Problems Of Chemical Kinetics Research Articles

Autonomous Kinetic Invariants of Linear Chemical Reactions

A method has been developed for calculating the exact autonomous kinetic invariants of multistage linear chemical reactions occurring in a gradientless isothermal reactor via any number of elementary stages and involving an arbitrary number of reagents within the framework of mass-action law. These invariants are functions that remain strictly constant throughout the reaction (i.e., are time-independent), although they depend on the nonequilibrium reagent concentrations measured in two experiments with different initial conditions (dual experiments). For the dynamic models of linear reactions, one can always find general and particular analytical solutions expressed explicitly in terms of the initial conditions. The idea of the method is based on choosing the initial conditions that allow the nonequilibrium reagent concentrations to be expressed in terms of constant kinetic parameters (stage rate and flow rate constants). An applicability criterion of the method was formulated. The number of different invariants was shown to be equal to the number of reagents. The relations obtained in this study were used to investigate the relaxation properties of multistage linear reactions occurring in closed and open isothermal systems. The invariant curves found for these reactions were compared with the concentration curves measured in two nonequilibrium experiments throughout the transition process. For these reactions, the time dependences of the invariants remained strictly constant, while the reagent concentrations continuously changed (including nonmonotonously) during the reaction. The results give new insight into the relaxation of linear chemical reactions and can be used to solve the inverse problems of chemical kinetics under the conditions of an isothermal perfect-mixing reactor.

Optimization algorithms for the conditions of complex catalytic reactions

A procedure was developed for the investigation of complex catalytic reactions using optimization algorithms. The first stage is a solution of the inverse problem of chemical kinetics involving the recovery of unknown parameters by single-objective optimization algorithms. In the second stage, based on the kinetic model, we solved the problem of determining the optimal conditions for the reaction resorting to multi-objective optimization algorithms. Parallelization models of computing for solving the multi-objective optimization problem are considered.

KINETICS QUASIINVARIANTS OF CHEMICAL REACTIONS IN OPEN SYSTEMS

The methods of nonequilibrium multi-experiments are one of the new approaches to solving inverse problems of chemical kinetics and optimization of chemical reactors. Currently, these methods are developed only for closed isothermal systems. In this paper, a generalization of the dual-experiment method and its extended version of the multi-experiment method for open systems is obtained, which allows to determine the approximate kinetic invariants (quasiinvariants) of chemical reactions in open continuous stirred tank reactor. The multi-experiment method for open systems is based on conducting two or more special nonequilibrium (unsteady) experiments under certain conditions. For nonlinear reactions of arbitrary complexity (multi-step, multi-equilibria), simple relations are obtained that allow to calculate the conditions for nonequilibrium experiments necessary for the identification of the reaction mechanism under study. The method allows to use any permissible values, except equilibrium ones, as initial values of reagent concentrations. The technique of carrying out multi-experiments and performing the necessary numerical calculations based on the multiple integration of systems of ordinary differential equations under different initial conditions is developed. The examples of using the developed method for one-stage linear and two-stage nonlinear reactions with two and three reagents are given. Found with the help of this method, the kinetic curves of the nonequilibrium quasiinvariants compared with the nonequilibrium curves of change of concentrations during the whole reaction. It is shown that quasiinvariant curves change within narrower limits than concentrations in different experiments, i.e. remain practically constant in time. The obtained results are also applicable for open nonisothermal systems.

Development and automation of algorithm for determining basis of nonlinear parameter functions of kinetic constants

Mathematical modeling of catalytic processes is necessary for the complete and accurate description, as well as for controlling the quality and physicochemical studied of catalysts. In the paper, theoretical issues of industrial catalysis are discussed. The work is devoted to theoretical graph analysis of informativity of kinetic parameters of the model of a complex chemical reaction. The aim is the development and automation of algorithm for determining basis of nonlinear parameter functions in solving inverse problems of chemical kinetics in order to define the number and form of independent combinations of rate constants of elementary stages. A program package for analysis of informativity of kinetic parameters of the mathematical model of a complex catalytic reaction is developed and described. The obtained functional relations between the kinetic parameters can be useful for experimentalists in physicochemical interpretation and analysis of mechanisms of chemical reactions. In other words, the proposed method allows independent combinations of kinetic constants to be distinguished that results in shortening the number of the model parameters and, as a consequence, enhance the accuracy of the mathematical model. The mechanism of hydrogen oxidation over a platinum catalyst is given as an example of the use of the software.

Adaptive Interpolation Algorithm Based on a kd-Tree for the Problems of Chemical Kinetics with Interval Parameters

The questions of simulating chemical processes with uncertainty in their parameters are considered. A new approach consisting in the construction of a dynamic structured grid based on a kd-tree over the space formed by the interval parameters of a problem is proposed. When the algorithm is executed, a piecewise polynomial function interpolating the dependence of a solution on certain values of interval parameters is built at every step of integrating the initial system of ordinary differential equations. The results of testing the algorithm on the problems of chemical kinetics, including the combustion processes, demonstrate its efficiency and broad range of applications.

Investigating Student Understanding of Rate Constants: When is a Constant “Constant”?

The themes discussed in this study relate to how students reason about the information encoded in rate constants, which is important for developing a deep understanding of chemical kinetics at the molecular level. This study is part of a larger project centered more generally on students’ understanding and use of mathematics in chemical kinetics. During semistructured interviews, students—36 general chemistry students, five physical chemistry students, and three chemical engineering students—worked through chemical kinetics problems and were asked to discuss equations used in chemical kinetics, yielding rich data that provided insight regarding students’ understanding of rate constants. Analysis revealed that students often conflated ideas from chemical kinetics and equilibrium, such as rate constants and equilibrium constants. Furthermore, students demonstrated varying levels of sophistication regarding the distinction and relationship between rate and rate constants. Finally, students conveyed different ideas about the mathematical nature of the rate constant quantity. These findings suggest students need more support in order to develop a more sophisticated understanding of rate constants.

Synthesis of hydrated furans: Analysis of the Prins reaction mechanism

Mechanistic analysis of hydrogenated furan synthesis by the Prins reaction was carried out using mathematical modeling. The direct and inverse chemical kinetic problems are solved for the rate constants of all elementary steps. A probable mechanism is derived from the obtained quantitative kinetic parameters and the kinetic patterns of the reaction. These include kinetic curves from the concentration changes of all species in the reaction.

Specification and simulation of material science problems in ISMA

The model types supported by a modeling and simulation environment called ISMA are presented. A fragment of the grammar of the general-purpose modeling language LISMA used in ISMA is given. A typical material science and engineering problem, namely modeling and simulation of the heat transfer within a beam, is solved using ISMA. A grammar for specifying chemical kinetics problems is briefly presented in order to demonstrate how the developed mathematical software suite can be extended to a new domain.

Interval Algorithm for the Global Search of Uncertainty Domains in Simulating the Kinetics of Complex Chemical Reactions

This study is devoted to the mathematical description of a kinetic experiment under the setting of multiple solutions to the inverse problems of chemical kinetics. A new algorithm for the global search of uncertainty domains is proposed, which represents a symbiosis of the methods of global optimization with an interval evolutionary approach. Chemical reaction models built using the kinetic parameters of these areas describe experimental data within an acceptable measurement error specified by the researcher. The use of this algorithm allows a complex multidimensional domain in the global space of kinetic parameters, whereas the local solution sets describing the experimental data with near equal accuracy are searched or the one-, two-, or three-dimensional projections of uncertainty domains are separately built in many other studies. The proposed algorithm has been tested using a reaction simulation of n-decane inhibited oxidation. Since this structure has a formal parallelism, parallel computing techniques are applied to speed up calculations.

Productive features of problem solving in chemical kinetics: more than just algorithmic manipulation of variables

Problem solving is a critical feature of highly quantitative physical science topics, such as chemical kinetics. In order to solve a problem, students must cue into relevant features, ignore irrelevant features, and choose among potential problem-solving approaches. However, what is considered appropriate or productive for problem solving is highly context-dependent. This study is part of a larger project centered on students’ integration of chemistry and mathematics knowledge and skills. The data for this study came from semi-structured interviews with 40 general chemistry students using a think-aloud protocol. Interview prompts involved students working through two chemical kinetics problems, one involving a second-order system and one involving a zero-order system. In both cases, students could solve the problem using the data provided and relevant equations, or by taking a conceptual approach and considering the relationship between quantities. Using the resource-based model of cognition as our theoretical framework, analysis focused on characterizing the productive and unproductive problem-solving routes used by students. Findings emphasize the role of using conceptual reasoning and reflecting on one's work during problem solving, which have implications for instructors as they guide students to think about chemical kinetics and to solve problems across quantitative topics in science, technology, engineering, and mathematics.

KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters

KD-Tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters

Fitness functions for evolutionary optimization of rate parameters in chemically reacting systems

Evolutionary optimization studies are mainly focused on the type of methodologies and controlling parameters of the algorithms. However, the choice of fitness functions which gives a measure on how well a dataset correlates with reference values is often not justified in a rigorous way. The work in this paper addresses this issue by investigating the performance of a set of fitness functions and their suitability according to the input data used for training the optimization algorithms. A Genetic Algorithm was applied to a chemical kinetics problem with the aim of matching the measured species concentrations through optimization of a set of Arrhenius reaction rate parameters. The challenge with this type of problem lies in the wide range of species concentrations used as the training dataset, where they can be up to fifteen orders in magnitude. This revealed large differences in optimized values arising from the various fitness function formulations, thus highlighting the need for detailed knowledge of fitness functions for a specific optimization problem. Furthermore, the quality of training dataset through error propagation and a scaling parameter were assessed bringing further insight on differences fitness functions. Such information is crucial for the appropriate selection of fitness functions for the effective optimisation of reaction rate parameters.

Mathematical Simulation of the Process of Nickel Oxide Recovery in a Tube-Type Rotary Kiln

Solid-phase reduction of nickel oxide is an important process stage in nickel production. The mathematical model of the steady-state operation of the apparatus is an important component, which allows determining the optimal operating mode of the rotary tube furnace (TRK). The automated software application ReactOp is capable of generating a mathematical model of the reactor on the basis of hydrodynamic and thermal models supplemented by the mechanism of chemical transformations, kinetic parameters, thermal and physical properties, and heat transfer parameters. The software complex is equipped with a user-friendly interface, which allows to interactively generate a runtime program based on the mathematical model. The software application ReactOp has built-in programs enabling numerical solution of differential equations and solution of inverse chemical kinetics problems, and is capable of providing statistical performance analysis and finding the optimal control solutions based on the methods of non-linear programming. The paper is dedicated to the development of a mathematical model of the steady-state TRK operation. Production of nickel from ores involves several stages of processing raw materials, wherein a corresponding intermediate product is obtained at each stage. As a result of the oxidizing roasting of the nickel flotation concentrate, a roasted product is obtained mainly comprising nickel oxide and other impurities, such as compounds of copper, cobalt, iron, sulfur, as well as rare and precious metals. This stage is followed by the process of solid-phase reduction of nickel oxide in the rotary tube furnaces aimed at obtaining a coarsened, mostly reduced product, which can be further subject to anode electric melting. The latter process allows reducing the consumption of electricity and a reducing agent. The paper further provides the results of mathematical modeling of the reduction process of oxidized copper-nickel ore concentrate in the rotary tube furnace.

Multi-objective optimization of chemical reaction conditions based on a kinetic model

The main purpose of the study is to introduce the multi-objective optimization using Pareto approximations to problems of chemical kinetics. We report the setting up and solution of the multi-objective optimization problem for conditions of a chemical reaction on the basis of a kinetic model. The study addresses the reaction of alcohols with dimethyl carbonate catalyzed by cobalt or tungsten carbonyl. The objective functions for optimization of chemical reaction conditions based on a kinetic model are presented. The NSGA-II algorithm was applied to determine the Pareto set and front for the multi-objective optimization problem applied to the reaction of alcohols with dimethyl carbonate for two catalysts, which make it possible to find the compromise values of variable parameters providing extrema of the objective functions.

Binary correspondences and the inverse problem of chemical kinetics

We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters. As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of the composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics.

Using Symbolic and Graphical Forms To Analyze Students’ Mathematical Reasoning in Chemical Kinetics

This work is part of a larger project that seeks to understand how students blend (integrate) chemistry and mathematics as they work through chemical kinetics problems. Here we focus on four students from our larger sample: two students that demonstrated more instances of blending chemistry and mathematics in their interviews (“high-frequency blenders”) and two students that did not have any instances of blending chemistry and mathematics in their interviews (“nonblenders”). In this study, we characterized the intuitive mathematical ideas the students associated with equations (symbolic forms) and graphs (graphical forms), focusing on how the students’ mathematical reasoning influenced their understanding of chemical phenomena. The findings suggest that, in order for students to integrate chemistry and mathematics in a way that supports their understanding of the chemical processes in chemical kinetics, they must be able to reason using symbolic and graphical forms.

Improvement of a nonnegative preserved efficient solver for atmospheric chemical kinetic equations

Generally, chemical reactions from atmospheric chemistry models are described by a strongly coupled, stiff and nonlinear system of ordinary differential equations, which requires a good numerical solver. Several articles published about the solvers of chemical equations, during the numerical simulation, indicate that one renders the concentration null when it becomes negative. In order to preserve the positivity of the exact solutions, recent works have proposed a new solver called Modified-Backward-Euler (MBE). To improve this solver, we propose in this paper an iterative numerical scheme witch is better fitted to stiff problems. This new approach, called Iterative-Modified-Backward-Euler (IMBE), is based on iterative solution of the P-L structure of the implicit nonlinear ordinary differential equations on each time step. The efficiency of the iteration process is increased by using the Gauss and Successive-Over-Relaxation (SOR). In the case of fast/slow chemical kinetic reactions, we proposed an other variant called Iterative-Quasi-Steady-State-Approximation (IQSSA). The numerical exploration of stiff test problem shows clearly that this formalism is applicable to a wide range of chemical kinetics problems and give a good approximation compared to the recent solver. The numerical procedures give reasonable accurate solutions when compared to exact solution.Generally, chemical reactions from atmospheric chemistry models are described by a strongly coupled, stiff and nonlinear system of ordinary differential equations, which requires a good numerical solver. Several articles published about the solvers of chemical equations, during the numerical simulation, indicate that one renders the concentration null when it becomes negative. In order to preserve the positivity of the exact solutions, recent works have proposed a new solver called Modified-Backward-Euler (MBE). To improve this solver, we propose in this paper an iterative numerical scheme witch is better fitted to stiff problems. This new approach, called Iterative-Modified-Backward-Euler (IMBE), is based on iterative solution of the P-L structure of the implicit nonlinear ordinary differential equations on each time step. The efficiency of the iteration process is increased by using the Gauss and Successive-Over-Relaxation (SOR). In the case of fast/slow chemical kinetic reactions, we proposed an other variant called Iterative-Quasi-Steady-State-Approximation (IQSSA). The numerical exploration of stiff test problem shows clearly that this formalism is applicable to a wide range of chemical kinetics problems and give a good approximation compared to the recent solver. The numerical procedures give reasonable accurate solutions when compared to exact solution.

Mathematical model of obtaining a hydrocarbon fuel based on the fischer­tropsch pathway in a stationary layer of the cobalt­based catalyst

Studying the kinetics of the Fischer-Tropsch process is a rather important task, since this process is very sensitive to the temperature mode, as well as to the characteristics of the catalyst. In addition, a given process is accompanied by many side reactions that negatively affect the rate and selectivity of the reaction. The Fischer-Tropsch synthesis is an alternative source for obtaining high-quality fuel from coal or biomass rather than petroleum. Therefore, investigating the kinetics of the Fischer-Tropsch reaction, in order to improve the selectivity and activity of catalysts, and to determine the rate constants of chemical reactions, is a relevant problem. The choice of the catalyst is one of the main factors affecting the quality and product yield for the Fischer-Tropsch synthesis. We fabricated two samples of cobalt catalysts for conducting the experiments. The first sample of the catalyst Со/γ-Al 2 O 3 contains cobalt nanoparticles of the same size. The second sample of the catalyst (Со)/γ-Al 2 O 3 was obtained by the method of impregnating the carrier with a solution of cobalt nitrate. The catalyst, obtained through the method of impregnation of (Со)/γ-Al 2 O 3 , demonstrated a higher activity, larger by an order of magnitude than the monodispersed catalysts. However, the monodispersed catalyst showed high selectivity for the lower hydrocarbons. In order to calculate the kinetics of the Fischer-Tropsch process and to find the reaction rate constants, we developed a software module in the programming environment MS Visual Studio 2017 in the programming language C# using the .NET Framework v4.6 technologies. By using the developed program module, we calculated reaction rate constants of the Fischer-Tropsch process. After analyzing the data obtained, one can see that the relative error is within 2…3 %, demonstrating the adequacy of the proposed model to solve the inverse problem of chemical kinetics. Therefore, we can state that a given model for the calculation of rate constants could be applied to study the Fischer-Tropsch process.

STUDY OF CARBON DIOXIDE ADSORPTION ON CHROMOXIDE CATALYST ON NON-STATIONARY CONCENTRATIONS

Investigation of the regularities of chemical processes, not only near but also far from the stationary state, gives additional information on their mechanisms. In this paper, we present a new method for estimating rate constants of adsorption-desorption processes from the experimentally measured values of the nonstationary concentrations of an adsorbed substance, based on calculating the instantaneous rates of the adsorption (or desorption) process. This method allows to connect unknown kinetic parameters of adsorption (desorption) of a substance on the catalyst surface for various most probable assumed mechanisms with the calculated values of the instantaneous rates of adsorption-desorption processes. As a consequence, the method makes it possible to solve two types of inverse problems of chemical kinetics: calculate point and interval values of rates constants of adsorption and desorption; determine the most likely mechanism from several proposed mechanisms of implementation of these processes. Using this method, point and interval values of the rates constants of adsorption and desorption of carbon dioxide were determined on the base of nonstationary experimental data on adsorption on the assumption of carbon dioxide adsorption on a chromoxide catalyst to three proposed mechanisms: linear, bimolecular and dissociative. Based on the results of calculations, the corresponding non-stationary dependences of carbon dioxide adsorption were restored, which were compared with the experimental data. The obtained results confirm that the previously established dissociative mechanism of adsorption of carbon dioxide on the chromoside catalyst is the most probable. The developed simple method does not require the use of complex optimization calculations and can be used to solve the inverse problem of chemical kinetics associated with the determination of mechanisms and the estimation of the rates constants of adsorption and desorption of substances on various catalysts.Forcitation:Kol’tsov N.I., Fedotov V.Kh. Study of carbon dioxide adsorption on chromoxide catalyst on non-stationary concentrations. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2018. V. 61. N 7. P. 37-42

Generation and application of multivariate polynomial quadrature rules

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.