Introduction

Currently, there is a tendency to improve the environmental characteristics of automobile fuel while maintaining a high octane number. Sulfuric acid alkylation of isobutane with olefins makes it possible to obtain a high-octane component of gasoline with a minimum content of aromatic hydrocarbons. Therefore, it is a significant process of the modern refinery. The main advantage of alkylate is the high octane number: according to the research method, it is 96, according to the motor method, it is 92. The alkylation product is non-toxic, does not contain benzene, and has a low saturated vapor pressure, hence it does not evaporate much during storage and transportation [1]. In the middle of the twentieth century, at the design and construction stages of most industrial installations, including reactors, insufficient attention was paid to in-depth study of the mechanism of most processes. Now the intensive development of information technologies, such as the theory of analysis and storage of large amounts of data, parallel computing technology, artificial neural networks, etc., has led to the creation of universal software systems that allow you to develop detailed kinetic models of production processes [2, 3]. The most complete account of thermodynamic and kinetic regularities of target and side reactions during modeling will allow predicting the yield of the target product with the highest degree of confidence. What can be used in the future to find optimal technological conditions and intensify the process [4]. When modeling this process, it is necessary to take into account the properties of acid catalysts with high corrosion activity and low selectivity, which leads to the fact that along with the target alkylation reactions, a large number of side reactions occur. They lead to a loss of activity and an increase in the consumption of the catalyst, as well as to the formation of lighter and heavier hydrocarbons than the target product, which lowers the octane number of the alkylate [1, 5].

Experimental section

A mathematical model of sulfuric acid alkylation of isobutane with olefins, presented in the work of the East China University of Science and Technology, was chosen as a prototype [4]. The model used includes targeted reactions to obtain the alkylate product, as well as side reactions leading to the formation of heavy fractions. The formation of the latter can be regulated by changing the temperature and composition of the raw materials. The experiments were carried out at the batch process at the temperature range of 3–12 °C and 0.5 MPa.

The object of the study is the reactor unit of the sulfuric acid alkylation of isobutane with olefins. Butane-butylene fraction was used as a raw material. The pressure in the reaction zone was 0.5 MPa. The inlet temperature was selected in the range of 3–12 °C. Sulfuric acid with a concentration of 95–98% was used as a catalyst for the process. The feedstock compositions and reaction conditions of alkylation experiments are listed in Table 1.

Table 1 Initial data. Composition of raw materials
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The initial stage of the mathematical description of the object is to draw up a scheme of reactions of conversion of hydrocarbons during the process. It should be noted that the accuracy of calculations and the adequacy of the mathematical model to the real process directly depends on the degree of detail of chemical transformations. Therefore, the transformation scheme should sufficiently reflect the physicochemical essence of the process, in our case, a combined graph compiled using graph theory was constructed (Fig. 1).

Fig. 1
figure 1

Combined graph of reactions for the process of sulfuric acid alkylation of isobutane with olefins, where x1 is iC4H8; x2 is iC4, x3 is iC4+; x4 is TMPs+; x5 are DMHs+; x6 are TMPs; x7 are DMHs; x8 are HEs; x9 is iCx+; x10 is iCy = ; x11 is 2-C4H8; x12 is 1-C4H8 and TMP are trimethylpentanes; DMH are dimethylhexanes; HEs are heavy ends

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It is known that reactions in the process of alkylation of olefin isobutane in the presence of sulfuric acid proceed according to the classical carbonium-ion mechanism. In this case, the thermodynamics of the process is the basis. The result of thermodynamic analysis is a list of reactions that are likely to occur under given conditions. The probability of reactions occurring under the technological conditions of the process is estimated by the value of the Gibbs energy ΔG. The resulting combined graph includes 12 individual components and 15 reaction stages.

The mathematical model of the process is a system of nonlinear differential equations:

$$\frac{{dx}_{1}}{dt}=-{k}_{1}{x}_{1}+{k}_{2}{x}_{3}-{k}_{3}{x}_{1}{x}_{3}-{k}_{7}{x}_{1}{x}_{2}{x}_{4}-{k}_{11}{x}_{1}+{k}_{14}{x}_{11}$$
$$\frac{{dx}_{2}}{dt}=-{k}_{4}{x}_{2}{x}_{4}-{k}_{6}{x}_{2}{x}_{5}-{k}_{7}{x}_{1}{x}_{2}{x}_{4}-{k}_{15}{x}_{11}{x}_{2}{x}_{4}$$
$$\frac{{dx}_{3}}{dt}={k}_{1}{x}_{1}+{k}_{4}{x}_{2}{x}_{4}+{k}_{6}{x}_{2}{x}_{5}-{k}_{3}\left({x}_{1}+{x}_{11}\right){x}_{3}-{k}_{5}{x}_{12}{x}_{3}-{k}_{2}{x}_{3}+{k}_{7}{x}_{1}{x}_{2}{x}_{4}+{k}_{15}{x}_{11}{x}_{2}{x}_{4}$$
$$\frac{{dx}_{4}}{dt}={k}_{3}\left({x}_{1}+{x}_{11}\right){x}_{3}-{k}_{4}{x}_{2}{x}_{4}-{k}_{7}{x}_{1}{x}_{2}{x}_{4}-{k}_{15}{x}_{11}{x}_{2}{x}_{4}$$
$$\frac{{dx}_{5}}{dt}={k}_{5}{x}_{12}{x}_{3}-{k}_{6}{x}_{2}{x}_{5}$$
$$\frac{{dx}_{6}}{dt}={k}_{4}{x}_{2}{x}_{4}$$
$$\frac{{dx}_{7}}{dt}={k}_{6}{x}_{2}{x}_{5}-{k}_{10}{x}_{7}$$
$$\frac{{dx}_{8}}{dt}={k}_{7}{x}_{1}{x}_{2}{x}_{4}+{k}_{15}{x}_{11}{x}_{2}{x}_{4}+{k}_{9}{x}_{9}{x}_{10}-{k}_{8}{x}_{8}$$
$$\frac{{dx}_{9}}{dt}={k}_{8}{x}_{8}-{k}_{9}{x}_{9}{x}_{10}$$
$$\frac{{dx}_{10}}{dt}={k}_{8}{x}_{8}-{k}_{9}{x}_{9}{x}_{10}$$
$$\frac{{dx}_{11}}{dt}=-{k}_{3}{x}_{11}{x}_{3}-{k}_{15}{x}_{11}{x}_{2}{x}_{4}+{k}_{11}{x}_{1}+{k}_{12}{x}_{12}-{k}_{13}{x}_{11}-{k}_{14}{x}_{11}$$
$$\frac{{dx}_{12}}{dt}=-{k}_{5}{x}_{12}{x}_{3}+{k}_{13}{x}_{11}-{k}_{12}{x}_{12}$$

here xi are concentrations of substances involved in the reaction in mole fractions; kj are reaction rate constants. The initial conditions are t = 0, x1 = x10; x2 = x20; x3 = 0; x4 = 0; x5 = 0; x6 = 0; x7 = 0; x8 = 0; x9 = 0; x10 = 0; x11 = x110; x12 = x120. The corresponding species are (1) iC4H8; (2) iC4, (3) iC4+; (4) TMPs+; (5) DMHs+; (6) TMPs; (7) DMHs; (8) HEs; (9) iCx+; (10) iCy = ; (11) 2-C4H8; (12) 1-C4H8. The mathematical model is constructed according to the mass-action law [6, 7].

In the Python programming language, a program code was written to develop a kinetic model of the process of sulfuric acid alkylation of isobutane with olefins, the system of differential equations was solved using the method of the Radau-II A method of the fifth order.

To determine the rate constants kj of reactions occurring during the process of sulfuric acid alkylation of isobutane with olefins, a genetic algorithm was chosen that is used in solving optimization and modeling problems by sequential selection. This algorithm is chosen based on the following advantages: no derivative information is required, speed, efficiency, parallel calculation capabilities and optimization of continuous multi-purpose tasks. The solution of the inverse kinetic problem is determined by the objective function:

$$F = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {\left| {x_{ij}^{\exp } - x_{ij}^{calc} } \right|} } \to \min$$
(1)

here N is the number of observed components of the reaction (N = 2); M is number of experiments (M = 12); \({\mathrm{x}}_{\mathrm{ij}}^{\mathrm{e}xp}\) and \({\mathrm{x}}_{\mathrm{ij}}^{calc}\) are the experimental and calculated values of i-th observed component at j-th experiment.

The number of optimization parameters was 15 (n = 15). The estimated parameters of k1 − k15 are listed in Table 2. To solve the optimization problem, a gravitational algorithm was used [8].

Table 2 Optimal values of rate constants under different temperature conditions
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Further, using the constants of the calculated reaction rate constants, the direct kinetic problem was solved, thus the change in the concentrations of the reaction components over time was calculated. Figs. 2, 3, 4 and 5 show the calculated concentration profiles of the target components, the markers indicate the literature data. At the beginning of the process, HEs is formed by the reaction of isobutylene or 2-butene with TMP+ or DMH+, followed by the transfer of the hydride by isobutene. Thus, the initial content of TMP and DMH is small, while HEs sharply increases to the maximum value in tens of seconds. HEs cations are then converted to LEs in the reaction of cleavage and hydride transfer, which corresponds to a sharp decrease in the HEs content. This observation corresponds to the rapid consumption of olefins, and also demonstrates the nature of the very rapid alkylation reaction of isobutane.

Fig. 2
figure 2

Concentration profile of the components involved in the alkylation process when using feedstock #1 (3 °C). TMP are trimethylpentanes; DMH are dimethylhexanes

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Fig. 3
figure 3

Concentration profile of the components involved in the alkylation process when using feedstock-#1 (6 °C). TMP are trimethylpentanes; DMH are dimethylhexanes

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Fig. 4
figure 4

Concentration profile of the components involved in the alkylation process when using feedstock-#1 (9 °C). TMP are trimethylpentanes; DMH are dimethylhexanes

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Fig. 5
figure 5

Concentration profile of the components involved in the alkylation process when using feedstock-#1 (12 °C). TMP are trimethylpentanes; DMH are dimethylhexanes

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DMH are mainly formed at the initial stage of the reaction and subsequently decreases slightly. The concentration of TMP gradually increases over time. In general, the reaction reaches a stable state after 5 min, which indicates an equilibrium reaction time of sulfuric acid alkylation within 5 min.

Optimization of the process of sulfuric acid alkylation of isobutane with olefins

The improvement of industrial technological processes is one of the most important tasks due to the fact that such improvement allows to increase the output of target products with minimal financial costs. As mentioned earlier, the developed mathematical model allows us to make predictable calculations and select optimal rate constants.

Based on the purpose of the process of sulfuric acid alkylation of isobutane with olefins, which consists in obtaining a high-octane component of gasoline, the primary criterion for optimizing this process should be such an indicator as the yield of the target product having the highest octane value. In our case, such a product is the sum of trimethylpentanes.

During the process, side polymerization reactions occur, so the next optimization criterion is also the output of by-products—the sum of dimethylhexanes and heavy fractions.

Thus, the optimization criteria are: the total yield of trimethylpentanes and the total yield of by-products. To increase the efficiency of the process, it is necessary that the first criterion tends to the maximum, the second one—to the minimum.

The optimization problem for the process of sulfuric acid alkylation of isobutane with olefins was solved by the “Differential evolution base" method [9]. This is a method of multidimensional mathematical optimization, which belongs to the class of stochastic optimization algorithms (works using random numbers). Earlier, the authors of the article developed kinetic models of other chemical processes using various heuristic algorithms [8, 10,11,12,13,14,15,16,17], which have proven their effectiveness.

According to the literature, it is necessary that the ratio of isobutane:olefin is within (6–10):1. This is due to the fact that an excess of isobutane intensifies the target alkylation reactions and suppresses the occurrence of side effects, but at the same time, a too large excess of isobutane can contribute to the occurrence of side reactions of self-alkylation.

$$\left\{ \begin{gathered} x_{4} + x_{6} \to \max \hfill \\ x_{5} + x_{7} + x_{8} \to \min \hfill \\ x_{1} + x_{2} + x_{3} + x_{4} = 1 \hfill \\ 6 \le \left( {\frac{{x_{2} }}{{x_{2} + x_{3} + x_{4} }}} \right) \le 10 \hfill \\ \end{gathered} \right.$$

The results of the calculations carried out to optimize the process of sulfuric acid alkylation are presented in Table 3. The optimal ratio of isobutane is selected for different temperature conditions:olefin. Its effect on the yield of target high-octane trimethylpentanes and the yield of by-products is presented.

Table 3 Dependence of the yield of the target alkylation products on the isobutane ratio:olefin in feedstock
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Based on Table 3, it can be seen that despite the use of raw materials with an optimal ratio of isobutane:olefin, the highest yield of trimethylpentanes is observed at a temperature of 12 °C and decreases with decreasing temperature. Changes in the concentrations of products of the alkylation process when using raw materials with an optimal isobutane ratio:olefins are shown in Fig. 6.

Fig. 6
figure 6

Concentration profiles of the components involved in the alkylation process under different temperature conditions: a 3, b 6, c 9, d 12 °C. TMP are trimethylpentanes; HEs are heavy fractions

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Fig. 6 shows the calculated concentration profiles of the target components when using the composition of raw materials with an optimal ratio of isobutane:olefin, markers indicate experimental data. At the beginning of the process, HEs is formed by the reaction of isobutylene or 2-butene with TMP+ or DMG+, followed by the transfer of the hydride by isobutene. Thus, the initial content of TMP and DMH is small, while HEs sharply increases to the maximum value in tens of seconds. HEs cations are then converted to LEs in the reaction of cleavage and hydride transfer, which corresponds to a sharp decrease in the HEs content. This observation corresponds to the rapid consumption of olefins, and also demonstrates the nature of the very rapid alkylation reaction of isobutane.

DMH is mainly formed at the initial stage of the reaction and subsequently decreases slightly. The concentration of TMP gradually increases over time. In general, the reaction reaches a stable state after 5 min, which indicates an optimized reaction time of sulfuric acid alkylation within 5 min. Fig. 6 shows that the yield of the TMP reaction product has increased compared to the experimental data (dots), and the yield of the DMH and HEs reaction byproducts has decreased in total as a result of process optimization.

Conclusions

Thus, the article presents a mathematical model of the reaction of sulfuric acid alkylation of isobutane by olefins in the form of a system of ordinary nonlinear differential equations. Software that allows us to determine the numerical values of the reaction rate constants at temperatures of 3, 6, 9 and 12 °C has been developed. With the help of the developed kinetic model, the process of sulfuric acid alkylation was optimized, where the total yield of trimethylpentanes and the total yield of by-products were used as optimization criteria. To increase the efficiency of the process, it is necessary that the first criterion tends to the maximum, the second to the minimum. The ratio of isobutane to olefin ranged 6–10 to one. As a result of solving the problem of conditional global optimization, the optimal ratio of isobutane:olefin was selected for different temperature conditions.