Decomposition of the problem in the numerical solution of differential-algebraic systems for chemical reactions with partial equilibria

Decomposition of the problem in the numerical solution of differential-algebraic systems for chemical reactions with partial equilibria

Автор: Donskoy I.G.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp

Рубрика: Программирование

Статья в выпуске: 4 т.15, 2022 года.

Бесплатный доступ

The paper considers two simple systems of differential-algebraic equations that appear in the study of chemical kinetics problems with partial equilibria: some of the variables are determined from the condition argmin for some system function state, which depends on all variables of the problem. For such a statement, we can write a differential-algebraic system of equations where the algebraic subproblem expresses the conditions for the minimality of the state function at each moment. It is convenient to use splitting methods in numerical solving, i.e. to solve dynamic and optimization subproblems separately. In this work, we investigate the applicability of differential-algebraic splitting using two simplified systems. The convergence and order of accuracy of the numerical method are determined. Different decomposition options are considered. Calculations show that the numerical solution of the split system of equations has the same order of accuracy as the numerical solution of the joint problem. The constraints are fulfilled with sufficient accuracy if the procedure of the numerical method ends with the solution of the optimization subproblem. The results obtained can be used in the numerical solving of more complex chemical kinetics problems.

Еще

Differential-algebraic systems, optimization, numerical methods

Короткий адрес: https://sciup.org/147240330

IDR: 147240330   |   DOI: 10.14529/mmp220405

Список литературы Decomposition of the problem in the numerical solution of differential-algebraic systems for chemical reactions with partial equilibria

  • Gorban A.N., Karlin I.V. Method of Invariant Manifolds for Chemical Kinetics. Chemical Engineering Science, 2003, vol. 58, pp. 4751–4768. DOI: 10.1016/j.ces.2002.12.001
  • Maas U., Pope S.B. Simplifying Chemical Kinetics: Intrinsic Low-Dimensional Manifolds in Composition Space. Combustion and Flame, 1992, vol. 88, pp. 239–264. DOI: 10.1016/0010-2180(92)90034-M
  • Chen Yulin, Chen Jyh-Yuan. Towards improved Automatic Chemical Kinetic Model Reduction Regarding Ignition Delays and Flame Speeds. Combustion and Flame, 2018, vol. 190, pp. 293–301. DOI: 10.1016/j.combustflame.2017.11.024
  • Turanyi T. Applications of Sensivity Analysis to Combustion Chemistry. Reliability Engineering and System Safety, 1997, vol. 57, no. 1, pp. 41–48. DOI: 10.1016/S0951-8320(97)00016-1
  • Prigogine I. Introduction in Thermodynamics of Irreversible Processes. Izhevsk, Regular and Chaotic Dynamics, 2001. (in Russian)
  • Keck J.C. Rate-Controlled Constrained-Equilibrium Theory of Chemical Reactions in Complex Systems. Progress in Energy and Combustion Science, 1990, vol. 30, pp. 125–154. DOI: 10.1016/0360-1285(90)90046-6
  • Jones W.P., Rigopoulos S. Reduction of Comprehensive Chemistry Via Constraint Potentials. Proceedings of the Combustion Institute, 2005, vol. 30, pp. 1325–1331. DOI: 10.1016/j.proci.2004.08.198
  • Popkov Yu.S. Positive Dynamic Systems with Entropic Operator. Automation and Remote Control, 2003, no. 3, pp. 104–113. (in Russian)
  • Popkov Yu.S. Basics of a Theory of Dynamic Systems with Entropic Operator and Its Applications. Automation and Remote Control, 2006, no. 6, pp. 75–105. (in Russian)
  • Koukkari P., Pajarre R. Introducing Mechanistic Kinetics to the Lagrangian Gibbs Energy Calculation. Computers and Chemical Engineering, 2006, vol. 30, pp. 1189–1196. DOI: 10.1016/j.compchemeng.2006.03.001
  • Kaganovich B.M., Filippov S.P., Keiko A.V., Shamanskii V.A. Thermodynamic Models of Extreme Intermediate States and their Applications in Power Engineering. Thermal Engineering, 2011, vol. 58, pp. 143–152. DOI: 10.1134/S0040601511020054
  • Messerle A.V., Messerle V.E., Ustimenko A.B. Plasma Thermochemical Preparation for Combustion of Pulverized Coal. High Temperature, 2017, vol. 55, pp. 352–360. DOI: 10.1134/S0018151X17030142
  • Donskoi I.G. Mathematical Modeling of the Reaction Zone of a Shell-Prenflo Gasifier with the Use of the Models of Sequential Equilibrium. Solid Fuel Chemistry, 2016, vol. 50, pp. 191–196. DOI: 10.3103/S0361521916030034
  • Currier N.G., Hyams D.G. A Hybrid Method for Flows in Local Chemical Equilibrium and Nonequilibrium. 50th AIAA Aerospace Sciences Meeting including the new Horizon Forum and Aerospace Exposition, Nashville, Tennessee, 2012, pp. 2012–1239. DOI: 10.2514/6.2012-1239
  • Rodrigues R. Modelagem Cinetics e de Equilibrio Combunadas para Simulacao de Processes de Gaseificacao. Tese de Doutorado, Porto Alegre, 2015.
  • Koniavitis P. Rigopoulos S., Jones W.P. Reduction of a Detailed Chemical Mechanism for a Kerosene Surrogate Via RCCE-CSP. Combustion and Flame, 2018, vol. 194, pp. 85–106. DOI: 10.1016/j.combustflame.2018.04.004
  • Lovas T., Navarro-Martinez S., Rigopoulos S. On the Adaptively Reduced Chemistry in Large Eddy Simulations. Proceedings of the Combustion Institute, 2011, vol. 33, pp. 1339–1346. DOI: 10.1016/j.proci.2010.05.089
  • Zhuyin Ren, Zhen Lu, Yang Gao, Tianfeng Lu, Lingyun Hou. A Kinetics-Based Method for Constraint Selection in Rate-Controlled Constrained Equilibrium. Combustion Theory and Modelling, 2017, vol. 21, pp. 159–182. DOI: 10.1080/13647830.2016.1201596
  • Hiremath V., Pope S.B. A Study of the Rate-Controlled Constrained Equlibrium Dimension Reduction Method and its Different Implementations. Combustion Theory and Modelling, 2013, vol. 17, pp. 260–293. DOI: 10.1080/13647830.2012.752109
  • Mohammad Janbozorgia, Wang Haib. Bottom-Up Modeling Using the Rate-Controlled Constrained-Equilibrium Theory: The n-Butane Combustion Chemistry. Combustion and Flame, 2018, vol. 194, pp. 223–232. DOI: 10.1016/j.combustflame.2018.04.026
  • Koniavitis P., Rigopoulos S., Jones W.P. A Methodology for Derivation of RCCEReduced Mechanisms Via CSP. Combustion and Flame, 2017, vol. 183, pp. 126–143. DOI: 10.1016/j.combustflame.2017.05.010
  • Kaganovich B.M., Keiko A.V., Shamansky V.A., Shirkalin I.A., Zarodnyuk M.S. Technology of Thermodynamic Modelling. Reduction of Dynamic Models to Static Model. Novosibirsk, Nauka, 2010 (in Russian)
  • Neron A., Lantagne G., Marcos B. Computation of Complex and Constrained Equilibria by Minimization of the Gibbs Free Energy. Chemical Engineering Science, 2012, vol. 82, pp. 260-271. DOI: 10.1016/j.ces.2012.07.041
  • Pope S.B. Gibbs Function Continuation for the Stable Computation of Chemical Equilibrium. Combustion and Flame, 2004, vol. 139, pp. 222–226. DOI: 10.1016/j.combustflame.2004.07.008
  • Scoggins J.B., Magin T.E. Gibbs Function Continuation for Linearly Constrained Multiphase Equilibria. Combustion and Flame, 2015, vol. 162, pp. 4514–4522. DOI: 10.1016/j.combustflame.2015.08.027
  • Feinberg M. Necessary and Sufficient Conditions for Detailed Balancing in Mass Action Systems of Arbitrary Complexity. Chemical Engineering Science, 1989, vol. 4, pp. 1819–1827. DOI: 10.1016/0009-2509(89)85124-3
  • Chistyakov V.F., Tairov E.A., Chistyakova E.V., Levin A.A. On Decomposition of Difference Schemes for Numerical Solution of Differential Algebraic Equations. Bulletin of the South Ural University. Series: Mathematical Modelling, Programming and Computer Software, 2012, vol. 11, pp. 88–100.
  • Chistyakov V.F. Preservation of Stability Type of Difference Schemes when Solving Stiff Differential Algebraic Equations. Numerical Analysis and its Applications, 2011, vol. 4, pp. 363–375. DOI: 10.1134/S1995423911040082
  • Bulatov M.V., Chistyakova E.V. Numerical Solution of Integro-Differential Systems with a Degenerate Matrix Multiplying the Derivative by Multistep Methods. Differential Equations, 2006, vol. 42, pp. 1317–1325. DOI: 10.1134/S0012266106090102
  • Bulatov M.V., Tygliyan A.V., Filippov S.S. A Class of One-Step One-Stage Methods for Stiff Systems of Ordinary Differential Equations. Computational Mathematics and Mathematical Physics, 2011, vol. 51, pp. 1167–1180. DOI: 10.1134/S0965542511070050
  • Bulatov M.V., Solovarova L.S. On the Loss of L-Stability of the Implicit Euler Method for a Linear Problem. The Bulletin of Irkutsk State University. Series: Mathematics, 2015, vol. 12, pp. 3–11.
  • Snegirev A.Yu. Perfectly Stirred Reactor Model to Evaluate Extinction of Diffusion Flame. Combustion and Flame, 2015, vol. 162, pp. 3622–3631. DOI: 10.1016/j.combustflame.2015.06.019
  • McBride B.J., Zehe M.J., Gordon S. NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species. Cleveland, Glenn Research Center, 2002.
  • Tsanas C., Stenby E.H., Wei Yan. Calculation of Multiphase Chemical Equilibrium by the Modified RAND Method. Industrial and Engineering Chemistry Research, 2017, vol. 56, pp. 11983–11995. DOI: 10.1016/j.ces.2017.08.033
  • Donskoy I.G., Shamansky V.A., Kozlov A.N., Svishchev D.A. Coal Gasification Process Simulations Using Combined Kinetic-Thermodynamic Models in One-Dimensional Approximation. Combustion Theory and Modelling, 2017, vol. 21, pp. 529–559. DOI: 10.1080/13647830.2016.1259505
Еще
Статья научная
QR-код для установки приложения SciUp Наведите камеру и скачайте бесплатное приложение SciUp