williams 1992 theory

8/2/2019 Williams 1992 Theory

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Twenty-Fourth Sym posium (International) on Combustion/The Combustion Institute, 199 2/pp . 1-17

H O T I ' E L P L E N A RY L E C T U R E

T H E R O L E O F T H E O R Y I N C O M B U S T I O N S C I E N C EF . A . W I L L I A M S

C e n t e r f o r E n e r g y a n d C o m b u s t io n R e s e ar c hDep ar tme n t o f App l i ed Mechan ics and Eng ineer ing Sc iencesUnive rs i ty of Cal i fornia , San D iegoLa Jolla, CA 92093-0310 U SA

P ro m e t h e u s h a s b e e n i d e n t i f ie d a s th e m y t h o l o g i c a l d e l i v e r e r o f c o m b u s t i o n s c i e n c e t om a n k i n d . 1 S c i en c e s t r a d it i o na l ly h a v e b e e n d i v i d e d i n to t w o p a r t s - - e x p e r i m e n t a n d t h e o r y . .A q u e s t i o n t h e r e fo re a r i se s n a t u r a l l y : W a s P ro m e t h e u s m o re n e a r l y a n e x p e r i m e n t e r o r at h e o re t i c i a n ? M y p u rp o s e i s t o a t t e m p t t o r e v i e w fo r y o u t h e h i s t o ry o f th e c o n t r i b u t i o n o ft h e o ry t o t h e s c i e n c e o f c o m b u s t i o n . To d o s o I n e e d t o d e s c r i b e " ' t he o ry " a n d " s c i e n c e "a n d t o t r y t o t r a c e c o m b u s t i o n f rom a n t i q u i t y o n i n t o t h e fu t u r e . A c o n c l u s i o n w i l l b e t h a tt o d a y m o s t sc i e n c e s ; a n d n o t a b l y th a t o f c o m b u s t i o n , m u s t b e d i v i d e d i n t o t h r e e r a t h e r t h a nthe t rad i t iona l two par t s . B u t fi rs t, l e t ' s addre ss tha t b u r n in g que s t ion ra i sed above .

M y t h o l o g yW h e n I w a s a c h il d , m y p a re n t s b ro u g h t m y si s -

t e r a n d m e b y r a i l r o a d o n e d a y f ro m o u r h o m e i nru ra l N e w J e r s e y t o N e w Y o rk C i t y , w h e re w e m a r -v e l e d a t t h e s k y s c ra p e r s a n d v i s i t e d R o c k e fe l l e rP laza . There , nex t to what i s now the ice-ska t ingr i n k , s t a n d s a s t a t u e , c o i n c i d e n t a l l y e r e c t e d i n t h em o n t h t h a t I w a s b o rn , s h o w i n g P ro m e t h e u s w i t ha ba ton o f f i re in h is r igh t ha nd , a t t rac t ive ly exe-c u t e d b y P a u l M a n s h i p . A l t h o u g h m y p a re n t s i n -s i s te d t h a t w e s t u d y t h e s t a t u e , I l e a rn e d o n l y t h a th e w a s s o m e s o rt o f G re e k g o d o r h e ro - - I w a s m u c hm o re i m p re s s e d b y t h e t r a i n r i d e , t h e g i g a n t i cbu i ld ings and the spec tacu lar show ins ide Radio Ci tyMu s ic Hal l . W hat a s t ra nge q u i rk o f fa te i t i s tha th e r e p re s e n t s t h e o l d e s t s y m b o l o f w h a t w a s to b e -c o m e m y c h o s e n p ro fe s s i o n .A n d n o m e a n p ro fe s s i o n ( a v o c a t i o n i n J a p a n ) i ti s, i f w e m a y j u d g e f ro m t h e q u a l i t ie s o f P ro m e -t h e u s . H i s n a m e t r a n s l a t e s a s f o r e t h o u g h t , t h e o p -p o s i te o f t h a t o f h i s b ro t h e r Ep i m e t h e u s , a f t e r-t h o u g h t . T h e y w e re s o n s o f o n e o f t h e T i t a n s , w h ow e re t h e f i r s t b e i n g s i n t h e w o r l d t h a t w e re n o tp u re l y d e s t ru c t i v e a n d t h e f a t h e rs o f t h e g o d s, s u c ha s Ze u s , t h e ru l e r . I n o n e a c c o u n t , t h e g o d s a s k e dt h e b ro t h e r s t o c r e a t e a n i m a l s a n d h u m a n s , a n dEp ime theus qu ick ly gave the bes t a t t r ibu tes he cou ldi m a g i n e t o v a r i o u s a n im a l s , t h e n r a n o u t o f i d ea sf or h u m a n i t y a n d r e q u e s t e d h e l p f r om P r o m e t h e u s ,w h o c o m p l e t e d t h e c r e a ti o n , e n d o w i n g u s w i th a n

9 9 2u p r i g ht , n o b l e s t a tu r e a n d p r e s e n c e o f m m d .W i s d o m i s t h e m o s t p r e v a l e n t c h a ra c t e r i s t i c o fP ro m e t h e u s i n m y t h o l o g y . A l w a y s a f r i e n d t o m a n ,

h e k n e w a n d u n d e r s t o o d t h i n g s t h a t n o t e v e n t h eru l i n g g od s c o u l d fa t h o m . H e e x h i b i t e d a m a z i n gd e t e rm i n a t i o n a n d a b i l i ty t o w i t h s t a n d t o r t u r e - - e v e nt h o u g h h e h a d h e l p e d Ze u s o v e r t h ro w t h e T i t a n s ,a s p u n i s h m e n t fo r b r i n g i n g f i re t o m a n k i n d Ze u sc h a i n e d h i m t o a C a u c a s u s m o u n t a i n t o p a t t h e e d g eof the wor ld , wh ere (F ig . 1 ) each day fo r th i r t ee ngenera t ions (o r , in d i f fe ren t accoun ts , 30 o r 30 ,000y e a rs ) h e e n d u re d t h e e x c ru c i a t i n g p a i n o f a n e a g l ed e v o u r i n g h i s l i v e r (w h i c h g r e w b a c k e a c h n i g h t ) ,r a t h e r t h a n r e v e a l t o Z e u s t h e n a m e o f t h e w o m a nw i t h w h o m Ze u s w o u l d s i r e a s o n w h o e v e n t u a l l yw o u l d o v e r t h r o w h i m . T h e q u a l it i e s o f P r o m e t h e u sm u s t s u r e l y b e a g r e a t a i d t o a n y o n e i n o u r p ro fe s -s ion .Ex p e r i m e n t e r o r t h e o re t i c i a n ? A s s u re d l y t h e l a t -t e r . Th e re i s n o i n d i c a t i o n t h a t P ro m e t h e u s e v e rb u i l t s o m e t h i n g w h i c h h e fo u n d t h a t h e d i d n ' t q u i t el i k e , t h e n t r i e d a g a i n a n d i m p ro v e d i t , n o r d i d h ee v e n n e e d t o o b s e r v e i n o r d e r t o u n d e r s t a n d t h ewor ld abou t h im. On the con t ra ry , he though t th ingso u t c o m p l e t e l y il l a d v a n c e , a n d w h e n h e t o o k a na c t i o n t h e r e s u l t c o n fo rm e d e x a c t l y t o h i s t h e o re t -i ca l concep t ion .M a n k i n d c r e a te d t h e m y t h s o f P r o m e t h e u s s o m e -t i m e b e t w e e n 1 0 0 0 a n d 7 0 0 B . ( ' . Th i s s e e m s a n a p -p ro p r i a t e a g e i n w h i c h t o b e g i n a n a r r a t i v e o f t h e -o ry i n a n y s c i e n c e , b e c a u s e i t i s t h e e a r l i e s t t i m eto wh ich w r i t t en reco rds o f genera l ized concep ts tha tu l t i m a t e l y l e a d t o e s t a b l i s h m e n t o f a s c i e n c e c a n b et r a c ed . T h e M e s o p o t a m i a n w r i t i n g , f r o m t h e c r a d leo f c iv i l i za t ion , wh ich da tes u p to 2000 years ear l i e r ,d o e s n o t a p p e a r t o e x h i b i t s u c h g e n e ra l i z a t i o n s . Tos a ti s fy t h e i r d e s i r e t o u n d e r s t a n d t h e u n d e r l y i n g r a -


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2 H O T I ' E L P L EN A R Y L E C T U R EAlchemy

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FIG. 1 . The pu nishm ent of Prometheus for g ivingfire to mankind (T. Velasquez).

t ionale for their existence, the ear ly Greeks con-nec ted theory wi th exper iment by c rea t ing spec ia lgods who guided the grand exper iments o f na ture .The Greek her i tage of mythology sure ly must haveinf luenced Aristot le , arou nd 350 B.C., in develo pinghis theory of matter . Aristot le 's theo ry led to thef irst science in which combustion played a promi-nent par t .

Although the pr incipal o bjective of the alchem istwas to turn base metals into gold, m any useful as-pects of metal lurg y were involved in alchemy. Th etheory, going back to Aristot le , took the hot-coldand moist-dry quali t ies and combined them into thefour elements, earth (cold , dry), water (cold , moist),air (hot, moist) and fire (hot, dry). Not a science ofcombust ion you say, and I agree , even thoughcombu st ion formed one-qu ar te r o f i ts founda tion.Yet st il l a science, in that th eory sugge sted exper-iments and exper iment cont r ibuted to re f inementof theory. T ransmutation was built in from the start,in tha t by v arying the propor t ions o f the qua li ties ,the four e lements and everything in be tween couldbe obtained. As the science evolved, i t a lso tende dto deter iorate , for about 300 to 900 years, becausetheory and exper iment went separa te ways , wi thincreasingly esoteric theoretical writings (for ex-ample, relat ing gold to the es sence of the soul) thatbecame total ly divorced from experiment,I t r emained for the A rabs , a ro und A.D. 100 0, inBaghdad and C6rdoba , to r euni te these two essen-t ials and to greatly advance the modern science ofalchemy. Books by the Spanish metal lurgist Ge be r(or Jabir) , a t th e b egin ning of the Renaissance,a round 1310 , se t for th foundat ions of the mo dernscience, for exam ple identifying me rcur y and sulfuras providing the mel t ing and burning ( rus t ing)pro per ties of metals, respectively. Ge ber 's treatisesestablished the experim ental ly based, hard-scienc eunderpinn ing of a lchem y for 300 years. 3 I t r e -maine d for An toine La nre nt L avoisier (France) , Jo -seph Priestley (England) and Karl Wilhelm Scheele(Sweden) , in the decade 1770-1780, to f inally com-ple te the exper iments tha t would replace a lchemyby a new and be t te r sc ience of chemis t ry .

The F i r s t Ha l f-Mi ll ion Years of Comb ust ionResearchFelix W einberg ' s ente r ta in ing descr ip tion 1 ofmankind's use o f f ire for 500,000 years, culm inatingin his ability to start fires, beginning about 30,000years ag o, demo nstrates clearly that experiment heldsway in the f ie ld of com bust ion for more than 99%of i ts h i story . The prehis tor ic Copp er , Bronze andIron Ages also we re eras of experiment, witho ut ex-ception. Exp er iment a lone does not a sc iencemake- -otherwise cookbooks should be t r ea ted asscientif ic documents. The classic Greek philosophy,

associated with Mesopotamian astrology and espe-cially Egyptian craftsmanship, resulted, in abou t A.D.100, in a true science, with both theoretical andexperimental com pon ents, the science of alchem y. 3

PhlogistonFo r near ly 100 years of the post-Renaissance pe-r iod ( including m ost of the e ighte enth centu ry) , be-tween the d ec l ine of a lchem y and the r ise of mo d-ern chem is t ry , the theory o f phlogis ton re igned,championed la rge ly by the German sc ient i s t GeorgErns t S tahl . Combust ion was even more a par t ofthis the ory than of alchem y. Fire (phlogiston) wasa fundamenta l subs tance , n ot jus t a combina t ion oftwo quali t ies, and during combustion a mater ial lostphlogiston. W he n i ts phlogiston was com pletelygone , the mate r ia l could no longer burn.This was a semi-quanti ta t ive theory in that therewas som e sor t o f a co ncep t of conservat ion of phlo-

giston, and i t enjoyed considerable success in ex-plaining experimental observations. H ow eve r, asmeasu rements of mass changes dur ing combu st ionwere ref ined, the theory encountered increasing


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T H E R O L E O F T H E O R Y I N C O M B U S T IO N S C I E N C E 3dif f icul ty . Some mate r ia ls (charcoa l) los t we ight andothers ( i ron) ga ined i t in combust ion. Al though thein i t i a l e xpe r im e n t s t ha t i so l a t e d e l e m e n t s suc h a shydr oge n a nd oxyge n ( " de ph log i s t i c a t e d a i r" ) we r ein t e r p r e t e d , f o r e xa m ple by He nr y C a ve nd i sh a ndby Jose ph P r i e s t l ey , i n t e r m s o f t he th e o r y o f ph lo -g i s ton , t h e qua n t i t a t i ve s c i e nc e o f c he m is t r y , w i thi t s pe r iod ic t a b l e o f e l e m e n t s ( fi na lly g ive n by M e n -d e l e y e v i n 1 8 6 9 ) e v e n t u a l l y e m e r g e d ( b e g in n i n ga r ound 1800) a s a m uc h m o r e use f u l a nd se l f -c on -s i s t e n t the o r y o f m a t t e r . A la s , w i th m or e tha n 100e le m e n t s , c om bus t ion i s no t e ve n one ( sm a l l c on -so la tion tha t P r om e th iu m , r a d ioa c t ive r a r e e a r th ,a tom ic num be r 61 , i s one ) . F e w i f a ny o the r p r o f e s -s ions can, have suf fe red a dec l in e and hum il ia t ionso g r e a t a s t ha t e xpe r i e n c e d by ou r s i n t he f i rs t pa r to f t he n in e t e e n th c e n tu r y !

Emergence of the Science of C o m b u s t i o nWe d id , howe ve r , be g in to bounc e ba c k in t hela t te r ha l f of tha t century . No longer a bui ld in g blockf o r t he o r i e s o f m a t t e r , c om bus t ion c ou ld now b es tudied jus t for i t se l f .I t i s r e l e va n t t o d i s t i ngu i sh be twe e n the sc i e nc ea nd th e t e c hno logy o f t he sub je c t . The m a r c h o ft e c hno logy has ne ve r he s i t a t e d . I t u se s s c i e nc ewh e ne ve r poss ib l e bu t o f t e n , e spe c ia l ly i n c om bus -t ion , forges ahead by t r ia l and e r ror , or for tu i tous ly

by a pp l i c a t ion o f s c ie n t i fi c m i sc onc e p t ion , b u t w i th -ou t s c i en t i fi c unde r s t a nd ing , a s i t d id du r ing thef i rs t ha l f- m i l li on ye a r s . S t e a m e ng ine s w e r e de ve l -ope d w h i l e ph log is ton wa s dom ina n t , a nd the in -dus t r i a l r e vo lu t ion , w i th i t s s t e a m boa t s a nd loc o -m ot ive s , su r ge d f o r wa r d du r ing the pe r iod o fd i sg r a c e o f c om bus t ion in s c i e nc e , be ne f i t i ng f romne w sc i e nc es suc h as c he m is t r y . M or e ove r , a t l e a s tby p r e se n t - da y s t a nda r ds , t he c o n t r ibu t ions o f c om -bus t ion sc i e nc e to t he in i t i a l de ve lopm e n t o f t heDie se l a nd Ot to e ng ine s , t ow a r ds the e nd o f t hen ine t e e n th c e n tu r y , r a nge d f r om p i t i fu l ly sm a l l t onone x i s t e n t . Espe c ia l ly w i th th e l a t t e r , p r e m ixe df la m e s c a m e i n to w i d e s p r e a d u s e - - W e i n b e r g 1 h a se m pha s i z e d tha t du r ing m os t o f h i s to r y on ly d i f f u -s ion fl a m e s we r e know n , a l though pa r t i a l l y p r e -mixed di f fus ion f lames , inc luding t rans ient hea tf e e dba c k f r om the c onde nse d pha se to t he c oo l i n -c om ing a i r , da t e ba c k a pp r e c i a b ly , t o t im e s whe nar t i sans ski l l fu l ly b lew a i r in to the i r charcoa l fur -na c e s to a c h ie ve supe r a d ia ba t i c t e m pe r a tu r e s f o rm e ta l lu r g i c a l p r oc e s s ing . M y top ic i s t he s c i e nc e ,n o t t he t e c hno logy ( howe ve r im por t a n t i t m a y be ) ,a nd so no m or e a bou t a pp l i c a t ions .W h a t a r e t h e e l e m e n t s o f t h e s c ie n c e o f c o m -b u s t io n ? H o w a n d w h e n w e r e t h e y f o u n d? D i v e r s ea nswe r s ha ve be e n g ive n to t he se que s t ions .M a ny ye a r s a go a s tude n t b r ough t t o m e a t ype -se t pa r a g r a ph p e r t a in ing to t h e de f in i ti on o f com -

bust ion sc ience, which im presse d me so grea t ly tha ta f t e r 30 ye a r s I s t i l l ha ve i t d i sp l a ye d on the wa l lof my of fice . I don ' t know who w rote i t , and I w ouldbe inde b te d to a nyone who c a n t e l l m e . A s l igh t lypa r a phr a se d v e r s ion o f t he pa s sa ge i s t he f ol lowing :" Al though c om bus t ion i s ba se d on sc i e n ti f ic p r in -c ip l e s , no one i s c e r t a in e xa c t ly wha t t h e y m a y be .To be c om e kn owle dge a b le a bou t t he b u r n ing o f f ue l,one m us t pe r sona l ly obse r ve e xpe r im e n t s t o r e a chreason able conclusions abou t th e hows, whats, w hysa nd whe ns o f c om bus t ion . B u t i n do ing so the r e i sa lwa ys the e x t r e m e h a z a r d o f m i s in t e r p r e t a tion ,wh ic h m a y l e a d the obse r ve r t o e r r o r s g r e a t e r t ha nthose a l r e a dy in t he l i t e r a tu r e . " A l though the r e c e r -t a in ly i s nonne g l ig ib l e t r u th in t h i s s t a t e m e n t , Isubsc r ibe to a l e s s pe s s im is t i c v i e w .The sc i e nc e o f c om b us t ion s t a nds on f ou r pe d -e s t a l s ( wh ic h do no t r e se m ble the f ou r e l e m e n t s o fa l c he m y) . The se f ou r suppor t ing s t r uc tu r e s a r ethermodynamics, chemical kinet ics , f luid mechanicsa n d t ranspor t processes , e a c h a m a tu r e s c i e nc e ini t se l f. I c l a im tha t " to be c om e kn owle dg e a b le a bou tthe bu r n ing o f f ue l " one m us t no t on ly obse r ve e x -pe r im e n t s bu t a l so f i r s t m a s t e r t he se f ou r s c i -e nc e s - so g r e a t a t a sk tha t i t i s r e m a r ka b le so m a nyof u s a r e he r e ! A l though th e m a in p r inc ip l e s o fthe r m odyna m ic s a nd f lu id m e c ha n ic s , ne e de d inc om bus t ion , ha d de ve lope d ove r m a ny c e n tu r i e s a ndwe r e a va i l a b l e e a r ly in t he n ine t e e n th c e n tu r y , t hesc i e nc e s o f c he m ic a l k ine t i c s a nd o f m o le c u la rt r a nspor t p r oc e s se s , w i th th e i r e s se n t i a l ba c kgr oundin the k ine t i c the o r y o f ga se s , d id no t be g in tof lou ri sh un t i l t he m idd le o f t ha t c e n tu r y . The r e f o r e ,the sc i e nc e o f c om b us t ion , a s we know i t , c a nno tha ve be e n b o r n un t i l t he l a s t ha l f o f t he n in e t e e n thc e n t u ry . T h e e x p e r i m e n t s o f S i r H u m p h r y D a v y(1815) , of Rober t Bunsen (1866) , of Claude LouisBer the lot and Paul M ar ie Eug ene Vie il le (1881, 1882)a nd o f E r ne s t M a l l a r d a nd He nr y Lou i s l e C ha te -l i e r ( 1881 , 1883), a nd the the o r i e s o f V la d im i r A le k -sa ndr ov ic h M ikhe l ' son ( 1890), o f Da v id Le onha r dCha pm an (1899) and of Emi le Jougu e t (1905) in fac tl e d to t he e m e r g e nc e o f t he s c i e nc e o f c om bus t iona r ound th e tu r n o f t he c e n tu r y . I n t r a c ing the r o l eo f t he o r y in t he m o de r n sc i e nc e o f c om bus t ion , t hes t a r t ing po in t shou ld be w i th the se f o r e be a r s . How-ever , be fore moving on to spec i f ics , I would l ike too f f e r som e ge ne r a l c om m e nt s a bou t t he na tu r e o fou r s c i e nc e i t s e lf a nd a bou t w ha t i s good the o r y .

C ha r a c t e r i s t i c s of the Science of C o m b u s t i o nAny de f in i ti on tha t a t t e m pt s t o c om p le t e ly de -sc r ibe c om bus t ion sc i e nc e shou ld e nc om pa ss i t s r e -

l a t i onsh ip to i t s f ou r suppor t ing sc i enc e s . An a spe c tof many combu st ion processes is the i r ab rupt changesin spa c e a nd /o r t im e o f va r ious p r ope r t i e s , no ta -b ly t e m pe r a tu r e , a nd th i s a spe c t ha s o f t e n be e n


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H O ' I q ' E L P L E N A R Y L E C T U R Esuggested as a sui table bas is for a def in i t ion . Fore xa m ple , F r a nk- Ka m e ne t sk i i , in h i s e xc e l l e n t e x -pos i t ion o f t he sub je c t , 4 p r e f e r r e d to i nc lude se l f-acceleration in the definit ion, call ing combustion thesc i e nce o f la r ge he a t r e l e a se a nd o f e i t he r c ha inb r a nc h ing o r l a r ge a c t iva tion e ne rgy . How e ve r , no ta l l c om bus t ion p r oc e s se s a r e a b r up t ; hom oge ne ousthe r m a l e xp los ions a r e no t a b r up t i n spa c e , s t a t i on -a r y l a m ina r f l a m e s no t i n t im e , a nd c om bus t ion inr oc ke t p lum e s , f o r e xa m p le , t yp ic a lly i s a b r up t ne i -the r i n spa c e no r i n t im e . F o r t he se r e a sons , 1 p r e -fe r to ca l l combust ion the sc ience of exothermicchemical reactions in flows with heat and masstransfer, or m or e b r i e f ly , exothermic chemistry inflows. So fa r as I can see , some form o f exoth er -mic i ty , somewhere , i s a lways essent ia l to the sub-j e c t , e xc e p t , pe r ha ps , f o r som e e ndo the r m icbranched-chain processes,4 and wi th "chemis t ry" th isc a n d r a w toge the r t he the r m odyna m ic s a nd c he m -ica l k ine t ics . The " f lows" a re des igned to br ing inthe f lu id m e c ha n ic s a nd t r a nspor t p r oc e s se s , a l -though th is c lea r ly i s imper fec t in tha t so l id-sol iddeflagrations involve f low only in a g eneraliz ed sense,a nd s t r ic t l y hom og e ne ous e xp los ions ne c e s s i ta t e a ne ve n g r e a t e r e x t e ns ion o f t he u sua l m e a n ing o f t hete rm. The di ff iculty , o f course , s tem s from t ry ingto express many aspec ts in few words . I t i s impor-t a n t t o f i nd a de f in i t i on tha t doe s no t e xc lude a nyof t he a t t e nde e s a t t he sym pos ia !I ha ve o f t e n he a r d c om bus t ion de sc r ibe d a s a" m a tu r e " s c i e nc e . I t s e e m s to m e tha t t h i s i s i n -a p p r o p r i a t e - - a n d n o t o n l y b ec a u s e o f t h e i m m a -tu r i t y o f i t s p r a c t i t i one r s . I t c e r t a in ly m us t be l e s smature than the fonr sc iences on which i t i s based.Jus t be c a use c om bus t ion i t s e l f ha s be e n a r ound f o rman kind 's ha l f -mi l l ion years does not m ean tha t i t ssc i e nc e ha s e x i s te d tha t l ong o r m us t ha ve b e e n de -p le t e d b y now. Th e r e a r e p l e n ty o f t h ings ye t t ob e d i s c o v e r e d i n o u r f i e l d - - w e a r e n o t a t t h e e n dof the u n t r a ve le d r oa d .A m o r e a p p r o p r i a t e d e s c r i p to r o f c o m b u s ti o nwou ld be a n a pp l i e d sc i e nc e , o r a de r ive d sc i e nc e .I ts founda t ion is der ived f rom other sc iences , andi t is a pp l i e d in t he se nse tha t t he f unda m e n ta le qua t ions tha t de sc r ibe c om bus t ion p r oc e s se s a r ebe l i e ve d to be known . The d i s t i nc tion he r e i s be -twe e n pu r e a nd a pp l i e d sc i e nc e s , s inc e in pu r e s c i -e nc e s , suc h a s pa r t i c l e phys i c s , t he unde r ly inge qua t ions a r e no t known . T he t a sk o f t he the o r e -t ic ian in a pure sc ience is to f ind the equa t ions ,wh i l e i n a n a pp l i e d sc i e nc e i t i s t o so lve the m .The l a s t de sc r ip t ion t e nds to m a ke ou r j ob a p -pe a r t oo m e c ha n ica l . A l l we ne e d to do i s to so lvee q u a ti o n s , b u t t h e p h y s i c i s t c a n i n v e nt t h e m - - o u rfield should be assigned to the dullards! W ell, thingsa r e ju s t no t a l l t ha t b l a c k a nd wh i t e . W e c e r t a in lyknow tha t t hose e qua t ions a r e n ' t pa r t i c u l a r ly e a syto solve . In fac t , wha t brought me in to th is f ie ldwa s a g r a dua te c om bus t ion c ou r se , g ive n by M a r t in

Summerfleld at Princeton in 1954, which two fr iendsa nd I t ook a s unde r g r a dua te s . He wr o te down theful l equa t ions descr ib ing chemica l ly reac t ing f lows,f rom the a r t ic le b y von K~trm~in and Penn er . ~ Ilooked at those equations and said to myself : "Surelythe r e i s e nough the r e to oc c upy m e f o r a l i f e t im e . "That obse r va t ion c e r t a in ly a ppe a r s t o ha ve be e nva l id .Our job i s no t j u s t t o so lve - - i t i s a l so to i n t e r -p r e t a nd to unde r s t a nd . I n do ing so we c a n be a sinve n t ive a s t he phys i c i s t . We c a n ide n t i f y ne w a p -p r ox im a t ions f o r so lv ing the S c h r od inge r e qua t ionto de sc r ibe c he m ic a l - k ine t i c p r oc e s se s i n c om bus -t ion. W e c a n c a l l on c onc e p t s o f c ha os to a dd r e s stu r bu le n t c om bus t ion p r oc e s se s . Our a pp l i e d sc i -e nc e ha s a s m uc h va r i e ty a s a ny pu r e s c i e nce . D u l l ,m e c ha n ic a l a c t ions , i n e i t he r t he o r y o r e x pe r im e n t ,a r e no t wha t we ne e d . The une xpe c te d e xpe r im e n-t a l obse r va t ion a nd the c onc e p tua l ly ne w the o r e t i -c al i de a a r e t he sou r c e s o f ou r a dva n c e m e n t .

C h a r a c t e r i s t i c s o f G o o d T h e o r yGive n the c om p le x i ty o f t he c onse r va t ion e qua -t ions o f c om bus t ion , i t c an b e t e m pt ing to c onc ludet h a t o n l y d e t a i l e d n u m e r i c a l t r e a tm e n t s c a n p r o -duc e goo d the o r y . I m a in t a in tha t t h i s i s a bso lu t e lyfa lse . Expanding a l i t t le on ideas tha t o thers have

e xpr e s se d be f o r e , I p r opose th r e e m a x im s f o r c om -bus t ion the o r y : Theory needn't be right to be good,theory ne edn 't be mathem atical to be right, andtheory needn't be incomprehensible to be mathe-matical. Ea c h o f t he se th r e e s t a t e m e n t s c l e a r ly r e -qu i r e s som e d i sc uss ion .Theory Needn't be Right to be Good:

The m a in po in t be h ind th i s s t a t e m e n t i s t ha t t hep r og r e s s i n t he o r y tha t im pr ove s hum a n unde r -s tandin g is achieved thro ugh s impl i f ica t ions tha t a renever exactly r ight. A classical example is the Burke-Schum ann theo ry of d i f fus ion f lames . 6 Oxygen asne ga t ive f ue l ? S ha de s o f ph log i s ton ! No t on ly doe sthe c he m ic a l k ine t i c s ne e d to be in f in it e ly f a st a ndproceed in jus t one s tep , but a lso the d i f fus ionc oe f fi c ie n t s o f fue l a nd oxyge n m us t b e e qua l . W ha tnonsenseT Yet , th is i s a remarkably useful way todescr ibe d i f fus ion f lames and grea t ly improves ourunde r s t a nd ing o f the m . R e su l t i ng a cc u r a c ie s in p r e -d i c t e d f l a m e sha pe s a nd f l a m e he igh t s t yp ic a l ly a r ebe t t e r t ha n 20%. I r e m e m be r puz z l ing t im e a nda ga in ove r t he que s t ion o f how th i s t he o r y c a n beso good. The var ious d i f fe rent ways to a r r ive a t thethe o r y , suc h a s po s tu l a t ing f ul l c he m ic a l e qu i l i b -r i u m o r s t ro n g l y te m p e r a t u r e - d e p e n d e n t r e ac t io nra tes , shed di f fe rent l ights on i t s meaning. Fur -the r m or e , s tud ie s o f t he w a ys in wh ic h th i s t he o r y


5/17

T H E R O L E O F T H E O RY I N C O M B U S T IO N S C I E N C E 5i s no t r i gh t ha ve g r e a t ly im pr ove d ou r unde r s t a nd -ing of diffusion f lames.M a ny e xc e l l e n t t he o r e t i c a l c on t r ibu t ions in c om -bus t ion a r e n o t e xa c t ly r igh t . C ons ide r , f o r e xa m -p le , a c t iva t ion - e ne r gy a sym pto t i c s , a s a p p l i e d byZe l ' dov ic h a nd F r a nk- K a m e n e t sk i i t o l a m ina r f la m epr opa ga t ion . 7 Th i s c a p tu r e s e s se n t i a l phys i c s , e ve nthough r e a l c om bus t ion c he m is t r y i s s e ldom e ve rone - s t e p , a nd i t m a ke s u s r e a l i z e tha t t he bu r n ingve loc i ty i t se l f i s an imp rec ise qua nt i ty , in tha t i t sva lue c a n be e xp r e s se d on ly a s a n a sym pto t i c e x -pa ns ion . Ano the r e xa m ple i s p r ov ide d by the r e -c e n t i n t e nse s tud ie s o f r e du c e d c h e m ic a l - k ine t i cm e c ha n i sm s . We c ou ld go on a nd on wi th c i t a t i onso f t he o r i e s t ha t a r e no t " r igh t , " b u t c lo se e noughto be ing r igh t , a nd suc c e s s f u l e nough in p r ov id ingne w v ie wpo in t s , t o be ve r y good the o r i e s . The be s tthe o r i e s a r e p r e t ty ne a r ly r igh t f o r s ome th ings , anda t a sk o f t he the o r e t i c i an i s t o po in t ou t w ha t t hoseth ings a r e . Howe ve r , e ve n the o r i e s t ha t a r e p r ove nqu i t e wr ong c a n be good in in sp i r ing r e se a r c h tha ti m p r o v e s u n d e r s t a n d i n g - - a n e x a m p l e is t h e e x c e ss -e n tha lpy the o r y , wh ic h he ld tha t t he to t a l e n tha lpyin a l a m ina r f l a m e m us t be pos i t i ve f o r t he f l a m eto p r opa ga te .The ge ne r a l i de a c a n e ve n be c a r r i e d to t he e x -t r e m e o f c l a im ing tha t t he o r i e s t ha t a r e t oo ne a r lyr igh t a r e no t ve r y good in tha t t he y do no t c on -t r ibu te t o im pr ove unde r s t a nd ing . " Tha t t he o r y i sno good . Why , i t ' s no t e ve n wr ong! " As a r e c e n te xa m ple , we ha ve be e n in t e r e s t e d in a x i sym m e t r i cf lows in la te ra l ly burn ing cyl ind r ica l so l id-p rope l-l a n t r oc ke t c ha m be r s , wonde r ing whe the r t he f l a t -tenin g o f the radia l prof i les of axia l ve loc i ty towa rdst h e d o w n s t r e a m e n d i s d u e t o t u r b u l e n c e o r c o m -pr e s s ib i l i t y . M or e tha n o ne g r oup o f i nve s t iga to r sha s suc c e s s fu l ly de ve lope d a nd r un Na v ie r - S toke sc ode s f o r t h is p r ob le m , w i th bo th k -E m ode l ing a ndc om p r e s s ib i l i ty i nc lude d ; p r o ud ly e xh ib i t i ng thef l a t t e n ing bu t , p r o f e s s ing the i r be l i e f in k -E tu r bu -lence and in i t s impor tance to the phenomenon, notbe ing w i l li ng to t a ke ou t on e o r t he o the r o f t heef fec ts to show un que st iona bly what causes i t . The-ory th a t i s exac t ly r ight n ot only wi l l conta in toom u c h t o c o m p r e h e n d b u t a l so u l t im a t e l y m a y b e -c o m e t a u t o l o g i c a l - - d e v o i d o f c o n t e n t - - t e s t i n g o n l ythe prac t i t ioner ' s abi l i ty to run the compu ter . I f youwant to make progress in theory , you 've got to makea t leas t a l i t t le e r ror (pre fe rab ly knowingly).Theory Needn ' t be Mathemat ical to be Right :

This and the fo l lowing s ta tement re fe r to math-e m a t i c s , a wor d s tud ious ly a vo ide d u n t i l now. Thed i sc uss ion o f m a the m a t i c a l t he o r y w i l l be r e s t r i c t e dhe r e to i t s r o l e i n c om bus t ion , a nd " r igh t " nowm e a ns p r e t ty ne a r ly r igh t , bu t a l i t t l e wr ong , so a sno t t o be no good . De sp i t e r e c e n t a ppa r e n t de -c l ine s i n m a the m a t i c a l a b i l i t i e s o f unde r g r a dua te s ,

t he young pe op le in ou r f i e ld , by a nd l a r ge , ha vebe t t e r m a the m a t i c a l ba c kgr ounds tha n the o lde rge ne r a t ion . M a ny f o r e f a the r s f e a r be ing d r ive n tode spa i r by the i r i na de qua c y in a da p t ive g r idd ing ,a sym pto t i c s , a nd so on , up the m a the m a t i c a l a l -pha be t . The poor o ld c om bus t ion e xpe r im e n te r , i nhis mo dest labora tory , may b egin to feel out o f touchwi th m ode r n r e a l i t y . To se e why , i t i s ne c e s sa r yon ly to s ca n the vo lum e s o f t he c om bus t ion sym -pos i a f o r t he i r m a the m a t i c a l c on te n t .Of f ou r t e e n pa pe r s i n t he f i r s t sym pos ium , on lyt h a t o f B u r k e a n d S c h u m a n n w a s m a t h e m a t i c a l - - i tinvolved par t ia l d i f fe rent ia l equa t ions , fa i r ly ad-va nc e d m a the m a t i c s f o r t he f i e ld in 1928 ( pe r ha psthe t r oub le ha s be e n wi th u s l onge r t ha n we r e -a l ize ) . Back then combust ion sc ient is ts genera l lykne w a r i thm e t i c , t r i gonom e t r y , a lge b r a a nd som ea spe c t s o f c a l c u lus a nd o f o r d ina r y d i f f er e n t i al e qua -t ions . M os t o f t he e a r l i e r a dva nc e s in c om bus t ionthe o r y , suc h a s t he d i s t i nc t ion be twe e n de f l a g r a -t i ons a nd de tona t ions ide n t i f i e d by C ha p m a n a ndJougue t , ha d m a in ly r e qu i r e d a t ho r ough knowl -e dge o f non l ine a r a lge b r a i c e qua t ions . I n t he se c -ond sym pos ium ( 1937) , f i ve ou t o f twe n ty - f ive pa -pe r s i nvo lve d o r d ina r y o r pa r t i a l d i f f e r e n t i a lequa t ions , a whopping 20%, up f rom less than 10%in the f i r s t . R e pr e se n ta t ive i s t he s e c ond- sym po-s i u m p a p e r b y B e r n a r d L e w i s a n d G u n t h e r v o nElbe , w h ic h a dd r e s se d de f l a g r a t ion the o r y , e m ploy -ing par t ia l d i f fe rent ia l equa t ions .

Par t ia l d i f fe rent ia l equa t ions ce r ta in ly a re centra lto combust ion sc ience ; a f te r a l l , the conserva t ionequations, s as f inally dev elop ed fully, largely throug hthe e f fo r ts o f Joe H i r sc h f e lde r a n d h i s c o l l a bo r a -tors , 9 an d of Th eodo re von K~irm~in and f r iends , 1~a r ound the t im e o f t he th i r d sym pos ium ( 1949), a r ee xpr e s se d m os t c onve n ie n t ly in pa r t i a l d i f f e r e n t i a lform. Asp ec ts of ea r ly advanc es in the f ie ld in-vo lve d o the r b r a nc he s o f m a the m a t i c s , suc h a s o r -d ina r y d i f f e r e n ti a l e qua t ions f o r C y r i l Nor m a n Hin -she lwood ' s a nd Nic o la i N ic o la ye v ic h S e m e nov ' sse m ina l de ve lopm e n t s i n c he m ic a l k ine t i c s 11'12 an din t e g r a l e qua t ions f o r Hoy t Ho t t e l ' s e x t e ns ive c on -t r ibut io ns to radia t ive t ran sfe r , 13 bu t par t ia l d i f fe r -ent ia l equa t ions sure ly can be sa id to form the core .T h e y h a v e b e c o m e a m u s t i n t h e r e p e r t o ir e o f t h ec om bus t ion sc i e n t i s t .Nowa da ys , m or e tha n o ne - th i r d o f t he sym po-s ium pa p e r s m a ke e s se n t i a l u se o f m a the m a t i c s a tt he l e ve l o f d i f f e re n t i a l e qua t ions o r be yon d . A ndwha t a va r i e ty o f m a the m a t i c a l top ic s we f ind ! M od-e l ing doe s no t a ppe a r i n t he t e n - vo lum e inde x o fthe t e n th sym pos ium bu t ha s num e r ous e n t r i e s i nthe t e n - vo lum e inde x o f t he twe n t i e th . Asym pto t i ca na lysi s ha s be gun to c r e e p in , a nd the twe n ty - se c -ond even has an ent i re sec t ion on f rac ta ls . I s ourm a the m a t i c s ge t ti ng ou t o f c on t ro l ? Ar e we he a de dtowa r ds the d ivo r c e o f t he o r y a nd e xpe r im e n t e x -p e r i e n c e d b y a l c h e m y ?


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H O T T E L P L E N A R Y L E C T U R ECer ta in ly not . As fa r back as the Rena issance wele a r ne d tha t t he o r y ha d to ke e p in t ouc h wi th e x -pe r im e n t t o m a in ta in he a l thy sc i e nc e , a nd we a r eno t a bou t t o f o r ge t . I nc r e a se in m a the m a t i c a l c on -t e n t w i th inc r e a s ing t im e i s a c ha r a c t e r is t i c o f e ve r y

sc i enc e . M o de m c he m is t r y wa s bo r n f r om the d e -s i r e t o ha ve som e th ing m or e qua n t i t a t i ve tha n thesc i e nc e s tha t p r e c e de d i t . The m a r c h f r om a lge b r a ,throug h di f fe rent ia l equa t io ns , to asymp tot ics , s to-chas tics a nd f rac ta ls i s a na tura l p rogress io n of ahe a l thy sc i e nc e . B u t t h i s doe sn ' t m e a n tha t we a l ln e e d t o b e c o m e m a t h e m a t i c i a n s . W e d o n e e d t or e a d to f ind ou t som e th ing a bou t t he se c onc e p t s ,b u t i f w e e n c o u n t e r a b o o k b a s e d o n t h e " t h e o r e m -pr oof " f o r m a t we c a n pu t i t down a nd f ind a no the rtha t spe a ks a m or e hum a n l a ngua ge . The e s se n t i a lth ing in t he the o r y i s t he c onc e p t , wh ic h o f t e n c a nbe e xh ib i t e d m os t c l e a r ly whe n s t r ippe d o f i t se qua t ions . E xc e ss ive ly m a the m a t i c a l t he o r i e s t e n dno t t o be ve r y r e ve a l ing .M or e ove r , i t shou ld no t be c onc lude d tha t i n t hef i r s t sym pos ium on ly the B ur ke - S c hum a nn pa pe rinvo lve d the o r y . M os t o f t he o the r pa pe r s a l so in -vo lve d good the o r y , bu t no t m a the m a t i c a l t he o r y .I t i s t rue tha t , a s t ime goes on, mathemat ica l as-pe c t s o f t he the o r i e s t ha t a r e r i gh t i nc r e a se . How -e v e r, t h o s e a sp e c ts n e e d n ' t d o m i n a t e - - t h e y m e r e l yr e f le c t the ge ne r a l p r og r e s s ion o f hum a n th ink ing .Adva nc e d m a the m a t i c s i s no t e s se n t i a l i n c o r r e c tthe o r y .Theory Needn't be Incomprehensible to beMathematical:

This s t a t e m e n t a dd r e s se s t he f o r m ida b i l i t y o f t oomany m athemat ica l ly couched pap ers . There can bea t e nde nc y to be l i e ve tha t m a the m a t i c a l c on te n t i sd i r e c t ly p r opor t iona l t o t he n um be r o f sym bo l s o rthe nu m b e r o f e qua t ions . On the c on t r a r y , a n in -ve r se p r op or t iona l i t y u sua l ly i s m or e ne a r ly c o r r e c t .M a the m a t i c s i s a l a ngua ge tha t shou ld be wr i t t e nso tha t i t c a n be r e a d , w i th the sa m e type o f ph r a s -ing a nd punc tua t ion a s a ny o the r l a ngua ge , a ndobf usc a t ing sym bo l s i n e xc e s s ive num be r de t r a c tf r om tha t o b je c t ive . I t i s unne c e ssa r y to wr i t e downe ve r y l i t t l e a lge b r a i c s t e p . Le t ' s on ly in t r oduc esym bo l s tha t we ne e d a nd use s ign i f ic a n t ly , a nd de -f ine the m c l e a r ly . Th i s u sua l ly r e qu i r e s r e r e a d inga nd r e wr i t i ng e a c h o f ou r pa p e r s a fe w t im e s . U n-l e s s a m a the m a t i c a l pa pe r i s c om pr e he ns ib l e , i tp r oba b ly doe s no t c on ta in a good m a the m a t i c a l t he -o r y . One e xa m ple o f a we l l - wr i t t e n m a the m a t i c a lp a p e r i s G r a h a m D i x o n - L e w i s' s i n v i te d p a p e r o ns t r uc tu r e s o f l a m ina r f l am e s in t he twe n ty - th i r dsym pos ium ; 14 m a ny o the r e xa m ple s c ou ld be c i t e df r om sym pos ium vo lum e s .

Tripartite ScienceA r e vo lu t ion in t he sc i e nc e o f c om bu s t ion be ga na bou t twe n ty ye a r s a go . The l a s t t he o r y - o r i e n te d

p le na r y l e c tu r e , t he e xc e l l e n t r e v i e w g ive n bH o w a r d E m m o n s a t t h e t h i r t e e n th s y m p o s i u m , L5m a r k e d t h e e n d o f t h e e r a i n w h i ch c o m p u t a t i o n - -the num e r i c a l so lu t ion o f d i f f e r e n t i a l e qua t ions - -c ou ld on ly be c o ns ide r e d to b e a t oo l o f t he o r y . Heshowe d a c a l c u la t ion by G . A . B a l l o f t he sha p e o fa th in p r e m ixe d f l a m e t r a ve l ing a long a g r a v i ty - f re ec ha nne l , ob ta ine d by a r e l a xa t ion m e tho d a nd r e p -r e se n t ing a s ign i fi c a n t a c c om p l i shm e n t o f num e r i c a lcomputa t ion a t the t ime tha t i t was per formed. Whyi s t he r e su l t i ng c onve x sha pe wha t wa s c a l c u la t e d?Tha t i s a que s t ion f o r t he o r y to a dd r e s s . We knoww h a t e q u a t i o n s p r e d i c t e d t h a t s h a p e , b u t w e d o n ' tha ve a good unde r s t a nd ing o f why tha t i s wha t t he ypr e d ic t e d . S udde n ly , t he o r y ne e ds no t on ly to t r yto e xp la in e xpe r im e n ta l obse r va t ions - - i t a l so ne e dsto e xp la in c om pu ta t iona l r e su l t s . Wi th the f ou r -t e e n t h s y m p o s i u m w e b e g i n t o s e e m o r e a n d m o r enum e r i c a l so lu t ions o f pa r t i a l d i f f er e n t ia l e qua t ionswi th h igh a c c u r a c y , a nd the se be c om e dom ina n tm e thod s f o r m o de l ing o f c om bu s t ion p r oc e s ses . Ex -p los ive c om pu te r a dva nc e s ha ve now m a de the m e -c ha n ic s o f t he B a l l c om pu ta t ion ne a r ly t r i v i a l a ndh a v e e n a b l e d u s t o i n c l u d e m a n y m o r e p h e n o m e n ain num e r i c a l i n t e g r a t ions ( a l though r e se a r c h on theB a l l p r ob le m c on t inue s , m a in ly c onc e r n ing a pp r o -p r i a t e bounda r y c ond i t i ons a t t he wa l l ) .Toda y i t c a n be sa id , w i thou t c on t r ove r sy , t ha tthe r e a r e t h r e e s ign i fi c a n t pa r t s t o c om b us t ion sc i -e n c e - e x p e r i m e n t , c o m p u ta ti o n a n d th e o ry . M u -tua l c om pa r i sons o f a l l t h r e e he lp to a dva nc e thesc i e nc e . S o m a ny phe nom e na a r e i nc lude d in t yp -i c a l c om pu ta t iona l r e su l t s t ha t a g r e e m e n t be twe e ne xpe r im e n t a nd c om pu ta t ion no longe r e xp la ins e x -pe r im e n ta l re su l t s . The or y i s ne e d e d to p r ov id e un -de r s t a nd ing o f the r e su l t s . N owa da ys , adva nc e s inthe o r y a r e a c h ie ve d a s m uc h by c om pa r i son wi thnum e r i c a l r e su l t s o f c om p u ta t iona l p r o j e c ts a s byc om pa r i son wi th e xpe r im e n ta l r e su l t s ob ta ine d inthe l a bo r a to r y , C om pute r l a bo r a to r i e s p r ov ide thene w d im e ns ion , The th r e e b r a nc he s o f t he s c i e nc ea r e now c om pa r a b ly im por t a n t a nd a r e de s t ine d tor e m a in so .

S p e c i f ic C o n t r i b u t i o n s o f T h e o r yUp u nt i l now, I have no t rea l ly s ta r ted to do whatwa s e xpe c te d o f m e , a nd a t t h i s po in t t he r e i sn ' ts p a ce t o d o i t - - y o u d e s e r v e a c o m b u s t io n t h e o r i s t' sa po logy . I wa n te d to t a ke a l ong v i e w a nd to p l a c esom e p r e jud ic e s be f o r e you . H a v ing done so , I t u r nto the spe c if i cs t ha t you ha d a n t i c ipa t ed . M y S a -t a n i c v e r s e i s h e r e b y c o n c l u d e d - - o n t o m o r e t e c h -nica l topics .Eve n twe n ty ye a r s a go , Em m on s 15 wa s a b le on lyto p r e se n t a s e l e c t ion o f t he m a ny c on t r ibu t ions o ftheory to comb ust ion sc ience . Although I have easedm y t a sk by d iv id ing the f i e ld in to th r e e pa r t s , so


7/17

T H E R O L E O F T t l E O R Y I N C O M B U S T I O N S C I E N C Ethat I can exclude modeling and simulation basedon computation, I am s t i l l faced with too ma ny con-t r ibut ions to theory to review in one paper ( evenhad I star ted from the beginning with specif ics) . Ishall therefore c onte nt my self with a few examples,asking forgiveness in advance from the many whomI over look- -would tha t I could c i te you a l l . At theend of each topic, I shall t ry to indicate curren tlyoutstanding research questions or future direct ions.

Such simple algebraic derivations are quick al-tho ug h subject to pitfalls identifiable from asymp-totic m ethod s o f greate r r igor.Fur m ethane f lames , com parable accuracy de-mand s a four-step approximation, which throu gh asimilar derivation can be written withCH4 ~- 2H + H2 0 ~ C O + 4H2, (5)

CO + H 20 ~- COz + t tz, (6)Reduced Che mi s t r y

At least for problems in no more than one spacedimension, com puters now enable ful l chem istry tobe handled rout ine ly in combust ion computa t ions .How ever , s impl if ied chem is t ry is needed for h igherdimensions or for turbulence, as well as for im-proved und ers tanding in one dimension. Wh i lesensitivity analysis and relate d tech niq ues 16'17 en -able m any steps of lesser imp ortance to be identi-f ied and discarded, mo re drast ic reductions areneeded in advancing theory. These reductions comesystematically from steady-state and partial-equilib-rium approximations, or from generalizations thereof.Although such approximations have been known forma ny years, ~ advances in asym ptotic analysis n owplace us in a much better posi t ion to identify theseapproximations and test the ir accuracy.IS ' l~ Im po r-tant r educed mechanisms have recent ly been de-. 20 21 22ve loped for many com bustm n problems. ' 'For hydrogen f lames, systematic simplif icat ionsthat result in the one-step approximation 2Ha + 022H 20 are of little use, except for correlating m ainradical concentrations, H, O H and O (not HOz) inthe hottest part of diffusion flames at low strain, 23'24but the two-step approximation

3H2 + O2 ~ 2H 20 + 2H (1)2 H + M ~ H 2 + M , (2)

with rates determined, respectively, mainly by therates of the e lementa ry s teps

add ed to (1) and (2), the rates o f (5) and (6) bein gdetermined, respectively, mainly by the rates of thee lementa ry s teps( ' t t4 + H ~ CI t3 + He , ( 7 )C O + O H ~ C O 2 + H. ( 8 )

Th e fuel-consumption (5) and oxyg en-consump tion(1) steps are different here, with the former re-moving radicals and the lat ter producing them. Ex-tensions to higher hydrocarbons and to alcohols arein progress; som e results for these mo re compli-cated problem s already are available , 21'25 and mo redefini tive resolutions ma y be exp ected in the nearfnture ( for example, this w eek) .W e may also anticipate extensive d evelo pm ent ofreduced chemistry for pollutant production (NO , CO,etc .) in combustion. Re duc ed mechanisms for pro-pellant combustion, especially solid propellants, alsoare progressing. Fo r example, for the propellant-related f lames of m ethan e with ni trogen dioxide, agoo d four-step mechanism com pos ed of (5) , (6),NO2 + H2 ~ NO + H20 , (9)

2NO----> N2 + 02 , (10)has rec ently been identif ied, 26 the rates of the lasttwo bein g controlled, respectively, mainly by thera tes of

NO2 + H- -~ NO + OH , (11)H + 0 2 ~ O H + O , (3) N O + H ~ N + O H . (12)

H + P c + M ~ H ( ) 2 + M , (4)has been found to be remarkably good for predic t -ing diffusion-flame extinction and profiles of tem -pera ture , of concent ra tions o f major species and o fconce ntrat ions of al l interm ediates except HO z andH202. z~ Th e two-step des criptio n involves steady-state approximations for O, OII , t lO2 and H20~ andarrives at (1) by adding to (3) rapid shuffle steps tocancel OIl and O algebraicaily;~" similarly, additionof rapid HO2 and shuffle steps to (4) gives (2) whoserate is that of the slowest step (4) in the sequen ce.

The rebi r th of r educed chemis t ry , which occur redabout f ive years ago, wil l be fundamental to ad-vances in theory for many years to come, supplant-ing the use of mo del chem is t ry not t i ed to spec if icreactants.

P r e mi xe d La mi na r F l a me sRestr ict ing at tention to steady, planar f lames, wenote truly remarkable advances in theory since thereview o f Emmons.15 T he signif icance o f the ear ly


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H O T T E L P L EN A I4 .Y L E C T U R ET J

SLOPESMATCH /Y F u ~ y F ~ . ~ t ' ~ Tb

i DIFFUSIVE ~ ~ ~ t " ~ - - REACTIVE-i L _ . Z O NT ,i-"~-~'~_~/(o..v.,cn ~ ' I ~)./(puVuCp,B)

0 0" x

I ; . ( ) , I ( , ~ V u C p ) - = I1 II I H 2 A N D C O O X I D A T I O NZ O N E T ,X/(~uC "1 I ~ i

i P R z E H E A Tl ~ . T " ' ~ : ~ - - - - - - - - F U E L - C ~ S U M P T I O N Z O N E

d ,

FIC. 2 . I l lus t ra t ion of zone s t ruc ture and match -ing for premixed laminar f lames according to AEA(Y~. = fue l mass f rac t ion , A = therm al co ndu c t iv i ty ,p , = unbu r n t ga s de ns i ty , v= = bu r n ing ve loc i ty ,= Ze r dov ic h num be r de f ine d " af te r e qua t ion 13 ).

wor k o f Ze l ' dov ic h a nd F r a nk- Ka m e ne t sk i i f o r de -sc r ip t ions ba se d on the one - s t e p , Ar r he n ius a p -p r ox im a t ion wa s no t f u l ly a pp r e c i a t e d the n , a l -t hough the ge ne r a l c ha r a c t e r o f t he p r ob le m a nd o fthe b urning-ve loc i ty behavior was known. I t was notr e a l i ze d tha t t h e i r 1938 r e su l t s7 r e p r e se n t t he l e a d -ing t e r m in a n a sym pto t i c e xpa ns ion f o r t he bu r n -ing velocity. Asymptotic analysis, especially matcheda sym pto t i c expa ns ions, now a m a ins t a y o f c om bu s -t ion the o r y a nd the p r inc ipa l m a the m a t i c s c ou r setha t g r a dua te s tude n t s i n t he f i e ld shou ld be e n -couraged to take a f te r par t ia l d i f fe rent ia l equa t ions ,wa s v i r tua l ly unknown to ou r f i e ld a t t ha t t im e . Theformal expansion of Bush and Fen de l l , 27 as fu r therexten ded and e luc ida te d in textboo ks , zs 'zg '3~ dem -ons t r a t e d the a sym p to t i c c ha r a c t e r o f t he r e su l t a ndled to ac t iva t ion -energy asymptot ics (AEA) . S incethe va lue o f t he sm a l l pa r a m e te r , /3 - ] o f F ig . 2 ,t yp ic a l ly i s on the o r d e r o f 0 . I , two- t e r m ( r a the rtha n l e a d ing - t er m ) e xpa ns ions a r e de s i r a b le f o r p r o -duc ing a c c u r a c y o f t he o r e t i c a l p r e d ic t ions c om pa -r a b le w i th e xpe r im e n ta l a c cu r a c y. 3~ The f unda -m e n ta l AEA too l o f t he c om bus t ion the o r e t i c i a nr e ve a le d the l a m ina r - f la m e s t r uc tu r e i l h l s t r a t e d inFig. 2.I t i s espec ia l ly in te res t in g to read I s f rom Em -m on ' s p l e na r y l e c tu r e o f 1970 tha t " f or e s se n t i a l lya l l in te res t in g fue ls , the de ta i l s of chem ica l k ine t icsa re so complex, i f know n a t a l l , tha t f lame spe edsa r e m uc h e a s i e r t o m e a su r e tha n to c a lc u la t e . " Th i ss t a t e m e n t e m bo d ie s i n a nu t she l l how a m a z ing ly f arwe ha ve c om e in twe n ty ye a r s . De ta i l s o f c he m ic a lk ine t i c s a nd e l e m e n ta r y r a t e s a r e known , f o r e x -a m ple f o r m e tha no l , we l l e nough to e na b le p l a na r ,a d i a ba t ic fl a m e spe e ds to b e c a l c u la t e d m or e a c -c u r a t e ly tha n the y c a n be m e a su r e d , a nd m or e ove r ,w i th toda y ' s c om p u te r s a nd p r og r a m s , t he se c a l -c u la t io n s ar e r o u t i n e - - t h e o p p o s i te o f th e q u o t e ds t a t e m e n t ha s c om e t r ue . A l though the o r y wa s in -vo lve d in de ve lop ing the num e r i c a l m e thods , p r e s -e n t - da y the o r y l a r ge ly i s d i r e c t e d towa r ds unde r -

F ie, . 3 . I l lus t ra t io n of the s t ruc ture o f the lamina rpremixed meth ane-a i r f lame according to RRA (samenota t ion as F ig . 2) .

s t a nd ing the c om pu ta t iona l ( a nd e xpe r im e n ta l )r e su l ts . A sym pto t i c m e tho ds , a p p l i e d to r e a l c he m -i s t ry , ha ve c l a r i f i e d the s t r uc tu r e o f t he oz one de -com posi t ion f lame. 32 'aa '34 Un ders ta ndin g of thes t r uc tu r e o f t he m e tha ne f l am e ha s be e n o b ta ine dby such meth ods , as 'a6 as i l lus t ra ted in F ig . 3 . Inth i s m or e r e c e n t wor k , r a t e - r a t io a sym pto t i cs (R R A)replaces AEA, th e p rincipal small param eters, (5 and~, repre sen ting the ratio of the rate constant for step(3) to that for step (7) a n d ( T / , - T ~ - T , ) ,r e spe c t ive ly , whe r e Th a nd Tu a r e t he bu r n t a ndunbu r n t ga s t e m pe r a tu r e s , a nd T O i s t he t e m pe r a -tu r e o f t he f ue l - c onsum pt ion z one , t he t e m p e r a tu r ea t wh ic h th e p r odu c t o f t he r a t e s o f s t e ps (4 ) a nd(7) is e f fec t ive ly equ a l to the square o f the ra te ofs t e p ( 3 ) . I n R R A the r a t io o f t he r a t e s o f two im -por tant reac t ions in a c r i t ica l zone , or the ra t io ofthe f low ra te to the ra te o f an imp or tant reac t ion ,i s t r e a t e d a s t he sm a l l pa r a m e te r i n de ve lop ing theso lu t ion by m a tc he d a sym pto t i c expansions . W e a r ein the m ids t o f a n e xp los ion o f t he o r e t i c a l e xp la -na t ions o f fl a m e s t r uc tu r e s a n d b u r n ing ve loc i t i es ,on the ba s i s o f unde r ly ing e l e m e n ta r y c he m ic a l k i -ne t i c s, b y c om bina t ions o f AEA a nd P tB A, f o r m a nyinte res t ing fuels. F ig ure 4 i l lus t ra tes buruin g-ve loc -9 . 37l t y a g r e e m e n t s ob ta ine d f o r m e tha ne f l a m e s .

Lam inar Diffusion Flam es

The same a r ray of asymptot ic tools tha t have b eenappl ied to premixed f lames has been brought to bearon lamina r d i f fus ion flames 9 A E A h a s d o n e w o n d e r sf or ou r unde r s t a n d ing o f p l ana r , l a m ina r d i f f u sion -f lame stru cture and extinction. 39 Th e S-shape d curveof the peak te mp era ture as a fimction of a D am kfhl e rnum be r tha t i s i nve r se ly p r opor t iona l t o t he e x te r -nally imposed strain rate was already available morethan twenty years ago, 15 but the know ledge tha t fourdi f fe rent regimes (d i f fus ion- f lame, premixed- f lame,pa r t i a l - bu r n ing a nd ign i t i on r e g im e s ) oc c u r a long i t sb r a nc he s wa s no t. On e o r a no the r o f t he se f ou r r e -g im e s o f AE A r e a c t ion - z one s t ruc tu r e s ha s be e n e n -


9/17

T H E R O L E O F T H E O R Y I N C O M B U S T I O N S C I E N C E 95040

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EQUIVALENCE RATIOF IG . 4 . T h e bu r n i ng ve l oc it y o f me t ha ne - a i r fl a me s

a t 1 a tm . a nd a n i n i t ia l t e m pe r a t u r e o f 300 K , a c -c o r d i n g t o a s y m p t o t ic a n a l y s is w i t h r e d u c e d c h e m -i s tr y , 37 num e r i c a l i n t e g r a t i on w i t h f u l l c he m i s t r y , 7a nd e xpe r i m e n t . 38

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F IG . 6 . D e p e n d e n c e o f t h e m a x i m u m t e m p e r a -t u r e on t he e f f e c t i ve i nve r s e s t r a i n r a t e f o r me t h -a n e - a i r d i f f u s i o n f l a m e s , a c c o r d i n g t o a s y m p t o t i ca na l y s is w i t h a t h r e e - s t e p me c ha n i s m ( c u rve ) , nu -me r i c a l i n t e g r a t i on w i t h a f ou r - s t e p me c ha n i s m ( t r i-angles ), and exp er im ent ( squares ), f rom Seshadr i andPete rs . 4~

c o u n t e r e d t i m e a n d a g a i n i n w i d e l y d i v e r s e p r o b -l e ms i n c om bus t i on t he o r y . B a s ic a ll y , g i ve n a f ue la nd ox i d i z e r , t he r e a r e f ou r d i f f e r e n t w a ys i n w h i c ht h e y c a n b u r n , t h r e e t h i n - s h e e t a n d o n e n o t : T h e yc a n b o t h b e u s e d u p i n a t h i n r e a c t i o n z o n e , o n ec a n b e u s e d u p t h e r e b u t n o t t h e o t h e r , n e i t h e rm a y b e u s e d u p , o r t h e r e a c ti o n m a y b e d i s t r i b u t e dove r a t h ic k r e g i on . I n m a ny s i t ua t i ons , t he r e s u l t -i n g a s y m p t o t i c d e s c r i p t i o n s h a v e d o n e m o r e t h a ni m p r o v e o u r u n d e r s ta n d i n g o f th e p h e n o m e n a - - t h e yha ve a l s o g i ve n r e a s ona b l y a c c u r a t e f o r mu l a s f o r i g -n i t i o n a n d e x t in c t i o n c o n d i t io n s .4~M e t hods o f R R A ha ve b e e n s uc c e s sf u l f o r d if f u -s i on f la me s a s w e ll . T he me t ha n e - a i r d i f f u s ion - f l a mes t r uc t u r e a c c o r d i ng t o R R A , i n t he c oo r d i na t e o fmi x t u r e f r a c ti on Z , i s i l l u s t r a t e d i n F i g . 5 , w h i c hs h o w s t h e s e p a r a te f u e l - c o n s u m p t i o n a n d o x y g e n -

~OXYGEN-CONSUMPTION ZONE, I" _ , THICKNESS(()~FUEL-CONSUMPTIONZONE iT T cr THICKNESS(8 )

0 Z Z

T o10

F IG . 5. I l l u s t r a t i on o f t he s t r uc t u r e o f t he me t h -a ne - a i r d i f f u s i on f l a me a c c o r d i ng t o R R A .

c o n s u m p t i o n z o n es . T h e e x t i n c t io n p o r t io n o f t h e S -c u r ve i s s how n i n F i g . 6 . R e s e a r c h i s i n p r og r e s st o o b t a i n t h e i g n i t i o n p o r t i o n b y t h e s e t e c h n i q u e s( w h i c h i s o f i n t e r e s t f o r a pp l i c a t i on t o d i e s e l c om -b u s t i o n , f o r e x a m p l e ) a n d t o e x t e n d t h e s e m e t h o d st o o t he r f ue l s. W e m a y a n t i c i pa t e a dva nc e s i n de -s c r i p t ions o f s t ruc t u r e s o f ma ny r e a l d i f f u s ion f la me sb y a s y m p t o t i c m e t h o d s .

A u t o i g n i t i o n , D e t o n a t i o n a n d E x p l o s i o nA s y m p t o t i c m e t h o d s a r e a l so b e c o m i n g c o rr e -s p o n d i n g l y u s e f u l f o r a n a l y s i s o f t i m e - d e p e n d e n ta u t o i gn i t i on w i t h r e a l c he m i c a l k ine t i c s , no t on l yf o r h y d r o g e n 42 bu t a l s o f o r hyd r oc a r b ons ;4z they

c l e a r ly de f i ne d i f f e r e n t s t a ge s i n t he i gn i t i on p r o -c e s s. A l t houg h e m pi r i c i s m s t i l l i s i nvo l ve d f o r hy -d r o c a r b o n s, f u r t h e r a d v a n c e m e n t to w a r d s f u n d a -m e n t a l d e s cr i p ti o n s m a y b e e x p e c t e d i n th e f u t u re .P r o g r e ss t o w a r d s u n d e r s t a n d i n g t h e Z N D( Z e r d o v i c h , y o n N e u m a n n , D 6 r i n g ) d e t o n a t i o ns t ruc ture 44 '45 '46 an d i t s im pl ica t io ns for b oth t rans -v e r s e a n d l o n g i t u d i n a l e x o t h e r m i c g a s - d y n a m i c i n -s t a b i l i t i e s i n de t ona t i ons w a s i n f u l l s w i ng t w e n t yy e a r s ag o. C o n s e q u e n t l y w e c u r r e n t l y h a v e a r e a -s o n a b l e k n o w l e d g e o f t h e o m n i p r e s e n c e o f c e l lu l a rd e t o n a t i o n s Y H o w e v e r , n e w e r m a t h e m a ti c a l m e t h -ods , ba s e d on A E A , b i f u r c a t i on a na ly s i s a nd m u l -t i p l e - s c a l e e xpa ns i ons u s e f u l f o r i mpr ove d nume r i -c a l i n t e g r a t i ons , a r e l e a d i ng t o f u r t he ri m p r o v e m e n t s . 4 s'4 9 W e n e e d t o a p p l y s u c h m e t h -o d s t o a d d r e s s t h e i m p o r t a n t q u e s t i o n s o f d e t o n a -t i o n t r a n s m i s s io n f r om o n e r e g i o n t o a n o t h e r a n d o fd e t o n a t i o n f a i l u r e i n n a r r o w t u b e s o r i n e o n f i g u r a -


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10 H O T T E L P L E N A R Y L E C T U R Etions bou nded b y inerts , for which we currently havequali ta t ive understanding and empir ical correla-tions. 5~Development of explosions f rom autoignit ionprocesses, in some cases leading eventually to det-onation, exhibits complicated dynamics providingcha llenges in ad vancem ent of unders tanding, notonly with respect to the com plex gas-dynamic as-pec ts but a l so in de te rmining where t r anspor t andchemical-kinetic effects occ ur, scales of differe nt re-gions and influences o f different chem istry. T he oryin this area co ntinues to p rogress, 4s'51'52'53again withextensive help f rom asymptotic analysis.

I gn i t i on The or yMainly in the decade of the 1970s, AEA was ap-plied extensively to t ime-dependent ignit ion prob-lems, 54'55'56 prov iding on e of the main arenas inwhich the pow er of tha t technique was dem on-strated. Solutions to many ignition problems are nowavailable, 3~ illustrating juxtapositions of differe ntspatial zones and temporal stages in ignit ion pro-cesses. Recent worko~'Ss is applying asymptoticmethods to address inf luences of more co mpl ica tedchemical kinetics on ignition times and on ignitiondistances in boun dary- layer f lows. Fur ther r esearcha lso i s needed concerning e ffec ts of geometry , suchas sharp corners, 59 on ignition.

Deflagration of Sol ids and LiquidsAlthough Lewis and Prandt l numbers genera l lyare of order uni ty for gases, they both tend to belarge for con den sed phases. This and var iat ions oftrans port coefficients can affect reaction-zone struc -tures, 6~ in on e limit yield ing difffusion-free defla-gration. Instabilities an d bifurcations, lead ing tooscillatory propagation with supera-diabatic temper-atures, approaching chaos, and to spinning def la-

grations, then occur . 61 Und erstandin g o f these andre la ted phenomena has been deve loped by theo-retical analyses . 62-e5Cond ensed-phase ch emica l k inet ics genera l ly be-come important in propellant def lagrat ion, and in-teractions between chemistry, transport and flow canbe comp lex. M oreo ver, different propellants ty pi-cally behave differently. ~ Theoretical analyses forspecific propellants provide insights on deflagrationmechanisms.For example, for ni tramines melt ing forms a l iq-uid layer in which d ecompo si t ion occurs, but the reis also vaporization and furth er exothermicity in b otha pr imary and a secondary gas-phase f lame zone.O n l y t he c he m i s tr y o f the c onde nse d pha se a ndprimary zone affects the def lagrat ion rate appreci-ably. Analyses applying AEA to one-step approxi-

mat ions for the ch em is t ry in each of these zonesand treat ing the l iquid-gas interface as planar canproduce exce l lent agreement wi th measured def la -grat ion rates and their pressure and temperaturesensit ivi ties. 6s Altho ugh this ignores bub bling ob-served in the l iqu id layer , study of this two-ph aseflow69 reveals that , desp ite th e necess i ty of intro-ducing a sublayer of nonequilibrium vaporization andthe in t rus ion of a num ber of physica l phenom ena ,notably Marangoni convection, in inf luencing thetwo-phase behavior , the def lagrat ion rates pre-dicted by th e planar-interface mo del are modifiedonly sl ightly. Outstanding challenges to theoreti-c lans inc lude deve lo pm ent of r educed chem is t ry toexplain the two-stage gas-phase combustion and toidentify the source o f the values o f the en ergeticand rate constants that are successful in the one-step approximations.

Flame SpreadThe or y ha s c on t r i bu t e d t o und e r s ta nd i ng o f p r o -cesses o f f lame spreading through conden sed fue lsin recent years. Emmons discussed this subjecttwe nty yea rs ago, 15 bu t fur th er advances have bee nmade. 7~ In particular, for sprea d along solid fuelsthere are im prov ed ke rnel decompositions, 72-7s ex-tend ing th e or iginal analysis of John de Ris.76 In-t roduc ing an Ose en approximat ion for counte rcu r -

ren t gas f low over the surface o f a thermally th ickor thermally thin solid fuel , de Ris76 der ive d a setof linear partial differential equations with con stan tcoeff icients having different boundary condit ions inthe two semi-infinite dom ains at the fuel surfaceahead of and b ehind the point of f lame a t tachment .That s et is equiv alent to a set of integral eq uationsthat is suited for solution through Fourier t rans-f or ms by t he W i e ne r - H op f t ec hn i que o f ke r nalsplit ting in complex-var iable theory . Althoug h hesplit th e k ernel exactly fo r his therm ally thick fuel,de Ris76 accom plished this only approximately forther m ally th in fuels. Delichatsios 75 was able t o splitthe kernel exactly for the thermally thin fuel andthereby remo ved a d isc repancy o f about a f actor oftwo betw een theo retical and experimental spreadrates. Wichman74 replaced the Oseen approxima-t ion by a l inear ve loc i ty gradient and succeeded insolving the result ing system having var iable coeff i-cients, again by the W einer-H opf technique, therebyobta ining improv ed cor re la tions of measured ra tesof creeping spread along therm ally thick fuels. 72Integral equations ar ise in an essential way indescr ibing concurrent or upward f lame spread. Al-thou gh approximate solutions have bee n obtaine dtha t a f ford reasonable agre em ent w i th exper imenta lspread rates, 77'78 m ore research is need ed, espe-cial ly concerning behaviors of spread along charr ingfuels. Further theoretical studies also could be


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T H E R O L E O F T H E O R Y I N C O M B U S T I O N S C I E N C E 11helpful o n spread o ve r surfaces of liquid fuels, oninfluences o f gravity in flame spread, on spre adthrough porous fue l beds and on spread throughfuel partic le clouds, in clud ing effects of radiant e n-ergy transfer . Although the re h ave bee n significantrec en t advances in thes e areas, appreciable addi-t ional progress ma y be anticipated in the near fu-ture.

F l a m e I n s t a b i l i t i e s

Premixed laminar flames are especially rich in in-s tabi l i ty phenomena , our theore t ica l unders tandingof which has progressed continually, beginning m orethan a half-century ago. The Darr ieus-Landau hy-dro dy nam ic instability 79 of the planar flame, alwayspresent when the dens i ty of the f resh mixture ex-ceeds that of the bu rnt gas, necessi tates stabil iz ingeffects f rom diffusive- thermal phenomena at shortwavelengths and from bo dy-fo rce (buoyancy) or ap-para tus-boundary phenomena a t long wave lengthsfor the planar flame to exist. D esp ite excellent ear-l ier work, s~ the f inal co m preh ens ion that enablesthis statement to be made def ini t ively stems mainlyfrom the research of Paul Clavin and hiscollaborators 79 in th e 197 5-85 decade. Tho se stud -ies succeeded in assimilat ing ear l ier theory and indeve loping new theory to a r r ive a t the genera l con-clusion stated above, which self-consistently ex-plains observed flame instabilities.Diffusive- thermal p hen om ena and body-forceph eno m ena can b e destabil iz ing (as for fuel- lean hy-drogen f lames and upward-propagating f lames, re-spectively), in which cas es, instead of planar flames,cellular o r curv ed f lames are observed. W hen h y-drodynamic instabil i ty and a body-force effectthrough a stabilizing gravitational acceleration g arebo th included, the n a dispersion relation for awav enum ber k and a growth ra te proport iona l toe~ is found to be

(13)

w he r e V u and vb a re the ve loc i ty wi th respec t tothe wave in the unburnt and burnt gas , r espec-tively, (the burning velocity), and r is the ratio ofthe burnt -gas dens i ty to the unburnt -gas dens i ty .Since ~" < 1, (13) shows th at o" > 0 whe n g < 0,wh ich is gravitational instability. Diffusive-thermalinstabil i ty is r icher and is exhibited most clear ly bypu tt ing ~" = 1 to el iminate the oth er two instabil-i t ies. Th e result, der ived by Joulin an d C lavin, sl isa m ore complicated dispersion relation, 3~ the im-

4(1+~3)21/2

~(L-1) )LSATIN

i i , ,

I

6 m

18/5--

o-TRAVELNG,PULSATNG~I \,.k

ffITRAVELNG~ [ / " ' " , = k

(7

t ' . 2 ~ ~ k~n (tlF )FIG. 7. R egions of diffusive-thermal instability ina Lewis-number , heat- loss plane, with dispersionrelations illustrated by insets. 8'

plications of wh ich are illustrated in Fig. 7, whe reL is the Lewis nu m be r of the def icient reactant ( theratio o f the therm al diffusivity to its molecular d if-fusion coefficient), /3 = E ( Tb - T , ) / ( R T ~ ) i s theZel 'dov ich num ber , and the horizontal scale (with/x the ratio of the b urn ing velocity of a nonadiabaticplanar f lame to that of the adiabatic planar f lame)is a measure of the ra te of hea t loss . Al though thetheoretical analysis employed AEA, recent studiesby RRA indicate that, with prop er interpretat ion,the AEA results apply well to real f lames.Diffusive-thermal instabilities hav e be en ana-lyze d w ith variable-density effects, s2's3's4 with in-f luences of an u pstream f lame-holder, sS-s9 and withinfluences of stretch, 9~ for example. The theo ry hasamassed an extensive b od y of information in recentyears. 2s '29,3~ Th e diverse character of f lame in-stabilities has, in fact, attracted wide attention be-yond t he c ombus t i on c ommuni t yb in the genera lphysics o f nonlinea r phen om ena. 2 Besides p oly-hedral , cel lular structures, spinning structures andoscil la t ions with p er iod doubling, approach to chaosis obs erved in cer tain f lames, as described, for ex-am ple, by the Kuram oto-Sivashinsky equation. 91'92Since f lame chaos is simpler th an the chaos o f Na-vier-Stokes turbulence in that it can occur (from highactivation ene rgy and unequal diffusion) eve n in justone space dimension, for example, study of f lameinstabil i t ies provides a useful approach to the de-ve l opme n t o f unde rs t a nd ing o f non li ne ar phe n om-ena . Much more research remains to be per formedon the non linear the ory of f lame instabili t ies.

F l a m e s i n N o n u n i f o r m F l o w sA closely related topic concerns the structure anddynam ics of premixed f lames in no nuni form f lows.


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12 H O T T E L P L E N A R Y L E C T U R EStra in a long a f lame she e t mod if ies the f lame s t ruc- z.5 / 'ture , making i t th in ner . Kar lovi tz 3 '94 ca l led th is" f l a m e s t r e t c h , " a nd i t s e f f e c t c a n be m e a su r e d bya nond im e ns iona l pa r a m e te r K , o f t en c a l l e d the z .o~K arl ov itz nu m be r. F o r w ea k st re tc h an d cu rv at u re h ~ SOPERADIABATICof a near ly p lana r f lame, the inf luence of bot h of t5 ~ / r0f -rthese per tu rba t io ns a r ise throug h a s ingle para m e- ~ H.13~ - -~- -t e r , t he nond im e ns iona l f l am e s t r e tc h , k~

ADIABATIC 01 f I.o SUBADIABATIC , . , ~ ' . I - - ~ " - -

K = - - } I)u(V " n ) / [ .~ E X T I N C T I O NVu (14) ~ #\u~1760 5 T R A N T , O .- ~ n " ( V u ) + ( V v ) r 9 n , ~ _ _ _ _ _ . ~ . _ ~ . ~ . . J . . . . . . . .

I I

whe r e 1 i s the und i s tu r b e d f l a m e th i c kness , n i s aun i t ve c to r no r m a l t o t he f l a m e she e t , po in t ing to -wa r ds the f r e sh m ix tu r e , a nd ( V v ) i s the d vadic gra -dien t of the loca l ve loc i ty f ie ld , wi th (Vv) i t s t rans-pose , t he sum be ing tw ic e the s t r a in - r a t e t e nso r .The f i rs t t e r m in ( 14 ) de f ine s t he s t r e t c h d ue toc u r va tu r e . We a k s t r e t c h c o r r e sponds to sm a l l K ,m o de r a t e s t r e t c h to K o f o r de r u n i ty a nd s t r ongs t r e t c h to K l ar ge ; s t r ain a nd c u r va tu r e e xe r t qua n -t i ta t ive ly d i f fe rent inf luences for modera te or s t rongst re tch .AEA a na lyse s ha ve be e n c om ple t e d f o r we a k ,m o de r a t e a nd s t r ong s t r e t c h o f p l a na r f la m e s incounte r f lows, inc lud ing inf luences of var iable den -s i ty , o f Le wis num be r s d i f f e re n t fr om un i ty , o f non -a d ia ba t i c i ty a r i s ing f r om p r oduc t - ga s t e m pe r a tu r e sT~ be ing d i f f e r e n t f r om the a d ia ba t i c f l a m e t e m -pe r a tu r e T,~f, of d i f fe rent s t ra in ra tes in d i f fe rent or -thogona l d i rec t ions , and o f swir l having vor t ic i tyno r m a l t o t he p l a ne o f t he f l am e . 95- 99 R e pr e se n -t a t ive r e su l t s f o r i n f lue nc e s o f nona d ia ba t i c i t y w i thLe wis num be r s L o f un i ty a r e shown in F ig . 8 ,wh ic h de m ons t r a t e s a b r up t e x t inc t ion f o r su f f i -ciently subad iabatic conditions. Fo r adiabatic f lames,AE A shows a b r up t e x t inc t ion to be a c h ie va b le on lywith d i f f icul ty and o nly for L > 1; a t la rge tr grad-ua l ext inc t ion occurs for a l l H and L. The f i r s t the -o r e t i c a l a na lys i s o f t h i s t y pe o f p r ob le m wa s g ive nby Kl im ov ,100 who d id no t u se f o r m a l AEA e xp a n-s ions but o bta in ed lead ing -ord er e f fec ts . Fo r suff i-ciently weak stretch , the re are indications from R RAtha t AEA r e su l t s ca n be m a de to a pp ly w i th r e a lc he m is t r y , bu t t h i s w i l l no t a lways e x te nd to l a r ge rs t r e t c h , 1~ a nd m or e r e se a r c h i s ne e de d on th i squest ion . For weak s t re tch , the b urning ve loc i ty canbe a pp r ox im a te d a s

V = v u e - b ' , (15)

~ lot t to1/KFIG. 8 . T he ra t io / .t of the ra te o f hea t re leasepe r un i t a r e a o f a s t r a ine d p r e m ix e d f la m e to tha to f a n u ns t r a ine d f l a m e a s a f unc tion o f t he inv e r senond im e ns iona l s t r a in r a t e f o r va r ious nona d ia ba t -ic i ty para me te rs H (/3 = Ze l 'dov ich num ber ) .

be der iv ed. 3~176176176 Because of the res t r ic t io nto we a k s t r e t c h , a l i n e a r e xpa ns ion o f t he e xpone n-t i a l in ( 15 ) i s e qua l ly a c c u r a t e a nd o f t e n m os t c on -venient , but in ca lcula t ions such as s imula t ions tha toccas iona l ly pro du ce bK > 1 , (15) has the v i r tu e ofm a in ta in ing pos i t i ve bu r n ing ve loc i t i e s .

Vor t i c e s a r e f l ow nonun i f o r m i t i e s r e l a t e d to t u r -bu le nc e , a nd p r og r e s s ha s be e n m a de in de sc r ib inglaminar - f lame in te rac t ions wi th vor t ices . F la t f lamesc a n be spun in to su f f i c ie n t ly s t r ong uns t r e t c he dvor t e x c o r es , r e su l t i ng in a c o r e o f bu r n t ga s w i tha f lame pro pag a t ing ou tward f rom i t . lo~ S tre tch ofthe vo r t e x c a n a r r e s t t he ou twa r d p r opa ga t ion , g iv -ing c y l ind r i c a l f l a m e s wr a pp e d a r ound vo r t i ce s t ha ta r e good c a nd ida te s f o r e l e m e n t s i n t u r bu le n t c om -bus t ion . A l so , s inc e the r a d ia l p r e s su r e f i e ld o f avo r t e x de pe nds on th e ga s de ns i ty , i f pa r t o f t hevor t e x c o r e i s f r e sh m ix tu r e a nd a no the r pa r t bu r n tga s , t he p r oduc t s w i l l be d r ive n a long the c o r e bythe p r e s su r e g r a d ie n t a t ve loc i t ie s t ha t c a n g r e a t lye xc e e d the l a m ina r bu r n ing ve loc i ty . ]~ Diffusionf l am e s a lso c a n be wr a p pe d in to uns t r e t c he d v o r t e xc o r e s , a l t hough th i s r e qu i r e s ve r y s t r ong vo r t i c e ss e l d o m e n c o u n t e r e d i n t u r b u l e n t c o m b u s t i o n J ~T h e r e h a s b e e n a n a p p r e c i a b l e a m o u n t o f r e c e n tthe o r e t i c a l a na lys i s o f f l a m e - vor t e x in t e ra c t ions , b u tm or e r e m a ins to be done in t he f u tu r e , f o r e xa m pleconcerning e f fec ts of vor tex s t re tching and of rea lc he m is t r y .

w h e r e V u r e f e r s t o t he uns t r a ine d f l a m e , a nd b l isa M a r ks t e in l e ng th , wh ic h c a n be c a l c u la t e d f r om79 83the f lame s t ruc ture . ' F rom these resul ts , for weaks t r e t c h , a ge ne r a l e qua t ion f o r f l a m e dyna m ic s c a n

T u r b u l e n t C o m b u s t i o nThe or i e s f o r t u r bu le n t c om bus t ion d i f f e r f o r p r e -m ixe d a nd nonpr e m ixe d sys t e m s a nd f o r r e a c t ion -


13/17

T H E R O L E O F T H E O R Y I N C O M B U S T I O N S C I E N C E 134 0 1 L A M r N '~ R 1 T U R B L J t.E / T T

\ ~ - ~

o V I t / I N - - ~ - - , U ~"~ 4 o 8 oE X I T V E I . 0 C I ' P fm l s )F IG. 9 . D e pe nd e nc e o f t he he igh t o f a t u r bu le n t -je t d i f fus ion f lame on je t ve loc i ty .

she e t a nd d i s t r i bu te d - r e a c t ion l im i ts . 3~ F igu r e 9 ,ba se d on a d i a g r a m f i r s t p r e se n te d by Ho t t e l a ndHa w thor ne in t he th i r d sym po s ium , 1~ i l l u s tr a t e s av a r i et y o f p h e n o m e n a a n d p r o b l e m s t h a t h a v e b e e nstud ied theore t ica l ly for tur bu len t d i ffusion f lames .A l though th i s i s a c on t inu ing top ic o f t he o r e t ic a l i n -ve s t iga t ion , d i sc uss ion i s r e s t r i c t e d he r e to p r e -m ixe d tu r bu le n t f la m es .I n th e r e a c t ion - she e t r e g im e , a sm o o th f unc tionG c a n be in t r oduc e d , suc h tha t G < 0 in f r e sh m ix -ture , G > 0 in bu rnt gas , and G = 0 a t the f lame.An e vo lu t ion e qua t ion f o r t he f l a m e in the tu r bu -l e n t f l ow c a n the n be wr i t t e n a s8~176176

g G- - + v - V G = v I V G I , ( 1 6 )8 t

i n wh ic h ( 14 ) a nd ( 15 ) m a y be use d f o r t he l a m ina rburning ve loc i ty V. Theore t ica l s tudy of (16) i sp l a y ing a s ign i fi c an t r o l e i n a t t e m pt s t o unde r s t a ndpr e m ixe d tu r bu le n t - f l a m e p r opa ga t ion . S inc e the r ea r e n o r e q u i r e m e n t s o n G b e y o n d t h o s e ju s t s t a t e d ,i t m a y be g ive n d i f f e r e n t phys i c a l i n t e r p r e t a t ions ;e ve n i t s un i t s a r e a r b i t r a r y . I n one in t e r p r e t a t ion ,i t s r oo t - m e a n- squa r e f luc tua t ion g ive s the tu r bu le n tf l a m e th i c kne ss , u lF o r su f f ic i e n tly we a k tu r bu le nc e , pe r tu r ba t ionm e t h o d s h a v e b e e n e m p l o y e d t o o b t a i n b u r n i n gve loc i t i e s a nd c ha nge s in t u r bu le n c e a c r os s t hef l am e , n~ d l3 On the ba s i s o f t he se r e su l ts , i t ha sbe e n wide ly a c c e p te d tha t f o r sm a l l va lue s o f t her a t io o f t he r oo t - m e a n- squa r e tu r bu le n t f l uc tua t ionve loc i ty u ' t o vu , t he tu r bu le n t b u r n ing ve loc i ty d i f -fers from V u by a n a m oun t p r opor t iona l t o ( u ' / V u ) 2 .How e ve r , t he r e ha s be e n a r e c e n t c ha l l e nge to t h i sresul t , n4 in e f fec t based on the observa t ion tha t th eF our i e r de c om pos i t i ons do no t a c h ie ve s t a t i s t i c a ls t a t iona r i ty ; t h i s s tudy , ba se d on ( 16 ) , de duc e s b yba la nc ing f l a m e - a r ea g r owth th r o ugh tu r bu le n t d i f -f u s ion wi th i t s c onsum pt ion by t r a nsve r se l a m ina r -f l a m e p r opa ga t ion tha t t he p r opor t iona l i t y i n s t e a d i sto ( u ' / v u ) 4 / 3 . This obse r va t ion m a ke s the inve n t iver e nor m a l i z a t ion m e thods , wh ic h ha ve be e n a p -p l i e d to ( 16 ) t o g ive in t r igu ing r e su l t s f o r t u r bu le n tbu r n in g ve loc i t i e s a t h igh tu r bu le nc e in t e ns i -

t ies , nS"n6 'n7 even mo re controvers ia l than they wereini t ia l ly .Equa t ion (16) appl ies a t a l l sca les in the reac t ion-she e t r e g im e a nd f o r l a r ge va lue s o f ( u ' / v , ) in thisr e g im e p r e d ic t s m u l t ip l e f l a m e le t s , e nc los ing poc k-e t s o f unb ur n t ga s (bu t none o f bu r n t ga s) . Thewr ink le d f l am e su r f a c e in t h i s m u l t ip l e - she e t sub -regim e m ay be des cr ibe d as a fracta l, ns having af r ac t a l d im e ns ion o f 21 , b u t on ly f o r s i z es be tw e e nthe in t e g r a l s c a l e o f t u r bu le nc e ( the l a r ge - e ddy d i -m e ns ion ) a nd the G ibson sc a le , n9 the e d dy s i ze a twh ic h vu e qua l s t he e ddy tu r nove r ve loc i ty . I fR e yno lds num be r i s i nc r e a se d a t f i xe d Da m k6h le rnum be r un t i l t he l a m ina r - fl a m e th i c kne ss i s c om -pa r a b le t o t he Ko lm ogor ov sca l e , the n th e G ibsonsc a le be c om e s e qua l t o t he Ko lm ogor ov sc a l e , a ndf lamele ts begin to q uench in the smal les t edd ies ( theKl im ov c r i t e r ion ) , g iv ing r i s e t o a b r oke n- f l a m e le tr e g im e . A t h ighe r R e yno lds num be r s i n t he b r o -ke n - f l a m e le t re g im e , ( 16 ) c a n a p p ly on ly in e dd ie sha v ing s i z e s be twe e n the in t e g r a l s c a l e a nd the P e -t e r s s c a l e , n9 the e dd y s i ze be low whic h the s t ra inr a t e i s t oo h igh f o r l a m ina r f l a m e le t s t o e x i s t i n t hee dd ie s . Th e r e c e n t i de n t i f i c a t ion o f t he se im por t a n tl e ng th sc a l e s i n p r e m ixe d tu r bu le n t c om bus t ion , a swe l l a s o f o the r s I 2~ tha t a r e c lo se ly r e l a t e d a nd tha tinc lude c ons ide r a t ions o f hydr odyna m ic in s t a b i l i t y ,h e l p s t o m a r k t h e r a p i d d e v e l o p m e n t t h a t t h e s u b -j e c t ha s e xpe r i e nc e d in r e e e n t ye a r s .No t a l l r e c e n t de ve lo pm e n t s i n t he o r i e s o f p r e -m ixe d tu r bu le n t c om bus t ion a r e ba se d on ( 16 ) . F o re xa m ple , t he B r a y - L ibby - M oss a pp r oa c h , wh ic hbe a r s som e r e l a t ionsh ip to ( 16 ) i n p r e sum ing tha tp r im a r i ly on ly r e a c t a n t s a nd p r oduc t s a r e p r e se n t ,e m ploys d i f f e r e n t e qua t ions ;121 a more de ta i ledpresenta t ion of these and o ther approaches soon wi llbe pub l i she d . 122 I n t e nse r e se a r c h c u r r e n t ly i s i np r og r e s s , f o r e xa m ple r e l a t e d to ( 16 ) , a nd im por -t a n t , ne w the o r e t i c a l re su l t s w i l l soon be se e n f r omva r ious g r oups . Toda y i s a ve r y e xc i ti ng t im e in thet h e o r y o f t u r b u l e n t c o m b u s t io n .

C o n c l u d i n g C o m m e n t sThe se l a s t f e w se c t ions c l e a r ly ha ve be e n r ushe da nd c om pr e s se d ne a r ly to t he po in t o f i nc ohe r e nc e .A m o r e l e i su r e ly m ono gr a ph on the sub je c t , o f l e s sthan 200 pag es , i s jus t now beco min g ava i lable . 123

T h e l a s t c h a p t e r o f t h a t m o n o g r a p h i s e n ti t l e d " T h eF u tu r e " a nd o f f e r s m or e e x te ns ive spe c u la t ions onwha t m a y l i e a he a d tha n c a n be f ound in the b r i e fcomments made in the preceding sec t ions . Not onlyhave I been unable to g ive thorough c i ta t ions here - -I ha ve e ve n ha d to e xc lude som e e n t i r e t op ic s( d r op le t s, sp r a ys , c om bus t ion c ha m be r s , e t c . ) . A tl e a s t you c a nno t he lp bu t l e a ve w i th the se nsa t iontha t t he o r y ha s be e n , i s a nd wi l l be a s ign i fi c a nt


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1 4 H O T F E L P L E N A R Y L E C T U R Ep a r t o f c o m b u s t i o n s c i e n c e . A b e t t e r p r e s e n t a t i o nw o u l d h a v e b e e n m a d e b y P r o m e t h e u s .

AcknowledgmentI w i sh t o t h a n k E d L a w f o r t h e i n v i t a t i o n t h a t

g a v e m e t h e o p p o r t u n i t y t o p r e p a r e t h i s m a n u s c r i p ta n d a l s o f o r s e c u r i n g c o m m e n t s o n m y f i r s t d r a f tf r om a l ar g e n u m b e r o f r e v i e w e r s . M y s i n c e r e g r a t -i t u d e g o es t o t h e d o z e n o r s o r e v i e w e r s , m o s t o fw h o s e i d e n t i ti e s a r e u n k n o w n t o m e , w h o r e t u r n e dr e p o r t s f o r m y c o n s i d e r a t i o n . T h e m o s t c r i t i c a l a n dl e a s t k i n d r e p o r t s w e r e t h e m o s t u s e f u l a n d l e d t om a n y r e v i s i o n s, a l t h o u g h I a m f u ll y a w a r e t h a t m a n ys t i l l w i l l r e m a i n d i s s a t i s f i e d w i t h t h i s f i n a l p r o d u c t .T h a n k s a ls o g o t o t h e N S F , N A S A , D O E a n d I ) O Df u n d i n g ag e n c i e s th a t h a v e s u p p o r t e d m y r e s e a r c ha n d t o s t u d e n t s , w h o s e q u e s t i o n i n g a n d e n t h u s i a s mh e l p g r e a t l y i n s h a r p e n i n g i d e a s .

R E F E R E N C E Si. WEINBER(,, F . J . : F i t ~ e e n th S y m p o s i u m ( I n -

t e r n a t io n a l ) o n C o m b u s t i o n , p . 1 , T h e C o m -b u s t i o n I n s t i t u t e , 1 9 7 5 .

2. HAMILTON, E.: M y t h o l o g y , L i t t l e , B r o w n &Co . , 1 9 4 2 .

3 . LEICESTER, H. M . : A lc h e m y , En c y c lo p e -d i a Am e r i c a n a , V o l . 1 , p . 5 1 0 , Gro l i e r , 1 9 83 .4. FRANK-KAMENETSKII, D . A.: Dif fusio n an d H ea tT r a n s f e r i n C h e m i c a l K i n e t ic s , P l e n u m P r e s s ,1947.

5. YON KAtlMAN, Th . AND PENNER, S. S.: Se

williams 1992 theory

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