INFLUENCE OF DISPERSION, EXCLUSION, AND IIETATIETICAL SORPTION
ON THE TRANSPORT OF INORGANIC SOLUTES IN A CALCILISATURATED
POROUS IHEDIUM
By
NARAINE PERSAUD
A DISSERTATION PRESSETED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLIIENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1978
my ^arientS
ACKNOWLEDGEMENTS
The author wishes to express his sincere gratitude to Dr.
J. M. Davidson, chairperson of his supervisory committee, for the
unique and conscientious guidance and help provided throughout the
duration of this study.
The helpful suggestions made by members of the committee, viz.
Dr. J. G. A. Fiskell, Dr. L. Hammond and Dr. S. J. Locascio, during
their review of this manuscript are highly appreciated. The interest
shown by Dr. R. S. Mansell who was also a member of the co nittee until
he left on sabbatical leave is also highly esteemed.
A special word of thanks is reserved for Dr. P. S. C. Rao and
IMr. Ron Jessup for the useful advice given during many discussions on
diverse subjects pertinent to this study.
The cheerful companionship of Mr. Rick Janka did much to relieve
the tedium of the long hours spent in the laboratory.
The financial assistance provided by the Soil Science Department
is gratefully acknowledged.
Superlatives fail to express the author's deep feelings towards
his wife Savi whose constant encouragement and support were sin qua
non to the successful completion of this study.
TABLE OF CONTENTS
Page
ACKNOWLEDEMENTS . . . . . . . . iii
LIST OF FIGURES . . .. .. . . . . . . .. . vi
ABSTRACT . . .. . . . . . . . . . . ix
INTRODUCTION . . . .. .. . . . . . .. . 1
CHAPTER
1 A REVIEW OF PERTINENT CONCEPTS ON SOLUTE TRANSPORT DURING
MISCIBLE DISPLACEIIENT IN DISCRETE POROUS 11EDIA AND ON
INORGANIC ION EXCHANGE EQUILIBRIA AND KINETICS . . . 3
Miscible Displacement Processes in Discrete
Porous Media . . . . . .... . . ... 3
Dispersion of Solutes During Miscible Displacement
in Discrete Porous MIedia . . . . .... .. 4
The ConvectiveDispersion Equation Including
Assumptions, Auxiliary Conditions, Analysis,
and Limitations .. ... .. . .. .. ... .. 5
Extension of the Continuum Mlass Balance Approach
for Single Interacting Solutes . . . . .. 11
Equilibria and Kinetics of Inorganic Ion Exchange
Adsorption . . . . . . . . . . 15
2 SYNTHESIS OF PERTINENT THEORY ON INORGANIC ION EXCHANGE
AND TRANSPORT PROCESSES. . . . . . . . . ... 23
Thermodynamic Conceptualization of IonExchange
Processes . . . . . . . . . . 23
Disposition of Charged Species in Solution/
Exchanger Systems at Equilibrium . . . . .. 24
Physical Basis for Exchanger "Selectivity" ..... 28
Concepts on Inorganic Ion Transport During
Miscible Displacement. . . . ... . . 31
3 SOLUTION OF TIE CONVECTIVEDISPERSION EQUATION FOR A "PULSE
INPUT" AND ITS EXTENSION FOR A SOLUTE FOLLOWING A LINEAR
ISOTIIERII . .. .. .. . ...... . . . 332
4 EXPERIMENTAL OBJECTIVES, MATERIALS AND METHODS . . . 36
CHAPTER
5 RESULTS AND DISCUSSION OF STUDIES ON EXCHANGE EQUILIBRIA
AND TRANSPORT OF Na Ll ^5Ca2+ AND C]L. . . . .. 42
Exchange Adsorption Isotherns. ... ...... . 42
Miscible Displacement Experiments with Na .... 51
Miscible Displacement Experiments with Li.+ .... 70
Miscible Displacenent Experiments with 45Ca2. . 83
6 SUMPIARY AND CONCLUSIONS . . . . . . . . 87
LITERATURE CITED ......................... .. . 91
BIOGRAPHICAL SKETCH .................. ...... 97
LIST OF FIGURES
Figure Pae
L Schematic of the flow system. . . . . . . 39
2 Exchange adsorption isotherms for Na+ in 0.05 II, 0.02
M, and 0.005 M Ca(NO3)2. ............... 43
3 Exchange adsorption isotherms for Li in 0.05 M, 0.02
M, and 0.005 M Ca(NO3)2 ................ 44
4 Dependence of the isotherm K values for Na and Li on
the concentration (C) of Ca in the equilibrium solu
tion. . . . . . ... ... . . ...... 45
45 2+
5 Exchange adsorption isotherms for Ca2+ in 0.075 I1
and 0.05 11 Ca(O3)2 ................... 49
6 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 1 Ca(NO )2 at a porewater velocity be
tween 14 and 15 cm/r. . . . . . . . 52
7 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.05 M Ca(NO )2 at a porewater velocity
between 7 and 8 cm/r . . . . . . . . 53
+ 
8 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 11 Ca(NO3)2 at a porewater velocity
between 1 and 2 cm/hr . . . . . . . . 54
9 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.02 H Ca(NO ), at a porewater velocity
between 14 and 15 cu/hr. . . . . . . . ... 55
10 Elution curves for a "pulse input" of Na Cl and
HTO in 0.02 I Ca(NO ) at a porewater velocity
between 7 and 8 cm/r. ................. 56
11 Elution curves for a "pulse input" of Na+, Cl, and
HTO in 0.02 M Ca(NO ) at a porewater velocity
between 1 and 2 cm/r ................ 57
LIST OF FIGURES (continued)
gurc Page
I? Elution curves for a "pulse input" of Na C , and
HTO in 0.005 M Ca(NO)2 at a porewater velocity
between 14 and 15 ch ................. 58
between 14 and 15 m/hr. . . . . . . . . 58
13 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 11 Ca(N3 )2 at a porewater velocity be
tween 7 and 8 cm/hr................... 59
14 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 M Ca(NO )2 at a porewater velocity
between 1 and 2 cm/hr. ............... 60
15 Linear dependence of the dispersion coefficient on
the porewater velocity. . . . . . . . .. 62
16 Repeated elution curves for "pulse inputs" of Na
and HTO in 0.05 H1 and 0.005 11 Ca(NO3 )2 in a short
column at porewater velocities between 7 and 8
cm/hr . . . . . . . . . . . . . 64
17 Elution curves for a "pulse input" of Na+, Cl and
HTO in 0.005 N Ca(NO3)2 in a column packed with
unsulphonated, macroporous, polystyrene beads .... 67
18 Elution curves for a "pulse input" of Na Cl and
HTO in 0.05 NI Ca(NO3)2 under steadystate unsaturated
water flow conditions. . . . . . . ... 68
19 Elution curves for a "pulse input" of Na Cl and
HTO in 0.02 M Ca(NO3)2 under steadystate unsaturated
water flow conditions. .. . ................. 69
20 Elution curves for a "pulse input" of Li and HTO
in 0.05 M Ca(NO3)2 at a porewater velocity between
7 and 8 cm/hr. . . . . . . . . . 71
21 Elution curves for a "pulse input" of Li+ and HTO
in 0.02 11 Ca(NO )2 at a porewater velocity between
7 and 8 cm/hr and without adjustment of the ionic
strength of the eluting solution. . . . . ... 72
22 Elution curves for a "pulse input" of Li and HTO
in 0.005 1t Ca(NO3)2 at a porewater velocity between
7 and 8 cm/hr and without adjustment of the ionic
strength of the eluting solution. . . . .... 73
LIST OF FIGURES (continued)
Figure Page
23 Elution curves for a "pulse input" of Li and HTO
in 0.02 M Ca(NO )2 at a porewater velocity between
7 and 8 cm/hr and with adjustment of the ionic
strength of the eluting solution. . . . . ... 75
24 Elution curves for a "pulse input" of Li and 11TO
in 0.005 M Ca(NO ), at a porewater velocity between
7 and 8 cm/hr and with adjustment of the ionic
strength of the eluting solution. . . . . ... 76
25 Elution curves for "pulse inputs" of Li at 50 to 55
ppm in 0.02 M and 0.01 M Ca(NO3)2 at porewater velo
cities between 7 and 8 cm/hr and without adjustment
of the ionic strength of the eluting solution. .. 78
26 Obseryations on the elution pattern for displacement
of Li in 0.005 M Ca(N03)2 by deionized H10 followed
by 0.005 1 Ca(NO3)2 .. . . .. . 80
27 Breakthrough curve for a "step input" of Na in
0.005 M Ca(N03)2. ................... 84
45 2+
28 Elution curves for "pulse inputs" of 11TO and Ca
in 0.075 11 and 0.05 _i Ca(N03)2 ........... 86
viii
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
INFLUENCE OF DISPERSION, EXCLUSION, AND METATHETICAL SORPTION
ON THE TRANSPORT OF INORGANIC SOLUTES IN A CALCIUMSATURATED
POROUS MEDIUM
By
Naraine Persaud
June 1978
Chairperson: J. M. Davidson
Major Department: Soil Science
Classical thermodynamic concepts were used to derive general rela
tionships to describe the equilibrium disposition of charged species in
aqueous contact with a cation exchanger. It was shown through these
relationships that anions would be excluded by the exchanger and solu
tion phase cations of the same type as those initially saturating the
exchanger influenced the adsorption of counterions. The latter result
was verified with exchange adsorption isotherms for Na+ and Li+ in solu
tions of 0.05 M, 0.02 M, and 0.005 M Ca(NO3)2 and a Casaturated ex
changer. These results illustrated that Na4 and Li adsorption de
creased with increasing concentrations of Ca2+ in the equilibrium solu
tion. Similar results were observed for the adsorption of 45Ca2+ in
solutions of 0.075 M and 0.05 M Ca(NO )2. The adsorption isotherms were
linear over the range of concentrations studied for a given Ca(NO3)2
cu.ncontrationl. The thermodynamic relations in conjunction with the
DLbyellucke] theory were used to predict the slopes for these isotherms.
The computed values agreed reasonably well with their experimental
counterparts. These results demonstrated inductively, that electro
static ionion interactions as conceived and quantified by the Dehye
Huckel theory, constituted the physical basis fox the observed equi
librium behaviour.
The consequences of the foregoing results on the transport be
haviour of Na+, Li+, 45Ca2+ and C" were investigated using miscible
displacement experiments in laboratory columns packed with the exchanger
material. These studies were conducted using pulse inputs of tracer
solutions containing various concentrations of Ca(NO3)2. Tritiated
water (HTO) was introduced with all tracer pulses to evaluate hydro
dynamic dispersion. An analytical solution to the convectivedispersion
mass transport equation for a reactive solute (linear adsorption iso
thern) was used to describe the experimental data.
As expected, the elution curves for Cl were displaced to the left
of those for HTO illustrating exclusion of Cl" by the exchanger. The
elution curves for Na, Li+ and 45Ca21 were displaced to the right of
those for HTO, the shift increasing with decreasing concentration of
Ca2+ in the tracer solution. Anomalous and unusual patterns were ob
served in some of the elution curves. The elution curves for Na+ in
0.005 M Ca(NO )2 and for 45Ca2+ in 0.05 N Ca(NO3)2 showed deviations
that simulated the effect of kinetic mass transfer processes. Inflec
tions were observed on the desorption side of the pulse elution data
for Li+ in 0.02 1! and 0.005 M Ca(NO )2 and for 45Ca2+ in 0.075 M
Ca(No ) 2. These elution curves were not described by the analytical
3 2
solution. Further experimentation demonstrated that these data could
be explained on the basis of differences between the parameters
characterizing exchange adsorption and desorption. A quantitative
treatment, based on the thermodynamic relations coupled with the
DebyeHuckel theory, showed that these differences were a direct con
sequence of the metathetical nature of the sorption process.
Some cursory observations were made regarding the transport be
havior of Na IITO and Cl under steadystate unsaturated water flow
conditions. The exclusion of Cl was again evident. Discrepancies
between the experimental and analytical curves for Na indicated that
portions of the waterunsaturated exchanger phase became either inacces
sible to Na+ or was accessible only by diffusion.
INTRODUCTION
The classic work of Thonpson and Way, reported during 18501852
on exchange reaction involving soil materials Created with inorganic
salt solutions, marked the beginning of systematic investigations of
ion exchange. Their experiments were conducted decades before Arrenhius
proposed the theory of electrolytic dissociation and many years before
the law of mass action was enunciated. There was, therefore, no exist
ing theoretical framework within which the two English agricultural
chemists could interpret their observations. The principle that sub
stances did not react except in a dissolved state was generally accepted
by chemists of their era. Whether the reactions reported were purely
chemical or physical became a debated question. This dichotomy in
views has still not been fully resolved.
For a period after its discovery, ion exchange remained of
academic interest until about 1907 when it was used to soften water.
Instability under nonneutral pH conditions and the low capacity of the
available natural and artificial siliceous ion exchangers provided the
impetus for the discovery and synthesis of organic ion exchangers. These
synthetic materials are now utilized in most industrial and laboratory
applications involving ion exchange. Extensive research concurrent
with their increasingly widespread use has resulted in the evolution
of more comprehensive and refined ion exchange theories.
The miscible displacement technique has in the past few decades
become an increasingly valuable experimental tool for studying
n'i:3icochcmical processes associated with solute transport in porous
nedia. Theoretical concepts on physicochemical interactions of partic
ular solutes have been successfully incorporated into classical mass
transport models. Comparisons between experimental and theoretical
results on solute transport have served to verity concepts and to
evaluate the influence of physical factors. The diversity of possible
physicochemical interactions of inorganic ions with various soil
components makes natural soil materials a complex medium in which to
study ion exchange. This difficulty can be circumvented somewhat by
the use of synthetic exchangers.
The primary objective of this study was to evaluate some theore
tical concepts on ion exchange equilibria and kinetics using the mis
cible displacement technique and a porous medium prepared from a syn
thetic organic ion exchanger.
CHAPTER 1
A REVIEW OF PERTINENT CONCEPTS ON SOLUTE TRANSPORT DURING MISCIBLE
DISPLACEMENT IN DISCRETE POROUS MEDIA AND ON INORGANIC ION EXCHANGE
EQUILIBRIA AND KINETICS
Miscible Displacement Processes in Discrete
Porous Media
Miscible displacement is the term used to describe a process
whereby one fluid is displaced by another, both fluids being miscible
in each other. This process occurs in a soil when a solution is dis
placed downwards by incoming rain or irrigation water or when sea water
displaces fresh water during drawdown in coastal aquifers. In the
petroleum industry, fluids miscible with crude oil are used to increase
the efficiency with which the crude oil is displaced from oil bearing
strata during secondary operations. Separation and recovery processes
in industrial packed towers or in chromatographic columns are other
examples of miscible displacement processes in porous media.
The transport behavior of materials dissolved in the displacing
or displaced fluids is influenced by the physical properties of the
medium (particle size, shape, and manner of packing), theological pro
perties of the fluids, and both the equilibria and rates of reactions
between the solute and solid matrix of the porous medium. Miscible
displacement experiments where there is no interaction between the
medium and the solute can be used to evaluate the effect of the first
two factors. The combined effect of the first two factors has been
grouped under the term "dispersion" and a considerable body of literature
has grown out of these studies. Excellent reviews on dispersion have
been presented by Fried and Combarnous (25) and Bear (6).
Dispersion of Solutes During Miscible
Displacement in Discrete Porous Media
Dispersion is the result of a physical mixing between displacing
and displaced fluids. Its effect can be studied by measuring the rela
tive concentration gradients between the displacing and displaced
fluids. Gradients are usually determined experimentally by withdrawing
samples at suitable time intervals at a fixed location in the flow region
under consideration. This technique and the relative concentration
versus time plots, termed breakthrough curves, have been described by
Nielsen and Biggar (53).
Dispersion in many practical situations involving solute transport,
especially in chromatographic and industrial separations, has provided
much of the initial motivation to quantify and describe the process.
Several theoretical approaches have been taken in this connection.
Attempts have been made to predict macroscopic dispersion effects based
on various geometrical models of the microstructure of the channels in
the porous matrix (23,33,64). These, however, require complex mathe
matical analyses, are rather academic in nature, and have not proven to
be useful. Limitations of this approach have been discussed by Scheidegger
(66). Analysis of solute transport based on a physical description of
the flow region into theoretical plates has proved quite useful in eval
uating dispersion in chromatographic separations (28,68).
An approach based on differential massbalance equations for the
solute in the fluid flow field has been successful and has gained wide
acceptance. Taylor (67) used this approach to study solute dispersion
during laminar flow in straight capillary tubes. Analysis of the
differential massbalance equation with appropriate initial and boundary
conditions yielded expressions which satisfactorily explained his experi
mental results. These results showed that dispersion was dependent on
the fluid velocity and the distance travelled by the displacing front.
During laminar flow of Newtonian fluids in straight tubes,a typical
parabolic velocity distribution is produced. This would not be the case
for a discrete porous medium with geometrically complex, tortuous channels
between the particles. For a porous medium it is assumed that the micro
scopic velocities between particles would fluctuate continuously in a
random manner about some mean value. This viewpoint led to a concept
that was developed almost simultaneously by Scheidegger (65) and
Danckwerts (16). Although they used somewhat different approaches in
their arguments, they agreed that dispersion could be regarded as a
quasidiffusion process. With this concept, the classical differential
ecnation for mass transport by simultaneous convection and diffusion
could be applied if an appropriate change was made in the physical
meaning attached to the diffusion coefficient. The validity of this
approach was demonstrated immediately by other investigators (4,19,52).
The ConvectiveDispersion Equation Including
Assumptions, Auxiliary Conditions, Analysis,
and Limitations
Hass balance considerations for transport by simultaneous diffu
sion and convection in a homogenous, isotropic porous medium yield
the classical convectivediffusion equation. In onedimension this
equation (39) is
c = { vC} (1)
at ax ax 
where. C(x,t) is concentration of solute in the fluid of the flow re
gion (M/L3), t is time (T), x is distance (L), D is the diffusion
coefficient (L2/T), and v is vector velocity of the fluid at any point
in the flow region (L/T).
Because of the pore size distribution, complex fluid velocity
fields arise within the channels of the porous medium. It is assumed,
however, that these velocity vectors are statistically distributed
about some fixed mean velocity vector v*. The probability distribu
tion of the deviations about this average is then considered by the
introduction of a quasidiffusion coefficient, D*, which characterizes
the unidirectional transport of the solute. This quasidiffusion co
efficient has been termed the "dispersion coefficient" by Scheidegger
(65). If it is also assumed that the dispersion coefficient is inde
pendent of concentration, then equation (1) reduces to
a= D* a2C *a (2)
at wx7 ax
As shown by Scheidegger (65) and in a somewhat more heuristic manner
by Rifai et al. (59), this equation can also be obtained by arguments
based entirely on probability calculus. This equation has been appro
priately called the "convectivedispersion" equation.
It is at once apparent from its description that v is the ex
perimentally measurable average porewater velocity of the fluid in
the porous medium. This is given by the relationship
v* =
X At (3)
where, Q is volume of fluid introduced (L3), A is the cross sectional
area of the flow region (L2), 0 is the fractional fluid filled poro
sity of tne medium (L /L ), and t is time (T).
It is also clear that D is related to D* in the following manner:
lim D* = D.
vD, * 0 (4)
For a fixed X = L, the solution of the convectivedispersion
equation would be C = C(t, D*, L, v*) where D*, L, and v* are para
meters. It is advantageous to formulate the differential equation
and auxiliary conditions in dimensionless form. The problem then
becomes independent of units and the parameters are reduced to dimen
sionless groupings. In addition, interactions among the parameters
become more obvious. The dimensionless variables are C* = C
Co
X = and T = which, when substituted into equation (2) give
L
ac* D* a2C* ac*
T v*L X .' (5)
D*
The reciprocal of the dimensionless grouping L has been termed
the Peclet number, usually denoted by P, and appears as a single
parameter in solutions of the transformed equation. The dimension
less time T corresponds physically to the number of fluidfilled
void volumes introduced into the flow region.
The convectivedispersion equation in either form [equation
(2) or (5)] has been analyzed for various sets of boundary and ini
tial conditions appropriate to different physical situations. These
have been surmarized by Nir and Gershon (54). The analyses are based
on the theory of equations in partial derivatives, a very difficult
and still incomplete branch of higher mathematics. Analytical solu
tions can be obtained with comparative ease for simple sets of boundary
and initial conditions; however, for complex conditions or for finite
domains of definition, one is forced to resort to numerical integration
methods.
Analytic solutions to equation (2) have been obtained for two
simple sets of conditions. These can be written as
I. C(x > 0, t = 0) = 0 C(x = 0, t > 0) = Co,
C(x = t > 0) = 0 (6)
II. C(x > 0, t = 0) = 0 C(x = 0, 0 < t < tl) = Co
C(x = 0, t > tl) = 0 C(x = t > 0) = 0 (7)
Physically, the first set of conditions corresponds to a solute
of concentration Co continuously displacing the solvent in a semi
infinite medium. In the second case, the same displacement is allowed
to proceed for a time t and for all times thereafter the displacing
solution is replaced by pure solvent. These two situations have been
termed miscible displacement with a "step input" and "pulse input,"
respectively.
The solution to equation (2) for a "step input" (59) is
given by
C(xt) I r xvt v x+vt
C = (erfc[ l + exp() erfc[2 (8)
where the superscripts have been omitted from D* and v".
Except near the inlet (x = 0) or for very small tines, unless
D is large, the second term in equation (8) is small and can be
neglected. Applying this condition and the identity orfc(x) =
1 erC(x), equation (8) reduces to
C(xt) ie vt (9)
c = Ii erf[ I] (9)
The cumulative distribution function P(::) for a standardized
Gaussian random variable of mean x and standard deviation o is given
by
P(x) = ( + erf[ 2 (10)
From the identity erf(x) = erf(x), it is apparent that equa
tion (9) represents, for a given value of x = L, a Gaussian cumula
tive distribution function of mean L and standard deviation '2Dt.
The solution for the "pulse input" is obtained by shifting the
solution for the "step input" by the time width of the pulse and sub
tracting this from the original solution. Applying this rather in
tuitive procedure to equation (9) gives
c(x t) 1 xv(tt xvt
Co erf[ U(ttl)erf[X 2v ]} (11)
Co1
where U(tt ) is the Heaviside unit function.
The previous equations have been used to analyze experimental
results from finite laboratory columns. As shown by Nir and Gershon
(54) only a small error is involved by assuming a finite rather than
a semiinfinite column.
The boundary conditions appropriate for a defined finite domain
are obtained by imposing the conservation of mass on the fluxes at the
boundaries. This results in the following boundary conditions for a
step input and a space domain of definition 0 < x < L
x= 0; t > 0; vCo = vCIx = D (12a)
x = L; t > 0; vCe = vCx= L D x = 12b)
where Ce is the exit concentration at a given time.
however, as argued by Danckwerts (16), if at x = L, xL 0
then CIx=L < C x
9 a=1 > 0. Hence x=L must be equal to zero otherwise a maximum
yy x=L hx
or a minimum exists in the interior of the column. The appropriate
condition at x = L must therefore be x= = 0.
ax xL
An approximate solution to the convectivedispersion equation
in dimensionless form for the above boundary conditions has been pre
sented by Brenner (14). In dimensionless form these conditions become
1 aC=3
S= 0; T > 0; C* = 1 (13a)
X = 1; T > 0; 1 = O (13b)
ehnler and Wilhelm (75) and Lindstrom et al. (46) have showm that,
for large values of P, the condition at x 0 reduces to C* = 1 for
t > 0.
In order to make the condition at x = 0 homogeneous, Brenner
defines C = where C is the initial concentration in the
fluid at t = 0. The initial concentration was taken to be zero
for his solution. Brenner tabulated numerical values of his solu
tion for a wide range of values of P/4. It was found that, with
increasing values of P, his solution asvyptotically approached
the simplified solution given by equation (9). He also discusses the
limiting behavior of his solution for P = 0 and P = . These corre
spond physically to complete instantaneous mixing and to no dispersion,
respectively. His solution has been used frequently for analysis of
experimental data from miscible displacement experiments (61).
Numerical procedures for integration of the convectivedispersion
equation are all based on methods whereby continuous systems are
matlhelatically reduced to equivalent discrete systems. Suitable algo
iU
rithms can then be developed and solved iteratively using high speed
computers. Serious and difficult mathematical questions of adequacy,
accuracy, convergence and stability arise for numerical solutions and
adequate answers are often not available. Procedures based on finite
differences and finite elements are discussed by Ames (1) and Finder
and Gray (55).
From the assumptions used in its development, the convective
dispersion equation represents an idealized conception of the miscible
displacement process. Its shortcomings were demonstrated early by
its inability to describe nonreactive solute behavior in unsaturated
porous media (53). Also, it fails to predict the effects of density
differences between displacing and displaced fluids (62). Its major
weakness probably lies in the assumption of a fluid continuum. The
presence of deadend pores and regions where the fluid is hydro
dynamically immobile would clearly lead to unpredicted results. Re
finements made to handle such cases (15,69) have resulted in better
agreement between predicted and experimental results.
Extension of the Continuum Mass
Balance Approach for Single Interacting
Solutes
The convectivedispersion equation can be extended to describe
the transport of a single interacting solute through a porous medium.
This requires additional terms to cover the time rate of change of
solute concentration in the fluid as a result of mass transfer be
tween the fluid and solid matrix of the medium. For a general case,
these terms constitute the difference between the instantaneous adsorp
tion and desorption rates. Equivalently and more conveniently, this
can bc expressed in terls of the instantaneous net time rate of
accumulation of the solute on the particles of the medium. The
concentration of the solute in the matrix expressed as mass of
solute/unit mass of matrix can be denoted by S(x,t). The instan
taneous net time rate of change of concentration in the fluid due
to mass transfer is then in which p is the dry bulk density
ov @t
of the matrix and 6 is the fractional waterfilled porosity. Con
sidering only one mass transfer process, the resulting differential
equation is
DC ..@2C v P_ p S
21 a22 2 _as
= D 7 x at (14)
The functional form of S(x,t) has profound consequences on the analysis
of the resulting differential equation. In general, the mass transfer
processes may involve purely diffusion kinetics, irreversible and
reversible chemical kinetics or both. Hence, it would be expected
that in the general case
cS
t = f[C, S, 2, (1' 2"' n diffusion,!1' 2" )chemical (15)
where T is temperature, and A ..Xn are parameters characterizing
diffusion and chemical kinetics (57). The case of mass transfer
involving only irreversible chemical kinetics has been analyzed by
Amundson (2).
Processes involving only reversible chemical kinetics can be
represented in the general case as
a = 1 (S C, i X2f2(S, C, T( (16)
whlre At and A2 are constants characterizing the kinetics of surp
Lion and desoiption, respectively.
An important simplification of chemical kinetics is based on the
assumption that, at a constant temperature, the processes occur fast
enough to insure that equilibrium is instantaneous. The functional
form of S(x,t) is then given by an equation which describes the equi
librium isotherm. This equation can be experimentally determined
or deduced from kinetic equations, if these are known, by setting
IS
= 0. In general, these isotherms can be represented as
S = f(C, 1.l ,n) (17)
where A 1... are parameters characterizing the equilibrium distri
bution of the solute between the fluid and solid matrix of the medium.
as aS Cin the d e
With this assumption, introducing = in the differen
at at at
tial equation gives
IC D* 12C v* IC
It RT R x (18)
where, R 1 +  S
0v 7
From this equation, some useful deductions can be made regarding
the transport behavior of the solute for certain generalized func
tional forms for the isotherm.
In the case where the isotherm is a singlevalued concave down
ward function S > 2 and thus c < for C1 < C2
Icc c Ri ReL 2
Similarly the reverse is true for C1 < C2 when an adsorption isotherm is
represented by a concave upward function. For a linear isotherm,
V*'.
i s independent of C. Typically, for a pulse input of a nonreactive
solute, dispersion causes the elution curves to have a gradual increase
in concentration from zero to a maximum on the front and falling again
to zero on the back side of the elution data. For a concave downward
isotherm, the concentration effect on the velocity would result in
a nullification of dispersion on the front and an enhancement on the
back side of the elution curve. This effect would not appear in the
case of a linear isotherm and the reverse would be true for a concave
upward isotherm. For a linear isotherm, the front and rear of the
elution curve would match exactly, but will not match for nonlinear
isotherms. If the assumption of instantaneous equilibrium was valid
for a particular situation, the shape of the elution curve can pro
vide a valuable indication of the type of adsorption isotherm in
volved in the displacement process. These arguments have been con
firmed by analysis of the differential equation for various functional
forms of the isotherm, including cases where it was not singlevalued
(45, 46,70).
As would be expected, the assumption of instantaneous equili
brium is not valid for all situations. Exact solutions of the
differential equation incorporating two important models of rever
sible chemical kinetics have been presented by Amundson(3). These
models are
klC k2S (19)
for linear adsorption kinetics, and
S klC(Smax S) k2S (20)
for "Langmuir" kinetics, where Smax is the saturation capacity.
Numerical procedures have increased the range of possible theoretical
models that can be used.
Except for a few cases (24, 69), less effort has been devoted
to studying diffusion kinetics. In soils, however, where particles
exist as aggregates and dead end pores are present, diffusion kine
tics may be important (69).
Investigation of the simultaneous transport of a number of
solutes which influence the mass transfer processes of each other
have not been attempted. In addition, comprehensive reviews have
not been made of the various equilibrium and kinetic models commonly
used by investigators. The equations derived from theory reflect
the effects of only those interactions for which the models have
been developed. Different theories may well attribute experimental
results to quite different causes. This constitutes the deficiency
in maay studies since it is often extremely difficult if not impossible
to independently measure the theoretical parameters introduced in
sorption kinetic models.
Equilibria and Kinetics of Inorganic
Ion Exchange Adsorption
Inorganic ion exchange was discovered more than a century ago.
For soils, ionexchange properties were traced to their alumino
silicate fractions (73,74). For many silicates, only ions from the
exposed surface layers are in a position to exchange. Typically
grinding increases the exchange capacity of these minerals (41).
For others, especially the zeolites, the lattice structure is open,
permeated by waterfilled channels, and possess internally accessible
exchange sites. Industrial applications of these and other synthetic
aluminosilicates were limited by their low exchange capacity and
instability under acid or alkaline conditions. The recent intro
duction of synthetic organic ion exchangers with superior properties
led to widespread industrial and laboratory exploitation of ion
exchange and a resultant surge in research on ion exchange equilibria
and kinetics. With these exchangers, it was possible for the first
time to vary their properties systematically. Much of the theore
tical advances made were due to this fact.
Ion exchange equilibria and kinetics are fundamental to an
understanding of ion transport and have been studied intensively.
As yet, no single comprehensive theory exists to explain all of the
results involving ion exchange. Certain characteristics are normally
common to all ionexchange reactions. The reaction is usually revers
ible and always involves an equivalent exchange of ions. In addition,
the exchanger preferentially adsorbs one ion over another, a property
appropriately termed "selectivity."
Electrical doublelayer theory (71) advanced by Helmholtz and
modified by Couy and Stern as an explanation of the electrokinetic
properties of colloids has been utilized to explain the phenomena
associated with ion exchange by soil clays (7,22,37). This theory
adequately accounts for the preferential sorption of ions of higher
valence over those of lower valence, but fails to explain observed
exchanger selectivity among ions of equal valence. Bolt (8) attempted,
with limited success, to extend the double layer theory to account for
such behavior.
The overall ion exchange process can formally be represented as
a reverse ible chemical reaction. The exchange of ions 1 and J of
valence + zi and + zj respectively can be written as
zj i lil + zi T. I zj mizil + 'i Jlzjl
where the bar signifies the ion associated with the exchanger. As
demonstrated by Kerr (42), the ion in combination with the exchanger
could not be considered as a precipitate with unit activity, He
assumed that the combination behaved as a solid solution, a view that
was subsequently supported (5,11,72), and has gained wide acceptance
by investigators studying ion exchange.
A useful quantity, the selectivity coefficient, characterizing
the relative preference of the exchanger for the ions I and J was
obtained by applying the law of mass action without activity correc
tions. Thus
iC.zi C.j zi
K = l (21)
j zil Cilz j
where C represents molar concentrations.
Except when zi = zj, the numerical value of K depends on the
choice of concentration units. The total equivalent concentration
Co = z C + zj Cj of ions I and J in the solution and the corre
1i J J
spending quantity Qo = z C.+ z C associated with the exchanger
must remain unchanged throughout the reaction. Defining Xi = ziCi/C
and Xi = ziCi/Qo as the "equivalent ionic fractions" of the ion I
in solution and exchanger respectively, and similar quantities for
the ion J, yields
K jl Xj i ( l zil zil
3 Qo
xlzjl. xjlzil (22)
The selectivity coefficient as defined is thus dependent on the total
equivalent concentration of the exchanging ions in solution and the
capacity of the exchanger. In addition, as shown by Bonner and Bonner
and Payne (9,10), it is also a function of the extent of exchange and
assumes its highest value when the exchanger is completely saturated
with ion J. The selectivity coefficient is thus not a particularly
useful quantity for predicting ion exchange equilibria.
A thermodynanically rigorous application of the mass action law
results in an expression for the thermodynamic equilibrium constant.
This expression is
K* ailZJI ajziL
ailzj aj zi (23)
where "a" represents ion activity. Determination of K* requires
quantitative values for the activity coefficients of the exchanging
ions in solution and in combination with the exchanger. This con
stitutes the difficulty in using the classical thermodynamic approach.
The DebyeHuckel theory and its extensions to predict activities of
electrolytes in solutions are applicable for concentration ranges
below those normally associated with the solid solution concept.
This difficulty is not insurmountable,however, because methods based
on classical thermodynamics are available to determine the activity
coefficients of ions in combination with the exchanger and the
equilibrium constant (17,26). This treatment is of more theoretical
than practical interest because it requires numerous measurements
before the equilibrium constant can be determined (21,30).
Ion exchange as a Donnan system was introduced by Mattson and
Larsson (47) and probably represents the most powerful concept in
explaining ion exchange. It was mainly through the work of Baunan
(5), Boyd (11) and Glueckauf (27) that this important concept has
gained general acceptance. The quantitative principle involved in
the Donnan approach is essentially a generalization of the double
layer concept, which is of universal occurrence whenever electrical
charge is confined within a definite region of space. The Donnan
approach imposes the thermodynamic condition that all other novable
charges must adjust themselves accordingly to produce a minimum in
the free energy of the system at equilibrium. The Donnan treatment
thus represents a partial fusion of the electrostatic and thermo
dynamic aspects of the ionexchange phenomenon.
Another approach likens the ionexchange process to the phys
ical adsorption of gases. Both the Langnuir and Freundlich equa
tions have been used to describe data on ion exchange equilibria
(11,37). There is a general consensus that ion exchange processes
involve strong, long range electrostatic forces which were not con
sidered in the conceptual development of the Langnuir and Freundlich
equations. Only vague physical meanings can be attached to the para
meters obtained from the application of these equations, and there
fore, they have not provided nuch insight into the adsorption pro
cesses involved.
A novel and interesting simplified statistical approach has
been used by Jenny (38) and Davis (18) to derive general theoretical
equations for ionexchange equilibrium. Although the results appear
interesting, these concepts have not been used widely.
Although ion exchange can be represented formally as a chemical
reaction, the physical processes involved have little in common with
true chemical reactions. Evidence for this has appeared in equilibrium
studies (42,72) where the ion in combination with the exchanger was
treated as a solid solution. In addition, standard enthalpy changes
for ionexchange reactions are often less than two kilocalories per
mole which is typical of the orientation energies involved in dipole
dipole interactions. Such evidence indicates that ion exchange is
essentially a statistical redistribution of the exchanging ions be
tween the exchanger and the solution.
Further support for this concept was provided by the pioneering
ionexchange kinetic studies of Boyd, Adamson and Meyers (12). They
obtained close agreement between their experimental data on exchange
kiineics and predictions from theoretical equations based on the con
cept of ion exchange as a diffusion process. They hypothesized that
either diffusion through a stagnant film around the particle or
diffusion into the particle were rate controlling. Integrated rate
equations for both cases were obtained by application of Ficks' laws
for constant diffusion coefficients. The hypothetical stagnant film
was assumed to have a finite thickness and was regarded as similar to
the 'Nernst' film encountered in reactions at electrode surfaces. They
found that either one or both mechanisms were ratecontrolling and
depended upon the experimental conditions. Their findings were
immediately confirmed by other investigators (32,44,58). These
studies showed that, in general, particle diffusion kinetics were
favored for solution concentrations greater than 102 11,efficient
mixing (which reduces film thickness), large particle size and low diffu
sion coefficients for ions in the exchanger. Opposite conditions were
conducive to filmdiffusion kinetics. measurements of selfdiffusion
coefficients of cations in synthetic organic exchangers were made by
Boyd and Soldano (13). In general, these were an order of magnitude
less than the corresponding values in solution and decreased with
increasing valence.
A significant improvement in the analysis of the two step
diffusion concept was introduced by Helfferich and his coworkers
(34,36,56). They observed that Fick's first law did not take into
account electrokinetic forces involved in the interdiffusion of two
charged species. They introduced the more appropriate NernstPlanck
flux equation, which contained an additional electrical transference
term. This equation is
1* 3C z a10
J*  DC (24)
where J* is flux of any charged species, F is the Faraday constant
and 4 is electrical potential. Equation (24) was derived for diffu
sion of charged particles in an electrical field assuming ideal
systems and is widely used in the analysis of electrochemical reactions.
The requirement that electroneutrality be maintained at all points in
the system implies a rigid coupling of the concentrations and fluxes
of the exchanging ions. Thus, for two ions I and 3, electroneutrality
requires
Izil Ci + 2zj Cj = constant (25a)
and
ziJ + z J = 0. (25b)
These conditions allow the derivation of a single flux equation
for either I or J. This equation is given (36) by
i DiD. (zi2C. + z2C) aCi
zi2CiDi + z2CjDj ax (26)
An immediate observation from this equation is that the interdiffu
sion flux is dependent on the relative concentrations of the inter
diffusing ions. With vanishing concentration of either I or J
the interdiffusion flux is controlled by the diffusion coefficient
of the ion in the minority. Further analysis of the kinetic be
havior of ionexchange reactions by these investigators for both
film and particle diffusion (36) led to the conclusion that the
rates of forward and reverse exchange reactions were not equal.
This conclusion was confirmed experimentally (34,35).
Incorporation of the simple diffusionkinetics model based on
Fick's law into the differential masstransport equations pro
duces a mathematical problem of extreme complexity (40). As a
result simpler equations based on the linear diffusion concept
introduced by Glueckauf (29) have been utilized (63). As pointed
out by Helfferich in his comprehensive monograph on ion exchange
(35), for practical situations, the gain in accuracy does not warrant
the time and effort expended in solving the problem of greater com
plexity.
CHAPTER 2
SYNTHESIS OF PERTINENT THEORY ON
INORGANIC ION EXCHANGE AND TRANSPORT PROCESSES
Thermodynamic Conceptualization of
IonExchange Processes
A system is defined thermodynamically as a body or group of in
teracting bodies intended for separate study. Any physically homo
geneous body or set of identical homogeneous bodies is called a phase.
Phases are either pure or mixed, depending on whether they consist of a
single or several chemically individual species.
Systems are either homogeneous or heterogeneous depending on
whether they consist of a single or several phases. The existence of
physical boundaries (interfaces) and interphase regions are necessary
features of polyphase systems. A solution/exchanger system can
therefore be conceived thermodynamically as a heterogeneous system con
sisting of two mixed phases.
A heterogeneous system may exist either in an equilibrium or non
equilibrium state. In the former state all thermodynamic state
variables remain constant with time. If the system is nonequilibrium,
spontaneous phase interactions occur resulting in the establishment of
an equilibrium state, characterized by definite compositions of all the
phases. Interactions that do not involve the production of new phases
or new chemical compounds result in material or energy exchanges
across the interfaces. Such interactions are considered as physical
sorption processes and involve atomic and molecular interaction
energies distinct from those involved in chemical bonds. The term
adsorption refers to physical sorption in which the species trans
ferred becomes either concentrated at the interface or distributed in
the bulk of the phase. If instead of being transferred, the species
is displaced by interactive forces back into the same phase it is
termed negative adsorption or loosely as exclusion.
A unique property of exchanger phases in the presence of fixed
electrical charge sites, which may be either restricted to the ex
changer surface or distributed throughout its bulk volume. The
quantity of fixed charges, conveniently expressed as equivalents,
defines the absolute capacity of the exchanger. This property,
coupled with the restriction that electroneutrality be satisfied at
all points in either phase of the system, forms the basis for the
metathetical sorption phenomena in solution/exchanger systems. Were it
not for its fixed charges, the exchanger would lose its identity as a
distinct phase in the system at equilibrium. Ion exchange processes
occurring during equilibration of a solution/exchanger system can be
considered as physical adsorption, if it is hypothesised that no in
teractions occur involving the formation of chemical bonds and produc
tion of new chemical species or phases in the system.
Disposition of Charged Species in
Solution/Exchanger Systems at Equilibrium
For charged species in a heterogeneous system, a necessary con
dition at equilibrium is equality of the electrochemical potential of
each species in the various phases. The electrochemical potential n
for a species in a phase is defined (31) by
n = u + zF (27)
in which p is the chemical potential, z the valence, F the Faraday
con,;tant and 0 the inner potential of the phase, The electrochemical
potential can be conceived as the sum of the reversible chemical and
electrical work required to transfer a charged particle from infinity
to any point in the interior of the phase. If the particle is un
charged no electrical work is involved and n =.p.
Consider two ions I, J of valence zi, zj in the solution/
exchanger system at equilibrium. Then, ni = in and nj = ni, where the
bar signifies the exchanger phase. The chemical potential for a
species in a phase is given by
V = po + RT in a (28)
where p is the chemical potential in an arbitrary reference state, R
is the gas constant, T is absolute temperature and "a" is activity.
Substituting equation (28) into equation (27) and equating the
electrochemical potentials of I and J gives
S+ RT in ai + zi F + = + RT En ai + zi F (29a)
vi + RT in a. + z. F v = up + RT an a. + z. F i (29b)
The difference in the inner potential of the phases O i, at
equilibrium is invariable, and p = p., p = p'. Thus
RT T a; RT (29c)
$ = 7 n i= n (29C)
z.F i F a
and
1 n Zi = 1 n (29d)
zi a.i zj a
flultiplying by z. zj equation (29d) becomes
z an ni = z. inA (29e)
3 ai 1 a
( )zj= (d)zi (29f)
a. aj
Similar reasoning for an ion X of valence zx yields
(X)zi (a)zx and (Aj)zj ()zx (30)
x i x j
Insight into the usefulness of the above relations can be ob
tained by considering some specific cases.
Consider an exchanger with fixed negative charges and absolute
capacity Qo satisfied by J ions in equilibrium with a solution of an
electrolyte J X Electroneutrality requires
Zx zj
zjC = Qo + zxCx and zjCj = zxCx (31a)
where C represents the molar concentration. From above equation
(29c) shows that
RI
z.F a.
3 3
The potential difference 9 i is the equilibrium Donnan potential
across the interphase, and increases away from the solution/exchanger
interface. The interphase functions as a Donnan membrane in a ther
modynanic sense because it is impermeable to the fixed exchanger
charges. This macroscopic potential is immediately established and is
the Force preventing the net transfer of J ions out of the exchanger
and of X ions into the exchanger despite concentration differences that
exist between the two phases. Increasing concentrations of J ions in
solution causes a lowering of the Donnan potential while increasing
the exchanger capacity would result in larger potentials. It is ob
vious that lim (Ai) = 0. If a dilute solution Jx Xj is used for
equilibration then aj : C. and aj = Y jC, where y denotes the activity
confficient. Iroa equation (31a) it follows that
C = Qo + z Cx 9 (31b)
j zj zj
3 z
From above equation (30) is
x j
Substituting C. for a. and 4' for a gives
3 zj j3
(J J = () l zx)x (31c)
x 'Y Q0o
Since y Q is much larger than Cjzj, the concentration of X in the
exchanger is lower than in solution. This effect is greater the more
dilute the solution of J xXj., and is enhanced with an increasing
valence of the ion X. It is commonly called "Donnan exclusion."
If a salt I XXzi is now introduced into the system described
above, exchange of I and J occurs. At equilibrium equation (29f) shows
that
(9i)z ( zi
a. a.
Rearranging gives
aj) 'I" (a.i (32)
3 i 3 ^
If I is introduced in a trace quantity, then a. is approximately
constant. With this condition, application of Le Chatelier's Principle
to equation (32) shows that if the concentration of J is increased in
the equilibrium solution then ai decreases and viceversa. Thus, for
dilute solutions of I xXzi the presence of J ions in the solution
phase will lower the selectivity of the exchanger for I ions. This
effect will be greater the larger the difference between zj and zi.
From the general relation given by equation (29f)
(i)j= () zi
1i
Dividing throughout by (dif)zi results in
5j
(i)z (a i = 1. (33a)
ai a
Introducing activity coefficients gives
Cizj Y izj .
ci Ci i 1 (33b)
CiZj C.~i Yij I.
The term on the left side of equation (33b) is the definition
of the "selectivity coefficient" obtained by application of the mass
action law without activity corrections for the exchange of I and J.
Since the activity coefficients are functions of species concentration,
the selectivity coefficient depends on the experimental conditions.
Physical Basis for Exchanger "Selectivity"
The power of thermodynamics is its ability to produce general
relations among system variables without detailed knowledge of the
specific physical forces involved in the phase interactions. Those
are concealed in the thermodynamic activities of the system components.
The equilibrium composition of the phases is governed by the require
ment that the free energy of the system be minimized. A charged
species in a nonequilibrium solution/exchanger system can lower its
free energy by interactions with the exchanger, solvent, and other ions
in the system. Quantitative theories do not exist to predict exactly
the effect of such interactions, generally termed solvation processes,
on the thermodynamic properties of the phases. The sum total of all
interactions reveals itself in the observed selectivity property of
the exchanger. In effect, selectivity is a measure of the relative in
teractive effect of two exchanging ions on the thermodynamic properties
of the exchanger phase.
In an aqueous solution/exchanger system, electroneutrality re
quires that the fixed charge on the exchanger be satisfied, at all
times, by an equivalent quantity of charges of opposite sign. This
fact determines the minimum equivalent concentration of the intersti
tial solution of porous exchangers. If Qo represents the absolute
capacity in equivalents per gram, p the dry bulk density, and f the
fractional internal porosity of the exchanger particles, then pQ /f
equivalents/cm is the concentration of the interstitial solution. An
exchanger with Qo = 100 meq/100g, p = 1 g/cm3, and f = 0.5 gives a
concentration of 2 N for the interstitial solution. Similar concen
trations would occur in the interphase regions of exchangers with sur
face charge sites. In high capacity synthetic organic exchangers,
values as high as 10 N are encountered. Such concentration ranges are
beyond the scope of the DebyeHUckel theory for obtaining the thermo
dynamic effect of ionion interactions.
Ionsolvent effects can be evaluated by the Born theory. This
theory gives the solvation freeenergy per mole of ions in solution
as
G = NA(ze)2 (1 ) (34a)
2(r + 0.85)
where NA is Avogadro's number, z is the valence, r the crystal radius,
e the electron charge, e the dielectric constant, and 0.85 A an em
pirical correction factor. The change in free energy in transferring
one mole of I ions from solution to an exchanger in the J form due to
ionsolvent interactions and electrical work would be
G = (2( 0) + zi F( ) (34b)
1
Since the exchanger solution is more concentrated e < e and the
solvent interaction term is positive. Thus the transfer does not
occur spontaneously unless the second term, which is negative, has a
greater absolute value.
The transfer of I into the exchanger must be accompanied by an
equivalent transfer of zi/zj moles of J out of the exchanger. For
this process
AG = 2r .85) ( ) + zF( ) (34c)
2(rt + GM5) t t j
In this case the leading term is negative and the electrical work term
is positive. The total free energy change for the metathetical reac
tion is
AG = AG + AG. (34d)
When zi = zj, AG is negative if rj < ri. Thus, considering only ion
solvent interactions, an exchanger with fixed charges neutralized
by J has preference among ions of equal valence which would increase
in order of increasing crystal radius. In part, ionsolvent interac
tions serve to explain selectivity among ions of equal valence. When
zi i z., the electrical work term dominates the solvation term and
thus regardless of radius, the exchanger prefers the ion of higher
valence.
The concept of exchanger selectivity as competitive solvation
cannot be developed further because a complete understanding of these
processes is far from being realized. However, the above does
illustrate the complexity of the physical interactions involved in
solution/oxchanger systems,
Concepts on Inorganic Ion Transport
During Hiscible Displacement
Established concepts on ionexchange equilibria and kinetics
allow some qualitative deductions regarding the transport behavior
of inorganic ions during miscible displacements.
If instantaneous exchange equilibrium is assumed, the transport
behavior of an ion depends on the shape and characteristics of the
adsorption isotherm. These would rest heavily upon the selectivity
properties of the exchanger. Over a small range of very dilute con
centrations of the equilibrating solution phase, it can be expected
that the isotherms would be linear. In such ranges, for an exchanger
saturated with J ions, the presence of J ions in the equilibrating
solution would influence the sorption of another counterion in the
system. As a result, variations in the concentration of J ions in a
displacing solution of I ions would produce variations in the elution
tines for the ion I.
The twostep diffusion concept of exchange kinetics suggests that
for small diameter exchanger particles and trace concentrations of I
in the displacing solution, filmdiffusion kinetics may control mass
transfer of I. Since the film is considered as a stagnant hydro
dynamic boundary layer around the exchanger particle, its thickness
would be inversely influenced by flow velocity of the solution. Also,
as discussed in the foregoing review [equation (26)], mass transfer of
I by diffusion would be influenced by the presence of J ions. The
existence of such ratecontrolled diffusion processes would reflect
in deviations from the predicted equilibrium shapes of elution curves
for I.
CHAPTER 3
SOLUTION OF THE CONVECTIVEDISPERSION
EQUATION FOR A "PULSE INPUT" AND ITS EXTENSION
FOR A SOLUTE FOLLOWING A LINEAR ISOTHER1I
Although solutions to the convectivedispersion equation are
often quoted and used, detailed derivations of these solutions are not
usually presented. Details of an asymptotic solution to the convective
dispersion equation for "pulse input" boundary conditions are given
below for a nonreactive and a reactive solute (linear adsorption
isotherm).
The mathematical formulation of the problem is given by the
equation
ac C aC
t = x for 0 < x < and t > 0 (35)
with initial and boundary conditions
C(x,o) = 0, for 0 < x < m (36a)
C(0,t) = Co[U(t) U(t tl)], for t > 0 (36b)
C(x,t) = 0, for t > 0 (36c)
where U(t) is the Hieaviside unit function and t1 is the timewidth of
the pulse.
Let the Laplace transform of C(x,t) be denoted by u(x,s), The
above equation and auxiliary conditions under the Laplace transforma
tion become
su(x,s) C(x,0) = D v (37)
Dd dx
u(0,s) = (1 eStl). (38a)
u(x,s) = 0. (38b)
The characteristic equation of the ordinary differential equation is
Dm2 vm s = 0. (39a)
with roots v v+ 4Ds.
2D
Denoting v_ + 4Ds by R, the general solution is
2D
u(x,s) = Cle( + R)x + Ce R)x (39b)
Applying the above initial and boundary conditions
u(0,s) = C1 + C2 = (1 eStl). (39c)
u(x,s) = 0=: C1 = 0. (39d)
x '
Thus
u(x,s) = (l eStl)e( R)x (39e)
S S
vx vx
C e2 eR Ce etl eRx (39g)
s s
Consulting a table of Laplace transforms (60), transform pair #3.280 is
listed as
ea(s + b2 1/2 ab bt/2
a 2 erfc(2t 1/2
+ e erfc( + bt1/2
valid for real s >0
Since
e,= e sx = e + s)
2D
e 'D (40)
S v
letting a = and b the first term inverts to
[if 2/UD
vx vx t1/2 vx 1/2
C_ e2D e 2 erfc( t ) + eM erfc( + vt )}(41a)
S2/t 2 2/i 2/T
= {erfc( ) + e erfc( X )} (41b)
S2/t 2/Dt
The second term is inverted by noting that it is equal to the first
term x est By the shifting property of the Laplace transforms, if
L{f(t)} = f(s), and g(t) = f(t tl). U(t tl) then, L{g(t)}
= est f(s) The second term inverts to
C {erfc[x v(t tl)] + e erfc[X + v t tl) U(t tl) (42)
2 2(t t) 2D(t tl)
The second term in both inverse transforms is small except near the
inlet where x = 0 and for small values of t unless D is large. As a
result it can be ignored without introducing a serious error. The
solution reduces to
C /2{erfc( t) erfc [ (t U(t t) (43a)
Co 2/ Dt 2/D(t t1)
Using the identity erfc (x) = 1 erf (x), this can be written as
C = 1/2{erf[ v(t t)] U(t t) orf( Vt)} (43b)
C 2/D(t tl) 2/Dt
It is convenient to use a transformed variable 6 = which is
physically equivalent to the number of pore volumes, for experiments
with a fixed value of x = L. Setting x = L and dividing the arguments
of the error functions in equation (43b) top and bottom by L, the
solution given by equation (43b) transforms into
= 1/2{erf[l ] U(6 ) erf( I)} (43c)
Co 2(e e) 2 DP
SVL vL
For a solute following a linear isotherm, S = KC, and = K The
It at
differential equation becomes
S= a C C pK DC
a3t 2 v 9 wt (44a)
IC
Transposing L and rearranging
ac D Ba2 v ac
at R ax2 R ax (44b)
where R = 1 + p
0
v
For the same initial and boundary conditions it is obvious that the
solution to equation (44b) can be obtained by setting D = D/R and
v = v/R in the solution given by equation (43b), This yields after
vt
introduction of 8 = and rearrangement
= 1/2{erf[(R 8 + 61) ( 0)
o 2/D( )R(45)
erf[ ) 
2/ D6R
Equation (45) reduces to equation (43c) when R = 1. The former
equation can therefore be used to generate theoretical elution curves
for both reactive and nonreactive solutes, with given values of the
parameters v, L, D, R and 01. Equation (45) was obtained assuming a
semiinfinite space domain but it can be used to analyse displacement
experiments in finite columns. It has been shown (14, 46, 54) that
for large values of v and L no serious error is involved in using equa
tion (45) for this purpose.
CHAPTER 4
EXPERIMENTAL OBJECTIVES, MATERIALS AND METHODS
The primary objective of this study was to examine the transport
behavior of selected inorganic cations during steady saturated or un
saturated flow in a reactive porous medium. It was anticipated that
these results would provide insight into the nature of the metathetical
mass transfer processes.
From the onset, it was clear that soils were too heterogeneous to
study exchange processes in detail. Therefore, a mixture was prepared
with a synthetic exchange material and sand. Synthetic exchangers
were utilized previously in miscible displacement studies by Day and
Forsythe (20). In addition to providing the required homogeneity, the
overall exchange capacity could be controlled.
A rigid, analytical grade, iacroporous, granular (50 100 mesh),
organic exchange resin of a highly crosslinked sulphonated copolymer
of styrene with divinylbenzene was purchased (BioRad Laboratories).
This material is both thermally and chemically stable and has a cation
exchange capacity of 4.9 meq/g. In addition, special treatment during
polymerisation results in a low resistance to intraparticle mass trans
fer by diffusion. Preparation of synthetic organic exchangers in gen
eral has been described by lelfferich (35). Their general physical prop
erties have been reviewed by Heyers et al. (48) and the special physical
and chemical properties of the macroporous resins have been described
by Miller et al. (49,50). Because of its high exchange capacity, it
36
was necessary to dilute this resin with an inert material. Soil from
the I m 1.3 m horizon of a Lakeland sand (Typic Quartzipsamment)
was passed through a nest of sieves and the fractions retained on the
500, 250 and 105micron sieves were combined in the ratio of 25 : 50 :
25, respectively. This material was then treated with hydrogen peroxide
to destroy any organic matter present. The resin was treated repeatedly
with I N calcium acetate solution until no further pH change was observed,
and then packed into a plastic column and eluted with 1 N Ca(NO3 )
solution. This procedure was considered sufficient to Casaturate
the resin. The resin was then dried and sieved and the fraction between
200 and 105 microns combined with the sand to yield a computed exchange
capacity of 30 40 meq/100 g. This porous exchange material was used
in all subsequent studies.
Cations selected for investigation were Li+ and Na+ because of
their ease of detection at low concentrations by flame spectrophoto
metry. Calcium was chosen as the common ion because the Casaturated
exchanger was expected to exhibit low selectivity for Li and Na
resulting in low residence times. Because of this, longer columns
could be used. The exclusion of Cl by the exchanger was also studied
and a cursory investigation was made of the transport behavior of
45Ca2+
The columns to contain the exchange medium were prepared from a
single length of rubbercast plexiglass tubing with an internal dia
meter of 5.0 cm. The columns were designed to provide unsaturated
flow conditions. To achieve unsaturation, 3mm holes were drilled
in the walls of the column, and small pieces of wire gauze (less than
100micron mesh) were placed on the inside to retain the material in
the column. Porous endplates with an airentry pressure of 30 40
cm of water were used to retain the material tightly in the columns.
The dead volume of the end plates did not exceed 7 cm The holes in
the column were plugged during saturated flow studies. For unsaturated
flow studies the holes were not plugged and the column was sealed into
a 10cm diameter column. The outer column was pressurized to achieve
the desired unsaturated soilwater potential and the pressure was kept
constant with a bubble tower.
The solid matrix material was packed into the column under water
to insure complete water saturation. An adjustable peristaltic pump
was used to deliver solution at predetermined rates to the columns.
A fraction collector was used to sample the effluent at equal time
intervals. A schematic of the experimental apparatus is given in
Figure 1.
Molar stock solutions of Ca(NO0) NaC1, LiCI, LiNO and NaNO
were prepared. Tracer solutions were prepared by combining suitable
portions of the required stock solutions and making up to volume.
The concentration of Na+, Li+ and Cl in the tracer solutions were
kept between 80 85 ppm. Concentrations of the common ion (Ca2+)
used were 0.05 M, 0.02 M and 0.005 M. Tritiated water (HTO) was added
is a nonreactive tracer to evaluate hydrodynamic dispersion. The
transport of 45Ca2+ was studied using tracer solutions prepared by
dissolving one gram of 45CS04 in 2 liters of 0.075 It and 0.05 M
Ca(NO( )
32
Exchange isotherms for Na+ and Li+ in 0.05 1. 0.02 N and 0.005 M
Ca(:KO3)2 were determined over a 0 200 ppm concentration range. Ten
C) P
01 .0:
0. C
C U C
C) )
U 0
C) C)
4 C
C) 0
.01 .,.
440
C) 'C C)
0 0
.r00
U CC
C) '
CD CC.
'C, C
00 '
.0 CC
C)" CC
a0 C
CC )
H CC
0 x
C)Is .
00
C) C).C
0.
*s4 C
C) .
C) 0.
''C)"
CCC)C)
0.cCi
C)o rl
..C)0
rI C)
OUCCO
0.0
Fflo4
solutions in increments of 20 ppm were prepared by weighing out the
required amounts of 2000 ppm stock solutions and making it up to volume
with the appropriate Ca(NO3)2 solution. Tengram portions of the dry
solid matrix material, previously washed with deionized water, were
mixed with 10ml portions of each solution and shaken at frequent
intervals during a 12hour period. This time was shown to be sufficient
for equilibration. A sample of the supernatant was then withdrawn and
analyzed. The amount adsorbed was calculated from the concentration
difference between the sample and original solution. A similar tech
nique was used to determine exchange isotherms for Ca2+ in 0.075 M
and 0.05 M Ca(NO )2.
The 'ulse input' boundary condition was utilized in all miscible
displacement studies. The columns were leached with appropriate
tracerfree solutions of Ca(NO3)2 and then a pulse of the tracer solu
tion containing the cation plus chloride and HTO was introduced. This
pulse was subsequently eluted with the tracerfree Ca(NO3 )2 solution.
Changeover of solutions was achieved in approximately 2 minutes during
which time the outlet was sealed, and the front endplate and delivery
tubing were flushed and refilled with the new solution. The total
amount of solution introduced during a displacement was determined by
weight differences in the bottles containing the solutions. These
together with the measured time, were used in computing an average
porewater velocity.
Analysis for Li and Na in the effluent samples were made using
a Beckman model B flame spectrophotometer. Sets of standard solutions
for these analyses were prepared by accurate dilution of a 2000 ppm
41
stock with the appropriate Ca(NO )2 solutions. Analyses for Cl were
made using an "Orion" specificion electrode. Activity of HTO and
45Ca2+ was determined by liquid scintillation counting of lml samples,
in 10 ml of a commercial phosphor (Aquasol II).
The pore volume of the columns was determined by drying the material
in the column at the end of a series of displacements. The dispersion
coefficients were extracted from the breakthrough data for HTO using
a least squares curvefitting procedure and the simplified asymptotic
solution [equation (43c)]. These coefficients were then used to gen
erate analytical curves for the reactive solutes using the sorption
parameters from the adsorption isotherms.
CHAPTER 5
RESULTS AND DISCUSSION OF STUDIES ON EXCHANGE EQUILIBRIA
AND TRANSPORT OF Na+, Li+, 45Ca2+ AND Cl
Exchange Adsorption Isotherms
The adsorption isotherms for Na+ and Li+ in 0.05 i, 0.02 M and
0.005 M Ca(NO3)2 on the exchange mixture are given in Figures 2 and 3.
The adsorption isotherms were linear for both Na and Li over the 0 to
200 ppm concentration range. The slope of the adsorption isothern in
creased as the Ca2+ concentration in the equilibrating solution de
creased.
The data in Figures 2 and 3 were fitted by the least squares pro
cedure to the equation for a linear isotherm, S = K C. The resulting K
values for Na+ in 0.05 M, 0.02 M and 0.05 M Ca(NO3)2 were 0.1324,
0.2196 and 0.3770, respectively. The corresponding values for Li were
0.0822, 0.1122 and 0.1534. These K values increase in a nonlinear
fashion with decreasing Ca2+ concentration in the equilibrating solu
tion. As shown in Figure 4, an assumed general exponential relation
ship of the form K = Ko exp [P CCa] fitted the data reasonably well.
For Na the values of K and B were 0.3883 and 9.7 and for Li the
corresponding values were 0.1568 and 5.80. These values provide use
ful quantitative insight into the damping effect of the Ca2+ on the
adsorption of Na+ and Li .
From the theoretical considerations discussed previously [equation
(32)] for equilibrium conditions the following relationship holds:
/43
0
0
Co
\ (D
00
"4
0
o, O
0
00 00
E
o o
IC
\ t \\ 0 c
.0
RI
*ot
\\00 000 0
44
UCM
UC
0 CC
N 0 s
LO 0
03
2 M I.w
0 0 0
0
N N
(I)e II
4 tJ) c~
:3 5
.0.
Do, C
U
.0,J
C Q.
+ 0
U II
U .H
0 0
c3 
4,'
^ Oi
.004
c
0 G
I H
if
a0
C
0 0
+ 4
00
~rl U
0r
.0
0 H
00
0
00
0.0
crt
.0
wo
uEC(
c>
0
4,0
0.u
2 2
a4 L (46)
aCa aCa
For the concentrations used in this study, both aNa and aCa are accessi
ble from the extended DebyeHickel theory. This theory allows calcula
tion of the activity coefficient from the following relationship:
Az2
log y = (47)
1 2
where y is the activity coefficient, I = 2 EC.z. and is the ionic
i i
strength of the solution, A = 0.507 at 20 C, B = 0.328 at 20 C and a is
the ionsize parameter. The value of a is 6 for Li+ and Ca2+ and 4 for
Na (43). If only small quantities of Na are adsorbed in the exchanger
phase one can assume that aCa is constant. If it is further assumed
that yN remains constant, then CN = pp may be substituted into
equation (46) to give
a= Y S = K S (48a)
Na 1 Na
Ca Ca
Rearranging
YN = SNa (48b)
Ca Ca
or
S =  a C(48)
Na Na (48c)
CaCa
If K1 is known, the K value for each adsorption isotherm can be computed.
It is recognized that since I varies with the Na+ concentration in
the equilibrating solution, both yNa and aCa are not true constants;
however, their range of variation can be determined. For the experimental
isotherm with Na+ in 0.05 11 Ca(NO3)2 over the 0 200 ppm Na range,
a minimum calculated value of 0.15 M for I is obtained using the above
formula. From the measured data for this isotherm a maximum value of I
can be obtained. When the highest concentration of 200 pg/ml was
used, the measured equilibrium values were S = 22 pg/g and CNa = 178
pg/ml. Assuming that anions are excluded completely by the exchanger,
the concentration of ionic species in the solution phase was Na+ =
0.0077 1, Ca2+ = 0.0505 M, Cl = 0.0057 M and NO& = 0.10 M giving
1 = 0.161 M. The computed values for yNa for I = 0.15 M and I = 0.161 M
are 0.740 and 0.735, respectively. Corresponding values for iCa are
0.114 and 0.132. Thus, the values of yNa and /aa do not vary appre
ciably between the maximum and minimum I values. However, it is clear
that with decreasing concentration of Ca(NO3)2 in the equilibrating
solutions, the gap between the maximum and minimum values of yNa and
aCa does increase. As a realistic approximation, an average value of
I = 0.155 M can be used to compute values of yNa and aCa with the ex
tended DebyeHUckel formula. These calculations yield yNa = 0.7381 and
aCa = 0.1331. Utilizing these values and the experimental K value of
0.1324 for this isotherm, a value of K1 = YNa /K Ca can be calculated.
If the above reasoning and assumptions are correct, this value of K1
may be used to predict the experimental K values for the adsorption
isotherms using 0.02 M and 0.005 1 Ca(NO3)2. Similar arguments can be
advanced for the equilibrium isotherms of Li +
With the above approach, K values for the isotherms of Na+ in
0.02 11 and 0.005 N Ca(NO3)2 were predicted using a calculated K value
of 41.9 and mean I values of 0.064 M and 0.020 M. These predicted K
values were 0.201 and 0.347 which compares favorably with the measured
values of 0.2196 and 0.3770, respectively. Similar computations for
Li+ using K1 = 70.18 and mean I values of 0.079 I and 0.025 1,
yielded predicted K values for the Li+ adsorption isotherms in 0.02 M
and 0.05 M Ca(NO3)2 of 0,121 and 0.199 which compared reasonably well
with the measured values of 0.112 and 0.153.
Implicit in the use of the DebyeHiickel theory is the assumption
that the activities of the ions in the exchanger and solution phases
are predominantly the result of electrostatic ionion interactions.
The foregoing theoretical results confirm the validity of this assump
tion and underscores the importance of such interactions in ion
exchange equilibria.
The exchange adsorption isotherms for Ca2+ in 0.075 M and 0.05 M
45 2+
Ca(N03)2 are given in Figure 5. These were linear over the Ca2+ con
centrations used and illustrate a similar decrease in the K value with
an increase in 40Ca2+concentration in the equilibrating solution.
Assuming that 45Ca2+ and 40Ca2+ are indistinguishable, then y45Ca
74Ca and y45Ca = Y40Ca This implies that there are no differences in
their physical interactions to produce selectivity. With the previous
theoretical considerations [equation (33b)], and using a dagger to dis
45 2+ 40 2+
tinguish between properties of 4Ca2 and 4Ca2
CC 1 (49n)
whence
Ct (49b)
C T
At low concentrations of 4Ca2 both C and C can be regarded as con
stant.
Then
c* C(49e)
at = C c)
and
S = C t
C
(49d)
(0
U'
II0
V0 0
a
0
0
\ \0 o
o
0
Qo o
o o
0
0O ,
0 O\O 0 0
u
U
\ \ 0 "
\ \ "
in wh;ch K embraces C and all factors involved in adjustment of units.
Employing the same approach as that used for Na and Li+, it is possible
to compute K1 using the experimental K value of one adsorption isotherm.
This value can then be used to predict the K value of other isotherms.
This approach provides an evaluation of the validity of equation (49d).
The experimental K value of the isotherm for 45Ca2+ in 0.075 1 Ca(NO)2
is 0.764 which gives K1 = 0.057. The predicted K value of the isotherm
for 45Ca2+ in 0.05 M Ca(NO3)2 was 1.146 which compares favorably with
the measured value of 1.041.
45 2+
The exchange adsorption isotherms for 4Ca2 are of interest in
40 2+
explaining the basis for the effect of increasing Ca concentrations
on the K values. Consider specifically the exchanger in equilibrium
with 0.05 M Ca(NO3)2 solution. If a Donnan electrostatic potential
difference is set up across the interphase region, this essentially
equalizes the diffusion rates of Ca2+ ions into and out of the exchanger
allowing the continued existence of a higher concentration of Ca2+ in
the exchanger phase to maintain electroneutrality. Although macro
scopically the composition of either phase remains fixed, microscopic
exchange of Ca2+ ions continues to occur at a fixed rate across the
45 2+
phase boundary. If some fixed amount of 4Ca2+ ions is introduced in
45 2+
the solution, it is reasonable to expect that at equilibrium the 4Ca2
ions entering the exchanger depend solely on their relative abundance
to 40Ca2+ in the solution. Increasing the solution concentration of
Ca2+ to 0.075 1 would reduce the relative concentration of Ca2+ to
40C2+ on the exchanger and thus the macroscopic adsorption of Ca2
is decreased. It is therefore reasonable to assume that the ratio of
Ca2+ in the exchanger and Ca2+ in the solution is inversely propor
tional to the concentration of 40Ca2+ in the solution. Expressed
quantitatively, K.05/K.075 = 0.075/0.05 which is what was concluded
above [equation (49d)] using a different approach. The foregoing re
sults serve to give credence to the concept of ion exchange as a
Donnantype redistribution of ions between the solution and exchanger
phases.
It was not possible to determine the negative adsorption isotherms
for Cl in batch studies because the increases in C1 concentration were
too small to detect above random variations associated with the specific
ion electrode.
Miscible Displacement Experiments with Na'
Elution curves are presented in Figures 6 through 14 for a series
of input pulses containing Cl, HTO and Na+ in 0.05 M, 0.02 H1 or
0.005 M Ca(NO3)2,using three porewater velocities ranging from 1.5 to
15 cm/hr for each Ca(NO3)2 concentration. These displacements were
made using a 30.4cm long column. The medium was packed to a bulk
density (p) of 1.786 g/cm3 and had a saturated, fractional volumetric
water content, 6 of 0.344. Areas under the breakthrough curves in
Figures 6 through 14 were determined by trapezoid rule integration and
in each case indicated complete recovery of the material injected.
Thus, complete reversibility of the mass transfer processes was achieved
in these column studies.
It was not possible to maintain identical values of the three
porewater velocities for each concentration of Ca(NO3)2. However, in
no case did the measured porewater velocity deviate by more than 2% of
the mean values (14.69, 7.29 and 1.52 cm/hr) taken over the three
Ca(N03)2 concentrations, Dispersion coefficients were determined from
YrY
cP
.0 0Ii
'oo
I U
0
LO
,co
0 0 0
cUlu
II I I
.20
(JIZ
000 I u1
00I 0
~~0L~tC~C\J'
010,
.^
0
c i
So0
06 6
* >n >
i II .0 o
0. 0 0
[]
&, 0' 1"
.4 O
0.
r3
33
0)
Ii '1
z 
' o
441
__ I I I I 1 0 o
>r
n OI L
J
a0 ~ 000
o
0+
SIZ
000
u0
Ulu
00
ri II
''C
0
Lh
C
u)
o I
0
2
C;)
0
cy"J
LO
rr
0. 0
0
U g
LO
00
56
c
00
.00
0
CH
CI C
:4 OCC
0~
00
C'~ I 0C
0 0
044 0
OCJc 0C
C,
10.
0~
r C)X
0 00
00 u
uu
ii II 4
a 4 0
9qcr( y C cs 'O
Cu
4C
g
Cu
m
0
O Cu
00.
(40
Cu
E0
0I i
VI Cu
Cu
0
0r0
0
Ct
*H
rt
0 '
El
~ 0
C ^
rl
Cu 0
U CO
U
0i
m ^
C 3
u u
C ( .,
I co g
0 0 0 oco
0
I C' 0C0
,3
C5C
F 0
0+0
F ci
(5 Q jLFL N
flUl
I I 2 2 1 1
U
e
E
u
o
rr
17C
3 II
II i
>P
c
3
Ci
O
10
II N
m II
m
3 yj
=
4
"E
EU
U
L1
Jr
O
3
rO
II II
rl"
the HTO Clution curves and were not markedly affected by the Ca(NO3)2
concentration. The dispersion coefficients averaged over the three
Ca(NO )2 concentrations and corresponding to the mean porewater
velocity values given above were 0.453 + 0.025, 0.263 0.040, and
9
0.082 0.007 cm/hr, respectively. These values plotted against the
average velocity, as shown in Figure 15, gave a linear relationship.
Extrapolation to zero porewater velocity gave an intercept of 0.0458
cm2/hr (1.3 x 105 cm2/sec) which represents the diffusion coefficient
of HTO. Recent values reported by Mills (51) for the selfdiffusion
coefficient of HTO are 1.724 0.003 x 105 cm2/sec at 15 C and 2.236
5 9
0,004 x 10 cm2/sec at 25 C. The experiments reported in this study
were conducted at approximately 20 C. A linear interpolation yields a
5 2
value of 1.88 x 10 cm /see which compares favorably with the experi
mental value when tortuosity factors are considered.
The elution curves for Cl appeared, in all cases, slightly to the
left of those for HTO confirming the expected exclusion of Cl by the
exchanger. Because no exclusion isotherm was measured for Cl, the
excluded volumes determined by the magnitude of the lefthand shift
of the Cl curve from C/Co = 0.5 at V = V for these nine elution
curves were averaged. The excluded volume was calculated to be 12.8
cm using the Cl data. This information was used to calculate a
"retardation" coefficient of R = 0.938 for C1 from the relationship
R = 1 12.8/Vo. As shown in the figures 6 through 14 this value gave
analytical curves which described the elution curves for Cl.
Analytical curves for Na utilizing R values (1 + ) calculated
with the K values from the appropriate isotherms, described the elution
curves for a+ in 0.05 M and 0.02 1 Ca(N03)2 reasonably well. These
(QQ
Ca1
+ L
cd oU o
FO)L
0I0
oo
> ;
cC
c10
Lo)
E 5
u
results indicate that equilibrium was instantaneous and followed the
experimental isotherm. The somewhat poorer agreement between the
analytical and experimental curves for Na in 0.02 M Ca(N03)2 was
probably the result, in this case only, of not using the same batch
of 0.02 M Ca(N03)2 solution as that used to determine the adsorption
isotherm. Concentration differences between the two batches would ex
plain the consistent left displacements observed in this case. It also
underscores the sensitivity of the system to changes in the concen
2+
tration of Ca2
The linear equilibrium model failed completely to describe the
elution curves for Na+ in 0.005 It Ca(NO3)2. This was unexpected and the
displacements in 0.05 M and 0.005 1 Ca(NO3)2 were repeated using a
shorter column with p = 1.764 g/cm3 and 0v = 0.344. These elution
curves are presented in Figure 16 and are shown to behave in an identi
cal manner as that observed for the longer column. This rules out the
possibility that the observed results were experimental artifacts.
Failure of the equilibrium model to describe experimental data is
usually construed as an indication of nonequilibrium. As previously
discussed, it was expected that, at the concentrations of Na+ used,
film diffusion would constitute the main resistance to mass transfer.
According to this concept, a stagnant fluid film is thought to exist
around the exchanger particles across which interphase mass transfer
takes place by molecular diffusion. The thickness of this film would
vary inversely with fluid velocity. However, no marked shifts or
changes were observed in the shapes of the elution curves for Na or
HTO in 0.05 N, 0.02 1 or 0.005 i Ca(NO3)2 although the pore velocity
was varied by almost an order of magnitude.
I I
00
r r
9 L
z z 00 (D0
voo
01LO
Li
N.
4141
44
00l
Whatever the process associated with the observed results in
Figure 16, it clearly depends in some fashion upon the concentration of
Ca2+ in the system, since the concentration of Na was held constant at
80 85 ppm in all cases. The theoretical and experimental studies of
Helfferich and his coworkers, reviewed previously, indicated that the
interdiffusion coefficient of Na in either phase would depend upon
the relative concentrations of Na and Ca2+ in that phase [equation
(26)]. Their theory showed that at low Na to Ca2+ ratios, the inter
diffusion coefficient is close to the diffusion coefficient of Na+ and
2+
approaches that of Ca as the ratio increases. However, it was in
conceivable that the diffusion coefficients of Na and Ca2+ differed
by an amount large enough to account for the drastic changes in the
elution curves observed when the concentration of Ca2+ was decreased
from 0.02 M to 0.005 M. This argument coupled with the observed null
effect of variations in pore velocity leads to the conclusion that in
terdiffusion in films, as conceived above, was not the process
responsible for the observed behavior.
The possibility existed that intraparticle diffusion nay be
the factor responsible for the results presented in Figures 12 to 14.
However, if this were true, it was difficult to explain why such
results appear only during the displacement of Na in 0.005 M Ca(N03)2
and not with the two higher concentrations. In order to investigate
the resistance to intraparticle mass transfer of the exchanger par
ticles, unsulphonated, spherical, 20 50 mesh, macroporous copolymer
beads were used. Their mean diameter was at least 5 times larger than
that of the sulphonated exchanger particles used in the previous ex
periments. The identical tracer solution of Na in 0.005 i Ca(NO )2
was displaced through a 20.5ca long column packed with these beads. The
elution curves for Cl, HTO and Na are shown in Figure 17. These
curves showed no indication of diffusional masstransfer processes thus
negating the possibility that intraparticle diffusion kinetics were
responsible for the previously observed results. The results in Figure
17 also confirm the assumption that a low resistance to intraparticle
mass transfer existed for the macroporous, polystyrene exchanger
materials.
From the foregoing studies, it was now clear that diffusional
mass transfer kinetics was not the major factor responsible for the
failure of the equilibrium model to fit the elution data for Na in
0.005 M Ca(N03)2. Experiments for the transport of Li+ were next used
in an effort to gain a deeper insight and provide a reasonable explana
tion to this anomaly.
Before proceeding onto these experiments, displacements of two
tracer pulses consisting of Na Cl and HTO in 0.05 1 and 0.02 H
Ca(NO3)2 were done for steadystate unsaturated flow conditions in the
same column used in the foregoing experiments. The column was kept
under a constant pressure of 23 cm of water using the method described
previously. The fractional volumetric water content, v was reduced
to 0.2740. The column was positioned vertically in order to give the
highest possible flow rate without incurring large watercontent
gradients in the column. The flow rate was adjusted so that a constant
pressure head of 5 6 cm of water was maintained at the inlet.
The elution curves for these displacements are presented in
Figures 18 and ]9. Areas under these curves by the trapezoid rule in
tegration indicated that the material injected with the pulse was
00
0 
0 3 0
^^i~~J~^
0 +
 a
I z
* 0
I I I I I I I I I 
Ocr ron
Ulu
if)
on
u
c
0r4
u,
4
C)
,,
m
440
*H 40
04.
ifl"' o
0 ill
r!
C"M
0 *
04
Sou
oli
u ^
3R
0 rl 3
30.
C.
r:
E
Li
m
c
I!
'
E
u
rr
m
s
c II
II rl
a
,1
h
i
II
ii
>,
69)
0 
C 0,
00 0
'So I I
o y SC,
000
OG.
0 (Y
0
0,r
=" o,0
02
.o> t
Q ) p O j T q C\
> uu
recovered completely in the effluent. The R values used for the
saturated flow experiments were suitably modified to take into account
the reduction in 0 The equilibrium model [equation (45)] was used to
generate the analytical curves shown in Figures 18 and 19.
Rather poor agreement was obtained between the analytical and
experimental curves for Na in 0.05 N Ca(N03)2 and was worse for Na
in 0,02 M Ca(N03)2. The general tendency was a displacement of the
experimental data to the left of the predicted analytical curve.
It is clear that better agreement is obtained if a lower R value were
used. It is possible that with unsaturation a portion of the porous
matrix becomes either inaccessible or was accessible only at an ex
tremely slow rate. Subtle arguments are required to justify a shift to
the left of the predicted equilibrium curve in the second case. These
fortunately were discussed elsewhere (69, 70) and are not presented
here because it was the intention of this study to make only cursory
observations of Na+ transport under unsaturated flow conditions.
Miscible Displacement Experiments with Li+
As a consequence of the foregoing results, it was decided to ob
serve the elution behavior of pulse inputs of Li+ and HTO in 0.05 M,
0.02 M, and 0.005 M Ca(NO3)2 at a single fluid pore velocity. The
experimental and analytical curves for these displacements are presented
in Figures 20, 21, and 22, respectively. Areas under these curves by
the trapezoid rule indicated complete recovery of materials injected
with the pulse. A 20.5cm long column packed to a bulk density (p) of
1.764 g/cm3 and with a fractional volumetric water content (0 ) of 0.344,
was used in these and all subsequent displacements involving Li .
71
Lo
0
U C)
_05
I I1u
00 0
o 0 0
LD
r 0
0 u
u ua
I 0
0 0
C6)
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^0 & ^ ~ ^ ^ ^ ~ =
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ul
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003
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00
0
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0 0
o.
If)
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c
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uc
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nCC
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rCC
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aj o
The behavior of Li in 0.05 t Ca(NO3)2 agreed with the theoretical
curve, but the linear equilibrium model failed to describe the elution
patterns of Li+ in 0.02 M and 0.005 I Ca(NO3)2. A mild discrepancy was
apparent in the front portions of these latter curves becoming more
pronounced as the Ca2+ concentration was decreased. In order to obtain
a concentration of 80 85 ppm Li in the tracer solution, it was
necessary to add 12.5 cm3 of 1 M LiCl/liter of tracer solution. This
resulted in density differences and may have caused the observed dis
crepancies which were similar to those reported by Rose and Passioura
(62).
The results on the back side of the breakthrough data were in
disagreement with the predicted analytical curves. A clue to the
reason for this anomalous behavior was provided by the observation that
the deviations commenced at approximately one pore volume after change
over to the tracer free eluting solution.
As discussed above, for analytical purposes, it was necessary to
keep the concentration of Li+ in the tracer solutions between 80 85
ppm. This concentration of Li+ resulted in a larger numerical contri
bution to the ionic strength of these solutions than the same concen
tration of Na+. It was, therefore, reasonable to assume that
differences in ionic strengths between the eluting and tracer solutions
were the reason for the observed behavior. Figures 23 and 24 present
elution curves for the displacement of the same tracer solutions of Li
in 0.02 1 and 0.005 I Ca(NO3)2, but with the ionic strength of the eluting
solution adjusted to match that of the Li+ plus Ca(NO3)2 tracer solu
tion. The observed inflections in both cases were altered, but did not
disappear completely; however, the change was towards better agreement
S0
 0
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with the calculated curves. In order to pursue this idea further and
to confirm that the concentration of Li+ was a contributing factor,
displacements were conducted using 50 55 ppm Li in 0.02 It Ca(NO0)2
and 0.01 M Ca(N03)2. These elution curves are shown in Figure 25. The
analytical curves were generated using a value of D obtained from a pre
vious displacement at almost the same porewater velocity. The R value
for the displacement of Li+ at 50 55 ppm in 0.01 M Ca(N03)2 was com
puted using the empirical exponential relationship for the dependence
2+
of K on the concentration of Ca (Figure 4). The inflection in the
elution curve for the displacement of Li+ in 0.02 M Ca(N0)2 was less
pronounced than for the displacement of Li+ at 80 85 ppm. The in
flection in the elution curve for the displacement in 0.01 M Ca(N03)2
was more pronounced than in 0.02 ti Ca(NO )2; showing similar enhance
Iment with decreasing Ca2+ concentrations as those observed previously
(Figures 21 and 22).
The foregoing experiments served to isolate some factors associated
with the observed results, but did not provide any insight into the
mechanism. This was needed to explain why the effect appeared only
on the desorption portions of the elution curves and why the inflections
did not disappear with ionic strength corrections. In addition, cal
culations utilizing the extended DebyeHiUckel formula show that the
activity of Ca2+ in the tracer solutions containing 80 85 ppm Li
in 0.05 M, 0.02 M and 0.005 11 Ca(N03)2 were respectively 2.4, 4.8 and
11.4% less than the corresponding activities in the pure solutions.
It was shown above [equation (48c)] that the K value is inversely re
lated to the activity of Ca2+. The above results therefore imply a
depression in adsorption with a resulting increase instead of a decrease
in the Li+ concentration of the solution phase.
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Ulu
4J
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These facts led to the following experiment which provided a
reasonable explanation for both the observed inflections and the
anomalous behavior of Na in 0.005 rl Ca(NO3)2. The solution in the
column was displaced with a tracer solution containing 80 85 ppm of
Li+ in 0.005 M Ca(NO3)2 until the concentration of Li+ in the effluent
was equal to its concentration in the influent solution. This im
plies that equilibrium was achieved at all points within the system.
This solution was then displaced with deionized water and the con
centrations of all components in the effluent were monitored. The
appearance of Ca2+ and Cl was monitored qualitatively by precipitation
with Naoxalate and AgNO3, respectively. Nitrate was identified by the
brown ring test. At exactly one pore volume, the concentrations of all
components including Li+ fell sharply to zero. It thus became clear
that the electroneutrality requirement peculiar to ion exchange adsorp
tion processes resulted in all components behaving as nonreactive
solutes once equilibrium was achieved. This result implies that an
i+ equivalent toK
amount of Li equivalent to VoCo remained adsorbed by the exchanger.
The adsorbed Li+ was then eluted with a 0.005 M Ca(NO3)2 solution.
The elution curve showed the appearance of Li shortly before one
pore volume, rising sharply to a steady maximum concentration and then
falling sharply to zero at approximately 2.56 pore volumes. This be
havior is analogous to that observed on the back side of the pulse
elution data for Li+ in 0.005 1 Ca(NO3)2 (Figure 22). These results
are presented in Figure 26, and indicate that the exchange adsorption
for Li+ in 0.005 M Ca(NO3)2 was distinct from its exchange desorption.
If the same K value characterized both processes, the elution volume for
desorption should have been RL = 1.79 pore volumes instead of observed
0
cC~
LO
~, T
>J>1
0
Of)
0+ +
0 0
0 @ 0
0 0 0
LO
1 F) Iz I I I L
* uj6
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Ulu
9Q )
* CM
f,,
Ci
0
cb CO
0
+
L 21
c uj
Ci
Ci
0
0 C
Ci
[C) Z
0 *
o m
value of 2.56. This indicates that a higher K value was associated with
the desorption process.
The samples collected from the desorption experiment were tested
2+ 2+
for Ca with Naoxalate solution. The tests showed that Ca2+ appeared
in the effluent with the Li+. It was also observed by comparing the
turbidity developed in the effluent samples with a standard (0.005 IH
2+
Ca(N03)2) that the concentration of Ca rose sharply to a maximum
concentration which was less than 0.005 11. The Ca2+ concentration
remained at the lower level until the elution of Li+ was complete,
after which it rose to 0.005 M. This was expected because electo
neutrality necessitates that an equivalent amount of Ca2+ replace the
Li+ desorbed. These results indicate that the desorption of Li+
occurs with a concentration of Ca2+ in the solution phase less than
0.005 M. This implies a lower Ca2+ activity and therefore a higher K
value [Equation (48c)].
The concentration of Li in the saturating solution was 0.0122 iH and
from the exchange desorption curve the maximum value reached was 0.0067
M. Applying the electroneutrality requirement, the concentration of
Ca2+ would have to decrease from 0.005 M to 0.0016 M. Thus, during
desorption the solution phase constitution is Li = 0.0067 1, Ca2
0.0016 M, NO3 = 0.01 M. This yields a value for the ionic strength of
0.0117 M. With this value the extended DebyeHUckel formula gives
values for YLi = 0.9012 and aaCa = 0.0325. The empirical formula K =
y i/70.1837 'i4C, used previously to calculate K values of the ad
sorption isotherms for Li+, yields a value of K = 0.3952 which gives
P = 2.03. This value is greater than the experimental value of 1.56
By
(Figure 26) but it is in the right direction. Observations from the
elution curve for Li+ in 0.02 M Ca(NO3)2 given in Figure 21 suggest
that if a similar experiment as that described above were conducted
with 0.02 M Ca(NO3)2, a maximum concentration of Li+ = 0.82 Co would
result for the desorption process. By the same reasoning, the con
centration of the various species in the solution phase would be Li =
2+ 
0.01 M, Ca2 = 0.015 M, NO3 = 0.04 i yielding an ionic strength of
0.055 M. Similar calculations yield values of yLi = 0.8292, 4 Ca
0.0842, a predicted K value of 0.1403, and s K = 0.7205. The value of
vy
pK v 0 g
compares favorably with the observed value of 0.60 (Figure 21).
The inflections observed in the elution curves for Li (Figures
21 through 25) become clear in light of the above results. When lower
concentrations of Ca(N03)2 were used in the tracer pulse, the quantity
of Li+ adsorbed is increased. The desorption process is characterized
by a K value that is dependent on the activity coefficient of Li and
the activity of Ca2+ in the desorbing solution. If a low concentration
of Ca(NO3)2 was used in this solution, the equivalence of exchange re
sults in a reduced value for the activity of Ca2+ and hence a corres
pondingly larger K value. When the concentration of the desorbing
Ca(NO3)2 solution is increased, the K value decreases. In addition, at
higher concentrations of Ca(NO3)2 the effect of the adsorption of
2+
Ca would be less pronounced. The desorption K value then approaches
more closely the K value for adsorption. These facts explain the re
duced degree of inflection when the concentration of the eluting
Ca(NO3)2 was increased to match the ionic strength of the tracer solu
tion, or when the concentration of Li in the latter was lowered. In
addition, it explains why the inflections did not disappear after the
ionic strength adjustment and why they were not observed for the
displacement of L.+ in 0.05 1 Ca(NO3)2'
Thi foregoing results also clarify the apparent kinetic effect
observed for displacements of Na in 0.005 II Ca(NO3)2. Examination of
the condition under which the data in Figures 11 through 14 were obtained
reveals that because of large R values, the tracer front for Na in
all cases, had not reached the Covalue in the effluent at the time of
changeover to the eluting solution. It is obvious from the foregoing
that this would result in a change in the ionic composition of the
solution phase before the column was completely equilibrated with Na.
Thus, instead of reaching a maximum concentration of Co, the effluent
would reach a maximum lower than Co. This idea was tested by a step
input displacement experiment with Na in 0.005 M Ca(NO3)2. The result
given in Figure 27 shows that the above is indeed a reasonable ex
planation since the effluent now shows none of the previous apparent
kinetic effect. A further point in favor of the correctness of this
hypothesis is that it explains why the apparent kinetic effect was
associated with only the front portion of the curves. If true
kinetic processes were operating and the adsorption isotherm was linear
the effect would have been symmetrical.
In light of the foregoing, the data for Na+ in 0.02 I Ca(NO3)2
may seem anomalous since they did not show inflections. It is likely
that inflections such as those for Li in 0.02 I Ca(NO3)2 may have
existed but were overlooked because they were too small.
Miscible Displacement Experiments with 45Ca2+
Two displacements of pulse inputs were conducted with 45Ca2+ in
0.05 M and 0.075 M Ca(NO3)2 to verify that the adsorption isotherns for
ACa2+ do reflect appropriately in their transport behavior. Because
of large R values, these displacements were conducted in a shorter
814
'I
I I I I
Cc C
UK)> '
YC
it I ~c, I 1. C
5.7cm long column packed to a bulk density (p) of 1.769 g/cm3 and
saturated to a fractional volumetric water content (6 ) of 0.347.
The results of these displacements are given in Figure 28. Because
of the shorter column, experimental error was greater. In both cases,
if higher R values were used, closer agreement would have been obtained
between the experimental and analytical curves.
It is interesting to observe that for the displacement in 0.05 M
Ca(NO3)2 changeover to the eluting solution before the tracer front had
completely appeared in the effluent, resulted in a similar disagreement
between the analytical and experimental curve as was obtained for Na
in 0.005 M Ca(NO3)2. Further, the use of a much larger pulse for the
displacement in 0.075 M Ca(NO3)2 produced the expected inflection at
approximately one pore volume after changeover and a shift to the
right of the analytical curve.
These results confirm that the behavior of Na in 0.005 M
Ca(NO3)2 cannot be attributed to the effect of relative concentrations
+ 2+
of Na to Ca on the interdiffusion coefficient of either ion. Exa
mination of the relevant equation of Helfferich and his coworkers
[equation (26)],reveals that in the case of isotapic exchange, the
interdiffusion coefficient is equal to the diffusion coefficient of
Ca2+ and independent of relative concentrations. Also, increasing
the relative concentration of Na to Ca2 should result in the inter
diffusion coefficient approaching the diffusion coefficient of Ca2+
If this were indeed responsible for the behavior of Na in 0.005 M
Ca(NO3)2, then similar effects should have appeared in both the elution
curves for 45Ca2+ This however was not observed.
0 In
2!
> C,
II 2
o C
000
C
r0
0
C,
L1
=I
CM
o
e
00
00
c
t
o
o
cl
c"
11
>0r
C C
uri
3
140
0
cc
,3
CHAPTER 6
SUMMARY AND CONCLUSIONS
Theoretical approaches to describe cation exchange processes are
fairly recent although the topic spans a period of over a century.
Despite the acknowledged importance of this process in soil systems,
there are only a few references which deal with transport of ionic
species. In this study, the miscible displacement technique was
utilized to investigate the influence of exchange adsorption on the
transport of selected inorganic ions in a porous medium. The influence
of dispersion and exclusion was also studied. A Casaturated organic.
cation exchanger was used as the medium, which eliminated the possi
bility of interactions other than ion exchange.
Solution and exchanger was conceived as a heterogeneous system
consisting of two mixed phases. A thermodynamic treatment based on
this concept predicted the exclusion of anions. It was shown that, in
general, solutionphase ions common to those initially saturating the
exchanger influence the adsorption of other charged species. This was
adequately verified by experimental exchange adsorption isotherms deter
mined for Na+ and Li+ in 0.05 _, 0.02 I and 0.005 M Ca(NO3) and for
45Ca2+ In 0.075 M and 0.05 Ii Ca(N03)2. For the range of concentrations
studied, the adsorption isotherms for Li+ and Na+ were linear and
their slopes increased in a nonlinear fashion with decreasing Ca2+
concentration in the equilibrium solutions. Similar trends were evi
dent from the 45Ca2+ exchange isotherms. It was demonstrated inductively
87
thnt these results could be attributed to ionion interactions as
conceived and quantified by the Debyelluckel theory.
Miscible displacement experiments involving Na+, Li+, 4Ca2
and C1 were conducted to examine the consequences of the foregoing
results, and to determine the presence of kinetic masstransfer pro
cesses. All experiments were performed using pulse inputs of the
tracer solution. Tritiated water (HITO) was present in all tracer
solutions in order to evaluate dispersion. An asymptotic solution
to the convectivedispersion transport equation for a reactive
solute (linear adsorption isotherm) was used to predict the experi
mental breakthrough data.
A series of displacements involving pulses of Na+, HTO, and C1
in 0.05 M, 0.02 M or 0.005 i Ca(NO3)2 were conducted under steadystate
watersaturated flow conditions at three porewater velocities ranging
from 1.5 to 15 cm/hr. Dispersion coefficients obtained from the HTO
elution data were not influenced by Ca(NO3)2 concentration and were
linearly related to porewater velocity. The elution curves for Cl
were all displaced to the left of those for HTO confirming its exclu
sion by the exchanger. Reasonable agreement was obtained between the
computed and experimental elution curves for Na+ in 0.05 1i and 0.02
M Ca(N03)2 at all porewater velocities studied. The analytical
solution, however, failed to describe any of the experimental elution
curves for NaF in 0.005 1 Ca(NO3)2. These results could have been
used to support the concept of diffusional masstransfer kinetics, but
it was shown experimentally that such processes were not involved.
A series of similar steadystate, watersaturated column experi
ments were conducted to study the transport behavior of Li+ and JITO
