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C O M B U S T I O N A N D F L A M E 3 4 , 8 5 - 9 8 ( 1 9 7 9 ) 8 5
S t a n d - O f f D i st a n c e s o n a F l a t F l a m e B u r n e r
C O L I N R . F E R G U S O N
De pa rtm ent o f Mechanical Engineering, Purdue University , West Lafa yette , Indiana 47 90 7
a n d
J A M E S C . K E C K
De par tme nt of Mechanical Engineering, Massachusetts Ins ti tut e o f Technology, Cambridge, Massachusetts 0 213 9
Fo r a g iven s tan d-of f d is tance of a laminar f lame on a porous meta l burner , i t has been shown b oth exper i -
menta l ly and theore t ica l ly tha t the re exis ts two solut ions , a low-speed flame and a h igh-speed f lame. Fo r
small enough s tand-of f d is tance the re i s no solu t ion . The minim um s tand-of f d is tance is ident i f ied as the
quasi - s teady approx imat ion to the ext inc t io n length for f lames quenching in f lows perpendicula r to a hea t
sink.
Measurements of f lame speed, maxim um f lame tempera ture , and s tand -of f d is tance have been cor re la ted
f o r hyd r oge n , e thy l e ne , a nd m e tha ne f l am e s by a P e c l e t num be r d e pe nde n t on ly on the r a t i o o f t he he a t o f
com bust ion to the he a t loss. The cor re la t ion agrees quant i ta t ive ly wi th solu t ion o f one-dimensiona l f lame
equa t ions where a D irac -de l ta func t ion m ode ls the reac t ion ra te .
I . I N T R O D U C T I O N
T h i s p a p e r p r e s e n t s r e s u l ts o f a n e x p e r i m e n t a l a n d
t h e o r e t i c a l s t u d y o f n o n - a d i a b a t i c l a m i n a r f l a m e s
[ 1 ] . I t e x t e n d s a n e a rl i e r w o r k [ 2 ] w h i c h p re -
d i c t e d t h a t f o r s u f f i c i e n tl y l ar g e s t a n d - o f f d i s t a n c e s
o f a f l a m e o n a f la t f l am e b u r n e r t h e r e w o u l d b e
b o t h a h i g h - t e m p e r a t u r e an d a l o w - t e m p e r a t u r e
s o l u t i o n . I t w a s a l s o p r e d i c t e d t h a t t h e r e w o u l d b e
a m i n i m u m d i s t an c e f o r w h i c h s o l u t i o n s e x i s t . I n
t h e p r e s e n t w o r k t h o s e p r e d i c t i o n s a re e x p e r i-
m e n t a l l y v e r i f i e d a n d t h e e a r l i e r a n a l y s i s i s i m -
p r o v e d s o t h a t q u a n t i t a t i v e a g r e e m e n t i s a c h i e v e d
b e t w e e n t h e o r y a n d e x p e r i m e n t .
B o t h s t u di e s w e re m o t i v a t e d b y a n d c o n t r i b u t e
t o t h e p r a c ti c a l p r o b l e m o f m o d e l i n g t h e h y d r o -
c a r b o n a n d c a r b o n m o n o x i d e e m i s s io n fr o m
i n t e r n a l c o m b u s t i o n e n g i n e s . T h e a n a l y s i s i s
r e l e v a n t t o t h e p r o b l e m o f a t r a n s i e n t f l a m e
p r o p a g a t i n g t o w a r d s a w a l l a n d e x t i n g u i s h i n g a t
t h e q u e n c h d i s ta n c e . T h e m i n i m u m s t a n d - o f f
d i s t a n c e i s a q u a s i - s t e a d y a p p r o x i m a t i o n t o t h a t
q u e n c h d i s t an c e . A c o m p l e m e n t a r y a n a l y s is o f
t r a n s i e n t f l a m e q u e n c h i n g h a s b e e n m a d e b y
K u r k o v a n d M i r s k y [ 3 ] . T h e q u e n c h d i s t a n c e
c o m p u t e d b y t h e q u a s i - s t e a d y m o d e l d e s c r i b e d
l a t e r a n d t h e i r a n a l y s i s i s a b o u t t h e s a m e . A n
i m p o r t a n t d i f f e r e n c e a p p e a r s, h o w e v e r , i n t h e
a m o u n t o f u n b u r n e d f u e l l e f t at t h e w a l l a f t e r
q u e n c h i n g h a s o c c u r r e d , t h e q u a s i - s te a d y m o d e l
p r e d i c t i n g n e a r l y t w i c e as m u c h f u e l i n t h e q u e n c h
l a y e r .
T h e p r o b l e m a n a l y z e d i s a o n e - d i m e n s i o n a l
l a m i n a r f l a m e w i t h h e a t l o ss a t t h e c o l d b o u n d a r y .
T h e e n e r g y a n d d i f f u s i o n e q u a t i o n s s o l v e d a r e
t h o s e g i v e n b y S p a l d i n g [ 4 ] w i t h a D i r a c - d e l t a
C opyr igh t © 1979 by The C om bus t ion I n s t i tu t e
Publ ished by Elsevier Nor th-H ol land, Inc . 0010-2180 /79/010085+ 14 $01.75
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8 6 C O L I N R . F E R G U S O N a n d J A M E S C . K E C K
f u n c t i o n m o d e l i n g t h e r e a c t i o n r a t e . T h e a n a l y s i s
r e l a t e s t h e f l a m e s p e e d , t h e b u r n e d g a s t e m p e r a -
t u r e , a n d t h e s t a n d - o f f d i s t a n c e . T h e e x p e r i m e n t s
w e r e d e s i g n e d t o m e a s u r e t h e s e t h r e e q u a n t i t ie s .
1 I . T H E O R Y
A s s u m i n g t h a t a l l s p e c i e s h a v e e q u a l a n d c o n s t a n t
s p e c i f i c h e a t s a t c o n s t a n t p r e s s u r e , t h e e n e r g y
e q u a t i o n f o r s t e a d y l a m i n a r f l a m e p r o p a g a t i o n i s
t a k e n t o b e [ 4 ]
d f x d T ~ d T
d x +
x [_ d r _ ] - p , , S , ,c p - p u S u q ~ D - x ) = O
(1)
w h e r e t h e c o o r d i n a t e s y s t e m is c h o s e n s o t h a t t h e
po rou s bu rne r su r f ac e i s a t x --- 0 a nd the f l am e i s
a t x - -- D . T h e s y m b o l s in E q . ( 1 ) a r e d e f i n e d b e l o w :
?, - t h e r m a l c o n d u c t i v i t y
T - t e m p e r a t u r e
p - d e n s i t y
Su
- f l a m e s p e e d
q - e n t h a l p y o f c o m b u s t i o n p e r u n it m a s s
6 ( D - x ) - D i r a c - d e l t a f u n c t i o n w i t h u n i t y i n te g r al
S u b s c r i p t s u a n d b r e f e r t o t h e u n b u r n e d a n d
b u r n e d s t a t e s , r e s p e c t i v e l y . T h e b o u n d a r y c o n -
d i t ions a re :
7 (0) = 7 u
d
T °°) = Tb dx oo) = 0
(2 )
I n d i m e n s i o n l e s s v a r ia b l e s th e e n e r g y e q u a t i o n
m a y b e w r i t t e n a s :
d2T d¢
- - - - - - + 6 ( P e - - ~ ) = 0 ( 3 )
d ~ 2 a t
w h e r e i n t e r m s o f t h e a d i a b a ti c f la m e t e m p e r a t u r e
Tb o = T~, + q/ c,,
T - - T ~
T b ° - T u
f o x d r l
p , ,S , ,cp - -
f o D d r '
Pe =
puSucp k
T h e s p a c e v a r i a b l e ~ i s a P e c l e t n u m b e r b a s e d o n
d i s t a n c e x f r o m t h e w a l l a n d P e i s t h e P e c l e t
n u m b e r b a s e d o n t h e s t a n d - o f f d i st a n ce D . T h e
b o u n d a r y c o n d i t i o n s a r e:
r (0) = 0
d~-
r(°° ) = rb ~ (oo) = 0 (4)
T h e s o l u t i o n t o E q . ( 3 ) s u b j e c t t o t h e b o u n d a r y
c o n d i t i o n s ( 4 ) i s
( e x p (~ ) - 1
r b \ e x p ( P e ) - - 1 / 0 < ~ < P e
rb ~ ~> Pe
(5)
a n d t h e P e c l e t n u m b e r b a s e d o n s t a n d - o f f d i st a n c e
is
P e = I n (1 _ % ) - 1 = l n \ T b ° ( 6 )
A s i m i l a r r e s u l t h a s b e e n d e r i v e d b y K i h a r a e t a l .
[ 5 ] u s i n g a n i g n it i o n t e m p e r a t u r e m o d e l .
T h e c h a r a c t e r i s ti c a n a l y s i s fo r p e r p e n d i c u l a r
f l o w p r e s e n t e d p r e v i s o u l y ( 2 ) i s , a c c o r d i n g t o E q .
( 6 ) , s t r i c tl y v a l id o n l y f o r s m a l l P e c l e t n u m b e r s , i n
w h i c h c a se t h e l e f t si d e m a y b e i n t e g r a t e d a p p r o x i -
m a t e l y a n d t h e r i g h t - h a n d s i d e e x p a n d e d i n t o a
T a y l o r se ri es . T h e z e r o o r d e r a p p r o x i m a t i o n i s
Pe ----PuSucp----------D_~- b - - Tu ( 7 )
~ , ( D ) T ~ o _ r ~
i d e n t i c a l t o t h a t g i v e n i n t h e p r e v i o u s p a p e r .
T h e e f f e c t o f c h e m i c a l r e a c t i o n s o n t h e s t a n d -
o f f d i s t a n c e g i v e n b y E q . ( 6 ) i s e n t i r e l y c o n t a i n e d
i n t h e l a m i n a r f l a m e s p e e d , a n d s i n c e t h i s c a n b e
m e a s u r e d e x p e r i m e n t a l l y , t h e i n c l u s io n o f a
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S T A N D - O F F D I S T A N C E S O N A F L A T F L A M E B U R N E R 8 7
d e t a i l e d c h e m i c a l k i n e t i c m o d e l i s u n n e c e s s a r y .
H o w e v e r , t o c h e c k t h e s e n i t i v i t y o f t h e r e s u l ts t o
i 'm i t e r eac t ion r a t es , a more genera l ana lys i s was
car r i ed ou t [ 1 ] i n wh ich the hea t r e l ease r a t e was
m o d e l e d b y a n o v e r a l l A r r h e n i u s r a t e e q u a t i o n .
T h e r e s u lt s s h o w e d t h a t E q . ( 6 ) i s a g o o d a p p r o x i -
m a t i o n p r o v i d e d t h e
E / R T b °
1> 7, where E is a
charac t e r i s t i c ac t iva t ion en ergy an d R i s t he gas
c o n s t a n t .
T o p r o c e e d f u r t h e r , i t i s n e c e s s a r y t o k n o w t h e
d e p e n d e n c e o f t h e f l a m e s p e e d o n t h e f l a m e t e m -
p e r a t u r e . K a s k a n [ 6 ] h a s s h o w n t h a t t h e f l a m e
s p e e d o n p o r o u s m e t a l b u r n e r s o f t e n c o r r e l a t e s
emp i r i ca l ly as
s o = e x p
( 8 )
wh ere Su ° i s t he ad iaba t i c f l ame speed and E A is
a n a p p a r e n t a c t i v a t i o n e n e r g y . C o m b i n i n g E q s .
( 6 ) a n d ( 8 ) y i e l d s a P e c l e t n u m b e r b a s e d o n t h e
a d i a b a ti c f l a m e s p e e d a n d t h e s t a n d - o f f d is t a n c e
fo D 1
e o =
p ~S u Oc v - -d x
I E A ( 1 1 ) 1
{ T b ° - T u )
= e x p ~ -~ T b T b ° I n
\T b ° T b
( 9 )
E q u a t i o n ( 9 ) c a n b e u s e d f o r c o m p u t i n g s ta n d -
o f f d i s t ance g iven the ad iaba t i c f l ame speed , t he
a p p a r e n t a c t i v a t i o n e n e r g y , a n d t h e b u r n e d g a s
t e m p e r a t u r e . A t y p i c a l s o l u t i o n o f E q . ( 9 ) is
s h o w n i n F i g . 1 . T h e P e c l e t n u m b e r is a U - s h a p e d
f u n c t i o n o f t e m p e r a t u r e . I t c a n b e s e e n t h a t f o r
su f f i c i en t ly l a rge d i s t ances the re i s bo th a h igh-
t e m p e r a t u r e a n d a l o w - t e m p e r a t u r e s o l u t i o n .
T h e r e i s a l s o a m i n i m u m d i s t a n c e f o r w h i c h
so lu t ions ex i s t .
T h e p h y s i c a l r e a s o n f o r t h is b e h a v i o r is t h a t f o r
t e m p e r a t u r e s n e a r t h e a d i a b a t i c f l a m e t e m p e r a t u r e
o n l y a s m a ll a m o u n t o f h e a t n e e d s t o b e tr a n s-
f e r r e d t o t h e b u r n e r t o m a i n t a i n a s t e a d y - s t a t e
e n e r g y b a l a n c e , h e n c e t h e s t a n d - o f f d i s t a n c e w i l l
be l a rge . As the f l ame t empera tu re decreases , t he
hea t l oss pe r un i t mass inc reases m on o to n ica l ly ,
s o t h a t t h e h e a t t r a n s f e r t o t h e b u r n e r m u s t a l s o
i n c re a s e a n d t h e s t a n d o f f d i s ta n c e m u s t d e c r e a s e .
H o w e v e r , a t l o w e n o u g h t e m p e r a t u r e s , t h e e n e r g y
g e n e r a t i o n d e c r e a s e s m o r e r a p i d l y w i t h f l a m e
t e m p e r a t u r e t h a n t h e h e a t l o s s p e r u n i t m a s s
i n c r e a s e s , s o t h a t t h e h e a t t r a n s f e r t o t h e b u r n e r
m u s t d e c r e a s e a n d t h e s t a n d - o f f d i s t a n c e m u s t
inc rease .
T o c o m p u t e t h e m i n i m u m s t a n d - o ff d i st a n ce ,
w h i c h i s t h e q u a s i - s t e a d y a p p r o x i m a t i o n t o t h e
q u e n c h d i s ta n c e D 1 f o r t h e p e r p e n d i c u l a r f l o w ,
o n e m u s t d e t e r m i n e t h e t e m p e r a t u r e a t w h i c h
~D/OTo
= 0 s o th a t b y E q . ( 9 )
Tb O~ EA Tb 1
~ - - - . = 0
T b
OTb 2R Tb Tb ° Tb ln
( T b °
|
/
Tb ° T,,
( 1 0 )
w h e r e
f o ° l
T h e a p p a r e n t a c t i v a ti o n e n e r g y o f m o s t f la m e s is
l a r g e e n o u g h t h a t a n a s y m p t o t i c s o l u t i o n t o E q .
( 1 0 ) i s a d e q u a t e . T h e z e r o o r d e r e q u a t i o n f o r t h e
b u r n e d t e m p e r a t u r e a t q u e n c h i n g T b * is
{ r b ° - r b * ~ 2 R r b ° ( 1 1 )
e
T h e s o l u t i o n o f E q . ( 1 1 ) i s g i ve n in F i g. 2 f o r t h e
spec ia l an d m os t p r ac t i ca l case Tb ° >> Tu . Com -
p a r i s o n t o a n e x a c t s o l u t i o n c a n b e m a d e w i t h
F i g . 1 , w h i c h s h o w s t h e P e e l e r n u m b e r i s m i n i m u m ,
f o r
E A / 2 R T b 0
- - 5 , when
T b 0 - - T b ) /T b ° =
0 . 0 7 2 . T h e c o r r e s p o n d i n g v a lu e o b t a i n e d f r o m F i g .
2 is
Tb ° - - Tb ) /Tb °
= 0 . 0 7 6 . N o t e t h a t f o r l a r g e
E A / 2 R T b °
i t t a k e s v e r y l i t t l e h e a t t o q u e n c h a
f l a m e s o t h a t t e m p o r a l t e m p e r a t u r e c h a n g e s d u r in g
f l ame quench ing a r e smal l and a quas i - s t eady
approx imat ion i s va l id .
A n o t h e r p a r a m e t e r o f i n t e r e s t i n m o d e l i n g
h y d r o c a r b o n e m i s s i o n f r o m i n t e r n a l c o m b u s t i o n
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8 8 C O L I N R . F E R G U S O N a n d J A M E S C . K E C K
1
. .~
I
Z
~ 5
W
. , J
U
a .
I 0 - -
1 I 1
~ . = 5
2 R T b
Tu _
- 0 . 2 5
o I I I
.6 O. 8 I.O
F L A M E T E M P - T b
T b o
Fig. 1. Stand-o ff Peclet number based on adiabatic flame speed as a function of flame
temperature.
e n g i n e s is t h e m a s s o f f u e l l e f t a t t h e w a l l w h e n t h e
f l a m e is q u e n c h e d . T o c o m p u t e t h is p a r a m e t e r , a
c o n s e r v a t i o n e q u a t i o n f o r th e f u e l m a s s f ra c t i o n
CA
is i n t r o d u c e d :
d / dCa\ dCA
= 0
( 1 2 )
w h e r e D is t h e d i f f u s i o n c o e f f i c i e n t f o r t h e f u e l .
E q u a t i o n ( 1 2 ) c a n b e d e r i v e d w h e n t h e b i n a r y
d i f fu s i on c oe f f i c i e n t s o f a ll pa i r s o f spe c i e s are
e q u a l o r b y p o s t u l a t i n g t h a t a ll d i f f u s i o n v e l o c i ti e s
o b e y F i c k ' s l a w [ 7 ] .
I n d i m e n s i o n l e s s v a r ia b l e s, E q . ( 1 2 ) i s :
1 d 2 y d y
8 ( P c - - ~ ) = 0 ( 1 3 )
w h e r e L = )~/pl)cp i s t h e L e w i s n u m b e r a n d y =
CA/CA u i s t h e n o r m a l i z e d f u e l m a s s f r a c t i o n . T h e
b o u n d a r y c o n d i t i o n s t o i m p o s e o n t h e f u e l a r e :
y (oo ) = 0 d~ (oo) = 0 (14 )
a n d t h e s o l u t i o n i s
y = 1 - e xp [L(~ - P e ) ]
(15)
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S T A N D - O F F D I S T A N C E S O N A F L A T F L A M E B U R N E R 8 9
1.0
.8
.6
. 4
e
i
W
B _
O
' . 0 8 ~
I I :
I - -
~ . 0 6 - -
l ad
ILl
I - - . 0 4
.02
.01
Fig. 2.
I 1 I I I
I I I I I I
2 4 6 8 I 0 20
A C T I V A T I O N E N E R G Y -
E A
2 R T b
Leading order asy mpto tic solut ion fo r the tem perature at quenching.
m
4O
N o t i c e t h a t y ( 0 ) = ~ 1 a n d
dy/d~) O) 4=
0 ; r a t h e r
y ( 0 ) - l / L ) d y /d ~ ) O ) = 1 , w h i c h s h o w s t h a t the
r e a c t a n t m a s s f l u x f r a c t i o n a t t h e b u r n e r e x i t
p l a n e i s u n i t y b u t t h a t t h e c o m p o s i t i o n i s n o t
p u r e r e a c t a n t s . T h u s p r o d u c t s a r e d i f f u s i n g i n t o
t h e b u r n e r a t t h e s a m e r a t e t h a t t h e y a r e c o n -
v e c t e d o u t . T h i s i s e q u i v a l e n t t o t h e f l a m e h o l d e r
m o d e l o f H i r s c h f e l d e r e t a l. [ 8 ] .
T h e m a s s o f f u e l p e r u n i t w a l l ar e a , a s s u m i n g
pX = PuXu
is
m q ~
fo D CA , ,______~
p C a d x = S ucp
× P e - - L [1 - - e x p ( - -L P e ) ]
= P ~ C A= D x - - ~ e [ 1 - - e x p ( - L P e ) I
( 1 6 )
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90 COLIN R. FERGUSON and JAMES C. KECK
The mass of fuel in a quench layer is computed by
evaluating the Peclet number and flame speed,
Eqs. (6) and (8), at the temperature of quenching
T b*. Such calculations have been done under the
conditions specified by Kurkov and Mirsky [3] for
comparing their transient solution to the present
quasi-steady model. When differences of definition
are accounted for, good agreement is realized for
the Peclet number but the quasi-steady model
predicts about 1.75 times as much fuel as the
quench layer.
I I I . E X P E R I M E N T S
The theoretical analysis relates the flame speed,
the burned gas temperature, and the stand-off
distance by a Peclet number dependent only on
the ratio of the heat of combustion to the heat
loss. The experiments were designed to check that
relationship. The basic apparatus used to study the
stand-off distance experimentally is a porous
metal flat flame burner. The design is similar to
Kaskan's [6] with three important differences.
One, the 5.1-cm diameter burner sits atop a 15-
cm diameter by 46-cm long plenum that creates a
steady uniform velocity upstream of the porous
metal. Two, the cooling coils were located in the
middle of the porous plug, seven diameters from
the exit plane of the burner, in order to minimize
velocity nonuniformities at the exit plane. The
velocity of gas issuing from the burner in the
absence of combustion was measured with a hot-
wire anemometer and found to be uniform over
the burner surface to within -+5 . The third
difference was the inclusion of an outer annulus of
porous metal through which nitrogen flowed. This
served to eliminate entrainment o f air by the
burned gases downstream of the flame and im-
proved the flame stability for low-velocity flames,
so that a stabilizing screen ia the burned gases was
not necessary.
The temperature of the exit plane of the
burner was measured by two 250-/~ iron-con-
stantan thermocouples in 1.6-mm diameter protec-
tive sheaths imbedded in the porous metal 1 mm
from the surface. One was located at the center
and the other at the edge of the burner. The
burner temperature was controlled by varying
either (or both) the temperature of the cooling
water or of the nitrogen. The two thermocouple
temperatures differed by less than 30 K. This is an
important advantage of Kaskan's design over the
edge cooled burner discussed by Kihara et al. [5].
The flow rates of fuel and air were controlled
by critical flow orifices calibrated by either a
mercury-sealed piston diplacement meter or a
liquid displacement technique. The reactants were
premixed in a mixing chamber upstream of the
plenum.
Flame temperatures were measured by moni-
toting the radiation emitted from a fine wire held
in the flame, a technique called hot-wire pyrom-
etry [9] . The hot-wire pyrometer and the burner
are shown in Fig. 3. A 25-micron Pt-13 percent
Rh wire was held in tension parallel to the burner
surface by the spring loaded probe shown in the
figure. This was the smallest wire that could be
held in tension and survive in the flame for many
hours. A thermocouple was not used, because the
welded junction of fine wire thermocouples is
twice the wire diameter; hence spatial resolution
would have been reduced by a factor of two. The
wire was coated with a film of quartz by immer-
sion into the hot gases from a flame in which a
small amount of hexamethyldisiloxane had been
burned [6] . The diameter of the wire after coating
was measured under a microscope and typically
was 35 /l. The purpose of the coating is to kill
catalytic heating of the wire.
An image of a 3-ram long section of the wire
was focused by a cylindrical lens on the entrance
slit of a photomultiplier. The image on the cathode
is diffuse and covers the whole photosensitive area.
An orange filter was used to minimize flame
radiation seen by the photomultiplier. With an
electrically heated wire it was determined that the
overall sensitivity of the hot-wire pyrometer varied
less than 2 over a wire translation of 5 mm.
The photomultiplier signal V depends on wire
temperature Tw to a power like 14. The calibra-
tion curve is given by:
V = ATw 7/4 exp --BTw -112 ) (17)
where A and B are the calibration constants that
are discussed in reference [9]. Because the wire is
8/18/2019 FergusonKeck CombustionFlame 34-85-1979
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S T A N D - O F F D I S T A N C E S O N A F L A T F LA M E B U R N E R 9 1
°1
.J
J l
t , t
t . i
z
o_
z ~ D
~ o
o ~
0
o
z \ \
u
~ _ z w
w . .. .. .. . z ~ x
/ ~ _
i :
T i
I
~
I
/ / ~
g :
l
I
e~
d ~
.=
8/18/2019 FergusonKeck CombustionFlame 34-85-1979
8/14
9 2 C O L I N R . F E R G U S O N a n d J A M E S C . K E C K
to I > I I I
C 2 H 4
~=1.6
~,= 37 6
8 Tu : :575 K
S u = 0 e r a / s ee
> P : I e lm
- ~ -->
3
o o l _£ 1 2 1
t ~ 0 -V
2
o
o
Q.
2
F i g . 4 ( a ) I n t e n s i t y s i g n a l v e r s us d i s t a n c e f o r e t b y ] e n e - a i r f l a m e . ~ is t h e f u e l - a i r e q u i v -
a l e n ce ra t i o , a n d ~ i s t h e m o l a r n i t r o g e n t o o x y g e n r a t i o o f t h e r e a c t a n ts .
r a d i a t i ng , i t s t e m pe r a t u r e i s s l i gh t l y l e ss t ha n t he
g a s t e m p e r a t u r e . T o e s t i m a t e h e a t t r a n s f e r t o t h e
w i r e t h e f o l lo w i n g c o r r e l a ti o n o f N u s s el t n u m b e r
N = a n d P e c l e t n u m b e r P o b a s e d o n w i r e d i a m e t e r
w a s u s e d [ 1 0 ] :
2
N u / 4 . 4 9 2 \ ( Po ~ 0 ) ( 1 8 )
In \--pT-o}
T h i s g iv e s a r a d i a t i o n c o r r e c t i o n t o t h e b u r n e d g a s
t e m p e r a t u r e :
e o T w 4 d l n
4.492~,b
Tb - - Tw = 2X b \ puSucpb -~d~ l
( 1 9 )
w he re e i s w i re e mi s s i v i t y , o i s t he S t e fa n -Bo l t z -
m a n n c o n s t a n t , a n d d is t h e w i r e d i a m e t e r . T h e
e m i s s i v i t y o f t h e c o a t e d w i r e i s e = 0 . 2 2 ( 6 ) . T h e
e q u i li b r iu m b u r n e d t h e r m a l c o n d u c t i v i t y a n d
s p e c if i c h e a t w e r e c a lc u l a t e d b y t h e c o m p u t e r
p r o g r a m o f S v e h l a a n d M c B r i d e [ 1 1 ] a t th e w ir e
t e m p e r a t u r e . T h e s e c o r r e c t i o n s w e r e t y p i c a l l y
5 0 K .
T h e t e m p e r a t u r e p r o f i l e in t h e f la m e w a s
m e a s u r e d b y m o v i n g t h e c a n t il e v e r p r o b e d o w n t o
t h e b u r n e r s u r fa c e . T h e c e r a m i c p o s ts o f t h e p r o b e
w e r e d e s i g n e d t o s l i p w h e n c o n t a c t o c c u r r e d w i t h
t h e b u r n e r s u r f a c e . S c r e w i n g t h e m i c r o p o s i t i o n e r
d o w n u n t i l t h e p o s t s s l i p e s t a b l i s h e s t h e z e r o
c o o r d i n a t e . T h e w i re is t h e n m o v e d a w a y f r o m t h e
b u r n e r a n d a n
x - y
r e c o r d e r s l a v e d t o t h e m i c r o -
p o s i t i o n e r p l o t s t h e p h o t o m u l t i p l i e r s i g n a l v e r s u s
w i r e p o s i t i o n . T h e m e a s u r e m e n t i s u s u a l l y r e p e a t e d
a t fou r d i f fe re n t s e ns i t i v i t i e s on t he
x - y
r e c o r d e r ,
i n o r d e r t o l o c a t e t h e i n f l e c t i o n p o i n t o f t h e te m -
p e r a t u r e p r o f i l e ( t h e s t a n d - o f f d is t a n c e ) w i t h
p r e c i s i o n . T h e r e p e a t e d m e a s u r e m e n t s a l s o s e r v e t o
e s t a b l i s h a n a v e r a g e r e a d i n g f o r t h e z e r o c o o r d i -
n a t e . T o a c c o u n t f o r t h e p r o b e ' s i n f l u e n c e o n t h e
a p p a r e n t f l a m e p o s i t i o n , f iv e w i r e d i a m e t e r s h a v e
b e e n a d d e d t o th e d i st a n ce m e a s u r e m e n t s [ 1 2 ] .
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S T A N D - O F F D I S T A N C E S O N A F L A T F L A M E B U R N E R 9 3
1600
I l , I
15OO
5 ~
1400
I - -
n ~
w
I - - -
1 5 0 0
/
/
/
/
/ /
5WIRE DIAMETERS
1200
IIOO L__
I I I I
i
I
05 I0 15 20 25
WIRE POSITION-mm
F i g . 4 ( b ) W i r e t e m p e r a t u r e v e r s u s d i s t a n c e f o r t h e s a m e . T h e r e a c t i o n z o n e t h i c k n e s s
is
A = T b _ T i ) / d T [ d x ) i ;
w h e r e i d e n o t e s t h e i n f l e c t i o n p o i n t .
5 0
T h e i n f l e c t io n p o i n t o f t h e t e m p e r a t u r e p r o f il e
w a s c h o s e n t o d e f i n e t h e s t a n d - o f f d i s t a n c e i n s te a d
o f th e m a x i m u m b e c a u se t h e m a x i m u m o c c u rs in a
r e g i o n o f s l o w l y v a r y i n g t e m p e r a t u r e , s o t h a t i t s
p o s i t i o n i s d e t e r m i n e d i n p a r t b y t h e r a d i a t i v e
l o s s e s f r o m t h e b u r n e d g a s e s a n d i n p a r t b y t h e
sm al l v a r i a t ions in de tec to r s ens it iv ity~ w i t h w i re
p o s i t i o n .
A n i n t e n s i t y d i s t a n c e d i a g r a m i s s h o w n i n F i g . 4
t o g e t h e r w i t h t h e w i r e t e m p e r a t u r e d i s t a n c e p l o t .
I t i s c l e a r f r o m t h e s e c u r v e s t h a t t h e z e r o c o o r d i -
n a t e i s a c c u r a t e o n l y t o - +0 .1 m m . N o t e t h a t t h e
r e a c t io n z o n e t h ic k n e s s i s n o t s m a l l c o m p a r e d t o
t h e s t a n d - o f f d i s t a n c e . E v e n m o d e l i n g t h e h e a t
r e l e a s e b y a n o v e r a l l A r r h e n i u s r a t e e q u a t i o n d o e s
n o t a d m i t t h i s p o s s i b i li t y [ 1 ] . T h e p r o b l e m i s t h a t
t h e r e a r e m a n y r e a c t i o n s c h a r a c t e r i z i n g t h e h e a t
r e le a se . I n a h y d r o c a r b o n f l a m e , o n e -h a l f t o t w o -
t h i r d s o f t h e h e a t r e l e a s e i s a s s o c i a t e d w i t h t h e
f o r m a t i o n o f C O a n d H 2 0 i n a z o n e l a b e le d t h e
p r i m a r y r e a c t i o n z o n e b y F r i s t r o m a n d W e st en -
b e r g [ 1 2 ] . T h e r e i s a s e c o n d a r y r e a c t i o n z o n e
w h e r e C O b u r n s u p , r e c o m b i n a t i o n t a k e s p l a c e,
a n d t h e r e m a i n i n g h e a t i s r e l e a se d . I n a h y d r o g e n
f l a m e , o n e e x p e c t s m o s t o f t h e h e a t r e l e a s e t o b e
b y t h r e e - b o d y r e c o m b i n a t i o n r e a c t i o n s, s o w h a t
w a s t h e s e c o n d a r y r e a c t i o n z o n e i n a h y d r o c a r b o n
f l a m e i s t h e p r i m a r y z o n e o f h e a t r e l e a se in a
h y d r o g e n f l a m e . T h e s e c o n s i d e r a t i o n s a re n o t
i m p o r t a n t f o r c o m p u t i n g s t a n d - o f f d i s t a n c e ,
s i nc e t h e e x p e r i r n e n t a l r e s u l t s t h a t f o l l o w in d i c a t e
t h a t m o s t o f t h e h e a t r e le a s e o f a l l t h e s e fl a m e s i s
f a s t e n o u g h t o b e c h a r a c t e r i z e d b y a s i n g l e t e m -
p e r a t u r e T b .
T w o s e t s o f e x p e r i m e n t s w e r e p e r f o r m e d : o n e
a t c o n s t a n t e q u i v a l e n c e r a t i o a n d o n e a t c o n s t a n t
f l a m e s p e e d . F o r e a c h f l a m e , t h e w a l l t e m p e r a t u r e
w a s h e l d c o n s t a n t . B e c a u s e t h e c o o l i n g c o i l s w e r e
2 2 m m b e l o w t h e e x i t p l an e o f t h e b u r n e r , t h e
s u rf a c e t e m p e r a t u r e w a s a s t ro n g f u n c t i o n o f co n -
d i t i o n s a n d c o u l d b e v a r i e d b y o n l y - +2 5 K . T h e
c o n d i t i o n s s t u d i e d w e r e c h o s e n t o k e e p t h a t
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94 COLIN R. FERGUSON and JAMES C. KECK
I 0 0
8 0
60
40
20
E
I
a
,,, 10
I J . I
t . O
'
8
<
.J
u_
6
m
m
m
r 1 1 I
EA= 6 7 5 8 5 2 6 1 -
FUEL ~ @ Tu K)
A H2 0.4 3.76 548
0 C 2 H 4 O. 6 3 6 3 7 3
-~ C 2 H 4 1 .0 7 . 0 0 3 4 8
[ ] C H4 0 . 7 3 .7 6 3 4 8
i t I J I
5 6 7 8
R E C I P R O C A l_ F L A M E T E M P - 1 0-4K - I
Fig. 5. The dependence of flame temperature on burning velocity. The apparent
activation energies in kcal/mole are given at the bottom of each curve; they agree with
Kaskan's [6] within -+18 .
temperature constant so that only one parameter
was varied at a time. A better burner would use
smaller cooling coils.
The data at con stan t equivalence ratio are given
in Figs. 5 and 6. In agreement with theoretical
predi ction, it was foun d that for a given stand- off
distance based on the i nflec tion poi nt in the tem-
perature profile there is indeed both a high-tem-
perature and a low-temperature solution. These
results should not be confused with Spalding's
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I I I I 1 I
E
E
I
I. iJ
Z
0 3
I..1_
i ,
r - ,
z
O 3
1 . 0 - -
I
r-
0 . 5 ; -
I100
S T A N D - O F F D I S T A N C E S O N A F L A T F L A M E B U R N E R 9 5
I I
1200 1300
l 1 I I I
1400 1500 1600 1700 1800
F L A M E T E M P E R A T U R E - K
F i g. 6 . T h e d e p e n d e n c e o f s t a n d - o f f d i s ta n c e o n f l a m e t e m p e r a t u r e . T h e d a t a k e y i s
g i v e n in F i g . 5 . T h e s o l id c u r v e s a re t h e t h e o r e t i c a l f u n c t i o n g i v e n i n E q . ( 9 ) .
1 9 0 0
p r e d i c t i o n [ 1 3 ] t h a t i n t h e p r e s e n c e o f d i s tr i b u t e d
hea t l o ss and hea t r e l ease the re a r e two so lu t ions
for a g iven heat loss per uni t mass ( i .e f lame
t e m p e r a t u r e ) ; r a t h e r t h e r e i s a h i g h - t e m p e r a t u r e
a n d a l o w - t e m p e r a t u r e s o l u t i o n f o r a g i v e n h e a t
l o ss t o t h e b u r n e r .
S i m i la r t o o t h e r r e p o r t s [ 6 , 1 4 , 1 5 ] , i n s t a b i l i t ie s
a n d t h r e e - d i m e n s i o n a l p h e n o m e n a h a v e b e e n
o b s e r v e d . F o r f l a m e sp e e d s c o m p a r a b l e t o v e l o c i t y
f l u c t u a t i o n s i n t h e r o o m , a f li c k e r in g f l am e w a s
o b s e r v e d . A f u r t h e r r e d u c t i o n i n s p e e d c a u s e d
e x t i n g u i s h m e n t .
F o r h y d r o c a r b o n - a i r f l a m e s a c o r r u g a t e d o r
w r i n k l e d f l a m e a p p e a r e d f o r t e m p e r a t u r e s s l i g h t l y
l a rger t han
To*.
I f t he f low r a t e i s i nc r eased
f u r t h e r t h e t e m p e r a t u r e b e c o m e s i n d e p e n d e n t o f
f l a m e s p e e d , a n d e v e n t u a l l y t h e f l a m e i s b l o w n
o f f . T h e w r i n k l in g m e c h a n i s m c a n b e e x p l a i n e d b y
a r g u in g t h a t t h e s t a n d - o f f d i s t a n c e - t e m p e r a t u r e
r e l a t ionsh ip i l l u s t r a t ed by F ig . 1 i s va l id fo r any
s t r e a m t u b e . I t c a n b e s e e n t h a t f o r n e a r l y a d i a b a ti c
f la m e s , t h e s m all v e l o c i t y ( f la m e t e m p e r a t u r e )
v a r i a t i o n s f r o m s t r e a m t u b e t o s t r e a m t u b e c a u s e
l arge va r i a t ions in l oca l s t an d-o f f d i s t ance .
T h e h y d r o g e n u s e d i n th e s e e x p e r i m e n t s h a d
e n o u g h h y d r o c a r b o n i m p u r i t i e s s o t h a t i n a d a r k
r o o m t h e f l a m e w a s l u m i n o u s . I t a p p e a r s t h a t t h e
h igh- t em pera tu re so lu t ion fo r t h is f l ame i s un -
s t ab le , s ince a ce l lu l a r s t ruc tu re was obse rv ed fo r
t e m p e r a t u r e s g r e a t e r t h a n t h e t e m p e r a t u r e T b* o f
m i n i m u m s t a n d - o f f d i st a nc e . T h e m a x i m u m t e m -
p e r a t u r e o f t h e s e f l a m e s i n c r e a s e d m o n o t o n i c a l l y
( a l t h o u g h n o t l i n e a r l y o n t h e A r r h e n i u s p l o t )
w i t h f l a m e s p e e d u p t o t h e h i g h e s t s p e e d m e a s -
u r e d , a b o u t t w i c e t h e a p p a r e n t a d i a b a t i c f l a m e
speed . The appearance o f ce l lu l a r f l ames ind ica t es
the occurence o f se l ec t ive d i f fus ion . S ince these
ce ll s a re t h r e e d im ens iona l , t em pera tu res la rge r
t h a n t h e e q u i l i b r i u m a d i a b a t i c f l a m e t e m p e r a t u r e
c o u l d b e e x p l a i n e d b y s h i f t s o f c o m p o s i t i o n d u e t o
se l ec tive d i f fus io n .
T h e h y d r o g e n - a i r f l a m e s a l s o s h o w e d a n o m -
a l o u s b e h a v i o r r e g a r d i n g c a t a l y t i c h e a t i n g o f t h e
w i r e . F o r b a r e w i r e s , u n l i k e h y d r o c a r b o n f l a m e s
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9 6 C O L I N R . F E R G U S O N a n d J A M E S C . K E C K
E
E
LIJ
¢-)
Z
0 3
E 3
L L
14.
0
( :3
Z
1.0
0.5
CH 4 + AIR
= 3 4 8 K
/ ~
C 2 H 4 + A I R J
Tu =
3 7 3
K
Su=lO cm /sec
P=I arm
0 i 1 f I t I I , I J I I I J ~
.4 .6 .8 1.0 1.5
EQUIVALENCE RATIO
Fig. 7. The depe nden ce of stand-o ff distance on equivalence ratio. The solid curves
are the theore t ica l func t ion g iven in Eq. (6 ) ; they a re k inky because measured Tb have
been used.
2 .0
w h e r e o n l y o n e m a x i m u m o f w i r e t e m p e r a t u r e is
o b s e r v e d , t h e r e w e r e t w o , t h e l a r g e s t o f w h i c h
o c c u r s i s t h e p r e h e a t z o n e .
T h e s t a n d - o f f d i s t a n c e as a f u n c t i o n o f e q u i v -
a l e n c e r a t i o a t c o n s t a n t f l a m e s p e e d i s g i v e n i n
F i g . 7 . T h e s t a n d - o f f d i s ta n c e a s a f u n c t i o n o f
e q u i v a l e n c e r a t i o h a s a m i n i m u m n e a r ¢ = 1 ,
b e c a u s e t h e h e a t r e l e a s e d i n t h e r e a c t i o n z o n e is a
m a x i m u m a t t h i s p o i n t . T h e s t a n d - o f f d i s t a n c e i s
d e t e r m i n e d b y a b a la n c e o f c o n d u c t i o n a n d c o n -
v e c t i o n , a n d s i n c e t h e f l a m e s p e e d i s h e l d c o n s t a n t
t h e s t a n d - o f f d i s t a n c e m u s t g e t s m a l l e r a s t h e h e a t
r e l e a s e d i n c r e a s e s .
T h e l e a n l i m i t a t t h e s e s p e e d s a p p e a r s d e t e r -
m i n e d b y t h e o n s e t o f T a y l o r i n s t a b i l it i e s i n t h e
b u r n e d g a se s w h i c h a r e c h a r a c t e r iz e d b y t h e
f o r m a t i o n o f h o l e s i n t h e f l a m e [ 1 4 ] . T h e r i ch
l i m i t f o r m e t h a n e w a s c a u se d b y f l o w - o f f , w h e r e a s
f o r e th y l e n e m e a s u r e m e n t s a t ¢ > 1 .8 c o u l d n o t b e
m a d e b e c a u s e r a d i a t i n g s o o t p a r t ic l e s m a s k e d t h e
w i r e . A n o t h e r i n s t a b i l i ty a p p e a r e d f o r t h e m e t h a n e -
a i r f l a m e . A t e q u i v a l e n c e r a t i o s o f ¢ = 1 . 2, 1 . 3, a n d
1 .4 , s u r f a c e w a v e s t r a v e l l e d a l o n g t h e f l a m e c a u s i n g
p r e s s u r e f l u c t u a t i o n s t h a t w e r e a u d i b l e . T h e d i s -
t u r b a n c e s w e r e s m a l l e n o u g h , h o w e v e r , t h a t t h e
m e a n s t a n d - o f f d i s ta n c e h a s b e e n p r e s e n t e d i n t h e
f igu r e s .
A c c o r d i n g t o t h e t h e o r y f o r a ll f la m e s w i t h a
8/18/2019 FergusonKeck CombustionFlame 34-85-1979
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S T A N D - O F F D I S T A N C E S O N A F L A T F L A M E B U R N E R 9 7
6
tic
z
.._I
fo
bJ
O_
,,3
Or)
I
I I I I I
Pe =̧
Pu Su CPu+C~)D
X u + k b
0
[ ] - -
o
[]
I I 1 I L
4 6 8 I0 15
Tb -Tu
Tb -Tb
F i g. 8 . S t a n d - o f f P e cl e t n u m b e r a s a f u n c t i o n o f t h e r a t io o f t h e h e a t o f c o m b u s t i o n
t o t h e h e a t l o s s . T h e l i n e is t h e t h e o r e t i c a l c u r v e g i v e n a s E q . ( 6) . T h e d a t a k e y i s g i v e n
i n F i g s . 6 a n d 7 .
2 0
d i m e n s i o n l e s s a c t i v a t i o n e n e r g y
E / R T ~ , °
g r e a t e r
t h a n a b o u t 7 , t h e r e s h o u l d b e a u n i v e l s a l p l o t o f
P e c l et n u m b e r v e r su s t h e r a t i o o f t h e h e a t o f c o m -
b u s t i o n t o t h e h e a t l o s s . T o t e s t t h e t h e o r y a n
e x p e r i m e n t a l P e c le t n u m b e r h a s b e e n d e f i n e d a s:
p , ,Su c . . + cpb )D
Pe = (20 )
Xu + Xb
w h e r e t h e t h e r m o c h e m i c a l a n d t r a n s p o r t d a t a
n e e d e d f o r t h e d e f i n i ti o n h a v e b e e n c o m p u t e d
u s i n g th e p r o g r a m o f S v e h l a a n d M c B r i d e [ 1 1 ] .
F i gu r e 8 s h o w s q u a n t it a t iv e a g r e e m e n t b e t w e e n
t h e t h e o r e t i c a l P e c l e t n u m b e r a n d t h e e x p e r i -
m e n t a l d a t a . I n d e e d . t h e g r o s s b e h a v i o r o f t h e
f l am es s tud ied i s desc r ibed by a s ing le cu rve wh ich
i s th e l o g a r i th m o f th e r a t i o o f h e a t o f c o m b u s t i o n
t o t h e h e a t l o s s . T h e m e a s u r e m e n t s a r e n o t a c c u -
r a te e n o u g h t o s t u d y t h e s m a l l d e v i a ti o n s t h a t m a y
b e d u e t o t h e f i n it e h e a t r e le a s e z o n e . I t i s c l e a r
t h a t t h e h e a t f l u x a t t h e w a l l n o r m a l i z e d b y t h e
t e m p e r a t u r e r is e , th e s t a n d - o f f d i s t a n c e , a n d th e
t h e r m a l c o n d u c t i v i t y o f t h e b u r n e d g a se s i s n o t a
c o n s t a n t a s a s s u m e d p r e v i o u s l y [ 1 ] .
F o r v a l u e s o f
T b ° - - T u ) / T b ° - T b )
g r e a t e r
t h a n a b o u t 1 0 t h e r e a r e l a r g e d e v i a t i o n s f r o m t h e
p l o t c a u s e d b y t h e s m a ll n es s o f t h e q u a n t i t y
T b ° - T b . T h e s e e r r o r s a r e j u s t l a r g e e n o u g h t h a t
t h e y c a n n o t b e d i sm i s se d a s o n l y e x p e r i m e n t a l
u n c e r t a i n t y . R a t h e r i t a p p e a r s t h a t h y d r o c a r b o n
f l a m e s d o n o t b u r n t o t h e r m o d y n a m i c eq u i li b -
r i u m . F o r e x a m p l e , a n y c a r b o n m o n o x i d e i n t h e
p r o d u c t s t h a t is n e g le c t ed i n c o m p u t i n g t h e
a d i a b a t i c f l a m e t e m p e r a t u r e s c a u s e s T b ° , h e n c e
T b ° - T b , t o b e o v e r p r e d i c t e d .
F i n a l l y , t h e c o r r e l a t i o n w o r k s f o r t h e h y d r o g e n -
a i r f l a m e d e s p i t e t h e f a c t t h a t t h e s p e c i fi c h e a t a n d
d i f fu s i v it y o f h y d r o g e n a r e a n o r d e r o f m a g n i t u d e
l a r g er t h a n t h o s e o f t h e o t h e r s t a b l e s p e ci e s i n t h e
f l a m e . T h e t h e o r y i s a d e q u a t e b e c a u s e t h e r e i s s o
m u c h n i t r o g e n i n t h e f l a m e t h a t v e r y l it t le o f t h e
s e n s i b l e e n t h a l p y i s t i e d u p i n t h e h y d r o g e n . T h e
8/18/2019 FergusonKeck CombustionFlame 34-85-1979
14/14
9 8 C O L I N R . F E R G U S O N a n d J A M E S C . K E C K
a s s u m p t i o n t h a t a l l s p e c i e s i n t h e f l a m e h av e e q u a l
s p e c i f ic h e a t s a n d d i f f u s i v i ti e s i s n o t e x p e c t e d t o
b e v a l i d i n a h y d r o g e n - o x y g e n f l a m e .
I V . C O N C L U S I O N S
1 . T h e P e c l e t n u m b e r b a s e d o n s t a n d - o f f
d i s t a n ce d e p e n d s o n l y o n t h e r a t i o o f t h e h e a t
c o m b u s t i o n t o t h e h e a t l o s s p r o v i d e d t h a t ( 1 ) t h e
d i m e n s i o n l e s s a c t i v a t i o n e n e r g y /~ =
E/RTb
is
g r e a t e r t h a n a b o u t 7 a n d ( 2 ) t h a t w h e n h y d r o g e n
i s a r e a c t a n t i t c a r r ie s l it t l e o f t h e s e n s i b le e n t h a l p y .
2 . T h e r e i s a m i n i m u m s t a n d - o f f d i s t a n c e t h a t ,
b e c a u s e s o l i t t l e h e a t i s r e q u i r e d t o q u e n c h a f l a m e ,
i s a g o o d a p p r o x i m a t i o n t o t h e q u e n c h d i s t a n c e
f o r a n u n s t e a d y f l o w .
3 . A n e x p r e s s i o n h a s b e e n d e r i v e d f o r t h e f u e l
p e r u n i t a r e a i n a q u e n c h l a y e r . I t n e e d s t o b e
c h e c k e d e x p e r i m e n t a l l y b e c a u s e th e d i f f u s io n o f
p r o d u c t s i n t o a fl a t f l am e b u r n e r h a s n e v e r b e e n
o b s e r v e d .
The authors are grateful to the General Motors
Corporation fo r fund ing this work.
R E F E R E N C E S
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Burner, Ph.D. Thesis, Dept. Mech. Eng., M.I.T.,
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2. Fergu son, C. R. , and Keck, J. C. ,
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(L.
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7056 (1973).
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Structure,
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Received 1 7 February 1978; revised 16 May 19 78