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8. Numerical scheme
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8.3.3 Two sites kinetic adsorption
For a pore throat with two sites kinetic, the mass balance equation may be
written as
d
(
)
V ij,k
dt (c ij,k) = |q ij,k | c CU,j − |q ij,k | c ij,k − V ij,k αij
sw KD,ijc sw
ij,k
− s sw
ij,k
− V ij,k αij
aw (
K
aw
D,ij c ij,k − s aw )
ij,k
(8.30)
From this point, we drop the superscript of t+∆t, and only we show superscript
for time t, such as c t ij,k . kinetic adsorption for corner kth of a drained pore body
may be written as
ds β ij,k
dt
= α β ij
(
)
K β D,ij c ij,k − s β ij,k
; β = sw, aw (8.31)
solving for concentration of adsorbed mass
s β ij,k = αβ ij Kβ D,ij ∆t
1 + α β ij ∆t c ij,k +
1
1 + α β ij ∆tsβ,t ij,k
; β = sw, aw (8.32)
we substitute the Equation (8.32) into pore throat mass balance Equation (8.30)
to get
( )
c ij,k
− c t ( )
ij,k
V ij,k = q ij,k c
∆t
j
− c ij,k −
(
V ij,k α sw
ij
− V ij,k α aw
ij
KD,ijc sw
ij,k − αsw ij Ksw D,ij ∆t
(
1 + αij sw∆t
c ij,k −
KD,ijc aw
ij,k − αaw ij Kaw D,ij ∆t
solving for c ij,k and rearranging gives
[
c ij,k = 1 q ij,k ∆t
c
B ij,k V j
+ ∆tαsw ij
ij,k 1 + α sw
where
B ij,k = 1 + q ij,k∆t
V ij,n
+ ∆tα sw
ij K sw
1 + αij aw∆t
c ij,k −
ij ∆tssw,t ij,k +
D,ij − αsw2 ij
K sw
D,ij ∆t2
1 + α sw
190
ij ∆t
)
1
1 + αij sw∆tssw,t
ij,k
1
1 + αij aw∆tsaw,t
ij,k
)
]
∆tαaw ij
1 + αij aw∆tsaw,t
ij,k
+ ct ij,k
+ ∆tα aw
ij K aw
D,ij − αaw2 ij
(8.33)
(8.34)
K aw
D,ij ∆t2
1 + α aw
ij ∆t