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8. Numerical scheme
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mass balance equation for a corner unit i with equilibrium adsorption may be
written as
V CU,i
d
dt (c CU,i) =
where N tube
in
Nin∑
tube
j=1
N ij
edge
∑
k=1
q ij,k c ij,k +
N CU,i
in,edge
∑
n=1
V CU,i KD,i
sw dc CU,i
− V CU,i K aw dc CU,i
D,i
dt
dt
q i,n c CU,n − Q CU,i c CU,i −
(8.24)
is the number of pore throats flowing into the corner unit i; N ij
edge is
the number of edges within the angular pore throat (for example, for a tube with
triangular cross section, N ij
edge = 3), each with the volumetric flow rate of q ij,k.
N CU,i
in,edge
is the number of pore body edges, within the same pore body, flowing
into corner unit i, each with the volumetric flow rate of q i,n (for example, the
maximum value of N CU,i
in,edge
for the case of a cubic pore body is equal to three).
KD,i sw and Kaw D,i [-] are upscaled adsorption distribution coefficients at the solidwater
(SW) and air-water (AW) interfaces within corner unit i, respectively,
where
K αw
D,i = k αw
d,i a αw
i ; where α = s, a (8.25)
where a αw
i is the specific surface area within corner unit i. discretization of
Equation (8.24) in fully implicit scheme, and rearranging, gives
( )
V CU,i 1 + K
sw
D,i + KD,i
aw cCU,i −c t CU,i
∆t
=
Nin∑
tube
j=1
Nij∑
edge
k=1
q ij,k c ij,k +
Nin∑
CU,i
n=1
solving for c CU,i , we will have
q i,n c CU,n − Q CU,i c CU,i (8.26)
c CU,i = 1
B CU,i
1
B CU,i
∆t
V CU,i
(1 + K sw
D,i + Kaw D,i
)
∆t
)
V CU,i
(1 + KD,i sw + Kaw D,i
Nin∑
tube
j=1
N CU,i
in,edge
∑
n=1
N ij
edge
∑
k=1
q ij,k c ij,k +
q i,n c CU,n + 1
B CU,i
c t CU,i (8.27)
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