Kinetic models based on the detailed mechanisms of complex chemical reactions, represent a systems of differential equations. In these systems the number of the unknowns equals the number of substances involved in the reaction. Hypothetical schemes of complex chemical reactions contain a large number of substances and reactions between them. However, direct measurement is available only for some of these substances. Precise description of the behavior of only a few substances are required for the analysis of the reaction mechanism. In this connection there is a need to replace an original system with a system of a smaller dimension, preserving the dynamics of concentrations of the selected substances. As a result of the reduction of the kinetic mechanism of the reaction scheme there would be defined by an equivalent scheme containing less substances and stages than an original one. Therefore, the construction of mathematical models of reduced reaction schemes results in solving the problem of an identification of a mathematical model of the reaction, i.e, solving the inverse problem of chemical kinetics.

Kinetic model of the reduced scheme of α-methylstyrene dimerization reaction

Let’s construct the kinetic model of the reduced scheme of α-methylstyrene dimerization reaction. The products of this reaction (the linear and cyclic dimers) have been in practical use as plasticizers, polymer modifiers, rubber in the manufacture of synthetic lubricants, etc. A number of chemical reactions describing the same reaction, and the corresponding kinetic equations are as like [1]:

(1)

where the following designations were entered X1 – α-methylstyrene; X2 – α-dimer; X3 – β- dimer; X4 – cyclic dimer; X5 – trimers, where ωi(t, x) – velocity of the i-th stage (kmol/(m3·h)) (i = 1,…,9); C = (C1, …, C5) – vector of concentration of the components (kmol/m3); k = (k1,…,k12) – vector of kinetic rate constants of the j-th reaction (m3/(kmol·h))(j = 1, …, 12).

The values of the kinetic constants and activation energies are shown in Table 1. The rate constant of the j-th reaction is calculated with the use of the selected basic temperature Tbase = 373 K defined by the formula

Table 1

Kinetic parameters of the process of α-methylstyrene dimerization in the presence of a catalyst NaHY of at a temperature 373 K

Number |
ki(373 К), m3/(kgcat∙h) |
Ei, kJ/mol |
Number |
ki(373 К), m3/(kgcat∙h) |
Ei, kJ/mol |

1 |
61,357 |
196 |
7 |
0,019308 |
247 |

2 |
8,9534 |
263 |
8 |
41,556 |
194 |

3 |
7,7916 |
259 |
9 |
0,03662 |
115 |

4 |
1,1693 |
238 |
10 |
0,04547 |
279 |

5 |
0,11922 |
275 |
11 |
0,0995 |
204 |

6 |
0,12041 |
127 |
12 |
0,05132 |
138 |

Kinetic model of α-methylstyrene dimerization with the changes in the number of moles in the course of a chemical reaction is a system [1]:

with initial conditions: , i = 1, ..., 5, N(0) = 1, where xi – concentration of i-th component (mole fraction); N = C/C0 – concentration of i-th component (mole fraction); С0 – initial total concentration of reactants (kmol/m3); (νik) – matrix of stoichiometric coefficients, Wj = ωj/C0 – values of the chemical reaction rate (j = 1,..,9) (1/h).

The reduced scheme of this reaction, obtained in [2, 3] based on the combined algorithm of reduction of the reaction scheme in the time and temperature range of the reaction course, and its kinetic equations have the following form:

(2)

where C = (C1, C2, C3, C4) – vector of concentration of components, k = (k1, …, k9) – vector of kinetic rate constants of the reaction stages (2).

Solution of the inverse kinetic problem for the reduced reaction scheme

An inverse kinetic problem is a problem of minimizing the functional deviations between the calculated and experimental data:

(3)

where – calculated and experimental values for substances respectively; l – number of measurements; n – number of substances.

To solve the problem of identification of a mathematical model of the reduced reaction scheme it is necessary to calculate the values of kinetic constants k0j, minimizing the functional (3), and the values of activation energies Ej.

With the application of the algorithm for solving the inverse problem of chemical kinetics, constructed in [2] on the basis of Hooke-Jeeves’ method, kinetic parameters of the reduced scheme of α-methylstyrene dimerization reaction have been calculated (Table 2).

Results and Discussion

As a result of solving the inverse kinetic problem the values of activation energies Ej and kinetic constants k0j (j = 1, …, 9) for the reduced scheme of α-methylstyrene dimerization reaction have been calculated. On the basis of the obtained values a direct kinetic problem has been solved. The relative difference between the calculated and experimental values of the concentration of substances has been more than 11 %, which is within the measurement error in the experiment. The reduction of the reaction scheme (1) has not changed the overall dynamics of changes in the concentrations of substances with time. Relative error vectors concentrations of substances X1, X2, X3, X4 for the reduced scheme of α-methylstyrene dimerization reaction amounted: δ(x′1) = 1,35 %, δ(x′2) = 1,68 %, δ(x′3) = 10,24 %, δ(x′4) = 7,93 %. This shows that the accuracy of the description of the dynamics of concentrations of target substances of the reduced scheme of α-methylstyrene dimerization reaction is within the error limits of quantitative analysis. Therefore, the reduced scheme of α-methylstyrene dimerization reaction (2) can be used to solve other problems based on the analysis of the kinetic model of the reaction scheme.

Table 2

Kinetic parameters of the reduced scheme of α-methylstyrene dimerization reaction in the presence of a catalyst NaHY at a temperature of 373 K

Number |
ki(373 К), m3/(kgcat∙h) |
Ei, kJ/mol |
Number |
ki(373 К), m3/(kgcat∙h) |
Ei, kJ/mol |

1 |
62,788 |
197,6 |
6 |
0,70168 |
320,9 |

2 |
6,037 |
231,4 |
7 |
0,00121 |
301,3 |

3 |
9,055 |
263,3 |
8 |
0,00847 |
242,4 |

4 |
1,092 |
311,9 |
9 |
0,00468 |
184,0 |

5 |
0,0012 |
573,3 |