Biomedical Engineering Reference
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k
∂C
C
∂t
(
D
C
C
C
)+
k
AB
C
A
C
B
(
C
C
=
∇·
∇
−
−
C
eq
)
δ
(
x
−
x
f
)
da,
x
∈
Ω
p
(7.37)
F
where
D
I
>
0(
I
=
A, B,C
) are the molecular diffusion coecients of species
I
in the solvent,
k
AB
>
0 and
k>
0 are the rate coecients of the homogeneous
and heterogeneous reactions,
x
f
is a point on the interface,
F
, and
da
is an
infinitesimally small surface element of
F
. It is important to recognize that the
formulation (7.36-7.37) reflects several physical assumptions. First, it neglects
diffusion in the solid phase, which is usually many orders of magnitude slower
than its counterpart in the liquid phase. Second, it disregards the reverse reac-
tion
C
(aqueous)
A
+
B
. Third, (7.37) implies that precipitation/dissolution
of the soluble reaction product,
C
, is described by a first-order kinetic-reaction
model at the fluid-solid interface,
D
C
→
C
C
·
n
=
k
(
C
C
∇
−
C
eq
)
(7.38)
where
C
eq
is the concentration of
C
in equilibrium with the solid matrix, and
n
is the unit vector in the direction normal to the interface pointing toward
the fluid. The normal velocity,
v
n
, with which the fluid-solid interface at a
point
x
s
advances into the liquid, is given by
n
,θ
=
C
A,
0
+
C
B,
0
ρ
s
v
n
(
x
s
)=
θD
C
C
C
∇
·
(7.39)
where
ρ
s
>
0 is the density of the precipitated solid phase. The parameters
D
I
(
I
=
A, B, C
),
k
AB
,
k
, and
ρ
s
are measurable quantities that are assumed
to be known.
7.5.3 SPH Representation of the Pore-Scale RDEs
Using the SPH interpolation scheme, equations (7.36-7.37) can be discretized
as (Tartakovsky et al. 2008b):
∂C
a
∂t
=4
b∈
Ω
p
D
a
D
b
D
a
+
D
b
C
a
−
C
b
x
ab
x
ab
·∇
a
x
ab
∂W
(
x
ab
,h
)
∂x
ab
k
AB
C
a
C
a
,I
=
A, B
V
b
−
(7.40)
=4
b∈
Ω
p
∂C
a
∂t
D
a
D
b
D
a
+
D
b
C
a
C
b
x
ab
−
x
ab
·∇
a
x
ab
∂W
(
x
ab
,h
)
∂x
ab
+
k
AB
C
a
C
a
V
b
k
b∈
Ω
s
b
=
j∈
Ω
p
∆
b
(
C
a
C
eq
)
W
−
1
b
−
−
W
(
x
ab
,h
r
)
,
V
J
W
(
x
jb
,h
r
)
(7.41)
Here, subscripts
a
and
b
denote properties and positions associated with
SPH particles
a
and
b
,
x
ab
=
x
a
−
;
b∈
Ω
p
; and
b∈
Ω
s
indi-
cate summation over fluid and solid particles;
V
a
is the volume (area in two-
dimensional simulations) of particle
a
and ∆
b
denotes the reactive surface
x
b
;
x
ab
=
|
x
ab
|
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