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PREDICTION OF A TURBULENT REACTING DUCT FLOW WITH SIGNIFICANT RADIATION
F. Lockwood, D. Spalding
To cite this version:
F. Lockwood, D. Spalding. PREDICTION OF A TURBULENT REACTING DUCT FLOW WITH SIGNIFICANT RADIATION. Journal de Physique Colloques, 1971, 32 (C5), pp.C5b-27-C5b-31.
�10.1051/jphyscol:1971563�. �jpa-00214782�
PREDICTION OF A TURBULENT REACTING DUCT FLOW WITH SIGNIFICANT RADIATION F. C. Lockwood and D. B. Spalding
Mechanical Engineering Department, Imperial College, London, S.W.7, England
On p r s s e n t e une f i t h o d e pour prPdire un Gcoulement t u r b u l e n t dans un conduit 3 s p g t r i e a x i a l e ou une f l a m e d i f f u s i v e s e produit, e t on developpe un modale pour p r d d i r e larayonnementconsiddrable de chaleur.
Ce modsle e s t incorpor'e a' une p r o d d u r e disponible pour rssoudre l e t r a n s f e r t convectif e t d i f f u s i f d e moment, de c h a l e u r , e t de masse.
A method i s presented f o r p r e d i c t i n g a confined, t u r b u l e n t , axisymmetrical duct flow i n which a d i f f u s i o n flame occurs. Radiation i s important and a model i s developed f o r i t s p r e d i c t i o n which i s incorporated i n t o an e x i s t i n g s o l u t i o n procedure f o r t h e convective and d i f f u s i v e t r a n s f e r s of momentum, h e a t and matter.
1. Introduction
1.1 The problem. I n t h i s paper we present a method s u f f i c i e n t t o employ t h e "conduction approximation"
f o r p r e d i c t i n g t h e hydrodynamic and thermal behaviour of r a d i a t i o n . Usually, however, t h e flame emisiviky of an axi-symmetrical t u r b u l e n t d i f f u s i o n flame con- i s too small t o j u s t i f y t h i s step.
f i n e d i n a duct (Fig. 1). We suppose t h a t r a d i a t i o n The best-known method of solving t h e r a d i a t i v e - makes a s i g n i f i c a n t c o n t r i b u t i o n t o t h e h e a t f l u x e s t r a n s f e r problems of flames i s t h a t of Hottel and i n t h e r a d i a l d i r e c t i o n , b u t t h a t t h e a x i a l t r a n s f e r Sarofim 121. This r e q u i r e s t h e s o l u t i o n of an i n t e - of h e a t , by both r a d i a t i o n and c o n d ~ ~ c t i o n , may be g r o - d i f f e r e n t i a l equation, i n f i n i t e - d i f f e r e n c e f o n ; neglected. The v e l o c i t y i n t h e a x i a l d i r e c t i o n i s t h e computational t a s k i s a heavy one, because each supposed everywhere t o be of t h e same sign. sub-domain of t h e space i n f l u e n c e s each o t h e r sub-
domain, and t h e i n f l u e n c e c o e f f i c i e n t s a r e burdensome t o compute. We have t h e r e f o r e p r e f e r r e d t o employ a
diffusion -
formulation introduced by Schuster [3] and elab-flame
orated by Hamaker [ A ] , which focusses a t t e n t i o n~ i r 0 - - - - on t h e f l u x e s of r a d i a t i o n i n two opposed d i r e c t i o n s , and l e a d s t o d i f f e r e n t i a l equations which can be Fig. 1 Geometry and flow conditions
of t h e problem solved
There a r e some simple i n d u s t r i a l flames of t h i s kind; f o r them t h e method of p r e d i c t i o n w i l l be use- f u l t o design engineers. Most i n d u s t r i a l flames a r e however complicated by three-dimensionality and by r e c i r c u l a t i o n of gas flow; f o r t h e s e , a more elabor- a t e p r e d i c t i o n procedure i s necessary; however, t h e present treatment of r a d i a t i o n can be extended t o t h e s e cases, and methods of handling t h e convective processes e i t h e r e x i s t o r a r e under development.
1.2 The method of solution. I n t h e absence of radi- a t i o n , t h e problem can b e solved straightforwardly by a p p l i c a t i o n of t h e f i n i t e - d i f f e r e n c e procedure of Patankar and Spalding [I], which solves coupled non- l i n e a r p a r a b o l i c equations, and which has a b u i l t - i n model of t u r b u l e n t t r a n s f e r .
This method i s used i n t h e present work f o r hand- l i n g t h e equations describing t h e convective and d i f f u s i v e t r a n s f e r s of momentum, h e a t and matter;
solved f a i r l y simply.
Two developments of t h i s l a t t e r method have had t o be made. F i r s t , t h e equations have required form- u l a t i o n f o r a n axi-symmetrical flow; it emerges t h a t new terms, of non-obvious form, e n t e r t o a f f e c t appreciably t h e behaviour. Secondly, it i s necessary t o express t h e new equations i n a manner which allows t h e i r s o l u t i o n , simultaneously with t h o s e f o r veloc- i t y , stagnation enthalpy and concentration, by means of t h e Patankar-Spalding computer program.
I n s e c t i o n 2, d e t a i l s of t h e p r e d i c t i o n procedure a r e given; t h e g r e a t e s t a t t e n t i o n i s devoted t o t h e r a d i a t i o n t r a n s f e r because of i t s novelty; t h e re- maining matters a r e t r e a t e d f l e e t i n g l y because of t h e a v a i l a b i l i t y o f [I]. Section 3 of t h e paper i s de- voted t o t h e d e s c r i p t i o n and discussion of t h e r e s u l t s of p a r t i c u l a r computations; t h e s e have been provided mainly t o i l l u s t r a t e t h e new p r e d i c t i v e p o s s i b i l i t i e s , n o t f o r comparison with any p a r t i c u l a r experiments.
Such comparisons have not y e t been made.
however we have had t o extend it so a s t o account 2. General Analysis
a l s o f o r t h e important r a d i a t i v e exchanges. Of 2.1 Assumptions. The chemical k i n e t i c s a r e supposed course, i f t h e mean f r e e path of r a d i a t i o n were very f a s t enough f o r t h e r e a c t i o n t o b e p h y s i c a l l y con- small i n comparison with t h e flame width, it would b e t r o l l e d . The thermodynatnics a r e those of a simple-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971563
F.
C. LOCKWOOD ET D . B . SPALDINGc h e m i c a l l y r e a c t i n g system. Three a s s u m p t i o n s d e f i n e t h i s s y s t e m , t h e y a r e : t h a t t h e o n l y r e a c t i o n i s b e t - ween f u e l and o x i d a n t which u n i t e i n f i x e d propor- t i o n s ; t h a t t h e c o n s t a n t - p r e s s u r e s p e c i f i c h e a t s o f t h e f u e l , o x i d a n t and p r o d u c t s a r e c o n s t a n t and e q u a l ; and t h a t t h e d i f f u s i v e t r a n s p o r t p r o p e r t i e s :
rh, rfu,
Ex
andtrod
a r e a l l e q u a l a t a n y p o i n t i n t h e f l u i d . P r a n d t l ' s m i x i n g - l e n g t h h y p o t h e s i s i s employed t o de- t e r m i n e t h e e f f e c t i v e v i s c o s i t y o f t h e t u r b u l e n t f l o w , and c o n s t a n t t u r b u l e n t P r a n d t l / S c h m i d t nurrhers a r e p r e s c r i b e d .The r a d i a t i o n h e a t t r a n s f e r i s d e t e r m i n e d from a two-flux model, t h e s e f l u x e s b e i n g opposed and i n t h e r a d i a l d i r e c t i o n . The a n c u l a r d i s t r i b u t i o n of b o t h i n c i d e n t and s c a t t e r e d r a d i a t i o n i s i s o t r o p i c . The a b s o r p t i o n and s c a t t e r i n g c o e f f i c i e n t s a r e supposed p r o p o r t i o n a l t o t h e mass c o n c e n t r a t i o n o f f u e l .
2.2 D i f f e r e n t i a l e q u a t i o n s . With t h e p r e c e e d i n g a s s u m p t i o n s a b o u t t h e c h e m i c a l k i n e t i c s and thermo- dynamics, t h r e e s i m u l t a n e o u s c o n s e r v a t i o n e q u a t i o n s f o r momentum, e n t h a l p y and m i x t u r e f r a c t i o n e n a b l e t h e c o n v e c t i v e h e a t and mass t r a n s f e r and combustion t o b e determined. These e q u a t i o n s a r e :
a f a f
p x + p y ; =
C).
r ar e f f 3 r
One f u r t h e r d i f f e r e n t i a l e q u a t i o n i s r e q u i r e d t o pre- d i c t t h e r a d i a t i o n t r a n s f e r ; t h i s e q u a t i o n i s now d e r i v e d .
C o n s i d e r an a n n u l a r r i n g o f t h i c k n e s s d r - a t a d i s - t a n c e r from an a x i s o f symmetry i n an a b s o r b i n g and s c a t t e r i n g medium ( s e e Fig. 2 ) . L e t t h e r a d i a t i o n f l u x t r a v e l l i n g i n t h e d i r e c f i o n o f p o s i t i v e r b e d e s i g n a t - ed by I ; and t h a t t r a v e l l i n g i n t h e d i r e c t i o n o f n e g a t i v e r b y J . L e t t h e a b s o r p t i o n and s c a t t e r i n g c o e f f i c i e n t s o f t h e medium h a v e t h e symbols a and s , r e s p e c t i v e l y .
C o n s i d e r t h e r a d i a t i o n e n e r g y f l o w 2jfrI t r a v e r s i n g t h e t h i c k n e s s d r ef t h e a n n u l a r r i n g . A f r a c t i o n 2 r r a I d r i s a b s o r b e d ; a n d , i f t h e s c a t t e r i n g i s i s o - t r o p i c , a f r a d n 2 t r r z 1 d r i s e l i m i n a t e d b y backward
2
s c a t t e r i n g i n t h e d i r e c t i o n o f n e g a t i v e r. The e n e r g y f l o w 2 r r 1 i s augmented by a f r a c t i o n 2 8 r
$
Jdc d u e t o backward s c a t t e r i n g o f t h e J f l u x . I f t h e i n c i d e n t r a d i a t i o n i s i s o t r o p i c , i t i s f u r t h e r augmented by ad r
f r a c t i o n
-
o f t h e f l u x which e n t e r s t h e r i n g o f r a d i u s r + d r , b u t which d o e s n o t l e a v e t h e r i n g o fr a d i u s r. F i n a l l y , i t i s augmented by a f r a c t i o n d a r E d r due t o e m i s s i o n from t h e g a s e s and p a r t i c l e s w i t h i n t h e r i n g .
L
2
J f l u x w h i c h
joins I
f l u x F i g . 2. S k e t c h showing r a d i a t i o n f l u x e sSummation o f a l l o f t h e s e i n f l u e n c e s on t h e f l o w 2 r r 1 y i e l d s a d i f f e r e n t i a l e q u a t i o n f o r t h e v a r i a t i o n o f t h e f l u x I w i t h r a d i u s , namely:
The c o e f f i c i e n t a which m u l t i p l i e s E c a n b e v e r i f i e d a s t h e c o r r e c t o n e by c o n s i d e r a t i o n o f t h e c a s e o f t h e r m a l e q u i l i b r i u m where I = Z = E and d ( r I ) / d r
= J/r. The v a r i a t i o n o f t h e J f l u x c a n b e deduced from a n a l o g o u s arguments; t h e r e s u l t i s :
E q u a t i o n 4 and 5 a r e t h e f o u n d a t i o n on which o u r t h e o r y rests It i s t h e t e r m s J/r, and t h e p r e s e n c e o f r t o t h e r i g h t o f t h e d i f f e r e n t i o n s i g n , t h a t d i s t i n g u i s h them from t h o s e o f [2,3]. The two e q u a t i o n s c a n b e combined t o y i e l d a second o r d e r d i f f e r e n t i a l e q u a t i o n :
where F s t a n d s f o r t h e q u a n t i t y ( I + J ) . T h i s i s t h e e q u a t i o n d e s c r i b i n g t h e r a d i a t i o n t r a n s f e r which we s o l v e s i m u l t a n e o u s l y ~ w i t h e q u a t i o n s 1 t o 3. The r a d i a t i o n h e a t f l u x Q g I
-
J i s c a l c u l a b l e from t h e r e l a t i o n :Q = l + ( a + s ) r d r 2 r ( 7 2.3 The s o l u t i o n procedure. The computer-based s o l u t i o n p r o c e d u r e o f P a t a n k a r and S p a l d i n g i s d e s - c r i b e d i n d e t a i l i n [I]. I t s o l v e s p a r a b o l i c equa- t i o n s of t h e form:
PREDICTION OF A TURBULENT REACTING DUCT FLOW WITH SIGNIFICANT RADIATION C5b-29
where d s t a n d s f o r a dependent v a r i a b l e ;
w
f (W-vI)/(Y - Y
) i s a d i m e n s i o n l e s s stream I f u n c t i o n :f u n c t i o n which r a n g e s a r -(d$!I/dx)/qE-YI); b E -(d(W E
-Y
I ) / ~ X ) / ( V ~ - Y ! ) ; cd E ' r 2 ~ ~ ~ e f f / I . ( $ E-
$ I ) aaeff
and d i s a s o u r c e term.
d
Couette-flow formulae a r e i n c o r p o r a t e d which d e t e r - mine t h e f l u x e s of h e a t , mass and momentum n e a r walls. Marching i n t e g r a t i o n i s employed, i t e r a t i o n b e i n g e n t i r e l y avoided. The procedure p e r m i t s a s e t of simultaneous e q u a t i o n s which p o s s e s s t h e common form o f 8 t o b e solved v e r y r a p i d l y .
E q u a t i o n s 1 , 2 , 3 and 6 a l l p o s s e s s t h e form o f e q u a t i o n 8 , with $ s t a n d i n g r e s p e c t i v e l y f o r u , h , f and F; b u t convection t e r m s , r e p r e s e n t e d by t h o s e on t h e l e f t - h a n d s i d e of e q u a t i o n 8 , a r e a b s e n t from e q u a t i o n 6. This d i f f e r e n c e i s , however, e a s i l y r e s o l v e d by m u l t i p l y i n g t h e terms on t h e right-hand s i d e of e q u a t i o n 8 , when d s t a n d s f o r F, by a f a c t o r s u f f i c i e n t l y l a r g e t o make n e g l i g i b l e t h e i n f l u e n c e of t h o s e t o t h e l e f t of t h e equal sign.
The Patankar-Spalding s o l u t i o n procedure p e r m i t s l i n e a r i z a t i o n of t h e s o u r c e term o v e r t h e forward step. We have a v a i l e d o f t h i s f e a t u r e i n t h e c a s e of e q u a t i o n 6 and have expressed i t s source term, t h e second one, a s :
d =
-
a ( E u - FD)F PU ( 9 )
where t h e ~ u b s c r i p t s U and D r e f e r t o t h e upstream and downstream l o c a t i o n s between which a forward s t e p i s made. The source term of e q u a t i o n 2, d =
2 - ,
i s n o t l i n e a r i z e d and i s expressedh pur d r
from e q u a t i o n s 6 and 7 a s :
These f o r m u l a t i o n s f o r t h e s o u r c e terms e n s u r e a c o r r e c t e n t h a l p y b a l a n c e and r e s u l t i n good compu- t a t i o n a l s t a b i l i t y .
The r a d i a t i o n f l u x streaming from a w a l l i s t h e summation of t h e r e f l e c t e d i n c i d e n t f l u x p l u s t h e w a l l emission:
J W = ( 1
-
c w ) Iw+
EWEw (11)Using e q u a t i o n 7 and t h e d e f i n i t i o n s o f F and Q, t h i s r e l a t i o n can b e r e c a s t a s :
T h i s i s t h e boundary c o n d i t i o n which must b e a p p l i e d
t o e q u a t i o n 6 a t t h e d u c t wall. It i s unusual i n t h a t b o t h t h e g r a d i e n t and magnitude o f t h e depend- e n t v a r i a b l e F a r e p r e s c r i b e d . We have i n c o r p o r a t e d i t i n t o t h e f i n i t e - d i f f e r e n c e scheme 'of t h e s o l u t i o n procedure u s i n g an i m p l i c i t formulation.
3. P a r t i c u l a r Computations
3.1 Problem s p e c i f i c a t i o n . The d u c t o f Fig. 1 i s 1 m i n d i a m e t e r and i s cooled t o a uniform tempera- t u r e of 363O~. The f u e l i s o i l and t h e f i r i n g r a t e i s 3 . 1 ~ 1 0 7 W. The a i r i s a t ambient c o n d i t i o n s and i t s flow r a t e i s 20% i n e x c e s s of t h e s t o i c h i o m e t r i c requirements. The d e n s i t y of t h e gaseous m i x t u r e i s c a l c u l a t e d from t h e e q u a t i o n o f s t a t e ; i t s l a m i n a r v i s c o s i t y i s presumed t o vary a s t h e t e m p e r a t u r e r a i s e d t o t h e power one h a l f . The mixing l e n g t h v a r i e s i n accordance w i t h f o l l o w i n g s p e c i f i c a t i o n :
A D / ~ x < Y / ( D / ~ ,
R = h W 2 ; (13b) where y i s t h e d i s t a n c e from t h e w a l l and6
and/\
a r e c o n s t a n t s a s s i g n e d t h e v a l u e s 0.44 and 0.09 r e s - p e c t i v e l y . The t u r b u l e n t Prandtl/Schmidt numbers a r e e q u a l t o 0.86.
It i s n o t t h e i n t e n t i o n h e r e t o suppose formulae f o r a b s o r p t i o n and s c a t t e r i n g c o e f f i c i e n t s i n keeping with t h e most r e c e n t knowledge. An a d e q u a t e demon- s t r a t i o n of t h e c a p a b i l i t i e s of t h e flux-model can b e e f f e c t e d u s i n g v e r y simple formulae which never- t h e l e s s a r e n o t wholly u n r e a l i s t i c . I n t h i s s p i r i t , we pressume t h a t t h e a b s o r p t i o n and s c a t t e r i n g c o e f f i c i e n t s a r e p r o p o r t i o n a l t o t h e mass concentra- t i o n of f u e l , hence:
a = h m
a f u ' ( 1 4 a )
and s = & m
s f u ' (14b)
where t h e
8's
a r e t h e c o n s t a n t s of p r o p o r t i o n a l i t y . 3.2 Some r e s u l t s Fig. 3 shows t h e d i s t r i b u t i o n o f w a l l h e a t f l u x o v e r a l e n g t h o f 10 d u c t d i a m e t e r s downstream o f t h e f u e l i n j e c t i o n l o c a t i o n . Curves a r e shown f o r v a l u e s o f8
r a n g i n g from 2 t o 40 i n t h e absence o f s c a t t e r i n g . The maximum v a l u e o f t h e a b s o r p t i o n c o e f f i c i e n t space-averaged a c r o s s t h e d u c tamax
i s i n d i c a t e d f o r e a c h v a l u e o faa.
Thesevary from 0.14 t o 2.8, a range which i n c l u d e s v a l u e s t y p i c a l l y encountered i n o i l - f i r e d furnaces. The shapes o f t h e h e a t - f l u x d i s t r i b u t i o n s a r e a l s o t y p i c a l o f t h o s e measured i n f u r n a c e t r i a l s . The maximum h e a t f l u x o c c u r s a t an x/D o f about 1.7 i n each c a s e
.
The r a t i o o f t h e maximum t o averaged h e a t f l u x e s i n c r e a s e s with i n c r e a s i n g absorption.F. C . LOCKWOOD ET D . B. SPALDING
The i n c r e a s e i n t h e h e a t f l u x r e l a t i v e t o an i n c r e a s e i n
ga
becomes p r o g r e s s i v e l y l e s s s i n c e t h e absorb- t i v i t y cannot exceed u n i t y . Although n o t shown on Fig. 3, t h e c o n v e c t i v e wall h e a t t r a n s f e r i s a l s o an outcome of t h e s o l u t i o n . When&a i s 40, f o r example, it r i s e s slowly with x reaching t h e s i g n i f i c a n t value4 2
of 5.7 x 1 0 W/m a t x/D = 10.
Fig. 3. D i s t r i b u t i o n of \ $ a l l h e a t f l u x
Fig. 4 e x h i b i t s temperature p r o f i l e s a t two down- stream s t a t i o n s : x/D = 1.9 and 8.2. P r o f i l e s a r e shown a t b o t h s t a t i o n s f o r two c a s e s : without r a d i a t i o n t r a n s f e r , and with r a d i a t i o n corresponding t o
8
= 40 and no s c a t t e r i n g . T h i s m e a n t of r a d i a - t i o n lowers t h e maximum t e m p e r a t u r e s , ~ h i c h ofc o u r s e o c c u r a t t h e flame l o c a t i o n , by about 2 6 0 ~ ~ a t x/D = 1.9 and by about 4 2 0 ~ ~ a t x/D = 8.2. The maximum t e m p e r a t u r e s a r e s t i l l h i g h e r than would o c c u r i n p r a c t i c e , t h e y exceed t h e a d i a b a t i c flame t e m p e r a t u r e , b u t a more j u d i c i o u s s p e c i f i c a t i o n o f s p e c i f i c h e a t would c o r r e c t t h i s . The d e c r e a s e i n flame temperature caused by r a d i a t i o n has t h e con- sequence of d i m i n i s h i n g t h e importance of d i s s o c i a - t i o n which cannot b e p r o p e r l y accounted f o r by simple t h e o r e t i c a l models of chemical r e a c t i o n . Note t h a t t h e r a d i a t i o n removes h e a t from t h e c e n t r a l r e g i o n of t h e flow with t h e r e s u l t t h a t t h e tempera- t u r e r i s e t h e r e due t o conduction from t h e flame be- tween t h e two s t a t i o n s i s o n l y small.
Fig. 5 d i s p l a y s p r o f i l e s o f h e a t flux a t t h e s t a t i o n x/D = 1.9 f o r
8
r a n g i n g from 2 t o 40 and no s c a t t e r i n g . The p r o f i l e s have t h e expected shapes.They a r e i n c l u d e d because t h e y demonstrate t h a t l o c a l i n f o r m a t i o n a b o u t t h e h e a t f l u x can r e a d i l y b e o b t a i n e d u s i n g t h e p r e s e n t procedure. Such i n f o r m a t i o n i s of g r e a t v a l u e when t h e l o c a t i o n o f ,
r ( m )
Fig. 4: Temperature p r o f i l e s The s o o t p a r t i c l e s formed i n o i l flames a r e smll enough f o r s c a t t e r i n g t o b e n e g l i g i b l e . The p a r t i c l e s may, however, aglomerate; and c e n o s p h ~ r e s may b e p r e s e n t . Also, t h e p a r t i c l e s in a pulverized- c o a l flame a r e n o t s m a l l enough for s c a t t e r i n g t o b e i n s i g n i f i c a n t . The complete r a d i a t i o n theory must, t h e r e f o r e , b e c a p a b l e o f h a n d l i n g s c a t t e r i n g . I n Fig. 6 two h e a t - f l u x p r o f i l e s a r e p r e s e n t e d f o r
(r_
= 40 a t x / D = 1.9; t h e broken c u r v e i s o b t a i n e dr ( m )
Fig. 5 Heat-flux p r o f i l e s
PREDICTION OF A TURBULENT REACTING DUCT FLOW WITH SIGNIFICANT RADIATION C5b-3 1
when t h e r e i s no s c a t t e r i n g , while t h e s o l i d curve i s t h e result h e n t h e r a t i o of s c a t t e r e d t o absorb- ed r a d i a t i o n i s one h a l f . The e f f e c t of t h i s r a t h e r l a r g e amount of s c a t t e r i s n e g l i g i b l e . ?his was a l s o found t o b e t h e case f o r smaller values of%,.
Of course, s c a t t e r i n g i s not, i n general, i s o t r o p i c a s assumed i n t h e present analysis. This assumption i s , however, not a requirement o f the f l u x model.
5. Conclusions
A numerLca1 method has been presented which enables t h e e n t i r e hydrodynamic and thermal n a t u r e of a chemically-reacting, absorbing and s c a t t e r i n g flow t o be predicted. The f e a s i b i l i t y of t h e method has been denonstrated through i t s applica- t i o n t o t h e p r e d i c t i o n of a t u r b u l e n t duct flow containing a d i f f u s i o n flame. The method i s rapid.
Using 20 g r i d d i v i s i o n s i n t h e r a d i a l d i r e c t i o n and with a forward s t e p s i z e equal t o 1/50 of t h e duct
r a d i u s , t h e time required by an IBM 7094 t o compute
0 0.1 0.2 0.3
0.40.5
s o l u t i o n s i n t h e range 0 Q d D 10 was t y p i c a l l y
r ( m )
2 minutes. The development of t h e method i s , however, i n i t s e a r l y stages. I n t e n s i v e t e s t i n g i s y e t required before t h e s t a t u s of a r e l i a b l e design t o o l f o r t h e p r a c t i s i n g engineer can be accorded t o it.
6. Nomenclature
a
-
s c a t t e r i n g c o e f f i c i e n t-
a maxirnum value of a space-averaged across duct maxD duct diameter
E emissive power
f mixture f r a c t i o n defined a s mass of f u e l - bearing stream 141ich has mixed with oxidant- bearing stream t o produce u n i t mass of products.
F (I+J), t h e sum of t h e r a d i a t i o n f l u x e s h s p e c i f i c enthalpy
m
mass f r a c t i o n p p r e s s u r e r r a d i a l d i s t a n c e s s c a t t e r i n g c o e f f i c i e n tFig. 6: E f f e c t of s c a t t e r i n g streamwise v e l o c i t y
r a d i a l v e l o c i t y
streamwise d i s t a n c e from f u e l - i n j e c t i o n s t a t i o n
d i s t a n c e from and normal t o t h e duct wall p/u, a d i f f u s i v e t r a n s p o r t c o e f f i c i e n t a/mfu, an absorption constant
d m f u , a s c a t t e r i n g constant e m i s s i v i t y
a mixing-length constant a mixing-length constant f l u i d v i s c o s i t y
f l u i d d e n s i t y
Prandtl/Schmidt number a dependent v a r i a b l e S u b s c r i p t s
e f f e f f c t i v e sum of laminar and t u r b u l e n t conzributions
f u , ox, prod fuel, oxidant, products
w wall value
REFERENCES
[I] Patankar (S.V. and Spalding (D.B. ), 'Radiative t r a n s f ~ r ' . McGraw H i l l , 1967.
'Heat and Mass Transfer i n boundary l a y e r s ' .
I n t e r t e x t Books. London, 1970. [31 s c h u s t e r ( A e ) , Astrophys. J., 1905,
2,
1.[21 H o t t e l (H.C.) and Sarofim (A.F. 1, [4I Hamaker (H.c.), P h i l i p s Res. Rpt., 1947,.
