1b10-070rk-0048

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Paper # 070RK-0048 Topic: Reaction Kinetics

1

8th U. S. National Combustion Meeting

Organized by the Western States Section of the Combustion Institute

and hosted by the University of Utah

May 19-22, 2013

Chemical Kinetic Uncertainty Minimization through Laminar FlameSpeeds of Mixtures of Air with C1-C4-Hydrocarbons

Okjoo Park,1 Peter S. Veloo,

2 David A. Sheen,

3 Fokion N. Egolfopoulos,

1 Hai Wang

1

1 Department of Aerospace and Mechanical Engineering,

University of Southern California, Los Angeles, California 90089-1453, USA

2 Department of Mechanical and Aerospace Engineering,

Princeton University, Princeton, NJ 08544, USA

3Department of Chemical Sciences Division,

National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Abstract

Performing high-fidelity simulations of complex reacting flows requires that the uncertainties associated with predictions

of fundamental flame properties are characterized and minimized. Although the kinetics of C1-C4 hydrocarbon flames

have been studied extensively in numerous past investigations, notable uncertainties exist. In the present study, the types

of hydrocarbon fuels with the greatest impact on model uncertainty reduction are identified along with the attendant

accuracy that is needed in flame measurements to facilitate better reaction model development. The results demonstrated

that a reaction model constrained only by laminar flame speeds of methane/air flames reduces notably the uncertainty inthe predictions of the laminar flame speeds of C3 and C4 alkanes, because the key chemical pathways of all of these

flames are similar to each other. However, the uncertainty in the model predictions for flames of unsaturated C3-C4

hydrocarbons would remain to be significant without considering their laminar flames speeds in the constraining target

data set, because the secondary rate controlling reaction steps can be different from those in methane flames. It was also

demonstrated that the constraints provided by the laminar flame speeds of C4 unsaturated hydrocarbons fuels could

reduce notably the uncertainties in the predictions of laminar flame speeds of C4 alcohol/air mixtures. To obtain an

accurate prediction of the laminar flame speed of a particular C4 alcohol/air mixture, the best data to acquire is not

necessarily those of that particular mixture. In many cases, a consistent set of experimental flame measurements of key

molecular intermediates formed during the pyrolysis and oxidation of the parent fuel are more critical to achieving a

better prediction for the flame propagation rates of C4 alcohol flames.

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1. Introduction

Because of the hierarchical nature of combustion kinetics of hydrocarbons, the high-temperature reaction

kinetics of H2, CO, and C1-C4 hydrocarbons are foundations to an accurate description of the elementary

combustion kinetics of higher hydrocarbons. For this reason, the combustion chemistry of hydrogen, carbon

monoxide, and C1-C4 hydrocarbon has been studied extensively over the last several decades. In recent years,

the chemical kinetics of straight and branched-chain C3-C4 alkenes have received particular attention. These

compounds are key stable intermediates produced in the oxidation of larger hydrocarbons (e.g., Dryer and

Brezinsky 1986) and bio-derived alcohols (e.g., Veloo and Egolfopoulos 2011). The oxidation of these

alkene intermediates is usually the rate limiting steps towards the overall oxidation rate of the larger, parent

hydrocarbons. To isolate effectively the uncertainties in the fuel specific kinetics of large hydrocarbons and

oxygenated fuels, it is essential that the uncertainties of the foundational C0-C4 model be reduced first.

The intent of the present study was to understand better the role of accurately measured laminar flame

speeds, S u

o , in constraining reaction model uncertainty of C0-C4 hydrocarbon combustion. To achieve this

goal, an uncertainty propagation and minimization method was used to propagate the kinetic uncertainty into

S u

o

predictions and to constrain the coupled model uncertainty using a set of well-characterized S

u

o data. The

present study uses the method of uncertainty minimization using polynomial chaos expansions (MUM-PCE)

(Sheen et al. 2009, Sheen and Wang 2011a). This method addresses the impact of uncertainties in the model

predictions of combustion properties and provides additional, coupled constraints to rate coefficients and their

uncertainties through an inverse problem approach utilizing reliable experimental flame property data as the

targets. By using experimental data with well-quantified uncertainties, it is possible to constrain the

uncertainties in the reaction model and as a result to reduce the overall prediction uncertainties. More

recently, MUM-PCE has utilized multispecies time histories from shock tubes to constrain kinetic

uncertainties (Sheen and Wang 2011b). The study concluded that accurate measurements of global

combustion properties such as S u

o , are still necessary for obtaining a predictive kinetic model for hydrocarbon

combustion.

In the present study MUM-PCE was employed to demonstrate how systematically determined S u

o ’s with

well-defined measurement uncertainties can be utilized to constrain the model uncertainty, using USC Mech

II (Wang et al. 2007) as an example. For this purpose, a comprehensive set of S u

o ’s of C1-C4 hydrocarbon

flames was utilized to address several questions critical to a better utilization of the S u

o

data. The required

accuracy of S u

o measurements that would best facilitate model development was assessed and the questions

concerning the type of hydrocarbon fuels that have the greatest impact on model uncertainty reduction were

explored. Discussion is also provided on the second part of our continuing study on the minimization of

uncertainties in chemical models utilizing S u

o (Park et al. 2012).

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2. Approach

2.1 Reaction Model

The chemical kinetic model used here is based on USC Mech II (Wang et al. 2007), which consists of 111

species and 784 elementary reactions. Although a comprehensive optimization study was conducted for the

H2/CO kinetic subset (Davis et al. 2005), USC Mech II as a whole is un-tuned. For the current optimization

study, the rate parameters of the H2/CO subset were restored to their nominal values listed in (Sheen et al.

2009). Additionally, the rate parameter for the OH + HO2 → H2O + O2 reaction was revised using the

expression of Baulch et al. (1992), which was shown to represent more accurately the recent measurements by

Hong et al. (2010) and Srinivasan et al. (2006). For the reaction H + O2 + M = HO2 + M, the collision

efficiency of H2O is quite uncertain and was taken as an optimization parameter.

The reaction model described above is referred to as the unconstrained or prior model. Further details of

this unconstrained model can be found in Sheen and Wang (2011a). Uncertainty factors for the Arrhenius

pre-factors in USC Mech II are provided in the supplementary materials of Sheen, et al. (2009). These factors

were taken either in consultation with literature compilations (e.g., Baulch et al. 1992; Baulch et al. 2005) or

through evaluation (Sheen et al. 2009, Sheen and Wang 2011). The uncertainty in the activation energy and

the pressure fall-off parameters was not considered in the present work.

Nominal model predictions using several available kinetic models are presented also along with USC

Mech II. These models are the Galway Mech, C1-C5 model of Curran and coworkers (Healy et al. 2010a,

2010b), the San Diego Mech for C1-C3 hydrocarbons (San Diego Mechanism webpage, 2009), and the four

butanol isomers model, or the Nancy Model, by Battin-Leclerc and coworkers (Moss et al. 2008).

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2.2 Computational Details

MUM-PCE evaluates the prior model against experimental datasets of S u

o . The reaction model was

represented by a multivariate polynomial response surface at each selected experimental condition. The

polynomials were obtained from sensitivity-analysis based method (Davis et al. 2004). Active parameters

were determined using a first-order sensitivity analysis by ranking the uncertainty-weighted sensitivity

coefficients. S u

o ’s were computed using the PREMIX code (Kee et al. 1986) along with the Sandia

CHEMKIN (Kee et al. 1989) and the transport (Kee et al. 1983) subroutine libraries. The transport

parameters were based on recent updates (Middha and Wang 2005, Dong et al. 2005).

Prediction uncertainty was quantified by stochastic spectral expansion (e.g., Reagan et al. 2003) and the

model was constrained by the experimental data using MUM-PCE. It should be noted that the model

posterior uncertainty space depends strongly on the uncertainties of the experimental measurements (Sheen

and Wang 2011). The posterior model uncertainties were quantified by MUM-PCE. It was assumed that the

rate parameters were distributed lognormally.

Table 1 lists the experimental flame speed targets for model optimization and uncertainty minimization.

Table 1. Experimental data set considered for optimization.

Series mixtures equivalence ratio,

1 CH4/air 0.7, 1.0, 1.3

2 C2H4/air 0.7, 0.8, 1.0, 1.2, 1.4

3 C2H6/air 0.7, 0.8, 1.0, 1.2, 1.4

4 C2H2/O2/N2 0.7, 0.8, 1.0, 1.2, 1.4,1.6

5 C3H6/air 0.7, 0.8, 1.0, 1.2, 1.4

6 C3H8/air 0.7, 0.8, 1.0, 1.2, 1.4

7 n-C4H10/air 0.7, 0.8, 1.0, 1.2, 1.4

8 iso-C4H10/air 0.7, 0.8, 1.0, 1.2, 1.4

9 1-C4H8/air 0.7, 0.8, 1.0, 1.2, 1.4

10 2-C4H8/air 0.7, 0.8, 1.0, 1.2, 1.4

11 iso-C4H8/air 0.7, 0.8, 1.0, 1.2, 1.4

12 1,3-C4H6/air 0.7, 0.8, 1.0, 1.2, 1.4

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3 Results and Discussion

Advances in the flame diagnostic methods over the last two decades allow us now to measure S u

o ’s of

hydrocarbon flames, especially for C1-C4 ones, to within an accuracy of ±2 cm/s or better. The prediction

uncertainty of the reaction model is larger, and is typically ±4~5 cm/s or larger (Sheen and Wang 2011b). To

illustrate this point, the top panels of Fig. 1 depict the probability density function and the 2 uncertainty

band (95% confidence level) predicted for S u

o ’s of methane/air and propane/air mixtures using the

unconstrained, prior model (top panels). Figure 1 depicts also selected experimental data and nominal

predictions of several kinetic models. It is seen that the combined uncertainty in the individual rate

coefficients results in an uncertainty in the model predictions which is substantially larger than the scatter in

the selected experimental data and the uncertainty of the most recent systematic measurement by Park et al.

(2011a & 2011b). These S u

o values, therefore, could pose some constraints to the reaction model and its

uncertainty. This observation underscores the purpose and utility of the S u

o data on model accuracy

improvements. As shown in the bottom panels of Fig. 1, the posterior Model I results in notably improvedS

u

o predictions, both in terms of the nominal values and reduced uncertainty bands. Additional work is being

conducted to better understand the utility of the current set of flame speed data and to formulate rational

strategies for model optimization and uncertainty minimization.

Figure 1. Laminar flame speeds of methane-air (left panels) and propane-air (right panels) mixtures at 1 atm pressure

and 298 K unburned temperature. Top panel: nominal prediction (solid line) and its 2 uncertainty band of the

unoptimized USC Mech II compared with experimental data (symbols) and predictions of Galway, Nancy and UCSD

models. Bottom panel: 2 uncertainty band of optimized USC Mech II (see text). Symbols are experimental data. The

grey intensity represents the normalized probability density function; the dashed lines correspond to predictions at ±2

uncertainty.

Galway

USC

UCSD

Bosschaart and de Goey (2004)Park et al. (2011a)

CH4 /air 0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.30

0.20

0.10

0.50

0.40

0.60

Park et al. (2011)

LminFameS

S

cm/s

30

20

10

50

40

40

30

20

10

0

0.7 0.8 0.9 1.0 1.1 1.2 1.3

Equivalence Ratio

0.16

0.14

0.12

0.10

0.06

0.04

0.02

0.08

GalwayUSC

UCSD

Bosschaart and de Goey (2004)Park et al. (2011b)

(b) C3H8 /air

Nancy

30

20

40

0.60

0.40

0.20

0.80

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Equivalence Ratio

50

40

30

20

10

Park et al. (2011b)

LminFameS

S

cm/s

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Two additional models, Galway Mech (Galway), and San Diego Mech (UCSD), predict S u

o ’s of CH4/air

mixtures within 2 uncertainty band of USC Mech II. Hence, major fundamental differences in these

reaction models, if any, are not exhibited in the reaction pathways critical to methane flames. For the C3H8

flames, discrepancies among the reaction models tested become substantially larger than the CH4 flames. For

example, the prediction of the Galway model lies outside of the uncertainty band of USC Mech II towards

high equivalence ratios, indicating that there exist fundamental differences between the two models.

Although prediction of S u

o ’s using the Galway model are in better agreement with the experimental data of

Bosschaart and de Goey (2004), notable discrepancies are observed when compared to data from the present

work.

The 2 model uncertainty of the prior model has also been performed for S u

o ’s of n-butane- and i-butane-

air flames (not shown here). For i-butane flames, there exist notable uncertainties in both the S u

o data and in

the reaction models. For example, both data from Davis and Law (1998) and Bosschaart and de Goey (2004)

lie outside of 2 uncertainty band of USC Mech II. A greater uncertainty is associated with unsaturated

hydrocarbons owing to the fact that S u

o ’s of these hydrocarbons can be sensitive to fuel specific reactions and

these rates parameters can be quite uncertain. For this reason, the 2 standard deviation predicted for S u

o ’s of

C3H6/air mixtures is notably larger than that of C3H8/air mixtures. Yet, the nominal prediction of S u

o ’s of

C3H6/air from the Galway model is still outside the 2 uncertainty bands of USC Mech II under fuel-rich

conditions. The problem stems from differences in the two models about the pathways of propene

consumption. In the Galway model, a larger fraction of propene forms ethylene than USC Mech II via

C3H6 + H C2H4 + CH3 . (R362)

This difference causes the Galway model to predict larger S u

o values for fuel-rich mixtures than USC Mech

II. An additional factor is that the experimental data of unsaturated fuel remain more scattered than their

saturated counterparts. For example, the S u

o values of propene-air mixtures reported by Davis and Law

(1998) differ from the recent measurements at USC by as much as 7-8 cm/s.

To understand the hierarchical nature of the reaction model and the impact of a particular set of the S u

o

data on constraining the model predictions and their uncertainties, several testing cases were designed in the

framework of MUM-PCE. Table 2 lists these test cases. The goal of these tests is to compare how the

predictions of S u

o of a particular fuel are improved as data pertaining to the kinetic subset are added to the

constraining target set: Test Ia is constrained only by the S u

o ’s of methane-air mixtures; Test Ib considers the

experimental S u

o ’s of both methane and all C2-hydrocarbons targets; Test Ic* represents a test in which only

the data for the specific fuel in question are considered and of course, there is a test Ic* for each fuel; Test Id

is constrained by the S u

o ’s of all C1- to C3- hydrocarbons as targets; Test I uses data of all C1- to C4-

hydrocarbons. Lastly, Test V uses S u

o ’s of butanol isomers with air mixtures in addition to all targets

employed in Test I as the targets.

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Table 2. List of the various test cases

Test No. Constraints

Ia CH4/air

Ib All C1- to C2- hydrocarbons with air

Ic* Single fuel only with air (see text)

Id All C1- to C3- hydrocarbons with air

I All C1- to C4- hydrocarbons with air

V Model I + butanol isomers with air

Regarding S u

o ’s of propene flames, the model uncertainty would remain large without considering its

own data as the targets, as shown in Fig. 2. The standard deviations of the models were computed to be 3 to

4 cm/s for the prior model and decrease only marginally for Models Ia and Ib. Only when the experimental

S u

o ’s of propene-air mixtures are introduced as targets does the standard deviation reduces to below 1 cm/s.

As expected, models ic* and 1d gave roughly the same flame speed uncertainties. In contrast, the model

uncertainties for propane can be largely constrained without using S u

o ’s of propane/air flames themselves as

the targets, as shown in Fig. 2. In this case, the standard deviation was reduced from 3 to 4 cm/s in the prior

model to 2 cm/s in Model Ia. When S u

o ’s of the propane/air mixtures themselves are considered as targets

(i.e., test Ic*), the standard deviation reduces to about 1 cm/s or lower.

Figure 2. Uncertainties (one-standard deviation) predicted for laminar flame speeds of (a) propane-air and (b) propene-

air mixtures at T u = 298 K and p = 1 atm.

(a) C3H8/air (b) C3H6/air

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Table 3. Effect of flame speed measurement uncertainty on the prediction uncertainty of the model

(Test case I for C3H8-air mixture at =1.2). All values are 2 values in cm/s.

Experimental Posterior Model I

∞ 7.4

2.5 3.12.0 2.6

1.0 1.70.5 1.2

The impact of the experimental uncertainty on the degree with which the measurement constrains the

reaction model was examined. The 2 uncertainties of the experimentally measured S u

o ’s of C3H8/air

mixtures are 1.0~1.2 cm/s. Reducing this uncertainty further would, of course, enhance the impact of the

measurement on model accuracy. Table 3 illustrate the variation of the prediction uncertainty of posterior

model I as a function of the measurement uncertainty. Two observations may be made. First, within the

framework of MUM-PCE, the prediction uncertainty of the model would always be larger than the

uncertainty of the experiment. Second and as expected, an improvement of the experimental accuracy

improves the model accuracy.

Finally, a reaction model was compiled by combining USC Mech II with the butanol reaction model of

Moss et al. (2008). The purpose was to explore how the model predictions for S u

o ’s of butanol isomers

flames respond to the foundational fuel chemistry constrained by small hydrocarbon S u

o ’s. A conservative

uncertainty span of 2.5 was assigned to all butanol related reactions and the true uncertainty for some of the

rate parameters can be larger than those assigned. The predicted model uncertainties for S uo ’s are shown in

Fig. 3 for the various butanol isomers flames. For n-butanol/air flames, the largest uncertainty of the priormodel is at = 1.0, whereas the prediction uncertainties are fairly uniform with respect to stoichiometry for

both i- and t -C4H9OH-air flames. The standard deviation of are shown in S u

o ’s of n-butanol/air flames in the

prior model is ~5 cm/s near stoichiometry, compared to ~4 cm/s under the fuel-lean and rich conditions. This

uncertainty was constrained quite notably by test I, to ~3 cm/s at = 1.0, and ~1 cm/s at = 0.8 and 1.4. For

i-C4H9OH/air flames the prior model uncertainty is ~5.5 cm/s at = 1.4, compared to Model I, which is

~1 cm/s. For t -C4H9OH/air flames, the model uncertainty was constrained to an even larger extent with

< 1 cm/s for all equivalence ratios considered. Meanwhile, the experimental data of the butanol isomers

themselves have little impact on the model predictions and their uncertainties.

The tests just discussed illustrate an important point in reaction model development. Often and especially

beyond C4 hydrocarbons, what impacts the model development and accuracy of a particular fuel is not always

in the availability of the measurement of the target fuel. Because of the hierarchical nature of reaction

kinetics, what matters is usually the availability of accurate fundamental combustion data for foundational,

small hydrocarbon fuels. This is especially the case for conditions where high-temperature chemistry

constitutes the rate limiting processes.

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Figure 3. Uncertainties (one-standard deviations) predicted for ’s of (a) n-C4H9OH-air, (b) i-C4H9OH-air, and (c) t -

C4H9OH-air mixtures at T u = 298 K and p = 1 atm.

4 Concluding remarksThe method of uncertainty minimization using polynomial chaos expansions (MUM-PCE) and a large set

of laminar flame speed data with systematically quantified uncertainties were utilized to examine the

underlying uncertainty of USC Mech II for laminar flame speed predictions. It is shown that the reaction

model accuracy and precision can be greatly improved through uncertainty minimization against the

aforementioned flame dataset.

An important portion of this study quantified the value, in a hierarchical manner, of laminar flame speed

data in a multi-parameter kinetic model optimization problem. The laminar flame speeds constrain more

strongly the heat release and chain branching reactions. It was demonstrated that a model constrained only by

the laminar flame speeds of methane/air flames, notably reduces the uncertainty in the predictions of the

laminar flame speeds of C3 and C4 alkane with air mixtures. The uncertainty in the model predictions for

flames of unsaturated C3-C4 hydrocarbons with air flames however remains largely unconstrained without

including the fuel specific laminar flames speeds in the experimental target data set. The constraint provided

by the laminar flame speeds of C3-C4 unsaturated hydrocarbons fuels with air mixtures could reduce notably

the uncertainties in the predictions of laminar flame speeds of C4 alcohols/air mixtures.

Finally, this study demonstrated that the application of MUM-PCE to a reaction model within a narrow

range of constraint conditions can lead to improvements of the accuracy of the model outside the conditions in

which the model is constrained.

Acknowledgements

This material is based upon work supported as part of the CEFRC, an Energy Frontier Research Center

funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award

Number DE-SC0001198.

(a) n-C4H9OH/air

(b) i -C4H9OH/air

(c) t -C4H9OH/air

S uo

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http://ignis.usc.edu/Mechanisms/USC-Mech%20II/USC_Mech%20II.htm

http://ignis.usc.edu/Mechanisms/USC-Mech%20II/USC_Mech%20II.htm



1b10-070rk-0048

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