Improved prediction of buoyancy effects on flame length and combustion properties of flares

An improved integral method is presented for determining the flame length and combustion properties of buoyant jet flames, including flares, that avoids the Morton entrainment hypothesis entirely and thereby removes the ad hoc "entrainment modeling" required in most other integral approaches. Rather than working with an integral equation for the entrainment rate dm/dx, we use the momentum flux to develop the equation for the local centerline velocity µ,(x). This allows modeling to be done in terms of the local flow width δ(x) ~ x in both the momentum-dominated jet limit and the buoyancy-dominated plume limit, and experiments show the proportionality constant ç to be constant between these limits. The 'entrainment modeling" required in traditional integral methods is thus replaced by the observed constant ç valuse in the integral equation for the buoyancy flux gives an expression for B(x) in terms of the centerline velocity, and provides a simple integral equation for µ (x) that can be readily sloved for arbitrary flare exit conditions by a simple spreadsheet calulation. The resulting µ(x) determines the local mass flux m(x), momentum flux J(x) and buoyancy flux B(x) throughout the flow, as well as the centerline mixture fraction ξ(x) and the flame length L. Comparisons with flame length data show excellent agreement over a wide range of flame conditions, and provides the parameter that characterizes the extent to which buoyancy effects are significant throughout the flame.

Presented at the AFRC-JFRC 2004 Joint International Combustion Symposium, 10-13 October 2004, Maui, HI Im p r o v e d P re d ic tio n o f B u o y a n c y E ffe c ts o n F la m e L e n g th a n d C o m b u s tio n P ro p e rtie s o f F la re s Francisco J. Diez1 and Wemer J.A. Dahm2 Laboratory for Turbulence & Combustion (LTC) Department of Aerospace Engineering, The University of Michigan Ann Arbor, MI 48109-2140 USA http://www.engin.umich.edu/dept/aero/ltc Abstract An improved integral method is presented for determining the flame length and combustion properties of buoyant jet flames, including flares, that avoids the Morton entrainment hypothesis entirely and thereby removes the ad hoc "entrainment modeling" required in most other integral approaches. Rather than working with an integral equation for the entrainment rate dm/dx, we use the momentum flux to develop the equation for the local centerline velocity uc(x). This allows modeling to be done in terms of the local flow width 5(x), for which dimensional considerations show 6(x) ~ x in both the momentum-dominated jet limit and the buoyancydominated plume limit, and experiments show the proportionality constant cb to be constant between these limits. The "entrainment modeling" required in traditional integral methods is thus replaced by the observed constant cb value in the present method. In the nonreacting case, this new integral approach gives an exact solution for uc(x) that provides excellent agreement with experimental data, as well as an expression for the virtual origin. In the exothermically reacting case, the constant cb value in the integral equation for the buoyancy flux gives an expression for B(x) in terms of the centerline velocity, and provides a simple integral equation for uc(x) that can be readily solved for arbitrary flare exit conditions by a simple spreadsheet calculation. The resulting uc(x) determines the local mass flux m(x), momentum flux J(x) and buoyancy flux B(x) throughout the flow, as well as the centerline mixture fraction ^c(x) and the flame length L. Comparisons with flame length data show excellent agreement over a wide range of flame conditions, and provides the parameter that characterizes the extent to which buoyancy effects are significant throughout the flame. 1. I n tr o d u c tio n Although computational simulations are being increasingly used to predict flame characteristics of flares and other combustion systems, in practice combustion engineers more commonly use simple scaling laws or other approximate methods to estimate such properties as flame lengths, radiation levels, and emissions levels, as well as the conditions that produce onset of sooting and many other combustion properties. Flares differ from most other combustion systems due to the 1. Postdoctoral Researcher. 2. Professor of Aerospace Engineering and Head, Laboratory for Turbulence & Combustion (LTC); author to whom correspondence should be addressed. large b u o y a n c y fo r c e s that a c t o n th e fla m e due to h ea t r e le a se . U n lik e sim p le p lu m e s, in fla res th e b u o y a n c y flu x t y p ic a lly in c r e a se s d ra m a tically d u e to c o m b u stio n h ea t relea se a lo n g th e len gth o f th e fla m e, an d th e r e su ltin g variation in b u o y a n c y flu x p r o d u c e s im p o rtan t ch a n g es in th e flo w p r o p ertie s an d in th e fla m e len gth . P rop er m o d e lin g o f th is v a ria tio n in th e b u o y a n c y flu x alon g th e flam e is e s se n tia l to d e v e lo p in g sim p le y e t a ccu ra te m e th o d s fo r p red ic tin g th e co m b u stio n p ro p ertie s o f fla r e s. V a rio u s em p irica l and se m i-e m p irica l sca lin g la w s are w id e ly u s e d fo r estim a tin g th e fla m e le n g th an d o th er c o m b u stio n p ro p ertie s o f fla re s. In tegral m e th o d s b a se d o n v a r io u s a p p ro x im a tio n s are a lso u se d to p r o v id e a h ig h er le v e l o f fid e lity in m o d e lin g fla re s (e.g., G o ttg e n s 1991 ; B la k e & M c D o n a ld 1995; B la k e & C o te 1 9 9 9 ). H e sk e sta d 1981; P e te r s & In all integral m e th o d s, th e original p artial d iffer en tia l e q u a tio n s fo r c o n se r v a tio n o f m a s s , m o m e n tu m , and en er g y are in teg ra ted a cro ss th e lateral d ir e c tio n o f th e flo w , to g e th e r w ith a ssu m e d se lf-sim ila r fo rm s fo r th e lateral p r o file s o f v e lo c it y a n d d e n sity , to p r o v id e o rd in a ry d iffer en tia l eq u a tio n s fo r v a rio u s in teg ra l p ro p erties o f th e f lo w su c h a s th e m a ss flu x B(x). m(x), m o m e n tu m T h e s e integral e q u a tio n s ca n b e rea d ily s o lv e d b y J(x) and b u o y a n c y flu x flu x n u m erica l in tegration w ith certain m o d elin g a p p r o x im a tio n s to d eterm in e th e resu ltin g ch a n g es in v a r io u s flo w p r o p ertie s w ith in crea sin g d o w n strea m d ista n c e x . M o s t integral m e th o d s to d a te fo r b u o y a n t j e ts and fla res h a v e b e e n b a se d o n an a p p r o x im a tio n that is w id e ly referred to a s th e "M o r to n en train m en t h y p o t h e s is " , in tro d u ced b y M o r to n , T a y lo r & T u rn er ( 1 9 5 6 ) ( s e e a ls o M o r to n 1 9 5 8 , L ist 1 9 7 7 , an d T u rn er 1 9 8 6 ). lo ca l m a ss en train m en t rate in to th e f lo w a s sim ilar p r o file w id th an d a is term ed dmldx = a T h is m o d e ls th e m (x )/S (x ), w h e r e 6 (x ) is th e lo c a l s e lf ­ an "en tra in m en t c o e ffic ie n t" . C o m p a riso n s w ith exp erim en tal d ata fro m n o n r ea ctin g flo w s h a v e sh o w n that su c h an en tra in m en t c o e ffic ie n t w o u ld n e e d to va ry fro m a = 0 .0 5 7 in m o m e n tu m -d o m in a te d j e t f lo w s h a v in g n e g lig ib le b u o y a n cy , to a *= 0 .0 8 2 in p lu m e f lo w s th at are d r iv e n en tire ly b y b u o y a n c y . A s a re su lt, a m u st b e ch a n g ed as th e rela tiv e im p o rta n ce o f m o m e n tu m and b u o y a n c y e f f e c ts v a r ie s a lon g th e flo w . L ack in g a n y fu n d am en tal b a sis o n w h ic h to m o d e l th is en tra in m en t c o e ffic ie n t, integral m eth o d s fo r b u o y a n t je ts an d fla res b a se d o n th e M o r to n en tra in m en t h y p o th e sis in ste a d re p r esen t th e lo c a l v a lu e o f a b y v a rio u s ad hoc e x p r e ssio n s in term s o f a F rou d e n u m b er th a t ch a ra cterizes th e lo ca l ra tio o f in ertial and b u o y a n c y fo r c e s. A s su m m a rized b y G ep h a rt et al ( 1 9 8 8 ) , n u m ero u s su ch ad hoc m o d e ls fo r a h a v e b e e n p r o p o se d , an d th e v a rio u s in tegral m e th o d s in u s e to d a y d iffer la r g e ly o n th e b a sis o f th eir c h o ic e fo r th e " en tra in m en t m o d e l" fo r a ( s e e a ls o T e ix e ir a & M iran da 1 9 9 7 ). H ere w e d e v e lo p a d iffe r e n t a p p r o a ch that a v o id s th e M o r to n en tra in m e n t h y p o th e sis a lto g eth e r, ad hoc "en train m en t m o d e lin g " . an d th ereb y r e m o v e s th e n e e d fo r a n y su ch in term s o f an integral eq u a tio n fo r dmldx, w e u s e th e integral eq u a tio n fo r co rresp o n d in g e q u a tio n fo r th e lo c a l cen terlin e v e lo c it y wc(x ). m o d elin g can n o w b e d o n e in ter m s o f th e lo c a l flo w c o n sid er a tio n s a lo n e s h o w th a t 6 (x ) ~ x R ath er than w o r k in g dJIdx to d e v e lo p a T h is h a s th e ad vantage th a t th e w id th S (x ), fo r w h ic h d im en sio n a l in b o th th e m o m e n tu m -d o m in a te d j e t lim it an d th e b u o y a n c y -d o m in a te d p lu m e lim it, an d ex p erim en tal d ata fu rth er s h o w th a t th e p r o p o r tio n a lity co n sta n t c 6 is id en tica l in b o th th e j e t and p lu m e lim its and re m a in s co n sta n t b e tw e e n th e s e tw o lim its as w e ll (e.g., P a p a n ic o la o u & L ist 1 9 8 8 ). T h u s, w h e re a s a v a r ie s b e tw e e n th e se tw o lim its an d th erefo re m u st b e m o d e le d , cs is in variant b e tw e e n th e se lim its an d th u s ca n b e rep r esen te d 2 b y a co n sta n t v a lu e w ith o u t a n y further m o d e lin g . In e ffe c t, th e ad hoc "en tra in m en t m o d e lin g " that is requ ired in trad ition al integral m eth o d s is r e p la c e d in th e p r e se n t a p p ro a ch b y th e o b ser v ed co n sta n t v a lu e o f c6. In th e c a se o f n o n r ea ctin g b u o y a n t j e t s , for w h ic h B(x) is co n sta n t, th is n e w in te g ra l a p p roach is fou n d to g iv e an e x a c t so lu tio n fo r th e cen terlin e v e lo c it y uc(x) that p r o v id e s e x c e lle n t agreem en t w ith th e b u o y a n t j e t d ata o f P a p a n ic o la o u & L ist ( 1 9 8 8 ) . T h is ex a c t so lu tio n is far sim p le r than that o b ta in ed b y M o r to n (1 9 5 8 ) from th e en tra in m en t h y p o th e s is , an d w ill b e m o re accu rate sin ce it a v o id s a n ad hoc "en train m en t m o d e l" . M o r e o v e r , th e p resen t s o lu tio n a lso p r o v id e s an exact e x p r e ssio n fo r th e virtu al origin o f n o n rea ctin g b u o y a n t je ts . U n lik e th e v irtu a l origin m o d e ls o f M o r to n ( 1 9 5 8 ), M o rto n & M id d le to n ( 1 9 7 3 ) , an d m o re r e c e n tly H u n t & K a y e (2 0 0 1 ), th e p r esen t r e su lt fo r th e virtu al origin is o b ta in ed w ith o u t a n y re so r t to m o d e lin g o f an en train m en t c o e f fic ie n t a . B(x) cb v a lu e is in tro d u ced in th e integral e x p r e s s io n fo r B(x) in ter m s o f th e cen terlin e In th e c a se o f e x o th e r m ic a lly rea ctin g b u o y a n t j e t fla m e s, fo r w h ic h th e to ta l b u o y a n c y flu x in crea ses alon g th e len g th o f th e fla m e, th e sa m e c o n sta n t dBldx. T h is is fo u n d to g iv e a sim p le v e lo c ity uc(x). In tro d u cin g th is e x p r e ssio n in th e in tegral p r o v id e s a s im p le integral eq u a tio n fo r uc(x) th a t ca n b e eq u a tio n fo r c o n d itio n s v ia a s im p le sp re a d sh ee t ca lcu la tio n . m a ss flu x m(x), th e m o m en tu m flu x J(x) and fo rm o f th e m o m e n tu m e q u a tio n th en re a d ily s o lv e d fo r arbitrary flare ex it T h e re su ltin g th e b u o y a n c y flu x uc{x) th e n d e ter m in es th e B(x) th r o u g h o u t th e f lo w . loca l The resu ltin g m a ss flu x in turn d eterm in es th e ce n te rlin e m ix tu re fra ctio n £ c(x ), w h ic h d e te r m in e s th e fla m e len g th L. C o m p a r iso n s w ith m easu red fla m e le n g th d ata sh o w e x c e lle n t agreem en t o v e r a w id e range o f fla m e c o n d itio n s. M o r e o v e r , th e re su ltin g fla m e len g th e x p r e ssio n p r o v id e s a natural p a ra m eter th a t ch a ra cterizes th e e x te n t to w h ic h b u o y a n c y e f f e c ts are sig n ific a n t th ro u g h o u t th e fla m e . T h e p r e se n ta tio n is o rg a n ized a s f o llo w s . In §2 w e first la y o u t certain b a sic a sp e c ts o f integral m e th o d s, and r e v ie w th e required sca lin g in th e j e t an d p lu m e lim its. In §3 w e th e n d e v e lo p th e integral m e th o d a n d th e p rin cip a l m o d elin g a ssu m p tio n fo r b u o y a n t je t s , an d o b ta in th e exa ct so lu tio n o f th e n o n rea ctin g ca se . In § 4 w e th e n ex te n d th e integral m e th o d to th e ex oth erm ic reactin g c a se an d d esc rib e its p ra ctica l a p p lic a tio n . In §6 w e d eriv e th e fla m e len g th sc a lin g im p lied b y th is in te g ra l m eth o d , an d m ak e c o m p a r iso n s w ith m easu red fla m e le n g th data. In §7 w e d is c u s s th e im p lic a tio n s o f th is n e w integral m e th o d fo r im p r o v ed m o d e lin g o f th e flam e le n g th an d c o m b u s tio n p ro p erties o f p ra ctic a l fla res. 2. S c a lin g o f t h e J e t a n d P lu m e L im its B(x) in cr ea se s a lo n g th e le n g th o f th e dBldx is se t b y th e rate o f h ea t relea se, F la res are tu rb u len t b u o y a n t j e t s in w h ic h th e b u o y a n c y flu x flam e d u e to c o m b u s tio n h ea t relea se. T h e lo c a l v a lu e o f w h ic h in turn is s e t b y th e rate o f en tra in m en t dmldx in to th e flo w . T h e r e su ltin g n o n lin ea r c o u p lin g m a k e s th e rea ctin g flo w d iffic u lt to a n a ly z e d ir e c tly , h o w e v e r th ere are se v er a l lim its in w h ic h e x a c t s o lu tio n s c a n b e o b ta in ed an d w h ic h p r o v id e th e b a sis fo r th e p resen t in te g ra l m eth o d fo r th e c o m p le te p r o b le m . In th is se c tio n w e th u s c o n sid e r th e n o n rea ctin g lim it, an d r e v ie w b o th th e m o m e n tu m -d o m in a te d j e t lim it and th e b u o y a n c y -d o m in a te d p lu m e lim it. T h e s e r e su lts th en 3 form th e b a sis for a n e w in te g ra l m eth o d fo r n o n rea ctin g tu rb u len t b u o y a n t j e ts d e v e lo p e d in §3. T h o se re su lts in turn p r o v id e th e b a sis fo r th e c o m p le te in tegral m eth o d fo r ex o th erm ic reactin g tu rb u len t b u o y a n t j e ts d e v e lo p e d in § 4 . 2 .1 M a s s , m o m e n tu m , a n d b u o y a n c y f lu x e s A s in d ica ted in F ig s. 1 an d 2 , th e f lo w in th e se lf-sim ila r fa r-field p r o d u c e d b y a n y fin ite -a r ea mE, m o m e n tu m flu x JEan d b u o y a n c y flu x BEca n b e x = 0 h a v in g so u r c e m o m e n tu m flu x J0 and b u o y a n c y flu x B0, b u t w ith th e so u rc e m a ss flu x m0 = 0 . T h ro u g h p ro p er c h o ic e o f th e so u rc e v a lu e s J0and B0, and th e e x it v a lu e xE, th e f lo w at su ffic ie n tly la rg e d o w n strea m d ista n c e s x w ill b e so u rce issu in g ex it v a lu e s o f th e m a ss flu x eq u iv a le n tly rep resen ted b y a p o in t so u r c e at id e n tic a l to that p ro d u ced b y th e a ctu a l fin ite -a r e a so u rce. A t a n y d o w n strea m lo c a tio n b u o y a n c y flu x B(x) are x fro m th e so u r c e , th e lo c a l m a ss flu x m(x), m o m e n tu m flu x J(x), and g iv e n b y m ( x) = J pu(x,r)2nrdr (la ) 0 00 J(x) ■ f p u 2(x,r) 2 nr dr (1 6 ) 0 00 B(x)=j~ gApu(x,r)2nr dr , (lc ) o w h e r e A p (x ,r ) ■ p „ - p (x ,r ) . T h e co r re sp o n d in g ex it v a lu e s are d e n o te d e x a m p le, for a so u rc e w ith e x it area AE, u n ifo rm e x it v e lo c it y UEan d mE, JE, an d u n ifo r m e x it d e n sity BE; fo r pE, th e so u rce v a lu e s are ™E ~PE^E^E ( 2 ^) JE= pEU2EAE (2b) BE= gApEUEAE (2c) A t s u ffic ie n tly large d o w n str e a m d ista n c e s x fro m th e so u r c e , m(x) » mE and th u s m o s t o f th e flu id m o v in g in th e f lo w h a s b e e n en train ed fro m th e a m b ien t. A s a c o n s e q u e n c e , th e d e n s ity p is a p p r o x im a tely equal to th e a m b ien t flu id d e n sity p „ . M o r e o v e r , in th is "far field " th e m ean str e a m w ise v e lo c it y an d d e n s ity -d iffe r e n c e p r o file s are se lf-sim ila r in th e sc a le d radial co o rd in a te t] ■ r/b(x), w ith th e p r o file sh a p e s b e in g a p p r o x im a tely g a u ssia n an d g iv e n b y U^X,l\ UC\ X ) - f ( ‘n ) w h ere / ( r t f - e x p - c ^ i f = /i(ti) w h e re h(r\) = e x p - a A x\ (3 a ) . (3b) A p c( x ) H ere ujx) d e n o te s th e lo c a l m ea n cen terlin e v e lo c it y , 6 (x ) is th e lo c a l p r o file w id th , an d Apc is th e m ea n c e n te r lin e d e n sity d iffe r e n c e . T h e co n sta n t 4 ay in ( 3 a ) r e su lts fro m th e c h o ic e o f 6 , h ere d efin ed as th e fu ll w id th at w h ic h th e m ea n v e lo c ity p r o file h as d e c r e a se d to 5% o f its ce n te r lin e (3b) is v a lu e , and th e co r re sp o n d in g a , in ob ta in ed from P a p a n ic o la o u & L ist (1 9 9 8 ), g iv in g af = 1 2 .0 (4 a ) ( 4 b) a , - 1 0 .7 . F rom (3 ) and (4 ) v a rio u s in te g ra ls that a rise in th e a n a ly sis b e lo w ca n b e e v a lu a te d as /j = J f(y\)2m\ dx\ = - « 0 .2 6 2 (5 a ) oo h - 2<x/ 00 I3 =Jh(r])2ivr] dr\ = h uc(x), b(x) an d » 0 .2 9 4 s J /W K T l) 2 j t T i f / r ] - F rom (1 ) - (5 ), th e lo c a l m a ss flu x term s o f (5b) f / 20 l ) 2 :n:Tl rfTl = r ^ - « 0 . 1 3 1 o m(x), m o m en tu m (5 c ) ( 5 d) - 0 .1 3 8 flu x J(x), and b u o y a n c y flu x B(x) are g iv e n in Apc (jc) as m(x) - uc(x)b2(x) (6a) J(x) « 12 p„ u2 c(x)b2(x) (6b) B (x )-I 4 g&pc(x)uc(x)b2(x) ( 6c ) /] p„ T h e e x p r e ssio n s in ( 6 a - c ) p r o v id e th e sta rtin g p o in t fo r th e a n a ly sis p r e s e n te d b e lo w . all o th er lo ca l p r o p e r tie s fo r th e f lo w , su ch a s th e m a ss en tra in m en t rate scalar m ix in g rate, are in turn d eterm in ed b y uc(x) and dm/dx, N o te that th e c o n se r v e d 5 (x ). 2.2 The jet limit 2 .2 .1 Scaling laws for b(x) and uc(x) T h e term "j e t " r e fe r s to th e f lo w p r o d u c e d b y a p o in t so u r c e o f m o m e n tu m fo r w h ich th e so u rc e m a ss flu x is z e r o , n a m e ly m0 = 0 , a n d th e b u o y a n c y flu x o n th e flu id , th e lo c a l m o m e n tu m flu x its so u rc e v a lu e J0. J(x) at B(x) = 0. S in c e n o b u o y a n c y fo r c e s a ct a ll d o w n strea m d ista n c e s J0 is T h e so u rc e m o m en tu m flu x x m u st rem ain co n sta n t at th e o n ly in tegra l in variant o f th e f lo w , and th u s in th e se lf-sim ila r far fie ld th e lo c a l f lo w w id th 5 (x ) and th e lo c a l m ea n cen terlin e v e lo c it y uc(x) ca n d ep en d o n ly o n J0, th e d o w n str e a m lo c a tio n x, resu lt, o n d im e n sio n a l g ro u n d s a lo n e th e sc a lin g s for 5 a n d an d th e a m b ie n t flu id d e n sity p „ . A s a uc in j e t s m u st th u s b e b~x «c ~ ( ^ o / p . ) V , w h ere th e p r o p o r tio n a lity c o n sta n ts cb and cuin 5 (7 ) m u st b e th e sa m e fo r all je ts. (la) (lb) F ro m carefu l e x p er im en ts (e.g., P a p a n ic o la o u & L ist 1 9 8 8 ) th e se h a v e b een fo u n d to b e (c5),«0.36 ( c „ ) ,~ 7 .2 w h ere th e su b sc rip t (8a) (86) , (6a), d e n o te s co n sta n ts a p p lic a b le to th e je t-lim it sca lin g . F ro m ( 7 ) and ( 8), th e re su ltin g m a ss flu x sc a lin g in je ts is th u s m(x) = 2 .2 .2 /, c„ c 52 (p „ J 0 )'/2 x . (9 ) The virtual origin xE T h e r e su lts in ( 7 )- (9 ) are fo r th e f lo w p ro d u c ed b y a n id e a l p o in t so u rc e at x ■ 0 , n a m e ly o n e that in tr o d u ce s m o m e n tu m flu x J0 b u t in tr o d u ce s n o so u rc e m a ss flux; i.e., m0 = 0. R ea l j e t s are mEan d a n ex it mE an d JEto th e id eal ty p ic a lly p r o d u c e d b y fin ite-a rea so u rc es th a t in tr o d u c e a n o n -z e r o ex it m a ss flu x JE. P ro p er a cc o u n t m u st b e m0an d J0v ia th e virtu al o rig in . m o m e n tu m flu x so u rc e v a lu e s ta k e n to relate th e e x it v a lu e s F igu re 1 s h o w s th e b a sic p rin cip le in v o lv e d in d eterm in in g th e virtu al origin from th e id e a l p o in t so u rc e, th en xEis JE . th e re su ltin g m o m e n tu m flu x J(x E) = m o m e n tu m flu x J(x) a J0 for all x, x is m ea su red m(xE) = mE and If T h e la tter is a u to m a tica lly sa tis fie d , s in c e in j e t s th e an d th u s th e a p p r o p r ia te v a lu e o f p ro d u c e th e sa m e f lo w a s th e actu al so u rc e is fro m (9 ) to th e actu al e x it m a ss flu x xE. th e v a lu e at w h ic h th e resu ltin g m a ss flu x mEth en J0 = JE. J0 fo r th e p o in t so u r c e to M a tc h in g th e re su ltin g m a s s flu x tm(jce) req u ires mE = h cucl (pco^o T (10 ) xe from w h ic h 2 mE (11) F o r th e v a lu e s o f / , , (N o te th a t d* cuand cbin ( 5 a ) an d (8a,b), th is g iv e s th e virtu a l o rig in fo r j e t s a s xE» 3 .6 d*. is o fte n c a lle d th e "fa r -fie ld e q u iv a le n t so u rce d iam eter"; from its d e fin itio n in (1 1 ) th e e q u iv a le n t d ia m eter fo r a circular n o z z le w ith u n ifo rm ex it d e n sity p£ an d v e lo c it y th e c la ssic a l r e su lt d* = (p J p y /2dE .) UE g iv e s If, a s is co m m o n in p ra ctice, jc is u s e d to d e n o te th e d o w n str e a m d ista n c e m easu red fro m th e e x it o f th e fin ite-a rea so u rce, rath er th a n th e d ista n c e fro m th e id e a l p o in t so u r c e , th en x in ( 7 )- (9 ) m u st b e re p la ce d b y x + xEas F ig . 1 in d ic a te s. 2.3 The plume limit 2 .3 .1 Scaling laws for 6 ( x ) and uc(x) In co n tra st, a "p lu m e " is p ro d u c ed b y a p o in t so u r c e fo r w h ic h th e so u rc e m a ss a n d m o m e n tu m flu x e s are b o th z e r o , n a m e ly b u o y a n c y flu x B0 in tro d u ced m0 ■ 0 an d J0 = 0 , w ith th e flo w created in ste a d b y an in itia l b y th e so u rc e. In th is ca se , th e resu ltin g b u o y a n c y fo r c e s th at a ct o n th e flu id c a u se th e m o m en tu m flu x J(x) to in cr ea se w ith d o w n strea m d ista n c e x. In th e a b sen ce o f c o m b u stio n h ea t r e le a se or o th e r e f f e c ts that chan ge th e to ta l b u o y a n c y flu x , th e 6 sou rce b u o y a n cy flu x B0 is th e o n ly integral in variant o f th e flo w . T h u s in th e se lf-sim ila r far fie ld the lo c a l flo w w id th 5 (x ) an d th e local m ean c e n te r lin e v e lo c ity wc(x) ca n d ep en d o n ly o n the d ow n stream lo c a tio n sca lin g s for 6 and uc in x, B0, an d th e am bient flu id d e n sity p„. O n d im e n sio n a l g rou n d s a lo n e , th e p lu m e s m u st th erefore b e 6~ x ( 12a ) uc ~ ( B j p J V3x~113 . T h e p ro p o rtio n a lity c o n sta n ts c 8 an d cuin ( 1 2 b) th e fa r-field sc a lin g s in (1 2 ) m u st, fu rth erm ore, b e th e sa m e fo r all p lu m es, an d fro m P a p a n ic o la o u & L ist ( 1 9 8 8 ) are fo u n d to b e ( c6) , ~ 0 . 3 6 ( c . ) , - 4 .0 w h ere th e su b sc rip t "p " d e n o te s (1 3 a ) , (1 3 6 ) co n sta n ts a p p lica b le to th e p lu m e -lim it sc a lin g . F ro m (6a,b), ( 12) and (1 3 ), th e r e su ltin g m a ss an d m om en tu m flu x s c a lin g s in p lu m es are th u s m(x)/p„ = /, cuc\ (b 0/ p„ )1/3 x5n (1 4 a ) J( x)Ip„ ~ h cl cl (Bo/ p T * 13 ■ 2 .3 .2 ( 14fe) The virtual origin xE T h e resu lts in ( 1 2 )- (1 4 ) are fo r p lu m e s p ro d u ced b y an id e a l p o in t so u rc e at b u o y a n c y flu x B0 b u t n o m a ss flux; i.e., m0 = 0. A p ro p erly c h o s e n v ir tu a l o r ig in 0 that in tr o d u c e s R ea l p lu m e s are m ore t y p ic a lly p r o d u c e d b y mEin fin ite-a rea so u rc es that in tr o d u ce an ex it m a ss flu x Be. x■ xe m a tch es th e co n ju n c tio n w ith th e e x it b u o y a n c y flu x m a ss flu x m(xE) to th e e x it m a ss flu x mE. x is m ea su red fro m th e id e a l p o in t m(xE) = mEan d B(xf) = BE. T h e la tter is a u to m a tica lly s a tisfie d , s in c e in p lu m e s B(x) = B0fo r a ll x, and th u s th e b u o y a n c y flu x requ ired for th e p o in t so u rc e to p r o d u c e th e sam e flo w a s th e a ctu al so u rc e is B0 = BE. F ro m ( 1 4 a ) th e m ass flu x m atch in g m(xE) = mEreq u ires F igure 1 again se r v e s to s h o w th e b a sic p rin cip le in v o lv e d . I f so u rce, th en xEis th e v a lu e at w h ic h th e resu ltin g m a ss flu x (1 5 ) from w h ic h th e v irtu a l o r ig in is ,3/5 ' 3/5 r (™ */P«) (1 6 ) (ws) (beIp T I f x is u se d to d e n o te th e d o w n str e a m d istan ce m ea su red fro m th e e x it o f th e fin ite so u r c e , rather than th e d ista n c e fro m th e id e a l p o in t so u rc e, th en x in ( 1 2 ) - ( 1 4 ) m u st b e r e p la c e d b y 3. B u o y a n t J e t s in t h e N o n r e a c tin g L im it B (x ) « x + xB. B0 T h e term "b u o y a n t j e t " is u s e d fo r th e flo w p ro d u c ed b y a p o in t so u rc e th a t in tr o d u c e s b o th an in itial m o m e n tu m flu x J 0 an d in itia l b u o y a n c y flu x B0; M o r to n (1 9 5 8 ) re fer s to su c h f lo w s as "fo rc ed p lu m e s " . In th e a b se n c e o f co m b u stio n h ea t r e le a se o r o th er e ffe c ts that c h a n g e th e to ta l 7 b u o y a n c y flu x , th e so u rc e b u o y a n c y flu x x , but th e m o m en tu m flu x J(x) B(x) rem a in s co n sta n t at B0fo r a ll d o w n str e a m d ista n c e s w ill in cr ea se a s a r e su lt o f th e b u o y a n c y b o d y fo r c e s th a t a ct o n th e flu id . In th is se c tio n , w e o b ta in th e sc a lin g la w s fo r th e lo c a l flo w w id th 5 (x ) and cen terlin e v e lo c it y 3.1 uc(x) for su ch n o n rea ctin g b u o y a n t je ts. F u n d a m e n t a l f o r m o f th e sc a lin g la w s f o r b u o y a n t je ts B0o r uc m u st O w in g to th e fa ct that b u o y a n t j e t s h a v e o n e a d d ition a l so u rc e p aram eter (e ith e r J 0) rela tiv e to th e j e t an d p lu m e lim its in § 2 , o n d im e n sio n a l g ro u n d s th e sc a lin g s fo r 6 a n d in v o lv e an a d d itio n a l le n g th s c a le /* and an a d d itio n a l v e lo c it y s c a le that ca n b e fo rm ed from J0, B0, a n d u*. T h ese in turn are th e o n ly q u a n tities p m, a n d th e re fo r e m u st b e I. (17o) W pJ K/p.r . lr I V/ 4 Uo / P* / N o te that u* I* is ( 176) ' s o m e tim e s referred to a s th e "M o rto n len g th sc a le" , and w e w ill th u s a ls o refer to as th e "M o r to n v e lo c it y sc a le " . O n d im e n sio n a l g ro u n d s, th e r e su ltin g s c a lin g s fo r 5 an d uc in b u o y a n t j e t s m u st th u s b e T -/,© 7 (18o) (1" ) • T h e f u n c t i o n s / an d f m u st re co v e r th e je t-lim it sca lin g a s p lu m e -lim it sc a lin g as -*■ 0 , fo r w h ic h 1* -* Ofc) B0 - > 0 , fo r w h ic h /* -* oo, an d th e oo. T h u s in th e je t-lim it | - > 0 , fro m /i( l)-^ (c » )7 - |. / 2( i ) ^ ( c „ ) j - r 1 , (la, b) (1 9 a ) (19b ) w h ile in th e p lu m e -lim it | - » oo, fro m ( 12a, b) /( D - ^ V S m ) - * ( c u)p- ^ m . ( 20 a ) (2 0 6 ) In § 3 .2 an d § 3 .3 , th e sc a lin g fu n ctio n s_ /J (|) an d / ( | ) are ob ta in ed fo r n o n r e a c tin g b u o y a n t je t s , for w h ic h B(x) s B0. In § 4 th e co r re sp o n d in g sca lin g fu n c tio n s are th e n o b ta in ed fo r exoth erm ic r e a ctin g b u o y a n t j e t s , fo r w h ic h 3.2 B(x) in c r e a se s d u e to co m b u stio n h e a t r e le a se . T h e flo w w i d t h scalin g ( 8 / /*) = / , ( | ) C o m p a rin g ( 1 9 a ) an d (2 0 a ) it is a p p a ren t th a t in b u o y a n t je ts th e p ro p er f lo w w id th scalin g 8 f(x / / * ) h a s th e sa m e fu n ctio n a l fo rm in b o th th e je t-lik e an d p lu m e -lik e lim its. M o r eo v e r, from ( 8 a ) an d ( 1 3 a ), it is ap p aren t that th e s c a lin g co n sta n t c 6 in b oth th e j e t -lik e an d p lu m e -lik e lim its is a lso th e sa m e. F rom th e s e o b se r v a tio n s it is e v id e n t that f o r a ll | ( 5 / / * ) » c 6-| (2 1 ) , an d th u s 5«c. T h e co n sta n t v a lu e o f cb in x c. « w ith 0 .3 6 fo r a ll x. (22a, (2 2 ) th r o u g h o u t th e flo w w ill b e s e e n b e lo w to r e p la c e th e b) ad hoc m o d elin g o f an "entrain m en t c o e ffic ie n t" a ( ^ ) required in m o st o th e r in teg ra l m eth o d s fo r b u o y a n t j e t s an d fla res. 3 .3 T h e c e n t e r lin e v e lo c it y s c a lin g ( uc / u*)= / 2(§ ) uc(x) W h erea s § (x ) is re p r esen te d b y ( 2 2 ), th e ce n te rlin e v e lo c it y e q u a tio n fo r th e m o m en tu m flu x 3 .3 .1 is o b ta in ed fro m th e integral J(x). Integralform of the momentum equation T h e lo c a l rate o f ch a n g e o f a x ia l m o m e n tu m is eq u a l to th e b u o y a n c y b o d y fo r c e a s d_ [pM 2( x ,r ) ] = g A p ( x ,r ) dx , (2 3 ) fro m w h ic h •J*pw 2 2jit d r-J gAp(x,r)2nrdr dx (2 4 ) F ro m (1 b,c) th is b e c o m e s J 7 i(r i)2 jtr | ± T( dx dx\ >_ uc(x) (2 5 ) ^ f(r\)h (r\)2 n y\d t\ 0 fro m w h ic h L ± L = ( t n ) 1 B(x) J dx 1 3 / 4,J(x) uc(x) F ro m (6b) an d (2 6 ) ■ (2 2 ) J(x) = I2c lp auc(x)x2 , (2 7 ) a n d th u s th e le ft sid e o f ( 2 6 ) ca n b e e x p r e s s e d a s dJ J dx 1 2 duc u dx 2 x (2 8 ) an d th e rig h t s id e o f (2 6 ) ca n b e e x p r e sse d a s (/,//.)■1 J(x ) u ix) w here 9 = o (b/ p . ) ,3 J 2 u: X 5 (2 9 ) • ( 3 0 ) S u b stitu tin g (2 8 ) an d ( 2 9 ) in ( 2 6 ) g iv e s a sim p le d iffe r e n tia l eq u a tio n fo r dx 3 .3 .2 x uc(x) in term s o f B(x) as (31) ucx Integration to determine uc(x) when B(x) = B0 U p to th is p o in t, th ere h a s b e e n n o restrictio n o n th e b u o y a n c y flu x B(x) in ob ta in in g th e fo r m o f th e m o m en tu m e q u a tio n in (3 1 ). In th is se c tio n w e o b ta in an e x a c t so lu tio n to (3 1 ) fo r th e c a s e w hen B(x) is co n sta n t at its so u rc e v a lu e B0, w h ic h c o r r e sp o n d s to th e lim it in w h ic h th e a d d ed b u o y a n cy d u e to c o m b u stio n h ea t r e le a se is n e g lig ib le . C h a n g in g v a ria b les to th e d im e n s io n le ss ce n te rlin e v e lo c it y an d d o w n strea m co o rd in a te / 2f e ) - f 5 r w e n o te that w h e n B(x) = B0( 3 1 ) 5 -7 7 , an d 02a,b) b ecom es /I / \ _? f a f e / , r (3 3 ) . T h e form in (3 3 ) is e q u iv a le n t to d ( l f j = 3 a |r f | (3 4 ) t e / 2)3 = f ° i 2 + C , (3 5 ) w h ic h in teg ra tes tr iv ia lly to an d th u s ■3 2 < * 2 T h e in teg ra tio n co n sta n t C is uc an d x v ia 3 fo u n d b y tak in g th e j e t-lim it | f 2-> R ew ritin g (3 7 ) in f +cl (3 6 ) 0 , fo r w h ic h ( 3 6 ) b e c o m e s C 1/3 1 ' 1 . (3 7 ) (3 2 ) g iv e s uc ^ C m( j 0lp x)V2x~l , and co m p a rin g w ith th e j e t sc a lin g in (lb) and (3 8 ) (8 6 ) s h o w s th at C = ( c u)j . (3 9 ) A s a c h e c k , n o te that ta k in g th e p lu m e -lim it ^ - > 00 in ( 3 6 ) g iv e s /,- ( f ° ) V 10 \ (40) w h ic h c a n b e r e w r it t e n in uc a n d x v ia (3 2 ) to g iv e (4 1 ) C om p arin g (4 1 ) w ith th e p lu m e sc a lin g in (1 0 6 ) an d (1 1 6 ) sh o w s that (3 \ U3 ( °) (4 2 ) 2 F rom ( 3 0 ) w ith ( 5 b-d) an d ( 2 2 b), th e le ft sid e o f ( 4 2 ) is 4 .5 , w h ic h co m p a re s a c c e p ta b ly w ith th e valu e 4.0 in (1 3 6 ); th e d iffer en ce s are d u e to th e r e la tiv e ly sm all tu rb u len t flu x c o n tr ib u tio n s to th e Inin (5 ), w h ic h h a v e h ere b een ig n o re d b ut c o u ld b e rea d ily in clu d ed . S u b stitu tin g (3 9 ) an d (4 2 ) in (3 6 ) g iv e s th e c e n te r lin e v e lo c ity in b u o y a n t j e t s a s (4 3 ) w h ere (cu)j ( c j p are and from ( 86 ) an d (1 3 6 ). N o te that u sin g (cu) p in ( 4 3 ) in p la c e o f (3o/2) in (3 6 ) im p lic itly ta k e s a cco u n t o f th e tu rb u len t flu x co n trib u tio n s that w e r e ig n o red in th e in tegrals uc(x) in T h e re su lt in (4 3 ) p r o v id e s an e x a c t c lo se d -fo r m e x p r e ssio n fo r th e c e n te r lin e v e lo c ity tu rb u len t b u o y a n t j e t s w ith and ( 2 2 ). B(x) = B0, su b je c t to th e o b se r v e d co n sta n t v a lu e o f cdin (2 1 ) T h is is co m p a red in F ig . 3 w ith th e d a ta o f P a p a n ico la o u & L ist ( 1 9 8 8 ) , w h e r e it is apparent that it p r o v id e s e x c e lle n t a g re em e n t w ith m ea su red v a lu e s. T h e r e su lt in (4 3 ) m a y b e co m p a red to th a t o f M o r to n et al (1 9 5 6 ) an d M o r to n ( 1 9 5 8 ) , th o u g h th eir a p p ro a ch req u ired an a ssu m e d m o d e l fo r th e "en train m en t c o e ffic ie n t" a fo r w h ic h th ere is little ex p erim en ta l su p p o rt e x c e p t in th e j e t an d p lu m e lim its. rep la ces th is ad hoc m o d e lin g T h e p r e se n t r e su lt e f f e c tiv e ly o f an en tra in m en t c o e ffic ie n t w ith th e o b se r v e d c o n sta n t v a lu e o f c 6 b e tw e e n th e j e t an d p lu m e lim its, fo r w h ic h th ere is stro n g su p p ort in th e d ata o f P a p a n ic o la o u & L ist (1 9 8 8 ). 3 .4 T h e s o u r c e m o m e n t u m f lu x J0 a n d t h e v ir t u a l o r ig in xE T h e r e su lt in (4 3 ) is fo r th e f lo w p ro d u c ed b y an id eal p o in t sou rce at m o m e n tu m flu x J0 an d p ro d u c ed b y fin ite-a rea so u r c e s in tro d u ce a n o n -z e r o ex it m a ss ex it m o m e n tu m flu x i.e., m0 ■ flu x mE in b u o y a n c y f l u x 5 0 b u t n o so u rc e m a ss flu x; JEand ex it b u o y a n c y flu x BE. x ■ 0 th a t in tr o d u ce s 0 . R e a l b u o y a n t je t s c o n ju n c tio n w ith th e A p ro p erly c h o se n v irtu a l o rig in x E m a tc h e s m(xE) an d m o m e n tu m flu x J(xE) p r o d u c e d b y th e p o in t so u rce to th e e x it m a ss and mEand JE. In th is c a s e , h o w e v e r , b o th th e virtu al o r ig in x E a n d th e a p p r o p r ia te so u rce m o m e n tu m flu x J 0 m u st b e d eterm in ed , s in c e J(xE) £ J0. th e m a ss flu x m o m e n tu m flu x e s x is m easu red fr o m th e id e a l p o in t m(xE) = mE, th e m o m e n tu m flu x J (x E) F igu re 1 aga in se r v e s to s h o w th e b a sic p r in c ip le in v o lv e d . I f so u rc e, th e n = JE, xEis th e v a lu e at w h ic h th e re su ltin g m a ss flu x an d th e b u o y a n c y flu x B(xE) -BE. T h e la tter requ irem en t is a u to m a tic a lly s a tis fie d , sin c e th e b u o y a n c y flu x rem ain s co n sta n t in th e a b se n c e o f co m b u stio n h eat r e le a se, an d th u s T h e m a ss flu x m a tch in g m(xE) = mEreq u ires fr o m (6a) to g eth e r w ith 11 B0 = BE. (1 7 ), ( 2 2 ), ( 3 2 ) a n d ( 4 3 ) that Aci P . - ? e [ ( c. ) j + ( O U s f 1 J W ft.) (4 4 ) ' w h ich m a y b e rearranged as £ f )rpz> ^EjT S £\ [(^cu;? y +i ',cu w h ere w e h a v e a lso m a d e u s e o f th e fa c t that 12 V»c# B0= BE. (7 I (4 5 ) \ 5/4 Uo/P-j T h e so u r c e m o m e n tu m flu x J 0 o n th e righ t sid e o f (4 5 ) is u n k n o w n , b u t c a n b e o b ta in ed fro m th e m o m en tu m flu x m a tch in g J(xE) =JE. T h is req u ires from ( 66 ) to g e th e r w ith ( 1 7 ) , ( 22 ), (3 2 ) an d ( 43 ) that JE-^J0[(cfj+(cu)X] 12 /3 (4 6 ) S u b stitu tin g fo r th e b ra ck eted term in (4 6 ) fro m (4 5 ) and rearranging th e n g iv e s ___ (V P .)- Ufe \ il . ^ -4 /3 w h ic h m a y b e su b stitu ted in (4 5 ) an d rearranged to d eterm in e / ' 2 /, > (Je / p T 1 %E~j U 5^ w ; J (4 8 ) is first u s e d to fin d th e v irtu a l origin | £. J0o f origin xE. W h en , u*, v ia ( 1 7 a,b). ((c u)p \ 3 ‘ 1 ' 1 (4 8 ) U o J mE, JE and BEo f th e fin ite -a r ea so u rce, W ith th is | £, ( 4 7 ) th e n g iv e s th e required th e p o in t so u r c e , w h ic h to g eth e r w ith and v e lo c it y s c a le s , /* an d (4 7 ) %E a s | (™e/ p»T( be/ p ») F or th e g iv e n e x it m a ss, m o m e n tu m , and b u o y a n c y flu x e s m om en tu m flu x '>/'1 {Je !p«) =>£ 6 . (mg/Poo) (^/Poo) 1 B0m BEth e n T h e re su ltin g /* to g e th e r w ith g iv e s th e M o r to n le n g th th en g iv e s th e virtu al a s is c o m m o n ly d o n e in p r a ctic e, jc is u se d to d e n o te th e d o w n str e a m d ista n c e m easu red fro m th e e x it o f th e fin ite so u rc e, rather th an th e d ista n c e fro m th e id ea l p o in t so u rce, then x in (2 2 ) fo r S (x ) an d in (4 3 ) fo r uc(x) m u st b e re p la ce d b y x + xE. 4. R e a c tin g B u o y a n t J e ts : B (x) d u e to C o m b u s tio n H e a t R e le a s e F rom th e ex a c t so lu tio n fo r n o n rea ctin g b u o y a n t j e t s fro m § 3 , w e n o w c o n sid e r th e c a se o f ex o th er m ic a lly rea ctin g b u o y a n t j e t fla m es. E v e n in th e a b se n c e o f b u o y a n c y e ffe c ts , th e red u ctio n in d e n s ity d u e to c o m b u stio n h ea t relea se alters th e in ertia in th e flo w , an d th ereb y a ffe c ts th e sc a lin g la w s. T a cin a & D a h m (2 0 0 0 ) h a v e sh o w n that th e p ie c e w is e lin ear v a ria tio n s o f tem p era tu re w ith m o le fra c tio n d em an d ed b y en th a lp y c o n se r v a tio n a llo w s th e d e n sity ch a n g es d u e to ex o th er m ic rea ctio n to b e rela ted to an eq u iv a len t n o n r ea ctin g flo w , in w h ic h th e d e n sity o f o n e o f th e flu id s is red u ced to an e ffe c tiv e v a lu e d eterm in ed b y th e p ea k tem p eratu re and o v e r a ll sto ic h io m e tr y . T h is le a d s to a gen era l eq u iv a le n c e p rin cip le b y w h ic h th e sc a lin g la w s for n o n r ea ctin g flo w s ca n b e e x te n d e d to p red ict e ffe c ts o f h ea t r e le a se b y ex o th e r m ic rea ctio n . In p articu lar, th e sca lin g la w s fo r n o n rea ctin g f lo w s m a y b e ex ten d ed to e x o th er m ic a lly reactin g flo w s i f th e a m b ie n t d e n sity is e v e r y w h e r e re p la ce d w ith 12 pef , w h e re T f = T0 + T‘_ J ° w h ere T0 and Tm th e w ith so u rce and am b ien t flu id tem p e ra tu r es, and , (4 9 ) Ts and Xs th e sto ic h io m e tr ic tem p eratu re and m o le fra ctio n s. T a cin a & D a h m ( 2 0 0 0 ) an d D ah m (2 0 0 3 ) h a v e s h o w n th a t th is general e q u iv a le n c e p rin cip le a cc u r a tely p red icts th e e f f e c t s o f h eat relea se o n th e sc a lin g la w s in b o th th e near- an d fa r -fie ld s o f p lan ar an d a x isy m m e tr ic tu rb u len t je ts , as w e ll a s th e h ea t r e le a se e ffe c ts in p lan ar tu rb u len t m ix in g la y e rs, in th e a b se n c e o f sig n ific a n t b u o y a n c y . H ere w e u s e th is e q u iv a le n c e to rela te th e n o n r ea ctin g b u o y a n t j e t resu lts in § 3 to e x o th e r m ic a lly reacting b u o y a n t j e t fla m e s. T h e flo w is aga in p ro d u ced b y a p o in t so u rc e th a t in tr o d u c e s b o th an in itia l m o m e n tu m flu x J0 and in itia l b u o y a n c y f lu x 5 0, as in th e p r e v io u s s e c t io n , b u t n o w th e h eat relea sed b y c o m b u stio n in th e f lo w lea d s to an in cr ea se in th e b u o y a n c y flu x w e u se th e in tegral e q u a tio n fo r th e b u o y a n c y flu x to e x p r e ss v e lo c ity 4 .1 uc(x), a n d B(x) th e n u se th is in (3 1 ) to o b ta in th e in te g r a l eq u ation for T h e b u o y a n c y f lu x B(x). In §4.1 in ter m s o f th e cen terlin e uc(x) B(x) U n d er th e c o n d itio n s o f m ix in g -lim ite d co m b u stio n an d en train m en t-lim ited m ix in g th a t a p p ly in m o st p ra ctica l c o m b u stio n situ a tio n s, th e lo cal rate o f h ea t release is se t b y th e lo c a l rate o f en train m ent dm(x)/dx in to th e flo w . T h is g iv e s th e c la s s ic a l in tegral eq u a tio n fo r B(x) a s (5 0 ) w h ere (5 1 ) and w h e re g is th e g ra v ita tio n a l a cc eler a tio n , q is th e m a s s -s p e c ific h ea t o f r e a c tio n o f th e fu e l, is th e sp e c ific h e a t o f th e sto ic h io m e tr ic fu el-a ir m ix tu re, and S u b stitu tin g ( 6 a ) fo r th e lo c a l m a ss flu x m(x) in d_ B(x) = dx cp is th e a m b ie n t tem p era tu re. (5 0 ), a n d u s in g 6 (x ) from (2 2 a ), g iv e s ( x 2 uc ) , /j c j n (5 2 ) w h ich in teg ra tes d ir e c tly to g iv e (5 3 ) w h ere Ys A s ab ove, 4 .2 xEd e n o te s h cl n • th e x -lo c a tio n o f th e e x it, w h e re (5 4 ) B(xE) = BEand uc(xE) ■ ucE. T h e c e n t e r lin e v e lo c it y s c a lin g ( uc /u*)= / 2 ( | ) T h e m o m e n tu m e q u a tio n in ( 3 1 ) m a y b e m u ltip lie d b y x2an d rearranged to g iv e (5 5 ) 13 N o w su b stitu tin g th e r e su lt in (53) fo r th e b u o y a n c y flu x an d rea rra n g in g fu rth er g iv e s ~^(x ucf = 3av x 3uc - x[3ay x\ uc - 3a (b e/ p f )] , (5 6 ) w h ere from (4 2 ) (5 7 ) In term s o f th e d im e n s io n le ss c e n te r lin e v e lo c ity and d o w n str e a m c o o r d in a te in ( 3 2 ) th is b e c o m e s : ( i / 2)3 = A | 3/ 2 - | [ M | / 2( ^ ) - 3a] (5 8 ) w h ere (Jo/P f ) 5" (59) , 3/2 K /p f) T h is m a y b e p u t in a m o re co m p a c t fo rm b y d e fin in g W i (6 0 a ) (6 0 6 ) in term s o f w h ic h (5 8 ) b e c o m e s dW - AAfcZ - l w - 4tfc ? (6 1 ) T h e re su lt in (6 1 ) is th e integral eq u a tio n fo r th e ce n te rlin e v e lo c it y reactin g a x isy m m etric tu rb u len t b u o y a n t je t. uc(x) in an ex o th er m ic a lly T h is eq u a tio n ca n b e r e a d ily so lv e d b y num erical in teg ra tion fro m in itia l v a lu e s an d wE, w h ic h are o b ta in ed fr o m th e e x it m a ss, m o m e n tu m , and b u o y a n c y flu x e s , mE, JEan d BEa s fo llo w s : 1) is first o b ta in ed d ir e c tly fro m (4 9 ), w h ic h ca n b e a p p lie d b e tw e e n th e id e a l p o in t so u rc e an d th e e x it o f th e a ctu al fin ite -a r ea so u rce, n a m e ly 0 < x < xE, b e c a u se n o h ea t re le a se o cc u r s b e fo r e th e flo w is s u e s fro m th e a ctu a l so u rce. 2 ) F ro m th e req u ired ^ ( | £) is o b ta in ed fro m (4 3 ), w h ic h a ls o a p p lie s b e tw e e n 0 th e sa m e rea so n ; th e c o n sta n ts 3 ) T h e in itia l v a lu e o f v ia (6 0 a ). w (cu)j and (cu) p are g iv e n in ( 86 ) an d (1 3 6 ). th at co r r e sp o n d s to th e e x it c o n d itio n s is o b ta in ed from 4 ) F ro m th e requ ired m o m e n tu m flu x d ir e c tly fro m (4 7 ). J0 o f 0 < x < x Eb e fo r e B0 = Be, sin c e th e f lo w is s u e s fro m th e actu al so u rce. 14 a n d _ ^ (|£) th e e q u iv a le n t id ea l p o in t so u rc e is o b ta in ed 5 ) T h e b u o y a n c y f l u x 5 0 o f th e eq u iv a len t id ea l p o in t so u r c e is g iv e n b y r e le a se o cc u r s b e tw e e n <x < xE fo r n o h ea t 6) a in (5 9 ) an d (6 0 6 ) is o b ta in ed from (5 7 ), w h ere ( c j p is g iv e n in (1 3 6 ). 7 ) y in (5 9 ) is o b ta in ed from ( 5 4 ), w h ere / , is g iv e n b y (5 a ), c 6 is g iv e n b y ( 2 2 6 ), an d I I is ev a lu a ted for th e g iv e n fu e l v ia (5 1 ). 8) A in (6 0 6 ) an d (6 1 ) is th en o b ta in ed from (5 9 ), w ith p‘f fro m (4 9 ). 9 ) T h e integral e q u a tio n in (6 0 ) is th en in tegrated fo rw a rd fro m th e in itia l c o n d itio n ( | £, w £), w ith th e in teg ra tio n co n tin u in g u n til th e fla m e tip is rea ch ed (s e e § 5 ). 10) B e y o n d th e ^ -lo c a tio n co r re sp o n d in g to th e fla m e tip , n o further h ea t r e le a se o c c u r s, and th u s p a st th is p o in t I I in (6 0 ) is se t to z e r o , w ith th e re su lt that th e b u o y a n c y flu x B(x) rem a in s co n sta n t fro m th ereo n . 11) T h e re su ltin g w ( | ) th en g iv e s th e ce n te rlin e v e lo c it y uc(x) v ia (5 9 a ), ( 3 2 a ,6 ) a n d ( 1 7 a ,6); th e lo c a l flo w w id th 5 (x ) is o b ta in ed tr iv ia lly fro m ( 22 a , 6 ). 12) T h e re su ltin g uc(x) and b(x) can th en b e u se d to o b ta in th e m a ss flu x m o m en tu m flu x J(x) v ia ( 66 ), and th e b u o y a n c y flu x re p la ce d b y p'f fro m (4 9 ). B(x) m(x) v ia ( 6 a ) , th e v ia ( 6 c ), w ith th e d e n s ity p„ F igu res 4 - 6 s h o w ex a m p le r e su lts o b ta in ed from th is in tegral m e th o d fo r th e e f f e c t o f v a rio u s so u rce p a ram eters o n th e c e n te r lin e v e lo c it y uc(x) in ex o th e r m ic rea ctin g b u o y a n t je ts . E a ch cu rve in th e se figu res s h o w s r e su lts from th e ex it lo c a tio n xE to th e flam e tip c o r r e sp o n d in g to the criterion in (6 9 ) b e lo w , w ith F ig . 4 sh o w in g th e e ffe c t o f th e so u rc e e x it v e lo c it y 5 sh o w in g th e e f f e c t o f Ae, and F ig . 6 sh o w in g th e e f f e c t o f th e fu e l ty p e . ea ch c a s e is th e n o n r ea ctin g resu lt in (4 3 ). UEin (2 a -c ), F ig . T h e d a sh e d cu rv e in N o te th a t in each c a s e th e cen terlin e v e lo c it y in itia lly f o llo w s th e n o n rea ctin g c u r v e , a n d th en as th e b u o y a n c y flu x B(x) in cr ea se s sig n ific a n tly b e y o n d its so u rce v a lu e Be d u e to c o m b u stio n h ea t relea se, th e s o lid c u rv e s d ep art fro m th e n on reactin g cu rv e. In m o s t c a s e s s h o w n , th e cen terlin e v e lo c ity re a ch es a m in im u m at so m e d o w n strea m lo c a tio n , b e y o n d w h ic h it in c r e a se s a s th e ad d ed b u o y a n c y d u e to h e a t r e le a se a ccelera tes th e flu id fa ste r th a n th e in crea se in 6 (x ) ca n sp rea d th e a d ded m o m e n tu m a c r o ss th e lateral d irectio n . F ig u res 7 -9 s h o w th e co r r e sp o n d in g v a ria tio n in th e m a ss flu x th e b u o y a n c y flu x 5. B(x) fro m ( 6a -c ) m(x), th e m o m e n tu m flu x J(x), and a lo n g th e le n g th o f th e fla m e. F la m e L e n g t h o f R e a c tin g B u o y a n t J e t s W h ile (6 1 ) p r o v id e s an integral m e th o d fo r th e e v o lu tio n o f ex o th er m ic a lly rea ctin g b u o y a n t je t fla m es a lo n g th e ir d o w n str e a m d ire ctio n , th e flam e len g th can b e o b ta in ed b y d e t e r m i n i n g th e d o w n strea m lo c a tio n ^ at w h ic h th e ce n te rlin e v a lu e o f th e so u r c e -flu id m ix tu re fra ctio n £ c(jt) has reach ed th e sto ic h io m e tr ic v a lu e t*. 15 5 .1 T h e f la m e tip c r it e r io n in t h e in t e g r a l m e t h o d S in c e th e m ix tu re fraction is a c o n se r v e d scalar, th e to tal sca la r flu x in teg ra l oo = J pu£,(x,r)2xrdr (6 1 ) o is an in va ria n t o f th e sca la r fie ld . In a m a n n er sim ila r to ( 6 a ) th is ca n b e w ritten a s /Mj.(x) « / 4 p f t,c(x)uc(x)b2(x ) , (6 2 ) w h e r e w e h a v e a d d itio n a lly m a d e u se o f th e fa c t th at *,r) w ith h(r\) a s g iv e n in (3b). a On) (6 3 ) » T h is g iv e s th e c e n te r lin e m ixtu re fra ctio n as 1 £ ,(* )an d d e fin in g th e m ixtu re fra ction I4 p f u c(x)b2(x) (6 4 ) ' at th e e x it o f th e actu al so u rc e a s rriy (6 5 ) g iv e s 1 ________________ (6 6 ) X>E h P e£ Uc(x)b2(x) S u b stitu tin g fro m (22a,b) an d (32a,b), w ith k (1 7 a , b), then g iv e s {mEl p f )(be I p‘£ T eff\M hcl (JE/P?) 1 (6 7 ) i / 2( i ) W e n o w d e fin e qp as th e a m b ie n t-to -e x it flu id m a ss ratio in a sto ic h io m e tr ic m ixtu re; for ex a m p le, fo r a so u r c e issu in g p ure CHi in to air, th e re su ltin g sto ic h io m e tr ic m a ss ra tio is tp = 1 7 .2 . In term s o f th e so u r c e -flu id m ix tu re fra ctio n , th e sto ic h io m e tr ic re q u irem en t is th en (6 8 ) l + <p ' w h e r e ^ is th e sto ic h io m e tr ic m ixtu re fra c tio n . ce n te rlin e m ix tu re fraction v a lu e £ c(x ) = k criterio n fo r th e fla m e len g th t,s. T h e fla m e tip lo c a tio n , x = L, o cc u r s w h e r e th e S u b stitu tin g in (6 7 ) an d rearranging th en g iv e s the L as 1 j 4 2 6 K (mElp t) { BE/p e£ ) 112 Cff\5/4 16 ( l + <p) . (6 9 ) T h e in tegra tio n o f ( 6 1 ) thu s c o n tin u e s u n til th e le ft s id e o f (6 9 ) rea ch es th e v a lu e o n th e right, at w h ic h p o in t th e fla m e tip lo c a tio n b LII* h a s b een reach ed . T h e v a lu e o f k n e e d e d to e v a lu a te th e right sid e o f (6 9 ) ca n b e ob ta in ed from th e je t-lim it fla m e len g th sc a lin g L !d + = 1 0 ( l+ q p ) (T acin a & D a h m 2 0 0 0 ) b y ta k in g th e lim it (7 0 ) ► 0 in ( 6 9 ), u sin g (3 7 ) a n d ( 3 9 ) w ith (8b), to g e th e r w ith (1 8 c ) an d (1 7 a ), w h ic h g iv e (7 1 ) w h ere (7 2 ) and d is d e fin e d in ( 11). T h is req u ires k ■Jk = 2 0 / 4 c 52 ( c „ ) . (7 3 ) ' and from th e v a lu e s a b o v e fo r th e v a rio u s co n sta n ts th is g iv e s k <= 0 .6 9 . 5 .2 T h e f la m e le n g t h s c a lin g p a r a m e t e r £2 F rom (6 9 ), to g e th e r w ith (6 1 ), (6 0 ), (5 9 ), ( 5 7 ), (5 4 ) an d (4 8 ), it is ap p aren t th a t th e fla m e length %Lin e x o th er m ic a lly rea ctin g b u o y a n t je t fla m e s is a fu n c tio n o f th e righ t sid e in ( 6 9 ), a s w e ll as th e h ea t r e le a se p a ra m eter A , n a m e ly K/pf)K/pfr (jE/p?r F rom (1 8 c ) to g e th e r w ith I* fro m (1 7 a ), d+ fro m (7 2 ) an d b e ex p re sse d in term s o f th e je t-lim it fla m e le n g th sc a lin g as L (£ (l + qp) = g ( l + cp); A <£ (7 4 ) fro m ( 1 1 ), th e le ft sid e in (7 4 ) ca n {q ; a } (7 5 a ) ( jjp ? Y (7 5 b) ( l + qp)' K /p f f K / p f )2 w h ere Q is th e p ro p e r p a ra m eter that d eter m in es th e r e la tiv e in flu e n c e o f b u o y a n c y o n th e fla m e len gth L. F ro m ( 7 5 a ) , w h e n th e ex it m a ss, m o m en tu m , an d b u o y a n c y flu x e s, mE, JEand BEare su ffic ie n tly large re la tiv e to th e sto ic h io m e tr ic m a ss ra tio qp fo r Q to b e in th e b u o y a n c y -fr e e lim it, th en fro m ( 7 0 ) th e fla m e le n g th m u st fo llo w t h e j e t - l i m i t sc a lin g G {Q ;A } = 1 0 . W h en mE, JE, BE an d (7 6 ) qp are su ch th a t Q is b e lo w th is lim it, th en b u o y a n c y w ill ca u se th e fla m e len gth to d ep a rt fro m th e je t-lim it scalin g in (7 6 ). 17 T h e v a lu e o f Q at w h ic h th e flam e len g th in (7 5 ) d ep a rts s ig n ific a n tly fr o m ( 7 6 ) th u s m a r k s th e o n s e t o f b u o y a n c y e f f e c t s in th e fla m e . 5 .3 C o m p a r is o n s w it h m e a s u r e d b u o y a n t j e t f la m e le n g t h s F lam e len gth d ata from th e literature (W o h l et al 1 9 4 9 , H a w th o r n e et al 1 9 4 9 , B e c k e r & L iang 1 9 7 8 , B e c k e r & Y a m a z a k i 1 9 7 8 ) are sh o w n b y s y m b o ls in F ig . 10, w h ic h is p r esen te d in th e fo rm o f (7 5 a,b) a s in d ica ted b y th e p r esen t in teg ra l m eth o d . It is a p p aren t th a t th e d ata c o v e r a w id e range o f th e b u o y a n c y p aram eter Q , an d th a t th e fo rm in (7 5 ) c o r re la tes th e d ata w e ll., p r o v id in g ad d ition al su p p o rt fo r th e p resen t in tegral a p p r o a ch T h e figure a lso s h o w s lin e s o b ta in e d fo r each fu e l ty p e b y so lv in g th e integral e q u a tio n in ( 6 1 ) to d eterm in e th e fla m e le n g th v ia th e criterion in (6 9 ). It is ap paren t that th e e f f e c t o f th e h e a t r e le a se p a ram eter A o n th e fla m e len g th is m u ch sm a ller th an th e e ffe c t o f th e b u o y a n c y p a ra m eter Q . M o r e o v e r , th e r e su lts fro m th e integral m e th o d in (6 1 ) agree w e ll w ith th e ex p erim en ta l fla m e len g th m ea su re m e n ts o v e r th e en tire ran ge o f Q , p ro v id in g further su p p o rt fo r th e p r e se n t in te g ra l m eth o d . T h e je t-lim it sc a lin g in (7 6 ) is s e e n in F ig . 10 to h o ld fo r Q > 1 0 6, an d fo r Q < 10 4 th e fla m e le n g th s are s e e n to sc a le w ith Q a s £2",/9. 6 . C o n c lu d in g R e m a rk s T h e p r e se n t stu d y h a s d e v e lo p e d a s u b sta n tia lly d iffer en t and im p r o v e d in tegral m e th o d fo r d eterm in in g th e fla m e len g th and c o m b u stio n p r o p e r tie s o f e x o th er m ic a lly rea ctin g b u o y a n t j e t fla m e s. T h e ap p ro ach a v o id s th e M o r to n e n tra in m e n t h y p o th e s is e n tire ly an d th e r e b y r e m o v e s th e ad hoc " en train m en t m od elin g " req u ired in m o s t o th e r integral a p p r o a c h e s. w o rk in g w ith an integral eq u a tio n fo r th e en tra in m en t rate flu x to d e v e lo p th e eq u a tio n fo r th e lo c a l c e n te r lin e v e lo c it y R a th er than dm/dx, w e h a v e u s e d th e m o m e n tu m u(x). T h is h a s th e a d v a n ta g e th a t th e m o d elin g ca n n o w b e d o n e in term s o f th e lo c a l f lo w w id th 6 (x ), fo r w h ic h d im e n sio n a l co n sid e r a tio n s s h o w that S (x) ~ x in b o th th e m o m e n tu m -d o m in a te d j e t lim it an d th e b u o y a n c y d o m in a ted p lu m e lim it. E xp erim en tal d a ta fu rth er s h o w that th e p r o p o r tio n a lity c o n sta n t c 6 is id en tica l in b o th th e je t and p lu m e lim its an d rem a in s co n sta n t b e tw e e n th e s e tw o lim its a s w e ll {e.g., P a p a n ic o la o u & L ist 1 9 8 8 ). T h u s, w h e r e a s th e M o r to n "en tra in m en t c o e f fic ie n t " a u s e d in m o s t in teg ra l m eth o d s v a ries b e tw e e n th e s e t w o lim its, and th erefo re m u st b e m o d e le d o n an hoc b a s is ad in term s o f a lo ca l F ro u d e n u m b er, c 6 in th e p r e se n t m e th o d is s h o w n to b e in varian t b e tw e e n th e s e lim its and th u s ca n b e r e p r e se n te d b y a co n sta n t v a lu e w ith o u t a n y fu rth er m o d e lin g . In e ffe c t, th e "en train m en t m o d e lin g " that is req u ired in tra d itio n a l in te g ra l a p p ro a ch es is r e p la c e d in th e p resen t m eth o d b y th e o b s e r v e d co n sta n t v a lu e o f c 6. F o r n o n rea ctin g b u o y a n t j e t s , th is n e w in teg ral a p p ro a ch g iv e s an ex a ct s o lu tio n fo r m c( x ) th a t p r o v id e s e x c e lle n t agreem ent w ith ex p er im en ta l d ata, a s w e ll a s an e x p r e ssio n fo r th e v irtu a l origin . In th e ex o th er m ic a lly reactin g b u o y a n t je t s , th e co n sta n t c 5 v a lu e in th e in tegral eq u a tio n fo r th e b u o y a n c y flu x g iv e s an e x p r e s s io n fo r B(x) in term s o f th e ce n te rlin e v e lo c it y , an d p r o v id e s a s im p le integral eq u a tio n fo r wc(x ) th a t ca n b e rea d ily s o lv e d fo r arbitrary flare ex it uc(x) d eterm in es th e lo c a l m a ss flu x B(x) th ro u g h o u t th e flo w , a s w e ll a s th e L. C o m p a riso n s w ith fla m e len g th d a ta c o n d itio n s v ia a sim p le sp rea d sh eet c a lc u la tio n . T h e r e su ltin g m(x), th e m o m e n tu m flu x J(x) and th e b u o y a n c y flu x ce n te rlin e m ix tu re fra ctio n £ c(x ) an d th e fla m e len g th s h o w e x c e lle n t agreem ent o v e r a w id e ra n g e o f fla m e c o n d itio n s, an d p r o v id e s th e p a ra m eter th a t 18 c h a r a c te r iz e s th e e x te n t t o w h ic h b u o y a n c y e f f e c t s a r e s ig n if ic a n t t h r o u g h o u t t h e fla m e . T h e p resen t in tegral m eth o d p r o v id e s a sim p le m ea n s fo r co m b u stio n e n g in e e r s to ca lcu la te su ch p ro p erties a s flam e len g th s, ra d ia tio n le v e ls , and e m is s io n s le v e ls , a s w e ll as th e c o n d itio n s that p ro d u ce o n se t o f so o tin g and m a n y o th e r co m b u stio n p r o p e r tie s, o f fla res. W h ile th e p resen t w o rk h as n o t a d d ressed th e im p o rta n t e ffe c ts o f a c r o ssw in d o n th e d o w n str e a m e v o lu tio n o f th e flo w and co m b u stio n p r o c e s s e s in th e flare, th e se ca n b e re a d ily in co r p o r a te d b y exten d in g th e m eth o d d escrib ed h ere. References B eck er, H .A . and L iang, D . ( 1 9 7 8 ) V isib le len g th o f v ertica l free tu rb u len t d iffu sio n fla m es. Combustion and Flame, V o l. 3 2 , p p . 1 1 5 -1 3 7 . B eck er, H . and Y a m a sa k i, S. (1 9 7 8 ) E n train m ent, m o m en tu m flu x an d tem p era tu re in v er tica l free turbu len t d iffu sio n fla m e s. Combustion and Flame, V o l. 3 3 , p p . 1 2 3 -1 4 9 . B la k e, T .R . and C o te , J .B . ( 1 9 9 9 ) M a s s en train m ent, m o m e n tu m flu x , and len g th o f b u o y a n t g a s d iffu sio n fla m e s. Combustion and Flame, V o l. 1 1 7 , p p . 5 8 9 -5 9 9 . B la k e, T .R . and M c D o n a ld , M . ( 1 9 9 5 ) S im ilitu d e and th e in te rp re ta tio n o f tu rb u len t d iffu sio n fla m e s. Combustion and Flame, V o l. 1 9 1 , pp. 1 7 5 -1 8 4 . C eteg en , B .M ., Z u k o sk i, E .E . an d K u b o ta , T . (1 9 8 4 ) E n train m en t in th e n ea r an d far fie ld o f fire p lu m es. Combustion Science and Technology, V o l. 3 9 , p p . 3 0 5 -3 3 1 . D a h m , W .J .A . ( 2 0 0 3 ) E ffe c ts o f h e a t re le a se o n tu rb u len t sh ea r flo w s . P art 2 . T u rb u len t m ixin g la y ers and th e e q u iv a le n c e p r in c ip le . S u b m itted to Journal of Fluid Mechanics. G ep h art, B ., Jaluria, Y ., M a h a ja n , R .L . and Sam m akia, B . (1 9 8 8 ) transport. Buoyancy-induced flows and H e m isp h er e P u b lis h in g C o r p ., N e w Y ork . H aw th o rn e, W .R ., W e d d e ll, D .S . an d H o tte l, H .C . (1 9 4 9 ) M ix in g a n d c o m b u stio n in tu rb u len t g a s je ts. Proceedings of the Combustion Institute, V o l. 3 , pp. 2 6 6 -2 8 8 . H e sk e sta d , G . (1 9 8 1 ) P e a k g a s v e lo c it ie s and fla m e h e ig h ts o f b u o y a n c y -c o n tr o lle d tu rb u len t d iffu sio n fla m e s. Proceedings of the Combustion Institute, V o l. 18, p p . 9 5 1 -9 6 0 . H u n t, G .R . an d K a y e , N .G . ( 2 0 0 1 ) V irtu a l origin co rrectio n fo r la z y tu rb u len t p lu m e s. of Fluid Mechanics, V o l. Journal 4 3 5 , pp. 3 7 7 -3 9 6 . K o ts o v in o s , N . E . & L ist, E . J. 1 9 7 7 P lan e tu rb u len t b u o y a n t j e t s . P art 1. Integral p ro p ertie s. Journal of Fluid Mechanics, V o l. 8 1 ,2 5 - 4 4 . L ist, E . J. 1 9 8 2 T u rb u len t j e t s an d p lu m e s . Annual Reviews of Fluid Mechanics, V o l. 19 1 4 , p p. 1 8 9 - 212. M o rton , B .R . (1 9 5 8 ) F o r c e d p lu m e s . Journal of Fluid Mechanics, V o l. 5 , p p . 1 5 1 -1 6 3 . M o r to n , B .R . an d M id d le to n , J. (1 9 7 3 ) S ca le d iagram s fo r fo rc ed p lu m e s . Mechanics, V o l. Journal of Fluid 5 8 , p p . 1 6 5 -1 7 6 . M o r to n , B . R ., T a y lo r , G . I. and T urner, J. S. 1 9 5 6 T u rb u len t g rav ita tio n a l c o n v e c tio n from m a in ta in ed and in sta n ta n e o u s so u r c e s. 1 -2 3 . Proceedings of the Royal Society of London, V o l. P a p a n ico la o u , P. N . & L ist, E . J. 1 9 8 8 In v e stig a tio n o f ro u n d tu rb u len t b u o y a n t je t s . Fluid Mechanics, V o l. Journal of 1 9 5 , p p . 3 4 1 -3 9 1 . P eters, N . an d G o ttg e n s, J. (1 9 9 1 ) S c a lin g o f b u o y a n t tu rb u len t j e t d iffu s io n fla m e s. and Flame, V o l. 23 4 , pp. Combustion 8 5 , p p . 2 0 6 -2 1 4 . S tew a rd , F .R . ( 1 9 7 0 ) P r e d ic tio n o f th e h eig h t o f tu rb u len t d iffu s io n b u o y a n t fla m e s. Science and Technology, V o l. Combustion 2 , pp. 2 0 3 -2 1 2 . T eix eira, M .A .C . a n d M ira n d a , P .M .A . ( 1 9 9 7 ) O n th e en tra in m en t a ssu m p tio n in S c h a tz m a n n 's in tegral p lu m e m o d e l. Applied Scientific Research, V o l. 5 7 , p p. 1 5 -4 2 . T acin a, K .M . an d D a h m , W .J .A . (2 0 0 0 ) E ffe c ts o f h e a t r e le a se o n tu rb u len t sh ea r flo w s . P art 1. A gen eral eq u iv a len ce p r in c ip le fo r n o n b u o y a n t f lo w s an d its a p p lic a tio n to tu rb u len t j e t fla m e s. Journal of Fluid Mechanics, V o l. 4 1 5 , pp. 23 - 4 4 . T urner, J. S . 1 9 8 6 T u rb u len t entrain m ent: th e d e v e lo p m e n t o f th e en tra in m en t a ssu m p tio n . Journal of Fluid Mechanics, V o l. 1 7 3 , p p . 4 3 1 -4 7 2 . W o h l, K ., G a z le y , C . ad K a p p , N . ( 1 9 4 9 ) D iffu s io n fla m e s. T h ird S y m p o siu m (In tern a tio n a l) on C o m b u stio n , W illia m s & W ilk in s, B a ltim o r e, p p . 2 8 8 -3 0 0 . Z u k o s k i, E .E ., K u b o ta , T . and C eteg en , B . ( 1 9 8 1 ) E n train m en t in fire p lu m e s . 3 , p p . 1 0 7 -1 2 1 . Journal, V o l. 20 Fire Safety (a) F ig u r e 1. at x m 0, A ctu a l fin ite -a r ea so u rc e. (b) E q u iv a len t id e a l p o in t so u rc e R ep re sen ta tio n o f a fin ite -a r e a b u o y a n t j e t so u r c e b y an e q u iv a le n t id e a l p o in t so u rc e lo c a te d a d ista n ce so u rce m o m en tu m flu x xEu p strea m JQand m0 s 0 , at xE. o f th e actu a l so u rc e w ith so u rc e m a ss flu x b u o y a n c y flu x B0, c h o se n to m a tch th e e x it v a lu e s w ith F ig u r e 2 . N o ta tio n in th e s e lf-s im ila r far fie ld o f a x isy m m e tr ic tu rb u len t b u o y a n t j e t s , s h o w in g th e lo c a l flo w w id th S(jc) and c e n te r lin e v e lo c ity uc(x) fro m th e v e lo c it y p r o file y (r |). F ig u r e 3 . V e r ific a tio n o f e x a c t so lu tio n fo r ce n te r lin e v e lo c it y for n on reactin g (B(x) = B0) c a se , com p a rin g th e uc(x) fro m p resen t in te g ra l m e th o d (line) w ith e x p e r i m e n ta l je t-lim it sc a lin g / 2( | ) - * (cu)j ■ r e su lt in ( 4 3 ) m ea su rem en ts o f P a p a n ic o la o u & L ist ( 1 9 8 8 ) (symbols). N o te a s 5 - > 0 and p lu m e -lim it sc a lin g / 2( | ) - * (cu) p ■ ^~l,3r a s | oo. Figure 4. Centerline velocity uc(x) from present integral method for exothermic reacting buoyant jets, showing effect of increasing exit velocity UEfor fixed exit area AEfor C3Hs issuing into air. Shown are results for five different UE values from jet exit to flame tip (solid lines) and nonreacting curve from Fig. 2 (dashed line). Exit values mE9JEand BEin (2a-c) differ for each UE value, leading to different initial conditions in (48). Velocity initially decays for all cases, but for strongly buoyant cases subsequently increases as buoyancy accelerates fluid. Figure 5. Centerline velocity « (x) from present integral method for exothermic reacting buoyant jets, showing effect of decreasing exit area AE for fixed exit velocity UE for CJfg issuing into air. Shown are results for five different AE values from jet exit to flame tip (solid lines) and nonreacting curve from Fig. 2 (dashed line). Exit values mE, JEand BEin (2a-c) differ for each AE value, leading to different initial conditions %Ein (48). Velocity initially decays for all cases, but for strongly buoyant cases subsequently increases as buoyancy accelerates fluid. Figure 6. Centerline velocity uc(x) from present integral method for exothermic reacting buoyant jets, showing effect o f fuel type for various fuels issuing into air with fixed exit area AEand exit velocity UE. Shown are results from jet exit to flame tip for eight different fuels (solid lines), together with nonreacting curve from Fig. 2 (dashed line). Exit values mE, JE and BE in (2a-c) differ for each curve due to differing exit densities pEin (2), leading to different initial conditions in (48). Velocity increases after initially decreasing as buoyancy accelerates fluid. Figure 7. Mass flux m(x) from present integral method via (6a) for exothermic reacting buoyant jets, showing effect of fuel type for various fuels issuing into air with fixed exit area^£ and exit velocity UE. Shown are results from jet exit to flame tip for eight different fuels (solid lines). Exit values mE, JEand Be in (2a-c) differ for each curve due to differing exit densities p£ in (2), leading to different initial conditions in (48). Mass flux m(x) increases linearly in x for jet-limit cases, in accordance with (9), and then transitions to jc5'3 in plume-limit as buoyancy accelerates the flow. £ . «£ q r Figure 8. Momentum flux J(x) from present integral method via (66) for exothermic reacting buoyant jets, showing effect of fuel type for various fuels issuing into air with fixed exit area AE and exit velocity UE. Shown are results from jet exit to flame tip for eight different fuels (solid lines). Exit values mE, JEand BEin (2a-c) differ for each curve due to differing exit densities p£ in (2), leading to different initial conditions %Ein (48). Momentum flux J(x) is independent o f x for jet-limit cases, and then transitions to xm in plume-limit as buoyancy accelerates the flow. Figure 9. Buoyancy flux B(x) from present integral method via (6c) for exothermic reacting buoyant jets, showing effect of fuel type for various fuels issuing into air with fixed exit area AE and exit velocity UE. Shown are results from jet exit to flame tip for eight different fuels (solid lines). Exit values mE, JEand BEin (2a-c) differ for each curve due to differing exit densities p£ in (2), leading to different initial conditions in (48). Momentum flux B(x) increases linearly with x for jet-limit cases, in accordance with (50) and the linearly increasing m(x) in jet-limit, and then transitions to x513 in plume-limit as buoyancy accelerates the flow. 1 x 1 tr 1x10 1x1 O'8 1x1 O'4 1 x1 0° 1 x 1 0 4 1 x 1 0 " 1 x 1 0 12 (Jj9 f ) 5 ( l + q p )' K / p f ) 4(5£/ p f ) 2 Figure 10. Flame length Z from experimental measurements (symbols) and from present integral method for exothermic reacting buoyant jets (lines), shown in the form o f (15a, b), where Q is the parameter characterizing the relative importance of buoyancy in the flame. Note that the effect of the heat release parameter II is relatively small. Buoyancy-free flames corresponding to the jet-limit scaling in (76) apply for Q > 10*. For Q < 104, flame length scales as Q l/9.