Udc 17. 977+644. Bashkir State University, Republic of Bashkortostan

Step 4. Mutation - the transformation of descendant genes in order to prevent hit
(7)
Computational experiment. Using the developed algorithm, we will solve the problem of optimal control for
an industrially significant reaction for obtaining phthalic
dx1
= ÿk1x1 ÿ k3x1 ÿ k4x1; dt
,
where k0j is the pre-exponential factor (1/h), Ej is the value of the activation energy of the jth stage (J/mol), R is the
universal gas constant (8.31 J/(mol K)).
If S(u ÿ (t), zk ) ÿ, then stop searching. Solution of the optimal control problem ÿ (t). niya will be the last best
individual u
where X1 is naphthalene, X2 is naphthoquinone, X3 is phthalic anhydride, X4 is carbon dioxide, X5 is maleic
anhydride. The dynamics of the concentrations of substances in this reaction can be described by a system of
ordinary differential equations composed according to the law of mass action [13]:
In the system of differential equations (7), the variables xi (i = 1, 5 ) (substance concentrations) are state
variables, T is the absolute temperature of the control variable. Based on technological considerations, the choice of
the optimal value
where xi is the concentration of the i-th substance (i = 1.5) (mole fraction), t is the time, kj is the rate constant of the j-
th stage of the reaction (j = 1.6) (1/h), depending on the temperature T according to the Arrhenius equation:
Step 6. Checking the conditions for ending the search. If
S(u ÿ (t), zk ) > ÿ, then z = Gzk (t) = u ÿ (t), j = 1, Rp, q = 0,
k = k + 1, go to step
2.
temperature limits are imposed:
Step 5. Updating the population and checking the termination condition of the algorithm. From the
descendants exposed to the mutation operator, we randomly choose one and place it in the current population
instead of the individual with the worst value of the fitness function. If q Max P, then q = q+1, and go to step 2,
otherwise choose ÿ (t) from the last one. population, the individual with the best value of the fitness function u
(6)
dt
by replacing with a random value from the range of valid values.
Phthalic anhydride is one of the most important raw materials for the production of plasticizers, paints and
varnishes, medicines, dyes, lubricant additives, rubber vulcanization accelerators, jet fuel additives, and insecticides
[19]. The scheme of the chemical reaction for obtaining phthalic anhydride is a set of stages:
dx5
= k6x3,
dt
620 KT 644 K.
(8)
Machine Translated by Google

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E. V. Antipina, S. I. Mustafina, A. F. Antipin, S. A. Mustafina
137
x4(t1) + x5(t1) < 0.2.
xi(0) = 0, i = 1, 5.
Let the concentrations of by-products at the end of the reaction not exceed 20%:
x1(0) = 1,
In table. Figure 1 shows the solution of the system of differential equations (7) with initial conditions
(11) for some admissible temperatures. As can be seen from the table, the solution of the optimal control
problem obtained using the developed algorithm satisfies the constraints of the problem and ensures the
highest yield of the reaction product.
The initial concentrations of substances are given by the values
In reaction (6), the starting material is naphthalene (X1). The target product of the reaction is
phthalic anhydride (X3), the side products of the reaction are carbon dioxide (X4) and maleic anhydride
(X5). Therefore, let us formulate the problem of finding the optimal temperature regime of the chemical
reaction (6) in order to obtain the maximum value of the target reaction product with restrictions on the
concentrations of side products and the conversion of the initial substance.
Let us determine the optimal temperature regime T(t) of reaction (6) taking into account control
constraints (8) and terminal constraints (9), (10) to obtain the maximum value of the reaction product,
phthalic anhydride:
x3(t1) ÿ max .
k is the value of the penalty
parameter z is the parameter for terminating the search
for a solution ÿ = 0.01. As a result of the computational experiment, it was established that in order
to obtain the maximum value of phthalic anhydride, it is necessary to maintain a constant temperature
of 620 K. 1, 2 shows the dynamics of reagent concentrations corresponding to the optimal temperature
regime. The maximum concentration of the reaction product X3 at the end of the reaction will be 0.716
mole fractions, the conversion of the starting material will be 98.9%, and the yield of side products will be
19.4%.
(10)
The numerical values of the kinetic parameters of the reaction for the preparation of phthalic
anhydride are presented in [19]. The reaction time is t1 = 0.8 h. The system of differential equations (7)
with initial conditions (11) was solved numerically using the 4th order Runge–Kutta method. The
parameters of the genetic algorithm for finding an unconditional extremum are: the number of individuals
in a population is 60, the number of populations is 3000, the selection operator is tournament
selection, the crossing operator is an arithmetic crossover, and the mutation operator is a random
mutation. The solution was obtained with the following parameters of the penalty method algorithm: initial
= 0.01, penalty parameter increase constant G = 10,
ditch genetic algorithm.
1 ÿ x1(t1) = 0.99.
We require that the conversion of the starting material at the end of the reaction is 99%:
To automate the search for a solution to the problem in the Delphi visual programming environment,
a program was developed based on an algorithm for solving the optimal control problem with terminal
constraints. The program allows the user to enter the parameters of the process of obtaining phthalic
anhydride and parameters
(9)
(11)
Machine Translated by Google

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4
2
x4(t1) + x5(t1),
1 ÿ x1(t1),
640
Table 1
1
644
they say share
0,716
0,715
0,713
0,712
0,707
AUTOMETRY. 2020. V. 56, No. 6
they say share
138
0,194
0,230
0,237
0,247
0,262
they say share
0,989
0,996
0,998
0,999
0,999
ÿ
620
630
635
5
T,
K
3
x3(t1),