two-phase, three-component
alcohol–water–organic system, the inter-phase mass transfer equations can be writ-
ten as a series of equations in the form Fx ¼ 0 to give two equations for the molar fluxes in the wetting phase
55
R
w
; c
tw
[ k
• w
] x
b w
¹ x
I w
þ N
t
x
b w
¹ N
¼ 20
where R
w
is the residual vector for the wetting phase inter-phase mass transfer equations, [k
w •
] is the finite flux mass transfer coefficient matrix,
x
b w
¹ x
I w
is the vector containing the differences in mole fractions of each com-
ponent in the bulk wetting phase and at the interface in the wetting phase, N
t
is the total molar flux, and N is the vector of molar fluxes. Two equations for the molar
fluxes in the non-wetting phase can be written as R
nw
; c
tnw
[ k
• nw
] x
I nw
¹ x
b nw
þ N
t
x
b nw
¹ N
¼ 21
where the corresponding variables are defined as above. The equation set is completed using equations of state.
The independent equations are ordered into a vector of functions F as follows modified after Ref.
55
F
T
¼ R
w 1
, R
w 2
, R
nw 1
, R
nw 2
22 The unknown variables corresponding to this set of
equations are ordered into a vector x as follows modified after Ref.
55
x
T
¼ N
1
, N
2
, N
3
, x
I p
23 where x
I
is one mole fraction at the interface. Note that for a Type 1 ternary system, the equilibrium composition of the
two phases are distinctly defined by the mole fraction of one component in one phase from which all other mole
fractions can be determined. The mole fraction at the inter- face, x
I
, has been chosen to be the mole fraction of water in the wetting phase at the interface, x
1w I
, in this work. The Newton–Raphson structure to eqn 20 and eqn 21
is given by ]F
1
]N
1
]F
1
]N
2
]F
1
]N
3
]F
1
]x
I 1w
]F
2
]N
1
]F
2
]N
2
]F
2
]N
3
]F
2
]x
I 1w
]F
3
]N
1
]F
3
]N
2
]F
3
]N
3
]F
3
]x
I 1w
]F
4
]N
1
]F
4
]N
2
]F
4
]N
3
]F
4
]x
I 1w
2 6
6 6
6 6
6 6
6 6
6 6
6 6
4 3
7 7
7 7
7 7
7 7
7 7
7 7
7 5
DN
1
DN
2
DN
3
Dx
I 1w
B B
B B
B 1
C C
C C
C A
¼ ¹ R
w 1
R
w 2
R
nw 1
R
nw 1
B B
B B
B 1
C C
C C
C A
24 The Jacobian terms in eqn 24 are calculated numerically.
Several terms in eqn 24 may be approximated as sug- gested by Ref.
55
to give x
b 1w
¹ 1
x
b 1w
x
b 1w
]F
1
=]x
I 1w
x
b 2w
x
b 2w
¹ 1
x
b 2w
]F
2
=]x
I 1w
x
b 1nw
¹ 1
x
w 1n
x
b 1nw
]F
2
=]x
I 1w
x
b 2nw
x
b 2nw
¹ 1
x
b 2nw
]F
2
=]x
I 1w
2 6
6 6
6 6
4 3
7 7
7 7
7 5
25 During development of the mass transfer algorithms, both
numerically determined and estimated Jacobian terms have been used for solution to eqn 20 and eqn 21. The com-
putational savings afforded by using the estimated Jacobian matrix were found to be offset by the additional iterations
required for convergence and decreased algorithm robust- ness for the three-component systems tested here. While
both methods may be used in the non-equilibrium model, all subsequent simulations shown here use the Jacobian
matrix given by eqn 24.
4 MODEL TESTING
Due to the complexity of physical phenomena occurring during an alcohol flood and the non-linear equations
required for numerical simulation of this system, only partial model verification can be accomplished. Partially
verification was carried out using exact analytical solutions for two-phase flow
33
and for mass transport.
21
In both cases, agreement between the numerical model and the analytical
solution was excellent. Both Cauchy-type inlet conditions and free exit boundary conditions were shown to be properly
solved.
4.1 Mass transfer algorithm testing using Lewis Cell simulations
A series of simulations was conducted to test the developed inter-phase mass transfer algorithm using a configuration
similar to the experimental apparatus employed in Ref.
31
, thus referred to as the Lewis Cell. The ternary systems used
in the simulations were the 2-propanol IPA–water– tetrachlorethene PCE system shown in Fig. 1 and the 1-
propanol–water–trichloroethene TCE system shown in
Fig. 1. Ternary phase behavior for the 2-propanol IPA–water– tetrachlorethene PCE system with Hand Plot fit to experimental
data from Ref.
4
. Several experimental tie line data have been removed for clarity.
668 S. Reitsma, B. H. Kueper
Fig. 2. The simulations, representing a closed system containing a water-rich phase and an organic-rich phase
initially having non-equilibrium conditions, were completed using various different initial bulk phase compositions. The
bulk phases were assumed to be well mixed. A constant film thickness of 10
¹ 4
m for each phase was used in all simulations.
The NRTL
46
thermodynamic properties model was used for calculation of the diffusion coefficients for the IPA–
water–PCE system. The UNIQUAC
2
thermodynamic properties model was used for the 1-propanol–water–TCE
system because it was found to be more robust than the NRTL model for this system. A set of thermodynamic para-
meters for the two ternary systems can be found in Tables 4–8, below. A specific interfacial area of 10 m
¹ 1
was used in the simulations. This translates to a cube-shaped con-
tainer with sides measuring 0.1 m, where the interfacial area between the two phases is equal to the cross-sectional
area of the container. A simulation time of 10
5
seconds was chosen to allow phases to approach equilibrium.
The results of three different simulations are shown in Fig. 3. These simulations were performed to illustrate pro-
cesses occurring during inter-phase mass transfer for the IPA–water–PCE system. The figures present the initial
phase compositions, the overall system composition, and the phase composition pathways followed during the simu-
lations. The simulation represented in Fig. 3a was per- formed because it closely represents the introduction of an
alcohol-rich wetting phase to a porous medium containing PCE. Note that the composition path of the organic-rich
phase starts at pure PCE and follows along the binodal curve to its final equilibrium value. The water-rich phase
composition path curves sharply rather than following a straight line to the final equilibrium composition. This
illustrates the strong molecular interaction between the different components during diffusion.
54
The simulation represented in Fig. 3b is initiated with a high alcohol concentration in the organic-rich phase. The
composition path trends compare favorably to those observed experimentally in Ref.
29
for a similar system. Again, curvature in the composition traces indicates
molecular interaction during diffusion. Fig. 3c is used to illustrate the composition path for
initial phase compositions that both lie close to the binodal curve. It might be expected that the composition paths
would follow a straight line from the initial composition to the final equilibrium composition. If this were to occur,
either a water-rich phase would precipitate from the organic-rich phase, or the organic-rich phase would
become super-saturated in response to the composition of the phase falling below the binodal curve. However, due to
the change in diffusion coefficient with composition, the composition paths follow along the binodal curve such
that neither phase becomes super-saturated.
Fig. 2. Ternary phase behavior for the 1-propanol–water– trichlorethene system with Hand Plot fit to experimental data
from Ref.
35
.
Fig. 3. Phase composition histories from Lewis Cell simulations for: a high alcohol concentration in the water-rich phase, b
high alcohol concentration in the PCE-rich phase, and c initial phase compositions on the binodal curve.
Non-equilibrium alcohol flooding model for immiscible phase remediation: 2. Model development and application 669
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
