Modeling in Membranes and Membrane-Based Processes: Industrial Scale Separations [1 ed.] 1119536065, 9781119536062

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Modeling in Membranes and Membrane-Based Processes

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Modeling in Membranes and Membrane-Based Processes

Edited by

Anirban Roy, Siddhartha Moulik, Reddi Kamesh, and Aditi Mullick

This edition first published 2020 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2020 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no rep­ resentations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchant-­ ability or fitness for a particular purpose. No warranty may be created or extended by sales representa­ tives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further informa­ tion does not mean that the publisher and authors endorse the information or services the organiza­ tion, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-53606-2 Cover image: Industrial Membrane Device, Kondou | Dreamstime.com Cover design by Kris Hackerott Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Acknowledgement xiii 1 Introduction: Modeling and Simulation for Membrane Processes Anirban Roy, Aditi Mullick, Anupam Mukherjee and Siddhartha Moulik References 2 Thermodynamics of Casting Solution in Membrane Synthesis Shubham Lanjewar, Anupam Mukherjee, Lubna Rehman, Amira Abdelrasoul and Anirban Roy 2.1 Introduction 2.2 Liquid Mixture Theories 2.2.1 Theories of Lattices 2.2.1.1 The Flory-Huggins Theory 2.2.1.2 The Equation of State Theory 2.2.1.3 The Gas-Lattice Theory 2.2.2 Non-Lattice Theories 2.2.2.1 The Strong Interaction Model 2.2.2.2 The Heat of Mixing Approach 2.2.2.3 The Solubility Parameter Approach 2.2.3 The Flory–Huggins Model 2.3 Solubility Parameter and Its Application 2.3.1 Scatchard-Hildebrand Theory 2.3.1.1 The Regular Solution Model 2.3.1.2 Application of Hildebrand Equation to Regular Solutions 2.3.2 Solubility Scales 2.3.3 Role of Molecular Interactions 2.3.3.1 Types of Intermolecular Forces

1 6 9 10 11 11 11 12 13 13 13 13 14 15 18 18 18 19 20 21 21

v

vi  Contents 2.3.4 Intermolecular Forces: Effect on Solubility 2.3.5 Interrelation Between Heat of Vaporization and Solubility Parameter 2.3.6 Measuring Units of Solubility Parameter 2.4 Dilute Solution Viscometry 2.4.1 Types of Viscosities 2.4.2 Viscosity Determination and Analysis 2.5 Ternary Composition Triangle 2.5.1 Typical Ternary Phase Diagram 2.5.2 Binodal Line 2.5.2.1 Non-Solvent/Solvent Interaction 2.5.2.2 Non-Solvent/Polymer Interaction 2.5.2.3 Solvent/Polymer Interaction 2.5.3 Spinodal Line 2.5.4 Critical Point 2.5.5 Thermodynamic Boundaries and Phase Diagram 2.6 Conclusion 2.7 Acknowledgment List of Abbreviations and Symbols Greek Symbols References 3 Computational Fluid Dynamics (CFD) Modeling in Membrane-Based Desalination Technologies Pelin Yazgan-Birgi, Mohamed I. Hassan Ali and Hassan A. Arafat 3.1 Desalination Technologies and Modeling Tools 3.1.1 Desalination Technologies 3.1.2 Tools in Desalination Processes Modeling 3.1.3 CFD Modeling Tool in Desalination Processes 3.2 General Principles of CFD Modeling in Desalination Processes 3.2.1 Reverse Osmosis (RO) Technology 3.2.2 Forward Osmosis (FO) Technology 3.2.3 Membrane Distillation (MD) Technology 3.2.4 Electrodialysis and Electrodialysis Reversal (ED/EDR) Technologies 3.3 Application of CFD Modeling in Desalination 3.3.1 Applications in Reverse Osmosis (RO) Technology 3.3.2 Applications in Forward Osmosis (FO) Technology

23 24 25 26 27 28 32 33 34 36 36 36 36 37 38 40 40 40 42 42 47 48 48 49 55 56 61 65 68 73 77 77 95

Contents  vii 3.3.3 Applications in Membrane Distillation (MD) Technology 108 3.3.4 Applications in Electrodialysis and Electrodialysis Reversal (ED/EDR) Technologies 121 3.4 Commercial Software Used in Desalination Process Modeling 122 Conclusion 132 References 133 4 Role of Thermodynamics and Membrane Separations in Water-Energy Nexus Anupam Mukherjee, Shubham Lanjewar, Ridhish Kumar, Arijit Chakraborty, Amira Abdelrasoul and Anirban Roy 4.1 Introduction: 1st and 2nd Laws of Thermodynamics 4.2 Thermodynamic Properties 4.2.1 Measured Properties 4.2.2 Fundamental Properties 4.2.3 Derived Properties 4.2.4 Gibbs Energy 4.2.5 1st and 2nd Law for Open Systems 4.3 Minimum Energy of Separation Calculation: A Thermodynamic Approach 4.3.1 Non-Idealities in Electrolyte Solutions 4.3.2 Solution Thermodynamics 4.3.2.1 Solvent 4.3.2.2 Solute 4.3.2.3 Electrolyte 4.3.3 Models for Evaluating Properties 4.3.3.1 Evaluation of Activity Coefficients Using Electrolyte Models 4.3.4 Generalized Least Work of Separation 4.3.4.1 Derivation 4.4 Desalination and Related Energetics 4.4.1 Evaporation Techniques 4.4.2 Membrane-Based New Technologies 4.5 Forward Osmosis for Water Treatment: Thermodynamic Modelling 4.5.1 Osmotic Processes 4.5.1.1 Osmosis 4.5.1.2 Draw Solutions

145 146 148 148 149 149 149 152 153 154 154 155 155 156 157 157 159 160 164 166 167 173 173 174 175

viii  Contents 4.5.2 Concentration Polarization in Osmotic Process 4.5.2.1 External Concentration Polarization 4.5.2.2 Internal Concentration Polarization 4.5.3 Forward Osmosis Membranes 4.5.4 Modern Applications of Forward Osmosis 4.5.4.1 Wastewater Treatment and Water Purification 4.5.4.2 Concentrating Dilute Industrial Wastewater 4.5.4.3 Concentration of Landfill Leachate 4.5.4.4 Concentrating Sludge Liquids 4.5.4.5 Hydration Bags 4.5.4.6 Water Reuse in Space Missions 4.6 Pressure Retarded Osmosis for Power Generation: A Thermodynamic Analysis 4.6.1 What Is Pressure Retarded Osmosis? 4.6.2 Pressure Retarded Osmosis for Power Generation 4.6.3 Mixing Thermodynamics 4.6.3.1 Gibbs Energy of Solutions 4.6.3.2 Gibbs Free Energy of Mixing 4.6.4 Thermodynamics of Pressure Retarded Osmosis 4.6.5 Role of Membranes in Pressure Retarded Osmosis 4.6.6 Future Prospects of Pressure Retarded Osmosis 4.7 Conclusion 4.8 Acknowledgment Nomenclature 1. Roman Symbols 2. Greek Symbols 3. Subscripts 4. Superscripts 5. Acronyms References 5 Modeling and Simulation for Membrane Gas Separation Processes Samaneh Bandehali, Hamidreza Sanaeepur, Abtin Ebadi Amooghin and Abdolreza Moghadassi Abbreviations Nomenclatures Subscripts 5.1 Introduction 5.2 Industrial Applications of Membrane Gas Separation

177 177 178 180 180 181 181 181 182 182 182 183 183 184 186 186 187 188 190 191 192 192 192 192 193 194 194 194 195 201 201 202 203 203 205

Contents  ix 5.2.1 Air Separation or Production of Oxygen and Nitrogen 5.2.2 Hydrogen Recovery 5.2.3 Carbon Dioxide Removal from Natural Gas and Syn Gas Purification 5.3 Modeling in Membrane Gas Separation Processes 5.3.1 Mathematical Modeling for Membrane Separation of a Gas Mixture 5.3.2 Modeling in Acid Gas Separation 5.4 Process Simulation 5.4.1 Gas Treatment Modeling in Aspen HYSYS 5.5 Modeling of Gas Separation by Hollow-Fiber Membranes 5.6 CFD Simulation 5.6.1 Hollow Fiber Membrane Contactors (HFMCs) 5.7 Conclusions References 6 Gas Transport through Mixed Matrix Membranes (MMMs): Fundamentals and Modeling Rizwan Nasir, Hafiz Abdul Mannan, Danial Qadir, Hilmi Mukhtar, Dzeti Farhah Mohshim and Aymn Abdulrahman 6.1 History of Membrane Technology 6.2 Separation Mechanisms for Gases through Membranes 6.3 Overview of Mixed Matrix Membranes 6.3.1 Material and Synthesis of Mixed Matrix Membrane 6.3.2 Performance Analysis of Mixed Matrix Membranes 6.4 MMMs Performance Prediction Models 6.4.1 New Approaches for Performance Prediction of MMMs 6.5 Future Trends and Conclusions 6.6 Acknowledgment References 7 Application of Molecular Dynamics Simulation to Study the Transport Properties of Carbon Nanotubes-Based Membranes Maryam Ahmadzadeh Tofighy and Toraj Mohammadi 7.1 Introduction 7.2 Carbon Nanotubes (CNTs) 7.3 CNTs Membranes 7.4 MD Simulations of CNTs and CNTs Membranes

205 206 210 210 210 218 221 222 225 227 227 228 229 237

237 238 242 242 242 243 246 246 253 253

257 258 259 263 265

x  Contents 7.5 Conclusions References

271 272

8 Modeling of Sorption Behaviour of Ethylene Glycol-Water Mixture Using Flory-Huggins Theory 277 Haresh K Dave and Kaushik Nath 8.1 Introduction 278 281 8.2 Materials and Method 8.2.1 Chemicals 281 8.2.2 Preparation and Cross-Linking of Membrane 281 8.2.3 Determination of Membrane Density 281 8.2.4 Sorption of Pure Ethylene Glycol and Water 282 in the Membrane 8.2.5 Sorption of Binary Solution in the Membrane 282 8.2.6 Model for Pure Solvent in PVA/PES Membrane Using F-H Equation 283 8.2.7 Model for Binary EG-Water Sorption 285 Using F-H Equation 8.3 Results and Discussion 289 8.3.1 Sorption in the PVA-PES Membrane 289 8.3.2 Determination of F-H Parameters Between Water and Ethylene Glycol (Xw−EG) 290 8.3.3 Determination of F-H Parameters for Solvent and Membrane (χwm and χEGm) 292 8.3.4 Modeling of Sorption Behaviour Using F-H Parameters 293 296 8.4 Conclusions 297 Nomenclature Greek Letters 298 Acknowledgement 298 References 298 9 Artificial Intelligence Model for Forecasting of Membrane Fouling in Wastewater Treatment by Membrane Technology Khac-Uan Do and Félix Schmitt 9.1 Introduction 9.1.1 Membrane Filtration in Wastewater Treatment 9.1.2 Membrane Fouling in Membrane Bioreactors and its Control

301 302 302 302

Contents  xi 304 9.1.3 Models for Membrane Fouling Control 9.1.4 Objectives of the Study 305 9.2 Materials and Methods 305 9.2.1 AO-MBR System 305 305 9.2.2 The AI Modeling in this Study 9.2.3 Analysis Methods 307 9.3 Results and Discussion 308 9.3.1 Membrane Fouling Prediction Based on AI Model 308 9.3.2 Discussion on Using AI Model to Predict Membrane Fouling 316 9.4 Conclusion 320 Acknowledgements 321 References 321 10 Membrane Technology: Transport Models and Application in Desalination Process 327 Lubna Muzamil Rehman, Anupam Mukherjee, Zhiping Lai and Anirban Roy 10.1 Introduction 328 331 10.2 Historical Background 10.3 Theoretical Background and Transport Models 335 10.3.1 Classical Solution Diffusion Model 336 10.3.2 Extended Solution-Diffusion Model 339 10.3.3 Modified Solution-Diffusion-Convection Model 341 342 10.3.4 Pore Flow Model (PFM) 10.3.5 Electrolyte Transport and Electrokinetic Models 344 10.3.6 Kedem–Katchalsky Model – An Irreversible 346 Thermodynamics Model 10.3.7 Spiegler–Kedem Model 346 10.3.8 Mixed-Matrix Membrane Models 347 10.3.9 Thin Film Composite Membrane Transport Models 348 10.3.10 Membrane Distillation 349 10.4 Limitations of Current Membrane Technology 351 10.4.1 External Concentration Polarisation 351 10.4.2 Internal Concentration Polarisation 352 10.4.3 External Concentration Polarisation Due to Membrane Biofouling 354

xii  Contents 10.5 Recent Advances of Membrane Technology in RO, FO, and PRO 355 10.5.1 Hybrids 358 10.5.2 Other Membrane Desalination Technologies 359 10.5.2.1 Membrane Distillation 359 10.5.2.2 Reverse Electrodialysis (RED) 360 10.6 Techno-Economical Analysis 360 362 10.7 Conclusion List of Abbreviations and Symbols 363 Greek Symbols 365 Suffix 366 References 366

Index 375

Acknowledgement Dr. Roy would like to acknowledge RIG and OPERA grants from BITS Pilani for carrying out the work.

xiii

1 Introduction: Modeling and Simulation for Membrane Processes Anirban Roy1*, Aditi Mullick2, Anupam Mukherjee1 and Siddhartha Moulik2† Department of Chemical Engineering, BITS Pilani Goa Campus, Goa India Cavitation and Dynamics Lab, CSIR-Indian Institute of Chemical Technology, Hyderabad, India 1

2

Abstract

The chapter introduces the book to the reader. This chapter discusses about the evolution of membrane technology as well as related mathematical modeling. It is needless to state that mathematical modeling is imperative as far as industrial scale up or process feasibility analysis is concerned. However, the interplay of various mathematical modeling has contributed significantly to the development of membrane technology. From molecular interaction to transport models to computational fluid dynamics models to thermodynamic perspectives, mathematical modeling has been an “inseparable” ingredient to one of the most advanced ­“separation” technology devised by man. Keywords:  Mathematical modeling, simulation, membrane technology

Membrane Separation Process is a frontier area of research with diversified portfolio of applications [1]. The history of membrane based separation process can be traced back to the discovery by Thomas Graham (1805-1869) where he observed solute transported through a vegetable parchment to water. He was the first person to coin the term ‘dialysis’ for the phenomenon [2]. However, experimental inquisitiveness and industrial translation is a long road to transverse with innumerable challenges to overcome. Two world wars did not serve any good too, but definitely changed the demographic sensitivities as well as did the unthinkable [3]. *Corresponding author: [email protected] † Corresponding author: [email protected] Anirban Roy, Siddhartha Moulik, Reddi Kamesh, and Aditi Mullick (eds.) Modeling in Membranes and Membrane-Based Processes, (1–8) © 2020 Scrivener Publishing LLC

1

2  Modeling in Membranes and Membrane-Based Processes The wars pushed the human civilizations to look for solutions which challenged the framework of contemporary thought processes. Biomedical engineering to nuclear technology, tremendous advances made in short periods to vanquish the enemy, laid the path for posterity. In this whole journey,mankind witnessed and experienced scarce resources become a plenty and resources, otherwise thought to be inexhaustible became challenged. Water is one such example. Fast forward to the 1960’s , the revolutionary discovery by Sidney Loeb and S.Souirajan changed the complete scenario with invention of phase inversion technology [4-5]. The feasibility of obtaining drinking water from sea became a reality and mankind took a giant leap to it’s sustenance. Suddenly it seemed that challenges posed by nature could be overcome by technological advances. Soon the dry lands were dryno more and agriculturebloomed, civilizations prospered and humankind advanced [4]. Similar is the story of biomedical sectors. From the world war II, “Surgeon Hero” era, where collaborative knowledge enhancement between section became restricted, this sector experienced exponential growth [3]. During World War II, the government regulations were minimum with regard to human protection from medical trials. The doctors enjoyed tremendous freedom but on the other hand, were continually pressurized to preserve a resource which ran cant life of a soldier. The doctors had to resort to desperate measures in order to preserve a dying soldier’s life and often took unthinkable risks in order to try various avenues to restore an organ/ organs for a soldier. Thus the term “Surgeon Hero” was coined as they were the indeed the less celebrated heroes of a deadly war. However during these years, a number of solutions were either tried or their seeds were sown to reap benefits later. From dental implants to intralocular lenses to vascular grafts as well as pacemakers- all were either conceived or tried, attributed to the “Surgeon Hero” era [3, 7, 8]. However, the field of membranes also had its foundation laid due to successful trials of an artificial kidney during these years, which laid to the foundation of Hemodialysis. Hemodialysis had an interesting history as during 1913-1944, as a consequence of two wars, the technological development went on simultaneously in the respective nations involved in the conflict [7-11]. However, one was oblivious of the development of other, so much so that the research of John Abel at Jokhns Hopkins was halted as anticoagulant obtained from leeches were not available. Good quality leeches were soured from Hungary which the WW I stopped to be imported to USA, thereby inhibiting development. Fast forward 1970’s, with development of capillary membranes, and Seattle groups “1 m2 hypothesis”, membranes for artificial kidney became a lifesaving technology [5].

Introduction  3 The two most important fluids in human life- water and blood- in today’s world has some relation or the other with membrane technology. Both the reverse osmosis and hemodialysis technology enjoy the major share of a membrane market. Thus, market driven needs of two most important needs for human survival has led to both maturity of technological development as well as customer segmentation. Now, membranes find application in oxygenation, hemoconcentration, artificial kidney, reverse osmosis for desalination, ultrafiltration for general water treatment, as well as for applications like bioreactor systems [6]. In fact, state of art of membranes are being researched and developed for specialized applications like generating power from salinity gradients. Technologies like Pressure Retarded Osmosis (PRO) is the next challenge where the Gibbs Energy of  mixing  of rivers and sea water is harvested to run turbines [7]. The membrane market is projected to reach a USD 2.8 billion by 2020 [8]. It is thus a great success story for the human race to be able to conceive, prototype, build and sustain a technology and eventually make it a commercial success. However, the most important aspect to note is that such a scale of application as well as commercial maturity took time. It took almost a century for simple “ideas” to find their way, meandering through a plethora of challenges to reach this stage. For any process or technological development at the laboratory scale, there lies innumerable hindrances towards its successful implementation at the commercial level. For developing proper understanding and related challenges for scaling up, mathematical modeling is a very important tool [9]. It provides quick insights in the parameters like flux, fouling and resistance building in membrane system [10] [11]. Modeling not only provides scaling up insights, but also helps understand the irreversibility’s occurring in modules. Membrane coupon scale results are often misleading when one tries to understand phenomenon like fouling and pressure loses [12]. Flat sheet membrane coupon scale experiments can yield certain results which can either underpredict or overpredict real life scaled up results. This can, more often than not, give rise to false expectations, thereby giving encouragement or discouragement which is false placed. There are generally three broad kinds of mathematical modeling encountered in literature. The first is modeling for transport process which involves first principle based models and simulation of results. This is the oldest approach which membrane engineers have been resorting to. From simple to fairly complicated systems can easily be solved using this approach. From liquid filtration to gas permeation, first principle based modeling approach has proven to be a versatile approach to understand membrane separation. The second type of modeling approach is based on classical thermodynamics. This approach is extremely useful

4  Modeling in Membranes and Membrane-Based Processes for modeling systems like phase inversion and pore formation in polymer membrane synthesis [13]. Thermodynamics also helps us in understanding the entropy generation and thus related irreversibilities in processes, which in itself an indication on the probable steps which could be taken to mitigate them. Thermodynamic approach also helps us in understanding feasibility of processes and thus gives an idea on how membrane technology intervention can improve efficiencies. The third kind of modeling approach is more recent and has gained popularity over the years due to (i) advent of computers and (ii) robust algorithms to solve non-linear fluid flow equations. This is called Computational Fluid Dynamics (CFD) modeling and is now extensively used in membrane related applications [14] [15]. A schematic representation is shown in Figure 1.1. CFD is now being implemented in areas like membrane module design, packing efficiency calculations, flow phenomena understanding and various other domains which was previously unexplored. A classic example of mathematical modeling in membrane systems is design of reverse osmosis (RO) modules [16]. While first principle based modeling and calculations were used previously to understand flux and fouling, thermodynamic modeling has been used to understand the minimum energies of desalination [17]. The first principle modeling and thermodynamic modeling gave an idea on the deviation from theoretical limits and ideas started developing on how to actually engineer systems so that minimum energies for desalination can be obtained [18]. CFD modeling of flow in commercial modules and design of modules were implemented to get better

yp Permeate out

Vp =0Lf

dz

z=0 Low-pressure side (pf )

V+dV, y+dy

plug flow dq

V, y

xn Lf =(1-0)Lf

Lf , xf , pf Feed in

(pf )

L+dL, x+dx

L, x

Reject out

plug flow dAn

differential volume element

Thermodynamic modelling

Schematic of membrane

Wsep

Transport modelling

yf

CFD simulations on a RO module

Q0, T0

Feed water Splitter Permeale

Booster pump Pressure Exchanger

Black box concept on typical RO plant

Figure 1.1  Schematic representation.

yp < yf

Rejected water

High pressure pump

Brine

yc > yf

CFD modelling

Introduction  5 hydrodynamic flow patterns evolving better results in minimized fouling and greater fluxes. This coupled with energy recovery devices have significantly improved the energies of separation in desalination applications. Another practical example is design of dialyzers [19]. The artificial kidney or a hemodialyzer is the example of a wonderful engineering design which has elements of first principle modeling coupled with CFD simulations. These have helped industries pack more surface area in a given dialyzer volume without compromising on separation efficiencies. Hemodialyzer design involves a complicated set of components assembled to give rise to an optimum clearance of toxins from blood. The components include space fibers and hollow fibers packed in a particular efficiency such that the dialysate fluid can flow within the filters to wash out the toxins being filtered. Around 10000 to 15000 hollow fine fibers are packed in a dialyzer yielding surface areas of 1.5-2 m2 in a cartridge of length 30 cms and diameter of 5-6 cms [20]. CFD modeling has helped immensely in recent years towards achieving this perfection. With the evolution of new membrane technologies, mathematical modeling has a major role to play to make them feasible industrially applicable and economically operable. Thus, membrane based solution or “ideas” which seemingly is infeasible now, can definitely be a solution to several decades into the future. Hence any technological development in this field is of prime importance for our progeny and sustenance of the species. In this regard, the current book has been designed to focus on the understanding of existing matured technologies, their challenges as well as technologies which have the potential to impact the membrane market in the future. The book starts of with the understanding of thermodynamics of casting solutions which impact the morphology of polymer membranes. The chapter lays the foundation of the underlying mechanism and governing principles which determines the pore formation in phase inversion technology. The next chapter deals with a “state of art” computational fluid dynamic modeling of membrane based desalination technologies. In this, the authors have developed in detail the modeling and simulation related to desalination technologies like Reverse Osmosis, Forward Osmosis, Membrane Distillation and Electrodialysis/Reverse Electrodialysis. Thus a comprehensive understanding of CFD in desalination is dealt with. The next chapter is dedicated on the role of thermodynamics in water-energy nexus, where the authors have dealt with the thermodynamic benchmarks and feasibility of various membrane based technologies which finds application in the “Water-Energy Nexus”. The next chapter deals with one of the most challenging aspects of membranes, i.e., gas separation. The authors have delved in detail the modeling of various gas purification technologies,

6  Modeling in Membranes and Membrane-Based Processes as well as technologies for CO2 removal. In continuation the next chapter is on state of art Mixed Matrix membrane based solutions and understanding the mechanism of gas transport and modeling of the same. Traversing from bulk scale modeling to molecular modeling, the next chapter explains molecular dynamics and simulation in relation to study the transport properties of carbon nanotubes based membranes. This is followed by a chapter on modeling of sorption behavior of water-ethylene glycol mixtures in composite membranes. This gives an insight into polymer thermodynamics and application in membrane synthesis and related properties. The next chapter is onapplication of Artificial Intelligence models to understand and predict membrane fouling in waste water treatment technologies.The last chapter is a niched studyon transport modeling in desalination processes involving membranes. Thus, the book gives a glimpse to the readers on “State of Art” existing matured membrane technologies as well as future direction of membranes. As discussed earlier, the seemingly impossible today can be a life savior tomorrow. Technological innovation, leading to industrial revolution has been the benchmark of human existence. To gainer the positive side of every industrial revolution depends on the thought and sensitivities of the contemporary generation. As John F Kennedy said “A revolution is coming: a revolution which will be peaceful if we are wise enough, compassionate if we care enough, successful if we are fortunate enough-but a revolution is coming whether we like it or not. We can affect it’s character, we cannot alter its inevitability.”

References 1. Baker, R.W., Membrane Technology and Application, John Wiley & Sons, Ltd, California, 2004. 2. Graham, T., Liquid diffusion applied to analysis. Philos. Trans. R. Soc. Lond., 151, 183–224, 1861. 3. De, S. and Roy, A., Hemodialysis Membranes For Engineers to Medical Practitioners, CRC Press, New York, 2017. 4. Tal, A., Rethinking the sustainability of Israel’s irrigation practices in the Drylands. Water Res., 90, 387–394, 2016. 5. Babb, A.L., Popovich, R.P., Christopher, T.G., Scribner, B.H., The genesis of square-metre-hour hypothesis. Trans. Am. Soc. Artif. Intern. Organs, 17, 81–91, 1971. 6. Stamatialis, D.F., Papenburg, B.J., Girones, M., Saiful, S., Bettahalli, S.N.M., Schmitmeier, S., Wessling, M., Medical applications of membranes: Drug delivery, artificial organs and tissue engineering. J. Membr. Sci., 308, 1–34, 2008.

Introduction  7 7. Chakraborty, A. and Roy, A., Seasonal Variations in River Water Properties and Their Impact on Mixing Energies and Pressure Retarded Osmosis. Environ. Eng. Sci., 35, 1075–1086, 2018. 8. Sridhar, S., and Moulik, S., Membrane Processes: Pervaporation Vapor Permeation and Membrane Distillation for Industrial Scale Separations, Wiley, USA, 2018. 9. Nazia, S., Moulik, S., Jegatheesan, J., Bhargava, S.K., Sridhar, S., Molecular Dynamics Simulation for Prediction of Structure-Property Relationships of Pervaporation Membranes, in: Membrane Processes: Pervaporation, Vapor Permeation and Membrane Distillation for Industrial Scale Separations, pp. 211–225, John Wiley & Sons, Inc, USA, 2018. 10. Roy, A. and De, S., Resistance-in-series model for flux decline and optimal conditions of Stevia extract during ultrafiltration using novel CAP-PAN blend membranes. Food Bioprod. Process., 94, 489–499, 2015. 11. Roy, K., Mukherjee, A., Maddela, N.R., Chakraborty, S., Shen, B., Li, M., Du, D., Peng, Y., Lu, F., Cruzatty, L.C.G., Outlook on the bottleneck of carbon nanotube in desalination and membrane-based water treatment—A Review. J. Environ. Chem. Eng., 8, 103572, 2019. 12. Roy, A., Moulik, S., Sridhar, S., De, S., Potential of extraction of Steviol glycosides using cellulose acetate phthalate (CAP)–polyacrylonitrile (PAN) blend hollow fiber membranes. J. Food Sci. Technol., 52, 7081–7091, 2015. 13. Roy, A., Bhunia, P., De, S., Solvent effect and macrovoid formation in cel­ lulose acetate phthalate (CAP)–polyacrylonitrile (PAN) blend hollow fiber membranes. J. Appl. Polym. Sci., 134, 1–12, 2017. 14. Jana, D.K., Roy, K., Dey, S., Comparative assessment on lead removal using micellar-enhanced ultrafiltration (MEUF) based on a type-2 fuzzy logic and response surface methodology. Sep. Purif. Technol., 207, 28–41, 2018. 15. Sarkar, A., Moulik, S., Sarkar, D., Roy, C.A., Performance characterization and CFD analysis of a novel shear enhanced membrane module in ultrafiltration of Bovine Serum Albumin (BSA). Desalin., 292, 53–63, 2012. 16. Jamal, K., Khan, M.A., Kamil, M., Mathematical modeling of reverse osmosis systems. Desalin., 160, 29–42, 2004. 17. He, W., Yang, H., Wen, T., Han, D., Thermodynamic and economic investigation of a humidification dehumidification desalination system driven by low grade waste heat. Energy Convers. Manage., 183, 848–858, 2019. 18. Semiat, R., Energy issues in desalination processes. Environ. Sci. Technol., 42, 8193–8201, 2008. 19. A. Roy, S. De, L. Vincent, S.V. Rao, Low cost spinning and fabrication of high  efficiency (he) haemodialysis fibers and method thereof. USA Patent 14598697, 141, 1–8, 14598697, 2016. 20. Roy, A., Dadhich, P., Dhara, S., De, S., Understanding and tuning of polymer surfaces for dialysis applications. Polym. Adv. Technol., 28, 174–187, 2017. 21. Loeb, S. and Souirajan, S., Sea Water Demineralization by Means of an Osmotic Membrane. Adv. Chem., 38, 117–132, 1963.

8  Modeling in Membranes and Membrane-Based Processes 22. Cohen, Y. and Glater, J., A tribute to Sidney Loeb – The pioneer of reverse osmosis desalination research. Desalin. Water Treat., 15, 222–227, 2010. 23. Ratner, B.D., Hoffman, A.S., Schoen, F.J., Lemons, J.E., Biomaterials Science: An Introduction to Materials in Medicine, Academic Press, Cambridge, 2004. 24. Egdahl, R.H., Hume, D.M., Schlang, H.A., Plastic venous prostheses. Surg. Forum, 5, 235–241, 1954. 25. Crubezy, E., Murail, P., Girard, L., Bernadou, J.P., False teeth of the Roman world. Nature, 391, 29–30, 1998. 26. Piccinini, E., Sadr, N., Martin, I., Ceramic materials lead to underestimated DNA quantifications: A method for reliable measurements. Eur. Cell Mater., 20, 38–44, 2010. 27. Cui, L., Liu, B., Liu, G., Zhang, W., Cen, L., Sun, J., Yin, S., Liu, W., Cao, Y., Repair of cranial bone defects with adipose derived stem cells and coral scaffold in a canine model. Biomaterials, 28, 5477–5486, 2007.

2 Thermodynamics of Casting Solution in Membrane Synthesis Shubham Lanjewar1, Anupam Mukherjee1, Lubna Rehman1, Amira Abdelrasoul2,3 and Anirban Roy1* Department of Chemical Engineering, BITS-Pilani K K Birla Goa Campus, Goa, India 2 Department of Chemical and Biological Engineering, University of Saskatchewan, Saskatoon, Canada 3 Global Institute of Water Security, Innovation Blvd, Saskatoon, Saskatchewan, Canada 1

Abstract

Membrane science and engineering is a frontier area of research in the field of separation and purification technology. The technology has reached maturity in certain applications like desalination and hemodialysis whereas technology is still evolving in other areas of application. The art of producing consistent membranes with uniform pore structure is a difficult and often proprietary technology. The phenomenon of formation of pore structure can be explained by the thermodynamics of the solution and kinetics of phase inversion mechanism. In this chapter, the authors have focused on the relevant thermodynamics involved which dominate the properties of the solution and therefore, the nature of pores which can be expected from the system. The chapter introduces basic concepts related to polymer thermodynamics and polymer solutions. Then the basic concepts of solubility of polymers and factors affecting them are discussed, ultimately delving into the details regarding phase inversion techniques and related thermodynamics. Keywords:  Phase inversion, thermodynamics, polymer, solubility

*Corresponding author: [email protected] Anirban Roy, Siddhartha Moulik, Reddi Kamesh, and Aditi Mullick (eds.) Modeling in Membranes and Membrane-Based Processes, (9–46) © 2020 Scrivener Publishing LLC

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2.1 Introduction Membrane synthesis and its application became popular after Loeb and Sourirajan [1] developed the first desalination membranes utilizing the phase inversion technique [2]. The technology of phase inversion became attractive for membrane scientists and engineers as the greatest appeal was the easy scalability of the manufacturing process. Phase inversion is fundamentally a precipitation method where a polymer is dissolved in a solvent to form a homogenous solution [3]. Then the solution is either cast or extruded to form a flat sheet or hollow fiber membrane, respectively. The cast sheet or extruded hollow fiber is brought in contact with a non-solvent (typically water) which results in the exchange of solvent and non-solvent. Needless to state, this path-breaking invention revolutionized the water treatment segment as well as, more importantly so, the desalination market [4]. Gradually, the phase inversion mechanism popularized to an extent which helped membrane manufacturers design porous structures for a variety of application [5]. With technological maturity, the fundamental questions asked by the community of researchers was: “How to understand the phase inversion process?”, and “Which parameters control the phase inversion phenomenon?” Over the years, researchers from Israel, the Netherlands and the United States have worked extensively to understand this phenomenon via thermodynamics and kinetics. The incentive to understand this concept came from the need to control the factors affecting the membrane pore sizes. One of the drawbacks of the phase inversion technology is that the pore sizes are non-uniform and have a distribution. While this diffused pore size distribution can be acceptable in certain applications (like microfiltration), the same is deemed ineffective and, maybe dangerous, in applications like hemoconcentration. Thus, it became quite evident for membrane researchers to understand the specifics of pore size formation and controlling parameters. The efforts started from the mid-1980s when Reuver et al. developed a model for understanding the thermodynamics of ternary (polymer-solventnon-solvent) systems [6] [7]. This was soon followed by the model of Boom et al. [8] to understand quaternary systems involving a blend of polymers. Later on a comprehensive understanding of the various polymer, polymer blends and their respective interactions with solvent, non-solvent were studied by various membrane engineering groups worldwide [9]. With this perspective, it is of utmost importance that any membrane or membrane-based technology should always be understood in light of

Thermodynamics of Casting Solution  11 the fundamental phase inversion process. The current chapter is an effort in a particular direction. The authors have introduced the basic concepts of thermodynamics related to polymers and polymer solutions and then have delved into details regarding the thermodynamics of polymer blends with regards to the membrane synthesis and fabrication technique. The book chapter is envisaged to be helpful to engineers working in the field of polymers and membranes.

2.2 Liquid Mixture Theories The theories of liquid mixtures can extensively elucidate and predict the behavior of the substance in a vast range of independent variables. The understanding of these theories is significant in determining the miscibility of gases or liquids in polymers and will assist in computing phase diagrams of polymer blends, and other data.

2.2.1 Theories of Lattices The lattice theory is based on the principles of statistical mechanics which utilizes a pseudo-crystalline model of properly arranged elements on a “lattice”. The theories, such as free volume, cell-hole, tunnel, Monte Carlo, or molecular dynamics are grouped under the name of lattice theories. Among all the mentioned theories, only two will be discussed in depth. The first and pivotal is the well-recognized theory originated by Flory and Huggins in 1941. It was extended by authors like Utracki and Koningsveld, etc. [10] [11]. The next one is the cell-hole theory developed by SimhaSomcynsky [12] which has been constantly evolving over the past few decades.

2.2.1.1 The Flory-Huggins Theory Binary systems are given the notation of i = 1 or 2 for each of the components (traditionally, the subscript 1 and 2 indicates solvent and polymer, respectively). The Flory-Huggins relation has been expressed in several equivalent forms. Its initial assumption was a major drawback, where all cells of the lattice are consumed by solvent molecules or polymeric segments of an equal size. Therefore, the contribution of free volume was neglected throughout. However, with all drawbacks, this theory is well accepted and a benchmark to describe polymer thermodynamics.

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2.2.1.2 The Equation of State Theory The first equation of state was proposed in 1873 by Van der Waals in his Ph.D. thesis. It is also termed as PVT relationships. These relationships can also be written in the reduced variables form indicating the expected observance of the related principle of states such as

P=

Where,

RT a 8T r 3 − 2    or     P r = r − 2 V −b V 3V − 1 V r Pr =

(2.1)

a 8a ; V r = 3b   ; T r = 2 9 Rb 9b

Eq. 2.1 also emphasizes the freely available volume concept, i.e., as T ≈ 0, V ≈ b. In this relation, Van der Waals explained the movement of molecules in “cells”. A uniform potential revolves around these surrounding molecules. The free volume is the space where free movement of molecules takes place within the cell. Thus, the following terms are defined: 1. 2. 3. 4. 5.

The total volume as V The volume occupied by molecules as Vo Freely available volume as Vf Free volume fraction explained by Doolittle is fD = Vf/Vo Free volume fraction as f = Vf/V

For any chemical structure, the detailed computation methods of Van der Waals excluded volume have been developed [13]. Thermodynamically, the free volume can also be represented in the form of entropy of vaporization as shown in Eq. 2.4

 −∆H v  RT exp  P=  Vf  RT 



P=



(2.2)



 −∆S v  RT RT exp  = V  R  V

(2.3)

 V ∆S v = R ln    Vf 

(2.4)

Thermodynamics of Casting Solution  13

2.2.1.3 The Gas-Lattice Theory As per the gas-lattice theory, a liquid is designated as a binary mixture of molecules which are randomly distributed and occupies vacant sites. The concentration of holes can be varied on changing P and T. However, the size cannot be varied in a similar way; one molecule may utilize a different number of sites. To determine the polymer/polymer miscibility as per the gas-lattice theory the factors responsible are: 1. 2. 3. 4.

Surface areas of interacting polymer segments. The dimension of coils. Molecular weight polydispersity. Free volume fraction.

2.2.2 Non-Lattice Theories 2.2.2.1 The Strong Interaction Model It is the directive model for strong segmental interaction systems, such as incompressible acid-base type system [14] [15]. The following expression was derived:



U  ∆Gm =  2  + ln(1 − λ ) + 1 + q −1 VRTφ1φ2  RT 

(2.5)



 1 U  = 1 + q × exp U1 − 2  ; 0 ≤ λ ≤ 1 λ RT  

(2.6)

(

)

The attractive and repulsive energies are represented as U1 and U2, q indicates degeneracy number. The choice between UCST and LCST depends on the corresponding magnitudes of U1 and U2.

2.2.2.2 The Heat of Mixing Approach When the configurational entropy is infinitesimally small, the enthalpy effect of a polymer blend dominates. The polymer/polymer miscibility can be appropriately predicted by adiabatic calorimetry method.

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 V ∆G m∆H mBφ1φ2 ; B ≡ χ12RT    V1 



(2.7)

After the validation of this ideology through experiments, the main authors have attempted to explain the concept of “miscibility windows”. The term refers to the limited range of the copolymer composition where polymer/copolymer blends show miscibility.

2.2.2.3 The Solubility Parameter Approach The solubility parameter approach was given by Joel Henry Hildebrand in 1936, based on enthalpy of non-ideal solutions which describes that molecular interactions should be nonspecific in nature. Hence, it should not form associations or orientation, including hydrogen and polar bonding [15] [16]. Another preliminary assumption was that the intermolecular interactions 1-2 and the geometric mean of the intramolecular interactions, 2-2 are identical in nature. The solubility parameter, δi, can be determined from the molecules without polar groups: 1. The term can be defined as:



δ 22 =

∆E iv = ∆Hiv − PVi / Vi ≅ ∆Hiv − RT /Vi V

(

)

(

)

(2.8)

2. The surface tension coefficient Vi can be empirically correlated with δi as,



δ i2 =

14 v i Vi1/3

3. The knowledge of experimentally determined values of ∆Hm for material 1 in different solvents with a known value of δI is beneficial for the studies. The molecules which are small in size and without strong interactions, the solubility parameter value ranges from 5.9 (for C6F14) to 14.1 (for I2) and 23.8 (for H2O). The standard deviation of these estimates ± 0.2. By the measurement of behavior in a solvent determines the

(2.9)

Thermodynamics of Casting Solution  15 solubility parameter of a polymer. The biggest drawback of the solubility parameter approach is neglecting the entropic and specific interaction effects. Though minimizing the solubility parameter values which are included in those parameters are unfavorable to miscibility, but help the miscibility. Hansen suggested one way of modification of the solubility parameter into account for the presence of specific interactions within solvent and polymer and the solubility parameter is considered as a vector in this concept made up of three components including due to hydrogen bonding, due to dipole interactions, and due to dispersive forces. Values of each component for different polymers and solvents are determined based on the experimental as well as theoretical modeling.

2.2.3 The Flory–Huggins Model The Flory-Huggins theory is basically a thermodynamic model based on mathematical correlations for polymer solution which considers high dissimilarity in molecular size ranges during entropy generation, resulted in the change of Gibbs free energy ΔGm at the time of mixing a polymer in solvent. At constant pressure and temperature the change in Gibbs free energy accompanying mixing can be written as [17]:

ΔGm = ΔHm − TΔSm

(2.10)

The goal is to find an explicit relationship between ΔHm and ΔSm, i.e., the enthalpy and entropy during mixing and the correlation obtained can be written as

ΔGm = RT[n1lnϕ1 + n2lnϕ2 + n1ϕ2χ12]

(2.11)

Where n1 and ϕ1 represents the number of moles and volume fraction of component 1 (Solvent). While n2 ϕ2 represents number of moles and volume fraction of component 2 (Polymer). χ represents the interdispersing polymer and solvent energies. R is the universal gas constant. T represents the absolute temperature. Firstly, the entropy of mixing is usually calculated to understand the rise in the uncertainty about the site while interspersion of the molecules. The correlation of mixing entropy for smaller molecules in terms of mole fractions is not reliable while the solute is a macromolecular

16  Modeling in Membranes and Membrane-Based Processes chain. Therefore, the dissimilarity in molecular sizes is taken into account for the experimental purpose by considering the individual polymer segments as well as individual solvent molecules occupying the sites on a lattice. The total number of sites during utilization of each site by a single molecule of the solvent or by single monomer of the polymer chain is



N = N1 + xN2

(2.12)

Where, N1 and N2 represents the solvent and polymer molecules. x represents the polymer segment. The rise in spatial uncertainty due to the mixing of polymer and solvent can be termed as entropy change on mixing.



 N   xN   ∆S m = − k  N1 ln  1  + N 2 ln  2    N  N  

(2.13)

The lattice volume fractions ϕ1 and ϕ2 can be defined as

ϕ1 = N1/N

(2.14)

ϕ2 = xN2/N

(2.15)

This represents the probability of occupying a lattice site by the molecules of solvent or segments of polymer which was chosen randomly and therefore

ΔSm = −k[N1lnϕ1 + N2lnϕ2]

(2.16)

If the polymer chain is having small dimensions and its molecule utilizes just a single lattice site, i.e., x = 1, then the volume fractions will be equivalent the mole fractions. The following molecular interactions to consider: 1. Solvent-Solvent ω11 2. Segment-Segment ω22 3. Solvent-Segment ω12. Each of these interactions depends on the expense of the other two interactions, so the rise in energy per segment-solvent contact is

Thermodynamics of Casting Solution  17



∆ω = ω12 −

1 2(ω 22 + ω11 )

(2.17)

The overall contacts can be written as

xN2zϕ1 = N1ϕ2z

(2.18)

while z represents the coordination number which is the number of closest neighboring lattice site, xN2 is the overall polymer segments and xN2z is the number of closest neighbor site to all the polymer segments The total ­polymer-solvent interactions can be obtained by multiplying the probability factor ϕ1 represents any solvent molecule is utilizing such sites. The change in enthalpy can be defined as the product of change in energy per polymer segment-solvent interaction and the number of interactions

ΔHm = N1ϕ2zΔω

(2.19)

The polymer-solvent interaction parameter χ can be termed as

χ12 = zΔω/kT

(2.20)

This parameter depends on the role of the solvent and solute and is the only parameter in the model which specifically depends on material. The change in enthalpy becomes

ΔHm = kTN1ϕ2χ12

(2.21)

On accumulating all the terms, the total change in Gibbs free energy becomes

ΔGm = RT[n1lnϕ1 + n2lnϕ2 + n1ϕ2χ12]

(2.22)

Here, the expression is converted from N1 and N2 molecules to n1 and n2 moles respectively by shifting the Avogadro’s number NA into R = kNA. The interaction parameter can be estimated from the Hildebrand solubility parameter δa and δab

χ12 = Vseg(δa – δb)2/RT Where Vseg is the volume of a polymer segment?

(2.23)

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2.3 Solubility Parameter and Its Application The understanding of the degree of solubility of polymers is key to the evaluation of interactions between polymers and solvents. This is determined by the extent of miscibility of a solvent, which is governed by a parameter called the solubility parameter [18]. Several solubility theories have been proposed such as Hildebrand, Hansen, Fedors and Van Krevelen. The common base for these theories is the Cohesive Energy Density [19]. It is defined as the ratio of the Heat of Vaporisation to the molar volume. The most widely used parameters are Hildebrand and Hansen. Several computational tools have been cited in the literature for the determination of these parameters. Choi and Kavasallis first used atomistic simulations to evaluate the solubility parameters of alkyl phenol ethoxylates and went on to apply it to the estimation of the 3D Hansen solubility parameters [20].

2.3.1 Scatchard-Hildebrand Theory 2.3.1.1 The Regular Solution Model It is a quantitative explanation of the non-ideal behavior of a solution. The basic assumption of this model is, the entropy of mixing of components is identical to the ideal mixing. Whereas, the enthalpy of mixing is non zero. A mixture of A and B is considered, where the average coordination number is z. Suppose that, there is a presence of B as nB molecules in the mixture and these are interacting with znB molecules, and with molecules A contacts are xAznB. Before mixing, there would be BB contacts existing in pure B. Therefore, the resultant change in mixing energy for the B molecules can be

x A znB (BA − BB )



(2.24)

Similarly, the change in mixing energy for the A molecules can be

x B znA (BA − BB )



(2.25)

and the overall change in mixing energy can be



E=

nAnB 1 znAnB  (2AB − AA − BB ) = β (nA + nB ) 2 (nA + nB )

(2.26)

Thermodynamics of Casting Solution  19 In order to avoid counting of interactions twice, A ½ factor has been introduced. z (2AB − AA − BB )/2 replaced with β, which is termed as interaction parameter. It is assumed that the energy of mixing is identical to the enthalpy of mixing. The non-zero enthalpy or energy of mixing leads to another term in the chemical potential which is, μregular, and can be obtained from enthalpy of mixing.



 ∂E   nB nAnB  µ regularA =  = β x B2 (2.27) =β −  2 ∂ n n n + ( ) n n +  A T . P .nB A B  A B  Therefore, the chemical potential of a regular solution is given by



µ A = µθA + RTlnx A + β x B2 = µθA + RTlnx A + RTlnf A

(2.28)

Where a new quantity, the activity coefficient, given by RTlnf A = β x B2 θ Combining the last two terms we get: µ A = µ A + RTln( x A f A )

µ A = µθA + RTln(aA )



(2.29)

where, µθA represents the chemical potential of A in the standard state, aA is the activity of component A [21]. The interaction energy depends on concentration. However, the deviations from Raoult’s Law must be evaluated before examining the dependence on concentration. By designating the solvent as A, xA is approximated to 1 at the Raoult’s Law limit and xB is approximated to 0. Therefore, BxA2 is approximated to 0 and by doing so, the ideal value of the chemical potential will be achieved at this limit. Thus, Raoult’s Law will always be similar to the ideal limit [22].

2.3.1.2 Application of Hildebrand Equation to Regular Solutions The Scatchard-Hildebrand expression for solubility of a solute in a regular solution can be written as:



−logX 2 =

T ∆Smf V ∅2 log m + 2 1 (δ1 − δ 2 )2 R T 2.303RT

(2.30)

20  Modeling in Membranes and Membrane-Based Processes Where X2 represents solute’s mole fraction solubility, ∆Smf  as the entropy of fusion of the solid at its melting point, Tm, in Kelvin, V2 as molar volume of the solute, φ1 as solvent’s volume fraction, δ2 and δ1 as solubility parameter of the solute and solvent, respectively. The first term on the right of Eq. 2.30 is commonly written as:



T (∆H  /2.303RT ) ∆Smf log m = (l/T – 1/Tm ) R T

(2.31)

As per Hildebrand and Scott, the solubility parameter of the solvent, δ1, can be estimated using Eq. 2.32.



 ∆E1v  δ1 =   V  1

1/ 2

 ∆H1v − RT  =  V  1

1/ 2



(2.32)

where ΔE1 represents molar energy of vaporization, ∆H as the heat of vaporization, and V1 as molar volume of the solvent. The square of δ1, can be termed as cohesive energy density of the solvent. Similarly, different methods for evaluating δ1 have been provided by Hildebrand and Scott [23].

2.3.2 Solubility Scales Several indicators are available to judge the solubility of a solvent. Some include: 1. K  aouri-Butanol number – This value is used as a measure of the solvent power of hydrocarbon solvents. A high kauributanol value results in strong solvency [24]. 2. S olubility grade – The Kaouri-Butanol test is used to measure the strength of a solvent. However, the cloud-point test is helpful in determining the solubility grade of a polymer. 3. A  romatic character – The aromatic character of a solvent is the percentage of the molecule that has benzene-structure; can also be obtained on addition of the atomic weights of the same. It is basically, the total percentage of ring-like structures present in the solution.

Thermodynamics of Casting Solution  21 4. S olubility scales such as the aniline cloud-point, the heptane number, the wax number, comes under other empirical scales [25]. These diverse solubility scales give information about the relative strengths of solvents. This makes the determination of the type of solvent to be used to dissolve a material easier. Many of the solubility scales can be related to the Hildebrand solubility parameter directly. It includes parameters such as Hildebrand number, Hansen parameter, hydrogen bonding value, and fractional parameter. When only numerical values for the above-mentioned terms are observed, a Teas graph also known as a triangular graph is found to be more helpful due to its accuracy and clarity in results. Solvents containing distinct chemical groups and different solubility parameter zones have given fractional solubility parameters, which are given to every solvent and a fixed position on the graph which provides a visual indication of interaction type based on the dispersion forces, the polar forces, and hydrogen bonding of the solvent [26]. However, non-ionic liquid interactions with extended polymer interactions, the system which involves water-based systems such as reactions involved acid and base cannot be determined by only simplified solubility parameter method.

2.3.3 Role of Molecular Interactions The attractive forces between the molecules of two or more species are called intermolecular forces. These forces are highly responsible for the relative boiling points and solubility properties of molecules.

2.3.3.1 Types of Intermolecular Forces 1. L  ondon Dispersion: The weakest intermolecular force can be termed as London dispersion force or temporary attractive forces because, when the electrons inside two adjacent atoms utilize postitions which make the atoms form temporary dipoles [27]. Unequal sharing of electrons causes rapid polarization and counter-polarization of the electron cloud, forming short-lived dipoles Figure 2.1. These dipoles interact with the electron clouds of neighbouring molecules forming more dipoles. The weak forces as compared to intermolecular forces are observed under this type and these

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Figure 2.1  Polarization caused due to London Dispersion.

do not extend over long distances. The ease of movement of molecules defines the strength of these interactions within a given molecule. For example: In between hexane molecules the forces lies are London dispersion forces.  ipole-Dipole: The interaction between two dipolar mol2. D ecules is termed as Dipole-Dipole interaction. At this instance, a partially negative portion one polar molecule gets attracted towards the partially positive portion of other polar molecules [28] Figure 2.2. Several physical and biological phenomena can be explained with this type of molecular

Uneven Distribution of electrons

Instantaneous

Figure 2.2  Induced dipole moments in He atoms.

Figure 2.3  Hydrogen bonding between two water molecules.

Dipole-Induced Dipole

Thermodynamics of Casting Solution  23 interactions. For example: dipole-dipole force between two He molecules, results in elevation of boiling point of water.  ydrogen Bonding: When two strongly electronegative atoms 3. H bonded with a lone pair of electrons, there exist a weak type of force called a hydrogen bond which creates a special type of dipole-dipole attraction. These are relatively stronger than other dipole-dipole as well as dispersion forces. However, they are weaker than covalent and ionic bonds [29]. The two polar molecular attractions, specifically one is Hydrogen bonded to an electronegative atom (e.g., H-O-, H-N-, HF), and another with a nonbonded electron pair on an electronegative atom. The special type of dipole-dipole bond is particularly stronger in nature [30]. For example,,hydrogen bonding between ethanol and water and two water molecules Figure 2.3. 4. I on-Dipole: The electrostatic attraction between an ion and a neutral molecule which has a dipole is called as an iondipole attractive force. The ionic compounds in polar liquids carry these type of forces. A cation gets attracted to the partially negative end of a polar molecule with neutral charge. Similarly, an anion gets attracted to the partially positive end of the neutral polar molecule [29]. For example, sodium ion and water holds ion-dipole force. 5. H  ydrophilic and Hydrophobic Solutes: Hydrophilic and Hydrophobic are the classifications of solutes based on their affinity towards water, meaning that, hydrophilic solutes have electrostatic attraction to water. While hydrophobic solutes repel water. A hydrophilic compound is polar in nature and usually contains O–H or N–H groups which form hydrogen bonds with water molecules. For example, five O–H groups glucose is hydrophilic. A hydrophobic substance may be polar, however, contains C–H bonds which do not interact favorably with water molecules, e.g., naphthalene and n-octane. Hydrophilic substances are highly soluble in water and other strong polar solvents, but hydrophobic substances are insoluble in water and highly soluble in nonpolar solvents. For example, benzene and cyclohexane.

2.3.4 Intermolecular Forces: Effect on Solubility The intermolecular forces between solute and solvent molecules govern the extent of solubility and hence the solubility parameter. The stronger the

24  Modeling in Membranes and Membrane-Based Processes intermolecular forces between the solute and solvent molecules, the higher is the solute’s solubility in the solvent. In polar solvents, the dipole-dipole attraction between the polar solute and solvent molecules are predominant intermolecular forces. Therefore, solubility of polar molecules are higher in polar solvents, whereas solubility of nonpolar molecules are lesser in nonpolar solvents as there exist London dispersion attraction between the nonpolar solute and nonpolar solvent molecules. However, polar and nonpolar molecules do not mix in each other [27]. The ionic compound solubility can be highly determined not with the polarity of the solvent but with its dielectric constant, which is a measure of its ability to separate ions in the solution [28].

2.3.5 Interrelation Between Heat of Vaporization and Solubility Parameter Cohesive energy density is defined as the measure which determines interaction strength between the molecules of a substance. A potential energy E can be possessed by all molecules of liquid which are identical in magnitude and opposite in sign to the required energy to separate the molecules an infinite distance. The attraction between atoms and molecules give rise to this potential energy. The energy of individual molecules decreases to zero when evaporation occurs. The parameter used to define this energy is the cohesive energy, expressed as (-E/V), where V represents molar volume which is numerically identical to the potential energy of the substance per cm3. An experimental evaluation of this energy which is required to separate molecules gives rise to the heat of vaporization of the liquid, Hv, which balances change in potential energy as well as work done in the expansion. According to Eq. 2.30, A molecule of a vapour obeys the ideal gas law, which then equal to RT. Therefore,



 ∆H1v − RT  CED (Cohesive Energy Density ) =   V1 



(2.33)

It is relevant for experimental purposes to represent the cohesive energy density in the form of solubility parameter (δ) where δ2 = C.E.D. Thus, the cohesive energy density can be measured for the compounds which have ability to vaporize [30]. However, it is not directly applicable to polymers and dyes as indirect methods of evaluation are available.

Thermodynamics of Casting Solution  25 In order to understand the correlation between these parameters, we have to revisit topic 3.1.2. The Hildebrand solubility parameter for a pure liquid substance is defined as the square root of the cohesive energy density [20]. As mentioned in Eq. 2.32:



 ∆H1v − RT  δ =  V  1

1/ 2

(2.34)



2.3.6 Measuring Units of Solubility Parameter After calculating the solubility parameter, the value obtained must be represented carefully by specifying the correct units. Solubility parameter is in turn a function of cohesive energy density in calories/cc, and as per the standard international units (SI units), it is derived from cohesive pressures. The SI unit of pressure is Pascal, therefore in SI system Hildebrand solubility parameters can be represented in megapascals.

∂/cal½cm-3/2 = 0.48888 × ∂ /MPa1/2

(2.35)

∂/MPa½ = 2.0455 × ∂ /cal½cm-3/2

(2.36)

The common form, designated as ∂ is in use for decades now. Hence, the SI units are used now are designated as ∂/MPa½ or ∂ (SI). The use of the units would depend on the form chosen to derive the solubility parameter. Both the systems of measurement are called Hildebrand parameters [25]. The section presented here gives a brief overview of the various methods of determining the solubility parameter, its significance, and the units of measurement. The concept revolves around the calculation of the Cohesive Energy Density, which is in turn used to find the solubility parameter. The significance of molecular interactions is well established and must be taken into consideration while analysing the solubility of molecules. The calculation of this quantity can be done by molecular dynamics simulations. Various software is available for this task such as ASPEN Plus and Material Studio. A clear understanding of the system would require an in-depth analysis of the interactions of the molecules followed by the application of correct operating conditions, for the simulation to give acceptable values. Also, the Hansen solubility parameter gives corrections and accommodates the shortcomings of the Hildebrand Solubility Parameter. Finally, the critical analysis of the interaction of a mixture of solvents, the relative

26  Modeling in Membranes and Membrane-Based Processes solubility parameters of the individual components as well as the type of Solubility parameter used to define the system, ends up playing a major role in determining the separation technology and methodology that would be applied later.

2.4 Dilute Solution Viscometry Rheology is used as a quantitative tool in polymer science and engineering. This tool is used to define the relationship between the structure and rheology of a polymer or complex fluids. It is significant because rheological properties are very sensitive to the type of structure and are simpler to use than analytical methods. These properties govern the flow behaviour of polymers as well as polymer blends [31]. When two structurally different polymers form a physical mixture which shares secondary forces with non-covalent bonding is called as polymer blends. These are characterized based on the degree of molecular mixing into totally miscible (compatible blends); semi-miscible (semi-­compatible blends) and immiscible (incompatible blends) [32]. While immiscible blend usually follows a rheology pattern which can be predictable (since they exhibit single-phase behavior), the rheology of the phase-inverted system is complex in nature. The need to produce polymer blends has increased in the past few years due to the rise in demand for materials with better and more desirable properties as compared to the individual component polymers. It has also proven to be advantageous as cost-effective; improvement in processing and enhancement of properties has been observed. However, the interpretation of superior properties depends upon compatibility or the miscibility of homopolymers at molecular levels [32]. Compatible polymer pairs are generally described as polymer mixtures that have desirable properties when blended. Compatibility is the term used to cover a wide range of situations to describe good adhesion, useful and improved properties [33], whereas the molecular level mixing phenomena is expressed as “miscibility”. There is a need of quantifying miscibility or interaction parameters experimentally for blended system. Therefore, the relationship between structure and properties can be established in a better way. There are both qualitative and quantitative experimental methods available for measuring miscibility. The most common and well-known techniques used for such investigations are Thermal analysis, Electronic microscopy, Spectroscopy, and Viscometry. Among these, Viscometry is an attractive method for studying the compatibility of polymer blends due

Thermodynamics of Casting Solution  27 to various reasons. It is easy and provides useful information about the relationship between dilute solution properties and bulk structure of polymer blends [33]. Two criteria have been proposed about the miscibility of polymer-polymer in solution by Viscometry. (i) Based on intrinsic viscosity, the plot of intrinsic viscosity vs. composition should come linear indicates miscible blends, while deviation from linearity is an indication of incompatible blends. (ii) A criteria [6] for ternary system containing a solvent and two polymers [34]. An assumption of additivity law is the basis for considering dilute solution viscosity as a parameter for assessment of compatibility of polymer blend, where, the shrinkage of polymer coils in solution happens due to the repulsive interaction which results in a viscosity of polymer mixture that is lower than the value evaluated from viscosities of the pure species [33]. The phase morphology of the blend critically affects the size and shape changes in shear flow, as observed by viscosity. The continuous phase of the phase-separated blend is determined by two important factors: the volume fraction and the viscosity of the components. High volume fraction and low viscosity favour phase continuity [31].

2.4.1 Types of Viscosities In order to understand Viscometry, several terms such as relative viscosity, intrinsic viscosity, and specific viscosity must be understood. The ratio of the viscosities of the polymer solution (of stated concentration) and of the pure solvent at the same temperature is called Relative viscosity. It is also known as Solution Solvent Viscosity Ratio. Relative viscosity (ηr) is expressed in the following equation



ηr =

ρ1 (tc1t1 − E1t12 )

ρ0 (tc0t0 − E0t02 )

(2.37)



Where: ρ = density, tc = tube calibration constant (cSt/s), E = kinetic energy correction constant (cSt·s2), t = flow time (seconds), 1 cSt = 1 × 10-6m2/s. At relatively low concentrations, the relative viscosity is determined of a known concentration of polymer in polymer solution, e.g., 0.2 g per 100 ml of solution, or less. Similarly, the increment in the solution viscosity, η with respect to that of the pure solvent, η0 can be related to the relative viscosity as the ratio ηr = η/η0. The specific viscosity of a solution of concentration C is

28  Modeling in Membranes and Membrane-Based Processes



ηsp =

(η − η0 ) = ηr − 1 η0

(2.38)

This equation gives a relation between the specific viscosity and relative viscosity of the polymer. The intrinsic viscosity can be defined as



ηsp η −1 = lim r C →0 C C →0 C

[η] = Lim

(2.39)

For polymer solutions, calculation of intrinsic viscosity is of prime importance. A classical procedure for the determination of viscosities of solutions of various concentrations, followed by extrapolation of ηsp/C to zero concentration [35].

2.4.2 Viscosity Determination and Analysis The viscosity of a dilute polymer solution, measured under prescribed conditions, is an indication of the molecular weight of the polymer and can be ­ uggins used to calculate the degree of polymerization. It hinges on the H equation, where the specific viscosity, ηsp of a single-solute solution is ­expressed as a function of the concentration C, represented as,



ηsp = [η]C + bC2

(2.40)

Where [η] is the intrinsic viscosity, and b is related to the Huggins coefficient kH by



b = kH [η]2

(2.41)

kH has values ranging from 0.3 in good solvents to 0.5 in poor solvents. It contains information about hydrodynamic and thermodynamic interactions between coils in the polymer solution. For all non-electrolyte dilute solutions, a graph of ηsp/C vs C should come up as a straight line having intercept and gradient equal to [η] and b respectively. Theoretically, the parameter [η] is very useful in measuring the actual hydrodynamic specific volume of an isolated polymer; however, the quantity b represents the binary interactions between polymer segments [7]. Eq. 2.37 can be adapted readily to a ternary system containing a solvent (1) and polymers (2 and 3) by equating as follows,

Thermodynamics of Casting Solution  29



c = c2 + c3

[η] = w2 [η]2+w3[η]3 b = w 22 b22 + w32 b33 + 2w 2w3 b23



(2.42) (2.43) (2.44)

where w designates the normalized weight fraction of polymer (i.e., wi = ci/c; i = 2, 3), and the subscripts refer to the component numbers. (b − b ) Krigbaum and walls suggested an interaction parameter ∆B = is 2  w 2 w 3 defined which can be found experimentally. In this relation,

b = w 2b22 + w3b33 .



(2.45)

Here, the b22 and b33 coefficients can be obtained from the solutions of polymers 2 and 3 respectively. ΔB can be used to predict miscibility of polymers. If ΔB is greater than 0, the polymers are miscible and if it is lesser than 0, they are immiscible. However, if [η]2 and [η]3 are sufficiently far apart, the factors w2 and w3 may be expressed in terms of [η]s. As a result, a more effective parameter is defined:

µ=



∆B ([η]3 − [η]2 )2

(2.46)

Therefore,



 b − b22 b −b  − 33 22  ÷ [2([η]3 − [η])] µ= [η] − [η]2 [η]3 − [η]2 

(2.47)

This equation is applicable when [η]3 ≠ [η]2. It can be used to find the polymer-polymer miscibility quite easily from the computed data of dilute-solution viscometry (DSV) [36]. A summary of various regression models has been provided below and can be chosen, depending on the type of polymer investigated [37] [38]: 1. Huggins

[η] = Lim(ηr ) (2.48)

2. Kraemer

[η]K = lim(ηi ) (2.49)

C →0

C →0

30  Modeling in Membranes and Membrane-Based Processes 3. Schulz-Blaschke

[η]K = lim(ηi ) (2.50)

4. Martin

[η]M = lim(ln ηr ) (2.51)

C →0

C →0

Since the type of viscosity of importance in studying the characteristics of polymers is intrinsic viscosity, an Ubbelohde viscometer. Viscosity measurements can be made with this apparatus at 30 ± 0.02°C. Dilute solution viscosity measurement of the polymer samples can be conducted to determine the intrinsic viscosity. Viscosity can also be calculated using standard ASTM methods. The inherent viscosity must be controlled so that the processability and end properties of the resin remain in the desired range. The ASTM D4603-18 [39] allows the determination of the intrinsic viscosity of a polymer with a single concentration, and ASTM D2857–16 [40] carries out successive dilutions of the original solution. The former is only applicable at 30°C, whereas the latter can be used at other temperatures as well. Tube 1

Start mark

Tube 2

Polymer solution is drawn into Tube 1 after applying suction at Tube 1

Stop mark Time taken for fluid to flow from start to stop mark is measured

Figure 2.4  Ubbelholde Viscometer ASTM D445-18 [41] [42].

Thermodynamics of Casting Solution  31 Firstly, solvent mixtures of polymer and solvent are made as per desired concentration requirements. Once the solutes are completely solubilized in the solvent, a primary filtration step is conducted, after which the accurate value of the solute concentration is determined. This can be done by evaporating a known volume of the solution [43]. Henceforth, parameters such as relative (ηr), inherent (ηinh), reduced (ηred), and intrinsic ([η]) viscosities can be advertently determined from the predetermined values of concentrations of polymer solutions by the following equations:

ηr =



ηinh = ln ηr



t t0

ηred =

ηr −1 C

[η] = 0.25 ηr -1+ 3ln ηr

(2.52) (2.53) (2.54) (2.55)

where t is the flow time of polymer solution (s); t0 is the flow time of pure solvent mixture (s); C is the polymer solution concentration (g/dL) [42]. The intrinsic viscosities for solutions which have been infinitely diluted and the solute concentrations of which tend to zero, can be determined by utilising Huggins Eq. (2.37) and Kraemer Eq. (2.49) as follows:

ln ηr = [η]C + kk[η]2C

(2.56)

where ηr is the relative viscosity, and kh and kk are the Huggins and Kraemer constants respectively. ηsp and ηr can be calculated from the following equations:





η t = η0 t0

(2.57)

(η − η0 ) = ηr − 1 η0

(2.58)

ηr = ηsp =

where ηr and η0 are the values of the polymer solution and pure solvent viscosities, respectively, and t and t0 are the recorded time readings during

32  Modeling in Membranes and Membrane-Based Processes which the polymer solution and pure solvent, respectively, passed through the Ubbelohde viscometer. The error in flux measurements is of the order of ±0.05s. Upon obtaining a linear curve from Eq. (2.37) or (2.49), the linear slope so obtained [η] represents the intrinsic viscosity. As stated by the additivity law, the [η]cal – experimentally determined viscosity of the polymer system can be written as:

[η]cal = w1[η]1 + w2[η]2

(2.59)

Where [η]1 and [η]2 are the viscosities of the polymers of the blend, which are obtained open Viscosity measurement, the mass fractions of the polymers are represented by w1 and w2 Also,

Δ[η] = [η]exp – [η]cal

(2.60)

Then, Δ[η] can be obtained from Eq. (2.52). It is the difference between the calculated and experimental viscosity values [44]. In this type of viscometer, the solution flows through a vertical capillary as a result of gravitational force from start mark as shown in Figure 2.4, the time required for a specific volume of solution to flow through the capillary and to reach stop mark is measured. This technique becomes very insensitive for small viscosities that barely deviate from the viscosity of the pure solvent. Furthermore, capillary forces demand certain corrections which are incorporated later. Therefore, capillary viscometer can be used instead. This instrument measures a difference in pressure at either side of a capillary, through which the solution is pressed with arbitrary inlet pressure. According to the Hagen-Poiseuille Law, viscosity can be calculated from the pressure decay over the capillary. This viscometer has several advantages. It has a much higher accuracy of pressure measurement. It is not limited to gravitational force. The inlet pressure can vary so that the variable shear rate gives access to phenomena of thixotropy and shear dilution. Also, this instrument does not require many samples for analysis.

2.5 Ternary Composition Triangle To represent a binary phase diagram, the horizontal axis is enough to specify the composition because the amount of only one component is required to know and other can be obtained by subtracting it from one (i.e., unit

Thermodynamics of Casting Solution  33 mole fraction). But in case of ternary systems, the amounts of two components need to be specified by two different axes and the amount of the third can be obtained by subtracting the sum of these two from one (unit mole fraction). In a ternary system, the sum of the mole fractions is unity, i.e., xA + xB + xC = 1 (for components A-B-C) and therefore, there are two independent composition variables. A representation of composition, symmetrical with respect to all three components, may be obtained with the equilateral composition triangle. Generally, a space model is required to illustrate the complete phase equilibria of a ternary system at constant pressure at the three-dimensional pattern [45]. For the representation of composition in ternary diagrams, it requires two dimensions, and that of temperature as a third dimension. A ternary temperature-composition phase diagram at constant total pressure may be plotted as a three-­dimensional “space model” within a right triangular prism with the equilateral composition triangle as base and temperature as the vertical axis.

2.5.1 Typical Ternary Phase Diagram From a thermodynamic point of view, the study of polymer-solvent– nonsolvent system can be well depicted in a ternary phase diagram. The pure components are represented at the corners of the triangle while boundary lines between any two corners of the triangle represent mixtures of two components, and any point inside the triangle diagram represents a mixture of all three components. Generally, a typical ternary phase diagram contains binodal and spinodal curve, critical point, and a tie line as shown in Figure 2.5. Binodal curves are obtained by performing experiments whereas spinodal curves are obtained theoretically. Region left to the binodal curve represents the thermodynamically stable region or homogeneous one-phase region and the region to the right represents the thermodynamically unstable region or liquid-liquid two-phase region. The region between binodal and spinodal line is a type of metastable state, known as binodal mixing zone where nucleation of polymer poor phase occurs. Ternary mixtures whose compositions are in the left a portion of the binodal as depicted in Figure 2.5 are homogeneous and stable. The mixtures whose compositions are in between binodal and spinodal curves are metastable. The mixtures whose compositions are in the right portion of the binodal curve are unstable. The evaluation of interaction parameters for the binary mixture has allowed a remarkably accurate prediction of the ternary phase diagram. For thermodynamic evaluation of a polymer solution, ternary phase diagrams are drawn using the Flory-Huggins theory, which gives us the free

34  Modeling in Membranes and Membrane-Based Processes Polymer

Binodal curve Spinodal curve

Metastable region Polymer-rich phase Stable region Critical point

Solvent

Unstable region Tie line

Polymer-lean phase

Metastable region

Non-solvent

Figure 2.5  A Typical ternary phase diagram.

energy of mixing for a ternary system as a function of the concentrations. The magnitude and concentration dependency of these interaction parameters have a large effect on the binodal curve, spinodal curve, and critical point positions of a phase ternary diagram. The preparation-mechanism of polymeric membranes with asymmetric structure is generally followed by the phase inversion technique and this whole process can be explained by a change in the composition of the cast membrane, along a line on a ternary nonsolvent-solvent-polymer composition diagram, i.e., ternary phase diagram. Since the separation of a given composition into two phases of different compositions occurs at thermodynamic equilibrium, hence thermodynamic consideration is also taken to draw phase boundary line and this approach is not used to describe the formation of membrane microstructure because it deals with the compositions at bulk phase [22]. In such a ternary system, two types of the mechanism may exist during the development of the membrane. One is instantaneous liquid-liquid demixing in which membrane forms as precipitation instantly after immersion in the nonsolvent system and the other one is delayed liquid-liquid demixing in which precipitation comes after some time of immersion.

2.5.2 Binodal Line In thermodynamics, the binodal line indicates the condition at which two distinct phases may coexist, i.e., the thermodynamic condition which is preferable to separate the distinct phases. It consists of polymer-rich phase and polymer lean phase and during the phase separation of the mixture,

Thermodynamics of Casting Solution  35 two phases should be in equilibrium, i.e., the chemical potentials of the components in each phase should be in equal. Gibb’s free energy of mixing follows the following Flory-Huggins relation [46]:



∆G m = ni lnχi + g ijni χ j RT

(2.61)

Now the chemical potential for a ternary system is



∂  ∆G m  ∆µ i =   RT ∂ni  RT  P ,T ,n j

(2.62)





 ∆µ1  v v =  ln χ1 + 1 − χ1 − 1 χ 2 − 1 χ3 + (g12 χ 2 + g13χ3 ) RT  v2 v3  (2.63)   dg12   v1 × (χ 2 + χ3 ) − g 23 χ 2 χ3 − u1u 2 χ 2   v2  du 2   



 v  ∆µ 2  v v =  ln χ 2 + 1 − χ 2 − 2 χ1 − 2 χ3 +  g12 2 χ1 + g 23χ3   v1 RT  v1 v3   (2.64)  v 2  dg12   v2 × (χ1 + χ3 ) − g13 χ1χ3 + u1u 2 χ1   v 1  du 2   v1 

  ∆µ 3  v v v v =  ln χ3 + 1 − χ3 − 3 χ1 − 3 χ 2 +  g13 3 χ1 + g 23 3 χ3   RT  v1 v2 v1 v2      v (2.65) × (χ1 + χ 2 ) − g12 3 χ1χ 2  v 1   Where; 1, 2, 3 indicates the non-solvent, solvent and polymer, respectively, and gij refers to the binary interaction parameter of the components i and j.

36  Modeling in Membranes and Membrane-Based Processes

2.5.2.1 Non-Solvent/Solvent Interaction Generally, these interaction parameters are calculated using Gibbs free energy data [47]:



g 12 =

 x1   x2  G E  1  ln ln x x + +  1    2  x1φ2   φ2   φ2  RT 

(2.66)



Where GE is usually calculated by experimental data [48]. The relation between GE and ΔGm is given by [49]:

GE = ΔGm – RT(x1lnx1 + x2lnx2)

(2.67)

2.5.2.2 Non-Solvent/Polymer Interaction This interaction factor is experimentally analyzed using equilibrium swelling value [50]:



g 13 = −

ln(1 − φ3 ) + φ3 φ32

(2.68)

Apart from this, the interaction parameters could be theoretically evaluated also by applying the Hansen solubility method which is discussed above.

2.5.2.3 Solvent/Polymer Interaction Some well-known experimental methods of calculating solvent/polymer interaction parameter are osmotic pressure, vapor sorption, inverse gas chromatography, etc.

2.5.3 Spinodal Line Thermodynamically, the limit of local stability with respect to small fluctuations is clearly defined by the condition that the second derivative of free

Thermodynamics of Casting Solution  37 energy is zero. The locus of these points (the inflection point) is known as the spinodal curve. Following relations are used to calculate the spinodal line [51]

∂ 2∆G =0 ∂χ 2



G22G23 = (G23)2







G 22 =

G33 =

G 23 =

(2.69)

(2.70)

 dg   dg  v 1 + 1 − 2g12 + 2(u1 − u 2 ) 12  + u1u 2  122  (2.71) χ1 v 2 χ 2  du 2   du 2 

 d 2 g12   dg  v 1 (2.72) + 1 − 2g13 + 2u 22 (1 − u1 ) 12  + u1u 32  χ1 v 3χ3  du 2   du 22 

 d 2 g12   dg  1 v1 + g 23 − (g12 + g13 ) + u 2 (u1 − 2u 2 )  12  + u1u 22  χ1 v 2  du 2   du 22 



(2.73)

2.5.4 Critical Point The extremum of a binodal curve in temperature coincides with one of the spinodal curves and is known as a critical point which is expressed by the following relation



∂ 2∆G ∂3∆G = =0 ∂χ 2 ∂χ3

(2.74)

38  Modeling in Membranes and Membrane-Based Processes

2.5.5 Thermodynamic Boundaries and Phase Diagram Before doing any thermodynamic study of any system to check the reliability and feasibility of that process, a consistent thermodynamic model must beprepared. In most of the case, to study equilibrium and develop the thermodynamic model in the ternary system of polymer, non-­solvent and solvent systems, the Flory-Huggins’s theory has been considered. Basically, the Flory-Huggins theory has some drawbacks related to the determination of experimental data for binary interaction parameter gij [52] and therefore, by Flory-Huggins theory, the value of these parameters, correlation between available data are critically predicted. Accordingly, it would be more logical to develop such a thermodynamic model which would be helpful to calculate activity coefficients, chemical potentials, different thermodynamic regions recognition, etc. As per the previous research and available literature, Compressible Regular Solution (CRS) theory is the most acceptable and well-developed theory for the prediction of ternary phase diagrams due to the minimum requirement of component properties in the calculation [53]. According to this CRS theory, the relation for the calculation of Gibb’s free energy of mixing per unit volume (in ternary phase) is following for compressible and incompressible conditions, respectively.

   χρ  χρ χρ ∆G m = K B T  1 1 ln χ1 + 2 2 ln χ 2 + 3 3 ln χ3  N2 v 2 N3 v 3  N1v 1 

1δ1,0 − ρ 3δ 3,0 ) (2.75) 1δ1,0 − ρ  2δ 2 ,0 ) + χ1χ3 ( ρ + χ1χ 2 ( ρ 2



 2δ 2 ,0 − ρ 3δ 3,0 ) + χ2χ2 (ρ

2

2



   χρ  χρ χρ ∆G m = K B T  1 1 ln χ1 + 2 2 ln χ 2 + 3 3 ln χ3  N2 v 2 N3 v 3  N1 v 1 

( (  )(δ −ρ

) ) − δ )

1ρ  2 (δ1,0 − δ 2 ,0 )2 + χ1χ 2 (ρ 1 − ρ  2 ) δ12 − δ 22 (2.76) + χ1χ 2ρ 1ρ 3 (δ1,0 − δ 3,0 )2 + χ1χ3 (ρ 1 − ρ 3 ) δ12 − δ 32 + χ1χ3ρ



 2ρ 3 (δ 2 ,0 − δ 3,0 )2 + χ 2 χ3 (ρ 2 +  χ 2 χ 3ρ

3

2 2

2 3

Thermodynamics of Casting Solution  39 where reduced density and hard-core solubility parameter are





i = ρ

ρi = exp(−α i T ) ρ0i

δ i2 (T ) = δ i2 (298)

ρi (T ) ρi0 (T )

(2.77)

(2.78)

At the point of intersection between the binodal layer and glass transition boundary layer (i.e., glass transition temperature/vitrification boundary, Tg) of a system, gelation takes place illustrated in Figure 2.6. For a membrane forming system, Tg can be described by the following relation



Tg =

Rχ 2 Tg 2 + χ3Tg 3 Rχ 2 + χ3

(2.79)

Using equilibrium data from the binodal and spinodal boundary of the given system, the intersecting point of the binodal and vitrification boundary can be predicted at a glass transition temperature (Tg) to obtain gelation boundary shown in Figure 2.6. During the calculation of these boundary layers in the ternary system, the first binodal curve should be determined followed by spinodal boundary calculation and that curve has to be validated by cloud point data measured experimentally. Then the intersection of binodal and Berghmans point should be determined to obtain gelation boundary, i.e., the common Polymer Vitrificationboundary Berghmans point

Spinodal boundary

Binodal boundary

Solvent

Gelation-boundary

Non-solvent

Figure 2.6  Ternary phase diagram of an amorphous polymer having a ternary system containing displaying gelation boundary-layer, Berghmans point, bimodal and spinodal point, and Vitrification boundary, i.e., glass transition boundary of the system.

40  Modeling in Membranes and Membrane-Based Processes point of binodal boundary, glass transition boundary and gelation boundary as depicted in Figure 2.6. After the determination of this point, the remaining boundaries can be calculated.

2.6 Conclusion The science of pore formation continues to intrigue engineers and scientists alike. While engineering pores is not possible without understanding the science, the science of thermodynamics, in turn, sheds light onto the membrane structure formation. As more and more synthetic polymers find their way into the market catering to the needs of both w ­ ater sector as well as the biomedical domain, there is a continuous need for more theoretical development and fundamental understanding for polymers. In order to apply the polymers for specific separation performances, knowledge of polymers, their interaction with solvents and r­elated thermodynamics is mandatory. Hence, it can be expected that more ­generalized models, including multiscale models, can be developed for various applications and theoretical development with regard to pore size distribution and polymer thermodynamics can be the “holy grail” in this regard.

2.7 Acknowledgment Anirban would like to thank the Research Initiation Grant (BPGC/ RIG/2017-2018, Dt. 01/08/2017) and OPERA award FR/SCM/230117/ CHE, Dt. 19/08/2017) by BITS Pilani Goa for carrying out the work.

List of Abbreviations and Symbols i,c V Tr Vr P r R T

Component Volume (cm3) Reduced Temperature Reduced Volume Reduced Pressure Universal gas constant (J/mol.K) Temperature (K)

Thermodynamics of Casting Solution  41 w x H l A ΔGm ΔHm GE B ni gij P KB vi Ni Tg z n wi ΔHv ΔE Vf Vseg ΔSv V T Tm ∆Smf X2 aA fA N U1 U2 q tc C E t ΔB

Weight (g) Mole fraction Enthalpy (J/kg) Length of HF membrane (cm) Cross sectional Area (cm2) Gibb’s free energy for polymer mixing, J/kg. Change in Enthalpy for polymer mixing, J/kg. Excess Gibb’s Free Energy, J/kg. Parameter indicating polymer Number of moles of component, Mol. Binary interaction parameter Pressure, Pa. Boltzmann Constant Molar volume, cm3/mol. Number of segments in vi. Glass temperature, K. Average coordination number Number of molecules Mass fraction Heat of vaporization Molar energy of vaporization Free Volume Actual volume of a polymer segment Entropy of Vaporisation Molar volume of the solvent Temperature in Kelvin at which the solubility is determined Melting point; on the Kelvin scale Entropy of fusion of the solid Mole fraction solubility of the solute Activity of component A Activity coefficient Number of molecules Attractive energies Repulsive energies Degeneracy number Tube calibration constant (cSt/s) Concentration of polymer solution concentration (g/dL) Kinetic energy correction constant (cSt·s2) Flow time (s) Interaction parameter

42  Modeling in Membranes and Membrane-Based Processes

Greek Symbols δ Solubility parameter (MPa0.5) χ Binomial interaction parameter Θ Thermodynamic enhancement parameter  Energy Directional specific interaction λ Surface tension coefficient γi Relative viscosity ηr Specific viscosity ηsp [η]exp Experimental intrinsic viscosity [η]cal Calculated intrinsic viscosity ∂ Unit of solubility parameter ϕ Volume fraction of the solvent β Interaction parameter µ Chemical potential µθA Chemical potential of A in the standard state a Activity ε Porosity (%) ω Binary Interactions Chemical potential, J/mol. µi i ρ Reduced density of component, kg/m3. Density of species, kg/m3. ρi Hard-core solubility parameter at 0 K. δi,0 Solubility parameter at particular temperature δi Volume coefficient of thermal expansion αi µθA Represents the chemical potential of A in the standard state Volume fraction of component i. ϕ i Density of water (g/cm3) ρ

References 1. Loeb, S. and Sourirajan, S., High-flow semipermeable membranes for separation of water from saline solutions. Adv. Chem., 38, 1, 117–132, 1962. 2. Cheryan, M., Ultrafiltration and microfiltration handbook, Technomic Publishing Co. Inc, Lancaster, Pennsylvania, 1998. 3. Loeb, S. and Sourirajan, S., Seawater demineralization by means of a semipermeable membrane, UCLA Department of Engineering, 1960. 4. Mok, S., Worsfold, D.J., Fouda, A.E., Matsuura, T., Wang, S., Chan, K., Study on the effect of spinning conditions and surface treatment on the geometry

Thermodynamics of Casting Solution  43 and performance of polymeric hollow-fiber membranes. J. Membr. Sci., 100, 3, 183–92, 1995. 5. Bakeri, G., Ismail, A.F., Shariaty-Niassar, M., Matsuura, T., Effect of polymer concentration on the structure and performance of polyetherimide hollow fiber membranes. J. Membr. Sci., 363, 1–2, 103–111, 2010. 6. Reuvers, A.J. and Smolders, C.A., Formation of membranes by means of immersion precipitation: Part II. The mechanism of formation of membranes prepared from the system cellulose acetate-acetone-water. J. Membr. Sci., 34, 1, 67–86, 1987. 7. Reuvers, A.J., Altena, F.W., Smolders, C.A., Demixing and gelation behavior of ternary cellulose acetate solutions. J. Polym. Sci., 24, 4, 793–804, 1986. 8. Boom, R., d. Boomgaard, T.V., Smolders, C., Mass transfer and Immersion Precipitation for a two-polymer system PES-PVP-NMP-water. J. Membr. Sci., 90, 3, 231–249, 1994. 9. d. Witte, P.V., Dijkstra, P.J., V. d. Berg, J.W.A., Feijen, J., Phase separation processes in polymer solutions in relation to membrane formation. J. Membr. Sci., 117, 1, 1–31, 1996. 10. Koningsveld, R. and Staverman, A.J., Determination of critical points in multicomponent polymer solutions. J. Polym. Sci. C, 16, 3, 1775–1786, 1967. 11. Utracki, L., Investigations of the phenomena of coacervation. Part III. Phase equilibrium in the three-component system toluene–ethanol–polystyrene. J. Appl. Polym. Sci., 6, 22, 399–403, 1962. 12. Simha, R. and Somcynsky, T., On the Statistical Thermodynamics of Spherical and Chain Molecule Fluids. Macromolecules, 2, 4, 342–350, 1969. 13. Krevelen, D., Properties of Polymers, Elsevier, Amsterdam, 1976. 14. Brinke, G.T. and Karasz, F.E., Lower critical solution temperature behavior in polymer blends: Compressibility and directional-specific interactions. Macromolecules, 17, 4, 815–820, 1984. 15. Manias, E. and Utracki, L.A., Thermodynamics of Polymer Blends, in: Polymer Blends Handbook, pp. 171–289, Springer, Dordrecht, 2014. 16. Huggins, M.L., Solutions of Long Chain Compounds. J. Chem. Phys., 9, 440, 1941. 17. Flory, P.J., Thermodynamics of High Polymer Solutions. J. Chem. Phys., 9, 8, 660, 2004. 18. Belmares, M., Blanco, M., Goddard, W. 3, Ross, R.B., Caldwell, G., Chou, S.H., Pham, J., Olofson, P.M., Thomas, C., Hildebrand and Hansen solubility parameters from molecular dynamics with applications to electronic nose polymer sensors. J. Comput. Chem., 25, 15, 1814–1826, 2004. 19. Li, C. and Strachan, A., Cohesive energy density and solubility parameter evolution during the curing of thermoset. Polymer, 135, 162–170, 2018. 20. Kavassalis, T.A., Choi, P., Rudin, A., The Calculation of 3D Solubility Parameters Using Molecular Models. Mol. Simul., 11, 2–4, 229–241, 1993. 21. Hildebrand, J.H. and Scott, R.L., The Solubility of Non-Electrolytes, 3rd ed., Reinhold Publishing Corporation, New York, 1950.

44  Modeling in Membranes and Membrane-Based Processes 22. Matsuura, T., Synthetic Membranes and Membrane Separation Processes, 1st ed., CRC Press, 1993. 23. Martin, A., Newburger, J., Adjei, A., Extended Hildebrand Solubility Approach: Solubility of Theophylline in Polar Binary Solvents. J. Pharm. Sci., 69, 5, 487–491, 1980. 24. Standard Test Method for Kauri-Butanol Value of Hydrocarbon Solvents, 2002. 25. Burke, J., Solubility Parameters: Theory and Application, The American Institute for Conservation, California, 1984. 26. Luo, C.J., Stride, E., Edirisinghe, M., Mapping the Influence of Solubility and Dielectric Constant on Electrospinning Polycaprolactone Solutions. Macromolecules, 45, 4669–4680, 2012. 27. Peter, A. and de Paula, P.J., Physical Chemistry for the Life Sciences, Oxford University Press, United Kingdom, 2011. 28. Fevre, R.J.W., Dipole Moments, Their Measurement and Application in Chemistry, Methuen & Co. Ltd, United Kingdom, 1953. 29. Peter, A., Loretta, J., Leroy, L., Chemical Principles, W.H. Freeman & Co Ltd, United Kingdom, 2016. 30. Chen, X., Yuan, C., Wong, C.K., Zhang, G., Molecular modeling of the temperature dependence of solubility. J. Mol. Model., 18, 2333–2341, 2012. 31. Sadiku-Agboola, O., Sadiku, E.R., Adegbola, A.T., Biotidara, O.F., Rheological Properties of Polymers: Structure and Morphology of Molten Polymer Blends. Mater. Sci. Appl., 2, 30–41, 2011. 32. Singh, Y.P. and Singh, R.P., Compatibility Studies on Solutions of Polymers Blends by Viscometric and Ultrasonic Techniques. Eur. Polym. J., 19, 6, 535– 541, 1983. 33. Barani, H. and Bahrami, S.H., Investigation on Polyacrylonitrile/Cellulose Acetate Blends. Macromol. Res., 15, 7, 605–609, 2007. 34. Sun, Z., Wang, W., Feng, Z., Criterion of Polymer-Polymer Miscibility Determined by Viscometry. Eur. Polym. J., 28, 10, 1259–1261, 1992. 35. Pamies, R., Cifre, J.G.H., Martínez, M. d. C. L., Torre, J. G. d. l., Determination of intrinsic viscosities of macromolecules and nanoparticles. Comparison of single-point and dilution procedures. Colloid Polym. Sci., 286, 11, 1223– 1231, 2008. 36. Chee, K.K., Determination of Polymer-Polymer Miscibility by Viscometry. Eur. Polym. J., 26, 4, 423–426, 1990. 37. Kulicke, W. and Clasen, C., Viscosimetry of Polymers and Polyelectrolytes, Springer-Verlag, Berlin, 2004. 38. Nicholson, J., The chemistry of polymers, CPI Group, Croydon, 2017. 39. ASTM D4603-18, Standard Test Method for Determining Inherent Viscosity of Poly(Ethylene Terephthalate) (PET) by Glass Capillary Viscometer, ASTM International, West Conshohocken, 2018. 40. ASTM D2857-16, Standard Practice for Dilute Solution Viscosity of Polymers, ASTM International, West Conshohocken, 2016.

Thermodynamics of Casting Solution  45 41. ASTM D445-18, Standard Test Method for Kinematic Viscosity of Transparent and Opaque Liquids (and Calculation of Dynamic Viscosity), ASTM International, West Conshohocken, 2018. 42. Farah, S., Kunduru, K.R., Basu, A., Molecular Weight Determination of Polyethylene Terephthalate, in: Poly(Ethylene Terephthalate) Based Blends, Composites and Nanocomposites, pp. 143–165, Elsevier, 2015. 43. Kim, N., Bang, J., Choi, S., Kim, E., Determination of the molecular weight distributions from rheological properties of viscoelastic polymers. J. Ind. Eng. Chem., 2, 97–105, 1996. 44. Mei, S., Xiao, C., Hu, X., Interfacial Microvoid Formation of Poly(vinyl chloride)/ Polyacrylonitrile Blend Hollow-Fiber Membranes. J. Appl. Polym. Sci., E9– E16, 124(S1), 2011. 45. Campbell, F.C., Ternary Phase Diagrams, in: Phase Diagrams - Understanding the Basics, pp. 191–200, 2012. 46. Mohsenpour, S., Esmaeilzadeh, F., Safekordi, A., Tavakolmoghadam, M., Rekabdar, F., Hemmati, M., The role of the thermodynamic parameter on membrane morphology based on the phase diagram. J. Mol. Liq., 224, 776– 785, 2016. 47. Mulder, J., Basic Principles of Membrane Technology, Springer Science & Business Media, USA, 2012. 48. Cheng, L.P., Dwan, A.H., Gryte, C.C., Isothermal phase behavior of nylon6,-66, and -610 polyamides in formic acid-water systems. J. Polym. Sci. B Polym. Phys., 32, 1183–1190, 1994. 49. Wei, Y.M., Xu, Z.L., Yang, X.T., Liu, H.L., Mathematical calculation of binodal curves of a polymer/solvent/nonsolvent system in the phase inversion phase. Desalin., 192, 91–104, 2006. 50. Yilmaz, L. and McHugh, A., Analysis of nonsolvent-solvent-polymer phase diagrams and their relevance to membrane formation modeling. J. Appl. Polym. Sci., 31, 997–1018, 1986. 51. Koningsveld, R., Stockmayer, W.H., Nies, E., Polymer Phase Diagrams: A Textbook, Oxford University Press, UK, 2001. 52. Geveke, D.J. and Danner, R.P., Application of the Flory-Huggins theory to the system chloroform-polystyrene-butadiene rubber. Polym. Eng. Sci., 31, 21, 1527–1532, 1991. 53. Keshavarz, L., Khansary, A.M., Shirazian, S., Phase diagram of ternary solutions containing nonsolvent/solvent/polymer: Theoretical calculation and experimental validation. Polymer, 73, 1–8, 2015.

3 Computational Fluid Dynamics (CFD) Modeling in Membrane-Based Desalination Technologies Pelin Yazgan-Birgi1,2, Mohamed I. Hassan Ali1,3* and Hassan A. Arafat1,2 Center for Membrane and Advanced Water Technology, Khalifa University of Science and Technology, Abu Dhabi, United Arab Emirates 2 Department of Chemical Engineering, Masdar Institute, Khalifa University of Science and Technology, Abu Dhabi, United Arab Emirates 3 Department of Mechanical Engineering, Masdar Institute, Khalifa University of Science and Technology, Abu Dhabi, United Arab Emirates 1

Abstract

One solution to the water scarcity problem is membrane-based ­desalination technologies. However, these technologies still need further improvement. Modeling tools are helpful in this regard to investigate and enhance the transport phenomena through the membrane or within the module flow channels. Various simulation tools, which depend on specific time and spatial scales of the target system domain, have been used to model membrane processes. Simultaneous solution of the governing conservation equations while considering the irregularities in a model geometry yields the local velocity, temperature, and concentration values. Process development and scale-up strategies are performed for novel membrane module designs in desalination field via Computational Fluid Dynamics (CFD) tools. The aim of this chapter is to provide a review of the implementation of the CFD modeling method in membrane-based desalination processes such as reverse osmosis, membrane distillation, forward osmosis, electrodialysis, and electrodialysis reversal. For this purpose, conventional modeling approaches, CFD modeling tools and the application of the CFD tools in selected membrane desalination technologies will be summarized. The governing model equations, which are used to investigate the transport phenomena, will be briefly explained. Additionally,

*Corresponding author: [email protected] Anirban Roy, Siddhartha Moulik, Reddi Kamesh, and Aditi Mullick (eds.) Modeling in Membranes and Membrane-Based Processes, (47–144) © 2020 Scrivener Publishing LLC

47

48  Modeling in Membranes and Membrane-Based Processes the limitations, current and future trends of CFD models in membrane desalination processes will be emphasized. Keywords:  Computational fluid dynamics (CFD), desalination, reverse osmosis (RO), nanofiltration (NF), membrane distillation (MD), forward osmosis (FO), electrodialysis (ED), electrodialysis reversal (EDR)

3.1 Desalination Technologies and Modeling Tools 3.1.1 Desalination Technologies The consequences of climate change, population growth, and poor irrigation approaches and the stress over the available water resources steer researchers to search for powerful mitigation strategies against water scarcity. During the last decade, 4.3 billion people, around 71% of the world population, faced medium to high levels of water scarcity for a minimum of one month a year [1]. Global freshwater demand is expected to increase by 55% compared to the global freshwater demand in 2000 [2]. Estimates show that by 2050, water scarcity will have an impact on an additional 0.5 billion people’s lives [3]. Therefore, the management of accessible water resources and the development of alternative technologies are critical to attaining sustainable development goals [2]. It is no surprise, then, that at the beginning of 2018, more than 19,000 desalination plants and projects were available globally to produce potable water [4]. These desalination plants extract freshwater from seawater and brackish water containing dissolved salts and other impurities. Sodium chloride, magnesium salts, bromide salts, and potassium chloride are also being extracted from seawater [5]. It was reported that the total capacity of global desalination plants was around 66.4 million m3/day in 2013 [6] and exceeded 103.7 million m3/ day at the beginning of 2018 [4]. The available commercial plants mainly include thermal-based (multistage flash distillation, MSF, and multi-effect distillation, MED) and membrane-based (reverse osmosis, RO) desalination processes which are classified based on the type of separation process. Thermal technologies trigger a phase change to extract pure water from the salty feed solution and rely on heat to evaporate water prior to condensation. Membrane desalination technologies, on the other hand, utilize selective porous membranes for separating pure water from the salty stream. Currently, RO dominates the desalination market with a share of 67.31%, whereas MSF and MED contribute 19.29% and 7.04%, respectively [4, 7].

Computational Fluid Dynamics (CFD) Modeling  49 Thermal desalination technologies (MSF and MED) have high energy demands and generally rely on non-renewable energy sources. Therefore, they are considered as first-generation desalination technologies [8, 9]. Gradually, membrane-based desalination technologies (reverse osmosis/ nanofiltration (RO/NF), electrodialysis (ED)) gained popularity and are considered as second-generation desalination technologies. Even though membrane-based processes provide significant advantages over their thermal-based competitors (in terms of lowered energy intensity), the improvement still does not offer a comprehensive solution for high energy consumption and brine management issues [8–10]. Therefore, thirdgeneration desalination technologies, such as membrane distillation (MD), forward osmosis (FO), and electrodialysis reversal (EDR) have emerged recently [11]. Although the application of these emerging technologies is currently limited to the laboratory and pilot-scale, they are promising for achieving energy-efficient and sustainable desalination [5]. Additionally, hybrid alternatives such as RO/MD and RO/FO have also been proposed to minimize the overall energy requirements further. A comparison of four main membrane-based desalination technologies is given in Table 3.1. It should be noted that the membrane characteristics as well as the hydrodynamics within the module have to be improved to achieve better permeate flux and quality. Various module configurations have been developed for ­membrane-​ based desalination technologies [8, 10, 12]. In order to investigate the transport mechanisms through the membrane or within the process module and explain the characteristics of the local phenomena in novel module design, several computational modeling tools are available and will be discussed in this chapter. While modeling membrane separation, simultaneous transport processes, such as momentum, mass and also heat transfer (in case of MD) within the module must be included in the mathematical model [13–16]. The complexity of a process from small to large scale results in a set of nonlinear partial differential equations which are mostly impossible to solve analytically. Nevertheless, progress in numerical ­methods and simulation tools are helpful in solving such model equations and ­monitoring the transport phenomena of membrane desalination processes [17].

3.1.2 Tools in Desalination Processes Modeling Various simulation methods based on the target system-specific temporal and spatial scales are presented in Fig. 3.1. Individual disciplines have been focused on specific length scales. For example, chemistry and physics problems are

+ Well-known technology + Lower energy requirement compared to its thermal competitors + Lower maintenance and water production costs compared to the thermal technologies + Common in solar-powered desalination + Possible to remove undesired solutes almost completely from the feed stream (such as ammonia and pesticides)

+ Lower electric energy requirement compared to RO process + Possible to utilize low-grade thermal energy + Lower CP compared to RO + Less membrane fouling problem at a high salt concentration + Less restriction on the type of treated feed (capable to treat highly concentrated brine) + Highest salt rejection in theory (theoretically 100% removal of salt)

Hydraulic pressure difference

Vapor pressure difference (due to the temperature difference)

Reverse osmosis (RO) [5, 18–21]

Membrane distillation (MD) [11, 13, 22–24]

Advantages

Driving force

Technology

Table 3.1  Comparison of membrane-based desalination technologies.

(Continued)

− Expensive membrane replacement − Low permeate flux − Unpredictable membrane durability − Temperature polarization (TP) issue

− Critical concentration polarization (CP) issue that can cause:   − Fouling (such as particulate fouling, organic fouling, and biofouling) and scaling   − Sudden water (permeate) flux decline at constant pressure   − Increase pressure at constant flux results in high energy consumption − Membrane durability − Further deionization process requirement after RO (as a polishing step) process to eliminate ions, silica, and boron completely

Limitations and issues

50  Modeling in Membranes and Membrane-Based Processes

Driving force

Osmotic pressure difference

Technology

Forward osmosis (FO) [20, 25–27]

+ Utilizes natural osmotic pressure gradient + Very low or no hydraulic pressure requirement + Lower energy consumption compared to conventional desalination processes + High potential to employ low-cost energy sources + High salt rejection and water recovery + Reversible and lower membrane fouling compared to RO + Good permeate flux stability + Possible removal of foulants by physical cleaning without the use of chemicals + Effective rejection of a wide range of contaminants

Advantages

Table 3.1  Comparison of membrane-based desalination technologies. (Continued)

(Continued)

− Difficult to select suitable draw solution − Effective regeneration requirement for the draw solution − A significant amount of energy need for the draw solution regeneration − CP issue − Solute diffusion and membrane fouling problems − Higher fouling propensity compared to RO under certain conditions

Limitations and issues

Computational Fluid Dynamics (CFD) Modeling  51

Driving force

Electrical potential difference

Technology

Electrodialysis / Electrodialysis reversal (ED/ EDR) [5, 16, 28, 29]

+ Low pressure requirement + A flexible process with a broad range of applications + High water recovery + Low membrane fouling and scaling via the periodic reversal of applied voltage + Easy to clean the membrane through the chemical cleaning and a change of polarity + Common for solar-powered desalination

Advantages

Table 3.1  Comparison of membrane-based desalination technologies. (Continued) − High cost of electrodes and membranes − Usage of electricity as a primary energy source − Short membrane life in desalination applications − Limitation in the type of treated feed (Low TDS brackish waters 10, collisions between gas molecules and the pore wall dominate the mechanism, rather than the molecule-molecule collisions. This is called Knudsen type flow, in which the B value (BK) can be calculated as [86]



2 ε r  8mw  Bk =   3 δτ  π RT 

0.5



(Eq. 3.19)

where mw is the molecular weight of water (kg/mol), ε is the membrane porosity, r is the pore radius (m), τ is the tortuosity and δ is the hydrophobic membrane thickness (m). In the case of Kn < 0.01, molecular diffusion is considered a suitable mechanism. The trapped air molecules dominate the mass transport

Computational Fluid Dynamics (CFD) Modeling  71 mechanism, and the molecular diffusion B value (Bm) is calculated from the following equation [86]:



Bm =

mw PDw ε RT Pa δτ 

(Eq. 3.20)

where Dw is the diffusion coefficient of the water vapor and Pa is the partial pressure of the air within the membrane pores. PDw, which depends on T, can be calculated from [86]:



PDw = 1.89510−5 T 2.072

(Eq. 3.21)

Besides BK and Bm, viscous (Poiseuille) type flow also exists. There, the B value for the viscous type flow (Bv) is calculated as [87]:



Bv =

P εr 4 8 µ RT δτ

(Eq. 3.22)

The air flow along the membrane pores is small compared to the vapor flux in MD. Therefore, viscous flow is generally neglected except in the VMD configuration [85]. The dusty gas model (DGM), which combines Knudsen flow (Eq. 3.19), molecular diffusion (Eq. 3.20) and viscous flow (Eq. 3.22), is frequently employed to model water vapor flow in porous media [23]. In DGM, water vapor and the air trapped in the pores are assumed as ideal gas and the solubility of air molecules is considered negligible in the feed/permeate liquid. The water flux (Jw) in MD can be calculated by using the following equation:



(

)

vap vap J w = B. Pfvap ,m − Pp ,m = B.∆P



(Eq. 3.23)

vap vap where Pfvap are the partial pressure on the membrane feed ,m, Pp ,m and ∆P side (Pa), the partial pressure on the membrane permeate side (Pa) and the vapor pressure difference across the membrane, respectively. Pfvap ,m and Ppvap ,m depend on the feed and permeate stream temperatures, Tf and

72  Modeling in Membranes and Membrane-Based Processes Tp, respectively. The vapor pressure (Pvap) of water is calculated from the Antoine equation [73]:



 3816.44  P vap = exp  23.1964 −   T − 46.13 



(Eq. 3.24)

Mass and heat transfer occur simultaneously during the MD process and heat flux is also a function of vapor flux. In DCMD configuration, heat transfer involves transport through: (i)  the feed side thermal boundary layer by convection (ii)  the thickness of the membrane (δm) via conduction and the latent heat of vaporization (iii)  the permeate side thermal boundary layer by convection Heat transfer fluxes are equal under steady-state condition as given below:



q = q BL , f = q BL , p = qm





q BL , f = h f (T f − T f ,m )





q BL , p = hp (Tp ,m − Tp )



(Eq. 3.25) (Eq. 3.26) (Eq. 3.27)

where q BL , f is the heat flux through the feed side boundary layer, q BL , p is the heat flux through the permeate side boundary layer, qm is the heat flux through the membrane thickness, hf is the heat transfer coefficient at the feed-side boundary layers and hp is the heat transfer coefficients at ­permeate-side boundary layer. Increasing hf and hp improves the permeate flux. The heat flux across the membrane, qm, is defined as:

qm = q vap + q cond



q vap = J . h fg





q cond =





km .(T f ,m − Tp ,m ) = hm ,cond .∆T δm

(Eq. 3.28) (Eq. 3.29) (Eq. 3.30)

Computational Fluid Dynamics (CFD) Modeling  73 where q vap is the heat flux due to phase change (evaporation), q cond is the heat flux due to the conductive heat loss through the membrane, hfg is the latent heat of evaporation, km is thermal conductivity of the membrane, hm,cond is the conductive heat transfer coefficient of the membrane and ΔT is the temperature difference across the membrane. Temperature polarization (TP) is a phenomenon that happens in MD, which has a substantial effect on the local heat and mass fluxes during the process. In order to assess the contribution of the boundary layer resistance to the total heat transfer resistance, temperature polarization coefficient (TPC) is calculated as below [85]



TPC =

(T f ,m − Tp ,m ) (T f − Tp )

(Eq. 3.31)



The following scenarios are possible: (i) TPC → 0, overwhelming effect of TP (ii) TPC ≤ 0.2, the heat transfer limits the process, which has a poor design. (iii) TPC ≥ 0.6, the mass transfer process limits the process and TP does not have a dramatic effect on the process performance. (iv) TPC → 1, no effect of TP The major factors limiting the MD process performance are heat losses, TP phenomena and membrane pore wetting. TP phenomena is related to the loss of driving force because of limited heat conductivity and efforts have been devoted to minimizing it in various simulations approaches. Since mass flux is mainly influenced by membrane properties, module design parameters, and process operating conditions, MD modeling has focused on heat and mass transfer analysis of various MD configurations and on studying the influence of membrane characteristics on its performance.

3.2.4 Electrodialysis and Electrodialysis Reversal (ED/EDR) Technologies Ion exchange membranes (IEMs) are the key components of electrodialysis (ED) and electrodialysis reversal (EDR) processes. Both are electrochemical separation processes in which anion exchange membranes (AEMs) and cation exchange membranes (CEMs) are placed parallel to each other between an anode and a cathode to remove salt ions and charged organic

74  Modeling in Membranes and Membrane-Based Processes molecules form the feed solution (Fig. 3.7). The AEMs and CEMs have positive and negative fixed groups, respectively. Saline solution is circulated through an electrodialysis stack and a potential difference is applied between two electrodes. The potential difference is created between the two electrodes by utilizing a direct current (DC). The potential difference triggers the movement of cations to migrate toward the cathode and anions toward the anode. At each electrode, oxidation (loss of an electron) and

Concentrate Diluate

(a)

CEM

Na Oxidation

A N O D E

Reduction

e–

CEM

Cl Na Cl AEM

C e– A T H O D E

repeating unit

Saline Feed Solution (b) Current Density (A/m2)

1

2

3

iLCD

Voltage

Figure 3.7  (a) Schematic representation of the electrodialysis (ED) process (AEM and CEM represent the anion exchange membrane and the cation exchange membrane, respectively.) and (b) s-shaped current-voltage curve which includes a limiting current density and the presence of three distinct regions which are (1) ohmic, (2) limiting and (3)  overlimiting.

Computational Fluid Dynamics (CFD) Modeling  75 reduction (gain of an electron) occur and each reaction is called as half cell reaction. During the process, AEMs allow only anions to pass through, while CEMs only let cations to cross. As a result, dissolved solids are selectively removed, and dilute and concentrate streams can be generated separately at the channel exits. The accumulated ions are discharged from the compartment as the brine. The product water with reduced salt concentration is collected in a neighboring compartment (dilute channel). A repeating unit includes an AEM, a CEM, a dilute channel and a concentrate channel as represented in Fig. 3.7. An ED stack includes several repeating units, the number of which depends on the scale of the process. For instance, few units are enough at the laboratory scale but hundreds of them are required for an industrial ED process. Like other membrane processes, spacers can be placed between membranes not only to keep a fixed space between these membranes but also to promote mixing along the flow channels. Periodically, the polarity is reversed to prevent salt deposition and membrane fouling in the membranes. The mechanism of the EDR process is similar to that of the ED process. However, the ion flow direction is periodically reversed by reversing the membrane stack DC electric field. The reversed ion flow reduces the fouling and scaling on the membrane and hence the need for acid and antiscalants. Therefore, EDR is a suitable process to treat feed solutions with high scaling potential. Transport mechanism and governing equations The momentum and convective mass transport can be modeled using the Navier-Stokes and convective-diffusion equations in the flow channels. Besides these equations, mass transport and CP near the interfaces need to be analyzed. The Nernst film model, which assumes a Nernst diffusion boundary layer between the membrane-solution interface, can be coupled with the transport equations [29]. The Nernst-Planck equation is widely used to model mass transport in IEMs and electrolyte solutions. The total ion flux (Ji), which is the sum of diffusive, migrative and convective fluxes, is expressed as [29, 88]:



  J i = − Di ∇Ci − z i FCDiCi ∇ϕ + Ci u

(Eq. 3.32)

where zi, F, φ and Di terms represent the valence, the Faraday’s constant, the electric potential and the ionic diffusion coefficient, respectively. In Eq. 3.32, the terms on the right side represent diffusion (concentration

76  Modeling in Membranes and Membrane-Based Processes gradient), migration (electrical potential gradient), and convection (pressure gradient), respectively. The presence of CP near the charged membrane surface generates a potential difference which is called Donnan potential at the interface. The Donnan potential drop (ηD) on the CEM is calculated from the following equation [29, 89]:



ηD =

RT Cs ,m ln z + F Cs ,b



(Eq. 3.33)

where Cs,m represents the salt concentration at the membrane-solution interface and Cs,b is the salt concentration in the bulk solution. IEMs are in contact with an electrolyte solution. The fixed charges of the IEM attract counter-ions by Coulomb forces. As a result, an electrical field can be generated. During the process, Donnan exclusion results in a dramatic variation in concentration on the IEM/solution interface. As a result, the electrical double layer is generated as a very thin charged region. In this region, the counter-ions are able to neutralize the fixed charges. Operational efficiency of an IEM can be characterized by its currentvoltage curve which is shown in Fig. 3.7(b). The curve has three distinct regions: (1) region of low current Ohmic regime, (2) region of plateaulimiting regime, and (3) region of over-limiting current regime. Limiting current density (LCD) is the value at which the concentration of ion becomes zero near the electrode (in the depleted layer) and the maximum rate of deposition is achieved [90]. Channel hydrodynamic condition, geometric properties of the module and physical properties of the solution are necessary for the LCD. The current depends on the concentration of bulk solution and is limited by the diffusion near the electrode. LCD is proportional to the bulk concentration and for an electrode, it is calculated from the equation below [29]:



iLCD = ±

Ci ,b z i FD δ (Ti − ti )

(Eq. 3.34)

where Ci,b is the bulk concentration if ion (i) (mol/m3), zi is the valence of ion, F is the Faraday’s constant (C/mol), D is the electrolyte diffusion coefficient (m2/s), δ is the diffusion boundary layer thickness (m), ti is the migration transport number, and Ti the integral transport number within the membrane includes ionic diffusion and migration.

Computational Fluid Dynamics (CFD) Modeling  77 There are two main categories of problems that diminish the productivity of the ED process. The first is related to the deterioration of AEMs and CEMs. The second is associated with the hydrodynamic properties of the module which directly affect the transport phenomena within the module. Separation efficiency is limited by the CP that hinders the mass transport and results in a lower salt concentration and higher electric field adjacent to the membrane surface [91]. Due to the CP, membrane fouling and scaling can be observed. The CFD studies on the use of spacers [16, 88, 91–95], post structures [96] and corrugated membrane surfaces [89] have been reporting promising results to minimize the thickness of diffusion boundary layer and the effect of CP.

3.3 Application of CFD Modeling in Desalination The CFD studies in the early 2000s modeled spacer-filled channels in membrane processes. Recently, there is a trend in applying CFD for developing, improving and implementing desalination processes. Several CFD studies have been conducted to determine the values of dimensionless groups and link those numbers to the critical process parameters. The CFD topics in general can be summarized as listed below: • Investigation of local hydrodynamic properties and boundary layer • Optimization of module configurations • Optimization of spacer geometries • Modeling and optimization of spacer-filled feed channels • Investigation of fouling mechanisms • Investigation of CP and TP phenomena • Enhancement of mass transport

3.3.1 Applications in Reverse Osmosis (RO) Technology Several RO modeling studies have focused on the design of novel membrane modules and spacer geometries to minimize the CP phenomena and the severity of membrane fouling as summarized in Table 3.3. Monitoring the velocity field is critical to explore the CP effects along the flow channel and mitigate the negative impact of CP, which appears at the boundary layer adjacent to the feed/membrane interface. Since the disturbance of the

Usta et al. (2018) [109]

Publication reference

Ansys CFX 14.5 (commercial) FVM

CFD tool

Module-scale model 3D hollow fiber membranes Steady-state laminar flow condition (ReH = 500-1500) Salt concentration gradient was accounted for SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Geometry and highlighted model properties Membrane module geometry (Twisted and straight)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools.

CP phenomena and process performance in terms of transmembrane permeate flux

Keywords

(Continued)

The twisted hollow fiber membrane geometry enhanced the momentum mixing along the module. As a result, CP effect had less influence on process performance and transmembrane permeate flux was increased by 5-9%.

Key conclusions

78  Modeling in Membranes and Membrane-Based Processes

Su et al. (2018) [108]

Publication reference

ANSY Fluent (commercial) FVM

CFD tool

Module-scale model 2D PFM module with ‘zigzag’ spacers Transient large eddy simulation turbulence model (ReH = 344- 688) Solute transport of a non-reacting solution in the RO membrane channel governed by the convection-diffusion equation SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Geometry and highlighted model properties Vibration frequency (twelve vibration cases)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena (NaCl and CaSO4), shear rate and process performance in terms of local transmembrane permeate flux

Keywords

(Continued)

The effect of CP was decreased by imposing vibration to the module (such as local gypsum fouling rate reduction). Increasing the vibration frequency (while keeping its amplitude constant) enhanced the permeate flux. It was important to employ a critical frequency level above which paybacks were insignificant.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  79

Lim et al. (2018) [107]

Publication reference

ANSYS CFX 16.2 (commercial) FVM

CFD tool

ReS = 2) Solute concentration was calculated based on flux balance condition at the membrane surface SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Module-scale model 2D narrow channel geometry with spacer Transient laminar flow condition (ReH = 408 and

Geometry and highlighted model properties The resonant frequency of unsteady forced slip velocity

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena, permeate flux and recovery rate

Keywords

(Continued)

Forced slip condition reduced the negative impact of fouling. CP was reduced and the permeate flux was improved at high membrane permeance up to 23%. However, 5% to 7% higher pumping energy was required.

Key conclusions

80  Modeling in Membranes and Membrane-Based Processes

Li et al. (2018) [54]

Publication reference

ANSY Fluent 16.2 (commercial) FVM

CFD tool

geometry Steady-state and laminar flow condition (ReH = 50-150) Transmembrane flux was included in the model by using “mass jump” approach. SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Module-scale model 3D tubular membrane

Geometry and highlighted model properties Varying operating pressures, feed velocities and different fiber lengths

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena, permeate flux and salt concentration profile

Keywords

(Continued)

Permeate flux decreased along the axial location of tubular membranes because of the CP (increase in osmotic pressure as well).

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  81

Gu et al. (2017) [102]

Publication reference

COMSOL Multiphysics 5.0 (commercial) FEM

CFD tool

Module-scale model 3D SWM geometry with spacer Steady-state and laminar flow condition (ReH = 224) Solute fluxes through the membrane walls are also included

Geometry and highlighted model properties Different feed spacer designs with a total of 20 geometric difference (filament configuration, mesh angle and attack angle)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena, permeate flux, solute flux, and pressure drop

Keywords

(Continued)

Fully woven spacer geometries resulted in higher water flux and lower CP. However, pressure drops were higher than the nonwoven spacer geometries. The decrease in mesh angle resulted in lower water flux. Additionally, variations in attack angle resulted in a significant influence on pressure drop.

Key conclusions

82  Modeling in Membranes and Membrane-Based Processes

Haaksman et al. (2017) [52]

Publication reference

COMSOL Multiphysics 5.1 (commercial) FEM

CFD tool

Solute transport and deposition of particulate and bacterial fouling in spacer-filled channels are investigated

Module-scale model 3D SWM module geometry with spacer Steady-state laminar flow condition (ReH = 50-200)

Geometry and highlighted model properties Various spacer designs from X-ray CT scans

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Fouling patterns

Keywords

(Continued)

Since the local membrane shear stress distribution and local velocity vectors are critical to monitor the pattern of particle deposition on the membrane, CFD modeling was important. CFD modeling results showed good agreement with the pressure drop measurements using CT scans.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  83

ANSYS Fluent (commercial) FVM

In-house code MATLAB (commercial)

Park et al. (2017) [104]

CFD tool

Kavianipour et al. (2017) [103]

Publication reference

Module-scale model 2D SWM module geometry Steady-state laminar flow condition (ReH ≤ 400)

SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Module-scale model 3D SWM module geometry with spacer Steady-state laminar flow condition (ReH = 50-200)

Geometry and highlighted model properties

Two different cases in terms of module properties, operating conditions and spacer properties

Various feed spacer configurations (Laddertype, Triple, Wavy and Submerged)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

The performance of the proposed model was compared with the data from the literature.

Spacer configuration efficacy which combines the mass transfer and the energy consumption

Keywords

(Continued)

A new 2D model and a solution method were proposed by developing the own in-house code.

For Re > 120, wavy spacer simulations showed the highest performance. On the other hand, the ladder-type model showed better performance when Re < 120.

Key conclusions

84  Modeling in Membranes and Membrane-Based Processes

Usta et al. (2017) [61]

Publication reference

Ansys CFX (commercial) FVM

CFD tool

Module-scale model 3D flat membrane module geometry Steady-state laminar flow condition (ReH = 100) Steady-state turbulence flow condition (ReH =400-1000, k-ω shear stress transport turbulence model) Solution-diffusion modelbased membrane flux model was applied. SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Geometry and highlighted model properties Corrugated membranes with chevrons on both top and bottom membrane (also called as membrane-based micromixers)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena, fouling and permeate flux

Keywords

(Continued)

Corrugation with triangular chevrons had a significant influence on flow separation at all flow rates. Corrugated membranes showed better performance. Water permeation rate was increased, and CP phenomena were reduced. Also, the potential fouling was minimized in the RO module.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  85

Anqi et al. (2016) [105]

Publication reference

Ansys CFX 14.5 (commercial) FVM

CFD tool

Steady-state turbulence flow condition (ReH = 400-800, k-ω shear stress transport turbulence model) Solution-diffusion modelbased membrane flux model was applied. SIMPLE algorithm for pressure-velocity coupling First order upwind algorithm for discretization of the conservation equations

Module-scale model 2D and 3D flat membrane module geometry with spacer Steady-state laminar flow condition (ReH = 100)

Geometry and highlighted model properties 2D and 3D modeling strategy with and without spacer under various flow conditions

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena, fouling and permeate flux

Keywords

(Continued)

Both the spacer usage and the flow rate increase improved the process performance in terms of permeate flux and minimized the impact of CP. The averaged membrane property predictions of the 2D models were in good agreement with the 3D model results at low Reynolds number value. However, the flow field properties became necessary in 3D when Re ≥ 400. When the intensity of the CP increased, also the fouling tendency increased on the same sites.

Key conclusions

86  Modeling in Membranes and Membrane-Based Processes

CFD tool

Not reported FVM

Publication reference

Amokrane et al. (2016) [106]

Module-scale model 2D SWM module geometry with spacer Time-dependent laminar flow (ReH ≤ 380)

Geometry and highlighted model properties Various spacer designs (ellipse, circle, and oval shapes)

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

CP phenomena and fouling

Keywords

(Continued)

Permeate flux was improved and the fouling propensity was reduced with oval spacers (which were tilted at 20°) compare to the conventional spacers.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  87

CFD tool

OpenFOAM (open-source)

Publication reference

Gruber et al. (2016) [15]

simulations were run with a steady-state solver. Then the transient solver was used to monitor if any transient effects arise over time. Solute flux was balanced by using the diffusive and convective solute fluxes at the membrane. PISO algorithm to solve coupled transient continuity and NavierStokes equations SIMPLE algorithm for the steady-state cases

Module-scale model 2D and 3D PFM module geometries with spacers Both steady-state and transient simulations (ReH = 4.5-450). Firstly,

Geometry and highlighted model properties Various flow and module geometry parameters such as the inlet angles, the number of inlets, various spacer types, densities and configurations

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Model development

Keywords

(Continued)

It was concluded that there was no noteworthy transient effect on the average membrane fluxes in any simulations (including the cases with spacers). The optimized solver code converged within 24 h using just a single CPU for the cases containing millions of cells.

Key conclusions

88  Modeling in Membranes and Membrane-Based Processes

Bucs et al. (2014) [100]

Publication reference

Combination of COMSOL Multiphysics, MATLAB and Java codes (Commercial + in-house coding)

CFD tool

Module-scale (micro-scale) model 3D spacer filled feed channel geometry Steady-state laminar flow condition (ReH ≤ 200) Impermeable membrane assumption Coupled fluid dynamics, solutes transport and biofouling by biofilm formation (Fluid flow and biofilm development) Biomass growth was calculated from a CDR equation

Geometry and highlighted model properties The effect of flow velocity, bacterial cell load, biomass attachment location and various feed spacer geometries on biofouling and channel pressure drop were evaluated.

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Biofouling in feed channels filled with spacer

Keywords

(Continued)

The biofilm formation over the spacer geometry presented substantial effect on the pressure drop in the flow channel. This formation could be reduced by utilizing thicker and spacer geometry under a lower feed flow rate in the channel.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  89

Radu et al. (2012) [99]

Publication reference

Combination of COMSOL Multiphysics, MATLAB and Java codes (Commercial + in-house coding)

CFD tool Module-scale (micro-scale) model 2D spacer filled feed channel geometry Steady-state laminar flow condition (ReH ≤ 200) Coupled fluid dynamics, solutes transport and biofouling by biofilm formation (Fluid flow and biofilm development) includes membrane permeability

Geometry and highlighted model properties Cross-flow velocity, substrate concentration and different hydraulic cleaning strategies were investigated.

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Biofouling in feed channels filled with spacer

Keywords

(Continued)

Permeate flux had a significant effect on the rate of biomass growth because feed stream provided further substrate for the biofilm near the membrane surface. Biofilm formation tendency was higher around the low liquid shear stress regions and higher shear caused more detachment and thinner biofilm layers at high flow rates.

Key conclusions

90  Modeling in Membranes and Membrane-Based Processes

Radu et al. (2010) [97]

Publication reference

Combination of COMSOL Multiphysics, MATLAB and Java codes (Commercial + in-house coding)

CFD tool

Module-scale (micro-scale) model 2D spacer filled feed channel geometry Steady-state laminar flow condition (ReH ≤ 200) Coupled fluid dynamics, solutes transport and biofouling by biofilm formation (Fluid flow and biofilm development) includes membrane permeability

Geometry and highlighted model properties The effect of biofilm growth on flux decline and pressure drop increase in time, and the salt rejection decline with various feed spacer geometries.

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Biofouling in feed channels filled with spacer

Keywords

(Continued)

The developed model demonstrated that the spacer configuration has direct impact on the amount of biomass formation in the feed channel and clarified the experimental results that the permeate flux declined and the salt passage increased in time because of the biofilm formation. Besides these, the biofilmenhanced CP resulted in the most critical effect on the permeate flux decline compared to the effect of increased hydraulic resistance to transmembrane flow; and increased pressure drop in the feed channel.

Key conclusions

Computational Fluid Dynamics (CFD) Modeling  91

Vrouwenvelder et al. (2010) [98]

Publication reference

Combination of COMSOL Multiphysics, MATLAB and Java codes (Commercial + in-house coding)

CFD tool

Module-scale model 3D spacer filled feed channel geometry in SWM Steady-state laminar flow condition Impermeable membrane assumption Coupled fluid dynamics, solutes transport and biofouling by biofilm formation (Fluid flow and biofilm development)

Geometry and highlighted model properties The qualitative comparison of biofouling data from the developed model and experiments

Research focus and selected parameters

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Biofouling in feed channels filled with spacer

Keywords

(Continued)

Feed channel pressure drop, biomass accumulation and velocity distribution profile were investigated, and biomass growth was observed on the feed spacer. It was highlighted that the fluid flow pattern is affected by the location of the biofilm and the biofilm growth is also triggered by the water flow. The feed spacer fouling is more critical than membrane fouling.

Key conclusions

92  Modeling in Membranes and Membrane-Based Processes

Combination of COMSOL Multiphysics, MATLAB and Java codes (Commercial + in-house coding)

CFD tool

Module-scale model 3D spacer filled feed channel geometry Steady-state laminar flow condition (ReH = 127) Coupled fluid dynamics, solutes transport and biofouling by biofilm formation (Fluid flow and biofilm development) Impermeable membrane assumption

Geometry and highlighted model properties First 3D CFD study that focused on modeling RO membrane biofouling in the spacer filled feed channel geometry.

Research focus and selected parameters Biofouling in feed channels filled with spacer

Keywords

Pressure drop, liquid velocity distribution and residence time distribution The main pressure drop is encountered by the flow passing over the spacer filaments The real impact of biofouling is on the flow regime leading to quasi-stagnant zones and an increase in the dispersion of the residence time distribution.

Key conclusions

CDR: convection-diffusion-reaction equation; CP: concentration polarization; CT: computed tomography; ReH: hydraulic Reynolds number; ReS: slip Reynolds number; SWM: spiral wound membrane; 2D: two-dimensional; 3D: three-dimensional; CDR equation [100]: Ds∇2Cs − u∇Cs + ∇rs = 0 where Ds is the diffusion coefficient of substrate, Cs is the concentration of substrate and rs is the substrate consumption rate by biomass.

Picioreanu et al. (2009) [101]

Publication reference

Table 3.3  Summary of selected RO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  93

94  Modeling in Membranes and Membrane-Based Processes hydrodynamic boundary layer (due to liquid recirculation) has a significant effect on solute transport, scaling, fouling and water permeation processes need to be coupled while modeling a spacer design. At high feed flow rates, the level of mixing is high enough to assume the concentration near the wall boundary to be equal to that of the bulk. However, the concentration difference between the bulk stream and the boundary layer would become significant at lower feed flow rates. In the latter case, the wall concentration has to be monitored and accounted for in the calculations. Even though the experimental approaches are critical for the appropriate evaluation of the developed CFD model at the model validation step, it is challenging to monitor the concentration gradient experimentally. Thus, the CP phenomenon further complicates the modeling of transport phenomena within the membrane module. On the other hand, several 2D and 3D modeling studies investigated the effect of bioscaling and biofouling on process performances and most of these models were validated against experimental data [97–100]. An advanced combination of MATLAB (main algorithm script), COMSOL (solution of the partial differential equations governing the flow field and the solute mass balances) and Java codes (cellular automata biofilm model) were employed in these studies. As an initial attempt Picioreanu et al. developed a novel 3D CFD model representing biofouling in RO membrane [101]. Then, another study focused on the same topic and the data from the developed 3D biofouling were compared with experimental data [98]. The simulation results indicated that the pressure drop in the feed channels with simplified spacer geometries showed significantly different results compared to the spacer geometry with filament thinning. In both steady-state models, the RO membrane was assumed impermeable, so the CP phenomenon was not explored [98, 101]. Radu et al. evaluated the impact of spacer filament configuration and process parameters on the development of biofilm (attachment, growth and detachment steps) in the spacer filled channel [97, 99]. In order to calculate the fluid flow in the liquid sub-domain and porous biofilm sub-domain, Navier-Stokes equations and Brinkman flow equations were applied, respectively. In one of these studies, permeate flux decline and pressure drop increase with time were investigated using the 2D model, by coupling the permeability of RO membrane, mass transport of solutes and the biofilm development in the model [97]. A significant pressure drop increase and formation of various flow patterns were observed due to the biofouling in the RO module. On the other hand, the CP layer necessitated further mesh refinement, demanding much higher computational requirements compared to the previous models which did not include RO membrane permeability. In the next study, the cross-flow

Computational Fluid Dynamics (CFD) Modeling  95 velocity and substrate concentration on biofilm formation were selected as parameters [99]. During last decade, the CFD modeling tool has been increasingly used to model feed channels filled with spacer, and to investigate velocity, concentration and pressure distributions in simplified module geometries [15, 102–106]. Beside these studies, the recent modeling studies have explored novel and innovative approaches to minimize CP in the module [107–109]. For instance, Su et al. imposed vibration to the RO module and investigated the effect of increased vibration frequency on the process performance in terms of local gypsum fouling rate and permeate flux [108]. After testing twelve vibration cases, a critical frequency level was found as a limit, above which paybacks were insignificant. In another study, twisted hollow fiber membrane modules were utilized to enhance the momentum mixing along the module [109]. As a result, the severity of CP was reduced and transmembrane permeate flux was increased by 5–9%.

3.3.2 Applications in Forward Osmosis (FO) Technology Several CFD modeling studies focused on the 2D standard mass transfer models which were used to evaluate the hydrodynamic properties of FO modules [110]. CFD models employing the fully coupled hydrodynamics and solute mass transfer equations can be found in the literature. The porous layer was not considered as a separate region in some of the CFD studies and the layer was applied as a flux boundary condition which was combined with the film model [111]. 2D steady-state FO models were also developed and coupled Navier-Stokes equation, Brinkman equation and convection-diffusion are solved via the FEM for the feed channel, draw channel and support layer [110, 112, 113]. “Mass jump” solution strategy is also available for modeling FO module mass transport across the porous layer [55]. Besides this method, Kang et al. developed an appropriate methodology to include mass transfer across the active membrane layer and the support layer in their pore scale CFD model [114]. In the model, boundary constraints were defined initially for the solution-diffusion model for Jw and Js. Then these constraints were applied on the interface of the active membrane layer and the support layer. Gruber et al. developed 2D models using an open-source modeling tool OpenFOAM [14, 15]. In their model, the solver was set for the governing equations (conservation of mass, conservation of momentum and conservation of solute mass fraction) and the boundary conditions were utilized to define the transport of water and solute across the membrane. The boundary conditions were set by the water flux (Jw) (Eq. 3.35) and the

96  Modeling in Membranes and Membrane-Based Processes solute flux (Js) (Eq. 3.36), where Js has to be balanced by the diffusive and convective solute fluxes at the membrane (Eq. 3.37) as given below

Jw =

− ρDAB

(Eq. 3.35)



B Jw φ .A

(Eq. 3.36)

∂mA nd + ρmmA ,m J w = J s ∂nd

(Eq. 3.37)

Js = −





1  B + Aπ d ,m  ln K  B + J w + Aπ f ,m 

Selected CFD modeling studies were summarized in Table 3.4. Most of the models were in module scale and only few studies were ­performed at pore scale, which were developed by employing OpenFOAM [114, 115]. The porous support layer properties, such as its porosity, tortuosity, and thickness control the ICP phenomena. Besides the properties of the support layer, the porosity at the interface between the selective layer and support layer is another important factor which determines the process performance in terms of water flux [114]. Recently, Lee et al. modeled various pore geometries (straight, trapezoidal converging, trapezoidal diverging, concave and convex shapes) of the support layer with different combinations of surface and bulk porosities and compared the resultant water flux [115]. It was concluded that water flux was most sensitive to surface porosity. An increase in the latter led to the highest improvement in FO process performance. Besides this, ICP could be reduced by increasing the bulk porosity. The displacement of membrane and contraction of a flow channel depend on FO process operating conditions such as flow rate and pressure within the channel. Lian et al. developed 3D CFD models of spiral wound module (SWM) and plate-and-frame membrane module (PFM) with different draw channel spacers and validated these models using the experimental pressure drop (ΔP) results [55]. The CFD model results follow the trend obtained experimentally. For instance, ΔP data from the CFD model for SWM were in good agreement with experimental ΔP results (less than 5% error).

[55]

Lian et al. (2018)

reference

Publication

FVM

(commercial)

Ansys Fluent

CFD tool

channel height. The structural

conservation equations

for discretization of the

First order upwind algorithm

velocity coupling

(Continued)

pressure drop in the SWM.

properties caused a significant

had lower porosity and flow

at the two sides of the channel

SIMPLE algorithm for pressure-

draw side carrier spacer which

The symmetry boundary condition

compared to the PFM. This was

sensitive to the CFV variation

Pressure drop in SWM was more

Key conclusions

mainly due to the nature of the

water flux

Pressure drop and

Focus

No slip wall condition

mm; PFM: 0.5 to 1 mm),

heights (SWM: 0.7 to 1.3

Various CFV and channel

Selected parameters

CFVs: 0.1-0.4 m/s

channel spacers

3D SWM and PFM with draw

Module-scale model

properties

Geometry and highlighted model

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools.

Computational Fluid Dynamics (CFD) Modeling  97

Lee (2018) [115]

reference

Publication

improvement in FO process performance. Besides this, ICP could be reduced by increasing the bulk porosity. This led a

the support layer with different combinations of surface and bulk porosities

and steady-state turbulent flow for advection-diffusion scalar transport (u=0.0043 m/s)

advection-dispersion equation

layer was defined using the

Mass transfer in the porous support

system

periodicity of the support layer

(Continued)

osmotic pressure.

significant increase in effective

surface porosity led the highest

and convex shapes) of

until reaching steady state)

A unit cell was modeled due to the

surface porosity. An increase in

diverging, concave

continuity equations (solution

flux was most sensitive to

converging, trapezoidal

Transient Navier-Stokes and

FVM

It was concluded that the water

Water flux and ICP

(straight, trapezoidal

2D pore geometries

Various pore geometries

Pore-scale model

Key conclusions

Focus

Selected parameters

(open-source)

properties

Geometry and highlighted model

OpenFOAM

CFD tool

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

98  Modeling in Membranes and Membrane-Based Processes

(2018) [116]

Zhang et al.

reference

Publication

conservation equations

for discretization of the

First order upwind algorithm

velocity coupling

SIMPLE algorithm for pressure-

(Continued)

observed.

to mitigate ECP significantly, so water flux increase was

Corrugated surface helped

water flux

Non-permeable wall condition surfaces

and the mixing of the solution.

transmembrane

ReH = 500-1500 was adopted at the membrane

enhanced the vortex formation

in terms of

Steady-state laminar flow condition

Corrugated wall channel module

Key conclusions

FVM

ECP and process

Focus

performance

Corrugated wall channel

Selected parameters

3D flat membrane module

Module-scale model

properties

Geometry and highlighted model

(commercial)

Ansys Fluent

CFD tool

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  99

concentration resulted in a

Numerical solver UMFPACK

(Continued)

flux profile due to the CP.

nonlinear increase in water

An increase in draw solution

solutions, respectively

in water flux improvement.

feed/draw flow rates resulted

rate

Steady-state laminar flow

FEM

[41]

NaCl/Sugar for feed/draw

concentration, overall ∆π and

∆π and feed/draw flow

Increases in draw solution

structural parameters.

intrinsic and effective

2D flat membrane module

Module-scale model

porosities

Water flux

which was one of the reasons for disagreement between

layer is an important factor

diffusing solutes

Identical surface and bulk

selective layer and support

flow for the transport of

condition. The porosity at

under a steady-state FO

passive transport of solutes

developed to simulate the

A novel pore-scale model was

Key conclusions

the interface between the

parameters

structural

and effective

between intrinsic

Disagreement

Focus

under the influence of a steady

Draw solution concentration,

Porosity

Selected parameters

transient variation of concentration

m/s)

Steady-state laminar flow (u=0.001

geometries

2D rectangular straight pore

Pore-scale model

properties

Geometry and highlighted model

(commercial)

COMSOL

FVM

(open-source)

OpenFOAM

CFD tool

et al. (2017)

Madhumala

[114]

Kang et al. (2017)

reference

Publication

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

100  Modeling in Membranes and Membrane-Based Processes

(2016) [15]

Gruber et al.

reference

Publication

FVM

(open-source)

OpenFOAM

CFD tool

in any simulations including the cases with spacers. The optimized solver code

configurations

Firstly, simulations were run

the average membrane fluxes

spacer types, densities and

containing millions of cells.

monitor if any transient effects

state cases

SIMPLE algorithm for the steady-

Stokes equations

transient continuity and Navier-

PISO algorithm to solve coupled

solute fluxes at the membrane.

the diffusive and convective

Solute flux was balanced by using

(Continued)

just a single CPU for the cases

the transient solver was used to arise over time.

converged within 24 h using

with a steady-state solver. Then

Both steady-state and transient simulations (ReH = 4.5-450).

noteworthy transient effect on

It was concluded that there was no

Key conclusions

as the inlet angles, the

Model development

Focus

geometry parameters such

Various flow and module

Selected parameters

number of inlets, various

geometries with spacers

2D and 3D PFM module

Module-scale model

properties

Geometry and highlighted model

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  101

[110]

Sagiv et al. (2015)

reference

Publication

FEM

Not reported

CFD tool

membrane selective layer

hydrodynamics and reduced

convection-diffusion equation

(Continued)

and feed channels.

the level of ECP in both draw

critical to improve channel

channels is described by the

the module design was

porosity. Additionally,

Solute diffusion in the flow

support layers with increased

compared to the use of thinner

was found more important

permeability improvement

flux during the FO process,

ECP

In order to enhance the water

Key conclusions

of ICP versus

Relative significance

Focus

BRE

common film model)

Modeling tools (CFD vs.

Selected parameters

support was applied by using

The flow field within the porous

feed and draw channels

Crossflow of the solutions along the

Steady-state laminar flow

2D flat membrane module

Module-scale model

properties

Geometry and highlighted model

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

102  Modeling in Membranes and Membrane-Based Processes

water flux and significantly lower cross migration of feed

SFD)

Numerical solver UMFPACK

entrance

Parabolic velocity profile at the

convection-diffusion equation

channels is described by the

Solute diffusion in the flow

BRE

support was applied by using

The flow field within the porous

(Continued)

and draw solutes.

in terms of improvement in

operation, and SFF or

0.006-0.15 m/s)

countercurrent cross

Steady-state laminar flow (CFVs:

showed the best performance

Counter-current/ SFD mode

Key conclusions

FEM

velocity profile

CP, water flux and

Focus

(Co-current or

Module configuration

Selected parameters

2D flat membrane module

Module-scale model

properties

Geometry and highlighted model

(commercial)

COMSOL

Sagiv et al. (2014)

[112]

CFD tool

reference

Publication

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  103

(2012) [117]

Gruber et al.

reference

Publication

FVM

(open-source)

OpenFOAM

CFD tool

Navier-Stokes equations

couple transient continuity and

PISO algorithm to solve coupled

solute fluxes at the membrane.

the diffusive and convective

Solute flux was balanced by using

simulations (ReH ≈ 300)

Both steady-state and transient

solutions

NaCl for both draw and feed

spacers

3D PFM module geometry with

Module-scale model

properties

Geometry and highlighted model 3D model development

Selected parameters

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

agreement with the ECP data from the experiments.

the chamber and

(Continued)

model estimations are in good

flow fields within ECP

spatial effects visually. CFD

visualization of the

The developed model captured the

Key conclusions

model developed,

Validation of CFD

Focus

104  Modeling in Membranes and Membrane-Based Processes

(2011) [14]

Gruber et al.

reference

Publication

FVM

(open-source)

OpenFOAM

CFD tool

at high flow velocities without

ECP, fouling, and flux

Navier-Stokes equations

couple transient continuity and

parameters.

(Continued)

stability depend on geometry

in mass transfer resistance.

solute fluxes at the membrane.

PISO algorithm to solve coupled

flux because of an increase

the diffusive and convective

caused a decline in water

ECP. On the other hand, ECP

Tangential slip flow reduced

feed and draw spacers.

included in the CFD models

water flux

difference and slip velocity

The influence of ECP should be

Key conclusions

transmembrane

ECP and

Focus

osmotic pressure

Cross-flow velocity, bulk

Selected parameters

Solute flux was balanced by using

simulations (ReH ≈ 300)

Both steady-state and transient

solutions

NaCl for both draw and feed

spacers

2D PFM module geometry with

Module-scale model

properties

Geometry and highlighted model

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  105

Water/NaCl for feed/draw solutions,

FEM

(Continued)

channel CP layer.

resistance. This was mainly

an increase in mass transfer

membrane was observed with

Water flux decline through the

improvement in water flux.

performance in terms of

SFD mode showed the best

Key conclusions

due to the dilution at the draw

water flux

transmembrane

CP and

Focus

BRE

and SFD)

Membrane orientation (SFF

Selected parameters

support was applied by using

The flow field within the porous

0.1-0.4 m/s)

Steady-state laminar flow (CFV ≈

respectively

2D flat membrane module

(commercial)

properties

Module-scale model

COMSOL

Sagiv et al. (2011)

Geometry and highlighted model

CFD tool

reference

Publication

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

106  Modeling in Membranes and Membrane-Based Processes

CFD tool

Numerical solver UMFPACK

entrance

Parabolic velocity profile at the

layers was recommended.

thinner membrane support

with shorter membranes and

flux, narrower channel modules

resistance. To increase water

minimization of support layer

effect on flux than the

resulted in a more significant

in the selective layer thickness

For this reason, a reduction

the porous membrane support.

effect of ICP which is due to

Additionally, the effect of ECP

Key conclusions is more significant than the

Focus

diffusion equation

Selected parameters

is described by the convection-

Solute diffusion in the flow channels

properties

Geometry and highlighted model

κ

ηp

  ηp 2 = ∇.  − p p I + ∇u p  where ∇.up = 0 , ηp is the dynamic viscosity (Pa s), κ is the solution permeability (m ) and εp is the porosity of the support layer. ε p  

dC where -Ds is the effective diffusivity in the support layer. dx

Solute diffusion in the flow channels convection-diffusion equation: ∇.(Di∇Ci) = uiCi where Di is the solute diffusion coefficient in the the support layer solution phase. Di can be calculated from Di = εD/τ where τ is the support layer pore tortuosity.

BRE:

Advection-dispersion equation: J s = J w C − Ds

BRE: Brinkman equation; CFV: cross-flow velocity; CP: concentration polarization; ECP: external concentration polarization; NS: Navier-Stokes equation; PFM: plateand-frame membrane module; ReH: hydraulic Reynolds number; ICP: internal concentration polarization; SFF: membrane selective layer faces the feed solution; SFD: membrane selective layer faces the draw solution; SIMPLE: semi-implicit method for pressure linked equations; SWM: spiral wound module;

reference

Publication

Table 3.4  Summary of selected FO modeling studies utilizing CFD tools. (Continued)

Computational Fluid Dynamics (CFD) Modeling  107

108  Modeling in Membranes and Membrane-Based Processes However, the errors from the PFM simulations were very high (within the 10-50% range). The discrepancy was further investigated by monitoring the pressure drop data. The data for PFM from the simulations were in good agreement with experimental data only for specific channel heights. For instance, ΔP data of feed side fitted for higher channel height and the draw side ΔP fitted for lower channel heights. These results were mainly because of the relative displacement of the membrane and the experimental observations showed that the existing channel height was changed in PFM during the experiments. For this reason, the hypothetical membrane displacement was also modeled and it was found that a 100 μm displacement of the membrane towards the draw side resulted in a decrease of ΔP by 0.11 bar in the feed channel and increase of ΔP by 0.08 bar in the draw channel.

3.3.3 Applications in Membrane Distillation (MD) Technology Several MD studies focused on the development of accurate CFD models to investigate the hydrodynamic properties of various types of MD configurations (Table 3.5). In these models, velocity, concentration and temperature distributions were mapped and the MD process performances were evaluated in terms of the productivity (permeate flux) and/or the energy efficiency (thermal efficiency or gained output ratio). Some of these studies were conducted to determine the effect of spacer on heat and mass transfer enhancement in different membrane systems and investigation of spacer filled channel mostly in DCMD configuration. Several module scale CFD studies available in the literature to investigate fouling and TP issues in MD process. Even though CP is expected to have much lower impact on VMD process performance compared to the RO, Julian et al. claimed that neglecting the CP phenomena resulted in significant discrepancy between the actual and calculated deposition rates at high feed concentrations [118]. Even though module scale CFD studies available in the literature, only a limited number of pore-scale CFD modeling studies is available to investigate the hydrodynamic properties at pore scale [119]. In the study, wetting phenomena was modeled in hydrophobic MD membranes to estimate the liquid entry pressure at which membrane pore wetting begins [119]. The conductive heat loss is challenging to minimize in DCMD which has lower thermal efficiency than AGMD. However, AGMD process has an additional mass transfer resistance for the water vapor transport through the membrane thickness due to the presence of an air gap on the permeate side. For this reason, permeate flux of the AGMD is significantly lower than that

(2018) [131]

Perfilov et al.

Focus

Key conclusions

modules with or without spacers. An increase in

assessment

membrane properties

model

and length)

(membrane thickness

gas velocity) and

described by the k-ε

salinity, and sweeping

temperature and

(feed velocity,

the literature

transient flows

ReH < 400: Laminar and

channels:

Operating conditions

(Continued)

minor impact.

temperature showed

but permeate

process performance,

a critical role on the

Feed temperature played

reduction of vapor flux.

caused a dramatic

Strategy 1 for spacer-filled

in feed concentration

model

5.9 for HFM. Increase

the water vapor flux of

and length)

35 to 55 °C) enhanced

(membrane thickness

feed temperature (from

for both FSM and HFM

permeate flux

Developed model is suitable

development and

Accurate model

properties

and membrane

feed temperature)

feed flow rate and

(feed concentration,

ReH > 400: Turbulent flow

several

Selected parameters Operating parameters

described by the k-ε

ReH > 2100: Turbulent flow

transient flows

ReH < 2100: Laminar and

without spacers:

For empty channels

modules

2D FSM and HFM

Module-scale model

important properties

Geometry and

publication in

results from

Experimental

FEM

SGMD

data

(commercial)

COMSOL

experiments and literature

data from

Vapor flux

method

Validation

FEM

DCMD

module

MD

(commercial)

COMSOL

Perfilov et al.

(2018) [130]

CFD tool

reference

Publication

Table 3.5  Summary of selected MD modeling studies utilizing CFD tools.

Computational Fluid Dynamics (CFD) Modeling  109

reference

Publication

CFD tool

module

MD method

Validation

interface

the feed/membrane

as an average over

flux was evaluated

Calculated water vapor

transport.

transmembrane mass

DGM was used to define

Plummer equation

(Continued)

decrease in vapor flux.

thickness both result in

membrane length or

ReH > 400: Burke-

effect. An increase in

Forchheimer equation

and permeate velocities showed no significant

threshold after that feed

porous medium. 10< ReH Sf

Fig. 4.4  A control volume desalination system.

Thermodynamics and Membrane Separations  161

 sep = G p + G b − G f + Ta s gen W





(4.46)

Considering the process is reversible, entropy generation will be zero and the work of separation becomes the reversible work of separation, also called least work of separation. rev  sep = W  sep W = G p + G b − G f



(4.47)



In actual applications and in the existent desalination systems, entropy is mainly generated due to viscous losses occurring in membrane systems and heat losses occurring in thermal systems [9]. Although the evaluation of compositional changes is significant in mixing processes, that of entropy generation is not. Hence, it is fair to deduce that the least work is a germane factor for estimating the impact of non-idealities on the performance of the system. 1. Mass Basis To write properties per unit mass of solution, Eq. 4.47 is best written on mass flow rate basis as

 least = m  pg p − m  cgc − m  f gf W



(4.48)

where gj is specific Gibbs free energy per kilogram of solution. The recovery ratio is defined as the ratio of the mass flow rate of product water to the mass flow rate of feed seawater and it can be written as



r=

 p the mass flow rare of product m = f m the mass flow rare of feed

(4.49)

The conservation of mass for the mixture and the salts can be written as



 f =m  p +m  b m

(4.50)



 f Sf = m  pS p + m  bS b m

(4.51)

Where, m j is the mass flow rate in kg/s. The least work of separation per unit mass of product can be obtained by using Eqs. 4.48, 4.49 and 4.50



 least W 1 = (g p − g c ) − (g f − g c ) p m r



(4.52)

162  Modeling in Membranes and Membrane-Based Processes The Gibbs free energy of each stream in Eq. 4.52 can be evaluated using seawater properties as a function of temperature and salinity [4]. The feed Sf and product Sp salinities are known properties, the brine salinity Sb can be evaluated as



Sb =

rS p Sf − 1− r 1− r

(4.53)

2. Mole Basis Physical properties can also be defined on a mole basis, hence Eq. 4.48 on substituting Eq. 4.17 can be written as



 least =  n H Oµ H O + n NaClµ NaCl  +  n H Oµ H O + n NaClµ NaCl  − W  2 p  2 c 2 2  n H2 Oµ H2 O + n NaClµ NaCl  f





(4.54)

Conservation of moles for H2O and NaCl can be written as



n H2 O ,f = n H2 O ,p + n H2 O ,c

(4.55)



n H2 O ,f = n H2 O ,p + n H2 O ,c

(4.56)

where, n j is the mole flow rate. On substituting Eqs.4.18, 4.55 and 4.56 into Eq. 4.54

 least =  n H ORTlna H O + n NaClRTlna NaCl  W  2  2



p

+  n H2ORTlna H2O + n NaClRTlna NaCl 

c

−  n H2ORTlna H2O + n NaClRTlna NaCl 

f

(4.57)



The molar recovery ratio is defined as



r=

n H2 O ,p mole flow rate of water in the product = mole flow rate of water in the feed n H2 O ,f



(4.58)

Thermodynamics and Membrane Separations  163 Using Eq. 4.55, 4.56 and 4.58 the following result can be obtained



n NaCl ,J = m NaCl ,J M H2 O n H2 O ,J



(4.59)

And normalizing the least work by n H 2O , p RT Eq. 4.57 becomes

 least  a H O ,p a NaCl ,p  W =  ln 2 + m NaCl ,p M H2 O ln n H2 O ,pRT  a H2 O ,c a NaCl ,c 

 a 1  a H O ,f −  ln 2 + m NaCl ,f M H2 O ln NaCl ,f  r  a H2 O ,c a NaCl ,c 

(4.60)



Hence the generalized least work of separation for the separation of various salts present in feed water or seawater or the water that has to be treated can be written as

 least  a H O ,p W =  ln 2 + n H2 O ,pRT  a H2 O ,c

∑

1  a H O ,f −  ln 2 + r  a H2 O ,c

s

m s ,p M H2 O ln

∑m s

s ,f

a s ,p  a s ,c 

M H2 O ln

a s ,f  a s ,c 

(4.61)



The fact that can be noted down for Eq. 4.61 is that the least work of separation is a function of temperature, feed molality, product molality, molar recovery ratio and activities of solute as well as solvent. To examine the non-ideal solution behavior on least work of separation, Eq. 4.61 is broken down into two parts the ideal part that is a function of mole fraction and the non-ideal part that is a function of activity coefficients [3],



ideal nonideal  least = W  least  least W (x i ) + W ( Yi )

(4.62)

164  Modeling in Membranes and Membrane-Based Processes The ideal part can be written as ideal  least  x H O ,p W =  ln 2 + n H2 O ,pRT  x H2 O ,c



∑v s

1  x H O ,f −  ln 2 + r  x H2 O ,c

s ,p

∑

m s ,p M H2 O ln

x s ,p  x s ,c 

x  v s ,f m s ,f M H2 O ln s ,f  s x s ,c 

(4.63)



The non-ideal part can be written as nonideal  least  γ H O ,p W =  ln 2 + n H2 O ,pRT  γ H2 O ,c



∑v

1  γ H O ,f −  ln 2 + r  γ H2 O ,c

s

s ,p

∑

m s ,p M H2 O ln

s

γ s ,p  γ s ,c 

v s ,f m s ,f M H2 O ln

(4.64) γ s ,f  γ s ,c 

4.4 Desalination and Related Energetics The global population is faced with the impending doom of water shortage. One of the major challenges of present-day governance is the issue of water scarcity. Several contributing factors such as population growth, industrialization, desertification and resultant climate change are responsible for this situation [10, 11]. Most of the water which is available on the earth’s surface is saline in nature [12]. Due to the requirement for clean water to check pollution, the most significant requirement is water reuse. Not only for irrigation and drinking purposes all over the world, treated wastewater is also used in the manufacture of semiconductors which requires pure water [13]. A brief assessment on the current state-of-the-art technologies, their energy efficiency as well as the environmental impact which these technologies might have in the context of seawater and brackish-water desalination have been discussed in this topic. Also, advances and limitations of current technology and an insight into the future prospects of desalination techniques are highlighted in this context. Most of the desalination processes are energy demanding and require a minimum energy consumption for separating saline solution to pure water and concentrated brine which never follows the particular technology or process mechanism. The theory behind minimal energy of separation is

Thermodynamics and Membrane Separations  165 well examined in thermodynamics [14]. The minimum energy of desalination is equivalent to the energy produced when salts are dissolved in pure water [15].



Energy produces = RT ln aw;

(4.65)

Where, R = Gas Constant; T = Absolute temperature; aw = Activity of water. In the present scenario, the amount of energy used is about 5 to 26 times that of the theoretical minimum. Therefore, it is essential to make desalination processes as energy efficient as possible while keeping in mind the sustainability and finances involved [14]. The knowledge of the minimum amount of energy which has to be supplied to separate pure water from seawater, gives a basis for comparing the energies and efficiencies of current processes and can assist in guiding future work in desalination. While this ideal value of theoretical minimum energy can be achieved, it is possible only when the separation follows the principle of a reversible thermodynamic process [16]. Thus, the energy of separation will be identical in magnitude to the energy of mixing; however, the sign convention will be opposite [10]. There is a close relationship between the free energy of mixing and the osmotic pressure:



−d(∆G mix ) = −RT ln a w dn w = π sw Vw dn w

(4.66)

where ΔGmix is the free energy of mixing, R is the ideal gas constant, T is the absolute temperature, aw is the activity of water, nw is the number of moles of water, Psw is the osmotic pressure of the seawater, and Vw is the partial molar volume of water. The fundamental understanding of reverse osmosis is further collaborated for by the relation obtained between the minimum energy and the osmotic pressure of the system. The driving force for transferring an infinitesimally small quantity of water across a semipermeable membrane,Vw dnw , the hydraulic pressure applied is equivalent to the osmotic pressure of seawater. With the advances in research and production of newer technologies, the energy requirements for desalination in SWRO plants has seen a significant decline over the last 40 years [17, 10]. Seawater RO process configuration significantly differs from the brackish water RO process. Traditional seawater RO consists of a single-stage membrane process with a single feed, single product, and single brine streams, whereas brackish RO can have two or even three stages with a much higher number of streams. For these types of reasons, brackish-water should be designed for a wide range

166  Modeling in Membranes and Membrane-Based Processes of flow rates [18]. Even though less contaminated water is easier to treat, the main source of water is that from large water bodies such as oceans, which are being purified by several feasible desalination techniques over the years. Among them, RO is presently the fastest-growing technique of treatment while Multistage flash distillation remains the most frequently used technology, where consumption of energy does not present any negative issue [13]. In the present era, we are facing the continuous energy crisis at every moment of engineering applications. In this respect, Multieffect distillation (MED) has a higher potential as an evaporation technique rather than vapor compression technique; however, methods such as freezing of water have been proven to be highly inefficient [19].

4.4.1 Evaporation Techniques An incredibly large amount of energy of the order of 650 (kW-h)/m3 of product is required in processes such as the Single-Stage evaporation of seawater process. However, it is considered to be the simplest form of techniques available for desalination purposes. Therefore, to provide this energy in a sustainable manner, engineers set up a multistage process which involved evaporation techniques, MSF and MED [14] for seawater desalination. In the Persian Gulf, about half of the global desalination capacity runs on the MSF technique. Also, a term known as the GOR (Gained Output Ratio) is used to define the number of times during which the heat of evaporation is reused [20]. This is nothing but the ratio between the mass of desalinated water to the energy consumed by desalination at a fixed amount. Another parameter, PR (Performance Ratio) is defined as the quantity of water (pounds) produced by desalination per 1,000 BTU of heat provided to the process [21]. The SI equivalent of this formulation is the number of kg of water produced per 2,326 kJ of heat [22]. MSF distillation is basically an energy-intensive process due to its requirement of both thermal and electrical energy. An MSF unit has ideal values which range from 10 to 35,000 m3/day and involves a sequence of phases (from 4 to 40 each), with consecutively lower temperature and pressure that will cause flash evaporation of the hot brine present in the system. This will then give freshwater upon condensation [14]. The components of a typical MSF system will include many flashing stages, a brine heater, pumping units, venting system, as well as a cooling water control loop. The number of flashing units are increased because it will correspond to a higher internal energy recovery. Firstly, the feed which is seawater is allowed to pass through a suitable heat exchanger where a gradual increase in temperature occurs. The heat is supplied in the form of steam from an

Thermodynamics and Membrane Separations  167 external heat source and the seawater is gradually heated up to 90-100oC [14]. Next, the heated feed water is allowed to pass into a series of vacuum chambers. This is followed by the condensation of water vapour into fresh water which is produced by the cooling water control loop, and this operation is repeated for other stages [23, 24]. Several factors which are responsible for energy consumption in MSF have a maximum temperature of the heat source (energy input), the temperature of the heat sink, number of stages, and salt concentration in the flashing brine solution, the geometrical configuration of the flashing stage, construction materials, and design configuration of heat-exchangers. Hence, the energy consumption of the MSF unit can be reduced by increasing the GOR (or PR), number of stages, and the heat-transfer area [25, 14, 24, 13]. Commercial manufacturers of MSF provide a GOR design range between 8 and 12 kg distillate/kg of steam depending on the steam feed temperature [20]. As in MSF, a multi-effect distillation plant involves a series of columns, wherein the pressures are maintained at decreasing pressure levels in order to fascilitate heating of one column as cooling of another column occurs [14, 26]. MED units run based upon the configuration of the tubes placed within the tubes. Therefore capacities of 600 to 30,000 m3/day are accommodated based upon vertical or horizontal arrangements of the tubes. The earliest distillation plants used MED, but MSF displaced it due to its lower cost and less tendency to scale [27]. Generally, two types of energy are required for the MED process – a low temperature heat supply and electricity. The process takes place at operating temperatures of 64° to 70° C [25]. The total equivalent energy consumption of the MED units ranges from 14.45 to 21.35 kW-hr/m3 [14].

4.4.2 Membrane-Based New Technologies Recent advances in technology have resulted in the development of various types of membrane processes like Electro-dialysis (ED), Reverse osmosis (RO), and Forward osmosis (FO). The energy supplied for these processes is mainly in the form of electricity. As an example, AC electricity is consumed to drive the different pumps in RO or FO process as well as in membrane distillation process, whereas ED electrodes consume DC electricity and to drive the ED pumps, AC/DC electricity is required. Electro-dialysis (ED) is an electrochemical separation process that operates at atmospheric pressure and uses direct electrical current to move salt ions selectively through a membrane, leaving fresh water behind. The principle of ED process has been known for about 120 years; however, the large-scale industrial utilization of ED did not begin until about 50

168  Modeling in Membranes and Membrane-Based Processes years ago [28, 29]. As shown below in Fig. 4.5, the various components which form the ED system are: a pre-treatment system (which ensures the removal of biological and fouling contaminants), membrane stack (across which the ionic transfer occurs), low-pressure circulation pump (installed to run the set up and to ensure that sufficient flux and pressure conditions are maintained), direct-current power supply (this may be of the form of a rectifier or a photovoltaic system), and a post-treatment system. A schematic of the ED system has been shown in Fig. 4.5. An ED plant’s typical capacity ranges from 2 to 145,000 m3/day [14, 30]. It has been in commercial use for desalination of brackish water for the past three decades, particularly for small and medium-scale processes [31]. Most of the current technologies have applications pertaining to very low concentrations only (less than 500 ppm) [32]. Generally, ED is a low-cost process, provided the concentrations are within 500-2000 ppm. The following relation, expressing the Donnan potential (ψDonnan), is obtained when a likeness between the electrochemical potentials of ions in the electrolyte solution and MEM in equilibrium with it is obtained, [33]:

power source permeate cathode +

– anode

+

+

+

+

+

–

–

–

+

Feed water

Fig. 4.5  Schematic diagram of ED system.

Membranes

–

–

–

Thermodynamics and Membrane Separations  169



∆ψ Donnan = ψ mem − ψ sol =

  a SOL  1  i RT ∆ ln + V π   1  a mem  z iF   i

(4.67)



where ψmem is the electrical potential on the membrane side, ψsol is the electrical potential on the solution side, F is the Faraday constant, z is the valence, R is the universal gas constant, T is the absolute temperature, a is the activity of the ion, V is the partial molar volume, Δπ is the osmotic pressure difference between the two phases, the subscript i refers to the salt ion i and the superscripts mem and sol indicate the membrane and the solution, respectively. The Donnan potentials for cation and anion are equated and the osmotic pressure term Vi ∆π is neglected in regard with the RT-logarithmic one [28]. The activity coefficients are assumed to be identical in both phases, and this gives the Donnan equilibrium as follows for a particular concentration [34, 35]: 2



C

mem CO

C  C =  fix  + C Sol − fix  2  2



(4.68)

mem where, C CO is the co-ion concentration in the membrane, Cfix is the fixed charge concentration and CSol is the salt concentration in the solution. The type of water desalination chosen depends upon the concentration of the salt in the feed solution. For instance, brackish water having a salinity which is lower than 10,000 ppm, and that with a higher TDS value, may be treated either by RO or ED or both. Although RO or ED can be chosen as a suitable treatment option, the economic feasibility of either, will be the final deciding factor. Therefore, it has been observed that for a low concentration salinity water and higher recovery rate requirement, ED is the more cost-effective desalination system. The cost of water production of BWRO plants having a large capacity (40,000 to 46,000 m3/day) is in the 0.26 to 0.54 US$/m3 bracket, while the cost of water production of ED plants lies in the 0.6 to 1.05 US$/m3 bracket. The cost of water production mainly depends upon the salinity of the feed water [14, 36, 37]. The electricity consumption of an ED unit for low solute concentrations of below 2500 ppm, lies in the range of 0.7 to 2.5 and that for a solute concentration of 2000-5000 ppm, is about 2.64 to 5.5 kWh/m3 [38]. Although RO is an energy-intensive process, the amount of energy needed for this process is significantly less in comparison to thermal methods and the schematic of the RO process is shown in Fig. 4.6, [39] as the

170  Modeling in Membranes and Membrane-Based Processes process does not require very high temperatures which are needed for a phase change to occur. Over the past decade the specific energy for RO desalination has diminished considerably and is moving towards the minimum theoretical thermodynamic limit [39]. Development of large pumps with an efficiency of around 90% was achieved by using modern efficient turbines and other energy recovery devices. At high pressures, energy requirement of the pumps is generally high and this is for ensuring that the osmotic pressure barrier is overcome. It depends upon the feed solute concentration and temperature. For instance, the osmotic pressure is around 7.9 bars for every 1% of NaCl concentration [40]. The development in this technique enabled considerable reduction in consumption of power which is used to pressurize the feed; for example, in desalinating 3.5% seawater at a recovery rate of 50% the energy consumption is as low as 2.7 (kWh)/m3of product. Other desalinating applications such as brackish water streams or slightly polluted source, the energy consumption can be as low as 1 (kWh)/m3 or less which generally depends on water salinity and energy recovery availability. In RO desalination, a parameter called the specific energy consumption (SEC) is used to quantify the amount of power required by the entire process and can be defined as the amount of energy being used to produce a unit volume of permeate (J/m3). It is expressed in SI units of kWh/m3 or kJ/kg. As has been said previously, the most energy-intensive pressure-driven membrane desalination process is RO, with the operating pressure in the range of 50-80 bar and 20-40 bar, respectively. Energy consumed in the RO process is mainly by a high-pressure pump which can be practically estimated using the following relation

Membrane Feed

Concentrate Pump

Recycle stream

Fig. 4.6  Schematic of RO process.

Permeate

Thermodynamics and Membrane Separations  171



Es =

Q f (Pf − Pi ) ηpQ p



(4.69)

Where, Qf is the volumetric feed rate, Pf and Pi is the hydraulic pressure of the pump output and input respectively and p is the pump ­conversion efficiency. Depending upon the type of pump this pump efficiency varies and has an overall efficiency of 85-95% for positive displacement pumps and less than 60% for centrifugal pump. When the above equation is defined in terms of process recovery rate R = Q p /Q f :



Es =

Pf − Pi ηpR



(4.70)

A key role is played by the process recovery rate as it helps in determining the capital and the operating cost of the RO system. About 50% reduction in the total energy consumed, is achievable with the addition of Energy Recovery Systems (ERS) into the process framework. There is a visible improvement on the system recovery rate as well as the systems conversion efficiency. At higher recovery rates, a reduction in recovered energy is witnessed which makes it less viable. This happens due to the fact that a smaller volume of the concentrate is associated with higher recoveries. At present, a recovery rate of 35–50% and 75% is achievable in RO plants for seawater and brackish water desalination. The equation governing Osmotic Pressure at constant temperature, of the feed and concentrate, assuming a high solute rejection rate, can be expressed in terms of R as follows:



π c = π f /1 − R

(4.71)

From literature, the amount of energy supplied for the seawater pressure to be increased to the osmotic pressure, is equal to the theoretical minimum energy required for desalination. In practice, to ensure the equal distribution of water molecules across the membrane channel during its permeation, the applied pressure must be equal to the osmotic pressure on the membrane channel during its leaving (Pf = πc) [10]. But a staged operation always can minimise energy requirement in RO separation process because here in every stage operation, a desired percentage recovery is achieved. Another membrane-based desalination process is existent which is called Forward Osmosis and the schematic of the FO process is shown in Figure 4.7 [26]. Its functioning mainly depends on the permeability of

172  Modeling in Membranes and Membrane-Based Processes thermolytic draw solution

Heat Industry permeate

sea-water pumping

draw solution recovery

Fig. 4.7  Schematic diagram of FO using industrial heat energy.

the specialized membrane. Even though it is similar in application to RO, the driving force here for separation is the natural osmotic pressure of the system, not hydraulic pressure. Recent technology based on the FO principle has the capacity of using an edible solute, such as concentrated glucose or ammonium salts [26]. The formation of mixtures of ammonium bicarbonate, carbonate, and carbamate, are obtained upon reaction of ammonia and carbon dioxide gasses. These salts are easily rejected by the specialized FO semi-permeable membrane which then leads to a high osmotic pressure [26]. For the single stage FO, the theoretical global minimum for energy consumption occurs at a fractional recovery of 50%, as claimed in literature [41]. The specific energy consumption (SEC) of the pump was calculated as follows:



SEC pump =

 pump Q ∆P W = f Qf Qp



(4.72)

where, SECpump is specific energy consumption of pump used in FO, W pump is pump work, Qp is permeate flow-rate, Qf is feed flow-rate, and ΔP is pressure drop. The energy consumption for the FO stage itself is very low, e.g., ~0.11 kWh/m3 at 50% recovery because the process is driven by osmotic pressure instead of hydraulic pressure difference and the low pressure pump only needs to overcome the pressure drop in the feed channel. At 50% product recovery, the order of membrane processes with the lowest to the highest SEC is two-stage RO ≤ single-stage RO < FO with two-stage NF recovery ≤ FO with NF recovery. There is no significant difference in SEC between the FO and RO at 50% recovery [42].

Thermodynamics and Membrane Separations  173

4.5 Forward Osmosis for Water Treatment: Thermodynamic Modelling Forward osmosis (FO) is an emerging separation process in which water in the feed solution (low osmotic pressure) spontaneously flows through a selectively permeable membrane to draw solution (high osmotic pressure) under the osmotic pressure difference [43]. The schematic diagram of FO process is shown in (Fig. 4.8). The draw solution is relatively concentrated on one side of membrane than the feed solution on other side. FO has attracted growing interest recently in water treatment at lower energy consumption. Modern applications of FO are water treatment, desalination, food processing, concentrating landfill leachate and controlled drug release [44]. People often get confused between Forward Osmosis (FO) and Reverse Osmosis (RO). FO is a process which requires low or no hydraulic pressure or separation of water from dissolved solutes, contrary to that RO process is completely dependent on hydraulic pressure for separation [72]. Now we will try to understand in depth of the specific terms associated with forward osmosis.

4.5.1 Osmotic Processes Osmotic process find its application in many fields including purification of low salinity water to desalination of sea and brackish water. Osmotic processes can be best described by thermodynamic equations for salt and water fluxes. Reverse osmosis (RO), forward osmosis (FO), pressure enhanced osmosis (PEO) and pressure retarded osmosis (PRO) forms the major osmotic processes.

FORWARD OSMOSIS

OSMOTIC AGENT WATER MOLECULE SOLUTES SEMI-PERMEABLE MEMBRANE

Fig. 4.8  Schematic diagram of FO process.

174  Modeling in Membranes and Membrane-Based Processes

4.5.1.1 Osmosis The transport of water molecules across a selectively permeable membrane from a region of higher chemical potential to a region of lower chemical potential is termed Osmosis. The driving force behind this phenomena is the difference in solute concentration across the membrane surface that allows selective passage of water molecules rejecting most of the solute molecules or ions. Osmotic pressure (π) is the pressure which is applied to the more concentrated solution to prevent the movement of pure solvent across the membrane. The physical representation of osmotic pressure is shown in (Fig. 4.9). According to the Morse equation which is derived from the van’t Hoff equation by considering dilute ionic solutions [46], the osmotic pressure of a solution, can be expressed as in Eq. 4.73.

 n π = icRT = i   RT  V



(4.73)



Where i is the van’t Hoff factor, c is the molarity of the solute which is equal to the ration of the number of solute moles, n, to the volume of the solution, V, R is the gas constant of 8.314K-1mol-1, and T is the absolute temperature. The main driving force of FO process is osmotic pressure differential (Δπ) across the membrane which is contrary to RO which utilizes hydraulic pressure differential as driving force for transport of water across membrane. The FO process results in concentration of a feed stream and dilution of a highly concentrated stream (referred as draw solution). However, there is one another osmotic process also which is still gaining widespread acquaintance and is known as Pressure-retarded osmosis OSMOTIC SEMI-PERMEABLE MEMBRANE PRESSURE (π)

FLUX

PURE SOLVENT

Fig. 4.9  Physical representation of osmotic pressure (Adapted from ref. [45]).

SOLUTION

Thermodynamics and Membrane Separations  175 (PRO). This PRO is considered to be an in-between process of FO and RO. It utilizes the hydraulic pressure in the opposite direction of osmotic pressure gradient (similar to RO) but the net water flux is still in the direction of the concentrated draw solution (similar to FO). The general equation describing the transport of water in FO, RO and PRO is shown in Eq. 4.74:

Jw = Aw (Δπ – ΔP)

(4.74)

where Jw is water flux, Aw is the water permeability constant of membrane, σ the reflection coefficient and ΔP is the applied pressure. All the three osmosis processes FO, RO and PRO can be distinguished simply by ΔP parameter, i.e., for FO, ΔP = 0; for RO, (ΔP > Δπ) and for PRO, (ΔP < Δπ). The flux directions and driving forces for all three processes were characterized in the early 1980s by Lee et al. The direction of the permeating water in FO, RO and PRO is illustrated in (Fig. 4.10). The illustration in Fig. 4.11 gives respective PRO zone, RO zone and flux reversal point (Fig. 4.11).

4.5.1.2 Draw Solutions Draw solution is the concentrated solution which lies on the permeate side of the membrane acting as the driving force for FO process. However, in literature different terms have been proposed like draw solution, osmotic agent, osmotic media, driving solution, osmotic agent, sample solution, or just brine. The primary criterion for selecting the draw solution is that it should have higher osmotic pressure than feed solution. The osmotic FEED

BRINE

FORCE (∆P)

FORCE (∆P)

PRO

RO

∆π

FO

Fig. 4.10  Solvent flow in FO, PRO, and RO. For FO, water diffuses to more saline side of membrane. For PRO, water diffuses to more saline liquid that is under positive pressure (ΔP < Δπ). For RO, water diffuses to less saline side due to hydraulic pressure (ΔP < Δπ)[44].

176  Modeling in Membranes and Membrane-Based Processes

Water Flux

Reverse Osmosis (∆P > ∆π)

Flux Reversal point (∆P = ∆π)

∆P

0 ∆π

Pressure-Retarded Osmosis (∆P < ∆π)

Forward Osmosis ∆P = 0

Fig. 4.11  Direction and magnitude of water flux as function of applied pressure in FO, PRO and RO [44].

pressures of several solutions being considered as draw solution are shown in [Table 1] [46]. In FO, the selection of a suitable process for reconcentrating the draw solution is a significant benchmark. Although, NaCl is used quite often for this purpose because of its high solubility and relative simplicity in reconcentrating to high concentration by using RO without any risk of scale up. Certain application where the requirement is of high rejection rate, researchers tend to go for multivalent solutions. Although, apart from salt solution other chemicals were also suggested and tested as drawing agent Table 4.1  Osmotic pressure for different draw solution on basis of their concentration. Draw solute(s)

Concentration

Osmotic pressure (atm)

NaCl

0.60 M

28

MgCl2

0.36 M

28

KCl

2M

89.3

NH4HCO3

0.67 M

28

Sucrose

1M

26.7

Thermodynamics and Membrane Separations  177 for seawater desalination like Sulphur dioxide, aluminiumsulphate solution, glucose solution and many more. There have been novel achievements attained by researchers in formulating draw solution like combination of ammonia and carbon dioxide gases in specific ratio has the capability of generating osmotic pressure of around 250 atm. This approach from concentrated saline feeds witnesses exceptionally high recovery of potable water and not limited to this only, ammonium salts are also thermally removable from system. In a new Nano technological approach, naturally occurring non-toxic magneto ferritin is tested as new potential draw solutions in which magneto ferritin can be rapidly separated from aqueous streams under influence of magnetic field [44].

4.5.2 Concentration Polarization in Osmotic Process Concentration polarization (CP) is basically build-up of concentration gradient both inside and around membrane during operations which affects the effective osmotic pressure across the membrane active layer and ultimately, limiting the achievable water flux. Specifically, two types of CP phenomena can take place in osmotic driven membranes, i.e., internal CP and external CP.

4.5.2.1 External Concentration Polarization Reverse osmosis is a pressure-driven process and there is convective flow of permeates across porous membrane which leads to deposition of solutes at the active layer of membrane. This deposition ultimately mitigates the water flux as osmotic pressure is increased due to concentration polarization. Although the issue of concentration polarization (CP) is not limited to membrane processes associated with pressure-driven system it can also be observed in osmotic-driven membrane processes. When the feed solution flows on the active layer of membrane then solute concentration build-up at the active layer which may be called concentrative external CP. Simultaneously, at the membrane-permeate interface the draw solution lying at the permeate side of membrane gets diluted by the permeating water and they are called dilutive external CP. The effects of osmotic driving force is strongly hampered by both the concentrative and dilutive external CP. However, membrane fouling induced by external CP has very minimal effect on water flux in FO process due to low hydraulic pressure requirement in it [44].

178  Modeling in Membranes and Membrane-Based Processes

4.5.2.2 Internal Concentration Polarization When the porous support layer of membrane faces feed solution (like in PRO), a polarized layer is established along the inside of the dense active layer as water and solute propagate the porous layer which is referred as concentrative internal CP. In FO application for desalination and water treatment, the active layer of membrane faces the feed solution and the porous support layer faces draw solution. As water permeates the active layer, the draw solution within the porous substructure becomes diluted, which is referred to as dilutive internal CP. The osmotic pressure difference between the bulk feed and bulk draw solution (Δπbulk) is higher than the osmotic pressure difference across the membrane (Δπm) due to external CP and that the effective osmotic pressure driving force (Δπeff ) is even lower due to internal CP [44]. 1. Mathematical modelling equation for concentrative internal concentration polarization In FO, mass transfer of water across membrane is strongly affected by internal CP in porous support layer. From the concentration profile illustrated in (Fig. 4.12), C1 and C5 denotes the concentrations of the bulk feed and draw solution, respectively, whereas C2 and C4 denotes the concentration of feed-membrane and draw solution-membrane interfaces,

C5

C4

Js Jw

Feed

∆πbulk

∆πm C3

C2

Jw

Feed

∆πbulk

∆πm ∆πeff

C1

C5

C4

Draw Solution

Draw Solution ∆πeff

Js

C1

(a)

C3 C2

active layer

support layer

support layer

active layer

(b)

Fig. 4.12  (a) Concentration internal CP across asymmetric membrane in FO (b) Dilutive internal CP across asymmetric membrane in FO.

Thermodynamics and Membrane Separations  179 respectively, and C3 is the concentration at the active layer-support layer interface A simplified equation to describe the water flux during FO without considerations for membrane orientation was introduced by Loeb et al. [47], is shown as:

Jw =



1 π High ln K π low

(4.75)

Where K is the resistance to solute diffusion within the membrane porous support layer, and (πHigh) and (πlow) are the osmotic pressures of the bulk draw solutions (C5) and feed solution (C1) respectively. K is defined as

K=



tτ ∈Ds

(4.76)

where t, τ and ∈ are membrane thickness, tortuosity and porosity respectively and Ds is the diffusion coefficient of the solute. However, there is limitation of the equation demonstrated in Eq. 4.76 as it can be used only for very low water fluxes [48], although, further development of Eq. 4.76 has led to more general governing equation for concentrative internal CP which is shown as:



 1   As + Aw π High − J w  K =   ln   Jw   As + Aw π Low 



(4.77)

where As is the solute permeability coefficient of the active layer of the membrane, which can be determined from a RO-type experiment [47] using:



As =

(1 − R s )Aw (∆P − ∆π ) Rs

(4.78)

where, Rs is the salt rejection. Eq. 4.78 is used to quantify the intensity of internal CP on system; higher K denotes more severe internal CP [44]. 2. Mathematical modelling equation for dilutive internal concentration polarization The mathematical modelling equation for dilutive internal CP was presented by Loeb et al. [47]

180  Modeling in Membranes and Membrane-Based Processes



 1   As + Aw π High  K =   ln   Jw   As + Jw + Aw π Low 



(4.79)

The concentration profile across an asymmetric or composite FO membrane for dilutive internal CP is also shown in (Fig. 4.12 (b)). It has been shown in recent studies that Eq. 4.77 and Eq. 4.79 are the actual modelling equations with capability of successfully predicting the results which are obtained during experiment with an FO-designed membrane [49].

4.5.3 Forward Osmosis Membranes FO membrane doesn’t have any extremely specific specification. Any dense, non-porous and selectively permeable membrane can be used as a membrane for FO. Nowadays, most of the membranes used for FO process are commercially available RO membranes. Early development of membranes saw experimentation with every type of possible membrane material available like animal bladders, nitrocellulose and rubber. Until 1960, membrane science marked an era of major development in fabrication of membranes. During that period important discoveries were made which led to transformation of membrane separation from being laboratory project to an industrial phenomenon. As industrial sector started gaining interest in this field, the best quality of membranes were also demanded like making defect-free, high-flux, anisotropic membranes. During the 1970s, researchers started using commercially available RO membranes and in-house cellulose acetate for treating dilute wastewater by FO process using simulated seawater as draw solution. Researchers also used flat sheet RO membranes and cellulose acetate for desalinating the seawater using glucose as draw solution by FO process [44]. During 1990s, a special FO membrane was developed by Osmotek Inc. (Albany, Oregon) which was quite successful in commercial application like water purification for military, emergency and recreational purposes [50]. Some of the membranes which are used nowadays for FO process are Polyamide imide polyether sulphone blend (PAI/PES), Polyamide imide-polyethersulphone polyether Imide blend (PAI-PES/PEI), Cellulose acetate phthalate (CAP), Polyacronitryl (PAN), etc.

4.5.4 Modern Applications of Forward Osmosis Commercial application of forward osmosis is primarily in the water purification field and pharmaceutical industry. Forward osmosis has stretched its reach into seawater desalination, food processing, pharmaceutical

Thermodynamics and Membrane Separations  181 application and power generation. But in this chapter, we will focus on wastewater treatment and water purification only.

4.5.4.1 Wastewater Treatment and Water Purification Lots of literature has been published since the invention of FO technology for wastewater treatment. In this chapter a few applications on Forward Osmosis technology in wastewater treatment are explained briefly.

4.5.4.2 Concentrating Dilute Industrial Wastewater The first feasible industrial application of FO process was concentration of dilute industrial wastewater. The sole aim of this experiment was to use low-energy process to treat industrial wastewater containing very low concentrations of heavy metals for possible reuse. For studying the feasibility of cellulose RO membrane to concentrate the synthetic wastewater containing copper and chromium a bench-scale system was installed. In a coastal region, being quite inexpensive, simulated seawater was used vastly as draw solution. But destitute performance of RO membrane led to cancellation of further pilot-scaling testing. According to researchers, a more improvised version of membrane is required with higher water flux and salt rejection to ascertain the future of industrial application of forward osmosis [44].

4.5.4.3 Concentration of Landfill Leachate During early 1998, FO found another application in concentrating landfill leachate. Landfill leachate is composed of quite variable feed solution with generally four types of pollutant like organic compounds, dissolved heavy metals, organic and inorganic nitrogen, and total dissolved solids (TDS). The most commercially feasible process to treat the TDS from wastewater stream is either membrane process or mechanical evaporation. In the treatment facility, first collection and pre-treatment of the raw leachate took place. After this the water is extracted in six stages of FO cells. Finally, a three-pass RO system is used which actually produces stream of purified water which has its application in land activities and a draw solution of reconcentrated stream. The concentrated leachate is solidified before disposal. Between 1998 and 1999, the treatment plant treated over 18,500 m3 of leachate with the average water recovery of 91.9%. Many contaminants even achieved greater rejection of up to 99% and final effluent concentrations were under permissible National Pollutant Discharge Elimination System (NPDES) total maximum daily load (TMDL) levels [44].

182  Modeling in Membranes and Membrane-Based Processes

4.5.4.4 Concentrating Sludge Liquids Wastewater treatment facilities produces sludges which are treated in anaerobic digesters for degradation of organic solids. Not limited to degradation, digesters also play a role in stabilizations of the sludge. After the digestion process of sludge is over, it is sent into a centrifuge. The centrifuge serves the purpose of dewatering the sludge and ultimately producing concentrated biosolids and liquid stream which is called centrate. The centrate contains high nutrients contents which need to be extracted for purpose of fertilizers. The FO process has shown the potential of concentrating the centrate, which is why it is studied at Truckee Meadows Water Reclamation Facility (TMWRF) in Reno, Nevada. The primary goal of this study is to find out the advantages and limitations of using FO for concentrating the centrate, for developing an economic model and determining cost effectiveness, for development of a hybrid system with integration of FO and RO system and analysing potential agricultural use of concentrated ­fertilizer [51].

4.5.4.5 Hydration Bags The idea of the hydration bag was conceived to serve the purpose of creating a reliable drinking water source during emergency situations. The prime consumers targeted for hydration bags are military, trekkers and people affected by natural catastrophes like flood, hurricane, etc. The biggest advantage of FO hydration bags is that they don’t require any power for the purification process. Chemical potential is used as the driving force for the purification process. Researchers have tested its feasibility [52] by carrying out experiments and they have successfully tested this concept. Currently, studies are going on to optimize the FO bag and develop an economically viable product for consumers. An analytical study [52] also shows that the FO bag made of PAI/PES membrane is capable of collecting 500-600 ml of water in 8-10 minutes with the given membrane area 0.12474 m2. Experiments are going on to develop membrane which is capable of delivering higher flux and putting Oral Rehydration Solution (ORS) as draw solution to serve the purpose of emergency water.

4.5.4.6 Water Reuse in Space Missions The most remarkable achievement of the FO process was when the National Aeronautics and Space Administration (NASA) decided to evaluate this technology for water reuse in space for long-term human missions. During the period 1994–99, NASA and Osmotek Inc., in a joint collaboration, did the

Thermodynamics and Membrane Separations  183 first phase of their study on design, construction and testing of basic functions of Direct osmotic concentration (DOC) systems. Later on, NASA transferred its DOC unit to the University of Nevada, Reno, to carry out further research for detailed investigation on the operating conditions of the system [44].

4.6 Pressure Retarded Osmosis for Power Generation: A Thermodynamic Analysis First conceptualized by Pattle in 1954 [53], Pressure Retarded Osmosis (PRO) is a membrane-driven process similar to Forward Osmosis (FO). It provided information for the first time on how to generate energy by virtue of a salinity gradient. For the next couple of decades, due to cheap fuel prices, there was no research carried out down this avenue. With the oil crisis in 1973, focus was shifted to study such a process for the purpose of generating energy [54]. Loeb in 1976 [55] performed experiments with h ­ ollow fibre membranes for Pressure Retarded Osmosis, which greatly varied from theoretical calculations. The explanation was that state-of-art ­hollow fibre membranes suited to the demand at the time were not available – a challenge, as we shall see, that persists to this day. After the energy crisis of the 1970s, alternative forms of producing sustainable energy were extensively researched and novel methods discovered. Towards the last quarter of the 20th century, interest piqued amongst researchers about potential energy generation through salinity gradient. This led them to try to understand, model and analyse its potential [54]. Studies to estimate the potential of energy generated through PRO process for two hypersaline systems have been carried out by Loeb – one in the Dead Sea in Israel and the other in the Great Salt Lake in the United States [55-56]. Economic and engineering analyses were carried out which provided promising results regarding the potential of the PRO process. A pioneer in salinity gradient energy, Norwegian company Statkraft built its first PRO plant with a nominal capacity of 10 kW in 2009. The area of membranes used was estimated to be approximately 2000 m2. Inherent inefficiencies led to production of 5 kW of electric power [54]. However, the plant was shut down citing unavailability of high-performing and cheap bulk PRO membranes, among others.

4.6.1 What Is Pressure Retarded Osmosis? Having defined osmotic pressure (π) in section 4.5, with its physical representation depicted in Figure 4.9, it is essential to understand the key

184  Modeling in Membranes and Membrane-Based Processes difference between forward osmosis (FO) and pressure retarded osmosis (PRO) processes at a fundamental level. From Eq. 4.73, the osmotic pressure of a solution is dependent on the concentration. As previously discussed in section 4.5.2, forward osmosis is the process by which molecules of solvent pass across a semi-permeable membrane from a less concentrated solution into a more concentrated one, thereby creating additional liquid head equal to the accumulation of solvent transferred, i.e., when the osmotic pressure gradient between the feed and draw solutions is positive (Δπ > 0), and the solutions are at the same hydrostatic pressure. This is driven by the difference in the solute concentrations across both sides of the membrane, which is manifested by the different chemical potential across both sides of the membrane. In pressure retarded osmosis, a hydrostatic pressure ΔP is applied on the draw solution such that its value is less than the osmotic pressure differential Δπ, thereby the resulting osmotic gradient (Δπ – ΔP) still remains > 0, which means that the solvent transfer continues from the feed to the draw solution but at a slower rate than would be without the application of the hydrostatic pressure ΔP on the draw solution. Thus, while for the FO process, the driving force can be represented in terms of pressure as Δπ, that for PRO process is reduced due to the applied hydrostatic pressure to Δπ – ΔP. As stated above, in both the processes, water permeates across the membrane and continues to dilute the draw solution – solution which is higher in concentration, and thus higher in osmotic pressure. Water continues to permeate from the region of low concentration to that of higher concentration solution, but the flux is lower than that of the FO process – and therefore “retarded”. This can be further understood from Eq. 4.74, where the water flux is proportional to the driving force across the membrane – lower for PRO, higher for FO. The relation between the direction and magnitude of flux, along with the hydrostatic pressure applied on the draw solution has previously been depicted in Figure 4.13 [67]. As can be interpreted from Figure 4.11, PRO is an intermediate process between FO and reverse osmosis (RO) due to its driving force lying in the middle range of the two processes.

4.6.2 Pressure Retarded Osmosis for Power Generation A simplified PRO process plant setup has been depicted in Figure 4.14. It consists of a membrane module with two solutions – a feed solution which is the low salinity river water, while the other is the draw solution

Thermodynamics and Membrane Separations  185 consisting of high salinity seawater. The draw solution is pressurised as depicted, with a pressure less than that of the osmotic pressure differential. For purposes of simplicity, we consider that the river and seawater channels contain varying concentration of Sodium Chloride (NaCl) salt. Thus, the solvent is water, which permeates across the membrane from the feed to the draw solution. This results in a hydrostatic head being generated at the draw solution side, while the concentration of this solution reduces and more solvent permeates into it. 0< P< π

π

Fig. 4.13  Pressure retarded osmosis process –a simplified schematic.

River Water Bleed 0