MODELING OF ELECTROLYTE SORPTION FROM PHASE EQUILIBRIA TO DYNAMIC SEPARATION SYSTEMS

MODELING OF ELECTROLYTE SORPTION

– FROM PHASE EQUILIBRIA TO DYNAMIC

SEPARATION SYSTEMS

Acta Universitatis

Thesis for the degree of Doctor of Science (Technology) to be presented with due permission for public examination and criticism in the Auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 2nd of December, 2011, at noon.

Supervisor Associate Prof. Tuomo Sainio Laboratory of Industrial Chemistry Department of Chemical Technology Faculty of Technology

Lappeenranta University of Technology Finland

Reviewers Prof. Dmitri Muraviev

Analytical Chemistry Division Autonomous University of Barcelona Spain

Dr. Anatoli Kalinitchev Institute of Physical Chemistry Russian Academy of Sciences Russian Federation

Opponent Prof. Dmitri Muraviev

Analytical Chemistry Division Autonomous University of Barcelona Spain

ISBN 978-952-265-162-4

ISBN 978-952-265-163-1 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto

Digipaino 2011

ABSTRACT

Markku Laatikainen

Modeling of electrolyte sorption – from phase equilibria to dynamic separation systems

Lappeenranta 2011 121 pages

Acta Universitatis Lappeenrantaensis 450 Diss. Lappeenranta University of Technology

ISBN 978-952-265-162-4, ISBN 978-952-265-163-1 (PDF), ISSN 1456-4491

In this thesis, general approach is devised to model electrolyte sorption from aqueous solutions on solid materials. Electrolyte sorption is often considered as unwanted phenomenon in ion exchange and its potential as an independent separation method has not been fully explored. The solid sorbents studied here are porous and non-porous organic or inorganic materials with or without specific functional groups attached on the solid matrix. Accordingly, the sorption mechanisms include physical adsorption, chemisorption on the functional groups and partition restricted by electrostatic or steric factors. The model is tested in four Cases Studies dealing with chelating adsorption of transition metal mixtures, physical adsorption of metal and metalloid complexes from chloride solutions, size exclusion of electrolytes in nano-porous materials and electrolyte exclusion of electrolyte/non-electrolyte mixtures. The model parameters are estimated using experimental data from equilibrium and batch kinetic measurements, and they are used to simulate actual single-column fixed-bed separations.

Phase equilibrium between the solution and solid phases is described using thermodynamic Gibbs-Donnan model and various adsorption models depending on the properties of the sorbent. The 3-dimensional thermodynamic approach is used for volume sorption in gel-type ion exchangers and in nano-porous adsorbents, and satisfactory correlation is obtained provided that both mixing and exclusion effects are adequately taken into account. 2-Dimensional surface adsorption models are successfully applied to physical adsorption of complex species and to chelating adsorption of transition metal salts. In the latter case, comparison is also made with complex formation models.

Results of the mass transport studies show that uptake rates even in a competitive high-affinity system can be described by constant diffusion coefficients, when the adsorbent structure and the phase equilibrium conditions are adequately included in the model. Furthermore, a simplified solution based on the linear driving force approximation and the shrinking-core model is developed for very non-linear adsorption systems.

In each Case Study, the actual separation is carried out batch-wise in fixed-beds and the experimental data are simulated/correlated using the parameters derived from equilibrium and kinetic data. Good agreement between the calculated and experimental break-through curves is usually obtained indicating that the proposed approach is useful in systems, which at first sight are very different. For example, the important improvement in copper separation from

concentrated zinc sulfate solution at elevated temperatures can be correctly predicted by the model. In some cases, however, re-adjustment of model parameters is needed due to e.g. high solution viscosity.

Keywords: Electrolyte sorption, Modeling, Chelating adsorption, Size exclusion, Electrolyte exclusion

FOREWORD

This thesis is based on studies carried out in the Laboratory of Industrial Chemistry at Lappeenranta University of Technology over the period of 2005-2011. During this time, I have had the privilege to work with several colleagues. In particular, I wish to thank my supervisor, Associate Prof. Tuomo Sainio and my benefactor, Prof. Erkki Paatero, for creating auspicious conditions to fulfill this thesis. Moreover, I am indebted to Prof. Dmitri Muraviev from the Autonomous University of Barcelona and Dr. Anatoli Kalinitchev from the Russian Academy of Sciences for reviewing this thesis.

A number of persons in the Laboratory of Industrial Chemistry have participated in the thesis and their contribution is highly appreciated. Special thanks go to Dr. Katri Laatikainen, M.Sc. Jari Heinonen and Ms. Anne Hyrkkänen for their skilful assistance in experimental work. I am also grateful to Mr. Markku Levomäki for his help in practical matters. Part of the experimental data was obtained from the laboratory of Prof. Vadim Davankov at the Russian Academy of Sciences and the cooperation is gratefully acknowledged.

Some of the studies included in the thesis have been carried out within projects financed by Academy of Finland and TEKES (Finnish Funding Agency for Technology and Innovation). The financial support is gratefully acknowledged. I am also indebted to the Research Foundation of Lappeenranta University of Technology for partial funding of the thesis preparation.

Final motivation to this task came, however, from elsewhere. Without the encouragement and urge of my dear wife Katri, I hardly took this step. The last catalysts needed for this undertaking were the birth of our son Juho, and the endless joy and energy radiating from him.

Lappeenranta, 2011

TABLE OF CONTENTS

AUTHOR’S CONTRIBUTION... 9

LIST OF SYMBOLS ... 11

1 INTRODUCTION ... 15

2 SCOPE AND LIMITATIONS... 19

3 PHASE EQUILIBRIA ... 20

3.1 Material Structure: from Pore Solution to Homogeneous Polymer Gel ... 21

3.2 General Thermodynamic Relations... 22

3.3 Evaluation of the Thermodynamic Quantities ... 25

3.3.1 Activity Models... 25

3.3.2 Donnan Potential ... 26

3.3.2 Swelling Pressure ... 27

3.3.3 Steric Exclusion... 28

3.3.4 Surface Adsorption Models... 29

3.4 Complex Formation Models... 30

3.4.1 Surface Complexation Models (SCM)... 30

3.4.2 Chelation Model... 32

3.4 Speciation ... 33

4 MASS TRANSPORT ... 35

4.1 Intra-Particle and Film Transport ... 35

4.1.1 General Equations ... 35 4.1.2 Approximate Models... 38 4.2 Fluid Dynamics ... 41 5 METHODS ... 43 5.1 Experimental Methods ... 43 5.2 Calculations... 43

6.1 Electrolyte Adsorption on Chelating Adsorbents... 46

6.1.1 Background ... 46

6.1.2 System Description ... 48

6.1.3 Correlation of the Equilibrium Data with NICA... 50

6.1.4 Comparison of the Equilibrium Models... 56

6.1.5 Sorption Kinetics... 58

6.1.6 Break-Through Curves ... 65

6.1.7 Summary ... 74

6.2 Sorption of Metal Complexes on Non-Ionic Adsorbents... 74

6.2.1 Background ... 74

6.2.2 System Description ... 75

6.2.3 Correlation of the Equilibrium Data... 77

6.2.4. Calculation of the Break-Through Curves ... 87

6.2.6 Summary ... 89

6.3 Electrolyte Sorption in Non–Ionic Nano–Porous Materials ... 89

6.3.1 Background ... 89

6.3.2 System Description ... 90

6.3.3 Correlation of Binary Electrolyte Mixture Separation... 92

6.3.4 Summary ... 98

6.4 Sulfuric Acid Sorption in Cation Exchangers ... 99

6.4.1 Background ... 99

6.4.2 System Description ... 100

6.4.3. Modeling of Equilibrium Sorption... 103

6.4.4. Correlation of Chromatographic Separation ... 105

6.4.5 Summary ... 109

7 CONCLUSIONS... 110

AUTHOR’S CONTRIBUTION

Although written in the form of a monograph, this thesis is largely based on results published in Refs [46], [63], [74] and [88-93]. In some papers, writer of this thesis is a co-author and there his contribution is in model development and parameter estimation. Except in Ref. [91], experimental data have been measured by others.

Most of the figures shown in this thesis are identical with those in the original papers. This is in agreement with the policy of the publisher (Elsevier).

LIST OF SYMBOLS

a activity, -

â average activity, -

Asp specific surface area, m2/kg

B equilibrium constant for the proton-metal exchange reaction, (L/mol)n

BV bed volume, m3

c concentration in solution, mol/L

c concentration in solid, mol/L

*

c concentration at solid–liquid interface, mol/L

cp concentration in pore solution, mol/L

C capacitance, F/m2

dc correction factor, -

dp average particle diameter, m

D diffusion coefficient, m2/s

Dapp apparent diffusion coefficient, m2/s Dax axial dispersion coefficient, m2/s

D Stefan-Maxwell diffusion coefficient, m2/s

Eact activation energy, J/mol

f friction factor, kg/mol/s

F Faraday constant, C/equiv

G Gibbs energy, J/mol

h NICA parameter, -

H apparent bed height, m

∆Hads adsorption enthalpy, J/mol

I ionic strength, mol/L or mol/kg

Is supporting ionic strength, mol/L

J flux, mol/(m2s)

kf film mass transfer coefficient, m/s

kp mass transfer coefficient in pore solution, m/s ks mass transfer coefficient in homogeneous solid, m/s

kso salting-out coefficient, L/mol

K binding constant, L/mol

Ka thermodynamic dissociation constant, -

Kc apparent dissociation constant, -

KD distribution coefficient, L/kg

Kel elastic parameter, Pa

KF Freundlich constant, L/mol

KL Langmuir constant, L/mol

KLF Langmuir-Freundlich constant, L/mol

Ks solubility product, -

Kst steric distribution coefficient, -

KAB adsorption equilibrium constant of electrolyte AB, -

Lion ion diameter, nm

Lp average pore diameter, m

mi chelation model parameter, - m(i,j) lumped capacitance parameter, -

M molar mass, g/mol

n number of moles, mol

nc coordination number, -

nF inverse of the Freundlich isotherm exponent, -

nLF inverse of the Langmuir-Freundlich isotherm exponent, - nstat elastic parameter, -

N normality, equiv/L

p pressure, Pa

q amount bound in the solid matrix, mol/kg

qmax total amount of sorption sites, mol/kg q* value of q at resin surface, mol/kg

Q total uptake, mol/kg

r radial coordinate, m

Rp average radius of the adsorbent particle, m

Rg gas constant, 8.314 J/(molK)

S entropy, J/(molK)

T temperature, K

u interstitial velocity, m/s

v superficial velocity, m/s

V volume, L

Veq equivalent volume, L/equiv

Vm partial molar volume, L/mol

Vɺ volumetric flow rate, m3/s

w stoichiometric ratio, -

x axial coordinate, m

x* mole fraction of the adsorbable complex, -

y lumped concentration variable, -

z ion charge, -

Greek Letters

α degree of dissociation, -

β stability constant,

βw water sorption parameter,

γ ionic activity coefficient, -

γ± mean activity coefficient, -

σ charge density, C/m2

θ coverage, -

εb bed void fraction, -

εp pore void fraction, -

φp polymer volume fraction, -

µ chemical potential, J/mol

ν stoichiometric coefficient, -

κ NICA affinity coefficient, L/mol

π spreading pressure, Pa/m2

πsw swelling pressure, Pa

φ electrical potential, V

φD Donnan potential, V

χ solvent-free mole fraction in adsorbed phase, -

η dynamic viscosity, Pas

ρs solid concentration in the wetted adsorbent particle, kg/L Subscripts and Superscripts

0 initial or pure component value, infinite dilution

∅ unit value

a sulfuric acid

b base (NaOH)

el elastic contribution

f liquid film

feed feed value

i, j component H proton k site population L liquid phase p pore R functional group s surface S solid phase SC stoichiometric component st steric contribution AC actual component

tot total concentration

w water

Abbreviations

A anion

ARD average relative deviation

M proton or metal cation

Me metal cation

NC nano-porous carbon

NL ligand group (N-donor type)

PGM platinum group metals

1

INTRODUCTION

Treatment of electrolyte solutions using solid ion exchange materials in batch or fixed-bed operations is well-established technology in chemical engineering and several books on this topic are available [1-4]. Large-scale water treatment with cation and anion exchangers is a typical example. Much less attention has been paid on electrolyte sorption as a separation method. Fundamental difference between electrolyte sorption and ion exchange is binding of equivalent amount of cations and anions rather than cations or anions.

Besides water treatment and some large-scale applications in food industry [5], exploitation of ion exchange and electrolyte sorption in several potential areas is less developed. Use in hydrometallurgy, for example, is still rather limited when compared with solvent extraction [6, 7]. Reasons for such situation stem partly from some inherent limitations of these techniques but also from insufficient understanding of the basic factors, which dictate the overall performance in various applications. The key factors are the uptake capacity and selectivity, and in most cases they are controlled by the equilibrium distribution of target species between the solution phase and the solid material. At practical separation condition, mass transport in and around the solid particles may also become the controlling step in the overall efficiency. Understanding and mathematical modeling of the physicochemical processes are thus of utmost importance in assessing the potential of this technology in a given separation problem.

Solid separation materials considered here can be classified in several ways. For example, distinction is made on the basis of the presence or absence of fixed charge on the material surface. This criterion is, however, unclear for materials containing weakly acidic or basic functionalities, because charging is dependent on the operation conditions. In this study, systems are therefore classified on the basis of the binding mechanism. Sorption of electrolytes is rather ambiguous term and as will be shown in this thesis, it can mean different things depending on the properties of the separation material. In this thesis the term covers physical adsorption, chemisorption and partition. Moreover, chemisorption is limited to formation of coordinative bonds.

Physical adsorption is characterized by relatively weak interactions between the electrolyte and the solid surface. It is normally used in separation of non-ionic gaseous or liquid mixtures [8, 9], and applications for electrolyte solutions are quite rare. Some complex ions form, however, rather hydrophobic structures in aqueous solutions and they have substantial affinity on solid surfaces. Typical examples include chloro complexes of noble metals and the complex acid of gold, H+[AuCl4]-, formed in chloride solutions can be separated quite selectively using various hydrophobic surfaces [10-12]. The adsorbents are typically macro-porous polymer resins and this system is considered in detail in Case Study 2 of the experimental part (Section 6.2).

Physical adsorption is useful for electrolyte separation in few special cases only, but adsorbents containing chelating functional groups are suitable for binding a wide variety of ions at different conditions [13-17]. In such cases, coordination complexes are formed and the interactions are typically very strong approaching the region of chemisorption. It is important to note that such cases are frequently considered as ion exchange, because the chelating groups often act also as weak anion exchangers. Binding mechanism in electrolyte adsorption is, however, based on exchange of central atoms rather than counter-ions and thus the name chelating adsorption is used here. Chelating adsorbents consist of ligand groups covalently anchored on porous organic or inorganic supports. A large number of different ligands have been tested in laboratory-scale and their use in selective sorption of metal salts has been reviewed [18-20]. Relatively few of such materials are, however, available commercially and Dowex 4195 containing bis-(2-pyridylmethyl)amino groups in a PS-DVB matrix is a well-known representative [15]. Analogous materials that have meso-porous silica as the support are also available commercially [17] and the data obtained with them are used in Case Study 1 (Section 6.1).

Chelating adsorbents have been mainly used in similar applications as chelating ion exchangers; namely in selective uptake of transition metals. In both cases the functional groups comprise of complex-forming moieties but in contrast to adsorbents, the ion exchangers contain fixed negative charges to bind the metal cations. Number of such studies is huge and reviews covering both ion exchange and adsorption systems are available [18, 19, 21]. In this thesis, focus is on investigations related to the zinc sulfate solution (Case Study 1) and they are considered in detail in Section 6.1.1.

In contrast to physical and chelating adsorption, partition means non−specific uptake of electrolytes in various materials. In case of gel-type ion exchangers, partition is considered to be controlled mainly by electrostatic effects and electrolyte exclusion has been known for long time as a separation method for electrolytes and non-electrolytes [22, 23]. This system is discussed in detail in Case Study 4 (Section 6.4). It may be also noted that in ion exchange literature partition is called electrolyte sorption (or invasion) and it refers to undesirable uptake at high salt concentrations. Studies with nano-porous uncharged materials have shown [24-26] that steric exclusion may also be important and Case Study 3 of this thesis (Section 6.3) deals with separation of concentrated electrolyte solution with such materials.

Behavior of ion exchangers and adsorbents has been studied theoretically and experimentally for several decades and guidelines for their utilization can be found in classical textbooks [e.g. 1, 27-28]. At the same time, large number of theoretical, semi-theoretical and empirical models has been derived. Consequently, ingenious way of combining the existing models rather than development of new ones was chosen for the purposes of this study.

The main objective of this thesis is to give a systematic, phenomenological and practice-oriented treatment to explain all the relevant aspects observed in electrolyte adsorption. Systematic means that general principles are used as a starting point in formulation of phase equilibrium conditions

and dynamics of the studied systems. Development of the model equations is presented in Sections 3 and 4, and their application in actual separation problems is discussed in Case Studies 1-4. Phenomenological means that all observations are described in terms of real physical and chemical phenomena, and the pertinent model parameters are estimated from independent data as far as possible. This point is emphasized in treatment of the experimental data of the Case Studies. The final goal is, however, to offer tools for practical engineering calculations and therefore balance between rigorousness and usefulness has to be found.

Key word of the thesis is modeling and the author’s view on the role of the modeling “tool” in the scientific methodology is illustrated in Fig. 1.

Figure 1. Linking of modeling with various steps of experimental research.

The important point here is that modeling is not mere means to rationalize the experimental data for further utilization. Model calculations should be made continuously in preliminary testing of the hypothesis, in experimental planning as well as in interpretation of the results. This means that modeling itself is a dynamic process starting from a primitive version and ending to a useful tool for further utilization in even more complex systems.

Interpretation Results Experimental planning Hypothesis Experiments Utilization Modeling

2

SCOPE AND LIMITATIONS

The topic of this thesis is electrolyte sorption in solid materials but because of the close connection with ion exchange, some overlap is inevitable. Same materials can be used in both cases and some systems, which are treated here as electrolyte adsorption, are even called in the literature ion exchange. However, no systematic treatment for ion exchange is presented. All systems studied are aqueous solutions. Systems, where the oxidation state of the adsorbed species changes, are not considered in this thesis. For example, recovery of soluble Au3+ complexes with activated carbon is accompanied by reduction to metallic gold and such systems are thus excluded from this study.

The experimental data used in this thesis include results from equilibrium measurements, batch kinetic data measured in stirred tank, and batch pulse or frontal data obtained in a fixed-bed. Calculations of continuous processes and process design or optimization are beyond the scope of this thesis. Moreover, the breakthrough curves are correlated/simulated using a general rate model that includes contributions of phase equilibrium, intra-particle and film diffusion as well as axial dispersion. No systematic comparison with other standard models of chromatographic theory is made. Theoretical analysis of linear and moderately non-linear systems is well-established (so called equilibrium theory of chromatography) but in cases of extreme non-linearity and/or complex equilibrium conditions met in this thesis, solution of the general mass balances was deemed to be a more adequate approach.

Standard mathematical methods were used in solution of differential and arithmetic equations, and no attempts were made to develop new calculation procedures.

Finally, the solid separation materials studied here are in the form of spherical particles with a diameter of 0.2-1 mm. Other geometries, e.g. membranes, are not considered explicitly but same principles are applicable to such systems, too.

3

PHASE EQUILIBRIA

The topic of this Section is illustrated schematically in Fig. 2. The system under study is defined in terms of number (NSC) and concentrations (cSC,i, i = 1… NSC) of stoichiometric components. In electrolyte solutions these are the cations and anions from fully dissociated electrolytes. Next, the actual species existing at the studied conditions are identified and their concentrations (ci, i = 1… Nac) are estimated. In this step, all association/dissociation equilibria involved in the solution phase are taken into account, and the actual forms and amounts of the target components are calculated. In hydrochloric acid solutions considered later in Case Study 2, for example, most metals and metalloids exist as negatively charged chloro complexes rather than as hydrated cations, and the species distribution depends on e.g. acid concentration. Finally, phase equilibrium conditions (Eq. (1)) are constructed for each component to relate the solution concentrations with the amounts sorbed by the solid phase, Qi (i = 1…NAC).

Figure 2. Schematic illustration of phase equilibrium between a solution phase and a solid separation material. 1 2 ( , ... ) AC i N Q = f c c c (1)

Formulation of the function f depends on the structure of the separation material and, therefore, these factors are first discussed in Section 3.1. Mathematical formulation of the general phase equilibrium condition is considered in Section 3.2. Speciation is briefly discussed in Section 3.4.

Stoichiometric components: 1…Nsc Actual components: 1…Nac Speciation Phase equilibrium Invaded species Bound species Solution phase Solid phase

3.1 Material Structure: from Pore Solution to Homogeneous Polymer Gel

Commercially available materials can be divided roughly in three categories; homogeneous polymer gels, macro-porous polymeric resins and functionalized or un-functionalized rigid materials. Gel-type resins are composed of three-dimensional polymer network with or without covalently attached functional groups. Poly(styrene) and polyacrylates are most often used and cross-linking is typically achieved by co-polymerization with divinylbenzene. Water content is 50 wt-% or more and thus the structure allows reasonably fast diffusion of ions and other small components in the resin phase. Macro-porous resins are prepared in a similar way, except that porogens are used in the polymerization step to create permanent pores in the structure and that the degree of cross-linking is higher. In these resins, majority of the functional groups is still inside the polymer matrix but their accessibility is improved by the porous internal structure. The materials of the third group have an impenetrable porous support and the functional groups, if present, are exclusively on the solid surface. Functionalized silica and hyper-crosslinked poly(styrene) are typical examples. A summary of the materials considered in this thesis is given in Table 1.

The gel structure is simplest to model because only two mutually insoluble phases are present. The 3-dimensional gel phase can be considered as a combination of homogeneous polymer solution and elastic network [29]. Overall description of the resin phase thus requires suitable polymer solution model and a network model. Moreover, formulations of the electrostatic interactions and steric exclusion factors are needed. These are the essential contributions in the Gibbs-Donnan model considered in Section 3.2.

Table 1. Comparison of solid materials considered in this thesis.

Gel-type resins Macro-porous resins Rigid-pore materials

Structure Homogeneous polymer

gel

Polymer gel + permanent pores

Permanent pores

Diffusion medium Polymer solution Pore solution +

polymer solution

Pore solution

Location of solid-liquid phase boundary

Particle surface Pore surface Pore surface

In rigid-pore materials, on the other hand, the adsorbent phase can be considered 2-dimensional and it is located on the surface of the internal pores. Surface complexation models (Section 3.4.1) and surface adsorption models (Section 3.3.4) are frequently applied to model such materials. The solution phase that fills the pores is usually assumed to have similar properties as the bulk solution and can be treated as continuation of the bulk phase. This assumption may be reasonable for macro-porous materials but its validity is highly questionable, when the pore dimensions become similar to ion diameters. In that case, an additional phase boundary is located at the pore

opening. Such system will be met in Case Study 3 (Section 6.3), where size exclusion of ions in nano-porous materials is investigated.

As shown in Table 1, macro-porous resins have features of both gel-type and rigid-pore structures. Depending on the system under investigation, either homogeneous sorption or surface adsorption may be considered dominant and the equilibrium model is constructed accordingly. In Case Studies 1 and 2 (Sections 6.1 and 6.2), the surface adsorption approach is used.

Difference between homogeneous and porous structures can be further illustrated as shown in Fig. 3. In a gel–type resins, the general phase equilibrium condition qi = f(c1, c2, c3…cN) needs to be taken into account only at the outer particle surface as a boundary condition for diffusion. In a rigid-pore case, on the other hand, diffusion takes place in the pore solution and the phase equilibrium condition must be considered at every radial position. Mathematical formulation of the two cases is discussed in Section 4.1.

Fig. 3. Coupling of phase equilibrium and diffusion in gel-type resin and in porous material. Jdiff

indicates diffusion flux and N is number of adsorbing species. Small circles indicate positions where the local phase equilibrium is established.

3.2

General Thermodynamic Relations

As discussed in the previous Section, the solid phase is either 3-dimensional or 2-dimensional depending on the structure. In the first case the solid phase consists of homogeneous polymer

Gel-type resin Porous material

J

_diff

J

diff

q

i

= f(c

1

,c

2

,c

3

…c

N

)

solution and elastic polymer network. The second case represents surface adsorption and equilibrium is established between the liquid phase and the 2-dimensional surface phase. Derivation of the basic equations from the thermodynamic principles is shown only in that extent, which is necessary for understanding of the various contributions important for this study. More complete treatments can be found elsewhere [29, 32].

Three–Dimensional Geometry

General condition for isothermal phase equilibrium in the three-dimensional system can be written as shown in Eq. (2). Here, G is Gibbs energy, T is temperature, p is pressure, n is number of moles and S is entropy. Superscripts L and S refer to liquid and solid phases, respectively. Subscripts el and st indicate elastic and steric contributions.

( , , ) ( , , ) 0

L L L S S S

el st

dG T p n +dG T p n −TdS −TdS = (2)

The Gibbs energies of the liquid and solid phases are given by Eq. (3), where µ is chemical potential, z is the ion charge, F is Faraday constant and ϕS is the electrical potential of the solid phase. By definition, ϕL = 0 when no electrical field is imposed on the liquid phase.

1 1 1 ( , , ) ( , , ) N L L L L L j j j N N S S S S S S S j j i j j j dG T p n dn dG T p n dn z F dn µ µ ϕ = = = = = −

∑

∑

∑

(3)

The entropy terms in Eq. (2) stem from the effect of resin swelling on the statistical distribution of the polymer chains (Sel) and from the steric hindrances experienced by the sorbed components (Sst). The former term can be interpreted in terms of pressure of the swollen resin phase [29] and thus

( S L) S S

sw sw

TdS p p dV π dV

− = − = (4)

Here, p is pressure and V is volume. The quantity πsw is called swelling pressure.

In an analogous way, the steric contribution, –TdSst, can be related to the geometric characteristics of the system and on the basis of statistical thermodynamics, Giddings et al. [30] have shown that –TdSst = RTΣlnKst,j, where Kst is steric distribution coefficient.

The mixing terms are written in terms of activities, a, and the general Gibbs-Donnan phase equilibrium condition becomes as shown in Eq. (5).

, , ln ln 0 S i g L m i sw i S g st i i a R T V z F R T K a + π + ϕ − = (5)

Here z is the ion valence, and Rg and T have their usual meanings. Vm represents the partial molar volume. Eq. (5) forms the basis for equilibrium calculations and relations of the thermodynamic quantities to measurable variables are discussed in Section 3.3.

Two–Dimensional Case

In this thesis, the ideal adsorbed solution theory (IAST) originally developed for gas adsorption [31] and later extended to solution phase adsorption [32] was selected as the general framework for 2-dimensional adsorption. Application of IAST to competitive electrolyte adsorption, which often is characterized by very strong interactions, is justified only by analogy. Nevertheless, several authors [e.g. 33-35] have used IAST in metal binding studies.

Assuming that both the bulk solution and the adsorbed phase behave ideally, thermodynamic considerations lead to Eq. (6) [32], which can be considered as an analogy for the Raoult law. In other words, Eq. (6) relates the solution concentration, ci, of component i in a multi-component solution to the concentration, ci0, in a hypothetical single-component system giving the same spreading pressure, π, on the surface. The corresponding adsorbed amounts in multi-component and single−component systems are qi and qi0, respectively. The solvent-free mole fraction in the adsorbed phase is represented by χi.

0 1 ( , ) ( , ) i i i i i N j j c T c T q q π π χ χ = = =

∑

(6)

The spreading pressure is obtained from the single-component data using Eq. (7), which can be considered as integrated form of the Gibbs adsorption isotherm. Here Asp is the specific surface area of the adsorbent. No specific functional form is assumed and each component can have different expressions relating qi0 and ci0. The only constraint is that the spreading pressure is equal for all adsorbed components.

0 0 0 0 0 i c sp i i g i A q dc R T c π =

_∫

(7)

Moreover, ideal behavior of the adsorbed phase implies that the molar areas of the adsorbed components do not change upon mixing and thus

1 0 1 1 N N j j j j j q q χ − = =   = _ _  

∑

∑

(8)

3.3 Evaluation of the Thermodynamic Quantities

The main problem in using the thermodynamic expressions lies in finding suitable models that relate the thermodynamic quantities to experimentally accessible values. Calculation of activities in electrolyte solutions, for example, is well-established but extension of the existing models to concentrated polyelectrolyte gels is quite challenging. In a similar way, suitable models are needed for calculation of the swelling pressure and the steric exclusion coefficients. Mathematical formulation of these is discussed in the following Sections.

3.3.1 Activity Models

In most electrolyte solution models, the activity is defined on the molality scale and the standard state is a hypothetical ideal solution at 1 mol/kg. Moreover, the activities of ions are not experimentally accessible, and correlations are made using mean ionic activity, a±. Similar definitions can be made using molar concentration, c, and the relationships between the different values are given in Eq. (9) for a fully dissociated electrolyte A BA B ( )

A B z z

A B

ν ν ν ν= +ν . The quantities with the asterisk refer to the molality scale and the unit concentrations are given by m∅ and c∅, respectively. 1/ 1/ ( / ) ( / ) ( / )( ) [( ) ( ) ] ( / ) /( / ) A B A B A A A B B B A B A B a m m a m m a m m m m c c ν ν ν ν ν ν γ γ γ ν ν γ γ γ γ γ ∗ ∗ ∅ ∗ ∗ ∅ ∗ ∗ ∅ ± ± ∗ ∗ ∗ ± ∗ ∅ ∅ ± ± = = = = = (9)

The molar concentrations are systematically used in this thesis, because they are more convenient to use in mass balance calculations of dynamic modeling. Conversion between m and c requires knowledge on solution density. Experimental data are rarely available for multi-component systems and estimation methods must be used. Li and Lee [36] have proposed a procedure to calculate density from the ionic radii and osmotic pressure of the solution. Values for the optimized ionic radii are available for a number of ions [36].

In very dilute solutions, the long–range interactions dominate and the activity coefficients can be calculated using the Debye-Hückel equation in its original or modified form. Short–range forces become important in more concentrated solutions and several models have been proposed for their evaluation. For single electrolytes, extended Debye-Hückel equations have been used and the formulation of Lietzke and Stoughton [37] is shown in Eq. (10). S is the Debye-Hückel slope and A-D are adjustable parameters. It is also important to note that for uncharged species (z = 0)

and by retaining only the linear term, this expression reduces to the empirical Setchenov equation [27] commonly used to describe the salting-out or salting-in effects.

2 3 , 2 1 3 4 ln 2 2 3 1 C A a N j j j z z S I BI CI DI A I I z m γ_± = = − + + + + =

∑

(10)

In mixed electrolyte solutions, the Pitzer model [38] is most often used and the short–range interactions are described by means of binary and ternary interactions. Extensive tabulations of the model parameters are available [39, 40]. Thomsen [41] have used the UNIQUAC model to describe the binary short-range interactions between ions. A simpler approach has been proposed by Kusik and Meisner [42] and according to their mixing rule, mean activity coefficients in the mixture are derived from the values of pure electrolytes at the same ionic strength.

In the resin phase, similar arguments apply and the solution models have been used; UNIQUAC has been applied to gel-type resins [43, 44] and Mumford et al. [45] have used the Wilson equation Special attention must be given, however, to the polymer backbone, which is not included in the activity model but which occupies substantial fraction of the total volume of the solid phase. When no distinction is made between the various forms of water in the resin, molality of component i gives the number of moles of i per total weight of sorbed water. Molal concentrations are, however, not convenient in formulation of the mass balances in dynamic models and molar concentrations are preferred. On the molarity scale, concentration is calculated per total resin volume, VS, and the numerical values thus are much smaller than on the molality scale. A simple correction for the “excluded volume” is achieved by subtracting the polymer volume fraction, φp, and using (1–φp)VS as the solid phase volume [46]. The volume fraction of the polymer network is obtained from Eq. (11), where Veq,R is the equivalent volume of the resin matrix. , , , 1 eq R p N eq R m j j j V V V q φ = = +

∑

(11) 3.3.2 Donnan Potential

In principle, the electrical potential appearing in Eq. (5) can be calculated using the Poisson equation, which relates the potential and the charge density. Most often, however, the potential is eliminated by writing the phase equilibrium conditions for electrolytes rather than for individual ions [28] or by considering an exchange reaction rather than binding of individual species. When using the general equilibrium condition of Eq. (5), the following expression is obtained for an

electrolyte A BA B ( )

A B z z

A B

ν ν ν ν= +ν . Here the quantities with over-bar refer to the resin phase and the concentrations may be calculated for the total volume (gel resin) or pore volume.

, , , , , , , 1/ , , , ( ) exp( ) [( ) ( ) ] A B A B A B m AB sw A B AB A B AB st AB g m AB A m A B m B st AB st A st B V c c c c K R T V V V K K K ν ν ν ν ν ν ν ν ν ν π γ γ ν ν ± = ± − = + = (12)

At the same time, the electroneutrality condition for both phases given in Eq. (13) is taken into account. Here, zR and qR are the valence and concentration of the functional groups in the resin phase. 1 1 0 0 N j j j N R R j j j z c z q z c = = = + =

∑

∑

(13) 3.3.2 Swelling Pressure

Influence of swelling pressure on phase equilibrium is usually rather small despite the fact that the pressure may attain quite high values in densely cross-linked gel-type resins [48]. Moreover, the swelling pressure effect can be totally ignored in materials, where the functional groups are attached on the surface of a rigid support. Conventionally, the swelling pressure in ion exchangers has been modeled using an empirical linear correlation between πsw and the resin volume [47]. Simple theoretical models, on the other hand, assume Gaussian or near-Gaussian chain length distribution, which is too restrictive for densely cross-linked materials. Therefore, they give very poor correlation with the experimentally determined swelling pressures [48]. More realistic representation is obtained with non-ideal polymer network models and Eq. (14) has been used in our laboratory [48]. Volume swelling is given by φp–1/3, where φp is the polymer volume fraction in the swollen resin defined in Eq. (11). L-1 is the inverse Langevin function.

1/ 3 2/ 3 1/ 2 1 sw 1/ 3 1/ 2 1 ( ) 3 ( ) el c p stat c p stat K d n w w d n π φ φ − − − − = = L (14)

The parameters Kel and nstat depend on properties of the polymer chains and the degree of cross-linking. They can be evaluated from the existing swelling pressure data and, with some reservations, from mechanical measurements [48]. In principle, Kel is equal to the shear modulus of the network in the unstrained state characterized byφ0_p. Normally φ0_p is defined at conditions,

where the cross-links are introduced in the network but no such well-defined reference state exists for materials considered here. Thus dry polymer is taken here as the reference state ( 0

p

φ =

1) and Kel is treated as an adjustable parameter.

The correction factor dπ is used to account for network imperfections and it formally gives the fraction of elastically active chains. The physical significance is less clear in densely cross-linked resins and dπ should be considered as empirical parameter. The mathematical form of w in Eq. (14) is taken from Zaroslav et al. [49], who used the correction factor for polyelectrolyte gels to improve the agreement between the experimental data of and the theoretical predictions. Attempts have been also made to improve consistency between the shear modulus and swelling pressure data by assuming different values for dG anf dπ [48].

3.3.3 Steric Exclusion

The steric exclusion term can be viewed as entropy increase stemming from restricted distribution of the sorbed species in the solid phase. In adsorbents having impenetrable support, only the volume fraction filled with liquid is available. In later Sections, this quantity is referred to as particle porosity εp. Distinction between the available and non-available volume is not so clear-cut in gel-type and macro-porous polymer resins. As discussed earlier, the available volume fraction in a polymer gel may be approximated by 1–φp. In this case, the dimensions of the available space are not considered and the same excluded volume is experienced by all adsorbed components.

When the dimensions of the available space and the sorbed species are of the same order of magnitude, the exclusion term can be written more specifically. Giddings et al. [30] have derived expressions for various pore geometries and molecule shapes, and the following expression is obtained for spherical ions or molecules and cylindrical pores.

2 1 ln(1 ) N j st j p L TdS RT L = − =

∑

− (15)

Here, the summation goes over all adsorbed components and L is diameter. For ions and small molecules, Eq. (15) becomes important only at nano-scale. Evaluation of the ion dimension is difficult but as a first assumption, hydrated radii are used. However, behavior of ions in nano-tubes and nano-pores suggests that partial dehydration may also occur and the dimension depends on e.g. concentration [27, 50].

3.3.4 Surface Adsorption Models

Thermodynamically consistent multi–component surface adsorption model can be constructed using IAST. Integration of Eq. (7) for some commonly used isotherms gives Eq. (16), where qimax is the saturation capacity of component i. KL, KF and KLF are the corresponding equilibrium coefficients and 1/nF is the Freundlich exponent.

, , max 0 , 1/ 0 , , 1/ max 0 , , ln(1 ) (Langmuir) ( ) (Freundlich) ln[1 ( ) ] (Langmuir-Freundlich) F i LF i sp i L i i g n sp F i F i i g n sp i LF i LF i i g A q K c R T A K n c R T A q n K c R T π π π = + = = + (16)

It is obvious that in general cases iterative procedures are needed to solve the set of equations. For example, Crittenden et al. [51] have applied the IAST approach for a system composed of

Freundlich single-component isotherms ( 0 0 1/ ,

,( ) F i n i F i i

q = K c ) and they obtained Eq. (17).

, , 1 , , 1 F i n N F j j j i i N F i F i j j n q q c n K q = =       =      

∑

∑

(17)

Some explicit forms have been derived using approximate formulations or additional assumptions. LeVan and Vermeulen [52] used Freundlich and Langmuir isotherms for single components and derived explicit binary isotherms by applying Taylor series expansions. Binary cases have been studied also by Gritti and Guiochon [53]. They used an extended BET isotherm and derived an explicit expression for the competitive case provided that the saturation capacity is same for both components.

Another thermodynamically near-consistent adsorption model is provided by the non-ideal competitive adsorption (NICA) equation [54]. The thermodynamic consistency has been discussed by Koopal et al. [55], and because of the scaling with respect of a given component, it can be shown to be consistent with the Gibbs adsorption equation. However, the behavior at vanishing adsorption does not comply with the Henry law, but the limiting value is θi,L(ci→ 0) = (κici)hi.

NICA is non-ideal in two senses. Different binding stoichiometry for each component is allowed. On the other hand, site heterogeneity is included by using the local isotherm together with the Sips distribution [25, 32] to obtain an affinity distribution for the sites. If several sites (k = 1…S) are assumed to be present in the solid phase, the isotherm equation of a N-component system can

be written as shown in Eq. (18). κi,k is the median value of the affinity distribution for component i at site k and the parameter hi depends mainly on the binding stoichiometry of the adsorbed species. The heterogeneity of site k is characterized by the value of pk (0 < pk ≤ 1) and fraction of the k-type sites is given by ωk (Σωk = 1).

, , , 1 , , , max 1 , , ( ) ( ) 1 ( ) k j k i k k j k p h h i k i j k j S j i k i k p k H k h j k j j c c h q q h c κ κ ω κ − =       =   +    

∑

∑

∑

(i,j = 1…N, k = 1…S) (18)

Usually the maximum proton binding capacity is assumed is equal to the total amount of sites represented by qmax.

NICA-Donnan model [25] is an advanced version, where Eq. (18) is empirically combined with the Donnan equilibrium. In a general case, Eq. (18) or any other isotherm is combined with the equilibrium condition (Eq. (5)) and the set of equations is solved using suitable simplifying assumptions. The bulk concentration c in Eq. (18) is thus replaced by c and total uptake of component i is obtained from Q_i = c_i/ρ_s+ . q_i

3.4 Complex Formation Models

In contrast to the foregoing discussion, these models attempt to focus on the actual binding process taking place at the adsorption sites. Therefore, such models are more suitable in describing behavior in high-affinity systems as will be shown in Case Study 1 (Section 6.1). Two approaches are considered here; surface complexation of the electrolytes at surface sites (Section 3.4.1) and chelate formation by polymeric ligands (Section 3.4.2).

3.4.1 Surface Complexation Models (SCM)

In these models, sorption in the solid phase is described by considering distribution of ions within the electrical double-layer (EDL) formed at the solid/liquid interface. A simplified structure for EDL is shown in Fig. 4.

Only proton and hydroxide ion are assumed to react at the actual surface (0-plane), while other ions are located either in the Stern layer or in the diffuse part of EDL. In more sophisticated SCM versions, several Stern layers are assumed allowing binding at different distances from the surface [56, 57]. When the SCM is applied to ion exchange resins or adsorbents, the available functional groups are assumed to be distributed on a plane surface. The assumption is plausible for macro−porous and rigid materials but less realistic for gel-type ion exchangers [58].

For the purposes of this thesis, the SCM version developed by Höll and co-workers [59-61] is most suitable. Their derivation follows same principles as in conventional SCM models but each bound ion is located at a specific Stern plane characterized by its own potential φi and capacitance C0,i. Moreover, the diffuse double layer (DDL) is omitted and electroneutrality is secured by locating the charge-compensating ions at a given plane. At the outer edge of the multiple Stern layers, potential thus reaches the bulk solution value.

ϕ0 ϕ1 = ϕDDL σ0 σ1

Figure 4. Electrical double-layer at the solid-liquid interface. ϕ and σ represent the electrical potential and the charge density, respectively.

Formulation of the SCM model for competitive adsorption of electrolytes follows the treatment of Stöhr et al. [60], except that the binding stoichiometry of the metal salts is not fixed a priori but the coordination number is an adjustable parameter. The unknown surface potentials are eliminated by considering simultaneous binding of cation and anion. Consequently, the equations for a binary mixture of acid (H+)νH(Xz-)νHX and metal salt (Mez+)νMe(Xz-)νHMe containing only a common anion Xz- shown in Eq. (19) can be derived. Here KAB is the equilibrium constant for adsorption of electrolyte AB, θ is coverage and m(0,i) is a lumped capacitance parameter.

2 0, log[ ] log [ ( 0, ) (0, )] (0, ) ( ) log[ ] log [ (0, ) ( 0, )] ( ) ( ) (0, ) ( , ) ln10 HX MeX H HX X H w S H X H HX HX Me MeX X w n S Me X Me MeX MeX tot g sp i K m X m Me m Me a a w K m X m Me a a w F q m i i Me X R TA C θ θ θ θ ν ν θ _θ θ ν ν = + − + = = + − = = = (19) S u rf a ce (S ) D iff u se d o u b le la y e r (D D L) S te rn la y e r

The model contains two adjustable parameters (K and m) for each electrolyte and can be easily extended to systems containing an acid and several metal salts with a common anion [60]. The general expressions for the acid and NMe metal salts are given in Eq. (20).

1 1 log[ ] log (0, ) (0, ) ( ) log[ ] log (0, ) (0, )( ) (0, ) ( ) ( ) Me HX Me Me iX N H HX X j w j S H X N N i iX X H j j w n j j i S i X K m X m j a a K m X m i y m j a a θ _{θ θ} θ θ _{θ θ θ} θ = = > = + − = + − + −

∑

∑

∑

(20) 3.4.2 Chelation Model

This model was derived for the case of a polymeric ligand, because the situation in a polymeric system is fundamentally different due to the high local ligand concentration. This important difference has been pointed out also by Skurlatov et al. [62] in studies on metal complexation by poly(4-vinylpyridine). Detailed derivation of the model has been reported elsewhere [63] and only the equations describing competitive adsorption of acid and NMe metal salts are given in Eq. (21), when wo different sites (k = 1, 2) are assumed. Ka is the dissociation constant of the ligand and B is the equilibrium constant for the metal-proton exchange. θ is coverage calculated with respect of qtot. Number of protons displaced from sites 1 and 2 by metal j are given by nj,1 and nj,2. ,1 ,2 ,1 ,2 1 , , 1 ,1 ,2 , , , , 1 ,1 ,2 ,1 ,2 ,1 ,2 max ,1 ,2 ( ) ( 1, 2) ( ) ( ) ( 1... ) k k i i i i m N j k H k H k j j j j a k m N j k H k H k j j j j n n i H i n n i i i H H i i i i n c n n K k n n n c B n n c i N q q n n ω θ θ θ θ θ θ θ θ θ + = = +   − −  ₊      = =   +  ₊      = + = = +

∑

∑

(21)

The exponent mk is an adjustable parameter including the charging effects within the site population k. It is related to the well-known Henderson-Hasselbalch parameter n by mk = n – 1. It should be noted that mk is a purely empirical parameter [64]. The theoretical formulation that was originally developed by Katchalsky and Lifson has been applied to sorption in branched poly(ethylene imine) by Shephard and Kitchener [65].

3.4 Speciation

In simple cases, all electrolytes are fully dissociated and only hydrated cations and anions need to be considered. Actual concentrations are thus equal to stoichiometric concentrations. Unfortunately, this is rarely true in practice and several dissociation/association/precipitation equilibria must be considered for adequate description of the system. In extreme cases, some or all of the components exist exclusively in the form of complexes. Consider, for example, dissolution of metals in HCl. All noble metals are in the form of complex acids (e.g. H[AuCl4] and H2[PtCl6]), which in turn may or may not be fully dissociated. Base metals, too, form chloro complexes and the structures depend on acid concentration. Metalloids, like Sb and Te, have complicated reaction schemes including partially hydrolyzed species.

When considering all processes as association reactions, they can be described by Eq. (22). Here M represents proton or a metal cation and A anion or some other ligand. βn is the cumulative stability constant of [MAn] and for acids (Mz+ = H+) it is related to the dissociation constant Ka,n by log Ka,n = – (log βn – log βn-1).

( ) ( )

[ ]

z z z n z

n

M ++nA−  MA + − − (22)

The actual composition of the liquid phase is obtained by combining the expressions written for the stability constants with appropriate mass balances. Extensive tabulations of the βn values are available [66, 67] and empirical rules to correct for the ionic strength exist [68]. Complexation equilibria of metal and metalloid cations in chloride matrix are considered in detail in Case Study 2 (Section 6.2.3).

Occasionally ion exchangers or adsorbents are used at conditions, where precipitation of some components occurs. Typical example is formation of basic sulfates or chlorides in concentrated solutions of transition metals, when the solution pH exceeds the precipitation limit. Such reactions can be treated mathematically similarly as the complexation reactions, except that solubility products rather than stability constants are used. Example of such systems is given in Case Study 1 (Section 6.1.2).

Formally similar procedure applies also to the solid phase. In porous materials, the same stability constants can be used for the pore solution as first approximation. The situation is more complex in polymeric gels and no general guidelines can be given. The problem is related to the applicability of the solution-side constants and to the evaluation of the activity coefficients. Nelson and Kraus [69], for example, have shown that the second dissociation constant of sulfuric acid is much higher in a sulfate-form anion exchanger than in ordinary aqueous solution.

4

MASS TRANSPORT

4.1

Intra-Particle and Film Transport

Mass transport within the solid particles takes place by diffusion. The diffusion rate depends on the properties of the migrating species, on the structure of the solid particle and on the interactions between the migrating species and the functional groups of the ion exchanger or adsorbent. External mass transport in the liquid film surrounding the particles is normally described using simplified linear correlations irrespective of the above-mentioned factors. The size and shape of the diffusing species are reflected in the mass transport parameters but they need not to be taken explicitly into account in the model formulation. If ions are involved, however, the electrostatic contribution in the driving force must be considered.

The interactions of the diffusing species with the solid phase can range from weak hydrophobic or electrostatic interactions to highly specific chelate-forming reactions. If strong interactions are present, the apparent mass transfer rate may decrease several orders of magnitude. In such cases adequate description of the equilibrium properties is vitally important in order to obtain meaningful mass transfer parameters.

4.1.1 General Equations

Models used to describe mass transport within the solid particles can be divided in three groups. Exact models involve formulation and solution of the intra-particle concentration profiles. This approach is usually adopted in theoretical and mechanistic studies, because physically relevant values for the mass transport parameters can be obtained. Use of these models in process simulations is, however, seriously limited by the complexity of the computing procedures. Therefore, much effort has been paid on development of approximate solution schemes and here they are referred to as approximate models. The third group includes various empirical models based e.g. on analogies with reaction kinetics. Their generalization to multi-component systems and to various equilibrium conditions is, however, difficult and they are not considered in this thesis. In principle, same concepts apply to film diffusion around the particles, too, but normally only approximate models are used.

The most rigorous treatment of intra-particle mass transport is based on the Stefan-Maxwell equation (Eq. (23)), where the chemical potential gradient is used as the driving force and coupling of the diffusion fluxes is fully accounted for. When using the generalized formulation of the chemical potential discussed in Section 3.2, influence of swelling, electrostatic interactions and steric hindrances on diffusion rate can be taken into account. Eq. (23) can be considered as a general force balance; the thermodynamic driving force is balanced by the friction forces and the

friction coefficient is related to diffusion coefficient by the Einstein equation, fij = RgT/Dij. Dij is the mutual Stefan-Maxwell diffusion coefficient for components i and j. For swelling materials, change of the coordinate system is advantageous and the velocities are taken with respect of the mass of the dry polymer [70].

(

)

1 R R 1 N i if j i j j j i f u u R µ + _φ = ≠ ∂ − = − ∂

∑

(23) 1 R 1 0 N i i i u φ + = =

∑

In Eq. (23), φ is the volume fraction and uR is the velocity with respect of fixed coordinates. Normally, however, a less ambiguous approach is adopted and the model is formulated in terms of directly measurable quantities. Following assumptions are thus made.

1. The solid phase is fixed in the laboratory coordinates and the velocities are counted with respect of the solid. Resin swelling is thus ignored or included in the apparent diffusion coefficients.

2. Cross-terms between the diffusing species are neglected.

3. Activity coefficients of the diffusing species are incorporated in the apparent diffusion coefficients.

When only the mixing and electrostatic terms in the chemical potential are retained, the Nernst– Planck equation shown in Eq. (24) is obtained. Here Js is intra-particle flux and depending on the solid phase structure, different expressions result in. Concentration in the solid phase is given by q and cp is the concentration in the pore solution. The diffusion coefficients Ds,i and Dp,i have different meaning in the two cases; Ds is related to movement in a homogeneous polymer or polyelectrolyte gel, while Dp represents movement in a solution phase confined in narrow pores. It is important to note that both these quantities are apparent values with much less physical relevance than Dij. They are, however, frequently used in ion exchange and adsorption studies and the prefix “apparent” is omitted in the following discussion.

The first term in the flux expressions is due to the concentration gradient and for uncharged components, Eq. (24) is identical to the Fick’s first law. The second term stems from the fact that the charged species cannot migrate independently but the electroneutrality condition must be fulfilled at every radial point. As a result, the fluxes are coupled and movement of faster ions is retarded by the slower ones and vice versa. The potential gradient is obtained from the condition of zero current flow, i.e. ΣzjJj = 0.

, , , 1 2 , 1 , , , , , , 1 2 , , 1 : ( ) : ( ) j j i i i s i s i S N j j s j j N s j j j p i i p i s i p i S N p j j p j j N p j p j j Gel type q z Fq J D r RT r q RT z D r r F z D q Macroporous c z Fc J D r RT r c RT z D r r F z D c ϕ ρ ϕ ϕ ρ ϕ = = = = − ∂ ∂ = − + ∂ ∂ ∂ ∂ ∂ _{= −} ∂ ∂ ∂ = − + ∂ ∂ ∂ ∂ ∂ _{= −} ∂

∑

∑

∑

∑

(24)

Attempts have been made to estimate the apparent solid-phase coefficients from the diffusion coefficients in the bulk solution. Mackie and Meares [71] have proposed Eq. (25) for the dependence of Ds on the polymer volume fraction φp. Ds0 is the value at φp → 0, i.e, in bulk solution. 2 0 , , (1 ) (1 ) p s i s i p D D φ φ  −  =  ₊      (25)

Diffusion in porous structures is conventionally explained using the tortuosity concept and the pore diffusion coefficient is related to the bulk value simply by Dp,i = Dp,i0/τ, where τ is the tortuosity factor.

The mass balance equation for a gel-type and macroporous particle is given in Eq. (26). Spherical symmetry with r representing the radial coordinate is used because most commercial resins and adsorbents can be taken as regularly-shaped spherical beads. Js and Jf are the diffusion fluxes in the particle and in the external liquid film. Concentrations in the bulk solution and at the solid-liquid interface are designated by c and c*. Mass transport through the film is characterized by the mass transport coefficient kf. The particle radius, diameter and porosity (pore volume fraction) are represented by Rp, dp and εp.