Environmental Engineering Reference
In-Depth Information
Within the reactor the compound reacts at a rate
r
. The overall material balance is
[
accumulation
]=
input
(
bulk flow
+
dispersion
)
]−[
output
(
bulk flow
+
dispersion
)
]+[
reaction
]
.
At steady state, accumulation is zero, Hence,
uA
c
C
A
|
x
−
x
uA
c
C
A
|
x
+Δ
x
−
x
+Δ
x
D
ax
A
c
d
C
A
d
x
D
ax
A
c
d
C
A
d
x
0
=
−
+
v
A
rA
c
Δ
x
.
(6.32)
Rearranging the terms and dividing by the volume
A
c
Δ
x
gives
u
C
A
|
x
−
D
ax
(
d
C
A
/
d
x)
C
A
|
x
+Δ
x
Δ
|
x
−
(
d
C
A
/
d
x)
|
x
+Δ
x
−
+ ν
A
r
=
0.
(6.33)
x
Δ
x
Taking limit as
Δ
x
→
0, we obtain
∂
2
C
A
∂x
2
u
∂C
A
−
∂x
+
D
ax
+
v
A
r
=
0.
(6.34)
The above equation and its variations appear in many cases in environmental engi-
neering.The first term on the left-hand side is called the
advection
or
convection
term,
the second is the
dispersion
term, and the last is the
reaction
term. For a first-order
reaction A
→
B, the equation can be written in the following form:
D
ax
d
2
C
A
d
x
2
u
d
C
A
−
d
x
−
kC
A
=
0.
(6.35)
To make the above equation dimensionless, we use the following transformations:
z
=
x/L
and d
z
=
d
x/L
with
L
being the length of the tube. Hence, we have
d
2
C
A
d
z
2
Pe
L
d
C
A
−
d
z
−
Da
L
C
A
=
0,
(6.36)
kL
2
/D
ax
is called the
Damköhler number
. If mixing is rapid, that is,
D
ax
/uL
is very small or
uL/D
ax
is
large, the axial dispersion term will be negligible and the system will approach plug-
flow behavior. This is a good approximation for analyzing contaminant dynamics in
surface waters. In the atmosphere, since the dispersion is large, we have to consider
the entire advective-dispersion equation.
where
Pe
L
=
uL/D
ax
is called the
Peclet number
and
Da
L
=
6.1.4 R
EACTION IN A
H
ETEROGENEOUS
M
EDIUM
In many cases, the reactant is transported from one medium to another where it reacts.
A good example is gaseous NH
3
that dissolves and reacts with an aqueous acidic
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