Laminar Flow Transitions in a 2D Channel with Circular Spacers

Ind. Eng. Chem. Res. 2007, 46, 5387-5396

5387

Laminar Flow Transitions in a 2D Channel with Circular Spacers A. Alexiadis,*,† D. E. Wiley,† D. F. Fletcher,‡ and J. Bao† School of Chemical Sciences and Engineering, The UniVersity of New South Wales, Sydney NSW 2052, Australia, and School of Chemical and Biomolecular Engineering, The UniVersity of Sydney, Sydney NSW 2006, Australia

A computational fluid dynamics (CFD) model was used to simulate the flow in a narrow spacer-filled channel, which represents the longitudinal section of a spiral-wound module membrane. The flow was computed for different geometries and Reynolds numbers. Calculations show the presence of different flow patterns in the laminar regime. Equations describing the critical Reynolds numbers at which the transition between different patterns occur were derived. It is shown that the critical Reynolds number determines the point at which good mixing conditions are achieved without excessive pressure drop. Introduction Spiral-wound modules (SWMs) are widely used in industrial applications ranging from reverse osmosis (RO) to ultra-filtration (UF) because they offer advantages in terms of packing density and fouling control. Alternating layers of membrane leaves and spacer sheets are rolled up together, as shown in Figure 1a, in order to fabricate the module. The spacers separate the membrane leaves in the module and can enhance wall shear stress and mixing and, consequently, reduce concentration polarization and fouling. The effect of the spacer filament on the velocity pattern in the channel has been the subject of many studies both from the theoretical and experimental points of view.1 More recently, computational fluid dynamics (CFD) modeling has been used extensively1-11 in order to gain insights into the hydrodynamics of the system. These works show, in particular, that the flow inside the channel is strongly affected by the spacer geometry. At low Reynolds numbers, the velocity profile is regular and not time-dependent, while above a certain critical Reynolds number (ReCR), the spacers perturb the flow, producing a typical vortex-shredding pattern in the laminar regime. Work has concentrated on laminar flow regimes as industrial spiralwound modules are unlikely to operate in the turbulent regime because the pressure drops associated with turbulent flow can cause telescoping and, therefore, destruction of the module. Under these conditions, steady-state simulations do not give accurate results and transient modeling is required. The presence of vortices enhances the mixing in the channel and, consequently, reduces concentration polarization and fouling. The increased mixing and mass transfer and the reduced fouling, however, come at the expense of higher pressure drops and operating costs. The pressure drop in the SWM is proportional to Rem (with m values that range from 1.69 to 1.82)12 and therefore, increasing Re above ReCR can lead to higher operational costs without necessarily comparable benefits to the mixing. It is clear, from this perspective, that knowledge of ReCR is useful for improving the performance of a SWM because it * To whom correspondence should be addressed. Current Address: Marie Curie Transfer of Knowledge Centre for the Computational Sciences, University of Cyprus, Department of Mechanical and Manufacturing Engineering, 75 Kallipoleos Street, P.O. Box 20537, 1678 Nicosia, Cyprus. E-mail: [email protected]. † The University of New South Wales. ‡ The University of Sydney.

might permit operation of the membrane process in the most beneficial flow pattern while avoiding excessive pressure drop. A similar configuration to the one in this paper has already been investigated but with square instead of circular spacers.13 That study, however, focused on determination of the critical Reynolds numbers at which laminar-to-turbulent transition occurs. Other articles have been devoted to the analysis of the flow behind an unbounded circular cylinder. In this case, different flow patterns in the laminar regime have been characterized and transition between these patterns has been investigated.14-16 This paper investigates different configurations of circular spacers in a two-dimensional channel in order to determine the effect of the spacer diameter and of the distance between spacers on ReCR. Potential operational advantages and disadvantages of the different regimes are also presented. Transient effects (i.e., frequency of oscillations or Strouhal number) of the flow were investigated in previous papers.2,3 The question of validation of such simulations is an important one. The results presented here have been checked to ensure that they are independent of numerical parameters, such as grid density and time-step. Simulation results have been compared previously with flow around cylinders in unbounded domains, where experimental data are available, and have been shown to be accurate. No experimental data were available to validate the results presented here directly, but because the NavierStokes equations govern this flow, provided the numerical practices are correct, the results represent the true flow, i.e., they do not depend on model assumptions as in, for example, turbulent flow. The development of particle image velocimetry (PIV) techniques may allow such flow to be studied experimentally in the future, but there are significant challenges to overcome in constructing a model channel that gives good visual access, represents a section of a module with fully developed flow patterns, and allows data acquisition on the ms time scale. Geometry Commercial spacer nets are available in various geometries and sizes. In this paper, cylindrical spacer filaments, as shown in Figure 1b, are considered. The respective positions of the filaments in the net can change. The two most common options, which are studied in this work, are called “cavity” (Figure 1c) and “zigzag” (Figure 1d). Numerical Methods It is currently not possible to simulate the flow in an entire membrane module. The computer memory usage and the

10.1021/ie0607797 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/10/2007

5388

Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007

Figure 1. Schematic picture of membrane spiral-wound modules.

computational times required would be far beyond the capacity of both common desktop PCs and large supercomputers. Practical calculations are limited to a small portion of the module as shown in Figure 1. It must be ensured, however, that the channel is long enough for the flow to reach the fully developed profile. We found that, for multispacer channels, this is usually achieved after five to six spacers; therefore, simulations were carried out in a channel with nine spacers. The CFD model described in early works17,18 was used with a two-dimensional channel. The membrane was not specifically modeled but was treated as an impermeable wall. Usually, the order of magnitude of the velocity though the membrane is between 10-4-10-6 m s-1; therefore, it does not affect the bulk flow, which has characteristic velocities between 1 and 5 m s-1. The use of a 2-D model instead of a 3-D model presents advantages and disadvantages at the same time. The adequacy of 2-D simulations has recently been experimentally verified for a channel with square spacers.13 In three-dimensional channels,11 calculations have shown that the velocities in the z-direction must be taken into account in some cases, and consequently, 3-D simulations will potentially produce more accurate results than 2-D simulations for these cases. However, calculations in three-dimensional domains are much more expensive in terms of both memory usage and computational time. They can, therefore, only be carried out for a very small portion of the membrane channel generally limited to not more than two consecutive spacers and must be modeled by use of periodic boundary conditions for which it has not yet been possible to quantify the accuracy. In this paper, simulations were carried out for relatively long 2-D channels with up to nine spacers. Usually the channel height is 2 orders of magnitude lower than the curvature diameter. For this reason, it is generally agreed that the curvature does not play a significant role in the channel flow. This is particularly true in the presence of spacers, where the flow is disturbed and a degree of mixing is introduced every few millimeters. The curvature can have some effect if

Figure 2. Mesh density in the region of a spacer.

the flow is azimuthal (and without spacers) instead of axial, as in the case under study.19 In Figure 2, a picture of the mesh used for the simulations is shown. Triangles with size in the y-direction between 0.05 and 0.01 mm were used. The triangles were stretched with a factor of 1.8 in the x-direction. Near the membrane and around the spacers, an inflated boundary was used in order to resolve the near-wall velocities to suitably high accuracy in these regions. The commercial code ANSYS CFX 10.0 was used for the simulations. Laminar calculations using second order in time and space schemes were carried out. Grid independence was tested by repeating selected simulations with meshes of different size. Transient features related to the number, position, and frequency of the vortices were checked qualitatively. Timeaveraged velocity profiles obtained with different grids were compared quantitatively in order to ensure that the average error was within 1%. The convergence criterion for root mean square (rms) residuals was fixed to 10-4 with a time-step size of 5 × 10-5 s. Under these conditions, the calculations required more than 25 MB of memory and usually 5 000 time steps to calculate one residence time. The resulting computational times were between 20 and 40 h for each simulation with a Xenon 2.4 GHz CPU.

Ind. Eng. Chem. Res., Vol. 46, No. 16, 2007 5389

Figure 3. Velocity patterns at different Reynolds numbers for a single spacer (d/h ) 0.45). (Note that the scale is different for each case to better highlight the flow structure.)

Figure 4. Velocity patterns at different dimensionless diameters for a single spacer (Re ) 1600).

Single Spacer The simulations were carried out for laminar flow (Re < 2000)20 and spacer diameters from 0.4 to 0.63 mm. The definition of Re for a SWM is given in ref 21 as

Re ) FU0dh/µ

(1)

where µ, F, and U0 are the viscosity, the density, and the average inlet velocity, respectively, and dh is the hydraulic diameter defined as

dh ) 4(VTOT - VSP)/(SC + SSP)

(2)

VTOT is the channel volume without the spacers, VSP is the volume of the spacers, SC is the surface area of the channel, and SSP is the surface area of the spacers. The three typical laminar velocity patterns for a single spacer configuration are reported in Figure 3, where the results, computed at different Re for a spacer with a dimensionless

Figure 5. Critical Reynolds numbers for various Reynolds and spacer sizes: comparison between eqs 3 and 4 and CFD calculations.

diameter (d/h) of 0.45 (d is the diameter of the spacer and h is the height of the channel), are shown. These results are

5390


Figure 6. Developing and fully developed flow for the case of d/h ) 0.45, ∆L/h ) 3, and Re ) 1200.

Figure 7. Simulated flow patters for d/h ) 0.58, ∆L/h ) 3, and Re ) 100.

Figure 8. Simulated flow patterns for d/h ) 0.58, ∆L/h ) 3, and Re ) 800.

analogous to the ones reported in our previous work.2,3,5 At Re ) 800, the velocities in the channel can be computed assuming steady flow. The spacer perturbs the flow initially, but the perturbation tends to diminish along the channel. At Re ) 1600, the “tail” of the velocity streamlines oscillates with time, and consequently, the numerical simulations have to be carried out for the transient flow. At Re ) 1900, there is a new hydrodynamic regime. The tail of the streamlines breaks, and vortices appear in the flow. The transition between the first and second regimes can be considered as smooth, in the sense that timedependent oscillations start to appear gradually with increasing Reynolds number. However, above a certain critical Reynolds number (ReCR), the flow pattern changes dramatically from pattern two to pattern three. Thus, as Re increases, the flow pattern changes from “steady” to “oscillation” to “vortex” flow. The transition between the regimes depends not only on Re but

also on the spacer diameter. Large diameters increase the flow perturbation and reduce ReCR. Figure 4 shows the velocity patterns at Re ) 1600 for two different dimensionless diameters d/h ) 0.4 and d/h ) 0.58, which can be compared with the flow at d/h ) 0.45 in Figure 3. The critical Re can be approximated by the following correlation

ReCR ) 1276.1

(dh - 1)

1.11

(3)

for the steady-oscillation transition and

ReCR ) 1534.6

(dh - 1)

0.77

(4)

for the oscillation-vortex transition. Figure 5 shows a comparison between the values for the transition estimated using


Figure 9. Schematic of the method of determining ReCR from Ω profiles.

Figure 11. Complete picture of ReCR from eqs 3, 4, 5, 6, 9, and 10.

Figure 10. Critical Reynolds numbers for different ∆L/h and d/h.

eqs 3 and 4 and the results obtained from the CFD simulations. The fit of eqs 3 and 4 to the data is very good with a maximum error below 1%. Multiple Spacers The same approach was used to describe the regime transition for a channel with multiple spacers. In this case, the system

has an extra degree of freedom: the distance between spacers (∆L) must be specified in addition to the spacer diameter. Two of the possible geometric configurations were studied in this paper: cavity (Figure 1c), with all the spacer cylinders located on one of the two sides of the channel, and zigzag (Figure 1d), with the spacers located alternately between the two sides of the channel. Different simulations with four, six, eight, or nine spacers were carried out. At low laminar Reynolds number values (Re < ReCR), the flow can be considered fully developed after the first spacer. (ReCR, in this case, refers to the critical Reynolds number for multispacer channels.) At high laminar Reynolds number values (Re > ReCR), however, the fully developed flow was reached after five or six spacers. Consequently, simulations completed with a nine-spacer geometry were adequate to establish fully developed flow for all the conditions investigated. The flow is defined to be fully developed when the influence of the inlet of the channel on the flow becomes negligible and, basically, the flow repeats the same pattern along the channel. Figure 6 shows the difference between the developing and the fully developed flow for the case d/h ) 0.45, ∆L/h ) 3, and Re ) 1200. The critical ReCR increases if the distance between the spacers is reduced. A minimum distance LMIN was determined, such that when the distance of the centers between two consecutive spacers is below LMIN, the flow is steady regardless of the

5392


Figure 12. Friction factor results for various Reynolds numbers and spacer geometries: comparison between eq 12 and CFD calculations. Table 1. Values of A and n in Equation 12 at Different Values of d/h for Both Cavity and Zigzag Configurations for Re Values above and below the Critical Reynolds Number ReCR both spacers

cavity spacer

d/h

n(Re < ReCR)

n(Re > ReCR)

A(Re < ReCR)

0 0.3 0.4 0.5 0.6 0.7

1 0.86 0.81 0.75 0.68 0.61

0.25 0.25 0.25 0.25 0.25 0.25

16 16.24 16.43 16.67 18.30 23.32

Reynolds number. This value was found to be

(LMIN/h) ≈ 1.6

(5)

for the cavity configuration and

(LMIN/h) ≈ 1.4

(6)

for the zigzag configuration. There is an effect of d/h and Re on LMIN, but it can be neglected since it was found to be (0.2. There is also a maximum distance LMAX beyond which the flow is independent of the preceding spacer. In this case, the system can be simulated by n autonomous spacers and eqs 3 and 4 hold. The value of LMAX is of the order of 20-50h. A

zigzag spacer

A(Re > ReCR)

A(Re < ReCR)

A(Re > ReCR)

0.51 0.68 0.90 1.66 7.23

16 17.07 17.54 18.12 20.54 60.8

0.35 0.47 0.69 1.46 6.74

specific correlation for LMAX is not provided since this value can be described by the more general eqs 9 and 10, as discussed later where long channel simulations were carried out. The oscillating flow pattern observed for the single spacer is absent when multiple spacers are present in the channel. If ∆L < LMAX, the presence of the second spacer does not allow the tail to oscillate as in the case of single spacer. As a consequence, the definition of critical Reynolds number (ReCR) for the multispacer channel is different from the previous one. In this case, it represents the Reynolds number at which the transition between the steady and the vortex-shedding behavior occurs. The transition between the two regimes for multiple spacers is also not as sharp as for the single-spacer case. This is due to


Figure 13. Mixing parameter versus Reynolds number: comparison between eq 13 and CFD calculations (cavity configuration). Table 2. Values of a, b, and c in Equation 13 at Different Values of d/h and ∆L/h for Both the Cavity and Zigzag Configurations Cavity Spacer d/h ) 0.40

d/h ) 0.45

d/h ) 0.58

d/h ) 0.63

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

6.97 6.83 6.63

values of a 8.02 7.52 7.39

9.73 9.29 9.13

10.90 10.79 10.60

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

1.74 1.48 1.37

values of b 3.19 2.74 2.50

4.83 4.09 3.74

6.51 5.62 5.14

0.994 0.997 0.998

values of c 0.995 0.999 0.999

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

in Figure 7 is not very high because the velocities in that region of the flow are relatively low if compared with the main stream located in the upper half of the channel. The transition to the vortex-shedding regime occurs when the vortices “absorb” enough energy from the main stream and the flows in the two halves of the channel interact with each other, exchanging momentum (Figure 8). To determine the exact point of transition, we defined the variable Ω, called the mixing parameter, to estimate the extent of mixing.

Ω) 0.994 0.998 0.999

0.993 0.997 0.997

Zigzag Spacer d/h ) 0.40

d/h ) 0.45

d/h ) 0.58

d/h ) 0.63

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

8.29 7.21 6.63

values of a 9.14 7.83 7.21

11.26 9.70 8.93

12.87 11.28 10.39

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

3.51 2.48 1.32

values of b 4.09 2.68 1.74

6.08 3.66 2.37

7.88 4.36 2.82

∆L/h ) 3 ∆L/h ) 6 ∆L/h ) 9

0.998 0.998 0.998

values of c 0.999 0.998 0.998

0.998 0.997 0.998

0.997 0.996 0.997

the fact that small vortices are present in the flow when Re < ReCR (Figure 7). Initially, the energy associated with the vortices

h U0

∂V - | dx dy ∫∫S | ∂u ∂y ∂x

(7)

The definition of the mixing parameter Ω depends on the flow vorticity averaged over a section S between two spacers in the fully developed region. The absolute value is required in order to take into account zones of both negative and positive vorticity, and the ratio h/U0 is used to make Ω dimensionless. Since the value of the mixing parameter oscillates with time for Re > ReCR, the time-average is calculated. The value of the mixing parameter was determined at different Re. In Figure 9, for example, the profile of the mixing parameter for a zigzag configuration with d/h ) 0.58 and ∆L/h ) 3 is shown. Figure 9 also shows the graphical approach used to determine ReCR. The same technique was used to find the critical Reynolds number under laminar flow (Re < 2000), for 0.4 < d/h < 0.63 and 3 < ∆L/h < 9, with the aim of determining a simple correlation based on a power-law fitting of the CFD data (shown as triangles in Figure 10). The

5394


Figure 14. Mixing parameter versus Reynolds number: comparison between eq 13 and CFD calculations (zigzag configuration).

data were correlated by introducing a new dimensionless variable

ξ)

[1 -φ2φ] + 1 3

(8)

with φ ) d/h. Thus, the critical Reynolds number can be correlated using:

ReCR ) 193.5ξ2.74

(∆Lh)

0.54

(∆Lh)

0.385

Pressure Drop The presence of different regimes has important consequences on the channel pressure drop. In this section, the relationship between the pressure drop and the Reynolds number is studied by means of the friction factor

(9) f)

for the cavity configuration and

ReCR ) 382.8ξ2.81

point, eqs 3 and 4 should be used instead of eqs 9 and 10, as shown in Figure 11.

(10)

for the zigzag configuration. Figure 10 compares the values of ReCR estimated using the technique shown in Figure 9 (triangles) and the output of eqs 9 and 10 (continuous line). The maximum error is within 5%. Figure 11 shows the effect of (∆L/h) on ReCR for both the cavity and zigzag configurations. If ∆L/h < LMIN, the flow is always steady regardless of the value of Re. The value of LMIN is given by eq 5 for the cavity configuration or by eq 6 for the zigzag configuration. If ∆L/h > LMIN, ReCR is given by eq 9 in the case of the cavity spacer or by eq 10 in the case of the zigzag configuration. When eqs 9 and 10 and eqs 3 and 4 give the same ReCR, the value of LMAX is reached: starting from this

dh 2FU0

2

(∆PL)

(11)

where ∆P is the channel pressure drop and L is the length of the channel. Results are correlated using the law

f ) A/Ren

(12)

where A and n depend on the geometry and the hydrodynamic conditions. The theoretical value for the exponent n for laminar flow in a channel without spacers is 1.0. For the simulations presented in this paper, however, the values of n are ReCR, the value of n does not depend on d/h and it is approximately constant for both configurations at 0.25. Experimental values of n for different types of spacers were found12 to lie in the range 0.15-0.30. Interestingly, 0.25 is the same value of the exponent n in the Blasius equation, which can be used to calculate the friction factor for turbulent flow in smooth pipes. Figure 12 shows the comparison between the friction factor values computed by the CFD model and those calculated with eq 12; the average error is ReCR) and the continuous line (Re < ReCR). This is a common characteristic of the friction factor that is observed near regime transitions. It is also observed around the transition between laminar and turbulent flow in smooth pipes. Therefore, it is not surprising that it emerges also for regime transitions in spacer-filled channels even if, as in the case under study, both the regimes correspond to laminar conditions. Within the range of values investigated (0.4 < d/h < 0.63 and 3 < ∆L/h < 9), the parameters A and n were found to be a function of only d/h. The other geometric parameter (∆L/h) does not have a significant direct effect on f; it plays, however, an indirect role, as shown in Figure 12. When Re > ReCR, the value of the friction factor jumps from the continuous to the dotted line; while these lines do not depend on (∆L/h), ReCR does. The triangular points in Figure 12, for example, are calculated for ∆L/h ) 9; in this case, the critical Reynolds number is higher than in the case of ∆L/h ) 6 (squares) or ∆L/h ) 3 (diamonds) (see Figure 10). Figure 12 shows that the triangles continue to be positioned on the continuous line at higher Re values than the squares or the diamonds, which jump from the first to the second line at lower Re values. Mixing Parameter Versus Re The dimensionless number to describe the mixing parameter Ω was introduced in eq 7 in order to explain the method used to determine ReCR, as shown in Figure 9. In this section, a correlation between the mixing parameter and Re is proposed.

The complete optimization of spacer performance should be based on economic evaluation, which depends on parameters including the cost of membrane installation and maintenance, as well as the commercial value of the permeate and the retentate.12,20,21 The mass-transfer coefficient should also be evaluated in order to determine the relationship between the transmembrane pressure and the permeate flux. Economic evaluation is beyond the scope of this paper; in this section, however, it is shown that eqs 9 and 10 alone can provide useful information about the optimal Re in order to achieve optimal conditions of high mixing and low power consumption. The specific power consumption (SPC) dissipated per m3 given by

SPC )

∆P U L 0

(14)

is used to define a new dimensionless parameter called the power number11 (Pn)

F2dh4 ) 2fRe3 Pn ) SPC µ

(15)

Using eqs 12, 14, and 15, it is possible to combine the power number and the mixing parameter. Figure 15 shows the plot of the power number Pn versus the mixing parameter, Ω: the power number provides a measure of the amount of energy dissipated, while the mixing parameter provides an estimate of the mixing in the channel. Although Figure 15 shows results for a specific case (a cavity spacer with d/h ) 0.4 and ∆L/h ) 3), analogous results were obtained across the entire range of cases studied (0.4 < d/h < 0.63 and 3 < ∆L/h < 9) and for both cavity and zigzag spacers. The curve in Figure 15 is divided into two sections by the point PCR, whose coordinates are given by the values of Pn and Ω calculated at ReCR evaluated as shown previously in Figure 9. (The inflection point PCR in Figure 15 is not a new point but simply the corresponding position of ReCR. The aim of the section is to explain that the critical Reynolds number falls in the inflection area and that the critical value falls within the expected range of “normal” operating conditions.) The left section is characterized by low values of the derivative dPn/dΩ, which means that small increments in the power number lead to large increments in the mixing. The right section, in contrast, requires large increments in the dissipated power to produce small increments in mixing. The critical point is at the junction of the two sections and represents a good compromise between required mixing and the power consumption of the operation. Economic calculations take into account a larger number of parameters, and consequently, the

5396


optimal operating conditions are likely to be located at a different location than PCR. These calculations, however, are based on specific costs including those for the membrane and power. If these costs or the commercial values of the permeate and the retentate change, so do the optimal conditions. The critical Reynolds number, on the other hand, is based only on the hydrodynamics of the flow in the channel. It can provide good operating conditions with high mixing and low power consumption independent of the costs or the residual values. Conclusions In this paper, the critical Reynolds number at which the transition between two distinct laminar flow regimes in a spacerfilled channel occurs is studied using a CFD model. Three correlations, fitted to the CFD numerical data, are provided. The first correlation describes the relationship between the critical Reynolds number and the geometrical properties of the channel. The second correlation quantifies the effect of Reynolds number on the friction factor. The third correlates the extent of mixing in the channel with the Reynolds number. Using these three correlations together, it is possible to relate the power dissipation with the mixing in the channel. The critical Reynolds number alone cannot determine the optimal spacer geometry and operating conditions, which would require a more complete economic analysis. For a given spacer, however, the critical Reynolds number can be used to determine the flux of the channel for that particular geometry, which ensures good mixing conditions without excessive pressure drop. Acknowledgment This research was supported by an Australian Research Council Discovery Grant. Literature Cited (1) Schwinge, J.; Neal, P. R.; Wiley, D. E.; Fletcher, D. F.; Fane, A. G. Spiral Wound Modules and Spacers: Review and Analysis. J. Membr. Sci. 2004, 242, 129. (2) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. Simulations of Unsteady Flow and Vortex Shedding for Narrow Spacer-Filled Channels. Ind. Eng. Chem. Res. 2003, 42, 4962. (3) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. Simulations of Flow around Spacer-Filaments between Narrow Channel Walls. 1. Hydrodynamics. Ind. Eng. Chem. Res. 2002, 41, 2977. (4) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. Simulations of Flow

around Spacer-Filaments between Narrow Channel Walls. 2. Mass Transfer Enhancement. Ind. Eng. Chem. Res. 2002, 41, 4879. (5) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. A CFD Study of Unsteady Flow in Narrow Spacer-Filled Channels for Spiral-Wound Membrane Modules. Desalination 2002, 146, 195. (6) Ahmad, A. L.; Lau, K. K.; Abu Bakar, M. Z. Impact of Different Spacer Filament Geometries on Concentration Polarization Control in Narrow Membrane Channel. J. Membr. Sci. 2005, 262, 138. (7) Koutsou, C. P.; Yiantsios, S. G.; Karabelas, A. J. Numerical Simulation of the Flow in a Plane-Channel Containing a Periodic Array of Cylindrical Turbulence Promoters. J. Membr. Sci. 2004, 231, 81. (8) Dendukuri, D.; Karode, S. K.; Kumar, A. Flow Visualization through Spacer Filled Channels by Computational Fluid Dynamics. II: Improved Feed Spacer Designs. J. Membr. Sci. 2005, 249, 41. (9) Ranade, V. V.; Kumar, A. Fluid Dynamics of Spacer Filled Rectangular and Curvilinear Channels. J. Membr. Sci. 2006, 271, 1. (10) Song, L. F.; Ma, S. W. Numerical Studies of the Impact of Spacer Geometry on Concentration Polarization in Spiral Wound Membrane Modules. Ind. Eng. Chem. Res. 2005, 44, 7638. (11) Li, F.; Meindersma, W.; de Haan, A. B.; Reith T. Optimization of Commercial Net Spacers in Spiral Wound Membrane Modules. J. Membr. Sci. 2002, 208, 289. (12) Da Costa, A. R.; Fane, A. G.; Fell, C. J. D.; Franken, A. C. M. Optimal Channel Spacer Design for Ultrafiltration. J. Membr. Sci. 1991, 62, 275. (13) Geraldes, V.; Semia˜o, V.; de Pinho, M. R. Flow Management in Nanofiltration Spiral Wound Modules with Ladder-Type Spacers. J. Membr. Sci. 2002, 20, 87. (14) Zhang, H. Q.; Fey, U.; Noack, B. R.; Koenig, M.; Eckelmann, H. On the Transition of the Cylinder Wake. Phys. Fluids 1995, 7, 779. (15) Williamson, C. H. K. Mode A secondary Instability in Wake Transition. Phys. Fluids 1996, 8, 1680. (16) Barkley, D.; Henderson R. D. Three Dimensional Floquet Stability Analysis of the Wake of a Circular Cylinder. J. Fluid Mech. 1996, 322, 215. (17) Fletcher, D. F.; Wiley, D. E. A Computational Fluid Dynamics Study of Buoyancy Effects in Reverse Osmosis. J. Membr. Sci. 2004, 245, 175. (18) Wiley, D. E.; Fletcher, D. F. Computational Fluid Dynamics Modelling of Flow and Permeation for Pressure-Driven Membrane Processes. Desalination 2002, 145, 183. (19) Brewster, M. E.; Chung, K. Y.; Belfort, G. Dean Vortices with Wall Flux in a Curved Channel Membrane System. 1. A New Approach to Membrane Module Design. J. Membr. Sci. 1993, 81, 127. (20) Wiley, D. E.; Fell, C. J. T.; Fane, A. G. Optimisation of Membrane Module Design for Brackish Water Desalination. Desalination 1985, 52, 249. (21) Schock, G.; Miquel, A. Mass Transfer and Pressure Loss in Spiral Wound Modules. Desalination 1987, 64, 339.

ReceiVed for reView June 19, 2006 ReVised manuscript receiVed March 21, 2007 Accepted April 30, 2007 IE0607797

Laminar Flow Transitions in a 2D Channel with Circular Spacers

Recommend Documents