Environmental Engineering Reference
In-Depth Information
2 Intra-particle void space (assuming cylindrical pores):
R
D
i
∂
c
i
∂
k
p
(
c
−
c
i
R
)
=−
(boundary condition)
r
D
i
=
β
D
AB
q
=
effective diffusion coefficient
where
β
=
internal intra-particle void fraction
q
=
tortuosity
D
AB
=
molecular diffusion coefficient.
t
=−∇·
N
r
+
β
∂
c
i
∂
intra-particle fluid differential mass balance
N
r
D
i
∂
c
i
=−
neglect convection in pores
∂
r
=
ρ
p
∂
c
ads
∂
=
mass transfer between fluid and solid phases in particle,
t
ρ
p
=
where
particle density
mass adsorbed
mass adsorbent (particle)
c
ads
=
c
ads
c
i
,
eq
=
k
a
k
d
=
rate of adsorption
rate of desorption
⇒
K
a
=
first-order process
k
a
c
i
−
,
∂
c
ads
∂
c
ads
K
a
=
k
a
(
c
i
−
c
i
,
eq
)
=
t
where
K
a
is the equilibrium constant and
c
i
,
eq
is the concentration in the intra-particle
void space that would be in equilibrium with
c
i
.
3 Laplace transform of equation for
c
ads
:
k
a
c
ads
K
a
.
Using the initial condition:
c
ads
(
r
s c
ads
=
k
a
c
i
−
≥
0
,
t
=
0)
=
0 the solution for
c
ads
is:
k
a
c
i
c
ads
=
K
a
.
s
+
k
a
/
Now, solving the equations for
c
and
c
i
:
Boundary conditions
Initial conditions
c
i
(
r
=
0)
=
finite
c
(
z
>
0
,
t
=
0)
=
0
r
=
0
=
∂
c
i
∂
r
0
c
i
(
r
≥
0
,
t
=
0)
=
0
c
(
z
=
0,
t
)
=
c
0
(
t
)
c
(
z
→∞
,
t
)
=
finite
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