Calibration of the CO + OH CO 2 + H rate constant with direct and indirect information

Paper # 070RK-0246 Topic: Reaction Kinetics 8 th US National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013. Calibration of the CO + OH CO 2 + H rate constant with direct and indirect information S. Plessis PECOS, ICES, University of Austin, TX While the literature contains a vast array of kinetics parameterizations of chemical reactions, identical (or nearly identical) physical models have reported parameter values that can vary over several orders of magnitude. In this work, the investigation of the kinetics description of the reaction CO + OH CO 2 + H is presented. Multiple models have been calibrated and compared using bayesian techniques with data of two different types: measurements of (k, T ) and laminar flamespeeds. First, measurements of (k, T ) in the range of 100 2500 K were used. These data are taken from several different sources. Three different models were calibrated and their results were compared. All three model formulations performed equally well when evaluated in the bayesian context. These direct measurements of (k, T ) were not as informative as one might expect resulting in highly-correlated posterior distributions that vary over several orders of magnitude. These results can be used to explain the large discrepancies found in the literature. In addition to using the above data, a second set of calibrations were performed using indirect measurements: laminar flamespeed in a CO/H 2 mixture. By starting with the GRI-Mech 3.0 [1] mechanism and using very large prior distributions on the model parameters (signifying a lack of knowledge), the flamespeed data were used to inform the model parameters via a statistical inverse problem. Calibrating against this information did not lead to agreement with direct measurements, therefore yielding unrealistic descriptions of the rate constants. Thus, it is reasonable to conclude that the previous calibration alone is not sufficient for reliable flamespeed calculations. In conclusion, this work proposes a methodology to parameterize rate constants in low-pressure conditions that appears to be consistent with all available experimental data of (k, T ). However, rate constant measurements alone do not sufficiently constrain kinetics parameters to reliably predict laminar flamespeeds and additional information is required in order to parameterize a model that is both consistent with (k, T ) experimental data that also accurately predicts laminar flamespeeds. 1 Introduction The bimolecular CO + OH CO 2 + H reaction plays an important role in hydrogen combustion systems [2] and has two competing reactive channels [3, Fig. 3], which makes it difficult to represent by a general kinetics model. This reaction is known to depend heavily on pressure [4, 5]. In this work, we will only consider atmospheric pressure for combustion in order to be able to neglect this dependence [6]. Mechanisms and kinetics models used in the literature [6 9] usually derive from already compiled databases [10] and then update all or a subset of the kinetics parameters [9] by integrating indirect Corresponding author splessis@ices.utexas.edu 1

information. The GRI-Mech 3.0 mechanism [1] is built by calibrating all kinetics parameters of an extended chemical mechanism against numerous experiments (direct and indirect information) in order to obtain the most general model for gas-phase chemistry, while Davis et al. updated only the pre-exponential parameters (see Eq. 1) by using a large set of experimental measurements [9, Fig. 1 and Tab. 1]. One of the downsides of this approach is that it provides model-dependent kinetics characterization, which translates for the CO + OH CO 2 + H chemical reaction to orders of magnitude discrepancies on kinetics parameters (Tab. 1). A reaction is modeled by an elementary process or a duplicate process when the pressure conditions allow one to neglect pressure dependent models as in this work. These processes are represented by one or the sum of several Kooij equations, respectively (Eq. 1). n 1 ( ) βi ( T k(t ) = A i exp E ) ai T ref RT i=1 with k the rate constant, T the temperature, T ref a reference temperature (in this work T ref = 300 K), A the pre-exponential factor, β the power parameter, E a the activation energy and R the ideal gas constant. In this work, we use different kinetics models found in the literature, i.e. n = 1, 2 and 3, respectively called M 1, M 2 and M 3. The M 1 model is used for elementary processes while the M 2 and M 3 models add degrees of freedom to account for chemical processes not well approximated by an elementary process. The most common kinetics model for this reaction is the simple Kooij equation M 1 [1, 2, 7, 8, 10, 11], while Davis et al. [9] optimized the rate constant using a sum of two Kooij equations (M 2 ), and Sun et al. [6] used the M 3 form (Tab. 1) following the low pressure expression of Troe et al. [5]. See Tab. 1 for detailed values. In this work, we used bayesian calibration methods to inform the kinetics parameters for these rate constants using direct measurements values of rate constant at low pressures with respect to temperature and indirect information flamespeed measurements, thus obtaining a kinetics description relevant to a combustion model. To inform the model with indirect information, we used a laminar flamespeed as the Quantity of Interest (QoI) on a CO/H 2 combustion model, starting from the GRI-Mech 3.0 [1] reactive scheme. We used the CANTERA library 1 to perform the calculations. Combustion conditions are represented by the parameter Φ. This parameter is used to represent the type of combustion: Φ = 1 is stoichiometric combustion, a Φ < 1 is a lean combustion and a Φ > 1 is a rich combustion. We have reproduced the experimental conditions of McLean et al. [7] on the 5% H 2-95% CO mixture, Φ = 1.148 and Φ = 3.895, to compare to their results and to the larger calibration of Davis et al. [9], who used a mechanism of 30 chemical reactions and optimized this mechanism over 35 different experimental measurements, including the two considered in this paper. 1 http://code.google.com/p/cantera/ (1) 2

Source A β E a T range T ref (cm 3 mol 1 s 1 ) GRI-Mech 3.0 [1] 4.76 10 7 1.228 70 (cal mol 1 ) > 500 1 Baulch et al. [10] 1.05 10 17 1.5 250 ( Ea in K) 300 2000 1 R Mueller et al. [8] 1.40 10 5 1.95 1.35 10 3 (cal mol 1 ) 750 1100 1 Li et al. [2] 10 5.35( 0.39 +0.48 ) 1.89( +0.11 0.14) 583( +82 106) ( Ea R (K) in K) 300 2500 1 Davis et al. [9] 7.968 10 11 0.14 7352 (cal mol 1 ) > 300 1 8.784 10 10 0.03 16 Sun et al. [6] 1.0 10 13 0 8050 from Troe et al. [5], 9.0 10 11 0 2300 (cal mol 1 ) 100 3000 low-pressure limit. 1.01 10 11 0 30 Table 1: Values of OH + CO H + CO 2 rate constant parameters with some parameterizations available in the literature. The uncertainty reported from Li et al. [2] are the 95% confidence intervals. : in cm 3 molecule 1 s 1. 2 Bayesian calibration 2.1 Parameter calibration In a bayesian framework, parameters are represented by probability density functions (pdf) to account for uncertainties. Bayesian inference is based on Bayes theorem [12 14] (Eq. 2) which formalizes the concept of updating knowledge by integrating new information. p(x y) = p(x)p(y x) p(y) (2) A parameter x, which prior knowledge is represented by the pdf p(x) is updated by the information contained in the hypothesis y. The pdf p(x y) denotes the updated knowledge of x knowing y. For a vector θ of the kinetics parameters of a chemical model, Bayes theorem is expressed as in Eq. 3. π(θ; D) p post (θ D) = p prior (θ) pprior (θ)π(θ; D)dθ (3) with D the vector of information one wishes to use to inform the parameters; p prior the prior pdf, which denotes the state of knowledge of the parameters independently of the data; p post the posterior pdf, which is the state of knowledge after having integrated the information contained in the data D. π is the likelihood function, measure of the agreement between the data D and the model prevision for specific values of the parameters. 3

2.2 Model calibration This method is generalizable to models, and thus allow model comparison. Let s consider a set M = {M 1, M 2,..., M K } of K candidate models. Eq. 3 rewritten for a model class M i gives: π(θ i, M i ; D) p post (θ i D, M i ) = p prior (θ i M i ) (4) pprior (θ i M i )π(θ i, M i ; D)dθ i with θ i the parameters of the model class M i. To compare different competitive model classes, one needs to compare the posterior distributions. Using Bayes theorem (Eq. 3), we obtain as posterior pdf for model M i π evid (M i ; D) p post (M i D, M) = p prior (M i M) K k=1 p prior(m k M)π evid (M k ; D) The prior distributions are the probabilities of model classes M i, i [1,K] before integrating the data D. Usually, little information is available, therefore equiprobability is appropriate, i.e. uniform probability (p prior (M i M) = 1/K) to all model classes M i. π evid is the evidence function, analog to the likelihood function in (3). The evidence function measures the consistency of the concerned model and the data considering the entire parameter space. It is given by the following integral over parameter space. π evid (M i ; D) = p prior (θ i M i )π(θ i, M i ; D)dθ i (6) which corresponds to the denominator of (4). It can be rewritten as: log(π evid (M i ; D)) = log(π like (θ, M i ; D))p post (θ D, M i )dθ ( ) ppost (θ D, M i ) log p post (θ D, M i )dθ p prior (θ M i ) The first term is the posterior expectation of the log-likelihood which measures how well the model is able to fit the data, averaged over the posterior pdf for the parameters. The second term is the relative information entropy (or Kullback-Leibler divergence [15]) between the posterior and the prior pdfs. It measures the information about the parameters that is gained from the data. For two model classes that fit the data equally well, as measured by the first term in (7), the model that requires the least tuning, as measured by information gain, is preferred. For more details and discussion, see Jaynes [14] (Chapter 20) and Muto and Beck [16] (Section 4). (5) (7) 2.3 Monte Carlo method Eq. 3 and 4 are usually not solvable analytically, and their estimations are computationally very demanding. Computing the posterior pdfs or the evidence for instance requires solving highdimensional integrals. To evaluate such quantities, we use a Monte Carlo approach [17] in which the pdfs are represented by samples, which enables to approximate the integrals using those samples. In this work, the stochastic method referred as the adaptive multi-level algorithm [18] is used, implemented in the QUESO library [19]. 4

2.4 Calibration cycles The kinetics parameters are characterized through two different calibration cycles which enable to integrate different type of information. Calibrating over measurements of a reaction rate will inform directly the kinetics parameters and therefore will provide the most accurate representation of the rate constant considering available knowledge. This calibration will provide a model-agnostic description: an unbiased database, that is, containing only the kinetics measurements biases and uncertainties. Any model calculation using this description will contain all the model inadequacies. This cycle will be called the C I cycle thereafter. A calibration process using a combustion model and measured values of a chosen QoI will inform the kinetics parameters with both the measurements information and the model characteristics. This calibration contains a large amount of information as every chemical reaction has an impact on every other ones. Thus the chosen mechanism (the considered chemical reactions), the kinetics model chosen for every reaction, and, in this case, the transport model have all an impact on the posterior pdfs of the informed quantities. The effect of such calibration is a tuning of the informed parameters to the model and QoI, which will adapt the said-informed parameters to the model and measurements biases. Therefore this calibration will produce the most reliable model in terms of accuracy to the QoI. This cycle will be called the C II cycle thereafter. These two calibration processes are pictured in Fig. 1. Database on kinetics parameters of reaction: (k, T ) surface Reaction alone Kinetics models: M 1 : one Kooij equation, M 2 : two Kooij equations, M 3 : three Kooij equations. Full mechanism Bayes Calibrated kinetics parameters Combustion model Combustion data Figure 1: To obtain information on a reaction, we can infer from measurements on the reaction rate (black path); to obtain the most accurate model, we can infer from measurements on a model s output (red path). 5

2.5 Likelihood For the calibration with direct data, they are available in the form of points in the (k, T ) space, k being the rate constant (unit varies) and T the temperature (in K). The uncertainties are considered in the k and T directions. A rate constant is a positive parameter over which measurements disagreements can be large (> 30%). This is well represented by a lognormal distribution [20], leading to use a lognormal expression in the k direction (Eq. 8). The uncertainty on the temperatures is smaller, we chose a normal expression for the T dimension (Eq. 9). For a data point (k data, T data ) with uncertainty ( k data, T data ). The likelihood for a modeled data point, (k M, T M ), will therefore be expressed in the k and T directions. For the k direction, we define the lognormal deviation as F = 1 + k data /k data. π km k data,f = LogN (k M ; k data, F ) ( 1 = 2π kdata ln(f ) exp 1 ( ) ) 2 ln(kdata ) ln(k M ) 2 ln(f ) (8) The likelihood in the T dimension is of the form: π TM T data, T data = Norm(T M ; T data, T data ) ( 1 = exp 1 ( ) ) 2 TM T data 2π Tdata 2 T data (9) For each (k M, T M ) point, we thus define the likelihood as: π (km,t M ) data = 1 N data data i π km k i,f i π TM T i, Ti (10) When the data uncertainty is not provided, representative values are used: 10% uncertainty over k ( k data = 0.1 k data ), and 1% uncertainty over the temperature ( T data = 0.01 T data ). The rate constant is calculated along a interval of temperature. To estimate a likelihood, it is necessary to define a temperature grid on which to calculate the likelihood at each grid point along a k(t ) curve. The arithmetic average will provide the likelihood value at each sampled point. In the indirect information case, we have a measured flamespeed associated to its uncertainty. In such case, a gaussian likelihood is adapted (Eq. 11): π fm f meas, f meas = Norm(f M ; f meas, f meas ) ( 1 = exp 1 ( ) ) 2 fm f meas 2π fmeas 2 f meas (11) with f meas the measured flamespeed, f meas its associated uncertainty and f M the modeled flamespeed. 6

Parameters C I C II A U(0, 20) in log 10 (cm 3 mol 1 s 1 ) N (11, 3) in ln(m 3 kmol 1 s 1 ) β U(0, 2) N (1.1, 0.45) E a U( 50, 50) in kj mol 1 N ( 560, 410) in cal mol 1 Table 2: A priori pdfs for the calibration cycles. In the C II cycle, we add the constrains 2 10 9 k 300 K 5 10 13 and 10 10 k 2500 K 10 14 in cm 3 mol 1 s 1. 2.6 Priors The default priors were chosen as large as possible, in order to simulate at best a complete lack of information. In the C I calibration cycle, the prior considered knowledge was: the pre-exponential factor is positive and lower than 10 20 cm 3 mol 1 s 1, the temperature dependence is positive but never above the quadratic, and the activation energy is either positive or negative, below an absolute value of 50 kj mol 1. Following the maximum entropy method [21, 22], we thus define a uniform pdf on the log 10 values of the pre-exponential factor, the power parameters and the activation energy (Tab. 2). It was not possible to use as large priors as previously defined for the C II cycle, as this description defines some physically unreasonable rate constants, and prevent the flamespeed calculations to converge in such cases. Thus the priors were changed considering the C I cycle posterior pdfs, and filters were added to ensure convergence, see Tab. 2. 3 Results and comparisons 3.1 Calculations conditions The rate constant measurements are taken from seven different sources (Fig. 2), in atmospheric pressure conditions, in agreement to the flamespeed measurements used in cycle C II [7, 9]. The grid used is defined by T start = 100 K, T end = 2500 K and T step = 150 K. The combustion model used is based on the Gri-Mech 3.0 [1] mechanism, in which only the reaction CO + OH H + CO 2 was parametrized by probabilistic values. The transport model used in these calculations is a mixture-averaged model. The laminar flamespeed calculations are performed with the CANTERA chemistry program [23]. The flamespeed measurements and conditions of calculations are taken from McLean et al. [7] to enable comparison with Davis et al. [9]. Results (Tab. 3, 4 and Fig. 3) are interpreted in terms of posterior rate constants (Fig. 4) and flamespeeds (Fig. 5). The different models are compared in terms of performances within the C I cycle (log(ev) column of table 3a), and only the best one in terms of bayesian evidence ( 2.2), will be used in the C II cycle, to limit computational time. 7

10 12 k (cm 3 mol 1 s 1 ) 10 11 Wooldridge et al., 1994 Wooldridge et al., 1996 Westenberg and de Haas, 1973 Jonah et al., 1984 Golden et al., 1998 Frost et al., 1991 Frost et al., 1993 500 1,000 1,500 2,000 T (K) Figure 2: Experimental data for the calibration of CO + OH CO 2 + H. Units have been harmonized to cm 3 mol 1 s 1. 3.2 Kinetics parameters posterior pdfs The C I inversion cycle provides highly-correlated posterior pdfs that spreads over several orders of magnitude, especially for the M 2 and M 3 models. The posterior pdfs of A have a lognormal deviation between 3 and 7.5, E a s sign is not even characterized. This can explain the widely different parameterizations found in the literature (Fig. 3), as these results agree with all considered comparative results. Yet these results show that direct measurements alone are not sufficient to characterize the kinetics parameters of CO+OH CO 2 +H. As all tested model classes perform equally well in term of bayesian evidence (Tab. 3a), the chosen model for the C II cycle was the simplest, M 1. Posterior pdfs obtained with the C II inversion cycle are more constrained: β is set to 0, which, in agreement to the high-correlation observed in the previous calibration, constrain A to a relatively small interval. E a is still not characterized. 3.3 Resulting rate constants While the C I cycle ensures realistic rate constants description, the posterior pdfs of the kinetics parameters define rate constants that are in blatant disagreement with the direct measurements (Fig. 4, red and green curves). It appears that the information contained in the laminar flamespeed is not accurate with respect to the rate constant. The information integrated in this inversion cycle contain the model inadequacies, which could partly explain the disagreements between calibrated and measured values. 8

Φ Model A(F ) β (σ) E a (σ) cm 3 /mol/s cal/mol log(ev) M 1 7.0 10 9 (4.7) 1.1 (0.5) -525 (400) -45 1.3 10 M 11 (7.5) 1.0 (0.5) 880 (3150) 2 3.4 10 10 (4.5) 1.2 (0.5) 800 (3750) M 3 1.1 10 11 (5.3) 1.0 (0.5) 1550 (3650) 1.8 10 10 (3.1) 1.1 (0.5) 3700 (4250) 1.1 10 11 (4.1) 1.0 (0.6) 3000 (4150) (a) C I calibration cycle results A (F ) β (σ) E a (σ) cm 3 /mol/s cal/mol -45-45 Flamespeed (cm s 1 ) 1.148 1.9 10 11 (1.3) 0 (0) 600 (530) 40 ± 6 3.895 1.4 10 11 (1.3) 0 (0) 600 (550) 50 ± 10 (b) C II calibration cycle results Table 3: Results summarized for the bayesian inversion cycles. The uncertainties of inverse parameters are given as standard deviation (σ) or relative multiplicative factor (F = 1 + σ A /A). Above: direct information inverse on the temperature grid T start = 100 K, T end = 2500 K and T step = 150 K. Below: indirect information inverse on the GRI-Mech 3.0 mechanism [1], flamespeed calculations performed by the CANTERA software [23] on a 5% H 2-95% CO mixture. A β E a A 1-0.99 0.83 β 1-0.80 E a 1 (a) C I calibration cycle A β E a A 1-0.79 β - - E a 1 (b) C II calibration cycle with Φ = 1.148 A β E a A 1-0.69 β - - E a 1 (c) C II calibration cycle with Φ = 3.895 Table 4: Rank correlation of the results for the bayesian inversion for a temperature grid of T start = 100 K, T end = 2500 K and T step = 150 K. 9

10 6 10 7 10 8 10 9 10 10 10 11 10 12 A (cm 3 mol 1 s 1 ) 6,000 4,000 2,000 0 E a (J mol 1 ) 0 0.5 1 1.5 2 β GRI 3.0-Mech Baulch et al, 1992 Mueller et al., 1999 Li et al., 2007 This study, dir. cal. This study, ind. cal. Φ = 1.148 This study, ind. cal. Φ = 3.895 Figure 3: Visual comparison of M 1 parametrizations used in the literature and this study results for T ref = 300 K. Li et al., 2007: 95% confidence intervals; this study: 90% confidence intervals. 3.4 Corresponding flamespeed Inversely, the C II cycle provides the most accurate flamespeeds, which is expected, as the flamespeed and model information are integrated in the kinetics model. The combustion model, using the posterior pdfs of cycle C I provides overestimated flamespeed, but not inconsistent, which can be ascribed to model inadequacies, notably the transport model. 3.5 Comparison with Davis et al. Davis et al. [9] used a chemical mechanism close to the Gri-Mech 3.0 [1] (14 species and 30 chemical reactions), a M 2 kinetics model on the CO+OH CO 2 +H reaction and a multi-component transport model. As seen in 3.2, the difference between the kinetics models is negligible: they both provides very similar rate constants (Fig. 4). The mechanisms are also quite similar, as the GRI 3.0 [1] was used as a base for their mechanism. The main expected source of discrepancies is therefore the transport model. The calibration was performed only on A parameters, thus acting like a multiplicative coefficient on the rate constant curves, which translates as a calibration on constrained kinetics models. These constrains enable to ensure physical consistence of the kinetics models (Fig. 4b) without a lost of the flamespeed accuracy (Fig. 5), but necessitate strong prior information on the activation energy and the power parameter. The combustion model used in this paper, while overestimating the flamespeed using the C I cycle results, do not provide inconsistent results. Measurements from McLean and al. [7] and results of Davis et al. [9] are within the flamespeed prediction uncertainty. 10

k (cm 3 mol 1 s 1 ) 10 12 10 11 10 10 0 500 1,000 1,500 2,000 2,500 3,000 T (K) (a) Inversions results 10 12 k (cm 3 mol 1 s 1 ) 10 11 GRI-Mech 3.0 Baulch & al., 1992 Mueller & al., 1999 Li & al., 2007 Davis & al., 2005 Sun & al., 2007 500 1,000 1,500 2,000 2,500 3,000 T (K) (b) Models in the literature Figure 4: Above: generated rate constants. Blue: direct calibration; red: indirect calibration with Φ = 1.148; green: indirect calibration with Φ = 3.895. Below: available models in the literature. 11

Davis et al., 2005 Davis et al., 2005 0.2 0.4 0.6 0.8 Flamespeed (m s 1 ) (a) Φ = 1.148 0.2 0.4 0.6 0.8 Flamespeed (m s 1 ) (b) Φ = 3.895 Figure 5: Histograms of flamespeed, N sample = 1000, the black solid curve is the target measured by McLean et al. [7]. Blue: direct calibration; red: indirect calibration. Davis et al. [9] is given for comparison. 4 Conclusion This paper proposes a bayesian methodology to calibrate, through a statistical inverse problem, the kinetics parameters for the reaction CO + OH CO 2 + H using two types of information: direct measurements of type (k, T ) and indirect measurements of laminar flamespeed. The first inversion did not constrain the parameters while the second provides unrealistic rate constants. It therefore seems not possible to obtain, in one characterization, realistic kinetics parameters and a reliable flamespeed description for a combustion model. Thus an accurate and reliable description would require more information than kinetics or flamespeed measurements alone. Combining the two calibration cycles into a double-calibrated kinetics model by (i) constraining the kinetics parameters to a physical domain by a C I calibration cycle, and (ii) improving the model s reliability by applying a C II calibration cycle on obtained pdfs by step (i) could constrain the kinetics parameters while ensuring physical consistency and flamespeed calculation accuracy. This is pictured in Fig. 1. Another improvement would be to integrate more information, either more specifically on the kinetics parameters, as done in Davis et al. [9], or on the rate constants by integrating pressure dependence, using measurements at higher temperature, etc. Other interesting possibilities reside in using quantum calculations to provide additional calibration data, which are not constrained by experimental limitations. 12

Database on kinetics parameters of reaction: (k, T ) surface Reaction alone Kinetics models: M 1 : one Kooij equation, M 2 : two Kooij equations, M 3 : three Kooij equations. Bayes Full mechanism Calibrated kinetics parameters Combustion data Combustion model Figure 6: Proposed scheme for estimating kinetics parameters. A first calibration cycle (black path) against direct information ensures a physical description of the rate constants. The second calibration cycle (red path) would then integrate the model information to ensure the best flamespeed predictive capabilities of the model. 13

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