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Automatic generation of reduced reaction

mechanisms for hydrocarbon oxidation withapplication to autoignition boundary prediction for

explosion hazards mitigation

R. Portera, M. Fairweathera, J.F. Griffithsb, K.J. Hughesb, A.S. TomlinaaSchool of Process, Environment and Materials Engineering andbSchool of Chemistry,

University of Leeds, Leeds LS2 9JT, UK

Abstract

In this work we present an automatic method for removing species and reactions from

comprehensive reaction mechanisms without significant detriment to model performance. Numerical methods are applied to a lean n-butane - air closed vesselsystem. A method for the automatic construction of closed vessel ambient temperature -composition (Ta ) ignition diagrams is presented, which is used to evaluate thecomprehensive and reduced models. Application of the quasi-steady stateapproximation to the reduced mechanism has been proven to significantly reduce thenumber of species with very little loss of output accuracy.

Keywords: Combustion, autoignition, lean n-butane-air, QSSA, sensitivity analysis.

1. Introduction

Despite our considerable knowledge of the potential hazards associated with thechemical process industries, explosion hazards continue to occur during hydrocarbonprocessing under partial oxidation conditions. Among the reasons for this is the changeof conditions that arise from process intensification, combined with an incompleteknowledge of the oxidation characteristics of the processed materials. The ability tocouple chemical kinetics with fluid dynamics and simulate these processes in reactivemulti-dimensional flows would be a powerful process engineering tool that wouldconstitute a significant advance in methodologies available to predict such hazards.

Detailed combustion kinetic mechanisms contain hundreds of chemical species andthousands of reactions, making them too computationally expensive to be solved incomputational fluid dynamics (CFD) codes. By adopting formal mathematical

procedures, more compact and computationally efficient kinetic models can begenerated by reducing the numbers of species and reactions from the detailedmechanisms. Currently, this involves running full kinetic models with multiple initialconditions in a non CFD-based environment, interpreting the results using localsensitivity methods, identifying and removing redundant species and reactions, and thentesting the reduced mechanisms. Many hours can be saved by automating these tasksusing programming techniques.

In this paper we describe software which can be used to automatically minimise thenumbers of chemical species and reactions without loss of important kinetic detail. Thecodes are based on the use of UNIX shell scripts to completely automate the utilisationof numerical integration and local sensitivity analysis software. Reduced chemicalmodels which can be used in higher dimensional simulations are obtained as output.

and 9th International Symposium on Process Systems EngineeringW. Marquardt, C. Pantelides (Editors)

2006 Published by Elsevier B.V.

16th European Symposium on Computer Aided Process Engineering

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The bench-mark is set by the performance of the full scheme and the criteria forperformance of the reduced models are matched to this.

As well as being fundamental to the potential hazards, an important basis forvalidation of the models is the ignition diagram as a function of ambient temperatureversus composition or pressure, in which is mapped a wide range of combustionregimes. The construction of the numerically predicted ignition diagram is also alaborious process which is amenable to automatic generation. This new software,encompassing automation in both areas, is applied in the present work to illustrate the

accurate reproduction of ignition and cool flame boundaries over a range of operatingconditions using significantly reduced kinetic schemes when compared with the fullmodels adopted at the outset.

2. Methodology and Models

The comprehensive model to which the methods were applied was derived at CNRS-DCPR, Nancy [1] for n-butane oxidation, comprising 125 species in 314 irreversiblereactions and 417 reversible reactions. The reversible reactions can be expressed asirreversible pairs equivalent to a total of 1148 irreversible reactions for the full scheme.

The resulting system of ordinary differential equations was solved using the SPRINTintegration package [2] for a closed vessel system with spatial uniformity assumed. Anambient temperature composition (Ta ) ignition diagram was automaticallyconstructed using developed software which can categorise the various non-isothermal behaviour such as 2-stage autoignition, cool flames, and slow reaction by monitioringtemperature and gradient changes in the predicted temperature profiles. The softwareworks by conducting a series of simulations over the selected temperature range of 550 750 K at specified intervals of 5 K and at a fixed pressure and composition where

exclusively 2-stage ignition occurs. Then a bisection method is employed in which thepartial fuel pressure is initially halved (while maintaining the total pressure), and thenprogressively adjusted in order to locate the boundary between ignition and cool flameor slow reaction behaviour, and similarly for the cases where cool flame behaviour isobserved, to locate the cool flame/slow reaction boundary. These calculations proceeduntil the desired level of accuracy is obtained, in this case to 0.5 torr. Similar softwarehas been developed to compute the pressure ambient temperature ignition diagram.The resulting Ta ignition diagram was used as the benchmark against which thereduced models were tested.

Using the ignition diagram as reference, a number of different operating conditionswere selected covering a representative range of the temperature/composition space atwhich sensitivity analysis and mechanism reduction are to be performed. A shell scriptwas set up to run the integration code at each chosen condition, and manipulate theoutput data files. Time points from the calculated temperature profiles at the chosenoperating conditions were automatically selected on the basis ofT and the gradient ofeach trajectory, as shown in fig. 1. Information related to these conditions and rate datafrom the mechanism were used to identify necessary species via the investigation of theJacobian matrix [3] using algorithms incorporated into the SPRINT code originallyimplemented in the KINALC package [4, 5]. The necessary species include selectedimportant species as defined by the user, and other species for which realistic

concentrations are required in order to reproduce the concentrations of importantspecies or important reaction features. The union of identified necessary species wastaken at the selected time points and the irreversible consuming and reversible reactions

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of all redundant species removed. The resulting mechanism was then converted toirreversible form for further analysis.

Via a similar process, techniques were then used to identify reactions that can beeliminated. Local sensitivity analysis was used to identify redundant reactions by

consideration of the rate sensitivity matrix F~

:

where kj is the rate parameter of the jth reaction and fi is the rate of production ofspecies i. The effect of a change of each rate parameter on the rates of production ofnecessary species is given by a least-squares objective function:

A reaction is considered important if it has a Bj value above a user specified threshold.Finally, principal component analysis based on the eigenvalue-eigenvector

decomposition of the cross-product matrix FFT~~

, was used to identify redundantreactions. Each eigenvector represents a set of coupled reactions whose relative

contributions are shown by the relative size of the eigenvector elements. Thresholdswere defined for the significant magnitudes of the eigenvalues and eigenvectors and thisprovided an automatic way of deciding which reactions can be eliminated [6-8].

Considerable improvement in the performance of the reduced models can be

achieved by using subsets of necessary species relevant for each specific time pointwithin the objective function, rather than the combined set of necessary species acquiredfrom the species reduction. This is illustrated in Fig. 2 by comparing reduced

mechanisms obtained using Equation 2, with either the full set of species included in thesummation i, or time point specific sets as identified by the local Jacobian matrix. Asimilar result would follow from principal component analysis.

Fig. 1. Automatically selected time pointsduring simulated 2-stage ignition in the viscinity

of the transition from cool flame to ignition.

The first time point was automatically selected

at 0.003 seconds.

)1(,~

j

i

i

j

k

f

f

kF

=

)2(.

2

=

ij

i

i

j

jk

f

f

kB

Fig. 2. Comparison of using all necessaryspecies or a subset at each time point in the

objective function. Unbroken line species

reduced, 715 reaction mechanism. Dotted line

subset reduced, 449 reaction mechanism.

Dashed line all necessary species reduced, 449

reaction mechanism.

0.35 0.40 0.45 0.50600

800

1000

1200

T

/K

t / s

0.35 0.40 0.45 0.50600

800

1000

1200

T

/K

t / s

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3. Application of the Quasi-Steady State Approximation

The application of the above sensitivity methods leads to a skeleton mechanism withall redundant species and reactions removed. However, in many cases the level ofreduction achieved by such methods is not sufficient for application of the chemicalmodel within complex flow computations. Subsequent reduction may be based onexploiting the time-scales present in the mechanism, with a range of reductiontechniques falling into this category including intrinsic low dimensional manifold(ILDM) based methods [9] and methods based on the application of the quasi-steady

state approximation (QSSA). QSSA based methods are commonly used in kineticmodel reduction by assuming that fast reacting species locally equilibrate with respectto the slower species within the system. The concentration of the QSSA species can

then be approximated via the algebraic expression 0,=qif rather than a differential

equation, where the superscript q denotes a QSSA species. In many cases QSSA speciescan be removed via simple reaction lumping. Alternatively, the concentration of species

cican be expressed in terms of the concentrations of other species in the system and the

rate parameters. Such expressions can be solved either analytically or via iterativetechniques for sets of highly coupled species. The choice species suitable for applicationof the QSSA can be determined in a variety of ways including using perturbationmethods. The instantaneous QSSA error for a single species, was defined in [10] using alocal linear perturbation method as:

where Jii is the diagonal element of the chemical Jacobian for species i. Although the

QSSA errors vary throughout the simulations, peaking during ignition, for many speciesthe errors remain below a certain threshold throughout. Using a tolerance of 1% acrossall selected time-points for the QSSA error, 31 QSSA species can be automaticallyidentified. Many have fairly simple reaction structures and therefore can be removed viathe methods illustrated in the following example.

In the reaction sequence shown in Fig. 3, solving the algebraic expressions resultingfrom the application of the QSSA for the highlighted species can be demonstrated to be

equivalent to the lumping of several of the individual reaction steps resulting in theremoval of RO2, QOOH and O2QOOH. The central part of the reaction sequence canthen be replaced by:

where

RH R RO2 QOOH O2QOOH

OH + productR' + alkene

1 2

-2

6

3

-3

4

-4

5

(3),J

1=

ii

i

i

s

i

f

cc

product,OHR2'

+

( )( )( )( ).1 544443333222'

2 kkkkkkkkkkkkk +++=

Fig. 3. Reaction sequence to which the QSSA was applied.

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Then R can be removed to leave the final reaction sequence:

where

In the simplest approach, k2is assumed to be a constant fraction ofk2, and set at thefraction calculated in the region of maximum flux through R to OH + product. A rate of production analysis of the full scheme shows this to be a good approximation in this

instance, and applying it gives simulated temperature profiles in excellent agreementwith those obtained from the original scheme. The ratio of k7 to k8 is not constant, andchanges significantly with temperature, favouring k8 at low temperatures and switchingover to k7 at high temperatures. Even so, assuming a constant ratio based on thatapplicable at low temperatures still gives very good agreement in the simulatedtemperature profiles, with only slight deviation at the later times and highertemperatures where this approximation is no longer valid. A more rigorous approach isto program directly the variation ofk2, k7 and k8 with temperature, although this resultsin a loss of compatibility of the reduced mechanism with commercial simulationpackages such as CHEMKIN. Of the QSSA species identified, 14 were easily removedby applying the method highlighted above resulting in a final mechanism of 58 speciesand 270 reactions.

4. Model Validation and Application of Sensitivity Analysis

Fig. 4 shows the experimental and simulated Ta ignition diagrams for n-butane +air. The qualitative features of the experimental Ta ignition boundary [11], shown inFig. 4, are captured by the numerical models showing both cool flame and two stageignition behaviour. The reverse s shape of the ignition boundary is displayed by the

+=

6

'

2

617

kk

kkk .

6

'

2

'

218

+=

kk

kkkand

product,OHRH

alkeneRRH

8

'7

+

+

0.0 0.5 1.0 1.5 2.0 2.5550

600

650

700

750

Slow

Reaction

Ta

/K

% n-C4H

10by volume in air

Cool flame

Slow

Reaction

Experiment

2-stage

ignition

1.6 1.8 2.0 2.2550

600

650

700

750

2-stage

ignition

Cool flame

Ta

/K

% n-C4H

10by volume in air

Fig. 5. Comparison of full scheme (solid line),

species reduced (dotted line) and QSSA reduced

(dashed line) Ta ignition boundaries.

Fig. 4. Comparison of experimental and full

scheme Ta ignition diagrams.

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models and this is an important validation. However, quantitative disagreements withthe experiment remain, especially at higher temperatures where the model over-predicts

the autoignition temperatures. This may imply a shortcoming in the way that theintermediate molecular products that lead to high-temperature reactions are interpreted.

There may also be some discrepancy due to inhomegeneities of temperature in theunstirred vessel [11].

Comparison of the model results shows that both reduced mechanisms illustrated inFig.5 reproduce the behaviour of the full scheme extremely well. The scheme produced by removal of redundant species from the full scheme produced a mechanismcomprising of 72 necessary species and 713 irreversible reactions, generated a Ta

ignition diagram that matched that of the full scheme very closely. Further reduction byremoval of redundant reactions and applying the QSSA to remove a further 14 species,giving a scheme of 58 necessary species and 270 reactions, also behaved very well, withonly minor deviations to the full scheme prediction. It is possible to apply different cut

off values in these methods in order to reduce the mechanisms still further but at a costof a reduced level of agreement with the full scheme. By specifying higher thresholdsfor the eigenvalues and eigenvectors of principal component analysis, prior to QSSA, itis possible to reduce the numbers of reactions even further. However, the increasingerror induced by this reduction was considered to be unsatisfactory since it gave littleextra computational saving.

5. Conclusions

Software for the automatic construction of ignition diagrams has been developed.

Programming techniques have allowed the automatic and systematic reduction of a leann-butane - air kinetic model, simulated in a closed vessel. Comparisons of the predictions of full and reduced schemes have shown that the numbers of species andreactions have been successfully reduced. Further reductions have been achieved using

the quasi-steady state approximation to lump reactions and further reduce species.

AcknowledgementThe authors gratefully acknowledge financial support from the EU (EVG1-CT-2002-

00072-SAFEKINEX) and from EPSRC (GR/R42726/01).

References[1] www.ensic.u-nancy.fr/DCPR/Anglais/GCR/softwares.htm[2] M. Berzins, R.M. Furzland, Shell Research Ltd., TNER 85058, 1985.

[3] T. Turnyi, New J. Chem. 14 (1990) 795-803.[4] www.chem.leeds.ac.uk/Combustion/kinalc.htm[5] T. Turnyi, Reliab. Eng. Syst. Safe., 57 (1997) 41-48.[6] S. Vajda, P. Valk, T. Turnyi, Int. J. Chem. Kinet., 17 (1985) 55-81.[7] A.C. Heard, M.J. Pilling, A.S. Tomlin, Atmos. Environ. 32 (1998) 1059-1073.

[8] A.S. Tomlin, T. Turnyi, M.J. Pilling, in: M.J. Pilling (Ed.), Low TemperatureCombustion and Autoignition, Elsevier, Amsterdam, 1997, p. 293.[9] U. Maas, S.B. Pope, Combust. Flame 88 (1992) 239-264.[10] T. Turnyi, A.S. Tomlin, M.J. Pilling, J. Phys. Chem. 97 (1993) 163-172.

[11] M.R. Chandraratna, J.F. Griffiths, Combust. Flame 99 (1994) 626-634.

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