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Optimal control of a two-stage reactor system

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Abstract

Optimal operation policies were investigated for a batch reactor system with two different operation stages. At the end of the first nonisothermal stage one of the reactants was added. Since that moment the reactor was operated isothermally. In each stage behavior of the reactor was described by a set of differential equations. The maximum conversion problem was investigated subject to various operating constraints. Dynamic optimization based on the control vector parametrization was used to find the optimal control profile. Gradients of the resulting nonlinear programming problem were obtained by adjoint method based on the optimal control theory.

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Abbreviations

c A-F :

concentration of components A-F mol m−3

c sB :

concentration of added component B at switching times mol m−3

c WD :

required minimum value of the concentration of the component D at final time mol m−3

f:

system of state equations

F:

component of the cost function or constraints evaluated over a period of time

G:

component of the cost function or constraints evaluated at the final conditions

H:

Hamiltonian function

I:

identity matrix

J:

cost function or constraints

k:

number of constraints

k 1 :

first rate constant m3mol−1min−1

k 2 :

second rate constant min−1

L:

integral part of gradients

n:

number of iterations

p:

vector of time-independent parameters

P:

number of intervals

S:

amount of the solution of the component B added after the first stage with concentration c sB m3

t:

time min

t P :

processing time for both reaction stages min

t s :

switching time min

u:

profile of the reactor temperature for the first reaction stage K

u(t):

control vector

V 1 :

volume of the material loaded into the first reactor m3

V 2 :

volume of the material loaded into the second reactor m3

x(t):

state vector

Δ:

discontinuity of the state vector at switching times

λ:

vector of Lagrange multipliers

0:

initial time

i:

i-th segment

j:

j-th function of iteration

−:

time immediately before the switching time

+:

time immediately after the switching time

*:

optimum value

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Authors and Affiliations

  1. Department of Information Engineering and Process Control, Institute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology, SK-812 37, Bratislava, Slovakia

    T. Hirmajer & M. Fikar

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  1. T. Hirmajer
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  2. M. Fikar
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Correspondence to M. Fikar.

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Hirmajer, T., Fikar, M. Optimal control of a two-stage reactor system. Chem. Pap. 60, 381–387 (2006). https://doi.org/10.2478/s11696-006-0069-x

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