Biomedical Engineering Reference
In-Depth Information
1, the specific reaction rate is much greater than the diffusion rate
distribution and is said to be diffusion limited. In other words, diffusion is slowest
diffusion characteristics dominate, and the reaction is assumed to be instantane-
ously in equilibrium.
For
N
Da
For
N
Da
>>
1, the reaction rate is much slower than the diffusion rate distribu-
tion and is said to be reaction limited. In other words, diffusion occurs much faster
than the reaction; thus, diffusion reaches equilibrium well before the reaction is at
equilibrium.
By estimating the
N
Da
, one can get an intuitive idea about what process domi-
nates a chemical distribution. In a problem where the reaction rate is proportional
to concentration in the bulk of the solution,
N
Da
is equivalent to
kL
2
/
<<
α
D
AB
, where
k
is the rate constant,
D
AB
is the diffusivity, and
α
is a parameter representing the
geometry of the problem.
is 1 in Cartesian coordinates [2].
The generalized Navier-Stokes equation for diffusion-reaction models is writ-
ten in vector notation as:
α
∂
C
A
+∇⋅
NR
−
=
0
A
A
∂
t
where
R
A
is the rate of the chemical reaction and
N
A
is the mass transfer flux. The
expanded terms in different coordinates are given here.
∂
N
∂
N
∂
N
∂
C
⎛
⎞
Ay
,
Ax
,
Az
,
Cartesian coordinates :
A
+
+
+
−
R
=
0
⎜
⎟
A
∂
t
⎝
∂
x
∂
y
z
NN
∂
⎠
⎡
∂
∂
⎤
∂
C
1
∂
1
(
)
A
,
θ
A z
,
Cylindrical coordinates :
A
+
rN
+
+
−
R
=
0
⎢
⎥
Ar
,
A
∂
t
r
∂
r
r
∂
θ
∂
z
⎣
⎦
∂
N
∂
C
⎡
1
∂
1
∂
1
⎤
−=
(
)
(
)
A
,
ϕ
2
Spherical coordinates :
A
+
rN
+
N
sin
θ
+
R
0
⎢
⎥
Ar
,
A
,
θ
2
∂
t
r
∂
r
r
sin
θθ
∂
r
sin
θ
∂
ϕ
⎣
⎦
In order to solve the set of nonlinear partial differential equations, initial and
boundary conditions for the domain are necessary. Often, properties of diffusion
are derived from free diffusion (Chapter 2), and proposed as generic diffusion
properties while the actual boundary conditions shape the concentration profiles.
Knowing these shapes is important when interpreting experimental results or when
selecting curves (i.e., solutions of model problems for fitting experimental data
to estimate (apparent) diffusion parameters). Choosing the appropriate boundary
conditions is also important. Free diffusion is of use to model interactions in infinite
space or in a bulk volume on appropriate timescales (i.e., situations where diffusion
is used as a simple passive transport or dissipative mechanism). However, when
diffusion is used as a transport mechanism to link cascading processes that are
spatially separated or when diffusion is used in conjunction with processes that are
triggered at certain concentration thresholds, a detailed description of the geometry
of the system is very important. Then the complex geometry imposes the major-
ity of the boundary conditions. These systems of partial differential equations are
