WOLFRAM|DEMONSTRATIONS PROJECT

Optimal Temperature Policy for a Reversible Reaction

​
reaction time
500
T
maximum
600
E
1
-
E
2
R
2500
A
2
E
2
A
1
E
1
2500
This Demonstration shows the temperature trajectory that maximizes the reaction rate of a reversible reaction.
For the reaction
A+B⇌Q+S
, the rate is
r
A
=
A
1
-
E
1
/RT
e
2
c
A0
(1-
x
A
)(M-
x
A
)-
A
2
E
2
/RT
e
2
c
A0
2
x
A
,
with
M=
c
B0
c
A0
and
c
Q0
=
c
S0
=0
,
where
A
1
and
A
2
are the pre-exponential Arrhenius constants for the forward and reverse reactions,
E
1
and
E
2
are the energies of activation,
R
is the universal gas constant,
T
is the absolute temperature,
c
A0
,
c
B0
,
c
Q0
, and
c
S0
are the initial concentration of the reactants, and
x
A
is the conversion of species
A
. The temperature function that gives the maximum reaction rate satisfies the condition
∂
r
A
∂T
=0
at each point in time; this function has an analytical solution for this reaction [1]
T
optimum
=
-1
-
1
B
1
ln(
B
2
B
3
)
,
with
B
1
=
E
1
-
E
2
R
,
B
2
=
A
2
E
2
A
1
E
1
, and
B
3
=
2
x
A
(1-
x
A
)(M-
x
A
)
;
the initial concentrations of
A
and
B
are taken equal to 0.5, and 1.0.
One complication can occur: for low conversions,
B
3
may have a value sufficiently small enough to make
B
2
B
3
≤1
; then the equation for
T
optimum
gives a value
T
optimum
→∞
(or negative); in practice the temperature is limited by the reactor materials or the catalyst's physical properties. The optimum temperature profile and the concentration of the reactants as a function of time are shown for user-set values of reaction time, maximum allowable temperature, and parameters
B
1
and
B
2
.