Environmental Engineering Reference
In-Depth Information
(3) The porous matrix is idealized consisting in a periodic array of infinitely long,
rounded solid rods, allowing to reduce the dimensionality to 2D (see Fig.
2
).
(4) A first-order combustion reaction is assumed.
(5) The combustion reaction is isothermal, avoiding the solution of the energy
balance.
The last issue indicates that the conditions for simulations are close to those
encountered at the vicinity of the combustion front, where the combustion reaction
takes relevance.
The existence of only gas in the system and an initial coke distribution is sup-
ported on the idea that these conditions are mainly met behind the combustion front.
Indeed, as seen in Fig.
1
, the assumptions mentioned above mean that our simula-
tions correspond for the clean zone, where only there are sand and gas, and the coke
zone represents the vicinity of the combustion front where simultaneously there are
generation and consumption of coke.
2.1 Governing Equations, Initial and Boundary Conditions
The mathematical model includes the oxygen mass balance in the gas-phase
∂
C
O
2
∂
2
C
O
2
+
u
·∇
C
O
2
=
D
O
2
∇
(1)
t
where
C
O
2
is the oxygen molar concentration and
D
O
2
is the oxygen diffusion coef-
ficient inside the gas-phase. The field velocity
u
is computed through the solution of
the gas momentum balance:
I
T
ˁ
∂
u
2
3
μ (
∇·
t
+
ˁ (
u
·∇
)
u
=−∇·
(
p
I
)
+∇·
μ
∇
u
+
(
∇
u
)
−
u
)
(2)
∂
Here
ˁ
is the mixture density,
p
is pressure and
μ
viscosity. In addition, the
gas-phase continuity equation is required,
∂ˁ
∂
t
+∇·
(ˁ
)
=
u
0
(3)
Equations (
2
) and (
3
) are enforced to satisfy the following initial and boundary
conditions (see Fig.
2
to identify each boundary):
At
t
=
0
,
C
O
2
=
0
,
u
=
0
,
p
=
p
atm
(4)
At the inlet boundary:
F
inj
ˁ
C
inj
C
O
2
=
O
2
,
u
=−
n
(5)
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