CJ Rallis AM Garforth PECS 1980
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Transcript of CJ Rallis AM Garforth PECS 1980
7/29/2019 CJ Rallis AM Garforth PECS 1980
http://slidepdf.com/reader/full/cj-rallis-am-garforth-pecs-1980 1/27
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
7/29/2019 CJ Rallis AM Garforth PECS 1980
http://slidepdf.com/reader/full/cj-rallis-am-garforth-pecs-1980 2/27
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
r io _
)-
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.
7/29/2019 CJ Rallis AM Garforth PECS 1980
http://slidepdf.com/reader/full/cj-rallis-am-garforth-pecs-1980 3/27
The determin ation of lamina r burning veloci ty 305
IS0q
140q
1001
Lu
%o
i r I i I [--E TH Y L E N E I O X Y G E N ~ =
I I [ f ' - T - ~ ' x ' i l : M P f i R A T U R E
0 , 2 0
g,18O0C '~ 0 ,~O O
2C - - 0 , 14 >
0,12
O,lO0 , 8 - ~ i ~ . . . . . . . . . ~ A R E A R A T I O
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)
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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.
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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
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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 ,
7/29/2019 CJ Rallis AM Garforth PECS 1980
http://slidepdf.com/reader/full/cj-rallis-am-garforth-pecs-1980 7/27
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
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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
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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 .
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2 s \ \ I \ \ R \ \
t Ve loci ty ector,200 cm. per second
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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) .
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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).
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The determin ation of lamina r burning veloci ty 313
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 .
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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
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The determin ation of lamina r burning veloci ty 315
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 ).
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316 C.J. RALLISand A. M. GARFORTH
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-
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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
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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
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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
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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 ]
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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 .
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322 C.J. RALLISand A. M. GARFORTH
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
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The determin ation of lamina r burning veloci ty 323
~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 .
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324 C.J. RALLISand A. M. GARFORTH
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
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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 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 .
F l a m e 18, 133-153 (1972).A N D R E W S , G . E . a n d B R A DL E Y, n . T h e b u r n i n g
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
19, 275-288 (1972).A N D R E W S , G . E . a n d B R A D L E Y , n . D e te rmin a t io n o fburn ing ve loc i ty by double ign i t ion in a c losedvessel, Combus t . F lame 20, 77-89 (1973).ADAMS,G. R. and COOK,G. B. The effect of pressureon the mechan ism and speed of the hydraz inedecompos i t ion f lame , Combus t . F lame 4, 9-18(1960).ANDERSON,J . W. and FEIN, R. S . M easurem ents o fnormal burn ing ve loc i t ie s and f lame tempera tu resof Bunsen f lames , J . Chem. Phys . 17, 1268-1273(1949).A N D E RS O N , J . W . a n d F E I N , R . S . Me a s u re me n t o fnorm al burn ing ve loc i t ie s o f p rop ane /a i r f lamesf ro m s h a d o w p h o to g ra p h s , J . Chem. Phys . 19, 441 -443 (1950).A G N E W , J . T . an d G R A I F F , L . B . The pressured e p e n d e n c e o f l a min a r b u rn in g v e lo c i ty b y th es p h e r i c a l b o mb me th o d , C o m b u s t . F l a m e 5 , 209-219 (1961).BOLZ, R. E . and BURLAGE,H . Je t Propu ls ion 25, 265(1955).BOLZ, R. E. an d BURLAGE,H . P ro p a g a t io n o f fr eef lames in lamin ar and tu rbu len t f low f ie lds , N A S AT e c h . N o t e D-551 (1960).BRADLEY, D. and HUNDY, G. F. Bu rning velocities
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
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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)
p r o p a n e / a i r m i x t ur e . Dokl. Akad. Sci. SSSR 146, (3)6 2 5 - 6 2 7 S e p t e m b e r (1 9 62 ); E n g l i s h t r a n s l a t i o n ,
Proc. Acad. Sci. USSR, Phys . Chem . Soc. 146 , 677 -
679 (1962).BABKIN, V. S., KOZACHENKO,L. S. and KUZNETSOV,
I . L . T h e i n f lu e n c e o f p r e s s u r e o n t h e b u r n i n g ( C P 3 7 )v e l o c i t y o f m e t h a n e / a i r m i x t u r e s . Zh. Prike
Mekhan. Tekn. Fiz. 145-149 (1964) . (CW6 3)BABKIN, V. S. and KOZACHENKO,L. S. E n e r g y l o s se sin exp los ions in a sphere . Fiz. Gored Vzryva 1 (2) (De49 )114-117 (1965); E ng l i sh t r ans la t io n , Combustion,Explosion and Shock Waves 1, 81-83 (1965).BABKIN, V. S. an d KOZACHENKO, L. S. Study o f (Di53)n o r m a l b u r n i n g v e l o c it y in m e t h a n e / a i r m i x t u r e s a t
h igh p ressures . Fiz. GoreD Vzryva 2 (3) 77-86
( 1 9 6 6 ); E n g l i s h t r a n s l a t i o n , Combustion, Explosion
and Shock Waves 2 , 46-5 2 (1966) . (Di67)BABKIN, V. S. an d KONONENKO, V. G . Eq ua tio ns for
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 ty i n a s p h e r i c a lc o n s t a n t v o l u m e b o m b . Fiz. GoreD Vzryva 3 (2)
268-27 5 (1967) . (DG 75)BLEDJIAN,L . C o m p u t a t i o n o f t im e - d e p e n d e n t l a -
m i n a r f l a m e s t r u c t u r e . Combust. Flame 20 , 5 -17(1973).BROEZE, J. Third Symp. on Combustion, p. 146,
W i l l i ams and Wi lk ins , Ba l t im ore (1949) . (DS57)
BOTHA, J. P. an d SPALDING,D . B . T h e l a m i n a r f l am es p e e d o f p r o p a n e / a i r m i x t u r e s w i t h h e a t e x t r a c t i o n
f rom the f l ame . Proc. Roy. Soc. A225, 71-96 (1954).BOLLINGER, L. E., STRAUSS, W . A. an d EDSE, R. Ind. (DS74)
engng Chem. 49, 768 (1957).BALIN, V. A ., SHORIN, S. N. an d ERMOLAEV, O. N.
Teploenergetika 16 , 75 (1969) ; Eng l i sh t r ans la t io n ,
Thermal Engng 4, 109 (1969).BONE, W. A. an d TOWNEND, D. T. A . Flame and ( D u 5 2 )
Combustion in Gases, C h a p t e r s 1 1 - 1 4 , L o n g m a n s ,
L o n d o n ( 1 92 7 ).
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
i n v e s t i g a t e d b y t h e i n i t i a l -s e c t i o n m e t h o d i n ac o n s t a n t v o l u m e v e ss el . Fiz. Gored Vzryva 2 (2) 52 -
6 0 ( 1 9 6 6 ) ; E n g l i s h t r a n s l a t i o n , Combustion, ( D W 6 7 )
Explosion and Shock Waves 2 (2 ) 32-37 , Apr i l
(1966).
BURGOYNE, J . H . a nd WEINBERG, F. A m eth od of
a n a l y s i s o f a p l a n e c o m b u s t i o n w a v e , Fourth Symp.on Combustion, pp . 294-302 , W i l l i ams and Wi lk ins , (EA58)
Bal t imore (1953) .BURGOYNE,J . H . and WEINBERG,F. Proc. Roy. Soc.A224, 216-308 (1954).COWARD, H. F. a n d HARTWELL, F. J. Stud ies in the (EH69)m e c h a n i s m o f f l a m e m o v e m e n t . P a r t I : T h e u n i -
f o r m m o v e m e n t o f fl a m e i n m i x t u re s o f m e t h a n ea n d a i r , i n r e l a t i o n t o t u b e d i a m e t e r . J. Chem. So c. ( E H 7 0 )
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328 C .J . RALLISand A. M. GARFORTH
( 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 )
(LV61)
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(RM71)
(RO62)
(RP62)
(RT63)
(SA57)
(SE59)
(Se61)
(SG56)
(Sh73)
(SH76)
(Si59)
(SL48)
(Sm37)
(Sp56)
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