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Numerical Optimisation for the Evaluation of Combustion Kinetic Models
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 66 (2015) 145 – 148
The 12th International Conference on Combustion & Energy Utilisation – 12ICCEU
Numerical optimisation for the evaluation of combustion kinetic models M. Fischer, X. Jiang Engineering Department, Lancaster University, Lancaster LA1 4YR, UK.
Abstract Four optimisation methods have been chosen and implemented for optimising kinetic parameters with respect to a set of experimental data. They were first successfully validated via specifically tailored minimisation problems where kinetic coefficients had been varied so as to produce discrepancies with the initial predictions of the GRI (Gas Research Institute) mechanism 3.0. Three of them could retrieve an almost perfect agreement whereas the fourth approach found a slightly sub-optimal solution. Afterwards, a set of CH3íO2 and C2H6íO2 oxidation experiments inconsistent with the initial values of GRI 3.0 were considered. It could be shown that the parameters of the most sensitive reactions could not be optimised under reasonable limits, thereby indicating that these experiments are probably not predictable by GRI 3.0. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of 12ICCEU Peer-review under responsibility of the Engineering Department, Lancaster University Keywords: Chemical Kinetics, Combustion, Reaction Mechanism, Optimisation, Model Evaluation
1. Introduction Chemical kinetic models are necessary for simulating combustion processes. Their sensitive parameters having a considerable influence on numerical predictions must be accurately determined. Until now, the traditional method has consisted of designing experimental conditions isolating the reactions of interest in such a way that optimal parameter values can be analytically estimated [1]. These can provide in turn approximations for the coefficients of other reactions involving similar molecular changes. During the last decades, methods from quantum chemistry have also been increasingly employed to compute realistic values on a more theoretical foundation [2]. While sufficient in several situations, this combined approach proves too limited in numerous cases where the reactions can neither be isolated nor approximated and computational chemistry lacks the precision needed for guaranteeing reliable results or involves an enormous computational burden. Under such circumstances, the utilisation of numerical optimisation methods turns out to be indispensable [3]. Generally, the search for an optimal set of parameters p can be defined as the minimisation of the chi-square norm (equation 1).
X
2
i= N ni
§ mi, j p ei, j ¨ ¨ ı i, j j=1 ©
p = ¦¦ i=1
· ¸ ¸ ¹
2
(1)
N is the number of experiments, ni the number of points for the i-th experiment, mi,j and ei,j the model prediction and experimental measurement, respectively. ıi,j is the standard deviation. Since the last one is
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Engineering Department, Lancaster University doi:10.1016/j.egypro.2015.02.081
146
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
often unknown, it can be approximated as ıi,j = ıei,j where ı is a relative standard deviation (also called coefficient of variation) assumed to be constant over the measurements. The software Homrea [4] for the simulation of homogeneous chemistry coupled with the optimisation program Kinefit [5] were used. Kinefit contains four optimisation methods: an adaptive Random Search (RS), a Genetic Algorithm (GA), and two algorithms which mimic quasi-Newton methods through the indirect estimation of derivatives, namely Condor [6] and Bobyqa [7]. In section 2, Kinefit is validated on constructed problems where sensitive parameters have been perturbed to such an extent that great discrepancies with the original predictions were caused. In section 3, it was determined whether or not the GRI-mechanism 3.0 [8] can be optimised in relation to experimental data in conflict with its initial results. Owing to space limitation, in this short description of our endeavour only few representative results will be shown. 2. Validation of the optimisation techniques To evaluate the capacities and performance of Kinefit, the H2 - O2 sub-system of the GRI-mechanism 3.0 was considered. For given combustion conditions, numerical experimental data were first generated using the initial values of the parameters p0. Some of the most influential parameters were then varied to the values pvar in such a way to produce a significant distance d(pvar) from the results obtained with the initial parameter values (playing the role of experimental data). The solution of such a parameter estimation problem d(p0) = 0 is exactly known, thereby enabling the evaluation and comparison of the different minimisation techniques employed by Kinefit. Many optimisation situations with various numbers of parameters and local minima were created. Among them, a realistic but particularly hard problem was defined through 6 experiments spanning the ranges 1200-1700 K and 3-300 bar for a duration of 2E-03 s whereby the profiles of OH, H2O and H2O2 were targeted and had to be reproduced. The pre-exponential factors, temperature coefficients and activation energies of 7 reactions were varied within two orders of magnitude (in relation to the resulting reaction coefficients k(T) ) and optimised with the four optimisation methods. The adjusted profiles from Condor, RS and the genetic algorithm are almost identical to the numerically generated experimental values. The optimised profiles from Bobyqa are nearly indistinguishable from the solution except for the second experiment where small discrepancies exist. Therefore it can be concluded that the methods implemented in Kinefit are both reliable and efficient for the optimisation of kinetic coefficients. 3. Evaluation of the GRI-mechanism with respect to C2H6 and CH3 oxidation.
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
Saito et al. [9] carried out a series of 12 experiments in shock tubes where temporal profiles of O and H free radicals were measured during the oxidation of either C2H6 or CH3. ARAS (Atomic Resonance Adsorption Spectroscopy) was employed for quantifying the concentrations. If the discrepancies are still too high with optimal coefficient values, it can be concluded that the model is probably not compatible with the data. A thorough optimisation was performed via Kinefit and an optimal set of parameter values was determined. It included all sensitive reactions. It could be shown that the model predictions can only be consistent with the measurements if their relative standard deviations are superior to 20 %. The simulated results are shown for the 3-th and 4-th experiment in Figure 1 as an illustration.
Figure 1. Results of the optimisation Adsorption curves given in [9] show that for H the relative standard deviations can be at most 17% and for O at most 6%. This entails that the best predictions achievable with the GRI-mechanism fall largely outside the uncertainty ranges of the measured free radicals. Therefore it is likely that the GRI-mechanism does not provide us with a correct description of the oxidation of ethane and methyl under the conditions of Saito et al. While this technique can be used to indicate that a reaction mechanism is likely inconsistent with measurements and should therefore be modified, a successful optimisation does not prove its adequacy until all other experiments have been simulated again using the new values. This methodology is non-Bayesian and related to the error-statistical approach of Mayo [10] for model evaluation. The traditional Bayesian methodology relies on assigning non-informal uniform prior probabilities to all set of parameter values (using the so-called principle of indifference) and updating them through the information stemming from experimental results. As Salmon [11] pointed out, this is extremely problematic because while such prior probabilities are meant to represent the absence of information, they end up being handled as involving a very specific knowledge which is not actually there [12]. This can lead to meaningless results which are pure numerical artefacts of this assumption and lack any relation to experimental and empirical reality [13]. Consequently, it was deemed preferable to leave aside non-physical prior probability distributions while assessing the agreement of a model with measurements. 4. Conclusion and outlooks. The increasing complexity of reaction mechanisms in combustion goes hand in hand with a greater use of numerical optimisation techniques for the estimation of kinetic parameters. In the present study, numerical validation tests demonstrated the ability of the program Kinefit to determine optimal solutions of problems involving the minimisation of the distance between model predictions and measurements.
147
148
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
Further, the GRI-mechanism 3.0 was optimised with respect to an ensemble of experiments at variance with its initial predictions. These concerned the oxidation of ethane and of the free radical methyl. The prediction of the optimised model remained inconsistent with the measurements within their uncertainties. Overall, the optimisation methods utilised here have thus been shown to be efficient for the practical adjustment of kinetic parameters. In many complex situations, such an optimisation cannot provide us with warranted true parameter values but allows us to reasonably conclude that a reaction mechanism cannot describe a set of experimental data within their uncertainty ranges [14]. This in turn indicates that new reaction paths have to be envisaged. Acknowledgement We are thankful to professor Uwe Riedel and Dr. Elke Goos from the German Aerospace Centre (DLR) for having put the software Homrea to our disposal and assisted us in the development of the program Kinefit.
References [1] [2] [3] [4] [5] [6] [7] [8]
S. Scherer, T. Just, P. Frank, High-temeprature investigations on pyrolytic reactions of propargyl radicals, Proceedings of the Combustion Institute 28 (2000) 1511–1518. A. Violi, A.F. Sarofim, and T.N. Truong Quantum mechanical study of molecular weight growth process by combination of aromatic molecules. Combustion and Flame 126 (2001) 1506. M. Frenklach. Process informatics for combustion chemistry. 31-th International Symposium on Combustion, Heidelberg, 2006. C. Heghes, Soot Formation Modeling during Hydrocarbon Pyrolysis and Oxidation behind Shock Waves, Ph.D. thesis, University of Heidelberg, 2006. M. Fischer, On the relevance of optimization methods for the systematic improvement of combustion rection mechanisms, Ph.D. thesis, University of Heidelberg, 2011. F. Vanden Berghen, H. Bersini, CONDOR, a new parallel, constrained extension of powell’s BOBYQA algorithm: Experimental results and comparison with the DFO algorithm, Journal of Computational and Applied Mathematics 181 (2005) 157–175. M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives, Technical Report, Cambridge University, 2009. G. Smith, D. Golden, M. Frenklach, N. Moriarty, B. Eiteneer, M. Goldenberg, C. Bowman, R. Hanson, S. Song, W. Gardiner, V. Lissianski, Z. Qin, Gri – mechanism 3.0, Website. http://www.me.berkeley.edu/grimech/version30/text30.html
[9] [10] [11] [12] [13] [14]
K. Saito, R. Ito, T. Kakumoto, A. Imamura, The initial process of the oxidation of the methyl radical in reflected shock waves, The Journal of Physical Chemistry 90 (1986) 1422. D. Mayo, Statistical science meets philosophy of science part 2: Shallow versus deep explorations, RMM 3 (2012) 71–107. W. Salmon, The Foundations of Scientific Inference, 1967. J. D. Norton, "Probability Disassembled," British Journal for the Philosophy of Science, 58 (2007), 41171. J. D. Norton, Cosmic confusions: Not supporting versus supporting not, Philosophy of Science 77 (2010) 501– 523. A. B. Singer, J. W. Taylor, P. I. Barton, and W. H. Green. Global dynamic optimization for parameter estimation in chemical kinetics. J. Phys. Chem. A. 110 (2006) : 971.
ScienceDirect Energy Procedia 66 (2015) 145 – 148
The 12th International Conference on Combustion & Energy Utilisation – 12ICCEU
Numerical optimisation for the evaluation of combustion kinetic models M. Fischer, X. Jiang Engineering Department, Lancaster University, Lancaster LA1 4YR, UK.
Abstract Four optimisation methods have been chosen and implemented for optimising kinetic parameters with respect to a set of experimental data. They were first successfully validated via specifically tailored minimisation problems where kinetic coefficients had been varied so as to produce discrepancies with the initial predictions of the GRI (Gas Research Institute) mechanism 3.0. Three of them could retrieve an almost perfect agreement whereas the fourth approach found a slightly sub-optimal solution. Afterwards, a set of CH3íO2 and C2H6íO2 oxidation experiments inconsistent with the initial values of GRI 3.0 were considered. It could be shown that the parameters of the most sensitive reactions could not be optimised under reasonable limits, thereby indicating that these experiments are probably not predictable by GRI 3.0. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of 12ICCEU Peer-review under responsibility of the Engineering Department, Lancaster University Keywords: Chemical Kinetics, Combustion, Reaction Mechanism, Optimisation, Model Evaluation
1. Introduction Chemical kinetic models are necessary for simulating combustion processes. Their sensitive parameters having a considerable influence on numerical predictions must be accurately determined. Until now, the traditional method has consisted of designing experimental conditions isolating the reactions of interest in such a way that optimal parameter values can be analytically estimated [1]. These can provide in turn approximations for the coefficients of other reactions involving similar molecular changes. During the last decades, methods from quantum chemistry have also been increasingly employed to compute realistic values on a more theoretical foundation [2]. While sufficient in several situations, this combined approach proves too limited in numerous cases where the reactions can neither be isolated nor approximated and computational chemistry lacks the precision needed for guaranteeing reliable results or involves an enormous computational burden. Under such circumstances, the utilisation of numerical optimisation methods turns out to be indispensable [3]. Generally, the search for an optimal set of parameters p can be defined as the minimisation of the chi-square norm (equation 1).
X
2
i= N ni
§ mi, j p ei, j ¨ ¨ ı i, j j=1 ©
p = ¦¦ i=1
· ¸ ¸ ¹
2
(1)
N is the number of experiments, ni the number of points for the i-th experiment, mi,j and ei,j the model prediction and experimental measurement, respectively. ıi,j is the standard deviation. Since the last one is
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Engineering Department, Lancaster University doi:10.1016/j.egypro.2015.02.081
146
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
often unknown, it can be approximated as ıi,j = ıei,j where ı is a relative standard deviation (also called coefficient of variation) assumed to be constant over the measurements. The software Homrea [4] for the simulation of homogeneous chemistry coupled with the optimisation program Kinefit [5] were used. Kinefit contains four optimisation methods: an adaptive Random Search (RS), a Genetic Algorithm (GA), and two algorithms which mimic quasi-Newton methods through the indirect estimation of derivatives, namely Condor [6] and Bobyqa [7]. In section 2, Kinefit is validated on constructed problems where sensitive parameters have been perturbed to such an extent that great discrepancies with the original predictions were caused. In section 3, it was determined whether or not the GRI-mechanism 3.0 [8] can be optimised in relation to experimental data in conflict with its initial results. Owing to space limitation, in this short description of our endeavour only few representative results will be shown. 2. Validation of the optimisation techniques To evaluate the capacities and performance of Kinefit, the H2 - O2 sub-system of the GRI-mechanism 3.0 was considered. For given combustion conditions, numerical experimental data were first generated using the initial values of the parameters p0. Some of the most influential parameters were then varied to the values pvar in such a way to produce a significant distance d(pvar) from the results obtained with the initial parameter values (playing the role of experimental data). The solution of such a parameter estimation problem d(p0) = 0 is exactly known, thereby enabling the evaluation and comparison of the different minimisation techniques employed by Kinefit. Many optimisation situations with various numbers of parameters and local minima were created. Among them, a realistic but particularly hard problem was defined through 6 experiments spanning the ranges 1200-1700 K and 3-300 bar for a duration of 2E-03 s whereby the profiles of OH, H2O and H2O2 were targeted and had to be reproduced. The pre-exponential factors, temperature coefficients and activation energies of 7 reactions were varied within two orders of magnitude (in relation to the resulting reaction coefficients k(T) ) and optimised with the four optimisation methods. The adjusted profiles from Condor, RS and the genetic algorithm are almost identical to the numerically generated experimental values. The optimised profiles from Bobyqa are nearly indistinguishable from the solution except for the second experiment where small discrepancies exist. Therefore it can be concluded that the methods implemented in Kinefit are both reliable and efficient for the optimisation of kinetic coefficients. 3. Evaluation of the GRI-mechanism with respect to C2H6 and CH3 oxidation.
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
Saito et al. [9] carried out a series of 12 experiments in shock tubes where temporal profiles of O and H free radicals were measured during the oxidation of either C2H6 or CH3. ARAS (Atomic Resonance Adsorption Spectroscopy) was employed for quantifying the concentrations. If the discrepancies are still too high with optimal coefficient values, it can be concluded that the model is probably not compatible with the data. A thorough optimisation was performed via Kinefit and an optimal set of parameter values was determined. It included all sensitive reactions. It could be shown that the model predictions can only be consistent with the measurements if their relative standard deviations are superior to 20 %. The simulated results are shown for the 3-th and 4-th experiment in Figure 1 as an illustration.
Figure 1. Results of the optimisation Adsorption curves given in [9] show that for H the relative standard deviations can be at most 17% and for O at most 6%. This entails that the best predictions achievable with the GRI-mechanism fall largely outside the uncertainty ranges of the measured free radicals. Therefore it is likely that the GRI-mechanism does not provide us with a correct description of the oxidation of ethane and methyl under the conditions of Saito et al. While this technique can be used to indicate that a reaction mechanism is likely inconsistent with measurements and should therefore be modified, a successful optimisation does not prove its adequacy until all other experiments have been simulated again using the new values. This methodology is non-Bayesian and related to the error-statistical approach of Mayo [10] for model evaluation. The traditional Bayesian methodology relies on assigning non-informal uniform prior probabilities to all set of parameter values (using the so-called principle of indifference) and updating them through the information stemming from experimental results. As Salmon [11] pointed out, this is extremely problematic because while such prior probabilities are meant to represent the absence of information, they end up being handled as involving a very specific knowledge which is not actually there [12]. This can lead to meaningless results which are pure numerical artefacts of this assumption and lack any relation to experimental and empirical reality [13]. Consequently, it was deemed preferable to leave aside non-physical prior probability distributions while assessing the agreement of a model with measurements. 4. Conclusion and outlooks. The increasing complexity of reaction mechanisms in combustion goes hand in hand with a greater use of numerical optimisation techniques for the estimation of kinetic parameters. In the present study, numerical validation tests demonstrated the ability of the program Kinefit to determine optimal solutions of problems involving the minimisation of the distance between model predictions and measurements.
147
148
M. Fischer and X. Jiang / Energy Procedia 66 (2015) 145 – 148
Further, the GRI-mechanism 3.0 was optimised with respect to an ensemble of experiments at variance with its initial predictions. These concerned the oxidation of ethane and of the free radical methyl. The prediction of the optimised model remained inconsistent with the measurements within their uncertainties. Overall, the optimisation methods utilised here have thus been shown to be efficient for the practical adjustment of kinetic parameters. In many complex situations, such an optimisation cannot provide us with warranted true parameter values but allows us to reasonably conclude that a reaction mechanism cannot describe a set of experimental data within their uncertainty ranges [14]. This in turn indicates that new reaction paths have to be envisaged. Acknowledgement We are thankful to professor Uwe Riedel and Dr. Elke Goos from the German Aerospace Centre (DLR) for having put the software Homrea to our disposal and assisted us in the development of the program Kinefit.
References [1] [2] [3] [4] [5] [6] [7] [8]
S. Scherer, T. Just, P. Frank, High-temeprature investigations on pyrolytic reactions of propargyl radicals, Proceedings of the Combustion Institute 28 (2000) 1511–1518. A. Violi, A.F. Sarofim, and T.N. Truong Quantum mechanical study of molecular weight growth process by combination of aromatic molecules. Combustion and Flame 126 (2001) 1506. M. Frenklach. Process informatics for combustion chemistry. 31-th International Symposium on Combustion, Heidelberg, 2006. C. Heghes, Soot Formation Modeling during Hydrocarbon Pyrolysis and Oxidation behind Shock Waves, Ph.D. thesis, University of Heidelberg, 2006. M. Fischer, On the relevance of optimization methods for the systematic improvement of combustion rection mechanisms, Ph.D. thesis, University of Heidelberg, 2011. F. Vanden Berghen, H. Bersini, CONDOR, a new parallel, constrained extension of powell’s BOBYQA algorithm: Experimental results and comparison with the DFO algorithm, Journal of Computational and Applied Mathematics 181 (2005) 157–175. M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives, Technical Report, Cambridge University, 2009. G. Smith, D. Golden, M. Frenklach, N. Moriarty, B. Eiteneer, M. Goldenberg, C. Bowman, R. Hanson, S. Song, W. Gardiner, V. Lissianski, Z. Qin, Gri – mechanism 3.0, Website. http://www.me.berkeley.edu/grimech/version30/text30.html
[9] [10] [11] [12] [13] [14]
K. Saito, R. Ito, T. Kakumoto, A. Imamura, The initial process of the oxidation of the methyl radical in reflected shock waves, The Journal of Physical Chemistry 90 (1986) 1422. D. Mayo, Statistical science meets philosophy of science part 2: Shallow versus deep explorations, RMM 3 (2012) 71–107. W. Salmon, The Foundations of Scientific Inference, 1967. J. D. Norton, "Probability Disassembled," British Journal for the Philosophy of Science, 58 (2007), 41171. J. D. Norton, Cosmic confusions: Not supporting versus supporting not, Philosophy of Science 77 (2010) 501– 523. A. B. Singer, J. W. Taylor, P. I. Barton, and W. H. Green. Global dynamic optimization for parameter estimation in chemical kinetics. J. Phys. Chem. A. 110 (2006) : 971.