
HOMOGENEOUS BED ION EXCHANGE COLUMN
MODELS FOR ULTRAPURE WATER APPLICATIO S
AND SIMULATIO OF ION EXCHANGE BEDS
fN SERIES
By
ASHWIN P GRAMOPADHYE
Bachelor of Engineering
University of Pune
Maharashtra, India
1993
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 1996
HOMOGENEOUS BED fO EXCH GE COL
MODELS FOR ULTRAPURE WATER APPUC TIO S
AND SIMLTLATIO OF 10 EXCHANGE BEDS
IN SERIES
Thesis Approved
Dean of Graduate College
11
PREFACE
This thesis deals with modeling of homogeneousbed ion exchange under
conditions of filmdiffusion control for a multicomponent system of ions. A computer
program to simulate ionexchange columns in series is developed and. used to compare the
performance of homogeneousbed trains with a mixed bed for equal ionexchange
capacltles.
I am grateful to my advisor, Dr. Gary L. Foutch for his guidance, patience and
inspiration throughout my study at Oklahoma State University I would also like thank
Dr Arland H. Johannes and Dr. Randy Lewis for serving on my committee. I am indebted
to Dr J D Carlson (Department of Biosystems and Agricultural Engineering) for his
encouragement and the interest he has shown in my wellbeing and progress. Working
under Dr. 1. D. Carlson and Dr. M. A. Kizer has been a great pleasure and has given me
exposure to research in Agricultural Engineering  a field new to me. Financial support
from the School of Chemical Engineering and the Department of Biosystems and
Agricultural Engineering is gratefully appreciated.
Special mention is due to my sister Chitra and parents Prabhakar and Geeta
Gramopadhye for their love and emotional support. This study would not have been
possible without their motivation and help. I also want to thank my friends Ram, Tara,
Parag and Mandy for their encouragement and backing.
111
Chapter
1.
TABLE OF CONTENTS
Page
INTRODUCTION . . ... .. . . . . .. . ... .. . . .. .. . .. . . .
II
Ultrapure Water  Role of Mixed Bed Ion Exchangers and
Homogeneous Bed Ion Exchangers .
Objective ._. .. .. . . . . .. . . .. ... . .
LITERATURE REVIEW '" .
2
4
6
Applications of Homogeneous Bed Ion Exchangers..... 6
Classification of IonExchange Processes and Modeling
Approaches _ _.. _... ..... .. 7
Mechanism for IonExchange Processes Involving Reactions II
Models for Homogeneous Bed Ion Exchange at Low Solution
Concentrations .. . .. .. . . .. . .. .... .. ... .... 12
III
IV.
HOMOGENEOUS BED ION EXCHANGE MODEL .
Introduction.. .. . .
Assumptions . _.. _.. .. . .
IonExchange Equilibria . . .
Flux Expressions for LiquidFilm Diffusion .
Material Balance Equation for the Column .
Calculation of Temperature Dependent Parameters .
Comparison of Material Balance Equations in MBIE and
HEIE.... .. ..
Simulation of Homogeneous Beds in Series... . '"
HOMOGENEOUS BEDS VERSUS MIXED BEDS .
Abstract . . .. . . .
Introduction . . .. . .
Feed Water Conditions, Resin Properties and Bed Parameters
Results and Discussion ._ .. _.. .. . .. . . .. . .. ...
Discussion on Breakthrough Curves .. _.. ...... _..... ..
Discussion on pH of Effluents.. .. .. .. . .. .. .. . .. .
Conclusions _. .. ' ..
IV
17
17
18
21
23
26
27
29
'0
32
.? J_
32
36
37
50
60
BIBLIOGRAPHy................................................................ 62
APPENDICES. .. .. 65
APPENDIX A INTERFACIAL CO CENTRATIONS AND
IONEXCHANGE EQUILIBRIA. .. ... ...... . .. ... ... ... 65
APPENDIX B  MODEL EQUATIONS............................ 68
APPENDIX C  MATERJAL BALANCE EQUATIONS 77
APPENDIX D  NUMERICAL METHODS 82
APPENDIX E  SIMULATION OF HOMOGENEOUS BEDS
IN SERlES .. 84
APPENDIX F  COMPUTER CODE... .. 86
v
Table
I.
II
III.
IV
V.
VI.
VII.
VIII.
IX.
x.
LIST OF TABLES
Classification of IonExchange Processes Based on Nature of Interacting
Resin and Electrolyte . . .. ... .. .. . , .. . . . .
Classification of IonExchange Processes Involving Reactions .
Assumptions for Derivation ofNemstPlanck Equation from 1\1axwell
Stefan Equation ,.. . . o'
Assumptions Made in the HBIE Model... .. .. ..
Algorithm for Interfacial Concentrations and Rates of Film Diffusion .,.
Water Viscosity and Water Dissociation ...
Conductance as Function of Temperature .
Bed Parameters , ..
Resin Properties ..
FeedWater Condition used in the Simulations..................... ,... .......
VI
Page
8
10
20
25
27
28
35
35
36
Figure
2
...
'.
4
5.
6.
7.
8.
9.
10.
11.
12.
13
14
15.
16.
LIST OF FIGURES
Solution Strategies for Differential Material Balance......... .
Schematic Diagram of Homogeneous Beds in Series and Mixed Bed .
Sodium Breakthrough for ACB, CAB and !\.1B (High Concentrations) ....
Calcium Breakthrough for ACB, CAB and MB (High Concentrations) ...
Sodium Breakthrough for ACB, CAB and MB (Low Concentrations) ......
Calcium Breakthrough for ACB, CAB and ME (Low Concentrations) ....
Chloride Breakthrough for ACB, CAB and Iv1B (High Concentrations) "
Sulfate Breakthrough for ACB, CAB and Iv1B (High Concentrations)
Chloride Breakthrough for ACB, CAB and MB (Low Concentrations) ....
Sulfate Breakthrough for ACB, CAB and MB (Low Concentrations) .
pH of Effluent for ACB, CAB and Iv1B (High Concentrations) .
pH History for ACB (High Concentrations) .
pH History for CAB (High Concentrations) .
pH of Effluent for ACB, CAB and Iv1B (Low Concentrations) .
pH History for ACB (Low Concentrations) .
pH History for CAB (Low Concentrations) .
VIl
Page
30
34
38
40
42
43
45
46
47
48
5 J
53
54
56
58
59
NOMENCLATURE
as surface area per unit volume ofresin (L 1
)
B; any parameter B related to ionic species i
Cj concentration of ionic species i (meqlL~)
Ci' concentration of ionic species i at the resinliquid interface (meq/L·')
CO concentration of ionic species i in bulk liquid (meq/L~)
Cr total equivalent concentration (meqlL3)
dp particle diameter (L)
Oi selfdiffusivity of ionic species i (L2/T)
Dc effective diffusivity (L2/T)
F Faraday's constant (Coulombs/mole)
FR volumetric flowrate (L3/T)
FAR fraction of anionic resin in a mixed bed of ion exchange resin
FeR fraction of cationic resin in a mixed bed of ion exchange resin
Ji flux of ionic species i in the liquid film (meq/T L2)
k mass transfer coefficient (LIT)
KAB selectivity coefficient for ion A in resin replaced by B from solution
Kw equilibrium constant for water dissociation
m number of coions
Nj relative valence of counterion i
Vlll
n number of counterions
P an exponent derived from solution ofthe flux expressions
qi concentration of ionic species i in the resin (meq/L')
Q total capacity of resin (meqlL')
R universal gas constant
Re Reynolds number
Sc Schmidt number
T Temperature (K)
time (T)
u,
v
Xi
y,
Zj
Z·J
p
superficial fluid velocity of bulk liquid (LIT)
volume of resin in resin bed (L3)
fractional concentration of ionic species i in solution phase
fractional concentration of ionic species i in resin phase
charge (valence) on counterion i
charge (valence) on coion j
mean valence of coion i
GREEK LETTERS
thickness of liquid film surrounding resin bead (L)
void fraction in the resin bed
electric potential (ergs/coulomb)
conductance (Sm/mole)
viscosity of water (cp)
density (MIL')
lX
A
B
r
over bar
>I<
f
o
dimensionless time coordinate
parameter set to +1for cations and 1 for anions
dimensionless space coordinate
SUBSCRIPTS
ion leaving resin phase
ion entering the resin phase
counterions
cOlons
reference ion
SUPERSCRIPTS
any quantity related to species in the resin phase
any quantity at the interface between resin and liquid
any quantity in the feed
any quantity in the bulk liquid
x
CHAPTER I
INTRODUCTION
Ion exchange is the partition of charged species between different phases of a
system. It is a stoichiometric process. Every ion removed from one phase is replaced by
an equivalent amount of another ionic species resulting in an exchange of equal charges
between the two phases. Each phase maintains its electroneutrality.
Ion exchange shares traits with other methods of separation as illustrated by the
similarity between the methods used to quantify and study ionexchange phenomena and
the methods used in other separation sciences. For example, equilibrium curves can be
used to calculate distribution between phases for ion exchange in a fashion similar to
liquidliquid extraction. Blumberg (1984) examined the application by analogy of
concepts from liquidliquid extraction to resinliquid systems.
Ionexchange materials may be classified on the basis of the matrix that carries
the fixed charges. Thus we have the following types (Helfferich, 1962):
1. Mineral ion exchangers, e.g. zeolites,
2. Ionexchange resins, e.g. phenol sulfonic resins,
3. Ionexchange coals  they can be used as cation exchangers due to the
carboxylic acid groups. Some coals can be sulfonated into cation exchangers.
4. Liquid ion exchangers, where two immiscible liquids exchange ions, e.g. long
chain aliphatic amines dissolved in liquid xylene can act as an anion exchanger
when the solvent is dispersed in aqueous phase having exchangeable ions.
5. Other materials with ion exchange properties, e.g. keratin, alumina, etc. Some
substances such as nut shell and olive pits can be sulfonated to make cation
exchangers.
Ionexchange resins consist of a crosslinked hydrocarbon matrix, Vvith bonded
acidic or basic groups. The matrix can be fonned by polycondensation or addition
polymerization reaction and the fixed ionic groups can be introduced in the monomer or
in the polymer after crosslinking. Ionexchange resins are used more often than the other
materials listed above, due to their superior chemical and mechanical stability, higher ionexchange
capacity and higher rate of ion exchange as compared to the other materials.
Ultrapure Water  Role of Mixed Bed Ion Exchangers and Homogeneous Bed Ion
Exchangers
The term 'Ultrapure Water' denotes water with I ppb or less of ionic
contaminants (Sadler, 1993). Equally low levels of particulate and microbial impurities
are expected. Such water is required by power plants (Harfst, 1995), paper and pulp
manufacturers, petroleum refineries, dialysis units in hospitals, phannaceutical
manufacturers (Golden, 1986), compact disc manufacturers, semiconductor
manufacturing industry, etc. (Okouchi et aI., 1994).
2
Mixed bed ion exchange (MBIE) uses a mixture of cationexchange resin and
anionexchange resin. A fixed bed of such a mixture, with the aqueous phase flowing
down the bed, has been the favorite method in industry for achieving ultrapurity. In this
mode, cations and anions are removed simultaneously. The alternative is to use
homogeneous bed ion exchange (HBIE) where beds of cation and anion exchange resins
are separate stages.
The ad","antage ofMBIE over HBIE is that MBIE provides more separation zones
in less volume  very similar to having several stages of homogeneous cation and anion
exchange beds of very small depth alternating with each other in series. The cation and
anion resins replace the cations and anions, with hydrogen ions and hydroxide ions
respectively, which then combine to form water. The simultaneous removal of cations
and anions leads to a net reduction in ionic concentration in the bulk solution. Thus,
there is a localized equilibrium within the bed, and consequently we get a very high
separation efficiency. The net reduction in ionic concentration offers a distinctive
advantage over HBIE where demineralization becomes progressively difficult with the
removal ofjust cations or anions in a bed. For example, consider a cationexchange bed
followed by an anionexchange bed. The effluent from the cationexchange bed is acidic
and goes to the anionexchange bed. Anion exchange is thus highly favorable at the inlet.
However, with progressing anion exchange, equilibrium becomes limiting and exchange
is less favorable. Thus, the separation factor decreases sharply with progressive anion
exchange.
3
Net reduction in ionic charge was the primary justification for use ofMBIE as
opposed to HBIE in ultrapurewater facilities in industry. However, resin regeneration
presents two problems for MBIE, namely,
1. separation of cation exchange resin from anion exchange resin prior to
regeneration and
2. remixing the resins unifonnly after regeneration.
No such separation and remixing are required in HBIE. This represents one of the
biggest advantages of HBlE over MBIE. The other problem encountered in MBIE is
deterioration of anion exchange resin due to organic [oulants in the feed water (Fisher,
1993). Cation resin is less susceptible to organic fouling as compared with anion resin
and the cation resin can adsorb organic foulants without affecting resin perfonnance
drastically. Thus, the problem of anion resin fouling by organics in feed water can be
ameliorated in HBIE by installing the cation resin bed ahead of the anion bed. These
advantages have stimulated redevelopment of homogeneous bed techniques where cationanion
cation beds are used. A model that can predict the perfonnance of such a
homogeneous bed train will be an important tool in design and operation of water
treatment units for ultrapure water, and will complement MBIE programs in use.
Objective
The purpose of this thesis is to develop a model that can predict the breakthrough
for a homogeneous bed of resin at very low ionic concentrations. The model for MBIE at
very low ionic concentrations developed by Haub and Foutch (1986 a,b) considered the
4

effect of water dissociation, the ratio of cationic to anionic resin, differing resin exchange
rates and differing resin exchange capacities. Temperature effects on resin selectivity
coefficients, ionic diffusion coefficients, ionization constant for water and viscosity of
bulk solution were also accounted for (Divekar et al., 1987). Capability and complexity
of the model were increased through subsequent efforts by Zecchini (1990), Pondugula
(1994) and Bulusu (1994). The model is now capable of handling a multicomponent
system of ions having arbitrary valences in a mixed bed. This model will be used as a
foundation for the current endeavor. The model developed in this thesis will be used to
simulate a train of homogeneous beds. Performance of the homogeneous bed train will
be compared to that of a mixed bed.
5
CHAPTER II
LITERATURE REVIEW
A thorough literature review of ion exchange in general, mixed bed ion exchange
modeling and homogeneous bed ion exchange modeling (for weak electrolytes) has been
perfonned by Haub (1984), Yoon (1990), Zecchini (1990), Lou (1993) and Chowdiah
(1996). This chapter will focus on previous efforts in modeling of homogeneous bed ion
exchange at low concentrations (around 103 M).
Applications of Homogeneous Bed Ion Exchange
Homogeneous bed ion exchange (HBIE) may find use in a wide variety of
operations. For example, anion resin in hydroxide form is used to catalyze the reaction of
oxygen scavengers such as hydrazine (N2H4) and carbohydrazide ((N2HJ)2CO) with
oxygen at ambient temperature (Cutler and Covey, 1995). Golden (1986) lists other uses
like color removal from organic solutions, chromatographic separations and 'controlled
release.' The concept oecontrolled release' is used to administer macronutrients and
trace elements in hydrocultures and to administer drugs at correct levels in medicine.
6
Homogeneous beds of ionexchange resins are also used in industry for
demineralization of water. Besides homogeneous bed ion exchange, several methods are
available for demineralization of water. Design of an ultrapure water facility will be
guided by the economics and applicability of these methods with respect to their strengths
and weaknesses. Beardsley et al. (1995) compared the following systems for
demineralization of water:
1. three bed ion exchange (anioncationmixed bed)
2. double pass Reverse Osmosis (RO)
3. RO followed by mixed bed ion exchange
As total dissolved solids (TDS) in feed go up, cost of demineralization increases for all
three systems. However, there is a breakpoint below which the ionexchange system is
cheaper. This breakpoint was found to be at 75 ppm ofTDS as calcium carbonate in
1987. The breakpoint rose to 130 ppm ofTDS as calcium carbonate in the year 1994.
Though most waters demineralized in the USA are still above this level, the results of
Beardsley et al. (1995) indicate that ionexchange systems are becoming more
economical than RO systems.
Classification of IonExchange Processes and Modeling Approaches
A survey of modeling efforts in ion exchange is incomplete without reference to
the classification of ionexchange processes. As borne out by the following discussion,
classification is an integral part of ionexchange modeling. The assumptions made in
7
developing the models, their limitations and their applications will also be studied in this
survey.
Classification Based on Nature of Interacting Resin and Electrolyte
Ionexchange processes can be classified on the basis of the nature of interacting
resin and the nature of electrolyte. Based on this approach, we can have eight different
cases (see Table I). Lou (1993) developed a model capable of handling cases IV and
VIII. His model simulated sorption of boric acid at very low inlet concentration, when
film diffusion becomes the rate controlling step.
Table I
Classification Based on Nature of Interacting Resin and Electrolyte.
Case Number Resin Type Electrolyte Type
I Strong Base Resin Strong Acid
II Strong Base Resin Weak. Acid
III Weak. Base Resin Strong Acid
IV Weak Base Resin Weak Acid
V Strong Acid Resin Strong Base
VI Strong Acid Resin Weak Base
VII Weak Acid Resin Strong Base
VIII Weak Acid Resin Weak Base
8
Classification Based on Rate Determining Step
Typically, the ratedetermining step for ionexchange processes is diffusion of
counter ions. The chemical reaction is instantaneous compared with the rate of ionic
diffusion. An exception occurs in resins with chelating groups, which form reaction
complexes that react very slowly (Helfferich, 1962). The rate determining step also
depends on the concentration of the electrolyte in the bulk solution and either particle or
film diffusion can be rate determining depending on the solution concentration.
Helfferich (1962) gives the following criterion for identifying the rate determining step:
affect the rate of ion exchange
film diffusion is controlling
particle diffusion is controlling
in the intennediate range, particle as well as film diffusion
where,
Q = resin phase concentration of ions
C =bulk phase concentration of ions
o = diffusivity of ions in liquid film
9
D = diffusivity of ions in resin phase
ro= radius of the resin bead
8 =film thickness
U BA = resin selectivity
The scope of this thesis is limited to ultrapure water applications (ionic
concentrations in the range of ppm or less). The above criterion indicates that,
controlling mechanism for such applications is film diffusion, because of very low values
of ionic concentrations involved.
Classification Based on the Nature of Reaction between Participating Ions and Coions
Helfferich (1965) classified ionexchange processes involving reactions into four
types (see Table II). Helfferich also proposed rate laws for each type of process, under
conditions of both film and particlediffusion control. Blickenstaffet al. (1967a)
provided experimental evidence to support Helfferich's model for Type I, under
conditions of filmdiffusion control. Their study (B lickenstaff et aI., 1967b) of Type I,
with particlediffusion controlled neutralization, gave verification of Helfferich's model
for this case too, but only under the condition that the concentration ofthe exchanging
electrolyte in the solution phase does not fall with time.
10
I
Table II
Classification of IonExchange Processes Involving Reactions
Number Process Description
I Counterions from the ion exchanger react with coions from the solution
II Counterions from the solution react with fixed ionic groups of the ion
exchanger
III Undissociated fixed ionic groups of the ion exchanger react with coions from
the solution and form salts that can dissociate into ions
IV Undissociated fixed ionic groups of the ion exchanger react with counterions
from the solution and form new unclssociated ionic groups
Mechanism for IonExchange Processes Involving Reactions
The re::tction between weak base anion exchanger and acid proceeds via
protonation of the ionic sites of the resin by the acid (Helfferich and Hwang, 1985,
Bhandari et al., 1992 a,b). Helfferich and Hwang (1985) studied the kinetics of acid
sorption by weak base anion exchangers and they assumed that the acid sorption by most
weak base resins is irreversible under most conditions. They applied the shrinking core
model along with this assumption to describe the above process. However, Bhandari et
al. (1992a) hold that neither the shrinking core model nor the assumption of irreversible
sorption is valid for most conditions, especially so at lower concentrations than those
studied by Helfferich and Hwang (1985) and when the reacting resin has a lower basicity
than the ones studied by Helfferich and Hwang (1985). Bhandari et al. (1992a) propose
11
the existence of a '"charged double layer" at the pore walls in the resin. Their reversible
sorption model for sorption of strong acids on weak base resins yields a concentration
profile in the bead which is similar to the one given by the shrinking core model, but the
boundary between reacted shell and wrreaeted core is diffused.
Bhandari et al. (1992b) extended the "double layer theory" to sorption of weak
acids on weak base resins and found that :
1. for weak acids like fonnie acid, the contribution of undissociated acid to net
flux of acid into the resin is higher than the contribution of ionic fluxes, and the
rate controlling model of Helfferich and Hwang (1985) is valid only in this case
2. for stronger acids, contribution of ionic flux to net flux of acid into the resin is
higher
The "double layer theory" postulates a much weaker Donnan exclusion of coions
than that assumed by the rate controlling model. However, Bhandari et al. (1993) found
that the results from the "double layer theory" applied to the sorption of dibasic acid on
weak base resin agree very well with those from the shrinking core model proposed by
Helfferich and Hwang (1985). Bhandari et al. (1993) postulate that, reversibility of
sorption is much lower for dibasic acids than monobasic acids. Bhandari et al. (1993) also
postulate that, hydrogen coions in a solution of dibasic acid have more access to the resin
pores than in the case of monobasic acids. The increased access for hydrogen coions, to
resin pores in a solution of dibasic acid, is due to weaker Donnan exclusion in dibasic
acids as compared to monobasic acids. A weaker Donnan exclusion ofcoions in dibasic
acids, as compared to monobasic acids, is in tum attributed to stronger sorption of
divalent anions by neutralization with two sites in anion resin pores. Thus, both the
12
"double layer theory" and shrinking core model predict more access for hydrogen coions,
to anion resin pores, when bulk solution has polybasic acids. Both models also agree on
irreversibility of sorption of polybasic acids on weak base resins.
Models for Homogeneous Bed Ion Exchange at Low Solution Concentrations
All the HBIE models found in the course of this literature survey apply to solution
concentrations of 0.001M or more. Kraaijeveld and Wesselingh (1992) studied ion
exchange under conditions of filmdiffusion control. Their experiments (at solution
concentrations of 0.001 M to 0.1 M) show that under conditions of film diffusion control,
the ionexchange processes between sodium  hydrogen and calcium  hydrogen are faster
when the hydrogen ion moves from solution to the resin than the case where hydrogen
ion moves from resin to solution. They conclude that under the conditions studied by
them ionexchange kinetics depend upon the direction in which exchange is taking place.
Petruzzelli, Liberti et al. (1987) studied binary ion exchange (chloride  sulfate) in
the forward and reverse directions and isotopic exchange for sulfate. They concluded that
film and particle diffusion both played a role in the exchange kinetics at the concentration
range (0.006 M chloride and 0.003 M sulfate) studied by them. The results from their
computer simulation combining film and particle diffusion resistance yielded results
close to pure film diffusion up to a fractional approach to equilibrium of 0.4. The results
were close to pure particlediffusion control only when complete equilibrium was
approached.
13
~q
Investigation by Liberti, Petruzzelli et al. (1987) into chloride  sulfate exchange
at high concentration (0.9 M sulfate, 1.8 M chloride) show that kinetics of ion exchange
is particlediffusion controlled at those concentrations. They also evaluated the results
obtained by using NernstPlanck equations to model the kinetics at high concentration.
They found that actual rates at low conversion of the resin were lower than those
predicted and the actual rates at high conversion were greater than those predicted. At
low conversions, the authors regard retardation by film diffusion as the cause for model
inaccuracy and at high conversion, they attribute the inaccuracy to an appreciable change
in swelling of the acrylate based resin.
Applicability ofNernstPlanck Equation and Applicability of Complete Donnan
Exclusion at Low Solution Concentrations
The NernstPlanck equation is a special case of the MaxwellStefan equation and
the former can be derived form the latter by making assumptions listed in Table III. The
NernstPlanck equation neglects all the terms except the term for friction between the
solvent and the ions, and the term for resistance to mass transfer by electrical gradient
(Kraaijeveld and Wesselingh, 1992). Kraaijeveld and Wesselingh (1992) modeled the
ionexchange process using MaxwellStefan transport equations and used the film
thickness as the fitting parameter. Even at the concentration range (0.001 M  0.1 M)
studied by them, the difference in film thickness fitted for MaxwellStefan and Nemst
Planck equations was found to be less than three percent. The difference between the
14
predictions made by MaxwellStefan and NemstPlanck ,equations is expected to reduce
further at the low concentration ranges encountered in ultrapure water applications.
Therefore, use of NemstPlanck equations instead of MaxwellStefan equations is
justified for work at these concentrations.
Table III
Assumptions for Derivation of NemstPlanck Equation from MaxwellStefan Equation
Assumptions
1. Constant coefficient of diffusion for each ion.
2. All activity coefficients equal unity.
3. No convection (diffusion across a static film).
4. No gradients of temperature and pressure.
5. Electric potential changes only in the radial direction for the resin bead and liquid
film.
Complete Donnan exclusion of coions from an ionexchange resin at low solution
concentrations is a well knoVlIl and widely accepted principal. However, complete
Donnan exclusion of COlons from the resin does not occur at high solution concentrations
and some accumulation of coions is observed in the resin at these conditions (Femandez
et al., 1994). Fernandez et al. (1994) have studied the kinetics and equilibrium of the
phenomenon.
Fernandez et al. (1995) have evaluated the pore diffusion model and unreacted
core model for chelating ion exchange and cationic exchange under conditions of
15
reaction control. Helfferich (1984) described a new approach to modeling of
multicomponent ion exchange which differs from the usual theoretical approach
involving differential material balances and flux equations. The new approach is
intended for application to columns with variable feed conditions so as to yield
predictions with little calculation effort. The concentration variations along the length of
the column are viewed as "waves" and the disturbances created by variations in feed
concentrations are compared to interference between waves.
16
. r..

CHAPTER III
Homogeneous Bed Ion Exchange Model
Introduction
This chapter deals with the development of a model for muiticomponent,
homogeneous bed ion exchange at very low solution concentrations (film diffusion
controlling). The mixed bed ion exchange (MBIE) model (Bulusu, 1994) is modified for
homogeneous bed ion exchange (HBIE) when either cation or anion exchange resins are
used.
There are very few studies of ionexchange kinetics at very low solution
concentrations (Jess than 104 M) available in the literature. Experimental as well as
modeling efforts in ion exchange have traditionally focused on operations at higher
concentrations for homogeneous beds as well as mixed beds. Haub and Foutch (I 986a,b)
and other workers (mentioned in Chapter I) developed simulation tools for
demineralization plants employing mixed bed ion exchange at ultrapure water
concentrations and filled the gap. The current model is intended to be an extension of
this work into the area of HBIE. Minimization of the computational cost of the
17
q


simulation is also of high priority in the current effort. The model development will
follow the sequence and nomenclature described by Bulusu (1994).
Asswnptions
Table IV lists the assumptions made in this model. These asswnptions have been
used and justified by earlier workers in MBIE with success. In the course of the literature
review (chapter II) new studies were found which have bearing on some of these
assumptions and these assumptions will be discussed here. Those assumptions, which
have special bearing on HBIE modeling (as opposed to the MBIE models for which these
have been justified and tested), will be also discussed.
L BulkPhase Neutralization
Consider a bed of cationexchange resin followed by a bed of anionexchange
resm. The effluent from the cationic bed has an acidic pH and this effluent is fed to the
anionic bed. In the anionic bed, the pH of the solution increases from an acidic pH at the
inlet to a basic pH at the exit. Thus, at the inlet the reaction plane for the neutralization
reaction will be close to the surface of the resin bead in the liquid film surrounding the
bead. The reaction plane will shift away from the resin bead and towards the bulk liquid
as the pH increases down the column. Thus, bulk phase neutralization may not be a good
assumption for all the conditions encountered in HBlE. However, complexity of the
problem increases greatly if liquidfilm neutralization is assumed (Haub and Foutch,
18
q

1986a). Haub and Foutch (1986a) also conclude that the reaction plane approaches the
bulk phase as the ionic concentrations fall to 1x107 M(at 25°C). Therefore, a small error
is expected due to the assumption of bulk phase neutralization at the concentration levels
encountered in ultrapure water applications. In keeping with the objective to minimize
model complexity and consequently the computational effort, the assumption of bulk
phase neutralization is justified.
2. LiquidFilm Diffusion is Rate Controlling
Resistance from particle diffusion is negligible compared with film diffusion.
Haub and Foutch (1986a) first justified and successfully applied this assumption under
the conditions of high flow rate and low ionic concentrations encountered in the
demineralization plants. Petruzzelli et a1. (1987) combined particle and film diffusion in
their model for binary exchange of ions. Their results indicate that the solution for
combined resistance coincides with that for film diffusion up to 40% conversion of resin
at concentration of 0.006 M and particle diffusion becomes important only after 75%
fractional approach to equilibrium. Studies by Kataoka et al. (1976) indicate that particle
diffusion becomes important only after 80% conversion is reached in the resin at
concentration of 0.0025 M. Thus, fi 1m diffusion resistance remains the controlling
resistance for high conversion of the resin at low solution concentrations.
19
• .AiII
Table IV
Asswnptions Made in the HBIE Model
1. Bulk phase neutralization
2. Liquidfilm diffusion is rate controlling
3. NemstPlanck equation is adequate to model all the interactions between the
diffusing ions
4. Pseudo steady state ion exchange
5. Local equilibrium at solidliquid interface
6. Total Donnan exclusion of coions from the resin beads (no coion flux across the
particle surface)
7. No net coion flux in the liquid film
8. No net current flow
9. Electroneutrality is always maintained in the resin, film and bulk liquid
10. Selectivity coefficients are constant throughout the column, and with temperature
11. Binary selectivity coefficients can be applied to multicomponent ion exchange
12. Reaction step is instantaneous when compared to the rate offilm diffusion
13. Particle diffusion resistance is negligible
14. Uniform bulk and resin phase compositions
15. Curvature of the liquid film is negligible
16. Activity coefficients are always unity
17. Plug flow, negligible axial dispersion
18. Isothermal, Isobaric operation
20
.
 ...91
An ionexchange resin placed in an electrolyte solution will take up ions from the
demineralization can be achieved with a given ionexchange resin bed. Consider an ion
"
(32)
(31 )
entity with over bar signifies that ion is in resin phase
ZA and ZB are charges on ion A and B
KBA= resin selectivity for ion A in resin phase replaced by ion B from bulk phase
IonExchange Equilibria
solution in exchange for counterions from its own ionic groups embedded in the
hydrocarbon matrix. After equilibrium is reached, no net exchange of ions is possible
between the resin and the solution phases. This equilibrium represents the limit to which
where,
exchange reaction,
The law of mass action for this reaction is written as
where,
qA and ~ are resin phase concentrations of A and B respectively
CA' and CB' are interfacial concentrations of A and B respectively
The equilibrium constant for the ionexchange reaction is called the selectivity
coefficient. Assumption of local equilibrium at the resin  liquid interface lets us write
21
.~

the equilibriwn relationship in terms of resin phase concentration and interfacial ionic
concentrations. The selectivity can be written in terms of equivalent fractions as,
where, we know initial resin loadings YB and YA> the resin capacity Qand selectivity
independent equations for equilibria. The material balance at the resin  liquid interface
(33)
(34)
If there are a total of n counterions involved in exchange, we can write nl such
YA and YB are resinphase fractional concentrations ofA and B respectively
XA' and xB' are interfacial fractional concentrations ofA and B respectively
Cr' is total interfacial ionic concentration
Qis total resinphase ionic concentration (resin capacity)
where,
The interfacial equivalent fraction of an ionic species B can now be written as,
coefficient (properties of the resin determined by experiment or known from
manufacturer's specification).
is the nIh equation,
n
LX~ 1 = 0
i=l
(35)
Substitute the n1 equilibrium equations (for nl counterions, of form identical to
Equation 34) into the material balance (Equation 35) to obtain a polynomial in terms of
22

 .I!IA
described here.
find the correct value by iterations.
Flux Expressions for LiquidFilm Diffusion
(36) dC,
dtI   J i as
polynomial can be evaluated and the polynomial can be solved to obtain the interfacial
The ionexchange equilibrium is a function of total interfacial concentration
We have assumed that liquidfilm diffusion is the ratecontrolling step for the ionwhich
is unknown and must be found from the ionic flux rates (as shown in the following
XA•• If the total interfacial concentration CT· is known, then the coefficients of the
equivalent fractions. Refer to Appendix A for a detailed discussion on the methods
discussion). The flux rates in turn depend on the interfacial and bulk concentrations.
Thus, we must start with a guess value for the total interfacial ionic concentration and
exchange process. Therefore, the rate of change of concentration of an ionic species in
the resin is same as the rate of diffusion of that species of ion throu'gh the liquid film.
Thus,
where,
as = surface area of resin beads per unit volume
Jj = flux of ionic species in the liquid film
Cj = resin phase concentration of ions i
23
The term for gradient in electrical potential (d~/dr) in the NernstPlanck equation is
The flux expressions are then integrated with appropriate boundary conditions to find a
(37)
(38)
R = universal gas constant
<I> = electric potential
F = Faraday's constant
r = radial distance
where,
the ions in the resin yields
In order to find the rate of resin loading, the flux of counterions across the liquid
Replacing the resinphase concentration of ions in Equation 36 by equivalent fraction of
film Ji must be known. Ji is modeled by the NernstPlanck equation as :
gradient in electrical potential is replaced by a function of total equivalent concentration.
eliminated because the gradient in electrical potential is difficult to evaluate. The
relation between the total equivalent concentration CT and the individual ionic
concentration Cj • The method proposed by Franzreb et al. (1993) is used for this
purpose. This method yields an exact solution of the flux expression for the special case
of counterions with equal valences and yields an approximation for the case of arbitrary
counterion valences (Refer Appendix B for a detailed derivation of the methods described
24
here). The total interfacial concentration of ions (on the resinliquid interface) is given
P = a parameter in the solution of the flux expression and Pis detennined by
(39)
coion valence) and
c~=
Nj = the relative valence for ion i (ratio of valence of ion i to the average value of
by,
where,
integration of the flux expression between appropriate boundary conditions.
CT
D = total bulkphase concentration of ions
Ionic concentrations within the liquid film are expressed in terms of
concentrations at the resinliquid interface and the bulk concentrations. Ionic flux rates
(JJ for each species of counterions in the liquid film are then given by,
(310)
This expression for Jj is valid for any arbitrary number of counterions and coions with
arbitrary valences. The film thickness in Equation 310 is substituted by
8 = De
k
(31 I)
25
and k is calculated by the correlation of Dwivedi and Upadhyay (1977). The rate at
which ions load on the resin is found by substituting value of Jj from Equation 310 into

The effective diffusivity Dc in Equation 311 is given by,
n
L:ILO!
D e = _.:....=i..;..1__
f Ic;  c~1
1=1
Equation 37 as described earlier.
(312)
Algorithm to Calculate Interfacial Concentrations and Flux Rates As described earlier.
the total interfacial concentration must be found by iteration. For this purpose the
strategy outlined in Table V is adopted.
Table V
Algoritlun for Interfacial Concentrations and Rates of Film Diffusion
1.
2. Evaluate the coefficients for the polynomial obtained by substituting Equations 34
3.
into Equation 35
Solve the polynomial for xA ' and calculate<'s from the equilibrium Equations 34 !o
4. Find the value of CT' using Equation 3.9.
5. Compare old value of CT' with new value. Return to 2 and iterate till CT' has
converged to a value within the desired tolerance.
6. Calculate the flux rates for ions in the liquid film with Equation 310
26

Material Balance Equation for the Column
In order to predict the effluent concentration with time, a material balance for the
column must be setup. A differential material balance can be applied to a very small slice
of the resin column. The net increase in the amount of an ionic species present in the
slice equals the difference between the influx and efflux of the ionic species. In a film
diffusion controlled rate model, the net rate of accumulation (or exhaustion) of an ion in
the slice equals the rate of ion transport through the film. The material balance equation
for exchange of ions between the resin bed and the solution phase is written as,
Us aCi + aCi + (1~) aqj =0
~ az at f: at
where,
Us = superficial fluid velocity of bulk liquid
£ = void fraction in the resin bed
Z = bed depth
(313)
With dimensionless distance (~) and dimensionless time (T), this equation can be
reduced to the following fonn (refer to Appendix C for definition of dimensionless
variables and derivation of the equation)
_ax_'+ _ay_.1=0
a~ Or:
The material balance for the column is integrated using the method of
(314)
characteristics (Haub, 1984). Appendix 0 discusses the approach used for solution of the
material balance equations and lists the numerical methods applied for the integration.
27

Calculation of Temperature Dependent Parameters
The HBIE model uses the following temperature sensitive parameters:
1. Viscosity of water. The viscosity of water is estimated at the desired temperature
with the relation given in Table VI.
2. Equilibriwn Coefficients. Water dissociation equilibrium is a weak. function of
temperature (see Table VI). The model assumes the ion selectivities are constant with
temperature on account oflack of data to compute the temperature effects on this
parameter.
Table VI
water viscosity
water dissociation
Water Viscosity and Water Dissociation
!J. =1.43123 + (T  273.15)[0.000127065(T  273.15)  0.0241537]
pKw = 4470.99 _ 6.0875 + 0.0176T
(T)
3. Diffusion Coefficients The diffusion coefficients and electrical conductivitie of ions
are interrelated (Helfferich, 1962). If the conductivities (AoJ are available as a
function of temperature, diffusion coefficients can be calculated at the desired
temperature with the use of Nemst equation,
28
(315)

Table VII lists the conductivities as functions of temperature (Divekar et al.,
1987). The conductivities estimated by these relationships are used for calculation of
diffusion coefficients in this model.
Table VII
Conductance as Function of Temperature
Ion Conductance
Hydrogen ).°H =221.7134+5.52964T0.OI4445T1
Sodium ).°Na =23.00498+1.06416T+0.0033196T2
Calcium /..0Ca =(23.27+ 1.575T)/2.0
Hydroxide /..°OH =1.0474113+3.807544T
Chloride /..°Cl =39.6493+1.39176T+O.0033196T2
Sulfate ).°504 =(35.76+2.079T)/2.0
Comparison of Material Balance Equations in MBIE and HBlE
So far, the treatment for HBIE modeling has been the same as MBIE with the
exception of material balance equations used to describe the column. The material
balance used in the MBIE model (Bulusu, 1994) reduces to the following fonn after
applying the dimensionless time and distance variables:
~
29
(316)
......

The derivatives in Equation 316 are with respect to dimensionless variables defined for a
common reference ion. In both the models, HBrE as well as MBIE, the column is
divided into infinitesimal slices (perpendicular to the column axis) and integration of the
column material balance is carried out by finite difference methods. In the HBrE
simulation program, effluent from one slice forms the feed for the next slice. A
subroutine carries out the algorithm described in Table V for each slice and returns the
flux rates to the main program which carries out the task of integrating the column
material balance.
Figure I
Solution Strategies For Differential Material Balance
Homogeneous Bed
Water In
~
Water Out
+Slice of bed .
Mixed Bed
30
Water In
~
/~l
~~
~
Water Out

The solution strategy used in the MBIE model, to solve the colwnn material
balance, is different from the solution strategy used in HBIE. The MBIE model further
divides each slice into four zones ofcationic and anionic resins alternating with one
another (see Figure l). The HBIE strategy does not require this division. In the MBIE
model, each zone is treated like a slice of homogeneous resin bed and calculation of flux
rates, interfacial concentrations, etc. is carried out for each zone. Thus, effluent from the
first zone of cationic resin forms the feed for the next zone of anionic resin and so on.
This alternating pattern of homogeneous subslices is used to simulate a mixed bed in
MBIE.
Simulation of Homoger:eous Beds in Series
The HBIE model can be adapted for simulation of homogeneous ionexchange
beds in series. For this purpose, the integration of the column material balance forthe
second column must be done under the condition of variable feed concentration. The step
sizes in time used for integration of column material balances for two columns in series
must be equal.
Ionexchange trains consisting of one cationicresin bed and one anionicresin bed
in series are simulated and the effect of the order of the beds on the effluent water quality
is studied. The performance of a mixed bed unit (predicted by MBlE model) is also
compared with the performance of homogeneous beds in series. The procedure used for
these simulations is outlined in Appendix E.
31
CHAPTER IV
HOMOGENEOUS BEDS VS. MIXED BEDS
Abstract
The perfonnance of a mixed bed is compared with that of two homogeneous beds
in series for different configurations. Simulations are run for feed water having ionic
contaminants at two levels, namely, at concentrations equivalent to city water (several
ppm) and at lower concentrations (several ppb). The order of the beds in the
homogeneousbed train is found to affect the effluent pH and the ionic concentration in
the effluent.
Introduction
An ultrapure water facility may employ a train of ionexchange beds consisting of
two homogeneous beds followed by a mixed bed (it may also have other units like
reverse osmosis included in this train). The order in which the ionexchange beds are
placed in such a train will be decided by feedwater chemistry and the effects of the order
of the beds on effluent quality, bed performance and resin life. For instance, consider
32

feed water contaminated by calcium salts. Calcium hydroxide tends to precipitate on the
ionexchange resin beads at high pH. Therefore, the cation bed must be placed upstream
of the anion bed to remove the calcium cations before the water reaches a high pH in the
anion bed. If the anion bed is placed upstream for a feed contaminated by calcium, then
the calcium ions will precipitate on the resin beads at the high pH encountered in the
anion beds. On the other hand, if the cation bed is placed before the anion bed, the cation
bed will "slough" off benzene sulfonic acids. These products are fonned due to
degradation of the cation resin with the passage of time, and can foul the anionexchange
resins in the following bed (Fisher, 1993). If city water is being used as feed, chloride
radicals from the city water will decrosslink. resins by oxidizing the bonds (Fisher, 1993).
In this case, if we have an anionexchange bed in the lead, then chloride ions can be
removed from the feed before they decrosslink resins in subsequent beds.
Thus. feed water chemistry, desired effluent composition, etc., will decide the
order in which beds should be placed. Therefore, it is desirable to predict the effect of the
order of the beds on effluent quality. How will a mixed bed unit compare with a cationic
bed followed by anionic bed or anionic bed followed by cationic bed, if resins in all the
beds have the same exchange capacity and resin properties?
Feed Water Conditions, Resin Properties and Bed Parameters
In this chapter, the bed configurations sho'Ml in Figure 2 are compared with
respect to their effluent quality. The configurations will be referred to by the
abbreviations  ACB for anion bed followed by cation bed, CAB for cation bed followed
33

by anion bed and MB for mixed bed. Same anion and same cation resin is used in all the
configurations. Resin properties are listed in Table IX. The volume of cation resin in all
the configurations is same and so is the anionresin volume. Thus, the total ionexchange
capacity of the anion resin in the ACB, CAB and MB configurations is same. Similarly,
total ionexchange capacity of cation resin in all the configurations is also equal. The bed
parameters are listed in Table VIII. Cationexchange capacity is not equal to anionexchange
capacity in any configuration.
Figure 2
Schematic Diagram of Homogeneous Beds in Series and Mixed Bed
(White for Anion Bed, Black for Cation Bed, Gray for Mixed Bed)
./.~.
""
ACB CAB MB

34
Table VIII

Bed Height
Bed Parameters
Bed Diameter Void Fraction
ACB
CAB
MB
46.84 em each bed 152 cm
46.84 em each bed 152 cm
93.68 em (cation/anion volwne = 1: I) 152 cm
0.35 in each bed
0.35 in each bed
0.35
Table IX
Resin Properties
Cation Resin Anion Resin )1'1
~:.c
Dowex Monosphere 650C Dowex Monosphere 550A ~
.:>..
Diameter of Resin Bead 0.0625 cm 0.055 cm 2:
::>
Capacity of Resin 1.9 1.1 ~
Initial Resin Loading 1% for all ions in feed 1% for all ions in feed ~
~ Resin Selectivity: sodium 1.61 chloride 22
calcium 41.44 sulfate 60 ~
35


Table X
FeedWater Condition used in the Simulations
Case  I CaseII
sodium 37 ppm 20.24 ppb
calcium 10 ppm 22.4 ppb
chloride 13 ppm 31.24ppb
sulfate 83 ppm 53.76 ppb
Temperature 25°C 25 °c
Feed Rate 6lxlO3 eels 61 xlO> eels
The performance of different bed configurations is studied at two levels of ionic
concentrations, namely, at concentrations equivalent to city water (several ppm) and at
lower concentrations (several ppb). The feed water compositions, feed rates and
temperatures for both cases are given in Table X.
Results and Discussion
In a mixedbed ionexchange column, cations and anions are removed
simultaneously and replaced with hydrogen and hydroxide ions, respectively, which then
combine to form water. Now consider a homogeneous cationexchange bed. Here, with
progressing cation exchange the hydrogenion concentration increases down the bed.
Thus, the simultaneous removal of cations and anions in a mixed bed facilitates
immediate neutralization and the pH in the mixed bed remains closer to neutral than the
36


homogeneous bed. In the homogeneous cation bed, pH turns extremely acidic (or basic
in an anion bed) down the bed from the inlet to the outlet. Thus, the equilibrium becomes
adverse to the ionexchange process in the homogeneous bed. Consequently, we expect a
higher separation efficiency in a mixed bed than a homogeneous bed.
Discussion on Breakthrough Curves
Figure 3 shows sodium breakthrough curves for the case of higher feed
concentrations, for all the bed configurations studied. ACB (anion followed by cation
bed) gives a sharp breakthrough for sodium, very similar to the MB (mixed bed), while
CAB (cation followed by anion bed) gives a comparatively gradual breakthrough.
Effluent sodium concentrations before breakthrough are higher for the CAB than the MB
and ACB. These differences may be explained by the effect of pH and removal of
coions.
Effect of pH. In the ACB configuration, basic effluent of the anion bed is fed to the
cation bed. With progressing cation exchange, the pH of the bulk solution in the cation
bed of ACB turns increasingly acidic. In the mixed bed, the simultaneous removal of
cations and anions facilitates immediate neutralization and the pH in the mixed bed
remains closer to neutral than the homogeneous bed. Thus, the pH of the bulk solution in
the mixed bed is close to neutral and it changes from basic to acidic in the cation bed of
the ACB configuration. These pH conditions are more favorable for cation exchange
than the pH in the lead cationresin bed of the CAB configuration, where the feed is
37
1
r
1.E+02 ... "' .
1
\ i '_."''  I I
w
00
1.E+01
1.E+OO
....
E
0.. .e 1.E01
co
.~ rn ~ .......
~ 1.E02 uco
U
1.E03
1.E04
1.E05
o
  // .......
50
,,
100 150 200
MBNa
..... ·ACB Na
 CAB Na
250 300
Time (minutes)
Figure 3. Sodium Breakthrough for ACB, CAB and MB (high concentrations)
OKLAHOMASTAl'l!; UN1V~n'I


neutral and turns acidic from inlet to outlet with progressing cation exchange. An acidic
pH in the lead cation bed of CAB results in an unfavorable equilibrium for cation
exchange. Consequently, the effluent cation concentration before breakthrough is higher
than ACB and MB.
Removal of coions. The anion concentration is constant with time for any fixed point in
the cation bed of CAB, while in the other two cases, i.e., MB and ACB the anion
concentration is changing with time at any given distance in the bed. However, the effect
of coion removal I is expected to be very weak (Franzreb, 1993).
Figure 4 shows calcium breakthrough for ACB, CAB and MB for higher feed
concentration. The calcium leakage for CAB again stands apart from the ACB and MB,
but the differences are ~ess pronounced for calcium as compared to those for sodium
(Figure 3). The difference in performance of CAS as compared to ACB and MB may be
attributed to an unfavorable pH (unfavorable equilibrium) and the effect of coion removal
as in the case of sodium.
I This effect of coion concentration, on the rate of ion exchange, is incorporated in
the model through the use of Franzreb's (1993) method to solve the NemstPlanck
equation for the ionic flux in the static film assumed to exist around a resin bead. This
method uses the coion concentrations to find the mean value of the coion valences. The
mean coion valence is used to find the relative valence of the counterions for the solution
of the NemstPlanck equation.
39
r
1.E+02 ,  ..  
1
1.E+01
1.E+00
,;.
E
c...
S 1.E01
c0
+'
~
+' cQ) 1.E02
~ ()
0 c0u
1.E03
1.E04
MBCa
···_··ACBCa
   CAB Ca
1.E05
o 50 100 150
Time (minutes)
200 250 300
Figure 4. Calcium Breakthrough for ACB, CAB and MB (high concentrations)
OKLAHOMA ~TATlS UNlV~u. 1


The discontinuities in the breakthrough curves (Figure 3 and Figure 4) are
ascribed to the following factors:
Change in the Cation being Replaced from the Cation Bed. Sodium breakthrough for
ACB, CAB and MB occurs between 10 and 50 days (Figure 3). The sodium
breakthrough signifies that all hydrogen in the cation bed has been exhausted. Coincident
with the sodium breakthrough, effluent calcium concentration also rises sharply for all the
configurations because the calciwn now begins to replace sodium in the bed instead of
hydrogen. Sodium has higher selectivity than hydrogen, which makes the former more
difficult to replace than hydrogen.
Logscale on the Vaxis of the plot. The concentrations in the figures are plotted on
logarithmic scale. This magnifies small changes in the effluent concentration. These
small changes in effluent concentrations may occur on account of:
1. changes in feed pH and coion concentrations (when the cation bed is
downstream of anion bed  ACB configuration),
2. instability in the numerical integration.
Figure 5 shows that, even at lower concentrations (several ppb), sodium leakage
from the CAB is higher than ACB and MB. Calcium leakage (Figure 6), however, does
not show pronounced differences for different configurations, at lower concentrations.
These differences between the breakthrough curves for different configurations at higher
concentrations (Figures 3 and 4) were explained on the basis of effect of pH and removal
41
r 1
1.E+01
1.E+00
.......... ..c
c.. e:
c0
:.t:=' r..o... c
~ ~ N
C
0u
1.E01

I ,.
,
,
:
~MBNa
   .  ACB Na
~
"
 CAB Na
200 400 600 800 1000 1200 1400 1600 1800 2000
1.E02 \ I ,
o
Time (days)
Figure 5. Sodium Breakthrough for ACB, CAB and MB (low concentrations)
OKLAHUMA ~rl\n!; UN1vr.d\I.')lll
r ,
1.E+01
.... ··ACB Ca
UKL..'U1UMA ~TJ\l.t. UNlvM\A.")111
1800 2000
CABCa
MBCa
400 600 800 1000 1200 1400 1600
Time (days)
Figure 6. Calcium Breakthrough for ACB. CAB and MB (low concentrations)
200
1.E05
o
1.E04 L.....~
1.E+00
. 1.E01
.c
Q.
Q. C0
'..tJ 1.E02 CO
L... C
~
Q)
w () C
0
(j
1.E03
~
of coions. At lower concentrations, these factors are expected to have a smaller effect
because the magnitude of changes (in pH as well as coion concentrations) is smaller at
lower concentrations. Moreover, coion removal is expected to have a smaller effect, as
the concentrations approach the ideal case of infinite dilution (at infinite dilution the
coions will have no effect at all on the exchange of counterions).
The initial leakage of chloride anions for different configurations is shown in
Figure 7 for the case of higher feed concentrations. In this case, the ACB configuration is
seen to have the highest chloride leakage before breakthrollgh. MB gives the least
chloride leakage, while CAB gives an initial chloride leakage intermediate to ACB and
MB. The high leakage of chloride from ACB can be explained in a manner similar to the
higher leakage of sodium from CAB, that is, on the basis of the pH and the effect of coion
removal.
Effect of pH. Th~ pH conditions in the mixed bed and in the anion bed of the CAB
configuration are more favorable than the pH in the lead anion resin bed in the ACB
configuration for anion exchange. An adverse pH (basic) in the lead anion bed of ACE
results in an unfavorable equilibrium and consequently a higher effluent anion
concentration before breakthrough.
Removal of coions. The cation (coion) concentration is constant with time for any fixed
point in the anion bed of ACB, while in the other two cases, i.e., MB and CAB the cation
(coion) concentration is changing with time at any given distance in the bed. As stated
earlier, the effect of coion removal is expected to be very weak (Franzreb, 1993).
44
......
i
r 1
1.E+02
80 90
 CABCI
MBCI
..... ·ACB CI
10 20 30 40 50 60 70
UlUJ\11UNt.1\ "1J\!~ Ul"lV~l~ 1
Time (minutes)
Figure 7. Chloride Breakthrough for ACB, CAB and MB (high concentrations)
    
I" . " ,;' , "
/' .'
     / .'
  .    .       .                 / (.::/ 
"
////
/
//
,,""
/"
1.E+01
1.E04
1.E06
o
1.E+00
1.E05
co~
1.E02 .:= c~
co
1.E03
()
i 1.E01
Q. .........
~
V'l
r 1
1.E+03  "_. _. ....._._.__..._..
Time (minutes)
Figure 8. Sulfate Breakthrough for ACB, CAB and MB (high concentrations)
90 100
...
 CAB S04
MBS04
 .... ·ACB S04
40 50 50 70 80
• 1$
.....,J..... .". ......,; ,
.'''; ,..
~v
/..'
.... 4""'"
... ..;,,:,. ;'..
...............
.".,,;" J"
..."'
30
... ......
.......
...
20
....... ...
U.lil.1\11UNtJ\ ~ 11\l.£.. U 1'41 v£t.n,:u J. 1
.... ...10
.... ,. 
1.E05 ,',r,
o
1.E+02
1.E+01
.. 1.E+00
E
aa.. CJ) 1.E01 c0
..::;
ro
"  1.E02 c
Q)
()
c0
u 1.E03
1.E04
1.E05
~
0\
r ,
I •...•.•. _.. I
1200
"'"~. ~
 CAB CI
MBCI
·· .... ACB CI
.o;;;;:~,
200 400 600 800 1000
U1'\LfU1V~.J\ ~.1J\.1J:. U1U vc..1\6,)111
Time (days)
Figure 9. Chloride Breakthrough for ACB, CAB and MB (low concentration)
I :
/ :'
I :
I '
I :
/ ,.,
I '
I
I
I
I "
..    .. ...  .  .  .. ..    .)~::
I
I /' ;'
~"
1.E03
o
1.E02
1.E+01 ._..__ .__... 
1.E+00
....
.c
0.
0. C0
+:;
CO 1.E01 ' +'" c::
Q)
~ ()
..) c0U
r ,
1.E+01 ....  .... _.
V1\Lf\11VIUt\ ~ 11\.1 £I V!..... ....:..a.\tJu. .a.
1200
 CAB S04
..... ·ACB S04
200 400 600 800 1000
Time (days)
Figure 10. Sulfate Breakthrough for ACB, CAB and MB (low concentrations)
 ",
......: ....:
,/ 7,'
,/ /' , "
,/
,/
,/
/'
,/
,/
./
,,"
,/
.,.,, ./ .,.,, .,.,,.,.,,
/ .,.,, .,.,,
./
.,.,, ./
,/
./
,, .
..."'/ .V MB S04
1.E06 ! 
o
1.E05
1.E04
1.E+00
1.E01
:0
c..
~ 1.E02
o
"§
......
c
~ 1.E03
co
(j
~
00

Figure 9 shows the history of chloride concentration in the effluents for the three
configurations for lower feed concentrations. At lower concentrations (several ppb) also,
chloride leakage from the ACB is higher than CAB and MB. A small dip is seen between
200 days and 400 days in the chloride throw from MB. This dip is attributed to
instability in the numerical integration of the column material balance.
The sulfate leakages from the beds (for higher feed concentration) are shown in
figure 8. Sulfate leakage from ACB is not higher than sulfate leakage from CAB, as one
would expect after studying chloride leakages. MB gives the lowest sulfate leakage of all
three configurations. The same observations are valid for sulfate leakages at lower feed
concentrations (Figure 10), namely, sulfate leakage from ACB is not higher than sulfate
leakage from CAB and, MB gives the lowest sulfate leakage of all three configurations.
This difference between chloride and sulfate leakages may be due to the binary charge on
the sulfate ion (as opposed to the unit charge on the chloride radical). Consequently, the
effect of feed pH on the sulfate breakthrough is not similar to the effect of feed pH on
chloride breakthrough.
The question now is: How can the effect of feed pH on monovalention
breakthrough be different from the effect of feed pH on divalention breakthrough? The
answer to this question may lie in the effect of the feed pH on the masstransfer
coefficients of the counterions. The masstransfer coefficient of the counterions is a weak
function of the bulkphase pH. The feed pH may have different effects on masstransfer
rates of monovalent and divalent ions and may thus affect their breakthrough.
Leakages of calcium from MB and ACB are lower than those from CAB (figures
4 and 6). The calcium breakthrough curves are similar to those of sodium though the
49
calcium is a divalent cation while sodium is monovalent. Thus, feed pH does not seem
affect the calcium breakthrough (and masstransfer coefficient) to the same extent as it
affects the sulfate breakthrough.
Discussion on pH of Effluents
Figures 11 to 16 show pH histories of the effluents for different configurations at
high and low feed concentration levels studied here. The pH ofthe effluent from
different configurations shows roughly similar trends at low as well as high feed
concentrations, with the difference that, the magnitude of changes in effluent pH (with
time) is smaller at lower feed concentrations for all configurations.
As seen in Figure 11, the pH of effluent from MB is close to neutral for a neutral
feed, before breakthrough is reached. The effluent pH is acidic for the ACB and basic for
the CAB, before any breakthrough is reached. Thus, for ACB or CAB, before either
cation or anion breakthrough is reached, the effluent pH depends on which bed is placed
at the end of the ionexchange train.
A comparison of breakthrough curves for cations and anions at higher feed
concentrations (Figure 4 and Figure 8) shows that the anion resin reaches saturation at
approximately 80 minutes while cation resin reaches saturation at 250 minutes. Thus,
anion resin reaches saturation before the cation resin. This leads to a fall in effluent pH
for ail the configurations (Figure 11) after the anion breakthrough has started at around 55
minutes. At 80 minutes, anion resin has reached saturation and effluent pH is at a
minimum, because the anion resin no longer exchanges anions while the cation resin
50
r 1
10
200 250
Effluent pH For MB
     . Effluent pH after Cation Bed for ACB
   Effluent pH after Anion Bed for CAB
50 100 150
VnL..'1.!lV1'.!...n tJln~J;' Vl·...... U4....,~.&. ~
Time (minutes)
Figure 11. pH of Effluent for ACB, CAB and MB (high concentrations)
'.
" ",
"
\\,
\,I
I
I
I
I
I
\
\
\
I
8
9
7
6 t·············.
I
c.. 5
VI
...... 4
3
2
1
0
0
continues to replace cations in the solution with hydrogen ions leading to an acidic
effluent. As the cation breakthrough occurs, the effluent pH rises and reaches a peak
(around 170 minutes). This peak can be attributed to sodium throw from the cation bed
as follows:
The effluent pH, is a function of the charge balance in the bulk solution. If the
cation concentration (sodium plus calcium) is greater than the anion concentration
(chloride and sulfate), effluent pH will be basic. Thus, a rise in cation
concentration (sodium plus calcium) results in a rise in effluent pH.
As the throw of sodium decreases, the charge balance favors a lower pH and the effluent
pH falls again. At approximately 250 minutes, the cation bed is saturated with calcium
(Figure 4) and the effluent pH reaches the same value as the feed pH.
Figure 12 and 13 show the history of effluent pH for the ACB and CAB
configurations respectively. The intennediate stream between two beds of an ionexchange
train is shown as a broken line, while the final effluent from the train is shown
as an unbroken line in both figures.
Refer to Figure 12 for the following discussion. The anionbed effluent has a very
high pH till the anionexchange bed is replacing anions with hydroxide ions. But, once
the anion bed reaches saturation (at 80 minutes), the effluent from anion bed is same as
the feed to the anion bed. Thus, in Figure 13 also, after 80 minutes, the effluent from
CAB is sarne as the effluent from the cation bed (the first bed in the CAB train).
Compare Figure 13 with Figure 12. We observe a fall in the pH of the effluent from
ACB as well as CAB after the anion bed gets saturated. The history of effluent pH from
the ACB, CAB and MB are same after this point. As the cation breakthrough occurs, the
52
1
..
4l3
r l
12 ~_._ .... _. _ .., ' .. .
............................ __ .. _ .....  ..
v~ru~vnl.n ...,~n.£ ....• "'~ ........
150 200 250
I I .......~ I I
100
Effluent pH after Cation Bed for ACe
    . Effluent pH after Anion Bed for ACB
\
I
Time (minutes)
Figure 12. pH History for ACB (high concentrations)
50
4
2
o
o
6
8
10
I
0..
V.
1..)
1
10
9
200 250 300
Effluent pH after Anion Bed for CAB
100 150
.  .... Effluent pH after Cation Bed for CAB
Time (minutes)
Figure 13. pH History for CAB (high concentrations)
V.Lu.OUlvn.tn V.1.n~"'· "" ....... ..,.....
50
...... _. ..... _.._ .. .\,."".. 
o
o
1
3
5
4
2
6
7
8
I
C.
VI
.f:>.

effluent pH rises and reaches a peak (around 170 minutes). This peak can be attributed,
to sodium throw from the cation bed, as follows:
The effluent pH, is a function of the charge balance in the bulk solution. If the
cation concentration (sodium plus calcium) is greater than the anion concentration
(chloride and sulfate), effluent pH will be basic. Thus, a rise in cation
concentration (sodium plus calcium) results in a rise in effluent pH.
As the throw of sodium decreases, the charge balance favors a lower pH and the effluent
pH falls again. At approximately 250 minutes, the cation bed is saturated with calcium
(Figure 4) and the effluent pH reaches the same value as the feed pH.
Refer to Figure 14 for the pH history of effluents from ACB, CAB and MB at
lower feed concentrations. The pH changes involved in this case are smaller than those at
higher feed concentrations. Also, for very low feed concentrations, the differences in
effluent pH between ACB, CAB and MB are less prominent than at higher feed
concentrations. As seen in Figure 14 the pH of effluents from ACB, CAB and MB are
close to neutral for a neutral feed, before breakthrough is reached. The effluent pH, is a
function of the charge balance in the bulk solution. Therefore, at about 400 days, the pH
of the effluent from all the configurations begins to fall as the anion beds approach
saturation and anion breakthrough begins. This trend continues up to 600 days of column
operation and the pH of the effluents reaches a minimum value of6.63 (approximately) at
600 days. The effluent pH begins to rise after 600 days due to cation breakthrough and
sodium throw from the cation bed. At 1000 days of column operation, the throw of
sodium from Cation bed reaches a peak and stabilizes at that value till 1600 days. This
results in a constant pH of the effluent during this period (1000 days to 1600 days). After
55
.~
...
)
c
7.05 .
Vl
0\
I
a..
7
6.95
6.9
6.85
6.8
6.75
6,7
6.65
===_....:":".~~
\\
I ~
'\,I
I
\
\
\~
~,i
:\
~
~
:1:,
~
~
:1
'I
",
I
,\
~ .... "
~ <";•• 
MBpH
.....  Effluent pH after Cation Bed for ACB
   Effluent pH after Anion Bed for CAB
500 1000 1500 2000
Time (days)
Figure 14. pH of Effluent for ACB, CAB and MB (low concentrations)
6.6 I i
o
V!u...."U.1VH~J U.l~ !~ .... ..., ....... _•••

the sodium throw from Cation bed falls to feed concentrations, the Cation bed can be said
to be saturated with calcium and the effluent pH rises as calcium concentration in the
effluent rises to feed concentration of calcium. At 2000 days the effluent pH is same as
feed pH because both beds are now saturated.
Figure 15 and Figure 16 show the history ofeffluent pH for the ACB and CAB
configurations respectively. The intermediate stream between two beds of an ionexchange
train is shown as a broken line, while the tinal effluent from the train is show as
an unbroken line in both figures. The anion breakthrough (Figure 15) begins first as seen
from the drop in pH of the anionbed effluent after 400 days of operation. Comparison of
Figure 9 with Figure 15 reveals that the throw of chloride ions from anion bed stabilizes
at 600 days to a peak value. This leads to a stable effluent pH (for anion bed, seen as
broken line) from 600 days till the cWoride throw stops (anion bed saturated with sulfate)
at 1000 days and the effluent pH falls again to stabilize at feed pH. The anion bed is
saturated with sulfate (at 1000 days) before the cation bed is saturated with calcium (at
approximately 2000 days). Comparison of Figure 16 with Figure 9 similarly reveals that
anion breakthrough begins at 400 days and this leads to a drop in pH of the CAB effluent
till it reaches a minimum at 600 days. Thereafter. it follows the same curve as the pH of
effluent from the cation bed. We observe that the pH histories of effluents from ACB,
CAB and MB follow identical paths after this point (600 days) and have identical
interpretation (given in discussion on Figure 14).
57
•)
.~......,
::l
'..
:1
~
.:>.
~
r 1
7.4 _.__._  ~  .. ~
2000
Effluent pH after Cation Bed for ACe
...... Effluent pH before Cation Bed for ACe
._.......... _ ......... __ ....._  
,,
I
,
I
,.
500 1000 1500
Time (days)
Figure 15. pH History for ACB (low concentraions)
7 I ,
V.L~U..LV~'~" IV"~ ~.",.' ......_.
6.6 I I I
a
6.7
6.8
6.9
7.2
7.1
7.3
I
c..
VI
00
7.05 ..__  ._.
7
6.95
1
~ "'=_.,1
U\
\0
6.9
~ 6.85
6.8
6.75
6.7
6.65
...... Effluent pH before Anion Bed for CAB
Effluent pH after Anion Bed for CAB
I   , .. • . ~ _. _ ..  ~ i 6.6
o 500 1000 1500 2000
Time (days)
Figure 16. pH History for CAB (low concentrations)
UJ.\,/J1'~4 U.l":.e.~;e  ••  ~
The step changes in pH at low feed concentrations are attributed to the following :
1. increased resolution of changes in concentration (ofHydrogen ion) when
these changes are viewed on the logscale (pH is negative logarithm of
Hydrogen ion concentration to base 10).
2. changes in pH get further magnified because a small range ofpH values is
stretched over a large area in the Y direction (in other words, the minimum
and maximum pH values on the Y axis differ by only 0.45 or 0.8).
Conclusions
Before breakthrough, effluent concentrations of monovalent ions from the mixed
bed are lower than the effluent concentrations of monovalent ions from any configuration
of homogeneous beds studied. The results seem to indicate that, in the case of
monovalent cOll..'1terions, performance close to the mixedbed perfonnance is achieved if
the ionexchange bed exchanging the counterions in question is placed downstream from
the bed that exchanges ions of opposite charge. For instance, in the case of sodium,
initial leakages as low as those from mixedbed are achieved if the cationexchange bed is
placed down stream of the anionexchange bed. Similarly, in the case of chloride, initial
leakage as low as that from mixedbed are achieved if anionexchange bed is placed
down stream of cationexchange bed. However, the same cannot be said of calcium or
sulfate ions.
At lower concentrations, the differences between the leakages from different
configurations observed for monovalent ions, are less pronounced as compared to the
60
1
,
1
~
.)
·f
1~
4
)
differences at high concentrations. For instance, at higher feed concentrations, the
sodium leakage from CAB is four orders of magnitude higher than the sodium leakage
from MB. Similarly, at higher concentrations, chloride leakage from ACB is four orders
of magnitude higher than the chloride leakage from MB. However, these differences are
not as pronounced for the case of lower feeds concentrations. This could be because of
the progressively reducing effect of the coions on the rate of ion exchange as the solution
reaches the ideal state of infinite dilution where the coions will have no effect at all on the
ionexchange rate. The pH changes involved at low concentrations are also much smaller
than those at higher concentrations and therefore, are expected to have a smaller effect on
the equilibrium at low concentrations.
The order of the beds in the homogeneousbed train affects the effluent pH also
(besides the effluent concentrations). Before breakthrough, the pH of the effluent from
the mixed bed is found to be closer to neutral than the pH of effluent from homogeneousbed
trains.
61
).,
.~
14
)
BIBLIOGRAPHY
Beardsley S. S., Coker S. D. and Whipple S. S (1995). Demineralization: The economics
of reverse osmosis and ion exchange Ultrapure Water, 12 (2), March 1995.
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exchange resins. L. Sorption of strong acids on weak base resins. Ind. Eng. Chern.
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Bhandari, V M., Juvekar, Y. A, Patwardhan, SA., ( 1992 b) Sorption studies on ion
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Bhandari, V. M., Juvekar, V A, Patwardhan, S. A, (1993) Sorption ofdibasic acids on
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Blickenstaff, R. A, Wagner, J. D and Dranoff, 1 S. (L967 a). The kinetics of ion
exchange accompanied by irreversible reaction. I. Film diffusion controlled
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Blickenstaff, R. A, Wagner, 1 D. and Dranoff, 1. S. (1967 b) The kinetics of ion
exchange accompanied by irreversible reaction. II. Intraparticle diffusion
controlled neutralization of a strong acid exchanger by strong bases. The Journal
of Physical Chemistry, 71 (6), )6701675
Blumberg, R. (J 984) Chemical Processing: acidbase salt systems. Ion Exchange
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Bulusu, R. (1994). Development of a column model to predict muLticomponent mixed bed
ion exchange breakthrough. M.S. thesis, Oklahoma State University
Chowdiah, V. N., (1996). Liquidfilm diffusion controlled ionexchange modeling  Study
of weak electrolyte mass transport and film masstransfer kinetics. PhD.
Dissertation, Oklahoma State University.
Cutler, F. M. and Covey, J. N. (1995). Ion Exchange: Deoxygenation at ambient
temperature Ultrapure Water, 12 (4).
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Divekar, S. v., Foutch, G. L. and Haub, C. E. (1987). Mixed bed ion exchange at
concentrations approaching the dissociation of water. Temperature effects. Ind.
Eng. Chern. Res, 26 (9), 1906  1909.
Dwivedi, P. N. and Upadhyay, S. N. (1977). ParticleFluid Mass Transfer in Fixed and
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high concentration cationic ion exchange. Ind. Eng. Chern. Res., 33 (11) , 27892794.
Fernandez, A., Diaz, M. and Rodrigues, A (1995). Kinetic mechanisms in ion exchange
processes. The Chemical Engineering Journal, 57 , 1725
Fisher, S. (1993). Track resin health to gage feedwaterdemineralizer performance Power,
March 1993, 8088.
Franzreb, M., Holl, W. H. and Sontheimer, H. (1993) Liquidphase Mass Transfer In
Multicompenent ion exchange. I. Systems Without Chemical Reactions In The
Film. Reactive Polymers, 21, 11 7133.
Golden, L. (1986). Industrial use of ion exchange resins. The Chemical Engineer, October
1986, 3134
Harfst, W F (1995). Back to basics: Controlling condensate corrosion without
chemicals. Ultrapure Water, 12 (4), ] 995.
Haub, C. E. (1984). Model development for liquid resistancecontrolled reactive ion
exchange at low solution concentrations with applications to mixed bed ion
exchange. M.S. Thesis, Oklahoma State University.
Haub, C. E. and Foutch, G. L. (1986 a). Mixedbed ion exchange at concentrations
approaching the dissociation of water. I. Model development Ind. Eng. Chern.
Fundam., 25, 373  381
Haub, C. E. and Foutch, G. L. (1986 b). Mixedbed ion exchange at concentrations
approaching the dissociation of water. 2. Column model applications. Ind. Eng
Chern. Fundam., 25, 381  385.
Helfferich, F. G. (1962) Ion Exchange. McGraw Hill Book Company, New York.
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)•s
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)
Helfferich, F. G. and Hwang. Y. L (1985). Kinetics of acid uptake by weakbase anion
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Exchange Accompanied By Chemical Reaction. Journal Of Chemical Engineering
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exchange. Chemical Engineering Science, 48 (3),467473.
Liberti, L, Petruzelli, D, Helfferich, F. G. and Passino, R. (1987). Chloride/Sulfate ion
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64
),
sI4
)

APPENDIX A
INTERFACI.AL CONCENTRATIONS AND IONEXCHA.l\,fGE EQUILIBRlA
The assumption of local equilibrium at the solidliquid interface allows us to apply
the law of mass action to resinphase and interfacial concentrations. Thus, for an ionexchange
reaction,
c
 
ZsA+ ZA B <=> ZA B+ ZsA
the law of mass action is written as,
(AI)
(A2)
The equilibrium constant for the ionexchange reaction is called the selectivity coefficient.
Define the resinphase and interfacial ionic equivalent fractions for ion A as,
y. = q.'.
Q
(A3)
C....
.
X.... = CT
.
Writing Equation A2 in terms of equivalent fractions gives,
(A4)
65
The interfacial equivalent fraction of an ionic species B can now be written as,
(A5)
where, initial resin loadings YB and YA, the resin capacity Q and selectivity coefficient are
known (properties of the resin determined by experiment or known from manufacturer's
specification).
We can also write relations similar to Equation :\5 for additional exchange
reactions. Consider the following reactions,
where, A is the counterion in the resin phase and counterions B, C, 0, E from the bulk
phase are exchanging with A. Then, selectivity relationships for these reactions will give,
[
.Jzn{, ( JI_(ZD~.\)
• = y (KE )X, x... 7z 9. /z
X£ E A y.", Cr
66
(A6)
(A7)
(A8)
For any counterion i exchanging with A,
..
Z
A =Y(K'J~'~(Y )~~i[_. Q.Jlz.:
1 I'~ A CT
The material balance at the resinliquid interface is,
n * I Xi I =0
i=l
Thus, there are nl equilibrium relations for n coumerions involved in the
(A9)
(AIO)
(Al l)
exchange and one material balance equation. Substitute the nl equilibrium equations
(Equations A5 to A8 in this case) into the material balance Equation A9 to obtain a
polynomial in terms of XA:.
Ze Zc ZD ~E.
X: + AB(X:)z" + Ac(X:J z, + AD(X~ )ZA + AE(X:)ZA = I (A12)
The coefficients of the terms of the polynomial are defined in Equation AI O. If
the total interfacial concentration Cr· IS known, then the coefficients of the polynomial
can be evaluated and the polynomial can be solved to obtain the interi"acial equivalent
fractions. Ionexchange equilibrium is a function of total interfacial concentration which is
unknown and must be found from the ionic flux rates. The flux rates in turn depend on
the interfacial concentration and bulk concentration (as shown in Appendix B). Thus, it is
necessary to start with a guess value for the total interfacial ionic concentration and find
the correct value by iteration (as described in Table V)
67
A.PPENDIX B
MODEL EQUATIONS
We have assumed that liquid film diffusion is the rate controlling step for the ion
exchange process. Therefore, the rate of change of concentration of an ionic species in
the resin is same as the rate of diffusion of that species of ion through the liquid film.
Thus,
The equivalent fraction of ions in the resin is given by
V. = ZiCi
J 1 Q
(B1)
(B2)
Replacing the resin phase concentration of ions in Equation 81 by equivalent fraction of
the ions in the resin (Equation B2) yields
d Y· I
· = Zj J las ..:~..::..
dt Q
(B3)
The flux of counterions across the liquid film J, is modeled by the NernstPlanck
equation as :
68
(84)
Assumption of pseudo steady state lets us use ordinary differentials instead of partial
differentials. Thus, the NemstPlanck equation for the flux of counterions across the
liquid film is now written as
Similarly, for coion flux
J =  0 (d Cj + CI zlF d¢J
I .1 dr RT dr
We can substitute the flux of counterions Ji in Equation B3 by the NernstPlanck
(B5)
(B6)
Equation (85) However, we first eliminate the term for gradient in electrical potential
(d¢/dr) in Equation 85.
Elimination of Term for Gradient in Electrical Potential
Assumption of electroneutrality in the liquid film gives equal concentration of
counterions and coions in the tilm:
n m
LZiC = Iz,c 1 i =1 j::l . .
Assumption of no coion flux in the liquid film yields
z·J = 0 J J
(B7)
(B8)
Also, assumption of no net current flow in the film can be written mathematically as :
n m
Lzdi = I ZjJJ
i :: J j:: 1
69
(B9)
From Equation B8 and Equation B9,
n m
LZ,Ji = LzJJ j = 0
i=l j=]'
(BI0)
Thus, the sum of equivalent fluxes of counterions and the sum of equivalent fluxes of
COlons IS zero. From Equation B6 and Equation B1 0, the gradient of electrical potential
IS
~ dCj
~Zl
d<t> RT j=l dr =
dr F tz.2 C.
j= 1 J .I
(B1 I)
The total equivalent concentration of counterions is equal to that of coions because
we assume eJectroneutrality in the liquid film The total equivalent concentration is given
by
n m
Cr =W LZiCI =W LZJCJ
i = I J= 1
The mean value for coion valence is defined as .
(8) 2)
Zv =
m 2 LZ C .1 J
i=l
In
LZjCj
j= I . .
From Equation 812 and Equation B13
m
LZ/ Cj =zvC T
j=l
70
(814)
~4
I
Differentiating Equation B12 with respect to distance in the film,
de m de_
T =LZJ
dr j=l J dr
Substitute Equations B15 and B14 into Equation Bl1 to get
d~ RT de = T
dr Zy FeT dr
(B15)
(B16)
Now, the NernstPlanck equation for counterion flux (Equation B5) can be written as
(BI7)
Thus, the term for the gradi,ent in electrical potential has been replaced by a
function of total equivalent concentration. In order to evaluate the ionic flux rates Jj • it is
now necessary to find a relation between the total equivalent concentration Cr and the
individual ionic concentration C;
Finding Relation between Total Equivalent Concentration and Individual Ionic
Concentrations
Define the relative valences for the counterions as
N =~
I "4
Equation B1 0 can be combined with Equation B17 to get
n n d C1 dCT 1 n _
LZ1Ji =LZiDld+d 
C LZi Oi N,Ci  0
i=l i=l r r Ti=]
71
(B18)
(B19)
For the case of a monovalent system of ions involved in ion exchange this equation can be
integrated to obtain a relation between Cj and CT However, integration is not possible in
the case of arbitrary valences. So we use the method of Franzreb et al. (1993) This
method yields a solution which is exact for the case of monovalent system of ions. The
solution given by this method (Franzreb et a!., 1993) for the case of counterions with
arbitrary valences is only approximate
Case I (Counterions having equal valences)
Differentiate Equation B17 with respect to distance in the film, and substitute
Equation B18. The flux of an ionic species does not change with distance in the film
because of mass balance and from our assumption that curvature of the film is negligible
Therefore. derivative of Ii with respect to distance in the film is zero
(820)
Summation of Equation B20 for all counterions leads to the following equation
(B21 )
On applying Equation B12 and its derivatives, Equation B20 reduces to
(822)
72
From Equation B22, we know that, the total equivalent concentration of ions in
the liquid film varies linearly with distance in the film for the case of counterions with
equal valences. Therefore,
dC r=m
dr g
(B23 )
where mg is a constant. We can now express the derivatives of Cj with respect to distance
in the film in terms of derivatives with respect to CT and we can write Equation 820 as
(B24)
This expression is of the same form as Euler's differential equation and its solution is
(B25)
where P is Nj for the case of equal valences of counterions. The values of the parameters
Ai and Bi in Equation B25 are found by applying the boundary conditions for the liquid
film,
• at r := 0, CT =CT
at r := 8, CT =C~~
The values of Ai and Bi are,
(B26)
(B27)
From Equations 825 and B27 we now know the individual ionic concentration
C; in terms of total equivalent concentration CT. Substitute Cj and its derivative in the
73
modified NernstPlanck flux expression (Equation 817) with Equation 825 and its
derivative,
(B28)
Substitute P by Ni in Equation 828 for the case of counterions with equal valences. For
case I, Franzreb has derived the expression for total equivalent concentration at the resin
surface to be,
C*T ::::
n *
IDiXi
i=1
1
P+l
(829)
Case II (Counterions having unequal valences)
In this case, the total equivalent concentration in the liquid film does not vary
linearly with distance in the film. We may still apply Equation 825 developed for the case
of equal coumerion valences, but, with a different value for P as given by Franzreb
Substitute Equation 828 into Equation B1 0,
(830)
For Equation B30 to be true, both the terms in the parentheses must be zero. Thus,
n
I(J + NJOiAi:::: °
i=l
74
(B3 1)
(B32)
For case I (equal counterion valences), an equation similar to B31 can be derived
(B33)
Comparison of Equations B3 1 and B33 shows that (1 +N) D; in case II replaces Dj of
case I Therefore, we can replace D; with ( 1+ i) Dj in the expression for total equivalent
concentration at the resin surface (Equation B29) as follows:
n
I(l + Na Oi x~)
i=l
I
P+I
c~ (B34)
.AJso, P for Case II is obtained by substituting for B, (Equation B27) in Equation B32,
n
I Nl Di(x~  x:J
)
p = ..:...i=....o.l _
n I Oi (x~  x~)
i=1
(B35)
Similar to case 1, integration of the modified ernstPlanck Equation (B] 7) with
boundary conditions (Equation B26) and substitution of P (Equation 835) gives,
(836)
We now know the individual ionic flux rates in the liquid film (Ij) in terms of
interfacial and bulk liquid concentrations (Equation B36). The liquid film thickness in
Equation B36 is still an unknown and is eliminated as follows:
75
Therefore,
where the effective diffusivity Dc is defined as,
n :LIJj 61
Dto: =n,ic=,,I
Llc~  c~1
1=]
(B37)
(B38)
(B39)
1
and the mass transfer coefficient k is given by the correlation of Dwivedi and Upadhyay
(1977),
k = (De SCi. 3 R[ 0.765 + 0.365 ]
. df' ) (eRf82 (eR/· 386 (B40)
The Ji for each species is found using Equations B38 and 840 The rate ofloading of
the resin beads can then be evaluated from Equation 8:1
76
APPENDIX C
MATERIAL BALAJ'\jCE EQUAnONS
A material balance for the ion exchange column must be setup in order to predict
the effluent concentration with time The differential column material balance can be
viewed as a material balance applied to a very small slice of the resin column. The net
Increase or decrease in the amount of an ionic species present in the resin in the slice
equals the net influx or efflux of the ionic species from the bulk solution. The material
balance equation for exchange of ion i between the resin bed and the solution phase is
written as,
Us aCi + aCi + (1 £) aq\ =0
soZ Or s Or
Define dimensionless variables for any ion i as,
1: = k l C~ l(t  EZJ
dpQ Us
and,
77
(CI)
(C2)
In order to write the column material balance in terms ofdimensionless variables,
the differentials of ordinary variables must expressed in tenns of differentials of
dimensionless variables Differentiating the expressions in Equation C2,
az =
Ol; =0
Ot:
(C3)
Evaluate the differentials of ordinary variables in terms of differentials of dimensionless
variables using the chain rule and equations C3,
DC l DC I ~ 8C j eTc =+ az a; az &r az
DC 1 =ac j (k] (1  £ )J + ac j (_ ki CtEJ
az a~ dpUs Ot dpQUs
DC i DC j oc, 8C j Ct
=+
at o~ Ot: Ct at
DC] =8C j (kl C~J
at 8L dpQ
oq 1 Bq i a~ oq j Ot
=+
at o~Ot: mOt:
~' ~ a;. (~~~J
78
((4)
Substitute the differentials of ordinary ariables Z and t in terms of differentials of or and ~
from equations C4 to get,
(C5)
Define equivalent fractions in resin and bulk phase as,
Substitute equivalent fractions for concentrations into Equation C5.
ax Oy.
_1+1=0
at, Or
(C6)
(C7)
The material balance equation C7 is written for any ion i Similar equations may
be written for all ions passing through the column However, solution of all the material
balance equations must proceed \vith the same step sizes in time and distance because the
flux expressions for all the ions are linked to the concentrations of other species and
determination of fluxes requires bulk concentration and resin loadings of all species of ions
at that time and depth in the column Therefore, it is necessary to write the material
balance equations for all the ions in terms of the same dimensionless variables L and~. A
reference ion is chosen for this purpose and the material balance equations are written in
terms of the dimensionJess variables defined for the reference ion. In this study, chloride
was chosen as the reference ion for the anion exchange bed and sodium was chosen for
cation exchange beds.
79
Dimensionless variables for the reference ion rare,
1: r = krC~ (t ezJ
dpQ Us
and,
e = kr(l E)Z .::..::.!...''
r usd p
Writing the differentials for any ion i in terms of the dimensionless variables for the
reference ion r,
Th' .;, column material balance equation for the ion i is now,
Divide Equation Cl 0 by (kj ki),
~ cy oX, + __i = 0
a~r C1: r
(C8)
(C9)
(C10)
(C) I)
Modification of Flux Expressions to Incorporate the Dimensionless Variables
The rate of resin loading for ion i is related to its flux across the liquid film as,
(C12)
Substituting dimensionless variable for ion i,
80
d y, =(_ zlaJ ~
dT, Q dT,
d y, =(_ Z 1, aJ dpQ
dT, Q k,Cr '
Now. substitute dimensionless variable for reference ion r,
Nate that as dp = 6, jf we assume spherical resin beads Thus,
(C13 )
(C14)
(CIS)
The flux of ions across the liquid film is evaluated and substituted into Equation C
15 to find the rate of resin loading. The dimensionless material balance combined with
Equation C15 wiH enable us to predict the effluent concentration histories and also the
profile of resin loading along the length of the column at any time.
81
APPENDIX D
NUMERICAL METHODS
The column material balance equation must be applied to each ionic species in
order to obtain the effluent concentration history or the resin loading profile across the
length of the column. This requires integration of a system of partial differential equations
of the form,
ax av __' + _.1_, =0
or at '"::lr r
(DI)
The differential equation involves two independent variables, land 1;. The methou
of characteristics is used to integrate the system of material balance equations. Thi
method involves integratlOn with respect to one variable, keeping the other variable
constant Thus, the partial differential equation is considered to be an ordinary differential
equation in terms of one variable and integrated while the other variable is being kept
constant. The integration of the material balance can be divided into two tasks:
1. integration to obtain liquid phase concentrations along the length of the column
at constant time
2. integration to obtain resin loading at the next step in time for the same depth in
the column
82
Zechhini (1990) has conducted a thorough survey of numerical methods for
carrying out integration of the differential equations and Pondugula (1992) has justified
the choice of numerical methods used in the current model. The explicit Euler method is
used to predict resin loading in the HEIE model, and Gear's backward difference method
of fourth order is used for the liquid phase concentrations.
83
.A.PPENDIX E
SIMULATION OF HOMOGENEOUS BEDS fN SERIES
The code for simulating HBIE can be adapted for simulation of ionexchange beds in
series. Ionexchange trains consisting of one cationicresin bed and one anionicresin bed
in series are simulated and the effect of the order of the beds on the etlluent water quality
is studied. The performance of a mixedbed unit (predicted by rvrnrE model) is also
compared with the performance of homogeneous beds in series. The procedure used for
these simulations is outlined here.
For the simulation of beds in series, separate programs are run for each bed. The
history of effluent ionic concentrations predicted by the first program is written to an
interface file which is read by the second program In this regard, one has to keep in mind
the following points
1. The second program must be capable of handling a variable feed condition
because the output from one bed is being fed to the second bed.
2 The step sizes used to integrate the material balances for the two beds must be
the same in terms of time However, the step size in terms of distance can vary
between the two codes for beds in series.
84
Thus, the time interval (delta t) between two consecutive effluent concentrations
generated by the first program must be the same as the step size in time used by the
second program. l The integration procedure is very sensitive to the step size. Therefore,
both programs must be run with a very sman step size in time.
! Note that the time mentioned here is not the program run time, but real time for the bed
runs.
85
APPENDIX F
CONlPUTER CODE
*******************************************************************
*
*
****
THIS PROGRAM IS USED FOR PREDICTING THE EFFLUE T
CONCENTRATIONS FOR A MULTICOt\1PONENT SYSTEM OF
IONS EXCHANGING IN A HOMOGENeOUS BED OF ANIONEXCHANGE
RESIN
*****
***
*
OUTPUT FROM THIS CODE GOES TO INTERFACE.DAT WHICH IS *
USED AS INPUT FOR CATIONIC RESIN BED. THE 2 BEDS ARE IN *
SERIES. **
****
DEVELOPED BY.
ASHWIN P G&
DR. GARY L. FOUTCH
****
*******************************************************************
***
C  PREFIX FOR CATONS; A  PREFIX FOR ANIONS
***
*******************************************************************
IMPLICIT INTEGER (I ), REAL*8 (AH,OZ)
CO:MM:ON CBC(20), CBA(20)
CO:MM:ON TKC(20), TKA(20), DC(20), DA(20), lC(20), lA(20)
CO:MM:ON RC(20), RA(20), CF, QC, QA, DRC, ORA, SUMYC, SUMYA
COMJ..10N DISS, CBH, CBOH, DH, DOH, Z1, Z2,NC,NA
REAL*8 RTC, RTA(20,4,6500), MC(20), MA(20),
1 CFC(20) CFA(20),YCO(20), YAO(20), COEC(20), COEA(20),
1 PPBC(20), PPBA(20),KREF,YACUR(20),
1 XACUR(20),YA{20,4,6500),XC(20),
86
***
*
XA(20,4,6500)
Correlations Dwivedi and Upadhyay
F2(DR,R,S) = (DRlPDA) * (S**(1.013.0» * R *
1 «0.765/«VD*R)**O.82» + (0 365/«VD*R)**OJ 86»)
READING THE DATA
OPEN(UNIT=9,FILE='animult.dat',STATUS = 'UNKNOWN')
OPEN(UNIT=8,FILE='interface.dat',STATUS='UNKNOWN')
READ(9, *)NC,NA,Z I ,Z2,DH,DOH
READ(9, *)KPBK, KPPR, THv1E
READ(9,*)PDA, VD
READ(9, *)FR, DIA, CHT
RE.t\D(9, *)TAU, XI, TMP
READ(9, *)DEN, QA
DO 2 II = I,NC
READ(9, *)YCOCH), CFC(II),ZC(U), MC(II), TKC(II)
WRITE(*, *)YCO(II),CFC(II),ZC(II),MC(II),TKC(II)
2 CONTINUE
DO 3 ]J = I,NA
READ(9, *)YAO(JJ),CFA(JJ),ZA(JJ),MA(JJ),TKA(JJ)
WRITE(*, *)YAO(JJ),CFA(JJ),ZA(JJ),MA(JJ),TKA(H)
3 CONTINUE
CP = I 43123+TMP*(0.OOOI27065*TMP00241537)
ALOGKW = 447099/(TMP+27315)60875+001706*(TrvtP+273 15)
DrSS = 10. **(ALOGKW)
CALL EQB(CFC,CFA)
PH = 14.0 + LOGIO(CBOH)
WRITE (*,*) 'PH of inlet soin =', PH
CF = 0.0
D08JJ=I,NA
CF = CF + CFA(JJ)
8 CONTINUE
CF = CF+CBOH
WRITE (*,*) 'CF =',CF
RTF = (8.931DIO)*(TMP+273.J6)
87
C
C SELF DIFFUSIVITIES OF IONS
C
C DC(I) = SODIUM, DC(2) = CALCIUM
C DA(I) = CHLORIDE, DA(2) = SULFATE
C
dc(1) =(RTF)*(23 00498+ I.06416*TMP+0.0033196*TMP**2)
dc(2) =(RTF)*(23.27+ 1.575*TMP)/2.0
da(l) =(RTF)*(39.6493+1.39176*TMP+O.0033 I96*TMP**2)
da(2) =(RTF)*(35.76+2.079*TMP)/2.0
C
C kref is set to Cl ion.
C
kref= 0.01937466
AREA = 3.1415927*(DIA**2)/4.
VS = FRJAREA
RPA = PDA*IOO.*VS*DEN/«VD)*CP)
SCA =(CP/IOO.)/DENIDA(I)
*
>I< CALCULATE TOTAL NUMBER OF STEPS [N DISTANCE (NT) DOWN
* COLUMN:SLICES
*
CHTD = KREF*(l.VD)*CHT/(VS*PDA) !distance dimensionless
T = CHTD/XI
WRITE (*, *)chtd.xi,nt
>I< SET INITIAL RESIN LOADING THROUGHOUT THE ENTIRE COLUMN
>I<
MT = NT + 1
DO 4 M=l,MT
D06JJ=1,NA
YA(JJ, I,M) = YAO(JJ)
6 CONTINUE
4 CONTfNUE
>I<
>I< CALCULATE DIMENSIONLESS PROGRAM TIME LIMIT
* BASED ON INLET CONDITIONS (AT Z=O)
*
TMAX = 6.0*QA*3.142*(DIN2.)**2*CHT/(FR*CF*60.)
TAUMAX = KREF*CF*(TMAX*60)/(PDA*QA)
DMAX=TMAXl1440.
88
WRITE(6,*)
WRITE(6,*)
WRITE(6,222)
WRITE(6,223)DMAX
WRITE(6,224)
222 FORMATC PROGRAM RUN TIME IS BASED ON TOTAL RESIN CAPACITY')
223 FORMATC AJ'\ID FLOW CONDITIONS. THE PROGRAM WILL RUN
FOR',F12.1)
224 FORMATe DAYS OF COLUMN OPERATIO FOR THE CURRE T
CONDITIONS.')
* INITIALIZE VALVES PRIOR TO ITERATIVE LOOPS
*
J = 1
JK = 1
TAUTOT=O.
JFLAG = 0
KK= 1
KPRlNT = 100
CONS = 6./KREF/CF
Dtime = TAU*PDA*QA/(KREF*CF*60.)/1440.
write(8, *) Dtime ttirne in days between 2 consecutive output values
**
TIME STEP LOOP WITHIN WHICH ALL COLUMN CALCULAnONS ARE
* IMPLErvtENTED TIME IS INCREMENTED AND OUTLET CONCENTRATION
* CHECKED
1 CO TINUE
IF (TAUTOT.GT.TAUMAX) GOTO 138
IF (J.EQ.4) THEN
ID= 1
ELSE
JO=J+1
ENDIF
**
SET ll\TLET LIQUID PHASE FRACTIONAL CONCENTRATIO S FOR EACH
* SPECIES IN THE MATRIX
*
007II=1,NC
XC(II) = CFC(II)/CF
7 CONTINUE
89
DO 10 JJ = I,NA
XA(JJ,J,l) = CFA(JJ)/CF
10 CONTINUE
*
;,< LOOP TO INCREIv1ENT DISTAi'\JCE (BED LENGTH) AT A FIXED TI1vfE
*
DO 400 K=I,(NT+4)
If (K.EQ 1) then
DO II II = I,NC
CBC(II)=XC(II)*CF
11 CONTINUE
Endif
DO 12 J] = ],NA
CBA(JJ)=XA(JJ,J,K)*CF
12 CONTINUE
**
CALL ROUTINES TO CALCULATE RN, RB, C I, CBI(INTERFACIAL
CONCENTRATIONS
* & COEFFICIENTS)
*
DO 15 IJ = I,NC
RC(II) = 0.0
15 CO TINUE
SUMYA = 0.0
DO 1411 = 1, A
SUMYA = SUMYA+YA(JJ),K)
14 CO TTNUE
DO 29 JJ = I,NA
YACUR(JJ) = YA(11,J,K)
29 CONTINUE
DO 30 JJ = I,NA
XACUR(JJ) = XA(JJ,J,K)
30 CONTINUE
IF(SUMYA.LT. O.999)THEN
CALL ANION(YACUR,XACUR,k)
90
ELSE
DO 16 JJ = I NA
RA(JJ) = 00
16 CONTINUE
ENDIF
SCA = (cpn OO.)/OE /ORA
AKA = F2(DRA,RPA,SCA)
DO 17 II = I, C
RTC = 00
17 CONTINUE
DO 18 JJ = I,NA
RTA(JJ,J,K) = RA(JJ)*AKA*CONS
18 CONTINUE
DO 20 JJ = I,NA
YA(JJ,m,K) = YA(JJ,J,K)+TAU"RTA(JJ,J,K)
20 CONTINUE
" IMPLEME T ITvrPLlCIT PORTIO OF THE GEARS BACKWARD DIFFERENCE
J\t1ETHOD
* FROM THE PREVIOUS FUNCTlON VALUES. FOR THE FIRST THREE STEPS
* USE FOURTHORDER RUNGE KUTTA I'v1ETHOD
IF(1< LE. 3)THEN
D022JJ=I, A
XA(JJ,J,KT]) = XA(JJ,J,K)(XI*RTA(1J,1,K»
22 CONTINUE
ELSE
DO 24 JJ = I,NA
COEA(JJ) =3 *XA(JJ,J,K3)/25.16. *XA(JJ,J,K2)/25. +
] 36*XA(JJ,J,Kl )/25. 48*XA(JJ,1,K)/25.
24 CONTINUE
9]
DO 26 JJ = 1, A
XA(JJ,J,K+I) = XI*12*RTA(JJ,J,K)/25.COEA(JJ)
26 CONTINUE
E:NUIF
*
* DETERMINE CONCENTRAnONS FOR THE DISTANCE STEP AND
RECALCULATE
* BULK PHASE EQUILIBRIA
*
DO 32 JJ = I,NA
CBA(JJ)=XA(JJ,J,K+ 1)*CF
32 CONTINUE
CALL EQB(CBC.CBA)
400 CONTINUE
*
* PRINT BREAKTHROUGH CURYES
*
IF (KPBK.NE.I) GO TO 450
DO 45 II = I,NC
PPBC(lI) = CBC(II)*MC(II)/I.E6
45 CONTINUE
DO 46 JJ = I, A
PPBACJJ) = CBA(JJ)*MA(JJ)/l.E6
46 CONTINUE
TAUTIM = TAUTOT*PDA*QA./(KREF*CF*60.)
PH = 14. + LOG1O(CBOH)
write(8,48)CBC( 1),CBC(2),CBH.CBA(1 ),CBA(2),CBOH.PH
48 FORMAT(1 x,E 12 7,2x,E 12.7,2x,E12. 7,2x,E12.7,2x,EI2.7,2x,E 127
1 ,2x,F5.2)
IF (KPRINT.NE. 100) GOTO 450
WRITE(*,47)TAUTIM,PPBC( I),PPBC(2),PPBA( 1),PPBA(2),PH
92
47 FORMAT(lx,Fll.6,2X,E12.7,2X,E12.7,2X,E12.7,4X,E12.7,4XF5.2)
**
STORE E"VERY TENTH ITERATIO TO THE PRINT FILE
*
KPRINT = 0
450 CONTINUE
KPRINT = KPRINT+ I
JK = J
IF (J.EQA) THEN
J = 1
ELSE
J = 1+1
ENDIF
*
* END OF LOOP RETURN TO BEGINNING AND STEP IN TIME
*
IF (JFLAG.EQ 1) STOP
TAUTOT = TAUTaT + TAU
GOIO I
138 STOP
END
SUBROUTINE EQB(CC,CA)
IIv1PLICIT INTEGER (lN), REAL*8 (AH,OZ)
COMMON CBC(20), CBA(20)
COM:MON TKC(20), TKA(20), OC(20), OA(20), ZC(20), ZA(20)
COM:MON RC(20), RA.(20), CF, QC, QA, ORC, ORA, SUMYC, SUMYA
COM:MON DISS, CBH, CBOH, DH, DOH, Z1, Z2,NC,NA
REAL*8 CC(50),CA(50)
SUMC =00
DO I I1= I,NC
SUMC = SUMC + CC(I1)
CONTINUE
SUMA=OO
93
DO 2 11= 1,NA
SUMA = SUMA+CA(11)
2 CONTINUE
VI = SUMCSUMA
V2 = VI **2.+4. *OISS
CBOH =(VI +(V2**05»/2.
CBH = DISS/CBOH
RETlJRJ"J
END
SUBROUTINE ANIO '(YY,XX,k)
Il'v1PLICIT INTEGER (IN), REAL*8 (AH,OZ)
COMMON CBC(20), CBA(20)
COMMON TKC(20), TKA(20), DC(20), DA(20), ZC(20), 2A(20)
COMMON RC(20), RA(20), CF, QC, QA, ORC, DRA, SUMYC, SUMYA
COMMON DISS, CBH, CBOH, OH, DOH, 21, Z2,NC NA
REAL*8 YY(50),XX(50),XXN(50),CCO(SO),N(50),LAM(50),XXI(50),
1 BB(50),AA{50),CBN(50),CI(50),R 1(SO),NOH
W= 10
YOH = 1 SUMYA
CTO = 0.0
DO I II = 1, A
CTO = CTO+CBA(II)
CONTIl\;1JE
CTO = CTO+CBOH
DO 2 II = I,NA
XXN(Il) = XX(II)*CF/CTO
2 CONTINUE
SUMXB=O.O
DO 3 II = I,NA
SUMXB = SUMXB+XXN(II)
94
3 CONTINUE
XBOH = 1. SUMXB
DO 4 JJ = I,NC
CCO(n) = CBC(JJ)/ABS(lC(JJ)
4 CONTINUE
CH =CBHlABS(ll)
SUMZN = 0.0
DO 5 11= I,NC
SUMZN = SUMZN+(ZC(JJ)**2.)*CCO(JJ)
5 CONTINUE
SUMZN = SUMZN + (ll **2)*CH
SUMZD = 0.0
DO 6 JJ = I,NC
SUMZD = SUMZD+(ZC(JJ)*CCO(JJ»
6 CONTINUE
SUMZD = SU1v1ZD + (Zl *CH)
Zy = SUMZN/SUMZD
D07II=1,NA
J (II) = ZA(II)/lY
7 CONTINUE
. OH = Z2/ZY
CTI = CTO
8 CO TrNUE
DO 9 II = I, A
LAM(II) = YY(II)*TKA(II)**(I./ABS(Z2))*YOH**(ZA(II)/Z2)
1 *(QNCTI)**(1.lA(II)/Z2)
9 CONTINUE
C NEWTO RAPHSON SOLVIR
EPS = I.E07
X = XBOH
SUMFN = 0.0
DO 10 II = 1,NA
SUMFN = SUMFN + (LAM(II)*X**(ZA(II)/Z2»
10 CONTINUE
SUMFN = SUMFN+XI.O
95
SUMFD = 1,0
DO 11 II = 1,NA
SUMFD = SU1vlFO,(ZA(1I)/Z2)*LAM(II)*X**(ZA(II)/Z21 »
11 CONTINUE
XOHI = XSUMF /SUMFD
DO WHILE «(ABS(XOHIX)/XOHI)GT.EPS)
x = XOHI
SU1\1FN = 0.0
DO 12 II = 1,NA
Sillv1FN = SUMFN + (LAJ\.1(II)*X**(ZA(II)/Z2»
12 CONTINUE
Sillv1FN = SUMFN+Xl 0
SUMFD = 10
DO 13 II = I,NA
SU1vfFD = SUMFD+«ZA(II)/Z2)*L~\1(II)*X**(ZA(II)/Z21))
13 CONTINUE
XOHI = XSU1\1FN/SLJTv1FD
END DO
DO 14 II = I,NA
XXI(II) = LAM(II)*(XOHI**(ZA(II)/Z2»)
14 CO TINUE
C CALCULATIO OF TOTAL INTERFACIAL CONCENTRATIO CTI
SUMPN = 0.0
DO 15 II = I,NA
SUMPN = SUMPN+ABS( (H)*DA(II)*(XXI(II)XXN(II»)
15 CONTINUE
SUMPN = SU1v1PN + ABS(NOH*DOH*(XOHIXBOH)
SUMPD = 0.0
DO 16 II = 1,NA
SUMPD = SUMPD+ABS(DA(II)"'(XXI(II)XXN(II))
l6 CONTINUE
SUMPD = SUMPD + ABS(DOH*(XOHIXBOH»
96
P = SUMPN/SUMPD
SUMTN== 0.0
DO 17 II == l,NA
SUMTN =SUMTN + (l.+N(II»*DA(II)*XXN(II)
17 CONTINUE
SUMTN =SUMTN+(1+NOH)*DOH*XBOH
SUI\1TD =00
DO l8 II = I,NA
SUMTD == SUMTD + (1.+N(II»*DA(II)*XXI(II)
18 CONTINUE
SUI\1TD == SUMTD+(I +NOH)*DOH*XOHI
CTIN = (StJMTN/SUMTD)**(l./(P+l»*CTO
IF«ABS(CTINCTI)/CTIN).GTEPS)THEN
CTI=CTIN
GOT08
ELSE
CTI == CTIN
ENDIF
C CALCULATION OF Ri's OF THE IONS B, C, D, E.
DO 19 II = 1,NA
BB(II) == W*(XXI(II)XXN(II»)/(CTI**(Pl)CTO**(P J »
19 CONTINlJE
BOH = W*(XOHIXBOH)/(CTI**(Pl.)CTO**(PI »
DO 20 II = I,NA
CBN(II) == W*CBA(Il)/ZA(II)
20 CONTTj\;UE
DO 21 II = I,NA
AA(II) =(ZA(II)*CBN(II)BB(II)*CTO**(P»)/CTO
21 CONTINUE
AOH = (Z2*CBOHBOH*CTO**(P»/CTO
DO 22 II = I,NA
CI(ll) = W*XXI(ll)*CTVZA(II)
22 CONTINUE
97
•
COHI = W*XOHI*CTIIZ2
DO 23 11= 1,NA
Rl(II) = DA(II)*«1. (II)fP)*(CI(II)CBN(II))
I +N(II)* (AA(Il)/ZA(II») *( 1.+1./P)*(CTICTO»)
23 CONTINUE
ROHI = OOH*«1.NOHIP)*(COHICBOH)+NOH*(AOHlZ2)*
1 (l.+l.fP)*(CTICTO))
SIGR = 0.0
DO 24 II = 1,NA
SIGR = SIGR + ABS(RI (II))
24 CONTINlTE
SIGR = SIGR + ABS(ROHl)
SIGO = 00
DO 25 II = 1,NA
SIGD = SIGO + ABS(CI(II)CBN(II)
25 CONTINUE
SIGD = SIGD + ABS(COHICBOH)
DRA = SIGRlSIGD
DO 26 II = 1,NA
RA(II) = W*ZA(II)*R I(II)/DRA
26 CONTINUE
RETURN
END
98
Thesis:
.'\
I
VITA
Ashwin P Gramopadhye
Candidate for the Degree of
Master of Science
HOMOGENEOUS BED ION EXCHANGE COLUMN MODELS FOR
ULTRAPURE WATER APPLICATIONS AND SIMULATION OF ION
EXCHANGE BEDS IN SERIES
Major Field: Chemical Engineering
Biographical:
Education: Graduated from S.LE.S. College of Arts and Science, Bombay, India
in March 1989; received Bachelor of Engineering degree in Petroleum and
Petrochemical Engineering from University of Pune, India in May 1993.
Completed requirements for Master of Science degree in Chemical
Engineering at Oklahoma State University in December 1996.
Experience: Employed by 1.1.1. (Powai), Bombay, India as Research Associate;
Oklahoma State University, School of Chemical Engineering as Graduate
Teaching Assistant and Graduate Research Assistant; Oklahoma State
University, Department of Biosystems and Agricultural Engineering as
Graduate Research Assistant and Computer Programmer.