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8.3 Numerical scheme; partially saturated conditions
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where B ij is defined as
B ij = 1 +
q ij,k ∆t
V ij,k
(
1 + K D,ij
) + ∆tαβ ij Kβ D,ij
(1 + K D,ij
) +
∆t 2 α β2
ij Kβ2 D,ij
) )
(1 + K D,ij
(1 + α β ij ∆t
The mass balance equation for a corner unit within a drained pore body, with
one site kinetic and one site equilibrium, may be written as
V i
d
dt (c CU,i) =
Nin∑
tube
j=1
Nij∑
edge
k=1
− V CU,i α β CU,i
q ij,k c ij,k +
N CU,i
in,edge
∑
n=1
(
)
K β D,i c CU,i − s β CU,i
q i,n c CU,n − Q CU,i c CU,i
− V CU,i KD,i
α d
dt (c CU,i) (8.42)
The mass balance equation for adsorbed mass due to kinetic adsorption, s β CU,i ,
is similar to Equation (8.36). Substitution for s β CU,i
in mass balance equation
for the pore throat, and solving for c CU,i results in
c CU,i =
−
+
−
∆tQ CU,i
V CU,i
(
1 + K D,i
)c CU,i +
∆t
V CU,i
(
1 + K D,i
)
∆tαβ CU,i
(1 + K D,i
)
(
where B i is defined as
B i = 1 +
N CU,i
in,edge
∑
n=1
∆t
V CU,i
(
1 + K D,i
)
Nin∑
tube
j=1
Nij∑
edge
k=1
q ij,k c ij,k
q i,n c CU,n (8.43)
K β D,i c CU,i − αβ CU,i Kβ D,i ∆t
)
1 + αCU,i sw ∆t c 1
CU,i −
1 + αCU,i sw ∆tssw,t CU,i
+ c t CU,i
∆tQ i
V i
(
1 + K D,i
) + ∆tαβ i Kβ D,i
(1 + K D,i
) −
α β2
i ∆t 2 K β D,i
) )
(1 + α β i
(1 ∆t + K D,i
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