New Quasi-Steady-State And Partial-Equilibrium Methods For Integrating Chemically Reacting Systems

NEW QUASI-STEADY-STATE AND PARTIAL-EQUILIBRIUM METHODS FOR INTEGRATING CHEMICALLY REACTING SYSTEMS by David Ray Mott Chairperson: Bram van Leer We present new quasi-steady-state (QSS) and partial-equilibrium (PE) methods for integrating systems of ordinary di erential equations (ODEs) that arise from chemical reactions. These methods were developed for use in process-split reactingow simulations. The new QSS integrator is a second-order predictor-corrector method that is Astable for linear equations. The method is accurate regardless of the timescales of the individual ODEs in the system and works well for problems typical of hydrocarbon combustion. The method has very low start-up costs, making it ideal for processsplit reactingow simulations which require the solution of an initial-value problem in each computational cell in the ow eld for every global timestep. For problems of extreme sti ness, PE tools can be used in combination with the QSS integrator. PE methods remove the fastest reactions in the mechanism from the kinetic integration when their e ects can be calculated using algebraic equilibrium constraints. Conservation constraints are used to write an ODE for the reaction's progress variable. The solution of this equation provides a new method for identifying reactions in equilibrium. A systematic method for nding a set of conserved scalars for an arbitrary group of reactions is presented, and this method is used to eliminate reactions that produce redundant equilibrium constraints. Since the equilibrium reactions must compensate for changes in the system that disturb their equilibrium, the equilibrium source terms are not forced identically to zero. Equilibrium is imposed by driving these source terms to the average value required to compensate for the perturbations caused by the other processes. Integration results for a cesium-air mechanism, a hydrogen-air mechanism, and a thermonuclear mechanism used in astrophysics are presented. One-dimensional ame and detonation results are presented for a single-step hydrogen mechanism and the thermonuclear mechanism, respectively. The new QSS method outperforms the integrator CHEMEQ in speed, accuracy, and stability, and is recommended for reactingow applications in the sti ness range of typical hydrocarbon combustion problems. The new PE tools broaden the range for which a partial-equilibrium calculation will maintain the accuracy of the solution.

[1]  S. H. Lam,et al.  Singular Perturbation for Stiff Equations Using Numerical Methods , 1985 .

[2]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics ∗ , 1992 .

[3]  S. Lam,et al.  The CSP method for simplifying kinetics , 1994 .

[4]  D. Faddeev,et al.  Computational methods of linear algebra , 1981 .

[5]  Elaine S. Oran,et al.  Multilevel Structure of Cellular Detonations in Type Ia Supernovae , 1999 .

[6]  T. Young,et al.  CHEMEQ - A Subroutine for Solving Stiff Ordinary Differential Equations , 1980 .

[7]  S. H. Lam,et al.  Understanding complex chemical kinetics with computational singular perturbation , 1989 .

[8]  J. Boris,et al.  Solution of continuity equations by the method of flux-corrected transport , 1976 .

[9]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[10]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[11]  F. Spellman.  Combustion Theory , 2020 .

[12]  Elaine S. Oran,et al.  A Numerical Study of a Two-Dimensional H2-O2-Ar Detonation Using a Detailed Chemical Reaction Model , 1998 .

[13]  A. Khokhlov.  Stability of detonations in supernovae , 1993 .

[14]  Sanford Gordon,et al.  Computer program for calculation of complex chemical equilibrium compositions , 1972 .

[15]  Krishnan Radhakrishnan,et al.  LSENS: A General Chemical Kinetics and Sensitivity Analysis Code for homogeneous gas-phase reactions. Part 1: Theory and numerical solution procedures , 1994 .

[16]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[17]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[18]  Jan Verwer,et al.  An evaluation of explicit pseudo-steady-state approximation schemes for stiff ODE systems from chemical kinetics , 1993 .

[19]  K. Radhakrishnan.  New integration techniques for chemical kinetic rate equations. II - Accuracy comparison , 1986 .

[20]  Elaine S. Oran,et al.  Numerical Simulation of Reactive Flow , 1987 .

[21]  S. H. Lam,et al.  A study of homogeneous methanol oxidation kinetics using CSP , 1992 .

[22]  Stephen B. Pope,et al.  Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation , 1997 .

[23]  A. C. Hindmarsh,et al.  LINEAR MULTISTEP METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS: METHOD FORMULATIONS, STABILITY, AND THE METHODS OF NORDSIECK AND GEAR. , 1972 .

[24]  J. Boris,et al.  A numerical technique for solving stiff ordinary differential equations associated with the chemical kinetics of reactive-flow problems , 1977 .

[25]  David T. Pratt,et al.  Fast Algorithm for Calculating Chemical Kinetics in Turbulent Reacting Flow , 1988 .

[26]  C. Curtis.  Linear Algebra: An Introductory Approach , 1984 .

[27]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[28]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[29]  Shi Jin.  Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1995 .

[30]  Mohit Arora,et al.  Explicit characteristic-based high-resolution algorithms for hyperbolic conservation laws with stiff source terms. , 1996 .

[31]  S. H. Lam.  Reduced Chemistry Modeling and Sensitivity Analysis , 1995 .

[32]  William H. Press,et al.  Numerical recipes : the art of scientific computing : FORTRAN version , 1989 .

[33]  S. H. Lam,et al.  CONVENTIONAL ASYMPTOTICS AND COMPUTATIONAL SINGULAR PERTURBATION FOR SIMPLIFIED KINETICS MODELLING , 1999 .

[34]  J Crank,et al.  C.R.C. Standard Mathematical Tables , 1962 .

[35]  Ulrich Maas,et al.  Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds , 1992 .

[36]  K. Kuo.  Principles of combustion , 1986 .

[37]  W. H. Vandevender,et al.  The SLATEC mathematical subroutine library , 1982, SGNM.

[38]  U. Maas.  Simplifying Chemical Kinetics Using Intrinsic Low-Dimensional Manifolds , 1996 .

[39]  Martin Rein.  Erratum: ‘‘The partial‐equilibrium approximation in reacting flows’’ [Phys. Fluids A 4, 873 (1992)] , 1992 .

[40]  J. D. Ramshaw.  Partial chemical equilibrium in fluid dynamics , 1980 .