Environmental Engineering Reference
In-Depth Information
=
,
,
,...,
ʱ
k
)isa
constituent. We assume thermal equilibrium among the species in a given phase and
among all phases so that
T
k
ʱ
=
1
2
3
n
, for the species (or components). Therefore, each pair (
k
,
T
.
We start by stating that each constituent (
k
,
ʱ
) has its own intrinsic mass density
ˁ
k
ʱ
, measured as the mass of component
k
per unit volume of phase
ʱ
, and its own
velocity
v
k
ʱ
. By analogy with Eq. (
4
) we can write the mass balance equation for
each constituent (
k
,
ʱ
)as
∂ (ˆ
ʱ
ˁ
k
ʱ
)
∂
+∇·
(ˁ
k
ʱ
v
k
ʱ
)
=
I
k
ʱ
,
(75)
t
where
I
k
ʱ
represents the sources and sinks of component
k
in phase
. In the absence
of injection and production of this component by external means, the exchange terms
I
k
ʱ
ʱ
must obey the restriction
n
I
k
ʱ
=
0
.
(76)
k
=
1
ʱ
=
R
Using relation (
2
) together with the definitions
n
ˁ
ʱ
=
1
ˁ
k
ʱ
,
(77)
k
=
ˁ
k
ʱ
ˁ
ʱ
,
c
k
ʱ
=
(78)
1
ˁ
c
k
=
S
ʱ
ˁ
ʱ
c
k
ʱ
,
(79)
ʱ
=
R
n
1
ˁ
ʱ
v
ʱ
=
1
ˁ
k
ʱ
v
k
ʱ
,
(80)
k
=
u
k
ʱ
=
v
k
ʱ
−
v
ʱ
,
(81)
d
k
ʱ
=
ˁ
ʱ
c
k
ʱ
u
k
ʱ
,
(82)
where
ˁ
ʱ
is the intrinsic mass density of phase
ʱ
,
c
k
ʱ
is the concentration of species
k
in phase
ʱ
,
c
k
is the total volumetric concentration of species
k
in the mixture,
v
ʱ
is the barycentric velocity of phase
ʱ
,
u
k
ʱ
is the diffusion velocity of species
k
in
phase
ʱ
, and
d
k
ʱ
is the diffusive flux of constituent (
k
,
ʱ
), the mass balance equation
(
75
) can be rewritten in the more convenient form
∂ (ˆ
S
ʱ
ˁ
ʱ
c
k
ʱ
)
+∇·
(ˁ
ʱ
c
k
ʱ
v
ʱ
)
=−∇·
d
k
ʱ
+
I
k
ʱ
.
(83)
∂
t
The effective diffusive flux
d
k
ʱ
can be written in Fickian formby simply extending
Eq. (
58
) to multiphase flows, i.e.,
d
k
ʱ
=−
D
k
ʱ
·∇
c
k
ʱ
,
(84)
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