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8. Numerical scheme
substituting for c ij,k (Equation 8.41) in Equation (8.43), we get
( )
V CU,i 1 + KD,i Bi
c CU,i =
∆t
⎛
Nin∑
tube Nin∑
edge
1
q ij,k
⎝
B ij,k
j=1
+
k=1
N CU,i
in,edge
∑
n=1
q i,n c CU,n −
q ij,k ∆t
( )c CU,j +
V ij,k 1 + K D,ij
V CU,iα β CU,i
(
)s sw,t
1 + αCU,i sw ∆t CU,i + V CU,i
∆tα β ij
(1 + K D,ij
) (
1 + α β ij ∆t )s β,t
(
1 + KD,i
)
∆t
c t CU,i (8.44)
Rearranging Equation (8.44) and collecting unknowns on the l.h.s. and know
terms on the r.h.s, we obtain the final form of system of equations to be solved
for the concentrations within corner units
( )
V CUi 1 + KD,i Bi
c CU,i −
∆t
−
Nin∑
tube
N CU,i
in,edge
j=1
∑
n=1
N ij
edge
∑
k=1
q i,n c CU,n = −
⎛
q ij,k
⎝
B ij,k
Nin∑
tube
j=1
N ij
edge
∑
k=1
V CU,iα β CU,i
q 2 ij,k ∆t
B ij,k V ij,k
(
1 + K D,ij
)c CU,j
(
)s sw,t
1 + αCU,i sw ∆t CU,i + V CU,i
∆tα β ij
(1 + K D,ij
) (
1 + α β ij ∆t )s β,t
ij,k + ct ij,k
(
1 + KD,i
)
∆t
⎞
c t CU,i
⎠ (8.45)
having concentration of the pore units calculated, we can calculate concentrations
within pore throat edges (c ij,k ), using Equation (8.41).
ij,k + ct ij,k
⎞
⎠
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