Environmental Engineering Reference
In-Depth Information
where the diffusion-dispersion tensor
D
k
ʱ
is defined as
E
↥
(
D
k
ʱ
(
v
ʱ
)
=
ˆ
ʱ
ˁ
ʱ
ʶ
k
ʱ
I
+
ˁ
ʱ
|
v
ʱ
|
ʶ
l
ʱ
E
(
v
ʱ
)
+
ʶ
t
ʱ
v
ʱ
)
,
(85)
where now
ʶ
k
ʱ
is the molecular diffusivity of component
k
in phase
ʱ
,
ʶ
l
ʱ
and
ʶ
t
ʱ
are, respectively, the longitudinal and transversal dispersivities of phase
ʱ
|
is, as before, the Euclidean norm of the phase velocity. The orthogonal projections
along the phase velocity are as given by relation (
60
) with
v
ʱ
, and
|
v
.
Summing Eq. (
83
) over all phases, using relations (
29
) and (
84
), and the correction
factor for species advection
ₒ
v
ʱ
ʳ
kc
=
ˁ
ʱ
ʻ
ʱ
c
k
ʱ
ʱ
ˁ
ʱ
S
ʱ
c
ʱ
,
(86)
where we have made use of relation (
79
), we obtain the species mass conservation
equation for the multiphase mixture
∂ (ˆˁ
c
k
)
+∇·
(ʳ
kc
ˁ
v
c
k
)
=∇·
D
k
ʱ
·∇
c
k
ʱ
−∇·
c
k
ʱ
J
ʱ
.
(87)
∂
t
ʱ
ʱ
In passing from Eq. (
83
)to(
87
), we have assumed that
I
k
ʱ
=
0
,
(88)
ʱ
since the production of component
k
in phase
must be accompanied by destruction
of this component in other phases. However, if there is an external generation of
components due to chemical or biological reactions the sum of
I
k
ʱ
over all phases
does not vanish and so this term should appear in Eq. (
87
). Moreover, since
v
ʱ
and
v
are needed in Eqs. (
85
) and (
87
), the momentum conservation equation is given by
Darcy's law in the form given by Eqs. (
10
) and (
15
).
When heat transfer is important, we must write a further equation for the specific
internal energy. In analogy with Eqs. (
25
) and (
63
), the energy balance equation for
each constituent (
k
,
ʱ
ʱ
) will read as follows
v
k
ʱ
)
=∇·
ˆ
T
−
ʵ˃
SB
T
4
∂(ˆ
ʱ
ˁ
k
ʱ
U
k
ʱ
)
+∇·
(ˁ
k
ʱ
U
k
ʱ
v
k
ʱ
)
+∇·
(
p
k
ʱ
k
T
,
k
ʱ
∇
∂
t
+
Q
k
ʱ
.
(89)
Using relations (
2
), (
29
), (
78
), (
81
), and (
82
) into the above equation and combining
the result with the mass balance equation (
83
), we obtain the energy equation in
terms of the common temperature
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