Impact of Reaction System Turbulence on the Dispersity and Activity of Heterogeneous Ziegler–Natta Catalytic Systems for Polydiene Production: Insights from Kinetic and CFD Analyses

Abstract

An analysis was conducted to investigate how reaction system turbulence affects the butadiene-isoprene copolymerization in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system. A model was developed, which integrates CFD simulations of TiCl₄ + Al(i-Bu)₃ particle breakage based on population balance equations with the kinetic modeling of the butadiene-isoprene copolymerization. It was established that an increase in turbulent kinetic energy leads to a reduction in catalyst particle size, an increase in active site concentration, an acceleration of the copolymerization process, and a decrease in the average molecular weights of the copolymer. Furthermore, catalytic activity correlates with both the average and maximum values of turbulent kinetic energy in the reaction system, whereas the effect of the average residence time of catalytic particles under turbulent conditions is insignificant. Based on these results, recommendations were provided for optimizing the impact of reaction system turbulence on TiCl₄ + Al(i-Bu)₃ particles to enhance the butadiene-isoprene copolymerization rate and achieve precise control over the molecular weight characteristics of the copolymer. The findings of this study can be applied to optimize the synthesis technology of the cis-1,4 butadiene-isoprene copolymer, which is used in the production of frost-resistant rubber.

1. Introduction

A common industrial method for producing polydienes—polyisoprene (isoprene rubber, IR), polybutadiene (butadiene rubber, BR), and the copolymer of butadiene and isoprene (butadiene-isoprene rubber, BIR)—is coordination (co)polymerization in the presence of Ziegler–Natta catalytic systems, whose key components are catalysts consisting of complexes of transition and rare-earth metals, as well as lithium [1,2]. A distinctive feature of coordination (co)polymerization is its specific chain propagation mechanism. According to current understanding, this mechanism (the Cossee–Arlman mechanism) involves the coordination of the monomer at the transition metal atom and the migratory insertion stage of the monomer molecule into the (co)polymer chain [3]. Such a mechanism leads to the formation of stereoregular (co)polymer chains (chains that consist almost entirely of monomer units of the same configuration) [4]. The physical properties of the (co)polymers depend on the configuration of the monomer units [4]. For example, cis-1,4-polybutadiene is an amorphous polymer with a glass transition temperature of −100 °C, whereas syndiotactic 1,2-polybutadiene is a crystalline polymer with a melting temperature of 80–140 °C [4]. The configuration of the monomer units depends on the conditions of coordination (co)polymerization, including the structure of the catalysts and the monomer molecules. Therefore, by purposefully selecting the coordination (co)polymerization conditions, it is possible to control stereoregularity and, consequently, tailor the properties of the (co)polymers [4]. In the case of diene (co)polymerization in the presence of heterogeneous Ziegler–Natta catalytic systems, chain propagation occurs at the so-called active sites (places or surface atoms (with one or more atoms) capable of adsorbing and/or transforming reactants) [3]. Active sites of heterogeneous Ziegler–Natta catalytic systems are formed in the course of the interaction between solid catalyst components and organometallic compounds, usually organoaluminum compounds [5,6].

Thus, the mechanisms of coordination (co)polymerization of dienes and the physical properties of the obtained (co)polymers (IR, BR, and BIR) are determined by the patterns of formation and functioning of the active sites of Ziegler–Natta catalytic systems. These phenomena can be described by kinetic models, which are essential for optimizing the synthesis technology of IR, BR, and BIR—specifically, for selecting synthesis conditions that yield the desired molecular characteristics and physical properties at rates suitable for industrial production.

Difficulties arise in constructing such a model if it is required to take into account the influence of external non-chemical factors on the (co)polymerization process, for example, the effect of turbulence in the reaction system. This influence is currently poorly studied and can be described using a CFD model (CFD—computational fluid dynamics). This work is devoted to the analysis of turbulence in the reaction system as a factor controlling the dispersity and activity of heterogeneous Ziegler–Natta catalytic systems for polydiene synthesis using kinetic and CFD modeling. The models developed by these methods rely on a priori information about the patterns of the modeled object. Therefore, in Section 1.1, we present an overview of the main patterns of coordination polymerization of dienes in the presence of Ziegler–Natta catalytic systems. In Section 1.2, the aim and tasks of this work are formulated.

1.1. Modern State of Research in the Field of Coordination (Co)Polymerization of Dienes in the Presence of Ziegler–Natta Catalytic Systems

1.1.1. Ziegler–Natta Catalytic Systems and the Stereoregulating Ability of Their Active Sites in Coordination (Co)Polymerization of Dienes

Titanium catalytic systems were the first to be used for coordination (co)polymerization of dienes [4,7]. These systems, which can generally be represented as TiCl₄ + AlR₃ (where R is an alkyl, halogen atom, or hydrogen), continue to play a significant role in the (co)polymerization of dienes [4]. At the same time, the qualitative composition and structure of Ziegler–Natta catalytic systems, which determine the qualitative composition and structure of the active sites, can vary widely. For example, the catalytic properties of supported catalytic systems TiCl₄/MgCl₂ + Al(i-Bu)₃ [8], complexes of Ti(II) and Ti(IV) with bidentate dialkylphosphine ligands [4], neodymium complexes (e.g., neodymium trisphosphates [9]), ligand-stabilized bis(alkyl) yttrium complexes (pincer-ligated organoyttrium bis(alkyl) complexes) [4,10], the molybdenum catalytic system MoO₂Cl₂ + m-cresol + tris(nonylphenyl)phosphite, and cobalt phosphine complexes have been demonstrated for the coordination (co)polymerization of dienes. In some cases, catalyst ligands can be changed directly during the (co)polymerization process, affecting the resulting (co)polymer structure [4]. For example, if a monodentate aromatic phosphine is added directly to the reaction system during coordination polymerization of butadiene in the presence of a cobalt catalytic system containing at least one phosphine ligand (a hindered aliphatic phosphine or a bidentate phosphine), the stereoregularity of the synthesized polybutadiene changes from cis-1,4 to 1,2-syndiotactic [4]. This provides the possibility to obtain (co)polymers with alternating blocks of units having the same stereoregularity [4]. An increase in the stereoregulating ability of the active sites of Ziegler–Natta catalytic systems can also be observed with increasing (co)polymerization time (due to the transition of active sites of one type into active sites of another type) [8] and with the introduction of electron donors (Lewis bases) into the reaction system [1,8]. In general, different Ziegler–Natta catalytic systems exhibit different chemoselectivity (selectivity in the rate of coordination and incorporation of 1,2-, 1,4-, 3,4-, or other dienes into the (co)polymer chain) and different stereoselectivity (selectivity in the rate of formation of trans-1,4, cis-1,4, 1,2-isotactic, 1,2-syndiotactic, or other configurations of monomer units) [11,12]. The structure of the diene, as a rule, does not affect the stereoregulating ability of Ziegler–Natta catalytic systems in the polymerization of that diene [13]. The only exception, according to the authors of [13], is the yttrium bis(alkyl) complex bearing the pyridylmethylene functionalized fluorenyl ligand, in the presence of which polyisoprene enriched in 3,4-units and polybutadiene enriched in cis-1,4 units are obtained.

1.1.2. Cocatalysts in Ziegler–Natta Catalytic Systems and Their Role and Influence on Coordination (Co)Polymerization of Dienes

As mentioned above, cocatalysts in Ziegler–Natta catalytic systems are organic compounds of non-transition metals from groups I–III, most often aluminum (for example, diethylaluminum chloride, triethylaluminum, and others) [6,14]. Cocatalyst molecules participate in the reduction and alkylation reactions of the catalyst metal atom, which leads to the formation of a bond between the catalyst metal atom and the carbon atom of the cocatalyst [14]. During chain propagation, the migratory insertion of the monomer molecule into the (co)polymer chain occurs via this bond. In supported Ziegler–Natta catalytic systems (for example, TiCl₄/MgCl₂ + Al(i-Bu)₃ [8]), cocatalyst molecules adsorb onto the surface of the catalytic system particles, increasing steric hindrance near the active sites, which further enhances their stereoselectivity [14]. The increase in the size of the alkyl groups of the cocatalyst further intensifies these steric hindrances [6] and reduces the rates of reduction and alkylation reactions of the catalyst metal atom, thereby decreasing the concentration of active sites [14], as well as lowering the molecular weight and polydispersity index of the (co)polymer [15]. Cocatalyst molecules act as chain transfer agents; therefore, with an increase in cocatalyst concentration, the molecular weight of the resulting (co)polymer decreases [15]. The structure of the cocatalyst molecule also affects the relative reactivity of the monomers and, consequently, the composition of the synthesized (co)polymers [15].

1.1.3. Multi-Site Nature of Ziegler–Natta Catalytic Systems and Methods for Determining the Structure and Number of Types of Their Active Sites

Ziegler–Natta catalytic systems are generally polynuclear, meaning they contain multiple types of active sites with diverse structures [16]. The multi-site nature of Ziegler–Natta catalytic systems results from the interaction of the components of these catalytic systems (catalysts and cocatalysts), which leads to the formation of catalytically active products (active sites) of varying structures. For heterogeneous Ziegler–Natta catalytic systems, the surface heterogeneity of catalyst particles further enhances the diversity of active site structures. For example, in previous work [16], it was shown using DFT calculations that during the formation of active sites in supported catalytic systems based on TiCl₄/MgCl₂ catalysts, adsorption of TiCl₄ molecules can occur on different crystallographic planes of the MgCl₂ support. The activity and stereoselectivity of these active sites are predicted to be higher when TiCl₄ molecules adsorb on the MgCl₂ surface in the (110) crystallographic plane compared to the (104) plane [16]. Moreover, in other work [16], it was also shown that the stereoselectivity of the active sites is influenced by the relative arrangement of TiCl₄ molecules adsorbed on the MgCl₂ surface with respect to each other. According to the relative positioning of the TiCl₄ molecules adsorbed on the MgCl₂ surface in previous work [16], isolated mononuclear active sites (where there are no other TiCl₄ molecules near one adsorbed TiCl₄ molecule), binuclear active sites (adsorbed Ti₂Cl₈ molecules), and clustered mononuclear active sites (aggregates of adsorbed TiCl₄ molecules) of various sizes were distinguished. The highest stereoselectivity predicted by DFT calculations in isoprene polymerization was exhibited by isolated mononuclear active sites [16].

Although quantum chemical calculations predict the existence of several types of active sites, the simultaneous presence of all these types in a Ziegler–Natta catalytic system is not necessarily guaranteed. The actual number of active site types present in a Ziegler–Natta catalytic system is determined by the conditions of catalyst preparation and the conditions of (co)polymerization [17]. The real number of active site types in a Ziegler–Natta catalytic system can be determined by analyzing the molecular weight distribution of (co)polymers obtained in the coordination (co)polymerization in the presence of these catalytic systems [17]. This approach relies on the fact that differences in active site structures lead to differences in their reactivity and, consequently, to variations in molecular weights of polymer chains produced by these sites [12]. This analysis is known as the solution of the inverse problem of the molecular weight distribution of (co)polymers. As a result of this solution, the kernel of the integral equation is determined

$$MWD\left(M\right)=\underset{0}{\overset{\infty }{\int }}\Psi \left(\lambda \right)Kernel\left(\lambda ,M\right)d\lambda ,$$

where $MWD\left(M\right)$ denotes the experimentally measured molecular weight distribution of the (co)polymer; M is the molecular weight of an individual (co)polymer chain; $\lambda$ is the ratio of the chain propagation limitation rate (the sum of chain termination and chain transfer rates) to the product of the molecular weight of the repeating unit and the chain propagation rate; $\Psi \left(\lambda \right)$ is the sought distribution function of the catalytic system activity by a parameter $\lambda$; and $Kernel\left(\lambda ,M\right)$ is the molecular weight distribution of chains produced by the active site with a specific value of the parameter $\lambda$ ($Kernel\left(\lambda ,M\right)$ is chosen based on the mechanism of (co)polymerization; in the case of polydiene synthesis, it is usually the Flory molecular weight distribution) [18]. The number of peaks in the computed distribution corresponds to the number of active site types, and the area under each peak corresponds to their relative kinetic activities, i.e., the ratio of the kinetic activities of active sites of each type to the total kinetic activity of the catalytic system [17,18]. In previous work [17], such an analysis established that polybutadiene, polyisoprene, and their copolymer are synthesized in the presence of the catalytic system TiCl₄ + Al(i-Bu) on active sites of four, two, and two types, respectively. A disadvantage of this method for determining the number of active site types is that it does not allow the determination of their structure [17]. The structure of active sites can be established using spectroscopic methods: NMR spectroscopy [4], infrared spectroscopy [2,8], Raman spectroscopy [8], and X-ray photoelectron spectroscopy [8]. In general, factors influencing the number of active site types include the reagents used for the synthesis of catalytic systems and the order of their addition during catalytic systems synthesis, as well as synthesis time, temperature, and method [12].

1.1.4. Influence of Diene Structure on Its Reactivity in Coordination (Co)Polymerization

Both conjugated and non-conjugated dienes participate in coordination (co)polymerization, but unlike conjugated dienes, non-conjugated dienes tend to form macrocycles during the polymerization process [4]. The most industrially significant diene monomers, from which large-tonnage rubbers are obtained, are 1,3-butadiene (also simply called butadiene or divinyl), 2-methyl-1,3-butadiene (isoprene), 2-chloro-1,3-butadiene (chloroprene), and 1,3-pentadiene (piperilene) [4]. Laboratory synthesis of (co)polymers from 3-methyl-1,3-pentadiene, 2,3-dimethyl-1,3-butadiene, 4-methyl-1,3-pentadiene, 1,3-hexadiene, 5-methyl-1,3-hexadiene, 1,3-heptadiene, and 1,3-octadiene has also been reported [4]. Dienes copolymerize with other dienes [19,20] (including functionalized ones containing (RO)₃Si-, R₂N-, RSO₂-, RSO₂NH-, and (RO)₂B-groups, where R is an alkyl or aryl group [19]), as well as with olefins [4].

The relative reactivity of dienes in their coordination copolymerization substantially affects the composition and molecular weight of the copolymer [21]. This should be taken into account when synthesizing BIR—the most industrially demanded product of copolymerization of two dienes: butadiene and isoprene. Often, butadiene is more reactive than isoprene, and its concentration decreases faster than that of isoprene during their copolymerization, which can lead to the formation of a gradient copolymer [22]. The ratio of the non-cross-chain propagation reaction (chain propagation where the monomer and the active chain end are of the same type) and the cross-chain propagation reaction (chain propagation where the monomer and the active chain end are different) for butadiene and isoprene depends on the structure of the catalytic system [2,23]. For example, during copolymerization in the presence of the catalytic system MoO₂Cl₂·TNPP:Al(OPhCH₃)(i-Bu)₂ (where TNPP is tris(nonyl phenyl) phosphate), butadiene is more likely to undergo a non-cross-chain propagation reaction, while isoprene tends to undergo a cross-chain propagation reaction [2]. In contrast, during copolymerization in the presence of an yttrium complex with an unsymmetrical pincer ligand, isoprene is more likely to participate in a non-cross-chain propagation reaction, and butadiene in a cross-chain propagation reaction [23]. The reactivity of dienes also depends on the structure of their substituents [24]. For example, in the copolymerization of butadiene with oxygen-containing dienes, the reaction rate significantly decreases [24]. It was established that oxygen-containing dienes are not incorporated into the copolymer chain [24], which is presumably related to the direct interaction of oxygen atoms with the metal atoms in the active sites, thereby deactivating these sites.

1.1.5. Methods for Studying the Kinetics of Coordination (Co)Polymerization of Dienes and the Molecular Characteristics of the Resulting (Co)Polymers

One of the most widely used methods for studying the processes of coordination (co)polymerization of dienes is conducting experimental kinetic studies that allow for an in-depth analysis of the mechanisms and specific features of the reactions [20]. These studies include the analysis of the time dependencies of reactant concentrations, reaction rates, and molecular weights of (co)polymers [20]. It has been established that the types and concentrations of active sites may change over the course of (co)polymerization, especially at the initial stage [21].

Configurations of monomer units in the (co)polymers and the sequence of their alternation are determined by methods such as NMR ¹H [20,21], NMR ¹³C [24], and infrared spectroscopy [22]. This not only allows for the identification of the composition of diene (co)polymer monomer units but also reveals the stereoregular structure of these units. In particular, in the NMR ¹H spectrum, the peak at δ = 5.40 ppm corresponds to the trans-1,4-butadiene monomer unit, while the peak at δ = 5.10 ppm corresponds to the trans-1,4-isoprene unit [21]. Moreover, NMR spectroscopy enables the determination of the dyad composition of (co)polymers, based on which the average numerical length of blocks of units with a certain structure is calculated [25]. Previous research [20] demonstrated that the configurations of monomer units in (co)polymers and the order of their alternation can evolve during the course of (co)polymerization.

The molecular weight distribution of (co)polymers is determined by the gel permeation chromatography method [20,26]. Diene (co)polymers generally exhibit a broad molecular weight distribution [26], which is a consequence of the multi-site nature of the Ziegler–Natta catalytic systems used in their synthesis. Depending on the mechanism of coordination (co)polymerization and the number of types of active sites in the Ziegler–Natta catalytic system, the breadth of the molecular weight distribution of (co)polymers may vary. For example, the butadiene-isoprene copolymer synthesized in the presence of the TiCl₄/MgCl₂ + Al(i-Bu)₃ catalytic system via the classical mechanism of coordination copolymerization has a weight-average molecular weight of 88.2 × 10⁴–94.7 × 10⁴ and a polydispersity index of 2.2 to 2.7, depending on the synthesis time [25]. Meanwhile, the butadiene-isoprene copolymer synthesized in the presence of the Nd(Oi-Pr)₃ + Al(i-Bu)₂H catalytic system (where Oi-Pr is the isopropoxide group (OCH(CH₃)₂)) via the Coordinative Chain Transfer Copolymerization mechanism has a number-average molecular weight of approximately 10.0 × 10³ and a polydispersity index of 1.34 to 1.59 [22]. Such a low value of the polydispersity index is explained by the reversible chain transfer occurring on Al(i-Bu)₂H, due to which the copolymerization via the Coordinative Chain Transfer Copolymerization mechanism is close in nature to living copolymerization [22].

1.1.6. Influence of Mechanical Effects on the Kinetics of Coordination (Co)Polymerization of Dienes and the Molecular Characteristics of the Resulting (Co)Polymers

Coordination (co)polymerization of dienes, like any other (co)polymerization process, is subject to the influence of external non-chemical factors such as mechanical effects, light, differences in electric potentials, and magnetism [27,28,29]. Among these external factors, mechanical effects are of particular interest because they can initiate the breaking of chemical bonds, the formation of new radicals, and even (co)polymerization without the presence of reagents specifically sensitive to these effects. Mechanical effects are divided into two groups depending on the nature of the applied forces: shear forces or impact forces [29]. For example, mechanical plastification—continuous shear in viscous (co)polymer melts—leads to decreased molecular weights [29]. Impact forces arise as a result of wave effects (ultrasound) and impulses [29].

The coordination (co)polymerization of dienes is sensitive to mechanical effects, which are related to the heterogeneous nature of most Ziegler–Natta catalytic systems. Under mechanical influence, particles of the catalytic system may break, leading to changes in the morphology of their surfaces and, consequently, to alterations in the quantity and nature of active sites [17]. Such an effect of mechanical influences is illustrated [30] by the dependence revealed in work on the composition and glass transition temperature of the butadiene-isoprene copolymer on the turbulence conditions of the reaction system at the stage of its formation. The change in turbulence of the reaction system was achieved by introducing an additional stage in the copolymerization process, during which the reaction mixture was passed through a tubular turbulent apparatus of a diffuser–confusor design before the onset of copolymerization [30]. It was shown that with this additional stage, the resulting copolymer contains a higher proportion of cis-1,4-butadiene units and exhibits a lower glass transition temperature compared to the absence of this stage [30]. A lower glass transition temperature indicates greater mobility of the copolymer chains and a larger free volume of the copolymer [31]. The decrease in the glass transition temperature of the butadiene-isoprene copolymer was explained by an increase in the length of isoprene blocks within the chains of this copolymer [30].

1.1.7. Butadiene-Isoprene Copolymer (The Most Demanded Product of Coordination Copolymerization of Two Dienes) and Its Structure, Properties, and Applications

The classic copolymer of dienes is BIR. In terms of stereoregularity, BIR exists in two types: trans-1,4-BIR (trans-1,4-poly(butadiene-co-isoprene)) [32,33] and cis-1,4-BIR (cis-1,4-poly(butadiene-co-isoprene)) [34].

Trans-1,4-BIR has a wide range of advantages compared to other rubbers. It exhibits good resistance to fatigue failure under bending and low heat generation, making it a suitable material for use in tires subjected to intense dynamic loads [33,35]. Trans-1,4-BIR is less susceptible to aging through various mechanisms (thermal oxidative or thermomechanical) compared to natural rubber [36]. The benefits of BIR are especially well revealed when it is used as part of blends with other rubbers. Blending trans-1,4-BIR with other rubbers such as natural rubber, butadiene rubber, chloroprene rubber, or styrene-butadiene rubber significantly affects the properties of the blends (the ability of the blend to retain its shape and withstand stretching or deformation before vulcanization completion—green strength) and the properties of the resulting co-vulcanizates (tensile strength, hardness, modulus, rebound, abrasion resistance, flexural fatigue properties, and tear strength) [33,35,37]. For example, the addition of multiblock trans-1,4-BIR to a blend of natural and butadiene rubbers can increase the abrasion resistance of the blend by as much as 30% compared to the original blend [38]. In this case, BIR acts as a compatibilizer, improving the compatibility between different components of the blend [37,38]. This is due to the presence of both isoprene and butadiene units in BIR, which allows it to blend well with the corresponding components of the mixture [38].

Cis-1,4-BIR has a lower degree of crystallinity than trans-1,4-BIR and, consequently, greater elasticity than trans-1,4-BIR. Therefore, cis-1,4-BIR is recommended for producing frost-resistant vulcanizates that can operate at temperatures as low as −70 °C [34]. At these temperatures, vulcanizates made from cis-1,4-BIR exhibit good fatigue endurance and excellent resistance to crack growth and cuts [34]. Cis-1,4-BIR is available in two industrial grades developed by the Scientific Research Institute of Synthetic Rubber, named after Academician S.V. Lebedev (St. Petersburg)—BIR-15 and BIR-24, containing 15 mol% and 24 mol% isoprene units, respectively [34].

The composition of BIR (the relative content of butadiene and isoprene units) follows a first-order Markov statistical model, which indicates that the reactivity of the propagation copolymer chains depends only on the structure of their terminal unit [32]. Important molecular characteristics of BIR include its composition and the number-average block lengths of butadiene and isoprene units [32], as well as its number-average and weight-average molecular weights [33]. These characteristics influence the degree of crystallinity of BIR and its phase transition temperatures between physical states [32], as well as the above-mentioned properties of its vulcanizates and co-vulcanizates [33]. For example, with an increase in molecular weight of trans-1,4-BIR, the tensile strength, hardness, modulus, rebound, abrasion resistance, and flexural fatigue properties of trans-1,4-BIR co-vulcanizates with natural rubber increase [33]. The molecular characteristics of BIR themselves depend on the synthesis temperature and the initial concentrations of butadiene and isoprene [32]. At relatively low initial butadiene concentration, increasing the synthesis temperature of BIR leads to a decrease in the block lengths of butadiene and isoprene units in BIR chains, resulting in a copolymer with a more uniform composition distribution along the chain length [32]. As the initial concentration of butadiene increases, the block length of butadiene units in BIR chains increases, while the block length of isoprene units decreases [32].

1.2. The Aims and Objectives of This Study

The rate of (co)polymerization of dienes, the molecular characteristics, and the properties of the resulting (co)polymers depend on the composition of the Ziegler–Natta catalytic system used for the (co)polymerization, the concentration, and the number of types of its active sites, which, in the case of heterogeneous catalytic systems, depend on the particle size of the catalyst components [39]. Previous studies [17,30,39,40,41,42,43] have described a method of influencing the regularities of diene (co)polymerization in the presence of Ziegler–Natta catalytic systems by breaking catalyst particles through turbulence within the reaction system. This effect is achieved by passing the reaction mixture through a tubular turbulent apparatus of diffuser–confuser design during the formation stage of the reaction system. This method allows an increase in the activity of Ziegler–Natta catalytic systems and targeted control of the molecular characteristics of the resulting (co)polymers (IR, BR, and BIR) [39,40,41,42,43]. Additionally, this method is chemically clean and can be implemented within existing synthesis technologies for IR, BR, and BIR without significant modification or cost increase. Thus, it is relevant to analyze the turbulence of the reaction system as a factor for controlling the dispersity and activity of heterogeneous Ziegler–Natta catalytic systems for polydiene synthesis.

To study the effect of turbulence in the reaction system on the particles of Ziegler–Natta catalytic systems, it is advisable to select a simple, well-investigated heterogeneous catalytic system, such as TiCl₄ + Al(i-Bu)₃. (Co)polymers such as BR, IR, and BIR are well-suited as subjects for such studies, as all are industrially demanded rubbers. The experimental investigation of the effect of turbulence in the reaction system on the particles of the TiCl₄ + Al(i-Bu)₃ catalytic system and the influence of this effect on the kinetics of synthesis and molecular characteristics of BR, IR, and BIR has been extensively studied in many works [17,30,39,40,41,42,43]. The accumulated body of experimental data enables a comprehensive interpretation and theoretical description not only of the physicochemical process of forming new active sites during the impact of turbulence on TiCl₄ + Al(i-Bu)₃ particles (using a CFD model) but also of the kinetics of the (co)polymer synthesis process itself (using a kinetic model). The most comprehensive theoretical description can be constructed by considering the most complex research object among those listed in terms of its kinetics—the synthesis of BIR (butadiene-isoprene copolymerization). In previous work [39], the effect of turbulence in the reaction system on the particles of the TiCl₄ + Al(i-Bu)₃ catalytic system was experimentally studied using the synthesis of BIR grades BIR-15 and BIR-24 as examples. These data can be used for the experimental verification of the constructed theoretical description; therefore, the direct research objects of the present work are the synthesis processes of BIR-15 and BIR-24.

Based on the foregoing, the purpose of the study was to establish the fundamental patterns of the influence of breakage particles of the TiCl₄ + Al(i-Bu)₃ catalytic system in a tubular turbulent apparatus of diffuser–confuser design on the formation of types and concentrations of active sites, the kinetics, and the molecular weight characteristics of the butadiene-isoprene copolymer (BIR-15 and BIR-24). Particular attention was paid to the quantitative assessment of the impact of the reaction system’s turbulence conditions (the average and maximum turbulent kinetic energy in the tubular turbulent apparatus and the residence time of catalytic system particles within it) on the dispersity of catalyst particles, their activity, and the molecular weights of the copolymer.

To achieve the research aim, the following tasks were set:

1.

Development of a kinetic model of the butadiene-isoprene copolymerization, taking into account the multi-site nature of the TiCl₄ + Al(i-Bu)₃ catalytic system and the temporal dynamics of changes in concentrations of its active sites of different types. Verification of the kinetic model based on experimental data of the butadiene-isoprene copolymerization kinetics.

2.

Development of a CFD model for the breakage of TiCl₄ + Al(i-Bu)₃ particles in a tubular turbulent apparatus of a diffuser–confuser design, which allows the description of changes in the distribution of catalytic system particles by equivalent radius as a result of their breakage. Verification of the CFD model based on experimental data on the distributions of TiCl₄ + Al(i-Bu)₃ catalytic system particles by equivalent radius before and after breakage.

3.

Quantitative description of the relationship between the turbulence conditions of the reaction system, sizes of TiCl₄ + Al(i-Bu)₃ catalytic system particles after turbulent exposure, concentration of active sites, kinetics of butadiene-isoprene copolymerization, and molecular weight characteristics of the copolymer. Development of an integrated model of the butadiene-isoprene copolymerization that combines CFD model calculations and kinetic model calculations using auxiliary equations that link the output parameters of the CFD model with the input parameters of the kinetic model.

4.

Conducting computational experiments. Accumulating data on the relationship between reaction system turbulence conditions, particle sizes TiCl₄ + Al(i-Bu)₃ of catalytic system after turbulent exposure, concentration of active sites, copolymerization kinetics, and copolymer molecular characteristics as predicted by the integrated model under various geometric parameters of the tubular turbulent apparatus and linear feed rates of the reaction system.

5.

Analyzing the influence of turbulence conditions of the reaction system—average and maximum turbulent kinetic energy, and average residence time of TiCl₄ + Al(i-Bu)₃ catalyst particles in the tubular turbulent apparatus—on their dispersity, the copolymerization rate, and the molecular weight characteristics of the copolymer.

6.

Formulating scientifically justified recommendations for controlling reaction system turbulence conditions to purposefully synthesize copolymers with desired molecular weight characteristics in the presence of heterogeneous Ziegler–Natta catalytic systems.

2. Materials and Methods

2.1. Experimental Data Used for the Development and Verification of the CFD Model of Breakage of TiCl₄ + Al(i-Bu)₃ Catalytic System Particles and the Kinetic Model of Butadiene-Isoprene Copolymerization in the Presence of TiCl₄ + Al(i-Bu)₃ Catalytic System

Experimental results from the study were used to develop the CFD model of breakage of TiCl₄ + Al(i-Bu)₃ catalytic system particles and the kinetic model of butadiene-isoprene copolymerization in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system.

The CFD model was verified based on particle size distributions of the TiCl₄ + Al(i-Bu)₃ catalytic system, characterized by equivalent radius, before and after breakage, and determined by the sedimentation method [39].

The kinetic model was verified using experimental time-dependent data of copolymer yield, relative activities of different active site types, and the number-average and weight-average molecular weights of the copolymer [39]. In a previous study [39], these experimental data were obtained for copolymerization carried out using two methods.

In Method 1, the TiCl₄ + Al(i-Bu)₃ catalytic system was prepared separately and aged at 0 °C for 30 min, after which it was introduced into a 500 cm³ flask containing a solution of the monomer mixture in toluene. A magnetic stirrer was used for constant mixing of the reaction system during copolymerization. Toluene served as the solvent in the preparation of the catalytic system and the butadiene-isoprene copolymerization. Copolymerization was conducted at the same initial total monomer concentration, while the molar ratio of the monomers was varied in different experiments.

In Method 2, the TiCl₄ + Al(i-Bu) ₃ catalytic system was prepared separately and aged at 0 °C for 30 min (toluene served as the solvent in the preparation of the catalytic system). Afterwards, the suspension of catalyst particles TiCl₄ + Al(i-Bu)₃ in toluene, together with the solution of the monomer mixture in toluene, was subjected to turbulent mixing. For this purpose, the suspension and solution were simultaneously fed into a tubular turbulent apparatus of diffuser–confusor design (the schematic of the setup is shown in Figure 1a) at a velocity of v = 0.9 m/s. After exiting the tubular turbulent apparatus, the reaction system was transferred to a 500 cm³ flask, where copolymerization subsequently took place. A magnetic stirrer was used for constant mixing of the reaction system during polymerization.

The initial conditions for copolymerization, carried out by both method 1 and method 2, were as follows: total monomer concentration [M]₀ = 1.5 mol/L; catalytic system preparation conditions: [TiCl₄]₀ = 5 mmol/L, [Al(i-Bu)₃]₀/[TiCl₄]₀ = 1.4; copolymerization temperature 25 °C.

Samples of the reaction system were taken at certain time intervals by precipitating the copolymer with methanol containing 1% ionol [39].

The copolymer yield was determined gravimetrically: the copolymer was washed with methanol and then dried at 40–45 °C in a vacuum oven until constant weight [39].

Molecular weight characteristics of copolymer (number-average molecular weight M_n, weight-average molecular weight M_w, and polydispersity index M_w/M_n) were determined by gel permeation chromatography using a Waters GPC-2000 chromatograph (4 columns packed with styrogel having pore sizes of 5.0 × 10³–1.5 × 10⁴ Å; temperature of 80 °C; solvent—toluene; elution rate of 1 mL/min), calibrated with polystyrene standards with M_w/M_n = 1.01 [39].

2.2. CFD Model of Breakage of Particles of the TiCl₄ + Al(i-Bu)₃ Catalytic System

The breakage of particles of the TiCl₄ + Al(i-Bu)₃ catalytic system in a tubular turbulent apparatus of a diffuser–confuser design was simulated using the Fluent module of the ANSYS Workbench 17.1 platform with a 2D axisymmetric formulation of the problem (the computational domain had the geometry shown in Figure 1b, with dimensions varied in computational experiments). The number of mesh cells was chosen such that further refinement no longer affected the calculation results. During the calculation, at each mesh node, the system of equations presented in

Table 1. CFD model of breakage of particles of the TiCl₄ + Al(i-Bu)₃ catalytic system in a tubular turbulent apparatus of diffuser–confuser design.

Equations	Explanation of Symbols Used in the Equations
1. Law of conservation of mass for the reaction system [44,45] $\nabla \cdot {\overrightarrow{\nu }}_m=0.$	${\overrightarrow{\nu }}_m={\displaystyle \frac{{\sum }_{k=1}^2{\alpha }_k{\rho }_k{\overrightarrow{\nu }}_k}{{\rho }_m}}$—average mass velocity of the reaction system, m/s (k = 1—reaction system; k = 2—particles of the TiCl4 + Al(i-Bu)3 catalytic system) [46]; αk—volume fraction of the k-th phase; ρk—density of the k-th phase, kg/m3; ${\overrightarrow{\nu }}_k$—velocity of the k-th phase, m/s; ${\rho }_m={\sum }_{k=1}^2{\alpha }_k{\rho }_k$—density of the reaction system, kg/m3 [46];
2. Momentum conservation equation for the reaction system [44,45] ${\displaystyle \frac{\partial }{\partial t}}({\rho }_m{\overrightarrow{v}}_m)+\nabla \cdot ({\rho }_m{\overrightarrow{v}}_m{\overrightarrow{v}}_m)=-\nabla p+$ $+\nabla \cdot \left[({\mu }_m+{\mu }_T)(\nabla {\overrightarrow{v}}_m+(\nabla {\overrightarrow{v}}_m{)}^T)\right]+$ $+{\rho }_m\overrightarrow{g}+\nabla \cdot {\displaystyle \sum _{k=1}^2}{\alpha }_k{\rho }_k({\overrightarrow{v}}_k-{\overrightarrow{v}}_m{)}^2.$	t—time, s; p—pressure of the reaction system, Pa; ${\mu }_m={\sum }_{k=1}^2{\alpha }_k{\mu }_k$—viscosity of the reaction system, Pa·s [46]; µk—viscosity of the k-th phase, Pa·s; ${\mu }_T=C_{\mu }{\rho }_m{K}^2/\epsilon$—turbulent viscosity of the reaction system, Pa·s [46,47]; The subscript T refers to quantities caused by turbulent fluctuations; The superscript T indicates matrix transpose; $C_{\mu }=0.09$—constant; K—turbulent kinetic energy, J/kg; ε—turbulent kinetic energy dissipation rate, m2/s3; $\overrightarrow{g}$—gravitational acceleration, m/s2;
3. Energy conservation equation for the reaction system [44,48] ${\displaystyle \frac{\partial }{\partial t}}{\displaystyle \sum _{k=1}^2}({\alpha }_k{\rho }_k{E}_k)+$ $+\nabla \cdot {\displaystyle \sum _{k=1}^2}({\alpha }_k{\overrightarrow{v}}_k({\rho }_k{E}_k+p))=$ $=\nabla \cdot \left({\lambda }_{eff}\nabla T\right).$	Ek—enthalpy of the k-th phase, J/kg; ${\lambda }_{eff}={\displaystyle \frac{c_{pm}{\mu }_T}{{Pr}_T}}+{\sum }_{k=1}^2{\alpha }_k{\lambda }_k$—effective thermal conductivity, W/(m2·K); λk—thermal conductivity of the k-th phase, W/(m2·K); $c_{pm}={\displaystyle \frac{{\sum }_{k=1}^2{\alpha }_k{\rho }_k{c}_{pk}}{{\rho }_m}}$—molar heat capacity of the reaction system, J/(kg·K); cpk—molar heat capacity of the k-th phase, J/(kg·K); ${\mathit{Pr}}_T = 0.85$—turbulent Prandtl number; T—temperature of the reaction system, K;
4. Equations for calculating the relative velocity of phases [44] ${\overrightarrow{v}}_2-{\overrightarrow{v}}_1={\displaystyle \frac{\left({\rho }_2-{\rho }_m\right)d_2^2}{18{\mu }_1{f}_{drag}}}\overrightarrow{a}-$ $-{\displaystyle \frac{{\eta }_T}{{Pr}_T}}\left({\displaystyle \frac{\nabla {\alpha }_1}{{\alpha }_1}}-{\displaystyle \frac{\nabla {\alpha }_2}{{\alpha }_2}}\right)$, $f_{drag}=1+0.15{\mathit{Re}}^{0.687}at \mathit{Re}<1000$, $f_{drag}=0.0183\mathit{Re} at \mathit{Re}>1000$, $\overrightarrow{a}=\overrightarrow{g}-({\overrightarrow{v}}_m\cdot \nabla ){\overrightarrow{v}}_m-{\displaystyle \frac{\partial {\overrightarrow{v}}_m}{\partial t}}$, ${\eta }_T=C_{\mu }{\displaystyle \frac{K^2}{\epsilon }}\left({\displaystyle \frac{{\gamma }_{\gamma }}{1+{\gamma }_{\gamma }}}\right){\left(1+C_{\beta }{\zeta }_{\gamma }^2\right)}^{-0.5}$, $C_{\beta }=1.8-1.35{\displaystyle \frac{\left({\overrightarrow{v}}_2-{\overrightarrow{v}}_1\right)\cdot {\overrightarrow{v}}_2}{\left|{\overrightarrow{v}}_2-{\overrightarrow{v}}_1\right|\cdot \left|{\overrightarrow{v}}_2\right|}},$, ${\zeta }_{\gamma }={\displaystyle \frac{\left|{\overrightarrow{v}}_2-{\overrightarrow{v}}_1\right|}{\sqrt{\frac{2K}{3}}}}$.	d2—diameter of dispersed phase particles (catalytic system particles), m; fdrag—drag function; Re = ρmvmd2/μm—Reynolds number; $\overrightarrow{a}$—acceleration of dispersed phase particles, m/s2; ${\eta }_T$—turbulent diffusion coefficient, m2/s; γγ—parameter reflecting the inertia of the catalytic system particles;
5. Equation for calculating the change in volume fraction of the dispersed phase [44,49] ${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_2{\rho }_2\right)+\nabla \cdot \left({\alpha }_2{\rho }_2{\overrightarrow{v}}_2\right)=0.$
6. K-ε turbulence model equations (these equations are necessary to close the system of equations, specifically for calculating µT according to the formula ${\mu }_T=C_{\mu }{\rho }_m{K}^2/\epsilon$) [44,46,47,50] ${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_{m}K\right)+\nabla \cdot ({\rho }_m{\overrightarrow{v}}_{m}K)=$ $=\nabla \cdot \left[\left({\mu }_m+{\displaystyle \frac{{\mu }_T}{{\sigma }_K}}\right)\nabla K\right]+$ $+G_{K,m}+{\Pi }_{K_m}-{\rho }_m\epsilon ,$ ${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_m\epsilon \right)+\nabla \cdot ({\rho }_m{\overrightarrow{v}}_m\epsilon )=$ $=\nabla \cdot \left[\left({\mu }_m+{\displaystyle \frac{{\mu }_T}{{\sigma }_{\epsilon }}}\right)\nabla \epsilon \right]+$ $+C_{\epsilon 1}{\displaystyle \frac{\epsilon }{K}}{G}_{K,m}-C_{\epsilon 2}{\rho }_m{\displaystyle \frac{{\epsilon }^2}{K}}+{\Pi }_{{\epsilon }_m}.$	${\Pi }_{K_m}$(W/m3) and ${\Pi }_{{\epsilon }_m}$(kg/(m·s4))—source terms caused by turbulent interaction between the dispersed phase and the dispersion medium; $G_{K,m}={\mu }_T\left(\nabla {\overrightarrow{v}}_m+{\left(\nabla {\overrightarrow{v}}_m\right)}^T\right){\nabla }^{-1}{{\overrightarrow{v}}_m}^{-1}$—generation of turbulent kinetic energy, W/m3 [46]; ${\sigma }_K=1$ and ${\sigma }_{\epsilon }=1.3$—turbulent Prandtl numbers for K and ε, respectively [46,47]; $C_{\epsilon 1}=1.44$ and $C_{\epsilon 2}=1.92$—standard constants of the K-ε turbulence model [46,47];
7. Population balance model equations (these equations are necessary for calculating the concentrations N of the catalytic system particles with volumes V) [49,50,51,52] ${\displaystyle \frac{\partial N}{\partial t}}+\nabla \cdot \left({\overrightarrow{v}}_{m}N\right)=$ $={\int }_V^{\infty }ch\cdot g\left(V\right)\beta \left(V\right|V^{\prime })Nd{V}^{\prime }-g\left(V\right)N,$ $g\left(V\right)=C_1{\displaystyle \frac{{\epsilon }^{1/3}}{\left(1+{\alpha }_2\right)V^{2/9}}}{e}^{-{\displaystyle \frac{C_2{\left(1+{\alpha }_2\right)}^2}{{\rho }_1{\epsilon }^{2/3}{V}^{5/9}}}},$ $\beta (V|V^{\prime })={\displaystyle \frac{F_S}{2V^{\prime }}}+$ $+{\displaystyle \frac{1-F_S/2}{2V^{\prime }}}\left\{24{\left({\displaystyle \frac{V}{V^{\prime }}}\right)}^2-24\left({\displaystyle \frac{V}{V^{\prime }}}\right)+6\right\}$	N—concentration of catalytic system particles, m−3; ch—number of particles formed by breakage from one original particle (assumed equal to 2); g(V)—frequency of breakage of catalytic system particles; ${\rho }_1$—density of the dispersion medium; C1 and C2 (kg/s2)—constants; $\beta \left(V\right|V^{\prime })$—probability density function of breakage from a particle of volume V’ to a particle of volume V, m−3; FS—particle shape factor.

was numerically solved using built-in algorithms [44] of the Fluent module.

2.3. Kinetic Model of Butadiene-Isoprene Copolymerization in the Presence of the TiCl₄ + Al(i-Bu)₃ Catalytic System

The kinetic model equations for the butadiene-isoprene copolymerization were written using the method of generating functions. A detailed derivation of the kinetic model equations is presented in Supporting Information S1 (SI1). The values of the reaction rate constants used in the kinetic model calculations were determined as a result of solving the inverse kinetic problem. The solution of the inverse kinetic problem was divided into stages (solution decomposition). The rate constants at each stage were obtained using a semi-analytical method. Within this method, analytical solutions of the differential equations of the kinetic model were first obtained. Then, the parameters of the analytical equations were determined by minimizing the discrepancy between calculated and experimental data, using an optimization algorithm implemented in the FindMinimum operator in the Mathematica 12.0 computer algebra system. A detailed algorithm for solving the inverse kinetic problem is provided in SI2. Since this algorithm is suitable for solving the inverse kinetic problem of any multi-site coordination copolymerization, it represents a new significant result of this study. Therefore, a brief description of this algorithm is presented in Section 3.1.

The butadiene-isoprene copolymerization was described according to the information from a previous study [39] regarding the number of types of active sites in the catalytic system TiCl₄ + Al(i-Bu)₃ operating in this copolymerization. In a previous study [39], by solving the inverse problem of the molecular weight distribution problem, the researchers determined, at most, four types of active sites function during the butadiene-isoprene copolymerization, differing in structure and, consequently, in the reaction rate constants characteristic of each. The number of active site types depends on the composition of the monomer mixture q = [M₁]/([M₁] + [M₂]), where [M₁] is the concentration of butadiene in the monomer mixture and [M₂] is the concentration of isoprene in the monomer mixture; q = 1 corresponds to the homopolymerization of butadiene, and q = 0 corresponds to the homopolymerization of isoprene (concentrations are denoted by square brackets […] throughout). It was found in the study [39] that at q = 1, all four types of active sites are functional; at q < 1, only the active sites of the 2nd and 3rd types are active, meaning that even a small amount of isoprene apparently deactivates the active sites of the 1st and 4th types.

The kinetic model construction also relied on theoretical results from previous studies [42,43]. One study [43] developed a kinetic model for the homopolymerization of butadiene in the presence of the catalytic system TiCl₄ + Al(i-Bu)₃. Another study [42] developed a kinetic model for the homopolymerization of isoprene under the same catalytic system TiCl₄ + Al(i-Bu)₃. The classical concept regarding the reactivity of compounds in homo- and copolymerization is the terminal unit model [53], according to which the rate constants of reactions involving the active polymer chains are determined by the type of the terminal unit of the active polymer chain. Therefore, rate constants for chain propagation in butadiene and isoprene homopolymerization, established in previous studies [42,43], were used here to calculate the rates of non-cross propagation reactions in butadiene-isoprene copolymerization. The rate constants of cross-propagation reactions were calculated using the copolymerization constants $r_1=k_{px11}/k_{px12},$ $r_2=k_{px22}/k_{px21}$. In another study [39], the calculation of copolymerization constants using the Fineman–Ross method [54] showed that r₁ = r₂ = 1.

3. Results

3.1. Heuristic Solution of the Inverse Kinetic Problem of Butadiene-Isoprene Copolymerization in the Presence of TiCl₄ + Al(i-Bu)₃ Catalytic System

The equations of the butadiene-isoprene copolymerization kinetic model were formulated based on the mass action law and classical kinetic schemes of coordination copolymerization [39], which include the following reactions.

1.

Chain propagation:

$$R_{\mathit{xy}}(n,m) +{ M}_1\stackrel{k_{\mathit{pxy}1}}{\to }{R}_{x1}(n+1,m),$$

(1)

$$R_{\mathit{xy}}(n,m) +M_2\stackrel{k_{\mathit{pxy}2}}{\to }{R}_{x2}(n,m+1).$$

(2)

2.

Chain transfer to monomer:

$$R_{\mathit{xy}}(n,m) +{ M}_1\stackrel{k_{\mathit{Mxy}1}}{\to }{P(n,m) +R}_{x1}(1,0),$$

(3)

$$R_{\mathit{xy}}(n,m) +M_2\stackrel{k_{\mathit{Mxy}2}}{\to }{P(n,m) +R}_{x2}(0,1).$$

(4)

3.

Chain transfer to cocatalyst:

$$R_{\mathit{xy}}(n,m)+C\stackrel{k_{\mathit{Cxy}}}{\to }{P(n,m) +R}_x(0,0).$$

(5)

4.

Deactivation of active sites:

$$R_{\mathit{xy}}(n,m)\stackrel{k_{\mathit{txy}}}{\to }P(n,m).$$

(6)

5.

Interconversion of active sites of different types:

$$R_{\mathit{xy}}(n,m)\stackrel{k_{c\mathit{xwy}}}{\to }{R}_{\mathrm{wy}}(n,m),$$

(7)

where R is the active copolymer chains (n and m are the numbers of butadiene and isoprene units in the copolymer chains, respectively), M₁ and M₂ are the butadiene and isoprene molecules, respectively, P is the inactive copolymer chains, C is the cocatalyst molecule (Al(i-Bu)₃), k is the rate constant of the corresponding reaction, w and s = 1, 2, 3, 4 represent the type of active site at the active chain end; y = 1, 2 represents indices reflecting the type of the chain terminal unit or the type of monomer molecule involved in the reaction: 1 corresponds to a butadiene terminal unit and 2 corresponds to an isoprene terminal unit. The equations of this kinetic model and their derivation are presented in SI1.

The equations of the kinetic model, written with respect to the concentrations of low-molecular-weight (${ M}_1, M_2, C$) and high-molecular-weight ($R_{\mathit{xy}}, P$) components of the reaction system, were transformed using the method of generating functions. As a result, a kinetic model was obtained, formulated in terms of the concentrations of the low-molecular-weight components of the reaction system (${ M}_1, M_2, C$) and the molecular weight distribution moments of the high-molecular-weight components of the reaction system:

$${\mu }_{x100}={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\left[R_{x1}\left(n,m\right)\right],{\mu }_{x110}= {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}n\left[R_{x1}\left(n,m\right)\right],$$

$${\mu }_{x101}={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}m\left[R_{x1}\left(n,m\right)\right],{\mu }_{x120}={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}{n}^2\left[R_{x1}\left(n,m\right)\right],$$

$${\mu }_{x102}={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}{m}^2\left[R_{x1}\left(n,m\right)\right],{\mu }_{x111}={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}nm\left[R_{x1}\left(n,m\right)\right],$$

$${\mu }_{x200}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}\left[R_{x2}\left(n,m\right)\right],{\mu }_{x210}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}n\left[R_{x2}\left(n,m\right)\right],$$

$${\mu }_{x201}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}m\left[R_{x2}\left(n,m\right)\right],{\mu }_{x220}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}{n}^2\left[R_{x2}\left(n,m\right)\right],$$

$${\mu }_{x202}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}{m}^2\left[R_{x2}\left(n,m\right)\right],{\mu }_{x211}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=1}^{\infty }}nm\left[R_{x2}\left(n,m\right)\right],$$

$${\lambda }_{00}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}\left[P\left(n,m\right)\right],{\lambda }_{10}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}n\left[P\left(n,m\right)\right],{\lambda }_{01}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}m\left[P\left(n,m\right)\right],$$

$${\lambda }_{20}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}{n}^2\left[P\left(n,m\right)\right], {\lambda }_{02}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}{m}^2\left[P\left(n,m\right)\right],{\lambda }_{11}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{m=0}^{\infty }}nm\left[P\left(n,m\right)\right].$$

The kinetic model equations presented in SI1 allow for the calculation of the time dependencies of the relative activities of active sites of each type S_x, the copolymer yield U, the number-average molecular weight M_n, and the weight-average molecular weight M_w of the copolymer, respectively:

$$S_x={\displaystyle \frac{{\sum }_{z=1}^2\sum _{y=1}^2{k}_{pxyz}{\mu }_{xy}\left[M_z\right] }{{\sum }_{x=1}^4{\sum }_{z=1}^2{\sum }_{y=1}^2{k}_{p2yz}{\mu }_{2y}\left[M_z\right] }},$$

$$U={\displaystyle \frac{[M_1{]}_0+[M_2{]}_0-\left[M_1\right]-\left[M_2\right]}{[M_1{]}_0+[M_2{]}_0}},$$

$$M_n={\displaystyle \frac{m_1\left({\sum }_{x=1}^4{\mu }_{x110}+{\sum }_{x=1}^4{\mu }_{x210}+{\lambda }_{10}\right)+m_2\left({\sum }_{x=1}^4{\mu }_{x101}+{\sum }_{x=1}^4{\mu }_{x201}+{\lambda }_{01}\right)}{\left({\sum }_{x=1}^4{\mu }_{x100}+{\sum }_{x=1}^4{\mu }_{x200}+{\lambda }_{00}\right)}},$$

$$\begin{array}{cc}{M}_w=& {\displaystyle \frac{m_1^2\left({\sum }_{x=1}^4{\mu }_{x120}+{\sum }_{x=1}^4{\mu }_{x220}+{\lambda }_{20}\right)+m_2^2\left({\sum }_{x=1}^4{\mu }_{x102}+{\sum }_{x=1}^4{\mu }_{x202}+{\lambda }_{02}\right)}{m_1\left({\sum }_{x=1}^4{\mu }_{x110}+{\sum }_{x=1}^4{\mu }_{x210}+{\lambda }_{10}\right)+m_2\left({\sum }_{x=1}^4{\mu }_{x101}+{\sum }_{x=1}^4{\mu }_{x201}+{\lambda }_{01}\right)}}\\ & +{\displaystyle \frac{2m_1{m}_2\left({\sum }_{x=1}^4{\mu }_{x111}+{\sum }_{x=1}^4{\mu }_{x211}+{\lambda }_{11}\right)}{m_1\left({\sum }_{x=1}^4{\mu }_{x110}+{\sum }_{x=1}^4{\mu }_{x210}+{\lambda }_{10}\right)+m_2\left({\sum }_{x=1}^4{\mu }_{x101}+{\sum }_{x=1}^4{\mu }_{x201}+{\lambda }_{01}\right)}},\end{array}$$

where m₁ and m₂ are the molecular weights of butadiene and isoprene, respectively.

The kinetic model of butadiene-isoprene copolymerization presented in this work has 36 unknown reaction rate constants (6 rate constants for chain propagation reactions, 14 rate constants for transitions between active sites, 6 rate constants for the deactivation of active sites, and 10 rate constants for chain transfer reactions). Simultaneous determination for all these unknown values by an optimization algorithm minimizing the discrepancy between calculated and experimental data is not the best strategy for solving the inverse kinetic problem for the following reasons:

1.

Large computational volume, since the size of the solution search space is proportional to d^X, where d is a certain constant (the conditional range within which parameter values are sought) and X is the number of unknowns. Large computational volume leads to significant calculation time.

2.

High solution uncertainty. Due to the incomplete nature of experimental data regarding the process, different sets of rate constant values can enable the kinetic model to satisfactorily describe the experimental data. This is particularly observed because the concentrations of intermediate unstable compounds often cannot be measured. A classic example is a chemical process at equilibrium. If time-dependent concentrations during the system’s transition to equilibrium are not experimentally measured, then only the equilibrium constant—the ratio of the forward and reverse reaction rate constants—can be determined precisely, but not their individual values. In other words, an infinite number of individual forward and reverse reaction rate constant pairs can equally well describe the chemical process at equilibrium. Unfortunately, the number of measurable process regularities usually grows more slowly than the number of unknown rate constants in the process as its mechanism complexity increases.

Therefore, heuristic methods become advantageous for solving inverse kinetic problems with many unknowns. One such method is the decomposition of the solution. Decomposition of the inverse kinetic problem solution refers to dividing it into separate stages, each of which determines a limited number of unknowns based on a limited set of experimental data. The methodological basis for the decomposition is the differing parametric sensitivity of the kinetic model’s output variables (in this case, copolymer yield U, relative activities of active sites S_x, number-average molecular weight M_n, and weight-average molecular weight M_w of the copolymer) to changes in various rate constants. However, this parametric sensitivity itself depends on the values of rate constants, which are initially unknown. Thus, the decomposition method for solving the inverse kinetic problem is heuristic since it relies on the researcher’s experience to a priori estimate the parametric sensitivity of the kinetic model to certain rate constants.

The classical method for solving the inverse kinetic problem involves the simultaneous determination of all reaction rate constants of the process through numerical minimization of the discrepancy between calculated and experimental data. Based on our own research experience [40,41,42,43], we divided the solution of the inverse kinetic problem into several stages. A detailed justification of all equations of this algorithm is given in SI2. A brief outline of the sequence of stages and the logic of this algorithm is presented in

Table 2. Algorithm for solving the inverse kinetic problem.

Stage Number	Rate Constants Determined at This Stage	Experimental Data on the Basis of Which the Rate Constants Are Determined	Equations Relating the Values of the Rate Constants to the Experimental Data
1	$k_{\mathit{pxy}1},$$k_{\mathit{pxy}2}$	the initial slope of the time dependence of the copolymer’s Mn at the beginning of copolymerization	${\left.{\displaystyle \frac{{dM}_n}{dt}}\right|}_{t=0}\approx {\left.{\displaystyle \frac{{\sum }_{z=1}^2{\sum }_{y=1}^2{\sum }_{x=1}^4{m_{z}k}_{pxyz}{\mu }_{xy00}\left[M_z\right]}{\sum _{y=1}^2\sum _{x=1}^4{\mu }_{xy00}}}\right|}_{t=0}$
2	$k_{cx\mathrm{wy}}$	The time dependences of the relative activities of active sites of various types, Sx	$k_{c23}^{app}+k_{c32}^{app}={\displaystyle \frac{1}{t}}\mathit{ln}{\displaystyle \frac{\frac{1}{\frac{{\left.k_{p2}^{app}{S}_3\right|}_{t=0}}{{\left.k_{p3}^{app}{S}_2\right|}_{t=0}}+1}-\frac{1}{\frac{k_{p2}^{app}{S}_3^{eq}}{k_{p3}^{app}{S}_2^{eq}}+1}}{\frac{1}{\frac{k_{p2}^{app}{S}_3}{k_{p3}^{app}{S}_2}+1}-\frac{1}{\frac{k_{p2}^{app}{S}_3^{eq}}{k_{p3}^{app}{S}_2^{eq}}+1}}},$ ${\displaystyle \frac{{\left.S_3^{eq}\right|}_{q=0}}{{\left.S_2^{eq}\right|}_{q=0}}}={\displaystyle \frac{k_{p322}{k}_{c232}}{k_{p222}{k}_{c321}}}, {\displaystyle \frac{{\left.S_3^{eq}\right|}_{q=0.9}}{{\left.S_2^{eq}\right|}_{q=0.9}}}={\displaystyle \frac{k_{c232}\frac{0.1}{k_{p222}}+k_{c231}\frac{0.9}{k_{p211}}}{k_{c321}\frac{0.9}{k_{p311}}+k_{c322}\frac{0.1}{k_{p322}}}}.$
3	$k_{\mathit{txy}}$	$\mathrm{The} \mathrm{limiting} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{copolymer} U_{lim}$	$k_{t31}=k_{t21}, k_{t32}=k_{t22},$ $k_{t21}q\left({\displaystyle \frac{S_{2m}}{k_{p211}}}+{\displaystyle \frac{S_{3m}}{k_{p311}}}\right)+k_{t22}\left(1-q\right)\left({\displaystyle \frac{S_{2m}}{k_{p222}}}+{\displaystyle \frac{S_{3m}}{k_{p322}}}\right)={\displaystyle \frac{{\left.{\mu }_{00}\right|}_{t=0}}{\mathit{ln}\frac{1}{1-U_{lim}}}}.$
4	$k_{\mathit{Mxy}1}, k_{\mathit{Mxy}2}$	the limiting values of the number-average molecular weight Mn and weight-average molecular weight Mw of the copolymer	$M_{nx}={\displaystyle \frac{\left(m_{1}q+m_2\left(1-q\right)\right)}{k_{Mx11}\frac{q^2}{k_{px11}}+k_{Mx12}\frac{\left(1-q\right)q}{k_{px11}}+k_{Mx21}\frac{\left(1-q\right)q}{k_{px22}}+k_{Mx22}\frac{{\left(1-q\right)}^2}{k_{px22}}}},$ $M_{wx}={\displaystyle \frac{2\left(m_{1}q+m_2\left(1-q\right)\right)}{k_{Mx11}\frac{q^2}{k_{px11}}+k_{Mx12}\frac{\left(1-q\right)q}{k_{px11}}+k_{Mx21}\frac{\left(1-q\right)q}{k_{px22}}+k_{Mx22}\frac{{\left(1-q\right)}^2}{k_{px22}}}},$ $M_n={\displaystyle \frac{1}{\frac{S_2}{M_{n2}}+\frac{S_3}{M_{n3}}}}, M_w=S_2{M}_{w2}+S_3{M}_{w3}.$

. The distinguishing feature of this solution to the inverse kinetic problem lies in the novel proposal, for the first time within the class of multi-site coordination copolymerization processes, of a simple semi-analytical method for calculating the reaction rate constants. This proposed method possesses a significant advantage over the classical approach, as the latter entails lengthy computation times due to the large number of unknown reaction rate constants when applied to multi-site coordination copolymerization.

In this table, the following index notations are used: app is the apparent value of the reaction rate constant (values of reaction rate constants within the kinetic model of single-site homopolymerization or multi-site homopolymerization that allow this kinetic model to satisfactorily reproduce the experimental kinetic patterns of multi-site copolymerization); eq is the value of the quantity at chemical equilibrium; and lim is the value of the quantity, which it approaches as the copolymerization time tends to infinity, i.e., t → ∞; m is average integral values of the quantity (integration was performed over the copolymer yield).

Since the initial concentration of the cocatalyst $\left[C\right]$ was constant during the acquisition of experimental data in the previous study [39], which is used to solve the inverse kinetic problem, it is not possible to separate the contribution of chain transfer to monomers from the contribution of chain transfer to the cocatalyst in the overall rate of chain growth termination. Therefore, it was assumed that $k_{Cx2}=0.$ Comparisons of experimental and calculated time dependencies of the copolymer yield U, number-average molecular weight M_n, and weight-average molecular weight M_w of the copolymer are presented in Figure 2, Figure 3 and Figure 4 and Figures S1–S3 in SI2. The values of the rate constants found as a result of solving the inverse kinetic problem are provided in

Table 3. Values of reaction rate constants obtained from solving the inverse kinetic problem, with their uncertainty intervals.

Notation of Reaction Rate Constants	Active Sites Type Numbers x
	1	2	3	4
Chain propagation reactions 1
$k_{px11}, k_{px12}$	$6_{-4.2}^{+5.7}$	${1.63}_{-0.36}^{+0.95}\times {10}^2$	${7.68}_{-2.07}^{+0.92}\times {10}^2$	${3.23}_{-1.00}^{+1.75}\times {10}^3$
$k_{px21}, k_{px22}$	0	${7.20}_{-1.44}^{+8.64}\times {10}^2$	${4.40}_{-1.06}^{+0.18}\times {10}^3$	0
Active sites interconversion reactions 2
$k_{\mathrm{cx}11}$	–	0	0	0
$k_{\mathrm{cx}21}$	0	–	${1.77}_{-0.44}^{+3.19}\times {10}^{-2}$	$1.46\times {10}^{-3}$
$k_{\mathrm{cx}31}$	$5.53\times {10}^{-5}$	${2.63}^{+1.03}\times {10}^{-3}$	–	${4.12}_{-1.77}^{+2.64}\times {10}^{-1}$
$k_{\mathrm{cx}41}$	${1.49}_{-1.07}^{+4.47}\times {10}^{-3}$	${3.28}_{-2.13}^{+9.84}\times {10}^{-3}$	$2.17\times {10}^{-6}$	–
$k_{\mathrm{cx}12}$	–	0	0	0
$k_{\mathrm{cx}22}$	0	–	$9.62\times {10}^{-3}$	0
$k_{\mathrm{cx}32}$	0	${2.07}_{-1.32}^{+0.31}\times {10}^{-2}$	–	0
$k_{\mathrm{cx}42}$	0	0	0	–
Active sites deactivation reactions 2
$k_{tx1}$	${2.11}^{+19.00}\times {10}^{-2}$	${2.11}_{-1.46}^{+0.72}\times {10}^{-2}$	${2.11}_{-0.46}^{+2.95}\times {10}^{-2}$	${2.11}^{+12.24}\times {10}^{-2}$
$k_{tx2}$	0	${6.36}_{-4.20}^{+3.94}\times {10}^{-2}$	${6.36}_{-1.27}^{+8.90}\times {10}^{-2}$	0
Chain transfer reactions to monomers 1
$k_{Mx11}$	$2.00\times {10}^{-2}$	${1.81}_{-0.74}^{+3.98}\times {10}^{-1}$	${2.26}_{-1.33}^{+4.07}\times {10}^{-2}$	$5.00\times {10}^{-4}$
$k_{Mx12}$	0	${2.82}_{-2.59}^{+14.1}\times {10}^{-1}$	${2.05}_{-0.37}^{+2.62}\times {10}^{-1}$	0
$k_{Mx21}$	0	${1.25}_{-1.15}^{+6.25}$	${1.18}_{-0.21}^{+1.47}$	0
$k_{Mx22}$	0	${9.82}_{-3.93}^{+21.60}\times {10}^{-1}$	${7.81}_{-0.51}^{+3.59}\times {10}^{-1}$	0

¹ the units of the reaction rate constants are L/(mol·min); ² the units of the reaction rate constants are min⁻¹.

.

3.2. Assessment of the Adequacy of the Kinetic Model and the Accuracy of Its Parameter Estimation

The adequacy of the constructed kinetic model was confirmed by evaluating how accurately the kinetic model reproduces the experimental data used to solve the inverse kinetic problem. A quantitative measure of this comparison was the mean absolute percentage error (MAPE). The MAPE values for the copolymer yield δU, the number-average molecular weight of the copolymer δM_n, and the weight-average molecular weight of the copolymer δM_n are, respectively,

$$\delta U={\displaystyle \frac{1}{J_U}}{\displaystyle \sum _j}\left|{\displaystyle \frac{U_j^{calc}-U_j^{exp}}{U_j^{exp}}}\right|=11.2\%,$$

$$\delta M_n={\displaystyle \frac{1}{J_{M_n}}}{\displaystyle \sum _j}\left|{\displaystyle \frac{M_{n,j}^{calc}-M_{n,j}^{exp}}{M_{n,j}^{exp}}}\right|=18.6\%,$$

$$\delta M_w={\displaystyle \frac{1}{J_{M_w}}}{\displaystyle \sum _j}\left|{\displaystyle \frac{M_{w,j}^{calc}-M_{w,j}^{exp}}{M_{w,j}^{exp}}}\right|=16.5\%,$$

where $J_U$, $J_{M_n}$, and $J_{M_w}$ are total number of the corresponding experimental points, and j is the ordinal number of the experimental point.

In a previous study [39], from which the experimental data used to solve the inverse kinetic problem were taken, no metrological processing of the results was presented, and the measurement errors were not assessed (this also precluded the use of statistical analysis methods to evaluate the adequacy of the constructed kinetic model). Therefore, the MAPE values were compared with the corresponding relative experimental errors δU^exp, δM_n^exp, and δM_w^exp, characteristic of the experimental methods by which the copolymer yield, number-average molecular weight M_n, and weight-average molecular weight M_w were determined in the study [39]. According to estimates from previous studies [55,56], δU^exp = 15%, δM_n^exp = 33.4%, and δM_w^exp = 17.7%.

Thus, the constructed kinetic model was recognized as adequate because all the conditions were met

$$\delta U < \delta U^{\mathit{exp}}, \delta M_n < \delta M_n^{\mathit{exp}}, \delta M_w < \delta M_w^{\mathit{exp}}.$$

(8)

The accuracy of predicting the relative activities of active sites δS was

$$\delta S={\displaystyle \sum _x}{\displaystyle \frac{1}{J_{Sx}}}{\displaystyle \sum _j}\left|{\displaystyle \frac{S_{x,j}^{calc}-S_{x,j}^{exp}}{S_{x,j}^{exp}}}\right|=33.4\%,$$

but this value was not used to verify the adequacy of the kinetic model since the values of S_x are determined as a result of solving the inverse molecular weight distribution problem and are largely calculated rather than being an experimental result.

The values of the reaction rate constants obtained as a result of solving the inverse kinetic problem are presented in Table 3.

For each reaction rate constant in Table 3, an uncertainty interval is indicated—the range of values of this constant within which conditions (8) are satisfied. When determining the uncertainty interval for each constant, its value was varied with a fixed step (approximately 5% of the final width of the uncertainty interval), while the values of the other reaction rate constants were kept unchanged (equal to the values found from the solution of the inverse kinetic problem). The left and right boundaries of the uncertainty intervals were sought separately by gradually deviating the values of the reaction rate constants from the value obtained by solving the inverse kinetic problem, downward and upward, respectively. The search for each boundary was stopped either when at least one of the conditions (8) was violated or when conditional limiting values of reaction rate constants were reached, beyond which further searching for the uncertainty interval boundaries was considered meaningless. For the lower boundary, this value is zero. For the upper boundary, this value is ten times the parameter value obtained from the solution of the inverse kinetic problem. If the uncertainty interval boundaries are not indicated in Table 3, they correspond to these limiting values.

3.3. Algorithm for Conducting Computational Experiments

The influence of turbulence conditions in the reaction system during its formation stage in a tubular turbulent apparatus of diffuser–confuser design on the butadiene-isoprene copolymerization rate and the molecular weights of the copolymer was evaluated using computational experiments. The calculations within these computational experiments were divided into stages, guided by the cause-and-effect relationships presented in

Table 4. Cause-and-effect relationships established as a result of computational experiments (non-italic), laws governing the cause-and-effect relationships (italic, left column), and the mathematical models that made it possible to describe these relationships (italic, right column).

A complex concept that can be quantitatively expressed by characteristics of varying detail (scalar characteristics, distribution functions, fields, etc.)	Simple scalar quantitative characteristic used in the calculation
Conditions for the formation of the reaction system	Geometric parameters of the tubular turbulent apparatus and the linear feed velocity of the reaction system into it
Conservation laws of mass, momentum, and energy of the reaction system considering turbulent fluctuations of its velocity, pressure, and temperature	CFD model including the two-parameter K-ε turbulence model with specified boundary conditions
Turbulence of the reaction system	Turbulent kinetic energy K, J/kg
Equations of solid mechanics and strength of materials	Population balance model equations for particles, based on the dependence of particle breakage frequency on the turbulent kinetic energy dissipation rate ε, m2/s3
Dispersity of particles of the heterogeneous catalytic system TiCl4 + Al(i-Bu)3	Specific surface area of these particles—particle surface area normalized to their mass Surfsp, m2/g
Various theories of heterogeneous catalysis	Semi-empirical dependence reflecting the kinetics of active sites formation according to the Langmuir monomolecular adsorption theory
Activity and kinetic heterogeneity of the TiCl4-Al(i-Bu)3 catalytic system	Concentrations of active sites of various types μxy00, mol/L.
The law of mass action, the method of statistical moments, the method of generating functions	Kinetic model of butadiene-isoprene copolymerization developed in this work
Kinetic patterns of butadiene-isoprene copolymerization	Copolymerization rate, number-average Mn and weight-average molecular weight Mw of the copolymer

. In Table 4, each subsequent row is a consequence of the previous one, and the italicized text between the rows indicates the laws describing these relationships and the mathematical models enabling their calculation.

In the computational experiments, calculations were performed under conditions where the value of, at most, one parameter differed from the experimental conditions reported in [39] (copolymerization conducted using Method 2). The values assumed for the parameters are given in

Table 5. Parameters varied in the computational experiments.

Parameter Name	Value of the Parameter
Number of sections of the tubular turbulent apparatus	1, 2, 3, 4, 5, 6, 7, 8
Feed velocity of the reaction system into the tubular turbulent apparatus v, m/s	0.3, 0.6, 0.9, 1.2, 1.5
Ratio of the section length L to the diffuser diameter dd at constant diffuser diameter	4/3, 5/3, 6/3, 7/3, 8/3
Ratio of the section length L to the diffuser diameter dd at constant section length and constant ratio of confuser diameter to diffuser diameter	12/4, 12/5, 12/6, 12/7, 12/8
Ratio of the diffuser diameter dd to the confuser diameter dc at constant diffuser diameter	8/3, 8/4, 8/5, 8/6, 8/7
Diffuser opening angle α, °	15, 30, 45, 60, 75, 90

.

The values of the constants $C_1=0.6$, $C_2=6\times 10^{-5} \mathrm{kg}/s^2$ and the particle shape factor of the catalytic system $F_f=3$, which appear in the population balance equations, were selected so that the particle size distribution of the TiCl₄ + Al(i-Bu)₃ catalytic system after breakage computed in the Fluent module of the ANSYS Workbench 17.1 platform matched the corresponding experimental distribution from the previous study [39] (Figure 5).

The density of TiCl₄ + Al(i-Bu)₃ catalytic system particles was assumed to be constant; therefore, their mass is proportional to their volume. Under this assumption, the following relationship between the specific surface area of TiCl₄ + Al(i-Bu)₃ catalytic system particles with equivalent radius r and r holds:

$$Sur{f}_{sp}={\displaystyle \frac{Surf}{{\rho }_{2}V}}={\displaystyle \frac{4\pi r^2}{\frac{4}{3}\pi r^3{\rho }_2}}={\displaystyle \frac{3}{r{\rho }_2}}.$$

(9)

According to the G.K. Boreskov rule [57] (the rule of constant specific catalytic activity), the concentrations of active sites of the TiCl₄ + Al(i-Bu)₃ catalytic system must be proportional to their specific surface area. Therefore, for the TiCl₄ + Al(i-Bu)₃ catalytic system particles distributed by equivalent radius, taking Equation (9) into account, the following proportionality must hold:

$${\mu }_{00}\sim {\int }_0^{\infty }{\displaystyle \frac{dq}{dr}}{\displaystyle \frac{1}{r}}dr.$$

From the proportionality

$${\displaystyle \frac{{\mu }_{00}}{{\mu }_{00}^{exp}}}={\displaystyle \frac{{\int }_0^{\infty }\frac{dq}{dr}\frac{1}{r}dr}{{\int }_0^{\infty }\frac{d{q}^{exp}}{dr}\frac{1}{r}dr}}$$

the equation for calculating the concentration of active sites during computational experiments was obtained

$${\mu }_{00}={\displaystyle \frac{{\int }_0^{\infty }\frac{dq}{dr}\frac{1}{r}dr}{{\int }_0^{\infty }\frac{d{q}^{exp}}{dr}\frac{1}{r}dr}}{\mu }_{00}^{exp}.$$

(10)

Here, the superscript “exp” denotes the values of the total concentration of active sites and the distribution of TiCl₄ + Al(i-Bu)₃ catalytic system particles by the equivalent radius during copolymerization conducted according to Method 2. dq/dr is the distribution of TiCl₄ + Al(i-Bu)₃ catalytic system particles by the equivalent radius obtained from calculations performed in the Fluent module of the ANSYS Workbench 17.1 platform.

The dependence of active site concentrations on the monomer mixture composition q was described using our kinetic model of active site formation [58]. In a previous study [58], active sites of the TiCl₄ + Al(i-Bu)₃ catalytic system were defined as catalytically active sites on the surface of TiCl₄ + Al(i-Bu)₃ particles containing a Ti–C bond—a bond between a titanium atom and a carbon atom originally present in the butadiene or isoprene molecule. The existence of such a bond is a prerequisite for active site formation, as it is necessary for the chain propagation to proceed via the migratory insertion mechanism [3]. Therefore, the kinetic model of active site formation was formulated based on a mechanism comprising the following two stages [58].

Adsorption of monomer molecules at adsorption sites (Stage 1, physical) [58]:

$$M_1+A \begin{array}{c}{k}_{f1}\\ \rightleftarrows \\ k_{-f1}\end{array} M_1^{*},$$

(11)

$$M_2+A \begin{array}{c}{k}_{f2}\\ \rightleftarrows \\ k_{-f2}\end{array}{ M}_2^{*}.$$

(12)

Formation of the Ti-C bond (Stage 2, chemical) [58]:

$$M_1^{*} \begin{array}{c}{k}_1\\ \rightleftarrows \\ k_{-1}\end{array}{ R}_1,$$

(13)

$$M_2^{*} \begin{array}{c}{k}_2\\ \rightleftarrows \\ k_{-2}\end{array} R_2,$$

(14)

where $M_1^{*}$ and $M_2^{*}$ are the adsorbed molecules of butadiene and isoprene, respectively.

In a previous study [58], the kinetic model of active site formation was formulated based on mechanisms (11)–(14), and the kinetic model equations were written according to the law of mass action. The kinetic model equations were solved analytically under the assumption that the active sites reach their equilibrium concentration during their formation [58]. As a result, the following equation was obtained to predict the concentration of active sites depending on the monomer mixture composition q [58]:

$${\mu }_{00}^q=(1+aq+{bq}^2){\displaystyle \frac{K_1{K}_{f1}q+K_2{K}_{f2}\left(1-q\right)}{\left(1+K_1\right)K_{f1}q+\left(1+K_2\right)K_{f2}\left(1-q\right)+\frac{1}{\left[M\right]}}}[A{]}_0,$$

(15)

where $[A{]}_0$ is the initial concentration of adsorption sites and $K_{f1}=k_{f1}/k_{-f1},$ $K_{f2}=k_{f2}/k_{-f2},$ $K_1=k_1/k_{-1},$ and $K_2=k_2/k_{-2}$ are the equilibrium constants correspond to individual stages of the active sites formation mechanism. The multiplier $(1+aq+{bq}^2)$ is a correction factor that accounts for influences on the active site formation process unrelated to the stages (11)–(14). Such factors may include, for example, the breakage of TiCl₄ + Al(i-Bu)₃ catalytic system particles caused by propagation copolymer chains or the dependence of adsorption center concentrations (potential active sites) on q. Experimentally, in a previous study [39], it was established that ${\mu }_{00}^q$ does not depend on the turbulence of the reaction system during active site formation at q = 0, but it does depend on turbulence when q ≠ 0. Therefore, in multiplier F, the first term is constant and equal to 1, while the coefficients in the terms dependent on q must vary with changes in reaction system turbulence during active site formation.

Thus, during the computational experiments, calculations were performed according to the following algorithm:

1.

Based on the CFD model of breakage of TiCl₄ + Al(i-Bu)₃ catalytic system particles (the equations of which are presented in Table 1), calculations of the particle size distribution by equivalent radius dq/dr after exiting the tubular turbulent apparatus with the diffuser–confuser design were performed using the Fluent module of the ANSYS Workbench 17.1 platform. The composition of the reaction system in the calculations corresponded to the case of butadiene homopolymerization (q = 1, the extreme right point of the dependence ${\mu }_{00}^q$). For each new turbulence condition of the reaction system arising from changes in the geometric parameters of the tubular turbulent apparatus and the feed velocity of the reaction system, a separate CFD model calculation was carried out. Each such calculation yielded its own particle size distribution dq/dr.

2.

Using Equation (10), the concentrations ${\mu }_{00}$ (at q = 1) were calculated for each obtained dq/dr distribution.

3.

For each obtained concentration value ${\mu }_{00}$ (at q = 1), new parameters a and b of the multiplier were adjusted so that the discrepancy between the value obtained at step 2 of this algorithm and the value obtained from Equation (15) at q = 1 was minimized. This minimization was performed using the optimization algorithm embedded in the FindMinimum operator of the Mathematica 12.0 computer algebra system. Since for each concentration, two parameters (a and b) were sought based on a single value ${\mu }_{00}$, an additional constraint was required to identify a and b. This condition was the minimal deviation between the new values a and b and the original values a₀ and b₀ from Equation (15). Therefore, at this stage of the algorithm, the following objective function was minimized to determine a and b:

$${\left(1-{\displaystyle \frac{{\mu }_{00}^q}{{\mu }_{00}}}\right)}^2+0.1{\left(1-{\displaystyle \frac{a}{a_0}}\right)}^2+0.1{\left(1-{\displaystyle \frac{b}{b_0}}\right)}^2.$$

4.

For each obtained pair of a and b values, Equation (15) was used to calculate μ at q = 0.76 and q = 0.85. These monomer mixture compositions correspond to the production of BIR grades BIR-24 and BIR-15, respectively, and thus, these q values were chosen for the computational experiments.

5.

For each obtained value ${\mu }_{00}^q$, the initial concentrations of active sites of all types ${\mu }_{2100}, {\mu }_{2200}, {\mu }_{3100}, {\mu }_{3200}$ were calculated. From the equation

$${\mu }_{00}^q={\mu }_{2100}+{\mu }_{2200}+{\mu }_{3100}+{\mu }_{3200}$$

and Equations (S63) and (S65) presented in SI2, the following equations for this calculation were derived:

$${\mu }_{2100}={\displaystyle \frac{S_2\frac{q}{k_{p211}}{\mu }_{00}^q}{(\frac{q}{k_{p211}}+\frac{1-q}{k_{p222}})S_2+(\frac{q}{k_{p311}}+\frac{1-q}{k_{p322}})S_3}},$$

$${\mu }_{2200}={\displaystyle \frac{S_2\frac{1-q}{k_{p222}}{\mu }_{00}^q}{(\frac{q}{k_{p211}}+\frac{1-q}{k_{p222}})S_2+(\frac{q}{k_{p311}}+\frac{1-q}{k_{p322}})S_3}},$$

$${\mu }_{3100}={\displaystyle \frac{S_3\frac{q}{k_{p311}}{\mu }_{00}^q}{(\frac{q}{k_{p211}}+\frac{1-q}{k_{p222}})S_2+(\frac{q}{k_{p311}}+\frac{1-q}{k_{p322}})S_3}},$$

$${\mu }_{3200}={\displaystyle \frac{S_3\frac{1-q}{k_{p322}}{\mu }_{00}^q}{(\frac{q}{k_{p211}}+\frac{1-q}{k_{p222}})S_2+(\frac{q}{k_{p311}}+\frac{1-q}{k_{p322}})S_3}}.$$

(16)

6.

Using the kinetic model equations of butadiene-isoprene copolymerization in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system, presented in SI1, the time dependencies of copolymer yield U, number-average molecular weight M_n, and weight-average molecular weight M_w of the copolymer were calculated for each obtained set of parameter values ${\mu }_{2100}, {\mu }_{2200}, {\mu }_{3100}, {\mu }_{3200}.$

3.4. Results of Computational Experiments

All computational experiment results are presented in SI3 (see Tables S1–S6 and Figures S4–S15 in SI3). The main text of the article includes only those results that illustrate the strongest influence of changes in turbulence conditions of the reaction system during its formation stage on the butadiene-isoprene copolymerization process (

Table 6. Influence of the feed velocity of the reaction system into the tubular turbulent apparatus (v) on the maximum (K_max) and average (K_m) turbulent kinetic energy values therein, average residence time of TiCl₄ + Al(i-Bu)₃ catalytic system particles in the tubular turbulent apparatus (τ), and concentrations of active sites of the catalytic system (μ₀₀).

v, m/s	Kmax × 101, J/kg	Km × 101, J/kg	τ, s	(1 + a + b)[A]0 × 104, mol/L	μ00 × 105, mol/L at q = 0.76	μ00 × 105, mol/L at q = 0.85
0.3	0.45	0.14	0.463	3.90	5.35	7.63
0.6	0.75	0.35	0.232	4.84	6.87	9.70
0.9	1.05	0.68	0.154	6.91	10.19	14.23
1.2	1.80	1.12	0.116	9.60	14.50	20.10
1.5	3.00	1.64	0.093	12.35	18.93	26.13

and

Table 7. Influence of the ratio of diffuser diameter to confuser diameter at constant diffuser diameter (d_d/d_c) on the maximum (K_max) and average (K_m) turbulent kinetic energy values therein, average residence time of TiCl₄ + Al(i-Bu)₃ catalytic system particles in the tubular turbulent apparatus (τ), and concentrations of active sites of the catalytic system (μ₀₀).

dd/dc	Kmax × 101, J/kg	Km × 101, J/kg	τ, s	(1 + a + b)[A]0 × 104, mol/L	μ00 × 105, mol/L at q = 0.76	μ00 × 105, mol/L at q = 0.85
8/3	3.80	1.82	0.144	12.99	19.95	27.52
8/4	1.80	1.02	0.150	8.81	13.24	18.39
8/5	1.20	0.68	0.154	6.97	10.29	14.36
8/6	0.80	0.48	0.158	6.03	8.78	12.30
8/7	0.40	0.30	0.159	5.21	7.47	10.51

and Figure 6, Figure 7, Figure 8 and Figure 9).

Based on the obtained results, it can be stated that the turbulence of the reaction system is differently sensitive to variations of different geometric parameters of the tubular turbulent apparatus and to the feed velocity of the reaction system into it (i.e., changes in the CFD model input parameters). According to Tables S1–S6 and Figures S4–S15 in SI3, the CFD model input parameters can be arranged in the following sequence, ranked by decreasing influence on the CFD model output parameters (K_max, K_m, τ, dq/dr) and on the concentrations of active sites of the catalytic system ${\mu }_{00}$ calculated based on dq/dr using Equations (10) and (15) (here, K_max is the maximum turbulent kinetic energy in the tubular turbulent apparatus, K_m is the average turbulent kinetic energy, and τ is the average residence time of catalytic system particles in the tubular turbulent apparatus):

1.

Ratio of diffuser diameter d_d to confuser diameter d_c at constant diffuser diameter

$$\mathit{Sens}\left(K_{\max }\right)=2.98, \mathit{Sens}\left(K_m\right)=2.35, \mathit{Sens}\left(\tau \right)=0.86, \left({\mu }_{00}\right) = 1.26;$$

2.

Feed velocity v of the reaction system into the tubular turbulent apparatus

$$\mathit{Sens}\left(K_{\max }\right)=1.82, \mathit{Sens}\left(K_m\right)=1.65, \mathit{Sens}\left(\tau \right)=1.80, \left({\mu }_{00}\right) = 0.99;$$

3.

Diffuser opening angle α

$$\mathit{Sens}\left(K_{\max }\right)=0.45, \mathit{Sens}\left(K_m\right)=0.45, \mathit{Sens}\left(\tau \right)=0.08, \left({\mu }_{00}\right) = 0.35;$$

4.

Ratio of section length L to diffuser diameter d_d at constant diffuser diameter

$$\mathit{Sens}\left(K_{\max }\right)=0.46, \mathit{Sens}\left(K_m\right)=0.49, \mathit{Sens}\left(\tau \right)=1.04, \left({\mu }_{00}\right) = 0.27;$$

5.

Number of sections of the tubular turbulent apparatus

$$\mathit{Sens}\left(K_{\max }\right)=0.40, \mathit{Sens}\left(K_m\right)=0.29, \mathit{Sens}\left(\tau \right)=1.00, \left({\mu }_{00}\right) = 0.24;$$

6.

Ratio of section length L to diffuser diameter d_d at constant section length and constant ratio of confuser to diffuser diameter

$$\mathit{Sens}\left(K_{\max }\right)=0.73, \mathit{Sens}\left(K_m\right)=0.80, \mathit{Sens}\left(\tau \right)=0.03, \left({\mu }_{00}\right) = 0.22.$$

Here, for each CFD model input parameter X, the sensitivity coefficients of the output parameters Y (K_max, K_m, τ, ${\mu }_{00})$ to changes in that input parameter are provided. These coefficients were determined by the formula

$$Sens\left(Y\right)=\left|{\displaystyle \frac{∆Y}{∆X}}{\displaystyle \frac{X}{Y}}\right|,$$

where X and Y represent the values of the input and output parameters of the CFD model during copolymerization according to Method 2, where $∆X$ and $∆Y$ are the differences between the maximum and minimum values of the input and output parameters, respectively, obtained in experiments following copolymerization in Method 2 (see Table 5).

From the presented sequence, it is evident that the two CFD model input parameters exerting the greatest influence on the concentrations of active sites of the TiCl₄ + Al(i-Bu)₃ catalytic system are the ratio of diffuser diameter d_d to confuser diameter d_c at constant diffuser diameter and the feed velocity of the reaction system into the tubular turbulent apparatus. These two cases exhibit high sensitivity coefficients across all CFD model output parameters and active site concentrations ${\mu }_{00}.$ Increasing the values of these two input parameters results in increased values of K_max and K_m, decreased values of τ, and increased concentrations of active sites. This indicates that despite the reduction in the exposure time of turbulence on catalytic particles, their dispersity increases (see Table 6 and Table 7 and Figure 7a and Figure 9a). In other words, the effect of increasing turbulence intensity outweighs that of decreasing exposure time to turbulence in determining the dispersity of the particle.

Other computational experiments also demonstrate that the turbulence exposure time of the catalytic system particles is an insignificant parameter. For example, varying the diffuser opening angle α and the ratio of section length L to diffuser diameter d_d at a constant section length and a constant ratio of confuser-to-diffuser diameter results in almost no change in the average residence time of particles in the tubular turbulent apparatus (Sens(τ) = 0.08 and Sens(τ) = 0.03, respectively). Meanwhile, changes in turbulent kinetic energy in these experimental series yield changes in active site concentrations (Sens(${\mu }_{00}$) = 0.35 and Sens(${\mu }_{00}$) = 0.22, respectively). Comparing the influence on butadiene-isoprene copolymerization of changing the diffuser opening angle α versus changing the ratio of section length L to diffuser diameter d_d at a constant diffuser diameter reveals practically identical values of Sens(K_max), Sens(K_m), and Sens(${\mu }_{00}$) but substantially different values of Sens(τ) = 0.08 and 1.04, respectively, further confirming the weak influence of τ on the catalytic active site concentrations of the TiCl₄ + Al(i-Bu)₃ catalytic system.

To confirm the hypothesis that τ is a “weak” factor, while K_max and K_m are “strong” influencing factors on ${\mu }_{00}$, all K_max, K_m, τ, and ${\mu }_{00}$ data from Tables S1–S6 (presented in SI3) were combined into a unified dataset, as shown in Figure 10.

The Pearson linear correlation coefficients r were calculated between the CFD model output parameters K_max, K_m, τ, and the concentration of active sites ${\mu }_{00}$ according to the equation:

$$r_{Y,{\mu }_{00}}=1-{\displaystyle \frac{{\sum }_{i=1}^n\left(Y_i-\overline{Y}\right)\left({\mu }_{00,i}-\overline{{\mu }_{00}}\right)}{\sqrt{{\sum }_{i=1}^n{\left(Y_i-\overline{Y}\right)}^2{\sum }_{i=1}^n{\left({\mu }_{00,i}-\overline{{\mu }_{00}}\right)}^2}}},$$

where $\overline{Y}={\sum }_{i=1}^n{Y}_i,$ $\overline{{\mu }_{00}}={\sum }_{i=1}^n{\mu }_{00,i}.$

The following values of the Pearson correlation coefficient r were obtained:

For q = 0.76

$$r_{K_{max},{\mu }_{00}}=0.964,{ r}_{K_m,{\mu }_{00}}=0.981,{ r}_{\tau ,{\mu }_{00}}=-0.117.$$

For q = 0.85

$$r_{K_{max},{\mu }_{00}}{=0.952, r}_{K_m,{\mu }_{00}}=0.972, { r}_{\tau ,{\mu }_{00}}=-0.105.$$

Thus, it was clearly demonstrated for the first time that there is practically no correlation between τ and ${\mu }_{00}$ and the active site concentration (r ≈ 0), while there is a strong correlation between K_max, K_m, and ${\mu }_{00}$ and the active site concentration (r ≈ 1).

The absence of correlation between τ and ${\mu }_{00}$ can be explained by several reasons. Based on the influence of the number of sections of the tubular turbulent apparatus on the butadiene-isoprene copolymerization (Figures S4 and Table S1 in SI3), it can be established that the fields of turbulent kinetic energy and its dissipation rate inside each apparatus section begin to qualitatively replicate each other starting from the second section and quantitatively become identical from the fourth section onward (Table S1 in SI3). At the same time, increasing the mean residence time of catalytic system particles in the apparatus does not contribute to an increase in the turbulent kinetic energy acting on them (Table S1 in SI3).

Analyzing the equation used in this work to calculate the particle breakage frequency g(V) of the catalytic system particles

$$g\left(V\right)=C_1{\displaystyle \frac{{\epsilon }^{1/3}}{\left(1+{\alpha }_2\right)V^{2/9}}}{e}^{-{\displaystyle \frac{C_2{\left(1+{\alpha }_2\right)}^2}{{\rho }_1{\epsilon }^{2/3}{V}^{5/9}}}},$$

it becomes clear that the frequency g(V) changes very sharply with variations in the turbulent kinetic energy dissipation rate and particle volume. For certain values of ε and V, particle breakage occurs almost instantaneously; for others, the breakage frequency is practically 0. Moreover, for the values of ε and V corresponding to the boundary between these two particle breakage regimes, the following rule applies: the higher the boundary value of ε, the lower the boundary value of V. Therefore, it is evident that the particle breakage time in the tubular turbulent apparatus is not large and is most likely much shorter than the average residence time τ (≈0.5 s), and the dq/dr distributions are determined based on the value of ε in the tubular turbulent apparatus and the corresponding boundary V value (particles with volumes smaller than this boundary do not break at a given ε level). The values of ε are proportional to K, which explains why dq/dr depends on K and is practically independent of τ.

The logic described above fits well into modern understandings of continuum mechanics and solid mechanics [44,51]. According to these concepts, turbulent kinetic energy K and its dissipation rate determine the magnitude of turbulent velocity fluctuations of the reaction system, which in turn determine the shear stresses in the reaction system. These shear stresses induce tangential stresses between the surface and internal layers of the catalytic system particles. Particle breakage occurs when these tangential stresses exceed a certain critical value, and not as a result of prolonged stress application (i.e., breakage does not occur via a fatigue mechanism).

The increase in these critical stresses with decreasing particle volume V (manifested by the increase in the boundary value of ε as the boundary value V decreases) is explained by the heterogeneity of the TiCl₄ + Al(i-Bu)₃ catalytic system particles, which is a consequence of their formation mechanism. During the reaction between TiCl₄ and Al(i-Bu)₃, needle-like TiCl₃ crystals are released, which arises from their formation mechanism. During the reaction between TiCl₄ and Al(i-Bu)₃, needle-like TiCl₃ crystals are released, which subsequently begin to agglomerate [59]. As a result, the catalytic system particles consist of stronger crystalline regions and weaker amorphous regions between them [59].

Thus, the following general pattern holds true for the results of the computational experiments. The greater the values of turbulent kinetic energy K and its dissipation rate ε (due to the design features of the tubular turbulent apparatus or the feed velocity of the reaction system into it), the more the turbulent kinetic energy K dissipates into heat through viscous friction and is spent on work of breaking the catalytic system particles.

Since the average dissipation rate of turbulent kinetic energy ε (Figure 6b and Figure 8b) is practically directly proportional to the average turbulent kinetic energy K (Figure 6a and Figure 8a), the breakage frequency of TiCl₄ + Al(i-Bu)₃ particles g(V) also increases with increasing average turbulent kinetic energy K. In other words, the greater the turbulent kinetic energy K, the stronger the turbulent velocity fluctuations of the reaction system, and thus the greater the stresses caused by viscous friction. Consequently, the dispersity of catalytic system particles TiCl₄ + Al(i-Bu)₃ increases, meaning their sizes decrease more strongly compared to their sizes before passing through the tubular turbulent apparatus (Figure 7a and Figure 9a). At the same time, the specific surface area of the particles (the total surface area of the particles normalized to their mass), Surf_sp, increases. Because of this, the active sites “preserved” within the pores of the catalytic system particles before breakage become accessible to monomer molecules (butadiene and isoprene), resulting in an increase in the concentration of active sites (Table 6 and Table 7). Consequently, the copolymerization rate also increases (characterized by the tangent of the slope angle on the time dependence of the copolymer yield).

With the increase in active site concentration (Table 6 and Table 7) and the copolymerization rate (Figure 7b and Figure 9b), a decrease in number-average (Figure 7c and Figure 9c) and weight-average (Figure 7d and Figure 9d) molecular weights of the copolymer occurs. This is because, as the copolymerization rate increases, the monomer concentration decreases more rapidly. The chain propagation rate is directly proportional to the monomer concentration (according to Equation (S102) from SI2), and the chain termination rate is proportional to the monomer concentration only if all chain propagation limitation reactions other than chain transfer to monomers are neglected (according to Equation (S104) from SI2). Otherwise, if the deactivation reactions of active sites are not neglected (and they cannot be neglected in the process under consideration), the chain propagation limitation rate decreases more slowly than the monomer concentration decreases. Therefore, at lower monomer concentrations observed at higher copolymerization rates, the number-average and weight-average molecular weights of the chains will be smaller.

Hence, with an increase in the ratio of diffuser diameter d_d to confuser diameter d_c, at a constant diffuser diameter and feed velocity of the reaction system into the tubular turbulent apparatus, a decrease in the number-average and weight-average molecular weights of the butadiene-isoprene copolymer is observed due to increased concentrations of active sites of the TiCl₄ + Al(i-Bu)₃ catalytic system. Since, as shown in a previous study [39], concentrations of active sites increase with increasing butadiene fraction q in the monomer mixture, all computational experiments exhibited a higher synthesis rate and lower molecular weights for BIR-15 (dashed lines in Figure 7b–d and Figure 9b–d) than for BIR-24 (solid lines in Figure 7b–d and Figure 9b–d).

4. Discussion

This study is primarily a theoretical study, parts of which have been verified based on experimental data. Therefore, the shortcomings of this research arise from the imperfections of the constructed kinetic model of butadiene-isoprene copolymerization and the CFD model. The imperfections of the models are largely dictated by the limited scope of the experimental data used for their development.

The kinetic model has the following limitations:

1.

The kinetic model did not allow for tracking the influence of copolymerization conditions on the copolymer composition. Technically, the kinetic model permits calculating the copolymer composition (mole fraction of butadiene units in the copolymer, Q) using the equation:

$$Q={\displaystyle \frac{{\sum }_{x=1}^4{\mu }_{x110}+{\sum }_{x=1}^4{\mu }_{x210}+{\lambda }_{10}}{{\sum }_{x=1}^4{\mu }_{x110}+{\sum }_{x=1}^4{\mu }_{x210}+{\lambda }_{10}+{\sum }_{x=1}^4{\mu }_{x101}+{\sum }_{x=1}^4{\mu }_{x201}+{\lambda }_{01}}}.$$

However, all experimental compositions of butadiene-isoprene copolymers obtained in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system were equal to the monomer mixture composition. Therefore, tracking the influence of copolymerization conditions on copolymer composition within kinetic modeling was deemed meaningless. Yet, when studying the synthesis of another polydiene in the presence of a different catalytic system, the equations of this kinetic model with reidentified reaction rate constants could allow tracking this influence.

2.

The kinetic model equations do not allow for the calculation of the isomeric composition of copolymer units and their stereoregularity. This was done intentionally to avoid complicating the kinetic model equations. If such calculations become necessary, the kinetic model equations can be modified. Each chain propagation reaction can be split into separate reactions, each leading to the formation of a monomer unit with a specific structure. The rate constants of these reactions can be identified to describe the experimental isomeric composition of the copolymer units. The fractions of triads of units in the copolymer composition can be calculated based on the isomeric composition of its units within Markov statistics.

3.

The kinetic model is formulated in a pseudo-homogeneous approximation, i.e., it does not take into account the diffusion stage of monomer molecules to the surface of catalytic system particles and inside their pores. Nevertheless, the kinetic model was able to describe the experimental data on the kinetics of butadiene-isoprene copolymerization, which is sufficient justification for the validity of using this pseudo-homogeneous approximation. In practice, this means that if monomer diffusion processes influence the copolymerization rate, then the obtained values of the reaction rate constants should be regarded as apparent values.

The CFD model has the following limitations:

1.

The verification of the CFD model was carried out on a very limited set of experimental data. Essentially, this consists of two distributions of particle size of the TiCl₄ + Al(i-Bu)₃ catalytic system by the equivalent radius dq/dr (before and after passing the reaction system through the tubular turbulent apparatus). The choice of equations for calculating the particle breakage frequency g(V) and the probability density function for particle breakage from V  $\beta \left(V\right|V^{\prime })$ depended on the shape of these distributions. After testing various equations, it was established that the available experimental distributions dq/dr are most accurately described by the equations from study [52] (these equations are shown in Table 1). However, there are many CFD models for the breakage of solid particles in liquids [51,60,61,62]. With the accumulation of more experimental data on the influence of turbulence on dq/dr, the choice of the CFD model that most adequately describes this influence may change.

2.

In the calculations, catalytic system particles are represented very simply as spheres, and the surface area considered does not take into account the surface area of their pores. This allowed us to demonstrate that the specific surface area of the particles, which determines the catalytic system activity, is inversely proportional to their equivalent radius r. This generally reproduces the experimentally observed trend (the smaller the particles, the more active they should be). However, quantitatively, taking into account active sites in the pores of catalytic system particles, the activity of these particles may be proportional to 1/rⁿ, where n ≠ 1.

3.

The cause of the destruction of solid catalytic system particles should be the mechanical stresses arising in them, caused by shear stresses in the liquid phase (reaction system). Mechanical stresses do not explicitly appear in the developed CFD model. Their influence on the particle breakage rate in the developed CFD model is effectively taken into account by the values of the constants $C_1=0.6$ and $C_2=6\times 10^{-5} \mathrm{kg}/s^2$ and the particle shape factor $C=3$ in the population balance equation. The values of C, C₁, and C₂ should themselves depend on the properties of the reaction system to avoid paradoxes. For example, with an increase in the viscosity of the reaction system, its turbulence should decrease (turbulence arises when the inertial forces of laminar flow tubes exceed the viscous friction forces between these flow tubes), but at the same time, the shear stresses in the liquid increase, which should increase the breakage rate of catalytic system particles. That is, with an increase in the viscosity of the reaction system (for example, with an increase in copolymer yield), the turbulent kinetic energy and its dissipation rate should decrease, while the particle breakage rate of the catalytic system should increase. This statement is based on experimental observation—in a previous study [39], it was shown that if, during butadiene polymerization in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system, the reaction system is passed through the tubular turbulent apparatus but not immediately, but after a certain polymerization duration, the catalytic system activity increases more strongly, and the longer the polymerization time, the stronger the increase. Unfortunately, this effect cannot be described by the developed CFD model with constant values of the constants F_S, C₁, and C₂. The values of C, C₁, and C₂ are not universal.

Despite the listed shortcomings, we believe that they have not reduced the reliability and significance of the key research results, which carry practical value for a wide range of researchers. These key results are as follows:

1.

An original method for solving the inverse kinetic problem for the butadiene-isoprene copolymerization was proposed, based on a heuristic decomposition of the solution. This method allowed the determination of the rate constants of the reactions in the considered process at a relatively low computational cost. Since each reaction rate constant was determined based on the experimental data on which it had the greatest influence, the obtained solution has a low degree of uncertainty. Equations were obtained linking the apparent values of the reaction rate constants (the values of the reaction rate constants within the framework of the kinetic model of single-site homopolymerization that allow this kinetic model to satisfactorily reproduce the experimental kinetic patterns of multi-site copolymerization) to their true values for multi-site copolymerization. The presented heuristic decomposition method for solving the inverse kinetic problem is a universal method by which reaction rate constants for other multi-site copolymerizations can be found. The preliminary description of experimental data using a single-site homopolymerization kinetic model is a convenient method for finding apparent reaction rate constants, which can then be used as initial approximations for the true reaction rate constants of multi-site copolymerization.

2.

The influence of turbulence on the distribution of catalytic system particles by equivalent radius as a result of their breakage, the activity of the catalytic system, and the kinetic patterns of multi-site copolymerization in the presence of this catalytic system for one specific research object—the butadiene-isoprene copolymerization in the presence of the TiCl₄ + Al(i-Bu)₃ catalytic system—was theoretically described. The cause-and-effect relationships describing this influence are presented in Table 4. Despite the aforementioned limitations of the kinetic and CFD models, the forecasts obtained using this theoretical framework can be considered, as it is grounded in fundamental physical and chemical laws. In subsequent studies, this theoretical description may be extended to other research objects—(co)polymerization processes in the presence of heterogeneous catalytic systems—enabling a more detailed determination of the role of reaction system turbulence as a controlling factor of the particle size distribution and activity of heterogeneous Ziegler–Natta catalytic systems. Taking into account the specifics of each research object, individual parameters and equations in this theoretical framework may accordingly be modified. The limitations identified in the developed theoretical description will aid in understanding how it should be adapted when generalized.

3.

It has been established that the influence of turbulence on the distribution of catalytic system particles by equivalent radius and on their activity is determined by two factors: the duration of turbulent exposure to the particles and the intensity of this exposure (which, in the present study, was characterized by the turbulence kinetic energy). For the research object considered, turbulence intensity proved to be a more significant factor than the exposure duration. Considering that the turbulent impact on catalytic system particles in the examined copolymerization process can be implemented without significant modifications or cost increases to the process technology, a tubular turbulent apparatus of a diffuser–confuser design is recommended to achieve this impact. Based on computational experiment results, to enhance the intensity of catalytic particle breakage and thereby increase catalytic activity, it is primarily recommended to increase the ratio of the diffuser diameter d_d to the confuser diameter d_c while maintaining a constant diffuser diameter, as well as to increase the feed velocity of the reaction system into the tubular turbulent apparatus. The stronger the catalytic system particles are dispersed, the more the molecular weights of the copolymer decrease (this effect not only follows from calculations but is also confirmed by experimental data shown in Figure 4). Consequently, to obtain a copolymer with high molecular weight, turbulent impact on catalytic system particles should be avoided.

5. Conclusions

1.

A kinetic model of the butadiene-isoprene copolymerization has been developed, taking into account the multi-site nature of the TiCl₄ + Al(i-Bu)₃ catalytic system and the time changes in concentrations of different types of active sites. For the first time, a semi-analytical method for solving the inverse kinetic problem has been developed for the class of multi-site coordination copolymerization processes. The method is based on decomposing the solution of the inverse kinetic problem into four stages: determination of the chain propagation reaction rate constants k_p (Stage I), interconversion of active sites $k_c$ (Stage II), deactivation of active sites k_t (Stage III), and chain transfer to monomer molecules k_M (Stage IV). As a result of solving the inverse kinetic problem, 32 nonzero values of the reaction rate constants for the considered copolymerization at 25 °C were identified.

2.

A CFD model of breakage of TiCl₄ + Al(i-Bu)₃ catalytic system particles in a tubular turbulent apparatus of a diffuser–confuser design at the reaction system formation stage was developed using the Fluent module of ANSYS Workbench 17.1. This CFD model involves the numerical solution of the conservation equations for the mass, momentum, and energy of the reaction system, the K-ε turbulence model equations, and population balance equations for the catalytic system particles. The developed CFD model can be applied to describe the breakage of particles of other catalytic systems, provided that new parameters of the population balance equations are identified.

3.

The integrated model of the butadiene-isoprene copolymerization has been developed, combining calculations from the CFD model and the kinetic model through auxiliary equations that link the output parameters of the CFD model with the input parameters of the kinetic model. Within these auxiliary equations, the concentrations of active sites are considered proportional to the specific surface area of the catalytic system particles, which is calculated based on the particle size distribution of the TiCl₄ + Al(i-Bu)₃ catalytic system by equivalent radius. The dependence of the concentrations of different types of active sites on the monomer mixture composition q is described using a kinetic model of active site formation via a two-stage mechanism (stage 1—adsorption of monomer molecules on adsorption sites; stage 2—formation of the Ti–C bond).

4.

Using the integrated model of butadiene and isoprene copolymerization, it was established that the greater the turbulent kinetic energy K (attributable to the design features of the tubular turbulent apparatus or the magnitude of the feed velocity of the reaction system), the higher the dispersity of the TiCl₄ + Al(i-Bu)₃ catalytic system particles, the higher the concentration of active sites of this catalytic system, the higher the copolymerization rate, and the lower the number-average and weight-average molecular weights of the copolymer.

5.

It was established for the first time that the rates of butadiene and isoprene copolymerization, and consequently the activity values of the TiCl₄-Al(i-Bu)₃ catalytic system, correlate with the average or maximum turbulent kinetic energy of the reaction system in the tubular turbulent apparatus, but do not correlate with the mean residence time of the reaction system in the tubular turbulent apparatus. In other words, this study demonstrates for the first time that to achieve maximum efficiency in increasing the activity of the TiCl₄-Al(i-Bu)₃ catalytic system through turbulence exposure of the reaction system, it is necessary to increase the level of turbulent kinetic energy in the tubular turbulent apparatus rather than its volume.

6.

Based on the results of computational experiments, it was established that to achieve more intensive breakage of catalytic system particles and increase its activity, it is primarily recommended to increase the ratio of the diffuser diameter d_d to the confuser diameter d_c while maintaining a constant diffuser diameter, along with increasing the feed velocity of the reaction system into the tubular turbulent apparatus.