Chairman - Virginia Tech

REVERSE OSMOSIS TRANSPORT PHENOMENAIN THE PRESENCE OF STRONG

SOLUTE—MEMBRANE AFFINITY

by

JAMES MORLEY“D[CKSONH

Dissertation submitted to the Faculty of theVirginia Polytechnic institute and State University

in partial fulfillment of the requirements for the degree ofDOCTOR OF PHILOSOPHY

inChemical Engineering

APPROVED: ;_

Chairman/ . .

.R. Lloyd, Supervi. r G.H. Beyer

gfx?./ J.P. Wightman H.M. McNair

April, 1985

Blacksburg, Virginia

YREVERSE OSMOSIS TRANSPORT PHENOMENA

L? IN THE PRESENCE OF STRONGSOLUTE-MEMBRANE AFFINITYé

by

JAMES MORLEY DICKSON ‘

(ABSTRACT) _The reverse osmosis performance of cellulose acetate membranes has been

examined and analyzed for several aqueous systems where there is a strong attractionbetween the organic solute and the membrane material.

The systems investigated included the aromatic hydrocarbons benzene, toluene,and cumene in single-solute aqueous solutions. Six cellulose acetate membranes, modified byannealing at different temperatures, were studied.

Experiments were performed at four pressures (690, 1725, 3450, and 6900 kPa) and' at several concentrations (in the range 5 to 260 ppm). The results were found to be markedlydifferent than those observed in the absence of strong solute-membrane affinity. Inparticular, the solute~water separation decreased rather than increased with increasingpressure and the flux decreased with increasing concentration even though lowconcentrations, with low osmotic pressures, were studied. Qualitatively, the behavior wasexplained in terms ofa porous membrane mechanism with both solute-membrane affinity andsolute mobility varying as a function of solute position with respect to the membrane. Theobserved reduction in flux was expressed by an empirical equation as a function ofconcentration of solute in the boundary layer.

The experimental results were analyzed quantitatively by several transportmodels. The irreversible thermodynamics phenomenological transport, solution—diffusionimperfection and extended solution-diffusion relationships generated parameters that wereinconsistent with the original formulations of the models. The irreversible thermodynamicsKedem—Spiegler model, solution·diffusion model, Kimura—Sourirajan analysis, and the threeparameter finely—porous model were functionally unable to represent the data. Only the fourparameter finely—porous model and the surface force—pore flow model were consistent withexperimental results. From the finely—porous model the partition coefficient was found to bedifferent on the high and low pressure sides of the membrane and this difference was afunction of both pore size and solute. For the surface force-pore flow model, the agreementbetween the model and data was excellent. However, the surface force—pore flow model wasconsiderably more difficult to use.

Dedicated to the memory of my father.

V

TABLE OF CONTENTS

ABSTRACT iiACKNOWLEDGEMENTS ivTABLE OF CONTENTS viLIST OF TABLES viiiLIST OF TABLES IN APPENDICES ixLIST OF FIGURES xiii

Chapter 1 INTRODUCTION 11.1 Historical Background 51.2 Theoretical Background 8

1.2.1 General Considerations 81.2.1.1 Concentration Polarization 91.2.1.2 Definition ofSeparation 111.2.1.3 Osmotic Pressure Effects 121.2.1.4 Membrane Structure 131.2.1.5 Membrane Pore Size 151.2.1.6 Membrane Compaction 16

1.2.2 Phenomenological Transport Models 171.2.2.1 Irreversible Thermodynamics — Phenomenological 17

Transport Relationship1.2.2.2 Irreversible Thermodynamics — Kedem Spiegler 19

Relationship1.2.3 Nonporous Membrane Transport Models 20

1.2.3.1 Solution—Diffusion Relationship 201.2.3.2 Solution·Diffusi0n Imperfection Relationship 221.2.3.3 Extended Solution-Diffusion Relationship 23

1.2.4 Porous Membrane Transport Models 24- 1.2.4.1 Preferential Sorption—Capillary Flow Model 241.2.4.2 Kimura-Sourirajan Analysis 251.2.4.3 Finely—Por0us Model 271.2.4.4 Surface Force—Pore Flow Model 30

1.2.5 Restricted Diffusion in Pores 351.2.5.1 Steric Exclusion 40

1.2.6 Summary and Comparison ofTransport Models 421.2.7 Physicochemical Nature of the Solutes Studied 46

Chapter 2 EXPERIMENTAL 51

2.1 Membrane Fabrication 512.2 Reverse Osmosis Experiments 522.3 Chemical Analysis 552.4 Chemicals 562.5 Equipment 56

vi

Chapter 3 RESULTS AND DISCUSSION 583.1 Membrane and Cell Characterization 593.2 Hydrocarbon—Water-Cellulose Acetate Reverse Osmosis System 613.3 Interpretation of the Hydrocarbon Solute Data 773.4 Membrane Compaction Analysis 823.5 Parameter Estimation Methods 823.6 Phenomenological Transport Models - 843.6.1 Irreversible Thermodynamics — Phenomenological 84Transport Relationship

3.6.1.1 Method A 853.6.1.2 Method B 933.6.1.3 Method C 1023.6.2 Irreversible Thermodynamics — Kedem Spiegler Relationship 1033.6.3 Interpretation of the Results for the Phenomenological 105Transport Models3.7 Nonporous Membrane Transport Models 1063.7.1 Solution—Diffusion Relationship 1063.7.2 Solution—Diffusion Imperfection Relationship 1113.7.3 Extended Solution—Diffusion Relationship 1193.8 Porous Membrane Transport Models 1223.8.1 Kimura-Sourirajan Analysis 1223.8.2 Finely-Porous Model 1233.8.2.1 Three Parameter Finely-Porous Model 1233.8.2.2 Four Parameter Finely-Porous Model 1243.8.3 Surface Force—Pore Flow Model 1393.8.3.1 Pressure Dependence of the Solute Chemical Potential 1403.8.3.2 Friction Function 1433.8.3.3 Steric Exclusion 1443.8.3.4 Flux versus Pore Size for Different Membranes 1443.8.3.5 Numerical Analysis 1453.8.3.6 NaCl-Water Data 1493.8.3.7 Toluene-Water Data 150

CONCLUSIONS 159RECOMMENDATION 162NOMENCLATURE 166APPENDIX A: Derivation of the Solution~Diffusion Imperfection Model 171for Low Osmotic Pressure and Dilute SolutionsAPPEN DIX B: Raw Data for the NaCl-Water~Cellulose Acetate System 173APPENDIX C: Raw Data for the Aromatic Hydrocarbon—Water—Cellulose 191Acetate SystemAPPENDIX D: Integration of the Flux Equation in the z—direction for the 220Surface Force—Pore Flow ModelREFERENCES 223VITA 228

vii

LIST OF TABLES

Table Page1 Physicochemical Parameters for the Solutes Studied 42 Summary of Several Reverse Osmosis Transport Models 433 Comparison of the Form ofSeveral Reverse Osmosis Transport Models 454 Relationship between Polarizability and Molar Attraction Constant 48

for Some Aromatic Hydrocarbon Solutes

5 Characterization and Performance ofthe Cellulose Acetate Membranes 60and ofthe Radial Flow Cells

6 Proportionality Constants from the Correlation of XA; and the Extent 76ofPore Blocking

7 Parameters for the Phenomenological Transport (IT-PT-/\) and 86Extended Solution—Diffusion (ESI)) Models

8 Parameters for the lrreversible Thermodynamics — Phenomenological 95Transport (IT—PT—B) Model

9 Parameters for the lrreversible Thermodynamics — Kedem Spiegler (IT-KS) 104and the Three Parameter Finely-Porous Model (FPM—3) Relationships

10 Parameters for the Solution-Diffusion (SD) and Kimura—Sourirajan 108Analysis (KSA) Relationships

11 Parameters for the Solution—Diffusion lmperfection (SDI) Model 112

12 Evaluation of the Significance of Pressure Induced Solute Transport 121

13 Parameters for the Four Parameter Finely—Porous (FPM—4) Model 126

14 Parameters for the Four Parameter Finely—Porous (FPM—4) Model 134with the Pore Size Parameter (t/6) Forced to be Constant for all Solutes

15 Parameters for the Surface Force-Pore Flow (SF—PF) Model for the 156Solute Toluene

viii

LIST OF TABLES IN APPENDICES

EMG; @22B—l Data for Experiment Number 93 174B-2 Data for Experiment Number 97 175B—3 Data for Experiment Number 98 176B—4 Data for Experiment Number 101 177B—5 Data for Experiment Number 103 178B—6 Data for Experiment Number 109 179B—7 Data for Experiment Number 114 180B—8 Data for Experiment Number 119 181B-9 Data for Experiment Number 122 182B—l0 Data for Experiment Number 127 183B~11 Data for Experiment Number 132 184B—12 Data for Experiment Number 137 185B—13 Data for Experiment Number 141 186B—4 Data for Experiment Number 145 187BÜ15 Data for Experiment Number 149 188B-16 Data for Experiment Number 155 189B—17 Data for Experiment Number 160 190C—1 Summary of the Data Collected for the Solute Benzene 192C-2 Summary of the Data Collected For the Solute Toluene 193C—3 Summary of the Data Collected for the Solute Cumene 194C—4 Data for Experiment Number 88 195

C-5 Data for Experiment Number 89 195C-6 Data for Experiment Number 90 196

ix

EEssC-7 Data for Experiment Number 91A 196C—8 Data for Experiment Number 91B 197C-9 Data for Experiment Number 94 197C-10 Data for Experiment Number 95 198

C·1 1 Data for Experiment Number 96 198C-12 Data for Experiment Number 99 199C-13 Data for Experiment Number 102 199C-14 Data for Experiment Number 104 200

C-15 Data for Experiment Number 105 200C-16 Data for Experiment Number 106 201C-17 Data for Experiment Number 108 201C-18 Data for Experiment Number 110 202

C-19 Data for Experiment Number 111 202

C-20 Data for Experiment Number 112 203E

C-21 Data for Experiment Number 113 203C-22 Data for Experiment Number 115 p 204C-23 Data for Experiment Number 116 204

C—24 Data for Experiment Number 117 205

C-25 Data for Experiment Number 1 18 205


C-27 Data for Experiment Number 121 206C—28 Data for Experiment Number 123 207



C—31 Data for Experiment Number 128 208

C—32 Data for Experiment Number 129 209C—33 Data for Experiment Number 133 209C—34 Data for Experiment Number 134 210C-35 Data for Experiment Number 136 210C—36 Data for Experiment Number 138 211C—37 Data for Experiment Number 139 211C-38 Data for Experiment Number 140 212C-39 Data for Experiment Number 142 212C—40 Data for Experiment Number 143 213C—41 Data For Experiment Number 144 213C—42 Data For Experiment Number 146 214C—43 Data For Experiment Number 147 214C—44 Data for Experiment Number 148 215C—45 Data for Experiment Number 150 215C—46 Data For Experiment Number 151 216C—47 Data for Experiment Number 152 216C—48 Data for Experiment Number 153 217C-49 Data for Experiment Number 154 217C-50 Data for Experiment Number 156 218C—51 Data for Experiment Number 157 218C—52 Data for Experiment Number 158 219C-53 Data for Experiment Number 159 219

xi

LIST OF FIGURES

Figure Page1 Concentration polarization at the high pressure surface ofa reverse osmosis 10membrane.

2 A Sutherland potential function. 343 Drag coefficient as a function of the radial position in a pore.- 374 Restricted diffusion as a function of the ratio ofsolute radius to pore radius. 415 The relationship between polarizability and the molar attraction constant 49

for several aromatic hydrocarbons.

6 A schematic of the reverse osmosis test equipment. 537 A schematic ofa radial flow test cell (cross—section). 548 Effect of feed concentration on the reverse osmosis performance for the 62

benzene—water system at four different pressures.

9 Effect of feed concentration on the reverse osmosis performance for the 64toluene—water system at four different pressures.

10 Effect of feed concentration on the reverse osmosis performance for the 66cumene—water system at four different pressures.

11 Effect ofoperating pressure on the separation of benzene, toluene, and 70cumene in single—s0lute aqueous systems for the six cellulose acetate

- membranes annealed at different temperatures.

12 Correlation of extent of pore blocking (1 — n1-/np) and boundary layer 73concentration of benzene, XA2, for two representative membranes ofdifferent pore size.

13 Correlation of extent of pore blocking (1 — n·p/np) and boundary layer 74concentration oftoluene, XA2, for two representative membranes ofdifferent pore size.

14 Correlation ofextent of pore blocking (l — nT/np) and boundary layer 75concentration ofcumene, XA2, for two representative membranes ofdifferent pore size.

15 Correlation ofproportionality constant, Z, and pore size parameter ln C*NuC}. 78

16 Membrane compaction as represented by the decrease in the pure water 83permeability coefficient, A, as a function of increasing operatingpressure for the six cellulose acetate membranes.

_ xii

FigureI

Page17 The effect of volume flux on separation for the benzene—water—cellulose 87acetate system at 25°C. Experimental points; irreversible thermodynamics-

phenomenological transport model and the extended solution—diffusionmodel; irreversible thermodynamics — Kedem Spiegler model and thethree parameter ünely—porous model; solution-diffusion model, and theKimura Sourirajan analysis model.

18 The effect of volume flux on separation for the toluene—water—cellulose 89acetate system at 25°C. Experimental points; irreversible thermodynamics-phenomenological transport model and the extended solution—diffusionmodel; irreversible thermodynamics — Kedem Spiegler model and thethree parameter finely—porous model; solution-diffusion model, and theKimura Sourirajan analysis model.

19 The effect of volume flux on separation for the cumene-water-cellulose 91acetate system at 25°C. Experimental points; irreversible thermodynamics-phenomenological transport model and the extended solution-diffusionmodel; irreversible thermodynamics — Kedem Spiegler model and thethree parameter finely·porous model; solution—diffusion model, and theKimura Sourirajan analysis model.

20 The effect of the volume flux to osmotic pressure ratio, JV/ng, on separation 96for the benzene—water—cellulose acetate system at 25°C. Experimentalpoints; irreversible thermodynamics-phenomenological transport model.

21 The effect of the volume flux to osmotic pressure ratio, JV/ng, on separation 98for the toluene—water—cellulose acetate system at 25°C. Experimentalpoints; irreversible thermodynamics-phenomenological transport model.

22 The effect of the volume flux to osmotic pressure ratio, JV/ug, on separation 100for the cumene-water-cellulose acetate system at 25°C. Experimental_points; irreversible thermodynamics-phenomenological transport model.

23 The effect ofoperating pressure on separation for the benzene—water— 113cellulose acetate system at 25°C. Experimental points; solution-diffusion imperfection model.

24 The effect ofoperating pressure on separation for the toluene-water- 115cellulose acetate system at 25°C. Experimental points; solution-diffusion imperfection model.

25 The effect ofoperating pressure on separation for the cumene-water- 117cellulose acetate system at 25°C. Experimental points; solution-diffusion imperfection model.

26 The effect of volume flux on separation for the benzene—water—cellulose 127acetate system at 25°C. Experimental points; the four parameter finely-porous model; the four parameter finely-porous model with the pore sizeparameter, c/e, forced to he constant for all three solutes.

xiii

Figure Page27 The effect of volume flux on separation for the toluene—water-cellulose 129acetate system at 25°C. Experimental points; the four parameter finely-porous model; the four parameter finely—porous model with the pore size

parameter, t/e, forced to be constant for all three solutes.

28 The effect of volume flux on separation for the cumene-water—cellulose 131 -acetate system at 25°C. Experimental points; the four parameter finely-porous model; the four parameter finely—porous model with the pore sizeparameter, t/e, forced to be constant for all three solutes.

29 The relationship between the pore size parameters, t/s and ln C*Na(;;. 13630 The relationship between the ratio of the partition coefficients on the low 137and high pressure side ofthe membrane, K3/K2, and the pore size

parameter, ln C*NaC;.

31 Comparison ofthe separation measured and the separation calculated byl

152the surface force—pore flow model for the solute toluene and membrane 1.32 Comparison ofthe extent of pore blocking, (l — nq-/np), measured and 153calculated by the surface force—pore flow model for the solute tolueneand membrane 1.

33 Comparison of the separation measured and the separation calculated hy 154the surface force-pore flow model for the solute toluene and membrane 3.34 Comparison of the extent of pore blocking, (1 — n·p/np), measured and 155calculated by the surface force-pore flow model for the solute toluene

and membrane 3.

xiv

Chapter 1

INTRODUCTION

This dissertation deals with mass transport across reverse osmosis membranes inthe presence of strong solute-membrane affinity. ln reverse osmosis, a feed solution istransported under pressure through a polymeric membrane. The rate at which individualsolution components emerge in the permeate stream under specified operating conditionsdepends upon the morphology of the membrane and upon the nature and magnitude ofinteractions between solvent, solute, and membrane material. Either the solvent or thesolute may exhibit strong affinity for the membrane. Depending on which affinity is greater,markedly different separation behavior may occur in the reverse osmosis process.

The specific systems studied were a series of three aromatic hydrocarbon solutes(benzene, toluene, and cumene) in single—solute aqueous solution with six cellulose acetatemembranes with different flux and separation characteristics. The three aromatichydrocarbon solutes exhibit strong affinity for cellulose acetate membranes due to variousphysicochemical characteristics, whereas the solvent, water, is less attracted to the surface ofthe membrane. These solutes represent a large number oforganics which have industrial andenvironmental significance. Selection was based upon the range of nonpolar or dispersivecharacter, solubility in water, and societal import. Cellulose acetate was chosen for themembrane material because its performance has been well characterized for many systems,and because many commercially available membranes are composed ofthis polymer.

Over the last 20 years, much effort has been spent in the investigation of thereverse osmosis membrane separation process as a practical unit operation in chemical

engineering. One major result of these efforts was the emergence of several mathematical

l

2

models to describe transport in reverse osmosis membranes. Notwithstanding these efforts tounderstand and to apply reverse osmosis, the modelling of transport through reverse osmosismembranes is still controversial. Each of these models has been successful to some extent,even though the underlying premises of the models vary greatly. Agreement between dataand each of these models has been demonstrated for inorganic solutes and many organicsolutes in the absence of strong solute-membrane attraction. However, for organic soluteswith strong solute-membrane attraction, the reverse osmosis behavior is significantly alteredand no definitive model exists to describe this case. A primary goal of this research projectwas to evaluate the ability of existing models to describe the transport in the presence ofstrong solute-membrane affinity.

The difference in membrane performance between the case in which water-membrane affinity dominates and where solute-membrane affinity dominates aresummarized below. ln this work, separation is defined as;

F: mA1—mA3 (1)mm

where mA, and mm are the feed and permeate molalities, respectively. The case in whichstrong solvent—membrane affinity dominates includes most inorganic and many polar orionized organic solutes with cellulose acetate membranes and the solvent water. Thesesystems have been reported extensively in the literature (see for instance, (ll). Some generalstatements can be made concerning reverse osmosis transport:

1. lncreasing the operating pressure usually increases separation.2. The decrease in permeate flux with increasing feed concentration is due to the

osmotic pressure effects.

3. Positive separation is observed.

3

However, when solute-membrane affinity dominates these statements do not holdtrue. Systems involving strong solute-membrane affinity (2) are characterized by:l. lncreasing the operating pressure tends to decrease separation.

2. The permeate Flux is significantly less than the pure water permeation Flux, evenwhen osmotic pressure effects are negligible.

3. The separation may be positive, zero, or negative depending on the specificoperating conditions.

Note that negative separation implies that the solute is enriched in the permeate stream.The physicochemical criteria for the reverse osmosis separation ofseveral classes of

organic solutes in aqueous solutions using l„oeb—Sourirajan type cellulose acetate memhraneshave been discussed extensively (LQ). Cellulose acetate memhranes have both polar and non-polar characteristics. The polar character of the membrane is attributed to the hydroxyl andester groups present in the polymer, and the nonpolar character is due to the backbone carbonchain. Therefore, both the polar solvent (water) and nonpolar solute (hydrocarbon) can haveaffinity toward the membrane material. The relative affinity of the solute and solvent isimportant in determining the behavior ofthe system. The nonpolar or dispersive character ofthe aromatic hydrocarbon solute can be represented by Small’s molar attraction constant

(ZA). An increase in the dispersive character of a series of homologous compounds asindicated by an increase in the molar attraction constant results in an increase in the solute-membrane attractive Forces. Values of the molar attraction constant and other data for thesolutes under investigation are presented in Table l.

Both a theoretical and an experimental approach to reverse osmosis transport are

included in this study. A review of several existing transport models is presented withemphasis on the applicability of these models to systems involving strong solute-membrane

41 Table 1Ph sicochemical Parameters for the Solutes Studied

Benzene Toluene CumeneStructure 06 H6 06 H6 - 0H6 C6115 — CH<CH3>2

Molecular Mass, kg/kmol 78.11 92.15 120.20

Molar attraction constanta, 53.4 61.4 77.1(kJ m0)1/2/kmolSolubility in watera, ppm 1780 515 50

Vapor Pressureb, kPa 0.69 0.36 0.03

TLV¢, ppm of vapor in air 10 200 50

Molal Volume at the normal 0.0960 0.118 0.163boiling point<1,m0/kmol

Partial Molar 0.0889 0.106 0.140Volumee, ml!/kmol

Diffusivity in 1.096 0.968 0.799WaterV, mz/s, x 109

Solute radiusg, m, x 1010 2.23 2.53 3.06

" as reported in reference (8) at 25°C.1> as reported in reference (Q) at 20°C.¢ Threshold Limit Values (TLV) as reported in reference (8).<1 as estimated by the method of LeBas (8).<·> estimated from pure solute density as suggested by reference (Z).V estimated by the method of Wilke—Chang (8).rs calculated based on the Stokes—Einstein equation.

5

affinity. Papers in which authors have included solutes that exhibit strong solute—membraneattraction are discussed. The experimental data are presented and discussed. These data arecorrelated by a number of the available models and the results are interpreted based on thephysical significance of the parameters generated. Finally, conclusions and recom-mendations for further study, based on the presented results, are made.

1.1 Historical Background

The purpose of this section is to provide the reader with some historical perspectiveon reverse osmosis systems involving strong solute—membrane attraction. lncluded arereferences in which data have been collected and interpreted qualitatively for these systems.A review of the work done on quantitative modelling of these systems is presented in theTheoretical Background section.

The first experimental evidence of unusual reverse osmosis behavior due to strongsolute—membrane affinity was observed by Blunk (Q) and by Keilin (LQ). Their resultsinspired Lonsdale et al. (Z) to investigate the behavior of the phenol—water—cellulose acetatesystem in more detail. Lonsdale et al. (Z) observed negative separation for phenol and notedthat separation decreased with increasing pressure. These results are characteristic ofsystems with strong solute—membrane affinity. Equilibrium sorption and unsteady statediffusion experiments were conducted to determine the partition coefficient and the diffusioncoefficient for phenol in the membrane material. ’l‘he partition coefficient was large, 42 (kgphenol/m3 cellulose acetate)/(kg phenol/m3 solution), compared to 0.03 for sodium chloride.The diffusion coefficients for phenol and sodium chloride were both about 9.6x10°l4 m2/s.Subsequent work by Lonsdale and coworkers(Z,L1,Q)confirmed the earlier results for phenoland attempted to analyze the data in terms of the solution·diffusion model and a frictionalmodel. The negative separation of phenol was attributed to the large partition coefficient and

6

to "flow coupling" (in which the solute and solvent transport are linked rather thanindependent, as is usually assumed).

The separation of phenol-water solutions has dominated the literature on systemsinvolving strong solute-membrane affinity, probably because this was the first system studiedand because phenol-water separation is environmentally important. The early work byLonsdale and coworkers has been repeated and re—analyzed by several authors includingMatsuura and Sourirajan (@—@), Pusch and coworkers (@,11), Tone et al. (E, Q), and others(@-2]). The qualitative features of the phenol-water—cellulose acetate system were similar inall these papers. Several of these authors also noted that the flux was recluced for theirmembranes in the presence of phenol compared to pure water flux. 'f‘his effect is the same asthe flux reduction effect mentioned above and is characteristic of systems with strong solute-membrane affinity. Notwithstanding the similarities in the data, interpretation of resultswas often different. Lonsdale and coworkers (1,Q) favored a solution-diffusion approach,while Matsuura and Sourirajan (2,@,@) used a pore flow model with surface forces acting onthe solute and solvent. Pusch and coworkers used the original (@) and a modified version ofthe solution-diffusion model, as well as irreversible thermodynamics (11) to describe their

data. These models are discussed in more detail in the next section.

ft was soon found that these results were not unique to the phenol-water—cellulose

acetate system. Phenol derivatives, such as chlorophenols and nitrophenols, also exhibited

strong affinity for cellulose acetate membranes (g,1§_,1<1,@,§). Although negative

separation was usually observed, for some experimental conditions positive separations were

reported (12,21). Hydrocarbon solutes, such as benzene and hexene, were also found to havestrong affinity for the membrane material (2). For the hydrocarbon solutes, positive

separations (2,22) were usually observed, but under certain experimental conditions,negative separations (211,251) occurred. Therefore, it became evident that strong solute-

7

membrane affinity was not restricted to phenol, or to cellulose acetate membranes, andpositive, zero, or negative separation could be effected by experimental conditions.

The existence of solute—membrane attraction was also observed for othermembrane materials. In particular for phenols and substituted phenols the characteristicbehavior of strong solute-membrane affinity was found for aromatic polyamide (gg),cellulose acetate butyrate, and North Star (NS—l00) membranes (gg). The NS—l00membrane is a polyethylenimine coated on ai polysulfone support and cross—linkecl withtoluene 2,4—diisocyanate. For a given solute and solvent, the strength and direction of thesolute-membrane interaction is a function of the membrane material used. One example isthe separation of p—chlorophenol—water with modified NS—l0() membranes. The modifiedmembrane (fabricated with twice the normal amount of cross—linking agent) exhibited anincrease in p—chlorophenol—water separation with increasing pressure (gß). But, for theunmodified NS—100 membrane it was observed that the p—chlorophenol—water separationdecreased with increasing pressure (g). In this case, for the modified membrane the solvent-membrane attraction dominates and for the unmodified membrane the solute—membraneattraction dominates. ()f course, douhling the amount of cross—linking agent changes thephysical structure as well as the chemical nature ofthe membrane. However, it is primarilythe chemical nature that determines the direction of change in separation as a function ofincreasing operating pressure. This example illustrates that even a relatively small changein membrane character can result in a large change in the nature of the solute-solvent-

membrane interactions.

That the observed behavior for the phenol-water—cellulose acetate system is aresult of the strong solute—membrane aflinity is evidenced by the following experiment.

When the pH of the solution is increased, phenol dissociates to a phenoxide anion.Experimental results (gg) indicate that the phenol-water separation increases directly as

8

the degree of dissociation for phenol increases. Apparently, the phenoxide ion is repelled bythe membrane. Therefore, it is only the neutral phenol molecule which experiences a strongattraction to the membrane material.

Strong solute-membrane attraction effects are important for many solutes andmembrane materials. The strength of the interaction between solute and the membrane areimportant in determining the overall reverse osmosis performance. The application of theseprinciples to the data in the literature for a number of quantitative transport models isdiscussed in the next section.

1.2 Theoretical Background

A number of models have been developed over the years to describe reverse osmosistransport. ln most cases, these models have been developed to describe the .transport ofsolvent and solute where the solute (usually NaCl) is repelled by the membrane material.Several papers published recently have reviewed these different models and have discussedthe ability of these models to describe the transport for various inorganic and organic solutes(Q,@,§,@). The following review focuses on how several of these models apply to solutesystems involving strong solute-membrane attraction. The discussion will include a briefdescription of the basis, physical significance, and assumptions of each model as well as thefinal form of the equations. lnstances where systems involving strong solute-membraneattraction have been modelled are reviewed.

1.2.1 General Considerations

Before the transport models are discussed, some general aspects of membrane

transport will be discussed that are often considered to be independent of the particularmembrane model under consideration.

·

Concentration Polarization. When solute is rejected by the membrane, the soluteconcentration near the membrane surface increases. The build—up in concentration in thisboundary layer region is referred to as concentration polarization. ln order to descrihe thisconcentration polarization, the film theory (Q) is used. At steady state, the flux of solute tothe membrane, (CA/C) (NA + NB), the flux of solute through the membrane, NA, and thesolute back diffusion, — DAB dCA/dx, are balanced as illustrated in Figure l. Mathematically:

dC C (2*which is a form of Fick’s first law (Q). All symbols are defined in the N()MENCl.ATUREsection. Solving this equation with appropriate boundary conditions ( Qi), Q) gives:

CA2 = CAB + (CM — C^3)exp(n„[„/kC MB) (3)where nT is the mass flux through the membrane, and CM, CAQ, and CAB are the feed,boundary layer, and permeate concentrations, respectively. As the mixing on the highpressure side of the membrane is increased, the mass transfer coefficient, k, increases andEquation (3) predicts that concentration polarization will decrease (CAB decreases).

The mass transfer coefficient is a function of feed flow rate, cell geometry, andsolute system. At fixed feed flow rate and cell geometry, k lvaries as a function of thediffusivity ofthe solute (Q):

k G Di; (4)That k varies as the 2/3 power of diffusivity comes from generalized correlations of masstransfer which have been proposed by several authors (Q,Q—Q4). lüquation (4) implies thatk/DABW3 is a constant. Writing Equation (4) for a reference solute at the same experimentalconditions and taking the ratio of the equations gives:

<5>

lO

u.1 2g OI (1) XO(Q (D .IUJ— LU DUJI Q: mu.

CL

inQ4 >< Zts ‘¤+ >- (I‘° Z4 ä LU

Q Z(I (_)<I U ..1<evOUJ

Z<IE5LU an < LuZ Z I2; ¤J; U) LU E3 <¤ EUJ CE0: Q LUQ. Q &

Figure 1. Concentrution polarization at the high pressure surface 0f‘ a reverse osmosismembrane.

ll

lf k is known for a reference solute, then k for any other solute can be estimated usingEquation (5) if the diffusivities ofthe solute and reference solute are known.

Other approaches have been used to describe the concentration polarization layermore accurately (see for instance, (äh. flowever, for most practical purposes, Equation (3) issufficiently accurate. For each of the models presented below, the original reference should bechecked to determine how the authors modelled concentration polarization.

Definition of Separation. Separation, f, was defined above in terms of the feed and permeatemolal concentrations, mA, and mA3, respectively:

f: (I)For moderately dilute solutions, the molal concentration, mAi, can be approximated by themolar concentration, CAP and Equation ( I) can be rewritten as:

(6)Alternatively, separation can be defined in terms of the concentration of the boundarysolution near the membrane surface, CAQ, as calculated by Equation (3). The separationbased on the boundary layer concentration, f', can be written as:

f' = (CA2 —— CA3)/CA2 " (7)The separation calculated in this manner represents the separation that would be measuredwith no concentration polarization (that is, CA, = CA2, and k : infinite). Some authorschoose to ignore the difference between fand f' by claiming that if the mixing is sufficientlythorough, the boundary layer concentration will approach the bulk concentration. This maynot be the case in large scale systems or even in many laboratory scale apparatus.

ln reverse osmosis single-solute systems, the concentration of the solute in thepermeate is related to the solvent and the solute fluxes by a simple material balance as:

12 V

NXA3 (8)

A BAt moderately low concentrations, where NB > > NA and the difference between molal andmolar concentration may be ignored (CM = mAi), Equations (7) and (8) can be combined togive:

<9>CA2 Jv CA2 NB

where C is the molar density of water and JV is the solvent volume Flux. Equation (9) is usedFrequently in the derivation ofthe models discussed below

Osmotic Pressure Effects. When an ideal semi-permeable membrane (one that is permeableto solvent but not solute) is placed between two compartments, one containing pure solvent

and the other containing a solution (solvent plus solute), the solvent will pass through themembrane to the solution side. This phenomenon is called osmosis. Transport occurs due tothe chemical potential driving force which is caused by the presence of the solute. The exactpressure that must be applied to the solution side to stop the solvent flux is called the osmoticpressure. In reverse osmosis, a pressure greater than the osmotic pressure is applied to thesolution to reverse the flow and drive solvent from the solution side to the pure solvent side;hence, the name reverse osmosis. To model the flux through a membrane, the influence of theosmotic pressure driving force must be considered. For a real membrane, some solute existsin the permeate and therefore the osmotic pressure of the solution on each side of the mem-brane must be considered. An effective pressure driving force across the membrane can bedefined as the applied pressure difference minus the osmotic pressure difference. For mostmodels, the water flux is considered to be proportional to the effective pressure driving force.

The manner in which each of the models to be discussed below handles the effect ofIosmotic pressure contributions varies. Each section briefiy discusses the method used in that

13

particular model. Usually these equations take on the form shown here (written for volumeflux, JV, and for molar flux of solvent, NB) so that the flux is directly proportional to theeffective pressure driving force:

u JV = Ep (AP — An) (10)

Ng=A(AP—A¤) (11)where Cp and A are the appropriate proportionality constants. For the systems reported inthis dissertation, the concentrations being considered are sufficiently low that osmoticpressure contributions are usually considered to be negligible.

For dilute systems, the osmotic pressure ofa solution can be estimated by the van’tIHoffequation (Q) as:

l Hi Z CM RT (12)Therefore, the osmotic pressure difference, An, across a membrane is related linearly to theconcentration difference, CA2 — CA;. I

Membrane Structure. In order to facilitate the description of the various membrane models, abrief discussion of membrane structure is included here. The inquiry into the exact

Telationship between membrane structure and performance is an on-going concern (see forinstance references (Q, Ql) and the present discussion is limited to some of what is knownabout cellulose acetate membranes. The asymmetry of the membrane, the nature of the skinlayer, and the influence ofan annealing treatment are discussed.

The success of the reverse osmosis process is due in part to the development of theasymmetric membrane. An asymmetric membrane has a relatively dense surface layersupported by a porous layer underneath. Such a structure greatly reduces the resistance to

flow through the membrane compared to a homogeneous dense membrane of the same overallthickness. For cellulose acetate membranes, such as those used in this study, the asymmetric

14

structure is a direct consequence of the casting procedure used. When a polymer solution iscast on a flat surface, the evaporation of the solvent produces a surface skin. Subsequentgelation in cold water fixes the structure; the porous substructure is formed by thereplacement of solvent by the nonsolvent water. Scanning electron microscope examinationof membranes made in this manner (Q) indicates that three layers exist; a relatively densesurface skin, a transition layer, and an open porous support layer. The transition layer isintermediate in both density and position with respect to the other two layers. Most of theresistance to mass transfer through the membrane exists in the surface skin. Therefore, itmay be assumed that the performance of the membrane is dependent primarily on thechemical nature, thickness, and structure ofthe surface skin.

For the skin layer, the basic question is whether it is porous. Membranes withpores sufficiently large that they can be seen with a scanning electron microscope are usuallyconsidered to be ultrafiltration membranes. These membranes are clearly porous. As the poresize becomes progressively smaller, there is no clear point at which the pores disappear. Onepossible definition ofthis point would be when the pore volume fraction becomes equivalent tothe free volume of the polymer. This point is difficult to determine with a water—swollenpolymer. The degree ofcrystallinity, for instance, will change thefree volume. At what pointdoes a void space in the polymer structure become a pore? With the technology availabletoday, the existence of pores in the membranes such as those used in this study cannot betested. Therefore, mechanism—independent models, nonporous models, and porous modelshave been included in the analysis of the data in this dissertation.

The performance of reverse osmosis membranes can be influenced by a thermalpretreatment (@, Q). This annealing or shrinking step involves exposing the wet

membrane to an elevated temperature for a short period of time (approximately 10 minutes),after which the membrane is cooled rapidly back to room temperature. Annealing has been

15

found to be effective at modifying the membrane performance, particularly for celluloseacetate membranes. The sodium chloride—water separation can be increased from less than50% to more than 99%, with a corresponding decrease in permeate flux. Separation increaseswith increasing annealing temperature. Temperatures are typically in the range60°C — 90°C. lt is agreed generally that elevated temperatures increase the kinetic energy ofthe polymer molecules, allowing a rearrangement in the membrane structure to take place(Q). The nature of the change depends on whether the membrane is considered to benonporous or porous. lf the membrane is nonporous, then the annealing step is a way ofreducing any imperfections that may exist in the surface layer. In addition, the degree ofcyrstallinity is increased, leading to relatively smaller amorphous regions through whichtransport can take place. Alternatively, if the membrane is porous, the annealing procedureshrinks the existing pores to a smaller size. lt is not known which of these mechanisms iscorrect. However, annealing membranes at different temperatures does provide a convenientmethod for generating a series of membranes with gradated properties.

Membrane Pore Size. lf the membranes are assumed porous, then it is desirable to have ameasure ofthe pore size. Since, as cliscussed above, the pores cannot be seen, there is no directway of measuring the size. An alternative is to use some relative measure of pore size basedon experimental data. ()ne such method has been suggested by Sourirajan and coworkers(Q). They found that data for several single so|ute—water-cellulose acetate membranes, whenwater—membrane affinity dominates, could be correlated by the following equation:

ln(DAM K/1:) = lnCQIHCIwhere(DAMK/r;) is the solute transport parameter (discussed below), and AAG is the freeenergy required to bring a solute molecule from free solution to the membrane surface. Freeenergy parameters, (—AAG/RT)i, for various ions have been tabulated elsewhere (Q). The

16

wremaining term is interpreted as a membrane pore size parameter, ln C*NaCl, which isconstant for a given membrane sample. Substituting the known values of(— AAC}/RT)i for thereference solute sodium chloride, Equation (13) may be rearranged as:

ln Cälacl = ln( DAMK/t cm/s) — 1.37 (14)where (DAMK/t) must be in cgs units, to be consistent with the original paper, rather than inthe Sl units used in the rest of this dissertation.

The value of ln C*NuCl calculated by Equation (14) is a relative measure of themembrane pore size, based on experimental data for sodium chloride—water separation. Aspore size increases, ln C“°'NuCl increases. This approach is consistent with other workpublished previously (ggg).

Membrane Compaction. Theoretically, when pressure is increased the pure water permeationflux increases proportionally. This behavior is predicted by Equation (11) (or equivalently byEquation (10)) for pure water in which case An = () and NB : A AP. llowever, at higherpressures the flux will usually be less than that prediced by Equation (10). 'l‘he difference isattributed to compaction of the asymmetric membrane. One possible explanation of thisphenomenon is as follows. The asymmetric membrane consists of three layers (a surface skin,a transition layer and a porous support) as discussed above. The primary resistance to flow isin the surface skin. When the operating pressure is increased the membrane is compressed bythis force. The result is that some of the transition layer may be effectively incorporated intothe dense surface skin. Therefore, the resistance to permeation is increased with increasingpressure as progressively more of the transition layer resistance is included in the overallmembrane resistance. It is convenient to represent this compaction effect as a pressuredependence of the pure water permeability coefficient, A, by the empirical equation (@3):

A:A0exp[—yAP] (15)

17

Equation (15) suggests a linear relationship between A and AP on a semilog coordinatesystem.

1.2.2 Phenomenological Transport Models

In this section, models which are independent oFthe mechanism oF transport will bediscussed. These models are called phenomenological transport models and are based on thetheory ofirreversible thermodynamics.

Irreversible Thermodynamics Phenomenological Transport Relationship. In the absence oFany knowledge oF the mechanism oF transport or the nature oF the membrane structure, it ispossible to apply the theory oF irreversible thermodynamics (IT) to membrane systems (Q2).

In IT, the membrane is treated as a "black box". Models stating the relationship betweenForces acting on the system and the Flux oF material through the membrane are Formulated.For systems that are not Far From equilibrium, IT suggests reasonable choices For Forces andFluxes. The phenomenological relationships are manageable ways oF expressing therelationships between the observed Fluxes and the applied Forces. Onsager (Q) suggested thatthe Fluxes and Forces could be expressed by the Following linear equations:

Ji : Lii Fi LiiFi Fori: l,...,n (I6)] ¢ I

where the Fluxes, Ji, are related to the Forces, Fi, by the phenomenological coeFFicients, LU.

For membrane systems, the driving Forces can be related to the pressure andconcentration diFFerences across the membrane, and the Fluxes are solvent and solutepermeate Fluxes. This equation can be simplified by assuming that cross coeFficients areequal, as First proposed by Onsager (Q):

LU: Lji For ixj (17)

18

The above Onsager reciprocal relationship is valid when the system is close to equilibrium,the linear laws (Equation (16)) are valid, and the correct choice oFFluxes and Forces has beenmade. For systems that are Far From equilibrium, as is oFten the case in reverse osmosis,Equation (17) may not be correct. ’I‘he validity oF the Onsager reciprocal relations has beendiscussed by Soltanieh and Gill (Q).

Kedem and Katchalsky (Q) used Equations (16) and (17) to derive what are knownas the phenomenological transport equations (IT — PT):

JV = €p(AP — oA¤) (18)

NA=wAri+(1—o)(CAM)ln•IV (lg)where the adjustable parameters €p, ro, and o are simple Functions oF the originalphenomenological coeFficients, LU . (CAM)ln is the logarithmic mean solute concentration inthe membrane. Equation (18) is similar to Equation (10) with the addition oF the reFlectioncoeFFicient, o, as originally proposed by Staverman (Q). Pusch (Q) has shown that Equation(19) can be rewritten to relate separation, F', and Flux, JV, as:

é: é + — o2)(€p/o)¤2(}) Q(2)))

[) V

The above equation predicts a linear relationship between 1/F' and 1/JV. The osmoticpermeability, €n, is related to w as:

co = (En/€p — o2)(CAM)ln FP (21)The parameters in the model are the solvent and osmotic permeabilities, FP and Fn, and thereFlection coeFFicient, o. Theseparameters can be determined For a given solute andmembrane by applying Equations (18) and (20) simultaneously using data collected at severaldiFFerent pressures and concentrations.

19

Jonsson (Q) and Burghoff et al. (Q) Found this treatment to be satisfactory Fordescribing the data For phenol—water separation For cellulose acetate membranes. Theseresults seem promising; however, there are limitations. First, Equation (20) is onlyapplicable when the linear laws apply across the whole membrane. For reverse osmosissystems, the concentration differences across the membrane are often large enough that thelinear laws are not valid. As a result the Lg coefficients will be concentration dependent tovarious extents (Q). However, For many systems, the coefficients €p, o, and w are nearlyconstant provided that the concentration changes are not too great. This assumption isrelaxed in the Kedem-Spiegler relationship, to be discussed next. Second, by considering themembrane as a "black box", the resulting analysis does not give any insight into the transportmechanism.

lrreversible Thermodynamics — Kedem Spiegler Relationship. One critical assumption in theIT- PT relationship is that the linear laws were assumed to apply over the whole thickness oFthe membrane. Spiegler and Kedem (Q) resolved the problem by rewriting the originallinear IT equations in diFFerential Form and then integrating them over the thickness oF themembrane. The equations in differential Form For the solvent and solute Flux, respectivelyare:

(22)

(23)where PA is the solute permeability, pg is the water permeability, and x is the coordinatedirection perpendicular to the membrane.

20

If pA, pB, and 0 are constant, Equation (22) can be integrated to give Equation (24)

below and Equation (23) can be integrated and combined with Equation (9) to give (@,43)

Equation (25):

JV = (pB/Ax) (Ap - 0 An)i

(24)

L — 1-0exp[-(1-0)(Ax/pA)JVl(25)

f' 0{1-exp[-(l-0)(Ax/pA)JvI}where Ax is the membrane thickness. The result is a three—parameter model described byEquations (24) and (25), similar to the IT- PT relationship but which should have coeflicientsthat are independent of concentration and pressure. The three parameters in the Kedem—Spiegler relationship (IT- KS) are pn/Ax, pA/Ax, and 0.

Similar to the IT- PT relationship, the IT- KS model was found by Burghoff et al.(Q) and Jonsson (gl) to describe negative separation for phenol—water-cellulose acetate

systems. Again, this approach lends no insight into the mechanism of membrane transport.

1.2.3 Nonporous Membrane Transport ModelsV

In this section, models in which it is specifically assumed that the membrane is

nonporous are described. First, the solution—diffusion model and then two modifications ofthesolution—diffusion model are presented.

Solution—Diffusion Relationship. The solution-diffusion (SD) model was originally applied to

reverse osmosis by Merten and coworkers(ä1_,@). The membrane surface layer is consideredto be homogeneous and nonporous. '1‘ransport of both solvent and solute occurs by the

molecules dissolving in the membrane phase and then diffusing through the membrane. The

permeability ofa species is equal to the product of the solubility and the diffusivity for that

species. Theoretically, the solubility and the diffusivity of the solute can be determined for a

21

membrane material by performing equilibrium sorption and unsteady statesorption/desorption studies, respectively. The water flux is proportional to the solventchemical potential difference (usually expressed as the effective pressure difference across themembrane), and the solute flux is proportional to the solute chemical potential difference(usually given as the solute concentration difference across the membrane). The solvent andsolute fluxes, respecitvely are:

JV : (AP - Am (26)

NA <¤AA - <>AA> (27)Note that Equation (26) is identical to Equation (10), except that Fp has been

replaced by more physically meaningful terms, DAM and DBM are the diffusivities of thesolute and the solvent in the membrane, respectively; CBM is the membrane water content;Vß is the partial molar volume of water; R is the universal gas constant; ’l‘ is the temperature;Ax is the actual thickness of the membrane skin; and K is the partition coefficient defined asfollows:

K — kg solute/m3 membrane _ (28)_kg solute/m3 solution

K is a measure of the relative solute affinity to (K > 1.0) or repulsion from (K < 1.0) themembrane material.

As illustrated by Pusch (Q), Equations (26) and (27) may be combined withEquation (9) and rearranged to give:

A)Equation(29) predicts a linear relationship between 1/f' and 1/JV. Equations (26)and (29) can be fit to experimental data to generate the two parameters (DBM CBM VB/RT Ax)

22

and (DAMK/Ax), both of which are treated as single quantities. In order to resolve either ofthese terms into its component parts, it is necessary to have an independent measure of someofthe terms (see for example reference (ß) on how to measure DAM, DBM, and K, separately).

One restriction of the SD model is that the separation obtained at infinite flux isalways equal to 1.0. However, this limit is not reached for many organic solutcs; inparticular, the model is not applicable to negative separation cases. Also, for negativeseparation, Equation (29) produces a negative value for (DAM K/Ax), which is physicallyimpossible, since none of the quantities in (DAM K/Ax) can be negative. For these reasons, theSD model is inappropriate for systems involving strong solute—membrane attraction.

Two attempts to improve on this original model are the solution—diffusion

imperfection model (SI)I) and the extended solution—diffusion model (ESl)).

Solution—Diffusion Imperfection Relationship. 'l‘he solution—diffusion imperfection model(SDI), derived by Sherwood et al. (@), allows for small imperfections which can exist in the

membrane surface. lt was assumed that imperfections will transport solution from the

boundary layer (CA2) to the permeate side, without any change in concentration. Therefore,an additional term to both solvent and solute fluxes, that is proportional to the pressure

driving force appears in the model as follows:

JV = x' (AP -Au) + x"' AP (30)NA=x"An+x"'APCA2 (31)

23

As illustrated by Jonsson and Boesen (Q), Equations (9), (30), and (31) can berearranged to give:

(AP - An) + (K"'/K')AP (32)I —f' (K"/K')(Ar1/CA2) + (K"'/K')APFor the particular case when AP > > An and solutions are dilute, then Equation (32) can berearranged as:

I K"' K" RT .(33)which suggests a linear relationship between 1/f' and 1/AP. The derivation of Equation (33)

from Equation (32) is presented in Appendix A. The SDI model has three adjustable

parameters, K', K", and K"'. The K' and K" are the transport parameters for diffusive water

flux and diffusive solute flux, respectively. The K"’ transport parameter is for pore flow.

Therefore, the ratio ofK"' to K' is a measure ofthe relative contribution ofpore flow compared

to diffusive flow. The K" value is proportional to the product of the solute diffusivity in the

membrane phase and the solute partition coefficient. Jonsson (Q) found that this model

failed in the negative separation region, which implies that the SDI model is not appropriate

for handling the case of strong solute—membrane attraction.

Extended Solution—Diffusion Relationship. Burghoff et al. (LZ) and Jonsson (Q) pointed out

that in the original SD model, a pressure term in the solute chemical potential equation was

neglected. 'I‘he complete expression for chemical potential, including the pressure term is:

ApA == RT ln(CA2/CAB) + VA AP (34)where ApA is the solute chemical potential difference across the membrane and VA is thesolute partial molar volume. In general, to neglect the pressure term, ln(CA2/CA3) must be

significantly greater than VA AP/RT. For sodium chloride—water separation (where

VA = 0.02 m3/kmol), as long as ln(CA2/CA3) >> 8.0 x I0“6 AP (with P in kPa), it is

24

acceptable to ignore the pressure term (Q). For organic—water systems that exhibit lowerseparation, and For higher operating pressures, it is important to include the pressure term inthe analysis.

[ncluding the pressure term in the analysis leads to the Following equation (Q): _

N,. = @1. — ¤.„„> + wp “‘5’where EAP is the pressure induced solute transport parameter. Equations (9), (10), and (35)can be combined (Q) to give:

1 F p AP 1 D K 1Z p v .

For the special case AP > > An then:

(37)l" (1- EM/CA2€p) 1 Ax (l—€AP/CA2€p) JVThe three parameters in the model are fp, (DAM K/Ax), and €AP. Again, a linear relationshipis predicted between 1/F' and 1/JV. Alternately, 1/(1 —€AP/CA2€p) may be used as a groupedparameter which may or may not be less dependent on the operating pressure andconcentration. BurghoFF et al. (Q/) Found good agreement between the ESI) model and theobserved negative separation For phenol with cellulose acetate membranes. For this case,negative separation is attributed to a large pressure contribution to the Flux oFsolute.

1.2.4 Porous Membrane Transport Models

ln this section, transport models in which it is speciFically assumed that themembrane is porous are presented.

Preferential Sorption-Capillary Flow Model. The preferential sorption—capillary Flow model

(PS-CF) was First proposed by Sourirajan (Q). According to this model, the membrane surFaceis microporous. In addition to the number and size oF pores, the membrane perFormance

25

depends on the chemical nature of the solute, solvent, and membrane material. In manyapplications, solvent-membrane attraction dominates; this situation is called waterpreferential sorption. A layer of almost pure water which exists at the membrane solutioninterface is continuously withdrawn through the membrane pores under the pressure drivingforce. The solute (for example, sodium chloride) is preferentially repelled. Only pores thathave diameters larger than twice the thickness of the sorbed water layer allow solute to pass.On the other hand, for some solutes the solute-membrane attraction dominates over thewater—membrane attraction. In this case it is possible to have preferential transport of thesolute, leading to negative separation. Whether water—membrane or solute—membrane{affinity dominates, the basic premise of the PS~CF model is that membrane performance canbe described in terms of the number and size of pores and the physicochemical interactionsbetween solute, solvent, and membrane.

The model provides a qualitative description of the reverse osmosis process. To beuseful, the model must be translated into a quantitative transport relationship. Sourirajan

and coworkers have proposed two transport models: the Kimura-Sourirajan analysis and thesurface force—pore flow model. 'I‘hese two models are discussed in separate sections below.

Kimura—Sourirajan Analysis. The Kimura-Sourirajan analysis (KSA) (Qt}, QL) was

developed from the PS-CF model to describe the transport through reverse osmosismembranes when solvent-membrane affinity dominates. According to the KSA relationship,

the membrane surface is microporous and transport occurs only through the pores. Themembrane has a preferential attraction for water and the resulting sorbed layer of almost

pure water is forced through the membrane pores by pressure. The solvent flux is viscous in

nature and therefore the driving force for solvent transport is given by the effective pressureas in Equation (11). The solute flux is diffusive in nature and is driven by the concentration

26

gradient:

NA Z (C2 XA2 “C3 XA3)i (38)

Equations (3), (8), (11), and (38) together make up the Kimura—Sourira_jan analysis.For a pure water permeation experiment, n2 = riß = 0, and A can be calculated directly fromthe pressure and observed flux using Equation (11). Feed and permeate concentrations andflux data obtained with solute present can then be used to determine the transportparameters using the KSA equations. Using physical data relating the osmotic pressure tosolution concentration, Equation (11) can be used to calculate XA2. 'l‘his value of XA2 can beused in Equations (3) and (38) to calculate k and (DAMK/L), respectively. Therefore, the threetransport parameters, A, (DAMK/L), and k, for a given solute and operating conditions, can becalculated from data obtained for a single experiment.

Equations (9), (11), and (38) can be combined to give the following relationshipbetween f' and JV:

Note that this equation is functionally the same as the SD model. The two parameters are A(from Equation (10)) and (DAMK/L). Even though Equation (39) is similar to Equation (29) forthe solution-diffusion model, the coefficients are interpreted differently. ln the KSA model,

DAM is the diffusivity of the solute in the membrane pore rather than in the polymermaterial; K is the partition coefficient defined based on the amount of solute in the poresrather than in the membrane material; and L is the effective length ofa pore, rather than theactual thickness of the membrane surface, Ax. As in the SD model, Equation (39) predictsthat f' approaches 1.0 for infinite flux and (DAMK/L) is negative for negative separation cases.

Neither of these characteristics of Equation (38) is realistic for the case of strong solute-membrane affinity.

27

Recently, attempts have been made to modify this approach to include cases involving

' strong solute-membrane attraction. For many solutes, it has been found that flux through the

membrane is significantly decreased by the presence of solute in the feed solution. Several

authors (2,Q,Q) have hypothesized that this decrease in flux is due to blocking of membranepores by solute molecules. The '°extent of pore blocking" (Q) may be represented by

(1 — nT/np). IfnT = np, no pore blocking has occurred and the extent of pore blocking is zero.lf nT = 0, the pores are totally blocked and the extent of pore blocking is 1.0. For the solutes

p—chlorophenol (Q), benzene, toluene, cumene (Q), cyclohexane, cyclohexene, and 1,3-

cyclohexadiene (Q), the following empirical relationship was found to describe the extent of

pore blocking as a function ofconccntration:

(l—n/n)=ZXC (40)'1 P A2

The parameters Z and C, for a given solute and membrane, were found to be independent of

pressure. For p—chlorophenol (Q), the solute fiux was found to be related to concentration and

pressure as:

NA : z' xf(2AP‘” (41)These relationships are potentially useful to describe a specific system. llowever, due to the

empirical nature of Equations (40) and (41), it is not possible to extrapolate to other systems

nor to interpret physically the parameters generated.

Finely—Por0us Model. The finely—porous model (FPM), developed by Merten (Q), is based on a

balance of applied and frictional forces in a one~dimensional pore, as first proposed by

Spiegler (Q). A complete derivation of the model has been given by Jonsson and Boesen (Q)

and by Soltanieh and Gill (Q). The general form of this model relates the volume flux, JV, and

the separation, f', as follows:

1 b K3 _ b C Jv (42)vl2 2 AB

28

Algebraic rearrangement allows Equation (42) to be written in terms of 1/f' and JV as:

L : 1—(l—K3/b)exp[—(t/cDAB)JVl (43)f' (1-K/b)—(1—K3/b)exp[—(deDAB)„JVlThe solvent flux is represented by Equation (11). The parameters in the FPM relationship arethe pure water permeability, A, the partition coefficients on the high and low pressures sidesof the membrane, K2 and K3, respectively, the friction parameter, b, the effective membranethickness, t, and the fractional pore area of the membrane surface, e. DAB is the diffusivity ofthe solute in free solution.

The pure water permeability was discussed in Section 1.2.1. The partitioncoefficients, K2 and K3, are defined in a manner similar to that given earlier in Equation (28),with one difference. ln this case, the concentration of solute in the membrane is interpretedas the concentration ofsolute in the membrane pore. The friction parameter, b, is defined (Q4)

as:

b = (XAM + XAB)/XAB (44)where XAB represents friction between the solute and solvent and XAM represents friction

hbetween the solute and membrane material. Therefore, b can be thought ofas the ratio of thetotal friction of the solvent plus membrane upon the solute to the friction between solute andsolvent. The frictional forces can be represented (inversely) by the diffusivity of solute withinthe membrane phase, DAM, and the diffusivity of the solute in the free solvent, DAB, so thatEquation (44) can be rewritten as:

b = DAB/DAM (45)

A detailed discussion of the physical significance of b and how it can be estimated is givenbelow in Section 1.2.5. The effective thickness of the membrane, t, is a product of the actual

thickness of the membrane surface layer (membrane "skin" layer) multiplied by the

29

tortuosity of the membrane pore. The tortuosity factor corrects the actual membrane skinthickness to an effective thickness that includes the nonlinearity of the pore geometry. 6 is

the fractional pore area of the membrane surface. For an asymmetric membrane, the value of

6 will be much less than that calculated from the water content of the whole membrane.

The FPM relationship, as represented by Equations (1 1) and (43), is a four parameter

model (referred to as FPM-4). The four grouped parameters are A, b/K2, K3/K2, and t/6, whichcan be obtained by litting experimental reverse osmosis data to the model. The individual

parameters, b, K2, K3, c, and 6 cannot be obtained unless independent information isavailable for some of them. The parameter ratio, t/6, is a measure of the size and number of

pores only, and should be a constant for a given membrane sample.

ln principle, K2 and K3 may be different, but it is often assumed (L(_§,Q,§) that

K2 = K3 = K. ln order for this to be true, K should be independent of concentration,

pressure, and membrane structure. Whether K is constant should be determined for each new

system studied. When the above assumption is made, Equation (43) reduces to:

L Il—(l—K/b)exp[”—(•;/6DAB)JVI (46)

f' (l—K/b){l—exp[—(t/6DAB)JVl}which is a three—parameter model (referred to as FPM—3). The three parameters are A, b/K,and L/6.

Several authors (Q,@_,Q1_,@) have successfully used this model (in the FPM—3 form)

to describe the transport of various electrolyte and nonelectrolyte solutes through reverse

osmosis membranes. Few cases involving strong solute-membrane attraction have been

examined and analyzed by the FPM relationships. Most notable were the efforts by Jonsson

(_2Q,Q) and by Burghoff et al. (Q). Jonsson (2_Q) examined three solutes (phenol, octanol, and

hexanol) in which the nature of the strong solute—membrane attraction produced negative

separation. The data for the solutes phenol and octanol were well fitted by the model.

30

However, the hexanol data were not represented well in that the parameter estimation

routine did not converge. The b/K terms for phenol and octanol were found to be less than 1.0,which reflects the large value of K that is associated with strong solute-membrane attraction.Burghoffet al. (Q) reported similar results for phenol and once again the FPM—3 relationship

did a good job of describing the observed results. By independently measuring the K value(using equilibrium sorption experiments), values ofb were calculated to be in the range 25 to

35 (indicating that the diffusivity of phenol in the pore is about 1/30th of the free solutiondiffusivity). More recently (@) the FPM—4 model was used to describe the decrease in flux as

a function of concentration for cyclic hydrocarbon solutes in water with cellulose acetate

membranes.

Surface Force—Pore l·‘low Model. 'l‘he surface force—pore flow (SF-PF) model has been proposed

recently by Matsuura and Sourirajan (_¢§,@). 'l‘his model mathematically extends the

physical concepts of the preferential sorption—capillary flow mechanism. The flow of solute ‘

and solvent takes place through pores which are modelled as perfect cylinders of radius RW

and length L. Similar to the FPM relationships, a balance of applied and frictional forces is

made on the solute in the pore. However, this balance is written as a function of both radial

and axial positions in the pore. ln the FPM relationship, only the axial dependence is

considered. 'I‘he two—dimensi0nal description makes it possible to include a surface potential

function which represents the attraction (or repulsion) of the solute by the membrane

material as a function ofradial position. The resulting equations are shown below.

The following dimensionless quantities are defined (4Q,@) as:

p : r/Re (47)c1(P) = uB(p) L/DAB (48)

‘ CA(p) = CA3(p)/CA2 (49)

31

ßl = q DAB/RT CA2 R2e (50)[32 = AP/RT CA2 (51)

where r is the radial position in the pore, uB is the solvent velocity at position p, and n is thesolution viscosity. The dimensionless symbols defined have meanings as follows: p is thedimensionless radial position, (lfp) is the dimensionless solvent velocity, CA(p) is thedimensionless concentration at the pore exit, ßl is a dimensionless ratio of physical constantsfor a s0lute—solvent—membrane system, and recognizing that RT CA2 = 112 (from Equation(12)), [$2 is the ratio of the applied pressure to the osmotic pressure at the pore entrance. Theeffective pore radius, Re, is related to the actual pore radius, RW, as follows:

Re : RW — RB (52)Using the effective pore radius takes into account the finite size of a water molecule. The

water molecule can get no closer than within one molecular radius ofthe pore wall. Therefore,

flow through the pore should be calculated based on RP rather than RW.The performance ofa inembrane can be represented by the pure water mass flux, np,

the solution mass flux, n,l., and the separation, f'. For the SF—Pl*‘ model, the pure water flux is

represented by Iiquation (1 1), and:

NB : np/MB (53)The solution mass flux, nT, and the separation, f', are given by the following two equations:

(l—E)=1—16·)&llo(p)pdp (54)HP [$2 0

1

{,:1- lo CA(p)c1(p)pdp (55)1

<1(p) p d p0

CA(p) and ¤(p) are determined as a function ofp by solving the following differential equation:

32

d2c1(p) 1 d<1(p) B2 1T- + — — + — + — (1 — exp(—<l>(p))(CA(p) — 1)dp B dp B, B,

l— —(b(p)— l) ¤(p)CA(p)=0 (56)B1

where

exp((1(p))C ( ) = (57)—— ex (1 —

Ap1+ mp) (((1 11exp(—<l>(p)) P p

with boundary conditions:

d <1(p) .l : 0 when p = 0 (58)d Band

<1(p) = 0 when p = 1 (59)

In the above equations, the form of the potential function <|>(p) is different for ionized

inorganic and nonionized organic solutes. For inorganic solutes:A

(60)( ) = —Q) P d( p)and for organic solutes:

10 when d(p) S D(Mp) I l B (611-5 when d(p) > Dd(p)

where the distance from the pore wall d(p) is given by:

d(p)= RB+ Re(l—p) (62)

The parameters in the model are the p11re water permeability coefficient, A, the pore radius,

RW, the solute—membrane interaction parameter, B (or A for inorganics), the steric repulsion

parameter, D, and the friction parameter, b. The significance of the pure water permeability

coefficient was discussed in the section on osmotic pressure effects. Pores on the surface are

33

different in size and shape so that RW is an effective average pore radius. The potentialfunction, cp(p), is a Sutherland—type function (Q), represented by Equation (61) andillustrated in Figure 2. B and D are characteristic parameters describing the potentialfunction. As a solute molecule approaches the membrane surface (decreasing d), attractionbetween solute and membrane increases (negative q>, corresponding to attractive forces) untila minimum at d = D is reached. At d = D, solute is repelled strongly by steric forces. As afirst approximation, D can be set equal to the solute radius, RA. The solute-membranefriction parameter b (defined the same as in the FPM relationship) may be written afunction of radial position. A more detailed description of the friction parameter is presentedbelow in Section 1.2.5.

'I‘heref0re, the SF—PF relationship is a five—parameter model. 'l‘he parameters are A,A or B, D, RW and b. ln principle, according to the original authors (Q), data for severalmembranes of different pore size are analyzed together for a reference solute, such as NaCl,which does not involve strong solute—membrane attraction. 'l‘he values of A (for NaCl), D,and RW are determined simultaneously by litting the data using a nonlinear least squaresanalysis. The known pore sizes can then be applied to data for an attracting solute (such astoluene) to determine the B and D parameters for that solute.

The SF-PF model has been applied successfully to the solutes p-chlorophenol,benzene, and cumene (Q) with cellulose acetate membranes. It would be expected that themodel should work well for the similar solute systems under consideration in thisdissertation. One problem with the SF—PF model approach is that it is considerably morecomplex than the other models discussed here. It is essential to have good numerical routinesto solve the equations accurately to provide meaningful results.

ati

fd=Ü® O d

Figure 2. A Sutherland potential function

35

1.2.5 Restricted Diffusion in Pores

The functions b and b(p), which appear in the FPM and the SF—PF relationships,

respectively, describe friction between the solute molecule and the pore wall. ln free solution,

friction between solute and solvent is represented by the diffusivity. However, in a pore,

there is additional friction between the solute and pore wall. As presented in the section on

the FPM relationship, b is equal to DAB/DAM. For relatively large pores, this ratio is 1.0. Asthe pore size gets progressively smaller, the size of the solute becomes signilicant compared to

the size of the pore and friction becomes a function of the ratio of the solute radius to pore

radius, A (where A = RA/RW). This situation can be compared to the hydrodynamic problem of

describing a sphere moving in a narrow cylinder. What is needed is a mathematical model

which describes the drag exerted on the sphere by the wall as a function of position in the

cylinder. Since solute molecules are seldom spherical and pores are usually not cylindrical it

is also desirable to account for the shape ofthe solute and the pore.

Faxen originally approached the problem of flow past a sphere in a narrow cylinder in

1923 (Q). The analytical solution obtained for the particular case in which the sphere is on

the center line of the cylinder is:

1/b = 1 -2.104A + 2.09 A3 -0.95 A5 (63)

This problem was reanalyzed by Bohlin in 1959 (Q), and again by Faxen in 1959 (Q).

The idea of applying the Faxen equation to describe the friction of a solute in a

cylindrical pore immediately suggests itself. Lane and Riggle (55) successfully used the

Faxen equation to describe the restricted diffusion of solutes in dialysis membranes. Bean(55) utilized the Faxen equation in his elaborate work on the physics of porous membranes.

Jonsson and Boesen (Q) applied the Faxen equation to data for transport of single solute

aqueous solutions (solutes: raffinose, sucrose, glucose, pentaerythritol, glycerol, and urea)across two cellulose acetate membranes. They found that variation in separation as a function

36

of solute size was well represented by the Faxen equation for the larger pore membrane.However, the pore radius, estimated by the Faxen equation for the smaller pore membrane,was significantly too small based on the solvent permeability.

The Faxen equation is simple in form and hence easy to use. However, it would bemore correct to allow the solute to achieve any radial position when calculating the solute-membrane friction. Bean (Q) pointed out that there is no analytica] solution for theequations describing the drag exerted on a sphere by the wall of a cylinder except the centerline solution given by Faxen. The radial dependence of the friction function b was estimatednumerically by Bean (Q), based on the work of Famularo (Q). Figure 3, reproduced from

Bean’s paper (Q), illustrates how the drag coefficient and hence the friction function, b,increases as the sphere approaches the cylinder wall. An average b can be obtained by

integrating b(p) as a function of p. The average values of b were found to be signiflcantly

higher than those calculated based on the Faxen equation alone.

Satterfield et al. (Q) have also reviewed this problem. They found that the Faxen

Equation (63) was not adequate to fit their experimental data for effective diffusion

coefficients in silica—alumina catalyst beads for a variety of binary systems of aromatic and

paraffinic hydrocarbons and with aqueous solutions ofsugars and salts. instead, they arrived

at the empirical correlation:

logl0(l/b) = -0.37 -2.0 Ä (64)Why the Faxen equation was unsuccessful is discussed in the Satterfleld paper; however, theproblem is not resolved.

More recently, Anderson and Quinn (Q) have reexamined the problem of restricted

diffusion in pores. The conditions necessary in order to assume a one-dimensional flow model

are reviewed. They strongly endorse the position of Bean (Q) that the radial dependence ofbshould be taken into account, using a two-dimensional flow model.

37

6

5

Eni; 49u.

3O020

I

0 0 0.2 0.4 0.6 0.8 I.0RADIAL POS|TION, r/RW

Figure 3. Drag coefficient as a function of the radial position in a pore (adapted fromreference (_Q§)). The friction function, b(p), increases with the drag coeflicientas the solute approaches the pore wall.

38

Matsuura and Sourirajan have discussed the applicability of the Faxen equation todescribing the b parameter in the SF—PF model. The approach they suggested in their firstpaper was modified in subsequent papers. In the first paper (Q), the friction is a function ofthe radial position in the pore. As discussed by Anderson and Quinn (Q), from a theoreticalpoint of view this approach is probably better than using an average value over the wholepore radius. The friction function is given as:

mp) : {exp(10) when d(p) S D (65)exp(E/d(p)) when d(p) > I)

so that the friction decreased as an inverse function of distance, d(p), from the memhrane,

where E is an adjustable parameter and d(p) is given by Equation (62). However, it is simplerto use a constant b value (essentially a b value averaged, in some manner, over the pore

radius). Also, most other authors have treatecl h as a constant value. Therefore, in subsequent

papers (Q,Q) they used an average b value. They reasoned that for relatively larger pores

(small A), the Faxen equation would be sufficient. But for relatively smaller pores (larger A), a

different value would be necessary. An empirical equation was determined by finding the b

values that best fit the reverse osmosis data for each of several solutes with the SF—PF modeland then these b values were correlated with the known A values (Q,Q). As illustrated

below, the Faxen equation was used for A values less than 0.22 and the empirical equation for

values above A of0.22:

1 — 2.104A + 2.09 A3 — 0.95 A5 when A S 0.22 (66)1/b Zl 1/(44.57 — 416.2 A + 934.9 A2 + 302.4 A3) whenA > 0.22

In order to keep the two equations consistent with each other at the transition point, it was

necessary to use a value of A = 0.227531, which was the calculated intersection of the two

curves. The form of the function is consistent (Q) with the data obtained by Satterfield (Q).

More recent results published by Deen et al. (Q) for Ficoll solute (Ficoll is a cross-linked

39

polymer of sucrose and epichlorohydrin) in polycarbonate track-etched membranes closely

· follow Equation (66) (see Figure 5 in Deen’s paper).

In addition to the radial dependence of the friction parameter, it is also possible to

adjust the model to account for different pore and solute geometries. These effects can include

any of: nonspherical solute, noncylindrical pore, pore size distribution, and nonrigid solute.

Including these factors can make the model more realistic but also much more complicated.

For the aromatic solutes under consideration in this work it is reasonable to assume that they

can be characterized as rigid spheres. For the membranes used, no firm information is

available on pore shape or pore size distribution, so that uniform rigid cylinders are a

reasonable first choice. Therefore, to include the above additional factors for the particular

case under consideration could not be justified based on the physical knowledge ofthe system.

Some ofthe work done in this area is referenced here for future use. The effect ofsolute shape

has been considered by Anderson and Quinn (Q) and by l)een et al. (Q). The choice of pore

shape was examined by Bean (Q) and hy Beck and Schultz (Q). Both Beck and Schultz (Q)

and Matsuura and Sourirajan (Q) have examined the influence of pore size distributions on

membrane performance. Munch et al. (Q) have compared the results for rigid and nonrigid

macrosolutes in membranes ofwell characterized pore size. These modifications to the porous

model are usually onlyjustifiable when there is physical evidence for a particular system that

such effects are important.

()ut ofthis review of the literature on restricted diffusion in pores, several significant

pieces of information arise. The problem of modelling the friction as a function of radial

position in the pore is a formidable one that has been an issue in the literature for more than

50 years. The temptation to use an average b value should be resisted until it is determined

that it is not possible to find a reasonable representation of the radial variation of the friction

function. In light of the results of Satterfield discussed above, it may be that the Faxen

40

equation will not work in the present case at all. It is possible to account for different solute

and pore geometries, but the analysis is more complex.

Steric Exclusion. In addition to friction between solute and the membrane pore, steric

exclusion of solute from pores may influence membrane transport. Since the solute radius is

of the same order of magnitude as the pore radius, there is significant exclusion just due to the

size of the solute. By simple geometric arguments, the reduction in effective diffusivity in thepore by steric exclusion is given by (1 - A)2. '[‘his idea was first proposed by Ferry (Q) and

has been discussed more recently by Ackers (Q) and Beck and Schultz (Q). The nature of this

exclusion is similar to that proposed as the separation mechanism in size exclusion

chromatography. The coupling of the steric exclusion with the frictional drag (given by the

Faxen equation) leads to the following equation:

1/b = (1 -A)2(1—2.104A + 2.09 A3 -0.95 A5) (67)

which was first given by Renkin (Q). Thus, Equation (67) is known as the Ferry—Faxen

equation or the Renkin equation. This equation was determined by Beck and Schultz (Q) to

he consistent with their experimental results on diffusion in track·etched mica membranes.

Bean (Q) also uses the additional (1 - A)2 term as in Equation (67) to account for the exclusion

or as he calls it, the steric restrictions. Figure 4, reproduced from Bean’s paper (Q),

illustrates the trends for the case of (a) steric exclusion alone, (b) steric exclusion and

frictional drag from Equation (67), and (c) with the radial variation of frictional drag

included. When the solute is small compared to the pore size, A = 0, and all three cases predict

no restricted diffusion, b = 1.0. As A increases, 1/b decreases, corresponding to increasing

friction. Adding frictional drag (curve (b)) and radial variation of the frictional drag (curve

(c)) to the model compared to the case of steric exclusion alone (curve(a)) results in the

Hl

|.O _x

Ä x

xO.8 ‘~\ x\ x

‘ x\ x\ x\ sl x \\b ‘

~ (¤lx .O 4 \ X!. x

x \\ sxx0 2 ‘ ‘=

• \\ s\\ \\x ~„

0 02 0.4 0.6 0.8 I.0

Figure 4. Restricted diffusion as a function of ratio of solute radius to pore radius(adapted from reference (Q§_)). Curve a) illustrates the influence of stericexclusion alone. Curve bl illustrates the effect of steric exclusion plusfrictional drag calculated based on the center line value from the Faxenequation. Curve c) is a numerical approximation including both stericexclusion and the radial variation ofthe frictional drag.

42

prediction ofa larger friction parameter. Therefore, when solute size is relatively large thenit is important to include the influence of steric exclusion and restricted diffusion in themodel. Jonsson and Boesen (Q) have successfully used the Ferry~Faxen equation incombination with the l•‘PM—3 relationship to model the behavior of organic solute-water-cellulose acetate systems.

1.2.6 Summary and Comparison ofTransport ModelsIn the preceding sections, several models which can be used to represent reverse

osmosis transport have been presented. At this point, the models will be summarized andcompared to each other. Table 2 furnishes a summary of the models and the appropriateequations and parameters. As indicated in Table 2, knowledge of the mass transfercoefficient, k, or some other means of representing concentration polarization is also required.'l‘he calculation of k and the modelling of the concentration polarization are discussed inSection 1.2.1. (

Although the models presented have been developed based on different conceptualpoints ofview, the final models are in some cases similar in mathematical form. 'l‘his is par-

ticularly true ofthe relationship between JV (or AP) and f' predicted by each model. With the

exception ofthe SF—PF model, all ofthe models may be written in one of the following forms:

Form a)

gj = EO + E1<£—) (68)Form b)

1 l—E0exp(—El-IV) (69)E0[1—exp(—ElJV)l

Form c)

(70)

43

Table 2

Summary of Several Reverse Osmosis Transgort Models

Equation Number ofModel Number Parameters ul Parameters

PhenomenologicalIT-PT 18, 20 3 Ep, C", 0IT—KS 24, 25 3 pB/Ax, pA/Ax, 0

NongorousSI) 26, 29 2 €p, (l)AM K/Ax)SDI 30,33 3 k',K",k"'ESD 10,37 3 CP, €M,,(DAM K/Ax)

PorousKSA 11,39 2 A,(DAMK/c)I•‘PM—3 4 11,46 3 A, b/K, t/eFPM—4 11,43 4 A, b/K2, K3/K2, t/2;SF-PF 11,54-57 5 A, RW, B, I), h

M) In calculating the number of parameters, it has been assumed that the mass transfercoefllcient, k, is known.

44

Form d)

L — 1-EO exp(—E2JV) (71)f' Ei—E0exp(-EZJV)

Form c) is similar to Form a), the difference being that f' is related to JV in Form a) and to APin Form c). Also, Form b) and d) are similar. lf EO = Ei in Form d), then Form b) and d) areidentical.

The Ei coefficients are generalized coefficients. Coefficients with the same Ei symbolare not the same from one equation to the next. The values of Ei for each of the appropriate

models are summarized in Table 3. Four of the models, [T-PT, Sl), ESD, and KSA, predict a

linear relationship between l/f' and [/JV, as in Equation (68). Both the SD and KSA

relationships predict that EO should be [.0, while for the [T-PT and ESD models EO is notnecessarily [.0. The significance of EO in Equation (68) is that at infinite flux through the

membrane, [/f' should approach Ei). For the Sl), ESD, and KSA models, the Ei parametershould be relatively concentration independent. However, for the lT~PT relationship, the Ei

parameter contains a 112 term, which implies a concentration dependence.

Both the l'l‘—KS and FPM-3 relationships can be represented by Form b). This

si1nilarity has been noted previously by several authors (ggg). 'l‘his result is significant in

that a connection can be made between the phenomenological coefficients of the lT—KS modeland the physical parameters ofthe FPM-3 model.

The FPM—4 relationship is represented by Form d). As pointed out above, when the

simplification that K2:-K3: K is imposed upon the FPM·4 model, it reduces to the FPM-3

model. The above simplification corresponds to Form d) reducing to Form b). 'I‘he SF—PF

model does bear some resemblance to the FPM-3 relationship. Equation (46) in the FPM-3

relationship can be arranged as (Q):

DAB)JV| (72)ACA2

—[ +(b/K){exp[(1/eI)AB)JV_]—1}

45

Table 3

Comgarison ofthe Form of Several Reverse Osmosis Transgort Models

orm a — = —F ( ) 1 E + E1f'0 1 JV

1 1-E exp(—E J )Form (b)f E0[1-exp(—E1Jv)l

1Form (c) F = E0 + El (RT/AP)

1 1-E exp(—E_J')ZF El-EOexp(-E2Jv)I

Model Form EO El E2

PhenomenologicalIT- PT a l/0 H2 -IT- KS b 0 (1 -0) (Ax/pA) —

Nongorous 'SD a l (DAMK/Ax) -SDI c l+x"'/x' K"/K' -ESD a 1/(1—€Ap/CA2€p) (DAMK/Ax) E0 -

Porous3 I (DAMK/E) *‘

FPM—3 b (1- K/b) (E/6 DAB) —FPM-4 d (1- K3/b) (1- K2/b) (L/2: DAB)

46

which has the same form as Equation (57) in the SF—PF relationship. The main differencesare that the SF—PF model applies over an annular region, between pand(p+dp), andtherefore is written as a function of p, and the K term is explicitly replaced as a function of thepotential function, <p(p). The similarity in form of the SF—PF and FPM—3 models is due to theparallelism in the initial assumption ofthe models.

Table 3 provides a summary of all but one of the models discussed. Dne conclusionwhich can be drawn from Table 3 is that simple agreement between the experimental dataand a model does not necessarily mean that the model is conceptually correct, since several ofthe models have the same mathematical form (Q). How well these models perlormed atfitting the data obtained in the current work is presented in the RICSUI/I‘S ANl)DISCUSSION section.

1.2.7 Physicochemical Nature ofthe Solutes Studied

The solutes which were chosen for study are the three aromatic hydrocarhonsbenzene, toluene, and cumene. As discussed above, these solutes were selected on the hases ofrange of nonpolar character, solubility in water, and societal impact. The structure of thesolutes and some relevent physicochemical data are illustrated in Table I. Most of the

information in Table l is self-explanatory. Vapor pressure values for these solutes are

significant which indicates that care should be exercised in handling samples so that littlesolute is lost by evaporation. The Threshold Limit Value (TLV) indicates that care should hetaken so that the vapors are not inhaled. The vapor pressure and TLV values were obtainedfrom reference (5).

Previous work by Matsuura and Sourirajan (Z) indicated that the qualitative

behavior of hydrocarbon-water separation could be expressed by the nonpolar character of thesolute. For the purpose of delining a nonpolar parameter, l\/latsuura and Sourirajan (2) used

47

the molar attraction constant defined by Small (4). Experimental data was collected forseveral different classes of hydrocarbon solutes including aliphatic, alicyclic, and aromatichydrocarbons. They found that separation could be related to the molar attraction constantfor a given class of hydrocarbon. However, the correlation of separation versus molarattraction constant resulted in a different line for each class ofsolute. This result is less thanoptimal. lt would be better ultimately to find one parameter that would correlate all the data.

lt is important to examine exactly what is meant by the nonpolar character of thesolute. This term refers to the dispersive character of the solute, which is related to theLondon dispersion forces (Q). London dispersion forces are the result of induced dipolemoments which can be represented by the polarizability of the solute (Q). Polarizability, P,can be calculated by the Lorentz—Lorenz equation (Q):

P Z ..L ww[.,2+2 g 4 rx NUwhere n is the refractive index, g is the density, and NO is Avogadro’s number. Refractiveindex values were obtained from reference (Q). For the 12 aromatic hydrocarbons listed inTable 4 the polarizability was calculated and compared to the molar attraction constant as

reported by Matsuura and Sourirajan (2). That the molar attraction constant is related to the

polarizability is illustrated in Figure 5. Therefore, when the term "nonpolar character of thesolute" is used in this work, it refers to the dispersive character or polarizability ofthe solute.

48 I

Table 4

Relationshig between Polarizability and Modified Sma1l’s Number for SomeAromatic Hydrocarbon Solutes

Solute Molecular Solute Refractive Polarizabilityh MolarWeighta Densityß Indexü x 1030 Attractionkg/kmol kg/m3 m3 Constantß

Benzene 78.11 879 1.5011 10.38 53.4Toluene 92.15 867 1.4961 12.31 61.4Ethylbenzene 106.17 867 1.4959 14.18 70.0o—Xylene 106.17 880 1.5055 14.19 70.3m-Xylene 106.17 864 1.4972 14.26 70.3p—Xylene 106.17 861 1.4958 14.27 70.3Cumene 120.20 862 1.4915 16.03 77.1Propylbenzene 120.20 862 1.4920 16.04 78.6Mesitylene 120.20 865 1.4994 16.18 79.1t-Butylbenzene 134.22 867 1.4927 17.84 83.1s—Butylbenzene 134.22 862 1.4920 17.90 85.7i—Butylbenzene 134.22 853 1.4866 17.92 85.7

8 Values obtained from reference (@1l> Polarizability calculated from Equation (72).¢ Molar attraction constants have units of (kJ m3)!/2/kmol which were converted from the

values in reference (2).

49

Figure 5. The relationship between polarizability and the molar attraction constant Forseveral aromatic hydrocarbons.

50

GruÄ

.Al

\<\I\

Of'\PO

QE">At°•

O u.:O•+*

S+*

U)C0

OOCNoliO0L

<¤<L(U

oOX

0 0 0 0 0 0'°O CD CD *4* <<I C;N Oi ? Ci Q** 1***

0gOl" [gw] ‘¥!|!9°Z!~'°|°d

Chapter 2

EXPERIMENTAL

This section briefly describes the experimental methods used in this study. More

details of membrane making and experimental procedures have been given elsewhere

(@,Q-EQ-Q). The methods used here have been adopted from those sources. Thechemicals and equipment used are listed separately. The reverse osmosis flow cell equipment

list is given in the laboratory equipment operating manual (32).

2.1 Membrane Fabrication

ln this study, l.oeb—Sourirajan type cellulose acetate membranes (@,31) were

fabricated. Briefly, the casting solution composition was 17.0 wt—% cellulose acetate, 69.2 wt.-

% acetone, 12.35 wt.—% water, and 1.45 wt.-% magnesium perchlorate (33). The casting

solution was cooled to 0°C and then cast, with a Gardner Knife, on a glass plate at room

temperature (25°C). An evaporation period of 1.0 minute was allowed before the membrane

was gelled by immersing it in ice water (0°C). After one hour, the membrane sheet was

removed from the glass plate and cut into circles of the appropriate diameter for mounting in

the reverse osmosis cell. Prior to mounting in the flow cell, the membrane was annealed in

water for 10 minutes. The annealing step has been shown to increase the separation and

reduce the flux of the membrane. The higher the annealing temperature used, the greater the

increase in separation. The temperatures used were 68.0, 72.0, 75.0, 79.0, 82.0, and 85.0°C.

'[‘his method produced a series of 6 membranes with gradated morphological properties.

After mounting the membranes in the cells, they were pressurized at 12,000 kPa for twohours to stabilize the behavior (QQAB).

51

52

2.2 Reverse Osmosis Experiments

The apparatus is illustrated schematically in Figure 6. A schematic ofa radial flowcell is presented in Figure 7. A detailed description of the equipment and operatingprocedures is given in reference (Q). Brieliy, the system consists of six radial flow cellsconnected in series, a high pressure diaphragm metering pump, a bladder accumulator, aback pressure regulator, and a pressure gauge. The feed solution was pumped through the sixcells, and after passing through the regulator, it was returned to the feed solution. 'l‘hemembrane permeate for each cell was collected separately, weighed, and analyzed to give theflux and permeate concentration. The feed concentration remained approximately constantduring the course of the experiment since the volume removed through each membrane wassmall compared to the total feed solution volume. The cells were placed in a constanttemperature chamber in order that the temperature could be held constant at 25 i* l°C. The

apparatus allows the feed flow rate, feed concentration, and operating pressure to be

controlled independently. By using the six cells, different membranes could be studied under

the same experimental conditions.

Experiments were performed at the following conditions. The temperature andfeed flow rate were fixed at 25 i 1°C and 400 i" 20 m€/min., respectively for all experiments.

The operating pressures used were 690, 1725, 3450, and 6900 kPa gauge (gauge pressures are

used throughout this dissertation). At each pressure, experiments were repeated at four ormore concentrations for each ofthe three aromatic hydrocarbon solutes. 'l‘he range ofthe feed

concentrations for the organic solutes was from 5 to 260 ppm, depending on the solubility

limit and the sensitivity of the chemical analysis. Data were collected for each of the six

celllulose acetate membranes which had been previously annealed at different temperatures.

The range of sodium chloride-water separation was from 58% to 98%. Thus the effects of

changing operating pressure, feed concentration, solute, and membrane were studied.

53

. N2 GASUNDER

PRESSUREDRAINIIIII""°V°'''''°'”'°''' ''''°“““I

I LMMQI II I' II II I{ Faso {, SOLUTION IIl :I„ sum ; "§§ää‘é'*‘L, ,,,,,,,__‘ TANK { «

III:M ’- ? *‘

° I PRESSURE{ CONTROLI PERMEATE I VALVEI{ iILCONSTANT TEMPERATURE BOX J

Figure 6. A schematic ot the reverse osmosis test equipment. ln the actual equipment,six cells were connected with the Feed streams in series.

MEMBRANE

D

FEEDour+—FFEEFDEDDEEFEDEDED PQROUS

55

Each experiment consisted of the following steps. First, the pure waterpermeability of the six membranes was measured by using a pure water feed, under a givenset of operating conditions (feed flow rate, pressure, and temperature). Then the feed wasswitched to a feed solution of specified concentration. After the system reached steady state,the permeate flux and permeate concentration for each cell were measured. This stepcorresponds to measuring the quantities np, nT and XA3 at a given AP, XM, and temperature,T. Steady state was insured by allowing at least 10 mL of permeate to pass through themembrane. Previous studies (2,&2_,@g) have indicated that this procedure was sufficient.

Sodium chl0ride~water experiments were repeated periodically to characterize themembranes and to monitor any changes in the membrane performance. 'l‘hese experimentswere performed at 6900 kPa operating pressure ancl a feed concentration of 10 000 ppm NaC€.

2.3 Chemical Analysis

Sodium chloride concentrations were measured using a YSl Model 31 ConductivityMeter. A calibration curve of conductance, in pmhos, versus sodium chloride concentration,in parts per million (ppm), was constructed over the range of 0.5 ppm to l2,()0() ppm. Allsamples were analyzed at 25( i0.5)°C. Details of the analysis and the calibration data aregiven in reference (LQ).

The organic solutes were analyzed using an Oceanography lnternationalCorporation Total Carbon System, Model 524C. This instrument allowed the dilute organicin water samples to be sealed into glass ampules for later analysis. lmmediately sealing thesamples reduced the loss of solute by evaporation. The sealed ampules, which containedpotassium persulphate as an oxiclating agent. were heatecl in an autoclave to oxidize all thecarbon to CO2. The ampules were broken in the analyzer and purged with nitrogen gas. Thepurging gas carried the CO2 to an infrared gas analyzer. Output from the gas analyzer was

56

integrated and peak areas were used to determine the carbon concentration in the sample.Details of the calibration and many precautions employed in the analysis are given inreference (IQ).

2.4 Chemicals

The chemicals used in this project are as follows:

1. Acetone (C31{6()); MA = 58.08; Fisher Scientific (A-18); Certified ACS; Lotunknown.

2. Benzene (Cöllö); MA = 78.12; Fisher Scientific (3-245); Certified ACS; Lot 773729.3. Cellulose Acetate; 39.8 wt.—% acetyl content; ASTM viscosity 3; lüastman Chemical

(E398-3); Lot A6F.

4. Cumene (CQHIZ); MA = 120.20; Fisher Scientific (()—2065); Certified; Lot 781931.— 5. Glycerol (C3l18()3); MA = 92.10; Matheson, Coleman and Bell (GX—185); Reagent

ACS; Lot A 4N28.

6. Magnesium Perchlorate (Mg(ClOA)2); MA = 223.23; Fisher Scientific (M—54);Reagent; Lot B106.

7. Potassium Persulphate (KZSZO8); MA = 270.33; Matheson, Coleman and Bell (PX-1560); Reagent ACS; Lot E8M04.

I

8. Sodium Chloride (NaCl); MA = 58.45; Fisher Scientific; Certified ACS; Lot 791337.9. Toluene (C7fI8); MA = 92.14; Fisher Scientific ('l‘—324); Certified ACS; Lot 70287.

2.5 Equipment

The equipment used in this project is documented in detail in reference (Q). Other

equipment is as follows:

1. Analytical Balance; Mettler AK160; 4 decimal digit accuracy.

57

2. Carbon Analyzer; Oceanography lnternational Corporation; Model 524C.3. Conductivity Meter; YSI; Model 31.

4. Pipetman; Gilson H-79-13637; Models P1000 and P5000.

Chapter 3’ RESULTS AND DISCUSSION

In this chapter, the experimental results obtained are presented and discussed.The results for the sodium chloride-water system are used to characterize the test cells andthe membrane samples. The data for the three aromatic hydrocarbon solutes are presentedand then intcrpreted in a qualitative manner. A description of the compaction analysis andthe numerical parameter estimation methods is given. Finally, a quantitative description ofthe hydrocarbon solute data, using the models presented in the lNTR()DUCTl()N, ispresented and the applicability of the various models is discussed. The models investigatedinclude the phenomenological transport models of irreversible thermodynamics, thenonporous membrane transport models based on a solution-diffusion approach and the porousmembrane transport models.

The exact structure of the membrane surface layer is unknown. The question ofwhether the membrane surface is porous has not been resolved. However,rin order to discussthe data in Sections 3.1-3.3, it has been assumed for the moment that the membranes areporous. This assumption allows a qualitative explanation of the data to be made, with poresize as one of the parameters. The data and most of the discussion in Sections 3.1-3.3 have

been published previously (Q). Subsequent work with cyclic hydrocarbon solutes was foundto be qualitatively similar (Q).

The raw data are tabulated in the Appendices. Appendix B summarizes the NaCl-

water performance data and Appendix C summarizes similar information for the aromatic

hydrocarbon solutes.

58

59

3.1 Membrane and Cell Characterization

The six cells and corresponding cellulose acetate membranes wel e characterized bypure water and sodium chloride—water performance. The data are summarized in Table 5.Characterization experiments were repeatccl at regular intervals to monitor membranechange, and the tabulated data represent the average of 17 tests.

The pure water permeability coefficient (a measure of the overall membraneporosity (@)) decreasedover the period of several experiments. The decrease, which isattributed to membrane compaction, varied from 10% for the most permeable membrane to5% for the least permeable membrane. The rate of membrane compaction during exposure tothe hydrocarbon solutions was slightly higher than is usually observed for salt solutionexperiments at similar pressures. When cellulose acetate is exposed to a high concentrationofan organic solute the polymer can swell or dissolve. Swelling the polymer would lead to anincreased rate of creep of the membrane structure which would be observcd as an increasedrate of compaction. l·‘or the dilute solutions used in this study no such detrimental effect wasexpected. llowever, repeated exposure to the relatively high interfacial concentrations mayhave caused a slight swelling of the polymer, which ultimately increased the rate ofcompaction. 7

The original membrane in cell 6 was damaged during the course of study and wasreplaced by a membrane annealed at the same temperature. Since these two membraneswere similar in performance, they were treated as one membrane (membrane 6) in allanalyses. _

The average separation and the mass flux for pure water and for aqueous NaClsolutions measured under the indicated conditions are listed in Table 5. The six membranesamples covered the range of sodium chloride-water separation from 58 to 98%

60

Table 5

Characterization and Performanceof the Cellulose Acetate Membranes and ofthe Radial Flow Cellsa

Film and Cell Number

1 2 3 4 5 6

Annealing temperature,T, °C 85.0 82.0 79.0 75.0 72.0 68.0llydraulic permeability,€p,

x 109, m/s kPa 1.277 1.560 2.193 3.285 4.014 4.421Pure water permeabilitycoefficient,A, x 107, kmol/mzs kPa 0.7068 0.8635 1.214 1.818 2.222 2.447NaCl - Water separationb,f, % 97.9 97.3 96.0 91.1 78.9 58.5

Total solution mass flux,nT, x 10*}, kg/m2 s 7.57 9.25 12.84 19.06 23.48 24.88

Solute transport parameter,(DAMK/t)NHCl, x 107, m/s 1.388 2.157 4.166 12.87 39.05 63.94ln C*NaCl -12.555 -12.114 -11.456 -10.328 -9.218 -8.725

Mass transfer coefficient,k, x 106, m/s 43.0c 50.9° 49.9° 51.8° 49.5c 25.6‘l

a Film area, 1.443 x 10’3 m2; operating pressure, 6900 kPa; feed concentration, 10 000ppm NaCl; temperature, 25°C; feed flow rate 400 ml/min. The experimental resultsreported are the average of 17 tests.

b Separation is calculated based on the feed concentration, f = (CM - CA3)/CM.° An average value ofk : 49.0 x 10’6 m/s is used.d A value ofk = 25.0 x 10“6 m/s is used.

61

(corresponding to a range of the pore size parameter, ln C*Na(,l, from -8.7 to -12.5)measured at 6900 kPa operating pressure and 10 C00 ppm feed concentration. To determinethe pore size parameter, the sodium chloride—water data were analyzed by the Kimura—

· Sourirajan Analysis. The solute transport parameter, DAMK/t, and the mass transfercoefficient, k, were calculated from the experimental data for each sodium chloride—waterexperiment using Equations (3), (8), (11), and (38) and the osmotic pressure data tabulated inAppendix II of reference (QQ). The ln C*NaCl values were calculated using Equation (14).Average values of DAM K/t, k, and ln C*NaCI are listed in Table 5. The solute transportparameter for sodium chloride, and hence ln remained essentially constant over theexperimental time period. When A decreases and ln C*NaCl remains constant as a function oftime, this indicates that the effective pore diameter has remained constant and the effectivepore length has increased (@). Since the separation is primarily determined by the effectivepore diameter the results from early and later experiments can still be compared.

Values of the sodium chloride mass transfer coefficient listed in Table 5 arereasonably constant and an average value of k = 49 x 1o·"m/S was used for cells 1 to 5.llowever, cell 6 had a marginally larger flow channel for feed which resulted in lower feedsolution velocities and a lower k value, 25 x 10'6 m/s. ’

3.2 Hydrocarbon—Water—Cellulose Acetate Reverse Osmosis SystemThe experimentally determined performance for the separation of benzene and

water at four different pressures is illustrated in Figure 8. The data obtained at eachoperating pressure indicate that as pore size increases the benzene—water separationdecreases, reaches a minimum, and then increases. For each pressure, the separation wasshown via an analysis of covariance (Q, Z4) to have no concentration dependence at a 0.99

62

Figure 8. Effect of feed concentration on the reverse osmosis performance for thebenzene·water system at four different pressures. membrane material =cellulose acetate; ln obtained at 6900 kPa; feed flow rate : 400mL/min; temperature = 25°C. .

63

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64

Figure 9. Effect of feed concentration on the reverse osmosis performance for thetoluene-water system at four different pressures. membrane material =cellulose acetate; ln C*N„C; obtained at 6900 kPa; feed flow rate = 400mL/min; temperature = 25°C.

65

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66

Figure l0. Effect of feed concentration on the reverse osmosis performance for thecumene-water system at four different pressures. membrane material =cellulose acetate; ln C*N,,g; obtained at 6900 kPa; feed flow rate = 400ml,/min; temperature = 25°C.

67

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68

confidence level. Comparing separation data over the range of pressures studied, it isobserved that as the operating pressure increases the separation decreases.

At each pressure and for each concentration, as the pore size increases the extent ofpore blocking decreases. The data have been fit to a first order (linear) relationship. Anumber of points are worthy of note. For any given operating pressure, the slope increases

with increasing feed concentration. This observation is in contrast to the previously stated

apparent concentration independence of separation. The effect of operating pressure on theextent of pore blocking is not immediately evident from Figure 8, due to the additional

influence of concentration. The influence of pressure on extent of pore blocking is discussed

below. By way of comparison, ifa plot of(l—n„[./np) versus ln C*NaCl were made for sodiumchloride—water systems in the concentration range discussed here, the result would be a

horizontal line at (l—nT/np) equals zero. At higher concentrations of sodium chloride in

water, there would be a reduction in the ratio n„[Jnp due to osmotic pressure effects ratherthan the blocking of pores by solute molecules. However, the results would not necessarilyfollow a simple relationship such as that shown in Figure 8.

The data for the single-solute systems of toluene-water and cumene—water are

illustrated in Figures 9 and 10, respectively. These results are similar to those for the

benzene—water system. The shape and direction of the separation curves as a function of the

pore size parameter ln C*NuCl are similar to those for benzene. Separation again decreases

with increasing pressure and remains independent ofconcentration. Similarly, the decrease

in extent of pore blocking with both increasing pore size and decreasing concentration are

observed. Since cumene is much less soluble in water than either toluene or benzene (seeTable 1), the concentrations of cumene were much lower. Even when these concentration

differences are taken into consideration, differences are observed between the three solute-water systems. For any given pressure, concentration, and membrane, the separation

69

observed for the benzene—water system is less than that of the toluene—water system, which isless than the corresponding value for cumene—water. For similar molar concentrations, theextent of pore blocking increases in the order benzene < toluene < cumene, and the slope ofthe (l — nT/np) versus ln C*NaCl plot increases in the same order.

For all three solutes there is a minimum in the separation versus ln C*NuCl curves.This minimum appears to occur, within experimental error, at an ln C*NaCl value of- 9.75 forall solutes. The rate at which separation decreases and then increases with increasing pore

size once again is ordered benzene < toluene < cumene.

By taking the concentration—independent average separation obtained for a given

solute and membrane, it is possible to plot the data for all membranes and solutes as a

function of pressure. These plots are illustrated in Figure ll. Note that the curves for

membranes 3 to 6 are similar. l\/lembranes l and 2 exhibit the same trend as the other

membranes although the position of the curves are different. ln particular, the curve tor

cumene—water with membrane 1 is concave downward whereas all the other curves are

concave upward. The actual lines given in Figure ll are third order curves which have been

hand fit to emphasize the parallelism in the data and are not hest~fit curves through the data

points. Each data point in Figure ll, represents the average of several points (either 4 or 6

points) at different concentrations. For the three solutes, all membranes exhibit a decrease in

separation as pressure is increased. As discussed in the INTRODUCTION, this decrease in

separation with increasing pressure is characteristic of strong solute—membrane attraction.

From a practical point of view, it is desirable to operate at low pressures to increase the

separation; however, this decrease in pressure will also result in a decrease in flux.

The blocking of the pores on the membrane surface by sorhed solute can he treated

in a more quantitative manner than presented above. The appropriate k value for the

70

Figure ll. Effect ofoperating pressure on the separation ofbenzene (Ü), toluene (O), andcumene (A) in single solute aqueous systems for the six cellulose acetatemembranes annealed at different temperatures.

7l

FD

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A 72

particular solute ofinterest was estimated for each cell (Q) using Equation (5) and the knownvalue of k for NaCl listed in Table 5. The values of DAB for the three organic solutes wereestimated by the method of Wilke and Chang (§). For the Wilke—Chang method it wasnecessary to estimate the molal volume of the solute at the normal boiling point, VA, usingthe method of LeBas as outlined in reference (Q). The diffusivity of sodium chloride in water is1.61 x 10“9 m2/s as reported in reference (Q). Calculated values of VA and DAB are listed inTable 1. The boundary layer concentration, CA2, and hence XA2, were calculated fromEquation (3) for each experiment.

For membranes 1 and 6, the extent of pore blocking versus XA2 is illustrated inFigures 12-14 for the three hydrocarbon solutes. These figures illustrate that extent of poreblocking is independent of pressure. Similar results were obtainecl for the other fourmembranes. Thus, pressure changes effect the same relative change both in solution flux andin pure water flux. Figures 12-14 suggest that a more quantitative treatment of the poreblocking data is possible. _As stated in the lNTR()l)UCTl()N and in previous publications (Q-Q), it is possible to relate the extent of pore blocking to the boundary concentration byEquation (40). This equation, which has the form of the Freundlich sorption isotherm, isconsistent with the sorption phenomenon taking place between the solute and the membranesurface. By taking the natural logarithm of both sides of Equation (40), a linear equation isproduced. Using a linear least—squares (it, the coefficients Z and Q were calculated. The Qvalues were all reasonably close to 1.0 (the average Q is 1.035 i 0.043 (95% confidence) basedon 18 estimates), so the Z values were recalculated with Q set equal to 1.0. Setting Q : 1.0 is

equivalent to saying that the (1—nT/np) versus XA2 relationship can be represented by astraight line through the origin as indicated in Figures 12-14. These new Z values for allmembranes are given in Table 6. The data in Table 6 demonstrate not only the dependence ofZ on the average pore size but also the dependence of Z on the nonpolar or dispersive

73

O¤ 6 E5 CZ6 19 °°¤ \

~ 5\ LD\(‘\ am4( G<—·E9 5 55U')EI El lg

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<l ~ ><6 \1 flI

%c<• CZ

S1 •'•

° Ä?\ O

O

9I °U ZI '0 80 ’0 V0 °0 00

Figure 12. Correlation of extent 0f” pore blocking, (1-nT/np), and bounclary layerCO!'!C€f1{!‘21{lO1°1 of b€f1Z€f1€, XAQ, For two l‘€pI‘€S€f1{zl{1V€ IT1€II1l)1°2l1'l€S Ol~(lll‘l‘€l‘€l1{pore size: ([:|) 690 kPa; (O) 1725 kPa, (A) 3450 kPa, (O) 6900 kPa; ( — )membrane 1; (---) membrane 6.

vu

i\ g9\ ü'

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O

ze *0 vz *0 91 *0 00 *0 00 *0°

Figure 13. Correlation of extent of pore blocking, (l—nq~/np), and boundary layerconcentration of toluene, XM, For two representative membranes of differentpore size: (I3) 690 kPa; (0) 1725 kPa, (A) 3450 kPa, (O) 6900 kPa; ( — )membrane 1; (———) membrane 6.

75

‘? 26;; Ö6 \ cO 1 2

1ll\ qG19 (D1 •„——l

El , 6 E El ‘<> „6 ‘ " XÄ) d(’\lO O ll6

(1 2 CE6 616 6 ><<>l ‘ @

l O

Q l cil\l O

Q

St ’0 ZI '0 80 °U vu '0 00 '0°

. Figure 14. Correlation of extent of pore hlocking, (1—n·p/np), and boundary layerconcentration of cumene, XAQ, For two representative membranes of dilTerentpore size: ([3) 690 kPa; (0) 1725 kPa, (A) 3450 kPa, (O) 6900 kPa; ( — )membrane I; (——-) membrane 6.

76

Table 6

Progortionality Constants from the Correlation of XA2 and the Extent of Pore Blocking

Z .

Membrane Benzene Toluene Cumene1 2944 7538 308952 2501 6535 275663 2122 5525 231204 1696 4527 187215 1348 3592 134036 935 2183 8072

77U

character of the solute. With the limited amount of data available, it is not possible tospeculate on the exact nature of the relationship between Z and the dispersive character otherthan to state that Z increases with increasing solute dispersive character. The physical

_ interpretation of this trend is that at identical molar concentrations benzene blocks the porethe least while cumene has the greatest ability to block the pore.

There are sufficient data to obtain the functionality between Z and In C*NaCl. InFigure 15, this functionality is shown to be adequately represented by a linear relationship ona semi-logarithmic basis. Within experimental error, the slopes of these three lines appear tobe identical. Figure 15 may be used to predict Z from the experimental ln C*NaCl data for anyof the three solutes studied. Subsequent work with cyclic hydrocarbon solutes also followedEquation (40) with (equal to 1.0. The cyclic hydrocarbon data when plotted as in Figure 15produced lines parallel to the aromatic hydrocarbon data (Z4). Therefore, in spite of theempirical nature of the treatment, Figure 15 and Equation (40) can be used to predict theextent of pore blocking when the solute is strongly attracted to the membrane material.

3.3 Interpretation of the Hydrocarbon Solute Data

The results presented above can be interpreted qualitatively based on a modifiedversion of the preferential sorption—capillary flow model (Z, @). The membranes areassumed to be porous. The flux and separation are determined by the number and size ofpores, by the interactions between solute, solvent, and membrane, and by the relativemobility of solute and solvent in pores.

The trends in Figures 8-11 are consistent with the qualitative features of strongsolute-membrane attraction. The solution flux is lower than the pure water flux due to poreblocking rather than to any significant osmotic pressure effects (the maximum osmoticpressure difference across the membrane in any of the experiments was about 3.5 kPa, which

78

caai

III BENZENEQ CD TULUENE_·:•_ A CUMENEO6"" Al‘*\|Acac°‘ A

·

Q606-13. 0 -12. 0 -11. 0 -10. 0 -9. 0

>|<l V1 C N QC l

Figure 15. Correlation 0f‘pr0p0rti0nality constant, Z, and pore size parameter ln C*N„(;l.

79

is much lower than the lowest operating pressure of690 kPa). This effect is enhanced eitherby decreasing the pore size or by increasing the feed concentration. Both of these factors leadto a relative increase in the solute content in the pore and thus to a restricted solutiontransport through the pore.

ln the highly concentrated region of solution near the membrane surface (referredto here as the interfacial region), the solute—membrane attraction forces decrease with

distance from the membrane surface. Thus, solute mobility increases and solute

concentration decreases with increasing distance from the membrane surface. ln the limit, apoint is reached at which solute mobility and concentration can be considered to be no longer

under the influence of interfacial attractive forces. When a portion of this interfacial layer istransported through the pores of a membrane, the average solute mobility and the averagesolute concentration of the transported portion determine the llux and separation. lf the

average pore on the membrane surface is extremely small, the portion of solution transported

is that region ofhigh concentration and low mobility. As the average pore size is increased,

the portion of the interfacial layer being transported increases to include less concentrated

but more mobile solution. Thus, the average concentration in the pore decreases and the

average solute mobility increases. Decreasing average concentration of the transported

solution indicates increasing separation. increasing solute mobility of the transported

solution indicates decreasing separation. These two factors present opposing influences on

separation as a function of pore size. Note that the solute in the region near the membrane is

strongly attracted to the membrane and is relatively immobilized while the water remains

relatively mobile. However, the velocity of the solution passing through the pore creates a

shear force which strips the sorbed solute molecules from the surface and transports them.

The factors discussed above can be used to explain the results observed in thepresent study. The opposing factors ofdecreasing concentration and increasing mobility with

80

increasing distance from the surface suggest that separation as a function of pore size canexhibit a local minimum as observed in Figures 8-10. The initial section of decreasingseparation with increasing pore size represents the transport of the lower portions of theinterfacial layer. ln this region, the mobility of the solute is a stronger function of distancethan is the concentration; thus, solute mobility determines the separation. The section of thecurve with increasing separation as pore size increases represents the transport of a largerportion of the interfacial layer. In this region, the solute concentration is a stronger functionofdistance than is the mobility; thus, the average solute concentration in the pore determinesthe separation. Between these two sections of the curves there is a minimum where the twoopposing factors cancel.

For a membrane with a given average porosity, increasing the pressure increasesthe velocity of the solution through the pore, resulting in an increased flux. The increasedvelocity causes increased solute mobility due to shear of the concentrated interfacial layer;thus, separation decreases with increasing pressure as observed in Figures 8-11. 'l‘he solute-water separation may be positive, negative, or zero depending upon the system and specificoperating conditions. When these factors are such that the solute—membrane interaction isrelatively strong, the solute is immobilized and the permeating water leads to positiveseparation. When these factors are such that the solute-membrane interactions are relativelyweak, the solute enriched solution in the interfacial region is carried through the pore withless resistance, resulting in negative separation. Balancing these two factors produces zeroseparation. The solution flux is significantly less than the pure water flux because the solutemolecules sorbed to the pore wall tend to decrease the effective pore diameter. Thisphenomenon is known as the pore blocking effect.

ln comparing the present work to aromatic hydrocarbon-water-cellulose acetatestudies reported in the literature (Z), a number of points are of interest. The present findings

81

are in agreement with the previously reported results in that for any given system theseparation is pressure dependent. However, Matsuura and Sourirajan (Z) also report apressure dependence of the extent of pore blocking; in the present study no such pressuredependence was found (see Figures 12-14). instead, the extent of pore blocking was found tobe concentration dependent (see Figures 8-10). The discrepancy can be explained by the factthat Matsuura and Sourirajan treated a range of feed concentrations (36 to 64 ppm benzene,19 to 53 ppm cumene) as one concentration. Evidently, a different concentration was used foreach pressure study; thus, the concentration effect was incorrectly interpreted as a pressureeffect. The present study demonstrates the signiticant role that concentration plays indetermining membrane performance in systems showing strong solute-membrane aflinity.Since the extent of pore blocking is primarily a function of the physicochemical interaction ofthe solute and the membrane, it is more logical that the extent of pore blocking beconcentration dependent as reported here rather than pressure dependent as reported earlier(Z).

The dependence ofthe extcnt of pore blocking on the physicochemical nature of thesolute is evident when Figures 12-14 are compared. All other parameters being equal, theextent of pore blocking is least with the more mobile solute (henzene) and greatest with theleast mobile solute (cumene).

In comparing the results in previous sections, certain differences in behavior werenoted for the three solutes. 'I‘he order of these differences is always benzene < toluene< cumene. This order is the same as the order of the dispersive character of the solute and,therefore, the order of solute-membrane affinity. Thus, the dispersive character of the soluteprovides a rough guide for predicting the reverse osmosis performance of the aromatic

hydrocarbon-water-cellulose acetate membrane system. In order to obtain a more

82

quantitative relationship between performance and dispersive character, data for moresolutes are required.

3.4 Membrane Compaction Analysis

The dependence of the pure water permeability coefficient, A, on the operatingpressure is illustrated in Figure 16 for each of the membranes. As predicted by Equation ( 15),a linear semilogarithmic relationship is obtained. Each point on the graph represents theaverage value of several data obtained at different times under the same conditions. Rangingfrom 1.2x 10“5 to 3.8x 10“5kPa“l, the values of y obtained are comparable with thosereported in the literature by Sourirajan (3@) which ranged from 3.0x l0’5 to 6.0 xl0“5kPa“l. Therefore, Equation (15) appears to be valid for the cellulose acetatemembranes used in this study.

Some of the models examined in this study make the assumption that nomembrane compaction is occurring. Over the range of pressures studied, the A valuesdecreased between 5% for the least permeable and 10% for the most permeable membrane. ltis clear that A is not independent of pressure for the membranes used in this study.

3.5 Parameter Estimation Methods

Before discussing the details of the transport analysis, some general information isgiven here about the parameter estimation methods. For the majority of the models, theInternational Mathematics and Statistical Library (E) IMSL routine ZXSSQ has been used

to estimate the unknown parameters. This routine is an unconstrained non—linear parameterestimation routine which uses a modified Levenberg—Marquardt algorithm (E, Q) that does

not require the user to supply explicit partial derivatives. In all cases the routine acts tominimize the sum of the square error between f' calculated and f°’ experimental by adjusting

83

Q

'E “‘0- IJ•

I

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I-O

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.6•-IC I

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QS•-O .I 0. 0 2. 0 4. 0 6. 0 8. 0

Pressure [kPoCI ä<lÜ’3

Figure I6. Membrane compaction as represented by the decrease in the pure waterpermeability coeFFicient, A, as a Function oF increasing operating pressure Forthe six cellulose acetate membranes.

([])membrane1; (O) membrane 2; (A) membrane 3; (O) membrane 4;(+) membrane 5; (X) membrane 6.

84

the model parameters. For specific models, the dependence of f' on system parameters is

written as 1/f', but the program always uses the error in f' for the minimization so that all

the models can be compared equally. The standard deviation of the error was calculated (Q)

as: _S = iz ir' - r'(_u,>2/d.r.>‘” mi

where d.f. is the number of degrees of freedom, which is given by the number of data minus

the number of parameters estimated.

Due to the sensitivity of the surface force-pore flow model to the adjustable

parameters, an unconstrained optimization code was found to be undesirable. Therefore, a

constrained optimization code called GRG2 (ßl was used. ’l‘his algorithm uses a Generalized

Reduced Gradient method (Z2) which allows equality and inequality constraints to be

incorporated into the optimization. Supplying partial derivatives is optional in this program.

With GRG2, the adjustable parameters could be bound to reasonable limits in order to

prevent the algorithm from choosing unreasonably small or large values.

3.6 Phenomenological Transport lVlodels

ln this section the results of analyzing the aromatic hydrocarbon data by

phenomenological transport models ofirreversible thermodynamics are discussed.

3.6.1 lrreversible Thermodvnamics — Phenomenological Transport Relationship

The form of the irreversible thermodynamies-phenomenological transport

relationship (IT—PT) as discussed in Section 1.2.2 is:

L _ L 2 fg L L (20)V · ; +(2 ·“ lf., lvl.,)p V

85

The three parameters in the model are Ep, 0, and En. The model is of Form al asrepresented by Equation (68), which suggests a linear relationship between 1/f' and 1/JV atconstant concentration (actually constant ng). For the aromatic hydrocarbon data it wasdifficult to maintain a constant concentration for different experiments. Therefore, it wasnecessary to take the concentration into account when analyzing the aromatic hydrocarbondata. The data were analyzed by three different but related methods. In the first method, anypossible concentration dependence was ignored and the separation f' and the volume flux JVwere correlated by l·]quation (68) with constant coefficients. ln the second approach, the ngterm was accounted for explicitly. Finally, in the third method, the concentration dependenceofEp was taken into account. These three methods are discussed below separately.

For all of the phenoinenological transport models, the Ep parameter was estimatedfrom the pure water data. For this purpose Equation (18), or equivalently for dilute solution

lüquation (10), was used. The results are listed in Table 5 for each of the six cellulose acetate

membranes.

Method A. When the linear relationship between 1/f' and 1/JV is assumed with constant EUand El coefficients, the model is designated as the lT—PT-A model. The values of EO, El, and sestimated for each solute and membrane are summarized in 'l‘able 7. The shape of the curves

thus calculated can be compared to the experimental data on a f' versus JV plot in Figures 17-

19 for the three solutes. 'l‘he model appears to do a reasonable job of representing the data.The 0 values presented in Table 7 are an estimate of the separation at infinite flux for each

solute and membrane. The 0 values increase, for the different solutes, in the order benzene< toluene < cumene, which is the order of increasing nonpolar or dispersive character for the

solutes (Q). For a given solute, 0 increases with increasing annealing temperature, except formembrane 6. The 0 values for membrane 6 are larger than would be expected based on the

86

Table 7Parameters for the Phenomenological Transgort Q IT—PT—A}

and Extended Solution—Dif‘fusion (ESD} Models

D KAM ‘Film Solute EO E1xl06 s 0k

1in/s kPa) m/smzs kPa

(1T-PT-A1 (IT-PT-A1 <ESD> <ESD>

1 Benzene 5.626 -3.128 0.114 0.1777 -1.444 1.632 -0.55601 Toluene 3.550 -1.745 0.120 0.2817 - 2.164 0.8407 -0.49151 Cumene 2.131 -0.8624 0.215 0.4693 -14.15 0.07826 -().40452 Benzenu 6.215 -4.517 0.090 0.1609 -1.887 2.035 -0.72682 Toluene 4.120 -2.686 0.120 0.2427 -2.871 1.083 -0.65192 Cumene 3.073 -1.934 0.184 0.3254 -22.()0 0.1215 -0.62943 Benzene 10.33 -11.56 0.()93 ().0968 - 2.906 3.079 - 1.1 193 Toluene 6.295 -6.683 0.103 0.1589 -4.676 1.691 - 1.0623 Cumene 4.845 -4.489 0.202 0.2064 -34.99 0.2010 - 1.001 ·4 Benzene 12.42 -20.26 0.071 0.08()5 -4.237 4.695 -1.6314 Toluene 8.249 -14.42 0.098 0.1212 -7.700 2.646 — 1.7484 Cumene 6.262 -10.32 ().135 0.1597 -57.62 0.3187 - 1.6485 Benzene 29.84 -84.10 0.095 0.()()34 -7.320 6.030 -2.8185 Toluene 12.88 -32.57 0.075 0.0776 -11.14 3.393 -2.5295 Cumene 5.609 -10.31 0.098 0.1783 -64.27 0.3809 - 1.8386 Benzene 7.538 -11.63 0.116 0.1327 -4.007 5.960 -1.5436 Toluene 4.372 -5.876 0.119 0.2287 -5.918 3.125 -1.3446 Cumene 2.796 - 2.299 0.110 0.3577 -28.74 0.3279 -0.8222

“Fn has been calculated assuming an average value ol' n2 = 3.85, 2.27, and 0.286 kPaFor benzene, toluene, and cumene, respectively.

h FAP has been calculated assuming an average value ofCA2 = 1.55 x 10 °3, 0.916 x10 “3, and 0.115 x 10 '8 kmol/m3 for benzene, toluene, and cumene, respectively.

87

Figure 17. The effect of volume flux on separation for the benzene~water-celluloseacetate system at 25°C. ([3) experimental points; (—) irreversiblethermodynamics-phenomenological transport model (IT—PT—A) and theextended solution—diffusion model (ESD); (— · — ·) irreversiblethermodynamics — Kedem Spiegler model (lT—KS) and the three parameterlinely—porous model (FPM—3); (—-—) solution-diffusion model (SD) and theKimura—S0urirajan analysis model (KS/\).

88

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89

Figure 18. The effect of volume flux on separation for the toluene-water—cellulose acetatesystem at 25°C. (Ü) experimental points; (—) irreversible thermodynamics—phenomenological transport model ([T—PT—A) and the extended solution-diffusion model (ESD); (— · — ·) irreversihle thermodynamics - KedemSpiegler model (lT—KS) and the three parameter finely—por0us model (FPM—3);(—--) solution—diffusion model (SD) and the Kimura-Sourirajan analysis model(KSA).

90

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91

Figure 19. The effect of volume flux on separation for the cumene—water—cellulose acetatesystem at 25°C. (|:]) experimental points; (—) irreversible thermodynamics—phenomenological transport model (IT-PT—A) and the extended solution-diffusion model (ESD); (— · — ·) irreversible thermodynamics - KedemSpiegler model (IT-KS) and the three parameter linely-porous model (FPM-3);(———) solution—diffusion model (SD) and the Kimura—Sourira_jan analysis model(KSA).

92

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93

trend observed for the other membranes. It is not known if the results for membrane 6represents a trend or if there is some other reason for the observed results. Further work is

required to clarify this point. lf the experiments had been performed such that CA2, and_ hence 1*12, were constant for all the experiments for each solute, then €n could be calculated

from the known €p, o, and El = (€n/€p —02)(€p/o)112 values. Due to the difficulty of making up

and 1naintaining a predetermined concentration of these solutions, it was not feasible to do

several experiments at the same concentration. Even if sufficient means were employed to

ensure that the same feed concentration was used for experiments at different operating

conditions, this effort would not guarantee a constant houndary layer concentration.

Therefore, strictly speaking, 112 cannot be assumed to be constant. However, for the purpose

of estimating €u for the IT-PT-A model, it was assumed that :12 was constant for each solute.The value of 112 11sed was the arithmetic average of the ::2 values calculated for all theexperiments for each solute. Calculated €H values, listed in Table 7, are all negative. 'I‘he

absolute value of Eu increases in the order benzene < toluene < cumene and decreases with

increasing annealing tempe1·ature (again membrane 6 is an exception). According to the

original definition ofthe IT model, it is impossible for the €H coefficient to be negative (E, Q).

Negative €H values imply that solute is being driven against the concentration gradient.

Since this result contradicts the original assumptions of the model, it casts doubt on the

interpretation of the results by the IT—PT—A model. Changing the value of n2 used to estimate

Eu only changes the magnitude and not the sign of the €n parameter. Therefore, the

conclusions drawn are not influenced by the assumption that 112 is constant.

Method B. A more logical approach to the application of the IT—P'I‘ model is to remove the

concentration effect of the nz term and group it with JV. Therefore, a plot off' versus (JV/ng) is

94

suggested with the three parameters €p, o, and (fn/€p — o2)(€p/0). This approach is designatedas the IT-PT-B model. _

The results for the IT-PT—B model, summarized in Table 8,are somewhat similar tothose obtained for the IT-PT-A model. The data plotted in Figures 20-22 as f' versus JV/nz aresimilar to Figures 17-19 for the IT-PT-A model. Comparing the plots for the IT-PT—A and IT-PT-B models, it may be stated that there is considerably more scatter in the data for the IT-

PT-B model. In fact, the scatter in the plots for the IT-PT-B model is large enough that it is

questionable whether a line should even be drawn through the data. 'l‘he o parameters aremarginally larger and follow the same trends as the values obtained for the I'l‘-PT—A model.The Cu values calculated for the I'I‘-PT-/\ and lT—P'l‘-B models are different but follow thesame trends. In this case, it is not necessary to assume that either CAQ or ug is constant sincethe effect of varying u2 is included in the (JV/112) term. 'l‘he s values are larger for the l'I‘-P'I‘—I§

model than they are for the I'I‘-P'I‘—A model, which is expected based on the scatter in the data

for the I'l‘-PT-!} model. This result implies that the grouped parameter (€H/€p — o2)(€p/ol ll? is

less concentration dependent than (€H/€p — 02)(€p/0). ()therwise, the o and €u values aresimilar for the I'I‘-PT-/\ and IT-PT-B model and the general conclusions are the same. The

negative €" values are inconsistent with the basic assumptions of the model.

It is apparent that neither the IT—PT-A nor the IT-PT—B model is satisfactory

because of the negative €nparameters calculated. In the estimation of the parameters it has

been implicitly assumed that the parameters are all constant, independent ofoperating pres-

sure and concentration. In fact, one might suspect that €p should decrease with increasing

concentration as indicated by the lowered flux in the presence of the solute. The next section

presents the results obtained when the concentration inlluence on {fp is included in the model.

95

Table 8F

Parameters for the lrreversible Thermodynamics — PhenomenologicalTransgort llT—PT—BQ Model

Film Solute EO 1•]lx106 s o = 1/EO €" x 107(m/s kPa) (m/s kPa)

1 Benzene 4.113 - 0.2332 0.163 0.2431 -0.56571 Toluene 2.871 -0.2784 0.180 0.3483 -0.96661 Cumene 2.002 -2.250 0.227 0.4995 -11.242 Benzene 4.532 -0.3373 0.141 0.2207 -0.74272 Toluene 3.272 -0.4109 0.178 0.3056 -1.2542 Cumene 2.720 -4.310 0.214 0.3676 -15.843 Benzene 6.971 -0.8082 0.134 0.1435 -1.1593 Toluene 4.755 -0.9857 0.159 0.2103 -2.0733 Cumene 4.011 -9.519 0.235 0.2493 -23.734 Benzene 9.234 -1.693 0.080 0.1083 -1.8334 Toluene 6.072 -2.081 0.137 0.1647 -3.4264 Cumene 5.019 -18.65 0.172 0.1992 -37.165 Benzene 15.15 -4.582 0.120 0.0660 -3.0245 Toluene 9.010 -4.671 0.108 0.1110 -5.1845 Cumene 4.686 -17.42 0.125 0.2134 -37.176 Benzene 6.494 -1.093 0.120 0.1540 - 1.6826 Toluene 3.893 -1.165 0.127 0.2569 -2.9896 Cumene 2.557 -1.724 0.117 0.3911 -6.735

96

Figure 20. The effect ofthe volume flux to osmotic pressure ratio, JV/ng, on separation forthe benzene—water—cellulose acetate system at 25°C. (|:]) experimental points;(—) irreversible thermodynamics—phenomenological transport model (IT-PT—B).

97

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98

Figure 21. The effect ofthe volume flux to osmotic pressure ratio, JV/ng, on separation forthe toluene—water-cellulose acetate system at 25°C. ([:]) experimental points;(—) irreversible thermodynamics—phenomenological transport model (I'I‘—PT~B).

99

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100

Figure 22. The effect ofthe volume flux to osmotic pressure ratio, JV/ng, on separation forthe cumene—water—cellul0se acetate system at 25°C. (Ü) experimental points;(—) irreversible thermodynamics-phenomenological transport model (lT—PT—B).

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102

Method C. ln order to investigate the possibility of incorporating a concentration dependentEp parameter into the [T—PT model, a further re—arrangement of the model was made. Thisapproach is designated as the lT—PT-C model. In the present case, An and f' values are suchthat AP > > 0Au (note that theoretically 0 is less than or equal to 1 @1)) and lüquation (18)

can be rewritten as:

JV : 6p AP (75)By substituting for €p in Equation (20) using Equation (75), the Ep parameter can be removedfrom the model. The resulting equation is:

1/f“'= 1/0 + (EV AP/JV - 02)(JV ng/0 AP)(1/JV)

= 1/0 — 0(n2/AP) +(€"/0)

(nz/JV) (76)

Data for F', JV, AP, and u2 may be used to estimate the two remaining parametcrs 0 and EV. lnessence, the experimental value For €p

lapproximated by JV/AP) is used For each experiment.Any concentration dependence that Ep may have is accounted for explicitly.

'l‘he data for each solute and membrane were fit to Equation (76). The resultsobtained were the same as the results for the lT—PT—B model within 3 digits accuracy. Thisresult occurs because the magnitude ofthe (TV and 02€p

terms in Equation (19) are such that:

6V > > 02 (rp (77)Because of the dominance of the En parameter, the €p term (or equivalently JV/AP term) has

little influence on the model. This means that including a description of the concentration

dependence of Ep will not improve the ability of the model to describe the aromatichydrocarbon data. _

As mentioned in the lNTR()DUCTl()N the [T-PT relationship contains the

assumption that the linear laws of [T may be applied across the thickness of the membrane.

This assumption may not be valid for reverse osmosis type membranes where there are large

103

concentration gradients across the memhrane. The model discussed in the next section avoidsthis problem.

3.6.2 Irreversible Thermodynamics — Kedem Spiegler RelationshipWhen the linear equations oF IT are written For a diFFerential element oF membrane

thickness and then integrated over the total membrane thickness, the equations known as theKedem-Spiegler model result. 'I‘he relationship between F' and JV predicted by the IT-KSmodel is:

L Z 1—oexp[——(1—o)(Ax/pA)JVI (25)F' 0{1—exp[—(1—0)(Ax/pA)JVI}

which is oF Form b) represented by Equation (69). The three adjustable parameters in themodel are CP, 0, and Ax/pA.

The IT-KS model was Found to be unable to describe Functionally the reverseosmosis data For aromatic hydrocarbon solutes. When the separation, F', is positive, the modelalways predicts F' will increase with increasing JV as shown by Equation (25). However, thedata indicate that F' decreases with increasing-IV. When the data are Fit to the IT-KS model, alarge value oF Ax/pA is predicted such that the exponential terms in the model approach zero.The calculated value oF 0 is then _just an average value oFF'. The 0 values and the standard

deviations, s, are listed in Table 9 For the IT-KS model. Similar to the IT—PT models, 0 is Foundto increase in the order benzene < toluene < cumene and increases with increasing

annealing temperature (except For membrane 6). 'l‘he average F' values calculated are illu-strated in Figures 17-19 For comparison. 'I‘hereFore, the IT—KS relationship must be rejected as

a possible method oFdescribing the aromatic hyclrocarbon-water—cellulose acetate data.

104

Table 9

Parameters for the Irreversible Thermodynamics Kedem—Sgiegler¤ QIT-KS} and theThree Parameter Finely—P0rous Modelb [FPM-3} Relationshigs

Film Solute EO s 0 b/K(IT—KS)1

Benzene 0.2857 0.199 0.2857 1.4001 Toluene 0.4048 0.212 0.4048 1.6801 Cumene 0.6104 0.278 » 0.6104 2.5672 Benzene 0.2612 0.174 0.2612 1.3542 Toluene 0.3566 0.202 0.3566 1.5542 Cumene 0.4593 0.258 0.4593 1.8503 Benzene 0.1738 0.161 0.1738 1.2103 Toluene 0.2518 0.180 0.2518 1.3373 Cumene 0.3178 0.263 0.3178 1.4664 Benzene 0.1318 0.100 0.1318 1.1524 Toluene 0.1993 0.153 0.1993 1.2494 Cumene 0.2528 0.193 0.2528 1.3385 Benzene 0_.0893 0.146 0.0893 1.0985 Toluene 0.1421 0.122 0.1421 1.1665 Cumene 0.2538 0.135 0.2538 1.3406 Benzene 0.1727 0.125 0.1727 1.1426 Toluene 0.2902 0.138 0.2902 1.4096 Cumene 0.4023 0.118 0.4023 1.673

a All of the Ax/pA parameters for the IT—KS model are infinity.b All of the t/6 parameters for the FPM-3 model are infinity.

105

3.6.3 Interpretation ofthe Results for the Phenomenologicaf Transport ModelsThese results presented above for the phenomenological transport models may be

explained in the following manner. ln order to estimate the transport parameters for any ofthe [T models, it is necessary that all adjustable parameters remain constant over the range

ofoperating conditions studied. For the present case, at least one of the transport parametersappears to be changing as a function of the operating conditions. The possibifity that 6P maybe a function of concentration was investigated. lt was determined that including aconcentration dependence in the €p term for the lT—PT model did not influence the €" valuessignificantly. Therefore, the influence of concentration and pressure on the transportparameters must be more complex. ldentifying possible concentration and pressuredependence has not been attempted for two reasons. First, there is little theoreticalbackground to predict the form of such a relationship. Second, it would be difficult toextrapolate the resulting equations to other solutes and membranes. ln any case, it may bestated that the nature of the solute and solvent transport is a strong function of the operating

conditions. Yet lT type models have been found previously to do an excellent job ofrepresenting the solute transport in the absence of strong solute—membrane affinity (Q). ltcan be hypothesized that the unsatisfactory behavior of the phenomenological models for thearomatic hydrocarbon data is related to the inability of the models to account for the strongsolute—membrane attraction.

More recently, two papers by Thiel et al. (Q, Q) have addressed the above problem.The phenomenological transport equations of irreversible thermodynamics were modified toaccount for possible interaction between solute and membrane. This interaction enters as anexternal force acting on the binary solvent/solute solution in the membrane phase. The

resulting transport equations are similar in form to the four parameter finely—porous model.

106

Good agreement has been Found between the single—solute aromatic hydrocarbon data and themodel. .

Since these models are phenomenological, the transport parameters are not linkedto physicochemical concepts such as solute diffusivity in the membrane or surface pore size.For example, it was noted that o For the lT—P'I‘ models varied as a Function of solute andmembrane annealing temperature. llowever, it would be incorrect to try to explain thebehavior in terms oFdiFFerences in the solute and membrane character. ln order to do this, itis necessary to turn to models which are dependent on a transport mechanism. 'l‘hese modelsare considered in the next sections.

3.7 Nonporous Membrane 'l‘ransport Models

When the membrane is considered to be nonporous, a solution-diffusion type ofmechanism is usually used to describe the reverse osmosis transport. In this section theresults of applying the solution—difFusion model and two modifications of the solution-diffusion model are discussed.

3.7.1 Solution-Diffusion Relationship

The solution—diFFusion model assumes that the membrane surface is homogeneous

and nonporous. Therefore, a given polymer should have a fixed (or intrinsic) separation and

Flux per unit thickness for a fixed solute—solvent system. These values are a Function of thesolubility and diffusivity of the solute and solvent in the polymer and the membrane

thickness. IF any imperfections (pinholes, pores, or void spaces) exist, the solute can '°leak"

with the solvent through these passageways and less than ideal separation (with a

corresponding increase in Flux) will be observed. lt has been Found that annealingasymmetric cellulose acetate membranes acts to increase salt—water separation and decrease

107

the Flux, as discussed in Section 1.2.1. According to the solution-diFFusion mechanism, theannealing step is thought to decrease the size and/or number of imperFections, possibly byincreasing crystallinity or by closing pinholes. The six cellulose acetate membranes used inthe current work may be considered to have various number and size oF imperFections.

Membrane 1 is closest to a true SD type membrane while membrane 6 has the most and/orlargest imperFections. In spite oFimperFections, it is not unusual to apply a solution—diFFusion

type mechanism to all reverse osmosis membranes; this approach is examined in this section.

For the SD type nonporous membrane transport models, the parameter (DBM CBMVB/RT Ax) was determined From the pure water permeation data. Since no independent

method oF measuring DBM, CBM, and Ax was employed, the values are treated as a singlegrouped parameter. Noting the similarity between lüquation (11)) and Equation (26), the

grouped parameter (DBM CBM VB/RT Ax) can be identified as being numerically the same asCP. Therefore, the CP values listed in Table 5 are also the (DBM CBM VB/RT Ax) parameters torthe SD type models.

Equation (28) predicts the relationship between volume Flux and separation For the

SD model:

The SD relationship is a two parameter model oF Form a) (Equation (68)). The unknown

parameters are CV = (DBM CBM VB/RT Ax) and (DAM K/Ax). 'I‘he parameters estimated For theSD model are listed in Table 10, and the curves oF an F' versus JV plot are illustrated in

Figures 17-19 For the solutes benzene, toluene, and cumene, respectively. As the dashed lines

in Figures 17-19 illustrate, the Form oF the SD model is not consistent with the data. In Fact,

the data Follow the opposite trend predicted by the SD model; that is, F' actually decreases,

rather than increases as predicted, as aFunction oFincreasing JV. The DAM K/Ax values

108

Table 10

Parameters for the Solution~Dif”f°usion“ {SD} and Kimura—SourirajanAnalysisb { KSAQ Relationships

El X 105 'Film Solute m/s s

1 Benzene 1.993 0.3111 Toluene 0.6388 0.3701 Cumene 0.07917 0.4242 Benzene 2.786 0.2792 Toluene 1.155 0.3422 Cumene 0.5992 0.4293 Benzene 9.678 0.2253 Toluene 3.886 0.2793 Cumene 3.276 0.3854 Benzene 14.47 0.1474 'l‘o1uene 8.070 0.2264 Cumene 6.734 0.2905 Benzene 65.86 0.1675 Toluene 16.01 0.1705 Cumene 6.207 0.2336 Benzene 10.41 0.1666 Toluene 4.588 0.2446 Cumene 2.156 0.240

" For the SD model, fil = DAM K/Ax and by definition B0 = 1.0.b For the KSA relationship E1 = DAM K/L and by definition EO = 1.0.

109

decrease in the order benzene > toluene > cumene and decrease with increasing annealing

temperature (except for membrane 6). The s values are larger than for any other model.

These results indicate that the SD model is not appropriate for the hydrocarbon solute data.

Similar results concerning the inapplicability of the SD model for phenol-water

separation have been reported previously (LQ). The conclusion is that either the parameters

are not independent ofoperating conditions or that at least one of the assumptions of the SD

model is incorrect. As discussed for the IT models, the possible dependence of the adjustable

parameters on operating conditions is difficult to identify and the resulting model is usually

less general. 'l‘herefore, only the validity of the assumptions in the model was investigated.

At least four different possibilities have been suggested to explain why the Sl) model is

inappropriate. One assumption questioned by previous authors (LQ) is the validity of

assuming that water and solute fiows were independent. 'l‘he existence of so-called "flow

coupling" between solvent and solute could qualitatively cxplain the observed decrease in

separation with increasing volume flux. Two methods of accounting for flow coupling have

been proposed. The first (1) of these is identical to the lT—KS approach, which has already

been discussed. The second method (Q) leads to the finely porous model, which is a porous

membrane transport model and is discussed later. The second possible explanation for the

inapplicability of the SD model is that the membrane surface is not homogeneous. This

potential explanation would suggest either a solution—diffusion imperfection model, which is

described elsewhere in this dissertation, or again a porous membrane model. The third

possibility is the validity of excluding the VA AP term in the chemical potential equation.

The modification of the SD model to account for the VA AP term is discussed later as the ESDmodel.

A fourth possibility is that the presence of the solute may cause the polymer to

swell, thereby changing the transport properties of the membrane (Q). Since the

110

performance of the membranes is reproducible, it must be concluded that any swelling that

does take place is reversible. This means that the membrane returns to its original state after

the solute is washed out with pure water. Since it has been implicitly assumed in the SD

model that the membrane structure is unchanged by the presence of the solute, then swelling

would lead to parameters that are dependent on operating conditions. ln fact, reversible

swelling may be the reason that the SD model as represented by Equations (26) and (29) does

not work. The possibility of identifying and quantifying swelling has not been pursued in this

work. Therefore, modification of the assumptions in the SD model lead to either the IT-KS,

SI)I, ESD, or to a porous membrane model. The SDI and ESD models are discussed next.

In a qualitative sense the following consideration may be made concerning solvent

flux. The SD model assumes that the solute and solvent are transported independently

through the membrane. Therefore, changes in the feed concentration should not affect the

solvent flux as long as the solute concentration is small so that AP > > Au. For the present

case, as has already been discussed, the solvent (lux is influenced strongly by the presence of

the solute. A possible explanation for this observed behavior is that the presence ofthe solute

in the membrane material alters the structure of the polymer to reduce the water flux. 'l‘he

nature of the alteration may be a change in the swelling of the polymeric material due to an

interference of the specific interactions between solvent and membrane caused by the solute-

membrane attraction (Q). Data for the swelling of the polymeric membrane by water as a

function of phenol concentration are approximately consistent with the observed

reduction in flux due to phenol in the feed solution of reverse osmosis experiments (QQ).

However, it is not clear from these results if the change in water content is due to a change in

the membrane structure, a change in the water—solute—membrane interactions, or both.

111

3.7.2 Solution-Diffusion Imperfection Relationship

As presented in the Section 1.2.3, the soIution—diffusion imperfection (SDI) modelwas developed to include the influence of transport through small imperfections which mayexist in the surface of the membrane. Equation (32) may be applied to the data directly.However, when the concentrations used are sufficiently low that AP > > An then Equation(32) may be rearranged as:

1 K"' K" ' RT (33)The SDI relationship is a 3 parameter model ofForm c) (Equation (70)). 'I‘he three parameters

are K', K", and K"'.

The aromatic hydrocarbon data were analyzed with Equation (33) and the results

are summarized in 'I‘able 11. That the data can be correlated by a f’versus AP plot is

Illustrated in Figures 23-25. The EO and E, values as well as the ratios K"/K' and K"'/K' aregiven in Table 11. Noting the similarity between Equation (l()) and (30) for the case of purewater (An = 0) leads to:

€p= K' + K"' (78)

With the €p values from Table 5 and Equation (78), the individual K', K", and K"' values may

be calculated and are listed in Table 11. The EO values for the IT—PT—A and SDI models aresimilar. El values for the two models follow similar trends although the magnitudes are

different. 'I‘herefore, the dependence of EO and E, on membrane annealing temperature andsolute is similar to that described for the I'I‘—P’I‘—A model. The s values are almost identicalfor the I'I‘—P’I‘—A and SDI models. The SDI model is actually quite similar in form to the IT—P'I‘—

A model. When AP > > An, then AP is related to JV, as illustrated earlier for the l'I‘—PT model:JV = Ep AP (75)

indicating that JV and AP are related to each other by a linear equation. Therefore, Form a)

and Form c) in Table 3are equivalent and it is not surprising that the IT—PT—A and SDI

112

Table 11

Parameters For the Solution—Di1Tusion1mgerf‘ecti0n(SD[[ Model

Film Solute EO El = K"/K' s K"'/K' K' x 10"' K" x 10"' K"' x 10‘0

*<m<>*/m3 m/s kPa m/s kPam skPa

1 Benzene 5.902 -1.130 0.113 4.902 2.164 -2.445 10.611 'l‘0luene 3.685 -0.6284 0.116 2.685 3.465 -2.178 9.3051 Cumene 2.127 -0.2716 0.215 1.127 6.004 -1.631 6.7662 Benzene 6.405 -1.221 0.088 5.405 2.436 -2.974 13.162 Toluenc 4.220 -0.7312 0.116 3.220 3.697 -2.703 11.902 Cumene 3.084 -0.4777 0.181 2.084 5.059 -2.416 10.543 Benzene 11.32 -2.426 0.089 10.32 1.937 -4.700 19.993 Toluene 6.642 -1.296 0.097 5.642 3.032 -4.279 18.633 Cumene 4.935 -0.8903 0.199 3.935 4.444 -3.956 17.494 Benzene 12.31 -2.266 0.072 11.31 2.669 -6.047 30.184 Toluene 8.458 -1.659 0.095 7.458 3.884 -6.443 28.974 Cumene 6.354 -1.171 0.132 5.354 5.170 -6.054 27.685 Benzene 35.13 -8.725 0.095 34.13 1.143 -9.969 39.005 Toluene 12.70 -2.622 0.076 11.70 3.161 -8.287 36.985 Cumene 5.570 -0.8522 0.097 4.570 7.206 -6.141 32.936 Benzene 7.527 -0.9531 0.116 6.527 5.874 -5.598 38.346 'Foluene 4.357 -0.5217 0.120 3.357 10.15 -5.294 34.066 Cumenc 2.789 -0.1839 0.109 1.789 15.85 -2.915 28.36

113

Figure 23. The effect ofoperating pressure on separation for the benzene—water—cellul0seacetate system at 25°C. ([:]) experimental points; (——) soluti0n—diffusi0nimperfection model (SDI).

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models give similar results.

The ratio of x"' to x' represents the relative contribution of flow throughimperfections to diffusive flow. This ratio (from Table 11) varies from 1.13 (for cumene andmembrane 1) to 34.1 (for benzene and membrane 5) which indicates that a large portion,between 53% and 97%, of the water flux is occurring through imperfections. Since annealingat higher temperatures decreases the number and/or size of imperfections, it would beexpected that x"’/x' should increase with decreasing annealing temperature. This resultis confirmed by experimental x"’/x' values. The x"'/x' ratio for a given membrane decreasesin the solute order benzene > toluene > cumene. Thus the pore flow contribution decreaseswith both increasing solute size and increasing solute—membrane afflnity. Perhaps the solutemolecules are partially blocking the imperfections and the extent of the blocking increasesboth with solute size and with solute—membrane afflnity. This explanation is consistent withthe qualitative discussion in Section 3.3. 'I‘his result indicates that the importance of the poreflow decreases in the order benzene > toluene > cumene. It would be expected from the basicpremise of the SDI model that K', x", and K"' should all be positive. Yet, the x" values arenegative in all cases. For K" to be negative, either the solute diffusivity in the membrane orthe solute partition coefflcient should be negative. Since it is impossible for either of thesephysicochemical parameters to be negative, it may be concluded that negative values of x"are inconsistent with the SDI model. Therefore, although the x' and x"’ parameters seemreasonable, the negative x" values cast doubt on the applicability of the SDI model.

3.7.3 Extended Solution-Diffusion Relationship

The extended solution—diffusion model (ESD) allows the influence of pressureinduced solute transport to be included in the model. The possibility of pressure inducedsolute transport occurring is dictated by the presence of the VA AP term in the solute chemical

120

potential equation (see Equation (34)). Usually it is safe to ignore the VA AP term since it is

much less than the concentration term. This simplification breaks down at high pressure, low (separation, and high VA. Before examining the results of the ESD model, it is appropriate to

determine if the VA AP term is important for the aromatic hydrocarbon·water-cellulose -

acetate data.

The criterion for ignoring the pressure induced solute transport is;

|ln(CA2/CA3)| >> VA AP/RT (79)Substituting in the definition of f' gives:

|ln(1/(1—f'))I > > VA AP/RT (80)

ln order to check this inequality, it is necessary to have values for VA and experimental data

for f' as a function of AP. Partial molar volumes, VA, have been estimated from the densityand molecular weight ofthe solutes (7) and are listed in Table 1. The dependence of f' on AP is

illustrated in Figures 23-25. 'l‘he results ofevaluating Equation (80) are shown in 'l‘able 12.

At one extreme, with large f' and small AP values, Equation (80) is valid and the VA AP/R'l‘

term may be ignored. At the other extreme, with small f' and large AP values, Equation (8())

is not valid and VA AP/RT may not be ignored. 'l‘he remaining results are between these

extremes. lt appears that the influence of pressure induced solute transport may be ignored

only for the highest separation data. In fact, the pressure induced solute term may dominate

over the concentration term. Therefore, the VA AP/RT term should be included in theanalysis for the aromatic hydrocarbon data.

Having established the importance of including the VA AP/RT term in the model,

the results of fitting the data to ESD model can be considered.

The form of the ESD model is identical to the l'I‘—PT—A model, so the EO and E,values in Table 7 and the curves in Figures 17-19 apply to both models. The definitions of EO

121

Table 12

Evaluation 0F the Significance 0F Pressure Induced Solute Transgort

Benzene Toluene Cumene

Case 1: Large F' and small AP = 690 kPa

F' 0.65 0.73 0.90ln(1/1- F') 1.050 1.309 2.303VA AP/RT 0.02475 0.02952 0.03898ls ln(1/1-F') > > VA AP/RT ? Yes Yes Yes

Case 2: Small F' and large AP = 6900 kPa

F' 0.01 0.01 0.03

ln(1/1—F') 0.0100 0.0100 0.0305VA AP/RT 0.2475 0.2952 0.3898lsln(1/1—F') >> VA AP/RT'? No No No

122

and El in Table 3 are used to calculate the CAP and DAM K/Ax values listed in Table 7.Similar to the case for the lT—PT—A model, CA2 is not constant, so an average CA2 for allexperiments for each solute was used to calculate an approximate value of CAP. The CAPvalues were found to decrease both with increasing annealing temperature and with solutenonpolar character: benzene < toluene < cumene. Therefore, the pressure-induced solutetransport decreases as the membrane permeability decreases, which is consistent with thepremises ofthe ESD model. Since CAP should be directly proportional to VA (listed in Table 1),

it would be expected that CAP should increase with the solute order benzene < toluene <cumene. ln fact, the opposite trend is observed. ’I‘he DAM K/Ax values calculated arenegative. This result is impossible since DAM, K, and Ax all must be positive. Therefore, theactual data are inconsistent with the ESD model.

3.8 Porous Membrane Transport Models

Once the membrane is considered to be porous, describing the transport through a

reverse osmosis membrane becomes more complicated. Three of these models, the Kimura-

Sourirajan analysis of the preferential sorption—capillary flow mechanism, the linely—porous

model, and surface force—pore flow model are discussed in this section with respect to the

aromatic hydrocarbon—water-cellulose acetate me mbrane system.

3.8.1 Kimura—Sourirajan Analysis

Kimura and Sourirajan assumed that the membrane is microporous and that all

transport takes place through membrane pores. The application of the KSA relationship to

the present data has been given in Sections 3.2 and 3.3 of the RESULTS AND

DISCUSSIONS. The KSA model in the original form is not capable of explaining theobserved data. However, the empirical Equation (40) can be used to represent the so—called

123

pore blocking effect. Both the KSA and SD models are similar in mathematical form, eventhrough the underlying concepts are different. Therefore, the parameters in Table 10 and thecurves in Figures 17-19 also apply to the KSA model. The trends in (DAM K/L) as a function ofannealing temperature and solute are the same as for the SD model. The KSA model is notable to describe the decrease in separation observed with increasing pressure. According tothe KSA model, the discrepancy between theory and experimental result is due to the strongsolute—membrane affinity. When the solute is strongly sorbed to the membrane material, thetransport in the pore is too complicated to be described by this simple one-dimensional model.Two other attempts at describing the data with a porous model are discussed below.

3.8.2 Finelv—Porous Model

As discussed in Section 1.2.4, the linely—porous model (FPM) is a one-dimensional

transport model based on a combined viscous flow and frictional model. ’I‘he model can be” used as either a three parameter model, FPM·3, or as a four parameter model, FPM—4. These

two approaches are presented separately below.

Three Parameter Finely—Porous Model. In the three parameter model it is assumed that thepartition coefficient is the same on the high and low pressure sides of a membrane for an

experiment. Mathematically, this assumption can be expressed as K2 = K3 = K. The

relationship between f' and JV is given as:

1 1—(l—K/b)exp[-(1/e1)AB)JV| (46)F Z (1—K/b){1—expl—(t/£DAB)JVl}

The FPM—3 relationship is a three parameter model of Form b) (see lüquation (69)). The three

parameters are A, b/K, and c/e.

124

The aromatic hydrocarbon data were analyzed by the FPM—3 model and results aresummarized in Table 9 and in Figures 17-19. Noting that FPM—3 model is of the same generalform as the [T-KS model, it is not surprising that the same Ei values and the same lines in

Figures 17-19 are generated. lfowever, as illustrated in Table 9, the parameters are

interpreted differently for the two models. In all cases the value of L/e was estimated as a

value sufficiently large that the exponential terms in the model approached zero. As a result,the (1 — K/b) term is simply an average of the experimental f' values. The b/K values increase

in the order benzene < toluene < cumene and decrease with increasing annealing

temperature (except for membrane 6). The b/K values seem reasonable, but the result that L/c

equals infinity is impossible. ln order for L/e to be infinite, either L should be infinite,

corresponding to an infinitely thick membrane, or the c should be zero, corresponding to no

pores in the membrane. Both of the above possibilities are unreasonable physically for the

FPM model.

The conclusion is that the FP)/I—3 model is not capable ofquantitatively describing

the observed decrease in separation with increasing volume flux. Therefore, the FPM-3 model

must be rejected as a possible model for describing the aromatic hydrocarbon-water-cellulose

acetate data. The exact reasons why this model failed are difficult to determine. llowever, it

was found that relaxing the assumption that the partition coefficient K was constant greatly

improved the model. ’I‘he results ofthis analysis are discussed in the next section.

Four Parameter Finely-Porous Model. When the assumption that the partition coefficient

remains constant is relaxed, the FPM-4 model results. As discussed in Section 1.2.4, therelationship between f' and JV for the FPM—4 model is:

1 1—(l—K3/b)exp[-(deDAB)JVI (43)F Z (1 —K2/b)- (1 —K8/b)expl—(L/e DAB)JVl

125

This model is a four parameter model of Form d) (see Equation (71)). The four parameters are

A, b/K2, K3/K2, and t/e.The aromatic hydrocarbon data were analyzed by the FPM-4 model and the results

are summarized in ’l‘able 13 and in Figures 26-28. The agreement between the model and

experimental data is good. The standard deviation values, s, are small. The model is capable

of predicting several different shapes for the f' versus JV relationship. These include a

concave upward curve which is the most usual case, a concave downward curve (see the case of

membrane 1 for the solute cumene) or a curve with an inflection point (see the case of

membranes 2 and 3 for the solute cumene). Also, the model predicts that separation may

become negative at higher volume fluxes (see the case of membrane 5 for the solute benzene

and of membrane 3 for the solute cumene). None of the other models examined have been able

to predict a concave downward curve or a continuous change from positive to negative

separation. lt is interesting to note that previously this model has been applied primarily to

cases where separation increases with increasing flux (for example, NaCl-water-cellulose

acetate system). However, the model does an excellent job of following the trend of the data

when the separation decreases with increasing pressure.

The estimated parameters are listed in Table 13. 'l‘he results for benzene and

toluene are similar to each other in that the b/K2 parameters appear to be independent of pore

size and solute. The average b/K2 value for benzene and toluene is 1.091 i 0.046 (95%confidence limit). Based on the Faxen Equation (63), it would be expected that b should be a

function of both pore size and solute. Also, since K8/K2 varies as a function of both pore size

and solute, it is expected that K2 could vary with both pore size and solute. Since we expect

both b and K2 to vary with pore size and solute, it is surprising to find that b/K2 is

approximately independent of pore size and solute, at least for benzene and toluene. lt is

possible that the variations in b and K2 are cancelling each other such that b/K2 is

126

Table 13

Parameters for the Four Parameter Finely-Porous Q FPM—4Q ModelFilm Solute EO El E2 s b/K2 K3/K2 t/2

(1-KO/b) (1-K2/b) (L/S DAB) x104x 102 x 10'5 m

Benzene -1.617 10.10 6.820 0.097 1.112 2.911 7.4751 Toluene -2.902 14.55 7.160 0.068 1.170 4.567 6.931

Cumene 39.39 657.0 1.967 0.061 -0.1797 6.893 1.572

Benzene -1.510 9.297 5.813 0.058 1.102 2.767 6.3712 Toluene -2.564 11.86 6.014 0.061 1.134 4.043 5.822

Cumene -3.985 -11.26 2.763 0.061 0.8988 4.481 2.208

Benzene -1.022 1.760 3.577 0.(l64 1.018 2.058 3.9203 Toluene -1.738 4.829 4.070 0.053 1.051 2.877 3.940

Cumene -3.161 -55.94 1.320 0.()97 0.6410 2.669 1.055

Benzene -0.3839 5.362 1.985 0.071 1.057 1.462 2.1764 Tolucne -1.151 5.294 2.689 0.079 1.056 2.271 2.603

Cumene -1.330 0.2086 1.617 0.090 1.002 2.334 1.292

Benzene -0.8303 -1.827 2.304 0.087 0.9821 1.798 2.5255 Toluene -0.7857 3.679 2.148 0.070 1.038 1.854 2.079

Cumcne -0.7144 10.50 1.227 0.087 1.117 1.916 0.980

Benzene -0.3377 13.16 2.286 0.119 1.151 1.540 2.5056 '1‘0luene -0.5746 17.96 1.340 0.114 1.219 1.919 1.297

Cumene -0.3870 35.53 1.687 0.112 1.551 2.151 1.348

127

Figure 26. The effect of volume flux on separation for the benzene~water—celluloseacetate system at 25°C. (Ü) experimental points; (—) the four parameter1inely—porous model (FPM—4); (———) the four parameter finely—porous modelwith the pore size parameter, t/2, forced to be constant for all three solutes.

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129

Figure 27. The effect of volume flux on separation for the toluene-water—cellulose acetatesystem at 25°C. (Ü) experimental points; (—) the four parameter finely-porous model (FPM—4); (-— —) the four parameter finely—por0us model with thepore size parameter, 1;/6, forced to be constant for all three solutes.

130

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131

Figure 28. The effect ofvolume flux on separation for the cumene—water—cellulose acetatesystem at 25°C. (Q) experimental points; (—) the four parameter finely—porous model (FPM—4); (— — —) the four parameter 1inely—porous model with thepore size parameter, t/2, forced to be constant for all three solutes.

l32

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133

approximately constant. From the estimated values for b/K2 and (1 — K3/b), the ratio of thepartition coefficient on the low and high pressure side of the membrane, K3/K2, listed in Table13, can be calculated. These ratios are clearly greater than one and are a function of pore sizeand solute. The overall trend indicates that K3/K2 increases in the solute order benzene <toluene < cumene and increases with increasing annealing temperature, although there issome scatter. This result is unusual in that K is often considered to be constant for a givensolute and membrane material. Here, the partition coefficient appears to be different on thehigh and low pressure sides of the membrane, and the difference is a function of pore size.These differences may reflect that K is a function of membrane structure, concentration,and/or pressure. Pressure should have a insignificant effect on K for the case of a liquidsolution, since liquids are almost incompressible. Therefore, the observed variation in K isprobably caused by the asymmetric structure of the membrane and/or a concentrationdependence. The parameter t/e is approximately the same for benzene and toluene for eachmembrane, but does increase with increasing annealing temperature, with some scatter.This result is encouraging since it was expected that t/e should increase with increasingannealing temperature. lncreasing annealing temperature causes a reduction in porediameter, which decreases the fractional pore area, e, and hence increases t/e.

Comparing the results for cumene with those for benzene and toluene, similarvalues for the various parameters are observed. However, there is more scatter in thenumerical values and the trends are not as clear.

As discussed above and earlier in this dissertation, assuming that the membrane isnot temporarily altered by the solute in the concentrated boundary layer, it is expected thatthe pore size parameter, t/e, should be independent of the solute used. Therefore, the datawere reanalyzed, assuming that u;/2 was constant for each membrane. ln this case 7parameters were estimated for each membrane (b/K2 and K8/K2 for each of the 3 solutes plus

134

Table 14Parameters For the Four Parameter Finely—Porous QFPM-4} Model with the Pore Size

Parameter ( L/2:} Forced to be Constant For A1lSo1utes

Film S0lU1L€ EQ E1 E2 S b/K2 Kg;/K2(1-K3/b) (1-K2/bl L/6: x 104x 102 m

Benzene -1.237 2.235 1.023 2.2891 Toluene -2.179 4.531 4.110 0.085 1.047 3.330Cumene -11.99 5.539 1.059 13.76Benzene -1.113 3.452 1.036 2.1882 Toluene -1.867 5.066 3.664 0.071 1.053 3.020Cumene -4.944 8.540 1.093 6.500Benzene -0.8110 -5.848 0.9447 1.7113 Toluene -1.215 -4.491 2.017 0.084 0.9570 2.120Cumene -2.728 -9.631 0.9122 3.400Benzene -0.3572 4.040 1.042 1.4144 'Foluene -0.8916 1.345 1.697 0.081 1.014 1.917Cumene -1.492 4.331 1.045 2.605Benzene -0.6723 -3.903 0.9624 1.6095 'Foluene -0.7158 2.751 1.807 0.081 1.028 1.764Cumene -0.9216 15.51 1.184 2.274Benzene -0.2456 12.26 1.140 1.4206 Toluene -0.6180 ' 19.05 1.555 0.113 1.235 1.999Cumene -0.4183 35.89 1.560 2.212

135

t/s for the membrane) based on 50 experimental data. The parameters can be estimated withmore confidence than previously with 3 parameters based on 16-18 data points. Therecalculated values are given in Table 14 and the f' versus JV relationships are illustrated inFigures 26-28. The solid lines represent the calculated correlation based on separateparameters for each solute and membrane combination. The dashed line is the modelrecalculated with t/2: forced to be constant for a given membrane. lt is clear from Figures 26-28 that the data are as equally well represented by the recalculated line as by the initialmodel. That a better estimate of the parameters was obtained is confirmed by the smaller s(standard deviation) values obtained (listed in 'Table 14). From this table, b/K2 is observed tobe approximately constant at 1.074 i' 0.075 (95% confidence) for all membranes and solutes.Even for cumene an average value seems to be appropriate. Again, it must be emphasizedthat b/K2 would not be expected to be a constant, however the experimental results indicatethat this is true. The ratio of partition coefficients K3/K2 decreases with increasing pore size

‘ and increases tor the three solutes (benzene < toluene < cumene). As before, the K3/K2terms are neither constant nor equal to 1.0. With the recalculated parameters, the trend inK3/K2 may be clearly seen for all three solutes. Finally, the pore size parameter t/c is foundto decrease with increasing pore size, as expected. The net result of the above analysis is thata better estimate of the unknown parameters is obtained. As a result, the trends in theparameters, which were pointed out above, are clearer.

Previously in this work, ln C*NuCl was used as a pore size parameter. For the FPM-4 relationship, the pore size parameter, t/c, has been calculated. Figure 29 illustrates the

relationship between the two pore size parameters ln C*NaCl and t/e. As would be expected,an inverse and unique relationship is found between t/e and ln C*NuCl for the six membrancs.The curve is a hand·drawn line through the data points and is not a least squares lit. In orderto support the shape ofthis curve, data collected independently (ZQ) for similar cellulose

136

Qui

Q-5<l‘

G1-%

X Q65

I-!E

L.IOLU O; III

Q..2

Qci

SIEl rw C N c1C l

Figure 29. The relationship between the pore size parameters, L/e and ln C*NnC[. (El)values determined for membranes 1-6; (O) values determined independentlyfor five different cellulose acetate membranes with the solutes cyclohexane,cyclohexene, and 1,3-cyclohexadiene.

167

ca-5

III B E N Z E N ECD T 0 L U E N EA C U M E N E

:-1 9c\: '°K\

CY') ¤ni

Kl_l

C ci OIll ,-4

II]

Elcaci-13.0 -12.0 -11.0 -10.0 -9.0

X1 rw C N c1C 1

Figure 30. The relationship between the ratio ol‘ the partition coellicients on the low andhigh pressure side of the membrane, K3/Ko, and the pore size parameter,In

_

138

acetate membranes and similar solutes (cyclic hydrocarbons) are also included in the figure.These additional data help confirm the shape of the curve although it is desirable to havemore data in the middle pore size range.

Figure 30 illustrates the relationship between ln[K3/KQI and the pore sizeparameter ln C*NuCI. This figure emphasizes the trends in K8/K2 as a function of solute andpore size as discussed above. Also, Figure 30 can be used to estimate the K3/K2 parameter forother cellulose acetate membranes for the solutes shown.

It can be concluded that the four parameter finely—porous model is successful atdescribing the separation ofaromatic hydrocarbons from water by reverse osmosis. Since thepore size parameter L/e for each membrane is independent of solute, this lends credibility tothe physical interpretation of the model. That b/K2 is constant (independent of pore size andsolute) is an unexpected result. Although this result simplifies the application of the model,care should be exercised in extrapolating this to other solutes without checking thecorrelation first. Also, the partition coefficient was found to be different on the high and lowpressure side of the membrane. Future work should include a theoretical and anexperimental examination of the behavior of the partition coefficient as a function ofconcentration, membrane structure, and possibly pressure. The possibility ofindependently

estimating the partition coefficient using high pressure liquid chromatography (QQ, 81) orequilibrium sorption studies (Z, ß) is being investigated.

To apply the model in a predictive sense for similar hydrocarbon solutes,concentrations, and pressures, it is necessary to have a relationship between the volume fluxand the operating pressure. As was discussed in Section 3.3, the flux of solution through themembrane is usually less than the pure water flux. One possible method for describing the

pore blocking effect is to use the empirical correlation represented by Equation (40) andFigure 15. The method will have limitations, due to the empirical nature of Equation (40).

139

Therefore, it would be difficult and dangerous to extrapolate these correlations beyond thelimits of the present study. Further work needs to be done with more solutes to aid ingeneralizing the correlations. A more fundamental description of how pore blocking takesplace needs to be incorporated into the model to describe the reduction in flux due to poreblocking. With additional information on the partition coefficients and a general descriptionof the pore blocking phenomenon, the FPM—4 relationship promises to be useful inrepresenting and explaining the separation ofhydrocarbon type solutes by reverse osmosis.

3.8.3 Surface Force — Pore Flow Model

The surface force—pore flow model (SF—PF) has been discussed in Section 1.2.4. Thismodel is more complex in several respects than any of the other models presented. Themathematical equations are more complex, the solution ofthe equations is more difficult, andmore adjustable parameters appear in the model. 'I‘o add to the complexity, the model can beused in various forms. ln their original paper, Matsuura and Sourirajan (49) used the modelin a different form than in their later papers (Q, Q). The major difference concerns the formin which the friction function is expressed. The friction function has been discussed in Section1.2.5 and is discussed again later.

Because of the mathematical complexity of the model, numerical solutions ratherthan analytical solutions are required. The numerical techniques used are discussed later inthis section. Suffice it to say at this point that the solutions on the computer were timeconsuming and expensive.

The net result of the above considerations is that only preliminary results can bepresented at this time. More work is required before results for all the membranes andsolutes can be presented with confidence.

140

The remainder of this section addresses the following points. Two modificationsthat have been made to the original SF-PF model are presented. The numerical programsused in the analysis are discussed. The model is applied to NaCl—water data presented byMatsuura and Sourirajan (Q) and to the NaCl-water data reported in this dissertation. Theexperimental results for the toluene-water system for two membranes are examined. Finally,suggestions are made concerning the use of the SF-PF model.

Pressure Degendence ()f the Solute Chemical Potential. As discussed for the [CSI) model, thepressure term in the solute chemical potential Equation (34) is signilicant compared to theconcentration term. These results are summarized in Table 12. ln the original developmentofthe SF—PF model (Q) the pressure induced solute transport was ignored. llow the pressureterm can be included in the model is presented in this section. The following mathematicaldevelopment parallels the approach presented by Matsuura and Sourirajan (Q) in theirderivation of Equation (57).

The solute in the membrane pore has three forces acting ()n it; a diffusion drivingforce, FA(r,z), the frictional force between the solute and solvent, F^B(r,z), and the frictionalforce between the solute and membrane, FAM(r,z). At steady state, these forces balance sothat:

FA(r,z) + FAB(r,z) + FAM(r,z) = 0 (81)The expressions describing the latter two of the three forces are unaffected by the change inthe chemical potential equation. From reference (Q), the expressions are given as:

FAB(r,z) = —XAB luA(r,z) — uB(r)l (82)FAM(r,z) = —XAM(r) N^(r)/CAM(r,z) (83)

However, the diffusional driving force now includes the pressure term;

FAM) Z _ (84)

141

RT 6C Ä (r,z) TPFAR,) Z (85)CAM(r,z) dz Oz

Equations (82), (83), (85), and the definition of h given in füquation (44) written as a functionof r, can he combined and rearranged to give:

RT ÖC L (RZ) VI C A (!‘,Z) O P( YZ)NA(r) = [—— — — ·;i) + CAw(r,z) ußtr) |/b(r) (86)Oz Oz

Note that, from reference (Q):

O P(r,z) P(r,L) — P(r,())——· = (87)Oz L

dP(r,z) _ AP R’1‘877··T*T,%·%“*) *·@ ‘8)

Defining AP(r) as:

AP(r‘) = —(f’(r,L) — P(r,0)) (89)

then:

AP(r) = AP — R'I‘(CA2 — CA3(r))[1— e"f""I (90)and Equation (86) may be rewritten as:

RT oC (r,z) V C L (r,z) AP(r)N^(r) = '— —l—· + + CAM(r,z)uB(r) I/h(r) (91)AB öz AB °

Using the boundary conditions (Q):

CAM(r,0) = CA2 e"f"”’(92)

and

CAM(r,L) = CA3(r)e“‘b"”) (93)and noting that:

CA3(r) : NA(r)/uB(r) (94)then Equation (91) can be integrated with respect to z to give;

142

r Bexp) RP XAB u*l L'.

XABL VAAP . (95)1+ ·) exp l u (r)+——) -1)exp(—rp(_r)) VAAP — RT B XABL ,uB(r)+ T-ABL

The integration of Equation (91) to give Equation (95) is presented in Appendix 1).

Definingz

V AP( )(, (9, : ij (96)3 RTand using the other previously defined dimensionless groups (Equations (47) to (51)) andEquation (96), then Equation (_95) can be rewritten as:

exp (c1(p) + [53(p))C ( )=^p b(p) ¤(p) (97)1+ -—- ——-——) exp(<1(p) + ß_(p))- 1)exp(—q>(p)) ~c1(p) + ß3(p) J

This result can be compared with Equation (57) as derived by Matsuura and Sourirajan (Q):. ( ( ))

^ b(p)1+ exp(<1(p))-1)exp(—<l>(p))

When c1(p) > > [5(,(p), Equation (97) correctly reduces to Equation (57).

For dilute solutions the pressure term will dominate over the concentration term inEquation (90). 1n this case then AP = AP and:

V AP(3 2 ^ (98)3 RT

Therefore, Equation (97) is further simplilied by using a constant; [53 value. This dilutesolution assumption must be used with caution. When the solute is attracted strongly to themembrane material, then CA3(p) may be large near the membrane surface. The bestapproach is to use Equation (97) where no dilute assumption has been made.

143

Friction Function. In the lNTROl)UCTl()N, several approaches to describe the frictionfunction, b(p), were discussed. Since the SF—PF model is a two-dimensional representation offlow in a pore, it is possible to allow the friction function to vary both as a function of radialposition in the pore and as a function of relative solute size, .\. Plquations (65) and (66) are thetwo forms used by Matsuura and coworkcrs (Q, Q, Q). For convenience these equations arerepeated here:

Mp) Z {exp (10) when d(p) S 1) (65)exp(E/d(p)) when d(p) > D

and

1- 2.10421 + 2.09A3 —0.95}\5 when i\ S 0.22751/b Z { (66)1/(44.57 — 416.2A + 934.9A2 + 302.4)i3) when .\ > 0.2275

The former is a function of both solute (D and E are different for each solute) and radialposition (given here as distance from the wall, d(p)). The latter relationship is a function ofonly relative solute size, .\. ln this work, a further modification of the form of the frictionfunction is suggested.

lt seems logical that as the solute moves away from the pore wall towards the porecenter, the value of b should approach the Faxen Equation solution (given by lüquation (63),

or equivalently, the first half of Equation (66)). For Equation (65), b approaches 1.0 as d isincreased instead of yielding the Faxen solution. On the other hand, Equation (66) has notincluded a radial dependence for b.

Based on the above observations, an approach which combines the feature of bothlüquations (63) and (65) is suggested. Let:

b(p) = bFum1 exp(E/d(p)) (99)where bFax€_n is given by Faxen in Equation (63). As the solute moves away from the pore

wall, b decreases exponentially, similar to Equation (65). However, b now approaches theFaxen solution near the pore center. Equation (99) appears to be more appropriate than

144

either equation used by Matsuura and Sourirajan. It must be recognized that Equation (99) isa semi-empirical representation of the friction forces acting on the solute in the pore and thatE must be determined experimentally. It is expected that E is a function ofthe solute only· fora given membrane material. .

Steric Exclusion. The influence of steric exclusion was presented in Section 1.2.5. Severalauthors have used the l*‘erry—Faxen Equation (67) to include the effect of the size of the solutemolecule on transport through membrane pores. Steric exclusion of the solute is included inthe model through an additional contribution to the friction function, b.

In the SF-PF model a different approach is used. 'l‘he flnite size of the solutemolecule is taken into account by the potential function given in Equation (61):

I() when d(p) S I)(Mp) Z

l B (61)— when d(p) > I)d(p)3The solute molecule is strongly repelled by the membrane when the solute is less than acritical distance D from the surface of the membrane, where I) is an effective radius for thesolute. Therefore, the steric exclusion ofsolute is taken into account by the potential functionrather than by the friction function.

I

Flux versus Pore Size for Different Membranes. 'l‘he SF-PF model predicts that the flux ofpure water should vary as a function ofpore radius to the fourth power ($9):

np ci RÜ4 (100)However, experimentally it is difficult to test this relationship. A series of membranes thathave been annealed at different temperatures have different pore sizes and different waterfluxes. However, there is no guarantee that the annealing process has altered only theaverage pore radius. In addition, the effective pore length, t, the shape of the pore size

145

distribution, and the number of pores must remain unchanged. Previously, Glueckauf (Q)found that the pure water flux varied ast

npo 1te‘·5 (101)The lower dependency on Re is probably caused by the above mentioned factors. Matsuuraand Sourirajan (42) used Equation (101) as an integral part of their application of the SF—PFmodel. The data for six membranes annealed at different temperatures were forced to followEquation (101) in their parameter estimation scheme. In subsequent work by Sourirajan andcoworkers (Q), it was found that:

np Q 1tem (102)was more appropriate. This dependency is closer to the predicted fourth power.

The conclusion which can be drawn from the previous w()rk summarized here isthat the relationship between ne, and Re is dependent on the mechanism of the annealingprocess. lf the annealing process is ideal in the sense that only the average pore size is

changed, then Equation (100) should be valid. l*‘or the membranes used in this work, thedependency between ne, and Re is unknown. Therefore, in the current project n() relationship

between ne, and Re is assumed. The Re values for each 1nen1brane are cleterminedindependently.Numerical

Analysis. Before continuing to the results of applying the Sl·‘—PF model, it is

appropriate to summarize the numerical methods. The equations ofinterest from a numericalmethods point of view are the differential Equation (56) and the integrals in Equations (54)

and (55).

Equation (56) is an ordinary nonlinear nonhomogeneous second order differential

equation:

146

d2o(p) 1 do(p) 52 1+ Ä T + + (1 — exp(—<b(p))(CA(p) — 1)1 1

1- —<b<p1-1>¤<p1c<p>= 0 (56)

where CA(p) is given by Equation (97). The manner in which the dimensionless velocity, c1(p),

varies as a function of dimensionless radial position, p, is described by this differential

equation. The two boundary conditions are given by:

= 0 when p :0 (58)d 0and

ofp) = 0 when p = 1 (59)which indicate that the ¤(p) is finite at the pore center and that there is no slip at the wall,

respectively.

· For pure water permeation, the fourth and fifth terms in Equation (56) are zero and

the analytical solution is:

(103)

Equation (103) can be integrated to average the flux in the radial direction which gives the

following relationship for the flow rate of pure water through a single pore: '

Qp : g n RB4 AP/8 r} L (104)

which is the well known Poiseuille equation (see for instance reference (22)). ln other words

the velocity distribution is parabolic. The fourth and lifth terms in lüquation (56) distort the

velocity distribution from the usual parabolic shape, reflecting the influence of friction lorces

and surface—solute interactions. No analytical solution is known for Equation (56), except as

above, and therefore a numerical solution is necessary.

Equation (56) can be rewritten as two simultaneous first order equations, and afourth order Runge—Kutta method (see for instance reference (QD may be used. flowever, to

147

apply the Runge—Kutta method, it is necessary that both houndary conditions be known at oneend of the interval. For the given problem, one houndary condition is known at each end ofthe interval producing a two point houndary value problem. 'I‘herefore, the procedure is asfollows:

1) Estimate ¤(p) at (p) = 0, noting that dc1(p)/dp = 0.

2) lntegrate with Runge—Kutta from p = 0 to p = 1.

3) Atp=1,checkifo(1)=0.

4) lf<1(1) ¢ 0, then update the estimate of¤(0) and repeat 2)—4).

5) lfn(l) = 0, then the solution is complete.

Several unidimensional search algorithms are possible to update o"(0), for the nm iteration,

based on o"“ [(1), for the previous iteration. The simplest method seems to work as well as

any, that is:

(1H(Ü): (1n—1(Ü)- o““l(1)(105)

A good initial estimate (step 1) above) for this algorithm is suggested by the approximate

solution given earlier as Equation (103) at p : 0:

o"(0) Z l52/4 [5, (106)The end result ofthe above application of the Runge—Kutta technique is that c1(p) is

known at various values of p over the interval 0 S p S 1. Fifty equal size steps were used

over the interval.

The evaluation of the integrals in lüquation (54) and hence in Hquation (55) is

relatively simple:

1[0 CA(p)o(p)pdp (55)f' = 1 — —

1

lo<1(p) p d p

148

A trapezoidal method (Q) can be used to integrate the above integrals from the o(p) valuesdetermined above by solving the differential Equation (56).

The above information summarizes the numerical techniques required to apply theSF-PF model. Ilowever, ifthe model parameters (A, A or B, I), E, and RW) are unknown thenthey need to be determined based on experimental data. The method of searching forparameters was discussed in Section 3.5. lt was often found that ZXSSQ (E) wasunsatisfactory as a search routine. The ZXSSQ algorithm would generate relatively largechanges in some of the parameters resulting in the overall calculation algorithm becomingunstable. 'l‘he GRG2 code (Q), which is a constrained nonlinear parameter estimationalgorithm, provided a useful alternative. By constraining the parameters to reasonablelimits, it was possible to find optimum parameters. Since the SF-PF model simultaneouslypredicts the separation, f', and the extent ofpore blocking, the sum of the squares error (SSQ)

in both of the performance variables was used to evaluate the model. 'l‘he optimizationroutine tries to minimize SSQ, calculated as:

ssewhere¤ 2SSQ1~= E1:1

G?. 2|(1(11:1The f’cul and (1 — n.lJn[,)cal represent the rnodel—calculated values corresponding to theexperimental value of fi' and (1 — nT/nP)i, respectively. It is possible to weigh one SSQi over

the other, but an equal weighting was used in this work. The absolute values of f' and(1 — nT/np) are of the same order ofmagnitude and so an even weighting seems reasonable.

149

From the above discussion it can be seen that the calculations involved in

estimating parameters for the SF-PF model are lengthy. This situation is caused by having

iterations within iterations within iterations. For example, it was not unusual to have the

EXP Fortran subroutine, which is required in Equations (56), (57), and (99), to be called more

than a million times. The computing time and cost were quite large. Unfortunately, this put

real limits on how much evaluation could be done with this model.

NaCl-Water l)ata. The computer programs developed to descrihe the SF-Pl·‘ model were

tested by reanalyzing the NaCl-water data reported by Matsuura and Sourirajan (Q). In

order to simulate the model used by Matsuura and Sourirajan, it was necessary to make the

following adjustments to the computer program. RB was set to zero, since Matsuura and

Sourirajan did not take into account the size of the water molecule. 'l‘he function b(p) was set

to 1.0, which is equivalent to stating that the cliffusivity of the solute in the pore is the same

as the bulk solution diffusivity. Equa-tion (101) was used to force the 1.5 power dependcnce of

Re on nl,. For an inorganic solute, Equation (60) was used to descrihe the potential function.

An iterative search was applied to find the best A value to fit the data for all six membranes.

A was found to be approximately 50 x 1040 m compared to the value of 21 x 1040 in reported

by Matsuura and Sourirajan (@1 However, when the calculation was redone based on the

data for their membranes 3 and 6 only (ßj), good agreement was found between their data and

the computer program developed. 'l‘he RW values agreed within 5% and the A value was 18 x1040 m. The minimum in the sum of the squares error, SSQ, as a function of A was broad,

which implies a relatively large uncertainty in A. 'l‘he analysis as described in this

paragraph has several questionable assumptions built into it and was only used to check the

computer subroutines against the results reported previously. ’l‘hese problems are discussed

later.

150

In order to use the SF—PF model on the NaCl—water data, it is useful to have an

estimate of the size of the NaCl molecule. The size of the solute molecule is required to use themodified Faxen Equation (99). Since NaCl exists in solution as hydrated Na+ and Cl' ions,

which differ in size, it is difficult to determine the correct solute size to use in the modified

Faxen Equation.

The application of the SF—PF model suggested by Sourirajan and coworkers (Q, Q,Q) depends on defining the membrane pore size from NaCl—water data alone. In order to be

successful with this approach, they had to carefully exclude some of the NaCl—water data on

an arbitrary basis. Also, one of Equations (100), (101), or (l()2) had to be chosen to describe

the dependence of water flux as a function of pore size. When the above approach was applied

to the NaCl—water data reported in this work, no minimum in SSQ could be found as a

function of A.

In the current work, a different approach which avoids the above difficulties was

employed. The method allowed the pore size to be determined simultaneously along with the

other unknowns for the organic solute data only. This approach avoids the problem of defining

the pore size based on NaCl—water data and the problems of specifying a relationship between

pore size and water flux for different membranes. The actual method used is discussed in the

next section.

Toluene-Water Data. As a preliminary evaluation of the SF—l·‘F model, the toluene—water

data for two of the membranes was evaluated. A summary of the equations used is given as

follows. The basic equations of the SF—PF model given by Iiquations (47)-(56), (58), (59), (62),

(97), and (98) were used. The surface potential for an organic solute was described by

Equation (61). For the friction function, the modified Faxen Equation (99) was applied. Theadjustable parameters for a given membrane are A, RW, B, D, and E. The pure water

151

permeability coefficient, A, has been calculated and is given for each membrane in Table 5.The distance of closest approach of the solute to the pore wall, D, can be approximated by thesolute radius, RA. There are several methods possible to calculate the solute radius (Q). lnthis work the Stokes—Einstein equation was used:

RA Z (110)6m] DAB NOTherefore the values for solute radius, listed in Table 1, were calculated based on the solutediffusivity and Equation (110). Preliminary studies indicate that the adjustable parameter inthe friction function, E, can be set to zero. Setting E Z 0.0 is equivalent to using only theFaxen equation to represent the friction function, b.

The above approximations simplify the model to two adjustable parameters, RWand B. The data for separation and pore blocking are used to estimate the unknown

parameters, using the numerical techniques outlined earlier in this section. Unfortunately,

even with only two adjustable parameters, numerical optimization is time consuming and

expensive.

'l‘he results obtained for membranes 1 and 3 are summarized in Figures 31-34.

Figures 31 and 33 illustrate the agreement between calculated and experimental separation

for membranes 1 and 3, respectively. Similarly, Figures 32 and 34 illustrate the agreement

between the calculated and the experimental extent of pore blocking. The calculated

separation corresponds well with the experimental separation in the low separation region.

However, in the high separation region, the model systematically underestimates theseparation. The agreement between calculated and experimental extent of pore blocking is

excellent. The model well represents the extent of pore blocking. This result is significant.()f all the models examined, only the SF—PF model is able to predict the extent of pore

blocking from first principles. The KSA model uses Equation (40) to describe the extent of

pore blocking but the equation is empirical and is not derived from the model.

152

cz

//

an /ci _ /

‘O /(D /

-1-ß (Ö /O , /•--• C7 /D /O /·—• / {Il IEci **1 @1 um E‘—’ ° @1 EFV_ /

/4- N / E]G H FQ

//

O /:50.0 0.2 0.4 0.6 0.8 1.0

f° Experimental

Figure 31. Comparison of the separation measured and the separation calculated hy thesurface force—pore flow model for the solute toluene and memhrane 1.

l53

<s•ci'O /Ü /

"" /Ö (7) /

D

Q Ö //

I? /0. Ü IIIC ·· E1\ C;I-CI„-1 zu., Q V

Q0. 0 0. 1 0. 2 0. 3 0. 4(1-nT/ np] E ><per‘ 1 ment c11

Figure 32. Comparison of the extent of pore hlocking (1-nT/np), measured andcalculated by the surface force—pore flow model for the solute toluene andmemhrane l.

15L+

·c>

//

Q /ci / /”OO /

/-•-ß (D /Ö I

.-1 O /0 /O /

•—| /C5/

/ UIIIDq_ mm/müm~ /ci EQ /

//

/Cf uca r: i’‘•l u 0.2 0.4 0.6 0.8 1.0f' Experimental

Figure 33. Comparison of” the separation measured and the separation calculated by thesurface l°orce—pore flow model For the solute toluene and membrane 3.

l55

Vd'O /(D /·'·’ /O (T)

/

:7 ° /O /

'+0 N // 111I? Ü0- /C ___

I.E é él? W

1-·I 1 "C, cgG0. 0 0. 1 0. 2 0. 3 0. 4{1-nT/ np] E ><per‘ 1 ment ol

Figure 34. Comparison of the extent of pore blocking (1-nT/np), measured andcalculated by the surface force—pore flow model for the solute toluene andmembrane 3.

156

Table 15

Parameters for the Surface F0rce—P0re Flow (SF-PF) Model for the Solute Toluene

Membrane 1 Membrane 3 .

Parameters

RW,x 1010,m 4.511 5.217

B, X 1030,1113 157.7 152.3Da, x 10m, m 2.52 2.52

E", m 0.0 0.0

Sum ofSquare Errors

SSQV 0.302 0.196SSQU _ „T;„p, 0.005 0.004

These parameters were fixed at the values shown.

157

The values of RW and B generated for the two membranes are compiled in Table 15.Since membrane 3 is the more open of the two membranes, it is expected that RW formembrane 3 would be larger than RW (or membrane 1. The results in 'l‘able 15 confirm that

_ RW is about 16% larger for membrane 3 than for membrane l. These values of RWare consistent in magnitude with those calculated with the SF-PF model by Sourirajan andcoworkers for similar cellulose acetate membranes (Q, Q). Similar magnitude values arealso predicted by the theories proposed by Glueckauf (Q) and Bean (Q). ()n the other hand,since the polymer and solute are fixed, B should be the same for both membranes. 'l‘he values

of B determined agree within 12%, which supports the above expectation. 'l‘he larger SSQvalue for separation compared to the extent of pore blocking reflects the relative deviationbetween the model and the experimental results.

Again it must be emphasized that the numerical difficulties in determining the

values of RW and B were large. lt was necessary to adjust the initial estimate of the param-

eters manually until the minimum in SSQ was almost reached. At this point the automated

search routine (either ZXSSQ or GRG2) would converge to the solution. lftbe initial estimate

was too far from the solution, ZXSSQ would generate a relatively large step in one of the

parameters and the search would diverge from the solution. For GRG2, an initial guess too

far from the solution would generate a search that adjusted the parameters by small

increments such that little or no change in SSQ resulted. The routine would then terminate

prematurely before the minimum was reached. Apparently neither ZXSSQ nor GRG2 are

doing a good job in this case. Future work should include trying other search routines.

Ratber than apply the SF—PF model to all the solutes and all of the membranes, this

part of the project was halted. It is now necessary to reevaluate each of the numerical

routines for accuracy and efliciency and try to improve the practical application of the SF-PFmodel.

158

In summary, the results for the SF—PF model are both encouraging anddiscouraging. On the positive side, the model is able to predict the trends in separationand the trends in flux which no other one model could do. The accuracy of the prediction of

separation was satisfactory in the low separation region, but deviated in the higher

separation region. For the extent of pore blocking, the prediction was excellent. On the

negative side, the application of the model is at the moment impractical. The numerical

analysis is expensive in terms of real time and computer time. Future work should center

around the reevaluation of the numerical program. ln addition, the exact form of the friction

function and the potential functions may be modified to improve the accuracy of the

prediction.

CONCLUSIONS

'l‘his dissertation has examined both the qualitative and quantitative aspects ofdescribing reverse osmosis transport phenomena in the presence of strong solute-membraneaffinity. The major findings and conclusions ofthis research are summarized here.

Experimental data for the flux and separation of aqueous solutions has beenreported for various solutes, membranes, and operating conditions. ’l‘he three organic soluteswere benzene, toluene, and cumene. Results have been reported at four pressures rangingfrom 670 to 6900 kPa and at several concentrations ranging from 5 to 260 ppm, for all threeorganic solutes and for all six cellulose acetate membranes. 'l‘he six membranes wereannealed at different temperatures in order to provide a series of membranes with gradatedproperties. Additional experiments to characterize and monitor the membrane performance

with the solute sodium chloride were also performed.

The performance for the organic solutes was found to be markedly different thanthat for the solute sodium chloride. The difference in behavior is the result of the strong

attraction between the solute and the membrane. These results have been explained, in a

qualitative sense, by the preferential sorption—capillary flow mechanism. Performance varied

systematically with the nonpolar character of the solute. An empirical equation, which

describes the decrease in flux as a function of concentration, has been proposed. 'l‘he results

obtained are consistent with similar experiments performed by others.

Several transport models have been examined with respect to their ability to

describe the observed data for the aromatic hydrocarbon-water-cellulose acetate systems.

[ncluded in this study are four phenomenological transport models (three forms of the

irreversible thermodynamics—phenomenological transport model, lT—PT-A, lT—PT—B, and [T-

PT—C, and the irreversible thermodynamics - Kedem Spiegler model, lT—KS), three nonporous

159

l60

models (the solution—diffusion, SD, solution—diffusion imperfection, SI)I, and extendedsolution—diffusion models, ESD), and four porous models (the Kimura-Sourirajan Analysis,KSA, two forms ofthe fincly porous model, FPM-3 and FPM—4, and the surface force—pore flowmodel, SI·‘—PI·‘). Of the models examined, the experimental data were best represented by theFPM-4 and SI*‘—I’l*‘ models.

The four parameter version of the fincly porous model, FPNl—4, was able toquantitatively describe the observed data. The parameters generated for the I·‘I’M—4 modelindicated that the pore size parameter, t/c, could be considered constant for a given membranefor all solutes. The partition coefficient on the high and the low pressure sides of themembrane differ. The partition coefficient is apparently changing as a function of membranestructure and/or concentration.

Considerable difficulty and expense (computer time) was encountered withapplying the SI·‘—PI·‘ model. Preliminary results for the solute toluene and two of themembranes were promising. The separation was well represented in the lower separationregion and the model did an excellentjob of predicting the flux characteristics. 'l‘he averagepore radii for the two membranes calculated by the SF-PF model were 4.51 x 1()“ lll m and5.22 x IO' 10 m, which are reasonable compared to estimates made by other researchers. Theparameter B in the potential function equation was found to be constant for the twomembranes analyzed by the SF—PF model.

The remaining models examined performed in less than a satisfactory manner.The lT—PT—A, ESD, and SDI relationships were able to curve fit the experimental data.However, the parameters generated were not consistent with basic premises of any of thesemodels. The forms of the SD, KSA, IT—KS, and l*‘I’M-3 models were inconsistent with theobserved data, and the IT—I"l‘—B and lT—P'l‘-C models were less satisfactory than the IT-P'l‘—Amodel.

I6!

The conclusion is that none of the phenomenological or nonporous transport modelsexamined are completely satisfactory at describing the reverse osmosis data for the aromatichydr0carbon·water—cellulose acetate systems. 'l‘he simple picture of solute and waterdissolving in and diffusing through a homogeneous membrane is not sufficient in this case. Inaddition, modification ofthe solution-diffusion model to account for surface imperfections andto correct the chemical potential driving force for pressure induced solute transport are notsufficient to explain the observed data. The coefficients of the SDI model suggest that a majorportion of the transport is occurring through imperfections, which implies that a porous modelmay be more correct. The mechanism—independent phenomenological transport modelsshould always be applicable. That the phenomenological models are not applicable has beendemonstrated. 'l‘he failure of the phenomenological models is probably related to the natureof the solute-membrane attraction. 'l‘he porous models were the most promising of all themodels examined. Ilowever, this result does not necessarily mean that a porous transportmechanism is correct. More work is required to further resolve the problem of modellingreverse osmosis transport in the presence ofstrong solute—membrane affinity.

162

RECOMMENDATIONS

A great deal of work still needs to be done before a full understanding of transportin reverse osmosis membranes is reached. This statement is particularly true for systemssuch as those addressed in this dissertation in which a strong attraction between the soluteand membrane exists. The ultimate goal of this research is to develop a transport modelwhich will describe the performance of a membrane under a variety of operating conditions

for any solute (or solutes). In addition, it should be possible to predict the performance of amembrane for a new solute by relating the parameters in the transport model to the

physicochemical nature of the solute. To realize these objectives both experimental and

theoretical work remains to be done. The following is a list of recommendations lor further

work:

1) More data on the phenomena of strong solute—membrane attraction need to be

collected. Although several authors have reported results, the breadth of systems

examined has been limited. The approach here has been to measure many data for

a few systems in order that the experimental behavior could be well established. lt

is now necessary to expand to other solutes. The purpose of this work would be to

examine the influence of different interactive forces between the solute and the

membrane on transport behavior as represented by the coefficients in the transport

model. The solutes studied in this dissertation are aromatic hydrocarbons which

are attracted to the membrane by nonpolar forces. Other solutes, such as aliphatic

and alicyclic hydrocarbons, could be used to test the extension of the model to other

cases where nonpolar forces dominate. Phenol and substituted phenols could beused to extend the model to include attraction to the membrane by polar forces.

Higher alcohols, such as hexanol, could be used to include the effect of both polarand nonpolar forces. From this broad base of data it should be possible to develop

163 A

a model that can predict the behavior of cellulose acetate membranes for anystrongly attracting solute.

2) Along the same idea as 1), new data needs to be collected for other membranematerials and compared with the results reported here. ()ther cellulosicmembranes and commercially available polyamide and polyamide derivativemembranes should be included in this study. The purpose of this research istwofold. First, the general form of the relationships developed in 1) could be testedfor other membranes. Second, the model could be extended to include the effect ofthe physicochemical nature of the membrane. The second obiective would requirethe independent development of a theory that can relate membrane structure to

membrane performance.

3) It would be interesting to perform some mixed solute experiments. 'I‘his work couldinclude systems with one solute attracted and one repelled by the membrane (for

example, toluene and NaCl) and with two solutes that both are attracted to themembrane (for example, benzene and toluene). The purpose of examining mixedsolute systems is as follows. Industrial separation problems are almost always

multicomponent systems. Therefore it is practically important to expand the

theoretical treatment to include more than one solute. From a theoretical point ofview, addition of another component increases the number of interaction forces

present. For some systems, particularly with dilute solutions, the solutes maybehave independently. With NaCl as one solute the influence of osmotic pressureon the model can be examined. This effect is difficult to study otherwise becausemost of the organic solutes have limited solubilities and hence limited osmotic

pressures.

164

4) Almost all the data reported are at room temperature. lt can be expected that thesolute—s0lvent-membrane interation forces will each vary as a function oftemperature. An Arrhenius type relationship can be used to describe thetemperature dependence of the interactions and apparent activation energies canbe calculated. ’l‘he magnitude of the activation energy is an indication of thenature of the interaction force. 'l‘his study could lead to a better understanding ofthe solute-solvent-membrane interactions.

5) The results reported here for the FPl\/1-4 model suggest that the partition coefficientis not the same on the high and low pressure sides of the membrane. Thedependence of the partition coefficient can be determined as a function ofmembrane structure by measuring equilibrium sorption of a solute on celluloseacetate powder, and on membrane samples annealed at different temperatures.’l‘he

concentration dependence can be determined by measuring equilibriumsorption at different concentrations. These data can then be compared with theratio of partition coefficients determined from the transport model. lt is alsopossible to incorporate an expected dependence into a reformulation of the finely-porous model.

6) Along with the experimental study in recommendation 5) it would be possible tomeasure the water content of membranes annealed at different temperatures andtreated at different solute concentrations. Measurements may be difficult as smallchanges in water content are expected. If no change is found then it is reasonableto assume that for a given membrane the pore size ( in the finely-porous and thesurface force~pore flow model) is independent ofsolute and solute concentration. lfthe water content does change then it may be possible to relate the water content tothe observed change in flux with solute concentration.

l65

7) Further work must be extended to improve the numerical methods used in the SF-PF model. At the moment the SF—PF model is impractical to apply on a routinebasis to reverse osmosis systems. A more efficient and reliable search routine isrequired to estimate the unknown parameters in the model. lt is necessary to

examine the relative importance of each of the terms in the differential equation

which describes the solution velocity in a pore (Flquation (56)). Elimination of someterms would reduce computation time. lt is possible to apply orthogonal collocation

methods to the solution of the differential equation. ()nce the computational

problems are reduced it is possible to modify the model by changing the form of thefriction function and the solute—membrane interaction potential to improve the

performance of the model. 'l‘he influence of pore size distribution, noncylindrical

pore geometry, and nonspherical solute geometry can also be included in the model.8) 'l‘he fact that the phenomenological models would not explain the experimental

data is disturbing. Reasonable arguements why this happens have been presented.

Yet it should be possible with further work to extend these models in a logical

manner so that they would be successful. A theorectical study should be

undertaken to examine the inclusion of attractive forces between the solute and

membrane, along with the usual relationship between fluxes and forces, to produce

a new transport model.

lt is hoped that the above recommendations will provide a useful guide to furtherresearch efforts.

166

NOMENCLATURE

A pure water permeability coefficient, kmol/m2 s kPa.A force constant defined by Equation (60), m.

AU pure water per111eability coefficient extrapolated to zero operating pressure asdefined by Equation (15), kmol/m2 s kPa.

B, constant defined by Equation (D-7) in Appendix 1), m°‘l.

B2 constant defined by Equation (l)—8) in Appendix 1), k111ol/1114.

B s0lute—membrane attraction constant defined hy Equation (61), 1113.b friction parameter defined by Equation (44).

bmxm] friction parameter calculated by Equation (63).

C molar density, kmol/m3.

Ci molar density at location i, l<ITl()l/1113.

CU molar concentration ofcomponent i in phasej, kmol/m3.

membrane pore size parameter with reference to the solute NaCl as defined byEquation (13).

CA dimensionless concentration defined by Equation (49).

1) closest distance ofapproach between solute and membrane, m.[)ij diffusivity ofcomponent i in componentj, mg/s.

d distance from the center ofthe solute molecule to the membrane, m.

d.f. degrees offreedom.

E constant defined by Equation (65), n1.EO, El, E2 generalized transport parameters in Equations (68)—(71 ) and Table 3.

FA driving force for diffusion ofthe solute in a pore, kJ/m kmol.

FAB friction force between the solute and solvent, kJ/111 kmol.FAM friction force between the solute and pore wall, kJ/m kmol.

167

Fi generalized thermodynamic Force deFined by Equation (16).F separation.

F' separation calculated based on the boundary layer concentration.(—AAG/R’l‘)i Free energy parameter For ion i.

g density, kg/m3.Ji generalized thermodynamic Flux defined by Equation (16).JV solvent volume Flux, mg/m2 s.K solute partition coeFficient.

Ki solute partition coeFlicient at location i.k mass transFer coeFficient, m/s.Lii generalized thermodynamic phenomenological coeFFicients deFined byliquation ( 16).€Ai,

pressure—induced solute transport parameter, kmol/mg s kPa.€ii

hydraulic permeability coeFlicient, m/s kl’a.€ii

osmotic permeability coeFlicient, m/s kl’a.

Ni molar Flux oFcomponent i, kmol/m2 s.

No Avogadro’s number, kmol°l.

n reFractive index.

np pure solvent mass Flux, kg/m2 s.

n„i. total solution mass Flux, kg/m2 s.

Mi molecular mass oFcomponent i, kg/kmol.

mAi molal concentration at location i, kmol/m3.AP pressure diFFerence across the membrane, kPa.AP eFFective pressure driving Force deFined by Equation (89), kPa.P polarizability, m3.

p^ solute permeability coeFFicient, mz/s.

l68

pß water permeability coeflicient, mz/s kPa.QP mass rate ofpure water through a single pore, kg/s.

R gas constant, kJ/kmol K.

RW radius ofa pore, m. _

Re effective radius ofa pore as defined by Equation (52), m.

Ri radius ofcomponent i, m.

r radial position in a pore, m.

SSQi residual sum ofsquares in response i.

s standard deviation.'l‘

temperature, K.

uß velocity ofsolvent in a pore, m/s.

Vi partial molar volume ofcomponent i, m3/kmol.

XM mole fraction at location i.x coordinate direction perpendicular to the membrane, m.

Ax membrane thickness, m.

Y term defined by Equation (l)—5) in /\ppendix D, kmol/m3.

Y' term defined by Equation (l)—6) in Appendix l), kmol/m4.

z axial position in a pore, m.

Greek Symbols

ci dimensionless solvent velocity in a pore as defined by Equation (48).[$1 dimensionless ratio defined by Equation (50).

[52 dimensionless ratio of the applied pressure to the osmotic pressure at the poreentrance as defined by füquation (5l ).

[53 dimensionless ratio defined by Plquation (96).

v constant defined by liquation (I5).

169

e fractional pore area.

Z, ( constants defined by Equation (40).

Z', C constants defined by lüquation (41).

r] solution viscosity,k1’a s.U

K’,i<",x"' transport parameters in the solution-diffusion imperfection model as definedby füquations (30) and (31).

,\ ratio ofsolute radius to pore radius.

pi chemical potential ofcomponent i, kJ/kmol.

Api chemical potential difference across the memhrane for component i, kJ/kmol.

ni osmotic pressure of solution i, k1’a.

An osmotic pressure difference across the membrane, k1’a.p dimensionless radial position in a pore as defined by liquation (47).o refiection coefficient.

t effective thickness ofa membrane, m.

<1> dimensionless potential function of force exerted on a solute molecule hy apore wall.

Xii friction coefficient between components i andj, kJ s/kmol mg.

w transport parameter defined by Equation (21), kmol/mz s k1’a.

Subscripts

A solute

B solvent

cal calculated

e effective

f' with respect to f'

In logarithmic average

M membrane

170

P pure water

ref reference’1‘

total solution

W wall

(1-nT/np) with respect to (1-n.l/np)1 feed solution

2 houndary layer solution

3 permeate solution

171

APPENDIX ADerivation ofthe Solution—Dif”f'usion lmperfection Model for

l.ow Osmotic Pressure and l)ilute SolutionsThe purpose of this appendix is to illustrate the simplifieation of the Sl)l

relationship for the case of low osmotic pressures (AP > > A11) and low concentrations. Thisanalysis uses the Sl)l model as presented by Equation (32) and van’t llof”l”s law given byEquation (12) to derive lüquation (33). The derivation of Equation (33) is not available in theliterature and is therefore presented here explicitly.

The Sl)l model as presented by Jonsson and Boesen (@1) is represented by:

Z ( 11 + K K (32)1 AP—A ) ( "’/ ')AP1—f" (K"/K')(A11/CAZ) + (K"'/K')AP

When AP > > A11 then the above equation becomes;III/ I-—Z1—f’ (K"/K’)(A11/APC^_,) + (K"'/K')

Substituting the van’t 11ofT lüquation (12) into the definition ()f‘A11 gives;

A11 = 112 — 113 (A-?)A11 =CA2 RT — CM RT (A—3)A11 = (CAQ — CA3)

R'[‘ (A—4)Substituting Equation (A-4) into Equation (A~1) gives:

1 _ 1 -1- K"'/K'CA2'CA:1 1

RT{x2Recognizing that f" = (CA2 — CM)/(CM (from lüquation (7)) and inverting Equation (A—5)leads to:

" ' f’(R'1‘/AP "’/ ’,_,.:;+..*;._ M,1 + K"'/K' 1 + K"'/K'

I72

Collecting terms amd rearramging Equa1tion(/\-6)gives:( "/ ')(R'I‘/AP) "'/ ’6·{1+l—):I-—‘l mml + K"'/x' I + x"'/x'

L : I + x"'/x' + x"/x’(l{'l‘/AP) (A-8)FI Kill/Kl

I l + K"'/K' + K"/x'(l{'l‘/AP) (33)which is the Form ofthe Sl)[ model used in this project.

173

APPENDIX B

Raw I)ata For the NaCl—Water—Cellulose Acetate Systemln this appendix the data obtained For the solute NaCl with the six cellulose acetate

membranes are presented. The water Flux reported, in kg/mgs, is the average oFmeasurements made with a pure water Feed before and aFter the solution experiment. Thesolution Flux, in kg/mgs, was measured with NaCl present in the Feed. 'l‘he Feed concentrationis the average ofsamples collected at the start and end oF the permeate sampling period. Theseparation is calculated based on the Feed concentration by Equation (1). 'I‘he other symbols

and the units are defined in the N()l\/ll•}NCl./\TURl*) section.

174

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191

APPENDIX C

Raw Data for the Aromatic Hydrocarbon-Water-Cellulose Acetate System

In this appendix the data obtained for the three aromatic hydrocarbon solutesbenzene, toluene, and cumene are presented. The data are tabulated in order by experiment

number which corresponds to the order in which the data were obtained. Tables C—1 to C·3summarize the information by solute, pressure, and concentration. The water flux, in kg/mzs,is the average of measurements made with a pure water feed before and after the solutionexperiment. The solution flux, in kg/m2s, was measured withthe organic solute present inthe feed. The feed concentration is the average of samples collected at the start and the end ofthe permeate sampling period. The separation is calculated based on the feed concentration

by Equation (6). "Pore-blocking" is the extent of pore blocking as defined on page 21. Other

symbols are defined in the NOlVlI*]NCLA’I‘URE section.

192

Table C-1

Summary of the Data Collected for the Solute Benzene

Solute Pressure Feed Experiment Table. Concentration Number Number

kPa ppm

Benzene 690 53.4 129 C-32116.7 133 C-33138.2 128 C-31259.2 136 C—35

1725 24.2 134 C-3440.4 125 C—30

124.8 124 C—29227.0 123 C-28

3450 38.4 118 C-2563.2 117 C-2490.7 116 C—23

252.8 115 C-22

6900 46.2 113 C-2177.9 112 C-20

149.4 111 C- 19243.9 110 C- 18

193

Table C-2

Summary ofthe Data Collected for the Solute Toluene

Solute Pressure Feed Experiment TableConcentration Number Number

kPa ppm

Toluene 690 11.2 102 C—1332.4 99 C-1276.7 96 C—l1

153.9 95 C-10

1725 17.2 91A C—718.5 91B C-835.4 90 C—685.1 89 C—597.5 88 C—4

214.1 94 C—9

3450 45.4 121 C~2751.6 139 C—37

136.7 138 C—36161.2 120 C-26

6900 26.1 108 C—1755.8 106 C—1693.3 105 C—15

211.4 104 C—14

194

Table C—3

Summary of the Data Collected for the Solute Cumene

Solute Pressure Feed Experiment TableConcentration Number Number

kPa ppm

Cumene 690 6.8 154 C—498.1 153 C—48

15.9 152 C-4717.3 151 C—46

1725 9.9 159 C—5312.2 158 C—5216.1 157 C—5125.9 156 C 50

3450 7.7 150 C—4510.2 148 C—4416.0 147 C—4326.5 146 C—42

6900 4.5 144 C—418.1 143 C—40

12.7 142 C—3924.6 140 C-38

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APPENDIX I)

Integration ofthe Flux Equation in the z—clirection for theSurface I•‘orce—I’ore Flow Model

The purpose of this appendix is to illustrate the integration of Equation (91) with

respect to z to yield Equation (95). In this analysis the effect of pressure induced solute

transport is added to the surface f0rce—pore flow model. 'l‘he integration, although relatively

straightforward, is significantly different than the original version (Q) and is therefore

presented here in detail.

lt is required to integrate lüquation (91 ):

RT OC! (r,z) V C A (r,z)AI’(r)NAI,-1 : I- ‘9"XAB Oz XM; L ·

with the boundary conditions (Q):

C^M(r,0) Z C^2e"L"'I’ (92)and

C^M(r,L) Z Cmtr) e (93)Noting that: '

CA3(r) Z N^(r)/uB(r) (94)and substituting Equation (94) into Equation (91) gives:

C\_,(r‘) UB(!')l)(!‘) Z — —·— —+·— + #T———· + CM/I(r‘,z) uB(r) II XM; ‘}‘ AB Lwhich can be arranged as:

OC/\M(r,z) · OC^M(r,z) uB(r) XM,+

VA CAM (r,z) AI’(r) CA3(r) uB(r) l>(r) XAB (D2)Oz RT RT L RT

Collecting CAM(r,z) terms:

6C XAB VA AI’(r) C^3(r) uB(r) b(r) XM, (I):CIIOz^M RT R'I‘ L RT

221

and moving them to the left hand side gives:XM lu 0+ XA ^""’| C XM L 0 ob,) MM·—··—· — —— ——l— r,z : - —— „ raz RT B XM, L AM RT A-) XXB X Xl)efining:

Y(z) : CAMöx

XVX A8 XXA8Z - —- - 1)-882 RT C^3(r) uB(r) h(r) l X

allows liquation (l)—4) to he rewritten as:Y'(x) — 81 Y(x) : 8_) XX)_9X

which is a first order linear nonhomogeneous differential equation. 8, and 82 are functions of

r only and may he considered as constants when integrating with respect to x. ’l‘he houndary

conditions, lüquation (92) and (93), can he rewritten as:. (1)-11))atx : 0, Y(()) :C^2e_‘XXXX

_ atx: L, ll)‘lll

Equation (I)-9) can be solved by multiplying through by the integration factor

exp( — Bl x):-8lx -812 -8lx ([)-|2)Y'(x)c — 81 e Y(z) : 8,,e

which is equivalent to:d -8 z -8 x—(Y(x)e X : 8, e X (Ü·l3)dz , Z

lntegration from x : () to x : L gives:

-81 L_B Z [:1X

I B 0 :-———eX’LX X B, Z = 0Y(())e

222

which can be rearranged as:—B L B. -B LY(L)e L -Y(0)=- l(e L -1) (D45)

BLiV1ultiplyinghyexp(BL L)/Y(0) gives:

L

B, .Y(L) :cBL“— Z (L (1)-16)Y(0) Y(0) BLSuhstituting iüquations (1)-7), (1)-8), (1)-10) and (D-11) into Equation (1)-16) gives:

L+

CALL(r) uLL(r) h(r) (LI BI LA (III?)—T —— — eC L V AP(r)AZ -Lb(1') _ ACA2eXAB L

Collecting CA3(r)/CM terms gives:B L

Z (1)-18)CAQ I b(¤‘) ( UHU) L LL; L)exp(-c|>(r)) VA AP

uLL(r)+ ·———·XALL L

and substitution for BL and rearrangement produces:

XALL L VA APCAII(L·) _ R1 XALL L „ ·CA2 h(r) uam XALL L VA M)

1+--- ————·——·————) exp — U1I')*)""") -1)exp( -Lpm) vA AP irr 1* XALLLu(r)+ —i AB XAB L

which is the desired solution of° Equation (91) integrated with respect to z and the appropriate

houndary conditions.

REFERENCES

1. Sourirajan, S.; Matsuura, T. In "Reverse Osmosis and Synthetic Membranes",Sourirajan, S., Ed.; National Research Council Canada: Ottawa, 1977; Chapters 2,3.

2. Matsuura, T.; Sourirajan, S. J. Appl. Polymer Sci. 1973, Q, 3683-3708.

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