Environmental Engineering Reference
In-Depth Information
can be considered as a kind of stirred pot. This type of ideal reactor works under the
assumption that the reaction volume is perfectly mixed, so the composition of the
mixture is exactly the same in the entire reactor. The concentration of species is time
dependent . By definition, a BR cannot operate in steady state since the system
evolves dynamically. A BR may be a constant-volume reactor (isochoric) or a
variable-volume reactor if it has a piston
cylinder arrangement.
Since no reactants are supplied and no products are released, the formation of
products and the disappearance of reactants will be translated as accumulation,
positive for formation and negative for disappearance. For reactant A, the molar
balance of Equation (3.5) then simplifies to
-
Rate of accumulation = rate of supply
rate of release + rate of production
ð
Eq
:
6
:
7
Þ
in which the rate of release and rate of production are zero, so
dn A
dt =V
or n A0 d X A
R ð Þ
dt =V
R ð Þ
ð
Eq
:
6
:
8
Þ
Since A is a reactant, the accumulation term (dn A /dt) must be negative, indicating that
the number of moles of A decreases in time.
Integration of Equation (6.8) gives the time necessary to obtain a conversion X A of
reactant A. If the reaction volume is constant,
ε A = 0, which means that the density is
also constant because the total mass is conserved, then
ð
;t=c A0 ð
c A
X A
dc A
d X A
t=
ð
Eq
:
6
:
9
Þ
R ð Þ
R ð Þ
c A0
0
For reactions where the volume of the reacting mixture changes with the conversion,
ε A
0, such as single-phase reactions with significant density changes, Equation (6.9)
becomes
t=n A0 ð
;t=c A0 ð
X A
X A
d X A
d X A
ð
Eq
:
6
:
10
Þ
R ð Þ
V 0 1+
ð
ε A X A
Þ
R ð Þ
ð
1+
ε A X A
Þ
0
0
in which t is the necessary time to reach a conversion X A . To solve Equations (6.9) and
(6.10), the reaction term (
R A ) must be substituted with a kinetic rate equation, and
then the equation must be integrated.
Example 6.1 Solution of the molar balance for a BR
Consider a first-order irreversible reaction with constant volume,
ε A =0:
A
!
b B+ c C+
R A =kc A
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