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C O MB US T IO N A N D FLA ME 76: 375-391 (1989) 375

Laminar Flame S pe e ds o f Me tha ne -Air Mixture s Unde r Re duce d and Ele vate d Pre s s ure s

F. N. EGOLFOP OULOS , P . CHO, and C. K. LAW*

De partm e nt o f Mechancial Engineering, Univers ity o f California, Davis , CA 95616

Us ing the counte rflow me thodology, the la mina r fla me s pe e ds of me tha ne -a ir mixture s ha ve be e n a ccura te ly de te rmine d ove r the pre s s ure ra nge o f 0 .25-3 a tm a nd ove r e xte ns ive le a n-to-rich conce ntra tion ra nge s . The s e fla me s pe e ds a re the n compa re d with the nume rica lly ca lcula te d va lue s obta ine d by us ing va rious publis he d kine tic s che me s of e ithe r the C1 me cha nis m or the full C2 me cha nis m. Two s uch s che me s s how ve ry clos e a gre e me nt with the e xpe rime nta l da ta . Howe ve r, a va ila ble informa tion ca nnot furthe r diffe re ntia te the re la tive s upe riority be twe e n the m for fla me s pe e d ca lcula tions , e s pe cia lly the importa nce of C2 re a ctions for mode ra te ly rich s itua tions . Two re duce d me cha nis ms a re a ls o de duce d through s e ns itivity a na lys is a nd a re e xpe cte d to be us e ful for fla me s pe e d ca lcula tions a nd a pproxima te fla me s tructure s tudie s .

INTR O DUC TIO N

The la mina r fla me s pe e d S ° is a n importa nt phys icoche mica l pa ra me te r o f a combus tible mixture be ca us e it conta ins the ba s ic informa tion re ga rding its d iffus ivity, e xothe rmicity, a nd re a c- tivity. Howe ve r, in s pite o f the e xte ns ive e ffort e xpe nde d to a ccura te ly de te rmine the ir va lue s , e s pe cia lly thos e o f the conve ntiona l hydroca rbon- a ir mixture s , wide s ys te ma tic s pre a ds in the re porte d e xpe rime nta l da ta s till e xis t e ve n though in ma ny ins ta nce s the e xpe rime nts a ppe a r to ha ve be e n ca re fully e xe cute d [1]. F igure 1, pa rtly re produce d from Re f. 1 a nd the n upda te d to include s ome re ce nt da ta [2, 3], c le a rly de mon- s tra te s this point for the la mina r fla me s pe e ds o f me tha ne -a ir mixture s a s a function o f the e quiva - le nce ra tio ~b.

Re ce ntly it ha s be e n s ugge s te d tha t the s e s ys - te ma tic dis cre pa ncie s a re proba bly ca us e d by the couple d e ffe cts o f fla me s tre tch a nd pre fe re ntia l

* P re s e nt a ddre s s e s : (P . C.) De pa rtme nt o f Me cha nica l Engi- ne e ring, Michiga n Te chnologica l Unive rs ity, Houghton, MI 49331. (C. K. L.) De pa rtme nt of Me cha nica l & Ae ros pa ce Engine e ring, P rince ton Unive rs ity, P rince ton, NJ 08544.

Copyright © 1989 by The Combus tion Ins titute P ublis he d by Els e vie r S cie nce P ublis hing Co., Inc. 655 Ave nue o f the Ame rica s , Ne w York, NY I0010

diffus ion [4, 5]. S pe cifica lly, it ha s be e n de mon- s tra te d both the ore tica lly a nd e xpe rime nta lly tha t the fla me re s pons e ca n be qua lita tive ly re ve rs e d whe n the fla me s tre tch cha nge s from pos itive to ne ga tive , a s in the ca s e s o f e xpa nding s phe rica l fla me ve rs us the Buns e n fla me , a nd whe n the mixture 's e ffe ctive Le wis numbe r cros s e s a critica l va lue typica lly a round unity, a s in the ca s e s o f le a n me tha ne -a ir a nd rich propa ne -a ir mixture s ve rs us rich me tha ne -a ir a nd le a n propa ne -a ir mixture s .

A pa rticula rly s e rious implica tion o f the s tre tch- induce d fla me re s pons e is the pote ntia l fa ls ifica - tion o f the kine tics informa tion de te rmine d or va lida te d through compa ris on be twe e n the nume rica lly ca lcula te d a nd e xpe rime nta lly de te rmine d re s ults [6, 7]. It is cle a r tha t if cons ide ra ble unce rta inty a nd/or ina ccura cy e xis t in the e xpe rime nta l da ta , the ir us e fulne s s a nd fide lity for kine tics s tudie s ca n be s e rious ly compromis e d.

In vie w o f the a bove cons ide ra tion, La w a nd co- worke rs [8-10] ha ve propos e d a counte rflow- ba s e d me thodology through which s tre tch e ffe cts ca n be s ys te ma tica lly s ubtra cte d out s uch tha t S ° ca n be una mbiguous ly de te rmine d. The la mina r fla me s pe e ds o f me tha ne -a ir a nd propa ne -a ir

001~21801~I$03.50

376 F . N. EGOLFOPOULOS

50

45

40

35

25

20

15

10 i t i = I 0.7 1.0 1.1 1.2 1.4

d~

- i - Andrews/Bradley (1972) -e- Gunther/Janisch (1972)

Sharma et aL (1981) Babkin/Kozachenko (1966)

.o- IijimalTakeno (1986) -~- Yarnaoka/Tsuji (1984)

= ! = ! = 0.8 0.9

CH 4/AIR p= 1 ATM

| I | 1 .3

F ig . 1. Va rio u s re p o rte d e xp e rim e n ta l fla m e s p e e d s S °(0 ) , fo r m e th a n e -a ir m ixtu re s a t p = 1 a tm .

mixtures unde r a tmospheric pressure ha ve thus been de te rmined [9, 10].

The present inves tiga tion continues our s tudy of the s tructure a nd propaga tion o f la mina r premixed flames with two specific objectives . Firs t, we sha ll apply the counte rflow method to de te rmine S ° o f me tha ne -a ir mixtures unde r reduced as well as e leva ted pressures , ranging from 0.25 to 3 a tm. Although these da ta a re o f practica l inte res t in the ir own right, especia lly in te rms of high- pressure combus tion, we note tha t the y a lso ca rry s ignificant kine tics informa tion because pressure not only influences the fre que ncy o f molecula r collis ion but a lso diffe rentia tes the re la tive e ffl- ciencies o f two-body branching reactions versus three-body te rmina tion reactions . Thus the y provide additiona l cons tra ints on the va lida tion o f the

kine tic schemes . This the n leads to the second objective , which is to compare our experimenta l da ta with those de te rmined numerica lly by us ing the kine tic schemes proposed by diffe re nt research groups , us ing the same fla me code a nd transport properties code . It will be shown tha t through such comparisons , a nd because o f the fide lity o f the present da ta , useful ins ights ha ve been ga ined on these kine tic schemes . S implified schemes a re a lso proposed for accura te ca lcula tion of the fla me speeds within the parametric ranges s tudied here in.

The experimenta l me thodology is brie fly speci- fied in the next section, which is followed by presenta tion o f the experimenta l results . These results a re then compared with the numerica lly ca lcula ted va lues in the following section.

LAMIN AR F LAME S P E E D S

f O-IAMBER SOLENO1D VALVE ~ PRESSURE GAUGE

EXHAUST ~ ?

. . . . . . ROTAMI~II~R FLOW

MIXINQ FLOW ~ VALVE PRESSURE

I ~ ON/O~ I y

NITROGEN INLET

Fig. 2, Schematic of the experimenta l se tup.

377

E XP E R IME NTAL ME THO D O LO G Y

Figure 2 s hows the s che ma tic o f the e xpe rime nta l s e tup a nd the va rious flow va ria ble s a re de fine d in Fig. 3. The e xpe rime nt ba s ica lly involve s the

1.1

1.0

0.9'

0.8

,~ 0.7

d 0.6

0 . 5

0 . 4

es tablishment o f two s ymme trica l, pla na r, ne a rly adiaba tic fla me s in a nozzle -ge ne ra te d counte r- flow, the de te rmina tion o f the axia l ve locity profile a long the ce nte rline o f the flow by us ing la s e r Dopple r ve locime try, a nd the identifica tion

Rames =

K= - / dx

u = S L CH4 /AIR

0.3 • = 0.8 0.2 i t t . t ~ I , ,

0 1 2 3 4 5 6 7 X~ 1111111

Fig. 3. Typica l axia l ve locity profile across a s tagna tion flame, showing the definitions of K and SL.

378 F. N. E G O LF O P O ULO S

4 0

3 0

2 0

10

O = 1.0

O = 1.3

I~ ' ~ 0 = 0.6

i I i I i I i 0 2 0 0 4 0 0 6 0 0

K, S "1

C H 4 / AIR p = 2 ATM

I 8 0 0

Fig. 4 . Va ria tion o f the fla me s pe e d S L(K) with the s tra in ra te K.

1000

o f the minimum point o f the ve loc ity p ro fd e a s a re ference ups tre a m fla me s pe e d SL corre s ponding to the impos e d s tre tch ra te K, which by de finition [5] is s imply the ve loc ity gra die nt a he a d o f it. Thus by plotting SL ve rs us K, the la mina r fla me s pe e d without s tre tch, S ° , ca n be de te rmine d through line a r e xtra pola tion to K = 0 (Fig. 4) in a ccord- a nce with the ore tica l pre s crip tions [4, 5]. The line a r e xtra pola tion is me a ningful a nd a ccura te only fo r s ma ll va lue s o f the nondime ns iona l s tre tch ra te

DK Ka = ,~ 1, (s o)

whe re Ka is the Ka rlovitz numbe r a nd D a re pre s e nta tive ma s s diffus ivity. Re ce nt s tudie s on fla me e xtinction [5] ha ve s hown tha t Ka inde e d a s s ume s s ma ll va lue s p rio r to e xtinction. It ma y a ls o be e mpha s ize d tha t SL is on ly a re ference ups tre a m fla me s pe e d be ca us e the ga s te mpe ra ture a lre a dy s ta rts to incre a s e s lightly be fo re the minimum point is re a che d. The SL s o de fine d turns out to incre a s e with incre a s ing K. Ho we ve r, SL doe s de ge ne ra te to the true S O in the limit o f va nis hing s tre tch.

The dia me te rs o f the nozzle s us e d in the pre s e nt inve s tiga tion a re 7 a nd 14 mm. The y ha ve high contra ction ra tios a nd a re wa te r-coole d a nd nitro- ge n-s hroude d. The e ntire bu rne r a s s e mbly is

hous e d in a la rge s ta inle s s -s te e l h igh-pre s s ure cha mbe r with continuous ve ntila tion, the re by a l- lowing fo r pre s s ure a djus tme nt. The cros s s e ction o f the te s t portion o f the cha mbe r is 14.5 by 14.5 c m .

Ignition is a chie ve d by a continuous high- e ne rgy s pa rk p roduce d by re tra cta ble e le c trode s . The LDV ope ra te s in the fo rwa rd s ca tte ring mode with 0.3-/~m a lumina pa rtic le s e e ding.

EXP ERIMENTAL RES ULTS AND DIS CUS S IONS

Figure 5 s hows the me a s ure d S O a s functions o f the e quiva le nce ra tio ~ a nd s ys te m pre s s ure p fo r me tha ne -a ir mixture s . The re s ults s how tha t the fla me s pe e d de cre a s e s with incre a s ing pre s s ure , which is in qua lita tive a gre e me nt with the pre vious ly re porte d be ha vior [11]. The s e re s ults , how- e ve r, a re not a s funda me nta lly s ignifica nt a s the ma s s burning ra te

O _ 0 m L-- puS L,

which a ccounts fo r the de ns ity va ria tion with pre s s ure a nd is the p rope r e ige nva lue fo r la mina r fla me propa ga tion, whe re Pu is the de ns ity o f the unburne d mixture . Thus the da ta o f Fig. 5 a re re plotte d in Fig. 6 a s m ° ve rs us ~b a n d p . It is s e e n tha t m ° incre a s e s with incre a s ing pre s s ure . Be -

LAMINAR F LAME S P EEDS 379

7 0

5 0

4 0

r J

°3= r,/3 30

2 0

10

4 - :

0 , I , I , I , I i I i 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 , 6

@ Fig. 5. Experimental laminar tiame speeds S°(O, p) for pressures between 0.25 and 3 a tm.

ca us e (m0L) 2 is proportiona l to the ove ra ll re a ction ra te a s e xpe rie nce d by the fla me , the incre a s ing tre nd o f m0L with pre s s ure implie s tha t the ove ra ll re a ction orde r n , a t a give n ~ , is pos itive for a ll s toichiome try a nd pre s s ure ra nge s s tudie d he re in. Furthe rmore , s ince the diffe re nce in m ° is la rge r for ne a r-s toichiome tric fla me s a nd s ma lle r for ne a r-limit fla me s , the s pe cific va lue s o f n s hould a ls o va ry a ccordingly. The de cre a s ing tre nd o f n for s lowe r burning fla me s note d a bove is e xpe cte d to be due to the incre a s ing importa nce o f the thre e - body, te mpe ra ture -ins e ns itive , te rmina tion re a c- tion

H + O2 + M-*HO 2 + M

re la tive to the two-body, te mpe ra ture -s e ns itive , bra nching re a ction

H + O2--'O + OH,

a s dis cus s e d, s a y, in Re fs . 6, 7, a nd 12 a nd a nticipa te d for the pre s e nt inve s tiga tion.

Figure 7 plots the pre s e nt da ta a nd the ba nd cove ring the e xis ting e xpe rime nta l da ta from the lite ra ture for the me tha ne -a ir fla me s pe e ds a t 1 a tm. Figure 8 compa re s the s toichiome tric da ta for va rious pre s s ure s . The s ignifica nt s pre a d in the e xis ting da ta a nd the re by the ne e d for the pre s e nt a dia ba tic, s tre tch-fre e re s ults a re a ga in e mpha - s ize d.

NUME R C IAL R E S ULTS AND DIS CUS S IONS

(R1) In orde r to nume rica lly s imula te the fre e propa ga - tion o f a dia ba tic, s te a dy, one -dime ns iona l pla na r me tha ne -a ir fla me s , we ne e d a nume rica l code for fla me propa ga tion a nd a kine tic s che me de s cribing me tha ne -a ir re a ctions . The s e a re pre s e nte d in the

(R2) following.

3 8 0 F . N. E G O LF O P O U LO S

0.08

@1

-0- 0.25 ATM • .e- 0.5 ATM

1.0 ATM -e- 2.0 ATM

0.06 3.0 ATM

E Y

0.04 / ~ 4

\, 0.02

0 . 0 0 , I , I , I ' ' 0 . 4 0 . 6 0 . 8 1 .0 1 .2 1 .4 1 .6

@ Fig. 6. Experimental laminar mass burning rates m°(~, p) for pressures between 0.25 and 3 a r m .

50

4 0

~ 3O o . I

/ t "x% / /~Z • % / %

/ % /

2O " - Ba n d o f S e le c te d Lite ra ture Va lue s • P re s e n t Expe rime n ta l Da ta at P re s e n t Nume rica l Da ta

10 ' ' ' ' ' 0.6 0.7 0.8 " 0.9 1.0 1.1

@

C H 4 / AIR p = 1 AT M

l i I .

1.2 1.3 1.4 1.5

Fig. 7. Compa ris on be twe e n pre s e nt e xpe rime nta l a nd nume rica l S °(~) for me tha ne -a ir fla me s , a t p = 1 a tm, with the "b a n d " o f va lue s re porte d in the lite ra ture a nd nume rica l da ta ca lcula ted by us ing the Ke e e t a l.-C~ s che me [15].


50

m Garforth/Rallis • • Babkin et al.

• Andrews/Bradley 40 ~ , ~ • lijima/Takeno lit ~ 4 Present Data

~ 30 ~ • ~ • 0%

20 • C H 4 / AIR

• = 1.0 10 , I , I , I ,

0 1 2 3 Pressure , Atm

Fig. 8. C o m p a ris o n be twe e n p re s e n t e xpe rime n ta l S ° fo r s to ich iome tric m e th a n e -a ir fla m e s , with s e le c te d va lue s fro m the lite ra ture , a t d iffe re n t p re s s u re s .

Flame Code

For the fla me code we choos e the one de ve lope d by Ke e a nd co-worke rs [13-16], which is ge ne r- a lly a cce pte d to be a mong the mos t a ccura te a nd e fficie nt. We ha ve , howe ve r, ma de one modifica - tion in the de s cription o f the tra ns port prope rtie s . Tha t is , by a s s uming tra ce s pe cie s diffus ion, Re f. 14 a pproxima te s the ma s s diffus ion coe ffic ie nt o f s pe cie s k a s

1 -Yk Dk-- N

j~k

a nd its ma s s diffus ion ve locity a s

Vk= - D__~ VXk, Xk

whe re Xk a nd Yk a re , re s pe ctive ly, the mola r a nd ma s s fra ctions o f k, Djk the ma s s diffus ivity be twe e n j a nd k, a nd N the tota l numbe r o f s pe cie s .

Be ca us e Eq. 1 is not e xa ct, ma s s cons e rva tion

N

I

is not s a tis fie d. The re fore in orde r to s a tis fy Eq. 3, Vk a s give n by Eq. 2 is corre cte d [14] by a n a mount e qua l to - Z IN Vk Yr.

In obta ining the fina l re s ults in the pre s e nt ca lcula tions , we dire ctly us e the more rigorous re s ult for multicompone nt diffus ion [18]

N

= 1, 2 , . . . , N , (4 ) (1)

whe re diffus ion due to pre s s ure gra die nt, body force , a nd S ore t e ffe c t a re ne gle cte d. The N unknowns Vk ca n be s olve d from Eq. 3 a nd the N - 1 line a r e qua tions give n by Eq. 4. Ove ra ll diffus ive ma s s flux cons e rva tion, Eq. 3, is a uto- ma tica lly s a tis fie d.

(2) In a ctua l ca lcula tions we firs t s olve the proble m by us ing the tra ce s pe cie s formula tion. This s olution is the n us e d to s ta rt the ca lcula tion with the rigorous d iffus ion formula tion. For a s toichiome tric me tha ne -a ir fla me a t 1 a tm, us ing a C1 kine tic s che me with 17 s pe cie s a nd 58 re a ctions [15], the CP U time for the comple te double pre cis ion s olution is a pproxima te ly 130 min on a VAX 11/785 compute r. The a dditiona l CP U time

(3) to a llow for the e xa ct d iffus ion formula tion is a bout 15 ra in.

382 F. N. EGOLFOP OULOS

For the fla me s pe e ds ca lcula te d in the pre s e nt s tudy, the rigorous diffus ivity formula tion yie lds only a ma rgina l diffe re nce from thos e obta ine d by a s s uming tra ce s pe cie s diffus ion. Furthe rmore , the diffe re nce is within the 0 .5 -2 cm/s unce rta inty a s s ocia te d with the e xpe rime nta l da ta . Thus this rigorous formula tion is a ctua lly not ne e de d for the pre s e nt ca lcula tion. It is dis cus s e d he re be ca us e o f its pote ntia l utility for mixture s with h ighe r re a c- ta nt conce ntra tions .

It is importa nt to point out tha t we ha ve us e d the s a me tra ns port prope rtie s code a nd fla me code in the following compa ris ons o f the va rious kine tic s che me s . Thus diffe re nce s in the ca lcula tions a re s ole ly cons e que nce s o f the diffe re nt kine tic s che me s us e d. S ys te ma tic compa ris ons o f this na ture ha ve a ls o be e n conducte d by Coffe e a nd He ime rl [19] for the tra ns port a lgorithms a nd by Coffe e [20] for the kine tic me cha nis ms . The pre s e nt s tudy provide s a dditiona l contributions in te rms o f nume rica l e va lua tions o f the kine tic s che me s ove r e xte ns ive ra nge s o f conce ntra tion a nd pre s s ure a nd compa ris ons o f the ca lcula te d re s ults with the pre s e nt e xpe rime nta l da ta .

Kinetic S che me s

For me tha ne oxida tion, the C2 che mis try which include s the C1 re a ctions is s ufficie nt for comple te mode ling [7, 17, 21, 22]. The C1 pa th is the principa l cha nne l for fue l-le a n fla me s , whe re a s the comple te C2 che mis try is e xpe cte d to be ne e de d to de s cribe fue l-rich fla me s . In the pre s e nt inve s tiga - tion we ha ve te s te d quite a numbe r o f publis he d kine tic s che me s a ga ins t our da ta a s we ll a s a ga ins t e a ch othe r. The s e compa ris ons a re pre s e nte d be low. Be fore doing s o, howe ve r, it ma y be e mpha s ize d a t this point tha t be ca us e the fla me s pe e d is only a bulk prope rty o f the la mina r pre mixe d fla me , a gre e me nt be twe e n the pre s e nt da ta a nd ca lcula te d va lue s a s s uming a ce rta in kine tic s che me is only a ne ce s s a ry but not s ufficie nt re quire me nt for the va lida tion o f the s che me . A comple te va lida tion would re quire a gre e me nt in the me a s ure d a nd compute d te mpe ra ture a nd s pe cie s profile s a s we ll. The pre s e nt compa ris on, howe ve r, is s till quite us e ful be ca us e o f the following re a s ons : (1) Be ca us e a gre e me nt in the

la mina r fla me s pe e d is the min imum re quire me nt o f a propos e d s che me , the la ck o f a gre e me nt is the n a cle a r indica tion tha t the s che me re quire s re vis ion. (2) Be ca us e o f the a ccura cy o f the pre s e nt e xpe rime nta l da ta , a tighte r re quire me nt in the de gre e o f a gre e me nt ca n be impos e d, the re by le nding gre a te r confide nce to the va lida tion. (3) Accura te e xpe rime nta l de te rmina tions o f the te m- pe ra ture a nd conce ntra tion profile s for adiabatic uns tre tche d fla me s a re a s ye t una va ila ble . Be fore the s e da ta be come a va ila ble , va lida tion through fla me s pe e d compa ris on is a via ble inte rim s olution.

We now pre s e nt the compa ris on be twe e n our e xpe rime nta l da ta a nd the compute d one s a s s uming va rious kine tic s che me s . Mos t o f the s e s che me s a re ta bula te d in the ir re s pe ctive re fe r- e nce s a nd the re fore a re not s e pa ra te ly lis te d he re .

C1 S che me o f Ke e e t ai [15[ Figure 9 compa re s the e xpe rime nta l fla me s pe e ds with ca lcula te d va lue s us ing this s che me , which include s only the C1 me cha nis m with 17 s pe cie s a nd 58 re a ctions . Ca lcula tion wa s te rmina te d whe n conve rge nce be ca me e xce s s ive ly s low, typica lly for high-pre s s ure a nd low-te mpe ra ture fla me s . The e xpe rime nta l fla me s pe e ds a re s hown a s dotte d line s which a re dire ct re productions o f the s olid line s in Fig. 5. It is s e e n tha t clos e a gre e me nt e xis ts , to within the e xpe rime nta l unce rta inty, for a ll s itua tions compa re d, including a ll le a n fla me s a s we ll a s rich fla me s up to e quiva - le nce ra tios o f a bout 1.2 to 1.3. Although the clos e a gre e me nt with the rich fla me s ma y be s ome wha t s urpris ing cons ide ring tha t only the C1 me cha nis m is us e d, it ha s be e n s ugge s te d [23] tha t the a dditiona l C2 re a ctions ma y not be too importa nt for the de te rmina tion o f me tha ne -a ir fla me s pe e ds for e quiva le nce ra tio a s high a s 1.3.

C1 S che me o f Pe te rs and Ke e I241 This is a re duce d Cl s che me tha t include s 14 s pe cie s a nd 18 re a ctions . Figure s 10 a nd 11, re s pe ctive ly, compa re the fla me s pe e d a s a function o f e quiva le nce ra tio a t 1 a tm, a nd a s a function o f pre s s ure a t ¢ = 1. It is s e e n tha t this s che me uniformly ove rpre dicts the la mina r fla me s pe e d by a s ubs ta ntia l a mount.

LAMINAR FLAME S PEEDS 383

70

I

60 ~

50

40

¢J

r,/3 30

20

10

- - - - EXPERIMENT

qUMERICAL (KEE ET AL.-C1) l a

P /~Z , f ,~ / / , f

/ - ~ ' . , # //g

0.25 ATM N 0.5 ATM / \ 1.0 ATM / \ • \ 2.0 ATM ,/ 3.0 ATM / / \ \

/ / ~ - ~ , \ • ¶ '~ / / 1 4 f \ \

/ / - / \ \ = / ,'= \ \

I \ \ / / ~--* \ s / ,s ~ ~ \ / , / / \ " \

/ / ~/ \ \ k \

/ , / / j , ~ - s - . x \ / ,m ,,,/ ~ " \ X \

• "/ / / / / :\ \\ X / / s \ \ \ = \

/ , ~ / \ \ \

0 , I i I i I t I , I i

0,4 0.6 0.8 1,0 1.2 1.4 1.6

Fig. 9. Compa ris on be twe e n e xpe rime nta lly a nd nume rica lly de te rmine d S°L(~, p ) for me tha ne -a ir mixture s , us ing the C, kine tic s che me of Kee e t a l. [15].

60

5O

4O

~ 30

20

10

0 0.4

-- -- EXPERIMENT CH 4 / A I R • PETERS/KEE-C1 • • • p = l ATM • ECL-Cl(19) • • ECL-C1(23)

* ..4r~ - - i t . . / 0 0 "~ l

v\ /

s \ \ / o \ \

/1l' N \ /

I i I I i I

0.6 0.8 1.0 1.2 1.4 1,6

Fig. 10. Compa ris on be twe e n e xpe rime nta lly a nd nume rica lly de te rmine d S°L(~b) for me th- a ne -a ir mixture s a tp = 1 a tm, us ing P e te rs -K•• [24] a nd the pre s e nt ECL-CI(19) a nd ECL- Cj (23) kine tic s che me s .

384 F . N . EGOLFOP OULOS

° ~

70

60

50

40

30

20

10

- - - EXPERIMENT ,D PETERS/KEE-Cl

\ . • ECL-C1(19} o \ • ECL-Cl(23)

IN ,o. \ ~'

\ •

O, ~ ,O

C H 4 / AIR . @ =1.0 *

0 1 2 3 4 Pressure , A~

Fig. 11. Compa ris on be twe e n e xpe rime nta lly a nd nume rica lly de te rmine d S o for s toichiome - tric me tha ne -a ir mixture s a t d iffe re nt p re s s ure s , us ing P e te rs -Ke e [24] a nd the pre s e n t ECL- Ci (19) a nd ECL-C1 (23) kine tic s che me s .

Pre s e nt C~ S che me s In orde r to e xpla in the la ck o f a gre e me nt for the s che me o f P e te rs a nd Ke e [24], a s e ns itivity a na lys is wa s pe rforme d us ing the C~ s che me o f Ke e e t a l. [15]. The re s ults s how tha t the ra dica l re combina tion re a ction

CH3 + H + M~ C H4 + M,

which is ne gle cte d in the s che me o f P e te rs a nd Ke e [24], is a ctua lly o f le a ding orde r importa nce . Its omis s ion ca n a ffe ct the fla me s pe e d by a s much a s 8 cm/s for s toichiome tric me tha ne -a ir fla me s a t 1 a tm. Figure s 10 a nd 11 s how tha t much improve - me nt re s ults by including this s te p. This s che me o f 14 s pe cie s a nd 19 s te ps , de s igna te d a s ECL- C~ (19), is lis te d a s the re a ctions without a s te ris ks in Ta ble 1.

The s e ns itivity a na lys is furthe r s hows tha t the re a re four a dditiona l re a ctions o f s e conda ry importa nce not include d in ECL-CI(19). By including the m, Figs . 10 a nd 11 s how ve ry clos e a gre e me nt with the e xpe rime nta l da ta . This s che me o f 23 s te ps with 15 s pe cie s , de s igna te d a s ECL-C1 (23), is lis te d a s re a ctions 1 through 23 in Ta ble 1.

Although ECL-C1 (23) s e e ms to provide a clos e r a gre e me nt with the e xpe rime nta l da ta tha n ECL- C~(19), we ca nnot conclude tha t it is a be tte r s che me . This is be ca us e o f the s ma ll unce rta intie s

tha t s till e xis t in our e xpe rime nta l da ta a s we ll a s the C 1 s che me o f Ke e e t a l. [15] ba s e d on which the s e ns itivity a na lys is is pe rforme d. Tha t is , if the e xpe rime nta l fla me s pe e ds a round s toichiome try we re lowe r tha n our re porte d va lue s by, s a y, 2 cm/ s , which is s tre tching our unce rta inty limit but is ne ve rthe le s s a rgua bly pos s ible , the n the e xpe rime nta l da ta would mos tly lie e ithe r midwa y be twe e n ca lcula te d va lue s give n by the s e two s che me s or a ctua lly be clos e r to thos e o f the ECL- C1 (19) s che me .

C2 S che me o f Glarborg e t ai. [251 This s che me , de s igna te d a s Gla rborg e t a l.-C2, cons ide rs 26 s pe cie s with 123 re a ctions . Figure 12 s hows tha t a t 1 a tm it pre dicts the le a n fla me s we ll but ove rpre dicts the rich fla me s . Figure 13 s hows tha t the pre dicte d va lue s ha ve lowe r fla me s pe e ds unde r lowe r pre s s ure s but highe r fla me s pe e ds unde r highe r pre s s ure s . This diffe re nce is be lie ve d [23] to be ca us e d by the us e o f a s ome wha t s lowe r ra te for the fa llo ff be ha vior

CH3 + CH3 + M-*C2H6 + M.

Modifica tions a re curre ntly be ing imple me nte d by Mille r [23] to re ctify this e ffe ct.

Figure 12 a ls o s hows the ca lcula te d fla me s pe e ds us ing only the C1 pa rt o f the full C2 s che me , de s igna te d a s Gla rborg e t a l.-C~. Al-

LAMINAR FLAME S PEEDS

TABLE 1

E C L-C I (19) An d ECL-CI (23) S che me s °

385

Re a c tion .4 /~ Eo

1. CH3 + H + M = C I-h + M 8 .0E26 - 3 , 0 0. 2 . CI-L + H = CH3 + H2 2.2E,4 3 .0 8750.

*3. Cl'Lt + O = CH3 + O H 1.6E6 2 .36 7400. 4. CI'L + O H = CH3 + H2 0 1 .6E6 2.1 2460. 5. C Ha + O - C H 2 0 + H 6 .8E13 0 .0 0.

*6. CHa + O H = CH2 + H2 0 1.5E13 0 .0 5000. *7. C H 2 + O 2 = C O 2 + H + H 1.6E12 0 .0 1000. *8. C H 2 + 02 = CO2 + H2 6 .9E11 0 .0 500.

9. CH~O + H = HC O + H2 2 .1 9 E 8 1.77 3000, 10. C H2 0 + O H = HC O + H2 0 3 .43E9 l. 18 - 477. 11. H C O + H = C O + H 2 4 .00E13 0 ,0 0, 12. H C O + M= H + C O + M 1 .6 0 E I4 0 ,0 14700. 13. HC O + O 2 =HO 2 + C O 3 .3E13 - 0 , 4 0, 14. C O + O H = C O 2 + H 1 .51E7 1,3 - 758. 15. H + O 2 = O H + O 5 .1 3 E l6 - 0 , 8 1 6 16507. 16. O + H 2 = O H + H 1.8E10 1 .0 8826. 17. O H + H2 ~ H2 0 + H 1 .17E9 1.3 3626. 18. O H + O H = H 2 0 + O 6 .0E8 1.3 0. 19. H + O 2 + M= H O 2 + M b 3 .61E17 - 0 . 7 2 0, 20 . H + O H + M= H 2 0 + M c 1 .6E22 - 2 . 0 0. 21. H + H O e ~ 2 O H 1.4E14 0 ,0 1073. 22 . H + H O 2 ~ H 2 + O 2 1.25E13 0 ,0 0. 23 . O H + HO2 = HzO + 02 7 .5E12 0 ,0 0.

o Re a c tion m e c h a n is m ra te coe ffic ie n ts in the fo rm k~ = AT# e xp(-E,/R T) (units a re mo le s , cub ic ce n time te rs , s e conds , Ke lvins a n d ca lorie s /too l, ta ke n fro m Ke e e t a l. [15]). An a s te ris k indica te s the re a c tion is no t inc lude d in the ECL-C~ (19) s c h e m e .

b Th ird -body e ffic ie nc ie s : k(02) = 1 ,2 6 k(Ar), k(N:) = 1 .2 6 k(Ar), k(H:O ) = 1 8 .6 k(Ar), k(C O ) = 2 .1 l k(Ar), k(CO2) = 4 .2 k(Ar), k(H2) = 2 ,8 6 k(Ar).

c Th ird body e ffic ie nc ie s : k(O2) = k(N2) = k(C O ) = k(COz) = k(Hz) = 1, k(H2 0 ) = 5 .0 k(Ar).

60

50

4O

30 o~

20

10

0 L m ~ I _ •

0.4 0.6

- - - - EXPERIMENT • GLARBORG et aL-C2 o GLARBORG et al.-C1 • COFFEE.C1

/ ,11 /

/ 0

\ • ta a \ '~

a N A

CH 4 / AIR p= ! AT M

\\m a\

"x \

I . . _1 . . . . | • I

0.8 1 .O 1.2 1.4 1.6

Fig. 12. C o m p a ris o n be twe e n e xpe rime nta lly a nd nume rica lly de te rmine d S °t(~) fo r me th - a n e -a ir m ixtu re s at p = 1 a tm , u s in g the C2 a nd C~ kine tic s c h e m e s o f Gla rborg e t a l. [25] a nd the C~ kine tic s c h e m e o f Coffe e [20].

386 F . N . E G O LF O P O ULO S

70

60

50

• 4o

r./3 30

20

10

\ \

• \ \ , , ,q ,

CH 4 / AIR ~ = 1.0

-- - - EXPERIMENT • GLARBORG et al.-C2 m COFFEE-C1

, I t I i I i

0 1 2 3 4 Pressure, A i m

Fig. 13. Comparison between experimentally and numerically determined S ° for stoichiome- tric methane-air mixtures at different pressures, using the C2 kinetic scheme of Glarborg e t al. [25] and the C, kinetic scheme of Coffee [20].

though ide a lly one would e xpe c t tha t the ca lcula te d re s ults s hould a gre e with thos e obta ine d from the C~ s che me o fKe e e t a l. [15] s hown in Fig. 9 , Fig. 12 s hows tha t the y a re un iformly lowe r tha n the pre vious one s o ve r the e ntire ra nge o f s to ichiome - try compa re d.

In o rde r to ide ntify the ca us e fo r s uch a d iffe re nce , Fig. 14 compa re s the individua l re a c- tion ra te s from the two s che me s tha t s how s ome diffe re nce , with kc2 a nd kc l, re s pe ctive ly, de s igna ting the s che me s o f Gla rborg e t a l. [25] a nd Ke e e t a l. [15]. F igure 15 s hows the s e ns itivity o f the s e individua l re a ction ra te s on the fla me s pe e d. The compa ris on s hows tha t pe rha ps e xce p t fo r the

C H2 0 + H - ' H C O + H2 0

re a ction, the re is no s ingle re a ction tha t is dra s tica lly d iffe re nt in the two s che me s . Thus the d iffe re nt fla me s pe e ds pre dic te d a re the cumula - tive d iffe re nce s from a numbe r o f re a ctions .

The a bove re s ult illus tra te s the s ubtle ty o f code ca libra tion. Tha t is , in de ve loping kine tic s che me s , it is a fre que nt pra c tice to a djus t s ome kine tic cons ta nts s uch tha t the ca lcula te d re s ults a gre e with the e xpe rime nta l da ta a t one point, typica lly the s ta te o f s to ichiome try. The re fo re if ca libra tion ha s be e n s e pa ra te ly conducte d fo r the C~ a nd the full C2 s che me s , it is the n re a s ona ble to

e xpe ct tha t re s ults from the C1 pa rt o f the (?2 S che me s hould de via te from the ca libra tion point.

C2 S che me o f Coffe e [2011 This s che me cons ide rs 21 s pe cie s with 57 re a c- tions . A de ta ile d compa ris on ha s not be e n conducte d, a lthough a s ingle point che ck a t q~ = 1 s hows a ca lcula te d fla me s pe e d o f 45 .7 e ra /s , which is cons ide re d to be high re la tive to the e xpe rime nta l va lue o f 40 cm/s . Figure s 12 a nd 13 s how compa ris ons by us ing the C1 pa rt o f this s che me . Good a gre e me nt is obta ine d up to the s ta te o f s to ichiome try. F o r riche r fla me s Fig. 12 s hows h ighe r ca lcula te d va lue s a t 1 a ttn.

Although both the C1 s che me s o fKe e e t a l. [15] a nd o f Coffe e [20] yie ld good a gre e me nt fo r le a n fla me s , Co ffe e 's s che me give s h ighe r va lue s fo r rich fla me s . A de ta ile d s e ns itivity s tudy s hows tha t this d iffe re nce is ca us e d by the s lightly d iffe re nt ra te s o f the crucia l re a c tion (R2).

We note in pa s s ing tha t ou r ca lcula te d re s ults a re cons is te ntly h ighe r tha n Co ffe e 's publis he d va lue s . Thus his C2 ca lcula tion a ctua lly yie lds a fla me s pe e d o f 40 cm/s a t ~b = 1, which a gre e s with the e xpe rime nta l va lue . Be ca us e both his

The value of Eo/R for the reaction CH3 + H = CH2 + H2 in Ref. 20 should be + 1500 K instead of - 1500 K; this is only a misprint.


1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0 1200

• • • • • • • • • • • • • • • • m [] • • •

.ii,,i; ||lllll

• H+OH+M,=H20+M • H+O2+M,.HO2+M x HO2+H=H2+O2 • HO244,,OA+OH • HO~+O-OH*O2

i I m I i I i I i

: : : : : ; : ; : | l l l! .

÷ 4- 4- 4- 4- 4- 4- 4" 4- 4- 4- 4- 4- 4- 4-

t I m I t 1400 1600

T, K

- ~ + H , ÷

1800 2000 2200

F ig . 14 . C o m p a r is o n b e twe e n th e s p e c ific re a c t io n ra te s , k2 a n d k l, o f s e le c te d C i- im p o r ta n t re a c t io n s fr o m th e kin e tic s c h e m e s o f G la r b o r g e t a l. [2 5 ] a n d Ke e e t a l. [1 5 ], re s p e c t ive ly.

tra ns port a lgorithm a nd fla me code a re be lie ve d [26] to be a ccura te , the ca us e o f this dis a gre e me nt is not known a t pre s e nt.

C2 Scheme o f Warnatz 122] This s che me cons ide rs 26 s pe cie s with 74 re a c- tions . Figure s 16 a nd 17 s how compa ris ons for the full (?2 s che me a s we ll a s its Ci compone nt. The compa ris on ca n be cons ide re d s a tis fa ctory for the following re a s ons : (a ) The (2?2 s che me a gre e s we ll ove r the e ntire ra nge o f s toichiome try compute d, with the la rge s t d iffe re nce occurring a round s toichiome try. Co) The pre s s ure de pe nde nce o f the ca lcula te d va lue s a gre e we ll with the e xpe rime nta l tre nd. (c) Contra ry to the (:72 me cha nis m o f Gla rborg e t a l. [25], pre dictions o f the le a n fla me s by us ing the Cx compone nt o f Wa rna tz 's C2 s che me a gre e we ll with thos e obta ine d from the

full (?2 s che me a s we ll a s the e xpe rime nta l da ta , indica ting inte rna l cons is te ncy o f the s che me .

Present C2 Scheme A s e ns itivity a na lys is o f Wa rna tz 's C2 s che me a llows us to propos e a C2 s che me with 22 s pe cie s a nd 40 re a ctions , a s lis te d in Ta ble 2. Figure s 16 a nd 17 s how tha t the ca lcula te d va lue s a gre e we ll with the e xpe rime nta l da ta . Compa re d with the C2 s che me o f Wa rna tz [22], the ca lcula tion time -for the pre s e nt s che me is re duce d.

C2 Scheme o f We s tbrook [27] This s che me cons ide rs 24 s pe cie s with 74 re a c- tions . The ca lcula te d fla me s pe e d for the s toichiome tric me tha ne -a ir fla me a t 1 a t • is 27.5 cm/ s . Cons ide ring this va lue to be low, we did no t


CISCHEMEOFKEEETAL. CH4/AIR

O=I.0 p=IATM

HCO-tO--CO2+H I CH20+H=HCO+H2 I

CH4+OH=CH3+H201 HO2+OH=H20¢O21

HO2+O=OH+O2 I

HO2+H=H2-¢O2 I

I CH2+O2=CH20+O

-0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 Fig. 15. Normalized sensitivity on the flame speeds of the reactions shown in Fig. 14.

continue ca lcula tions a t d iffe re n t conce ntra tions a nd pre s s ure s .

C2 S c h e m e o f P itz and We s tbrook [28]

By e xcluding fro m the lis te d s che me re a c tions involving CH3OH, CH3CHO, CH2CO, a nd HC C O , which a re be lie ve d [22] to be min ima lly importa n t fo r fla me s pe e d ca lcula tions , the re - duce d s che me cons ide rs 24 s pe cie s a nd 82 re a c- tions . The ca lcula te d fla me s pe e d o f s to ich iome tric m e th a n e -a ir mixtu re a t 1 a tm is 23 .0 cm/s , which re duce s to 21 .0 c m /s whe n on ly the C1 pa rt o f the s che me is us e d.

We note tha t the ca lcula te d fla me s pe e d o f s to ichiome tric m e th a n e -a ir mixtu re a t 1 a tm b y We s tb rook [29] is 38 .0 cm/s , which is cons ide ra - b ly h ighe r tha n the va lue s ca lcula te d he re . Be ca us e the s a me kine tic s che me is us e d, the d iffe re nce is be lie ve d [30, 31] to be ca us e d b y the d iffe re nt tra ns port code s us e d in Re fs . 2 7 -2 9 a nd the pre s e nt ca lcula tions .

C O N C LU D IN G R E MAR KS

The pre s e nt s tudy ha s de mons tra te d the importa nce o f a ccura te de te rmina tion o f the la mina r

¢/3

o ~

6O

5O

4O

30

2O

10

0 0.4

- - - - EXPERIMENT CH 4 / A I R a WARNARTZ-C2 p = 1 A T M + WARNARTZ-C1 x ECL-C2(40)

/ ÷ \ ~

+ \+ /

I I I I I I

0.6 0.8 1.0 1.2 1.4 1.6

Fig. 16. Comparison between experimentally and numerically determined S t(O) for methane-a ir mixtures a t p = 1 a tm, us ing the C2 and C1 kinetic schemes of Warnatz [22], and the present ECL-C2(40) kinetic scheme.

LAMINAR FLAME S PEEDS 389

TABLE 2

ECL-C2 (40) S che me °

Re a ction A /3 Eo

1. H + O 2 + O H + O 1.20E17 - 0 . 9 1650~ 2. O + H2 = OH + H 1.50E07 2.0 7548. 3. OH + H2 = H20 + H 1.00E08 1.6 3296. 4. H + H + M= H 2 + M b 9.70E16 - 0 . 6 0. 5. H + O H + M= H 2 0 + M ~' 2.20E22 - 2 . 0 0. 6. H + O 2 + M= H O 2 + M b 2.00E18 --0 .8 0. 7. H + H O 2 = O H + O H 1.50E14 0 .0 1003. 8. H + HO2 = H2 + 02 2.50E13 0.0 693. 9. OH + HO2= H20 + O2 2.00E13 0 .0 0.

10. C O + H + M= H C O + M b 6.90E14 0 .0 1672. 1 l. C O + O + M= C O 2 + M b 7.10E13 0 .0 -4 5 3 8 . 12. CO + OH = CO2 + H 4.40E06 1.5 - 740.4 13. CI-L + H = CH3 + H2 2.20E04 3.0 8742. 14. CI'L + O = CH3 + OH 1.20E07 2.1 7619. 15. CI-L + OH = CH3+ H20 1.60E06 2.1 2460. 16. C H3 +H=C I-L c 6.00E16 - 1.0 0. 17. CH3 + O = C H2 0 + H 7.00E13 0 .0 0. 18. CH3 + 02 = CH20 + H + O 1.50E13 0 .0 28662. 19. CH20 + H = HCO + H2 2.50E13 0.0 3989. 20. CH20 + OH = HCO + H20 3.00E13 0 .0 1194. 21. H C O + H = C O + H 2 2.00E14 0 .0 0. 22. H C O + O = C O + O H 3.00E13 0 .0 0. 23. H C O + O = C O 2 + H 3.00E13 0 .0 0. 24. HCO + OH = CO + H20 5.00E13 0.0 0. 25. HCO + 02 = CO + HO2 3.00E12 0 .0 0. 26. CH2 + 02 = CO2 + H + H 1.30E13 0 .0 1505. 27. C H + O = C O + H 4.00E13 0 .0 0. 28. CH3 + CH3 = C2H6 c 2.40E14 - 0 .4 0. 29. CH3 + CH3 = C2H5 + H 8.00E14 0 .0 26512. 30. CH3 + CH3 = C21"L + H2 1 .00El6 0.0 32005. 31. C2H6 + H = C2Hs + H2 5.40E02 3.5 5207. 32. C2H6+ OH = C2H5 + H20 6.30E06 2 .0 645. 33. C2H~ = C2I~ + H c 2.00E13 0 .0 39648. 34. C2I'L + OH = C2H3 + H20 3.00E13 0 .0 2986. 35. C 2 H3 +H=C 2 H2 +H2 2.00E13 0 .0 0. 36. C2H3 + O2 = C2H2 + HO2 1.00El2 0 .0 0. 37. C2H3 =C 2 H2 +H c 1.60E14 0 .0 37976. 38. C2H2 + O = CH2 + CO 4.10E08 1.5 1696. 39. C2H2 + O = HCCO + H 4.30E14 0 .0 12110. 40. HC C O + H = C H2 + C O 3.00E13 0.0 0.

a Re a ction me cha nis m ra te coe fficie nts in the fo rm k~ = A T ~ e x p ( - E . / R T ) (units a re mole s , cubic ce ntime te rs , s e conds , Ke lvins a nd ca lorie s /tool, ta ke n from Wa rna tz [22]).

b Third body e fficiencies : k(O2) = 0.4k(H2), k(N2) = 0.4k(H2), k(H20), = 6.5k(H2), k(CO) = 0.75k(H2), k(CO2) = 1.5k(H2).

c High pre s s ure va lue ; corre ction for fa ll-off be ha vior mus t be included.


70

60

50 t~

, o © ~ -,,

30

20

10

\ \

\

÷

- - - EXPERIMENT " WARNARTZ-C2 + WARNART'Z-C1 x ECL-C2(40)

CH 4 / AIR + @ = 1.0 +

i l m I m I

I 2 3 Pressure, Arm

Fig. 17. Compa ris on be twe e n e xpe rime nta lly a nd nume rica lly de te rmine d S O for s toichiome - tric me tha ne -a ir mixture s a t diffe re nt pre s s ure s , us ing the C2 a nd CI kine tic s che me s o f Wa rna tz [22], a nd the pre s e nt ECL-C2(40) kine tic s che me .

fla me s pe e ds fo r the (pa rtia l) va lida tion o f kine tic s che me s . It is c le a r tha t critica l compa ris ons o f the va rious kine tic s che me s could not ha ve be e n conducte d without a s s uring tha t on ly s ma ll unce r- ta intie s a re a s s ocia te d with the e xpe rime nta l da ta . Cons e que ntly, ce rta in s che me s ha ve e me rge d to be more pre dic tive tha n othe rs a s fa r a s the fla me s pe e d is conce rne d. S pe cifica lly, the C1 s che me o f Ke e e t a l. [15] a nd the C2 s che me o f Wa rna tz [22] both a ccura te ly re produce ou r e xpe rime nta l da ta ove r the e xte ns ive conce ntra tion a nd pre s s ure ra nge s te s te d.

The C~ s che me o f Ke e e t a l. indica te s tha t re a ctions involving C2 s pe cie s ma y not be importa nt fo r fla me s a s rich a s ~b ~ 1 .2 -1 .3 a nd pre s s ure s a s high a s 3 a tm, while the C2 s che me o f Wa rna tz s hows tha t the y do ha ve s ubs ta ntia l influe nce e ve n fo r the s to ichiome tric , a tmos phe ric fla me . This d iffe re nce ca nnot be e a s ily re s o lve d be ca us e it is pos s ible tha t the C~ s che me o f Ke e e t a l. ca n be e xpa nde d to a n improve d , full C2 s che me tha t inde e d doe s not s how s ignifica nt influe nce o f the C2 re a ctions up to , s a y, ~b ~- 1.3. Furthe r cons tra ints in the compa ris ons be twe e n the e xpe rime nta l da ta a nd the fla me code ne e d to be de ve lope d to re s olve this s ma ll, but importa nt, d iffe re nce .

Fina lly, we note tha t a s fa r a s fla me s pe e d ca lcula tion is conce rne d , the Cl s che me s (Ta ble 1) propos e d he re in a re e ffic ie nt a nd a ccura te . Fur-

the rmore , a n a pproxima te fla me s tructure ca n a ls o be obta ine d by us ing the (272 s che me (Ta ble 2) de ve lope d in this s tudy.

FNE and P C were s upporte d by the A ir Force Office o f S cientific Research unde r the technical m anage m e nt o f Dr. J. M. T is hko ff and CKL was s upporte d by the Office o f Bas ic Ene rgy S ciences o f the De partm e nt o f Ene rgy unde r the technical m anage m e nt o f Dr. J. Welty. Additional com pute r tim e was provide d by the N S F S upe rcom pute r Center at the Univers ity o f California at S an Diego. W e appreciate the technical discuss ions with Drs . T. P. Coffe e , R . J. Kee , J. A . Miller, and C. K. W e s tbrook on this proble m .

REFERENCES

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5. Law, C. K., Twe n ty -S e cond S ym pos ium (Inte rnational) on Com bus tion , The Combus tion Ins titute , P itts burgh, in pre s s .

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6. Wes tbrook, C. K., and Drye r, F. L., Eighte e nth S ym pos ium (International) on Com bus tion, The Combustion Ins titute , P ittsburgh, 1981, p. 749.

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8. Wu, C. K., and La w, C. K., Twe ntie th S ym pos ium (International) on Com bus tion, The Combustion Ins titute , P ittsburgh, 1985, p. 1941.

9. Yu, G., La w, C. K., and Wu, C. K., Com bus t. Flam e 63:339 (1986).

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23. Mille r, J . A., Persona l communica tions . 24. Pe te rs , N., and Ke e , R. J ., Sandia Re port SAND86-

8842, 1986. 25. Glarborg, P ., Mille r, J . A., and Ke e , R. J ., Com bus t.

Flam e 65:177 (1986). 26. Coffe e , T. P ., Persona l communica tions . 27. Wes tbrook, C. K., Com bus t. S ci. Te chnol. 20:5

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R e ce ive d 11 March 1988; revised 28 Augus t 1988

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