CJ Rallis AM Garforth PECS 1980

7/29/2019 CJ Rallis AM Garforth PECS 1980

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Prog. Energy Com bust . Sci . ,Vol. 6, pp. 303-329. 0361~1285/80/12014)303505.00/0Pergamon Press Ltd., 1980. Printed in Gre at Britain.

T H E D E T E R M I N A T I O N O F L A M I N A R

B U R N I N G V E L O C IT Y

C . J . R A L L IS a n d A . M . G A R F O R T H

S c h o o l o f M e c h a n i c a l E n g i n e e r in g , U n i v e r s it y o f th e W i t w a t e r sr a n d , J a n S m u t s A v e n u e ,J o h a n n e s b u r g 2001, S o u t h A f r ic a

Abstrac t T h e r e l e v a n c e o f d a t a o n l a m i n a r b u r n i n g v e l o c it ie s , b o t h f r o m t h e i r v a l u e t o i n d u s t r y a n d r e l a t e dr e s e a r c h a r e a s , a n d f o r t h e v a l i d a t i o n o f t h e o r e t ic a l c h e m i c a l k i n e t i c m o d e l s , i s d is c u s se d , a s a r e c e r t a i n

f u n d a m e n t a l p r o b l e m s a s s o ci a te d w i t h t h e m e a s u r e m e n t o f t h i s i n t ri n s ic p r o p e rt y . V a r i o u s m e t h o d s w h i c hh a v e b e e n u s e d o v e r t h e y e a r s t o e x p e r i m e n t a l l y d e t e r m i n e t h i s p r o p e r t y a r e r e v i ew e d a n d i t is c o n c l u d e d t h a t

t h e s p h e r i c a l c o n s t a n t - v o l u m e ve ss el m e t h o d i s b o t h t h e m o s t v e r s a t i l e a n d a c c u r a t e . A s a c o n s e q u e n c e , a

s u m m a r y o f th e e q u a t i o n s r e q u i r e d f o r i ts c o r r e c t u s e a r e p r e s e n t e d . F o r t h e p u r p o s e o f c o m p a r i n g t h e r e s u l t so f t h e m o r e r e l i a b le t e c h n i q u e s , a s w e l l a s c o m p a r i n g t h e s e w i t h r e c e n t c o m p u t e r p r e d i c t i o n s , d a t a o n t h e

e ff ec ts o f e q u i v a l e n c e r a t io , p r e s s u r e a n d u n b u r n t g a s t e m p e r a t u r e f o r m e t h a n e - a i r m i x t u r e s a r e r e p o r t e d , a sa r e a n u m b e r o f e m p i r i c a l e q u a t i o n s c o r r e l a t i n g t h e s e v a r i a b l e s w i t h b u r n i n g v e l o c it y .

NOTATION

A a r e a ;

B d e f i n e d a s ( 1 / f l X g - ~XI - F 2) ;

C de f ined as - ( 1 / f l X r f l 3 X l - F 3 ), spec i f i c hea t capac i ty ;

D def ined as - C , d i f fus ion coef f ic ien t ;E de f ined as B F 2 / ( 1 - F 2 ) ;

F d e f i n e d a s ( 1 - ~ ;G def ined as ~F 3 + g (1 - F3) ;

h e n t h a l p y ;

k c o n s t a n t ;

K c o r r e c t i o n f a c t o r ;

m m a s s ;M m o l e c u l a r w e i g h t ;

n m a s s f r a c t i o n ;

p a b s o l u t e p r e s s u r e ;

r r a d i u s ;R b o m b i n n e r -w a l l r a d i u s , t u b e r a d i u s ;

R u n i v e r sa l g a s c o n s t a n t ;S ve loc i ty ;

t t ime ;

T a b s o l u t e t e m p e r a t u r e ;v v e l o c i t y ;W r e a c t io n r a t e ;

x d i s t a n c e ;

y d i s t a n c e ;

de f ined as (PdPo) , a n g l e ;

de f ined as ( P J P o ) ;

7 r a t i o o f h e a t c a p a c i t i e s ;

de f ined a s ( P / P o ) ;p d e n s i t y ;

r f l a m e - f r o n t t h i c k n e s s ;~b equ iva lence ra t io ;

S u b s c r i p t sb b u r n t g a s ;c c e n t r e , c o r r e c t i o n ;e a t e n d o f c o m b u s t i o n ;

f f l a m e f r o n t ;

i ign i t io n ;2 t h e r m a l c o n d u c t i v i t y ;o in i t i a l ;p c o n s t a n t p r e s s u r e ;p r p r e h e a t z o n e ;

s s p a t i a l ;t t r a n s f o r m a t i o n ( o r b u r n i n g ) , t h r o a t ;u u n b u r n t g a s ;u g u n b u r n t g a s ;w a t t h e b o m b w a l l ;

A b a r o v e r a n y s y m b o l i n d i c a t e s a n i n s t a n t a n e o u s l y s p a ti a l l y -

a v e r a g e d v a lu e . T h e s y m b o l ~ d e n o t e s n o n d i m e n s i o n a iquan t i t i e s .

1. INTRODUCTION

T h e m a i n a i m o f c o m b u s t i o n r e s e a r c h i s t h e a c-

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

i s m s o f i g n i t io n , s p e c ie s d i s t r i b u t i o n , f l a m e p r o p a g a -

t i o n a n d e n e r g y r e l e a se o f c o m b u s t i b l e m i x t u r e s. T h e

p r a c t i c a l r e s u lt s o f s u c h k n o w l e d g e a r e e v i d e n t l y t h e

c o n t r o l o f t h e c o m b u s t i o n p r o ce s s, b o t h f r o m t h e p o i n t

o f v ie w o f s a fe t y a n d i t s u t i l i z a t i o n a s a s o u r c e o f

e n e r g y .

U l t im a t e l y , su c h u n d e r s t a n d i n g c a n o n l y b e

a c h i e v e d t h r o u g h c h e m i c a l k i n e t i c s - - t h e m a s s , s p e ci e s

a n d e n e r g y e q u a t i o n s b e i n g u s e d t o p r e d i c t th e c o n -

d i t i o n s n e c e s s a r y f o r i g n i t i o n , a s w e l l a s t h e o v e r a l l

r a t e s o f r e a c t i o n ( H C 5 3 , H C 5 4 , S p 5 6 ). U n f o r t u n a t e l y ,

d u e t o t h e c o m p l e x i t y o f t h e p r o b l e m , t h i s m e t h o d h a s

o n l y r e c e n t l y s t a r t e d y i e l d i n g re s u lt s , a n d t h e n o n l y f o r

t h e s i m p l e r c o m b u s t i b l e m i x t u r e s ( S H 7 6 , T s7 8 ). I n a n y

e v e n t , t h e v a l i d a t i o n o f a n y t h e o r e t i c a l m o d e l r e q u i r e s

t h e a v a i l a b i l i ty o f r e li a b l e e x p e r i m e n t a l d a t a a g a i n s t

w h i c h i ts p r e d ic t i o n c a n b e c o m p a r e d ( S H 76 , H K 7 7 ,T s 7 8 ) .

T h e p r o x i m a t e g o a l o f g r a d u a ll y a c c u m u l a t i n g

k n o w l e d g e v i a a d h o c e x p e r i m e n t a t i o n i s t h u s s t il l

n e c e s s a r y - - b o t h i n th e s h o r t a n d l o n g t e r m s.

F u r t h e r m o r e , c o r r e l at i o n o f s u c h d a t a c a n p r o v i d e

v a l u a b l e c lu e s f o r e l u c i d a t i o n b y m o r e c o m p r e h e n s i v e

t h e o r e t i c a l t r e a t m e n t s . T o q u o t e L e w i s (L e 5 9) , " . . . i t

i s p o s s i b le t o d e v e l o p n e w c o n c e p t i o n a l u n d e r s t a n d -

i n g s f r o m e x p e r i m e n t a l o b s e r v a t i o n s a n d s i m p l e c o r r e -

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

s o m e p h y s i c a l b a s is in c o m m o n " .

T w o c o m p l e m e n t a r y e m p i r i ca l a p p r o a c h e s t o th e

s t u d y o f c o m b u s t i o n p r o c es s e s a r e a v a i l a b l e - -

m i c r o s c o p i c a n d m a c r o s c o p i c . T h e a i m o f t h e fo r m e r ,

w h i c h u s e m e a s u r i n g i n s t r u m e n t s o f h i g h s p a t i al

r e s o l u t i o n a n d s h o r t r e s p o n s e t i m e , i s t o p r o v i d e

~Pzcs 6:~ A 30 3



304 C .J . RALLIS nd A. M . GARFORTH

d e t a i l e d d e s c r i p t i o n s o f f l am e s t r u c t u r e t h r o u g h i n f o r-

m a t i o n o f t h e f lu i d f lo w , t e m p e r a t u r e a n d c o m p o s i t i o n

p r o f i le s , r e a c t i o n r a t e s a n d t r a n s p o r t c o e f f i c ie n t s o f t h e

v a r i o u s s p e c i e s p r e s e n t i n t h e f l a m e ( G o 7 6 ) . T h e l a t t e r

p r o v i d e d a t a o n t h e e ffe cts o f t h e r m o d y n a m i c v a r i a b l e s

s u c h a s c o m p o s i t i o n , t e m p e r a t u r e , a n d p r e s s u r e , o n t h e

g r o s s b e h a v i o r o f c o m b u s t i b l e m i x t u r e s : v i z . i n f l a m -

m a b i l i t y l i m i t s ; i g n i t i o n e n e r g i e s , te m p e r a t u r e s , a n d

d e l a y s ; q u e n c h i n g d i s t a n c e s ; i n s ta b i l i t y p h e n o m e n a ;

b u r n i n g v e l o c it i e s o f b o t h l a m i n a r a n d t u r b u l e n t

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

k n o c k ; a n d t h e l i k e .

T h i s r e v i e w w i ll b e p r i m a r i l y c o n c e r n e d w i t h t h e

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

p e r ti e s o f a n y c o m b u s t i b l e m i x t u r e - - i t s l a m i n a r b u r n -

i n g v e l o c i t y .

A n u m b e r o f ex c e l le n t r e v ie w s h a v e b e e n p u b l i s h e d

o v e r t h e y e a r s o n t h i s t o p ic . N o t a b l e a m o n g s t t h e s e a r e

t h o s e o f L e w i s a n d v o n E l b e ( L V 5 6 , L V 6 1 ) , L i n n e t t( L i 5 3 , L i 5 4 ) , F i o c k ( F i 4 3 , F i 5 5 ) , S i m o n a n d W o n g

( S W 5 3 ) , D u g g e r e t a l . ( D S 5 7 ) , G a y d o n a n d W o l f h a r d

( G W 6 0 ) , L af fi te a n d C o m b o u r i e u ( L a 62 , C o 6 2 a n d

L C 6 4 ), F r i s t r o m a n d W e s t e n b e r g (F W 6 5 ) , L i d l o w

( Li 6 7) a n d , m o r e r e c e n tl y , o f A n d r e w s a n d B r a d l e y

(ABV2).

W h a t i s p e r h a p s s u r p r i s i n g i s t h a t a l t h o u g h t h e

s u b j e c t o f th e b u r n i n g v e l o c i t y o f c o m b u s t i b l e m i x t u r e s

h a s b e e n s t u d i e d f o r c l o s e o n a c e n t u r y , t h e r e i s s t i ll a

l a c k o f c o n s e n s u s b o t h a s t o t h e m o s t e f f e ct iv e m e t h o d s

o f m e a s u r e m e n t a n d o n t h e r e l i a b i l i ty o f t h e p u b l i s h e d

d a t a f o r v a r i o u s m i x t u r e s.

2. FLAME-FR ONT STR UC TUR E

A s a p r e r e q u i s i te t o t h e r e l i a b le d e t e r m i n a t i o n o f

b u r n i n g v e l o c i t y i t is n e c e s s a ry t o h a v e s o m e u n d e r -

s t a n d i n g o f f l a m e - f ro n t s t r u c t u r e .

A f l a m e i s t h e r e s u l t o f a s e l f - s u s ta i n i n g c h e m i c a l

r e a c t i o n u s u a l l y m a d e v i s ib l e b y t h e l u m i n o s i t y o f t h e

b u r n i n g g a s e s . A s s o c i a t e d w i t h a f l a m e i s a f la m e f r o n t ,

i n w h ic h t h e u n b u r n t g a s is h e a te d a n d c o n v e r t e d i n t o

p r o d u c t s . W h e t h e r t h e f l a m e is s t a t i o n a r y o r m o v i n g i n

Flame

t.-

i#a

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O

s p a c e , t h e f l a m e f r o n t , w h i c h i s o f fi n i t e th i c k n e s s , i s

t a k e n a s a n i n d i c a t i o n o f t h e p r o g r e s s o f t h e f l am e .

T h e f l a m e fr o n t i s g e n e r a l l y c o n s i d e r e d t o c o n s i s t o f

t w o r e g i o n s , r e f e r r e d t o a s t h e p r e h e a t a n d r e a c t i o n

zones ( F ig . 1 ).

T h e p r e h e a t z o n e o c c u r s b e t w e e n t h e c o l d b o u n -

d a r y , a t t e m p e r a t u r e , T ,, a n d t h e l o c a t i o n o f t h ei g n i t io n t e m p e r a t u r e , Tv I n t h i s r e g i o n , t h e t e m p e r a -

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

c o n d u c t i o n a n d s o m e c o n v e c t i o n f r o m t h e r e a c t i o n

z o n e - - r a d i a t i o n h e a t t r a n s f e r b e i n g n e g l ig i b le . S i n ce

e a c h e l e m e n t i n t h i s r e g i o n a c t s a s a h e a t s i n k , t h e

t e m p e r a t u r e p r o f i l e i s c o n c a v e u p w a r d s ( 0 2 T / S x 2 > 0 ) .

A l s o , b e c a u s e o f t h e t e m p e r a t u r e i n c re a s e , t h e u n b u r n t

g a s e x p a n d s a n d i s a c c e l e r a t e d (F i g . 2) . N o s i g n i f i c a n t

c h e m i c a l r e a c t i o n o c c u r s i n t h i s z o n e ( D S 5 7 ).

O n r e a c h i n g i t s i g n i t i o n t e m p e r a t u r e , T~, e a c h

e l e m e n t o f g as s t a r t s u n d e r g o i n g c h e m i c a l r e a c t io n

w i t h a c o n s e q u e n t e v o l u t i o n o f h e a t , r e s u l t in g i n at e m p e r a t u r e p r o f il e w h i c h i s c o n c a v e d o w n w a r d s

( ~ 2 T / O x 2 < 0 ) . T h e t e m p e r a t u r e c o n t i n u e s t o r i s e u n t i l

i t s e q u i l i b r i u m v a l u e o f T i s r e a c h e d . T h e r e g i o n

b e t w e e n t h e l o c a t i o n o f t h e i g n i t i o n t e m p e r a t u r e , T~,

a n d t h e h o t b o u n d a r y a t t h e e q u i l i b r i u m t e m p e r a t u r e ,

Tr , i s r e f e r r ed to as the r e ac t ion z one ( F ig . 1 ).

T h e r e a c t i o n z o n e i s f u r t h e r d i v i s i b l e i n t o t w o p a r t s :

t h e p r i m a r y a n d s e c o n d a r y r e a c t i o n z o n e s . T h e

p r i m a r y z o n e i s a p p r o x i m a t e l y c o i n c i d e n t w i t h t h e

l u m i n o u s z o n e ( W F 6 1 ) , w h i l e t h e s e c o n d a r y z o n e i s

a s s o c i a t e d w i t h a n a r e a o f w e a k s e c o n d a r y l u m i n o s i t y

d u e t o C O o x i d a t i o n ( F W 6 5 ) .T h e w h o l e r e g i o n c o m p r i s i n g t h e p r e h e a t a n d

r e a c t i o n z o n e s - - c h a r a c t e r i z e d b y t h e t e r m f l a m e

f r o n t g e n e r a l l y h a s a s i g n i f i c a n t t h i c k n e s s . I d e a l l y ,

t h i s t h ic k n e s s i s t h e d i s t a n c e b e t w e e n t h e c o l d a n d h o t

b o u n d a r i e s o f th e f r o n t . H o w e v e r , s i n c e th e t e m p e r a -

t u r e p r o f il e a p p r o a c h e s b o t h T , a n d T a s y m p t o t i c a l l y ,

i t s e x t r e m i t i e s a r e u s u a l l y s p e c i f i e d b y t h e a r b i t r a r y

c o n d i t i o n s :( T ~ - - T ' ) / ( T ~ - - T,) = 0.99

f ron t thickness, T

( T ) - T 3 / ( T y - T~) = 0.99 (1)

" i

Ix ; x f

Preheat zone I [ React ion zone

D i st a nc e - x ~

FIG. 1. Ty pical tempe rature profile throu gh a f lame front.



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0,12

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0 ,6 ~ ~ ~ ~ 0 ,0 8

1 2 3 4 5 6 7 8

D I S T A N C E ( r n m )

FIG. 2. Tem perature, velocity, an d stream-tube are a profiles (Fr61).

w h e r e T " a n d T } a r e t h e t e m p e r a t u r e s a t t h e s e a r b i t r a r y

e x t r e m e s .

A p p l i c a t i o n o f e q . (1 ) t o t h e t e m p e r a t u r e d i s t r i b u t i o n

t h r o u g h t h e p r e h e a t z o n e v i a t h e c o n s e r v a t i o n e q u a -

t i o n s p r e s e n t e d i n S e c t i o n 3 y i e l d s a n e s t i m a t e o f t h e

t h i c k n e s s o f t h i s z o n e , g i v e n b y

r pr = 4 . 6 X / C p p , S , . (2 )

E v i d e n t l y ( F W 6 5 ) , t h i s r e l a t i o n s h i p i s r e p r e s e n t a t i v e

o f a t o t a l f l a m e t h i c k n e s s , z ' , d e f i n e d i n t e r m s o f t h e

a v e r a g e t e m p e r a t u r e g r a d i e n t t h r o u g h t h e f r o n t , a s

i n d i c a t e d i n F i g . 3 .

T ~ _.(_ _d / d x l a v e r a g e

T f ~ , , ~ / . _

- - J ' ~ - - . . . . T u xoTe

FIG. 3. Flam e-front thickness based on preh eat zone analysis.

T h e f o r e g o i n g d e s c r i p t i o n i s r e l e v a n t i n t h i s r e v i e w

f o r t w o r e a s o n s . A s w i l l b e d i s c u s s e d i n S e c t i o n 4 , t h e

d e t e r m i n a t i o n o f b u r n i n g v e l o c i t y i n e x p e r i m e n t s i n

w h i c h t h e f l a m e f r o n t i s n o t p l a n e , i s c r i t i c a l l y d e -

p e n d e n t o n t h e d e c i s i o n t a k e n r e g a r d i n g t h e l o c a t i o n

o f t h i s f r o n t ( L i5 3 ). A l s o , s i n c e a t l o w d e n s i t i e s ( p ,) a n d

b u r n i n g v e l o c i t i e s ( St ) t h e t h i c k n e s s o f t h e f l a m e f r o n t

c a n b e c o m e s i g n i f i c a n t , t h i s e ff ec t m u s t b e i n c o r p o -

r a t e d i n t o t h e e q u a t i o n s u s e d t o c a l c u l a t e b u r n i n g

ve lo c i ty ( Sec t ion 7 .2).

A s w i t h t e m p e r a t u r e a n d v e l o c it y , s p e c ie s c o n c e n -

t r a t i o n p r o f i l e s a l s o e x i s t w i t h i n t h e f l a m e f r o n t ( F i g . 4 ).

T h e l a t t e r a r e m u c h m o r e c o m p l e x t h a n e i t h e r t h e

t e m p e r a t u r e o r v e l o c i ty p r o fi l es , a n d a r e l a r g e l y d e -p e n d e n t u p o n t h e c o m b u s t i b l e m i x tu r e c o n c e rn e d .

T h e s e p r o f i l e s a r e n o t o f d i r e c t r e l e v a n c e i n t h i s r e v i e w

p r o v i d e d t h a t , a t a n y s t a g e t h r o u g h t h e f l a m e , t h e

s y s t e m m a y b e c o n s i d e r e d a s a m i x t u r e o f p e rf e c t g a s e s

f o r t h e p u r p o s e s o f d e n s i ty c a l c u l a t i o n .

T h e i n t e r e s t e d r e a d e r i s r e f e r re d t o t h e l i t e r a t u r e f o r

m o r e d e t a i l e d t r e a t m e n t s o f fl a m e s t r u c t u r e ( F r 6 1 ,

F W 6 5 ) .

3. CONSERVATIONEQUATIONS FOR APROPAGATING FLAME-FRONT

T w o m a i n m e t h o d s f o r c a l c u l a t i n g l a m i n a r b u r n i n g

v e l o c i ty a n d f l a m e s tr u c t u r e h a v e b e e n e v o l v e d d u r i n g

t h e p a s t t h r e e d e c a d e s . T h e o r d i n a r y d i f fe r e n ti a l e q u a -

t i o n s f o r h e a t c o n d u c t i o n , d i f f u s i o n a n d s p e c i e s c o n -

t i n u i t y d e r i v e d b y H i r s c h f e l d e r e t a l . ( H C 5 3 ) a n d

s o l v e d b y t h e m b y t h e s i m p l e s h o o t i n g m e t h o d r e -

p r e s e n t s o n e a p p r o a c h . H e r e , t h e f l a m e s p e e d i s t h ee i g e n v a lu e o f t h e t w o - p o i n t b o u n d a r y v a l u e p r o b l e m

s p e c if i ed b y t h e u n b u r n t a n d b u r n t g a s c o n d i t i o n s .

O f t e n , h o w e v e r , a n a s s u m e d b u r n i n g v e l o c i t y i s n o t

s u f fi c i en t t o i n i t i a t e t h e s o l u t i o n , a n d m u l t i - e i g e n v a l u e

s o l u t i o n s a r e n e c e s s a r y (C H 6 3 ). N o a c c e p t a b l e c o n v e r -

g e n c e c r i t e r i a e x i s t f o r s u c h s o l u t i o n s a n d , a l t h o u g h

o t h e r m e t h o d s h a v e b e e n d e v i s e d (K e 6 8 ), n o s t a n d a r d i -

z a t i o n h a s b e e n a c h i e v e d .

T h e o t h e r m a i n a p p r o a c h , u s e d b y S p a l d i n g ( S p5 6 ),

i n v o l v es s e tt i n g u p t h e e q u a t i o n s f o r c o n t i n u i t y a n d t h e

t i m e - d e p e n d e n t e n e r g y c o n s e r v a t i o n e q u a t i o n s . T h e s e

a r e s o l v e d b y f i n i te - d i ff e r e n c e m e t h o d s , w i t h a r b i t r a r y

i n i t i a l p r o f i l e a s s u m p t i o n s . T h e y a r e m o r e e a s i l y

a p p l i e d t o t h e u n s t e a d y p r o p a g a t i o n o f l a m i n a r fl a m e s

t h a n a r e t h o s e o f H i r s ch f e l d e r e t a l . A s u se d b y A d a m s

a n d C o o k ( A C 6 0 ), Z e l d o v i c h a n d B a r e n b l a t t ( Z B 5 9 ),

D i x o n - L e w i s ( D L 6 7 ) a n d S p a l d i n g e t a l . (SS71) , these

e q u a t i o n s h a v e b e e n a p p l i e d t o t h e s o l u t i o n o f th e o n e -

d i m e n s i o n a l fl a m e p r o b l e m . R e c e n t ly , t h e m e t h o d h a s

b e e n e x t e n d e d t o h a n d l e f la m e s i n c y l i n d r ic a l a n d

s p h e r i c a l c o - o r d i n a t e s b y B l e d j i a n ( B1 7 3 ) a n d D i x o n -

L e w i s a n d S h e p h e r d ( D S7 4 ).

P e r t i n e n t r e c e n t r e f er e n ce s c o v e r i n g t h e p r o c e d u r e s

a n d p r o b l e m s e n c o u n t e r e d i n c l u d e D S 7 4 , D G 7 5 ,

S H 7 6 a n d T s 7 8.I n t h i s f o rm t h e e q u a t i o n s a r e o n l y a p p l i c a b l e t o t h e

a d i a b a t i c f l a m e c a se . T h e s e e q u a t i o n s a r e :

(1) The m a s s c o n t i n u i t y e q u a t i o n - -

( c 3p /c 3 t) + ( 1 / r k ) ( c 3 / c ~ r ) ( r k p v ) = 0. (3)



306 C.J. RALLISand A. M. GARFORTH

0 1 0

0 0 9 0 9 4

0 0 8 0 9 2

0 . 0 7 0 9 0 o Z

g ~~- 0 06 08 8 ~-

0 0 5 ~ 6 ~o

0 0 4 0 8 4

0 05 D 82

0 0 2

0 0 l

O

0 I 2 3 4 5 6 7

Z ( m m )

F1G. 4. Composition profiles of lean ethylene-oxygen flame (gr61).

(2) The s p e c i e s c o n t i n u i t y equation--

(~yi / t~t ) + v( t~yi / t~r

= ( 1 / p r k ) ( O / O r ) [ r k D i p ( O y J d r ) ] + W ~ p . (4)

(3) The e n e r g y c o n s e r v a t i o n equation, in terms of static

temperature--

C . ( O T / ~ 3 0 + v C p ( c 3 T / O r ) = ( 1 / p r k ) ( ~ / ? ~ r ) [ r k ) j ~ T / O r ) ]

- - ~ ( W ~ h ] p ) (5)i= 1

+ .~ Di (~3yi /Or ) (~h, /Or )i- 1

where by setting k = 0, 1 or 2, the equations may be

written for cartesian, polar or spherical co-ordinates,

respectively.

Provided the detailed kinetics of the reaction pro-

cesses are k nown, or some simple relationsh ip between

both the overall reaction rate and the diffusion coef-

ficients and the gas properties of pressure and tempera-

ture may be assumed, a solu tion is possible (Sp57). The

method allows, e.g., the t ime-history of the develop-

ment of a steady-state flame to be computed (B173).

Vance and Krier (VK74) have presented a model for

spherical and cylindrical flames using a modified formof the above relationships which attempts to predict

the behavior of flames in the presence of conductive

and convective heat loss. They repor t a fair degree of

success with their method.

As is evident from the foregoing discussion, a

comprehensive solution to the problem of burningvelocity and flame structure calculations requires a

detailed knowledge of the reaction mechanism for

laminar flames. Even then, the success of the ana lyti cal

approach may on ly be assessed by comparison with

reliable experimental data. In most instances, such

data is only available over a limited range of pressures

and temperatures, albeit usually for the full range of

mixtures between the flammability limits. At the

present time, confusion still exists with the burning

velocity data of most fuels as "conflicting results

continue to be published" (AB72). Most authors who

currently approach the problem from the experimentalpoint of view attempt a correl ation with the analytical

results based on some simplified reaction scheme

(BH71) with limi ted success.

These two methods of studying flame propa-

gation-analytical and experimental--are clearly

complimentary and interdependent. The analytical

approach requires reliable experimental data both for

its execution and for comparison to assess its success.

On the other hand, the achievement of good experi-

mental values relies on some of the results of the

theoretical studies for its accuracy. In this context, the

corrections to burning velocity for flame thickness,

reported in Section 7.2, depend materially on thetemperature profiles through flames at pressures

where it is most difficult to measure them, due to the

small thickness of the flame front. Analytical models

can provide valuable information in the determi nation

of such correction factors.



The determin ation of lamin ar burning veloci ty 307

4. LAMINARBURN ING VELOCITY

O n e o f t h e m o s t i m p o r t a n t i n t r i n s i c p r o p e r t i e s o f

a n y c o m b u s t i b l e m i x t u r e i s i t s l a m i n a r b u r n i n g v e -

l o c i t y a n d t h e d e p e n d e n c e o f t h i s p r o p e r t y o n s u c h

v a r i a b l e s a s m i x t u r e c o m p o s i t i o n , t e m p e r a t u r e a n d

p r e s s u r e . H o p e f u l l y , a s h a s b e e n s u g g e s t e d i n t h ep r e v i o u s s e c t i o n , t h i s o b s e r v a b l e m a c r o s c o p i c e f f e c t

w i ll u l ti m a t e l y y i e l d to a n a l y t i c a l s t u d i e s - - a s m a y s u c h

a l t o g e th e r m o r e" c o m p l e x p h e n o m e n a a s t u r b u l e n t

f la m e v e l o c i t y a n d k n o c k i n s p a r k - i g n i t i o n e n g i n e s. I n

t h e m e a n t i m e , h o w e v e r , t h e r e i s a w e a l t h o f e x p e r i -

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

v e l o c i ty i s a n i m p o r t a n t v a r i a b l e i n c o r r e l a t i o n e q u a -

t i o n s fo r su c h p h e n o m e n a a s f la s h b a c k a n d f l a m e - ti l t

i n b u r n e r s , m i n i m u m i g n i t i o n e n e r g ie s o f e l ec t r ic

s p a rk s , a n d t u r b u l e n t f l am e v e l o c i t y - - t o m e n t i o n b u t

a f ew ( L V 6 1 ). E v i d e n t l y , t h e r e f o r e , a n a c c u r a t e k n o w -

l e d g e o f t h i s p r o p e r t y , t o g e t h e r w i t h t h e i n f l u e n c e o f

o t h e r v a r i a b l e s o n i t , i s i m p o r t a n t i n a n y c o m b u s t i o n

s t u d y .

T h e i g n i t i o n o f a c o m b u s t i b l e h o m o g e n e o u s m i x t u r e

f r o m s o m e p o i n t w i t h i n i t r e s u lt s in t h e p r o p a g a t i o n o f

a f l a m e . T h e v e l o c i t y o f s p r e a d o f s u c h a f l a m e r e l a t i v e

t o t h e p o i n t o f i g n i t i o n i s r e a d i l y m e a s u r a b l e . H o w e v e r ,

t h i s s p a t i a l v e l o c i t y ( S ~) i s n o t a u n i q u e p r o p e r t y o f s u c h

a m i x t u r e . I t c a n b e s h o w n t o b e t h e s u m o f tw o

v e l o c i t i e s , n a m e l y t h e b u r n i n g o r t r a n s f o r m a t i o n

v e l o c i t y ( S t ) a n d t h e u n b u r n t g a s v e l o c i t y i m m e d i a t e l y

adj ace nt to the f lame f r o n t ( S~g) : i .e .

S~ = S t + S , o . (6 )

T h e s e c o n d a n d g e n e r a l l y l a r g e r c o m p o n e n t (S ug ) i s

a p p a r e n t l y a f u n c t io n o f t h e r e l a t i v e d e n s i t ie s o f t h e

b u r n t a n d u n b u r n t g a s a t a n y i n s t a n t , a s w e ll a s o f t h e

p r e s e n c e o r a b s e n c e o f a n y c o n s t r a i n i n g b o u n d a r y .

T h e b u r n i n g v e l o c i t y (S t) i s d e f in e d a s t h e r e l a t i v e

v e l o c i t y , n o r m a l t o t h e f l a m e f r o n t , w i t h w h i c h t h e

u n b u r n t g a s m o v e s i n t o t h i s f ro n t a n d i s t r a n s f o rm e d .

I t i s c o n s i d e r e d t o b e a n i n t r i n s i c p r o p e r t y o f t h e

s p e ci fi c c o m b u s t i b l e m i x t u r e . S y m b o l i c a l l y , i t c a n b e

e x p r e s s e d a s ( R P 6 2 )

S t = - - ( 1 / A s p u ) ( d m d d t )

- - ( 1 / A r p ~ ) ( d m J d t ) (7 )

w h e r e A s i s t h e f l a m e f r o n t a r e a , p ~ i s t h e u n b u r n t g a s

d e n s i t y i m m e d i a t e l y a d j a c e n t t o t h i s a r e a , a n d ( d m d d t )

i s th e m a s s r a t e o f fl o w o f u n b u r n t g a s i n t o t h e f l a m e

f r on t , w h i c h i s e q u a l t o t h e m a s s r a t e o f f o r m a t i o n o f

b u r n t p r o d u c t s (drab~dr ) .

T h e m e t h o d s u s e d f o r d e t e r m i n i n g l a m i n a r v e l o c i t y

f a ll i n t o t w o c a t e g o r i e s : t h o s e i n w h i c h t h e f l a m e f r o n t

r e m a i n s s t a t i o n a r y i n s p a c e , i .e . S s = 0 o r S , = - - S , g ,

a n d t h o s e i n w h i c h i t m o v e s w i t h r e s p e c t t o s o m e f ix e d

p o i n t - - u s u a l l y t h e p o i n t o f ig n i ti o n .

T h e f i r s t c a t e g o r y c a n b e s u b d i v i d e d i n t o t w oc l a s s es : n a m e l y , d i f fu s i o n f l a m e s a n d p r e m i x e d f l a m e s .

E x a m p l e s o f t h e f o r m e r a r e t h e b u r n i n g o f ca n d l e s , o i l

w i c k s, w o o d o r c o a l . T h e b e s t e x a m p l e o f a p r e m i x e d

f l am e is t h a t o b t a i n e d i n a B u n s e n b u r n e r . W h e n t h e

c o m b u s t i b l e g a s e s a r e p r e m i x e d a n d t h e f l o w i s

l a m i n a r t h i s y i e ld s th e c h a r a c t e r i s ti c i n n e r c o n e a n d

m a n t l e o f a l l b u r n e r f l a m e s .

P r o p a g a t i n g f l a m e s m a y a l s o c o n v e n i e n t l y b e s u b -

d i v i d e d i n to t w o c l a s s es : n a m e l y c o n s t a n t v o l u m e a n d

c o n s t a n t p r e s s u r e . E x a m p l e s h e r e a r e t h e s p h e r i c a l

c o n s t a n t v o l u m e v e ss e l, s o a p b u b b l e a n d c y l i n d r i c a l

t u b e m e t h o d s . T h e i r s t u d y r e q u i r e s a m e a n s o f m e a s u r -i n g t h e v e l o c i t y o f p r o p a g a t i o n o f t h e f l a m e - f r o n t

r e l a t i v e t o t h e f i x e d p o i n t o f i g n i t i o n ( Ss ), a s w e l l a s a

m e a n s o f d e t e r m i n i n g t h e u n b u r n t g a s v e l o c i t y (S ug ).

T h e f o r m e r m a y b e a c h i e v e d e i t h e r p h o t o g r a p h i -

c a l l y u s i n g d i r e c t , S c h l i e r e n , s h a d o w o r i n t e r f e r o m e t r i c

m e t h o d s ; o r b y t h e u s e o f i o n i z a t i o n g a p s , f u si n g w i re s ,

t e m p e r a t u r e p r o b e s o r s i m i l a r d e v i c e s c a p a b l e o f

d e t e c t i n g t h e f l a m e f r o n t . T h e l a t t e r (S~o) m a y e i t h e r b e

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

m e a s u r i n g d e v i ce , s u c h a s a h o t - w i r e a n e m o m e t e r . I f

s u i t a b l e p a r ti c l e t r a c k i n g t e c h n i q u e s f o r o b s e r v i n g t h e

g a s m o t i o n i n b o t h t h e u n b u r n t a n d b u r n t r e g io n s ca nb e d e v e l o p e d , t h e n S t fo l l o w s d i r e c t l y f r o m e q . ( 6) . I n

t h e c o n s t a n t v o l u m e m e t h o d a n a d d i t i o n a l d e s i r a b l e

o b s e r v a t i o n - - i f f ul l a d v a n t a g e i s t o b e t a k e n o f t h i s

m e t h o d - - i s t h e c h a n g e i n p r e s s u r e in t h e v e s se l d u r i n g

t h e c o u r s e o f t h e f la m e .

E a c h o f t h e f o r e g o i n g m e t h o d s h a s c e r t a in a d -

v a n t a g e s a n d d i s a d v a n t a g e s . T h e s e w i l l b e d i s c u s s e d i n

g r e a t e r d e t a i l i n S e c t i o n 6 .

5. F U N D A M E N T A L P R O B L E M I N BURNINGVELOCITY MEASUREMENT

M a n y o f t h e d i f fi c u lt ie s a s s o c i a t e d w i t h t h e d e t e r m i -

n a t i o n o f l a m i n a r b u r n i n g v e l o c i ty c e n t r e a r o u n d t h e

p r e c is e m e a n i n g s g i v e n t o t h e v a r i a b l e s i n e q . ( 7 ) - - a n d

i n p a r t i c u l a r t o t h e f l a m e - f r o n t a r e a A f .

T h e c a s e o f a n i n f i n i t e p l a n e f l a m e , i n w h i c h t h e

f l am e f r o n t i s n o r m a l t o t h e d i r e c t i o n o f f lo w o f t h e

u n b u r n t g a s , i s t h e o n l y o n e i n w h i c h t h e s e d i f f i cu l t i es

d o n o t a r is e . I n a l l o t h e r s y s t e m s " . . . n o d e f i n i ti o n fr ee

f r o m a l l p o s s i b l e o b j e c t i o n s c a n b e f o r m u l a t e d " ( L i 53 ) .

U n f o r t u n a t e l y , t h e c l o se s t a p p r o x i m a t i o n t o a n i n f i n it e

p l a n e f l a m e , t h e f l a t - fl a m e b u r n e r o f P o w l i n g a n d

E d g e r t o n ( P o 4 9 , P o 6 1 , E T 5 2 ) , i s l i m i t e d i n t h i s a s w e l l

a s o t h e r r e s p e c t s .

T h e b a s i c q u e s t io n t o w h i c h a n a n s w e r i s re q u i r e d i s

t h e r e f o re : W h i c h p a r t o f t h e f l a m e s h o u l d b e s e l e ct e d

f o r m e a s u r e m e n t p u r p o s e s ? E v i d e n t l y , f r o m e q . ( 7),

a n y s u r f a c e w i t h i n t h e f l a m e f r o n t a t w h i c h t h e

c o r r e s p o n d i n g v a l u e s o f d e n s i t y ( p) a n d m a s s f lo w r a t e

( d m / d t ) c a n b e a c c u r a t e l y a n d r e p r o d u c i b l y d e t e r -

m i n e d w o u l d b e a c c e p t a b l e . U n d o u b t e d l y , t h e b e s t

s u r fa c e is t h a t a t w h i c h t h e t e m p e r a t u r e j u s t s t a r t s t o

d e v i a t e f r o m t h e u n b u r n t g a s v a l u e ( T ~ , F i g . I ) .

U n f o r t u n a t e l y , d u e t o t h e a s y m p t o t i c n a t u r e o f t h e

t e m p e r a t u r e p r o f i l e t h i s p o s i t i o n i s u n m e a s u r a b l e .

T e m p e r a t u r e o r d e n s i t y m e a s u r i n g d e v i c e s o f h i g hs p a t i a l r e s o l u t i o n a n d s e n s i t i v i ty , s u c h a s S c h l i e r e n

p h o t o g r a p h y o r l a s e r i n t e r f e ro m e t r y (G a 7 6 ), w o u l d

s e em t o b e i n d i c a t e d - - p r o v i d e d c o r r e sp o n d i n g v a l u e s

o f a r e a a n d t e m p e r a t u r e a r e u s e d i n e q . (7 ).

F r i s t r o m h a s s h o w n t h a t t h e i n n e r r e g i o n o f t h e



308 C .J . RALLIS nd A. M. GARFORTH

l u m i n o u s z o n e r e p r e s e n t s t h e b e s t l o c a t i o n a t w h i c h t o

m e a s u r e g a s v e l o c i t i e s a n d a r e a s f o r c u r v e d t h i c k

f lames ( F r 65) . U s ing eq . ( 7) he de f ines a bur n ing

v e l o c i ty , S * , i m m e d i a t e l y a d j a c e n t t o t h e r e a c t i o n

s u r f a c e o f a r e a A * , a n d d e n s i t y , p * , a s

S * = ( a ' p ' / A * p * ) S c

w h e r e A ' , p ' a n d S't a r e t h e v a l u e s m e a s u r e d a t s o m e

o t h e r s u r f a c e.

5.1. M e t h o d s o f O b s er v in 9 t h e F l a m e F r o n t

V a r i o u s m e t h o d s o f l o c a t i n g t h e p o s i t i o n o f th e

f l a m e f r o n t h a v e b e e n p r o p o s e d a n d u s e d ( D S5 7 ).

T h e s e i n c l u d e :

(1 ) D i r e c t p h o t o g r a p h y o f t h e l u m i n o u s f l am e (L P 5 1 ,

G W 5 3 , S E 5 9 ).

(2 ) S h a d o w p h o t o g r a p h y u s i n g a p o i n t s o u r c e o f l i g h t

( S L 48 , G L 4 9 , A F 4 9 , A F 5 0 ) .

(3 ) S c h l i e re n p h o t o g r a p h y u s i n g e i th e r a c o a r s e g r a t-

i n g i l l u m i n a t e d fr o m b e h i n d ( G W 5 3 , G i 5 7 ) o r a n

o p t i c a l s y s t e m u s i n g l e n s e s o r m i r r o r s ( C L 5 1 ,

P L 5 1 , B W 5 3 , D i 5 3 , B W 5 4 , S e6 1 ).

( 4) I n t e r f e r o m e t r y ( O I 4 9 , G W 5 3 ) .

( 5 ) P a r t i c l e t r a c k m e a s u r e m e n t s ( p h o t o g r a p h i c o r

l a s e r D o p p l e r ) ( S m 3 7 , V L 4 3 , A F 5 0 , F A 5 4 , F r 5 6 ,

G o 7 6 , E P 7 6 ).

( 6) I o n i z a t i o n g a p s ( E S 5 7 , S A 5 7 , E A 5 8 , G r 5 9 , A G 6 1 ) .

(7 ) T e m p e r a t u r e m e a s u r e m e n t s ( K W 4 8 , F r 5 3 , G L 5 3 ,

G W 5 3 , F B 5 4 ) .

S i n ce t h e i n t e n s i t y o f t h e l u m i n o u s z o n e i s g e n e r a l l y

s u ff ic i e nt l y h i g h , p a r t i c u l a r l y f o r h y d r o c a r b o n m i x -

t u r es , d i re c t p h o t o g r a p h y i s p o s s i b l e a n d h a s

f r e q u e n t l y b e e n u s e d . H o w e v e r , a s w a s d i s c u s s e d i n

S e c t i o n 2 , t h i s z o n e i s l o c a t e d s o m e d i s t a n c e b e h i n d t h e

i n i t ia l t e m p e r a t u r e r i s e a n d h e n c e d o e s n o t r e p r e s e n t

t h e s t a r t o f t h e i n t e r a c t i o n b e t w e e n t h e u n b u r n t g a s

a n d t h e c o m b u s t i o n w a v e . ( F i g s 1 a n d 2 ) . I n b u r n e r

f l a m e s , t h e t h i c k n e s s o f t h e p r e h e a t r e g i o n a h e a d o f th e

l u m i n o u s f r o n t is f o u n d t o v a r y f ro m a b o u t 1 m m a t

a t m o s p h e r i c p r e s s u r e ( A F 5 0 ) t o a b o u t 1 0 m m a t 1 / 2 0

a t m o s p h e r e ( K W 4 8 ) . T h i s s u r f a c e w o u l d t h e r e f o r ea p p e a r t o b e u n s u i t a b l e f o r d e t e r m i n i n g b u r n i n g

v e l o c i t y ( f or l o w - p r e s s u r e b u r n e r f l a m e s a t a n y r a t e )

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

b e m e a s u r e d .

S h a d o w p h o t o g r a p h y h a s b e e n s t u d i e d f a ir l y e x t e n -

s i v e l y ( S L 4 8 , W o 4 9 , A F 5 9 ) , a n d i t h a s b e e n e s t a b l i s h e d

t h a t t h e s h a r p i n n e r s h a d o w g r a p h e d g e i s d e p e n d e n t

o n t h e d i s t a n c e b e t w e e n t h e f la m e a n d t h e s c r ee n o r

p h o t o g r a p h i c p l a t e . F u r t h e r , t h is e d g e is l o c a t e d a h e a d

o f t h e p r e h e a t r e g i o n a n d a p p r o a c h e s t h e s t a r t o f t h i s

r e g i o n a s t h e d i s t a n c e b e t w e e n f l a m e a n d s c r e e n i s

d e c r e a s e d . T h i s w e l l - d e f i n e d e d g e m a y t h e r e f o r e o n l y

b e u s e d i f s u i t a b l e c o r re c t i o n s c a n b e m a d e . O n t h eo t h e r h a n d , t h e o u t e r s h a d o w g r a p h e d g e , w h i c h i s

c o i n c i d e n t w i t h t h e S c h l i e r e n e d g e , a n d i s n o t d e -

p e n d e n t o n d i s t an c e , i s n o t w e l l -d e f i n ed a n d i s h e n ce

d i f fi c u l t t o m e a s u r e . F o r t h e s e a n d o t h e r r e a s o n s ,

s h a d o w p h o t o g r a p h y i s c o n s i d e r e d u n r e l i a b l e , a n d

G a y d o n a n d W o l f h a r d r e c o m m e n d t h a t i t b e a b a n -

d o n e d ( G W 5 3 ).

O f t h e o p t i c a l m e t h o d s , t h e b e s t m e a s u r e m e n t s o f

t h e s t a r t o f t h e p r e h e a t z o n e a r e a f f o r d e d b y t h e

S c h l ie r e n , i n t e r fe r o m e t r i c a n d p a r t i c l e t r a c k m e t h o d s .

I n t e r f e r o m e t r y , a l t h o u g h p o s s e s s i n g m a n y a t t r a c t i v e

f e a t u r e s , c a n b e e x p e n s i v e a n d c o m p l i c a t e d t o u s e . I na n y c a s e , i t i s d o u b t f u l w h e t h e r f o r t h i s p u r p o s e i t c a n

p r o v i d e m u c h a d d i t i o n a l i n f o r m a t io n a s c o m p a r e d

w i t h S c h l i e r e n o r p a r t i c l e t r a c k t e c h n i q u e s .

S c h l i e r e n m e t h o d s y i e l d a f o c u s s e d i m a g e o f t h e

f la m e , e n a b l i n g t h e p o s i t i o n o f m a x i m u m i n t e n s it y ,

g i ve n a p p r o x i m a t e l y b y ( - 1 / T Z ) ( d T / d x ) , t o b e e a s i l y

l o c a t e d . I n b u r n e r f l a m e s , t h i s a p p a r e n t l y o c c u r s a t

a b o u t 2 0 0 ° C ( K W 4 8 ) . I t h a s , h o w e v e r , b e e n p o i n t e d

o u t t h a t " . . . b e c a u s e o f t h e o p t i c a l a r r a n g e m e n t f o r

p h o t o g r a p h y t h e S c h l ie r e n i n a c o n i c a l f l am e m a y b e a t

m u c h l o w e r t e m p e r a t u r e , a n d t h i s d i s p l a c e m e n t w i ll

d e p e n d o n t h e t h i c k n e s s o f t h e p r e - h e a t i n g z o n e , a n dt h e r e fo r e o n t h e b u r n i n g v e l o c it y . T h e S c h l ie r e n m a y

t h u s s e r v e t o l o c a t e t h e p o s i t i o n o f t h e f ir st t e m p e r a t u r e

i n c r e a se w h i c h i s r e q u i r e d f o r m e a s u r e m e n t o f b u r n i n g

v e l o c i t y " ( G W 5 3 , p . 6 7) . S m o k e ( B r4 9 ) a n d p a r t i c l e

t r a c k ( A F 5 0 , G L 5 2 , G e 5 3 ) m e a s u r e m e n t s i n b u r n e r

f l a m e s c o n f i r m t h a t t h e f l o w li n e s r e m a i n p a r a l l e l to t h e

b u r n e r a x i s u n t i l t h e S c h l i e r e n i m a g e i s r e a c h e d . T h e r e

s e e m s l i t t le d o u b t , t h e r e f o re , t h a t w h e r e v e r p o s s i b l e t h e

S c h l ie r e n e d g e s h o u l d b e u s e d f o r b u r n i n g - v e l o c i t y

s t u d i e s ( G H 5 0 ) ( F i g . 5 ).

FLOWLINES /V IS IBLE E DGE

, S CH L I E R E N\ ~ EDGE

~ , t l N N E RSHADOWE DGE

FIG. 5. Relation of f low lines to f lame images (GH50).

T h e p a r t i c l e t r a c k m e t h o d i s a n e x t r e m e l y v e r s at i let e c h n i q u e f o r t h e s t u d y o f f l am e s . B y i ts u s e , v o n E l b e

a n d L e w i s ( V L 4 3 ) w e r e a b l e t o s h o w t h a t b u r n i n g

v e l o c i t y i s a " g e n u i n e p h y s i c a l c o n s t a n t " ( G W 5 3 ) .

H o w e v e r , i t i s a p t t o b e r a t h e r t e d i o u s a s a m e a n s o f

d e t e r m i n i n g b u r n i n g v e l o c it i e s per se . C a r e h a s t o b e

e x e r c i s e d t h a t t h e p a r t i c l e s u s e d d o n o t h a v e a n y

c a t a l y t i c ef fe ct s o n t h e f l a m e a n d a l s o t h a t t h e y a r e

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

d i r e c t i o n o f t h e s t r e a m t u b e s . N e v e r t h e l e s s , i t i s o n e o f

t h e m o s t p o w e r f u l t e c h n i q u e s a v a i l a b l e (C W 6 3 , S G 5 6 ,

A F 4 9 , L i 7 6 , L i 6 8 , R M 7 1 , F W 6 5 , B S 6 9 , E H 6 9 ) .

I t h a s l o n g b e e n k n o w n t h a t t h e e l e c t ri c a l c o n d u c -

t i v i ty o f f l a m e g a s e s is h i g h a s c o m p a r e d w i t h t h eu n b u r n t g a s. A m e t h o d o f d e t e c t i n g th e p o s i t i o n o f t h e

f l a m e f r o n t a s a r e s u l t o f s u c h c h a n g e s i n c o n d u c t i v i t y

( b y m e a n s o f i o n i z a t i o n g a p s ) h a s b e e n u s e d b y A g n e w

a n d h i s c o w o r k e r s i n t h e i r st u d i es o f f l am e p r o p a g a t i o n

i n s p h e r i c a l c o n s t a n t - v o l u m e v e s s e l s ( E s5 7 , S a 5 7 ,



The determination of lamina r burning veloci ty 309

E a 5 8 , G r 5 9 , A G 6 1 ) . E v i d e n t l y s u c h i o n i z a t i o n g a p s

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

( C K 5 5 , K N 5 9 ) . I f t h e f l a m e - f r o n t t h i c k n e s s i s v a r i a b l e ,

a s i t m i g h t w e l l b e i n c o n s t a n t - v o l u m e e x p e r i m e n t s ,

t h i s t e c h n i q u e m a y i n t r o d u c e s i g n i f i c a n t e r r o r .

F u r t h e r , u n l e ss c a r e i s e x e rc i s e d re g a r d i n g t h e p r o b e

d i m e n s i o n s , t h e r e i s a p o s s i b i l i t y o f d e f o r m i n g t h ef l am e f r o n t d u e t o c o o l i n g a n d / o r q u e n c h i n g e ff ec ts

( P B 5 9 ). S i n c e it a p p e a r s t h a t t h e b e s t s u r f a c e a t w h i c h

t o l o c a t e t h e f la m e f r o n t i s t h a t a t w h i c h t h e t e m p e r a -

t u r e j u s t s t a r t s t o d e v i a t e f r o m t h e u n b u r n t g a s v a l ue ,

e v i d e n t l y t h e m o s t d i r e c t d e t e r m i n a t i o n i s v i a s o m e

f o r m o f d e n s i t y o r t e m p e r a t u r e m e a s u r i n g d e v i c e o f

h i g h s p a t i a l r e s o l u t i o n a n d s e n s it i v it y .

T o s u m m a r i z e , t h e re f o r e, i t w o u l d a p p e a r t h a t : (1 ) i f

a n o p t i c a l m e t h o d o f o b s e r v i n g t h e f l a m e fr o n t i s to b e

u s e d t h e n e i t h e r S c h l i e r e n o r p a r t i c l e t r a c k i n g t e c h -

n i q u e s a r e i n d i c a t e d , a n d (2 ) i f n o n o p t i c a l m e t h o d s a r e

n e c e s s ar y , a d e n s i t y o r t e m p e r a t u r e s e n s in g d e v i c e o fs r h a l l s i ze , h i g h s e n s i t i v i t y a n d s h o r t r e s p o n s e - t i m e

w o u l d s e e m t o b e p r e f er a b l e .

I t m u s t b e s t r e s s e d , h o w e v e r , t h a t t h e b u l k o f t h e

e x p e r i m e n t a l e v i d e n c e r e l a t i n g t o f l a m e - fr o n t s t r u c t u re

h a s b e e n o b t a i n e d o n l o w - v e l o c i t y , l o w - p r e s su r e ,

s t a t i o n a r y f l a m e s . F o r s u c h f l a m e s , t h e t h i c k n e s s o f t h e

p r e h e a t r e g i o n i s g i v e n a p p r o x i m a t e l y b y e q . (2 ). I f t h i s

e q u a t i o n i s e q u a l l y a p p l i c a b l e t o h i g h - v e l o c it y , h i g h -

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

p r e h e a t z o n e m a y n o t b e s i g n if ic a n t . A n y o f t h e a b o v e

m e t h o d s w o u l d t h e n b e e q u a l l y e f fe c ti v e. H o w e v e r ,

t h e r e i s a p o s s i b i l i ty t h a t t h e c o n t r o l l i n g m e c h a n i s m sa n d r e a c t i o n s c h e m e s d e d u c e d f r o m s t a t i o n a r y f l am e s

m a y n o t b e a p p l i c a b l e t o p r o p a g a t i n g f l am e s (G W 5 3 ,

p . 8 1 ) . W h e r e v e r p o s s i b l e t h e r e f o r e , a l t e r n a t i v e

m e t h o d s o f o b s e r v i n g th e f l a m e f r o n t w o u l d a p p e a r

t o b e d e s i r a b l e a s a c h e c k o n e a c h o t h e r .

6. M E T H O D S O F M E A S U R I N G L A M I N A R

BURNING VELOCITY

S i n c e l a m i n a r b u r n i n g v e l o c i ty i s k n o w n t o b e

d e p e n d e n t o n t h e t e m p e r a t u r e , p r e s s u r e a n d c o m -

p o s i t i o n o f t h e u n b u r n t m i x t u r e , a n y e x p e r i m e n t f o r it s

d e t e r m i n a t i o n s h o u l d e n s u r e a c c u r a t e c o n t r o l o r

m e a s u r e m e n t o f t h e s e p a r a m e t e r s . F u r t h e r , t h e f la m e

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

a n y s u r fa c e w h i c h c o u l d c a u s e h e a t i n g , q u e n c h i n g o r

c a t a l y t i c e ff ec ts . I f a p l a n e s u r f a c e i s u n a t t a i n a b l e , t h e n

t h e f l a m e s h o u l d a t l e a s t h a v e a l a r g e r a d i u s o f

c u r v a t u r e c o m p a r e d w i t h t h e a n t i c i p a t e d f l a m e - f r o n t

t h i c k n e s s . I n s u c h c a s e s , t h e s u r f a c e s h o u l d p r e f e r a b l y

b e o n e - d i m e n s i o n a l , s o t h a t i t is a m e n a b l e t o r i g o r o u s

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

c o r r e c t i o n s f or f l a m e - fr o n t t h i c k n e ss a n d c u r v a t u r e t o

b e a p p l i e d .

O n t h e e x p e r i m e n t a l si d e , t h e m e t h o d s h o u l d b e a se c o n o m i c a l a s p o s s i b l e o n f u el , p e r m i t w i d e v a r i a t i o n s

i n u n b u r n t g a s t e m p e r a t u r e , p r e s su r e a n d c o m p o -

s i t i o n , y i e l d r e p r o d u c i b l e r e s t i l t s a n d , i f p o s s i b l e , b e

s i m p l e t o u s e .

T h e m e t h o d s w h i c h h a v e b e e n u s e d t o d e t e r m i n e

l a m i n a r b u r n i n g v e l o c i ty h a v e b e e n c a t e g o r i z e d e i th e r

i n t e r m s o f t h e c o m p l e x i t y o f t h e i r f l a m e - f ro n t s h a p e s

( L i 5 3 , L i 5 4 ), o r i n t h e t w o c l a s se s o f s t a t i o n a r y a n d

p r o p a g a t i n g f l am e s (A B 72 ). A c o m b i n a t i o n o f th e s e

t w o a p p r o a c h e s w o u l d s e e m t o b e c o n v e n ie n t . W e w i ll

t h u s t r e a t e a c h o f th e s e t w o c l a s s e s - - s t a t i o n a r y a n d

p r o p a g a t i n g f l a m e s - - i n o r d e r o f d ec r e as i n g c o m -p l e x i t y o f t h e i r f l a m e - f r o n t s h a p e s .

6.1. S t a t i o n a r y F l a m e s

L a m i n a r b u r n i n g v e l o c i t ie s c a n b e m e a s u r e d b y

c a u s i n g a p r e m i x e d c o m b u s t i b l e m i x t u r e i n w h i c h t h e

f l o w i s l a m i n a r t o e n t e r a s t a t i o n a r y f l a m e f r o n t ( S ~

= 0 ) w i t h a v e l o c i t y e q u a l t o t h e b u r n i n g v e l o c i t y (S~o

= - S t ) . I d e a l l y , t h e fl a m e f r o n t s h o u l d b e p l a n e .

H o w e v e r , s i n c e a s t a t i o n a r y f l a t fl a m e i s u n s t a b l e , s o m e

m e a n s m u s t b e p r o v i d e d fo r a n c h o r i n g t h e f lo w a b o v e

t h e b u r n e r m o u t h . B y d e f i n it i o n , a l l s u c h m e t h o d sd i s t o r t t h e f l a m e f r o n t t o a g r e a t e r o r l e s s e r e x t e n t .

6.1.1. B u r n e r m e t h o d s

B u r n e r f l a m e s o f o n e k i n d o r a n o t h e r h a v e b e e n

s t u d i e d e x t e n s iv e l y a n d h a v e p r o v i d e d v a l u a b l e i n s ig h t

i n t o a h o s t o f c o m b u s t i o n p r o b l e m s (J o 4 6 , LV 6 1 ) . A s a

c o n s e q u e n c e, m o s t o f th e e a r l i e r b u r n i n g v e l o c i t y d a t a

a v a i l a b l e i n t h e l i t e r a t u r e h av e b e e n o b t a i n e d b y

b u r n e r m e t h o d s ( J o 5 6 , G W 5 3 , L i 5 3 , L V 6 1 , A B 7 2 ) .

D o u b t l e s s , t h e m a i n r e a s o n s f o r t h i s a r e t h a t i n i t s

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

s iv e , v e r s a t i l e , a n d s u p e r f i c i a l l y e a s y t o u s e . H o w e v e r ,

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

p a r a t u s m u s t b e c o n s i d e r e d o f d o u b t f u l r e l i a b i l it y .

V a r i o u s k i n d s o f b u r n e r h a v e b e e n t r i e d - - c i r c u l a r

t u b e , s h a p e d n o z z l e , o r if i c e , a n d r e c t a n g u l a r s l o t t y p e s .

I n t h e c i r c u l a r - t u b e a n d r e c t a n g u l a r - s l o t t y p e s , a t

a p p r o p r i a t e l y l o w R e y n o l d s n u m b e r s , th e u n b u r n t g a s

v e l o c i t y p r o f i l e i s p a r a b o l i c , w h i l s t s u i t a b l y d e s i g n e d

n o z z l e a n d o r i f i ce b u r n e r s y i e l d e s s e n t i a l l y u n i f o r m

v e l o c i t y p r o f i l e s ( e x c e p t i n t h e b o u n d a r y l a y e r ) a n d

s t r a i g h t - s i d e d f l a m e c o n e s ( F i g . 6).

I n e s s e n c e , t h e s e m e t h o d s e n t a i l e s t a b l i s h i n g l a -

m i n a r f l o w i n a v e r t i c a l c o n d u i t , t h e f l a m e b e i n g h e l d

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

g a s e s . S in c e t h e s p a t i a l v e l o c i t y , S s, i s t h e n z e r o , t h e

b u r n i n g v e l o c it y , S t, a t a n y p a r t i c u l a r p o i n t o n t h e

f l a m e - f r o n t c o n e i s n u m e r i c a l l y e q u a l t o t h e n o r m a l

c o m p o n e n t o f t h e g a s v e l o c i ty a t t h a t p o i n t . T h a t i s

( F ig . 7 )

St = Sug sinct. (9)

A l t e r n a t iv e l y , t h e a v e r a g e b u r n i n g v e l o c i t y o v e r t h e

w h o l e f l a m e - fr o n t c o n e c a n b e o b t a i n e d f ro m m a s s

c o n t i n u i t y , t h u s

S , = K n / i , A f . (10)A p r e f e ra b l e , b u t m o r e t e d i o u s , a p p r o a c h t o t h e

f o r e g o i n g i s t h e u s e o f p a r t i c l e t r a c k i n g , s o s u c c e s s f u l ly

d e m o n s t r a t e d b y L e w i s a n d v o n E l b e ( LV 6 1 ) a n d L e v y

a n d W e i n b e r g ( L W 5 9 ) ( s e e F i g . 8 ) . I n t h i s t e c h n i q u e ,

s t r o b o s c o p i c a l l y i l l u m i n a t e d p a r t i c l e s p r o v i d e d i r e c t



310 C .J. RALLIS nd A. M. GARFORa'n

FIG. 6. Nozzle bu rner w ith straight-sided flame cone (Fr57).

m e a s u r e m e n t o f b o t h t h e v e l o c i t y a n d d i r e c t i o n o f t h e

c o n t a i n i n g s t r e a m t u b e . U s e o f t h i s m e t h o d e n a b l e d

L e w i s a n d v o n E l b e (L V 6 1 ) t o s h o w t h a t t h e f l a m e

v e l o c i ty w a s c o n s t a n t o v e r m o s t o f th e f l am e f r o n t ( F i g .

9 ) . R e s u l t s u s i n g t h i s a p p r o a c h , o r m o d i f i c a t i o n

t h e r e o f , a r e r e p o r t e d i n R e f s A F 4 9 , S G 5 6 , C W 6 3 ,

L i 6 7 a , L i 6 8, R M 7 1 , F W 6 5 a n d B S 6 9.

O n e o t h e r m e t h o d w h i c h h a s b e e n u s e d t o d e t e rm i n e

b u r n i n g v e l o c i t i e s i n b u r n e r s i s t h e s o - c a l l e d f l a m e

t h r u s t m e th o d . F r o m t h e m o m e n t u m e q u a t i o n f or o n e -

d i m e n s i o n a l f lo w it c a n b e s h o w n t h a t t h e p r e s s u r e

~ a A f

S u g ~ ' ~ , ~ o A t

FIG. 7. Fla me cone geom etry.

d i f f e r e n c e a c r o s s a f l a m e f r o n t i s g i v e n b y ( L v 6 1 ,

G W 7 0 )

P , , - - P b = p u S 2 ( p , J P b - - 1). (11)M e a s u r e m e n t o f th i s p r e s s u r e d i f f e re n c e , u s i n g a

m i c r o p i t o t - t u b e p r o b e , t o g e t h e r w i t h c a l c u l a t io n o f p ,

a n d P b, c a n y i e l d t h e b u r n i n g v e l o c i t y , S ,. B e c a u s e o f t h ev e r y l o w p r e s s u r e d i f f e r e n c e s a v a i l a b l e t h e a t t a i n a b l e

a c c u r a c y i s p o o r . N e v e r th e l e s s, t h e a p p r o a c h h a s b e e n

u s e d w i t h s o m e s u c c e ss , p a r t i c u l a r l y f o r h i g h b u r n i n g -

v e l o c i ty m i x t u r e s ( E H 6 9 , F W 6 5 , B S 5 7 , G W 6 0 , E H 7 0 ).

A l t h o u g h s m a l l , i t i s th i s e ff ec t w h i c h c a u s e s d i v e r g e n c e

o f th e u n b u r n t g a s f l o w -l in e s i n t u b e b u r n e r s a n d

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

( F W 6 5 ) . T h i s d i v e r g e n c e i s q u i te p r o m i n e n t i n i n v e r t e d

f l a m e c o n e b u r n e r s ( F i g . 1 0 ) w h i c h , i n c i d e n t a l l y ,

a p p e a r t o h a v e a n u m b e r o f a t t r a c t i v e f e a t u re s w h e n

u s e d w i t h p a r t i c l e t r a c k i n g f o r l a m i n a r b u r n i n g v e -

l o c i t y m e a s u r e m e n t s .

A l t h o u g h e s s e n t ia l l y s i m p l e , b u r n e r m e t h o d s s u ff er

f ro m t h e f o l l o w i n g in h e r e n t d i s a d v a n t a g e s :

(1 ) L a c k o f u n i f o r m i t y o f t h e b u r n i n g v e l o c i ty o v e r t h e

f l a m e s u r fa c e , p a r t i c u l a r l y a t t h e f l a m e t i p a n d n e a r

t h e b u r n e r r i m ( L V 61 , G W 7 0 ) .

( 2) D i f f i c u l ty o f e s t a b l i s h i n g t h e r e l e v a n t u n b u r n t g a s

t e m p e r a t u r e p r o f i le t h r o u g h a n y s e c t i o n o f t h e

f l a m e c o n e . S i n c e b u r n i n g v e l o c i t i e s a r e s t r o n g l y

d e p e n d e n t o n u n b u r n t g a s t e m p e r a t u r e , t h i s i s a n

e s s e n t i a l p i e c e o f i n f o r m a t i o n ( G R 7 8 ) .

(3 ) N o n a d i a b a t i c n a t u r e o f t h e f la m e p a r t i c u l a r l y n e a r

t h e b a s e o f th e c o n e ( G W 7 0 ) .

( 4) A i r e n t r a i n m e n t i n t o t h e f l a m e - c o n e b a s e , e s p e -

c i a l l y w i t h r i c h m i x t u r e s ( K U 6 2 ) .

( 5) D i s t o r t i o n o f t h e f l a m e c o n e d u e t o f l a m e t h r u s t

( F r 57) .

( 6) E f fe c t s o f v a r i a b l e f l a m e - f r o n t t h i c k n e s s ( G W 7 0 ) .

( 7) C a t a l y t i c a n d i n e r t i a l ef fe ct s o f s o l i d p a r t i c l e s w h e n

t h e p a r t i c l e t ra c k i n g m e t h o d i s u se d ( A F 4 9 ) .

I f t h e u s e o f a b u r n e r m e t h o d i s i n e v i t a b l e , t h e n a

n o z z l e o r i n v e r t e d c o n e b u r n e r w i t h p a r t i c l e tr a c k i n g

a n d S c h li e re n p h o t o g r a p h y i s r e c o m m e n d e d - -

p a r t i c u l a r l y if th e S c h l i er e n c o n e c a n b e a s s o c i a t e d

w i t h a p a r t i c u l a r u n b u r n t g a s t e m p e r a t u r e ( K W 4 8 ,A B72) .

C o m p r e h e n s i v e r e v ie w s o f b u r n e r m e t h o d s a r e t o b e

f o u n d i n R e f s J o 4 6 , L V 61 , G W 7 0 a n d A B 7 2 .

6.1.2. Fla t - fl ame me tho d

T h i s m e t h o d , d u e t o P o w l i n g a n d E d g e r t o n ( Po 6 1 ),

p r o v i d e s a c l o s e a p p r o x i m a t i o n t o t h e i d e a l o n e -

d i m e n s i o n a l f l a t fl a m e , b u t i s u n f o r t u n a t e l y l i m i t e d t o

l o w b u r n i n g v e l o c i t ie s ( 0 . 1 5 - - 0 . 2 0 m / s e c ). T h e a p -

p a r a t u s u s e d i s s h o w n i n F i g . 1 1 . H e r e , t h e p r e - m i x e d

c o m b u s t i b l e g a s es e n t e r a c y l i n d r i c a l b u r n e r t u b e A , o f

a p p r o x i m a t e l y 6 0 m m d i a ro ~ t e r, f ro m b e l o w . A f t e rb e i n g e v e n l y d i s t r i b u t e d a c r o s s t h e w h o l e d i a m e t e r o f

t h e t u b e b y a m a t r i x E , g la s s b e a d p a c k i n g B , a n d f in e

d i f f u s io n s c re e n s C , t h e m i x t u r e e n t e r s t h e v e r t i c a l

c h a n n e l s o f t h e m a t r i x D . T h i s m a t r i x i s b u i lt u p o f

a l t e r n a t e l a y e r s o f p l a i n a n d c o r r u g a t e d m e t a l t a p e , a s



The determin ation of lamin ar burning veloci ty 311

1FIG. 8. Particle track ph oto grap h of a f lat f lame with sketch o f directio n of observed flow (LW59).

s h o w n , r e s u l t in g i n a u n i t o f h ig h d e n s i t y a n d l o w

p r e s s u r e d r o p . I n p a s s i n g t h r o u g h t h e c h a n n e l s o f D,

a n y t u r b u l e n c e i n t h e g a s is e li m i n a t e d a n d l a m i n a r

f l o w i s s e t u p i n e a c h c h a n n e l . T h e d i s t a n c e o f t h e t o p o f

D b e l o w t h e b u r n e r p o r t i s a d j u s t e d t o a v a l u e ( u s u a ll y

5 - 1 0 m m ) s u c h t h a t t h e s e i r r e g u l a r i t i e s i n v e l o c i t y a r e

e l i m i n a t e d , b u t p a r a b o l i c f l ow h a s n o t b e g u n t o b e

e s t a b l i s h e d i n t h e m a i n b u r n e r t u b e . A w i r e g a u z e o r

p e r f o r a t e d a s b e s t o s b o a r d p l a c e d a b o v e t h e f l a m e

s e r v e s t o s t a b i l i z e i t b y s e t t i n g u p a s y s t e m o f v o r t i c e s

o n t h e f l a m e r i m ( F i g . 8 ) ( L W 5 9 ) . T h e r e s u l t i n g f l a m e

h a s t h e a p p e a r a n c e o f a f ia t d i s k w i t h a s l i g h t l y c u r l e d -u p e d g e . T h e a r e a o f t h is d i s k d i v i d e d i n t o t h e

v o l u m e t r i c f l o w - r a te o f t h e m i x t u r e y i e l d s th e b u r n i n g

v e l o c i t y .

S i n c e f l o w r a t e s c a n g e n e r a l l y b e a c c u r a t e l y d e -

t e r m i n e d t h e a c c u r a c y o f t h is m a n n e r o f u si n g t h e

m e t h o d w o u l d a p p e a r t o d e p e n d o n t h e m e a s u r e m e n t

o f th e d i s k d i a m e t e r . H o w e v e r , t h e p a r t i c l e t r a c k

p h o t o g r a p h s o f L e v y a n d W e i n b e r g ( F i g . 8 ) ( L W 5 9 )

s u g g e s t t h a t s o m e u n b u r n t g a s m a y e s c a p e a t t h e f l am e

e d g e s , t h u s l o w e r i n g t h e c a l c u l a t e d v a l u e o f S t . A l s o ,

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

n i t r o g e n , t h e e x a c t p o s i t i o n o f th e f l a m e e d g e i s

u n c e r t a i n . I t i s c l a i m e d t h a t p r o v i d e d t h e b u r n e r

d i a m e t e r i s l a rg e , t h e e r r o r f r o m t h i s c a u s e i s s m a l l . B u t

f o r f la m e s t a b i l i t y r e a s o n s , la r g e d i a m e t e r s m a y o n l y b e

u s e d f o r s l o w b u r n i n g m i x t u r e s . F a s t e r f l a m e s r e q u i r e

n a r r o w e r b u r n e r s . A l t e r n a t i v e l y , c o n c e n t r i c n o z z l e

b u r n e r s m a y b e u s e d . P o w l i n g ( P o 6 1 ) c la i m s t h a t w i t h

s u c h a n a r r a n g e m e n t , e s s e n t i a l l y f l a t f l a m e s c a n b e

o b t a i n e d u p t o b u r n i n g v e l o c it i e s o f a b o u t 1 .0 m / s e e .

A n o t h e r d i s a d v a n t a g e o f th i s m e t h o d i s t h e h e a t l o s s

f r o m t h e f l a m e a n d a s s o c i a t e d h e a t i n g o f th e m a t r ix .

T h e f o r m e r r e n d e r s t h e f la m e n o n a d i a b a t i c , t h e l a t t e rp e h e a t s t h e u n b u r n t m i x t u r e - - l o w e r i n g a n d r a i s i n g S t,

r e sp e c t iv e l y . B o t h a a n d S p a l d i n g ( B S5 4 ) u s e d a p o r o u s

s i n t e r e d b r o n z e p l a t e f o r s t a b i l i z i n g t h e f lo w , a n d

m e a s u r e d t h e h e a t t r a n s f e r re d t o t h i s p l a t e a t v a r i o u s

f lo w r a te s . B y e x t r a p o l a t i n g t h e r a t i o o f t h e v o l u m e t r i c

f l o w r a t e t o f l a m e d i s k a r e a ( " a p p a r e n t b u r n i n g

v e l o c i t y " ) t o z e r o h e a t t r a n s f e r , i t i s c l a i m e d t h a t t r u e

a d i a b a t i c b u r n i n g v e l o c it i e s w e r e o b t a i n e d . T h i s s e e m s

a r e a s o n a b l e c l a i m , p a r t i c u l a r l y s i n ce t h e u n b u r n t g a s

t e m p e r a t u r e , T , , w o u l d b e c o n t r o l l e d b y t h e w a t e r

c o o l i n g p r o v i d e d .

2. 0

k.-Zt~

- 1 . 5

u .

O

~,1,0

2 s \ \ I \ \ R \ \

t Ve loci ty ector,200 cm. per second

o 1 I I1 . 5 1 , 0 0 , 5 2 0 :0.5

m e n t ~

f l

(c ~

1 . 0 1 . 5 0 2 0 4 0 6 0

CENTIMETERS BURNINGVELOCITY,CENTIMETERS ER SECOND

FIG. 9. Com bustion-zone f lame pa ttern a nd burning veloci ty determined through ver tical centre p lane of anatural gas-a ir flame on a rectang ular burner tube (LV61) .



312 C .J . RALLIS nd A. M. GARFORTH

FIG. 10. Inverte d flame-cone with particle tracks (LV61

B e c a u s e o f t h e f o r e g o i n g d i f f i c u l t i e s , L e v y a n d

W e i n b e r g ( L W 5 9 ) p r e f e r t h e u s e o f p a r t ic l e t r a c k i n g

t e c h n i q u e s w i t h fl a t - f la m e b u r n e r s . D i x o n - L e w i s a n dW i l l i a m s (D W 6 7 ) u s i n g t h i s a p p r o a c h h a v e r e p o r t e d

h i g h e r v a lu e s f o r m e t h a n e - a i r t h a n t h o s e o b t a i n e d b y

t h e a r e a m e t h o d ( E T 5 2) . B o t h B o t h a a n d S p a l d i n g ' s

r e s u lt s f o r p r o p a n e - a i r (B S 54 ) a n d E d m o n d s o n a n d

H e a p ' s f o r m e t h a n e - a i r ( E H 7 0 ) a r e l o w e r t h a n t h o s e

o b t a i n e d b y t h e p a r t ic l e - t ra c k - c o n e - a n g l e m e t h o d s

u s i n g n o z z le b u r n e r s . S o m e d o u b t m u s t , h o w e v e r , b e

e x p r e s s e d a b o u t t h e r e l e v a n t v a l u e o f T , i n t h e s e l a t t e r

e x p e r i m e n t s .

6.2. P r o p a g a t i n 9 F l a m e s

T h e i g n i t i o n o f a q u i e s c e n t, h o m o g e n e o u s c o m -b u s t i b le m i x t u r e f r o m s o m e p o i n t w i t h i n i t r e su l t s in

t h e p r o p a g a t i o n o f a f la m e . T h e s u b s e q u e n t s p r e a d o f

s u c h a f l a m e i s d e t e r m i n e d , inter alia, b y t h e n a t u r e o f

t h e b o u n d i n g s u r f a c e b e t w e e n t h e m i x t u r e a n d i t s

s u r r o u n d i n g s . T h r e e t y p e s o f b o u n d i n g s u r f a c e h a v e

b e e n u s e d : r i g i d c y l i n d r i c a l t u b e s , e i t h e r c l o s e d a t b o t h

e n d s o r o p e n a t o n e o r b o t h e n d s ; s o a p b u b b l e

s o l u t i o n s o r t h i n e l a s t ic m e m b r a n e s ; a n d , r i g i d

spher ica l ves se l s .

Wind(

f i o n

Sliding ou ter-

jacket

Coolingwater in

c ~ m b ~

Cos mixture

' -- Cool ing jack et

C o o ~ i n g water

FIG. 11. F lat-flam e burn er (Po61).




6.2.1. C y l i n d r i c a l t u b e m e t h o d s

T h e e a r l y l it e r a t u r e c o n t a i n s m a n y v a l u e s o f f la m e

s p e e d s m e a s u r e d i n t u b e s ( B T 2 7 , J o 4 6 , L V 6 1 ) .

H o w e v e r , i t i s d o u b t f u l w h e t h e r t h e s e r e s u l t s a r e o f a n y

s i g n i f i c a n c e i n d e t e r m i n i n g b u r n i n g v e l o c i t y , s i n c e t h is

m e t h o d i s s u b j e c t t o s e r i o u s w a l l i n t e r a c t i o n e f f e c t s ,o n e r e s u l t o f w h i c h i s t o d e f o r m t h e f l a m e f r o n t . T h u s ,

t h e s p a t i a l v e l o c i t y o f a n y g i v e n m i x t u r e m a y i n c r e a se

m a n y t i m e s - - a s a r e s u l t o f l e ss w a ll i n t e r a c t i o n - - a s t h e

t u b e d i a m e t e r i s in c r e a s e d ( F i4 3 ). H o w e v e r , v i b r a t i o n s

u s u a l l y t h e n s e t in , a n d t h e f l a m e - f r o n t s h a p e t e n d s t o

b e c o m e i r r e g u l a r a n d d i f fi c u l t t o m e a s u r e . A l s o , d i f -

f e r e n t v a l u e s a r e o b t a i n e d d e p e n d i n g o n w h e t h e r t h e

f la m e p r o p a g a t e s i n a n u p w a r d , d o w n w a r d o r h o r i z o n -

t a l d i re c t i o n . C e r ta i n l y , o t h e r t h a n f o r t u b e d i a m e t e r s

c l o se t o t h e q u e n c h i n g d i s t a n c e, f la m e - f r o n t v i b r a t i o n s

a p p e a r d i f f i c u l t t o a v o i d , p a r t i c u l a r l y f o r m i x t u r e s

i g n i t ed b y m e a n s o f a s p a r k r a t h e r t h a n a n o p e n f l a m e .

T h e e x c e l le n t se t o f p h o t o g r a p h s g i v e n b y F l o c k ( F i 4 3 )

i l l u s t r a t e s o m e o f t h e s e p o i n t s ( F i g . 1 2 ).

* D I R E C T I O N O F F I L M M O T I O N

m e t h o d ( C o 6 1 , G L 5 1 , C L 5 9 , H H 5 6 , E P 6 9 ) . T h e r e l e -

v a n t e q u a t io n n o w b e c o m e s (G M 4 8 )

S , = (S s - S u g ) n R 2 / A s . (13)

H e r e t h e u n b u r n t g a s v e l o c i t y , S ~g , s d e t e r m i n e d f r o m

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

o r if i ce a t t h e u n b u r n t - g a s e n d .I n a f u r t h e r a d a p t a t i o n o f t h is m e t h o d , F u l l e r e t a l .

( F P 6 9 ) i g n i te d t h e m i x t u r e a t b o t h e n d s , t h u s p r o d u c -

i n g a d o u b l e - f l a m e k e r n e l w i t h r e l a t i v e l y f l a t f l a m e -

f r o n t s - - t h u s r e d u c i n g t h e e r r o r i n d e t e r m i n i n g A z .

T h i s i s s i m i l a r t o t h e t e c h n i q u e u s e d b y R a e z e r a n d

O l s e n i n a s p h e r i c a l v e s se l ( R O 6 2 ) ( s ee a l s o F i g . 1 2 D) .

P r o v i d e d v i b r a t i o n s ca n b e a v o i d e d a n d t h e u n b u r n t

g a s b e t w e e n t h e f l a m e k e r n e l s i s s t a t i o n a r y , t h e n S t

= S~. H o w e v e r , Tu a n d p u a l s o n e e d t o b e m e a s u r e d i f

t h e y d o n o t r e m a i n c o n s t a n t .

E v e n w i t h a l l t h e a b o v e r e f i n e m e n t s , i t i s d o u b t f u l

w h e t h e r t h e w a l l i n t e r a c t i o n e f fe c ts c a n e v e r b e a d -

e q u a t e l y c o r r e c t e d f o r . O n t h e w h o l e , t h e r e f o r e , t h e

m e t h o d a p p e a r s t o b e i n h e r e n t l y u n s a ti s f a ct o r y .

F1G. 12. T ypic al record s of explosions in closed and open-ended tubes (Fi43). A, B, C --t ub e closed at b othends, initial pressure 1, 2/3, and 1/3 atm osph ere respectively. D -- tu be closed at both en ds, initial pressure 1atmosphere, f ired simultaneously a t both ends. E -- to p end o f tube open, pressure 1 atmosphere, f ired at

closed end. F -- to p end o f tube open, pressure 1 atmosphere, f ired at open end.

I n e s s e n c e , t h e m e t h o d a s u s e d b y C o w a r d a n d

H a r t w e l l ( C H 3 2 , C P 3 7 ) c o n s i s t s o f a l o n g c y l i n d r i c a l

t u b e c l o s e d a t o n e e n d a n d f il le d w i t h t h e g a s m i x t u r e

u n d e r t e s t . I g n i t i o n a t t h e o p e n e n d r e s u l t s i n a n

i n i t i a ll y u n i f o r m f l a m e t r a v e l l i n g to w a r d s t h e c l o s e d

e n d ( F i n F i g . 1 2 ) . M e a s u r e m e n t o f t h i s u n i f o r m

v e l o c i t y , S s, t o g e t h e r w i t h t h e a r e a , A f , o f th e p s e u d o -

h e m i s p h e r i c a l f l a m e - f ro n t y i e l d s an a v e r a g e b u r n i n g

v e l o c i ty v i a t h e f o l lo w i n g e q u a t i o n , d i r e c t l y d e r i v a b l e

f r om the de f in ing eq . ( 7 ) :

S t = S s T Z R 2 / A f (12)

w h e r e R i s t h e t u b e r a d i u s .T h e p r o v i s i o n o f a s m a l l h o l e a t t h e e n d o f t h e t u b e

t o w a r d s w h i c h t h e f l a m e tr a v e l s , a n d a l a r g e r o n e a t t h e

o t h e r e n d , a p p a r e n t l y r e n d e r s t h e f l a m e m o v e m e n t

u n i f o rm , s t a b l e a n d r e p r o d u c i b l e ( G M 4 8 , G L 5 1 ) . M o s t

s u b s e q u e n t w o r k e r s h a v e u s e d t h i s r e v i s e d t u b e

6.2.2. F l a m e k e r n e l m e t h o d

F o l l o w i n g o n t h e o r e t i c a l a n d e x p e r i m e n t a l w o r k b y

D e r y ( D e 4 9 ) o n t h e i g n i t io n , g r o w t h a n d t r a n s p o r t o f a

f l am e k e r n e l i n a l a m i n a r c o m b u s t i b l e g a s - s tr e a m , B o l z

a n d B u r l a g e ( B B 5 5 , B B 6 0 ) a t t e m p t e d t o u s e t h i s

t e c h n i q u e t o m e a s u r e b u r n i n g v e l o c i t y . M e a s u r e m e n t s

o f th e f l a m e a r e a a n d h e n c e t h e r a d i u s o f a n e q u i v a l e n t

s p h e r e a s t h e f l a m e k e r n e l w a s c a r r i e d d o w n s t r e a m ,

y i e l d e d S s , f r o m w h i c h S t w a s d e t e r m i n e d f r o m t h e

e q u a t i o n

S t = ( p b / p o ) S s = o rS s (16)

t h e d e n s i t y r a t i o , a , b e i n g d e t e r m i n e d f r o m t h e r m o -c h e m i c a l c a l c u l a t io n . T h e m e t h o d h a s t h e a d v a n t a g e o f

r e m o v i n g t h e f l a m e f r o m t h e i n fl u e n c e o f th e s p a r k

e l e c t r o d e s , b u t s u f fe r s f r o m t h e d i s a d v a n t a g e o f a

c o m p l e x f l a m e - f r o n t s h a p e s i n c e t h e k e r n e l i s n o t

s p h e r i c a l .



314 C .J . RALLIS nd A. M . GAm~ORTH

I n a n a d a p t a t i o n o f t h is a p p r o a c h , R a e z e r a n d O l s e n

( R O 6 2 ) p r o d u c e d t w o f l a m e k e rn e l s b y s i m u l t a n e o u s l y

i g n i t i n g a c o m b u s t i b l e m i x t u r e a t t w o s e p a r a t e p o i n t s

i n a c o m b u s t i o n v e s se l . S i n c e t h e u n b u r n t g a s v e l o c i t y

o n t h e a x i s j o i n i n g t h e t w o i g n i t i o n p o i n t s m u s t b e z e r o

a s t h e k e r n e l s p r o p a g a t e t o w a r d s e a c h o t h e r , t h e n ,

f r o m e q . (6 ), S , = S~. I n t h e o r y , p r o v i d e d t h a t i g n i t i o n i ss i m u l t a n e o u s , t h e p r e s s u r e r i s e i s s m a l l , a n d t h e g a s

b e t w e e n t h e k e r n e l s i s q u ie s c e n t, t h i s m e t h o d s h o u l d

y i e l d r e l i a b l e r e s u l ts .

A r e c e n t m o d i f i c a t i o n b y A n d r e w s a n d B r a d l e y

( A B7 3 ) o f th i s m e t h o d a p p e a r s t o h a v e y i e l d e d g o o d

r e s u l t s . T w o f l a m e k e r n e l s p r o p a g a t e t o w a r d s o n e

a n o t h e r s u c h t h a t t h e i r f l am e s p e e d s a s t h e y m e e t t e n d

t o w a r d s t h e b u r n i n g v e l o c it y . T h e k e r n e l s a re c o n -

t a i n e d i n a c y l i n d r i c a l v e s s e l w i t h d o u b l e i g n i t i o n , a n d

m e a s u r e m e n t s a r e t a k e n d u r i n g t h e c o n s t a n t p r e s s u r e

p e r i o d o f c o m b u s t io n . S p a r k s y n c h r o n i z a t io n p r o b -

l e m s g e n e r a l l y r e s u l t e d i n k e r n e l s o f d i ff e r e n t si z e s.S a t i s f a c t o r y r e s u l t s a r e c l a i m e d w h e n t h e s i z e d i s -

c r e p a n c y i s n o t t o o l a r g e . T h e m e t h o d i s a p p a r e n t l y

b e s e t w i t h c o n s i d e r a b l e e x p e r i m e n t a l d i f f ic u l t y , r e s u l t -

i n g i n a l o w s u c c e s s r a t e w i t h f i r i n g s . A n a d v a n t a g e o f

t h is t e c h n i q u e i s t h a t f l am e c u r v a t u r e a n d t h i c k n e s s

c o r r e c t i o n s a r e n o t r e q u i re d .

A p o s s i b le a d a p t a t i o n o f t h is a p p r o a c h m i g h t b e t o

u s e t w o p a r t i a l l y in t e r s e c ti n g s o a p b u b b l e s o r l a t e x

b a l l o o n s , w h i c h a r e s i m u l t a n e o u s l y i g n i te d .

6.2.3. S o a p - b u b b l e m e t h o d

O f t h e t e c h n i q u e s w h i c h a r e e s s e n t ia l l y i n d e p e n d e n to f s o l i d s u r f a c e s , p o s s i b l y t h e s i m p l e s t i s t h e s o a p -

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

S t e v en s (S t2 3 ), a n d d e v e l o p e d b y F i o c k a n d R o e d e r

ZO[ -.©

o {

Z ( m R ) aC E=

r ~

.~, = -g

' I ) 1 I { E C T I O N O O F F L A M E ~I ~ I O T I O N

! ~ ~ . . . . . . . . . . ~ . - - - - ~ S L I T|

I i

r , / > I

FIG. 13. Typical record of an explosion in a soap bubble{Fi43).

( F R 3 5 ) a n d L i n n e t t e t a l . ( L P 5 1 ) . H e r e t h e c o m b u s t i b l e

m i x t u r e t o b e t e s te d i s u se d t o b l o w a s p h e r i c a l b u b b l e

a r o u n d a c e n t r a l s p a r k g a p . P r o v i d e d s u c h a b u b b l e

o f f e r s n o s i g n i f i c a n t r e s i s t a n c e t o g a s e x p a n s i o n , i g -

n i t i o n o f t h e m i x t u r e r e s u l t s in t h e p r o p a g a t i o n o f a

s p h e r i c a l c o m b u s t i o n w a v e a t e s s e n t i a l ly c o n s t a n t

p r e s s u r e . ( F o r r e l a t i v e l y l o w v e l o c i t y f la m e s , t h e p r e s -s u r e f ie l d a c r o s s t h e f l a m e f r o n t m a y g e n e r a l l y b e

i g n o r e d . S i g n if i ca n t e r r o r m a y b e i n t r o d u c e d i n h i g h

v e l o c i t y fl a m e s i f t h e p r e s s u r e i s a s s u m e d c o n s t a n t . )

F o r p u r p o s e s o f a n a l y s is , t h e b u b b l e i s p h o t o g r a p h e d

t h r o u g h a n a r r o w h o r i z o n t a l s l i t u s i n g a d r u m c a m e r a .

A t y p i c a l r e c o r d i s s h o w n i n F i g . 1 3 , f r o m w h i c h i t i s

s e e n t h a t t h e s p a t i a l v e l o c i t y , S~ , i s e s s e n t i a l l y c o n s t a n t

t h r o u g h o u t t h e p r o c e ss .

N o w , f o r a s p h e r i c a l f l a m e f r o n t o f n e g l i g i b l e

t h i c k n e s s

A I = 4 n r ~

w h i l s t t h e m a s s o f b u r n t g a s , o f c o n s t a n t d e n s i t y p ~, i s

m b = 4 x r ~ p f f 3 .

H e n c e ,

A l s o ,

d m f f d t = 4 7 zr 2 p b ( d r f f d t ) .

p , = c o n s t a n t = P o .

H e n c e , f r o m o u r d e f i n i t i o n o f b u r n i n g v e l o c i t y (e q. ( 7) ),

S t = ( p f f p o ) ( d r f f d t ) = c tS s . (15)

T h u s , p r o v i d e d t h e d e n s i t y r a t i o , ~, c a n b e m e a s u r e d o r

c a l c u l a te d , a n d S s d e t e r m i n e d f r o m t h e p h o t o g r a p h i c

t r a c e , t h e n S t f o l lo w s . G e n e r a l l y , c ( i s o b t a i n e d f r o m

m e a s u r e m e n t s o f th e i n i t i a l d i a m e t e r o f t h e s p h e r i c a l

b u b b l e , 2 t o , a n d t h e f i n a l d i a m e t e r o f t h e b u r n t g a s , 2 re ,

s i n ce f o r c o n s e r v a t i o n o f m a s s

o t = ( ro / r e ) 3 . (16)

T h e a c c u r a c y o f th e m e t h o d i s t h u s e v i d e n t l y v e r y

s e n s i t iv e t o e r r o r s i n r o a n d r e. A l t h o u g h i t i s g e n e r a l l y

p o s s i b l e t o m e a s u r e b o t h S s a n d r o a c c u r a t e l y , t h e

d e t e r m i n a t i o n o f r e c a n p r e s e n t d i f f i c u lt i e s. A s a r e s u l t

o f a ft e r b u r n i n g i t m a y n o t b e p o s s i b l e to e s t a b l is h a

p r e c i s e v a l u e f o r t h e f i n a l d i a m e t e r ( L i 5 3 , S S 5 3 ) .I n e r t i a l e ff ec ts s h o u l d a l s o b e c o m e s i g n i f i c a n t a t h i g h e r

v e l o c i t i e s ( S i5 9 , S W 5 3 ) . S u r f a c e i r r e g u l a r i t i e s c e r t a i n l y

t h e n b e c o m e a p p a r e n t . A l s o , f o r l o w - v e l o c i t y f l a m e s ,

b u o y a n c y i n t r o d u c e s d i s t o r t i o n o f t h e b u r n i n g s p h e r e .

A n a l t e r n a t i v e m e t h o d o f d e t e r m i n i n g c ( i s v i a

t h e r m o d y n a m i c c a l c u l a t io n s ( G a7 3 ). C l o s e a g r e e m e n t

w i t h o b s e r v e d v a l u e s h a s b e e n r e p o r t e d ( S W 5 3 , SE 5 9 ).

I t i s s u g g e s t e d t h a t w h e r e v e r p o s s i b l e b o t h m e t h o d s b e

u s e d .

A d e c i s i o n a s t o t h e u s e o f S c h l i e r e n o r d i r e c t

p h o t o g r a p h y f o r o b s e r v i n g t h e f l a m e f r o n t d o e s n o t

a r i s e h e r e , s i n ce , b e c a u s e o f t h e c o n s t a n c y o f S~ , b o t ht e c h n i q u e s s h o u l d y i e l d t h e s a m e r e s u l t s ( L P 5 1 , P L 5 1 ) .

A l s o , s i n c e b u r n i n g v e l o c i t y e q u a t i o n s f o r t h i c k f l a m e -

f r o n t s a r e n o w a v a i l a b l e (G R 7 5 ) , i t is p o s s i b l e t o

cor r e c t f or the e f fec ts o f cur v a tur e .

A d i s a d v a n t a g e o f t h is m e t h o d i s t h a t i f w a t e r - b a s e d




s o a p s o l u t i o n s a r e u s e d , d r y m i x t u r e s c a n n o t b e t e s t e d

( S W 5 4 ) . H o w e v e r , n o n a q u e o u s s o a p s o l u t i o n s b a s e d

o n g l y s e r o l h a v e p r o v e d q u i t e s u c c e s s fu l ( S S 5 3, S E 5 9 ).

A l s o , i t h a s b e e n s u g g e s t e d t h a t d i f f u s i o n o f th e m i x t u r e

t h r o u g h t h e b u b b l e w a l l m a y b e s u f fi c ie n t ly g r e a t t o

a l t e r it s c o m p o s i t i o n ( L V 5 1 ). N o e v i d e n c e o f t h i s h a s

b e e n f o u n d w i t h e i t h e r a c e t y l e n e - a i r m i x t u r e s ( L P 5 1 )o r w i t h e t h y l e n e - o x y g e n - h e l i u m m i x t u r e s ( M L 5 3 ). I n

a n y e v e n t , t h e u s e o f t r a n s p a r e n t l a t e x r u b b e r b a l l o o n s

s h o u l d o b v i a t e t h i s d i f fi c u l t y ( P P 5 3 , S E 5 9 ).

T w o i m p o r t a n t a d v a n t a g e s o f t h i s m e t h o d a r e i t s u s e

o f o n l y s m a l l q u a n t i t i e s o f c o m b u s t i b l e m i x t u r e s a n d

t h e a b i l i t y p r o v i d e d t o v a r y b o t h i n i t i a l t e m p e r a t u r e

( w i t h i n n a r r o w l i m i t s ) a n d p r e s s u r e ( w i t h i n w i d e

l i m i ts ) . I n t e r m s o f t h e b u r n i n g v e l o c i t y ra n g e , l o w

v e l o c i t i e s l e a d t o b u o y a n c y a n d h e a t - l o s s e ff e ct s , w h i l s t

h i g h v e l o c i t i e s ( 25 m / s e c ) p r o d u c e s u r f a c e i r re g u l a r i t i e s

( Si 59 ). A t i n t e r m e d i a t e v e l o c i t i e s, t h i s m e t h o d y i e l d s

g o o d r e s u l t s ( L P 5 1 ).

6.2.4. S p h e r i c a l c o n s t a n t - v o l u m e v e s s e l m e t h o d

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

u s e d a s e x t e n s iv e l y a s m a n y o f t h e o t h e r s , a l t h o u g h i t

h a s b e e n d e s c r i b e d b y L i n n e t t a s " . . . p o t e n t i a l l y a

p o w e r f u l m e t h o d f o r d e t e r m i n i n g b u r n i n g v e l o c i t i e s"

( L i5 3 ) . I f it s f u l l p o t e n t i a l i t i e s a r e u s e d i t c a n y i e l d

c o n s i d e r a b l y m o r e i n f o r m a t i o n f r o m a s i n g l e e x p e r i -

m e n t t h a n c a n a n y o t h e r m e t h o d . A l s o , t o q u o t e L e w i s ,

" . . . t h e m e t h o d i s se l f - c o r ro b o r a t i n g , a n d s i n c e t h e

e x p e r i m e n t a l p r e c i s i o n i s v e r y h i g h , i t m u s t b e r e -

g a r d e d a s a p r e c i s i o n m e t h o d a n d , a t t h e s a m e t i m e , a

s t a n d a r d m e t h o d a g a i n s t w h i c h t h e v a l id i t y o f o t h e r

m e t h o d s c a n b e t e s t e d " (L e 54 ). I t s l i m i t e d u s e t o d a t e

h a s p e r h a p s b e e n d u e t o a l a c k o f u n d e r s t a n d i n g o f t h e

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

t h e p o s s i b l e e r r o r s i n t r o d u c e d d u r i n g t h e e a r l y s ta g e s

o f c o m b u s t i o n b y t h e i n f lu e n c e o f t h e s p a r k , o f f la m e

c u r v a t u r e , a n d o f fl a m e - f r o n t t h i c k n e s s , a s w e l l a s i t s

r e l a t i v e a n a l y t i c a l a n d e x p e r i m e n t a l c o m p l e x i t y .

H o p e f u l l y , m o s t o f th i s u n c e r t a i n t y h a s b e e n e l i m -

i n a t e d o v e r t h e y e a r s a s t h e s e p r o b l e m s h a v e b e e n

p r o g r e s s i v e ly o v e r c o m e o r s h o w n t o b e u n i m p o r t a n t

( O R 5 9 , R P 6 2 , R T 6 3 , R a 6 3 , R G 6 5 , R G 6 5 a , R G 6 5 b ,G a 7 3 , G a 7 4 , G R 7 5 , G R 7 5 a , G R 7 6 , G R 7 8 ) .

I n t h i s m e t h o d , a p r e m i x e d c o m b u s t i b l e m i x t u r e

c o n t a i n e d i n a t h i c k - w a l l e d s p h e r i c a l v e ss e l i s i g n i t e d

a t t h e c e n t r e . P r o v i d e d t h e d i f f e re n c e s i n c o n c e n t r a t i o n

a n d d i f f u si v i t y b e t w e e n t h e v a r i o u s c o n s t i t u e n t s a r e

n o t t o o l a r g e a n d t h e s p a t i a l v e l o c i ty i s n o t t o o l o w , t h e

r e s u l ti n g c o m b u s t i o n w a v e i s g e n e ra l l y is o t r o p i c . T h e

p r o p a g a t i o n o f t h e f l a m e t o w a r d s t h e w a l l o f t h e v e s se l

i s a t t e n d e d b y a p r e s s u r e r i s e w h i c h r e s u l t s i n a

t e m p e r a t u r e i n c r e a s e o f t h e u n b u r n t g a s. W h e n t h e s e

c h a n g e s i n p r e s s u r e a n d t e m p e r a t u r e , a s w e l l a s t h e

p o s i t i o n o f t h e f l a m e f r o n t , a r e a c c u r a t e l y m e a s u r e d ,t h e m e t h o d b e c o m e s e x t r e m e l y v e r s a ti l e . T h e e f fe c ts o f

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

c a n t h e n b e o b t a i n e d , o v e r a f a i rl y w i d e r a n g e , f r o m a

s i n g l e e x p e r i m e n t . A l i m i t e d s e t o f e x p e r i m e n t s c a n

t h u s p r o v i d e t h r e e - d i m e n s i o n a l S t p - T sur f aces f or

m i x t u r e s o f g i v e n e q u i v a l e n c e r a t i o (~ b) ( R G 6 5 , G R 7 8 ) .

A l t e r n a t iv e l y , a n y t h r e e o f t h e se f o u r p a r a m e t e r s c a n

b e r e p r e s e n t e d o n s u c h a m a p .

A l t h o u g h H o p k i n s o n ( H o0 6 ) u se d c o m b u s t i o n i n a

c o n s t a n t - v o l u m e v e s se l f o r d e t e r m i n i n g t h e m e a n h e a t

c a p a c i t i e s o f g a s e s a t h i g h t e m p e r a t u r e s , N a g e l ( N a 0 7 ),

a n d F l a m m a n d M a c h e ( F M 1 7 ) a p p e a r t o h a v e b e e nt h e f i rs t t o d e r i v e e q u a t i o n s r e l a t i n g t h e p r e s s u r e a t a n y

i n s t a n t t o t h e v o l u m e o f g a s b u r n t ( L V 5 1 , J o 4 6 ). L a t e r ,

E l l i s a n d W h e e l e r ( E W 2 7 , E 1 2 8) , w i t h t h e i r c l a s s i c a l

p h o t o g r a p h s o f c o m b u s t i o n i n a g l as s s p h e r e, s h o w e d

t h a t , p r o v i d e d t h e s p a t i a l v e l o c i ty w a s n o t t o o l o w , th e

f l am e p r o p a g a t e s i s o t r o p i c a l l y . H o w e v e r , t h e i n i t i a l

d e v e l o p m e n t o f t h e s p h e r i c a l c o n s t a n t - v o l u m e v e s s el

t e c h n i q u e f o r d e t e r m i n i n g l a m i n a r b u r n i n g v e l o c it i es

a p p e a r s t o h a v e b e e n d u e t o L e w i s a n d v o n E l b e , a n d

F i o c k a n d h i s a ss o c i a t e s ( L V 34 , L V 6 1, F K 3 5 , F M 3 7 ,

F M 4 0 ) . U s i n g a s i m p l i f i e d f o r m o f t h e F l a m m a n d

M a c h e e q u a t i o n , L e w i s a n d v o n E l b e d e r i v e d a na p p r o x i m a t e e x p r e s s i o n f or t h e m a s s - f r a c t io n b u r n t , i n

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

s t a g e s o f c o m b u s t i o n w h e n t h e p r e s s u r e r i s e i s s m a l l .

T h i s e n a b l e d t h e m t o d e t e r m i n e b u r n i n g v e l o c i t i e s

d u r i n g t h is p e r i o d . A s a c h e ck o n t h e i r a s s u m p t i o n s ,

t h e y c o m p a r e d c a l c u l a t e d a n d o b s e r v e d v a l u e s o f

f l a m e - f r o n t r a d i u s a n d f o u n d v e r y g o o d a g r e e m e n t

( M V 5 3 ). A t a b o u t t h e s a m e t i m e , F i o c k e t a l . , i n s o m e

e x c e l le n t p a p e r s , r e p o r t e d o n t h e i r a n a l y t i c a l a n d

e x p e r i m e n t a l w o r k u s i n g t h i s m e t h o d ( F K 3 5 , F M 3 7 ,

F M 4 0 ) . U n f o r t u n a t e l y , b o t h t h e s e g r o u p s u s e d e i t h e r

r e s t ri c t e d o r u n s a t i s f a c t o r y fo r m s o f b u r n i n g v e l o c i ty

e q u a t i o n s .

I n 1 9 5 3 , M a n t o n e t a l . ( M V 5 3 ) u s e d a n e q u a t i o n

a t t r i b u t e d t o D e r y , in w h i c h o n l y t h e t i m e d e r i v a t iv e o f

t h e f l a m e r a d i u s w a s r e q u i r e d . R e s u l t s o n s t o i c h i o m e t -

r i c p r o p a n e - a i r s h o w e d g o o d a g r e e m e n t ( f o r m a s s -

f r a c t i o n s b u r n t o f l e ss t h a n 0 .0 1 ) w i t h e q u a t i o n s

i n v o l v i n g b o t h p r e s s u r e a n d r a d i u s. A g n e w a n d c o -

w o r k e r s ( S A 5 7 , E s 5 7 , E A 5 8 ) p r o p o s e d a n d u s e d

s e v e r a l e q u a t i o n s , w h i c h i n c l u d e d a n a l l o w a n c e f o r

t h e f i n i t e v e l o c i t y o f p r o p a g a t i o n o f p r e s s u r e

w a v e s - - n e c e s s a r y f o r v e r y h i g h v e l o c i t y f l a m e s .

U n f o r t u n a t e l y , t h e s e a r e o n l y a p p l i c a b l e t o t h e e a r l y

s t a g e s o f c o m b u s t i o n . G r u m e r e t a l . ( G C 5 9 ) p r o d u c e da m o d i f i e d fo r m o f e q u a t i o n f o r m a s s - fr a c t i o n b u r n t ,

f o r w h i c h t h e y c l a i m g r e a t e r a c c u r a c y t h a n t h a t

p r o p o s e d b y L e w i s a n d v o n E l be . O ' D o n o v a n a n d

R a l l i s d e r i v e d a n e q u a t i o n f o r m a s s - f r a c t i o n b u r n t

w h i c h i s v a l i d t h r o u g h o u t t h e c o m b u s t i o n p r o c e s s .

S u b s e q u e n t t o t h i s , R a l l i s a n d c o w o r k e r s d e r i v e d a

c o m p l e t e s e t o f e q u a t i o n s , a s w e l l a s a v a r i e t y o f

c o r r o b o r a t i v e r e la t i o n s , a n d s h o w e d t h a t a l l p r e v i o u s

e q u a t i o n s w e r e p a r t i c u l a r c a s e s o f t h i s g e n e r a l s e t

( R T 6 3 , R a 6 3 , R a 6 4 ). T h e y u s e d a p a r t i c u l a r l y s t a b l e

f o rm o f e q u a t i o n - - r e f e r r e d t o a s t h e c o m b i n e d

e q u a t i o n - - t o d e t e r m i n e t h e e ffe cts o f e q u i v a l en c er a t i o , p r e s s u r e , a n d t e m p e r a t u r e o n t h e b u r n i n g v e -

l o c i t y o f a c e t y l e n ~ a i r m i x t u r e s , a n d s h o w e d t h a t t h e

n a t u r e o f t h e p r e s su r e a n d t e m p e r a t u r e d e p e n d e n c e

a p p e a r e d t o b e m o r e c o m p l e x t h a n h a d h i t h e rt o b e e n

s u p p o s e d ( R G 6 5 ).




At this period in time, concern still existed regarding

the use of certain assumptions in the theory of this

method, as well as with the comput atio nal complexity

associa ted wi th its correct use. Specifically, these were

that (GR78) :

(1) the flame front remains smooth, spherical andcentered on the point of ignition ;

(2) the pressure at any instant is uniform throughout

the vessel--i.e, there are no pressure gradients or

time lag effects in the measu remen t of pressure ;

(3) both the burnt- and unburnt-gas regions are

adiabatic systems--i. e, there is no heat loss or gain

from these regions;

(4) the effects of flame-front thickness and curvature

are negligible

(5) the burnt-gas density immediately behind the

flame front, p f , is an accurate approximation to

the i nstanta neous spatially-averaged density of the

burn t gas,/~b ;

(6) chemical equilibrium is achieved immediately

behind the flame front--i.e, the temperature at this

point corresponds to the theoretical equilibrium

adiabatic value;

(7) no dissocia tion or preflame reactions occur in the

unbu rnt gas region;

(8) the required methods of calcula ting gas properties

are complex and tedious.

We will now deal with each of these in turn.

6.2.4.1. Fl a me- f ro n t i r reg u la r i t i e s . For low burning

velocity mixtures, the effects of buoyancy, especially in

large vessels, can introduce errors. A check on this is

provided by photographing the flame through a

vertical slit and measu ring the flame radii both above

and below the spark. Care has to be taken to correct for

any slit-width effects (Ra63). Where any discrepancies

are found, approximate corrections for the flame

propagating as an oblate or prolate spheroid can be

applied (Ra63). In any event, checks can be made

between the observed and calculated radii (MV53,

Ra63). F lame- front irregularities at high velocities and

pressures, as observed in constant-pressure experi-

ments by Strauss and Edse (SE59) and Simon and

Won g (SW54), are a possibility. These may be caused

by acoustic waves set up by the flame and reflected by

the walls. No evidence of such oscillation has been

observed to date, even with acetylene-air mixtures

(Ra63, RG65, Ga75). This is no guarantee that they

may no t occur at higher velocities and pressures.

6.2.4.2. P r e s s u r e u n i f o r m i t y . Various writers have

examined the pressure distribution through a flame

front (LV61, Es57, EA58, Gr59, Ra63, Ra64). The

pressure profile in a spherical constan t-vo lume vessel

is var ious ly dependent on Pu, P~, S~ and St (Ra63, Ra64).At low velocities and densities its effects are insignifi-

cant. Agnew and coworkers have developed

approximate burning-velocity equations which sug-

gest that corrections become necessary when Ss is

greater than one-tenth the velocity of sound in the

unburnt gas (Es57, EA58). Rallis has shown that

correcting for the time-lag for information to be

transmitted from the flame front to a pressure trans-

ducer on the wall significantlyreduces any discrepancy

between the observed an d calculated pressures, except

close to the bomb wall (Ra63).

Babkin and coworkers have derived an approxi-mate equa tion for correcting the final pressure, p~, for

heat-loss and quenching effects when the flame is in

close prox imity to the wall (BK65, BV66). Thus, both

from theoretical and experimental points of view (of

accurately observing the flame-front radius close to the

wall due to internal reflections (GR78)), it would

appear undesirable to place undue reliability on results

obtained for flame-front radii, r b, greater than about

909/oo of the bomb radius, R.

6.2.4.3. Hea t l o s s . Transfer of heat may occur

during the combu stion process by radiation from theburnt gas to the unburnt gas and to the containing

wall; by radiation and conduction from the unburnt

gas to the wall; and by conduction along the spark

electrodes. Rallis estimated that for acetylene-air

mixtures the heat lost by radiation from the flame to

the wall and by conduction along the electrodes was

negligible (Ra63). Ca lculat ions indicate that this is still

likely to be the case, even for much lower velocity

mixtures. Garforth developed an infinite--fring e aser

interferometer, with which he measured the density

close to the wall of the un bur nt gas during com bustion

of methane-air mixtures (Ga74, Ga75). He showed

that for a stoichiometric mixture, the calculated adia-

batic temperature and that determined from measure-

ments of density and pressure never deviated by more

than 1 "o at the early stages of co mbu sti on an d 2~10~oat

the end. F or slower-burning mixtures this difference

may be larger. There is a remote possibility that this

small difference may be the result of an equilibrium

situation prevailing between heat loss and gain by the

end gas. The corroborative relations used in this

method (Ra63, GR78) suggest that this is unlikely, but

an u nambigu ous answer will have to await con tinuous

measurements of the flame front temperatu re, T (see

sub-Sect ion 6.2.4.6).

6.2.4.4. F l a m e - f r o n t t h i c k n e s s a n d c u r v a t u r e would

appear to go hand in hand, since the effect of the latter

is negligible when the flame front is considered "thin".

As mentioned in the introduction to Section 5, any

surface within the flame front, at which corres ponding

values of area and density are measured, can in

principle be used as a reference to specify buring

velocities. However, with "thick" flames in a spherical

bomb, the mass of gas contained in the flame front,

particularly at the early stages of combustion, can

account for a significant proportion of the mass ofburni ng and bur nt gas. If the more stable burnt gas or

combined forms of equa tion are used (RG65), the

assumptio n that this mass has all burn t can introduce

an appreciable error (AB72, GR78) (see also DS74).

Babkin et al . (BK62) derived an approximate re-



The determination of lamina r burning veloci ty 317

l a t i o n s h i p f o r t h e b u r n i n g v e l o c i t y o f a s p h e r i c a l g a s -

p o c k e t w i t h i n w a r d f l a m e p r o p a g a t i o n , i n t e r m s o f th e

p l a n e - f l a m e v a l u e a n d t h e f l a m e - f r o n t t h ic k n e s s , a n d

s h o w e d t h a t f l a m e - c u r v a t u r e e ff ec ts a r e s i g n i f i c a n t fo r

r j - < 3 m m . A n d r e w s a n d B r a d l e y ( A B 7 2 a ) d i d a n

a p p r o x i m a t e a n a l y s i s o f c o n s t a n t - p r e s s u r e s p h e r i c a l -

f l am e p r o p a g a t i o n w i t h c e n t r a l i g n i t i o n , i n c l u d i n g t h em a s s o f g a s c o n t a i n e d i n t h e f l a m e fr o n t , a n d s h o w e d

t h a t a t a f l am e r a d i u s o f 2 5 m m w i t h p = 1 a t m o s -

p h e r e , T~ = 30 0 K a n d a f l a m e - f r o n t t h i c k n e s s o f

1 .1 m m , t h e t h i n - f l a m e e q u a t i o n s w e r e p r o b a b l y i n

e r r o r b y a s m u c h a s 2 2~ o. G a r f o r t h a n d R a l l i s d e r i v e d a

m o r e r i g o r o u s s e t o f " t h i c k f l a m e " e q u a t i o n s ( G R 7 5 ) ,

w i t h t h e a i d o f w h ic h t h e y s h o w e d t h a t t h e c o r r e c t i o n s

f o r m e t h a n e - a i r m i x t u r e s a r e n o t a s l a r g e a s s u g g e s te d

b y A n d r e w s a n d B r a d l e y ( G R 7 8) . I n a n y e v e n t, i t i s

d u r i n g t h e e a r l y s ta g e s o f c o m b u s t i o n , w h e n t h e

p r e s s u r e i s e s s e n t i a l l y c o n s t a n t , t h a t t h e r e s u l t s a r e

m o s t u n c e r t a in . Y e t t h i s is t h e r e g i o n m o s t f a v o r e d b yp r e v i o u s i n v e s t i g a t o r s f o r d e t e r m i n i n g b u r n i n g v e l o -

c it ie s . A l t h o u g h t h e r e s u l ts r e p o r t e d o n s t o i c h i o m e t r i c

m e t h a n e - a i r m i x t u r e s ( G R 7 8 ) m u s t b e r e g a r d e d a s

t e n t a t i v e ( i n t h a t t h e c o r r e c t i o n s f o r f la m e - f r o n t t h i c k -

n e s s a r e b a s e d o n a n e q u a t i o n d e r i v e d f ro m r e s u l t s o n

l o w - p r e s s u r e b u r n e r f l a m e s (D W 5 1 ) ) , i t is c o n f i d e n t l y

e x p e c t e d t h a t t h e se v a l u e s o f b u r n i n g v e l o c i t y a r e n o t

l i k e ly t o b e m o r e t h a n 5 ~ l o w i n t h e r a n g e

1.1 < (P/Po) < 6 .0 , n o r m o r e t h a n 9 ~ i n e r r o r o u t s i d e

t h i s ra n g e ( G a 7 5 , G a 7 7 ) .

6.2.4.5. Average burnt - gas dens i ty . L e w i s a n d v o nE l b e r e c o g n i z e d t h a t d u e t o r e c o m p r e s s i o n a n d g a s

m o v e m e n t t h e r e w o u l d e x i s t a t e m p e r a t u r e , a n d h e n c e

d e n s i t y , d i s t r i b u t i o n i n t h e b u r n t g a s ( L V6 1 ). U s i n g a n

a p p r o x i m a t e e x p r e s s i o n f o r t h e m a s s - f r a c t i o n b u r n t

( b u t i g n o r i n g g a s m o v e m e n t ) t h e y c a l c u l a t e d th i s

d i s t ri b u t i o n f o r h y d r o g e n o x y g e n a n d o z o n e - o x y g e n

c o m b u s t i o n i n a s p h e r i c a l v e s s e l . R a l l i s et al . ( R a 5 9 ,

T r 6 2 , R a 6 3 ) p r o v i d e d a s i m p l e e x p r e s s i o n fo r c a l c u l a t -

i n g t h e a v e r a g e b u r n t - g a s d e n s i ty , a s s u m i n g t h e d i s t r i-

b u t i o n o f d e n s i t y w i t h r a d i u s w a s k n o w n , b u t , b e c a u s e

o f t h e l a c k o f a n a c c u r a t e m e t h o d o f a n a l y z i n g t h e

l a t te r , d i d n o t a p p l y i t . T h e y i n s t e a d a s s e s se d t h e e r r o r

i n b u r n i n g v e l o c i t y d u e t o t h e u s e o f P z i n p l a c e o f/ ~b -

T h e d e v e l o p m e n t b y G a r f o r t h o f a c o m p u t e r p r o -

g r a m m e f o r d e t e r m i n i n g a d i a b a t i c e q u i l i b r i u m f l a m e

t e m p e r a t u r e s w a s t h e f i r s t s t e p i n r e s o l v i n g t h i s

p r o b l e m ( G a 73 ) . T h i s w a s f o l l o w e d b y a n a n a l y s i s o f

t h e g a s m o v e m e n t d u r i n g f l a m e p r o p a g a t i o n , w h i c h

t h u s p e r m i t t e d d e t e r m i n a t i o n o f t h e d i s t r i b u t i o n o f g a s

p r o p e r t i e s b e h i n d t h e f l a m e fr o n t ( G R 7 5 a ). F o r t u i t o u s

v a l i d a t i o n o f th i s a n a l y s i s w a s p r o v i d e d b y s o m e

p a r t i c l e t r a c k s a c c i d e n t a l l y o b t a i n e d d u r i n g c e r t a in

t e s t s ( G R 7 5 a , G a 7 5 ) . T h e n e t e f f ec t o f t h e c o r r e c t i o n f o r

b o t h t h e t e m p e r a t u re d i s t ri b u t i o n a n d g a s m o v e m e n t

o n t h e v a l u e s f o r b u r n i n g v e l o c i t y i s o f th e o r d e r o f 1 2 ~o( G R 7 8 ) .

6.2.4.6. Chem ical equi l ibr ium . T h e r e i s s o m e e v i -

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

c o m p l e t e l y a c h i e v e d i n t h e f l a m e f r o n t ( R T 6 3 , R a 6 3 ) .

F i r s t ly , th e r e i s a l w a y s a n a f t e rg l o w o r r e i l l u m i n a t i o n

o f th e b u r n t g a s , w h i c h s t a r t s f r o m t h e c e n t r e o f t h e

v e s s e l b e f o r e t h e f l a m e f r o n t r e a c h e s t h e w a l l , a n d

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

a b o u t t h e s a m e t i m e t h a t t h e l a t t e r r e a c h e s t h e w a l l

( F i g . 1 4 ) . T h i s s t r o n g l u m i n e s c e n c e p e r s i s t s f o r s o m e

c o n s i d e r a b l e t i m e a f t e r th e e n d o f th e p r o c e s s , g r a d u -a l l y c o l l a p s i n g t o w a r d s t h e c e n t r e a s t h e s y s t e m c o o l s

d o w n ( R a 6 3 , R G 6 5 , G a 7 5 ) . T h e r e a s o n f o r t h i s r e -

i l l u m i n a t i o n h a s n e v e r b e e n e x p l a i n e d t o o u r s a t is f a c-

t i o n , a l t h o u g h L e w i s a n d v o n E l b e c o n s i d e r th a t i t m a y

b e d u e t o t e m p e r a t u r e g r a d i e n t s t h a t e x i st in t h e b u r n t

g a s w i th c o n s t a n t - v o l u m e c o m b u s t i o n , s i n ce th e s a m e

e ff e ct i s n o t o b s e r v e d i n c o n s t a n t - p r e s s u r e o r o p e n

v e s s el e x p e r i m e n t s ( LV 6 1) . S e c o n d l y , i n a l l o u r e x p e r i -

m e n t s t o d a t e , w e h a v e o b s e r v e d t h a t t h e m a x i m u m

p r e s s u r e a l w a y s o c c u r s a s h o r t t i m e a f t e r t h e f l a m e h a s

r e a c h e d t h e w a l l o f t h e v e s s e l ( R a 6 3 , G a 7 5 ) . W h e n

i n c o r p o r a t e d i n t o t h e e x p r e s s i o n f o r ~ e = P J P o , t h ed e n s i t y r a t i o a t t h e e n d o f t h e p r o c e s s , t h i s l e a d s t o

v a l u e s w h i c h a r e a l w a y s l e s s t h a n t h e v a l u e u n i t y ,

w h i c h i s t h e o r e t i c a l l y r e q u i r e d ( R T 6 3 , G a 7 5 ) .

F u r t h e r m o r e , t h e v a l u e o f ~ a p p e a r s t o c o r r e l a te i n t h e

c o r r e c t m a n n e r w i t h s p a t i a l v e lo c i t y a n d e q u i v a l e n c e

r a t i o f o r a c e t y l e n e - a i r m i x t u r e s ( R T 6 3) . T h e c u r r e n t

a n t i c ip a t e d e rr o r o f s o m e 5 ~ i n o u r m e t h a n e - a i r

r e s u l t s i s p r o b a b l y m a i n l y d u e t o t h i s e f f e c t ( G R 7 8 ) .

T h e q u e s t i o n o f c h e m i c a l e q u i l i b r i u m w i l l h a v e t o

r e m a i n u n r e s o l v e d u n t i l a n a d e q u a t e m e a n s o f m e a s u r -

i n g t r a n s i e n t b u r n t - g a s t e m p e r a t u r e i s d e v i s e d .

P o s s i b l y t h e u s e o f a f a s t - re s p o n s e i n f r a r e d - r a d i a t i o np y r o m e t e r s u c h a s t h a t r e p o r t e d b y P e n z i a s et al .

( P D 6 6 ) a n d S h i m u z u ( S h 7 3 ), o r a f a s t - r e s p o n s e s o d i u m

D - l i n e r e v e rs a l m e t h o d m i g h t p r o v e s u cc e ss fu l .

6.2.4.7. D issoc ia t ion or pre f lam e reac t ions in the

unburnt gas reg ions d o n o t a p p e a r t o b e s i g n i fi c a n t a t

t h e u n b u r n t g a s t e m p e r a t u r e s , a n d f o r t h e r a n g e o f

s p a t i a l v e lo c i ti e s f o r w h ic h t h e b o m b m e t h o d h a s b e e n

u s e d t o d a t e . T h e u s e o f a n e x t e r n a l h e a t e r t o r a i s e t h e

i n i ti a l u n b u r n t g a s t e m p e r a t u r e o f t h e m i x t u r e , t h e r e b y

i n c r e a si n g t h e r a n g e o f b u r n i n g v e l o c i ty v a l u e s - - a s

s u g g e s t e d b y R a l li s ( R a 6 3) a n d u s e d b y B a b k i n a n d

K a z a c h e n k o ( B K 6 6 ) - - c o u l d i n t r o d u c e p r o b l e m s i n

t h i s r e g a r d .

6.2.4.8. C o m p u t a t i o n a l c o m p l e x i t y . T h a t t h e c o m -

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

t h e c o n s t a n t - v o l u m e b o m b i s la r g e c a n n o t b e g a i n s a i d .

H o w e v e r , t h e u s e o f c e r t a i n s i m p l i f i c a t i o n s a r i s i n g o u t

o f t h e c o r r o b o r a t i v e n a t u r e o f t h e m e t h o d c a n m a t e r i -

a l l y r e d u c e t h i s c o m p u t a t i o n a l e f f o r t w i t h o u t s i g n i f i -

c a n t l y a f f e c t i n g i t s a c c u r a c y ( R a 6 3 , G a 7 5 , G R 7 8 ) . A s

w i l l b e s h o w n i n t h e n e x t s e c t i o n , b e c a u s e o f t h e n a t u r e

o f t h e s y s te m , a n u m b e r o f c o r r o b o r a t i v e e q u a t i o n s c a n

b e d e r i v e d - - u s u a l l y i n t e r m s o f t h e p r o p e r t i e s o f t h eu n b u r n t a n d b u r n t ( o r b u r n t a n d b u r n i n g ) ga s , r e s p e c-

t i v e ly . S t r i c t l y , s u c h a l t e r n a t i v e f o r m s s h o u l d y i e l d t h e

s a m e r e s u l t s w h e n a p p l i e d t o a n y g i v e n s e t o f e x p e r i -

m e n t a l d a t a . H o w e v e r , b e c a u s e o f t h e d i f f e r i n g s e n -

s i ti v i ti e s o f th e e q u a t i o n s t o s y s t e m a t i c a n d o t h e r



318 C.J . RALUSand A. M. GARrORTH

( a ) F l a m e T ra c e

( b ) P r e s s u r e o nd i n t e r f e r o m e t e r t r o c e s

FIG. 14 . Typica l te s t records in a spher ica l cons tan t vo lume vesse l . (a ) F lame t race . (b ) Pressure andin te rfe romete r t races.

e r r o r s i n h e r e n t i n t h e o b s e r v a t i o n s , t h e y c a n l e a d t o

s l i g h tl y d i ff e r e n t r e s u l ts o v e r t h e r a n g e o f t h e c o m b u s -

t i o n p r o c e s s . T h i s i s u s e d t o a d v a n t a g e , b o t h f o r

p r o v i d i n g i n d i c a t i o n s o f th e p r o b a b l e s o u r c e a n d

e x t e n t o f s u c h e r r o r s , a n d f o r s i m p l i f y in g t h e a n a l y s i s

( R a 6 3 , G a 7 5 , G R 7 8 ) .

C o n s i d e r t h e b u r n t - g a s d e n s i t y r a t i o

= # b / P o . (17)

S i n c e t h e a v e r a g e d e n s i t y o f th e e n t i r e s y s t e m r e m a i n s

c o n s t a n t t h r o u g h o u t t h e p r o c e s s, a n d in p a r t i c u l a r a t

t h e e n d o f t h e p r o c e s s ~ e = P o , t h i s c a n a l s o b e

e x p r e s s e d a s

= f d f e ( 1 8 )

w h i c h m u s t e v i d e n t l y y i e l d t h e c o r r e c t e n d v a l u e c ~= 1. C o m p a r i s o n o f t h e v a l u e s o f ~ f r o m t h e s e t w o

e q u a t i o n s t h u s g i v e s a n i n d i c a t i o n o f t h e m a g n i t u d e o f

a n y e r r o r s m a d e i n c a l c u l a t i n g fib -

F i g u r e 1 5 s h o w s t h e v a r i a t i o n s o f ~ w i t h p r e s s u r e

r a t i o ( P / P o ) , a s d e t e r m i n e d f r o m e q s ( 1 7) a n d ( 18 ),

c a l c u l a t e d f o r a s to i c h i o m e t r i c m e t h a n e - a i r t e s t w i t h

P o = 0 . 1 0 1 3 M P a , (1 a t m o s p h e r e) , i n c l u d in g b o t h

d e n s i t y - d i s t r i b u t i o n a n d g a s - m o v e m e n t e ff ec ts . T h e s e

t w o c u r v e s c a n b e c o n s i d e r e d a s t h e u p p e r a n d l o w e r

e r r o r - b o u n d s i n th e d e t e r m i n a t i o n o f t h is i m p o r t a n t

v a r i a b l e . F o r t h i s t e s t , t h i s e r r o r i s e s s e n t i a l l y c o n s t a n t

a t 4 . 0 ~ o t h r o u g h o u t t h e p r e s s u r e r a n g e . A s t r a i g h t - l i n e

f i t b e t w e e n s o f ro m e q . (1 7 ) a n d ~t = 1 a t p = P e r e s u l t s

i n t h e s am e v a l u e fo r th e m a x i m u m a p p a r e n t e r r o r

b a s e d o n ~ f r o m e q . ( 17 ). T h e i m p o r t a n c e o f t h is l ie s i n

t h e f a c t t h a t , p r o v i d e d c to s k n o w n o r c a n b e c a l c u la t e d ,

s u b s e q u e n t v a l u e s o f ~ c a n b e o b t a i n e d f r o m t h e

e q u a t i o n

= s o -k (1 - - c t o ) [ ( p / p o ) - 1 ] / [ ( p e / p o ) - - 13 .

( 19 )

I t t h e r e f o r e b e c o m e s u n n e c e s s a r y f o r p r o s p e c t i v e u s e rs

o f t h i s m e t h o d t o e n t e r i n t o t h e c o m p l e x i t y o f c a l c u l a t -

i n g t h e f la m e - f r o n t a d i a b a t i c t e m p e r a t u r e s , b u r n t - g a s

d e n s it y d i s tr i b u ti o n , o r b u r n t - g a s m o v e m e n t d u r i n g

t h e p r o c e s s . E q u a t i o n ( 19 ) w i ll y i e l d ~ v a l u e s w e l l



Th e d e t e rmin a t i o n o f l am in ar b u rn in g v e lo c it y 3 19

1.0 F I ! / I

! , 4 . / / / !. 9

: a ' l B ' = P b / P e l ~ / / ~ ( 1 7 > ~ 5 b / 0 o I

t I A,,,o a , v s / = ~ A r ~ . p ,

: P e P o ~

, ~ r i F IG 1 5 VARIATIO NS O F g ERSUS

I 2 3 4 5 6 7 8P / P o

FIG. 15 . Var iat ions fo r a versus p ressure rat io p /p o fo rs t o ich io met r ic meth an e-a i r .

w i t h i n t h e e x p e r i m e n t a l a c c u r a c y o f t h e m e t h o d .

H o w e v e r , a c c u r a t e v a l u e s o f ~o a n d p a r e r e q u i r e d .

A s c u r r e n t l y u s e d ( G R 7 6 , G R 7 8 ) , t h e c o n s t a n t -

v o l u m e m e t h o d h aq t h e f o ll o w i n g i m p o r t a n t

a d v a n t a g e s :

(1 ) O n l y s m a l l q u a n t i t i e s o f c o m b u s t i b l e m i x t u r e a r e

r e q u i r e d .

(2 ) T h e m i x t u r e c o m p o s i t i o n a n d i n i t i a l t e m p e r a t u r e

a n d p r e s s u r e c a n b e a c c u r a t e l y c o n t r o l l e d .

( 3) O n l y a f e w e x p e r i m e n t s a t d i f f e r e n t i n i t i a l p r e s -

s u r e s , o r t e m p e r a t u r e s , a r e r e q u i r e d i n o r d e r t o

e s t a b l i s h t h e s e p a r a t e e f fe cts o f t h e s e p a r a m e t e r s

o n t h e l a m i n a r b u r n i n g v e l o c it y o f a p a r t i c u l a r

m i x t u r e .

(4 ) T r u e f r e e - s p a c e a d i a b a t i c b u r n i n g v e lo c i ti e s c a n b e

d e t e r m i n e d . T h a t i s , t h e r e a r e n o s u r f a c e i n t e r a c -

t i o n e f f ec t s a n d t h e h e a t l o s s i s n e g l i g i b l e .

(5 ) S i n c e i t h a s b e e n e x p e r i m e n t a l l y c o n f i r m e d t h a t

t h e v a r i a t i o n in m e a n d e n s i ty r a t io ~ t h r o u g h o u t

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

d e n s i t y d i s t r i b u t i o n i n th e b u r n t g a s , a n d g a s

m o v e m e n t - - i s e s s e n ti a ll y a l i n e a r f u n c t io n o f

i n s t a n t a n e o u s p r e s s u r e r a t io , t h e t e d i u m i n c a r ry -

i n g o u t t h e c o m p l e x c a l c u l a t i o n s r e q u i r e d i s d r a s t i -

c a l l y r e d u c e d .

( 6) T h e m e t h o d i s s e l f - c o r r o b o r a t i v e , i n t h a t a t l e a s t

t w o e q u a t i o n s o r m e t h o d s o f c a l c u l a ti n g th e

r e q u i r e d v a r i a b l e s a r e a v a i la b l e . T h i s p e r m i t s

c h e ck s o n a c c u r a c y - - n o t p o s s i b le w i th m o s t o t h e r

m e t h o d s . A l s o , b y p r o v i d i n g a " c a r p e t " o f d a ta ,r e p r e s e n t e d b y t h e t e s t c u r v e s a t d i f f e r e n t i n i t i a l

p r e s s u r e s w i t h s u p e r i m p o s e d i s o b a r s a n d i s o -

t h e r m s , t r e n d s b e c o m e a p p a r e n t w h i c h a s s i s t c o n -

s i d e r a b l y i n p r o p e r e v a l u a t i o n o f th e r e s u l t s

( G R 7 8 ) .

I t s d i s a d v a n t a g e s a r e :

(1 ) T h e a p p a r a t u s r e q u i r e d i s r e l a t iv e l y c o m p l e x a n d

e x p e n s i v e a n d t h e t i m e r e q u i r e d t o d o a t e s t is q u i te

l o n g - - o f t h e o r d e r o f l h r f o r te s t s in w h i c h t h e

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

(2 ) T r a n s c r i p t i o n o f t h e o b s e r v a t i o n s c a n b e t e d i o u su n l e s s r e la t i v e ly s o p h i s ti c a t e d a p p a r a t u s i s

a v a i l a b l e .

7 . BURNING VELOCITY EQUATIONS--SPHERICALCONSTANT-VOLUME VESSEL METH OD

U n f o r t u n a t e l y , t h e m a j o r i t y o f u s e r s o f t h is m e t h o d

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

o n l y a p p l i c a b l e t o t h e e a r l y s t a g e s o f th e p r o c e s s , w h e n

t h e p r e s s u r e r i s e is s m a l l . T h i s h a s s e r i o u s l y r e s t r i c t e d

i ts v a l u e , v i r t u a l l y r e l e g a t i n g i t to t h e s t a t u s o f a

c o n s t a n t - p r e s s u r e t e c h n i q u e , a l b e i t w i t h s e v e r a l a d -

v a n t a g e s o v e r t he s o a p - b u b b l e m e t h o d .S i n c e w e b e l i e v e t h is m e t h o d t o b e t h e m o s t p r e c is e

a v a i l a b l e t o d a te , a s u m m a r y o f t h e p e r t i n e n t e q u a t i o n s

w o u l d s e e m t o b e a p p r o p r i a t e . T h e s e w i l l b e p r e s e n t e d

u n d e r t w o h e a d i n g s : ( 1) t h i n - f l a m e e q u a t i o n s i n w h i c h

t h e f l a m e f r o n t i s c o n s i d e r e d a s a s u r f a c e o f d i s -

c o n t i n u i t y a n d h e n c e i n f i n it e l y t h i n , a n d ( 2) t h i c k - fl a m e

e q u a t i o n s f o r fl a m e f r o n t s h a v i n g a f i n it e t h i c k n e s s z .

7 .1 . T h i n - F l a m e E q u a t i o n s ( R a 6 3 , R a 6 4 )

7.1.1. B a s i c r e l a t i o n s

F o r m a s s c o n s e r v a t i o n

m o = m u + m b . (20)

H e n c e , f o r m a s s c o n t i n u i t y a c ro s s t h e f l a m e f r o n t ,

S , = - - ( 1 / A i p u ) ( d m J d t ) = ( 1 / A ip u ) ( dm b / d t ) . (7 )

D e f i n i n g th e m a s s - f r a c t i o n b u r n t a s n = ( m b / m o ) y i e l d s

( d n / d t ) = ( 1 / m o )( d m b / d t) = - ( 1 / m o ) ( d m J d t ) .

H e n c e

S t = ( m o / A f p u )( d n / d t ) . (21)

F o r i s o tr o p i c f la m e p r o p a g a t i o n i n a c o n s t a n tv o l u m e s p h e r i c a l v e s s e l:

m o = ( 4 /3 ) T z R a p o (22)

m , = ( 4 / 3 ) n ( R 3 - r 3 ) f iu (23)

m b = (4 /3)nr~f ib (24)

A f = 4grg (25)

w h e r e

f t , = [3 / ( R 3 - - r~)3 p 'u r 2 d r (26)b

p ' b e i n g s o m e f u n c t i o n o f r a n d r b < r < R , a n d

f := (3 / r~) p'br 2 d r (2 7 )

p ; b e in g so m e fu n c t i o n o f r an d 0 _< r _< r b .

Ev id e n t l y , a t r b = R , P b = P e = P o.

JPECS 6:4 B



320 C . J . RALLIS nd A . M. GARFORTH

7.1 .2 . M a s s - f r a c t i o n b u rn t

V a r i o u s f o r m s o f e q u a t i o n a r e a v a i l a b l e fo r t h e

m a s s - f r a c t i o n b u r n t . T h u s , i n t e r m s o f th e p r o p e r t i e s o f

t h e u n b u r n t g a s ,

n u = 1 - - ( m u / m o ) = 1 - - fl + ~ ( r J R ) 3 (28)

w h i l s t in t e r m s o f t h e p r o p e r t i e s o f t h e b u r n t g a s

no = (rob~too) = ~( rb /R ) 3 . (29)

A l s o , s i n c e s t r i c t l y n b = n u , t h e y c a n b e e q u a t e d t o y i e ld

a c o m b i n e d f o r m

nt = o ~ ( f l - 1 ) / ( f l - a ). ( 3 0)

7 .1 .3 . L a m i n a r b u r n i n g v e lo c i t y e q u a t i o n s

D i f f e r e n t i a t i n g e a c h o f t h e e x p r e s s i o n s f o r m a s s -

f r a c t i o n b u r n t a n d s u b s t i t u t i n g i n t o e q . ( 21 ) i n t u r n

y i e l d s t h r e e f o r m s o f b u r n i n g v e l o c i t y e q u a t i o n : i .e . t h eu n b u r n t g a s e q u a t i o n - -

S ~ = ( f l / f l ) [ ( d r b / d t ) - { ( g 3 - r 3 ) / ( 3 r Z f l ) } ( d f l / d t ) ] ( 3 1 )

t h e b u r n t g a s e q u a t i o n - -

S b = ( ~ / f l ) [ ( d r b / d t + ( G / 3 ~ ) ( d ~ / d t ) ] ( 3 2 )

a n d a c o m b i n e d e q u a t io n , a p a r t i c u l a r f o r m o f w h i c h

i s - -

S t = ~ ( f l / f l ) [ ( d r b / d t )

+ { r b f l ( 1 - - 0~)/307(fl- ~ ) } ( d / d t ) ( 8 / f l ) ] ( 3 3 )

w h i c h c o n t a i n s t h e s p a t i a l v e l o c i t y S~ = d r b / d t e x -

p l i ci t ly a n d i s f o u n d n o t t o m a g n i f y e x p e r i m e n t a l

e r r o r s .

I t s h o u l d b e n o t e d t h a t e x c e p t f o r v e r y h ig h b u r n i n g -

v e l o c i ty f la m e s , t h e p r e s s u r e g r a d i e n t i n t h e u n b u r n t

g a s i s n e g l i g i b l e . T h u s , f o r a l l p r a c t i c a l c a s e s , ~ = f l

= ( P . / P o ) .

7 .1 .4 . C o r r o b o r a t i v e r e l a t i o n s

O n e o f t h e i m p o r t a n t c h a r a c t e r i st i c s o f t h e c o n s t a n t -

v o l u m e m e t h o d i s i ts s e lf - c o r r o b o r a t i v e n a t u r e . T h u s ,

i t w a s s h o w n i n th e f o r e g o i n g t h a t t w o b a s i c e q u a t i o n s

a r e a v a i l a b l e f o r t h e m a s s - f r a c t i o n b u r n t ( e q s 2 8 a n d

2 9 ) a n d f o r th e b u r n i n g v e l o c i t y ( e q s 31 a n d 3 2) . O t h e r

c o r r o b o r a t i v e t y p e e q u a t i o n s f o l lo w s .

S i n c e

P e = P o = ( ) Q l~ p ~ )/ (R T " e ) = ( M o P o ) / ( g T o )

t h e d e n s i t y r a t i o , 07, c a n b e e x p r e s s e d a s

= ( P b / P o ) = (¢3b/C3e) (3 4)

a n d t h e m a x i m u m p r e s s u re a s

p ¢ = ( M o / M e ) ( T ~ / T o ) P o ( 3 5 )

w h i c h p r o v i d e s a u s ef u l c h e c k b e t w e e n t h e m e a s u r e d

m a x i m u m p r e s s u r e a n d t h a t d e t e r m i n e d f r o m t h e

c a l c u l a t i o n s o f ( Tb /M b ).

S e t t i n g e q . ( 2 9 ) e q u a l t o e q . ( 2 8 ) a n d s o l v i n g f o r r b

p r o v i d e s a n e q u a t i o n f o r c a l c u l a t i n g v a l u e s o f f la m e -

f r o n t r a d i u s , w h i c h c a n b e c o m p a r e d w i t h t h e o b s e r v e d

v a l u e s t o c h e c k o n t h e d e n s i t y r a t i o s 07 a n d ft. T h u s

r b = g [ ( f l - 1) / ( f l - - ~ ) ]1 /3 . (36)

F i n a l l y , i f i t i s a s s u m e d t h a t t h e u n b u r n t g a s i s

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

e x i s t s t h r o u g h t h e v e s se l , t h e n i t c a n b e s h o w n t h a t( R a 6 3 , R a 6 4 )

( p / p e ) ( r b / R ) 3 + ( p / p o )l /% [ 1 - - ( r b / R ) 3 ] , ~ 1

(37)

f r o m w h i c h p c a n b e c a lc u l a t e d a n d c o m p a r e d w i t h th e

m e a s u r e d v a l u e s .

T h e v a r i o u s f o r m s o f b u r n i n g - v e l o c i t y e q u a t i o n s

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

s h o w n t o b e s p e c i a l c a s e s o f t h e f o r e g o i n g .

7 .2 . T h i c k - F l a m e E q u a t io n s ( G R 7 5 , G R 7 5 a )

7 .2 .1 . B a s i c r e l a t i o n s

F o r m a s s c o n s e r v a t i o n o f a fl a m e w i t h a f i ni te f la m e -

f r o n t t h i c k n e s s z ,

m o = m u + m b 4- m f ( 3 8 )

w h e r e t h e s u b s c r i p t s u , b a n d f d e n o t e u n b u r n t g a s ,

b u r n t g a s , a n d f l a m e - f r o n t , r e s p e c t iv e l y .

H e n c e , f o r m a s s c o n t i n u i t y a c r o s s t h e f l a m e f r o n t ,

S , = - - ( 1 / A ~ p u) (d m J d t )

= ( 1 / A , p , ) [ ( d m b / d t ) + ( d m s / d t ) ] . ( 3 9 )

D e f i n i n g t h e m a s s - f r a c t i o n " b u r n t a n d b u r n i n g " a s

n = ( m b + m y ) / m o = 1 - ( m u / m o ) (40)

y i e l d s

S , = ( m o / A u P u ) ( d n / d t ) . (41)

I t i s c o n v e n i e n t f o r l a t e r a n a l y s i s t o e x p r e s s t h e

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

t h e f o l l o w i n g r e f e r e n c e q u a n t i t i e s :

(1 ) i n s t a n t a n e o u s f l a m e - f r o n t r a d i u s , - r s ;

( 2 ) v e s s e l i n t e r n a l r a d i u s , - - R ;(3 ) t i m e f r o m i g n i t i o n t o t h e f l a m e r e a c h i n g t h e w a l l ,

- t e.

H e n c e

= ( z / r f ) ; ~ = ( r / R ) ; ~ = ( t / te ) ( 4 2 )

w h e r e t h e s i g n ~ i n d i c a te s a n o n d i m e n s i o n a l q u a n t i t y .

T h u s , f o r i s o t r o p i c f la m e p r o p a g a t i o n i n a c o n s t a n t -

v o l u m e s p h e r i c a l v e s s e l :

m o = ( 4 / 3 ) ~ z R 3 p o (43)

m u = ( 4 / 3 ) n ( R 3 - - r } ) f i .

= ( 4 / 3 ) T z R 3 p o f l ( 1 _ f3 ) (44)

m b = ( 4 / 3 ) T r R a p o F 3 ~ ( 1 - - z-33

= ( 4 / 3 ) ~ t R 3 p o F 3 ~ F 3 (45)

m f = ( 4 / 3 ) n R 3 p o F 3 g [ 1 - - ( 1 - - ~ 3 ]



Th e d e t e rmin a t i o n o f l amin ar b u rn in g v e lo c it y 3 21

= ( 4 /3 ) n R 3 p o 7 3 g ( 1 - F 3) (46)

Au = 4 n R 2 r 2 (4 7 )

w h e r e

F = (1 - z) a nd g = P y / P o . (48)

S u b s t i t u t i n g t h e a p p r o p r i a t e t e r m s i n e q . (4 1 ) y i e ld s

S , = ( R / 3 F 2 f l t e X d n / d ~ . (49)

7.2.2. M a s s - f r a c t i o n b u r n t a n d b u r n i n g

I n t e r m s o f t h e p r o p e r t i e s o f th e u n b u r n t g a s

n , = 1 - ( m d m o ) = 1 - / ~ + p ~3 (5 0 )

w h i l s t i n t e r m s o f t h e p r o p e r t i e s o f t h e b u r n t a n d

b u r n i n g g a s es

n b = ( m J m o ) + ( m y / m o )

= r a [ ~ F 3 + g(1 - - F3) ] (51)

o r e q u a t i n g t h e s e to y i e l d a c o m b i n e d f o r m y i e ld s

n, = (~ - 1)G/(fl - G) (52)

w h e r e

G = a F 3 q- g (1 - - F3) . (53)

7.2.3. L a m i n a r b u r n i n g v e l o c i ty e q u a t i o n s

D i f f e r e n t i a t in g e a c h o f t h e e x p r e s s i o n s f o r m a s s -

f r a c t i o n b u r n t a n d s u b s t i t u t i n g in t o e q . (4 9) in t u r n

y i e ld s t h r e e f o r m s o f t h ic k - f l a m e b u r n i n g v e l o c i ty

e q u a t i o n : i. e. t h e u n b u r n t g a s e q u a t i o n - -

S ~ = ( R / t e X f l / f l ) [ ( d f / d / )

- {(1 - F 3 ) / 3 k a ~ ( d f l / d ? ) ] (54)

th e b u r n t g a s e q u a t i o n - -

S ~ = ( R / t e ) ( 1 / f l ) [ G ( d f / d i )

+ ( f F 3 / 3 ) ( d ~ / d / )

+ {~(1 - F 3 ) / 3 } ( d g / d f )

- f ( ~ - 0 f 2 ( d f / d / ) ] ( 55 )

a n d o n e f o r m o f c o m b i n e d e q u a t i o n

S t = [ { ? ( g / t e ) ( f l / f l ) } / 3 ( f l - G )]

x [ F 3 ( d S / d f ) + {G(1 - - G ) / f l ( f l - 1)}(d j~/d~

+ (1 - F 3X dg/d /) - 3(c~ - e-)F2(d?/d/')]. (56)

H e r e a g a i n , e x c e p t f o r v e r y h i g h - v e l o c i t y fl a m e s i t is

p e r m i s s i b l e t o s e t [1 = f l . -- - P, , /Po .

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

r e s p o n d i n g t h i n - fl a m e c o u n t e r p a r t s w h e n

f = 0 = (d r / d r ) = (d e /d / )

a n d h e n c e ,

F = 1 a n d G = 5 .

7.2.4. C o r r e c t i o n s f o r f l a r n e - f r o n t t h i c k n e s s ( G R 7 5 )

I t i s g e n e r a l l y c o n v e n i e n t t o c a l c u l a t e b u r n i n g

v e l oc i ti e s u s i n g t h e t h i n - f l a m e e q u a t i o n s a n d t h e n

a p p l y a c o r r e c t i o n f o r t h e e f fe c ts o f f la m e t h i c k n e s s .

I f t h i s c o r r e c t i o n i s w r i t t e n i n t h e f o r m

K c = S , - S ' , (57)

w h e r e S ' i s th e t h i n - f l a m e b u r n i n g v e l o c it y , th e f o r e go -

i n g a n a l y s i s y i e l d s

K c = B ( d r f / d t ) + C ( d £ ( d t )

+ D ( d g / d t ) + E ( d z / d t ) (58)

w h e r e

B = ( 1 / f l ) ( g - ~)(1 - F 2 ) = E(1 - F 2 ) / F 2

C = - ( 1 / f l ) ( r y / 3 ) ( 1 - - F 3 ) = _ D. (59)

A s i m i l a r, b u t u n f o r t u n a t e l y i n a c c u r a t e , e q u a t i o n

h a s b e e n d e r i v e d b y A n d r e w s a n d B r a d l e y ( A B 72 ).

T h e p a r t i c u l a r f o r m o f th e t h i n - f l a m e c o m b i n e d

e q u a t i o n , g i v e n b y e q . (3 3), h a s a d v a n t a g e s f r o m a d a t a

h a n d l i n g p o i n t o f v i ew , a n d i s g e n e r a l l y u s e d t o p r o v i d e

v a l u e s o f S', w h i c h a r e t h e n c o r r e c t e d b y t h e u s e o f e q.

(58).

T h e v a l u e s o f fl a m e - f r o n t t h i c k n e s s , z, a n d a v e r a g e

f l a m e - f r o n t d e n s i t y , g, u s e d t o c o r r e c t t h e r e s u l t s o f t h e

s t o i c h i o m e t r i c m e t h a n e - a i r t e s ts ( d is c u s s ed i n S e c t i o n

8 ) w e r e d e t e r m i n e d a s f o l l o w s :

( 1) A s s u m i n g t h a t z o c 1 / p . S ~ w a s a n a c c u r a t e e n o u g h

r e p r e s e n t a t i o n o f t h e v a r i a t i o n o f f la m e t h i c k n e s s

f or m e t h a n e - a i r , a n a p p r o x i m a t e r e l a t io n s h i p fo r

S t i n t e r m s o f p a n d T~ w a s d e t e r m i n e d f r o m t h e

t h i n - f l a m e r e su l ts , w h i c h o n s u b s t i t u t i o n i n t o t h ea b o v e y i e l d e d

z = 1.848 x 10 7 ( T o / P ° o ' Z t S ) { p - 1 . 0 4 ) (57)

t h e c o n s t a n t b e i n g c h o s e n t o y i e ld z = 1.1 m m a t

To = 3 0 0 K a n d p = P o = 1 a t m o s p h e r e t o c o r -

r e s o n d t o J a n i s c h ' s r e s u l t s ( J a 7 1 ) .

( 2 ) T h e m e a n g a s d e n s i t y r a t i o s , g = P y / P o i n t h e

s p h e r i c a l s h el ls o f b u r n i n g g a s, w e r e t h e n o b t a i n e d

u s i n g t h e t e m p e r a t u r e d i s t r i b u t i o n p r o f il e s o f

D i x o n - L e w i s a n d W i l s o n ( D W 5 1 ) b y a s s u m i n g

t h a t t h e s e d o n o t v a r y s i g n i f i c a n t l y f o r s p h e r i c a l

f la m e s u n d e r t h e s a m e u n b u r n t - g a s c o n d i t i o n s ,a n d r e m a i n s i m i l a r f o r a ll f la m e s o f t he s a m e

c o m b u s t i b l e m i x t u r e ( G R 7 5 ).

S t r i c tl y , t h is p r o c e s s o f St c o r r e c t i o n f o r f l a m e -

f r o n t t h i c k n e s s e f f e c t s s h o u l d b e a n i t e r a t i v e o n e .

H o w e v e r , w i t h t h e p r e s e n t u n c e r t a i n t y i n z v a l u e s,

t h is d o e s n o t p r o v i d e r e l i a b l e i m p r o v e m e n t o v e r

t h e c u r r e n t l y r e p o r t e d d a t a .

7.2.5. C o r r o b o r a t i v e r e l a ti o n s

A s w i t h t h e t h i n - f l a m e e q u a t i o n s , c o r r o b o r a t i v e

r e l a t i o n s a r e p r o v i d e d b y t h e v a r i o u s f o r m s o f m a s s -f r a c t i o n b u r n t e q s ( 5 0 ) , ( 5 1 ) , ( 5 2 ) , a n d t h e b u r n i n g -

v e lo c i t y eq u a t i o n s (5 4 ) , ( 5 5 ) , ( 5 6 ) . Th e d en s i t y r a t i o , ~ ,

a n d m a x i m u m p r e s s u re , P c, c a n b e c h e c k e d u s i n g e q s

( 3 4 ) a n d ( 3 5 ) r e s p e c t i v e l y . S i m i l a r e q u a t i o n s t o t h o s e

g i v e n b y e q s ( 3 6 ) a n d ( 3 7 ) c a n a l s o b e d e r i v e d .




8 , C OM P AR IS ON OF E XP E R IM E NT A L R E S UL T S F OR

S T O I C H I O M E T RI C M E T H A N E A I R M I X T U R E S

For the purpose of comparing the experimental

results of various methods used to determine bur ning

velocity, as well as comparing these results with

theoretical prediction which have recently become

available (Ts78), we will confine ourselves to me thane-air mixtures, and in particular to the pressure and

temperature dependence of stoichiometric mixtures

since these have been well reported (AB72, GR78,

Ts78).

Figure 16 shows plots of burning velocity versus

equivalence ratio for meth ane-a ir at p = 1 atmosphere

(0.101 MPa) and various values of T~ as repor ted by

different workers. F igures 17, 18 and 19 present results

for stoichiometric methane-air for the ranges

0.06 < p < 2.04 MP a and 290 < T, < 525 K, as ob-

tained in a spherical constant-v olum e vessel (Ga75,

GR78). In Fig. 17 the dependence of St on pressure can

be seen to be initi ally very large at low unburnt-gas

temperatures, becoming practically independent of

pressure at higher pressures and temperatures. Figure18 shows the variation of S~ with T~ for a range of

pressures. Evidently, the isobars (particularly after

being corrected for flame-front thickness), are essen-

tially straight at lower pressures, and can in con-

sequence be described by relatively simple equations,

as will be discussed later. The three-dimensional

representation of Fig. 19 illustrates the combined

effects of both unburn t-gas temperature and pressure.

0,8

f,7 / "~.

/

/

/

0 . 6 /

x x

x ~ .~ ~x

x )'x

o,2 "~ :~ ' ~ ~ . . +

°J o , 6 o ,z 0 ,8 0 ,9 Lo ~ ,~ ~ ,2 ~ ,3 ~ , " ~ ,5 ~ ,6

Req

FIG. 16. Comparison of burning velocity versus equivalence ratio data for methane-air.

Pressure Temp.Curve No. Ref. Method (MPa) (K)

1 AB73 cylindrical bomb, double kernel 0.101 293

2 AB 7 2 cylindrical bomb, hot-wireanemometer 0.101 293

3 G J73 nozzle burner, button flame, 0.101 293particle tracks

4 RM71 nozzle burner, Schlieren cone, 0.101 293particle tracks

5 2896 spherical bomb with heater, 3137 BK 64 Schlierenlame photography, 0.101 3438 constant pressure region only 4139 493

10 323BK66 As for BK64 0.101

11 423

12 T s 7 8 computer prediction 0.101 298




~n

%

ll~7

TN tS SrUOV

. . . . . . . . U N C O a R I C t l O

a 2

p ( M P a )

FIG. 17. Burnin g velocity pressure depend ence for stoichio-metr ic methane-air .

O 6

SO.4

FIG.

0~ 3T u ( K )

18. Burning-velocity tem pera ture dependen ce forstoichiometric methane-air .

A n d r e w s a n d B r a d l e y h a v e , fa i r l y r e c e n t l y , p r e -

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

m e t h a n e - a i r m i x t u r e s ( A B 7 2 , A B 7 2 a ) . I t i s t h u s u n -

n e c e s s a r y t o r e p e a t a l l th e s e d a t a h e r e . S u ff ic e i t t o s a y

t h a t t h e v a l u e s o f m a x i m u m b u r n i n g v e l o c it i e s r e p -

o r t e d o v e r t h e y e a r s r a n g e f r o m a s l o w a s 0 .3 2 m / s e c t o

a s h i g h a s 0 .5 0 m / s e c w i t h a v a l u e , r e c o m m e n d e d b y

A n d r e w s a n d B r a d l e y (A B 7 2) , o f 0 .4 5 _ 0 . 02 m / s e c a t 1

a t m o s p h e r e a n d 2 98 K . W e b e l i e v e t h i s t o b e t o o h i g h

b y a b o u t 0 . 0 8 m / s e e , f o r r e a s o n s t o b e d i s c u s s e d i n

w h a t f o l l o w s .

8.1. E q u i v a l e n c e R a t i o

F i g u r e 1 6 g i v es s o m e i d e a o f t h e d i s c r e p a n c i e s w h i c h

s t il l e x i st b e t w e e n t h e m o r e r e l ia b l e d a t a r e p o r t e d .

A l t h o u g h t h e r e i s g o o d c o r r e s p o n d e n c e b e t w e e n t h e

v a l u e s o f A n d r e w s a n d B r a d l e y ( A B 7 2 , A B 7 3 ) , R e e d e t

al . ( R M 7 1 ) a n d G u n t h e r a n d J a n i s h ( G J 72 ), t h e r e a r e

q u i t e l a r g e d i f f e re n c e s b e t w e e n t h e s e r e s u l t s a n d t h o s e

o f B a b k i n a n d c o w o r k e r s o b t a i n e d a t d i ff e re n t t i m e s

( B K 6 4 , B K 6 6) . H o w e v e r , r e c o g n i z i n g t h a t B a b k i n ' s

r e s u l t s w e r e d e t e r m i n e d f r o m o b s e r v a t i o n s t a k e n

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

v e s s el , t h a t n o c o r r e c t i o n s f o r fl a m e - f r o n t t h i c k n e s s

w e r e a p p l i e d , a n d t h a t s i m p l i f i e d e q u a t i o n s w e r e u s e d ,

w e c o n c e d e t h a t t h e i r r e s u l ts a r e l i k e l y t o b e l o w b u tn o t t o t h e e x t e n t s u g g e s te d b y A n d r e w s a n d B r a d l e y .

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

t i o n s p u b l i s h e d b y S m o o t et a l . ( S H 7 6 ) a n d m o r e

r e c e n t l y b y T s a t s a r o n i s ( Ts 78 ). T h e l a t t e r a r e b a s e d o n

a m e t h a n e - o x y g e n r e a c t i o n m e c h a n i s m c o n s i s t in g o f

2 9 e l e m e n t a r y r e a c t i o n s . E v e n t h o u g h T s a t s a r o n i s

a d m i t s s e l ec t in g o r a d j u s t i n g t h e c h e m i c a l k i n e ti c d a t a

t o i m p r o v e t h e a g r e e m e n t b e t w e e n c a l c u l a t e d a n d

m e a s u r e d v a l u e s , h i s p r e d i c t i o n , s h o w n o n F i g . 1 6 , i s

a b o u t 8 ~ b e l o w t h o se o f A n d r ew s a n d B r a d l e y (A B 7 2,

A B 7 3) a n d a b o u t 3 ~ h i g h er t h an t h a t r e c o m m e n d e d

b y G a r f o r t h a n d R a l l is (G R 7 8 ) , a l l a t a n e q u i v a l e n c er a t i o o f 1 .0 . T h i s p o i n t w i l l b e t a k e n u p a g a i n e l s e w h e r e

i n t h i s s ec t io n . T h e g r a d u a l c o n v e r g e n c e o f m e a s u r e d

a n d p r e d i c t e d d a t a i s c e rt a i n l y a m o s t e n c o u r a g i n g

d e v e l o p m e n t .

8.2. P r e s s u r e D e p e n d e n c y

F i g u r e 1 7 s h o w s a f a m i l y o f i s o t h e r m s i l l u s t r a t i n g

t h e p r e s s u r e d e p e n d e n c y o f s t o i c h i o m e t r i c m e t h a n e -

a i r m i x t u r e s c o r r e c t e d f o r t h e e f fe ct s o f f l a m e - f r o n t

~ . ~ I

/ S

F I G . 19. Three-dimensional repres entatlo n of burn ing velocity dependence on pressure and tem pera ture forstoichiometric methane-air .




I

2 0 0 i

"~ ~ _ _ 5 1 = 4 6 p -O '5 1(Ts 7 8 )

!

1 0 0 ~ •9 0 ~ I •8 0 . . . .

7 0

~ 40

/

30

2O

! [!

I

I I

/ 1 1 ! T S A TS A R O NI S P R ED IC T IO N I T s 7 8 )

, , Q G A R F O R T H & R A L L I S ( G a 7 5 G R 7 8~

i i i • B R A D L E Y & H U N D Y ( B H 7 1 )

. A G U N T H E R & J A N I S C H ( G J 7 2 )

J t . F R A N C E P R , ' C H A , 0E ' 0 ,_ N B A B K I N e t a l ( B K 6 6 )

m A N D R E W S & B R A D L E Y ( A B 7 3 ]

1 0

0, 1

T S A T S A R O N I S ~ ' I ~

'R ED IC TleN . ~ [~ E A D O F F- , H IS IF IG UR E 12 )

0.2

L

. . . . . . . . . . . . . .

0 , 4 0 , 6 0 ; 8 1 , 0 P a t m o s 6 9 10

FIG. 20. Comparison of pressure dependence results for stoichiometric methane-air.

thickness (Ga75, GR78). F or purposes of compariso n

with the results of other workers, some of these

isotherms have been reproduced on a log-log plot in

Fig. 20. Also included on this plot are so me of the more

recently published data. Of particular interest again is

the plot of Tsatsaronis' predicted curve read off Fig. 12

of his paper (Ts78).

For the purpose of comparing the results obtainedby various workers, consider the burning velocity at a

pressure of 1 atmosphere (0.101 MPa) and an

unburnt-gas temperature of 298 K. Our corrected

results yield a value under these conditions of

0.353 m/sec, which we concede may be up to 5~ low

due to the a ssumption of adiabatic equilibrium flame-

front temperature s. Thus, the probab le value may be as

high as 0.37 m/sec--essentially the value obtained by

Tsatsaronis' prediction. However, the data of Bradley

and Hundy, using hot-wire anemometry in closed

vessel explosions (BH71) ; that of Andrews and Bradley

from the double-kernel closed-vessel experiments

(AB73); the measurements of Gunther and Janisch on

button flames above a Mach e-H ebr a burner (G J72);

and the laser-doppler anemometer data from the

nozzle-burner studies of France and Pritchard (FP76),

all point to a value of St at 1 atmosphere and 293 K of

between 0.4 and 0.5 m/sec.

For reasons discussed elsewhere (GR78) we believe

that even the more reliable of these values, of just over

0.4 m/se c (GJ72, FP76), are some what on the high side.

Certainly, however, the target range appears to be

being bracketed to a value of St of between 0.37 and

0.4 m/sec at a pressure of 1 atmosphere and at an

unburn t-gas tempe ratur e of between 293 and 298 K.Consider next the form o f the pressure dependency

curves for stoichiometric methane-air mixtures.

Tsats aroni s predicts a 298 K iso therm of progress ively

increasing negative slope with increasing pressure on a

log -lo g plot (Fig. 20). This shows r emark able simi-

larity with B abkin's 298 K iso therm, the absolute

values of which we believe to be low. Our 300 K curve

is in closer agreement numerically with Tsatsaronis',

but does not "bend" to the same extent over the range

0.6 < p < 3 atmo spheres. Tsatsa ronis suggests re-

lationships of the form :

St = 46 p o.sl (cm/sec) for p > 4 atmospheres

and

S = 39.6 p-O.~2 (cm/sec) for p < 0.6 atmosp heres.

Somewhat better approximation s for the range 0.1-

10 atmospheres would appear to result from :

S, = 38 p-0.145 (cm/sec) for p < 0.6 atmospheres

St = 36 p o.26s (cm/sec) for 0.6 < p < 3 atmosp heres

and

St = 46 p 0.51 (cm/sec) for 3 < p < 10 atmospheres.

In the light of the foregoing, the values reported byBradley et al . at pressures below 1 atmosp here mus t be

considered too high, since they do not follow what are

considered to be more reliable trends.

Still on the subject of trends, it will be noted from

Figs 17 and 20 that according to our results, pressure

dependency appears to diminish at higher pressures

and temperatures, as sh own by a flattening off of the

higher temperature isotherms at higher pressures. This

is in conflict both with Babkin's results and

Tsatsaronis' prediction. No obvious reasons for these

anomalies come to mind at present and their elucid-

ation will have to ~wait further work.

8.3. T e m p e r a t u r e D e p e n d e n c y

Figure 18 shows isobars (uncorrected for flame-

fron t thickness) o f St versus T~ (Ga75, GR78) . Again,

for purposes of comparison with the results of other



T h e d e te rmin a t io n o f l a min a r b u rn in g v e lo c i ty 3 25

w o r k e r s , s o m e o f t h e s e c o r r e c t e d i s o b a r s h a v e b e e n

r e p l o t t e d o n a l o g - l o g b a s i s i n F i g . 2 1 .

W i t h t h e e x c e p t i o n o f t h e i s o b a r s l y i n g i n t h e r e g i o n

T , h i g h e r t h a n a b o u t 4 3 0 K , t h e n a t u r e o f t h e t e m p e r a -

t u r e d e p e n d e n c e c u r v e s a p p e a r s t o b e l e ss c o m p l e x

t h a n t h e i r p r e s s u r e c o u n t e r p a r t s . T h u s , a t t e m p e r a -

t u r e s b e l o w a b o u t 4 3 0 K t h e y r e d u c e t o s t r a ig h t b u t

s l i g h t ly d i v e r g e n t l i ne s , w h i c h a r e a p p r o x i m a t e l y d e -

s c r i b e d b y a n e q u a t i o n o f th e f o r m S t = c T f , t h e

c o n s t a n t s c a n d d h a v i n g t h e v a l u e s g i v e n o n F i g . 2 1 .

E v i d e n t l y , e m p i r i c a l c u r v e f i t t i n g t o d a t a s u c h a s

t h a t d e p i c t e d o n F i g . 21 i s a r e l a t i v e ly a r b i t r a r y p r o c e s s

w h e n a p p l i e d t o t h e l im i t e d r a n g e o f d a t a a v a i l a b l e.

T h u s , e .g ., o u r m e a s u r e d 0 . 1 M P a ( 0. 99 a t m o s p h e r e s )

i s o b a r c a n b e m a d e t o c o i n c i d e w i t h T s a t s ar o n i s ' c u r v e

b y p l o t t i n g v a l u e s o f ( S - 7 ) r a t h e r t h a n ( S - 1 0)

c m / s e c . U n t i l r e l i a b l e t h e o r e t i c a l r e a s o n s f o r u s i n g a

p a r t i c u l a r e x p o n e n t f o r Tu a r e p r o p o s e d ( s u c h a s t h e

d e t a il e d r e a c t i o n s c h e m e s o f S m o o t e t a l . a n dT s a t s a r o n is ) , i t i s p e r h a p s a d v i s a b l e t o c o n f i n e o n e s e l f

i n a n y a p p l i c a t i o n c o n t e x t t o t h e u s e o f r e l i a b l e

e x p e r i m e n t a l d a t a .

100

90

80

7 0

60 .

50

G A R F O R T H & R A L L I SE X P E R I M E N T A L R E S U L TS ~ - . / ° h ' n ~x

( G a 7 5 G R 7 8 } / / / ' ~

dS t = c T u ~ 3O O K g T u g 4 3 0 K )

p ( a t m o s ) c d

0 , 9 9 1 ,8 7 x 1 0 3 1 , 72 6 ~ '

1, 48 2.09 ~ ~ 0 3 1 , 6 8 81 , 9 7 3 , 4 5 = 1 0 3 1 , 5 8 6

40 2,96 . 5 .6 1 x 1 0 - 3 1 , 4 8 2

3 , 9 6 9 . 8 8 s 1 0 3 1 , 3 6 6 / /

/ /i 30 /

20

/ j / ~',-," "~"

I , / )T S A T S A R O N I S ( T s 7 8 ~ /St - 10 =3 '15 x 10 -4 Tu I

~ ] ~ J ~ D G G E R (D u 5 2 )A N D R E W S B R A D L E Y { A B 7 " 2 0 ) S t - l O = 3 , 7 4 x I O -S T u 2 ' 3 3

,o s , , o 4 5 9 x , O T u ' 3 ' I J200 300 4 0 0 5 0 0

T K

FIG. 21 . C om pariso n o f tempera tu r e dependence resu l ts fo rs to i c h io me t r i c me th a n e -a i r .

s t o i c h io m e t r i c m e t h a n e - a i r a t p = 1 a t m o s p h e r e a n d

T , = 2 9 8 K . T h e p o s i t i o n r e g a r d i n g t h e p r e s s u r e a n d

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

m i x t u r e i s e v i d e n t l y r e l a t i v e l y c o m p l e x a n d r e q u i r e s

f u r t h e r i n v e s t i g a t i o n b o t h f r o m t h e t h e o r e t i c a l a n d

e x p e r i m e n t a l v i e w p o i n t s . H o w e v e r , b e c a u s e o f t h e

c o m p a t i b i l i t y o f t h e c o m p u t e r p r e d i c te d t r e n d s a n d

t h o s e o b t a i n e d b y b o m b m e t h o d s , i t i s p o s s i b l e to

p r o p o s e l i m i t ed e m p i r i c a l c o r r el a t i o n e q u a t i o n s o v e r

l i m i t e d r a n g e s o f T , a n d p .

T h e c o n c l u s i o n i s r e a c h e d t h a t o f t h e e x p e r i m e n t a l

m e t h o d s i n v o l v i n g s t a t i o n a r y f l a m e s , n o z z l e - b u r n e r

m e t h o d s w i t h e i t h e r o p t i c a l o r l a s e r - d o p p l e r p a r t i c l e -

t r a c k i n g t e ch n i q u e s a p p e a r m o s t e f f e c t i v e - - p r o v i d e d

u n b u r n t - g a s t e m p e r a t u r e s c a n b e a c c u ra t e l y m o n i -

t o r e d . A l t e r n a t iv e l y , i n v e r t e d - c o n e b u r n e r s w i t h t h e

s a m e m e a s u r i n g t e c h n i q u e s m i g h t b e w o r t h

i n v e s t i g a t i n g .

O f t h e p r o p a g a t i n g f l a m e m e t h o d s , t h e r e s e e m s l i tt l ed o u b t t h a t t h e c o n s t a n t - v o l u m e v e s se l t e c h n i q u e is t h e

m o s t v e r s a t i l e a n d a c c u r a t e . A l s o , b y u s i n g i t s s e lf -

c o r r o b o r a t i v e c h a r a ct e r i s ti c s , i ts c o m p u t a t i o n a l c o m -

p l e x i t y c a n b e c o n s i d e r a b l y r e d u c e d .

A c k n o w l e d g e m e n t - - T h i s work has been pa r t ia l ly sponsoredover the yea rs by the South Afr ican Counc i l fo r Sc ient i fic andIndus tr ia l Resea rch , to whom the au thors express the irapprec ia t ion .

(AB72)

(AB72a)

(AB73)

(AC60)

(AF49)

(AF50)

(AG61)

9 . SUMM ARY AND CONCLUSIONS (BB55)

T h e v a r i o u s m e t h o d s w h i c h h a v e b e e n u s e d f o r t h e ( B B 6 0 )

e x p e r i m e n t a l d e t e r m i n a t i o n o f l a m i n a r b u r n i n g v e -

l o c i t y h a v e b e e n r e v i e w e d . A v a i l a b l e r e s u l ts f ro m s o m e (B H 71 )

o f t h e m o r e r e c e n t e x p e r i m e n t s o n m e t h a n e - a i r m i x -t u re s h a v e b e e n c o m p a r e d b o t h a g a i n st e a c h o t h e r a n d

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

b e c o m e a v a i l a b l e ( T s 7 8) . T h e s e i n d i c a t e a p r o g r e s s i v e ( BK 6 2)

n a r r o w i n g o f t h e b o u n d s o f u n c e r t a i n ty a n d s u g g es t a

b e n c h m a r k v a l u e o f S t = 0 .3 7 + 0 .0 2 m / s e c f o r

R E F E R E N C E S

ANDREWS,G. E . and BRADLEY,D . D e te rmin a t io n o fburn ing ve loc i t ie s : a c r i t ica l rev iew, Combus t .

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ve loc i ty o f me thane /a i r mix tures , C o m b u s t . F l a m e

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o f me th a n e /a i r m ix tu res u s in g h o t -w i re a n e mo -mete rs in c losed vessel exp los ions , T h i r t e e n t h S y m p .(Int .) on Combustion, pp . 575-581, The Com bus t ionInstitute , Pittsburgh (1971).BABK1N, V. S., KUZNE TSOV, I. L. an dKOZANCHENKO,L. S . In f luence o f curva ture o n th era te o f p ropa ga t ion o f a lamina r f lame in a lean



326 C .J . RALLIS an d A. M. GARFORTH

( B K 6 4 )

( B K 6 5 )

( B K 6 6 )

( B K 6 7 )

(B173)

(Br49)

(BS54)

(BS57)

(BS69)

(BT27)

(BV66)

( B W 5 3 )

( B W 5 4 )

( C H 3 2 )

( C H 6 3 )

( C K 5 5 )

(CL51)

(CL59)

(Co62)

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BABKaN, V. S., V'YuN , A. V. an d KOZACHENKO, L. S. (D W 51)T h e e ff ec t o f p r e s s u r e o n n o r m a l f l a m e v e l o c i t y

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BURGOYNE, J . H . a nd WEINBERG, F. A m eth od of

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DIXO N-LEw is, G. , GOLDSWORTHY, F. A. a ndGREENBERG,J . B . F l a m e s t r u c t u r e a n d f l a m e r e a c -

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DUGGER, G. L . Ef fec t o f in i t i a l t emp era tu re o n f l ame

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(F i55)

( F K 3 5 )

( F M 1 7 )

( F M 3 7 )

( F M 4 0 )

( F P 6 9 )

( F P 7 6 )

( F R 3 5 )

(Fr53)

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( H C 5 4 )

( H H 5 6 )

( H K 7 7 )

( H o 0 6 )

( Ja71)

( Jo46)

(Ke68)

( K N 5 9 )

( K U 6 2 )

( K W 4 8 )

(La62)

(LC64)

(Le54)

(Le59)

(Li53)

(Li54)

(Li67)(Li673)

(Li68)

( L P 5 1 )

( L V 3 4 )

(LV51)

( L V 5 6 )

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

v e l o c i t y o f a g a s m i x t u r e i n a s p h e r i c a l c o n s t a n t

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C o m p a r a t i v e s t u d y o f i o n i z a t i o n i n a c e t y l e n ~

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g a s e s b y i n f r a r e d e m i s s i o n a n d a b s o r p t i o n

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(1961).

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T h e d e te rmin a t io n o f l a min a r b u rn in g v e lo c ity 3 29

(RM71)

(RO62)

(RP62)

(RT63)

(SA57)

(SE59)

(Se61)

(SG56)

(Sh73)

(SH76)

(Si59)

(SL48)

(Sm37)

(Sp56)

REED, S. B., MINEOR , J. a nd MCNA UCHTON ,J. P.The e ffec t on the burn ing ve loc i ty o f me thane o fv i t i a t io n o f c o mb u s t io n a i r , J . Ins t . F ue l 44 , 149-155 (Sp57)(1971).RAEZER, S . n . and OLSEN, H. L . M easurem ent o fl a min a r f l a me s pe ed s o f e th y le n e -a i r a n d p ro p a n e - (S S 5 3 )a i r m ix tu re s b y th e d o u b le k e rn e l me th o d , Combus t .

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me th o d , Seventh (Int .) Syrup. on Combustion, pp .377-385 , B ut te rworths , L ondo n (1959).SENIOR, D. A. B urn ing ve loc i t ie s o f hy drog en-a ira n d h y d ro g e n -o x y g e n mix tu re s. D e te rm in a t io n b y (T r6 2 )b u rn e r me th o d w i th S c h l i e re n p h o to g ra p h y .C o m b u s t . F l a m e 5 (1), 7-10 (1961).SINGER, . M. ,GRUMER, . and COOK,E. B. Proc . GasDynamics Symp. on Aero thermochemis t ry , p. 139,No rthw es te rn Univers i ty , Evans ton , I l l ino is (1956) . (Ts78)SHIMIZU, S . Temp era tu re measu remen t o f p re -mix e d fu e l - a i r m ix tu re s b y a n in f ra re d r a d ia t io np y ro me te r , B u ll . J S M E 16 (92), 333-344 (1 97 3) . (VK74)SNOOT, L. n ., HECKER,W. C . and WILLIAMS,G . A .

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with a cha in reac t ion , Phil . Trans. Roy. Soc.(London) A249, 1-25 (1956).

SPALDING,D. B. Pred ic t ing the lam inar f lam e speedin gases with tempera tu re -exp l ic i t reac t ion ra tes ,Combus t . F lame 1, 287-295 (1957).STREHLOW, R. A. an d STUART, . G . A n im pro vedsoap bubble m ethod of measuring f lame ve loc it ies ,

Fourth (Int .) Syrup. on Combustion, pp. 329-336,W il l iams and W ilk ins , Ba l t imore (1953).SPALDING,D. B., STEPHENSON, . L. and TAYLOR,R .G . A c a lc u la t io n p ro c e d u re fo r t h e p re d ic t io n o flaminar f lame speeds , C o m b u s t . F l a m e 17, 55-64(1971).STEVENS, F. W . N A C A R e p o r t s , 176 (1923); 280(1927); 305 (1929); 337 (1929); 372 (1930).STEVENS, F . W. The ra te o f f lame p ropag a t ion ingaseous exp los ive reac t ions , J. Am. Chem. Soc. 48,1896 (1926); ibid. 50, 3244 (1928).SIMON,D. and WON G, E . L . Burn ing ve loc i tyme a s u re me n t s , J . Chem. Phys . 21, 936 (1953).

SIMON, D. i . and WONG, E . L . An eva lua t ion o f thes o a p b u b b le m e th o d fo r b u rn in g v e lo c i ty me a s u re -

me n t s u s in g e th y le n e -o x y g e n -n i t ro g e n a n dme th a n e -o x y g e n -n i t ro g e n mix tu re s , N A C A T e c h .N o t e 3106, February (1954).TREMEER,G . E . B . T h e d e te rmin a t io n o f t h e b u rn in gve loc i t ie s o f gaseous mix tures us ing a spher ica lcons tan t vo lum e combu s t ion vesse l, M.Sc. Thesis,U n iv e r s i ty o f t h e Wi tw a te r s ra n d , J o h a n n e s b u rg ,August (1962).TSATSARONIS,G . P re d ic t io n o f p ro p a g a t in g l a min a rf lames in methane , oxygen , n i t rogen mix tures ,C o m b u s t . F l a m e 33, 217-239 (1978).VANCE, G. M. and KRIER, H. A theo ry f or sph ericaland cy l indr ica l lamin ar p rem ixed f lames , Combus t .F l a m e 22, 365-375 (1974).VON ELBE,G. and LEWIS, B. Stab ility and s tructu re

of burner f lames , J . Chem. Phys . 11, 75-97 (1943).WESTENBERG,A. A. an d FRISTROM,R . i . M e t h a n e -oxygen f lame s t ruc tu re . IV. Chem ica l k ine t ic con-s ide ra t ions . J . Phys . Chem. 65, 591 (1961).WOHL, K. Third Syrup . on Combus t ion , p. 203,W il l iams and W ilk ins , Ba l t imore (1949).ZELDOVICH,Y. B. and BARENBLATT,G . I . T h e o ry o ff l a me p ro p a g a t io n , C o m b u s t . F l a m e 3, 61-74 (1959).

( M a n u s c r i p t r e c e i v e d 8 F e b r u a r y 1 9 8 0 )

CJ Rallis AM Garforth PECS 1980

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