Environmental Engineering Reference
In-Depth Information
the fluids. Furthermore, heat conduction may also occur between the coexisting fluid
phases. For simplicity in the exposition, we may invoke local thermal equilibrium in
the fluid and assume that the temperature is the same in all fluid phases. In addition,
some variables such as porosity, density, and viscosity may depend on temperature.
We start by defining the internal energy of the composite system, consisting of the
flowing multiphase mixture and the solid matrix, as
ˆˁ
U
=
ˆ
ˁ
ʱ
S
ʱ
U
ʱ
+
(
1
−
ˆ)ˁ
R
C
R
T
,
(23)
ʱ
where
U
, the heat capacity of
the rock, and the common temperature, respectively. The overall density is given by
,
C
R
, and
T
are the specific internal energy of phase
ʱ
ʱ
ˆˁ
=
ˆ
ˁ
ʱ
S
ʱ
+
(
1
−
ˆ)ˁ
R
.
(24)
ʱ
From the statement of the first law of thermodynamics in a differential volume
occupied by phase
ʱ
, we can derive the internal energy balance equation of phase
ʱ
as
+∇·
(ˁ
ʱ
H
ʱ
v
ʱ
)
=∇·
ˆ
T
−
ʵ
r
˃
SB
T
4
∂(ˆˁ
ʱ
S
ʱ
U
ʱ
)
∂
S
ʱ
k
T
,ʱ
∇
+
Q
ʱ
,
(25)
t
ʱ
where
H
ʱ
is the specific enthalpy of phase
given by
p
ʱ
ˁ
ʱ
,
H
ʱ
=
U
ʱ
+
(26)
k
T
,ʱ
is the thermal conductivity of phase
˃
SB
is
the Stefan-Boltzmann constant, and
Q
ʱ
is the interphase heat transfer rate associated
with phase
ʱ
,
ʵ
r
is a radiation emissivity factor,
ʱ
. Here,
Q
=
Q
ʱ
,
(27)
ʱ
where
Q
includes all external volumetric heat sources and sinks. Noting that
U
ʱ
=
C
ʱ
T
, where
C
ʱ
, and using the mass conservation
equation (
4
) and the enthalpy definition (
26
), Eq. (
25
) can be rewritten in terms of
the temperature
T
as follows
is the heat capacity of fluid phase
ʱ
∂
T
∂
ˆˁ
ʱ
S
C
t
+
ˁ
ʱ
C
v
ʱ
·∇
T
+
C
I
T
=−∇·
(
p
v
ʱ
)
ʱ
ʱ
ʱ
ʱ
ʱ
ʱ
+∇·
ˆ
T
−
ʵ
r
˃
SB
T
4
S
ʱ
k
T
,ʱ
∇
+
Q
ʱ
.
(28)
Sometimes it is useful to define the diffusive mass flux of phase
ʱ
within the multi-
phase mixture as
J
ʱ
=
ˁ
ʱ
v
ʱ
−
ʻ
ʱ
ˁ
v
,
(29)
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