refined membrane characterization and modeling in forward ... · draw solute flux was found to...

Refined Membrane Characterization and Modeling in Forward Osmosis

by

Jeffrey Martin

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Department of Chemical Engineering and Applied Chemistry University of Toronto

© Copyright by Jeffrey Martin 2019

ii

Refined Membrane Characterization and Modeling in Forward Osmosis

Jeffrey T. Martin

Master of Applied Science

Graduate Department of Chemical Engineering and Applied Chemistry

University of Toronto

2019

Abstract

This work builds upon recent FO transport models and presents an improved FO-based

membrane characterization method that addresses the non-ideality of concentrated draw

solutions, physical properties that are not based on the bulk draw solution, and all instances of

concentration polarization. Using the FO-based characterization method of this work,

consistent water permeability and structural parameter values are obtained for multiple

inorganic draw solutions. When compared to additional experimental FO transport data,

improvements of 18-107% were observed over the existing FO transport models. Using the

transport model of this work both the water and reverse draw solute flux were found to increase

with increasing crossflow velocities. Further, through experimentation, the water and reverse

draw solute flux was found to decrease with decreasing temperature. Using the modeling

approach developed in this work, the assessment for a potential FO application is refined by

providing the means for accurate process modeling and membrane characterization.

iii

Acknowledgments

I would first like to thank my supervisor, Dr. Vladimiros Papangelakis for his mentorship and

patience throughout my MASc degree program. He was more than supportive when my initial

project transitioned to the work of this current thesis. Through his guidance I have gained an

immense appreciation for attention to detail and the need to theoretically justify results; it is

not enough to simply present data. Secondly, I would like to thank Dr. Georgios Kolliopoulos

for his support and friendship throughout my degree. My thesis wouldn’t be half of what it is

without his guidance. I would also like to thank Dr. Arun Ramchandran for his expert advice

regarding the mass transfer model in this work.

I would also like to thank the entire Aqueous Process and Engineering Chemistry (APEC)

group for their support, especially Georgiana Moldoveanu, as they were always willing to

assist and provide constructive criticism when necessary. Further, I would like to thank those

who have made my time enjoyable during my degree: Joe Brazda for our many culinary

adventures and for surviving the writing classes with me; Laís Mazullo for our Toronto Maple

Leafs outings and for her essential ICP support; Amir Esmaili for our detailed discussions in

the water lab; Elliot Pai and Adam Heins for our weekly climbing outings and for their

friendship throughout my academic career; and Christine Pham for her unwavering support

with all of my endeavours.

Finally, I thank my parents and family members for being my biggest supporters throughout

all of my life, all that I have accomplished is because of them.

iv

Table of Contents Abstract ..................................................................................................................................... ii

Acknowledgments ................................................................................................................... iii

List of Figures .......................................................................................................................... vi

List of Tables ......................................................................................................................... viii

List of Appendices……………………………………………………………………………ix

List of Nomenclature .................................................................................................................x

List of Abbreviations ............................................................................................................... ix

1 Introduction.............................................................................................................................1

1.1 Overview ..........................................................................................................................1

1.2 Research Scope ................................................................................................................3

1.3 Objectives ........................................................................................................................3

1.4 Thesis Organization .........................................................................................................4

2 Literature Review ...................................................................................................................5

2.1 Forward Osmosis Overview ............................................................................................5

2.2 Forward Osmosis Draw Solutions ...................................................................................6

2.3 Forward Osmosis Membranes .........................................................................................7

2.4 Previous Forward Osmosis Transport Modeling .............................................................7

2.5 Effects of Temperature on Forward Osmosis ..................................................................8

2.6 Effects of Hydrodynamic Conditions on Forward Osmosis ............................................9

3 Development of Improved Forward Osmosis Model ...........................................................10

4.1 Membrane Experimental Method and Materials ...........................................................16

4.2 Low Temperature Experimental Methodology ..............................................................18

4.3 Hydrodynamic Condition Analysis Methodology .........................................................19

4.4 Draw Solution Physical Properties ................................................................................19

5 Results and Discussion .........................................................................................................21

5.1 Determination of Steady State .......................................................................................21

5.2 Experimental FO Transport Results ...............................................................................21

5.3 Membrane Parameterization ..........................................................................................23

5.4 Modelling Results and Validation .................................................................................27

5.5 Effect of Low Temperatures on FO Performance ..........................................................32

5.5.1 Experimental Water Flux Results at Low Temperatures ........................................32

v

5.5.2 Experimental Reverse Draw Solute Flux at Low Temperatures .............................34

5.5.3 Experimental Specific Water Flux at Low Temperatures .......................................36

5.6 Effect of Hydrodynamic Conditions on FO performance ..............................................38

5.6.1 Effect of Crossflow Velocity on FO Water Flux ....................................................38

5.6.2 Effect of Crossflow Velocity on FO Reverse Draw Solute Flux ............................40

6 Conclusion ............................................................................................................................42

6.1 Recommendations and Future Work .............................................................................43

References................................................................................................................................44

Appendix A: Data for Membrane Parameterization and Model Validation Experiments.......52

Appendix B: Data for Low Temperature Experiments............................................................56

Appendix C: MATLAB Membrane Parameterization Codes .................................................59

C1: Intrinsic Parameter Fitting Codes ..................................................................................59

C2: Model Validation Code .................................................................................................61

C3: Crossflow Velocity Analysis Code ...............................................................................65

Appendix D: Modeling Initial Guess Values ..........................................................................68

vi

List of Figures

Figure 1 - Forward Osmosis Transport Mechanisms................................................................2

Figure 2 - Forward Osmosis Transport Model Algorithm ......................................................15

Figure 3 - Forward Osmosis Experimental Apparatus ...........................................................16

Figure 4 - Low Temperature FO Apparatus ...........................................................................18

Figure 5 - Steady State Draw Solution Flow Rate Profile ......................................................21

Figure 6 - Experimental Water Flux Results ..........................................................................22

Figure 7 - Experimental Reverse Draw Solute Flux ...............................................................22

Figure 8 - Activity Coefficients of Water in the mole fraction scale for Various Draw

Solutions ..................................................................................................................................25

Figure 9 - Draw Solute Permeability Correlated with the Hydrated Cation Radius...............26

Figure 10 - Draw Solute Permeability Correlated with the Absolute Cation Hydration

Enthalpy ...................................................................................................................................26

Figure 11 - Water Flux Model Comparison – NaCl Draw Solution .......................................29

Figure 12 - Reverse Draw Solute Flux Model Comparison - NaCl Draw Solution ...............29

Figure 13 - Water Flux Model Comparison - MgCl2 Draw Solution .....................................30

Figure 14 - Reverse Draw Solute Flux Model Comparison - MgCl2 Draw Solution .............30

Figure 15 - Water Flux Model Comparison – CaCl2 Draw Solution......................................31

Figure 16 - Reverse Draw Solute Flux Model Comparison - CaCl2 Draw Solution ..............31

Figure 17 - Effect of Temperature on the Experimental Water Flux using 1 Molal Draw

Solutions ..................................................................................................................................33


Solutions ..................................................................................................................................34

Figure 19 - Effect of Temperature on the Experimental Reverse Draw Solute Flux using 1

Molal Draw Solutions ..............................................................................................................35


Molal Draw Solutions ..............................................................................................................36

Figure 21 - Effect of Temperature on the Specific Water flux using 1 Molal Draw Solutions

.................................................................................................................................................37

Figure 22 - Effect of Temperature on the Specific Water Flux using 3 Molal Draw Solutions

.................................................................................................................................................38

vii

Figure 23 - Effects of Crossflow Velocity on the Water Flux - MgCl2 Draw Solution .........39

Figure 24 - Effects of Crossflow Velocity on the Water Flux - NaCl Draw Solution............39

Figure 25 - Effects of Crossflow Velocity on the Reverse Draw Solute Flux - MgCl2 Draw

Solution ....................................................................................................................................40

Figure 26 - Effects of Crossflow Velocity on the Reverse Draw Solute Flux - NaCl Draw

Solution ....................................................................................................................................41

viii

List of Tables

Table 1 – Draw Solution Materials .........................................................................................16

Table 2 - Model Parameterization and Validation Experiments .............................................17

Table 3 – Draw Solution Osmotic Pressure Empirical Parameters ........................................19

Table 4 - Draw Solute Diffusivity Empirical Parameters .......................................................20

Table 5 - Draw Solution Density Empirical Parameters .........................................................20

Table 6 – Draw Solution Dynamic Viscosity Empirical Parameters ......................................20

Table 7 - Membrane Intrinsic Parameter Regression Results .................................................24

Table 8 - Hydrated Radii and Hydration Enthalpy of Draw Solute Cations ..........................26

Table 9 - Summary of Non-Ideal FO Transport Model Comparison .....................................32

Table 10 - Experimental FO Modeling Data - NaCl Draw Solute .........................................52

Table 11 - Experimental FO Modeling Data – CaCl2 Draw Solute ........................................53

Table 12 - Experimental FO Modeling Data – MgCl2 Draw Solute .......................................54

Table 13 – Low Temperature Experimental Data – NaCl Draw Solute .................................56

Table 14 – Low Temperature Experimental Data – CaCl2 Draw Solute ................................57

Table 15 – Low Temperature Experimental Data – MgCl2 Draw Solute ...............................58

Table 16 – Initial Guess Values for Proposed FO Transport Model ......................................68

ix

List of Appendices

Appendix A: Data for Membrane Parameterization and Model Validation Experiments.......52

Appendix B: Data for Low Temperature Experiments............................................................56

Appendix C: MATLAB Membrane Parameterization Codes .................................................59

C1: Intrinsic Parameter Fitting Codes ..................................................................................59

C2: Model Validation Code .................................................................................................61

C3: Crossflow Velocity Analysis Code ...............................................................................65

Appendix D: Modeling Initial Guess Values ..........................................................................68

x

List of Nomenclature

Subscripts

a, active layer interface

b, bulk solution

d, draw solution side of the membrane

f, feed side of the membrane

w, referring to water

s, referring to the draw solute

p, porous support and draw solution interface

Parameters

A, water permeability (L/m2/bar/h)

B, salt (draw solute) permeability (L/m2/h)

ci, concentration of solute i (mol solute i/ L solvent)

Di, Diffusivity of solute i (m2s)

E, Error (dimensionless)

Ji, Flux of species i (L/bar/m2/h) or (mol/m2/h)

K, Sorption coefficient into the active layer of the membrane (dimensionless)

k, Exterior mass transfer coefficient (m/s)

R, gas constant (L/bar/mol/K)

Re, Reynolds number (dimensionless)

S, Structural parameter (m)

Sc, Schmidt number (dimensionless)

Sh, Sherwood number (dimensionless)

T, Temperature (°K or °C)

t, Thickness (m)

v, Crossflow velocity (m/s)

X, Solution physical property polynomial parameter (variable units)

xw, Mole fraction of water (dimensionless)

xi

z, Distance across the membrane (m)

γi, Activity coefficient of species i (dimensionless)

δ, Exterior mass transfer boundary layer thickness (m)

ε, Porous support porosity (dimensionless)

µ, Solution dynamic viscosity (kg/m/s)

νw, Molar volume of water (L/mol)

π, Osmotic pressure (bar)

ρ, Solution density (kg/m3)

τ, Porous support tortuosity (dimensionless)

xii

List of Abbreviations

FO Forward Osmosis

FECP Feed-side External Concentration Polarization

DECP Draw solution-side External Concentration Polarization

ECP External Concentration Polarization

ICP Internal Concentration Polarization

RO Reverse Osmosis

CTA Cellulose Triacetate

ICP-OES Inductively Coupled Plasma – Optical Emission Spectrometry

1

1 Introduction

1.1 Overview

Industrial wastewater treatment poses a substantial economic and environmental challenge, for

it is necessary to recover fresh water for reuse while reducing the volume of the effluent.

Compared to traditional water recovery technologies, forward osmosis (FO) is an attractive

solution due to its reduced membrane fouling potential [1]–[8] and favourable energy

consumption [9], [10]. The potential for FO use is widespread, with current research focusing

in hydrometallurgical [11]–[16], petrochemical [17]–[24], and seawater desalination [25]–

[29] applications.

Driven by an osmotic pressure differential, forward osmosis uses a concentrated draw solution

to spontaneously pull water across a semipermeable membrane from an effluent, rejecting and

concentrating its solutes. Conjunctly, the draw solute leaks in the opposite direction of the

water flux at a much slower rate, driven by the concentration gradient across the membrane

[30]. The transport profile for FO is illustrated in Figure 1.

2

Figure 1 - Forward Osmosis Transport Mechanisms

Due to the selective transport of water across the membrane, solutes are rejected by the

membrane and accumulated at the membrane-solution interface. This results in a boundary

layer defined as concentration polarization. For traditional membrane processes like reverse

osmosis, concentration polarization occurs only at the feed solution side. However, for forward

osmosis (FO), concentration polarization occurs at three interfaces: external concentration

polarization (ECP) at the feed solution and active membrane interface (FECP); external

concentration polarization at the draw solution and support membrane interface (DECP); and

internal concentration polarization (ICP) within the porous membrane support layer [31].

These polarizations are illustrated in Figure 1 by the decline in the draw solute concentration,

cs, between that of the bulk draw-side solution and that of the active layer-porous support

interface. To accurately model the osmotic process, it is vital to estimate concentration

Variables: ta – Active layer thickness δf – Feed-side boundary layer thickness ts – Support layer thickness

δd – Draw-side external boundary layer thickness cs – Draw solute concentration z – Distance across the membrane

Subscripts: a – Active layer interface b – Bulk solution d – Draw solution side of the membrane

f – Feed solution side of the membrane p – Porous support layer and draw solution interface

3

dependent physical properties (i.e., diffusivity, dynamic viscosity, and density) within each

boundary layer formed.

1.2 Research Scope

As an emerging technology for water treatment, FO requires a model for accurate process

optimization and scale-up assessment, yet current models fail to consider the non-ideality of

the concentrated draw solution, all instances of concentration polarization, and deviations from

the bulk solution physical properties. Extending previous FO models [32], [33], in this work

we present an improved FO transport model, derived from first principles, that addresses both

the inaccuracy of assuming bulk solution properties throughout the membrane and all instances

of concentration polarization. It follows similar methodology with the work presented

previously in [32], [33]. It consists of 3 nonlinear ordinary differential equations (ODE’s),

which are solved using an in-house developed algorithm and intrinsic membrane parameters

obtained solely from FO experimental data.

The corresponding membrane characterization method considers the non-ideal osmotic

pressure and all forms of concentration polarization. When validated against the results from

a second set of experimental data and compared to the results from previous FO modeling

attempts in literature, this improved membrane characterization method and modeling

approach produced significantly better agreement with experimental measurements for three

inorganic draw solutions.

Further, the experimental effects of temperature below 25 °C on the forward osmosis

performance (water and reverse draw solute flux) are examined for three draw solutions.

Finally, using the model developed in this work, the simulated effects of different

hydrodynamic conditions are investigated on the FO performance.

1.3 Objectives

The main objectives of this study are as follows:

1. To develop an improved water and draw solute transport model for the forward

osmosis process.

4

2. To develop a membrane characterization method based solely on forward

osmosis experimental data using the improved model of this work.

3. To evaluate the effects of temperatures below 25 °C on the experimental

forward osmosis performance.

4. To evaluate the simulated effects of varying crossflow velocities on the forward

osmosis performance.

1.4 Thesis Organization

Chapter 2 provides an overview of forward osmosis technology and highlights previous

modeling attempts in literature. Chapter 3 details the derivation of this work’s model from first

principles and the corresponding model algorithm. Chapter 4 outlines the experimental

procedures used for the lab-scale forward osmosis testing. Chapter 5 discusses the

experimental and calculated results and the agreement between the two, as well as the effects

of temperature and crossflow velocity on FO performance. Chapter 6 concludes with the

important findings from this work.

5

2 Literature Review

2.1 Forward Osmosis Overview

Taking advantage of an osmotic pressure differential, forward osmosis uses a concentrated

draw solution to spontaneously pull water across a semipermeable membrane from an effluent,

rejecting and concentrating effluent solutes. Conjunctly, the draw solute permeates in the

opposite direction of the water flux, driven by a concentration gradient across the membrane.

While the effluent solution is concentrated, the draw solution is diluted until an established

dilution ratio or water recovery is achieved.

While the removal of water from the effluent occurs spontaneously, a secondary energy-

intensive process is required to recover water from the diluted draw solution and regenerate

the draw solution for reuse. The energy-intensive draw solution regeneration stage is

dependent on the type of draw solution used, including thermal separation [34]–[36], reverse

osmosis (RO) [27], [37], [38], nanofiltration [39], [40], or freeze crystallization [41]. On a

volumetric basis, the theoretical minimum energy per unit volume to recover pure water from

a solution was found to be equal to the osmotic pressure of the solution; therefore, draw

solutions with greater osmotic pressure required greater energy to regenerate the draw solution

[42]. As such, based solely on the minimum theoretical energy, reverse osmosis processes

energetically outperformed forward osmosis and for FO to achieve the same energy

consumption as RO, a draw solution regeneration efficiency of 70 % was required, much

greater than the 6-8 % currently observed [42]. However, forward osmosis has displayed

significantly-less energy consumption than evaporative processes, when thermolytic draw

solutions were used [10], [36]. When considering equivalent specific work, FO (6.8-16.7

kWh/m3 water produced) was comparable to reverse osmosis (4-8 kWh/m3 water produced),

while producing water of a greater purity (150 ppm compared to 400-500 ppm) [36].

As no pressure is applied to the membrane surface in forward osmosis, the fouling potential

was notably low, particularly for organic foulants, and fully reversible by controlling the

hydrodynamic conditions of the cell (backwash and applied shear force) [3], [5], [6], [43]. The

fouling potential was further reduced when operating at higher temperatures [44] and with the

active layer facing the feed (foulant) solution [3], [5], [6].

6

2.2 Forward Osmosis Draw Solutions

As the transport of water in forward osmosis is driven by the osmotic pressure gradient

between the effluent and draw solution, the optimization of the draw solution is critical to the

FO process. The ideal draw solution must produce a sufficient osmotic driving force to

facilitate water transport, be efficiently separated from the recovered water, and have minimal

reverse draw solution flux. Additionally, the draw solute should be soluble for all operating

conditions, inexpensive, and of minimal risk to process operators and the environment. The

draw solutions currently prominent in the literature include inorganic salt solutions (NaCl,

MgCl2, etc.) and thermolytic/switchable salt solutions (TMA-CO2-H2O, NH3-CO2, etc.). These

draw solutions were stated as the most effective due to their high osmotic pressures, low

viscosities, and high diffusivities which increase the driving force for water transport and

mitigate concentration polarization effects [38].

Inorganic salts, predominately NaCl, were used for their high solubility, diffusivity, osmotic

pressure and availability [37], [38]. In particular, the draw solute diffusivity was found to have

an exponential effect on the water flux, highlighting the value of inorganic draw solutions [37],

[38]. Reconcentration of inorganic draw solutions was performed via reverse osmosis [27],

[38] or nanofiltration [39], [40] processes. Compared to monovalent inorganic salts,

multivalent inorganic salts were found to produce a higher water flux with less reverse draw

solute flux at the same draw solute concentration [37].

In contrast to inorganic salts, thermolytic salts were used due their energy-efficient

regeneration method; as they undergo a phase change, from the aqueous to gas phase, with the

application of mild heat (60-70 °C) [36]. This enabled the removal of the draw solute at

temperatures lower than traditional thermal-driven separation processes, as the draw solute is

evaporated and recovered at lower temperatures than the solvent, water. Additionally, the cost

of heat energy is stated to be up to 10x less than that of electrical energy, as used in reverse

osmosis, [45] and many applications can take advantage existing waste heat or geothermal

sources [38]. Thermolytic salts investigated for FO applications include aqueous-carbonated

carbon dioxide (NH3-CO2) [10], [17], [35], [46]–[48] and aqueous carbonated trimethylamine

(TMA-CO2-H2O) [34], [36], [49], [50].

7

2.3 Forward Osmosis Membranes

A typical FO membrane consists of a thin dense active layer, similar to that of a reverse

osmosis membrane, and a porous support. The ideal forward osmosis membrane is one that

maximizes the water flux while minimizing the reverse draw solution flux (draw solution loss).

The ideal active layer was defined as uniform, of minimal thickness, and provide high

selectivity for water by using a hydrophilic polymer [51]. Additionally, the ideal porous

support layer was stated as one that minimizes the resistance to mass transfer so as to mitigate

internal concentration polarization by decreasing the support layer tortuosity and thickness and

increasing the support layer porosity [51]. These properties are combined in the membrane

structural parameter S (the ratio of the support layer thickness, corrected by the tortuosity, to

the support layer porosity), and for RO membranes, which are typically less porous and thicker

than those of FO, the structural parameter was greater than 1000 μm [52], however, in FO

applications, the structural parameter was within the range of 300-500 μm [51]. The first

commercially available FO membranes were made of cellulose triacetate (CTA) [53], however

recently developed thin-film composite membranes (polyamide active layer with a polysulfone

support) displayed greater water flux, solute rejection, and extended pH range (2-12) [54],

[55].

2.4 Previous Forward Osmosis Transport Modeling

The transport of water across the membrane in FO has previously been modelled to provide

estimates of the power generation capacity in pressure-retarded osmosis systems [56]–[58].

However, these early models assumed that the reverse draw solute flux was negligible [56],

[57], taking into account only the ICP, neglecting thereby any external concentration

polarization. Subsequent improvements to FO modeling considered the reverse draw solute

flux and additional forms of concentration polarization [31], [58]–[62]. However, these models

did not take into account the nonideality of concentrated draw solutions [63] and were derived

using a tacit assumption that Van’t Hoff’s law remains valid under all conditions. As a result,

existing FO models miscalculate the osmotic pressure and inaccurately predict systems with

draw solutions of high concentrations [64]. Further improvements excluded the Van’t Hoff

assumption, requiring iterative models to determine the water flux [65], [66] and reverse draw

solute flux [32], [33]. However, these updated models did not consider all forms of external

8

concentration polarization [65], [66], and do not properly take into account the draw solute

diffusivity corrected for the thermodynamic driving force [32], [33]. Additionally, it was

shown that the assumption of constant solution physical properties can cause substantial errors

when modeling the FO process [67].

The permeability values of both the water and the draw solute were estimated using a

methodology under pressurised conditions via reverse osmosis [62], followed by an

osmotically-driven experiment to estimate the membrane structural parameter, S (the ratio of

the support layer thickness, corrected by the tortuosity, to the support layer porosity). The

validity of this approach has been questioned [62], [68], [69], as FO membranes behave

differently under pressurized conditions [70], thus affecting their transport properties. The only

attempt to develop a specific to FO approach by Tiraferri et al. [62], consisted of four FO

experiments at different draw solution concentrations, assumed constant draw solute

diffusivity, assumed Van’t Hoff’s law holds across all draw solute concentrations, and also

assumed negligible external concentration polarization.

2.5 Effects of Temperature on Forward Osmosis

With the emergence of hybrid FO-thermal processes (both stripping and freezing based) the

temperature of the FO process is a critical parameter for process optimization. Generally, it

was found that increasing the draw solution temperature, within the range of 20 to 45 °C,

resulted in an increased water flux [7], [33], [71]–[75]. This was due to the positive correlation

between the water and draw solute diffusivity with temperature and the inverse relationship

between the solution viscosity and temperature [7], [71], [75], which decreased the resistance

to mass transfer across the membrane. Additionally, increased temperatures were found to

reduce the severity of both ICP [7], [73] and ECP, contributing to the increased water flux

observed in the literature. However, conflicting results regarding the relationship of the reverse

draw solute flux with temperature were reported, as both reduced [73] and increased [75]

reverse draw solute flux were observed at elevated system temperatures. The effects of

operating the FO cell under a temperature gradient were also investigated and conflicting

results were presented on whether the draw solution [73] or feed temperature [33] had a greater

affect on the FO performance (water flux).

9

In addition to critically affecting the mass transfer across the membrane, the system

temperature was found to affect the degree of membrane fouling and rejection of feed solutes.

For feed solutions containing trace organic components, increased feed temperatures (up to 50

°C) reduced the amount of organic fouling due to increased mass transfer of the foulants away

from the active layer surface [7]. However, in the presence of a brackish water feed, increased

temperatures (up to 45 °C) were found to increase membrane scaling due to more prominent

and compact crystallization on the membrane surface [74]. Additionally, increasing the system

temperature from 20 to 40 °C was found to increase the rejection of rare earth elements from

a simulated acid mine drainage stream; however, the rejection decreased in the presence of a

temperature gradient (feed at 40 °C and draw solution at 20 °C) [12].

2.6 Effects of Hydrodynamic Conditions on Forward Osmosis

The impact of hydrodynamic conditions on forward osmosis performance has been analysed

in previous studies [31], [65], [76]–[79]; however, limited research has been performed

regarding the interaction effects between the feed and draw solution flow rates and the

hydrodynamic effects for concentrated draw solutions (> 2 mol/L). Through FO modeling, it

was previously determined that increased crossflow rates will have a positive effect on the

water flux and water recovery, due to a reduction in the thickness of the ECP boundary layer

on either side of the membrane [65], [77], [78]. Further, increased draw solution flow rates

were found to reduce the thickness of the (ICP) boundary layer in FO experiments, due to an

increased mass transfer of the draw solution into the pores of the support layer [31], [80].

Additional modeling determined that the exergetic efficiency of the FO process could be

enhanced by 3-21% through its optimization [76]. Experimentally, for a 50 g/L (0.86 mol/L)

NaCl draw solution, it was determined that optimal water flux was obtained at both high draw

solution and feed solution crossflow velocities (110 cm/s). However, high crossflow velocities

also produced a greater reverse draw solute flux, and the optimal specific reverse solute flux

(ratio of reverse draw solute flux to water flux) was determined to be at high draw solution

crossflow velocity and low feed solution velocity [79].

10

3 Development of Improved Forward Osmosis Model

The water flux across the membrane active layer Jw (L/m2/h) for an osmotically-driven process

at steady state, is calculated by Equation (1) [81], [82]:

𝐽𝑤 = 𝐴(∆𝜋) = 𝐴(𝜋𝑑𝑎 − 𝜋𝑓𝑎), (1)

where A is the permeability of water across the membrane (L/m2/h/bar), π is the osmotic

pressure (bar) of the feed or the draw solution at the interface of the active layer solution side

(Figure 1). Subscripts d, f, a refer to the draw side of the system, feed side, and the active layer

interface solution side.

Similarly, the salt flux at steady state, Js (mol/m2/h), is calculated by Equation (2) [81], [82]:

𝐽𝑠 = 𝐵(𝑐𝑠,𝑑𝑎 − 𝑐𝑠,𝑓𝑎), (2)

Where B is the salt membrane permeability (L/m2/h), and cs,fa is the concentration of solutes

(mol/L) at the active layer interface solution/feed side and cs,da at the interface support layer

side (Figure 1). A linear gradient is assumed in the active layer according to the solution

diffusion model, as the active layer is a dense polymer where mass transport occurs only

through the diffusion of the species through spaces in the polymer chains [81], [82]. This is

further justified by the Peclet number, shown in Equation (3) [81], which is less than 0.08,

indicating that diffusion forces dominate, and convection is negligible in the active layer.

𝑃𝑒 =𝐽𝑤

𝐷𝑠𝑡(3)

To consider internal concentration polarization in the porous support boundary layer, the draw

solute flux can be described by the sum of the concentration driven diffusion component, and

the convection component resulting from the water flux in the opposite direction (Peclet

number of 0.06 to 1.4 for the conditions examined). This is represented by Equation (4),

𝐽𝑠 = 𝐷𝑠,𝑑(𝑐𝑠)휀𝑑𝑐𝑠

𝑑𝑧−𝐽𝑤𝑐𝑠, (4)

11

where Ds represents the diffusivity of the draw solute in the draw solution (m2/s). By “solute”

it is meant an average value for both anion and cation in case of a strong electrolyte, corrected

by the thermodynamic driving force [83]. In turn, ε represents the support layer porosity

(dimensionless), and z represents the distance (m) across the support layer from the active layer

to the draw solution. The support layer porosity defines the fraction of the volume of voids

(volume for the solution to flow through), over the total volume of the support layer, and is

included with the tortuosity to detail the actual transport path through the porous support. At

steady state the water and draw solute fluxes across the entire membrane are constant, and

therefore:

𝐽𝑠 = 𝐵(𝑐𝑠,𝑑𝑎 − 𝑐𝑠,𝑓𝑎) = 𝐷𝑠,𝑑𝑎(𝑐𝑠)휀𝑑𝑐𝑠

𝑑𝑧−𝐽𝑤𝑐𝑠. (5)

Rearranging Equation (4) results in:

𝑑𝑐𝑠

𝑑𝑧=

𝐽𝑠 + 𝐽𝑤𝑐𝑠

𝐷𝑠,𝑑(𝑐𝑠)휀. (6)

Which can be solved using the following boundary conditions where the membrane support

layer length is corrected by its tortuosity, τ (dimensionless), the ratio of the actual flow path

length to that of the thickness of the porous support:

However, as the porous support layer tortuosity is unknown, a variable transform is required:

�̂� =𝑧 − 𝑡𝑎

휀, (7)

resulting in:

𝑑𝑐𝑠

𝑑�̂�=


𝐷𝑠,𝑑(𝑐𝑠), (8)

with the following adjusted boundary conditions:

𝑐𝑠 = 𝑐𝑠,𝑑𝑎 at 𝑧 = 𝑡𝑎

𝑐𝑠 = 𝑐𝑠,𝑑𝑝 at 𝑧 = 𝑡𝑠𝜏 + 𝑡𝑎

12

where S is the membrane structural parameter (µm) defined by:

𝑆 =𝑡𝑠𝜏

휀 . (9)

Similarly, a mass balance is performed for the DECP at the draw solution-porous support

boundary layer (Peclet number 0.26 to 3), using the following boundary conditions to obtain

cs,dp.

𝐽𝑠 = 𝐷𝑠,𝑑(𝑐𝑠)𝑑𝑐𝑠


𝑑𝑐𝑠

𝑑𝑧=


𝐷𝑠,𝑑(𝑐𝑠), (11)

where δd is the thickness of the draw-side external boundary layer (m), defined by:

𝛿𝑑 =𝐷𝑠,𝑑𝑏

𝑘𝑑, (12)

and kd is the draw-side mass transfer coefficient obtained from Equation 16. The effect of the

water flux was found to have little effect on the boundary layer thickness as it is 4 orders of

magnitude less than that of the crossflow velocity (5x10-6 m/s vs. 0.021 m/s).

Similarly, the external feed-side boundary layer is represented by the following mass balance

(Peclet number 0.25 to 0.86):

𝑐𝑠 = 𝑐𝑠,𝑑𝑎 at �̂� = 0

𝑐𝑠 = 𝑐𝑠,𝑑𝑝 at �̂� =𝑡𝑠𝜏

휀+ 𝑡𝑎 − 𝑡𝑎 =

𝑡𝑠𝜏

휀= 𝑆

𝑐(𝑧) = 𝑐𝑠,𝑑𝑏 at 𝑧 = 𝑡𝑎 + 𝑡𝑠 + 𝛿𝑑

𝑐(𝑧) = 𝑐𝑠,𝑑𝑝 at 𝑧 = 𝑡𝑎 + 𝑡𝑠

𝑐(𝑧) = 𝑐𝑠,𝑓𝑏 at 𝑧 = 0

𝑐(𝑧) = 𝑐𝑠,𝑓𝑎 at 𝑧 = 𝛿𝑓

13

𝐽𝑠 = 𝐷𝑠,𝑓(𝑐𝑠)𝑑𝑐𝑠


𝑑𝑐𝑠

𝑑𝑧=


𝐷𝑠,𝑓(𝑐𝑠), (14)

where δf is the feed-side external boundary layer thickness (m), defined by:

𝛿𝑓 =𝐷𝑠,𝑓

𝑘𝑓. (15)

The mass transfer coefficients kd and kf (m/s) of a solute in the draw solution or feed stream

respectively, is obtained from the Sherwood number equation,

𝑘𝑑 =𝑆ℎ𝑑𝐷𝑠,𝑑𝑏

𝑑ℎ, (16)

𝑘𝑓 =𝑆ℎ𝑓𝐷𝑠,𝑓𝑏

𝑑ℎ, (17)

where Sh refers to the Sherwood number and dh the hydraulic diameter (m) of the respective

channel through which the fluid is flowing. The Sherwood number is obtained from the

following correlation for laminar flow [78]:

𝑆ℎ = 1.85 (𝑅𝑒𝑆𝑐𝑑ℎ

𝐿)

0.33

, (18)

where Re is the Reynolds number, Sc is the Schmidt number, and L is the characteristic length

(m) of the respective channel. The Reynolds number is determined using the following

equation:

𝑅𝑒 =𝜌𝑣𝑑ℎ

𝜇, (19)

where µ represents the dynamic viscosity (kg/m/s), ρ the density (kg/m3), and v the velocity

(m/s) of the solution respectively. The Reynolds numbers for the conditions used were < 4.3.

Additionally, the Sc Schmidt number is calculated using Equation (20):

14

𝑆𝑐 =𝜇

𝜌𝐷𝑠. (20)

All draw solute and solution physical properties (µ, ρ, D, and π) are determined using

concentration dependent empirical equations obtained from OLI Studio 9.6 simulation data (as

explained in Section 4.4). To improve the accuracy of the model, the empirical correlation for

the draw solute diffusivity, Ds, is substituted into each of the 3 ODE’s presented (Equations 8,

11, and 14) and solved over their respective boundary conditions.

The proposed model for this work is a system of 3 ODE’s, to obtain the active-layer interfacial

draw solute concentrations from the bulk solution conditions, and two additional equations for

the water and draw solute flux across the active layer. The system of equations is as follows:

𝐽𝑤 = 𝐴∆𝜋 = 𝐴(𝜋𝑑𝑎 − 𝜋𝑓𝑎),

𝐽𝑠 = 𝐵∆𝑐𝑠,𝑎 = 𝐵(𝑐𝑠,𝑑𝑎 − 𝑐𝑠,𝑓𝑎),

ICP 𝑑𝑐𝑠

𝑑�̂�=


𝐷𝑠,𝑑(𝑐𝑠),

FECP 𝑑𝑐𝑠

𝑑𝑧=


𝐷𝑠,𝑑(𝑐𝑠),

DECP 𝑑𝑐𝑠

𝑑𝑧=


𝐷𝑠,𝑓(𝑐𝑠),

with additional empirical equations used to determine Ds,i and πi as functions of the solute

concentration in the respective stream i (listed in Section 4.4). This system of equations is

solved using the algorithm presented in Figure 2, with each ODE solved using the ode45

function in MATLAB R2019b, using initial guesses for Js and Jw. The MATLAB codes used

are presented in Appendix C and the initial guess values in Appendix D.

15

Figure 2 - Forward Osmosis Transport Model Algorithm

16

4 Methodology

4.1 Membrane Experimental Method and Materials

The properties of the draw solution chemicals used are shown in Table 1.

Table 1 – Draw Solution Materials

Chemical Name CAS No. Source Purity-Assay(a)

CaCl2.2H2O 10035-04-8 ACROS ORGANICS 99+%

NaCl Certified ACS

Crystalline 7647-14-5 Fisher Sci. 99.6%

MgCl2 (anhydrous) 7786-30-3 Afla Aesar 99%

DI water 7732-18-5 Milli-Q Reference 18.2 MΩcm

(a) as stated by the supplier

Figure 3 provides a schematic of the forward osmosis membrane apparatus, featuring a

rectangular stainless steel CF042-FO cell, obtained from Sterlitech (active area of 42 cm2). An

asymmetric cellulose triacetate (CTA) membrane was used, obtained from Fluid Technology

Solutions, with an active layer thickness of 8-18 µm and a porous support thickness of 50 µm

[53], [84]. The membrane was oriented with the active layer facing the feed solution side. The

FO cell was run counter-currently, and the flow rates were controlled by a Thermo-Fisher Easy

Load II peristaltic pump.

Figure 3 - Forward Osmosis Experimental Apparatus

17

A summary of the draw solution concentrations used for the model parameterization and

validation is shown in Table 2. The intrinsic membrane parameters were obtained using draw

solute concentrations of 1, 2, 3, and 3.5 mol/L. The subsequent model validation was

performed using additional experimental data including two concentrations outside of the

fitting range (i.e. 0.5, 2.5, and 4 mol/L). The model parameterization and validation data are

listed in Appendix A.

Table 2 - Model Parameterization and Validation Experiments

Type of

Experiment

NaCl Draw Solution

Concentrations

(mol/L)

MgCl2 Draw Solution

Concentrations

(mol/L)

CaCl2 Draw

Solution

Concentrations

(mol/L)

Model

Parameterization 1, 2, 3, 3.5 1, 2, 3, 3.5 1, 2, 3, 3.5

Model

Validation 0.5, 2.5, 4 0.5, 2.5, 4 0.5, 2.5, 4

For each experiment DI water was used as the feed stream. The draw solutions were prepared

by dissolving the appropriate amount of the draw solute in DI water. The draw and feed

solutions were recirculated each in closed loops using a solution reservoir of 2 L to ensure an

approximately constant solution concentration during the experiment duration. A crossflow

velocity of 2.1 cm/s was used for both the feed and draw solutions. Every 30 sec the mass of

each reservoir was recorded digitally by Mettler-Toledo balances (NewClassic MF MS4002S).

Samples of both the feed and draw solutions were taken initially at steady state (achieved

within 15 min as explained in Section 4.1), and again at the conclusion of the experiment (after

1 h). Metal ion concentrations in the feed and draw solutions were measured using inductively

coupled plasma optical emission spectrometry (Agilent 700 series ICP-OES). Prior to analysis,

each sample was diluted with 5% nitric acid: 50x for the feed samples, and 1000x for the draw

solution samples. The water flux was calculated based on the change of the draw solution mass,

while the reverse draw solute flux was determined based on the change in the solute

concentration in the feed. The experiments were carried out in triplicates to ensure the

reproducibility of the results.

18

4.2 Low Temperature Experimental Methodology

For the low temperature experiments, the same FO cell and membrane were used as in Section

4.1. Experiments were performed at temperatures of 5, 15, and 25 °C and the temperature was

controlled by placing the FO cell into a Thermo Scientific Forma FRGL1204A laboratory

refrigerator. For the 5 °C test, the entire FO set-up was placed in the refrigerator. For the 15

°C test, the draw and feed solutions were placed outside of the refrigerator, while the cell was

placed inside. The 25 °C tests were performed outside of the refrigerator.

Figure 4 - Low Temperature FO Apparatus

Draw solutions of 1 and 3 mol/kg H2O NaCl, CaCl2, and MgCl2 were used with DI water as

the feed solution. The draw solutions were prepared by dissolving the appropriate amount of

the salt in DI water. The draw and feed solutions were recirculated in a closed loop, and both

the draw and feed solution reservoirs were of a large enough volume to ensure an

approximately constant concentration for the experiment duration. Similar to the methodology

in Section 4.1, a mass flow rate of 8,000 g/h (2.1 cm/s crossflow velocity) was used for both

streams. The mass of each reservoir was digitally recorded every 30 s using Mettler-Toledo

balances (NewClassic MF MS4002S), with draw and feed solution samples taken at steady

state (within 15 min), and again after the duration of the experiment (30 min). The low

temperature experimental data is listed in Appendix B.

19

4.3 Hydrodynamic Condition Analysis Methodology

To analyze the impact of hydrodynamic conditions on forward osmosis performance,

simulations were performed using the model presented in this work in Section 3.0, varying

both the draw and feed solution crossflow rates. The impacts of varying crossflow velocities

on the water flux and reverse draw solute flux were examined using both 1 and 3 mol/L MgCl2

and NaCl draw solutions with a feed solution of 0.5 mol/L NaCl. The feed and draw solution

crossflow velocities were varied between 1 to 10 cm/s at a temperature of 25 °C.

4.4 Draw Solution Physical Properties

The physical properties of each draw solution (osmotic pressure, diffusivity of the draw solute

in the draw solution, density, and dynamic viscosity) were determined using the Mixed-

Solvent Electrolyte (MSE) model in OLI Studio 9.6. For ease of calculation, empirical

polynomial equations were fitted to the OLI data as a function of the draw solute concentration

(mol/L) for each physical property dataset. All data was fitted with an R2 > 0.98 for draw solute

concentrations of 0 mol/kg to their respective solubility limits at 25 °C and 1 atm. The

parameters and equations for osmotic pressure, diffusivity, density, and dynamic viscosity are

detailed below in Equations 21-24 and Table 3 to Table 6 respectively. The coefficients of the

polynomials are represented by Xn,p, where the subscripts n and p refer to the number of the

coefficient and the corresponding physical property respectively.

𝜋 = 𝑋1,𝜋𝑐𝑠2 + 𝑋2,𝜋𝑐𝑠 (21)

Table 3 – Draw Solution Osmotic Pressure Empirical Parameters

Draw Solution X1,π X2,π

NaCl 6.4248 39.258

CaCl2 52.546 13.013

MgCl2 60.996 6.7275

𝐷 = 𝑋1,𝐷𝑐𝑠2 + 𝑋2,𝐷𝑐𝑠 + 𝑋3,𝐷 (22)

20

Table 4 - Draw Solute Diffusivity Empirical Parameters

Draw Solution X1,D X2,D X3,D

NaCl -6.92E-12 -9.95E-11 1.51E09

CaCl2 1.50E-11 -3.13E-10 1.34E-09

MgCl2 3.14E-11 -3.82E-10 1.24E-09

𝜌 = 𝑋1,𝜌𝑐𝑠 + 𝑋2,𝜌 (23)

Table 5 - Draw Solution Density Empirical Parameters

Draw Solution X1,ρ X2,ρ

NaCl 0.0392 1

CaCl2 0.0796 1

MgCl2 0.0685 1

𝜇 = 𝑋1,𝜇𝑐𝑠2 + 𝑋2,𝜇𝑐𝑠 + 𝑋3,𝜇 (24)

Table 6 – Draw Solution Dynamic Viscosity Empirical Parameters

Draw Solution X1,µ X2,µ X3,µ

NaCl 0.0173 0.0640 0.8907

CaCl2 0.3136 -0.3430 0.8907

MgCl2 0.4151 -0.3182 0.8907

21

5 Results and Discussion

5.1 Determination of Steady State

Prior to the sampling of the feed and draw solutions, steady state across the membrane had to

be achieved. This was determined by plotting the change in the mass flow rate of the draw

solution against time, until a constant mass flow rate was observed (±5%), indicating a constant

water flux. As seen in Figure 5, steady state was typically achieved within 5 min, however the

sampling was performed after 15 min to ensure steady state in all experiments.

Figure 5 - Steady State Draw Solution Flow Rate Profile

(2 M NaCl Draw Solution and DI Water Feed at 25 °C)

5.2 Experimental FO Transport Results

Comparing the experimental water flux results for each draw solute in Figure 6, it is observed

that using a MgCl2 draw solution provides the greatest water flux, closely followed by CaCl2.

This is expected due to the greater osmotic pressures exhibited by divalent electrolytes when

compared to the monovalent NaCl.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

Wat

er F

lux

(L/m

2/h

)

Time (min)

22

Figure 6 - Experimental Water Flux Results

Contrary to the water flux, the NaCl draw solution produces the highest reverse draw solute

flux (Figure 7), followed by CaCl2 and MgCl2. This trend follows the established draw solute

permeability trends in Section 5.3, which are explained by the hydrated ion size and hydration

enthalpy. It appears that MgCl2 produces the highest water flux with the lowest reverse draw

solute flux.

Figure 7 - Experimental Reverse Draw Solute Flux

5

7

9

11

13

15

17

19

21

0 1 2 3 4 5

Wat

er F

Lux

(L/m

2/h

)

Bulk Draw Solute Concentration (mol/L)

MgCl₂CaCl₂NaCl

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5

Rev

erse

Dra

w S

olu

te F

lux

(mo

l/m

2 /h

)


NaClCaCl₂MgCl₂

23

To ensure the quality of the experimental data, the coefficient of variation (CV) for the ratio

of the water flux to the reverse draw solute flux, Jw/Js, is determined. The ratio of the two fluxes

is only dependent on the selectivity of the active layer and is an indicator of the quality of the

experimental FO data. The CV is defined as the standard deviation of the experimental ratio

of Jw/Js divided by the arithmetic mean. The closer the value of CV to zero, the better the fit

of data, with a CV < 10% recommended [62]. The coefficient of variation values for NaCl,

CaCl2, and MgCl2 are 2.8%, 1.5%, and 3.9% respectively. These are far below the

recommended 10%, justifying the integrity of the presented experimental results.

5.3 Membrane Parameterization

Building upon the method of Tiraferri [62], 4 FO experiments were performed each at different

draw solute concentrations, and from these experiments 8 data points were obtained (4 water

flux and 4 reverse draw solute flux corresponding measurements). These 8 data points were

then used to regress the 3 intrinsic membrane parameters; water permeability (A), draw solute

permeability (B), and membrane structural parameter (S) by a least-squares minimization of

the global error (E):

𝐸 = 𝐸𝐽𝑤+ 𝐸𝐽𝑠

= ∑ (𝐽𝑤,𝑖

𝐸𝑥𝑝− 𝐽𝑤,𝑖

𝐶𝑎𝑙𝑐

𝐽 ̅𝑤,𝑖𝐸𝑥𝑝,𝑛 )

2

+ ∑ (𝐽𝑠,𝑖

𝐸𝑥𝑝− 𝐽𝑠,𝑖

𝐶𝑎𝑙𝑐

𝐽 ̅𝑠,𝑖𝐸𝑥𝑝,𝑛 )

2

,

𝑛

𝑖=1

𝑛

𝑖=1

(25)

where n refers to the number of FO experiments performed and superscripts Exp and Calc refer

to calculated and experimental values respectively. The error for both Jw and Js are scaled by

the average experimental value of Jw and Js respectively, to avoid a biased global error. The

calculated values were obtained using the model and algorithm presented in Chapter 3.

However, unlike the method in Tiraferri [62], this method does not assume constant Ds, ideal

osmotic pressure, and negligible external concentration polarization. The initial guesses for A,

B, and S were 0.5 L/m2/h/bar, 0.4 L/m2/h, and 200 µm respectively. To justify the wide-spread

applicability of this model and membrane characterization approach, intrinsic parameter

regressions were performed for each draw solute. To ensure the permeability values obtained

for the CTA membrane are within a realistic range, the results of the minimization were

constrained by the range of those observed in literature (0.44 to 1.34 for A, 0.24 to 1.36 for B

respectively) [33], [62], [85]. The values of the structural parameter S were not constrained as

24

the porous support layer structure may have changed with different membrane producers.

Table 7 displays the intrinsic parameters obtained using this method:

Table 7 - Membrane Intrinsic Parameter Regression Results

Draw

Solution

A

(L/m2/h/bar) B (L/m2/h) S (µm) Etotal

NaCl 0.724 0.651 397.9 0.074

CaCl2 1.092 0.376 221.21 0.055

MgCl2 0.739 0.385 215.99 0.047

Average 0.851 ± 0.09 N/A 278.4 ± 48.8 0.059 ± 0.006

Using the proposed model and FO experimental method, consistent water permeability and

structural parameter values were obtained, varying by ±11.5% and ±17.5% respectively

between multiple draw solutes, which is within the ±7.5 to ±19% seen in the literature [62],

[86]. With minor variations between the water permeability and structural parameters obtained

using different draw solutes, it is therefore sufficient to characterize the membrane using only

a single draw solute at a minimum of 4 different concentrations. However, it is recommended

that multiple draw solutes be examined so a more comprehensive characterization is obtained.

It is acknowledged that the water permeability A has been shown to be dependent on the draw

solute concentration [87], however the reported degree to which the water permeability is

concentration dependent is questionable. Based on the definition of water permeability,

Equations (26) and (27) [30], this concentration dependence is minor as the mole fractions of

water, xw,d and xw,f, vary less than 10% up to the point of draw solute saturation. Further, the

sorption coefficient Kw defined in Equation (27), will vary only by up to 35% for less ideal

draw solutions (CaCl2 and MgCl2) at concentrations up to 4 mol/L. This is illustrated by the

water activity data shown in Figure 8 as a function of draw solute concentration (obtained from

OLI Studio 9.6). Additionally, due to mass transfer limitations, the concentration of the draw

solute at the active layer interface is substantially lower, up to 84 %, than that of the bulk, and

outside of the range where a significant change in the water permeability would be observed.

For the range of bulk draw solute concentrations examined, the draw solute concentration at

the active layer interface ranges from 0 to 1 mol/L, at which point there is a < 1% change in

25

the water activity coefficient. It is therefore confidently assumed in this work that the water

permeability obtained is averaged across all concentration values with little error.

𝐴 =𝐷𝑤𝐾𝑤𝑥𝑤,𝑑𝑣𝑤

𝑡𝑎𝑅𝑇, (26)

𝐾𝑤 =𝛾𝑤

𝛾𝑤(𝑎), (27)

Figure 8 - Activity Coefficients of Water in the mole fraction scale for Various Draw

Solutions

The draw solute permeability, B, values in this work follow the established trends based on the

hydrated radius of the draw solute cation [88], [89] in Table 8. Further, the draw solute

permeability can also be correlated with the hydration enthalpy of the cation [90]–[92], which

represents the energy barrier for the cation to become fully or partially dehydrated [93]. It

stands that a more strongly hydrated draw solution cation (more negative hydration enthalpy)

will less favourably transition from an aqueous phase to the membrane phase, inhibiting its

transport through the membrane according to the solution-diffusion model [30]. Figures 9 and

10 illustrate the correlation between the draw solute permeability and the hydrated cation

radius and cation hydration enthalpy respectively.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Wat

er A

ctiv

ity

Co

effi

cien

t (m

ol-

frac

b

ased

)

Draw Solute Concentration (mol/L)

NaCl

MgCl₂

CaCl₂

26

Table 8 - Hydrated Radii and Hydration Enthalpy of Draw Solute Cations

Ion B (L/m2/h)

Hydrated Radius

[94]

(Å)

Hydration Enthalpy [95]

(kJ/mol)

Na+ 0.651 3.58 -416

Ca2+ 0.376 4.12 -1602

Mg2+ 0.385 4.28 -1949

Figure 9 - Draw Solute Permeability Correlated with the Hydrated Cation Radius

Figure 10 - Draw Solute Permeability Correlated with the Absolute Cation Hydration

Enthalpy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3 3.5 4 4.5

Dra

w S

olu

te P

erm

eab

ility

(L

/m2 /

h)

Hydrated Ion Radius (Å)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 500 1000 1500 2000 2500

Dra

w S

olu

te P

erm

eab

ility

(L

/m2 /

h)

Absolute Hydration Enthalpy (kJ/mol)

27

5.4 Modelling Results and Validation

While the benefits of using a solely FO-based membrane characterization approach over the

traditional FO-RO method have been demonstrated by Tiraferri et al. [62], improvements are

needed to provide robustness to this approach when applied to concentrated draw solutions

that deviate from thermodynamic ideality. To demonstrate the improvements of the model and

characterization approach of our work, the experimental water and reverse draw solute flux

were compared to calculated values from 4 models: that of this work, of Chowdhury &

McCutcheon [33], of Tiraferri et al. [62], and of Bui et al. [86]. Our model and that of

Chowdhury & McCutcheon [33] do not have an explicit solution, but the latter two models do.

The water and reverse draw solute flux models of Tiraferri et al. [62] are presented in Equations

28 and 29,

𝐽𝑊 = 𝐴 {

𝜋𝑑𝑏exp [−𝐽𝑤 (𝑆

𝐷𝑠)] − 𝜋𝑓𝑏exp (

𝐽𝑤𝑘𝑓

)

1 +𝐵𝐽𝑤

{exp (𝐽𝑤𝑘𝑓

) − exp [−𝐽𝑤 (𝑆

𝐷𝑠)]}

} , (28)

𝐽𝑠 = 𝐵 {

𝑐𝑠,𝑑𝑏exp [−𝐽𝑤 (𝑆

𝐷𝑠)] − 𝑐𝑠,𝑓𝑏exp (

𝐽𝑤𝑘𝑓

)

1 +𝐵𝐽𝑤


) − exp [−𝐽𝑤 (𝑆

𝐷𝑠)]}

} . (29)

The model presented in Bui et al. [86], improves on that of Tiraferri et al. [62] by taking into

account the external concentration polarization on the draw solution side,

𝐽𝑊 = 𝐴 {

𝜋𝑑𝑏exp [−𝐽𝑤 (1

𝑘𝑑+

𝑆𝐷𝑠

)] − 𝜋𝑓𝑏exp (𝐽𝑤𝑘𝑓

)

1 +𝐵𝐽𝑤


) − exp [−𝐽𝑤 (1

𝑘𝑑+

𝑆𝐷𝑠

)]}} , (30)

𝐽𝑠 = 𝐵 {

𝑐𝑠,𝑑𝑏exp [−𝐽𝑤 (1

𝑘𝑑+

𝑆𝐷𝑠

)] − 𝑐𝑠,𝑓𝑏exp (𝐽𝑤𝑘𝑓

)

1 +𝐵𝐽𝑤


) − exp [−𝐽𝑤 (1

𝑘𝑑+

𝑆𝐷𝑠

)]}} . (31)

The intrinsic membrane parameters, A, B, and S for the latter two models were determined

using the method described in Tiraferri et al. [62], with all the corresponding assumptions and

the intrinsic parameters for the Chowdhury & McCutcheon [33] model were determined using

28

the method of this work. The intrinsic membrane parameters used for the model presented in

our work are listed in Table 7. The intrinsic membrane parameters obtained were then used

with their respective models to predict the water flux and reverse draw solute flux for NaCl,

MgCl2, and CaCl2 draw solutions. To properly address the non-ideal FO system, it was

assumed that the diffusivity of the draw solute is a function of the draw solute concentration

and that all forms of concentration polarization were present for all simulated results. All

subsequent R2 values were calculated using the experimental data not used in the membrane

parameterization (model validation data), as represented in Table 2.

Figures 11 and 12 display the simulated water flux and reverse draw solute flux results for a

NaCl draw solute. While the results produced using the model of Bui et al. [86] show high

agreement with the experimental water flux values (R2 = 0.94), they diverge substantially from

the experimental reverse draw solute flux at draw solute concentrations greater than 2 mol/L.

However, our model is in excellent agreement with the experimental water flux and the reverse

draw solute flux data (R2 of 0.99 for both Jw and Js respectively), demonstrating a better fit

across a wider range of NaCl concentrations as draw solution. In Figure 13-16, the models of

Tiraferri et al. [62] and Bui et al. [86] diverge even further from the experimental data for the

divalent draw solutes, MgCl2 and CaCl2. This is likely due to their assumptions that the draw

solute diffusivity is constant, that Van’t Hoff’s Law validity holds at high draw solute

concentrations, and that the external concentration polarization is negligible. By removing

these assumptions in our work, the water and reverse draw solute flux predictions were

improved by 27-107% and 24-87% respectively over that of Tiraferri et al. [62] and Bui et al.

[86]. The largest improvement was observed for the MgCl2 draw solute. The coefficients of

determination (R2) for the water flux and reverse draw solute flux, using this model, are 0.97,

0.99 for MgCl2 and 0.95, and 0.98 for CaCl2 respectively.

29

Figure 11 - Water Flux Model Comparison – NaCl Draw Solution

Figure 12 - Reverse Draw Solute Flux Model Comparison - NaCl Draw Solution

0

5

10

15

20

25

30

0 1 2 3 4 5

Wat

er F

lux

(L/m

2 /h

)


This Work

Chowdhury &McCutcheon 2018

Bui et al. 2015

Tiraferri et al. 2013

Experiment

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5

Rev

erse

Dra

w s

olu

te F

lux

(mo

l/m

2 /h

)


This Work

Chowdhury & McCutcheon 2018

Bui et al. 2015


Experiment

30

Figure 13 - Water Flux Model Comparison - MgCl2 Draw Solution

Figure 14 - Reverse Draw Solute Flux Model Comparison - MgCl2 Draw Solution

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5

Wat

er F

lux

(L/m

2 /h

)


This Work


Bui et al. 2015


Experiment

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5

Wat

er F

lux

(L/m

2/h

)


This Work


Bui et al. 2015


Experiment

31

Figure 15 - Water Flux Model Comparison – CaCl2 Draw Solution

Figure 16 - Reverse Draw Solute Flux Model Comparison - CaCl2 Draw Solution

Additionally, a comparison of the two iterative models shows that our model improves upon

that of Chowdhury & McCutcheon [33] by considering the concentration dependence of the

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5

Wat

er F

lux

(L/m

2 /h

)


This Work

Chowdhury &McCutcheon 2018Bui et al. 2015


Experiment

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5

Rev

erse

Dra

w s

olu

te F

lux

(mo

l/m

2 /h

)


This Work


Bui et al. 2015


Experiment

32

draw solute diffusivity, corrected for the thermodynamic driving force, within each boundary

layer.

For the reverse draw solute flux, both models have high agreement with the experimental

results, with R2 values of 0.99, 0.97, and 0.93 for NaCl, MgCl2, and CaCl2 respectively for the

work of Chowdhury & McCutcheon [33], compared to R2 values of 0.99, 0.99, and 0.98 for

NaCl, MgCl2, and CaCl2 respectively for this work. However, Chowdhury & McCutcheon [33]

deviates by 20 % from the experimental results of CaCl2 at high concentrations (4 M), and our

model provides better agreement with experimental measurements for the water flux of all

draw solutions. This indicates that the model presented in this work is better able to take into

account the concentration polarization and provide accurate estimates of both the water and

reverse draw solute flux for multiple draw solutions.

Table 9 - Summary of Non-Ideal FO Transport Model Comparison

This Work

Chowdhury &

McCutcheon [33]

Draw Solution

Species R2

Jw R2Js R2

Jw R2Js

NaCl 0.99 0.99 0.96 0.99

MgCl2 0.97 0.99 0.87 0.97

CaCl2 0.95 0.98 0.88 0.93

5.5 Effect of Low Temperatures on FO Performance

5.5.1 Experimental Water Flux Results at Low Temperatures

As previously described in Section 4.1, the water flux was determined from the rate of change

of the draw solution mass per unit of membrane area, expressed in units of L/m2/h. The effect

of temperature on the experimental water flux, using NaCl, CaCl2, and MgCl2 draw solutions

at concentrations of 1 and 3 molal, is presented in Figure 17 and Figure 18, respectively.

33


Solutions

The water flux is observed to decrease by 31.6 to 46.8% from 5 to 25 °C and is exacerbated

for more concentrated draw solutions. This is due to the decreased draw solute diffusivity and

the increased draw solution viscosity at lower temperatures, which hinders the transport of the

draw solute to the active layer of the membrane, thus reducing the osmotic driving force for

water transport. Of the three draw solutions examined, NaCl displays the largest reduction at

both 1 and 3 molal concentrations due to a greater decrease in the NaCl diffusivity with

decreasing temperature when compared to MgCl2 and CaCl2 solutions. With the exception of

1 molal CaCl2 at 15 °C, the water flux produced by the MgCl2 draw solution is the highest

across all temperatures and concentrations, corresponding with the room temperature results

observed in Figure 6.

0

2

4

6

8

10

12

0 5 10 15 20 25 30

Wat

er F

lux

(L/m

2 /h

)

System Temperature (°C)

1 Molal NaCl

1 Molal CaCl₂

1 Molal MgCl₂

34


Solutions

5.5.2 Experimental Reverse Draw Solute Flux at Low Temperatures

The reverse draw solute flux was determined from the rate of change of the feed solute

concentration per unit of membrane area, expressed in units of mol/m2/h. Similar to the

previous section, the effect of temperature on the experimental reverse draw solution flux data

using the NaCl, CaCl2, and MgCl2 draw solutions were compared at concentrations of 1 and 3

molal in Figure 19 and Figure 20, respectively.

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30

Wat

er F

lux

(L/m

2 /h

)


3 Molal NaCl

3 Molal CaCl₂

3 Molal MgCl₂

35


Molal Draw Solutions

As observed for the water flux, lower system temperatures reduced the reverse draw solute

flux by 35.3 to 64.5%, due to increased mass transfer resistance for the draw solute. Comparing

each draw solute at all temperatures studied, MgCl2 and NaCl display the lowest and highest

reverse draw solute flux, respectively, agreeing with the results in Figure 7. Notably, the NaCl

draw solution undergoes the largest reduction in reverse draw solute flux with decreasing

temperature, eventually producing similar values to those of the divalent salts at 5°C. As

reverse draw solute flux can influence scaling on the feed-side of the active layer [37], lower

temperatures may reduce the extent of scaling on the active layer; however, the solubility of

all potential scaling species at lower temperatures must also be considered.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25 30

Rev

erse

Dra

w S

olu

te F

lux

(mo

l/m

2/h

)


1 Molal NaCl

1 Molal CaCl₂

1 Molal MgCl₂

36


Molal Draw Solutions

5.5.3 Experimental Specific Water Flux at Low Temperatures

The specific water flux was determined from the ratio of the water flux to that of the reverse

draw solute flux for a given draw solute, expressed in units of L of water/mol draw solute.

Similar to the previous sections, the effect of temperature on the specific water flux data using

the NaCl, CaCl2, and MgCl2 draw solutions were compared at concentrations of 1 and 3 molal

in Figure 21 and Figure 22, respectively.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20 25 30

Rev

erse

DR

aw S

olu

te F

lux

(mo

l/m

2 /h

)


3 Molal NaCl

3 Molal CaCl₂

3 Molal MgCl₂

37

Figure 21 - Effect of Temperature on the Specific Water flux using 1 Molal Draw Solutions

For each draw solution, the specific water flux displays an inverse relationship with

temperature, indicating an increase in the selectivity of the membrane for water over the draw

solute at lower temperatures. Therefore, while lower temperatures may reduce the amount of

water recovered per unit of membrane area, increasing the capital expenditure of the process,

less draw solute will be released into the feed stream during operation, reducing operating

expenditures. Comparing each draw solute, NaCl and MgCl2 display the lowest and highest

specific water flux, respectively, with MgCl2 exhibiting the largest increase (51.3%) in specific

water flux with decreasing temperature. Notably there is only a minor increase (5.7%) in the

specific water flux for CaCl2 at lower temperatures, suggesting negligible increases in

selectivity when operating at temperatures below 25 °C. Further, the specific water flux for the

1 molal NaCl draw solution decreases between 15 and 5 °C due to the minor decrease in reverse

draw solute flux over this temperature range. Overall, based on the water and the reverse draw

solute flux results, it is recommended to use a MgCl2 draw solution as it produces a higher

water flux with less draw solution permeation across all temperatures and concentrations

examined.

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30

Spec

ific

Wat

er F

lux

(L w

ater

/mo

l dra

w

solu

te)


1 Molal NaCl1 Molal CaCl₂1 Molal MgCl₂

38

Figure 22 - Effect of Temperature on the Specific Water Flux using 3 Molal Draw Solutions

5.6 Effect of Hydrodynamic Conditions on FO performance

5.6.1 Effect of Crossflow Velocity on FO Water Flux

The effect of the feed and draw solution crossflow velocities on the water flux were simulated

using the improved model presented in Section 3.0. The results using MgCl2 and NaCl draw

solutions at 3 mol/L are presented in Figure 23 and Figure 24, respectively. Crossflow

velocities of 0.01 to 1 m/s and a feed solution of 0.5 mol/L NaCl were assumed.

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30

Spec

ific

Wat

er F

lux

(L w

ater

/ m

ol d

raw

so

luti

on

)


3 Molal NaCl

3 Molal CaCl₂

3 Molal MgCl₂

39

Figure 23 - Effects of Crossflow Velocity on the Water Flux - MgCl2 Draw Solution

Figure 24 - Effects of Crossflow Velocity on the Water Flux - NaCl Draw Solution

Both the feed and draw solution crossflow velocities have a positive effect on the water flux

and the maximum water flux is observed at 1 m/s feed and draw solution crossflow velocities,

agreeing with previous studies [65], [77], [78], [79]. Further, the draw solution crossflow

40

velocity has a greater effect on the water flux, highlighting the importance of reducing the

draw solution side ECP. Overall the water flux is found to increase by 53 to 67% by increasing

both stream crossflow velocities due to the increased turbulence in the streams and decreased

severity of ECP on both sides of the membrane. Comparing both the MgCl2 and NaCl draw

solutions, the increased crossflow velocities have a greater effect on the water flux and ECP

reduction for the MgCl2 draw solution due to its greater viscosity and lower diffusivity.

5.6.2 Effect of Crossflow Velocity on FO Reverse Draw Solute Flux

Additionally, the effect of the feed and draw solution crossflow velocities on the reverse draw

solute flux were simulated using the improved model presented in Section 3.0. The results

using MgCl2 and NaCl draw solutions at 3 mol/L are presented in Figure 25 and Figure 26,

respectively. Crossflow velocities of 0.01 to 1 m/s and a feed solution of 0.5 mol/L NaCl were

assumed.

Figure 25 - Effects of Crossflow Velocity on the Reverse Draw Solute Flux - MgCl2 Draw

Solution

41

Figure 26 - Effects of Crossflow Velocity on the Reverse Draw Solute Flux - NaCl Draw

Solution

Similar to the results in Section 5.6.1, the feed and draw solution crossflow velocities have a

positive effect on the reverse draw solute flux and the maximum flux is observed at 1 m/s feed

and draw solution crossflow velocities, agreeing with previous experimental and simulated

results [65], [77], [78], [79]. However, for the MgCl2 solution the feed solution crossflow

velocity has a greater effect on the reverse draw solute flux, highlighting the need to address

both forms of ECP for process optimization. Overall the reverse draw solute flux is found to

increase by 58 to 181% by increasing both stream crossflow velocities due to the decreased

severity of ECP on both sides of the membrane. Comparing both the MgCl2 and NaCl draw

solutions, the increased crossflow velocities have a greater effect on the reverse draw solute

flux and ECP reduction for the MgCl2 draw solution due to its greater viscosity and lower

diffusivity.

42

6 Conclusion

This work details a rigorous and improved forward osmosis water and reverse draw solute flux

model as well as a corresponding membrane characterization method, by addressing the non-

ideality of the concentrated draw solutions. Based solely on experimental forward osmosis

measurements, the proposed membrane characterization method produces consistent water

permeability and structural parameter estimates (±11.5% and ±17.5%) for three inorganic draw

solutions, NaCl, CaCl2 and MgCl2. Conjunctly, the draw solute permeabilities obtained follow

established theoretical trends, with draw solute permeability increasing with decreasing ionic

radius and increasing hydration enthalpy (less negative). This method improves upon the

existing FO-based membrane characterization method, up to 107% in certain cases, by taking

into account the non-ideal behaviour of draw solutions regarding their osmotic pressure, the

concentration dependence of draw solute diffusivity, as well as all forms of concentration

polarization.

Further, by considering the concentration dependence of the draw solute diffusivity corrected

for the thermodynamic driving force the FO transport model of this work improves upon

existing non-ideal solution transport models up to 18.7%. The proposed FO model and

characterization method allow a more accurate evaluation of draw solutes, membranes, and

FO applications, producing R2 > 0.95 for the water and reverse draw solute flux for three

inorganic draw solutions from 0 to 4 mol/L. As novel applications for FO emerge, it is essential

that a standard and rigorous FO model be implemented to accurately assess the efficacy of FO

for concentrated effluents and draw solutions.

Using the model in this work, it was found that increasing both the feed and draw solution

crossflow velocities reduced the ECP boundary layer, increasing the water and reverse draw

solute flux by 53 to 67% and 58 to 181% respectively. The draw solution crossflow velocity

was found to have a greater impact on both the water and reverse draw solute flux for the

conditions examined. Provided that draw solution losses aren’t at a critical level, the optimal

water flux was obtained at a high draw solution crossflow velocity and high feed crossflow

velocity.

43

When analysing the FO performance at temperatures below 25 °C, it was found that decreasing

the system temperature from 25 to 5 °C decreased both the water and reverse draw solute flux

by 31.6 to 46.8% and 35.3 to 64.5%, respectively. Comparing the specific water flux at various

temperatures, it was determined that the specific water flux increased up to 51.7% with

decreasing temperature. Further the MgCl2 draw solution was identified as producing the

greatest water flux with the lowest reverse draw solute flux for all temperatures and

concentrations examined.

6.1 Recommendations and Future Work

The results presented in this work provide a better understanding of species transport in

forward osmosis and the effects of critical process parameters yet additional studies may be

pursued. The model and characterization method in this work should be expanded to take into

account different operating temperatures and the membrane permeability changes with

temperature should be quantified using the provided characterization method. Additionally,

interactive effects between the temperature and hydrodynamic conditions should be

investigated, as the temperature dependence of physical properties (solute diffusivity and

solution viscosity) must be considered.

The current model in this work neglects the effects of convection caused by the diffusion of

the draw solute in a concentrated solution, i.e. density greater than that of a dilute solution (1

g/mL). This additional convective component is likely to occur in the external boundary layers,

particularly on the draw solution side of the membrane, and must be accounted for in future

model development.

Further, the model in this work should be applied to estimate the potential of forward osmosis

to recover water from real effluents and verified experimentally for said applications. The

characterization method in this work should also be used with different types of forward

osmosis membranes, i.e. aquaporin and thin-film composites.

44

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52

Appendix A: Data for Membrane Parameterization and

Model Validation Experiments

The experimental data for the transport modeling experiments are listed in Table 10 to Table

12 for NaCl, MgCl2, CaCl2 draw solutes respectively.

Table 10 - Experimental FO Modeling Data - NaCl Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/L)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

NaCl-1 0.37 5.62 0.135 9.263 25.3

NaCl-2 0.36 6.60 0.143 10.649 25.6

NaCl-3 0.35 6.96 0.118 6.457 25.6

NaCl-4 1.20 10.32 0.229 4.813 24.4

NaCl-5 1.17 10.79 0.225 7.488 25.1

NaCl-6 1.21 9.23 0.171 21.508 25.3

NaCl-7 1.84 12.74 0.224 0.042 26.4

NaCl-8 1.94 11.44 0.238 7.78 25.6

NaCl-9 1.91 11.47 0.269 3.44 25.3

NaCl-10 2.74 14.61 0.282 7.36 25.5

NaCl-11 2.63 12.88 0.285 1.88 25.5

NaCl-12 2.56 14.14 0.271 9.55 25.5

NaCl-13 3.15 14.90 0.295 5.79 25.6

NaCl-14 3.24 15.27 0.340 3.87 25.4

53

NaCl-15 2.81 13.37 0.320 5.17 24.9

NaCl-16 3.80 16.13 0.363 5.43 21.1

NaCl-17 3.60 15.24 0.449 8.63 21.1

NaCl-18 3.63 17.30 0.318 7.34 25.3

NaCl-19 4.17 16.14 0.383 4.77 25.3

NaCl-20 3.96 17.26 0.326 4.90 24.7

NaCl-21 4.06 15.50 0.372 10.34 24.5

Table 11 - Experimental FO Modeling Data – CaCl2 Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/L)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

CaCl2-1 0.49 7.86 0.083 1.93 22.3

CaCl2-2 0.47 8.06 0.066 11.00 25.1

CaCl2-3 0.44 7.76 0.067 5.26 25.2

CaCl2-4 0.96 9.34 0.036 10.97 24.0

CaCl2-5 0.87 9.32 0.159 14.42 24.0

CaCl2-6 0.80 10.01 0.164 2.46 25.3

CaCl2-7 1.85 15.32 0.163 22.64 24.5

CaCl2-8 1.90 13.01 0.155 6.14 24.6

CaCl2-9 1.69 15.19 0.169 0.24 25.1

CaCl2-10 2.36 14.99 0.171 15.01 24.6

54

CaCl2-11 1.91 14.62 0.205 21.01 25.4

CaCl2-12 2.12 12.97 0.185 5.35 24.9

CaCl2-13 3.17 16.75 0.228 3.30 25.5

CaCl2-14 3.26 14.92 0.207 2.96 24.7

CaCl2-15 3.28 15.73 0.185 20.56 24.5

CaCl2-16 3.42 17.82 0.183 3.57 25.3

CaCl2-17 3.18 16.56 0.195 22.16 21.1

CaCl2-18 3.44 16.28 0.248 2.35 22.3

CaCl2-19 4.65 17.35 0.245 0.94 26.4

CaCl2-20 3.93 16.29 0.251 9.60 25.6

CaCl2-21 3.82 16.76 0.202 2.30 24.6

Table 12 - Experimental FO Modeling Data – MgCl2 Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/L)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

MgCl2-1 0.67 8.80 0.092 6.89 24.2

MgCl2-2 0.50 8.83 0.081 6.27 25.8

MgCl2-3 0.47 8.07 0.081 6.65 25.9

MgCl2-4 0.91 9.53 0.086 1.87 24.7

MgCl2-5 0.86 9.93 0.118 1.03 24.0

MgCl2-6 0.81 9.60 0.075 2.18 24.6

55

MgCl2-7 1.94 15.23 0.160 4.69 25.2

MgCl2-8 1.66 14.21 0.146 3.66 24.6

MgCl2-9 1.76 12.18 0.112 0.36 23.4

MgCl2-10 2.86 16.29 0.145 4.80 25.5

MgCl2-11 2.58 13.64 0.165 9.75 24.7

MgCl2-12 2.50 15.15 0.142 6.60 24.5

MgCl2-13 2.78 15.30 0.162 0.19 24.6

MgCl2-14 2.92 14.86 0.159 9.90 25.5

MgCl2-15 3.37 17.58 0.167 2.54 25.3

MgCl2-16 3.77 17.44 0.198 17.44 23.9

MgCl2-17 3.44 16.20 0.174 16.20 23.9

MgCl2-18 3.51 16.98 0.171 16.98 24.2

MgCl2-19 4.31 17.19 0.183 0.02 26.4

MgCl2-20 4.10 19.42 0.188 7.00 25.6

MgCl2-21 4.08 17.09 0.192 4.76 25.4

56

Appendix B: Data for Low Temperature Experiments

Table 13 – Low Temperature Experimental Data – NaCl Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/kg H2O)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

NaCl-1 1.0 5.76 0.123 5.76 5.1

NaCl-2 1.0 5.14 0.100 5.14 5.0

NaCl-3 1.0 5.95 0.064 5.95 4.8

NaCl-4 1.0 6.63 0.090 6.63 14.9

NaCl-5 1.0 6.42 0.100 6.42 15.1

NaCl-6 1.0 8.04 0.120 8.04 16.0

NaCl-7 3.0 8.80 0.131 8.80 4.7

NaCl-8 3.0 7.10 0.104 7.10 4.7

NaCl-9 3.0 7.28 0.104 7.28 5.0

NaCl-10 3.0 9.55 0.194 9.55 16.0

NaCl-11 3.0 9.29 0.204 9.29 15.0

NaCl-12 3.0 8.81 0.173 8.81 15.7

57

Table 14 – Low Temperature Experimental Data – CaCl2 Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/kg H2O)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

CaCl2-1 1.0 6.571 0.041 15.22 4.8

CaCl2-2 1.0 6.231 0.108 13.98 6.2

CaCl2-3 1.0 6.805 0.085 12.02 5.6

CaCl2-4 1.0 8.290 0.137 17.89 13.5

CaCl2-5 1.0 6.910 0.058 26.37 14.6

CaCl2-6 1.0 8.495 0.094 17.83 15.4

CaCl2-7 3.0 8.571 0.096 18.71 5.0

CaCl2-8 3.0 8.690 0.125 38.56 5.1

CaCl2-9 3.0 8.405 0.104 25.25 5.0

CaCl2-10 3.0 11.486 0.140 3.23 14.7

CaCl2-11 3.0 11.667 0.150 2.62 14.9

CaCl2-12 3.0 11.095 0.165 3.48 15.3

58

Table 15 – Low Temperature Experimental Data – MgCl2 Draw Solute

Experiment

Number

Draw Solute

Concentration

(mol/kg H2O)

Water Flux

(L/m2/h)

Reverse Draw

Solute Flux

(mol/m2/h)

Mass

Balance

Error (%)

Temperature

(°C)

MgCl2-1 1.0 6.68 0.025 5.82 5.5

MgCl2-2 1.0 6.48 0.051 1.46 5.3

MgCl2-3 1.0 6.34 0.043 17.84 4.8

MgCl2-4 1.0 7.00 0.057 57.89 16.1

MgCl2-5 1.0 7.77 0.050 20.70 15.2

MgCl2-6 1.0 8.50 0.055 4.76 15.3

MgCl2-7 3.0 10.29 0.061 2.72 6.8

MgCl2-8 3.0 10.53 0.068 21.66 5.4

MgCl2-9 3.0 10.73 0.069 11.15 5.0

MgCl2-10 3.0 13.51 0.118 9.92 14.8

MgCl2-11 3.0 12.08 0.081 25.11 14.5

MgCl2-12 3.0 12.24 0.131 11.93 14.8

59

Appendix C: MATLAB Membrane Parameterization

Codes

C1: Intrinsic Parameter Fitting Codes

function [] = Parameter_Regression_Martin_fmin

x=menu('Draw Solution?', 'NaCl', 'MgCl2', 'CaCl2');

A0 = 0.5; %Water Permeability A (L/m2/h/bar)

B0 = 0.4; %Draw Solution Permeability (L/m2/h)

S0 = 240; %Structural Parameter (micron)

S0=S0*1E-6; %Conversion from micron to m

y0=[A0 B0 S0]; %Initial Guess Matrix

options=optimset('PlotFcns',@optimplotfval, 'TolX', 1E-10);

f=@(y)ABS_Fit_V2(y,x);

%[y,fval,exitflag,output]=fminsearch(f,y0, options);

[y,fval]=fmincon(f,y0,[],[],[],[],[0.5,0.24,0],[1.34,1.36,800*1E-6]);

A=y(1)

B=y(2)

S=y(3)*1E6

fval

end

function Etotal = ABS_Fit_V2(y,x)

global Jw Js v_d v_f Para_D Cd Cf

% x=menu('Draw Solution?', 'NaCl', 'MgCl2', 'CaCl2');

A0=y(1);

B0=y(2);

S0=y(3);

if x == 1 %NaCl

Para_D = [-6.921e-12 -9.952e-11 1.509e-09]; %Self-Diffusivity Parameters

NaCl

Para_Pi = [6.4248 39.258]; %Osmotic Pressure Paramaters NaCl

Cd=[1.2 1.9 3.1 3.68]; %(M)

Cf=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw=[10.111 11.886 14.513 16.223]; %(L/m2/h)

Jw=Jw./3600;

Js=[0.208 0.244 0.318 0.341]; %(mol/m^-2/h^-1)

Js=Js./3600;

err_Jw=[0.652 0.872 0.820 0.236];

err_Js=[0.026 0.0189 0.018 0.022];

elseif x == 2 %MgCl2

Para_D = [3.142e-11 -3.155e-10 1.234e-09]; %MgCl2

Para_Pi = [60.996 6.7275]; %MgCl2

60

%MgCl2 Experimental Data for Regression

Cd=[0.86 1.8 3.0 3.6]; %(M)

Cf=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw=[9.687 13.874 15.913 16.871]; %(L/m2/h)

Jw=Jw./3600;

Js=[92 139 162 181]./1000; %(mmol/m^-2/h^-1)

Js=Js./3600;

err_Jw=[0.172 1.269 1.19 0.511];

err_Js=[0.0236 0.0198 0.0034 0.0123];

elseif x==3 %CaCl2

Para_D = [1.9e-11 -3.133e-10 1.337e-09]; %CaCl2

Para_Pi = [52.546 13.013]; %CaCl2

%CaCl2

Cd=[0.89 1.82 3.24 3.53]; %(M)

Cf=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw=[9.558 12.912 15.795 16.880]; %(L/m2/h)

Jw=Jw./3600;

Js=[119 164 203 219]./1000; %(mol/m^-2/h^-1)

Js=Js./3600;

err_Jw=[0.32 1.249 0.749 0.672];

err_Js=[0.0591 0.0051 0.0117 0.0243];

end

v_d=0.022; %m/s

v_f=0.022; %m/s

%Membrane Cell Dimensions

W = 4.57; % Membrane cell width (cm)

H = 0.23; %Membrane cell height (cm)

L = 9.2; % Membrane cell length (cm)

dh= 4*W*H/(W+H)/100; %Hydraulic diameter of membrane channel (m)

tp=50*1e-6; %Support layer thickness (m)

tau_eps=S0/tp; %Ratio of the tortuosity to porosity of the porous support

Jw_avg=mean(Jw);

Js_avg=mean(Js);

for n = 1:length(Cd)

deltad(n)=dh./(1.85.*(dh.*v_d./(Para_D(1).*Cd(n).^2+Para_D(2).*Cd(n)+Para_

D(3)).*dh./(L/100)).^0.33);

deltaf(n)=dh./(1.85.*(dh.*v_f./(Para_D(1).*Cf(n).^2+Para_D(2).*Cf(n)+Para_

D(3)).*dh./(L/100)).^0.33);

zspan_DSECP = [deltad(n) 0];

c0 = Cd(n);

[z,c] = ode45(@(z,c)DSECP(z,c,Jw(n),Js(n),Para_D), zspan_DSECP, c0);

csdp(n)=c(end);

zspan_FECP = [0 deltaf(n)];

61

c0 = Cf(n);

[z,c] = ode45(@(z,c)FECP(z,c,Jw(n),Js(n),Para_D), zspan_FECP, c0);

csfa(n)=c(end);

zspan_ICP = [S0 0];

c0 = csdp(n);

[z,c] = ode45(@(z,c)ICP(z,c,Jw(n),Js(n),Para_D), zspan_ICP, c0);

csda(n)=c(end);

end

Jw_calc=A0.*(Para_Pi(1).*csda.^2+Para_Pi(2).*csda-Para_Pi(1).*csfa.^2-

Para_Pi(2).*csfa).*1.01325./3600;

Js_calc=B0.*(csda-csfa)./3600;

Ew=sum(((Jw-Jw_calc)./Jw_avg).^2);

Es=sum(((Js-Js_calc)./Js_avg).^2);

Etotal=Ew+Es;

csdp

csda

csfa

SSE_w=sum(Jw_calc([1 2 3 4])-Jw([1 2 3 4])).^2;

SSE_s=sum(Js_calc-Js).^2;

SST_w=sum((Jw([1 2 3 4])-mean(Jw)).^2);

SST_s=sum((Js-mean(Js)).^2);

R2_w=1-SSE_w./SST_w

R2_s=1-SSE_s/SST_s

end

function dcdz = DSECP(z,c,Jw,Js,Para_D) % Draw Solution Side ECP

dcdz=(Js+Jw*c)/(Para_D(1)*c^2+Para_D(2)*c+Para_D(3))/1000;

end

function dcdz = FECP(z,c,Jw,Js,Para_D) % Draw Solution Side ECP


end

function dcdz = ICP(z,c,Jw,Js,Para_D) % Draw Solution Side ECP


end

C2: Model Validation Code

function [] = FO_Model_V2


Cd=0.01:0.1:5; %Bulk Draw Solute Concentration (mol/L)

Cf=0.001; %Bulk Feed Concentration (mol/L)

Jw0=10; %Initial Water Flux Guess (L/m2/h)

Js0=0.01; %Initial Reverse Draw Solute Flux Guess (mol/m2/h)

y0=[Jw0 Js0]; %Initial Guess Matrix

62

Jw=zeros(1,length(Cd));

Js=zeros(1,length(Cd));

for n=1:length(Cd)

% options=optimset('PlotFcns',@optimplotfval, 'TolX', 1E-10);

f=@(y)ODE_Model(y,x,Cd(n),Cf);

[y,fval]=fminsearch(f,y0);

% [y,fval]=fmincon(f,y0,[],[],[],[],[0.5,0,0],[1.1,1,800*1E-6]);

Jw(n)=y(1);

Js(n)=y(2);

fval(n)=fval;

end

% Jw

% Js

if x == 1 %NaCl

Cd_exp=[0.36 1.2 1.9 2.6 3.1 3.68 4.1]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[6.390 10.111 11.886 13.88 14.513 16.223 16.637]; %(L/m2/h)

Js_exp=[0.132 0.208 0.244 0.279 0.318 0.341 0.36]; %(mol/m^-2/h^-1)

err_Jw=[0.567 0.319 0.607 0.731 0.82 0.844 0.467];

err_Js=[0.0106 0.0260 0.0189 0.00596 0.0181 0.0219 0.0249];



Cd_exp=[0.6 0.9 1.8 2.6 3 3.6 4.3]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[8.569 9.687 13.874 15.026 15.913 16.871 18.295]; %(L/m2/h)

Js_exp=[0.086 0.092 0.139 0.151 0.162 0.181 0.187]; %(mmol/m^-2/h^-1)

err_Jw=[0.351 0.172 1.269 0.511 1.19 0.954 1.028];

err_Js=[0.0014 0.0236 0.0198 0.0101 0.0034 0.0123 0.0041];

elseif x==3 %CaCl2

%CaCl2

Cd_exp=[0.46 0.89 1.82 2.24 3.24 3.53 4.00]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[7.892 9.558 12.912 14.195 15.795 16.378 16.765]; %(L/m2/h)

Js_exp=[0.072 0.119 0.164 0.187 0.203 0.219 0.225]; %(mol/m^-2/h^-1)

err_Jw=[0.128 0.32 1.249 0.904 0.749 0.129 0.543];

err_Js=[0.0079 0.0591 0.0051 0.014 0.021 0.0183 0.02];

end

Jw

SSE_w=sum(Jw([6 23 41])-Jw_exp([1 4 7])).^2;

SSE_s=sum(Js([6 23 41])-Js_exp([1 4 7])).^2;

SST_w=sum((Jw_exp([1 4 7])-mean(Jw_exp)).^2);

SST_s=sum((Js_exp([1 4 7])-mean(Js_exp)).^2);

R2_w=1-SSE_w./SST_w

R2_s=1-SSE_s/SST_s

% plot(Cd,Jw(:),'k')

% hold on

63

% errorbar(Cd_exp, Jw_exp,err_Jw,'or','Linestyle','none')

plot(Cd,Js(:),'k')

hold on

errorbar(Cd_exp, Js_exp,err_Js,'or','Linestyle','none')

end

function Etotal = ODE_Model(y,x,Cd,Cf)

Jw0=y(1)./3600;

Js0=y(2)./3600;

if x == 1 %NaCl


NaCl

Para_Pi = [6.4248 39.258]; %Osmotic Pressure Paramaters NaCl

Cd_exp=[1.2 1.9 3.1 3.68]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[10.111 11.886 14.513 16.223]; %(L/m2/h)

Jw_exp=Jw_exp./3600;

Js_exp=[0.208 0.244 0.318 0.341]; %(mol/m^-2/h^-1)

Js_exp=Js_exp./3600;

err_Jw=[0.652 0.872 0.820 0.236];

err_Js=[0.026 0.0189 0.018 0.022];

A=0.5899; %L/m2/h/bar

B=0.5332; %L/m2/h

S=340.14; %micro m

S=S*1E-6;


Para_D = [3.142e-11 -3.155e-10 1.234e-09]; %MgCl2

Para_Pi = [60.996 6.7275]; %MgCl2


Cd_exp=[0.86 1.8 3.0 3.6]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[9.687 13.874 15.913 16.871]; %(L/m2/h)


Js_exp=[92 139 162 181]./1000; %(mmol/m^-2/h^-1)


err_Jw=[0.172 1.269 1.19 0.511];

err_Js=[0.0236 0.0198 0.0034 0.0123];

A=1.092; %L/m2/h/bar

B=0.3674; %L/m2/h

S=222.22; %micro m

S=S*1E-6;

elseif x==3 %CaCl2

Para_D = [1.9e-11 -3.133e-10 1.337e-09]; %CaCl2

Para_Pi = [52.546 13.013]; %CaCl2

64

%CaCl2

Cd_exp=[0.89 1.82 3.24 3.53]; %(M)

Cf_exp=[1.26 2.12 0.868 1.04]./1000; %(M)

Jw_exp=[9.558 12.912 15.795 16.880]; %(L/m2/h)


Js_exp=[119 164 203 219]./1000; %(mol/m^-2/h^-1)


err_Jw=[0.32 1.249 0.749 0.672];

err_Js=[0.0591 0.0051 0.0117 0.0243];

A=0.7422; %L/m2/h/bar

B=0.3783; %L/m2/h

S=218.00; %micro m

S=S*1E-6;

end

v_d=0.022; %m/s

v_f=0.022; %m/s







tau_eps=S/tp; %Ratio of the tortuosity to porosity of the porous support

Jw_avg=mean(Jw_exp);

Js_avg=mean(Js_exp);

deltad=dh./(1.85.*(dh.*v_d./(Para_D(1).*Cd.^2+Para_D(2).*Cd+Para_D(3)).*dh

./(L/100)).^0.33);

deltaf=dh./(1.85.*(dh.*v_f./(Para_D(1).*Cf.^2+Para_D(2).*Cf+Para_D(3)).*dh

./(L/100)).^0.33);

zspan_DSECP = [deltad 0];

c0 = Cd;

[z,c] = ode45(@(z,c)DSECP(z,c,Jw0,Js0,Para_D), zspan_DSECP, c0);

csdp=c(end);

zspan_FECP = [0 deltaf];

c0 = Cf;

[z,c] = ode45(@(z,c)FECP(z,c,Jw0,Js0,Para_D), zspan_FECP, c0);

csfa=c(end);

zspan_ICP = [S 0];

c0 = csdp;

[z,c] = ode45(@(z,c)ICP(z,c,Jw0,Js0,Para_D), zspan_ICP, c0);

csda=c(end);

Jw_calc=A.*(Para_Pi(1).*csda.^2+Para_Pi(2).*csda-Para_Pi(1).*csfa.^2-

Para_Pi(2).*csfa).*1.01325./3600;

Js_calc=B.*(csda-csfa)./3600;

65

Ew=sum(((Jw0-Jw_calc)./Jw0).^2);

Es=sum(((Js0-Js_calc)./Js0).^2);

Etotal=Ew+Es;

csdp;

csda;

csfa;

end

function dcdz = DSECP(z,c,Jw0,Js0,Para_D) % Draw Solution Side ECP

dcdz=(Js0+Jw0*c)/(Para_D(1)*c^2+Para_D(2)*c+Para_D(3))/1000;

end

function dcdz = FECP(z,c,Jw0,Js0,Para_D) % Draw Solution Side ECP


end

function dcdz = ICP(z,c,Jw0,Js0,Para_D) % Draw Solution Side ECP


end

C3: Crossflow Velocity Analysis Code

function [] = CFV_V2


tic

v_f = 0.01:0.03:1; %Feed solution velocity (m/s)

v_d = 0.01:0.03:1; %Draw solution velocity (m/s)

Jw0=10; %Initial Water Flux Guess (L/m2/h)

Js0=0.01; %Initial Reverse Draw Solute Flux Guess (mol/m2/h)

y0=[Jw0 Js0]; %Initial Guess Matrix

Jw=zeros(length(v_d),length(v_f));

Js=zeros(length(v_d),length(v_f));

fval=zeros(length(v_d),length(v_f));

for n=1:length(v_d)

for i=1:length(v_f)

% options=optimset('PlotFcns',@optimplotfval, 'TolX', 1E-10);

f=@(y)ODE_Model(y,x,v_d(n),v_f(i));

[y,fval]=fminsearch(f,y0);

% [y,fval]=fmincon(f,y0,[],[],[],[],[0.5,0,0],[1.1,1,800*1E-6]);

Jw(n,i)=y(1);

Js(n,i)=y(2);

fval(n,i)=fval;

end

end

Js

Jw

66

Jw_specific=Jw./Js;

surf(v_d,v_f,Jw_specific')

xlabel('Draw Solution Velocity (m/s)')

yticks([0 0.2 0.4 0.6 0.8 1])

ylabel('Feed Velocity (m/s)')

xticks([0 0.2 0.4 0.6 0.8 1])

zlabel('Specific Water Flux (L/mol)')

%zlim([0 20])

toc

end

function Etotal = ODE_Model(y,x,v_d,v_f)

Jw0=y(1)./3600;

Js0=y(2)./3600;

Cd=3; %Bulk draw solution concentration (mol/L)

Cf=0.5; %Bulk feed concentration (mol/L)

if x == 1 %NaCl


NaCl

Para_Pi = [6.4248 39.258]; %Osmotic Pressure Parameters NaCl

A=0.7239; %L/m2/h/bar

B=0.6509; %L/m2/h

S=397.9; %micro m

S=S*1E-6;


Para_D = [3.142e-11 -3.155e-10 1.234e-09]; %MgCl2

Para_Pi = [60.996 6.7275]; %MgCl2

A=1.092; %L/m2/h/bar

B=0.3758; %L/m2/h

S=221.21; %micro m

S=S*1E-6;

elseif x==3 %CaCl2

Para_D = [1.9e-11 -3.133e-10 1.337e-09]; %CaCl2

Para_Pi = [52.546 13.013]; %CaCl2

A=0.7394; %L/m2/h/bar

B=0.3851; %L/m2/h

S=215.99; %micro m

S=S*1E-6;

end





67



tau_eps=S/tp; %Ratio of the tortuosity to porosity of the porous support

deltad=dh./(1.85.*(dh.*v_d./(Para_D(1).*Cd.^2+Para_D(2).*Cd+Para_D(3)).*dh

./(L/100)).^0.33);

deltaf=dh./(1.85.*(dh.*v_f./(-6.921e-12.*Cf.^2-9.952e-11.*Cf+1.509e-

09).*dh./(L/100)).^0.33);

zspan_DSECP = [deltad 0];

c0 = Cd;

[z,c] = ode45(@(z,c)DSECP(z,c,Jw0,Js0,Para_D), zspan_DSECP, c0);

csdp=c(end);

zspan_FECP = [0 deltaf];

c0 = Cf;

[z,c] = ode45(@(z,c)FECP(z,c,Jw0,Js0), zspan_FECP, c0);

csfa=c(end);

zspan_ICP = [S 0];

c0 = csdp;

[z,c] = ode45(@(z,c)ICP(z,c,Jw0,Js0,Para_D), zspan_ICP, c0);

csda=c(end);

Jw_calc=A.*(Para_Pi(1).*csda.^2+Para_Pi(2).*csda-6.4248.*csfa.^2-

39.258.*csfa).*1.01325./3600;

Js_calc=B.*(csda-csfa)./3600;

Ew=sum(((Jw0-Jw_calc)./Jw0).^2);

Es=sum(((Js0-Js_calc)./Js0).^2);

Etotal=Ew+Es;

end

function dcdz = DSECP(z,c,Jw0,Js0,Para_D) % Draw Solution Side ECP


end

function dcdz = FECP(z,c,Jw0,Js0) % Feed Solution Side ECP

dcdz=(Js0+Jw0*c)/(-6.921e-12*c^2 -9.952e-11*c+1.509e-09)/1000;

end

function dcdz = ICP(z,c,Jw0,Js0,Para_D) % Draw Solution Side ICP


end

68

Appendix D: Modeling Initial Guess Values

The initial guesses for the model presented in Section 3.0 for the variables are presented in

Table 16. When varied by ±50%, the initial model guesses were found to produce constant

water flux and reverse draw solute flux.

Table 16 – Initial Guess Values for Proposed FO Transport Model

Variable Initial Guess

Jw (L/m2/h) 10

Js (mol/m2/h) 0.01

The initial guess values for the intrinsic membrane parameters were 0.5 L/m2/h/bar for the

water permeability, 0.4 L/m2/h for the draw solute permeability, and 200 µm for the

structural parameter. When varied by ±50%, the initial model guesses were found to produce

constant intrinsic parameter estimates (±0.01%).

refined membrane characterization and modeling in forward ... · draw solute flux was found to...

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