therefore provide a better representation of the viscosity of a mixture than will eqn 20. Ref.
45
gives detail on the polynomial fits to the alcohol–water systems considered
in this work. Mass density and molar densities are calculated assuming
that volume is additive. The molar density is therefore given by
c
tb
¼ X
n
c
i ¼ 1
x
ib
c
p i
¹ 1
23 where c
i
is the molar density of a pure phase consisting entirely of component i. The range in pressures and
changes in pressures encountered at shallow depths less than 100 m are not great so that liquid phases can essen-
tially be considered as incompressible. Assuming that the phases are incompressible and the volumes from each
component are additive, the mass density is given by
r
b
¼ c
tb
X
n
c
i ¼ 1
x
ib
W
i
24 where W
i
is the molecular weight of component i.
5 EQUATIONS OF STATE
Typical ternary phase behavior for a Type 1 system is shown in Fig. 2. The ternary behavior can be represented
in various ways. Thermodynamic properties models, such as UNIQUAC
3
or NRTL,
46
can be used to calculate phase compositions. These models require thermodynamic data
to calculate activity coefficients. Ref.
51
summarizes the calculation of activity coefficients for the NRTL and
UNIQUAC thermodynamic properties models. Ref.
42
pre- sents a method of representing the ternary behavior using
UNIQUAC. Type 1 ternary behavior has also been repre- sented by various functional fits, e.g. Refs.
18,26,57
. Func- tional fits can be utilized for both the binodal curve and
the distribution of components between the phases tie- line behavior. Ref.
18
represented two-phase ternary behavior based on the observation that certain ratios of
equilibrium phase concentrations are straight lines on a log–log or a Hand Plot.
The Hand Plot representation for ternary phase behavior was chosen for this work because it lends itself to
straightforward calculation of miscibility of bulk phases. This calculation is not necessary when the assumption of
local equilibrium is used because the overall composition, given by the sum of the two phase compositions, need only
be above the binodal curve for the phases to be miscible. When not using the assumption of local equilibrium, how-
ever, an overall composition above the binodal curve does not necessarily indicate miscibility. All possible composi-
tions made up of two bulk phases must be miscible for certain miscibility of the two bulk phases. These composi-
tions are represented by a line drawn between the bulk phase compositions on a ternary phase diagram. If the line tying
the two bulk phase compositions intersects the binodal curve, the two phases are immiscible.
Thermodynamic properties model representation of ternary phase behavior requires a complex iterative routine
to determine miscibility between the bulk phases and there- fore was not adopted in this work. To be consistent with
diffusion coefficients calculated using thermodynamic properties models, however, ternary phase data can be
generated using the appropriate thermodynamic properties model. In turn, the generated ternary phase data can be
represented using the Hand Plot equations and the fitting parameters can be used in the compositional model. The
Hand Plot representation was also chosen because of its utility and goodness of fit to experimental data. Details of
Hand Plot equations, fitting, and numerical solution are included as an appendix and examples are given in Ref.
45
. 6 MASS TRANSFER CONSIDERATIONS
Mass transfer between phases is of central importance when developing a non-equilibrium compositional model. Calcu-
lation of inter-phase mass transfer has been completed in various ways by different authors, e.g. Refs.
20,40,43
. A pop- ular technique is using a single-film model where mass
transfer rates are governed by the film thickness and the concentration gradients in the film. Although appropriate
under certain circumstances, this technique may be invalid when significant mass transfer may occur in two directions,
both into and out of a given phase. For an alcohol–water– DNAPL system, for example, where significant amounts of
alcohol may partition into the DNAPL phase, a more involved inter-phase mass transfer model is required. The
following sections, based on Ref.
51
, present the equations and assumptions used to solve mass transfer in a two-phase,
multicomponent system.
6.1 Mass transfer in a non-ideal liquid multicomponent system
The molar flux of component i, N
i
, is given by N
i
¼ c
i
u
i
25 where c
i
is the molar density of component i and u
i
is the velocity of component i with respect to a stationary
coordinate reference frame. The total molar flux, N
t
, is given by
N
t
¼ X
n
c
i ¼ 1
N
i
¼ c
i
u 26
where u is the molar average velocity given by u ¼
X
n
c
i ¼ 1
x
i
u
i
27 The diffusion flux, J
i
, which is the flux of component i with 654
S. Reitsma, B. H. Kueper
respect to the flux of the mixture as a whole, is given by J
i
¼ c
i
u
i
¹ u
28 The sum of the component diffusion flux is always equal to
zero. Therefore, eqn 28 represents n
c
¹ 1 independent
equations. The molar flux is related to the diffusion flux by N
i
¼ J
i
þ c
i
u ¼ J
i
þ x
i
N
t
29 eqn 29 also represents n
c
¹ 1 independent equations and
can be rewritten in matrix form as N
¼ J
þ x
N
t
30 The
Maxwell–Stefan equations
for multicomponent
systems are used to determine diffusive flux as a function of concentration gradient. The Maxwell–Stefan equations
representing the driving force for diffusion of component i, d
i
, in non-ideal fluids is given by
47
d
i
¼ ¹ X
n
c
j ¼ 1
x
i
x
j
u
i
¹ u
j
Ð
ij
¼ X
n
c
j ¼ 1
x
i
N
j
¹ x
j
N
i
c
t
Ð
ij
¼ X
n
c
j ¼ 1
x
i
J
j
¹ x
j
J
i
c
t
Ð
ij
ð 31Þ
where Ð
ij
is an inverse drag coefficient of component j on component i, known as Maxwell–Stefan diffusivity,
and c
t
is the molar density. Eqn 31 represents n
c
¹ 1
independent equations since X
n
c
i ¼ 1
=x
i
¼ 32
For non-ideal fluids, d
i
can also be defined as d
i
; x
i
RT =
T ,
P
m
i
33 where R is the gas constant, T is temperature, P is pressure,
and =
T ,
P
m
i
is the chemical potential gradient at constant temperature and pressure. Since it is difficult to measure
chemical potential, a more practical expression for d
i
can be given as
51
d
i
¼ x
i
RT =
T ,
P
m
i
¼ x
i
RT X
n
c
¹ 1
j ¼ 1
]u
i
]x
j T
, P
, S
=x
j
¼ x
i
RT X
n
c
¹ 1
j ¼ 1
RT ]
ln g
i
x
i
]x
j T
, P
, S
=x
j
¼ x
i
X
n
c
¹ 1
j ¼ 1
] lnx
i
]x
j
þ ]
lng
i
]x
j T
, P
, S
=x
j
¼ X
n
c
¹ 1
j ¼ 1
d
ij
þ x
i
] lng
i
]x
j T
, P
, S
=x
j
¼ X
n
c
¹ 1
j ¼ 1
G
ij
=x
j
34 where g
i
is the activity coefficient of component i. The thermodynamic factor, G
ij
, given by G
ij
¼ d
ij
þ x
i
dlng
i
]x
j T
, P
, S
i ,
j ¼ 1
, :::
, n
c
¹ 1
35 can be calculated using thermodynamic properties models
such as NRTL or UNIQUAC. The symbol S is used to indicate that the differentiation of ln g
i
with respect to frac- tion x
j
must be completed using constant values of all the other components except for the nth component. The sum
of x
i
must continue to be unity. eqn 31 can be rewritten in matrix form as
c
t
d ¼ ¹
[ B
] J
36 where:
B
ij
¼ x
i
Ð
in
c
þ X
n
c
k ¼ 1
k Þ i x
k
Ð
ik
i ¼ 1
, :::
, n
c
¹ 1
37a
B
ij
¼ ¹ x
i
1 Ð
ij
¹ 1
Ð
in
c
i ,
j ¼ 1
, :::
, n
c
¹ 1
37b eqn 36 can rearranged to give
J ¼ ¹
c
t
[ B
]
¹ 1
d 38
eqn 34 can also be written in matrix form to give d
¼ [
G ]
=x 39
eqn 39 can be combined with eqn 38 to express diffu- sion flux as a function of concentration gradient with the
result
J ¼ ¹
c
t
[ B
]
¹ 1
[ G
] =x
40 A more familiar expression for diffusion is given by Fick’s
Law, written in a general form as J
i
¼ ¹ c
t
X
n
c
¹ 1
k ¼ 1
D
ik
=x
k
i ¼ 1
, :::
, n
c
¹ 1
41 where D
ik
are diffusion coefficients. Eqn 41 can be written in matrix form as
J ¼ ¹
c
t
[ D
] =x
42 From eqn 40 and eqn 42, although not mathematically
rigorous,
51
the Fick diffusion coefficient matrix can be given by
[ D
] ¼
[ B
]
¹ 1
[ G
] 43
eqn 43 allows one to predict [D] from the binary Maxwell–Stefan diffusivities, Ð
ij
, and activity coefficients. Maxwell–Stefan diffusivities are a function of composi-
tion and must also be estimated. Ref.
55
suggested that, for concentrated liquid binary systems, the compositional
dependence of Ð can be expressed by the relation of the form
Ð
12
¼ Ð
+ 12
x
2
Ð
+ 21
x
1
44 Non-equilibrium alcohol flooding model for immiscible phase remediation: 1. Equation development
655
where Ð
12 +
is the Maxwell–Stefan diffusivity of component 1 infinitely diluted in solvent composed of component 2.
Eqn 44 has also been recommended by Ref.
44
. Ð
12 +
has been estimated in several ways with one of the best
51
given by
59
Ð
+ 12
¼ 7·4 3 10
¹ 12
J
2
M
2
3 10
3
ÿ ÿ
0:5
T m
2
3 10
3
ÿ V
1
3 10
6
ÿ
0:6
45 where M
2
is the molar mass of the solvent, T is the tem- perature, m
2
is the viscosity of the solvent, V
1
is the molar volume of solute 1 at its normal boiling point, and J
2
is the association factor for the solvent e.g. 2·26 for water, 1·9
for methanol, 1·5 for ethanol, and 1.0 for unassociated solvents. The value of 2·26 for water was found in
Ref.
18
to provide better results than the value of 2·6 suggested by Ref.
59
. Multiplication factors have been added to eqn 45 to be consistent with units used elsewhere
in this paper. For a multicomponent system, eqn 44 can be general-
ized as follows
22,58
Ð
ij
¼ Y
n
c
k ¼ 1
Ð
ij ,
x
k
→1
ÿ
x
k
46 where
Ð
ij ,
x
j
→1
¼ Ð
+ ij
Ð
ij ,
x
i
→1
¼ Ð
+ ji
47 Substituting the limiting diffusivities from eqn 47 into
eqn 46 gives Ð
ij
¼ Ð
+ ij
x
j
Ð
+ ji
x
i
Y
n
c
k ¼ 1
k Þ i ,
j Ð
ij ,
x
k
→1 x
k
48 Finally, Ref.
22
proposed the following model for the limiting diffusivities
Ð
ij ,
x
k
→1
¼ Ð
+ ik
Ð
+ jk
0:5
49 which can be placed in eqn 48 to calculate Maxwell–
Stefan diffusivities at various compositions. Estimates of
diffusivities now
allow for
calculation of
[B] using eqns 37a and 37b, and when combined with esti-
mates of [G] using a thermodynamic properties model, [B] can be used to calculate the diffusion coefficient
matrix [D].
6.2 Mass transfer coefficients and the two-film model
