Decomposition of the Problem in the Numerical Solution of Differential-Algebraic Systems for Chemical Reactions with Partial Equilibria

В работе рассматриваются две простые системы дифференциально-алгебраических уравнений, которые появляются при исследовании задач химической кинетики с частичными равновесиями: часть переменных определяется из условия argmin для некоторой функции состояния системы, которая зависит от всех переменных задачи. Для такой постановки можно записать дифференциально-алгебраическую систему уравнений, в которой алгебраическая подзадача выражает условия минимальности функции состояния в каждый момент времени. При численном решении удобно провести декомпозицию (расщепление) задачи, т.е. решать динамическую и оптимизационную задачи последовательно. В работе на двух примерах исследуется применимость такой декомпозиции: определяется сходимость и порядок точности численного метода, а также предложены другие варианты декомпозиции. Показано, что численное решение расщепленной системы уравнений имеет такой же порядок точности, как и численное решение совместной задачи. Выполнение ограничений удовлетворяется с достаточной точностью, если временной шаг численного метода заканчивается решением оптимизационной задачи. Полученные результаты могут быть использованы при разработке численных алгоритмов для решения более сложных задач химической кинетики.

Ключевые слова

дифференциально-алгебраические системы; оптимизация; численные методы.

Литература

1. 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
2. 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
3. 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
4. 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
5. Prigogine I. Introduction in Thermodynamics of Irreversible Processes. Izhevsk, Regular and Chaotic Dynamics, 2001. (in Russian)
6. 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
7. 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
8. Popkov Yu.S. Positive Dynamic Systems with Entropic Operator. Automation and Remote Control, 2003, no. 3, pp. 104-113. (in Russian)
9. 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)
10. 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
11. 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
12. 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
13. 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
14. 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
15. Rodrigues R. Modelagem Cinetics e de Equilibrio Combunadas para Simulacao de Processes de Gaseificacao. Tese de Doutorado, Porto Alegre, 2015.
16. 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
17. 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
18. 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
19. 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
20. 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
21. Koniavitis P., Rigopoulos S., Jones W.P. A Methodology for Derivation of RCCE-Reduced Mechanisms Via CSP. Combustion and Flame, 2017, vol. 183, pp. 126-143. DOI: 10.1016/j.combustflame.2017.05.010
22. 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)
23. 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
24. 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
25. 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
26. 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
27. 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.
28. 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
29. 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
30. 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
31. 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.
32. Snegirev A.Yu. Perfectly Stirred Reactor Model to Evaluate Extinction of Diffusion Flame. tCombustion and Flame, 2015, vol. 162, pp. 3622-3631. DOI: 10.1016/j.combustflame.2015.06.019
33. McBride B.J., Zehe M.J., Gordon S. NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species. Cleveland, Glenn Research Center, 2002.
34. 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
35. 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