Complex CSP for Chemistry Reduction and Analysis

Transcription

1 Complex CSP for Chemstry Reducton and Analyss TIANFENG LU, YIGUANG JU, and CHUNG K. LAW* Department of Mechancal and Aerospace Engneerng, Prnceton Unversty, Prnceton, NJ 08544, USA The method of computatonal sngular perturbaton for the analyss and reducton of complcated chemcal mechansms has been extended to the complex egensystem. The characterstc tme scale for each speces was defned by usng the tme scales of the ndependent modes weghted by radcal ponters, and the tme scale of each speces normalzed by a characterstc tme scale of the system was used as a crteron n determnng the quas-steady-state speces. Furthermore, for oscllatory modes the radcal ponter and the mportance ndex of the prevous computatonal sngular perturbaton theory were redefned. Results show that the tme scales of chemcal speces change dramatcally and non-monotoncally, and the oscllatory modes appear frequently n large chemcal reacton mechansms. The present method was then employed to generate a 4-step and a 10-step reduced mechansm for the hgh-temperature H 2 /ar and CH 4 /ar oxdaton, respectvely. The valdty of these reduced mechansms were evaluated based on the responses of the perfectly strred reactors and the one-dmensonal planar propagatng premxed flames. Comparsons between the reduced and detaled chemstres over a wde range of pressures and equvalence ratos show good agreement on the flame speed, flame temperature, and flame structure. A software package based on the present algorthm was compled to generate reduced mechansms for complex chemcal mechansms. The valdty and effcency of the present algorthm s demonstrated by The Combuston Insttute INTRODUCTION It s well establshed that, even for the smplest hydrocarbon fuels, the detaled chemstry for combuston stll nvolves tens of speces and hundreds of elementary reactons. Ths large number of speces and reactons results n varous modes and tme scales for the destructon and producton of speces durng combuston, and render the computaton of complex combuston phenomena, such as turbulent flames, ether very dffcult or bascally untenable. Recently, efforts have been made n dervng local and/or global reduced chemstres for understandng complex chemcal processes and for numercal smulaton by employng partal equlbrum and quas-steady-state (QSS) assumptons [1]. Reduced mechansms for hydrogen and smple hydrocarbon fuels have been developed by several groups usng the methods of reacton rate analyss [2 6], computatonal sngular perturbaton (CSP) [7 11], and ntrnsc low-dmensonal manfolds [12]. The key step n dervng a reduced chemstry s to dentfy and remove the QSS speces wth small tme scales. The conventonal method s to use a crteron based on small mole fractons, normalzed net producton rates, and senstvty *Correspondng author. E-mal: cklaw@prnceton.edu analyss [2 6]. However, ths method does not always work well because t does not ensure that all the smallest tme scales are dentfed. In addton, the senstvty analyss cannot provde any nformaton on the destructon or producton modes. The CSP method [7 10] can successfully dentfy the steady-state speces and the tme scales of dfferent modes. However, f hgher than leadng order accuracy s requred, the refnement procedure for the tme-dependent Jacoban mght result n dffcultes n the accurate calculaton of the dervatves of the bass vectors. Furthermore, oscllatory modes are not consdered. Because knetc and thermal-dffusonal oscllatons frequently occur n flames [13, 14], predcton and dentfcaton of the ampltude and frequency of the oscllatory modes are also of mportance. In addton, n determnng the QSS wth CSP, a reference tme crteron s needed to separate the fast and slow subspaces and to dentfy the speces projected mostly to the fast subspace,.e., the fast modes. In a recent study of the one-dmensonal planar propagatng premxed flame [10 11], the thermal dffuson tme was used as the crteron. Although the thermal dffuson tme s meanngful to ndcate the overall tme scale of reactons across the flame zone, t s system dependent. Furthermore, the dramatc change COMBUSTION AND FLAME 126: (2001) 2001 by The Combuston Insttute /01/$ see front matter Publshed by Elsever Scence Inc. PII S0010-(01)

2 1446 T. LU ET AL. of temperature and radcal concentratons across the flame could lead to sgnfcant dffcultes n choosng the proper weghtng functons to ntegrate the CSP data across the flame. As such, t appears preferable to use a homogenous reactng system such as the perfectly strred reactor (PSR) to defne a parameter that s more lkely to be the chemcal property of the mxture. In vew of the above dscussons, the frst objectve of the present nvestgaton was to present an extended CSP algorthm to determne the reacton modes, tme scales, oscllaton frequency and QSS speces, and to obtan a comprehensve reduced chemstry by usng complex egensystem. The second objectve was to develop a software package that can be almost fully automatc to generate reduced chemstres. In the followng secton, the algorthm of the complex CSP method s formulated. Then, tme scales and reduced mechansms of hydrogen/ar and methane/ar flames were obtaned by usng the PSR analyss for a wde range of pressures and equvalence ratos. Ths was followed by the generaton of the reduced chemstry usng the newly developed software package. Fnally, the reduced chemstry was appled to the one-dmensonal planar propagatng flame, and the flame speeds, flame temperatures, and flame structures obtaned by the reduced and detaled chemstres, respectvely, are compared. Formulaton and Algorthm of Complex CSP Basc Formulaton A general chemcal reacton system can be expressed as: g(y) dy S F(y) (1) dt where y s the concentraton vector of all the speces, S the stochometrc coeffcent matrx, and F(y) the rate vector of the elementary reactons. By takng the tme dervatve of Eq. 1, we obtaned: dg J g (2) dt where J dg/dy s the tme-dependent Jacoban matrx. By usng the decomposton of J A B, Eq. 2 can be wrtten as: df f, f B g (3) dt where A s the matrx of bass vectors and B the nverse matrx of A. In Eq. 3, f A s a matrx bult up by deal bass vectors, than reduces to a dagonal matrx and ts dagonal elements are the egenvalues of J. CSP Data n Complex Space Because the Jacoban matrx J s actually tme dependent, a refnement procedure was used prevously [8] to obtan a group of real number deal base vectors. Ths s necessary for certan tme-ntegraton processes n whch the tmedependent propertes of the Jacoban matrx J must be consdered. Another advantage of the refnement procedure s that only the fast subspace needs to be decomposed [10]. However, f hgher than leadng order accuracy s requred, the refnement procedure requres evaluaton of the tme dervatve of the deal base vectors. Ths mght be dffcult n some cases. If only the tme scales of the modes or speces are of nterest, the Jacoban matrx can be assumed to be locally tme ndependent, and the refnement procedure can be replaced by performng a full decomposton of the Jacoban matrx. Furthermore, because oscllatory modes frequently appear n a large chemcal reacton system, t s necessary to decompose J n complex space n order to strctly dagonalze such that the frequences and ampltudes of the oscllatory modes can be obtaned accurately. As such, the defntons of the modes, ther tme scales, the radcal ponter, and the partcpaton ndex n the orgnal CSP theory [7 11] need to be modfed accordngly. Because A and B are complex matrces when oscllatory modes appear, complex conjugate pars of columns and rows n each of them need to be treated n the defntons of CSP data: A A R A I A 1 A R A I B B R B R (4a) (4b) (4c)

3 COMPLEX CSP FOR CHEMISTRY REDUCTION AND ANALYSIS 1447 B 1 B R B R (4d) where the th and ( 1)th columns of A and the th and ( 1)th row of B are complex conjugate pars, and the subscrpt R and I desgnate the real and magnary parts, respectvely. After smple algebrac manpulaton, each par of complex conjugate modes n Eq. 3 can be converted nto a par of real modes: f B R g f 1 B I g (5a) (5b) These two modes were coupled wth each other through the complex conjugate egenvalue par R I and 1. Ths mples that these two modes have the same characterstc tme scale of 1/ R and oscllatory frequency I wth a phase dfference of. The correspondng projecton matrces assocated wth the par of oscllatory modes n Eq. 5 are: Q 2A R B R Q 1 2A I B I (6a) (6b) The radcal ponters R,r and R 1,r are the rth dagonal elements of Q and Q 1, respectvely. The radcal ponter R,r s normalzed wth respect to ether or r. It ndcates how parallel the axs of the rth speces s to the th mode, and can be used to dentfy the QSS speces [7 11]. The partcpaton ndex for the par of oscllatory modes are gven as: P r II r 1 P r 1 B S rr F r B S rr F r II r 1 B S ri F r B S ri F r (7a) (7b) The partcpaton ndex P r s the normalzed contrbuton of the rth elementary reacton to the th mode [7 9]. For non-oscllatory modes, the radcal ponters are the dagonal elements of A B and the partcpaton ndex can be obtaned by Eq. 7a. The mportance ndex s the same as that n the orgnal CSP algorthm [7 11] I r S r F r II S r F r r 1 (8) where I r ndcates the normalzed contrbuton of the rth reacton to the reacton rate of th speces. Identfcaton of QSS Speces Eq. 3 shows that f the th egenvalue of J s a large negatve number, the th mode wll be quckly exhausted. Therefore, f the radcal ponter assocated wth a speces and ths fast mode s almost unty,.e., f the speces s almost only assocated wth one exhausted mode, ths speces can be dentfed as a QSS speces [7 11]. However, n most cases, each speces s projected onto several modes of dfferent tme scales. Therefore, specal procedures were requred to obtan an overall parameter that can be used to dentfy the QSS speces n such cases. The projecton of a speces onto the fast subspace s used n Refs. 10 to 11. When a speces s projected more onto the fast space than another speces, t s a better QSS canddate. The concentratons of the speces are also weghted nto the ndcator based on the common phenomenon that QSS speces generally have lower concentratons than other major speces. Although most QSS speces can be successfully dentfed by usng ths ndcator, a problem stll remans. That s, f speces A s projected onto several fast modes n the fast subspace and speces B s projected to other modes wth longer tme scales than those of speces A, and the two speces have the same radcal ponter to the fast subspace, speces A should be a better QSS canddate than speces B. Ths happens because ths ndcator does not reflect the nternal dstrbuton of how the speces s projected onto the fast subspace. To resolve ths problem, the tme scales of ndependent modes can be summed to an ndcator weghed by suffcently large radcal ponters, and ths ndcator can be defned to be the recprocal tme scale of the speces. That s, a speces belongng to a faster subspace, whch conssts of a group of faster modes, can be exhausted faster than a speces belongng to a

4 1448 T. LU ET AL. less fast subspace. The defnton s provded n Fg. 9, n whch a crtcal tme scale s used to normalze the tme scales of those major speces that control the overall reacton of the system to be around O(1). It s dffcult to fnd the crtcal tme scale from the Jacoban matrx tself and, hence, t has to be chosen from the many tme scales assocated wth the reactng system. Once ths crtcal tme scale s found, the normalzed tme scale of the speces can be defned. In the present study, speces were consdered to be QSS f ther tme scales satsfy: 1 KK ch p, p R r r, R r (9) r 1 where ch s the crtcal tme scale for the progress of chemcal reacton and wll be gven later; KK s the total number of speces; R r, whch s the rth dagonal elements of the projecton matrx Q, s the radcal ponter assocated wth the th mode and the rth speces; s a small number below whch the speces and the mode are almost orthogonal, as ndcated by the a radcal ponter, and the weak projecton from the speces to the mode can be consdered neglgble subsequently. For example, 0.01 was used n the present study. The parameter s the threshold factor, whch should be chosen accordng to the accuracy requrement. For example, 100 means that speces wth tme scales at least 100 tmes shorter than that of the crtcal tme scale, or the tme scale of the controllng speces as stated above, can be selected as QSS speces. As wll be shown later, the crteron of Eq. 9 works very well for H 2 /ar and CH 4 /ar mxtures. However, to avod lmtng cases, verfcaton s needed for the reduced mechansm obtaned from Eq. 9 before t s appled. Of course, a more accurate reduced mechansm can always be obtaned by specfyng a suffcently large value for. Obtanng Global QSS Speces It s shown n CSP theory that dfferent reduced mechansms are requred for dfferent ntal condtons and reacton tmes. A good QSS speces canddate at a certan tme may become a poor QSS speces canddate as tme progresses or as a result of other changes n the ntal condtons. The CSP data above are only the local chemstry nformaton at a certan tme step under a specfed condton, and the QSS speces dentfed by the above nformaton s therefore only vald locally. In many cases, however, a comprehensve reduced mechansm wth fxed numbers of QSS speces and global steps that can work for a certan range of ntal condtons s more convenent to use. In order to dentfy the QSS speces that s vald for a wde range of condtons, global ndcators need to be generated from the local CSP data under varous temperatures and pressures and equvalence ratos. In prevous work [10 11], ntegraton of the radcal ponters across a flame was used to choose the global QSS speces. The ntegrated ndcator therefore reflects the overall behavor of the speces across the entre flame. However, by usng ths ntegrated ndcator wthout a suffcently comprehensve weghtng functon, whch may be very dffcult to fnd, the possblty that a speces could be a poor QSS canddate n only a very narrow zone can be overlooked. As such, naccurate results may be obtaned f ths zone s very mportant even although t s very narrow. In the present study, a QSS speces s selected f t can satsfy the condtons n Eq. 9 everywhere wthn the range of nterest. However, t s not applcable f the global data are obtaned across a flame. Ths s because the temperature and radcal concentratons change dramatcally across the flame, leadng to correspondngly dramatc changes n the tme scales of the speces. Consequently, t may not be adequate to normalze the tme scales of the speces wth a sngle crtcal tme scale across the entre flame zone. A preferred choce s to use a homogeneous reactng system wthout transport processes. The advantage s that the reactvty of a speces, whch s a chemcal property determnng whether a speces can be selected as a QSS speces, can be assessed more clearly wthout the nfluence of transport. PSR was used n the present study to generate the data source for CSP analyss. The crtcal tme scale ch must be selected for each system accordng to Eq. 9. A well-defned tme scale n PSR s obvously the extncton

5 COMPLEX CSP FOR CHEMISTRY REDUCTION AND ANALYSIS 1449 resdence tme, whch s the shortest tme for the major radcal pool to buld up gven a suffcently hgh temperature and radcal concentratons such that the reactons can sustan themselves. For gnton phenomena, the gnton tme s the approprate choce, whch measures the tme requred for the controllng speces to buld up. Obvously, a comprehensve reducton should take nto account of both the gnton and extncton phenomena. For smplcty, we shall nevertheless conduct the followng llustraton based only on strongly burnng and near-extncton stuatons. Identfcaton of the Fast Elementary Reactons For each QSS speces, a fast elementary reacton can be elmnated so that the reacton rate of ths fast elementary reacton wll not appear n the global reacton rates n the reduced mechansm [3, 10, 11]. Ths fast reacton can be chosen by CSP analyss usng the mportance ndex defned n Eq. 8, and the elementary reacton wth the largest mportance ndex for each QSS speces s selected and elmnated. In the prevous CSP theory [10, 11], an algorthm to dentfy the fast elementary reacton by usng the ntegrated mportance ndex across the flames was provded. In the present study, the mnmum values of the mportance ndex of the elementary reactons for each speces n the entre range of ntal condtons are used as a global mportance ndex. For reversble reactons, the forward and backward reactons are consdered separately so that reactons n partally equlbrum state wll not be overlooked. The elementary reacton wth the largest global mportance ndex for each speces s the best canddate to be elmnated for ths speces. The advantage of usng the mnmum value of the mportance ndex, rather than the ntegrated value, s that t can account for effects due to the exstence of narrow regmes wthn whch the mportance ndex could be very small. By usng Eq. 8, a skeletal mechansm can also be obtaned by elmnatng the unmportant elementary reactons that have neglgble mportance ndex for every speces. The accuracy of the resultng skeletal mechansm depends on the small, user-specfed threshold value under whch an mportance ndex can be consdered neglgble. Reduced mechansms can then be generated from these skeletal mechansms wth fewer speces and global reacton numbers. These two steps can also be performed together [10 11]. Once the QSS speces and ther correspondng fast elementary reactons are dentfed, an algebrac procedure n matrx form can be used to generate the mnmum set of lnearly ndependent global reactons and to evaluate the global reacton rates [3]. Applyng the Reduced Mechansms When the tme scale of transport s slower than that of chemcal extncton, the QSS speces obtaned from PSR data by usng the present method should be vald n general. For those applcatons wth transport tme scales faster than the chemcal extncton tme scales, such as those assocated wth dstrbuted turbulent combuston, an approprate choce of and pretests of the reduced mechansm have to be conducted. In lamnar premxed flames, the chemcal extncton tme ( ext ) s expected to be shorter than the global dffuson tme ( dff ). They assume smlar order only when the flame approaches the extncton state. Therefore, n most cases, t s adequate to apply the reduced chemstry generated from PSR to lamnar premxed flames. For weak turbulent flames, the stuaton s almost the same as that for lamnar flames. For strong turbulent flames, however, a suffcently large should be used to obtan the reduced mechansm. In addton, n applyng the reduced mechansms n flame calculatons, specal procedures are frequently needed to deal wth the ncorrect concentratons calculated for some QSS speces. Specfcally, abnormally hgh concentratons of QSS speces may be obtaned n the low- and ntermedate-temperature zones of flames, mplyng that the QSS assumpton s not applcable there. A prevous method used to solve ths problem s to truncate some elementary reactons [11]. The advantage of ths method s that the concentratons of the QSS speces n lowtemperature zone wll change smoothly. However, problems wll arse when the elementary reactons are mportant n both hgh- and low-

6 1450 T. LU ET AL. temperature zones. Another method s to set upper and lower bounds for the concentratons of the QSS speces such that truncaton of the concentraton of QSS speces wll be effected only when they are out of the bounds. The second method s used n the present study because the abnormally hgh concentratons should not occur where the reduced mechansm s generated f the QSS speces are dentfed successfully, and the concentratons of the QSS speces n the low-temperature flame zone are always lower than those of the non-qss radcals and have a weaker effect on flames. Fg. 1. Dependence of the recprocal of tme scale of speces normalzed by the extncton resdence tme of PSR for H 2 /ar oxdaton. Software Package ARC-CSP A software package based on the PSR data, and termed automatc reducton of chemstry wth CSP (ARC-CSP), was developed by usng the present algorthm. It can automatcally create the skeletal mechansm, analyze the CSP data from PSR outputs, determne the QSS speces, and create the global reduced chemstry. It s compatble wth CHEMKIN-II and uses Wndows nterface. Detaled nformaton on ths package s avalable by contactng the authors. Here, we only gve a bref descrpton of ts nput and output. The nputs of the package are the detaled chemstry, the threshold error of the skeletal mechansm, the threshold factor defned n Eq. 9, and the range of pressure, temperature, and equvalence rato. The package frst automatcally creates the extncton profles of PSR at dfferent pressures, temperatures, and equvalence ratos, and generates the skeletal mechansm and bulds the CSP data by usng the specfed error threshold and Eqs. 4 9, respectvely. Second, the steady-state speces are selected and the reduced chemstry was created automatcally by usng the threshold factor. The package also provdes a choce for users to dsable the recommended selecton and to select ther own QSS speces and preferred reduced chemstres. Specfcally, after dentfyng the QSS speces, all the possble lnearly ndependent global reactons are shown to the user. The user then sequentally selects hs preferred global reactons, beng aded by dsplayng the remanng lnearly ndependent reactons after the selecton. Thrd, a set of CHEMKIN-II compatble FORTRAN subroutnes are generated for applcaton purpose, and comparson of the reduced chemstry wth the detaled chemstry are made based on the PSR and SENKIN data. RESULTS AND DISCUSSION Tme Scales of Chemcal Speces The recprocals of the tme scales of speces n the stochometrc H 2 /ar reacton under 1 atm calculated from PSR data are plotted n Fg. 1 as a functon of the resdence tme. Here, the tme scale of each speces s normalzed by the extncton resdence tme. It s seen that normalzed tme scales of the major speces such as H 2, O 2, and H 2 O are close to O(1), whereas other hghly reactve radcals such as H 2 O 2 and HO 2 have much shorter tme scales. Ths result ndcates that the extncton resdence tme of PSR s a good ndcator of the characterstc tme for the progress of the chemcal reacton. It shows that the tme scale of H s smaller than those of O, OH, and HO 2 when the resdence tme s long. However, as the resdence tme decreases the tme scale of H becomes larger than those of O, OH, and HO 2, partcularly near the extncton lmt. Ths result mples that the local tme scales of the reactng speces should not be used alone to derve a comprehensve reduced chemstry. Furthermore, Fg. 1 shows that the tme scales of H 2 O 2, OH, and O are much shorter than those of the reactants and products, and thus can be assumed to be QSS. In addton, t s also seen that OH and HO 2 are better QSS than

7 COMPLEX CSP FOR CHEMISTRY REDUCTION AND ANALYSIS 1451 Fg. 2. Dependence of the recprocal of tme scale of speces normalzed by the extncton resdence tme of PSR for CH 4 /ar oxdaton. O and H. These results agree well wth those obtaned by the analyses of net producton rates [5, 15]. Near the extncton lmt, for short resdence tmes, Fg. 1 shows that the tme scales of the reactants and man radcals decrease rapdly as the mxture approaches the extncton state. Ths mples that the QSS assumpton becomes poor at the extncton state, as there s not enough tme for the radcal pool to buld up. By truncatng a very short resdence tme segment next to the extncton pont, a reduced mechansm wth a much smaller sze can be obtaned whle t s stll adequate for most flame applcatons for whch the operatng condtons are generally suffcently far from the extncton lmt. On the other hand, when the resdence tme approaches nfnty, the extncton tme wll no longer be a good crtcal tme scale to normalze the tme scales of speces because there s no more controllng speces n ths lmt, and most speces are actually n quas-steady state here. As such, only the data ponts wth resdence tmes less than a crtcal value, say 100 tmes the extncton tme, need to be consdered to obtan the global reduced mechansm. The tme scales of speces n stochometrc CH 4 /ar mxture calculated from PSR data are plotted n Fg. 2. Smlar to the H 2 /ar mxture, the tme scales of reactants and products are longer than those of the radcals and are close to the extncton resdence tme. Numercal results at hgher and lower pressures as well as for near-lmt mxtures also support ths observaton. Therefore, t can be concluded that the Fg. 3. Number of complex egenvalues of Jacoban matrx J dg/dy as a functon of resdence tme scale of PSR n H 2 /ar and CH 4 /ar system. extncton resdence tme s a reasonable choce as the characterstc tme for the progress of chemcal reactons. Ths characterstc tme wll be used as the tme crteron defned n Eq. 9 to determne the QSS. It can be clearly seen that speces such as C 2 H, C 2 H 5, and CH are the best canddates for QSS. The results also agree wth those of prevous works [1, 2, 6]. Therefore, by checkng the crteron n Eq. 9 for all the mxtures of nterest, comprehensve reduced chemstres for H 2 /ar and CH 4 /ar flames can be obtaned wthn 1 mn on a modest PC by usng the current software package. To demonstrate the exstence of oscllatory modes, the number of oscllatory modes appearng n stochometrc H 2 /ar and CH 4 /ar flames at varous equvalence ratos s plotted n Fg. 3 as a functon of the resdence tme. It s clearly seen that oscllatory modes frequently appear n chemcal reactons. In addton, as the flame approaches the extncton lmt, the number of oscllatory modes ncreases. Ths result suggests that t s mportant to nclude the oscllatory modes n CSP analyss. It may be noted that the present algorthm can also be drectly extended to dentfy the oscllatory modes caused by thermal-dffuson n flame or heat loss to the chamber wall n PSR f the temperature term s added to the Jacoban matrx computaton.

8 1452 T. LU ET AL. QSS speces canddates have normalzed tme scales close to or well above 100. There are only a few speces, wth tme scales less than or close to 50, whch can be consdered to be margnal QSS speces. These margnal QSS speces such as O and OH should be selected wth care and subsequent verfcaton. Perhaps the best strategy s not to nclude the margnal speces as QSS speces because there are only a few of them n a large mechansm anyway. Fg. 4. Sze of reduced mechansms as a functon of the value of n H 2 /ar mxture. The number of QSS speces dentfed depends on the choce of the threshold value as shown n Fgs. 4 and 5, n whch we have plotted the number of speces n the reduced mechansm for the value used, wth ndcaton of the specfc speces that s ether removed ( QSS, for larger ) or added ( QSS, for smaller ). Inspectng Fg. 4, t appears that a regme of nsenstvty n the choce of would be n that smaller values of could result n unacceptable degradaton n the accuracy n reducton, whereas the larger value of would lead to too many speces n the reduced mechansm. As such, we have put HO 2 and H 2 O 2 as the QSS speces and used 100 to generate the reduced mechansm relevant for hgh-temperature hydrogen oxdaton. For methane/ar mxture, Fg. 5 shows that most Fg. 5. Sze of reduced mechansms as a functon of the value of n CH 4 /ar mxture. Reduced Mechansms for H 2 /Ar and CH 4 /Ar Flames The detaled H 2 /ar mechansm [16] used here nvolves nne speces: H 2,O 2,N 2,H 2 O, O, OH, H, HO 2, and H 2 O 2. The choce of n Eq. 9 depends on the requrement of the accuracy for the resultng reduced chemstry. The larger the value of, the hgher s the accuracy of the reduced chemstry. By choosng 100, wth ntal pressure between 0.2 atm and 20 atm and equvalence rato from 0.7 to 2, for example, the software package ARC-CSP automatcally recommends two speces, HO 2 and H 2 O 2,asthe QSS, fnds and elmnates the correspondng fast reactons, and yelds the followng four-step reduced mechansm: 1: H 2 O OH H 2: O 2 H O OH 3: H 2 OH H 2 O H 4: H H M H 2 M Ths reduced mechansm resembles the threestep reduced mechansm n Refs. 10 to 11, whch ncludes OH as an addtonal QSS speces. Clearly, the same three-step reduced mechansm could be obtaned by the current algorthm f a smaller value of, 40, s used to nclude OH as a QSS speces. For CH 4 /ar flames, we used the GRI MECH 1.2 as the detaled mechansm. It s shown n Fg. 5 that there are about fve speces whch have global normalzed tme scales very close to 100. They are very dffcult to be dfferentated and should be ncluded as QSS speces as a group. By decreasng slghtly to 80, wth the same pressure range as above and the equvalence rato range between 0.7 and 1.3, for example, ARC-CSP generates a 10-step reduced mechansm. Reduced mechansms wth smaller szes requre much smaller values of,

9 COMPLEX CSP FOR CHEMISTRY REDUCTION AND ANALYSIS 1453 whch would result n substantal degradaton of the results because the mportant radcals such as O and H wll be ncluded as QSS speces. The selected QSS n the 10-step reduced mechansm are CH 3,CH 2,CH,CH 2 O, HCO, HO 2,H 2 O 2, C 2 H 6, C 2 H 5, C 2 H 3, C 2 H, HCCO, CH 3 OH, CH 2 OH, CH 3 O, CH 2 (S), C, and the global reactons are: 1: CH 4 3O CO H 2 2OH 2: O O M O 2 M 3: H OH H 2 O 4: CO O CO 2 5: 3O C 2 H 2 2CO H OH 6: H 2 O H OH 7: H O OH 8: H 3O C 2 H 4 2CO 2H 2 OH 9: H CH 2 CO OH C 2 H 2 10: HCCOH CH 2 CO Ths reduced mechansm resembles the 12- step reduced mechansm of Ref. 6, whch ncludes HO 2 and H 2 O 2 as non-qss speces. These two speces are found to have slow tme scales only n gnton and low-temperature/ hgh-pressure cases and are very good QSS canddates for extncton and hgh-temperature/ low-pressure phenomena. The radcal O s found to be a margnal QSS speces wth a global normalzed tme scale 70, and has a large effect on the PSR temperature f t s ncluded as QSS speces. Assessment of the Reduced Mechansms The fdelty of these two reduced mechansms was evaluated by comparng them wth the detaled mechansms usng the PSR n the pressure range between 0.2 and 20 atm, and equvalence rato range between 0.7 and 1.3. The comparson of the temperature dependence on the flow resdence tme n PSR for H 2 /ar and CH 4 /ar mxtures are shown n Fgs. 6 and 7. For H 2 /ar mxtures, t s seen that the results of the reduced chemstry agree very well wth those of the detaled chemstry over the entre range of pressure and equvalence rato. The results for CH 4 /ar show smlar agreement wth the detaled chemstry. In addton, comparson of the mole fractons of the remanng speces also shows equally good agreement. Therefore, t can be concluded that the reduced chemstry Fg. 6. Comparson of dependence of temperature on the resdence tme of PSR between the reduced and detaled mechansms for varous pressures and equvalence ratos n H 2 /ar flame. generated by the present CSP algorthm can well reproduce the PSR data. We next assess the reduced mechansm by usng the one-dmensonal planar propagatng flame at the same pressure and equvalence rato ranges. The code used was SANDIA s PREMIX. For hydrogen/ar flames, Fg. 8 compares the flame structure at 1 atm, whereas Fg. 9 shows the dependence of the flame speed on the equvalence rato and pressure. It s seen that the flame speed obtaned by the reduced mechansm agrees qute well wth the detaled chemstry on the lean sde or at pressures less than 1 atm. The relatve error ncreases slghtly wth ncreasng pressure on the rch sde. At 20 atm, the relatve error of flame speed s about 5%. The reason s that HO 2 and H 2 O 2, whle beng selected as the QSS at 100, become ncreasngly mportant wth ncreasng pressure. By comparng Fgs. 8 and 9, we conclude that the QSS assumpton for HO 2 and H 2 O 2 only slghtly affects the flame speed of rch hydrogen flames at hgh pressures. We note agan that

10 1454 T. LU ET AL. Fg. 9. Comparson of the dependence of flame speed on the equvalence rato between reduced and detaled mechansms for H 2 /ar flames at varous pressures. Fg. 7. Comparson of dependence of temperature on the resdence tme of PSR between the reduced and detaled mechansms for varous pressures and equvalence ratos n CH 4 /ar flame. HO 2 and H 2 O 2 have been found to be mportant n such lower temperature phenomenon as gnton [6], and as such cannot be treated as QSS n these cases. Comparson of the structure of the stochometrc CH 4 /ar flame at 1 atm, obtaned wth the reduced as well as the detaled mechansms, s gven n Fg. 10. It s seen that the dstrbutons of temperature and mole fractons of the reactants and products calculated by usng the reduced 10-step mechansm agree very well wth those of the detaled chemstry. Fgure 10b also shows that the reduced chemstry reproduces well the dstrbutons of OH, O, H, HCCOH, and CH 2 CO, and slghtly under-predcts the peak mole fractons of C 2 H 4 and C 2 H 2. The comparson of flame speed as a functon of equvalence rato and pressure for CH 4 /ar Fg. 8. Comparson of the flame structures of temperature and speces predcted by the reduced and detaled mechansms, respectvely, for stochometrc H 2 /ar flames at 1 atm. Fg. 10. Comparson of the flame structures predcted respectvely by the reduced and detaled mechansms for stochometrc CH 4 /ar flames at 1 atm. 10a: temperature, reactants, and products; 10b: radcals.

11 COMPLEX CSP FOR CHEMISTRY REDUCTION AND ANALYSIS 1455 obtaned by the reduced and detaled chemstres, respectvely, over a wde range of pressures and equvalence ratos. The authors are grateful to Professor S. H. Lam of Prnceton Unversty for many enlghtenng dscussons on the CSP algorthm, and to Professor D. A. Gousss of the Unversty of Patras for helpful comments. Ths work was supported by the Ar Force Offce of Scentfc Research under the techncal montorng of Dr. Julan M. Tshkoff. Fg. 11. Comparson of the dependence of flame speed on the equvalence rato between reduced and detaled mechansms for CH 4 /ar flames at varous pressures. flames s shown n Fg. 11. It s seen that the 10-step reduced chemstry agrees very well wth the detaled chemstry at pressures greater than 1 atm. At pressures below 1 atm, the reduced mechansm slghtly over-predcts the flame speed near 1, but stll shows good agreement near both the lean and rch lmts. CONCLUSION The prevous CSP method for analyss and reducton of large chemcal reacton mechansms s extended to the complex egensystem. An algorthm to dentfy the QSS speces s presented by defnng the characterstc tme scale for each chemcal speces. The results show that the tme scales of the chemcal speces change dramatcally and non-monotoncally, and oscllatory modes appear frequently n chemcal systems. It s also shown that the extncton resdence tme of PSR s an approprate characterstc tme responsble for the progress of the chemcal reacton for strongly burnng and near-extncton stuatons. By usng the present method, 4-step and 10-step reduced mechansms of H 2 /ar and CH 4 /ar mxture were obtaned and evaluated by usng PSR and the one-dmensonal planar propagatng flames. The valdty of the present algorthm s demonstrated through good agreements on the flame speed, flame temperature, and flame structure REFERENCES 1. Peters, N., Lecture Notes n Physcs Sprnger, Berln, 1985, pp Peters, N., and Kee, R. J., Combust. Flame, 68:17 29 (1987). 3. Chen, J. Y., Combust. Sc. Technol. 57:89 94 (1988). 4. Smooke, M. D., Lecture Notes n Physcs, 384, Sprnger- Verlag, New York, 1991, p Ju, Y., and Noka, T., Combust. Flame, 99: (1994). 6. Sung, C. J., Law, C. K., and Chen, J. Y., Proceedngs of the Twenty-Seventh Internatonal Symposum on Combuston, The Combuston Insttute, Pttsburgh, 1998, pp Lam, S. H., and Gousss, D. A., Proceedngs of the Twenty-Second Internatonal Symposum on Combuston, The Combuston Insttute, Pttsburgh, 1988, pp Gousss, D., and Lam, S. H., Proceedngs of the Twenty- Fourth Internatonal Symposum on Combuston, The Combuston Insttute, Pttsburgh, 1992, pp Lam, S. H., Combust. Sc. Technol. 89: (1993). 10. Massas, A., Damants D., Mastorakos, E., and Gousss, D., Combust. Flame 117: (1999). 11. Masss, A., Damants, D., Mastorakos, E., and Gousss, D. A., Combust. Theory Model. 3: (1999). 12. Maas, U., and Pope, S. B., Combust. Flame 88: (1992). 13. Tomln, A. S., Pllng, M. J., Turany, T., Merkn, J. H., and Brndley, J., Combust. Flame 91: (1992). 14. Jouln, G., and Clavn, P., Combust. Flame 35: (1979). 15. Sanchez, A. L., Balakrshnan, G., Lnan, A., and Wllams, F. A., Combust. Flame 105: (1996). 16. Km, T. J., Yetter, R. A., and Dryer, F. L., Proceedngs of the Twenty-Ffth Internatonal Symposum on Combuston, The Combuston Insttute, Pttsburgh, 1994, pp Receved 9 May 2000; revsed 1 Aprl 2001; accepted 5 Aprl 2001