Environmental Engineering Reference
In-Depth Information
Knowing that
u
=
φ
n
,
A
0
S
c
A0
and using Arrhenius law for the temperature dependence
of the reaction rate constant,
1
dT
dL
=
−
Δ
r
H
ð
Þ
u
c
p
k
0
exp
E
a
R
u
T
4
h
d
u
c
A0
c
p
T
f
−
−
X
ð Þ
+
ð Þ ð
T
Eq
:
6
:
40
Þ
−
The molar balance is given by Equation (6.24)
d
φ
n
,
A
dV
=
R
ð Þ
−
Substituting the reaction rate term by the first-order kinetics, we obtain
c
A0
1
d
φ
n
,
A
dL
E
a
R
u
T
=
S
k
0
exp
−
X
ð Þ
−
ð
Eq
:
6
:
41
Þ
−
φ
V
c
A0
d
X
A
and
u
=
φ
V
φ
n
,
A
=
φ
V
dc
A
=
Since d
S
,
1
d
X
A
dL
=
1
E
a
R
u
T
u
k
0
exp
−
X
ð Þ
−
ð
Eq
:
6
:
42
Þ
The resultant system is a system of two differential equations with two unknown
variables, temperature and conversion. The solutions of the balances for different
coolant temperatures are shown in Figures 6.5 and 6.6.
The conclusion is that the reactor needs to be cooled. If the temperature of the
coolant is 340Kor higher, the conversionofAis greater than90%(Figure6.6), a value
that satisfies the requirements of the company. At higher coolant temperatures,
there is a temperature peak above 500 K that will cause the degradation of product
B (Figure 6.5). If the temperature of the coolant is lower than 340 K, the temperature
of the whole reactor will be kept below 500 K; however, at these coolant tempera-
tures, the conversion obtained is too low. Consequently, the temperature of the coolant
that best matches the cooling and conversion requirements is around 340 K.
6.5 RESIDENCE TIME AND SPACE TIME FOR FLOW REACTORS
In Section 6.3, it was explained that the parameter
φ
V
0
used in the balances of
CSTRs and PFRs is not exactly the same as the residence time t. The next example,
extracted from Levenspiel (1998), serves to illustrate the differences.
Consider a PFR for the production of popcorn (Figure 6.7) with a capacity of 1 L.
In this reactor, 1 L
τ
=V/
min
−1
of popcorn is obtained.
The volume flow of the products is larger than the flow of the reactants due to the
expansion.
min
−1
of raw corn is fed and 28 L
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