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
A set of equations can be developed for multicomponent inter-phase mass transfer using the concepts of mass transfer
coefficients and a two-film model. Consider the two-film model shown in Fig. 3. The composition of the bulk
phases w and nw are given by x
iw b
and x
inw b
respectively. If the interface is assumed to have negligible resistance to
mass transfer, it will be at equilibrium at all times.
4,11,33
Thus, at the interface, the composition of the wetting phase is given by x
iw I
and of the non-wetting phase by the corresponding equilibrium compositions, x
inw I
. Equations of State govern the relationship between x
iw I
and x
inw I
. The film thickness in phase b is given by l
b
. If the interface between the phases is assumed to be
stationary, molar fluxes N
i b
can be taken in reference to stationary coordinates. In addition, most inter-phase mass
transfer takes place in a direction normal to the interface. Thus the scalar quantity of N
ib
is adequate to describe the inter-phase mass transfer. The interface is assumed to be a
flat surface. Considering first one film in the two-film model, sub-
scripts b are dropped for clarity. For a binary system, the mass transfer coefficient, k
b
, can be defined as
9
k
b
¼ lim
N
1
→0
N
1
¹ x
b 1
N
1
c
t
x
b 1
¹ x
I 1
¼ J
1
c
t
Dx
1
50 where Dx
1
is taken to be the difference between the bulk phase mole fraction, x
1 b
, and inter-phase mole fraction, x
1 I
. Under conditions where mass transfer rates are not small,
eqn 50 must be rewritten to account for the flow due to diffusion of components 1 and 2 across the interface. For
high mass transfer rates, the mass transfer coefficient, k
b •
, can be written as
9
k
• b
¼ N
1
¹ x
b 1
N
1
c
t
x
b 1
¹ x
I 1
¼ J
1
c
t
Dx
1
51 The finite rate mass transfer coefficient, k
b •
, can generally be related to the mass transfer coefficient, k
b
, by
51
k
• b
¼ k
b
Y
b
52 where Y
b
is a correction factor to account for the effect of finite fluxes.
A similar development for finite mass transfer coeffi- cients can be made for a multicomponent system. Eqn 51
can be rearranged to give
51
J ¼
c
t
[ k
• b
] x
b
¹ x
I
¼ c
t
[ k
• b
] Dx
53 where the finite flux mass transfer coefficient matrix of
order n
c
¹ 1 is related to the near-zero flux mass transfer
Fig. 3. Two-film model for inter-phase mass transfer modified
after Ref.
51
.
656 S. Reitsma, B. H. Kueper
coefficient matrix, [k
b
], by [
k
• b
] ¼
[ k
b
][ Y
b
] 54
where Y
b
is the correction factor matrix of order n
c
¹ 1.
The diffusion process in the diffusion layer is determined by the following conditions. The diffusion process is
assumed to be at steady state such that dN
i
dr ¼
55a dN
t
dr ¼
55b where r is the distance into the diffusion film from the
bulk phase. The diffusion process follows Fick’s Law of diffusion eqn 41. The boundary conditions for the film
model are
x
i
¼ x
b i
r ¼ r x
i
¼ x
I i
r ¼ r
d
56 It is convenient to define a dimensionless distance, h, as
h ¼ r ¹ r
r
d
¹ r
¼ r ¹ r
l 57
The compositional profiles across the diffusion film must first be determined to calculate the fluxes N
i
. A constant diffusion coefficient is assumed for solution
to the diffusion process in the diffusion layer. Ref.
49
and Ref.
53
showed independently that solutions to the multi- component diffusion problem using the assumption of a
constant diffusion coefficient matrix led to adequate results. This assumption allows for linearization of the
equations governing diffusion flux and greatly simplifies the solution to multicomponent diffusion problems. Several
other investigators
23,48,50
have also shown the linearized equations to be of excellent accuracy, even in situations of
high composition gradients. For practical purposes, this implies that an average diffusion coefficient, [D
av
], must be evaluated using suitable average mole fractions in the
definition of [B
av
] and [G
av
]. The arithmetic average mole fraction x
i ,
av
¼ x
I i
þ x
b i
=2, is normally recommended for calculation of [B
av
] and Ð
ij
,
6,48,49
and of [G
av
].
51
Thus for non-ideal liquid mixtures
[ D
av
] ¼
[ B
av
]
¹ 1
[ G
av
] 58
The compositional profiles are given by
15,50,52
x ¹ x ¼
[ exp
[ w
] h ¹
[ I
][ exp
[ w
] ¹
[ I
]]
¹ 1
x
d
¹ x
59 where the matrix of mass transfer factors, [w], is defined as
[ w
] ¼
N
t
l c
t
[ D
av
]
¹ 1
60 The composition gradients at the surface at r ¼ r
h ¼ 0 and at r ¼ r
d
h ¼ 1 can be obtained by differentiating eqn 59 to get
d x
dh
h ¼ 0
¼ ¹ [
w ][
exp [
w ]
¹ [
I ]]
¹ 1
x ¹
x
d
61a d
x dh
h ¼ 1
¼ ¹ [
w ]
exp [
w ][
exp [
w ]
¹ [
I ]]
¹ 1
x ¹
x
d
61b Evaluating eqn 40 at the surface at r ¼ r
h ¼ 0 and at r ¼ r
d
h ¼ 1 to obtain the diffusion fluxes J and J
d
, respectively, gives
J ¼ ¹
c
t
l [
D
av
] d
x dh
h ¼ 0
62a
J
d
¼ ¹ c
t
l [
D
av
] d
x dh
h ¼ 1
62b Combining eqns 61a and 61b and eqns 62a and 62b
gives:
51
J ¼
c
t
l [
D
av
][ w
][ exp
[ w
] ¹
[ I
]]
¹ 1
x ¹
x
d
63a
J
d
¼ c
t
l [
D
av
][ w
] exp
[ w
][ exp
[ w
] ¹
[ I
]]
¹ 1
x ¹
x
d
63b The zero-flux mass transfer coefficient matrices are given
by
51
[ k
] ¼
[ D
av
] =l ¼
[ k
d
] ¼
[ k
av
] 64
The correction factor matrices used in eqn 54 are given by [
Y ]
¼ [
w ][
exp [
w ]
¹ [
I ]]
¹ 1
65a [
Y
d
] ¼
[ w
] exp
[ w
][ exp
[ w
] ¹
[ I
]]
¹ 1
65b and the finite flux mass transfer coefficient matrices are
given by [
k
•
] ¼
[ Y
][ k
av
] 66a
[ k
• d
] ¼
[ Y
d
][ k
av
] 66b
Now, the correction factor matrix may be calculated from an application of Sylvestor’s expansion formula
51
[ Y
] ¼
X ˆ
Y
i
n Y
m j ¼
1
j Þ i [[
w ]
¹ ˆ w
j
[ I
]] =
Y
m j ¼
1
j Þ i ˆ
w
i
¹ ˆ w
j
o 67
where m is the number of distinct eigenvalues of [w]m n
c
¹ 1. The eigenvalue functions are given by
ˆ Y
i
¼ ˆ
w
i
exp ˆ w
i
¹ 1
68a
ˆ Y
id
¼ ˆ
w
i
exp ˆ w
i
exp ˆ w
i
¹ 1
68b Non-equilibrium alcohol flooding model for immiscible phase remediation: 1. Equation development
657
depending on whether the flux correction is to be evaluated at h ¼ 0 or h ¼ 1.
For the ternary case where there are two distinct Eigenvalues, eqns 68a and 68b gives
[ Y
] ¼
ˆ Y
1
[[ w
] ¹ ˆ
w
2
[ I
]] ˆ
w
1
¹ ˆ w
2
þ ˆ
Y
2
[[ w
] ¹ ˆ
w
1
[ I
]] ˆ
w
2
¹ ˆ w
1
69 The two eigenvalues ˆ
w
1
and ˆ w
2
are roots of the quadratic equation
ˆ w
2
¹ w
11
þ w
22
ˆ w þ
w
11
w
22
¹ w
12
w
21
¼ 70
so that ˆ
w
1
¼ 1
2 tr
[ w
] þ
disc
[ w
] p
n o
71a
ˆ w
2
¼ 1
2 tr
[ w
] ¹
disc
[ w
] p
n o
71b where
tr [
w ]
¼ w
11
þ w
22
72a lwl ¼ w
11
w
22
¹ w
12
w
21
72b disc
[ w
] ¼
tr [
w ]
2
¹ 4
lwl 72c
With [Y] evaluated using eqn 70 and [k] evaluated using eqn 64, [k
•
] can now be calculated using eqns 66a and 66b. Eqn 40 is then used to evaluate the diffusion flux
matrix J which can then be used in eqn 30 to evaluate the molar flux matrix N.
6.3 Computation of fluxes
