Numerical Simulation Studies of Mass Transfer Under Steady and Unsteady

Numerical Simulation Studies of Mass Transfer

under Steady and Unsteady Fluid Flow in Two- and Three-Dimensional Spacer-Filled Channels

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

by Gustavo Adolfo Fimbres Weihs

School of Chemical Sciences and Engineering UNESCO Centre for Membrane Science and Technology The University of New South Wales Sydney, Australia

July, 2008 Abstract

Hollow fibre and spiral wound membrane (SWM) modules are the most common commercially available membrane modules. The latter dominate especially for RO, NF and UF and are the focus of this study. The main difficulty these types of modules face is concentration polarisation. In SWM modules, the spacer meshes that keep the membrane leaves apart also help reduce the effects of concentration polarisation. The spacer filaments act as flow obstructions, and thus encourage flow destabilisation and increase mass transfer enhancement. One of the detrimental aspects of the use of spacers is an increase of pressure losses in SWM modules. This study analyses the mechanisms that give rise to mass transfer enhancement in narrow spacer-filled channels, and investigates the relationship between flow destabilisation, energy losses and mass transfer. It shows that the regions of high mass transfer on the membrane surface correlate mainly with those regions where the fluid flow is towards the membrane. Based on the insights gained from this analysis, a series of multi-layer spacer designs are proposed and evaluated. In this thesis, a Computational Fluid Dynamics (CFD) model was used to simulate steady and unsteady flows with mass transfer in two- and three-dimensional narrow channels containing spacers. A solute with a Schmidt number of 600 dissolving from the wall and channel Reynolds numbers up to 1683 were considered. A fully-developed concentration profile boundary condition was utilised in order to reduce the computational costs of the simulations. Time averaging and Fourier analysis were performed to gain insight into the dynamics of the different flow regimes encountered, ranging from steady flow to vortex shedding behind the spacer filaments. The relationships between 3D flow effects, vortical flow, pressure drop and mass transfer enhancement were explored. Greater mass transfer enhancement was found for the 3D geometries modelled, when compared with 2D geometries, due to wall shear perpendicular to the bulk flow and streamwise vortices. Form drag was identified as the main component of energy loss for the flow conditions analysed. Implications for the design of improved spacer meshes, such as extra layers of spacer filaments to direct the bulk flow towards the membrane walls, and filament profiles to reduce form drag are discussed.

i Acknowledgements

This thesis would have been impossible to produce without the direct and indirect help and support of many people. Firstly, I would like to express my most sincere gratitude to my supervisor, Professor Dianne Wiley, who provided me with much needed encouragement and advice from day one. I am deeply grateful for her support and the belief she has shown in me and in this project, which was invaluable in keeping me motivated to see this thesis through. I would like to thank Adjunct A/Professor David Fletcher from the University of Sydney for his collaboration in chapters 3 and 4 and appendix A of this thesis, and for his helpful advice and support with the CFX software. Thanks are also extended to Dr. Alessio Alexiadis for his insightful discussions of transient phenomena. The computational aspects of this thesis would not have been possible without the help and support of many University staff. I would like to particularly thank Craig Howie, Ee Meen Iliffe and Wendi O’Shea-Ryan for their IT support, and Dr. Deyan Guang for his help with the acquisition and maintenance of the computer equipment used for this study. I would like to acknowledge the Australian Research Council for funding this project through a Discovery grant. I would also like to thank the University of New South Wales and the Faculty of Engineering for scholarship funding through the “University International Postgraduate Award” and the “Supplementary Engineering Postgraduate Award”. I could not go on without mentioning my fellow students, co-workers and friends at the UNESCO centre, some of whom have come and gone, and some who are still there. Thanks to Minh, Regina, Eisham, Pierre, Paul, Alessio, Olga, Yulita, Vera, Kevin, Javeed, Ebrahim, John, Dong, Ilio and Alex, for welcoming me into the group, and making it a fun and friendly place to work in. I would like to thank my parents. Although they are far away, they have provided me with encouragement and support as if they were not on the other side of the world. Finally, I would like to especially thank my wife, Kate, whose patience has been endless, and her support everlasting, and my son, Javier, for being a source of everyday motivation for all of my efforts put into this thesis.

ii Publications emanating from this thesis include the following:

Journal Papers:

Fimbres-Weihs, G.A.; Wiley, D.E.; Fletcher, D.F. Unsteady Flows with Mass Transfer in Narrow Zigzag Spacer-Filled Channels: A Numerical Study. Ind. Eng. Chem. Res. 2006, 45, 6594-6603. Fimbres-Weihs, G.A.; Wiley, D.E. Numerical Study of Mass Transfer in Three- Dimensional Spacer-Filled Narrow Channels with Steady Flow. J. Membr. Sci. 2007, 306, 228-243.

Conference Poster:

Fimbres-Weihs, G.A.; Wiley, D.E. Unsteady Flows with Mass Transfer in Narrow Spacer-Filled Channels: A Numerical Study, International Congress on Membranes and Membrane Processes (ICOM) 2005, Seoul, Korea, August 21-26, 2005.

Conference Presentation:

Fimbres-Weihs, G.A.; Wiley, D.E. Three-Dimensional Spacer-Filled Narrow Channels with Steady Flow, 6th International Membrane Science and Technology Conference (IMSTEC) 2007, Sydney, Australia, November 5-9, 2007

iii Table of Contents

Abstract i Acknowledgements ii Table of Contents iv Nomenclature viii List of Symbols viii Greek Symbols xiii Subscripts xiv

Chapter 1 1. Introduction 1

Chapter 2 2. Literature Review 5 2.1. Basic Concepts 6 2.1.1. Diffusion in Systems with Multiple Ionic Components 6 2.1.2. Mass Transfer Coefficient 9 2.1.3. Energy losses and the Friction Factor 12 2.2. Concentration Polarisation 14 2.3. Turbulence promoters and Spacer Studies 22 2.3.1. Heat transfer enhancement 22 2.3.2. Mass transfer enhancement 25 2.3.2.1. Flow regimes 27 2.3.2.2. Mechanisms for mass transfer enhancement 29 2.3.2.3. Energy loss trade-off 31 2.3.2.4. Variation of geometric characteristics 35 2.3.2.5. Effect of turbulence promoters on fouling 36 2.3.2.6. Other mass transfer enhancement techniques 38 2.4. Computational Fluid Dynamics in Membrane studies 39 2.4.1. Two-dimensional studies 39 2.4.2. Three-dimensional studies 46 2.5. Conclusion 51

iv Chapter 3 3. Methodology 54 3.1. CFD Theory 54 3.1.1. Transport Equations 54 3.1.2. Mass Transport 56 3.1.3. The Finite Volume Method 57 3.1.4. Discretisation schemes 59 3.2. Verification and Validation 62

Chapter 4 4. Unsteady Flows with Mass Transfer in Narrow zig-zag Spacer-Filled Channels 65 4.1. Introduction 65 4.2. Problem Description, Assumptions and Methods 65 4.3. Results and Discussion 68 4.3.1. Steady flow 68 4.3.2. Moderately unsteady flow 71 4.3.3. Highly unsteady flow 76 4.3.4. Friction Factor and Sherwood number 80 4.3.5. Comparison of velocity and concentration effects 82 4.3.6. Fourier Analysis 85 4.4. Conclusions 87

Chapter 5 5. Mass Transfer in Three-Dimensional Spacer-Filled Narrow Channels with Steady Flow 89 5.1. Introduction 89 5.2. Problem Description, Assumptions and Methods 90 5.2.1. Geometry description 93 5.2.2. Characterization of the projection of a vortex onto a plane 95 5.2.3. Vortices in 2D flow 96 5.3. Results and Discussion 97 5.3.1. Mesh Independence Study 97 5.3.2. Hydrodynamics 98 5.3.2.1. Validation 98

v 5.3.2.2. 90° Orientation 99 5.3.2.3. 45° Orientation 104 5.3.3. Mass Transfer effects 108 5.3.3.1. Validation 108 5.3.3.2. 90° Orientation 109 5.3.3.3. 45° Orientation 113 5.3.3.4. Comparisons of Mass Transfer effects between different geometries 115 5.4. Conclusions 118

Chapter 6 6. Multi-layer spacer designs for minimum drag and maximum mass transfer 120 6.1. Introduction 120 6.2. Problem Description, Assumptions and Methods 121 6.2.1. Channel Interpolation 125 6.2.2. Cost Estimation 127 6.3. Results and Discussion 131 6.3.1. Two-layer spacer geometries 131 6.3.2. Multi-layer spacer geometries 134 6.3.3. Unsteady flow 141 6.3.4. Economic analysis of spacer performance 143 6.3.4.1. Base membrane cost of $100/m2 145 6.3.4.2. Effect of changes in membrane cost 151 6.4. Conclusions 153

Chapter 7 7. Conclusions 157

References 165

Appendices 177

Appendix A A. Multiple Ionic Components 178 A.1. Introduction 178

vi A.2. Theory 178 A.2.1. Mass Transfer 178 A.2.2. Maxwell-Stefan diffusion model 182 A.2.3. Nernst-Planck diffusion model 185 A.3. Incorporation of Multiple Ionic Effects into CFD 188 A.4. Problem description 189 A.5. Results and Discussion 191 A.6. Conclusions 193

Appendix B B. Videos produced in the course of this thesis 194 B.1. Videos from Chapter 4 194 B.2. Videos from Chapter 5 195 B.3. Videos from Chapter 6 196

vii Nomenclature

Symbol Description Units a , , c , db Empirical constants aE Ellipse major axis m ae Ellipse minor axis m A Area m2 A Area vector m2

2 Am Membrane area m

2 Amt Mass transfer area m

2 AT Cross-sectional area m ,,, Empirical constants a,aij Coefficient matrix kgmol kg

A,aij Coefficient matrix kgmol kg

2 b,bij Coefficient matrix sm

2 B, Bij Coefficient matrix sm

3 Cc Capital cost $ m

Ce Energy cost $ kWh

2 Cm Membrane cost $ m

3 Cop Operating cost $ m

3 Ctot Total permeate processing cost $ m c Total concentration of mixture kgmol m3

3 ci Molar concentration of species i kgmol m J c Specific heat capacity p kg4 K

D Diffusion coefficient ms2

2 Dij Onsager cross-diffusion coefficient ms Maxwell-Stefan cross-diffusion ms2 ij coefficient

viii Diffusion coefficient of component i in D ms2 im a mixture Maxwell-Stefan diffusion coefficient of ms2 im component i in a mixture de Equivalent diameter m d f Filament diameter m

Driving force in Maxwell-Stefan J d diffusion model kgmol4 m dh Hydraulic diameter m Relative error in the calculation of an e integral function F Vector field, body force F Vortex shedding frequency Hz Faraday constant A4 s gmol Integral function

Fa Amortisation factor 1 yr d p f h 2 Fanning friction factor 2 uLeff f w loc 1 2 Local friction factor 2 ueff GCI Grid Convergence Index gmol GrE Electrical potential gradient term mgeq4 v Hn Normalized helicity v hch Channel height m he Height to ellipse m hf Filament height m

Iht Heat transfer enhancement integral W

Imt Mass transfer enhancement integral kg s kgmol j Molar flux of component i relative to v i ms2 4

ix Molar flux of component i kgmol j i relative to v ms2 4 kg J Mass flux of component i relative to v i ms2 4 kg J Mass flux of component i relative to v i ms2 4 kg J Flux of solute through the membrane slt ms2 4 kg J Flux of solution through the membrane sln ms2 4 K Empirical constant k f Friction factor coefficient kmt Mass transfer coefficient ms L Channel length m kg L Specific permeate flux of membrane, p permeability coefficient Pa44 m2 s l Length vector m lcv Control volume characteristic length m l f Filament length m lm Mesh length m M Average molecular weight of mixture kg kgmol

M i Molecular weight of component i kg kgmol m Eddy or mixing constant mi Mass of component i kg Distance in the direction normal to a n m surface ni Moles of component i kgmol kgmol n Absolute molar flux of component i i ms2 4 N Number of mesh elements

NC Number of components in mixture

Ni Shape function

x kg N Absolute mass flux of component i i ms2 4 , Empirical constants

Nsp Number of spacers

Ps Spacer pitch m p Pressure Pa pch Channel pressure drop Pa Pe Peclet number Pn Re3 f Power number Q Volumetric flow ms3 r Radius m J R Universal gas constant kgmol4 K Mesh refinement ratio kg R Rate of generation of component i via i reaction ms3 4 Re Reynolds number uh Re avg ch Channel Reynolds number ch

ud Re avg f Cylinder Reynolds number cyl

ud Re eff h Hydraulic Reynolds number h Re V Vortex Reynolds number

SS , Source of transported property

S Average source of transported property over a control volume kg S Source of component i in the fluid i ms3 4

2 Sw Wetted surface m Distance vector from nodal value at s m up, ip upwind cell to integration point

Sc Schmidt number D kd Sh mt h Sherwood number D

CSd CSY Sh DTh DT loc DTLocal Sherwood number EUYYwavgEU yw F d Sr f Strouhal number u t Time s top Operating time hr yr tst Time step size s T Temperature K u Velocity in x direction ms uueff avg Effective velocity ms

V Volume m3

3 Vi Partial molar volume of component i mkgmol

3 Vtot Volume of fluid in the channel m v Velocity vector ms v Molar velocity vector ms vi Velocity vector of component i ms v Velocity in y direction ms w Velocity in z direction ms wch Channel width m

Ws Pumping energy W x Distance in bulk flow direction m

X i Molar fraction of component i

L x Dimensionless friction distance dReh

L x* Dimensionless mass transfer distance dReSch Y Salt mass fraction

xii Yi Mass fraction of component i Distance in direction perpendicular to y m bulk flow Distance in the direction perpendicular m z to both x and y

Zi Valence or charge of ionic component i geq gmol

Greek symbol Description Units High resolution differencing scheme blending parameter Circulation ms2

2 Diffusion coefficient for variable ms i Activity coefficient for component i Boundary layer thickness m conc Concentration boundary layer thickness m vel Velocity boundary layer thickness m ij Kronecker delta function Void fraction, porosity Dimensional number (number of dimensions in flow field) pump Pump efficiency e Angle of attack of elliptical filament ° Am4 2 Ionic conductance of component i i Vgeq4 kg Dynamic viscosity of the fluid ms4

i Chemical potential of component i J kgmol

i Electrochemical potential of component i J kgmol Scaled solute mass fraction Osmotic pressure Pa Density of the fluid kg m3

3 i Mass concentration of component i kg m

xiii Reflection coefficient w Wall shear rate Pa

Ratio of permeable to impermeable Sherwood number ! Electric potential Vgmolgeq4 Transported scalar Osmotic pressure coefficient Pa Transported property " Ratio of permeation Peclet number to impermeable Sherwood number .v Vorticity 1 s Vorticity component in direction normal to plane 1 s n of vortex % Integration volume m3

Subscript Description 1, 2 Value at a specific set of conditions AM Arithmetic mean value avg Average value integrated over channel cross-section b Mass flow average or bulk flow value coarse Value for a coarse mesh cv Value for the control volume end Value at the end of the current timestep fd Value at fully developed boundary layer conditions fine Value for a fine mesh G Value in gel layer glob Integrated or global value for the channel conditions as a whole i Value for component i in a mixture in Value at the domain inlet ip Value at an integration point j Value for component j in a mixture LM Logarithmic mean value loc Value at the local flow conditions max Maximum value

xiv out Value at the domain outlet p Value for the permeate pm Value at the permeate-membrane interphase start Value at the start of the current timestep tm Trans-membrane value up Nodal value at the upwind cell V Value for a vortex w Value at the channel wall wm Value at the feed-membrane interphase y, z Value at a location on the yz plane

xv Chapter 1

Introduction

Over the past decades, membrane processes have become mainstream industrial technologies and are nowadays being used in many types of large scale separations [1]. These technologies have come a long way since the work of Loeb and Sourirajan [2] in the late 1950s, and are now considered a cost-effective industrial separation technique, in many cases having lower energy consumption costs when compared to other traditional separation processes such as distillation. Membrane operations or membrane processes define a broad category that covers a large range of unit operations that utilise an artificial or synthetic membrane to separate the components of a mixture. Many are physical separation processes, meaning that if the two outlet streams were remixed, the resulting stream would be identical in composition to the inlet stream. Some membrane processes rely in part or in whole on chemical effects. These include liquid membranes and many gas separation membranes for which the separation involves chemical interaction without chemical conversion. In addition, a driving force is required to promote the separation. For example, an electric potential gradient is used for electrodialysis, and a temperature gradient is used for membrane distillation. However, most common large scale industrial applications of membrane processes currently in operation are pressure driven. A range of pressure driven membrane separation processes cover the particle size spectrum. Depending on the size of the particles that need to be separated, different types of membranes are used. Microfiltration (MF), ultrafiltration (UF), nanofiltration (NF) and reverse osmosis (RO) or hyperfiltration, are located at the lower end of the particle size spectrum (below 5 m), and are currently considered amongst the best available technologies for water treatment. At the upper end of the spectrum (above 1 m) are conventional particle filtration operations such as cartridge and bag filtration. From an operational perspective, MF, UF, NF and RO are related to conventional particle filtration, making use of increasingly tighter filters or membranes. Due to the similarities between heat and mass transfer phenomena, the types of modules used in membrane operations have often started as adaptations of their heat transfer unit operation counterparts. In this manner, the “plate-and-frame” and “shell-in-

1 tube” units are analogous to hollow fibre and spiral wound membranes (SWM), that are now the most common commercially available membrane modules. The latter are often the better choice for industrial applications from RO to UF [3, 4], and are the focus of this work. The main difficulty these types of modules face is the development of a high- concentration layer near the membrane surface, an effect referred to as “concentration polarisation”. Apart from reducing the flux of solvent through the membrane, concentration polarisation also accelerates the onset of fouling. Concentration polarisation arises due to the convective transport of solutes from the bulk and their rejection at the membrane surface being faster than the diffusive transport back to the bulk. The effect of concentration polarisation in SWM modules is reduced by the spacer nets also used to keep the membrane leaves apart. It has been shown [5] that the presence of spacers in narrow channels promotes flow instabilities and increases mixing. Moreover, since SWM modules operate at low to moderate Reynolds numbers, the flow instabilities generated by spacer meshes are usually laminar rather than turbulent. Computational Fluid Dynamics (CFD) is a technique that allows simulation and subsequent analysis of fluid systems by solving conservation equations for mass, momentum and energy using numerical methods. Its results have become accurate and dependable enough to enable its application in areas such as aerospace engineering [6]. CFD has recently also become a more widely used tool in the field of membrane science [7], with more and more research groups utilising this technique in order to assist the design process, improve the performance, and gain insights into the phenomena taking place inside membrane modules. In spite of recent CFD studies of the behaviour of fluid flow in spacer-filled channels, the transient nature of mass transfer enhancement in unsteady flow, especially in the near wall region, is not yet fully understood. Moreover, three-dimensional (3D) CFD studies have focused on spacer performance, but failed to analyse the mechanisms giving rise to mass transfer enhancement in 3D flow. Since real-world SWM units comprise 3D flows by nature, and operate in a flow regime where unsteady flow is likely to be present, understanding of the effect of these flow conditions on mass transfer enhancement is critical for membrane unit design and optimisation. In addition, modelling of multiple component mixtures with interacting components, such as multiple ionic component mixtures, is not readily available in most

2 CFD software packages. Traditionally, the problem of diffusion in multiple component mixtures has been simplified by treating each component as diffusing independently, irrespective of the other components. However, for many solutions used in membrane separation systems, such as seawater and brackish water among others, this is not a reasonable assumption. In those cases the interactions amongst solutes have a significant effect on mass transfer. Clearly, a diffusion model which takes into account the interactions amongst dissolved ionic species must be used when modelling the flow of these types of solutions in membrane systems. The work reported in this thesis uses CFD to simulate and produce data for the analysis of the dynamics of fluid flow and mass transfer in narrow spacer filled channels such as those encountered in SWM modules. The thesis aims to: Understand the impact of unsteady flow on mass transfer, particularly the effect of vortices near the membrane wall on the developing boundary layer. Understand the effect of three-dimensional (3D) spacer geometries on flow, and consequently on mass transfer. Understand the effect of multiple-layer spacer meshes on membrane performance. In keeping with these aims, the structure of this thesis is as follows. Chapter 2 reviews previous work which focused on the effect of flow behaviour in membrane modules and their effect on mass transfer. Particular emphasis is given to studies analysing the performance of different spacer designs and their effect on the flow field, as well as the interactions between hydrodynamics and mass transfer. Chapter 3 details the methodology used throughout this work for generating and analysing data: CFD. The numerical procedure utilised in obtaining converged solutions for the Navier- Stokes transport equations is described. In Chapter 4, the work of Schwinge et al. [8], who studied two-dimensional (2D) steady flow with and without mass transfer and 2D unsteady flow without mass transfer, is extended to 2D unsteady flow with mass transfer. The simulation results are validated against published experimental results. The relationship between vortex shedding, pressure drop and mass transfer enhancement is explored, and a Fourier analysis is performed to gain insight into the dynamics of the different flow regimes encountered. Chapter 5 deals with the 3D steady state laminar flows encountered in a typical spacer mesh configuration. In order to analyse these 3D steady flows, a model for fully-developed velocity and concentration

3 fields is implemented. Validation of these 3D simulation results is carried out against published experimental results of 3D geometries similar to the ones under consideration in Chapter 5. The implications of the findings of this work for the design of improved spacer meshes are also discussed. Based on results from Chapters 4 and 5, designs of multi-layer spacers that might improve mass transfer are proposed, simulated and analysed, in Chapter 6. Final conclusions are presented in Chapter 7. Two appendices are included in this thesis: Appendix A explores the effect of multiple ionic components and, Appendix B gives an index to the videos in the disc provided with the thesis.

4 Chapter 2

Literature Review

Given that Hollow fibre and spiral wound membrane (SWM) modules are the most common commercially available membrane modules, and that the latter dominate for UF, NF and RO, this thesis focuses on SWM. The cut-away of a typical SWM is shown in Figure 2.1.

Figure 2.1: Schematic of a typical SWM, showing the direction of each flow and the wound layers of membrane and spacers (Source: [1]).

The need to better understand and perfect membrane separation systems has encouraged the publication a vast amount of literature related to membrane processes. Early studies led to a good understanding of the mechanisms that give rise to concentration polarisation. However, they all used approximations for the velocity profiles, and neglected the interactions between multiple ionic components in the feed solution, which limited their applicability for design purposes. In recent years, new developments in the fields of numerical simulation have provided the tools necessary for predicting velocity profiles and pressure losses for systems with complex geometries [6]. The capability of those tools for incorporating mass transfer into numerical

5 simulations has opened many prediction, design and performance evaluation possibilities for the field of membrane science. This chapter reviews the main research dealing with the SWM modules commonly used in pressure driven membrane processes, ranging from early numerical and experimental studies to recent studies employing Computational Fluid Dynamics. An overview of the theory of diffusion models for multiple interacting species is also provided.

2.1. Basic Concepts The following sections present the basic concepts for mass transfer and energy losses that are used throughout the published literature and research dealing with membrane systems and SWM modules. In addition, most of the work and conclusions for this thesis are based on the understanding of these concepts and terms.

2.1.1. Diffusion in Systems with Multiple Ionic Components One of the most basic equations used to describe mass transfer is Fick’s law [9]: mass is transferred from a region of higher concentration to a region of lower concentration. The rate at which mass is transferred per unit area is called the mass flux, and it is proportional to the concentration gradient. The proportionality constant is the diffusivity of the media. The three-dimensional form of Fick’s law for a binary mixture can be expressed in vector notation by: . J AADYBA (2.1)

One of the downfalls of this representation of Fick’s law is that it does not provide an obvious way of treating a multiple component system. A traditional and simple way to deal with multiple components is to treat the diffusion of each component in the mixture independently [9-11]. This approach is valid for the diffusion of a dilute component in a multiple component mixture, and yields the following multiple- component version of Fick’s law: . JiiDYmi (2.2)

Nonetheless, there are many cases in which the underlying assumption of independent diffusion in equation (2.2) is not valid, such that the diffusion rate of one

6 component affects that of the other components. Onsager [12] realised this, and proposed the following reformulation of Fick’s law for a system of n components:

n . JiiB DYjj (2.3) j1

In equation (2.3) the term Dij represents a matrix of diffusion coefficients. In this matrix, the diagonal terms (Dij, i=j) are analogous to the diffusivity coefficients in equation (2.2), which relate the diffusive flux of a component to its own mass fraction gradient. The remaining off-diagonal terms (Dij, ij) relate the flux of one component to the concentration gradients of other components and, hence, non-zero values for these off-diagonal coefficients indicate that the fluxes are coupled. Onsager [12] also suggested that the driving force for diffusion, which in equation (2.3) is the mass fraction gradient, does not need to be in the form of concentration or mass fraction, but may be described by other composition dependent parameters, e.g. chemical potential. Although this diffusion model of Onsager is general enough to describe most systems encountered in real life, the determination of the diffusion matrix coefficients presents a difficulty. Felmy and Weare [13] experimentally calculated these coefficients for common seawater components, but mention that the lack of experimental data for many other systems is one of the main obstacles of this approach. In their view, accurate models using the Onsager transport coefficients must rely on the chemical potential derivatives cancelling out. Moreover, theoretical calculation of these coefficients is not an easy task. An alternative way of describing diffusive phenomena was originally proposed by Maxwell [14] and Stefan [15], and later extended to liquids and other media by Spiegler [16], Lightfoot et al. [17], Standart et al. [18], Krishna [19] and Graham and Dranoff [20]. This model was thoroughly described in a review by Krishna and Wesselingh [10] and is generally referred to as the Maxwell-Stefan (MS) diffusion model, although some authors call it the Stefan-Maxwell model. In essence, this formulation considers diffusion as the result of the various frictional forces exerted between individual component particles. As a result, it proposes a force balance between each species, in which a driving force (d) is counter-balanced by the frictional forces. In its most general form, the MS diffusion model is described by:

7 d XXij(vv i j) i B (2.4) RT j ij

A generalised driving force which is applicable for most systems is given by [18, 21]:

cV Y CSn . ii i . 1 diiiXpcYD iiijFFBc j T (2.5) ccEUj

However, according to Krishna [19], for electrolyte systems such as those found in membrane applications the pressure term in equation (2.5) can be safely neglected. Moreover, the only relevant body force is due to the electrical potential gradient and, given the electroneutrality condition, the driving force simplifies to: ( . .!) . diiiiXZ X ii (2.6)

After appropriate matrix manipulation [10, 19], it is possible to relate the coefficients of the MS diffusion model (ij) to those of the Onsager model (Dij). Despite the higher degree of complexity inherent to the MS formulation, Graham and Dranoff [20] found that the MS coefficients are less concentration dependent than their Onsager counterparts, as well as being independent of the components present in the system. One-dimensional MS type models have been used in the membrane field for modelling transport through the membrane, particularly in electrodialysis [22], electrolysis [23], UF [24, 25] and NF [26-29]. Another commonly used diffusion model for systems with dissolved ionic components is the Nernst-Planck (NP) model [30]. The NP model is a simplification of the MS model which, unlike Fick’s law, takes electrical effects into account and therefore is widely used in the field of electrochemistry [31]. However, the NP model makes other more restrictive assumptions that limit its applicability. It neglects friction amongst solutes and assumes an ideal solution, effectively ignoring the effect of the activity coefficient. As a consequence of these assumptions it is only applicable for dilute solutions, and may only be used as a qualitative tool for concentrated solutions. Keeping a form similar to that of the MP model given in equation (2.4), the NP model is described by the following relationship:

8 X . vv ii i (2.7) RT im

One of the main advantages of the NP model is that the values for the diffusion coefficients (im) are readily found in the literature [32], or can be calculated from published ionic conductance data [33].

2.1.2. Mass Transfer Coefficient The mass transfer coefficient is a parameter used to correlate mass transfer rates, contact area and concentration differences. It can be defined in many different ways, but the most common is by analogy to one-dimensional diffusional solute transport, which is given by Fick’s law [9] :

dY JD (2.8) slt dy

Equation (2.8) relates mass flow of a solute per unit area, or solute mass flux, to a mass fraction gradient. However, the mass fraction gradient is a local one, and it can present large variations within a flow field. It is therefore more practical to relate solute mass fluxes to concentration or mass fraction differences between bulk flows or between a bulk flow and a surface. For mass transfer at a fluid-solid interface, this relationship is given by:

() JkYYslt mt, loc w b (2.9)

By combining equations (2.8) and (2.9), the local mass transfer coefficient is defined as:

DYCS k DT (2.10) mt, loc YYbwEU nw

For general applications, it is useful to know the area-averaged value of the mass transfer coefficient over the mass transfer area, which is calculated by:

1 kk O dA (2.11) mt, avg A mt, loc mt Amt

9 However, it is often the case that the local mass fraction values, and hence the local mass transfer coefficient values, are not available for every point on the mass transfer surface. Therefore, for simplicity reasons, the use of average concentration differences has been proposed. The mean-logarithmic concentration is defined by:

()()YY YY Y wb22 wb 11 (2.12) LM CSYY ln DTwb22 EUYYwb11

By making use of mean-logarithmic concentration differences, a global coefficient of mass transfer can be defined such that:

JkYslt,, avg mt glob LM (2.13)

Although the global and area-averaged mass transfer coefficients will not necessarily have the same value, they will generally be reasonably close to each other.

Hence, the global mass transfer coefficient (kmt,glob) provides a suitable approximation to the area-averaged mass transfer coefficient (kmt,avg). Some researchers prefer the use of arithmetic-mean concentration differences, i.e. a simple average, for defining the mass transfer coefficient. The arithmetic-mean concentration difference is given by:

()()YY YY Y wb22 wb 11 (2.14) AM 2

The use of the arithmetic mean is reasonable in the cases where the concentration differences change little over the length of the channel, for example, for short channel lengths with low relative permeation compared with the bulk flow. For large variations in concentration difference, the use of an arithmetic mean will yield a lower mass transfer coefficient than that obtained using a logarithmic-mean concentration difference, and will also under-represent the area-averaged mass transfer coefficient. Nonetheless, both the arithmetic-mean and logarithmic-mean appear in mass transfer coefficient correlations found in the literature for various flows and geometric configurations [1, 34-38]. Published analytical and empirical correlations for the calculation of the mass transfer coefficient are often expressed as a dependence of the Sherwood number (Sh)

10 on the Reynolds number (Re), Schmidt number (Sc), channel length (L) and other geometrical parameters. Skelland [35] obtained an analytical solution, in the form of an infinite series, to the mass transport equation for the case of 2D flow between parallel plates. He assumed a fully developed (parabolic) velocity profile, no permeation, and either constant wall concentration or constant wall solute flux. The graphical form of these solutions is shown in Figure 2.2. According to the results of Skelland [35], the ( ) Sherwood number only depends on the dimensionless group x* LdReSch . As this dimensionless parameter increases, the Sherwood number decreases, and reaches a lower limit at approximately x* 0.05. This result shows that, for flow with a fully developed concentration boundary layer in a 2D rectangular channel, the Sherwood number is constant and does not depend on the Reynolds number. The value of the Sherwood number for a fully developed concentration profile is 7.6 for a constant wall concentration, and 8.23 for a constant wall solute mass flux.

Figure 2.2: Channel local Sherwood number distribution for a 2D rectangular channel, with constant wall solute flux and constant wall concentration (derived from [35]).

An approximation to the result of Skelland [35] was presented by Blatt et al. [36], and is given by the following expression, which is only valid for short channel lengths, using an arithmetic concentration difference:

11 13 kh, ()2 CS uh2 C dReScS13 , mt, glob ch DTavg ch h Shavg 2.354DT1.483D T (2.15) DDEULEUL

For turbulent flow conditions, entrance effects are not as significant as for the laminar regime, and are usually neglected. The Deissler analogy [37] is a commonly cited expression for calculating the Sherwood number in this regime:

kd Sh mt h 0.023Re0.875 Sc 0.25 (2.16) D

McCabe et al. [38] suggest the following alternate empirical correlation for turbulent flows:

Sh 0.0096 Re0.913 Sc 0.346 (2.17)

In broad terms, for the same channel length the Sherwood number (and hence the mass transfer coefficient) becomes larger with increases in Reynolds number. However, as is evident by the exponent on the Reynolds number in equations (2.15) and (2.17), the rate of increase of the Sherwood when the Reynolds number is increased is higher in the turbulent regime than in laminar flow.

2.1.3. Energy losses and the Friction Factor Increasing the Reynolds number in order to increase the mass transfer coefficient does not come without an added cost, as energy losses also increase with increasing Reynolds number. Energy losses can be calculated if the pressure drop for a given channel length is known [38]:

WQps ch (2.18)

Pressure drop data in channel flow is normally reported in terms of the dimensionless Fanning friction factor which, for conduits without obstacles, is defined by [9]:

dp f hch (2.19) 2 2 uLavg

12 For laminar flow in 2D rectangular channels, the friction factor can be calculated analytically, and is given by [9]:

24 f (2.20) Re

Empirical correlations are typically used for calculating the friction factor in the turbulent regime. A commonly used expression is the Blasius equation [9], which takes the following form:

0.0791 f (2.21) Re0.25

Comparisons of different mass transfer enhancement techniques are usually made at the same Reynolds number [39]. However, this may not be the best option when energy losses are significant. Huang and Tao [40] proposed making comparisons at the same pressure drop and at the same pumping power. From the definition of the friction factor for flow with obstacles in equation (2.55), pressure drop per unit length can be calculated by:

p 2 fu 2 2 2 ch eff Ref2 (2.22) 3 h Ldhh d

From equation (2.22), for two different flows of the same fluid to have the same pressure drop per unit length, the following must be satisfied:

Ref22 Ref hh11 2 2 33 (2.23) ddhh12

The necessary pumping power per unit length can be calculated by:

WQp 2 fA u3 2A 3 schTeff T Ref3 (2.24) 24 h LL dhh d

Similarly, from equation (2.24), for two different flows of the same fluid to have the same pumping power per unit length, the following must be satisfied:

13 Ref33 A Ref A hT111 1 h 2 22 T 2 44 (2.25) ddhh12

In general, comparisons at the same pressure drop or pumping pressure for different geometries cannot be made by comparing a single dimensionless number because both of pressure drop and pumping power are dimensional quantities. However, flows of the same fluid with the same hydraulic diameter will have the same pressure

2 drop if the dimensionless factor Refh is the same. Analogously, for flows of the same fluid which have the same hydraulic diameter, porosity and transversal area, energy

3 losses will be the same at the same value of the dimensionless factor Refh . Other researchers [41] have used a modified friction factor which is proportional to the cubic

3 root of this last dimensionless group. Moreover, the factor Refh has also been referred to [42] as Power Number (Pn).

2.2. Concentration Polarisation

Figure 2.3: Filtration spectrum showing typical particle size ranges of common materials (Source: Osmonics Inc.).

Membrane separation processes such as MF, UF, NF and RO have many things in common. In all of these processes a driving force in the form of a pressure differential is

14 applied to a solution in contact with a semi-permeable membrane. The membrane allows the passage of the solvent or solvents, but selectively rejects the rest of the components of the solution, which can be particles or aggregates (MF), macromolecules or colloids (UF) or small solutes such as dissolved ions (NF and RO), as shown in Figure 2.3. Rejection of components by the membrane means that concentrations near the interface become greater than in the bulk, generating what is known as “concentration polarisation”. Concentration polarisation reduces the separation performance of membrane systems by adding an extra layer of resistance [43, 44] and/or counteracting the effect of the driving force in the region near the membrane [45]. Moreover, an increased concentration of solutes near the membrane will also increase the likelihood of precipitation and fouling. A fouled membrane will present a higher resistance to the passage of solvent, or even prevent it altogether, further decreasing the performance of the membrane system. Since concentration polarisation occurs due to solute rejection, it is inevitable and inherent to membrane separation processes. Hence, as higher water fluxes are achieved by means of better performing membranes, the effect of concentration polarisation will only become greater [45]. Nevertheless, it is possible to reduce its extent by boundary layer control or promoting the mixing of low and high concentration fluid in the feed channel [45, 46], thereby increasing the separation performance of membrane systems.

Figure 2.4: Schematic representation of the development of a concentration polarisation region in UF (Source: Porter [47])

Early studies [43, 44, 47, 48] focused on developing ways for predicting the effect of concentration polarisation on solvent flux. Kimura and Sourirajan [49] analysed

15 experimental RO data using a simple one-dimensional (1D) mass transfer model of the form:

D JXXmm( ) (2.26) slt wm pm

Their analysis of the mass transfer coefficient between the membrane (Xw) and the feed solution (Xb) showed that mass transfer coefficients obtained for flows without permeation could be used for RO systems. Using this 1D model, Kimura and Sourirajan developed a methodology [50] to predict the effects of concentration polarisation in RO making use of the following expressions for the calculation of solvent and solute flux:

FV() () JLpXslv pHX w w Xp (2.27)

D J m ( XX) (2.28) slt K w p

Equations (2.27) and (2.28) are analogous to those developed by Kedem and Katchalsky [51] and Merten [52], and make use of two experimentally determined parameters: the pure water permeability constant Lp, and the solute transport parameter, DKm . For a given operating pressure, these parameters are independent of feed concentration and flow rate. The variation of the above mentioned parameters was found to correlate with the operating pressure through the following relationships:

ap Lep (2.29)

D m p b (2.30) K

Kimura and Sourirajan [53] further expanded this methodology for the prediction of the performance of RO systems (i.e. salt concentration in the product, throughput rate and concentration polarisation) by making use of appropriate local and average mass transfer coefficients. The benefit of this approach is that it can be used for both laminar and turbulent flow in flat and tubular channels.

16 DeFilippi and Goldsmith [48] applied basic mass transfer theory to membrane processes in order to relate concentration polarisation to the mass transfer coefficient, kmt, by the following expression:

CScc Jk ln D wPT (2.31) mt EUccbP

Making use of the mass transfer model used to derive equation (2.31), Goldsmith [43] analysed the mechanisms of macromolecular UF. He determined that the main parameter limiting the flux throughput was mass transfer on the feed side of the membrane. Mass transport away from the membrane was also the determining factor for concentration polarisation. Baker and Strathmann [44] studied solute accumulation at the membrane barrier for high-flux UF membranes using batch and recirculation cells. Their experiments showed that at lower pressures the solvent flux for different macromolecular solutions was approximately the same as the pure solvent flux, and increased as the pressure increased. However, at higher pressures the flux reached a maximum and remained constant, becoming independent of any further increases in pressure. The magnitude of the flux maximum was nevertheless dependent on other variables: it increased with increases in temperature and agitation, and decreased with increasing solute concentration. Their explanation for this phenomenon was the formation of a gel layer adjacent to the membrane, which acts as a barrier for the flow of solvent and lower molecular weight solutes. On the other hand, NF and RO do not present gel layer formation, i.e. it is characteristic of UF. Porter [47] further explained the theory of the formation of a gel layer in UF, as well as its effect on solute flux. He assumed the solute concentration in the whole gel layer to be constant (cG), but the gel layer was free to vary in thickness and porosity. By this reasoning, increases in pressure would not increase the flux of solute, but instead increase the thickness of the gel layer. In light of this, the analogous of equation (2.31) for UF with gel layer formation is as follows:

CSc G Jkmt ln D T (2.32) EUcb

17 Using either equation (2.31) or equation (2.32), it therefore became possible to calculate the effect of concentration polarisation. However, these equations require the value of the mass transfer coefficient for their successful application, which varies from case to case and can only be estimated beforehand by means of correlations. In addition, the underlying mechanisms and factors affecting mass transfer and the formation of the concentration polarisation layer were not fully understood. In an attempt to gain better understanding of those mechanisms, several studies used numerical analysis to predict and examine the concentration and velocity profiles at every point inside the membrane module. The first numerical studies of concentration polarisation followed the approach of Brian [54], by solving the mass transport equation in two dimensions, but neglecting diffusion in the direction of the bulk flow. Therefore, the mass transport equation solved had the following form:

CS c ()ucDT vc D 0 (2.33) xyEU y

In order to solve equation (2.33), a velocity field was needed. Berman [55] obtained a second order perturbation solution for the continuity and momentum transport equations for the case of 2D flow with porous walls. The truncated velocity field for this solution, for the case of uniform constant wall permeation, is given by:

FV22IY F 4 V 3 Cvx S CSyyLLhv CS CSy uuDin wc TGW11277 DTJZ hw G DT DTW (2.34) avg GW G W 2Ehhch UHX EUch K[LL 420 HEUhhch EUch X

FVF226V vyCS yvy CSCS y y GWG32DT w 3DTDT W (2.35) GWG W vhwchch2 HXHEU h280 EUEU hchch h X

Karode [56] extended the solution of Berman [55] for channels and tubes where the permeation is not constant but depends on the channel pressure, which is often the case for UF and MF, although not as usual in RO. He presented his solution in the form of expressions for pressure drop. Those expressions were intended to be used for quick engineering calculations or as benchmarks for other numerical studies.

18 Kozinski et al. [57] also extended the solution of Berman [55], for the general case of non-uniform wall permeation. However, because the analytical expressions obtained by Kozinski et al. were relatively complicated, some researchers [50, 58, 59] opted to use a simplified form of these expressions, based on the assumption that the parabolic velocity profile in the direction of the bulk flow is not significantly distorted by the removal of fluid at the membrane wall. The simplified velocity profiles are given by:

FV2 u 3 CSy ()11GWDT (2.36) in GW uhavg2 HXEU ch

FV2 vyCS y GW3 DT (2.37) GW vhwchch2 HXEU h

where represents the fractional solvent removal, and is given by:

1 x O vdx (2.38) uhin w avg ch xin

Brian [58] applied the velocity field defined by equations (2.36) and (2.37) to the partial differential equation for mass transport (2.33), in order to obtain an approximate numerical solution for the mass transfer problem in a RO membrane channel. By using this method, he was therefore taking into account concentration polarisation. He studied cases with constant flux and complete salt rejection, and a variable flux case with incomplete rejection. In the latter case, flux was dependent on the concentration at the membrane wall. He found that the concentration polarisation effect was greater near the channel entrance for the variable flux case, but it was lower near the outlet. He concluded that an average concentration polarisation could be utilised for design purposes, using the constant flux solution. Shah [59] extended the results of Brian [58] to the case of variable diffusivity. He found that if diffusivity increases with increasing concentration, solute build-up at the membrane increases when compared to the case of constant diffusivity, thus worsening the negative effects of concentration polarisation. His assessment agrees with that of

19 Doshi et al. [60], who also mention that this effect is further worsened as the osmotic pressure of the feed solution approaches the value of the transmembrane pressure. Kleinstreuer and Paller [61] made a further extension to the numerical modelling approach by coupling the solvent flux through the membrane with the solution of the concentration transport equation. They used the velocity profiles obtained by a perturbation solution to the momentum transport equations, in a manner similar to that of Berman [55], but allowing for variable wall permeation at one of the channel walls, thus resulting in asymmetric velocity profiles. Following the models of Kedem and Katchalsky [51] and Merten [52], wall permeation was assumed to depend on the transmembrane pressure as well as on the osmotic pressure differential between the feed and permeate channels, thus taking the following form:

L vpp ( ) (2.39) wtm tm

Since the osmotic pressure differential in equation (2.39) depends on the solute concentration at the wall, an iterative method was used to solve the coupled concentration and velocity transport equations. Their results showed greater wall shear rates than for flow without permeation, and correctly predicted a decline in solvent flux due to concentration polarisation. However, the authors remarked that the applicability of this model was limited. In addition, further development on modelling various solute interactions, laminar and turbulent flow patterns, as well as fouling and other time dependent phenomena, was needed in order to develop a more flexible model. The advantages of the numerical technique are exemplified by the study of Shaw et al. [62], who assessed the use of non-rejecting (i.e. completely permeable) membrane sections as a technique for reducing concentration polarisation. For the rejecting section they numerically solved the mass transport equation assuming constant rejection. For the non-rejecting section they neglected solute transport due to diffusion, and used the velocity field defined by equations (2.36) and (2.37). They found that this technique not only increased the productivity of the system by increasing solvent flux due to the reduction of concentration polarisation, but also decreased the product salinity. Moreover, the productivity of this type of system was increased more significantly for higher feed solute concentrations, when compared to conventional single membrane systems.

20 Despite the benefits of the numerical approach, the use of analytical expressions for the velocity profiles restricted its applicability to those cases for which those expressions were already known or could be easily calculated. Moreover, such analytical expressions are only available for a handful of cases, neither of which incorporate the use of flow obstructions such as the spacers commonly used to keep membranes apart. In order to include the mixing effect of these spacers, Miyoshi et al. [63] developed the following equation for describing the velocity profile:

dp hv2 LLIYFVCSy CSy CSx ummmch JZ()1lnGWDT 1 1 DT 1D 1 w T (2.40) 2 D in T dx m LLK[HXEUhch EUhch EU uavg h ch

The term m in equation (2.40) is an eddy or mixing constant related to the geometry of the spacer, and can be approximated by:

0.8 2.4 CS1 mNhd2.1 105 FV() DT (2.41) HXsp ch f EU 3

Although the model described by equations (2.40) and (2.41) accounts for mixing due to the presence of the spacers, it ignores the increase in pressure losses due to increased drag. Bhattacharyya et al. [11] combined the models used by Shah [59], Kleinstreuer and Paller [61] and Miyoshi et al. [63], in order to calculate the concentration distribution inside an RO module and thus predict its performance. They obtained a reasonable agreement with experimental data, with no more than 10% relative error in concentration polarisation. However, due to the lack of consideration of the effect of spacers on pressure losses, this approach cannot be used for optimising the geometry of membrane modules. A model similar to that of Bhattacharyya et al. [11] which also incorporates pressure losses was later developed by Zhou et al. [46]. They showed that under the in given flow conditions ( hmmch 0.6 , umsavg 0.15 ) mixing due to flow obstacles could reduce the effect of concentration polarisation by as much as 50%, thus doubling the permeate flux. These results provided substantial motivation for the investigation of means to increase mixing inside the feed channels of membrane modules.

21 2.3. Turbulence promoters and Spacer Studies One of the components of cost in a membrane system is energy losses [64], which can also be measured as pressure losses. This fact, coupled with the knowledge that increased mass transfer reduces concentration polarisation and therefore increases the productivity of membrane systems [48], has prompted many researchers to develop strategies for enhancing mass transfer while maintaining low energy losses. Increases in mass transfer associated with turbulent and unsteady flow are generally believed to be caused by the higher rate of mixing and interaction of the bulk flow and the boundary layer. Due to this, the insertion of flow obstructions has become a common practice for improving mass transfer performance in narrow channels. These obstructions are usually referred to by the term “turbulence promoters” [65] despite the fact that the mechanism for enhancing mass transfer when using these devices is not fully understood, and that in many cases the flow regime presents characteristics of laminar flow, such as recirculation regions. Some researchers, therefore, refer to the obstructions as “eddy promoters” [66], which is a more appropriate term.

2.3.1. Heat transfer enhancement Analogies to extrapolate heat transfer concepts into the field of mass transfer are commonly cited [40, 67]. This comes from the similarities between the mass and heat transfer phenomena and transport equations [9]. Moreover, certain analyses and concepts in this thesis draw on the analogy to heat transfer. For these reasons, a brief summary of heat transfer enhancement techniques, and their relationship to mass transfer enhancement, are presented in this section. In a review of heat transfer enhancement technologies, Bergles [68] states that enhancement is achieved by increasing the heat transfer coefficient, which is analogous to the mass transfer coefficient. In heat transfer, increasing this coefficient is beneficial because it can either: a) increase heat fluxes, b) reduce the heat transfer area needed for the same heat flux, or c) reduced the temperature difference needed for the same heat flux. All these consequences can potentially reduce the associated costs of heat transfer unit operations, and are mirrored in membrane mass transport units by: a) increased solvent flux, thus increasing productivity, b) reduced membrane area required, thus reducing capital costs, and c) decreased concentration polarisation, thus reducing the potential for fouling. Of the representative heat enhancement techniques listed by

22 Bergles [68], the following have the potential to be applied for mass transfer in membrane operations: Rough surfaces Extended surfaces Displaced enhancement devices Swirl flow devices Coiled tubes Surface vibration Fluid vibration Injection or suction

Generation and promotion of vortices and vortical flow has also been identified as one of the possible alternatives for enhancing heat and mass transfer [68-71]. In particular, the work of Fiebig, Grosse-Gorgemann and co-workers [69-71] was significant in identifying that the vortices themselves do not enhance heat transfer, but their influence on the flow field (i.e. flow separation, re-attachment, increased wall shear and boundary layer thinning) does cause increased heat transport as well as a pressure loss penalty. Since heat and mass transport follow analogous mechanisms, it follows that the presence of vortices in the flow should enhance mass transfer, while mass transfer inside a vortex would be minimal. Icoz and Jaluria [72] analysed three different vortex promoter profiles, circular, square and hexagonal, and compared their relative impacts on heat transfer and pressure drop. Their data showed that the hexagonal and square filaments enhanced heat transfer to a greater extent than the circular filaments, but the former were also found to cause a larger increase in pressure drop. They also found the pressure drop to be increased as the promoter size was increased. They concluded that if pressure drop was one of the main design criteria, then the circular filaments were the better choice. Since pumping energy along with membrane costs are two of the main design considerations for membrane systems [73], circular filaments are a common vortex promoter type used for membrane applications [39]. Fiebig [70] pointed out that, for the same energy losses, longitudinal vortices (i.e. streamwise vortices) enhance heat transfer to a greater extent than transverse vortices (i.e. spanwise vortices). He recommended winglets as the most effective type of vortex

23 generator, although he advised that this might depend on the heat transfer problem and more research could lead to better or optimal vortex generators. While winglets are not common in mass transfer operations, other types of vortex generators which produce streamwise vortices might prove to be the optimal type of vortex generator for mass transfer.

Figure 2.5: Streamline plots for flow in narrow channels with (a) circular filament and (b) delta winglet obstructions, illustrating the presence of (a) transversal/spanwise and (b) longitudinal/streamwise vortices (Source: Li et al. [67]).

Patera and Miki [74] identified resonance as another heat transfer enhancement mechanism with the potential to be utilised in heat transfer operations. Greiner et al. [75] visually analysed the heat transfer enhancement mechanisms present in a channel with a saw-tooth shaped wall, and concluded that flow oscillations were the main mechanism responsible for the enhancement of heat transfer. A numerical investigation of heat and momentum transport by oscillatory flows was carried out by Majumdar and Amon [76], who found that after a critical Reynolds number was reached, a flow bifurcation led to self-sustained periodic oscillatory flow. The oscillations predicted by their direct numerical simulation promoted thinner boundary layers and a high degree of

24 mixing which, in turn, increased heat transfer. Since high degrees of mixing and thin concentration boundary layers are known to enhance mass transfer [65], some researchers have attempted to utilise resonance in the form of pulsatile flow [77, 78] as a mass transfer enhancement mechanism. Guo et al. [79] and Tao et al. [80] proposed a method for analysing heat transfer enhancement as an alternative to calculating a heat transfer coefficient. According to this method, heat transfer is enhanced if the value of the integral over the whole volumetric flow domain % as shown in equation (2.42) is increased.

.% () Iht O cTdp v (2.42) %

It can be seen from examination of equation (2.42) that the value of the integral is greater when the velocity and temperature gradient are aligned. Using this principle, Li et al. [81] proposed a channel geometry which makes use of secondary flows in the form of streamwise vortices as a means for reducing the angle between the velocity and temperature gradients. Extrapolation of the method of Guo et al. [79] and Tao et al. [80] to mass transfer yields the definition of following mass transfer integral:

.% () Imt O v Yd (2.43) %

Due to the similarities between equations (2.42) and (2.43), mass transfer is enhanced when the velocity and concentration gradients are aligned. This means that secondary flows such as Taylor [82], Dean [83, 84], and streamwise vortices are not only beneficial for heat transfer enhancement, but will also enhance mass transfer.

2.3.2. Mass transfer enhancement In a review dealing with the application of the principles of Fluid Mechanics in the field of membrane science, Belfort [85] points out that it is well-known that fluid flow and its mechanics affect the performance of pressure-driven membrane units. Moreover, he states that understanding the mechanisms by which these two are related can help predict the extent of concentration polarisation and fouling, two of the main phenomena that hinder the performance of membrane units. For these reasons, many

25 experimental studies have focused on analysing the impact on mass transfer enhancement of the hydrodynamic conditions in narrow channels. Experimental studies have successfully identified boundary layer destabilisation as one of the main mechanisms for enhancing mass transfer [65, 66, 86-93]. The main method for achieving destabilisation is through the use of flow obstructions [87, 94], although other methods have also been proposed [84]. Despite the fact that increases in mass transfer are generally accompanied by an increase of energy losses [39, 40], improvements to mass transfer performance have been shown to translate to economic benefits for membrane separations systems [64, 93, 95]. In order to increase the ratio of mass transfer enhancement to increases in energy losses, new designs for flow obstructions have been proposed [67, 96, 97]. A summary of these contributions is presented in Table 2.1.

Table 2.1: Summary of important contributions to experimental study of mass transfer enhancement in membrane systems.

Researcher Analysis methodology and significance Thomas [65, 86] Effect on mass transfer of cylinders placed on outer edge of boundary layer. Cylinders increase mass transfer up to 190%, but also increased pressure losses, up to 230% Hicks and Mandersloot [98] Developed correlations between pressure drop and heat and mass transfer for systems with turbulence promoters. Divided pressure drop into viscous and kinetic losses. Feron and Solt [87] Visualisation study of use of suspended and attached circular rods as turbulence promoters. Identified three main flow regimes: laminar steady, unsteady and vortex shedding. Belfort and Guter [94] Comprehensive study of effect of spacers on fluid dynamics and mass transfer. Concluded that different types of spacers had significantly different effects on flow. Mass transfer enhancement was more significant at higher Reynolds numbers. Solan, Winograd and co-workers Quantification of the effect of mixing via turbulence promoters on [88-90] mass transfer. Developed the “mesh step” model. Thomas et al. [64] Cost analysis of the impact of the use of turbulence promoters in RO. Promoters allow operation at lower velocities and pressure drops, and require less membrane area. Kuroda et al. [99] Related pressure drop and mass transfer to geometric parameters of spacers. Identified possibility of optimising the mesh size in order to provide higher mass transfer at lower pressure drops. Kim et al. [91, 92] Identified the formation of recirculation regions for 2D promoters, and the interruption of the development of the concentration boundary layer for 3D promoters, as the main mechanisms for mass transfer enhancement. Focke [66] Analysed semicircular promoters attached to the channel walls. Identified recirculation regions as main flow features. Found mass transfer maxima at channel inlet (entrance effect) and at boundary layer reattachment regions. Schock and Miquel [39] Developed correlations for friction factor and mass transfer coefficient for use with several commercial spacers.

26 Da Costa et al. [93, 95, 100, 101] Studied UF spacer pressure drop and mass transfer enhancement characteristics. Significant performance differences were found between the spacers used. The best spacers for reducing concentration polarisation were also the best for hindering fouling. Identified form drag as main component of pressure drop. Economical analysis revealed that systems with spacers performed better than empty channels. Li et al. [102] Studied influence of spacer geometric parameters on mass transfer. Determined optimal geometric parameters for typical non-woven spacer nets, which agreed with numerical predictions. Li et al. [103] Studied the behaviour of particles in MF using the DOTM technique. Found that flux variations on membrane surface can lead to local particle deposition variations. Neal et al. [3] Studied effect of spacer geometry on MF deposition patterns using DOTM. Found the 45° orientation to be optimal for SWM modules. Hendricks et al. [104] Found buoyancy induced instabilities to be beneficial for RO performance. Drezansky and Gill [105] Analysed the mass transfer mechanisms in tubular RO. Found that buoyancy had a notable impact on mass transfer rates for laminar flows. Youm et al. [106] Investigated the effect of destabilisations due to natural convection caused by density variations. Observed 3 to 5 fold UF permeate flux increases when membrane was located on top of the channel, for Reynolds numbers below 90. Schwinge et al. [96] Proposed 3-layer spacer which despite increasing pressure drop, produced higher fluxes which led to lower capital and total processing costs. Li et al. [67] Proposed and tested a novel multi-layer spacer mesh. Found 20% and Balster et al. [97] higher mass transfer than traditional 2-layer spacers, at the same power consumption. Gimmelshtein and Semiat [107] Utilised PIV for visualising flow conditions in spacer-filled channels. Proposed use of Mixing Index to estimate regions of enhanced mass transfer. Also analysed fouling patterns.

2.3.2.1. Flow regimes

Figure 2.6: Bulk flow streamlines inside a spacer-filled narrow channel, as observed by Da Costa et al. [101].

27 In order to recognise the effect hydrodynamics have on mass transfer, the behaviour of the fluid inside spacer-filled channels must first be understood. The following features have been identified [66] for flow over an obstacle: (a) a reversed flow region (recirculation) behind the obstacle, (b) a free shear layer between the reversed flow region and the bulk flow, (c) a corner eddy located inside the reversed flow region, and (d) additional flow separation regions downstream of the obstacles, on both channel walls. For 3D spacer geometries, Da Costa et al. [101] observed that at lower Reynolds numbers the bulk of the flow follows a zigzag path, as shown in Figure 2.6. As the Reynolds number is increased, the behaviour of the flow is altered. Feron and Solt [87] observed three main flow regimes for the promoters tested as they varied the Reynolds number: a steady laminar regime with recirculation zones behind the rods, an unsteady regime characterised by moderate oscillations of the flow in free shear layers (i.e. at the boundaries between the bulk flow and recirculation areas), and a vortex shedding regime at the highest Reynolds numbers. Neither of these flow regimes is turbulent, which is a common misunderstanding [94]. This agrees with the observations of Kang and Chang [108] (see Figure 2.7) and Belfort and Guter [94]. .

Figure 2.7: Flow visualisation for a cavity type spacer as observed by Kang and Chang [108], showing three distinct flow regimes: steady laminar flow (top), unsteady with moderate oscillations (middle) and vortex shedding (bottom).

28 The critical Reynolds number (based on the hydraulic diameter of the empty channel and the superficial velocity) at which flow becomes unsteady varies with the geometric characteristics of the system, but usually falls in the range of 180 to 550 [87, 94, 108], which is significantly lower than the critical Reynolds number for an empty channel. In the vortex shedding regime, for rods attached to both walls, vortices appeared behind each rod, only to disappear below the following rod and then reappear shortly before hitting the next rod. For the geometry with suspended rods, vortices only appear near the channel walls for unsteady flows [87]

2.3.2.2. Mechanisms for mass transfer enhancement Several studies have focused on identifying the main mechanisms which give rise to mass transfer enhancement in membrane systems. It is commonly known that increased Reynolds numbers tend to increase mass transfer. In order to characterise the mass transfer performance of the spacers, an expression of the following form is commonly used:

CSd d Sh aRebc Sc DTh (2.44) EUL

Typical values for the exponent of the Reynolds number (b) in equation (2.44) are summarised in Table 2.2. The value for the exponent b is higher for the transient and turbulent regimes. This means that the mass transfer enhancing effect of the Reynolds number is accentuated when flow is unsteady.

Table 2.2: Summary of results of the effect of Reynolds number on Sherwood number; exponent for the Reynolds number (b) in equation (2.44)

Flow Regime and Exponent for the Reynolds Researcher configuration number, b Schock and Miquel [39] Laminar, commercial spacers 0.333 Kim et al. [91] Laminar, zigzag 0.376 Kang and Chang [108] Laminar to transient, zigzag 0.475 Kim et al. [92] Laminar, 3D at 90° 0.475 Kuroda et al. [99] Laminar to transient, 3D spacers 0.5 Laminar to turbulent, Da Costa et al. [93] 0.43 to 0.66 commercial spacers Hicks and Mandersloot [98] Turbulent 0.5 Focke [66] Turbulent, zigzag 0.65 Schock and Miquel [39] Turbulent, commercial spacers 0.875

29 Many researchers [65, 66, 86, 95] have identified an increased velocity gradient (i.e. viscous friction) and the interaction of the wakes of obstacles (i.e. recirculation regions) with the boundary layer as the main mechanisms for mass transfer enhancement. Although the work of Thomas [65, 86] was done using a fluid in the gas phase, a follow up study by Watson and Thomas [109] showed that mass transfer enhancement for liquid flows using cylindrical promoters followed similar tendencies to those obtained for the gas phase. Kim et al. [91] also attributed the increase in mass transfer rates when using promoters to the interruption of the development of the concentration boundary layer. Since their data supported the idea that this interruption is caused by the recirculation regions that appear in front and behind the flow obstacles, this agrees with the ideas of other researchers [65, 66, 86, 95]. For 3D promoters, however, Kim et al. [92] were unable to detect any recirculation regions and therefore concluded that, unlike for the 2D case, recirculation was not an important mass transfer enhancement mechanism in 3D. Enhanced mixing has also been recognised as one of the causes of increased mass transfer. A series of studies into the quantification of the effect mixing and turbulence promoters on mass transfer in electrodialysis processes was carried out by Solan and Winograd [88, 89] and Winograd et al. [90]. They considered that the solution of the flow problem under the assumption of perfect mixing is the upper performance limit, i.e. the ideal situation that is sought with the introduction of turbulence promoters. Building on this idea Solan et al. [89] developed a “mesh step” model which assumes that each repetition of the turbulence promoter pattern causes partial mixing of the boundary layer and the bulk flow. A notable characteristic of the “mesh step” model is that it separates the effect of Reynolds number on mass transfer from the purely geometrical effects of the turbulence promoter, which is accounted for in the mixing efficiency. However, a downfall of this modelling approach is that the mixing efficiency must be determined by experimentation, and cannot be calculated a priori. Gimmelshtein and Semiat [107] utilised particle image velocimetry (PIV) for visualising the flow conditions inside spacer-filled membrane channels. This technique allowed them to measure the magnitude and direction of the fluid velocity with a relatively high resolution within each unit cell of the turbulence promoting spacer nets. Although the PIV technique does not measure mass transfer, the use of the mixing index

30 (MI) allowed them to estimate the regions in which the channel mass transfer is enhanced. They found that at higher Reynolds numbers than typically used in membrane operation, the high MI regions extended further from the filaments. Analysis of the membranes after use showed significant fouling beneath the filaments, which was attributed to flow stagnation in the areas near the filaments and to precipitation onto the membrane due to inertial forces. Therefore, PIV showed potential for providing further insights into the mass transfer phenomena taking place inside membrane modules.

2.3.2.3. Energy loss trade-off Thomas [65, 86] found that although cylinders placed on the outer edge of the boundary layer increased mass transfer significantly, by as much as 190%, they also increased the pressure losses by approximately 230%. Da Costa et al. [93] also found that, using spacer meshes, UF and RO solvent fluxes increased between 3 to 5 times when compared to the flux in an empty channel, but pressure drops also increased between 5 to 160 times. In light of this, Belfort and Guter [94] stipulate that in order for a spacer to be considered “good” it must promote mixing without excessively increasing the pressure drop. In addition, the spacer must generate as few regions of stagnant flow as possible, as those regions may encourage scale formation. Hicks and Mandersloot [98] assumed that only viscous energy losses contribute to heat and mass transfer enhancement, whereas kinetic losses such as form drag do not. Using those assumptions, they developed a correlation based on an equation first proposed by Reynolds [110] which describes the pressure drop as a quadratic function of fluid velocity:

p au bu2 (2.45) L avg avg

In equation (2.45), the coefficient for the linear term (a) is a measure of the viscous losses and the coefficient for the quadratic term (b) is a measure of the kinetic losses. They applied this concept to develop a modified Reynolds number:

Su Re* wavg (2.46) a

31 This modified Reynolds number is in turn used in a generalised Chilton-Colburn analogy:

12 jKRe ()* (2.47)

Although this type of correlation appears to be quite general, the K constant in equation (2.47) is still dependent on the geometry used. Moreover, since some of the assumptions made in the development of this correlation are only valid in the turbulent regime, equation (2.47) would not be valid for laminar flow. A similar attempt at relating pressure losses to geometric parameters was carried out by Kuroda et al. [99]. They found that the following expression reasonably fitted their friction factor to data:

12 f kRef (2.48)

Due to the Reynolds number exponent in equation (2.48) being greater than -1, the value for steady state laminar flow in an empty channel, Kuroda et al. inferred that the flow was transient. In addition, they attributed pressure losses primarily to the flow expansion caused by the obstructions, and proposed the following correlation for relating the coefficient kf to geometric parameters of each spacer:

d k 0.002 e (2.49) f ()hdch f P s

The spacer pitch (Ps) in equation (2.49) was defined as shown in Figure 2.8.

Figure 2.8: Spacer-filled channel schematic showing the spacer dimensions used in defining the spacer

pitch, Ps (Source: [99])

The equivalent diameter (de) in equation (2.49) was defined by:

2hw d ch ch (2.50) e hwch ch

As opposed to most previous studies, which used the channel height based Reynolds number, Schock and Miquel [39] used a definition of the Reynolds number based on the hydraulic diameter of the channel as the characteristic length. The hydraulic diameter is given by four times the volume of fluid in the channel divided by the wetted surface (the area in contact with the flowing liquid):

V tot dh 4 (2.51) Sw

The hydraulic Reynolds number is then defined as:

ud Re eff h (2.52) h

The effective velocity (ueff) in equation (2.52) is the characteristic velocity of the system, and is defined as the volumetric flow divided by the effective area. The effective area is equal to the product of the height and width of the channel times the porosity (channel voidage). Therefore, the effective velocity is given by:

Q u (2.53) eff whch ch

The porosity is defined as the volume of fluid flow in the channel, divided by the volume of the empty channel (without a spacer):

V 1 sp (2.54) Vtot

Using the hydraulic diameter and the effective velocity as characteristic length and velocity respectively, the friction factor then becomes:

33 dp f hch (2.55) 2 2 uLeff

However, it must be mentioned that equation (2.55) uses the definition of the Fanning friction factor while, in their work, Schock and Miquel [39] used the definition of the Moody friction factor, which is equivalent to four times the Fanning friction factor. This choice, however, does not affect the results of their study, as the Moody and Fanning definitions of the friction factor are directly proportional and conversion from one definition to the other is trivial. In terms of the Fanning friction factor (which is the definition used throughout this thesis), they found that despite their geometric differences, the pressure drop characteristics of all of the spacers tested could be approximated by the following correlation, within the Reynolds number range of 100 <

Reh < 1000:

0.3 f 1.5575Reh (2.56)

A more complex correlation for calculating the pressure drop in a spacer-filled channel was developed by Da Costa et al. [101]. As opposed to other correlations [39, 98, 99], this one uses spacer characteristics such as mesh size and angle, and therefore can be used in designing netlike spacers. In addition, the study of Da Costa et al. [101] showed that pressure drop was mainly due to form drag, as viscous drag on the membrane walls and on the spacers was minimal at the flow conditions studied. Due to the fact that mass transfer enhancements attained due to the use of spacer meshes go hand in hand with increased energy losses [65, 86, 93], the benefit of increased mass transfer can only be quantified by means of an economic study. Thomas et al. [64] carried out a cost analysis of the impact of turbulence promoters in RO, which concluded that their effect is relatively small if velocity and tube length are optimal. However, if a specified water quality is required, water cost is reduced due to the reduction in concentration polarisation in units with promoters. Moreover, an optimised unit with promoters will always operate at a lower velocity and lower pressure drop, and will require a shorter length than one without promoters. It will also be able to maintain higher fluxes for a longer time due to a reduction in the fouling potential. These results agree with the economic assessment of Da Costa et al. [93], which showed that systems with spacers performed better than empty channels, mainly

34 due to the increased membrane costs for empty channels. Da Costa et al. [95] also analysed the economic impact of the angle of attack of the spacer, and found that spacers which did not change the direction of the bulk flow had the best performance.

2.3.2.4. Variation of geometric characteristics Belfort and Guter [94] found that different types of spacers had markedly different effects on the flow. Da Costa et al. [93] also found significant performance differences between the various spacers tested, especially when varying the characteristic angle. Kuroda et al. [99] compared the relative performances of different spacers and concluded that in order for spacer meshes to provide higher mass transfer enhancement at lower pressure losses, the mesh size must be optimised.

Figure 2.9: Illustration of the typical commercial spacer meshes tested by Da Costa et al. [93]. From left to right: CONWED-1, CONWED-2, NALTEX-124, NALTEX-51-1, NALTEX-51-2, UF1, UF2, UF3 and UF4. (Source: Da Costa et al. [93])

Li et al. [102] varied the spacer geometric parameters in order to determine their value for optimal mass transfer. They found that a spacer with an lm/hch of 4, angle of attack of 30º and an internal angle of 120º presented the best performance out of the geometries analysed. This result agreed with their numerical predictions [42]. Moreover, their experiments showed that the entry region only occupied the first 3 to 5 flow cells of the spacer-filled channel, which is considerably shorter than in empty channels. This means that the effect of the entry region for spacer-filled channels can be safely neglected.

35 Alternative spacer configurations have also been investigated. It is known that in the turbulent flow regime, free stream turbulence does not substantially increase mass transfer until turbulence levels reach a threshold value, at which point turbulent eddies penetrate the boundary layer [111]. In analogy, for unsteady flows such as those encountered in membrane applications, it is believed that vortices near the membrane wall should enhance mass transfer to a greater extent than those in the bulk. Therefore, triple layer spacer configurations have been proposed by Schwinge et al. [96] and by Li et al. [67] as alternatives to the common 2-layer spacer, for reduced pressure drop and increased mass transfer. Smaller filaments are placed near the membrane walls, while a larger filament is placed in the middle of the channel in order to support the wall filaments.

Figure 2.10: Triple-layer spacer mesh configuration proposed by Schwinge et al. [96].

Schwinge et al. [96] observed higher solvent fluxes for a 3-layer spacer at the same flow rate as traditional 2-layer spacers; despite increasing the pressure drops, the 3-layer spacer did not cover any additional membrane area, which led to lower capital and total processing costs. Balster et al. [97] tested the multi-layer spacer proposed by Li et al. [67], and found that it increased mass transfer by 20% compared to traditional spacers operating at the same power consumption.

2.3.2.5. Effect of turbulence promoters on fouling Another effect commonly associated with turbulence promoters is their ability to reduce fouling tendency. Da Costa et al. [100] found that the best spacers for reducing

36 concentration polarisation were also the spacers that best hindered fouling. However, fouling was not completely prevented, as solute is still carried towards the membrane via convection. In addition, fouling tendencies were similar for all the spacers studied, and cleaning was also unaffected by the choice of spacer. Using the direct observation through the membrane (DOTM) technique Li et al. [103] noticed that the magnitude of the solvent flux affected the manner in which particles deposited on the membrane. They also observed that particles were more likely to become attached to the membrane in locations where deposits were already present, and pointed out that flux variations on the membrane surface would lead to some places on the membrane surface being more prone to particle deposition than others.

Figure 2.11: Typical particle deposition patterns obtained using the DOTM technique [103] for the 90º and 45º attack angles of a typical commercial spacer mesh, as observed by Neal et al. [3].

Using the same technique as Li et al. [103], Neal et al. [3] studied the changes in deposition patterns and flux enhancement when varying the orientation angle of a typical spacer mesh in MF. They found better performance (higher flux enhancement and lower fouling potential) the larger the angle of attack. Nevertheless, since the spacer tested had an internal angle of 90º, positioning the spacer for the highest possible angle of attack (90º) on one membrane wall meant the opposite membrane wall had the lowest possible angle of attack (0º). Hence, they concluded that an angle of 45º was the optimum orientation for SWM modules. This differs from the results of Da Costa et al. [93], but the differences can be explained when taking into account the different processes (MF vs. UF) being analysed by these two studies, since the predominant mass transfer mechanism away from the membrane wall is not the same: surface renewal and Brownian diffusion in UF, as opposed to shear-induced diffusion and inertial lift in MF. In regards to fouling patterns, Neal et al. [3] found different patterns for the different orientations. The 90º orientation characterised by a deposition field parallel to the

37 filament and located roughly in between successive filaments. The 45º orientation showed deposition predominantly in the centre of the cell. The shape of the deposits for the 0º was less defined. The patterns for the 90º and 45º orientations are shown in Figure 2.11.

2.3.2.6. Other mass transfer enhancement techniques Other methods for destabilising boundary layer flow and thus enhancing mass transfer in membrane modules have been proposed [84]. The most significant ones are pulsatile flow [77, 78], secondary flows such as Taylor [82] and Dean [83, 84] vortices, and natural convection [106] owing to the changes in density caused by the concentration changes in the boundary layer. Hendricks et al. [104] performed RO experiments which revealed the existence of flow instabilities caused by buoyancy effects. Variations in density caused by the differences in concentration within and outside the boundary layer were the source of the buoyancy effects. These instabilities appeared to be beneficial to the performance of RO systems since they promoted mixing and worked against concentration polarisation. In light of this, Drezansky and Gill [105] analysed the mass transfer mechanisms taking place inside a tubular RO membrane, and found that free convection (i.e. buoyancy) had a noticeable impact on the mass transfer rates for laminar flows after a distance of approximately 16 tube diameters away from the tube inlet. At locations less than 16 tube diameters from the inlet, the mass transfer rates could be correctly predicted by assuming that only forced convection took place. Youm et al. [106] investigated the effect of flow destabilisations due to natural convection caused by density variations in the solution near the membrane. They observed that when the membrane is located in the top of the flow channel, natural convection instabilities caused a 3 to 5 fold in permeate fluxes for the UF cases studied. However, these instabilities are overshadowed by forced convection at Reynolds numbers higher than approximately 90. They proposed the criterion that when the ratio of Grashof to Reynolds number is above 3 for an empty channel, or above 500 for a spacer-filled channel, then natural convection instabilities due to density variations will have a significant impact on mass transfer.

38 2.4. Computational Fluid Dynamics in Membrane studies In spite of the numerous improvements to mass transfer enhancement and the increased understanding of mass transfer phenomena that experimental studies have brought about, local and time dependent phenomena occurring inside membrane units are still not fully understood. In addition, traditional experimental techniques tend to alter the flow field when trying to make measurements close to the membrane wall, and present difficulties when trying to isolate the different enhancement mechanisms. Therefore, a different approach is needed for obtaining further understanding of mass transfer enhancement in membrane separation systems. Computational techniques present potential for improving the understanding of mass transfer in membrane separation systems. They have the capability of providing information on flow conditions at any point of the geometry without disturbing the flow. Moreover, the use of numerical modelling significantly reduces the time, costs, and risks associated with running repeated experiments. The numerical technique used for simulating fluid flow is called Computational Fluid Dynamics (CFD) [6]. This section reviews the use and main contributions of CFD to the field of membrane science. CFD has become a more widely used tool in the field of membrane science [7], with more and more research groups utilising this technique in order to gain insight into the phenomena taking place inside membrane modules, to assist the design processes and improve the performance of these modules. Due to its simplicity and smaller computational demands, many researchers have opted for modelling spacer filled channels in two dimensions, as is evidenced by the amount of work undertaken in this area [5, 112-130]. As a result, CFD is now accepted as a reliable tool, and gradually three-dimensional (3D) [42, 131-134] and transient flow studies [124, 131, 133, 135] are beginning to emerge, most of them finding good agreement between predicted and experimental results.

2.4.1. Two-dimensional studies Despite the fact that the flow inside SWM modules is 3D by nature, 2D studies have provided many insights into the mechanisms affecting the mass transfer performance of real-world membrane systems. Recirculation regions forming on both the upstream and downstream side of spacer filaments have been identified [108, 113,

39 114, 121], as well as regions of high and low wall shear rate caused by these filaments [113, 114, 119]. The relationship between increased wall shear and enhanced mass transfer has also been quantified [115, 117]. Gravity has been excluded as a significant parameter for RO [123], transient responses to sudden changes in pressure have been studied [128, 129], and the effect of vortex shedding has been identified as a potential mass transfer enhancing mechanism [5, 124]. In addition, important considerations regarding the CFD solution methodology have been identified, such as the fine spatial resolution needed near the membrane wall when solving the mass transport equation [115], and a fine temporal resolution when solving for unsteady flows [5, 124]

Table 2.3: Summary of important contributions to Membrane studies utilising 2D CFD

Geometry Incorporation of Researcher Flow regime Significance analysed mass transfer Kang and Chang 2D channel, Laminar steady, Fully-developed Early numerical [108] rectangular Reh = 90–910 concentration, study of flow filaments, zigzag impermeable wall patterns and effect & cavity with constant of flow on mass concentration transfer. Pellerin et al. 2D empty channel Steady laminar Developing Early CFD study [112] and turbulent, k- concentration, incorporating turbulence model, permeable wall permeation and Reh = 240–50000 mass transfer. Cao et al. [113] 2D channel, Turbulent, RNG None, Early qualitative circular filaments, k- turbulence hydrodynamics study using submerged, model, only commercial CFD zigzag & cavity Reh = 210–840 software. Schwinge et al. 2D channel, Laminar steady Developing Effect of various [5, 114, 115] circular filaments, and unsteady, concentration, filament submerged, Reh = 90–2000 impermeable wall configurations zigzag & cavity with constant and vortex concentration for shedding. steady cases Geraldes et al. 2D empty channel Laminar steady, Developing Concentration [116-118] Reh = 500–2000 concentration, dependent permeable wall properties; calculates permeate flux. Geraldes et al. 2D channel, Laminar steady, Quasi-periodic Calculates [119-121] rectangular Reh = 79–368 concentration, permeate flux for filaments, cavity permeable wall obstructed channel. Koutsou et al. 2D channel with Laminar None, periodic Effect of unsteady [124] submerged unsteady, unit cell flow on wall shear circular filaments Reh = 174–633 hydrodynamics rate and flow only patterns. Wiley and 2D empty channel Laminar steady, Developing Models both feed Fletcher [122, Reh = 8–400 concentration, and permeate 123] permeable wall channels. Studies effect of gravity.

40 Ahmad et al. 2D empty channel Steady laminar Developing Incorporation of [125, 126] and with circular, and turbulent, concentration, permeation to triangular and RNG k- permeable wall commercial CFD square filaments, turbulence model, software, analysis cavity Reh = 100–1300 of filament profile. Alexiadis et al. 2D empty channel Unsteady laminar, Developing Effect of changes [128, 129] Reh = 2500–7500 concentration, in transmembrane permeable wall pressure on concentration polarisation.

One of the main discussion points when simulating membrane systems with the aid of CFD is the choice of boundary condition at the membrane surface [126]. While some authors have chosen to implement an impermeable wall [108, 115], others have used a permeable wall in order to account for the physical effects that give rise to concentration polarisation [116, 122, 126]. Geraldes and Afonso [136] have recently demonstrated that Sherwood numbers obtained for cases without permeation can be used to predict the wall conditions in cases where permeation is present. This is possible due to the permeate velocity being a few orders of magnitude smaller than the velocity of the bulk fluid in the channel. As a result, data obtained by either methodology can provide valuable information for understanding the processes inside membrane modules. The impermeable wall approach, also known as the “dissolving wall” approach [115], requires less computational resources and is therefore the method of choice when computing power is at a premium. However, as the magnitude of permeate flux relative to the bulk fluid velocity is increased, the assumptions behind this approach will no longer hold, and modelling will need to include the effect of permeation. Flow patterns were studied by Kang and Chang [108], who found a good correlation between the velocity gradients at the wall and the mass transfer rates; i.e. higher mass transfer rates where the velocity gradients were larger. Both their numerical and visual experiments revealed the formation of two main recirculation regions: one downstream and one upstream of each obstacle, the downstream one being the larger of the two. These findings were later confirmed by Schwinge et al. [5, 114, 115]. For the zigzag geometry, both eddies grew in size, and the centre of the larger one moved downstream, as the Reynolds number was increased (see Figure 2.12). In the cavity case, the larger downstream eddy followed a similar tendency as in the zigzag case, with

41 the difference being that the larger eddy continued to grow until it occupied the whole of the cavity region, obliterating the second smaller eddy (see Figure 2.12).

Figure 2.12: Streamlines for the cavity (left) and zigzag (right) spacer geometries, as presented by Kang and Chang [108].

Pellerin et al. [112] argued that since it is widely suggested in literature [137] that UF modules operate in the turbulent regime, a turbulence model must be used to account for the mixing effects of turbulence. However, turbulence models are only suited for high Reynolds numbers (i.e. Re > 30,000) at which turbulence can be assumed to be isotropic and fully developed [138, 139]. In addition, Pellerin et al. [112] failed to include the turbulent mixing effects on the apparent diffusion coefficient of the solute, as suggested by Mizushina et al. [140] and Rosén and Trägårdh [141, 142], and only included turbulent mixing effects on the momentum equation. In an effort to find an optimal spacer design, Cao et al. [113] used the commercial CFD code FLUENT to simulate the flow around two successive circular filaments in a narrow channel. Cavity, zigzag and submerged configurations, as shown in Figure 2.13, were investigated. They solved the time-averaged Navier-Stokes equations using the RNG k- turbulence model, and did not include concentration transport in their calculations. Since Cao et al. [113] studied the Reynolds number range of 210 to 840, which only covers the transition region, their results can only be considered as qualitative. Nevertheless, their determination of the positions of recirculation regions

42 and shear rate maxima were in good agreement with previous studies [66, 108]. The study of different spacer filament profiles conducted by Ahmad et al. [125] suffered the same drawbacks as that of Cao et al. [113] because of the use of the RNG k- turbulence model, despite the inclusion of turbulent mixing effects in the concentration transport equation.

Figure 2.13: Schematic of typical 2D spacer configurations. The configurations shown are repeated numerous times over the length of the channel.

Schwinge et al. [5, 114, 115] extended the work of Cao et al. [113] by performing steady state, two-dimensional numerical simulations with and without mass transfer, as well as transient simulations, but only without mass transfer. As opposed to Pellerin et al. [112] and Cao et al. [113], Schwinge et al. [5, 114, 115] did not utilise turbulence models, but solved the time-dependent Navier-Stokes equations. They varied the distance between spacers and the diameter of the filaments. For the steady laminar flow conditions, they found that the zigzag geometry performed better than both the cavity and submerged configurations, when taking both mass transfer enhancement and pressure loss characteristics into account. They also found that the onset of vortex shedding occurred at a hydraulic Reynolds number between 200 and 400 for the submerged geometry, and between 400 and 800 for the cavity and zigzag configurations. Evidence of vortex shedding for the zigzag geometry is shown in Figure 2.14, and agrees with the experimental findings of Kang and Chang [108], who found that the flow becomes unsteady at a hydraulic Reynolds number between 455 and 545 for the zigzag and cavity spacers.

Figure 2.14: Transition to vortex shedding for a zigzag geometry (df/hch = 0.5, lm/hch = 4; Source: Schwinge et al. [5]).

Similar work was carried out by Geraldes et al [116, 117], who performed steady- state two-dimensional laminar simulations of the flow in a slit. However, their model included a semi-permeable wall as one of the boundaries, unlike the dissolving wall employed by Schwinge et al [115], and covered a range of Schmidt numbers from 570 to 3200. Although they employed experimental values for their permeate flux, they used a model which assumes diffusional transport through the membrane to predict its intrinsic rejection coefficient. They found the hydrodynamic boundary layer thickness to be insensitive to the permeation velocity, but the concentration boundary layer thickness to be strongly correlated with the permeation velocity. They also performed simulations with the same parameters but in a channel filled with square filaments [119- 121], using quasi-periodic boundary conditions that accounted for loss of fluid through permeation. They came to the conclusion that the disruption of the concentration boundary layer has a strong effect on the concentration polarisation distribution along the channel, and therefore placing spacer filaments adjacent to both channel walls enhances mass transfer. Also using periodic boundary conditions, Koutsou et al. [124] employed a commercial CFD code (FLUENT, v. 6.0.12) to simulate unsteady two-dimensional fluid

44 flow without mass transfer around an array of submerged cylindrical spacers. They examined the effect of Reynolds number on pressure drop and wall shear, and found that for cylinder Reynolds numbers greater than 60 (hydraulic Reynolds number greater than 190) the flow becomes time-dependent, destabilised through a Hopf bifurcation. For Reynolds numbers higher than, but in the vicinity of this bifurcation, the flow becomes periodic in time, with an increasing degree of chaos as the flow increases above the destabilisation point. These results agree with the findings of Yang [135], who found that 2D obstructed flow becomes periodic after undergoing a Hopf bifurcation at a critical Reynolds number, and as Reynolds number is further increased a space-time symmetry-breaking bifurcation causes the flow to become unstable to 3D disturbances. Koutsou et al [124] corroborated their results by the frequency spectra of the flow oscillations, which showed the appearance of more harmonics as the Reynolds number increases. They also identified that the dominant frequency corresponds to the Strouhal number, the dimensionless shedding frequency. Wiley and Fletcher [122] developed an approach to take into account the conditions in the permeate channels when making calculations for the feed channel. They later utilised this approach [123] to quantify the effect of gravity on RO membrane separation performance. Their results showed that gravity only has an impact at very low cross-flow velocities, at Reynolds numbers below 20, when gravity was aligned with the bulk flow. Hence, the effect of gravity can be safely neglected in CFD simulations of RO systems for typical operational Reynolds numbers, which are in the range of 50 to 2000. However, this is not the case for UF systems, for which Youm et al. [106] have previously shown that gravitational effects are significant.

Figure 2.15: Evolution of solute concentration at the membrane surface as a response to a sudden change of inlet pressure (Source: Alexiadis et al. [128]).

45 Alexiadis et al. [128, 129] focused their study to the effect of time-dependent changes to TMP on membrane performance. They made use of data obtained from transient CFD simulations of a narrow membrane channel to develop transfer functions relating the TMP to permeate flux and solute concentration at the membrane surface. However, this is no simple task, as the dynamics of the TMP-permeate flux are highly non-linear. An increase in TMP causes an increase in permeate flux which, in turn, increases the transport of solute to the membrane, thus increasing the concentration of solute at the membrane surface. In addition, an increase in osmotic pressure caused by the increased surface concentration reduces the effective TMP, until equilibrium is reached. Simultaneously, a higher TMP caused by an increase of pressure at the channel inlet also causes the cross-flow velocity in the channel to increase which, in turn, promotes mixing and reduces surface concentration. The non-linearity of the process causes the system to go through a number of phases before reaching equilibrium, as shown in Figure 2.15. The ability to predict the behaviour of the surface concentration through transfer functions, as described by Alexiadis et al. [128, 129], provides a valuable tool for the real-time control of membrane operations, and for the mitigation of membrane fouling.

2.4.2. Three-dimensional studies Even though 2D studies have provided valuable insights into the interactions between the flow and mass transfer, SWM modules for real-world applications are 3D by nature. The work of Iwatsu et al., who simulated [143] and later analysed [144] flow in a 3D cavity, illustrates the differences encountered between 2D and 3D flows. They found that at low hydraulic Reynolds numbers (below 80) the flow on the vertical symmetry plane is qualitatively similar to the flow obtained by assuming the flow is 2D. However, for hydraulic Reynolds numbers of 800 and higher, the differences between 2D and 3D flow are significant, such that the characteristics of 3D flow cannot be extrapolated from 2D calculations, not even on the symmetry plane. An interesting finding of this work is that vorticity in the z direction (perpendicular to both the flow and the height of the cavity) is greater for the 2D case than for the 3D counterpart, due to the lateral end-wall effects. In 3D flow, the walls on the lateral sides of the cavity (parallel to the direction of the movement of the top wall) slow down the primary spanwise vortex and, as a consequence, the effect of the velocity at the top of the cavity

46 cannot reach as close to the bottom wall as in the 2D case. The analysis also concluded that 3D effects dominate the region close to the bottom wall of the cavity. Due to the fact that the extra dimension carries with it an increased computational burden not encountered in two-dimensional (2D) flow, memory and time requirements for 3D calculations of accuracy similar to previous 2D studies would have been prohibitive until recent times. Initial three-dimensional studies have been restricted to low spatial resolutions [132] or did not include mass transport [134].

Table 2.4: Summary of important contributions to Membrane studies utilising 3D CFD

Geometry Incorporation of Researcher Flow regime Significance analysed mass transfer Karode and Various common Steady laminar, None, only Early 3D CFD Kumar [132] commercial Reh = 225–2225 hydrodynamics study of spacer meshes hydrodynamics and pressure losses in membrane units. Li et al. [42] 3D channel with Laminar steady Periodic unit cell, Attempts to find circular filaments, and unsteady, impermeable wall optimum spacer ladder type spacer Reh = 441–2205 with constant mesh geometrical concentration characteristics; proposes power number (Pn). Li et al. [67] 3D channel with Laminar steady Periodic unit cell, Proposes and tests circular, twisted and unsteady, impermeable wall optimal multi- and modified Reh = 60–850 with constant layer spacer filaments, multi- concentration designs. layer spacer Ranade and 3D flat and Steady laminar None, only Effect of Kumar [133] curved channel, and turbulent, k- hydrodynamics, curvature, circular, concave turbulence model, periodic unit cell quantifies form and modified Reh = 50–1500 and viscous drag, filaments, ladder tests alternative type at 45° angle filament designs. of attack Koutsou et al. 3D channel with Unsteady laminar, None, only Effect of unsteady [131] ladder type spacer Reh = 174–633 hydrodynamics, flow on wall shear at varying internal periodic unit cell rate and flow and attack angles patterns. Santos et al. [145] 3D channel with Laminar steady Developing Effect of rectangular and unsteady, concentration, longitudinal filaments, ladder Reh =60–1000 impermeable wall filaments; type spacer with constant proposes modified concentration friction factor. Shakaib et al. 3D channel with Laminar steady, None, only Analysis of wall [146] ladder type spacer Reh =20–200 hydrodynamics, shear patterns at at varying internal periodic unit cell low Re, and and attack angles determination of critical Re at which flow becomes unsteady.

47 Karode and Kumar [132] carried out steady-state three-dimensional simulations of laminar fluid flow without mass transfer in a test cell filled with non-woven net type spacers. The spacers were simulated as cylindrical rods, whose geometric characteristics (diameter, angle, etc., as seen in Figure 2.16) were taken from commercially available spacers for membrane modules. They modelled the whole test cell with a flat velocity profile at the cell entrance and constant pressure at the outlet. Depending on the characteristics of the spacers, between 8 and 12 filaments were placed in the flow domain. They concluded that a higher degree of mixing in the bulk does not necessarily translate to higher shear at the membrane walls. Their simulations showed a similar zigzag path of the bulk of the fluid to the one observed experimentally by Da Costa et al [101], but only for spacers with large inter-filament distance to filament diameter ratios, which agrees with the results of Shakaib et al. [146]. However, for spacers with low inter-filament distance to filament diameter ratio, they noticed that the bulk of the fluid flows parallel to the spacer filaments, changing direction only when the flow reaches the lateral walls of the test cell. In this latter case, they found that most of the pressure drop across the channel was due to the shear between the top and bottom layers of fluid moving in different directions.

Figure 2.16: Schematic of a 3D ladder spacer with circular filaments, illustrating the attack (;) and

internal () angles and inter-filament distances (l1 and l2) (Source: Li et al. [42]).

Yuan et al. [147] pointed out that for periodically disturbed duct flow with heat transfer, entrance effects are much shorter than for flows without the disturbances. Often, the heat and fluid flow become fully developed after about 5 repetitions of the periodic section. Since heat transfer units may consist of dozens or even hundreds of these repeating sections, the analysis of just one periodic section can give a clear picture

48 of the transfer phenomena dictating the overall performance of the unit. Therefore, for this and other types of spatially-repeating geometries many heat transfer studies [81, 147, 148] have utilised a fully developed temperature profile, i.e. a spatially-periodic heat transport boundary condition. These concepts can be extrapolated to mass transfer by utilising a fully developed mass fraction profile as a boundary condition. Ranade and Kumar [133] used a periodic “unit cell” approach to model rectangular and curvilinear three dimensional spacer filled channels, and found that curvature did not influence fluid behaviour significantly, unlike the shape of the spacer filaments (see Figure 2.17). Although they also included mass transfer effects in their calculations, it is unclear how they coupled this to the periodic boundary condition of their unit cell, in particular for their transient simulations. In addition, their results regarding the lack of influence of curved channels on fluid flow differs from many experimental [149, 150] and numerical studies [83, 151] of conduits without spacers, which show an enhancement of membrane performance due to curvature by means of secondary flows, such as Dean vortices. This discrepancy can be explained by considering that the effect of the spacers on fluid flow is larger than the effect of secondary flows due to curvature [39, 152].

Figure 2.17: Modified spacer filament types simulated by Ranade and Kumar [133].

In order to obtain better spatial resolution, Li et al. [42] utilised a spatially- periodic cell to model a non-woven spacer geometry with filaments parallel and perpendicular to the bulk flow. Spatially-periodic boundary conditions are common in

49 3D fluid mechanics studies [153, 154], and have also been used in 2D studies [124] to reduce computational requirements and increase spatial resolution. The calculations of Li et al. also included the mass transfer for a solute with a Schmidt number of 1278, and for hydraulic Reynolds numbers from 441 to 2205. However, the method used for the implementation of the periodic boundary condition for solute transport was not specified. In the geometry analysed, vortex shedding occurred for hydraulic Reynolds numbers greater than 662, which agrees with previous 2D studies [5, 108]. They concluded that the optimal geometry for non-woven spacers is at lm/hch = 4, which maximises the Sherwood number while keeping energy losses due to pressure drop low.

Figure 2.18: Modified multi-layer spacer meshes tested by Li et al. [67].

In a later work using the same mass transport model, Li et al. [67] assessed different conventional spacer configurations as well as novel spacer types (see Figure 2.18) by comparing their Sherwood and power numbers. Based on work for heat transfer, they proposed that an optimal spacer mesh should promote both longitudinal and transversal vortices. A multi-layer type spacer was found to be promising. However, their CFD and experimental results for the novel spacer designs presented

50 were not in good agreement; they concluded that the CFD simulations for those geometries are unreliable, despite their lack of spatial and time resolution information. More recently, Koutsou et al. [131] utilised Direct Numerical Simulations with spatially-periodic boundary conditions to analyse 3D spacer meshes in both the steady and the transient regimes, with cylinder Reynolds numbers ranging from 35 up to about 300. A range of spacer geometries with varying internal angle and mesh length to diameter ratio was analysed, though none with filaments parallel or perpendicular to the bulk flow direction. As opposed to 2D flow, the flow in 3D does not move in closed streamlines but in spiral paths along the spacer filaments. They also report the existence of a central free vortex, and that the interaction of this free vortex with the vortices attached to the filaments generates regions of closed recirculation. In their studies, they encountered the transition to the transient regime in the cylinder Reynolds number range of 35 to 45. Santos et al. [145] extended the work of Geraldes et al. [119-121] on rectangular spacer filaments to 3D flows. In doing so, they analysed the effect of longitudinal filaments on the 3D flow profile and found that it was minimal. They also found a direct correlation between the membrane wall distribution of a “modified” friction factor , 2 ( flocfRe loc ) and the local Sherwood number distribution. Their results were in excellent agreement with their experimental data, and correctly predicted the transition to unsteady flow at a hydraulic Reynolds number between 200 and 420, depending on the geometry, which agrees with the findings of Shakaib et al. [146] for a similar spacer type. However, since their concentration profiles were not periodically wrapped at the end of the simulated channel, their mass transfer results are only valid for the entry region and are not representative of the average conditions in a membrane module.

2.5. Conclusion The general understanding of the processes taking place during pressure-drive membrane separation operations has come a long ways since the early work of Loeb and Sourirajan [2]. Concentration polarisation has been identified as one of the main problems reducing the performance of membrane separation units [43-45]. Many numerical [11, 58, 59, 108, 115] and experimental [65, 66, 86, 87, 111] studies have helped identify the main mechanisms that reduce the effect of concentration polarisation and enhance mass transfer, therefore increasing the performance of membrane units

51 such as SWM modules. Novel means for enhancing the performance of these units have been proposed and tested [67, 96, 97]. Recently, CFD has become an widespread tool used for the numerical study of the conditions inside membrane modules [7]. Although the analogous heat transfer problem of flow in narrow conduits with obstructions has been the focus of much research [68], the mass transfer phenomena encountered in membrane operations are of a much smaller scale as a consequence of the higher Schmidt number, which can be several orders of magnitude higher than the typical Prandtl number encountered in heat transfer applications. This also makes adequate numerical resolution of concentration boundary layers particularly difficult and, until recent times, placed prohibitive computational costs on their calculation. Despite the numerous CFD studies of the behaviour of fluid flow in spacer-filled channels, the transient nature of mass transfer enhancement in unsteady flow, especially in the near wall region, is not yet fully understood. Given that membrane modules operate in a flow regime where unsteady flow may be present, it is important to understand the effect of transient flow. This thesis aims firstly to use CFD to simulate and allow the analysis of the dynamics of fluid flow and mass transfer in narrow spacer filled channels such as those encountered in SWM modules. Thus, Chapter 4 of this thesis extends the work of Schwinge et al. [5] to incorporate mass transfer in unsteady flows. Since high concentration gradients in the near-wall region require a large number of grid nodes to achieve suitable convergence of the mass transport equation in that region, Chapter 4 focuses on a two-dimensional case as an approximation to more complex geometries. This approach not only provides insights into the fluid behaviour inside real modules but also allows us to establish some baseline conditions in order to develop a suitable approach to 3D modelling. Three-dimensional CFD studies have either focused on the hydrodynamics of flow in spacer-filled narrow channels neglecting mass transfer altogether [131-133, 146], or focused on spacer performance without analysing the mechanisms giving rise to mass transfer enhancement in 3D flow [42, 67]. Only the recent study of Santos et al. [145] has made an attempt to analyse the mass transfer enhancement mechanisms characteristic of 3D flow, but their work only covered the case of an attack angle of 90°. In order to tackle these issues, Chapter 5 presents a description of the spatially-periodic mass transport boundary condition, and its application to the analysis of the flow features and performance of a simple 3D non-woven spacer filament mesh. In principle,

52 it extends the work of Schwinge et al. [115] to 3D geometries, focusing on the effects of form and viscous drag on mass transfer. Moreover, it explores the connection between mass transfer enhancement and flow features exclusive to 3D flow under steady flow conditions. Finally, in Chapter 6, insights gained from the results of Chapters 4 and 5 will be used to propose novel spacer configurations for enhancing mass transfer with a minimal impact on energy loses, and these proposed spacer designs will be tested using CFD.

53 Chapter 3

Methodology

This chapter presents the concepts and fundamentals behind Computational Fluid Dynamics (CFD), which is the main technique used for obtaining the data analysed in this thesis. In addition, it explains the procedure used for transforming the partial differential equations that describe fluid dynamics into algebraic equations that can be solved via numerical methods. Lastly, the methodology for ensuring that the results obtained are reliable and realistic is described.

3.1. CFD Theory The following section gives a brief introduction to the basic theory behind CFD. The main transport equations are presented, as well as the methods used for solving them in the CFD software package used in this thesis: ANSYS CFX.

3.1.1. Transport Equations Transport processes can be described by a set of equations generally known as the Navier-Stokes equations [155]. These equations can be obtained from mass, momentum and energy balances [9], and are partial differential equations which have analytical solution only for simple cases [156, 157]. For more general flows involving complex geometries and/or boundary conditions, it is possible to solve the system of Navier- Stokes equations making use of numerical methods [6]. CFD is the name given to techniques used to numerically solve the partial differential equations of continuity, momentum, energy and species transport. The general form of a transport equation in vector notation is [6]:

() .()v .() . S (3.1) t

In equation (3.1), the symbol represents any transported quantity, which could be a scalar (e.g. temperature, mass fraction, etc.), a vector (e.g. velocity, the turbulent flux of a scalar, etc.), or a second order tensor (e.g. Reynolds stress tensor). is the diffusion coefficient of ; S is the generation of by a source, or consumption of by

54 a sink. The first term represents the accumulation of , the second term represents the transport of due to convection (i.e. due to the velocity of the fluid), and the third term represents the transport due to diffusion. Any transport equation can be rearranged into the form of the general transport equation. Previous work by Wiley and Fletcher [122, 123] has shown that for CFD simulations of flow in narrow channels such as those found in SWM modules, neither gravity nor density variation have a significant effect on the solutions obtained. Therefore, for all the simulations carried out throughout this thesis constant density is employed and the effect of gravity was excluded. In addition, and in order to simplify the complex numerical problem and isolate mass transfer enhancement effects due to the hydrodynamic characteristics of the flow in narrow channels, constant properties are used and the fluid is assumed to be Newtonian and isothermal. SWM modules typically operate at Reynolds numbers below the transition to the turbulent flow regime [65]. Although unsteady flow conditions are not uncommon in this type of membrane systems, the time variations encountered in those cases are nevertheless laminar in nature, as they do not present the chaotic variations commonly associated with turbulence [5, 65]. Hence, it is possible to simulate the unsteady flows encountered in SWM modules by directly solving the transport equations, without the need to employ turbulence models. The continuity equation for constant density fluids can be expressed in Eulerian form by: .v 0 (3.2)

The momentum transport equation for an incompressible Newtonian fluid is as follows [9]:

() v FVT .()vv .HX () . v . v .p F (3.3) t

The species transport equation is [9]:

()Y i .()Y vJ . S (3.4) t iii

55 These equations are valid at every point of the flow field; however, they need to be solved for given boundary conditions, which are problem specific. All the boundary conditions used in the simulations presented in this thesis, as well as their validity, are discussed in the problem description section of each of the chapters involving CFD simulations. There are many numerical methods available to solve a system of partial differential equations [158]. The most widely used are the finite difference, finite volume, finite element, and spectral methods. They all involve a transformation of the system of differential equations to a system of algebraic equations, which are subsequently solved numerically. Most commercial CFD codes (including ANSYS CFX, the software used for this thesis) use the finite volume method [6], which consists of dividing up the flow domain into small volumes or elements, integrating the transport equations over these control volumes, discretising the partial derivatives (thus converting the equations into an algebraic system), and finally solving this system of algebraic equations by means of an iterative numerical method.

3.1.2. Mass Transport Mass transport in CFD is incorporated by means of the species transport equation (3.4). Before this equation can be solved by the Finite Volume Method, an expression for the mass diffusive flux is needed. ANSYS CFX makes use of the multiple component extrapolation of Fick’s law for calculating the diffusive fluxes of species. The species transport equation solved by ANSYS CFX is, therefore:

()Y i .()(YDYv . . ) S (3.5) t iimii

Equation (3.5) has the form of the general transport equation described by equation (3.1), and can therefore be solved by the Finite Volume Method. This is the form of the species transport equation used for most cases in this thesis, in which only one dissolved species is simulated. However, due to the inability of Fick’s law to correctly describe the cases with multiple interactive species diffusion, equation (3.5) must be modified when dealing with systems with multiple dissolved ionic species. This topic is discussed in Appendix A.

56 3.1.3. The Finite Volume Method This section describes the application of the Finite Volume Method for the solution of the Navier-Stokes equations as employed by ANSYS CFX [159]. In the following sections, the discussion explicitly addresses the transport of a scalar . However, one must keep in mind that the formulations presented here are also applicable to the Cartesian components of vectors or elements of a higher order tensor, which in turn are scalars. Integration of the general transport equation (3.1) over a control volume (%) can be expressed for a scalar as:

() dd% .() v % .() . d %Sd % (3.6) OO O O %%t % %

At this point it is convenient to introduce Gauss’ Divergence Theorem. This theorem states that the volume integral of the divergence of a vector over a control volume is equal to the surface integral of that vector over the area that encloses the control volume [160]. This can be expressed by: OO().%FFd dA (3.7) % A

Applying equation (3.7) to the second and third terms in equation (3.6) and rearranging yields:

() vA . dSd %() d % (3.8) OOO A %%t

Figure 3.1. Control volume used to illustrate discretisation of a transport equation [159].

57 Under the Finite Volume Method, equation (3.8) must hold for every division of the flow domain. These divisions are called control volumes. A general control volume in 2D can be represented as shown in Figure 3.1. In order to convert the system of partial differential equations to a system of algebraic equations, each term in equation (3.8) needs to be discretised. For that purpose, the volume integrals are broken up by sectors, and the surface integrals are evaluated at integration points located at the centre of each surface segment, as shown in Figure 3.2.

Figure 3.2. Mesh element used to illustrate discretisation of a transport equation [159].

The discrete form of the volumetric terms in equation (3.8) is obtained by calculating an approximation to their value in each sector, and then integrating those values for the sectors that contribute to a control volume. Likewise, surface terms are discretised by approximating the value of the necessary variables at the integration points, and integrating those values for the surfaces of each control volume. The method used to obtain the approximations will depend on the discretisation scheme, and will be discussed in section 3.1.4. The discrete form of equation (3.8) is as follows:

FV() () B FV vA () . S % % GWend start (3.9) HXip ip ip , ip ip ip cv cv ip HXGWt

In equation (3.9), the first term in the summation is the advective term, and the second term is the diffusive term. The term Aip refers to the discrete surface vector pointing away from the node at the centre of the control volume, and Vcv is the volume

58 of the control volume. For the time discretisation, the start and end of timestep values depend on the discretisation scheme used. In ANSYS CFX, the values of and its diffusion coefficients are stored at the nodal points. This is a type of co-located (non-staggered) grid layout, since both the velocity and pressure values are stored at the same locations. Patankar [161] showed that co-located methods can lead to what is known as a “checkerboard” pressure field. In order to avoid this effect, ANSYS CFX [159] utilises an interpolation scheme similar to that proposed by Rhie and Chow [162] and later modified by Majumdar [76] for obtaining the values of the advective velocity at the integration points in equation (3.9).

3.1.4. Discretisation schemes For the unsteady simulations in this thesis, the Second Order Backward Euler 1 scheme is used, which assumes the start and end of the timestep to be at t 2 t and

1 t 2 t respectively. This scheme utilises the values at the two previous timesteps for interpolating the values at the start and end of the current timestep. This is done by applying the following approximations:

() () 1 FV() () (3.10) start tt 2 HXtt t2 t

() 1 FV () (3.11) end 2 HXt t

All the values for the variables in equations (3.9), (3.10) and (3.11) are for the current timestep being solved, unless otherwise noted. The Second Order Backward Euler scheme for time discretisation avoids the truncation errors associated with the First Order Backwards Euler scheme, while retaining the qualities of being robust, implicit and conservative with respect to time. As evidenced by equation (3.9), the values of several variables are needed not only at the nodal points, but also at the integration points. ANSYS CFX makes use of shape functions (Ni) analogous to those used in Finite Element methods [158], in order to approximate the solution and its variation within each element. The shape functions utilised are linear in terms of parametric coordinates within each element, and can be found in [159]. The value of a variable within a mesh element is then obtained from:

59 nmax B Nii (3.12) in 1

Variable gradients can be calculated in a manner similar to that used in Finite Element methods [158], by making use of these shape functions. Thus, the gradients needed at the integration points for the diffusive term in equation (3.9) can be calculated from the following expression:

nmax (). ( .N ) (3.13) ipB i ip i in 1

Shape functions are particularly useful for the calculation of the volume integral of the pressure gradient in the momentum transport equation. According to the Green- Gauss theorem [163]: OO.%pd pdA (3.14) % A

Discretisation of equation (3.14) gives the following expression for the integral of the pressure gradient over the control volume, which is then calculated by making use of the shape functions: .% O pdB pipA ip (3.15) % ip cv

Variable values at the integration points for use in the advective term in equation (3.9) can also be calculated by making use of the shape functions represented by equation (3.12). This practice is analogous to the central differencing scheme [6], which is second order accurate in regards to Taylor series truncation error. However, it is known to produce unbounded solutions and non-physical wiggles that may prevent the numerical procedure converging [159]. The scheme will only be stable and accurate if the control volume Peclet number is less than 2 [6], where the control volume Peclet number is defined as follows:

v l cv Pecv, (3.16)

60 Due to the stability problem associated with central differencing, this differencing scheme is rarely used. A more adequate scheme for high cell Peclet number flows is the upwind differencing scheme. The first order upwind scheme assumes that the nodal values hold through the entire cell, and sets the face values equal to the nodal values in the upstream cell. The second order upwind scheme calculates the integration point values by taking the scalar product of the gradient of the quantity and the displacement vector between the upstream cell and the cell face, and adding it to the value at the upstream cell [159]. The first and second order upwind schemes values of at that face are calculated by equations (3.17) and (3.18) respectively.

ip up (3.17)

.() s (3.18) ip up up up, ip

Due to its simplicity, calculations using the first order upwind scheme are faster than those using its second order counterpart. However, upwind discretisation schemes introduce numerical error to the calculations. This error comes from the truncation error in the Taylor expansion of the upwind scheme, which is proportional to the second derivative, and adds to the diffusion coefficient. This inaccuracy is generally referred to as “false diffusion”, and its magnitude is higher if the grid is not aligned with the flow [6]. Second order discretisation reduces the numerical diffusion and, although slightly slower, it provides more accurate results, especially on unstructured meshes. Nonetheless, the second order upwind differencing scheme also introduces discretisation errors, which may lead to non-physical oscillations near the locations of steep gradients [159]. The hybrid differencing scheme [6] was developed in an attempt to combine the advantages of the central scheme at low cell Peclet numbers, with those of the upwind scheme at high cell Peclet numbers. It applies the central differencing scheme for those cells whose absolute value of the cell Peclet number is less than 2, and uses first order upwind differencing for the rest of the cells. In general terms, this scheme proves very useful and stable, its only drawback being the fact that it is only first-order accurate in terms of Taylor series truncation error.

61 ANSYS CFX makes use of a type of hybrid differencing scheme for the advective term of equation (3.9), called the “High Resolution Scheme” [159]. Unlike the hybrid differencing scheme described previously, the High Resolution Scheme computes a blending parameter () based on the boundedness principles described by Barth and Jesperson [164]. The blending parameter is chosen to be as close to 1 as possible while avoiding the generation of non-physical oscillations. Variable values at the integration point are then calculated by:

.( ) s (3.19) ip up up up, ip

3.2. Verification and Validation Verification is the process of comparing a computational model with the conceptual model it is meant to represent, whereas validation is the process of comparing it to the real world it is trying to model (see Figure 3.3). According to Oberkampf and Trucano [165] verification and validation (V&V) of numerical models is the main way to make sure that results obtained from simulations are to be trusted; they also state that verification deals with the mathematics of the model, while validation deals with its physics.

Figure 3.3. Phases of modelling and simulation and the role of V&V [165, as in 166].

In the CFD environment, the purpose of verification is to ensure that the computational code is accurately solving the system of partial differential equations that the transport equations defined, and not a different set of equations brought about by numerical errors. The five major sources of these types of errors are [165]: Insufficient spatial discretisation convergence

62 Insufficient temporal discretisation convergence Insufficient convergence of an iterative procedure Computer round-off Computer programming errors.

When utilizing a commercially available CFD package the last source of error cannot be corrected by the user, but still represents a potential source of error. Therefore, the user only needs to focus on the first four. Round-off error can be reduced by utilising double precision for real numbers. Insufficient spatial resolution is eliminated by making sure that the solution does not change as more grid points are introduced. Insufficient temporal resolution is tackled by reducing the time step in unsteady simulations, and insufficient convergence of the iterative procedure is dealt with by reducing the minimum allowed iteration residual. Often, the main result of a CFD simulation is the calculation of an integral function, such as a drag coefficient or an outlet mass fraction. Grid independence, which is related to verification, can be checked by making use of the Grid Convergence Index (GCI) to estimate the amount of error associated with the grid. The GCI is based on a generalization of Richardson’s extrapolation [167] and was developed by Roache [168]. If the refinement ratio and relative error are defined by:

N fine (3.20) Ncoarse

e coarse fine (3.21) fine

then the GCI for the fine and coarse grids can be calculated by [167]:

3 e GCI (3.22) fine 1

3 e GCI (3.23) coarse 1

In a few cases where the “correct answer” is available, verification can be carried out by comparing the solution of the numerical model against this benchmark. Such

63 solutions, however, are rarely available. For membrane systems, analytical and semi- analytical solutions only exist for 2D unobstructed round and flat channels (applicable to hollow fibres and SWM) with [50, 54-56, 58, 59, 61] and without permeation [35]. Only semi-empirical solutions are available for cases with flow obstructions [11, 46, 63]. Validation of a numerical model consists of comparing the computational solution with experimental results. Although this is often assumed to be a straightforward process, a series of questions must be raised before this comparison is made [169]: Are the conditions of the simulation the same as those of the experiment? Do the simulation boundary conditions match those of the experiment? Are the boundaries in the correct location? Are the experimental data reliable?

The strategy, according to Oberkampf and Trucano [165], is to first identify and quantify or estimate the error in both the numerical and experimental results, and then compare the actual data. The fact that experimental measurements may not be more accurate than their numerical counterparts must also be considered. Once these questions can be answered affirmatively, then validation can proceed. In this thesis, data generated through CFD are validated against the experimental and numerical data available in literature. Although a large number of experimental studies are reported in the literature [65, 66, 86-94], the geometries analysed in those works often differ significantly from the cases studied in this thesis, such that a strict quantitative validation is not possible and only a qualitative comparison can be made. Among the variables available for comparison during validation are the friction factor [39, 114], Sherwood number distributions [91, 92, 115], Reynolds number at the onset of unstable flow [5, 108], and vortex location and shedding patterns [5, 108, 124, 131], among others.

64 Chapter 4

Unsteady Flows with Mass Transfer in Narrow zig-zag Spacer- Filled Channels

4.1. Introduction Despite the number of CFD studies of the behaviour of fluid flow in spacer-filled channels, the transient nature of mass transfer enhancement in unsteady flow, especially in the near wall region, is not yet fully understood. Given that membrane modules operate in a flow regime where unsteady flow is likely to be present, it is important to understand the effect of transient flow. This chapter uses CFD to simulate and analyse the dynamics of fluid flow and mass transfer in narrow spacer filled channels such as those encountered in SWM modules. The effect of the Reynolds number on the flow regime and mass transfer are also assessed. This chapter extends the work of Schwinge et al. [5] who studied two-dimensional unsteady flow without mass transfer and steady flow with mass transfer. In the latter type of flow, the high concentration gradients in the near-wall region require a large number of grid nodes to achieve suitable convergence of the mass transport equation in that region. Therefore, the focus of this chapter is on a two-dimensional case as an approximation to more complex geometries. Moreover, a dissolving wall approach to mass transfer is utilised, following the work of Schwinge et al. [115] for steady state flows with mass transfer. Such an approach not only provides insights into the fluid behaviour inside real modules but facilitates the establishment of baseline conditions in order to develop a suitable approach to three- dimensional modelling.

4.2. Problem Description, Assumptions and Methods The commercial CFD code ANSYS CFX-10.0 is used to solve the continuity, momentum and mass transport equations [9] in a spacer-filled channel. For the reasons outlined in chapter 3, constant properties are employed and the effect of gravity is excluded. In order to further simplify this system the fluid is assumed to be Newtonian, the flow two-dimensional, and a binary mixture of water and salt is considered, with no sources of salt in the fluid. Mass transfer is incorporated in the form of a dissolving wall

65 boundary condition. Although a more accurate boundary condition for real life membrane applications would be a permeable wall [123, 127], the dissolving wall approximation is still capable of providing valuable insights into the transient mass transfer phenomena taking place within the boundary layer, without adding complexity to an already large computational problem. Schwinge et al. [5] indicated that large computational times (between 50 and 150 hours on a NEC SX5 supercomputer) and memory (more than 2 GB) were required for each simulation with no mass transfer. When mass transfer is included, even larger convergence times would be anticipated, since an additional partial differential equation needs to be solved. The simulations for this chapter were completed on an x86 Linux cluster, and the computational times were between 150 and 400 hours per run (depending on the Reynolds number). Due to the large computational times required for each parametric run, and given that the purpose of this chapter is to study the mass transfer mechanisms encountered in membrane modules and not the optimisation of spacer design, only one geometry type was modelled. The geometry analysed is the zigzag spacer arrangement [5] with df/hch =

0.5 and lm/hch = 4. This geometry was chosen because it presents the most similarities with spacers used in real membrane modules, and was also found to be better performing than the other geometries studied by Schwinge et al. [115], in regards to its mass transfer and pressure loss characteristics. Long entrance and exit lengths were necessary to fully develop the flow before the beginning of the spacer array, and to avoid the outlet condition to interfere with the recirculation regions after the last filament. Preliminary simulations revealed that at least six spacer filaments were necessary for vortices to appear in unsteady flow, and that the flow around the last spacer filament differed from the rest of the filaments due to the exit region. In light of this, an array of 10 cylindrical spacers, preceded by an entrance length of 20 times the filament diameter and followed by an exit length twice as long as the entrance length was chosen as the flow domain. The entrance and exit length dimensions were chosen following the considerations stated by Schwinge et al. [115]. Each set of two successive spacers comprises a “unit cell” for this array, as seen in Figure 4.1. The second last unit cell, which consists of the 7th and 8th filaments from the channel entrance, was chosen as the best representation of the flow in a spatially

66 periodic channel, which would in practice consist of over a hundred filaments per module. This unit cell is far enough downstream so that vortices are being shed, and far enough in front of the exit region so as to avoid any interference of the outlet. Measurements of wall shear, mass transfer coefficient, velocity vectors and other monitored variables were made mostly in this unit cell, and the results presented in this chapter focus on this region of the computational flow domain.

Figure 4.1: Geometry of the spacer unit cell. This pattern is repeated 5 times in the computational fluid domain.

At the inlet of the flow domain, a flat velocity profile with u = uavg, v = 0 and Y = 0 is specified. The channel walls over the entrance and exit lengths as well as the spacer surfaces are treated as non-slip walls with no mass transfer, where both velocity components and the mass fraction gradient normal to the boundary are set to zero (u = v = 0 and Yy0 ). The membrane walls in the unit cells are also treated as non-slip walls (u = v = 0), but the mass fraction at the wall is fixed at a constant value (Y = Yw). At the outlet, an average reference pressure of zero is specified. Given that only the pressure gradient appears in the momentum transport equation, as opposed to the value of the pressure, the reference pressure is therefore hydrodynamic, and not thermodynamic [123]. This chapter follows the work of Schwinge et al. [5] and therefore uses the hydraulic Reynolds number definition used by Schwinge et al. [114] and Schock and Miquel [39]. The hydraulic Reynolds number was varied between 210 and 1683. A Schmidt number of 600 was used in this chapter, which is the same used by Schwinge et al. [115]. This Schmidt number is characteristic of typical monovalent salts, such as sodium chloride, which are commonly encountered in many membrane applications, particularly in RO and NF. Mesh independent solutions were obtained using an unstructured mesh with element sizes of the order of 10% of the filament diameter in the bulk of the channel,

67 and 5% of the filament diameter in the vicinity of the spacer. This was determined after a series of runs with increasingly finer meshes. As a high solute Schmidt number leads to a thin concentration boundary layer [141], it was of the utmost importance that the boundary layer was properly resolved. Inflated boundaries were used for all the boundaries with a non-slip condition, where the thickness of the first grid element layer was of the order of 0.3% of the filament diameter. Angular resolution on the surfaces of the filaments was set to 17, and to 10 for the “corners” between the filaments and the membrane wall. Schwinge et al. [5] reported that time steps smaller than 0.1 ms were needed for Reynolds numbers in range 400 to 800, and that they had to be reduced to as small as 1 s for the range 800 to 2000. Koutsou et al. [124] used a dimensionless time step -3 (tustavgf d) smaller than 1.5×10 for all of their simulations. The time steps used in this chapter were determined by decreasing the time step size until the solution did not depend on this parameter. Dimensionless time steps smaller than 8.9×10-3 were employed for all but the highest Reynolds number simulated in this chapter, for which the dimensionless time step was 8.6×10-4. An approximate steady-state solution was chosen as the initial state (t = 0) for the transient simulations. This was selected on the basis that the mass transfer process is very slow relative to the momentum transfer process. If the initial state had been chosen as zero velocity and concentration it would have taken more time steps to reach a state where the statistical quantities of the transient simulation stabilized. In particular, the concentration time series would have steadily increased throughout the simulation. Using the initial condition chosen here, the statistical quantities of all the variables stabilized after approximately two residence times.

4.3. Results and Discussion

4.3.1. Steady flow For the simulations up to a Reynolds number of 526 the flow was steady, with no variation of the flow variables at any point in the channel as time progressed. For these steady simulations, the calculated flow was the same as that described by Schwinge et al. [114], where the zigzag pattern of the spacers forces the flow to follow that same pattern. Small recirculation regions were observed upstream of each filament and larger

68 recirculation regions were located downstream of each filament. At a Reynolds number of 526, as seen in Figure 4.2, the larger recirculation region reattaches to the membrane wall just upstream of the next filament. As noted by Schwinge et al. [115], concentration was higher within the recirculation regions than in the bulk flow due to the flow recirculation. Development of the concentration boundary layer was disturbed by the higher shear caused by the reduction of the channel cross section due to the filament on the opposite wall.

Figure 4.2: Streamline (top), solute mass fraction (middle), local friction factor and Sherwood number plots along the bottom channel wall at a Reynolds number of 526, showing separation (S) and reattachment (R) points.

As seen in Figure 4.2, the maximum local Sherwood number along the membrane wall is located near the point of reattachment of the larger recirculation region, and the minima are located at the separation point of both recirculation regions. Since mass transfer at the membrane wall is directly proportional to the local Sherwood number, these are also the points of maximum and minimum mass transfer, respectively. This suggests that separation points are associated with regions of low mass transfer, as are reattachment points with regions of high mass transfer. At a point of reattachment there is flow of low concentration fluid towards the channel wall, where a high salt concentration is located. At the same time, the direction of the flow within the boundary layer is away from the point of reattachment. These two

69 behaviours combine to create a low salt concentration and, therefore, a high salt concentration gradient at the point of reattachment. Likewise, at a point of separation the high concentration at the membrane wall means there is convection of high concentration fluid along the boundary layer towards that point, and then away from the wall, towards the bulk. Therefore, a high salt concentration region is formed at the reattachment point, which in turn slows diffusion from the wall. The boundary layer can be thought of as a layer of fluid near the boundary in which only molecular transport is taking place on the plane perpendicular to this layer, and transport due to convection in this direction is negligible [170]. Using this definition, the velocity and concentration boundary layers can be calculated respectively by:

u eff (4.1) vel CSu abs DT EUy w

YY bw (4.2) conc CSY DT EUy w

The dimensionless boundary layer thicknesses along the bottom channel wall can be seen in Figure 4.3. Comparing Figures 4.2 and 4.3, it can be seen that the regions of high shear rate correspond to regions where the developing concentration boundary layer is thinner. In particular, the shear rate maxima are located at the points on the channel wall opposite the filaments. From equation 4.1 it can also be seen that the velocity boundary layer thickness is inversely proportional to the wall shear rate. Moreover, a thinner velocity boundary layer allows the lower concentration bulk flow to come closer to the wall. Given that the shear maxima are located just downstream of the reattachment point, this reasoning explains why the region of high mass transfer is shifted slightly downstream of the reattachment point, where the velocity boundary layer is thinner.

Figure 4.3: Dimensionless concentration and velocity boundary layer thickness along the bottom channel wall at a Reynolds number of 526. The thicknesses were calculated using equations 4.1 and 4.2.

4.3.2. Moderately unsteady flow

Table 4.1. Hydraulic Reynolds numbers reported in literature for transition from steady to unsteady flow in narrow spacer-filled channels.

Hydraulic Reynolds number Researcher Geometry at Transition 2D zigzag rectangular spacers, Kang and Chang. [108] 455 < Reh < 545 lm/hch = 5, df/hch = 0.5 3D ladder type spacers, Li et al. [42] Reh > 662 lm/hch = 4, df/hch = 0.5 2D Submerged cylindrical spacers, Koutsou et al. [124] Reh > 190 lm/hch = 3, df/hch = 0.5 2D zigzag cylindrical spacers, Schwinge et al. [5] 400 < Reh < 800 lm/hch = 4, df/hch = 0.5 2D zigzag cylindrical spacers, This work 526 < Reh < 841 lm/hch = 4, df/hch = 0.5

At a Reynolds number of 841 the flow is unsteady. From Table 4.1 it can be seen that the transition from steady to unsteady flow for the type of geometry modelled in this chapter falls within the common ranges found in literature. Vortices form behind each filament and at the membrane walls, and move downstream with the bulk flow. As opposed to the case at lower Reynolds numbers where the flow was steady and there were only two recirculation regions located around each spacer (a smaller one upstream

71 and a larger one downstream), at this Reynolds number there can be as many as three vortices between successive filaments at any one time, both upstream and downstream of each spacer along the membrane wall.

Figure 4.4: Paths followed by the “eye” of shed vortices at Reynolds number 841 for the fourth unit cell of the computational domain.

Figure 4.4 depicts the paths that the “eyes” of the shed vortices take through the channel. These paths were recorded over a period of approximately 0.6 s. It can be seen that there are mainly two locations where vortices are formed: behind a spacer, and at the wall approximately two thirds of the way between successive spacers. Two vortices with opposite rotation form behind each filament; however, the vortex that is formed closer to the wall is smaller in size than the one near the centre of the channel, and the former usually decays or is forced against the wall by the latter. It should also be pointed out that although the paths followed by eddies closer to the centre of the channel are similar, they all differ slightly. This last observation is evidence that the flow has been through more than one bifurcation, as it has lost its time periodicity. Figure 4.5 shows snapshots of the velocity vectors in the channel at successive time steps 6.1 ms apart, and numerically identifies the shed vortices. As opposed to Figure 4.4, Figure 4.5 shows the position of the vortices relative to each other at different time steps. The smaller vortices are not labelled in Figure 4.5 because the velocity of the circulating fluid is too small compared with the larger vortices, and cannot be distinguished in a vector plot of the scale used. The reader is also referred to video V4.4, located in the disc in the back cover of this thesis (see Appendix B), which depicts the evolution of the velocity vectors in this channel section at a Reynolds number of 841. The shedding pattern shown in Figure 4.5 and video V4.4 is similar to the one described by Schwinge et al. [5] for this spacer configuration. Vortices are formed at approximately regular intervals behind the spacer filaments (~12.2 ms) and at the membrane walls (~24.4 ms). It is interesting to notice how successive shed vortices

72 behind the upstream spacer “merge” or “coalesce” before continuing downstream, as is evidenced by vortices 1 and 5, as depicted in Figure 4.5. However, such coalescence is not observed for every pair of eddies and, although the vortex shedding is quite regular and follows similar patterns, no two vortices follow the exact same path, as evidenced by Figure 4.4.

Figure 4.5: Velocity vector plots at a Reynolds number of 841 showing the 7th and 8th filaments from the channel inlet. Successive images are 6.1 ms apart. Shed vortices are identified by numbers 1 through 8.

Due to the fact that the larger vortices near the bottom wall always rotate clockwise, and the ones near the top wall rotate counter-clockwise, the wall shear resulting from the larger vortices scouring the channel walls is always of opposite sign to that resulting from the bulk flow. As they move downstream, all vortices take on an elliptical shape and grow in size. They grow to approximately half of the channel height, until they reach half of the way to the downstream filament. After that, the size of the vortices decreases until they

73 are ultimately squeezed between the downstream filament and the membrane wall. At that point they disappear momentarily only to later reappear downstream. This squeezing phenomenon can be clearly observed in Figure 4.6, which shows how the vorticity contours are squeezed between each filament and the opposite wall. The reader is also referred to videos V4.5 and V4.6 for sequences of vorticity evolution in the channel. Video V4.6 only illustrates the sign of the vorticity.

Figure 4.6: Snapshot of the vorticity contour plot at a Reynolds number of 841 showing the 7th through 10th filaments from the channel inlet.

As the vortices move downstream, Figure 4.7 shows how they move high concentration fluid from the near-wall region to the bulk of the fluid, and low concentration fluid from the bulk to the wall. They also mix the low and high concentration fluids in a rotating motion. When a vortex scours the channel wall, the salt from the near-wall region is first carried upstream and then moves towards the centre of the channel following the rotation of the eddy carrying it. These phenomena can also be observed in videos V4.1, V4.2 and V4.3. A region of high relative salt concentration can be observed behind the upstream filament in Figure 4.7. By comparison with the vector plot (Figure 4.5), this is also a region of low velocity or stagnant fluid. Therefore, the salt concentration builds up at that point for the same reasons as in the steady flow cases at lower Reynolds numbers. As for the high concentration regions present in front of the filaments in steady flow, they are not present at this Reynolds number. As the counter-clockwise eddies formed along the channel wall (number 3 in Figure 4.7) come closer to the upstream filament they move lower concentration fluid from the middle of the channel to the region in front of the filament, and simultaneously eject high concentration fluid from this region into the bulk flow.

Figure 4.7: Evolution of solute mass fraction at a Reynolds number 841. Successive images are 6.1 ms apart and correspond to the times and filaments shown in Figure 4.5. Shed vortices are identified by numbers 1 through 8.

The local Sherwood number and friction factor distribution along the bottom wall region between successive filaments is depicted in Figure 4.8. For reasons mentioned before, local negative friction factor minima indicate the position of a vortex near the channel wall. It can be seen that the maxima in local Sherwood number, identified by lines A through F in Figure 4.8, mostly correspond to the front of the vortex which is scouring the wall (eddy number 2 in Figures 4.5 and 4.7). This supports the earlier contention that high mass transfer regions should be expected near the reattachment point of a recirculation region as, in that vicinity, low concentration flow is being forced closer to the channel wall, thus increasing the concentration gradient. This relationship is loosely confirmed in Figure 4.8, except that the local Sherwood number maxima at a

75 given instant are always slightly behind the reattachment points. This is understandable if one takes into account the fact that the timescales for momentum transfer are smaller than those for mass transfer for Schmidt numbers above unity as evidenced by the lag in the local Sherwood number distribution when compared with the local friction factor at any point in time.

Figure 4.8: Local Sherwood number and friction factor along bottom channel wall at a Reynolds number of 841, corresponding to the times shown in Figure 4.5. Lines A, B, C and E identify the maximum local Sherwood number at each time. Lines D and F identify local maxima of local Sherwood number for the last two times presented.

4.3.3. Highly unsteady flow At a Reynolds number of 1683 the flow pattern is unsteady, as was the case at the Reynolds number of 841. Again, there is vortex shedding behind the spacer filaments and from the channel walls, and a maximum of three vortices can be found on opposite sides of each spacer filament. However, at this higher Reynolds number the vortices found in the channel are visibly larger in size than those found at the lower Reynolds number of 841, and as they flow downstream they continue to grow in size until they

76 are approximately 90% of the way to the downstream filament. Moreover, their shape is round rather than elliptical, and they usually occupy more than half of the channel height.

Figure 4.9: Velocity vector plots at a Reynolds number of 1683 showing the 7th and 8th filaments from the channel inlet. Successive images are 4.58 ms apart. Shed vortices are identified by numbers 1 through 8.

Videos V4.7 and V4.8 present the evolution of velocity vectors, salt concentration and vorticity for a Reynolds number of 1683. The vortex shedding pattern at this Reynolds number presents characteristics not found at the lower Reynolds number of 841, even though they are similar. At the higher Reynolds number, the eddy shedding pattern is not one per period, but rather two vortices shed successively within a short period (~2 ms), followed by a longer period (~5 ms) before the cycle repeats itself. After the first vortex is shed (see eddy number 2 in Figure 4.9) it remains relatively stationary for a few milliseconds, until the second vortex is shed (eddy number 1) and merges with the first. Subsequently, the newly formed vortex (eddy number 3) continues downstream. As opposed to the observed behaviour at the lower Reynolds number, in this case vortex merging occurs for every eddy pair. However, the merging pattern is by no means periodic in time. Some of the variations observed are:

77 A longer time between the first and second eddy shedding (~12 ms) The eddy resulting from the first merger remaining stationary until a third eddy joins it. A third eddy catching up with the moving vortex that resulted from the first merger. These two vortices merge approximately halfway to the downstream spacer.

Figure 4.10: Evolution of solute mass fraction at Reynolds number 1683. Successive images are 4.58 ms apart and correspond to the times and filaments shown in Figure 4.9. Shed vortices are identified by numbers 1 through 8.

The mechanisms by which salt is convected from the near-wall region into the bulk flow are the same as for the lower Reynolds number cases. However, due to the larger size of the vortices at this Reynolds number, the high concentration fluid ejected from one wall region can potentially reach the opposite channel wall and reduce the concentration gradient there, thus reducing the local Sherwood number. This phenomenon can be observed in Figure 4.10 for eddy numbers 4 and 5.

Figure 4.11: Local Sherwood number and friction factor along bottom channel wall at a Reynolds number of 841, corresponding to Figure 4.9. Lines A through E identify local maxima of the local Sherwood number associated with wall scouring vortices.

Figure 4.11 shows the local Sherwood number and friction factor profiles along the bottom channel wall. Lines A and C correspond to eddy number 2, line E to eddy number 3, and lines B and D to eddy number 5. As was the case at a Reynolds number of 841, Sherwood number maxima can be associated with corresponding vortices scouring the wall, and lag slightly behind the reattachment point. However, at this Reynolds number local Sherwood number maxima caused by passing eddies can be found before the halfway mark between successive spacer filaments (Line A), which was not the case in the flow with a Reynolds number of 841. Moreover, it can be observed that there is a local minimum just downstream of Line C at the 4.58 ms curve, which is caused by the high concentration fluid moving towards the bottom wall due to eddy number 4.

79 4.3.4. Friction Factor and Sherwood number The average friction factor and Sherwood number for the spacer unit cell that comprises filaments numbers 7 and 8 were calculated for each of the runs, for both the channel with the zigzag spacer, and for an empty channel at the same distance downstream where the previously mentioned filaments would be located. The results obtained are summarized in Figures 4.12 and 4.13.

Figure 4.12: Friction Factor dependency on Reynolds number.

Figure 4.12 shows that between Reynolds numbers of 841 and 1683 the friction factor increases, which indicates that the flow over that Reynolds number range is in the transition region between laminar and turbulent flow. This agrees with the results obtained by Schwinge et al. [5, 115] using a different CFD software package (CFX 4.3, AEA Technology) and a structured quadrilateral mesh, as opposed to the unstructured meshing approach used in this thesis. Their data also shows an increase in the friction factor as the Reynolds number goes from 800 to 1600.

Figure 4.13: Sherwood number dependency on Reynolds number at a Schmidt number of 600.

Figure 4.13 shows that, as expected, the Sherwood number increased with increasing Reynolds number. Assuming a power law dependence of the Sherwood number with respect to the Reynolds number of the form [171]:

CSd d Sh c Reab Sc DTh (4.3) EUL

The exponent (a) for the Reynolds number dependence of the Sherwood number, for both the empty channel and the channel with the zigzag spacer, was calculated from the data shown in Figure 4.13. At steady state, the exponent was 0.332 for the empty channel, and 0.605 for the zigzag spacer. Under unsteady flow conditions, the exponent increased to 0.92 for the zigzag spacer, which indicates a higher degree of mixing caused by the vortices, which in turn enhances mass transfer. These findings are similar to the experimental results of Schock and Miquel [39], who calculated a value for the Reynolds number exponent of 0.33 for the empty channel case, and 0.875 for the spacer-filled channel case. They also agree with the results of Da Costa [171], who found values for the exponent in the range of 0.40 to 0.60 for the laminar regime, using Reynolds numbers below 100.

81 4.3.5. Comparison of velocity and concentration effects

Figure 4.14. Location of the wall monitoring points in the spacer unit cell.

Monitoring points were placed at several locations on the top membrane wall of the spacer unit cell, in regions where the highest and lowest wall shear rates were expected, as well as at some other random points along the wall. At these points, the salt concentration and velocity gradients were recorded for each of the runs. The data was used to calculate the Root Mean Square (RMS) velocity and concentration boundary layer thickness at each wall monitoring point. The results of this calculation, given in Figure 4.15, show the correlation between the velocity (shear) and the concentration (mass transfer) boundary layer thicknesses. Although a correlation coefficient of 0.701 does not indicate a strong dependence, it does confirm that a thinner velocity boundary layer generally results in a thinner concentration boundary layer. Figure 4.15 also shows that the concentration boundary layer is approximately one order of magnitude smaller than its corresponding velocity boundary layer, which agrees with the predictions of Bird et al. [9]. Velocity and mass fraction data was also collected every 100 time steps along the channel walls. Using this information, the RMS of the local friction factor as well as the Sherwood number profiles along the bottom channel wall for various Reynolds numbers were calculated and are depicted in Figures 4.16 and 4.17 respectively. Along the top wall, these profiles are approximately the same, only shifted 8 mm to the right. These graphs also show the range that these variables cover, which is represented by the shaded regions. As the Reynolds number increases the local friction factor maximum decreases and moves from being closer to the downstream spacer ( xhch 40 at Re 210, 526 & 841) towards the upstream filament ( xhch 37.8 at Re 1683). Moreover, in the laminar regime there is zero friction at the location of the reattachment points, which leads to there being two distinct local maxima for the local friction factor. It can also be seen

82 that the region in front of the upstream filament has higher friction in the unsteady cases than in the steady ones, which agrees with the general trends for friction factor.

Figure 4.15. Correlation between Friction factor and Sherwood number at the Wall Monitoring Points shown in Figure 4.14.

Figure 4.16. Friction factor distribution along bottom channel wall for the fourth unit cell of the computational domain. Shaded areas show the range of this variable for the transient runs.

83 The behaviour of the local friction factor over the wall region opposite the downstream spacer is also different for Reynolds numbers 841 and 1683. In the former << case, no vortices scour the wall in the region of 40xhch 41, as evidenced by the fact that the local friction factor does not change sign in that section. However, at the higher Reynolds number the whole of the channel wall experiences both positive and negative local friction factors, which means that eddies sweep the entire wall region at some time or another. It also means that a vortex shed upstream affects the shedding of vortices further downstream at a Reynolds number of 1683, but not at 841, since the eddy has already disappeared before it reaches the region where the next vortex is shed. From Figure 4.17 it can be seen that, as the Reynolds number increases, the region of higher mass transfer moves upstream, from being closer to the downstream filament to the region in between the spacers. This corresponds with the upstream movement of the friction factor maximum. Moreover, a region of high mass transfer also appears near both sides of the filament as the Reynolds number increases. This latter phenomenon is caused by vortices renewing the high concentration fluid with lower concentration fluid as they come near the filaments.

Figure 4.17. Local Sherwood number distribution along bottom channel wall for the fourth unit cell of the computational domain. Shaded areas show the range of this variable for the transient runs.

84 4.3.6. Fourier Analysis In order to obtain further information about the dynamics of the momentum and mass transport processes in the membrane channel, a Fourier analysis was carried out. A 1 monitoring point was placed in the middle of the channel ( y 2 hch ) in between successive spacers. Velocity and salt concentration were recorded at that point for each time step, the Fast Fourier Transform (FFT) was applied to this time data, and the frequency spectra were obtained by calculating the square of the absolute value of the FFT. However, only the velocity perpendicular to the bulk flow gave meaningful information about the flow. The time series for this variable can be seen in Figure 4.18. Due to the monitoring point being placed halfway between successive spacers, it can be seen from Figures 4.5 and 4.9 that it is mainly the vortices shed by the upstream spacer that affect this point. For both unsteady cases, at Reynolds numbers of 841 and 1683, the flow was not periodic in time nor did it follow a time repetitive vortex shedding pattern. This agrees with observations made previously in relation to Figures 4.5 and 4.9. Moreover, at the higher Reynolds number the oscillations have larger amplitude, and their frequency is also slightly higher.

Figure 4.18. Time evolution of Velocity in the y direction in between successive spacers, at the middle of the channel.

85 From Figure 4.19 it can be seen that there are two main frequencies for both of the transient runs presented there. For the run at Reynolds number 841, the strongest peak is at a frequency of 81.5 Hz (period of 12.3 ms). There is a second strong peak at 43.7 Hz (period of 22.9 ms), and frequencies in the range of 37 to 43.5 Hz also contribute significantly to the spectrum. The higher frequency corresponds to the vortex shedding frequency, and the lower frequency correspond to the frequency with which large eddies go past the monitoring point (see Figure 4.5). The latter also loosely correspond to the frequency with which local positive v velocity maxima appear in Figure 4.18. The run at a Reynolds number of 1683 also presents two strong frequency regions. However, the lower frequency region is broader at this Reynolds number than at the lower one. In this case it covers the range from 60 to 100 Hz, with the strongest peak at 87.8 Hz (period of 11.4 ms). The higher frequency region is markedly weaker than the lower frequency region, extending from 160 to 187 Hz, with the strongest peak at 178.5 Hz (period of 5.6 ms). Once again, the higher frequency corresponds to the average eddy shedding frequency, and the lower region represents the passing of vortices through the monitoring point.

Figure 4.19. Power spectrum for Velocity in the y direction in between successive spacers, at the middle of the channel.

86 As the Reynolds number was increased, the vortex shedding frequency also increased, as did the rate at which vortices went past the monitoring point. The ratio of shedding frequency (in Hz) to Reynolds number was about constant with a value of about 0.1. Experimental [172] and numerical [173] works suggest that the Strouhal number for circular cylinders increases slightly with Reynolds number, and stabilises at about 0.21 for cylinder Reynolds numbers above 500. Using 1.82 times the inlet velocity as the reference velocity (which is, in fact, the average velocity of the fluid in the gap between the filament and the opposite wall) and the filament diameter as the length scale, we obtain Strouhal number values of 0.199 and 0.218 for the flows at hydraulic Reynolds numbers of 841 and 1683 respectively. The cylinder Reynolds numbers for these flows are 417 and 915 respectively using the velocity of the fluid in the gap as the reference velocity, which represents a very good agreement (5.24% and 3.81% relative error) with the reported value. Moreover, the peak representing the shedding frequency in the power spectrum became weaker than the peak associated with the passing of eddies as the Reynolds number was increased. This corresponds with the appearance of larger eddies closer to the downstream spacer at the higher Reynolds number. It also suggests that the disturbances generated by the shedding of the vortices at the spacer filaments lose strength as they travel downstream.

4.4. Conclusions For the zigzag geometry simulated in this chapter, it was observed that the flow becomes unsteady at a hydraulic Reynolds number between 526 and 841. It was found that the regions of higher mass transfer on the membrane wall are not only correlated to those of high shear rate, but also to those where the fluid flow is towards the wall. Likewise, the wall regions in which the fluid flow is away from the wall correspond to those of low mass transfer. Knowledge of the location of the regions of low and high friction and mass transfer on the membrane surface is particularly important for spacer design. New filament profiles could be devised in an attempt to influence the fluid flow in order to increase wall shear rate in the regions where mass transfer is low, while retaining the current flow characteristics for the regions where it is already high. The exponents for the Reynolds number dependence of the Sherwood number for the empty channel and the channel with the zigzag spacer at steady state were 0.332 and 0.605, respectively. In unsteady flow, the exponent increased to 0.92 for the zigzag spacer due

87 to the higher density of wall vortices per channel unit length. Vortices near the membrane walls enhance mass transfer due to two main effects: an increased wall shear which decreases the thickness of the boundary layer, and an inflow of lower concentration fluid into the boundary layer. The results in this chapter indicate that inflow of lower concentration fluid into the boundary layer dominates unsteady mass transfer enhancement for membrane filtration of sodium chloride. For a constant Schmidt number, a thinner concentration boundary layer represents a higher degree of mass transfer. On the other hand, concentration boundary layer thickness is inversely proportional to Schmidt number. Therefore, for a higher Schmidt number (such as those encountered in UF and MF) the low concentration fluid moved towards the channel walls by the passing vortices must come closer to the wall to retain the same degree of mass transfer. This means that for higher Schmidt number solutes the efficiency of scouring vortices is lower. In order to maintain a high mass transfer coefficient at a high Schmidt number, mechanisms must be found to decrease the size of the velocity boundary layer, thus enabling velocity fluctuations to approach closer to the wall and disturb the concentration boundary layer.

88 Chapter 5

Mass Transfer in Three-Dimensional Spacer-Filled Narrow Channels with Steady Flow

5.1. Introduction Analogies to extrapolate heat transfer concepts into the field of mass transfer are commonly cited [40, 67]. Yuan et al. [147] pointed out that for periodically disturbed duct flow with heat transfer, entrance effects are much shorter than for flows without the disturbances. Often, the heat and fluid flow become fully-developed after about 5 repetitions of the periodic section. Since heat transfer units may consist of dozens or even hundreds of these repeating sections, the analysis of just one periodic section can give a clear picture of the transfer phenomena dictating the overall performance of the unit. Therefore, for this and other types of spatially-repeating geometries many heat transfer studies [81, 147, 148] have utilized a fully-developed temperature profile, i.e. a spatially-periodic heat transport boundary condition. These concepts can be extrapolated to mass transfer by utilizing a fully-developed mass fraction profile as a boundary condition. The method for obtaining a periodic velocity and temperature field was originally proposed by Patankar et al. [161], and involves an iterative solution of the flow field by setting a linear pressure gradient across the periodic unit and reapplying the outlet flow variables and their gradients at the inlet of the unit. However, constant fluid properties must be assumed and the outlet temperature field must be scaled appropriately before it can be applied at the inlet. Although the analogous heat transfer problem of flow in narrow conduits with obstructions has been the focus of much research [68], the mass transfer phenomena encountered in membrane operations are of a much smaller scale as a consequence of the higher Schmidt number, which can be several orders of magnitude higher than the typical Prandtl number encountered in heat transfer applications. This also makes adequate numerical resolution of concentration boundary layers particularly difficult and, until recent times, placed prohibitive computational costs on their calculation. This chapter presents a description of the spatially-periodic mass transport boundary condition, and its application to the analysis of the flow features and

89 performance of a simple 3D non-woven spacer filament mesh. In principle, it extends the work of Schwinge et al. [115] to 3D geometries, focusing on the effects of form and viscous drag on mass transfer. Moreover, it explores the connection between mass transfer enhancement and flow features exclusive to 3D flow under steady flow conditions.

5.2. Problem Description, Assumptions and Methods The commercial CFD code ANSYS CFX-10.0 is used to solve the continuity, momentum and mass transport equations [9] in a spacer-filled channel. As in chapter 4, constant properties are employed and the effect of gravity is excluded. In order to further simplify this system the fluid is assumed to be Newtonian, and a binary mixture of water and salt is considered, with no sources of salt in the fluid. Due to the high memory requirements for 3D simulation, a different approach to that used chapter 4 for 2D simulations is employed, bypassing the entrance region effects and focusing on the spatially periodic region of the spacer-filled channel. This approach reduces the computational requirements and enables accurate and reliable 3D simulations of the flow inside a SWM module, with applicability to real-world operations. Therefore, fully-developed (or periodic) boundary conditions for both the momentum and the mass transport equations are employed. Periodic momentum boundary conditions are commonly used for geometries in which a particular pattern is repeated over the length of the fluid domain. This repeating pattern, such as the spacer filament mesh, generates a spatially repeating velocity field. This type of momentum boundary condition is available in ANSYS CFX, and is used for the case studied in this chapter. However, even in spatially repeating fluid domains, the mass fraction field does not become spatially periodic in the same way that velocity does. Instead, the bulk mass fraction along the length of a SWM module gradually increases due to the permeation and removal of low concentration fluid. This is similar to the situation in a heat exchanger, where the geometry and velocity fields are spatially periodic, but the temperature field is not. Several authors [81, 148, 174] have modelled this situation by utilizing a dimensionless scaled temperature which remains spatially periodic, using the method originally developed by Patankar et al. [175] or variations of this method. In particular, Rosaguti et al. [148] applied a variation of the method of Patankar et al. using the CFX-5.7 package, in which only the flow variables but not

90 their gradients are wrapped from the outlet to the inlet. The lack of gradient wrapping causes the solution to deviate from the truly periodic solution, especially near the periodic boundaries. Rosaguti et al. [148] found that they could overcome this effect by simulating three repeating units and extracting all of their data from the central unit. Given the similarities between temperature and mass fraction transport it is possible to specify a non-dimensional scaled mass fraction which remains spatially periodic, i.e. the shape of its profile remains the same for downstream locations separated by a distance equal to the unit cell length, even though the actual mass fraction values are not the same. There are two ways to define a scaled mass fraction with such properties: the first is to assume a constant mass fraction gradient at the membrane wall surface, and the second way is by assuming a constant mass fraction value at the membrane wall. For many membrane applications the concentration at the membrane reaches a limiting value, such as when the channel becomes gel-polarized in the pressure independent region of the flux-pressure curve or due to a boundary layer of high fixed osmotic pressure or viscosity [123, 176, 177]. Therefore, the latter assumption of a constant wall mass fraction should be a suitable approximation. The scaled mass fraction for a constant wall mass fraction then takes the following form:

YY w (5.1) YYwb

It should be mentioned that because the value of the Sherwood number, and hence the mass transfer coefficient, approaches an asymptotic or limiting value as the flow becomes fully-developed both in terms of velocity and mass transfer [35], the choice of wall mass fraction and inlet bulk mass fraction values becomes irrelevant. These values still need to be specified in order to calculate the actual mass fraction values used in the calculations, but should have no impact on the results obtained. The use of a constant wall mass fraction means the boundary condition for the membrane wall used in this chapter is precisely the same dissolving wall boundary condition used in the previous chapter. Although a permeable wall boundary condition [123, 127] in which the wall mass fraction is dependent on the permeation velocity would be more physically accurate, the relative magnitude of the permeation velocity is usually a few orders of magnitude smaller than the average fluid velocity, such that the effect of permeation on wall shear and mass transfer enhancement is minimal [178]. In

91 addition, for channel geometries and flow conditions typically found in NF and RO, Geraldes and Afonso [136] have recently demonstrated that Sherwood numbers obtained for cases without permeation can be used to predict the wall conditions in cases where permeation is present. The impermeable-dissolving wall model is therefore capable of providing valuable insights into the flow field induced mass transfer phenomena taking place inside narrow channels, and the results obtained should still represent a good approximation of the flow and mass transfer in SWM modules. The ANSYS CFX-10.0 software does not implement the fully-developed mass transfer boundary condition. Therefore, an approach was developed to implement this condition on the inlet and outlet boundary profiles of the unit-cell domain. This approach is analogous to that of Rosaguti et al. [148], and consists of two steps. Firstly, the velocity field is obtained for the fluid domain in question, using the available periodic boundary conditions. This ensures that the velocity field solution includes the periodic wrapping of the velocity gradients. In the second step the mass fraction field is obtained. This is done using the velocity field solution and applying the scaled outlet mass fraction conditions to the inlet in an iterative manner, by means of the following relationship:

CSCSYY YY DTDwyz, wyz, T (5.2) YY YY EUEUwbin wbout

During the second step, the outlet is treated as an “opening” boundary condition and the inlet mass fraction is calculated using the current outlet mass fraction. This process is repeated until a desired convergence criterion is met, such as a non-changing Sherwood number. Since the mass fraction gradients are not wrapped, the converged solution will not be the exact fully-developed solution, but an approximation. In order to avoid deviation from the exact solution, and following the approach of Rosaguti et al. [148], three repetitions of the unit-cells were modelled for the 2D geometries, and the results were taken from the middle unit. Although less than three unit cells were used for the 3D geometries due to computational costs, the flow domains chosen for those cases were also larger than the minimal unit cell.

92 5.2.1. Geometry description A periodic 3D non-woven spacer filament mesh unit-cell that is representative of typical behaviour throughout a SWM module was analysed. The characteristics of this mesh are identical to those used in the previous chapter, except for the df/hch ratio, which was increased from 0.5 to 0.6 in order to cover the entire channel height. The lm/hch ratio was kept at 4. Orientations of 90° and 45° were modelled, as shown in Figures 5.1 and 5.2 respectively.

Figure 5.1. 90° Orientation of the non-woven spacer filament mesh. Flow is from left to right.

As in chapter 4, a Schmidt number of 600 was chosen. This Schmidt number is characteristic of typical monovalent salts, such as sodium chloride. The spacer surfaces were treated as no-slip walls with no mass transfer, where all velocity components and the mass fraction gradient normal to the boundary are set to zero (u = v = w = 0 and Yn0 ). The membrane walls in the unit cells were also treated as no-slip walls (u = v = w = 0), but the mass fraction at the wall was fixed at a constant value (Y = Yw). The rest of the boundaries were specified as translational periodic with the exception of the inlet (x = 0) and outlet (x = xmax) boundaries for salt mass fraction, which were treated as fully-developed velocity and mass fraction profiles, using the approach already described. The net mass flow from the inlet to the outlet was achieved by varying the pressure drop between these boundaries.

Figure 5.2. 45° Orientation of the non-woven spacer filament mesh. Flow is from left to right.

This chapter focuses on the steady-state, laminar flow regime. Given that flows in real world membrane operations fall in the Reynolds number range where flow is steady and laminar, the choice of such a flow regime is justified. However, it is well known that unsteady flows are also encountered in membrane systems, and time variations of the flow field can significantly enhance mass transfer. Nevertheless, the increased computational processing power required in order to carry out time-dependent simulations in 3D becomes prohibitive when considering that steady state simulations can take up to a month to achieve convergence. In addition, many of the insights gained by analysing the relationships between fluid flow and mass transfer in steady flow are still applicable in the vortex shedding flow regime, after taking into consideration the increased mixing and time dependent nature of such flows. This has been demonstrated in the previous chapter. Therefore, by focusing on the steady laminar flow regime, this chapter establishes a baseline approach for the subsequent analysis of time dependent flow regimes. Given that the hydraulic Reynolds number was the only characterization parameter related to the velocity of the fluid used in the calculations, it will be referred to as the “Reynolds number” in the rest of this chapter, and the term “hydraulic” will be implied. The Reynolds number at which the flow becomes unsteady for obstructed narrow channels depends on the geometry of the obstructions, and can range from 200 to 600 [5]. Kang and Chang [108] observed this transition for rectangular zigzag and

94 cavity 2D obstructions in the Reynolds number range of 455 to 545, and Li et al. [42] found vortex shedding at Reynolds numbers greater than 660. In the previous chapter, flow was found to be steady for a 2D zigzag spacer for Reynolds numbers up to 526. However, for 3D cases the Reynolds number at which flow becomes unsteady may be lower than for 2D flow, as Feron and Solt [87] observed this transition at a Reynolds number around 250. Transient simulations were carried out without mass transfer for the geometries studied and flow was found to be steady at a Reynolds number of 200 for the both the 45° and 90° orientation spacers, but became unsteady at a Reynolds number between 200 and 300. Therefore, the Reynolds number for the calculations in this chapter was limited to values equal to or less than 200 for the 3D cases, and 500 for the 2D cases. ANSYS CFX obtains steady-state solutions by making use of a “false” time step or time-scale. This was chosen to be one fifth of the residence time within the flow domain for the continuity and momentum transport equations. Due to the high Schmidt number, the time-scale for the mass transport equation was chosen to be at least one order of magnitude higher than that used for the momentum and continuity equations. The solutions obtained were considered to have converged once the root mean square of the normalized residuals was below 10-5, and the average Sherwood number had stabilized.

5.2.2. Characterization of the projection of a vortex onto a plane For a given projection of a vortex onto a plane, it is possible to calculate its area

(AV) and circulation ( ). These will depend on the definition of the boundaries of the vortex, a topic which is much debated even to this day [179-181]. Given that there is no single widely accepted method for vortex identification, a method following the guidelines of Portela [179] was used in this chapter, to locate both the spanwise and streamwise vortices. Tangential velocity vectors and streaklines were analysed in order to find the centres of the 2D projections of the vortical flow structures, i.e. the xy planes were scanned for spanwise vortices, and the yz planes were examined for streamwise vortices. The vortex radius (rV) was taken to be the distance from the vortex centre to the closest point of zero vorticity. Following the approach of Portela [179], it was assumed that each vortex was circular for reasons of simplicity, such that the vortex area can be represented by:

95 2 AVVr (5.3)

Circulation is equal to the integral within the vortex area of the vorticity component normal to that area. By applying the Kelvin-Stokes theorem [182], circulation can be calculated using:

OOnVdAvl d (5.4) AAVV

Taking circulation as the main defining vortex characterization parameter, the vortex Reynolds number is defined by:

Re (5.5) V

5.2.3. Vortices in 2D flow Assuming a constant density fluid, for any given closed surface the net fluid flow in or out of the volume enclosed by that surface must be zero. However, for the 2D projection of an open vortex to a plane, streamlines within the vortex region all start or finish at the vortex core, and all cross a closed line around that core. This means that for 2D flows, the only possible vortex structure that exists must be a closed vortex, in which all streamlines are closed curves. Tao et al. [80] analysed the heat transfer enhancement around a cylinder with and without fins, and found that the “synergy” between the heat and flow fields decreases as Reynolds number increases, especially in the region behind the cylinder where flow separation occurs and a vortex is formed. Given that there is no convective transport perpendicular to a streamline, any change in mass fraction inside a closed streamline for 2D constant density flow can only be due to a reaction inside that region or due to molecular diffusion in or out of that region. This means no convection occurs into or out of a vortex in 2D constant density flow. If the flow is further restricted to be non- reactive and at steady-state, then there cannot be any net diffusive flux in or out of a vortex. Although this seems to imply that generating spanwise 2D vortices is counter productive to mass transfer enhancement, the generation of a vortex in the flow domain affects the surrounding areas. This is evidenced by the work of Grosse-Georgemann et al. [71], who found that even though the vortices themselves do not enhance heat

96 transfer, their effects on the flow field do. A vortex near a wall increases wall shear and reduces boundary layer thickness. An attached vortex creates regions of boundary layer separation and reattachment. Both of these phenomena enhance mass transfer, but are usually located outside of the perimeter of the vortex.

5.3. Results and Discussion

5.3.1. Mesh Independence Study Before reliable results can be obtained, a sufficiently fine meshing of the fluid domain must be attained so that the solution of the transport equations does not introduce numerical error or artifacts. For that purpose, successively finer meshes were employed for both the 90° and the 45° spacer orientations. As a high solute Schmidt number leads to a thin concentration boundary layer [141], it is of the utmost importance that the boundary layer is properly resolved. Inflated boundaries were used for all the boundaries with a non-slip condition, where the thickness of the first grid element layer was of the order of 0.1% of the filament diameter. Angular resolution on the surfaces of the filaments was set to 15. It was found that meshes with just over 15.5 million elements for the 90° orientation, and with just under 19 million elements for the 45° orientation, had grid convergence indexes [168] below 2% for the friction factor, and below 4% for the Sherwood number. These were obtained using unstructured discretisation meshes with element sizes of the order of 5% of the filament diameter in the bulk of the channel, and 3% of the filament diameter in the vicinity of the spacers. Given than in the previous chapter the grid convergence indexes for friction factor were between 1.14% and 0.2%, and for Sherwood number were between 3.5% and 1.7%, our current errors are acceptable and the results can be considered reliable, especially given the extra degree of freedom in the 3D case. Moreover, the RMS error for the difference between dimensionless scaled mass fraction profile at the inlet and outlet of the flow domain for the 3D meshes was below 1%. This error, calculated for the work of the previous chapter, was between 2% and 0.94%. Therefore, our current simulation approach converges to within the same error limits as our previous approach. In a further comparison, 2D steady state periodic runs were compared to the results of the previous chapter, and it was found that the values of friction factor values obtained were the same, and the Sherwood number values were within 5%.

97 5.3.2. Hydrodynamics

5.3.2.1. Validation Figure 5.3 compares the friction factors obtained for our simulations with the experimental results of Schock and Miquel [39], who tested different commercially available spacers and developed a correlation which was in good agreement with their data in the range of 100 < Reh < 1000. Of the spacers they tested, the Toray PEC-1000 feed channel spacer is the most similar geometrically to our simulated spacer, with the main difference being that df/hch ratio for the commercial spacer is 0.5, compared with 0.6 for our geometry. This results in a lower porosity for our geometry, and helps explain why the friction factors obtained for our simulations are higher than those reported by Schock and Miquel for the Toray PEC-1000 feed spacer. Taking this into consideration, our results show good agreement with the experimental data. Moreover, the differences in friction factor between the geometries modelled are minimal for the same Reynolds number, which is to be expected given that the definition of Reynolds number used accounts for surface friction effects.

Figure 5.3. Dependence of friction factor on Reynolds number for different spacer geometries

98 5.3.2.2. 90° Orientation Various flow characteristics for the 90° orientation mesh were encountered for all of the Reynolds number for which runs were performed. The reader is referred to videos V5.4, V5.5 and V5.5 in the disc located in the back cover of this thesis (see Appendix B). These videos show a moving plane which illustrates the velocity vectors and solute concentration throughout the flow field for the 90° orientation mesh, at Reynolds numbers of 50, 100 and 200 respectively. It should be pointed out that the resulting flow field for this orientation, regardless of the Reynolds number, was symmetric about the xy plane that slices the flow field in half. The translational periodic conditions on the z-extremes of the unit-cell flow field were also found to be symmetry planes. This was corroborated by checking that both the velocity in the z direction and all the gradients normal to these planes were zero. In light of this, our analysis for this geometry will focus on only half of the flow field, as the other half is a mirror image. The main flow characteristic for the 90° orientation mesh is the appearance of a spanwise vortex located behind the upstream filament, even at Reynolds numbers as low as 1. This spanwise vortex increases in size as the Reynolds number increases. This is the main spanwise vortex (MSV), and has also been encountered in 2D simulations [114]. The MSV for the 90° orientation presents characteristics similar to its 2D counterpart, the cavity spacer. As the Reynolds number is increased, the boundary layer reattachment region associated with the MSV moves downstream until it reaches the downstream filament, and the MSV takes a more elliptic shape. Moreover, for the same Reynolds number, the reattachment point in the 2D cavity spacer and the reattachment line for the 3D 90° orientation spacer are at approximately the same distance from the upstream spacer (see Figure 5.4), although they are further downstream for the 3D case for reasons that will be discussed later. At a Reynolds number of 200, the MSV has formed a recirculation region that encompasses the whole cavity between the upstream and downstream filaments, such that the membrane between these filaments only experiences shear in the direction opposite to the bulk flow (see Figure 5.14) and the boundary layer reattachment region is adjacent to the downstream spacer filament. The centre of the MSV also moves downstream as the Reynolds number is increased.

Figure 5.4. Local friction factor values along the bottom membrane wall (y/hch = 0) for the central symmetry plane of the 3D 90° spacer and for the 2D cavity spacer.

Two other spanwise vortices present in 2D cavity simulations were also encountered in the 90° spacer orientation. A secondary spanwise vortex (SSV) forms in front of the downstream filament, and a third spanwise vortex (TSV) also appears between the MSV and the corner between the upstream filament and the membrane wall. The SSV rotates in the same direction as the MSV, but has a vortex Reynolds number an order of magnitude smaller than that of the MSV. The TSV has a vortex Reynolds number at least two orders or magnitude smaller than that of the MSV, and rotates in the opposite direction. For both of these vortices separation of the boundary layer occurs along the membrane wall, and the boundary layer becomes reattached onto the spacer filament to which the vortices are adjacent. Circulation for the spanwise vortices was higher in the 2D cavity simulations than on the symmetry plane for the 90° orientation spacer simulations. This agrees with the results of Iwatsu et al. [144], who found that the average z-Vorticity over the whole domain was greater for 2D cavity flow than for 3D cavity flow.

100

Figure 5.5. Local Sherwood number and wall shear vectors on bottom membrane wall (y/hch = 0) for the 90° orientation spacer at a Reynolds number of 100, showing flow separation and reattachment regions with zero wall shear.

Flow on the symmetry xy plane along the middle of the flow domain is qualitatively similar to the flow encountered in 2D cavity spacer simulations. This also agrees with the findings of Iwatsu et al [144]. Quantitatively, however, the average fluid velocity on the symmetry plane is about 35% to 40% faster than the average velocity for the 2D cavity spacer at the same Reynolds number. This is the main reason why the boundary layer reattachment point on the central symmetry plane caused by the MSV for the 3D 90° orientation spacer is further downstream than for the 2D cavity case (see Figure 5.4). The higher fluid velocity on the symmetry plane for the 3D case is partly caused by the reduction of the cross-sectional area, i.e. the hydraulic diameter for the 3D flow field is smaller than that for the 2D case; however, the void fraction () is also smaller for the 3D case, such that the purely geometrical effect accounts for about a quarter of the increase in velocity. Most of the difference is caused by the interaction of the side filaments with the fluid. The side filaments slow down the fluid and, as a

101 consequence, the fluid near the centre plane is accelerated in order to maintain the required total mass flow through the flow domain for the same overall Reynolds number. This effect is also evidenced by the reattachment line caused by the MSV, which is closer to the upstream filament in the area near the side filaments than in the centre plane, as is seen in Figure 5.5. The main qualitative difference between 2D and 3D spanwise vortices is that 3D spanwise vortices do not present closed streamlines. Flow inside a vortex in the 90° orientation spacer spirals toward the central symmetry plane. Since the spanwise vortices on the central symmetry plane are outward flowing, the fluid that gets to this central plane then spirals out until it joins the bulk flow and exits the unit cell. The spiralling fluid inside spanwise vortices is replaced by bulk fluid which, after flowing over the upstream filament and over the MSV, reaches the zone near the boundary layer reattachment line, flows outward towards the side filaments, and then commences the spiralling motion. This motion is the reason why the region near the centre of the MSV has a lower concentration, as opposed to the 2D cavity spacer, which can be seen in Figure 5.6.

Figure 5.6. Comparison of solute mass fraction for the central symmetry plane of the 3D 90° spacer and the 2D cavity spacer, at a Reynolds number of 50.

102 One of the main features of 3D flow which is absent in 2D simulations is the occurrence of streamwise vortices. The simulations from Reynolds number of 50 to 200 all presented streamwise vortices in the flow domain. There are two main regions where streamwise vortices were observed: near the area where the top and bottom filaments intersect, and below and adjacent to the filament parallel to the bulk flow. An interesting relationship was found for the ratio of volume average of x- vorticity magnitude to volume average of z-vorticity magnitude, as seen in Figure 5.7. This ratio increases up until a Reynolds number of approximately 100, and then decreases as the Reynolds number is increased further. This is an indication that once the MSV grows large enough to cover the whole cavity between successive filaments, spanwise circulation effects begin to dominate such that they slow down the rate of increase of streamwise circulation with Reynolds number. This is corroborated by the fact that the ratio of z to x-wall shear magnitude also decreases after a Reynolds number of 100.

Figure 5.7. Ratios of streamwise to spanwise effects for the 90° orientation spacer; a) ratio of volume average of x-vorticity magnitude to volume average of z-vorticity magnitude; b) ratio of average z-wall shear magnitude to average x-wall shear magnitude.

103 5.3.2.3. 45° Orientation As opposed to the 90° orientation spacer, the flow around the 45° orientation spacer did not present any symmetry planes. However, symmetry characteristics were still encountered. In particular, the shear and mass transfer patterns on both membrane walls were found to be identical if rotated 180° about the z-axis. Therefore, membrane wall analysis will only focus on one of the membrane walls.

Figure 5.8. Flow regions for the 45° orientation spacer geometry, showing some streamlines for flow at a Reynolds number of 100. Region I is the region near both membrane walls, hence it cannot be depicted in this figure. Streamlines a and b lie in region I, while streamlines c and d lie in regions II and III respectively.

The general flow pattern can be divided into three distinct regions: (I) the region near the membrane walls, in which flow is predominantly parallel to the spacer filament which touches that wall; (II) the region near the xy plane parallel to the bulk flow which goes through the point where the filaments intersect each other (roughly the middle of the channel along the z direction in Figure 5.2), in which flow follows a horizontal zigzag path avoiding the filament intersection; and (III) the rest of the fluid domain, for

104 which an xy plane intersects the filaments such that the cross section is similar to the 2D zigzag spacer but not identical, as the cross section of the filaments is elliptical. The last two regions are identified in Figure 5.8. Region III represents the general flow field observed experimentally by Da Costa et al. [101], in which the bulk of the fluid follows a zigzag path in both the y and z directions while moving in the x direction. Region I represents the flow pattern identified by Karode and Kumar [132] at low lm/df ratio, and its existence is evidenced by the wall shear vectors along the membrane wall, which tend to follow a direction parallel to the filament that touches the wall, as seen in Figure 5.9. According to the results Karode and Kumar [132], by increasing this ratio the extent of region I would decrease. However, for this study the lm/df ratio was kept constant.

Figure 5.9. Local Sherwood number and wall shear vectors on the bottom membrane wall (y/hch = 0) for the 45° orientation spacer at a Reynolds number of 100.

Also from Figure 5.9 it can be seen that, at a Reynolds number of 100, there are no clear boundary layer reattachment regions as there are in the 90° orientation spacer.

105 The reason for this is that, although flow separation downstream of the filaments does occur, the vortices are not closed recirculation regions, and flow behind the filaments follows a spiral path roughly parallel to the filament. This flow feature has also been reported by Koutsou et al. [131]. Their study also observed the formation of a streamwise free vortex in the region between successive filament intersections, that is, in the middle of region II. This free vortex was also encountered in our simulation results, but only for Reynolds numbers at and above 100, as is evidenced by Figures 5.10 and 5.11. The appearance and location of the free vortex can also be observed in videos V5.1, V5.2 and V5.3, which illustrate the velocity vectors and solute concentration throughout the flow field for the 45° orientation mesh, at Reynolds numbers of 50, 100 and 200 respectively.

Figure 5.10. Tangential velocity streaklines for the 45° orientation spacer at different Reynolds numbers,

on a plane at x/hch = 4. Rotation of the free vortex is counter-clockwise, i.e. it has negative x-vorticity.

Apart from the central free vortex, several other vortical structures can be observed when analysing the patterns of normalized helicity, as defined by Levy et al. [183], which are shown in Figure 5.11. In particular, vortical flow is present downstream and upstream of each spacer filament. This is analogous to the spanwise vortices present in the zigzag 2D geometry. However, these 3D vortices are not completely spanwise, but are not streamwise either, as they produce a corkscrew-like flow pattern parallel to the spacer filament. A third vortical structure can also be

106 observed, but only at the higher Reynolds number of 200. It follows the same path as streamline d in Figure 5.8.

Figure 5.11. Isosurface plots of regions with absolute value of normalized helicity of 0.9 for the 45° orientation spacer at different Reynolds numbers.

Figure 5.12. Ratios of streamwise to spanwise effects for the 45° orientation spacer; a) ratio of area average of w-velocity at the side boundaries to area average of u-velocity at the outlet; b) ratio of volume average of x-vorticity magnitude to volume average of z-vorticity magnitude; c) ratio of average z-wall shear magnitude to average x-wall shear magnitude.

107 Another flow characteristic of this geometry is flow parallel to the filaments. From Figure 5.12 it can be seen that as the Reynolds number is increased, the ratio of the magnitude of x-vorticity to the magnitude of z-vorticity also increases, as opposed to the 90° orientation case where this ratio decreases after a Reynolds number of 100. The ratio of average w-velocity magnitude at the side boundaries to u-velocity average at the outlet also increases as the Reynolds number is increased. This, coupled with the fact that the ratio of z to x-wall shear magnitude also increases, means that at higher Reynolds numbers the flow tends to align itself to the spacer filaments, despite these not being parallel to the main pressure gradient, which is in the x-direction.

5.3.3. Mass Transfer effects

5.3.3.1. Validation Assuming a power law dependence of the Sherwood number with respect to the Reynolds number of the form [171]:

CSd d Sh c Reab Sc DTh (5.6) EUL

the exponent (a) for the Reynolds number dependence of the Sherwood number for the 45° orientation spacer, calculated from the data shown in Figure 5.13 was 0.591. This value agrees with the results of Da Costa et al. [93], who found values for the exponent in the range of 0.49 to 0.66 for commercially available spacer meshes with geometries similar to the 45° orientation spacer geometry modelled in this chapter. It also falls close to the value of 0.5 found by Kuroda et al. [99] and the value of 0.62 reported by Schwager et al. [184], and is similar to the value of 0.475 reported by Kim et al. [92] using an experimental setup with a similar but simplified geometry of two layers of fluid at a 90° cross flow angle. The Sherwood number values for the similar commercial spacer reported by Schock and Miquel [39], the Toray PEC-1000 (see Figure 5.13), are lower than those obtained for the 45° orientation. Exact correspondence is not to be expected given that the spacer geometries are slightly different and, more importantly, the phenomena giving rise to mass transfer in these two cases are not the same: membrane permeation in the experimental case, as opposed to wall dissolution in the case modelled. Moreover,

108 as shown in Figure 5.3, the friction factor for the experimental case was also lower than for the modelled case, which can partly explain the lower Sherwood number observed in the experiments. Despite the differences in these two cases, and the assumptions used in the simulation such as constant properties, the fact that the Sherwood numbers are within the same order of magnitude is remarkable and represents reasonable agreement between simulation and experiments.

Figure 5.13. Overall Sherwood number dependence on Reynolds number for different geometries. Here, the Sherwood number is averaged over both top and bottom membrane walls.

5.3.3.2. 90° Orientation For this case the regions of higher mass transfer on the membrane surface, as in the previous chapter, correlate very well with the regions where flow is towards the membrane, i.e. mass transfer is higher near the boundary layer reattachment lines associated with the spanwise vortices. This is caused by the renewal of higher concentration fluid located in the boundary layer by lower concentration fluid coming from the bulk flow. Similarly, mass transfer on the membrane surface is lower near the boundary layer separation lines associated with the spanwise vortices, as is evident in Figure 5.5 for a MSV which does not cover the whole region between successive filaments and in Figure 5.14 for a MSV that does cover the whole region.

109

Figure 5.14. Local Sherwood number and wall shear vectors on bottom membrane wall (y/hch = 0) for the 90° orientation spacer at a Reynolds number of 200, showing flow separation and reattachment regions with zero wall shear.

The effect of streamwise vortices on mass transfer can be seen along the top membrane wall (y/hch = 1). Local Sherwood number contours on the top membrane surface (see Figure 5.15) show that the region closer to the central symmetry plane

(from approximately z/hch = 1 to z/hch = 2) exhibits almost no variation of local Sherwood number in the z direction. However, closer to the side filament there are regions of low and high mass transfer roughly aligned with the direction of the bulk flow. The region of high mass transfer is broken by a region of low mass transfer which cuts at an angle similar to that of the central rotation axis of a streamwise vortex located close to the side filament. As seen previously, the regions of low mass transfer are associated with flow away from the membrane wall caused by a vortex located near the wall. This is evidenced in Figure 5.16, where salt mass fraction contour lines closer to the centre of the channel indicate regions where the concentration boundary layer is thicker, and thus there is less mass transfer. On the other hand, in the places where transversal flow

110 pushes lower concentration bulk fluid towards the membrane wall, the concentration boundary layer is thinner and mass transfer is higher.

Figure 5.15. Contours of local Sherwood number along the top membrane wall (y/hch = 1) for the 90° orientation spacer at a Reynolds number of 200, showing the locations streamwise vortical flow.

Due to the differences in geometrical characteristics between the top and bottom membrane walls (the top membrane is touched by filaments parallel to the bulk flow, while the filaments touching the bottom membrane are perpendicular to the bulk flow), they also present different mass transfer trends as the Reynolds number is varied. From Figure 5.17 it can be observed that despite the wall shear rate always being lower on the bottom membrane wall, that surface always accounts for more than half of the mass transfer. The top membrane wall only experiences boundary layer separation and reattachment in the region near the spacer filaments parallel to the bulk flow, which are caused by the streamwise vortices present in that region. In contrast, the concentration boundary layer along the bottom membrane wall is strongly destabilized by the presence of the MSV, which generates regions of distinct boundary layer separation and reattachment. This highlights the importance of boundary layer destabilization by means of its separation and reattachment as a more efficient method for enhancing mass transfer than only increasing wall shear.

111

Figure 5.16. Channel cross-section of the 90° orientation spacer at x/hch = 0.1 and x/hch = 3, at Reynolds number of 200, showing solute mass fraction contour lines with tangential velocity vectors superimposed.

Figure 5.17. Mass transfer and wall friction comparison for top (y/hch = 1) and bottom (y/hch = 0) membrane walls for 90° orientation spacer.

112 5.3.3.3. 45° Orientation

Figure 5.18. Contours of local Sherwood number along the bottom membrane wall (y/hch = 0) for the 45° orientation spacer at a Reynolds number of 200, showing the locations of streamwise vortices. Bold lines show the vortices with positive z-vorticity.

The regions of high mass transfer for this geometry are located approximately on the membrane wall opposite a filament, roughly midway between successive filament intersections. This can be seen in Figure 5.9 for a Reynolds number 100 and in Figure 5.18 for a Reynolds number of 200. These regions are located slightly upstream of the wall shear maxima, and are associated with regions of flow towards the wall due to flow around the filaments and the spiralling motion of the free vortex. Likewise, the regions of lower mass transfer are located where the action of the free vortex pulls fluid away from the membrane wall. For Reynolds numbers below 100, despite the absence of a free vortex in the region downstream of the filament intersection, flow is nevertheless towards the membrane wall opposite the spacer filaments (see streamlines for Reynolds

113 number of 50 in Figure 5.10). Consequently, a region of higher mass transfer is located in that area. These results qualitatively agree with the experimental findings of Kim et al. [92], who found that the local Sherwood number was higher at the wall locations on the opposite side of the flow obstructions. In addition, their measurements for a geometry similar to our 45° orientation spacer also agree with our calculations, which show that the region in between spacers presents a lower Sherwood number relative to the rest of the membrane wall, for the Reynolds number range studied in this chapter. Another mass transfer feature of this geometry is an alternating low and high local Sherwood number pattern found upstream of each filament. This pattern is caused by vortices located on the upstream side of the filament, similar to those encountered for both the cavity and zigzag 2D spacers [114, 115], which can be observed when analysing the normalized helicity patterns (see Figure 5.11).

Figure 5.19. Qualitative comparison of schematic representation of depth of fouling coverage, particle deposition patterns [3, 4] and local Sherwood number for the top membrane wall of the 45° orientation spacer, at a Reynolds number of 100. Darker areas represent a deeper coverage and a higher local Sherwood number on the respective schematics.

The mass transfer patterns found for this orientation roughly agree with the particulate fouling patterns obtained by Neal et al. [3] using the direct observation through the membrane (DOTM) technique. As evidenced by Figure 5.19, the areas of higher mass transfer on the membrane are approximately aligned with those that are more prone to particulate fouling. A similar relationship can also be observed for the 90° orientation spacer, as shown in Figure 5.20. This might seem counter-intuitive at first, since higher mass transfer rates are generally associated with higher permeation fluxes, and fouling is known to lead to a decrease in permeation flux. However, since

114 the density of particles such as those used by Neal et al. [3] is greater than that of the surrounding fluid, they do not exactly follow the flow streamlines, but deviate due to inertial effects. In the 45° orientation, the free vortex located downstream of the filament intersection pushes fluid towards the membrane walls at locations where wall shear rates are low. Likewise, in the 90° orientation, the MSV acts in a similar manner. Although particulate fouling is not directly encouraged by high mass transfer rates, both of these phenomena are enhanced by flow towards the membrane wall, and low wall shear rates encourage the growth of the fouled layer. This helps explain the similar location of particulate fouling and high mass transfer regions. In addition, Li et al. [103] have pointed out that flux variations on the membrane surface would lead to some places on the membrane surface being more prone to particle deposition than others, given that they did not observe particulate fouling at lower permeation rates. Heterogeneous particulate fouling due to local permeation rate variations has also been reported by Kim et al. [185, 186].

Figure 5.20. Qualitative comparison of schematic representation of depth of fouling coverage, particle deposition patterns [3, 4] and local Sherwood number for the top membrane wall of the 90° orientation spacer, at a Reynolds number of 100. Darker areas represent a deeper coverage and a higher local Sherwood number on the respective schematics.

5.3.3.4. Comparisons of Mass Transfer effects between different geometries When comparing the different simulation results in Figure 5.13, we can see that mass transfer is higher for the 3D cases than for the 2D flows. This increased mass transfer is thought to be a product of a combination of 3D effects such as increased mixing due to streamwise vortices, spanwise vortices being open (which reduces the

115 residence time of fluid in the recirculation regions) and increased viscous shear on the membrane in the direction perpendicular to the bulk flow (which decreases the boundary layer thickness). In the case of the 90° orientation spacer, since the MSV is weaker for this 3D geometry than for 2D cavity flow, it means that the increased mixing and mass transfer seen for this case must come primarily from the streamwise vortices, which do not occur in the 2D cavity spacer. At a Reynolds numbers of 200, this difference is reduced due to the increased relative importance of spanwise vortices compared to streamwise vortices, as seen in Figure 5.7. It can also be seen in Figure 5.13 that the 45° orientation spacer has the highest Sherwood number at any given Reynolds number. Furthermore, the Reynolds number dependence of Sherwood number for the 45° orientation spacer follows a similar tendency to that of the zigzag spacer, and the 90° orientation spacer shows a trend similar to that of the cavity spacer; i.e. the rate of increase of Sherwood number as the Reynolds number is increased tends to decrease for both the cavity and 90° orientation spacers at higher Reynolds numbers, whereas it maintains its rate of change for the other two geometries. The deceleration of the increase in Sherwood number for the 90° orientation and cavity spacers appears to be related to the MSV covering the whole cavity in between successive filaments. The onset of this phenomenon can be observed when comparing the contribution of form drag to energy losses. Form drag is caused by the pressure difference on opposite sides of a flow obstacle. Since vortices are associated with regions of relatively lower pressure, a vortex downstream of an obstacle significantly increases the form drag on that obstacle. Spanwise vortices attached to the downstream side of the filaments were found in all of the geometries studied in this chapter. The effect of attached vortices on form drag as the Reynolds number increases (and vortices grow in both intensity and size) is an increase in the percentage of energy losses due to form drag (see Figure 5.21). However, when the MSV is large enough to reach the downstream filament, the pressure on both sides of each filament becomes relatively low and the increase of form drag is slowed. This occurs around a Reynolds number around 200, which is the same Reynolds number range where the difference in Sherwood number between the cavity and 90° orientation spacers begins to decrease.

116

Figure 5.21. Dependence of energy losses due to form drag on Reynolds number for different geometries.

As is evidenced in Figure 5.21, a higher percentage of energy losses are due to form drag in the 90° orientation than in the 45° orientation. At Reynolds numbers below 200, both geometries feature an increase in the percentage of form drag due to the formation of vortices downstream of the spacer filaments. However, as seen in Figure 5.22, viscous drag for the 45° orientation spacer remains higher than for its 90° counterpart. This is mainly due to the absence of a fully formed recirculation region (the MSV). Although z-wall shear does not contribute to viscous drag since its cumulative effect is symmetrical, it still decreases the thickness of the boundary layer, and therefore also increases mass transfer. It has been argued that energy losses due to form drag do not contribute to mass transfer enhancement, whereas viscous drag is believed to be beneficial to mass transfer [67, 133]. However, from Figures 5.13, 5.21 and 5.22 it is evident that the amount of form and viscous drag are not the only parameters that determine the extent of mass transfer. For example, the zigzag geometry has a consistently higher Sherwood number while also having a higher form drag percentage than the cavity geometry. Moreover, since the percentage of energy losses due to viscous drag for the cavity spacer stabilizes beyond a Reynolds number of 200, after having been consistently decreasing after a Reynolds number of 10, one could expect

117 that the Sherwood number would increase at a faster rate after a Reynolds number of 200. However, it actually increases at a slower rate.

Figure 5.22. Dependence of energy losses due to viscous drag on Reynolds number for different geometries.

In the case of the zigzag spacer, form drag is higher than for the cavity and 90° orientation spacer because of the appearance of the fully formed recirculation region in the latter. However, the boundary layer reattachment region associated with the MSV in both the cavity and 90° orientation spacers is on the opposite membrane wall from where the highest shear rate occurs. On the other hand, the recirculation region that increases form drag for the zigzag spacer causes the boundary layer to reattach near the point of highest wall shear rate and, as a result, it presents higher overall mass transfer than the cavity and 90° orientation. Therefore, when trying to optimize spacer geometry, it is important to consider the effect of the geometry on the flow, such as boundary layer separation and reattachment regions, and not just its effect on viscous and form drag, as exemplified by the fact that an increase in form drag in the zigzag geometry actually enhances mass transfer.

5.4. Conclusions Two orientations of a 3D non-woven cylindrical filament spacer mesh were simulated using a periodic unit-cell approach, and applying a fully-developed mass

118 fraction profile as the boundary condition for the solute. The simulations were restricted to the steady laminar flow regime. The locations where mass transfer is greater were found to be related to the position of both streamwise and spanwise vortices. The 45° orientation spacer presents geometric characteristics that promote mass transfer to a greater extent than the 90° orientation. These characteristics include the absence of a fully formed recirculation region and an increase of z-wall shear as the Reynolds number is increased. The 3D geometries modelled present greater mass transfer enhancement than their 2D counterparts. This is due to 3D effects such as the occurrence of streamwise vortices, open spanwise vortices, and higher wall shear rate perpendicular to the bulk flow. However, the magnitude of these 3D effects for the 90° orientation spacer is reduced after a Reynolds number of 100, such that at a Reynolds number of 200 its Sherwood number is approximately the same as that for the 2D zigzag spacer. At higher Reynolds numbers, where flow is unsteady but oscillates periodically, it is expected that the flow conditions that lead to mass transfer enhancement will remain. However, the flow variations will cause mass transfer to oscillate in a way such that the time averaged enhancement will be higher than in the steady laminar case analysed in this chapter. At Reynolds numbers such as those encountered in real world SWM modules, the main component of energy losses for the geometries analysed is form drag. Therefore, greater energy savings would be possible by reducing this type of drag. However, not all types of form drag are detrimental to mass transfer. Vortices located downstream of flow obstacles increase form drag, but they also promote boundary layer renewal through flow separation and reattachment, which greatly enhances mass transfer. In contrast, increases in viscous drag generally increase mass transfer by decreasing the thickness of the boundary layer, thus reducing mass transfer resistance. A spacer design for optimal mass transfer should primarily encourage flow towards the membrane wall such that the fluid in the region close to the wall is constantly being renewed, while minimizing form drag that does not produce this renewal. Wall shear perpendicular to the bulk flow should also be encouraged when it does not produce form drag, due to its ability to reduce the boundary layer thickness. A combination of streamwise and spanwise vortices is a possible means of achieving the above mentioned objectives.

119 Chapter 6

Multi-layer spacer designs for minimum drag and maximum mass transfer

6.1. Introduction Despite the beneficial impact of spacer meshes on mass transfer, some energy losses do not translate into enhanced mass transfer, as was seen in chapter 5. Different spacer configurations will lead to different mass transfer to energy losses ratios [95]. In addition, chapters 4 and 5 suggest that certain flow characteristics lead to increased mass transfer (e.g. flow towards the membrane wall and wall shear perpendicular to the bulk flow direction). Therefore, it is expected that spacer filament designs which promote these flow characteristics, while keeping increases in energy losses to a minimum, would result in further economic improvements for the operation membrane units. As explained in chapter 2, spacer configurations consisting of more than the typical 2 layers of filaments have been proposed [67, 96]. The efficacy of multi-layer spacers is due to the ability of smaller near-wall filaments to promote the formation of recirculation regions and vortex shedding in the vicinity of the membrane wall, thus disrupting the boundary layer. Additionally, the middle spacer filament layer or layers serve the purpose of re-directing the low-concentration bulk flow towards the membrane walls. Since form drag caused by the recirculation regions and vortex shedding behind the middle layers does not result in boundary layer separation or reattachment, it is therefore possible to reduce energy losses by optimising the filament profile of the middle layer spacer filaments. Kim and Kim [187] developed a numerical algorithm to calculate the minimum drag profile in 2D flow. They found that this profile depends on the Reynolds number. They also found that elliptical profiles are very close to the optimal profiles, with a total drag never more than 0.1% higher. The total drag of a circular profile was 4.54% higher at a Reynolds number of 1, but this ratio quickly increased to above 30% higher at a Reynolds number of 40. As elliptical filaments present desirable drag characteristics, they appear as the obvious alternative for a middle-layer spacer filament.

120 This chapter considers possible multi-layer spacer configuration designs for improving the mass transfer enhancement characteristics of typical zigzag spacers. For this purpose, various configurations incorporating elliptical filaments as the middle spacer layers are simulated and analysed. The main aim of this chapter is to understand the effect of these multi-layer spacer designs on mass transfer. Laminar steady and unsteady flow conditions are investigated. In addition, an economic analysis is carried out in order to compare the relative cost-effectiveness performance of the designs tested.

6.2. Problem Description, Assumptions and Methods As in the preceding chapters, the commercial CFD code ANSYS CFX is used to solve the continuity, momentum and mass transport equations [9] in a spacer-filled channel. Constant solution properties are employed and the effect of gravity was excluded. In order to further simplify this system the fluid is assumed to be Newtonian, the flow two-dimensional, and a binary mixture of water and salt is considered, with no sources of salt in the fluid. Mass transfer is incorporated in the form of a dissolving wall boundary condition.

Figure 6.1. Solute mass fraction contours for different spacer filament diameters at a Reynolds number of 100.

The geometries analysed in this chapter are variations of the 2D zigzag spacer arrangement studied in chapters 4 and 5. The zigzag geometry was chosen because it presents the most similarities to spacers used in real membrane modules, and was also found to be better performing than the other geometries studied by Schwinge et al. [115], with regards to its mass transfer and pressure loss characteristics. The configuration in chapter 5 (df/hch = 0.6 and lm/hch = 4) was altered by decreasing the

121 df/hch ratio while keeping the ratio of lm/df constant, and by introducing a third spacer layer consisting of submerged elliptical filaments with various angles of attack (e). The lm/df ratio was kept constant in order to maintain the same membrane area covered by the spacer mesh. As a consequence of keeping the lm/df ratio constant, changes in filament diameter for a given Reynolds number do not affect the aspect ratio (length to height) of the recirculation regions formed due to the presence of the obstruction, as seen in Figure 6.1. In addition, the geometries considered were chosen to have the same 4 dh ratio. Therefore, as explained in chapter 2, comparisons at the same Power

3 number ( Refh ) value would have the same pumping power requirements per unit volume of the respective channel. As a result, increases in mass transfer performance at

3 the same Refh value are directly related to changes in spacer arrangement, and would represent a better performing spacer configuration regardless of the economic impact of pumping costs.

Figure 6.2. Generic unit cell geometry of the spacer type modelled in this Chapter.

A general unit cell for this family of geometries is shown in Figure 6.2. The aspect ratio of the elliptical filaments (aE/ae) was chosen such that it was as close as possible as that of the optimal ellipse for minimum drag, as reported by Kim and Kim

[187]. This gave an aE/ae ratio of 5 for the cases with a df/hch ratio of 0.4, and an aE/ae ratio of 4 for the cases with a df/hch ratio of 0.3. For the cases where the angle of attack of the elliptical filaments was not zero, the elliptical filaments were tilted in opposite directions in order for the lift generated by these filaments to cancel each other. The distance from the centre of the elliptical filaments to the channel walls was chosen to be either half of the channel height or as given by equation (6.1).

122 1 () hhadeche3 0.3 f (6.1)

The configurations considered are portrayed in Figure 6.3. In the 3-layer (3L) configuration, the elliptical filaments are placed equi-distant from both membrane walls. In the 4-layer (4L) configurations, successive elliptical filaments are placed at a distance from one of the membrane walls as given by equation (6.1). This was done in order to allow the reduction in elliptical filament size while keeping the spacer configuration representative of a real-world spacer mesh, i.e. without leaving gaps in the channel height. For each 4L spacer there are two possible locations for the first elliptical filament, depending on which membrane wall it is closer to. If the first elliptical filament is closer to the membrane wall to which the upstream circular filament is attached, the configuration is designated as “Low” (-L). Conversely, if the filament is closer to the membrane wall to which the downstream circular filament is attached, then it is designated as “High” (-H). The designation by which each spacer configuration will be referred to in this chapter is indicated in parentheses in Figure 6.3.

Figure 6.3. Schematic of the unit cells for the different spacer configurations analysed in this chapter.

It must be noted that geometries 2L04 and 2L03 are not representative of typical spacer geometries because the spacer height is smaller than the channel height, and a gap is present in between the spacer filaments, as seen Figure 6.3. In real world

123 membrane modules, the spacer filaments usually touch each other, as they are placed in order to keep the membrane leaves apart. Nonetheless, these geometries are realisable if a third filament parallel to the flow is introduced, or via the use of spacer protuberances. For steady laminar flow at low Reynolds numbers, it generally takes several repetitions of the unit cell before the Sherwood number reaches a limiting value. In order to avoid the need to simulate an excessively long channel at low Reynolds numbers, two separate simulations were carried out. The first consisted of an array of at least 8 repetitions of the unit cell, preceded by an entrance length of 10 times the channel height and followed by an exit length twice as long as the entrance length. These long entrance and exit lengths were necessary to fully develop the velocity profiles before the beginning of the spacer array, and to prevent the outlet condition interfering with the recirculation regions after the last filament. The entrance and exit length dimensions were chosen following the same considerations outlined in chapter 4. The second simulation involved three repetitions of the unit cell and the application of the methodology for periodic wrapping described in chapter 5. The first simulation allows quantification of the entrance effects in the channel, while the second simulation permits the calculation of the characteristics far away from the channel inlet. For unsteady flow, the Sherwood number achieves its limiting value much closer to the channel entrance than in the case of steady laminar flow, often after as few as 5 repetitions of the periodic unit cell [147]. The results in chapter 4 revealed that at least six spacer filaments were necessary for vortices to appear in unsteady flow. Therefore, the same channel characteristics were used for the unsteady runs as for the entrance region runs under steady laminar flow. As in the previous chapters, a Schmidt number of 600 was chosen, which is characteristic of typical monovalent salts, such as sodium chloride. The channel walls over the entrance and exit lengths as well as the spacer surfaces are treated as non-slip walls with no mass transfer, where both velocity components and the mass fraction gradient normal to the boundary are set to zero (u = v = 0 and Ynw 0 ). The membrane walls are also treated as non-slip walls (u = v = 0), but the mass fraction at the wall is fixed at a constant value (Y = Yw).

For the entrance region and unsteady runs, a flat velocity profile with u = uavg, v = 0 and Y = 0 is specified at the inlet of the flow domain. At the outlet, an average reference pressure of zero is specified. For the steady laminar runs under fully

124 developed conditions, the inlet and outlet boundaries were treated as fully developed velocity and mass fraction profiles, using the approach described in Chapter 5. The results were taken from the middle unit cell, following the approach of Rosaguti et al. [148], in order to avoid the error introduced due to the lack of gradient information at the periodic boundaries. Mesh independent solutions were obtained using an unstructured mesh with element sizes of the order of at most 3% of the circular filament diameter. This element size was determined after a series of runs with increasingly finer meshes. Inflated boundaries were used for all the boundaries with a non-slip condition, where the thickness of the first grid element layer was of the order of at most 0.15% of the circular filament diameter. Angular resolution on the surfaces of the filaments was set to 17, and to 15 for the “corners” between the filaments and the membrane wall. The time steps used for the unsteady runs were determined by decreasing the time step size until the solution did not depend of this parameter. Typically, the RMS Courant number for the unsteady runs was less than 0.2, and the maximum Courant number was less than 3. Following the approach used in the previous chapter, an approximate steady-state solution was chosen as the initial state (t = 0) for the transient simulations. Using this initial condition, the statistical quantities of all the variables stabilized after approximately one residence time.

6.2.1. Channel Interpolation In this chapter, solutions are obtained for the entrance region and for conditions of fully developed velocity and mass fraction profiles. These two results can be used to estimate the mass transfer and pressure drop conditions for the region where entrance effects are not dominant, but the mass transfer conditions are not yet fully developed. Churchill and Usagi [188] developed a simple correlation to approximate intermediate solutions to transfer processes for which asymptotic solutions are known at large and small values of the independent variable. Mass transfer to a narrow channel under laminar flow fits this category of problems, with the Sherwood number as the dependent variable, and the dimensionless channel length as the independent variable. Following the approach of Churchill and Usagi, the following is a suitable interpolation expression:

125 1 FV Sh() x Sh (6.2) HXGW* fd

A similar expression is also applicable for the friction factor:

1 FV fxf() (6.3) HXGW fd

Entrance region effects are quantified by parameters and in equation (6.2), and parameters and in equation (6.3). Parameters and quantify the transition from entrance region to fully developed flow, as seen in Figure 6.4; i.e. a very high or would signify a sharp transition, which would follow the entrance region trend until it reaches with the fully developed value, and then follow the latter. A lower or would result in a smoother transition, similar to a hyperbolic function with the entrance region and fully developed curves as asymptotes. The parameters in equations (6.2) and (6.3) were estimated via a non-linear regression using the results from the channel simulations in the laminar regime.

Figure 6.4. Conceptual diagram of interpolation functions, as defined by equations (6.2) and (6.3), showing asymptotes and interpolation curves.

126 6.2.2. Cost Estimation In order to effectively compare the relative performance of different spacer geometries in real-world operations, an economic analysis is needed. However, a thorough economic analysis would have to take into account many factors that vary greatly with geographical region and also over time including salaries, equipment cost and interest rates. Such a complete analysis is outside the scope of this study. However, it is possible to obtain an approximation of total costs by analysing the direct costs for the production of permeate, without taking into account pre-treatment costs, which are independent of the spacer geometry in the SWM units. In addition, Maskan et al. [189] explain that cleaning costs are largely proportional to the membrane area, and can therefore be considered as being incorporated with the membrane cost. Moreover, due to the nature of the spacer prototypes considered in this chapter, estimation of their production costs would be outside the scope of this work. Therefore, the cost analysis carried out in this chapter does not directly take into account membrane cleaning costs, and ignores pre-treatment and spacer costs, focusing instead on direct permeate processing costs. Permeate production is directly proportional to the average permeate flux. The local permeate flux can be calculated following the approach of Kedem and Katchalsky [51] and Merten [52], which yields the following expression:

() JLpsln p tm tm (6.4)

The osmotic pressure of the solute can be approximated by a linear expression [2]:

Y (6.5) where the value for the osmotic pressure coefficient () used was the value reported by Geraldes et al. [116], and is shown in Table 6.1. Substituting equation (6.5) into (6.4) gives the following expression for the local permeate flux in terms of the local wall solute mass fraction:

FV () JLpsln pHX tm YYw p (6.6)

127 The local solute mass fraction at the membrane wall can be directly obtained from the CFD simulation results. However, a more convenient approach is to use the mass transfer coefficient data obtained from the simulations, where the local mass transfer coefficient is defined as:

DYCS k DT (6.7) mt () YYwbEU yw

From the mass balance of solute at the membrane surface, it is also known that:

CSY JY DDT JY (6.8) sln w sln p EUy w

Combining equations (6.7) and (6.8) yields the following expression for the permeate flux:

(YY ) Jk wb (6.9) sln mt ()YYwp

Equating (6.6) and (6.9), and subsequently solving for the solute mass fraction at the wall (Yw) yields:

2 11CSCSkkkFV YYDTDT p mt GWmt p mt () YY (6.10) wp DTDTtm tm bp 22EUEULLppHXGW L p

Finally, substituting (6.10) into (6.6) gives the following expression for the permeate flux:

FV2 11 JLpksln() p tm mt GW() kLpkLYYmt p tm mt p () b p (6.11) 22HX

Equation (6.11) can be used to calculate the local permeate flux in terms of the operating parameters and the mass transfer coefficient. However, the mass transfer coefficient obtained from the CFD simulation results is calculated using impermeable wall conditions, i.e. a dissolving wall where there is no fluid permeating through the wall. In the real case, the permeation of fluid through the membrane wall alters the mass

128 transfer conditions, and hence the mass transfer coefficient, when compared to the dissolving wall case. Geraldes and Afonso [136] have shown that the mass transfer coefficients for permeable wall conditions can be related to the impermeable (dissolving wall) mass transfer coefficients using the following relationship.

Sh k 1.7 per mt, per (10.261.4 ) (6.12) Shimp k mt, imp where the variable is the ratio of permeation Peclet number to the impermeable Sherwood number, and can be calculated using the following:

J sln (6.13) kmt, imp

Permeate flow is calculated using the following expression:

AJ Q mslnavg, (6.14) p

where the membrane area (Am) is the product of the width and length of the channel, and the average permeate flux (Jsln,avg) can be estimated as the average of the permeate flux values calculated for the inlet and outlet flow conditions of the membrane module. In this chapter, the permeate flux at the inlet flow conditions is calculated using equation (6.11) and the inlet conditions (pin, kmt,in,per and Yb,in). Since the permeate flux is an input for calculating the permeable wall mass transfer coefficient, an iterative procedure is followed where an initial estimate of the permeate flux is used to calculate the permeable wall mass transfer coefficient using equations (6.12) and (6.13), and this value is in turn used in equation (6.11) to calculate a revised estimate of the permeate flux. This procedure is repeated until the initial and revised estimates of the permeate flux are equal to 6 significant figures. Likewise, the permeate flux at the outlet is calculated using equation (6.11) and the outlet conditions (pout, kmt,out,per and Yb,out) in an iterative manner. In order to calculate the outlet operating conditions, the following relationships are used:

Quhwin avg, in ch ch (6.15)

129 QQQout in p (6.16)

QY QY in b, in p p Yb, out (6.17) Qout

Q out uavg, out (6.18) hwch ch

ptm,, outpp tm in ch (6.19)

2 2 L p Re2 f (6.20) ch 3 h, avg dh

In equation (6.20), the average Reynolds number is calculated as the average of the inlet and outlet Reynolds numbers. The iterative procedure for the calculation of the permeate flux at the outlet is slightly more complex than that for the permeate flux at the inlet. This is because the outlet permeate flux estimate is also used for calculating the permeate flow (Qp) through equation (6.14), which in turn is used for calculating the outlet Reynolds number through equations (6.16) and (6.18). The value for the permeate flow is given by equation (6.14) once converged values for both the inlet and outlet permeate fluxes are obtained. The permeate flow value is then used for calculating the capital cost for the membrane unit per cubic metre of permeate flow using the following expression:

AC F mma Cc (6.21) Qtpop

Operating costs are proportional to the pumping energy, which is equal to:

WpQs ch in (6.22)

The operating cost per cubic metre of permeate flow is then given by:

WC C se (6.23) op Qp pump

130 Finally, the total processing cost per cubic metre of permeate flow is calculated as the sum of the operating and capital costs:

CCCtot op c (6.24)

6.3. Results and Discussion

6.3.1. Two-layer spacer geometries The first series of runs conducted was for 2-layer (2L) spacers with circular filaments, i.e. without elliptical filament layers. The df/hch ratio was varied from 0.6 down to 0.3, while keeping the ratio of lm/df at a constant value of 6.667. As the filament diameter is decreased, the cross-section of the channel that is obstructed is also decreased. It is therefore expected that for lower filament diameter to channel height ratios the friction factor will be lower, approaching the value for an empty channel. Figure 6.5 shows that this is the case.

Figure 6.5. Friction factor dependence on Reynolds number for 2-layer spacer-filled channels with

varying df/hch ratios.

The effect of filament diameter on Sherwood number is more complex than its effect on the friction factor. For empty channels, the Sherwood number for flow with a fully developed concentration profile does not depend on the Reynolds number.

131 However, as shown in Figure 6.6, for spacer-filled channels the Sherwood number increases as the Reynolds and Power numbers are increased, as a consequence of increased boundary layer destabilisation at higher Reynolds numbers. At low Reynolds numbers, where no boundary layer separation occurs and creeping flow conditions are prevalent, the Sherwood number will be lower for the spacer-filled channels than for the empty channel as a result of the spacer filaments covering the membrane surface and reducing the available mass transfer area, i.e. “membrane coverage”. This effect is evidenced for Power numbers below 105 in Figure 6.6.

Figure 6.6. Sherwood number dependence on Power number for 2-layer spacer-filled channels with

varying df/hch ratios.

As mentioned in chapter 2 for empty channels, entrance region effects caused by the developing boundary layer result in a higher Sherwood number near the channel inlet. As the distance from the channel inlet is increased, the Sherwood number eventually reaches its asymptotic value. Since further increases in channel length do not affect the Sherwood number, the concentration profile is considered to be “fully- developed”. This phenomenon is also encountered in spacer filled channels as can be observed in Figure 6.7, which was obtained by making use of the interpolation functions described by equations (6.2) and (6.3). As a consequence of the entrance region effect,

132 the ratio of energy losses to mass transfer is increased as the channel length is increased. This suggests that short and wide channels will result in lower energy needs for given mass transfer needs.

Figure 6.7. Sherwood number distribution along channel length for spacer-filled and empty channels, at a Reynolds number of 200.

As mentioned in section 6.2. the geometries in this chapter were chosen to have 4 3 the same dh in order for comparisons at the same Refh value to have the same pumping power requirements per unit volume of the respective channel. The group

3 RefLh is therefore a measure of accumulated energy losses at a specified channel length (L). In order to make this value dimensionless, it can be multiplied by 1 4 dh (which was kept constant for all the geometries modelled in this chapter), thus 1 3 4 giving the dimensionless group Refh Ldh . From Figure 6.7 it can be seen that for a given energy loss, the Sherwood number is always higher for spacer filled channels than for an empty channel. In addition the empty channel requires less energy to attain a fully-developed concentration profile than the spacer-filled geometries tested. This is to be expected given that pressure drop for the empty channel is significantly lower than for the spacer-filled cases, as was shown in Figure 6.5. In addition, Figure 6.7 shows that the geometries with a higher Sherwood number under fully-developed

133 concentration profile conditions will usually have a higher Sherwood number at any given value for the energy loss throughout the entire length of the channel. This is confirmed in Figure 6.7, where it is shown that the lines for the different spacers do not intersect each other, and the asymptotic value of the Sherwood number for the 2L03 geometry is slightly higher than that for the empty channel.

6.3.2. Multi-layer spacer geometries In order to assess the effect of the angle at which the elliptical filaments intersect the oncoming bulk flow, simulations modelling configurations with different angles of attack were undertaken. For the purposes of isolating the effect of the angle of attack, the Reynolds number was not varied and kept constant at a value of 200. This Reynolds number is representative of typical operating conditions in real-world membrane systems which operate in the laminar flow regime.

Figure 6.8. Velocity contour plots for the 3L04 spacer configurations at various angles of attack, and a Reynolds number of 200.

134 Aerodynamic studies show [190] that for airfoils, form drag and its variation are small at small angles of attack. Moreover, as the angle of attack is increased above a threshold value (usually in the range of 5° to 15° for Reynolds numbers of the order of 100,000) flow separation causes form drag to increase drastically (usually doubling or tripling its value). Therefore, the angle of attack of the elliptical filaments in this study was kept within the range of +10° to -10°. Due to the relative positioning of the circular and elliptical spacer filaments and the definition of the angle of attack, a negative angle of attack is expected to align the elliptical spacer filament with the bulk flow such that form drag is reduced. Figure 6.8 shows that as the attack angle increases, a larger region of stagnant fluid forms downstream of the elliptical filaments, thus increasing form drag. It can also be seen that this region of stagnant flow, caused by boundary layer separation, is smaller for the negative angles of attack than for the positive values. At an angle of -5° the velocity of the fluid flowing above and below the elliptical filament is approximately equal and the stagnant region is smaller than at the other angles of attack. Therefore, for the 3L04 spacer, form drag is expected to be lower for an attack angle of -5° than for the other values.

Figure 6.9. Variation of the percentage of energy losses due to form drag with angle of attack for multi- layer spacer geometries, at a Reynolds number of 200.

135 The effect of the angle of attack on form drag for all spacer configurations and angles of attack can be seen in Figure 6.9. For all these configurations the lowest percentage of energy losses due to form drag is attained at an angle in the range of -5° to -7.5°. In the case of the 3L04 configuration, form drag is reduced as the angle of attack is reduced from 0° to -5°, and increases again as the angle is reduced further to -10°. This suggests that, as expected from Figure 6.8, an angle of -10° overshoots the negative angle value necessary to align the elliptical filament with the bulk flow. As a result, this angle causes a larger region of stagnant fluid behind the obstacle, thus leading to a larger percentage of form drag than at a -5° angle.

Figure 6.10. Dependence of friction factor on angle of attack for multi-layer spacer geometries, at a Reynolds number of 200.

Since form drag is the main component of pressure drop, it can provide an indication of the behaviour of the friction factor as the angle of attack is changed. As can be seen in Figure 6.10, however, the pattern of lower friction factor for negative angles of attack is only followed by the 4L04L and 4L03L configurations. For the 3L04, 4L04H and 4L03H configurations the friction factor is at its lowest value at an angle of attack between -5° and 0°. The increase in friction factor at negative angles of attack for the –L configurations is due to an increase in the viscous drag. Figure 6.9 confirms that

136 the percentage of pressure drop due to form drag for these cases is lower than for the other configurations.

Figure 6.11. Velocity contour plots for the 4L03H, 4L03L, 4L04H and 4L04L spacer configurations at an attack angle of -10°, and a Reynolds number of 200.

The reason for the increase in viscous drag is visualised in Figure 6.11. It can be seen in this figure that for the 4L03L and 4L04L configurations the relative positioning of the elliptical filament to the circular filaments causes most of the fluid to flow on one side of the elliptical filaments. This channelling of the fluid creates a large area of stagnant or slow moving fluid downstream of the circular filaments, between the elliptical filaments and the membrane wall. The relatively slow flowing fluid does not contribute significantly to viscous drag on the elliptical filament and the membrane wall surfaces. In contrast, for the 4L03H and 4L04H configurations the flow is more evenly distributed on both sides of the elliptical filament. Therefore, both sides of the elliptical filaments and membrane walls experience higher viscous drag in the –H configurations than in the –L configurations. This results in a higher friction factor for the –H configurations at the -10° angle of attack than at the 0° angle, despite form drag being lower for the negative angle.

137

Figure 6.12. Dependence of Sherwood number on angle of attack for multi-layer spacer geometries, at a Reynolds number of 200.

For the effect of the angle of attack on the Sherwood number, it is expected that both positive and negative angles of attack would increase the average Sherwood number over the unit cell by directing flow away from the bulk and towards the membrane walls. As seen in Figure 6.12, this is only the case for the 3L04 and (to a lesser extent) 4L04H configurations. For those cases, the non-zero angles of attack cause low concentration fluid to be diverted from the bulk flow towards the membrane surface, causing a larger degree of boundary layer renewal and thus a higher Sherwood number. In contrast, the 4L03H and 4L03L configurations show a local minima for the average Sherwood number when the angle of attack is -10° or +10°. In these latter cases, the large angles of attack reduce the velocity of the fluid flowing between the elliptical filament and the membrane wall as shown previously in Figure 6.11. This in turn reduces the wall shear rate and increases the concentration boundary layer thickness. Since less fluid is being forced to make contact with the membrane surface, less boundary layer renewal occurs and, as a result, the average Sherwood number is also reduced. This effect is evidenced by comparing the velocity and mass fraction contour plots in Figure 6.13, which show that the bulk of the fluid channels past the

138 flow obstructions following a zigzag path, and generates large stagnant areas where the solute concentration builds up and mass transfer into the bulk fluid is low.

Figure 6.13. Velocity and solute mass fraction contour plots for the 4L03L spacer configuration for various angles of attack, at a Reynolds number of 200.

For the case of the 4L04L configuration, the Sherwood number tends to increase as the angle of attack is increased. This is because the negative angle of attack reduces the amount of fluid that flows near the membrane wall, whereas the positive angle of attack increases the amount of low concentration bulk fluid that approaches the membrane wall.

Figure 6.14. Dependence of the Sherwood number to friction factor ratio on angle of attack for multi- layer spacer geometries, at a Reynolds number of 200.

139 It is well known that mass transfer can be enhanced by boundary layer destabilisation through the use of flow obstructions [65, 66, 87, 94]. These obstructions, in turn, increase the channel pressure drop, i.e. energy losses. Since different angles of attack result in different combinations of mass transfer enhancement and energy losses, the Sherwood number to friction factor ratio can be used to identify the angle of attack that achieves the best balance between mass transfer enhancement and energy losses. This comparison is shown in Figure 6.14. For all but the 4L04L and 4L03L configurations, the Sherwood number to friction factor ratio was higher at a zero angle of attack. This means that for all but the 4L04L and 4L03L configurations, the benefit of an increased Sherwood number is outweighed by the increase in energy losses caused by a non-zero angle of attack. Both the 4L04L and 4L03L configurations show a higher Sherwood number to friction factor ratio at the -5° angle of attack. It is interesting to note that, although both the 4L04L and 4L03L spacer configurations showed a lower friction factor due to the negative angle of attack, the 4L03L configuration prevented flow from reaching the region near the membrane wall, whereas the 4L04L configuration did not exhibit this behaviour.

Figure 6.15. Local Sherwood number for the bottom membrane wall of 2L04 and 3L04 spacer geometries with varying angles of attack, at a Reynolds number of 200.

For the 3L04 spacers, the configuration with a -10º angle of attack is aligned with the flow, and has lower friction factor than the configuration with a +10º angle of attack. From Figure 6.15 it can be seen that the configuration with a -10º angle of attack also has more evenly distributed local Sherwood number values along the membrane

140 wall than the other angles of attack, which results in one of the highest overall Sherwood numbers of this series of spacers. However, these overall Sherwood number values are still lower than for the configuration without an elliptical filament (2L04). In general, it can be said that increases in the angle of attack lead to increases in both the friction factor and Sherwood number. Although there are specific situations in which due to the geometrical alignment of the spacer filaments this relationship might not hold, it has been shown that this trend is generally followed. In addition, positioning the elliptical filaments in the –L configuration will result in higher Sherwood numbers.

It must be said, however, that this trend was found at a constant lm/df ratio of 6.667, and there is the possibility of a different trend if the lm/df ratio is changed significantly from this value.

6.3.3. Unsteady flow For the multi-layer spacer geometries considered in this chapter, flow becomes unsteady for Reynolds numbers above 200 to 500. In order to asses the effect of unsteady flow on spacer performance for multi-layer spacer geometries, two simulations were run at a Reynolds number of 800, for the 4L03H spacer geometries at attack angles of 0° and 10°. The simulation time for one residence time was one week for the angle of attack of 0°, and one month for the angle of attack of 10°. The data presented here was collected after more than one residence time had passed and the time averaged values for the flow variables had stabilised. The reader is also referred to videos V6.1 and V6.2 included in the disc located in the back cover of this thesis (see Appendix B). These videos depict the evolution of the salt concentration with time, for the 4L03H spacer geometries at attack angles of 0° and 10° respectively. In Figure 6.16 it can be seen that the friction factor converges to its fully developed value after only 3 repetitions of the spacer pattern for both angles of attack. In addition, the friction factor, and hence energy losses, are lower for the configuration with an angle of attack of 0° than for the attack angle of 10°. Variations of the friction factor with time, caused primarily by vortex shedding and flow oscillations, are very low for both geometries. Nonetheless, they are slightly larger for the attack angle of 10°, and are also larger near the channel entrance. The entrance region effect can be clearly observed in Figure 6.16, and it is more dramatic for the zero attack angle than for the non-zero case.

141

Figure 6.16. Friction factor time average and range along channel length for 4L03H spacer geometries at 0° and 10° angles of attack, and a Reynolds number of 800.

Although mass transfer entrance region effects are prevalent in the steady laminar flow regime, where they usually extend up to a dimensionless mass transfer length ( x* ) around 0.01, unsteady flow reaches time-averaged fully-developed mass transfer conditions much closer to the channel inlet. This effect can be seen in Figure 6.17, which shows that the channel Sherwood number stabilises after around 3 repetitions of the spacer geometry for both angles of attack simulated. As with the friction factor, the entrance region effect is more pronounced for the case with an angle of attack of zero than for the non-zero case. Moreover, the time average of the Sherwood number for both geometries is approximately the same, and both show similar degrees of variability due to unsteady flow and vortex shedding. Given that the friction factor and energy losses obtained for the case with an angle of attack of zero were lower than for the non- zero case, an angle of attack of 0° would be preferred for the flow conditions used in this set of simulations. The unsteady state results presented in this section agree with the steady state simulation results for the 4L03H geometry at a Reynolds number of 200. In particular, an angle of attack of +10° resulted in a higher friction factor but a slightly lower Sherwood number than the case at an attack angle of 0°. This suggests that the effect of the angle of attack on mass transfer and energy losses in the steady laminar flow regime can be extrapolated to the unsteady laminar flow regime.

142

Figure 6.17. Sherwood number time average and range along channel length for 4L03H spacer geometries at 0° and 10° angles of attack, and a Reynolds number of 800.

6.3.4. Economic analysis of spacer performance

Table 6.1. Case study parameters for cost analysis.

Module length (L) 90 cm

Channel width (wch) 90 cm Salt rejection 99.6%

Feed mass fraction (Yb,in) 0.025

Inlet transmembrane pressure (ptm,in) 60 atm Reflection coefficient ( ) 1 Osmotic pressure coefficient () 8.051107 Pa

-6 -1 -1 Membrane permeability (Lp) 3.9410 m s atm

Channel height (hch) 0.001 m -1 Energy cost (Ce) $0.10 kWh -2 Membrane cost (Cm) $100 m -1 Amortisation factor (Fa) 0.4 yr

Pump efficiency (pump) 0.6 -1 Operation time (top) 8000 hr yr

The economic analysis of the different spacer configurations was carried out following the approach described in section 6.2.2. In order to isolate the impact of the spacer configuration used, only processing costs were calculated; i.e. feed pre-treatment

143 costs and operating costs other than pumping through the membrane module (e.g. cleaning costs) were neglected. Therefore, the costs reported in this section do not represent the total cost of permeate production, and should only be considered as a comparative indicator of spacer performance, as explained in section 6.2.2.

Table 6.2. Sample values for various operational variables and cost components for the 4L03H spacer geometry at an angle of attack of 0°.

(kg/m·s) (kg/m·s) (m3/s) (m3/s) (Pa) (W) ($/m3) ($/m3) ($/m3) Reh Jsln,in Jsln,out Qp Qin pch Ws Cc Cop Ctot 50 0.0169 0.0108 1.12×10-5 3.9×10-5 2362 0.092 0.100 0.000 0.101 100 0.0228 0.0163 1.58×10-5 7.8×10-5 6110 0.476 0.071 0.001 0.073 200 0.0325 0.0254 2.34×10-5 1.56×10-4 16009 2.496 0.048 0.005 0.053 300 0.0449 0.0343 3.21×10-5 2.34×10-4 27869 6.518 0.035 0.009 0.045 800 0.1049 0.0913 7.95×10-5 6.24×10-4 120306 75.034 0.014 0.044 0.058

A set of cost parameters, representative of what is currently found in literature [93, 95, 176], was utilised for the cost analysis case study. These parameters are summarised in Table 6.1. In addition, three different membrane costs were considered: a case study base cost of $100 per square metre, a higher cost of $500 per square metre, and a lower cost of $10 per square metre of membrane. The high membrane cost is representative of a membrane system which is very prone to fouling, and thus includes the cost of cleaning and replacement in case of blockage. The low membrane cost is representative of the possible future membrane costs, since current trends indicate that membrane production costs will continue to decrease. A sample cost breakdown is presented in Table 6.2. As the Reynolds number is increased, the components of the total cost (Cc and Cop) vary in opposite directions. Given that energy losses increase as the Reynolds number increases, the operating costs also increase. On the other hand, increases in Reynolds number lead to increases in Sherwood number, which in turn leads to decreased membrane area needs due to increased mixing and mass transfer. Therefore, typical cost curves follow a “U” shaped trend (see Figure 6.18). At low Reynolds numbers, well within the laminar flow regime, capital costs dominate due to low mass transfer and low energy losses. As the Reynolds number is increased and capital costs are reduced, operating costs are increased until they become the dominant component. In the region where operating costs dominate, the total cost increases as the Reynolds number is increased further. Therefore, there exists a point in any total cost curve at which the total cost is minimised, and either an increase or decrease of Reynolds number would lead to an increase in total cost. The

144 Reynolds number at which this minimum total cost occurs will be referred to as the “optimal” Reynolds number for operation. For the data presented in Figure 6.18, the optimal Reynolds number for the empty channel is between 1000 and 2000, and around 400 for the 4L03H spacer geometry.

Figure 6.18: Dependence of cost components on Reynolds number for two sample geometries. A membrane cost of $100 per square metre was assumed.

6.3.4.1. Base membrane cost of $100/m2

Figure 6.19. Dependence of total processing cost per cubic metre of permeate on Reynolds number, for empty, 2-, 3- and 4-layer spacer-filled channels. The graph on the left shows effect of filament diameter

145 on cost. The graph on the right compares costs for selected multi-layer spacers against empty channel and 2L06 spacer. A membrane cost of $100 per square metre was assumed.

Figure 6.19 depicts the effect of filament diameter and Reynolds number on permeate processing costs for the geometries analysed in this chapter. Despite energy losses encountered in spacer filled channels being larger than for empty channels, the cost of permeate production using spacers is lower than the cost without using spacers. Modules using spacers have smaller membrane area requirements due to the higher Sherwood numbers attained, thus offsetting the higher energy costs required for their operation. This result agrees with previous studies [64, 93]. However, at Reynolds numbers below 100, spacer geometries 2L03 and 2L04 do not present significant cost benefits over the empty channel. As explained in section 6.3.1. this is due to the amount of membrane area covered by the spacer filaments in these configurations. Membrane coverage results in lower Sherwood numbers, and hence higher permeate processing costs than for the channel without spacers. In any case, operating at Reynolds numbers below 100 would not be recommended under the conditions of this study, as Figure 6.19 shows that lower permeate processing costs can be achieved by operating in the Reynolds number range of 300 to 1000. In addition, operating at low Reynolds numbers also increases the potential impact of fouling. The results presented in Figure 6.19 show that lower processing costs are achieved by the 2L04 and 2L06 spacer geometries. For both of these geometries, as well as for the 2L03 spacer geometry, the trends indicate lower costs as the Reynolds number is increased. Given that it is not clear whether a minimum in the cost curve has been reached at the highest Reynolds number for which data was collected, it is unclear which filament diameter to channel height ratio will achieve the lowest permeate processing cost. Moreover, flow becomes unsteady for these geometries at Reynolds numbers higher than the ones presented in Figure 6.19. Therefore, transient data is needed before the best performing spacer configuration for these cost parameters and flow conditions can be determined. Nonetheless, one conclusion can be drawn from these simulations results: for 2D 2L zigzag spacer meshes operating in the laminar flow regime, significant permeate cost reductions cannot be achieved by reducing the filament diameter to channel height ratio from 0.6 to 0.4 or 0.3. The multi-layer spacer geometries proposed in this chapter show lower permeate processing costs than the 2L06 geometry for Reynolds numbers below 200. This is

146 partially due to the decrease in open volume inside the channel (due to the added spacer layers) which leads to an increase in local velocity and wall shear rates. It is also due to the increased mass transfer enhancement caused by the elliptical filaments directing low concentration fluid towards the membrane wall. However, the added filament layers also increase energy losses and, thus, operating costs. At Reynolds numbers above 300 the ratio of energy costs to membrane costs is higher for the multi-layer configurations than for the 2L06 spacer and, as a result, lower permeate processing costs are achieved with the 2L06 spacer at Reynolds numbers above 300.

Figure 6.20 shows a comparison of the different geometries at a df/hch ratio of 0.4. The results shown in Figure 6.20 are similar to those seen in Figure 6.19, in that multi- layer spacers present lower permeate processing costs at Reynolds numbers below 200. In addition, the 3L04 spacer at an angle of attack of 0° results in lower costs than its 4L04H counterpart. From the trends shown in Figure 6.19 it is difficult to predict which geometry would be the better overall performer as the Reynolds number is increased beyond 200, which would lead to unsteady flow. Transients simulations would be necessary in order to determine which configuration would result in the lowest permeate processing cost.

Figure 6.20. Dependence of total processing cost per cubic metre of permeate on Reynolds number, for spacer-filled channels with a df/hch ratio of 0.4. A membrane cost of $100 per square metre was assumed.

147 The scenario for the geometries at a df/hch ratio of 0.3 is somewhat similar to that for the geometries at a df/hch ratio of 0.4. As opposed to the cases at df/hch ratio of 0.4 and 0.6, transient simulations were carried out for the 4L03H geometries, and these results are presented for the Reynolds number of 800 in Figure 6.21. It can be seen that the use of 4-layer spacer geometries resulted in lower permeate processing costs for all of the Reynolds numbers tested in the laminar flow regime, but the use of the 2L03 spacer resulted in the lowest processing costs in the unsteady flow regime, at a Reynolds number of 800. From the data collected from the transient simulations, it can be concluded that under the assumptions of this study, operating at such a high Reynolds number is not recommended for these 4-layer geometries as the increased energy costs associated with the increased energy losses outweigh the benefits of increased permeate production due to increased mass transfer. However, operating at a high Reynolds number may be justified for other reasons not considered here, such as reduced cleaning costs due to reduced fouling. Nevertheless, the low cost achieved by using the 2L03 spacer at a Reynolds number of 800 cannot be reduced by using the 4- layer spacers at a df/hch ratio of 0.3.

Figure 6.21. Dependence of total processing cost per cubic metre of permeate on Reynolds number, for spacer-filled channels with a df/hch ratio of 0.3. A membrane cost of $100 per square metre was assumed.

The economic impact of configuration variations was also analysed. The effect of filament positioning and angle of attack on permeate processing cost for multi-layer

148 spacer meshes is shown in Figure 6.22. From these results it can be seen that changes in positioning of the elliptical filaments (i.e. –H or –L) have a larger effect on permeate processing costs than changes to the angle of attack. Regardless of the angle of attack, the –L configurations for the 4-layer spacers always resulted in lower permeate processing costs than the –H configurations. This result agrees with the predictions using the Sherwood number to friction number ratio (see Figure 6.14), and can be attributed to the fluid channelling effects seen in Figure 6.11 and Figure 6.13. The positioning of the elliptical spacers in the –L configurations generates channels of faster flowing low concentration fluid, which cause larger velocity and concentration gradients at the membrane wall, thus resulting in higher mass transfer and lower permeate processing costs. Changes in the angle of attack for the 4L04H spacer do not appear to have a significant effect on permeate production costs (less than 1%). In contrast, increases to the angle of attack for the 4L04L configuration decrease the total permeate processing cost, albeit only by a small amount (4.1%). In the case of the 3L04 spacer, both positive and negative angles of attack result in a significant reduction of the total permeate processing cost (16% and 13% respectively). However, from the data presented in this chapter, it is not possible to determine either the value or the direction of the angle of attack that would produce the lowest processing cost.

For multi-layer spacer meshes with a df/hch ratio of 0.3, the angle of attack has a significant effect on permeate processing costs, with the difference between the lowest and highest processing costs being over 30% in the angle range studied. Figure 6.22 suggests that the “optimal” angle of attack is between +10° and -10°. This is because, as seen in Figure 6.12, mass transfer is lower for the geometries at attack angles of +10° and -10° than for 0° and 5°. Despite energy losses being lower for the 4L03L case with an angle of attack of -10° than for 0° and 5° (see Figure 6.10), the lower Sherwood number results in a lower permeate flux, which in turn increases the operating cost per cubic metre of permeate. Angles above +10° or below -10° are unlikely to reduce processing costs, as aerodynamic studies [190] suggest that flow separation would cause significant increases to form drag, and hence to operating costs. Given that the low Sherwood numbers observed for the attack angles of +10° and -10° imply a low degree of mixing and boundary layer renewal, it is unlikely that these configurations will achieve lower processing costs for higher Schmidt number solutes.

149

Figure 6.22. Dependence of total processing cost per cubic metre of permeate on angle of attack of elliptical filaments, for multi-layer spacer-filled channels at a Reynolds number of 200. A membrane cost of $100 per square metre was assumed.

For the df/hch ratios of 0.4 and 0.3, the –L spacer configurations resulted in lower processing costs than the –H configurations. Therefore it can be concluded that for 2D 4-layer zigzag spacers with elliptical filaments as middle layers, the –L configuration will result in lower processing costs than the –H configuration, for the cost parameter conditions used in this chapter. Since the attack angles which produced the lowest permeate processing cost for each spacer configuration also presented the highest Sherwood numbers, it can be said that mass transfer had the greatest impact in determining the permeate processing costs. The data collected suggests that positioning the middle elliptical filament layers such that they cause a change in the direction of the bulk flow (via positive angles of attack) will result in lower processing costs for the geometries analysed, despite lower energy losses occurring when the filaments are aligned with the flow (at negative angles of attack). However, more data points are needed before optimal operating angles of attack can be determined for these geometries.

150 6.3.4.2. Effect of changes in membrane cost

Figure 6.23. Dependence of total processing cost per cubic metre of permeate on Reynolds number, for empty, 2-, 3- and 4-layer spacer-filled channels. A membrane cost of $500 per square metre was assumed.

When higher membrane costs per unit area are assumed, the difference between spacer-filled channels and empty channels becomes more evident. As seen in Figure 6.23 for a membrane cost of $500 per square metre, the permeate production cost when utilising an empty channel is higher than for any of the spacer-filled channel geometries tested within the Reynolds number range of this study. This is because energy losses in the empty channel are up to an order of magnitude lower than for the spacer-filled channels, and therefore membrane costs account for most of the empty channel processing costs. In addition, as membrane costs are increased, the Reynolds number for minimum total cost is also increased. This effect is clearly seen for the 4L03H spacer geometries, for which the processing costs in the unsteady flow regime (i.e. at a Reynolds number of 800), are lower than in the steady laminar regime. In contrast, at lower membrane costs, processing costs were higher at a Reynolds number of 800 than at the lower Reynolds number of 200 (see Figure 6.19). Therefore, as membrane costs are increased, it becomes desirable from the economic viewpoint to operate in the unsteady flow regime. This is because the increased energy losses and associated costs of operating in the unsteady flow regime are easily offset by the higher mass transfer rates and permeate fluxes attained at higher Reynolds numbers.

151

Figure 6.24. Dependence of total processing cost per cubic metre of permeate on Reynolds number, for empty, 2-, 3- and 4-layer spacer-filled channels. A membrane cost of $10 per square metre was assumed.

Conversely, at the lower membrane cost of $10 per square metre, the optimal operating Reynolds number for every spacer configuration is lower than the optimal Reynolds number at higher membrane costs. Figure 6.24 shows that the advantage of spacer-filled channels against the empty channel is reduced at this lower membrane cost, and that the optimal Reynolds number for the empty channel is higher than the optimal Reynolds number for spacer-filled channels. The reason for this change is that at low membrane costs, energy costs become the main contributor to the total processing cost, and the total cost curve approaches the operating cost curve. Moreover, it can also be seen that the optimal Reynolds number for the multi-layer spacer configurations is lower than the optimal Reynolds number for the 2L06 spacer. This suggests that operating costs dominate in multi-layer spacers, mainly due to the increased pressure losses and form drag caused by the extra filament layers in the multi- layer configurations. In other words, a geometry with a larger amount of form drag will have a lower optimal operating Reynolds number, while a geometry with less form drag will have a higher optimal operating Reynolds number. However, the value of the minimum permeate processing cost is not solely dependent on the value of optimal Reynolds number. Therefore, form drag alone cannot determine which geometry will attain the lowest permeate processing costs. For this scenario, the best performing

152 (lowest processing cost) spacer geometry is the 2L06 spacer, followed by the 3L04 at an angle of attack of 0°. The geometry with the highest processing cost at the optimal Reynolds number is, as in the previous scenarios, the empty channel.

6.4. Conclusions A design study for novel multi-layer spacer configurations including elliptical filament spacer layers was conducted. Single and double layers of elliptical spacer filaments were incorporated to 2D zigzag configurations of circular filaments. The expected effect of the elliptical filaments was to direct low concentration fluid from the bulk to the near-wall region, and to increase the disruption of boundary layer formation caused by the circular filaments. In order to asses the benefits of this type of spacer configurations, an economic analysis was also carried out using cost parameters representative of those quoted in literature. The results presented in this chapter show that, although a comparison of different spacer geometries via power consumption (Re3f) and mass transfer (Sh) is useful, an economic analysis incorporating permeate production provides more realistic cost information. This is because permeate flux does not follow a linear dependence with Sherwood number. In addition, although operating costs are linearly dependent on energy losses, increases in permeate production by means of improved mass transfer also reduce the operating costs per unit volume of permeate. Therefore, in order to determine which spacer geometries will yield lower permeate processing costs, an economic analysis is recommended over a comparison of mass transfer at similar energy losses. According to the economic analysis, the optimal operating Reynolds number (at which total processing costs were minimised for each spacer geometry) was found to be closely related to the amount of form drag generated by the spacer filaments. For geometries with a higher amount of form drag the optimal Reynolds number was lower than for those with a lower amount of form drag. As such, the empty channel resulted in the highest optimal Reynolds number. It was also found that for higher membrane costs, the optimal operating Reynolds number was increased. For the highest membrane cost assumed in this chapter ($500 per square metre), the optimal operating Reynolds number fell in the unsteady flow regime. For low membrane costs of $10 per square

153 metre, operating in the laminar flow regime (below a Reynolds number of 300 for spacer-filled channels) resulted in the lowest processing costs. One of the main components of processing cost which was not incorporated into the economic analysis carried out in this chapter was that of fouling and cleaning. Since this cost is usually proportional to the membrane area, it was therefore assumed to be included in the membrane cost. If fouling and cleaning costs are significant, the analysis of this chapter suggests that increasing the operating Reynolds number will reduce total processing costs. This is achieved in two ways. Since it is known that membrane fouling is hindered at higher Reynolds numbers due to increased shear rate, operating at a higher Reynolds number will reduce the extent of fouling observed. In addition, higher mass transfer rates will result in a smaller membrane area requirement, which would also reduce cleaning costs. The direct effect of multiple spacer layers on fouling was not studied in this chapter. However, based on the study of Schwinge et al. [96] and on the results presented in this chapter, it is expected that the higher local velocities and shear rates at the membrane wall will contribute to reduce the extent of fouling. Another cost component not taken into account in the analysis carried out in this chapter was the cost of production of the novel spacer geometries. Although new techniques are available for the production of new spacer designs [67, 97], common mass produced spacers are currently available at much lower costs, which will have an impact on the economic results presented here. Incorporation of the capital costs of spacers would give an economic advantage to traditional 2-layer spacers over new multi-layer designs, which are more expensive to produce. Therefore, new designs must increase permeate production substantially in order to provide an economic incentive for their production. The multi-layer configurations with elliptical filaments showed potential for increased productivity due to mass transfer enhancement. However, lower permeate processing costs were obtained by using a simple 2D zigzag spacer (2L06), particularly under the assumption of membrane costs equal to or below $100 per square metre. Overall, the 2L04 configuration at a Reynolds number of 500 showed the lowest permeate processing cost, closely followed by the 2L06 configuration. The middle layers of elliptical filaments in multi-layer spacer configurations promoted mass transfer enhancement when compared to 2-layer spacer filled channels with the same df/hch ratios. However, stagnant fluid regions near the membrane walls

154 caused by re-direction of flow towards the middle of the channel by the elliptical filaments, resulted in lower mass transfer for the multi-layer spacer configurations than for the 2L06 and 2L04 spacers. Variation of the angle of attack of the elliptical filaments showed potential to increase mass transfer enhancement at the same or lower energy losses by optimising the profile of the middle spacer layer, such that little or no recirculation regions are formed on its downstream side. This is possible because form drag caused by the middle spacer layer does not contribute significantly to mass transfer, since it does not generate regions of boundary layer detachment and reattachment. In addition, multi-layer spacer showed potential for reducing processing costs in high membrane cost scenarios. Although negative attack angles usually reduced energy losses, they failed to direct flow towards the membrane wall, and thus resulted in lower mass transfer. Since the amount of permeate produced was proportional to the Sherwood number, lower mass transfer conditions also resulted in higher operating costs per unit volume of permeate. On the other hand, positive angles of attack generally resulted in lower total permeate processing costs. This was mainly due to increased mass transfer, caused by the change of direction of the low concentration bulk flow. Thus, mass transfer performance was the main factor in determining the total processing costs for the conditions of this study. As such, improvements in mass transfer enhancement have the potential to reduce both capital and operating processing costs per unit volume of permeate produced. This is possible even if energy losses are increased, since larger permeate fluxes will reduce the operating cost per unit volume of permeate. Another geometric characteristic varied in this chapter was the relative positioning of circular and elliptical filaments for 4-layer spacers. It was found that the “low” (–L) configuration resulted in higher mass transfer enhancement, and hence lower permeate processing costs. In the –L configuration, the elliptical filament was placed closer to the recirculation region downstream of the circular filaments, and therefore disrupted the formation of this recirculation region, and directed flow towards the opposite membrane wall. 4L spacers in the –L configuration, coupled with positive angles of attack for the elliptical filaments were found to be the most effective in disrupting flow and redirecting low concentration fluid towards the membrane walls. However, further studies are required in order to optimise the hydrodynamic profile and angle of attack of the submerged layers, and thus reduce the energy losses incurred by the addition of

155 further flow obstacles. Variation of the filament length to channel height ratio (lm/hch) should also be explored. In addition, three-dimensional simulations of real-world multi- layer spacer designs are needed in order to investigate the 3D flow and mass transfer effects that have not been taken into account in this chapter.

156 Chapter 7

Conclusions

The work presented in this thesis investigated the fluid flow and mass transfer characteristics of narrow spacer-filled channels with the aid of Computational Fluid Dynamics (CFD). Analysis of the dynamics and relationships between fluid flow and mass transfer in narrow spacer-filled channels provided insights into the prevailing mechanisms and transport processes taking place inside spiral wound membrane (SWM) modules. These insights can, in turn, be used to develop guidelines for improved designs of spacer meshes and improved operation of SWM modules. As an analysis tool, CFD provides the ability to change operating conditions, fluid properties and geometric characteristics of the flow channels in a flexible and practical way. The geometric parameters of the channels simulated can be varied without the need to construct and test new spacer meshes. This represents a significant advantage of the CFD technique over traditional experimental methods. For example, the fluid velocity, solute diffusivity and inlet concentration can be set at the start of the simulation, and there is no need for control of these variables. In addition, flow data can be reported at any position and time during the simulation, and the evaluation of these flow variables is possible without any disturbance to the underlying flow. Unlike many of the previously reported CFD studies for spacer and SWM module performance, this thesis focused on the effects on mass transfer enhancement of the different fluid flow patterns caused by the spacer meshes in membrane modules. The study of 2D and 3D simulations provided evidence that the regions of higher mass transfer on the membrane wall are not only correlated with regions of high shear rate, but also with the regions where the fluid flow is towards the wall. Similarly, the wall regions in which the fluid flow is away from the wall presented low mass transfer. Although the transition from laminar steady to unsteady flow produced changes in the mass transfer conditions, it was found that the correlation between flow in the direction of the wall and enhanced mass transfer was also observed in unsteady flow. The thesis also includes work carried out for developing an approach to the simulation of multiple ionic component mixtures in CFD. The results of this sub-project are presented in Appendix A. This objective was partially achieved, as a multi-

157 component diffusion model was adapted for the conditions of a 3D membrane channel. The model was successfully incorporated into the commercial CFD software used in this thesis. However, converged simulations were not obtained due to instabilities in the numerical solution process. In order to obtain converged solutions, a linearisation of the source terms in the transport equations is necessary. However, this requires access to the programming code related to the solution of the transport equations, which is unavailable due to the commercial nature of the CFD software used. Nevertheless, the model presented in Appendix A is valuable for researchers with access to the source code of their numerical solvers. One of the aims of this thesis was to understand the impact of unsteady flow on mass transfer, particularly the effect of vortices near the membrane wall on the developing boundary layer. Analysis of unsteady 2D simulations of flow inside narrow zigzag spacer-filled channels revealed that for the case of membrane filtration of sodium chloride, inflow of lower concentration fluid into the boundary layer dominates mass transfer enhancement in the unsteady flow regime. The size of shed vortices as well as the frequency of shedding were dependent on the Reynolds number: at higher Reynolds numbers the shed vortices were larger and were shed more frequently. Although the increase in shed frequency improves mass transfer enhancement, the increase in size is not necessarily beneficial, as large vortices were shown to push high concentration fluid from one membrane wall to the other, thus decreasing the concentration gradient and mass transfer coefficient at the membrane. This thesis also aimed to understand the effect of 3D spacer geometries on flow, and consequently on mass transfer. In order to overcome the high computational and memory demands of 3D simulations, an approach was developed to implement fully- developed velocity and scaled concentration profiles at the inlet and outlet boundaries of the 3D flow domain. This approach bypasses the entrance region and models flow and mass transfer characteristics representative of the average conditions encountered in the membrane module. Increased understanding of the influence of 3D spacer geometries on flow and mass transfer was achieved via the CFD simulation of two orientations of a 3D non- woven cylindrical filament spacer mesh. In general terms it was found that, as expected, the 3D geometries modelled attain greater mass transfer enhancement than their 2D counterparts. The main 3D flow characteristics which had a significant effect on mass

158 transfer enhancement were the occurrence of streamwise vortices, open spanwise vortices, and higher wall shear rate perpendicular to the bulk flow. In particular, the locations of higher mass transfer were found to be related to the position of both streamwise and spanwise vortices. It was also found that the geometric characteristics of the 45° orientation spacer promote mass transfer to a greater extent than those of the 90° orientation. Characteristics beneficial to mass transfer enhancement in the 45° orientation spacer include the absence of a fully formed recirculation region and an increase of z-wall shear as the Reynolds number is increased. Increased wall shear is of particular importance for high Schmidt number solutes, for which the concentration boundary layer is thinner, and therefore a thinner velocity boundary layer is required to effectively enhance mass transfer. Although the 3D simulations were restricted to the steady laminar flow regime, in similarity to 2D simulations it is expected that the 3D specific flow conditions encountered in this flow regime, and which were shown to be beneficial to mass transfer, will also be present in the unsteady flow regime. The results from Chapters 4 and 5 indicated that knowledge of the location of the regions of low and high friction and mass transfer on the membrane surface, as well as the mechanisms that give rise to these regions, are particularly important for spacer design. Moreover, form drag was identified as the main component of energy losses for the geometries analysed. Some types of form drag are not detrimental to mass transfer, such as form drag due to recirculation regions behind obstacles adjacent to the membrane surface. However, form drag in the bulk of the channel does not contribute to increased mass transfer to or from the membrane wall, and therefore only increases energy losses. Therefore, an elliptical filament profile was proposed as the middle layer or layers of a multi-layer spacer meshes. Elliptical filament profiles have lower form drag coefficients than circular filaments, and therefore generate lower total drag and energy losses. By varying the angle of attack of these elliptical filaments, low concentration can be encouraged to flow towards the membrane surface, and therefore generate regions of high mass transfer. In order to understand the effect of multiple-layer spacer meshes on membrane performance, a study of 2D multi-layer spacer configurations incorporating elliptical filament spacer layers was conducted. An economic analysis with typical cost parameters was also carried out to asses the benefits of this type of spacer configuration. Although the multi-layer spacer configurations resulted in higher costs than simple 2-

159 layer 2D zigzag spacers at Reynolds numbers equal to and higher than 200, these novel designs showed potential for increased productivity. By varying the angle of attack of the middle elliptical filament layers, improvements in productivity were obtained for the configuration with a df/hch value of 0.3. In addition, the generation of slow-moving fluid regions due to the positioning of the elliptical filament layers was found to contribute to the decrease in mass transfer and increase in processing costs. In general terms, improved mass transfer was observed when the middle spacer layers were positioned such that they would change the direction that the bulk flow would normally adapt in 2- layer spacer-filled channels. In other words, mass transfer enhancement was observed when the elliptical filaments were not aligned with the bulk flow. Despite the increased energy losses that these configurations generated, the operating cost increases were offset by the increased permeate production. This design study provides a starting point for the development of advanced multi-layer spacer geometries. Comparisons of the energy losses and mass transfer generated by the use of different spacer designs provide valuable insights into the performance of SWM modules. However, an economic analysis taking into account capital and operating costs as well as permeate production will yield a clearer picture of which spacer geometries will achieve higher performance in a SWM module. The main difference between these two approaches is that the economic analysis takes into account the reduction in operating costs caused by an increase in mass transfer enhancement. A higher Sherwood number will result in a higher permeate throughput, and thus lower operating costs per unit volume of permeate. In addition, the cost analysis can also help in determining the most beneficial operating flow regime from an economic point of view. Therefore, in order to determine which spacer geometries will yield lower permeate processing costs, an economic analysis is recommended over a comparison of mass transfer at similar energy losses. This also highlights the importance of obtaining mass transfer information from the CFD simulations, as opposed to only hydrodynamic simulations, given that geometries with higher energy losses due to form drag were found to result in lower processing costs. It must also be mentioned that it is very unlikely that there will be a single spacer geometry which results in the lowest processing costs for all possible applications and operating modes, that is, a “universally best” spacer geometry. In this thesis, the Reynolds number was varied, but the Schmidt number was kept constant. There is no

160 guarantee that the geometries which resulted in the lowest permeate processing costs under the conditions analysed in this study will also result in the lowest costs under a different set of conditions, such as a different Schmidt number solute or for a multiple component solution. Therefore, it is expected that the “best” spacer design will vary depending on the specific circumstances of each membrane system. However, in light of the data presented in this thesis, it can be expected that as the Schmidt number is increased, the better performing spacers will be those that can direct low concentration bulk flow towards the membrane walls, thus causing a larger degree of boundary layer renewal. In addition, the low concentration bulk fluid must be able to penetrate deeper into the boundary layer as the Schmidt number is increased, in order to keep the Sherwood number high and processing costs low. Membrane costs were found to have a significant impact on which spacer configuration would produce the lowest total costs. Due to the increased energy losses caused by the middle spacer layer, multi-layer spacer configurations were better placed on the performance scale when high membrane costs (above $100 per square metre) were assumed. By definition, obstacles in 2D flow are perpendicular to the flow, and hence result in higher total drag than obstacles placed at an angle to the oncoming flow. This effect is obvious when comparing 45° and 90° orientation 3D spacers. Since the energy loss to mass transfer ratio can be significantly reduced by placing the filaments at an angle, further 3D studies are therefore required in order to optimise the hydrodynamic profile and angle of attack of the middle spacer filament layers. Among the costs not considered directly in the economic analyses conducted for this thesis were the cost of fouling and cleaning and the cost of the spacer meshes. Since the fouling phenomena was outside the scope of this study, and hence not incorporated into the CFD simulations, the cost of fouling and cleaning was assumed to be proportional to membrane area, following the approach of Maskan et al. [189]. Thus, the results of the economic analysis for higher membrane costs can also be interpreted as a case of high fouling and cleaning costs. This study found that for the case of high membrane costs, the operation of membrane modules at high Reynolds numbers, in the unsteady flow regime, was economically justified. This agrees with the current belief that operating membrane modules at higher cross-flow velocities will reduce the impact of fouling on the performance of membrane systems.

161 The cost of the spacer meshes will only have a significant impact if the cost of production of new spacer designs is considerably higher than that for commonly available commercial spacers. Due to the complex nature of multi-layer spacers, it is likely that the mass production costs of these novel designs will be higher than those for common 2-layer spacers. Therefore, new designs must increase permeate production substantially in order to provide an economic incentive for their production. Current trends indicate that energy costs are increasing, and that membrane production costs are decreasing. This accentuates the need for improved spacer meshes which reduce energy losses in general, and detrimental form drag in particular, in order to reduce processing costs for membrane operations. Multi-layer spacer meshes represent a potential alternative to traditional 2-layer spacer meshes, but further optimisation of geometric parameters and filament profiles are required. Although none of the proposed multi-layer designs resulted in lower processing costs than traditional 2- layer configurations, especially at lower membrane costs, further study into the 3D flow and mass transfer effects of the middle layer spacer filaments would provide invaluable information to either support or rule out multi-layer spacers as a real alternative to traditional spacer meshes. The results of the design study in Chapter 6, as well as aerodynamic studies [190] can provide starting points for the optimisation of spacer filament profiles, since the study of flow separation and drag has been the focus of much research in the airfoil design field. In addition, variation of other geometric characteristics in multi-layer spacers, such as the filament length to channel height ratio (lm/hch) should be considered, as the effect of the middle spacer layers introduces a variation not considered in previous spacer studies [42, 93, 115]. For example, the lm/hch ratio influences the size of the recirculation regions downstream of the attached filaments, and therefore affects the position of the boundary layer reattachment point. Since it has been shown in this work that the position of the boundary layer reattachment greatly affects mass transfer performance, the consideration of these variations is justified. One of the main difficulties most CFD studies face, particularly in the field of membrane science, is the lack of experimental data appropriate for validation purposes. Although no experiments intended for validation were conducted in this thesis, the results obtained using CFD were successfully validated against the body of published data. Good agreement was found for the relationships under consideration, namely the

162 variation of friction factor and Sherwood number with Reynolds number. In particular, remarkable agreement was found between the 3D simulation data and experimentally obtained data for a similar spacer mesh geometry, despite the differences in conditions at the membrane surface. This demonstrates the reliability and flexibility of CFD studies, and their potential for further studies in the field of membrane science. The CFD model used throughout this thesis assumed a membrane boundary condition of an impermeable dissolving wall. This boundary condition has been shown [136] to predict mass transfer conditions comparable to those obtained for the permeable wall case. Due to its simplicity, the computational resource requirement of the impermeable dissolving wall boundary condition was low, while providing a good approximation of the flow and mass transfer in SWM modules. The results and conclusions presented in this thesis apply to low permeation rates such as those currently encountered in real-world membrane applications. However, the assumptions made under the dissolving wall approach do not hold for permeation rates that approach the magnitude of the bulk flow velocity. In order to increase the applicability of CFD results to cases with higher permeation rates, a more realistic permeable wall boundary condition can be incorporated into membrane channel simulations. Although these types of models were not utilised in this study due to their computational complexity, fast semiconductor technology improvements seen in recent years have seen the computational cost of complex and numerically intensive problems decrease at a rapid pace. This may allow later studies to include more complex mechanisms such as the permeable wall condition in 3D flow. Given the flexibility of CFD, incorporation of a permeable membrane boundary condition might be carried out with relative ease in the future. In conclusion, CFD has proven to be a valuable tool for the performance analysis and design of spacer meshes. Mass transfer enhancement mechanisms have been identified, as well as the spacer characteristics which give rise to enhanced mass transfer by altering the flow conditions inside narrow membrane channels. Vortices play a primary role in mass transfer enhancement by promoting mixing of low and high concentration fluid, increasing wall shear, and reducing the thickness of the boundary layer. Vortices can either be oscillating, shed vortex streets, or stationary, such as recirculation regions or streamwise vortices in 3D flow (corkscrew-like flow). Alternative spacer mesh designs, which are intended to promote these mass transfer

163 enhancing mechanisms, were tested with the aid of CFD. These tests included the analysis of variations in various geometric characteristics, and have taken a fraction of the time that an experimental analysis would have required. The volume of information provided by this numerical technique has also allowed the economic assessment of the proposed spacer geometries. With the results of the economic analysis, it is possible to determine the best operating flow regime to minimise processing costs. Despite the investigated spacer designs not providing enough improvements in membrane system performance to justify the increased energy losses when compared to traditional spacer designs, they provide a valuable starting point for further improvements in energy loss reduction and mass transfer enhancement. Improvements in filament profile designs and spacer geometric parameters are the next logical steps in the design process. Future studies following the line of work presented in this thesis should address the issue of the difficulty of validation against experimental data. For this purpose, Particle Image Velocimetry is a logical technology for acquiring a type of data that could be compared to data obtained from CFD simulations. Permeation and fouling phenomena should also be incorporated in more detailed design studies of 3D spacer geometries for improving membrane system performance while keeping energy losses at a minimum. In addition, several of the assumptions made in the CFD model can be relaxed in subsequent studies, such as the constant fluid properties. This should enable the design of a new generation of spacer meshes with the ability to improve mass transfer enhancement while keeping energy losses at a minimum, thus improving the performance of membrane systems.

164

References

165 References

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174 174. Carluccio, E.; Starace, G.; Ficarella, A.; Laforgia, D. Numerical analysis of a cross-flow compact heat exchanger for vehicle applications. Appl. Therm. Eng. 2005, 25, 1995-2013. 175. Patankar, S.V.; Liu, C.H.; Sparrow, E.M. Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area. J. Heat Transf. 1977, 99(2), 180-186. 176. Wiley, D.E.; Fell, C.J.D.; Fane, A.G. Optimisation of membrane module design for brackish water desalination. Desalination 1985, 52(3), 249-265. 177. Gill, W.N.; Wiley, D.E.; Fell, C.J.D.; Fane, A.G. Effect of Viscosity on Concentration Polarization in Ultrafiltration. AIChE J. 1988, 34(9), 1563-1567. 178. Ndinisa, N.V.; Wiley, D.E.; Fletcher, D.F. Computational Fluid Dynamics Simulations of Taylor Bubbles in Tubular Membranes - Model Validation and Application to Laminar Flow Systems. Chem. Eng. Res. & Des. 2005, 83(A1), 40-49. 179. Portela, L.M. Identification and characterization of vortices in the turbulent boundary layer, Department of Mechanical Engineering. Stanford University, 1997, p. 456. 180. Finn, L.I.; Boghosian, B.M.; Kottke, C.N. Vortex core identification in viscous hydrodynamics. Phil. Trans. R. Soc. A 2005, 363, 1937-1948. 181. Vollmers, H. Detection of vortices and quantitative evaluation of their main parameters from experimental velocity data. Meas. Sci. Technol. 2001, 12, 1199- 1207. 182. Pozrikidis, C. Fluid dynamics: theory, computation and numerical simulation; Kluwer Academic Publishers: Norwell, 2001. 183. Levy, Y.; Degani, D.; Seginer, A. Graphical visualization of vortical flows by means of helicity. AIAA J. 1990, 28(8), 1347-1352. 184. Schwager, F.; Robertson, P.M.; Ibl, N. The use of Eddy Promoters for the enhancement of Mass Transport in Electrolytic Cells. Electrochim. Acta 1980, 25(12), 1655-1665. 185. Kim, K.J.; Chen, V.; Fane, A.G. Ultrafiltration of Colloidal Silver Particles: Flux, Rejection, and Fouling. J. Colloid. Interface Sci. 1993, 155, 347-359. 186. Kim, K.J.; Chen, V.; Fane, A.G. Characterization of clean and fouled membranes using metal colloids. J. Membr. Sci. 1994, 88, 93-101. 187. Kim, D.W.; Kim, M.-U. Minimum Drag Shape in Two-Dimensional Viscous Flow. Int. J. Numer. Meth. Fluids 1995, 21(2), 93-111. 188. Churchill, S.W.; Usagi, R. A General Expression for the Correlation of Rates of Transfer and Other Phenomena. AIChE J. 1972, 18(6), 1121-1128. 189. Maskan, F.; Wiley, D.E.; Johnston, L.P.; Clements, D.J. Optimal Design of Reverse Osmosis Module Networks. AIChE J. 2000, 46(5), 946-954. 190. Anderson, J.D. A History of Aerodynamics; Cambridge University Press: Cambridge, 1997. 191. Anderson, D.E.; Graf, D.L. Multicomponent Electrolyte Diffusion. Annu. Rev. Earth Pl. Sc. 1976, 4, 95-121. 192. Callen, H.B. Thermodynamics and an Introduction to Thermostatistics. 2nd ed; John Wiley & Sons: New York, 1985. 193. Robinson, R.A.; Stokes, R.H. Electrolyte Solutions; Butterworths: London, 1965. 194. Reid, R.C.; Prausnitz, J.M.; Sherwood, T.K. The properties of gases and liquids. 3rd ed; McGraw-Hill: USA, 1997.

175 195. Harned, H.S.; Owen, B.B. The Physical Chemistry of Electrolytic Solutions. 2nd ed. Am. Chem. Soc. Monograph Vol. 95; Reinhold Pub. Corp.: New York, 1950.

176

Appendices

177

Appendix A

Multiple Ionic Components

A.1. Introduction In most systems in the field of mass transfer, be it geochemistry, petrochemistry or electrochemistry among others, there is more than one diffusing component in a fluid solution [191]. In particular, it is very rare for the solutions used in membrane operations to consist of only one dissolved component. As the number of components increases, the number of possible interactions amongst solutes also increases which, in turn, increases the complexity of the system, such that Fick’s law [9] is no longer the best way for describing diffusion phenomena taking place. An overview of multiple component diffusion modelling was presented in Chapter 2. This appendix deals with the application of the three-dimensional (3D) version of a multiple component diffusion model to describe the mass transfer phenomena encountered in membrane applications such as nanofiltration (NF) and reverse osmosis (RO). The model is to be incorporated into the commercial CFD software ANSYS CFX, in order to obtain converged solutions of the continuity, momentum and mass transport equations. The ability to integrate a multiple component diffusion model with CFD software will enable its application in complex geometries such as those of spacer-filled channels, which are often encountered in membrane unit operations.

A.2. Theory

A.2.1. Mass Transfer

Table A.1. Definitions of various flow variables related to mass transfer.

Total Mean local Flux of Concentration Fraction Total flux concentration velocity species i mi i 1 Mass B i Y vv B Nv vNB i i i ii iii V i i i i ni ci 1 Bcc i cvnB i Molar ci X i vvBcii nviic i M i V i c c i i

178

The mass transfer nomenclature followed throughout this thesis is similar to that used in [9], which can be summarized in Table A.1 and Table A.2

Table A.2. Definitions of the mass and molar diffusive fluxes.

Relative to v Relative to v () Mass Jvviii Jvviii() () Molar jiiic vv jiiic ()vv

It’s important to mention a few relationships that can be derived from the definitions in Table A.1 and Table A.2. Firstly, the sum of the mass fractions for all components must add to unity. This summation relationship also applies to the molar fractions. Moreover, for the diffusive fluxes:

NC B ji 0 (A.1) i1

NC BJi 0 (A.2) i1

The summation relationships shown in equations (A.1) and (A.2) are also true for the other diffusive fluxes, but the two diffusive fluxes mentioned above are the most commonly used. Another useful definition is the average molecular weight, which is defined as:

NC MXMB ii (A.3) i1

Equation (A.3) leads to the following relationships:

M X i Yi (A.4) M i

c (A.5) M

179

In addition, the relationship between the mass and molar diffusive fluxes can be described by:

NC JiiiiMYMjjB( jj) (A.6) j1

However, because the flux for one component can be obtained from equations (A.1) and (A.2) if all other flux components are known, it is sometimes useful to define the fluxes for the first (NC –1) components without referring to the flux of the last component. Thus:

NC 1 J MYMMjjB FV( ) (A.7) iiiiNjjHXC j1

Equation (A.7) can be more conveniently expressed by:

NC 1 JiiB ( Ajjj ) (A.8) j1

The Aij coefficients in equation (A.8) are given by:

A MYM() M (A.9) ij ij i i NC j

Where the symbol ij is the Kronecker delta function, defined by:

I1,ij J (A.10) ij K0,ij

Another useful way to express equation (A.8) is in the matrix form: ()JAj > ?() (A.11)

Where the flux vectors are given by:

180

CSJ DT1 () J D T (A.12) DT J EUNC 1

CSj DT1 () j D T (A.13) DT j EUNC 1

And the coefficient matrix:

FVAA GW1,1 1,NC 1 >?A GW (A.14) GW AA HXNNCC1,1 1,NC 1

Equation (A.11) can be reversed, to give: ( j ) >aJ?( ) (A.15)

In equation (A.15), the matrix [a] is the inverse of matrix [A]. However, a simpler expression for calculating the components of matrix [a] is given by:

CS11 aXij D T (A.16) ijMMM i D T iNEUC j

Finally, another useful relationship relates the molar and mass fraction gradients:

().X MY>a?() . (A.17)

where the gradient vectors are given by:

CS.X DT1 ().X D T (A.18) DT .X EUNC 1

181

CS.Y DT1 ().Y D T (A.19) DT .Y EUNC 1

A.2.2. Maxwell-Stefan diffusion model As explained in Chapter 2, the MS diffusion model can be thought of as a force balance for a component in solution. It equates the driving force for each component in the solution to the friction forces between that component and all other components. For electrolyte systems such as those found in membrane applications, the driving force . simplifies to the electrochemical potential gradient ( i ). Therefore, the MS diffusion model for electrolyte systems is given by:

. X (vv ) i B ji j (A.20) RT j ij

However, the form of the MS model presented in equation (A.20) is not useful for determining the diffusive fluxes of the solutes. Multiplying both sides of equation

(A.20) by the molar fraction Xi, and multiplying both the numerator and denominator of the right hand side by the molar concentration c, yields:

X . ()XXjinn i j iiB (A.21) RTj c ij

Equation (A.21) is more useful than equation (A.20) as it is expressed in terms of the flux of the solutes. However, an even more useful expression is obtained after making use of the following equality:

** X jinnXXX i j jijj ij (A.22)

Substitution of equation (A.22) into equation (A.21) gives the following expression in terms of the diffusive fluxes:

182

** X . ()XXjij ij j iiB (A.23) RTj c ij

Taking into consideration that the sum of all the molar fractions is 1, and that the sum of the diffusive fluxes is zero, an expression which involves only the diffusive fluxes of all but one of the solution components can be obtained:

IY NNCC11FVFCS CSV c. j LL1111 1 iiJZ GWGDT D TW BBX kjDTj DT (A.24) RT X GWG W iiNLLCCkj11HXHEUikiN EijiNC UX K[kiji

The solution component that is commonly overlooked is the solvent, as it is usually the most abundant component. The mass fraction and diffusive flux for the solvent can be obtained by making use of the summation relationships mentioned previously. Thus, the system of equations given by expressing equation (A.24) for each solute can be thought of as a matrix system:

c (). >?B ()j (A.25) RT

Where the components of the matrix of coefficients, [B], are given by:

I CS N 1 FV L C CS 11DTGWDT 1 1 L DTB X k DT, ij X DT GW LEUiiNCCk 1 HXikiN E ki U Bij J (A.26) L CS11 L DT, ij L DT K EUij iNC

At this point it becomes useful to define matrix [b] as the matrix inverse of [B].

>bB? > ? 1 (A.27)

By multiplying both sides of the matrix equation (A.25) by [b], an explicit form for the diffusive fluxes in terms of the MS diffusion model is obtained:

183

NC 1 .c jiiB b jj (A.28) RT j1

In order to incorporate equation (A.28) into a CFD software package, we must be able to express the diffusive mass flux in terms of the available flow variables, such as the mass fraction of the solutes and their gradients. The relationship between the mass and molar diffusive fluxes is expressed in equation (A.7) The electrochemical potential is composed of the chemical potential and an electrostatic force. This relationship is given by:

! iiiZ (A.29)

In equation (A.29), is the electric potential. In terms of gradients, this relationship is expressed as:

.. .! iiiZ (A.30)

The electric potential gradient can be calculated if we take into consideration the zero-current condition, which is stated by: BZiij 0 (A.31) i

th Moreover, since the NC solution component is defined as the solvent, its charge is zero and equation (A.31) becomes:

NC 1 B Ziij 0 (A.32) i1

Substituting equation (A.28) into equation (A.32), and expanding the electrochemical potential gradient using equation (A.30), yields:

NNCC11 . .! BBZbiij( j Z j ) 0 (A.33) ij11

Solving equation (A.33) for the electric potential gradient yields:

184

NNCC11 . BBZbiij j .! ij (A.34) NNCC11 BBZijijZb ij

Finally, from thermodynamics the chemical potential gradient is known to be [192]:

FV NC 1 GW CS ln ... VpST RTB . XGWij DTi (A.35) ii i j DT j1 XXTpX,,k GWijEUkj @ HXkNC

As previously mentioned, for membrane applications the temperature and pressure gradients can be neglected [19]. This yields the following expression for the chemical potential gradient:

FV NC 1 GW CS ln . RTB . X GWij DTi (A.36) ijDT j1 XXTpX,,k GWijEUkj @ HXkNC

Therefore, using equations (A.28), (A.7), (A.30), (A.34) and (A.36) it is possible to express the mass diffusive flux given by the MS diffusion model in terms of the molar fractions of the solutes.

A.2.3. Nernst-Planck diffusion model Despite the generality of the MS diffusion model, many of the model parameters, particularly the diffusion coefficients, are difficult to calculate and are not readily available from literature. As mentioned in Chapter 2, the NP diffusion model is a simplification of the MS model, where the only interaction amongst solutes is through the electric potential gradient generated by the charges of the solutes. This simplification is valid for dilute solutions, but restricts the applicability of this model for concentrated solutions, such as those encountered close to the membrane surface in membrane systems. Since the friction effects between solutes are neglected in the NP diffusion model,

185 the only relevant values of the diffusion coefficients are those between each solute and the solvent. These diffusion coefficients are the coefficients of each solute in the mixture (im). Thus, equation (A.20) for the NP diffusion model takes the following form:

. vv* ii (A.37) RT im

Equation (A.37) can be rearranged in order to give a more useful form, analogous to equation (A.28), which relates the flux of each solute to the driving forces:

c j iim( . Z .!) (A.38) iiiRT

Substituting equation (A.38) into the zero current condition of equation (A.32) yields:

NC 1 .c F ( .!)V B HZXiiim i Z i X 0 (A.39) RT i1

Equation (A.39) can be solved for the electric potential gradient, which gives the following expression:

NC 1 .() B ZXiiimi .! i1 (A.40) NC 1 2 B ()ZXiiim i1

Finally, under the assumption of an ideal solution the activity coefficient becomes constant and equal to 1. Thus, the expression for the chemical potential gradient in equation (A.36) becomes:

.X . i i RT (A.41) X i

Therefore, using equations (A.7), (A.38), (A.40) and (A.41) it is possible to

186 express the mass diffusive flux given by the NP diffusion model in terms of the molar fractions of the solutes. However, as opposed to the case of the MS diffusion model, the diffusion coefficients used for the NP model (im) can be readily calculated by making use of the single solute form of the Nernst-Haskell equation [193], which states that:

RT 0 0 i (A.42) im 2 Zi

The limiting ionic conductances needed for equation (A.42) are reported in many sources for the cases when the solvent is water [32, 33, 193, 194], which is the common case for membrane operations. Values of this variable for ions commonly occurring in membrane operations are reported in Table A.3, along with the values of the NP diffusivity coefficient.

Table A.3. Limiting Ionic Conductances, Ionic valences and Limiting Ionic Diffusivity in Water at 25°C [194, as in 195]

0 Z 0 Ion i i im (A-m2/V-geq) (geq/gmol) (m2/s) H+ 0.03498 1 9.314710-9 Li+ 0.00387 1 1.030510-9

+ -9 NH4 0.00734 1 1.954510 Na+ 0.00501 1 1.334110-9 K+ 0.00735 1 1.957210-9 Mg+2 0.00531 2 7.069910-10 Sr+2 0.00505 2 6.723710-10 OH- 0.01976 -1 5.261810-9 Cl- 0.00763 -1 2.031810-9

- -9 HCO3 0.00445 -1 1.18510 - -9 NO3 0.00714 -1 1.901310 Br- 0.00783 -1 2.085010-9

- -9 ClO4 0.0068 -1 1.810710 I- 0.00768 -1 2.045110-9 Ca+2 0.00595 -2 7.92210-10

-2 -9 SO4 0.008 -2 1.065110

187

A.3. Incorporation of Multiple Ionic Effects into CFD The MS diffusion model provides a generalised view of diffusion which is valid for any concentration range, whereas the NP model is only valid for the case of dilute solutes. Nevertheless, the NP model represents a good option for a first approximation to the simulation of 3D diffusive phenomena in multiple ionic component solutions. Therefore, obtaining converged solutions using the NP model can be seen as a first step towards incorporating the more complex MS diffusion model into CFD software. The mass fraction transport equation is given by [9]:

()Y i .()Y vJ . RS (A.43) t iiii

However, most CFD packages assume Fick’s law for mass transport, and only allow one diffusion coefficient, such that the equation solved takes the following form:

()Y i .()(YDYRv . . ) S (A.44) t iimiii

For multiple component ionic solutions, the mass flux can be calculated using equation (A.7). Therefore, to be able to specify the correct diffusive flux term in a CFD package we need to define a mass fraction source term such that the following relationship holds: . .() . JiiD miiYS (A.45)

In other words, the source term must be calculated by using the following relationship: . .() . SiiJ DY imi (A.46)

The Fick’s law diffusion coefficient (Dim) is required, but should not affect the calculations. This because the term including Dim is being added in equation (A.44) and later subtracted in the source term given by equation (A.46). Therefore, the terms cancel each other out, such that a dummy value can be used for this variable and the results should not depend on the value chosen.

188

The description of the approach for calculating the source term specified by equation (A.46) is as follows: first, the expression for calculating the chemical potential gradient for ideal solutions, equation (A.41), is substituted into the NP model expression for calculating the molar diffusive flux, equation (A.38), which yields:

FV CS.X .! i jiiimcZGWDT i (A.47) HXEUXRTi

The electric potential gradient term (GrE) is defined as the expression associated with the ionic valence in equation (A.47). This term can be calculated using equations (A.40) and (A.41), and is given by:

NC 1 B ()Z .X .! iimi i1 GrE (A.48) NC 1 RT 2 B ()ZiimiX i1

Using the calculated GrE, the molar fluxes for each component are calculated using equation (A.47), and then converted to mass fluxes by means of equation (A.7). Finally, the source term is specified using equation (A.46).

A.4. Problem description The commercial CFD code ANSYS CFX-10.0 was used to solve the continuity, momentum and mass transport equations [9] in an empty rectangular channel. This geometry was chosen for the sake of simplicity, as the 3D diffusion model will be expanded to more complex geometries at a later stage. For these same reasons, and in order to further simplify this system, constant properties were employed, the effect of gravity was excluded, and the fluid was assumed to be Newtonian, at steady state (no time variations), and in the laminar flow regime. The commercial CFD code used, ANSYS CFX-10.0, applies the Finite Volume Method [6] to a co-located (non-staggered) grid in order to solve the Navier-Stokes equations. It uses a coupled solver, which means that the continuity and momentum transport equations are treated as a single system and are solved simultaneously. To avoid decoupling, velocity and pressure are connected using a Rhie-Chow [162]

189 interpolation scheme. Space is discretised using a bounded second order upwind differencing scheme. A multiple component mixture consisting of an aqueous solution of sodium (Na+), potassium (K+) and chloride ions (Cl-) was used for the first set of simulations. These components were chosen due to their simplicity (their valences are either positive or negative unity), and because they are commonly encountered in membrane unit operations such as RO and NF. In the first instance, the diffusivity coefficients for each solute were set to zero, as the diffusive fluxes will be included in the source term of the transport equation of each component.

Figure A.1. Geometry of the channel used for testing the incorporation of the NP diffusion model to ANSYS CFX.

For initial testing purposes, a 2D rectangular empty channel was chosen as the fluid domain. Even though the test geometry is two-dimensional, the user subroutines written are also valid for 3D geometries. As can be seen in Figure A.1, entrance and exit lengths separate the membrane wall section from the inlet and outlet respectively. The channel entrance and exit region wall surfaces were treated as non-slip walls with no mass transfer, where all velocity components and the mass fraction gradient normal to the boundary are set to zero (u = v = 0 and Yyi 0 ). The membrane walls in the unit cells were also treated as non-slip walls (u = v = 0), but the mass fraction at the wall was either fixed at a constant value (Yi = Yi,w), or a wall mass flux was specified ( JJii,w ). Therefore, mass transfer was included by means of a dissolving wall. At the inlet of the flow domain, a flat velocity profile with u = uavg, v = 0 and Yi = 0 is specified. At the outlet, an average reference pressure of zero is specified.

190

A.5. Results and Discussion

For the first simulation runs, the dummy diffusion coefficient (Dim) was set to zero such that all the diffusive fluxes were only included in the source term. However, when this approach was used the simulations did not converge and continually crashed due to numerical errors such as floating point errors and overflows. Therefore it was decided to -9 2 specify a small value for the diffusion coefficient Dim (e.g. 110 m /s) for equations (A.44) and (A.46). Moreover, user routines were written in order to specify the molar fraction of each solute at the membrane wall by varying the wall flux. Using this latter approach, converged solutions were successfully obtained.

Figure A.2. Effect of ionic interaction on predicted molar fractions of each of the solution components.

As shown in Figure A.2, the results obtained using the Nernst-Planck diffusion model differ from the predictions using a simple salt diffusion model based on Fick’s law and neglecting salt dissociation into ions. The Nernst-Plank diffusion model predicts higher concentrations of potassium and lower concentrations of sodium and chloride ions within the concentration boundary layer. Since boundary layer concentrations have a significant effect on membrane performance in NF and RO systems, this means ionic interactions cannot be neglected when modeling multiple ionic component solutions in membrane systems.

191

It was expected that the results would be insensitive to the magnitude of the dummy diffusion coefficient. However, as shown in Figure A.3, it was found that the value did affect the result of the simulations. The predicted molar fractions for the sodium ion were slightly higher when using the lower coefficient 110-10 m2/s, as compared to those predicted using a value of 110-9 m2/s for the dummy coefficient. For the potassium and chloride ions, the predicted molar fractions were between 5% and 10% lower in some parts of the channel when using the higher dummy coefficient. These differences are almost an order of magnitude higher than the differences observed between simple salt diffusion and the diffusion predicted by the Nernst-Planck model. Moreover, the value of the dummy coefficient had a significant effect on the calculated value for the solute wall flux necessary to obtain a specified solute wall mass fraction. Converged solutions could not be obtained if the dummy coefficient was reduced below 110-11 m2/s. Therefore, the results obtained in the converged solutions cannot be considered as “verified”, possibly due to numerical errors in the programming code.

Figure A.3. Effect of dummy diffusion coefficient (Dim) on predicted molar fractions of each of the solution components.

Due to these problems, CFX staff were contacted in order to gain further insights into the possible causes of the observed convergence problems. After examination of the user defined routines it was concluded that, as coded, the source term on the mass

192 fraction transport equations made the equations extremely stiff. In order to reduce this stiffness, a way needs to be found to linearise the source term so that any term containing a negative multiple of the concentration is treated implicitly. Such an approach was deemed to be beyond the scope of this thesis.

A.6. Conclusions An approach for incorporating a simplified version of the Maxwell-Stefan diffusion model, namely the Nernst-Planck model, into the ANSYS CFX software package was attempted. This model is useful for predicting the interactions of dissolved ionic solutes in multiple component situations, for which the simple model of Fick’s law is no longer applicable. The inclusion of a source term as a means for incorporating the NP diffusion model into the commercial ANSYS CFX-10.0 code caused the mass fraction transport equations to become extremely stiff, particularly in the region near the membrane wall. In order to reduce this stiffness, the source term must be linearised so that any term containing a negative multiple of the concentration is treated implicitly.

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Appendix B

Videos produced in the course of this thesis

The purpose of this appendix is to serve as an index to the videos included in the disc in the back cover of this thesis. The videos are organised into folders depending on the chapter of this thesis that they relate to. In addition, the filename for each video begins with VX.Y, which is the quick identification number for each video within this thesis. The DivX compression algorithm was used to encode these videos. The installation executable for this codec (DivX521XP2K.exe for Windows users and DivXInstaller.dmg for Mac users) can be found in the Codec folder. Once the codec is installed, it should be possible to play the videos with any video player, such as Apple Quicktime or Windows Media Player.

B.1. Videos from Chapter 4 Chapter 4 deals mainly with unsteady flows. Therefore, the videos generated from the simulated data produced for Chapter 4 show flow variations in time within a section of the narrow spacer-filled channel. In these videos, vortex shedding is particularly evident, as well as their effect on salt concentration in the bulk of the flow and on the boundary layer. The squeezing of vortical flow regions between the spacer filaments and the membrane walls is also easily observed in the vorticity videos. The following videos show the evolution of salt concentration, at a Reynolds number of 841, as time progresses. The first of these also illustrates the relative magnitude of the wall shear rate at different positions along the membrane walls, with blue and red representing positive and negative wall shear rates, and green representing a wall shear rate of zero. The third of these shows a presents a longer section of the channel. V4.1_TwoLayer_df0p5_Re841_NaCl.avi V4.2_TwoLayer_df0p5_Re841_NaCl_2.avi V4.3_TwoLayer_df0p5_Re841_NaCl_Channel.avi

The following video also illustrates the value of local wall shear rate at the

194 membrane walls, as well as the velocity vectors for a Reynolds number of 841. V4.4_TwoLayer_df0p5_Re841_Velocity.avi

The following two videos present the evolution with time of the magnitude of the vorticity perpendicular to the plane of the channel, which allows the identification of vortex streets. The second video only illustrates the sign of the vorticity. V4.5_TwoLayer_df0p5_Re841_zVort_Channel.avi V4.6_TwoLayer_df0p5_Re841_zVort_sign.avi

Finally the last two videos deal with a Reynolds number of 1683. The first one shows salt concentration information, as well as velocity vectors and wall shear rates. The second video deals with the magnitude of the vorticity perpendicular to the plane of the channel. V4.7_TwoLayer_df0p5_Re1683.avi V4.8_TwoLayer_df0p5_Re1683_zVorticity.avi

B.2. Videos from Chapter 5 In Chapter 5, 3D steady laminar flows are analysed. Therefore, the following videos do not present variation of flow conditions in time, but illustrate the flow conditions on a plane of the fluid domain, which is then moved along the bulk flow direction as time elapses in each video. Each video relates to a different Reynolds number and spacer orientation (both of which are given in each file name). The moving plane displays salt concentration as well as tangential velocity vectors, i.e. the velocity components parallel to the moving plane. The magnitude of the velocity normal to the moving plane is illustrated by the colour of these vectors, with red being higher and blue being zero. V5.1_3D_45deg_df0p6_Re50_NaCl_TangentialVel.avi V5.2_3D_45deg_df0p6_Re100_NaCl_TangentialVel.avi V5.3_3D_45deg_df0p6_Re200_NaCl_TangentialVel.avi V5.4_3D_90deg_df0p6_Re50_NaCl_TangentialVel.avi V5.5_3D_90deg_df0p6_Re100_NaCl_TangentialVel.avi V5.6_3D_90deg_df0p6_Re200_NaCl_TangentialVel.avi

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In these videos, the main flow features (streamwise vortices, vortical flow, flow towards the membrane wall) as well as the thickness of the concentration boundary layer, can be observed. This complements the information in the figures presented Chapter 5, and aids in the understanding of the flow conditions inside 3D spacer geometries.

B.3. Videos from Chapter 6 The only videos generated for Chapter 6 were for those simulations dealing with unsteady flows. Only two of such transient runs were carried out. These videos show the salt concentration as time progresses, at a Reynolds number of 800, for the 4-layer

High-Low configurations with a df/hch ratio of 0.3 and angles of attack for the elliptical filaments of 0° and 10° respectively. V6.1_Ellipse_df0p3_HL_0deg_Re800.avi V6.2_Ellipse_df0p3_HL_10deg_Re800.avi

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