Biomedical Engineering Reference
In-Depth Information
When the effects of precipitation and/or dissolution cannot be ignored,
it is common to resort to ad hoc constitutive relationships that describe the
dependence of macroscopic parameters, such as the permeability on the chang-
ing porosity,
φ
=
φ
(
x
,t
). The hybrid algorithm described later obviates the
necessity of using constitutive relationships of this kind, since it relies on
the Darcy-scale description only for the portion of the porous medium where
changes in the pore geometry are insignificant and the parameters
φ
,
k
AB
eff
,
and
k
eff
remain constant.
7.5.5 SPH Representation of Averaged Darcy-Scale RDEs
Since the Darcy-scale description does not explicitly account for the solid and
liquid phases, only one kind of particle is used to discretize the computational
domain, Ω
c
. The SPH discretization of the Darcy-scale RDEs is given by
=4
b∈
Ω
c
∂φ
a
C
a
∂t
φ
a
φ
b
D
a
D
b
φ
a
D
a
+
φ
b
D
b
C
a
−
C
b
x
ab
x
ab
·∇
a
x
ab
∂W
(
x
ab
,h
)
∂x
ab
V
b
φ
a
k
AB
C
a
C
a
,I
=
A, B
−
(7.45)
eff
and
=4
b∈
Ω
c
∂φ
a
C
a
∂t
φ
a
φ
b
D
a
D
b
φ
a
D
a
+
φ
b
D
b
C
a
C
b
x
ab
−
x
ab
·∇
a
x
ab
∂W
(
x
ab
,h
)
∂x
ab
V
b
+
φ
a
k
AB
C
a
C
a
φ
a
k
eff
(
C
a
−
−
C
eq
)
(7.46)
eff
Darcy-scale (continuum) descriptions of reaction-diffusion processes in
porous media are based on the averaging of microscopic RDEs over a rep-
resentative volume of the porous media. Darcy-scale descriptions break down
when the concentration gradients are large enough for the concentrations to
change significantly on the scale of the representative volume (Whitaker 1999).
If this occurs in only a small region, Ω
p
, of the computational domain, Ω
T
,
hybrid simulations, which combine pore-scale simulations in Ω
p
with Darcy-
scale simulations in Ω
c
, become attractive. The eciency of a typical hybrid
algorithm increases as the ratio
Ω
T
indicates
the volume of the domain, Ω. Whether a small region in which pore scale
simulation is required, Ω
d
develops and whether the ratio
||
||
/
||
Ω
d
||
increases, where
Ω
Ω
T
is large
is determined by the physical process(es) under consideration, and by the ini-
tial and boundary conditions for the advection-diffusion-reaction equations.
In this study we are concerned with reaction-diffusion processes with localized
reaction fronts, which are formed, for example, when a solution with concen-
trations
C
A
= 1 and
C
B
= 0 is brought instantaneously into contact with a
solution with concentrations
C
A
= 0 and
C
B
= 1 in the same solvent.
||
||
/
||
Ω
d
||
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