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8.2 Numerical scheme; saturated conditions
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discretization of Equation (8.1) in fully implicit scheme gives
V ij
(
cij t+∆t − c ij
t )
∆t
=q ij
(
cj t+∆t − c ij
t+∆t ) −V ij α ij
(
Kd,ij c ij t+∆t − s ij
t+∆t )
the kinetic adsorption within a pore throat we be written as
(8.2)
ds ij
dt
= k att,ij c ij − k det,ij s ij (8.3)
discretization of Equation (8.3) gives
(
sij t+∆t − s ij
t )
∆t
solving for s ij t+∆t we will have
= α ij
(
Kd,ij c ij t+∆t − s ij
t+∆t ) (8.4)
s t+∆t ij = α ij∆tK d,ij
(1 + α ij ∆t) c ij t+∆t 1
+
(1 + α ij ∆t) s ij t (8.5)
substituting Equation (8.5) into Equation (8.2) gives:
c ij t+∆t = 1 B c ij t +
∆tα ij
B (1 + α ij ∆t) S t 1
ij +
B
q ij ∆t t+∆t
c j
V ij
(8.6)
where
B = 1 + q ij∆t
V ij
+ ∆tα ij K d,ij − ∆t2 α 2 ij K d,ij
(1 + α ij ∆t)
mass balance for a saturated pore throat may be written as
N
dc
in
i
V i
dt = ∑
q ij c ij − Q i c i (8.7)
discretization of Equation (8.7) using a fully implicit scheme results in
j=1
c t+∆t
i − c t i
V i
∆t
∑N in
=
j=1
q ij c t+∆t
ij
− Q i c t+∆t
i (8.8)
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