Environmental Engineering Reference
In-Depth Information
Then with Equation (6.9),
ð
c
A
dc
A
kc
A
=
1
k
c
A
c
A0
t=
−
−
lnc
A
j
c
A0
resulting in
kt = ln
c
A0
1
c
A
=ln
X
A
)
c
A
=c
A0
exp
ðÞ
−
kt
ð
Eq
:
6
:
11
Þ
1
−
From Equation (6.11), it is possible to obtain the variation of the concentration of
A in time or the time needed to obtain a certain conversion. Note that A is related to
B, C, and other products through the stoichiometric coefficients. Because only a
single reaction is considered, if we know how A changes in time, we know how the
concentrations of all reactants and products change in time. A solution of the molar
balance for BR and different reaction types and orders can be found in Levenspiel
(1998, 2002) and Schmidt (2004).
6.2.2 Non-isothermal Batch Reactors
Sometimes, it is not possible or even desirable to carry out a reaction under isothermal
conditions. One reason is that if there are several reactions taking place in parallel, it is
possible to vary the product distribution playing with the temperature. According to
the Arrhenius equation (Atkins and de Paula, 2009), the temperature affects the reac-
tion exponentially depending on the activation energy E
a
. If there are several reactions
taking place in parallel and they have different E
a
values, increasing the temperature
will favor the reactions with higher values of E
a
.
Assuming that the volume of the BR is constant, the molar balance Equation (6.8)
for the non-isothermal reaction of a reactant A reads
n
A0
d
X
A
dt
=V
ð
−
R
A
X
A
,T
ð Þ
Þ
ð
Eq
:
6
:
12
Þ
In non-isothermal cases, the temperature must be accounted for in the reaction rate.
Considering the first-order reaction of Example 6.1, the solution to the non-isothermal
molar balance would be
ð
c
A
dc
A
c
A
t=
−
ð
Eq
:
6
:
13
Þ
E
a
R
u
T
k
0
exp
−
c
A0
There are two unknown terms, T and c
A
, so an additional equation is necessary
to solve the molar balance. That additional equation comes from the energy
balance
Search WWH ::
Custom Search