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On the structure of laminar diffusion flames

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(1)Thesis by. I n P a r t i a l R l l f i l l r n e n t o f t h e Requirements For t h e Degree of'. Aeronautical Engineer. C a l i f o r n i a Institute o f Technology Pasadena, C a l i f o r n i a 1963.

(2) The author w i s h e s t o express h i s a p p r e c i a t i o n t o P r o f ' e s s o ~Frank E. Marble for his c r i t i c i s m and p;laa--cse. i n p r e p a r i n g t h i s work.. Thanks are due t o ldiss Sari(-iy. A s c h e r f o r h e r t y p i n g o f t h e manuscript.. This s t u d y was. i n i t i a t e d a t t h e Institute .:aci;,.al d e TBcnica A e r o n d u t i c a ,. Madrid, a s p a r t of a & r e n e r a i. investigation on laminar flames,partially spor,sorae6 by. the IISAi7 Office of Aerospace hesearch thraough i t s Ehrsp e a n O f f i c e u n d e r Grant AF EOAH 62-91. The o p p o r t u n i t y for t h e a u t h o r ' s graduate. st^?:: zit. Cal Tech was p r o v i d e d by a joint NASA and COPd3S. (..L;;.*;-. pean P r e p a r a t o r y Commission for Space ~ e s e a r c h )fellowsh;p..

(3) The structure of l a m i r i a r diffusion f l a n e s is. :~:IL-. lyzed in the lj- niti in^ case of l ~ i r p ; e ,a l t h o u g h Y i r i i t e , r e a c t i - ~ nr a t e s .. It is shown that the chemical. reaction tal e s place only in a very thin region or "chemical boundary l a y e r v. where convection effects may be n e g l e c t e d .. Then t h e. t e r n p e r z t u r e and mass fraction distributions within the. r e : > c t i o n zone are obtained analytically, ~. The fl ame position, rates. or. fuel consu.rnptlon, a z d. temperature and ~oncentrationdistributions outside of' the r t l a c t l o n zone may be obtained by ~ z s i n gthe assum2zlofi of infinite reaction rates,. For large Reynolds numbers mixing and c o ~ b u s t i o nt b k e place i n boundary layers and free nixing layers.. And. again analytical solutfons are obtained for the tempera t u r e and mass TractSon d i s t r i b u t i o n s outsI.de of' the. reaction zone..

(4) PART. PAGb. Acknowledgements Abstract. ii iii. Table of Contents. iv. Yomenclature I.. 11.. Introduction General E q u a t i o n s. 5. 11.. 1.. Equation o f S t a t e. 6. 11.. 2.. E q u a t i o n of C o n t i n u i t y flor t h e mixture. 6. 11.. 3,. E q u a t i o n of' Mass C o n s e r v a t i o n f o r the Species. 6. 11.. 4.. Momentum E q u a t i o n. 11.. 5.. Energy E q u a t i o n. IT.. 6.. Dimensionless Form of t h e E q u a t i o n s. D i s c u s s i o n of t h e E q u a t i o n s and L i m i t i n g Cases. 111.. 1.. General D i s c u s s i o n. 111.. 2.. Burke-Schumann S o l u t i o n. S t r u c t u r e of t h e R e a c t i o n Zone IV.. 1.. The Chemical Boundary Layer. IV.. 2.. T h e S o l u t i o n o f - t h e Chemical Boundary Equations. IV.. 3.. D i s c u s s i o n of t h e S o l u t i o n. The E x t e r n a l S t r u c t u r e of Laminar D i f f u s i o n FIame s V.. 1.. Large Reynolds Number Case. I n v i s c i d Equations.

(5) VI.. V.. 2.. Mixing Gayer Equations. V.. 3.. L o c a l Similarity Approximation. V.. 4.. Particular Cases. R6s1xrn6 References Appendix. A. Appendix B. Appendix C. Egures.

(6) The f o l l o v ~ i n gi s a list o f t h e most i m p o r t a n t s y m b o l s used i n t h i s pap.er. Parameter g i v e n by 44 , t h a t measures t h e d e v i a t i o n s from t h e 3urke-Schumann s o l u t i o n Specf f i c h e a t a t c o n s t a n t p r e s s u r e Diffusion coefficient A c t i v a t i o n e n e r g y of t h e chemical r e a c t i o n Dimensionless s t r e a m f i n c t i o n Mass f r a c t i o n of s p e c i e s. i. Some o v e r a l l c h a r a c t e r i s t i c l e n g t h Mean m o l e c u l a r mass Mass r a t e of' f u e l consumption p e r u n i t flame surface Frand t l number Pre ss u r e Heat r e l e a s e d p e r u n i t mass of f u e l TTniversal gas c o n s t a n t. Re. Reynolds number. Sc. Schmidt number. 7-. Temperature. 74 T C. A d i a b a t i c flame t e m p e r a t u r e g i v e n by 24 Temperature a t t h e i d e a l flame s u r f a c e 5. k. Characteristic chemical time defined by 15. tm. Characteristic mixing time, t m =. s&/,. L.

(7) U. C h a r a c t e r i s t i c o v e r a l l velocity. U,V. Velocity components in boundary l a y e r c o o r -. <. dinates &. V. Velocity v e c t o r. A. Diffusion velocity of species i. Vdi. wL Idass production rate,per u n i t volume,of species i. X,Y . I r . ,. X. Mixing b o u n d a r y layer c o o r d i n a t e s Position vector. X,y. Chemical boundary l a y e r coordinates. f3. Universal f u n c t i o n g i v i n g %he t e m p e r a t u r e d i s t r i b u t i o n w i t h i n the r e a c t i o n zone. Defined by 39. &a SG. Mixlng length. ,=s,. ?,DO/&. Characteristic t h i c k n e s s of the r e a c t i o n zone given by 41. 7,.. D i m e n s i o n l e s s d i s t a n c e normal t o t h e mixing layer. 8. Non-dimensional t e m p e r a t u r e. e, =. E/U(~-T~). /A. Viscosity coefficient. 3. , 0 S(T--T~)/(%~~). Stoichiometric ratio species 2 t o P.

(8) 5. Xon-dimensional d i s t a n c e a l o n g the mixing layer Density. C CI L. S t r e s s tensor. Subscripts : i,2,3. I n d i c a t e f u e l , o x i d i z e r and p r o d u c t s r e s p e c tively. F. I n d i c a t e s c o n d i t i ~ o n sa t t h e f u e l e x i t. 0. I n d i c a t e s c o n d i t i o n s on t h e o x i d i z e r s i d e of the f'lame, f a r from t h e flame. f. I n d i c a t e s c o n d i t i o n s a t t h e flame s u r f a c e f o r infinite reaction rates. The a s t e r i s k i s used f o r t h e non-dimensional v a r i a b l e s i n t r o d u c e d i n s e c t i o n 11, 6 ,.

(9) . D i f f u s i o n flames a r e obtained when t h e r e a c t i n g species a r e i n i t i a l l y separated.. Combustion and mixing. takes place simultaneously. I n t h e s e flames t h e r e a c t i o n zone s e p a r a t e s t h e t v ~ r e a c t i n g s p e c i e s which d i f f u s e , through i n e r t g a s e s and combustion p r o d u c t s , from e a c h s i d e towards t h e f l a m e . The r e a c t i n g s p e c i e s burn very r a p i d l y a s t h e y r e a c h t h e r e a c t i o n zone; t h e r e b y t h e combustion v e l o c i t y i s g e n e r a l l y c o n d i t i o n e d t o t h e a c c e s i b i l i t y of t h e. s p e c l e s t o t h e r e a c t i o n zone; o r i n o t h e r words, t o t h e i r f a c i l i t y t o d i f f u s e a c r o s s t h e i n e r t g a s e s and combustion p r o d u c t s ,. I t seems t h a t we can a r r i v e a t a d e s c r i p t i o n of some of' t h e most important f e a t u r e s of d i f f u s i o n flames by u s i n g t h e assumption, f i r s t i n t r o d u c e d by Wlrke and. ~ c h u r n a n n ( l ) ,o f i n f i n i t e l y f a s t r e a c t i o n r a t e s .. Then. t h e a c t u a l zones of' eombust%on become i n f i n i t e l y t h i n ,. and t h e mixing p r o c e s s a l o n e becomes r e s p o n s i b l e f o r. t h e r a t e of' burning and f o r flame l o c a t i o n and s i z e . Burke and Schumann have s u c c e s s f u l l y used t h e i r. assumption f o r t h e calcuILation of t h e shape and l e n g t h o f t h e laminar d i f f u s i o n flame formed when a f u e l j e t. discharges within a tube.. I n t h i s t u b e a n a i r stream. moves w i t h t h e same v e l o c i t y a s t h e f u e l j e t .. T h e same.

(10) assumption h a s been u t i l i z e d by B o t t e l and 3awthorne ( 2), Wbhl, Casley and ~ a p p ( ~ Yagi ) , and s a j d 4 ) , and ~arr(5),. f o r t h e p r e d i c t i o n o f t h e l e n g t h of onen f l a m e s , b o t h l a m i n a r and t u r b u l e n t .. Through rudimentary approxima-. t i o n s they o b t a i n an e x p r e s s i o n f o r t h e flame l e n g t h c o n t a i n i n g a n unknown f'unction; t h i s t h e y determine e m p i r i c a l l y from t h e r e s u l t s of t h e i r experiments. ~. a. ~ h a(s ~ c a l)c u l a t e d , by u s i n g Burke Schumann assump-. t i o n , t h e shape and c h a r a c t e r i s t i c s of t h e l a m i n a r d i f f'usion flame obtained when a f u e l j e t d i s c h a r g e s i n t o t h e open atmosphere. The i n f i n i t e r e a c t i o n r a t e assumption h a s a l s o been u t i l i z e d ( 7 ) 9 ( 8 ) FOP t h e s t u d y of d i f f b s i o n flames i n boundary l a y e r s .. I n a d d i t i o n , a n e x t e n s i v e l i t e r a t u r e e x f s t s on che a p p l i c a t i o n of t h e assumption t o f u e l d r o p l e t combustion. The Burke-Schumann assumption e l i m i n a t e s chemical. k i n e t i c s from t h e p r o c e s s , s i m p l i f y i n g t h e governing e q u a t i o n s and t h e i r s o l u t i o n .. However, t h i s s o l u t i o n. does n o t provide t h e c r i t e r i o n Tor t h e e x t i n c t i o n of t h e flame, o r f o r t h e v a l i d i t y of t h e assumption and solutLon. ~eldovich(h ~ a)s t a k e n i n t o c o n s i d e r a t i o n t h e f i n i t e t h i c k n e s s o f t h e r e a c t i o n zone t o e x p l a i n t h e blowing-off phenomenon.. Similar s t u d i e s have been performed by.

(11) s p a l d i n g ( l O )9 (11)9 ( I 2 )w i t h t h e purpose of r e l a t i n g t h e f ~ z e lconsumption r a t e p e r u n i t a r e a a t e x t i n c t i o n and t h e f u e l consumption r a t e p e r unlt a r e a i-, a p r d -. mixed flame.. For a g e n e r a l d e s c r i p t i o n o f t h e d i f f u s i o n f'lanes s e e , f o r example, t h e review p a p e r s by ~ a r r ( l 3 )and VJohl and ~ h i ~ m a n n ( l where ~), d a t a and b i b l i o g r a p h y on t h e s u b j e c t can be found.. We aim I n t h i s work t o ahow t h e e f f e c t s of f i n i t e. chemical r e a c t i o n r a t e s on t h e s t r u c t u r e of l a m i n a r d i f f u s i o n flames.. I n o r d e r t o d o s o , we w i l l s t u d y. c e r t a i n l i m i t i n g c a s e s , i n which simple a n a l y t i c a l s o l u t i o n s can be o b t a i n e d ,. i'Ve w i l l l i m i t o u r s e l v e s t o t h e. s t u d y of one s t e p chemical r e a c t i o n s i n which t h e forward r e a c t i o n i s dominant,. We s h a l l show t h a t f o r l a r g e r e a c t i o n r a t e s t h e chemical r e a c t i o n t a k e s p l a c e only in a very t h i n r e g i o n. or "chemical boundary l a y e r " .. T h i s h a s already been. shown(l5) i n t h e simple case o f t h e mixing and combustion of t w o parallel streams o f f u e l a n d o x i d i z e r s moving w i t h . t h e same v e l o c i t y .. There convection e f f e c t s may be. n e g l e c t e d compared w i t h t h e much more i m p o r t a n t d i f f ' u s i o n conduction and chemical r e a c t i o n e f f e c t s .. The governing. equations reduce in this case t o ordinary d i f f e r e n t i a l equations.. The k i n e t i c s of the r e a c t i o n appear i n t h e.

(12) s o l u t i o n ; but the temperatures are c l o s e t o t h e adlab a t i c flame t e m p e r a t u r e , and i n t h i s range of temperat u r e s t h e concept of an o v e r a l l k i n e t i c scheme has been. found by Levy and ~ e i n b e r ~ ( l 6 t o) be v a l i d . The s o l u t i o n w i t h t h e assumption of i n f i n i t e r e a c t i o n r a t e s (which we s h a l l c a l l t h e Burke-Schumann s o l u t i o n ) r e p r e s e n t s t h e t r u e s o l u t i o n o u t s i d e of t h e r e a c t i n n zone.. I t mag a l s o be used t o c a l c u l a t e t h e flame p o s i -. t i o n and f u e l consumption p e r u n i t flame a r e a , If' t h e Reynolds number, based on some o v e r a l l char-. a c t e r i s t i c l e n g t h , i s l a r g e ; mixing and combustion w i l l t a k e p l a c e o n l y i n a v e r y t h i n r e g i o n o r mixing l a y e r , where. boundary l a y e r a p p r o x i m a t i o n s may be u s e d ( 1 7 ) (18)9 ( 1 9 ) .. The mixing l a y e r l o c a t i o n a n d g e n e r a l f l o w c h a r a c t e r i s t l c s o u t s i d e of t h e mixing l a y e r may be determined by u s i n g t h e invf s c i d f l o w e q u a t i o n s ,. However, we must. a l l o w f o r t h e e x i s t e n c e of discontinuities i n t h e v e l o -. c i t y , d e n s i t y , t e m p e r a t u r e , and mass f r a c t i o n d i s t r i b u t i o n s within the flow f i e l d , I n F i g u r e 1 t h e t e m p e r a t u r e and mass f r a c t i o n d i s t r i b u t i o n s , a s o b t a i n e d by different l i m i t f n g a s s u m p t i o n s , a r e schematicalhy represented.

(13) Vie s h a l l b e g i n by w r i t i n g t h e g e n e r a l e q u a t i o n s J. g o v e r n i n g t h e s t e a d y laminar flow of a r e a c t i n g g a s. mixture (. ). .. We w i l l use t h e assumption t h a t. t h e f l u i d may be considered a s a continuous medium formed by a mixture of p e r f e c t g a s e s ,. Only t h r e e s p e c i e s w i l l be considered:. &%el, oxi-. For t h e sake of s i m p l i c i t y , any. d i z e r s , and p r o d u c t s .. i n e r t s p e c i e s p r e s e n t w i l l be considered a s p r o d u c t s . Besides t h e u s u a l dependent v a r i a b l e s of o r d i n a r y f l u i d mechanics, i . e .. and t e m p e r a t u r e + ,. d. velocity V, pressure. density9. ,. t h r e e new v a r i a b l e s , t h e mass f r a c t i o n s. of t h e r e a c t a n t s p e c i e s , e n t e r .. T h e r e f o r e , t h r e e new. e q u a t i o n s , s t a t i n g t h e mass conservation law for each of t h e s p e c i e s , must be added t o t h e fundamental equa-. En a d d i t i o n , t h e r e P a t f ons. f l u i d mechanics.. tions. between t h e t r a n s p o r t parameters and mass f r a c t i o n s , t e m p e r a t u r e , and p r e s s u r e of' t h e mixture w i l l be r e q u i r e d . We s h a l l u s e s u b s c r i p t 1 f o r f u e l , 2 f o r o x i d i z e r ,. and 3 f o r t h e p r o d u c t s . w i l l be w r i t t e n. The mass f r a c t i o n s of s p e c l e s. Ki= 5;/$'. The t h r e e mass f r a c t i o n s o b v i o u s l y s a t i s f y t h e r e l a t f on.

(14) 11.. 1.. Equation o f S t a t e. If' t h e f l u i d i s considered a s a mixture of p e r f e c t. g a s e s t h e e q u a t i o n of s t a t e i s a s f o l l o w s. where. R. i s t h e u n i v e r s a l c o n s t a n t of t h e g a s e s , and. i s t h e mean molecular mass.. I n order t o simplify the calculations w e w i l l use a mean c o n s t a n t v a l u e f o r. M. .. This approximation i s. j u s t i f i e d when t h e molecular masses of t h e s p e c i e s a r e n o t very d i f f e r e n t o r when t h e r e a c t a n t s a r e very d i l u t e . Then. M. M3. I n any c a s e t h e r e s u l t s w i l l n o t be e s s e n t i a l l y changed by c o n s i d e r i n g. 11.. 2.. M. as v a r i a b l e .. Equation of C o n t i n u i t y f o r t h e Mixture. This sFrnply s t a t e s t h e law o f mass c o n s e r v a t i o n .. 11.. 3.. Equations of Mass Conservation f o r t h e S p e c i e s. These s t a t e t h a t the mass q u a n t i t y of' each c o n s t i t u e n t. e n t e r i n g u n i t volume p e r u n i t time, e i t h e r due t o conv e c t i o n o r d i f f u s i o n , e q u a l s t h e mass q u a n t i t y of t h e c o n s t i t u e n t d i s a p p e a r i n g a s a consequence of t h e chemi c a l reaction, These e q u a t i o n s a r e a s f o l l o w s.

(15) 4. where \/ai. i s t h e d i f f u s i o n v e l o c i t i of s p e c i e s i and cJL. i. i s t h e mass p r o d u c t i o n r a t e p e r u n i t volume o f s p e c i e s. '8e c o n s i d e r a one s t e p chemical r e a c t i o n i n which t h e forward r e a c t i o n i s dominant, t h e backward r e a c t i o n being n e g l i g i b l e .. For an Arrhenius t y p e r e a c t i o n w i t h. second o r d e r chemical k i n e t i c s , w e may w r i t e. where. E. i s t h e a c t i v a t i o n energy of t h e r e a c t i o n and. i s t h e frequency f a c t o r .. b. Also i f $ i s t h e s t o i c h i o m e t r i c. ratio ox~dizer-he1. We shall u s e r e l a t i o n 6a through m o s t o f t h i s s t u d y . T h e e x t e n s i o n t o more g e n e r a l r e a c t i o n r a t e s o f t h e form. i s e a s i l y made.. The d i f f u s i o n v e l o c i t i e s depend on p r e s s u r e , ternpe r a t u r e and s p e c i e s c o n c e n t r a t i o n g r a d i e n t s .. Usually. t h e p r e s s u r e g r a d i e n t e f f e c t on d i f ' f u s i a n v e l o c i t i e s i s s m a l l compared t o t h o s e due t o mass r r a c t f on g r a d i e n t s .. This i s e s p e c i a l l y t r u e when m i x i n g t a k e s p l a c e i n t h i n mixing regions and boundary l a y e r s .. Thermal diffusion. w i l l be n e g l e c t e d because d i f f u s i o n v e l o c i t i e s due t o. g r a d i e n t s o f t e m p e r a t u r e a r e g e n e r a l l y a small f r a c t i o n of t h e velocities due t o c o n c e n t r a t i o n g r a d i e n t s , I f t h e molecular masses o f t h e s p e c l e s are approx-.

(16) i m a t e l y e q u a l w e may u s e Fick's l a w f o r the determination J. 0f vdi A. KiVdi=-D V where. Ki. (8). i s a n average d i f f u s l o n c o e f f i c i e n t .. If t h e c o n c e n t r a t i o n o f one of' t h e s p e c i e s is small,. F i c k T s l a w i s v a l i d f o r t h e o t h e r two s p e c i e s .. This. always happens in d i f f u s i o n flames where o x i d i z e r. cot-. c e n t r a t i o n , f o r example, i s v e r y s m a l l i n &he r e a c t i o n zone, o r i n t h e f l u i d s i d e o f t h e flame.. Then w e may u s e. F i c k T s Paw f o r f u e l and oxidizer with the d i f f u s i o n c o e f -. f i c f e n t s d e t e r m i n e d by t h e b i n a r y mixtures:. and o x i d i z e r - p r o d u c t s r e s p e c t i v e l y .. fuel-products. I n t h i s s t u d y we. w i l l u s e a single average d i f f u s i o n c o e f f i c i e n t. D.. I n s e r t i n g 8 i n t o 5 we o b t a i n. 11, 4.. Momentum e q u a t i o n. G . 0 3 =-"vP S. ! ? =. =-3T&T/'(h av. Where (r - i s the s t r e s s t e n s o r C.. av.. C1. We n e g l e c t t h e d i f f u s i o n s t r e s s t e n s o r .. Gravity forces. will a l s o be neglected f o r simplicity, although they can only be n e g l e c t e d f o r l a r g e Froude numbers,and t h i s. i s n o t always t h e c a s e I n d i f f u s i o n flames,. (10). au.

(17) 11,. 5,. Energy E q u a t i o n. If t h e s p e c i f i c h e a t s. CpL of t h e s p e c i e s a r e. assumed t o be e q u a l and c o n s t a n t t h e e n e r g y e q u a t i o n may be w r i t t e n. where and. Af. Q .-.-. : V ~. i s t h e Rayleigh d i s s i p a t i o n f u n c t i o n. i s t h e chemical e n e r g y t h a t a c o m b u s t i b l e m i x t u r e. containing's u n i t mass of f u e l and 3 u n i t s of o x i d i z e r has a v a i l a b l e f o r c o n v e r s f o n i n t o t h e r m a l e n e r g y. pp' i s t h e P r a n d t a number w h i c h . w f l 1 be assumed c o n s t a n t ,. Thermal r a d i a t i o n 1 s. not taken i n t o account, E q u a t i o n s 2,4,10,. 11 and 9 ( f o r i = L 2. ),. together. w i t h r e h t l o n s 6 and t h e f u n c t f o n a P r e l a t f o n s between t h e t r a n s p o r t p a r a m e t e r s and. p ,T and. Ki,. constitute. t h e system of d i f f e r e n t i a l e q u a t i o n s g o v e r n i n g t h e s t r u c -. t u r e of the d i f f u s i o n flames, I n a d d i t i o n w e must i n c l u d e t h e a p p r o p r i a t e bound-. ary c o n d i t i o n s , Without l o s i n g much g e n e r a l i t y w e c a n s t a t e a s boundary c o n d i t i o n s f o r t h e t e m p e r a t u r e s and mass f r a c t i o n s t h a t t h e y be c o n s t a n t s a t some s u r f a c e s o r zone I. of t h e flow f i e l d . For example, on some s u r f a c e o r r e g i o n a t t h e f u e l.

(18) s i d e o f t h e flame. - the f u e l e x i t Ki= KiF. T =TF. and. Ki= K i o. 9,. T. = To. on some surface or region far from t h e flame on t h e \. o x i d i z e r s i d e o f t h e flame.. 11.. 6.. Dfmensfonless Form of t h e Equations -. Let us introduce the followfng non-dimensional variables.. Re is. the non-dimensional Reynolds number, and. S c is. the Schmidt number t h a t will be assumed t o be c o n s t a n t. and equal to t h e Prandltl! number Subscripts. and F' will Indicate boundary conditions. f a r from the flame on the oxidizer and fuel s f d e o f the. flame respectively. The c h a r a c t e r i s t i c l e n g t h. L and v e l o c i t y 0 a r e. some o v e r a l l c h a r a c t e r i s t i c magnitudes.. In terms of these non-dimensional variables the governing e q u a t i o n s take t h e form.

(19) [$. = 9. and.. Here. will be t h e adiabatic temperature o f the flame.. 4 T;=q. @r. Then. t,. K,~(T~+K,F). as we shall see later, Kt, is a characteristic chemical time, such that K,+S. w i l l be o f order unity if the mass fractions are. not small and the temperature I s close t o the adiabatfc flame temperature..

(20) 1 1. DISCT7SSI OV OF TXE EQTJATI O N S h n J U L I M I ' I ' I I S G CASES. 0. *. 111,. 1.. General D i s c u s s i o n. By c h o o s i n g a p p r o p r i a t e l y t h e c h a r a c t e r i s t i c mag-. nitudes t h e non-dimensional f a c t o r s and t e r m s i n equat i o n 1 4 should be o f o r d e r u n i t y e x c e p t , a t most, a t r e -. gions such as boundary l a y e r s , shock waves, f r e e mixing layers.. I n these regions t h e f u n c t i o n s ? ,. t h e i r d e r i v a t i v e s may change v e r y r a p i d l y .. Ki , T. or. If such l a y e r s. do n o t e x i s t , o r i n any o t h e r r e g i o n t h e r e l a t i v e impor-. t a n c e o f t h e d f f f e r e n t terms i n e q u a t i o n 1 4 i s m e a s u r e d by t h e v a l u e s sf t h e non-dimensional parameters. T h i s i s n o t e x a c t l y t r u e f o r t h e term. because o f the large variations o f. with tern-. p e r a t u r e and mass f r a c t i o n s , From e q u a t i o n s 6b and 9 we deduce. Also i f. V' << j. 4. 0k4<1 KO. q I. and the energy e q u a t i o n may b e w r i t t e n. And if t h e f u e l d i f r u s i o n e q u a t i o n i s added t o 17. we o b t a i n. Tiwe v*(K,. =[J-]RePr. I -rm). *.o~, 3. 4 v.0 [pa $. *h.

(21) If the form 17 o f t h e e n e r g y e q u a t i o n i s u s e d , a n d. t a k i n g i n t o account t h e d i ] ' f u s i o n equations, t h e function. s a t i s f i e s the d i f f e r e n t i a l equation. that when s o l v e d wi t h t h e boundary c o n d i t i o n s 1 2 g i v e s. S o l u t i o n 2 8 i s independent of t h e chemical k i n e t i c s . There a r e c a s e s , however, i n which t h e boundary condit i o n s a s given i n 1 2 a r e n o t known "a p r i o r i " because. they depend on t h e chemical k i n e t i c s .. For example, i n. t h e case of a f u e l d r o p l e t burning i n a n o x i d i z i n g medium,. t h e o x i d i z e r mass f r a c t i o n a t t h e d r o p l e t s u r f a c e (which i s z e r o f o r i n f i n i t e o r v e r y l a r g e r e a c t i o n r a t e s ) may. b u i l d up t o some unknown v a l u e when t h e r e a c t i o n r a t e becomes low,. 111.. 2,. Burke-Schumann S o l u t i o n. For t h e s t u d y of d i f f u s i o n f l a m e s , Burke and Schumann i n t r o d u c e d t h e adsumption t h a t t h e r e g i o n where. '3. i s d i f f e r e n t from z e r o 1 s i n f i n i t e l y t h i n s a n d. on one s i d e o f t h e flame,and T h i s should be t r u e when both. and. Kz=O. M,=O. on t h e o t h e r .. L/~t,i s. very large.. K2were d i f f e r e n t from zero i n a r e g i o n. If.

(22) .. rf term -. ;. where t h e temperature i s n o t low compared with h. e. (-%would J. [&a$. be of o r d e r unity and t h e. would be very l a r g e compared w i t h. v**k* v*KJ,. -4-. V. *v*K;. t h e n we o b t a i n t h e r e s u l t. K,=. 0. ,. where k t , % ,,j= 2. =. 3. a nd. t h a t a r e o r o r d e r unity.. Also if i n system 1 4 we t a k e t h e l i m i t. K,:D o r. ki~gy. ,j. .. --r t, L. -. So e i t h e r. and system 1 4 t a k e s the f o l l o w i n g form, on t h e f u e l r i d e of t h e flame and on t h e o x i d i z e r s i d e of t h e flame,. W e can u s e w i t h system 21 t h e same boundary cond i t i o n s 12% o f system 1 4 , i f we a l l o w f o r d i s c o n t i n u i t i e s. i n t h e mass and t e m p e r a t u r e d i s t r i b u t i o n s a t t h e zero t h i c k n e s s flame* I n t h e flame t h e e q u a t f o n s of c o n s e r v a t i o n of mass and energy i n d i c a t e t h a t , 1) f u e l and o x i d i z e r d i f f u s e towards t h e flame In stoichiornetrFc p r o p o r t i o n s ; 2 ) t h a t %. K,,. and &must. be z e r o if h/Ut,4=.

(23) *he. h e a t l e a v i n g t h e f h me due t o coliduction e q u a l s t h e. h e a t r e l e a s e d by t h e r e a c t i n g s p e c l e s when r e a c h i n g t h e flame,. That i s. 3%. tively.. and. a-. I n d i c a t e d e r i v a t i v e s normal t o t h e 3% f'lame s u r f a c e toward t h e f u e l and o x i d i z e r s i d e s r e s p e c -. where. They must be e v a l u a t e d a t the flame.. The t e m p e r a t u r e , mass f r a c t i o ~ l s , and t h e r e f o r e t h e d e n s i t y and v e l o c i t y , a r e continuous f u n c t i o n s a t t h e flame,. For t h i s r e a s o n , mass and h e a t t r a n s p o r t by con-. v e c t i o n i s n o t t a k e n i n t o account when w r i t i n g t h e cons e r v a t i o n e q u a t i o n s through the flame, B-y s o l v i n g t h e system o f e q u a t i o n s 21,with bound-. a r y c o n d i t i o n s 1 2 and 22, w e o b t a i n t h e Burke-Schumann velocity,mass fractions,and. temperature d i s t r i b u t i o n s .. I n particular, l e t. be the s o l u t i o n s o f e q u a t i o n s 16 and 18 ( v a l i d o n l y.

(24) The e q u a t i o n of t h e flame s u r f a c e i s obtained by J. writing. K,= K2=. o r. f,Jz)=o. Also a c c o r d i n g t o 20 a t t h e flame. T*=T'=. k*KIF i +~T;). Kzo. +. KLO. and. By w r i t i n g s u r f a c e and. KL=O. K,rO. on t h e f u e l s i d e of t h e flame. on t h e o x i d i ' z e r side,we. obtain the. temperature and mass f r a c t i o n d i s t r i b u t i o n s .. It i s i n t e r e s t i n g t o p o i n t out t h a t t h e Burke-Schumann s o l u t i o n s a t i s f i e s t h e complete system of e q u a t i o n s 1 4 and a l s o i t $ - boundary c o n d i t i o n s .. This s o l u t i o n i s. n o t t h e c o r r e c t one,only because the f i r s t d e r i v a t i v e s of t h e temperature and mass f r a c t i o n d i s t r i b u t i o n s have d i s c o n t i n u o u s f i r s t d e r i v a t i v e s w i t h l n t h e flow f i e l d , S o l u t i o n s 23 of e q u a t i o n s 1 6 and 18 a r e modified when f i n i t e v a l u e s of. L/U. t,. a r e considered.. The. reason f o r these modirications i s t h a t , although t h e r e a c t i o n r a t e s d o n o t a p p e a r explEcitPy I n equatfons.

(25) C. 9 ,v. t h a t appear i n and t h o s e e q u a t i o n s w i l l depend on t h e r e a c t i o n r a t e s .. 1 6 and 18, t h e v a r i a b l e s. /U. Yowever, w e may expect t h a t , f o r s u f f i c i e n t l y l a r g e. v a l u e s of. L/otc ,. t h e r e a c t i o n zone ( o r r e g i o n where. \"lif 0 ) w i l l be v e r y t h i n .. Hence t h e Burke-Schumann. s o l u t i o n 25, f o r which t h e r e a c t i o n zone has zero t h i c k n e s s , w i l l be a v e r y good approximation i n t h e case of l a r g e but f i n i t e. .. L/(&. This w i l l be e s p e c i a l l y. t r u e o u t s i d e of t h e r e a c t i o n zone. Equations 16 and 1 8 , in p a r t i c u l a r , should remain p r a c t i c a l l y unchanged.. T h i s i s e x a c t l y r i g h t when mixing. and r e a c t i o n t a k e s p l a c e i n c o n s t a n t p r e s s u r e r e g i o n s. and boGndary l a y e r s i f. g/u. i s assumed t o be c o n s t a n t .. I n such c a s e s e q u a t i o n s 16 and 18 a s w e l l as t h e bound-. a r y conditions ( f o r l a r g e d e n t of' t h e r e a c t i o n r a t e s .. L/u~,. ), w i l l be indepen-. The same w i l l happen t h e n. t o their solution. Snmming up:. If the reaction r a t e i s sufficiently. l a r g e t h e r e a c t i o n zone w i l l be v e r y t h i n compared w i t h. any o t h e r important l e n g t h ( a s for example, t h e wldth of t h e mixing r e g i o n ) .. Then, i n o r d e r t o o b t a i n t h e v e l o -. c i t y , temperature and mass f r a c t i o n d i s t r i b u t i o n s o u t s i d e of t h e r e a c t i o n zone, i . e .. f o r t h e s t u d y of t h e e x t e r n a l. s t r u c t u r e of' t h e diff'usion. flame, we may u s e t h e assump-. tion of f n f i n i t e r e a c t i o n rates,.

(26) IV. IV.. 1.. STRTJCTURE OF THE REACTION ZOXE:. The "chemical Boundary Lager". The f a c t t h a t i n t h e l i m i t i n g c a s e of infinite r e a c t i o n r a t e s t h e t h i c k n e s s of t h e r e a c t i o n zone i s zero,. K;. and t h a t t h e f i r s t d e r i v a t i v e s of. and. T normal. t o t h e flame a r e d i s c o n t i n u o u s t h e r e , s u g g e s t s t h a t f o r large, although f i n i t e ,. .. L/C)t,. a). The t h i c k n e s s of t h e r e a c t i o n zone w i l l be s m a l l. b). The d i f f u s i o n terms. a n (pc 3an. ) 7 a-. and. w i l l balance t h e chemrcal p r o d u c t i o n terms d;/f,. -the. t h r e e terms being very barge compared w i t h a l l t h e o t h e r terms of t h e e q u a t i o n s .. (Here. a a-. indicate differen-. t i a t i o n normal t o the f l a m e ) I n o t h e r words, f o r l a r g e v a l u e s of t h e chemical r e a c t f o n r a t e s t h e t h i ckness of t h e r e a c t i o n zone will be a very t h i n r e g i o n o r "chemical boundary l a y e r " . There, due t o t h e r a p i d l y v a r y i n g g r a d i e n t s of tempera t u r e and mass f r a c t i o n s normal t o t h e flame, mass d i f f u s i o n and h e a t conduction normal t o t h e flame c o n s t i t u t e t h e only t r a n s p o r t mechanism r e q u i r e d t o balance t h e chemical p r o d u c t i o n terms.. T r a n s p o r t by d i f f u s i o n o r. conduction i n o t h e r d i r e c t i o n s o r convection may be '. n e g l e c t e d w i t h i n t h e r e a c t i o n zone,. I n o r d e r t o show t h i s , l e t u s assume t h a t w e know t h e Burke-Schumann solution 25.. Hence, w e know the flame.

(27) surface location f o r i n f i n i t e. L/ut,. , and. therefore. t h e approximate l o c a t i o n of t h e flame r e g i o n f o r large. L/LItG. .. F o r s i m p l i c i t y w e w i l l l i m i t o u r s e l v e s t o t h e two-. dinenslonal case.. The r e s u l t s , however, a r e c o m p l e t e l y. We s h a l l write the e q u a t i o n s of motion,and. general.. mass and e n e r g y c o n s e r v a t i o n e q u a t i m s i n a c u r v i l i n e a r. system o f c o o r d i n a t e s .. I n t h i s system, s e e Fig. 1,X will. be t h e d i s t a n c e measured a l o n g the flame s u r f a c e , a s determined by t h e Burke-Schumann s o l u t i o n .. The d i s t a n c e. normal t o t h i s s u r f a c e w i l l be i n d i c a t e d by V. will be t h e v e l o c i t y components i n t h e. tions;. %. ; U, and and. y. direc-. 1 / ~ i s t h e radius of c u r v a t u r e o f the flame. a t point X. .. Liniting o u r s e l v e s t o a r e g i o n where K y i s. small compared with. ('+~~)d%. and. the l i n e element h a s components. d ,and t h e e q u a t i o n s a r e a s follows:. 3. con ti nu it^ f o r the m i x t u r e.

(28) Equations of motion. 4heve. clr. a&. V*\I =: I + K J S T BE,= +~~. 1t~3. Continuity e q u a t i o n f o r each of the s p e c i e s. Energy e q u a t i o n. '~'Vhere. and.

(29) Now l e t. velocities.. and Let. 6,. be o v e r a l l c h a r a c t e r i s t i c l e n g t h and and s,be. t h e t h i c k n e s s of t h e renc-. t i o n zone and of t h e mixing r e g i o n r e s p e c t i v e l y .. co1.1ld show t h a t f o r low Reynolds numbers,. i'ie. SmuL. ,. .. while. Tor la r g c Reynolds n~irnbers. The mass f r a c t i o n s j u s t a t t h e o u t e r edge of t h e r e a c t i o n zone w i l l be of o r d e r. .. s,/&*. The same. o r d e r of magnitude w i l l be v a l i d i n s i d e of t h e r e a c t i o n zone,. L e t u s now i n t r o d u c e t h e non-dimensional v a r i a b l e s. /. /. K ~ S ~ /,,t.*g & ~P 8: 3,. by s t r e t c h i n g. t h e c o o r d i n a t e s , mass f r a c t i o n s and r e a c t i o n r a t e s , s o. a s t o make the non-dimensional f a c t o r s and t e r m s of o r -. Z. d e m i t y w i t h i n t h e r e a c t i o n zone.. For t h e remaining. v a r i a b l e s w e mag u s e t h e same non-dimensional v a r i a b l e s used I n s e c t i o n 11,. We w i l l w r i t e t h e governing e q u a t i o n s i n terms o f. t h e s e non-dimensional v a r f a b l e s .. Now, i f t h e terms ac-. c o u n t i n g f o r t h e conduction and d i f f u s i o n normal t o t h e flame a r e g o i n g t o be o f t h e o r d e r of t h e chemical prod u c t i o n term I f we now t a k e t h e l i m i t p i n t h e energy and d i f f u s i o n equations,. s,/6,40. e q u a t i o n s drop out.. .. , most. of t h e terms i n t h e s e. W e are left with t h e following. d i f f e r e n t i a l equations:.

(30) t h a t we have w r i t t e n i n d i a e n s i o n a l form. From t h e momentum e q u a t i o n w e deduce t h a t t h e v a r -. i a t i o n s o f . p r e s s u r e a c r o s s t h e r e a c t i o n zone a r e of. s,/~. order. Hence we may assume,when w r i t i n g t h e e q u a t i o n of s t a t e f t h a t the pressure i s constant across the reaction r e g i o n and e q u a l t o t h e v a l u e obtained a t t h e flame w i t h t h e Burke-Schumann assumption.. Therefore i n a d d i t i o n t o t h e above e q u a t i o n s f o r. K,, K1. where. and T w e have t h e e q u a t i o n. P@). 1s a known f u n c t i o n o f %. .. o r w e may u s e t h e more g e n e r a l e x p r e s s i o n. 7. for. tJi. a s i s done i n appendix A. 140 d e r i v a t i v e s w i t h r e s p e c t t o X appear i n system 26 t o 29.. Therefore t h e s e e q u a t i o n s may be s o l v e d a s. o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i n which % , stands as 8. parameter,.

(31) A s boundary c o n d i t i o n s we w i l l w r i t e t h a t when. Y -3-. t h e temperature and mass f r a c t i o n d i s t r i b u t i o l l s. c o i n c i d e w i t h t h o s e o b t a i n e d by assuming t h e r e a c t i o n r a t e t o be i n f i n i t e . IV.. 2.. The S o l u t i o n o f t h e Chemical Boundary. Lager Equations By i n t r o d u c i n g t h e new v a r i a b l e i f w e assume t h a t. may be w r i t t e n. YP =. !?or0. 8, = ~ ~ & / r ~ ) d g 9. e q u a t i o n s 27 and 28. From- e q u a t i o n 30 i f w e t a k e i n t o account t h a t Pd,=31JI we get.. K ~ - + K ~ - A,y,+Br = S i m i l a r l y , from 30 and 31 w e obtain. R e l a t i o n s 33 t o 35 a r e independent of t h e chemical. reactim rates.. r e a c t i o n zone.. However they a r e o n l y valid within t h e. The c o n s t a n t s. ai. D;. must be chosen.

(32) . s o t h a t these r e l a t i o n s coincide with the s i m i l a r r e l a -. t i o n s obtained from t h e Burke-Schumann s o l u t i o n , a t least f o r l o w values of. .. Then r e l a t i o n s 33 t o 34 may be. written.. Where sometiqes c a l l e d flame s t r e n g t h ,. is t h e mass r a t e of f u e l consumption p e r u n i t flame s u r face.. Also w e may w r i t e. $. o. ~. e. /. ~. SCX) ~ = where. % ,Cg). i s a mixing l a y e r t h i c k n e s s .. Now, by u s i n g e x p r e s s i o n 6a f o r t h e f u e l p r o d u c t i o n. rate, e q u a t i o n 31 t a k e s t h e f o l l o w i n g form:. 8,s. Where. s 4a , and. ea= % Rq. i s t h e non-dimensional. a c t i v a t i o n energy o f t h e r e a c t i o n . By s o l v i n g e q u a t i o n 36 w i t h t h e boundary c o n d i t i o n s. KLrO for. 4-. 71. and. 6<1=6for. -. we o b t a i n. t h e temperature d i s t r i b u t f o n w f t h i n t h e r e a c t i o n zone, If t h e r e a c t f o n r a t e i s s u f f i c i e n t l y lar&,. the. temperature w i l l no% d e v i a t e a p p r e c i a b l y , w i t h i n t h e r e a c t i o n z o n e , from i t s limiting v a l u e ,when % +,. @ = A at. yl=s. 0. 9.

(33) Then a good app~oximate s o l u t i o n of 36 may be ob-. tained by substituting the factor by its value at. & = o . Let 8=0$~)for. gso. '-"f. Then we approximate equation 36 by. This we may write in the form. And the boundary conditions are:. pFZ=o. for. z*-. +&. Here. Equation 39 was solved numerically(15). imate solution is presented in Appendix A.. p=pc"). An approxIts solution. ~ 3 8 ~ is) represented in Fig. 2 with a solid = 0.866 In particular P). line.. P(O). Fig. 3 shows the variation with Z of. 43L_zP9 which. is proportional to the fuel mass consumption rate per 0. unit volume.. For ~ = 3 . 2its value is roughly one per. cant of its maximum value at. Z=0. Hence we may.

(34) conclude t h a t t h e t h i c k n e s s of t h e r e a c t i o n zone o r themi c a l boundary l a y e r i s of t h e o r d e r of. 64 .. According t o 38 t h e temperature a t t h e i d e a l ( k r k e ~charnann)flame s u r f a c e p o s i t i o n ,. y,=O. ,. i s g i v e n by t h e. following r e l a t i o n. A f i r s t approximation f o r $= i s o b t a i n e d by w r i t i n g {. when e v a l u a t i n g. A. .. A second approximation f o r. IV.. Then. ec. 8, i s g i v e n by. v a l i d f o r v a l u e s of ec>0.8is. Discussion of the s o l u t i o n. R e l a t i o n 45 shows t h a t d e v i a t i o n s of t h e temperat u r e from i t s asymptotic Burke-Schumann value depend on t h e parameter. A,. .. eG=h. This parameter i n c o r p o r a t e s. b o t h t h e chemical k i n e t i c parameters of t h e c h a r a c t e r i s t i c chemical time. -. t,. through t h e value. -. and t h e f l u i d.

(35) mechanical p a r a m e t e r s. -. 2. t h r o u g h t h e mixing time. S,c.(, ern@)=. Do. The obvious r e s u . l t i s t h a t t h e d e v i a t i o n s of the t e m p e r a t u r e and mass f r a c t i o n s from t h e i r l i m i t i n g asympt o t i c v a l u e s i n c r e a s e w i t h d e c r e a s i n g v a l u e s of t h e r a t i o. t,/cC. .. Now. t ,. i s inversely proportional t o the f u e l. r a t e of s u p p l y t o the flame m. .. Hence w e deduce t h a t. by i n c r e a s i n g t h e f u e l r a t e of supply, t h e flame temper-. ature w i l l decrease. The i n i t i a l f u e l and o x i d i z e r mass f r a c t i o n s i n f l u -. ence t h e r e s u l t t h r o u g h t h e f a c t o r q 3 t h a t a p p e a r s i n 44.. By d i l u t i n g t h e f u e l o r t h e o x i d i z e r w e g e t l a r g e r. d e v i - a t i o n s from t h e Burke-Schumann r e s u l t . R e l a t i o n 46 i n d i c a t e s t h a t t h e t h i c k n e s s o f t h e r e a c t i o n r e g i o n d e c r e a s e s w i t h i n c r e a s i n g v a l u e s of t h e mass r a t e of f u e l s u p p l y . For t h i s chemical boundary l a y e r scheme t o be v a l i d. sc/sm must be. s m a l l compared t o u n i t y , hence t h e. same c r i t e r i o n *my be used f o r t h e v a l i d i t y of t h e BurkeSchumann assumption and s o l u t i o n , and f o r t h e v a l i d i t y of t h e boundary l a y e r s o l u t i o n . Obviously, t h e m o s t i m p o r t a n t c h a r a c t e r f s t f c s o f t h f s chernlcal boundary l a t e r s o l u t i o n and t h o s e of t h e Burke-Schumann s o l u t i o n c o i n c i d e .. For example, flame. l o c a t i o n and t h e q u a n t i t y of f u e l b u r n i n g p e r u n i t flame a r e a a r e the same i n b o t h s o l u t i o n s .. The chemical bound-.

(36) a r y l a y e r s o l u t i o n , however, g i v e s a f i n i t e thickness f o r t h e r e a c t i o n zone, and a small c o r r e c t i o n t o t h e t e n p e r a t u r e and mass f r a c t i o n d i s t r i b u t i o n s . .. This correc-. t i o n can be e v a l u a t e d v e r y e a s i l y and a c c u r a t e l y i n terms of t h e chemical k i n e t i c p a r a m e t e r s of t h e r e a c t i o n . So, t h e chemical boundary l a y e r s o l u t i o n can be o f h e l p f o r t h e s t u d y of chemical k i n e t i c s by means of experimen-. t a t i o n i n d i f f u s i o n flames. ( 1 2 ) ( 2 3 ) The p a r a m e t e r )!+,,that measures t h e d e v i a t i o n s from t h e i n f i n i t e p e a k t i o n r a t e solution,may be u s e d f o r t h e. d e t e r m i n a t i o n of an e x t i n c t i o n criterion.. T h i s is sup-. p o r t e d by t h e f o l l o w i n g r e a s o n s : a). Due to t h e high v a l u e s of t h e a c t i v a t i o n energy. of many of t h e chemical r e a c t i o n s , t h e chemical product i o n term i s very s e n s i t i v e ' t o temperature v a r i a t i o n s . T h i s accounts f o r the f a c t t h a t flame e x t i n c t i o n o c c u r s. i n a r a t h e r shapply d e f i n e d way.. This may also be due. t o t h e e x i s t e n c e of some i g n i t i o n temperature f o r t h e reaction, \. b). The concept o f an o v e r a l l r e a c t i o n r a t e i s n o t. i n g e n e r a l v a l i d through a l a r g e temperature range. Then, t h e i d e a o f s o l v i n g t h e complete e x a c t equations,. for o b t a i n i n g an "exact" e x t i n c t i o n c r i t e r i o n , l o s e s p a r t o f i t s i n t e r e s t i f u s e has t o be made of' some assumed o v e r a l l r e a c t i o n r a t e e x p r e s s i o n throughout the.

(37) whole t e m p e r a t u r e range. c). The chemical boundary l a y e r s o l u t i o n , a l t h o u g h. i t cannot explain e x t i n c t i o n except i n a q u a l i t a t i v e wag, can provide a c r i t e r i o n f o r e x t i n c t i o n n o t t o o c c u r. i f t h e overaall r e a c t i o n r a t e i s known t o be v a l i d i n a. g i v e n temperature range.. A,is. Whenever t h e t o make. T,40.8>. s u f f i c i e n t l y low as. (and this occurs f o r , roughly,. A,~80),. t h e t h i c k n e s s o f t h e r e a c t i o n r e g i o n b e g i n s t o be compar-. a b l e with t h e t h i c k n e s s of t h e mixing r e g i o n .. Then t h e. r a t e o f h e 1 consumption b e g i n s t o d i m i n i s h and hence TG. w i l l begin t o d e c r e a s e even f a s t e r w i t h d e c r e a s i n g A,. T h e r e f o r e , A p 8 0 m a y be used a s an approximate extinction c r i t e r i o n as w e l l as a c r i t e r i o n f o r the v a l i d i t y o f t h e Wlrke-Schumann s o l u t i o n .. \Ve have seen t h a t. Aac?$~O-. +c. .. NOW, this sane. parameter a p p e a r s i n t h e t h e o r y of premixed l a m i n a r flames.. There i t t a k e s a v a l u e o f t h e o r d e r o f 100,. t h a t depends on t h e i n i t L a l mass f r a c t i o n s and energy o f \. a c t i v a t i o n o f t h e r e a c t i o n ( 2 4 ) , when by t h e f i e 1 consumption r a t e. th i s. substituted. 6 per u n i t area.. There-. f o r e a n "approximates9 r e l a t i o n may be e s t a b l i s h e d (919 ( l o ). between t h e v a l u e of s t r e n g t h ) and 6 :. a t e x t i n c t i o n (maximum flame.

(38) V. .. TNE EXTERNAL STHTJCTUHE OF' LA'i\."iT:'TAR DI?'.'TiTiTSI03 f"l,kT;r:S. V,. 1,. Large Reynolds Yurnber Case.. I n v i s c i d idqua-. tions The chemical boundary l a y e r equationS,governing t h e t e m p e r a t u r e and mass f r a c t i o n s d i s t r i b u t i o n s w i t h i n t h e r e a c t i o n zone,have been solved i n t h e most g e n e r a l c a s e . However, f o r t h e d e t a i l e d e v a l u a t i o n of t h e s o l u t i o n we must know some p a r a m e t e r s a p p e a r i n g t h e r e . t h e mass r a t e. They i n c l n d e. & of f u e l consumption p e r u n i t flame s u r -. f a c e , and t h e i d e a l flame l o c a t i o n . I n o r d e r t o e v a l u a t e t h e s e p a r a m e t e r s a s well a s t h e mass f r a c t i o n s and . t e m p e r a t u r e d i s t r i b u t i o n s o u t s i d e o f t h e r e a c t f o n zone, t h e Burke-Schumann s o l u t i o n must be obtained f i r s t .. A s mentioned i n t h e i n t r o d u c t f o n t h i s s o l u t i o n h a s been o b t a i n e d i n some p a r t i c u l a r c a s e s . .. Unfortu-. n a t e l y , even when u s i n g t h e Burke-Schumann assumption of i n f i n i t e r e a c t i o n r a t e s , t h e r e s u l t i n g system o f ,. e q u a t i o n s 21 and boundary c o n d i t i o n s 1 2 and 22 i s s o complicated t h a t o n l y a f e w a p p r o x i m a t e s ' o l . u t i o n s -e'xf a t .. Marble and ~ d a r n s o n ( l 7 )have p o i n t e d o u t t h a t a numb e r of i m p o r t a n t combust€on problems may be i n v e s t i g a t e d a n a l y t i c a l l y w i t h t h e h e l p o f boundary l a y e r approximations,. Most of t h e soPutPons s o f a r o b t a i n e d make u s e.

(39) of these a p p r o x i m a t i o n s .. These may be used whenever. t h e R e y n o l d s number, based on some o v e r a a l l di-nension of t h e flow f i e l d , i s s u f f i c i e n t l y l a r g e . We w i l l show h e r e t h a t , b y u s i n g some a d d i t i o n a l. assumptions, a f a i r l y simply s o l u t i o n of t h e Burke-Schumann. mixing problem i s o b t a i n e d . If in system 21 w e take t h e l i x i t. Re-+*. we. o b t a i n t h e f o l l o w i n g system o f d i f f e r e n t i a l e q u a t i o n s , t h a t we s h a l l w r i t e i n dimensf o n a l form:. The bouhdary c o n d i t i o n s 22 cannot be r e t a i n e d because, i n t h e p r o c e s s of t a k i n g t h e l i m i t. Re---,,we. dropped. t h e higher order d e r i v a t i v e s i n ' t h e equations. Bowever, t h e boundary c o n d i t i o n s 1 2 can be s a t i s f i e d. if w e a l l o w f o r t h e e x i s t e n c e of' d i s c o n t i n u i t i e s i n temp e r a t u r e , mass f r a c t i o n s , d e n s i t y and v e l o c i t y a t some stream surface.. The p o s i t f o n 0 f . t h i . s s u r f a c e i s d e t e r -. mined by r e q u i r i n g t h a t a l l t h e boundary c o n d i t i o n s 1 2 be s a t i s f i e d.

(40) If w e c o n s i d e r o n l y low Mach number f l o w s , t h e n :. 1) The d e n s i t y , ternperatllre and mass f r a c t i o n s w i l l be c o n s t a n t , a l t h o u g h p o s s i b l y w i t h d i f f e r e n t vaI.ues, on e a c h s i d e of t h e d i s c o n t i n u i t y s u r f a c e . 2). See F i g . 1.. Equation 47 reduced t o t h e system. &nd t a n g e n t i a l d i s c o n t i n u i t i e s of. V. a r e allowed. f o r a t some s u r f a c e , s o a s t o s a t i s f y t h e r e q u i r e d bounda r y c o n d i t i o n s on. 3. .. The p r e s s u r e must, of c o u r s e ,. be c o n t i n u o u s a t t h e s u r f a c e ,. A s an example,the s o l u t i o n of this problem f o r t h e. low speed s o u r c e f l o w i s p r e s e n t e d i n ~ p p e n d f xC.. V.. 2,. Mixing Layer E q u a t i o n s. For l a r g e b u t f i n i t e Re. ,. the ideal discontinuity. s u r f a c e i s s u b s t i t u t e d by a t h i n mixing l a y e r w i t h t h e same approximate l o c a t i o n . \. Although t h e d i s c o n t i n u i t i e s. i n t h e t e m p e r a t u r e , mass f r a c t i o n s , e t c . ,. no l o n g e r e x i s t ,. t h e d e r i v a t i v e s of t h e s e v a r i a b l e s normal t o t h e mixing l a y e r w i l l be v e r y l a r g e compared t o t h e d e r i v a t i v e s i n the surface direction,. '. I n o r d e r t o s t u d y t h e s t r u c t u r e o f t h e mixing r e g i o n , w e w i l l w r i t e t h e system 29 i n boundary l a y e r c o o r d i n a t e s , Then we can o b t a i n t h e mixing l a y e r e q u a t i o n s by u s i n g.

(41) t h e w e l l I n o v ~ nboundary l a y e r l i m i t i n g p r o c e s s .. L i m i t i n g o u r s e l v e s t o t h e two-dimensional o r a x i a l -. l y - s y n r n e t r i c l o w Mach number f l o w c a s e s , we will w r i t e t h e s e e q u a t i o n s i n t h e Corm g i v e n by Lees ( 2 5 ). where. h,zo. f o r two-dimensional f l o w s and k = i f o r a x i a l l y -. symmetric f l o w ;. u#)is. t h e v e l o c i t y at t h e o x i d i z e r s i d e ,. j u s t o u t s i d e o f the mixing l a y e r . By i n t r o d u c i n g t h e stream f u n c t i o n. 3/ such. that. The c o n t i n u i t y e q u a t i o n i s automatically s a t i s f i e d .. v @ , ~J-isK~~5) )= -. T~O'L. Let. U/UO= ( 5 1~ ) I~~. where the primes d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o. 'lie will assume \ .. PP= y e p .. Then t h e mixfng l a y e r. e q u a t i o n s t a k e t h e form. Where. and. k3,3. i=1,3 ,,. ,, pi=. 4. on t h e f u e l s i d e of t h e flame on t h e o t h e r s i d e .. Thet.L.d.sljin. the.

(42) t e r m s involving r i g h t hand s i d e of t h e e q u a t i o n s i n d i c a m r i v a t i v e s. 5.. w i t h respect to. A s boundary c o n d i t i o n s w e may w r i t e. Yt~@. Where. gi-ves t h e i d e a l flame p o s i t i o n .. I n addition. For t h e s o l u t f on o f the above mixing problem a t h i r d boundary c o n d i t i o n f o r f. i s required.. T h i s should. be d e r i v e d from t h e cornpatibxlity c o n d i t i o n of t h e h i g h e r o r d e r approximation. ( 2 6. However, f o r o u r p u r p o s e s w e. 0. mag w r i t e a s t h i r d boundary c o n d i t i o n t. because t h e only e f f e c t o f changing t h e v a l u e of i s a displacement of t h e mixing l a y e r i n t h e V.. 3.. f(O,3). Y direction.. Local S i m i l a r i t v A ~ ~ r o x i m a t i o n. A s the boundary c o n d i t i o n s a r e independent of. t h e f u n c t i o n s ), K;. 5. and T would be f u n c t i o n s o n l y o f. and s i m i l a r i t y would e x i s t , if t h e p r e s s u r e g r a d i e n t parameter Uo. $ 3were d$. constant.. 7.

(43) T h i s o c c u r s a t t h e s t a g n a t i o n p o i n t where t h e p a r a -. ,. m e t e r has t h e v a l u e s 1/2 and. f o r the axially-symmetric use,. f o r t h e two-dimensional one. Similarity a l s o e x i s t s i n the constant pressure. c a s e c o r r e s p o n d i n g t o t h e mixing of two p a r a l l e l s t r e a m s . I n t h i s c a s e t h e p r e s s u r e g r a d i e n t term i s o b v i o u s l y zero. when. The l o c a l s i m i l a r i t y a p p r o x i m a t i o n may be used. 1 5 due i s a s l o w l y v a r y i n g f u n c t i o n of. m. Then we may n e g l e c t t h e of e q u a t i o n 52.. 5. .. t.i.d*~/S i n t h e r i g h t hand s i d e. The r e s u l t i n g system o f d i f f e r e n t i a l. e q u a t i o n s may t h e n be i n t e g r a t e d a s a system of o r d i n a r y d i f f e r e n t i a l equations, 1t'I.s i n t e r e s t i n g t o point out t h a t the f a c t o r. [. ). i n t h e p r e s s u r e g r a d i e n t term approaches z e r o. a t b o t h edges of t h e mixing l a y e r . is. the p r e s s u r e g r a d i e n t termYssirnilarg. The neglect-. of'. the. t o y n e g l e c t a r r s e of. f r e e convection i n d i f f u s i o n flames. The p r e s s u r e g r a d i e n t term is n e g l e c t e d , w i t h o u t much j u s t i f i c a t i o n ,. by S p a l d i n g when s t u d y i n g t h e opposed-. j e t d i r f u s i o n flame. ( 1 2 ) If' o u r main purp7se i s a s s e s s i n g t h e e f f e c t s of. chemical k i n e t i c s i n d i f f u s f o n f l a m e s , o n l y an approxi m a t e a n a l y t i c a l s o l u t i o n of t h e e q u a t i o n s i s r e q u i r e d .. This may be o b t a i n e d e a s i l y i f we n e g l e c t t h e p r e s s u r e g r a d i e n t t e r m i n t h e momentum e q u a t i o n 52.. Then i t.

(44) reduces t o the Rlassius equation. f"' + if" =*. w i t h t h e boundary c o n d i t i o n s. ~ c ! ) = otl An approximate a n a l y t i c a l s o l u t i o n o f t h i s e q u a t i o n. i s g i v e n i n Appendix B.. The f i r s t a ~ p r o x i m a t i o ni s. From e q u a t i o n s 5 2 we deduce t h e f o l l o w i n g system o f equations. Taking i n t o account t h e boundary c o n d i t i o n s 53b. we obtain. And t h e boundary c o n d i t i o n s a t. 7.6. a r e s a t i s f i e d fden-.

(45) The flame s u r f a c e i s l o c a t e d a t t h e p o i n t where. and. a. 'Y. T h e p a r a m e t e r A 6 , measuring t h e d e v i a t i o n s f r o m t h e Burke-Schumann s o l u t i o n , c a n now be e v a l u a t e d .. Also. t h e t e m p e r a t u r e a n d mass f r a c t i o n d i s t r i b ~ ~ t i o nosu t s i d e o f t h e r e a c t p i o n zone may be o b t a i n e d from 57 and 5 R by. putting. KI=Ofor. +. and 5 - 0. I n t h e p a r t i c u l a r case. for. h =1 ,. 24ri. we g e t. and t h e equatiorls may be solved even f o r obtain. V.. 4-,. Particular Cases. I n p a r t i c u l a r let u s c o n s i d e r. O. f s =i. Pr* 1. .. Vie.

(46) a). Axially-symmetric s t a g n a t i o n p o i n t. And a c c o r d i n g t o 4 4. b). Mixing of two p a r a l l e l s t r e a m s. Therefore. I n Reference 1 5 a n u m e r i c a l i n t e g r a t i o n of' t h e e q u a t i o n s f o r t h e mlxing and s i m u l t a n e o u s chemical r e a c t i o n of' two p a r a l l e l s t r e a m s of f u e l and o x i d i z e r , m o v i n g was c a r r i d out. w i t h t h e same v e l o c i m n o r d e r t o compare w i t h t h e chemical boundary l a y e r s o l u t i o n . shown i n F i g s . 4, 5 and 6.. The r e s u l t s a r e.

(47) VI, RESITME ~ x ~ e r i r n e n t s and ( ~ ) the s c c c e s s ~f t h e e x i s t i n g t h e o r i e s on l a m i n a r d i f f u s i o n flames have c l e a r l y s h w m t h a t , i n t h o s e c a s e s where a s t a b l e l a m i n a r diffusion flame has been o b t a i n e d , Burke-Schumann assumption o f i n f i n i t e l y fast reaction rate applies.. However, t h e. Burke-Schumann s o l u t i o r l i s i n d e p e n d e n t of chemical k i n e t i c s , a n d d o e s n o t g i v e any c r i t e r i o n e i t h e r f o r t h e flame e x t i n c t i o n o r f o r t h e v a l i d i t y of t h e s o l u t i o n . The f a c t t h a t i n t h i s s o l u t i o n t h e f l a m e t h i c k n e s s i s zero suggests t h a t i n p r a c t i c a l cases the r e a c t i o n z o n e must be o f n e g l i g L b l e t h i c k n e s s , making i t p o s s i b l e. t o o b t a i n a s o l u t i o n o f t h e boundary l a y e r t y p e , i n c l - u d i n g t h e e f f e c t s o r chemical k i n e t i c s .. A t e a c h s i d e o f t h e e x t r e m e l y thin r e a c t i o n zone, c h e n i c a l r e a c t i o n e f f e c t s a r e n e g l e c t e d a s compared w i t h I. c o n v e c t i o n , c o n d u c t i o n and d i f f u s i o n e f f e c t s ,. The r e a c -. t i o n zone r e d u c e s t o a flame f r o n t of n e g l i g i b l e t h i c k n e s s , which a c t s a s a s i n k f o r t h e r e a c t a n t s and a s a s o u r c e. f o r t h e h e a t and p r o d u c t s evolved i n t h e chemical r e a c t i o n . The l o c a t i o n of t h e flame frong r a t e of b u r n i n g , and tem-. p e r a t u r e and c o n c e n t r a t i o n d i s t r i b u t i o n s i n t h e e x t e r i o r. o f t h e r e a c t i o n zone a r e determined by u s i n g t h e i3urke-. Schumann assumption,.

(48) I n o r d e r t o a n a l z g e t h e s t r u c t u r e of t h e burning zone we may n e g l e c t i n i t c o n v e c t i o n e f f e c t s a s compared. w i t h c o n d u c t i o n , d i f f ' u s i o n and chemical r e a c t i o n e f f e c t s . The equations governing t h i s chemical boundary layer a r e o r d i n a r y d i f f e r e n t i a l e q u a t i o n s w i t h boundary condit i o n s determined by t h e Burke-Schumann s o l u t i o n .. Tem-. p e r a t u r e s t h e r e a r e c l o s e t o t h e a d i a b a t i c flame tempera t u r e , a n d t h e n a n o v e r a l l k i n e t i c soheme a p p l i e s . The c r i t e r i a f o r e x t i ~ c t f o no f the flame and f o r t h e v a l i d i t 7 of Burke-Schumann assumption a p p r o x i m a t e l y c o i n c i d e , and may be o b t a i n e d by s o l v i n g , once t h e Burke-. Schumann s o l u t i o n i s knowp, t h e chemical boundary k y e r equations.. T h i s s o l u t i o n a l s o p r o v i d e s t h e tem-. p e r a t u r e and c o n c e n t r a t i o n d i s t r i b u t i o n s i n t h e r e a c t i o n zone, If o u r main purpose i s t h e e v a l u a t i o n of t h e chem-. i c a l k i n e t i c e f f e c t s i n d i f f u s i o n f l a m e s , an approxfmate s o l u t i o n of t h e Burke-Schumann mixl ng problem w i l l be sufficient.. T h i s w e may o b t a i n e a s i l y f o r l a r g e Reynolds. numbers by u s i n g w e l l known boundary l a y e r a p p r o x i m a t i o n s ..

(49) Burlte, S.P. and Schumann, T.E.W. : "1)iff'usion ~'larnes", I n d u s t r i a l and E n g i n e e r i n g Chemistry, Vol 20, Yo. 10, pp. 998-1004, Oct. 1928.. .. H o t t e l , B.G., and I-Iawthorne, G.R. : iffusi us ion i n Laminar Flame ~e t s , T h i r d Symposium on Combustion, Fla me and E x p l o s i o n Phenomena, pp. 254-66, iiiilliams and ; , ? i l k i n s , B a l t i m o r e , 1949.. ". Wohl, K., Ganley, C., and Kapp, N.M.: i iff us ion Flames'?, T h i r d Symp~siumon Combustion, Flame and E x p l o s i o n Phenomena, pp. 288-301.. Yagi, S., and S a j i , K.:. "Problems of Turbulent D i f f u s i o n and Flame J e t s " , Fourth Symposium (intern a t i o n a l ) on Combustion, pp. 771-81, \Villiams and WilLins, B a l t i m o r e , 1953. B a r r , J.: "Length of C y l i n d r i c a l Laminar Diffusion ~ l a n e s " , b?uel, Jan. 1954, pp. 51-59.. Fay, J . A . : "The D i s t r i b u t i o n o f c 3 n c e n t r a t i o n and Temperature in a Laminar J e t D i f f u s i o n Flame", J o u r n a l of' t h e A e r o n a u t i c a l S c i e n c e s , Oct. 1954, pp. 681-89e Cohen, C.B., Bromberg, R. and L i p k i s , R.P.: "~ounda r y Layers Biith Chemical R e a c t i o n s Due t o Mass Addit i o n " , J e t P r o p u l s i o n , Vol. 28, 30. 10, Oct. 1958, pp. 659-668 ljloore, J . A . and Z l o t n i c k , M. : ombu bust ion i n a n A i r stream", ARS J o u r n a l , Vole 31, No. 10, Oct. 1961, pp. 1388-970 Zeldovich, Y.B.: "on t h e Theory of Combustion o f I n i t i a l l y Unmixed ~ a s e s " , NACA Tech. Man., No. 1296, June 1961, 10.. S p a l d i n g , D.B.: ''A Theory o f t h e E x t i n c t i o n of' D i f f u s i o n Flamesvq, Phel, Mar. 1954, pp. 255-73.. 11.. S p a l d i n g , D.B. and J a i n , V.K.: "A T h e o r e t i c a l Study of t h e E f f e c t s o f Chemical K i n e t i c s on a One-dimensional D i f f u s i o n name", Combustion and Flame, Vol. 6 , :Joe 4 , Dec, 1962, pp, 265-73,.

(50) S p a l d i n g , D.R. : "The Theory of' 7';ixiiig an3 Chemical R e a c t i o n i n t h e Opposed-Jet D i f f u s i o n ~ l a r n e " , AKS J o u r n a l , Vol. 31, June 1961, pp. 763-71. S a r r , J . : " D i f f u s i o n Flames i n t h e I 1 a b o r . a t o r y f t , AGARD I k n . , 30. AG ll/M7, 1954. Wohl, I ( . , and Shiprnann, CO1;J.: " ~ l f f u s i o n lames", Combllstion P r o c e s s e s , Vol. 2, High Speed Aerodynamics and J e t P r o p u l s i o n , pp. 365-404, P r i n c e t o n ijniv. Press, 1956, Tlii?Qn, A. : "On t h e I n t e r n a l S t r i ~ c t u r eof Laminar D i f f u s i o n Fla mes", OSR]EOAR TH 62-24, IiTTAET, ?Ladrid, 1961,. Levy, S., and d e i n b e r g , F. J. : " o p t i c a l Flame Stru-ct u r e S t u d i e s : Examination of' H e a c t i o n Rate L a w s i n Lean -tithylene-Air Flames", Combustion and Flame, pp. 2 2 9 - 5 3 , June 1959. .. ?larble, i?.E., and Adamson, T . C . , J r . : Combustion i n a Laminar Mixing zone", pp. 85-94, Karch-Apr. 1954.. " ~ g n i t i o nand J e t Propulsion,. Pai, S.I.: "Laminar J e t Mixing ?f Two Compressible F l u i d s w i t h Heat m el ease ", J o u r n a l of t h e A e r o n a u t i c a l S c i e n c e s , pp. 1012-18, N O V ~1956. Doole y, D.A. : "combustion i n Laminar Dlixing Regions a n d Boundary ~ a y e r s " , PhoDe Thesis., C a l i f o r n i a I n s t i t u t e of Technology, 1956. Von Kgrrntin, Th. : " ~ o r b o n n eL e c t u r e s on Aerothermoc h e m i s t r y 1951-195:!", See a l s o " ~ e r o t h e r m o c h e n i s t r ~ " , A r e p o r t based on t h e c o u r s e conducted by P r o f . Th. von ~ 6 r r n d na t t h e Univ. of P a r i s , by M i l l Q n , G., IXTA ARDC C o n t r a c t No. AF61 ( 5 1 4 ) 441, 1958. Hirschfelder, J.C., C u r t i s s , C.F., and Bird, R. B. ; " F ~ ~ o l e c u l Theory ar of Gases and ~ i ~ u i d s "John , 'ulile y and Sons, Inc., Wew York, 1954. 22.. Penner, S.S.: " ~ n t r o d u c t i o nt o t h e Study of Chemical R e a c t i o n s i n Fl.ow s y s t e m s H , B u t t e r w o r t b London, 1955.. 23.. P o t t e r , A.E., H e i n e l , S., and B u t l e r , J.N.: "~pparent Fb me S t r e n g t h : A Measure of 7\4axirnurn R e a c t i o n Rate i n D i f f u s i o n Flames", E i g h t h Symposiurn'on Combustion, Pasadena, C a l i k o r n f a, 1960,.

(51) 24.. Sendagorta, J . N . d e , a n d DaHiva, I . : ompa par is an of A n a l y t i c a l Xethods f o r t h e Calculation. l';Iill&n, G . ,. of Laminar Flame V e l o c i t y t ' , I?Jl'A AItDC C o n t r a c t 60. A P 6 1 (514)-997, Madrid, Aug. 1957. 25.. Lees, L.: " ~ o n v e c t i v e ileat Transfer with ?Aass k d d i t i o n m d Chemical ~ e a c t i o n ", s Third AGARD Combustion and P r o p u l s i o n Panel Colloquium, Palerrno, S i c i l - y , March 17-21: Pergamon P r e s s , 1958, pp. 451-98.. 26.. Ting, I..:. 27.. Tdeksyn, D. : "Yew h'lethods i n Boundary Layer Pergamon P r e s s , 1961.. 2F3... Penner, S.S.:. "on t h e Mixing of' Two P a r a l l e l s t r e a m s " , PIBAL Report, 70. 441, J u l y , 1958.. heo or^",. "Chemistry Problems in J e t ~ r o p u l s i o n " ..

(52) I n t h i s a p p e n d t x a n i n t e g r a l method for t h e a p p r o x imate s o l u t i o n o f e q u a t i o n 31 i s p r e s e n t e d .. We will. u s e t h e same a p p r o x i m a t i o n s a s t h o s e u s e d i n o b t a i n i n g 37 from 3 6 .. ?Ye. will c o n s i d e r t h e more g e n e r a l mass p r o -. d u c t i o n r a t e e x p r e s s i o n g i v e n by 7.. Then i f r e l a t i o n s. 35 a r e talcen i n t o a c c o u n t , b y t a k i n g 7 i n t o 31, w e o b t a i n. S e r e , however, we w i l l n o t c h o o s e T e & ) a s t h e t e m perature a t. 712 0 ,. but a s t h e temperature a t a p o i n t t h a t. w e w i l l determine P a t e r on.. By i n t r o d u c i n g t h e v a r i a b l e s. equation A . 1 transforms t o.

(53) That m u s t be s o l v e d w i t h t h e boundary c o n d i t i o n s. for. Z=O. z-+%-. I n ~ r d e rt o s o l v e e q u a t i o n A . 5 i n a n approximate way, w e w i l l u s e esk t h e f o l l o w i n g integral method, If 2, i s t h e p o i n t where. P. r e a c h e s i t s minimum. v a l u e , t h e r i g h t hand s i d e of e q u a t i o n A . 4 may be w r i t t e n i n t h e form. N e g l e c t i n g h i g h e r o r d e r terms i n t h e e x p a n s i o n of'. i n powers of' 2 - L ,. , equation. q~). A . 5 may be w r i t t e n i n. t h e approximate f orrn. The c o n s t a n t s. PI. =I. and. k. w i l l be determined. a s follows: a). The s o l u t i o n. E@)of. boundary c o n d i t i o n s f o r f.OL. A . 6 should s a t i s f y t h e. ~(1);namely. $TZ=0. for Z. ~. Z. or. b). If by means of t h e r e l a t i o n. W.

(54) we d e f i n e PI 7 =-I. p=F(z) , and. t h e n -we w i l l c h o o s e t h e parameters. k so that. t h e f o l l o w i n g r e l a t i o r ~ shold:. From 1 . e l a t i m A . 8 we d e d u c e. = gP-1) b(~j'+\) p-z jr,z. -1 kZ(z-z,). +. In particular fop. z=Z,. we o b t a i n. From A.10 w e deduce. Then from A . 9 ,. A.ll and A.12 we get the r e l a t i o n. t h a t has the s o l u t i o n. Novr Z 1 and )e. may be determined in terms of. means o f the r e l a t i o n s. p,. by.

(55) -kv-z ,)'. T(Z)= pi4-21erf ) k@ozJ-&ki'-. l. ( ~ ~ 1 5 ). As approximate s o l u t i o n of e q u a t i o n A . 1 we may use. J. e i t h e r the function 2,=. -. b-a. (A.16). o b t a i n e d by i n t e g r a t i n g e q u a t i o n A . G , o r. t h e more approx-. imate, b u t a l s o more d i f f i c u l t t o e v a l u a t e , f u n c t i o n ~ ( 2 ). s o l u t i o n o f t h e a l g e b r a i c e q u a t i o n A.8.. the parameters. p, ,. Z,. I n both c a s e s. a r e a s g t v e n by A.14. and. and. A.15.. I n o r d e r t o compare t h e above approximate s o l u t i o n s of e q u a t i o n A . 5 w i t h t h e e x a c t numerical s o l u t i o n , w e. ,b = L. cagsider the p a r t i c u l a r case. tl=0.g7. and. I, Z , = 0. jig) = 0,87+ z. ir. .. Then. k = 0.67. (A.17). ~0 . 6f7 2 - 0 . 8 4 1 1 - e ~ ~ < a b 7 ~ ) ~ ] (A.18). The approximate s o l u t i o n. p(',',)has. been p l o t t e d. w i t h a d a s h e d line i n Figs 2 t o compare with t h e e x a c t. solution drawn w i t h a solid l i n e ,. Going back t o t h e g e n e r a l c a s e in which. (r. and. b. n o t necessarily one, l e t us choose t h e t e m p e r a t u r e a t. are.

(56) the p o i n t Z, a s t h e ternperaturae u s e d in approximlat ing e q u a t i o n 31.. Then from r e l a t i o n A . 3 by making. This i s a n a l g e b r a i c e q u a t i o n f o r. z = z I , we. get. eG. Again a first approximation t o t h e s o l u t i o n is obt a i n e d by making r e l a t i o n A.20. .. 8:1 ,. when e v a l u a t i n g. 4.. XCxr@. Then. transforms t o. A second approxi'mation, s i m i l a r t o 45, may be o b t a i n e d for. Bc. without m a j o r d i f f i c u l t i e s ..

(57) APPENDIX B. I n o r d e r t o a p p r o x i m a t e l y s o l v e t h e B l a s s i u s equa-. w i t h t h e boundary c o n d i t i o n s. -fk)= ,, f bw~)='hfCO)= Q , tr. f o l l o w i n p Meksyn ( 2 7 ) l e t u s w r i t e e q u a t i o n B.1 i n the form. where a,=fto). and. a,=f"~). Then by i n t e g r a t i n g R.2. a r e assumed t o be known. t w i c e we g e t. A s the shear s t r e s s decreases very rapidly w i t h. i n c r e a s i n g v a l u e s of. \TI,. w e may e x p e c t t h a t most o f t h e. c o n t r i b u t i o n t o t h e i n t e g r a l B.3 v a l u e s of. I2'. 7. f o r which. f. comes from t h o s e low. can be w e l l approximated. irsing o n l y t h e f i r s t terms i n t h e plwer s e r f e s e x p a n s i o n B.2.. If we w r i t e. &&+%2 6. then. 3 to-. ='t:.

(58) Then t h e integral B.3 may be e v a l u a t e d an3 w e obtain. If we. ~OTI. make. The s e r i e s i n B.6 rapidly.. ~--+S'W. w e obtain. mag be expected t o converge very. Peeping only t h e f i r s t two terms o f t h e s e r i e s. expansions lappearing w i t h i n b r a c k e t s i n B.6, w e g e t. and. That may be approximated w i t h l e s s t h a n 4% e r r o r by. 6.5. Wow by means o f r e l a t i o n s B.8 t h e s o l u t i o n % a y. be. w r i t t e n up t o t h e second approxi-mation..: .A f i r s t approx-.

(59) Low Speed S o ~ l r c em. .0~. The l a m i n a r d i f f u s i o n flame produced when a p o i n t s o u r c e o f fuel i s immersed i n an ox-idizer stream has been q u a l i t a t i v e d i s c u s s e d by Penner. (28) L e t t h e d e n s i t y and temperature o f t h e fuel l e a v j n g. t h e s o u r c e be e q u a l t o t h e density a n d t e m p e r a t u r e o f t h e 8. o x i d i z e r stream.. I f M i s t h e mass r a t e of supply of. f u e l due t o t h e s o u r c e , t h e v e l o c i t y d i s t r f b u t i ~ nI s g i v e n. where. 3:i-i. * '5. and & ~ - l ( l a x i s i s i n t h e f r e e , o x i d i z e r ,. stream d i r e c t i o n . T h e stream s u r f a c e t h a t s e p a r a t e s f u e l and o x i d i z e r. i s easy t o calculate,and. t h a t h a s been done elsewhere.. I n p a r t i c u l a r , the radius of curvature a t the stagnation point, i s. Then. A, a t t h e s t a g n a t i o n p o i n t may be e v a l u a t e d by. u s i n g r e l a t i o n s 44 and 6 3 ,. We o b t a i n. Therefore t h e c r i t e r i o n f o r e i t h e r t h e v a l i d i t y o f Burke-Schumann assumption o r f o r e x t i n c t i o n i s p r a c t i c a l l y.

(60) i n d e p e n d e n t o f the t r a n s p o r t p r o p e r t i e s .. o n l y t h r o u g h t h e v a h l e s o f Pr and S c. .. They e n t e r.

(61) FIG. 1.

(62) FIG. 2.

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(64) -=?. .rn. .. a. c. *. pkB. -4. FIG. 4. eo. 4. -. 0.

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(66) FIG. 6.

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