Environmental Engineering Reference
In-Depth Information
v
A
=
v
M
if the convective flow is dominant and
v
A
=
v
A
D
It can be readily observed that
if there is no convection. An equation can be written for
v
A
D
analogous to the equation
for
v
A
:
J
A
C
A
;
v
A
D
=
J
A
=
molar diffusion flux
.
(3.73)
Returning to the definition of
v
A
:
N
A
C
A
⇒
v
A
=
N
A
=
C
A
v
A
(3.74)
C
A
C
N
A
=
v
M
+
v
A
D
)
=
v
M
+
C
A
v
A
D
C
A
(
C
(3.75)
=
x
A
(
N
A
+
N
B
)
+
J
A
.
(3.76)
For one-dimensional planar transport, substituting Fick's Law for
J
A
yields
D
AB
d
C
A
N
A
=
x
A
(
N
A
+
N
B
)
−
d
z
.
(3.77)
To simplify this equation, ask the following:
1Is
N
A
constant (from mass balance)?
2Is
C
a constant? d
C
A
/
d
z
=
C
(d
x
A
/
d
z
)?
3 What is flux ratio (
N
A
/
N
B
)? For example, is
N
B
=
0?
4Is
x
A
1? If so, the first term on the right can be neglected.
The mass balance equation plus the equation for
N
A
combine to produce a differential
equation. To solve it, boundary conditions are needed.
Concentrations are specified
(a) At an interface (such as phase equilibrium with surrounding fluid);
(b) for an instantaneous irreversible reaction at boundary (
C
A
=
0);
(c) at infinity if the mass transfer does not reach the opposite boundary (
C
A
=
initial
concentration).
Flux is specified
(a) At an interface;
(b) at a solid surface;
(c) as a heterogeneous reaction at a surface.
There are references [12, 13] that discuss diffusional processes in detail.
An alternative analysis to a differential mass balance is based on
mass transfer
coefficients
. This analysis is useful when detailed analysis of the concentration in the
system is not needed. This approach can be used in terms of resistances to mass
transfer.
There can often be more than one resistance to mass transfer. These resistances can
include boundary layer and diffusion effects. For example, in adsorption a solute must
diffuse through a fluid, cross the boundary layer between the fluid and the solid sorbent,
Search WWH ::
Custom Search