Environmental Engineering Reference
In-Depth Information
of the gas in the channels of the porous medium is related to the area-averaged
velocity via
u
g
=
u
ε
:
:
ð
Eq
4
61
Þ
The energy conservation equation can be written as
∂
∂
ρ
B
h
B
+
ρ
C
h
C
+
ε
ρ
g
Y
G
h
G
+Y
I
h
I
+Y
T
h
T
ð
Þ
t
=
+
1
r
n
∂
r
n
ε
ρ
g
u
Y
G
h
G
+Y
I
h
I
+Y
T
h
T
ð
Þ
∂
r
0
0
1
1
r
n
∂
1
r
n
k
eff
ρ
g
∂
T
∂
ε
ρ
g
D
eff
,
G
∂
Y
G
∂
h
G
+
D
eff
,
I
∂
Y
I
∂
r
h
I
+
D
eff
,
T
∂
Y
T
∂
@
@
A
A
r
+r
n
h
T
∂
r
r
r
ð
Eq
:
4
:
62
Þ
where h
i
is the mass-specific enthalpy of component i:
h
i
=h
i
+
ð
T
T
0
c
p
,
i
dT
ð
Eq
:
4
:
63
Þ
The effective diffusivity of the gas species in the particle depends on the pore size
relative to the mean free path of the molecules. In wide pores, where the mean free
path of is much larger than the pore diameter, a standard diffusivity
D
m
for the gas
mixture applies. In narrow pores, where the mean free path is of the same order of
magnitude as the diameter of the pores, the molecules collide with the wall more often
than with other molecules. In the limit of negligible molecule
molecule collisions,
this type of diffusion is called Knudsen diffusion. The diffusivity for Knudsen diffu-
sion
D
K
can be obtained from free-molecule flow theory. The general case for any
pore diameter can be covered by simply assuming additive resistances (Mills,
1999) and adding the inverses of the diffusivities for both regimes:
-
1
D
=
1
D
m
+
1
D
K
ð
Eq
:
4
:
64
Þ
The diffusivity of a gaseous component in a volume with both solid and gas is given by
D
eff
=
ε
D
τ
ð
Eq
:
4
:
65
Þ
ε
Here, the porosity
accounts for the reduction in cross-sectional area for diffusion
posed by the solid material, and the tortuosity factor
τ
accounts for the increased dif-
fusion length due to the tortuous paths of real pores and for the effects of constrictions
and dead-end pores (Mills, 1999).
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