DEVELOPMENT OF A COLUMN MODEL TO PREDICT SILICA
BREAKTHROUGH IN A MIXED BED ION EXCHANGER
By
SHARMAPAMARTHY
Bachelor of Technology
Osmania University
Hyderabad, AP, India
1992
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
May. 1996
DEVELOPMENT OF A COLUMN MODEL TO PREDICT SILICA
BREAKTHROUGH IN A MIXED BED ION EXCHANGER
Thesis Approved:
Dean of the Graduate College
11
Chapter
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
II. LITERATURE REVIEW 3
Basic Principles ofIon Exchange 3
Ion Exchange Equilibria and Selectivity.............................................................. 4
Silica Chemistry..................................... 5
Silica Solubility.................................................................................................... 8
Ion Exchange Kinetics................................................. 9
Particle Diffusion Control 10
Film Diffusion Controls 11
III. MODEL DEVELOPMENT 13
Basis and Assumptions........................................................................................ 13
Ion Flux Expression 15
Particle Rates 16
Material Balances 17
IV. RESULTS AND DISCUSSION 19
V. CONCLUSIONS 54
BIBLIOGRAPHY 56
APPENDIX A SILICA EQUILIBRIUM
APPENDIX B MODEL PARAMETER VALUES
APPENDIX C COMPUTER CODE FOR SILICA BREAKTHROUGH
IV
PREFACE
A computer model was developed to predict silica breakthrough in an ion
exchange column. Column material balance equations are derived using the assumptions
of film diffusion control and bulk phase neutralization. The variations in outlet
concentrations due to various model parameters were studied. Most of the literature on
ion exchange modeling addressing the removal of silica is qualitative in nature. This
study extends this qualitative background and makes a first attempt at quantitative silica
ion exchange issues. The silica particulates are accounted for in the model.
I wish to express my sincere gratitude to my advisor, Dr. Gary L. Foutch, for the
continuous support and needful assistance throughout the course of this work. I would
also like to thank Dr. Karen High and Dr. A. H. Johannes for serving on the advisory
committee and for their technical help.
I would like to express special thanks to my parents, my sister and my younger
brother for their moral support and continuous encouragement throughout my stay in the
U.S.A I wish to express my appreciation to Vikram Choudiah, Doctoral student at
Oklahoma State University, for his technical assistance. I am also thankful to all my
friends in the U.S.A and in India.
111
Chapter
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
II. LITERATURE REVIEW 3
Basic Principles of Ion Exchange 3
Ion Exchange Equilibria and Selectivity.............................................................. 4
Silica Chemistry................................................................................................... 5
Silica Solubility.................................................................................................... 8
Ion Exchange Kinetics......................................................................................... 9
Particle Diffusion Control 10
Film Diffusion Controls 11
III. MODEL DEVELOPMENT 13
Basis and Ass'Uffiptions 13
Ion Flux Expression 15
Particle Rates 16
Material Balances 17
IV. RESULTS AND DISCUSSION 19
V. CONCLUSIONS 54
BIBLIOGRAPHY 56
APPENDIX A SILICA EQUILIBRIUM
APPENDIX B MODEL PARAMETER VALUES
APPENDIX C COMPUTER CODE FOR SILICA BREAKTHROUGH
IV
Table
LIST OF TABLES
Page
I. Model Assumptions 19
II. Base Case Table 23
v
Figure
LIST OF FIGURES
Page
1. Effect of resin ratios on silica effluent concentrations 20
2. Effect of flow rate on silica breakthrough 22
3. Effect of temperature on silica effluent concentrations 25
4. Effect of particle size on silica breakthrough 26
5. Effect of the resin loadings on silica effluent concentrations 27
6. Effect of distance increment on silica breakthrough 29
7. Effect of time increment on silica effluent concentrations 30
8. Effect of Freundlich Coefficient on silica breakthrough 31
9. Effect of Freundlich index on silica effluent concentrations 33
10. Effect of Freundlich Adsorption Coefficient on particulate silica breakthrough 34
11. Effect of Freundlich Adsorption Coefficient on ionic silica breakthrough 35
12. Effect of Freundlich Adsorption Index on ionic silica effluent concentrations 36
13. Effect of Freundlich Adsorption Coefficient on colloidal silica breakthrough 37
14. Effect of Freundlich Adsorption Index on colloidal silica breakthrough 38
15. Effect of diffusivity on silica breakthrough 39
16. Effect of rate constant for polymarization on silica breakthrough 43
17. Effect of rate constant for hydrolysis on silica effluent concentrations 44
18. Effect of selectivity coefficient of silica on silica breakthrough 45
19. Effect of amorphous silica on particulate silica effluent concentrations 48
20. Effect of amorphous silica on colloidal silica breakthrough 49
VI
21. Effect of amorphous silica on particulate silica effluent concentrations 50
22. Effect of Equilibrium Constant on particulate silica breakthrough 51
23. Breakthrough of all the paticipating ions 52
24. Effect of inlet concentration of ionic silica on silica breakthrough 53
VII
NOMENCLATURE
as = interfacial surface area (L2/L3)
Cj = concentration of species I (meq/L3
)
C*i = concentration of resin in the resin (meq/L3
)
COi = concentration of resin in the bulk (meq/L3
)
dp = particle diameter (L)
Di = diffusivity of species i (L2IT)
De = effective diffusivity for species i (L2IT)
F = Faraday's constant (C/mole)
FAR = anionic resin volume fraction
FCR = cationic resin volume fraction
Ki = mass transfer coefficient (nonionic) (LIT)
,
Ki = mass transfer coefficient (ionic) (LIT)
KA
B
= resin selectivity for B compared to A
Kw= dissociation constant of water (mol/L3
)
Q= capacity of the resim (meq/L3
)
R = universal gas constant
Ri = ratio of mass transfer coefficients
T = temperature
t = time(T)
u = superficial velocity(L/T)
Xi = bulk phase concentration of species I
Yi = fraction of species I on the resin
Zi = charge on species I
8 = film thickness (L)
viii
Superscripts
Subscripts
E = void fraction
~ = electric potential (ML2fTC)
J.l = viscosity of the bulk liquid (MILT)
~ = dimensionless combined timedistance variable
~ = dimensionless distance variable
o = bulk phase value
* = interfacial value
A = species A
B = species B
c = chloride ion
h = hydrogen ion
i = species i
j = species j
n = sodium ion
o = hydroxide ion
p = silica ion
x = potassium ion
w=water
ix
Chapter I
Introduction
Deionization is a separation science. It differs from other separation processes
such as distillation or leaching in that the matter being separated is charged ionic species.
The process is one of stoichiometric reactioni'{Kunin, 1950). Electroneutrality of the
system is maintained constant, implying that ionic species are exchanged on an
equivalent basis. Ions are exchanged with ionic groups attached to the cationic and
anionic exchangers. These exchangers are basically resins with fixed groups attached,
such as sulfonic groups, etc. The charge on the resins determines whether they can be
classified as cationic or anionic. The fixed charged groups have mobile ionic species
bound weakly to them by Van der Waals forces to maintain electroneutrality with the
resin. These mobile ions are exchanged freely with ions in solution having a like charge.
The basic skeleton of an organic exchanger (anionic or cationic resin) is polystyrene with
highly crosslinked divinyl bezene. Crosslinking increases the resin structure rigidity
and effects the exchange kinetics.
Ion exchange can occur with cationic and anionic resin separately or intimately
mixed together as a 'Mixed Bed.' In separate beds one ion type is removed followed by
the other type. In a mixed bed, exchange occurs simultaneously. The effluent produced
2
by a mixed bed has less dissolved gases, ionic species, microbial impurities and
particulates, including silica.
Cationic resin in the hydrogen form and anionic resin in the hydroxyl form
exchange hydrogen and hydroxyl ions with the respective cations and anions of the feed.
The hydrogen and hydroxyl ions equilibrate to form water. This, in tum, increases the
driving force for exchange.
Ion exchange has many applications, primarily in the power, semiconductor and
pharmaceutical industries. These industries have very stringent water quality standards.
Mixedbeds produce water which meets these requirements and is commonly referred to
as "UltraPure Water." Ultrapure water has ionic impurities less than 1fJ.g/Kg (Ppb),
with correspondingly lower levels of particulates and microbial contaminants. The
current impurity limit in many industries is in the range ofng/Kg (ppt). This demand is
being met by employing membrane separation processes, such as electrodialysis and
reverse osmosis, in combination with ion exchange. Some of these methods pretreat feed
water to the ion exchangers, or polish the ultrapure water produced by ion exchange
(Sadler, 1993).
The importance ofparticulate removal has increased tremendously in recent years.
Silica and iron particulates must be removed from industrial waters to achieve ultra high
purity. Silica is found to deposit on turbine blades and thereby reduce power plant
efficiency (Iler, 1955). Particulates also adsorb onto microchips during machining of
silicon wafers. This study addresses silica removal in its different forms from water.
Soluble silica can be removed by ion exchange, but colloidal silica can be removed only
by adsorption mechanisms.
3
Chapter II
Literature Review
This chapter is divided into two sections; the basic principles of ion exchange and
the kinetic theories employed, and the chemistry of silica, its colloidal nature and the
methods for its removal from water other than ion exchange. The kinetic discussion is
similar to that of Haub (1984) and Zecchini (1990). The terminology used in this thesis is
from Kunin (1950)and Helfferich (1962). This section follows the work done by Iler
(1970), Hausen (19), and Owens (1985) in content and style. Silica removal by ion
exchange is the main objective ofthis thesis.
Basic Principles of Ion Exchange
The resins used in ion exchange are prepared by copolymerization of a monomer,
say styrene, with divinyl benzene (commonly referred to as DVB). Copolymerization
causes DVB to crosslink with styrene. Crosslinking increases the rigidity of the resin.
This is important because the resin should be resistant to "swelling," "osmotic shock,"
and other problems encountered in ion exchange. One significant resin classification is
based on DVB mole percent. Desirable resin characteristics are listed below. They
should:
• be negligibly soluble in solution (aqueous or nonaqueous),
• maintain structural rigidity and chemical stability,
• have consistent porosity and particle size, and
4
• be able to withstand fouling and corrosion.
The fixed charged groups determine the ion exchange behavior of the resins.
Resin selectivity depends on the nature of these groups (Zuyi, 1993). Acidic or basic
(charged) groups are fixed in the resin hydrocarbon matrix by heating in acids or bases
for prolonged times. For instance, sulphonic groups are introduced by heating resin in
sulfuric acid. These groups attract exchangeable ions of opposite charge. These
"counterions" are exchanged for ions of like charge on an equivalent basis. The exchange
of counterions ensures that electroneutrality is maintained within the exchanger and the
bulk liquid. The number of sites available for exchange determines the resin "capacity,"
expressed as meq/ml. The exchange equilibrium depends on resin selectivity.
Ion Exchange Equilibria and Selectivity
When an ion exchanger contacts an electrolyte solution, exchange will continue
until equilibrium between the two phases is attained (Samuelson, 1963). Equilibrium
depends on the properties and the quantities ofthe components in the system. Both the
resin and liquid phase will have all components, but in different concentrations.
The exchange of two monovalent ions is represented by:
(111 )
A and B refer to the counterions whereas Ar and Br refer to the mobile ions
attached to the resin. The forward reaction will not be favorable if the concentration of
component B is large in the bulk liquid phase. For a simple exchange, the selectivity
coefficient, K, is defined by the law of mass action as:
(112)
where the brackets refer to the concentration in the bulk and the resin phases.
5
If the exchanging species have different charges, then the corresponding relation
IS:
(113)
where a and b are the absolute values of exchanging ion valence. The selectivity
coefficient has no units compared to the quantities used to calculate it (meq/ml., etc.) for
univalent exchange.
The selectivity coefficient describes resin behavior. For KBA > 1, the resin has a
strong affinity for A compared to B. IfKBA < 1, the resin prefers B to A. IfKBA = 1.0, the
resin has no preference. Resins prefer counterions with higher valance, smaller
equivalent volume, greater polarity and stronger association with the fixed charged
groups in the matrix (Helfferich, 1962).
Silica Chemistry
Although silica is one of the most abundantly available materials in the earth's
crust, not much is known about its chemistry. Recently its importance has been
recognized and efforts are being directed at understanding its chemical nature. The
solubility of silica in water is not yet completely understood since its dissolution in water
is not simple. It involves a hydrolysis reaction:
(114)
where Si(OH)4 is known as silicic acid. Iler (1970) discusses silica in solution, similar to
that of sugar in water, where some molecules exist in a crystalline state. Low
condensation polymers such as [[(OH)2SiO]4]n appear to be water miscible fluids, soluble
silica or Si(OH)4, and would probably be a clear liquid in an anhydrous condition.
6
The dissolution and deposition of silica in water involves hydration and
dehydration reactions which are evidently catalyzed by the hydroxyl ions.
(115)
where the forward reaction is hydration and the reverse is dehydration. The different
silica forms participating in Reaction 115 require explanation.
Amorphous silica: This form is most commonly seen. It exists as clays, massive
dense amorphous glass, powders, gels, colloids, etc. Amorphous silica is soluble in
water. The solubility is both temperature and pressure dependent (Iler, 1955). There are
many correlations which describe the solubility of amorphous silica in water (Iler, 1970).
Amorphous silica is more soluble in seawater at great depths due to the higher pressure.
The solubility of "sands of seashore" is less compared to the amorphous silica that
dissolves and enters into the water due to the weathering of minerals which leave behind
amorphous silica residues which then dissolve as time goes by.
Amorphous silica can be broadly categorized as porous and nonporous. The
solubility of nonporous amorphous silica is less than that of its porous counterpart.
Porous amorphous silica has more sites for water to react, hydrate, and solubilize.
Solubility is measured as Si02 (in parts per million) at a particular temperature. For nonporous
amorphous silica, the equilibrium concentration at 25°C corresponds to 70 ppm as
Si02• The most common forms of amorphous silica consist of porous aggregates which
have SiOH groups on their surfaces due to hydration. This form shows a higher
solubility than the nonporous form. The equilibrium concentration of porous amorphous
silica is in the range of 100130 ppm as Si02• This is the reason for the higher solubility
of gels and most powders.
Crystalline silica: The crystalline form is denser and more compact. The
common forms are: quartz, flint, etc. Quartz is present almost universally as "sand."
7
Quartz can exist in different phases depending on temperature; quartz, tridymite,
cristobalite, keatite, coesite, and stishovite. The latter three are formed under high
temperature and pressure. The solubility of crystalline silica is very low compared to
other forms.
Soluble silica: The weathering of minerals causes silica to dissolve into water.
Silica solubility is a function of temperature, pressure and pH (Iler, 1970). It is
commonly referred to as monosilicic or orthosilicic acid, the chemical formula being
Si(OH)4. Monosilicic acid has a tendency to polymerize at high concentrations with time,
giving polysilicic acid, colloids, gels, sols, etc. These are amorphous forms. Monosilicic
acid is a very weak acid and does not ionize in the presence of weak acids, but in an
alkaline medium it does form ionic silica. Silicic acid is completely soluble at high pH.
Monosilicic acid ionizes to give H3Si04 above pH 9. At pH 11, it ionizes further to yield
H2SiOf. The reactions are:
(116)
(117)
The equilibrium constants at 2SoC for these reactions are known.
or
[H3SiO;][H+] = 109.8
[Si(OH)4l
The equlibrium constant for Equation 117 is:
(118)
(119)
(1110)
8
or
[HzSiOi][H+] = 1O1Z.l6
[H3SiO;]
(1111 )
Silica is completely soluble above pH 11. Soluble silica is present in almost all
water and found up to a few parts per million in animals. Silica is removed from water
by biochemical activity and stored in these organisms. Silica reacts readily with metallic
hydroxides like AI, Mg, etc. and is precipitated. This property allows precipitation as a
pretreatment method for silica removal.
Silica Solubility:
Colloid science is needed to explain the most important phenomena of highmolecular
organic compounds. Colloid science is the basis on which the properties of
siliceous compounds can be explained (Hausen, 1960). To understand the colloidal
nature of silica, it is important to know the solubility characteristics of silica. A detailed
review of the solubilities of different phases of silica is provided by Iler (1970). An
overview is presented here.
Solubility of crystalline silica: Originally it was thought that quartz showed no
true solubility equilibrium when allowed to dissolve in water. Many investigations were
made to determine this equilibrium. Van Lier (1965) determined a definite solubility
equilibrium for pretreated, cleaned quartz surfaces. The Van Lier equation is:
loge = 0.151 1162
T
(1112)
where C is the molar concentration of soluble silica in parts per million and T is the
absolute temperature (K). At ordinary temperatures, the solubility of quartz is around 10
ppm. Solubility of quartz under hydrothermal conditions was discussed by Kennedy
(1950).
9
Solubility of amorphous silica: The equilibrium solubility for amorphous silica is
not definite since studies vary (Iler, 1970). The reason for such results may be due to
different particle sizes and impurities in the system. Different forms of amorphous silica
have different equilibrium solubilities. This is because of the variations in the surface
area per unit volume of the water available for the hydration reaction. Also, the
attainment of equilibrium may take a very long time, since the hydration step is slow.
Solubility is a function of temperature, pressure and pH. For near neutral
solutions solubility increases, and after a few days attains the equilibrium solubility
asymptotically at that temperature and pressure. For higher pH, solubility increases
rapidly with time. The solution might become supersaturated with silica which is
relieved by the formation of colloidal silica. Thus, an equilibrium solubility of silica is
attained. At lower pH, the solution increases linearly with an increase in pH of the
solution. Silica solubility was found to be constant from pH 2 to 8 and then equilibrium
solubility increased rapidly with pH. No particular number was agreed upon to account
for the equilibrium solubility at standard temperature and pressure. The value is
arbitrarily chosen owing to the different forms of amorphous silica available.
Ion Exchange Kinetics
Ion exchange kinetics can be described by three steps. They are:
• A film diffusion process, where mass transfer takes place from the
bulk fluid to the resin surface.
• A particle diffusion process, where the mass transferred to the resin surface diffuses
inside the resin phase.
• An exchange ofthe mass diffused inside the resin phase with that attached to the
fixed groups.
10
In each of these steps, equivalent amounts of exchanging and exchanged species
are moving counter to each other (Samuelson, 1963). Equilibrium is usually achieved
quickly in ion exchange, depending on the degree of resin crosslinking. For a rigidly
packed resin, the time required for equilibrium is significantly higher than for a resin
which has a lower crosslinking. Column performance depends on ion exchange kinetics.
It is therefore important to guess the right combination of ratelimiting reactions for the
specific situation under consideration. Most ion exchange processes are diffusionlimited
(Zecchini, 1990). The diffusion step can be charge transfer from the bulk liquid to the
liquid film surrounding the resin, diffusion within the particle, or a combination of both
processes.
Particle Diffusion Control
An essential assumption is that electroneutrality be maintained within the ion
exchanger. Therefore, the concentration of the fixed charged groups should equal the
total counterion concentration.
Mathematically, this can be expressed as:
where,
C· = total concentration of fixed charged groups in the resin(meq/cm3
)
Writing a similar expression for the counterion fluxes,
ZAJ: + ZBJ; = 0
where,
(1113)
(1114)
J; =flux of species I (meq/sec'cm2
)
Flux can be defined by either Fick's first law of diffusion or the NemstPlanck
equation. The NemstPlanck equation has an additional term, electric potential due to the
11
different ionic mobilities, which affects the exchange rate. The flux is related to the
concentration gradient by using a diffusion coefficient. The NemstPlanck equation for
species i is:
where,
J. = D.(VC. + Z;FC; VJ.)
I I I RT 'fI (1115)
Di = Diffusion coefficient of species I (cm2/s)
F = Faraday's constant (coulombs/mole)
R = universal gas constant (ergs/mole.K)
~ = Electric potential (ergs/coulomb)
T = absolute temperature (K)
The first part of the equation is Fick's first law, the second part is the flux due to
an external driving force. It should be understood that there is no external electric field
applied to the exchanger, only an induced electric field is produced due to the differing
ionic mobilities and at low concentrations, the dissociative nature of water. The above
equation is used in conjunction with the staticfilm model (Zecchini, 1991).
Combining the NemstPlanck equation with the flux expression and
electroneutrality gives:
(1116)
where, the effective diffusivity is defined as:
(1117)
Film Diffusion Controls
The assumptions of film diffusion are:
12
• a stagnant film of uniform thickness surrounds the resin bead,
• the curvature of the film is neglected, and
• a sharp boundary exists separating the bulk liquid and the liquid film.
The liquid film can be modeled by employing various concepts such as a
hydraulic radius film model, boundary layer model and the Nemst static film model. The
NemstPlanck equation can describe the diffusion process in the liquid film. In order to
apply the NemstPlanck equation for ionic flux, we need to use the hydraulic radius film
model for the liquid film.
In order to calculate the exchange rates, we need a mass transfer coefficient for
packed beds. Correlations of Carberry (1960) and Kataoka (1987) are used to obtain the
nonionic mass transfer coefficients.
Carberry's equation is:
KI =U5(1:!:.)(Scr2/3(Rer1/2
E
where,
K1 = nonionic mass transfer coefficient (cm/s)
E = bed void fraction
f.l = superficial liquid velocity (cm/s)
Sc = Schmidt number
Re = Reynolds number
Kataoka's equation is:
K
I =1.85(1:!:.)(_E_)l/3(SC)2/3(Re)2/3
E lE
(1118)
(1119)
In the model, we used Kataoka's correlation for low Reynolds number region (less than
20) and Carberry's relation otherwise.
13
Chapter III
Model Development
Basis and Assumptions
Although extensive literature on the modeling of ion exchange columns is
available, no universal model can be applied to any exchange column under
consideration. It is easier to model an ion exchanger for specific problems with a great
emphasis on all the details of the problems. However, accounting for all the details
increases the model complexity. In order to solve this problem, we need to make
assumptions which simplify the model. Since employing assumptions may lead to
erroneous results in predictions, caution is recommended before using any assumptions
that might affect the desired accuracy of the model. The following model development is
different from previous models in that it is more rigorous in nature and requires greater
calculational effort. There is a need to optimize the present model in order to reduce the
calculational load. The model also lacks supporting data required to obtain reasonably
accurate predictions. The model is a continuation of the modeling effort started by Raub
and Foutch (1984) and continued by Zecchini and Foutch (1990). The present model
gives insight into the behavior of soluble and colloidal silica in industrial waters at very
low concentrations. The model developed by Zecchini (1990) has a limited number of
assumptions to give as general a model as possible. The present model is also fairly
general but can be employed only for Mixed Bed Ion Exchange (MBIE) columns. This
14
restriction was used because only MBIE columns produce water with minimal amounts
of contaminants.
The chief assumption is that the model is applicable only for processes which are
film diffusion controlled where the NemstPlanck equation can be used. Particle
diffusion in the resin is neglected in this development. The rate of neutralization is quite
high compared to the rate of exchange. So, neutralization is considered to be
instantaneous. Other assumptions included in the model development are: uniform bulkliquid
and resinphase compositions for a given ion exchanger, bulkphase neutralization,
equilibrium at the particlefilm interface, activity coefficients equal to unity, pseudosteady
state mass transfer, isothermal operation, plug flow, and negligible axial
dispersion. If activity coefficients are not taken as unity then we may need to calculate
them using the DebyeHuckle equation (Stumm and Morgan, 1981). The above
assumptions do not appreciably affect the predictions for an MBIE column expect for the
assumption of negligible axial dispersion which may have a pronounced effect for low
flow rates through the ion exchanger. The additional term in the column material balance
equation accounting for axial dispersion is negligible for high flow rates. For a more
general model, we may include axial dispersion to get more accurate estimates at lower
flow rates.
The assumption that the resinphase and bulkliquid composition near a resin
surface does not change may not be true. Concentration profiles are bound to develop
both in the bulk liquid and in the liquid film adjacent to the resin. We cannot account for
these complexities in the model. This assumption greatly simplifies the model and
solutions.
15
Table 1
Model Assumptions
1) Film diffusion limited
2) Pseudo steady state process
3) No coion flux across the particle surface
4) A static film surrounds the film adhering to the resin surface
?) All univalent exchange
6) Neutralization reactions are instantaneous compared to the ion exchange steps
7) Resinfilm interface is maintained at equilibrium
8) The flux can be expressed using NemstPlanck equation
9) Curvature of the film can be neglected
10) No net coion flux within the film
11) Uniform bulk liquid and resin phase compositions
12) Activity coefficients equal to unity
13) Plug flow
14) Isothermal operation
15) Negligible axial dispersion
Ion Flux Expressions
TheNemstPlanck equation for ion i can be written mathematically as:
J. =D.('1C. + ZjFCj V"')
1 1 1 RT 'f' (1111 )
Equation 1111, when used in conjunction with the static film model, gives us a
relationship for the anionic flux in terms of the diffusivities and the bulk phase
concentrations of the anions. The detailed derivation is shown in Appendix A. The final
equation for the anionic flux is:
(1112)
a similar expression can be derived for cationic flux and both may be combined to define
the effective diffusivity for each species.
16
Particle Rates
The static film model can be used to define the particle rate as:
8(C;) =Ka (C~  C~) at 1 S I I
(1113)
where <Ci> is the resin phase concentration of species I, and Ki ' is the mass transfer
coefficient for the ion exchange process. An Ri factor relates this mass transfer
coefficient to the nonionic mass transfer coefficient determined by either Kataoka's
(1987) or Carberry's correlation (1960). The factor can be determined from the effective
diffusivity for each ion. It is related to the effective diffusivity by:
Let us represent the total resin capacity as Q and the total counterion as CT.
Hence, we can define the resin and liquid phase fractions.
The resin phase fraction is defined as:
(C.) Y;=Q
Similarly, the liquid phase fractional concentration is:
C.
X.=_I
I C
T
Equations 1113 to 1116 can be combined to give:
By; =K.R.a C (x;x;) at liST Q
(1114)
(1115)
(1116)
(1117)
17
Material Balances
The column material balance determines the concentration profile of the ion
exchange column and estimates the outlet concentration for each ionic species. The
material balance equation is:
(1118)
The Right Hand Side (R.H.S) of the above equation becomes zero when
axial dispersion is neglected. For the model, therefore, R.H.S is equal to zero. The above
equation is for one ion exchanger only. Kataoka (1987) has modified Equation 1118 to
incorporate the volume fractions ofboth the cationic and anionic resins into the material
balance equation. Certain dimensionless quantities are employed for this purpose. The
details of this derivation are given in Appendix D. 
Freundlich equations are used to estimate a rough value for the interficial
concentration of ionic silica. The formula employed is given below:
(1119)
where, CPI is the interficial concentration for ionic silica, YP is the resin loading
for silica, e1 and al are the Freundlich index and Freundlich coefficient, respectively.
Freundlich adsorption isotherms are used to calculate the amounts of particulate
silica, amorphous silica, colloidal silica adsorbed onto the anionic resin. Mathematically
this can be expressed as:
SXADS = SXADS + Ax(SXSILxxB)
PARADS =PARADS + Ax(PARSILxxB)
(11110)
(11111 )
COLADS =COLADS + Ax(COLSILxxB)
18
(11112)
where, SXADS,PARADS, COLADS are the adsorbed forms of amorphous,
particulate and colloidal silica, respectively and SXSIL, PARSIL, COLSIL are the
amorphous, particulate and colloidal forms of silica, respectively. A and B are the
Freundlich coefficient and Freundlich index, respectively.
Chapter IV
RESULTS AND DISCUSSION
The model was developed to account for breakthrough of~onoy_alen~ionic sili~Jl,
and estimate the amount of particulate silica removed by adsorption on anion exchange
resin employing adsorption isotherms. Also, the present model can predict the colloidal
silica formed in the ion exchange column and the amount of colloidal silica that can be
removed by adsorption. The various model parameters considered during the
development are listed in Appendix C.
The volume fraction ofthe anion exchange resin has a remarkable effect on the
monovalent ionic silica breakthrough from the exchanger. For an anion exchange resin
fraction of 0.6, the breakthrough for monovalent silica was found to occur at around 45
days, but, for an anion exchange resin fraction of 0.7, breakthrough occurred after 52
days. An earlier breakthrough was observed when the anionic resin fraction is equal to
0.5. The observed breakthrough was after 35 days. These observations were expected
because the number of ion exchange sites available increased with an increase in the
anion resin ratio and viceversa. With an increase in the number of sites, the resin could
exchange more ionic silica than in the previous case, causing a delay in ionic
breakthrough. Similarly, with a decrease in the anionic resin ratio, the number of sites
available for exchange of ionic silica have decreased proportionately, leading to an earlier
breakthrough. The plots for these simulations are shown in Figure 1.
20
I
I
,I
I
I
I
I
I
I
I
,I
I
I
I
I
I
I
/
.
....
.:
Anionic Resin Fraction =0.5
 Anionic Resin Fraction =0.6
. _.  Anionic Resin Fraction =0.7
01........L.~:I:....&.....s....:....=..L. ...1.....__ o 10 20 30 40 50 6C
TIME (DAYS)
0.2
zo
i«= a: 0.6
I zw
ozo0
0.4
x 10.6
~1.2
W
t
1 (f)
IZ
W
.J
~
>
:::J aw 0.8
Figure 1. Effect of anionic resin fraction on silica effluent concentrations
21
The column dimensions that were considered in this study are:
column diameter: 100 cm
column height: 250 cm
flow rate of water: 1.25xl05 mUs
Anionic resin fraction: 0.6
The breakthrough time of monovalent anionic silica was found to increase with a
decrease in the influent flow rate of the service water and decrease with an increase in the
flow rate entering the ion exchange column. Three different cases with different influent
flow rates are considered. With a flow rate of 1.25xl05 mUs, the breakthrough was found
to occur after 45 days. But, with flow rates ofO.75xl05 mIls and 1.75xl05 mIls, the
breakthrough of ionic silica is estimated to occur after 75 days and 32 days, respectively.
Considering the latter case, this is also a reasonable estimate as the Reynolds number
increases in direct proportion with the flow rate of the influent. This increase in Reynolds
number causes a decrease in the mass transfer coefficient of the exchanger, as Reynolds
number appears in the denominator of the formulas used in both Carberry's and
Kataoka's mass transfer models. This decrease in the mass transfer coefficient causes
less ionic silica to be taken up by the anionic resin. This causes an earlier breakthrough
of ionic silica. Similar reasoning can also be applied to the former case where the flow
rate was decreased (0.5xl05 mIls). In this case the Reynolds number is lower compared
to the base case (1.25x105 mIls). So, the mass transfer coefficient is correspondingly
higher. With an increase in the mass transfer coefficient, the amount of ionic silica being
taken up by the anionic resin increases correspondingly. So, a delayed breakthrough is
expected. The results confirm this. The effect of flow rate on the breakthrough of ionic
silica is shown in Figure 2.
Flow Rate = 1.75E5 mils
 Flow Rate = 1.25E5 mils
. .  Flow Rate = O.75E5 mils
X 10.6
oc w
~ 1.2
.J
U5
~z
W
.J «>
a=> w
~0.8
o
~
a:
~ 0.6
w
ozo0
0.4
0.2
10 20 30 40 50 60
TIME (DAYS)
I,,,
I
f
r
r
I
I
,I
I
I
I
I
I
I
70 80 90
22
Figure 2. Effect of flow rate on silica breakthrough
23
MODEL PARAMETER VALUES
FOR THE BASE CASE
The various model parameters and their numerical values used for the base case are
given belo.w.
0.2
2.0
4.0
50
15 PPM
16.0
2.5
0.05
0.20
0.6
2.0
0.00001
0.0001
0.00001
0.06
0.06
0.42
1.25x105
2.5x102
2 1.0x10
0.002
0.004
0.40
1.0
2.0
1.2
Initial fractions of sodium and potassium on the cationic resin
Initial fractions of chloride on the anionic resin
Initial fractions of silica on the anionic resin
Particle diameter of the anionic resin (cm)
Particle diameter of the cationic resin (cm)
Void fraction of the bed
Volumetric Flow rate ofthe column influent (mVsec)
Diameter of the column (cm)
Height of the column (cm)
Dimensionless time increment
Dimensionless distance increment
Cationic resin volume fraction
Density of the solution (g/cc)
Capacity of the Cationic resin
Capacity of the Anionic resin
Anionic resin volume fraction
Influent pH 7.0
Temperature of the influent (C) 25
Initial Concentrations of sodium and potassium entering the column 1.0x10·6
Initial Concentrations of chloride and silica entering the column 1.0x106
Equilibrium constant for the hydrolysis of amorphous silica 1.995x10·6
Equilibrium constant for the ionization of soluble silica 1.85x104
Freundlich coefficient for calculating interfacial concentration
of monovalent ionic silica
Freundlich index for calculating interfacial concentration of
monovalent ionic silica
Freundlich adsorption coefficient
Freundlich adsorption index
Ionic conductivity of monovalent silica
Initial concentration of amorphous silica entering into the column
Selectivity coefficient for chloridehydroxide exchange
Selectivity coefficient for silicahydroxide exchange
Rate constant for polymerization of colloidal silica (l/day)
Rate constant for hydrolysis of colloidal silica (l/day)
24
Changes in temperature ofthe influent effects many physical properties of the
fluid flowing through the bed. The various parameters effected by temperature: density,
and viscosity of service water, diffusivity of the ions, dissociation constant for water,
equilibrium constants, selectivity coefficients, rate constants for polymerization and
hydrolysis of polymeric silica. Silica tends to polymerize on the anionic exchanger with
time. But, when the temperature of the service water is high, silica polymerized on to the
anionic exchanger is desorbed back into the water as soluble silica. Thus, the amount of
soluble silica increases with an increase in the service water temperature. Silica in the
lower band of the ion exchanger enters into the water and is seen as 'silica leakage' in the
effluent water stream. Therefore, it takes less time for silica to break through as the
temperature is increased.
Three cases are considered to explicitly show the temperature effect on 5
breakthrough of ionic silica. The base case is at 25° C. The ionic silica breakthrough is
18 days. For 30° C and 50° C, the breakthrough times were found to be 13 and 11 days,
respectively. The predictions from the simulations are in accordance with the above
discussion. The plots for these simulations are shown in Figure 3. The anionic resin
fraction in this case was 0.4.
One of the most important properties of an ion exchanger is the size of the resin
bead. The particle diameter of the anion exchanger was found to effect ionic silica
breakthrough. Three different cases are considered here. The base case is with an anion
particle diameter of 0.06 cm. Particle diameters of 0.08 cm and 0.10 cm are used for the
second and the third cases, respectively. The simulations for these three cases show that
the breakthrough for ionic silica increased with an increase in the particle diameter. The
breakthrough curves for the three cases are 45, 46 and 47 days. An increase in the resin
diameter increases the particle Reynolds number which decreases the particle mass
transfer coefficient. A decrease in the particle mass transfer coefficient decreases the rate
X 10.6
1.2
0.8
zo
i=
~a:
~ 0.6
w
ozo
()
0.4
0.2
10
.I .• I:
.I.:
I :
1: ,;.:
J:
'I.: I: ;:
I·
~
E
it
I
15 20
TIME (DAYS)
TEMP =25 C
..... TEMP = 30 C
.. TEMP =50 C
25 30
25
Figure3. Effect of temperature on ionic silica effluent concentrations
26
x 10.6
a:1.2
w
I
.J en
IZW
.J «>
:::> aw 0.8
zo
~«g: 0.6
zw
() zo
00.4
Anionic Resin Diameter
 PDA =0.06 CM
PDA =0.08 CM
. _.  PDA =0.10 CM
,/
/
I
/
/
: /
.: ./
: I
.: /'
:. /'
:. /'
:. /'
.:1' ..~.
'I.
45 50 55
 .......
O~==~:::::=::::==_.l..__.L.__.J
40
0.2
TIME (DAYS)
Figure 4. Effect of particle size on ionic silica breakthrough
1.2 X 10.
6
0.8' •
zo
~«a:
~ 0.6
w
ozo
o
0.4
0.2
_.
,.. /
I
I
I
I
I
I
I
I
I
I
I
I
I
I
/
I
I
I
I
I
I
I
I
I
/
/
/
,..'''''''  . :::::==:=
YPO = 0.00001
.. YPO = 0.0001
OC I I I
10 15 20 25
TIME (DAYS)
tv
.)
Figure 5. Effect of the resin loadings on silica effluent concentrations
28
exchange of ionic silica onto the resin. This produces a lag in breakthrough curves of
smaller particle diameters. Figure 4 shows the variation of the breakthrough curves with
the particle size.
The resin loading effects breakthrough. as observed from the breakthrough curves
for two column simulations with different anionic resin loading. The first case is with a
silica loading of 105
. The second case is with a silica loading of 103
. The rest of the
parameters were the same for both cases. The anionic resin fraction in both the cases was
0.4. Column simulations show in the former case, that the onset of breakthrough
occurred after 20 days. In the latter case, breakthrough occurred after 12 days. The
prediction is reasonable because with an increase in the initial resin loading of the anionic
resin, the resin can no longer exchange the same amount of ionic silica as in the earlier
case with lesser initial resin loading. That is, the resin's capacity to exchange ionic silica
decreases with an increase in the resin loading. The column simulations are, therefore,
reasonable. The simulations are shown in Figure 5.
Figures 6 and 7 show us the variation of breakthroughs with variations in the
distance and time increments, respectively. These two parameters were found to effect
the stability of the system. Variation in the time increments was found to have a more
pronounced effect on the output concentrations of ionic silica than the variation in the
distance increments.
The effect of the Freundlich coefficient used in the calculation of the anionic
interfacial concentrations for ionic silica is shown in Figure 8. Three different cases are
considered here. The base case uses a Freundlich coefficient (a1) of 4.0. The other two
cases are for 6.0 and 8.0. The breakthrough curves for the three cases are: 45 days, 68
days, and 105 days. It is obvious that the breakthrough times for ionic silica are directly
proportional to the Freundlich coefficient. The reason for this type of behavior is due to
the presence of the Freundlich coefficient in the denominator of the formula used to
estimate the interfacial concentration of ionic silica. With an increase in the value of the
29

F " ._._..•.•.
~
IONIC SILICA
x 10.6
_ 1.2:..:........:..=,r,,,,.,r.,
a:
w
t:
1  (f)
~ 1"
w
' «>
::J
@0.8" zo
i=«~
O.6~
z
w
()
Zo
() 0.4" 
0.2 
,) . _.  DISTANCE INCREMENT = 0.006
 DISTANCE INCREMENT = 0.010
a I I I , I a 20 40 60 80 100 120 140
TIME (DAYS)
Figure 6. Effect of distance increment on silica breakthrough
30
...,.."", _.'lit .•".,.... , '\..•"", "..,.r .
I ~ ...........,.t
I,.
;f
;:
.I.:
!:
,.I .::
,..:
i :
!j
,i
; i
,. f
I
I
I
I
IONIC SILICA
 DISTANCE INCREMENT = 0.04
DISTANCE INCREMENT = 0.08
 DISTANCE INCREMENT=0.12
0.2
ool....1...L.o2..L.o3..L.o4~O~::.50=~60=~70
TIME (DAYS)
x 106
_ 1.2 ~~~r,....r,
a:
w
t=
....J
""'
Cf)
rz
w
~«
>
::J
~O.8
zo
~«~
0.6
zw
ozo
00.4
Figure 7. Effect of time increment on silica effluent concentrations
31
Freundlich Coefficients
120 140
r·_·
lI
!
I
I
;
I
{
J,
J
J
I
I
I
,j
I
I
60 80 100
TIME (DAYS)
20 40
 a1 = 4.0
a1 = 6.0
' a1 = 8.0
X 10.6
0.2
~w
t: 1.2
1  (f)
IZW
1 «>
::J a
w
0.8
za
i= «a:
I 0.6 z
w
ozo
00.4
Figure 8. Effect of Freundlich Coefficient on silica ~reakthrough
32
Freundlich coefficient the interfacial concentration decreases. This decrease in the
interfacial concentration causes the rate of ion exchange to increase. An increases in the
rate of ion exchange causes more ionic silica to be exchanged by the anionic resin and
this causes a delayed breakthrough for ionic silica. We selected a value of 4.0, as we
know that ionic silica is the first to show up in the column effluent as 'silica leakage'.
This is because, as was mentioned earlier, silica forms the weakest known acid, silicic
acid, which is not exchanged with great affinity onto the anionic resin when compared to
other acids such as hydrochloric acid (from chloride ions) and sulfuric acid (from sulfate
ions).
Similarly, we note the effect of the Freundlich index on the breakthrough of ionic
silica. Two cases are presented here. The value of the Freundlich index used for the base
case is "el" = 0.2. For values higher than 0.2, lot of instability was observed in the plots.
For other case, the value of"el" was set equal to 0.15. The rest ofthe parameters are the
same for both cases. For "el" = 0.15, we observe that the breakthrough was delayed to
85 days, compared to the base case of 45 days. This is evident from the Freundlich
formula, shown earlier, used to calculate the interfacial concentration of ionic silica. The
same reasoning as in the case of Freundlich coefficient can be applied here. As "e1"
appears in the denominator of the formula used to calculate the interfacial concentration,
the interfacial concentration would decrease with an increase in the value of "e1". This
decrease in the interfacial concentration leads to an increase in the rate of ion exchange,
and ultimately to a delay in breakthrough. The simulations can be seen in Figure 9.
Figures 10, 11, and 13 show the effect of Freundlich Adsorption Coefficient on
particulate, ionic, and colloidal forms of silica, respectively. Similarly, Figures 12 and 14
describe the variation of breakthrough curves for ionic, and colloidal forms of silica with
variation in the Freundlich Adsorption Index. The effects were not as pronounced as in
the earlier cases of Freundlich Coefficient, and Freundlich Index (Figures 8 and 9). This
is because in the former case, only the bulk concentrations of different forms of silica
33
a:
wr
::i 1.8
(j)
t
~ 1.6
1 «>
S 1.4 awz
1.2
o
i= «a: rzw
0 0.8
zo0
0.6
,•
,·
•
!.•
i
!
e1 =0.20
e1 =0.15
Freundlich Indices
°OL2.J..04...t.O:......:L60:~~~~~~
TIME
0.2
0.4
Figure 9. Effect of Freundlich index on silica effluent concentrations
34
x 10.7
3.6
(23.5
w
t:..:..
J...
(f)
t
~ 3.4
l
<X:
~
::::> a
~3.3
zo
~
<X: a:
t 3.2
z
w
ozo
() 3.1
PARTICULATE SILICA
Freundlich Adsorption Coefficients
A1 =2.0
A1 =4.0
._. A1 = 6.0
20 40 60 80 100
TIME (DAYS)
140 160 180
Figure 10. Effect of Freundlich Adsorption Coefficient on particulate silica breakthrough
x 10.6
1.2
a:
w
I.
J
U5
I dJ 0.8
.J «>
::l a
~0.6
zo
i«=
a:
I 0.4
z
wozo0
0.2
IONIC SILICA
Freundlich Adsorption Coefficients
A1 =2.0
A1 =4.0
._. A1 =6.0
35
°OL_2..L.O111111!!40~=6~O:8:;O::~~12~0'~140
TIME (DAYS)
Figure 11. Effect ofFreundlich Adsorption Coefficient on ionic silica breakthrough
36
x 10.6
1.2
IONIC SILICA
Freundlich Adsorption Indices
8=4.0
B = 6.0
'' B = 8.0
a:
w
I
.J
U5
IdJ
0.8
1«>
:J a
~0.6
zo
i=«
0:
I 0.4 zwozo0
0.2
20 40 60 80
TIME (DAYS)
100 120
Figure 12. Effect of Freundlich Adsorption Index on ionic silica effluent concentrations
37
140 160 180
A1 =2.0
A1 = 4.0
''A1 =6.0
Freundlich Adsorption Coefficients
80 100 120
TIME (DAYS)
20 40 60
O .l__.L..__L__L__L__L__L__l._.J
o
COLLOIDAL SILICA
z
03
~«a:
rz
w2
()
zo
()
1
_6
a:
w
t::
...J ens rzw
...J «
~4
:::> aw
Figure 13. Effect of Freundlich Adsorption Coefficient on colloidal silica break.1:hrough
38
140 160 180
 8=4.0
B = 6.0
._. B = 8.0
Freundlich Adsorption Indices
80 100 120
TIME (DAYS)
40 60
COLLOIDAL SILICA
20
O..I....I....I....I.....__..I.....__..I.....__.&__..L...._I
o
_6
a:
w
J
J
005
J z
W
J «
~4
::J a
~
z
03
~
a:
J
~2
ozoo
1
Figure 14. Effect of Freundlich Adsorption Index on colloidal silica breakthrough
39
Diffusivity of Silica
 XLAMP =50
XLAMP =75
.  .  XLAMP == 30
43
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
/
I
I
/
42
/
..
......
39 40 41
TIME (DAYS)
36 37 38
0.2
0.4
~ 1.2
i= «a:
l
ZW
oz
0.8
oo
0.6
x 108
C? 2
W
l
~ 1.8
Cf)
lZ
~ 1.6
«
>6
1.4
w
Figure 15. Effect of diffusivity on silica breakthrough
40
were found to change with changes in the Adsorption Coefficient, and Adsorption Index,
whereas in the latter case the interfacial concentrations were found to vary. The rate of
exchange is more strongly dependent on the interfacial concentration. Hence, our
observation of large variations in the latter case are justified.
The effect of the diffusion coefficient on the breakthrough curves for ionic silica
is shown in Fig. 15. Different values for ionic conductivity of ionic silica are used for the
purpose of simulation. For the base case, we used a value of 50. The other two values
are 75 and 30. For a diffusivity of75, the breakthrough for ionic silica was observed
after 38 days. This is lower than the breakthrough for the base case. This is anomalous
behavior, because, with an increase in the ionic conductivity, the diffusion coefficient for
silica increases. An increase in the diffusion coefficient decreases the Schmidt number,
as diffusion coefficient is in the denominator of the formula used for the calculation of
the Schmidt number. This should increase the liquid phase mass transfer coefficient.
With an increase in the liquid phase mass transfer coefficient, the anionic resin should
take in more ionic silica, giving a delayed breakthrough. But, an increase in the
diffusivity of silica increases the interfacial concentration. This increase in the interfacial
concentration decreases the rate of ion exchange for silica. The decrease in the rate of
exchange offsets the increase in the liquid phase mass transfer coefficient. Therefore, we
observe an earlier breakthrough with an increase in the diffusivity. The value of ionic
conductivity for ionic silica was set at 50 for the base case, as the selfdiffusivity of silica
was estimated to be approximately 1.3Xl05 cm2/s by WilkeChang(19).
Colloidal silica is formed in the ion exchange column with time. The
concentration of colloidal silica entering into the ion exchange column through service
water is set equal to zero. Colloidal silica is assumed to form within the column due to
aging. Colloidal silica is formed from amorphous silica by polymerization of the
individual silica molecules into long polymeric chains. The rate of polymerization affects
the rate of colloidal silica formed in the ion exchanger. The rate of polymerization is a
41
first order reaction. So, the units for Kp, the rate constant for polymerization, are timeI.
The value ofKp used for estimating the amount of colloidal silica generated in the
column is 0.05 dayI, for the base. The amount of silica formed for a column operation of
175 days is 0.12 PPM. For the other case where Kp was set equal to 0.1 0 day1, the
amount of colloidal silica formed was equal to 0.075 PPM or 75 PPB. These two results
are shown in Figure 16.
Colloidal silica is formed by the rate of polymerization of amorphous silica and a
part of it is hydrolyzed back into amorphous silica, by hydrolysis. The amount of
colloidal silica at any instant of time is a net amount of colloidal silica formed in the
exchanger due to the rate of polymerization and the removal due to the rate of hydrolysis.
Therefore, the rate of hydrolysis is also important in estimating the colloidal silica
content at any instant. The rate constant of hydrolysis, Kh, has the same units of rate of
polymerization, Le., timeI. Typical values used in estimating the rate of hydrolysis are
0.15 day"tcfor the base case), 0.10 day"l, 0.20 day"l. IfKh equals Kp, then, the amount of
colloidal silica formed in the exchanger would be equal to zero. Therefore, that particular
case is avoided. The variation in the concentration of colloidal silica with respect to the
rate of hydrolysis is shown in Fig. 17.
The selectivity coefficient for monovalent ionic silica over hydroxyl ion is not
known precisely. So, different values of randomly chosen selectivity coefficients for
monovalent ionic silica were simulated. The results show that the breakthrough for
monovalent ionic silica occurred after 45 days, when the selectivity coefficient was equal
to 1.0. For a selectivity coefficient of 3.0, the breakthrough was delayed until 85 days.
42
The value ofKp used for estimating the amount of colloidal silica generated in the
column is 0.05 dayI, for the base. The amount of silica formed for a column operation of
175 days is 0.12 PPM. For the other case where Kp was set equal to 0.1 0 day1, the
amount of colloidal silica formed was equal to 0.075 PPM or 75 PPB. These two results
are shown in Figure 16.
Colloidal silica is formed by the rate of polymerization of amorphous silica and a
part of it is hydrolyzed back into amorphous silica, by hydrolysis. The amount of
colloidal silica at any instant of time is a net amount of colloidal silica formed in the
exchanger due to the rate of polymerization and the removal due to the rate of hydrolysis.
Therefore, the rate of hydrolysis is also important in estimating the colloidal silica
content at any instant. The rate constant of hydrolysis, Kh, has the same units of rate of
polymerization, i.e., timeI. Typical values used in estimating the rate of hydrolysis are
0.15 day"l(for the base case), 0.10 day"l, 0.20 day"l. IfKh equals Kp, then, the amount of
colloidal silica formed in the exchanger would be equal to zero. Therefore, that particular
case is avoided. The variation in the concentration of colloidal silica with respect to the
rate of hydrolysis is shown in Fig. 17.
The selectivity coefficient for monovalent ionic silica over hydroxyl ion is not
known precisely. So, different values of randomly chosen selectivity coefficients for
monovalent ionic silica were simulated. The results show that the breakthrough for
monovalent ionic silica occurred after 45 days, when the selectivity coefficient was equal
to 1.0. For a selectivity coefficient of 3.0, the breakthrough was delayed until 85 days.
x 10.7
_1.2
ffi COLLOIDAL SILICA .
::::i
.e"n
z
W
......J «~
::::)
~O.8
zo
~
~ 0.6
z
w
ozo
() 0.4
0.2
Rate Constant of Polymerization
 KP =0.05 DAY"1
KP =0.10 DAY"1
43
or:...L..L..L.L...L.....L:~___:_~~
o 20 40 60 80 100 120
TIME (DAYS)
Figure 16. Effect of rate constant for polymarization on silica breakthrough
x 10·a
9 a:
wr
~8
(J) rzw
;i7
>
:J
@S zo
i=5
<{
a: rdj4
ozo
03
2
COLLOIDAL SILICA
  ..."'
Rate of Hydrolysis
 KH =0.15 DAY"1
KH =0.10 DAY"1
._. KH =0.20 DAY"1
44
O~_..L__...L__l__.....l...__....J...._~~::::==:
o 20 40 SO 80 100
TIME (DAYS)
Figure 17. Effect of rate constant for hydrolysis on silica effluent concentrations
x 10.6
_1.2
a:
w
I
:=..i
(f)
IZ
W
.J
<t:
>
::l
@0.8 zo
;::
~ 0.6
Iz
Woz8
0.4
0.2
I
fI
,
I
I
I
SELECTIVITY OF SILICA
 SELECTIVITY = 1.0
.  .  SELECTIVITY = 3.0
45
Figure 18. Effect of selectivity coefficient of silica on silica breakthrough
46
When the selectivity coefficient was set equal to 3.0, we have ionic silica displacing
hydroxyl ions in the ion exchanger. Therefore, the hydroxyl ions are pushed further
down the column. The band of monovalent ionic silica is seen above the band of
hydroxyl ions. This implies that more of monovalent ionic silica is exchanged in
preference to the hydroxyl ions. As more silica is taken up by the anionic resin, we see a
delayed ionic silica breakthrough. This is what the simulation with a selectivity
coefficient of 3.0 shows us in Figure 18. In a case where the selectivity coefficient is
lower than 1.0, we expect a much earlier breakthrough.
The effect of initial amount of amorphous silica present in the influent service
water on the breakthroughs of particulate, colloidal, and ionic silica can be seen in
Figures 19,20, and 21. Two different cases are considered here. The initial amount of
amorphous silica entering into the exchanger for the base case is 15 PPM. Another
simulation is run with a value of 30 PPM amorphous silica. The results show that the
amount of amorphous silica entering into the mixed bed is not a strong parameter
effecting the breakthroughs of particulate, colloidal, and ionic silica.
The concentration of particulate silica produced in the ion exchange column is
dependent on the amount of amorphous silica entering into the column and the
equilibrium rate constant for eq. (115). The value of the rate constant for this reaction is
1.995xlO·6
, for the base case. The other two cases considered were for rate constant of
1.995xlO·3
, 1.995xlO·s. The variation in the concentration of particulate silica with
changes in the equilibrium constant for this reaction can be seen in Figure 22.
The breakthrough curves for all the participating ions in the mixed bed are plotted
in Figure 23. The ions that participated are: sodium, potassium, chloride, monovalent
ionic silica, hydrogen, and hydroxyl ions. Ionic silica was found to breakthrough earlier
than sodium, potassium, and chloride ions. This is because silica is known to form the
weakest acid known, silicic acid, which is loosely held onto the anionic resin. Therefore,
47
silica is the first anion to show up in the effluent water as "silica leakage." The column
conditions are the same as those provided in the base case table (Table II).
The effect of ionic silica feed concentration has a significant impact on the
breakthrough curves as seen in Figures 24. The base case was for an inlet silica
concentration of 1 PPM. Two other cases are studied here, one at influent concentration
of 5 PPM and the other with an influent concentration of 10 PPM. For the first case,
silica breakthrough was observed after 45 days. The breakthrough curves for the other
two cases are 13 and 8 days, respectively. When the concentration of the ionic silica
entering into the ion exchange column increases, we expect more silica to be exchanged
onto the anionic resin. This makes the resin lose its capacity rather early. This causes
lesser silica to be exchanged on to the anionic resin as the resin is no longer in a position
to exchange and has to be regenerated. Therefore, we observe an earlier silica
breakthrough.
48
 ... ''
 PARTICULATE SILICA
x 10.7
~ 3.6ir..rr..r.
W
t:
~..
(J)
~z 3.5
w
~«
~
~aw 3.4 zo
~
~ 3.3
zw
ozo
03.2
.  .  Initial Amorphous Silica = 15 ppm
 Initial Amorphous Silica =30ppm
3.1
180
30;~=:::=:~=L.l....L~
120 140 160
Figure 19. Effect of amorphous silica on particulate silica effluent concentrations
x 108
[£7
w
f
~ COLLOIDAL SILICA
(J)
~6 zw
...J
<{
>
55 awz
§4 «a:
fz
~3
zoo
2

 Initial Amorphous Silica =15 ppm
.  .  Initial Amorphous Silica =30ppm
49
20 40 60 80 100
TIME (DAYS)
120 140 160 180
Figure 20. Effect of amorphous silica on colloidal silica breakthrough
50

PARTICULATE SILICA
x 10.7
 3.6 ~~ry......rr..rI
a:
w
t::
.J. .en z 3.5
w
.......J «
~
::J aw 3.4 zo
i=«~
3.3
zw
() zo
() 3.2
. .  Initial Amorphous Silica =15 ppm
 Initial Amorphous Silica =30ppm
3.1
301.J20~l406..LO8LO1OLO1~2:=O:1~4=O:1~60:~180
TIME (DAYS)
Figure 21. Effect of amorphous silica on particulate silica effluent concentrations
zQ
10.5
f «a:
f
Z
W 10.6
oZoo
10.7
· _
..  .  .  .  .  ..  .  .  .  .  .  .  _.    
Equilibrium Constants
 K1 =1.995E06
K1 =1.995E03
.. K1 =1.995E05
10°
TIME (DAYS)
Figure 22. Effect of Equilibrium Constant on particulate silica breakthrough
51
x 10.7
9
8
a:
w
C 7
.J
'""
(f)
t
Z 6 w
.J «
> :::l 5 aw
z 4
0
~«
a: 3 t
ZW
0z 2
0u
o 50 100 150
TIME (DAYS)
SODIUM
POTASSIUM
.. CHLORIDE
+ IONIC SILICA
200 250
52
Figure 23. Breakthrough of all the paticipating ions
x 10.5
1.2 a:
w
~
1
U5
IZW
1 «
>
:J aw 0.8 zo
~«
g: 0.6
zw
ozo
00.4
0.2
Inlet Concentrations of Silica
..~~..   CPF = 1.0E06
CPF = 5.0E06
.   CPF =10.0EOS
I
!i
53
O~ .........« __L_... .L....._ ____L.__~_......L._ ____J
o 1a 20 30 40 50 60
TIME (DAYS)
Figure 24. Effect of inlet concentration of ionic sili~~ OP. silica breakthrough
CHAPTER V
CONCLUSIONS
The model developed in this study deals with a topic which has not
received great attention in the past. An equilibrium subroutine was written which
accounts for the effect of soluble silica on the concentration of H+ ions, other than
the dissociation constant of water. This subroutine differs from the equilibrium
subroutine for ternary exchange developed by Zecchini (1990). A subroutine was
developed to predict the interfacial concentrations for monovalent anionic silica
where Freundlich equation was used. The model accounts for the removal of
ionic, particulate, and colloidal forms of silica. Particle loading of ionic silica on
the anionic exchange resin is required at each slice in the mixed bed.
A Freundlich adsorption isotherm was employed to account for particulate
adsorption of colloidal and amorphous silica in the bulk liquid phase. This
approach gives better results compared to Langmuir adsorption isotherm.
The ternary exchange model was used to predict the outlet concentrations
of monovalent anionic silica, chloride, sodium and potassium. Monovalent ionic
silica showed earlier breakthrough than other ions. Various parameters that effect
the breakthrough of ionic silica were considered and simulations were run to
understand their role.
Test simulations were made for a typical industrial ion exchange column.
Values of dimensionless timedistance increment (,;) and dimensionless distance
55
increment(~) are 0.002 and 0.004. The model was found to be more sensitive to
changes in ~ than to changes in 1;. According to Haub (1984), there would be a
rapid decrease in the impurity concentrations as the influent moves down the
column, but the active exchange zone moves much slowly down the column.
The breakthrough predictions from the model for monovalent ionic silica
were in very good agreement with the works of Kataoka and Muto (1993). Most
of the predictions were qualitative since little data is available in the literature.
Recommendations
The present model suffers from lack of experimental data on the exchange
kinetics of ionic and particulate silica. Experimental data is necessary to evaluate
specific performance of ion exchange system with the present model.
The amount of divalent ionic silica produced in the column due to further
reaction of monovalent ionic silica, was assumed to be negligible. This
assumption is valid for only those operations where the pH of the inlet water is
less than 10.7. For higher pH, we need to account for divalent form of ionic
silica. The rate equations for a divalent species will be different (Pondugula,
1994).
The effect of temperature on the selectivity coefficients and adsorption
isotherms is not known. The values used for coefficients and adsorption indices
are approximated at 25° C. Variation of these values with changes in service
water temperature has to be determined and incorporated into the model. The
temperature effects on other parameters, such as the solution viscosity, diffusivity,
water dissociation constant are incorporated in the model.
56
BIBLIOGRAPHY
Bajpai, R. K. , Gupta, A. K. and Gopala Rao, M. (1974). Single particle studies of
binary and ternary cation exchange kinetics. AIChE J., 20 (5), 989995.
Blume, R. (1987). Preparing ultrapure water. Chern. En~. Pro~. (Dec), 5557.
Calmon, C. (1986). Recent developments in water treatment by ion exchange. Reactive
polymers, 4, 131146.
Copeland, J. P., Henderson, C. L. and Marchello, J. M. (1967). Influence of resin
selectivity on film diffusion controlled ion exchange. AIChE J., 13, 449452.
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.
En~. Chern. Res., 26(9), 19061909.
Dranoff, J. S. and Lapidus, ,L. (1961). Ion exchange in ternary systems. Ind. En~.
Chern., 53, 7176.
Harries, R. R. (1991). Ion exchange kinetics in ultrapure water systems. J. Chern. Tech.
Biotechnol., 51, 437447.
Haub, C. E. (1984). M.S. Thesis, Oklahoma State University, Stillwater, OK.
Haub, C. E. and Foutch, G. L. (1986a). Mixedbed ion exchange at concentrations
approaching the dissociation of water 1. Model development. Ind. En~. Chern.
Fund., 25, 373381.
Haub, C. E. and Foutch, G. L. (1986a). Mixedbed ion exchange at concentrations
approaching the dissociation of water 2. Column model applications. Ind. En~.
Chern. Fund., 25, 381385.
Helfferich, F. G. (1962). Ion Exchange. McGraw Hill Book Company, New York.
57
Helfferich, F. G. (1984). Conceptual view of column behavior in multicomponent
adsorption or ion exchange systems. AIChE Symposium Series, 80, (233), 113.
Helfferich, F. G. (1990). Models and physical Reality in ionexchange kinetics.
Reactive Polymers, 13, 191194
Iler. R. K. (1955). Colloid Chemisrty of Silica and silicates. Cornell University Press,
Ithaca, N.Y.
Kataoka, T. and Muto, A. (1991). Removal of Dilute Silicate by Ion Exchange MethodAdsorption
Equilibrium Relation of silicate and an OHtype Resin. Proceedings of /
the International Conference on Ion Exchange. Tokyo, 1991, 329334.
Kataoka, T. and Muto, A. (1993). The Removal of Soluble Dilute Silicate from water by
Ion Exchange Resins. Prodeedings ofthe IonEx '93 Conference. U.K, 1993,
159166
Kataoka, T., Yoshida, H. and uemura T. (1987). Liquidside ion exchange mass transfer
in a ternary system. AIChE J., 33, 202210.
Kunin, R. (1960). Elements of ion exchan~e. Reinhold publishing corporation, N.Y.
McCartney, B. (1987). Ultrapure water for the silicon chip industry. The Chemical
En~ineer (Jan.), 2426.
Omatete, 0.0., Vermeulen, T. and Clazie, R. N. (1980a). Column dynamics of ternary
ion exchange. Part I. Diffusion and mass transfer relations. Chern. En~ineerini:
J., 19, 229240.
Omatete, 0.0., Vermeulen, T. and Clazie, R. N. (1980b). Column dynamics ofternary
ion exchange. Part II. Solution mass transfer controlling. Chern. Engineering J.,
19, 241250.
Owens, L. Dean. Practical Principles of Ion Exchange Water Treatment, 1985, Tall Oaks
Publishing, inc. NJ.
Pondugula, S. K. (1994). Mixed bed ion exchange modeling for divalent ions in a
ternary system. M.S. thesis, Oklahoma State University, Stillwater, OK.
58
Sadler, M.A. (1993). Developments in the production and control of ultrapure water.
Ion Exchange Processes: Advances and Applications, 1528.
Schlogl, R. and Helfferich, F.G. (1957). comment on the significance of diffusion
potentials in ion exchange. J. Chern. Phys., 26,57.
Yoon, T. (1990). Ph.D. Dissertation, Oklahoma State University, Stillwater, OK.
Yoshida, H., and Kataoka, T.,(1987). Intraparticle ion exchange mass transfer in ternary
system. Ind. En~. Chern., 26, 11791184.
Zecchini, E. J. (1990). Solutions to selected problems in multicomponent mixed bed ion
exchange modeling. Ph.D. Thesis, Oklahoma State University.
Zecchini, E.J. (1990). M.S. Thesis, Oklahoma State University, Stillwater, OK.
Zecchini, E. J., and Foutch, G. L. (1991). Mixed bed ionexchange modeling with amine
form cation resins. Ind. En~. Chern. Res., 30 (8), 1886.
APPENDIX A
SILICA EQUILffiRIUM
The various forms of silica considered in the model development are:
[(Si02)J Amorphous form
[(Si0211l Particulate form
[Si(OH)4l Soluble form
[H3Si041 Monovalent form
[H2Si041 Divalent form
The different reactions considered are given below:
Dissolution:
59
(eq. AI)
The equilibrium constant for the above reaction, K1= 1.995 X 103
Ionization:
(eq. A2)
The equilibrium constant for the above reaction, K2 = 1.85 X 104
Water Dissociation:
(eq. A3)
where, Kw is the dissociation constant for water. The water dissociation constant is a
function oftemperature. Its value at 25°C is 10 14.
60
The terminology used for each form of silica in the development ofthe subroutine is given
below:
[SX] Amorphous silica
[SXl] Particulate silica
[HS] Soluble silica
[S 1] Monovalent silica
" ;
[S 2] Divalent silica
Applying the electroneutrality in the bulk liquid, we have,
Applying the mass balance equation for silica in the bulk phase,
[SX] = [SXl] + [HS]
Equation A4 can be written as:
From Equation A2, we get,
(eq. A4)
(eq. A5)
(eq. A6)
(eq. A7)
Equating the concentration ofwater to unity (because ofits existence in large quantites),
the concentration ofthe monovalent form can be expressed by:
[Sl = K2 [HS] [Olf] (eq. A8)
61
Substituting this equation in Equation A6,
Using the dissociation constant for 'water to eliminate [Olll, and simplifYing the resulting
equation,
Further simplification gives us a quadratic equation in terms of [H+],
(eq. A9)
+ 2 + + + [H] + {[Na ] + [X ]  [CI1}[H ]  {K2 [HS] +l}Kw == 0 (eq. AlO)
The solution for the above quadratic equation is:
[H+] == { {[Na+] + [X+]  [Cll} + sqrt({[Na+] + [X+]  [Cll}2 +
4 Kw {K2 [HS] +I})}/2.0
Therefore,
The pH ofthe solution is obtained from
pH == log(H+)
From Equation AI, Particulate silica can be written as:
[SXI] == K 1 [SX]/[HS]
Solving Equation A5, for Particulate silica,
(eq.AII)
(eq. AI2)
[SXl] = [SX]  [HS]
Eliminating [SXl] from the above two equations, we have
K1 [SX]/[HS] = [SX]  [HS]
Simplifying the above equation,
[HS]2  [SX][HS] + Kt[SX] = 0
The solution ofthis quadratic equation in [HS] yields
[HS] = {[SX] ± sqrt{[SX]2  4 K1 [SX]} }/2.0
(eq. A13)
(eq. A14)
62
Therefore, ifgiven [SX], we can calculate [H+] using [HS] calculated above and
substituting in equation El1. The new bulk concentration for ionic silica can be obtained
from the following equation:
[Sl = K2 [HS][OHl
Similarly, Particulate silica is calculated from Equation A5
[SXl] = [SX] [HS]
(eq. E15)
(eq. E16)
The amount ofparticulate silica adsorbed is determined using Freundlich
adsol]>tion isotherm Similarly, the amount of soluble silica adsorbed can also be
63
estimated by Freundlich isotherm Also, divalent form of silica can be seen in the system
only ifthe pH ofthe solution is higher than 10.7. The reaction for the formation of
divalent form of silica from monovalent silica can be written down as:
(eq. A17)
The rate equation for this reaction is expressed as:
(eq. AI8)
The particulate silica adsorbed onto the resin is determined by Freundlich adsorption
isotherms. Mathematically,
SXADS = Ax(SXxxB) (eq. A19)
where, SX is the amorphous silica in the seIVice water, SXADS is the adsorbed
amorphous silica, A and B are the Freundlich coefficients.
Similar equations can be written for particulate and colloidal silica.
PARADS = Ax(PARSILxxB) (eq. A20)
COLADS = Ax(COLSILxxB) (eq. A2I)
where, PARSIL and COLSIL are the amounts ofparticulate and colloidal silica in seIVice
water, PARADS and COLADS are the adsorbed amounts ofparticulate and colloidal
silica, respectively.
Freundlich equation was used to estimate the interficial concentration ofionic silica.
Mathematically, this can be expressed as:
CPI = «1.0/al)xVP)xx(I.0/el) (eq. A22)
where, CPI is the interficial concentration ofionic silica, VP is the particle loading ofionic
silica on the anionic resin, al and el are the Freundlich coefficient and Freundlich index,
respectively.
APPENDIXB
MODEL PARAMETER VALUES
The various model parameters and their numerical values used for the base case
are given below.
Volumetric flow rate: 1.25£5 cm3/s
Fraction ofcationic resin 0.4
Fraction ofanionic resin 0.6
Cationic particle diameter 0.06 cm
Anionic particle diameter 0.06 cm
Temperature 25oC
Selectivity coefficient for silica 2.5
Freundlich adsorption coefficient, 1 2.0
(for interfacial concentrations near the resin)
Freundlich adsorption index, 1 0.2
Freundlich adsorption coefficient, 2 2.0
Freundlich adsorption index, 2 4.0
(for bulk phase adsorption ofparticulates)
Initial concentration ofAmorphous silica 15 PPM
Initial resin concentrations:
a)YNO: 0.00001 b) YXO: 0.00001
c)YCO: 0.0001 d) YPO : 0.00001
Initial bulk phase concentrations:
a)CNO: IPPM b) CXO: IPPM
64
c)CCO: lPPM d) CPO: lPPM
pH 7.0
KP : 0.05 dayl KH : 0.15 dayl
(rate constants for the polymerization and hydolysis reactions of colloidal silica)
Kl : 1.995E03 K2 : 1.85E4
Tau : 0.002 Xi : 0.004
XLAMP : 50
(used for the calculation ofdiffusivity coefficient for silica)
65
APPENDIXC
COMPUTER CODE FOR SILICA BREAKTHROUGH
66
********************************
* TIllS COMPUTER CODE IS DEVELOPED TO *
* SIMULATE THE BREAKTHROUGH CURVES *
* FOR SILICA [[ SILICA CODE ]] *
* CODE DEVELOPED BY ZECCHINI, HAUB, *
* SHARMA PAMARTHY AND Dr. GARY L. FOUTCH *
********************************
IMPLICIT INTEGER (IN), REAL*8 (AH,OZ)
*
* Variable and array declaration
*
REAL KLN, KLX,YNC(4, 11000),XNC(4, 11000),XXC(4,11000),
I RATN(4,11000), YXC(4,11000), RATX(4,II000),
2 RATC(4, I 1000), YCA(4,11000), XCA(4,II000),
3 RATP(4, 11000), yPA(4, 11000), XPA(4, 11000),
4 KLC, KLP, Kl, K2, B2, HS, KP, KH
*
* Function statements for determining nonionic mass transfer
* coefficients based on system parameters
*
67
**
*
Carberry's Correlation
FI(R,S) = 1.15*VS/(VD*(S**(2./3.»*(R**0.5»
* Kataoka's Correlation
*
**
*
**
*
F2(R,S) = 1.85*VS*«VD/(1.VD»**(I./3.»/
1 (VD*(S**(2./3.»*(R**(2./3.»)
solubility correlation for amorphous silica
F3(P) = 10.**(2.440.053*(P»
Initial conditions and bed properties
OPEN (9, FILE='s15.d', STATUS='UNKNOWN)
READ(9,*) KPBK, KPPR, TIME
READ(9,*) YNO, YXO, YCO, YPO
READ(9,*) PDC, PD~ VD
READ(9,*) FR, DIA, CHT
READ(9,*) TAU, XI, FCR
READ(9,*) DEN, QC, Q~ FAR
READ(9,*) pH,TMP
READ(9,*) CNF, CXF, CCF, CPF
READ(9,*) KI, K2
READ(9,*) aI, el, A, B
READ(9,*) XLAMP,SX
READ(9,*) TKCO,TKPO
READ(9,*) KP, KH
*
*
* Concentrations and dissociation constant
*
*
CP = 1.43123DO+TMP*(0.000127065DO*TMPO.0241537DO)
ALOGKW = 4470.99/(TMP+273. 15)6.0875+0.01706*(TMP+273. 15)
DISS = 10.**(ALOGKW)
ClllI = 10.**(pH)
COIl = DISS/ClllI
CFCAT = CNF+CXF+ClllI
CFANI = CCF+CPF+COll
IF(ABS(CFCATCFANI).LE.(CFCAT/IOOO.» GO TO 600
IF(CFCAT.GT.CFANI) THEN
WRITE(6,550)
550 FORMAT(30X,'TOTAL CATIONS IS GREATER THAN TOTAL ANIONS')
GO TO 570
ELSE
WRITE(6,560)
560 FORMAT(30X,'TOTALANIONS IS GREATER THAN TOTAL CATIONS')
ENDIF
570 WRITE(6,580)COll, ClllI
580 FORMAT(30X,'COII =',EI0.5,3X,'ClllI =',EI0.5)
600 CONTINUE
IF(CFCAT.GE.CFANI) THEN
CF=CFCAT
ELSE
CF = CFANI
ENDIF
*
*
68
*
*
Calculate ionic diffusion coefficients based on temperature
using limiting ionic conductivities (Robinson and Stokes, 1959)
*
*
RTF = (8.931DI0)*(TMP+273.16)
XLAMH = 221.7134+5.52964*TMP0.014445*TMP*TMP
XLAMX = 1.40549*TMP+39.1537
XLAMN = 23.00498+1.06416*TMP+O.0033196*TMP*TMP
XLAMO = 104.74113+3.807544*TMP
XLAMC = 39.6493+1.39176*TMP+0.0033196*TMP*TMP
DN = RTF*XLAMN
DX = RTF*XLAMX
DO = RTF*XLAMO
DC = RTF*XLAMC
DR = RTF*XLAMH
DP = RTF*XLAMP
*
*
*Calculate Reynolds and Schmidt numbers and nonionic
*mass transfer coefficients
*
*
AREA = 3. 1415927*(DIA**2)/4.
VS=FRlAREA
REC = PDC*lOO.*VS*DEN/«l.VD)*CP)
REA = PDA*lOO.*VS*DEN/«l.VD)*CP)
SCX = (CP/IOO.)/DEN/DX
SCN = (CP/IOO.)/DEN/DN
SCC = (CP/IOO.)/DEN/DC
SCP = (CP/IOO.)/DEN/DP
IF (REC.LT.20.) THEN
KLN = F2(REC,SCN)
KLX = F2(REC,SCX)
ELSE
KLN = Fl(REC,SCN)
KLX = Fl(REC,SCX)
ENDIF
IF (REA.LT.20.) THEN
KLC = F2(RE~SCC)
KLP = F2(RE~SCP)
ELSE
KLC = Fl(RE~SCC)
KLP = Fl(RE~SCP)
ENDIF
69
***
***
*
Calculate the amount ofAmorphous Silica initially present
SX =F3(pH)
Calculate total number of steps in distance (NT) down column
CHTD = KLC*(l.VD)*CHT/(VS*PDA)
wrlte(6,45) CHTD
45 FORMAT(30X,'VALUE OF CHTD IS =',F7.3)
NT=CHTD/XI
*
*Print system parameters and calculated parameters
*
WRITE (6,10)
WRITE (6,11)
WRITE (6,12) YNO,YXO
WRITE (6,86) YCO,YPO
WRITE (6,13) PDC,PDA,VD
WRITE (6,14) QC,QA,FCR,FAR
WRITE (6,15) FR,DIA,CHT
WRITE (6,89) CNF,CXF
WRITE (6,90) CCF,CPF,pH
WRITE (6,92) XLAMP, SX
WRITE (6,94) TKCO, TKPO
WRITE (6,95) KP, KH
WRITE (6,16) DEN,TMP
WRITE (6,17) TAU,XI,NT
WRITE (6,91) al,el,A,B
WRITE (6,18)
WRITE (6,19)
WRITE (6,20)
WRITE (6,22) REC,KLN
WRITE (6,88) KLX,KLC
WRITE (6,23) VS,CP,CF
WRITE (6,80)
WRITE (6,81) DX,DN,DH
WRITE (6,82) DO,DC,DP
*
*
70
* Print breakthrough cwve headings
*
*
*
**
*
*
IF (KPBKNE.l) GO TO 50
WRITE (6,24)
WRITE (6,25)
WRITE (6,26)
WRITE (6,27)
WRITE (6,28)
50 CONTINUE
*
*Print concentration profile headings
*
T=O.
TAUPR = KLC*CF*(TIME*60.)/(PDA*QA)
IF (KPPR.NE.l) GO TO 60
WRITE (6,30)
WRITE (6,31) TIME
WRITE (6,32)
WRITE (6,33)
WRITE (6,34)
60 CONTINUE
*
*Set initial resin loading throughout the entire column
*
MT=NT+ 1
DO 100 M=I,MT
YNC(l,M)=YNO
YXC(I,M)=YXO
YCA(I,M)=YCO
YPA(l,M)=YPO
100 CONTINUE
*
* Calculate dimensionless program time limit
* based on inlet conditions (at Z=O)
*
TMAXC = QC*3.142*(DIN2.)**2.*Cm*FCR/(FR*CF*60.)
TMAXA = QA*3.142*(DIN2.)**2.*Cm*FAR/(FR*CF*60.)
TMAX = MAX(TMAXC,TMAXA)
TAUMAX = KLC*CF*(TMAX*60.)/(PDA*QA)
TMAX = TMAXl1440.
WRITE(6,222)
WRITE(6,223)TMAX
WRITE(6,224)
222 FORMAT(' Program run time is based on total resin capacity')
223 FORMAT(' and flow conditions. The program will run for',F12.1)
224 FORMAT(' days of column operation for the current conditions.')
*
* Initialize values prior to iterative loops
*
J= 1
JI(= 1
TAUTOT=O.
JFLAG= 0
XNC(JK,NT) = O.
KK= 1
KPRINT = 1000
SXADS=O.O
PARADS=O.O
COLADS=O.O
71
*
*
* Time stepping loop within which all column calculations are
* implemented, time is incremented and outlet concentrations checked
*
1 CONTINUE
IF (TAUTOT.GT.TAUMAX) GOTO 138
*
72
*
*
Correction oftime step value for AdamsBashforth Method
IF (J.EQ.4) THEN
JD= 1
ELSE
JD=J+ 1
ENDIF
* 
**
Set inlet liquid phase fractional concentrations for each
species in the matrix
* 
XNC(J,I) = CNF/CF
XXC(J,l) = CXF/CF
CHO = ClllI
COO = COIl
XCA(J,l) = CCF/CF
XPA(J,l) = CPF/CF
* 
* Loop to increment distance (bed length) at a fixed time
*
DO 400 K=I,NT
*
* Define bulk phase concentrations for subroutines
*
CXO = XXC(J,K)*CF
CNO = XNC(J,K)*CF
CCO = XCA(J,K)*CF
CPO = XPA(J,K)*CF
CCT2= CCO
CNT2= CNO
CXT2=CXO
CPT2 = CPO
CHI=CHO
COl = COO
YN = YNC(J,K)
YX = YXC(J,K)
YC = YCA(J,K)
yP = YPA(J,K)
*
*Integrate X using AdamsBashforth method
*
XXL = XXC(J,K)
XNL = XNC(J,K)
XPL = XPA(J,K)
XCL = XCA(J,K)
*
*Call subroutines to calculate RN, RX, CNI, CXI
*
IF (YX .LT. 1.0) THEN
IN=O
CALL CR (IN,CHO,CNO,CXO,DH,DN,DX,YN,YX,CNI,CXI,RN,RX)
XXI=CXI/CF
XNI= CNI/CF
ELSE
XXI = 1.0
XNI=O.O
RN=O.O
RX=O.O
ENDIF
*
IF (YP .LT. 1.0) THEN
IN= 1
CALL AR (IN,TKCO,TKPO,COO,CCO,CPO,DO,DC,DP,
1 YC,YP,CCI,CPI,RC,RP,a1,e1)
XCI = CCI/CF
XPI = CPI/CF
ELSE
XPI = 1.0
XCI = 0.0
RC = 0.0
RP=O.O
ENDIF
*
*Evaluate the particle rate of exchange
*
RATEN = 6.*RN*(XNL  XNI)*KLNIKLC*PDAlPDC
RATEX = 6.*RX*(XXL  XXI)*KLXlKLC*PDAlPDC
RATEC = 6.*RC*(XCLXCI)
RATEP = 6.*RP*(XPL  XPI)*KLPIKLC
*
*First step calculation across the bed inlet
*
73
IF (K .EQ. 1) THEN
RATN(J,I) = RATEN
RATX(J,l) = RATEX
RATC(J,l) = RATEC
RATP(J,l) = RATEP
*
*Calculate next time particle loading
*
YNC(JD,l) = YNC(J,l)+TAU*RATN(J,l)*QA/QC
YXC(JD,l) = YXC(J,l)+TAU*RATX(J,l)*QA/QC
YCA(JD,l) = YCA(J,l)+TAU*RATC(J,1)
YPA(JD,I) = YPA(J,l)+TAU*RATP(J,I)
*
*Check that values are within bounds
*
IF(YXC(JD,l)+YNC(JD,l) .GT. 1.0) THEN
CALL CORRECT(YXC(JD,l),YNC(JD,l),YXC(J,l),YNC(J,I),TAU,
RATEX,RATEN)
RATX(J,I) = RATEX*QC/QA
RATN(J,l) = RATEN*QC/QA
ENDIF
IF(YCA(JD,l)+YPA(JD,1) .GT. 1.0)THEN
CALL CORRECT(YCA(JD,l),YPA(JD,l),YCA(J,l),YPA(J,l),TAU,
1 RATEC,RATEP)
RATC(J,l) = RATEC
RATP(J,l) = RATEP
ENDIF
ENDIF
* 
* Impliment implicit portion ofthe GEARS BACKWARD DIFFERENCE method
* from the previous function values. For the first three steps
* use fourthorder Runge Kutta
* 
IF (KLE.3) THEN
F1N=XI*6.*RN*(XNC(J,K)XNI)*FCR*KLN/KLC
F2N=XI*6.*RN*«XNC(J,K)+F1N/2.)XNI)*FCR*KLN/KLC
F3N=XI*6.*RN*«XNC(J,K)+F2N/2.)XNI)*FCR*KLN/KLC
F4N=XI*6.*RN*(XNC(J,K)+F3NXNI)*FCR*KLN/KLC
XNC(J,K+l) = XNC(J,K)  (FIN+2.*F2N+2.*F3N+F4N)/6.
FIX=XI*6.*RX*(XXC(J,K)XXI)*FCR*KLX/KLC
F2X=XI*6.*RX*«XXC(J,K)+FIX/2.)XXI)*FCR*KLX/KLC
F3X=XI*6.*RX*«XXC(J,K)+F2X/2.)XXI)*FCR*KLX/KLC
F4X=XI*6.*RX*(XXC(J,K)+F3XXXI)*FCR*KLX/KLC
XXC(J,K+l) = XXC(J,K)  (F1X+2.*F2X+2.*F3X+F4X)/6.
FIC=XI*6.*RC*(XCA(J,K)XCI)*FAR*PDC/PDA
14
75
F2C=XI*6.*RC*«XCA(J,K)+FIC/2.)XCI)*FAR*PDCIPDA
F3C=XI*6.*RC*«XCA(J,K)+F2C/2.)XCI)*FAR*PDCIPDA
F4C=XI*6.*RC*(XCA(J,K)+F3CXCI)*FAR*PDCIPDA
XCA(J,K+l) = XCA(J,K)  (FIC+2.*F2C+2.*F3C+F4C)/6.
FIP=XI*6.*RP*(XPA(J,K)XPI)*FAR*KLP/KLC*PDCIPDA
F2P=XI*6.*RP*((XPA(J,K)+FIP/2.)XPI)*FAR*KLP/KLC*PDCIPDA
F3P=XI*6.*RP*((XPA(J,K)+F2P/2. )XPI)*FAR*KLP/KLC*PDCIPDA
F4P=XI*6.*RP*(XPA(J,K)+F3PXPI)*FAR*KLP/KLC*PDCIPDA
XPA(J,K+I) = XPA(J,K)  (FIP+2.*F2P+2.*F3P+F4P)/6.
ELSE
COEN = 3./25.*XNC(J,K3)16./25.*XNC(J,K2)+
1 36./25.*XNC(J,KI )48./25.*XNC(J,K)
XNC(J,K+I) = (XI*12./25.)*FCR*RATN(J,K)COEN
COEX= 3./25.*XXC(J,K3)16./25.*XXC(J,K2)+
1 36./25.*XXC(J,Kl)48./25.*XXC(J,K)
XXC(J,K+1) = (XI*12./25.)*FCR*RATX(J,K)COEX
COEC= 3./25. *XCA(J,K3)16./25.*XCA(J,K2)+
1 36./25.*XCA(J,KI )48./25.*XCA(J,K)
XCA(J,K+1) = (XI*12./25.)*FAR*RATC(J,K)COEC
COEP= 3./25.*XPA(J,K3)16./25. *XPA(J,K2)+
1 36./25.*XPA(J,KI)48./25.*XPA(J,K)
XPA(J,K+1) =(XI*12./25.)*FAR*RATP(J,K)COEP
ENDIF
* 
* Determine concentrations for next distance step and recalculate
* bulk phase equilibria
* 
CXO = XXC(J,K+I) * CF
CNO = XNC(J,K+l) * CF
CCO = XCA(J,K+l) * CF
CPO = XPA(J,K+l) * CF
CHO = CHICNOCXO+CXT2+CNT2
COO = COlCCO+CCT2CPO+CPT2
* 
* SILICA EQUILffiRIUM
* 
SXADS = SXADS + A*(SX**B)
SX = SXSXADS
Bl = SX**2.0
Cl=Kl*SX
Dl = ABS(BI4.0*Cl)
IF(DI .LT. 0.0) Dl = 0.0
HS = ABS(SXSQRT(DI»/2.0
COLSIL = SX*KP/(ABS(KHKP»*(EXP(KP*TAUTOT)EXP(KH*TAUTOT»
SX = ABS(SXCOLSIL)
*
*
*
*
PARSIL = ABS(SXCOLSIL)
PARADS = PARADS+A*(PARSIL**B)
PARSIL = (PARSILPARADS)
COLADS = A*(COLSIL**B)
COLSIL = COLSILCOLADS
TOTADS=SXADS+PARADS+COLADS
TOTSIL = PARSIL + CPO + HS + COLSIL
Particulate Silica for this step is the amorphous silica for the next step
CALL EQB(DISS,CNO,CXO,CCO,CPO,COO,CHO,K2,HS)
YX = YXC(J,K+l)
YN = YNC(J,K+l)
YC = YCA(J,K+l)
VP = VPA(J,K+l)
76
* 
* Determine rates at constant xi for solution ofthe tau
* material balance
* 
IF (YX.LT.l.0) THEN
IN=O
CALL CR (IN,CHO,CNO,CXO,DH,DN,DX,YN,YX,CNI,CXI,RN,RX)
XXI=CXI/CF
XNI=CNI/CF
ELSE
XXI = 1.0
XNI=O.O
RN= 0.0
RX=O.O
ENDIF
*
IF (VP.LT.I.O) THEN
IN= 1
CALL AR(IN,TKCO,TKPO,COO,CCO,CPO,DO,DC,DP,
1 YC,VP,CCI,CPI,RC,RP,al,el)
XCI = CCIICF
XPI= CPIICF
ELSE
XPI = 1.0
XCI = 0.0
RP=O.O
RC=O.O
ENDIF
*
RATN(J,K+I) = 6.*RN*«XNC(J,K+I»  XNI)*KLNIKLC*PDNPDC
RATX(J,K+I) = 6.*RX*«XXC(J,K+I»  XXI)*KLXlKLC*PDNPDC
RATC(J,K+I) = 6.*RC*«XCA(J,K+I»XCI)
RATP(J,K+I) = 6.*RP*«XPA(J,K+l»XPI)*KLPIKLC + XPL*(YPOLYP)
* 
* Integrate Y using AdamsBashforth (calculate next particle loading)
* 
YNC(JD,K+I) = YNC(J,K+I) + TAU*RATN(J,K+I)*QAlQC
YXC(JD,K+I) = YXC(J,K+1) + TAU*RATX(J,K+1)*QAlQC
YCA(JD,K+I) = YCA(J,K+l) + TAU*RATC(J,K+I)
YPA(JD,K+l) = YPA(J,K+l) + TAU*RATP(J,K+I)
* 
* Check values within bounds
* 
IF «YNC(JD,K+l)+YXC(JD,K+I».GT.I.O) THEN
CALL CORRECT(YXC(JD,K+I),YNC(JD,K+I),YXC(J,K+I),YNC(J,K+I),TAU,
1 RATX(J,K+1),RATN(J,K+I»
RATX(J,K+I) = RATX(J,K+I)*QC/QA
RATN(J,K+l) = RATN(J,K+l)*QC/QA
ENDIF
IF «YPA(JD,K+I)+YCA(JD,K+l».GT.l.0) THEN
CALL CORRECT(YCA(JD,K+I),YPA(JD,K+I),YCA(J,K+ I),YPA(J,K+I),TAU,
1 RATC(J,K+I),RATP(J,K+ I»
ENDIF
* 
* Print concentration profiles
* 
IF (KPPR.NE.I) GO TO 350
IF (TAUTOT.LT.TAUPR) GO TO 350
JFLAG= 1
ZA = FLOAT(NT)
ZB = FLOAT(KI)
Z = ZB*CHT/ZA
KOUNT = KOUNT+1
IF (KOUNT.NE.(KOUNT/IO*IO» GOTO 350
* 
* Open data file
* 
cOPEN (6, FILE='NEWMUL6.DAT', STATUS='UNKNOWN')
c WRITE (6,35) Z,XNC(J,K),XXC(J,K),YNC(J,K),YXC(J,K)
c CLOSE (6)
350 CONTINUE
400 CONTINUE
* 
* Print breakthrough curves
77
* 
IF (KPBKNE.l) GOTO 450
* write(*,137)
* 137 format(' TIME(MIN)',8X,'SODIUM',8X,'AMINE',7X,'CHLORIDE',
* 1 7X,'2ND ANION,6X,'pH)
TAUTIM == TAUTOT*PDA*QN(KLC*CF*60.)/1440.
pH==14.+LOG1O(COO)
WRITE(6,139) TAUTIM,CNO,CXO,CCO,CPO,pH
139 FORMAT(Ix,F9.3,2x,EI2.5,4X,EI2.5,3x,EI2.5,3x,EI2.5,2x,F6.3)
* 
* Store every tenth iteration to the print file
* 
IF(KPRINT.NE. 1000) GOTO 500
KPRINT=O
* 
* Open data file
* 
* WRITE (*,29) TAUTIM,CNO,CXO,CCO,CPO,pH
450 IF(KPBK .NE. 2) GO TO 500
TAUTIM == TAUTOT*PDA*QN(KLC*CF*60.)
WRITE(6,145) SX,PARSIL,HS,CPO,TOTADS,TOTSIL
145 FORMAT(Ix,F9.3,2x,EI2.5,4X,EI2.5,3x,EI2.5,EI2.5,EI2.5)
* 
* Store every tenth iteration to the print file
* 
IF(KPRINT.NE.I000) GOTO 500
KPRINT=O
500 CONTINUE
KPRINT=KPRINT+l
KK=KK+l
JK=J
IF (J.EQ.4) THEN
J=1
ELSE
J= J+l
ENDIF
* 
* End ofloop, return to beginning and step in time
* 
IF (JFLAG.EQ.l) STOP
TAUTOT = TAUTOT + TAU
GOTOI
* 
78
* .....printout formats.....
* 
10 FORMAT (' MIXED BED SYSTEM PARAMETERS')
11 FORMAT ('    ')
12 FORMAT (' RESIN REGENERATION,3X,': YNO =',F7.5,6X,'YXO =',F7.5)
86 FORMAT (25X,'YCO ==',F7.5,6X,'YPO ==',F7.5)
13 FORMAT (' RESIN PROPERTIES',8X,'PDC ==',F6.4,'em',5x,'PDA =',F6.4,
1 'em',7X,'VD =',F5.3)
14 FORMAT (' RESIN CONSTANTS(eq/l): QC =',F5.3,3X,'QA =',F5.3,3X,
1 'FCR =',F5.3,3X,'FAR =',F5.3)
15 FORMAT (25X,'FR =',F9.1,' ml/s',2X,'DIA =',F6.1,
1 ' em'"4X 'em =',F6.1,' em')
16 FORMAT (25X,'DEN =',F6.4,'g/ml',5X,'TEMP ==',F4.1,' C')
80 FORMAT (' DIFFUSNITIES :')
81 FORMAT (25X,'DX =',EI0.4,2X,'DN =',EI0.4,2X,'DH =',EI0.4)
82 FORMAT (25X,'DO =',EI0.4,2X,'DC =',EI0.4,2X,'DP =',EI0.4)
17 FORMAT (' INTEGRATION INCREMENTS: TAU =',F7.5,8X,'XI =',F7.5,
1 6X,'NT =',16)
18 FORMAT (' ')
19 FORMAT (' CALCULATED PARAMETERS')
20 FORMAT ('   ')
22 FORMAT (' TRANSFER COEFFICIENTS: REC =',E9.4,' KLN =',E9.4)
23 FORMAT (' SUPERFICIAL VELOCIlY:',2X,'VS =',F7.3,' em!s',2X,
1 'VISCOSIlY =',F7.5,' cp',' CF =',FIO.8)
88 FORMAT (25X,'KLX =',E9.4,' KLA =',E9.4)
89 FORMAT (' INPUT CONCENTRATIONS :',2X,'CNF =',F9.7,6X,'CXF =',F9.7)
90 FORMAT (25X,'CCF =',F9.7,6X,'CPF =',FIO.8,3X,'pH =',F5.2)
91 FORMAT (' MODEL PARAMETERS : al=',F5.3,2X,'el=',F4.3,4X,
1 'A=',F4.2,4X,'B=',F4.2)
92 FORMAT (25X,'XLAMP == ',F5.2,6X,' SX == ',F9.7)
* 93 FORMAT (25X,'TKNH = ',F5.2,8X,' TKXH =',F5.2)
94 FORMAT (25X,'TKCO =',F5.2,8X,' TKPO =',F5.2)
95 FORMAT (25X,'KP =',F5.2,IIX,'KH = ',F5.2)
24 FORMAT (' ')
25 FORMAT (' BREAKTHROUGH CURVE RESULTS:')
26 FORMAT ('   ')
27 FORMAT (' ',5X,'T(MIN)',6X,'SODIUM',6X,'2ND CAT,7X,'CHLORIDE',4X,
1 'SILICA',7X,'pH)
28 FORMAT (' ')
* 29 FORMAT (' ',5(4X,E8.3),5X,F4.2)
30 FORMAT (' ')
31 FORMAT (' CONCENTRATION PROFILES AFTER ',F5. 0,' MINUTES')
32 FORMAT (' ')
33 FORMAT (' ',5X,'Z',7X,'XNC',7X,'XXC',7X,'YNC',
1 7X,'YCA')
34 FORMAT (' ')
35 FORMAT (' ',6(2X,E8.3»
79
CLOSE (6)
138 STOP
END
*
SUBROUTINE CR (IN,CHO,CNO,CXO,DH,DN,DX,YN,YX,CNI,CXI,RN,RX)
**
Subroutine to calculate cation rates for ternary exchange
*
IMPLICIT REAL*8 (AH,OZ)
INTEGER IN
IF (IN.NE.l) THEN
TKNH== 1.5
TKXH==2.4
TKNX = TKNH/TKXH
ELSE
TKNH= 15.
TKXH=4.6
TKNX == TKNH/TKXH
ENDIF
AH==DHlDN
AX==DX/DN
CTO == CNO+CXO+CHO
S == (CHO+CNO+CXO)*(AH*CHO+CNo+AX*CXO)
*
DENOMI == TKNH+(ITKNH)*YN+(TKNXTKNH)*YX
DENOM2 == AH*TKNH+(IAH*TKNH)*YN+(AX*TKNXAH*TKNH)*YX
**
Calculate Interfacial Concentrations
*
CNI == YN*(S/DENOMI/DENOM2)**0.5
IF (CNI.LE.O.O) CNI==O.O
CXI == CNI*TKNX*YXNN
IF (CXI.LE.O.O) CXI==O.O
CID == CNI*TKNH*(IYNYX)NN
IF (CID.LE.O.O) CID==O.O
CTI == CNI+CID+CXI
CNR == CNI/CNO
CXR == CXIICXO
CTR == CTIICTO
BBB == 1.+CTR
**
Calculate Ternary Effective Diffusivities
*
CCC == CNO  CNI
80
IF (ABS(CCC).GE.(CNO/IOOO.» GOTO 57
DENN=O.O
GO TO 58
57 DENN = 2.*(CTR*CNRl.)
BBN= l.CNR
DENN = DENN/(BBN*BBB)
58 CCX = CXOCXI
IF (ABS(CCX).GE.(CXO/IOOO.» GOTO 59
DEX=O.O
GOT061
59 DEX = 2.*(CTR*CXRl.)
BBX = 1.CXR
DEX = DEX/(BBX*BBB)
61 CONTINUE
*
* Calculate Ri's for components
*
EPN= 2./3.
RN = (ABS(DENN»**(EPN)
RX = (ABS(DEX»**(EPN)
*
RETURN
END
*
81
*
*
*
*
*
SUBROUTINE AR (IN,TKCO,TKPO,COO,CCO,CPO,DO,DC,DP,
1 YC,YP,CCI,CPI,RC,RP,al,el)
Subroutine to calculate anion rates for ternary exchange
IMPLICIT REAL*8 (AH,QZ)
INTEGER IN
IF (IN.NE.l) THEN
TKNH= 1.5DO
TKXH=2.4DO
TKNX = TKNH/TKXH
ELSE
TKCP = TKCO/TKPO
ENDIF
AO=DOIDC
AP=DPIDC
CTO = CCO+CPO+COO
S = (COO+CCO+CPO)*(AO*COO+CCO+AP*CPO)
DENOMI = TKCP+(1TKCP)*YC+(TKCPTKCO)*YP
DENOM2 = AO*TKCP+(IAO*TKCP)*YC+(AP*TKCPAO*TKCO)*YP
CCI = YC*(S/DENOMI/DEN0M2)**O.5
IF(CCI .LE. 0.0) CCI=O.O
* CPI = CCI*TKCP*YPNN
CPI = «l/al)*YP)**(I/el)
IF (CPI .LE. 0.0) CPI=O.O
COl = CCI*TKCO*(1YCYP)NC
IF(COI .LE. 0.0) COI=O.O
CTI = CCI+COI+CPI
CCR = CCI/CCO
CPR = CPI/CPO
CTR = CTI/CTO
BBB = 1.+CTR
*
* Calculate Ternary Effective Diffusivities
*
CCP = CPOCPI
IF (ABS(CCP).GE.(CPO/I000.» GOTO 62
DEP=O.O
GOT063
62 DEP = 2.*(CTR*CPRl.)
BBP = I.CPR
DEP = DEP/(BBP*BBB)
63 CCC = CCoCCI
IF (ABS(CCC).GE.(CCO/IOOO.» GOTO 64
DEC = 0.0
GOT065
64 DEC = 2.*(CTR*CCRl.)
BBC = l.CCR
DEC = DEC/(BBC*BBB)
65 CONTINUE
*
* CALCULATE Rits FOR SILICA and CHLORIDE
*
EPN = 2./3.
RP = (ABS(DEP»**(EPN)
RC = (ABS(DEC»**(EPN)
*
RETURN
END
*
82
***
SUBROUTINE EQB(DISS,CNO,CXO,CCO,CPO,COO,CHO,K2,HS)
Subroutine to calculate bulk phase concentrations based on silica equilibrium
83
IMPLICIT REAL*8 (AH,OZ)
REAL Kl, K2
*
B2 = (CNO+CXoCCO)
C2 = DISS*(1.0+K2*HS)
B3 = SQRT(B2**2.+4.*C2)
CHO = ABS(B2+B3)/2.0
COO = DISS/CHO
* pH = 14.+logI0(COO)
* IF (pH .LT. 9.0) THEN
* CPO=CPO
* ELSE
* CPO = K2*HS*COO
* ENDIF
* IF( pH .GT. 11.0) THEN
* CPOO= 20000.*CPO*COO
* CPO = CPO CPOO
* ELSE
* CPO = CPO
* ENDIF
RETURN
END
*
****
SUBROUTINE CORRECT(YCA,VPA,YCAOLD,VPAOLD,TAU,
1 RATEC,RATEP)
subroutine to correct the bulk concentration when the
calculated loading exceeds unity on the resin
IMPLICIT REAL*8 (AH,OZ)
YVY = YCA+VPAl.0
IF(YCA.GE. VPA)THEN
IF(YCA .GT. 1.0) THEN
YCA = 0.999999
VPA = 0.0000005
GOTO 420
ELSE
VPA = VPAYVY
IF(VPA .LE. 0.0) THEN
VPA = 0.0000005
YCA = 0.999999
ENDIF
ENDIF
IF(VPA .GT. 1.0) THEN
VPA = 0.999999
YCA == 0.0000005
GOTO 420
ELSE
YCA==YCAYVY
IF(YCA .LE. 0.0) THEN
YCA == 0.0000005
YPA ==0.999999
ENDIF
ENDIF
ENDIF
420 RATEP == (YPAYPAOLD)/TAU
RATEC == (YCAYCAOLD)/TAU
RETURN
END
D
84
VITA
Sharma Pamarthy
Candidate for the Degree of
Master of Science
Thesis:
Major Field:
Biographical:
DEVELOPMENT OF A COLUMN MODEL TO PREDICT
SILICA BREAKTHROUGH FOR MIXED BED ION
EXCHANGER
Chemical Engineering
Personal Data: Born in Kottagudem, AP, India, November 12, 1971, the son of
Lakshmi and Lakshmi Narayana Pamarthy.
Educational: Graduated from Chaithanya Kalasala Jr. College, Hyderabad,
AP, India, in May 1988; received Bachelor of Technology Degree in
Chemical Engineering from Osmania University in May 1992; completed
requirements for the Master of Science degree with a major in Chemical
Engineering at Oklahoma State University in May, 1996.
Experience: Employed as a teaching assistant, School of Chemical
Engineering, Oklahoma State University, January 1993 to May 1993.
Employed as a research assistant., School of Chemical Engineering,
Oklahoma State University, June 1993 to August 1994. Employed
as a teaching assistant, School of Chemical Engineering, Oklahoma
State University, August 1994 to December 1994. Currently employed as
a Manufacturing Technologist, Chemical Vapor Deposition, Applied Materials,
Austin, Texas.