Automation of the Formation of a Mathematical Formulation of Kinetics for Multistage Chemical Reactions and Numerical Solution to a Direct Problem

Introduction. Studies on the kinetics of multistage chemical processes is of theoretical and practical importance, since it allows for revealing the mechanism of reactions and carrying out control. At the same time, it becomes possible to accelerate target or slow down side chemical reactions. Kinetic analysis, being an important task of chemical engineering, contributes to the optimization of synthesis processes and the prediction of their effectiveness [1–3]. A great many chemical processes are multistage reactions. For their effective design and optimization, it is required to have a mathematical model describing all stages [4][5]. Manual formation of such a model can be very time-consuming and costly, since it implies processing a large amount of information [6].

The object of the study is the industrial process of catalytic reforming of gasoline. The target product of the reaction — reformate — provides obtaining the bulk of commercial gasoline. The remaining components are products of isomerization, catalytic cracking, and alkylation. Most oil refineries in Russia and worldwide use catalytic reforming units [7][8]. Catalytic reforming of gasoline makes it possible to obtain aromatic hydrocarbons (xylene, toluene, benzene), which can then be used in the petrochemical industry [9][10]. However, for commercial gasoline, the presence of aromatic hydrocarbons in the product is limited by environmental criteria. Therefore, detailed complex kinetics, demonstrating the step-by-step formation of components, is able to operate the process conditions to control the yield of the product and limit the content of aromatic hydrocarbons. The process of catalytic reforming of gasoline is carried out in a reactor unit consisting of three consecutive reactors, between which furnaces are installed to heat the mixture. This is a necessity since the reaction is endothermic. In the mathematical formulation of the reactor unit, it is required to take into account the variation of temperature and volume of the reaction mixture along with changes in the concentrations of the components from the conditional contact time of the catalyst and the reaction mixture.

Automation of modeling of such complex processes makes it possible to significantly speed up obtaining a mathematical formulation of the kinetics of multistage reactions, reduce the probability of errors, and simplify the process of working with data [11]. In general, automation of the formation of a mathematical formulation for multistage chemical reactions is an important tool for effective process management in various industries, as well as for achieving more accurate and efficient implementation of chemical reactions in the laboratory.

Thus, the study aimed at developing a software product to help automate the formation of a mathematical formulation of multistage chemical reactions, taking into account the kinetics, thermodynamics and changes in the volume of the reaction mixture, as well as obtaining a numerical solution of the compiled mathematical model.

To achieve the research goal, the following tasks were solved in this work:

• development of a software module for the formation of a mathematical formulation of kinetics using the Python programming language;
• development of a software module for numerical solution to a direct problem according to a compiled mathematical formulation using the Python programming language;
• obtaining the results of the software product for the reaction of catalytic reforming of gasoline.

Materials and Methods. Mathematically, this process can be described by a system of ordinary nonlinear differential equations with initial data, which is a mathematical model. Considering that the catalytic reforming of gasoline is an exothermic reaction with a change in the volume of the reaction mixture, the model of multistage chemical kinetics can be considered in the following context:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

with initial conditions at

[11].

In this formula of reaction kinetics there are a number of variables and coefficients that determine the concentration of reagents, contact time and reaction rate. Specifically, Q and (concentrations of reaction reagents) can be expressed in mol/l or in fractions, and variable , showing the conditional contact time of the reaction mixture with the catalyst, in kg×min/mol or kg of the catalyst. The formula also includes the following indicators: the number of stages M (173 stages) and the number of substances N (38 substances), as well as stoichiometric matrix . — these are all negative elements of matrix at the intersection of the -th row and the j-th column, and — all positive elements of matrix at the intersection of the -th row and the j-th column. The rate of the j-th reaction stage can be expressed in 1/min or in mol/(kg×min), and the rate constants of stages — in 1/min. In addition, the formula specifies pre-exponential factors , , expressed in 1/min, the activation energies of the forward and reverse reactions , , expressed in kcal/mol, gas constant R and temperature T, measured in Kelvins. The period of reaction t* is determined by the time required to complete it in minutes, and the enthalpy of the formation of the i-th component is measured in J/mol. The specific heat capacity of the i-th component, measured in J/(mol×K), can be expressed in terms of the coefficients of the temperature dependence of the heat capacity of this component, which are indicated by letters [12]. i — serial number of the substance involved in the reaction.

When modeling chemical transformations of catalytic reforming of gasoline, it is required to take into account a large number of individual hydrocarbons. Therefore, in the article, group components are used in the model for the separation of types of hydrocarbons and the number of carbon atoms in the structure of the molecule: A_l — aromatic hydrocarbons; ACP_l — five-membered naphthenes; ACH_l — six-membered naphthenes; iP_l — paraffin isomers; nP_l — normal paraffins. Here, l — number of carbon atoms in the molecular structure. Table 1 shows the average coefficients of the temperature dependence of the heat capacity of the group components that are used in the calculations.

Equations (1), (3) and (4) describe changes in reagent concentrations in the process, and in system (2), molar fractions are used to describe changes in the molar flow rate of the mixture. In (5), the dependence of the reaction temperature change on the thermodynamic characteristics of substances is considered. In (6), (7), the dependence for calculating the enthalpy of formation of the i-th component and the specific heat capacity is given.

In this model, the direct problem (solving a system of differential equations) is both rigid and non-rigid. The Gear method used to solve the catalytic reforming model is based on backward differentiation [13–15]. To compare the results and identify the optimal method, an explicit one-step Runge-Kutta numerical integration fourth-order method and an explicit numerical integration method scipy.odeint() of the Python programming language were used.

The concept of automating the formation of a mathematical formulation of kinetics for multistage chemical reactions and a numerical solution method is illustrated using the IDEF0 methodology.

Figure 1 shows a contextual diagram of the process of automating the formation of a mathematical formulation of kinetics for multistage chemical reactions and solutions by numerical method.

The input parameters are the stoichiometric reaction matrix and experimental data. These include initial data on the concentration of substances, temperature, pre-exponential multiplier, activation energies. The result of solving the problem is a numerical solution to the direct problem. The controls are numerical methods for comparing the results, as well as the method of forming a mathematical description. The mechanism is the developed software package for automating the mathematical formulation of the kinetics of multistage reactions and solving a direct problem. Figure 2 shows the decomposition of the process of automating the formation of a mathematical formulation of kinetics for multistage chemical reactions and solutions by numerical method into stages:

• formation of mathematical formulation;
• direct problem solution.

The developed software package capabilities are represented by the features described below.

With the help of the program, you can create a kinetic model of a multistage chemical reaction and present its solution by the previously described methods, using a given stoichiometric matrix. If the reaction is exothermic and there is a change in the volume of the reaction mixture, two complementary equations are added to the system of differential equations to describe these changes.

The mathematical model of catalytic reforming of gasoline is a system of differential equations (DES) of form (1)–(7), consisting of 40 differential equations with respect to the number of group components, temperature dependence, and the mole change equation.

The basic functionality of the program is:

• main application window with the output of the mathematical formulation according to the stoichiometric matrix entered by the user;
• window with visualization of the solution: for each numerical method (Python-odeint, fourth-order Runge-Kutta, Gear method), the operating time, an array of time t and the solution of a system of differential equations at each moment of time are displayed, which are graphs of changes in the concentration of substances, changes in the reaction temperature or the volume of the reaction mixture.

Research Results. The catalytic reforming reaction took place in three reactors, between which a heating furnace was located. The initial temperature of the first reactor was 766 K (493°C). For the first 9.6 conditional hours, the reaction proceeded in the first reactor, the temperature dropped. After that, heating took place in the furnace between the two reactors to 763 K (490°C), and the chemical process passed into the second reactor. In the second reactor, the process occurred in the time interval from 9.6 to 32.3 conventional hours, the temperature decreased. Before proceeding to reactor 3, the mixture was heated in the second intermediate furnace to 768 K (495°C) and flew in this reactor in the time interval from 32.3 to 60 conventional hours.

Figures 3–5 show the kinetic curves of the hydrocarbon groups involved in the reaction and obtained using the Gear method. The intermittent transitions on the graphs indicate that the mixture is moving from one reactor to another with intermediate heating.

The curves of changes in the concentrations of the group components are given depending on the time of contact of the reaction mixture with the catalyst (Fig. 3–5). As an example, symbol nP1 stands for normal paraffin with one carbon atom, and symbol A9 — stands for aromatic hydrocarbon with nine carbon atoms. Figures 6 and 7 show, respectively, the curves of changes in temperature and volume of the reaction mixture of the catalytic reforming reaction of gasoline.

Discussion and Conclusion. In the presented work, a software module for the formation of a mathematical formulation of kinetics for complex multistage reactions has been developed. This makes it possible to analyze complex processes, including industrial, to consider various possible mechanisms in an acceptable time.

The developed software module for the numerical solution of a direct kinetic problem provides monitoring the dynamic change in the concentrations of components and the process temperatures in the case of its non-isothermality. In addition to the presented graphs of changes in component concentrations, measurements of the operating time of each of the three implemented methods were made. According to the scipy.odeint() method, the solution to the direct problem lasted 80.5 seconds; according to the Runge-Kutta fourth-order method — 111.8 seconds. The most optimal time, equal to 39.8 seconds, was shown by the Gear method. This confirms that Gear's method is optimized for working with rigid systems of differential equations.

The implemented approach was applied to modeling the industrial process of catalytic reforming of gasoline. Automation of the formation of a mathematical formulation of kinetics and the implementation of a numerical solution to a direct problem allowed us to develop a kinetic model of a detailed scheme of chemical transformations in an acceptable time, and obtain concentration and temperature profiles in the process reactors corresponding to industrial data.