# Optimal Temperature Policy for a Reversible Reaction

Optimal Temperature Policy for a Reversible Reaction

This Demonstration shows the temperature trajectory that maximizes the reaction rate of a reversible reaction.

For the reaction

A+B⇌Q+S

r

A

A

1

-/RT

E

1

e

2

c

A0

x

A

x

A

A

2

E

2

e

2

c

A0

2

x

A

with and ==0,

M=

c

B0

c

A0

c

Q0

c

S0

where and are the pre-exponential Arrhenius constants for the forward and reverse reactions, and are the energies of activation, is the universal gas constant, is the absolute temperature, , , , and are the initial concentration of the reactants, and is the conversion of species . The temperature function that gives the maximum reaction rate satisfies the condition =0 at each point in time; this function has an analytical solution for this reaction [1]

A

1

A

2

E

1

E

2

R

T

c

A0

c

B0

c

Q0

c

S0

x

A

A

∂

r

A

∂T

T

optimum

-1

-ln()

1

B

1

B

2

B

3

with

B

1

E

1

E

2

R

B

2

A

2

E

2

A

1

E

1

B

3

2

x

A

(1-)(M-)

x

A

x

A

the initial concentrations of and are taken equal to 0.5, and 1.0.

A

B

One complication can occur: for low conversions, may have a value sufficiently small enough to make ≤1; then the equation for gives a value →∞ (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 and .

B

3

B

2

B

3

T

optimum

T

optimum

B

1

B

2