Desalination by salt replacement and ultrafiltration.
Item Type Thesis-Reproduction (electronic); text
Authors Muller, Anthony B.
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Link to Item http://hdl.handle.net/10150/191593 DESALINATION BY SALT REPLACEMENT AND ULTRAFILTRATION
by
Anthony Barton Muller
A Thesis Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY AND WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA
1974 STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of require- ments for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Request for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGN:
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
D. D. EVANS Date Professor and Head of Hydrology & Water Resources; and Professor of Soils, Water & Engineering ACKNOWLEDGMENTS
The author wishes to thank Dr. Daniel D. Evans, Head of the
Department of Hydrology and Water Resources, for advice and guidance in both the undergraduate and graduate programs which have led to this thesis. My thanks also go to Dr. Simon Ince and Dr. Eugene S. Simpson of the Department of Hydrology and Water Resources for serving on my thesis committee. In addition, further thanks are due to Dr. Cornelius Stellink and Dr. Leslie S. Forester of the Department of Chemistry for their patient assistance with the organic chemistry involved, and to
Dr. William R. Salzman and Dr. Roald K. Wangsness of the Departments of
Chemistry and Physics, respectively, for reviewing the physical chemistry
and thermodynamics presented. Dr. Hasan K. Qashu of the Department of
Hydrology and Water Resources, my mentor for many years, deserves great
credit for helping provide the scholarly atmosphere from which this
thesis has come, as does Dr. A. Richard Kassander, Vice President for
Research, for providing the financial aid which allowed all aspects of
this thesis to be realized. TABLE OF CONTENTS
Page
LIST OF TABLES vi
LIST OF ILLUSTRATIONS vii
ABSTRACT ix
INTRODUCTION 1
Major Traditional Desalination Methods Having Received Attention as Potential Large-Scale Fresh Water Sources 5 Distillation Methods 5 Crystallization (Freezing) Methods 9 Solvent Extraction Methods 12 Chemical Separation Methods 14 Membrane Separation Methods 16 Salt Replacement: A New Method of Low-Energy Desalination Proposed for the Large-Scale Production of Fresh Water • • 24 Enzyme-Catalyzed Osmotic Pressure Reduction 25 Low-Pressure Step Ultrafiltration 32
THEORY 37
Development of Osmotic Pressure Relationships for a System in Membrane Equilibrium by Classical Thermodynamics 38 General Equilibrium Conditions 38 Membrane Equilibrium 40 Osmotic Pressure Equation 43 Development of Fundamental Equations of Flow Across a Semipermeable Membrane by Irreversible Thermodynamics 44 Entropy Production 44 Derivation of Entropy Change in Vector Notation 46 Development of Phenomenological Equations 48 Phenomenological Coefficients 51 Phenomenological Equations of an Osmotic System 53 Relationship of Derived and Empirical Expressions . . 57 Development of First-Order Transport Equations and Corresponding Coefficients Describing Intrinsic Membrane Characteristics 58 Transformation of Uni-Component Flux Equations 58 Membrane Description by First-Order Transport Coefficients 62 Evaluation of First-Order Transport Coefficients 64
iv ••
TABLE OF CONTENTS--Continued
Page
Permeation Models for Solute and Solvent Transport Kinetics of Diffusive and Microporous Osmotic Membranes 67 Solution-Diffusion Model 67 Pore Flow Model 73 Theoretical Treatment of Concentration-Polarization at Phase Boundaries in Membrane Separation Systems with Laminar Flow Regimes 81 Differential Equation for Solute Balance 83 Treatment of Constant and Variable Permeation Fluxes . . 88
EXPERIMENTATION 91
Experimental Apparatus Used in Ultrafiltration of Sucrose in this Study 92 Ultrafiltration Experiments with Pure Water Solvent which Determine Hydraulic Permeability and Transport Mechanism . 99 Determination of Hydraulic Permeability 99 Determination of Transport Mechanism 104 Ultrafiltration Experiments with Variable Sucrose Concentrations Determining UM Series Membrane Fluxes and Separation at Operating Concentrations 107 Osmotic Pressure of Sucrose Solutions 107 Permeation Flux as a Function of Solute Concentration . 111 Membrane Rejection as a Function of Solute Concentration 115 Reduction of Experimental Sucrose Separation Data to Generate Solute Rejection Coefficient from Permiate-Retinate Concentration Relationships 121 Permiate-Retinate Concentration Relationships 122 Evaluation of the Solute Rejection Coefficient R 125 Reduction of Experimental Sucrose Separation Data to Generate Reflection and Solute Permeability Coefficients from Concentration-Polarization Relationship 129 Concentration-Polarization Along UM Series Membranes . 129 Evaluation of the Reflection Coefficient 140 Evaluation of the Solute Permeability Coefficient . . . 144
CONCLUSIONS 147
NOTATION 151 SELECTED BIBLIOGRAPHY 159 LIST OF TABLES
Table Page
1. Principal Traditional Desalination Methods 4
2. Published UM Membrane Series Characteristics 95
3. Experimental Results and Determination of L 100
4. Comparison of Published and Observed L Values 103
S. Comparison of Pure Solvent Fluxes 105
6. Osmotic Pressure of Sucrose Solutions 109
7. Solute Reflection Coefficients 126
8. Boundary Layer Concentration 131
9. Bulk Permeation Flux Prediction 137
10. Concentration-Polarization Permeation Flux Prediction . . 139
ll. Reflection Coefficients 142
12. Reflection Coefficient Adjusted Permeation Flux Prediction . . 143
13. Solute Permeability Coefficient Evaluation 146
14. First-Order Transport Coefficients for UM Series Membranes . . 148
vi LIST OF ILLUSTRATIONS
Figure Page
1. Solute-Solvent Separation Processes Useful in Various Solute Size Ranges, and the Primary Underlying Separation Principal of Each 26
2. Salt Replacement Desalination 27
3. Desalination by Salt Replacement and Enzyme-Catalyzed Osmotic Pressure Reduction with Ultrafiltration 28
4. The Molecular Configuration of the Principal Disaccharides: Sucrose, Maltose, Cellobiose and Lactose 30
5. Concentration Profiles in a Solution-Diffusion Membrane 72
6. Angular Deformation of a Fluid Element 74
7. Laminar Flow Through a Cylindrical Tube 75
8. Concentration Profiles in Pore-Flow Membranes 80
9. Two-Dimensional Channel Between Flat Parallel Osmotic Membranes 83
10. Profile of Concentration-Polarization at Channel Boundary for Laminar Flow Along a Membrane Surface 84
11. Schematic Representation of Experimental Apparatus Used in the Ultrafiltration of Sucrose in this Study 93
12. Thin-Channel Ultrafiltration Cell 94
13. Volumetric Flux Versus Applied Pressure, Results of Pure Solvent Runs at Various Pressures Used to Determine L for the UM Series Membranes. 1 01
14. UM05 Permeation Flux Versus Sucrose Molality, in ml of Solution Per 5 Minutes from the Ultrafiltration Cell at 40 psi 113
15. UM2 Permeation Flux Versus Sucrose Molality, in ml of Solution Per 5 Minutes from the Ultrafiltration Cell at 40 Psi 114
vii vi ii
LIST OF ILLUSTRATIONS--Continued
Figure Page
16. UM10 Permeation Flux Versus Sucrose Molality in ml of Solution Per 5 Minutes from the Ultrafiltration Cell at 40 psi 116
17. Retinate Versus Permiate Molalities for UM05 (-), UM2 (+) and UM10 (o) Ultrafiltration Membranes -119
18. Retinate Versus Permiate Molalities for UM Series Membrane at Low Concentrations 120
19. Separation Relationship Observed by 'Kimura and Sourirajan (1968d) and Adjusted for Comparison to this Study, at Various Pressures and with Various Membranes 124
20. Membrane Performance Diagram for the UM Series Membranes, with Rejection Coefficient Plotted Against Retinate Concentrations 127
21. UM10 Permeation Flux Predictions Versus Sucrose Molality, Considering Hydraulic Permeability (o), Concentration- Polarization (+) and Reflection Coefficient (-) 133
22. UM2 Permeation Flux Predictions Versus Sucrose Molality, Considering Hydraulic Permeability (o), Concentration- Polarization (+) and Reflection Coefficient (•) 134
23. UM05 Permeation Flux Predictions Versus Sucrose Molality, Considering Hydraulic Permeability (o), Concentration- Polarization (+) and Reflection Coefficient (•) 135 ABSTRACT
The replacement of solutes in a saline solution by a replacer chemical across an osmotic membrane, and the subsequent removal of the chemical by virtue of its special removal characteristics, comprises salt replacement desalination. Any of a number of removal processes may be coupled to the replacement step, the process being determined by charac- teristics of the replacer. Ultrafiltration is examined as a removal process with sucrose as the replacer chemical. A theoretical treatment of osmotic flow across semipermeable membranes is presented in terms of phenomenological and first-order transport equations. The pore flow and
solution-diffusion models of the transport kinetics of such osmotic flows
are derived.
An experimental examination of the ultrafiltration of sucrose
shows that higher flux membranes operated at lower pressures than reverse
osmosis offer comparable product fluxes. Such a system would, thus, not
require the high-pressure apparatus required for reverse osmosis.
Experimental results also show that a constant separation relationship
exists between concentrate and permiate at operating pressures above the
osmotic pressure of the retinate, and that the separation characteristics
of membranes of different cutoff levels with solutes of molecular weight These findings well below the cutoff level being filtered are similar.
indicate that salt replacement with ultrafiltration has strong possibil-
ities for development as a large-scale desalination method. ix INTRODUCTION
"Water, water everywhere, Nor any drop to drink," is not only the plight of Samuel Taylor Coleridge's (1959) ancient mariner of 1798, but has become a major world problem. Approximately
1,365 million cubic kilometers of water, or more than 97 percent of the water on earth, is stored in the oceans (Yasso, 1965; McLellan, 1968).
Another 2 percent is in the form of ice in glaciers and ice caps or more than 4,000 meters below the surface of the earth (Nace, 1967). Only the remaining one percent is potentially available for the immediate uses of man. This remaining water exists in lakes and rivers, in aquifers and soil, and in the atmosphere. The amount of water yielded annually through the atmosphere in the form of precipitation which falls on land areas is more than sufficient to satisfy the needs of the earth's popu-
lation (Gillam and McCoy, 1966). Since the world annual average of 28
inches of precipitation (Ward, 1967) should be adequate for all human requirements and yet large areas of the world suffer from shortages of
fresh water, it is apparent that the distribution of available water does not correspond to the logistics of water use. Depending on definitions used, between 30 (qcGinnies, Goldman and Paylore; 1968) and 60 percent
(Gillam and McCoy, 1966) of the earth's land surface is considered arid,
with insufficient fresh water to support its portion of the world popu-
lation. The world is currently experiencing a tremendous drive to open
such areas for large-scale settlement as a result of the rapidly 1 2 increasing population. Moreover, another repercussion of this inexorable growth is the appearance of occasional fresh water shortages in highly industrialized areas in zones of moderate climate and abundant precipi- tation. Bradley (1962) estimates from his calculations of per capita water consumption in the United States that well before the year 2000 such water shortages shall occur to the extent that they will force a serious reduction in the current standard of living if radical solutions are not found to our water problems.
The problems of the availability of water in sufficient quantity, and of suitable quality, available when and where needed, at reasonable cost, have acquired a world-wide importance far in excess of that in the time of Coleridge. Such problems are so numerous and diversified that no one solution appears capable of alleviating them. Principally, solutions consist either of using a given water supply more efficiently, or of
increasing the quantity and/or improving the quality of the supply avail-
able. The first group of solutions involves the reduction of per capita
consumption, the reduction of storage, transit and usage losses, the
development of more efficient industrial practices as they relate to development water use, the growing of crops that consume less water and
of others that tolerate water of poorer quality, and the multiple- or The second re-use of existing water supplies (Gillam and McCoy, 1966). sources, the group of solutions involves development of new groundwater development of new employment of artificial recharge (Simpson, 1968), the
surface water sources, the importation of fresh water from water-rich sources regions, and the reclamation of local saline or polluted water
(Spiegler, 1962). Any of the areas experiencing water shortages, such as 3 the Caribbean Islands, the islands of the South Pacific, the Near East and North Africa, and Australia, have only ocean water or brackish groundwater available as sources of the increasing amounts of water required in the future (Spiegler, 1962). Of necessity, saline water conversion becomes the only plausible solution to the water scarcity problem in many such areas.
Saline water conversion (desalination or demineralization) is the purification process which reclaims fresh water from water more saline than acceptable for its specific use. All purification processes require
a separation phenomenon and an energy source for operation. Traditionally,
such processes are classified by these two conventions, as has been done
by Jenkins (1952); Gillam and McCoy (1966); Anderson, Sturga and Strobel
(1970); and many others. Here, a less rigorous breakdown of processes
than the above shall be presented, grouping similar methods having
received extensive attention as potential large-scale desalination
schemes. Obscure processes, such as electro-gravitational methods of
separation, and obscure energy sources, such as marine thermal gradients, shall not be treated. Rather, phenomenon-energy combinations which have
had the most attention in recent years shall be examined. Table 1
presents these principal methods for the demineralization of saline
waters, compiled from publications of the National Academy of Science
(Desalination Research and the Water Problem, 1962), of the University of
Maryland (Fresh Water from the Sea, 1958), of the Department of the
Interior (Jenkins, 1952), and of the United Nations (Water Desalination,
1965). 4
Table 1. Principal Traditional Desalination Methods.
Distillation: Vapor compression Flash Critical pressure Humidification (solar) Multiple effect
Crystallization: Direct freezing Indirect freezing
Solvent Extraction
Chemical Separation: Ion exchange Precipitation
Membrane Separation: Osmionis Electrodialysis Reverse osmosis
It should be noted that a principal consideration in the evaluation of a desalination method is the minimization of the energy requirement for the production of a unit of pure water. A thermodynamic and economic analysis of this consideration is presented by Evans,
Crellin and Tribus (1966). This paper is intended to introduce and examine several aspects of salt replacement desalination, a new saline water conversion method which shows great potential in optimizing this important criterion. Salt replacement entails the replacement of salts in saline solution by a replacer chemical across a semipermeable membrane and the subsequent low-energy removal of that replacer by virtue of its special removal characteristics. 5
Major Traditional Desalination Methods Having Received Attention as Potential Large-Scale Fresh Water Sources
Distillation Methods
All distillation methods are based on the phenomenon that at temperatures below 300 ° C water, and the gases dissolved in it, are volatile while salts are not. The distillation process itself consists
of the vaporization of part or all of the water from the saline solution
and the subsequent condensation of the resultant mineral-free vapor as
fresh water. The products of the process are concentrated saline
solution, possibly containing crystalline salts, and pure water. Since,
in this process, the water is removed from the salts, rather than the
converse, energy requirements are independent of salt concentration in
the feed solution. Several methods employing this process have been
extensively examined in recent years, of which the principal ones are
described below. Vapor-Compression Distillation. In vapor-compression, or thermo-
compression, saline water is evaporated at atmospheric pressure and
slightly above 100 ° C. The resultant water vapor mixture is compressed to just above 1.03 atmospheres, undergoing a temperature rise to almost 105 °
or 106 ° C. The high temperature vapor is returned to heat more salt water
from which the original vapor was formed. The temperature difference of solution 5 ° or 6 ° C between the compressed vapor and the evaporating
allows the required heat transfer. Substantially all of the latent heat
of the compressed vapor is transferred in this manner and condensation of
the vapor occurs in the same unit as does the vaporization. No separate
condensing unit or coolant is required in the system. It is apparent 6 that the energy in the process is supplied by the vapor compressor and, thus, the energy requirements are mechanical or electrical rather than thermal (ClerfaYt, 1967).
Flash Evaporation. In flash evaporation, saline water is heated in conventional heat exchangers and maintained slightly above its vapor pressure at 100 ° C so that no vaporization will occur. This liquid, at
100 ° C, is introduced into a chamber of slightly lower pressure where
flash vaporization is induced. Most of the dissolved salts are separated
out at this point and, thus, scale formation is virtually eliminated in
these processes. The resultant vapor is then condensed to obtain fresh water by either the introduction of a coolant or by the combination of
flash vaporization with the self-condensing system just described. Such
a combination is an efficient distillation unit operating on a simple
compressor. It is important that in the strict sense the system is
designed for 100 ° C, atmospheric pressure vaporization since, if it were
not so contrained, temperature and pressure reduction would lessen power
requirements and scale formation. Variations such as centrifugal
compression stills provide a radical improvement in efficiency by employing such reductions (Jenkins, 1952; Spiegler, 1962).
Critical Pressure Distillation. In critical pressure distilla-
tion, salt water is vaporized near the critical point of water (374 ° C,
218 atm), where the density of pure water and that of water vapor are
both 0.4 gm/ml, and where the heat of vaporization of sea water is
minimal. It is necessary to compress the feed solution to very high
pressure and to heat it to a very high temperature, but the energy
dissipation of the phase change is very small and part of the pumping 7 energy can be recovered by discharging products through turbines coupled to the compression mechanism. The disadvantages of the system are the high capital cost of high-pressure equipment and the extensive insulation required to minimize heat loss from the system. Some heat may be
reclaimed in the condensation process as in vapor-compression. Energy
requirements are thermal and mechanical or electrical (Spiegler, 1962).
Humidification (Solar Distillation). In humidification, vapor-
ization of water occurs at temperatures below boiling. This distillation
method is usually coupled to a solar energy source. Solar energy is
transmitted through a cover into a shallow basin of salt water. Part of
the energy is absorbed directly by the water, but most of it is absorbed
by the basin bottom and transmitted by conduction into the salt water.
This raises the temperature of the solution and causes partial vaporiza-
tion of the water into the air space below the cover. Condensation
occurs as the vapor strikes the cooler cover and the pure water collects
by gravity in troughs on either side of the salt water basin. Variations
on this method, having in common solar energy as the driving force and
partial vaporization as the mechanism, are employed in various humidifi- cation processes. It should be noted that the efficiency of such systems
is proportional to the amount of incident radiation and, thus, they are
best suited to tropical or subtropical zones (Leif, 1966).
Multiple-Effect Distillation. In multiple-effect distillation, is condensed in a the vapor produced in an initial evaporator, or effect, second unit. Here, the heat of condensation of the vapor serves to boil the second effect acts as the sea water of the second evaporator. Thus,
a condenser for the first, and vapor in the second acts as the heating 8 system of the first. Similarly, a third evaporator acts as a condenser for the second, and so on. A series of such evaporators comprises a multiple-effect system. The boiling pressures and temperatures of subse- quent effects must be lower for the heat transfer to occur. This requires pumps to be used between stages to maintain the pressure difference. In theory, the number of effects joined in this manner is virtually limitless, but in practice ten to twelve effects have been
found optimum. The number is constrained by the total thermal drop
available, the thermal drop across each effect and the energy cost for
the system. The energy requirements of the system are both thermal and - mechanical or electrical (Silver, 1966; Clerfayt, 1967).
Distillation is the most thoroughly studied desalination process.
The principles involved, as noted by Silver (1961), were recognized by
Aristotle in the fourth century B.C. Moynihan (1915) appears to be the
first proponent of sea water distillation as a source of fresh water in
modern times. By 1938, patents had been granted in the United States,
Great Britain, France and Germany for sea water distillation apparatus
(Jenkins, 1952). Then, during the second World War, distillation
received a great deal of attention when life rafts and naval vessels
required new sources of fresh water at sea. During this period, waste heat utilization (Kleinschmidt, 1942; Bunnel, 1944), vacuum evaporation
(Delahanty, 1943), flash distillation (Aiton, 1944) and solar distillation
(Schenk, 1944; Kain, 1944; Barnes, 1946; and many others) were extensively
studied. In recent years the use of nuclear energy for fresh-water production (Baron, 1964; Post and Seale, 1966) has been investigated, and
large-scale distillation plants have been put into operation throughout 9 the world, including a 3.4 million gallons per day, flash plant in
Curacao (Caribbean), and a 5.1 million gpd, flash plant in Kuwait
(Persian Gulf) (Gillam and McCoy, 1966). Since the establishment of the
Office of Saline Water, U. S. Department of the Interior, in the early
1950's, this organization has led research in saline water conversion on
all fronts, including distillation. In 1972-73, the Office of Saline
Water dispensed a significant portion of its 22.5 million dollars on the
most modern research developments in distillation (Saline Water Conversion
Summary Report, 1972-73, 1973).
Crystallization (Freezing) Methods
Freezing methods employ the phenomenon that essentially all salt
is excluded from ice crystals which develop when saline water is suffi-
ciently cooled. The crystallization process itself involves this phase
change, which removes the salts in solution, and the subsequent return of
the crystals to the liquid phase. As opposed to distillation, such a
process must be insulated from "cold loss." Also, in distillation, only
liquid and vapor need be transported and purified while in freezing
processes a solid phase must also be considered.
The advantage of freezing methods, on the other hand, are in the of relative reduction in scale and corrosion problems. In all methods
freezing separation, the three steps of crystallization, crystal separa-
tion (wash), and melting of the ice crystals are fundamental. Methods is formed. Both direct, differ in the manner in which the saline solution have been or self-cooling, and indirect, or induced, freezing processes
studied extensively. 1 0 Direct Freezing. In direct freezing, sea water is initially precooled in a heat exchange unit and then introduced into a freezing tower, or crystallizer, in which the pressure is approximately 0.005 atm.
Rapid evaporation takes place here, and since this requires heat, the sea water cools to freezing. This is approximately -1.9 ° C for undiluted sea water. The product must be washed since the ice crystallizes as a fine slush containing up to 50 percent (by weight) saline solution in the inter-crystal spaces. Counter-current washing has been found to be the most efficient technique in terms of fresh water lost in the process.
Brine concentrate and used wash water are discharged through the initial heat exchanger to minimize "cold loss." The vapor removed from the crystallizer to maintain low pressure is pumped into the melter with the product ice slush. There, the vapor condenses on the cold ice, both releasing fresh water and heat which melts the product ice. The melted ice is also discharged through the heat exchanger, reclaiming energy from the product water. The energy requirements of the system are essentially all mechanical or electrical because of the large reclamation of thermal
energy in the precooler (Spiegler, 1962).
Indirect Freezing. In indirect freezing, a refrigerant of high pressure and vapor pressure is used to overcome the problems of low vapor the large density of water at the freezing temperature which necessitates is vapor transfer and vacuum-tight apparatus of direct freezing. Butane butane and sea often used as such a refrigerant in these systems. Liquid which is maintained slightly water are introduced into a freezing unit
under atmospheric pressure. Here, the butane boils, deriving its heat of
vaporization from the heat of fusion of the water. The ice slurry is 11 counter washed as in direct freezing while the butane vapor is compressed to slightly above atmospheric pressure. This vapor and the slurry are introduced into the melter where the reaction opposite to tha -_ in the freezer occurs. Butane vapor condenses at this pressure and temperature and the ice melts Because of the density difference of 0.40 between the products in the melter, and because the two substances are immiscible, they may be separated in a decanter unit. The liquid butane is returned to the freezer for reuse while produce water and concentrated brine are passed out of the system through the preheating heat exchanger conserving thermal energy (Spiegler, 1962).
The salt exclusion phenomenon behind freezing desalination has long been exploited as it occurs in the environment. Eskimos and Lap- landers have used fresh sea ice for water, as well as sailors long at sea and early arctic explorers (Clerfayt, 1967). The first laboratory inves- tigation of the phenomenon was conducted by Whitman (1926) in 1925. The first patented apparatus for freezing desalination appeared in Japan in
1938 (Tanaka and Yosibumi, 1940), while the first United States patent was issued to General Electric Company in early 1944 (Whitney, 1944). A cen- trifugal separator was developed by Dacino and Visintin (1946) in 1945 with improvements by Borgerd and Palmer (1947) in 1947. The fundamental research upon which current freezing plants are based was conducted by
Steinback (1951) in 1950 and 1951, and a rash of engineering improvements were developed at approximately the same time. One of the very first projects of the Office of Saline Water after its inception in 1952 was a thorough study of freezing desalination by Rose and Hoover (1955). Fif- teen thousand gallons-per-day pilot plants were being proposed by 1960 12
(Bosworth, Barduhn and Sandell; 1960), and 250,000 gpd plants by 1966
(Snyder, 1966). The 200,000 gpd pilot plant at Wrightsville Beach, North
Carolina, in the mid-1960's was unsatisfactory (Bridge et al., 1970) mainly because of poor design and poor engineering. Although freezing as a desalination process has had some problems due to the poor results at
Wrightsville Beach, in recent years it again is being seriously considered as a source of fresh water and new freezing plants are being constructed in the United States, Israel and the Caribbean Islands (Sherwood, Brian and Sarofim, 1969; Bridge et al., 1970).
Solvent Extraction Methods
Solvent extraction methods employ the phenomenon that certain solvents which contain strong electronegative atoms in their molecular structure have the ability of forming hydrogen bonds with water molecules and yet due to their hydrophobic side chain they draw the water-solvent
couple into the solvent phase. Substitution of alkyl groups on or near
the nitrogen of the solvent, as in branched tertiary amines, results in extreme sensitivity of solubility to temperature, making the separation Thus, of solvent and water by a small change in temperature possible. water may be extracted from a saline solution with the exclusion of salt
(Hood and Davison, 1960).
In an apparatus using this method separation, feed sea water is
introduced into an extraction column where it is contacted with solvent
in counter-current flow. The resultant low-salinity extract is then
channeled through a heat exchanger and separated into two streams. One
stream is used in heat exchange with stripped solvent, and the other with 13 product water. Hot extract from the exchanger is further heated and then introduced into a phase separator. Here, the solvent is stripped of the water, yielding pure water and reusable solvent. Stripped solvent is returned to the exchanger and through an additional cooler to feed the
exchanger, and product water is discharged from the system through the heat exchanger (Spiegler, 1962; Hood and Davison, 1960).
Solvent extraction is principally a mechanism for the separation
of inorganic compounds such as metals or minerals from solution, with the
intent of reclaiming the solute rather than the solvent (Kertes and Marcus,
1969). Although the phenomenon has been recognized for this purpose for
many years, it was first proposed as a method for saline water conversion
by Hood and Harwell in 1953 (Bosworth et al., 1959). The initial research
was performed by this group at Texas A & M University, under the sponsor-
ship of the Office of Saline Water. Here, over 400 solvents were inves-
tigated for suitability in the solvent extraction process of water. Compounds with strong electronegative oxygen, nitrogen and phosphorous
groups, such as ethylisopropylamine, triethylamine and n-methy1-1,3-
dimethylbutylamine, were found to be the best solvents (Hood and Davison,
1960). More recently, O'Laughlin and Banks (1967) have found methylenebis (dialkylphosphine oxides), methylenebis (dialkylphosphonates) and
compounds with polymethylene bridging groups to have the properties of a
good solvent in water extraction.
Although Hood and others have constructed equipment that
successfully desalinates sea water on the laboratory scale, and even
small pilot scale, less attention has been payed solvent extraction than
the other methods presented here. According to ClerfaYt (1967), this is 14 due both to the complex chemistry involved in solvent extraction and the high capital cost of the equipment involved, which maintains the system of organic solvents.
Chemical Separation Methods
Fundamentally, chemical separation methods remove salt from saline
solutions either by the addition of a reagent which combines with the
salt, yielding a product which precipitates, or by exchanging the salts
with hydrogen and hydroxyl ions which pose no problem in fresh water.
Chemical precipitation and ion exchange techniques both require amounts
of the removal chemical roughly equal to the amount of salt removed. For
these systems to operate effectively, the product compounds must be
restored to their original states and reused in the separation process.
Even when this is done there is a high capital cost for the reagents
involved. Ion Exchange Separation. In ion exchange separation of fresh
water, the saline solution is first passed through a column of solid
cation exchanger. This organic resin exchanges hydrogen ions for the
positive ions in the solution, yielding a weak acid solution which con-
tains anions. This water is then introduced into a solid anion exchanger
column where the hydroxyl ions of the resin are exchanged for the anions
in solution (Samuelson, 1963).
Ion exchange phenomena were first observed in the early nineteenth
century as they related to retention of fertilizers in agricultural soils.
Thompson and Way (Applebaum, 1968) first demonstrated the ion exchange The between a fertilizer containing ammonia and a soil column in 1850. 15 reversibility of such an exchange was demonstrated eight years later by
Eichhorn, but no application of this pair of phenomena was made until
1905 when Gans developed a water "softening" process using synthetic cation exchange materials. This method was used in boiler, launderies and textile plants in Europe and the United States in subsequent years. The first successful exchanger with no residue or pH side effects was devel- oped in Holland by Smit in the early nineteen-thirties. Complete anion-
cation demineralization by ion exchange was developed by Adams and Holms
soon after, with large capacity units containing resins developed by
D'Alelio appearing by 1944. The units constitute the fundamentals of the
ion exchange systems employed today (Osborn, 1961; Applebaum, 1968).
Recent developments have made large-scale production of ultrapure water
possible (ielfferich, 1962) due to imrpovements in both exchange and
regeneration techniques (Calmon and Kingsbury, 1966). Regardless of
these improvements, ion exchange desalination is limited in practice to
low-salinity waters (1,000 ppm total dissolved solids maximum) because of
the high rates of chemical use and regeneration involved (ClerfaYt, 1967). currently Ion exchange is, thus, of questionable value as a method for
the desalination of sea water for agricultural, industrial or domestic
use. Chemical Precipitation. Methods which utilize the chemical
precipitation of dissolved minerals in saline water employ quantities of precipitating Chemicals approximately equal to the quantities of salt
removed. Although it has been recognized for many years that certain
chemicals will precipitate salts from solution, it had not received
attention as a method of desalination until the Second World War. At 16 that time, in a review of possible methods of desalination for lifeboats,
Parker (1942) noted that the precipitation of sodium and potassium salts
from sea water is a difficult process often involving toxic reagents.
During this period, silver nitrate (Goetz, 1944), silver oxide (Spealman,
1945) and silver carbonate (Nishihara, 1948) were the principal chemicals
studied. Some precipitation-type desalination methods were developed for
downed pilots (Ingleson, 1944) but no large-scale treatments were
considered (Jenkins, 1952). In post-war years, little attention has been
paid to precipitation methods due to the extremely high cost for single-
use chemical treatments of this sort, and the lack of development of
large-scale reusable desalting reagents. Until a feasible conservative
chemical reagent system is developed, chemical precipitation may not be
considered a valid possibility for the large-scale production of low-cost
fresh water.
Membrane Separation Methods
Unlike distillation or crystallization methods, membrane separa-
tion methods do not all have one common underlying phenomenon but rather
have in common that semipermeable membranes are used to separate phases
in the system. Membrane desalination methods have developed very closely
with the membranes used within them. As higher flux and higher retention
membranes are developed, these methods become of greater significance as
potential sources of fresh water. The three principal membrane separation
methods are osmionis, electrodialysis and reverse osmosis. The first two
methods employ membranes selectively permeable to only anions or only 17 cations, while reverse osmosis uses membranes which are permeable only to water and not to dissolved salts.
Osmionis. Osmionis, from "osmosis" and "ions," is a separation process in which osmotic diffusion of ions from high salinity solutions into brackish water cause the desalination of another brackish water ele- ment in the system (Spiegler, 1962). In a simple osmionic system, five fluid cells are linearly connected by alternately anion and cation per- meable membranes, and with end cells connected by a fluid transfer provision. Initially, the two outside cells are filled with brackish water, and the three internal ones with brine solution. Under the natural tendency of ions to move from high to low concentration (osmosis), ions in the end cells tend to migrate into the adjacent brine cells. This migration is limited by the ion selective membranes which allow only anions to pass out of one cell and cations out of the other. This causes the brine solution to lose electrical equilibrium and the corresponding ions to migrate out of the central cell in an attempt to reestablish the balance. In this manner, the central cell is desalted, with anions leaving through one membrane and cations through the other (Murphy, 1958).
It should be noted that when the unit is operational there is a positive current flow through the cells due to the net ion migration. Thus, the unit may be considered to consist of a cell combination which provides electrical power, end salt cells and brine cells, and one in which electrodialysis takes place. The advantage of the system is that the only power required is in the pumping requirements, while the disadvantages are that a large plant is required due to the slow ion diffusion process 18 involved and that large amounts of brackish water must be used to obtain the fresh water yield.
Electrodialysis. In electrodialysis, as is osmionis, alternate anion permeable and cation permeable membranes are used. Stacks of such membranes, forming from ten to one hundred compartments (Spiegler, 1962), have electrodes at either end. The compartments are filled with saline water and the electrodes are connected to a direct current source for operation. In a given compartment set, positive ions travel toward the cathode side of the stack and negative ions travel toward the anode stack.
Upon encountering a membrane not permeable to ions or that particular
charge, the migrating ions are stopped, but because of the alternating membranes every other compartment has a decrease in ion concentration while adjacent compartments have a proportional increase (Wilson, 1960).
Electrodialysis has been recognized as a method of desalination
since the early part of this century. Several electrodialysis-based
devices were patented in Germany in 1922 (Jenkins, 1952), with descrip-
tions of operation of "electro-osmosis" by Behrman (1927, 1929) appearing
in subsequent years. Bartow and Perisho (1931), and Bartow and Bartow
(1930), experimented with the energy requirements and treatment rates of
the system while Billiter (1931) worked with ceramic and asbestos
diaphragms for cell separations. Billiter (1936) appears to be the first
to use the term "electrodialysis" rather than "electro-osmosis." World all desalination War II caused the rush of research in this field as in methods, with Streicher (1946) conducting substantial research in electro-
dialysis and electrolytic precipitation. Electrodialysis has developed employing this method to the extent that large-scale desalination plants 19 are currently in operation, including the 650,000 gpd plant in Buckeye,
Arizona, constructed by Ionic, Inc (Gillam and McCoy, 1966). It should be noted that since the power requirement for the desalination of saline
solution is directly proportional to the salt content, unlike distilla-
tion, electrodialysis becomes less effective with increasing salinity and
therefore is mainly applied to waters less saline than sea water. The
other principal problem with this method is the increase in membrane
stack resistivity with operation time (Furukawa, 1967), but both of these
problems are being currently investigated and electrodialysis has great
potential for being a method of large-scale fresh water production in the
future (Shaffer and Mintz, 1966).
Reverse Osmosis. Osmosis is the natural tendency for solvent to
flow from low to high solute concentration across a semi-permeable mem-
brane in a system which is unhomogeneous with respect to concentration
due to that membrane (Selkurt, 1971). Osmosis is a direct result of the
existence of the membrane, since without the membrane simple diffusion of
solute would return the system to equilibrium. If pure solvent is placed
on one side of the membrane and solution on the other, the membrane is
said to be perfect, or completely selective, if solute does not pass into
the pure solvent phase. If pressure is applied to the solution side of
the membrane, the osmotic flow through the membrane is impeded. If this
pressure is applied to the extent that the flow across the membrane stops,
the system is said to achieve osmotic equilibrium (Reid, 1966). The
pressure required to achieve osmotic equilibrium in a system with a
completely selective membrane is the osmotic pressure of the solution.
This is actually the difference between the osmotic pressures on either 20 side of the membrane, but since the membrane is completely permeable to the solvent, the pure solvent phase has no osmotic pressure. With imperfect, or leaky, membranes such a pressure achieves a dynamic flow equilibrium representing this osmotic pressure difference. The osmotic pressure is proportional to the concentration of a solution (Staverman,
1951) and at equal concentrations is approximately proportional to the molecular wieght of the solute (Wake and Posner, 1967).
If the pressure applied to the solution side of the membrane exceeds that required to achieve equilibrium, solvent will flow in the direction opposite to that of osmosis -- or from high to low solute con- centration (Gentry, 1967). This flow in the opposite direction from osmotic flow is called reverse osmosis (Sourirajan, 1970). The pressure required to achieve this reverse flow is proportional to the solute con- centration on the high pressure side of the membrane and, thus, the energy required to move solvent out of solution is proportional to the amount of solute present.
The terminology describing this reverse osmotic flow has been extremely varied and confused in early literature. In general, the terns "ultrafiltration" (Reid and Spencer, 1960), "hyperfiltration" (Vofsi and
Kedem, 1971) and "reverse osmosis" have been used interchangeably. In recent years, conventions in definition and usage have developed. Ultra-
filtration refers to the removal of large molecules and colloids from difference solution by the process described above. The osmotic pressure such filtration between the solutions on either side of the membrane in can generally be considered negligible, and this relationship is consid-
ered characteristic of ultrafiltration (Blatt et al., 1970). This 21 assumption is valid in ultrafiltration applications such as the frac- tionation of protein solutions (Blatt et al., 1967) or the preparation of ultrapure water (Smith and DiGregorio, 1970; Smith, 1972). Hyperfiltra- tion, on the other hand, deals with the case where the osmotic pressure difference across the membrane is not negligible (Spiegler, 1966). In usage, "reverse osmosis" has come to mean hyperfiltration, while ultra- filtration has come to mean the pressure separation of macromolecules.
At the point where these terms may both apply, the choice is still ambiguous, but is generally made in accordance with the molecular weights of the solutes involved. The theoretical treatment of these two methods is extremely similar, as shall be demonstrated. For further elaboration on the reverse osmosis process, the reader is referred to the excellent text by Sourirajan (1970) and the volume edited by Merten (1966a).
It appears that Reid was the first to recognize that salt rejec- tion characteristics of membranes may be used as a desalination mechanism, in his 1953 proposal to the Office of Saline Water (Breton, 1957). The rejection phenomena itself had been recognized by colloid chemist from filtration experiments for many years (McBain and Stuewer, 1936; Ferry,
1936). Initially, a wide variety of membrane materials were examined, both by the Reid group at the University of Florida (Reid, 1957; Reid and Breton, 1959; Reid and Kuppers, 1959; Reid and Spencer, 1960), and by
Sourirajan and others at the University of California (Yuster, Sourirajan and Bernstein, 1958; Loeb and Sourirajan, 1959), but only materials with
low permeation rates were found. The development of cellulose acetate membranes cast from solutions containing perchlorate salts by Loeb and
Sourirajan (1960) was the monumental breakthrough for reverse osmosis 22 systems, since such membranes had high hydraulic permeability, while maintaining high retention characteristics. McRae (1961) outlined the research needs in membrane demineralization techniques and Loeb-Sourirajan type membranes were extensively examined along those lines (Gilman, 1961;
Loeb and Manjikiah, 1963). General Dynamics Corporation (Lonsdale et al.,
1964) and Aerojet General Corporation (Keilin, 1964) led research in this
field under the support of the Office of Saline Water. In the mid-1960's
reverse osmosis was becoming recognized as_a possible source of fresh
water, with membrane separation of inorganic salts being examined by
Sourirajan (1964), Merten (1966b), Rosenbaum, Mahon and Cotton (1967) and
other leaders in the field. With the First International Symposium on
Water Desalination in 1965, reverse osmosis received world-wide attention.
Various aspects of reverse osmosis were discussed in six key papers at
the Symposium, with Glueckauf (1965) presenting new interpretations of the
desalting mechanism, and Sourirajan and Govindan (1965) showing membrane
performance with various solutes. In 1966, the University of California
began operation of a reverse osmosis plant at Coalinga, California (Loeb,
1966), where plant operation costs (Stevens and Loeb, 1967) could be
studied. The plant also gave a field laboratory for the testing of the Loeb-Sourirajan type membranes (Johnson, 1967) and offered information on
the problems of plant operation (Johnson, McCutchan and Bennion; 1969).
In recent years, cellulose acetate continues to be studied for higher work of flux and better retention at lower pressures, as indicated by the
Gulf General Atomic, Inc. (Lonsdale et al., 1970), Gulf South Research
Institute (Smith, Morton and Klein; 1970) and by the University of
California (Johnson, McCutchan and Bennion; 197 )4. 23 Another major recent development is that of spiral-wound membrane modules which compress large membrane areas into a small volume (Kremen and Reidinger, 1971). These elements are being studied, and commercially produced, by Gulf Energy and Environmental Systems (Foreman, Kremen and
Loose, 1971; Foreman et al., 1971). Porous tubes with external cellulose acetate membranes have also been studied in recent years for reverse osmosis applications (Havens, 1970; Harris and Humphreys, 1971). Although this is a new advancement, hollow fiber membranes (Mahon, 1961) have been proposed for many years. Kaup (1973) has proposed a set of criteria for
the design optimization of reverse osmosis plants which includes the minimization of energy requirements through the reduction of operating pressure. This low-pressure reverse osmosis is being made possible by
improvements made in high-flux membranes as those being developed by
Kunst and Sourirajan (1970).
Reverse osmosis is not limited strictly to desalination applica-
tions. Reverse osmosis has been proposed as a treatment for various
water pollution problems, including water recovery and reuse of industrial
waters (Leitner, 1969). Since the products of such a purification process
include a high concentrate stream, as well as the purified water, reverse
osmosis may be also applied to the recovery of valuable industrial or
mining by-products, as proposed by Channabasappa (1970). With recent
developments in membrane and pressure systems, and with the numerous
applications of reverse osmosis in industry and in fresh water production,
reverse osmosis has great possibilities for the future. As the energy
requirements are lowered, and efficiency is increased, this desalination
technique should see wider applications. 24 Salt Replacement: A New Method of Low-Energy Desalination Proposed for the Large-Scale Production of Fresh Water
Salt replacement entails the substitution of the salts dissolved in sea water by a replacing chemical of special removal characteristics, and the subsequent low-energy removal of that chemical. Since osmotic flow does occur not only between a solution and pure solvent but also between two solutions of different osmotic pressure, if a solution of higher osmotic pressure than that of sea water is placed opposite sea water in a membrane system, water will flow from the sea water into the high osmotic pressure solution. In essence, the salts in solution in the sea water are replaced by the solutes on the opposite side of the membrane, since the water solvent on both sides is the same. This first step in replacement desalination was first proposed by Moody and Kessler (1971) of The University of Arizona. They propose that sea salts be replaced by
chemicals which are normally added to agricultural waters by this method
to produce a water to be employed in irrigation. The osmotic pressure of
this water must be greater than 25.1 atmospheres (atm), which is the osmotic pressure of sea water (Reid, 1966). Even if caused by a benefi-
cial or neutral chemical, the osmotic pressure of the irrigation water
cannot much exceed 2 atm (Slatyer, 1967), even on salt-tolerant crops,
since the osmotic limit of the plant will be reached. This means that the
agricultural additive solution must either be diluted to a usable level,
or some of the chemical causing the osmotic pressure must be removed.
This leads to the second step in salt replacement desalination
the removal of the replacing chemical. The replacing chemical is chosen
with respect to the specific removal method to be employed. Moody and 25 Kessler (1971) suggest that the replacing chemical could be separated
from the fresh water solvent "by precipitation or evaporation caused by
another compound, by adding or removing heat or by inducing fermentation."
The principal considerations in the selection of the separation method to be employed are, first, for the method to use the minimum energy possible,
and second, for the method to be conservative -- in that the continual
addition of chemicals should not be required.
Figure 1 shows the solute size ranges in which certain separation
methods are the most useful. Filtration methods are seen to be most
applicable to larger molecules and have high product flux rates (Purchas,
1967). The direct osmotic replacement step of salt replacement desalina-
tion may be coupled with any number of separation steps, as illustrated
in Figure 2. The principal separation methods currently considered to
have greatest potential are filtration methods. Enzyme-catalyzed osmotic
pressure reduction with filtration and low-pressure step-ultrafiltration
are the two methods considered. It is the principal purpose of this
paper to theoretically and practically examine the latter.
Enzyme-Catalyzed Osmotic Pressure Reduction
The removal of solutes from solution has an energy requirement
proportional to the osmotic pressure of the solution. The driving solu-
tion, as Moody and Kessler (1971) call it (since it replaces the driving
pressure of reverse osmosis) would ideally have a high osmotic pressure
in the salt replacement step and low osmotic pressure in the separation
step. Since the molecular size of the solute is proportional to its
- •
26
ro
-.1n1••n•n•
0 'o
_J -
>- o - o
0 D = 7- 0 _ - -
- cl) ---
Ct.
=>-= CL
g •1-4 n ▪ .0 Cl) 4-(• • •-n cn c.) al g IL I o a) t44 4-) • o tn o c o r_- O P. 0 a. g 1-4 g O s-4 _J • P. 1-1 0 +.1 cd• og cd n-1 r-4 o • 4-) a) o X ai cd cf) cd c.) 4 - 4 •r-I g (1) X o Cr) r •-l bO g O g czt Cl) •r-1 I • v-i a) 4-1 • a) $-1 .1:3 o g a c/3
a.)
(r) $-1 tu) 27
/ r / /
Step Ultrafiltration
SALT REPLACEMENT Enzyme-Catalyzed Pressure \ Reduction and Filtration
Figure 2. Salt Replacement Desalination. Direct osmotic replacement step of this method may be coupled to any of a number of separation techniques, and therefore any number of replacing chemicals. 28
osmotic pressure, if the size were reduced for the separation step and
returned for the replacement step, the energy requirement of the mechanism
would be reduced.
Enzyme-catalyzed osmotic pressure reduction and filtration, or
simply enzyme catalysis is schematically illustrated in Figure 3. The
REPLACEMENT CELL
LOW SUCROSE' LOW SYNTHESIZING CONVERTER \ POLYSACCHARIDE
L PUMP
ULTRAFILTRATION CELL,. [ HIGH I I HIGH HYDROLYZING CONVERTER SUCROSE I I ôLYSACCHARID EL.J
SEA WATER
FRESH WATER
Figure 3. Desalination by Salt Replacement and Enzyme-Catalyzed Osmotic Pressure Reduction with Ultrafiltration.
driving solution has an osmotic pressure greater than sea water in the
replacement step. Here, water passes across the membrane from the sea passes through a water and into this solution. The driving solution then
converter which chemically joins solute molecules to form fewer, larger
molecules, thereby reducing the osmotic pressure of the solution. This 29 new solution is then passed through a low-pressure ultrafiltration unit.
The reduction in the pressure requirement is proportional to the extent to which the original solute molecules can be combined, and a ten-fold reduction is plausible. The ultrafiltration step yields a pure water stream, and one of secondary solute solution. This second stream then passes through a converter which returns the solution to its original solute, at high concentration. This high concentration solution is then injected onto the surface of the replacement membrane to encourage replacement flux. Although the total energy requirement of the system is not reduced, the applied pressure required is partially replaced by the
energy of the reactions in the converters, which is derived from the
temperature difference between the solution and the environment.
Yudkin, Edelman and Hough (1971) show that the osmotic pressure
of sucrose is approximately half that of glucose, at the same weight
percentage. This osmotic pressure relationship of a disaccharide to its
constituent sugars is the same for all similar molecules (Washburn, 1928). Figure 4 shows the principal disaccharides and their constituent mono-
saccharides. By joining two such monosaccharides in the synthesizing
converter and separating them in the hydrolyzing converter, a filtration
pressure reduction of 50 percent can be achieved. Pringsheim (1932), in
his excellent review of early saccharide chemistry, mentions that the
enzymatic synthesis of disaccharides is possible by using the hydrolyzing
enzymes of each sugar. Percival (1962) also makes this observation, but
notes that the reaction rate is extremely slow and that very high con-
centrations of the constituent monosaccharides are required. This would
cast doubt on the applicability of this reaction. 30
HO Sucrose
HO H Maltose CH 0H 2 H OH
H
H H HO OH HO Cellobose
OH Lactose
Figure 4. The Molecular Configuration of the Principal Disaccharides: Sucrose, Maltose, Cellobiose and Lactose.
CCram and Hammond, 1964; Levi and Purves, 1950) 31 Hassid and Doudoroff (1950), in their article on the enzymatic synthesis of sucrose, indicate that a-D-Glucopyroanosyl-B-D fructo- furanoside (sucrose form) can be synthesized from the monosaccharides
D-glucose-l-phosphate and D-fructose by phosphorolysis using the enzyme phosphorylase. Regrettably, the reaction yields inorganic phosphate, and the hydrolysis of sucrose yields a-D-glucose and -D-fructose (Edelman,
1971), making the process simply irreversible. It is possible that modifications of this reaction can be employed in the system, but more
research is required.
The formation of oligosaccharides from simpler sugars also offers
a possible mechanism for the osmotic pressure reduction required.
Edelman (1956) shows the formation of several trisaccharides by the
enzymic transglycosylation of disaccharides. Edelman -- and for the
general enzyme reaction of this type, Gray (1971) -- shows that a mono-
saccharide is also produced in the reaction. Entire saccharide chains
can be produced in this manner, but the hydrolysis of such chains does
not yield the original disaccharides as Westley (1969) points out in his
text on enzymic catalysis.
It is not the purpose of this paper to explore in depth all the
possible chemicals that may be used in enzyme-catalyzed osmotic pressure
reduction and ultrafiltration, but simply to demonstrate the possibilities
and advantages of such a system. An important advantage is that when the
proper chemical is found the corresponding enzymes could be immobilized
onto a membrane or resin surface in the manner suggested by Falb (1972)
or Manecke (1972). In this way, the enzyme need not be separated from
solution by another membrane filtration, as done by Bowski et al. (1972), 32 before the fluid undergoes the next step in the cycle. The collection of
papers edited by Porcellati and di Jeso (1971) describes membrane-bound
enzyme performance which must be considered in a system such as the one
proposed here.
Enzyme-catalyzed osmotic pressure alteration has a great deal of
potential as a separation step for salt replacement desalination. A
great deal of research is required in the selection of the appropriate
reversible reaction to be employed, and in the development of a system
using the methods outlined here. Such research is heartily recommended
so that the full potential of the system may be recognized.
Low-Pressure Step Ultrafiltration
As described above, ultrafiltration is the pressure separation of
macromolecules from solution. Amicon (Ultrafiltration, 1970) and
Millipore (Ultrafiltration Systems, 1972) are currently the leaders in
the field of ultrafiltration equipment and membrane production, with
Amicon producing the finer membranes and better stirred ultrafiltration
cells. The UM series of Amicon Diaflo (registered trademark) membranes
are the ultrafiltration membranes with the smallest pore size currently
available. According to Amicon's publication, Concentrating, Desalting,
Separating, Solutions and Suspensions (1972), these membranes are rated
by "cutoff" levels. A membrane is capable of completely retaining all
solutes of molecular weight equal or above the cutoff of the membrane
(Michaels, 1968). This does not mean that solutes of molecular weights
below this level are not retained, only that they are retained imper-
fectly. Amicon produces membranes with SOO, 1,000 and 10,000 molecular 33 weight cutoffs, which are UM05, UM2 and UM10 membranes, respectively
(Buyer's Guide, 1972). The UM05 membrane is the smallest pore size ultrafiltration membrane currently known.
As shown in Figure 1, ultrafiltration removes solutes from solu-
tion by virtue of their molecular size. The ideal salt substitute for
ultrafiltration removal must have sufficiently high osmotic pressure in
the replacement, and yet be of sufficient molecular size to be retained
by the polyion complex membranes (Michaels, 1965) used in ultrafiltration.
Fallick (1972) shows that sucrose, although below the cutoff level, is
significantly retained by UM05 membranes, but molecules of lower molecular
weight are not. On the other hand, the osmotic pressure data of Robinson
and Stokes (1968) shows sucrose to have a pressure of 26.6 atm in a one-
molal solution. An extensive literature search has shown that no chemical
commonly available has higher osmotic pressure with equal or greater
molecular size (i.e., Whistler and Smart, 1953; Jirgensons, 1962; Coutts
and Small, 1966). This search has also shown that the other principal v v disaccharides -- maltose, cellobiose and lactose -- (Stanek, Cerny and
Pack; 1965) have very similar osmotic properties and molecular sizes and
configurations. The structure of these sugars is shown in Figure 4. It
has been observed that chemicals of the same molecular weight need not
behave similarly in ultrafiltration. For example, a linear molecule of a
given molecular weight will pass through a membrane more easily than a
globular molecule of the same molecular weight (Eastman Membranes for
Reverse Osmosis and Ultrafiltration, 1972). This is the the reason Carbowax molecules (registered trademark) [Carbowax polyethylene glycols,
1972] of the same molecular weight as the disaccharies would pass through 34 the UM series membranes. Since both the structure and molecular weight of the disaccharides mentioned are extremely similar, they are believed to behave similarly in ultrafiltration. Since sucrose is the most common of the sugars considered, it was selected as the replacing chemical to be used in the ultrafiltration experiments in this study. The availability of the chemical is a consideration in design of the large-scale plants to
employ this treatment.
Since sucrose is metabolized by both plants and microorganisms,
the hydrolysis of sucrose may be a consideration in the use of the
chemical for practical desalination methods. Sucrose is hydrolyzed, or
broken into the Œ-D-glucose and f3-D-fructose constituents (Edelman, 1971)
by either the enzyme invertase or by the addition of an acid (Aspinall,
1970). To avoid the hydrolysis due to invertase, the system must be
disinfected of all organisms containing the enzyme when the system is
initiated. Recontamination probably will not be a problem, since the
replacement membrane excludes everything in that size range. The
hydrolysis of sucrose due to hydrogen ions is a more complex considera-
tion. Pigman and Goepp (1948) give the expression:
-d[S] _ k [s] [H+ ] dt
as the rate equation governing the acid hydrolysis of sucrose. The rate -1 constant, k, is equal to 0.0149 sec , and the terms in brackets repre-
sent the molarities of sucrose and hydrogen ion, respectively. The pH of
the fluid in the desalination system should be equal to that of sea water,
since the membrane in the replacement step is completely permeable to
hydrogen and hydroxyl ions. Skirrow (1965) shows the pH of sea water to 35 be 8.3, or a basic solution. Since pH is the negative of the logarithm of the hydrogen ion concentration when activity coefficient is unity as it may be assumed to be in this case, [H t ] = 5.0 x l0 for this level. If a sucrose concentration of 1.0 molal is used, then -d[S] _ 7.5 x 10 dt -11 moles/sec . -11 This means that 7.5 x 10 moles of sucrose would by hydrolyzed per second, or it would take approximately 378 years for 10 percent of the sucrose in the system to hydrolyze. It is felt that this is more than enough to make the acid hydrolysis of sucrose a minor problem in large- scale plants employing the system.
The ultrafiltration process is a function of solute type and quantity, membrane type, temperature and pressure. Sucrose, of variable concentration, has been selected as the solute in this study, for the reasons previously cited. The smallest pore-size membranes of the UM series have been found optimum and shall be used in this study. Further
information on the structure and behavior of these membranes is presented by van Oss (1970). Temperature is maintained constant, at 25 °C. The pressure at which the ultrafiltration study shall be made should corre-
spond to that used in a desalination method. The high pressure filtration
of sucrose, and its removal by reverse osmosis, has been extensively
studied by Kimura and Sourirajan (1968a, b, c and d). The pressures
dealt with in this study are much smaller, requiring no large capital
expenditures for equipment to contain the high pressures of reverse
osmosis. At 40 psi ultrafiltration is effective, yet the pressure may be
contained in low-pressure tubing and even household plumbing with good 36 seals. Due to the requirements of the method at this pressure, the mem- brane must be far from completely retentive. This requires the complete filtration of the solution to be accomplished by repetitive steps, rather than in one step. The actual number of steps is determined by the char- acteristics of the membrane and the sucrose concentration of the initial solution.
It is the purpose of this study to theoretically and practically examine the ultrafiltration of sucrose as a removal system to be coupled with salt replacement in a desalination process. The theoretical inves- tigation is to examine osmotic pressure and its flow behavior across osmotic membranes, to examine these membranes in terms of intrinsic mem- brane characteristics and transport kinetics, and to examine concentra-
tion-polarization problems at the membrane surface. The practical inves-
tigation is to experimentally examine membrane flux and retention at various sucrose concentrations and constant temperature and pressure, as
they relate to the adaption of the system to salt replacement
desalination. THEORY
Before a competent examination of the practical aspects of the ultrafiltration of sucrose may be performed, an understanding of the theory involved is necessary. There are five basic topics which lend themselves to a theoretical treatment. The first is the fundamental thermodynamic rationale behind expressions for osmotic pressure. The second is the development of the equations describing solute and solvent
flux through the membrane due to hydraulic and osmotic pressure. The
third is the development of coefficients describing osmotic membranes in
terms of membrane transport equations. The fourth is the development and
comparison of transport kinetic models applicable to the membrane systems
involved. And last, there is an analysis of the concentration polariza-
tion which occurs at the membrane interface due to phase separation.
Fundamentally, each of these topics as they are treated here may be applied to reverse osmosis systems, since osmotic pressure is not
neglected. If, on the other hand, the hydraulic pressure terms are
neglected, the developments may be applied to the salt replacement step
of the desalination method. The developments are not limited to any
specific solvent, and no physical properties are held constant unless
otherwise noted.
The development found here is fundamentally that found in the
literature, although far fewer assumptions are made, and often
independently developed concepts are combined to yield a novel interpre-
tation or development to accepted conclusions. In areas where conflicting 37 38 theories exist, the one most compatible with the development is chosen, and all theories and arguments for each are not presented although all assumptions are stated.
Development of Osmotic Pressure Relationships for a System in Membrane Equilibrium by Classical Thermodynamics
General Equilibrium Conditions
Let a closed system be considered consisting of m components and
0 phases. External field, boundary effects, chemical reactions and
electrolytic solutions are not included. The phases are at equilibrium
with each other and have no constraints in change of entropy, volume and
mole number. The energy representation of Gibb's fundamental equation is
dU = TdS - PdV + Z II. dn. [1] i=1 1 1
for the system, with variables having the usual thermodynamic definitions
presented in the list of symbols (Sears, 1959). This equation, applied
to each phase, indicated by the superscript, gives
(a) (a) (a) (a) (a) (a) + m (a) 8n. (SU = T 6S - P (SV ' i=1 1
(00 TO3 ) (ss( 0 E(13)6n1 (a) ' i=1
„(8) T o) p(o) v (o)p.(0611.(0) . [2] i=1
Here, 5 denotes a virtual change in the sense used in analytical mechanics
39
and represents a time independent change of state which is infinitesimally
of the first order. According to the general equilibrium condition, there
is no change in the internal energy of the system and (MUnster, 1970)
(k) E SU = [3] k=a
together with the secondary conditions
E 6S (1 ) = 0 [4] k=a
(k) E SIT = 0 [51 k=a
and
(k) cSn 0 (i = 1, 2, .. , m) [6] k=a i =
These secondary conditions may be treated by means of Lagrange's method
of undetermined multipliers. Equations [4], [5] and [6] are multiplied
by X 1 , X 2 and X 3 , respectively. They are then subtracted from
Equation [3] and with Equation [2] we get m e (k) (k) (k) (k) )6V (k) +E E (p i E (T - X )6S + E (P - X 2 k=a k=a i=1 k=a
(k) - X) = 0 . [7 ] 31 1
If the multiplicands are fixed such that if in each of the sums over k
one bracket vanishes, the remaining variations are freely adjusted and
Equation [7] is only satisfied if all the remaining brackets also vanish.
Thus, for each k and i
40
[8]
It follows that
T (a) = T U3) = . . = T (8) [91
p (a) p(3) . = P(e) , [10]
P. (a) = p. 03) = . . p. (e) (i = 1, 2, ..., m) . [11]
Thus, at equilibrium, all phases of the sytem must have the same tempera-
ture and pressure, and the chemical potential of each component must be
the same in all phases.
Membrane Equilibrium
Let the system be further defined such that all phases are
separated by rigid semipermeable membranes permeable only to s components
(s < m). The volume of each phase is fixed, and the variability of (m-s)
mole numbers vanishes for all phases. The secondary conditions, thus,
become
(k) E SS = 0 [ 12] k=a
dV (a) = (SV M = = = o , [13]
E Sn. (k) = 0 (i = 1, 2, . , m) k=a 1
Sn. (a) = = = Sn. (8) = 0 (j = s+1, s+2, m). [14]
41 The secondary equilibrium conditions in Equations [9] and [11] still
hold. Since a semipermeable membrane has been introduced, the same
pressure need not occur on either side of the membrane and only those
constituents which can pass through the membrane are restricted to equal
chemical potentials in all phases.
Consider the case of osmotic equilibrium, where 0 = 2, one phase
of pure solvent designated by i = 1, and one phase of a simple binary
solution with i = 2. Conditions on opposite sides of the membrane are
denoted by primes and double primes. Let the membrane be permeable only
to the solvent. Equilibrium conditions in Equations [9] and [11] become
T' = T" = T [16]
and p '(T,P') = p "(T,P",x ) [17] 1 1 1
where x is the mole fraction of the solvent, described by the partial 1 molal expression for a component i
n. 1 X. [18] 1 E n. i=1 1
Since the phases are not in equilibrium without the membrane, i.e.,
p 1 (T,P,1) p 1 (T,P,x 1 ), Equation [17] can only be satisfied by
P" PI . [19] Equation [19] does not indicate whether P" > P° or P" < P'. It can be
shown that, for the pure solvent 1 (Reid, 1966):
[20] 01T = v i t
42 where y 1 ' is the partial molal volume of the pure solvent, defined in the
manner of Equation [18], and that for any component i
[21]
(Reid, 1960). Since Equation [21] indicates p l ' > p i " for P' = P",
the condition in Equation [17] can only be satisfied by P" > P', because
of the constraint of Equation [20]. The excess pressure
n = P" - P' [22]
under which the solution is in equilibrium is called the osmotic pressure
(Prigogine and Defay, 1954).
The quantity Lip, or free energy of dilution, is defined by
"1 E 111 - 1110 [23]
where p the chemical potential of the pure solvent at standard - pressure and temperature (Munster, 1970). Free energy of dilution is
related to osmotic pressure by the relationship
P +7 o (P +7,x ) = p (P ,x ) + f v dP . [24] o 2 1 0 p 1
This is obtained by integrating Equation [21] from Po to P o+n and applying
Equation [23]. Further, since p i = p1 (P 0 +11. ,x2 ) and p lo = p1 (P 0 ,x2 ), with
Equation [23] we have
P +7 0 = f v i dP [25] o equal to the mean value T in the range P or, assuming v l is constant, 1 o to P+7, we get 43
[26]
Osmotic Pressure Equation
The definition of chemical potential of component i of an ideal gaseous mixture is (Haase, 1963)
P i (T) = P i ° (T) + RT ln Pi [27] atpartialpressureP.wherell.°(T) is the standard value chemical potential at temperature T. Thus, starting with Equation [23] and employing Equations [11] and [27], for component 1 we get
Ap i = p i - p lo = (RT in P 1 + 11 1 *(T)) - (RT ln P l ° + p i ° (T))
= RT ln 1 [28] r l where P is the vapor pressure of the solution and P1 ° is the vapor 1 pressure of the pure solvent. If this expression is substituted into
Equation [26], we have
RT / 1 iT = - - [29] V 1 p0 1 But since
a 1 [30] 1 0 1
(Lacey and Loeb, 1972), where a l is the activity of the solvent in the given thermodynamic environment, we obtain
ET. [31] 7 = - ln a v i 44 which is the fundamental expression for osmotic pressure in terms of the properties of the solvent.
The osmotic coefficient cp is defined by the expression (Robinson and Stokes, 1968)
1000 ln a 1 w.m.W [32] 111 where W1 is the molecular weight of the solvent, m is the molality of the constituent i in solution, and w. the total number of moles into 1 whichconstituentidisassociatesinthesolventloerll
for non-electrolytes). Rearranged to solve for -ln a l , we have
w . m.W -ln a - lii [33] 1 1000
or, for several (n) constituents in solution,
(11+1 E w.m.] W1 i=2 -ln a - [34] 1 1000 (i)
Equation [34], substituted into Equation [31], gives the practical
expression
n+1 RTW1 Tr = E w.m. [35] 1 1 _ (1) i=2 1000v1
for osmotic pressure.
Development of Fundamental Equations of Flow Across a Semipermeable Membrane by Irreversible Thermodynamics
Entropy Production
Classical thermodynamics tell us that irreversible processes in
isolated systems are characterized by an increase in entropy, which 45 reaches a maximum at equilibrium (Sears, 1959). The thermodynamics of
processes deals with this increase in entropy, or more precisely, the
change of entropy with time, dS/dt. For an open system, dS may be
decomposed into a part d eS derived from external sources, and into a part
d.S produced by internal irreversible processes, or
dS deS S • [36] 1
ThedeSmaybeeithernetpositiveornegativewhiled.Sis always
positive, or zero, so that
d.S > 0 [37] 1 —
for any part of the given system. The local time derivative of this part,
called inner entropy product d iS/dt, is used to characterize irreversible
processes. For isothermal processes the dissipation function ,T is useful,
where
d.S [38] E T dt
at temperature T (Katchalsky and Kedem, 1962).
It should be noted that the change of entropy with time d iS/dt,
applies to the entire system. To describe processes at a point in the
system, the local product entropy a is used. The term a is related to
the rate of increase in entropy within the system as a whole by a volume
integral
d. S - f a dv [39] dt1 V
(Katchalsky and Curran, 1965). Let us now consider the change in entropy
in terms which lend themselves to the description of a membrane system. 46 Derivation of Entropy Change in Vector Notation
Let the state of an adiabatically isolated system (d eS = 0) be characterized by the set of parameters, E, such that the members of the set, E l , ..., En , are general thermodynamic variables which have values at what de Groot (1952) calls thermostatic equilibrium. The fluctuation of entropy about its equilibrium value, as
AS = S - S may be described in terms of the fluctuation of defined by o' the parameter Ei from the value y, which is expressed by the deviation a. E E. - E. ° for i = 1, 2, n.
The second law of thermodynamics states that entropy is maximal at equilibrium, therefore, any fluctuation in S yields values of S smaller than S and AS < O. If the assumption is made that large spontaneous o fluctuations in S will be generally improbable, an expression for entropy may be obtained by expanding S in a Taylor's series near the equilibrium value (Katchalsky and Curran, 1965). First, a general Taylor's series expansion is made in one variable set, E (Hadamard and Mandelbrojt, 1926): 2 3 (E-E ) (E-E ) o o + (E-E )S ' + S " + S ", S = S o o 2! oo 3! o ) (E - E m (m) + + S [40] m!
Here, E o represents the set E with values at equilibrium, and the subscript o on the derivatives of S indicates differentiation at equilib- rium. Then derivatives in each term are replaced by the summation of partials which reflects the members of E, and the set differences by the
corresponding E i deviations (Bromwich, 1908). Thus, S becomes
47
DS n r 2 S = S o 1=1E i,j=1E S °
+ 1— r„,1 ( DE.DE.DE3 3S 1 0 ( i _ ci o)(Ei _ Ej o) kk _ Eko] 31 . . , 1,3,K=1 1 3 k) m 1 S
• , E , 1,3,.,...,m=, (DE.DE.DE...a 1 3 m
[41] -E,°) (Ek - Ek°)
in which the subscript o indicates that the partial derivatives are taken
at equilibrium (Ross, 1965). Since the maximal value of S is at
equilibrium,
[42] [Dk]o9 = o
for all The procedure is confined to sufficiently small deviations
to justify neglecting cubic and higher terms of the expansion.
Equation [41 ] , thus, reduces to
2 1 n D S AS = („ - [43] ` i,j=1 d i d j ° 1
Let the coefficients gii be defined such that
3 2 S -g. [44] (DE.DE.)0 ij j
Substituting this relationship into Equation [43] and replacing (.-E. ° ) get and(C,E.°)bythedeviationsa.anda,we1
1 AS = - gijaiai . [45] i,3=1
48 This may be represented in vector notation by the expression 1 + -± AS = - — aga. 2 — [4 6]
wherectisacolumvectorwiththedeviationsa.as elements, the tilde
indicates a transpose and g is the matrix with elements g . (Lacey and 13• Loeb, 1972). The conversion to this notation may be shown by (Bargmann,
1970)
--g g ... a 11 12 g 1 1 a g21 g22 g2n 2
l÷ - aga - [ a —- 2 1 a2 an II •
g g g a n1 n2 nn n
1 - — g..a.a. [47] 2 . . 1313
thus showing Equations [45] and [46] to be equivalent. This may now be
used in the development of equations describing component flux through a
membrane.
Development of Phenomenological Equations
Thetimederivativesofthestatevariablesa.are1 termed fluxes, J., of the ith parameter of the system, such that 1 • J. E a. (i = 1, 2, ..., n) [48] 1 1
where the dot indicates differentiation with respect to time. This
relationship may be readily seen in the example of the diffusion flux of
a constituent k at a point in the system, defined by the change in the
deviation of the number of moles of K at the point with respect to time
49
dak d( k -Ek°) J - [49] k dt dt
or by the change in volume with respect to time of a fluid, where the
volumetric flux (or flow rate) is defined by
dV J f [50] dt
Thedrivingforce,X., ofagivenflux,J., termed its conjugate 1 1 force, is defined by the following linear combination of state variables,
a. (de Groot, 1952)
X. = 3(AS) = E g. .a. (i = 1, 2, . [51] I Da. • n) 1 j=1 13
where AS is used as defined in Equation [45]. This leads to the time
derivative of Equation [45] in terms of the fluxes and forces involved, or
IS = - E 1313 i,j=1
= E J.X. . [52] 11 i=1
Since Equations [48] and [51] are • [53] —J = a and
X = - ga [54]
respectively, in vector notation, Equation [52] (or the time derivative
of Equation [46]) may be written as (Lacey and Loeb, 1972) 50 • = - + itg;;,)
4- = - aga
=JX=XJ [55] since the matrix g can always be chosen to be symmetric. The term As
represents the rate at which the irreversible process produces entropy,
and is often called entropy production or entropy source strength. Since
entropy production can be shown to be a linear combination of the fluxes
and forces in the system, this indicates /IS to be determined by purely
thermodynamic considerations. To identify the manner in which the forces
drive the fluxes, however, requires the introduction of kinetic coeffi-
cients which express the susceptibility of the system to the forces.
In order to employ the derivations above, relations between fluxes
and forces must be established. If the assumption in the statement of
Equation [43] (limiting the values of ai for all i to small increments)
is maintained, the fluxes should depend linearly on conjugate forces.
Further, it is plausible that every flux is linearly dependent on all the
forces in the system (Katchalsky and Kedem, 1962). The resulting set of
equations, known as phenomenological equations, may be written as + L X J1 = L 11 X 1 L12X2 L13X3 in n ' = L X + L X + L X + + L X J2 21 1 22 2 23 3 2n n ' J 3 = L 31 X 1 + L 32 X 3 + L 33 X 3 + + L 3nXn
• •
J = L X + L X + L X + +L X [56] n ni n2 2 n3 3 nnn 51 Or
= E L. X.. [57] 1 ik (i = 1, 2, ..., n) k=1 k
and
J= LX. [58]
The phenomenological coefficients L. constitute the matrix L. These 13 expressions describe a system in which n simultaneous processes or
fluxes, J 1 , J 2 , ..., Jn , occur. Each flux has a conjugate force, X 1 , X2 ,
restricted —,XII: TheselectionofconjugateforceX.tothefluxJ.is1 1 by the requirements that the product J iXi should have the dimensions of
entropy production, to comply with Equation [52].
Phenomenological Coefficients
If Equation [58] is substituted into the last forms of Equation
[55], we get
AS =XLX [59]
- -1 = J L J . [60]
-1 i The existence of L s assured since L is restricted to a positive
defined matrix by entropy production being positive for any natural
process. This means all main diagonal minors of all orders are positive
defined (Kreyszig, 1972); in particular, all diagonal elements, L ii , are
positive, and the determinant (for i j)
> 0 . [ 61]
The matrix L is symmetric, as shown by the classic papers of Onsager
(1931a, b); that is to say 52 L.. = L.. . 13 31 [62]
This relationship has subsequently been shown by many, including de Groot
(1951, 1952), Staverman (1951) and Katchalsky and Curran (1965), and has been empirically verified by Miller (1960). The immediate consequences
of Equation [62], Onsager's theorem, is that Equation [61] becomes
2 L..L.. > L.. [59] 11 33 13
andL...=OHL—orl,..equals zero. In terms of Equation [56], this 13 11 3) is to say that if a force, Xi , does not cause its conjugate flux, J i , to
occur (i.e., Lii = 0), it will cause no other flux to occur (since then
L . = 0 for all n). ni So far this development does not limit forces and fluxes. Let
the set of forces, then, be replaced by the linear combination
X' E BX [64]
where B is a nonsingular coefficient matrix. This leads to
-1 BJ=BLBBX [65]
by simple matrix algebra. Further, let us define the terms J' and L' as
J' E B J
L' E B L B . [66]
Thus, L' is symmetric, and with Equations [64] and [66], Equation [65]
acquires the form of Equation [58]. It is a fundamental theorem of matrix
algebra that a real symmetric matrix may be diagonalized by a transfor-
mation such as the one in Equation [65]. This indicates that forces may
be selected such that the cross-terms vanish and the force influences no
flux other than its conjugate (Lacey and Loeb, 1972). Thus, if Equation
53 [57] is expressed in the form
J. = L. .X. + E L. .X. [67] 1 111 . 13 3
after Merten (1966a), it may be reduced to
J. = L. .X. [68] 1 111
since the selection of forces may limit the second term in Equation [67]
to zero.
Phenomenological Equations of an Osmotic System
If the system considered is now restricted to one solvent (1)
and one solute (2), the equation for thermodynamic equilibrium, Equation
[1], becomes
TdS = dU + PdV - dn - dn . [69] 1 1 2 2
Fluxes from one side of the membrane (') to the other (") cause an
entropy change of
" dU" dU' P"dV" P'dV' T" dn " d(S" + S') = T" T' T" T' 1
u ' 1 2 2 dn - dn " dn ' [70] T' 1 T" 2 T1 2
from the application of Equation [69] to both sides. The dU', dU", dS',
and dS" quantities may be expressed in terms of internal and external
contributions in the manner of Equation [36]. Conservation of mass and
of energy in the system requires that
d.U" = -d.U' = d.0 1 1 1 dn " = - dn ' = dn 1 1 1 dn " = - dn ' = dn . [69] 2 2 2 54 Further, the external entropy change for a given side, as developed by Lakshminarayanaiah (1969) is
deU' + d S' - e T'
d eU" + P"dV" d S" - [72] e T",
By subtracting these relations from Equation [70] and employing Equation
[71], it is apparent that the internal change in entropy, d iS, is
described by
1 d.S = A (--) - A [—ill ] dn - A dn [73] 1 T T 1 T 2
where A designates the difference in the indicated quantities between the
two sides. If this expression is divided by dt to obtain time deriva-
tives, we get
d.S p ] dn rp2) dn2 1 = p (1] dU q 1 1 [74] dt 7 dt - T dt A' Ti dt
which is in the form of Equation [52]. The dn/dt terms are mass fluxes,
and dU/dt is energy flux. These elements, rewritten in terms of fluxes
and forces, give (Reid, 1972)
dn J 1 1 = dt
d.0 1 [75] J3 a'
and
(p AT - TAp 1-11 1 1 ) = - ( ) 2 A T T 55
11 (P T - TAp ) X = - A( 2) _ 2 2 T' 2 T
AT X = A(1) = - [76] T 2 • 3 T
If the temperature is uniform (AT = 0), and if only mass fluxes are considered, we obtain
L (Ap L (Ap ) T _ 111 ) 12 2 T
L21 (i)L (Ap ) J = 22 2 [77] 2 T when the format of Equation [56] is employed. Since T is constant, let the simplification be made that
L.. 1 13 [78] ij T and with Onsager's theorem we get
J = - 1 11 p 1 - 1 Vp 1 11 1
All - 1 Vp J2 = - 122 2 22 2 [79] with conditions of Equation [68] being fulfilled. The substitution for
All is made to account for the change in chemical potential across the membrane (Merten, 1965).
It is shown by Fitts (1962) and Merten (1966a) that the defini- tion of chemical potential of constituent i in Equation [27] may be written as
TI. = v.P + RT ln a. [80] 1 1 which indicates change in chemical potential to be 56 a." Ap. = v.AP + RT ln 1 [81] 1 1
With this substitution, Equation [79] becomes
a 1 v i a + RT ln [82] 1 - 1 11 a l ' for the solvent. If only the flux of the solvent is considered, we may employ Equation [31] as an expression for osmotic pressure and find ET ATT = - . in [83] v 1 a '
The solvent flux expression may be rearranged, with Equation [83] substituted, to read
1 11 - (AP - air) . [84] 1 vi
The solute flux expression may now be approached in a similar manner. With the approximation (Merten, 1966a)
Dp. grad pi .= (-5-El) p T grad ci + vi grad P
(T)p T grad ci (i = 2) [85] i where 11 2 and c 2 are the chemical potential and concentration of solute in the membrane, respectively, the assumption is made that the pressure gradient in the chemical potential gradient term is negligible. This has been shown by Clark (1962). If Pick's law (Michels, 1956) is assumed valid for a solute in the membrane, Equation [79], for the solute, becomes 57
412 Ac2 [86] J 2 = 1 22 9c 2 P,T Ax
Relationship of Derived and Empirical Expressions
The classical equations to describe the flux of solvent and
solute through a membrane are (King, 1971):
J 1 = K (AP - Aff)
J2 = K 2 (C2 " - c2 1 ) . [87]
The equation describing the flux of water through a given membrane with
the empirical coefficient K1 is generally accepted as giving a good
approximation of system behavior within normal ranges of temperature and
concentration. Comparing Ki with the corresponding term in Equation [84],
we see
1 L K . _ 11 . 11 1 _ [88] v 1 TV).
This indicates K1 is inversely related to temperature and mean partial
molal volume between the pressures for which Equation [87] is applied.
The term L 11 remains a constant dependent on the interaction of the
specific membrane and the given solvent.
The equation for solute flux, employing the empirical coefficient
1(2 is an acceptable model of membrane behavior. Equations [86] and [87] show
1 ap 22 2 K 2 - [89] Ax ) PT 58 This shows K 2 to be a function of the properties of the solute within the membrane, and of the effective thickness of the diffusion barrier.
Development of First-Order Transport Equations and Corresponding Coefficients Describing Intrinsic Membrane Characteristics
The flux of pure solvent and pure solute through a membrane are
not immediately useful quantities in the determination of membrane
performance characteristics. Total volumetric flux across a membrane
interface is a quantity easily determined experimentally, and more
useful in analysis of membrane characteristics in separation applications.
Transformation of Uni-Component Flux Equations
Equation [1] may be expressed as
Td.S = dU + PdV - E .dn. [90] . 1 1 1 1
for an isolated system. In this case, the dissipation function defined
in Equation [38] may be shown (Katchalsky, 1970) to consist of the sum of
products of fluxes J. and conjugate forces X.
diS (1) = T - E J.X. . [91] dt .
The driving forces may be separated into forces due to hydrostatic
pressure, and concentration gradient, respectively (Katchalsky and Kedem,
1962) 3p. 3P [92] 1 —Dx 3x
Equation [86] can be expressed as 59
2..1._2 J2 = - 1 22 (Dx -) [93] where changes in e 2 and x are considered differential quantities. The
Gibbs equilibrium condition, as demonstrated by Elata (1968), yields
41 1 = - v — [94] 9x 1 Dx ' where 971-/Dx is the total osmotic gradient across the membrane due to all solute constituents. Further, from the Gibbs-Duhen equation, we find
Dp Bp 1 2 + — = 0 . [95] ax C Cl Dx 2
Introducing c lv i = 1, from Equations [94] and [95] we see
Dp 2 _ 1 s [96] Dx c 2 bc where D .rr /3x is the osmotic pressure gradient due to the salt which is s permeable through the membrane. Thus, Equation [93] can be written 37 = 1 s [97] '-'2 22 c ax • 2
In accordance with the justification of Equation [56], the total may be expressed by volumetric flux Jv [98] Jv = v 1 J1 + v2J2 .
The total volumetric flux and solute flux may then be expressed from
Equations [84], [97] and [98] as the system of linear equations
Tr DP Dir 1 s [99] + 1 1Z Z;..— I] , "Tv = -I 1 11 px - Dx Tc
1 37s11 - 21 + 1 [100] J = -1121 3x Dx22 c 2 x 60 or
1 21 J = J - 1 1 121 21 1 9ffs s 1 v 22 1 c Dx [101] 11 11 / 2 as verified by Elata (1968, 1969); Katchalsky and Kedem (1962); Johnson,
Dresner and Kraus (1966) and others.
The assumption that 1..' are analytical functions of c 2 and P 13 s around c 2 = 0, P = 0 has been shown valid by Matz and Elata (1970).
Thus, 1.. may be expressed 13 (c) (P) (c) 2 1.. = 1.. + 1.. c 2 + 1 .. P + 1.. c 13 130 131 2 13 1 13 2 2
2 + 1. (13) P + [102] lj 2 wherealll. are independent of c and P. The osmotic pressure ij 2 n gradient may also be written as a function of concentration by use of the
osmotic coefficient g (Spiegler and Kedem, 1966)
Dir Dc s 2 - gwRT [103] 9x 9x
The osmotic coefficient (Robinson and Stokes, 1968), g = 1 for
an ideal fluid, may be expanded as
2 [104] g = go g1 c 2 g c jr
Note the additional constraint that as c 2 ± 0
9w [I s [105] lim 12 c 9x c 40 2 2
and
•
61
Dn 1 11m [11 — = 0 . [106] 22 c bc c =o 2 2
This implies the independent terms of the 1. expansion in Equation [102] i are zero, or 1 12 = 0 and 1 = O. 0 22o Substituting Equations [102], [103] and [104], and the constraints
of [105] and [106] into Equations [99] through [101], and remembering
Onsager's relationships, we find
(c) ;Tr J = - El + 1 c + 1 (P) P + 11 11 2 11 ( 9x - )c) 0 1 1
(c) (c) (c) + ( 1 g + 1 g c g0 12 1 12 1 1 2 2
(c,P) 3c2 + 1 go P + wRT [107] 12 2
(c) c 3P 37 (c) J = - s El 12 Z ÷ • — 1 (Bx - 9x) Ii22 gO, 1 1 (c) ( 1 22 (c) + 122 g0 c2 1 2 3c2 (c,P) + 1 go P + wRT -3)7- [108] 22 2 or (c) 132 (c) / 1 11 c2 • jv - [122 g0 = - [ 1 1
(c) (c) (c) /1 + 1 22 g1 4. 1 22 g0 112 / 111 0 g0 1 2 1 Dc 2 2Z (c,P) [109] + 1 goP + • - 2 62 Membrane Description by First-Order Transport Coefficients
To the first order for small c and P values, fluxes may be
expressed in terms of driving gradients, keeping only the lowest order,
non-zero coefficients of expansions. The equations above, thus, become
(c) 1 12 97r i] 313 D7 1 s Jv = - 111 [110] Dx 9x) 1 9x 0 [I( 11 0
(c) 12 9P 97 [111] 1 1 ) C 2 (3x - Dx) 122 (c) a xTrs = L111 0 11 0
or
1 (c) 12 37 1 (c) s J =- -1c J [112] s 1 1 1 2 v 22 Dx 0 1
where 7 = g wRTc. Note that Equations [111] and [112] are no longer 5 o identical, due to the neglecting of second-order terms.
Here, let us substitute the membrane coefficients, 1 (hydraulic
permeability), a (reflection coefficient) and w (solute permeability),
where (Elata, 1968)
1 [113] p E 1 11 0
1 12 (c) 1 a = 1 [114] 1 11 0
(c) , [115] w E 122
into the set of equations above, to yield what are commonly called the
first-order transport equations: 63
[116] jv = 1 __ (1 - ) 9xD7s--
9Tr DTrs- - El (1 - [117] a) c 2 (TX - Xj and
Birs J = (1- a) c2J - w [118] s v 9x
The coefficients defined in Equations [113] through [115] are internationally accepted in the description of intrinsic membrane per- formance characteristics, and may be integrated for steady-state condi- tions, yielding
J = - L [(AP - ATr) + (1 - u)Aff ] [119] v s where 1 [120] Lp = -EAx ' and
c2' c2" exP [-Jv (1 a)/-6 j (1 - a)J [121] s v 1 - exp [-J11 i5(1 - a)/] or 2 1 ( jv (1 - a) I 1 + J = (1 - o- )J jc..2 - flAc2 s 3 2b-
4 Jii(1 - a) 1 + [122] - 45 217 where w wRTg ° - w wRTg . [123] Ax Ax o 64 An alternative expression for J s has been proposed in a paper by Kedem and Katchalsky (1963) where, although the development is slightly differ- ent, the coefficients are defined in similar manners. Differences in assumptions in integration and constant of integration determination yield
J =-2(1 - a)J + w(Aff ) . [124] s v s
Note the similarity of this equation to Equation [118]. The sign differ-
ence arises from the difference in sign between the two definitions of w (c) this sign convention (w = - 1 22 /Ax). For the purpose of this paper, 1 and definition shall be adopted.
Equation [119] may be expressed in its most general form for all
Aff = Aff . The first-order linear transport solutes considered with s equations for this case, thus, become
J = L (AP - aArr) , [125] v p — J = c (1 - cr)J + w(Aff ) . [126] 2 v s
Evaluation of First-Order Transport Coefficients
The following development is based on placing constraints on the
first-order transport equations to yield special forms which may be
evaluated for the transport coefficients. The principal constraints
feasible for experimental determination are:
1. No solutes present,
2. No volume flow exists and no solutes impermeable to the
membrane are present, 65
3. No hydrostatic pressure difference exists and no impermeable
solutes are present,
4. No hydrostatic pressure difference and volume flow exists, as suggested by Elata (1969).
For the case of no solutes, permeable or impermeable, present on either side of the membrane, 47 = 0, since Mr = 47. + Ax and 47. = 1M s 1111 MT O. Equation [119] or [125] can be written as
[127] Jv = - L AP for this condition. Thus,
Jv L .= - — [128] ( AP )c. ,c im s
that the constraint is not expressed in the form 7 t = 7", 7. t , Note s S 1M which is a theoretically possible condition. The condition of balancing
osmotic pressures on either side of a semipermeable membrane is more
difficult to attain than using no solutes at all in the system.
For the case where no volumetric flow exists, Equation [119] may
-J /L , and expressed be solved for VP
AP - Ax + (1 - a)Affs = 0 . [129]
Expanding 47 for permeable and impermeable salts and solving for the
reflection coefficient we obtain
( AP - 47 im [130] 6 = J • Affs v
If the entire constraint in case 2 is applied, we find
AP [131] a = (—)47 Jv, im 66
where Aff becomes the total osmotic second s pressure difference A7. A result from case 2 may be obtained by applying J v = 0 to Equation [126]
and solving for the solute permeability coefficient
J s [132] w = 6)Jv •
Case 3 specifies that AP = 0 and c im = O. As in the previous
development, the second constraint implies A7 = Aff s . Equation [119],
solves for L with this substitution, yields
J v L - [133] p (aAn s1AP,Ac.im
The final case has the constraints AP = 0 and Jv = O. Equation with -J /L = 0, is [119], expanded for An.im + Ans v p
AP - (An im + Ans ) + (1 - a)Ans = 0 [134]
or, solved for the reflection coefficient, yields
[135]
The same constraints, applied to Equation [126], yield
J s [136] w = [nA s) A J .
It should be noted that in Equations [128], [131], [132] and
[133], Aff and A7 are interchangeable due to the application of case 2 s constraints.
The three first-order transport coefficients -- hydraulic perme-
ability (sometimes called filtration coefficient), reflection coefficient 67 and solute permeability coefficient -- are sufficient to characterize membranes governed by the first-order transport kinetics described in
Equation [119] and [126] Matz and Elata, 1970).
Permeation Models for Solute and Solvent Transport Kinetics of Diffusive and Microporous Osmotic Membranes
Theoretical models for the transport kinetics of ultrafiltration membranes have been described in current literature. A pore flow model has been found to best describe the transport mechanism of membranes which retain large macromolecules, those with molecular weight greater than 30,000 (Baker and Strathmann, 1970). Membranes which reject micro- solutes to the ionic level have been described by a solution diffusion model (Lonsdale, Merten and Riley; 1965), by the pore flow model of
coarser membranes (Meares, 1966), or by a combination of the two
(Michaels, Bixler and Hodges; 1965). This second type of membrane
includes those used in the ultrafiltration of low molecular weight macro- molecules, with molecular weight around 500. Before a detailed discussion
of systems involving such ultrafiltration may be considered, these models
of membrane transport kinetics must be examined.
Solution-Diffusion Model
Solution-diffusion type models apply to systems with membranes in
which the movement of each constituent can be described as solution of the constituent in the membrane and diffusion of the constituent through
the membrane. The solution of constituents is governed by an equilibrium
distribution law while diffusion occurs in response to concentration and
pressure gradients. 68
Equation [79] relates fluxes to chemical potential differences across the membrane through phenomenological coefficients, 1... Let 11 these coefficients be replaced by terms of more concrete physical signif- icance (Merten, 1966a). If the substitution
1.. = c.M. [137] 11 11 is made, we may write
all 1 "1 c M - c M [138] J 1 = - 1 1 dx 1 1 Ax
represents the concentration of the water for solvent flux. Here, c1 solvent in the membrane and is analogous to c i in the development of
Equation [85]. Further, M 1 is defined as the mobility of water in the membrane (Jost, 1960), and has units of velocity per unit force on one mole of particles, rather than a particle as conventionally employed
(Shewmon, 1963). Simplifying assumptions have been made that solvent concentration is constant within the membrane, and that since the range of p i is extremely small [i.e., the activity of water in a 5% NaCl solution at room temperature is 0.97 times the activity of pure water
(Washburn, 1928)], the chemical potential gradient may be replaced by the change in chemical potential across the membrane Ap i , divided by the membrane thickness Ax.
If we integrate Equation [85] for the two-component, water-solute system being considered, we get
4 1 pp, f b4 p,T dc, + f vl dP
[139]
69 where
I dP = AP [140] 1 1 '
By definition (Glasstone, 1946), when Ap i = 0, AP = An, so
131111 dc = - vAîr . [141] c 2 J. 2 1
Thus, combining the last two equations, we get
Ap i v (AP - An) . [142]
This expression, when substituted into Equation [138], yields
(AP - An) 1 [143] J1 = c M 1v 1 Ax •
Since Fick's law is still assumed to hold, we may use the Fick's law
diffusion coefficient (Weast, 1972), defined
D. = M.RT , [144] 1 1
to express Equation [143] as
-D c v 1 1 1 (AP - An) J = [145] ld RT Ax
which is the standard form of the solution-diffusion equation, where the
subscript d identifies the underlying model.
The flux of solute through the membrane may be similarly described
from Equation [79] and [85] as
dc 2 2 + V 2- . [146] 2 = M2C2 Dc2)P,T dx 2 dx 1 -- (41
If the solute mobility and concentration are considered constant through
the membrane, and we make the substitution dx Ax, we get
70 r@p 2) dc Dp 2 1 r 2 [147] / Bc 2J13 ,T dx Ax (Bc 2)13 ' T dc
and, since for an ideal gas
311 2 _ RT [148] (3c 21P,T c 2
Equation [147] becomes
RTAc 1 RT RT 2 dc, = f dc2 [149] Ax f c2 - Axc2 Axc2
With integration, Equation [146] becomes
Ac J2 = - M 2RT Ax2 [150]
and in terms of Fick's law diffusion coefficient,
Ac2 J = - D [151] d ? Ax •
It should be noted that the sign of the equation is dependent on the defi-
nition of the coefficient, and there is some variability in the literature
regarding such definitions. For example, Kedem and Katchalsky (1961)
prefer to express J 2d in terms of friction between solute and water, with
f ° where the coefficient sw
0 . 1 [152] f sw D42
Solute flux may be expressed in terms of solute concentration
difference between the solutions divided by the membrane, rather than the
concentration difference across and within the membrane. Let the distri-
bution coefficient, K, be defined
c. K (i = 2) [153] c. is 71
is the where c 2s solute concentration in the solution. This assumes that concentration is in equilibrium across the membrane interface. The solute
flux thus becomes
Ac 2s J = - D K [154] 2d 2 Ax
The solute reflection coefficient (R) describes membrane perfor-
mance in terms of these flux expressions. The solute rejection coeffi-
cient is defined (Jagur-Grodzinski and Kedem, 1966) as
c " 2s R E 1 [155] c ' 2s
where ' and " denote the feed concentration and permiate concentration,
respectively. The solute rejection coefficient for a solution-diffusion
model may be expressed using the relationship (Lacey and Loeb, 1972)
J 2 c " = c 1 [1 56] 2s J i is
as D KRTc " -1 J ,c 2 1 2441 is 1 + [157 ] R = 1 - J D c v (AP-An) d ldc2s" 1 1 1
Rejection improves with increasing net pressure difference, because sol-
vent flow is linearly dependent on pressure difference, while solute flow
is independent of pressure (Lonsdale, 1970). The profiles of solute and
solvent concentration in a solution-diffusion system are presented in
Figure 5.
A second coefficient used to describe membrane performance is the
concentration-reduction factor (CRF) which is defined (Jagur-Grodzinski
and Kedem, 1966) as 72
C
- ...... ,..... - ....""-"••••n•
1 C I S
n 1 1
— — —CI2
.--- C2 8 ....• u .....• • c. — 11— ...... • 2 C2
...... - - ..,,, PX
Figure 5. Concentration Profiles in a Solution-Diffusion Membrane.
(After Merten, 1966a; Bean, 1972; and Sourirajan, 1970) 73
C 2" CRF :. . [158] C2'
The CRF of a solution-diffusion system may thus be expressed
D K (AP-Aff) CRF = I I v I [159] d D K RT 2 2 for both osmotic and ultrafiltration systems.
Pore Flow Model
Solution-diffusion membranes represent the extreme case of no coupling of solvent and solute flux. Pore-flow, or viscous-flow, membranes represent the other extreme, where such coupling is character- istic of the transport kinetics of the system. Such membranes, as suggested by Gregor (1958) are of the coarser variety.
Consider a semipermeable membrane consisting of large, cylindrical pores. Let these pores initially be represented as narrow, cylindrical tubes through the membrane. A fluid element, in proximity to the cylinder boundary, travels at a mean velocity u in the x direction (as shown in
Figure 6). The element will be deformed at an angular rate equal to du/dy, from which arises the shear stress T along plane a-a. This rela- tionship is expressed by Newton's law of viscosity (Hillel, 1971)
du T = - n — [ 1 60] dy where n is the dynamic viscosity of the fluid. Note that the negative sign appears because u decreases with increasing y away from the center- line of the pore. 74
Y
X
Figure 6. Angular Deformation of a Fluid Element.
(After Rouse, 1950; and Bayley, 1958) 75
The fluid velocity is zero at the wall, and is maximal at the flow axis. Adjacent cylindrical laminae of the fluid move at different velocities and slide over each other in laminar flow.
Consider the co-axial fluid cylinder of length 1 and radius y within a pore cylinder of radius r. To maintain constant velocity the 2 pressure force acting on the face of the cylinder (Try ) must be equal to
the frictional resistance due to shear acting on the circumferential area
of the element (27)71). Let AP express the pressure difference P l -P 2
shown in Figure 7. This relationship between pressure and shear may be
Figure 7. Laminar Flow Through a Cylindrical Tube.
(After Hillel, 1971) 76
expressed (Hillel, 1971)
2 APTry = 27rylt [161]
or
AP y _ [162] - 1 2 •
Substituting this equation into Equation [160], we get
du _ AP y [163] dy nl 2 •
Integration yields
[AP ) 2 1, u(y) = - [164] 4n1 r
where k is a constant of integration. Evaluated at the boundary condition
u(y) = 0, at y = r, we find 2 APy k = n [165] 4 1
and so Equation [164] becomes
AP 2 2 [166] u(Y) = 4n1 (r )
(Bird, Stewart and Lightfoot, 1960). The flux through the cylinder per
unit time may be determined by integrating the velocity distribution over
the cross-sectional area as done by Polubarinova-Kochina (1962): 4 ffr AP Q = 27r f u(y)y dy = 8711 . [167]
This is the classical Hagen-Poiseuille law equation. If we let the
equation apply across the membrane, then 1 = Ax. The equation may be
written for the membrane, rather than the pore, by multiplying Equation
[167] by N (the number of pores per unit area). Further, in terms of 2 membrane porosity, c (where c = N7rr , pore area per unit area), the 77 equation may be written as (Lonsdale, 1972)
2 Er AP J = 1 8-11Ax • [168]
The additional parameter, t, representing the mean tortuosity of the pores in the membrane, may be introduced as suggested by Baker, Pinch and Strathmann (1972) to yield the complete form of the pore flow model of solvent flux
2 cr AP J = [169] ip 8ntAx where the subscript p identifies the underlying pore-flow model. It should be noted that this equation is analogous to pore-flow models used in groundwater flow theory by Muskat (1946), Davis and De Wiest (1966), and others. It should also be noted that the development above assumes that only solvent flows through the pore. If other than solvent flows,
J becomes J ., the volumetric total flux through the pore. Further, ip vp the additional driving component of osmotic pressure difference must be . added when membrane selectivity precludes osmotic pressure equilibrium across the membrane. Equation [169] becomes
2 cr (AP-A r) J = [170] ip 8ntAx as derived by Blunk (1964).
The flux of solute through the membrane in a pore-flow system may be expressed as the sum of two solute contributions. The first contri- bution is analogous to the diffusion solute flux expressed in Equation
[151], describing the solute flux in a solution-diffusion system (Michaels and Bixler, 1968). The second contribution is due to the transport of 78 solute through the pores in the membrane by center-of-mass motion of the fluid (Merten, 1966a). The latter term may be expressed as the product of solute concentration in the membrane and the velocity through the membrane. Solute flux may then be expressed in the form Ac
- D [171] J 2p = 2 Ax 2 + c 2u(y) .
Lonsdale (1972) further corrects this expression to account for the interactions of solute, solvent and membrane, in terms of the coeffi- cient b, where
f 0 b E + SM . [172] f 0 SW
Here, fsw ° is the frictional coefficient for the solute and water inter-
f ° is a similar coefficient action as expressed in Equation [152], and sm equal to the force acting on the solute as a result of its interaction with the membrane, divided by the solute velocity within the pore. As membrane behavior approaches purely pore-flow transport kinetics, solute- membrane interaction becomes negligible, and b approaches unity. The
corrected equation for real membranes (in differential form)
- D dc c u(y) 2 2 2 =----+ [173] 2P b dx thus becomes Equation [171] under these conditions.
If the distribution coefficient K replaces c2 in Equation [173]
(conforming to its definition in Equation [153]), we may integrate the
equation to find 79
J p 2 C - + B exp ) 2 u(y) Y [174] D2 where B is a constant of integration (Merten, 1966a). The boundary condi- tions of this equation in this case are (1) that Equation [153] is satis- fied at x = 0 and x = -tAx (to comply with the assumption of the equation) and (2) that the concentration of the permiate is proportional to the relative magnitude of solute and solvent fluxes as expressed in Equation
[156] (Merten, 1966a).
With these boundary conditions, and Equations [164] and [169],
Equation [174] becomes
K'c exp (u(y)tAx/D ) 2S 2 " [175] 2s K"_ - be + be exp (u(y)tAx/D2) in terms of Equations [172] and [173] (Merten, 1966a). The concentration profiles for solute and solvent for a pore-flow system are presented in
Figure 8. The solute rejection coefficient, as defined by Equation [155], becomes
K' exp (u(y)tAx/D2 ) = 1 - [176] K" - be + bE exp (u(y)t&/D2 ) and concentration reduction factor, after Equation [158], becomes
K" - be + be exp (u(y)tAx/D2 ) CRF - [177] K' exp (u(y)tAx/D2 ) for pore-flow kinetic systems. It should be noted that at high pore- fluid velocities diffusion becomes negligible, and with the additional simplifying assumption that b = 1, Equations [176] and [177] are
K' R = 1 - — [178] 80
C2
II C2S
0 5
Figure 8. Concentration Profiles in Pore-Flow Membranes.
The three curves are 0, 1 and 5 values of (u(y)tAx/1012 ). The solute c has been set distribution coefficient for the 2/C2s at 0.56 at both membrane-solution interfaces (after Michaels and Bixler, 1968; and Merten, 1966a). 81 an d
CRF = K' • [179]
Theoretical Treatment of Concentration-Polarization at Phase Boundaries in Membrane Separation Systems with Laminar Flow Regimes
Solute-solvent separation techniques, of necessity, require the
feed mixture to ultimately undergo a phase transition to product phases
of opposite characteristics with respect to the original solution. At
the feed-product phase boundary, where the actual separation is effected,
a solute-rich boundary layer is established corresponding to the conju-
gate solvent just separated. This solute-concentration boundary layer
has negligible influence on several desalination methods, such as
distillation, but is a significant consideration for membrane separation
techniques, such as electrodialysis, reverse osmosis, or the techniques
discussed in this paper. In such methods of separation, where a membrane
is employed to establish the phase boundary, this polarization of solute
concentration builds up to an extent exceeding that of the bulk feed
mixture. Since the osmotic pressure difference across the phase inter-
face must be overcome by the applied hydraulic pressure, the polarization-
concentration of the solution reduces the effective separation of a given
applied pressure. Not only is the energy requirement to maintain a given
volume flux increased, but the solute flux across the phase interface is
also increased, as demonstrated by Equations [151] and [171.
Modeling of the concentration-polarization phenomena in reverse osmosis systems (dealing with hydraulic pressures in excess of osmotic pressure differences, complete solute rejection and separation of 82 microsolute and ionic size materials) has been extensively examined by
Brian (1965b); Gill, Tien and Zeh (1965); Gill, Zeh and Tien (1966);
Johnson and McCutchan (1972); Merten, Lonsdale and Riley (1964); Winograd
and Solan (1969/70) and other leading researchers in reverse osmosis. Fouling of reverse osmosis membranes by solutes precipitated from solu-
tion, due to saturation concentrations achieved by polarization-concen-
tration, has also been investigated by Jackson and Landolt (1972), Loeb
and Johnson (1966) and others. Concentration-polarization with respect
to the ultrafiltration of macrosolutes and colloids has also been
thoroughly investigated by Fallick (1972), Blatt et al. (1970), Michaels
(1968) and others. The system under consideration here involves the
separation of small macromolecules by high flux ultrafiltration membranes with incomplete solute rejection. This aspect of concentration-polariza-
tion has not had wide theoretical treatment, although the treatment of
incomplete salt rejection in reverse osmosis as treated primarily by
Brian (1965c, 1966b) may be considered a parallel mechanism.
As suggested by Bray (1966) and others, a reasonable membrane
configuration for efficient separation design is the passage of feed
solution across a flat membrane surface, or more specifically, across two
such surfaces by injecting the solution into the channel between pairs of parallel flat membranes. The development presented here deals with such
a configuration, shown in Figure 9, which has membrane width much larger
than membrane spacing. Product solvent is assumed to permeate through both membranes with equal magnitude. These conditions suggest feed solution flow may be assumed two-dimensional and symmetric about the 83 midplane of the system. This implies a laminar flow regime may be consid-
ered to exist between the membranes, and convection may be considered negligible.
v,V fy,Y u,U x,X 2h
tvsf f f f Solvent Removal
Figure 9. Two-Dimensional Channel Between Flat Parallel Osmotic Membranes.
The half-channel thickness is designated h.
Differential Equation for Solute Balance
At the channel inlet a uniform solute concentration is assumed, equal to that of the feed concentration (c2 1). As the solution progresses down the channel, product water is removed at the membrane surface, and a salt-concentration boundary layer will develop with concentration at the interface (c2 ') increasing with distance along the direction of flow.
A solute balance on a differential element of volume may be made by differentiating solute flux along the directions in which flow is considered to change. Equation [177] may be considered to describe solute flux along one dimension in the system since transport through the 84
cylindrical pore is analogous to one component of flow through the two-
dimensional channel considered here. The continuity equation in two
dimensions and steady state (time derivatives absent) is
m- 0 a j 0 2x 2y 9x By [180]
where J2n ° is the solute flux on the element in the n direction, or
Dc -97 (uc 2) + (vc - D = 0 [181] ay 2 2 D Y from Equation [171] where u and v are the fluid-velocity components in
the x and y directions, respectively. This is shown in Figure 10. The
J2 C2
pc2 D2 a y
BULK SOLUTION 'BOUNDARY LAYER
Figure 10. Profile of Concentration-Polarization at Channel Boundary for Laminar Flow Along a Membrane Surface. 85 boundary conditions which apply are
at x = 0, any y: C = C 2 2 1' [182]
Dc at y = 0, any x: 2 = 0 [183] Dy
3c 2 at y = h, any x > 0: vc2 - D -57r— = J 2 [184]
The first constraint expresses the inlet distribution condition expressed above. Equation [183] expresses the assumption of symmetry about the midplane of the system. Equation [184] shows that, at the membrane surface J = J, the flow of solute in the transverse direction at any 2y ° point on the membrane surface is equal to the solute permeation of the membrane at that point (Brian, 1965b).
It should be noted that a convenient and equivalent way to express solute rejection introduced in Equation [155] is as unity minus the ratio of the solute flux through the membrane to the convective flux of salt to the membrane, or
R = 1 - ( [185] ucjy=h •
Combining this with the third boundary condition we find Dc D2 = Ruc . [186] at y = h, any x > 0: 2
Before Equation [181] may be solved, a description of the velocity field must be known. Berman (1953) has found a solution .for the velocity distribution for a homogeneous fluid flowing in a channel between two parallel porous plates with uniform withdrawal of fluid through the plates to be 86
— 1E N E 2 4 u = (u) [1 - 1 - 2 420 [2 - 7 -Y-- - 7 -:—)Y [187] h h2 h4 (3 y2I NFy r 3y2 y6)-1 = [188] h2j 280h 2 2 • h h6 Here, v' is the transverse velocity adjacent to the membrane surface and may be considered equal to the volumetric flux across the membrane per unit area. This assumtion breaks down with solutions where the solvent concentration is not close to unity. The NF is a permeation of Reynolds' number and is equal to hv'/v, where v is the kinematic viscosity of the solution. This term is generally so small (Brian, 1966b) that terms involving NF shall be neglected. This also breaks down with high solute concentrations. With this simplification, Equations [186] and [187] become
r 2 - [189] u (;-) (I-1) h
= (v') (k-) r3 _ _r!_) [190] h2 J •
These equations deal with constant permeation velocity, but our application has v' varying with longitudinal position. Equation [188] shows that constant permeation velocity along the channel simply decreases u as solution is withdrawn through the membrane, and the parabolic velocity profile is not disturbed by a constant vl. It is thus reasonable to assume a varying v' would affect Ti similarly and Equation [188] adequately describes this case also. Similarly, Equation [189] may be assumed adequate for varying v' since transverse velocity at any point is equal to -0 times a cubic function of transverse position. Thus, 87
Equations [188] and [189] may be used to describe the velocity field for both constant and varying v'. but the application of these equations over the entire channel implies the additional assumption that the parabolic longitudinal distribution is already developed at the channel inlet
(Brian, 1966b).
Equations [181], [182], [183], [186], [189] and [190] may be expressed in the dimensionless form
D „ DC, — LIC)t_ + — - a = BY o DY [191] where the dimensionless quantities are defined as (Sherwood et al., 1965) _ u U = [192] f
[193]
D 2 a [197] o v f' h •
The boundary conditions of Equation [191] then become
at X = 0, any Y: C = 1 [198]
DC at Y = 0, any X: .-T= 0 [199]
@C a ( RVC at Y = 1, any X > 0: o [200] 88 and
3 2 U = (1 - A) (1 - Y ) 2 [201]
Y 2 V = V' — (3 - Y ) . 2 [202]
Here, A represents the volumetric fraction of the feed stream that has been removed through the membrane at a given position along the channel.
The conversion from Equations [189] and [190] to [201] and [202] involves relating il- to —uf in terms of A, and noting that V' = vVvf '. The rela- tionship between V' and A is given by Brian (1966b) as the complete solute solvent balance
X A = f V'dX [203] o where X is a dummy variable of the definite integral.
Treatment of Constant and Variable Permeation Fluxes
From Equations [145] and [170], is it apparent that the permeation velocity of the solvent may be expressed
v' = A(AP - An) [204] independent of transport method. Brian (1965c) shows that if osmotic pressure is assumed directly proportional to salt concentration, and AP is constant, Equation [204] may be normalized with respect to permeation velocity at the inlet to be
V' = 1 - f3r [205] where
rfff = [206] AP - rTrf 89 which is the fraction of the driving force caused by the feed solution osmotic pressure, and where
r E C I - 1 [207] which is the concentration-polarization. Thus, r represents the decrease in permeation velocity due to concentration-polarization at the membrane surface.
Equation [205] expresses a nonlinear coupling between the concen- tration-polarization at the membrane surface and the permeation flux that produces the polarization. This nonlinear problem constituted by
Equations [191], [198] through [203], and [205] has been solved by digital computer using a finite difference method by Brian (1965a) and by Srinivasan and Tien (1969/70). It should be noted that the treatment presented above applies only to a solution containing only one solute.
Multi-component systems should rather be modeled using the treatment suggested by Srinivasan and Tien (1970). There is some discussion in the literature concerning the validity of the finite difference solution (Gill and Tien, 1966), but the solution shows very good correlation with experimental findings (Brian, 1966a).
Dresner (1964) has obtained an approximate analytical solution to
Equation [191] for the case of constant permeation flux which avoids the need of computer solutions, and yet yields fairly accurate results
(Sherwood et al., 1965). Near the channel entrance the approximate solution is
r = + 5 [1 - exp (- IZTg )] [208] where 90
A 2 3ao
X 2 3ao
v ix —f 2 [209] 3u ha o from Equations [203] and [194]. For far downstream solutions,
r - 2 [210] 3ao
It has been shown (Sherwood, Brian and Fisher; 1963) that
1/3 r = 1.536 () [211] is a better model of concentration-polarization than Equation [208] in the range c < 0.02 (Sourirajan, 1970). Equations [208], [210] and [211] may, thus, be used to describe concentration-polarization along the chan- nel in the x direction. Similar expressions expressing concentration- polarization in terms of Schmidt number are presented by Gill, Derzanski and Doshi (1969). A comparison of the finite-difference solution of
Brian compared to the results of Dresner show that for any value of ao , r is seen to increase with A and then to level out at an asymptotic value of r. The asymptote corresponds to the constant far downstream value for the polarization, and thus the Dresner equations may be considered an envelope for the finite difference values. EXPERIMENTATION
The theoretical treatment of the ultrafiltration system to be used in step filtration must now be experimentally examined to obtain a complete description of the system. This combined evaluation may be used to judge its adaptability to salt replacement desalination.
All experimentation was conducted at 25 ° C in a modified Amicon
TCF10 ultrafiltration cell. Pure water ultrafiltration runs were per- formed to evaluate the hydraulic permeability of the membranes, and to determine the membrane transport mechanisms. Ultrafiltration runs were also conducted with various concentrations of Baker reagent grade sucrose to determine separation and permeation characteristics of the membranes.
These latter experiments were used to determine the solute reflection coefficient of the membranes at various feed concentrations, the permiate- retinate relationships and the extent of the concentration-polarization along the membrane surface. The first-order transport coefficients were determined from these experiments, enabling the membrane system to be completely described in the manner developed above.
In the actual ultrafiltration mechanisms to be used with salt replacement the hydrostatic pressure required is to be supplied with pumps, or elevation head, rather than by compressed nitrogen as used in the laboratory apparatus employed in these experiments.
91 92
Experimental Apparatus Used in Ultrafiltration of Sucrose in this Study
The experimental apparatus used in the ultrafiltration of sucrose in this study is illustrated in Figure 11. The equipment primarily con- sists of a pressurized ultrafiltration cell, a recirculating pump mechanism and a thermal control mechanism.
The ultrafiltration cell is a TCF10 model furnished by Amicon
Corporation. The cell itself is illustrated in Figure 12. The cell consists of a 600-ml reservoir constructed of chemically inert plexiglas resting on a Delrin [registered trademark], thin-channel, membrane-fixing apparatus. The thin-channel system produced exclusively by Amicon offers high polarization control. The fluid from the reservoir sweeps across the membrane through a spiral, shallow channel, with accompanying shear at the membrane surface. This method has been shown to yield trans- membrane fluxes several times higher than magnetically agitated non- recirculating ultrafiltration cells (Buyer's Guide, 1972). The fluid is introduced at the center of the membrane and progresses for 40 cm through a 0.4-mm deep channel, 1 cm wide, to a withdrawal point at the perimeter of the membrane. Although the 90-mm (diameter) membrane which fits the 2 cell has 63.6 cm of surface area, because of the channeling apparatus, 2 only 40 cm of the membrane is used.
In contrast to homogeneous cellulosic membranes, Amicon's DIAFLO
[registered trademark] membranes are of inert polymers in the form of a skin on a highly porous substructure making an anisotropic membrane.
Amicon's smallest-pored membranes, the UM series, have cutoffs of 500,
1,000 and 10,000 molecular weight. It should be remembered that the 93
THERMAL CONTROL UNIT NITROGEN REGULATOR
HOT WATER COLD WATER
THERMOMETE I VENT WELL
0 0 VALVES
MAIO _ _
ULTRAFILTRATION CELL
DRAIN
SELECTOR VALVE 01 Il
PERISTALTIC PUMP
INPUT/OUTPUT PERM I ATE
Figure 11. Schematic Representation of Experimental Apparatus Used in the Ultrafiltration of Sucrose in this Study. 94
PRESSURIZED NITROGEN
Figure 12. Thin-Channel Ultrafiltration Cell.
Recirculation of fluid through shallow channels minimizes concentration-polarization and maintains high solute fluxes.
95
cutoff level refers to the smallest molecular weight solute which is
absolutely retained by the membrane. The published data on these mem-
branes are presented in Table 2.
Table 2. Published UM Membrane Series Characteristics.
(Buyer's Guide, 1972)
De-ionized Water MW Sucrose Retention Flow Rate Pore Radius Membrane Cutoff (at 55 psi, pH 10) (cm/sec-atm) (cm)
-5 -4 -8 UM 05 500 80% 8.91x10 - 1.78x10 5.0x10
-4 -4 -8 UM 2 1,000 50% 1.78x10 - 3.56x10 5.5x10 -3 -8 UM 10 10,000 25% 4.45x10-4 - 1.34x10 7.0x10
The membrane rests upon a high hydraulic permeability membrane
support wafer of inert plastic fibers. The upper surface is smooth, and
the wafer opens to a loose matrix on the bottom. The fluid which passes
through the membrane skin flows through the backing and into the support.
Fluid then is collected from beneath the support in a deep, circular
channel which is subsequently drained by a tube into the permiate recep-
tacle. The channel, support and moist membrane retain 3 ml of permiate
solution. As shown in Figure 12, a circular rubber "O-ring" rests on the
edge of the membrane with which this section is sealed to the channel
guides. Similar rings seal the top and bottom of the reservoir. When
the four bolts are tightened, these rings are deformed, making the system
pressure-tight. 96
The fluid not passed through the membrane, that which is removed at the end of the thin channel, passes out to the master selector valve.
This valve determines the operating status of the unit. As indicated in
Figure 11, the valve occupies a "Y" connection where each of the limbs may be closed, one at a time. The system may, thus, be set to input solu- tion, to output it, or to recirculate it. In each case, the limb with the corresponding letter designation in Figure 11 is closed, and the other two are left open. To input fluid into the system, the pump moves fluid from the feed receptacle through the pump and thermal control unit into the reservoir. To output fluid, the pump may be shut down, and the fluid is pressed out along the membrane by the pressure in the reservoir.
In recirculation, fluid is removed from the membrane surface through the pump and thermal control unit back into the reservoir.
The feed solution is drawn across the surface of the membrane, through the valve, by a peristaltic pump. The pump operates by constric- ting a heavy-gauge Tygon [registered trademark] tube with rollers, attached to a 115V, 60 HZ, reversible motor. Some difficulty was encountered with the Tygon tubing. Extended high pressure, high flow rate pumping caused deterioration and failure of the tubing, but when the tubing was replaced by a heavier section, these problems were overcome. Although the pump and ultrafiltration cell are thermally insulated from each other, the heat of the pump reaches the fluid when the system is in operation for extended periods. This is one of the features of the system which makes a thermal control mechanism necessary.
Thermal control of the system is achieved with a temperature bath system. The fluid returning from the pump to the reservoir passes 97 through a 100-cm Tygon tubing coil immersed in the bath. Hot and cold water enter the styrofoam-insulated bath at the top of the mechanism and are drained at the bottom. A thermometer well is located on the top of the unit. The thermometer monitors the temperature of the system. Stop- cocks on the water input tubes were adjusted so that the thermometer read
25 ° C. All experiments of the study were conducted at this temperature.
A constant flow is maintained in the bath so that the water in the bath remains well mixed. It was found that no adjustments of the bath mixture were required at high bath flow rates, even on extended filtration runs.
For an ultrafiltration run, the reservoir must contain a minimum of 20 ml of solution for proper recirculation. The solution is introduced through a vent on the top of the ultrafiltration cell after the reservoir has been bolted closed. A safety blow-off, set to be triggered at SO psi, is used to close the vent when the solution has been introduced.
Next, the thermal control mechanism is activated, and temperature is stabilized at 25 °C. The selector valve is then checked to indicate recir- culation. The pump is turned on to minimum circulation so that all the tubing is filled with fluid, and so that this part of the system pressurizes along with the cell. All seals and valves are checked, and the system is slowly pressurized with compressed nitrogen from a cylinder to the desired level for the specific experiment. The pump is adjusted to the desired recirculation rate and the run is initiated.
The system is shut down with approximately the inverse procedure.
The last step in this operation is venting the system to atmospheric pressure through the safety blow-off. The remaining fluid in the reser- voir is drained through the selector while there remains some pressure in 98 the reservoir. Fast draining and venting are avoided, since they tend to blister some membranes.
Most of the experiments were conducted at 40 psi. At this pres- sure, the slip sleeves which attach the tubing to the ultrafiltration cell are not far below failure pressure. It was found that a nylon cord behind the sleeve secured connection. The remainder of the system, with the exception of the blow-off, is said to be able to sustain much higher pressure although this level is the recommended maximum.
After each run, all parts of the assembly must be thoroughly cleaned. A complete flushing and drying of the tubing through the pump and temperature bath was found important. While working with high molal sucrose solutions, if such precautions are not made, the system is easily contaminated. Further, severe clogging develops under these conditions.
The recirculation-type apparatus used in this study is not to be used in large-scale filtration plants. In recirculation, the retinate solution is concentrated as low concentrate solution is withdrawn. The sucrose concentration on the membrane surface is virtually constant. In a large-scale plant, solution would be introduced into long, narrow channels with permiate being withdrawn from one or both sides. Retinate would increase in concentration along the channel. Although the two procedures are not identical, information concerning the latter may be obtained when the similarity of the methods is recognized. 99
Ultrafiltration Experiments with Pure Water Solvent which Determine Hydraulic Permeability and Transport Mechanism
Several membrane characteristics can only be determined in an ultrafiltration system when solute concentration is zero. These include hydraulic permeability and transport mechanism. All the experimentation for these determinations were conducted with the apparatus described above.
Determination of Hydraulic Permeability
As demonstrated above, the hydraulic permeability of a membrane may be determined by evaluating Equation [128]. This is accomplished by measuring volumetric flux across the membrane at various applied pressures.
Experimentally, the simplest case is in which cim = cs = 0, where volu- metric flux equals solvent flux.
Each membrane of the UM series was tested at 20, 30 and 40 psi.
The volume of product was measured by graduated cylinder, and timing of runs was made with a stop-watch. Four runs were conducted with each mem- brane at these pressures. The flux data of these runs, reduced to milli- liters of product from the apparatus per minute, appear in Table 3.
These values are also plotted in Figure 13. The mean value of the four runs was calculated and used in Equation [128] to yield hydraulic perme- ability coefficients in terms of milliliters of water passing through the 2 40 cm of active membrane surface per minute per psi applied. The L values at various pressures for each membrane were then averaged to obtain a mean coefficient for the membrane. The development of these 100
Table 3. Experimental Results and Determination of L .
40 psi 30 psi 20 psi Membrane (ml/min) (ml/min) (ml/min)
UM10 11.70 8.80 5.30 12.56 8.58 5.64 12.50 8.96 5.72 12.06 9.10 6.04 12.20 8.86 5.68 --- mean L ( ml in2 2 0.3052 0.2954 0.2837 )/40 cm 0.2948 --- 17 p ` min lbs -3 1.81x10 cm/sec-atm = L um p( 10)
UM2 1.08 0.80 0.58 1.14 0.86 0.52 1.16 0.84 0.56 1.10 0.80 0.50 1.116 0.824 0.490 --- mean r ml in2 )/40 am2 0.0278 0.0275 0.0270 Lp "-;Ir7 lbs 0.0274 -4 cm/sec-atm = L um 1.68x10 p( 2)
UM05 0.632 0.432 0.280 0.566 0.416 0.300 0.632 0.382 0.310 0.600 0.450 0.300 -- mean 0.608 0.420 0.296 r in 2 2 0.01485 L cm 0.01520 0.01420 p ' min lbs )/40 0.01470 -- -5 um 9.00x10 cm/sec-atm = Lp( 05) 101
I 1 —
PRESSURE (PSI)
Figure 13. Volumetric Flux Versus Applied Pressure, Results of Pure Solvent Runs at Various Pressures Used to Determine L for the UM Series Membranes. 102
coefficients, in units of centimeters per second-atmospheres, are also presented in Table 3.
Figure 13 shows the linear relationship between solvent flux and
applied pressure which is demonstrated by Equation [127]. The vertical
bars in the figure represent the ranges of the four measured flux values
at the given pressure. The slope of the lines is determined by L from
Table 3. Brackets in Figure 13 represent the range of flux values at 50
psi based on the L values for the UM series membranes published by
Amicon in Concentrating, Desalting, Separating, Solutions and Suspensions
(1972). In each of the cases there does not seem to be an extremely high
correlation between the published and observed values. The UM10 membrane
showed fluxes much greater than expected from the published data, while
the UM2 and UM05 had lower yields than expected. This would indicate the
L range of the membranes is greater than suggested. The ranges of flux
observed in the study, however, were not large. The four flux tests in
each set at each pressure were conducted on different membranes, but membranes of the same production lot. Thus, there seems to be little variability within membrane lots, but greater variability than published, between them. A comparison of the hydraulic permeability coefficients obtained from experimentation and those derived from Amicon advertisements is presented in Table 4. A comparison of water flux from the system, projected at 50 psi from these coefficients, is also shown. The units in which the data from Amicon are presented have been adjusted to correspond to the "velocity" per unit pressure units used in this paper, and generally accepted in common usage. 103
ri .--1 11) 0) 01 cl-
0 C \I ri --I
Cr) tr) C•1 v:;)
"0 0 4-E P-1 • 0 •••Z ;•-n • t-I-1 O CD I--- r-• l'41 ;-1 N- C41 1"-- 0. 0 E X 0 -1- •-•1 0 • •r-i g ,--I
o' L.r)
V) cl- ct I I I O 0 0 0 ta0 r--i n—t r--1 g X X X cd d- .0 co a4 v)• u") t.--• . g-, r--1 tO ,--, .-Z + + -o + a) ..i- •td- V) .g I 1 I En o o o -1-1 I—I r-1 r--1 1--.1 k k k re. in CO •--1 P•• n• r-- 01 Cl. vd• r-I CO
V) cl. tf) I I I 0 0 0 •-i ,--I •--1 X X X ,--i CO 0 CO n0 0
O L.r) cs, 104
Determination of Transport Mechanism
The transport mechanism of water containing organic solutes, such as sucrose, across biological membranes has long been considered by plant physiologists. Mauro (1957) and Robbins and Mauro (1960) observed that
AP and Ay act equivalently in their effect on permeation flux, which supports either of the transport models previously discussed. Radioactive tracer studies (Durbin, Frank and Solomon, 1956; Villegas, Barton and
Solomon, 1958) show actual transport less than expected by the evaluation of the first-order transport equations. These results are not completely
accepted in the field, as shown in an excellent review by Dainty (1963).
Solution-diffusion kinetics (Chinard, 1952) and bulk or pore-flow kinetics
(Solomon, 1960) are the principal mechanisms considered.
Several criteria may be employed to distinguish between the
solution-diffusion and pore-flow transport mechanisms possible. One of
the simplest employs the hydraulic permeability for the membrane, which
is generally several orders of magnitude greater in pore flow membranes
than in those with solution-diffusion transport kinetics and equal thick- ness and porosity. Equation [145] describes solvent flow in diffusive membranes while Equation [170] describes the pore-flow membranes.
Theoretical flux values from these equations compared to experimental values should determine the mechanism of solvent transport in the mem- branes used in this study.
In Equation [145], the concentration of solvent in the membrane
is unity since the system contains only pure water. Robinson and (c 1 ) Stokes (1968) indicate the partial molar volume of water (v 1 ) to be 0.018
liters per mole and Daniels and Alberty (1966) give the universal gas 105 constant R as 0.0820 liter-atm per mole-degree Kelvin. Since all experi- ments were conducted at 25 ° C, the value of T in the expression was set at
298.1 ° K. It is difficult to determine a value for the Fick's law diffusion coefficient of water. Lonsdale et al. (1965) have reported D1 -6 2 -1 values as high as 1 x 10 am -sec . This value is criticized by
Meares (1966) as extremely high and points out that Thomas (1951) has -8 2 -1 obtained a more reasonable value of approximately 1.5 x 10 cm -sec .
This smaller D I is corroborated by Thomas and Barker (1963), and Long and
Thompson (1955) have obtained values of similar order. The Thomas (1951) value was selected for use in Equation [145]. The membrane thickness
(Ax) of all three membranes was 10 (Buyer's Guide, 1972). All calculations were made for a AP of 40 psi (2.7211 atm) and pure solvent, or Ar = 0. Since the only membrane characteristic used in Equation [145] is membrane thickness, which is constant for the three membranes, the
flux values obtained for each membrane are the same. These values appear in Table 5.
Table 5. Comparison of Pure Solvent Fluxes.
J ** J *** Pore Radius* id ip iexp Membrane (cm) (cm/sec) (cm/sec) (cm/sec)
-8 -7 -5 -4 UM10 7.0x10 3x10 9.46x10 49.25x10 -8 -7 -5 -4 UM2 5.5x10 3x10 5.83x10 4.57x10 -8 -7 -5 -4 UM05 5.0x10 3x10 4.83x10 2.45x10 *Concentrating, Desalting, Separating, 1972. **From Equation [145]. ***From Equation [170]. 106
Poiseuille flow, described by Equation [170], is more closely
associated to membrane characteristics than diffusive flow. The pore
radii of membrane-governed, pore-flow kinetics are the primary influences
determining flow rate. The pore radii of the UM series membranes are
presented in Table 5. Membrane porosity (c) of the UM membranes is shown by Baker, Eirich and Strathmann (1972) to be 0.2 and the tortuosity
factor (t) of 4.0 is suggested. The dynamic viscosity (n) of water at
25 ° C is given by Hillel (1971) as 8.9 x l0 The same -4 membrane thickness of 10 cm is used in this calculation, as is the same
AP and A7 values. The flux values obtained by these values from Equation
[170] appear in Table 5 for comparison.
The solution-diffusion and pore flow values of water flux across
the membrane must be compared to the observed flux to determine transport
mechanism. Representative water flux was determined by multiplying the
hydraulic permeability coefficients of each membrane by the applied pres-
sure of 2.7211 atm. This generates the last column of Table 5.
Comparison of the flux values in this table shows diffusive fluxes
to be several orders of magnitude smaller than observed. On the other
hand, experimental fluxes appear to be approximately an order of magni-
tude greater than the pore flow values. Similar findings were observed
by Baker, Eirich and Strathmann (1972) for the UM10 membranes. Baker, Eirich and Strathmann (1972) show Loeb-Sourirajan cellulose acetate
desalination membranes to primarily follow the slower solution-diffusion
transport kinetics. Thus, under similar operational conditions, the UM
membranes have water fluxes several orders of magnitude greater than
Loeb-Sourirajan membranes. It should also be noted that the experimental 107 fluxes decrease with membrane pore radius, as would be expected with a pore flow membrane. Pore flow transport kinetics shall be assumed to dominate in the UM series. This is not to suggest that membranes must either follow one mechanism or the other, or even that a high pore flux membrane does not have some diffusive transport. All membranes have both processes in operation to some extent. Membrane pore radius seems to determine which mechanism is dominant, with pore flow most significant in large-pored membranes and solution-diffusion in small. The UM05 membranes thus have more diffusive flow than the UM10 membranes, although since pore flow behavior is closely followed, both membranes shall be treated as microporous.
Ultrafiltration Experiments with Variable Sucrose Concentrations Determining UM Series Membrane Fluxes and Separation at Operating Concentrations
Although the hydraulic permeability coefficients obtained in the previous section describe membrane performance in pure solvent systems, they are insufficient to describe performance in sucrose solution systems since the osmotic pressure difference across the membrane is not known for Equation [125]. The retention and flux characteristics of the mem- brane must be experimentally determined at various sucrose concentrations
(and, thus, various osmotic pressures).
Osmotic Pressure of Sucrose Solutions
Sucrose concentrations with osmotic pressures equal and greater to that of sea water must be employed in a replacer unit. The mean osmotic pressure of sea water, 25.1 atm (Merten, 1966a), corresponds to a 108 sucrose concentration of approximately 0.91 molal (Sourirajan, 1970).
Solutions well above this concentration, to 1.4 molal (M) sucrose or 38.1 atm osmotic pressure (Sourirajan, 1967), were used in these experiments.
The relationship between the concentration of a solution and its osmotic pressure is not clear-cut. Equations [31] and [35] predict the osmotic pressure of solutions at various solute concentrations as a func- tion of activity and osmotic coefficients. Table 6 compares the results of these equations with experimental values for the osmotic pressure of sucrose solutions.
Equation [35] appears to very closely model the observed osmotic pressures, although Equation [31] shows no great divergences either. In this study, all calculations of osmotic pressures from solution molality are done with Equation [35]. Since only one solute is present in the solution, the reduced form is
w RTW su 1 rfr = M [212] - SU 1000V1 where m is the molality of sucrose in the solution, and w is unity su su since sucrose does not disassociate in water.
Although (I) is an empirical quantity, an equation which relates it to molality was developed for the concentration range in this study. A linear correlation analysis of the (1) values in Table 6 with respect to molality shows the linear equation
= 0.9969 + 0.0926m [213] to model (I) in the range of 0.0 to 1.4 molal with a T level of 44.78 and 2 an r value of 0.9950. This means Equation [213] very closely predicts cp
109
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tr) a) H 00 Cg 00 VD 00 01 01 r, Cg CD 0 -Q) CD qD 00 ri 00 00 U0 00 ri ri d' •H • • • • • • • • • • • 1 1 r, V) Lr) 00 CD r, r, 01 d- ri 00 ch ‘0 CD d- crlk cn Y., c•A oo C".3) CNI CD d- Ch LO CD r- 00 ri 00 V, LO VD V) 4-) • • • • • • • • • • • • • CD Cgcf) g cd r- CD Cg IJO 00 CD NO se cq 00 a) 0 ri ri ri ri Cg Cg Cg NI NI •H W CD se Lio NI r- Ch 01 ri CA ri CD C, CD CA CD0 •H 00 Cr) CA LIO 0) 0000 VD r, E E V) 0 gu CD VD CO CD 00 VD VD VD gD 00 cD r- r, 00 0k CO Cg %.0 CD •71- C7.1 r, VD ri ri ri Cg CA4-4 01 00 01 uo CNI r•-•1 • -•I tr) CD 0) LO CD r, CA CD r- LO CA • • • • • • • • • • • • • •—•• cr) g 4-, CD Cg d- r, 0) Cg U0 00 CD 00 s.0 CA 00 a) 0 cd ri ri ri Cg CA CA b0 N1 • H -H Ch C13 •H •H CD 00 00 CD VD 00 d- U0 Lo n.0 n.0 CD 4-) P.1 V) No y, oo cq \D CD d- 00 r, 0 g4 ri ri ri Cg Cg 00 01 00 d' U0 EE g 0 V) 0 CQ 0 et-1 Cd H • ,0 • •1.-1 -e- 00 H 7:$ %.c) -g a) cr, O .P g Ci) •H C.) g •H CD 4-) •H oo r- 00 ri CD CD 00 0) 00 00 Cr) M r-H r-i , r-1 0 O - CD ri CA 00 d- U0 gD VD r CO CD CA Ul E •H -E)- I CD CD CD CD CD CD CD CD CD CD ri ri NI NO Q) C.1.4 • • • • • • • • • • • • O u LnJ ri ri ri ri ri ri ri 0 C4-1 ri ri ri ri ri Q) cn g g ?. O 0 0 g 4-) +à cd C' cd cd 4-) g O u-) ;LI E 4-> U) 9-1 g H H H CD ri CA 00 'd* U0 •.0 r, 00 Ch CD CA et •H CD w =1, su R = 0.082 1-atm/mole ° K, T = 298 ° K, W = 18.0 g water/mole, and v 1= 0.018 1/mole (Robinson and Stokes, 1968; Stanek, Cerny; and Pa61,. 1965), we get 2 7 = 24.381 m + 2.266 m (atm) . [214] It is often convenient to express the concentration of a solution in terms of the weight percent of solute in the solution, rather than molality. Since molality is the number of moles of solute per thousand grams of solvent, there is no linear relationship between it and weight percent solute in solution. Equations [215] and [216] demonstrate the conversion of weight percent to molality, and vice versa, where M is the molecular weight of the solute and P is the weight percentage solute in solution: (1000)P [215] m = M(1 - mM [216] P = 1000 + mM Coutts and Smail (1966) show the molecular weight of sucrose to be 342.3, as it is for the other disaccharides in Figure 4. 111 Permeation Flux as a Function of Solute Concentration All three membranes in the Amicon UM series were evaluated for permeation flux rates at various sucrose concentrations. In each case, the appropriate sucrose solution was prepared in a 600- or 800-ml beaker. The ultrafiltration cell illustrated in Figure 11 was assembled without the safety blow-off and the solution was introduced through the blow-off receptacle. The pump was then set to very low circulation to fill all tubing with solution and to allow for equalization during cell pressuriza- tion. The blow-off was put in place and the cell was checked for seal. The cell was then pressurized to 40 psi while all seals and closures were monitored. Recirculation was then adjusted to the level desired in the specific run, and the thermal bath flows were adjusted to a constant 25 ° C. The 3 ml of cell storage were then allowed to fill, sometimes requiring extensive periods when low flux membranes were used with high sucrose concentrations. The runs were initiated when this volume was filled and the cell began to yield permiate solution. Timing was begun at that time and solution was collected in a graduated cylinder. The volume at five, ten and fifteen minutes was recorded and a mean five-minute flux was calculated. Only recirculation rates of 500 ml/min and 200 ml/min were used in this study. The 500 ml/min value represents the highest rate which is safely applicable to the UM membranes. Higher rates tend to blister the membrane skin from the substrate at the filtration channel inlet, especially with the UM10 membrane. The 200 ml/min value is a low recir- culation rate expected to allow some concentration-polarization in the 112 channel without building up a dead, highly polarized, boundary layer in the thin channel which would produce unrepresentative permiate fluxes. At the 40 psi pressure used in all the flux tests in this study, no membrane compaction due to applied pressure was noted. Such compac- tion is indicated by Bert (1969) to be apparent in all primarily porous membranes such as those in the UM series. The low pressures used in the study did not cause observable compaction although Sourirajan (1970) shows compaction to be significant at high pressures such as those in the reverse osmosis range. The UM05 membranes were tested at nine sucrose concentrations to 0.83 molal sucrose at the 500 ml/min rate and at eight concentrations to 1.00 molal at 200 ml/min. Concentrations higher than those indicated were not used because of the extremely low permiate yields expected. Figure 14 shows the observed mean volumetric permeation fluxes for the UM05 membrane. The 500 ml/min recirculation rate line is assumed to have minimal concentration-polarization since this rate is equivalent to a 2.1 meters per second mean lateral channel velocity in the maximum shear direction. On the other hand, the 200 ml/min rate is equivalent to only 0.8 meters per second. The UM2 membranes were tested at ten sucrose concentrations to 1.0 molal sucrose at the 500 ml/min rate and at five concentrations to 1.0 molal at the 200 ml/min rate. It should be remembered that the high rate values are considered characteristic of the system and that low rate values were obtained simply to determine the effect of concentration- polarization on flux range. The data in Figure 15, as that in Figure 14, represent the yield from the entire cell rather than from a unit area of 113 I.0 0.9 0.8 5.0 6.0 Figure 14. UM05 Permeation Flux Versus Sucrose Molality, in ml of Solution Per 5 Minutes from the Ultrafiltration Cell at 40 psi. 114 5.0 6.0 Figure 15. UM2 Permeation Flux Versus Sucrose Molality, in ml of Solu- tion Per 5 Minutes . from the Ultrafiltration Cell at 40 psi. 115 membrane surface. The flux value is expressed in these unconventional units to be more easily appreciated in terms of actual permiate produc- tion. The flux curves of the UM2 membranes have the same characteristics as those of the UM05 membranes, with the sharply decreasing flux as low sucrose concentration is increased and levels off at high concentrations. The primary difference in the curves is that the UM2 membrane fluxes are shifted in the higher yield direction. The UM10 membranes were tested at ten sucrose concentrations to 1.4 molal sucrose at both the 500 ml/min and 200 ml/min recirculation rates. The test results presented in Figure 16 show the same character- istically shaped flux curves, but in this case the curves indicate signi- ficantly higher permeation flux rates. The UM10 observations have greater scatter and have a larger range between recirculation rate curves than those of the other membranes. Membrane Rejection as a Function of Solute Concentration The extent of solute transport across a membrane may be determined by the rejection coefficient, defined in Equation [155]. The active transport of sucrose across biological membranes has been extensively studied by physiologists. Kaback (1970) reviews current research along these lines as it relates to bacterial membranes and Widdas (1971) reviews sucrose transport in intestinal membranes. This study is concerned with the inactive, or gradient transport of sucrose. The ultrafiltration of sucrose was first conducted by Flexner (1937) in a study of the process underlying the formation of cerebrospinal fluids. Flexner presented the fundamental thermodynamics which govern down-gradient transport of 116 1.5— 0 0 0 0 01 to 1.0 — 10 20 60 (ML. / MIN.) Jv Figure 16. UM10 Permeation Flux Versus Sucrose Molality in ml of Solution Per 5 Minutes from the Ultrafiltration Cell at 40 psi. The 50 ml/min recirculation rate flux lines of the UM05 and UM2 membranes are included for scale. 117 material, and conducted ultrafiltration experiments using collodion and cellodion membranes. These studies led eventually to the development of dialysis systems used in modern medicine. Craig and Pulley (1962) show the results of the dialysis of sucrose solutions in high pressure systems. Experiments involving the filtration by reverse osmosis, with the associated high pressures and cellulose acetate high rejection membranes, were conducted by Sourirajan (1967). The effect of operating pressure in this high range and feed concentration on sucrose separation and permiate rate were studied. In all cases, operating pressures were far in excess of the osmotic pressure differences across the membrane. This study deals with the low pressure filtration of such sucrose solutions with osmotic and operational pressures in the same range, as applicable to the repeated or step application of ultrafiltration. During the permeation flux experiments described in the previous section, permiate and retinate samples were taken at the initiation and termination of each filtration run. This provides two permiate-retinate concentration relationships which check each other for each run. Since a retinate recirculating system is used, the concentration of the feed solu- tion should increase. This increase is considered negligible since the well-stirred recirculating volume is much greater than the volume removed and since the membranes used in this study have very low retentions rather than the high retention of reverse osmosis membranes which would make recirculation a problem. The sample sucrose concentrations are gravi- metrically determined. All sample vials are initially weighed on an analytical balance. The solution is removed into the vial and the vial is weighed again. Then, each 3-ml sample is dried in a 50 ° C oven at one 118 atmosphere for 24 hours after all the solute has appeared to have vaporized. Caramelization is avoided at this temperature and the extended drying period assures that all bound water is driven off. The samples are then cooled to room temperature to assure that they do not affect the balance, and are weighed again. The percentage of the weight of solution due to solute is determined from the weights measured, and the solution molality is determined with Equation [215]. In the case of each of the three membranes, ten retinate molality (nr) versus permiate molality (m ) values were found to be free of large experimental or measurement errors. These values demonstrate the reten- tion performance of each of these membranes as a function of feed sucrose concentrations. Figure 17 graphically demonstrates the results of these experiments for the UM05, 1JM2 and UM10 membranes. The results for the UM10 membrane are supplemented by adjusted data obtained by Baker, Eirich and Strathmann (1972) in similar experiments at low sucrose concentrations. The line bisecting the diagram represents the situation where permiate and retinate concentrations are equal, which implies that the membrane is permeable to all components of the system. All points lying below this line represent a filtration action where permiate is of lower sucrose concentration than retinate. The points in Figure 17 appear to all fall on the same line. Figure 18 shows the low concentration ranges of Figure 17 in which slight separation of the curves is apparent in this lowest range. The UM10 curve is closest to the bisector with the UM2 and UM05 curves progressively further. Since the UM10 membrane is the most porous and UM05 the least, this is the configuration to be expected. Although the sequence of the 119 I. 3 1.2 1.0 09 / / 0.8 / m / P 0.7 / • / / • • 0.6 / / 0.5 / 0.4 / / / • 0.2 / OA / .44# '-p•Y 0.1. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 M r Figure 17. Retinate Versus Permiate Molalities for UM05 (•), UM2 (+) and UM10 (o) Ultrafiltration Membranes. 120 Figure 18. Retinate Versus Permiate Molalities for UM Series Membrane at Low Concentrations. 121 curves correspond to the theoretical, the actual separation between the curves, which is no greater than 0.01 molal along the m axis, is not sufficiently great to justify making a distinction between the membranes of the UM series. This unusual observation means that a UM10 membrane has the same separation capabilities as a UM05 membrane with sucrose in solution, a solute to which both membranes are highly permeable. The cutoff levels of these membranes indicate that at higher molecular weights the membranes demonstrate highly different separation characteristics, but in the sucrose-water system studied here a porous membrane such as the UM10 can separate as well as a fine one such as the UM05, at flux rates more than an order of magnitude higher. Reduction of Experimental Sucrose Separation Data to Generate Solute Rejection Coefficient from Permiate-Retinate Concentration Relationships Figure 17 may be used to calculate the number and efficiency of each step in the repeated filtration of sucrose solution. The concentra- tion of the initial feed solution is located on the mr axis and projected to the separation curve. The corresponding m value is determined from the opposite axis. This value may then be projected perpendicular to the bisector to the initial axis to identify the molality of the feed solu- tion of the next step. The process may be repeated for subsequent filtra- tions until the desired permiate concentration is reached. Although this procedure is effective and yields information about the system, the theoretical coefficients previously derived offer more complete and more thoroughly quantized information. 122 Permiate-Retinate Concentration Relationships The principal step in determining these coefficients is the iden- tification of the relationship between m r and m in Figure 17. The larger portion of the curve, the high concentration end, appears quite linear. A linear regression analysis was conducted on the experimental values with retinate molality coordinate greater than 0.3 M. The analysis shows that m = 1.00656 m -2 r - 6.76442 x 10 M [217] is the line of best fit for these points, with a T value of 248.5 and an 2 r value of 0.9997. Because of the extremely good fit of the line, the simplified equation m = m - 0.07 M p r [218] was used in this study to predict permiate concentration in the range involved, without appreciable loss in accuracy. This line appears dashed in Figures 17 and 18. Although a similar equation of higher order could be generated to describe the curvilinear portion of the curve, a different method of analysis was selected which describes the curve with more physical signi- ficance. The auxiliary (dotted) curve in Figure 18 represents (AP - A7), rather m , plotted against m . The (AP - A7) term is expressed in molal r units using the corresponding units for AP. This was accomplished by setting Equation [214] equal to the 2.7211 atm (40 psi) used in the experiments, and the resultant quadratic equation was solved for m, to find the pressure applied which is equivalent to the osmotic pressure of a 0.111 M solution. The ATT values obtained from the difference between 123 the separation curve and the bisector in the horizontal direction at each m are subtracted from the adjusted AP value and plotted. At the point where the separation curve of a specific membrane and the auxiliary curve intersect, 7 = ( AP - A7) by the commutative rule, since the 7 versus 7 r and 7 versus (AP - A7) relationships coincide. Since r AP = Pr - P and thus is positive, and A7 = 7 - 7 r P' Tr = (AP - A-n- ) = (AP - 'IT + ii ) [219] p (AP - Trr) = 0 [220] and AP = it [221] • at the point. This corresponds to the point of maximal curvature of the separation curve because at lower concentration 7r < AP and the function is approaching AP while at higher concentration 7r > AP and the function is diverging from AP, since A7 becomes constant at higher concentration. Kimura and Sourirajan (1968d) have conducted reverse osmosis experiments with aqueous sucrose solution, Loeb-Sourirajan CA-NRC-18 cellulose acetate membranes and a filtration apparatus described by Sourirajan and Govindan (1965). Selected results from these experiments appear in Figure 19 with the separation curve of Figure 17 included for scale comparison. The experiments of Kimura and Sourirajan were conducted at much higher operating pressures and sucrose concentrations, and with membranes of higher retention than those used in this study. The charac- teristic shape of the separation curves are apparent in these results, with the values tending to a linear curve parallel to the bisector. The 124 35 30 25 Mr 20 15 10 A — -0- -0-- 5 10 15 20 25 30 35 40 45 50 55 60 mP Figure 19. Separation Relationship Observed by Kimura and Sourirajan (1968d) and Adjusted for Comparison to this Study, at Various Pressures and with Various Membranes. 125 point of maximum curvature relationship discussed above is also apparent in this data, and m r levels which comply with the constraint in Equation [221] are indicated. Sufficiently high sucrose concentrations were not used which would have allowed the linear segment of the curve to develop, but the trend toward this segment is apparent in the figure. As operating pressure is increased, the separation of the mem- branes improves, as apparent from the shift of the curves away from the bisector. This is due to membrane compaction at these high pressures, and to the screening effect of the concentration-polarization boundary layer (Kimura and Sourirajan, 1968d). Kimura and Sourirajan used several membranes, or films, in their study. By comparing the separation curves of several films analyzed at the same operational pressure, it becomes apparent that similar films tend toward the same linear separation curve segment, as was observed in this study. Such a relationship from the data of Kimura and Sourirajan (1968d) is illustrated in Figure 19 with the dashed separation curve at the 500 psi level. Evaluation of the Solute Rejection Coefficient R The solute rejection coefficient defined in Equation [155] is simply a function of the ratio of permiate and retinate concentrations. Table 7 shows the values generated by this equation when applied to the points in Figure 17. Figure 20 presents the data of Table 7 in a membrane performance diagram for the films used in this study. The figure presents calculated rejection values against feed solution molality to show separation per- formance of the membrane at varying retinate concentrations. Lines of 126 Table 7. Solute Reflection Coefficients. m m m r (UM05) r (UM2) r (UM10) 0.010 0.80 0.056 0.55 0.107 0.39 0.081 0.62 0.090 0.50 0.210 0.28 0.110 0.44 0.107 0.47 0.344 0.20 0.143 0.42 0.111 0.40 0.466 0.16 0.219 0.32 0.211 0.35 0.550 0.14 0.291 0.25 0.331 0.21 0.701 0.10 0.292 0.24 0.398 0.18 0.891 0.08 0.351 0.21 0.490 0.13 1.032 0.07 0.541 0.13 0.524 0.14 1.127 0.06 0.727 0.10 1.212 0.06 1.440 0.01 (Baker, Eirich and Strathmann, 1972) 0.015 0.60 0.029 0.52 0.058 0.45 0.117 0.37 0.292 0.22 0.584 0.10 127 $-1 ti) 'ri 128 best fit for the data from each membrane have been also plotted. At zero concentration, permiate and retinate have the same molality, and there- fore the rejection coefficient at that point is zero. This is a trivial case and not reflected in the figure. Since even at very low concentra- tions there is an appreciable rejection coefficient, coefficient values were extrapolated to the rejection axis rather than taken to zero. Rejection coefficient for retinate concentrations corresponding to the linear portion of the separation curve in Figure 17 have a fixed rela- tionship between permiate and retinate concentrations, and thus have a rejection coefficient which is a function of only mr over that range. This function may be obtained from Equations [155] and [218] by R = 1 - -11 Cr c - 0.07 = 1 ( r cr c r 0.07 = 1 - — + 0.07 [222] cr Equation [222] is represented as the dashed line in Figure 20. The observed rejection curves join the dashed curve at approximately 3.5 m m . At concentrations higher than this value, Equation [222] can very r accurately predict the rejection coefficient for all three membranes. Henderson and Sliepcevich (1970) have conducted ultrafiltration experiments with sucrose solutions and du Pont No. 300 cellophane mem- branes at pressures from 52 to 395 psi. Only sucrose concentrations 129 below the juncture level were used. The results of these experiments show that rejection decreases with applied pressure, which is as expected. Based on their observed data, Henderson and Sliepcevich conclude that after the initial curvilinear decrease from the point extrapolated to the axis, the rejection curve of membrane performance diagram becomes linear, as demonstrated by the dotted line extending from the UM10 curve in Figure 20. The results of the current study suggest that this conclusion is inaccurate and that the rejection coefficient never goes to zero as would be implied by a linear relationship. These results rather suggest that at concentrations higher than those used by Henderson and Sliepcevich a constant difference in concentration is achieved between permiate and retinate, and thus rejection becomes asymptotic to zero with increasing sucrose concentration. Reduction of Experimental Sucrose Separation Data to Generate Reflection and Solute Permeability Coefficients from Concentration-Polarization Relationship The first-order transport equations not yet identified are a function of retinate solution concentration. This concentration is not identical to the bulk retinate concentration due to boundary layer effects. With the reflection (a) and solute permeability coefficients (w) identi- fied, sufficient parameters have been determined to completely describe the membrane system. Concentration-Polarization Along UM Series Membranes The far downstream solution for concentration-polarization presented in Equation [210] is applied over the entire channel since 130 half-channel thickness is more than three orders of magnitude less than channel length and the flow is thus fully developed very early in the system. Combining Equations [197] and [210], we obtain r D [223] 3 ( 2 12 vfh j which will be used to calculate the concentration-polarization in the system. Fick's law diffusion coefficients for the sucrose-water system at 25 °C used in Equation [223] were graphically obtained from the discreet diffusion coefficient data of Sourirajan (1970). Table 8 shows the calculation of sucrose concentration in the concentration-polarization boundary layer on the high pressure surface of the membrane. Permeation velocities, presented in column 3 of that table, were obtained from Figures 14 through 16 by dividing the 500 ml/min recirculation volumetric fluxes by the active membrane surface area. It should be remembered that C' in column 7 is defined as the dimensionless ratio between c' and cf'. The last column in Table 8 shows the actual concentrations of sucrose on the membrane surface as predicted by the far downstream solution of the concentration-polarization equation. This was obtained by c ' = c fC' [224] 2 2 from Equation [196]. Figures 21, 22 and 23 show sucrose retinate concentrations versus various permeation flux predictions for the UM series membranes. The solid lines in these figures represent the 500 ml/min recirculation rate observed flux relationships. The actual data points have been omitted 131 Table 8. Boundary Layer Concentration. (4) (2) Fick's Law Bulk Retinate (3) Diffusion Coefficient Concentration Permeation Velocity (1) 2 [V(i-lc x10 4)] [D2 (a x10 6)] Membrane (mr) UM10 0.000 52.3 5.26 UM10 0.100 22.1 5.09 UM10 0.200 17.3 4.99 UM10 0.300 13.7 4.96 UM10 0.400 11.7 4.83 UM10 0.500 10.7 4.77 UM10 0.600 10.3 4.72 UM10 0.700 8.7 4.67 UM10 0.800 8.0 4.63 UM10 0.900 8.2 4.59 UM10 1.000 7.5 4.55 UM10 1.200 7.2 4.48 UM10 1.400 6.7 4.41 UM2 0.000 4.65 5.26 UM2 0.010 3.95 5.22 UM2 0.050 2.62 5.15 UM2 0.100 2.10 5.09 UM2 0.150 1.60 5.04 UM2 0.200 1.42 4.99 UM2 0.250 1.29 4.94 UM2 0.300 1.21 4.96 UM2 0.500 1.08 4.77 UM2 0.800 1.00 4.63 UM2 1.000 0.96 4.55 UM05 0.000 2.54 5.26 UMOS 0.010 2.08 5.22 UMOS 0.050 1.67 5.15 UMOS 0.100 1.37 5.09 UM05 0.150 1.21 5.04 UM05 0.200 1.04 4.99 UM04 0.250 0.96 4.94 UMOS 0.300 0.92 4.96 UMOS 0.500 0.79 4.77 UM05 0.833 0.71 4.61 132 Table 8--continued (5) (6) (8) Dimensionless Concentration- (7) Boundary Layer Dimensionless System Factor Polarization Molality (1) Concentration (ao) (c2 ') Membrane {F (10)] (C') UM10 0.50 0.000000 UM10 1.15 2525.25 1.252525 0.125252 UM10 1.44 1610.30 1.161030 0.232206 UM10 1.80 1028.80 1.102880 0.330864 UM10 2.07 778.81 1.077881 0.431152 UM10 2.22 677.50 1.067750 0.533875 UM10 2.28 642.26 1.064226 0.638535 UM10 2.67 468.16 1.064226 0.732771 UM10 2.89 399.20 1.046816 0.831936 UM10 2.78 431.77 1.039920 0.938859 UM10 3.03 363.10 1.043177 1.036310 UM10 3.12 342.58 0.036310 1.241109 UM10 3.31 304.41 1.030441 1.442617 UM2 5.62 0.000000 UM2 6.61 76.29 1.007629 0.010076 UM2 9.81 34.63 1.003463 0.050173 UM2 12.12 22.69 1.002269 0.100226 UM2 15.75 13.43 1.001343 0.150201 UM2 17.61 10.74 1.001074 0.200214 UM2 19.12 9.11 1.000911 0.250227 UM2 20.53 7.90 1.000790 0.300237 UM2 22.02 6.87 1.000687 0.500343 UM2 23.15 6.21 1.000621 0.800496 UM2 23.75 5.90 1.000590 1.000590 UM05 10.35 0.000000 UM05 12.53 21.23 1.003111 0.010021 UM05 15.45 13.96 1.002123 0.050069 UM05 18.51 9.72 1.000972 0.100097 UM05 20.86 7.66 1.000766 0.150114 UM05 23.94 5.81 1.000581 0.200118 UM05 25.75 5.02 1.000502 0.250125 UM05 27.04 4.55 1.000455 0.300136 UM05 30.11 3.67 1.000367 0.500183 UM05 32.56 3.14 1.000314 0.833594 133 4-) Pl A *. Pl of 7 O. 6 o 0 I I I I I I I T 7 7 1 n r) t I 7 6 d 0 E6- E c.) g a)0 o Q • r-I 1-1 r—I 0 0 1-1 Csa ;-1 134 0.9 ----- 0.8 — mr 0.7— 0.6 — 0.5 — 0.4 — 0.3 Jv Figure 22. UM2 Permeation Flux Predictions Versus Sucrose Molality, Con- sidering Hydraulic Permeability (o), Concentration-Polariza- tion (+) and Reflection Coefficient (.). Solid line is observed 500 ml/min recirculation rate permea- tion flux curve. 135 1.0 0.9 0.6 Mr 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.0 2.0 3.0 4.0 5.0 6.0 Jv (ml. / 5 min.) Figure 23. UM05 Permeation Flux Predictions Versus Sucrose Molality, Considering Hydraulic Permeability (o), Concentration- Polarization (+) and Reflection Coefficient (•). Solid line is observed 500 ml/min recirculation rate permea- tion flux curve. 136 for clarity. The first prediction of permeation flux, adjusted to the units of Figures 14 through 16, is generated by JV =L (APp - A7r) [225] from Equation [86] where K = L (or 1 p from Equation [125] where a = 1). These data points appear as circles (o) in Figures 21 through 23. The points obtained by using measured concentration differences across the membrane (from Figure 17) to find Alr are designated by an additional point (o). The remaining values are obtained from column 1 of Table 8 using Equations [214] and [218] and Figure 18. These values are calcu- lated in Table 9. Figures 21 and 22 show the predicted fluxes to be greater than the observed values for the umla and UM2 membranes, while Figure 23 shows the predicted fluxes to be slightly less than the observed. It would be expected that in membranes with significant concentration-polarization layers, such as the UM10 and UM2 from Table 8, the Equation [225] prediction should be higher than observed while in the minimal polarization membranes, such as the UM05 film, the prediction should be nearer the observed value. This is due to the reduced osmotic gradient which accompanies the polarized boundary layers in the first type of membrane system. The second series of permeation flux predictions is based on the difference in osmotic pressures between the permiate solution and the solution making up the polarized boundary layer. Table 10 shows the calculations in obtaining the concentration-polarization adjusted system permiate flux of column 17. The Alr value of column 14 represents the osmotic pressure difference of the sucrose solution concentrations of 137 Table 9. Bulk Permeation Flux Prediction. (9) (10) (11) Bulk Bulk Osmotic Bulk Permiate Pressure Driving Concentration Difference Force (m ) [A7r(atm)] [AP-Air)(atm)] P 0.000 0.000000 2.72100 0.061 0.167316 2.553784 0.143 1.434019 1.287081 0.230 0.790738 0.930362 0.330 1.867556 0.853544 0.430 1.854186 0.866914 0.530 1.885910 0.835190 0.630 1.917634 0.803466 0.730 1.949358 0.771742 0.830 1.981082 0.740018 0.930 2.012806 0.708294 1.130 2.076254 0.644846 1.330 2.139702 0.581398 0.000 0.000000 2.721100 0.002 0.195265 2.525835 0.019 0.760860 1.960440 0.052 1.186820 1.534280 0.091 1.470699 1.244110 0.135 1.634107 1.086993 0.183 1.699265 1.021835 0.230 1.790738 0.930362 0.430 1.854186 0.866914 0.730 1.949358 0.771742 0.930 2.012806 0.708294 0.000 0.000000 2.721100 0.002 0.019721 2.701379 0.014 0.882936 1.838164 0.045 1.359026 1.362074 0.086 1.594609 1.125494 0.131 1.734042 0.987058 0.179 1.800071 0.921029 0.230 1.790738 0.930362 0.430 1.854186 0.866914 0.763 1.969578 0.751522 138 Table 9--continued (12) (13) Flux Flux From Equation [225] Pv ( Tl )J ml - cm sec FJv ( 5 min from system)] 4.925191x10 -3 59.102 4.622349x10 -3 55.464 2.329617x10 -3 27.955 1.683955x10 -3 20.207 1.544915x10 -3 18.539 1.569114x10 -3 19.097 1.511694x10 -3 18.140 1.454272x10 -3 17.451 1.396853x10 -3 16.762 1.339433x10 -3 16.073 1.282012x10 -3 15.384 1.167171x10 -3 14.006 1.052330x10 -3 12.628 4.571448x10 -4 5.486 4.243402x10-4 5.092 3.293539x10 -4 3.952 2.577590x10 -4 3.093 2.090104x10 -4 2.508 1.826148x10 -4 2.335 1.716682x10 -4 2.060 -4 1.563008x10 A 1.875 - ' 1.500 1.456415x10-4 1.296526x10 1.556 1.189933x10 -4 1.428 24.489000x10-5 2.938 24.312411x10 -5 2.917 16.543476x10 -5 1.985 12.258660x10 -5 1.471 10.129190x10 -5 1.215 8. 883522x10 5 1.066 8.289261x10 -5 0.995 8.373258x10 -5 1.005 7.802226x10 -5 0.936 6.763698x10 -5 0.812 139 Table 10. Concentration-Polarization Permeation Flux Prediction. (14) (17) Concentration- Flux From Polarization (15) Concentration Osmotic Concentration- (16) Flux Polarization Pressure Polarization ml „ Adjusted Equation [225] Difference Driving Force Pv( 2 II ml [Au(atm)] [AP-A7)(atm)] cm sec [-Tv ( 5 min from system )] 0.000000 2.721100 4.925191x10-3 59.102 1.598360 1.122740 2.032159x10 -3 24.386 2.263964 0.457136 0.827416x10 -3 9.929 2.610129 0.110971 0.200858x10 -3 2.410 2.670986 0.050114 0.090706x10 -3 1.088 2.798903 -0.077803 -0.140823x10 -3 -1.690 2.927092 -0.205992 -0.372846x10 -3 -4.474 2.878201 -0.157101 -0.284353x10 -3 -3.412 2.908822 -0.187722 -0.339777x10 -3 -4.077 3.166204 -0.445104 -0.805638x10 -3 -6.667 3.147985 -0.426885 -0.872662x10 -3 -6.272 3.410038 -0.688938 -0.846978x10 -3 -6.764 2.936233 -0.215133 -0.979391x10 -3 -8.273 0.000000 2.721100 4.571448x10 -4 5.486 0.197162 2.523938 4.240216x10 -4 5.088 0.764764 1.955336 3.284964x10 -4 4.485 1.195326 1.525774 3.205774x10 -4 4.406 1.481362 1.239738 2.082760x10 -4 2.489 1.648133 1.072967 1.802585x10 -4 2.163 1.004566 1.687671x10 -4 2.025 1.716534 -4 1.811512 0.909588 1.528108x10 1.834 1.875811 0.845289 1.420086x10 -4 1.704 2.005761 0.715339 1.201770x101 1.442 2.083560 0.637540 1.671067x10 ." 2.271 -5 0.000000 2.721100 24.439898x10 2.938 0.195815 2.525285 22.727565x10 -5 2.727 16.519951x10 -5 1.982 0.885545 1.835555 -5 1.364584 1.356516 12.308644x10 1.465 1.585697 1.135113 10.216017x10 -3 1.226 0.975057 8.775513x10 - 1.053 1.746043 -5 1.815288 0.905812 8.152308x10 0.978 0.912207 8.209863x10 -5 0.985 1.808893 -5 0.836315 7.526835x10 0.905 1.884785 -5 2.011490 0.709610 6.386490x10 0.766 140 columns 8 and 9 (Tables 8 and 9). The adjusted permeation fluxes from Table 10 appear as crosses (+) in Figures 21 through 23. In all cases, the concentration-polarization fluxes are lower than both the observed or bulk values. In the case of the UM10 membranes, this flux reduction is to the extent that the flows become negative above m r = 0.45 M. This would mean that if there are no other factors exerting influence on the system, the flow is in the opposite direction. Concentration-polarization increases the sucrose concentration on the high pressure side of the mem- brane, in this case, to the level that the osmotic gradient is in the opposite direction and flow direction reverses. In an actual system operating under this condition, reverse flow would never be realized, since permiate concentration is a function of retinate concentration at the membrane surface. In such a case, flow would reach equilibrium and stop as hydrostatic pressure is balanced, rather than allowing the concen- tration of the permiate to become lower than that of the boundary layer. It is apparent, though, that there is another factor exerting influence on the system. The reflection coefficient of the membrane has not yet been considered. Evaluation of the Reflection Coefficient Slatyer (1967) points out that the reflection coefficient (a) is the most important of the first-order transport coefficients since it relates specifically to influence of solute on solvent transport in a membrane system. Both Katchalsky (1961) and Dainty (1963) show that the reflection coefficient has the effective range 0 < a < 1. Several important implications about the system can be made from the value of a in 141 this range. First, a = 1 may be considered a good measure of the semi- permeability of a membrane, since a approaches unity only when solute permeability is minimized. Second, a .4- 0 for coarse membranes as this permeability is maximized. Ray (1960) shows that a < 0 for systems where solute transport rate is greater than solvent transport, but such a system is only of academic interest. It is of course apparent that a cannot be greater than unity, since a membrane cannot have better than complete retention. The reflection coefficients for the membranes used in this study may be obtained from the separation data obtained by solving Equation [125] for a and evaluating it for the known parameters. The expression used to determine the reflection coefficient values in this study is [226] obtained from columns 3 and 17 (Tables 8 and 10), where Jv and 47 are respectively; Lp is obtained from Table 3; and AP is a constant. Table 11 shows the values of a obtained in this manner, and gives the mean values and those which will be used to describe the systems. Equation [125] may now be completely evaluated as shown in Table 12, in a presentation which corresponds to the arrangement of Tables 8-10. The adjusted osmotic pressure differences are found from the reflection coefficients of Table 11 and the concentration-polarization corrected osmotic pressure difference values from column 14 of Table 10. The flux values from Table 12 in m1/5 min from the system are represented by the dots ( .) in Figures 21 to 23. In all three cases, the values from Table 12 lie very close to the observed flux curve, confirming that the 142 Table 11. Reflection Coefficients. UM10 UM2 UM05 0.9698 (2.0937) 0.9702 0.7797 0.9774 0.9982 0.7525 0.8785 0.9902 0.7767 0.8682 0.9839 0.7967 0.8966 0.9029 0.7352 0.9138 0.9982 0.7784 0.9318 1.1079 0.7835 0.9781 1.0598 0.7163 0.9687 0.9799 0.7296 0.9270 0.6814 0.8347 _ a 0.7698 0.9266 0.9991 a 0.770 0.927 0.999 143 Table 12. Reflection Coefficient Adjusted Permeation Flux Prediction. Adjusted Osmotic Flux Flux From Pressure Adjusted Equation [125] Difference Driving Force [Jv ( 31 A ml [aATqatm)] [(P-7)(atm)] cm sec [j(v 5 min from system)] 0.000000 2.721100 4.925191x10 -3 59.102 1.203536 1.517564 2.74679 x10 -3 24.889 1.720617 1.000488 1.810883x10 -3 19.091 1.983698 0.737402 1.334697x10 -3 16.122 2.029949 0.691151 1.250983x10 -3 14.425 2.127166 0.593934 1.075020x10 -3 12.958 -3 2.224589 0.496511 0.898684x10 10.489 2.187432 0.533668 0.965939x10 -3 11.049 2.210704 0.510396 0.923816x10 -3 10.698 2.406315 0.504785 0.889760x10 -3 8.748 2.392468 0.48736 0.694823x10 -3 7.957 2.591628 0.503611 0.894344x10 -3 8.952 2.231537 0.489562 0.886109x10 -3 8.384 0.000000 2.721100 4.571448x10 -4 5.486 0.182769 2.538331 4.264396x10 -4 3.953 -4 0.709863 2.011237 3.378878x10 3.516 1.108067 1.613033 2.709895x10 - 1 2.558 1.373222 1.347878 2.264435x10 - ' 1.887 1.193281 2.247120x10 -4 1.706 1.527819 -4 1.591227 1.129873 2.181806x10 1.628 -4 1.660271 1.041829 1.750272x10 1.358 1.738876 0.982224 1.650163x10i 1.175 1.447756x10 1.206 1.859340 0.861760 -4 1.931460 0.789640 1.326595x10 1.155 - 2.271100 2.448990x10 5 2.938 0.000000 -5 0.195619 2.525481 2.272932x10 2.740 0.884659 1.830441 1.652796x10 -5 2.000 1. 222092x10 5 1.708 1.363219 1.357881 -5 1.584111 1.136989 1.023290x10 1.453 1.744296 0.976804 0.879124x10 -5 1.233 1.813472 0.907628 0.816865x10 -5 1.141 1.807074 0.914016 0.822614x10: 1.085 0.854380x10 1.009 1.882900 0.838200 -5 2.009478 0.711622 0.640459x10 0.934 144 volumetric flux predicted by Equation [125] is an acceptable model of the observed flux values. Evaluation of the Solute Permeability Coefficient The solute permeability coefficient (w) is an indication of the transmissive properties of the membrane, with respect to solute rather than solvent. Equation [126] employs w to generate mass solute flux, in 2 2 gm/sec-cm units. This shows w may be resolved to units of gm/sec-cm - atm, or further to sec/cm with the appropriate conversion. Solving Equation [126] for w we get J - (1 - a)J S 2 w - v [227] A7s substituting Equation [125] for Jv and realizing that 1 73 = A7 in this system since all solutes are permeable to the membrane. Thus, the equation above becomes J - c (1 - a)L (AP - aA7) w - S 2 ATrs Js — AP =-- Lp c2 ( E7-fr - a) (1 - a) . [228] Further, J may be expressed as a function of experimental data by the expression Mc J Js = p v = Mc L (AP - aA7) [229] pp where M is a conversion factor for concentration and is equal to 0.3423 gm-l/mole-ml. With this substitution, Equation [228] becomes 145 mc I, (AP - 0A7) P P AP w = - L ( A7 2 p L_17 - a) =ML ( - a)(c - c (1 - a)) . p A7 p 2 [230] This expression is evaluated for w in Table 13, with the appropriate conversion to the basic units discussed above. The table is organized to correspond to the rows of Tables 8-10. The solute permeability coefficient calculated for the two fine' membranes, presented at the bottom of the table, are monotonically increasing in value with retinate concentration. This shows that w is not a completely independent variable for these membranes and in these concentration ranges. On the other hand, this dependence is not apparent for the UM10 membrane since the w values show no such relationship. Since a unique value of w is required completely describe the membrane system, the mean values of the last column of Table 13 were calculated. It was found that sec/cm [231] wUM10 = 85 wum2 = 45 sec/cm [232] and wUMOS = 0.2 sec/cm . [233] These values are considered representative of the membranes in the concentration ranges used. With this coefficient evaluated, the entire membrane systems may be completely described in terms of the first-order transport coefficients of the UM series membranes used. 146 Table 13. Solute Permeability Coefficient Evaluation. ML - a) p ATF gm 1 C-(1-a) gm 2) 2 x10 7) sec-atm-mole-cm (moles/1) sec-atm-cm ac) 0.000000 0.9790x10 -3 0.009821 9.6147 94.888 0.2484x10 -3 0.023023 5.7184 56.435 0.1567x10 - 0.037030 5.8030 57.270 0.1430x10 - ' 0.053130 7.6004 75.008 0.1163x10 -3 0.069230 8.0499 79.445 0.0918x10 -3 0.085330 7.8294 77.268 0.1009x10 -3 0.101430 10.2419 101.077 0.0915x10 -3 0.117530 10.7567 106.158 0.0514x10 -3 0.133630 6.8702 67.802 0.0543x10 -3 0.149730 8.1291 80.226 0.0161x10 -3 0.181930 2.9260 28.876 0.0901x10 -3 0.214130 19.2995 190.467 0.000000 0.2568x10 -4 0.000102 0.2619 2.585 -4 0.1627x10 0.000970 1.5784 15.577 0.0836x10 -4 0.002657 2.2214 21.923 -4 0.0563x10 0.004650 2.6439 26.092 0.0428x10 -4 0.66898 2.9549 29.162 0.0408x10 -4 0.009351 3.8138 37.639 0.0356x10 -4 0.011753 4.1878 41.329 -4 0.0324x10 0.021973 7.1285 70.351 0.0266x10 -4 0.037303 9.9297 97.996 0.0235x10 -4 0.047523 11.1584 110.122 0.000000 39.7326x10 - 0.000001 0.0040 0.039 6.3887x10 --) 0.000010 0.0065 0.063 3.0656x10 -5 0.000031 0.0095 0.094 2.2089x10 -5 0.000060 0.0132 0.131 1.7234x10 -5 0.000092 0.0158 0.156 1.5403x10 -5 0.000125 0.0193 0.180 1.5506x10 -5 0.000161 0.0251 0.247 1.3700x10 -5 0.000301 0.0412 0.407 1.0899x10 -5 0.000534 0.0520 0.513 CONCLUSIONS The replacement of salts in aqueous solution by a replacer chemical across an osmotic membrane, and the subsequent low-energy removal of that chemical by virtue of its special removal characteristics, comprises salt replacement desalination. The removal process, coupled to the replacement step in salt replacement desalination, is determined by the special removal characteristics of the replacer chemical, and, thus, may be any of a number of processes. Ultrafiltration, the membrane separation of macromolecules at low pressures, was found to be a separation process of great potential to be coupled to the replacer step. Sucrose was found to be the optimal replacer chemical of those examined, to be coupled with ultrafiltration, by virtue of its relatively high molecular weight and high osmotic pressure characteristics. Ultrafiltration with commercially available membranes, the UM series of Amicon Corporation, and 40 psi operating pressures was examined. These membranes were found to follow the transport kinetics predicted by a theoretical model of pore flow rather than solution-diffusion flow. This behavior indicates that permeation fluxes from ultrafiltration systems can be expected to be at least an order of magnitude greater than that from reverse osmosis systems when operating under similar conditions, since reverse osmosis follow$ the slower solution-diffusion kinetics. The low operating pressures used in ultrafiltration enable systems to be 147 148 designed with no special pressure containment considerations, as opposed to reverse osmosis systems which must be designed to contain pressures up to 1,500 psi. This greatly limits the initial capital investment required for this aspect of the construction of the salt replacement plant using ultrafiltration separation. It was developed that solute and solvent flux in a membrane system can be described by first-order phenomenological transport equations. The membranes themselves may be characterized by the corresponding first- order transport coefficients: L, hydraulic permeability; a, reflection coefficient; and w, solute permeability coefficient. These coefficients are presented in Table 14 to describe the three membranes examined in this study. Table 14. First-Order Transport Coefficients for UM Series Membranes. a Membrane (cm/sec-atm) (dimensionless) (sec/cm) -3 UM10 1.81x10 0.770 85 -4 UM2 1.68x10 0.927 45 -5 UM05 9.00x10 0.999 0.2 In a system with sucrose in solution, the retinate-permiate con- centration relationship was shown to be practically the same for all three membranes. This is to say that a relatively fine-pored membrane, such as the UM05, yields no lower concentration permiate than does a coarsely-pored membrane, such as the UM10, although they have greatly 149 different cutoff levels. This lack of improved separation with decreasing pore size is attributed to the solute being well below the cutoff, thus minimizing the sieving effect of the membranes. Since the separation behavior of the membranes is the same, the same permiate concentration may be expected at much higher flux rates with the coarsely-pored mem- branes. There is, thus, no advantage to using fine membranes under the conditions examined. The retinate-permiate concentration relationship shows permiate concentration to be 0.07 M lower than retinate concentra- tion for all retinate concentrations greater than 0.3 M. This indicates that, in this range, one ultrafiltration removes 0.07 moles of sucrose from 1,000 grams of solution. Reduction of 0.07 M per filtration between 0.3 and 0.1 M mr may be used as an estimate of the actual reduction, which is presented in Figure 8. Below the 0.1 M mr level, the concentra- tion reduction per filtration approaches zero with mr . For a solution to be an effective replacer, it must have an osmotic pressure greater than that of the saline solution which it intends to replace. If we consider sea water to be the saline solution to be desalted, this osmotic pressure must be greater than 25.1 atm. If we consider a 1.0 M sucrose solution, with osmotic pressure of 26.5 atm, water will move from the sea water into the sucrose solution in a replacer system. At higher sucrose concentrations, the flux of the water from the saline solution would be greater. Using an initial concentration of 1.0 M sucrose for the replacer solution to be purified, it would take 13 filtrations at 40 psi to reduce the concentration of the sucrose solution to 0.09 M, which is below the 0.1 M level to which the 0.07 M reduction per filtration relationship 150 holds. It is also below the 0.111 M level which is the sucrose concen- tration having an osmotic pressure of 40 psi. Thus, a solution of 0.09 M sucrose may be filtered in a reverse osmosis membrane system at 40 psi, yielding pure water filtrate. This procedure involved a total of 14 filtrations at 40 psi, or a total of 560 psi (37.4 atm) applied pressure used to remove 25.1 atm equivalent of salt. This constitutes a 66 percent effective efficiency for the system considered. This system operates at a pressure on the order of some of the most efficient reverse osmosis plants, but has the lower initical cost as discussed above. The operational pressure may be reduced by developing more effective separation methods to be coupled to the salt replacement step. Such a method may be the enzyme-catalyzed osmotic pressure reduction procedures discussed in this paper. In general, salt replacement desalination has shown itself to be a viable new method for the production of fresh water so far in its analysis. Ultrafiltration with a sucrose replacer has not demonstrated problems that would make it an unsuitable separation technique to be used in salt replacement. The construction and testing of a replacer unit should offer the remaining information needed to completely determine the values of such a combination. Additional research is also indicated for enzyme-catalyzed osmotic pressure reduction as a separation method to be coupled to the actual replacement step in salt replacement desalination. NOTATION a. Deviation of from its equilibrium i value. Fraction of the driving force caused by the feed solution osmotic pressure. Concentration polarization. A Volumetric fraction of feed stream removed through the membrane. Membrane porosity. Concentration-polarization factor developed in Equation [209]. Dynamic viscosity. O The number of phases in a system. X 1 ,X 2 ,X 3i Lagrangean multipliers used in Equation [7]. p. Chemical potential of component i. Chemical potential of component i at standard state and given temperature. .* Chemical potential of component i at reference state. of pure solvent at standard 10 Chemical potential temperature and pressure. Api Free energy of dilution. Kinematic viscosity. Ei General thermodynamic variable describing a system. The value of at the equilibrium value of S. 151 152 E The set of all E.. 1 The set E -0 where all members are at equilibrium. Osmotic pressure. a Local entropy production. a Reflection coefficient as defined in Equation [114]. Sheer stress. (1) Osmotic coefficient. (I) Dissipation function. Solute permeability coefficient as defined in Equation [115]. Partial solute permeability coefficient from Equation [123]. aco Dimensionless system factor as defined in Equation [97]. a. Activity of component i. A Permeation velocity constant as used in Equation [204]. Solute, solvent, membrane interaction coefficient as defined in Equation [172]. Constant of integration from Equation [174]. Non-singular coefficient matrix. c. Concentration of constituent i. 1 Dimensionless concentration as defined in Equation [196]. CRF Concentration reduction factor as described in Equation [111]. D. Fick's law diffusion coefficient. 1 Membrane diffusion coefficient as defined in Equation [123]. f ° Frictional coefficient between solute and membrane at SM standard conditions. -P ° Frictional coefficient between solute and water solvent at SW standard conditions. 153 Natural osmotic coefficient. gi Expanded term of natural osmotic coefficient. Matrix of all coefficients g... 13 -gij Coefficient defined in Equation [44]. h Half channel thickness. J. Flux 1 component i across a membrane. J o Solute flux on an element 2n in the n direction. Flux matrix. J' Flux matrix replacement as defined in Equation [64]. Constant of integration from Equation [164]. Concentration distribution coefficient. Empirical solvent flow coefficient as defined in 1 Equation [88]. Empirical solute flow coefficient as defined in Equation [88]. Length of cylindrical pore. Hydraulic permeability as defined in Equation [113]. Simplified phenomenological coefficient as defined in Equation [78]. L' Phenomenological coefficient matrix replacement as defined in Equation [66]. L.. Phenomenological coefficient for constituent i and conjugate 1 3 force X.. 3 Hydraulic permeability coefficient as defined in Equation [120]. 154 Phenomenological coefficient matrix. The number of components in a system. m. Molality of constituent i. Molecular weight of solute. M. Mobility of constituent i in the membrane. Number of constituents in solution. n. The number of moles of component i. Number of pores per unit area of membrane. N Permeation F of Reynold's number. Weight percentage of solute in solution. Total pressure. P. Vapor pressure of component i. P. ° Vapor pressure of pure component i. Volumetric flow rate. Radius of cylindrical pore. Gas constant. Solute rejection coefficient or percentage. Number of components to which membrane is permeable. Entropy. Entropy at equilibrium. So d S Entropy exchanged with the environment of a system. e d.S Entropy produced within a system. Time. Mean tortuosity. Absolute temperature. Lateral velocity as shown in Figure 9. 155 Velocity component in direction perpendicular to the membrane. Internal energy. Dimensionless lateral velocity as defined in Equation [192]. d.0 Internal energy produced within a system. Transverse velocity as shown in Figure 9. V' Transverse velocity adjacent to membrane surface. v. Partial molal volume of component i. V Volume. V Dimensionless transverse velocity as defined in Equation [193]. V' Dimensionless transverse velocity adjacent to membrane surface as defined in Equation [205]. w. Total number of moles into which constituent i disassociates 1 in solvent per mole of i. Molecular weight of solvent. 1 Direction perpendicular to the membrane. Lateral direction as shown in Figure 9. x. Mole fraction of component i. Ax Thickness of membrane. X Dimensionless lateral direction as defined in Equation [194]. X Dummy variable for integration. X. Driving force of component i across a membrane. X Force matrix. X' Force matrix replacement as defined in Equation [64]. 156 Y Transverse direction as shown in Figure 9. Y Direction tangent to the membrane. Y Dimensionless transverse direction as defined in Equation [195]. Operators da Differential quantity. da Infinitesimal time independent change in a. Ba Partial differential quantity a. La Change in a. Va Gradient in a. exp a Base of natural logarithms e to the a power. grad a Gradient of a. ln a Natural logarithm of a. Mean value of a. a. a defined per mole of component i. The first time derivative of a. -4- a The column vector a. a The transpose of a. a The matrix a. a! a factorial. a' First derivative of a (subsequent primes indicate subsequent derivatives). Subscri ts 1 Solvent, or water. 2 Solute. 157 Indicates original value of a changing variable. Indicates that partial derivatives were taken at equilibrium. d Designator value determined by solution-diffusion model. Indicates fluid is being considered. General index. im Designates solute impermeable through membrane. i,j,k,...m Index for the number of components in a system. Index of component to which membrane is permeable in Equation [15]). General index. Designates value determined by pore flow model. Denotes permiate, when in reference to variable defined across a phase boundary. Denotes retinate, when in reference to variables defined across a phase boundary. Designates solute permeable through membrane. Designates value associated with sucrose solute. su Designates value associated with sucrose solute. Designates volumetric value. 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