Environmental Engineering Reference
In-Depth Information
n
d
k
=
0
.
(61)
k
=
1
A common approach in oil reservoir simulations is to assume that hydrodynamic
dispersion is a small enough effect that the diffusion-dispersion fluxes in the com-
ponent mass balance equations are negligible (Allen
1985
).
In Eq. (
57
)theterm
I
k
is a source/sink term for the
k
th component. It can result
from injection and/or production of a particular component by external means. How-
ever, it can also stem from various other processes within the fluid, such as chemical
reactions among species, radioactive decay, biodegradation, and growth due to bac-
terial activities, that may cause the mass fraction of the components to increase or
decrease. In particular, for a reactive flow the term
I
k
can be expressed as
I
k
=
q
k
−
l
k
c
k
,
(62)
where
q
k
and
l
k
are the chemical production and loss rates, respectively, of the
k
th
species.
By analogy with Eq. (
25
), we can write the internal energy equation of component
k
as
p
k
v
k
)
=∇·
ˆ
T
−
ʵ
r
˃
SB
T
4
∂(ˆˁ
c
k
U
k
)
∂
+∇·
(ˁ
c
k
U
k
v
k
)
+∇·
(
k
T
,
k
∇
+
Q
k
,
(63)
t
where
k
T
,
k
is the heat conductivity associated with species
k
and
Q
k
is a heat source
or sink term for the
k
th component. We assume thermodynamic equilibrium among
all fluid components
k
so that
T
k
=
T
. Using relations (
53
), introducing the following
definitions:
n
U
=
c
k
U
k
,
(64)
k
=
1
n
p
=
p
k
,
(65)
k
=
1
p
k
ˁ
H
k
=
U
k
+
c
k
,
(66)
n
=
k
T
,
k
,
k
T
(67)
k
=
1
for the total specific energy, the total pressure, the specific enthalpy of species
k
,
and the total heat conductivity, respectively, summing up Eqs. (
57
) and (
63
) over
all components, and combining the resulting equations, we obtain the temperature
equation for the multicomponent fluid
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