Kaplan1994 - Absorption Equation Pag. 8

8/19/2019 Kaplan1994 - Absorption Equation Pag. 8

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C O M B U S T I O N A N D F L A M E

96: 1 -2 1 (1994) 1

ynam ics of a Strongly R adiat ing U nsteady Ethylene Jet

i f fus ion Flam e

C A R O L Y N R . K A P L A N

Chemistry Division, Naval Research Laboratory, Washington, D.C . 20375

S E U N G W . B A E K

Department o f Aerospace Engineering, Korea Advanced Institute of S cience and Technology, 373-1 Gusung-dong,

Yusung-ku, Ta ejon, Korea

E L A I N E S . O R A N

Laboratory fo r C omputational Ph ysics and Fluid Dynamics, Naval Research Laboratory, Washington, D.C . 20375

J A N E T L . E L L Z E Y

Department o f Mechanical Engineering, Universityof T exas, Austin, TX 78712

T im e - d e p e n d e n t n u m e r i c a l s im u la t io n s o f a n a x i s ym m e t r i c e t h y l e n e - a i r d i f fu s io n fl a m e a r e u s e d t o q u a n t i fy

the way in which rad ia t ion t ranspor t a f fec ts the deve lopment , s t ruc ture , and dynamics of the f lame. The

n u m e r i c a l m o d e l s o lv e s t h e t im e - d e p e n d e n t N a v i e r - S to k e s e q u a t i o n s c o u p l e d t o s u b m o d e l s f o r c h e m ic a l

reac t ion and hea t re lease ( e thy lene combust ion) , soo t fo rmat ion , and rad ia t ion t ranspor t . The soo t fo rmat ion

model inc ludes a lgor i thms for soo t nuc lea t ion , sur face growth , coagula t ion , thermophores is , and ox ida t ion .

The rad ia t ive hea t f lux is found by so lv ing the rad ia t ive t ransfer equa t io n us ing the D iscre te Ord ina tes

Me th o d a n d i n c lu d es r a d i a t i ve e f f e c t s f r o m s o o t , C O 2 a n d H 20 . T h e m o d e l i s t e s t ed b y c o m p a r in g

s imula t ion resu l ts w i th prev ious ly publ ished exp er imenta l da ta fo r a cof lowing laminar e th y le ne-a i r f lame.

S im u la t io n s o f a h ig h e r - s p e e d j e t a t 5 m / s s h o w th a t r a d i a t i v e h e a t l o s s e s re d u c e t h e f l a m e t e m p e r a tu r e ,

w h ic h d e c r e a s e s t h e c h e m ic a l h e a t r e l e a s e r a t e . T h e r e d u c t i o n i n h e a t r e l e a s e r a t e d e c r e a s e s t h e v o lu m e t r i c

expans ion , caus ing the f lame to shr ink cons iderab ly , and hence chang es the overa l l tempera tu re , spec ies

concentra t ion , and so o t vo lu me f rac t ion d is t r ibu t ions in the f lame. The co mpu ta t ions show tha t rad ia t ive

in tens i ty i s a t tenu a ted s ign i fican t ly wi th in the heav i ly soo t ing reg ion . Ra dia t ive hea t f lux vec tors a re p r im ar iy

d i rec ted in the rad ia l d i rec t ion ; howe ver , there i s a s ign if ican t ax ia l compo nent tha t fo l lows the curv a ture of

t h e s o o t i n g r e g io n . T h e c o m p u ta t i o n s f o r a n u n d i l u t e d f u e l j e t s h o w th a t h e a t t r a n s f er b y ra d i a t i o n d o m in a t e s

t ransfer by conduct ion and con vec t ion in the heav i ly soo t ing reg ions of the f lame.

N O M E N C L A T U R E

a

A

Cv

e

E

L

G

h k

a bso r p t ion c oe f f i c i e n t

s u r f a c e a r e a o f c o n t r o l v o l u m e

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

spe c i f i c i n t e r na l e ne r gy de ns i ty

f lu id e ne r gy de ns i ty

s o o t v o l u m e f r a c t io n

g r a v i t a t i ona l a c c e l e r a t i on

e n t h a l p y o f s p e c i e s k

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

* Corresp onding au thor : Dr . Caro lyn K aplan , C ode 6183 ,

Naval Research Labora tory , Washing ton , D .C . 20375

Copyr igh t © 1994 by The Combust ion Ins t i tu te

Publ ished by Elsev ie r Sc ience Inc .

I d i rec t ion a l in tens i ty

i m

i n t e ns i ty i n d i sc r e t e o r d ina t e s

I b b l a c k bod y in t e ns i ty

k B o l t z m a n n c o n s t a n t

k c t he r m a l c onduc t iv i t y

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

n k n u m b e r d e n s i ty o f s p e c i e s k

n d s o o t n u m b e r d e n s i t y

N o A v o g a d r o s n u m b e r

P p r e s su r e

Po2

pa r t i a l p r e s su r e o f oxyge n

Q e n e r g y r e l e a s e d f r o m c h e m i c a l

r e a c t i o n

qc the r m a l c ond uc t ive he a t f l ux

q r

r a d i a t i ve he a t f l ux

0 0 10 - 218 0 /94 /$ 6 . 0 0


2/21

2 C . R . K A P L A N E T A L .

r radia l d i rec t io n

R un ive r sa l ga s c ons t a n t

Rox r a t e o f soo t ox ida t ion

s s c a t t e r ing c oe f f i c i e n t

S op t i c a l pa th

t t im e

T t e m p e r a t u r e

w m

G a u s s i a n q u a d r a t u r e w e i g h t

A V v o l u m e o f c o n t r o l v o l u m e

V f lu id ve loc i ty

Uk

d i f f us ion ve loc i ty o f spe c i e s k

~ t t he r m oph or e t i c ve loc i ty

X k m o le f r a c t ion o f spe c i e s k

z axia l d i rec t ion

Greek Symbols

a in t e ns i ty f l ow t e r m in c u r ve d c o -

o r d i n a t e

/3 e x t inc t ion ( a bso r p t ion + sc a t t e r -

ing) coef f ic ient

I I d i r e c t ion o f r a d i a t i on

oJ~ p r o d u c t i o n / l o s s o f s p e c i e s k d u e

to c he m ic a l r e a c t ion

~ o n ~ p r o d u c t i o n / l o s s o f s o o t n u m b e r

de ns i ty

toL p r o d u c t i o n / l o s s o f s o o t v o l u m e

f r a c t ion

/~, ~:, r / radial , axial, an d azim uth al di-

r e c t ion c os ine s

~" in t e r p o la t i o n f a c to r u se d in

D O M

f lu id d e ns i ty

de ns i ty o f soo t pa r t i c l e

v i sc ous s t r e s s t e nso r

sc a t t e r ing pha se f unc t ion

a z im utha l a ng le

w a v e l e n g t h

spe c t r a l e m is s iv i ty

e m i s si v it y a t b o u n d a r y

k ine m a t i c v i sc os i ty

o p a c i t y o f l a y e r o f t h ic k n e s s S

P

Psoot

7"

¢ ( f V - -, f ~ )

¢

A

~

ew

P

K~(s)

I N T R O D U C T I O N

N u m e r i c a l s i m u l a t io n o f u n s t e a d y d i f f u si o n

f l a m e s i s a c ha l l e nge due to t he d i f f ic u l ty o f

r e so lv ing the ve r y d i spa r a t e t im e a n d spa c e

sc a l e s o f t he c o n t r o l l i ng phys i c a l a nd c he m ic a l

p r oc e s se s , a nd in su f f i c i e n t knowle dge o f t he

inpu t da t a . T he phys i c a l a nd c he m ic a l p r o -

c e s se s c a n c ove r t im e sc a l e s r a ng ing ove r n ine

o r d e r s o f m a g n i t u d e a n d s p a c e s c al e s ra n g i ng

ove r f i ve o r de r s o f m a gn i tude . I t i s no t p r a c t i -

c a l t o de ve lop a d i r e c t num e r i c a l s im u la t ion

tha t c a n r e so lve t he f u l l r a nge o f r e l e va n t t im e

a nd spa c e sc a l e s a pp l i c a b l e f o r uns t e a dy j e t

d i f f u s ion f la m e s . Ho we ve r , by t a k ing a dva n ta g e

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

i s t ry o f d i f fu s ion f l a m e s , one c a n c ho ose a pp r o -

p r i a t e op t im ize d a lgo r i t hm s wi th a da p t ive o r

va r i a b l e g r idd ing t e c hn ique s t o de ve lop a nu -

m e r i c a l m ode l t ha t i s c om pu ta t iona l ly f e a s ib l e .

T h e r e h a v e b e e n a n u m b e r o f n u m e r i c a l

s tud i e s o f s t e a dy- s t a t e l a m ina r d i f f u s ion f la m e s .

In som e s t e a dy- s t a t e c a se s , t he f l a m e in t e r f a c e

i s c ons t a n t i n spa c e a nd t im e a nd the f ue l a nd

ox id i ze r m ix th r ough d i f f u s ion o f t he r e a c t a n t s

in to t he f l a m e zone . M os t f l a m e s a r e , howe ve r ,

uns t e a dy o r f l uc tua t ing a nd the m ix ing p r oc e s s

i s m or e c om ple x . F i r s t , buoya nc y-d r ive n low-

f r e q u e n c y ( 1 0 - 2 0 H z ) s t r u c t u r e s [ 1 - 5 ] f o r m

ou t s ide t he f l a m e zone a nd r e su l t i n f l i c ke r ing .

A l so , whe n the j e t ve loc i ty i s h igh e n ough ,

the r e a r e sm a l l e r , h igh - f r e que nc y s t r uc tu r e s

(2 00 Hz ) a t t he i n t e r f a c e be twe e n the h igh -

ve loc i ty a nd low-ve loc i ty f l u id t ha t r e su l t f r om

K e l v i n - H e l m h o l t z i n s t a b i l i t i e s [ 2 - 5 ] . D u e t o

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

ox id i ze r m ix a s the y a r e e n t r a ine d by the

l a r ge - sc a l e s t r uc tu r e s a nd the n a r e c onve c t e d

in to t he h igh - t e m pe r a tu r e r e g ion . D i f f us ive

p r oc e s se s t he n m ix the r e a c t a n t s a t t he m o le c -

u l a r s c a l e whe r e c he m ic a l r e a c t ions c a n oc c u r .

R e c e n t l y t i m e - d e p e n d e n t a x i s y m m e t r i c n u -

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

[6 -8 ] a nd p r opa ne -a i r [ 9 ] d i f f u s ion f l a m e s ha ve

be e n r e po r t e d . S tud ie s [ 7 , 8 ] o f t he e f f e c t s o f

he a t r e l e a se , v i s c os i ty , a nd g r a v i ty on the dy -

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

he a t r e l e a se a nd v i sc os ity da m p the h igh -

f r e q u e n c y K e l v i n - H e l m h o l t z i n s ta b i li ti e s w h i l e

g r a v i ty ( buoya nc y ) i s r e spons ib l e f o r t he f o r m a -

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

( f l a m e f l i c ke r ) . C om bine d num e r i c a l a nd e x -

pe r im e n ta l i nve s t iga t ions [ 3 , 9 ] o f p r o pa ne -a i r

j e t d i f f u s ion f l a m e s a l so de m ons t r a t e d the im -

p o r t a n c e o f b u o y a n c y i n t h e f o r m a t i o n o f t h e

low - f r e que nc y osc i l la t i ons .

I n hydr oc a r bon f l a m e s , soo t u sua l ly f o r m s


3/21

S T R O N G L Y R A D I A T I N G E T H Y L E N E D I F F U S IO N F L A M E 3

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

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

p r o p e r t i e s o f w e a k l y r a d ia t i n g ( n o n l u m i n o u s )

d i f f u s ion f l a m e s c a n be c a l c u l a t e d w i th r e a son -

a b le a c c u r a c y [1 0 -1 2 ] . T he l a m ina r f l a m e le t

c onc e p t [ 1 3 , 1 4 ] f o r non lum inous f l a m e s i s

b a s e d o n t h e o b s e r v a t i o n t h a t s c a la r p r o p e r t i e s

in la m ina r d i f f u s ion f la m e s a r e n e a r ly un ive r sa l

f unc t ions ( s t a t e r e l a t i onsh ips ) o f m ix tu r e f r a c -

t io n . H e n c e , k n o w l e d g e o f t h e t r a n s i e n t b e h a v -

io r o f t he m ix tu r e f r a c t ion i s su f f i c i e n t t o ob -

t a in e s t im a te s o f al l t r a ns i e n t s c a l a r (ga s spe c i e s

c o n c e n t r a t io n s , t e m p e r a t u r e , d e n s i t y ) p r o p e r -

t i e s . T h i s c o n c e p t h a s b e e n e x t e n d e d t o n o n l u -

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

t h e m a s w r i n k l e d l a m i n a r f l a m e s h a v i n g

the s a m e p r op e r t i e s [ 11 , 1 5 ] . T he r e f o r e , d i r e c t

m e a s u r e m e n t s o f sc a la r p r o p e r t i e s i n n o n l u m i -

nous l a m ina r f l a m e s c a n p r ov ide the ne c e s sa r y

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

f lames.

P r e d i c t i o n o f t h e p r o p e r t i e s o f s t ro n g l y r a di -

a t i ng lum inous f l a m e s r e m a ins a g r e a t e r c ha l -

l e nge . Fo r t he se s t r ong ly r a d i a t i ng f l a m e s ,

r a d i a t ive he a t t r a ns f e r i s dom ina te d by c on t in -

uum r a d i a t i on f r om s oo t pa r t i c l e s [ 1 1 ]. A n im -

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

t i on p r ope r t i e s o f ve r t i c a l l y -up f lowing l a m ina r

a n d t u r b u l e n t e t h y l e n e - a i r d i f fu s i o n fl a m e s [1 6]

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

a nd f luc tua t ing ve loc i t i e s , m e a n c onc e n t r a t i ons

o f m a jo r ga s spe c i e s , soo t vo lum e f r a c t ion ,

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

a nd to t a l r a d i a t i ve he a t f l ux d i s t r i bu t ions . T he

r e su l t s showe d tha t t he m a jo r ga s spe c i e s f o l -

l owe d ne a r ly un ive r sa l s ta t e r e l a t i onsh ips f o r

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

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

a un ive r sa l s t a t e r e l a t i onsh ip due to t he e f f e c t s

o f hyd r odyna m ic s . P r e d i c t i ons o f f l a m e s t r uc -

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

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

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

H o w e v e r , s ig n i fi ca n t d i f f e r e n c e s b e t w e e n m e a n

p r o p e r t y a n d s t o c h a s ti c r a d i a t i o n e m i s s io n p r e -

d i c t i ons ( u s ing a na r r ow-ba nd m ode l ) i nd i c a t e d

s t ro n g t u r b u l e n c e - r a d i a t i o n i n t er a c ti o n s . T h e

in t e r na l r e d i s t r i bu t ion o f e ne r gy by r a d i a t i on

i s a pp r e c i a b l e i n t he se l um inous f l a m e s [1 6 ] .

O th e r e xpe r im e n ta l i nve s t iga t ions [1 7 , 1 8] ha ve

s tud ie d the e f f e c t s o f f l ow r a t e , f ue l t ype , a nd

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

e t h a n e , a n d m e t h a n e f l a m e s . M o r e r e c e n t e x -

pe r im e n ta l a nd the o r e t i c a l s t ud i e s [ 1 9 -2 1 ] ha ve

c o r r e l a t e d loc a l soo t f o r m a t ion r a t e s w i th m ix -

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

r e g ions o f l a m ina r e thy l e ne a nd e th a ne d i f f u -

s ion f l a m e s . I n a dd i t i on , de t a i l e d m e a su r e -

m e n t s o f m ix tu r e f r ac t ion , t e m pe r a tu r e , a n d

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

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

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

e t h y l e n e - a i r [ 22 ] a n d m e t h a n e - a i r [ 2 3 ] f la m e s .

O the r i nve s t iga t ions [2 4 -2 6 ] ha ve s t r e s se d

the n e c e s s i t y o f inc lud ing a c c u r a t e r a d i a t i on

m ode l s i n l um inous f l a m e s . A f ou r - f lux m ode l

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

d i s t ri b u t io n s o f t e m p e r a t u r e a n d r a d i a n t h e a t

t r a ns f e r i n a fu r na c e [2 4 ] , show e d tha t t h e

m ode l a c c u r a t e ly p r e d i c t e d the se qua n t i t i e s a t

f u r na c e wa l l s , bu t p r ov ide d poor p r e d i c t i ons

wi th in t he f l u id f l ow . A n o th e r s tudy [2 5] e va lu -

a t e d two m ode l s f o r p r e d i c t i ngs f l a m e r a d i a -

t i on in t u r bu le n t wa l l f i r e s : t he f i r s t m ode l

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

f r a c t ion o f t he e ne r gy l i be r a t e d pe r un i t t im e

by c he m ic a l r e a c t ion , wh i l e t he s e c ond m ode l

a s sum e d tha t r a d i a t i on i s e m i t t e d by a t h in ,

c o n s t a n t - t e m p e r a t u r e ( 1 40 0 K ) la y e r o f s o o t

pa r t i c l e s a t t he f l a m e f r on t . C om pa r i son w i th

e x p e r i m e n t a l d a t a f o r P M M A s h o w e d th a t t h e

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

to r o f py r o lys i s r a t e a nd f l a m e r a d i a nc e . A

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

f u l ly c oup l ing the r a d i a t i on c a l c u l a t i on to t he

c o n s e r v a t io n e q u a t i o n o f m e a n t o t a l e n th a l p y

[2 6 ], i nc lud ing the r a d i a t i ve h e a t l o s s /g a in

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

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

s ion t e m pe r a tu r e s a nd r a d i a t i on in t e ns it i e s fo r

lum inous f l a m e s .

T h i s a r t i c l e de sc r ibe s num e r i c a l s im u la t ions

o f t he nons t e a dy be ha v io r o f a s t r ong ly r a d i a t -

i ng a x i sym m e t r i c j e t d i f f u s ion f l a m e f o r m e d

b e t w e e n u n d i l u t e d e t h y l e n e a n d a c o f l o w i n g

s t r e a m o f air . T h e n u m e r i c al m o d e l i s b a s e d o n

one o r ig ina l ly de ve lope d f o r hyd r oge n j e t d i f -

f u s ion f l a m e s [6 -8 ] , bu t now inc lude s a c he m i -

c a l r e a c t ion a nd e ne r gy r e l e a se m ode l f o r e thy -

l e n e o x i d a t i o n a n d m o d e l s f o r s o o t f o r m a t i o n


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4 C . R . K A P L A N E T A L .

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

c a r r i e d ou t t o s t udy t he e f fe c t s o f r a d i a t i on

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

d y n a m i c s o f t h e f l am e .

N U M E R I C A L M E T H O D A N D

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

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

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

s i t y , mome n t um, e ne rgy , i nd i v i dua l spe c i e s

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

soo t vo l ume f ra c t i on :

Op

- - + V . p V ) = O , 1 )

at

ap V

cTt

- - + V . p W ) = - V P + p G - V . r ,

(2 )

a E

- - + V . E V ) = - V . P V - V . q ~ + q ~ )

Ot

- V ~,nkF;kh k + Q,

(3)

a n k

at

q - V ( n k V ) = - - V ( n k U k ) q- C a)k, (4)

a n d

Ot

-- + t7. (n d V) = --17. ('~tnd) + tO~d

(5)

aL

- - + V . f ~ V ) = - V . ~ , f ~ ) + to f,.. 6 )

a t

E q u a t i o n s 1 - 6 a r e c l o s e d b y t h e i d e a l g a s

re la t ions :

P = nkT,

(7)

de = p C v dT.

(8)

E q u a t i o n s 1 - 4 i n c l u d e t e r m s f o r c o n v e c t i o n ,

t he rma l c onduc t i on , mo l e c u l a r d i f fus i on , v i s -

c os i t y , c he mi c a l r e a c t i on a nd e ne rgy re l e a se ,

g ra v i t y , a nd ra d i a t i on t r a nspor t . T he soo t c on-

se rva t i on e qua t i ons , E qs . 5 a nd 6 , i nc l ude t e rms

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

t h e r m o p h o r e t i c v e l o c it y is d e f i n e d b y

0 l n T

vt = - 0.54v (9)

Or

Ou r so l u t i on t o E qs. 1 - 6 i nc l ude s bo t h ra d i a l

a nd a x i a l c ompone n t s o f t he c onve c t i ve a nd

d i f fusi ve t r a nspo r t ( t he rm a l c onduc t i on , mo l e c -

u l a r d i f fus i on , v i s c os i t y ) t e rms . H ow e ve r , w e

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

t h e r m o p h o r e t i c te r m i n E q s. 5 a n d 6 , a n d o n l y

t he a xi al c o m po ne n t o f the g ra v i t a t i ona l a c c e l-

e ra t i on t e rm i n E q . 2 .

T h e s e e q u a t i o n s a r e t h e n r e w r i t t e n i n t e r m s

of f i n i t e - vo l ume a pprox i ma t i ons on a n E u l e -

r i a n m e sh a nd so l ve d num e r i c a l l y fo r spe c i f ie d

bounda ry a nd i n i t i a l c ond i t i ons . A c ompl e t e

so l u t i on t o the se gove rn i ng e qua t i ons r e qu i re s

so l v i ng t he t e rms fo r e a c h o f t he i nd i v i dua l

p roc e s se s , a s w e l l a s a c c oun t i ng fo r t he i n t e r -

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

s i s t s o f s e pa ra t e a l go r i t hms fo r e a c h o f t he

i nd i v i dua l p roc e s se s , w h i c h a re t he n c oup l e d

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

[ 2 7] . T he a l go r i t hms fo r c onve c t i on , t he rma l

c onduc t i on , mo l e c u l a r d i f fus i on , v i s c os i t y a nd

t he c oup l i ng o f t he i nd i v i dua l p roc e s se s ha ve

be e n p re v i ous l y d i s cus se d i n d e t a i l [ 7 ] , a nd

t h e r e f o r e a r e o n l y b ri e fl y d e sc r i b e d h e r e . T h e

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

c a l r e a c t i on , soo t fo rma t i on , a nd ra d i a t i on

t r a n s p o r t a l g o ri th m s , a r e m o r e t h o r o u g h l y d e -

sc r ibed in th i s a r t ic le . In addi t ion, a de ta i led

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

ma l r a d i a t i on a l go r i t hm fo r t h i s p rob l e m i s

p r e s e n t e d i n a n A p p e n d i x .

C o n v e c t i o n

T he f l u id c onve c t i on i s so l ve d w i t h a h i g h - o rd e r

i mpl i c i t a l go r i t hm, Ba re l y I mpl i c i t Cor re c t i on

t o F l u x - C o r r e c t e d T r a n s p o r t ( B I C - F C T ) , t h a t

w a s de ve l ope d t o so l ve t he c onve c t i on e qua -

t ions for low-ve loc i ty f lows [28]. Th e Flux-C or-

re c t e d T ra n spor t ( FC T ) a l go r i t hm i t s e lf [ 2 7 ] i s

an expl ic it , f in i te -vo lum e a lgo r i thm th a t i s con-

s t r u c te d t o h a v e f o u r t h - o r d e r p h a s e a c c u ra c y .

T h r o u g h a t w o - st e p p r e d i c t o r - c o r r e c t o r al g o -

r i t hm, FC T e nsu re s t ha t a l l c onse rve d qua n t i -

t i e s r e m a i n m o n o t o n e a n d p o s i t i v e . H o w e v e r ,

be c a use FCT i s a n e xp l i c i t a l go r i t hm, t he nu -

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

s t a b i l i t y i s l i m i t e d by t he ve l oc i t y o f sound

a c c o r d i n g t o t h e C o u r a n t - F r i e d r ic h s - L e w y c o n -

d i t i on . T o f i l t e r ou t t he sound w a ve s f rom t he

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

s o u n d - s p e e d ( C o u r a n t ) li m i t at i o n o n t h e t i m e -

s t e p , t he c onve c t i on e qua t i ons a re u sua l l y


5/21

S T R O N G L Y R A D I A T I N G E T H Y L E N E D I F F U SI O N F L A M E 5

so lve d im p l i c i t l y . P a tna ik e t a l . [ 2 8 ] de ve lope d

B I C - F C T s o t h a t t h e t i m e s t e p i s l i m i te d b y t h e

f lu id ve loc i ty a nd no t t he sound spe e d . T h i s

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

p u t a t i o n s o f s l o w ly e v o lv i n g fl o w s b e c a u s e o n e

B I C - F C T t i m e s t e p r e q u i r e s t h e s a m e a m o u n t

o f c o m p u t e r t i m e a s o n e r e g u l a r F C T e x p l ic i t

t i m e s t e p , b u t t h e s i z e o f t h e t i m e s t e p m i g h t b e

a f a c t o r o f 5 0 - 1 0 0 t i m e s g r e a t e r.

D i f f u s i v e P r o c e s s e s

T h e e f f e ct s o f th e r m a l c o n d u c t i o n , m o l e c u l a r

d i f f u s ion , a nd v i sc os i ty a r e e va lua t e d us ing

two-d im e ns iona l f i n i t e -d i f f e r e nc ing a lgo r i t hm s

[ 7] . T e m p e r a t u r e - d e p e n d e n t t h e r m a l c o n d u c -

t i v it i e s a nd v i sc os i ty c oe f f i c i e n t s a r e c a l c u l a t e d

f r om k ine t i c t he o r y [2 9 ] ove r a 300-3000 K

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

a t h i r d -o r de r po lynom ia l f o r e a c h ind iv idua l

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

a pp l i e d to c a l c u l a t e m ix tu r e t he r m a l c onduc t iv -

i t ie s [30] and mixture v iscos i ty coef f ic ients [31]

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

de n t b ina r y d i f f u s ion c oe f f i c i e n t s a r e c a l c u -

l a t e d f r om k ine t i c t he o r y [2 9 ] . D i f f us ion c o -

e f f ic i e n t s o f e a c h ind iv idua l c om po ne n t i n a

m ix tu r e a r e t he n c a l c u l a t e d f o r e a c h c e l l [ 32 ] .

Subc yc l ing i s u se d in t he m o le c u la r d i f f u s ion

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

mer ica l s tabi l i ty [7] .

C h e m i c a l R e a c t i o n a n d E n e r g y R e l e a se

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

s e n t e d b y t h e s o u r c e t e r m i n E q . 4 . D u e t o t h e

l a rg e n u m b e r o f c o m p u t a t i o n a l c e ll s r e q u i r e d

t o r e s o l v e t h e c o m p l e x f l o w s t r u c t u r e o f j e t

d i f f u s ion f l a m e s , i t wou ld ha ve be e n p r o -

h ib i t i ve t o i nc lude the f u l l s e t o f e l e m e n ta r y

r e a c t ions f o r e thy l e ne ox ida t ion in t h i s m ode l .

I n s t e a d , w e d e s c r i b e t h e c h e m i c a l r e a c t i o n

a n d e n e r g y - r e l e a s e p r o c e s s p h e n o m e n o l o g i -

c a l ly ba se d on the s ing l e s t e p r e a c t ion ,

C2 H 4 -k 3 0 2 + ( N 2 ) ~ 2 C O 2

+ 2 H 2 0 + ( N e ) ,

(lO)

us ing a f i n i t e - r a t e , qua s i -g loba l A r r he n ius e x

press ion [33] ,

d [ C 2 H 4 ]

d t

4 . 3 X 1 0 1 2 e x p ( - 3 0 0 0 0 / R T )

x [C 2H 4 ]0.1 0 2 ]1.65

( m o l / c m 3 - s ) . ( 1 1 )

T he r a t e o f de p le t i on o f e thy l e ne i s c a l c u l a t e d

f r om E q . 1 1 . T he n , b a se d o n the r a t e o f f ue l

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

t i ons o f oxyge n , c a r bon d iox ide , wa te r , a nd

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

s to i c h iom e t r i c c oe f f i c i e n t s i n E q . 1 0 . T he he a t

r e l e a se r a t e , Q , i s de t e r m ine d f r om

d [ C 2 H 4 ]

Q - - - A H c d t (12)

A s d i sc usse d be low in t he Re su l t s s e c t ion , s im -

u l a t io n s w e r e c o n d u c t e d w i t h a n d w i t h o u t r a d i-

a t i on t r a nspo r t t o s tudy i t s e f f e c t s on the f l a m e

dyna m ic s a nd s t r uc tu r e . Fo r s im u la t ions c on -

d u c t e d

w i t h o u t

r a d i a t i on , i t wa s ne c e s sa r y t o

inc lude a c a l i b r a t i on f a c to r o f 0 . 9 i n t he A r r he -

n ius t ype r e a c t ion r a t e , E q . 1 1 , t o p r e ve n t i t

f r om inc r e a s ing w i thou t l im i t s a s t he t e m pe r a -

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

wi th r a d i a t i on , t h i s c a l i b r a t i on f a c to r wa s no t

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

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

inc r e a s ing w i thou t bonds .

S o o t F o r m a t i o n

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

de ns i ty a nd soo t vo lu m e f r a c t ion , E qs . 5 a nd 6 ,

i nc lude t e r m s f o r c onve c t ion , t he r m ophor e s i s ,

a nd sou r c e t e r m s , o )n~ a nd wL " T he s e s ou r c e

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

d i f f e r e n t i a l e qua t ions de r ive d by M oss e t a l .

[ 22 ] b a s e d o n e x p e r i m e n t a l m e a s u r e m e n t s o f

m ix tu r e f r a c t ion , t e m pe r a tu r e , a nd soo t vo l -

um e f r a c t ion in e thy le ne - a i r d i f f u s ion f l a m e s .

T h i s m o d e l i n c l u d e s t e r m s f o r s o o t n u c l e a ti o n ,

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

f o r m a t ion r a t e :

d n d

c lt = N ° C ~ p 2 T a / 2 X f u e l e - ~ ° / T

- C ~ T 1 / 2 n d 2 / N o , (1 3 )


6/21

6 C . R . K A P L A N E T A L .

- - = - - n a p T t /Z g f u e l e - Ty / r

d t #soot

C~ C a

+ - - p2 T 1/2 Xfu el e -

T . / T ,

(14)

#soot

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

b e 1 .8 g / c m 3 , a n d t h e c o e f f i c i e n t s a n d a c t iv a -

t i o n t e m p e r a t u r e s [ 2 2 ] a r e

C a = 1 .7 × 1 08 c m 3 / ( g 2 K 1 / 2 s ) ,

Ct3 = 1 x 1 01 5 c m 3 / ( K l / e s ) ,

C r = 4 . 2 x 1 0 -1 1 c m 3 / ( K l / Z s ) ,

C~ = 144 x 103 g,

T,~ = 46.1 × 103 K,

Tr = 12.6 × 103 K .

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

t h e o x i d a t io n m e c h a n i s m o f N a g l e a n d S t ri ck -

l a nd - C ons t a b l e [ 34 ] ,

R ox

= 12 1 + k z P % X + k B P o 2 ( 1 - - X )

( g / c m 3 - s ) , ( 1 5 )

w h e r e

X = (16)

1 + ( k r / k B ) P o 2

a n d w h e r e t h e c o e f f i c i e n t s a r e d e f i n e d [ 3 4 ] a s :

k A = 20 exp ( - 3 0 0 0 0 / R T ) ;

k n = 4 .4 6 × 1 0 - 3 e x p ( - 1 5 2 0 0 / R T ) ;

k r

= 1.51 ×

1 0 5 e x p ( - 9 7 0 0 0 / R T ) ;

k z = 21 .3 e x p ( 4 1 O O / R T )

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

u n i t s o f g / c m 2 - s i n t o a p p r o p r i a t e u n i t s o f so o t

n u m b e r d e n s i t y ( n u m b e r o f p a rt i cl e s o x i d i z e d /

c m a o f g a s -s ) a n d s o o t v o l u m e f r a c t io n ( c m 3

s o o t o x i d i z e d / c m 3 o f g a s- s) , a s s u m i n g s p h e r i-

c a l p a r t i c l e s w i t h a n a v e r a g e p a r t i c l e d i a m e t e r

o f 1 × 1 0 - 6 m . T h e s e o x i d a t i o n r a t e v a l u e s a r e

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

E q s . 1 3 a n d 1 4, s o t h a t t h e s o u r c e t e r m f o r

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

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

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

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

a t i o n , a n d o x i d a t i o n . T h e s o u r c e t e r m s a r e

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

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

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

c a l c u la t e n e w v a l u e s f o r s o o t n u m b e r d e n s i ty

a n d v o l u m e f r a c ti o n . T h e s e n e w v a l u e s a r e

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

s i s t e r m s i n t h e c o n s e r v a t i o n e q u a t i o n s , E q s . 5

a nd 6 .

adiation Transport

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

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

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

t i o n o r th e t r a n s p a r e n t g a s a p p r o x i m a t i o n h a v e

b e e n d e v e lo p e d . F o r s o m e n o n l u m i n o u s w e a k ly

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

m a t e r a d i a t i o n t r a n s p o r t w i t h a f i r s t - o r d e r o p t i -

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

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

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

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

m a t i o n . T h e M o n t e - C a r l o [ 3 5 ] a n d Z o n e [ 3 6]

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

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

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

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

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

w e m o d e l r a d i a t i o n t r a n s p o r t u s i n g t h e D i s -

c r e t e O r d i n a t e s M e t h o d ( D O M ) , f ir st p r o p o s e d

b y C h a n d r a s e k h a r [ 3 7 ], w h i c h w e b e l i e v e is a s

a c c u r a t e a s t h e M o n t e - C a r l o m e t h o d , b u t r e -

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

m e m o r y . D O M is a g e n e r a l a l g o r i th m t h a t c a n

d e s c r ib e r a d i a t i o n t r a n s p o r t i n m e d i a w h i c h

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

n o t n e c e s s a r y t o m a k e a p r e l i m i n a r y e s t i m a t e

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

s t u d i e d s i n c e t h e D O M a l g o r i t h m i s v a l id f o r

a n y l e ve l o f o p a c it y . A n o t h e r a d v a n t a g e o f t h e

D O M a l g o r i th m i s t h e e a s e w i t h w h i ch i t c a n

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

f i n it e -v o l u m e c o d es . M o s t r e c e n tl y , D O M h a s

b e e n u s e d t o s t u d y t h e e f f e c t s o f r a d i a t io n

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

l i z a t i on ove r a s o l i d f ue l p l a t e [ 38 , 39 ] .


7/21


8/21

8 C . R . K A P L A N E T A L .

TABLE 1

Gaussian Quadrat ure for the S 4 Approximation for

Axisymmetric Geomet ry

Radial Axial Azimuthal

Direction Component Component Component

(m ) /~,. ~:,. ~,.

1 - 0.2959 - 0.9082 0.2959

2 0.2959 - 0.9082 0.2959

3 - 0.9082 - 0.2959 0.2959

4 - 0.2959 - 0.2959 0.9082

5 0.2959 - 0.2959 0.9082

6 0.9082 - 0.2959 0.2959

7 - 0.9082 0.2959 0.2959

8 - 0.2959 0.2959 0.9082

9 0.2959 0.2959 0.9082

10 0.9082 0.2959 0.2959

11 - 0.2959 0.9082 0.2959

12 0.2959 0.9082 0.2959

so r p t ion c oe f f i c i e n t f o r a m ix tu r e o f

CO 2

a n d

H 2 0 f r o m t h e e x p e r i m e n t a l a n d th e o r e t i c a l

w o r k o f M a g n u s s e n a n d H j e r t a g e r [ 4 1]

a c oa + r ~a o = 0 . 001 (X c o= + X r ~o)

c m - l ) ,

(2])

w h e r e X c o r r e s p o n d s to m o l e f ra c t io n . B a s e d

o n t h e w o r k o f M a g n u s s e n a n d H j e r t a g e r [ 41 ],

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

s o o t a n d a m i x t u r e o f C O 2 a n d H a O t o o b t a i n

a n ove r a l l a bso r p t ion c oe f f i c i e n t , we c a l c u l a t e

the ove r a l l a bso r p t ion c oe f f i c i e n t f o r e a c h c e l l

a s a sum o f t he i nd iv idua l a bso r p t ion c oe f f i-

c i e n ts f o r s o o t a n d t h e m i x t u r e o f C O ~ a n d

H 2 0 .

B E N C H M A R K C A L C U LA T IO N S

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

r i t hm s f o r c he m ic a l r e a c t ion a nd e ne r gy r e -

l e a se , s o o t fo r m a t i o n a n d r a d i a t i o n t r a n s p o r t

u se d in t h i s m ode l , we s im u la t e d a l a m ina r

c o f io w i n g e t h y l e n e - a i r d i f fu s i o n f l am e a n d

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

G o r e a n d F a e t h [ 1 6] . T h e e x p e r i m e n t a l a p p a -

r a tu s o f G or e a nd Fa e th [1 6 ] c ons i s t e d o f e thy -

l e ne f ue l f lowing upw a r d th r ou gh a c e n t r a l

t u b e o f 1 . 4 3 c m d i a m e t e r w i t h R e y n o l d s n u m -

be r s i n t he r a nge o f 4 5 -6 3 , wh i l e a ir f l owe d

f r o m a c o n c e n t r i c o u t e r t u b e o f 1 0.2 c m d i a m e -

t e r . T he c o nd i t i ons o f t he c om pu ta t io ns d i s -

c usse d be low a r e v e r y s im i l a r.

T h e c o m p u t a t i o n a l d o m a i n a n d i n i ti a l co n d i -

t i ons a r e show n in F ig . 1. T he e n t i r e c o m p u ta -

t i ona l do m a in c ove r s a r e g ion o f 1 0 c m × 1 5

c m f o r a g r id c on ta in ing 6 4 × 88 c e l ls a nd

t i m e s t e p s a r e o n t h e o r d e r o f 1 0 × 10 -6 S.

U n d i l u t e d e t h y l e n e f u e l f lo w s a t 5 c m / s

th r ough a 1 . 4 - c m -d ia m e te r bu r ne r , r e su l t i ng in

a c o ld f low Re y no ld s num be r o f 45 . A c o f low-

ing s t r e a m o f a i r fl ows a t t he s a m e ve loc i ty

th r ough a 1 0 -c m -d ia m e te r ou t e r r i ng . T he

le f t - ha nd bounda r y i s a n a x i s o f sym m e t r y ,

wh i l e t he r i gh t -ha nd bounda r y i s a f r e e - s l i p

w a l l . T h e b o t t o m b o u n d a r y r e p r e s e n t s a n i n -

f lo w b o u n d a r y c o n d i ti o n , w h e r e t h e d e n s it y ,

ve loc i ty , a nd c o nc e n t r a t i on o f the i nc om ing

spe c i e s a r e spe c i f i e d . T he in f low bounda r y c o r -

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

15 cm

i l ll l1 1 1 1 1 1 1 1 |n o | | u | l l | | o | | a i m m

. . ~ . . . m o o | J m n l m | l . m m m . w l

. ~ . . . . m H a n f m H J g u N m , l a

. m m m m n n w J | | o | | l H a n i

. , . . u m ~ a J l J n m l | u n m m ~ m

, , . . u . m , l l m m n n m w u m a m l

, . . u m u m a m u | H H n l a

, , m m l m w o 0 n o a m m w w i | i m H

. , , , . j . . m l , j m n H m w m m u w m ~

. ~ m m m m a u o n o m m ~ i w n ~ H

, . . . m ~ m 0 0 n m n m m n n m ~ , n

, , . . . . . . . m n n ~ n w ~ w n

. . r m m m n n ~ N w a n m w ~ w n

. . . u m m H w ~ o a o ~ m ~ m W m ~ m a

' , ; ', l l ', ' , ', ' g g p l ' l I g l l ~ ~ I I I g l

liBFIB [ 1111101Hi i e [ [Ol l l l l I l l

I~ l l l l l l l l [ l [ l l l l l l l l l l l l l l l ] l l

I I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l

I I I l l l l l l l l [ l l l [ l l l l l l l l l l l l l l l [

I~ OII l l l l l l l l l l l l l l l l l l l l l l l l l [

I I H I I I I I I I I I I I I I I I I I I I I I I I I I I I I

Ilil I 100lilt i De 11 [[liil l In

II I l l l l l l l l l l l l l l l l l l l l l l l l l l l l

, l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

H I I I I I I I I I I I I I I I I I [ I I I I I I I I I I I

4 1 1 l l [ l l l l l l [ l l l l l l l l [ l l l l l l / l [

I il[ Dl0glllDl [ I 0 i ] [OI i i l [ I 1 1

. . . i , . m n n m m u o l m H m m W l

. . . , . . . . m m l ~ a m . . ~ n H ~ w n

, . . m . . n u l0 l N m u W l | |~ n a

i , . , . . . m l H l l | l n m n u n a

l . , Im m m H l l n 0 H ~ U l W | . l

I r ' , ' , l ' , l l l l l g l l g : l : [ | I I | ; ~ | - ' | | I

I I I I I I I I l ' ~ I g ~ | ' l I I : ' I g l ' , I ' , ' . I ~

li i} l I IIIUi l i II I lollin g i l[

FDIIIIII I I I III III III I I IIlll l I[I

I I , l l , l [ l l l l [ l l l l l l l [ [ [ l l l l l l l l l

Iq l~ l l l l lOl l i l l l l l 0 l l i l l l l l l l l l [

I I I I I I I l l l l l l l l l l l l l l l [ l l l l l l l l [

, l l l l l l l l l l l p l l l l l l l l l l l l l l l l l [ I

IIII lU Illll l Ill [ I IIII I I II 111

II[ l l ] l l l i l l l l l 0 i l l l i IOlOl l [ [ l l l

I I ] l l l l l l [ l l l l l l l l l l l l l i l l l l l l l [

I I I I I I I I I I I l l l l l l l l l l l l l l l l l l l l l

I H I I I I [ I I I I I I I I I I I [ I I [ I I I I I I ] I [

I?1~1111111111 ll III II II II I III II

~ , ' , l l I I ' , I II l ~ ' I I I ~ I | I I I | - g - I g l

I I 1 11FIII l l l I [ I [ l l [ l l l l l l l l l

i i i i I I l l l l l l l l l l l l l l l l l [ l l l l l l l l

I ] l l l l l l l l l l l [ l l [ l l l l l l l l l l l l l l l

1 4l rl l 0 0 0 1 1 1 i ig l l l [ [0 l l l l l l l l l [

Illlll 1111111 11 [[I III I [ l[ II II

' , ', ' ,' , l ; [ g g g I g . , I l l . ' ' | - I 1

FIH n i l I [ [ l l l l l l l I I [ [ [ l [ l l l I [ [

i,iid FIll I [ I II III I I I[ [ I Il l II I1 [

' , ' , ' , l l l l g l l g l : ' ; : : | | l p | | | g ~ - ' l

, , , , , . , , ,, , , , . ~ H o ~ . . . ~ ~ n m n ,

tUt t l a i l a lu l ee i | i i i eo m e~ lo ~ im ~ l

' , , , ' , , , l l ~ g l ' . ' . I g | l : l | g l | l t | | - ' - - ' I

Outflow

II

.m

y.

3 1 c m T T

5 c m / s 5 c m / s

C2H4 Air

Fig. 1. Computational domain and initial conditions for

benchmark simulation of low-velocity laminar ethylene

diffusion flame. Note that this figure only shows a region

of high resolution. The full computational domain covers a

radial distance of 10 cm.


9/21

S T R O N G L Y R A D I A T I N G E T H Y L E N E D I F F U S IO N F L A M E 9

nozz le e x i t ; t ha t i s , t he c om pu ta t iona l dom a in

doe s no t i nc lude the nozz l e i t s e l f . T he top

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

c o n d i t i o n . B o u n d a r y c o n d i t i o n s f o r t h e r a d i a -

t i on t r a nspo r t m ode l i nc lude d i ff u se ly e m i t -

t i ng - r e f l e c t i ng su r f a c e c ond i t i ons a t t he i n f low ,

o u t f lo w , a n d r i g h t - h an d b o u n d a r i e s ,

1 - E w

i, , = ewibw ~ [~m,lWm,im, ,

77" G,, < 0

~m > 0. (22 )

A t t he a x i s o f sym m e t r y , t he spe c u la r ly r e f l e c t -

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

ira,, / ,Zm, = --/Jt~m, ~m, = ~m , rim, ~- nm .

F i g u r e 2 s h o w s i n s ta n t a n e o u s c o n t o u r s o f

m ix tu r e f r a c t ion , oxyge n m ole f r a c t ion , t e m -

p e r a t u r e , C O 2 m o l e f r a ct i o n , H z O m o l e f r a c -

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

4 0000 . T he se r e su l t s a r e no t i n s t e a dy s t a t e ,

e ve n f o r suc h a l ow-ve loc i ty f l a m e ; how e ve r ,

t h i s t im e s t e p i s we l l pa s t a ny c r i t ic a l s t a ge s o f

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

so l id c i r c l e s t ha t a r e supe r im pose d on the c on -

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

f l a m e su r f a c e . M ix tu r e f r a c t ion i s de f ine d a s

the m a ss f r a c t ion o f f ue l e l e m e n t s i n the m ix -

tu r e a t a ny po in t i n t he f l ow . F igu r e 2 shows

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

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

t h e o u t e r r e g i o n o f t h e f la m e . A t im e s e q u e n c e

o f oxyge n m ole f r a c t ion c on tou r s [ 4 2 ] shows

tha t t he f l i c ke r f r e que nc y f o r t h i s f l a m e i s

a p p r o x i m a t e l y 1 6 H z . T h e m a x i m u m f l am e

te m pe r a tu r e , 2 000 K , i s l oc a t e d in t he r e g ion

w h e r e t h e C O 2 a n d H 2 0 m o l e f r a c ti o n s a r e

m a x i m u m . T h e m a x i m u m s o o t v o l u m e fr a c t io n

is 8

× 1 0 - 6 ,

a nd i s w i th in t he h igh t e m p e r a -

tu r e r e g ion in a n a r e a whe r e t he m ix tu r e i s

s l ight ly r ich of s to ichiometr ic .

R a d i a t i o n q u a n t i t i e s a t t h i s s a m e t i m e s t e p

a r e s h o w n i n F i g . 3 . T h e m a x i m u m a b s o r p t i o n

c oe f f i c i e n t i s a p p r ox im a te ly 0 . 4 c m -1 . T h e se

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

p r im a r i ly f r om the soo t ing r e g ion , so t ha t t he

s o o t , a n d n o t t h e

C O 2 o r

H 2 0 , i s t h e d o m i -

n a n t a b s o r b i n g - e m i tt i n g m e d i u m . T h e r a d ia t i v e

he a t f l ux ve c to r s show tha t t he r a d i a t i on t r a ns -

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

g ion , a nd f o l lows the c u r va tu r e o f the so o t ing

r e g ion . T he l e ng th o f t he ve c to r i s d i r e c t ly

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

he a t f l ux ; he nc e , t he s t r onge s t r a d i a t i ve f l ux i s

e m a n a t i n g f r o m t h e s o o t i n g r e gi o n .

1 4 c m

M i x t u re O z M o l e T e m p e r a t u r e

Fraction Fraction (K)

c m

C O 2 M o l e H 2 0 M o l e S o o t V o l u m e

F r a c t i o n F r a c t i o n F r a c t i o n

X 1 0 -7

,+

i

; L _ _ t _ _

F i g . 2 . I n s t a n t a n e o u s c o n t o u r s a t t i m e s t e p 4 0 0 0 0 f o r b e n c h m a r k s i m u l a t i o n . T h e l o c a t i o n o f t h e s t o i c h i o m e t r i c f l a m e

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


10/21

10 C . R . K A P L A N E T A L .

14 cm

Absorption

Coefficient

(cm-l)

I I

e

i

i

i

r

i

0

3 c m

Radiative

Intensi ty

( kW / m2)

I I

i

Magni tude

of Radiative

Hea t F lux

(kW/m 2)

Radiative H eat

Flux Vectors

I

~ , i t l

t i l t l l l l l l

, / / l l t l l l / I f

I t l l l l l l t / t l l

I l l l l l l l l t

~ t l l l l l l l l l /

I l l l l l l l i l l / .

/ / i / i l / . ~

I T

t t

Fig. 3. Instantaneous contours of radiation qu antities at timestep 40000 for b enchmark simulation. The location of

the stoichiometric lame surface is represen ted by the dash ed line with solid circles.

T h e s i m u l a t io n s w e r e a l so a n a l y z e d t o d e t e r -

m i n e i f a s t a t e r e l a t i o n s h i p e x i s ts b e t w e e n t h e

m a j o r g a s s p e c i e s a n d f u e l - e q u i v a l e n c e r a t i o

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

e x p e r i m e n t a l d a t a o f G o r e a n d F a e t h [ 1 6] .

S c a t t e r p lo t s w e r e p r e p a r e d i n w h i c h t h e m a s s

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

a s a f u n c t i o n o f lo c a l f u e l - e q u i v a l e n c e r a t i o a t

t h r e e a x i a l l o c a t i o n s i n t h e f l a m e : 5 , 8 , a n d 1 2

c m . F i g u r e 4 s h o w s t h e s t a t e r e l a t i o n s h i p s t h a t

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

t h e e x p e r i m e n t s o f G o r e a n d F a e t h [ 1 6] . I n

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

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

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

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

s t o i c h i o m e t r i c r e g i o n , ~b ~ 1 , b o t h t h e f u e l a n d

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

s i m u l a t i o n d a t a a n d t h e e x p e r i m e n t a l [ 1 6 ] d a t a .

I n t h e l e a n r e g i o n , ( t h < 1), t h e s i m u l a t i o n

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

m a t c h t h o s e o f th e e x p e r i m e n t s [ 1 6 ] ; i n t h e

f ue l r i c h r e g i on s , ma i n l y f o r ~b > 2 , t he s i mu l a -

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

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

a n d e x p e r i m e n t [ 1 6 ] i s q u i t e g o o d f o r t h e o x y -

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

h i g h e r d e g r e e o f s c a tt e r i n t h e C O 2 a n d H 2 0

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

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

s u r e m e n t s [ 1 6 ]. T h e m a x i m u m m a s s f r a c t io n s

o f C O 2 a n d H 2 0 a r e l o c a te d in th e r e g i o n o f

s t o i c h i om e t r y , a t ~b ~ 1 , a nd de c r e a s e i n t he

l e a n a n d r i c h r e g i o n s . I n t h e r i c h r e g i o n w h e r e

~b > 2 , t h e H 2 0 m a s s f r a c t i o n i s sl i g h tl y u n d e r -

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

F i g u r e 5 , w h i c h s h o w s t h e s t a t e r e l a t i o n s h i p

d a t a f o r s o o t v o l u m e f r a c t io n , s h o w s m o r e s c at -

t e r f o r b o t h t h e e x p e r i m e n t a l [ 1 6 ] d a t a a n d

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


11/21

S T R O N G L Y R A D I A T I N G E T H Y L E N E D I F F U S I O N F L A M E 11

3 10 '

1 l O

¢ .

O

3 I 0

~Q

O

o

. . . .

, 2 q e ~ ' L . e , m - ~

1

l ~

m.

O

eq

O

° = G o r e a n d F a e t h d a t a

A = S i m u l a t i o n

O

- . ° °

O o

o.

i l l l i

3 1 0

1 0 1 0 I

F u e l E q u i v a l e n c e R a t i o

O

O

I z l

O

glq

e

O

3 . 1 0 t

o

I i I S i I I l [ i I I I I I

t o * t o

Fig. 4. Mass fraction of C2H 4, 02, CO 2, and H 20 versus local fuel equivalence ratio shows state relati onships for

major gas species for benchmark simulation. Gore and Faeth data are extracted from Ref. 16.

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

i s tr y a n d h y d r o d y n a m i c s [ 1 6 ]. T h e s c a t t e r i n

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

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

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

t r a v e r se s . F i g u r e 5 s h o w s t h a t m o s t o f th e s o o t

i s f o r m e d i n t h e r e g i o n s l ig h t l y r i c h o f s t o i c h i o -

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

v o l u m e f r a c t io n is a r o u n d ( 8 - 1 0 ) × 1 0 - 6 ,

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

p e r i m e n t a l d a t a [ 1 6 ] .

U N S T E A D Y H I G H - V E L O C I T Y

E T H Y L E N E F L A M E

W e n o w p r o c e e d t o u s e t h e n u m e r i c al m o d e l

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

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

f o r m e d b e t w e e n a h i g h -v e l o ci ty (5 m / s ) f u e l

j e t f l o w i n g i n t o a 3 0 - c m / s a i r s tr e a m . T h i s j e t

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

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

l i ft o f f p h e n o m e n a , w h e r e t h e j e t v e l o c it ie s

r a n g e f r o m 5 to 5 0 m / s . S i m u l a t i o n s w e r e

c o n d u c t e d f o r t w o f u e l j e t m i x t u r e s: u n d i l u t e d

e t h y l e n e a n d a n i t r o g e n - d i l u t e d e t h y l e n e m i x -

t u r e ( C 2 H a : N 2 / 3 : l ) . T h e R e y n o l d s n u m b e r

( b a s e d o n c o l d f lo w c o n d i t i o n s a t t h e f u e l j e t )

f o r t h i s s y s t e m i s a p p r o x i m a t e l y 5 0 0 0 .

Co m p u ta t i o n a l G r i d a n d B o u n d a r y Co n d i t i o n s

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

g r i d w a s r e q u i r e d f o r t h i s h i g h - v e l o c i ty u n -

s t e a d y j e t d i f f u s i o n f l a m e i n o r d e r t o r e s o l v e

t h e i n s t a b il i ti e s i n t h e s h e a r l a y e r o f t h e j e t.

T h e r e g i o n n e a r t h e j e t a n d t h e i n i t i a l c o n d i -

t i o n s a r e s h o w n i n F i g . 6 . T h e f u l l d o m a i n i s

1 6 7 × 1 72 c m a nd i s mode l e d on a 1 2 8 × 2 2 4

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

0 .0 2 c m × 0 .0 2 c m a r e c o n c e n t r a t e d a r o u n d

t he j e t e x i t . T he g r i d i s t he n s t r e t c he d [ 7 , 8 ]


12/21

12 C . R . K A P L A N E T A L .

= G o r e a n d F a e th da t a

• S i m u l a t i o n

34 .1 cm

l l ~ l l r $ 1 f l l l l l l l l l l l l

r : [ l$1111111fl l l l l i l

i U i J J I J i i l l f l i l i l l l l l

l ; r l l [ l l l r l l l l l l l l l l l l l l

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f I i l r l l l l l l l l l [ l l l l l l l l

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i h l l P l l l l l l l l l l l l l l l l l

~ J ~ l l l J l l l l l l l l l l l l l l l l

i I I i I l l l l r l l l l l l l l l l l l l l

i J i i J i i i i i i i i i i i i i i i n

: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I l l l l l J I I I I I I l l l l l l l l

P I I H r l r l l l l l l l ll l l l l l l

i i l l l l l l l l l l l l l l l l l l l l

i l q l l e l t l l l U l l l l l l l l l

i l i l l l l l i l l P l i l i ~ l l l U l

i , ; i i i i i i f f f i i i i $ $ $ $ i

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. . . .. . . .. . , , , , , , , , , , , ,

. . .. . .. . , , , , , , , , , , , , , , ,

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l : l l ~ : l : ~ l l l l l l l l l l l

? ? i : , ? i l } i ] ] ] ii } ] ]~ ] ]

: : : : : : : : : : : : : : : : : : : : :

: = : : : : : : : : : : : : : : : :

O u t f l o w

}

a ~ "

~ o

0

0

10 ~ 10 '

F u e l E q u i v a l e n c e R a t i o

F ig . 5 . S t a t e r e l a t i o n s h i p f o r s o o t v o l u m e f r a c t i o n f o r

b e n c h m a r k s i m u l a t io n . G o r e a n d F a e t h d a t a a n d t h e s o li d

l ine ( f i t t o the da ta in Ref . 1 6 ) a r e ex t r ac t ed f rom Ref . 1 6 .

A A

* I I I I I I I

1#

b o t h r a d i a l l y a n d a x i al ly . T h e f u e l f l o w s t h r o u g h

a j e t o f ra d i u s 0 . 5 c m a t 5 m / s , w h i l e a i r f lo w s

t h r o u g h t h e o u t e r a n n u l a r r e g i o n a t 3 0 c m / s .

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

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

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

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

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

v e l o c it y at t h e o u t f l o w b o u n d a r y i s a d j u s t e d t o

a l l o w t h e f l o w t o r e l a x t h e a m b i e n t a t m o -

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

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

h e i g h t o f th i s f l a m e i s a p p r o x i m a t e l y 1 m , t h e

g r i d i s s t r e t c he d t o 1 .7 m i n t he a x i a l d i r e c t i on

t o p r e v e n t d i s t u r b a n c e s c r e a t e d a t t h e z e r o

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

f l a m e u p s t r e a m . A l t h o u g h m o r e e l e g a n t o u t -

f lo w b o u n d a r y c o n d i t i o n s [ 43 ] h a v e b e e n d e v e l-

o p e d f o r h i g h - v e l o c i ty j e t s u s i n g a n e x p l ic i t

F C T a l g o r i t h m [ 27 ], w e h a v e n o t y e t d e v e l o p e d

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

( B I C - F C T ) [ 2 8] a l g o r i t h m .

F l a m e S t r u c t u r e a n d R a d i a t i v e P r o p e r t i e s

F i g u r e 7 sh o w s th e i n s t a n t a n e o u s c o n t o u r s o f

f u e l m o l e f r a c t i o n , te m p e r a t u r e , s o o t v o l u m e

0 .0

0 .0

E

@

0.5

J

4.5

cm

?

/ A i r , 3 0 c m / s

F u e l M i x t u r e

5 m/ s

Fig. 6. Com putationaldom ain and initial conditions for a

5 m/s C2H4-N2 jet flowing nto a 30 cm/s air stream.

Note that the figure on ly shows the p art of the computa-

tional dom ain with high resolution. The full computational

dom ain covers a region of 167 × 172 cm.

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

r a d i a ti v e i n t e n si ty , a n d m a g n i t u d e o f t h e r a d i a-

t iv e h e a t f lu x a n d r a d i a t i v e f l u x v e c t o r s a t

t i m e s t e p 4 0 ,0 0 0 , is w e ll p a s t a n y t r a n s i e n t s t a g e s

o f th e f l a m e d e v e l o p m e n t . T h e d a s h e d l in e

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

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

f l a m e s u r fa c 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 he f l a me , 2 050 K , i s l oc a t e d a t a n a x i a l d i s -

t a n c e o f 4 .5 c m f r o m t h e b a s e o f t h e f l a m e .

F i g u r e 7 a l so s h o w s t h e b u o y a n c y - d r i v e n l o w -

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

t h e o u t e r r e g i o n o f t h is t r a n s i t io n a l f l a m e .

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

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

s l ig h t l y r i c h o f s t o i c h i o m e t r i c . S o o t v o l u m e

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

m a x i m u m v a l u e o f 9 ×

1 0 - 6

a t a he i gh t o f 1 1

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

a x i al p o s i t io n s . T h e r e g i o n i n t h e f l a m e w h e r e

V . q r a t t a i n s a m a x i m u m p o s i t iv e v a l u e l ie s i n

t h e s a m e r e g i o n a s t h e s o o t i n g z o n e , i n d i c a t i n g

a g a in , t h a t s o o t , a n d n o t t h e C O 2 o r H 2 0 , i s

t h e d o m i n a n t a b s o r b i n g - e m i tt i n g m e d i u m . A s


13/21


s h o w n i n t h e 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 ,

E q . 3 , a pos i t i ve va lue o f

V'qr

resul ts in a

de c r e a se i n t o t a l e ne r gy de ns i ty . He nc e , t he

r e g io n o f m a x im u m V . q r c o r r e sp o n d s t o t h e

r e g i o n w h e r e t h e e n e r g y l o s s e s d u e t o r a d i a -

t i on a r e g r e a t e s t .

F igu r e 7 show s tha t t h e r a d i a t i ve i n t e ns i ty i s

g r e a t e s t i n t he soo t ing r e g ion whe r e i t r e a c he s

a m a x i m u m v a l u e o f 2 × 1 0 z k W / m 2 an d t h e n

de c r e a se s w i th r a d i a l d i s t a nc e . T he se c on tou r s

show the s t r ong a t t e nu a t ion o f r a d i a t i ve i n te n -

s i t y i n t he he a v i ly soo t ing r e g ion . T he opa c i ty

o f a ga s i s a m e a su r e o f t he a b i l i t y o f a g ive n

p a t h l e n g t h o f g a s t o a t t e n u a t e r a d i a t io n o f a

g ive n wa ve le ng th a nd i s de f ine d [35 ] a s

iA(S)

= i a ( O ) e x p [ - K a ( S ) ] ,

( 2 3 )

wh e r e Ka (S ) i s t he opa c i ty o f a l a ye r o f t h i c k -

ne s s S . S im u la t ion r e su l t s show tha t t h e m a x i -

m um th i c kne ss o f t he soo t ing l a ye r i s a pp r ox -

im a te ly 0 . 5 c m . A f t e r pa s s ing th r ough th i s

soo t ing l a ye r , t he r a d i a t i ve i n t e ns i ty i s a t t e nu -

a t e d to a pp r ox im a te ly 50% o f i t s o r ig ina l va lue ,

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

0 . 7 . He nc e , i n t he he a v i ly soo t ing r e g ion , t he

m e d ium i s ne i the r op t i c a l l y t h in n o r t h i ck , bu t

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

t h in a n d t h i ck . T h e m e d i u m o u t s i d e o f th e

soo t ing r e g ion i s op t ic a l l y t h in .

F igu r e 7 a l so shows the r a d i a t i ve f l ux ve c to r s

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

2

rad iat iv e he at flux, de fin ed as (qr. rad~a~ +

q 2",1/2

r , ax~at S . Th e rad iativ e h ea t f lux ve cto rs poin t

p r e dom ina n t ly i n t he r a d i a l d i r e c t ion , no r m a l

to t he su r f a c e o f t he soo t ing r e g ion . T he l e ng th

o f e a c h o f t he r a d i a t i ve f lux ve c to r s i s p r opor -

t i ona l t o i t s m a gn i tude ; t he l a r ge s t f lux ve c to r s

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

c onc e n t r a t i on . A l thoug h the r a d i a t i ve fl ux ve c -

to r s po in t p r e dom ina n t ly i n t he r a d i a l d i r e c -

t ion , they do have a s igni f icant axia l compo-

n e n t n e a r t h e b o t t o m t ip o f t h e s o o t i n g r eg i o n .

F igu r e 8 shows the de c r e a se i n r a d i a t i ve f l ux

with radia l d is tance f rom the f lame a t an axia l

l oc a t ion o f 1 0 c m . Wi th in t he he a v i ly soo t ing

r e g ion , the s im u la t ions show r a d i a tve he a t

f lu x e s o n t h e o r d e r o f 90 k W / m 2. H o w e v e r , as

shown in Fig . 8 , the hea t f lux decreases s igni f -

i c a n t ly w i th r a d i a l d i s t a nc e f r om the soo t ing

r e g ion . A t 1 5 c m r a d i a l d i s t a nc e , t he r a d i a t i ve

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

m a g n i tude , a nd b y 70 c m r a d i a l d i s t a nc e , i t ha s

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

1 9 c m

C all4 M o l e T e m p e r a t u r e

F r a c t i o n ( K )

S o o t V o l u m e

F r a c t i o n x 1 0 "

M a g n i t u d e o f

R a d i a t iv e R a d i a t iv e H e a t

V.q,) x 102 I n t e n s i t y F l u x x 1 0 ~ R a d i a t i v e H e a t

( k W / r n 3) ( k W / m ~) ( k W / m 2) F l ux V e c t o r s

- - 1

i

~ L * ~ I ~ , \ \ , , \ N\ \ x x x \ ' ~ x '

l : 7 ; : 7 ' 2 7 7

: I

3

r , , , _ _

0 3 c m

F i g . 7 . I n s t a n t a n e o u s c o n t o u r s o f f la m e p r o p e r t i e s f o r 5 m / s f u e l j e t c o f l o w i n g i n t o a i r . T h e l o c a t i o n o f t h e s t o i c h i o m e t r i c

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


14/21

T e m p e r a t u r e ( K )

14

J I I J

0.0 15.0 30.0 45.0 60.0 75.0

R a d i a l D i s t a n c e ( er a)

Fig. 8. Radiative heat flux (kW /m 2) versus radial distance

from sooting layer.

v a l u e o f n e a r 1 k W / m 2. E x p e r i m e n t a l m e a -

s u r e m e n t s i n e t h y l e n e - a i r d i ff u si o n f l a m e s a ls o

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

f r o m a p p r o x i m a t e l y 3 to 1 k W / m 2 a t r a d i al

d i s t a n c e s r a n g i n g f r o m 1 0 t o 6 0 c m [ 1 6 ] .

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

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

r a d i a t i v e f lu x w i t h t h e c o r r e s p o n d i n g c r o s s s ec -

t i o n a l a r e a a l o n g t h e o u t e r b o u n d a r y ( r i g h t -

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

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

f l a m e is 6 .5 k W f o r t h e u n d i l u t e d f u e l j e t , a n d

5 k W f o r t h e n i t r o g e n - d i l u t e d c a s e

( C 2 H n : N 2 / 3 : l ) . T h e s e r a d i a t i v e h e a t l o s s v a l -

u e s r ep r e s e n t a p p r o x i m a te l y 3 5 % - 4 0 % o f th e

c h e m i c a l h e a t r e l e a s e d . A s e x p e c t e d , t h e r a d i a -

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

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

o

o

° ~

1 9 c m

'-o

¢ f i

C . R . K A P L A N E T A L .

S o o t V o l u m e

F r a c t i o n x l t Y ~

r

i t

f f ec t o f R a d i a t i o n o n F l a m e P r o p e r t i e s

F i g u r e 9 s h o w s c o n to u r s o f t e m p e r a t u r e a n d

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

w a s e x c l u d e d f r o m ( C a s e A ) a n d i n c l u d e d i n

( C a s e B ) t h e s i m u l a t i o n . W h e n r a d i a t i o n t r a n s -

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

p e r a t u r e i s 2 2 00 K , a n d s o o t v o l u m e f r a c t io n s

r e a c h v a l u e s o f 25 x 1 0 - 6 ( w h e n r a d i a t i o n i s

n o t i n c l u d e d i n th e c a l c u l a t io n , t h e f l a m e t e m -

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

t o t h e A r r h e n i u s t y p e r e a c t io n r a t e ; h o w e v e r ,

b y s c a l in g t h e r e a c t i o n r a t e b y a f a c t o r o f 0. 9,

2 c m

Fig. 9. Instantaneous contours of temperature and soot

volume fra ction at timestep 40000 for simulations con-

ducted with and without radiation. The contour interval is

deliberately maintained at the same value for each contour

type (200 K for temp erature contours, 20 × 10 7 for soot

volume fraction contours) to show the effect of radiation

on flam e sheet and sooting layer thickness.


15/21

STRONGLY RADIATING ETHYLENE DIFFUSION FLAME 15

the temperature leveled off at a value of 2200

K). However, when radiation is included, the

maximum flame temperature decreases to 2050

K and the maximum soot volume fraction at-

tained is 9

×

10 -6, a decrease by a factor of

three.

Figure 9 quantifies the fact that one of the

most significant effects of radiation transport is

to shrink the flame. When radiation is in-

cluded, the radiative heat losses reduce the

flame temperature, which reduces the chemical

heat release rate, which, in turn, reduces the

volumetric expansion, thus causing the flame

to shrink. As the flame shrinks, the overall

temperature distribution in the flame changes,

which, in turn, changes the distribution of

species concentrations and soot volume frac-

tion.

Figure 10 shows radial profiles of tempera-

ture and soot volume fraction at an axial loca-

tion of 10 cm. The high-temperature region of

the flame is significantly narrower when radia-

tion is included. Likewise, the sooting region is

narrower, and the quantity of soot is consider-

ably reduced due to the lower temperatures.

When radiation is included, the sooting region

is located closer to the flame centerline, as the

overall flame width is narrowed by the radia-

tive losses.

R a d i a t iv e V e r s u s C o n d u c t i v e H e a t F l u x e s

Figure 11 shows radial profiles of the magni-

tudes of radiative heat flux qr, as defined in

the radiation transport section) and conductive

heat flux qc -- - kc AT) for cases of an undi-

luted and nitrogen-diluted C2H4:N2/3:l ) fuel

jet, at axial locations of 4 cm below the soot-

ing region) and 10 cm within the heavy soot-

ing region). The magnitude of qc is maximum

inside of the flame sheet where the radial

temperature profile is sharply increasing. Then

qc decreases approximately two orders of mag-

nitude within the flame sheet itself, where the

maximum temperature is maintained and

therefore the thermal gradient is reduced, and

then increases immediately outside of the flame

sheet where the radial temperature profile

sharply declines. Further outside the flame

sheet toward the coflow region), there is a

very small thermal gradient and q~ is again

very small.

The behavior of the radiative heat flux, qr, is

very different. The value of qr is low at the

flame centerline, then sharply increases in the

sooting region, and then gradually decreases

with distance from the sooting region toward

the coflow region. For the undiluted fuel case,

more soot is generated and the radiatve flux is

o = w i t h o u t r a d i a t i o n

A = w i t h r a d i a t i o n

t l ,

0

i

)

0.2 0.6 1.0 1.4 l .B 2~. 0.2 0.6 1.0 1.4 1.8 2.2

R a d i a l D i s t a n c e c m ) R a d i a l D i s t a n c e c m )

Fig. 10. Radial profile of temperature and soot volume fraction at 10 cm axial distance for cases with and

without radiation.


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16 C . R . K A P L A N E T A L .

gr,

• ¢ , . , I

- ¢ 1

1 0 cm

I ~ 10 cm

0 . 0 0 . 6 1 . 2 1 . 8 2 . 4 3 . 0 ~ 0 . 0 0 . 6 1 . 2 1 .8 2 . 4 3 . 0

' ~ ~ 4 c m [ = 4 c m

r o, , 8 t o :

0 . 0 0 . 6 1 2 1 . 8 2 . 4 3 . 0 ~ 0 . 0 0 . 6 1 . 2 1 . 8 2 . 4 3 . 0

R a d i a l D i s t a n c e e m ) R a d i al D i s t a n c e

era)

Fig. 11. M agnitudeof radiative and con ductiveheat fluxesat heightsof 10 cm (within h e sooting ayer)and

4 cm (outside of sooting ayer).

g re a t e r t ha n fo r t he c a se w he re t he fue l mi x -

ture i s d i lu ted . Figure 11 shows tha t in the

heavi ly soot ing region (10 cm axia l he ight ) , the

ma x i m um va l ue o f q r i s s li gh tl y h i ghe r t ha n

t h e m a x i m u m v al u e o f qc f o r t h e u n d i l u t e d

fue l c a se , w h e re l a rge qua n t i t i e s o f soo t a re

g e n e r a t e d . F o r t h e n i t r o g e n - d il u t e d ca s e, w h e r e

less soot i s genera ted , q¢ > qr . Hence , the im-

por t a nc e o f r a d i a t i on t r a nspor t i nc re a se s ( i n

c ompa r i son t o c onduc t i ve he a t t r a nspor t ) a s

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

t ude o f q~ w i t h i n t he soo t i ng re g i on ( 1 0 c m

h e i g h t ) is a p p r o x i m a t e ly a n o r d e r o f m a g n i t u d e

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

t he f l a me , as show n a t a he i gh t o f 4 c m.

S U M M A R Y A N D D I S C U S S IO N

A s o l u t io n o f th e t i m e - d e p e n d e n t N a v i e r -

S t oke s e qua t i ons , c oup l e d w i t h submode l s fo r

e t hy l e ne c ombus t i on , soo t fo rma t i on , a nd ra d i -

a t i o n t r a n s p o rt , h a s b e e n p e r f o r m e d t o e v a lu -

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

t he dyna m i c s o f s t rong l y ra d i a t i ng l umi nou s

f l a me s . T he un i que fe a t u re o f th i s mod e l is t he

c oup l i ng o f mu l t i d i me ns i ona l r a d i a t i on t r a ns -

po r t , u s i ng t he D OM a l go r i t hm, t o one fo r

mu l t i d i me ns i ona l f l u i d dyna mi c s i n a x i symm e t -

r i c ge ome t ry . One o f t he mos t s i gn i f i c a n t a d -

v a n t a g e s o f t h e D O M m o d e l is th a t i t c a n b e

use d t o so l ve ge ne ra l r a d i a t i on t r a nspor t p rob -

l e ms fo r a ny l e ve l o f opa c i t y r a ng i ng f rom

opt ica l ly th in to th ick .

I n t h i s mode l , w e a s sum e t ha t t he f l a me c a n

b e r e p r e s e n t e d b y a x i sy m m e t ri c g e o m e t r y . T h e

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

as follows:

1 . Che mi c a l r e a c t i on mode l i s r e p re se n t e d by

a s i ng l e - s t e p ra t e Ar rhe n i us t ype a l go r i t hm,


17/21


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

C 2 H 4 , 0 2 , H 2 0 , C O 2 , a n d N 2.

2 . T h e s o o t f o r m a t i o n a l g o r it h m i s r e p r e s e n t e d

b y t w o c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a -

t i ons , wh ic h inc lude e m pi r i c a l l y de r ive d c o -

e f f i c ie n t s i n the r e p r e se n ta t i on o f su r f a c e

g r owth , nuc l e a t ion a nd c oa gu la t i on . T he

p h e n o m e n o l o g i c a l s o o t o x i d a t i o n m o d e l i s

b a s e d o n l y o n t e m p e r a t u r e a n d o x y g e n p a r -

t i a l p r e s su r e .

3 . T h e r a d i a t i o n t r a n s p o r t m o d e l a s su m e s :

a . R a d i a t i o n t r a n s p o r t i s i n d e p e n d e n t o f

wa ve le ng th (g r a y -ga s a pp r ox im a t ion to

t h e R T E ) .

b . Sc a t t e r ing i s ne g l ig ib l e c om pa r e d to a b -

so r p t ion .

c . W e c ons id e r t he r a d i a t ive e f f e c t s o f soo t ,

C O 2 , a n d H 2 0 o n l y , a n d n e g l e c t t h e

r a d i a t i ve e f f e c t s o f t he C 2 H 4 fue l .

d . T h e s o o t a b s o r p t i o n c o e f f i c i en t c a n b e

r e p r e s e n t e d b y a P la n c k m e a n . A l t h o u g h

th i s i s t he a pp r op r i a t e m e a n f o r de t e r -

m in ing the ou tgo ing r a d i a t i on in t he d i -

ve r ge nc e c a l c u l a t i on , i t c a n d i f f e r f r om

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

[44 , 45] , which is required for the incom-

ing r a d i a n t i n t e ns i ty . We m a ke the a s -

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

e qua l .

e . T he ga s a bso r p t ion c oe f f i c i e n t f o r t he

c o m b i n a t i o n o f C O 2 a n d H 2 0 i s a l s o a

P l a n c k m e a n .

Re su l t s f r om a s im u la t ion o f a c o f lowing

u n d i l u t e d l a m i n ar 5 c m / s e t h y l e n e - a i r d i ff u -

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

d a t a o f G o r e a n d F a e t h [ 1 6 ] t o p r o v i d e a

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

r e l e a se , a nd soo t f o r m a t ion a lgo r i t hm s . T he se

s im u la t ion r e su l t s sho we d tha t a un ive r sa l s t a t e

r e l at i o n s h ip e x is ts b e t w e e n e a c h o f th e m a j o r

ga s spe c i e s a nd loc a l f ue l - e qu iva l e nc e r a t i o ,

i nd i c a t ing tha t t he c he m ic a l r e a c t ion a nd e n -

e r g y r e l e a se a l g o ri t h m w a s a d e q u a t e l y d e s c r ib -

ing the spe c i e s c onve r s ion p r oc e s se s . T he s t a t e

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

m o r e s c a t t e r t h a n t h a t f o r t h e m a j o r g a s

s p e ci e s. S o o t w a s p r e d o m i n a t e l y f o r m e d i n t h e

r e g ion s l i gh tly r ic h o f s to i c h iom e t r i c , a nd the

q u a n t i ty o f s o o t g e n e r a t e d w a s a p p r o x i m a t e l y

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

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

c o r r e c t un ive r sa l s t a t e r e l a t i onsh ips f o r spe c i e s

c o n c e n t r a t i o n s a n d s o o t v o l u m e f r a ct i o n , t hi s

d o e s n o t p r o v i d e e x p e r i m e n t a l v a l i d a t i o n o f

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

la t ion .

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

v e l o c it y ( 5 m / s ) u n d i l u t e d e t h y l e n e j e t d if -

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

t r a nspor t . T h e r e su l t s show e d tha t r a d i a t i on

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

a t u r e a n d m a x i m u m s o o t v o l u m e fr a c ti o n i n

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

in t e m pe r a tu r e ( due to r a d i a t i ve he a t l o s s )

c a use s a de c r e a se i n t he c he m ic a l he a t r e l e a se

r a t e , wh ic h , i n t u r n r e duc e s t he vo lum e t r i c

e xpa ns ion , c a us ing the f l a m e to sh r ink . He nc e ,

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

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

f l a m e c ha nge d due to r a d i a t i on t r a ns f e r . Ra -

d i a ti v e lo s s e s w e r e a p p r o x i m a t e l y 3 5 % - 4 0 % o f

the c he m ic a l he a t r e l e a se d .

T he soo t ing r e g ion w i th in t he f l a m e i s qu i t e

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

r e g ion ne a r t he f ue l - r i c h s ide . T he r a d i a t i ve

f lux ve c to r s we r e d i r e c t e d a t a n a ng le no r m a l

to t he su r f a c e o f t he soo t ing r e g ion . T he f l ux

ve c to r s we r e d i r e c t e d p r im a r i ly i n t he r a d i a l

d i r e c t ion ; howe ve r , i n r e g ions whe r e t he soo t -

ing r e g ion c u r ve d , t he r a d i a t ive f lux ve c to r s

had a s igni f icant axia l component .

Ra d ia t i ve i n t e ns i ty wa s g r e a t e s t w i th in t he

he a v i ly soo t ing r e g ion , a nd wa s a t t e nua t e d to

a ppr ox im a te ly 50% o f it s o rig ina l va lue a f t e r

pa s s ing th r ough the 0 . 5 - c m - th i c k soo t ing l a ye r .

T he r e f o r e , t he opa c i ty i n the so o t ing r e g ion i s

a r o u n d 0 .7 , w h i c h c o r r e s p o n d s t o t h e i n t e r m e -

d ia t e r e g im e be twe e n op t i c a l l y t h in a nd th i c k .

Fo r a n und i lu t e d f ue l j e t i n t he r e g ions

w h e r e l a rg e q u a n t i t i es o f s o o t a

Kaplan1994 - Absorption Equation Pag. 8

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Transcript of Kaplan1994 - Absorption Equation Pag. 8