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8.3 Numerical scheme; partially saturated conditions
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for each corner unit. In the case of drained pore throats, we assign different
concentrations to each pore throat edge. For example, we assign three different
concentrations to each edge of a drained pore throat with triangular cross
section.
Throughout this section we assume flow from corner unit j to corner unit i (i.e.,
corner unit j is the upstream node) through an edge of drained pore throat
ij. We start the formulation for the case of a non-adsorptive solute and then
proceed with adsorptive solutes.
8.3.1 Non-adsorptive solute
Mass balance equation for edge of a drained pore throat may be written as
V ij,k
d
dt (c ij,k) = |q ij,k | c CU,j − |q ij,k | c ij,k (8.11)
we apply a fully implicit scheme to Equations (8.11), to get
c t+∆t − c t ij,k ij,k
V ij,k
∆t
= |q ij,k | c t+∆t
CU,j − |q ij,k| c t+∆t
ij,k
(8.12)
From this point forward, for the sake of simplicity in our notation, we drop the
t + ∆t superscript, and we only keep superscript of terms with time t, such as
c t ij,k .
the equation for c ij,k will be:
c ij,k = 1 ( )
∆tqij,n
c
B CU,j + c t ij,k
ij,k
V ij,k
where the constant coefficient, B ij,k , is defined as
(8.13)
B ij,k = 1 + ∆tq ij,k
V ij,k
(8.14)
The mass balance equation for corner units i, within a drained pore body, may
be written as
V CU,i
d
dt (c CU,i) =
Nin∑
tube
j=1
N ij
edge
∑
k=1
q ij,k c ij,k +
N CU,i
in,edge
∑
n=1
q i,n c CU,n − Q CU,i c CU,i (8.15)
for a drained triangular pore throat N ij
edge
= 3, and for a drained cubic pore
body N CU,i
in,edge
= 3. discretization of Equation (8.15) in a fully implicit scheme
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