June 1989
Agronomy and Soils
Departmental Series No. 135
Alabama Agricultural Experiment Station
Auburn University
Lowell T. Frobish, Director
Auburn University

DETERMINATION OF TRANSPORT PARAMETERS FROM STEP AND PULSE
DISPLACEMENT OF CATIONS AND ANIONS
Feike J. Leij and J.H. Dane
Graduate Student and Professor
of Agronomy and Soils
Alabama Agricultural Experiment Station
Auburn University
Auburn University, Alabama
Lowell T. Frobish, Director
ii
CONTENTS
LIST OF FIGURES......................... v
LIST OF TABLES........................ vii
ABSTRACT. ........................... ix
INTRODUCTION..........................1
THEORY............................. 5
Formulation of the Transport Equation............5
Methods to Determine Transport Parameters Reported in the
Literature...............................................10
Graphical Determination of R for Step and Pulse Input 11
Determination of Transport Parameters with
the Moment Method.................... .15
Theoretical Determination of Moments
According to Aris............... ..... 18
MATERIALS AND METHODS.............. ....... 22
RESULTS AND DISCUSSION.............. . ..... 25
Pilot Study.........................25
Transport in Layered Media with Binary Exchange. ..... 33
Transport in Homogeneous Media with Binary Exchange . . . 37
Transport in Homogeneous Media with Ternary Exchange . 55
SUMMARY AND CONCLUSIONS.....................65
LITERATURE CITED................... .... 69
APPENDIX A. Analytical Solution of the ADE for a Pulse Input
with a First- and Third-type Condition in a Two-Layer
medium.......................... 71
APPENDIX B. Derivation of Moments Based on Analytical Solutions
of the ADE in the Laplace Domain..... . ............. 75
iii
APPENDIX C. Time Moments m, p', and i for BTC's as a Result of
a Pulse Input During Steady Flow in a Medium with One or
Two Layers with a First- or Third-type Inlet Condition
APPENDIX D. Input Files for the Program CXTFIT Containing the
Experimentally Determined 
C/C -profiles . ..... 
.
0
APPENDIX E. Exchange Data
FIRST PRINTING, JUNE 1989
Information contained herein is available
to all without regard to race, color, sex,
or national origin
iv
77
82
92
LIST OF FIGURES
FIG.1. Schematic representation of solute displacement
experiments with step and pulse input ......... 9
FIG.2. Hypothetical concentration distribution as a
function of position at various times for a
pulse input ..................... 12
FIG.3. Determination of the mean breakthrough time based
on BTC's with step and pulse type input . ....... 14
FIG.4. Schematic of experimental setup ........... 23
FIG.5. BTC #1: Ca curve for Troup soil ........... 26
FIG.6. BTC #2: Ca curve for Troup soil ........... 26
FIG.7. BTC #3: Br and Cl curves for Lucedale soil ...... 27
FIG.8. BTC #4: Br and C1 curves for Lucedale soil ...... 27
FIG.9. BTC #5: Br and C1 curves for Wickham soil ...... 28
FIG.10. BTC #6: Br and Cl curves for Wickham soil ........ 28
FIG.11. BTC #11: Ca and K curves for Troup and Savannah soil . 35
FIG.12. BTC #12: Ca and K curves for Dothan soil ...... 35
FIG.13. BTC #13: Ca and K curves for Troup and Lucedale soil . 36
FIG.14. BTC #21: Ca, Na, Br, and C1 curves for Dothan I . . . 38
FIG.15. BTC #22: Ca, Na, Br, and Cl curves for Dothan II . . 39
FIG.16. BTC #23: Ca, Na, Br, and Cl curves for Wickham I . . 40
FIG.17. BTC #24: Ca, Na, Br, and Cl curves for Wickham II . . 41
FIG.18. BTC #25: Ca, Na, Br, and C1 curves for Troup soil . . 42
V
FIG. 19. BTC #26: Ca, Na, Br, and Cl curves for Lucedale I . . 43
FIG.20. BTC #27: Ca, Na, Br, and Cl curves for Lucedale II . 44
FIG.21. BTC #28: Ca, Na, Br, and Cl curves for Savannah I . . 45
FIG.22. Numerically predicted (solid lines) and
experimentally determined curves (symbols) for
cation displacement in binary systems .. ........ 54
FIG.23. BTC #31: Ca, K, and Na curves for Dothan I 56.
FIG.24. BTC #32: Ca, K, and Na curves for Dothan II .. .... 56
FIG.25. BTC #33: Ca, K, and Na curves for Wickham I ...... 57
FIG.26. BTC #34: Ca, K, and Na curves for Wickham II . .... 57
FIG.27. K and Na curves predicted with the moment method
(solid lines) and curve fitting (dashed lines), and
experimental curves (symbols) for displacement with
ternary exchange. . ........ .. .......... 61
vi
LIST OF TABLES
Table 1. Solutions of the ADE in the Laplace Domain for a
One- and a Two-layer System with a First- and
Third-type Inlet Condition ............ 19
Table 2. Expressions for 'l and p for Transport in a
One- and a Two-layer Medium Subject to a First-
and a Third-type Condition . ............ 20
Table 3. Classification of Soils .... .............. 22
Table 4. Experimental Conditions for Solute Displacement
During Pilot Study ..... ................ 25
Table 5. Determination of Dispersion Coefficients According
to Fried and Combarnous (7) ............ 29
Table 6. Determination of Dispersion Coefficients Using
ifc(2Ce) = a T + 3 . ............. 30
Table 7. Determination of Transport Parameters Using
CXTFIT (13) ...... ... ............ 31
Table 8. Experimental Conditions for Solute Displacement
in Layered Media. ........................... 33
Table 9. Determination of Transport Parameters for Step
Displacement in Layered Media with CXTFIT and
Column Holdup ....... .................. 34
Table 10. Experimental Conditions for Pulse Displacement
with Binary Exchange . ..................... 37
Table 11. Determination of Transport Parameters with Binary
Exchange Using CXTFIT ..... ............... 47
Table 12. Determination of Transport Parameters with Binary
Exchange Using Time Moments .................. .48
Table 13. Experimental Conditions for Solute Displacement
with Ternary Exchange ............ ........ 55
vii
Table 14. Determination of Transport Parameters with Ternary
Exchange Using CXTFIT................ 58
Table 15. Determination of Transport Parameters with Ternary
Exchange Using Time Moments............. 60
Table 16. Theoretical Moments Based on Linear Exchange and
Their Contribution to Experimental Time Moments
for BTC's with Ternary Exchange........... 
63
APPENDIX E. Exchange Properties for Ca/Na Soils........ 
92
viii
ABSTRACT
Although transport problems in the field usually involve a number
of solute species, which react in different ways with the soil, most
experimental work considers displacement in binary systems where the
two solutes are either non-reactive or obey a particular exchange
isotherm. This study reports theoretical and experimental findings for
displacement involving up to four solutes.
Experimentally determined 
breakthrough curves 
(BTC's) are
reported for pulse and step inputs, using Ca, K, Na, Br, and Cl as
so lutes. A1 though several methods were used to determine the
coefficient of longitudinal dispersion, D, and the retardation 
factor,
R, the most attention was given to the use of time moments.
Theoretical moments were derived from analytical solutions of the
advection-dispersion equation, ADE, in the Laplace domain to
investigate dispersion and retardation for pulse inputs. Solutions of
the ADE for a first- and a third-type boundary condition at the inlet
were used for a non-layered and a two-layer medium.
Step displacement experiments were conducted in various media
with layering transverse to the direction of flow. No increased
dispersion was found as a result of layering. Pulse displacement
experiments were carried out in homogeneous soils for binary systems, a
ix
NaCl-pulse in a CaBr2-saturated medium, and for ternary systems, a
2
pulse of K and Na in a Ca-saturated medium. The moment method was used
to determine values for D and R. The accuracy of the experiments was
found to be acceptable, based on various mass balances and the
electroneutrality principle. Values for D, obtained from anion
displacement experiments and measured cation exchange isotherms, were
used to numerically predict BTC's for cations. The role of non-linear
exchange was investigated by comparing experimental (non-linear
exchange) with theoretical (linear exchange) moments. Non-linear
exchange was shown to have a large influence on the value of D, in
particular for Na. The average pulse spread, determined by experimental
moments, resulting from hydrodynamic dispersion and non-linear
exchange, was in a number of instances five times the average pulse
spread determined by theoretical moments, based only on hydrodynamic
dispersion.
x
INTRODUCTION
Transport of dissolved chemicals in porous media has been studied
quite extensively in order to predict the fate of contaminants and
agricultural chemicals in soils. Such transport has traditionally been
described with the advection-dispersion equation (ADE) under the
assumption that the transport and flow properties are uniform with
respect to time and position. It is now widely recognized, however,
that under field conditions these properties can vary in time and space
(22). Therefore, deterministic approaches to model transport will
generally result in poor predictions of the fate of contaminants.
Various other methods to predict flow and transport have consequently
been employed, emphasizing the stochastic nature of transport and flow
(6) or abandoning any mechanistic description of the processes involved
(11).
Accepting the validity of the ADE under laboratory conditions,
the transport parameters to quantify the advective and dispersive flux
need to be known. A number of techniques are currently in use to
determine these transport parameters from experimental data. It should
be noted that in the older literature considerable attention was given
to the determination of D, assuming Fickian dispersion. Although this
assumption is incorrect for a number of situations, and even if it were
correct, it implies that the average solute displacement is unaltered
by dispersion. The value for R quantifies interactions between solute
2
and solid phase, which can greatly affect average solute displacement.
In order to predict the average location of the solute front, the value
for R is of more importance than D.
Techniques of determining D and R are based on relatively simple
analytical solutions, graphical methods, or fitting experimental data
with theoretical (non-equilibrium) models (e.g., 24). In particular,
curve fitting techniques are quite popular. As pointed out by Parker
and van Genuchten (13), uniqueness problems might arise if too many
unknown factors have to be considered to fit experimental concentration
profiles with theoretical solutions. In addition, the models used to
describe transport can not be properly validated with curve fitting
alone (15).
A method commonly used in chemical engineering to analyze
experimental BTC's is the method of moments (e.g., 16), although it has
al so been used in hydrology (9) and to some extent in soil science
(19). Values for D and R can be determined in a straightforward way
using time moments. of the BTC. The method was originally used. for
advection-dispersion problems by Aris (2). The total amount of solute,
the mean breakthrough time, and the degree of spreading and tailing are
easily obtained by determining various time moments of the BTC.
Our first main objective concerned the determination of transport
parameters. Fi rst, some well known' methods were evaluated using
expeimenall detrmind BC's.Secod, ome impl was todetemin
3
Aris (3) showed how analytical solutions of transport equations
in the Laplace domain can be used to obtain those moments
theoretically. Solutions are more easily obtained in the Laplace domain
than in the regular space-time domain. These solutions allow D and R to
be determined from results of solute transport experiments. From a
theoretical point of view, the technique can be used to obtain explicit
expressions for retardation factors and dispersion coefficients for a
particular transport model. Valocchi (21) applied the technique to
aggregated media in order to evaluate the validity of the local
equilibrium assumption and to predict spreading and tailing of the BTC.
Theoretical moments can also be used to study the effects- of
layering and the type of inlet condition on the transport of a solute
pulse. Our understanding of transport in heterogeneous media, as most
field soils are, might be enhanced by resorting to a relatively simple
medium such as a collection of uniform soil layers perpendicular to the
direction of flow (18). Regarding the effect of the type of inlet
boundary condition on the determination of D and R, van Genuchten and
Parker (23) showed that for flux- and volume-averaged concentrations a
first- and a third-type condition should be used, respectively. If the
results for a first- and third-type condition are similar, depending on
the values of D and v, one might prefer the use of the simpler
first-type condition.
Our second main objective was to derive theoretical moments based
on the ADE. First, these moments were used to obtain experimental
values for D and R. Second, theoretical moments were determined for
4
transport in a two-layer medium, with the interface perpendicular to
the direction of flow to demonstrate the effect of soil heterogeneity
on breakthrough time, spreading, 
and tailing. Third, theoretical
moments were derived based on a first- and a third-type inlet
condition. Again, this condition applies to both the inlet of a
homogeneous medium and individual layers of a layered medium.
Knowledge of the relationship between the solute concentration in
the adsorbed and solution phase is necessary to solve the transport
equation. In the case of transport of a non-linearly exchanging solute,
advection and dispersion terms depend on the concentration and
composition of solutes. Current methods used to determine transport
parameters do not account for this dependency because it is assumed
that the exchange is linear. This greatly simplifies the mathematical
solution of the transport equation. Although the assumption of
linearity is a convenient one, it is quite often not a realistic one.
Furthermore, the total amount of adsorbed cations depends on the
composition of the adsorbed cations. This affects the value of R.
Differences in the apparent cation exchange capacity (CEC) were
reported to be particularly large between Ca and Na soils (12).
Simultaneous cation and anion displacement experiments are helpful in
detecting changes in the total amount of solute. The latter is
accomplished using a mass balance for each individual ion as well as
total cation and anion balances. In binary systems with a constant
total electrolyte level, only one concentration can be chosen
independently. In ternary systems with a constant total electrolyte
level, the concentration of a particular cation depends on the two
other cation concentrations. This dependency allows us to illustrate
the difference in behavior of a favorably and an unfavorably
exchanging, incoming cation.
Our third main objective was to investigate the influence of ion
exchange on transport and to determine transport parameter values using
a pulse displacement. First, BTC's were determined in binary systems by
applying a NaCI pulse to a CaBr medium. Second, BTC's were determined
2
for a Na/K pulse applied to a Ca medium. Third, theoretical moments for
experiments with assumed linear exchange were compared with
experimental moments resulting from experiments with obvious non-linear
exchange.
THEORY
Formulation of the Transport Equation
One-dimensional transport of a reactive solute species during
steady flow in a homogeneous porous medium may be described by:
Pb as 
ac Da2c 
ac
-Dv- 
(1)
0 at 8at 2 8x
ax
where pb is the dry bulk density of the medium [ML-3], is the
3 -3
volumetric water content [L3L
-
I, S is the mass of solute in the
-1
adsorbed phase per mass of solid [MM-I, t is time [T], C is the solute
concentration in the liquid phase 
[ML-3], D is the effective dispersion
coefficient [L2T-I], x is the 
distance in the direction of 
flow [LI,
-1
and v is the average pore water 
velocity [LT I. It should be 
noted
6
that Eq. (1) is commonly rewritten as
ac Da c ac(2
R -= D - v _- (2)
aT a2 ax
8x
where R is the dimensionless retardation factor given by:
Pb 
as
R = 1+ b(3)
e ac
Persaud and Wierenga (14) discussed the influence of the slope of the
as
exchange isotherm, 8C on solute transport. This slope depends, among
other things, on the concentrations of solutes present. The
experimental determination of exchange isotherms is usually carried out
assuming that the sums of the solute concentrations in both the liquid
and the adsorbed phases are constant, no hysteresis occurs in the
exchange reaction, and the reactions reach instantaneous equilibrium.
In the case of linear exchange, the slope of the isotherm is constant
3 -1
and equal to the well known distribution coefficient K [L3M-I. The
d
retardation factor is then constant as well, which greatly simplifies
the mathematical solution of the ADE. For non-linear exchange, a
constant value for R can be used by considering an effective K value
d
(e.g., 20).
If physical non-equilibrium exists, the liquid phase is often
partitioned into a "mobile" and an "immobile" region. Solute transport
in the mobile region occurs by advection and dispersion, whereas the
amount of solute present in the immobile region is determined by
diffusive transport between the mobile and immobile region of the
liquid phase. Transport in the medium is now described by (13):
7
aC 8C. 82C ac
mo im mo mo
mo + bKd 8t 
(
im+(- bd t OmoDmo 2 8x
8x
aC.
(0. + (1-f)p ) im (C(5)
im bKd) at (Cmo m-C.
where the subscripts mo and im refer to mobile and immobile regions of
the liquid phase, respectively; f represents the fraction of exchange
sites that equilibrates with the mobile liquid phase; and a is a first
order rate constant, which determines the diffusive transfer of solute
-1
between mobile and immobile regions [T I. Note that 8 +8. =8 and
mo im
8. C. +8 C =OC. Furthermore, we assumed that the exchange process for
im im mo mo
mobile and immobile exchange sites is governed by the same distribution
coefficient.
Although our mathematical analysis concerns equilibrium transport
of a non-reactive solute (R=i1), it is easily extended to linearly
exchanging solutes by replacing v and D with v (=v/R) and D (=D/R),
respectively. The transport problem posed is subject to the following
conditions and assumptions: (1) the medium is initially free of the
displacing solute, (2) for the inlet boundary, both the concentration-
or first-type and the flux- or third-type condition will be used, (3)
the medium is semi-infinite in order to formulate the outlet condition,
and (4) both step and pulse type displacement will be used. These
conditions and assumptions can be mathematically described as:
8
C(x,t) = 0 O<x<oo t=O (6-a)
C(x,t) = f(t) (first-type) x=0 t>0 (6-b)
BC
(-Da + vC) = vf(t) (third-type) t>0 (6-c)
8x
x=0
C O0<t<t
0o 0
with f(t) = (6-d)
L0 
otherwise
BC
c =0 
t>0 (6-e)
8x
For the step type displacement t approaches infinity, whereas
0
for the pulse type displacement t is finite. Figure 1 schematically
0
illustrates the two types of experiments for displacement in a soil
column with length L, assuming a first-type inlet boundary condition.
The inlet concentration, C. at x=0, as well as the outlet
in
concentration, Cou
t 
at x=L, are shown as a function of time. For
out
D=0, the BTC is the same as the input curve. Also shown is front
spreading due to Fickian dispersion. Non-linear exchange will alter the
shape of the BTC, depending on the nature of the exchange isotherm.
inlet condition
t(0
0 (t 
(to
t )to0
x=0
x=0
x=0
C00
C =O
00
cout
x=0 x=L
PULSE
finite 
to
c
out
co
to
0
/
/
I
t
L/v* L/v* 
+ to
STEP
infinite to
cout)
Co '
0 L/v*
advection
- - advection and dispersion (equilibrium)
FIG.1. Schematic representation of solute displacement experiments with
step and pulse input.
Cin
co
.0
Cin
co
0
I I
30,
10
Methods to Determine Transport Parameters
Reported in the Literature
First, the method described by Fried and 
Combarnous (7) was used.
This method utilizes a simplified solution of Eq.(2) 
via transformation
into a diffusion type of equation, subject 
to a first-type inlet
condition for a step input. For a non-reactive 
solute the following
solution applies:
C
o [ 
x - v t
]
C(x,t) = 0e/ 
(7)
4Dt
The concentration profile obeys 
a normal distribution function. The
coefficient of dispersion can then be obtained from 
the BTC according
to:
D = (x-vt )/ - (x-vt )/ 
(8)
8.6 884
where t and t denote the time at which C/C equals 0.16 and
0.16 0.84 o
0.84, respectively.
Second, the simplified solution given by Eq. 
(7) can be used to
determine an explicit relationship between 
D and C(x,t). This
technique, first used by Rose and Passioura (17) was 
generalized by van
Genuchten and Wierenga (24), who expressed Eq.(7) 
in terms of the
following dimensionless variables:
T = vt/L 
(9)
P = vL/D 
(10)
where L is the length of the soil column [L], T is 
the number of pore
volumes leached through the column, and P is the 
column Peclet number.
ii
For a non-reactive solute, the following expression was derived for the
column outlet:
C 
P 1/
2  
1 -T 
.
-T(1/2
C - 2 1/2 2
0 T
Plotting the dimensionless exit concentration, C = C/C on probability
e o
paper as a function of Pa T should yield a straight line (see figure
44-9, 24). The slope a, of the curve 1mwte4c(2C ) versus n T, is
e
obtained by determining the value for inwui&(2C ) at two points which
e
are sufficiently far apart. The following relationship between P and a
was used:
P = 42 A (12)
where A is a correction factor. For further details, including the
determination of R, as well as values for A, one is referred to the
Chapter by van Genuchten and Wierenga (24).
Third, transport parameters in this study were determined with
the program CXTFIT, which fits analytical solutions of Eq. (2) or Eq. (4)
and (5) to observed concentration distributions obtained by step and
pulse displacement. The program, based on a least-squares inversion
method, was described by Parker and van Genuchten (13).
Graphical Determination of R for Step and Pulse Input
The determination of an effective retardation factor, assuming
zero dispersion and physical equilibrium, is illustrated schematically
in figure 2 for a pulse input, with concentration C and duration to
' 0 O
into a column of length L. The soil solute concentration, p,' expressed
as mass of solute in the liquid phase per volume of medium, is plotted
as a function of position at various times.
12
t=O
t=t
o
v t
0
t=t /2
o
vt /2
o
t=L/v
L-v t
0
t=t +(L/v )
o
0 L
L-(v t 
/2)
o
FIG.2. Hypothetical concentration distribution as a function of
position at various times for a pulse input.
The velocity of a linearly exchanging solute, v , is equal to
v/R, whereas for a non-reactive solute, the velocity, v =v, is equal to
the velocity of the solvent. From the average position of the pulse,
<x>, it can be seen that the mean residence time is equal to L/v and
that its mean breakthrough time, t, is 
equal to (L/v) + (t /2). The
O0
retardation factor, R, is the ratio of the mean residence time for
solute and solvent, respectively. This allows the determination of R
from BTC's either graphically (this section) or with the method of
moments (next section). An effective value for R can be obtained for
non-linearly exchanging solutes provided that the effluent
L xv
13
concentration reaches a maximum concentration C prior to complete
0
displacement of the pulse.
For most displacement experiments v is known, while for the
solute t can be obtained from the experimentally determined BTC. In 
the
case of a pulse displacement, R is determined according to:
t (t /2)
sol uteL/v 0(13)
R = (L/v)
The determination of t for the solute is illustrated in figure 3, which
shows an arbitrary BTC for a pulse input. A value for t can be obtained
via graphical or mathematical integration, satisfying the condition
that areas A and B are equal.
A similar method can be employed for a step type displacement.
The average residence time of the front, t, is equal to the average
breakthrough time and can be obtained graphically as illustrated 
in
figure 3. Using the step displacement, the retardation factor follows
from:
R = vt/L (14)
A very useful concept in BTC-analysis of a step input is that of
column holdup (4). The column holdup, H, can be defined as:
.00
v0
H = {(i - C dt (15)
where C is the concentration 
of the displacing solute in the 
effluent.
According to Eq. (15), H is the 
total amount of resident solute 
exiting
the column. In case the column 
is saturated with the displacing 
solute
(t- c), H is the total amount 
of solute that was present in 
the column
at t=O.
14
C
C 
STEP
co,- 
dt'
tt
C dt - (C-C) dt
PULSE
C
C fC dt
0
o . .
FIG.3. Determination 
of the mean 
breakthrough 
time based 
on BTC's with
step and pulse 
type input.
The column 
holdup is 
obtained rather 
easily from 
BTC data. The
relationship 
between H 
and R can 
be established 
from known 
analytical
solutions 
of the ADE. 
For conditions 
given by 
Eq.(6), the 
following
expressions 
were found 
(24):
SH 
(first-type)
R = 
-1(16)
H(1 + (D/vL))- 
(third-type)
15
Determination of Transport Parameters with the Moment Method
The last method to be discussed for determining transport
parameters is the moment method, which is applicable to BTC's in
response to a pulse feed. The p-th time moment, m, of a concentration
'p
distribution C(x,t), at a given location, is given by:
m (x) - t
p 
C(x,t) dt p = 0,1,2,.... (17)
P
0
The first, second, third, and fourth moments of the concentration
distribution can be used to characterize the mean, variance, skewness
and kurtosis, respectively, of the BTC. Moments can also be used to
characterize spatial distributions of the concentration (8). From the
time moments one can obtain absolute moments, defined by:
%00 
0
= p = t
p 
C(x,t) dt / C(x,t) dt (18)
p m (x) 
fcxt
0 j0
and central moments, defined by:
( =m ) (t Ll'(x))
p 
C(xt) dt (19)
gp(X=mO(x) (tg
0 J
Since the ADE was used to describe transport, we are particularly
interested in the use of the moment method to characterize advective
and dispersive transport.
The first absolute moment, ui(x), represents the mean
breakthrough time for the pulse at column position x. As was already
discussed, the ratio of the mean residence times of solute and solvent
in the columns can be used to determine an effective value for R. For
an arbitrary pulse in a column with length L, one can formally write:
16
R 1
L  
- l0)solute-(0
R = W t(0))(20)
(MI(L) - Pl(
0
)1solvent
which is a more general expression than 
Eq.(13). According to figure 2,
A(0)=t /2 for both the solvent and the solute for a pulse input.
Furthermore, the mean residence time 
for the solvent is determined by
the average pore water velocity; the 
denominator of Eq. (20) is
therefore equal to L/v. At this point 
the only unknown in Eq. (20) is
which can be obtained with the BTC using Eq.(17) 
and (18).
It is convenient to obtain the BTC as a function 
of dimensionless
time T (the number of pore volumes leached 
through the column), in
which case the p-th moment is defined as:
M= Tp C(x,T) dT p=0,1,2,.... 
(21)
p
0
Replacing m by M the following expression 
for R can be found
p p
according to Eqs.(18) and (20):
M T
R 1 0(22)
M
0
0
where the pulse duration is now expressed as T0 pore 
volumes.
The second central moment, p
2
(x), is a measure of the average
pulse spread relative to the mean breakthrough 
time, pl(x):
1 [2
A =m1 (t-9Q C(xt) dt 
(23)
g2 
( )  
mJ 1
0 
JO
0
Application of Eq. (17) allows Eq. (23) 
to be written as
(x 2-(M) (24)
The question then arises how experimentally 
determined moments of
the BTC can be used to quantify the transport 
parameters D and v (and
17
hence R). The answer lies in the determination of moments based on the
theoretical solutions of the transport equation. For instance, for a
Dirac type of initial condition in an infinite medium the solution 
of
the ADE, assuming Fickian dispersion, can be found with the help of
Eq.(2-6) of Crank (5). Substituting the 
solution into Eq.(17) and
evaluating the appropriate integrals leads to the following theoretical
expressions:
xR 2DR
1
(x )  
x + DR(25-a)
I~l -v 2
v
2DxR
2  
[DR 2
= v3 + 8 (25-b)
Equating these theoretical expressions for W'(x) and pi
2
(x) with their
experimental values, obtained from the BTC with the use of Eq.(18) and
(24), enables the subsequent determination of R and D for a known value
of v. The determination is facilitated if the BTC is determined at two
positions, x
I 
and x
2
, in the medium. The retardation and spreading
between these points can be characterized by:
W1(x2)-Pl(Xl) 
= 
(x2-xl) R 
(26-a)
1 2 112 v
2DR
2
A2(x2)-A2(x1) = (x2-x1)-3 (26-b)
v
The value for R is obtained from Eq. (26-a), whereas the value for D
follows from Eq. (26-b) using this R value.
However, in many cases the determination of moments by
substituting the appropriate solution for C(x,t) into Eq. (17) is not a
feasible approach. In the next section, a relatively simple method to
determine time moments will therefore be discussed.
18
Theoretical Determination of Moments According to Aris
Aris (3) showed how time moments can be determined from a
solution for the concentration in the Laplace domain, C(x,s). For a
theoretical BTC in the Laplace domain, the p-th moment of the
concentration distribution is given by:
P - L dsC(x
1
s) p =O012... (27)
s->)O ds
p  
''
The solution C(x,s) depends on the mathematical model used to describe
transport. This model is not necessarily restricted to the ADE. Because
it is generally easier to obtain C(x,s) than C(x,t), the use of moments
is convenient to determine transport parameters for more complicated
transport models (e.g., 21). By equating a sufficient number of
experimentally determined moments obtained from a particular BTC using
Eq.(17) to theoretical moments obtained according to Eq.(27), the
transport parameter values of the chosen mathematical model can be
determined.
We will consider the two inlet conditions (6-b) and (6-c) for a
homogeneous medium and for a medium consisting of two layers with a
well defined interface at x=L . Formulation of the problem and details
1
of the solution procedure are presented in Appendix A, while the
analytical solutions are given in table 1. The first layer of the
two-layer medium has the same properties as the homogeneous one-layer
medium. Although it was assumed that R=1, the analysis can be extended
to reactive solutes by using v (=v/R) and D (=D/R) instead of v and
D. With these solutions, time moments were determined according to
Eq. (27), as well as absolute moments according to Eq. (18) and central
19
moments according to Eq.(19).
Table 1. Solutions of the ADE in the Laplace Domain for a
One- and a Two-layer System with a First- and
Third-type Inlet Condition
Case Layer Condition Solution
A 1 first f ea (Alx)
vf
B 1 third -D X (A x)
1 1 2
C 2 first ea p(A L + A (x-L))
1 1 2 1
S v f
D 2 third ep(A L + X (x-L ))
0 v -D 2i A 11 2 1
2 2 2 2
A 1 and A are defined in Appendix A and the subscripts 1 and 2
denote first and second layers, respectively.
The procedure to determine these time moments is briefly outlined in
Appendix B, while the resulting expressions for moments up to order
three are presented in Appendix C. We will discuss the effects of the
inlet condition and layering on R and D in terms of these results.
The time moments most commonly used in BTC analysis are pl and 2
(10). The expressions for l and 2 for the four cases listed in table
1 are given in table 2.
The effect of the inlet condition can be evaluated by comparing
case A and B:
D
Breakthrough time (pl)B 
(  A  
2
v
1
D2
Spreading : (p2B- 
( 2 A 
= 3 4
1
20
Table 2. Expressions for g and WL for 
Transport in a One-
and a Two-layer- Medium2 Subject to a 
First- and a
Third-type 
Condition
case 'I
A
B 2
2
x
x
V
1
L
1
V
1
L
1
V
1
D
2
v
x-L
1
2
x-L
1
2
0
0
0
12
D
xD
1 1
3
v
t1
121
XD 
D 2
+2 
+ 3 N
3
v v
L D (x-L )D
+2 -+2
3 3
v v
1 2
L D 
(x-L)D 
D2
2 11+ 
1 2 3 2
3 3 4
1 2 2
It appears that. breakthrough time and 
spreading increase for a
third-type condition compared to 
a f irst-type condition. This
will be referred to as the "inlet effect."
The effect of layering will be evaluated 
by comparing case B and
D, i.e., for a third-type condition, 
assuming that v =v. At the
1 2
interface we'll let x approach L from x>L, 
i.e. , i Comparing the
1 1L
first absolute moment and the second central 
moment for case B and D we
get:
Spreading: 
(A)(P) = 
1(D -D)+-(D-D 
)
2D2B 3 2 1 4 2 1
v v
S1 3 22
( CD 2D
X2L L2D2BJ I 4 2 1
21
The difference in I indicative for the effect of layering on the
breakthrough time, can be viewed as the inlet effect for layer two,
2 2
D /v ,minus the inlet effect for layer one, D /v. As soon as the
2 2' 1 1
pulse enters layer two, the contribution of the inlet effect for layer
one will disappear. The breakthrough time in the two-layer medium can
therefore be derived from the breakthrough times in two homogeneous
media which have the same properties as the two layers. The fact that
an interface is present does not influence the breakthrough time. To
investigate whether layering introduces additional spreading, a similar
approach can be followed. Differences in p2 for the one- and two-layer
2 4
medium are caused by the "inlet effect", 3D2/v. It appears that there
is no particular "interface effect", the inlet effect for the first
layer is substituted by an inlet effect for the second layer at the
interface.
As mentioned earlier, theoretical moments can be used to
determine transport parameters. These moments depend on the physical
model chosen to describe solute transport. The expressions in Appendix
C were based on the ADE, applicable to equilibrium transport with a
constant R value, but other models could be used as well. Additional
transport parameters, for more complicated transport models, can also
be determined using theoretical moments and experimental data. It
should finally be noted that expressions for these moments also depend
on the mathematical model, i.e., the conditions at the interface,
chosen to describe solute transport.
MATERIALS AND METHODS
Four types of studies were conducted to study the problems
outlined in the introduction. First, a pilot study was carried out with
homogeneous soils to get acquainted with experimental procedures and to
evaluate the reliability of various techniques. Second, step-type
displacement experiments were carried out in layered soils with binary
exchange. The third and fourth study involved pulse-type displacement
for a binary system (different cations and anions) and a ternary system
(three different cations) in a homogeneous soil.
The experimental setup is illustrated in figure 4. Cylindrical
plexiglass columns of 15 or 30 cm length, with an internal diameter of
6 cm, were used for the displacement studies. The plexiglass cylinders
were enclosed by plexiglass endcaps. A piece of cheese cloth was placed
in each end cap to prevent loss of soil during the experiments. Each
column was carefully packed with air dried soil that had first been
passed through a 2-mm sieve and subsequently through another sieve to
obtain the desired size fraction. Soil types and their taxonomic
classification are listed in table 3. All soils are subsoils, which
were collected at Alabama Agricultural Experiment Station research
units. The dry soil weight and volume for each column were determined
Table 3. Classification of Soils
Soil series Family description
Dothan Fine-loamy, siliceous, thermic Plinthic Paleudults
Wickham Fine-loamy, mixed, thermic Typic Hapleudults
Savannah Fine-loamy, siliceous, thermic Typic Fragiudults
Lucedale Fine-loamy, siliceous, thermic Rhodic Paleudults
Troup Loamy, siliceous, thermic Grossarenic Paleudults
22
23
Masterfiex tubing
pump
supply bottles
top cap
plexiglass column
bottom cap
soil
cheese cloth
support rod
O-rings
fraction 
collector
FIG.4. Schematic of experimental setup.
gravimetrically to determine porosity, c, and dry bulk density, Pb The
soil was then slowly saturated from the bottom with water containing a
specific solute with concentration C . In the pilot study, a constant
o
.0
head device (Mariotte bottle) was used to apply the solution. However,
substantial variations in flow rate occurred. The Mariotte bottle
device was, therefore, replaced with a Masterflex constant volume pump
(Cole-Parmer, Chicago, Illinois) in all later studies.
24
After saturating the soil from the bottom and establishing steady
outflow, the columns were inverted to obtain vertical downward flow.
Solute concentrations of the effluent and, occasionally, of the eluent
were monitored to verify whether or not the soil was saturated with the
particular solute. Upon saturation with the resident solute, the input
line was switched, at time t=O, to the reservoir containing the
displacing solute(s). It is noted that the volume of solution in the
lines was kept at a minimum. Effluent solutions were collected in test
tubes by means of a fraction collector (Buchler, Fort Lee, New Jersey).
No tubing was used at the column outlet. For the pulse-type
displacement, the supply line was switched back to the original
solution at time t . At the end of each experiment, the weights of the
0
(saturated) soil column and air dry soil were determined to obtain
effective values for the volumetric water content and the pore volume.
The flow rate during the experiment was determined gravimetrically at
regular time intervals. Various solutes were used in the experiments,
namely Ca, Na, or K as cations and Cl or Br as anions. Ca and K
concentrations were determined with the ICAP, whereas Na concentrations
were determined with flame emission spectrophotometry. The Cl and Br
concentrations were determined with the HPLC.
RESULTS AND DISCUSSION
Pilot Study
BTC's for the pilot study are shown in figures 5 to 10, with the
experimental conditions given in table 4. In the figures, DBD and VWC
denote pb and 0. Al figures show the dimensionless concentration as a
function of dimensionless time (T=vt/L). The first two experiments
involved the displacement of a 0.1 M CaC12 solution from a Troup soil
by solute free water. The BTC's are rather steep and symmetrical.
Breakthrough of the front for C/C =0:.5 occurred after sligthly more
0
than one pore volume, indicating that some of the Ca was present as an
"adsorbed" phase.
Table 4. Experimental Conditions for Solute Displacement
During Pilot Study
Experiment Soil Size Solute Pb 8 v PV
# fraction Res. Dis.
-3 -1 3
tm g cm cm d cm
1 Troup 250-500 CaCl2 - 1.91 0.28 1304.6 175
2 Troup 250-500 CaCl - 1.92 0.28 1208.2 173
3 Lucedale <840 CaCl
2 
CaBr
2  
1.58 0.40 17.8 251
4 Lucedale <840 CaCl CaBr
2  
1. S9 0.40 67.8 249
5 Wickham <500 CaCl CaBr
2  
1.43 0.46 8.61 286
6 Wickham 500-1000 CaCl2 CaBr2 1.29 0.52 262.7 321
L = 30.6 cm.
Res. and Dis. denote resident and displacing solute, respectively,
and PV is pore volume.
25
26
BTC #1
Troup
DBD = 1.91 g/cm3
VWC = 0.28
v = 1305 cm/d
PV = 175 cm3
2.0
# PORE VOLUMES
FIG.5. BTC #1: Ca curve for Troup 
soil.
BTC #2 Troup
DBD = 1.92 g/cm3
VWC = 0.28
v = 1208 cm/d
PV = 173 cm3
calcium
0.0
.00 0.5 
1.0 1.5 
2.0
# PORE VOLUMES
FIG.6. BTC #2: Ca curve for Troup 
soil.
o O.
0
O0.,
0. o
0.(
0
0
0D
.
1.(
OJ
BTC -#3
27
Lucedale
bromide
DBD = 1.58 g/cm3
VWC = 0.40
v = 17.8 cm/d
PV= 251 cm3
chloride
QQ T, 4. *It, IT,
0.0 0.5 1.0 1.5 2.0 
2.53.
# PORE VOLUMES
FIG.7. BTC #3: Br and Cl curves for Lucedale soil.
BTC #4
1)
Lucedal e
bromide
DBD = 1.59 g/cm5
VWC = .0.40
v = 67.8 cm/d
PV = 249 cm3
chloride
1.5 2.0 2.5 3.0
#PORE VOLUMES
FIG.8. BTC #4: Br and Cl curves for Lucedale soil.
01.
0
C-)
C-)
04
0
C)
C-)
0.5 1.0
28
BTC #5
0.21
0.0 0.5. 1.0
i ckham
bromide
DBD = 1.43,g/cm3
VWC = 0.46
v = 8.61 cm/d
PV = -286 cm3
chloride
1.5 2.0 2.5 3.0
# PORE-VOLUMES
FIG.9. BTC #5: 
Br and Cl curves 
for Wickham soil.
BI-C #6
Wickham
bromide
DBD = 1.29 g/cm3
VWC = 0.52
v= 262.7 cm/d
PV = 321 cm3
chloride
1.5 2.0 2.5
3.0
# PORE VOLUMES
FIG.1O. BTC #6: Br and 
Cl curves for Wickham 
soil.
0A
0
C)
C)
C-)
0.5 1.0
29
Figures 7 to 10 show the results for the displacement of 0.01 M
CaCl
2 
by 0.01 M CaBr
. 
Both Cl and Br 
concentrations 
in the effluent
were determined. The BTC's were not symmetrical about C/C =0.5 for the
0
Lucedale soil, suggesting that non-equilibrium conditions existed. The
Wickham soil exhibits a more symmetric front, particularly at the lower
value for v (figure 9). For these experiments, breakthrough generally
occurred before one pore volume had passed, indicating the possibility
of anion exclusion. However, variations in flow rate were observed,
which affected the reliability of the number of pore volumes actually
leached through the column. As mentioned earlier, a constant volume
pump was used in later studies and the outflow rate was determined at
frequent intervals.
The transport parameters for experiments 1 to 6 were first
determined with some of the methods obtained from the literature (i.e.,
based on Eq.(7)). Table 5 contains the results obtained using Eq.(8),
Table 5. Determination of Dispersion Coefficients According
to Fried and Combarnous (7)
Experiment Solute t t v D
.0.16 0.84
-1 2 -1
d d cm d cm d
1 Ca 0.025 0.029 1304 104.1
2 Ca 0.026 0.030 1208 70.7
3 C1 0.928 2.647 17.8 76.7
Br 0.865 2.690 17.8 90.4
4 C 1 0. 230 0.668 67.8 303.4
Br 0. 225 0. 693 67.8 338.5S
5 Cl 2. 772 4.5B14 8.61 7.87
Br 2. 914 4. 371 8.61 5.14
6 Cl 0. 089 0. 204 262.7 712.9
Br 0. 088 0. 176 262.7 483.4
30
Table 6. Determination of Dispersion Coefficients Using
Liweafc(2C ) = MT +t
e
Experiment Solute a P D C 
(P) R
# e
cm d
1 Ca -9.640 370.7 107.6 0.50 1.12
2 Ca -9.589 366.8 100.5 0.50 
1.12
3 Cl 1.322 5.99 90.9 0.50 
0.92
Br -1.267 5.43 100.5 0.50 0.83
4'Cl 1.333 6.11 339.7 0.50 0.83
Br -1.405 6.90 300.8 0.50 0.83
5 Cl 2.746 29.2 9.03 0.49 
1.05
Br -3.295 42.4 6.21 0.48 1.00
6 Cl 1.710 10.7 751.5 0.50 1.11
Br -2.132 17.2 468.3 0.50 1.04
t Semi-infinite medium, third-type condition, 0.1<C <0.9 (24).
whereas table 6 shows the results derived using Eq. (12). 
The slope a
was determined by plotting C =C/C as a function of en T on 
probability
e o
paper.
To determine transport parameters via curve fitting, 
the
concentrations of resident and displacing anions were 
expressed as
(C -C)/C and C/C
0  
respectively. The input files, containing the
0 0 O
BTC's, for these experiments as well as all later 
experiments are
listed in Appendix D. Table 7 contains parameters obtained 
with the
program CXTFIT assuming equilibrium and physical non-equilibrium
conditions. Comparing the results obtained with the methods 
based on
Eq. (7) and the results from CXTFIT, it appears that D determined 
by the
first method (table 5), and D and R determined by the second method
(table 6), are of the same magnitude as D and R determined 
by the curve
fitting procedure assuming physical equilibrium (table 7). 
Insight into
31
Table 7. Determination of Transport Parameters Using CXTFIT (13)
Holdup Equilibrium Non-equilibrium
# CDMt H R D R 0 0. a D R
Solute 
mo im
2 -1 c
cm dd cm2d-
1 Ca r 1.155 105.4 1.127 0.28 0.00 1620 97.5 1.121
f 1.158 1.158 105.3 1.130 0.28 0.00 2630 97.4 1.124
2 Ca r 1.210 98.0 1.125 0.28 0.00 1863 89.0 1.118
f 1.213 1.213 97.9 1.128 0.27 0.01 4.09 57.4 1.131
3 C1 r 0.851 88.9 0.878 0.20 0.20 0.38 2.65 1.030
f 0.983 0.983 84.5 1.026 0.34 0.06 24.9 87.3 1.046
Br r 0.823 78.6 0.817 0.24 0.16 0.26 8.25 0.943
f 0.936 0.936 75.0 0.938 0.40 0.00 0 70.4 0.901
4 Cl r 0.795 377.0 0.839 0.21 0.29 1.52 73.7 0.955
f 0.931 0.931 354.8 0.996 0.22 0.28 1.49 85.5 0.989
Br r 0.814 328.0 0.820 0.17 0.23 2.26 0.96 0.946
f 0.935 0.935 309.5 0.952 0.17 0.23 2.26 0.99 0.947
5 C1 r 1.002 8.71 1.034 0.46 0.00 0 8.04 1.016
f 1.035 1.035 8.57 1.068 0.00 0.46 4.43 0.63 1.060
Br r 1.010 6.13 0.998 0.46 0.00 c 5.76 0.981
f 1.033 1.033 6.06 1.021 0.00 0.46 33.0 5.00 1.019
6 C1 r 1.123 719.1 1.108 0.45 0.07 0.49 496.2 1.208
f 1.220 1.220 694.9 1.209 0.34 0.18 0 703.9 1.216
Br r 1.067 508.5 1.051 0.52 0.00 o 470.0 1.026
f 1.133 1.133 495.6 1.118 0.52 0.00 W 495.6 1.118
t Concentration Detection Mode: r=resident, f=flux-averaged.
# is experiment number.
the nature of the exchange process can be gained from examining the
solute retardation. For the Troup soil (#1 and 2), R is slightly over
1, indicating that only a minor part of the displaced Ca was adsorbed,
as would be expected for the high initial solute concentration (0.1 M)
and sandy soil texture. For the Lucedale soil (#3 and 4) R<1, implying
that anion exclusion occurred. In contrast, some retardation of the
anion took place for the Wickham soil (# and 6), possibly due to
adsorption by Fe-oxides.
32
In examining the equilibrium versus non-equilibrium results
(table 7), it should be noted that the values for D according to the
equilibrium model are generally higher than those based on the
non-equilibrium model. This supports the idea that part of the
spreading can be explained on the basis of non-equilibrium.
It seems reasonable to assume that transport parameters
determined for Cl and Br by the same model should be approximately
equal for a given soil column if the anions are non-reactive. However,
especially for experiment 4, very different D values were found for Cl
and Br using the non-equilibrium model. The values for 0 and 0. also
mo im
show considerable differences. Experiment 6 can serve as an example; if
C1 is the tracer, the model predicts the existence of both mobile and
immobile regions in the liquid phase, whereas 0. =0 for Br. These
im
discrepancies may have been caused by poor initial estimates of the
transport parameters in the program CXTFIT and the high number of
unknowns (i.e., five), which can create uniqueness problems. For the
equilibrium model, the values for D are very much the same for C1 and
Br. It appears that fitting data with the non-equilibrium model does
not necessarily provide reliable parameters. The equilibrium model was,
therefore, used 
in the remaining 
part of this study, 
although
non-equilibrium conditions might have existed in some cases.
The holdup, according to Eq.(15), was determined from the BTC
using the trapezoidal rule. In a number of cases, the BTC was
extrapolated to asymptotically approach C/C =1 to "enure complete
0
displacement. The results are included in table 7. Subsequently, values
for R were determined according to Eq. (16), using "equilibrium"
33
D values for a third-type condition. The values for R based on H
correspond roughly to those found by the other methods. The
discrepancies between values for R for experiments 3 and 4, obtained
with H using a first- and a third-type condition, can be attributed to
the low values of the column Peclet number (cf. Eq.(16)).
Transport in Layered Media with Binary Exchange
The experimental conditions for the displacement studies in
layered media are listed in table 8. The soil materials possessed a
wide variety of chemical and physical properties. Note that v, 0, and
PV are averaged over all layeri. PV and 0 were found from the
difference in wet and dry weight of the column. The Darcy flux and v,
assuming steady flow, were obtained by determining the amount of
effluent over time.
Table 8. Experimental Conditions for
Layered Media
Solute Displacement in
Experiment Soil Depth Pb c 0 v PV
#
-3 -1 3
cm g cm cm d cm
11 Troup 0-12.5 1.64 0.38
Savannah I 12.5-17.5 1.37 0.48 0.38 132.63 232.7
Troup 17.5S-30 1.67 0.36
12 Dothan I 0-15 1.60 0.39
Dothan II 15-30 1.29 0.51
13 Troup 0-15 1.65 0.38
Lucedale II 
15-30 1.42 
0.470. 317 140.2 
191.4
I <250 pm.
II 5 600-840 pm.
PV is mean pore volume of combined
Resident solution : 0.01 M KBr.
Displacing 
solution 0.005S 
M CaBr
2
.
layers.
34
The resulting BTC's are shown in figures 11 to 13. The K
concentration decreases after a number of pore volumes has passed,
indicating that considerable retardation of Ca occurred. These curves
appear to be fairly symmetrical; the data were fitted with CXTFIT using
the equilibrium model, to determine D and R. The fitted parameters and
2
r , quantifying correlation between fitted and experimental curves, are
given in table 9. It appears that an excellent fit was obtained,
yielding values for the transport parameters which were averaged over
the layers. This illustrates that curve fitting can produce seemingly
reliable values, when in fact no physical basis exists to determine D
and R from matching a theoretical (based on a homogeneous soil) with an
experimental BTC for a layered medium. Also included in table 9 are
values for R based on H using a first- and third-type condition. It
should be noted that H is equal to R for a first-type condition. The
correspondence between the retardation factors obtained with curve
fitting and column holdup is excellent.
Table 9. Determination of Transport Parameters for Step
Displacement in Layered Media with CXTFIT and
Column Holdup
# Solute Dt Rt r R$
2 -1 Condition
cmd- First Third
11 Ca 57.858 4.120 0.996 4.136 4.077
K 60.513 4.088 0.996 4.208 4.145
12 Ca 55.266 10.884 0.998 10.881 10.688
K 53.667 10.818 0.996 11.072 10.881
13 Ca 21.017 9.220 0.996 9.325 9.278
K 24.930 9.237 0.994 9.323 9.268
#1
1
$
Experiment number.
CXTFIT.
Column holdup.
35
BTC #11 Troup & Savannah
calcium
0.0-12.5 cm TRO
12.5-17.5 cm SAV I
17.5-30.0 cm TRO
potassium
5.0 7.5 10.0 12.5 15.0
# PORE VOLUMES
FIG.11. BTC #11: Ca and K curves for Troup 
and Savannah soil.
BTC #12 Dothan
20
# PORE VOLUMES
FIG.12. BTC #12: Ca and K curves for Dothan soil.
0
(9
(9
2.5
++
0
C)
0)
36
BTC #13 Troup & Lucedale
0
(-)
C)
0 5 10 15
# PORE VOLUMES
20
FIG. 13. BTC #13: Ca and K curves for Troup 
and Lucedale soil.
The results suggest that the BTC for 
a layered medium might also
be obtained with an equivalent, uniform 
medium. The layering does not
seem to have a particular effect on 
the effluent concentration. This
was already concluded (i.e., page 20 and 21) 
based on the results of
Appendix C, which contain expressions for 
regular, absolute, and
central moments for the cases listed in table 
1. Therefore, no further
investigations of transport in layered media, e.g., an 
increase in the
number of layers, numerical predictions of the BTC, 
pulse studies, were
initiated.
37
Transport in Homogeneous Media with Binary Exchange
Simultaneous cation and anion displacement experiments were
carried out to investigate the effect of possible changes in CEC during
transport and to investigate the influence of non-linear exchange on
transport. These investigations were accomplished by temporarily
replacing the 0.005 M CaBr eluent (resident) solution with a 0.01 M
2
NaCl eluent (pulse) during continuous leaching.
The experimental conditions for the binary exchange study are
listed in table 10. Values for 0 and v are given as average values
because they vary along the column and over time, respectively. T
0
denotes the number of pore volumes at which the input line was switched
back to the original solution. Its value was obtained from the
gravimetrically determined effluent volumes. The BTC's are shown in
figures 14 through 21, the symbols denote experimentally determined
points and the solid lines are fitted by eye. From these curves it
appears that more spreading takes place for the cations than for the
Table 10. Experimental Conditions for Pulse Displacement
with Binary Exchange
Experiment Soil Pb c 0 v PV t T L
-3 -
gcm cm d cm d cm
21 Dothan I 1.15 0.57 0.55 100.2 167.1 0.208 1.416 14.8
22 Dothan II 1.23 0.54 0.48 100.0 151.3 0.208 1.376 15.5
23 Wickham I 1.21 0.54 0.43 128.1 132.3 0.208 1.760 15.1
24 Wickham II 1.29 0.51 0.41 116.5 128.4 0.208 1.584 15.3
25 Troup 1.67 0.37 0.31 146.0 93.0 0.347 3.414 14.8
26 Lucedale I 1. 32 0.54 0.49 74.1I 154.0 0.347 1. 165 15.5
27 Lucedale II 1.26 0.52 0.51 91.5 156.2 0.347 2. 080 15.1
28 Savannah I 1.42 0.47 0.42 118.8 130.1 0.347 2. 595 15.3
Resident solution: 0.005 N CaBr
2
.
Pulse solution : 0.01 N NaCl.
I : <250 rim.
II : 500-840 ilm.
38
BTC #21 Dotha2
calcium
DBD 1.15 g/cm3
VWC .= 0.55
v = 100.2 cm/d
PV =167.1 Qcm3
To =1. 416
sodium
5.0 7.5 10.0 12.5
15.0
#POHE VOLUMES
o
0*
0.
0.,
0.(
~0.
bromide
DD= 1.15 g/cm,3
VWC = 0.55
v = 100.2 cm/d
PV = 167.1 cm3
.To =1. 416
chloride
0.0 2.5 5.0 7.5 10.0 12.5 15.0
# PRE VOLUMES
FIG. 14. BTC 1#21: Ca. Na, Br, and Cl curves for Dothan I.
).0 2.5
BTC #'21 DothanI
I k
0
39
BTC #22, Dothan
Calcium
DBD = 1.23 g/cm3
VWC = 0.48
v =100.0. cm/d
PV = 151.3 cm3
To = 1.376
sodium
5.0 7.5 110.0 12.5
PRE VOLUMES
BIC #22 Dothan 1
DBD = 1.23 g/c
VWC = 0.48
v =100.0 cm/
PV = 151.3 crr
To = 1.376
chloride
0.0 2.5 5.0 7.5
10.0 12.5
#POPE VOLUMES
FIG.1.5. BTC #22: Ca, Na, Br, and Cl curves 
for Dothan II.
o )'
0..
2.5
1)
0'-"
00.
0.4
15.0
m3
d
15.0!
V
OJ
*
calcium
DBD = 121 g/cm3
VWC =0.43
v= 128.1 cm/d
PV =* 132.3 cm3
To =- 1.76
sodium
5.0 7.5 10.0 12.5.
#PORE VOLUMES
BIC #23 Wickham 'I
bromide
DBD = 1.21 g/c
VWC = 0.43
v 128.1 cm/
PV =132.3 or
To = 1.76
chloride
2.5 5.0 7.5 10.0 12.5
#PORE VOLUMES
FIG.16. BTC #23: Ca, Na, Br, and Cl curves for Wickham 
I.
40
BTC #23 W*ickhamI
1.4
0.(
.0
0.
2'.5
0A
0.:
0
15.0
~m
TM3
15.0
41
BTC #24 Wickham 11
calcium
DBD= 1.29 g/cm3
VWC = 0.41
v= 116.5 cm/d
PV =128.4 om3
To =1.584
sodium
0.Q /0
0.0 2.5 5.0 7.5 10.0
12.5 15.0
# PORE VOLUMES
BTC #24 Wickham
* bromide
DBD = 1.29 g/cm3
VWC = 0.41
v 116.5 cm/d
PV 128.4 cm3
To = 1.584.
chloride
W0 2.5
5.0 7.5 10.0 12.5
15.0
# PORE VOLUMES
FIG.17. BTC #124: Ca, Na, Br', and Cl curves for Wickham II.
0.(
\1
0
V
0c'
0.
42
BTC #25 Troup
5.0
1.
PORE VOLUMES
BIC #25 Troup
bromide
DBD 1.67 g/cm3
VWC = 0.31
v = 146.0 cm/d
PV = 93.0 cm3
T o =3.414
+chloride
ii PRE +
5.0 7.5 10.0 12.5 15.0
PORE VOLUMES
FIG.18. BTC 425 Ca, Na, Br, and Cl curves for Troup soil.
calcium
060 =1.67 'g/cm3
VWC = 0.31
v 146 cm/d
PV =95.0 omS
T o =3.414
sodium
7.5- 10.0 12L5 15.0
o ~
00
0.
2.5
0.
0
03
C)
) 2.5
43
BIC #26 Lucedale' 
I
calcium
DBD = 1.32 g/cm3
VWC = 0.49
v= 74.1 cm/d
PV =154.0 cm3
To =1.165
sodium
0.0 2 .5 5. 7.5 10.0 12.
15.0
RE VOLUMES
BTC #26 LucedaleI
** * bromide
DBD= 1.32 g/cm3
VWC =0.49
v 74.1 cm/d
PV= 154.0 cm3
To = 1.165
c choride
2.5 5.0 7.5 10.0, 12.5
# POHE VOLUMES
FIG.19. BTC #126: Ca, Na, Br, and Cl curves for Lucedale I.
0
0)
0)
0.
15.0
44
BTC #27 Lucedale 11
calcium
DBD 1.26 g/cm3
VWC = 0.51
v 91.5 cm/d
PV 156.2 cm3
To = 2.080
sodium
2.5 5.0 7.5 10.0
12.5 15.0
PORE VOLUMES
BTC #27
Lucedcle 1
Sbromide
DBD = 1.26 g/cm3
VWO = 0.51
v 91.5 cm/d
P V 15 6.2 cm 3'
To = 2.080
chloride
0.0 2.5 5.0 
7.5 10.0 12.5
15.0
# PORE VOLUMES
FIG.20. BTC #27: Ca, Na, 
Br, and Ci curves for Lucedale 
II.
1.0
0
0 19
0
C-)
C)
45
BTC #28 Savannah
calcium
DBD 1.42 g/cm3
VWC = 0.42
v = 118.8 cm/d
P V = 13 0.1 cm3
To = 2.595
s odium
7.5 10.0 12.5 15.0
#POFRE.VOLUMES
BTC #28- Savannah I
bromide
DBD 1.42 g/cm3
VWC = 0.42
v = 118.8 cm/d
PV = 130.1- cm3
To - 2.595
chloride
7.5 10.0
12.5 1500
PORE VOLUMES
FIG.21. BTC #28: Ca, Na, Br, and Cl curves for Savannah I.
00(
2,5
5.0
0
0.
2.5 5.0
46
anions. The peak values of C/C are higher for the anions than for the
0
cations. Both cations and anions were retarded, except for the
displacement in the Troup soil (figure 18). However, the anions were
retarded to a lesser extent than the cations. In a number of cases
tailing was observed (e.g., figures 14 and 17).
For further analysis, the C(T)/C relationship was used for Na
0
and C1 and the 1-C(T)/C relationship for Ca and Br. Values for D and R
0
were obtained with the program CXTFIT for a flux-averaged concentration
assuming equilibrium conditions. The results are listed in table 11.
2
Although the values for r2 are smaller than for the experiments
involving layered media, they still show a good correlation between
experimentally determined and fitted curves. The "inverse" Ca pulse
generally exhibits somewhat more retardation than the Na 
pulse,
presumably because more Ca than Na is sorbed by the soil. The
differences in R values (table 11) for the anions are rather small,
with the exception of experiment 21 where non-equilibrium conditions
seemed to exist. The larger values of D for the cations than the anions
(except experiment 21) confirm the impression that more spreading
occurred for the cations.
It should be noted that for binary systems with constant solute
concentrations in the liquid and the adsorbed phases, the
concentrations of pulse and resident solutions complement each other.
Theoretically, the values for D and R obtained from the displacing and
resident solute concentration curves (anions or cations) should,
therefore, be equal. This is not always the case, because of
non-equilibrium conditions and changes in apparent exchange capacity.
47
Table 11. Determination of
Binary Exchange
Transport Parameters with
Using CXTFIT
Experiment Solute D R r2AEC
r#E
21
22
23
24
25
26
27
28
Ca
Na
Br
Cl
Ca
Na
Br
Cl
Ca
Na
Br
Cl
Ca
Na
Br
C1l
Ca
Na
Br
Cl
Ca
Na
Br
C1l
Ca
Na
Br
C1l
Ca
Na
Br
C1l
cm d
-
781.56
618.43
1326.5
543.75
64.48
60.05
39.72
29.68
71.32
70.48
21.37
32.61
120.08
102.65
93.42
87.31
23.53
48.46
17.60
20.26
58.41
77.51
36.18
32.22
50.65
76.91
6.19
13.06
124.69
128.11
91.73
87.38
mol /kg
2.596
1.914
2.343
1.459
1.708
1.587
1.285
1.245
3.041
2.881
1.590
1.585
3.013
2.884
1.389
1.487
1.055
0.963
1.008
1.011
2.804
2. 525
1.406
1.364
3.474
3.425
1.173
1.196
2.229
2.227
1.003
0.9942
0.968
0.898
0.943
0. 927
0.967
0.964
0.977
0.983
0.967
0.985
0.973
0.962
0.988
0.952
0.981
0.982
0.994
0.952
0.991
0.948
0.964
0.972
0.964
0.977
0.918
0.914
0.983
0.962
0.943
0.948
0.953
0.937
0.642
0.220
0.111
0.096
0.210
0.208
0.124
0. 155
0.001
0. 002
0. 11
0.135
0.070
0. 079
0.001
-0.002
Second, the moment method was used to analyze the BTC' s and
determine values for D and R (table 12). Moments up to order three were
determined via numerical integration of the BTC according to Eq.(21).
The necessary integrations were performed with the trapezoidal rule
P, C)~3 C? 1 17
48
Table 12. Determination of Transport Parameters with Binary
Exchange Using Time Moments
#*Solute M
0
M 
1
M 
2
M 
3
Al 2 R D PEt
21 Ca
Na
Br
Cl
1.420
1.416
1.338
1.496
RMD* 2.72
22 Ca
Na
Br
Cl
1.373
1.595
1.430
1.249
3. 956
4.095
3.451
3. 073
14. 573
15. 702
14.086
7. 807
65. 245
72.694
83.286
23. 052
d 
d2
0.411 0.0546
0.427 0.0595
0. 381 0.0845
0.303 0.0218
1.64. 9.87 21.60
3.463
3.692
3.190
2.412
RMD[%] 0.73 4.30
23 Ca
Na
Br
Cl
1.830
1.983
2.090
1.604
7.436
8.202
6. 627
4.042
RMD[%] 4.44 6.91
24 Ca
Na
Br.
Cl
1.640
1.584
1.593
1.735
6.602
6.587
3. 900
4. 117
RMD[%] 1.31 0.95
25 Ca
Na
Br
Cl
3.825
3.319
3. 387
3. 391
RMD[%] 3.61
26 Ca
Na
Br
Cl
1.820
1.801
1.0 534
1.469
RMD[%] 1.27
27 Ca
Na
Br
Cl
2. 201
2.077
2. 268
1. 909*
14. 531
9. 204
9.372
9.869
10. 565
9.634
8.743
4.895
43.084
29.433
32. 169
10. 389
14.24 30.79
37. 225
38.691
27.514
10.883
256. 06
213.07
155. 57
30.884
13.32 25.58
30.983
31. 945
10. 886
10. 674
172.91
179.33
35.057
30.090
0. 390
0.358-
0.345
0.299
0.479
0.487
0.374
0.297
0.528
0.545
0.321
0.311
0.0319
0.0163
0. 0272
0.00454
0.0532
0.0334
0.0432
0.00603
0. 0462
0. 0494
0.0143
0.00896
0.89 0.35
95.293
29.064
29. 572
34. 218
763.61
100.27
103.59
138.76
0.385 0.1077
0.281. 0.0110
0.280 0.0110
0.295 0.0166
10.46 32.73 56.78
6.651
5.924
3.734
3.278
26.920
21.180
9. 817
7. 678
121.42
81. 185
27. 707
18. 767
6.04 12.01 19.74
9. 439
8. 696
5. 704
4. 231
43.470
38. 775
17.221
10. 009
211.90
181.95
63.429
24. 993
0.745
0. 671
0. 497
0. 455
0.708
0.691
0. 567
0. 575
0.0598
0. 0391
0. 0197
0.0103
0.0370
0. 0310
0.0345
0.0090
2.078
2.184
1.871
1.346
1.834
1.627
1.543
1.243
3. 183
3.256
2. 291
1.640
3.234
3. 366
1.656
1.581
2.093
1.068
1.062
1.205
2.822
2.457
1.602
1.399
3.249
3. 147
1.475
1. 176
cm2 
d-
401.4
398.3
785. 4
341.3
0.28
5.51
5.65
ENE? =0.04%
271.4 0.22
154.7 15.92
319.7 3.92
19.51 9.23
ENE=5. 12%
340.7
195.6
525. 1
62.8
3.98
12.67
18.75
8. 86
ENE=1. 59%
210.4
208.9
201.5
110.7
3.54
0.57
9.53
ENE= 1. 59%
2344
89.0
90.0
475. 4
12. 13
2.70
0.70
0.59
ENE=5. 26%
82.0
63.2
49.4
1.78
9.31
8.17
7.88
11.77
ENE=9. 33%
64.8
53.7
285.3
-41.0
5.82
0.14
9.04
8.22
RMD[%]5. 71 7.89 10.88 14.18
Cont 1 nued
ENE=1. 19%
49
Table 12. Determination of Transport Parameters with Binary
Exchange Using Time Moments
# Solute MO M
1  
M
2  
M
3  
R D PE
2 2 -1
d d cm d %
28 Ca 2.611 8.939 36.662 192.45 0.440 0.0384 2.126 343.9 0.62
Na 2.524 8.082 28.449 107.33 0.412 0.0168 1.905 102.2 2.74
Br 2.499 5.888 15.411 43.896 0.303 0.0102 1.069 8.1 3.70
Cl 2.338 5.295 13.178 35.240 0.291 0.0084 0.967 -92.6 9.90
RMD[%] 2.49 5.14 11.15 24.75 ENE=2.99%
t [ICT -M I/C0T]x100%.
$ (0.5/M) O.5(MCa+MBr)-MI + 0.5(MNa+MC1)-M x100% and
M = 0.25(Ma+MBr+MNa+M
C
1 
)
? I(MO)cat
- 
MOI/ 
Mx100%.
because the increments in T were not constant. The results (table 12)
include the values of several measures to evaluate the accuracy of the
experimentally determined BTC's. These measures, indicated as PE, RMD
and ENE, will be discussed first, after which attention will be focused
on the determination of D and R.
For each solute, a measure of the difference between the amount
of solute in the applied eluent pulse (C T ) and in the observed
o0o
effluent pulse (M
O ) 
was calculated. This measure, the pulse error PE,
is an indication of the reliability of the experiments due to errors in
analytical and experimental techniques and incomplete displacement. Its
value was less than 10% on average, which was considered acceptable.
Differences in values of the various moments of the BTC's for an
individual solute pulse (CaBr
2
or NaC1) compared to the average of that
50
moment for both pulses are given by the relative mean deviation, RMD.
The higher order moments exhibit greater differences, presumably
because of the inability to accurately determine the tail of the BTC,
resulting in larger values for RMD. The contribution of the tail to the
higher moments is relatively large. Errors in parameter estimation as a
result of experimental errors in the tailing portion of the BTC are a
well known disadvantage of the moment method although some
modifications can be made to overcome this problem (25).
The electroneutrality principle was evaluated by the relative
error, ENE, which was defined based on the difference between the
average mass of the cation pulse, (Mo,+Mo )/2, and the average mass
0,Ca O,Na
of the total solute pulse (M), (M +M +M +M Br)/4, for the
effluent. This relative error, ENE, is generally smaller than PE,
because it is averaged over two ionic pulses and does not account for
differences between the eluent and effluent pulse.
The sums of cations and anions were determined for the various
BTC's to investigate the hypothesis that the effective CEC is higher
for a Ca medium than for a Na medium. In such an event, the Na pulse
would decrease the cation equivalents in the adsorbed phase and,
therefore, temporarily increase the cation equivalents in the liquid
phase. Because of the electroneutrality principle, the anion
equivalents in the liquid phase should change correspondingly. Although
a slight increase in the total amount of cations in the effluent was
observed in some instances during the occurrence of the Na peak, the
low values for PE and ENE did not warrant the conclusion that the CEC
51
had changed. Such changes might not have happened anyhow, because the
pulse type displacement is of relatively short duration and the soil is
not saturated with the cation applied in the pulse. The BTC's, and
previously measured exchange curves (12), indicate that the majority of
adsorption sites in a Na/Ca system are always occupied by Ca during
these experiments, whereas changes in CEC presumably take place only
when highly favorably adsorbed Ca ions are actually displaced. Further
investigations are needed to pursue this issue, preferably by
determining solute concentrations in the adsorbed and liquid phase as a
function of distance rather than a function of time at the outlet.
Using moments allowed us to evaluate BTC's and calculate values
for D and R. First values for R in the binary systems (table 12) were
obtained according to Eq.(22). Second, values for g' (mean breakthrough
time) and p2 (average pulse spread) were determined according to
Eq.(18) and (24), respectively, where we used T instead of t (Eq.(9)).
These variables are useful to characterize the BTC. Third, the value of
the dispersion coefficient was determined by relating D and A2* Upon
substitution of v and D for v and D in the expression for 12 in
Appendix C, the following theoretical expression was obtained:
t
o  
LDR
W2
(L ) 
= 12 
+ 
2 3 (28)
v
where the "first-type" expression was used because the values of the
concentration are flux-averaged. This equation was solved for D, the
results of which are also included in table 12. The values for D vary
widely and include even some negative values. They do not seem very
reliable based on the BTC's shown in figures 14 to 21. This was
52
attributed to the small values of the experimentally determined W
2
(L)
as a result of the small amounts of solute spreading that occurred in
the soil columns. The part of the expression for p
2 
which is
22
independent of position, the inlet effect, t2/12, dominates and no
0
accurate prediction for D can therefore be made.
It should be noted that the more common approach is to obtain
BTC's at two positions, x
1 
and x
2 
(10). D is then determined according
to Eq. (26-b), which is the proper way to determine dispersion inside
the porous medium, particularly if boundary effects contribute
significantly to solute spreading (as in, e.g., short laboratory
columns).
Finally, an attempt was made to numerically predict the BTC's for
cation displacement using experimentally determined exchange isotherms
(12) and values for D determined from the simultaneous anion
displacement experiments using CXTFIT (table 11). Because the
dependency of the CEC value on the Ca and Na concentration was not
known, the CEC was assumed to be constant. Its value was determined
from the average R value for Ca and Na, presented in table 12, and pb
and 0, listed in table 10. Use of an average R value resulted in one
effective value for the CEC, i.e., the exchange was assumed to be
linear (20) over the observed jump in solute concentration. Eq. (2) and
(3) were solved numerically with a Crank-Nicolson scheme for a
third-type condition. Values for asac were obtained from Appendix E.
The predicted curves, along with the measured solute
concentrations, are shown in figure 22. The curves match the observed
53
BTC's with varying degrees of success. The average position of the
pulse is generally predicted fairly adequate as is the shape of the
pulse; Ca desorption is accompanied by spreading and Ca adsorption
accompanied by steepening of the front. In some instances, numerical
oscillations were encountered because of large variations in D and v
depending on the solute concentration in the liquid phase. Furthermore,
for some of the isotherms, S/a8C changed sign during passage of the
front. A scheme with a variable time step might have improved the
results. Also, some of the exchange isotherms were probably not highly
accurate, resulting in poor predictions of 8S/ac, and hence R according
to Eq. (3). However, most of the exchange isotherms seem to characterize
the exchange process reasonably well and could succesfully be used to
predict the BTC's numerically.
Values for the anion exchange capacity, AEC, were obtained by
using an average R value for C1 and Br from table 11 and values for pb
and 0 from table 10, i.e., similarly as for the CEC. It appears that
anion adsorption can generally not be ignored for these soil types.
Since values for the AEC are strongly dependent on pH, it should be
noted that the pH values of the effluent were generally close to that
of the 0.005 M CaBr
2
resident solution.
54
1.0
0.8-
o00.6-
"0.4-
0.2
0.0
0.(
BTC #21 Dothmn I
calcium
v 100.2
D -544.0
To = 1.4
:00CEO 
=0.705
+ sodium
0 2.5 5.0 7.5 10.0 12.5
# PORE VOLUMES
BTC #22 Dothan 11
cm/d
cm2/d
416
cmol/kg
0
15.0
v= 100.0 cm/d
0 34.7 cm2/d
To =1.376
CEC = 0.316 cmol/kg
sodium
#PORE VOLUMES
1.0~
0.81
BTC #23 WickhamrI
calcium.
4f.
0.6 + v 128.1 cm/d
+ 0D 2699 cm2/d
To-=1.76
0.4 + CECC 0.955 cmoi/kg
0.2. + sodium
0.0- + 
++ +
0.0 2.5 5.0 7.5 10.0 12.5 15.0
#.PORE VOLUMES
BDIG #27 Lucedole 11
1.0
calcium
0.8-
0.6- 4 v =91.5 cm/d
0= 9.624 cm2/d
0.4 CEOTo = 2.080
0.4- C = 0.991 cmol/kg
0.2 +
socflum
0.0- + _______________
0.0 2.5 5.0 .7.5 10.0 12.5 15.0
# PORE VOLUMES
#PORE VOLUMES
BTC #28 Savannah I
1.0 4
+ calcium
0.8 . ++
c)0.6- + + v 118.8 cm/d
+ D 89.56 cm /d
0.4 + + CEO -0,480 cmol/kg
0.2 
+ 
u
sodiumn
0.0.? -.. ~.4. + 4
0.0 ?.5 5.0 7.5 10.0 12.5 15.0
# PQRF VOLUMES
BTC #25 Trou~p
calcium
v= 146.0 cm/d
D =18.93 cn2/d
To = 3.414
CEO = 0.002 anol/k
sodiumn
0
.0L+__I,4 
J
4 
1 44 04--4 -----.
4-* 4
0.0 2.5 5.0 7.5 10.0 12.5 15.0
# PORE VOLUMES
FIG.22. Numerically predicted (solid lines), and experimentally
determined curves (symbols) for cation displacement in binary
systems.
0
0
0
55
Transport in Homogeneous Media with Ternary Exchange
To further investigate the effect of non-linear cation exchange,
BTC's were determined using three different cations. A pulse containing
a favorably (K) and an unfavorably (Na) exchanging solute was applied
to soils containing Ca as the resident cation. The experimental
conditions for these displacement studies are listed in table 13; the
same soil columns as for experiments 21-24 were used.
Table 13. Experimental Conditions for Solute Displacement
with Ternary Exchange
Experiment Soil Pb c 0 v PV t T L
-3 -1 3
gcm cm d cm d cm
31 Dothan I 1.15 0.57 0.55 99.24 167.1 0.434 2.838 14.8
32 Dothan II 1.23 0.53 0.48 98.25 151.3 0.434 2.699 15.4
33 Wickham I 1.36 0.49 0.47 98.55 145.8 0.306 1.999 15.1
34 Wickham II 1.29 0.51 0.45 120.0 138.4 0.306 2.392 15.3
Resident solution: 0.005 M
Pulse solution : 0.005 M
I : <250 pm.
II : 500-840 gm.
CaBr .
KBr and 0.005 M NaBr.
The BTC's for these experiments are shown in figures 23 to 26.
For all curves, Na appeared earlier in the effluent than K, because the
latter is favorably adsorbed. With the exception of maybe Dothan II
(figure 24), considerable tailing seemed to occur for K. Tailing was
less pronounced for Na. Based on this, and in light of the anion
displacement experiments which showed fairly symmetrical BTC's, we
conclude that tailing is primarily due to non-linear exchange and not
to physical non-equilibrium. The magnitude and location of the peaks
BTC #51
Dothan
*
Calcium
DBD - 1.15 g/cm3
VWC = 0. ,55
PV - 167.1 cm3
To = 2.838
potassium
7.5 10.0 12.5 15.0
FIG.23. BTC #31:
# ROFRE VOLUMES
Ca, Na, and K curves for Dothan 
I.
BTC #52 Dothan
calcium
DBD = 1.23 g/cm3
VWC = 0.48
v = 98.25 cm/d
PV = 151.3 c3
To =2.699
potassium
2.5 5.0 7.5 10.0 12.5 15.0
#ROFRE VOLUMES
FIG.24. BTC #32: Ca, K, and Na curves 
for'Dothan II.
56
1.(
0
2.5 5.0
0l
0.
57
BTC #33w ickhamI
calcium
DBD = 1.36 g/crn3
VWC =0.47
v= 98.55 cm/d
PV =145.8 om3
To -- 1.999
7.5 10.0 12.5 15.0
# PORE VOLUMES
FIG.25. BTC #33: Ca, K, and Na curves 
for Wickham I
BIC #
Wickham 11
0.0 2.5 5.0 7.5
DBD= 1.29 g/cm3
VWC = 0.45
v = 120.0 cm/d
PV = 138.4 om3
To = 2.392
potassium
10.0 12.5
15.0
J/PORE VOLUMES
FTG.26. BTC #34: Ca, K, and K curves 
for Wickham II.
WD
0.
5.0
0
C-)
C)
sodium
58
of the BTC's vary substantially, depending on the exchange process and
hydrodynamic dispersion. As for binary exchange, the Ca curve can be
viewed as complementary to the BTC for the other cations in the pulse,
viz. Na and K. In this ternary system the complementary Ca pulse can-
not be used for BTC analysis.
The transport parameters D and R were determined with the program
CXTFIT (table 14). The input files are given in Appendix D. In order to
use the program, C was chosen to be equal to the eluent concentration
of the solute, thus, C/C varies between 0 and 1. The differences in R
o
for the three cations are substantial. The retardation is largest for K
and smallest for Na. Ignoring the results for the complementary
Ca pulse, it appears that the values for D vary rather widely from soil
to soil. The D values for the binary systems showed considerably less
variation. This might be attributed to the effects of non-linear
Table 14. Determination of Transport Parameters
with Ternary Exchange Using CXTFIT
Experiment Solute 
D R 
r
cm2d
- 1
31 Ca 1315.8 3.978 0.973
K 639.72 5.489 0.902
Na 840.46 1.642 0.993
32 Ca 481.14 3.602 0.705
K 24.82 4.323 0.935
Na 44.79 1.325 0.982
33 Ca 1645.9 4.475 0.980
K 607.03 4.425 0.811
Na 1965.7 1.906 0.958
34 Ca 1022.6 5.886 0.577
K 81.84 11.022 0.836
Na 42.73 2.286 0.986
59
exchange, which occurred to a larger extent in the ternary experiments
than in the binary experiments (using the same columns).
Next, the moments of the BTC's were determined (table 15), as in
the section on binary exchange, to obtain values for D and R and to get
an impression of the reliability of the experimental BTC's. The errors
PE and RMD appear to be of similar magnitude as for the binary systems
(table 12). 'The relatively large amount of K n ot recovered in
experiments 31 and 33, indicated by the high values of the relative
pulse error, PE, was partly caused by incomplete displacement
(tailing). The values for R are roughly the same as those obtained with
CXTFIT (table 14). However, less outlying values for D were obtained
with the moment method than with CXTFIT (cf. #32 and 33).
It should be noted that a general advantage of the method of
moments is that the moments can be used to (explicitly) characterize
experimental concentration distributions in time or space, independent
of any model. Next, they can be used to determine parameters for a
certain model. With the curve fitting technique, experimental data are
directly fitted to the theoret ical solution of a particular model. This
does not give as much flexibility to determine transport parameters and
provides a less objective basis to characterize the concentration
distribution. Some disadvantages of the moment method are that the
method is not very accurate if the BTC exhibits substantial tailing and
60
Table 15. Determination of Transport Parameters with
Ternary Exchange Using Time Moments
# Solute M M M2 M3 2 R D PEt
d 
d cm
2
d
31 Ca 2.669 11.449 67.786 506.14 0.640 0.1556 2.871 560.4 5.95
K 1.236 7.116 50.458 423.35 0.859 0.1707 4.338 272.0 12.90
Na 1.432 4.240 16.416 78.77 0.442 0.0600 1.542 615.2 0.91
RMD* 0.02 0.41 0.68 0.40
32 Ca 2.615 10.554 49.563 257.13 0.633 0.0655 2.686 212.6 3.11
K 1.360 7.845 48.101 316.95 0.904 0.0514 4.419 56.3 0.78
Na 1.394 4.631 22.113 150.41 0.521 0.1186 1.973 814.0 3.30
RMD[%] 2.59 8.35 17.24 29 01
33 Ca 1.577 5.043 21.236 110.04 0.490 0.0761 2.198 448.0 21.11
K 0.689 2.900 15.564 104.14 0.645 0.1144 3.209 328.1 31.07
Na 0.940 2.300 9.071 54.54 0.375 0.0860 1.447 1184 5.98
RMD[%] 1.62 1.53 7.41 18.10
34 Ca 2.202 15.217 143.16 1597.87 0.881 0.2806 5.715 471.7 7.94
K 0.840 9.404 109.12 1306.00 1.427 0.0743 9.999 37.6 9.77
Na 1.296 4.669 18.304 78.46 0.459 0.0186 2.407 105.2 8.36
RMD[%] 1.52 3.91 . 5.82 7.16
t jCoT -M I/CoTo] x100%.
o o
* (0.5/M)MCa-MI+IMNa+MK-M 
x 100% and M = S(MCa+MKMNa
The utility of D and R values obtained with the moment method was
investigated by numerically solving the transport equation and using
the values listed in table 15. The resulting BTC's are shown in figure
27, along with curves by using fitted D and R values obtained with
CXTFIT. Since we used a constant value for R, the prediction assumes
linear exchange. Furthermore, the assumption was made that equilibrium
conditions existed and that the solute level in liquid and adsorbed
phases was constant. Obviously, this precludes 
a close fit with
experimental data. However, the position of the peak is predicted
%
#31 Dothan I
Potassium
CXTFtT
moment method
61
2.5 5.0 7.5 10.0 12.5 15.0
# PORE VOLUMES
#31 Dothan I
sodium
CXTFIT
Moment mthod
0
0
0
2.5 5.0 7.5 10.0
# PORE VOLUMES
#32 Dothan 11
potassium
CXTFIT
moment method
2.5 5.0 7.5 10.0 12.5 15.0
#I PORE VOLUMES
#32 Dothan 11
0.0 sodim CTFJ
0.0 2.5 50 7.5 10.0 
12.5 15.0
# PORE VOLUMES
#33 Wickham I
potassium CXT
moment method
2.5 5.0 7.5 10.0 
12.5 15.0
# PORE VOLUMES
#3 Wickham I
sodium CXTFtT
moment method
0
2.5 5.0 7.5 10.0
# PORE VOLUMES
12.5 15.0
0.6
#3-Wickhamn 11
potassium
0.4-
0.2-
0.0 2.5 5.0 7.5 10.0
# PORE VOLUMES
#34 Wckham 11
sodium CXTFtT
moment method
2.5 5.0 7.5 10.0
# PORE VOLUMES
12.5 15.0
FIG.27. K and Na curves predicted 
with the moment method (solid lines)
and curve fitting 
(dashed lines), 
and experimental 
curves
(symbols) for 
displacement 
with ternary exchange.
0
0
CXTFI
moment method
12.5 15.0
I? IC~ IU h
A rl -
06-
62
fairly accurately, along with the time interval for which solute
appears in the effluent. Although the curve fitting program is designed
to yield D and R values, which produce the 'best' fit to the
experimental data, a slightly better fit was occasionally obtained with
moment generated D and R values (experiments 31 and 33). As stated
earlier, the main advantage of -the moment method is the quantification
of the BTC independent of a particular transport model and its use in
theoretical analysis of solute transport. The moment method also seemed
more suitable for our ternary system (#31-34), for which the BTC's
exhibited considerably more spreading, due to exchange, and the inlet
effect was less important, than for the binary system (*21-28). The
importance of solute spreading relative to column inlet effects should
grow with increasingly longer columns for linearly exchanging and
non-reactive solutes, making the moment. method more suitable to
estimate transport parameters for longer columns.
The effect of non-linear exchange on spreading and tailing during
transport with ternary exchange was also investigated using theoretical
moments. Values of these moments were calculated in such a way that the
effects of non-linear exchange were eliminated as much as possible.
Therefore, we used D values obtained from anion displacement
experiments in the same soil columns (table 11), adjusted for
differences in v between the binary and ternary exchange experiments,
and ~ ~ ~ ~ ~ r valesforRNromtale 5.TheR alus-nd headjstd Dvaue
63
Table 16. Theoretical Moments Based on Linear Exchange and Their
Contribution-to Experimental Time Moments for BTC's with
Ternary Exchanget
# Solute D R M M m M ML NLRTt 
2
NLSP?
0 1 M2 3 2
2.-
cmd de d
31 Ca 544.0 2.871 2.910 15.684 118.33 957.42 0.804 -20.4 0.258 -39.7
K 544.0 4.338 1.455 10.767 112.25 1629.1 1.104 -22.2 0.497 -65.7
Na 544.0 1.542 1.455' 5.192 35.209 133.98 0.532 -17.0 0.255 -76.5
32 Ca 34.09 2.686 2.769 11.438 49.961 576.49 0.647 2.2 0.025 165.2
K 34.09 4.419 1.384 8.172 50.401 1479.49 0.926 2.4 0.037 38.2
Na 34.09 1. 973 1. 384 4. 710 17. 162 219. 28 0. 533 -2. 3 0. 021 475. 7
33 Ca 20.8 2.198 1.997 6.445 22.202 310.00 0.494 -0.8 0.017 347.7
K 20.8 3.209 0.999 4.246 19.175 86.35 0.651 -0.9 0.027 326.9
Na 20.8 1.447 0.999 2.462 6.563 18.18 0.378 -0.8 0.011 661.1
34 Ca 87.7 5.715 2.400 17.251 133.18 1102.6 0.916 -3.9 0.063 345.2
K 87.7 9.999 1.200 14.012 176.48 2396.0 1.489 -4.2 0.174 -57.3
Na 87.7 2.407 1.200 4.467 17.911 76.75 0.475 -3.4 0.017 6.2
t Theoretical moments according to Appendix C (third-type condition,
layer 1):
20 0'.tK2 j]
0 3 D x203 el D . xD
P, 2
+ R 
2
t 04D4+
U. t . V
2 tor
+ R22of1D4
2 
3IV
9x2D+ 
x3
V4 V311
4xD 
x 
2
1
v3 v211
xD x2
+ 122 
+ 
x-1+
3 V2J
V5 V3.1
~TT~T'.. 1experimental 1theoretical
NLRT=X100%/o
Sp 1 the o r e t c a 1 e
?NLSP=t
,2 )experimental -02 )theoretical l
2 ) theoret 
i c aIX10
C
0
2
1 
2
3
m2 
v3
L
4
m
64
assumed that anion exchange obeyed a linear isotherm, although
deviations from linear exchange would have a minimal effect on the
value of D because the AEC is relatively small.
Table 16 also contains values for the first absolute moment, i'
and the second central moment, g
2  
To assess the influence of
non-linear exchange on breakthrough time and spreading, the relative
differences between theoretical (table 16) and experimental (table 15)
values for W and p2 are included as NLRT and NLSP, respectively. One
can view this as the error made by obtaining D and R under the
assumption of linear exchange. Positive values of NLRT and NLSP
indicate that the experimentally obtained parameter value overestimates
the "correct" theoretical value.
The results for experiment 31 are not very reliable because of
the high values for D, which were attributed to non-equilibrium
conditions, possibly because of poor packing (cf. figure 14). For the
other experiments, no large deviations between experimental and
theoretical values were found for as indicated by the low values
for NLRT. Apparently, the difference in mean residence time for linear
and non-linear exchange is small. However, substantial differences in
values for p2 occurred, as pointed out by the high values for NLSP. The
theoretical values for p do not account for non-linear exchange,
whereas the experimental p 
2 
values (implicitly) include all mechanisms
contributing to spreading. Because the soil in each column was fairly
uniform, this difference was attributed to non-linear exchange. The
non-linearity increased solute spreading for all cations with the
65
exception of K during experiment 34. The W2 values for Na were most
affected, with the exception of experiment 34. A similar comparison can
be made for D. The adjusted (theoretical) values for D during ternary
exchange represent only hydrodynamic dispersion (table 16), whereas the
experimental values for D account for both hydrodynamic dispersion and
non-linear exchange (table 15).
The influence of non-linear exchange on R was already
demonstrated in tables 14 and 15. Because K is adsorbed more favorably
than Na, it is retarded two to four times more than Na. It seems that
this is the most important effect non-linear exchange has on solute
transport. This effect becomes more pronounced when the equivalents of
solute applied as a pulse are small relative to the total equivalents
in the adsorbed and liquid phases.
SUMMARY AND CONCLUSIONS
Breakthrough curves were determined in four types of experiments,
namely (1) a pilot study involving step displacement in homogeneous
soil columns of 30 cm length, (2) step displacement in layered soils,
(3) pulse displacement in soil columns of 15 cm length with binary
exchange, and (4) pulse displacement in soil columns of 15 cm length
for ternary exchange. Results of the pilot study indicated that
approximative methods based on Eq. (7) yield fairly reliable values for
D and R and that curve fitting may produce erroneous results if too
many unknowns need to be fitted (e.g., non-equilibrium model). The step
displacement experiments in layered media yielded symmetrical and steep
BTC' s.
66
Changes in total solute concentration in the effluent during
simultaneous anion and cation displacement were insufficient to
conclude that the CEC varied during the application of a Na pulse to a
Ca soil. Transport parameters determined from anion displacement were
used to study the effect of non-linear exchange on cation displacement.
Ternary displacement experiments demonstrated differences in
retardation and dispersion due to non-linear exchange. The Na pulse
moved two to four times faster through the soil than the K pulse.
The determination of R was illustrated graphically for pulse and
step input. Calculated results were presented using column holdup and
the ratio of residence time of solute and solvent. Statistical moments
with respect to time were used to analyze the BTC's for pulse
displacement. Theoretical moments, derived from the ADE for four cases,
seem to be very useful to determine parameters for arbitrary transport
models. The theoretical moments showed that more retardation and
dispersion occurred for a third-type condition than for a first-type
condition. Layering only affects spreading and dispersion by changes in
magnitude of the transport parameters. No interface effects occur,
however, and the layered medium is actually a series of homogeneous
media.
The theoretical moments were also used to interpret experimental
data. The determination of R yielded reliable results, whereas the
determination of D was inaccurate for binary systems. The reliability
can be improved by using longer columns or by determining BTC's at more
than one position. The values of the (theoretical) moments were
67
determined based on transport parameters obtained from anion
displacement. Comparison of theoretical and experimental moments for
cation displacement helped quantify the effect of non-linear exchange
on dispersion and retardation. In particular, the value for R, which
determines the residence time, was affected by non-linear exchange. If
solute movement needs to be simulated, it is worthwhile to first obtain
reliable values for R under similar circumstances. However, values for
D are affected as well by non-linear exchange: D values obtained from
experiments involving ternary (non-linear) exchange were up to ten
times larger than their corresponding values when hydrodynamic
dispersion was singled out as the sole process determining D.
The following guideline is offered for quick prediction of solute
movement during equilibrium conditions, beginning with the most
important steps. First, determine the R values for the various solutes.
If BTC's are available, this determinatioin is conveniently done with
the methods outlined in the section on the graphical determination of
R. Otherwise, values for the CEC, determined under conditions similar
to those for which transport needs to be predicted, can be used.
Second, the value for D must be determined to quantify hydrodynamic
dispersion. This determination can be done relatively fast with
non-reactive solutes, using approximative methods or with curve
fitting. If more time is available, experiments involving both
non-reactive and reactive solutes are to be preferred to determine D
with the BTC for the non-reactive solute, and an effective value for R
with the reactive solute. If the BTC is determined at two positions or
68
exhibits sufficient spreading, as for our ternary 
systems, the data are
conveniently analyzed with the method of moments. 
Third, for reactive
solutes, the exchange isotherm should be determined 
if a good
prediction of R is needed and sufficient time is available. 
This last
step is not needed for trace' amounts of solute or 
for incomplete
exchange.
LITERATURE CITED
(1) AGNEESSENS, J.-P., P. DREZE, and L. SINE. 1978. Mod61isation de la
Migration d'El6ments dans les Sols. II. D6termination du
Coefficient de Dispersion et de la Porosit6 et de la Porosit6
Effiace. Pedologie 27:373-388.
(2) ARIS, R. 1956. On the Dispersion of a Solute in a Fluid Flowing
through a Tube. Proc. Roy. Soc. A. 235:67-77.
(3) . 1958. On the Dispersion of Linear Kinematic Waves. Proc.
Roy. Soc. A. 235:268-277.
(4) BARRY, D.A. and G. SPOSITO. 1988. Application of the Convection-
Dispersion Model to Solute Transport in Finite Soil Columns. Soil
Sci. Soc. Amer. J. 52:3-9.
(5) CRANK, J. 1975. The Mathematics of Diffusion. Clarendon Press.
Oxford.
(6) DAGAN, G. and E. BRESLER. 1983. Unsaturated Flow in Spatially
Variable Fields. 1. Derivation of Models of Infiltration and
Redistribution. Water Resour. Res. 19:413-420.
(7) FRIED, J.J. and M.A. COMBARNOUS. 1971. Dispersion in Porous Media.
In Adv. Hydrosci. 9:169-282. Academic Press, New York.
(8) GOLTZ, M.N. and P.V. ROBERTS. 1987. Using the Method of Moments to
Analyze Three-dimensional Diffusion-limited Solute Transport from
Temporal and Spatial Perspectives. Water Resour. Res.
23:1575-1585.
(9) COVEN, 0., F.J. MOLZ, and J.G. MELVILLE. 1984. An Analysis of
Dispersion in a Stratified Aquifer. Water Resour. Res.
20:1337-1354.
(10) HAN, N.-W., J. BHAKTA, and R.G. CARBONELL. 1985. Longitudinal and
Lateral Dispersion in Packed Beds: Effects of Column Length and
Particle Size Distribution. AIChE J. 31:277-288.
(11) JURY, W.A., G. SPOSITO, and R.E. WHITE. 1986. A Transfer Function
Model of Solute Transport through Soil. 1. Fundamental Concepts.
Water Resour. Res. 22:243-247.
(12) LEIJ, F.J. and J.H. DANE. 1989. Evaluation of Methods to Determine
CEC and Exchange Isotherms. Ala. Agr. Exp. Sta. Dept. of Agronomy
and Soils Series 134.
69
70
(13) PARKER, J.C. and M.TH. VAN GENUCHTEN. 1984. Determining Transport
Parameters from Laboratory and Field Tracer Experiments. Va. Agr.
Exp. Stat. Bull. 84-3.
(14) PERSAUD, N. and P.J. WIERENGA. 1982. A Differential Model for One-
dimensional Cation Transport in Discrete Homo-ionic Ion-exchange
media. Soil Sci. Soc. Amer. J. 46:482-490.
(15) RAO, P.S.C., J.M. DAVIDSON, R.E. JESSUP, and H.M. SELIM. 1979.
Evaluation of Conceptual Models for Describing Non-equilibrium
Adsorption-desorption of Pesticides During Steady-flow in Soils.
Soil Sci. Soc. Amer. J. 43:.22-28.
(16) RAZAVI, M.-S., B.J. MCCOY, and R.G. CARBONELL. 1978. Moment Theory
of Breakthrough Curves for Fixed-bed Adsorbers and Reactors. Chem.
Eng. J. 16:211-222.
(17) ROSE, D.A. and J.B. PASSIOURA. 1971. The Analysis of Experiments
on Hydrodynamic Dispersion. Soil Sci. 111:252-257.
(18) SHAMIR, U.Y. and D.R.F. HARLEMAN. 1967. Dispersion in Layered
Porous Media. J.Hydraul. Div., Proc. ASCE 93(HY5):237-260.
(19) SKOPP, J. 1985. Analysis of Solute Movement in Structured Soils.
p.220-228. In J. Bouma and P.A.C. Raats (Eds.). Proc. ISSS Symp.
Water and Solute Movement in Heavy Clay Soils. ILRI. Wageningen,
the Netherlands.
(20) VALOCCHI, A.J. 1984. Describing the Transport of Ion-exchanging
Contaminants Using an Effective Kd Approach. Water Resour. Res.
20:499-503.
(21) . 1985. Validity of the Local Equilibrium Assumption
for Modeling Sorbing Solute Transport through Homogeneous Soils.
Water Resour. 21:808-820.
(22) VAN GENUCHTEN, M.TH. and W.A. JURY. 1987. Processes in Unsaturated
Flow and Transport Modeling. Rev. of Geophys. 25:135-140.
(23) and J.C. PARKER. 1984. Boundary Conditions
for Displacement Experiments through Short Laboratory Soil
Columns. Soil Sci. Soc Amer. J. 48:703-708.
(24) and P.J. WIERENGA. 1986. Solute Dispersion
Coefficients and Retardation Factors. In Methods of Soil Analysis
I. Physical and Mineralogical Methods. Soil Sci. Amer. Madison,
Wis.
(25) WAKAO, N and S. KAGUEI. 1982. Heat and Mass Transfer in Packed
Beds. Gordon and Breach. New York.
APPENDIX A
Analytical Solution of the ADE for a Pulse Input with a
First- and Third-type Condition in a Two-layer Medium
1. First layer
It is assumed that the first layer is in effect semi-infinite and
homogeneous with respect to the flow and transport properties. The
transport problem for which we seek a solution can be stated as
follows:
BC 82C BC
- D v t>0 0<x<L (A-i)
1t 1 2 1 x 
1
C(x,O) = 0 0<x<L (A-2-a)
C(O,t) = f(t) (first-type) t>O (A-2-b)
i -D + v C = v f(t) (third-type) t>0 (A-2-c)
Sx=
0
C O<t-<t
o o
f(t) = (A-2-d)
0 t>t
O
0 t>0 (A-2-e)
ax
Solution of Eq.(A-1), subject to Eq.(A-2), can be achieved with 
the
help of Laplace transforms. The transformed equation and boundary
conditions are:
2- v -
dC 1 dC s
- C -- = 0 
0<x<L (A-3)
2 D dx D 
1
dx 1
C(0,s) = fT(s) (A-4-a)
-D + v = v f(s) (A-4-b)
dC
X->O
71
72
where s is the transformation variable and C and f are given by:
so
C(x,s) = e
-st 
C(x,t) dt (A-5)
O
S-st 
o
f(s) = e f(t) dt - (1-eqxp(-st )) (A-6)
sJ0
0
The general solution of the ordinary 
differential equation, Eq.(A-3),
can be written as:
C(x,s) = a ep(Ax) + P e(Ax) (A-7)
1 1
+ v 2
1 + S
where A D 2 --, and a and R are coefficients depending
11 1
on the boundary conditions. According to Eq.(A-4-c) a is equal to 0 and
we denote A as A . The use of Eq. (A-4-a) or (A-4-b) allows the
1 1
evaluation of 3:
( = (s) (first-type) (A-8-a)
v f(s)
(v -D (third-type) (A-8-b)
Substitution of these expressions for ( into Eq. (A-7) results in the
following solutions:
C
C(x,s) - (1-exp(-st )) exp(A x) (first-type) (A-9-a)
s 0 1
v C (1-eXp(-st))
C(x,s) -10 vD 0 exp(A x) (third-type) (A-9-b)
1 1 1
73
2. Second layer
The solution of the concentration in a (uniform) second layer
(L<x<) is very similar to the solution for a non-layered uniform soil.
We assume that the concentration in the first layer (0<x<L) is
accurately described with Eq.(A-9-a) and (A-9-b) for a first- and
third-type inlet condition, respectively. We wish to solve:
8C 82C BC
- D - v - t> L <x< (A-10)
dt 2 2 2 8x 1
8x
subject to:
C(x,O) = 0 L <x<w (A-11-a)
1
C xL = Cx L (first-type) t>O (A-11-b)
-0 D -+ vC = - D + 0 vC (A-11-c)
S 2 5x 22 
xL 1 1 
x 1 1 L
1 1
(third-type) t>0
= 0 t>O 
(A-11-d)
The Laplace transform is used in a similar manner as for the
non-layered soil to obtain the solution of Eq. (A-10) subject to
Eq.(A-11). The transformed equations are:
2- v -
dC 2dC- = 0 
L <x< (A-12)
2 D dx D 
1
dx 2 2
C xL Cx
L  
(first-type) (A-13-a)
-D d+ v& = -eD -- + O vC (A-13-b)
.2dx 2 2 xL 
1 i dx 1 1 xtL
(third-type)
74
dC
X--)W
=0
where the concentration at the end of the f irst layer (xtL)
by:
CCL ,s)= f exrp(X L ) (first-type)
C(L ,s) = v-DX '(A Li (third-type)
The general solution of Eq. (A-12) is:
C(x,s) = a exp(AX) + (3exp(X)
2 2
(A- 13-0)
is given
(A-14-a)
(A-14-b)
(A- 15)
+ v 2
2 + v2 S
whe re A 2+ 5 an a and(3are again coefficients
22D J
2
determined by the boundary conditions. From Eq. (A-13-c) we conclude
that ax=0. Eq. (A-13-a) and (A-13-b) are used to evaluate ( for a f irst
and third-type condition, respectively:
(3 =felxp( (X -X
2
)L) (first-type) (A-16-a)
1 v if 
-6 b
f3 1 1 G X)L(third-type) (A-6b
0 v -D AX
2 2 2 2
where X =X. This leads to the following respective solutions:
2 2
C (x,s)= f exp(A L1+ X2
0 v f
C(x, S)= 1 1 e2X L +Af
6O V--DXA 11 2~
2 2 2 2
(first-type)
(third-type)
where =x-L1
(A- 17 -a)
(A- 17 -b)
APPENDIX B
Derivation of Moments Based on Analytical Solutions
of the ADE in the Laplace Domain
The p-th moment of a time dependent concentration distribution,
for a fixed position, is defined by:
M t
p 
C(xt) dt p=0,1,2, .... (B-i)
0
The concentration distribution, C(x,t), can 
be determined theoretically
or experimentally. However, explicit analytical solutions for C(x,t)
are sometimes not readily available. Aris (3) showed how m can be
p
obtained if a solution, C(x,s), in the Laplace domain is known. This
solution is obtained rather easily in comparison with the regular
solution. According to Aris (3), the p-th moment in the Laplace domain
is given by:
M = -
1 
1m [C(xs) (B-2)
p s->0 d
p
The procedure of differentiation and limitation is rather
straightforward, but becomes tedious for the higher moments as noted by
Valocchi (21). To illustrate the use of Eq.(B-2), we will determine m
for a non-layered soil using a third-type inlet condition. The solution
in the Laplace domain is given by Eq.(A-9-b). From Eq.(B-2) we get
(s=0: A =0):
75
76
_im 1 o o lim o
m - epr(A x) = C 
(B-3)
O sO s v -DA 4o s-90 
s
The indeterminate form is evaluated with ' Hopital's rule. Both
numerator and denominator tend to go to 0 if s->0. Because 
the ratio of
their derivatives is t for s->0, we have:
0
m = Ct (B-4)
The zeroth moment is proportional to the total amount of mass applied
during a pulse. The evaluation of moments of higher order is done 
in a
similar manner.
APPENDIX C
Time Moments m, , and pt for BTC's as a Result of a Pulse
Input During Steady Flow in aMedium with One or Two
Layeres with a First- or Third-type Inlet Conditiont
c .t
0 0
c t
0 0
c t
0 0
c t
0 0
0. x 1
+ -tI
V 01
1 -
D 
0
A
D 
o
B 
0[
D 
C0
T,
t
3
L D
23
1
+ t
I 1
I -
+
+ tot
L
1
V
1
D
2
2
V
2
2x
+t -+
t
3
1
L
1
V
1
t 2
+ 
2
D2
+1)
+2J
OL
r1
to(
+t1
2
x
2
+ 
+t 
4 
+ 4-
1
+ tt
2
2-0 1
D L
2 
L
v v
1 2
+ + t{14 2+ 2 .( A1 + +
2 2
1
+
tCases are listed in table 1.
77
mA
B
C
D
A 01
KD
1
+
3
2-
L,3
x2"
V
1
L
2111v
+
2
78
APPENDIX C. Continued.
t 2"xD
1
S2
x2
xD 
2
+o t[12 
v1
2
xD
1
+6
V
1
1
t t +?i+2rD
1 2
1 xD +
3r +0[2 
L6
4  
13 v
t v I vvvJv
2 2
L D1 L D
[ 12-5 +61
1 
1+ v
4 +311
1 11
v 2]' 2{6 11 3
L3
1
V
1 1
I L1 1
+6
1 2
+6L1
v v
1 2
L2
3-1
2
v v
1 2
+
2
v
2
D2
1252
2
2
2
~
+6 +
4
V
2
+3 
192J
v11
+ + 2 2
$3
12
t{30 
2-
2
L D
+61 
2+
2
v v
1 2
D2
+ 30 
5+
2
12- +
3
2
LD2
12 
1 2+
4
v v
.1 
2
2
4
v
+6 13122
1 2
L 1 
D1
6'3
1 2
mA C[
0
t3 x
+t -+
xD
2
+ 5D
t. 0 +3 0 1
v6 
v5
C o[
0
+t3[
L
1
V
1
3
L 
D2
3
1 2
t 
4
D C 
+ t 
3
D L
2 1
2
L
2
1.
L D
63
1
+61 +
v v
1 2
+121 
2
1 2
+9 
.4
2
2
1 2
V
2
L 2D
1 2
+32
1 2
L3
LD2
12 
1
12
79
APPENDIX C. Continued.
t
20
t2
30
30
t 
2
A 
0
B 
0
+ 
t
2
0
V
D
1
+ -+
2
V
1
L
1
+ -+
V
1
D
2
+ 2
V
2
x
V
1
V
2
L
1
V
1
toV
1
v
xD
2 
3
v
L1
V V
1 2
D
2
V
2
x
2 1
xD
41+
V
1
2
x
2
V
1
1
- 4--
4
V
1
v11
+2
2
t xD 2.
+ +
1 
2
+ 2
~7l 9 v
xD
+ 3
v
1
2
x
2
V
1
+3
2
L2
2
2
xD
2
125
.1
2
+2 
1
1 2
1
xD 2.
1 1 
1 +3xI
4 3 21
v V'
3
x
1 2
L D
+2_1 
2
2
v v
1 2
+2
3
v
1
1
3
x
V
1
3
6
1
xD
2
1
+ 30
V
1
+4
3
v
2
80
APPENDIX C.Continued.
0 20 1 1 2
L 2D
v1
L3
I L 
D2
6 
3
1 2
L
1
V
2
v
1
L 2D
4
v
L 1 D
+63
1 2
3L
2
1 2
2
12 
V
2
~3 
)
2
302
6
v
2
611 
2
+63 
2
v v
1 2
LD
+12 
5
v
1
D2
30 
2
2
L 2D
2 2
v v
1 2
L2
3 1. +
2
v v
1 2
L D
6.1 
2+
2
v Iv2
LD2
+12 
1 2
4
v v
1 2
3L2
2
1 2
63 
I
V.
D2
12 
+
2
12 
+
L D
3
v1
2 
D 
2
v4
2
6L 
D
1 1
1 2
2
1
+3 
+
1 2
12
4
v
2
V
2
D 
2 
+
4i
2
+6 L1,+
1 2
+12 
1 2
3
+ 
+ 6
3
v
81
APPENDIX C. Continued.
0
0
0
0
0
t 
2
0
C2
xD
1
+ 
2
31
4
1
t2 
D2
D0-3-+
12 4
v
2
3 34 2v
t3 
t2
t3 
t2
t t
C 4 1
D
1
V
1
L
1
V
1
xD
3
v
2
3
v
2
+ 
[(231 
+x2
1 1
XD 
2
12
1
+ + 0 
3-1 +2 
1
1 2
1
+23
6
v
LD2
11
. L 
ID
211
+23 +2
2
+ 17-- +
5
v
2
L 2D
1
D2
2
D L
2+
2 v
2
LD2
3 12 +
4
v v
1 2
LD2
12-5
v
2D2 
L1 
D2
- +2
4 3
v v v
2 1 2
2
vJ2)
2
L 
4
2D
34
2
2L 
D2
3
1 2
+2 2 +
3
v
2
+ 2 
13122
1 2
L D
+ 
1]2
22
LDv
1 2
D2
+23-2 +
6
v
2
L 1 D
3
v v
1 2
A
B
C
D
2
xD
1
+ 17
V
1
2
xD
1
+2
V
1
.2
L D
APPENDIX D
Input Files for the Program CXTFIT Containing the
Experimentally Determined C/C -profiles
The first column 
of each file 
contains values 
for C/C
0
, the second
column lists the 
value of the distance 
(column length, 
L) and the
third column contains 
the number of 
pore volumes, T. 
Length is
expressed in 
cm and time in 
days, with the 
exception of experiments 
1-6
where the time 
is given in minutes. 
Further details 
are provided by
Parker and van Genuchten 
(13).
Experiment 
1: Calcium
Experiment 
2: Calcium
1 2 
30 29 
0
SAND SATURATED 
WITH 0.1 N CACL2, 
DISPLACEHENT WITH 
SALT FREE WATER
DBD=1.92 g/cs3 
VMC=0.28 PV=173 
cs3 V=0.839 ca/in 
1-Ca
V... 0D.... 
R.... To... 
RXl.. RIO..
0.8388 
0.1 1.0 
0.0 0.5 
05
0 1 1 
0 1 1
1.0 0.0
0.0 30.6 
0.0549
0.0 30.6 
0.1919
0.0 30.6 
0.3290
0.0 30.6 
0.4661
0.0 30.6 
0.6032
0.0 30.6 
0.7403
0.0 30.6 
0.8225
0.0 30.6 
0.8774
0.0 30.6 
0.9048
0.012 30.6 
0.9871
0.056 30.6 
1.0145
0.132 30.6 
1.0420
0.234 30.6 
1.070
0.381 30.6 
1.097
0.519 30.6 
1.125
0.651 30.6 
1.152
0.753 30.6 
1.179
0,830 30.6 
1.207
0.882 30.6 
1.234
0.918 30.6 
1.262
0.943 30.6 
1.289
0.957 30.6 
1.317
0.967 30.6 
1.344
0.973 30.6 
1.371
0.979 30.6 
1.399
0.981 30.6 
1.426
0,984 30.6 
1.454
0.985 30.6 
1.481
0.987 30.6 
1.536
1 2 
30 24 
0
SAND SATURATED 
WITH 0.1 M CACL2, 
DISPLACEMENT 
WITH SALT FREE 
WATER
0DBD=1.91 q/cm3 
VMC=0.28 PV=175 
ca3 V=0.906 cmlsin
V .. .
... .R.. 
To... RXI.. 
RXO..
0.9056 
0.1 1.0 
0.0 0.5 
0.5
0 1 1 
0 1 I
1.0 0.0
0.0 30.6 
0.089
0,0 30.6 
0.237
0.0 30.6 
0.385
0.0 30.6 
0.533
0.0 30.6 
0.681
0.0 30.6 
0. 829
0.003 30.6 
0.977
0.031 30.6 
1.006
0.092 30.6 
1.036
0.213 30.6 
1.065
0.362 30.6 
1,095
0.507 30.6 
1.125
0.643 30.6 
1.154
0.758 30.6 
1.184
0.836 30.6 
1.213
0.890 30.6 
1.243
0.925 
30.6 1.273
0.946 30.6 
1.302
0.968 30.6 
1.332
0.976 
30.6 1.391
0.980 30.6 
1.450
0.983 30.6 
1.509
0.986 30.6 
1.568
0.987 30.6 1.598
82
0
0
Experiment 3: Bromide and Chloride
PRATTVILLd (upper)-' ISPLACEMENT OF 0A01 KCACL2'Y 0.01 A CABR2: PRATTVILLE (uppr); DISPLACEKED
OR- CONCENTRATbOfA PVu250.6 c.3 080:1.58 Q/ca3 VMCz0.402 (--CjDCONC NT 11 ON PV=?50.b
V.... 0.... Re.. X. I., MX..V . .. R...
0,012360 0.1 1.0 10.0 0.5 0.5 0.012360 0.1 1.0
0 . 1 1 0 1 1 0 1 1
0.0 1.0 .0.0 1.0
0.000 30.6 0.085 0.000 30.6 0.085
0.000 30.6 0.242 0.005 30.6 0.242
0.000 30.6 0.389 0.005 30.6 0.389
0.090 30.6 0.493 0.065 30.6 0.493
0.163 30.6 0.53? 0.149 30.6 0.537
0.291 30.6 0.610 0.265 30.6 0,610
0.390 30.6 0.679 0.375 30.6 0.679
0.547 30.6 0.820 0.431 30.6 0.820
0.658 30.6 0.966 0.533 30.6 0.966
0.695 30.6 1.089 0.649 30.6 1.09
0.726 30.6 1.122 0.699 30.6 1.122
0.794 30.6 1.296 0.767 30.6 1.296
0.852 30.6 1.478 0.821 30.6 1.478
0.861- 30.6 1:532 0.856 30.6 1.532
0.902 30.6 1.65? 0.859 30.6 1.657
0.910 30.6 1.829 0.885 30.6 1,829
0.947 30.6 1.996 0.913 30.6 1.996
0.938 30.6 .164
Experiment 4: Bromide and Chloride
10 
1 
2
4 1 2 30 15- 0 0
PRATTVILLE (upper) VDISPLACEMENT OF 0.01 M CACL2 BY 0.01 M CABR2: PRATTVILLE (upper) DISPLACEME,
BR CONCENTRATION ?=V249,0 cu3 080:1.59 g/cs3 VMC=0.4 units:cs~xin,q 1-Cl-CONCENTRATION V=249.0 cs3
V. ... 0.... R.... To... RXI.. RhO.. V.... D... R.
0.047330 0.1 1.0 10. 0.5 0.5 0.047330 0.1 1.
0 1 1 0 1 .1 0 11
0.0 1.0 0.0 1.0
0.0 30.6 0.083 0.0 30.6 0.083
0.000 30.6 0.242 0.006 30.6 0.242
0:045 30.6 0.399 0.065 30.6 0.399
0,155 30.6 0.480 0.132 30.6 0.480
0.245 30.6 0.557 0.229 30.6 0.557
0.395 30.6 0.714 .0.3,96 30.6 0.714
0.58 0.6 0.410.482 30.6 0.841
0.639 30.6 1.022 
0.62? 30.6 1.022
0.731 30.6 1.022 
0.709 30.6 1.175
0.789 30.6 1.330 
0.777 30.6 1.330
0.782 30.6 1.476 
0,832 30.6 1.476
0.982 30.6 1.461081 3. .2
0.958 30.6 1.771 0.8915 30.6 1.771
0.965 30.6 1.923 0.901 30.6 1.923
0.984 30.6 2.076 0.958 30.6 2.076
I Experiment 5: Bromide 
and Chloride
4 1 2 30 17 0
P8(1; DISPLACEMENT OF 0.01 M CACL2 BY 0.01 M CABR2 (units cm, min)
PV:285.7 Wm, 080:1,433 91Wm VMC=0.459 BR
V....o 0..., Ro. To.$. RhI.. RhO.
0.005972 0.1 1.0 1.925 0.5 0.5
0 1 1 0 1
0.0 1.0
0.0 
30.6 
0.055
0.0 30.6 0.167
0.000 30.6 0.281
0.000 30.6 0.394
0.009 30.6 0.505
0.01? 30.6 0.646
0.086 30.6 0.735
0.23? 30.6 0.855
0.440 30.6 0.974
0.668 30.6 1.092
0.821 30.6 1.214
0.894 30.6 1.313
0.96? 30.6 1.379
0.964 30.6 1.499
0.971 30.6 1.595
0:974 30.6 1.741
0,976 30.6 1.925
4 1 2
P8(1' DISPLACEMENT OF 0.01 M Ckl
PV= 85.7 co3
1 
080:1.433 g/cm3
v.... 0....: R.
0.005972 0. 1.
0 1 1
0.0 1.0
0.0 30.6 0.055
0.0 30.6 0.16?
0.006 30.6 0.281
0.001 30.6 0.394
0.015 30.6 0.505
0.063 30.6 0.646
0.121 30.6 0.735
0.222 30.6 0.855
0.392 30.6 0.974
0.542 30.6 1.092
0.747 30.6 1.214
0.852 30.6 1.313
0.894 30.6 1.379,
0.947 30.6 1.499
0.950 30.6 1.595
0.955 30.6 1.741
01.995 30.6 1.925
83
Experiment 6: Bromide and Chloride
3 1 2 30 14 0 0
PBU; DISPLACEMENT OF 0.01 n CACL2 BY 0.01M CABR2 (units cm, min, 
g)
PV=320.59 cm3i DBD=I.285 g/cW3, VMC=0.515 
BR
V.... D.... R.... To... RXI.. RXO..
0.18264 0.1 1.0 2.244 0.5 0.5
0 1 1 0
0.0 1.0
0.0 30.6 0.120
0.000 30.6 0.360
0.000 30.6 0.598
0.130 30.6 0.715
0.246 30.6 0.832
0.381 30.6 0.949
0.541 30.6 1.067
0.627 30.6 1.184
0.729 30.6 1.301
0.803 30.6 1.419
0.873 30.6 1.536
0.912 30.6 1.771
0.926 30.6 2.007
1.00 30.6 2.244
1 2
PBU DISPLACEMENT OF 0.01 M CAC
PV=20.59 ca3, DBD=1.285 
/ca3,
V ... D....
0,18 264 0.
0 1
0.0 1.0
0.0 30.6 0.120
0.024 30.6 0.360
0.053 30.6 0.598
0.131 30.6 0.715
0.221 30.6 0.832
0.337 30.6 0.949
0.461 30.6 1.067
0.564 30.6 1.184
0.671 30.6 1.301
0.739 30.6 1.419
0.798 30.6 1.536
0.850 30.6 1.771
0.899 30.6 2.007
0.934 30.6 2.244
Experiment 11: Calcium and Potassium
1 1 2 30 20 0
KBR REPLACED BY CABR2:SOIL: SAN-WINI-SAN;WINI:0-250;SAN:250-500
P.V.:232.7; CA 6AINST P.V.; THETA=0.380;LAR6E 
COLUMN I
V..... DO..... R ..... TO.... rxl... 
rx2...,
132.63 5.0 5.0 13.158 
0 0
0 1 1 0 0 
0
0.0 1.0
0.005 30.0 0.175
0.004 30.0 1.053
0.003 30.0 1.930
0.004 30.0 2.816
0.042 30.0 3.277
0.280 30.0 3.735
0.525 30.0 4.007
0.620 30.0 4.189
0.770 30.0 4.643
0.813 30.0 4.824
0.885 30.0 5.097
0.935 30.0 5.555
0.961 30.0 6.486
0.998 30.0 7.421
0.999 30.0 8.360
0.999 30.0 9.230
0.999 30.0 10.242
1.000 30.0 11.181
0.998 30.0 12.119
0.998 30.0 13.158
2 1
KBR REPLACED BY CABR2
P.V.:232.7; K
V..... D..... R
132.63 1.0
0 1
0.0 1.0
0.000 30.0 0.175
0.000 30.0 1.053
0.021 30.0 1.930
0.024 30.0 2.816
0.067 30.0 3.277
0.301 30.0 3.735
0.543 30.0 4.007
0.633 30.0 4.189
0.775 30.0 4.643
0.828 30.0 4.824
0.885 30.0 5.097
0.948 30.0 5.555
0.970 30.0 6.486
0., 980 30.0 7.421
0.985 30.0 8.360
0.987 30.0 9.230
0.989 30.0 10.242
0.991 30.0 11.181
0.993 30.0 12.119
0.994 30.0 13.158
84
Experiment 12: Calcium 
and Potassium
2 1 2 30 27 0 1
KBR REPLACED BY CABR2;SOIL: 2 LAYER HEAII-HEAI;II:500-840;:0-250; KBR REPLACED 
BY CABR2; S
P.V.:201.0; ca AGAINST P.V.; THETA=0.334;LARGE COLUMN 2 P.V.:201.0; I MINUS K AS
V D..... ..... R..... TO.... rx ... rx2... Va..... ..... R.
102.48 5.0 5.0 18.463 0 0 102.48 5.0
0 1 1 0 0 0 0 1
0.0 1.0 0.0 1.0
0.020 29.8 0.267 0.000 29.8 0.267
0.020 29,8 1.161 0.000 29.8 1.161
0.020 29.8 2.061 0.000 29.8 2.061
0.020 29.8 2.960 0.000 29.8 2.960
0.020 29.8 3.863 0.000 29.8 3.863
0.020 29.8 4.768 0.000 29.8 4.768
0.020 29.8 5.672 0.000 29.8 5.672
0.020 29.8 6.591 0.000 29.8 6.591
0.010 29.8 7.508 0.000 29.8 7.508
0.050 29.8 8.444 0.054 29.8 8.444
0.264 29.8 9.391 0.271 29.8 9.391
0.337 29.8 9.865 0.340 29.8 9.865
0.432 29.8 10.340 0.442 29.8 10.340
0.547 29.8 10.817 0.553 29.8 10.817
0.617 29.8 11.312 0.639 29.8 11.312
0.706 29.8 11.817 0.718 29.8 11.817
0.782 29.8 12.323 0.805 29.8 12.323
0.878 29.8 13.336 0.903 29.8 13.336
0.888 29.8 14.217 0.924 29.8 14.217
0.954 29.8 14.867 0.930 29.8 14.867
0.955 29.8 15.365 0.927 29.8 15.365
0.970 29.8 15.864 0.946 29.8 15.864
0.986 29.8 16.368 0.943 29.8 16.368
0.988 29.8 16.884 0.958 29.8 16.884
0.991 29.8 17.407 0.962 29.8 17.407
1.000 29.8 17.935 0.961 29.8 17.935
0.993 29.8 18.463 0.968 29.8 18.463
Experiment 13: Calcium and Potassium
2 1 2 30 18 0 
1
KBR REPLACED BY CABR2;SOIL: 2 LAYER PRtII-SAND;FR 500-840; THETA:.317
P.V.:191.4; CA A6AINST P.V.
V..... D..... R..... TO.... rxl... rx2...
140.20 5.0 5 13.837 0 0
0 1 1 0 0 0
0.0 1.0
0.010 29.7 .193
0.009 29.7 1.159
0.009 29.7 2.119
0.008 29.7 3.083
0.000 29.7 4.059
0.008 29.7 5.036
0.007 29.7 6.020
0.007 29.7 7.013
0.044 29.7 8.021
0.208 29.7 8.513
0.465 29.7 9.015
0.671 29.7 9.516
0.810 29.7 10.018
0.872 29.7 10.591
0.936 29.7 11.016
0.958 29.7 12.004
0.958 29.7 13.011
0.983 29.7 14.042
1
2 1
KBR REPLACED BY CABR2-S
P.V.:191.4; 1-k AS
V..... D..... R.
140.20 5.0
0 1
0.0 1.0
0.001 29.7 0.193
0.005 29.7 1.159
0.028 29.7 2.119
0.027 29.7 3.083
0.019 29.7 4.059
0.027 29.7 5.036
0.032 29.7 6.020
0.030 29.7 7.013
0.062 29.7 8.021
0.215 29.7 8.513
0.467 29.7 9.015
0, 673 29.7 9.516
0.787 29.7 10.018
0.858 29.7 10.591
0.916 29.7 11.016
0.945 29.7 12.004
0.961 29.7 13.011
0.968 29.7 14.042
85
Experiment 21: Calcium, Sodium, Bromide, and Chloride
1 2 30 1 
2
CABR2 DISPLACED BY NACL; short column nu CABR2 DISPLACED 
BY NACL;PI P.V.:167.1; 1-BR Al P.V.:167.1; 
CL A
P.V.:167.1; 1-ca AAINST P.V. P.V.:167.1; NA AI 
calculated with CTFIT pa calculating with CTFIT
V..... D..... R..... TO.... V..... D..... 
R. V..... D..... R V..... D.....
100.20 5.0 2.248 1.416 100.20 5.0 
1. 100.20 1326.5 2 100.20 543.75
0 1 1 0 0 1 0 
0 0 0
0.0 1.0 0.0 1.0 
0.0 1.0 0.0 1.0
0.000 14.8 0.118 0.01 14.8 0.118 
0.000 14.80 0.118 0.001 14.80 0.118
0.043 14.8 0.471 0.14 14.8 0.471 0.072 
14.80 0.471 0.149 14.80 0.471
0.192 14.8 0.824 0.28 14.8 0.824 0.230 
14.80 0.706 0.343 14.80 0.706
0.289 14.8 1.060 0.40 14.8 1.060 0.357 
14.80 0.943 0.460 14.80 0.942
0.316 14.8 1.178 0.41 14.8 1.178 0.424 14.80 
1.178 0.597 14.80 1.178
0.356 14.8 1.297 0.41 14.8 
1.297 0.489 14.80 1.298 0.640 14.80 1.298
0.392 14.8 1.416 0.54 14.8 1.416 0.556 14.80 
1.416 0.691 14.80 1.416
0.484 14.8 1.774 0.61 14.8 1.774 0.630 14.80 
1.655 0.686 14.80 1.656
0.503 14.8 1.894 0.62 14.8 1.894 0.615 14.80 1.755 
0.715 14.80 1.775
0.500 14.8 2.013 0.58 14.8 2.013 0.515 
14.80 1.894 0.608 14.80 1.894
0.486 14.8 2.132 0.58 14.8 2.132 0.414 14.80 2.013 0.509 
14.80 2.013
0.379 14.8 2.491 0.46 14.8 2.491 0.369 14.80 2.133 0.518 
14.80 2.133
0.260 14.8 2.611 0.34 14.8 2.611 0.284 14.80 2.250 0.400 14.80 
2.25
0.211 14.8 3.091 0.29 14.8 3.091 0.241 
14.80 2.371 0.360 14.80 2.371
0.184 14.8 3.331 0.28 14.8 3.331 0.178 14.80 
2.611 0.279 14.80 2.611
0.130 14.8 3.689 0.24 14.8 3.689 0.174 14.80 2.850 0.241 14.80 
2.853
0.107 14.8 4.283 0.22 14.8 4.283 0.151 14.80 
3.091 0.250 14.80 3.091
0.141 14.8 4.877 0.17 14.8 4.877 0.146 14.80 
3.331 0.241 14.80 3.331
0.122 14.8 5.478 0.16 14.8 5.478 0.085 14.80 3.689 
0.118 14.80 4.283
0.086 14.8 5.598 0.15 14.8 5,598 0.025 
14.80 4.283 0.000 14.80 4.877
0,049 14.8 5.717 0.12 14.8 5.717 
0.004 14.80 4.878 0.000 14.80 5.718
0.044 14.8 5.837 0.12 14.8 5.837 0.000 14.80 
5.478 0.000 14.80 5.957
0.033 14.8 6.074 0.12 14.8 6.074 0.000 
14.80 5.718 0.000 14.80 7.248
0.000 14.8 7.248 0.02 14.8 7.248 0.000 14.80 5.957 
0.000 14.80 8.432
0.000 14.8 8.431 0.01 14.8 8.431 0.000 
14.80 7.248 0.000 14.80 10.689
0.00 14.8 10.688 0.000 14.80 8.432
0.000 14.80 10.689
Experiment 22: Calcium, Sodinvm, Rrnmide, and 
Chloride
1 2 30 1 
12 1
CABR2 DISPLACED BY NACL; short column 2 CABR2 DISPLACED BY NACL;sh 
P.V.:151.3; 1-BR A( P.V.:151.3; CL Al
P.V.:151.3; 1-ca A6AINST P.V. 
P.V.:151.3; NA AG
V..... D..... R..... TO.... V..... 0..... 
R. V..... D... . R. V..... D..... R
100.00 5.0 1.7240 1.376 100,00 
5.0 1 100.00 1.0 1, 100.00 1.0
0 1 1 0 0 1 
0 1 0 1
0.0 1.0 0.0 
1.0 0.0 1.0 0.0 
1.0
0.008 15.46 0.335 0.0 
15.46 0.335 0.023 15.46 0.558 0.000 
15.46 0.558
0.006 15.46 0.558 0.005 15.46 
0.558 0.027 15.46 0.781 0.000 15.46 
0.781
0.006 15.46 0.781 0.005 
15.46 0.781 0.030 15.46 0.892 0.000 
15.46 0.892
0.006 15.46 1.004 0.13 
15.46 1.004 0.043 15.46 1.004 0.067 
15.46 1.004
0.188 15.46 1.228 0.41 
15.46 1.228 0.335 15.46 1.228 0.409 
15.46 1.228
0.440 15.46 1.454 0.57 
15.46 1.454 0.568 15.46 1.341 0.575 
15.46 1.341
0.558 15.46 1.681 0.73 
15.46 1.681 0.725 15.46 1.454 
0.777 15.46 1.454
0.654 15.46 1.907 0.77 
15.46 1.907 0.872 15.46 1.567 
0.910 15.46 1.567
0.713 15.46 2.133 0.87 
15.46 2.133 0.944 15.46 1.681 
1.000 15.46 1.681
0,754 15.46 2.358 0.91 
15.46 2.358 1.000 15.46 1.794 0.923 
15.46 1.794
0.781 15.46 2.470 0.86 
15.46 2.470 1.000 15.46 1.907 
0.978 15.46 1.907
0.802 15.46 2.581 0.84 
15.46 2.581 1.000 15.46 2.020 0.958 
15.46 2.020
0.813 15.46 2.693 0.78 15.46 
2.693 1.000 15.46 2.133 0.963 15.46 
2.133
0.759 15.46 2.805 0.72 
15.46 2.805 0.606 15.46 2.470 0.601 
15.46 2.470
0.537 15.46 3.029 0.48 
15.46 3.029 0.447 15.46 2.581 0.416 
15.46 2.581
0.284 15.46 3.252 0.32 
15.46 3.252 0.354 15.46 2.693 0.314 
15.46 2.693
0.047 15.46 3.477 0.10 
15.46 3.477 0.244 15.46 2.805 0.183 
15.46 2.805
0.022 15.46 3.704 0.05 15.46 
3.704 0.241 15.46 2.917 0.073 15.46 
2.917
0.020 15.46 3.931 0.03 
15.46 3.931 0.149 15.46 3.029 0.055 
15.46 3.029
0.010 15.46 4.158 0.03 
15.46 4.158 0.097 15.46 3.141 0.050 
15.46 3.141
0.011 15.46 4.385 0.02 15.46 4.385 0.090 15.46 
3.252 0.000 15.46 3.252
0.015 15.46 4.834 
0.01 15.46 4.834 0.038 15.46 
3.477 0.000 15.46 3.477
0.010 15.46 5.945 0.00 
15.46 5.945 0.026 15.46 3.704 
0.000 15.46 3.704
0.013 15.46 7.044 0.00 
15.46 7.044 0.057 15.46 4.159 
0.000 15.46 4.158
0.014 15.46 8.167 
0.00 15.46 8.167 0.000 15.46 
4.609 0.000 15.46 4.609
0.000 15.46 9.297 0.00 
15.46 9.297 0.023 15.46 5.945 
0.000 15.46 5.945
0.004 15.46 10,424 
0.00 15.46 10.424 0.013 15.46 
7.044 0.000 15.46 7.044
0.011 15.46 11.554 
0.00 15.46 11.554 0.002 15.46 
9.972 0.000 15.46 9.972
86
Experiment 23: Calcium, Sodium, Bromide, and Chloride
1 1 2 301 1
CABR? DISPLACED BY NACL;PULSE:300 MIN;SOI CABR2 DISPLACED BY NACL;F P.V.:132.3; 
1-BR
P.V.:132.3; 1 MINUS CA AGAINST P.V P.V.:132.3; NA
VIeals Dealt* Re 84 TO.... Vessel D. .V
128.14 5.0 3.6650 1.760 128.10 5.0 128.10
0 1 1 0 010
00 10 0.0 1,0 
0.0
000.0 15.1 1-. 0.147 
0.0 15.1 1. 0.147 
0.000 0 15.111
0.0 15.1 0.0 15.1 0.880 0.000 15.1
0.0 15.1 0.880 0.01 
15.1 .1.613 0.002 15.1
0.269 15.1 1.348 0.01 15.1 1.760 0.532 15.1
0.266 15.1 238 0.04 15.1 1.907 0.806 15.1
0.466 15.1 2.789 0.14 15.1 2.054 0.926 15.1
0.584 15.1 3.228 0.27 15.1 2.348 0.968 15.1
0.651 15.1 3.520 0., 15.1 2.789 1.000 15.1
0.651 15.1 3.813 0.70 15.1 3.082 1.000 15.1
0.726 15.1 4.105 0.70 15.1 3.229 1.000 15.1
0.60 15.1 4.542 0.72 15.1 3.520 1.000 15.1
0.640 15.1 4.542 0.73 15.1 3.813 1.000 15.1
0.513 15.1 4.688 0.69 15.1 4.105 0.567 15.1
0.267 15.1 4.980 0.62 15.1 4.252 0.243 15.1
0.118 15.1 5.271 0.56 15.1 4.542 0.120 15.1
0.034 15.1 5.999 0.36 15.1 4.688 0.159 15.1
0.006 15.1. 6.729 0.30 15.1 4.980 0.134 15.1
0.011 15.1 7.9461 0.19 15.1 5.271 0.096 15.1
0.013 15.1 80.494 0.08 15.1 5.999 0.111 15.1
0.005 15.1 10.9449 0.06 15.1 6.729 0.091 15.1
0.0 1. 3340.04 15.1 7.461 0.061 15.1
0.001 15.1 13.394 0.03 15.1 8.943 0.033 15.1
0.011 15.1 14.880 0.02 15.1 10.449 0.026 15.1
0.0 51 6340.0 15.1 
11.946 0. 000 15.1
0.0 15.11 13.394 0.052 
15..1
0.0 15.1 14.880
0.0 15.1 16.374
AS P.V.: 132.3; CL A(
I R. .V 0 De R
1. 128.10 1.0
0 1
0.0 1.0
0.147 0.000 15.1 0.147
1.173 0.000 15.1 1.173
1.320 0.000 15.1 1.320
1.613 0.501 15.1 1.613
1.760 0.730 15.1 1.760
1.907 0.873 15.1 1.90?
2.054 .1.000 15.1 2.054
2.201 0.972 15.1 
2.201
2.348 0.933 15.1 2.348
2.495 0.902 15.1 2.495
2.642 0.876 15.1 
2.642
2.789 0.895 15.1* 2.789
3.228 0.406 15.1 
3.228
3.520 0.223 15.1 
3.520
3.813 0.148 15.1 
3.813
4.105 0.087 15.1 4.105
4.251 0.000 
15.1 4.980
4.688 0.000 15.1 
5.271
4.980 0.000 15.1 5.999
5.271 0.000 15.1 
7.461
5.99 0.000 15.1 
10.499
7.461 0.000 15.1 
13.096
10.499 0.000 15.1 
14.434
13. 096
14. 434
Experiment 24: Calcium, Sodium, Bromide, 
and Chloride
1 2 30
CADR2 DISPLACED BY NACL;short column 4
P.V.:128.4;1I MINUS CA AGA1NST P.V.
V eso Dee Reee$ TO....
116.50 5.0 3.051 1.594
0 1 1 0
0.0 10.
0.018 15.27 0.132
0.018 15.27 0.791
0.018 15.27 1.394
0.066 15.2? 1.717
0.210 15.27 2.114
0.355 15.27 2.511
0.489 15.27 2.909
0.562 15.27 3.174
0.617 15.27 3.440
0.618 15.27 3.705
0.451 15.27 4:102
0.402 15.27 4.233
0.332 15.27 4.496
0.265 15.27 4.759
0.172 14.27 5.418
0.093 15.27 6.082
0.036 15.27 6.741
0.024 15.27 8.089
0.017 15.27 9.430
0,012 15.27 10.756
0.004 15.27 12.082
0.000 15.27 13.409
0.004 15.27 14.749
i I
CABR2 DISPLACED BY NACL;st
P. V. :128. 4; NA Al
V s ss Doo m R.
116.50 5.0 2.
0 1
0.0 1.0
0.000 15.27 0.132
0.000 15.27 0.791
0.040 15.27 1.394
0.130 15.27 1.717
0.330 15.27 2.114
0.440 15.27 2.511
0.540 15.27 2.909
0.630 15.27 3.174
0.660 15.27 3.440
0.650 15.27 3.573
0.650 15.27 3.705
0.630 15.27 3.873
0.480 15.27 4.102
0.440 15.27 4.233
0.350 15.27 4.496
0.290 15.27 4.759
0.280 15.27 5.418
0.140 15.27 6.082
0.100 15.27 6.741
0.050 15.27 8.069
0.040 15.27 9.430
0.020 15.27 11.802
0.010 15.27 14.454
P.V. :128.4; 1-BR AG P.V.:128.4; CL AS
Ve 00 Dev te Re VG 61 Dev te R.
116.50 1.0 1 116.50 1.0 1
0 1 0 1
0.0 1.0 0.0 1.0
0.000 15.27 0.132 0.000 15.27 
0.132
0.000 0.791 0.000 15.27 0.791
0.449 15.27 1.394 0.591 15.27 1.394
0.895 15.27 1.717 0.704 15.27 1.717
0.793 15.27 1.849 0.878 15.27 1.849
0.880 15.27 1.982 0.872 15.27 1.982
0.940 15.27 2.114 1.000 15.27 2.114
0.967 15.27 2.247 0.943 15.27 2.247
15.27 2.379 0.995 15.27 2.379
0.848 15.27 2.511 0.910 15.27 2.511
0.556 15.27 2.777 0.668 15.27 .2.777
0.422 15.27 2.909 0.575 15.27 2.909
0.366 15.27 3.042 0.475 15.27 3.042
0.283 15.27 3.174 0.407 15.27 3.174
0.178 15.27 3.307 0.361 15.27 3.307
0.156 15.27 3.572 0.214 15.27 3.572
0.047 15.27 3.837 0.118 15.27 3.137
0.070 15.27 4.102 0.074 15.27 4.102
0.004 15.27 4.365 0.045 15.27 4.365
0.083 15.27 4.496 0.043 15.27 4.496
0.023 15.27 5.419 0.000 15.27 5.419
0.055 15.27 6.741 0.000 15.27 6.741
0.051 15.2? .8.06? 0.000 15.27 8.089
0.000 15.27 9.430 0.000 15.27 9.430
87
I - --)1,311 13, \ 1Q 1Z
0 a 6
1.0
I
1.0
Experiment 25: Calcium, Sodium, Bromide, and Chloride
1 2 32 1 2 1
CABR2 DISPLACED BY NACL;PULSE:300 MIN,sh CABR2 DISPLACED BY NACL;P CABR2 DISPLACED BY NACL;PI CABR2 DISPLACED BY NACL;PI
P.Y.: 93.0; 1 - ca AGAINST P.V. P.V.: 93.0; na A P.V.: 93.0; 1-br P.V.: 93.0; cl Al
V..... Do..... R..... T1O.... V..... D 6..... R V..... ..... R V.... D..... R.
146.02 5.0 1.261 3.411 146.02 5.0 146.02 5.0 1 146.02 5.0
0 1 1 0 0 1 0 1 0 1
0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0
0.002 14.8 0.117 0.00 14.8 0.117 0.00 14.8 0.354 0.00 14.8 0.354
0.000 14.8 0.354 0.00 14.8 0.354 0.024 14.8 0.708 0.035 14.8 0.708
0.005 14.8 0.531 0.00 14.8 0.531 0.048 14.8 0.885 0.068 14.8 0.885
0.007 14.8 0.708 0.07 14.8 0.708 0.621 14.8 1.062 0.620 14.8 1.062
0.083 14.8 0.885 0.08 14.8 0.885 0.908 14.8 1.239 0.883 14.8 1.239
0.562 14.8 1.062 0.55 14.8 1.062 1.000 14.8 1.416 0.973 14.8 1.416
0.851 14.8 1.239 0.87 14.8 1.239 1.000 14.8 1.711 0.911 14.8 1.711
0.928 14.8 1.416 0.94 14.8 1.416 1.000 14.8 1.881 0.896 14.8 1.881
0.966 14.8 1.711 0.94 14.8 1.711 1.000 14.8 2.051 0.928 14.8 2.051
0.971 14.8 1.881 0.94 14.8 1.881 1.000 14.8 2.391 0.955 14.8 2.391
0.984 14.8 2.561 0.97 14.8 2.561 1.000 14.8 2.731 0.908 14.8 2.731
0.991 14.8 3.411 0.98 14.8 3.411 1.000 14.8 3.071 1.000 14.8 3.071
0.992 14.8 3.580 1.00 14.8 3.580 1.000 14.8 3.411 0.979 14.8 3.411
0.993 14.8 4.086 0.94 14.8 4.086 1.000 14.8 3.749 0.987 14.8 3.749
0.972 14.8 4.255 0.46 14.8 4.255 1.000 14.8 4.086 0.993 14.8 4.086
0.526 14.8 4.424 0.16 14.8 4.424 0.879 14.8 4.255 0.843 14.8 4.255
0.184 14.8 4.593 0.06 14.8 4.593 0.309 14.8 
4.424 0.356 14.8 4.424
0.098 14.8 4.761 0.01 14.8 4.761 0.130 
14.8 4.593 0.100 14.8 4.593
0.055 14.8 5.272 0.01 14.8 5.272 0.054 14.8 
4.761 0.039 14.8 4.761
0.054 14.8 5.447 0.00 14.8 5.447 0.019 14.8 
5.099 0.023 14.8 5.099
0.046 14.8 6.139 0.00 14.8 6.139 0.020 14.8 
5.447 0.000 14.8 5.447
0.037 14.8 7.000 0.00 14.8 7.000 0.016 
14.8 5.792 0.074 14.8 5.792
0.025 14.8 8.679 0.00 14.8 8.679 0.000 
14.8 7.504 0.043 14.8 7.504
0.038 14.8 10.360 0.00 14.8 10.360 0.005 
14.8 9.184 0.0 14.8 9.184
0.036 14.8 12.035 0.00 14.8 12.035 
0.0 14.8 10.860 0.0 14.8 10.860
0.032 14.8 13.787 0.00 14.8 13.787 0.0 
14.8 12.560 0.0 14.8 12.560
0.026 14.8 15.549 0.00 14.8 15.549
0.023 14.8 17.306 0.00 14.8 17.306
0.026 14.8 19.019 0.00 14.8 19.019
Experiment 26: Calcium, Sodium, Bromide, and Chloride
1 2 30
CABR2 DISPLACED BY NACL;short column 2
P.V.:154.0; 1-ca AGAINST P.V.*
V..... D..... R..... TO....
74.12 5.0 2.958 1.665
0 1 1 0
0.0 1.0
0.000 15.46 0.083
0.001 15.46 0.916
0.008 15.46 1.249
0.168 15.46 1.748
0.194 15.46 1.831
0.210 15.46 1.998
0.350 15.46 2.164
0.421 15.46 2.331
0.500 15.46 2.580
0.501 15.46 2.664
0.544 15.46 2.830
0.567 15.46 2.996
0.608 15.46 3.163
0.706 15.46 3.413
0.712 15.46 3.496
0.722 15.46 3.662
0.695 15.46 3.829
0.614 15.46 3.995
0.517 15.46 4.245
0.472 15.46 4.328
0.420 15.46 4.495
0.376 15.46 4.661
0.316 15.46 4.828
0.179 15.46 5.077
0.035 15.46 5.910
0.032 15.46 6.742
0.018 15.46 7.574
0.017 15.46 8.407
0.017 15.46 9.239
2 1 1 
2 1
CABR2 DISPLACED BY NACL;sh CABR2 
DISPLACED BY NACL;sl CABR2 DISPLACED 
BY NACL;s
P.V.:154.0; NA AG 
P.V.:154.0; 1-br Al 
P.V.:154.0; cl A
V..... D..... R. 
V..... D..... R V..... 
0..... R
74.12 5.0 2 74.12 
5.0 74.12 5.0
0 1 0 
1 0 1
0.0 1.0 0.0 
1.0 0.0 1.0
0.00 15.46 0.083 
0.018 15.46 0.083 
0.000 15.46 0.083
0.00 15.46 0.916 
0.000 15.46 0.916 
0.000 15.46 0.916
0.06 15.46 1.249 
0.188 15.46 1.249 
0.252 15.46 1.249
0.27 15.46 1.748 
0.836 15.46 1.748 
0.762 15.46 1.748
0.30 15.46 1.831 
1.000 15.46 1.831 
0.928 15.46 1.831
0.37 15.46 1.998 
1.000 15.46 2.164 
1.000 15.46 1.998
0.45 15.46 2.164 
0.920 15.46 2.580 
0.956 15.46 2.164
0.53 15.46 2.331 
0.834 15.46 
2.664 0.963 15.46 
2.331
0.63 15.46 2.580 
0.657 15.46 2.830 
0.860 15.46 2.580
0.64 15.46 
2.664 0.533 
15.46 2.996 
0.818 15.46 
2.664
0.66 15.46 2.830 
0.274 15.46 3.163 
0.630 15.46 2.830
0.67 15.46 
2.996 0.151 15.46 
3.413 0.505 15.46 
2.996
0.69 15.46 
3.163 0.153 
15.46 3.662 0.291 
15.46 3.163
0.66 15.46 
3.413 0.119 
15.46 3.829 
0.037 15.46 
3.413
0.63 15.46 
3.496 0.112 
15.46 3.995 0.008 
15.46 3.496
0.61 15.46 
3.662 0.072 
15.46 4.328 
0.000 15.46 
3.662
0.60 15.46 3.829 
0.021 15.46 4.495 
0.000 15.46 3.829
0.47 15.46 3.995 
0.000 15.46 5.910 
0.000 15.46 3.995
0.38 15.46 
4.245 0.000 
15.46 7.574 0.000 
15.46 4.245
0.36 15.46 
4.328 
0.000 15.46 
4.328
0.34 15.46 4.495
0.31 15.46 4.661
0.27 15.46 4.828
0.16 15.46 5.077
0.02 15.46 5.910
0.00 15.46 6.742
0.00 15.46 7.574
0.00 15.46 8.407
0.00 15.46 9.239
88
Experiment 27: Calcium, Sodium, Bromide, and Chloride
1
2 1 2 30
CABR2 DISPLACED BY NACI
P.V.:156.2; 1-ca A6AINST P.V.
91.51 5.0 3.201 2.080
0 1 1 0
0.0 1.0
0.000 15.1 0.104
0.053 15.1 1.141
0.03? 15.1 2.184
0.243 15.1 2.497
0.406 15.1 2.810
0.468 15.1 3.263
0.607 15.1 3.539
0.649 15.1 3.852
0.692 15.1 4.270
0.716 15.1 4.585
0.733 15.1 4.900
0.678 15.1 5.321
0.554 15.1 5.636
0.204 15.1 5.951
0.02? 15.1 6.371
0.012 15.1 8.585
0.000 15.1 9.658
0.008 15.1 9.732
0.004 15.1 10.781
21
CABR2 DISPLACED BY NACI
P.V.:156.2; NA AG
Vam D .es Re
91.51 5.0 3
0 1
0.0 1.0
0.00 15.1 0.104
0.00 - 15.1 1.141
0.14 15.1 2.184
0.34, 15.1 2.497
0.40 15.1 2.810
0.54 15.1 3.263
0.53 15.1 3.539
0.60 15.1 3.852
0.64 15.1 4.270
0.65 15.1 4.585
0.64 - 15.1 4.900
0.62 15.1 5.321
0.45 15.1 5.636
0.18 15.1 5.951
0.03 15.1 6.371
0.00 15.1 8.585
0.00 15.1 9.658
0.00 15.1 9.732
0.00 15.1 .10.781
.
11
CABR2 DISPLACED BY NACL;PI
P.Y.:156.2; br Al
V69 90 D.. o R.
91.51 5.0
0 1
0.0 1.0
0.02? 15.1 0.830
0.109 15.1 1.03?
0.246 15.1 1.141
0.886 15.1 1.350
1.000 15.1 1.558
1.000 15.1 1.663
1.000 15.1 1.767
1.000 15.1 1.871
1.000 15.1 1.915
1.000 15.1 2.080
1.000 15.1 2.184
1.000 15.1 2.288
1.000 15.1 2.392
1.000 15.1 2.601
1.000 15.1 2.705
1.000 15.1 2.810
0.901 15.1 2.914
0.888 15.1 3.018
0.769 15.1 3.122
0.101 15.1 3.331
0.000 15.1 3.539
0.028 15.1 3.750
0.044 15.1 3.956
0.070 15.1 4.585
0.013 15.1 5.636
0.004 15.1 6.685
Experiment 28: Calcium, 
Sodium, Bromide, and Chloride
1 2 30 2 1
CABR2 DISPLACED BY NACL;short column 4 CABR2 DISPLACED BY NACL;s
P.V. :130.1;, 1-ca AGAINST P.V. P.030.1; na A
vesel 0 ve R, aa TOO.a.. 9 V.....a 6 Do.to.. 9 R
118.76 5.0 2.13 2.595 118.76 5.0
0 1 1 0 0 1
0.0 1.0 0.0 1.0
0.001 15.27 0.135 0.00 15.27 0.135
0.000 15.27 0.540 0.00 15.2? 0.540
0.006 15.27 0.946 0.00 15.2? 0.946
0.102 15.2? 1.351 0.1? 15.2? 1.351
0.211 15.27 1.485 0.26 15.27 1.485
0.31? 15.2? 1.620 0.34 15.2? 1.620
0.460 15.27 1.754 0.40 15.27 1.754
0.491 15.2? 1.889 0.4? 15.27 1.889
0.548 15.27 2.023 0.52 15.27 2.023
0.591 15.27 2.15 0.57 15.27 2.15?
0.657 15.27 2.292 0.63 15.27 2.292
0.706 15.2? 2.595 0.74 15.27 2.595
0.723 15.2? 2.722 0.74 15.27 2.722
0.768 15.2? 3.103 0.80 15.2? 3.103
0.798 15.2? 3.495 0.82 15.27 3.495
0.834 15.27 3.866 0.82 15.2? 3.866
0.861 15.2? 4.104 0.85 15.2? 4.104
0.842 15.2? 4.141 0.?? 15.2? 4.141
0.848 15.27 4.279 0.82 15.2? 4.279
0.712 15.2? 4.416 0.75 15.2? 4.4116
0.5?? 15.2? 4.692 0.65 15.27 4.692
0.43? 15.2? 4.96? 0.37 15.2? 4.967
0.140 15.2? 5.378 0.13 15.2? 5.378
0.036 15.2? 6.736 0.00 15.2? 6.736
0.001 15.2? 8.11? 0.00 15.2? 8.11?
2 
1
CAMR DISPLACED BY NACL;st
P.V.:130 1,, br A(
V 0....,s R.
118.76 5.0
0 1
0.0 1.0
0.000 15.27 0.540
0.295 15.27 0.946
0.561 15.27 1.081
0.934 15.27 1.351
0.949 15.27 1.485
1.000 15.27 1.620
1.000 15.2? 1.754
1.000 15.27 1.889
1.000 15.27 2.023
1.000 -15.27 2.157
1.000 15.27 2.426
1.000 15.27 2,595
1.000 15.2? 2.722
1.000 15.27 2.848
1.000 15.2? 2.932
1.000 15.27 3.103
1.000 15.2? 3.231
0.914 15.2? 3.313
0.586 15.2? 3.495
0.320 15.27 3.408
0.130 15.2? 3.866
0.12? 15.27 4.141
0.075 15.27 4.416
0.034 15.27 4.692
0.012 15.2? 5.242
0.000 15.27 7.033
0.004 15.2? 9.89?
CABR2 DISPLACED BY NACL;P
P.V.:156.2; ci At
V 8 D s R
91.51 5.0
0 1
0.0 1.0
0.000 15.1 0.519
0.000 15.1 0.830
0.091 15.1 1.037
0.169 15.1 1.141
8 15.1 1.350
0.945 15.1 1.558
0.951 15.1 1.663
0.987 15.1 1.767
0.954 15.1 1.871
0.953 15.1 1.975
1.000 15.1 2.080
0.961 15.1 2:184
0.966 15.1 2.288
0.953 15.1 2.392
0.917 15.1 2.601
0.914 15.1 2.705
0.908 15.1 2.810
0.804 15.1 2.914
0.815 15.1 3.018
0.707 15.1 3.122
0.158 15.1 3.331
0.000 15.1 3.539
0.000 15.1 3.750
0.000 15.1 3.956
0.000 15.1 4.585
0.000 15.1 5.636
0.000 15.1 6.685
0.000 15.1 7.838
CABR2 DISPLACED BY NACL;sI
P.V.:130.1; ci At
vote% D R.
118.76 5.
0
0.0 1.0
0.000 15.2? 0.540
0.033 15.27 0.811
0.261 15.2? 0.946.
0.597 15.2? 1.081
0.891 15.27 1.351
0.952 15.27 1.485
0.969 15.27 1.620
0.991 15.27 1.754
1.000 15.2? 1.889
0.99? 15.27 2.023
0.958 15.2? 2.15?
0.958 15.2? 2.426
0.968 15.2? 2.595
0.953 15.27 2.722-
0.918 15.2? 2.848
0.913 15.27 2.932
0.919 15.2? 3.103
0.913 15.27 3.231
0.793 15.2? 3.313
0.529 15.27 3.495
0.266 15.2? 3.408
0.066 15.2? 3.866
89
Experiment 31: Calcium, Potassium, and Sodium
P.V.:167. 1; 1-ca AGAINST P.V. P.V.:167.1 K 
1
ye ss D aa a Ro *# T .. . ... D.. .. R. . O. ..
2 1 2 o
9 .0. . 141.. TO.... V . .0 R .1. TO.... 
CABR2 DISPLACED BY NACL AND KCL, short ci
99.24 5.0 3.141 2.838 99.28 5.0 5.139 2.838 P.V.:167.1; NA 
AGAINST P.V. C
0 1 1 0 0 1 1 0 V.....0 D. 
R..... TO....
0.0 1.0 0.0 1.0 99.24 B40.4 1.642 2.838
0.002 14.80 0.226 0.002 14.80 0.226 
0 0 0 0
0.064 14.80 0.452 0.004 14.80 0.452 0.0 
1.0
0.175 14.80 0.678 0.002 14.80 0.678 
0.0156 14.80 0.226
0.223 14.80 0.903 0.004 14.80 0.904 0.1716 
14.80 0.452
0.337 14.90 1.357 0.042 14.80 1.357 
0.390 14.80 0.678
0.382 14.80 1.584 0.070 14.80 1.584 0.4524 
14.80 0.904
0.431 14.80 1.811 0.104 14.80 1.811 0.5304 14.80 
1.130
0.492 14.80 2.153 0.198 14.80 2.153 0.6084 
14.80 1.357
0.514 14.80 2.495 0.238 14.80 2.495 0.702 
14.80 1.584
0.558 14.80 2.952 0.302 14.80 2.952 
0.7176 14.80 1.811
0.586 14.80 3.411 0.440 14.80 3.411 
0.7488 14.80 2.038
0.555 14.80 3.644 0.590 14.80 3.644 0.7488 
14.80 2.266
0.515 14.80 3.878 0.586 14.80 3.878 
0.7976 14.80 2.495
0.472 14.80 4.111 0.556 14.80 4.111 
0.8112 14.80 2.723
0.389 14.80 4.345 0.486 14.80 4.345 0.8268 
14.80 2.952
0.297 14.80 4.578 0.380 14.80 4.578 0.7956 
14.80 3.182
0.254 14.80 4.811 0.324 14.80 4.811 0.6864 
14.80 3.411
0.223 14.80 5.045 0.296 14.80 5.045 0.4992 
14.80 3.644
0.204 14.90 5.278 0.278 14.80 5.278 0.4524 14.80 
3.878
0.186 14.80 5.512 0.272 14.80 5.512 0.390 
14.80 4.111
0.177 14.80 5.745 0.270 14.80 5.745 0.3278 
14.80 4.345
0.166 14.80 5.978 0.272 14.80 5.978 0.2496 
14.80 4.578
0.160 14.80 6.211 0.270 14.80 6.211 0.1560 14.80 
5.161
0.149 14.80 6.443 0.260 14.80 6.443 0.1248 14.80 
5.512
0.131 14.80 6.910 0.232 14.80 6.910 0.0936 14.80 
5.978
0.129 14.80 7.370 0.214 14.80 7.370 0.078 
14.80 6.443
0.128 14.80 7.838 0.172 14.80 7.838 0.0624 14.80 
6.910
0.110 14.80 8.537 0.158 14.80 8.537 0.0312 14.80 
8.537
0.093 14.80 9,236 0.120 14.80 9.236 0.0156 14.80 
9.236
0.049 14.80 10.524 0.078 14.80 10.524 0.0 14.80 
10.524
0.036 14.80 11.699 0.058 14.80 11.699 0.0 
14.80 11.699
0.017 14.80 12.892 0.048 14.80 12.892 ^v.0 
^  
,4 .. ...C2
0.004 14.90 13.964 0.042 14.80 13.964 0.0 14.80 
13.964
Experiment 32: Calcium, Potassium, and Sodium
P.V.:151.3; CA AGAINST P.V. P.V.:151.3; K AGAINST P.V.
V..... D ..... 
R ..... TO .... 
I 
V .....0 D ..... 
Re...4 TO....
.2 Do020 
2.699 CABR22 
1 2 30 
98.25 53.50 
4.419 2.699
2 5.0 2.560 0CABR2 
DISPLACED BY NACL AND KCL 
0 0 0 0
0 11 P.V.:151.3;NA 
AGAINST P.V. 0.0 1.0
0.005 15.40 0.214 V ista* Dee. R..... TO.... 
0.002 15.40 0.214
0.014 15.40 0.429 98.25 5.0 2.411 2.699 
0.002 15.40 0.429
0.037 15.40 0.858 0 1 1 0 0.002 
15.40 0.858
0.114 15.40 1.072 0.0 1.0 0.002 
15.40 1.072
0.194 15.40 1.288 0.02 15.40 0.214 0.002 15.40 1.288
0.350 15.40 1.504 0.07 15.40 0.429 0.002 
15.40 1.504
0.422 15.40 1.719 0.11 15.40 0.858 
0.002 15.40 1.719
0.444 15.40 1.935 0.32 15.40 1.072 0.002 15.40 
1.935
0.456 15.40 2.151 0.58 15.40 1.288 0.002 
15.40 2.151
0.477 15.40 2.589 0.81 15.40 1.504 0.021 15.40 2.589
0.495 15.40 2.808 0.95 15.40 1.719 0.042 15.40 2.808
0.515 15.40 3.027 0.93 15.40 1.935 0.062 
15.40 3.027
0.531 15.40 3.246 0.95 15.40 2.151 0.080 
15.40 3.246
0.546 15.40 3.468 0.98 15.40 2.370 0.096 
15.40 3.468
0.552 15.40 3.692 1.00 15.40 2.589 0.136 
15.40 3.692
0.567 15.40 3.914 0.98 15.40 2.808 0.200 15.40 
3.914
0.474 15.40 4.137 0.96 15.40 3.027 0.272 15.40 
4.137
0.27 15.40 4.360 0.96 15.40 3.246 0.366 
15.40 4.360
0.317 15.40 4:583 0.96 15.40 3.468 0.532 15.40 
4.583
0.409 15.40 4.807 0.91 15.40 3.692 0.700 
15.40 4.807
0.481 15.40 5.030 0.74 15.40 3.914 0.820 15.40 5.030
0.48 1 5.40 5.254 0.42 15.40 4.137 0.916 
15.40 5.254
0.520 15.40 5.24 0.12 15.40 4.360 0.974 15.40 5.477
0.550 15.40 5.77 0.05 15.40 4.807 1.000 15.40 5.700
0.563 15.40 5.700 0.04 15.40 5.477 0.974 15.40 
5.924
0.459 15.40 5.924 0.04 15.40 7.488 0.822 15.40 6.147
0.49 15.40 6.147 0.03 15.40 10.052 0.620 15.40 6.371
0.246 15.40 6.594 0.03 15.40 11.178 0.442 15.40 6.594
0.086 15.40 7.041 0.02 15.40 12.370 0.230 15.40 7.041
0.048 15.40 7.488 0.01 15.40 13.370 0.134 15.40 7.488
0.020 15.40 7.932 0.066 15.40 7.932
0.000 15.40 8.824 0.048 15.40 8.824
0.00,0 15.40 10.052 0.026 15.40 10.052
0.000 15.40 11.178 0.020 15.40 11.178
0.000 15.40 12.370 0.014 15.40 12.370
0.000 15.40 13.370 90 0.012 15.40 13.370
Experiment 33: Calcium, Potassium, and Sodium
I 2 30
CABR2 DISPLACED BY NACL AND KCL
P.V.:145.8; CA AGAINST P.V.
V..... ..... R..... TO....
98.55 5.0 2.542 1.999
0 1 1 0
0.0 1.0
0.000 15.10 0.203
0.016 15.10 0.344
0.083 15.10 0.406
0.144 15.10 0.573
0.211 15.10 0.802
0.283 15.10 1.146
0.376 15.10 1.604
0.443 15.10 2.063
0.475 15.10 2.292
0.465 15.10 2.516
0.429 15.10 2.739
0.405 15.10 2.963
0.362 15.10 3.186
0.316 15.10 3.410
0.266 15.10 3.638
0.227 15.10 3.868
0.227 15.10 4.096
0.163 15.10 4.554
0.136 15.10 5.012
0.112 15.10 5.698
0.116 15.10 6.154
0.078 15.10 6.838
0.054 15.10 7.983
0.021 15.10 9.130
0.022 15.10 10.247
0.020 15.10 11.383
i 1 2 30
CABR2 DISPLACED BY NACL AND KCL
P..: 145.8; K AGAINST P.V,
V ..... ..... ..... TO....
98.55 5.0 3.88 1.999
0 1 1 0
0.0 1.0
0.004 15.10 0.203
0.002 15.10 0.344
0.004 15.10 0.406
0.004 15.10 0.573
0.004 15.10 0.802
0.014 15.10 1.146
0.110 15.10 1.604
0.198 15.10 2.063
0.232 15.10 2.292
0.332 15.10 2.516
0.502 15.10 2.739
0.570 15.10 2.963
0.530 15.10 3.186
0.438 15.10 3.410
0.348 15.10 3.638
0.280 15.10 3.868
0.218 15.10 4.096
0.154 15.10 4.554
0.122 15.10 5.012
0.088 15.10 5.698
0.086 15.10 6.154
0.058 15.10 6.838
0.044 15.10 7.983
0.038 15.10 9.130
0.034 15.10 10.247
0.032 15.10 11.383
2 1 2 30
CABR2 DISPLACED BY NACL AND KCL
P.V.:145.8; NA AGAINST P.V.
V ..... R ..... TO....
98.55 100.0 1.958 1.999
0 1 1 0
0.0 1.0
0.02 15.10 0.203
0.10 15.10 0.344
0.24 15.10 0.406
0.34 15.10 0.573
0.48 15.10 0.802
0.68 15.10 1.146
0.76 15.10 1.604
0.72 15.10 2.063
0.62 15.10 2.292
0.38 15.10 2.516
0.28 15.10 2.739
0.24 15.10 2.963
0.22 15.10 3.186
0.20 15.10 3.410
0.20 15.10 3.638
0.20 15.10 3.868
0.18 15.10 4.096
0.08 15.10 4.554
0.04 15.10 5.012
0.02 15.10 5.698
0.02 15.10 6.838
0.02 15.10 7.983
0.02 15.10 9.132
0.02 15.10 10.247
0.00 15.10 11.383
Experiment 34: Calcium, Potassium, and Sodium
P.V.:138.4; CA AGAINST P.V. P.V.:138.4; K AGAINST P.V.
V..... D0..... R..... TO.... 
1 2 30 V 
D . R . TO....
120.00 5.0 5.907 2.392 
CABR2 DISPLACED BY NACL AND KCL 
10.9 10. 232
0 1 
1 0 PV. :138.4; 
NA AGAINST 
P. 120.00 
5.0 10.775 2.392 
0
0.0 1.0 
V..... D..... 
R..... TO.... 
0 11.0
0.000 15.30 0.271 120.00 5.0 2.955 2.392 0.002 15.30 0.271
0.000 15.30 0.542 0 1 1 0 0.002 15.30 0.542
0.000 15.30 1.084 0.0 1.0 0.002 
15.30 1.084
0.000 15.30 
1.628 
0.00 15.30 
0.271 
0.002 15.30 
1.628
0.247 15.30 2.173 
0.00 15.30 0.542 
0.004 15.30 2.173
0.101 15.30 
1.901 
0.00 15.30 
1.084 
0.004 15.30 
1.901
0.347 15.30 
2.446 
0.04 15.30 
1.628 
0.004 15.30 
2.446
0.407 15.30 
2.719 
0.50 15.30 
2.173 
0.004 15.30 
2.719446
0.455 15.30 
3.268 
0.32 15.30 
1.901 
0.004 15.30 
3.268.719
0.522 15.30 
3.816 
0.56 15.30 
2.446 
0.004 15.30 
3.816
0.423 15.30 
4.370 
0.90 15.30 2.719 
0.004 15.30 
4.370816
0.135 15.30 
4.925 
1.00 15.30 
3.268 
0.00 15.30 
4.37925
0.533 15.30 4.092 
0.98 15.30 3.816 
0.004 15.30 4.92
0.052 15.30 
5.451 
0.78 15.30 
4.370 
0.004 15.30 
4.092
0.021 15.30 6.012 
0.28 15.30 4.925 
0.002 15.30 5.451
0.017 15.30 
6.849 
0.92 15.30 
4.092 
0.002 15.30 
6.012
0.000 15.30 
7.515 
0.10 15.30 
5.451 
0.002 15.30 
6.849
0.028 15.30 7.914 0.06 15.30 6.012 0.002 15.30 7.515
0.060 15.30 8.180 0.02 15.30 6.849 0.002 15.30 7.914
0.101 15.30 8.468 0.02 15.30 8.180 0.104 15.30 8.46180
0.128 15.30 8.755 0.00 
15.30 10.049 0200 15.30 
8468
0.145 15.30 
9.043 
0.00 15.30 
11.195 
0.254 15.30 
8.755
0.159 15.30 9.330 
0.00 15.30 12.818 
0.282 15.30 9.043
0.147 15.30 
9.474 
0.324 15.30 
9.334740.34 10.49
0.155 15.30 10.049 0.348 15.30 10.049
0.177 15.30 10.624 0.302 15.30 10.624
0.149 15.30 11.195 0.272 15.30 11.195
0.138 15.30 11.753 0.238 15.30 11.753
0.134 15.30 12.450 
0.216 15.30 12.450
0.118 15.30 12.818 
0.188 15.30 12.818
0.106 15.30 13.301 0.164 15.30 13.301
0.090 15.30 14.14 0.142 15.30 14.140
0.083 15.30 14.70 0.124 15.30 14.70
0.072 15.30 15.12 0.114 15.30 15.12
0.054 15.30 15.68 0.098 15.30 15.68
91
APPENDIX E
Exchange Data
Exchange isotherms in a binary system, A/B, were formulated in a
previous study (12) as:
2 3
YA = + fXA + X
+ 
ax (E-1)
where X and Y are the dimensionless concentrations in the liquid and
adsorbed phases, respectively. The isotherm for the competing cation B
follows from:
Y = 1-a-0-- + (P+27+33)X - (T+38)X + x
3
X (E-2)
B B B B
Parameter values for the exchange isotherms and CEC values for the soil
systems used in this study are listed below:
Exchange Properties for Ca/Na Soils
Soil CEC Cation a 7 8
cmol /kg
C
Dothan I 0.600 Ca 0 2.32 -2.24 0.94
Na -0.02 0.66 -0.58 0.94
Dothan II 0.253 Ca 0.02 3.90 -7.05 4.21
Na -0.08 2.43 -5.58 4.21
Wickham I 0.697 Ca 0.28 4.01 -7.66 4.43
Na -0.06 1.98 -5.63 4.43
Wickham II 0.619 Ca 0.31 4.06 -8.12 4.82
Na -0.07 0.18 -6.34 4.82
Troup 0.002 Ca 0.10 0.64 -0.52 0.86
Na -0.08 2.18 -2.06 0.86
Lucedale I 0.618 Ca 0.02 6.14 -11.48 6.37
Na -0.05 2.29 -7.63 6.37
Lucedale II 0.991 Ca 0.01 3.31 -4.63 2.35
Na -0.04 1.10 -2.42 2.35
Savannah I 0.363 Ca 0.16 4.67 -8.79 5.02
Na -0.06 2.15 -6.27 5.02
92