The results of two additional Lewis Cell simulations are shown in Fig. 4. These were performed to illustrate pro-
cesses occurring during inter-phase mass transfer for the IPA–water–TCE system. Fig. 4a illustrates a system
where the organic-rich phase consists entirely of TCE and the water-rich phase contains large amounts of alcohol. The
composition history illustrated in Fig. 4a closely resembles that which would occur during injection of an alcohol–
water solution in a porous medium containing separate phase TCE. The alcohol transfers from the water-rich
phase into the organic-rich phase, thus swelling the organic-rich phase. Note that the composition path of the
organic-rich phase closely follows the binodal curve.
Fig. 4b illustrates a system where the organic-phase contains a large amount of alcohol and the water-rich
phase is pure water. Alcohol transfers from the organic- rich phase to the water-rich phase. This emulates mass trans-
fer conditions at the tail end of an injected alcohol slug. Again, as for the IPA–water–PCE system, the composition
paths follow close to the binodal curve if the original com- positions lie near the binodal curve. Note that only small
differences in the overall composition cause large changes in the mass transfer direction.
4.2 Model calibration
Mass transfer coefficients are not available in the literature for systems involving alcohol, water, and organic in porous
media where significant amounts of alcohol partition into the organic-rich phase. Mass transfer coefficients may also
vary dramatically from one system to another. Because of the large uncertainty associated with extrapolating the mass
transfer coefficients determined for one system to another and the lack of published values, it is deemed necessary to
determine these coefficients through model calibration.
The experimental data used for model calibration in this study is taken from Ref.
8
. A laboratory experiment was performed in a one-dimensional vertical glass column.
The porous medium used in the experiment was made up of equal amounts by weight of 0.1 mm, 0.3 mm, and 0.5 mm
diameter glass beads. The column had an inside diameter of 2.5 cm and the porous medium length ranged from 60 to
65 cm. For the experiment of interest, the porous medium length was assumed to be 62.5 cm. The porous medium was
initially saturated with water. PCE was introduced into the top of the column at a gradient of 1.0 until approximately
1.5 pore volumes of PCE had been injected. Water was injected into the top of the column following the injection
of PCE at a rate of 100 ml hr
¹ 1
and was continued until no more separate phase PCE exited the bottom of the column.
Once residual PCE saturation was achieved, one pore volume of an IPA–water mixture, made up of 60 by
volume IPA, was injected into the bottom of the column at a gradient of 0.3. The IPA–water mixture was followed
by 3 pore volumes of water, also at a gradient of 0.3. Ref.
8
states that a spike in the PCE effluent concentration at approximately 2.25 pore volumes could have been due to
sample collection problems. As a result, these data points are not included in the calibration exercise.
Three parameters defining the mass transfer coefficients were determined by matching PCE effluent concentrations
from the numerical simulations to those observed experi- mentally. The three fitted parameters include constants
representing the film thickness in the wetting and non- wetting phase, l
w
and l
nw
, and the specific interfacial area fitting parameter, A
1
. A non-linear least squares fitting routine
58
was used to determine the best-fit parameters. The three-parameter calibration simulation was com-
pleted using the modeling parameters, soil properties, fluid properties, Hand Plot fit, and NRTL parameters shown in
Table 1, Table 2, Table 3 and Table 4. The longitudinal dispersivity, a
L
, has been set to 0.0 since numerical disper- sion, caused by upstream weighting of the advection terms,
results in dispersion of the concentration fronts. A uniform
Fig. 4. Phase composition histories from Lewis Cell simulations with initial phase compositions near the binodal curve for: a high
alcohol concentration in the water-rich phase, and b high alcohol concentration in the TCE-rich phase.
Table 1. Model parameters used in calibration simulations for the IPA–water–PCE system.
Nodal spacing 6.25 3 10
¹ 3
Maximum time step 100 s
Newton–Raphson convergence criterion DP
w
4.0 Pa DS
w
1.0 3 10
¹ 5
Dx
ib
1.0 3 10
¹ 6
670 S. Reitsma, B. H. Kueper
grid spacing equal to 6.25 3 10
¹ 3
m resulted in numerical dispersion similar to a
L
equal to 0.01 m. This was deter- mined by comparison of a single-phase one-component
numerical simulation to a one-dimensional analytical solution for mass transport.
34
Results from the three-parameter fit for two fitting runs, each using different initial estimates, are shown in Table 5.
The high degree of correlation between the film thickness in the wetting phase and the specific interfacial area constant
suggests that the chosen fitting parameters should be com- bined in some manner for this set of experimental data. It is
also clear from Runs 1A and 1B that the confidence in the best-fit values is low since different initial guesses result in
different best-fit values. This is likely due to the high degree of correlation between wetting phase film thickness and
specific interfacial area. Different combinations of A
1
and film thickness result in similar fits to the experimental data.
To further illustrate the non-uniqueness of these two parameters, the fits from Run 1A and 1B are compared in
Fig. 5. Although the values of l
w
and l
nw
are very different for the two simulations, the final effluent curve results are
similar. Because the best-fit values for the three fitting parameters
are uncertain, four fitting runs were completed using A
1
as the only fitting parameter while fixing l
w
and l
nw
. The film thicknesses were assumed to be equal. Model input para-
meters are shown in Table 1 Table 2 Tables 3 and 4. Runs 2 through 4 were completed to examine how A
1
varied as a function l
w
and l
nw
. Run 5, using identical input as Run 2 except for the initial guess of A
1
, was completed to ensure that the fitting run converged to similar best-fit values of A
1
. A summary of the results for Runs 2 through 5 are shown in
Table 6. The results from these runs show that A
1
is proportional to l
w
and l
nw
, and that the value of A
1
is not a function of the initial guess. A combination of mass transfer
terms, defined as a mass transfer ratio, k
o
, and given by k
o
¼ A
1
l
w
26 has been found to be constant regardless of the choice of l
w
, given that l
w
and l
nw
are equal to each other. It is then possible that the mass transfer ratio can be defined to
adequately represent mass transfer for a specific system. Similar expressions for mass transfer coefficients have
been found where interfacial area and film thickness are combined in single-film models, e.g. Refs.
22,41
. It should be noted that in systems where significant trans-
fer of all components takes place, the two-film model is required to properly account for chemical gradients that
exist on either side of the NAPL–water interface. Com- bining mass transfer parameters removes a number of fitting
parameters, but does not change the fact that a two-film approach is used for calculating inter-phase mass transfer.
While a single-film linear driving force model may provide an adequate fit to experimental data in a system where
significant inter-phase mass transfer of all components takes place, it is only through using a two-film modeling
approach that mass transfer phenomena of all components will be captured properly. However, a single-film model
may be appropriate in systems where one or more of the components does not experience significant inter-phase
mass transfer.
Runs 1A and 1B are compared to Run 2 in Fig. 5. The three-parameter formulation does not show significant
improvement in fit to the experimental data. The integral square error ISE is defined as
48
ISE ¼
X ˆy ¹ y
2
q X
y :100
27 where ˆy represents the model output and y represents the
observed experimental results. For the single-parameter fit from Run 2, the ISE is equal to 4.5, and for the three-
parameter fit from Run 1, the ISE is equal to 4.2. The ISE increases only slightly with removal of two of the fitting
parameters, further justifying the use of a single-parameter fit. For comparison, the ISE values for hydrograph
Table 2. Fluid properties used in calibration simulations for the IPA–water–PCE system.
Molecular weight Water
60
1.8 3 10
¹ 2
kg mol
¹ 1
Density Water
14
997 kgm
3
PCE
36
1.66 3 10
¹ 1
kg mol
¹ 1
PCE
36
1610 kgm
3
IPA
8
6.0096 3 10
¹ 2
kg mol
¹ 1
IPA
8
781 kgm
3
Association parameter Water
23
PCE
a
2.26 1.0
Molar volume at normal boiling point m
3
mol
¹ 1
a
Water PCE
1.87 3 10
¹ 5
1.03 3 10
¹ 4
IPA
a
1.0 IPA
7.69 3 10
¹ 5
Viscosity of pure fluid Water
14
8.9 3 10
¹ 4
Pa·s Solubility of water in PCE
37
9.67 3 10
¹ 4
PCE
36
9.0 3 10
¹ 4
Pa·s IPA
60
2.09 3 10
¹ 3
Pa·s Solubility of PCE in water
36
mole fractions 1.62 3 10
¹ 5
a
Indicates estimated parameters.
Table 3. Porous medium properties used in calibration simu- lations for the IPA–water–PCE system
S
rw 0a
0.1 f
8
0.33 S
rnw 8
0.16 ri
a
0.05 l
a
3.0 m
a
0.01 P
d a
3000 Pa a
L b
0.0 m s
¹ 1
j
a
4.5 3 10
¹ 2
N m
¹ 1
k
8
1.0 3 10
¹ 11
m
2
Temparature
a
258C
a
Indicates estimated parameters.
b
Refer to text for discussion.
Non-equilibrium alcohol flooding model for immiscible phase remediation: 2. Model development and application 671
modelling
48
can be examined to give an indication of the goodness of fit.
The values of m and ri, which influence the relative per- meability functions, are estimated in the model calibration.
Fig. 6 illustrates the impact of changing m. It can be seen that the model output is relatively insensitive to this
parameter. Fig. 7 illustrates the impact of changing ri. Although model output is more sensitive to ri than it is to
m, model output is again quite insensitive to ri in the range tested.
The sensitivity of A
1
on model output is shown in Fig. 8. The effluent concentrations are sensitive to the specific
interfacial area parameter, showing lower values in response to a decreased interfacial area available for mass transfer.
There is a stronger response when the interfacial area is reduced by a factor of two than when it is increased by a
factor of two. This indicates that the compositions of the phases near the exit of the column approached equilibrium
conditions for the performed experiment. Had the column been shorter, or flow rates higher, this condition may not
have occurred. Confidence in the predicted values of speci- fic interfacial area and film thickness is expected to decrease
as the experimental conditions used in the calibration approach equilibrium.
4.3 Analysis of mass transfer parameters
