Article No : a16_187
Membranes and Membrane
Separation Processes, 1. Principles
HEINRICH STRATHMANN, Institut f€ur chemische Verfahrenstechnik, Universit€at
Stuttgart, Germany
1.
1.1.
1.2.
1.3.
1.4.
2.
2.1.
2.1.1.
2.1.2.
2.1.3.
2.2.
2.2.1.
2.2.2.
2.2.3.
2.2.4.
2.2.5.
2.2.6.
2.3.
2.4.
2.5.
2.6.
2.6.1.
2.6.2.
3.
Introduction. . . . . . . . . . . . . . . . . . . . . . . .
Historical Development of Membranes and
Membrane Processes . . . . . . . . . . . . . . . . .
Advantages and Limitations of Membrane
Processes . . . . . . . . . . . . . . . . . . . . . . . . . .
The Membrane-Based Industry . . . . . . . . .
The Future of Membrane Science and
Technology. . . . . . . . . . . . . . . . . . . . . . . . .
Fundamentals . . . . . . . . . . . . . . . . . . . . . .
Definition of Terms . . . . . . . . . . . . . . . . . .
Definition of a Membrane and its Function . .
Definition of the Various Membrane Processes
The Membrane Transport Mechanism. . . . . .
Materials and Structures of Synthetic
Membranes . . . . . . . . . . . . . . . . . . . . . . . .
Symmetric and Asymmetric Membranes. . . .
Porous Membranes . . . . . . . . . . . . . . . . . . .
Homogeneous Membranes . . . . . . . . . . . . . .
Ion-Exchange Membranes . . . . . . . . . . . . . .
Liquid Membranes. . . . . . . . . . . . . . . . . . . .
Fixed-Carrier Membranes . . . . . . . . . . . . . .
Fluxes and Driving Forces in Membrane
Separation Processes . . . . . . . . . . . . . . . . .
Mathematical Description of Mass
Transport in Membrane Processes . . . . . .
Membrane Separation Capability . . . . . . .
Chemical and Electrochemical Equilibrium
in Membrane Processes . . . . . . . . . . . . . . .
Osmotic Equilibrium, Osmotic Pressure,
Osmosis, and Reverse Osmosis . . . . . . . . . .
Donnan Equilibrium and Donnan Potential . .
Principles . . . . . . . . . . . . . . . . . . . . . . . . . .
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421
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422
3.3.2.
3.4.
3.4.1.
3.4.2.
3.5.
3.6.
3.6.1.
3.6.2.
3.6.3.
3.7.
3.7.1.
3.7.2.
422
424
428
3.7.3.
3.7.4.
3.8.
3.9.
428
4.
429
430
431
1. Introduction
The separation, concentration, and purification
of molecular mixtures are major problems in the
chemical industries. Efficient separation processes are also needed to obtain high-grade products
in the food and pharmaceutical industries, to
supply communities and industry with highquality water, and to remove or recover toxic or
valuable components from industrial effluents.
2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
DOI: 10.1002/14356007.a16_187.pub3
3.1.
3.2.
3.3.
3.3.1.
Principle of Microfiltration . . . . . . . . . . . .
Principle of Ultrafiltration . . . . . . . . . . . . .
Principle of Reverse Osmosis. . . . . . . . . . .
Mass Transport in Reverse Osmosis Described
by Phenomenological Equations . . . . . . . . . .
Description of Mass Transport in Reverse
Osmosis by the Solution-Diffusion Model. . .
Principle of Gas Separation . . . . . . . . . . . .
Gas Separation by Knudsen Diffusion . . . . .
Gas Transport by the Solution- Diffusion
Mechanism in a Polymer Matrix . . . . . . . . .
Principle of Pervaporation. . . . . . . . . . . . .
Principle of Dialysis . . . . . . . . . . . . . . . . . .
Dialytic Mass Transport of Neutral
Components . . . . . . . . . . . . . . . . . . . . . . . .
Dialytic Mass Transport of Electrolytes in a
Membrane without Fixed Ions . . . . . . . . . . .
Dialysis of Electrolytes in IonExchange Membranes . . . . . . . . . . . . .
Principle of Electrodialysis . . . . . . . . . . . .
Electrical Current and Ion Fluxes . . . . . . . . .
Transport Number and Membrane
Permselectivity . . . . . . . . . . . . . . . . . . . . . .
Membrane Counterion Permselectivity . . . . .
Water Transport in Electrodialysis . . . . . . . .
Electrodialysis with Bipolar Membranes . .
Energy Requirements of Membrane
Processes . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Symbols . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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For this task a multitude of separation techniques
such as distillation, precipitation, crystallization,
extraction, adsorption, and ion exchange are used
today. More recently, these conventional separation methods have been supplemented by a family of processes that utilize semipermeable membranes as separation barriers. Membranes and
membrane processes were first introduced as an
analytical tool in chemical and biomedical laboratories, and they developed very rapidly into
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Membranes and Membrane Separation Processes, 1. Principles
industrial products and methods with significant
technical and commercial impact [1, 2]. Today,
membranes are used on a large scale to produce
potable water from seawater, to clean industrial
effluents and recover valuable constituents, to
concentrate, purify, or fractionate macromolecular mixtures in the food and drug industries,
and to separate gases and vapors. They are also
key components in energy conversion and storage systems, artificial organs, and drug delivery
devices.
The membranes used in the various applications differ widely in their structure, their function, and the way they are operated in a separation
process. However, all membranes and membrane
processes share several features that make them
particularly attractive tools for the separation of
molecular mixtures. The separation is performed
by physical means at ambient temperature without chemically altering the constituents. This is
important for applications in artificial organs and
in many drug delivery systems, as well as in the
food and drug industries or in downstream processing of bioproducts where temperaturesensitive substances must often be handled. Furthermore, membrane properties can be tailored
and adjusted to specific separation tasks, and
membrane processes are often technically simpler and more energy efficient than conventional
separation techniques and are equally well suited
for large-scale continuous operations as for
batchwise treatment of very small quantities.
A key property of a membrane is its permselectivity, which is determined by differences in
the transport rate of various components through
the membrane. The transport rate of a component
through a membrane is determined by the
structure of the membrane, the size of the permeating component, by the chemical nature and
the electrical charge of the membrane material
and permeating component, and by the driving
force due to the chemical or electrochemical
potential gradients, that is, concentration pressure and electrical potential differences. Some
driving forces such as concentration, pressure,
and temperature gradients act equally on all
components, in contrast to an electrical potential
as driving force, which is only effective with
charged components. The use of different membrane structures and driving forces has resulted in
a number of rather different membrane processes
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such as reverse osmosis, micro- and ultrafiltration, dialysis, electrodialysis, Donnan dialysis,
pervaporation, gas separation, and so on. The
versatility of membrane structures and functions
makes a precise and complete definition of a
membrane rather difficult. In the most general
sense a membrane can be described as a barrier
that separates two different regions and controls
the exchange of matter, energy, and information
between the regions in a very specific way. The
term ‘‘membrane’’ describes two very different
items, that is, biological membranes, which are
part of a living organism, and synthetic membranes, which are man-made. Biological membranes carry out very complex and specific transport tasks in organisms. They accomplish them
quickly, efficiently, and with minimal energy
expenditure, often utilizing carrier-facilitated
and active transport coupled to a chemical reaction. Synthetic membranes are not nearly as
complex in their structure or function as biological membranes. They are usually less selective
and energy-efficient but have significantly higher
chemical, thermal, and mechanical stability. In
practical separation processes biological membranes have no significance today and will therefore not be discussed in this article. Because
membranes are capable of separating molecular
mixtures very selectively and efficiently at room
temperature by physical means their potential
applications are extremely wide compared to
conventional separation processes. Today membranes are used on a large technical scale in three
distinct areas: The first area includes applications
in which the use of membranes is technically
feasible, but where they must compete with
conventional separation processes on the basis
of overall economy. This, for instance, is the case
in seawater desalination and the treatment of
certain wastewater streams. Here, membranes
must compete with distillation and biological
treatment, respectively. The second area includes
applications for which alternative techniques are
available, but membranes offer a clear technical
and commercial advantage. This is the case in the
production of ultrapure water and in the separation of certain food products. Finally, there are
applications where there is no reasonable alternative to membrane processes. This is the case in
certain drug delivery systems and artificial
organs.
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Membranes and Membrane Separation Processes, 1. Principles
1.1. Historical Development of
Membranes and Membrane Processes
Synthetic membranes are a rather recent development and the technical utilization of membrane processes on a large scale began just 40
years ago. The first recorded study of membrane
phenomena and the discovery of osmosis dates
back to NOLLET, who discovered in 1748 [3] that a
pig’s bladder passed preferentially ethanol when
it was brought in contact on one side with a
water – ethanol mixture and on the other side
with pure water. NOLLET was probably the first to
recognize the relation between a semipermeable
membrane, osmotic pressure, and volume flux.
More systematic studies on mass transport in
semipermeable membranes were carried out by
GRAHAM [4] in 1833, who studied the diffusion of
gases through different media and discovered
that rubber exhibits different permeabilities to
different gases. Most of the early studies on
membrane permeation were carried out with
natural materials such as animal intestines, bladders, and gum elastics. In 1867 TRAUBE [5] was
the first to introduce an artificially prepared
semipermeable membrane by precipitating cupric ferrocyanide in a thin layer of porous porcelain. This type of membrane was used by PFEFFER
[6] in his fundamental studies on osmosis. However, the theoretical treatment and much of the
interpretation of osmotic phenomena and mass
transport through membranes is based on the
studies of FICK [7] in 1877, who interpreted
diffusion in liquids as a function of concentration
gradients, and those of VAN T’ HOFF [8] in 1887,
who gave a thermodynamic explanation for the
osmotic pressure of dilute solutions. Later
NERNST [9] in 1889 and PLANCK [10] in 1890
introduced the flux equation for electrolytes under the driving force of a concentration or electrical potential gradient. With the classical publications of DONNAN [11] in 1911, describing the
theory of membrane equilibria and membrane
potentials in the presence of electrolytes, the
early history of membrane science ends with
most of the basic phenomena satisfactorily described and theoretically interpreted. With the
beginning of the 1900s membrane science and
technology entered a new phase. First, BECHHOLD
[12] developed in 1908 a method of making the
first synthetic membranes by impregnating a
415
filter paper with a solution of nitrocellulose in
glacial acetic acid. These membranes could be
prepared and accurately reproduced with different permeabilities by varying the ratio of acetic
acid to nitrocellulose.
Nitrocellulose membranes were also used in
the studies of ZSIGMONDY and BACHMANN [13] in
1918 as ultrafilters to separate macromolecules
and fine particles from an aqueous solution.
These studies were later continued by many
others, including ELFORD [14] in 1930, and
MCBAIN and KISTLER [15] in 1931. The relation
between the streaming potential, electroosmosis,
and electrodialysis was treated in 1931 in a
monograph of PRAUSNITZ and REITSTÖTTER [16].
Based on a 1922 patent of ZSIGMONDY [17],
Sartorius GmbH in Germany began in 1937 the
production of a series of nitrocellulose membranes with various pore sizes. These membranes
were used in microbiological laboratories in
analytical applications. The development of the
first successfully functioning hemodialyzer by
KOLFF and BERK [18] in 1944 was the key to the
large-scale application of membranes in the biomedical area. All synthetic membranes available
up to that time were microporous structures separating mixtures exclusively according to their
size. In the early days of membrane science and
technology membranes had been mainly a subject
of scientific interest with only a very few practical
applications. This changed drastically from 1950
on, when the practical use of membranes in
technically relevant applications became the main
focus of interest and a significant membranebased industry developed rapidly. Progress in
polymer chemistry resulted in a large number of
synthetic polymers which ultimately became
available for the preparation of new membranes
with specific transport properties plus excellent
mechanical and thermal stability. Membrane
transport properties were described by a comprehensive theory based on the thermodynamics of
irreversible processes by STAVERMAN [19] in 1952,
KEDEM and KATCHALSKY [20], SCHLÖGL [21], and
SPIEGLER [22] in 1961. A second route for describing membrane processes was based on postulating
certain membrane transport models, such as the
model of a diffusion-solution membrane by
MERTEN [23] in 1966. The properties of ionexchange membranes were studied extensively
by HELFFERICH [24] in 1962 and many others.
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Membranes and Membrane Separation Processes, 1. Principles
A milestone in membrane science and technology was the development of a reverse osmosis
membrane based on cellulose acetate which provided high salt rejection and high fluxes at moderate hydrostatic pressures by REID and BRETON
[25] in 1959 and LOEB and SOURIRAJAN [26] in
1962, which were major advances toward the
application of reverse osmosis membranes as an
effective tool for the production of potable water
from seawater.
The membrane developed by LOEB and SOURIRAJAN had an asymmetric structure with a dense
skin at the surface, which determined the membrane selectivity and flux, and highly porous
substructure, which provided the mechanical
strength. It was shown by KESTING [27] in 1971
that the preparation of asymmetric cellulose
acetate membrane was based on a phase inversion process in which a homogeneous polymer
solution is converted into a two-phase system,
that is, a solid polymer-rich phase providing the
solid polymer structure and a polymer-lean phase
forming the liquid-filled membrane pores.
STRATHMANN et al. [28] rationalized in 1971 the
formation mechanism of asymmetric membranes
and correlated different structures with the various preparation parameters. Soon, other synthetic polymers such as polyamides, polyacrylonitrile, polysulfone, and polyethylene were used as
basic materials for the preparation of synthetic
membranes. These polymers often showed better
mechanical strength, chemical stability, and thermal stability than the cellulose esters. However,
cellulose acetate remained the dominant material
for the preparation of reverse osmosis membranes until the development of the interfacialpolymerized composite membrane by CADOTTE
[29] and independently by RILEY [30] in 1980.
These membranes showed significantly higher
fluxes, higher rejection, and better chemical and
mechanical stability than the cellulose acetate
membranes.
The first membranes developed for reverseosmosis desalination and other applications were
manufactured as flat sheets and then installed in a
so-called spiral-wound module, based on the
work of BRAY [31] and WESTMORELAND [32] in
1982. A different approach to membrane geometry was taken in the early 1960s by MAHON [33],
who made self-supporting hollow-fiber membranes with a wall thickness of only 6 – 7 mm.
However, the permeate flux was rather low,
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presumably because the fibers were isotropic in
structure. This problem was overcome later by
Du Pont Corporation, which made an asymmetric
hollow-fiber membrane with main application in
desalination of brackish and seawater.
Soon after the development of efficient membranes, appropriate membrane housing assemblies, called modules, were devised. The criteria
for the design of such modules included high
membrane packing density, reliability, ease of
membrane or module replacement, control of
concentration polarization, and low cost. Membranes were produced in three different configurations: as flat sheets, as hollow fine fibers or
capillaries, and as tubes. In modern reverseosmosis desalination plants mainly spiral-wound
modules are used, while hollow-fiber membrane
modules are utilized in gas separation and pervaporation. In medical applications such as artificial kidneys and blood oxygenators, capillary
membranes play a dominant role today. Tubular
membranes are mainly used in micro- and
ultrafiltration.
Even earlier than the large-scale use of reverse
osmosis for desalination of sea- and brackish
water was the industrial-scale application of
electrodialysis. The history of electrodialysis
goes back to the work of MEYER and STRAUSS
[34], who developed the first multicell stack just
before World War II. However, modern electrodialysis became a practical reality with the work
of JUDA and MCRAE [35] in 1953, who developed
the first reliable ion-exchange membranes having both good electrolyte conductivity and ion
permselectivity. Electrodialysis was first commercially exploited for the desalination of brackish water by Ionics Inc. The commercial success
of Ionics was due to their membranes, their
compact stacking, and the mode of operation
referred to as electrodialysis reversal, which
provided a periodic self-cleaning mechanism for
the membrane stack and thus allowed long-term
continuous operation at high concentrations of
scaling materials without mechanical cleaning of
the stack [36].
While Ionics pioneered the development of
electrodialysis for water desalination a completely different use of electrodialysis was envisioned
in Japan. Here, electrodialysis was used for concentrating sodium chloride solutions from seawater to produce table salt, as described by
NISHIWAKI [37]. In the early 1980s a completely
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Membranes and Membrane Separation Processes, 1. Principles
new area for the application of electrodialysis
was opened up. At this time LIU et al. [38]
introduced bipolar membranes for the recovery
of acids and bases from the corresponding salts
by water dissociation induced by an electrical
potential.
The large-scale separation of gases and vapors is also a relevant industrial area for membrane applications. Gas separation was pioneered by HENIS and TRIPODI [39] in 1981.
Originally, the aim was to recover hydrogen
from off-gases and to produce oxygen- or nitrogen-enriched air. Today, however, a large number of other applications such as the removal of
CO2 from natural gas and the recovery of organic vapors from off-gases are typical applications for gas and vapor separation. Pervaporation, which in its basic principle is closely
related to vapor separation, was studied extensively by APTEL and NEEL [40] in 1968, who
pointed out a large number of interesting potential applications, but so far very few large commercial plants have been built.
Other applications of membranes which were
developed in more recent years and have reached
major technical and commercial significance
include the controlled release of drugs in therapeutic devices and the storage and conversion of
energy in fuel cells and batteries.
1.2. Advantages and Limitations of
Membrane Processes
In many applications membrane processes compete directly with the more conventional techniques. However, compared to these conventional
procedures membrane processes are often more
energy efficient, simpler to operate, and yield
higher quality products. Furthermore, the environmental impact of all membrane processes is
relatively low. There are no hazardous chemicals
used in the processes that have to be discharged
and there is no heat generation. However, there
are certain limitations to the application of membrane processes. A major disadvantage of membranes, especially in water- and wastewatertreatment processes, is that until today the
long-term reliability has not completely been
proven. Membrane processes sometimes require
excessive pretreatment due to their sensitivity to
concentration polarization, chemical interac-
417
tion with water constituents, and fouling. Membranes are mechanically not very robust and can
easily be destroyed by a malfunction in the
operating procedure. Another critical issue are
the process costs. In general, membrane processes are quite energy efficient. However, the
energy consumption is only part of the total
process costs. Other factors determining the
overall economics of a process include the
investment-related cost, which is determined
by the cost of the membranes and other process
equipment and their useful life under operating
conditions, and various pre- and post-treatment
procedures of the feed solutions and the products. Plant capacity may also play a role in total
cost. While, for example, in distillation processes usually a substantial cost reduction can be
achieved with an increase in the plant capacity,
the scale-up factor has only a relatively small
effect on the process cost in desalination of
water by reverse osmosis or electrodialysis.
Depending on the composition of the feed solution and the required quality of the product
water, a combination of processes may be appropriate. For example, if ultrapure water for
certain industrial applications is required, a
sequence of processes may be applied, such as
reverse osmosis, ion exchange, UV sterilization,
and microfiltration as a point-of-use-filter to
remove traces of particles.
The environmental impact of all membrane
processes is relatively low. There are no hazardous chemicals used in membrane processes
which have to be discharged. In certain applications such as the purification of industrial effluents and wastewaters or the desalination of
brackish water there may be a problem with the
disposal of the concentrate. In these applications brine post-treatment procedures may be
necessary.
1.3. The Membrane-Based Industry
Parallel to the development of membranes and
membrane processes was the development of a
membrane-based industry. The structure of this
industry is quite heterogeneous as far as the size
of the companies and their basic concept towards
the market is concerned. Several companies have
concentrated on the production of membranes
only. Membrane producers are frequently
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Membranes and Membrane Separation Processes, 1. Principles
divisions of major chemical companies. They
offer a range of membrane products as flat sheets,
hollow fibers, or capillaries with different properties and for different applications ranging from
seawater desalination and hemodialysis to
purification of wastewater and fuel cells. Other
companies buy membranes or modules as key
components from one or several membrane manufacturers, design and build the actual plant, and
often also operate it, guaranteeing the customer a
certain amount of product of a given quality.
Sometimes, these companies provide different
processes, such as reverse osmosis, nanofiltration, and electrodialysis according to the requirements of the separation task. These companies
provide a solution to a customer’s separation
problem in general, which might be a combination of separation processes such as ion exchange, carbon adsorption, flocculation and precipitation, or various chemical and biological
treatment procedures in addition to membrane
processes. Although the sales of membranes and
membrane modules to any one of these companies are often not very large they are of importance in the membrane industry because of their
specific application know-how in different markets. Finally, there are companies that provide
the membranes, the system or product design,
and the operation of the process. The companies
concentrate very often on a single, usually very
large application such as production of potable
water from sea- or brackish water or hemodialysis. They often not only provide the tools for
producing potable water in the case of seawater
desalination, but also operate the plant and
distribute the water. Companies producing artificial kidneys also operate dialysis stations. The
membrane market is characterized by a few
rather large market segments, such as sea- and
brackish-water desalination, production of ultrapure water, and hemodialysis and a large
number of small market segments in the food,
chemical, and pharmaceutical industries, analytical laboratories, and especially in the treatment and recycling of industrial wastewater
streams. The larger markets for water desalination and hemodialysis are dominated by a relatively small number of large companies. A
multitude of small companies are active in
market niches such as treating certain wastewater streams or providing services to the chemical
or food and drug industries.
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1.4. The Future of Membrane Science
and Technology
In many applications today’s membranes and
processes are quite satisfactory, while in other
applications there is a definite demand for further
improvements of both membranes and processes.
For desalination of sea- and brackish water by
reverse osmosis, for example, there are membranes available today that are quite satisfactory
as far as flux and salt rejection are concerned, and
the processes have been proven by many years of
operating experience. The same is true for hemodialyzers and hemofiltration. In these applications
only marginal improvements can be expected in
the near future. In micro- and ultrafiltration and
electrodialysis the situation is similar. The properties of current membranes are satisfactory.
However, there are other components such as the
process design, application know-how, and longterm operating experience that are of importance
in the use of micro- and ultrafiltration in the
chemical and food industries and in wastewater
treatment. Here, concentration polarization and
membrane fouling play a dominant role, and new
membrane modules and process design concepts
which provide a better control of membrane
fouling and thus result in a longer useful life of
the membranes are highly desirable. In other
membrane processes such as gas separation, pervaporation, fuel cell separators, and membrane
reactors, the situation is quite different. Here,
better membranes, improved process design and
extensive application know-how and long-term
experience are mandatory to establish membrane
processes as a proven and reliable technology.
2. Fundamentals
Membrane processes such as dialysis, electrodialysis, reverse osmosis, and micro- and ultrafiltration differ widely as far as the applied membrane structures and the driving forces for mass
transport are concerned. There are, however,
some common fundamentals, such as certain
thermodynamic and kinetic relations describing
mass transport in membranes and at the membrane/bulk solution interface. Phenomena such
as concentration polarization and membrane
fouling occur in all membrane processes and
have common causes and consequences. Thus,
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Membranes and Membrane Separation Processes, 1. Principles
before treating the different membrane processes
and their application in more detail, some common fundamentals are discussed.
419
determined by driving forces such as concentration, pressure, temperature, and electrical potential gradients, and the concentration and mobility
of the component in the membrane matrix.
2.1. Definition of Terms
The performance of membrane processes in a
certain application is determined by a number of
components and process parameters such as the
membranes, the membrane modules, the driving
forces applied to achieve the desired mass transport and certain chemical engineering aspects
relevant for designing processes. These components and process parameters and their function
must be defined.
2.1.1. Definition of a Membrane and its
Function
A precise and complete definition of a membrane
that covers all its aspects is rather difficult, even
when the discussion is limited to synthetic structures as in this article. In the most general sense, a
synthetic membrane is an interphase that separates two phases and restricts the transport of
various components in a specific manner [1]. A
membrane can be homogeneous or heterogeneous, symmetric or asymmetric in structure. It
may be solid or liquid and may consist of organic
or inorganic materials. It can be neutral or can
carry positive or negative charges, or functional
groups with specific binding or complexing abilities. Its thickness can be less than 100 nm or
more than one millimeter. The electrical resistance may vary from more than 106 W cm2 to less
than 1 W cm2. The term ‘‘membrane’’ therefore
includes a great variety of materials and structures, and a membrane can often be better described by its function rather than by its structure.
Some materials, such as protective coatings, or
packaging materials, though not meant to be
membranes, show typical membrane properties,
and are in fact membranes. All materials functioning as membranes have one characteristic
property in common: They restrict the passage
of different components in a very specific manner.
Separation of a mixture in a membrane process is
the result of different transport rates of different
components through the membrane. The transport
rate of a component through a membrane is
2.1.2. Definition of the Various Membrane
Processes
Membrane processes can be grouped according
to the applied driving forces into pressure-driven
processes such as reverse osmosis or ultra- and
microfiltration, concentration gradient driven
processes such as dialysis and Donnan dialysis,
and electrical potential driven processes such as
electrodialysis. The molecular mixture to be
separated is referred to as feed, the mixture
containing the components retained by the membrane is called the retentate, and the mixture
composed of the components that have permeated the membrane is referred to as permeate or
filtrate in micro- and ultrafiltration.
2.1.3. The Membrane Transport
Mechanism
The mechanism by which certain components are
transported through a membrane can also be very
different. In some membranes, for example, the
transport is based on viscous flow of a mixture
through individual pores in the membrane caused
by a hydrostatic pressure difference between the
two phases separated by the membrane. This type
of transport is referred to as viscous flow. It is the
dominant form of mass transport in micro- and
ultrafiltration but also occurs in other membrane
processes. If the transport through a membrane is
based on the dissolution and diffusion of individual molecules in the membrane matrix due to
a concentration or chemical potential gradient
the transport is referred to as diffusion. This form
of mass transport is dominant in reverse osmosis,
gas separation, pervaporation, and dialysis but
may occur in other processes, too. If an electrical
potential gradient across the membrane is applied
to achieve the desired transport of certain components through the membrane the transport is
referred to as migration. Migration occurs in
electrodialysis and related processes and is limited to the transport of components carrying
electrical charges, such as ions.
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Membranes and Membrane Separation Processes, 1. Principles
2.2. Materials and Structures of
Synthetic Membranes
Synthetic membranes show a large variety in
their physical structure and the materials they
are made from. Based on their structure they can
be classified into four groups:
.
.
.
.
Porous membranes
Homogeneous solid membranes
Solid membranes carrying electrical charges
Liquid or solid films containing selective
carriers
Furthermore, the structure of membranes may
be symmetric, that is, the structure is identical
over the entire cross section of the membrane, or
it may be asymmetric, that is, the structure varies
over the cross section of the membrane.
The materials used for the preparation of
membranes can be polymers, ceramics, glass,
metals, or liquids. The materials may be neutral
or carry electrical charges (i.e., fixed ions). The
schematic in Figure 1 illustrates the morphology
and materials of some technically relevant synthetic membranes.
2.2.1. Symmetric and Asymmetric
Membranes
As indicated earlier synthetic membranes may
have a symmetric or an asymmetric structure. In a
symmetric membrane the structure and the trans-
Vol. 22
port properties are identical over the entire cross
section, and the thickness of the entire membrane
determines the flux. Symmetric membranes are
used today mainly in dialysis and electrodialysis.
In asymmetric membranes structural and transport properties vary over the membrane cross
section. An asymmetric membrane consists of a
0.1 – 1 mm thick ‘‘skin’’ layer on a highly porous
100 – 200 mm thick substructure. The skin represents the actual selective barrier of the asymmetric membrane. Its separation characteristics
are determined by the nature of the material or the
size of its pores. The mass flux is determined
mainly by the skin thickness. The porous sublayer serves only as a support for the mostly thin
and fragile skin and has little effect on the
separation characteristics or the mass transfer
rate of the membrane. Asymmetric membranes
are used primarily in pressure-driven membrane
processes such as reverse osmosis, ultrafiltration, and gas and vapor separation, since here
the unique properties of asymmetric membranes, that is, high fluxes and good mechanical
stability, can best be utilized. Two techniques
are used to prepare asymmetric membranes: one
utilizes the phase-inversion process, which
leads to an integral structure with the skin and
the support structure made from the same material in a single process [27], and the other
resembles a composite structure where a thin
barrier layer is deposited on a porous substructure in a two-step process [29]. In this case
barrier and support structures are generally
made from different materials.
Figure 1. Schematic illustrating the various materials and structures of technically relevant synthetic membranes
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
2.2.2. Porous Membranes
A porous structure represents a very simple form
of a membrane, which closely resembles the
conventional fiber filter as far as the mode of
separation is concerned. These membranes consist of a solid matrix with defined holes or pores
which have diameters ranging from less than
1 nm to more than 10 mm. Separation of the
various components is achieved strictly by a
sieving mechanism with the pore diameters and
the particle sizes being the determining parameters. Porous membranes can be made from
various materials such as ceramics, graphite,
metals, metal oxides, and various polymers.
Their structure can be symmetric, that is, the
pore diameters do not vary over the membrane
cross section, or asymmetric, that is, the pore
diameters increase from one side of the membrane to the other, typically by a factor of 10 to
1000. The techniques for the preparation of
porous membranes can be rather different and
include simple pressing and sintering of polymer
or ceramic powders, irradiation and leaching of
templates, as well as phase-inversion and polymer-precipitation procedures or sol – gel conversion techniques. Porous membranes are used
to separate components that differ markedly in
size or molecular weight in processes such as
micro- and ultrafiltration or dialysis [41].
2.2.3. Homogeneous Membranes
A homogeneous membrane is merely a dense
film through which a mixture of molecules is
transported by a pressure, concentration, or electrical potential gradient. The separation of the
various components of a mixture is directly
related to their transport rates within the membrane phase, which is determined by their diffusivities and concentrations in the membrane
matrix. Therefore, homogeneous membranes are
referred to as solution – diffusion-type membranes [23]. They can be prepared from polymers, metals, metal alloys, and, in some cases,
ceramics, which may also carry positive or negative electrical charges. Since the mass transport
in homogeneous membranes is based on diffusion, their permeabilities are rather low. Homogeneous membranes are used mainly to separate
components which are similar in size but have
421
different chemical nature in processes such as
reverse osmosis, gas and vapor separation, and
pervaporation [42]. In these processes asymmetric membrane structures are used which consist
of a thin homogeneous skin supported by a
porous substructure.
2.2.4. Ion-Exchange Membranes
Films carrying charged groups are referred to as
ion-exchange membranes. They consist of highly
swollen gels carrying fixed positive or negative
charges. The properties and preparation procedures of ion-exchange membranes are closely
related to those of ion-exchange resins [24].
There are two different types of ion-exchange
membranes: 1) cation-exchange membranes,
which contain negatively charged groups fixed
to the polymer matrix, and 2) anion-exchange
membranes, which contain positively charged
groups fixed to the polymer matrix. In a cation-exchange membrane, the fixed anions are in
electrical equilibrium with mobile cations in the
interstices of the polymer. In contrast, the mobile
anions are more or less completely excluded
from the cation-exchange membrane because of
their electrical charge, which is identical to that
of the fixed ions. Due to this exclusion of the
anions, a cation-exchange membrane permits
transfer of cations only. Anion-exchange membranes carry positive charges fixed on the polymer matrix. Therefore, they exclude all cations
and are permeable only to anions. Although there
are a number of inorganic ion-exchange materials, mostly based on zeolites and bentonites,
these materials are rather unimportant in ionexchange membranes compared to polymer materials [24]. The main applications of ion-exchange membranes are in electrodialysis and
electrolysis. They are also used as ion-conducting separators in batteries and fuel cells.
2.2.5. Liquid Membranes
Liquid membranes are mainly used in combination with the so-called facilitated transport which
is based on ‘‘carriers’’ which transport certain
components such as metal ions selectively across
the liquid membrane interphase. Generally, it is
no problem to form a thin fluid film. It is difficult,
422
Membranes and Membrane Separation Processes, 1. Principles
however, to maintain and control this film and its
properties during a mass separation process. To
avoid breakup of the film, some type of reinforcement is necessary to support such a weak
membrane structure. Two different techniques
are used today for the preparation of liquid
membranes. In the first, the selective liquid barrier material is stabilized as a thin film by a
surfactant in an emulsion-type mixture [43]. In
the second, a porous structure is filled with the
liquid membrane phase. Both types of membranes are used today on a pilot-plant scale for
the selective removal of heavy metal ions or
certain organic solvents from industrial waste
streams [44]. They have also been used rather
effectively for the separation of oxygen and
nitrogen.
2.2.6. Fixed-Carrier Membranes
Fixed-carrier membranes consist of a homogeneous or porous structure with functional groups
which selectively transport certain chemical
compounds. Fixed-carrier membranes can have
a symmetric or asymmetric structure depending
on their application. They are used today, for
example, in co- and countercurrent transport and
in the separation of alkane – alkene mixtures.
2.3. Fluxes and Driving Forces in
Membrane Separation Processes
Separation in membrane processes is the result of
differences in the transport rates of different
chemical species through the membrane. The
transport rate is determined by the driving force
or forces acting on the individual components
and their mobility and concentration in the membrane. The mobility of a component in the membrane is primarily determined by its size and the
physical structure of the membrane material,
while the concentration of the solute in the
membrane is primarily determined by the chemical compatibility of the permeating component
and the membrane material. In membrane separation processes there are different forms of mass
transport as indicated in Figure 2, which show
schematically the mass transport through membranes separating two homogeneous phases.
Vol. 22
In the most general form mass transport
through a membrane is shown in Figure 2a, in
which the membrane acts as barrier that restricts
the flux of various components due to their
chemical nature, size, or electrical charge. The
transport of a component A is caused by a driving
force across the membrane acting on this component. The driving force is a difference in
energy of the component A in the two phases
separated by the membrane. Thus, the driving
force for the transport of a component through a
membrane can be expressed in a very general
form by the energy gradient, that is, the gradient
in its electrochemical potential across the membrane. Gradients in the electrochemical potential
of a component are the result of hydrostatic
pressure, concentration, electrical potential, or
temperature differences between the two phases
separated by the membrane [46].
In the example illustrated in Figure 2a it is
assumed that the membrane is just a passive
physical barrier without specific interactions between the permeating components and the membrane matrix. If the transport of certain components is facilitated due to specific interactions
with a carrier component in the membrane matrix
the process is referred to as facilitated transport.
There are two types of membranes with carrierfacilitated transport properties. The first type is a
membrane with the carriers fixed to the membrane matrix where the transported component
moves through the membrane by ‘‘hopping’’
from one carrier site in the membrane to the next
[45]. The second type is a liquid membrane with
the mobile carrier dissolved in a liquid film. The
process i illustrated in Figure 2b. Here, the
transported components are coupled to the carrier
on one side of the membrane, that is, the side
facing the feed solution. Then they are transported through the membrane coupled to the
carrier by diffusion. This facilitated transport is
generally more selective than transport through a
passive membrane. However, to achieve a flux of
a component an electrochemical potential gradient is also required in a membrane with facilitated transport properties.
There may also be a coupling of the fluxes of
individual components. One example of the coupling of fluxes is the transport of bound water
with an ion which is transported across a membrane by an electrical potential gradient. More
important in membrane processes is the coupling
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
423
Figure 2. Schematic illustrating various modes of mass transport in membranes
of ion fluxes which results when electrically
charged components have different permeability
in a membrane. Here, the coupling of fluxes is the
result of the electroneutrality condition, which
requires that on a macroscopic scale there is no
excess of electrical charges. If there is no electrical current, positively or negatively charged
components can only be transported across a
membrane when the same number of opposite
charges carried by counterions are transported in
the same direction, or if components with the
same charges carried by co-ions are transported
in the opposite direction. Thus two forms of flux
coupling can be distinguished: cocurrent coupled
transport when co- and counterion are transported in the same direction, and countercurrent
coupled transport when no counterions permeate
the membrane and co-ion fluxes are in opposite
direction. The transport of various ions through a
membrane by coupling with co- or counterions is
illustrated in Figure 3.
In the cocurrent transport shown schematically in Figure 3a component A is an anion, and Bþ
and Cþ are cations. The concentrations of the
components A and Bþ in phase0 is higher than in
phase00 and the concentration of the component
Cþ in phase0 is lower than that in phase00 . It is
assumed that the membrane is an anion-exchange
membrane and thus permeable for the anion A.
Cation Cþ can permeate the membrane only by
means of a specific carrier, and the cation Bþ can
not permeate the membrane at all. Due to the
concentration differences of the components in
the two phases there is a strong driving force for
anion A to permeate through the membrane
from phase0 into phase00 . However, because of
the electroneutrality requirement it must be accompanied by a cation. The only cation capable of
permeating the membrane due to a specific carrier
is Cþ, which will be transported against its chemical potential gradient from a lower concentration
in phase0 to a higher concentration in phase00 .
424
Membranes and Membrane Separation Processes, 1. Principles
Vol. 22
Figure 3. Schematic illustrating a) cocurrent and b) countercurrent coupled transport
In the countercurrent coupled transport illustrated in Figure 3b a negatively charged cationexchange membrane separates phase0 and
phase00 . The concentration of all components in
phase0 is assumed to be higher than that in phase00 .
Thus, there is a driving force for all components
to permeate the membrane from phase0 into
phase00 . Since anion A can not permeate the
cation-exchange membrane, cation Cþ can only
move from phase0 into phase00 if the same number
of electrical charges of component Bþ are transported in the opposite direction against its concentration gradient. Both cocurrent coupled
transport, which is usually referred to as diffusion
dialysis, and countercurrent coupled transport,
which is also called Donnan dialysis, are of
practical relevance in wastewater treatment.
2.4. Mathematical Description of Mass
Transport in Membrane Processes
Mass transfer in membrane processes is determined by the driving forces acting on the individual components in the membrane and by the
friction which the components must overcome
when moving through the membrane matrix. The
driving forces acting on the components of a
system can be expressed by gradients in their
electrochemical potential and by a hydrostatic
pressure gradient causing a bulk volume flow.
The friction that must be overcome by the driving
force or forces to achieve the transport of the
component is generally expressed by a hydrodynamic permeability coefficient, a diffusion coefficient, or, in electrolyte solutions, also by the
electrical resistance. To describe the mass transport through a membrane the thermodynamic
(i.e., driving forces) and kinetic parameters
(i.e., the friction coefficients) must be mathematically related.
In the most general form the mass transport
through a membrane can be described by:
Ji ¼ Pi
dXi
dz
ð1Þ
where J is the flux, P a general permeability
coefficient, gradient dX/dz represents the driving
force, and subscript i refers to a component.
The driving force for the transport of a component through the membrane can be expressed
by the gradient of its electrochemical potential in
the membrane matrix, which is given by:
dXi ¼ dhi ¼ dmi þzi Fdj ¼ V i dpþRTd ln ai þzi Fdj
ð2Þ
where dh and dm are the gradients in the electrochemical and the chemical potential; dp, da,
and dj the pressure, the activity, and electrical
potential gradients across the membrane; V is the
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
partial molar volume; F the Faraday constant; R
the gas constant; z the charge number, which is
also referred to as the valence of a charged
component; and subscript i refers to component i.
In the most general form the flux of a component through a membrane can be described by
inserting Equation (2) into Equation (1):
d ðV d pþRT ln ai þzi F d jÞ
dz i
Ji ¼ Pi
ð3Þ
Equation (3) describes the flux of a component i
through a membrane in reference to any other
component in the membrane. If the membrane is
used as frame of reference and there is convective
flow, for example, through defined membrane
pores this convective flow must be taken into
account and an additional term must be added in
Equation 3:
d ðV d pþRTlnai þzi F d jÞþJv Ci
dz i
Ji ¼ Pi
ð4Þ
where Jv is the convective flux in a membrane
pore, and Ci the concentration of component i in
the solution or mixture permeating the membrane
by convective flow.
Note that the gradient of the chemical or
electrochemical potential always refers to the
membrane phase. It is usually assumed that the
membrane phase is in equilibrium with the adjacent outside phases, that is, the electrochemical
potentials in the membrane hm and in the adjacent phase h0 are identical. This, however, is not
true for the concentration of a component, which
in the membrane is often very different from that
in the adjacent phase. The concentration in the
membrane is related to that in the adjacent phase
by:
C m ¼ kC
0
ð5Þ
where k is a distribution coefficient of a component between an outside phase and the membrane
matrix, which in homogeneous polymer membranes is identical with the solubility coefficient.
Equations (3) and (4) are complete but rather
complex descriptions of the transport of a component through a membrane in which three different
additive forms of transport can be distinguished,
that is, convection, diffusion, and migration.
Convection is a movement of mass due to a
mechanical force, generally a hydrostatic pressure difference. Convective transport is generally
425
less important in homogeneous solid materials
such as ion-exchange membranes. It is, however,
dominant in porous structures and stirred
solutions.
Diffusion is a flux of molecular components
due to a local gradient in the chemical potential.
Gradients in the chemical potential of a component under isothermal conditions can arise from
gradients in concentration or pressure. The dominant term for molecules or ions transported
through an ion-exchange membrane or in a solution by diffusion is the concentration gradient.
For the diffusion of ions it is mandatory that the
electroneutrality requirement is fulfilled, that is,
cations and anions have to diffuse in the same
direction.
Migration is a movement of ions due to an
electrical potential gradient. An electrical potential gradient is usually established by applying a
voltage difference between two electrodes dipping into an electrolyte solution. The positively
charged cations migrate towards the negatively
charged cathode and the negatively charged anions migrate towards the positively charged anode, that is, cations and anions are transported in
opposite directions.
Convection, diffusion, and migration are traditionally described by phenomenological equations that relate the fluxes to the corresponding
driving forces such as the Hagen – Poiseuille
law, Fick’s law, and Ohm’s law.
For instance, a flux of mass caused by a
hydrostatic pressure gradient is described by the
Hagen – Poiseuille law:
Jv ¼ Lp
dp
½ms1 dz
ð6Þ
A flux of individual components caused by a
concentration gradient is described by Fick’s
law:
Ji ¼ Di
dC mol m2 s1
dz
ð7Þ
A flux of electrical charges caused by an electrical potential gradient is described by Ohm’s law:
Je ¼ k
dj
½A m2 dz
ð8Þ
where C is the concentration, p the pressure, j the
electrical potential, D the diffusion coefficient,
426
Membranes and Membrane Separation Processes, 1. Principles
Lp the hydrodynamic permeability, k the electrical conductivity, and the subscripts v, i, and e
refer to volume, individual components, and
electrical charges.
Unfortunately, the simple linear relations expressed in Equations (6) to (8) are not suited to
describe the mass transport in many of the technically relevant membrane processes. First, the
conditions in the membrane are generally not
ideal, and the concentration of individual components must be replaced by their activities.
Furthermore, in many membrane processes mass
transport is the result of a combination of driving
forces; for example, in reverse osmosis and
ultrafiltration the mass flux is caused by pressure
and concentration gradients, and in electrodialysis the flux is due to an electrical potential and a
concentration gradient. In these processes transport must be described by more complex relations based on Equation (3). Only in membrane
processes such as microfiltration, where the mass
transport is the result of only a single driving
force (i.e., a hydrostatic pressure gradient), or in
dialysis, where the driving force is a concentration gradient, can Equation (3) be replaced by a
simpler linear relation between one driving force
and the corresponding flux, such as the Hagen –
Poiseuille law or Fick’s law.
Furthermore, in membrane processes driving
forces and fluxes may be interdependent, giving
rise to new effects. For instance, a concentration
or activity gradient across a membrane may not
only result in diffusion of certain components
through the membrane. Under certain conditions
it can also cause the buildup of a hydrostatic
pressure difference across the membrane. This
phenomenon is referred to as osmosis and the
generated pressure is referred to as osmotic
pressure. Similarly, a gradient in hydrostatic
pressure may not only lead to volume flux, but
may also result in the buildup of a concentration
difference across the membrane. This phenomenon is called reverse osmosis. An electrical potential gradient may not only result in an electric
current but also in the transport of individual ions
and buildup of a hydrostatic pressure. These
phenomena are referred to as electrodialysis and
electroosmosis.
In practice mass transport in membranes can
be described by postulating certain membrane
models, such as the pore model and the solutiondiffusion model. In these models any kinetic
Vol. 22
coupling between individual fluxes is neglected.
A more comprehensive treatment of the mass
transport in multicomponent mixtures is based on
the thermodynamics of irreversible processes.
Here, the fluxes of individual components are
described in a set of phenomenological equations
which relate the fluxes of electrical charges,
momentum, and individual components to the
driving forces and which also take a coupling of
different fluxes into account. Another approach
to describe mass transport in membranes, suggested by Stefan – Maxwell, also describes the
transport of the individual components in a set of
phenomenological equations.
Phenomenological Description of Mass
Transport. A flux of individual components,
heat, and electrical charges can be expressed in
the following general equation [47]:
Ji ¼
X
Lik Xk ði; k ¼ 1; 2; 3; . . . ; nÞ
ð9Þ
k
Here Ji refers to fluxes of individual components,
heat and electric charges, Xk refers to the driving
forces related to the fluxes, and Lik is a phenomenological coefficient relating flux and driving
force.
For multicomponent systems with fluxes of
heat, electric charges, and neutral components
Equation (9) can be written as a matrix. The
diagonal coefficients relate the fluxes to the
corresponding driving forces, and the cross-coefficients express the coupling of fluxes with indirectly conjugated driving forces [47].
Membrane processes are irreversible. Therefore, the entropy increases during the process.
Based on the thermodynamics of irreversible
processes it can be shown that the diagonal
coefficients are always positive, while the
cross-coefficients may be positive or negative
and their number is reduced by the so-called
Onsager relation [48].
To describe the transport in a multicomponent
system such as an electrolyte solution with gradients in hydrostatic pressure, chemical potential, and electrical potential as driving forces, a
very large number of phenomenological coefficients must be known which may be a function of
temperature, pressure, or composition but which
are by definition independent of the driving
forces. This limits the practical use of Equation
(9) substantially. However, under practical conditions many of the phenomenological coupling
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
coefficients can be neglected and the rather complex set of equations can be reduced to more
simple relations such as Fick’s law or Ohm’s law.
Another approach to describe mass transport
in membrane processes is based on a relation
developed by Maxwell and Stefan. This approach
is based on the assumption that at a constant flux
of a component i the driving forces acting on this
component are equal to the sum of the friction
forces between the component i and other components in the system [49]. If viscous flow (i.e.,
convection) is excluded the Stefan – Maxwell
equation is given by [50]:
Xi ¼
X fik
X RT
ðvi vk Þ ¼
ðvi vk Þ
C
Ðik Ci
i
k
k
ð10Þ
where X is the driving force, C the concentration,
v the linear velocity, f the friction coefficient, Ð
the Stefan – Maxwell diffusion coefficient, and
subscripts i and k refer to individual components.
The flux of an individual component is always
proportional to its velocity relative to the velocity
of another component which is used as a frame of
reference. Thus, the number of independent
fluxes in a given system is equal to the total
number of components in the system minus the
one that is used as reference. A system containing
a membrane with n components, including the
membrane, has n1 independent fluxes.
The flux of component i relative to that of
component k which serves as a frame of reference
is given by:
Ji ¼ Ci ðvi vk Þ
ð11Þ
In membrane processes the fluxes through the
membranes are of interest. Therefore, the membrane is used as frame of reference and its
velocity is assumed to be zero.
Combination of Equations (9) to (11) provides
the relation between the various terms of the
phenomenological equation and the Stefan –
Maxwell equation:
X
i
Lik ¼
X Ci X Ðik
¼
Ci
fik
RT
i
i
ð12Þ
Equation (11), that is, the Stefan–Maxwell relation, provides the same complete description of
transport processes through a membrane separating two homogeneous mixtures as does the
phenomenological Equation (9) based on the
thermodynamics of irreversible processes. The
427
only further boundary condition relevant for a
system which contains electrically charged components is the electroneutrality requirement and
the conservation of electrical charges. For all
phenomena observed in membrane systems, such
as osmosis, electro-osmosis, diffusion of individual components, viscous flow of a bulk solution,
electric current, or the buildup of an osmotic
pressure, a streaming and a diffusion potential
can be described by applying Equation (9) or
(10). However, as mentioned before, the practical value of these equations is rather limited since
they are only applicable close to equilibrium
because of the assumed linear relationships between fluxes and driving forces and the large
number of different coefficients that must be
known when multicomponent systems with viscous flow, diffusion, and migration are to be
treated. But even if the phenomenological treatment of mass transport in membrane processes is
of little practical use, it indicates how complex
the processes in multicomponent systems and
heterogeneous membranes really are. It shows
that through the coupling of fluxes a transport of
components can be obtained without a directly
related driving force. The magnitude of the
transport of certain components obtained through
coupling with other fluxes depends on the coupling coefficient. In many cases the coupling
between fluxes can be neglected. In some cases,
however, it can be significant. For instance, in
electrodialysis the coupling of water with an ion
flux can lead to significant water transport
through a membrane. A more simplified approach to describe mass transport in electrolyte
solution is provided by the so-called Nernst –
Planck flux equation which can be derived from
Equation (9) or (10) by introducing the electrochemical potential as driving force for mass
transport and by neglecting all other kinetic
coupling between individual components.
The electrochemical potential h is given by
the chemical potential m and the electrical potential j:
hi ¼ mi þzi F j ¼ moi þV i pþRT ln ai þzi F j
ð13Þ
where a is the activity, p the hydrostatic pressure,
V the partial molar volume, z the valence, F the
Faraday constant, R the gas constant, T the
absolute temperature, subscript i refers to a component, and superscript o to a standard state.
428
Membranes and Membrane Separation Processes, 1. Principles
Introducing the activity coefficient and the ion
concentration for the ion activity into Equation
(13) and differentiation in transport direction z,
that is, perpendicular to the membrane surface,
leads to:
dhi
dp
d ln ðCi g i Þ
dj
¼ V i
þRT
þzi F
dz
dz
dz
dz
ð14Þ
where C is the concentration, and g the activity
coefficient.
In electrodialysis and other ion-exchange
membrane separation processes the pressure difference across the membrane is kept as low as
possible to minimize viscous flow. Therefore, the
term Vi dp
dz in Equation 14 can generally be neigiÞ
can be expressed
glected and the term RT d lnðC
dz
d
ln
g
d Ci
i
by RT
Ci 1þCi d Ci
dz.
d hi
RT
d ln g i d Ci
dj
¼
þFzi
1þCi
dz
d Ci
dz
Ci
dz
ð15Þ
Introducing the gradient of the electrochemical
potential of a component in the membrane expressed in Equation (15) as driving force Xi into
Equations (10) and (11) gives the anion and
cation fluxes of a completely dissociated salt:
1þCi
dlng i
dCi
dCi zi Ci F d j
þ
þvk Ci
dz
RT dz
ð16Þ
If the solution is assumed to be ideal, that is, the
activity of a component is identical to its concentration, the Stefan – Maxwell diffusion coefficient can be replaced by Fick’s diffusion coefficient, and Equation (16) is simplified to:
Ji ¼ Di
d Ci
zi Ci F d j
Di
þvk Ci
dz
RT d z
ð17Þ
Equation (17) represents the extended Nernst –
Planck flux equation where Di is Fick’s diffusion
coefficient of component i relative to the water in
the membrane.The first term Di ddCzi in Equation
(17) represents diffusion, the second term
Ci F d j
Di ziRT
d z migration, and the third term vk Ci
convection.
Fick’s diffusion coefficient is related to the
Stefan – Maxwell diffusion coefficient and the
phenomenological coefficient by:
Di ¼ L i
and
d ln g i
Di ¼ Ði 1þ
d ln Ci
ð19Þ
The Nernst – Planck equation becomes identical
with the phenomenological equation of the thermodynamics of irreversible processes and the
Stefan – Maxwell equation if kinetic coupling
between individual ions is neglected and if the
activity coefficient is assumed to be unity. The
Nernst – Planck flux equation is a very convenient approach to describe the transport of ions in
ion-exchange membranes. It must, however, be
realized that a series of assumptions has been
made which in practical ion-exchange membrane
separation processes are not always applicable.
2.5. Membrane Separation Capability
Thus:
Ji ¼ Ði
Vol. 22
RT
d ln g i
1þ
d ln Ci
Ci
ð18Þ
In most technically relevant applications the
capability of a membrane to separate different
components from each other, that is, the membrane permselectivity, is the most important
membrane property. This separation capability
can be expressed by the selectivity or by the
rejection [51].
The membrane selectivity is given by:
SPA;B ¼
PA
PB
ð20Þ
and the membrane rejection by:
RA ¼
0 C
1 A00
CA
ð21Þ
where SPA;B is the permselectivity of a membrane
for the components A and B, and PA and PB are
their permeability; RA is the rejection of component A by the membrane; CA0 and CA00 are the
concentrations of component A in the two phases
separated by the membrane, for instance, a feed
solution and a filtrate when a filtration process is
considered.
2.6. Chemical and Electrochemical
Equilibrium in Membrane Processes
Equilibrium between two systems such as a
membrane and an adjacent solution or two solutions separated by a membrane is obtained when
the systems are in the same state, that is, when the
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
Gibbs free energies in both systems are identical.
The Gibbs free energy of a system is a function of
various state variables such as pressure, electrical
potential, and the activity of individual components. Consider a system separated by a membrane which is impermeable to some components
and permeable to others. These two systems may
well be in equilibrium even if they have differences in pressures, electrical potentials, or the
activities of individual components as long as
these differences compensate each other and the
sum of the state variables (i.e., the Gibbs free
energy) in both systems is identical. Examples
for two such equilibria are the osmotic equilibrium, in which pressure and activity differences
compensate each other, and the Donnan equilibrium which is attained when an electrical potential difference is compensated by an activity
difference. For membrane processes the osmotic
and Donnan equilibrium are both of importance
and will be discussed in more detail.
2.6.1. Osmotic Equilibrium, Osmotic
Pressure, Osmosis, and Reverse Osmosis
If two aqueous salt solutions of different concentrations are separated by a membrane which is
permeable for the solvent (e.g., water) but impermeable for the solute (e.g., salt), transport of
water from the more dilute solution to the more
concentrated solution is observed. This phenomenon (Fig. 4) is referred to as osmosis.
The figure shows a schematic diagram of a
membrane separating two solutions consisting of
429
a solvent and a solute, a salt. The two solutions
are indicated by 0 and 00 . The membrane is assumed to be permeable to the solvent, but impermeable to the solute. Depending on the concentrations and hydrostatic pressures in the two
phases separated by the membrane four different
situations can be distinguished:
1. The hydrostatic pressures in the two phases
separated by the membrane are equal but the
solute concentration in solution0 is higher than
that in solution00 . In this case the osmotic
pressure in solution0 is higher than that in
solution00 and flow of solvent occurs from the
more dilute solution into the more concentrated solution. This situation is referred to as
osmosis.
2. The two phases separated by the membrane
have different hydrostatic pressures, but the
difference in hydrostatic pressure is equal to
the difference in the osmotic pressures between the two solutions acting in opposite
direction. This situation is referred to as osmotic equilibrium and there will be no flow of
solvent through the membrane although the
concentrations in the two solutions are
different.
3. The two phases separated by the membrane
have different hydrostatic pressures, but the
difference in hydrostatic pressure across
the membrane is larger than the osmotic pressure and acts in the opposite direction. Thus,
solvent will flow through the membrane
from solution0 with the higher solute concentration into solution00 with the lower solute
Figure 4. Schematic illustrating the osmotic phenomenon. It shows two phases (0 and 00 ) separated by a semipermeable
membrane. The phases consist of a solvent and a solute, indicated by the subscripts l and s. C and m are concentration and
chemical potential, p is hydrostatic and p osmotic pressure, and J the membrane flux
430
Membranes and Membrane Separation Processes, 1. Principles
Vol. 22
If the solute is a salt, as is the case in many
practical applications, the concentration of the
individual ions must be considered when determining the osmotic pressure, that is, the degree of
dissociation and the stoichiometric coefficients
of the salt must be considered for determining the
osmotic pressure.
The osmotic pressure of an aqueous solution
with salts is given by:
p ¼ RT
X
gi ðvic þvia ÞCi s
ð24Þ
i
Figure 5. Solvent flux between two solutions of different
concentrations through a strictly semipermeable membrane
as function of hydrostatic pressure applied to the more
concentrated solution
concentration. This phenomenon is referred to
as reverse osmosis.
The flux of solvent between two homogeneous
solutions of different concentrations separated by
a semipermeable membrane which is only permeable for the solvent to the more concentrated
solution as a function of applied hydrostatic pressure is illustrated in Figure 5. Here, the solvent flux
between two solutions of different concentrations
through a strictly semipermeable membrane is
shown as a function of the hydrostatic pressure
applied to the more concentrated solution.
As long as the applied hydrostatic pressure is
lower than the osmotic pressure difference between the two solutions, solvent will flow from
the more dilute solution into the more concentrated solution by osmosis. When the hydrostatic
pressure exceeds the osmotic pressure difference
the flow is reversed and solvent will flow from the
more concentrated solution to the dilute solution.
This process is referred to as reverse osmosis.
The osmotic pressure p of a solution is proportional to the solute concentration Cs, but it can
be calculated from the chemical potential of the
solvent in a solution and is given by:
p¼
RT
ln al
V l
ð22Þ
For dilute solutions the osmotic pressure can be
expressed to a first approximation by the concentration of the individual components in the
solution [46]:
p ¼ RT
X
i
gi Ci
ð23Þ
where g is the osmotic coefficient and the subscripts s, c, and a respectively refer to a stoichiometric coefficient, cation, and anion.
2.6.2. Donnan Equilibrium and Donnan
Potential
In discussing the osmotic pressure it was assumed that two solutions separated by a membrane are in chemical equilibrium. If the solution
contains charged components (i.e., ions) and the
membrane is permeable for at least one ionic
component, the membrane will be in equilibrium
with the adjacent solution if the electrochemical
potential of all ions in the membrane and the
solution are equal. Thus, for each ion in equilibrium is:
s
m
m
s
s
hm
i ¼ hi ¼ mi þzi F j ¼ mi þzi F j
ð25Þ
where mi and hi are the chemical and the electrochemical potential, respectively of component
i, j is the electrical potential, F the Faraday
constant, and superscripts m and s refer to the
membrane and to the solution, respectively.
Thus, the electrochemical potential of an ion
is composed of two additive terms: the chemical
potential, and the electrical potential multiplied
by the Faraday constant and the valence of the
ion. Introducing the chemical potential mi as
defined in Equation (13) into Equation (25) and
rearranging gives the electrical potential difference between the membrane and the adjacent
solution as:
jm js ¼
1
as
RT ln mi þV i
ai
zi F
ðps pm Þ ¼ j Don
ð26Þ
where j is the electrical potential, a the ion
activity, V the partial molar volume, z the valence, F the Faraday constant, p the pressure, T
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
the absolute temperature, R the gas constant,
subscript i refers to individual components, and
superscripts m and s refer to the membrane and
the electrolyte solution, respectively. The potential difference between the membrane and the
solution is referred to as Donnan potential jDon.
[11].
The Donnan potential between an electrolyte
solution and an ion-exchange membrane cannot
be measured directly. It can, however, be calculated from the ion activities in the solution and
the membrane, and by the pressure difference
between the membrane phase and the adjacent
solution, pmps, which is referred to as swelling
pressure and is identical to the osmotic pressure difference between the solution and the
membrane.
Introducing the osmotic pressure into Equation (26) gives the Donnan potential as a function
of the ion and the water activities in the membrane and the solution:
1
as
RT ln mi V i Dp
ai
zi F
j Don ¼
ð27Þ
The numerical value of the Donnan potential
jDon can be calculated either from the cation or
anion activities. For a single salt it is:
1
as
1
as
RT ln mc V c Dp ¼
RT ln ma V a Dp
ac
aa
zc F
za F
j Don ¼
ð28Þ
Equation (28) gives a general relation for the
cation and anion distribution at the interface
between a solution and an ion-exchange
membrane.
Note that the value of the Donnan potential is
negative for an cation-exchange membrane and
positive for an anion-exchange membrane in
equilibrium with a dilute electrolyte solution.
The Donnan equilibrium describes the electrochemical equilibrium of an ion in an ionexchange membrane and an adjacent solution
and can be calculated for a single electrolyte by
rearranging Equation (27):
asa
am
a
z1 a
am
c
asc
z1
c
Dp V s
¼ e RT zc nc
ð29Þ
where z is the valence, n the stoichiometric
coefficient of the electrolyte, F the Faraday
constant, R the gas constant, T the absolute
431
temperature, V the partial molar volume, and a
the activity; the subscripts a and c refer to anion
and cation, and the superscripts s and m to the
solution and the membrane, respectively.
The Donnan equilibrium is an important relation in electro-membrane processes since it determines the distribution of co- and counterions
between the membrane and the adjacent solution.
3. Principles
Membrane separation processes can differ greatly with regard to membranes, driving forces,
areas of application, and industrial or economic
relevance. Table 1 summarizes industrially important membrane separation processes. The
driving force in these processes can be a hydrostatic pressure gradient as in micro- and ultrafiltration, reverse osmosis, or gas separation; a
concentration gradient as in dialysis; or an electrical potential gradient as in electrodialysis. In
some membrane separation processes, for example, pervaporation and membrane distillation,
partial pressure differences of individual components can be applied as driving force [51].
Membrane separation processes can conveniently be classified according to the driving
forces applied to achieve the transport of certain
components through the membrane into pressuredriven, concentration-driven, and electrical-potential-driven processes. Typical pressure-driven
membrane processes are microfiltration, ultrafiltration, and reverse osmosis. In all three processes
a feed solution is separated into a filtrate that is
depleted of particles or molecules and a retentate
in which these components are concentrated.
The term microfiltration is used when particles with a diameter of 0.1 – 10 mm are separated from a solvent or other low-molecular components. The separation mechanism is based on a
sieving effect, and particles are separated according to their dimensions. The membranes used for
microfiltration have pores of 0.1 – 10 mm in
diameter. The hydrostatic pressure differences
used are in the range of 0.05 – 0.2 MPa.
In ultrafiltration the components to be retained by the membrane are macromolecules or
submicrometer particles. Generally, the osmotic
pressure of the feed solution is negligibly small
and hydrostatic pressures of 0.1 – 5 MPa are
used. Ultrafiltration membranes are mostly
homogeneous asymmetric membrane
symmetric porous membrane
symmetric ion-exchange membrane
symmetric ion-exchange membrane
bipolar membrane
symmetric porous hydrophobic membrane
symmetric porous or liquid membrane
vapor pressure
concentration gradient
electrical potential
concentration of ions
electrical potential
vapor pressure
chemical potential
solution and diffusion
diffusion
Donnan exclusion
Donnan exclusion
Donnan exclusion
diffusion
diffusion and solution
separation of gases, vapors,
and isotopes
separation of azeotropic mixtures
artificial kidney
water desalination
water softening
acid and base production
liquid – solid separation
solvent extraction
diffusion, Knudsen diffusion
Pervaporation
Dialysis
Electrodialysis
Donnan dialysis
Electrodialytic water splitting
Membrane distillation
Membrane contactors
Reverse osmosis
Diafiltration
Ultrafiltration
Gas and vapor separation
hydrostatic pressure, 1 – 10 MPa
hydrostatic and vapor pressure
purification of molecular mixtures,
artificial kidney
sea- and brackish water desalination
filtration and dialysis
(size exclusion)
solution and diffusion
hydrostatic pressure, 0.1 – 0.5 MPa
separation of molecular mixtures
filtration (size exclusion)
hydrostatic pressure, 0.1 – 0.5 MPa
filtration (size exclusion)
symmetric porous membrane,
pore radius 0.1–10 mm
asymmetric porous membranes,
pore radius 2 – 10 nm
asymmetric porous structure,
pore radius 2 – 10 nm
asymmetric skin-type
solution-diffusion membrane
homogeneous or porous membrane
Microfiltration
hydrostatic pressure, 0.05 – 0.2 MPa
Mode of separation
Applied driving force
Membrane type used
Separation Process
Table 1. Technically relevant membrane separation processes, their operating principles, and their applications
water purification, sterilization
Membranes and Membrane Separation Processes, 1. Principles
Applications
432
Vol. 22
asymmetrically structured, with pores in the skin
layer having diameters of 2 – 10 nm.
Diafiltration is in principle identical to ultrafiltration. It is used to rinse out low molecular
weight components such as salts from a solution
with macromolecular components.
In reverse osmosis particles, macromolecules,
and low molecular mass compounds such as salts
and sugars are separated from a solvent, usually
water. Feed solutions, therefore, often have a
significant osmotic pressure, which must be overcome by the applied hydrostatic pressure. The
osmotic pressure of a solution containing low
molecular mass solutes can be quite high even at
low solute concentrations. Since the membrane
flux is a function of the effective pressure, which
is given by the hydrostatic pressure difference
between feed and filtrate solutions minus the
difference in osmotic pressure between these
solutions, and the osmotic pressure is relatively
high, hydrostatic pressures on the order of 1 –
10 MPa are used in practical applications.
In dialysis certain solutes are transferred from
a the feed solution to a receiving solution, which
is also referred to as stripping solution or dialysate, through a membrane due to a concentration
gradient across the membrane. The separation of
various compounds is the result of differences in
their diffusivity in the membrane matrix.
In electrodialysis an electrical potential driving force and anion- and cation-exchange membranes installed in alternating series between
electrodes are used to separate charged components such as ions from a solution. The separation
is the result of the permselectivity of the ionexchange membranes, which are permeable only
to counterions, that is, cation-exchange membranes are permeable to cations and anion-exchange membrane are permeable to anions.
Electrodialytic water dissociation is a process
similar to electrodialysis in which monopolar
ion-exchange membranes are used in combination with bipolar membranes which split water
into Hþ and OH ions to produce acids and bases
from the corresponding salt.
In gas and vapor separation two different
processes and membranes are used. One is similar to reverse osmosis using asymmetric solutiondiffusion membranes and a hydrostatic pressure
as driving force. The membrane selectivity is
determined by the solubility and diffusion of the
components in the membrane matrix. In the
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
second process, porous membranes are used
through which gases permeate at elevated temperature under a hydrostatic pressure as driving
force by Knudsen diffusion. The selectivity is
very low and a large number of stages are required to obtain significant separation of the
permeating components. Knudsen diffusion separation is used mainly for the separation of
radioactive isotopes.
In pervaporation volatile organic compounds
are removed from a liquid feed solution through a
solution-diffusion membrane into the gas phase
as a result of a difference in their partial vapor
pressure. Both hydrostatic pressure and temperature differences are used as driving force for the
transport of the different components through the
membrane.
Membrane distillation is similar to conventional distillation. The main difference is that the
feed and the product are separated by a porous
membrane.
Membrane contactors are used in solvent
extraction. Here, the membrane acts as barrier
between the feed and the stripping solution.
3.1. Principle of Microfiltration
The principle of microfiltration is depicted in
Figure 6. The membrane separating a feed solution from a permeate or filtrate has a symmetric
porous structure with an average pore size between 0.1 and 10 mm. The driving force for the
mass transport across the membrane is a pressure
gradient. Only large particles with diameters in
excess of about 0.1 mm are separated by the
membrane. The diffusion of particles and the
osmotic pressure difference between the feed
and filtrate solution are negligibly low and the
mass flux through a microfiltration membrane is
given to a first approximation by:
Jv ¼
X
Ji V i ffi Lv
i
Dp
Dz
ð30Þ
where Jv is the volume flux across the membrane,
the subscripts v and i refer to volume and components in the solution, V is the partial molar
volume, p the pressure, and Lv a phenomenological coefficient referring to hydrodynamic permeability of the membrane, Dp the pressure
difference between the feed and filtrate solution,
and Dz the thickness of the membrane.
The mass transport in microfiltration membranes takes place by viscous flow through the
pores. The solid membrane matrix can be considered as completely impermeable. Therefore,
the hydrodynamic permeability can be expressed
in terms of the membrane pore size and the
porosity and the solution viscosity according to
the Hagen – Poiseuille law, which is given by:
Jv ¼
er2 D p
8ht D z
ð31Þ
where Jv is the flux, e the membrane porosity, r
the pore radius, h the viscosity, t the tortuosity
factor, Dp the pressure difference across the
membrane, and Dz the membrane thickness.
The tortuosity factor is defined as the ratio of
the actual pore length to the thickness of the
membrane taking into account that the pore
length in general is somewhat longer than the
cross section of the membrane. Thus, t is always
greater than 1, and the porosity e is always less
than 1.
The flux through a microfiltration membrane,
which is often referred to as filtration rate, is
adequately described by Equation (31). Another
parameter which is of interest for the practical
application of microfiltration is the separation
characteristic of a membrane. This is generally
expressed by the retention or rejection of a
particle of a certain size, which is given by:
Ri ¼
Figure 6. Schematic illustrating the principle of microfiltration
433
Cp
1 if
Ci
ð32Þ
where R is the rejection coefficient, C is the
concentration, subscript i refers to a given
434
Membranes and Membrane Separation Processes, 1. Principles
component in the feed and the permeate, and
superscripts f and p refer to the feed and permeate
or filtrate solutions. R is always less than or equal
to 1 and a function of the particle and the pore size
and pore size distribution.
components between the feed and filtrate solutions. The volume flux (i.e., the filtration rate) is
given by the sum of the fluxes of the individual
components:
Jv ¼
X
Ji V i ¼
i
3.2. Principle of Ultrafiltration
In ultrafiltration the driving force again is a pressure gradient and the mass transport is also dominated by the convective flux through pores. The
principle of the process is depicted in Figure 7.
As in microfiltration a porous membrane and a
hydrostatic pressure difference are used to separate certain components from a feed solution.
However, the structure of an ultrafiltration membrane is asymmetric, with the smallest pores on
the surface facing the feed solution, and its pores
are significantly smaller than those of a microfiltration membrane. Their diameters at the feed
side surface of the membrane are between 2 and
10 nm and the components retained by an ultrafiltration membrane have a molecular weight
between 5000 and several million Dalton. Since
ultrafiltration membranes also retain some relatively low molecular weight solutes, osmotic
pressure differences between the feed and the
filtrate can be significant and diffusive fluxes of
the solutes across the membrane are no longer
negligibly low. Therefore, the flux of individual
components in ultrafiltration can be expressed as
a function of a hydrostatic pressure and a difference in the chemical potential of the different
Vol. 22
¼
X
i
X
i
V i Li
d mi
dp
þLv
dz
dz
d dp
ðV pþRT ln ai ÞþLv
dz i
dz
ð33Þ
where J is the flux, L a phenomenological coefficient referring to interactions of the permeating
components with the membrane matrix, Vi the
partial molar volume, m the chemical potential, p
the hydrostatic pressure, z a directional coordinate, a the activity, and the subscripts v and i refer
to volume flow and individual components. The
first term in Equation (33) describes the diffusive
fluxes of all components in the pores of the
membrane, and the second term the volume flow.
In ultrafiltration the total volume flux in a
dilute solution can be expressed to a first approximation by the flux of the solvent, that is, Jv Jw,
and the activity of the solvent in the solution aw
can be expressed by an osmotic pressure. Assuming a linear relation for the pressure and activity
gradients across the membrane, integration of
Equation (33) gives the flux through an ultrafiltration membrane as a function of pressure difference between feed and permeate solution, the
hydrodynamic permeability for the viscous flow,
the osmotic pressure difference between feed and
permeate solution, and the phenomenological
coefficient determining the diffusive flow of
water through the membrane pores:
Jv ffi Jw ¼ V w 2Lw
Figure 7. Schematic illustrating the principle of ultrafiltration
V i Li
D pD p
Dp
þLv
Dz
Dz
ð34Þ
where Jv and Jw are total volume and solvent
fluxes, respectively, Dp and Dp are the hydrostatic and the osmotic pressure gradients across
the membrane, Lv is the hydrodynamic permeability and Lw is the diffusive permeability of the
solvent, and Dz is the thickness of the selective
barrier of the membrane.
In most practical applications of ultrafiltration
the first term of Equation (34) can be neglected
since Lw Lv. The osmotic water transport will
effect the membrane volume flux only when the
difference in osmotic pressure between the feed
and permeate solution is significantly higher than
the hydrostatic pressure difference. This is only
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
the case for ultrafiltration of solutions containing a
high concentration of retained components of
relatively low molecular weight. Since generally
only macromolecular components are retained the
volume flux (i.e., the filtration rate in ultrafiltration) can be described to a first approximation by:
Dp
Jv ¼ Lv
Dz
ð35Þ
For the transport of solutes through an ultrafiltration membrane the diffusive flux can generally
not be neglected, and the flux of the solute
through the membrane can be expressed as the
sum of a diffusive flux due to a chemical potential
driving force and a convective flux due to the
concentration in the pore solution:
Ji ¼ V i Li
d dp
ðV pþRT ln ai ÞþLv m C i
dz i
dz
ð36Þ
where mCi is the concentration of the solute in the
pore solution.
It can be shown that for hydrostatic pressure
differences usually applied in ultrafiltration,
Vi p RTlnai . If it is assumed that the activity
coefficient of the solutes in the membrane is 1,
the phenomenological coefficient can be expressed by the Fick’s diffusion coefficient according to Equation (19). Introducing these approximations into Equation (36) leads to:
Ji ¼ Jv m C i m Di dm
Ci
dz
ð37Þ
Integration of Equation (37) over the membrane
thickness gives:
m
J i ¼ Jv
C fi exp
exp
Jv t D z m p
Ci
mD
i
Jv t D z
1
mD
i
ð38Þ
where J is the flux, C the concentration, D the
diffusion coefficient, t the tortuosity factor, and
Dz the thickness of the membrane; subscripts i
and v refer to solute and the volume; and superscripts m, f,and p refer to the membrane, the
solution in the membrane at the feed side, and the
solution in the membrane at the permeate side.
Equation (38) shows that the flux of a component i which is partly retained by the membrane in
ultrafiltration consists of two terms. The first term
represents the viscous flow and is a linear function
of the hydrostatic pressure. The second term represents the diffusion in the pore liquid and is an
exponential function of the hydrostatic pressure.
435
In ultrafiltration two limiting cases exist for
the transport mechanism. In the first case it is
assumed that the hydrostatic pressure and thus
the viscous flow is zero, so that the transport of a
solute is determined by diffusion only:
lim Ji ¼ m Di
Jv !0
m
C pi m C fi
tDz
ð39Þ
The second limiting case is obtained when the
pressure and thus the viscous flow is infinitely
high. Then the transport of the solute is determined by its concentration in the membrane:
lim Ji ¼ Jv m C pi
ð40Þ
Jv !¥
Since the flux of the solute in ultrafiltration is a
function of the viscous flux the retention of an
ultrafiltration membrane is obviously a function
of the applied hydrostatic pressure. It can be
described by combining Equations (32) and
(38) and by introducing partition coefficients to
relate the concentrations of the solute in the feed
and the permeate to that in the membrane:
Ri ¼ 1
kf exp JvmtDD z
i
k 1þ exp JvmtDDi z
p
ð41Þ
where
kf ¼
Cif
m Cf
i
Cp
and kp ¼ m i p
Ci
ð42Þ
where kf and kp are the partition coefficients
which relate the concentrations in the feed and
the permeate to that in the membrane.
The effect of the hydrostatic pressure on
retention in ultrafiltration becomes more clear
when two extreme cases are considered, that is, a
very large and a very small hydrostatic pressure
difference between feed and permeate solutions.
At large pressure differences the viscous flux is
high and the retention approaches a maximum
value which is determined by the distribution
coefficient of the solute between the membrane
and the solution.
lim Ri ¼ 1kf
Jv !¥
ð43Þ
At very low hydrostatic pressure the retention
goes to zero.
lim R ¼ 0
Jv !¥
ð44Þ
436
Membranes and Membrane Separation Processes, 1. Principles
3.3. Principle of Reverse Osmosis
In reverse osmosis the driving force is also a
hydrostatic pressure difference. The principle of
the process is illustrated in Figure 8, which shows
a solution containing a low molecular weight
component separated by a membrane from the
solvent and a hydrostatic pressure difference
applied across the membrane resulting in a flux
of solvent from the solution into the pure solvent.
The membrane used in reverse osmosis has an
asymmetric structure with a dense barrier layer
at the side facing the feed solution. It is assumed
that in this layer the individual components are
transported by diffusion and that viscous flow
through pores or pinholes can be neglected. For
describing the transport in reverse osmosis membranes two approaches are described in the literature [52]. One is the so-called solution-diffusion
model, which neglects kinetic coupling of different fluxes [23]. The second is based on the
phenomenological equation of the thermodynamics of irreversible processes [47]. This model
does consider kinetic coupling between different
fluxes. Both models describe the transport in
reverse osmosis membranes quite accurately,
and this indicates that kinetic coupling is not of
great importance in reverse osmosis.
3.3.1. Mass Transport in Reverse Osmosis
Described by Phenomenological Equations
In a reverse osmosis system which consists of a
membrane and a two-component feed solution,
i.e. a solvent and a solute, two independent fluxes
Vol. 22
will be obtained when the membrane is used as
frame of reference. These fluxes can be described
by:
Ji ¼
X
Lik Xk
ði; k ¼ 1; 2; 3; . . . ; nÞ
ð45Þ
k
where J is the flux of the different components, L
is the phenomenological coefficient, X the driving force, and subscripts i and k refer to different
components.
Assuming a system without viscous flow composed of a solute, a solvent, and a membrane as
frame of reference there are two independent
fluxes:
Jw ¼ Lw d mw þLws d ms
ð46Þ
and
Js ¼ Lsw d mw þLs d ms
ð47Þ
where J is the flux, m the chemical potential, L the
phenomenological coefficient, subscripts w and s
refer to the solvent, which is assumed to be water,
and the solute, which is assumed to be salt, and
subscripts ws and sw to the coupling between the
fluxes of water and salt and salt and water.
The coupling coefficients Lsw and Lws are
identical according to the Onsager relation
[48]. They are very difficult to measure in an
independent experiment. However, the coupling
coefficient can be converted to more easily measurable constants when the phenomenological
equations are slightly modified.
Usually in reverse osmosis the total volume
flux is measured. It is therefore possible to consider a volume flux Jv and a diffusion related flux
JD which are given by [20]:
Jv ¼ Jw V w þJs V s
ð48Þ
Js V w Jw
Cs
ð49Þ
JD ¼
The dissipation function for the above described
set of equations is:
Y ¼ Jw d mw þJs d ms
¼ Jw dðV w pþRT ln aw ÞþJs dðV s pþRT ln as Þ
Figure 8. Schematic illustrating the principle of reverse
osmosis
ð50Þ
where Y is the dissipation function, J the flux, C,
a, m, and V are the concentration, the activity, the
chemical potential, and the partial molar volume
in the membrane, p is the hydrostatic pressure,
and subscripts v, w, s, and D refer to volume,
solvent, solute, and diffusion.
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
Assuming a linear relation for the chemical
potential gradients, introducing the osmotic pressure for the activities of water and solute as
described in Equations (22) and (23), and inserting Equations (48) and (49) into Equation (50)
leads to a new set of phenomenological equations:
Js
Y ¼ ðJw V w þJs V s ÞD pþ
Jw V w D p ¼ Jv D pþJD D p
Cs
ð51Þ
The first term in Equation (51) describes the
total volume flux Jv as function of the applied
pressure and the second term describes the diffusion flux of the salt JD.
The new set of phenomenological equations
is:
Jv ¼ Lp D pþLpD D p
ð52Þ
JD ¼ LDp D pþLD D p
ð53Þ
By applying different experimental conditions
and Equations (52) and (53) the phenomenological coefficients can be determined from experimental measurements.
Assuming no pressure difference is applied
(Dp ¼ 0), then the diffusive flux is given by:
ðJD ÞD p¼0 ¼ LD D p
ð54Þ
and the volume flux is given by:
ðJv ÞD p¼0 ¼ LpD D p
ð55Þ
Assuming that there is no osmotic pressure difference (Dp ¼ 0), then the diffusive flux is given
by:
ðJD ÞD p¼0 ¼ LDp D p
ð56Þ
and the volume flux by:
ðJv ÞD p¼0 ¼ Lp D p
ð57Þ
The hydrostatic pressure difference at zero volume flux, that is, the osmotic equilibrium, is
given by:
ðD pÞJv ¼0 ¼ LpD
Dp ¼ sDp
Lp
ð58Þ
437
where s is the so-called Staverman reflection
coefficient [53], which is a measure for the
permselectivity of a membrane.
Equation (58) shows that in osmotic equilibrium the hydrostatic pressure difference between
two solutions separated by a membrane is equal
to the osmotic pressure difference if Lp ¼ LpD and
therefore s ¼ 1. This is only the case for a strictly
semipermeable membrane. Usually, the Staverman reflection coefficient has values between 0
and 1.
If the volume flux is 0 the ratio of salt flux to
osmotic pressure difference is given by:
Cs Lp LD L2pD
Js
¼
¼w
D p Jv ¼0
Lp
ð59Þ
where w is the permeability of the salt.
Combining Equations (49), (52), (53), (58),
and (59) leads to:
Jv ¼ Lp ðD ps D pÞ
ð60Þ
Js ¼ Cs ð1sÞJv þw D p
ð61Þ
Based on the phenomenological equations the
mass transport in reverse osmosis of a solution of
a single salt solution can be described by three
coefficients:the hydrodynamic permeability of
the membrane Lp, the salt permeability, w and
the reflection coefficient s. All three coefficients
can be experimentally determined.
3.3.2. Description of Mass Transport in
Reverse Osmosis by the Solution-Diffusion
Model
In the solution-diffusion model the flux of different components through a membrane is assumed
to be caused by diffusion and can be described by
the product of its concentration and mobility in
the membrane and the chemical potential gradient of the component in the membrane. Furthermore, it is assumed that there is no coupling
between fluxes, that the volume flux is approximately identical to the flux of the solvent, and that
the solute flux due to the pressure gradient is
negligibly small.
Thus, the flux of a component in the membrane is given by:
Ji ¼m C i mi ¼ Li d mi ¼ Li ðV i d pþRT dlnai Þ
ð62Þ
438
Membranes and Membrane Separation Processes, 1. Principles
where J is the flux, mC the concentration, m the
mobility, L a phenomenological coefficient composed of a concentration and a mobility term, m
the chemical potential, p the hydrostatic pressure,
a the activity, V the partial molar volume, and
subscript i refers to a component.
For i ¼ w it is assumed that Jv ffi Jw and thus
Cw Vw ¼ 1. For i ¼ s it is assumed that dms ffi
w
RTdlnas and p ffi RTlna
V w . Here the subscripts v,
w, and s refer to volume, solvent, and solute.
Integrating over the cross section of the
membrane and expressing the activity of the
solvent by the osmotic pressure and that of
the solute by its concentration gives the volume
flux as
Jv ¼ Lw V w ðD pD pÞ
ð63Þ
and the solute flux as:
Js ¼ Ls
RT m
D Cs
Cs
ð64Þ
The concentration of the different components in
the membrane is proportional to the concentration in the outside solutions and given by:
m
C i ¼ k i Ci
k w Dw C w
RT
V
w
ðD pD pÞ ¼
kw Dw
ðD pD pÞ
RT
ð66Þ
and
Js ¼ ks Ds D Cs
sure of the solutions, subscripts s and w refer to
solute and water, respectively, and the superscript m refers to the membrane. Furthermore it
is assumed that VwCw ffi 1.
In Equations (66) and (67) all kinetic coupling
between fluxes is neglected and activity coefficients in the membrane are assumed to be 1.
Nevertheless, Equations (66) and (67) describe
the mass transport in reverse osmosis for a membrane with a high solute rejection capability quite
well, as a comparison with Equations (60) and
(61) shows, since the reflection coefficient is very
close to 1 for a membrane with a high salt
rejection.
Besides the volume flux (filtration rate) and
the solute flux, the rejection of the membrane is
of interest. The rejection is given by Equation
(22). In reverse osmosis the concentration in the
permeate can be expressed by the solute flux
divided by the volume flux, that is, Csp ¼ JJvs .
Membrane rejection in reverse osmosis is given
by:
Rs ¼ 1
Csp
Js
ks m Ds RT D Cs
¼ 1
¼ 1
f
f
Cs
Jv Cs
kw Csfm Dw ðD pD pÞ
ð68Þ
ð65Þ
where ki is the partition coefficient of component
i between the membrane and the adjacent solution, and Ci and mCi are the concentrations in the
outside solution and in the membrane.
Introducing Equation (65) and Fick’s diffusion coefficient according to Equation (18) into
Equations (64) and (63) gives the volume flux
(i.e. the filtration rate) and the flux of the solute as
function of the applied hydrostatic pressure, the
osmotic pressure, the diffusion coefficient of the
components in the membrane, their concentration in the feed and permeate solution, and their
partition coefficient between the membrane and
the solutions:
Jv ¼
Vol. 22
ð67Þ
where Jv is the volume flux, Js the solute flux, D
the diffusion coefficient, k the partition coefficient of the components between the membrane
and the adjacent solutions, C the concentration, p
the hydrostatic pressure and p the osmotic pres-
Equation (68) shows that rejection of a component in reverse osmosis not only depends on the
diffusion coefficients of water and solute in the
membrane and on their partition coefficients
between the membrane and the adjacent solutions, but it is also a function of the applied
hydrostatic pressure and approaches asymptotically a maximum value. The rejection disappears
when the hydrostatic pressure is equal to the
osmotic pressure.
3.4. Principle of Gas Separation
The principle of gas separation is illustrated in
Figure 9, which shows two different gas mixtures
separated by a membrane. The driving force for
the gas to permeate the membrane is a pressure
gradient. In gas separation both porous and dense
membranes are used as selective barriers. In
porous membranes the transport of gases is based
on Knudsen diffusion, and in dense solid materials the gas transport is based on a solutiondiffusion mechanism. Both Knudsen diffusion
and solution-diffusion transport can result in
selective transport of gases and thus result in
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
439
Figure 10. Schematic illustrating viscous flow and Knudsen
diffusion
Figure 9. Schematic illustrating the principle of gas
separation
separation of gases. However, the extent of the
separation, that is, the separation factor, is much
higher in solution-diffusion transport than in
Knudsen diffusion.
3.4.1. Gas Separation by Knudsen
Diffusion
Knudsen diffusion can be considered as viscous
flow in narrow pores, that is, pores with a diameter that is smaller than the mean free path of the
diffusing gas molecules. The mean free path is
defined as the average distance a gas molecule
travels before it collides with another gas molecule. The mean free path of a gas molecule
depends on the nature of the gas, the temperature,
and the pressure. It is given by [54]:
l¼
kT
pffiffiffi
p d gas p 2
ð69Þ
where l is the mean free path of a gas molecule, k
the Boltzmann constant, dgas the diameter of the
gas molecule, and p the hydrostatic pressure.
The difference between viscous flow and
Knudsen diffusion is illustrated in Figure 10.
The figure shows two membrane pores of
different diameter with a number of gas molecules. In the pore with the larger diameter the gas
molecules have more interaction with each other
than with the pore wall since the mean free path
of the molecules is much shorter than the pore
diameter. In the pore with the smaller diameter
the gas molecules have more interactions with
the pore wall than with each other. In the case
of the larger pores the energy loss during transportation is mainly due to the interaction of the
molecules with each other. It is expressed in the
viscosity of the solution. Therefore, the flux is
referred to as viscous flow in which all molecules
are transported with the same rate. In the pores
with the smaller diameter the energy loss during
the transport of the gas molecule is due to their
interaction with the pore wall. The process is
similar to diffusion in a homogeneous solid phase
and is therefore referred to as Knudsen diffusion.
Since molecules of different molar mass have
different interaction energies when colliding
with the pore wall they are transported with
different rates. This can be utilized to separate
gases which differ in molecular weight. The
mean free path of gases at room temperature and
atmospheric pressure is on the order of several
nanometers. This means that only membranes
with a mean pore size below several nanometers
can separate gases by a Knudsen diffusion mechanism. Since the free path of a gas molecule is
pressure- and temperature-dependent, as Equation (69) indicates, membranes with larger pore
diameters can be used when the process is carried
out at higher temperature and low pressure.
The flux in Knudsen diffusion can be described by the following relation [55]:
Ji ¼
pnr 2 Dki D p
RTt D z
ð70Þ
where J is the flux through the membrane, n the
number of pores in the membrane, r the pore
radius, Dp the pressure difference across the
membrane, Dz the thickness of the membrane,
t the tortuosity factor, and Dk the Knudsen
diffusion coefficient.
440
Membranes and Membrane Separation Processes, 1. Principles
Vol. 22
The Knudsen diffusion coefficient is given by:
Dki ¼ 0:66r
rffiffiffiffiffiffiffiffiffiffiffiffiffi
8RT
p Mw
ð71Þ
Equation (71) shows that the Knudsen diffusion
coefficient of a gas molecule is inversely proportional to the square root of its molecular weight.
Thus, the separation of two gases based on
Knudsen diffusion is given by the ratio of the
square root of the molecular weights.
aj;k
sffiffiffiffiffiffiffiffi
Mk
/
Mj
ð72Þ
Because of the Knudsen diffusion transport mechanism the selectivity that can be achieved with
porous membranes is in general rather low when
gases differ only marginally in their molecular
mass. Nevertheless, porous membranes are used
to separate 235UF6 from 238UF6. Since the selectivity is only about 1.0043, hundreds of stages are
required to obtain significant separation.
3.4.2. Gas Transport by the SolutionDiffusion Mechanism in a Polymer Matrix
Significantly higher selectivity can be obtained in
homogeneous polymer membranes, where the
transport mechanism is based on the solution
and diffusion of the various components within
the membrane phase. The mass transport in a socalled solution-diffusion membrane is illustrated
in Figure 11. It consists of three steps:
1. Sorption of the various components from a
feed mixture according to their partition coefficients between the gas and polymer phase.
2. Diffusion of the individual components within the membrane phase according to their
activity gradients.
3. Desorption of the components from the membrane in the permeate gas phase.
If viscous flow is excluded the driving force
for the mass transport of gases in solution-diffusion membranes is the activity gradient of the
permeating components within the membrane
phase, which can be related to the difference
between the chemical potentials of the component in the feed and the permeate.
Figure 11. Schematic illustrating the three-step transport
mechanism in solution-diffusion membranes (m is the chemical potential, X the mole fraction, C the concentration, p the
partial pressure; the subscript i refers to a component, and the
superscripts f, p, g, and m to feed, permeate, gas phase, and
membrane)
The flux across the membrane can be described again by the phenomenological equation
as a function of the gradient of the chemical
potential of the permeating component:
Ji ¼ Li
dm mi
dz
ð73Þ
where J is the flux, L a phenomenological coefficient relating the flux to the corresponding driving force, m the chemical potential, z the directional coordinate perpendicular to the membrane
surface, subscript i refers to a component, and
superscript m to the membrane.
The chemical potential in the membrane at the
membrane surface is equal to the potential in the
adjacent gas phase. The change of chemical
potential of a component as function of pressure
and composition is given by:
d mi ¼ moi þd RT ln ai þV i d p
ð74Þ
where a is the activity, p is the hydrostatic
pressure, and V is the partial molar volume.
Introducing Equation (74) into Equation (73)
gives the flux of component i in the membrane:
Ji ¼ Li ðd RT ln m ai þm Vi d pÞ
ð75Þ
Since in the membrane the pressure is constant
(i.e., dp ¼ 0) and d ln a ¼ daa Equation (75) can
be written as:
Ji ¼ Li RT
dm ai
ma
i
ð76Þ
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
Integration of Equation (76) across the membrane gives the flux as function of the activity
of component i in the membrane at the interface
between the feed gas and the permeate gas:
J i ¼ Li
RT
ma
i
m p m f
ai ai
ð77Þ
Dz
where m a i is the average activity of component i
in the membrane, and superscripts f and p refer to
feed side and product side of the membrane.
Since at the interface between the membrane
and the gas the chemical potentials of component
i in the membrane and the gas are equal (i.e.,
m
mi ¼ gmi), the activity of component i in the
membrane at the interface can be expressed by
its activity in the gas. The chemical potential of
component i in the membrane and in the gas is
obtained by integration of Equation (74).
In the membrane the partial molar volume of
the component can be considered as constant and
equal to the molar volume of the condensed gas:
m
mi ¼ moi þRT lnm ag þm Vi ðm ppo Þ
ð78Þ
where the superscript o refers to a standard state.
In the gas the partial molar volume is a
function of the pressure and must, in accordance
with the ideal gas law, be replaced by RT
p.
The chemical potential in the gas is then given
by:
g
mi ¼ moi þRT lng ag þRT ln
g
p
ap
¼ moi þRT ln oi
poi
pi
ð79Þ
Since the chemical potentials in the membrane
and the gas at the gas membrane interface are
equal:
g
mi ¼m mi ¼ RT ln
¼
g
ai p
¼ RT lnm ai þm Vi ðm ppo Þ
poi
g
ai p m
¼ ai exp
poi
ð80Þ
RT
where pi is the saturation pressure of the gas.
It can be shown that the exponential term in
Equation (80) even at high pressure is always
close to 1. Therefore, the activity of component i
at the interface with the gas can be expressed by:
m f
ai ¼
g f f
ai p
poi
and m api ¼
g p p
ai p
poi
The activity of component i in the membrane
and in the gas can be related to its concentration
in the membrane and its molar fraction in the gas
by:
m
ai ¼m X i m g i and g ai ¼
ð81Þ
where the superscripts f and p refer to feed and
permeate side of the membrane.
X i ji
p0i
ð82Þ
where X is the molar fraction, g the activity
coefficient, j the fugacity coefficient, and superscripts m, g, and o refer to the membrane, the gas
phase, and the saturation pressure.
Introducing Equations (81) and (82) into
Equation (77) leads to:
Ji ¼ Li
RT
m g p0
mX
i i
i
Xip pp jpi Xif pf jfi
Dz
ð83Þ
where L is a phenomenological coefficient, X the
molar fraction in the gas, p the pressure, j the
fugacity coefficient, m Xi and m gi are the average
molar fraction and the average activity coefficient of component i in the membrane, Dz is the
thickness of the membrane, superscripts p, f, and
o refer to permeate, feed, and saturation pressure,
and subscript i refers to a component.
Introducing the relation between the phenomenological coefficient and the diffusion coefficient as expressed in Equation (18), that is,
i Ci
, the relation between the molar fracLi ¼ DRT
tion and the concentration given by
r
m
C i ¼ X i m M i , and assuming that to a first approxp
imation ji ¼ jif Equation (83) becomes:
Ji ¼ Di
ji m r
m
g i p 0i Mi
Xip pp Xif pf
Dz
ð84Þ
Introducing
a sorption coefficient given by
j mr
ki ¼ im gi p0i Mi into Equation (84) leads to:
Ji ¼ Di ki
m m
V i ð ppo Þ
441
Xip pp Xif pf
Dz
ð85Þ
where D is the diffusion coefficient in the membrane, ki the gas sorption coefficient of the membrane, r the density, M the molecular weight, and
subscript i refers to a component.
The product of diffusion and sorption coefficient is the membrane permeability given by
Pi ¼ Diki. Introducing the permeability Pi into
Equation (85) gives the flux Ji of a component
through the membrane as a function of its molar
fractions in the feed and permeate Xif and XiP , its
fugacity coefficient ji, its permeability in the
membrane Pi, the membrane thickness Dz, and
442
Membranes and Membrane Separation Processes, 1. Principles
the pressures in the feed and permeate pf and pp.
Ji ¼ Pi
Xip pp Xif pf
Dz
SPj;k ¼
Pj
Pk
aj;k ¼
ð87Þ
where SPj,k is the permeation selectivity of a
membrane for the components j and k, and Pj
and Pk are their permeabilities.
The permeation selectivity can be split into
two terms, namely, the selectivity due to different
diffusion coefficients of the components of the
gas mixture and that due to their solubility in the
membrane:
k
SPj;k ¼ SD
j;k Sj;k
more useful. For a binary mixture the separation
factor is defined by:
ð86Þ
In deriving Equation (86) several assumptions
have been made which are generally valid for
permanent gases. However, the assumption that
the diffusion and the solubility coefficient are
independent of concentration and pressure is not
valid for easily condensable vapors. In permeation of organic vapors, the diffusion coefficient
may vary by several orders of magnitude with the
concentration of the permeating component in
the membrane.
For the relationship between the diffusion
coefficient and the concentration of the diffusing
component different equations accounting for
dual sorption or free volume and structural
changes of the polymer are discussed in the
literature. For gas and especially vapor permeation it is generally not possible to predict the
mass transport behavior of various components in
a mixture from single-component measurements.
For a practical application the separation efficiency of the membrane is a crucial parameter. In
gas permeation the separation efficiency of a
membrane is expressed by its selectivity and/or
its separation factor. The selectivity of a membrane for various components of a mixture is
defined by the ratio of the permeabilities. For a
binary mixture of components k and j, the selectivity is:
ð88Þ
k
where SD
j;k and Sj;k are the diffusivity and solubility selectivity of a membrane for components j
and k.
The selectivity is a useful parameter to characterize a membrane. For the design of a membrane plant, however, the separation factor is
Vol. 22
Xjp Xkf
Xjf Xkp
ð89Þ
where a is the separation factor, X the mole
fraction, subscripts k and j refer to the two
components, and superscripts f and p refer to
feed and the permeate.
The separation factor is defined to be always
greater than 1. It is related to the membrane
selectivity by:
aj;k ¼ Sj;k
Xjp pp
pf
Xkp pp
pf
Xjf Xkf
f
X f Xj
ð90Þ
k
where S is the selectivity for a binary mixture, p
the pressure, X the molar fraction, superscripts f
and p refer to feed and the permeate, and subscripts k and j to the components in the mixture.
The separation factor is always smaller than the
selectivity and when the pressure ratio of feed to
permeate becomes infinitely high:
lim aj;k ¼ Sj:k
pp
!0
pf
ð91Þ
In vapor separation, the factor a is normally a
function of the feed concentration.
3.5. Principle of Pervaporation
Pervaporation is a membrane process in which
the permeation of certain components through a
membrane from a liquid feed mixture into a
vapor phase is combined with the evaporation
of these components. The principle of the process
is illustrated in Figure 12.
The membranes and transport mechanism of a
component through these membranes in pervaporation are the same as in gas separation. The
driving force for transport is the chemical potential gradient of the permeating components in the
membrane. The chemical potential gradient in
the membrane can be related to the partial vapor
pressures in the liquid and vapor phases. To
establish a partial pressure difference of a component in the liquid and the vapor phase, generally, a vacuum is applied at the permeate side of
the membrane. But differences in the partial
pressure of a component can also be established
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
443
brane. This dependency must be determined
experimentally and may be different for different
membrane materials and vapors.
The product of diffusion and distribution
coefficient is referred to as permeability coefficient, and the ratio of the permeability coefficients of two components determine the membrane selectivity for the two components.
Di k i ¼ P i
ð93Þ
Pj
Pk
ð94Þ
Sj;k ¼
Figure 12. Schematic illustrating the principle of pervaporation showing an asymmetric membrane with a dense solutiondiffusion barrier layer
by using a sweeping gas on the permeate side or
by a temperature difference between the liquid
feed and the vapor.
Mass transport in a pervaporation membrane
can be described by the same mathematical
relations as gas transport with the exception that
the chemical potential in the membrane on the
feed side of the membrane is not expressed by
the molar fraction, the hydrostatic pressure, and
the fugacity coefficient, but by the molar fraction
and the activity coefficient in the liquid phase,
and the saturation pressure of the component.
Thus, mass transport in pervaporation is given
by:
Ji ¼ Di ki
Xip pp Xil g li poi
Dz
ð92Þ
where Di is the diffusion coefficient of component i in the membrane, ki the sorption coefficient
of component i into the membrane matrix, X the
molar fraction, p the pressure, g the activity
coefficient, subscripts j and k refer to components
of the mixture, and the superscripts o, l, and p to
saturation pressure, feed, and permeate,
respectively.
Equation (92) describes the permeation flux of
a component i as a function of the feed and
permeate mixture composition and its diffusivity
and solubility in the membrane, which can easily
be determined by independent measurements.
The practical application of Equation (92), however, is rather limited since the diffusion coefficient and the distribution coefficient are not
constant but depend strongly on the concentration of the permeating component in the mem-
where P is the permeability of the membrane, S
the membrane selectivity, and j and k refer to the
components of the mixture.
The separation factor in pervaporation of a
two-component mixture is given by:
aj;k ¼
Xjp Xkp
Xjl Xkl
¼ Sj;k
Xjp jpj Xjl g lj poj Xkl
Xkp jpk Xkl g lk pok Xjl
ð95Þ
where a is the separation factor, S the membrane
selectivity, X the molar fraction, p the pressure,
and g the activity coefficient, subscripts j and k
refer to components of the mixture, and superscripts o, l, and p to saturation pressure, feed, and
permeate, respectively.
In pervaporation the partial pressure of the
components on the permeate side is kept as low as
possible by either using a sweeping gas or more
commonly by applying a vacuum, i.e.,
Xjp pp Xjl g lj Xj0 and Xkp pp Xkl g lk Xk0 . For a
pervaporation experiment using a vacuum the
separation factor is to a first approximation:
lim
aj;k ¼ Sj;k
p
p !0
g lj p0j
g lk p0k
ð96Þ
Thus, the separation factor consists of two terms:
Sj,k, which represents the membrane selectivity,
and the thermodynamic liquid/vapor equilibrium
g lj poj
,
g lk plk
which represents the separation that would
be achieved by distillation. In pervaporation the
membrane selectivity can increase or decrease
the distillation selectivity and eventually push the
overall separation factor in the opposite direction.
3.6. Principle of Dialysis
Dialysis is carried out under isobaric conditions,
and there is no temperature or electrical potential
444
Membranes and Membrane Separation Processes, 1. Principles
Vol. 22
where J is the flux, L a phenomenological coefficient, m the chemical potential, a the activity, C
the concentration, g the activity coefficient, z the
directional coordinate perpendicular to the membrane, and the subscript i refers to a component.
Assuming that the activity coefficient is 1 and
replacing the phenomenological coefficient by
the diffusion coefficient according to Equation
(18) leads to:
Ji ¼ L i
Figure 13. Schematic illustrating the principle of dialysis
with two neutral components of different size
gradient. The principle of the process is depicted
in Figure 13.
The schematic drawing shows a liquid
solution containing a solvent and molecular components of different molecular weight or size,
referred to as feed solution separated by a membrane from the solvent referred to as dialysate or
‘‘stripping’’ solution. Due to the driving force of a
concentration difference the components will
diffuse through the membrane from the feed
solution to the dialysate. Their fluxes through
the membrane and thus their separation is determined by their diffusion rate in the membrane
matrix.
There are different forms of dialysis depending on the constituents of the feed solution and
the structure of the membrane. The simplest form
is the dialysis of neutral components.
3.6.1. Dialytic Mass Transport of Neutral
Components
The mass flux in dialysis of a mixture, which
contains neutral components only, can be expressed by the phenomenological equation,
which can then be related to Fick’s law by several
approximations.
For a membrane and a feed solution with
uncharged components the flux of a component
at constant temperature and pressure is given by:
Ji ¼ Li
dmi
d ln ai
¼ Li RT
¼ Li RT d ln Ci g i
dz
dz
ð97Þ
RT dCi
dCi
¼ Di
dz
Ci dz
ð98Þ
Equations (97) and (98) indicate that the mass
transport in dialysis with feed solutions containing neutral components can be described adequately by Fick’s law if there is no coupling of
fluxes and if the activity coefficient of the components in the membrane is unity, which is the
case in many practical applications.
3.6.2. Dialytic Mass Transport of
Electrolytes in a Membrane without
Fixed Ions
If feed solutions with charged components (i.e.,
electrolyte solutions) are treated by dialysis, the
electroneutrality requirement must be satisfied.
For a system consisting of a membrane and
solutions with various ions the electroneutrality
condition requires:
X
jza jCa ¼
a
X
zc Cc
ð99Þ
c
where z is the valance, C the concentration, and
subscripts a and c refer to anion and cation,
respectively.
If the membrane contains fixed ions they must
be included in the electroneutrality requirement.
The fixed charges are negative in a cationexchange membrane and positive in an anionexchange membrane. If there is no electrical
current the conservation of electrical charge
requires that cations and anions move in the same
direction. Hence:
X
a
jza jJa ¼
X
zc Jc
ð100Þ
c
The concentration of the electrolyte (e.g., a single
salt) is related to that of the ions by:
Ca ¼ ana Cs and Cc ¼ anc Cs
ð101Þ
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
where a is the degree of dissociation of the
electrolyte, n a stoichiometric coefficient, and
subscript s refers to the electrolyte.
The stoichiometric coefficient relates the
number of ions to that of the salt molecules. For
a completely dissociated monovalent salt such as
NaCl a and n are 1. For a salt such as CaCl2 nc is 1
and na is 2.
In an electrolyte solution the fluxes of the ions
are the result of an electrochemical potential
gradient driving force. The electrochemical potential of ions is related to that of the electrolyte
by [56]:
hs ¼ na ha þnc hc
ð102Þ
where h is the electrochemical potential.
Since in dialysis no external potential and no
hydrostatic pressure difference is applied the
flux of the ions can be described as a function
of the diffusion coefficient and the concentration gradient by Equation (98) if the activity
coefficients are assumed to be 1. However, the
fluxes of the ions are related to each other by
the electroneutrality requirement and the conservation of charges, with the consequence that
in diffusion of an electrolyte which consists of
a cation and an anion with significantly different diffusivities the faster moving ion will
cause a minute charge imbalance and thus an
electrical potential gradient. This potential gradient is referred to as diffusion potential, and
although it may be very small it will slow down
the faster diffusing ion and speed up the slower
ion so that both ions diffuse with the same
velocity.
The flux of the completely dissociated monovalent electrolyte is related to that of the ions by:
Js ¼ Ja ¼ Jc ¼ Ds
dCs
dCa
dCc
¼ Da jza j
¼ Dc zc
dz
dz
dz
ð103Þ
where J is the flux, C the concentration, D the
diffusion coefficient, z the valence or the directional coordinate, and subscripts s, a, and c refer
to salt, anion, and cation, respectively.
Rearranging Equation (103) under consideration of Equation (102) gives the diffusion coefficient of the salt as a function of the diffusion
coefficients of the cations and anions:
Ds ¼
Da Dc ðjza jþzc Þ
Da jza jþDc zc
ð104Þ
445
Combination of Equations (103) and (104) gives
the salt and ion fluxes for a completely dissociated monovalent salt at constant temperature and
pressure in an ideal solution, that is, the activity
coefficients are 1:
Js ¼
Da Dc ðjza jþzc Þ dCs
Da jza jþDc zc dz
ð105Þ
where J is the flux, C the concentration, D the
diffusion coefficient; z the valence or the directional coordinate, and subscripts s, a, and c refer
to salt, anion, and cation, respectively.
Equation (105) describes the flux of an electrolyte as a function of the diffusion coefficients
of the individual ions. This is the consequence of
the electroneutrality requirement for the transport of ions which have different diffusivities.
Because the faster moving ion will cause the
buildup of a minute electrical potential gradient
which will slow down the faster ion and speed up
the slower ion, at steady state both ions move
with the same velocity. Therefore, an average
diffusion coefficient for each ion can be defined
to describe the fluxes of the different ions, which
are coupled by the conservation of charges and
the electroneutrality condition. The average dif given by:
fusion coefficient Dis
Dc ¼Da ¼ Ds ¼
Da Dc jza jþzc
Da jza jþzc Dc
ð106Þ
Equations (105) and (106) respectively describe
the transport of a single electrolyte under the
driving force of a concentration or activity gradient through a membrane that does not carry any
fixed charges. In this case all ions of the electrolyte are transported in the same direction even if
their diffusion coefficients are quite different.
The situation, however, becomes more complex
if a feed solution contains more than one electrolyte and if the membrane carries fixed charges.
In this case certain ions may be transported
against their concentration gradient.
3.6.3. Dialysis of Electrolytes in IonExchange Membranes
The fluxes of ions through ion-exchange membranes in dialysis are even more complex and
may result in transport of certain ions against
their concentration gradient, as illustrated in
446
Membranes and Membrane Separation Processes, 1. Principles
Vol. 22
the Naþ ion flux caused by the activity gradient is
identical to that caused by the Hþ ion flux
equilibrium is reached and there will be no
further ion transport through the membrane. This
equilibrium, which is referred to as Donnan
equilibrium, is given by:
dlnaHþ ¼ dlnaNaþ
ð108Þ
The Donnan equilibrium between two solutions
with different Naþ ion and Hþ ion concentrations
separated by a permselective cation-exchange
membrane is:
0
Figure 14. Schematic illustrating the fluxes across an ionexchange membrane which separates two electrolyte solutions of different concentrations containing HCl and NaCl.
a) Shows the fluxes through a cation-exchange membrane
and represents countercurrent coupled flux, i.e., Donnan
dialysis, and b) shows the fluxes through an anion-exchange
membrane and represents cocurrent coupled transport, i.e.,
diffusion dialysis
Figure 14. Figure 14a shows countercurrent coupled transport of Hþ and Naþ ions through a
cation-exchange membrane. This process is referred to as Donnan dialysis. Figure 14b depicts
cocurrent transport of Hþ ions and Cl ions
through an anion-exchange membrane.
In Donnan dialysis with a strictly permselective ion-exchange membrane the flux of the coion (i.e., in Figure 14a the Cl ions) is 0 and the
fluxes of the counterions (i.e., the Naþ and Hþ
ions) are equal but in opposite directions. Thus,
there is no net transport of electrical charges
across the membrane, that is, there is no electrical
current and the conservation of charges is fulfilled. For the system illustrated in Figure 14a the
transport of a counterion (i.e., the Naþ ion) is the
result of a flux of another counterion (i.e., the Hþ
ion) in the opposite direction under the driving
force of a concentration gradient. Since no coions (i.e., Cl ions) can permeate the cationexchange membrane, the flux of the Hþ ions
must be identical to that of the Naþ ions. The
fluxes are given by:
LHþ LNaþ RT dlnaHþ
¼ JNaþ
JHþ ¼ 2
LHþ þLNaþ
dz
ð107Þ
The transport of Naþ ions due to the Hþ ion flux
results in a buildup of an activity gradient of the
Naþ ions, which causes a backtransport. When
0
aNaþ aHþ
¼ 00
00
aNa
aHþ
þ
ð109Þ
where 0 and 00 refer to the two solutions separated
by the membrane.
Donnan dialysis plays an important role in
water softening, and Equation (109) determines
the minimum and maximum concentrations of a
counterion which can be achieved by a given flux
of another counterion.
Cocurrent coupled transport is illustrated in
Figure 14b, which shows a solution containing a
mixture of a salt (NaCl) and an acid (HCl),
separated by an anion-exchange membrane from
a solution containing no salt. The Cl ion concentration in the solution containing the salt –
acid mixture is assumed to be much higher than in
the solution containing only the acid. Due to a
concentration gradient the Cl ions permeate the
anion-exchange membrane towards the solution
containing only the acid. Because of the electroneutrality requirement an anion can permeate the
membrane only when accompanied by a cation.
The anion-exchange membrane, however, is impermeable to cations such as Naþ ions but it is to
some extent permeable to Hþ ions because protons are transported in aqueous solution and also
in highly swollen ion-exchange membranes by a
different transport mechanism. The Cl ion concentration gradient across the membrane acts as
driving force for the transport of Hþ ions, which
are transported against their concentration gradient. The minimum concentration of the acid in
the salt solution and the maximum concentration
of the acid that can be achieved is given when the
activity gradients of the Hþ ion and the Cl ion
across the membrane are identical. The cocurrent
transport of an anion with a Hþ ion is of practical
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
importance for the recovery of acids from spent
pickling solutions containing the acid in a mixture with salts.
3.7. Principle of Electrodialysis
The principle of the electrodialysis is illustrated
in Figure 15.
In electrodialysis cation-exchange membranes carrying fixed negative electric charges
and anion-exchange membranes carrying positive fixed charges are placed in alternating series
between two electrodes. Cation- and anion-exchange membranes form individual cells filled
with an electrolyte-containing feed solution. If an
electric potential between the electrodes is established the cations migrate towards the negatively
charged cathode while the anions migrate towards the positively charged anode. Cations can
easily permeate a cation-exchange membrane
but are retained by an anion-exchange membrane, while anions permeate the anionexchange membrane but are retained by the
cation-exchange membrane. Thus, ions are accumulated in alternating cells to form the
concentrate solution while the other cells are
depleted and form the diluate solution.
In electrodialysis mass transport is the result
of an electrochemical potential gradient across
the membrane, that is, a gradient in the chemical
and electrical potential. An electrical potential
driving force acts on charged components only
and in such a way that negatively charged com-
447
ponents (anions) migrate in the direction towards
the positively charged anode and the positively
charged cations migrate towards the negatively
charged cathode, that is, cations and anions move
in opposite directions.
To describe the mass transport of a salt in an
aqueous solution through a membrane which is
used as frame of reference three independent
fluxes must be considered: the flux of the cations,
the flux of anions, and the flux of the solvent. The
transport of ions result from an electrochemical
potential difference, and the transport of the
solvent through the membrane from osmotic
effects and coupling with the fluxes of the ions.
Under certain conditions the solvent flux can be
significant and must be taken into account.
Ion-exchange membranes carry positive or
negative electrical charges fixed to a solid matrix
and are therefore permselective as far as the
transport of ions is concerned, that is, the cation-exchange membranes are preferentially permeable to cations and the anion-exchange membranes are preferentially permeable to anions.
Mass transport in electrodialysis can be described by a phenomenological equation giving
the flux of the individual components by:
Ji ¼
X
k
Lik
dhk X
dlnak
dj
¼
þzi F
Lik RT
dz
dz
dz
k
ð110Þ
Assuming an ideal solution, that is, one in which
the activity of a component is identical to its
concentration and without kinetic coupling between individual components, and expressing
the phenomenological coefficient by the Fick’s
Figure 15. Schematic illustrating the principle of electrodialysis
448
Membranes and Membrane Separation Processes, 1. Principles
diffusion coefficient, Equation (110) becomes
identical to the Nernst – Planck flux equation,
which is given by [57]:
Ji ¼ Di
dCi zi FCi
þ
dz
RT
dj
dz
ð111Þ
where Di is the diffusion coefficient of component i with reference to the membrane.
i
The first term in Equation (111) Di dC
dz represents the diffusion, and the second term
dj
Ci F
Di ziRT
dz the migration.
Thus, the Nernst – Planck equation is an approximation of the more general phenomenological equation.
In ion-exchange membrane separation processes such as electrodialysis the flux of the
cations through the cation-exchange membrane
and the flux of the anions through the anionexchange membrane are interdependent because
of the electroneutrality requirement in the solution between the membranes, as can easily be
seen when considering the electrodialysis of a
solution of a monovalent single electrolyte such
as NaCl. If strictly permselective membranes are
assumed the fluxes of the cations through the
cation-exchange membrane and of the anions
through the anion-exchange membrane are
equal. Thus, in a solution containing a monovalent, completely dissociated electrolyte such as
NaCl and strictly permselective membranes it is
[22, 56]:
cm
cm
DNaþ am DCl
J Naþ¼ am J Ci ¼ Js ¼ 2 cm
DNaþ þam DCl
RT
dim C i
dj
þF
im C
dz
dz
i
ð112Þ
where cmJNaþ and amJCl are the fluxes of Naþ
ions through the cation-exchange membrane and
of Cl ions through the anion-exchange membrane, cmDNaþ and amJCl are the diffusion coefficients for the transport of Naþ and Cl ions
through the corresponding membrane, subscript i
refers to cation or anion, and superscript im to
cation- or anion-exchange membrane.
3.7.1. Electrical Current and Ion Fluxes
In electrodialysis it is assumed that the total
current through the membrane is carried by ions
Vol. 22
only:
i¼
X
I
jzi jJi
¼F
A
i
ð113Þ
where i is the current density, I the current, A the
membrane surface, F the Faraday constant, J the
flux, z the valence, and subscript i refers to
cations and anions.
The current density i can be related to the
specific conductivity k by:
i¼k
dj
dz
ð114Þ
where k is the specific conductivity, j the electrical potential, and z the directional coordinate.
The specific conductivity k can be expressed
by the specific electrical resistance, the equivalent conductivity, the ion migration velocity, or
the ion diffusivity:
k¼
X
X
1 X
Di
jzi jCi li ¼ F
jzi jCi ui ¼ F 2
jzi jCi
¼
RT
r
i
i
i
ð115Þ
where r is the specific resistance, C the concentration, F the Faraday constant, l the equivalent
conductivity, u the ion migration velocity, D the
ion diffusion coefficient, R the gas constant, T the
absolute temperature, and subscript i refers to
anions and cations.
Introducing Equations (114) and (115) into
Equation (113) and rearranging leads to:
i¼F
X
i
jzi jJi ¼ F 2
X
i
z2i
Ci Di RT
RT zi Ci F
dCi d j
þ
dz dz
ð116Þ
where i is the current density, C the concentration, F the Faraday constant, j the electrical
potential, z the valence, D the ion diffusivity,
R the gas constant, T the absolute temperature,
and subscript i refers to anions and cations.
dCi
The term ziRT
Ci F
d z has the dimensions of an
electrical potential gradient and represents the
concentration potential which is established between two electrolyte solutions of different concentrations. In electrodialysis this potential is
established between the concentrate and the diluate solution. It represents an electromotive force
which has to be overcome by the applied electrical potential.
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
3.7.2. Transport Number and Membrane
Permselectivity
In an electrolyte solution the current is carried by
both ions. However, cations and anions usually
carry different portions of the overall current. In
ion-exchange membranes the current is carried
preferentially by the counterions. The fraction of
the current that is carried by a certain ion is
expressed by the ion transport number, which is
given by:
jzi jJi
Ti ¼ P
j jzj jJj
ð117Þ
where Ti is the transport number of component i,
Ji its flux, zi its valence, and subscript j refers to
all ions involved in the charge transport.
The transport number Ti indicates the fraction of the total current that is carried by ion i;
the sum of the transport numbers of all ions in a
solution is 1.
The membrane permselectivity is an important parameter for determining the performance
of a membrane in a certain ion-exchange membrane separation process. It describes the degree
to which a membrane passes an ion of one
charge and retains an ion of the opposite charge.
The permselectivity of cation- and anionexchange membranes can be defined [58] by
the following relations:
449
in the membrane is identical to that in the electrolyte solution, that is, for Tccm ¼ Tc, Ycm ¼ 0.
For the anion-exchange membrane, the corresponding relation holds.
The transport number of a certain ion in the
membrane is proportional to its concentration in
the membrane, which again is a function of its
concentration in the solutions in equilibrium with
the membrane phase, due to Donnan exclusion,
as discussed above. The concentration of a co-ion
in an ion-exchange membrane can be calculated
from the Donnan equilibrium, which is given in
Equation (29). For a monovalent salt and a dilute
salt solution and assuming the activity coefficients of the salt in the membrane and the solution
to be 1, the co-ion concentration is given to a first
approximation by:
m
Cco ¼
s
C 2s
Cfix
ð120Þ
where C is the concentration, subscripts co, s, and
fix refer to co-ion, salt, and fixed ion of the
membrane, and superscripts s and m to membrane and solution.
Equation (120) indicates that the co-ion concentration in the membrane increases with increasing solution salt concentration, and the
membrane permselectivity decreases correspondingly.
Ycm ¼
Tccm Tc
Ta
ð118Þ
3.7.3. Membrane Counterion
Permselectivity
Yam ¼
Taam Ta
Tc
ð119Þ
The transport number of different counterions in
the membrane can be quite different. The transport rates of ions are determined by their concentration and their mobility in the membrane.
The concentration of the counterions is always
close to the concentration of the fixed charges of
the membrane. The mobility of the ions in the
membrane depends mainly on the radius of the
hydrated ions and the membrane structure. The
mobilities of different ions in an aqueous solution
do not differ very much from each other. An
exception are the Hþ and OH ions, the mobility
of which is higher than that of other ions by a
factor of about 5 to 6. The exceptionally high
mobility of the Hþ ion can be explained by the
transport mechanism of protons. Salt ions move
with their hydration shell through the solution.
Protons interact with water dipoles and form
where Y is the permselectivity of a membrane, T
the transport number, superscripts cm and am
refer to cation- and anion-exchange membranes,
and subscripts c and a to cation and anion,
respectively.
The permselectivity of an ion-exchange membrane relates the transport of electric charges by a
specific counterion to the total transport of electric charges through the membrane and the transport number of the ion in the solution. An ideal
permselective cation-exchange membrane
would transmit positively charged ions only, that
is, for a transport number of a counterion in a
cation-exchange membrane of Tccm ¼ 1 the
permselectivity Ycm ¼ 1. The permselectivity
approaches zero when the transport number with-
450
Membranes and Membrane Separation Processes, 1. Principles
hydronium ions, which are transported in the
membrane mostly by a so-called tunnel mechanism. The transport mechanism of the proton is
also the reason for the high permeability of
anion-exchange membranes to protons, while
these membranes generally have a very low
permeability for salt cations. The same mechanism as that described here for the transport of
protons also holds for the transport of hydroxide
ions, and thus the permeability of hydroxide ions
in a cation-exchange membrane is much higher
than that of other salt anions.
The permselectivity of an ion-exchange membrane for different salt counterions is determined
by the concentration and the mobility of the
different ions in the membrane. The concentrations of the different counterions in the ionexchange membrane are generally also different.
A typical counterion-exchange sequence of a
cation-exchange membrane containing SO3
groups as fixed charges is:
Ba2þ > Pb2þ > Sr2þ > Ca2þ > Mg2þ
þ
þ
> Agþ > Kþ > NHþ
4 > Na > Li
ð121Þ
A similar counterion-exchange sequence is obtained for anions in an anion-exchange membrane containing quaternary ammonium groups
as fixed charges [22, 56]:
2
2
I > NO
3 > Br > Cl > SO4 > SO4 > F
ð122Þ
Vol. 22
and the current density. In a highly permselective
membrane and for moderate differences in salt
concentration in the two solutions separated by
the membrane, the electroosmotic flux dominates
and is generally much higher than the osmotic
solvent flux. In electrodialysis the water flux due
to electroosmosis can be expressed by a solvent
transport number, which gives the number of
water molecules transported by one ion:
Jw ¼ mT w
X
Ji
ð123Þ
i
where mTw is the water transport number, Jw the
water flux, and Ji the flux of ions through a given
membrane.
The water transport number is thus:
m
¼ Jw
T wP
i Ji
ð124Þ
The water transport number is the number of
water molecules transferred by one ion through
a given membrane. It depends on the membrane
and on the electrolyte, that is, on the size of the
ions, their valence, and their concentration in
the solution. In aqueous salt solutions and commercial ion-exchange membranes the water
transport number is on the order of 4 to 8, that
is, 1 mol of ions transports ca. 4 – 8 mol of
water through a typical commercial ionexchange membrane.
3.7.4. Water Transport in Electrodialysis
3.8. Electrodialysis with Bipolar
Membranes
Water transport in electrodialysis from the diluate to the concentrate process stream can effect
the process efficiency significantly. If a convective flux as a result of pressure differences between flow streams can be excluded there are still
two sources for the transport of water from the
diluate to the concentrate solution. The first is the
result of osmotic pressure differences between
the two solutions, and the second is due to
electroosmosis, which results from the coupling
of water to the ions being transported through the
membrane due to the driving force of an electrical potential.
Each of the two fluxes may be dominant,
depending on the permselectivity of the ionexchange membrane, the concentration gradient,
Electrodialysis with bipolar membranes has
gained increasing attention as an efficient tool
for the production of acids and bases from the
corresponding salts [59]. This process is economically attractive and has many potential applications. A typical arrangement of an electrodialysis
stack with bipolar membranes is illustrated in
Figure 16, which shows the production of an acid
and a base from the corresponding salt in a
repeating cell unit which consists of three individual cells containing the salt solution, the acid,
and the base, and three membranes (a cationexchange, an anion-exchange, and a bipolar
membrane); 50–100 repeating cell units may be
placed between two electrodes in industrial-scale
stacks.
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
451
Figure 16. Schematic illustrating acid and base production from the corresponding salt by electrodialysis with bipolar
membranes
The key element in electrodialysis with bipolar membranes is the bipolar membrane itself. Its
function is illustrated in Figure 17, which shows a
bipolar membrane consisting of an anion- and a
cation-exchange layer arranged in parallel between two electrodes. If a potential difference is
established between the electrodes, all charged
components will be removed from the interface
between the two ion-exchange layers. If only
water is left between the membranes, further
transport of electrical charges can be accomplished only by protons and hydroxyl ions, which
have a concentrations of about 10–7 mol L1 in
completely deionized water. Protons and hydroxyl ions removed from the interface between the
cation- and anion-exchange membranes are replenished because of the water dissociation equilibrium:
2 H2 O H3 Oþ þOH
ð125Þ
The rate at which protons and hydroxide ions are
removed from the bipolar membrane, that is, the
Figure 17. Schematic illustrating the function of a bipolar
membrane
maximum electrical current, is limited by the rate
of water diffusion into the bipolar membrane and
the water dissociation rate. The water dissociation is given by:
JHþ ¼ JOH ¼ kd CH2 O 2l
ð126Þ
where J is the maximum ion flux from the bipolar
membrane into the outer phases, kd the water
dissociation rate constant, C the concentration in
the transition layer, and 2l is the thickness of the
transition layer.
The water dissociation rate constant kd at 25 C
is 2.5105 s1. The water concentration in pure
water is 55.5 mol L1. Assuming a thickness of
the transition layer of 1 nm, the maximum ion
fluxes JHþ and JOH , respectively, calculated by
Equation (127), would be 1.4109 mol m2
s1. The electrical current density i is given by:
i¼F
X
zi Ji :
ð127Þ
Thus, the ion fluxes JH and JOH would correspond to current density of about 1.4 104 A
m2. In practice, however, bipolar membranes
can be operated at current densities in excess of
1000 A m2 [60]. This means that the simple
model of a bipolar membrane is incorrect and
that water dissociation is at least 106 times faster
in bipolar membranes than in solution. The actual mechanism of water dissociation, which to a
large extent determines the overall efficiency of
the process, is still the subject of controversial
discussion, and various models are suggested in
the literature to explain the accelerated water
452
Membranes and Membrane Separation Processes, 1. Principles
dissociation which is observed in the practical
application of bipolar membranes [61–63].
The energy required for the production of
acids and bases in a bipolar membrane at constant
temperature and pressure can be calculated from
the concentration chain of solutions with different Hþ ion activities. The Gibbs free energy of
water dissociation in a bipolar membrane can be
calculated from the difference in the pH values of
the solution in the phases between the cation- and
anion-exchange layers of the bipolar membrane
and the solutions outside the bipolar membrane
and is given by:
DG ¼ RTln
aiHþ aiOH
Kw
¼ RTln o o ¼ 2:3RT DpH ¼ FDj
aoHþ aoOH
aHþ aOH
ð128Þ
where DG is the Gibbs free energy, R the gas
constant, T the absolute temperature, F the Faraday constant, a the activity of Hþ and OH ions,
Dj the electrical potential difference between the
solution in the bipolar membrane and the solutions outside the bipolar membrane, DpH the
difference in pH value of the two solutions
separated by the bipolar membrane, Kw the dissociation constant of water, and superscripts i and
o refer to the interface between the cation- and
anion-exchange layers of the bipolar membrane
and the outside solutions, respectively.
At 25 C log Kw is 13.997. The free energy
for the dissociation of 1 mol of water, and thus
for the production of 1 mol L1 acid and base at
25 C, is 79.887 kJ or 0.0222 kWh. The voltage
drop between the two outside solutions is
0.828 V. The actual potential drop across the
bipolar membrane is higher due to the electrical
resistance of the membranes and the solutions.
Vol. 22
ferent components permeating through a membrane consists of two parts. The first part is the
reversible energy given by the Gibbs free energy
obtained when the retentate and permeate solutions are mixed. It is the minimum energy required for the separation of components in a
mixture. It is identical for any separation process.
The second part is the energy required to overcome the friction between the transported components and the membrane matrix. It represents
the irreversible energy loss.
The minimum work that is required for the
separation of various components from a mixture
in a membrane process is given by the Gibbs free
energy of mixing. which is given at constant
pressure and temperature by:
0
00
ðdDGm Þp;T ¼ DG DG
0
X 0
X
ai
0
00
00
¼
ðm i Dn i m i Dn i Þ ffi RT
Dni ln 00
ai
i
i
where DGm is the Gibbs free energy of mixing
two solutions indicated by 0 and 00 , m the chemical
potential, a the activity, n the number of moles,
and subscript i refers to individual components.
For filtration processes such as reverse osmosis the energy requirement for removing the
solvent from the solution can be expressed by
the difference in the osmotic pressure of the two
solutions, which for a strictly semipermeable
membrane is given by:
Dp ¼
RT
Vw
0
lnaw
lna00w
The energy required in membrane processes is
the sum of two terms: 1) the energy to transfer the
components from the feed mixture through a
membrane into the permeate mixture, and 2) the
energy required to pump the solutions through
the membrane apparatus and to operate the process monitoring and control devices. The second
term depends on the plant capacity and process
design. The energy required for separating dif-
ð130Þ
where the subscript w refers to the solvent, for
example, water.
Introducing Equation (130) into Equation
(129) leads to:
DGm ¼ Dnw Vw Dp ffi Vf Dp
3.9. Energy Requirements of
Membrane Processes
ð129Þ
ð131Þ
where Dp is the osmotic pressure difference
between the two solutions 0 and 00 separated by
the membrane, and Vw and Vf are the partial
molar volumes of the solvent and the filtrate,
which for a dilute solution is to a first approximation given by DnVw ffi Vf.
In electrodialysis the change of the Gibbs free
energy for converting a feed solution into a
concentrate and a diluate is given by [56]:
Cc
Cd
DGtotal ¼ nd RT ncs ln sf þnds ln sf
Cs
Cs
ð132Þ
Vol. 22
Membranes and Membrane Separation Processes, 1. Principles
where DGtotal refers to the Gibbs free energy
change required for the production of the diluate,
C the salt concentration, nd the stoichiometric
dissociation coefficient, and superscripts f, c, and
d refer to the feed, the concentrate, and the
diluate.
The minimum energy requirement for the
separation of molecular mixtures such as salt
solutions, discussed above refers to an idealized,
completely reversible process. In practice, such
conditions never prevail and the energy requirements are usually significantly higher and rather
different in different membrane separation processes. They are also a function of the membrane
module construction and the overall process
design. The practical energy requirements in the
different processes are discussed in ! Membrane Separation Processes, 2. Design and
Operation.
4. List of Symbols
A
a
a
C
D
D
D
Ð
d
dH
d
E
F
f
DGm
g
g
h
I
i
ilim
J
Kw
k
k
L
Lpp
M
m
N
N
n
P
m2
mol m3
–
mol m3, mol L1
m
m s1
m2 s1
m2 s1
m
m
m
AVs
96485 A s eq1
N s m1 mol1
J
kg m s2
–
m
A
A m2
A m2
mole m2 s1
mol2 L2
various
J mol1 K1
mol2 N1 m2 s1
m
kg mol1
m s1
mol s1
–
–
mol m1 s1 J1
surface area
activity
constant
concentration
batch cell diameter
dialysance
diffusion coefficient
Stefan-Maxwell diffusion coefficient
distance
hydraulic diameter
stirrer length
energy
Faraday constant
friction coefficient
Gibbs free energy of mixing
acceleration
osmotic coefficient
height
current
current density
limiting current density
flux
water dissociation constant
coefficient
Boltzmann constant
phenomenological coefficient
process path length
molecular weight
mobility
flux
number
number of moles
permeability
p
Q
R
R
R
Rm
r
r
r
Re
S
Sc
Sh
T
T
t
U
u
u
V
V
v
w
X
X
X
x
Y
Zb
z
z
Pa
m3 s1
W
J mol1 K1
–
N m s1
W m2
N s1
m
–
–
–
–
K
–
s
V
m2 s1 V1
m s1
m3
m3 mol1
m s1
m
m
Nm mol1
–
m
m
m
m
eq mol1
453
pressure
volume flux
electrical resistance
gas constant
rejection
membrane resistance
area resistance
specific resistance
radius
Reynolds number
permselectivity
Schmidt number
Sherwood number
temperature
transport number
time
electrical potential
ion mobility
velocity
volume
partial molar volume
velocity
width
cell length
driving force
mole fraction
directional co-ordinate
cell width
boundary layer thickness
directional coordinate
valence
Greek letters
a
a
b
g
D
D
D
d
e
h
h
q
k
Leq
l
l
m
n
n
x
p
r
r
s
s
t
j
j
–
–
–
m3 mol1
m
–
–
–
–
kg m1 s1
A V s mol1, J mol1
–
W1 m1
m2 W1 eq1
m
m2 W1 eq1
J mol1
m2 s1
–
–
Pa, N m2
kg m3
Wm
mol N1 s1
N m1
–
V
separation factor
dissociation degree
enrichment factor
activity coefficient
cell thickness
difference
recovery rate
fractional product loss
porosity
dynamic viscosity
electrochemical potential
stage cut
specific conductivity
electrolyte equivalent conductivity
distance
ion equivalent conductivity
chemical potential
kinematic viscosity
stoichiometric coefficient
current utilization
osmotic pressure
density
specific resistance
reflection coefficient
surface tension
tortuosity factor
contact angle
electrical potential
454
j
f
Y
Y
w
w
w
Membranes and Membrane Separation Processes, 1. Principles
–
–
N m s1
–
s1
mol m1
–
fugacity coefficient
pressure ratio
dissipation function
membrane permselectivity
circular velocity
permeability
porosity
Subscripts
A
a
B
c
cell
co
con
D
Don
des
diff
e
fix
g
gas
i
j
k
l
m
p
p
pro
s
s
s
spec
st
tr
v
w
o
component
anion
component
cation
dialysis cell
co-ion
convection
diffusion
Donnan potential
desalination
diffusion
electric charge
fixed ion
gas
gas
component
component
component
feed
membrane
pore
pressure
product
salt
solute
solution
specific
stack
transition layer
volume
water, solvent
standard state
Superscripts
am
b
b
bml
c
c
cm
D
d
d
d
f
fc
anion-exchange membrane
bulk solution
constant
bipolar membrane
concentrate
constant
cation-exchange membrane
diffusion
constant
dialysate
diluate
feed
feed concentrate
fd
g
in
k
m
out
p
p
r
s
t
tr
w
_
o
Vol. 22
feed diluate
gas
inlet
Knudsen diffusion
membrane
outlet
product
permeate
retentate
solution
time
transition region
membrane wall
average number
standard state
References
General References
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9 W. Nernst, Z. Phys.Chem. 2 (1888) 613.
10 M. Planck, Ann. Phys. Chem. N.F. 39 (1890) 161.
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12 H. Bechhold: ‘‘Durchl€assigkeit von Ultrafilter’’, Z. Phys.
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13 R. Zsigmondy, W. Bachmann: ‘‘Über neue Filter’’, Z.
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14 W. J. Elford: ‘‘A New Series of Graded Collodion Membranes Suited for General Bacteriological Use, Especially
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Membranes and Membrane Separation Processes, 1. Principles
19 A. J. Staverman: ‘‘Non-Equilibrium Thermodynamics of
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24 F. Helfferich, Ion-Exchange, McGraw-Hill., New York,
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27 R. E. Kesting: Synthetic Polymeric Membranes, McGrawHill, New York, NY, 1971.
28 H. Strathmann, K. Kock, P. Amar, R. W. Baker: ‘‘The
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29 J. E. Cadotte, R. I. Petersen: ‘‘Thin Film Reverse Osmosis
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31 US 3 417 870, 1968 (T. D. Bray).
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34 K. H. Meyer, W. Strauss, Helv. Chim. Acta 23 (1940) 795.
35 W. Juda, W. A. McRae: ‘‘Coherent Ion-exchange Gels
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37 T. Nishiwaki: ‘‘Concentration of Electrolytes Prior to
Evaporation with an Electromembrane Process’’, in R.
E. Lacey, S. Loeb (eds.): Industrial Processing with
Membranes, Wiley Interscience, New York, NY, 1972,
pp. 83–106.
38 K. J. Lui, F. P. Chlanda, K. J. Nagasubramanian: ‘‘Use of
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455
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43 N. N. Li: ‘‘Permeation Through Liquid Surfactant Membranes’’, AIChE J. 17 (1971) 459.
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456
Membranes and Membrane Separation Processes, 1. Principles
61 S. Mafe, P. Ramirez, A. Alcaraz, V. Aguilella: ‘‘Ion
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Further Reading
E. Drioli, L. Giorno: Membrane operations, Wiley-VCH,
Weinheim 2009.
Vol. 22
B. D. Freeman, €e. P. €eIı̀Ampol’skii: Membrane gas separation, Wiley, Hoboken, NJ 2010.
S. P. Nunes, K.-V. Peinemann: Membrane technology in the
chemical industry, 2., rev. and extended ed., Wiley-VCH,
Weinheim 2006.
A. K. Pabby S. H. Rizvi (Editor), A M Sastre (Editor)
Handbook of Membrane Separations: Chemical, Pharmaceutical, Food, and Biotechnological Applications
CRC Press, Boca Raton 2007.
Y. P. Yampolskii: Materials science of membranes for gas
and vapor separation, Wiley, Chichester 2006
A. B. Koltuniewicz, E. Drioli: Membranes in clean technologies, Wiley-VCH, Weinheim 2008.