Fluid dynamics and mass transport in rotating channels with application to Centrifugal Membrane Separation

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Fluid Dynamics and Mass Transport in Rotating Channels

with Application to Centrifugal Membrane Separation

by

Jon George Pharoah

B.A.Sc.. University of W aterloo, 1994 M.A.Sc., University of Victoria, 1997

A D issertation Subm itted in P artial Fulfillment of the Requirements for the Degree of

Do c t o r o f Ph i l o s o p h y

in the

D epartm ent of Mechanical Engineering. We accept this dissertation as conforming

to the required stan d ard

upervisor (Dept, of Mechanical Engineering)

Dr. G.W.'A'ickers. Co-Supervisor (Dept, of Mechanical Engineering)

Dr. S. Dost, Member (D ept, of Mechanical Engineering)

______________________________ Dr. T.M . Fyles. O utside Member (D ept, of Chem istry)

Dr. K. N andakum ar, E xternal Examiner (University of A lberta)

© Jo n Ge o r g e Ph a r o a h. 2002

University of V ictoria

.Ml rights reserved. This dissertation may not be reproduced in whole or in p art, by photocopy or o th er means, w ithout th e permission of the author.

11

Supervisors: Dr. Ned Djilali, Dr. Geoff Vickers

A b stra ct

Centrifugal m em brane and density separation (CMS) is a novel technology pro­ posed for tre atm en t of waste water and industrial process stream s. This cross flow filtration process combines the energy recovery inherent to centrifugal reverse os­ mosis (CRO) w ith the potential alleviation of m em brane fouling and concentration polarization due to the favourable effects of centrifugal and Coriolis accelerations.

This dissertation presents a com putational stu d y of CMS undertaken to under­ stan d the basic hydrodynam ics and mass transfer of the processes and to provide in­ sight for the design of CMS devices. Two distinct m em brane models were developed, the porous wall model (PVVM) and the source term model (STM ), and incorporated into C om putational Fluid Dynamics (CFD) codes which solve the full Navier-Stokes equations coupled to a scalar tran sp o rt equation which accounts for dissolved species. These models are used to sim ulate two and three dim ensional lam inar flows in both non-rotating and ro tating reverse osmosis m em brane cartridges and to predict per­ m eate fluxes.

P late and frame geometries are first exam ined and it is determ ined th a t CMS benefits most from channels with stream wise directions directed radially. It is also shown th a t the benefits of CMS can be a ttrib u te d largely to the secondary flows and mixing associated with Coriolis acceleration, and the PW M and the STM are found to perform sim ilarly in the case of reverse osmosis. Next, the STM is used to perform a param etric study of the flow and mass transfer in rectangular and square rotating channels. It is shown th a t while normal ro tation is preferable to spanwise rotation, relatively smtdl deviations from the spanwise orientation are adequate to achieve most of the norm al ro tation performance, and th a t differences between the two orientations

in

are m inim al in the case of square channels. Also, the flow characteristics are again shown to correlate well w ith the the m agnitude of the Coriolis acceleration.

Flows in trian g u lar and circular channels are also considered, an d are shown to perform sim ilarly to rectangular channels. These channel orientations have applica­ tion in hollow fiber m em brane modules and potentially in spiral wound membrane modules.

Finally, the flow and mass transfer in channels with periodic stream w ise obstacles are considered. Such obstacles are related to feed spacers used in spiral wound mem­ brane elem ents and im pact considerably on the flow characteristics and mass transfer performance. Flow obstacles are shown to increase mass transfer perform ance in all cases, w ith altern atin g surface m ounted performing best. .-X. prelim inary investigation is undertaken into ro tatin g flows with periodic obstacles, and the flow fields are shown to depend strongly on the blockage ratio and on the Ross by num ber. In most cases, it is found th a t mass transfer perform ance does not necessarily correlate with either wall shear stress or the local flow field.

Several general conclusions regarding CMS can be drawn from this work. It is preferable to operate a CMS devices a t low flow rates, which is co n trary to conven­ tional wisdom in m em brane separation. Secondly, the mixing induced by channel rotation is bo th more effective and more efficient th an the m ixing induced by the feed spacers considered here. Finally, the m agnitude of the Coriolis acceleration is the dom inant p aram eter in determ ining CMS performance. This means th a t a CMS device can either operate a t relatively low rotational speeds with flow in the radial direction, or a t higher speeds but lower angles of inclination w ith respect to the rotational axis.

IV

Examiners:

Dr. N. p^ilali, Co-Supervisor (Dept, of Mechanical Engineering)

r. Vi

Dr. GW^. Vickers, Co-Supervisor (Dept, of Mechanical Engineering)

Dr. S. Dost, M ember (Dept, of Mechanical Engineering)

Dr. T.M . Fyles, O utside Member (D ept, of Chem istry)

Table o f C ontents

A b stract li

List o f T ables viii

List o f F igures x

N om en clatu re x ix

1 In tro d u ctio n to C M S 1

1.1 M embrane Separation ... 1

1.2 H istory-of C M S ... 4

1.3 Fluid Mechanical Param eters ... 9

1.4 O bjectives of the Present S t u d y ... 10

1.5 O utline of T h e s i s ... 11

2 Flow and M em brane M odeling 12 2.1 Flow Field E q u a tio n s ... 12

2.2 M embrane Boundarv' C o n d i t i o n ... 14

2.3 T he Porous Wall Model and the Source Term Model ... 15

2.4 Model Lim itations ... 17

2.5 Overview of Numerical Solutions ... 18

3 T h e E ffect o f M em brane O rientation 20 3.1 In tr o d u c tio n ... 20

3.1.1 Fluid Mechanical B a c k g ro u n d ... 21

3.2 Im p le m e n ta tio n ... 24

3.3 Model Validation: N on-rotating flow ... 26

3.3.1 Berm an S o l u t i o n ... 26

3.3.2 Conventional m em brane s e p a r a t i o n ... 28

3.4 T he Effect of R otation and O r ie n ta tio n ... 29 3.4.1 Secondary Flow P attern s and th eir Effect on Salt Concentration 31

T A B L E OF C O N T E N T S vi

3.4.2 Spanwise .\veraged Evolution of M embrane Surface Sait Con­

centration ... 37

3.5 Further Analysis of System R o t a t i o n ... 38

3.5.1 The Effect of Density V a r i a t i o n ... 38

3.5.2 Shear on the Membrane S u r f a c e ... 40

3.6 Com parison with Source Term M o d e l ... 42

3.7 C lo su re ... 45

4 R o ta tin g C h ann el Flow s 47 4.1 In tr o d u c tio n ... 47

4.2 Problem Setup ... 50

4.3 Grid Sensitivity S t u d y ... 53

4.4 Radial Flow Channels ... 57

4.4.1 Normal R o t a t i o n ... 57

4.4.2 Spanwise R o t a t i o n ... 66

4.4.3 From Streamwise to Normal R o ta tio n ... 75

4.5 From .-Vxial Flow to Radial F lo w ... 83

4.6 Mass T r a n s f e r ... 92

4.7 C lo su re... 105

5 R o ta tin g N on -R ectan gu lar C hannels 107 5.1 M o tiv a tio n ... 107 5.2 Relevant Previous S t u d i e s ... 108 5.3 T riangular Channels ... 108 5.3.1 H y d ro d y n a m ic s ... 109 5.3.2 Mass T r a n s f e r ... 119 5.4 Circular C h a n n e ls ... 125 5.4.1 H y d ro d y n a m ic s ... 126 5.4.2 Mass T r a n s f e r ... 131 5.5 C lo su re ... 137

6 Laminar O b stacle Flow 138 6.1 In tr o d u c tio n ... 138

6.2 C om putational P r o c e d u r e ... 142

6.3 G rid Study and V a lid a tio n ... 142

6.4 Periodic O bstacle F l o w s ... 147

6.5 M ultiple O bstacle F l o w s ... 156

6.6 Mass Transfer in M ultiple O bstacle Flows ... 162

6.7 Effect of R o ta tio n ... 167

T A B L E O F C O N T E N T S vii

7 S yn th esis and D esig n Im plications 178

7.1 M ethods of Flux Im p ro v e m e n t... 178

7.1.1 The influence of Reynolds N um ber ... 178

7.1.2 The Influence of Channel L e n g t h ... 182

7.1.3 The Influence of Feed S p a c e r s ... 182

7.2 Com parison of Channel G e o m e tr ie s ... 184

7.3 T he design of CMS a p p a r a t u s ... 185

7.3.1 Module Design for Open C hannels ... 185

7.3.2 Device D e s ig n ... 189

7.3.3 Sum m ary of Design R e c o m m e n d a tio n s ... 191

7.4 T he C orrelation between Membrane Shear R ate and Mass Transfer . 192 7.4.1 Mixing due to density g r a d i e n t s ... 192

7.4.2 R otating channel flow in rectangular c h a n n e l s ... 194

7.4.3 Spacer filled c h a n n e ls ... 196

7.5 Future W o rk ... 198

A C M S O p erating P aram eters 199 \ . l Fluid P r o p e rtie s ... 199

A.2 CMS O perating Conditions ... 200

A.2.1 P late and Frame S y s t e m ... 200

•A,.2.2 Com m ercial Spiral Wound Elem ents ... 202

B F lu x E quations 206 C R o ta tin g O bstacle Flow: Surface shear 210 C .l d /D = 0 . 2 5 ... 210

C.2 d /D = 0 . 5 0 ... 216

C.3 d /D = 0.75 ... 221

V lll

List o f Tables

3.1 Simulations perform ed to investigate the effect of system rotation for various m em brane orientations. The direction in which the additional acceleration term s act are given for each case. Centrifugal accelerations

are listed as 0 when no density gradients exist... 30

3.2 Param eters used for the numerical sim ulations... 30

4.1 Com puted Ross by and Ekman numbers. Not all cases were considered

for each geom etry and orientation... -52

4.2 Grids employed for numerical sim ulations... 55

4.3 Directions of th e Coriolis force generated by velocity com ponents in a

channel undergoing Normal ro ta tio n ... 64 4.4 Calculated Ross by and Ekman numbers in the case of ‘Streamwise to

N orm al’ channel ro ta tio n ... 84 5.1 R otational cases sim ulated and the corresponding Ross by and Ekm an

num bers... 109 6.1 Streamwise grids employed for numerical sim ulations. * = not con­

verged, - = not performed. UPS = upstream . O B J = o b j e c t ... 144

6.2 Rcu,, flux improvement and Cp predicted with the STM for various

m ulti-obstacle solutions. Reo = 1 0 0 ... 162 6.3 Re ^ and flux improvement predicted with the STM for various m ulti­

obstacle solutions. R e o = 692. Flux improvement is referenced to the no obstacle solution at Reo = 100... 163 6.4 Com puted Ross by and Ekman numbers assum ing Reo = 100...167

7.1 Sum m ary of potential CMS module designs... 188

A .l Properties of NaCl Solutions a t various concentrations and 25°C . . . 199

A.2 R otational speed required to develop various transm em brane pressures

L I S T O F T A B L E S Lx

A.3 Reynolds numbers achieved in the plate and frame assemblies . . . . 201 A.4 Rossby num bers a t various operating conditions in CMC plate and

frame assemblies. Rcssby numbers are for feed flows of (2 l/m in / 4 l/m in ) and assume nine open channels... 201 .\.5 Ekm an num bers a t various operating conditions in CMS plate and frame assemblies... 201 A.6 R otational speed required to develop various transm em brane pressures in CMS spiral wound assemblies... 202 -\.7 Dimensions of ladder type s p a c e r ... 203 •\.8 Dimensions of diam ond type s p a c e r ... 204 -\.9 Reynolds numbers and Rossby numbers in the diam ond spacer . . . . 204 -A.. 10 Reynolds numbers and Rossby numbers in the ladder s p a c e r .. 204 -A. 11 Summar}' of operating conditions for various CMS devices... 205

X

List o f Figures

1.1 Cross flow f iltr a tio n ... 2

1.2 Spiral wound m em brane e le m e n t... 3

1.3 F iltration s p e c tr u m ... 3

1.4 Schematic of Centrifugal Reverse Osmosis and Centrifugal M embrane S e p a r a tio n ... 5

1.5 The first CRO p r o to t y p e ... 6

1.6 The second CRO prototype ... 6

1.7 CMS experim ental a p p a ra tu s... 7

1.8 M embrane s ta c k ... 8

1.9 Feed spacers removed from modules at L*\'ic. Ladder type spacer on the left and diam ond type spacer on the rig h t... 9

2.1 T he porous wall model (PW M) and the source term model (STM) . . 16

3.1 Membrane orientation: membrane v ie w p o in t... 22

3.2 Membrane orientation: rotational axis v ie w p o in t... 23

3.3 Flux element on m em brane surface... 25

3.4 Geometry' used for initial validation against Berm an solution... 26

3.5 Profiles of U norm al to the porous wall at various distances along the channel. Rch = 250, R e ^ = 0 . 1 ... 27

3.6 Profiles of V normal to the porous wall a t various distances along the channel. Re^ = 250, R e ^ = 0 . 1 ... 27

3.7 Comparison of experim ental perm eate mass flux d a ta and CFD calcu­ lations for conventional membrane separation. C oncentrations are in parts per million X a C l ... 29

3.8 Surface salt concentration along the channel for various orientations. Dark shading indicates high concentrations, as per contour legend at to p ... 32

L I S T OF F IG U R E S xi

3.10 In plane velocity vectors a t x /h = 2 0 0 for various sim ulations. Velocity

vectors are magnified 10 tim es in the (0,0,90) case... 34

3.11 In plane salt concentration contours a t x /h = 2 0 0 for various sim ula­ tions. Fifteen evenly spaced contour lines are plotted between o = 0.0222 and <p = 0.026... 35 3.12 Relief plot of streamwise velocity com ponent for conventional (static)

m em brane sep aratio n ... 36 3.13 Evolution along the channel of the spanwise averaged salt concentrations. 38 3.14 Spanwise averaged increase in surface concentration w ith the addition

of centrifugal forces through variable density... 39

3.15 D istribution of m em brane surface concentration and shear rate across

the channel a t x /h = 2 0 0 41

3.16 Com parison of PW M and STM: streamwise velocity near the channel centrelines... 42 3.17 Com parison of PW M and STM: relief plots of stream w ise velocity in

the fully developed region... 43 3.18 Com parison of PW M and STM: spanwise averaged surface salt con­

cen tratio n s... 44

4.1 R otating channel geometry. Geometries with radial flow include 'span-

wise'. norm al' and general' rotation while stream w ise' and stream - wise to norm al' ro tation feature axial flow and flow inclined to the axis of ro ta tio n ... 51

4.2 Spanwise velocity in the Ekm an layer for various Ek... 53

4.3 Com parison of analytical and numerical wall shear stress for various Ek. 54

4.4 Spanwise average shear stress relative to B.A.SE grid for Q = 120s"'.

( r - T^-^sEy^BASE A) B) C) | r | ... 55

4.5 .Average shear stress as a function of grid spacing norm al to the long

wall... 56

4.6 Relief plots of stream wise velocity in the fully developed region for

various Ro. Normal ro ta tio n ... 58

4.7 Secondary' velocity vectors in the fully developed region. Ro = 0.3.

Norm al R o t a t i o n ... 59

4.8 Contours of u/Vav for various Ro: the effect of aspect ra tio ... 59

4.9 Streamwise velocity parallel to the axis of ro tatio n a t the channel cen­

treline in the case of norm al ro tation... 60 4.10 Streamwise velocity norm al to the axis of ro ta tio n a t the channel cen­

treline in the case of norm al ro tatio n... 61 4.11 Spanwise velocity normal to the axis of ro tation a t the channel centre­

L I S T OF F IG U R E S xii

4.12 Maximum streamwise velocity w ith increasing ro tatio n in the case of normal ro ta tio n ... 63 4.13 V ariation of friction coefficient in the fully developed region for various

Ro. Normal R o tation ... 66

4.14 Pressure and friction coefficients in the case of norm al rotation. . . . 67

4.15 Relief plots of streamwise velocity in the fully developed region for various Ro. Spanwise ro ta tio n ... 68 4.16 Secondary velocity vectors in the fully developed region. Ro = 0.3.

Spanwise R o t a t i o n ... 68 4.17 Streamwise velocity normal to the axis of ro tation a t the channel cen­

treline in the case of spanwise ro ta tio n ... 70 4.18 Streamwise velocity parallel to the axis of ro tation a t the channel cen­

treline in the case of spanwise ro ta tio n ... 70 4.19 Normal velocity parallel to the axis of ro tation a t the channel centreline

in the case of spanwise ro ta tio n ... 71 4.20 M aximum streamwise velocity with increasing rotation. Spanwise ro­

ta tio n ... 72 4.21 Variation of friction coefficient in the fully developed region for various

Ro. Spanwise ro tatio n ... 73 4.22 Pressure and friction coefficients in the fully developed region in the

case of spanwise ro ta tio n ... 74

4.23 General channel rotation with radial flow... 75

4.24 Relief plots of streamwise velocity in the fully developed region as rotation is varied between spanwise and norm al ro tatio n (0° < 3 < 90°). Ro = 0.37... 77 4.25 Contours of streamwise velocity and secondary stream lines in the fully

developed region as rotation is varied between spanwise and normal rotation (0° < 3 < 90°). Ro = 0.37. .A,R = 3 ... 78 4.26 Contours of streamwise velocity and secondât}' stream lines in the fully

developed region as rotation is varied between spanwise and normal rotation (0° < 3 < 90°). Ro = 0.37. AR = 1... 79 4.27 Contours of pressure in the fully developed region as ro tatio n is varied

between spanwise and norm al rotation (0° < d < 90°). Ro = 0.37 . . SO

4.28 Variation of friction coefficient in the fully developed region. Ro =

0.37. R otation is varied from spanwise to norm al... 81

4.29 Pressure and friction coefficients as rotation is varied from streamwise to normal. Ro = 0 .3 7 ... 82 4.30 Streamwise channel ro ta tio n ... 83 4.31 Contours of streamwise velocity and secondary- stream lines for 0° <

L I S T OF FIG U R E S xiii

4.32 Variation of friction coefficient in the fully developed region for various p 88

4.33 Com parison of the friction coefficient with Ro^ corresponding to Ro in

the case of norm al ro ta tio n ... 89 4.34 Com parison of the streamwise development of friction coefficient with

Ro^ corresponding to Ro in the case of norm al ro ta tio n ... 90

4.35 Pressure and friction coefficients in the fully developed region in the case of ’stream w ise to norm al’ ro tatio n ... 91 4.36 Average wall Reynolds number [Re^), as a function of rotation, de­ term ined w ith the source term model in the cases of: norm al rotation, spanwise. ro tation and streamwise to norm al’ (0" < ip < 90°) ro ta­

tion. Pt m = 400 psi, B: Pt m = 600 psi, C: Pt m = 600 psi. D: Pt m

= 1000 psi ... 93 4.37 Developing contours of NaCl concentration in the absence of channel

rotation. Pt m = 1000 psi. .Active membrane surfaces are located on

the top (NORTH) and bottom (SOUTH) of each fram e... 94

4.38 Developing contours of NaCl concentration in the cases of normal and

spanwise channel rotation. Pt m = 1000 psi... 96

4.39 Developing contours of NaCl concentration in the cases of normal and

spanwise channel rotation. .AR=1. Pt m = 1000 psi... 96

4.40 Surface NaCl concentration on the NORTH and SOUTH faces for var­ ious cases... 97 4.41 Spanwise averaged surface concentration over the NORTH and SOUTH

faces: the effects of orientation and aspect ratio. Pt m = 1000 psi. Ro

= 0 . 3 7 ... 99 4.42 Spanwise averaged surface concentration over the NORTH and SOUTH

faces. Pt m = 1000 psi... 100 4.43 Developing contours of NaCl concentration for cases in between normal

and spanwise rotation. Pt m = 1000 psi... 101 4.44 Total flux improvement over the non-rotating case for various Ro. Nor­

mal R otation. Aspect Ratio = 3... 102 4.45 Flux improvement over the non-rotating case for various Ro. Spanwise

R otation. .Aspect Ratio = 3... 103 4.46 Flux im provem ent over the non-rotating case for Ro = 0.37 as rotation

is varied from spanwise to normal. Pe = 1 .8 0 ... 104

5.1 Cross section of equilateral triangle duct. C oordinate system is as

shown, w ith the origin at the centroid... 109

5.2 Grid used to sim ulate flow in equilateral trian g u lar channels... 110

5.3 Percentage difference between analytic solution and sim ulation results (no ro ta tio n )... I l l

L IS T OF FIG U R E S xiv

5.4 Relief plot of streamwise velocity for various ro tatio n al speeds... I l l

5.5 Maximum streamwise velocity as a function of 1 /R o ... 112

5.6 Contours of streamwise and secondar\' velocity for 1.82 < Ro < oc. 113

5.7 Contours of streamwise and secondary’ velocity for 0.68 < Ro < 1.37. 114

5.8 Typical fully developed secondary' velocity vectors... 115

5.9 Spanwise velocity a t z/D = 0.5 for various ro tation ra te s... 116

5.10 Pressure coefficient versus 1/R o in the case of trian g u lar channels. Reo^ = 100... 117 5.11 Typical pressure contours in the fully developed regime. Lower pres­

sures are found at the leading side while higher pressures are found at the lagging s i d e ... 118

5.12 Rbu, as a function of Ro and applied pressure. T he dark surface corre­

sponds to a feed solution of 50.000 ppm while the light surface corre­

sponds to a feed solution of 35.000 ppm ... 119

5.13 Spanwise averaged NaCl concentrations along the channel. .A, - Pt m =

400 psi, B - Pt m = 600 psi.C - Pt m = 800 psi.D - Pt m = 1000 psi . 120

5.14 Surface NaCl concentrations with no channel ro tatio n for Pt m = 400

psi (lowest surface). 600 psi. 800 psi and 1000 psi (highest surface). . 121

5.15 Spanwise averaged NaCl surface concentrations along the channel for various Ro. Pt m = lOOOpsi ... 122 5.16 Spanwise distribution of surface NaCl concentration a t x /D = 40.8.

Pt m = lO O O p si... 122

5.17 NaCl concentration normal to the m em brane surface at x /D = 40.8 and z /D = 0 . 5 ... 123 5.18 Flux improvement over the non-rotating case as ro ta tio n is is increased 124

5.19 Five block grid used to sim ulate rotating flow in circular pipes . . . . 125

5.20 Relief plots of spanwise velocity in the fully developed region...126 5.21 Profiles of streamwise and azim uthal velocity parallel to the axis of

ro ta tio n ... 127 5.22 Streamwise velocity profile along the axis perpendicular to the axis of

ro ta tio n ... 128 5.23 Streamwise and azim uthal component of friction coefficient in circular

channels... 129 5.24 Streamwise and circumferential com ponent of friction coefficient in

square channels... 129 5.25 Circum ferentially averaged friction coefficients in circular and square

channels... 130

5.26 NaCl contours and secondary' velocities a t near the channel o utlet. . . 132

5.27 Relief plot of NaCl concentration with increasing channel ro tatio n . . . 133

L I S T O F F IG U R E S xv

5.29 Profiles of NaCl concentration along centrelines bo th perpendicular

and parallel to the axis of ro ta tio n ... 136

6.1 C oordinate system and flow over a single obstacle... 139

6.2 Periodic obstacle g e o m e tr ie s ... 140

6.3 Grid s p a c in g ... 143

6.4 Experim ental validation: Streamwise velocity profiles... 145

6.5 Cf on the top (y /D = 1.0) and bottom (y /D = 0) surfaces for R e p = 816. C ontour lines of zero shear are overlaid on the b ottom surface, clearly showing the reattachm ent line... 146

6.6 Pressure coefficient and reattachm ent length for various R e p ... 148

6.7 Friction coefficient on the top and bottom surfaces for various Rep. d /D = 0 . 2 5 ... 148

6.8 Stream lines patterns for secreted Rep. d /D = 0.25 ... 149

6.9 CpRcp versus Re p for various geometries... 151

6.10 Contours of streamwise velocity in the case of d /D = 0.25 surface m ounted obstacles... 152

6.11 Contours of streamwise velocity in the case of d /D = 0.25 channel centred obstacles... 153

6.12 Contours of streamwise velocity in the case of d /D = 0.50 surface m ounted obstacles... 154

6.13 Contours of streamwise velocity in the case of d /D = 0.75 surface m ounted obstacles... 155

6.14 Streamlines over an array of 10 surface m ounted obstacles. d /D = 0.25, Re p = 692... 156

6.15 Com parison of single obstacle solution with a periodic obstacle solution for R e p = 100 and Re p = 692. The vertical lines represent reattach­ ment in the single obstacle case... 157

6.16 Com parison of periodic solution with m ultiple obstacle solution. Rep = 692 ... 158

6.17 Friction coefficient vs x /d for various obstacle arrangem ents. R e p = 100. O bstacles are located at x /d =[0,20,40,60,80,100.120.140.160,180] 160 6.18 Friction coefficient vs x /d for two successive obstacles in the 10 obstacle array. R e p = 100. O bstacles are located a t x /d = [60,80.100] . . . . 160

6.19 Friction coefficient vs x /d for various obstacle arrangem ents. R e p = 692. O bstacles are located a t x /d =[0.20.40,60.80,100,120,140.160.180] 161 6.20 Friction coefficient vs x /d for two successive obstacles in the 10 obstacle array. R e p = 692. Obstacles are located a t x /d = [60,80.100] . . . . 161

L IS T OF F IG U R E S xvi

6.21 Surface NaCl concentration vs x /d for various obstacle arrangem ents. R e p = 100. Feed concentration = 35.000 ppm. O bstacles are located

a t x /d =[0.20,40,60,80,100,120,140,160,180]... 165

6.22 Surface NaCl concentration vs x /d for two successive obstacles in the 10 obstacle array. Re p — 100. Feed concentration = 35,000 ppm. Obstacles are located at x /d = [6 0 .8 0 ,1 0 0 ]... 165 6.23 Surface NaCl concentration vs x /d for various obstacle arrangem ents.

Re p = 692. Feed concentration = 35,000 ppm . Obstacles are located

a t x /d =[0,20,40.60,80,100,120.140.160,180]... 166

6.24 Surface NaCl concentration vs x /d for two successive obstacles in the 10 obstacle array. R e p = 692. Feed concentration = 35,000 ppm. O bstacles are located at x /d = [6 0 .8 0 .1 0 0 ]... 166

6.25 The effect of system rotation on R e p and Cp... 168

6.26 Friction coefficients on the y /D = 0 and y /D = 1 surface. Rep = 90. No R otation. d /D = 0 .2 5 ... 169 6.27 Friction coefficients on the y /D = 0 and y /D = 1 surface. Rep = 95.

Ro = 0.29. d /D = 0 . 2 5 ... 170 6.28 Position of the dow nstream reattachm ent line for various rates of ro­

tation. d /D = 0.25... 171 6.29 Contours of streamwise and spanwise velocity, d /D = 0.25. No R ota­

tion (top), Ro = 0.81 (middle) and Ro = 0.29 (b o tto m )... 172

6.30 Contours of streamwise and spanwise velocity, d /D = 0.50. No R ota­ tion (top). Ro = 1.82 (middle) and Ro = 0.!la91 (b o tto m )... 173 6.31 Contours of streamwise and spanwise velocity, d /D = 0.75. No R ota­

tion (top), Ro = 1.82 (middle), Ro = 0.91 (b o tto m )... 174

6.32 Contours of stream wise and spanwise velocity. d /D = 0.75. Ro = 0.61

(top). Ro = 0.46 (middle) and Ro = 0.36 (b o tto m )... 175

7.1 The effect of increasing Reynolds number on surface NaCl concentra­

tion. For reference, the case of Normal channel ro ta tio n a t Ro = 0.3 is also shown... 180

7.2 Flux improvement comparison between increasing flow rate and in­

creasing rotation. The subscript ’o’ refers to the n on-rotating R e p = 100 case... 181

7.3 Flux improvement comparison between increasing flow rate and in­

creasing rotation. The subscript o' refers to the ro tatin g Re p = 100 case... 183

7.4 Com parison of spanwise average surface concentration for different

channel cross sections. Pt m = 1000 psi. 35,000 ppm NaCl feed. Re p

L I S T OF FIG U R E S xvii

7.5 Hollow fibre module shown operating in the lum en to shell mode . . . 186

7.6 O rientation of channels in spiral wound m odule... 187

7.7 R otational speed required to develop 5 M Pa of transm em brane pressure at various perm eate release radii. Feed density has assum ed to be 1000 kg/m ^... 189 7.8 Total flux improvement over the non-rotating case for Various Ro. Nor­

mal R otation, .\sp ect Ratio = 3... 190

7.9 D istribution of m em brane surface concentration and shear rate across

the channel a t x /h = 2 0 0 193

7.10 Comparison of shear rates and flux im provem ents in CMS. Membrane

is located on the NORTH and SOUTH faces... 195

7.11 Surface shear coefficient and NaCl concentration vs x /d for two succes­ sive obstacles in a 10 obstacle array. Re p = 100. Feed concentration = 35,000 p p m ... 197 .-V.l Feed spacers removed from modules a t U \ ic. Ladder type spacer on

the left and diam ond type spacer on the rig h t... 203 C .l Friction coefficients on the y /D = 0 and y /D = 1 surface. R eo = 90.

No R otation, d /D = 0 .2 5 ... 211 C.2 Friction coefficients on the y /D = 0 and y /D = 1 surface. R e p = 110.

Ro = 2.01. d /D = 0.25 . . ' ... ' ... 211 C.3 Friction coefficients on the y /D = 0 and y /D = 1 surface. Re o = 89.

Ro = 0.81. d /D = 0.25 . . ' ... 212 C.4 Friction coefficients on the y /D = 0 and y /D = 1 surface. Re p = 77.

Ro = 0.47. d /D = 0 . 2 5 ... 212 C.5 Friction coefficients on the y /D = 0 and y /D = 1 surface. R ep = 89.

Ro = 0.40. d /D = 0 . 2 5 ... 213 C.6 Friction coefficients on the y /D = 0 and y /D = 1 surface. Re p = 89.

Ro = 0.33, d /D = 0.25 . ... 213

C.7 Friction coefficients on the y /D = 0 and y /D = 1 surface. R e p = 95.

Ro = 0.29, d /D = 0.25 . . ' ... ' ... 214

C.8 Friction coefficients on the y /D = 0 and y /D = 1 surface. Re p = 96.

Ro = 0.25. d /D = 0.25 . ... 214

C.9 Friction coefficients on the y /D = 0 and y /D = 1 surface. Re p = 93.

Ro = 0.21, d /D = 0.25 . . ' ... ' ... 215

C.IO Friction coefficients on the y /D = 0 and y /D = 1 surface. Re p = 93.

No R otation, d /D = 0 .5 0 ... 216 C . l l Friction coefficients on the y /D = 0 and y /D = 1 surface. R e p = 87,

L I S T O F F IG U R E S xviii

C.12 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e o = 100, Ro = 0.91, d /D = 0 . 5 0 ...217 C.13 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e o = 111,

Ro = 0.68, d /D = 0 . 5 0 ... 218 C.14 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e p = 100,

Ro = 0.46, d /D = 0 . 5 0 ... 218 C.15 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e p = 100,

Ro = 0.36, d /D = 0 . 5 0 ... 219 C.16 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. Re o = 100,

Ro = 0.30, d /D = 0 . 5 0 ... 219 C.17 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e o = 100.

Ro = 0.26. d /D = 0.50 ... 220 C.18 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e p = 100.

Ro = 0.23, d /D = 0.50 ... 220 C.19 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e o = 100,

No R otation, d /D = 0.75 ... 221 C.20 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. Re o = 100.

Ro = 1.82. d /D = 0.75 ... 222 C.21 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e p = 100.

Ro = 0.91. d /D = 0.75 ... 222 C.22 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. Re o = 100,

Ro = 0.61, d /D = 0.75 ... 223 C.23 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. R e p = 100.

Ro = 0.46, d /D = 0.75 ... 223 C.24 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. Rep = 100.

Ro = 0.36, d /D = 0.75 ... 224 C.25 Friction coefficients on tfie y /D = 0 and y /D = 1 surface. Re p = 100.

Ro = 0.30, d /D = 0.75 ... 224 C.26 Friction coefficients on tfie y /D = 0 and y /D = I surface. R e p — 100,

XIX

N om en clatu re

-^<mt Area of membrane associated with one control volume [m‘]

Ai Area of membrane associated with one control volume [m'-|

c Average concentration across membrane [kg/rri^]

Cf Feed concentration [kg/rn^]

Cf Local friction coefficient [-j

C f Circum ferentially averaged c/ [-]

C/x X com ponent of c / [-]

Cp Perm eate concentration [kg/rrA]

Cp Pressure coefficient [-]

V Diffusivity [m’/s]

D C hannel dimension [m]

Dh C hannel hydraulic diam eter [m]

E k Ekm an number, E k = u/D'~Q.

h Half height of channel [mj

i .j . k C oordinate directions in com putational space

Jy Solution volume flux [m/s]

solute mass flux [kg/{rn-s)]

J i,2 Com ponent mass fluxes [m o//(m ^s)|

I Vertical dimension of control volume

Lp Hydraulic perm eability [m /(s Pa)]

Lj: Streamw ise dimension of channel [m]

L . Spanwise dimension of channel [m]

P S tatic Pressure [Pa]

Pg Excess pressure [-]

Pi Local pressure [Pa]

Pfg Flux element averaged pressure [Pa]

N O M E N C L A T U R E xx

R Rejection based on concentration [-]

R' Rejection based on mass fraction [-]

Ri Accelerations due to rotational frame [m/s*]

Re^j Reynolds num ber Based on x. Re^j =

Re o Reynolds num ber based on D [-]

Reti Reynolds num ber based on channel half height [-]

Ro R otation num ber based on D [-]

S Circum ferential coordinate [-]

Sc Schm idt num ber [-]

S0 Volumetric Source term in o equation

S° S0 a t previous tim e step

t Tim e [s]

T T em perature [°C]

u, u. w Velocity com ponents in x.y and z directions [m/s]

U Bulk velocity [m /s]

Ui Velocity vector [m /s]

Uo Average streamwise velocity at channel inlet [m/s]

tv T ranspiration velocity [m/s]

V' Volume of control volume [m^]

V'ay Bulk velocity [m /s]

X j Position vector [m]

X . y. z C ordinate axes

N O M E N C L A T U R E xxi

G reek S y m b o ls

Sij Kronecker d elta [-]

St Tim e step [s]

Sj: Control volume dimension

Sy Control volume dimension

eum A lternating tensor [-]

^ Molecular viscosity [kg / m s]

u Kinem atic viscosity [m‘ /s]

n Osm otic Pressure [Pa]

0 Mass fraction [-]

0av Spanwise averaged mass fraction [-]

0° Mtiss fraction a t previous time step [-]

Oo Mass fraction a t channel inlet [-]

O, Local mass fraction [-]

p Density [kg/m^]

Pout Density of fluid passing through a membrane BC face [kg/m^]

Pi Local density [kg/m^]

pp Density of permeate[A-p/m ‘]

Pf Density of feed [kg/w3]

a Reflection coefficient [-]

uj Solute perm eability (osmotic pressure based) [s/m]

uj' Solute perm eability (concentration based [m/s]

D R otational speed [s“ ^]

A cron yn m s

AR Aspect R atio

CMS Centrifugal M embrane Separation

CRO Centrifugal Reverse Osmosis

PW M Porous Wall Model

RO Reverse Osmosis

X X ll

A ck n o w led g em en ts

I would like to sincerely thank Dr. Ned Djilali for his guidance, friendship and support over the past seven years. During this tim e, I have grown trem endously and I cannot imagine a b e tte r graduate experience than th a t which I have enjoyed under Dr. Djilali. I would also like to thank my co-supervisor, Dr. Geoff Vickers, and the m em bers of my exam ining com m ittee: Dr. T.M . Fyles, Dr. S. Dost and Dr. K. N andakum ar. I m ust also thank my fellow graduate stu d en ts including .\n o ta i Suksangpanom rung, Ian Spearing, Torsten Berning, .Juan C arretero and in particular Gonçalo Pedro who p u t up with an interm inable stream of senseless babble as the ideas and understanding were developing. Discussions with .\lv in Bergen were also instrum ental to the development of this work.

Nine years ago. I was introduced to C om putational Fluid Dynamics by a professor at the University of W aterloo. This same professor showed me the way to graduate studies and set me on the path th a t led to this point and th a t I will follow through my career. I am forever indebted. Dr. Stubley.

Finally, I must th an k my wife Stephanie and my son Luke who make it all worth­ while.

XXIU

C hapter 1

In trod u ction to C entrifugal

M em brane Separation (C M S)

1.1

M em b ran e Sep aration

Membrane separation processes are used in a wide variety of industries and processes including the production of potable water, the de-inking of recycled paper and the de­ watering of fruit juice. M embranes are finding increased use in all these industries due to the energy* efficiency of membrane processes which do not rely on phase changes. The current worldwide m em brane m arket is estim ated a t 6.55 billion USS and is anticipated to grow 8.3 % per year [1]. Membrane separation is a process which makes use of a semi-perm eable membrane in order to sep arate a feed stream into a perm eate stream and a concentrate, or retentate stream . W hile dead end separation, wherein all the feed passes through the membrane, is som etim es used, this work will focus on cross flow filtration, as depicted conceptually in figure 1.1.

C H A P T E R 1. IN T R O D U C T IO N T O CMS

Figure 1.1: Cross flow filtration

common arrangem ents: plate and frame, spiral wound, hollow fibre and tubular. In both plate and frame and spiral wound modules the m em brane is produced as a flat sheet. In the case of plate and frame modules the m em brane sheets are sim ply attached frames which are stacked together in such a fashion th at a feed flow channel is formed between the frames. Plate and frame m odules suffer from the fact th a t the packing density, or the am ount of m em brane area which can be packed into a given volume, is quite low and the m anufacturing costs tend to be high. Spiral wound elements neatly address both these problems. A typical spiral wound element is shown in figure 1.2. First two membrane sheets are placed back to back separated by a perm eate spacer and sealed with glue on three sides. Next, the rem aining side is connected to a porous perm eate tube which runs through the centre of the completed module. Finally, a feed spacer is placed adjacent to each active m em brane surface and the m em brane sheet is rolled around the perm eate tube to create a cylindrical module. The feed spacers create feed channels by insuring th a t the rolled up m em branes do not contact each other while the perm eate spacers provide a spiral p ath for the perm eate to reach the central tube.

Both hollow fibre and tubular membranes consist of tubes of m em brane m aterial. Hollow fibres are typically small diam eter ( < 1mm) and the entire tu b e is cast from the desired polym er while tubular membranes are much larger (diam eter % 2.5 cm) and are supported on a porous pressure vessel. Hollow fibre m em branes are typically packaged in large bundles and the feed flow can either be through the fibres

C H A P T E R 1. IN T R O D U C T IO N TO CM S Feed _Ret^eiitnte Pemiente Feed spncer Feed Flow Penuente sijacer Meiubiniie

Figure 1.2; Spiral wound m em brane element

themselves (lumen-shell), or around the bundles with perm eate flowing into the fibres (shell-lumen).

Irrespective of the module chosen, cross flow filtration processes may be classified into four subprocesses depending on the particle cut-off of the membrane: reverse- osmosis. nano-filtration, ultra-filtration and m icrofiltration 1.3. as shown in figure

1.3. Filtration sp ectru m ■pote stte' (rmcicxTSl Microfiltrallon 0.1 CZZ. 0.01 Ultra motion Nonomotion 0.001 I---Reverse osnnosis 0.0001 s u s p e n o e a soiics proteins molecules salt water

C H A P T E R 1. IN T R O D U C T IO N T O CM S 4

Two of the m ajor problems associated w ith m em brane separation processes are m em brane fouling and concentration polarization. Fouling occurs when the membrane is physically obstructed, either by a buildup of particulates on the surface or by m em brane com paction, whereas concentration polarization refers to the form ation of a high concentration boundary layer adjacent to the m em brane which results in a local increase in the osmotic pressure and a reduction of the perm eate production. Also, in the case of reverse osmosis and specifically w ith regards to desalination, the process energ>' requirem ents can be quite high since the feed stream pressure must be increased significantly beyond the osmotic pressure of the the feed solutions: typical feed pressures in sea water desalination plants are of the order of 5 M Pa (735 psi).

W hile the methodologies developed herein are applicable to the entire spectrum of filtration processes the focus of this dissertation will be reverse osmosis desalination since CMS was originally developed in this optic and the experim ental program has also focused on this application.

1.2

H isto ry o f C en trifugal M em b ran e S ep aration

Early CMS work focused on sea w ater desalination, which is a reverse osmosis process, and accordingly, was referred to as centrifugal reverse osmosis (CRO). This work was originally conceived prim arily to conserve energ}' by taking advantage of a rotating environm ent to recover the energ}- contained in the high pressure concentrate stream .

In conventional separation processes such as reverse osmosis, process pressures are achieved using high pressure pumps, and, typically, a turbine is required downstream to recover energ}* from the high pressure exhaust stream . Significant energ}- efficiency gains have been dem onstrated with CRO, shown schem atically in figure 1.4. in which process pressure is developed within a spinning centrifuge. T he feed stream in CRO

C H A P T E R 1. IN T R O D U C T IO N T O C M S

enters the axis a t low pressure and is pressurized as it flows radially outw ards to the m em brane. A fter exiting the membrane, the concentrate stream is depressurized as it retu rn s to th e axis and exits the rotor. Thus, the feed stream in CRO enters at low pressure and the exhaust stream leaves at low pressure, allowing inherent energ}' recover}' w ithout the addition of an auxiliary turbine. Reduction in specific energ}' consum ption of more th an 35% over non-rotating RO have been reported by Wild et al. [2] for a prototype producing 7.5 m ^/day of fresh water. It was also shown th at the theoretical energ}' efficiency of CRO increases with system capacity, and up to 70 % reduction in specific energ}' consum ption was predicted for units producing over 1000 m^/day.

Mi

Figure 1.4: Schem atic of Centrifugal Reverse Osmosis and Centrifugal Membrane Separation

Two prototypes, shown in figures 1.5 and 1.6, were built upon these ideas, and used conventional spiral wound reverse osmosis elements arranged in a g attlin g gun configuration around an axis of rotation. This means th a t while the bulk flow was in an axial direction, the membrane had active faces directed both towards and away from the axis of rotation. It was later proposed th a t the ro tatin g environm ent could

C H A P T E R 1. IN T R O D U C T IO N T O CM S 6

additionally effect membrane perform ance, but since the focus of the original pro­ totypes was to show energ}' efficiency, there was never any direct com parison of the m em brane performance in these devices with those of sim ilar non ro tatin g systems. If additional benefits exist then the concept of ro tatin g membrane separation extends beyond reverse osmosis to the entire filtration spectrum . This concept is a general­ ization of CRO. and is given the nam e Centrifugal Membrane Separation (CMS).

Figure 1.5: The first CRO prototype

C H A P T E R 1. IN T R O D U C T IO N T O C M S 7

In order to isolate the effect of m em brane orientation, a CMS test apparatus, shown in figure 1.7, was designed and built which could accom m odate custom plate and frame m em brane discs.

Figure 1.7: CMS experim ental apparatus.

Early CMS experim ents were carried out using these plate and frame membranes, depicted in figure 1.8. Lexan discs were machined such th a t when stacked, they formed feed flow channels of length 0.084m w ith a 0.0571m x 7.62 x lO” "* m (4.34 x 10“ '’m ‘ ) cross section. M embrane m aterial was then glued onto the face of the discs such th a t the resulting channels had an active m em brane on one face. T he m em brane stack could be rotated allowing various m em brane orientations with respect to the axis of rotation.

Typically, there were nine open channels stacked in the CMS ap p aratu s, four of which had active membrane on one face, and five of which were plain machined lexan channels.

This work, and the associated m odelling which will be presented in C hapter 3, clearly showed th a t the effect of m em brane orientation with respect to the axis of rotation was significant and th a t CMS performance could be enhanced by a judicious choice of m em brane orientation.

C H A P T E R 1. IN T R O D U C T IO N T O CM S

A A

T t Permeate Spacer

Active Membrane Surface

▼ V

Permeate Out Permeate Out

Section A-A

(A) (B)

Figure 1.8: M embrane stack

In order to capitalize on this additional benefit and to scale up the experim ents, a new CMS head, capable of housing a small radially oriented com m ercial spiral wound m em brane element, was constructed. The prim ary difference between the plate and frame mem branes (figure 1.8) and the spiral wound mem branes (figure 1.2) is th a t the spiral would elements require the use of a feed spacer in order to keep the membranes from contacting. Two representative spacers taken from modules used a t UVic are shown in figure 1.9. Clearly, these spacers have a profound effect on the flow in the resulting channels and, it is expected, on m em brane performance. This effect must be correctly modelled if m em brane perform ance is to be understood. In either case, though, it is im portant to quantify the flow regime of a given point in the operating space. A detailed account of the operating range of the various CMS devices is given in A ppendix A.

Since the focus of this work is the fluid dynam ics and mass transfer in rotating systems, it is instructive to set the stage by reviewing the param eters governing such types of flows.

C H A P T E R 1. I N T R O D U C T IO N T O C M S

Figure 1.9: Feed spacers removed from modules a t U \'ic. Ladder type spacer on the left and diam ond type spacer on the right.

1.3

F luid M ech an ical P aram eters

There are five basic non-dim ensional param eters which are im p o rtan t in characteriz­ ing the flow in a CMS device: the Reynolds num ber {Re), the wall Reynolds num ber {Rcuj) the Ross by num ber {Ro), the Ekman number {Ek) and the Schm idt num ber

(Sc).

The Reynolds num ber is a classical param eter characterizing non-rotating viscous flows, such as flow in conventional m em brane separation. This p aram eter is also relevant to ro tating cases and represents the ratio of inertial to viscous forces.

Reo = pUD

k- (1.1)

where p. U. D and p. represent respectively density, bulk velocity, hydraulic diam ­ eter and dynam ic viscosity. Since fluid perm eates through the m em brane, the normal com ponent of velocity at the m em brane surface is non-zero. T he wall Reynolds is de­ fined as above, with the velocity scale replaced with the normal com ponent of velocity.

C H A P T E R 1. IN T R O D U C T IO N T O C M S 10

The Rossby and Ekman numbers characterize flows in ro tatin g channels such as in CMS. T he Rossby num ber represents the ratio of convective acceleration to Coriolis acceleration.

R o = - § ^ (1.3)

and the Ekman num ber represents the ratio of viscous forces to Coriolis forces

E k = ^ (1.4)

where Q and u are the rotational speed and the kinematic viscosity. The Reynolds. Rossbv and Ekman num ber are related bv

(1.5)

Finally, the Schmidt num ber relates the viscous diffusion of mom entum to the mass diffusion of dissolved species (in this case N’aCl).

1.4

O b jectiv es o f th e P resen t S tu d y

The primary' objective of this work is to examine the flow and mass transfer in ge­ om etries of significance to both conventional membrane separation and to CMS. This will necessarily involve the application of m em brane models to flows in both rotating and non-rotating channels. The insight gained from this stu d y will both lead to a

C H A P T E R 1. IN T R O D U C T IO N TO C M S 11

b e tte r understanding of the tran sp o rt phenom ena involved in m em brane separation and will provide guidance in the design of future membrane separation devices.

1.5

O u tlin e o f T h esis

Each chapter in this dissertation is w ritten to stand alone as much as possible, and accordingly, an independent literature overview is presented in each chapter. C hapter 2 sets the stage for the dissertation by outlining the basic equations governing the tra n sp o rt phenom ena in membrane systems. Also presented and discussed are two different membrane models which can be applied to m em brane flows: the porous wall model (PW M ) and the source term model (STM). In chapter 3. the PW M is applied to reverse osmosis desalination in geometries representative of the plate and frame CMS device. These results indicate th a t radially outward flows are favourable and th a t the STM is adequate for param etric studies of reverse osmosis flows. C h ap ter 4 presents a param etric study of fluid flow and mass transfer in rectangular channels w ith flow in the radially outward direction while chapter 5 investigates sim ilar flows in channels of both triangular and circular cross sections. C hapter 6 presents a preliminaiy* investigation of fluid flows in geometries representative of spacer filled channels. Both non-rotating and ro tating flows are considered. Finally, in chapter 7. the im pact of the results on CMS is considered. Conclusions are draw n, and recom m endations made.

12

C hapter 2

Flow and M em brane M o d elin g

2.1

F low F ield E q u ation s

The flow in a CMS device is governed by the conservation of mass, the Navier-Stokes equations, and, when a dissolved second phase is present, a scalar tra n sp o rt equation. As shown in appendix A, the Reynolds number of the flow in a reverse osmosis m em brane channel is of the order of 10- and the flow may therefore be assum ed to be lam inar. The governing equations in a rotating frame of reference take the following conservation form:

d f ( d u .

C H A P T E R 2. F L O W A N D M E M B R .\N E M O D ELIN G 13

where o is the mass fraction of dissolved species. is the gravity vector and R,

is an acceleration term which can be used to solve the equations in rotating frames of reference.

or. alternativelv in vector notation, as

A = n X X r j + 2 Q x C ' (2.5) where the first term corresponds to centrifugal acceleration, the second term corre­ sponds to Coriolis acceleration, u.'/ ( Q ) is the rotation vector, z t ( R ) is the position

vector with respect to the rotational axis and is the altern atin g tensor, which

takes a value of 1 for cyclic perm utations of ijk. -L for acyclic perm utations and 0 otherwise. The centrifugal acceleration is often combined with the pressure through the definition of a reduced pressure.

p = P - Q x T ^ (2.6)

In the case of variable fluid properties, the fluid density and viscosity is determined based on the local concentration using empirical correlations developed in [3].

The governing equations can also be presented in non-dimensional form.

^ + Ro V ■ Vu + 2ÇÎ X V = —Vp + E k V^v (2.7)

C H A P T E R 2. F L O W A N D M E M B R A N E M O D ELIN G 14

where the fluid is assumed to be incompressible w ith constant properties and all variables have been non-dimensionalized. The governing equations in this form are the sta rtin g point for asym ptotic solutions, since it is clear th a t the Rossby number and the Ekm an num ber indicate the relative im portance of the inertial and viscous term s respectively.

In com ponent form, assuming steady fully developed conditions w ith flow in the

X direction(c)/ôx = 0), and an axis of rotation parallel to the y axis the governing

equations become

In addition to the stan d ard boundary conditions (inlet, o utlet, no-slip), a selective m em brane boundary condition must be specified for the CMS problem.

2.2

M em b ran e B ou n d ary C o n d itio n

The equations governing a perma-selective membrane can be derived using irreversible therm odynam ics [4|. Assuming th at dynam ic reversibility is valid for all cases con­

C H A P T E R 2. F L O W A N D M E M B R A N E M O D ELIN G 15

sidered, and th a t the m em brane is a perfect rejector w ith respect to convection, then the flux of solution and solvent are respectively given by [3].

ju = Lp(AP - AH) (2.12)

(2.13)

where Lp is the m em brane permeability. R is the rejection of the membrane, A P

and A fl are the hydrostatic and osmotic pressure differences across the membrane. Cf and Cp are the concentrations of the feed and perm eate respectively. In the above equations, the solution flux is presented as a volume flux and the solute flux is pre­ sented as a mass flux. F urther detail are given in appendix B.

Two different im plem entations of the above equations are outlined in the following section.

2.3

T h e P orou s W all M o d el and th e Source Term

M o d el

The Porous Wall Model (PW'M). shown on the left hand side of figure 2.1 allows perm eate to pass through the membrane, but selectively inhibits the transport of dis­ solved species. The solute which does not pass through the m em brane remains in the feed channel, adjacent to the membrane surface, and the bulk concentration increases. This model correctly accounts for the mass perm eating through the membrane, and hence the flow rate in the feed channel decreases along the flow direction and there is a com ponent of velocity normal to the m embrane. Since the perm eation velocity

C H A P T E R 2. F L O W A N D M E M B R A N E M O D ELING 16

is a function of bo th the local pressure and concentration through equation 2.12 the hydrodynam ics and the mass transfer are coupled.

Solute Source Region

PW M STM

Figure 2.1: T he porous wail model (PW M ) and the source term model (STM) Reverse osmosis affords two significant simplifications to the above model: i) The transm em brane pressure is much larger th an the pressure drop along the channel, and ii) the perm eate flux is vanishingly small such th a t the effect of perm eation velocity and of decreasing flow rate are u nim portant in the equations of motion. The perm eation is however critically im portant to the mass transfer equation, as it is solely responsible for the increase in bulk concentration in the feed channel.

The source term model, shown on the right hand side of figure 2.1. takes advantage of this and decouples the hydrodynam ics and the mass transfer. T he hydrodynam ics are solved using a conventional wall boundary condition and can accordingly be solved by any sta n d a rd ' CFD code w ithout modification. The mass transfer equation is then solved w ith the addition of a source term adjacent to the m em brane surface to account for the concentrating effect of the membrane. Equation 2.12 can be used with a constant transm em brane pressure to determ ine the local perm eate mass flux, and the am ount of solute which must be added locally in the source region is given by

^source — ^pA iju Js (2.14)

C H A P T E R 2. F L O W A N D M E M B R A N E M O D E L IN G 17

T he m ain benefit of the source term model is th a t the hydrodynam ics and the mass transfer are decoupled. This means th a t a hydrodynam ics solution can be calculated for a given geom etry and fiow rate and this solution can be used to predict membrane perform ance over a range of feed concentrations and operating pressures. Since the hydrodynam ics solution Is orders of m agnitude more costly th an the mass transfer solution, this greatly reduces the cost of the predictions. Also, a commercial CFD package can be used without modification for the hydrodynam ics solution and the mass transfer can be calculated either using the sam e package or separately.

2.4

M o d el L im itation s

T he PW M is applicable over the entire range of m em brane processes since the perme­ ation velocity is explicitly accounted for in the form ulation. Not only does this ensure the correct mass flow a t each section in a module, but it also ensures th a t the correct m odule pressure drop is calculated. This pressure drop is especially im portant when the process pressure is low since it results in a reduced driving force for the separation. .\n additional benefit of the PW M is th at it is possible to vary fluid properties with concentration in order to account for the effect of density stratification.

T he STM allows for a great increase in com putational efficiency by decoupling the mass transfer solution from the hydrodynam ics solution. T he compromises for this model are:

• The perm eation velocity a t the wall is neglected

• T he channel pressure drop is usually neglected in determ ining the trans-m em brane pressure

C H A P T E R 2. F L O W A N D M E M B R .A i\E M O D E L IN G 18

• T he solution properties m ust rem ain constant w ith respect to the hydrodynam ­ ics solution

For these reasons, the STM is lim ited to reverse-osmosis separations with trans­ m em brane pressures significantly higher than the channel pressure drop. An addi­ tional lim itation of the STM is th a t the m em brane area m ust be sufficiently small to ensure the m ass flow through the m em brane rem ains small com pared to the feed mass flow. Even when the perm eation velocity is everywhere small, if the membrane is large enough it is possible th a t the concentrate flow rate could decrease enough th a t the mass transfer would be affected. The validity of the STM is exam ined in detail in chapter 3.

Both models are capable of m odelling the effects of concentration polarization up to the point when gel layers begin to form adjacent to the m em brane surface. Once these layers form, a new phiise is present in the channel. At the very least, this imposes an additional resistance to flow through the m embrane. W hile the models will indicate th a t the gel concentration has been surpassed, additional physics and chem istry would have to be included to correctly model the flow beyond this point.

N either model as presented is suitable for modelling m em brane fouling. Fouling is inherently a tim e dependent process where the perm eate flux decreases with tim e due to pore narrowing, pore blocking or the build-up of a fouling layer adjacent to the m em brane surface. Again, the additional physico-chemical processes would have to be accounted for.

2.5

O v erv iew o f N u m erica l S o lu tio n s

The PW M was im plem ented in the com m ercial code TASCflow 2.6 [5|. TASCflow is a finite volume based finite elem ent code which solves the com plete Navier-Stokes

C H A P T E R 2. F L O W A N D M E M B R A N E M O D E L IN G 19

equations on collocated body fitted m ulti-block meshes. The continuity equation and the m om entum equations are solved in a fully coupled m anner which makes this code particularly well suited to both ro tatin g flows and to flows w ith transpiration. Steady sta te solutions are arrived at by a physical tim e m arching procedure and this means th a t unsteady solutions are possible provided a sm all enough tim e step is used. The im plem entation of this model required additional source code beyond the stan d ard user modifiable routines.

The STM was implemented in the commercial code cfx 4.2 [6]. Cfcc is a finite volume code which also solves the complete Xavier-Stokes equations on collocated body fitted multi-block meshes. The hydrodynam ics and the continuity equation are not solved in a coupled fashion and while the code has options for unsteady flow, steady flow solutions are arrived at through the use of relaxation techniques. The STM was im plem ented using only the stan d ard user modifiable Fortran routines provided w ith the Cfx code.