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3. Upscaling of Adsorbing Solutes; Pore Scale
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We introduce the following dimensionless variables and parameters:
c ∗ = c
c 0
; r ∗ =
r ; z ∗ = z ; t ∗ = ¯vt ; P e p = ¯vR 0
; s ∗ =
s ; κ = k D
R 0 R 0 R 0 D 0 c 0 R 0
R 0
(3.31)
where c 0 is concentration at the inlet boundary. Here P e p is the pore-scale
Peclet number, which expresses the ratio between the magnitude of the advective
and diffusive transport terms.
Substituting in Equation (3.28) gives the pore-scale dimensionless mass transport
equation
∂c ∗
∂t ∗ + 2(1 − r∗2 ) ∂c∗
∂z ∗ = 1 [ ∂ 2 c ∗
P e p ∂z ∗2 + 1 ( )]
∂
r ∗ ∂r ∗ r ∗ ∂c∗
∂r ∗
The dimensionless boundary conditions for our model are
(3.32)
at z ∗ = 0, c ∗ = 1.0 (3.33a)
at z ∗ = ∞,
at r ∗ = 0,
∂c ∗
∂z ∗ = 0
∂c ∗
∂r ∗ = 0
(3.33b)
(3.33c)
at r ∗ = 1,
∂c ∗
∂r ∗ = −P e ∂s ∗
p
∂t ∗
(3.33d)
Substitution of Equation (3.30) in the boundary condition (3.33d) results in:
∂c ∗
∂r ∗ = −P e p κ ∂c∗
∂t ∗ at r∗ = 1 (3.34)
It is evident that P e p and κ are the two parameters in this set of Equations
(3.32)-(3.34) which control the transport and reaction processes within the
tube.
The package, Flex-PDE (Flex, 2005) has been used to numerically solve this
set of equations. We have simulated solute transport for a range of values
of parameters P e p and κ. The solution of Equations (3.32)-(3.34) results in
a concentration field c ∗ (z ∗ , r ∗ , t ∗ ) for different values of Peclet number (P e p )
and dimensionless distribution coefficient (κ). This concentration field and its
cross-sectional average, ¯c ∗ (z ∗ , t ∗ ), may be considered to be equivalent to “observation
data”. We then use these observed data to obtain a relationship for
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