burning rate of liquid supplied through a wick

8/4/2019 Burning Rate of Liquid Supplied Through a Wick

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C O M B U S T I O N A N D F L A M E 8 6 : 4 7 - 6 1 ( 19 91 ) 4 7

B u r n i ng R a t e o f L i q u i d S u p p l ie d T h r o u g h a W i c kY . - X . T A O * a n d M . K A V I A N YDepartment o f M echanical Engineering and Applied Mechanics, U niversity of Michigan,Ann Arbor, M148109

Liquid fuel supplied through ceram ic or metallic wicks is com mon ly used as a simulant for bu rning of planar liquidor solid surfaces. Wh en the liquid flow through the w ick is at least partly controlled by capillarity, the partiallyexposed w ick matrix at the bu rning surface changes the concentration distribution adjacent to this surface. H ere weshow tha t when the characteristic length of the exposed matrix is not significantly smaller than the convectionboundary-layer thickness, the burning r ate from the w ick deviates (being smaller for low thermal conductivitywicks) from th at for p lanar liquid surfaces. The study is guided by experiments using a ceram ic wick, by aone-dimensional gas-side diffusion analysis, and by a wick-side two-phase flow and heat transfer analysis.

N O M E N C L A T U R Eb d e p e n d e n t v a r i a b l e i n E q . 2B m a s s t r a n s fe r n u m b e rB o B o n d n u m b e r , ( P l - P g ) g d 2 / ad p a r t ic l e d i a m e t e r ( m )D n~ b i n a r y m a s s d i t f u si v i ty ( m 2 / s )f s a t u r a t i o n f u n c t i o n i n t h e c a p i l l a r y p r e s -

s u r e e q u a t i o nh h e i g h t o f t h e l i q u id c o l u m n ( m )i e n t h a l p y ( J / k g ) c~i f g e n t h a lp y o f e v a p o r a t i o n ( J / k g ) /3g g r a v i t a t i o n a l c o n s t a n t ( m / s 2 ) 6k t h e r m a l c o n d u c t i v i ty ( W / m - K ) AK p e r m e a b i l i t y ( m E ) eK r ! l i q u i d r e l a t i v e p e r m e a b i l i t yK r g g a s r e l a ti v e p e r m e a b i l i ty XL b l o c k h e i g h t ( m )1 b l o c k l e n g t h ( m ) /z& " m a s s f lu x ( k g / m 2 - s )r h m a s s f l o w r a t e ( k g / m 2 ) /9n n o r m a l d i r e c t i o n trp p r e s su r e ( N / m 2 ) rc~" h e a t f l u x ( W / m 2 )Q h e a t o f r e a c t i o n ( J / k g )R g a s c o n s t a n t f o r g a s m i x t u r e ( J / k g - K )R e / R e y n o l d s n u m b e r , u = l / vs s a t u r a t i o n , e / / e *S (s - S im /(1 - S im) t

*Present address: Department of Mechanical Engineering,University of Saskatchewan, Sask atoon , Saskatchewan,Canada S7N 0W 0.Copy right 1991 by The C omb ustion InstitutePublished by Elsevier Science Publishing Co., Inc.

Tuuo ,xYY .z , Z

t e m p e r a t u r e ( K )v e l o c it y ( m / s )a m b i e n t , f r e e - s t r e a m v e l o c i t y ( m / s )h o r i z o n t a l c o o r d i n a t e a x i s ( m )m a s s f r a c t i o n o f t h e v a p o rm a s s f r a c t i o n o f th e n o n c o n d e n s i b l e g a sc o o r d i n a t e a x i s o p p o s i t e t o g r a v i t y ( m )

G reek S y m b o l st h e r m a l d i f fu s i v it y , k s t / ( p c ) t ( m 2 / s )s t o i c h i o m e t r i c f u e l - a i r r a t i ob o u n d a r y l a y e r t h i c k n e ss ( m )s u r f a c e r o u g h n e s s ( m )p o r o s i t yd e f i n e d i n E q . 1 2m e a n f r e e p a t h l e n g th o f g a s e o u s m o l e c u l e s(m )v i s c o s i t y ( P a - s )l e n g t h s c a l e v a r i a b l e ( m )d e n s it y ( k g / m 3 )s u r f a c e t e n s i o n ( N / m )t o r t u o s i t y

Superscr iptsa v e r a g ee v a l u a t e d a t t h e l i q u i d s u r f a c ee v a l u a t e d a t th e b o t t o m o f t he b o u n d a r yl a y e r

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1 e v a l u a t e d a t t h e s u r f a c e w i t h f l a m eoo ambien ta p p a p p a r e n tb bu rn ingc cap i l l a rye f t e f fec t i vef l iquid fi lmg g a s e o u s p h a s ei m i m m o b i l e o r i r r e d u c i b l e1 l i qu id phasen noncondens ib l er re l a t i ver e f r e f e r e n c es so l i d phas esa t sa tu ra t i onv v a p o r

I N T R O D U C T I O NIn add i t i on t o t he i r app l i ca t i ons i n hea t e rs , w icksa r e o f t e n u s e d i n e x p e r i m e n t s t h a t s i m u l a t e t h eb u r n i n g o f li q u id a n d s o l i d s u r f a c e s [ 1 - 3 ] . I nt h e s e a p p l i c a t i o n s a n d u n d e r s t e a d y - s t a t e c o n d i -t i ons , t he l i qu id i s supp l i ed t o t he bu rn ing su r -face . I t i s gene ra l l y a s sumed tha t a t h in l i qu idf i lm i s p re sen t on t h i s su r fa ce . I f cap i l l a r i t y i su sed fo r t he f l ow o f t he l i qu id , t hen a g rad i en t o fthe l oca l l i qu id sa tu ra t i on ex i s t s . Loca l l i qu idsa tu ra t i on i s de f ined a s t he f rac t i on o f t he l oca lv o i d v o l u m e i n t h e w i c k t h a t i s o c c u p i e d b y t h el i qu id . Grav i ty can a ss i s t o r oppose t he f l owd e p e n d i n g o n t h e s u r f a c e o r i e n t a t i o n . T h e p r e s -ence o f th i s l iqu id sa tu ra t i on g ra d i en t req u i re st h a t a t t h e b u r n i n g s u r f a c e t h e w i c k i s n o t c o m -p l e t e l y c o v e r e d b y a c o n t i n u o u s p l a n a r l a y e r o fl iq u i d. T h i s m a k e s f o r a n o n s m o o t h s u r f a c e w h e r et h e w i c k m a t r i x i s e x p o s e d a t v a r i o u s l o c a t i o n s .G e n e r a l l y , t h e s e s u r f a c e r o u g h n e s s e s d o n o t s u b -s t an t i a l l y i n f luence t he f l u id dynamics nea r t hes u r f a c e , e s p e c i a l ly i n t h e b u r n i n g o f l i q u id s ,whereas i n o the r d i f fu s ion f l ames , mass d i f fu s ioni s t h e d o m i n a n t m e c h a n i s m a d j a c e n t t o t h e b u r n -ing su r face . I t i s expec t ed t ha t t h i s d i f fu s ion a ta n d a d j a c e n t t o th e b u r n i n g s u r f a c e c a n b e n o t i c e -a b l y i n f lu e n c e d b y t h e s u r f a c e r o u g h n e s s r e s u l ti n gf r o m p a r t ia l s u r f a c e l i q u id s a t u r a ti o n . O f c o u r s e ,t he i n f luence i s no t i ceab l e on ly when the l eng ths c a l e a s s o c i a t e d w i t h t h e e x p o s e d d r y p a t c h e s i sm u c h l a r g e r t h a n th e m e a n f r e e p a t h o f th e e v a p o -r a t e d f u e l m o l e c u l e s [ 4 - 6 ] .

In a t rans i en t expe r imen t [7 ] , a cub i c , me ta l l i c

w ick was i n i t i a l l y fu l l y sa tu ra t ed w i th a l i qu idfue l , and t hen one ve r t i ca l s i de was exposed t oo the rwi se s t agnan t a i r and i gn i t ed . As t he fue lw a s c o n s u m e d o n t h i s s u r f a c e , t h e l i q u i d s a t u r a -t i on d i s t r i bu t ion changed in t he w ick and g radu -a l l y t h e s u r f a c e s a t u r a t i o n w a s l o w e r e d t o at h r e s h o l d v a l u e b e y o n d w h i c h t h e f l a m e q u en c h e d .A l t h o u g h t h e t im e - a v e r a g e d b u r n in g r a t e w a snea r ly t ha t p red i c t ed by [8 ] fo r a fu l l y l i qu idc o v e r e d p l a n a r s u r f a c e , t h e b u r n i n g r a t e d e -c r e a s e d m o n o t o n i c a l l y ( e x c e p t f o r a s h o r t p e r i o dw h e n t h e w i c k w a s w a r m i n g u p a n d d u r in g w h i cht h e b u r n i n g r a t e i n c r e a s e d w i t h t i m e ) . W h e t h e rt h is m o n o t o n i c d e c r e a s e o f b u r n i n g r a t e w a s a s s o -c i a t ed w i th a dec rease i n t he su r face l i qu id sa tu ra -t i o n w a s n o t s u b s t a n t i a t e d , h o w e v e r , b e c a u s e o ft h e l i m i t ed a c c u r a c y o f th e e x p e r i m e n t . I t s h o u l da l s o b e m e n t i o n e d t h a t t h r o u g h o u t t h e e x p e r i -m e n t , t h e s u r f a c e t e m p e r a t u r e w a s t h e s a t u r a t i o nt empera tu re o f t he l i qu id fue l and t ha t t he f l amew a s s t a b l e u p t o s h o r t l y b e f o r e q u e n c h i n g . T h i sind i ca t e s t ha t a s t he su r face l i qu id sa tu ra t i on de -c reas es , t he l i qu id d i s t r i bu t ion ove r t he su r face(wh ich i s pa r t i a l l y cove red w i th l i qu id ) changes ,l ead ing t o a dec rease i n t he bu rn ing ra t e . In -ve rs e ly s t a t ed , bu rn ing i s s tab l e even i f the l i qu idf low ra t e i s be low tha t co r re spond ing t o a f l ooded(th in l iquid f i lm) surface .

I n th e f o l l o w i n g , w e e x a m i n e b u r n i n g o f th efue l supp l i ed t h rough a w ick a t f l ow ra t e s f romf lood ing t o l ower bu t s t ab l e ra t e s where t he su r -face w i l l be pa r t i a l l y sa tu ra t ed . We repo r t t her e s u lt s o f e x p e r i m e n t s p e r f o r m e d w h e r e t h e f u e ls u p p l y t h r o u g h t h e w i c k i s c o n t r o l l e d . T h e n , w eana lyze t he mass d i f fu s ion ad j acen t t o t h i s su r faceu s i n g a s i m p l e m o d e l t o s h o w t h a t t h e s u r f a c el i qu id sa tu ra t i on , i . e . , t he d i s t r ibu t ion o f t he l i q -u i d i n t h e s u r f a c e p o r e s , d e v e l o p s i n a m a n n e rwh ich l eads t o sma l l e r bu rn ing ra t e s a s t he su r -f a c e s a t u r a t i o n d e c r e a s e s . T o d o t h i s , a n e s t i m a t eo f t he sa tu ra t i on d i s t r i bu t ion i n the w ick , i nc lud -ing t he su r face sa tu ra t i on , i s needed . Th i s i sp r o v i d e d b y a n a l y s i s o f t w o - p h a s e f l o w i n t h ew i c k . W e n o t e t h a t t h e s t u d y d o e s n o t a i m t op red i c t t he bu rn ing ra t e o f li qu id supp l i ed t h roug hw i c k s , b u t u s e s e x p e r i m e n t a l l y m e a s u r e d b u r n i n gra t e s t o po in t ou t t ha t s t ab l e bu rn ing can re su l t ,a l t h o u g h a t l o w e r r a t e s , w h e n t h e s u r f a c e i s n o tf l ooded . Th i s i s e spec i a l l y impor t an t i n t he des igno f e x p e r i m e n t s u s i n g l iq u id s u p p l ie d t h r o u g hwicks a s a s imu lan t fo r l i qu id o r so l i d su r faces .


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(d )F ig . 2 . (a) The measured abso lu te perm eab i l i ty us ing l iqu idethanol, (b) using air , (c) the me asured porosity, and (d) themeasured ambient air velocity.

v o i r w a s c o o l e d a n d k e p t a t t h e r o o m t e m p e r a t u r eTO T h i s i s n e c e s s a r y i n o r d e r t o a v o i d e v a p o r a -t i o n i n s id e t h e re se rv o i r .M A S S D I F F U SI O N A D J A C E N T T OP A R T I A L L Y L I Q U I D S A T U R A T E DS U R F A C E SB e f o r e c o n s i d e r i n g t h e b u r n i n g o f l iq u i d f u e l o np a r t i a l l y c o v e r e d w i c k s u r f a c e s , l e t u s c o n s i d e r as imi l a r p ro cess a s i t o ccu rs i n t h e we l l - s t u d i edc o n v e c t i v e d r y i n g o f p o r o u s m e d i a . I t h a s b e e nes t ab l i sh ed t h a t wh en t h e ch a rac t e r i s t i c l en g th o ft h e l i q u id d i sco n t i n u i t y o n t h e su r faces i s n o t v e rym u c h l a r g e r t h a n t h e m e a n f r e e p a t h o f t h e g a s e o u smo lecu l e s , X, t h en t h e d ry p a t ch es a re p rac t i ca l l yc o v e r e d b y t h e m o l e c u l e s d r i f t i n g f r o m t h e w e tp a t ch es . Th i s ch a rac t e r i s t i c l en g th i s t h e s i ze o ft h e d r y o r w e t p a t c h e s o r t h e d e p t h o f i n d e n t a -t i o n s (v a l l ey s ) z~ m easu red f ro m th e t o p o f t h eso l i d t o t h e l i q u id l ev e l . Th e re i s an ad d i t i o n a lreq u i remen t t h a t i s g en e ra l l y sa t i s f i ed i n d ry in gp r o c e s s e s . T h i s r e q u i r e m e n t i s t h a t t h e c o n c e n t r a -t io n b o u n d a r y - l a y e r t h i c k n e ss , 6 , m u s t b e m u c hl a r g e r t h a n A , i . e . , D / 6 ~ 0 . T h i s r e s u lt s i n n och an g e i n t h e ev ap o ra t i o n ra t e (o r d ry in g ra t e ) a st h e s u r f a c e s a t u r a t i o n d e c r e a s e s . T h e s e t r e n d sc o n t i n u e u n t i l t h e s u r f a c e b e c o m e s c o m p l e t e l yd r y . T h e n , d r y i n g p r o c e e d s a s a n i n t e r f a c e s e p a -r a t i n g t h e d r y a n d w e t r e g i o n s m o v i n g i n t o t h ep o r o u s m e d i a . T h e r e f o r e , w h e n t h e s u r f a c e l i q u i ds a t u r a t i o n s j i s n o t z e r o , A / X i s n o t v e r y l a r g ea n d 4 / 6 ~ O , t h en t h e ev ap o ra t i o n ra t e i s aco n s t an t ca l l ed t h e co n s t an t d ry in g ra t e .

T h e b o u n d s f o r m a x i m u m v a l u e s o f A / X a n dA / 6 fo r wh ich t h e ev ap o ra t i o n ra t e i s n ea r lyc o n s t a n t h a v e b e e n s t u d i e d i n R e f . 6 . T h e m o d e li n Re f . 6 i s b ased o n l i q u id p a t ch es (h av in g ah e m i s p h e r i c a l s h a p e ) p l a c e d o n a p l a n a r s u r f a c e .H o w e v e r , t h e li n e o f r e a s o n i n g u s e d b e l o w i si n d ep en d en t o f wh ich p h a se ( l iq u id o r so l id ) p ro -t ru d es . Th e ap p ro ach i n Re f . 6 u ses t h e ad d i t i o no f t h e K n u d s e n ( m o l e c u l a r ) d i f fu s i on a n d t h eFi ck i an d i f fu s io n ( i n se r i e s ) o v e r A an d t h e F i ck -i an d if f u s io n o v e r 6 - A . T h e m e a n f r e e p a t h[11] is

1 . 0 5 l k B Tk -21 / 2 7r dm 2p '

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1 0 - io m , T a n d p a r e s a tu r a t i o n t e m p e r a t u r e a n dp r e s s u r e , a n d k B i s th e B o l t z m a n n c o n s t a n t .W h i l e f o r l o w t e m p e r a t u r e ( s m a l l v a p o r p r e s s u r ed r y i n g ) , ~ is r e la t i v e l y l a r g e , f o r b u r n i n g w h e r et h e s u r f a c e i s a t t h e s a t u r a t i o n t e m p e r a t u r e a n dp res su re , ~ is r e l a t i v e ly sm a l l (X - - 7 x 1 0 -8 mf o r e th a n o l a t s a tu r a t i o n c o n d i ti o n s ) . T h e m o d e lu s e d i n R e f . 6 sh o w s th a t f o r s m a l l A / 5 ( A / 5 1 ,t h e b o u n d a r y - l a y e r t h i c k n e s s is m e a s u r e d f r o mZ = 0 ( t h i s req u i re s t h a t w e co n f in e t h e e f fec t o ft h e s u r f a c e r o u g h n e s s t o t h e t h e r m a l a n d c o n c e n -t r a t i o n b o u n d a r y l a y e r s ) . T h e s u r f a c e s a t u r a t i o n

o f t h e l i q u id i s d e f i n ed a s

S = v o l u m e o f l iq u i d i n t h e v o i dt o ta l v o l u m e o f th e v o i dF r o m t h e g e o m e t r y i n F i g . 3 , w e h a v e t h e f o l l o w -in g

s l = l - - - + 11 - 7 r /6 2 - "( 1 )

T h e p r o b l e m c a n b e d i v i d e d i n t o t w o p a r t s : ( a )c o n v e c t i o n a n d d i f f u s i o n t h r o u g h t h e b o u n d a r yl ay e r , an d (b ) d i f fu s io n o n ly i n t h e l ay e r mad e o ft h e e x p o s e d p a r t i c l e s ( h e r e c a l l e d t h e A d o m a i n ) .S i n c e 6 / A ~ , 1 , c o n v e c t i o n w i t h i n t h e A - d o m a i nis neg l ig ib le . A s za ~ 0 ( i .e . , s I = 1) , the pro b-l e m r e d u c e s t o ( a ) o n l y .

S t a g n a n t F i l m P r o b l e mT h e o n e - d i m e n s i o n a l t r a n s p o r t e q u a t i o n f o r as t ag n an t f i lm h as t h e fo rm [1 2 ]

d b d 2 bp U ~ - z = P D - d z 2 fo r 0 < Z _< 6, (2)w h ere b i s d e f i n ed i n Re f . 1 2 . Th e b o u n d a ryco n d i t i o n s a re

p U ( dd - -~ b z ) : . P s a t U s a t = p u ,

b ( z = 0 ) = a o ,b ( Z = 6 ) = b e . (3 )T h e m a s s t r a n s f e r f l u x c a n b e o b t a i n e d f r o m t h es o l u t io n o f th e a b o v e e q u a t i o n a n d i s

p D#/ ,, = /0sa tUsa t = -- ~- ln (1 + B ), (4 )w h e r e B i s t he M a s s T r a n s f e r N u m b e r . F o rc o m p l e t e c o m b u s t io n ,

~ Y o , o * " b ) ' s a tn - c p ( r ~ - ~ . , ) + ~ y o , ~ Q .1 - Y s a t i f g( 5 )

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s t r ea m . I t s hou l d be m e n t i one d t ha t B c a n bem od i f i e d t o i nc l ude t he he a t t r a n s f e r r e d t h r ought he l i qu i d a nd s o l i d by s i m p l y a dd i ng a n a dd i -t iona l t e rm Q c to i f g [9] . Our ca lcu la t ion , basedon t he e xpe r i m e n t a l da t a , i nd i c a t e s t ha t a dd i ngt h i s t e r m ha s a ne g l i g i b l e i n f l ue nc e on t he p r e -d ic ted mass t r ans fe r r a te , a s i s d i scussed la te r .

T he t o t a l m a s s t r a ns f e r r a t e a l ong t he w i c klength / isFn = N d l P ? l n ( 1 + B ) , (6 )6w he r e N i s t he num b e r o f c e ll s a l ong t he t h i r dd i m e ns i on pe r pe n d i c u l a r to the x a x i s a nd

1 l f o r d xg ; 6 ( x )

F o r A = 0 ( s 1 = 1 , i . e . , f u l l y l i qu i d - c ove r e ds u r f a c e ) , w e ha ver 'n f = N d l P D l n ( 1 + B * ) . (7 )

I f w e u s e t he s a m e a pp r ox i m a t i on u s e d i n R e f .12 t o f it t he E m m o ns p r e d i c t i on [ 9 ] , i . e . , w ed i v i de t he l e f t si de o f t he a bov e e qua t i on byB 1 5 / 2 , w e h av e

N d l P D _ l n ( 1 + B *)O t f = 6 B * ' ' 5 / 2 ( 8 )F u r t he r m or e , by e x t e nd i ng t h i s t r e a t m e n t t o c a s e swhere the sur face sa tura t ion i s l e s s than uni ty , theno r m a l i z e d bu r n i ng r a t e f o r a ny s u r f a c e r ough -ne s s ( 0 < A < d ) c a n be e xp r e s s e d a s

t// _ ln( 1 + B ' ) ( B * ] 0 ' 1 5r n f l n (1 + B * ) , B- T ] ' (9 )

w h e r e B ' a n d B * a r e d e f in e d in th e s a m e w a y a sby E q. 5 wi th T~a and Ysa t be ing rep lac ed by T '( o r T * ) a n d y ' ( o r y * ) r e s p e c t i v e ly , a n d r n y ist he E m m ons p r e d i c t i on f o r t h i n l i qu i d f i l m s .D i f f u s i o n i n a V a r i a b l e - A r e a P a s s a g e( z ~ D o m a i n )When the sur face l iqu id sa tura t ion i s l e s s thanun i t y , t he s o l i d m a t r i x be c om e s e xpos e d a nd t hee va po r a t e d va po r ha s t o d i f f u s e f r om t he l i qu i ds u r f a c e t o t he bounda r y l a ye r . T he n , t he s t a t e o ft he gas m i x t u r e a t l oc a t i on Z = 0 ( r e p r e s e n t e d byt he va po r p r ope r t i e s , P 'v , T ' , a nd y ' ) i s d i f f e re n t

t ha n at t he l iqu i d s u r fa c e ( p* , T * , y* , e t c . ) dueto the addi t iona l d i f fus ion length A . F or d i f fus iononly , the mass t r ans fe r r a te pe r un i t ce l l ( s ee F ig .3 ) c a n be e xp r e s s e d a s

dyr 'n ' = - A tpD --d - = cons tan t

f o r 0 < _ ~ < A , ( 10 )w h e r e ~ = Z + A ( ~ va r i e s f r om 0 t o A w he n Zva r i e s f r om - - A t o 0 ), a nd A t is the l iquidsur face a rea and i s a func t ion o f A (or s 1 in Eq.1) and i s g iven by

A t = 7rd 2 + C 1 q- C 2 , ( 1 1 )

c , = 1 - 2--7; c 2 - + .rr dT he bounda r y c ond i t i ons a r e

= 0 ) = y * ,= y ' .

I n t e g r a t ion o f E q . 10 y i e l d s* y ,

t h ' = p D d y -, ~ 'w h e r e

2 / r [= ~ t a n - 1j41( 12 )

1 - 2 a / d- t a n - ' d ,

T he t o t a l m a s s t r a ns f e r r a t e a l ong t he s u r f a c e o flength l ( in the f low d i rec t ion) i s

l y * - y 'm = N - ~ O D d ~ , ( 13 )W he n d i v i de d by E q . 8 , t he a bove e qua t i on be -c o m e s

r n y * - y ' B * " 5 / 2--7- = . (14 )m f ~ b d / 6 l n ( 1 + B *)


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5 4 Y . - X . T A O A N D M . K A V I A N Y

E qua t i on 14 g i ve s a r e l a t i on be t w e e n t he no r -ma l ized mass t r ans fe r r a te and the sur face sa tura -t i on ( ~ i s a func t i on o f s j t h r ough E qs . 1 a nd 12)f o r t he g i ve n t h in - l iqu i d - f il m p r ope r t i e s B * a ndy * , a n d 6 / d . H ow e ve r , w e s t i l l h a ve a n un -know n , t he m a s s f r a c t i on o f t he va po r a t Z =0 , y ' . W e c a n u s e t he s o l u t i on ob t a i ne d f r om t hes tagnant - f i lm ana lys i s (Eq. 9) to obta in y ' , s inceE q . 9 c on t a i n s a n unknow n B ' w h i c h i s a l s o af unc t i on o f y ' ( E q . 5 ) . T h i s i s done by u s i ng am a t c h i ng c ond i t i on ba s e d on m a s s c ons e r va t i on .S o l u t i o n f o r m/ l~ i f a s a F u n c t i o n o f s tT he m a t c h i ng c ond i t i on f o r t he s o l u ti ons f r om t het w o dom a i ns i s/ ~ /A d o m a i n = F ~ / b . I . ( 1 5 )

Fro m Eqs . 9 , 14 , and 15 , we havern l n ( l + B ' ) ( B * ) ' '- ' S ' - - =m y I n( 1 + B * ) - ~ -

y * - y , B * t s / 2' ~ d / 6 l n ( 1 + B *) ' ( 16 )

T h i s g i ve s t he r e l a t i on be t w e e n t he e va po r a t i onra te and the sa tura t ion wi th the r a t io 6 / d as apa r a m e t e r , i . e . ,r 'n /r h f= f ( s l , 6 / a ) .I n o r de r t o ob t a i n t he a bove r e l a t i on , w e r e w r i t eEq. 16 a s

y* (1 - r /) B ' ' ' 5In(1 + B ') - - - ,~ b a /6 2w he r e ~ = y ' / y * . Inse r t ing Eq. 5 in to the abovee qua t i on y i e l d s

r t = 1 - 2 ~ 7 g 3 L 2 + y r l

I n . . . . . ( 1 7 )1 - y * 7 /A t a g i ve n g / d , f o r a n i nde pe nde n t va r i a b l e s 1,A / d c a n be f ound f r om E q . 1 . N o t i ng t ha tP~ P~y * = - , ( 1 8 )p P

w he r e p * i s t he va po r p r e s s u r e a nd obe ysC l a p e y r o n e q u a t i o n , T * a n d y * c a n t h e n b esolved for by i t e ra t ion on Eqs . 18 and 5 , and ~ /c a n be ob t a i ne d by i t e r a t i ng on E q . 17 . T he r e -f o r e , a d ep e n d e n t v a r ia b l e r n / m f i s d e t e rm i n e df rom Eq. 16 us ing ~7 and y* .I n t he a bove , w e p r e s e n t e d a n a na ly s i s o f c on -vec t ion and mass d i f fus ion in the s tagnant f i lm( c ons i de r i ng t he a ve r a ge bounda r y - l a ye r t h i c k -ness ) in se r ie s w i th mass -d i f fus ion in a sur facei nde n t a t i on ( A dom a i n ) . M a s s c ons e r va t i on i sa pp l i e d t o t he t w o s o l u t i ons , w h i c h y i e l d s a ge n -e r a l r e l a ti on be t w e e n t he no r m a l i z e d bu r n i ng r a t eand the sur face sa tura t ion in an impl ic i t form,Eq. 16 . This equa t ion can be used to qua l i t a t ive lypredic t the e f fec t of the su r face sa tura t ion o f thel i qu i d on t he bu r n i ng r a t e f o r w i c ks w i t h l owt he r m a l c onduc t i v i ti e s .S U R F A C E L I Q U I D S A T U R A T I O NI n o r d e r t o d e t e r m i ne t he s u r f a c e l iqu i d s a t u r a ti onrequ i red fo r the ana lys i s of the mass d i f fus ionadjacent to the sur face , the f low of l iqu id e thanoli n t he w i c k c a us e d by c a p i l l a r y a nd hyd r os t a t i cf o r c e s i s a na l yz e d . T he m om e n t um e qua t i ons ( f o rt he l i qu id a nd va po r pha s e s) a r e c oup l e d w i t h t hee ne r gy e qua t i on be c a us e t he va po r p r e s s u r e d i s -t r i bu t i on de pe nds on t he t e m pe r a t u r e d i s t r i bu t i on( t h ro u g h C l a p e y r o n - K e l v i n e q u a t io n f o r t h e t h e r -m odyna m i c e qu i l i b r i um s t a t e s ) . F i gu r e 4 g i ve st he bounda r y c ond i t i ons u s e d . T he l i qu i d s a t u r a -t i on a t t he low e r s u r fa c e o f t he w i c k ( s ( z = 0 ) =s o) i s no t known. This fo l lows , un le s s r e la t ive lyhigh f low ra te s or h igh tempera ture s (but l e s sthan T s a r ) a r e us e d , be c a us e t he nonw e t t ing pha s ee n t r a ppe d i n t he po r e s a d j a c e n t t o t h i s l ow e rs u r f a c e c a nno t be r e m ove d . T he t r a ns i t i on f r om ap l a in m e d i um ( t he r e s e r vo i r ) t o a po r ous m e d i um ,and th i s d i scont inui ty in the sa tura t ion , have beena dd r e s s e d by L e ve r e t t [ 13 ] . A l t hough w e d i d no tm a ke l oc a l l i qu i d s a t u r a t i on m e a s u r e m e n t s , ou ra na l y s i s o f f l ow t h r ough t he w i c k a l s o s ugge s t stha t 0 .8 < s o < 1 , i . e . , a jum p in the sa tura t ionoc c u r s i n t r a ns i t i on f r om t he r e s e r vo i r t o t hew i c k .

T he vo l um e - a ve r a ge d c ons e r va t i on e qua t i onsf o r t he r m a l e ne r gy , l i ne a r m o m e n t um , a nd m a s sf o r one - d i m e ns i ona l , s t e a dy - s ta t e , t w o- pha s e f l owi n po r ous m e d i a a r e f ound by t he p r ope r r e duc -t i ons o f t he e qua t i ons g i ve n i n [ 14 ] . T he e ne r gy


9/15

B U R N I N G R A T E O F L I Q U I D 5 5

U

Flame z = l : T : T I 1 , ~ 1

3 y _ a T a s = 0a n a n a n

K , ~ . ~

= ~,+ , y1=o

a Ta n

\ oz=O: T=To , s=so , Y=Yo

I3S = 0a n

Fig. 4 . The bound ary condi t ions used insolving for the su rface l iquid sa tura t ion .

a n d c o n t i n u i ty e q u a t i o n s a r ed [ d T )~ Z [ ( p u i ) t + ( p u i ) o I = - ~ z [ k e f f - - ~ Z , ( 1 9 )

d- -~ Z ( p , u t + O v u m ) = O . ( 2 0 )N o w l e t t i n gl ;r t" = p Ua n d i n t e g r a t i n g E q s . 1 9 a n d 2 0 o n c e , t h e f o l l o w -i n g e m e r g e s : T h e e n e r g y e q u a ti o n is

d Tr h ? i , + r h ~ i v = keff - -~- - q - q " . ( 2 1 ) T h e m a s s c o n s e r v a t i o n e q u a t i o n is

" " " " " ( 2 2 )! + m v = m ,w h e r e q " a n d r n " a r e th e u n k n o w n h e a t a n dm a s s f l u x e s ( i n t e g r a t i o n c o n s t a n t s ) t h a t w i l l b ed e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n L

T h e li q u id m o m e n t u m e q u a t i o n isK ( d P e ~ P ' g - - - - ~L - h )m ' t ' = - P t - ~ t K + , ( 2 3 )

T h e g a s m o m e n t u m e q u a t io n isk r g d p d p gth '~ + rh ~ = - P g K - - , ( 24 )I ~ g d z

w h e r e p g i s t h e t o ta l p r e s s u r e o f th e g a s m i x -t u r e .I n E q s . 2 3 a n d 2 4 w e h a v e m o d i f i e d th e p e r m e -

a b i l i ty u s e d b y W h i t a k e r [ 1 4] i n o r d e r t o r e -c o v e r t h e D a r c y l a w f o r s i n g le p h a s e f lo w s T h i s i s a l s o n o te d b y W h i t a k e r [ 1 5] .

T h e g a s d i f f u s i o n e q u a t i o n i sdYn(en'; + y . = a++D o . a z ( 2 5 )

I n o r d e r t o s o l v e t h e g o v e r n i n g e q u a t i o n s , t h ec o n s t i t u t i v e e q u a t i o n s f o u n d i n R e f . 1 6 a r e u s e d .T h e s e a r e a s f o l l o w s : C l a p e y r o n - K e l v i n e q u a t i o n ( i n c lu d e s t h e e f f e c t

o f t h e m e n i s c u s c u r v a t u r e o n th e t h e r m o d y -n a m i c e q u i l i b r iu m s t at e ),

p g ~ - -Psat e x p [ ( - -1 - - y ~, _ P , ] .

P t R T ]

1 1 i i f e"T s a T ] R

( 2 6 )

T h e e q u a t i o n o f st a t e ( i d e a l g a s b e h a v i o r ) ,P e = p g R T . ( 2 7 )

T h e c a p i l l a r y p r e s s u r e ( a s a f u n c t i o n o f l i q u ids a t u r a t i o n ) i s g i v e n b y

(3"p g - p t = p e ( K / e ) , / 2 [ 1 . 4 2 ( 1 - S )

- 2 . 1 2 ( 1 - S ) 2 + 1 . 2 6 ( 1 - S ) 3]a- f ( K / e ) l / 2 , ( 2 8 )


10/15


w h e r eS -- Sim

S _ _ _ _ ,1 - Sire

a n d t h e i m m o b i l e s a t u r a t i o n Sim w a s e s t i m a t e da s 0 . 0 5 .

T h e e f f e c ti v e t h e r m a l c o n d u c t i v i t y i s g i v e n b yk e r r = k s , + s ' / 2 ( k s , + k s g ) . ( 2 9 )

T h e e f f e c ti v e m a s s d i f f u s i v i t y is t a k e n a se ( 1 - s ) D n oD e . - ( 3 0 )

T h e r e l a t iv e p e r m e a b i l i ty o f t h e p o r o u s m e d i u mi s d e t e r m i n e d f r o m t h e f o l l o w i n g a p p r o x i m a t i o n s :g r ! = S 3 , g r g = ( 1 - - S ) 3 . ( 31 )

T h e g r a d i e n t o f th e c a p i l la r y p r e s s u r e c a n b ef o u n d b y t a k i n g t h e t o t al d e r i v a t i v e o f E q . 2 8 ,i . e . ,

d f d s d o d T )az '/2

( 3 2 )I t s h o u l d b e n o t e d t h a t t h e s u r f a c e t e n s i o n g r a d i -e n t a l s o i n f l u e n c e s t h e l i q u i d f l o w b y i n f l u e n c i n gt h e s h e a r s tr e s s a t t h e l i q u i d - g a s i n t e r f a c e . T h i si n f l u e n c e i s n o t y e t s a t i s fa c t o r i l y t r e a t e d [ 1 7 ] a n di s n o t i n c l u d e d h e r e .

T h e b o u n d a r y c o n d i t i o n s a r es + s o , T = T o , a n d Y n = Y n . o , a tz = 0 , ( 3 3 )P g = Psa t , T = T , , , an d t / / " = m b" a tz = L , ( 3 4 )w h e r e P sa t i s t h e p r e s s u r e i n t h e w i n d t u n n e l( a s s u m e d t o b e t h e a t m o s p h e r i c p r e s s u r e ) . A ss h o w n i n R e f . 1 8 , f o u r o f t h e a b o v e b o u n d a r yc o n d i t i o n s a r e u s e d t o s o l v e t h e i n i t i a l v a l u ep r o b l e m s f o r T , s , y , , , a n d p g . T h e l a s t t w o a r eu s e d to d e te r m i n e 0 " a n d m " . T h e v a l u e o f s ow a s s e l e c t e d s u c h t h a t t h e c a l c u l a t e d t e m p e r a t u r ed i s tr ib u t io n w o u l d m a t c h t h e m e a s u r e d t e m p e r a -t u r e. T h i s w a s d o n e t h r o u g h i t e ra t io n , a n d , i ng e n e r a l , w e f o u n d s o t o b e a p p r o x i m a t e l y 0 . 8 3 .

F u r t h e r d i s c u s s i o n o n t w o - p h a s e f l o w a n d h e a tt r a n s f e r i n p o r o u s m e d i a c a n b e f o u n d i n R e f. 1 9 .

T h e g o v e r n i n g e q u a t i o n s c a n b e r e d u c e d t of o u r i n i t i a l v a l u e , f i r s t - o r d e r o r d i n a r y d i f f e r e n t i a le q u a t i o n s . T h e e q u a t i o n s a r e s o l v e d b y a f o u r t h -o r d e r R u n g e - K u t t a m e t h o d . T h e s p e c i f i c a t i o n o ft h e v a lu e s f o r s , T , y ~ , a n d p g a t z = 0 i sr e q u i r e d . T h e n e t m a s s f lo w r a te t h r o u g h t h e w i c k( w h i c h i s e q u a l t o t h e b u r n i n g r a te ) i s o b t a i n e df r o m t h e e x p e r i m e n t s , a n d a l i n e a r a p p r o x i m a t i o ni s m a d e t o th e s e r e s u l t s ( F i g . 5 ) . T h e d e t a i l s a n ds o l u t i o n s t e p s a r e g i v e n i n R e f . 1 8 . T h e s a t u r a t i o nd i s t r i b u t i o n [ i n c l u d i n g t h e s u r f a c e s a t u r a t i o n , i . e . ,s d z = L ) ] i s f o u n d f o r v a r i o u s r n ( h ) . T h i s l ea d st o t h e r e l a t i o n s b e t w e e n t h e b u r n i n g r a t e a n d t h es u r f a c e s a t u r a t i o n .

R E S U L T S A N D D I S C U S S I O NS i n c e w e u s e t h e c a s e o f t h e f lo o d e d s u r f a c e ( ap l a n a r , t h i n l i q u i d f i l m p r e s e n t o n t h e s u r f a c e ) a st h e b a se l in e , a n d a l s o i n o r d e r t o c o m p a r e o u rm e a s u r e d b u r n i n g r a t e w i t h s o m e e x i s t i n g m e a -s u r e m e n t s a n d p r e d i c t io n s , e x p e r i m e n t s w e r e a ls op e r f o r m e d f o r s ] = 1 . I n t h e f o l l o w i n g , t h e e x -p e r i m e n t a l r e s u l t s f o r s 1 a r e g i v e n f i r st . T h e n , t h ee x p e r i m e n t a l r e s u l t s f o r s n < 1 f o l l o w , w h e r e t h ep r e d i c t e d s u r f a c e l i q u i d s a t u r a t i o n a n d t h e e f f e c to f th i s s u r f a c e s a t u r a t i o n o n t h e b u r n i n g r a t e( o b t a i n e d t h r o u g h t h e d i f f u s i o n m o d e l d i s c u s s e di n t h e s e c t i o n o n m a s s d i f f u s i o n ) a r e p r e s e n t e d ,a n d t h e g e n e r a l b e h a v i o r o f r n ( s 1 A / X , A / c 5 , R e )

4.0

3.0

E

2 . 0 ~ u = 1 . 7 1 ( m / s )1 . 0 | I I I

0 5 1 0 1 5 2 Oh c mFig. 5. Variation of the burning rate with respect to thehydrostatic pressure (height), for u,~ = 1.71 m/s .


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B U R N I N G R A T E O F L IQ U I D 5 7

T A B L E IN u m e r i c a l V a l u e s f or P a r a m e t e r s A p p e a r i n g i n t h e A n a l y s e s

P a r a m e t e r s U n i t M a g n i t u d eB 3 . 2 6Dno m2/s 1 . 4 7 x 1 0 - 5d a / d T N / m - K - 8 . 6 0 2 1 0 5if g J / k g 8 . 8 x l 0 skg W / m - K 0 . 0 2 2k l W / m - K 0 . 1 5 5k s W / m - K 1 . 8 9ks g W / m - K 0 . 9 0ks/ W / m - K 1 . 0 12K m 2 8 . 5 1 0 - 1 21 m 0 . 0 4 3L m 0 . 0 3D e 7 6 . 2R J / k g - K 1 8 0 . 5TO K 2 9 8 . 2Tsa K 35 1 . 5T ~ K 2 9 7s i . 0 . 0 5u ~ m / s 1 . 7 1~ l m 2 / s 5 . 3 6 x 1 0 - 7

0 . 5# g P a - s 1 . 1 x 1 0 s~ / P a - s 5 X 1 0 - 4p / k g / m 3 7 7 0f f N / m 0 . 0 2~- 2

i s e x a m i n e d . T a b l e 1 g i v e s t h e v a l u e s o f th ep r o p e r t i e s a n d p a r a m e t e r s u s e d i n t h i s a n a l y s i s .

B u r n i n g R a t e o f P l a n a r L i q u i d S u r f a c e( s t = 1 )E x p e r i m e n t s w e r e p e r f o r m e d w h e r e t h e w i c k w a sp l a c e d i n a n a l u m i n u m c a s i n g w i t h i t s u p p e rs u r f a c e p l a c e d a b o u t 3 m m b e l o w t h e t o p e d g e o ft h i s c a s i n g . T h e n t h e l i q u i d f u e l w a s a l l o w e d t oto ta l l y f i l l th i s vo i d b y ap p l y i n g b ack p res s u re .T h e b u rn i n g ra te o f th i s p l an ar l i q u i d s u rface wast h e n m e a s u r e d . F i g u r e 6 s h o w s t h e e x p e r i m e n t a lr e s u l t s , a l o n g w i t h t h e e x p e r i m e n t a l r e s u l t s o fR e f . 2 0 a n d t h e p r e d ic t i o n o f R e f . 9 . T h e l at te rc a n b e e x p r e s s e d b y t h e G l a s s m a n a p p r o x i m a t i o n[12] as

rn" = t*= l n (1 + B ) ReO .5 . (35 )1 1 . 3 B 15

O u r m e a s u r e d 6z" i s h i g h e r t h a n t h e E m m o n sb o u n d a r y - l a y e r s o l u ti o n . T h i s s u p p o r t s t h e a r g u -

9O

8 0

~8 7060

50

40

u m / s1.0 1.5 2.0 2.5 3.0

, I I I I[]

_ / - { h = 1 1 . 2 c m ) [ 1 / U ( h - 9 . 0 era !

/ 0 (h=4 .0 cm)/ ~ ; I h . , 3 . s cm )= I I J I

2 0 0 0 5 0 0 0 8 0 0 0Ree

O Part~. l Saturat ion, Di f ferent H y d r o s t a t i c P r e s s u r e s[ ] Thin-Liquid-F ilm Combust ion, Exper ime nt Thin-Liquid-F i lm Combust ion, Exper iment (Hi rano) [19]Thin-Liquid-F i lm Predict ion (Emmons) [9]

F i g . 6 . C o m p a r i s o n o f t h e e x p e r i m e n t a l a n d p r e d i c t e d r e su l t sf o r b u r n i n g o f a p l a n a r , t h i n - l i q u i d - f i lm su r f a c e . A l s o g i v e na r e t h e e x p e r i m e n t a l r e s u l t s f o r v a r i o u s p a r t ia l l i q u i d c o v e r -a g e s a n d u = = 1 .7 1 m / s .

m e n t [ 2 1 ] t h a t t h e b u r n i n g r a t e o b t a i n e d f r o m t h es o lu t io n o f t h e c o m p l e t e N a v i e r - S t o k e s e q u a ti o n si s h i g h e r t h a n t h a t b a s e d o n t h e b o u n d a r y - l a y e ra p p r o x i m a t i o n s . T h i s i s d u e t o a x i a l d i f f u s i o no f sp ec i e s an d h ea t a t t h e l ead in g ed g e . In t h e ex -p e r imen t s , ad d i t i o n a l e f fec t s , n o t i n c lu d ed i nE m m o n s t r e a t m e n t , a r e a l s o p r e s e n t . T h e s e a r eb u o y a n c y , t h e s l i g h t s u r f a c e i n d e n t a t i o n w h e r eth e wic k i s im b ed d e d , an d t h e i n i t ia t i o n o f t h em o m e n t u m b o u n d a r y l a y e r u p s t r e a m f r o m t h ew i c k l o c a t i o n . N o t e t h a t t h e d i f f e r e n c e b e t w e e nt h e b u r n i n g r a t e o b t a i n e d f r o m t h e b o u n d a r y - l a y e rs o l u t i o n a n d t h a t o b t a i n e d f r o m t h e e x p e r i m e n t i sa l m o s t a c o n s t a n t .

B u r n i n g R a t e o f P a r t i a l l y L i q u i d C o v e r e dS u r f a c e ( s 1 < 1 )Also s h o w n in F ig . 6 a re t h e re su l t s fo r s~ < 1a n d a n a m b i e n t f r e e - s t r e a m v e l o c i t y o f 1 .7 1 m / s .At t h i s v e lo c i t y , t h e f l ame i s s t ab l e an d t h e b u o y -an cy e f fec t i s sma l l . Th e re su l t s sh o w th a t t h eb u r n i n g r a t e d e c r e a s e s a s t h e h y d r o s t a t i c p r e s s u r ed ec reases ( see a l so F ig . 5 ) .


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Th e b u rn in g ra t e i s co n t ro l l ed b y cap i l l a r i t y -g r a v i t y a n d b y t h e h y d r o s t a t i c p r e s s u r e . A d e -c rease i n t h e h y d ro s t a t i c p re ssu re re su l t s i n ad ec rease i n t h e su r face l i q u id sa tu ra t i o n , wh ichl ead s t o an i n c rease i n t h e cap i l l a ry p re ssu re . I ft h e i n c r e a s e i n t h e c a p i l l a r y p r e s s u r e c o m p e n s a t e sf o r t h e d e c r e a s e i n h y d r o s t a t i c p r e s s u r e , t h e n t h ef l o w r a t e r e m a i n s t h e s a m e . A s s h o w n i n F i g s . 5an d 6 , t h i s i s n o t fo u n d h e re . S in ce t h e cap i l l a ryp r e s s u r e i s s t r o n g l y i n f l u e n c e d b y t h e m i c r o s t r u c -t u r e o f p o r o u s m e d i a , o n e c a n e x p e c t t h a t t h em e a n p a r t i c l e s i z e o f t h e w i c k ( a s a c h a r a c t e r i s t icl e n g t h ) p l a y s a n i m p o r t a n t r o l e a m o n g a l l t h ei n f l u e n c i n g f a c t o r s . O n t h e o t h e r h a n d , v a p o rd i f fu s io n t h ro u g h t h e i n d en t a t i o n s an d t h e b o u n d -a r y - l a y e r r e g io n i s a l s o a f fe c t e d b y t h e m e a n w i c kp a r t ic l e s i z e . T h e c o m b i n e d e f f e c ts o f t he i n t e r n a lt w o - p h a s e f l o w a n d t h e v a p o r d i f f u s i o n n e a r t h es u r f a c e d e t e r m i n e t h e b u r n i n g r a t e . T h e s e a r ed i s c u s s e d b e l o w .

S u r f a c e L i q u i d S a t u r a t i o nT h e f o r m u l a t i o n f o r i n t e r n a l , t w o - p h a s e , m o m e n -t u m , h e a t a n d m a s s t r a n s f e r g i v e n i n t h e p r e v i o u ssec t i o n o n su r face l i q u id sa tu ra t i o n y i e ld s t h ed i s t r i b u t io n o f t h e li q u id sa tu ra t i o n s , g a s eo u sp r e s s u r e p , t e m p e r a t u r e T , a n d m a s s f r a ct io n o ft h e n o n c o n d e n s i b l e g a s Y n t h r o u g h o u t t h e w i c k ,i n c lu d in g t h e su r face . F ig u re 7 sh o ws t h ese d i s t r i -b u t i o n s fo r u oo = 1 .71 m / s . In t h e ca se wh ere t h eh y d r o s t a t i c p r e s s u r e d o m i n a t e s t h e l i q u i d f l o w(Fig . 7 a fo r h = 9 cm) , t h e su r face l i q u id sa tu ra -t ion s~ is jus t s l igh t ly less than sa tu ra t ion a t thel o w e r s u r f a c e s o (n o t e t h a t s~


13/15


M a s s D i f f u s i o n A d j a c e n t t o S u r f a c e f o r s I < 1I n t h e s e c t i o n o n m a s s d i f f u s i o n , w e s h o w e d t h a tv a p o r d i f f u s i o n a d j a c e n t t o t h e w i c k s u r f a c e i sg r e a t l y i n f l u en c e d b y t h e r a t i o o f t h e m e a n p a r t i -c l e s iz e t o t h e c o n c e n t r a t i o n b o u n d a r y - l a y e r t h ic k -n ess . Th e e f fec t o f za /X i s n o t s i g n i f i can t b eca u sef o r th e p r o b l e m c o n s i d e r e d , X i s m u c h s m a l l ert h a n A . T h e p r e d i c t i o n s b a s e d o n t h e d e v e l o p -m e n t m a d e a b o v e g i v e t h e v a r i a t i o n o f th e n o r -m a l i z e d m a s s t r a n s f e r r a te ( E q . 1 6) w i t h r e s p e c tt o th e su r fa ce sa tu ra t i o n d e f in ed i n Eq . 1 an dg/d. T h e r e s u l t s a r e s h o w n i n F i g s . 8 a n d 9 f o rt h e c o n d i t i o n s c o r r e s p o n d i n g t o t h e e x p e r i m e n t s .

0 .80 .70 . 60 . 5

Y ' 0 . 40 . 30 . 2

0 .10 . 0

0 . 0I I I I l I I I I

0 . 5 1 . 0S t

F i g . 8 . V a r i a t i o n o f t h e m a s s f r a c t io n o f e t h a n o l y ' (a tZ = 0 ) , w i t h r e s p e c t t o t h e s u r fa c e s a t u r a t i o n , f o r s e v e r a lgld.

1 .0 Em m ons' p r e d i c t i o ~

0 .8

0 . 6

0.5 ~ - - pred iction based onexternal boundary layer0.4 - - pred ict ion based oninternal two-phase f low0.3 = I I = 1 I I I =0 . 0 0 . 5 . 0

$ IF i g . 9 . T y p i c a l v a r i a t i o n o f th e b u r n i n g r a t e w i t h r e s p e c t t ot h e s u r f a c e s a t u r a t i o n , f o r v a r i o u s g /d .

A s e x p e c t e d , t h e m a g n i t u d e o f g /d g r e a t l yin f l u en ces t h e v a r i a t i o n o f t h e b u rn in g ra t e wi thre sp ec t t o t h e l i q u id sa tu ra t i o n . As sh o wn in F ig .8 , fo r g /d >_ 1 0 0 , t h e n o rm a l i zed b u rn in g ra t eh a s a m a g n i t u d e o f n e a r u n i ty , i . e . , n e a r l y i n d e -p en d en t o f s 1. As g/d --" oo, t h e c o n s t a n t b u r n -i n g r a t e i s r e a c h e d , i . e . , c o m p l e t e i n d e p e n d e n c eo f r n / r h f _ f r o m s ].

W h e n 5 /d i s f in i te (say less than 30 in thee x a m p l e s s h o w n i n F i g s . 8 a n d 9 ) , t h e d e p e n -d e n c e o f th e b u r n i n g r a t e o n t he s u r f a c e s a t u r a -t i o n b e c o m e s m o r e p r o n o u n c e d . T h e n o r m a l i z e dm a s s t r a n s f e r r a t e , r n / r h y , d e c r e a s e s r a p i d l y w i t hs t fo r g /d l e s s t h an 1 0 . Th i s i n d i ca t e s t h a t t h ead d i t i o n a l d i f fu s io n i n a sp ace (A d o main ) cau sedb y t h e s u r f a c e p r o t u b e r a n c e / i n d e n t a t i o n i s i m p o r -t a n t f o r s m a l l v a l u e s o f g /d an d can re su l t i n as i g n if i c an t r e d u c t i o n o f th e v a p o r m a s s f r a c t i o n a tt h e s u r f a c e , y'(Z = 0 ) . T h i s o b s e r v a t i o n i s s u p -p o r t e d b y a q u a l i t a ti v e c o m p a r i s o n o f th e p r e d i c -t i o n ( b a s e d o n o n e - d i m e n s i o n a l i n t e r n a l a n d e x -t e r n a l m o d e l s ) w i t h t h e e x p e r i m e n t a l r e s u l t s ,s h o w n i n F i g . 9 . T h e e x p e r i m e n t a l r e s u l t s a r ec lo se t o t h e p red i c t i o n s fo r ~ /d = 1 5 . I f wee s t i m a t e t h e c o n c e n t r a t i o n b o u n d a r y - l a y e r t h i c k -n e s s f o r t h e e x p e r i m e n t a l c o n d i t io n ( R e t = 4 2 0 0 ,l = 0 .0 4 3 m ) , u s i n g t h e B l a s i u s a p p r o x i m a t i o na n d a s s u m i n g S c = 1 , w e h a v e

15 .2g = 2 R e I - I / 2 ,

w h i ch g i ve s g - - 1 .7 m m . T h e C a r m a n - K o z e n yr e l a t i o n ( b a s e d o n n o n c o n s o l i d a t e d s p h e r i c a lp a r t i c l e s ) g i v e s t h e a v e r a g e p a r t i c l e s i z e o f t h ec e r a m i c w i c k a s 55 # m . T h e r e f o r e , f r o m t hee x p e r i m e n t s , w e h a v e g /d = 2 7 . Th e d i f fe ren cesin g /d b e t w e e n t h e p r e d i c t i o n s a n d t h e e x p e r i -m e n t s i s m a i n l y b e c a u s e t h e w i c k i s n o t m a d e o fn o n c o n s o l i d a t e d s p h e r i c a l p a r t i c l e s a n d m a y h a v ea l a r g e r e q u i v a l e n t m e a n p a r t i c l e s i z e t h a n t h a tp r e d ic t e d f r o m t he C a r m a n - K o z e n y r e l at io n . D e -s p i te t h is d i f f e r e n c e , t h e a b o v e c o m p a r i s o n p o i n t so u t , f r o m a m a s s d i f f u s i o n p o i n t o f v i e w , t h ei m p o r t a n c e o f t h e e f f e c t o f t h e s u r f a c e s a t u r a ti o no n t h e b u r n i n g r a t e f o r s m a l l g/d. Since g i si n v e r s e l y p r o p o r t i o n a l t o t h e R e y n o l d s n u m b e r ( t ot he p o w e r - 0 . 5 f o r l a m i n a r fl o w s ), th e li qu ids a t u r a t io n d e p e n d e n c e o f th e b u r n i n g r a t e w o u l db e m o r e p r o n o u n c e d a t h i g h R e y n o l d s n u m b e r s .

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


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s i o n in t he A d o m a i n o n t h e b u r n i n g r a t e i n t e r m so f a t o t a l r e s i s t an ce t o d i f fu s io n , R , w h e re R i sd e f in ed su ch t h a t m / f nf = ( R / R f ) - 1. F r o m E q .1 4 , w e a r r i v e a t t h e f o l l o w i n g a p p r o x i m a t i o n(a ssu m in g l n (1 + B ) = B ) :

R d f o r f i x e d B * a n d s 1R f t S 'an d

R t5- - - B * 8 5 , fo r f i x ed - - an d s 1R f dT h e n , a s t h e r a t i o ~ / d d e c r e a s e s o r a s B * i n -c reases , t h e t o t a l r e s i s t an ce t o t h e d i f fu s io n i n -c reases , r e su l t i n g i n a d ec rease i n t h e mass t ran s -fe r ra t e . Th e f i r s t r e l a t i o n ab o v e i s co n s i s t en t wi thth e re su l t s sh o wn in F ig . 9 . Th e seco n d re l a t i o ns u g g e s t s t h a t u s i n g f u e l s w i t h h i g h m a s s t r a n s f e rn u m b e r s m a y r e s u l t i n l a r g e r r e d u c t i o n i n t h eb u r n i n g r a t e ( f o r a g i v e n w i c k a n d s u r f a c e s a t u r a -t ion) .

I n t h e v a p o r d i f f u s i o n m o d e l g i v e n i n t h e m a s sd i f fu s io n sec t i o n , t h e h ea t p en e t ra t i o n i n to t h ew i c k , w h i c h i s a f o r m o f h e a t l o s s , w a s n e -g l ec t ed . Th i s h ea t l o s s , wh ich i s d u e t o t h e su b -co o l ed s t a t e o f t h e li q u id su p p l i ed a t t h e l o w erw i c k s u r f a c e , c a n b e a c c o u n t e d f o r b y m o d i f y i n gt h e m a s s t r a n s f e r n u m b e r a s [ 9 ]B ' = c p ( T o o - T ' ) + ~ Y o . ~ Q

ifg + Qcw h e r e Q c i s t h e h ea t p en e t ra t i o n t h ro u g h t h ewick , wh ich i s a fu n c t i o n o f th e su r face l i q u idsa tu ra t i o n . Ou r ca l cu l a t i o n i n d i ca t e s t h a t Qc i n -c rease s s l i g h t l y a s s 1 d ec re ases . In c lu s io n o f Qcr e d u c e s t h e v a l u e o f B ' , a n d , t h e r e f o r e , r e d u c e st h e p r e d i c t e d b u r n i n g r a t e . F o r t h e t y p i c a l c a s esh o wn in F ig . 5 , Q c i s l e s s th an 1 0 % o f th e h ea tf o r t he e v a p o r a t i o n o f e t h a n o l a t t h e u p p e r w i c ks u r f a c e . T h i s g i v e s a r e d u c t i o n o f 1 0 % i n B '.H o w e v e r , s i n ce t h e l iq u i d f u e l u s e d i n th e e x p e r i -m e n t f o r t h e f u l l y l i q u i d c o v e r e d s u r f a c e i s a l s os u b c o o l e d , w e h a v e a r e d u c t i o n i n B * t h a t i s t h es a m e o r d e r o f t h a t i n B ' . F r o m E q . 9 , i t c a n b es e e n t h a t t h e s e r e d u c t i o n s m a y c a n c e l e a c h o t h e r( i f t h e y a r e t h e s a m e ) i n t h e e x p r e s s i o n o f t h en o r m a l i z e d m a s s t ra n s f e r r a te r n / r n y . O u r c a l c u-l a t i o n i n d i ca t e s t h a t t h e d i f fe ren ce i n t h e p red i c t edt n / f n y cau se d b y i n c lu d in g Q i s l e s s t h an 2 %.T h i s s h o w s t h a t in o u r e x p e r i m e n t s , t h e e f f e c t o f

t h e s u r f a c e r o u g h n e s s i s m o r e i m p o r t a n t t h a n t h a to f Q c b e c a u s e t h e e f f e c ti v e t h e r m a l c o n d u c t i v it yo f t h e c e r a m i c w i c k u s e d i s r a t h e r s m a l l ( T a b l e1 ). Fo r wick s wi th l a rg e r k e j ( e . g . , m e t a l l i cwick s ) , t h i s su b co o l in g e f fec t b ec o m es s i g n i fi -c a n t . T h e e x p o s e d s o l i d m a t r i x t r a n s f e r s m o r eh e a t f r o m t h e a m b i e n t g a s t o t h e l i q u i d . T h e r e -f o r e , t h e b u r n i n g r a t e c a n r e m a i n t h e s a m e o rev en i n c rease a s t h e su r face sa tu ra t i o n d ec reases .T h i s e f f e c t o f t h e r m a l c o n d u c t i v i t y o f t h e s o l i dm a t r i x o n t h e b u r n i n g r a t e h a s b e e n o b s e r v e de x p e r i m e n t a l l y [ 1 8 ] .C O N C L U S I O N SFo r l i q u id fu e l s su p p l i ed t h ro u g h wick s , t h e v a -p o r p r e s s u r e a t t h e s u r f a c e b e i n g b u r n e d i s h i g hen o u g h so t h a t t h e ra t i o o f t h e ch a rac t e r i s t i cl e n g th o f th e e x p o s e d m a t r i x ( i . e . , t h e s u r f a c er o u g h n e s s ) i s m u c h l a r g e r t h a n th e m e a n f r e e p a t ho f th e v a p o r m o l e c u l e . H o w e v e r , t h e r a t io o f t h isc h a r a c t e r i s t i c l e n g t h t o t h e b o u n d a r y - l a y e r t h i c k -n e s s ( m o m e n t u m , t h e r m a l , a n d c o n c e n t r a t i o n )m a y n o t b e s m a l l , e s p e c i a l l y w h e n t h e f l o w i s a tl ea s t p a r t l y d u e t o cap i l l a r i t y . Th en , o u r re su l t ss h o w t h a t t h e b u r n i n g r a t e f o r p a r t i a l l y e x p o s e dwick ma t r i ce s i s l e s s t h an t h a t fo r fu l l y l i q u idc o v e r e d s u r f a c e s . T h e p e r m e a b i l i t y o f 8 .5 1 0 - 1 2 ( m 2 ) a n d p o r o s i t y o f 0 . 5 a r e t y p i c a l o fw i c k s t h a t o f f e r l a r g e e n o u g h c a p i l l a r y p r e s s u r e(e sp ec i a l l y , su i t ab l e fo r su r face o r i en t a t i o n s o th e rth an h o r i zo n t a l ) an d y e t t h e i r r e s i s t an ce t o l i q u idf l o w i s n o t so l a rg e a s t o req u i re an ex t e rn a ld r i v i n g p r e s s u r e . T h e r e f o r e , w e e x p e c t s o m e d e -v i a t io n s f r o m t h e i d e a l b e h a v i o r o f p la n a r ,l i q u i d - f il m - c o v e r e d s u r f a c e s w h e n t h e s e w i c k s a r eu sed a s s imu lan t s . We a l so men t io n ed t h a t t h ec e r a m i c w i c k u s e d h e r e h a s a n e f f e c t i v e t h e r m a lco n d u c t i v i t y t h a t i s mu ch c lo se r t o t h a t o f t h el iq u i d t h a n s o m e o f t h e m e t a l l ic w i c k s t h a t h a v eb een u sed a s s imu lan t s . In t h e ca se o f me ta l l i cw i c k s , t h e e x p o s e d m a t r i x o f h i g h th e r m a l c o n -d u c t i v i t y t en d s t o i n c rease t h e b u rn in g ra t e .T h e r e f o r e , p a r t i a l - l i q u i d - c o v e r e d m e t a l l i c w i c k sc a n , u n d e r c e r t a i n c o n d i t i o n s , h a v e b u r n i n g r a t e sa s h i g h a s o r e v e n h i g h e r t h a n t h a t f o r th e p l a n a r -t h in - l i q u id f i lm co v e red su r faces .REFERENCES

1 . A h m e d , T . , a n d F a e t h , G . M . , J . H e a t T r a n s f .1 0 0 : 1 1 2 - 1 1 9 ( 1 9 7 8 ) .


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2. Groff, E. G. , and Fae th , G. M ., Combust. Flame3 2 :1 3 9 -1 5 0 (1 9 7 8 ) .

3 . Ma la ry , S . F . , Awa d , J. K . , a n d Th a p a r , R . , Combust.Sci. Technol. 6 3 :1 -1 1 (1 9 8 9 ) .

4 . Ta o , Y. -X . , a n d Ka v ia n y , M. , J. Heat Transf. (t oappear) .5 . Ro g e r s , J . A . , a n d Ka v ia n y , M . , J . Heat Transf.1 1 2 : 6 6 8 - 6 7 4 (1990).6 . Schl i inder , E. U. , Chem. Eng. Sci. 4 3 : 2 6 8 5 - 2 6 8 8

(1988).7 . Ka v ia n y , M. , a n d Ta o , Y. -X. , J . Heat Transf.

1 1 0 : 4 3 1 - 4 3 6 (1988).8 . Kim, J . S . , de Ris , J . , and Kroes ser , F . W ., Thirteenth

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Received 24 May 1990; revised 18 February 1991

burning rate of liquid supplied through a wick

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