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8. Numerical scheme
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this equation can be rearranged to get
(
1 + ∆tQ )
i
c t+∆t
i − ∆t
V i i
∑
V i
N in
j=1
q ij c t+∆t
ij = c t i (8.9)
we use Equation (8.6) to substitute for c t+∆t
ij in Equation (8.9), and collect
unknown term on the l.h.s. and known terms on r.h.s. to get
where:
Ec t+∆t
i
∑N in
− I
j=1
F c t+∆t
j
∑N in
= c t (
i + I Gc
t t
ij + Hs ) ij (8.10)
E (Nnode ) =1 + ∆tQ i
; I (Nnode ) = ∆t ; F (Ntube ) = 1 V i V i B
G (Ntube ) = q ij
B ; H q ij ∆tα ij
(N tube ) =
B (1 + ∆tα ij )
j=1
q 2 ij ∆t
Note that, through substitution, we end up with Equation (8.10) in which
the number of unknowns is N node , concentration in pore bodies (i.e., c t+∆t
i
and c t+∆t
j ). In this way we could decrease the size of coefficient matrix by
about 3 times since we don’t need to solve simultaneously for concentration
of solute and adsorbed concentration in pore throats in Equation (8.10) (the
number of pore throats is much more than the number of pore bodies in the
network). After each time step the concentration of pore throats and adsorbed
concentrations can be calculated using Equations (8.6) and (8.5), respectively.
V ij
8.3 Numerical scheme; partially saturated conditions
For the case of partially saturated conditions, we write formulations for the
most general case, in which solute mass transport occurs through edges of
drained pore throats and edges of drained pore bodies. We divide the volume
of a drained pore body into element which we call as “corner unit”. Each corner
unit comprised of a corner domain together with half of the three neighboring
edges connected to it within the same pore body, as shown in Figure (6.3).
Therefore, in the case of cubic pore body, we will have eight corner units.
Thus, we assign eight different concentrations to a drained pore body, one
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