PDF Thesis by - California Institute of Technology

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 F o r 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

1 9 6 3

The author w i s h e s t o e x p r e s s 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 ; l a a - - c s e i n p r e p a r i n g t h i s work. _Thanks are d u e 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 s p o r , s o r a e 6 by the IISAi7 O f f i c e of Aerospace hesearch thraough i t s Ehrs- p 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 g r a d u a t e st^?:: zit Cal T e c h 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.

The structure of l a m i r i a r diffusion f l a n e s is : ~ : I L -

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 - ~ n r 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

o r

fuel consu.rnptlon, a z d temperature and ~ o n c e n t r a t i o n distributions outside of' the r t l a c t l o n zone may be obtained by ~ z s i n g the assum2zlofi of infinite reaction rates,

For large Reynolds numbers m i x i n g and c o ~ b u s t i o n t b k e

place i n boundary layers and free nixing layers. And again analytical solutfons are obtained for the temper- a t u r e and mass TractSon d i s t r i b u t i o n s o u t s I . d e of' the reaction zone.

PART P A G b

Acknowledgements ii

A b s t r a c t iii

Table of C o n t e n t s i v

Yomenclature I. I n t r o d u c t i o n

11. G e n e r a l E q u a t i o n s 5

11. 1. E q u a t i o n 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

t h e S p e c i e s 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 . D i m e n s i o n l e s s 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. G e n e r a l 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

I V . 1. The Chemical Boundary L a y e r

I V . 2. T h e S o l u t i o n o f - t h e Chemical Boundary E q u a t i o n s

I V . 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

V. 2. Mixing Gayer Equations

V. 3 . L o c a l Similarity Approximation

V. 4. Particular Cases VI. _R6s1xrn6

R e f e r e n c e s Appendix

A

Appendix B

Appendix C

E g u r e s

The f o l l o v ~ i n g i 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

D i f f u s i o n c o e f f i c i e n t

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 D i m e n s i o n l e s s 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 s u r f a c e

Frand t l number P r e s s 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 g a s c o n s t a n t

R e

Reynolds number

Sc Schmidt number

7-

Temperature

7 4 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

T C Temperature a t t h e i d e a l flame s u r f a c e

5

k

Characteristic chemical time d e f i n e d by 15

L

tm Characteristic mixing time,

t m = s,/&

U

C h a r a c t e r i s t i c o v e r a l l v e l o c i t y

<

U,V Velocity c o m p o n e n t s in boundary l a y e r c o o r - d i n a t e s

&

V V e l o c i t y v e c t o r

A

Vdi D i f f u s i o n v e l o c i t y o f s p e c i e s

i

wL Idass production rate,per u n i t volume,of species

i

X,Y

Mixing b o u n d a r y l a y e r c o o r d i n a t e s

.Ir.,

X P o s i t i o n v e c t o r

X,y Chemical boundary l a y e r c o o r d i n a t e s

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 Mixlng l e n g t h

, s,= ?,DO/&

S G 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 n o r m a l t o t h e mixing l a y e r

8 Non-dimensional t e m p e r a t u r e

,

⁰

S(T--T~)/(%~~) e, =

E / U ( ~ - T ~ )

/A Viscosity coefficient

3 Stoichiometric ratio s p e c i e s 2 t o P

5

Xon-dimensional d i s t a n c e a l o n g t h e mixing l a y e r

D e n s i t y

C S t r e s s t e n s o r

L CI

S u b s c r i p t s :

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 - t i v e l y

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 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 i n f i n i t e r e a c t i o n r a t e s

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 ,

.

D i f f u s i o n flames a r e o b t a i n e d when t h e r e a c t i n g s p e c i e s a r e i n i t i a l l y s e p a r a t e d . Combustion and mixing t a k e s p l a c e s i m u l t a n e o u s l y .

I n t h e s e f l a m e s 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 , t h r o u g h 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 i m p o r t a n t f e a t u r e s of d i f f u s i o n f l a m e s 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 z o n e s 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' b u r n i n g 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 u s e d 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 d i s c h a r g e s w i t h i n a t u b e . I n t h i s t u b e a n a i r s t r e a m 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

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 r u d i m e n t a r y approxima- t i o n s t h e y 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 d e t e r m i n e 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 e x p e r i m e n t s .

~ 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 o b t a i n e d 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 f l a m e s 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 g o v e r n i n g

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 p r o v i d e 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.

~ e l d o v i c h ( ~ ) 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

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 l consumption 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 c a n 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 , 'iVe 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 c a s e o f t h e mixing and combustion of t w o parallel s t r e a m s 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 c o n v e c t i o n 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 c o n d u c t i o n and chemical r e a c t i o n e f f e c t s . The g o v e r n i n g equations reduce in this case t o ordinary d i f f e r e n t i a l e q u a t i o n s . The k i n e t i c s of t h e r e a c t i o n appear i n t h e

s o l u t i o n ; b u t t h e t e m p e r a t u r e s a r e c l o s e t o t h e a d l a -

b a t i c flame t e m p e r a t u r e , and i n t h i s r a n g e of tempera- t u r e s t h e c o n c e p t 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 a s s u m p t i o n 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 o f 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 f l a m e 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 o f t h e mixing l a y e r may be d e t e r m i n e d by u s i n g t h e i n v f 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 w i t h i n t h e f l o w 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 s c h e m a t i c a l h y r e p r e s e n t e d

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 l a m i n a r flow of a r e a c t i n g g a s m i x t u r e ( )

.

^We w i l l u s e t h e assumption t h a t t h e f l u i d may be c o n s i d e r e d a s a c o n t i n u o u s medium formed b y 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 c o n s i d e r e d : &%el, o x i - d i z e r s , and p r o d u c t s . F o r t h e sake of s i m p l i c i t y , any i n e r t s p e c i e s p r e s e n t w i l l be c o n s i d e r e d 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

d

f l u i d mechanics, i . e . v e l o c i t y V , p r e s s u r e d e n s i t y 9

,

and t e m p e r a t u r e + , 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 e a c h o f t h e s p e c i e s , must be added t o t h e fundamental equa- t i o n s f l u i d mechanics. En a d d i t i o n , t h e r e P a t f ons between t h e t r a n s p o r t p a r a m e t e r s 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 m i x t u r e 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 . The mass f r a c t i o n s of s p e c l e s w i l l be w r i t t e n

K i = 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

11. 1. Equation o f S t a t e

If' t h e f l u i d i s c o n s i d e r e d a s a m i x t u r e 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 m o l e c u l a r mass.

I n o r d e r t o s i m p l i f y t h e c a l c u l a t i o n s w e w i l l u s e a mean c o n s t a n t v a l u e f o r

M .

T h i s a p p r o x i m a t i o n i s j u s t i f i e d when t h e m o l e c u l a r masses of t h e s p e c i e s a r e n o t v e r y 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 v e r y 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

M

a s v a r i a b l e .

11. 2. E q u a t i o n of C o n t i n u i t y f o r t h e Mixture T h i s 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. E q u a t i o n s of Mass C o n s e r v a t i o n 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' e a c h 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 con-

v 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 chem- i c a l r e a c t i o n ,

These e q u a t i o n s a r e a s f o l l o w s

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 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

i

'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 b e i n g 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

b

i s t h e f r e q u e n c y f a c t o r . Also i f $ i s t h e s t o i c h i o m e t r i c r a t i o o x ~ d i z e r - h e 1

We shall u s e r e l a t i o n 6a t h r o u g h 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 , ternp- e 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 . U s u a l l y 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 . T h i s 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 . T h e r m a l d i f f u s i o n 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 m o l e c u l a r masses o f t h e s p e c l e s are approx-

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 Ki

^{( 8 )}

where 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 . T h i s

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: f u e l - p r o d u c t s 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 . 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

! ? =

⁽¹⁰⁾

av av. au

Where (r

-

^{i s} t h e s t r e s s t e n s o r C.

_C1 =-3T&T/'(h.*5$i)

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 . G r a v i t y f o r c e s 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,

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 C p L 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

-

_.-.- ^{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 and

Af

i s t h e c h e m i c a l 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 o f 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 n o t t a k e n i n t o a c c o u n t ,

E q u a t i o n s 2,4,10, 11 and 9 ( f o r i = L 2 ), t o g e t h e r 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, c o n s t i t u t e 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 o f t h e d i f f u s i o n f l a m e s ,

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

o f 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

s i d e o f t h e flame

-

t h e f u e l e x i t

K i = K i F ^T

=TF and

T

=

To

Ki =

^{K i o 9,}

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 u s introduce the followf ng non-dimensional v a r i a b l e s .

Re

i s 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

S u b s c r i p t s 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

Here

9

and.

[$. ⁼

will be t h e adiabatic temperature o f the flame.

4

K,~(T~+K,F)

T;=q

K,,+S Kt, ^as we shall see later,

Then

t,

is a characteristic chemical time, such that

@r

^{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.

1 1 ⁰ DISCT7SSI OV OF TXE EQTJATI O N S h n J U L I M I ' I ' I I S G CASES

*

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- n i t u d e s t h e non-dimensional f a c t o r s a n d t e r m s i n equa- t 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 - g i o n s such a s boundary l a y e r s , shock w a v e s , f r e e mixing l a y e r s . I n t h e s e r e g i o n s t h e f u n c t i o n s ? ,

Ki , ^T

^{o r}

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 . 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 p a r a m e t e r s

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 6 b and 9 we d e d u c e

Also i f

V' <<

^j.

4

a n d

- -

^I

0 k 4 < 1

KO.

q

the e n e r g y 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, ^I

-rm) =[J-]

_RePr

⁴

_$

v.0 [pa *.o~, ^{*h 3}

If the form 1 7 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 t h e d i f f e r e n t i a l e q u a t i o n

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 condi- t i o n s a s g i v e n 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 c a s e 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 Schu- mann 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

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

M , = O

on one

'3

s i d e o f t h e flame,and K z = O on t h e o t h e r .

T h i s should be t r u e when

L/~t,

i s v e r y large. If b o t h and K2were d i f f e r e n t from zero i n a r e g i o n

.

where t h e t e m p e r a t u r e i s n o t low compared with

rf

^;

h e

(-%J

^{would be} of o r d e r unity and t h e term

-

-4-

ki~gy

w o u l d be v e r y l a r g e compared w i t h V *v*K; a nd

[&a$ ^{v**k* v*KJ,}

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

-

L

^{t ,}

^--r

-

t h e n we o b t a i n t h e r e s u l t

.

So e i t h e r K,:D o r

K,=

0

,

and system 1 4 t a k e s the f o l l o w i n g form, where k t , % ,,j= 2 on t h e f u e l r i d e of t h e flame and

=

3

, ^j

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 con- d 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 f l a m e *

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 e n e r g y 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

s t o i c h i o r n e t r F c 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=

*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 f l a m e , That i s

where 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% 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 - t i v e l y . 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 a n d v e l o c i t y , a r e c o n t i n u o u s f u n c t i o n s a t t h e f l a m e , 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 a c c o u n t when w r i t i n g t h e con- s e r v a t i o n e q u a t i o n s t h r o u g h t h e 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 v e l o c i t y , m a s s f r a c t i o n s , a n d t e m p e r a t u r e d i s t r i b u t i o n s .

I n p a r t i c u l a r , 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 1 6 and 18 ( v a l i d o n l y

The e q u a t i o n of t h e flame s u r f a c e i s o b t a i n e d by

J

w r i t i n g

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 i ~

^{+T;) K z o}

*KIF + KLO and

By w r i t i n g K L = O on t h e f u e l s i d e of t h e flame s u r f a c e and K , r O on t h e o x i d i ' z e r side,we o b t a i n 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 .

It i s i n t e r e s t i n g t o p o i n t o u t t h a t t h e Burke-Schu- mann 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 . T h i s 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 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 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 f l o w 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 c o n s i d e r e d . The r e a s o n f o r t h e s e m o d i r i c a t i o n s i s t h a t , a l t h o u g h 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 e x p l E c i t P y I n equatfons

1 6 and 18, t h e v a r i a b l e s

9 ,

C

v

and

/U

^{t h a t} a p p e a r i n 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 .

Yowever, w e may e x p e c t 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 z e r o t h i c k - n e s s , w i l l be a v e r y good a p p r o x i m a t i o n i n t h e c a s e of l a r g e b u t f i n i t e

L/(& .

^{T h i s} 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.

E q u a t i o n s 1 6 and 1 8 , in p a r t i c u l a r , s h o u l d 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 m i x i n g 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 a s t h e bound- a r y c o n d i t i o n s ( f o r l a r g e

L/u~,

^{), w i l l be} i n d e p e n - d e n t of' t h e r e a c t i o n r a t e s . The same w i l l happen t h e n t o t h e i r s o l u t i o n .

Snmming up: ^{I f} t h e r e a c t i o n r a t e i s s u f f i c i e n t l y 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 i m p o r t a n t 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 , 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 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 d i f f ' u s i o n 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,

IV. STRTJCTURE OF THE R E A C T I O N ZOXE:

IV. 1. 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 z e r o , 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

K;

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 l a r g e , a l t h o u g h 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 z o n e w i l l be s m a l l

a ( ^{3 )}

^and

--

b) The d i f f u s i o n terms

- _{7 a n} -

_pc

_an.

w i l l b a l a n c e t h e chemrcal p r o d u c t i o n terms d;/f, t h e t h r e e terms b e i n g v e r y barge compared w i t h a l l t h e o t h e r

a

i n d i c a t e d i f f e r e n - terms of t h e e q u a t i o n s . (Here

a -

t i a t i o n normal t o t h e 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 temper- a 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 c o n d u c t i o n 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 b a l a n c e t h e

chemical p r o d u c t i o n t e r m s . T r a n s p o r t by d i f f u s i o n o r c o n d u c t i o n 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 t h e flame

s u r f a c e l o c a t i o n f o r i n f i n i t e

L/ut, ,

^{and therefore}

t h e a p p r o x i m a t e 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- d i n e n s l o n a l c a s e . The r e s u l t s , however, a r e c o m p l e t e l y g e n e r a l . We s h a l l write the e q u a t i o n s of motion,and 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

d e t e r m i n e d 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 ; U, and V will be t h e v e l o c i t y components i n t h e % and

y

d i r e c - t i o n s ; 1 / ~ i s t h e radius o f c u r v a t u r e o f t h e f l a m e a t p o i n t 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 the l i n e e l e m e n t h a s components

('+~~)d%

^and

^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

Equations of motion

4 heve

V*\I clr =: I + K J S T + ~ ~

a& BE,= 1 t ~ 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

Now l e t and 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 v e l o c i t i e s . Let

6,

^{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 m i x i n g r e g i o n r e s p e c t i v e l y . i'ie

co1.1ld show t h a t f o r low Reynolds numbers,

SmuL ,

w h i l e 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

^{3 , by} s t r e t c h i n g P

8 :

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 - d e

Z

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 a c - c o u n t i n g f o r t h e c o n d u c t i o n 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 pro- d 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 e q u a t i o n s ,

s,/6,40

^.

,

most of t h e terms i n t h e s e e q u a t i o n s drop o u t . W e are left with t h e following 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 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 o r d e r

s , / ~

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 t h e p r e s s u r e i s c o n s t a n t a c r o s s t h e r e a c t i o n r e g i o n and e q u a l t o t h e v a l u e o b t a i n e d a t t h e f l a m e w i t h t h e Burke-Schumann assumption.

T h e r e f o r e 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

and T w e have t h e e q u a t i o n

where

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

f o r 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 a p p e a r i n system 26 t o ^29. T h e r e f o r e 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 ,% s t a n d s a s

8 p a r a m e t e r ,

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 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 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 .

I V . 2. The S o l u t i o n o f t h e Chemical Boundary Lager E q u a t i o n s

By i n t r o d u c i n g t h e new v a r i a b l e

8, = ~ ~ & / r ~ ) d g

⁹

i f w e assume t h a t

YP = !?or0

e q u a t i o n s 27 and 28 may be w r i t t e n

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 w e g e t .

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 c h e m i c a l r e a c t i m r a t e s . However t h e y a r e o n l y valid within t h e r e a c t i o n zone. The c o n s t a n t s

ai ^D;

^{must be chosen}

.

s o t h a t t h e s e r e l a t i o n s c o i n c i d e w i t h t h e s i m i l a r r e l a - t i o n s o b t a i n e d from t h e Burke-Schumann s o l u t i o n , a t least f o r l o w v a l u e s o f

.

^Then r e l a t i o n s 33 t o 34 may be w r i t t e n .

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 - f a c e . 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:

Where

8,s s a ,

^and

^{ea= %}

^{i s} t h e non-dimensional

4 Rq

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 K L r O f o r ^4- and 6 < 1 = 6 f o r

71

-

we o b t a i n

t h e t e m p e r a t u r e 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 t e m p e r a t u r e 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 %,+ 0 ⁹

@ = A a t

yl=s

Then a good app~oximate s o l u t i o n of 36 may be ob- tained by substituting the factor

'-"f

by its value at

& = o .

^{Let 8=0$~) for} g s o Then we approximate equation 36 by

This we may write in the form

And the boundary conditions are:

+ &

p F Z = o

f o r

z * -

Here

Equation 39 was solved numerically(15). An approx- imate solution is presented in Appendix

A.

Its solution

p=pc")

^P) ~ 3 8 ~ ) is represented in Fig. 2 with a solid line. I n particular

P(O) = ^0.866

F i g . 3 shows the variation with Z of

43L_zP9

which is proportional to the fuel mass consumption rate per unit volume. 0 For ~ = 3 . 2 its value is roughly one per

cant of its maximum value at Z = 0 Hence we may

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 them- i c a l boundary l a y e r i s o f t h e o r d e r of

64 .

According t o 38 t h e t e m p e r a t u r e a t t h e i d e a l ( k r k e -

~ c h a r n a n n ) 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 f o l l o w i n g r e l a t i o n

A f i r s t a p p r o x i m a t i o n 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 .

^{Then 8, i s} g i v e n by

A second approximation f o r

ec

v a l i d f o r v a l u e s of ec>0.8is

I V . D i s c u s s i o n 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 tempera- t u r e from i t s a s y m p t o t i c Burke-Schumann v a l u e

eG= h

depend on t h e parameter

A, .

T h i s 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 p a r a m e t e r s

-

t h r o u g h t h e v a l u e of t h e c h a r a c t e r i s t i c chemical time

t, -

^and t h e f l u i d

S,c.(,

2

m e c h a n i c a l p a r a m e t e r s

-

t h r o u g h t h e mixing time

ern@)= -

D o 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 asymp- t 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 i n v e r s e l y p r o p o r t i o n a l t o t h e 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- a t u r e w i l l d e c r e a s e .

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 , h e n c e t h e

same c r i t e r i o n *my be u s e d f o r t h e v a l i d i t y of t h e Burke- Schumann a s s u m p t i o n 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 o f t h e Burke-Schumann s o l u t i o n c o i n c i d e . F o r 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 t h e same i n b o t h s o l u t i o n s . The c h e m i c a l bound-

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 s m a l l 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 . T h i s c o r r e c -

. 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 o f 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 o f 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 ) D u e 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 e n e r g y of many of t h e chemical r e a c t i o n s , t h e chemical produc- t i o n term i s v e r y s e n s i t i v e ' t o t e m p e r a t u r e v a r i a t i o n s . T h i s a c c o u n t s f o r t h e 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 t e m p e r a t u r e f o r t h e r e a c t i o n ,

\ 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 t h r o u g h a l a r g e t e m p e r a t u r e 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 a n "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 as-

sumed 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 t h r o u g h o u t t h e

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 c a n n o t e x p l a i n e x t i n c t i o n e x c e p t i n a q u a l i t a t i v e wag, can p r o v i d e 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 t e m p e r a t u r e range.

Whenever t h e A,is s u f f i c i e n t l y low as

t o m a k e

T,40.8>

(and this o c c u r s f o r , r o u g h l y ,

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 b e g i n 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 a n approximate e x t i n c t i o n c r i t e r i o n a s w e l l a s a c r i t e r i o n f o r t h e 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 s e e n t h a t

A a c ? $ ~ 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

t h

i s s u b s t i t u t e d

\

by t h e f i e 1 consumption r a t e

6

p e r u n i t a r e a . 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 o f a t e x t i n c t i o n (maximum flame

s t r e n g t h ) and 6 ^:

V

.

^TNE EXTERNAL STHTJCTUHE OF' LA'i\."iT:'TAR DI ?'.'TiTiTSI 03 f"l,kT;r:S

V, 1, Large Reynolds Yurnber Case. I n v i s c i d idqua- t i o n s

The c h e m i c a l 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 s o l v e d 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 o f 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 . They i n c l n d e 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 - 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 o b t a i n e d 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 . U n f o r t u -

. n a t e l y , even when u s i n g t h e Burke-Schumann a s s u m p t i o n 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 c o m p l i c a t e d 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 num- b 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 approxima- t i o n s , Most of t h e s o P u t P o n s s o f a r o b t a i n e d make u s e

of t h e s e 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 d i - n e n s i o n 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

a s s u m p t i o n s , 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 t a k e 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 b o u h d a r y c o n d i t i o n s 22 c a n n o t be r e t a i n e d b e c a u s e , 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 h i g h e r o r d e r d e r i v a t i v e s i n ' t h e e q u a t i o n s .

B o w e v e r , 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 tem- p 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 s t r e a m s u r f a c e . 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

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 . See F i g . 1.

2 ) E q u a t i o n 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 bound- a r y c o n d i t i o n s on

3 .

The p r e s s u r e must, o f 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 e x a m p l e , t h e 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 x C.

V. 2 , Mixing L a y e r 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

,

t h e i d e a l d i s c o n t i n u i t y 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 m i x i n g 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 t h e s u r f a c e d i r e c t i o n , ^'

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

t h e w e l l I n o v ~ n boundary 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/

s u c h t h a t

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 . L e t

v @ , ~ ) = J-isK~~5) -

T ~ O ' L

U / U O = ~ I ~ ~ ~ )

^{( 5 1 )}

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

i=

^1,3

,,

on t h e f u e l s i d e of t h e flame a n d k 3 , 3

,, pi= 4

on t h e o t h e r s i d e . Thet.L.d.sljin

the

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 w i t h r e s p e c t t o

5 .

A s boundary c o n d i t i o n s w e may w r i t e

Where

Yt~@

^{gi-ves t h e} i d e a l flame p o s i t i o n . I n a d d i t i o n

For t h e s o l u t f on o f t h e above mixing p r o b l e m a t h i r d boundary c o n d i t i o n f o r

f

i s r e q u i r e d . T h i s should

be d e r i v e d from t h e c o r n p a t i b x l i t y 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 mag w r i t e a s t h i r d boundary c o n d i t i o n 0

t because t h e only e f f e c t o f changing t h e v a l u e of

f(O,3)

i s a displacement of t h e mixing l a y e r i n t h e

Y

d i r e c t i o n . V. 3. Local S i m i l a r i t v A ~ ~ r o x i m a t i o n

A s t h e boundary c o n d i t i o n s a r e independent of

5

t h e f u n c t i o n s

), K;

^and T would be f u n c t i o n s o n l y o f

7

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 p a r a m e t e r

$ 3

were c o n s t a n t .

U o

d$