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Steady three dimensional flow and mass transfer analyses for brackish water desalination by reverse osmosis membranes
International Journal of Heat and Mass Transfer 101 (2016) 399–411
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Steady three dimensional flow and mass transfer analyses for brackish water desalination by reverse osmosis membranes Ali E. Anqi a,b, Nawaf Alkhamis c, Alparslan Oztekin a,⇑ a
Lehigh University, Dept. of Mechanical Engineering & Mechanics, USA King Khalid University, Dept. of Mechanical Engineering, Saudi Arabia c King Abdulaziz University, Dept. of Mechanical Engineering, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 8 November 2015 Received in revised form 22 May 2016 Accepted 23 May 2016 Available online 2 June 2016 Keywords: Desalination Reverse osmosis Membrane performance Computational fluid dynamics Mass transfer
a b s t r a c t Reverse osmosis has been emerged as one of the most used technologies to desalinate water. Present study investigates steady three dimensional flows and mass transfer in the feed channel for a brackish water desalination process by using reverse osmosis membranes. Flow and the mass transfer in the feed channel are governed by Navier–Stokes and mass transport equations. The channel containing cylindrical shaped spacers is bounded by membranes. Computational fluid dynamics simulations are conducted for the range of the Reynolds number 100 6 Re 6 800. The laminar flow model is employed when Re ¼ 100 while SST k—x turbulence model is employed when Re P 400. Membranes are treated as a functional surface where water flux is determined from local concentration and pressure by employing the solution-diffusion model of the reverse osmosis. The influence of three dimensional flow structures on the concentration polarization and the potential fouling over the membrane surface is discussed. It is shown that high water flux regions and low concentration regions coincide with low wall shear regions. The high intensity concentration polarization sites correlates directly with the high potential fouling sites. Spacers enhance the membrane performance and they help to alleviate concentration polarization. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Reverse Osmosis (RO) is one of the leading technologies in fresh water production worldwide. The process requires an applied pressure higher than the osmotic pressure caused by dissolved salt. The semi-permeable membrane allows water to pass through while it blocks the passage of dissolved salt ions through the membrane. The membrane permeability and the salt rejection rate influence the membrane performance directly. Pure water production rate increases as the membrane permeability and the salt rejection rate increase. Recent RO membranes are built with higher permeability and nearly 100% salt rejection rate. Such a high water passage rate through membranes and the high salt rejection rates cause salt accumulations at the surface of the membrane. This well-known phenomenon is referred to as concentration polarization or concentration boundary layer. The concentration polarization at the membrane surface increases the osmotic pressure in the feed channel. As a result, the applied pressure needs to be increased to keep the water passage through the membrane at the same rate. Furthermore, it is well-documented that the concentration ⇑ Corresponding author. E-mail address: [email protected] (A. Oztekin). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.102 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
polarization is a leading cause of scaling and fouling at the membrane surface [1,2]. The scaling and fouling reduce the process efficiency and shorten the membrane life [1,2]. Spacers are placed between the membranes in the feed channel to keep the membranes a part and to enhance the membrane performance by promoting the momentum mixing and reducing the concentration polarization at the membrane surface. Flows past cylinders confined in a channel have been studied extensively. Some of the findings relevant to the present study are summarized here. Chakraborty et al. [3] documented that the drag coefficient decreases as the Reynolds number (Re) increases, and the drag coefficient increases as the blockage ratio increases at fixed Re. Kanaris et al. [4] reported that the transition from two-dimensional to three-dimensional flows occurs at about 180 < Re < 210 in flows past cylinder with a blockage ratio of 1/5, where Re is based on the centerline velocity and cylinder diameter. Rehimi et al. [5] reported that the onset of the vortex shedding occur at Re ¼ 108 and the onset of transition from two dimensional to three dimensional flow occurs at Re ¼ 159 in flows past a cylinder with a blockage ratio of 1/3. Griffith et al. [6] reported that the flow transitions occur at lower value of Re when the blockage ratio is increased to 1/2. The present paper focuses on the effect of such flow transitions on the membrane performance,
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Nomenclature (u; v ; w) ðx; y; zÞ hm CD F1; F2 U av e a1 c0 cb cp cw fx
vw
c f Dp l h A D Re Sh Sc d k
s t Sct p
components of velocity [m/s] spatial coordinates mass transfer coefficient [m/s] drag coefficient blending functions average velocity [m/s] turbulent model parameter inlet salt concentration [kg/m3] feed bulk concentration [kg/m3] production concentration [kg/m3] membrane salt concentration [kg/m3] force component in x-direction [N] water flux [m/s] salt concentration [kg/m3] pressure coefficient transmembrane pressure difference [Pa] distance between spacers [m] channel height [m] water permeability [m/s Pa] diffusivity [m2/s] Reynolds number, Re ¼ 2hU av e =m Sherwood number, Sh ¼ 2hhm =D Schmidt number, Sc ¼ m=D spacer diameter [m] turbulent kinetic energy [J/kg]
channel depth in z-direction [m] time [s] turbulence Schmidt number pressure [Pa]
Greek letters lt eddy viscosity [Pa s] mt kinematic eddy viscosity [m2/s] rk1 turbulent model parameter rx ; rx2 turbulent model parameters turbulent model parameters b; b c turbulent model parameter Dp osmotic pressure [Pa] l dynamic viscosity [Pa s] m kinematic viscosity [m2/s] q density [kg/m3] x specific dissipation rate [1/s] j osmotic coefficient [kPa m3/kg] s Wall shear stress magnitude [Pa] Subscripts i and j index notation w properties at the membrane surface
concentration polarization, and fouling for water desalination process by reverse osmosis. Many investigators have conducted computational study to simulate flow and mass transfer in the feed channel containing spacers for desalination process by reverse osmosis. Shakaib et al. [7] performed three-dimensional laminar flow simulations in the feed channel including mass transfer. Membrane was treated as an impermeable wall, and the salt concentration was assumed as a constant along the surface of the membrane. Shakaib et al. [7] reported that the local wall shear stress and the local mass transfer along the membrane surface are strongly influenced by the presence of the spacer. Similarly, Koutsou et al. [8]. Geraldes et al. [9] conducted two-dimensional flow simulations to investigate concentration polarization. Membrane was treated as an impermeable wall (no passage of water) and the salt concentration gradient at the membrane surface was fixed artificially. Geraldes et al. [9] concluded that the membrane performance can be enhanced by placing the spacers in touch with membranes. Srivathsan et al. [10] conducted three-dimensional flow simulations in a channel bounded by permeable walls serving as membranes. However, Srivathsan et al. [10] determined salt concentration along the
membrane surface using empirical relation for the local Sherwood number. As stated above, concentration polarization and fouling along the surface of membrane have undesirable influence on the membrane performance. These effects can be characterized accurately only when realistic flow, mass transfer and membrane flux model are employed. Ma and Song [11] and, recently, Anqi et al. [12] employed the membrane flux model along with the Navier–Stokes and mass transport equations to simulate a desalination process by reverse osmosis in two dimensional channel geometries. The solution-diffusion model imposed along the surface of the membrane determines the local water flux passing through the membrane as a function of local concentration and the pressure at the surface of the membrane. The present authors [12] also calculated the local Sherwood number directly at the membrane surface for two-dimensional flows in the feed channel containing uniformly spaced spacers. The fouling is another problem potentially related to the concentration polarization along the surface of the membrane. Koutsou et al. [8] documented that the fouling occurs at low shear regions along the surface of the membrane. Koutsou and his co-workers also reported that high water flux regions
Fig. 1. The schematic of flow geometry.
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401
Fig. 2. Profiles of stream-wise component of the velocity for mesh M1 and M2 (a) at x/h = 6 and (b) at x/h = 7.
coincide with the high shear regions. These studies reveal that accurate characterization of flow structures is necessary to determine membrane performance in desalination processes. Threedimensional flow and mass transfer simulations using realistic membrane flux model are necessary to properly design and optimize reverse osmosis desalination modules. Present study conducts simulations to characterize steady three dimensional flows and concentration profiles in the feed channel containing uniformly spaced, aligned circular cylinders. Profiles of the water flux, the Sherwood number and the wall shear stress along the membrane surface are presented to assess the membrane performance for a range of Re of 100 6 Re 6 800. Regions of concentration polarization and potential fouling are presented as well as correlations among the high water flux, concentration
polarization and fouling regions are established. The membrane in the present study is treated as the functional surface where the local water flux is determined by the local pressure and the concentration.
1.1. Governing equations The schematic of computational domain is illustrated in Fig. 1. The top and the bottom surface is an RO membrane. The gap between membranes is h ¼ 1.5 mm. Sixteen uniformly spaced circular cylinders of diameter, d ¼ 12 h, are placed in the mid-plane between the membranes to promote mixing in the feed channel. The axes of cylinders are parallel to the membranes and
Fig. 3. Contours of the stream-wise component of the velocity acquired at z-planes of z=h ¼ 0.43, 2, and 3.6 for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
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Fig. 4. Contours of the stream-wise component of the velocity acquired at the mid-plane between membranes, y/h = 0.5, for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
perpendicular to the primary flow direction. The length of the inlet and the outlet region is selected as 6 h and 15 h, respectively, to minimize the effect of the imposed inlet and outlet velocity boundary conditions on the characteristics of the flow in the spacer bank region. Flow simulations are conducted for Re = 100, 400, 600 and 800. The Reynolds number is defined as Re ¼ 2hU av e =m, where h is the channel height, U av e is the inlet average velocity, and m is the kinematic viscosity. Three dimensional steady flow simulations are performed in the feed channel containing spacers bounded by RO membranes. The brackish water is modeled as an incompressible fluid with constant physical properties, and the salt diffusivity, D, in the aqueous solution is assumed to be constant. The equations governing the velocity and salt concentration fields are: the continuity equation
@ui ¼ 0; @xi
ð1Þ
and transient turbulent flows in channels containing obstacles [12,15–17]. SST k—x turbulence model is used here to simulate steady three dimensional flows with mass transport in the feed channel for Re P 400. The k—x turbulence momentum transport model yields:
uj
uj
@ui 1 @p @ ui ¼ þm ; @xj q @xi @xj @xj
ð2Þ
and the mass transport equation
uj
@c @2c ¼D : @xj @xj @xj
ð3Þ
Here q is the fluid density, i and j are index placeholders. The spatial coordinates: x1 ¼ x, x2 ¼ y, and x3 ¼ z, the velocity components: u1 ¼ u, u2 ¼ v and u3 ¼ w and c is the salt concentration. Both Renormalized-group (RNG) k—e, shear stress transport (SST) k—x are successfully used to model turbulent flows in channels. It has been shown that turbulent flow structures and mass and heat transport in channels containing spacers can be characterized more accurately by using SST k—x model [13,14]. Recently, the present authors employ the time averaged baseline (BSL) k—x and SST k—x turbulence models successfully to simulate steady
ð4Þ
where l is the dynamic viscosity and lt is the eddy viscosity, lt ¼ q maxðaa11xk ;XF 2 Þ. The equations governing the turbulence kinetic energy, k, and the specific dissipation rate, x, are:
ui
ui
the conservation of momentum 2
@ui 1 @p 1 @ @ui ¼ þ ðl þ lt Þ @xj q @xi q @xj @xj
@ðqkÞ @ui @ @k ¼ sij b qxk þ ðl þ rk1 lt Þ ; @xi @xj @xj @xj @ðqxÞ c @ui @ @x ¼ sij bqx2 þ ðl þ rx lt Þ @xi @xj mt @xj @xj 1 @k @ x þ 2qð1 F 1 Þrx2 x @xj @xj
ð5Þ
ð6Þ
where X is the vorticity magnitude, a; a1; b; b ; rk , rx , rx2 and c are closure coefficients, and F 1 ; F 2 are the blending functions. Detailed description of the model can be found in Ref. [18]. The k—x turbulence mass transport model yields:
uj
@c @ ¼ @xj @xj
Dþ
lt @c qSct @xj
ð7Þ
where Sct is the turbulence Schmidt number. A parabolic profile is used for the stream-wise component of the 2
velocity at the inlet, u ¼ 6U av e ½y=h ðy=hÞ . The inlet velocity profile is assumed to be uniform in the z-direction (span-wise direction). The constant concentration, c0 ¼ 4000 ppm, is also specified at the inlet. At the outlet the zero pressure and constant concentration gradient are applied. At the side surfaces, z ¼ 0 and z ¼ 5 h, periodic boundary conditions are imposed on both velocity and concentration fields. Along the surface of cylinders, no-slip and
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403
Fig. 5. The iso-surfaces of the stream-wise component of the velocity for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
no-penetration conditions are imposed on the velocity field and zero flux condition is imposed on the concentration field. The solution-diffusion model was introduced in 1940’s and was used for membrane flux modeling since then. Wijmans and Baker [19] illustrated the derivation of the solution-diffusion model using chemical potential approach and proved that the model is applicable to determine the mass transport via the reverse osmosis process. By employing the solution-diffusion approach, the membrane in the present study is considered as a functional surface where the local water flux, the local salt concentration, and the local pressure are all coupled together. Hence, the local water flux is determined from the transmembrane pressure difference, Dp, and the osmotic pressure difference, Dp, between the feed and production channels. Simultaneously, the salt concentration gradient is calculated from the local water flux and local salt concentration at the membrane surface. The equations of the local water flux and local salt concentration gradient at the membrane surface are:
@y ¼ 0 and h : u ¼ 0;
9
v w ¼ A½Dp Dp > = @c D @y ¼ v w ðcw cp Þ ; > ;
Dp ¼ jðcw cp Þ ð8Þ
where v w is the local water flux through the membrane, cw is the local salt concentration at the membrane surface, A is the water permeability through the membrane, j is the osmotic coefficient, and cp is the salt concentration in the production side. The water flux through the membrane is normalized as v w =ðDpAÞ. The drag coefficient, C D , is determined from:
CD ¼
2f x
qU 2av e As
;
ð9Þ
where f x is the drag force exerted on the spacer and As is the projected area of the spacer perpendicular to the stream-wise direction. The Sherwood number, Sh, is evaluated from:
Sh ¼
ð2hÞhm ; D
hm ¼
@c D @y
y¼h
ðcb cw Þ
ð10Þ
where hm is the local mass transfer coefficient and cb is the bulk concentration. The pressure coefficient, f , is calculated as:
F¼
2hj dp j dx
ð1=2ÞqU 2av e
ð11Þ
where j dp j is the pressure gradient. dx 1.2. Numerical method, mesh optimization and validation The commercial software CFX 14.5 is utilized to conduct simulations. This software implements the finite volume discretization method. Equations are solved using Semi-Implicit Method Pressure Linked Equations (SIMPLE) algorithm that fully couples the velocity and pressure fields. The CEL expression language is deployed to implement the parabolic velocity profile at the inlet. Similarly, the water flux and salt concentration gradient equations at the membrane surface were modeled via the CEL expression language. The operating parameters and physical properties are selected so that the typical RO brackish water desalination process is simulated. The transverse pressure difference is fixed as 1.25 MPa since the pressure drop in the feed channel for the considered domain, <2 kPa, is negligible compared to the transmembrane pressure difference. The physical properties of the brackish water in the feed channel are assumed to be constant. The density is q ¼ 1000 kg/m3 and the dynamic viscosity is l ¼ 103 Pa s. The salt diffusivity in the water is assumed to be constant, D ¼ 1:5 109 m2/s with the Schmidt number, Sc ¼ m=D ¼ 667. The osmotic coefficient is j, is 75 kPa m3/kg, the water permeability is A ¼ 2:3 1011 m/(s Pa), and production side salt concentration cp ¼ 0. A mesh optimization study is performed using two mesh sizes: 20 million (M1) and 30 million (M2) elements. Profiles of the stream-wise component of the velocity are calculated across the gaps between membrane (1) across the first spacer at x=h ¼ 6 and z=h ¼ 2.5 and (2) in the wake of the first spacer at x=h ¼ 7 and z=h ¼ 2.5. Velocity profiles are shown in Fig. 2 for both meshes. The pressure coefficient, f , is also calculated for the mesh M1 and
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Fig. 6. The iso-surfaces of the z-component of the vorticity for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
mesh M2. The pressure coefficient is determined from Eq. (11) using average pressure drop across the part of spacer bank from x ¼ 38h to x ¼ 77h. The velocity profiles obtained for M1 and M2 are similar; implying that mesh size M2 is enough to ensure mesh independency. The pressure coefficient is calculated to be 0.320 and 0.323, respectively, for M1 and M2 mesh; confirming that the mesh independency might be achieved with M2. Stronger argument could be made by refining mesh further. The size of computation and the available computational resources make that to be a difficult task. Simulation results presented in the present study are obtained using M2 mesh. The membrane flux model used in the present study was validated previously by the present authors [12]. Two dimensional simulations in the feed channel containing spacers bounded by membranes were conducted using the solution-diffusion by both studies [11,12]. It was shown that results of water flux predicted by the present authors [12] and Ma and Song [11] agree well. Additionally, the drag coefficient, C D , of the 1st spacer at Re ¼ 800 is calculated and compared against that predicted by Chakraborty [3]. Drag coefficient, C D , predicted by the present three and two dimensional flow simulations is 4.482 and 3.995, respectively. Drag coefficient documented in [3] from two dimensional flow simulations at Re equivalent of 800 is 3.946. Agreement in the drag coefficient with previous work helps validating flow modeling and simulations presented here. 2. Results and discussion The results of steady three dimensional flow simulations are presented for Re ¼ 100, 400, 600 and 800. Comparison between results obtained from three dimensional and two dimensional simulations are made to determine the influence of three dimensional flow structures on the membrane performance. Two dimensional flow modeling consider velocity and concentration field in the
z-direction to be uniform. The laminar flow model is used for Re ¼ 100, and SST k—x turbulence model is used for Re ¼ 400, 600 and 800 to simulate mass and momentum transports in the feed channel. Fig. 3 illustrates contours of the stream-wise component velocity at Re ¼ 100, 400, 600 and 800. Images are shown at three z planes (z=h ¼ 0.4, 2 and 3.6) for each value of Re. The wake region behind spacers becomes larger as Re is increased. High speed flow region around the cylinder is also stretched in the stream-wise direction as Re is increased, as shown in Fig. 3. Three dimensional flow structures and their strength cannot easily be distinguished from these images. Primary flow in the stream-wise direction dominates the secondary flows induced by the flow transitions. These flows transitions are well-documented in the literature [20] and they are expected to onset for values of Re above the criticality. The secondary flows manifest themselves nearly periodic flow structures in the span-wise direction. Fig. 4 shows contours of the stream-wise components of the velocity at the mid-plane (y=h ¼ 0:5) between membranes at Re ¼ 100, 400, 600 and 800. The presence of secondary flows is very clear. At Re ¼ 100 the flow is nearly uniform in z-direction and it can be characterized as two dimensional, as shown in Fig. 4a. Velocity contour depicted in Fig. 4b for Re ¼ 400 shows vast contrast to the velocity contour obtained for Re ¼ 100. The flow becomes three dimensional as a result of induced secondary flows. It is clear that the critical value of Re for the onset of three dimensional flow is between 100 and 400. It has been reported that transition to the three dimensional flow occurs between Re of 100 and 200 for these geometries [4]. Velocity contours at Re ¼ 600 and 800 exhibit three dimensional flow structures with a lesser intensity relative to the primary flow, as shown in Fig. 4c and d. These three dimensional flow structures can have a strong influence on the membrane performance, the concentration polarization and the potential fouling along the surface of the membrane.
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Fig. 7. The iso-surfaces of x and y component of the vorticity for different values of the Reynolds number. Right and left column denote iso-surfaces of the x component of the vorticity and y component of the vorticity, respectively. Rows from top to bottom denote iso-surfaces at Re = 100, 400, 600 and 800.
The iso-surface images of the stream-wise component of the velocity are shown in Fig. 5. The iso-surfaces are consistent with images of the velocity contours shown in Fig. 4. Uniform isosurfaces in the span-wise direction for Re ¼ 100 while finger-like structures are realized in Fig. 5b–d for Re ¼ 400, 600 and 800, respectively. Flow structures are more regular for Re ¼ 400 compared to those for Re ¼ 600 and 800. Finger-like structures fill almost the entire region between spacers; indicating that three dimensional effects caused by flow transitions are strong and do not dissipate quickly away from spacers. Flows in the entire spacer bank region become three dimensional for Re P 400, as shown in Fig. 5. The iso-surface images of the z- component of the vorticity is depicted in Fig. 6 for Re ¼ 100, 400, 600 and 800. Flow is
two-dimensional for Re = 100, as indicated by uniformly distributed z-component of the vorticity in the span-wise direction. As the flow becomes three dimensional at Re = 400, z-component of the vortcity becomes spatially distributed in the span-wise direction. At higher flow rates (Re = 600 and 800) the primary flow is so dominant that the secondary flows are not visible from iso-surface images of the vorticity, see Fig. 6c and d. Fig. 7 illustrates the iso-surfaces of x and y component of the vorticity. Images on the left column denote iso-surfaces of x component of the vorticity and images on the right column denote y component of the vorticity for Re ¼ 100, 400, 600 and 800. The images reveal that there are weak and fragmented vorticities near spacers at Re ¼ 100, as shown in Fig. 7a. There is no indication of
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Fig. 8. Contours of normalized concentration (c=c0 ) acquired at z-planes of z=h ¼ 0.43, 2, and 3.6 for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
regular flow structure distributed along the span-wise direction. The vortices dissipate rapidly away from spacers. Counter rotating five pairs of vortices are uniformly spaced in the span-wise direction, as shown in Fig. 7a2 and b2 with images acquired at Re ¼ 400. Once again, these flow images clearly indicate that coherent three dimensional flow structures exist at this flow rate. The vortices are attached to the spacer and extend into the region between spacers. Flow becomes irregular as Re is increased to 600 and 800. Vortices break and form smaller eddies in the wake of spacers. The vortices extend longer, and they occupy the whole region between the spacers for Re ¼ 800, as depicted in Fig. 7. These vortices lead to the formation of streaks near the wall, as documented by several investigators [21–23]. The influence of such streaks can be profound on the membrane performance. Concentration polarization and the formation of fouling and scaling will be directly influenced by such flow structures. These effects are presented and discussed in the present study. Contours of normalized salt concentration (c=c0 ) at three z planes, z=h ¼ 0.4, 2 and 3.6, are shown in Fig. 8 for Re ¼ 100, 400, 600 and 800. There is a thin concentration boundary layer near membrane surface. The thickness of the boundary layer is greater for Re ¼ 100 and it decreases as Re is increased. The concentration boundary layer is disrupted by the presence of the spacer. The boundary layer grows away from the spacer. All three planes show similar concentration field at all flow rates. The concentration polarization at Re ¼ 100 is the highest while it is the lowest at Re ¼ 800, as depicted in Fig. 8. Contours of the water flux through the membrane and the concentration along the surface of the top membrane are shown in Fig. 9 for Re ¼ 100, 400, 600 and 800. Both water flux and the concentration are uniform in the span-wise direction for Re ¼ 100. The water flux through the membrane is profoundly increased by the presence of the spacers, as shown in Fig. 9a. The water flux is much higher above each spacer. The concentration along the membrane is much lower right above the spacers, but it increases rapidly away from the spacers. It is also noted that the presence of the spacers alleviate the concentration polarization occurring along the surface of the membrane. At Re ¼ 400 the distribution of the
water flux and the concentration along the surface of the membrane is strikingly different compared to those at Re ¼ 100. The effect of three dimensional flows in the feed channel influences the water flux and the local concentration along the membrane surface. Streaks of high and low regions of water flux and concentration are uniformly distributed in the span-wise direction. Streaks are more pronounced at Re ¼ 400 compared to those at Re ¼ 600 and 800; following the relative strength of the secondary flows over the primary flows as presented above. The water flux is greater and the concentration overall is lower near spacer at all values of Re. The intensity of the concentration polarization decreases with increasing flow rate. It is important to note that low water flux regions correlates to the high intensity concentration polarization regions at all flow rate considered here. Streaks of high concentration polarization region between spacers coincide with streaks of low water flux regions, as shown in Fig. 9. Results of three dimensional simulations confirm our conclusion drawn from two dimensional simulations reported earlier [12]. The present author documented that that spacers enhance the membrane performance while it helps to alleviate concentration polarization in two dimensional flows. The local value of the Sherwood number is calculated along the surface of membranes using Eq. (10). The bulk concentration, cb , in Eq. (10) is calculated from average of salt concentration at the inlet and the outlet. Contours of the local Sh along the surface of the top membrane are depicted in Fig. 10. Distribution of Sh is very similar to the distribution of the water flux over the membrane surface. Sh is distributed uniformly in the span-wise direction at Re ¼ 100 while strips of high and low regions of Sh are shown at Re P 400. Similar to the counter part of two dimensional flows presence of the spacers in three dimensional geometry leads to significant increase in Sh. Local value of Sh is much greater near spacer and decreases rapidly away from spacers. The Sherwood number also increases as the flow rate is increased; that is also consistent with the prediction of two dimensional simulations documented in [12]. Streaks of high Sh regions coincide with the regions of low intensity concentration polarization, as depicted in Figs. 10 and 9.
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Fig. 9. Contours of normalized water flux through the top membrane (right column) and contours of normalized concentration along the surface of the top membrane (left column) for different values of Re. Rows from top to bottom denote iso-surfaces at Re = 100, 400, 600 and 800.
Contours of the normalized wall shear stress along the surface of the top membrane are shown in Fig. 11. The local shear stress is normalized by the maximum value of the wall shear stress for each flow rate. The maximum value of the wall shear stress is calculated to be 0.9, 4.5, 8.5 and 12.6 Pa for Re ¼ 100, 400, 600 and 800, respectively. Similar to the flow properties presented above the wall shear stress is uniform in the span-wise direction at Re ¼ 100, but it becomes streaky as the flow becomes three dimensional at higher flow rates (Re ¼ 400, 600 and 800). Streaks induces by the secondary flow is not as pronounced as in the case of the water flux, concentration or Sh number distribution. At all flow rates, the maximum wall shear stress occur right above the spacers
where the fluid is accelerated passing through the spacers. The intensity of the wall shear stress increases as Re is increased. Koutsou et al. [8] experimentally determined regions where fouling occurs and showed that the fouling sites coincide with the low wall shear stress regions. Based on present results, potential fouling sites are situated in the regions between spacers and three dimensional flow structures in the feed channel induced by secondary flows help to alleviate potential fouling occurring along the surface of the membrane. The span-wise averages of normalized water flux, concentration and the Sherwood number at each x location are determined for three dimensional flows in the feed channel containing spacers.
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Fig. 10. Contours of the local Sherwood number (Sh) along the surface of the top membrane for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
Fig. 11. Contours of the wall shear stress normalized by the maximum value for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
These span-wised average profiles in the stream-wise direction are plotted in Figs. 12 and 13 in the spacer bank region for different values of Re. The profiles of the normalized water flux, concentration and local value of Sh predicted by two dimensional steady simulations are also plotted in Figs. 12 and 13 to assess the influence of the three dimensional effects on the span-wise averaged membrane performance properties for Re ¼ 100, 400, 600 and 800. In both figures, the solid lines denote profiles predicted by the three dimensional simulations while the dashed lines denote profiles predicted by two dimensional simulations. Overall distribution of the two dimensional and span-wised average three
dimensional profiles is similar for all membrane surface properties. Only the values of the peaks and valleys of these quantities differ at each flow rate, as shown in Figs. 12 and 13. The averaged three dimensional Sh profiles exhibit nearly the same distribution in the stream-wise direction as for the Sh obtained by the two dimensional simulations. Only noticeable difference in Sh profiles is observed near wake regions of the spacers. Sherwood number drops significantly right behind the spacers in the two dimensional geometry and rebound very quickly approaching the following spacers. The values of Sh is greater at the peak and is lower at the valley of the two dimensional profiles
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Fig. 12. Span-wised averaged water flux profiles (a1 and a2) and the concentration profiles (b1, b2) in the stream-wise direction for different values of Re. Solid lines denote span-wised averaged three dimensional profiles and dashed lines denote two dimensional profiles.
compared those of the span-wise averaged three dimensional profiles, as shown in Fig. 13. It has to be noted Sh varies greatly in the span-wise direction at Re P 400, as shown in Fig. 10. Membrane flow properties are influenced by the spatial characteristics of the flow in the feed channel. The average values of Sh and the normalized water flux in both span-wise and stream-wise direction in the spacer bank region are determined for Re ¼ 100, 400, 600 and 800. Spacer bank averaged values of Sh and the water flux are plotted in Fig. 14 for three and two dimensional geometry along with the averaged value of Sh and the normalized water flux determined in the three dimensional channel without spacers. Filled blue squares and hollow red circles denote results predicted by the three-dimensional and twodimensional simulations, respectively. Triangles represent the results of three-dimensional flow in a channel without spacers. To minimize the influence of the inlet and the outlet conditions averaging is performed in the spacer bank region between x=h ¼ 38 and x=h ¼ 77. The results reveal that two dimensional approximation predicts the total averages of the water flux and
the Sh reasonably well. Slight over-prediction at low Re and high Re by the two dimensional simulations are seen (see Fig. 14). It is proven that presence of spacers enhances the membrane performance greatly. Three dimensional effects induced by flow transition do not alter averaged membrane properties significantly. This conclusion is specific to the three dimensional spacer geometry considered in the present study. This geometry is simplified version of the ladder type spacer mesh configuration used in separation modules. The influence of the three dimensional flow structures could be profoundly different for other spacer mesh configurations (woven and non-woven spacers). Present authors are currently studying three dimensional flows in the feed channel containing mesh of spacer with different configurations. The results of those simulations are presented in a full paper. It is also important to note that transient effects in these flows can also have significant influence on the concentration polarization, fouling and the membrane performance. Investigating the influence of transient effects in three dimensional geometries is worthy of future work, but is not in the scope of the present study.
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Fig. 13. Span-wised averaged Sh profiles) in the stream-wise direction for different values of Re. Solid lines denote span-wised averaged three dimensional profiles and dashed lines denote two dimensional profiles.
Fig. 14. Span-wise and stream-wise averages of normalized water flux (1) and Sh (2) are plotted as a function of Re. Channel averages of membrane properties are shown for three dimensional channel with and without of spacers and two dimensional channel with spacers. The averages of membrane properties are calculated in the spacer bank region between x=h of 38 and 77.
3. Conclusion Steady three dimensional simulations have been conducted to study the mass transport through the reverse osmosis membrane in the feed channel of a brackish water desalination module. The membrane surface is treated as a functional surface where the water flux is determined from local salt concentration and local pressure along the membrane. The laminar flow model is employed at Re ¼ 100 while the SST k—x turbulence flow model is employed to simulate mass and momentum transport in the feed channel at Re P 400. The flow and concentration field is nearly uniform at Re ¼ 100, but it becomes strongly three dimensional for Re P 400. It is shown here that flow transition from two dimensional to three dimensional flows in a channel containing spacers
occur values of Re between 100 and 400 for flow past an array of cylinders with blockage ratio of 1/2. Even though two dimensional flow simulations predicts the averaged membrane properties reasonably well, the local water flux and Sh along the surface of the membrane varies strongly in the span-wise direction. Furthermore the concentration polarization is profoundly influenced by the three dimensional secondary flows induced by the flow transitions. Velocity and vorticity field indicate spatially periodic flow structures present in the feed channel. Streaks along the surface of the membrane for local properties such as Sh, concentration, water flux and wall shear stress are results from the vortical structures obtained in these three dimensional flows. High water flux regions and low concentration regions coincide with the low wall shear regions. The high intensity concentration polarization sites correlates
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directly with the high potential fouling sites. Spacers enhance the membrane performance, and it mitigates the concentration polarization. Increasing flow rate also enhance the membrane performance. This study proves that three dimensional effects have profound influence on the membrane performance, especially the polarization and fouling characteristics of the reverse osmosis membrane in desalination systems. Further study is needed to characterize flows in the feed channel with various mesh configurations of spacers used in the desalination processes. References [1] L. Malaeb, G.M. Ayoub, Reverse osmosis technology for water treatment: state of the art review, Desalination 267 (2011) 1–8, http://dx.doi.org/10.1016/j. desal.2010.09.001. [2] K.P. Lee, T.C. Arnot, D. Mattia, A review of reverse osmosis membrane materials for desalination—development to date and future potential, J. Membr. Sci. 370 (2011) 1–22, http://dx.doi.org/10.1016/j.memsci.2010.12.036. [3] J. Chakraborty, N. Verma, R.P. Chhabra, Wall effects in flow past a circular cylinder in a plane channel: a numerical study, Chem. Eng. Process. 43 (2004) 1529–1537, http://dx.doi.org/10.1016/j.cep.2004.02.004. [4] N. Kanaris, D. Grigoriadis, S. Kassinos, Three dimensional flow around a circular cylinder confined in a plane channel, Phys. Fluids 23 (2011) 064106, http://dx. doi.org/10.1063/1.3599703. [5] F. Rehimi, F. Aloui, S. Ben Nasrallah, L. Doubliez, J. Legrand, Experimental investigation of a confined flow downstream of a circular cylinder centred between two parallel walls, J. Fluids Struct. 24 (2008) 855–882, http://dx.doi. org/10.1016/j.jfluidstructs.2007.12.011. [6] M.D. Griffith, J. Leontini, M.C. Thompson, K. Hourigan, Vortex shedding and three-dimensional behaviour of flow past a cylinder confined in a channel, J. Fluids Struct. 27 (2011) 855–860, http://dx.doi.org/10.1016/j. jfluidstructs.2011.02.007. [7] M. Shakaib, S.M.F. Hasani, M. Mahmood, CFD modeling for flow and mass transfer in spacer-obstructed membrane feed channels, J. Membr. Sci. 326 (2009) 270–284, http://dx.doi.org/10.1016/j.memsci.2008.09.052. [8] C.P. Koutsou, S.G. Yiantsios, A.J. Karabelas, A.J. Karabelas, Membrane module performance optimization using CFD simulation of flow through narrow channels with spacers, in: Proc. IDA World Congr. Desalin. Water Reuse, 2007, pp. 21–26. [9] V. Geraldes, V. Semiao, M. Norberta de Pinho, Concentration polarisation and flow structure within nanofiltration spiral-wound modules with ladder-type
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spacers, Comput. Struct. 82 (2004) 1561–1568, http://dx.doi.org/10.1016/ j.compstruc.2004.03.052. G. Srivathsan, E.M. Sparrow, J.M. Gorman, Reverse osmosis issues relating to pressure drop, mass transfer, turbulence, and unsteadiness, Desalination 341 (2014) 83–86, http://dx.doi.org/10.1016/j.desal.2014.02.021. S. Ma, L. Song, Numerical study on permeate flux enhancement by spacers in a crossflow reverse osmosis channel, J. Membr. Sci. 284 (2006) 102–109, http:// dx.doi.org/10.1016/j.memsci.2006.07.022. A.E. Anqi, N. Alkhamis, A. Oztekin, Numerical simulation of brackish water desalination by a reverse osmosis membrane, Desalination 369 (2015) 156– 164, http://dx.doi.org/10.1016/j.desal.2015.05.007. W. Changcharoen, S. Eiamsa-ard, Numerical investigation of turbulent heat transfer in channels with detached rib-arrays, Heat Transf. Res. 40 (2011) 431– 447, http://dx.doi.org/10.1002/htj.20357. A. Chaube, P.K. Sahoo, S.C. Solanki, Analysis of heat transfer augmentation and flow characteristics due to rib roughness over absorber plate of a solar air heater, Renew. Energy 31 (2006) 317–331, http://dx.doi.org/10.1016/j. renene.2005.01.012. A.E. Anqi, N. Alkhamis, M.F. Alrehili, A. Oztekin, Numerical study of brackish water desalination using reverse osmosis, in: Proc. IDA World Congr. Desalin. Water Reuse, San Diego, 2015. N. Alkhamis, A.E. Anqi, A. Oztekin, Computational study of gas separation using a hollow fiber membrane, Int. J. Heat Mass Transfer 89 (2015) 749–759, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.090. N. Alkhamis, D.E. Oztekin, A.E. Anqi, A. Alsaiari, A. Oztekin, Numerical study of gas separation using a membrane, Int. J. Heat Mass Transfer 80 (2015) 835– 843, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.09.072. F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, Am. Inst. Aeronaut. Astronaut. J. 32 (1994) 1598–1605, http://dx. doi.org/10.2514/3.12149. J.G. Wijmans, R.W. Baker, The solution-diffusion model: a review, J. Membr. Sci. 107 (1995) 1–21, http://dx.doi.org/10.1016/0376-7388(95)00102-I. C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 477–539, http://dx.doi.org/10.1146/annurev. fl.28.010196.002401. M. Asai, Y. Konishi, Y. Oizumi, M. Nishioka, Growth and breakdown of lowspeed streaks leading to wall turbulence, J. Fluid Mech. 586 (2007) 371–396, http://dx.doi.org/10.1017/S002211200700688X. C.R. Smith, S.P. Metzler, The characteristics of low-speed streaks in the nearwall region of a turbulent boundary layer, J. Fluid Mech. 129 (1983) 27–54, http://dx.doi.org/10.1017/S0022112083000634. S.I. Chernyshenko, M.F. Baig, The mechanism of streak formation in near-wall turbulence, J. Fluid Mech. 544 (2005) 99–131, http://dx.doi.org/10.1017/ S0022112005006506.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Steady three dimensional flow and mass transfer analyses for brackish water desalination by reverse osmosis membranes Ali E. Anqi a,b, Nawaf Alkhamis c, Alparslan Oztekin a,⇑ a
Lehigh University, Dept. of Mechanical Engineering & Mechanics, USA King Khalid University, Dept. of Mechanical Engineering, Saudi Arabia c King Abdulaziz University, Dept. of Mechanical Engineering, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 8 November 2015 Received in revised form 22 May 2016 Accepted 23 May 2016 Available online 2 June 2016 Keywords: Desalination Reverse osmosis Membrane performance Computational fluid dynamics Mass transfer
a b s t r a c t Reverse osmosis has been emerged as one of the most used technologies to desalinate water. Present study investigates steady three dimensional flows and mass transfer in the feed channel for a brackish water desalination process by using reverse osmosis membranes. Flow and the mass transfer in the feed channel are governed by Navier–Stokes and mass transport equations. The channel containing cylindrical shaped spacers is bounded by membranes. Computational fluid dynamics simulations are conducted for the range of the Reynolds number 100 6 Re 6 800. The laminar flow model is employed when Re ¼ 100 while SST k—x turbulence model is employed when Re P 400. Membranes are treated as a functional surface where water flux is determined from local concentration and pressure by employing the solution-diffusion model of the reverse osmosis. The influence of three dimensional flow structures on the concentration polarization and the potential fouling over the membrane surface is discussed. It is shown that high water flux regions and low concentration regions coincide with low wall shear regions. The high intensity concentration polarization sites correlates directly with the high potential fouling sites. Spacers enhance the membrane performance and they help to alleviate concentration polarization. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Reverse Osmosis (RO) is one of the leading technologies in fresh water production worldwide. The process requires an applied pressure higher than the osmotic pressure caused by dissolved salt. The semi-permeable membrane allows water to pass through while it blocks the passage of dissolved salt ions through the membrane. The membrane permeability and the salt rejection rate influence the membrane performance directly. Pure water production rate increases as the membrane permeability and the salt rejection rate increase. Recent RO membranes are built with higher permeability and nearly 100% salt rejection rate. Such a high water passage rate through membranes and the high salt rejection rates cause salt accumulations at the surface of the membrane. This well-known phenomenon is referred to as concentration polarization or concentration boundary layer. The concentration polarization at the membrane surface increases the osmotic pressure in the feed channel. As a result, the applied pressure needs to be increased to keep the water passage through the membrane at the same rate. Furthermore, it is well-documented that the concentration ⇑ Corresponding author. E-mail address: [email protected] (A. Oztekin). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.102 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
polarization is a leading cause of scaling and fouling at the membrane surface [1,2]. The scaling and fouling reduce the process efficiency and shorten the membrane life [1,2]. Spacers are placed between the membranes in the feed channel to keep the membranes a part and to enhance the membrane performance by promoting the momentum mixing and reducing the concentration polarization at the membrane surface. Flows past cylinders confined in a channel have been studied extensively. Some of the findings relevant to the present study are summarized here. Chakraborty et al. [3] documented that the drag coefficient decreases as the Reynolds number (Re) increases, and the drag coefficient increases as the blockage ratio increases at fixed Re. Kanaris et al. [4] reported that the transition from two-dimensional to three-dimensional flows occurs at about 180 < Re < 210 in flows past cylinder with a blockage ratio of 1/5, where Re is based on the centerline velocity and cylinder diameter. Rehimi et al. [5] reported that the onset of the vortex shedding occur at Re ¼ 108 and the onset of transition from two dimensional to three dimensional flow occurs at Re ¼ 159 in flows past a cylinder with a blockage ratio of 1/3. Griffith et al. [6] reported that the flow transitions occur at lower value of Re when the blockage ratio is increased to 1/2. The present paper focuses on the effect of such flow transitions on the membrane performance,
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Nomenclature (u; v ; w) ðx; y; zÞ hm CD F1; F2 U av e a1 c0 cb cp cw fx
vw
c f Dp l h A D Re Sh Sc d k
s t Sct p
components of velocity [m/s] spatial coordinates mass transfer coefficient [m/s] drag coefficient blending functions average velocity [m/s] turbulent model parameter inlet salt concentration [kg/m3] feed bulk concentration [kg/m3] production concentration [kg/m3] membrane salt concentration [kg/m3] force component in x-direction [N] water flux [m/s] salt concentration [kg/m3] pressure coefficient transmembrane pressure difference [Pa] distance between spacers [m] channel height [m] water permeability [m/s Pa] diffusivity [m2/s] Reynolds number, Re ¼ 2hU av e =m Sherwood number, Sh ¼ 2hhm =D Schmidt number, Sc ¼ m=D spacer diameter [m] turbulent kinetic energy [J/kg]
channel depth in z-direction [m] time [s] turbulence Schmidt number pressure [Pa]
Greek letters lt eddy viscosity [Pa s] mt kinematic eddy viscosity [m2/s] rk1 turbulent model parameter rx ; rx2 turbulent model parameters turbulent model parameters b; b c turbulent model parameter Dp osmotic pressure [Pa] l dynamic viscosity [Pa s] m kinematic viscosity [m2/s] q density [kg/m3] x specific dissipation rate [1/s] j osmotic coefficient [kPa m3/kg] s Wall shear stress magnitude [Pa] Subscripts i and j index notation w properties at the membrane surface
concentration polarization, and fouling for water desalination process by reverse osmosis. Many investigators have conducted computational study to simulate flow and mass transfer in the feed channel containing spacers for desalination process by reverse osmosis. Shakaib et al. [7] performed three-dimensional laminar flow simulations in the feed channel including mass transfer. Membrane was treated as an impermeable wall, and the salt concentration was assumed as a constant along the surface of the membrane. Shakaib et al. [7] reported that the local wall shear stress and the local mass transfer along the membrane surface are strongly influenced by the presence of the spacer. Similarly, Koutsou et al. [8]. Geraldes et al. [9] conducted two-dimensional flow simulations to investigate concentration polarization. Membrane was treated as an impermeable wall (no passage of water) and the salt concentration gradient at the membrane surface was fixed artificially. Geraldes et al. [9] concluded that the membrane performance can be enhanced by placing the spacers in touch with membranes. Srivathsan et al. [10] conducted three-dimensional flow simulations in a channel bounded by permeable walls serving as membranes. However, Srivathsan et al. [10] determined salt concentration along the
membrane surface using empirical relation for the local Sherwood number. As stated above, concentration polarization and fouling along the surface of membrane have undesirable influence on the membrane performance. These effects can be characterized accurately only when realistic flow, mass transfer and membrane flux model are employed. Ma and Song [11] and, recently, Anqi et al. [12] employed the membrane flux model along with the Navier–Stokes and mass transport equations to simulate a desalination process by reverse osmosis in two dimensional channel geometries. The solution-diffusion model imposed along the surface of the membrane determines the local water flux passing through the membrane as a function of local concentration and the pressure at the surface of the membrane. The present authors [12] also calculated the local Sherwood number directly at the membrane surface for two-dimensional flows in the feed channel containing uniformly spaced spacers. The fouling is another problem potentially related to the concentration polarization along the surface of the membrane. Koutsou et al. [8] documented that the fouling occurs at low shear regions along the surface of the membrane. Koutsou and his co-workers also reported that high water flux regions
Fig. 1. The schematic of flow geometry.
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401
Fig. 2. Profiles of stream-wise component of the velocity for mesh M1 and M2 (a) at x/h = 6 and (b) at x/h = 7.
coincide with the high shear regions. These studies reveal that accurate characterization of flow structures is necessary to determine membrane performance in desalination processes. Threedimensional flow and mass transfer simulations using realistic membrane flux model are necessary to properly design and optimize reverse osmosis desalination modules. Present study conducts simulations to characterize steady three dimensional flows and concentration profiles in the feed channel containing uniformly spaced, aligned circular cylinders. Profiles of the water flux, the Sherwood number and the wall shear stress along the membrane surface are presented to assess the membrane performance for a range of Re of 100 6 Re 6 800. Regions of concentration polarization and potential fouling are presented as well as correlations among the high water flux, concentration
polarization and fouling regions are established. The membrane in the present study is treated as the functional surface where the local water flux is determined by the local pressure and the concentration.
1.1. Governing equations The schematic of computational domain is illustrated in Fig. 1. The top and the bottom surface is an RO membrane. The gap between membranes is h ¼ 1.5 mm. Sixteen uniformly spaced circular cylinders of diameter, d ¼ 12 h, are placed in the mid-plane between the membranes to promote mixing in the feed channel. The axes of cylinders are parallel to the membranes and
Fig. 3. Contours of the stream-wise component of the velocity acquired at z-planes of z=h ¼ 0.43, 2, and 3.6 for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
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Fig. 4. Contours of the stream-wise component of the velocity acquired at the mid-plane between membranes, y/h = 0.5, for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
perpendicular to the primary flow direction. The length of the inlet and the outlet region is selected as 6 h and 15 h, respectively, to minimize the effect of the imposed inlet and outlet velocity boundary conditions on the characteristics of the flow in the spacer bank region. Flow simulations are conducted for Re = 100, 400, 600 and 800. The Reynolds number is defined as Re ¼ 2hU av e =m, where h is the channel height, U av e is the inlet average velocity, and m is the kinematic viscosity. Three dimensional steady flow simulations are performed in the feed channel containing spacers bounded by RO membranes. The brackish water is modeled as an incompressible fluid with constant physical properties, and the salt diffusivity, D, in the aqueous solution is assumed to be constant. The equations governing the velocity and salt concentration fields are: the continuity equation
@ui ¼ 0; @xi
ð1Þ
and transient turbulent flows in channels containing obstacles [12,15–17]. SST k—x turbulence model is used here to simulate steady three dimensional flows with mass transport in the feed channel for Re P 400. The k—x turbulence momentum transport model yields:
uj
uj
@ui 1 @p @ ui ¼ þm ; @xj q @xi @xj @xj
ð2Þ
and the mass transport equation
uj
@c @2c ¼D : @xj @xj @xj
ð3Þ
Here q is the fluid density, i and j are index placeholders. The spatial coordinates: x1 ¼ x, x2 ¼ y, and x3 ¼ z, the velocity components: u1 ¼ u, u2 ¼ v and u3 ¼ w and c is the salt concentration. Both Renormalized-group (RNG) k—e, shear stress transport (SST) k—x are successfully used to model turbulent flows in channels. It has been shown that turbulent flow structures and mass and heat transport in channels containing spacers can be characterized more accurately by using SST k—x model [13,14]. Recently, the present authors employ the time averaged baseline (BSL) k—x and SST k—x turbulence models successfully to simulate steady
ð4Þ
where l is the dynamic viscosity and lt is the eddy viscosity, lt ¼ q maxðaa11xk ;XF 2 Þ. The equations governing the turbulence kinetic energy, k, and the specific dissipation rate, x, are:
ui
ui
the conservation of momentum 2
@ui 1 @p 1 @ @ui ¼ þ ðl þ lt Þ @xj q @xi q @xj @xj
@ðqkÞ @ui @ @k ¼ sij b qxk þ ðl þ rk1 lt Þ ; @xi @xj @xj @xj @ðqxÞ c @ui @ @x ¼ sij bqx2 þ ðl þ rx lt Þ @xi @xj mt @xj @xj 1 @k @ x þ 2qð1 F 1 Þrx2 x @xj @xj
ð5Þ
ð6Þ
where X is the vorticity magnitude, a; a1; b; b ; rk , rx , rx2 and c are closure coefficients, and F 1 ; F 2 are the blending functions. Detailed description of the model can be found in Ref. [18]. The k—x turbulence mass transport model yields:
uj
@c @ ¼ @xj @xj
Dþ
lt @c qSct @xj
ð7Þ
where Sct is the turbulence Schmidt number. A parabolic profile is used for the stream-wise component of the 2
velocity at the inlet, u ¼ 6U av e ½y=h ðy=hÞ . The inlet velocity profile is assumed to be uniform in the z-direction (span-wise direction). The constant concentration, c0 ¼ 4000 ppm, is also specified at the inlet. At the outlet the zero pressure and constant concentration gradient are applied. At the side surfaces, z ¼ 0 and z ¼ 5 h, periodic boundary conditions are imposed on both velocity and concentration fields. Along the surface of cylinders, no-slip and
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403
Fig. 5. The iso-surfaces of the stream-wise component of the velocity for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
no-penetration conditions are imposed on the velocity field and zero flux condition is imposed on the concentration field. The solution-diffusion model was introduced in 1940’s and was used for membrane flux modeling since then. Wijmans and Baker [19] illustrated the derivation of the solution-diffusion model using chemical potential approach and proved that the model is applicable to determine the mass transport via the reverse osmosis process. By employing the solution-diffusion approach, the membrane in the present study is considered as a functional surface where the local water flux, the local salt concentration, and the local pressure are all coupled together. Hence, the local water flux is determined from the transmembrane pressure difference, Dp, and the osmotic pressure difference, Dp, between the feed and production channels. Simultaneously, the salt concentration gradient is calculated from the local water flux and local salt concentration at the membrane surface. The equations of the local water flux and local salt concentration gradient at the membrane surface are:
@y ¼ 0 and h : u ¼ 0;
9
v w ¼ A½Dp Dp > = @c D @y ¼ v w ðcw cp Þ ; > ;
Dp ¼ jðcw cp Þ ð8Þ
where v w is the local water flux through the membrane, cw is the local salt concentration at the membrane surface, A is the water permeability through the membrane, j is the osmotic coefficient, and cp is the salt concentration in the production side. The water flux through the membrane is normalized as v w =ðDpAÞ. The drag coefficient, C D , is determined from:
CD ¼
2f x
qU 2av e As
;
ð9Þ
where f x is the drag force exerted on the spacer and As is the projected area of the spacer perpendicular to the stream-wise direction. The Sherwood number, Sh, is evaluated from:
Sh ¼
ð2hÞhm ; D
hm ¼
@c D @y
y¼h
ðcb cw Þ
ð10Þ
where hm is the local mass transfer coefficient and cb is the bulk concentration. The pressure coefficient, f , is calculated as:
F¼
2hj dp j dx
ð1=2ÞqU 2av e
ð11Þ
where j dp j is the pressure gradient. dx 1.2. Numerical method, mesh optimization and validation The commercial software CFX 14.5 is utilized to conduct simulations. This software implements the finite volume discretization method. Equations are solved using Semi-Implicit Method Pressure Linked Equations (SIMPLE) algorithm that fully couples the velocity and pressure fields. The CEL expression language is deployed to implement the parabolic velocity profile at the inlet. Similarly, the water flux and salt concentration gradient equations at the membrane surface were modeled via the CEL expression language. The operating parameters and physical properties are selected so that the typical RO brackish water desalination process is simulated. The transverse pressure difference is fixed as 1.25 MPa since the pressure drop in the feed channel for the considered domain, <2 kPa, is negligible compared to the transmembrane pressure difference. The physical properties of the brackish water in the feed channel are assumed to be constant. The density is q ¼ 1000 kg/m3 and the dynamic viscosity is l ¼ 103 Pa s. The salt diffusivity in the water is assumed to be constant, D ¼ 1:5 109 m2/s with the Schmidt number, Sc ¼ m=D ¼ 667. The osmotic coefficient is j, is 75 kPa m3/kg, the water permeability is A ¼ 2:3 1011 m/(s Pa), and production side salt concentration cp ¼ 0. A mesh optimization study is performed using two mesh sizes: 20 million (M1) and 30 million (M2) elements. Profiles of the stream-wise component of the velocity are calculated across the gaps between membrane (1) across the first spacer at x=h ¼ 6 and z=h ¼ 2.5 and (2) in the wake of the first spacer at x=h ¼ 7 and z=h ¼ 2.5. Velocity profiles are shown in Fig. 2 for both meshes. The pressure coefficient, f , is also calculated for the mesh M1 and
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Fig. 6. The iso-surfaces of the z-component of the vorticity for values of the Reynolds number of (a) 100, (b) 400, (c) 600, and (d) 800.
mesh M2. The pressure coefficient is determined from Eq. (11) using average pressure drop across the part of spacer bank from x ¼ 38h to x ¼ 77h. The velocity profiles obtained for M1 and M2 are similar; implying that mesh size M2 is enough to ensure mesh independency. The pressure coefficient is calculated to be 0.320 and 0.323, respectively, for M1 and M2 mesh; confirming that the mesh independency might be achieved with M2. Stronger argument could be made by refining mesh further. The size of computation and the available computational resources make that to be a difficult task. Simulation results presented in the present study are obtained using M2 mesh. The membrane flux model used in the present study was validated previously by the present authors [12]. Two dimensional simulations in the feed channel containing spacers bounded by membranes were conducted using the solution-diffusion by both studies [11,12]. It was shown that results of water flux predicted by the present authors [12] and Ma and Song [11] agree well. Additionally, the drag coefficient, C D , of the 1st spacer at Re ¼ 800 is calculated and compared against that predicted by Chakraborty [3]. Drag coefficient, C D , predicted by the present three and two dimensional flow simulations is 4.482 and 3.995, respectively. Drag coefficient documented in [3] from two dimensional flow simulations at Re equivalent of 800 is 3.946. Agreement in the drag coefficient with previous work helps validating flow modeling and simulations presented here. 2. Results and discussion The results of steady three dimensional flow simulations are presented for Re ¼ 100, 400, 600 and 800. Comparison between results obtained from three dimensional and two dimensional simulations are made to determine the influence of three dimensional flow structures on the membrane performance. Two dimensional flow modeling consider velocity and concentration field in the
z-direction to be uniform. The laminar flow model is used for Re ¼ 100, and SST k—x turbulence model is used for Re ¼ 400, 600 and 800 to simulate mass and momentum transports in the feed channel. Fig. 3 illustrates contours of the stream-wise component velocity at Re ¼ 100, 400, 600 and 800. Images are shown at three z planes (z=h ¼ 0.4, 2 and 3.6) for each value of Re. The wake region behind spacers becomes larger as Re is increased. High speed flow region around the cylinder is also stretched in the stream-wise direction as Re is increased, as shown in Fig. 3. Three dimensional flow structures and their strength cannot easily be distinguished from these images. Primary flow in the stream-wise direction dominates the secondary flows induced by the flow transitions. These flows transitions are well-documented in the literature [20] and they are expected to onset for values of Re above the criticality. The secondary flows manifest themselves nearly periodic flow structures in the span-wise direction. Fig. 4 shows contours of the stream-wise components of the velocity at the mid-plane (y=h ¼ 0:5) between membranes at Re ¼ 100, 400, 600 and 800. The presence of secondary flows is very clear. At Re ¼ 100 the flow is nearly uniform in z-direction and it can be characterized as two dimensional, as shown in Fig. 4a. Velocity contour depicted in Fig. 4b for Re ¼ 400 shows vast contrast to the velocity contour obtained for Re ¼ 100. The flow becomes three dimensional as a result of induced secondary flows. It is clear that the critical value of Re for the onset of three dimensional flow is between 100 and 400. It has been reported that transition to the three dimensional flow occurs between Re of 100 and 200 for these geometries [4]. Velocity contours at Re ¼ 600 and 800 exhibit three dimensional flow structures with a lesser intensity relative to the primary flow, as shown in Fig. 4c and d. These three dimensional flow structures can have a strong influence on the membrane performance, the concentration polarization and the potential fouling along the surface of the membrane.
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Fig. 7. The iso-surfaces of x and y component of the vorticity for different values of the Reynolds number. Right and left column denote iso-surfaces of the x component of the vorticity and y component of the vorticity, respectively. Rows from top to bottom denote iso-surfaces at Re = 100, 400, 600 and 800.
The iso-surface images of the stream-wise component of the velocity are shown in Fig. 5. The iso-surfaces are consistent with images of the velocity contours shown in Fig. 4. Uniform isosurfaces in the span-wise direction for Re ¼ 100 while finger-like structures are realized in Fig. 5b–d for Re ¼ 400, 600 and 800, respectively. Flow structures are more regular for Re ¼ 400 compared to those for Re ¼ 600 and 800. Finger-like structures fill almost the entire region between spacers; indicating that three dimensional effects caused by flow transitions are strong and do not dissipate quickly away from spacers. Flows in the entire spacer bank region become three dimensional for Re P 400, as shown in Fig. 5. The iso-surface images of the z- component of the vorticity is depicted in Fig. 6 for Re ¼ 100, 400, 600 and 800. Flow is
two-dimensional for Re = 100, as indicated by uniformly distributed z-component of the vorticity in the span-wise direction. As the flow becomes three dimensional at Re = 400, z-component of the vortcity becomes spatially distributed in the span-wise direction. At higher flow rates (Re = 600 and 800) the primary flow is so dominant that the secondary flows are not visible from iso-surface images of the vorticity, see Fig. 6c and d. Fig. 7 illustrates the iso-surfaces of x and y component of the vorticity. Images on the left column denote iso-surfaces of x component of the vorticity and images on the right column denote y component of the vorticity for Re ¼ 100, 400, 600 and 800. The images reveal that there are weak and fragmented vorticities near spacers at Re ¼ 100, as shown in Fig. 7a. There is no indication of
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Fig. 8. Contours of normalized concentration (c=c0 ) acquired at z-planes of z=h ¼ 0.43, 2, and 3.6 for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
regular flow structure distributed along the span-wise direction. The vortices dissipate rapidly away from spacers. Counter rotating five pairs of vortices are uniformly spaced in the span-wise direction, as shown in Fig. 7a2 and b2 with images acquired at Re ¼ 400. Once again, these flow images clearly indicate that coherent three dimensional flow structures exist at this flow rate. The vortices are attached to the spacer and extend into the region between spacers. Flow becomes irregular as Re is increased to 600 and 800. Vortices break and form smaller eddies in the wake of spacers. The vortices extend longer, and they occupy the whole region between the spacers for Re ¼ 800, as depicted in Fig. 7. These vortices lead to the formation of streaks near the wall, as documented by several investigators [21–23]. The influence of such streaks can be profound on the membrane performance. Concentration polarization and the formation of fouling and scaling will be directly influenced by such flow structures. These effects are presented and discussed in the present study. Contours of normalized salt concentration (c=c0 ) at three z planes, z=h ¼ 0.4, 2 and 3.6, are shown in Fig. 8 for Re ¼ 100, 400, 600 and 800. There is a thin concentration boundary layer near membrane surface. The thickness of the boundary layer is greater for Re ¼ 100 and it decreases as Re is increased. The concentration boundary layer is disrupted by the presence of the spacer. The boundary layer grows away from the spacer. All three planes show similar concentration field at all flow rates. The concentration polarization at Re ¼ 100 is the highest while it is the lowest at Re ¼ 800, as depicted in Fig. 8. Contours of the water flux through the membrane and the concentration along the surface of the top membrane are shown in Fig. 9 for Re ¼ 100, 400, 600 and 800. Both water flux and the concentration are uniform in the span-wise direction for Re ¼ 100. The water flux through the membrane is profoundly increased by the presence of the spacers, as shown in Fig. 9a. The water flux is much higher above each spacer. The concentration along the membrane is much lower right above the spacers, but it increases rapidly away from the spacers. It is also noted that the presence of the spacers alleviate the concentration polarization occurring along the surface of the membrane. At Re ¼ 400 the distribution of the
water flux and the concentration along the surface of the membrane is strikingly different compared to those at Re ¼ 100. The effect of three dimensional flows in the feed channel influences the water flux and the local concentration along the membrane surface. Streaks of high and low regions of water flux and concentration are uniformly distributed in the span-wise direction. Streaks are more pronounced at Re ¼ 400 compared to those at Re ¼ 600 and 800; following the relative strength of the secondary flows over the primary flows as presented above. The water flux is greater and the concentration overall is lower near spacer at all values of Re. The intensity of the concentration polarization decreases with increasing flow rate. It is important to note that low water flux regions correlates to the high intensity concentration polarization regions at all flow rate considered here. Streaks of high concentration polarization region between spacers coincide with streaks of low water flux regions, as shown in Fig. 9. Results of three dimensional simulations confirm our conclusion drawn from two dimensional simulations reported earlier [12]. The present author documented that that spacers enhance the membrane performance while it helps to alleviate concentration polarization in two dimensional flows. The local value of the Sherwood number is calculated along the surface of membranes using Eq. (10). The bulk concentration, cb , in Eq. (10) is calculated from average of salt concentration at the inlet and the outlet. Contours of the local Sh along the surface of the top membrane are depicted in Fig. 10. Distribution of Sh is very similar to the distribution of the water flux over the membrane surface. Sh is distributed uniformly in the span-wise direction at Re ¼ 100 while strips of high and low regions of Sh are shown at Re P 400. Similar to the counter part of two dimensional flows presence of the spacers in three dimensional geometry leads to significant increase in Sh. Local value of Sh is much greater near spacer and decreases rapidly away from spacers. The Sherwood number also increases as the flow rate is increased; that is also consistent with the prediction of two dimensional simulations documented in [12]. Streaks of high Sh regions coincide with the regions of low intensity concentration polarization, as depicted in Figs. 10 and 9.
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Fig. 9. Contours of normalized water flux through the top membrane (right column) and contours of normalized concentration along the surface of the top membrane (left column) for different values of Re. Rows from top to bottom denote iso-surfaces at Re = 100, 400, 600 and 800.
Contours of the normalized wall shear stress along the surface of the top membrane are shown in Fig. 11. The local shear stress is normalized by the maximum value of the wall shear stress for each flow rate. The maximum value of the wall shear stress is calculated to be 0.9, 4.5, 8.5 and 12.6 Pa for Re ¼ 100, 400, 600 and 800, respectively. Similar to the flow properties presented above the wall shear stress is uniform in the span-wise direction at Re ¼ 100, but it becomes streaky as the flow becomes three dimensional at higher flow rates (Re ¼ 400, 600 and 800). Streaks induces by the secondary flow is not as pronounced as in the case of the water flux, concentration or Sh number distribution. At all flow rates, the maximum wall shear stress occur right above the spacers
where the fluid is accelerated passing through the spacers. The intensity of the wall shear stress increases as Re is increased. Koutsou et al. [8] experimentally determined regions where fouling occurs and showed that the fouling sites coincide with the low wall shear stress regions. Based on present results, potential fouling sites are situated in the regions between spacers and three dimensional flow structures in the feed channel induced by secondary flows help to alleviate potential fouling occurring along the surface of the membrane. The span-wise averages of normalized water flux, concentration and the Sherwood number at each x location are determined for three dimensional flows in the feed channel containing spacers.
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Fig. 10. Contours of the local Sherwood number (Sh) along the surface of the top membrane for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
Fig. 11. Contours of the wall shear stress normalized by the maximum value for values of Re (a) 100, (b) 400, (c) 600 and (d) 800.
These span-wised average profiles in the stream-wise direction are plotted in Figs. 12 and 13 in the spacer bank region for different values of Re. The profiles of the normalized water flux, concentration and local value of Sh predicted by two dimensional steady simulations are also plotted in Figs. 12 and 13 to assess the influence of the three dimensional effects on the span-wise averaged membrane performance properties for Re ¼ 100, 400, 600 and 800. In both figures, the solid lines denote profiles predicted by the three dimensional simulations while the dashed lines denote profiles predicted by two dimensional simulations. Overall distribution of the two dimensional and span-wised average three
dimensional profiles is similar for all membrane surface properties. Only the values of the peaks and valleys of these quantities differ at each flow rate, as shown in Figs. 12 and 13. The averaged three dimensional Sh profiles exhibit nearly the same distribution in the stream-wise direction as for the Sh obtained by the two dimensional simulations. Only noticeable difference in Sh profiles is observed near wake regions of the spacers. Sherwood number drops significantly right behind the spacers in the two dimensional geometry and rebound very quickly approaching the following spacers. The values of Sh is greater at the peak and is lower at the valley of the two dimensional profiles
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Fig. 12. Span-wised averaged water flux profiles (a1 and a2) and the concentration profiles (b1, b2) in the stream-wise direction for different values of Re. Solid lines denote span-wised averaged three dimensional profiles and dashed lines denote two dimensional profiles.
compared those of the span-wise averaged three dimensional profiles, as shown in Fig. 13. It has to be noted Sh varies greatly in the span-wise direction at Re P 400, as shown in Fig. 10. Membrane flow properties are influenced by the spatial characteristics of the flow in the feed channel. The average values of Sh and the normalized water flux in both span-wise and stream-wise direction in the spacer bank region are determined for Re ¼ 100, 400, 600 and 800. Spacer bank averaged values of Sh and the water flux are plotted in Fig. 14 for three and two dimensional geometry along with the averaged value of Sh and the normalized water flux determined in the three dimensional channel without spacers. Filled blue squares and hollow red circles denote results predicted by the three-dimensional and twodimensional simulations, respectively. Triangles represent the results of three-dimensional flow in a channel without spacers. To minimize the influence of the inlet and the outlet conditions averaging is performed in the spacer bank region between x=h ¼ 38 and x=h ¼ 77. The results reveal that two dimensional approximation predicts the total averages of the water flux and
the Sh reasonably well. Slight over-prediction at low Re and high Re by the two dimensional simulations are seen (see Fig. 14). It is proven that presence of spacers enhances the membrane performance greatly. Three dimensional effects induced by flow transition do not alter averaged membrane properties significantly. This conclusion is specific to the three dimensional spacer geometry considered in the present study. This geometry is simplified version of the ladder type spacer mesh configuration used in separation modules. The influence of the three dimensional flow structures could be profoundly different for other spacer mesh configurations (woven and non-woven spacers). Present authors are currently studying three dimensional flows in the feed channel containing mesh of spacer with different configurations. The results of those simulations are presented in a full paper. It is also important to note that transient effects in these flows can also have significant influence on the concentration polarization, fouling and the membrane performance. Investigating the influence of transient effects in three dimensional geometries is worthy of future work, but is not in the scope of the present study.
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Fig. 13. Span-wised averaged Sh profiles) in the stream-wise direction for different values of Re. Solid lines denote span-wised averaged three dimensional profiles and dashed lines denote two dimensional profiles.
Fig. 14. Span-wise and stream-wise averages of normalized water flux (1) and Sh (2) are plotted as a function of Re. Channel averages of membrane properties are shown for three dimensional channel with and without of spacers and two dimensional channel with spacers. The averages of membrane properties are calculated in the spacer bank region between x=h of 38 and 77.
3. Conclusion Steady three dimensional simulations have been conducted to study the mass transport through the reverse osmosis membrane in the feed channel of a brackish water desalination module. The membrane surface is treated as a functional surface where the water flux is determined from local salt concentration and local pressure along the membrane. The laminar flow model is employed at Re ¼ 100 while the SST k—x turbulence flow model is employed to simulate mass and momentum transport in the feed channel at Re P 400. The flow and concentration field is nearly uniform at Re ¼ 100, but it becomes strongly three dimensional for Re P 400. It is shown here that flow transition from two dimensional to three dimensional flows in a channel containing spacers
occur values of Re between 100 and 400 for flow past an array of cylinders with blockage ratio of 1/2. Even though two dimensional flow simulations predicts the averaged membrane properties reasonably well, the local water flux and Sh along the surface of the membrane varies strongly in the span-wise direction. Furthermore the concentration polarization is profoundly influenced by the three dimensional secondary flows induced by the flow transitions. Velocity and vorticity field indicate spatially periodic flow structures present in the feed channel. Streaks along the surface of the membrane for local properties such as Sh, concentration, water flux and wall shear stress are results from the vortical structures obtained in these three dimensional flows. High water flux regions and low concentration regions coincide with the low wall shear regions. The high intensity concentration polarization sites correlates
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directly with the high potential fouling sites. Spacers enhance the membrane performance, and it mitigates the concentration polarization. Increasing flow rate also enhance the membrane performance. This study proves that three dimensional effects have profound influence on the membrane performance, especially the polarization and fouling characteristics of the reverse osmosis membrane in desalination systems. Further study is needed to characterize flows in the feed channel with various mesh configurations of spacers used in the desalination processes. References [1] L. Malaeb, G.M. Ayoub, Reverse osmosis technology for water treatment: state of the art review, Desalination 267 (2011) 1–8, http://dx.doi.org/10.1016/j. desal.2010.09.001. [2] K.P. Lee, T.C. Arnot, D. Mattia, A review of reverse osmosis membrane materials for desalination—development to date and future potential, J. Membr. Sci. 370 (2011) 1–22, http://dx.doi.org/10.1016/j.memsci.2010.12.036. [3] J. Chakraborty, N. Verma, R.P. Chhabra, Wall effects in flow past a circular cylinder in a plane channel: a numerical study, Chem. Eng. Process. 43 (2004) 1529–1537, http://dx.doi.org/10.1016/j.cep.2004.02.004. [4] N. Kanaris, D. Grigoriadis, S. Kassinos, Three dimensional flow around a circular cylinder confined in a plane channel, Phys. Fluids 23 (2011) 064106, http://dx. doi.org/10.1063/1.3599703. [5] F. Rehimi, F. Aloui, S. Ben Nasrallah, L. Doubliez, J. Legrand, Experimental investigation of a confined flow downstream of a circular cylinder centred between two parallel walls, J. Fluids Struct. 24 (2008) 855–882, http://dx.doi. org/10.1016/j.jfluidstructs.2007.12.011. [6] M.D. Griffith, J. Leontini, M.C. Thompson, K. Hourigan, Vortex shedding and three-dimensional behaviour of flow past a cylinder confined in a channel, J. Fluids Struct. 27 (2011) 855–860, http://dx.doi.org/10.1016/j. jfluidstructs.2011.02.007. [7] M. Shakaib, S.M.F. Hasani, M. Mahmood, CFD modeling for flow and mass transfer in spacer-obstructed membrane feed channels, J. Membr. Sci. 326 (2009) 270–284, http://dx.doi.org/10.1016/j.memsci.2008.09.052. [8] C.P. Koutsou, S.G. Yiantsios, A.J. Karabelas, A.J. Karabelas, Membrane module performance optimization using CFD simulation of flow through narrow channels with spacers, in: Proc. IDA World Congr. Desalin. Water Reuse, 2007, pp. 21–26. [9] V. Geraldes, V. Semiao, M. Norberta de Pinho, Concentration polarisation and flow structure within nanofiltration spiral-wound modules with ladder-type
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