Environmental Engineering Reference
In-Depth Information
A well-stirred tank reactor can be used to carry out this reaction in the liquid
phase. It can be assumed to operate in the steady state. Moreover, the density
of the reaction mixture can be assumed to be constant. Consider a case that only
the reactants are present in the feed stream in equimolar amounts. Derive an
expression for the conversion of ethanol as a function of the residence time,
τ
,
in the tank (
τ
=V
r
=φ
V
ð
Þ
=m
ð
=φ
m
Þ
, with m being the mass in the tank, V
r
the tank
volume, and
φ
V
the mass and volume flow, respectively). The reaction rate
expression for the esterification is
r
=k
1
c
1
c
2
−
φ
m
and
k
2
c
3
c
4.
The net rate of consumption
of ethanol,
R
1
, is related to
r
via
R
1
=
r
.
ν
1
=
r
(
−
1) =
−
r
.
Solution
Start by writing the overall mass balance over the control volume of the reac-
tor tank:
dm
dt
=0=
φ
m
,
in
−
φ
m
,
out
)
φ
m
,
in
=
φ
m
,
out
=
φ
m
Then an integral mass balance for species 1 (ethanol) can be set up:
+MW
1
R
1
V
r
dm
1
dt
=0=
φ
m1
,
in
Y
1
,
in
−
φ
m1
,
out
Y
1
,
out
+MW
1
R
1
V
r
=
φ
m
Y
1
,
in
−
Y
1
,
out
)
0=
φ
m
ξ
1
+MW
1
R
1
V
r
=
φ
m
Y
1
,
0
ζ
1
+MW
1
R
1
V
r
Divide by MW
1
and express the mass fraction as concentration:
)
0=
φ
V
c
1
,
0
ζ
1
+
R
1
V
r
0=c
1
,
0
ζ
1
+
R
1
τ
r
,
ζ
1
=
−
R
1
τ
c
1
,
0
)
Now, consider the expression for the conversion of ethanol, with all concentra-
tions herein expressed as a function of the conversion of species 1, taking into
account that density is constant throughout the reaction process (see Equation
(3.13), in which then
ρ
=
ρ
0
):
c
1
=c
1
,
0
1
ð
−
ζ
1
Þ
For reactant 2, as
ν
1
=
ν
2
and as we have an equimolar reactant mixture:
c
2
=c
2
,
0
1
ð
−
ζ
2
Þ
=c
1
,
0
1
ð
−
ζ
1
Þ
ν
4
=
−
ν
1
and c
3,0
=c
4,0
=0:
For products 3 and 4, as
ν
3
=
c
3
=c
3
,
0
+c
3
,
0
ζ
3
=c
1
,
0
ζ
1
c
4
=c
4
,
0
+c
4
,
0
ζ
4
=c
1
,
0
ζ
1
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