Environmental Engineering Reference
In-Depth Information
n
C
∂
T
∂
)
−
ʵ
r
˃
SB
T
4
ˆˁ
t
+
ˁ
C
v
·∇
T
=−∇·
H
k
d
k
−∇·
(
p
v
)
+∇·
(ˆ
k
T
∇
T
+
Q
,
k
=
1
(68)
where
C
is the multicomponent fluid heat capacity, defined as
n
1
ˁ
C
=
1
ˁ
k
C
k
.
(69)
k
=
The first term on the right-hand side of Eq. (
68
) describes the heat generated due to
diffusion of one species into another, while the last one accounts for the heat gen-
erated by chemical reactions and the kinetic energy generated when one component
chemically changes into another. For the composite system consisting of the multi-
component fluid and the solid porous matrix, Eq. (
68
) is still applicable by making
ˆˁ
C
ₒ
ˆˁ
C
+
(
1
−
ˆ)ˁ
R
C
R
,
(70)
and
n
ˆ
k
T
ₒ
(
1
−
ˆ)
k
T
,
R
+
ˆ
k
T
,
k
.
(71)
k
=
1
To complete the description of the multicomponent flow system, Eqs. (
57
) and (
68
)
are complemented by the mass balance equation for the single-phase fluid, namely
∂(ˆˁ)
∂
+∇·
(ˁ
v
)
=
0
,
(72)
t
in the absence of external sources or sinks. If Darcy's velocity (
56
) is replaced
into Eqs. (
57
), (
68
), and (
72
), we have a coupled set of
n
2 equations for the
n
concentrations
c
k
, the temperature
T
, and the pressure
p
, where the density and
dynamic viscosity are known functions of
p
,
T
, and
c
k
:
+
ˁ
=
ˁ(
p
,
T
,
c
1
,
c
2
,...,
c
n
),
(73)
μ
=
μ(
p
,
T
,
c
1
,
c
2
,...,
c
n
),
(74)
while the heat conductivity is a function of temperature and chemical concentration.
4 Compositional Flow in a Porous Medium
In this section, we consider multiphase and multicomponent flow, i.e., compositional
flow, in a porous medium. In this type of flows there may be several coexisting
fluid phases in which a finite number of chemical species may reside. As before,
the subscript
ʱ
is chosen for the fluid phases,
R
for the solid phase, and
k
, with
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