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8. Numerical scheme
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to be solved for the concentrations within corner units is
B CU,i V CU,i
c CU,i −
∆t
Nin∑
tube
j=1
V CU,i
(
N ij
edge
∑
k=1
Nin∑
tube
j=1
(
1
q ij,k
B ij,k
α
sw
CU,i
1 + α sw
CU,i ∆tssw,t
N ij
edge
∑
k=1
∆tα
sw
ij
q 2 ij,k ∆t
B ij,k V ij,k
c CU,j
−
ij,k +
1 + α sw
ij ∆tssw,t
CU,i +
∆tαaw ij
1 + α aw
αCU,i
aw
1 + αCU,i aw ∆tsaw,t CU,i
N CU,i
in,edge
∑
n=1
q CU,i,n c CU,n =
ij ∆tsaw,t ij,k
+ ct ij,k
)
)
+
+ V CU,i
∆t ct CU,i (8.39)
having concentration of the pore units calculated, we can calculate concentrations
within pore throat edges (c ij,k ), using Equation (8.34). Equation (8.32)
can be used to calculate the adsorbed mass concentrations, s sw
ij,k
and saw
ij,k , in
pore throat edges, and Equation (8.37) can be used to calculate the adsorbed
mass concentrations, s sw
CU,i and saw CU,i , in corner units.
8.3.4 One site equilibrium and one site kinetic adsorption
The following formulation is for a drained pore, in which the adsorption is
kinetic at either SW or AW interface, and is equilibrium at the other interface.
The mass balance equation for k th edge of a drained pore throat 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 α β ij,k
K β D,ij,k c − sβ ij,k ij,k
−
V ij,k KD,ij,k
α d
dt (c ij,k) (8.40)
where K β D,ij
, β = sw or aw shows the interface at which kinetic adsorption
occurs, and KD,ij α , α = sw or aw is the distribution coefficient for the interface
with equilibrium adsorption. The mass balance equation for adsorbed mass
due to kinetic adsorption is similar to Equation (8.31). Substitution for s β ij,k
in mass balance equation for the pore throat, and solving for c ij,k , we get
⎛
c ij,k = 1 ⎝
B ij,k
q ij,k ∆t
( )c j +
V ij,k 1 + K D,ij
∆tα β ij
(1 + K D,ij
) (
1 + α β ij ∆t )s β,t
ij,k + ct ij,k
⎞
⎠
(8.41)
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