a more concentrated solution than sea water in the opposite side of a semi-permeable membrane. As a result, water permeation occurs through the membrane between two solutions in trans-membrane direction [1,2]. Compared to the other desalination processes, the FO process has distinct advantages: it does not require thermal resources,

∗ Corresponding author. Tel.: +82 2 3290 3298; fax: +82 2 3290 3290. E-mail address: [email protected] (D.R. Yang). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.06.012

which makes the membrane-based method superior to the thermal ones. Moreover, unlike the RO process, it proceeds spontaneously without the application of hydraulic pressure by a high pressure pump; consequently, it also uses less energy than the RO process and the operational equipment used in the FO plant is much simpler because there is no need for additional pressure handling equipment. In addition, the merits of not applying hydraulic pressure include a lower propensity for membrane fouling and a higher rejection of salts [3–6,21]. Despite the advantages of the FO process, limitations still exist. One of the major concerns is the selection of the more concentrated solution than sea water, called “draw solution” [7,8]. After passing through the membrane process, the water product is recovered from the diluted draw solution through an extra separation process and the re-concentrated draw solution is sent to the membrane process again for recycling. Therefore, the draw solution should be carefully chosen with respect to non-toxicity and easy recovery. Furthermore, the separation process also needs to be attentively designed with consideration of the low energy use and the uncomplicated separation procedure for the draw solution. The development of the unique membrane and the membrane module to FO process should also be paid close attention [6,9–13,22]. In particular, concentration polarization that causes the flux reduction in the RO process also happens both at the external surface of the membrane and inside of the membrane structure (ECP and ICP). Since the internal concentration polarization (ICP) caused by asymmetric membranes plays significant role in the flux reduction in the FO process, unlike in the RO process, the desired characteris-

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tics of the FO membrane have been studied to manufacture a novel FO membrane that overcomes the ICP effect and enhances the performance [9,11,13]. The other difference from the RO process is the inlet flow streams, which consist of two different solutions. For these reasons, the design of the membrane module for the FO process should be either modified or newly suggested in order to utilize it exclusively. There are two possible types of membrane modules that can be applied to the FO module: flat sheet membranes and tubular membranes (tubes and hollow fibers) [3,14]. The tubular type membrane module has higher packing density than the modules packaged with the flat sheet membrane, but it is not as common as the flat sheet membrane in the membrane market due to excessive fouling and membrane integrity problems. Accordingly, many researchers have focused on flat sheet membrane modules for lab-scale and large-scale applications. The available candidates using the flat sheet membranes are the plate-and-frame and the spiral-wound modules, which has been modified especially for FO the process that involves two inlet streams in contrast to only one for RO process [3,15,16,23]. In this work, for the reasons stated earlier, modelling and simulations are conducted for FO modules with the flat sheet membrane and the effects of each module on performance are presented under various operating conditions. The models contain ECP and ICP equations for each membrane orientation so that the effects of CP’s and the membrane orientation are observed at the same time. Accordingly, it suggests a proper configuration for the desired performance that is less affected by CP phenomena. Especially for the modified spiral-wound module, the structural effect caused by winding the flat sheet membranes is investigated and reflected in the model. 2. Methods 2.1. Theoretical background 2.1.1. General flux equation The water flux through the membrane can be expressed as the proportionality of the difference in osmotic pressures as follows: Jw = A(D − F )

(1)

−Js = B(CD − CF )

(2)

where Jw is the water flux, Js the salt flux, A the water permeability, B the salt permeability, D and F the osmotic pressure of the draw solution and the feed sea water, CD and CF the concentrations of the draw solution and the feed sea water, respectively. The osmotic pressure is directly related to the concentration of each solution in the modified van’t Hoff formula [17]: Nion Rg TC  = ˛C = Mw

Fig. 1. Illustration of osmotic pressure profile caused by ECP for a dense symmetric membrane.

selective layer, it always happens in any membrane process, including FO [18]. Fig. 1 schematically shows how the osmotic pressure changes in the vicinity of the membrane surface. Even though the osmotic pressure difference of the two solutions is large, the effective difference decreases due to elevation of the feed sea water concentration and reduction of the draw solution concentration. These changes in the concentrations of the draw solution and the feed sea water are referred to as dilutive ECP and concentrative ECP, respectively. Based on boundary layer film theory, interface osmotic pressures are expressed as follows [18]: F,m = exp F,b

w



k

D,m Jw = exp − D,b k

(4)

 (5)

where Jw is the water flux, k the mass transfer coefficient, D,b and F,b the osmotic pressures of the draw solution and the feed in the bulk, respectively, and D,m and F,m the osmotic pressures on the membrane surface. The mass transfer coefficient, k, can be correlated with Sherwood number [18]: k=

(3)

J 

Sh · D dh



Sh = 1.85 ReSc

(6) dh L

0.33 (laminar flow)

(7)

Sh = 0.04Re0.75 Sc 0.33 (turbulent flow)

(8)

where Nion is the ionisation number in the water, Rg the ideal gas constant, T the temperature, Mw the molecular weight, and C the salt concentration. That is, the water flux has the amount proportional to the concentration difference of two solutions. Contrary to the expected water flux, the measure water flux is much lower due to the effects of external and internal concentration polarization (ECP and ICP). Hence, ECP and ICP phenomena need to be considered in the water flux equation.

where Sh is Sherwood number, Re is the Reynolds number, Sc is the Schmidt number, dh is the hydraulic diameter, L is the length of the channel. When the asymmetric membrane is used, ICP should be considered an important factor causing the flux reduction in the use of an asymmetric membrane.

2.1.2. External concentration polarization (ECP) equation Concentration polarization phenomena have been emphasized in membrane processes since they are the main reasons for flux reduction. As the permeation drag force in trans-membrane direction induces concentration polarization at the interface of the

2.1.3. Internal concentration polarization equation Since the support layer is much more porous than the active membrane, it allows both water and salts to penetrate the structure, but hinders the salts from diffusing freely. Therefore, there exists a gradient of concentrations inside. This phenomenon is called ICP.

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Fig. 2. Illustrations of dilutive ICP (left) and concentrative ICP (right) in an asymmetric membrane (AL-FW mode and AL-DS mode, respectively).

According to the asymmetric membrane orientation, two different phenomena can occur: concentrative ICP or dilutive ICP. Fig. 2 describes both ICP’s. Dilutive ICP occurs when the feed sea water is placed in contact with the active layer (AL-FW mode), while concentrative ICP occurs when the draw solution is in contact with it (AL-DS mode) [8,18]. At the interface of the support layer and the solution, ECP phenomenon can be negligible due to smaller resistance compared to others for the salts to diffuse into the porous structure. Thus, the effective osmotic pressure difference becomes much smaller than the original difference between the bulk solutions. In order to identify the flux behaviour in the presence of ICP, Loeb et al. [9,19] derived the equations by relating the water flux and introducing the property parameters of the membrane: K= K=

1 Jw

1 Jw

B + AD,b ln (Concentrative ICP) B + Jw + AF,m ln

B + AD,m − Jw (Dilutive ICP) B + AF,b

t K= Dε

Above all, the mass balance equation must be set before applying the flux and CP equations. For both modules, a very small control volume is considered. It comprises two channels split by the membrane as illustrated in Fig. 3. Using the general overall and component balance, the following partial differential equations will be used: −

∂uF,y ∂uF,z ∂uF,x Jw =0 − − − H ∂x ∂y ∂z

(12)

−

∂uD,y ∂uD,z ∂uD,x Jw =0 − − + H ∂x ∂y ∂z

(13)

 D

(9)



(10) D (11)

where K is the salt resistivity to diffusion inside the porous support layer. Here, D is the diffusivity, and t, , and ε are the thickness, tortuosity, and porosity of the porous support layer, respectively. Using Eqs. (9) and (10), the osmotic pressure at the interface of the active layer and the porous support layer can be determined. The final flux equation is very difficult to solve analytically since the water flux is dependent on itself because of inclusion ECP and ICP terms. In the next section, modelling for the numerical procedure based on these equations is presented in detail. 2.2. Modelling and numerical solution procedure In this section, the methods for modelling the plate-and-frame and the modified spiral-wound modules are discussed. The models are developed in two dimensions on the basis of the general flux equations considering ECP and ICP. The developed mathematical models should be solved iteratively and numerically for several reasons: first, the equations in the model are implicit and highly nonlinear since it considers ICP and ECP. Second, there is a crosscurrent in the feed and the draw solution flows because the FO process has two inlet streams.

∂2 CF ∂2 CF ∂2 CF + + 2 2 ∂x ∂y ∂z 2



∂2 CD ∂2 CD ∂2 CD + + 2 2 ∂x ∂y ∂z 2

− uF,x

∂CF ∂CF ∂CF Js − uF,y − uF,z − =0 H ∂x ∂y ∂z (14)

 − uD,x +

∂CD ∂CD ∂CD − uD,y − uD,z ∂x ∂y ∂z

Js =0 H

(15)

Since the channel height is small and the mixing is enhanced by spacers, a velocity profile in y-direction is assumed negligible. Also, diffusive transport happens much less than convective one, so that terms for the diffusive transport in Eqs. (14) and (15) are neglected in modelling. To solve the partial differential model equations, the membrane is divided into small segments, and their sizes are chosen to be small enough until the change of the calculated results is tolerable. Based on the bulk concentration obtained from Eqs. (14) to (15), ECP and ICP can be estimated. Fig. 4 describes the concentration profiles of the solutions according to the membrane orientation. As mentioned earlier, due to little resistance to penetrate the porous structure, ECP generated on the porous support layer side is regarded as negligible, so that the flat profiles are shown in the vicinity of the porous support layer in Fig. 4. They can be calculated by the modification of Eq. (4) and (9) as follows: CF,m = CF exp

J  w

k

(16)

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Fig. 3. Schematic diagram of a small segment for mass balance equation (top left: modified spiral-wound, bottom left: plate-and-frame module).

CD,m = CF,m −

CF,m − CD exp(−KJw ) 1 − (B/Jw )[exp(−KJw ) − 1]

(17)

Similarly, the concentrations by dilutive ECP and concentrative ICP in Fig. 4(b) are expressed as follows:

 J  w

CD,m = CD exp −

CF,m = CD,m −

k

CD,m − CF exp(KJw ) 1 + (B/Jw )[exp(KJw ) − 1]

(18)

(19)

Thus, the effective osmotic pressure can be calculated by Eq. (3) for each solution with the calculated concentration; accordingly,

the effective driving force is determined to estimate the water flux. The final flux equation is as follows: Jw = A(˛D CD,m − ˛F CF,m )

(20)

−Js = B(CD,m − CF,m )

(21)

As mentioned earlier, the model equations should be solved iteratively and numerically due to their implicit character and nonlinearity. Specifically speaking, Jw and Js are a function of CD,m and CF,m ; simultaneously, CD,m and CF,m are a function of Jw and Js . Based on the model equations stated up to this point, more specific description of modelling for the plate-and-frame and the modified spiral-wound module will be discussed.

Fig. 4. Concentration profiles considering ECP and ICP depending on the membrane orientation. (a) Concentrative ECP and dilutive ICP (AL-FW mode) and (b) dilutive ECP and concentrative ICP (AL-DS mode).

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Fig. 5. Schematic diagram of plate-and-frame module and possible flow directions of two solutions.

2.2.1. Modelling plate-and-frame module and modified spiral-wound module A plate-and-frame module is designed with flat sheet membranes and spacers alternately staked into a metal frame as shown in Fig. 5. The model also shows three possible modes of the plateand-frame module depending on the relation of two solutions’ directions. These three modes might perform differently under the same operating conditions. A spiral-wound module is typically used for the RO process and is manufactured by winding the flat sheet membranes around a central pipe. In order to utilize this module in the FO process, modification is required to make two inlet streams flow. Fig. 6 describes the modified spiral-wound module. The blocking in the middle of the central pipe and the additional glue line at the centerline of the membranes enable two inlet streams to pass through the module. That is, one stream flows outside of the envelope in the direction of length, and the other flows inside the envelope. Thus, the flow location should be decided when designing the FO process. Moreover, in the design of the asymmetric membrane module, it is also important to decide the orientation of the membrane. Once the membrane orientation is chosen for the process, either dilu-

tive or concentrative ECP and ICP are correspondingly determined according to the solution in contact with the active layer. Accordingly, by using the models, the dominant phenomena in the flux reduction can be determined for each orientation and the location of the streams. In order to solve the model equations for both modules, the algorithm for the numerical and iterative procedure is described in Fig. 7. For the numerical method, the width of the membrane is divided into m segments of the size, x, and the length is divided into n segments of the size, y. The calculation is progressing in the row vector form of the width segments along the length direction; that is, the i-th row of 1-by-m vector is solved as varying from i = 1 to i = n. To explain details, the fluxes of the first raw vector are initially guessed properly and used to calculate the flow velocities and the concentrations with Eqs. (12)–(15). After that, the effective concentrations reflecting ECP and ICP are calculated by Eqs. (16)–(19), and then the raw vector of the fluxes are recalculated with the effective ones by Eqs. (20) and (21). Finally, the whole procedure is repeated again until the discrepancies for every element are tolerable. After that, the next row vector is repeatedly solved in the same way up to the last row. More details on the solution procedure in each segment for both modules will be discussed in the next section. 2.2.1.1. Discretization procedure of plate-and-frame module. For the co-current flow direction in the plate-and-frame module, discretized equations of Eqs. (12)–(15) for each segment are as follows: uF (i, j) = uF (i, j − 1) − Jw ∗ (i, j)

 CF (i, j) =

CF (i, j − 1) · uF (i, j − 1) − Js∗ (i, j)(x/0.5H) uF (i, j)

∗ uD (i, j) = uD (i, j − 1) + Jw (i, j)

 CD (i, j) = Fig. 6. Schematic diagram of modified spiral-wound module.

x 0.5H

(22)



x 0.5H

CD (i, j − 1) · uD (i, j − 1) + Js∗ (i, j)(x/0.5H) uD (i, j)

(23)

(24)

 (25)

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Fig. 7. Algorithm for water flux calculation.

In the other flow directions, the same procedure is employed, but they require a slight modification to consider the different flow direction. In case of counter-current flow, Eqs. (22)–(25) can also be used by applying the shooting method that can solve a boundary value problem caused by counter-current flow of the draw solution. In case of cross-current flow, since the two steams do not meet in parallel but perpendicularly, the equations for the draw solution should be modified. In Eqs. (24) and (25), the previous information, uD (i,j − 1) and CD (i,j − 1) should be changed to uD (i − 1,j) and CD (i − 1,j), respectively. Also, x is converted to y as the cross sectional area changes due to the different flow direction of the draw solution.

2.2.1.2. Discretization procedure of modified spiral-wound module. First of all, the flow inside the modified spiral-wound module is assumed to have only one dominant direction. This assumption implies that the solution outside an envelope flows out along the length direction, and the one inside an envelope has the length and the width direction depending on the flow location inside the membrane as illustrated in Fig. 6. For the modified spiral-wound module, by winding the membrane along the -direction, the distinguished regions are built up outside the envelope, as depicted in Fig. 8. In regions A and C, permeation between the inside and outside of the envelope happens only through either the inner or

outer membrane, respectively. On the other hand, in region B, permeation can occur through both inner and outer membranes. If the membrane is divided into the same segment size in the direction of the width, it is not convenient to find the average of the fluxes through the inner and outer membranes along the segments because the length of the two membranes in contact with region B is different. To consider this effect, the width is divided into a vector of variant sizes by using an invariant segment size, . The segment width, x, is calculated by the relation to the radius of the wound membrane: x = r 

(26)

where r is the radius of the wound membrane at . Since r is varying along the membrane length, x is not constant. Moreover, because there exist several kinds of flow directions due to structural complexity of the modified spiral-module itself, further modifications are necessary from the model equations used in the plate-and-frame module. The model equations should differently be established in two regions: outside an envelope and inside an envelope. Fig. 9 illustrates the discretization of the stretched spiral-wound module for both ‘envelope-outside-zone’ and ‘envelope-inside-zone’. For ‘envelope-outside-zone’ in modified spiral-wound module: The stream flows only along the y-direction outside an envelope;

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Fig. 8. Distinguished regions formed by winding a pair of membranes.

consequently, it is only required to pay attention to the distinguished regions A, B, and C in Fig. 9. For the realization of the model equations in each region, two sheets of membranes which consist of one leaf are independently considered. In region A, only the inner membrane subscripted by 1 contributes to the water permeation. Thus, the discretized equations are expressed as followings: Ac,1 (i, j) = x(j)Hout

(27)

At,1 (i, j) = x(j)y(i)

(28)

∗ Qout,1 (i, j) = uout,1 (i − 1, j) · Ac,1 (i − 1, j) − Jw,1 (i, j) · At,1 (i, j)

(29)

uout,1 (i, j) =

Qout,1 (i, j) Ac,1 (i, j)

(30)

Cout,1 (i, j) =

Cout,1 (i − 1, j) · uout,1 (i − 1, j) · Ac,1 (i − 1, j) − Js,1 ∗ (i, j) · At,1 (i, j) Qout,1 (i, j) (31)

In region B where two sheets of membrane simultaneously contribute to produce the permeate flux, Eq. (27)–(31) are employed for the inner and the outer membrane in the same manner. The location of the outer membrane corresponding to the j-th inner membrane segment at the same angle is placed on the (j–ma )-th segment, where ma is an index where the region B starts. Thus, for the outer membrane subscripted by 2, the flow information at the (j–ma )-th segment should be solved simultaneously with the inner membrane at the j-th segment. In region C, the same equations as used in region A can be used only by changing the inner membrane into the outer membrane. For ‘envelope-inside-zone’ in modified spiral-wound module: In Fig. 9(b), the whole segments inside the envelope are classified into 5 parts according to the flow configuration as well as the change in flow direction. Since each segment will be calculated in sequential manner, the quantities of current and previous segment are required. The following equations are for Part I, for example: Ac (i, j) = y(i)Hin

(32)

At (i, j) = x(j)y(i)

(33)





Qin (i, j) = uin (i, j − 1) · Ac (i, j − 1) + Jw,1 ∗ (i, j) + Jw,2 ∗ (i, j) · At (i, j) (34)

uin (i, j) =

Qin (i, j) Ac (i, j)

(35)

Cin (i, j) =

Cin (i, j − 1) · uin (i, j − 1) · Ac (i, j − 1) + [Js,1 ∗ (i, j) + Js,2 ∗ (i, j)] · At (i, j) Qin (i, j)

(36)

Fig. 9. Discretization for modified spiral-wound module. (a) Envelope-outside-zone 2, (b) envelope-inside-zone, and (c) envelope-outside-zone 1.

As can be seen in Eqs. (32)–(36), the information of previous (i, j − 1)-th segment is required. The other parts can also adopt Eqs.

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(32)–(36), by modifying the indices of (i, j − 1) accordingly considering the flow directions. Using the calculated flow velocity and concentration in each segment, effects of ECP and ICP inside and outside an envelope can be predicted. Eqs. (16)–(19) can be modified according to operating conditions. The following equations are the examples of ECP and ICP concentration calculations when the feed flows inside and the active layer places inside the envelope: Cin,m (i, j) = Cin (i, j) · exp

Cout,m (i, j) = Cin,m (i, j) −

 J ∗ (i, j)  w k

(37)

Cin,m (i, j) − Cout (i, j) · exp(−K · Jw ∗ (i, j))

1 − (B/(Jw ∗ (i, j)))[exp(−K · Jw ∗ (i, j)) − 1] (38)

Again, the water flux and the salt flux should be calculated and the whole procedure should be conducted following the algorithm depicted in Fig. 7: Jw (i, j) = A(˛D Cout,m (i, j) − ˛F Cin,m (i, j))

(39)

−Js (i, j) = B(Cout,m (i, j) − Cin,m (i, j))

(40)

Finally, the flow behaviour and the permeate flux can be obtained by integrating all the segment equations explained above for both plate-and-frame and the modified spiral-wound modules. 3. Results and discussions The simulation results from the models and the methods in the previous sections are discussed. The results for each module are primarily discussed independently, and then the performances in both modules are compared under the same conditions. All the model equations are simulated using MATLAB software. 3.1. Plate-and-frame module The simulation for the plate-and-frame module was conducted by varying four conditions: volumetric flow rate of the feed and the draw solution, the concentration of the draw solution, flow direction, and the membrane orientation. The operation conditions and parameters for the plate-and-frame module are listed in Table 1. The parameters such as water permeability and salt permeability were found in the work by Achilli et al. [20]. In this simulation, the membrane dimensions are chosen as the same for fair comparisons of flow direction. The volumetric flow rate and the draw solution concentration are varying for the simulating effects of operating conditions. Table 2 shows the simulation results under various conditions with fixed feed concentration. As expected, the water flux increases as the volumetric flow gets higher due to the reduction in ECP effect. Also, the higher concentration of the draw solution contributes to the higher water flux since the osmotic pressure difference increases. The effects of flow direction are also shown in Table 2. In both AL-FW and AL-DS mode, two solutions flowing in countercurrent direction results in the highest water flux, whereas the lowest water flux is achieved with co-current direction flows. When the feed and the draw solution meet before dilution in the membrane cell, more water permeates from the feed to the draw solution because of their larger concentration difference. Consequently, the feed solution is highly concentrated and the draw solution is diluted by transfer of water through the membrane, and a drastic decrease in the concentration difference follows. Depending on the characteristic of the flow direction, the rate of reduction in the concentration difference is different. In the co-current direction

Fig. 10. Results in water permeate flux of AL-DS and AL-FW mode depending on the flow directions when CD = 75 kg/m3 . (a) Co-current, (b) cross-current, and (c) counter-current flow direction.

in which the two streams flow in the same direction, concentration of the feed and dilution of the draw solution happen relatively fast. Thus, the co-current direction causes a great reduction in the water flux. Interestingly, however, in the high volumetric flow rate, the discrepancy in the water flux among flow directions is much

B. Gu et al. / Journal of Membrane Science 379 (2011) 403–415

smaller. This is because ECP effects are lessened by the large bulk motion of the higher flow rate. In comparisons of AL-FW mode and AL-DS mode, it is also worth noting that the simulated water flux for the AL-DS mode is larger than for the AL-FW mode in the whole range of the investigated operating conditions. In the AL-FW mode where the porous support layer faces the draw solution, more severe ICP effects appear due to the high concentration of the draw solution, and accordingly, the flux reduction is more noticeable. Fig. 10 shows the comparison between AL-DS mode and AL-FW mode along the flow rate of the solutions at a fixed draw solution concentration. As can be seen, the gap between two modes widens as the flow rate increases and the gap width is almost the same for the flow directions. In the aspect of water flux, the AL-DS mode shows higher performance than the AL-FW mode for FO process. However, as the feed solution in desalination contains many kinds of particles, the AL-DS mode can cause severe fouling by large particles blocking the pores in the support layer. Therefore, for future work, it is necessary to optimize the structural properties of the porous support layer so that the ICP effect contributes less to the flux reduction. 3.2. Modified spiral-wound module Due to the use of high concentrations, the membrane orientation that decides whether ECP or ICP predominates should be considered. The location of the solutions flowing and the membrane orientation are factors that can be changed in the simulation of the various conditions. There are four cases of simulation for the modified spiral-wound module: the feed flowing inside the envelope in AL-DS mode, the feed flowing outside the envelope in AL-DS mode, the feed flowing inside the envelope in AL-FW mode, and the feed flowing outside the envelope in AL-FW mode. The module geometry and parameters are listed in Table 3. The parameters of required for modelling are found in reference [20]. The module size is selected from a commercial FO membrane manufactured by Hydration Technology Inc. (HTI, Albany, Oregon). In Fig. 11, as a representative example, the simulated results are presented in 2-dimension when the feed flows inside the envelope for AL-FW mode under the conditions presented in Table 3. The number subscripted, 1 and 2, means the inner and outer membrane of one leaf, respectively, and in and out mean inside and outside of the envelope. As seen in Fig. 11(a)–(d), (g), and (h), discontinuous behaviour is observed, especially for Jw , Js , Uout , and Cout . For the inner membrane denoted by 1, the discontinuous region is near the attached part of the central pipe and is illustrated as region A in Fig. 8. The other discontinuous region, C, appears in the outer membrane denoted by 2 near the end of the wound membrane. In order to explain the existence of the discontinuous behaviour, the modelling procedure needs to be reviewed. As mentioned earlier in the previous section, only one dominant flow direction is considered, and the flow velocity and the concentration are calculated using the simplified Eqs. (12)–(15). Consequently, Eqs. (12)–(15) become ordinary differential equations dependent on the one dominant spatial coordinate in each segment after discretization. For that reason, once regions A, B, and C are discriminated, uout and Cout are solved in each region under the assumption that there are no diffusive and convective transports between the regions. Thus, Jw and Js also have the discontinuity between the regions because they are calculated based on the uout and Cout . In regions A and C, higher water fluxes take place, because the channel is not shared by two membranes, unlike the rest of the region; consequently, it appears to have twice the channel height compared to the other region. Also, lower flow velocities in regions A and C are observed because the water permeates to the outside envelope by only one membrane, unlike in region B. As a result of the lower flow velocity with the low salt permeability, the concentrations are higher

411

than those in the region B. Unlike outside the envelope, the flow velocity and the concentration are continuous, since there are no distinguished regions inside. In addition, the simulations were conducted under various operating conditions. The volumetric flow rates of both the feed and draw solutions vary from 5.0 to 26.67 × 10−6 m3 /s (300, 500, 800, 1200, and 1600 cm3 /min), and the concentration of the draw solution is 50, 75, and 100 kg/m3 with a fixed feed concentration. The simulated results according to the operation conditions and the cases are shown in Table 4. For all cases, the water flux increases as the flow rate of the solutions and the concentration of the draw solution increase. In a comparison between AL-DS and AL-FW modes, the permeate flux tends to be smaller in AL-FW mode. It is thought that the ICP phenomenon plays a more significant role than ECP in AL-FW mode in which the draw solution of the higher concentration is in contact with the porous support layer. In other words, the concentration gradient is steeper than in AL-DS mode since more salts are inhibited from freely diffusing into the porous support layer. Moreover, the discrepancy in the water flux of AL-FW and AL-DS mode is larger at the higher concentration of the draw solution. This is because the ICP effect is increasing as the concentration becomes higher. Now, to compare the performance according to the feed flow location, the water flux is lower when the porous support layers are on the inside of the envelope (cases 1 and 4). This is caused by the different distances that the solutions travel inside and outside the envelope. Since the effective length that the solution flows inside the envelope is longer due to the curved path, the concentration change by ICP is much more remarkable. For the consideration of ECP effects, it is regarded that ECP does not differ much by fixing the flow rate for both solutions. Thus, for the modified spiral-wound module to obtain higher performance, the configuration of the membranes and the solutions is critical: the active layer of two membranes is suggested to face the inside of the envelope, and the lower concentration solution should be placed on the porous support layer. In future research, the operation conditions as well as the design of the modified spiral-wound module can be studied. In order to achieve the targeted performance, the optimal conditions of the design parameters can be found using the model developed in this work, such as the specifications of the membrane geometry, initial conditions of the solutions, and so on. 3.3. Comparisons of the plate-and-frame and the modified spiral-wound module The same order of magnitude in water flux for the both modules is obtained in the operating condition range studied in this work. As it turned out, the performance of AL-FW process is distinguished by a design parameter, the flow direction in the plate-and-frame module, and the location of the feed flowing in the modified spiralwound module. Therefore, the desired performance can be attained by finding optimized design parameters for FO module. Now, it is compared for the performance of the plate-and-frame and the modified spiral-wound module under the same parameters and geometry. In the plate-and-frame module, two types of geometry are simulated in the cross-current flow direction: a horizontal rectangle and a vertical rectangle. Fig. 12 shows the simulated results with the both membrane orientations and the feed location. In the investigated range of the flow rate, the modified spiral-wound module can generally obtain high water flux. However, the order of the magnitude in the flux is the same along the whole range. Since the plate-and-frame module is simulated with one sheet of membrane, the packing density is lower than the modified spiral-wound module with a pair of membranes. Even though the higher performance in the modified spiralwound module is obtained, there are still many factors to be

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Fig. 11. Simulation results of AL-FW mode when the feed sea water flowing inside envelope when QD = QF = 13.33 × 10−6 m3 /s (800 cm3 /min), CF = 36 kg/m3 , and CD = 75 kg/m3 . (a) Flow velocity outside an envelope on an inner membrane side (left). (b) Flow velocity outside an envelope on an outer membrane side (right). (c) Flow velocity inside an envelope. (d) Concentration inside an envelope. (e) Concentration outside an envelope on an inner membrane side (left). (f) Concentration outside an envelope on an outer membrane side (right). (g) Water flux through the inner membrane. (h) Water flux through the outer membrane. (i) Salt flux through an inner membrane. (j) Salt flux through an outer membrane.

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Table 1 Operation conditions and parameters for modelling plate-and-frame module. Symbol Length Channel height Water permeability Temperature Feed concentration

Value

L (m) H (m) A (m/s Pa) T (K) kg/m3

1 7.6 × 10−4 1.87 × 10−12 298 36

Width Structural parameter Salt permeability Types of solutions Draw solution concentration

Symbol

Value

W (m) t/ε (m) B (m/s) – kg/m3

1 5.0 × 10−4 1.1 × 10−7 NaCl solution Varying

Table 2 Data calculated for plate-and-frame module under various conditions. Flow direction

Q (10−6 m3 /s)

5.00

8.33

AL-DS mode

13.33

20.00

26.67

Co-current

CD (kg/m3 )

Jw (10−6 m/s)

50 75 100 50 75 100 50 75 100 50 75 100 50 75 100

0.43 1.00 1.43 0.50 1.18 1.70 0.55 1.31 1.89 0.58 1.39 2.02 0.62 1.50 2.19

Crosscurrent

Countercurrent

Flow direction

Q (10−6 m3 /s) 0.44 1.02 1.46 0.50 1.19 1.71 0.55 1.31 1.90 0.58 1.40 2.03 0.63 1.51 2.20

0.44 1.04 1.49 0.51 1.20 1.73 0.55 1.32 1.91 0.58 1.40 2.03 0.62 1.51 2.20

5.00

8.33

AL-FW mode

13.33

20.00

26.67

Co-current

CD (kg/m3 )

Jw (10−6 m/s)

50 75 100 50 75 100 50 75 100 50 75 100 50 75 100

0.42 0.96 1.35 0.48 1.11 1.57 0.53 1.21 1.72 0.56 1.28 1.81 0.59 1.35 1.91

Crosscurrent

Countercurrent

0.43 0.97 1.38 0.49 1.12 1.58 0.53 1.22 1.72 0.56 1.28 1.82 0.59 1.36 1.92

0.43 0.99 1.40 0.49 1.12 1.59 0.53 1.22 1.73 0.56 1.28 1.82 0.59 1.35 1.92

Table 3 Operating conditions and parameters for modified spiral-wound module. Symbol No. of leaves Width Channel height Central pipe radius Water permeability Temperature Feed concentration

Value

– W (m) H (m) rc (m) A (m/s Pa) T (K) kg/m3

1 2.25 7.6 × 10−4 0.03 1.87 × 10−12 298 36

considered to design the process and the module. As seen in Fig. 12, unlike the results in the high flow rate range, it is shown that the plate-and-frame has the higher water flux in some cases in the low flow rate range. Moreover, for each module, the performance in

Length Length of center glue line Membrane thickness Structural parameter Salt permeability Types of solutions Draw solution concentration

Symbol

Value

L (m) Lc (m) tm (m) t/ε (m) B (m/s) – kg/m3

0.32 2.09 8.8 × 10−5 5.0 × 10−4 1.1 × 10−7 NaCl solution Varying

respect to the water flux is highly dependent on the membrane orientation, the solution locations, and the membrane geometry. Thus, it is necessary to research the selection of a suitable module design and operating conditions for FO desalination process with respect

Table 4 Data calculated for modified spiral-wound module under various conditions. Case no.

Feed location

Q (10−6 m3 /s) 5.00

8.33

AL-DS mode

13.33

20.00

26.67

1.Inside envelope

CD (kg/m3 )

Jw (10−6 m/s)

50 75 100 50 75 100 50 75 100 50 75 100 50 75 100

0.36 0.83 1.18 0.45 1.03 1.47 0.52 1.19 1.71 0.57 1.31 1.88 0.60 1.36 1.94

2. Outside envelope

Feed location

Q (10−6 m3 /s) 0.38 0.88 1.24 0.46 1.10 1.59 0.52 1.27 1.85 0.55 1.38 2.02 0.55 1.43 2.11

5.00

8.33

AL-FW mode

13.33

20.00

26.67

3. Inside envelope

CD (kg/m3 )

Jw (10−6 m/s)

50 75 100 50 75 100 50 75 100 50 75 100 50 75 100

0.38 0.85 1.18 0.47 1.06 1.49 0.54 1.21 1.70 0.60 1.32 1.85 0.64 1.38 1.93

4. Outside envelope

0.35 0.80 1.12 0.41 0.97 1.37 0.46 1.08 1.55 0.48 1.16 1.66 0.48 1.18 1.69

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Fig. 12. Comparison of the plate-and-frame and the modified spiral wound modules at CD = 75 kg/m3 .

to the priority of concerns. Moreover, as mentioned in the previous section, optimization of the design parameters in the modules is necessary in future work. 3.4. Model validation The developed model for the modified spiral-wound module is validated with experimental data presented in Xu et al. paper [16]. The membrane module used for the experiments has the active layer facing the outside an envelope. DI water was used for the feed solution and NaCl solutions with various concentrations for the draw solution. As shown in Fig. 13, the experimental data and the simulated results using the model developed in the reference showed considerable discrepancy. For comparison, the developed model in this work is used to predict the experimental results under the same operating conditions and the membrane geometry. Fig. 14 shows the simulated results by our model and the experimental date extracted from the reference [16]. It is evident that the model here apparently follows the experimental data with the allowable error for measurements. From this result, the developed model in this study can predict the actual behaviour of FO membrane module for the various operating conditions.

Fig. 14. Simulation results in this study and the experimental data from the Ref. [16].

4. Conclusions In this study, modelling of a plate-and-frame and a modified spiral-wound module for FO process is carried out in 2-dimension. Since the actual behaviours of flow inside the modules are complicated, the balance models are simplified using the several assumptions. Based on the simplified balance equations, the water flux is predicted by considering ECP and ICP phenomena according to the membrane orientations. Thus, the developed model can simulate the various hydraulic conditions and the membrane orientations for the purpose of identifying the dominant phenomenon of flux reduction. Even if the model equations are relatively simplified, it is found out that the developed model matches better than the existing model used in the Ref. [16]. In addition, the performance of the FO modules under the various operating conditions is investigated by the developed model in this work. As expected, the water flux for both modules is higher as the flow rate and the concentration of the draw solution increase due to reduction in ECP effect and elevation in driving force, respectively. For both modules, the AL-FW mode results in less water permeation flux because of more severe ICP effects. For the plate-and-frame module, the difference in the performance according to the flow direction is investigated. It is reasoned that the flow direction influences the changes of concentration difference. For the modified spiral-wound module, since there exist several flow directions but only the dominant flow direction is considered in modelling, discrete profiles are distinctive characteristics compared to the plate-and-frame module. Due to its unique geometric structure, the feed stream location also decides the performance in a significant way. Furthermore, the superiority of performances of both modules is compared. In conclusion, the module and the operating condition should be carefully chosen according to the desired performance. Therefore, the developed model and the simulation results in this work can be applied to designing FO module and finding the optimal operating conditions for future research. Acknowledgements

Fig. 13. Experimental and simulation data from Ref. [16].

This research was financially supported by Ministry of Knowledge Economy (2008NFC12J0431502010) and Seawater Engineering & Architecture of High Efficiency Reverse Osmosis (SEAHERO) program funded by Ministry of Land, Transport and Maritime Affairs (10CPTA-A042522-05-000000).

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References Nomenclature Jw Js A B  C Nion Rg T Mw k Sh Sc Re dh L W H K D t  ε Q u C Ac At x y z  r ma

water flux (m/s) salt flux (kg/m2 s) water permeability (m/Pa s) salt permeability (m/s) osmotic pressure (Pa) concentration (kg/m3 ) ionisation number ideal gas constant (m3 Pa/mol K) temperature (K) molecular weight (kg/mol) mass transfer coefficient (m/s) Sherwood number Schmidt number Reynolds number hydraulic diameter (m) length of a channel (m) width of a channel (m) height of a channel (m) salt resistivity (s/m) diffusivity (m2 /s) thickness of porous support layer (m) tortuosity of porous support layer porosity of porous support layer volumetric flow rate (m3 /s) flow velocity (m/s) concentration (kg/m3 ) cross-sectional area (m2 ) trans-membrane area (m2 ) coordinate in channel width direction coordinate in channel length direction coordinate in channel height direction coordinate in membrane wound direction radius of the wound membrane at  (m) index where region A finishes

Subscript F feed seawater D draw solution b at bulk solution m at membrane eff effective in inlet or inside an envelope out outlet or outside an envelope

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