Environmental Engineering Reference
In-Depth Information
At the upper boundary:
T
n
T
n
n
n
−
n
·
N
O
2
=
0
,
u
·
n
=
0
,
∇
u
+
(
∇
u
)
−
∇
u
+
(
∇
u
)
·
=
0 )
At the lower boundary:
−
n
·
N
O
2
=
0
,
u
=
0
(7)
At the outlet boundary
I
n
·
∇
C
O
2
=
2
3
(
∇·
T
−
,
=
p
atm
,
∇
+
(
∇
−
=
n
0
p
u
u
)
u
)
0
(8)
At the particle surface:
−
n
·
N
O
2
=−
K
r
C
O
2
C
coke
,
u
=
0
(9)
In Eqs. (
4
)-(
9
)
p
atm
is the atmospheric pressure,
I
is the identity tensor,
C
inj
O
2
is
the oxygen concentration in the injected gas,
F
inj
is the mass flux of the injected
gas,
n
is the unit normal vector pointing outside the boundary,
K
r
is the reaction rate
constant,
C
coke
is the coke surface molar concentration at the particle surface, and
N
O
2
represents the oxygen total molar flux expressed in the next form
N
O
2
=−
D
O
2
∇
C
O
2
+
u
C
O
2
(10)
The units of
C
O
2
and
C
coke
are mole/m
3
and mole/m
2
, respectively. As initially
the oxygen concentration is 0, then the gas is only composed by nitrogen with a
molar concentration depending on temperature and pressure.
In turn, the coke concentration at the particle surface is computed through the
solution of the following differential equation
∂
C
coke
∂
=−
ʷ
K
r
C
O
2
C
coke
(11)
t
ʷ
where
is the stoichiometric coefficient between oxygen and coke. This equation
has the next initial condition
C
coke
At
t
=
0
,
C
coke
=
(12)
where
C
coke
is the initial coke concentration.
On the other hand, the density (
ˁ)
was calculated using the real gas equation, the
reaction rate constant (
K
r
)
was obtained using an Arrhenius-type expression, and
the gas viscosity was calculated with the Wilke approach (Wilke
1950
) for binary
mixtures at low pressures. The oxygen diffusivity was obtained from the work of
Bird et al. (
2010
).
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