Interval models of nonequilibrium physicochemical processes

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The paper discusses the application of the adaptive interpolation algorithm to problems of chemical kinetics and gas dynamics with interval uncertainties in reaction rate constants. The values of the functions describing the reaction rate may differ considerably if they have been obtained by different researchers. The difference may reach tens or hundreds of times. Interval uncertainties are proposed to account for these differences in models. Such problems with interval parameters are solved using the previously developed adaptive interpolation algorithm. On the example of modelling the combustion of a hydrogen-oxygen mixture, the effect of uncertainties on the reaction process is demonstrated. One-dimensional nonequilibrium flow in a rocket engine nozzle with different nozzle shapes, including a nozzle with two constrictions, in which a standing detonation wave can arise, is simulated. A numerical study of the effect of uncertainties on the structure of the detonation wave, as well as on steadyystate flow parameters, such as the ignition delay time and the concentration of harmful substances at the nozzle exit, is performed.

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1. Introduction In order to simulate gas-phase chemical transformations it is necessary to know the kinetic mechanism and the rates of the reactions taking place. As a rule, the dependences that describe the rates are obtained experimentally, often giving only approximate values [1]. The values of the functions approximating the rate of the same reaction, but obtained by different researchers, may differ by tens or hundreds of times. To account for these differences, we propose to use the interval apparatus [2-5]. In this case interval parameters are introduced into the model and simulation results are interval estimates for the values of interest. The previously developed adaptive interpolation algorithm [6-9] is used to solve such problems with interval parameters. The algorithm belongs to the methods that determine an explicit dependence of the solution of the problem on the values of interval parameters. Two subgroups can be distinguished in this group: methods using symbolic expressions [10-12] and methods representing the solution as a polynomial with respect to interval parameters [13, 14]. The adaptive interpolation algorithm belongs to the latter subgroup. This algorithm has a theoretical justification. It consists in constructing a polynomial for each moment of time which interpolates the dependence of the problem solution on the values of the parameters in a given area of uncertainty. The interpolation polynomial is constructed on the basis of a set of nodes that form a grid. At each step of the algorithm, values in the nodes of the grid are updated, and then adaptation is made depending on the interpolation error. New nodes are added in places with a large error, and nodes are removed in places with a small error. The classical version of the algorithm uses interpolation on complete meshes, which limits its application to systems with a small number of interval parameters. However, two approaches, sparse meshes [15-17] and tensor trains [18, 19], have been applied in [20-22], which extend the application of the algorithm to dynamic systems with a large number of interval parameters. The paper deals with the problems of chemical kinetics and gas dynamics. The simulation of combustion of a mixture of hydrogen and oxygen in the presence of interval uncertainties in the reaction rate constants has been carried out. A one-dimensional mathematical model describing chemical nonequilibrium flows in a nozzle of a given shape with uncertainties in the reaction rate constants is presented. Results of a numerical study of the effect of uncertainties on the structure of the detonation wave, as well as on steady-state flow parameters, such as the ignition delay time and the concentration of harmful substances at the nozzle exit, are presented. 2. Model of chemical kinetics Here is a description of the basic relations. A multicomponent system of a variable composition of
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About the authors

Alexander Yu. Morozov

Federal Research Center Computer Science and Control of the Russian Academy of Sciences; Moscow Aviation Institute (National Research University)

Email: morozov@infway.ru
ORCID iD: 0000-0003-0364-8665
Scopus Author ID: 57203389215
ResearcherId: ABC-7836-2021

Doctor of Physical and Mathematical Sciences, Senior Researcher, Department of Mathematical Modeling of Heterogeneous Systems, Federal Research Center Computer Science and Control of the Russian Academy of Sciences; Associate Professor of the Department of Computational Mathematics and Programming, Moscow Aviation Insti- tute (National Research University)

44-2 Vavilova St, Moscow, 119333, Russian Federation; 4 Volokolamskoe Highway, Moscow, 125993, Russian Federation

Dmitry L. Reviznikov

Federal Research Center Computer Science and Control of the Russian Academy of Sciences; Moscow Aviation Institute (National Research University)

Email: reviznikov@mai.ru
ORCID iD: 0000-0003-0998-7975
Scopus Author ID: 6602701797
ResearcherId: T-4571-2018

Doctor of Physical and Mathematical Sciences, Professor, Leading Researcher, Department of Math- ematical Modeling of Heterogeneous Systems, Federal Research Center Computer Science and Control of the Russian Academy of Sciences; Professor of the Department of Computational Mathematics and Programming, Moscow Aviation Institute (National Research University)

44-2 Vavilova St, Moscow, 119333, Russian Federation; 4 Volokolamskoe Highway, Moscow, 125993, Russian Federation

Vladimir Yu. Gidaspov

Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: gidaspov@mai.ru
ORCID iD: 0000-0002-5119-4488
Scopus Author ID: 6506396733
ResearcherId: B-4572-2019

Doctor of Physical and Mathematical Sciences, Associate Professor, Professor of the Department of Computational Mathematics and Programming

4 Volokolamskoe Highway, Moscow, 125993, Russian Federation

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