Environmental Engineering Reference
In-Depth Information
where
C
A
=
molar concentration of
A
N
A
=
total molar flux of
A
(convection
+
diffusion)
R
A
=
homogeneous reactions involving
A
within the control volume
∇=
del operator.
The term
R
A
does not account for reactions at the boundary of the control volume
(typically a surface where a heterogeneous reaction occurs). Reactions at boundaries are
accounted for by boundary conditions when solving the differential balance.
If the system is not changing with respect to time, then
∂
C
A
/∂
t
=
0. If no reactions
involving
A
are taking place within the control volume, then
R
A
=
0. Equation (3.33)
reduces to
∇·
N
A
=
0
.
(3.66)
If the physical meaning of this term is evaluated, in
=
out. For one-dimensional planar
transport (i.e.,
x
direction)
d
N
A
d
z
=
0
⇒
N
A
=
constant
.
(3.67)
This result indicates that the mass flux of
A
(and the rate since the cross-sectional area
is constant) remains constant as one moves in the
x
direction). For cylindrical systems
(tubes) where radial transport occurs,
d(
rN
A
)
d
r
constant
r
=
0
⇒
rN
A
=
constant
⇒
N
A
=
.
(3.68)
This result demonstrates that the flux is not constant but varies as
r
−
1
. The cross-sectional
area is 2
rL
(where
L
is axial length) so the rate is a constant. Prove this for yourself.
An equation is needed for
N
A
to substitute into the above mass balance. For a binary
system (
A
and
B
), the molar average velocity of flow (
π
v
M
) for both components is
N
A
+
N
B
v
M
=
;
C
=
total molar concentration
.
(3.69)
C
The velocity for component
A
is
N
A
C
A
.
v
A
=
(3.70)
Note that
x
A
=
C
A
/
C
.
The equation for
v
M
can be rewritten as
N
A
C
+
N
B
C
=
C
A
C
N
A
C
A
+
C
B
C
N
B
C
B
=
v
M
=
x
A
v
A
+
x
B
v
B
.
(3.71)
The velocity for component
A
has two contributions, the molar average velocity of the
system,
v
M
, plus the movement due to diffusion of
A
,
v
A
D
:
v
A
=
v
M
+
v
A
D
.
(3.72)
Search WWH ::
Custom Search