Skip to main content
Skip to article control options
No AccessFull-Length Paper

Rate-Controlled Constrained Equilibrium for Nozzle and Shock Flows

Published Online:https://doi.org/10.2514/1.B36226

The performance of different constraints for the rate-controlled constrained-equilibrium (RCCE) method is investigated in the context of modeling reacting flows characteristic of hypervelocity ground testing facilities and reentry conditions. Although the RCCE approach has been used widely in the past, its application in non-combustion-based reacting flows is rarely done; the flows being investigated in this work do not contain species that can easily be classified as reactants and/or products. The effectiveness of different constraints is investigated before running a full computational simulation, and new constraints not reported in the existing literature are introduced. A constraint based on the enthalpy of formation is shown to work well for the two gas models used for flows that involve both shocks and steady expansions.

References

  • [1] Hornung H., “Experimental Hypervelocity Flow Simulation, Needs, Achievements and Limitations,” Proceedings of the 1st Pacific International Conference on Aerospace Science and Technology, Taiwan, 1993. doi:https://doi.org/10.1017/jfm.2015.489 Google Scholar

  • [2] Anderson J. D., Hypersonic and High-Temperature Gas Dynamics, 2nd ed., AIAA, Resont, VA, 2006, Chap. 10. doi:https://doi.org/10.2514/4.861956 LinkGoogle Scholar

  • [3] Lu T. and Law C. K., “A Directed Relation Graph Method for Mechanism Reduction,” Proceedings of the Combustion Institute, Vol. 30, No. 1, 2005, pp. 1333–1341. doi:https://doi.org/10.1016/j.proci.2004.08.145 CrossrefGoogle Scholar

  • [4] Lu T. and Law C. K., “A Criterion Based on Computational Singular Perturbation for the Identification of Quasi Steady State Species: A Reduced Mechanism for Methane Oxidation with NO Chemistry,” Combustion and Flame, Vol. 154, No. 4, 2008, pp. 761–774. doi:https://doi.org/10.1016/j.combustflame.2008.04.025 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [5] Lu T., Law C. K., Yoo C. S. and Chen J. H., “Dynamic Stiffness Removal for Direct Numerical Simulations,” Combustion and Flame, Vol. 156, No. 8, 2009, pp. 1542–1551. doi:https://doi.org/10.1016/j.combustflame.2009.02.013 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [6] Menon S. K., Boettcher P. A. and Blanquart G., “Enthalpy Based Approach to Capture Heat Transfer Effects in Premixed Combustion,” Combustion and Flame, Vol. 160, No. 7, 2013, pp. 1242–1253. doi:https://doi.org/10.1016/j.combustflame.2013.02.008 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [7] Muller J.-M., Elementary Functions: Algorithms and Implementation, 2nd ed., Birkhäuser, Boston, MA, 2005, Chap 4. doi:https://doi.org/10.1007/b137928 Google Scholar

  • [8] Pepiot-Desjardins P. and Pitsch H., “An Efficient Error-Propagation-Based Reduction Method for Large Chemical Kinetic Mechanisms,” Combustion and Flame, Vol. 154, No. 12, 2008, pp. 67–81. doi:https://doi.org/10.1016/j.combustflame.2007.10.020 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [9] Peters N., “Reducing Mechanisms,” Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames, edited by Smooke M. D., Vol. 384, Lecture Notes in Physics, Springer, Berlin, 1991, pp. 48–67. doi:https://doi.org/10.1007/BFb0035362 CrossrefGoogle Scholar

  • [10] Goussis D. and Lam S., “A Study of Homogeneous Methanol Oxidation Kinetics Using CSP,” Symposium (International) on Combustion, Vol. 24, No. 1, 1992, pp. 113–120. doi:https://doi.org/10.1016/S0082-0784(06)80018-4 SYMCAQ CrossrefGoogle Scholar

  • [11] Maas U. and Pope S., “Implementation of Simplified Chemical Kinetics Based on Intrinsic Low-Dimensional Manifolds,” Symposium (International) on Combustion, Vol. 24, No. 1, 1992, pp. 103–112. doi:https://doi.org/10.1016/S0082-0784(06)80017-2 SYMCAQ CrossrefGoogle Scholar

  • [12] Maas U. and Pope S., “Simplifying Chemical Kinetics: Intrinsic Low-Dimensional Manifolds in Composition Space,” Combustion and Flame, Vol. 88, No. 34, 1992, pp. 239–264. doi:https://doi.org/10.1016/0010-2180(92)90034-M CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [13] Janbozorgi M., Ugarte S., Metghalchi H. and Keck J. C., “Combustion Modeling of Mono-Carbon Fuels Using the Rate-Controlled Constrained-Equilibrium Method,” Combustion and Flame, Vol. 156, No. 10, 2009, pp. 1871–1885. doi:https://doi.org/10.1016/j.combustflame.2009.05.013 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [14] Keck J. C., “Rate-Controlled Constrained-Equilibrium Theory of Chemical Reactions in Complex Systems,” Progress in Energy and Combustion Science, Vol. 16, No. 2, 1990, pp. 125–154. doi:https://doi.org/10.1016/0360-1285(90)90046-6 PECSDO 0360-1285 CrossrefGoogle Scholar

  • [15] Law R., Metghalchi M. and Keck J. C., “Rate-Controlled Constrained Equilibrium Calculation of Ignition Delay Times in Hydrogen-Oxygen Mixtures,” Symposium (International) on Combustion, Vol. 22, No. 1, 1989, pp. 1705–1713. doi:https://doi.org/10.1016/S0082-0784(89)80183-3 SYMCAQ CrossrefGoogle Scholar

  • [16] Beretta G. P., Keck J. C., Janbozorgi M. and Metghalchi H., “The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics,” Entropy, Vol. 14, No. 2, 2012, pp. 92–130. doi:https://doi.org/10.3390/e14020092 ENTRFG 1099-4300 CrossrefGoogle Scholar

  • [17] Hadi F. and Sheikhi M. R. H., “A Comparison of Constraint and Constraint Potential Forms of the Rate-Controlled Constraint-Equilibrium Method,” Journal of Energy Resources Technology, Vol. 138, No. 2, 2015, Paper 022202. doi:https://doi.org/10.1115/1.4031614 CrossrefGoogle Scholar

  • [18] Hadi F., Janbozorgi M., Sheikhi M. R. H. and Metghalchi H., “A Study of Interactions Between Mixing and Chemical Reaction Using the Rate-Controlled Constrained-Equilibrium Method,” Journal of Non-Equilibrium Thermodynamics, Vol. 41, No. 4, Oct. 2016, pp. 257–278. doi:https://doi.org/10.1515/jnet-2015-0052 JNETDY 0340-0204 CrossrefGoogle Scholar

  • [19] Hiremath V., Ren Z. and Pope S. B., “A Greedy Algorithm for Species Selection in Dimension Reduction of Combustion Chemistry,” Combustion Theory and Modelling, Vol. 14, No. 5, 2010, pp. 619–652. doi:https://doi.org/10.1080/13647830.2010.499964 CTMOFQ 1364-7830 CrossrefGoogle Scholar

  • [20] Hiremath V. and Pope S. B., “A Study of the Rate-Controlled Constrained-Equilibrium Dimension Reduction Method and its Different Implementations,” Combustion Theory and Modelling, Vol. 17, No. 2, 2013, pp. 260–293. doi:https://doi.org/10.1080/13647830.2012.752109 CTMOFQ 1364-7830 CrossrefGoogle Scholar

  • [21] Janbozorgi M. and Metghalchi H., “Rate-Controlled Constrained-Equilibrium Theory Applied to the Expansion of Combustion Products in the Power Stroke of an Internal Combustion Engine,” International Journal of Thermodynamics, Vol. 12, No. 1, 2009, pp. 44–50. Google Scholar

  • [22] Keck J. C. and Gillespie D., “Rate-Controlled Partial-Equilibrium Method for Treating Reacting Gas Mixtures,” Combustion and Flame, Vol. 17, No. 2, 1971, pp. 237–241. doi:https://doi.org/10.1016/S0010-2180(71)80166-9 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [23] Lapointe S., Bobbitt B. and Blanquart G., “Impact of Chemistry Models on Flame-Vortex Interaction,” Proceedings of the Combustion Institute, Vol. 35, No. 1, 2015, pp. 1033–1040. doi:https://doi.org/10.1016/j.proci.2014.06.091 CrossrefGoogle Scholar

  • [24] Nicolas G., Janbozorgi M. and Metghalchi H., “Constrained-Equilibrium Modeling of Methane Oxidation in Air,” Journal of Energy Resources Technology, Vol. 136, No. 3, 2014, pp. 032205-1–032205-7. doi:https://doi.org/10.1115/1.4027692 CrossrefGoogle Scholar

  • [25] Nicolas G. and Metghalchi H., “Comparison Between RCCE and Shock Tube Ignition Delay Times at Low Temperatures,” Journal of Energy Resources Technology, Vol. 137, No. 6, 2015, pp. 062203-1–062203-4. doi:https://doi.org/10.1115/1.4030493 CrossrefGoogle Scholar

  • [26] Nicolas G. and Metghalchi H., “Development of the Rate-Controlled Constrained-Equilibrium Method for Modeling of Ethanol Combustion,” Journal of Energy Resources Technology, Vol. 138, No. 2, 2015, pp. 022205-1–022205-11. doi:https://doi.org/10.1115/1.4031511 CrossrefGoogle Scholar

  • [27] Tang Q. and Pope S. B., “Implementation of Combustion Chemistry by In Situ Adaptive Tabulation of Rate-Controlled Constrained Equilibrium Manifolds,” Proceedings of the Combustion Institute, Vol. 29, No. 1, 2002, pp. 1411–1417. doi:https://doi.org/10.1016/S1540-7489(02)80173-0 CrossrefGoogle Scholar

  • [28] Tang Q. and Pope S. B., “A More Accurate Projection in the Rate-Controlled Constrained-Equilibrium Method for Dimension Reduction of Combustion Chemistry,” Combustion Theory and Modelling, Vol. 8, No. 2, 2004, pp. 255–279. doi:https://doi.org/10.1088/1364-7830/8/2/004 CTMOFQ 1364-7830 CrossrefGoogle Scholar

  • [29] Ugarte S., Gao Y. and Metghalchi H., “Application of the Maximum Entropy Principle in the Analysis of a Non-Equilibrium Chemically Reacting Mixture,” International Journal of Thermodynamics, Vol. 8, No. 1, 2005, pp. 43–53. Google Scholar

  • [30] Yousefian V., “A Rate-Controlled Constrained-Equilibrium Thermochemistry Algorithm for Complex Reacting Systems,” Combustion and Flame, Vol. 115, Nos. 1–2, 1998, pp. 66–80. doi:https://doi.org/10.1016/S0010-2180(97)00334-9 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [31] Beretta G. P., Janbozorgi M. and Metghalchi H., “Degree of Disequilibrium Analysis for Automatic Selection of Kinetic Constraints in the Rate-Controlled Constrained-Equilibrium Method,” Combustion and Flame, Vol. 168, June 2016, pp. 342–364. doi:https://doi.org/10.1016/j.combustflame.2016.02.005 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [32] Janbozorgi M. and Metghalchi H., “Rate-Controlled Constrained-Equilibrium Modeling of H/O Reacting Nozzle Flow,” Journal of Propulsion and Power, Vol. 28, No. 4, 2012, pp. 677–684. doi:https://doi.org/10.2514/1.B34545 JPPOEL 0748-4658 LinkGoogle Scholar

  • [33] Rabinovitch J. and Blanquart G., “A Computationally Efficient Approach to Hypersonic Reacting Flows,” 29th International Symposium on Shock Waves 1: Volume 1, Springer International Publ., Switzerland, 2015, pp. 173–178. doi:https://doi.org/10.1007/978-3-319-16835-725 CrossrefGoogle Scholar

  • [34] Morr A. R. and Heywood J. B., “Partial Equilibrium Model for Predicting Concentration of CO in Combustion,” Acta Astronautica, Vol. 1, No. 78, 1974, pp. 949–966. doi:https://doi.org/10.1016/0094-5765(74)90062-9 AASTCF 0094-5765 CrossrefGoogle Scholar

  • [35] Kao S. and Shepherd J. E., “Numerical Solution Methods for Control Volume Explosions and ZND Detonation Structure,” GALCIT FM2006-007, California Inst. of Technology, Pasadena, CA, 2006. Google Scholar

  • [36] Browne S., Ziegler J. and Shepherd J. E., “Numerical Solution Methods for Shock and Detonation Jump Conditions,” GALCIT FM2006-006, California Inst. of Technology, Pasadena, CA, 2006. Google Scholar

  • [37] Pope S. B., “Gibbs Function Continuation for the Stable Computation of Chemical Equilibrium,” Combustion and Flame, Vol. 139, No. 3, 2004, pp. 222–226. doi:https://doi.org/10.1016/j.combustflame.2004.07.008 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [38] Pope S., “CEQ: A Fortran Library to Compute Equilibrium Compositions Using Gibbs Function Continuation,” 2003, http://eccentric.mae.cornell.edu/pope/CEQ [retrieved June 2014] Google Scholar

  • [39] Pope S., “The Computation of Constrained and Unconstrained Equilibrium Compositions of Ideal Gas Mixtures Using Gibbs Function Continuation,” Cornell Univ. Rept. FDA 03-02, Ithaca, NY, 2003. Google Scholar

  • [40] Gou X., Sun W., Chen Z. and Ju Y., “A Dynamic Multi-Timescale Method for Combustion Modeling with Detailed and Reduced Chemical Kinetic Mechanisms,” Combustion and Flame, Vol. 157, No. 6, 2010, pp. 1111–1121. doi:https://doi.org/10.1016/j.combustflame.2010.02.020 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [41] Goodwin D. G., “An Open-Source, Extensible Software Suite for CVD Process Simulation,” Proceedings of Electrochemical Society (ECS), Vol. 14, Electrochemical Soc., Pennington, NJ, 2003, pp. 2003–2008. doi:https://doi.org/10.2514/1.J052389 Google Scholar

  • [42] Gordon S. and McBride B. J., “Thermodynamic Data to 20000 K for Monatomic Gases,” NASA TP-1999-208523, 1999. Google Scholar

  • [43] McBride B. J., Zehe M. J. and Gordon S., “NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species,” NASA TP-2002-211556, 2002. Google Scholar

  • [44] Gupta R. N., Yos J. M., Thompson R. A. and Lee K. P., “A Review of Reaction Rates and Thermodynamic and Transport Properties for an 11-Species Air Model for Chemical and Thermal Nonequilibrium Calculations to 30000 K,” NASA RP-1232, 1990. Google Scholar

  • [45] Parziale N. J., “Slender-Body Hypervelocity Boundary-Layer Instability,” Ph.D. Dissertation, California Inst. of Technology, Pasadena, CA, 2013. Google Scholar

  • [46] Rabinovitch J., “Advancing EDL Technologies for Future Space Missions: From Ground Testing Facilities to Ablative Heatshields,” Ph.D. Dissertation, California Inst. of Technology, Pasadena, CA, 2014. Google Scholar

  • [47] Pope S., “Computationally Efficient Implementation of Combustion Chemistry Using In Situ Adaptive Tabulation,” Combustion Theory and Modelling, Vol. 1, No. 1, 1997, pp. 41–63. doi:https://doi.org/10.1080/713665229 CTMOFQ 1364-7830 CrossrefGoogle Scholar

  • [48] Regele J. D., Knudsen E., Pitsch H. and Blanquart G., “A Two-Equation Model for Non-Unity Lewis Number Differential Diffusion in Lean Premixed Laminar Flames,” Combustion and Flame, Vol. 160, No. 2, 2013, pp. 240–250. doi:https://doi.org/10.1016/j.combustflame.2012.10.004 CBFMAO 0010-2180 CrossrefGoogle Scholar

  • [49] Smith G. P., Golden D. M., Frenklach M., Moriarty N. W., Eiteneer B., Goldenberg M., Bowman C. T., Hanson R. K., Song S., Gardiner W. C., Lissianski V. V. and Qin Z., “GRI-Mech 3.0,” 1999, http://www.me.berkeley.edu/gri_mech/ [retrieved 4 March 2014]. Google Scholar

  • [50] Mahaffy P. R., Webster C. R., Atreya S. K., Franz H., Wong M., Conrad P. G., Harpold D., Jones J. J., Leshin L. A., Manning H. and et al., “Abundance and Isotopic Composition of Gases in the Martian Atmosphere from the Curiosity Rover,” Science, Vol. 341, No. 6143, 2013, pp. 263–266. doi:https://doi.org/10.1126/science.1237966 SCIEAS 0036-8075 CrossrefGoogle Scholar

  • [51] Tonse S. R., Moriarty N. W., Brown N. J. and Frenklach M., “PRISM: Piecewise Reusable Implementation of Solution Mapping. An Economical Strategy for Chemical Kinetics,” Israel Journal of Chemistry, Vol. 39, No. 1, 1999, pp. 97–106. doi:https://doi.org/10.1002/ijch.199900010 ISJCAT 0021-2148 CrossrefGoogle Scholar