FergusonKeck CombustionFlame 34-85-1979

8/18/2019 FergusonKeck CombustionFlame 34-85-1979

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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


3/14

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


4/14


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)


5/14


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 )


6/14

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


7/14

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

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g :

l

I

e~

d ~

.=


8/14


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 ] .


9/14


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


10/14

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


11/14

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


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


12/14


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


13/14


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


14/14


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

1. Ferguson, C . R . , S tand-o f f Dis tances on a F la t F lam e

Burner, Ph.D. Thesis, Dept. Mech. Eng., M.I.T.,

Cam bridge, Mass., Sep t. 1977.

2. Fergu son, C. R. , and Keck, J. C. ,

Combust. Flame

28:197 -205 (1977) .

3. Ku rkov, A. P., and Mirsky, W ., Twelfth Symposium

Int.} on Combustion, The C om bus t ion I n s t i t u t e ,

Pit tsburgh, 1969, p. 615.

4. Spalding, D. B.,

Combust. Flame

1:296-307 (1957) .

5. Kiha ra, D. H. , F ox , J . S., and Kinosh ita , C. M.,

Combust. Sci. Technol.

11 : 239 -246 (1975).

6 . Kask an, W. T. , Sixth Symposium Int.} on Combus-

tion, W illiams and W ilkins, Baltimore, 1953 p. 126.

7. Williams, F. A., Combustion Theory, Addi son -

Wesley, Reading, Mass., 1965, p. 11.

8. Hirschfelder, J. O., Curtiss, C. F., and B ird, R. B.,

Molecular Theory of Gases and Liquids, John Wiley

& Sons, Inc. , New Yo rk, 1954, p. 761.

9. Fergu son, C. R. , and K eck, J . C. , Hot-Wire Pyro m-

etry, J . AppL Phys. 49:3031-3032 (1978) .

10. I l l ingwo rth, C. R. ,

Laminar Boundary Layers

(L.

Rosenhead, ed.) , Oxford University Press, Oxford,

England, 1963, p. 163.

11. Svehla, R. A. , and McBride, B. J . , NASA TN D-

7056 (1973).

12. Fristrom, R. M., and Westenberg, A. A. ,

Flame

Structure,

McGraw-Hill , New Y ork, 1965, p. 251

and 305.

13. Spalding, D. B., Combust. Flame 1:296-307 (1957).

14. K yd d, P. I -L, and Foss, W . I . , Combust. Flame 8:

267-273 (1964).

15. Bo tha, J. P., and Spalding, D . B.,

Proc. Royal Soc.

Ser. A 225 :71-96 (1954) .

Received 1 7 February 1978; revised 16 May 19 78

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