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ORIGINAL S. Litster J. G. Pharoah N. Djilali Convective mass transfer in helical pipes: effect of curvature and torsion Received: 17 August 2004 / Accepted: 22 July 2005 / Published online: 8 October 2005 Ó Springer-Verlag 2005 Abstract A 3D numerical analysis of the flow and mass transfer in helical pipes is presented. The interpretation of the flow patterns and their impact on mass transfer is shown to require a non-orthogonal pseudo-stream function based visualization. The strong coupling be- tween torsion and curvature effects, and the resulting secondary flow regimes are well characterized by a parameter combining both the Dean (Dn) and Germano numbers (Gn). For membrane separation applications, helical modules combining high curvature with low torsion would alleviate concentration polarization and yield appreciable flux improvement. 1 Introduction The introduction of helicity in pipes induces secondary Dean flows that have the potential to improve heat and mass transfer performance. This is in contrast to systems that employ straight pipes. There exists an abundance of possible applications, and the motivation for this study is membrane separation processes using hollow-fibre membranes. In cross-flow filtration processes, Fig. 1, the feed stream is separated into two effluent streams; the concentrate and the filtered permeate. These processes are limited by concentration polarization [12] (i.e., an increase of solute concentration near the active mem- brane surface that reduces the production of permeate). Various methods have been proposed to enhance mixing and alleviate the problem [11]. The focus of the paper is a systematic analysis of the 3D fully developed laminar flow regimes and associated mass transfer over a range of curvatures and torsions relevant to helically-wound hollow-fibre modules. A review of recent progress in the use of helical membranes to improve flux was recently compiled by Al-Bastaki and Abbas [1]. Prior to reviewing the rich body of literature on curved and helical duct flows, it is useful to summarize the param- eters characterizing flow in helical pipes. 1.1 Geometry of the problem Figure 2 depicts the coordinate and reference system used in the current study. The radius of curvature (coil) R, tangent T, normal N, and binormal B of a helix are defined by the following relationships: R ¼ R c cosðuÞ ^ i þ R c sinðuÞ ^ j þ P 2p u ^ k T ¼ dR ds N ¼ 1 k dT ds B ¼ T N ð1Þ where R is the position vector of the helix, R c is the radius of the helix coil, u is the angle of rotation about the helix centreline, P is the pitch between each coil, and s is the arc length. The angle of the helix can also be presented as a function of the arc length. u ¼ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 c þðP =2pÞ 2 q ð2Þ In addition, a helix can be described by two curve theory parameters; the curvature (k) of a helix, S. Litster J. G. Pharoah N. Djilali (&) Department of Mechanical Engineering, University of Victoria, Victoria, BC, V8W 3P6, Canada E-mail: [email protected] Tel.: +1-250-7216034 Fax: +1-250-7216323 Present address: J. G. Pharoah Department of Mechanical Engineering, Queen’s University at Kingston, Kingston, ON, K7L 3N6, Canada Heat Mass Transfer (2006) 42: 387–397 DOI 10.1007/s00231-005-0029-y
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Page 1: S. Litster N. Djilali Convective mass transfer in helical ...

ORIGINAL

S. Litster Æ J. G. Pharoah Æ N. Djilali

Convective mass transfer in helical pipes: effect of curvature and torsion

Received: 17 August 2004 / Accepted: 22 July 2005 / Published online: 8 October 2005� Springer-Verlag 2005

Abstract A 3D numerical analysis of the flow and masstransfer in helical pipes is presented. The interpretationof the flow patterns and their impact on mass transfer isshown to require a non-orthogonal pseudo-streamfunction based visualization. The strong coupling be-tween torsion and curvature effects, and the resultingsecondary flow regimes are well characterized by aparameter combining both the Dean (Dn) and Germanonumbers (Gn). For membrane separation applications,helical modules combining high curvature with lowtorsion would alleviate concentration polarization andyield appreciable flux improvement.

1 Introduction

The introduction of helicity in pipes induces secondaryDean flows that have the potential to improve heat andmass transfer performance. This is in contrast to systemsthat employ straight pipes. There exists an abundance ofpossible applications, and the motivation for this studyis membrane separation processes using hollow-fibremembranes. In cross-flow filtration processes, Fig. 1, thefeed stream is separated into two effluent streams; theconcentrate and the filtered permeate. These processesare limited by concentration polarization [12] (i.e., anincrease of solute concentration near the active mem-

brane surface that reduces the production of permeate).Various methods have been proposed to enhance mixingand alleviate the problem [11]. The focus of the paper isa systematic analysis of the 3D fully developed laminarflow regimes and associated mass transfer over a rangeof curvatures and torsions relevant to helically-woundhollow-fibre modules. A review of recent progress in theuse of helical membranes to improve flux was recentlycompiled by Al-Bastaki and Abbas [1]. Prior toreviewing the rich body of literature on curved andhelical duct flows, it is useful to summarize the param-eters characterizing flow in helical pipes.

1.1 Geometry of the problem

Figure 2 depicts the coordinate and reference systemused in the current study. The radius of curvature (coil)R, tangent T, normal N, and binormal B of a helix aredefined by the following relationships:

R ¼ Rc cosðuÞ̂iþ Rc sinðuÞ̂jþP2p

uk̂

T ¼ dR

ds

N ¼ 1

kdT

dsB ¼ T�N

ð1Þ

where R is the position vector of the helix, Rc is theradius of the helix coil, u is the angle of rotation aboutthe helix centreline, P is the pitch between each coil, ands is the arc length. The angle of the helix can also bepresented as a function of the arc length.

u ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2c þ ðP=2pÞ

2q ð2Þ

In addition, a helix can be described by two curvetheory parameters; the curvature (k) of a helix,

S. Litster Æ J. G. Pharoah Æ N. Djilali (&)Department of Mechanical Engineering, University of Victoria,Victoria, BC, V8W 3P6, CanadaE-mail: [email protected].: +1-250-7216034Fax: +1-250-7216323

Present address: J. G. PharoahDepartment of Mechanical Engineering,Queen’s University at Kingston, Kingston,ON, K7L 3N6, Canada

Heat Mass Transfer (2006) 42: 387–397DOI 10.1007/s00231-005-0029-y

Page 2: S. Litster N. Djilali Convective mass transfer in helical ...

k ¼ Rc

R2c þ ðP=2pÞ

2ð3Þ

and the torsion (g),

g ¼ P=2p

R2c þ ðP=2pÞ

2ð4Þ

1.2 Previous studies

An extensive body of literature exists regarding thetheoretical, experimental, and numerical aspects of flowsin curved duct and helices. Dean’s pioneering studies offlow in toroidal pipes were followed by numerous studiesfocusing on coils with a high radius Rc corresponding toa low Dean number Dn, traditionally defined as

Dn ¼ Rea

Rc

� �12

ð5Þ

where Re is the Reynolds number and a/Rc is the non-dimensional curvature of a bend when considering

negligible pitch. Patankar et al. [10] studied helicaltubes of large curvature and Dean numbers. The tor-sion (which later work identified as a critical charac-teristic parameter) was negligible in the casesinvestigated. Wang [13] examined the effect of curva-ture and torsion on the flow field in helical pipes bysolving the Navier–Stokes equations in non-orthogonalcoordinates, and proposed a modified definition of thetraditional Dean number by replacing the radius of thehelix with that of the non-dimensional curvature, de-fined as ak. Following this work, Germano [2] per-formed a Navier–Stokes based analysis and suggestedthat, contrary to Wang’s conclusion, torsion introducesa second-order effect opposed to the first-order effect ofcurvature. Further analysis led Germano [3] to proposea new parameter, (g/k)/Re, representing the ratio ofcurvature to torsion. Germano also provided a com-parison between solutions using various coordinatesystems.

Liu and Masliyah [7] (hereafter LM) presented acomprehensive analysis of laminar flow in helical pipesof a finite pitch. In addition to the systematic investi-gation over a broad range of geometric parameters,LM addressed the controversy over the appropriatenessof various coordinate systems for analyzing the velocityfield. They showed that though the flow field is inde-pendent of the reference system, the secondary flow canbe visualized differently depending on the adoptedcoordinate system. It was found that a correct repre-sentation and interpretation of the flow is only possiblewhen it is visualized using pseudo-secondary flowstream function isolines. Only a few works have sinceattempted to obtain the pseudo-stream function [15,16]. LM also defined a generalized Dean or helixnumber Dn,

Dn ¼ Rek12 ð6Þ

and a torsion parameter, the Germano number Gn,

Gn ¼ Reg ð7Þ

where k and g are the curvature and torsion as defined inthe previous subsection with the exception that the helixradius Rc and pitch P have been non-denationalized bythe tube radius a. This form of the Dean and Germanonumbers will be adopted in the parametric study pre-sented in this paper. LM found that in order for theGermano number to have a significant effect on the flow,GnDn�3/2 must be large, and consequently proposed thefollowing parameter:

c ¼ Gn

Dn32

¼ g

ðkDnÞ12

ð8Þ

c will be referred to as the LM number and will be usedto analyze the combined effect of curvature and torsion.A further observation made by LM was that the tran-sition from a dual vortex mode to a single swirlingvortex mode occurs when c . 0.2.

Feed

Permeate

Membrane

Concentrate

Fig. 1 Schematic of cross-flow filtration

r

aN

T

B

R

s

Y

X

Z

Rc

θ

ϕ

Fig. 2 The helix coordinates and reference system

388

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A number of hydrodynamic investigations have fo-cused specifically on helical hollow-fibre membranes.Kuakuvi et al. [6] compared woven hollow-fibres withhelical and straight hollow-fibres. It was found that for agiven pressure drop, the woven fibre module providedhigher levels of flux. The effect of curvature and torsionvariance was not explored. The performance of variousDean flow inducing geometries was examined experi-mentally by Ghogomu et al. [4]. Only one helix geome-try was considered, and the Dean number was varied bychanging the Reynolds numbers. As expected, for thesame curvature and Reynolds number the various Deanflow inducing geometries studied provided approxi-mately the same performances.

Moulin et al. [9] recently presented a computationalanalysis of the flow in hollow-fibre membrane. Tori ofvarying radii of curvature were considered, as well as ahelix and a straight pipe. The objective of the study wasto predict the wall shear stresses in these variousgeometries on the assumption that shear stress is anindicator of flux performance. However, recent work hasshown that shear and permeate flux do not always cor-relate [11, 12]. Moulin et al.’s work was extended withlaser visualization more recently [8].

The few studies reviewed here constitute only aportion of the extensive body of literature on Deanflows and flow in helices. It is, however, apparent thatthe membrane science community is not aware of therich fluid mechanics literature in the field. In particular,investigations of helicoidal membranes has been limitedby neglecting to investigate the influence of torsion.Further, little has been done on investigating the masstransfer directly. In this paper we focus on masstransfer characteristics associated with such flow inhelical pipes. In particular, simulations are performedand analyzed in the high torsion cases for which littledata is available in the literature at the Reynoldsnumbers of interest.

1.3 Computational method

The flows considered have a Reynolds number of order100. The flow is thus laminar and incompressible. Thegoverning continuity, Navier–Stokes and mass transportequations are:

@ui

@xi¼ 0 ð9Þ

qDui

Dt¼ � @p

@xiþ @

@xjl@ui

@xjþ @uj

@xi

� �� �

ð10Þ

@

@tðq/Þ þ @

@xiðqui/Þ ¼

@

@xiC@/@xi

� �

ð11Þ

where u is the velocity, x is the spatial coordinate, t istime, l is the viscosity, p is pressure, q is the density, / isthe mass fraction of transported species and G is the

diffusion coefficient. The walls of the helical hollow-fibremodules consist of a reverse-osmosis membrane. Thecorresponding boundary condition can be derived fromirreversible thermodynamics [12]. When the membraneis considered a perfect rejector, with respect to convec-tion, the flux of the solution jv and solvent Js are ex-pressed as:

jv ¼ LpðDP � DPÞ ð12Þ

Js ¼jvð1� RrÞ

Rrðcf � cpÞ ð13Þ

where Lp is the membrane permeability, DP and D P arethe hydrostatic and osmotic pressure differences acrossthe membrane, cfeed and cp are the concentrations of thefeed and the permeate respectively, and Rr is the rejec-tion ratio.

The computational model was implemented in thecommercial code cfx 4.3. This finite volume code solvesthe complete 3D Navier–Stokes equations at each cellcenter. The grid was constructed from a five-blockstructure, see Fig. 3, with the grid dimension of eachblock being [25·25·160]. In all cases, the grid was ex-truded in the third direction to a length of 100a. The gridwas smoothed using a global elliptic smoother to im-prove mesh orthogonality. The radial mesh size adjacentto the membrane surface was Dr/a=6e�3. A higher-order upwind difference scheme was used to discretizethe convective terms.

The membrane boundary condition was implementedwith a source term model (STM) developed by Pharoah[11] for the study of various rotating membrane geom-etries. The STM relies on the flux through a reverse-osmosis membrane being small enough as to not affectthe flow. The effect of permeation and the resulting localincrease in solute concentration is modeled via a sourceterm adjacent to the membrane wall /source, over a dis-crete area Ai, whereby solute is injected according to:

/source ¼ /qAijv � Js ð14Þ

1

2

3

4

5

Fig. 3 Computational grid and block arrangement

389

Page 4: S. Litster N. Djilali Convective mass transfer in helical ...

The STM methodology was validated by Pharoah[11] for the case of rectangular channel membranemodules, and was shown to yield flow field and masstransfer parameters within 1% of the more rigorousporous wall model. A key advantage to using the STM isits capability to decouple the hydrodynamics from themass transfer field. Thus, various feed concentrationsand transmembrane hydrodynamics pressures can besimulated starting from the same flow solution. Thisallows us to systematically explore a much broaderrange of parameters.

2 Results and discussion

The simulations were all performed for reverse osmosismembranes with a 35,000 ppm salt (NaCl) feed streamand a 6.89 MPa (1,000 psi) transmembrane pressure.The inlet boundary condition in all cases was set for aReynold’s number of 100. The Schmidt number for theinlet feed stream solution is 624.

The effect of curvature k and torsion g, the two keyparameters characterizing the flow field in helical tubes,were investigated for 0.05 £ k £ 0.25 and 0.05 £g £ 0.25. Figure 4 illustrates the spectrum of helix

geometry resulting from this range. Table 1 lists the non-dimensional hydrodynamic parameters that correspondto the present geometry range at a Reynold’s number of100. The planar solutions presented herein are at an arclength of s=90, where the hydrodynamics are fullydeveloped. Again, the arc length of all the helices studiedwas 100a.

2.1 Validation

The solutions were validated through comparison withsolutions presented by Liu and Masliyah [7]. Since thepresent study investigates helical coils with high torsionto curvature ratios, with a Reynolds number of 100, oneof the LM solutions with similar properties was bench-marked (Table 2). The pressure coefficient,Cp = �(aRe/q U2)(¶p/¶s) where U is the bulk velocity,has been computed in the present work to within 0.5%of the value obtained by LM. The analytic value of 0.16ðCpsÞ was also obtained for the limit of a straight pipe.

In order to examine the impact of the grid resolution,a grid sensitivity study was performed. Grid indepen-dence was verified by increasing the number of compu-tational planes in the tangential direction threefold andexamining the Cp. A change in the Cp of less than 1%was observed, indicating satisfactory grid resolution.

2.2 Analysis of the hydrodynamics

An interpretation of the flow field based solely on thevelocity vectors can lead to a distortion of secondaryflow effects [5, 14]. In the present study the secondaryflow structures will be presented with the pseudo-streamfunction as in Liu and Masliyah [7] and Yamamotoet al.’s [15, 16] work. This will be shown to be the mostappropriate way of visualizing the flow. In the hydro-dynamically fully-developed region, which is axiallyinvariant, the pseudo-stream function isolines depicttubes in which a particle, when placed inside, would notescape along the arc of the helix. This visualization willbe shown to be particularly useful for investigating theinfluence of the secondary flow on convection trans-ported properties.

The pseudo-stream function isolines are computed byapplying the stream function equation, dw ¼ ~U � ~ndA toeach non-tangential cell face, and then integrating acrossthe planar section (see Fig. 5). With this approach, amass conserving representation of the secondary flow isachieved, whereas in the case of an orthogonal velocityvector plot, mass sources and sinks are present due tothe 3D nature of the mesh.

0.05 0.1 0.15 0.2 0.250.050.1

0.150.2

0.250

0.01

0.02

0.03

0.04

0.05

Curvature (λ) and X [m]

Torsion (η)and Y [m]

Z [m

]

Fig. 4 Helix geometry for range of curvature and torsion

Table 1 The non-dimensional hydrodynamics parameters, theDean number Dn and the Germano number Gn, resulting from thehelix geometry parameters, curvature k and torsion g, at a Rey-nolds number of 100

k Dn g Gn

0.05 22.4 0.05 5.00.10 31.6 0.10 10.00.15 38.7 0.15 15.00.20 44.7 0.20 20.00.25 50.0 0.25 25.0

Table 2 Benchmark solution parameters

Re Dn Gn k g c Cp

80 20.0 18.8 0.0625 0.1875 0.1677 0.2095

390

Page 5: S. Litster N. Djilali Convective mass transfer in helical ...

Figure 6 illustrates the apparent contradiction in theflow structure when viewed with isolines of the pseudo-stream function and velocity vectors. When the Ger-mano number is equal to 25.0, the pseudo-streamfunction presents a single swirling vortex, while thevelocity vectors indicate two counter-rotating vortices.Moreover, in the high-curvature case, Dn=50.0, it canbe seen that the pseudo-stream function depicts asmaller lower vortex. This is in contrast to the smallerupper vortex shown by the orthogonal velocity vectors.Moll et al. [8] and Yamamoto et al. [16] together presentan experimental analogy to the present contradiction.Moll et al. presented planar laser visualization of sec-

ondary flow as the experimental analogy of the oforthogonal velocity vectors. Whereas, the flow structuredepicted by the non-orthogonal pseuodo-stream func-tion was revealed by Yamamoto et al. using a smokevisualization technique.

Figure 7 presents isolines of the pseudo-streamfunction for the entire range of curvature and torsion.Variation of curvature and torsion are each shown tohave two effects. Increases of curvature, from low tohigh Dean numbers Dn, is shown to increase the mag-nitude of the secondary flow and increase the symmetryof the dual-vortex secondary flow structure. Increases oftorsion, from low to high Germano numbers Gn, causesthe upper vortex to expand and twists the interface be-tween the two vortices in the counter-clockwise direc-tion.

The effect of curvature under low-torsion, (Gn=5.0),is presented in Fig. 8 with isolines of the pseudo-streamfunction in conjunction with the NaCl distributions atan arc length of s=100a. This plot clearly depicts a two-cell vortex structure that correlates very well with theconcentration contours. Increased secondary flowvelocities corresponding with increases in curvature arealso evident. The cell structure becomes more symmetricover the normal axis as the curvature increases. The saltdistribution contours exhibit increased mixing withhigher curvature.

Figure 9 presents the secondary flow structure andNaCl distribution for the high-torsion cases, Gn=25.0.A single swirling vortex is found in the lower Deannumber case (a), and again the flow pattern correlateswell with the corresponding salt distribution. Referringto the velocity vectors corresponding to this case inFig. 6, it is clear that mass transport is driven by the flowstructures presented with the pseudo-stream function. Asmall second vortex appears in the lower right quadrant

n

U

dA

B

N

Integration

Fig. 5 Calculation of the pseudo-stream function

a b

c d

Fig. 6 Comparison between isolines of the pseudo-stream functionand velocity vectors a Dn=22.4, Gn=25.0; b Dn=50.0, Gn=25.0

5.0

10.0

15.0

20.0

25.0

Gn

22.4 31.6 38.7 44.7 50.0 Dn

Fig. 7 Pseudo-stream function isolines for the entire range of Gnand Dn numbers

391

Page 6: S. Litster N. Djilali Convective mass transfer in helical ...

when the Dean number increases from 22.4 to 31.6. Witha further increase to a Dean number of 50.0, the flowpattern shifts toward two symmetric cells.

The effect of torsion for the low-curvature(Dn=22.4) cases is presented in Fig. 10. With the Ger-mano number equal to 5.0, an approximately asym-metric two-cell flow pattern forms. A further increase oftorsion results in the shrinkage and eventual vanishingof the lower vortex at a Germano number of 25.0. TheLM number for this case is c=0.236, in close agreementwith the value of 0.2 reported by Liu and Masliyah [7]for the transition to a single vortex pattern.

According to Liu and Masliyah [7], c captures thecombined effect of curvature and torsion on the helicalpipe’s flow structure when Dn‡20. A carpet plot of ccorresponding to this study’s parameter space is pre-sented in Fig. 11. The LM number c represents the rapidchange in the flow pattern for low-curvature cases (seeFig. 10) when the torsion is varied. In addition, c char-acterizes well the slower rate of transformation of theflow pattern for low-torsion cases when curvature ismodified (see Fig. 8).

The influence of curvature and torsion on the tan-gential velocity is presented in Fig. 12. As curvatureapproaches the limit of zero, the axisymmetric Poiseuillevelocity profile is recovered. When curvature is intro-duced, the bulk of the flow shifts to the outside of the

helix. In addition, two lobes form in the tangentialvelocity profile with a flattened central region. The effectof torsion on the tangential velocity is visualized as aclockwise rotation of the local maximum/maxima.

Figure 13 depicts the pressure field in a helical pipeillustrating the influence of curvature and torsion. In theGn=5.0 case, the pressure field is roughly symmetricacross the normal axis with the pressure gradient di-rected to the outside as a result of centrifugal forces. Thetorsional forces can be identified in the Gn=25.0 plot,where the magnitude of pressure difference remainsapproximately the same. However, the pressure field isdistorted in the counter-clockwise direction at the out-side of the helix.

The increase in the coefficient of pressure over that ofa straight pipe (Cp=0.16) is plotted in Fig. 14. A sig-nificant increase in the coefficient of pressure is found forthe high-curvature cases ðCpmax

� 1:4� CpsÞ: In addition,the coefficient of pressure increases approximately line-arly with curvature. Moreover, torsion has a minimaleffect on the coefficient of pressure.

2.3 Analysis of the mass transfer

We next examine mass transfer in straight and helicalhollow-fibre reverse-osmosis membranes. The masstransfer simulations were obtained using the STM de-

Secondary Flow

0.036

0.038

0.04

0.042

0.044

Salt Distribution

0.036

0.038

0.04

0.042

0.044

0.036

0.038

0.04

0.042

0.044

a

b

c

Fig. 8 Isolines of the pseudo-stream function and NaCl distribu-tions for low-torsion cases (Gn=5.0). a Dn=22.3, b Dn=31.6, cDn=50.0

Secondary Flow

0.036

0.038

0.04

0.042

0.044

Salt Distribution

0.036

0.038

0.04

0.042

0.044

0.036

0.038

0.04

0.042

0.044

a

b

c

Fig. 9 Isolines of the pseudo-stream function and NaCl distribu-tions for high-torsion cases (Gn=25.0). a Dn=22.3, b Dn=31.6, cDn=50.0

392

Page 7: S. Litster N. Djilali Convective mass transfer in helical ...

scribed in Sect. 2.3. In all cases, a feed flow concentra-tion of 35,000 ppm salt (NaCl) is prescribed, with atransmembrane pressure of 6.89 Mpa (1,000 psi). Un-like the hydrodynamics, the mass transfer is not axiallyinvariant in fully developed regions. The bulk NaClconcentration continues to increase along the length ofthe hollow-fibre. All planar salt concentrations in thepresent study are given at s=90.

Figure 15 presents the surface salt concentrationabout the circumference of the hollow-fibre for the high-torsion cases, Gn=25.0, with a variation of the curva-ture in the range 22.4 £ Dn £ 50.0. This range covers thehydrodynamic transition regime presented in Fig. 10.When Dn=22.4, which corresponds to the single vortexflow structure, the surface concentration is much higherthan that of the other cases. A higher surface concen-tration results in a lower permeate flux JH. When the thecurvature increases to Dn=31.6, a ‘jet’ of salt throughthe interface between the two vortices is evident. Thesurface concentration continues to decrease with cur-vature. However, the most significant change occurs atthe transition from one to two vortices.

To illustrate the effect of curvature and torsion on themass transfer, the circumferential average salt concen-tration is plotted against the length of the hollow-fibresin Figs. 16, 17, 18, 19. In all cases, profiles are comparedto that of a straight hollow-fibre of the same length. InFig. 16, the result of increasing the curvature from 22.4to 44.7 is presented for the low-torsion cases (Gn=5.0).The introduction of curvature lowers the salt concen-tration and reduces the rate of concentration increase. Itis apparent that further increase of the curvature has adiminishing effect on concentration levels. Comparedwith the straight pipe case when the monotonic increasein concentration results in a gradual flux reduction, thehigh-curvature hollow-fibres exhibit a significantlyslower rate of increase in the circumferential salt con-centration. In reverse-osmosis applications, this wouldresult in significant permeate flux enhancement com-pared to that of a straight hollow-fibre.

Figure 17 portrays the effect of curvature under theinfluence of high-torsion (Gn=25.0) on the salt con-centration along the hollow-fibre. Under high torsion, aDean number of 22.4 has significantly less impact on thesalt concentration than in the low-torsion cases. Thehigh-torsion case corresponds to the single swirlingvortex pattern shown in Fig. 9. With the next incre-mental increase in curvature in the present study’sparameter space, the concentration drops dramatically.Again, the diminishing returns of permeate with furtherincrease in the curvature is evident. The Dn=31.6 andGn=25.0 case marks a noticeable reduction in addi-tional permeate recovered from increasing the curvatureand corresponds to a LM number of 0.141.

The influence of torsion on the salt concentration forthe low-curvature cases (Dn=22.4) is illustrated inFig. 18. There is a significant salt concentration reduc-tion with a decrease in the torsion from Gn=25.0 toGn=15.0. However, below a Germano number of 15.0the decrease of torsion has a limited effect. The case(Dn=22.4, Gn=15.0) that locates the point of dimin-ishing returns corresponds to a LM number of 0.142.The impact of the variation of torsion between a Ger-mano number of 10 and 25 is shown to be negligible forthe high-curvature cases (Dn=50.0) depicted in Fig. 19.

The impact of curvature and torsion on the permeateflux is summarized in Fig. 20. The vertical axis is the

Secondary Flow

0.036

0.038

0.04

0.042

0.044

Salt Distribution

0.036

0.038

0.04

0.042

0.044

0.036

0.038

0.04

0.042

0.044

a

b

c

Fig. 10 Isolines of the pseudo-stream function and NaCl distribu-tions for low-curvature cases (Dn=22.4). a Gn=5.0, b Gn=20.0, cGn=25.0

2025

3035

4045

50

510

1520

250

0.05

0.1

0.15

0.2

0.25

DnGn

γ

Fig. 11 The LM number over the present study’s parameter space

393

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ratio of the permeate flux for helices to that of a straighthollow-fibre. A sharp reduction in the permeate flux canbe seen in the high-torsion/low-curvature portion of theparameter space. This region corresponds to the point atwhich the transition to a single swirling vortex regime

takes place. The influence of curvature on the permeateflux is limited for Dean numbers greater than 35. Inaddition, torsion appears to significantly affect the per-meate flux in region of Germano numbers greater than15 and Dean numbers less than 30.

The maximum increase in the permeate flux for thepresent study is 16.4%. However, as depicted inFigs. 16, 17, 18 and 19, the salt concentration of astraight pipe generally increases faster than in the case ofa helical pipe. In accordance with Eq. 12, the permeate

Fig. 12 Tangential velocityprofiles

Dn = 22.4, Gn = 25.0Dn = 22.4, Gn = 5.0

Fig. 13 Fully developed pressure fields

2025

3035

4045

50

510

1520

251

1.1

1.2

1.3

1.4

DnGn

Cp H

/ C

p S

Fig. 14 Coefficent of pressure for helical versus straight pipes

0 45 90 135 180 225 270 315 3600.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

Angle (Degrees)

NaC

lCon

cent

ratio

n

StraightDn = 22.4Dn = 31.6Dn = 38.7Dn = 44.7

Fig. 15 Surface NaCl concentration around the circumference at s/a=90.0

394

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flux decreases with increases in the osmotic pressure.The osmotic pressure being is proportional to the con-centration at the membrane wall. It is acknowledgedthat the relative permeate flux enhancement for

helically-wound modules increases with longer modules.These increased gains are generated because the con-centration increases faster along the length of a straighthollow-fibre than in a helical hollow-fibre. In the present

0 20 40 60 80 100

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

s

NaC

lCon

cent

ratio

nStraightDn = 22.4Dn = 31.6Dn = 38.7Dn = 44.7

Fig. 16 Circumferential average NaCl concentration for low-torsion cases (Gn=5.0)

0 20 40 60 80 100

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

s

NaC

lCon

cent

ratio

n

StraightDn = 22.4Dn = 31.6Dn = 38.7Dn = 44.7

Fig. 17 Circumferential average NaCl concentration for high-torsion cases (Gn=25.0)

0 20 40 60 80 100

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

sN

aClC

once

ntra

tion

StraightGn = 10.0Gn = 15.0Gn = 20.0Gn = 25.0

Fig. 18 Circumferential average NaCl concentration for low-curvature cases (Dn=22.4)

0 20 40 60 80 100

0.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052

s

NaC

l Con

cent

ratio

n

StraightGn = 10.0Gn = 15.0Gn = 20.0Gn = 25.0

Fig. 19 Circumferential average NaCl concentration for high-curvature cases (Dn=50.0)

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study the permeate flux enhancement is presented for a100a long hollow-fibre.

In Fig. 21 the profile of the LM number is comparedto the contour of the inverse of the permeate flux in-crease. Though, it cannot be said that these plots areproportional, or follow the same trend, it does provethat in this range the LM number can be employed as adesign guideline. The LM number should be below 0.14for optimal reduction of concentration polarization.

3 Conclusion

The hydrodynamics and mass transfer within helically-wound hollow-fibre membranes were investigated using

3D numerical simulations in order to analyze and betterunderstand the effect of the centrifugally induced Deanvortices on the mass transfer, and assess the effectivenessof the secondary swirling flow in reducing concentrationpolarization and membrane fouling. A STM was used tosimulate the mass transfer, and the effect of torsion andcurvature were explored independently over a range ofDean and Germano numbers.

A non-orthogonal pseudo-stream function was usedto visualize the flow. Vector plots were found to yield adistorted view of the secondary flow patterns, and donot properly convey the transition from the symmetricdual vortex pattern to the single vortex pattern. Thepseudo-stream function enables the visualization of thesecondary flow pattern that convects the salt. The saltconcentration distributions were found to correlate withthese patterns, and in particular the transformation froma dual vortex structure to a single swirling vortex wasshown to significantly lessen the effectiveness of helicalmodules in enhancing mass transfer.

This work has clearly shown that flux enhancementfor a helically-wound hollow-fibre membrane is not so-lely a function of the Dean number. The characteriza-tion of the performance of high-torsion modules, such asthose investigated by Kuakuvi et al. [6] and Ghogomuet al. [4], are a function of both Dean and Germanonumbers. Both numbers need to be specified to properlycharacterize the flow regime. Appreciable reduction inpolarization concentration and flux improvements werefound to result in modules combining a high Deannumber (high curvature) with a low Germano number(low torsion). The LM number was shown to fullycharacterize the transition between the dual and singlevortex regimes. Increasing curvature improves mixingand mass transfer performance only for LM numbers of0.14 or less. Beyond this there are no appreciable gainsin permeate recovery.

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2025

3035

4045

50

510

1520

251.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

DnGn

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Fig. 20 Permeate flux enhancement for helical versus straighthollow-fibres with a length of 100a

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