Combustion and Extinction in the Stagnation Point Boundary Layer of a Condensed Fuel 1978 Combustion and Flame

7/27/2019 Combustion and Extinction in the Stagnation Point Boundary Layer of a Condensed Fuel 1978 Combustion and Fl

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COMBUSTION AND FLAME 33, 55 - 68 (1978) 55

C o m b u s t io n a n d Ex t in c t io n in t h e S t a g n a tio n -Po in t Bo u n d a r y La y ero f a C o n d e n s ed F u e l

J. S. T' IEN, S. N. SINGHAL, D. P. HARROLD and J. M. PRAHLDepartment of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio 44106

This publication contains information derived from a research pro/ect sponsored by the Products ResearchCommittee. However, any conclusions drawn from the research prelect in this article are those of theauthor and not o f the PRCThe combustion and extinction phenomena in the stagnation point boundary layer of a condensed fuel iss tudi e d experimentally and theoretically with e m p h a s i s o n the near-limit flame. The numerical analysisa ssum e s a se c o nd-o r de r fo r wa r d overall chemical reaction in the gas phase, with gas-phase activation energyand modified frequency factor, determined by comparison with the experimental results. The effect ofexternal radiation on the extinction limit is computed using a simplified model. Burning rates and extinc-tion d a t a are determined from measurements taken o n p o l y m 0 t h y l m e t h a e r y l a t e samples in an opposed-jetdi f fus i o n f l a me a ppa r a tus . Fa v o r a bl e agreement between experimental extinction data and theoreticalpr e di c t i o ns i s o bta i ne d fo r a gas-phase activation energy of 30 kcal/mole and a modified frequency factor of5.2 X 107 see--l.

I. INTRODUCTIONBecause of its simple flame geometry, the op-posed-jet diffusion flame experiment has beenused many times in the past for studying extinc-tion and flame stabilization characteristics [1, 2].Application of the opposed-jet technique has alsobeen adapted for the combustion of condensedfuels [3, 4, 5]. Usually an oxidizer jet is deliveredto the fuel surface and a flame is established in itsstagnation-point boundary layer. For combustionaway from the extinction limit, it is expected thatheat and mass transport are the rate-controllingprocesses. Based on the bo undary layer thickness,the fuel-burning rate should be proportional to thesquare root of the j et velocity. This is found to bethe case for the burning experiments of solid fuelsin Refs. 3 and 4. However, as the extinction limitis approached, chemical kinetic rates become finitecompared with the transport processes and adeparture from the above-mentioned result isexpected. Since the extinction condition is theresult of interplay between the transport processesCopyright 1978 by The Combustion InstitutePublished by Elsevier North-Holland, Inc.

and the chemical kinetics, a precise extinctionanalysis should have a accurate description o f theburning rate and the velocity and temperaturefields in the near-limit region.

Previous theoretical works on opposed-jet dif-fusion flame extinction [6-10] have contributedgreatly to our understanding of the extinctionphenomena. However, all these papers haveassumed an incompressible potential flow in thecombustion field. For a bette r quantitative deter-mination of the extinction condition, compres-sible, viscous effects have to be included, espe-cially when a condensed fuel is involved. In Ref.11, boundary layer approximations are used andan asymptotic analysis based on a large activationenergy is employed. Extinction conditions aredetermined using expressions given in Ref. 9 andthe flow field calculation from previous fastkinetics computations [12]. Another extinctionanalysis in the stagnation point boundary layer[13] includes a two-step carbon combustionreaction where the carbon-surface oxidation rateis taken to be finite and the gas-phase carbon

0010-2180[78/0033-0055501.75


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5 6 J . S . T ' I E N E T A L .m o n o x i d e o x i d a t i o n r e a c t i o n i s a s su m e d t o b einf in i t e ly fa s t .

The presen t work presen t s a numer i ca l ana lys i so f t h e b o u n d a r y l a y e r c o m b u s t i o n m o d e l u s in g at ime-march ing f in i t e d i f fe rence scheme . In Sec -t ion I I , t he theore t i ca l fo rmula t ion and re su l t s a reg i v e n u s i n g n o n d i m e n s i o n a l p a r a m e t e r s . T h egenera l nea r . l imi t f l ame behavior , t he ex t inc t ionb o u n d a r i e s , a n d t h e e f f e c t o f external rad ia t ionon the ex t in c t ion l imi t a re g iven. In Sec t ion I I I ,t h e o p p o se d - j e t d i f f u s i o n f l a m e e x p e r i m e n t i sd e sc r i b e d . T h e b u r n i n g a n d e x t i n c t i o n d a t a o fp o l y m e t h y l m e t h a c r y l a t e a r e c o m p a r e d w i t h t h etheore t i ca l r e su l t s , and the ove ra l l chemica lk i n e t ic c o n s t a n t s a r e d e d u c e d .

H . T H E O R Y1. Governing E quat ionsT h e c o m b u s t i o n m o d e l a s su m e s a o n e - s te p f o r w a r dovera l l gas-phase chemica l reac t ion of secon d ord e ro c c u r i n g in t h e a x i sy m m e t r i c o r t w o - d i m e n s i o n a ls t a g n a t i o n - p o i n t b o u n d a r y l a y e r a d j a c e n t t o t h efue l sur face . The d i f fus ion coe f f i c i en t s o f t hereac tan t s a re a ssum ed to be equa l . The spec if i chea t s o f t h e d i f fe ren t specie s a re a ssumed to bee q u a l a n d c o n s t a n t . T h e P r a n d f l ( P r ) a n d S c h m i d t( S c ) n u m b e r s a r e c o n s t a n ts b u t d o n o t h a v e t o b euni ty . The idea l gas l aw is a ssumed for t he m ix turea n d c o m p o n e n t g ases . T h e p r o d u c t o f th e d e n s i t yand v i scos i ty , p / z , i s a ssumed cons t an t . The gov-e rn ing pa r t i a l d i f fe ren t i a l equa t ions for t heb o u n d a r y l a y e r a r e t r a n s f o r m e d i n t o th e f o l l o w i n gse t o f o rd ina ry d i f fe ren t i a l equa t ions us ing as imi l a r i t y va r i ab le :

1f ' " + f f f = - - [ 0 a ) z - - 4 0 ]l + e(1]P r )O" +f0 ' = - - -qw(I/Sc)YF" + f Y / = w( 1 / S c ) Y o " -I- Yo ' = NoW, ( I )w h e r e ' - d[drl

r e U e f O yr / = 2s a/---~z p dy

~ 0 r= P e l d e U e2 e dr{ ~ f o r a x i sy m m e t r i c f l o w

e = for two-d ime nsiona l f l ow .Ue = ar i s t h e v e l o c i t y a t t h e e d g e o f t h e b o u n d a r y ,l aye r , u/u, = f, 0 i s t h e t e m p e r a t u r e n o n d i m e n -s i o n a li z ed b y t h e f r e e- s t re a m t e m p e r a t u r e , Te, E isthe ac t iva t ion ene rgy of t he gas phase reac t ionn o n - d i m e n s i o n a li z e d b y RT,, q i s th e c o m b u s t i o nhea t re lease pe r un i t mass of fue l cons um ed non-d i m e n s i o n a li z e d b y cpT, and N O i s t he s to i ch io-me t r i c ox id i ze r t o fue l mass ra t io .

The chemica l - reac t ion ra t e fo r t he one -s t epsecond-o rde r reac t ion i s g iven byv = Wr(BT)p z Yr Yo e_~ /nT ' (2 )W~ Wow h e r e ~ is m a ss o f f u e l c o n su m e d p e r u n i t v o l u m ep e r u n i t t i m e . T h e a b o v e e q u a t i o n c a n b e r e w r i t te nas

_v= BYFY oe_ ~mT , (3)PAw h e r e B = (WM/Wo)(p[R)B h a s t h e u n i t o f 1 [

t ime .I n t r o d u c i n g E q . ( 3 ) i n t o t h e c o n se r v a t i o n e q u a -

t ions , t he non-d imensiona l reac t ion ra t e w in Eq .(1) i s given byw = DYFYo e-E/O , (4 )where D = [2/(l+e)](B/a) i s t h e D a m k o h l e rn u m b e r .

T h e b o u n d a r y c o n d i t i o n s a t th e e d g e o f th ebou nda ry l aye r ( ,/ --> oo) a re0 =1, f' =2, YF =0, Yo =Y o. . (5 )At the fu e l sur face r /= 0 :O = O w , fw'--OY/ = Sc fw(1 -- YFw)Y o ' = - - Sc fw Yow. ( 6 )


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B O U N D A R Y L A Y E R C O M B U S T I O N A N D E X T I N C T IO N 5 7T h e c o n d e n s e d p h a s e i s a s s u m e d t o b e a p u r e

f u e l : n o s u r f a ce a n d / o r c o n d e n s e d p h a s e r e a c t i o n sa n d n o o x y g e n p e n e t r a t io n i n t o t h e c o n d e n s edp h a s e . T h e c o n d i t i o n f w ' = 0 i s exac t f o r so l idf u e l ; m o d i f i c a t i o n i s n e e d e d i f i t i s a l i q u i d .

The ene r gy ba lance ac r os s the f ue l su r f ace i sg iven byOw' = - - fw P r Q + R t D a 120 w 4 _ R a D 1 /2 , ( 7 )w h e r e Q i s t h e n o n d i m e n s i o n a l h e a t l o s s t o t h es o l i d i n c l u d i n g l a t e n t h e a t a n d h e a t c o n d u c t i o ni n t o t h e s o l i d . I f t h e s o l i d t e m p e r a t u r e d i s t r ib u t i o ni s a s s u m e d t o b e o n e d i m e n s i o n a l , th e n t h e p r o f i lei s e x p o n e n t i a l a n d i t c a n b e s h o w n t h a tQ = O w - C s oc + L , (8 )Cpw h e r e L i s t h e n o n d i m e n s i o n a l l a t e n t h e a t a tr e f e r e n c e t e m p e r a t u r e a b s o l u t e z e r o . T h e l a stt w o t e r m s i n E q . ( 7 ) r e p r e s e n t r e s p e c t i v e l y t h er a d i a ti v e h e a t l o ss f r o m t h e h o t f u e l s u r f a c e t o t h es u r r o u n d i n g s a n d t h e r a d i a t i o n a b s o r b e d a t t h ef ue l su r f ace f r om ex te rn a l sources . 1 R a d i a t i v e h e a tt r a n s f e r i n t h e g a s p h a s e o f t h e f l a m e is n e g l e c t e d .T h e r a d i a t i o n t e r m s in E q . ( 7 ) a r e o n l y a p p r o x i m a -t i o n s a n d a r e g o o d o n l y w h e n t h e r a d i a t i o n a b s o r p -t i o n l e n g t h i s m u c h s h o r t e r t h a n t h e t h e r m a l d e p t hi n t h e c o n d e n s e d p h a s e d u e t o c o n d u c t i o n . D e f i n i -t i o n s e r R t a n d R a a re

x. Lp,aJ_ q a F P e 1 1 1 2

eo L p a j " (9 )

T h e s e t w o n o n d i m e n s i o n a l p a r a m e t e r s c a n b et h o u g h t o f a s r a d ia t i o n D a m k o h l e r n u m b e r s .

T h e n o n d i m e n s i o n a l b u r n i n g r a t e ( - - f w ) is1 By external sources, we mean the radiation from the

s u r r o u n d i n g s , n o t from th e gas phase of the flame itself.A polymer, ordinarily non-flammable, can be m a d eflammable in a r o o m o n fire because of the radiativeheat flux back from neighboring flames.

TABLE 1Values of the Nondi mensional Parameters for the

Reference CaseNondirnensional Numerical

Parameter Valuecs/cp 1.32L 4.32N O 1.92Pr 0.7O 5.5q 80R a 0R l 0Sc 0.70 c 10w 2.5e 1 (axisymmetric f low)

r e l a t e d t o t h e d i m e n s i o n a l m a s s b u r n i n g r a t e b y

= ( P e e a ) x / 2 ( - - f w ) . ( 1 0 )

T h e a b o v e s y s t e m o f E q u a t i o n s ( 1 , 4 - 9 ) i ss o l v ed b y a f i n it e d i f f e r e n c e m e t h o d u s i n g aU N I V A C 1 1 0 8 c o m p u t e r . F i r s t , a f i c t i t i o u su n s t e a d y t e r m i s a d d e d t o E q . ( 1 ) , a n i n i t i a lp r o f d e c h o s e n , a n d t h e s y s t e m t h e n t r e a t e d a s a ni n i ti a l- v a lu e p r o b l e m b y m a r c h i n g i n t i m e u n t i l t h es t e a d y - s t a t e s o l u t i o n r e su l t s. I f t h e p a r a m e t e r s a r es u c h t h a t a s t e a d y - s t a t e b u r n i n g s i t u a t i o n i s p o s -s ib le , a p r op e r cho ice o f the in i t i a l p r of i l e s p r o-d u c e s s u c h a s o l u t i o n . I f t h e p a r a m e t e r s a re s u c ht h a t s t e a d y - s t a t e b u r n i n g i s n o t p o s s i b l e , t h es teady - s ta te so lu t ion y ie lds ex t in c t ion , rega r d le s so f t h e i n i ti a l p ro f i l e s c h o s e n . T h e n u m e r i c a la n a l y s i s o f t h i s p r o b l e m i s p r e s e n t e d i n t h e A p -p e n d i x .2 . T h e o r e t i c a l R e s u l t sI n t h e n u m e r i c a l c o m p u t a t i o n , t h e v a l u e s o f t h en o n d i m e n s i o n a l p a r a m e t e r s e x c e p t t h o s e w h i c h a r espec i f i ed in each ind iv idua l f igur e a r e t aken f r omT a b l e 1 . F ig u r e 1 s h o w s t h e n o n d i m e n s i o n a lb u r n i n g r a te ( - f w ) a s a f u n c t i o n o f D a m k o h l e rn u m b e r o f s e v e r a l d i f f e r e n t a c t i v a t i o n e n er g i es a n da m b i e n t o x y g e n m a s s fr a c t i o n s . F o r a g iv e n v a l u eo f o x y g e n f r a c t i o n Y o e , ( - f w ) a p p r o a c h e s a n


4/14

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D x I 0 6Fig. 1. Nondimensional fuel burning rate vs. Damkohler number.

I 0 0

asymptotic value when the Damkohler number issufficiently large. This is the limit of fast chem-ical kinetics. For smaller Damkohler numbers, thevalue of ( - fw ) drops below its limiting value, sincenear the extinction limit, the heat and masstransfer rates in the flame zone become compa-rable to the chemical reaction rates. This reducesthe flame temperature (Fig. 2) which in turnproduces a small temperature gradient at the fuelsurface and a smaller nondimensional burning rate.In Fig. 1, the extinction points are those forwhich the burning rate curves become vertical.These points are static neutral stable points(static implies that the natural frequency is zero).We would expect a lower branch, statically-unstable, steady-state solution to exist, but wecannot determine this branch of the solutionbecause the gas phase equations are solved as aninitial value problem (see Appendix). On the otherhand, it is possible tha t part o f the steady solutionnear the extinction limit will be dynamically

unstable, as in the case in Refs. 14 and 15. Thistype of instability has not been investigated in thepresent paper.

Figure 2 shows the variation of the maximumflame temperature as a function of the Damkohlernumber for several different activation energies.Comparing Fig. 2 with Fig. 1, we note that thepercentage drop of 0max between the extinctionpoint and its adiabatic value is much larger thanthe corresponding drop of (-fw). Also the flametemperature approaches its limiting value in amore gradual manner than (-.fw), which indicatesthat the flame sheet model has a narrower range ofvalidity if the flame temperature, rather than theburning rate, is of interest.

Figure 3 presents the flame structures for twoDamkohler numbers; one far from the limit, theother at the extinction limit. When the flame is farfrom the limit, it can be seen from Fig. 3(a) thatthe flame temperature is higher and has a morepointed peak; the mass fraction of the fuel and the


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D x l O 5Fig. 2. Peak flame temperature vs. Damkohler number, Y O = .2324.

I 0 0 0

o x i d i z e r v a n i s h a t a b o u t t h e s a m e p o i n t . T h i s i sc l o se t o t h e p i c t u r e o f t h e f l a m e s h e e t m o d e l . A tt h e e x t i n c t i o n l i m i t , i t c a n b e s e e n f r o m F i g . 3 ( b )t h a t t h e f l am e t e m p e r a t u r e i s l o w e r a n d t h e p e a km o r e r o u n d e d . T h e r e g io n w h e r e b o t h f u el a n do x i d i z e r a m o u n t s a r e s ig n i f i ca n t is m u c h w i d e ri n d i c a t i n g a d i s tu r b e d r e a c t i o n z o n e .

F i g u r e 4 g i v e s t h e e x t i n c t i o n b o u n d a r y f o rseve r a l ac t iva t ion ene r g ie s and wi l l be used in then e x t s e c t i o n t o c o m p a r e w i t h t h e e x p e r i m e n t a ld a t a t o d e d u c e t h e v a l u e s o f t h e c h e m i c a l k i n e t i cp a r a m e t e r s o f t h e f l a m e .

F i g u r e s 5 a n d 6 s h o w t h e l i m i t i n g o x y g e n m a s sf r a c t i o n s a s a fu n c t i o n o f P ra n d t l a n d S c h m i d tn u m b e r s f o r a g i v e n D a m k o h l e r n u m b e r , a n di n d i c a t e t h e e f f e c t o f h e a t a n d m a s s tr a n s f e r o nt h e e x t i n c t i o n l im i t s. T h e e f f e c t o f t h e D a m k o h l e rn u m b e r v a r i a t i o n o n e x t i n c t i o n i s s h o w n in F i g . 4 .F o r a g i v en f u e l, h e n c e a f i x e d B , v a r y i n g t h eD a m k o h l e r n u m b e r c h a n g e s t h e p a r a m e t e r " a " .

C h a n g e s i n " a " a f f e c t b o t h h e a t a n d m a s s t r a n s f e rr a t e s s i m u l t a n e o u s l y t h r o u g h t h e r a t e o f c o n v e c -t i o n . I n o r d e r t o s t u d y i n d i v i d u a l l y t h e e f f e c t o ft h e h e a t t r a n s f e r r a t e , a l l o t h e r p a r a m e t e r s e x c e p tt h e P r a n d t l n u m b e r a r e f ix e d . D e c r e a s e d P r a n d t ln u m b e r s c a n b e r e g a r d e d a s i n c r e a s i n g t h e h e a tc o n d u c t i v i t y i f p a n d c p a r e h e l d c o n s t a n t . F r o mF ig . 5 i t can be s een tha t th i s inc r eases the l im i t ingo x y g e n i n d e x . C o m p u t e d r e s u l t s s h o w t h i s r e d u c e st h e f l a m e t e m p e r a t u r e a s w e ll . In c r e a s i n g t h e h e a tt r a n s f e r r a t e p a r a m e t e r a l o n e m a k e s t h e f l a m e le s sf l a m m a b l e . T h i s h a s t h e s a m e e f f e c t a s i n c r e a s i n gt h e h e a t l o ss p a r a m e t e r i n p r e m i x e d f l a m e t h e o r ya s s u g g e s te d ea r l ie r [ 1 6 ] . O n t h e o t h e r h a n d ,i n c r ea s i n g t h e m a s s t r a n s f e r r a t e b y i n c r e a si n g t h ed i f f u s i o n c o e f f i c i e n t ( d e c r e a s i n g t h e S c h m i d tn u m b e r ) , i s f o u n d t o i n c re a s e t h e b u r n i n g r a te ,i n c r e a s e t h e p e a k f l a m e t e m p e r a t u r e , a n d m a k e t h ef u e l m o r e f l a m m a b l e , se e F ig . 6 .

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a n d E x p e r i m e n t o f PM M Acp 0 .265 ca l /g rnc s 0 .35 ca l /gmL * 3 4 3 c a l / g mq * 6 3 0 0 c a l / g mT 300KT w 7 5 0 KUe 1.85 X 10 - -4 gm/c m-se cPe 1 .176 X 1 0 a g m / c m aPs 1 .18 gm/cm 3% 7 X 10 5 c a l / c m - s e c Kqa 0e 0

t o a f f e c t t h e f l a m m a b i l i t y o f f u e l s . U s i n g t h es i m p l i f i e d m o d e l d e s c r i b e d in t h e p r e v i o u s s e c t i o n ,t h e l i m i t i n g o x y g e n m a s s f r a c t i o n is c o m p u t e d a s af u n c t i o n o f th e R a d i a t i o n A b s o r p t i o n D a m k o h l e rn u m b e r , R a . A s e x p e c t e d , i n c r e a s i n g R a dec reasest h e l i m i t i n g o x y g e n i n d e x a s s h o w n i n F i g . 7 . T og iv e a p h y s i c a l f e e l f o r t h e r a d i a t iv e c o n t r i b u t i o n ,a t t h e D a m k o h l e r n u m b e r s p e c i f i e d i n F i g . 7 ,R a = 7 .7 1 X 1 0 4 r e p r e s e n t s a r a d i a ti v e c o n t r i b u -t i o n o f 3 5 % o f t h e t o t a l h e a t f l u x a b s o r b e d a t th ef u e l s u r f a c e . S u b s t i t u t i n g t h e d i m e n s i o n a l c o n -s t a n t s i n T a b l e 2 i n t o t h e d e f i n i t i o n o f R a a n dt a k i n g B t o b e 7 . 4 X 1 0 a 1 / s e e , t h e n R a = 7 . 7 1 X10- 4 c o r r e s p o n d s t o 1 c a l /c m Z s e c o f ra d i a t io nh e a t f l u x a b s o r b e d o n t h e s u r f a c e v s . 1 . 8 4 3 c a l /c m 2 s e c f r o m c o n v e c t io n .

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1 . E x p e r i m e n t a l S e t u pT h e e x p e r i m e n t a l s e t u p , s i m i la r t o t h e o n e u s e d inRef . 4 , i s shown schem at ica l ly in F ig . 8 . As com -b u s t i o n p r o c e e d s , t h e c y l i n d r i c a l l y s h a p e d P M M A( p o l y m e t h y i m e t h a c r y l a t e ) s a m p l e o f 1 2 . 7 m md i a m e t e r i s a u t o m a t i c a l l y f e d u p w a r d t o w a r d t h en o z z l e i n o r d e r t o m a i n t a i n a c o n s t a n t f u e l s u r f a c el e v el r e la t iv e t o t h e s u r r o u n d i n g p l a t e . T h e f u e lsur face d i s t ance above the p la te i s ad jus ted byv a r y i n g t h e l a s e r b e a m h e i g h t a b o v e t h e p l a t e .B l o c k a g e o f t h e l a s e r l i g h t s t r i k i n g t h e p h o t o -d i o d e b y t h e f u e l s u r f a c e c a u s e s t h e f e e d i n gp r o c e s s t o s t o p . A s f u e l p y r o l y s i s p r o c e e d s , th ef u e l s u r f a c e l e v e l d r o p s , t h e l a s e r b e a m i m p i n g e so n t h e p h o t o - d i o d e , a n d t h e f e e d i n g m e c h a n i s m i sa c t i v a t e d t o r e t u r n t h e f u e l t o i t s o r ig i n a l h e i g h t .

V e r t i c a l f u e l d i sp l a c e m e n t i n o n e o f t w o t i m ei n t e r v a l s i s d e t e r m i n e d b y m e a n s o f t h e s t e p p i n gm o t o r a n d a m p l i f i e r - c o u n t e r , e s t a b li s h i n g t h e f u e ls a m p l e l in e a r b u r n i n g r a t e .O n e r o t a m e t e r i s u s e d t o m e a s u r e ai r f l o w ra t e sa n d b o t h r o t a m e t e r s a r e u t i l i z e d t o c o n t r o l t h eo x y g e n m o l e f r a c t i o n o f t h e g a s m i x t u r e a n d j e tv e l o c i t y w h e n u s i n g m i x t u r e s o f n i t r o g e n a n do x y g e n .

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t h e n o z z l e e x i t . T h e s t a ti c p r e s s u re a t t h e n o z z l ee x i t i s t h e a m b i e n t p r e s su r e f o r a l l e x p e r i m e n t s .T h e n o z z e l - t o - p l a t e d i s t a n c e c a n b e v a r ie d t oa d m i t n e w f u e l sa m p l e s a n d t o s t u d y t h e e ff e c t o fj e t e n t r a i n m e n t ( f a r a w a y j e t ) a s w e l l a s e x t r e m ej e t - p l a t e i n t e r f e r e n c e ( c l o s e j e t ) .2 . E x p e r i m e n t a l P r o c e d u r eV a r i o u s i n v e s t i g a t io n s i n t o t h e n a t u r e o f t h es t a g n a t i o n f l o w f i e ld w e r e m a d e b e f o r e s t u d y i n g

t h e c o m b u s t i o n p r o p e r ti e s o f P M M A . B y r e p l a ci n gt h e f u e l f e e d i n g m e c h a n i s m b y a f la t p l a t e w i t hr a d i a l l y s p a c e d p r e s s u r e t a p s , t h e r a d i a l p r e s s u r ed i s t r ib u t i o n w a s m e a s u r e d a s a f u n c t i o n o f t h er a d i a l d i s t a n c e s q u a r e d , a l l o w i n g t h e d e t e r m i n a t i o no f t h e r e l a t i o n b e t w e e n j e t v e l o c i t y a n d t h es t a g n a t i o n - p o i n t v e l o c i t y g r a d i en t . It w a s f o u n dt h a t a = 0 . 8 1 V i e t w h e r e a i s i n 1 / s e e a n d l l j e t is i nc m / s e c . T h i s r e l a t i o n i s n e e d e d f o r c o m p a r i s o n o ft h e o r y a n d e x p e r i m e n t .


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A s p r e v i o u s l y m e n t i o n e d , t h e e f f e c t o f n o z z le -t o - p l a t e d i s t a n c e o n t h e f l o w f i e l d w a s s t u d i ed . F o rt h e r a n g e o f n o z z l e - t o - p l a te d i s t a n c e s , 1 . 2 7 c m t o3 . 8 1 c m , t h e a b o v e r e l a t i o n b e t w e e n a a n d V j e t ,and the burn ing ra te re su l t s , a re inva r ian t . An o m i n a l n o z z l e - t o - p l a t e d i s t a n c e o f 2 c m w a sc h o s e n f o r a ll t h e c o m b u s t i o n m e a s u r e m e n t s .

T h e j e t v e l o c i t y p r o f i l e a n d t h e t u r b u l e n c ei n t e n s it y a t t h e n o z z l e e x i t w e r e c h e c k e d b y m e a n so f a h o t - w i r e a n e m o m e t e r . T h e p r o f i l e w a s f la t t o1 % , a n d t h e t u r b u l e n c e l e v e l l e ss t h a n 1 % . U n i f o r mj e t v e l o c i t y is n e c e s s a r y t o p r o d u c e a p o t e n t i a lf l o w f i e l d o u t s i d e t h e p l a t e b o u n d a r y l a y e r , w h i c hi s a s su m e d i n t h e a n a l y si s . L o w t u r b u l e n c e l e v e l sa r e r e q u i r e d t o d e c r e a s e f l o w d i s t u r b a n c e s d u r i n gt h e P M M A e x t i n c t i o n d a t a c o l l e c t i o n .

B y a d j u s t i n g t h e h e i g h t o f t h e la s e r b e a m a b o v et h e p l a t e , a f l a t f la m e c a n b e e s t ab l i s h e d a b o v e t h e

f u e l s u r f a c e . A f l a t f l a m e i s p r e d i c t e d b y t h et h e o r e t i c a l a n a l y s is p r e s e n t e d i n t h e p r e v i o u s se c -t i o n s . F o r P M M A b u r n i n g i n a i r , t h e e n t i r e f l a m e i sb l u e i n c o l o r . A t h i g h e r o x y g e n m o l e f r a c t i o n s t h ec o l o r o f t h e f l a m e t u r n s t o y e l l o w a n d t h e r e i s c o n -s ide rab le loca l uns tead ines s in the f l am e zonew h i c h s e e m s t o b e c a u s e d b y f u e l s u r f a c e b u b b l i n ga n d b u b b l e b u r s t i n g .

T h e P M M A l i n e a r b u r n i n g r a t e i n a i r i s m e a s -u r e d a s a f u n c t i o n o f je t v e l o c i t y . A f t e r s t e a d ys t a t e c o m b u s t i o n i s a c h ie v e d a t v a r i o u s j e t v e l o c -i t i e s , t h e l i n e a r b u r n i n g r a t e s a r e o b t a i n e d b ym e a n s o f t h e a m p l i f ie r - c o u n t er s y s t e m .

T h e e x t i n c t i o n b o u n d a r y , F i g . 9 , i s f o u n d t od e p e n d o n j e t o x y g e n c o n t e n t a n d j e t v e l o c i t y .N u m e r o u s f l o w c o n d i t io n s ( o x y g e n m o l e f r a ct i o n sa n d j e t v e l o c i ti e s ) a re o b t a i n e d b y v a r y i n g th em a g n i t u d e a n d r a t io o f t h e o x y g e n a n d n i t r o g e n


10/14

6 4 J . S . T ' I E N E T A L .f l o w r a t e s . A n e x t i n c t i o n p o i n t i s a p p r o a c h e d b yi n c r e a si n g t h e j e t v e l o c i t y a t a g iv e n m o l e f r a c t i o n .F o r t h e p r e s e n t e x p e r i m e n t a l s e t u p th i s m e t h o dresu l t s in a fa s te r nozz le re sponse to changes in thef l o w r a t e s a t t h e r o t a m e t e r s a s c o n t r a s t e d t ov a r y i n g t h e m o l e f r a c t i o n a t a f i x e d j e t v e l o c i ty .T h e e x t i n c t i o n d a t a p o i n t s in a i r, c o r r e s p o n d i n g t oa n o x y g e n m o l e f r a c t i o n o f 0 . 2 1 , a r e a ls o s h o w n i nF i g . 9 . T h e a i r is s u p p l i e d f r o m a c y l i n d e r o f c o m -pres sed , pure a i r .3 . E x p e r i m e n t a l R e s u l t sT h e c o n s i d e r a b l e s c a t t e r i n t h e e x t i n c t i o n d a t as h o w n i n F i g . 9 i s n o t a t t r i b u t a b l e s o l e l y t o l a c k o fp r e c i s io n i n c o n t r o l l i n g t h e o x y g e n m o l e f r a c t i o no r j e t v o l o c i t y . F o r P M M A i n a i r t h e e x t i n c t i o nv e l o c i t y v a r i e s f r o m 1 5 0 t o 1 7 4 c m / s e c , a l t h o u g hg r e a t c a r e w a s t a k e n t o k e e p a l l e x t e r n a l c o n d i -t i o n s i d e n ti c a l a n d t h e a i r w a s t a k e n f r o m t h e s a m ec y l i n d e r . T h e s c a t t e r i s b e l i e v e d t o b e a r e s u l t o ft h e e x t r e m e s e n s i t i v i ty o f t h e e x t i n c t i o n e v e n t t od i s t u r b a n c e s [ 1 5 ] . T h e s e d i s t u r b a n c e s c o u l d b ed u e t o f l o w p e r t u r b a t i o n s , fu e l i n h o m o g e n e i t y ,a n d l i q u i d b u b b l e b u r s t s o n t h e P M M A b u r n i n gs u r f a c e d u r i n g t h e c o m b u s t i o n p r o c e s s .

F i g u r e 9 s h o w s t h e c o m p a r i s o n o f th e e x t i n c -t i o n d a t a o f P M M A a n d t h r e e t h e o r e t i c a l c u r v e sf o r v a r i o u s g a s . p h a s e a c t i v a t i o n e n e r g i e s t a k e nf ro m F ig . 4 . The theor e t i ca l re su l t s fo r E = 30k c a l / m o l e a p p e a r t o b e s t r e p r e s e n t t h e d a t a . N o t et h a t t h e s e d a t a p o i n t s l i e c lo s e t o t h i s c u r v e b u tm o s t l y o n t h e b u r n i n g s i d e o f th e c u r v e . T h i s isb a s e d o n t h e c o n s i d e r a t i o n t h a t a f l a m e c a n b eq u e n c h e d w i t h i n t h e s t e a d y - s t a t e e x t i n c t i o n l i m i tb y a d i s t u r b a n c e b u t i t c a n n o t s u s t a i n c o m b u s t i o no u t s i d e t h e l i m i t . I n p l o t t i n g t h e t h e o r e t i c a lc u r v e s , t h e r e f e r e n c e e x t i n c t i o n p o i n t i s c h o s e n t obe (V j )ex t = 170 cm /sec fo r a i r , so al l curves pas st h r o u g h t h i s p o i n t . D e s p i t e t h e d a t a s c a t t e r , g o o dcurve f i t t ing i s pos s ib le i f the da ta range i s widee n o u g h . T h e e x t i n c t i o n p o i n t s f o r t h e h i g h e ro x y g e n m o l e f r a c t i o n s d i f f e r e n t i a t e t h e t h r e et h e o r e t i c a l c u r v e s .S i n c e a l l b u t t h e a i r o x y g e n m o l e f r a c t i o n s a r ep r o d u c e d b y f l o w c o n t r o l f r o m s e p ar a t e o x y g e na n d n i t r o g e n g a s c y l i n d e r s , t h e m o l e f r a c t i o n s f o rt h e v a r i o u s e x t i n c t i o n p o i n t s h a v e e s t i m a t e d e r r o r sof ab ou t - -+0 .005 , sm a l le r than the obse rv ed s ca t t e r ,

which i s be l i eved to be a re su l t o f sm a l l d i s -t u r b a n c e s o n t h e e x t i n c t i o n p h e n o m e n o n i t s e l f .O n t h i s b a s is t h e a c t i v a t i o n e n e r g y i s a d j u s t e d s ot h a t t h e t h e o r e t i c a l e x t i n c t i o n l i n e l ie s o n t h e n o n -b u r n i n g e d g e o f th e e x t i n c t i o n d a t a , w i t h t h ee s t i m a t e d u n c e r t a i n t y o f t h e lo c a t i o n o f th i s ed g eb e i n g + 0 . 0 0 5 i n m o l e f r a c t i o n a n d a c o m p a r a b l yi n s i g n i f i c a n t u n c e r t a i n t y i n j e t v e l o c i t y , r e s u l t i n gi n a n a c t i v a t i o n e n e r g y o f 3 0 + - 5 k c a l / m o l e . T h ee x t i n c t i o n D a m k o h l e r n u m b e r i n a i r , D , a n d t h em o d i f i e d f r e q u e n c y f a c t o r , B , a r e f o u n d t o v a r yf r o m 1 0 .3 X 1 0 4 a n d 1 . 4 2 X 1 0 7 se c x t o 1 4 0 X104 and 19 . 3 X 107 see - 1 , r e sp e c t iv e l y , f o ra c t i v a t i o n e n e r g i e s f r o m 2 5 t o 3 5 k c a l / m o l e ,wi th va lues o f 38 . 3 X 104 and 5 . 27 X 107 see 1f o r E = 3 0 k c a l / m o l e . T h e c o n v e r s i o n t o c o n -v e n t i o n a l u n i ts o f v o l u m e / m o l e s e c f o r t h e f r e -q u e n c y f a c t o r c a n b e e a s i l y o b t a i n e d u s i n g E q . ( 3 ) .

F i g u r e 1 0 p r e s e n t s t h e P M M A b u r n i n g - r a te d a t aa n d t h e c o r r e sp o n d i n g t h e o r e t i c a l cu r v e ta k e nf r o m F i g . 1 . A c c o r d i n g t o F i g . 1 a n d E q . ( 1 0 ) , i f( - f w ) i s i n d e p e n d e n t o f D ( f l a m e s h e e t l i m i t) , th el i n e a r b u r n i n g r a t e i s p r o p o r t i o n a l t o t h e s q u a r er o o t o f " a " o r e q u i v a le n t ly t h e s q u a re r o o t o f t h eo x i d i z e r je t v e l o c i t y . T h e r a n g e o f d a t a i n F i g . 1 0l ie s o n t h e c u r v e d e n d o f t h e t h e o r e t i c a l c u r v e i nF i g . 1 , a n d b y t h e c o r r e s p o n d i n g p l o t i n F ig . 1 0 i sa l s o c u r v e d , e s p e c i a l l y n e a r t h e e x t i n c t i o n l i m i t .T h e e x p e r i m e n t a l d a t a n e a r t h e l i m i t a l s o s h o w as l ig h t d e p a r t u r e f r o m t h e s q u a r e r o o t r e l a t i o n s h ip .

I n c o n v e r t i n g t h e n o n d i m e n s i o n a l t h e o r e t i c a lr e s u lt s t o d i m e n s i o n a l o n e s , t h e v a l u e s o f t h ed i m e n s i o n a l q u a n t i t i e s a re t a k e n f r o m T a b l e 2 .4. Comparison with Previous StudiesA n u m b e r o f p r e v io u s s tu d i es e m p l o y e d P M M Aa s f u e l. T h e l i n e a r b u r n i n g r a t e s m e a s u r e d i n t h i ss t u d y a r e w i t h in + 1 5 % o f t h e v a l ue s r e p o r t e d i nR e f s . 3 a n d 4 . S o m e u n c e r t a i n t y i n t h e c o m -p a r i s o n w i t h t h e r e s u l t i n R e f . 4 e x i s t s b e c a u s e o ft h e d i f f e r e n t t y p e s o f j e t n o z z l e s u s e d a n d t h ef a c t t h a t t h e r e l a t i o n b e t w e e n t h e j e t v e l o c i t y a n dt h e b o u n d a r y - l a y e r v e l o c it y g r ad i e n t " a " h a s n o tb e e n r e p o r t e d i n R e f . 4 .

E x t i n c t i o n b o u n d a r y m e a s u r e m e n t s a n d t h ed e d u c t i o n o f t h e o n e - s t e p o v e r a ll g a s -p h a s e k i n e t i cc o n s t a n t s f o r P M M A h a v e b e e n r e p o r t e d i n R e f s .2 0 a n d 2 1 . A n e ar l ie r s t u d y [ 2 0 ] r e p o r t e d a n


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, . o ' : ' -

m~ / J E T V E L O C I T Y C M / S E C3 . 5 = m i , J8 5 I 0 0 I Z O 1 5 0 ; ) 0 0 2 5 0

I , , I7 o , ; o , ; o zooS T A G N A T I O N - P O I N T V E L O C I T Y G R A D I E N T~, ~/SEC

F i g . 1 0 . L i n e a r b u r n i n g r a t e o f P M M A v s s t a g n a t i o n - p o i n t v e l o c i t y g r a d i e n t : c o m p a r i s o no f e x p e r i m e n t a n d t h e o r y ( E = 3 0 k c a l / m o l e ) .

a c t i v a ti o n e n e r g y o f 2 0 k c a l / m o l e f o r a 5 0 / 5 0o x y g e n - n i t r o g e n m i x t u r e . A l a t e r s t u d y [ 2 1 ] ,us ing the opposed - j e t dev ice , r epo r t ed ac t iva t ione n e rg i es o f 4 2 k c a l / m o l e f o r o x y g e n m o l e f r a c t i o n sf r o m 1 7% t o 1 8 . 5 % a n d 5 5 k c a l/ m o l e fr o m 1 8 . 5 %t o 2 0 % . T h e d i s c r e p a n c y b e t w e e n t h e t w o s t u d i e swas a t t r i bu ted to th e neg lec t o f p rod uc t s d issoc ia -t i o n a t h i g h e r o x y g e n m o l e f r a c t io n s [ 2 1 ] i n t h etheory . The presen t i nves t iga t ion g ives 30 kca l /m o l e c o v e r in g t h e r a n g e o f 2 1 - 2 7 % o x y g e n m o l ef rac t ion . I t i s no t c l ea r whe the r t he neg lec t o fd i ssoc ia t ion i s t he on ly reaso n for the d i f fe ren tva lues of t he ove ra l l ac t iva t ion ene rgy repor t ed .I n c o m p l e t e c o m b u s t i o n m a y a l so b e i m p o r t a n t . Atheore t i ca l work inc lud ing mul t ip l e gas-phasereac t ions cou ld c l a r i fy th i s s i t ua t ion .I V . D I S C U S S I O NT h e t h e o r e t i c a l e x t i n c t i o n p r o b l e m o f a d i f f u s io nf l a m e i n t h e s t a g n a t i o n p o i n t b o u n d a r y l a y e r o f aso l id fue l has been so lved by a t ime-march ingf i n i t e d i f f e r e n c e s c h e m e . T h i s m e t h o d p r o v i d e sthe de t a i l ed s t ruc ture of t he f l ame and can bee x t e n d e d t o m u l t i p le - s t ep c h e m i c a l r ea c t i o n s .

I n t h e p r e se n t t h e o r y , th e e x t i n c t i o n b o u n d a r yi s d e t e r m i n e d a s a f u n c t i o n o f th e a m b i e n t o x y g e n

m o l e f r a c t i o n s a n d t h e D a m k o h l e r n u m b e r , d e -f ined a s the ra t io o f t he chemica l - reac t ion f re -q u e n c y f a c t o r a n d t h e b o u n d a r y - l a y e r v e l o c i t yg r a d ie n t " a " . T h e r e i s n o res idence t ime involvedin the theore t i ca l ana lys i s o f t he s t agna t ion po in tf low s ince the re i s no l ength sca l e in t roduced , andthe quench ing caused by inc reas ing the ve loc i tygrad ien t " a" i s t he re su l t o f t he rma l and d i f fu-s iona l e f fec t s .

Because the theo ry a ssumes one -s t ep chemica lk ine t i c s , i t is no t poss ib l e to t r e a t i ncom ple t e com-bus t ion . Gas samplings in Refs . 19 and 22 ind ica t etha t a number of chemica l spec ie s ex i s t i n t hePMMA gas f l ame , i n pa r t i cu la r l arge am oun t s o fCO. Thi s ra ise s the poss ib i l i t y t ha t i ncom ple t e con-v e r s i o n o f C O t o C O 2 m a y b e i m p o r t a n t i n d e t e r -min ing the ex t inc t ion l imi t . A mul t i - s t ep k ine t i cmode l i nc lud ing these chemica l s t eps i s t he nex tl e v e l o f c o m p l e x i t y t o w h i c h t h e a b o v e m o d e lsh o u l d b e e x t e n d e d t o u n r a v e l th e se u n c e r t a i n ti e s .

A P P E N D I XNum er ica l Ana lys i sT h e sy s t e m o f E q . ( 1 ) a n d t h e b o u n d a r y c o n -d i t i ons Eqs . (5 - 6) i s so lved by a march ing- in-


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6 6 J . S . T ' I E N E T A L .t i m e t e c h n i q u e t o o b t a i n t h e s t e a d y - s ta t e so lu -t ion . F i r s t , a f i c t i t i ous uns t eady t e rm i s added toEq. (1 ) , y i e ld inga0 1 a20 aoa t P r a n a f~ -~ - q D Y F Y e - E / = O ( A I )

a Y F 1 a 2 Y e f a Y F +"~"0a t S c a n z a n

( A 2 )a Y o 1 a 2 Y o a Y o- - " - - f ~ + N o D Y , Y o e -m / O = 0a t S c a n 2 a n

( A 3 )a U a z U a U 2f - - + ~ (U z - - 0 ) = 0 , (A4)a t a n 2 a n 1 + ew h e r e U==-f/2 t h a t

f o /f= 2Ud n +fw. ( A 5 )

A n e x p l i c it s c h e m e i s u se d f o r th e u n s t e a d yte rm s; a cen t ra l d i f fe re nce i s used for t he d i f fus ivet e r m s ; a n d a n u p w i n d s c h e m e i s u se d f o r t h e c o n -vec t ive t e rms. In the chemica l reac t ion t e rm, Y Fa n d Y o a re t rea t ed impl i c i t l y in Eqs . (A2) and(A3) bu t e xp l i c i t l y in E q . (A1) , 0 is t r ea t edexpl i c i t l y in a l l t h ree equa t ions (A1-A3) . Thei n t e g r a t i o n o f E q . ( A 5 ) i s p e r f o r m e d u s i n g t h eTrapezoida l ru l e fo r t he f i r s t s t ep and thenc h a n g e d t o t h e S i m p so n ' s r u le [ 1 7 ] f o r t h e r e st o fthe s t eps . The f in i t e d i f fe rence schem e for Eqs .(A1 , A2, A4) i s t he fo l lowing:

A t 1OlJ+ l m oiJ "1"Ar/a pr (01+11 -- 20i /+ 0 i_1 .t )A t

+ - - t J(OR - On)An+ A t q D ( Y e ) t /( Y o ) ~ i e x p ( - E / O J ) ( A 6 )

( Y F ) i J 1 = r A t 1' ( y F J + - - - - - - 2 ( Y FAt/z S cA t

+ ( r r ) t - l q +

- - ( Y F ) L ] } / ( I + A t D ( Y o ) ,X e x p ( - - E I O O } ( A 7 )

AtU i j + l = U i j + A n 2 ( U I + I - - 2 U i J -I- U / _ l J )

A t 2 ~ f l J ( U R - - U L ) - - ~ [ ( U i J ) - - O i J ] ,zaw l e

( A 8 )

where index i is fo r space gr id , ] i s fo r t empera lgrid. Also,O R --- Oi , 0 L = 01_1 / i f j ~ / < 0O R = O i + l j , 0 L = 0~ i f ~ / > 0.Express ions for (Ym)L, (YF )R, UR, UL a res imi la r. The f 'mi t e d i f fe rence schem e fo r Eq . (A 3)i s s imi la r t o Eq . (A7) . E qua t ion (5 ) i s expressed by0 2 J + 1 - - O o J + l

2 A nF= P r ( _ f j + i ) 101./+1

_RaD /z , ( A 9 )where i = 1 i s t he so l id sur face , 0o i s the ima gina ryt e m p e r a t u r e a t o n e s t e p b e y o n d t h e g as p h a seb o u n d a r y , w h i c h w a s e l im i n a t e d f r o m E q . (A 1 ) b yeva lua t ing Eq . (AI ) a t i = 1 . Thi s p rocedureenables us to eva lua te the grad ien t s a t t heb o u n d a r y u s i n g a c e n t r a l d i f f e r e n c e a n d i t i sb e l i e v e d t o y i e l d b e t t e r a c c u r a c y [ 1 8 ] . E q u a t i o n( 6 ) f o r Y F ' a n d Y o ' i s t r ea t ed in a s imi l a r way .E q u a t i o n ( A 9 ) e n a b le s u s t o f i n d f r + l w h i c h i s


13/14

B O U N D A R Y L A Y E R C O M B U S T I O N A N D E X T I N C T I O N 6 7u se d i n E q . ( A 5 ) f o r t h e i n t e g r a t i o n o f f z + t . T h em a r c h i n g s c h e m e is t h u s c o m p l e t e d .

In th i s work A~/= 0 .15 and At = 0 .005 . Ha lv ingthe spa t i a l s t ep s i ze was t e s t ed ' i n a couple of casesand fou nd to mak e l ess t han a 1% d i f fe rence in f wand a neg l ig ib l e e f fec t on the prof i l e s . The t em-pera l s t ep s ize used i s c lose to the m axim um tha t i sa l lo w e d f o r a s t ab l e n u m e r i c a l s c h e m e . S t u d y n o win progress ind ica t e s a much l a rge r t ime s t ep s i zec a n b e u se d i f t h e i m p l i c it s c h e m e i s e m p l o y e d .

T o s t a r t e a c h c o m p u t a t i o n , i n i t ia l p r o f d e s h a v et o b e c h o se n . F o r p o i n t s f a r a w a y f r o m t h e e x t i n c -t ion l im i t , a s t eady-s t a t e burn ing s o lu t ion can beo b t a i n e d i n d e p e n d e n t o f t h e d e ta i ls o f t h e i n it ia lp rof i l e s a s l ong as they a re ene rge t i c enough . Asthe l imi t i s approached , i n i t i a l p rofdes have to bec l ose e n o u g h t o t h e c o n v e r g e d b u r n i n g so l u t i o n ;o the rwise an ex t inc t ion so lu t ion wi l l r e su l t . Th i sre f l ec t s t he phys i ca l na ture of t he f l amm abi l i t yl imi t : t he nea r - l imi t f l ame i s ve ry sens i t i ve tod i s tu rbances .

Convergence to a s t eady-s t a t e so lu t ion ( f ivedigits in f w ) i s usua l ly ob ta ined a t t equa l t oa b o u t 1 .5 w h e n a w a y f r o m t h e e x t i n c t i o n li m i t .Near the l imi t , i t t akes a l onger t ime to reachconvergence , s ince the ex t inc t ion l imi t i s a neu-t raUy s t ab le po in t . Convergence nea r t he l imi tt e n d s t o b e o sc i l la t o r y r a t h e r t h a n m o n a t o n i c a s i ti s a w a y f r o m t h e l i m i t.

T h e a u t h o r s a r e i n d e b t e d t o D r . W . S . B l a z o w s k if o r h i s c o m m e n t s i n t h e c o u r s e o f th i s r e se a r ch , toD r . C C F e n g f o r h i s c a r e f u l c h e c k o f t h e a n a l y si sa n d t h e c o m p u t e r p r o g r a m , a n d t o D r s . P . C H . .C h e n a n d X P . S h e n g f o r t h e i r a d v i c e o n t h en u m e r i c a l s c h e m e .

N O M E N C L A T U R Ea S tagna t ion -poin t f l ow ve loc i ty grad ien t ,

Ue = ar , 1 / t i m eB Mo di f i ed f requ enc y fac to r fo r t he gas-phase

chem ica l reac t io n , see Eq . (3 ) , 1 / t ime/~ See Eq . (2 ) , ( /~T) i s t he convent io na l f re -

q u e n c y f a c t o r w h i c h h a s t h e u n i t v o l u m e /( m o l e ) ( t i m e )

cp Spec i f i c hea t o f t he gascs Spec i f i c hea t o f t he so l idD = ( 2 / l + e ) B / a , D a m k o h l e r n u m b e r , n o n -d i m e n s i o n a lE Ac t iva t ion ene rgy of t he gas phase reac t ion

n o n d i m e n s i o n a l i z e d b y R T ef M o d i f i e d s t r e a m f u n c t i o nf w N o n d i m e n s i o n a l b u r n i n g r a t e , s e e E q . ( 1 0 )L L a t e n t h e a t o f t h e c o n d e n se d f u e l n o n -

d i m e n s i o n a li z e d b y c p TeL * L a t e n t h e a t o f t h e c o n d e n se d f u e l a t r e f e r-

e n c e t e m p e r a t u r e a b so l u t e z e r o , h e a t /m a ssr h M a s s b u r n i n g ra t e o f t h e c o n d e n se d f u e l ,

m a ss / ( a r e a X t i m e )p P ressureP r = lacp/~,, P r a n d tl n u m b e rQ Non dimen siona l hea t l oss t o the so l id , see

E q . ( 8 )q = q * / c p T e , n o n d i m e n s i o n a l h e a t o f c o m -b u s t i o nqa H e a t f l u x a b so r b e d a t t h e f u e l su r f a c e f r o m

e x t e r n a l r a d i a t i o n so u r c e , h e a t / ( a r e a ) ( t i m e )R Unive rsal gas cons t an tN o S t o i c h i o m e t r i c o x i d i z e r t o f u e l m a ss r a t ioR a R a d i a t i o n a b so r p t i o n D a m k o h l e r n u m b e r ,

see Eq . (9 )R t R a d i a t i o n e m i s s io n D a m k o h l e r n u m b e r , s e eE q . ( 9 )r D i s t ance pa ra l le l t o t he fue l sur faces See Eq . (1 )Sc = # / p D , S c h m i d t n u m b e rt T i m e n o n d i m e n s i o n a l i z e d b y 1/aT T e m p e r a t u r eu Ve lo c i ty pa ra l l el t o t he fue l sur faceVjet Je t ve loc i ty a t t he noz z le ex i t i n t he expe r i -m e n tw Nond i rnens iona l reac t ion ra t e , see Eq . (4 )W M o l e c u l ar w e i g h t

y P e r p e n d i c u l a r d i s ta n c e m e a su r e d f r o m t h efue l sur faceYi Mass f rac t ion of spec ie ia Therm al d i f fus iv i tye See Eq . (1 )es Tota l hemisph er i ca l emiss iv ity o f t he fue lsur facer / Non dimen siona l d i s t ance norm al to the fue lsur face , see Eq . (1 )


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68 J. S. T'IEN ET AL.0 Temperature nondimensi onalizedby Te0w Nondimens ional wall temperature (fuel

surface)Oe Nond imen sion al tempera ture far inside the

condensed fuelp Viscosityp Densityo Stefan-Boltzmann consta ntD Diffusion coefficient

S u b s c r i p te Edge of boun dary layerF FuelM MixtureO Oxidizerw wallS u p e r s c d p t' - ~ c l / d ~* Dimension~ quantity

R E F E R E N C E S1. Pot ter, A. E . and B ut ler , J . N. , A R S Z 29, 54,1959.2. Tsuji, H. and Yamaoka, I., Thirteenth Symposium

on Combustion, The Combustion Institute, 1971,p. 273.3. Blazowski, W. S. and McAlevy, R. F., An Investiga-tion of the Combustion Characteristics of SomePolymers Using the Diffusion Flame Technique,Stevens Inst itute of Technology, Tech. Rep. ME-RT711004 (1971).4. Holve, D. J. and Sawyer, R. F., Fifteenth Interna-tional Symposium on Combustion, The CombustionInstitute, 1975, pp. 351-362.

5. Kent, J. H. and Williams, F. A., Fifteenth Sympo-sium on Combustion, The Combustion Institute,1975, p. 315.6. Zeldovich, Y. B., Tech. Phys., Moscow, 1949, 19,p. 1199; translated as U.S.N.A.C.A. Tech. Memo1296.7. Spalding, D. B.,ARSJ. 31,763, (1961).8. Fendell, F. E.,J. of Fluid Mechanics 21,281 (1965).9. Linan, A., Act a Astronautics 1, 1007 (1974).10. Ablow, C. M. and Wise, H. Combust. Flame 22,23-24 (1974).11. Krishnamurthy, L., Williams, F. A. and Seshadri,K., Combust. Flame 26, 363 (1976).12. Krishnamurthy, L. and Williams, F. A., Acta Astro-

nautics 1,711-736 (1974).13. Tsuji, H. and Matsui, K., Combust. Flame 26,283-297 (1976).14. Kirkby, L. L. and Schmitz, R. A., Combust. Flame10, 205 (1966).15. Baliga, B. R. and T'ien, J. S., AIAA J. 13, 1653-1656 (1975).16. T'ien, J. S.,J. Fire Flammability 6, 101-104 (1975).17. Lambert, J. D., Computational Method in OrdinaryDifferential Equations, John Wiley and Sons (1973).18. Smith, G. D., Numerical Solution in Partial Differen-tial Equations, Oxford University Press (1965).19. Fenimore, C. P. and Jones, G. W., Combust. Flame10, 245-301 (1966).20. Krishnamurthy, L. and Williams, F. A., FourteenthSymposium (International) on Combustion, TheCombustionInstitute, Pittsburgh, 1151-1164 (1975).21. Seshadri, K. and Williams, F. A., Effects of CFaBron Counterflow Combustion of Liquid Fuel withDiluted Oxygen, in Halogenated Fire Suppressants(R. G. Gann, Ed.) ACS Symposium Series 16,American Chemical Society, Washington, D. C.,1975, pp. 149-182.22. Williams, F. A., in Annual Conference on FireResearch, National Bureau of Standards, August1977.

Received 2 November 1976; revised 18 January 1978

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