A graph-based approach to developing adaptive representations of complex reaction mechanisms

Combustion and Flame 155 (2008) 585–604www.elsevier.com/locate/combustflame

A graph-based approach to developing adaptiverepresentations of complex reaction mechanisms

Kaiyuan He, Marianthi G. Ierapetritou, Ioannis P. Androulakis ∗

Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey,Piscataway, NJ 08854, USA

Received 9 January 2008; received in revised form 9 May 2008; accepted 12 May 2008

Available online 13 June 2008

Abstract

An effective adaptive mechanism reduction approach based on flux graph clustering is proposed in this paper.The instantaneous element flux is quantified and considered as a proxy for describing the reactive propensitiesof the system. Our underlying hypothesis is that even though particular conditions may be characterized by amultitude of combinations of species mass fraction, T , and P , the essential chemistry, and hence the reactionpropensity of the mixture that is active under this family of conditions, is the same. Therefore, we opt to usethe instantaneous fluxes through the active reactions as an intrinsic property of the system. Flux graphs are firstconstructed for the chemical reaction system under numerous conditions aiming at capturing the attainable region.Similarity between flux graphs is quantified through the distances between corresponding vectors, using the cosinecoefficient and a novel graph-distance metric taking into account the magnitude of each flux and the activitydistribution of different fluxes. A hierarchical clustering algorithm is implemented to group similar instantaneousflux graphs into clusters, and consequently a reduced mechanism is generated for each cluster. A search algorithmis defined afterward to assign a new query point to a particular flux graph cluster, and subsequently the reducedmechanism associated with this cluster is used to describe the system at this time point. Finally, the methodologyis demonstrated using n-pentane combustion in an adiabatic plug flow reactor model and a pairwise mixing stirredreactor model.© 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Flux graph; Graph clustering; Kinetic model; Adaptive reduction

1. Introduction

Fuel combustion is a very active research field,and a number of detailed kinetic mechanisms havebeen developed to model a variety of chemical pro-cesses [1–7]. Many important applications, including

* Corresponding author. Fax: +1 732 445 3753.E-mail address: [email protected]

(I.P. Androulakis).

0010-2180/$ – see front matter © 2008 The Combustion Institute.doi:10.1016/j.combustflame.2008.05.004

aerospace propulsion, car engine simulation and de-sign, and various manufacturing processes, require adetailed understanding of both fluid dynamics and ki-netics. However, detailed simulation of reactive flowsystems using complex kinetic mechanisms consist-ing of hundreds of species and thousands of reactionsis a computationally very demanding task. Henceconsiderable effort has been invested in the represen-tation of complex kinetic models by simpler, reducedmodels that can largely alleviate the computationalcomplexity while still retaining considerable accu-

Published by Elsevier Inc. All rights reserved.

586 K. He et al. / Combustion and Flame 155 (2008) 585–604

racy [8–15]. Significant effort has therefore been in-vested in developing reduced representations of com-plex reaction mechanisms. Among the first reductiontechniques were lumping approaches based on rig-orous mathematical formulations, such as those pro-posed by Wei and Kuo [9], that lump concentrationsof chemical species into a reduced species set. Thelumped concentrations are defined as linear combi-nation of the original concentrations. Alternatively, anumber of reduction methodologies based on iden-tifying redundant species that may be omitted fromthe mechanism without causing significant error inthe modeling targets were proposed by Turanyi [15].A sensitivity analysis proposed by Rabitz et al. [16]aims at calculating the derivatives, called sensitivitycoefficients or sensitivity densities, of quantities ofinterest with respect to specific parameters. Concen-tration sensitivity, rate sensitivity, and feature sensi-tivity are usually used to eliminate species or reac-tions that are not important to these parameters. Thesensitivity concept is widely used, because it is rela-tively simple to apply and provides critical informa-tion related to parameters and variables but does notprovide direct information about decoupling specieswith different timescales. The quasi-steady-state ap-proximation (QSSA) has been invoked by numerousresearchers [10,17–23] in an attempt to effectivelyutilize the time-scale separation inherently present ina reaction mechanism. Reduction is achieved by iden-tifying species in (quasi) steady state and certain re-actions in partial equilibrium, and replaces the ordi-nary differential equations (ODEs) of the steady statespecies by their corresponding algebraic equations.The transition from a system of ODEs to a system ofdifferential–algebraic equations (DAEs) reduces thestiffness of the original ODE system and solves itby explicit methods. A number of very successful al-ternative QSS implementations have been proposed,such as reaction rate analysis [10], augmented re-duced mechanism (ARM) [20] generation, intrinsiclow dimensional manifolds (ILDM) [19], and compu-tational singular perturbation (CSP) [17,18,22]. Con-ceptually all QSS approaches share the same under-lying characteristics. The dynamics of the reactionsystem are split into fast and slow modes. The slowmodes are considered to be dominant and the fastmodes are projected onto this slow manifold. Theoriginal system of differential equations is replacedby a set of coupled differential equations and alge-braic equations. Optimization-based approaches wereproposed in [13,24] to generate reduced mechanisms.The approaches first obtain species and temperatureprofiles with detailed kinetic mechanisms, and sub-sequently eliminate species and reactions that are oflesser importance while retaining desired accuracy.A genetic algorithm (GA) was introduced in [25],

which encodes N reactions as N binary variables. Thealgorithm first generates a large population of binarystrings, each of which represents a reduced reactionset. For each reduced set, the profiles are generatedby integrating corresponding ODEs and the accuracyis evaluated by comparing with profiles of detailedkinetic mechanism. The generated population of so-lutions is modified by the GA operators to create newand better population repeatedly until desired crite-ria is satisfied. A graph-based mechanism reductiontechnique was proposed by Lu and Law [26–28] andmodified by Pepiot and Pitsch [29]. This approachfirst constructs the directed relation graph (DRG) witheach node in the DRG uniquely mapped to a speciesin the detailed mechanism and edges weighted inproportion to the contribution of one species on theproduction rate of another (interaction coefficient).By applying a threshold value εR, edges with valuessmaller than the threshold are considered to be negli-gible. The species that are reachable from one speciesare defined as the dependent set of this species. Thus,once the principal species in the chemical system areidentified, a skeletal reduced mechanism is generatedas the union of the dependent sets of the principalspecies. The directed relation graph method is able toidentify a small set of species that is vigorously validfor their local conditions; however, the selection ofprincipal species and the threshold value, εR, are notdirected related to an error metric. Thus, in [29], theDRG method is modified and a directed relation graphmethod with error propagation (DRGEP) was pro-posed. In the DRGEP method, a path-dependent co-efficient, which evaluates error propagation in speciesselection, is used instead of the interaction coefficientin the DRG method.

An adaptive reduction method, in situ adaptivetabulation (ISAT), was introduced in [12]. The ra-tionale behind ISAT is to store the information ona reactive system in a table-structured database, andthe algorithm looks up similar entries for new querypoints during the simulation and uses the informationlinked to these similar entries to describe it. Any newreaction states that cannot be described or interpolatedon the basis of existing entries are integrated as newentries in the table. In [8,30], a set of reduced kineticmodels are used in a simulation where each reducedmodel is used only under reaction conditions where itcan predict the system within an allowable tolerance.In the adaptive simulation scheme, appropriate re-duced mechanisms are selected to address a differentrange of conditions through an effective search algo-rithm during the simulation. Most reduction schemesevaluate the appropriateness of a mechanism based onlocal temperature, pressure, and composition charac-teristics. However, reaction systems at different tem-peratures and with different compositions may acti-

K. He et al. / Combustion and Flame 155 (2008) 585–604 587

vate the same pathways and reactions [31]. Elementflux, introduced by Revel et al. [32], allows one tocapture the reactive propensity of the system morecomprehensively. In our previous work, we extendedthis concept in order to develop time-integrated el-ement flux graphs and explored their potential formechanism reduction, as well as exploring the hy-pothesis that the element flux graphs provide a betterdescription of the reactive characteristics of the re-action mixture [32,33]. The flux graph analysis andmechanism reduction share some intrinsic character-istics: they both identify active species and reactionsdominating the local reaction system at specific timepoints, which is the basis of the proposed reductionscheme. In [32,33] it was shown that the reactionsystem under different conditions, as would be de-fined by species compositions, temperature, pressure,or sensitivities, may have the same reactive propen-sities, which are directly represented by the chemicalflux graph. Based on our prior work, we introduce inthis paper a new instantaneous flux graph method foranalyzing and reducing complex kinetic mechanisms,and developing adaptively reduced representations.

Since our approach attempts to capture the reac-tion propensities of the mixture through the use offlux graphs, based on the hypothesis that similar fluxgraphs are associated with similar reaction mecha-nisms, it becomes important to be able to quickly andaccurately compare and cluster the graphs emanatingfrom the analysis of the multitude of possible reactionconditions.

A clustering algorithm is thus used so that simi-lar instantaneous flux graphs are grouped into clustersand consequently a different reduced mechanism cor-responding to each graph is generated. The approachfor developing reduced mechanisms implements theconcepts introduced in [33], where species participat-ing in the most active pathways are selected to estab-lish reduced mechanisms.

The result of this first stage is the constructionof a library of clusters and corresponding reducedmechanisms that can predict the reactive propensi-ties of each cluster. The next step, in order to developthe adaptive chemistry representation, is to define asearch algorithm that will assign a new query point toa particular cluster and subsequently a reduced mech-anism associated with this cluster. To achieve thisclassification, we implement a very effective, yet sim-ple, nearest neighbor classification scheme throughthe definition of a simple distance metric between thecenter of each cluster in the library and the flux de-fined based on the new query point. The cluster withcenter closest to the query point defines the currentactive mechanism.

The proposed methodology is demonstrated us-ing a detailed nC5 mechanism with 385 species and

1895 reactions [34] using the pairwise mixing stirredreactor (PMSR) model [12] to assess the significantcomputational enhancements and a plug-flow reac-tor (PFR) model to assess the ability of the adaptivescheme to predict autoignition delays.

We begin the presentation by first introducing thedefinition of elemental flux graphs followed by a de-scription of the flux graph clustering and search meth-ods in Section 3. The concept as applied to mech-anism reduction is then described in Section 4, fol-lowed by simulating detailed analysis of two reactormodels, the pairwise mixing stirred reactor (PMSR)and the plug flow reactor (PFR), for n-pentane com-bustion, described in Section 5. Finally, a detailed dis-cussion of the results and conclusions are presented inSection 6.

2. Flux graph construction

The concept of element flux analysis, first intro-duced by Revel et al. [32], allows one to identifycritical transitions in a reaction system. Accordingto [32], the instantaneous elemental flux of atom A

from species j to species k through reaction i is de-fined as

(1)Aijk(t) = qi(t)nA,j nA,k

NA,i,

(2)Ajk(t) =NR∑i=1

Aijk(t),

where qi is net production rate of reaction i (mol s−1),nA,j is the number of atoms A in species j , nA,k isthe number of atoms A in species k, NA,i is the to-tal number of atoms A in reaction i, and NR standsfor the number of reactions in which species j and k

are involved. Based on the instantaneous flux anal-ysis, we introduced the concept of time-integratedflux analysis to derive global information on criticalpathways and hence develop globally reduced kineticmechanisms [33]. By numerically integrating the in-stantaneous flux pointers, temporal dependencies canbe eliminated and time-independent reaction path-ways identified. The time-integrated flux indicator isdefined in

(3)

AFROM,TO =∫ τt=0 AFROM,TO(t) dt∑

FROM′∑

TO′∫ τt=0 AFROM′,TO′(t) dt

,

where AFROM,TO is the time-integrated flux pointer,

AFROM,TO is the instantaneous flux pointer defined inEq. (2), and τ is reaction time. This time-integratedflux pointer allows one to characterize the main trans-formations that take place during the reaction pe-riod. By sorting the flux values between all possible

588 K. He et al. / Combustion and Flame 155 (2008) 585–604

Fig. 1. Reduced flux graph for n-pentane oxidation.

source–sink pairs, one can set a high portion cut-off that can identify skeletal source–sinks. Once theskeletal source–sinks have been identified, a reducedspecies set and related reactions can be derived to rep-resent key pathways of the reaction system. The anal-ysis can be repeated for all the elements in a mecha-nism to obtain a complete picture characteristic of thechemical transformations.

In [33], n-pentane combustion is used to deriveskeletal pathways, which are represented in Fig. 1.The pathways summarized in this graph are composedof species identified through setting a cutoff at 90%of time-integrated carbon fluxes of pentane oxida-tion. This user-defined cutoff is a critical parameterfor capturing skeletal pathways on different compre-hensive levels. Forty-seven source–sink pairs repre-sented by forty-seven edges are included in this graph,which are weighted proportionally to the chemicalflux between corresponding nodes. If monitored in

a dynamic mode, the arc weights of the flux graphevolve as a function of reaction time. Hence, basedon different edges that are active under different ini-tial conditions and at different reaction times, a highportion cutoff will result in different pathways andconsequently different flux graphs. In other words,flux graphs composed of critical pathways identifiedby setting a high portion cutoff to time-integrated fluxanalysis are not able to predict reactive propensitiesfor diversified initial conditions and different reactiontimes. Therefore, a comprehensive flux graph includ-ing all the 1954 edges of the pentane oxidation systemis considered to provide a detailed characterization ofthe reaction propensity of the system. The mechanismproposed by Curran et al. [34] is used in the analysis,and the thermophysical properties and reaction ratedefinitions are provided by CHEMKIN [35].

In the present work, it was found that the existenceof quasi-steady-state species and partial equilibrium

K. He et al. / Combustion and Flame 155 (2008) 585–604 589

reactions induces some information loss in terms ofthe element flux calculated by Eq. (1) and Eq. (2).When the reaction rate qi is positive, which meansthat reaction i is proceeding in the forward direction,Aijk represents the flux-out of species j to speciesk through reaction i. When qi is negative, indicatingthat reaction i is proceeding in the reverse direction,Aijk represents the flux-in of species j from species k

through reaction i. For quasi-steady-state species, theflux-out and flux-in are large compared to the net flux,which is calculated by the sum of Aijk ’s over all thereactions. Thus the importance of quasi-steady-statespecies may be significantly underestimated basedon Eqs. (1) and (2). In addition to quasi-steady-statespecies, partial equilibrium reactions may also resultin information loss in calculating flux using Eq. (1).The reaction rate qi , in Eq. (1) is the net reaction rate,which is calculated as the forward reaction rate mi-nus the reverse reaction rate. For partial equilibriumreactions, the forward reaction rates and the reversereaction rates are much larger than the net reactionrates. Thus when reaction i is in partial equilibrium,the flux pointer Aijk is small, although the transitionof element A between species j and k through reac-tion i is very fast. To address the quasi-steady-statespecies and partial equilibrium reactions, Eq. (1) hasbeen improved as

(4)Aijk(t) = (∣∣qifwd(t)∣∣ + ∣∣qirev(t)

∣∣)nA,j nA,k

NA,i,

where qifwd and qirev are the reactions rates of for-ward and reverse reactions, respectively. The abso-lute values of both forward and reverse reaction ratesin Eq. (4) ensure two aspects: (a) the importance ofspecies involved in partial equilibrium reactions is es-timated appropriately by the flux pointer; (b) the flux-in and flux-out of species in quasi-steady state areevaluated separately and cancellation between flux-inand flux-out is avoided.

In the present work, the atomic fluxes for differ-ent elements (C, H, O, and N) are computed basedon reaction rates through Eqs. (4) and (2) and ac-tive source–sink pairs are identified. For a specificelement, flux analysis identifies reactive species andsource–sink pairs. Hence, a flux graph can be con-structed in which nodes represent species and arcs areweighted proportionally to the rate of element flux be-tween species. In the present work, carbon-atom fluxis analyzed to identify active species that describe ma-jor carbon pathways. However, species that are crucialto H/O chemistry are also included in the mechanismto describe the H/O chemistry as well as the carbonchemistry. Graph visualization software, GRAPHVIZ[36], is adopted in this work to construct graphs.

3. Graph clustering and search algorithm

Once the flux graphs have been derived, we moni-tor the graph in a dynamic way as the reaction systemevolves. Thousands of instantaneous flux graph snap-shots can be obtained during the reaction process. Inorder to identify similar snapshots and group theminto clusters to represent different stages of the re-action progress, a graph comparison technique andclustering algorithm, along with a search method, areneeded. In brief, a clustering algorithm involves thefollowing components: a set of objects, such as graphsin our work; a measure to assess similarity (or dis-similarity or distance) between pairs of objects; and aclustering method that identifies groups of both inter-nally homogeneous and externally distinct based onthe similarity data. Bhavsar et al. [37] introduced agraph similarity measure technique in an e-businessenvironment, where production and service descrip-tions of sellers and buyers are represented as node-labeled, arc-labeled, and arc-weighted trees. Similar-ity between trees is assessed by summing up similar-ities between each pair of weighted arcs in the trees.In [38], a chemical structure is described as a vectorin N -dimensional space, where N is the number of at-tributes that the chemical structure contains. The dis-tance or similarity between chemical structures can beevaluated through the distance between correspond-ing vectors using various distance or similarity coef-ficients. Later work [39] introduced a region-growingalgorithm to partition graph-based images. The algo-rithm first assigns each pixel of the image to a dif-ferent region. Then at each iterative step, adjacentregions are merged if their feature similarity satis-fies the homogeneity criteria. The criteria decreasewith the iteration and the whole algorithm stops whenno pair of regions satisfies the homogeneity criteria.Another two categories of graph comparison tech-niques, graph-based approach and fingerprint-basedapproach, were compared and adopted by Raymondet al. [40,41] and Yoshida et al. [42]. In the direct-comparison approach, similarity between two graphsis assessed using maximum common subgraphs [40].The maximum common subgraphs are identified andthe number of nodes and edges of these subgraphsare used to measure the similarity. In this approach,similarity can be measured directly based on graphs.However, identification of maximum common sub-graphs can be very expensive, and graphs with iden-tical structures cannot be compared using this ap-proach. In the fingerprint-based approach, importantfeatures of a graph are described as a bit string, andthe similarity between graphs is evaluated by com-paring corresponding bit strings. The difficulty withthis approach is the determination of key features; i.e.,graphs with components (nodes and arcs) of equal or

590 K. He et al. / Combustion and Flame 155 (2008) 585–604

similar importance cannot be grouped properly us-ing this approach. A novel approach that relies onthe Markov cluster (MCL) algorithm was proposedby Van Dongen and co-workers [43]. This approachuses a graph to represent a protein dataset, proteinsas nodes, and similarity between proteins as weightedarcs. The objective is to identify protein families fromthe protein dataset. Similarity between two proteins ismeasured through BLAST and hence a similarity ma-trix whose elements represent similarity metrics forany pair of proteins in the data set can be constructed.Based on the similarity matrix, the MCL algorithmis employed to identify nodes (proteins in this case)that are closely linked (i.e., with high similarity). Af-ter several iterations, the graph can be separated intosegments, each of which is considered as a proteinfamily. However, this approach is fast and accurateonly for limited size of the dataset, while for datasetswith tens of thousands of objects this approach con-sumes a substantial amount of calculation time.

In our work, based on the complicated nature ofthe flux graphs and equivalent importance of graphcomponents, each instantaneous flux graph is consid-ered as an N -dimensional vector, where N stands forthe total number of reactive fluxes in the system. Thefluxes are monitored over time and compared in orderto assess similarities among them, which will denotesimilarities in the underlying chemistry. Representinga flux graph G by a vector V , we qualify the similaritybetween flux graphs by evaluating the distance be-tween corresponding vectors using either cosine co-efficients [44] or normalized first-order Minkowskidistance. The cosine coefficient calculates the cosineof the angle between two vectors and considers theangle as an indication of vector similarity. The first-order Minkowski distance evaluates distance betweentwo vectors by summing up the difference betweenthe values for each dimension of the vectors.

Based on the wide range of conditions in a com-bustion system, a training data set of flux graphs isrequired to cover various reaction conditions to beextensively feasible. Therefore, n-pentane oxidationprocess is simulated using an adiabatic plug-flow re-actor (PFR) model under 18 different initial temper-atures, every 100 K from 600 to 1100 K, and dif-ferent fuel equivalence ratios, 0.5, 1.0, and 2.0. Of47,189 data points, each represents an instantaneousflux graph of chemical transformation at a specific re-action time; they are collected from 18 different initialconditions (6 different initial temperatures and 3 dif-ferent equivalence ratios) to establish the training set.The training data set, composed of 47,189 instanta-neous flux graphs, can be perceived as a data set of47,189 1954-dimensional vectors. Based on the mul-tidimensional vector data set, a hierarchical clusteringalgorithm is used to obtain k-way clustering, where k

represents the number of clusters to be exported in theclustering process. To implement hierarchical cluster-ing, first we need to define the distance (or similarity)coefficients to be used to assess the distance (or simi-larity) among vectors. Two coefficients are introducedin this work: one is cosine coefficient, as defined in

(5)SAB =∑j=n

j=1 xjAxjB[∑j=nj=1(xjA)2 ∑j=n

j=1(xjB)2]1/2

,

and the other is normalized 1-norm distance (Min-kowski distance of order 1) as defined in

(6)DAB =j=n∑j=1

∣∣∣∣ xjA∑i=ni=1 xiA

− xjB∑i=ni=1 xiB

∣∣∣∣.Ordinary 1-norm distance is also presented as

(7)DAB =j=n∑j=1

|xjA − xjB |,

where xjA and xjB express values of the j th elementof vectors A and B , respectively; and n is the dimen-sion of the vector. The cosine coefficient (Eq. (5))evaluates similarity among multidimensional vectorswhile the normalized first-order Minkowski distance(Eq. (6)) indicates distance among vectors. The onlydifference between normalized first-order Minkowskidistance (Eq. (6)) and ordinary first-order Minkowskidistance (Eq. (7)) is that the vectors are normalizedfirst and then subjected to comparison in Eq. (6),while raw data are compared directly in Eq. (7). Thereason for normalizing the vectors before calculatingdistances in this work is that normalization will cap-ture the pattern of magnitude distribution of vectors,which represents the activity distribution of edgesin corresponding flux graphs. Therefore, two vec-tors will be defined to be close to each other if theirnormalized values are similar, disregarding their realvalues, which could be on different scales. Thus thenormalized first-order Minkowski distance will groupvectors with similar patterns of distributions, ratherthan similar magnitudes, to the same cluster. As willbe shown in the next section, this process will be con-sistent with the methodology of developing reducedmechanisms. Fig. 2 represents two 47-dimensionalvectors (corresponding to a flux graph with 47 arcs,presented in Fig. 1) based on simulations with dif-ferent conditions at different reaction times; Fig. 2ais normalized, while Fig. 2b illustrates original data.One can see from these two figures that even fluxgraphs with different values may have a similar ac-tivity distribution. The two snapshots are taken fromdifferent conditions and reaction times; therefore theweights of edges are different. However, since the

K. He et al. / Combustion and Flame 155 (2008) 585–604 591

(a)

(b)

Fig. 2. Distribution patterns of two different flux graphs:(a) the distribution of original flux value and (b) the distribu-tion of normalized flux value.

distribution patterns are similar, these two instanta-neous flux graphs can be described by similar reducedmechanisms, and hence can be grouped into the samecluster.

Given the distance metrics (or any similarity met-ric), multiple trajectories obtained from reaction sim-ulations are used to generate alternative reaction fluxgraphs, which are subsequently clustered to identifysimilarities among them. There may be several dif-ferent algorithms that can be implemented for theclustering. One of these methods is the hierarchic ag-glomerative approach [45], which merges pairs of ob-jects that are most similar into a cluster and considersthis cluster as a single object in the following itera-tion. This step is repeated and hierarchies are builtup by progressively merging clusters. Another com-monly used clustering algorithm is the k-means algo-rithm [46]. The algorithm uses an iterative refinementthat starts by partitioning the objects in the datasetinto k initial sets, either at random or using someheuristic data. It then calculates the mean point, orcentroid, of each set and constructs a new partition

by associating each point with the closest centroid.Then the centroids are recalculated for the new clus-ters, and the algorithm continuously alternates thesetwo steps until the centroids are no longer changed.Many other methods, e.g., the Jarvis–Patrick sharednearest neighbor method [47], also have been exten-sively used in previous studies of clustering. Detailedreviews can be found in [41,48]. Due to the large scaleof the instantaneous flux graph dataset, the hierarchi-cal clustering algorithm is adopted and the CLUTOpackage [49] is used in this work to identify clustersbased on the generated distance matrix. Both cosinecoefficients and a normalized first-order Minkowskidistance function were implemented in this work, andsimilar results were obtained. Since the cosine coeffi-cient is optimized in CLUTO based on its properties,the remaining part of the methodology and results arepresented using cosine coefficient.

The question that arises then is how many clus-ters are needed to represent the dataset. A selectionfunction is provided in the CLUTO package and isemployed to assess the quality of clustering [50].The function, named H-2 in the package, is definedthrough the equations

(8)H-2 = I2

ε1,

(9)I2 =k∑

i=1

√ ∑ν,u∈Si

sim(ν,u),

(10)ε1 =k∑

i=1

ni

∑ν∈Si , u∈S sim(ν,u)√∑

ν,u∈Sisim(ν,u)

,

where sim(ν,u) represents the similarity betweenvectors ν and u and is evaluated by cosine coefficient,k is the number of vectors in the data set, ni is thenumber of vectors in cluster i, and Si denotes vectorsin cluster I , while S stands for the vectors in the dataset except for those included in cluster i. Hence, I2quantifies the internal similarity (similarity betweenvectors in the same cluster) while ε1 quantifies exter-nal similarity (similarity between vectors in differentclusters) based on internal similarity. High internalsimilarity and low external similarity represent a goodclustering. Therefore, the objective of the clustering isto maximize the function H-2.

The plot of H-2 for different numbers of clustersgenerated based on the training set is shown in Fig. 3.As seen from Fig. 3, the magnitude of H-2 increasesas the number of clusters increases and approaches amaximum, which is obtained when the cluster num-ber reaches 47,189 (i.e., every single vector is a clus-ter). This trend indicates that more clusters will re-sult in better classification. However, as the numberof clusters increases, more CPU time will be con-sumed in the following reduction and classification

592 K. He et al. / Combustion and Flame 155 (2008) 585–604

Fig. 3. Values of H-2 function versus different numbers ofclusters.

step. To determine the best compromise, which re-tains high quality of clustering while keeping clusternumber to a reasonable value, we set a criterion of95% of the maximum H-2, and as shown in Fig. 3,the minimum number of clusters to obtain 95% of themaximum H-2 is 30. Hence, 30 clusters are gener-ated in the hierarchical clustering algorithm using theCLUTO package. To obtain some knowledge aboutthe temperature and species composition regions thateach cluster covers, different temperatures and fuelmass fractions (n-pentane mass fraction in this work)covered by each cluster are plotted in Fig. 4, and thetemperatures and fuel mass fraction ranges coveredby six typical clusters are illustrated in Fig. 5. Twopoints can be interpreted from Figs. 4 and 5: (a) eachcluster, which is generated based on flux similarity,covers a wide range of temperatures and species com-

Fig. 4. Temperature and pentane mass fraction range covered by each cluster.

Fig. 5. Temperature and pentane mass fraction ranges covered by clusters 5, 13, 25, 12, 14, 15, and 18.

K. He et al. / Combustion and Flame 155 (2008) 585–604 593

Fig. 6. Multidimensional scaling by 2-D projection of the distribution of 30 cluster centers.

positions, and (b) some temperature and species com-position regions covered by different clusters overlapeach other, for example, cluster 12 and cluster 13 inFig. 5, which means that even for the same temper-ature and species composition, chemical fluxes aredifferent. These two points confirm one of the mo-tivations of this work: reaction systems with differenttemperatures and species compositions may have thesame reactive propensity, while reaction systems withthe same temperature and species composition mayhave different reactive propensities. The flux graphrepresentations can predict the reactive propensitiesof the system more directly and accurately than thetemperature and species composition.

After the clusters are generated, a search algorithmis needed to assign a new query point to a particularcluster. In this work, we introduce a search algorithmwith a rationale similar to that for nearest-neighbor(N-N) search [51]. N-N algorithm searches the objectin the dataset that is nearest to a new query object,and the new object is classified into the category ofthis nearest neighbor. However, instead of searchingamong all the data points in the training set to find thenearest neighbor, we only search among centers of theclusters. The mean vector of the cluster is consideredas its center. As introduced above, 30 clusters are gen-

erated, and therefore, 30 cluster centers are used torepresent the corresponding clusters. The distributionof 30 centers is assessed by multidimensional scaling(MDS) [52] and is shown in Fig. 6. MDS is a set ofmethods that allows one to visualize how close pointsare to each other by projecting them in lower dimen-sional space. MDS takes as an input a distance matrixof the dataset and creates a configuration of pointsin two or three dimensions, while the Euclidean dis-tances between them reproduce the original distancematrix. Thus, a scatterplot of the points created byMDS provides a visual representation of the originaldistances.

Before we implement the search method to makepredictions for new query points, we test the methodusing the training dataset itself. A commonly used sta-tistical technique named the n-fold cross-validationalgorithm [53] is used in this work. The n-fold cross-validation algorithm partitions a dataset into k subsetssuch that the required analysis is initially performedby taking one of the subsets as a testing dataset, whilethe other subset(s) are retained as the training datasetfor subsequent use in confirming and validating theinitial analysis. This test is repeated k times to en-sure that all the k subsets are tested. Since the vectorsin the testing set are already assigned to a cluster in

594 K. He et al. / Combustion and Flame 155 (2008) 585–604

Table 1Tenfold cross validation results

Cosine coefficient

1 201/2162 210/2163 204/2164 208/2165 201/2166 208/2167 209/2168 199/2169 199/216

10 208/216

Average 204.7/216Standard deviation 3.84Accuracy 94.77%

Note. 216 data points are selected each time as testing setand the number of data points that are classified into correctclusters is shown in the table.

the clustering step, we can validate the search methodby comparing the original assignments with the onesfrom the n-fold cross-validation process. Nearest-neighbor search and search among cluster centers areboth tested using a 10-fold cross validation. Sincethe 10-fold cross validation consumes large amountof CPU time, a random sample with size 2160 (200data points for each of 18 conditions) is used to testthe search method. The results are listed in Table 1.According to the results shown in Table 1, the accu-racy of the search method based on the cluster centersis approximately 95%; i.e., 95% of the test pointsare correctly assigned to clusters. The 5% inaccuracystems from the varying sizes of clusters. Points on theedge of a large cluster may have shorter distances toadjacent cluster centers than to their own cluster cen-ters. Although 5% inaccuracy exists in the statisticaltechnique, the searching algorithm employed in ourproposed methodology identifies the “nearest” mech-

anism, which is considered to be the most appropriateone to describe the system at that time point. Evenif the search algorithm assigns the query point to amechanism other than the right one, based on ourclustering and reduction technique, these two mech-anisms should be very similar. Thus using either oneto describe the system at this point is appropriate.

4. Development of reduced mechanisms

After the clusters are generated to represent thedataset and the search method is defined, we need toassign each cluster a reduced mechanism. The pur-pose is to establish a library of reduced mechanismsthat are linked to each cluster and can best character-ize the reactive propensities of the cluster. For eachflux graph in a cluster, flux values of all the edgesare sorted and a high portion cutoff is applied to eachgraph. By setting this cutoff, the most active edgescan be identified and the less active ones are aban-doned. This sort-and-cut procedure is done for everyflux graph in the training set. A set of active speciesthat are involved in the active edges can be identi-fied for each flux graph. Then a superset of speciesincluding all the active species of each flux graph ina particular cluster can be established for this cluster.Based on this set of species for each cluster, a reducedmechanism is developed. The sizes of reduced mech-anisms developed by setting a 99% cutoff are listedin Table 2. To verify that these reduced mechanismsare developed to describe different pathways, the dis-similarity of these mechanisms is quantified via a bi-nary string comparison. A string of binary variables isconstructed to represent the reduced mechanism, witheach variable in the string corresponding to a speciesin the detailed mechanism. The value is set as 1 to de-note the presence of a species in the reduced mecha-nism or as 0 to denote the absence of the species. Thus

Table 2Reduced mechanism size for n-pentane oxidation

Mechanism Species Reactions Mechanism Species Reactions Mechanism Species Reactions

1 33 137 11 49 238 21 105 6872 65 338 12 137 740 22 115 7553 124 520 13 151 968 23 65 4224 56 279 14 143 760 24 106 7295 139 640 15 143 768 25 142 9116 58 311 16 96 651 26 132 8167 69 410 17 73 480 27 86 5768 22 63 18 150 803 28 66 4219 61 345 19 145 809 29 151 843

10 80 526 20 147 941 30 67 417

Average size 99.2 576.8Detailed 385 1895

Note. Numbers of species and reactions of each reduced mechanism are listed in the table.

K. He et al. / Combustion and Flame 155 (2008) 585–604 595

Fig. 7. Dissimilarity of mechanisms in the same cluster and dissimilarity of mechanisms for different clusters. Dissimilaritybetween two mechanisms is quantified as the number of different species. The average number of species for all reduced mech-anisms is 99.2.

the dissimilarity between reduced mechanisms can bequantified as the difference between correspondingbinary strings. A binary string is constructed for eachreduced species set developed based on the flux graphin the training set. Then the dissimilarities betweenreduced species sets in the same cluster are quanti-fied for all the clusters and the mean value is used torepresent the “intracluster dissimilarity.” The dissim-ilarity of 30 reduced mechanisms corresponding to30 clusters is also quantified to represent the “inter-cluster dissimilarity.” Both intracluster dissimilarityand intercluster dissimilarity are illustrated in Fig. 7.The number of species in the detailed mechanism is385 and the average number of species in the reducedmechanisms is 99.2. As shown in Fig. 7, the intraclus-ter dissimilarity is 11.8 species, while the interclusterdissimilarity is 66.3 species. This result verifies thatthe mechanisms developed based on the flux graphsin the same cluster are similar, while mechanisms ofdifferent clusters are composed of different species,and thus describe different pathways. To get a fur-ther understanding of each mechanism, a flux graphis constructed for each mechanism. Some examplesof characteristic kinetic pathways can be identified inthese reduced graphs.

In n-pentane oxidation, three amyl radicals (C·CCCC, CC·CCC, CCC·CC) are formed first throughelementary dissociation reactions. At low tempera-ture, taking C·CCCC as an example, oxygen addsto the amyl radicals, forming amylperoxy radicals,and the peroxy radicals isomerize by hydrogen atomtransfer to hydroperoxide–amyl radicals:

C·CCCC + O2 → C(OO·)CCCC→ C(OOH)CC·CC. (10)

The hydroperoxide–amyl radicals then react with asecond oxygen, undergo similar isomerization, anddecompose through a chain-branching pathway:

C(OOH)CC·CC + O2 → C(OOH)CC(OO·)CC→ C·(OOH)CC(OOH)CC, (11)

C·(OOH)CC(OOH)CC→ C(=O)CC(O·)CC + ·OH + ·OH. (12)

This chain-branching pathway is responsible forthe increase of oxidation rate, which leads to low-temperature oxidation and cool flame formation.

As temperature increases, the decomposition ofthe initial hydroperoxide–amyl radicals becomesmore important, while the second oxygen additionbecomes less favorable:

C(OOH)CC·CC → cy(CCCO)–CC + ·OH. (13)

The result of this decomposition of initial hydro-peroxide–amyl radicals is that only one OH radical isformed, instead of the two OH radicals formed in thesecond oxygen addition pathways, which slows theoxidation rate. This is the phenomenon known as neg-ative temperature coefficient (NTC), which describesthe decrease of apparent oxidation rate with increaseof temperature.

When the system temperature increases further,even the first oxygen addition becomes disfavoredand unimolecular dissociation of the amyl radicals(β-scission) becomes the dominant pathway, whichdefines the high-temperature oxidation:

C·CCCC → C=C + CCC·. (14)

As can be seen from Fig. 8, the flux graph of re-duced mechanism 3 contains second oxygen additionpathways, as indicated by squares. This flux graph in-dicates that reduced mechanism 3 can describe theinitial stage of low-temperature oxidation. In Fig. 9,which represents the flux graph of reduced mecha-nism 14, second oxygen addition was replaced bydecomposition of the hydroperoxide–amyl radicals.Hence reduced mechanism 14 can describe medium-temperature oxidation, where the NTC phenomenonoccurs. In Fig. 10, the flux graph of reduced mecha-nism 12 is shown, where we can see that β-scission isthe dominant pathway, indicating that reduced mech-anism 12 can describe high-temperature oxidation.

596K

.He

etal./Com

bustionand

Flam

e155

(2008)585–604

s, which represent the low-temperature oxidation pathway.

K. He et al. / Combustion and Flame 155 (2008) 585–604 597

Fig.

9.Fl

uxgr

aph

ofre

duce

dm

echa

nism

14.

598 K. He et al. / Combustion and Flame 155 (2008) 585–604

Fig. 10. Flux graph of reduced mechanism 12.

Fig. 11. Flux graph of reduced mechanism 17.

In Figs. 8–10, the flux graphs start from n-pentane,go through different oxidation pathways, and termi-nate with C1–C4 intermediates. This indicates thatthese reduced mechanisms only capture the earlystage of the whole combustion process. However, inFigs. 11 and 12, which illustrate reduced mechanisms17 and 2, respectively, the flux graphs start from smallintermediates. Hence, reduced mechanisms 17 and 2may describe the intermediate and final stages of theentire oxidation process, respectively. Different stagesand pathways can also be identified from flux graphsof other reduced mechanisms. Therefore, the libraryof 30 reduced mechanisms is able to cover differentpathways and different stages.

5. Adaptive chemistry methodology

After the reduced mechanisms have been devel-oped and the search algorithm has been defined, weneed to implement these mechanisms in an adaptivescheme in the context of a reactor model to vali-date the feasibility and accuracy of the approach. In-stead of using a reduced mechanism in a specifiedrange of conditions, appropriate mechanisms are se-lected at every time step of the simulation throughthe search algorithm. The adaptive framework is illus-trated in Fig. 13. The simulation advances in discretetime steps. At the beginning of each time step, theinstantaneous atomic flux of the system is analyzed

K. He et al. / Combustion and Flame 155 (2008) 585–604 599

Fig. 12. Flux graph of reduced mechanism 2.

and the search algorithm is called on to search for thisnew query point among cluster centers and assign itto the nearest cluster. The reduced mechanism corre-sponding to this cluster is then used to describe thesystem at this particular time step. Although a skele-tal mechanism is used to describe the system at eachtime step, all the species in the detailed mechanismare tracked: species in the skeletal mechanism are in-tegrated based on corresponding ODEs, while speciesnot included in the skeletal mechanism are assumedto have zero production rates at that particular timestep. As the reaction system evolves, the procedure isrepeated for every time step selecting an appropriatereduced mechanism.

A pairwise mixing stirred reactor (PMSR) modelis used first to validate the reduced mechanisms andsearch algorithm in this work. In a PMSR model, a

reactive fluid composed of fuels, oxidizers, and dilu-ents is modeled as an even number of particles. Thesimulation is advanced in discrete time steps. Be-tween these time steps, these particles are arrangedin pairs and the composition of each particle under-goes a mixing-and-reaction evolution pattern, whichmeans the composition first evolves through a mix-ing fractional time step and a reaction fractional timestep afterward. The microscale mixing of each pair ofparticles (say p and q) is described by

dϕ(p)

dt= −(

ϕ(p) − ϕ(q))/τmix,

(11)dϕ(q)

dt= −(

ϕ(q) − ϕ(p))/τmix,

where τmix is the mixing timescale and ϕ(p) and ϕ(q)

are compositions of particle p and q , respectively.The PMSR simulation is performed with 100 par-

ticles and 100 steps, and the time step is �t = 3.1 ×10−3 s. The mixing time scale is τmix = 3.1×10−2 s,the paring time scale is τparing = 3.1×10−2 s, and the

residence time scale is τres = 3.1 × 10−1 s. Thirty re-duced mechanisms developed previously, along withthe detailed mechanism composed of 385 species and1895 reactions, are used in the simulation. Differentinitial temperatures and equivalence ratios (Φ) aretested (740 K, Φ = 1.0; 900 K, Φ = 2.0; 1050 K,Φ = 0.5) and the performance of the adaptive reduc-tion is illustrated in Figs. 14–16. Due to the stochasticnature of the PMSR model, fluctuations are expectedeven using the same conditions. To account for fluc-tuations, three simulations are launched for each con-dition and the average profiles are represented. Therelative error of the adaptive scheme compared to thedetailed simulations is shown in Table 3. The aver-age error of each condition is evaluated as the aver-

Fig. 13. Structure of the adaptive framework.

600 K. He et al. / Combustion and Flame 155 (2008) 585–604

Fig. 14. Performance of adaptive scheme in the prediction of temperature and composition profiles for 740 K, Φ = 1.0 in aPMSR.

Fig. 15. Performance of adaptive scheme in the prediction of temperature and composition profiles for 900 K, Φ = 2.0 in aPMSR.

K. He et al. / Combustion and Flame 155 (2008) 585–604 601

Fig. 16. Performance of adaptive scheme in the prediction of temperature and composition profiles for 1050 K, Φ = 0.5 in aPMSR.

Table 3Average error of adaptive scheme

Relative error (%)

740 KΦ = 1.0

900 KΦ = 2.0

1050 KΦ = 0.5

Temperature 3.25 2.18 2.31H2O mass fraction 5.75 4.17 4.62CO2 mass fraction 4.84 5.16 5.40n-C5H12 mass fraction 4.10 1.18 3.29Average error of T andall species

3.45 3.16 3.83

Note. The temperature profile and composition profiles ofall the species are compared between the detailed simulationand the adaptive scheme.

age of the relative errors of temperature and all thespecies compositions. While a considerable level ofaccuracy is retained in the adaptive scheme, the CPUtime of simulation is reduced by a factor of 60 us-ing the adaptive scheme. The CPU time includes fluxanalysis and searching as well as reactive flow simula-tion. The flux analysis is performed in a very efficientway [33] and the searching algorithm only searchesamong the cluster centers; thus only trivial computa-tional overhead (less than 3% of the total CPU time)is added. Since the flux analysis is implemented based

on all the species in the detailed mechanism, a fullset of species need to be traced during the simula-tion, and thus the required memory space is not re-duced.

The adaptive scheme is also tested in an adia-batic plug-flow reactor (PFR) model to illustrate theperformance of the adaptive scheme in the predic-tion of autoignition delay of n-pentane oxidation.Since the PFR model is a simple reactor model, evensimulation coupled with a detailed mechanism con-sumes less than 5 min. Therefore PFR is adopted inthis work only to validate the accuracy of profilesand autoignition delays. The autoignition delays pre-dicted by the adaptive scheme are calculated fromdifferent initial temperatures using the stoichiomet-ric ratio. The initial temperatures range from 600to 1100 K, which cover the low-temperature, NTCtemperature, and high-temperature oxidations. Theautoignition delays predicted by the adaptive schemeare compared with those predicted by the detailedmechanism in Fig. 17. Excellent agreement can beobserved between the two profiles, indicating that theproposed adaptive approach can accurately predictthe autoignition delay time of n-pentane combustionin low-temperature, NTC, and high-temperature re-gions. Temperature profiles of three different initial

602 K. He et al. / Combustion and Flame 155 (2008) 585–604

Fig. 17. Autoignition delay times of n-pentane combustionpredicted by adaptive scheme and detailed simulation. Au-toignition delay times of n-pentane combustion in a PFRusing different initial temperatures are measured in bothadaptive and detailed simulations. Three initial temperaturesare selected and temperature profiles are shown in Fig. 19.

temperatures, 750, 850, and 1050 K, which repre-sent low-temperature, medium-temperature (in theNTC region), and high-temperature oxidations, re-spectively, are shown in Fig. 18. As we can seefrom the temperature profiles, the two-step ignitionof low- and moderate-temperature oxidation and thesingle-step ignition of high-temperature oxidationcan be well predicted using the proposed adaptivescheme.

6. Conclusions

This study introduces a new approach for the de-velopment of adaptive reduced kinetic models basedon the flux graph concept. Several novel aspects in thereduction technique are developed in this study:

(1) Element flux is adopted to describe the reactionsystem and develop adaptive reduced mechanisms,instead of the temperature, pressure, and species com-position commonly used before. Reactive conditionscorresponding to different temperatures and fuel massfractions may be represented by the same flux value,which is directly associated with active reactions inthe system. Therefore, temperature, pressure, andspecies composition only provide limited informationabout the reactive propensities of the system, whilefluxes can capture the most intrinsic properties of thesystem. Evolution trajectories can be observed in theflux space illustrated by multidimensional scaling inFig. 19. As shown in Fig. 19, different operating con-ditions conform to different trajectories, indicatingthe role of different reduced mechanisms in the adap-tive scheme. Note that although different simulationsare initialized using different reduced mechanisms,

Fig. 18. Temperature profiles of n-pentane combustion in aPFR using (a) 700 K, Φ = 1.0, (b) 850 K, Φ = 1.0, and(c) 1050 K, Φ = 1.0.

they evolve through similar trajectories during the in-termediate and the final stage, where intermediatesare degraded further to form the final product, CO2.All trajectories represented in Fig. 19 are consistentwith the flux graphs illustrated in Figs. 8–12, indicat-ing that chemical flux is able to define the reactionsystem precisely.

(2) An efficient search algorithm is developedto accommodate the adaptive scheme. Rather than

K. He et al. / Combustion and Flame 155 (2008) 585–604 603

Fig. 19. Flux trajectories of n-pentane oxidation in a PFR. Trajectories of flux evolution are illustrated by connecting clustersthat are used sequentially during the combustion simulation. Scatterplots of cluster centers are illustrated by MDS.

searching among all the data points in the training set,cluster centers are used to represent the clusters andassign new query points proper mechanisms. In thisstudy, for example, 30 clusters are generated and thusthe search step only needs to check 30 cluster centersto select a mechanism.

(3) A new reduction technique is developed basedon fluxes. As described in Section 4, reduced mecha-nisms are developed by setting a high-portion cutofffor instantaneous fluxes, which is simple to realize,yet captures the most active species and reactions inthe reaction system. By verifying in the PMSR andPFR models, we can see that reduced mechanisms de-veloped under this methodology can predict the tem-perature, species composition, and autoignition delaywith excellent agreement to detailed simulations, andat the same time, simulation CPU time is largely re-duced.

Ongoing work is focused on two aspects: (a) veri-fying the methodology in other reaction systems andCFD models; and (b) developing an on-the-fly reduc-tion scheme based on the flux graph to achieve dy-namic reduction during the simulation.

Acknowledgments

The authors gratefully acknowledge financial sup-port from Exxon Mobil Corporation, NSF CBETGrant 0730582, and ONR Contract N00014-06-10835.

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