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8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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E L S E V I E R
ire Safe~ Journal
29 (1997) 235 258
~C: 1998 E lsevie r Sc ience Ltd . A l l r ights rese rved
P r i n t e d i n N o r t h e r n I r e l a n d
0379 7112/97/ 17.00
P I I S 0 3 7 9 - 7 1 1 2 ( 9 7 ) 0 0 0 6 3 - 5
S t ron g ly Bu oyan t P lu me S imi lar i t y an d
Smal l - f ire Vent i la t ion
G . G . R o o n e y * P . F .
Linden
Department of Applied Mathematics and Theoretical Physics, Silver Street,
Cambridge CB3 9EW, UK
Received 8 August 1996; revised version received 11 August 1997; accepted 10 September 1997)
A B S T R A C T
We re-exa mine the problem of natural venti la tion o f a room containing af i re
in the l ight o f recent resul t s obtained by R oon ey and Linde n
1996)
concern-
in9 the s imi lari ty so lu t ion for non-Boussinesq plumes. W e consider the case o f
a s te ad yf i r e in a com partm ent wi th openings a t f l oo r an d ceil ing levels and
obtain exp ressions for the depth and de nsi ty o f the hom ogeneous cei ling layer
maintained by the f i re p lume. Taking the l imi t o f a large lower opening area
we compare our resu l ts w i th exper imen t s per formed by Tho ma s
et al. 1963).
W e also perform a sa mp le calculation to est ima te the size o f the di fference
between the we akly and s t rongly buoyant cases. ©
1998
Elsevier Science Ltd.
NOT TION
A 'Ef f ec t iv e a r ea ' in Bo u ss in e s q v e n t i l a t io n m o d e l , s e e e q n ( 2 4)
,4 'Ef fe c t iv e a r ea ' in n o n - Bo u ss in e sq v e n t i la t io n m o d e l , s e e e q n (2 8)
A I A r e a o f f l o o r c o v e r e d b y b a se o f fir e
a A r e a o f v e n t
B P l u m e b u o y a n c y f lu x
b P l u m e r a d iu s
c r S p e c i f ic h e a t c a p a c i t y a t c o n s t a n t p r e s su r e
D D ia m e t e r o f fir e so u r c e
d U p p e r - l a y e r d e p t h
F S c a le d d e n s i t y d e f ic i t, a c o n se r v e d q u a n t i t y w i t h u n i t s o f b u o y a n c y f lu x ,
see eqn (15)
9 G r a v i t a t io n a l a c c e l e ra t i o n
9 ' R e d u c e d g r a v i ty
* Author to whom correspondence should be addressed.
235
8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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36 G . G . R oo ne v P . F . L in de n
H H e i g h t o f c o m p a r t m e n t c e il in g
h H e i g h t o f t w o - l a y e r i n te r f a c e
K U p p e r - v e n t d i s c h a r g e c o e f fi c ie n t le ss t h a n u n i ty )
k L o w e r - v e n t p r e s s u r e - l o s s c o e f f ic i e n t l es s t h a n u n i t y )
L D i m e n s i o n l e s s c o n s t a n t i n s i m i l a r it y s o l u t i o n , e q n 8 )
l F l a m e l e n g th
M D i m e n s i o n l e s s c o n s t a n t i n s i m i l a ri t y s o l u t i o n , e q n 9 )
N D i m e n s i o n l e s s c o n s t a n t i n s i m i l a r it y s o l u t i o n , e q n 1 0)
p L e n g t h o f p e r i m e t e r o f f ir e b a s e
) C o n v e c t i v e p o w e r o f f ir e
0 , T o t a l c o n v e c t i v e a n d r a d i a ti v e ) p o w e r o f f ir e
r R a d i a l d i s t a n c e f r o m p l u m e a x is
T o A m b i e n t t e m p e r a t u r e
V P l u m e v o l u m e f lu x
W P l u m e m a s s f lu x
w V e r t i c a l v e l o c i t y
z V e r t i c a l d i s t a n c e a b o v e f l o o r l ev e l
zv V i r t u a l - o r i g i n d e p t h
G r e e k
d p
0
P
1 1
Po
l 1
l e t t e r s
P l u m e e n t r a i n m e n t c o n s t a n t
P l u m e / a m b i e n t d e n s i ty d i f fe r en c e
R a t i o o f u p p e r - t o - l o w e r fl u id d e n s i t ie s , s e e e q n 2 6)
F r a c t i o n a l i n t e r f a c e h e i g h t , s e e e q n 2 3)
D e n s i ty o f p l u m e
D e n s i t y o f la r g e -f ir e p l u m e
D e n s i t y o f a m b i e n t l o w e r -l a ye r ) f lu i d
D e n s i t y o f u p p e r - l a y e r f l ui d
S u b s c r i p t s
A L o w e r v e n t
B I n t e r f a c e
C U p p e r v e n t
1 I N T R O D U C T I O N
T h e t o x ic c o m b u s t i o n p r o d u c t s f r o m a c c i d e n ta l f ir es in m o d e r n b u i l d in g s a re
a m a j o r h a z a r d , a n d e ff ic ie n t r e m o v a l o f s m o k e f r o m p o p u l a t e d s e c ti o n s o f
a b u i l d i n g is e s s e n t i a l f o r s af e e v a c u a t i o n i n t h e e v e n t o f s u c h a fir e. O n e m e a n s
o f s m o k e r e m o v a l e m p l o y e d is n a t u r a l p a ss iv e ) v e n t il a ti o n , w h e r e i n s m o k e
a n d f u m e s a r e v e n t e d t o t h e a t m o s p h e r e u s i n g t h e d r i v in g f or c e o f t h e ir o w n
8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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trongly buoya nt plum e similarity 37
b u o y a n c y . F o r t h e v e n t i n g o f s m o k e f r o m a b u i ld i n g , d i s p l a c e m e n t v e n ti la -
t i o n i s d e s i r ed .
T h e e s s e n t ia l p o i n t a b o u t d i s p l a c e m e n t v e n t i l a t i o n is t h a t l it tl e o r n o m i x i n g
t a k e s p l a c e b e t w e e n t h e c o n t a m i n a t e d a i r b e i n g r e m o v e d a n d t h e f r e s h a i r
w h i c h d i s p la c e s it, a n d t h is i s a c h i e v e d t h r o u g h t h e t h e r m a l b u o y a n c y o f t h e
c o n t a m i n a t e d a i r . T h i s b u o y a n c y , f i r s t l y , c a u s e s t h e c o n t a m i n a t e d a i r t o
c o l le c t i n a l a y e r i n t h e u p p e r p a r t o f a c o m p a r t m e n t ( w i th f re s h ai r b e l o w i t
f o r m i n g a s e c o n d l a y e r ) a n d , s e c o n d l y , i n h i b i t s m i x i n g b y s t a b i l i z i n g t h e
i n t e r f a c e b e t w e e n t h e l a y e r s . D i s p l a c e m e n t v e n t i l a t i o n t h e r e f o r e o p e r a t e s i n
a t w o - l a y e r s y s t e m , w i t h f re s h a i r e n t e r i n g t h r o u g h v e n t s l o w e r t h a n t h e
i n te r fa c e , a n d c o n t a m i n a t e d a i r l e a v i n g f r o m v e n t s h i g h e r t h a n t h e i n te r fa c e . I t
is cl e a r th a t , i n t h e a b s e n c e o f a n y o t h e r m e c h a n i s m , t h e i n t e r fa c e w i ll m o v e
u p w a r d a s t h e u p p e r l a y e r d r a i n s . ( T h e v e n t s s h o u l d t h e r e f o r e b e l o c a t e d a s
n e a r t o t h e f l o o r a n d c e il in g o f t h e c o m p a r t m e n t as p o s s ib l e , s o t h a t d i s p la c e -
m e n t m a y o c c u r f o r a r a n g e o f i n t e rf a c e h e ig h t s. ) I f a f ir e c o n t i n u e s t o b u r n i n
t h e c o m p a r t m e n t , h o w e v e r , t h e n t h e b u o y a n t p l u m e f r o m t h e f ire w ill p a s s
t h r o u g h t h e i nt e rf a c e a n d c o n t i n u a l l y r e p l e n is h t h e u p p e r l a y er w i t h c o m b u s -
t i o n p r o d u c t s . T h e e n t r a i n m e n t o f fr e sh a ir b y t h e p l u m e i n t h e l o w e r la y e r
t h u s p r o v i d e s a m e a n s o f m a s s t r a n s f e r a c r o s s t h e i n te r f a c e a n d , a s e n t r a i n -
m e n t c a u s e s t h e p l u m e m a s s f lu x to i n c r e a s e w i t h h e i g h t , t h e h e i g h t o f t h e
i n te r f a c e w i ll d e t e r m i n e t h e m a s s f l u x t o t h e u p p e r l a y e r.
T h e a b s e n c e o f t o x ic f u m e s a n d b l i n d in g s m o k e f r o m t h e l o w e r l a y e r m a k e s
it a u s e f u l c l e a r p a s s a g e f o r t h e e v a c u a t i o n o f p e r s o n n e l , a n d f a c il it a te s t h e
m o v e m e n t o f f ir e fi g h te r s. T h e s t a b le i n t e r f a c e b e t w e e n t h e l a y e rs , w h i c h
i n h ib i ts m i x i n g b e t w e e n t h e m , m a y b e d i s r u p t e d b y f l u id m o t i o n s o f s uf fi ci en t
k i n e t i c e n e r g y t o o v e r c o m e t h e s t a b il i zi n g i n f lu e n c e o f t h e b u o y a n c y f or ce . I n
t h is r e g a r d , i t h a s b e e n s u g g e s t e d 1 t h a t n a t u r a l v e n t i l a t i o n m a y b e b e t t e r t h a n
f o r c e d v e n t i l a t i o n , i n w h i c h t h e r e i s a d a n g e r t h a t t h e d i s p l a c i n g a ir m a y b e
p u m p e d i n a t t o o g r e a t a r a te , t h u s l o w e r i n g t h e s t a b i li ty o f t h e i n te r fa c e . I f
n a t u r a l d i s p l a c e m e n t v e n t i l a t i o n is t o b e s e r i o u sl y c o n s i d e r e d a s a n o p t i o n i n
b u i l d i n g d e s i g n f o r f i r e - s a f e t y , i t i s i m p o r t a n t t o h a v e a g o o d q u a n t i t a t i v e
u n d e r s t a n d i n g o f it s e ff ic ie n cy o f o p e r a t io n . T o c o n t r i b u t e t o t h i s u n d e r s t a n d -
in g , w e h e r e c o n s i d e r t h e a p p l i c a t i o n o f a n o n - B o u s s i n e s q p l u m e m o d e l t o
d i s p l a c e m e n t n a t u r a l v e n t i l a t i o n , a s a m o d e l o f th e v e n t i la t i o n b e h a v i o u r o f
f i r e s i n b u i l d i n g s .
2 S M A L L F IR E S A S S T R O N G L Y B U O Y A N T P L U M E S
2 1 La rge and sma l l fires
W e b e g i n b y b r i e f l y l o o k i n g a t t h e d i s t i n c t i o n b e t w e e n l a r g e a n d s m a l l f i r e s .
8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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38 G. G. Rooney P. F. Linden
I n th e w o r k b y T h o m a s e t a l . , z h e r e a f t e r re f e r r e d to a s P a p e r I , a d i s t i n c t i o n
is m a d e b e t w e e n ' s m a l l f ir e s' a n d ' la r g e f ir e s', d e p e n d i n g s o l e l y u p o n t h e a s p e c t
r a t i o o f t h e f i re p l u m e i n t h e l o w e r l a y e r . M o r e s p e c if ic a l ly , i f t h e f i r e c o v e r s a n
a r e a A f o f t h e f l oo r , a n d t h e f re e p l u m e e x t e n d s t o a h e i g h t h a b o v e t h e f l o o r
b e f o r e p l u n g i n g t h r o u g h t h e i n te r f ac e , t h e n P a p e r I st a te s t h a t s m a l l- f ir e
t h e o r y a p p l ie s f o r
A 1 / Z / h
< 0 '5 , a n d l a r g e - f ir e t h e o r y a p p l ie s f o r A ~ Z h 0"5.
2 .1 .1 L a r g e f i r e s
F o r l a r g e f ir e s, P a p e r g i ve s t h e t o t a l m a s s f lo w W o f e n t r a i n e d a i r i n t o a f ir e
o f s o u r c e p e r i m e t e r p a n d d e n s i t y p t, b e t w e e n t h e l ev e l o f t h e s o u r c e a n d h e i g h t
z as
w = o . 0 9 6 p p o ( g P 2 z 3 ' 2 1 1 t
\ P 0 /
b a s e d o n a s s u m p t i o n s a b o u t t h e m e a n v e rt ic a l v e lo c it y a n d e n t r a i n m e n t
b e h a v i o u r o f l a r g e f ir es .
F o l l o w i n g P a p e r I , t h e m a s s f lu x W i n k g s - ' f r o m a (l a rg e ) f ir e is w i d e l y
t a k e n t o d e p e n d o n t h e s o u r c e p e r i m e t e r l e n g t h p a n d t h e h e i g h t a b o v e t h e
s o u r c e z , b o t h i n m , a s
W = 0" 18 8p z s /2 (2)
o b t a i n e d f r o m e q n ( 1) b y s u b s t i t u t i n g v a l u e s f o r t h e fi re a n d a m b i e n t d e n s i t ie s ,
a n d f o r t h e g r a v i t a t i o n a l a c c e l e r a t i o n g . 3
H i n k l e y a p r e s e n t s c o m p a r i s o n o f th e l a r g e - f i r e e q u a t i o n , e q n (2), w i t h f iv e
s e ts o f e x p e r i m e n t a l r e s u lt s f r o m f o u r d i f f e r e n t e x p e r i m e n t s , w i t h e x c e l l e n t
a g r e e m e n t . I n d e e d , h e r e p o r t s t h e b e s t li n e f it t h r o u g h t h e d a t a c o n s i d e r e d a s
h a v i n g t h e e q u a t i o n
W = 0 " 1 8 9 p z l s (3 )
t h e d a t a t a k e n f r o m f ir es h a v i n g c o n v e c t i v e h e a t o u t p u t s p e r u n i t a r e a i n th e
r a n g e 3 4 1 80 0 k W m 2, a n d p e r i m e t e r s in th e r a n g e 0- 7 -1 6 - 2 m . T h i s p a p e r
r e m a r k s u p o n t h e e a s e o f a p p l i c a t i o n o f t h e l a rg e - fi r e e q u a t i o n d u e t o i ts
i n d e p e n d e n c e o f t h e s t r e n g t h o f t h e f ir e a n d s u g g es t s t h a t t h e l i m it o f a p p li c a -
t i o n o f t h e l a r g e - f ir e e q u a t i o n b e e x t e n d e d t o c o v e r fi re s i n t h e r a n g e
A l / 2 / h > 0-1.
T h o m a s s c o m m e n t s u p o n u s e o f t h e l a rg e - fi re e q u a t i o n t h a t 'a t h e o r e t i c a l
j u s t i f i c a t i o n i s s ti ll a w a i t e d f o r t h is w i d e l y e x p l o i t e d e x t e n s i o n o f a s i m p l e
f l a m e c o r r e l a t i o n ' .
I n c o n t r a s t t o H i n k l e y ' s 4 e x c e l le n t a g r e e m e n t , D e m b s e y e t
a l. ~'
c o m p a r e
d a t a f r o m e x p e r i m e n t s b y n i n e d i f f er e n t e x p e r i m e n t e r s w i t h a l a r g e -f ir e e q u a -
t i o n , o b t a i n e d f r o m t h e m a s s f l u x e x p r e s s i o n , e q n ( 1 ) , b y a s s u m i n g v a l u e s
o f t h e p l u m e a n d a m b i e n t d e n s it ie s s i m i l a r t o t h o s e g i v en b y D r y s d a l e . 3
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trongly buoyant plum e similarity 39
T h e y f i n d p o o r a g r e e m e n t b e t w e e n t h i s m o d e l a n d t h e e x p e r i m e n t s c o n -
s i d e re d , a n d s u g g e s t i n c l u d i n g a s u i t a b l e v i r tu a l o r i g i n in t h e m o d e l a s a m e a n s
o f c o r r e c t i o n . T h e s u g g e s t e d e x p r e s s i o n f o r t h e o f fs e t o f th e v i r t u a l o r i g i n
d e p e n d s o n b o t h t h e p o w e r o f t h e fir e a n d t h e s o u r c e d ia m e t e r , h o w e v e r .
2 .1 .2 Smal l f r e s
F o r s m a l l fire s, P a p e r I q u o t e s t h e p l u m e v o l u m e f lu x f r o m t h e p l u m e
s i m i l a r it y s o l u t i o n b y Y i h , v w h o g i ve s t h e v o l u m e f l ux f o r t h e B o u s s i n e s q
c a s e a s
v = 0 . 1 5 3 4 )
w h e r e
5 )
is t h e ( n o n -s p e c if ic ) b u o y a n c y f lu x . I n P a p e r I , t h e b u o y a n c y f l ux in t h e
B o u s s i n e s q c a s e is r e l a te d t o t h e t o t a l ( c o n v e c t iv e
and
r a d i a t i v e ) p o w e r 0 t o f
t h e f i r e b y
B _ Q t g (6)
p p o c p T o
w h e r e t h e s u b s c r i p t 0 d e n o t e s a m b i e n t v a l u e s , a s s u m e d c o n s t a n t . I t i s t h e n
c o n j e c t u r e d t h a t t h e n o n - B o u s s i n e s q v o l u m e f lu x m a y b e o b t a i n e d f r o m
t h e r e l a t i o n s h i p g i v e n b y Y i h 7 b y r e p l a c i n g t h e a m b i e n t d e n s i t y in e q n (6 ) b y
t h e p l u m e d e n si ty , a l th o u g h i t is a c k n o w l e d g e d t h a t t h e r e is s o m e u n c e r t a i n t y
a b o u t t h e e ff ec t o f s ig n if ic a n t d e p a r t u r e s f r o m t h e B o u s s i n e s q a p p r o x i m a t i o n .
T h i s r e p l a c e m e n t l e a d s t o a n e x p r e s s i o n f o r t h e m a s s f l u x ,
W = O 1 5 3 p (
JQ ~ t ~1/3 z5/3 7)
\ c p p T o J
w h e r e t h e h e i g h t z in c l u d e s t h e d e p t h o f t h e v i r t u a l o r i g in . T h i s m a s s f l ux
h a s t h e s a m e d e p e n d e n c e s o n h e i g h t a n d b u o y a n c y f lu x a s a c o n v e n t i o n a l
B o u s s i n e s q p l u m e m o d e l . 8
2.1.3
Near and far f i e ld
M o r e r e c e n tl y t h a n P a p e r I, th e d a t a p r e s e n t e d b y M c C a f f r e y 9 o r D e l ic h a t -
s i o s lo s u g g e s t t h a t a f i re b e g i n s t o s h o w p l u m e - l i k e ( sm a l l fi re ) b e h a v i o u r a t
t h e t o p o f t h e f l a m i n g r e g i o n , r a t h e r t h a n a t a f i x ed m u l t i p l e o f t h e b a s e
d i a m e t e r . T h e f l a m i n g r e g io n is c o m m o n l y t e r m e d t h e n e a r f ie ld , a n d t h e
r e g i o n a b o v e t h i s, t h e f a r f ie ld . T h e f l a m e l e n g t h in b u o y a n c y - d o m i n a t e d fir es
is a f u n c t i o n o f b o t h t h e s o u r c e d i a m e t e r a n d t h e p o w e r o u t p u t ( se e e q n ( 18 )),
8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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240
G . G . Roo ne y P . F . L in de n
a n d s o th e h e i g h t a b o v e w h i c h a fi r e m a y b e d e e m e d s m a ll w o u l d t h e n d e p e n d
o n t h e b u r n i n g r a t e a n d m a t e r i a l a s w e l l a s th e g e o m e t r y . T h i s f l a m i n g
r e g io n i n b u o y a n c y - d o m i n a t e d f ir es n o r m a l l y e x t e n d s a f e w so u r c e d i a m e t e r s
v e r t i c a l l y .
I n t h e n e a r f ie ld , t h e f ir e m a y d e p a r t f r o m t h e s i m i l a ri t y b e h a v i o u r p r e d i c t e d
b y t h e p l u m e m o d e l f o r se v e r al r e a s o n s . F o r e x a m p l e , t h e d i a m e t e r o f th e f ir e is
n o n - n e g l i g i b le c o m p a r e d w i t h t h e h e i g h t o f t h e n e a r f i el d a n d s o i n t r o d u c e s
a n e w l e n g t h s c al e i n t o t h e p r o b l e m , a n d in t h e c o m b u s t i n g r e g i o n t h e
b u o y a n c y f lu x is n o t a c o n s e r v e d q u a n t i t y b u t i n c r e a s e s w i t h h e i g h t. A t t e m p t s
t o m o d e l t h e f l o w a b o v e n o n - p o i n t s o u r c e s in t h e n e a r f ie ld u s i n g p l u m e
c o n s e r v a t i o n e q u a t i o n s 11 a r e u n r e l i a b l e b e c a u s e o f u n c e r t a i n t y a s t o p r o fi le
s h a p e a n d e n t r a i n m e n t b e h a v i o u r i n t h i s r e g i o n . M o r e s i m p l y , t h e p l u m e -
s i m i l a r i t y m o d e l is o f t e n a p p l i e d c l o s e t o t h e s o u r c e u s i n g a v i r t u a l - o r i g i n
h e i g h t c o r r e c t i o n , o f t h e o r d e r o f t h e s o u r c e l e n g t h s c al e, t o c o m p e n s a t e f o r
d i s p a r i t ie s f r o m t h e s i m i l a r i t y f o r m . T h e p o s i t i o n o f t h e v i r t u a l o r i g i n o f a f ir e
p l u m e h a s b e e n t h e s u b j e c t o f s e v e r a l p r e v i o u s s t u d i e s (s ee , e .g . t h e s u m m a r i e s
o f G u p t a 12 a n d C o x a n d C h i t t y 13). T h e v a r i e t y o f c o r r e l a t i o n s l e a d s G u p t a 12
t o s u g g e s t t h a t t h e r e i s n o s i n g l e c o r r e l a t i o n w h i c h i s a p p l i c a b l e t o a l l
s i tu a t i o n s, a n d t h a t t h e c o r r e l a t i o n s a v a i l a b le c a n n o t b e g e n e ra l iz e d . T h e s e
o b s e r v a t i o n s t o g e t h e r i n d ic a t e th a t m o r e w o r k n e e d s to b e d o n e t o a d e q u a t e l y
u n d e r s t a n d t h e b e h a v i o u r o f f ir es in t h e n e a r f ie ld .
2 2 P r e s e n t m o d e l
I t is s h o w n b y R o o n e y a n d L i n d e n ~4 t h a t t h e s i m i l a ri t y s o lu t i o n s f o r th e m e a n
v e r ti c a l v e l o c it y w , r a d i u s b a n d r e d u c e d g r a v i t y y o f a n o n - B o u s s i n e s q p l u m e
a r e g i v e n b y
w = L F l B z 1:3 (8)
b M z ( ~ ) 1/2
= 9 )
\ P o
~t = N F 2 / 3 z
5 , 3 ( 1 0 )
w h e r e z is t h e h e i g h t a b o v e t h e ( v i rt u a l) o r ig i n . F is a c o n s e r v e d q u a n t i t y w i t h
t h e u n i ts o f b u o y a n c y f lu x ( th e d i m e n s i o n s o f F a r e I F ] =
L 4 T
3) , and L, M
a n d N a r e c o n s t a n ts r e l a te d t o e ac h o t h e r a n d t o th e e n t r a i n m e n t c o n s t a n t
b y
N I L 2 4
- -3 (11)
L M 2 N = l,/Tr (12 )
M = 6~/5 (13)
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s o t h a t
N 1 = rc2/3 6~ (9 ~) 1 /3 (14)
T h i s s o l u t i o n m a y b e r e l a t e d t o t h e f a r- fi e ld p l u m e f r o m a f i re u s in g t h e p l u m e
e q u a t i o n s a4 t o g i v e
V - 9Q (15)
2 c p p o T o
w h e r e Q i s t h e c o n v e c t i v e p o w e r o f t h e f ir e.
T h e s i m i l a r i t y s o l u t i o n , e q n s ( 8 ) - ( 1 0 ) , t e n d s t o t h e B o u s s i n e s q s o l u t i o n a s
P / P o - - * 1, a n d f r o m t h e e x p r e s s i o n o f e q n (1 0) f or t h e r e d u c e d g r a v i t y
9 = 9 A p / p i t i s e a s i l y s e e n t h a t t h i s w i l l o c c u r w h e n
Z5/3 >~ N F 2/3 (16)
g
o r , f r o m eq n s (1 4 ) an d (1 5 ) ,
0 ~ -~ 1 /5 0 2 /5
z >> 0.0 03 = 0-01 (17)
f o r h e i g h t in m a n d 0 i n W , a n d t a k i n g t y p i c a l v a l u e s o f t h e o t h e r p a r a m e t e r s .
G i v e n t h a t t h e f la m e l e n g t h I i n m h a s b e e n c o r r e l a t e d w i th t h e p o w e r o u t p u t
i n W a n d t h e s o u r c e d i a m e t e r D i n m b y 3
1 ~ 0 01507/5 - - l ' 02 D (18)
w e s ee t h a t t h e f ir e p l u m e m a y n o t b e B o u s s i n e s q u n t i l a h e i g h t o f s e v e ra l
f l a m e l e n g t h s a b o v e t h e s o u r c e .
T h e s i m i l a r i t y s o l u t i o n s f o r r e d u c e d g r a v i t y a n d p l u m e v e l o c i t y i n t h e
s t r o n g l y b u o y a n t ( n o n - B o u s s i n e s q ) c a se a r e c o n s i s te n t w i t h e x p e r i m e n t a l
o b s e r v a t i o n s f or t h e c e n t r e l in e m e a n t e m p e r a t u r e s a n d v e lo c it ie s i n s t ro n g l y
b u o y a n t p l u m e s o u t s id e t h e b u r n i n g r e g i o n , s u m m a r i z e d b y D e l i c h a t s io s . l° I t
i s a l s o i n t e r e s t i n g t o n o t e t h a t t h e s o l u t i o n f o r t h e s t r o n g l y b u o y a n t p l u m e
r a d i u s a g r e e s w i t h H e s k e s t a d ' s 15 i n t e r p r e t a t i o n o f M o r t o n ' s 16 r a d i a l t r a n s -
f o r m a t i o n .
T h e m a s s f l u x i n t h e p l u m e i s t h e n g i v e n b y
z 5/3 (19)
W = r t p b Z w = N l p o 2 c ~ o T o
N o t e t h a t t h is e x p r e s s i o n i s d i ff e r en t in d e n s i t y d e p e n d e n c e f r o m t h e m a s s f lu x
o f e q n (7 ) c o n j e c t u r e d i n P a p e r I , b u t d o e s , h o w e v e r , a g r e e w i t h t h e e x p r e s s i o n
f o r t h e m a s s f l u x i n C e t e g e n et al . , 17
W 0 -2 1p o ( g o t ) 1 / 3
--- 7-5/3
(20)
\ ppo To
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G. G. Roon e y P . F . L i nde n
C o m p a r i n g t h e c o e f f i c ie n t s i n e q n s (7 ) a n d ( 20 ) w i t h e q n (19 ), w e f i n d u s i n g
e q n ( 14 ) t h a t t h e n u m e r i c a l c o n s t a n t i n e q n (7) c o r r e s p o n d s t o a v a l u e o f t h e
e n t r a i n m e n t c o n s t a n t i n th e p r e s e n t m o d e l o f ~ = 0 .1 24 , a n d t h e c o n s t a n t i n
e q n (2 0) ( i n c o r p o r a t i n g t h e e s t i m a t e d 2 5 - 3 0 % d i f fe r e n c e b e t w e e n t o t a l a n d
c o n v e c t iv e h e a t o u t p u t ) c o r r e s p o n d s t o a v a lu e o f t h e e n t r a i n m e n t c o n s t a n t o f
= 0 .2 0. D a t a f o r t h e B o u s s i n e s q c as e , a s s u m m a r i z e d b y T u r n e r ,
T
s u g g e s t
a v a l u e f o r t h e e n t r a i n m e n t c o n s t a n t o f ~ = 0 08 3. T h u s , w h i l e t h e v a r i o u s
v a l u e s o f ~ a r e a ll o f t h e s a m e o r d e r o f m a g n i t u d e , t h e n o n - B o u s s i n e s q / f ir e -
p l u m e e n t r a i n m e n t c o n s t a n t is o b s e r v e d to b e l a r g e r b y a f a c to r o f a p p r o x i m -
a t e l y 2 .
T h i s d i sc r e p a n c y m a y b e d u e t o t h e e x p e r im e n t a l m e t h o d o f C e t e g e n
e t a l 1 7
T h e y r e p o r t , f ir s tl y , t h a t f i r e - p l u m e e n t r a i n m e n t is s u b j e c t t o i n c r e a s e b y
a m b i e n t d i s t u r b a n c e s a n d , s e c o n d ly , t h a t t h e i r m e t h o d o f m e a s u r i n g p l u m e
m a s s f lu x is l ik e l y t o o v e r e s t i m a t e b e c a u s e o f t h e a d d i t i o n a l a i r e n t r a i n e d i n to
t h e u p p e r l a y e r i n t h e h o o d b y d i s t u r b a n c e s a t t h e i n t er f a ce , a s t h e f ir e p l u m e
p l u n g e s t h r o u g h t h e i n te r f a c e i n t o t h e u p p e r l a y e r. I t is w o r t h p o i n t i n g o u t
h e r e th a t , i n t h e m e a s u r e m e n t s o f e n t r a i n m e n t b y R i c o u a n d S p a l d in g , ~9 t h e
p l u m e / j e t w a s e n c l o s e d b y a p o r o u s c y l i n d e r , w i t h e n t r a i n m e n t b e i n g e s t i -
m a t e d f r o m t h e m a s s f lu x th r o u g h t h e c y l in d e r r e q u i r e d to r e m o v e t h e
p r e s s u r e d i f f e r e n c e a c r o s s i t. T h e s e e x p e r i m e n t s l e d t o a n e s t i m a t e d v a l u e o f
~ 0 ' 0 8 f o r j e t s , w i t h a h i g h e r ( u n s p e c i f i e d ) v a l u e f o r p l u m e s , a n d , s i g n i fi -
c a n t l y , r e d u c e d e n t r a i n m e n t f o r t h e c a se o f c o m b u s t i n g p l u m e s / j e ts ( m a i n l y in
t h e p r e - m i x e d ca se ). T h i s e x p e r i m e n t a l m e t h o d w o u l d p r e s u m a b l y l a c k m o s t
o f t h e e r r o r a s s o c i a t e d w i t h t h e h o o d m e t h o d , t h e p o r o u s c y l i n d e r s h i e ld i n g
t h e p l u m e f r o m a m b i e n t d i s tu r b a n c e s , a n d t h e a b se n c e o f a n u p p e r l a y e r
d i s c o u n t i n g a n y p o s s ib i li ty o f e x c es s e n t r a i n m e n t .
3 N A T U R A L V E N T I L A T I O N O F S M A L L F I R E S
3 1 P r e v i o u s w o r k
R e l e v a n t p r e v i o u s w o r k i n n a t u r a l v e n t i l a t io n h a s b e e n p e r f o r m e d in P a p e r I,
a n d b y L i n d e n e t a l . 2 ° h e r e a f t e r r e f e r r e d t o a s P a p e r I I. P a p e r I c o n s i d e r s t h e
n a t u r a l v e n t i l a ti o n o f a c o m p a r t m e n t a b o v e a s m a l l f ire . T h e c o m p a r t m e n t
c o n s i s t s o f a h o r i z o n t a l c e i li n g c o n t a i n i n g a s i n g le v e n t , w i t h a v e r t i c a l s c r e e n
a r o u n d its p e r i m e t e r e x t e n d i n g d o w n w a r d s f r o m c e i li n g l ev e l t o s o m e d i s ta n c e
a b o v e t h e f l o o r . T h e f i re is a t fl o o r l ev e l, a n d t h e b u o y a n t g a s e s i t p r o d u c e s r is e
a n d c o l le c t in t h e c o m p a r t m e n t , f o r m i n g a h o t l a ye r . F i g u r e 1 g iv e s a s c h e -
m a t i c o f t hi s c o n f i g u r a t i o n . T h e h y d r o s t a t i c p r e s su r e i m b a l a n c e b e t w e e n t h e
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F i g 1 P a p e r I e x p e r i m e n t a l c o n f i g u r a t io n
i ns id e a n d o u t s id e o f th e c o m p a r t m e n t , c a u s e d b y t h e l o w e r d e n s i ty o f th is
l a y e r, d r i v e s a f lo w t h r o u g h t h e c o m p a r t m e n t . A f t e r a s h o r t t i m e t h is f lo w
s e tt le s to a s t e a d y s ta t e , so t h a t t h e d e p t h o f t h e h o t l a y e r b e c o m e s c o n s t a n t .
P a p e r I c o n s i d e r s b o t h t h e c a s e w h e r e t h e l e v el o f t h e h o t l a y e r is h i g h e r t h a n
t h e e d g e o f t h e s c r e e n , s o t h a t a l l f lo w o u t o f t h e c o m p a r t m e n t is t h r o u g h t h e
v e n t a n d t h e f lo w b e t w e e n t h e s c r e e n a n d t h e f l o o r is i n w a r d o n l y , a n d t h e c a s e
w h e r e t h e l e ve l o f t h e h o t l a y e r is l o w e r t h a n t h e e d g e o f t h e s c r ee n , w i th f l u id
f r o m t h e h o t l a y e r le a v i n g th e c o m p a r t m e n t b e l o w t h e s c r e e n a s w e ll a s
t h r o u g h t h e v e n t. I n t h i s s e c o n d c a s e, th e f lo w b e t w e e n t h e s c r e e n a n d t h e f l o o r
is b i d i r e c ti o n a l , w i t h h o t g a s e s l e a v in g t h e c o m p a r t m e n t a t t h e t o p o f t h e g a p ,
a n d a m b i e n t a ir f lo w i n g i n to t h e c o m p a r t m e n t l o w e r d o w n . O n l y t h e f ir st c a se
is r e le v a n t t o o u r p r e s e n t s t u d y , s o w e w i ll n e g l ec t t h o s e p a r t s o f P a p e r I w h i c h
p e r t a i n t o t h e s e c o n d .
D a t a a r e p r e s e n t e d f r o m e x p e r i m e n t s p e r f o r m e d o n a r i g a s d e s c ri b e d
a b o v e , w i t h t h r e e s e t t i n g s o f t h e v e n t a r e a , a n d t h r e e s e t t i n g s o f t h e s c r e e n
d e p t h . T h e f ir e is f r o m a r i n g g a s b u r n e r . T h e d i m e n s i o n s o f t h e c e il in g a n d t h e
f ir e p a r a m e t e r s a r e s e t o u t i n T a b l e 1. A s s t a t e d p r e v i o u s l y , w e a r e o n l y
c o n c e r n e d w i t h t h e c a s e w h e r e t h e h o t l a y e r is a b o v e t h e e d g e o f t h e s c r e e n,
a n d h e n c e w e o m i t t h e s c re e n d e p t h f r o m o u r c o n s i d e r a t io n s .
T h e a n a l y s i s in P a p e r I m a y b e d e s c r i b e d a s f o ll ow s . A h y d r o s t a t i c b a l a n c e
is p e r f o r m e d o n a b o x p a r t i a l ly f il le d w i t h b u o y a n t f l u id t o o b t a i n t h e
h y d r o s t a t i c a l l y d r iv e n m a s s f lu x t h r o u g h t h e o p e n i n g . T h e i n t e r f a c e h e i g h t
m a y t h e n b e p r e d i c t e d b y m a t c h i n g t hi s f l ux to t h e m a s s f l ux o f a b u o y a n t
p l u m e w i t h in t h e b o x ( g iv e n b y e q n (7)), w h i c h is t h e o n l y m e c h a n i s m o f f lu i d
t r a n s p o r t a c r o s s t h e b u o y a n t / a m b i e n t i n t e r f a c e .
T h e e x p r e s s i o n f o r t h e h o t - l a y e r d e p t h d th u s o b t a i n e d ( e q n (3 1) i n P a p e r I),
Kad 1/2 =
0 043(Zv
H
- - d 5 /2 (21)
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G. G. Rooney, P. F. Linden
TABLE 1
Paper I Experimental Parameters
Sym bol Qu antity Paper I value S.I. equivalent
hc Ce iling height 18 in 0'46 m
Ceiling length x width 24 x 36 in-' 0 61 x 091 m z
A, Ve nt areas 16, 24, 40 in 2 0.0 10 ,0 016, 0.026 m 2
Qf H eat ou tput 2 1 Btu s 1 2 2 kW
rg De pth of virtual origin 6 in 0-15 m
d e p e n d s o n l y u p o n t h e e ff e ct iv e a r e a o f t h e t o p o p e n i n g K a , w h e r e K i s
a c o n s t a n t l es s t h a n u n i ty ) , t h e r o o m h e i g h t H , a n d t h e d e p t h Z v o f t h e v i r t u a l
o r ig i n o f t h e f ire . F r o m c o n s i d e r a t i o n o f t h e a s p e c t r a t i o o f a p u r e B o u s s i n e s q
p l u m e , i t is s t a t e d t h a t t h e d e p t h o f t h e v i r tu a l o r i g in m a y b e t a k e n a s
a c o n s t a n t p r o p o r t i o n o f t h e h o r i z o n t a l d i m e n s i o n o f t h e f ir e (i.e. t h e s q u a r e
r o o t o f t h e f l o o r a r e a c o v e r e d b y t h e f ire ). T h e r e f o r e , b a s e d o n t h is a n a l y s i s , th e
d e p t h o f t h e h o t l a y e r is g o v e r n e d o n l y b y g e o m e t r i c a l f a c to r s , a n d h a s n o
d e p e n d e n c e o n t h e p o w e r o f t h e f ire .
P a p e r I u s e s t h e p l u m e m a s s f lu x in c o n j u n c t i o n w i t h th e a s s u m p t i o n t h a t
h e a t is c o n s e r v e d i n t h e p l u m e t o o b t a i n a n e x p r e s s i o n f o r t h e t e m p e r a t u r e o f
t h e c e i li n g l a y e r in t e r m s o f h e a t o u t p u t , l a y e r d e p t h , c e i li n g h e ig h t , a n d d e p t h
o f t h e h o t l a y e r. I t a l s o s t a t e s t h e f o r m o f t h e e x p r e s s i o n ( s im i l a r t o e q n (2 4)) t o
r e p l a c e t h e v e n t a r e a a in e q n ( 21 ) i n t h e c a s e w h e r e t h e s c r e e n b e c o m e s s o l o w
t h a t t h e a r e a o f t h e g a p b e t w e e n t h e s c r e e n a n d t h e fl o o r a l s o b e c o m e s o n e o f
t h e c o n t r o l l i n g f a c t o r s i n t h e s y s t e m .
P a p e r I I ( a n d e x t e n s i o n s t h e r e o f 2 1'2 2) c o n s i d e r s n a t u r a l v e n t i l a t i o n i n t h e
B o u s s i n e s q c a se , f o r a b o x o f h e i g h t H w i th s m a l l o p e n i n g s t o p a n d b o t t o m
s u c h a s th a t s h o w n i n F ig . 2 . I t c o n t a i n s a n a l y s i s s i m i la r t o t h a t d e s c r i b e d
a b o v e , u s i n g B e r n o u l li 's t h e o r e m a n d t h e B o u s s i n e s q p l u m e s i m i l a ri ty s o l u -
t i o n t o o b t a i n a n e x p r e s s i o n f o r t h e i n te r f a c e h e i g h t h o f
N - 3 : 2 = _ _
_ H 2 2 2 )
w h e r e
= h / H 2 3 )
i s t h e f r a c t i o n a l h e i g h t o f t h e i n t e r f a c e , a n d
K6 A¢I C
A
[ ½ K 2 / k )
a 2
+
a ~ ) ] , / 2 2 4 )
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90
Strongly buoyant plume similarity
C ~ z = H
t
91
9
+
z h
z O
Fig. 2. Steady-state natural ventilation.
245
is th e e f fe c ti v e a r e a , w h i c h is a f u n c t i o n o f th e t o p a n d b o t t o m o p e n i n g
a r e a s , a c a n d aA . A a l s o d e p e n d s u p o n a p r e s s u r e - l o s s c o e f f i c i e n t k ,
w h i c h a c c o u n t s f o r v e n t - e d g e e ff e ct s o n t h e i n f lo w a t A , a n d a d i s c h a r g e
c o e f fi c ie n t K w h i c h p a r a m e t e r i z e s t h e e f fe c t o f th e vena con t rac ta a t t h e o u t l e t
C . E x p r e s s i o n o f e q n (2 2) is u s e d t o e x a m i n e t h e i n t e r f a c e b e h a v i o u r i n th e
c a s e o f a s i n g le p l u m e , m u l t i p l e p l u m e s , a n d p l u m e s o f d i f f e r e n t s t r e n g t h s .
V e r t i c a l ly d i s t r i b u t e d s o u r c e s o f b u o y a n c y w i t h in t h e b o x a r e a l s o c o n s i d e r e d .
T h e f l o w i s a g a i n a s s u m e d t o b e s t e a d y , a n d f l u i d l e a v e s t h e b o x b y t h e t o p
v e n t o nl y . C o m p a r i s o n s a r e m a d e w i th e x p e r i m e n t s r e c r e a t in g th e a b o v e
s i t u a t i o n s .
N o t i c e t h a t t h e i n t e r f a c e p o s i t i o n e x p r e s s i o n i n P a p e r I, e q n ( 21 ), is t h e s a m e
a s t h a t i n P a p e r I I , e q n (2 2), f o r t h e l i m i t o f l a r g e l o w e r - o p e n i n g a r e a , a A - ~ ,
i f t h e i n t e r f a c e h e i g h t i n c l u d e s t h e d e p t h o f th e v i r t u a l o r ig i n , a n d t h e
u p p e r - o p e n i n g a r e a i n c l u d e s a d i s c h a r g e f l o w - c o n t r a c t i o n c oe f fi ci e nt .
3 . 2 N a t u r a l v e n t i la t io n o f a n o n B o u s s i n e s q p l u m e
W e m a y u s e t h e m o d e l ~4 o f a n o n - B o u s s i n e s q f ir e p l u m e t o r e - e x a m i n e t h e
p r o b l e m o f n a t u r a l v e n t i l a t io n i n t h e n o n - B o u s s i n e s q c a se . W e r e fe r t o F i g . 2
f o r a s c h e m a t i c o f t h e f l o w c o n s i d e r e d . T h is c o n s i s t s o f a p l u m e i n a b o x o f
h e i g h t H w i t h o p e n i n g s a t t h e t o p a n d t h e b o t to m . T h e p l u m e w i t h i n t h e b o x
m a i n t a i n s a l a y e r o f f lu id l ig h t e r t h a n a m b i e n t ( in t h e p o s i ti v e ly b u o y a n t c a se )
in t h e u p p e r p a r t o f t h e b o x . A m b i e n t f lu id e n t e r s t h e b o x a t A , is e n t r a i n e d b y
t h e p l u m e , c r o s s e s t h e d e n s i t y i n t e r f a c e a s p l u m e f l u id a t B , a n d l e a v e s t h e b o x
a t C . T h e f l o w w e c o n s i d e r is s te a d y , s o t h a t t h e i n t e r f a c e i s a t a f i x e d h e i g h t
z = h a b o v e t h e fl oo r . U s i n g s u c h a m o d e l m e a n s w e ta c i t ly a s s u m e t h a t t h e
f lu id in t h e s y s t e m is i n h o m o g e n e o u s i n d e n s i t y a n d m a y c h a n g e its d e n s i t y b y
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246 G. G. Rooney, P. F. Linden
s i m p l e m i x i n g , b u t b y n o o t h e r m e a n s , i .e. t h e f lu id is i n c o m p r e s s i b l e e v e r y -
w h e r e e x c e p t a t t h e p o i n t o f t h e p l u m e s o u rc e .
A s d e t a i le d in t h e A p p e n d i x , w e m a y s h o w t h a t t h e h o t l a y e r is h o m o g e n e -
o u s , a n d o b t a i n e x p r e s s i o n s fo r t h e f r a c t i o n a l h e i g h t ~ o f t h e i n t er f ac e , a n d t h e
d e n s i t y P l o f t h e u p p e r l a y e r ,
1 - ~ = g ~ 2 5 )
a n d
w h e r e
a n d
( ) -
F2/3 5, 3 0 (~ ) 26)
-~ = 1 + ~ ~ - .
P o g H
= h / H 27)
O 1 / 2 g a A a c
= [½ ( ( K 2 / k ) a 2 + a 2 / 0 ) ] 1 / 2 28)
T h e r e l a t i o n s h i p , e q n 2 5), r e p r e s e n t s t h e d e p e n d e n c e o f t h e f r a c ti o n a l h e i g h t
o f t h e i n te r f a c e u p o n t h e p l u m e p r o p e r t i e s , t h e p o w e r o f t h e fi re a n d t h e
g e o m e t r y o f t h e b o x . T h e p l u m e p r o p e r t i e s a re r e p r e s e n t e d b y N , w h i c h
d e p e n d s u p o n t h e e n t r a i n m e n t c o n s t a n t c~. T h e d e p e n d e n c e u p o n t h e b o x
g e o m e t r y is c o n t a i n e d in A / H 2. T h e n o n - B o u s s i n e s q n a t u r e o f t h e f lo w , as
r e p r e s e n t e d b y t h e p o w e r o f t h e fir e, is c o n t a i n e d i n O , t h e n e w t e r m i n th i s
m o d e l c o m p a r e d w i t h th e B o u s s i n e s q ca s e, e q n s 2 2) - 2 4) . N o t e t h a t i n t h e
B o u s s i n e s q l im i t , O ---, 1, t h e p o w e r c e a s e s t o b e c o m e a f a c t o r a n d A / H 2 is
a p u r e l y g e o m e t r i c a l t e r m , r e p r e s e n t i n g th e d e p e n d e n c e o f t h e i n t e rf a c e h e i g h t
o n t h e b o x h e i g h t a n d t h e e ff ec ti ve a r e a s o f t h e t o p a n d b o t t o m o p e n i n g s . I n
t h i s l i m i t , t h e m o d e l i s i d e n t i c a l t o t h a t f o r o n e p l u m e b y C o o p e r a n d
L i n d en .2 1
3 . 3 C o m p a r i s o n w i t h P a p e r I
W e n o w l et aA --* oC t o c o m p a r e t h e a b o v e m o d e l w i t h t h e e x p e r i m e n t s i n
P a p e r I . In t h i s l i m i t w e h a v e t h a t
.~ = x ~ K O a c
s o t h a t e q n 2 5) b e c o m e s
29)
K a c
H 2
(30)
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trongly buoyant plume similarity
247
TABLE 2
Cases Conside red in Model Simula-
tions
H
0 46 m 0 61 m
0 083 1 2
0 124 3 4
0 20 5 6
This equation is different from the expression in Paper I, eqn (21), in tha t it
contains the term O which is a measure of the non-Boussinesq nature of the
fire plume.
We may solve eqns (30) and (26) simultaneously for the various values of the
entrainment constant ~ obtained from Paper I (~ = 0.124), Cetegen e t a l 17
(~ = 0 20), and Turner 18 (~ = 0 083), and the effective enclosure height, H, as
either the actual enclosure height in Paper I (H = 0.46 m), or the height used
in Paper I as obtained from the enclosure height plus the depth of the virtual
origin (H = 0 61 m). We cannot estimate the virtual-origin depth from the
expression given in Cetegen e t a l 1 7 as the flame height of the fire in Paper I is
not recorded. The cases we may therefore consider are listed in Table 2.
The best results for tempera ture vs. depth of the hot layer, and for hot-layer
depth vs. upper-opening area, as compared with the experimental results of
Paper I, are those for cases 2 and 3. These results are presented in Figs 3 and 4,
together with the results of the Paper I small-fire theory outlined above.
Case 2 shows good agreement with both temperature and depth for smaller
ceiling-layer depths, whereas case 3 accurately predicts the upper-layer
temperature, but underpredicts the layer depth. It may be seen from these
figures that the Boussinesq model of Paper I gives results of similar accuracy
to the non-Boussinesq model, and it is interesting to note from Fig. 4 that both
the present model and the model of Paper I predict a continual increase of
layer depth with vent area, whereas the data points seem to show a gradual
levelling-off of layer depth, despite the increase in vent area. This interpreta-
tion of the data is somewhat uncertain as the size of the possible errors are
unknown, however, it may indicate a more complicated behaviour at the vent
than the simple flow contraction parameterized by K.
3 4 P a r a m e t e r o p t i m i z a t io n
We may further consider optimizing the present model parameters to obtain
the best fit to the Paper I data. The results of solving the present model with
8/9/2019 61rl97 - Strongly Buoyant Plume Similarity and 'Small Fire' Ventilation
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2 4 8 G. G. Rooney P. F. L inden
2 5 0
5 0
0
2 0 0
o
I . { ; ,
I 0D
eO
£ 5
' 0 . 1 5
e-,
, .o
0 . 2
0 2 5
0 0 0 5
i
P a p e r I e x p e r i m e n t s
P a w r I t h e o r y • -
C a s e 2 - - -
C a s e 3
/
/
/
j ~
I I f I
0 0 5 0 , 1 ( I . 1 5 0 . 2 ( 1 ,2 5
Depth of ce i ling layer m )
F i g . 3 . C e i l i n g l a y e r t e m p e r a t u r e i n c r e a s e v s . d e p t h .
P a p e r I e x p e n m e n l s
< )
P a p e r
I l h e o r v -
C a s e 2 - - -
C a s e 3 - -
/ /
/
/ . '
, /
, /
i
0 0 1
f ~ f
j j .
f J . . - *
/
/ / ' ~ , ,
@ , ,
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o
k . t
, . )
[..
2 4 0
2 2 0
2 ( X 3
1 8 0
1 6 0
1 4 0
1 2 0
1 0 0
8 O
0 . 0 5
trongly buoy nt lu me imilarity
j J
/ / / ,
/ / / / , , / , / : / / i
/ / ~ , / / . /i
/ , ~ / : / / / /
< ......... j . ¸-
P t
0 1 0 . 1 5 0 . 2 0 . 2 5
Depth o f cei ling layer (m)
F i g . 5 . T e m p e r a t u r e v s . d e p t h v a r y i n g ~ .
a l p h a = 1 1
- -
a l p h a = . 1 2 . . . . .
a l p h a = . 1 3 . . . . .
alph~
= . 1 4 . . . .
a l p h a = . 1 5 . . . .
P a p c r l c x p t , O
I
0 . 3
2 4 9
n o v i r tu a l o r i g i n f o r v a r y i n g v a l u e s o f t h e e n t r a i n m e n t c o n s t a n t 0~ a r e p r e s e n t -
e d i n F i g s 5 a n d 6 . I t c a n b e s e e n t h a t a t t e m p t i n g t o o p t i m i z e t h e f i t w i t h t h e
o t h e r p a r a m e t e r s a t t h e i r p r e s e n t v a l u e s w i l l l e a d t o a b a d f i t o n b o t h g r a p h s .
T h a t is , t h e r a n g e o f e n t r a i n m e n t c o n s t a n t 0 .1 1 < ~ < 0 15 c o n t a i n s t h e
o p t i m u m f it t o t h e d a t a o f t e m p e r a t u r e d i ff er e n c e v s. l a y er d e p t h , b u t o n l y
b e g i n s t o t o u c h o n t h e d a t a o f l a y er d e p t h v s. ar e a. T o t w o s i gn i f ic a n t f ig u r es ,
t h e b e s t fit o n t e m p e r a t u r e d i ff e re n c e ( F i g u r e 5 ) c o m e s f r o m a v a l u e o f t h e
e n t r a i n m e n t c o n s t a n t o f e - - 0 - 1 3, w i t h z e r o v i r t u a l o ri g in .
A p o s s i b l e c a u s e f o r t h e p o o r f it o f t h e v e n t a r e a v s . l a y e r d e p t h a t t h i s v a l u e
o f e m a y b e t h e v a l u e o f t h e d i s c h a r g e c o e f fi c ie n t K . T h e v a l u e o f K = 0 .6 h a s
b e e n c h o s e n i n P a p e r s I a n d I I a s r e p r e s e n t i n g t h e v a l u e f o r a l a m i n a r v e n a
c o n t r a c t a b u t it i s l ik e l y t h a t a d d i t i o n a l t u r b u l e n t d i s s i p a t i o n a t t h e o u t l e t i n
t h e P a p e r I e x p e r i m e n t s i m p l y a l o w e r v a l u e t h a n t hi s. W e m a y t h e r e f o r e h o l d
t h e e n t r a i n m e n t c o n s t a n t a t t h e o p t i m u m v a l u e (f or t h e t em p e r a t u r e d a t a ) o f
= 0 - 1 3 a n d s e e k t o o b t a i n a b e t te r fit t o t h e v e n t - a r e a d a t a b y v a r y i n g K .
C l e a r l y , a d d i t i o n a l d i s s i p a t i o n w i l l l e a d t o a s m a l l e r o u t f l o w t h r o u g h t h e
v e n t , a n d s o w e s h o u l d c o n s i d e r v a l u e s o f K l es s th a n 0 -6 . A l o w e r v a l u e o f
K w i l l r e d u c e t h e e f f e c t i v e u p p e r - v e n t a r e a K a c w h i c h s h o u l d t h e r e f o r e l e a d
t o a g r e a t e r l a y e r d e p t h ( l o w e r i n t e r f a c e ) f o r t h e s a m e a c t u a l v e n t a r e a a c .
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2 5 0
0 . 0 5
~ 0 . 1
•~ 0 15
O
-5
~9
0 . 2
G. G. Rooney P. F. Linden
i i i i i i
/ / . ~ @
/ / / / • , ~
: . . ,
: , ,
¢ I 1
[ ) 0 1 0 . 0 2 5
( 1 2 5 I I I
0 . 0 0 5 0 . 0 1 5 0 . 9 2 0 0 3
Area of ceil ing vent (m2 )
F i g 6 D e p t h v s . a r ea v a y i n g ~
i
l p h = . 1 1 - -
l p h = . 1 2 - - -
l p h = . 1 3 . . . .
l p h
= . 1 4
l p h - . 1 5 . . . . .
P a p e r l e x p t .
T h e c u r v e f o r ~ = 0 1 3 o n F i g . 6 s h o w s t h a t a n i n c r e a s e i n d e p t h i s i n d e e d
r e q u i r e d t o p r o d u c e a b e t t e r fi t. W e f i n d t h a t , t o o n e s i g n i f ic a n t f i g u r e , t h e
o p t i m u m v a l u e o f t h e d i s c h a r g e c o e f f i c i e n t is f o u n d t o b e K = 0. 4, a n d t h e
r e s u l t s o f t h e s i m u l a t i o n w i t h t h e s e v a l u e s (0~ = 0 1 3 , K = 0 - 4, n o v i r t u a l
o r i g i n ) a r e p r e s e n t e d i n F i g s 7 a n d 8 .
W h e n c o n s i d e r i n g o p t i m i z a t i o n w .r .t , t h e v e n t -a r e a d a t a , g re a t er e m p h a s i s
h a s b e e n p l a c e d o n t h e d a t a p o i n t s f o r th e t w o l o w e r v e n t - a r e a v a l u e s. T h i s
i s b e c a u s e , b e t w e e n t h e m i d d l e a n d l a r g e s t - a r e a s e t s o f p o i n t s , t h e v a lu e o f
t h e v en t a r e a h a s i n c r e a s e d b y m o r e t h a n h a l f w i t h o u t t h e m e a s u r e d l ay e r
d e p t h c h a n g i n g b y a n y g r e a t a m o u n t . T h i s is p o s s i b l y d u e t o t h e a c t u a l f lo w
d e p a r t i n g f r o m o u r i d e a l p i c t u r e a s t h e la y e r d e p t h d e c r e a s e s a n d t h e v e n t a r e a
i n c r ea s e s , e .g . b y t h e m o m e n t u m w i t h w h i c h t h e p l u m e e n t e r s t h e u p p e r la y e r
h a v i n g s o m e e f f ec t o n t h e u p p e r - l a y e r f l o w . W e t h e r e f o r e e x p e c t t h e d a t a p o i n t s
a t l o w e r v e n t a r e a s t o b e t t e r r e p r e s e n t t h e f l o w i n t h e p r e s e n t m o d e l .
3 . 5 C o m p a r i s o n w i t h B o u s s i n e s q v e n t i la t i o n
W e m a y n o w m a k e a s a m p l e c o m p a r i s o n o f o u r m o d e l w i th it s B o u s s i n e s q
l im i t , i n o rd e r t o o b t a i n a n e s t im a t e f o r t h e m a g n i t u d e o f n o n - B o u s s i n e s q
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0
©
0
k~
¢
E
E-
2 2 0
2 0 0
1 8 0
1 6 0
1 4 0
1 2 9
1 0 0
8 0
0 , 0 5
trongly buoyant plume similarity
/
0 . 1 0 . 1 5 0 . 2 0 . 2 5
D e p t h o f c e i l in g l a ye r m )
F i g . 7 . T e m p e r a t u r e v s . d e p t h , o p t i m u m v a l u e s .
i
S i m u l a t i o n - -
P a p e r l e x p t . < ~
2 5 1
0 . 3
0
e~
0
0.1
0 . 1 2
0 , 1 4
0 . 1 6
0 . 1 8
0 . 2
0 . 2 2
0 . 2 4
0 . 0 0 5
i
S i m u l a t i o n - -
P a p e r I e x p t .
0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5
A r e a o f c e i l in g v e n t n ~ )
F i g . g . D e p t h v s . a r e a , o p t i m u m v a l u e s .
0 . 0 3
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trongly buoyant plum e similarity 2 5 3
r-
e-
0 . 36 i i i F i i ~ t i i i i i i i
N o n B o u s s i n e sq
B o u s s i n e s q . . . .
0 3 7
0 . 3 8 ~ ~ _
0 . 3 9
0 4
\
04
0 . 4 2
0 . 4 3
0 . 44 I I I I I I I I I I I I I I 1 I I
1 1 0 5 0
C o n v e c t i v e p o w e r o f f ir e k W )
F i g . 1 0 . D e p t h o f u p p e r l a y e r v s . c o n v e c t i v e p o w e r o f t h e fir e , c a s e 3 v a l u e s .
5 O
b
40
m
¢P
3
~. 20
g
E
1 0
U p p e r L a y e r
A m b i e n t . . . . .
/ /
f
J
i i f
I I 0 5 0
C o n v e c t i v e p o w e r o f f ir e k W )
F i g . 1 1 . T e m p e r a t u r e o f u p p e r la y e r v s. c o n v e c t i v e p o w e r o f t h e fir e , o p t i m i z e d v a l u e s .
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2 5 4 G. G. Rooney. P. F.
i n d e n
¢...
e ,
m
Q
0.52
0.53
0.54
0.55
0.56
0.57
0.58 ~-
0.59 [
0 6 ] I
i l
0.61
I
i i i i
N o n B o u s s m e s q
- -
Bouss lnesq - -
\ \
\
i i
I
I
5O
Convective power of fire kW )
Fig. 12. Dep th o f upper layer vs. convective power of the fire, optimized values.
Th e r e f o r e , w i t h e it h e r s et o f p a r a m e t e r s in t h i s c a se, t h e m e t h o d in P a p e r s
I a n d I I o f c a l c u la t in g t h e u p p e r - la y e r t e m p e r a t u r e f r o m c o n se r v a t io n o f h e a t
f lu x o r r e d u c e d g r a v i t y , a n d c a l c u la t in g t h e l a y e r d e p t h f r o m t h e Bo u ss in e sq
inter face express ion , ought to g ive reasonably accurate resu l t s .
4 C O N C L U S I O N S
W e h a v e c o n s id e r e d t h e n a t u r a l v e n t i l a t io n o f fi re s in e n c lo su r e s, u s in g t h e
n o n - B o u s s i n e s q p l u m e t h e o r y d e v e l o p e d e l s e w h e r e . T h i s i s a n i m p o r t a n t
c o n c e r n , as p r e v io u s m o d e l s o f n a t u r a l v e n t i l a t io n h a v e in c o r p o r a t e d n o n -
Bo uss in esq co rrec t io ns e i ther incorrec t ly based on conjec ture ) , or no t a t a ll .
W e h a v e t h e r e f o r e a p p l i e d o u r p r e se n t m o d e l t o th e p r o b le m , a n d c o m p a r e d i t
w i t h e x p e r i m e n t a l d a ta f r o m T h o m a s e t a l 2 F r o m t h i s c o m p a r i s o n , t h e m o s t
a p p r o p r ia t e v a lu e o f t h e e n t r a in m e n t c o n s t a n t h a s b e e n se l e c t ed , a n d u se d in
t h e s im u la t io n o f a m o d e l p r o b le m o f sm a l l f ir e s in a r o o m - s i z e d , v e n t i l a te d
c o m p a r t m e n t . W e h a v e a l s o o p t i m i z e d t h e e n t r a i n m e n t c o n s t a n t a n d t h e
o u t p u t d i s c h a r g e c o e f f i c ie n t t o o b t a in a b e s t fi t t o t h e se d a t a , a n d u se d t h e se
v a lu e s t o s im u la t e t h e m o d e l p r o b le m .
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trongly buoyant plume similarity 255
O p t i m i z i n g t h e e n t r a i n m e n t c o n s t a n t, r a t h e r t h a n u s in g t h e m o s t a c c u r a te
l a b o r a t o r y m e a s u r e m e n t , is p e r m i ss i b le g iv e n t h e o b s e r v a t i o n s o f C e t e g e n
et al . 17 a n d o t h e r s t h a t s m a l l a m b i e n t d i s t u r b a n c e s c a n g r e a t l y i n c r e a s e t h e
m a s s f l u x i n a p l u m e . A s i t is u n l i k e l y t h a t a n a c c i d e n t a l fi re in a v e n t i l a t e d
b u i l d i n g w il l p r o c e e d i n a c o m p l e t e l y q u i e s c e n t a m b i e n t , t h e b e s t p r o c e d u r e
w h e n m a k i n g p r e d i c t i o n s o f v e n t i la t i o n b e h a v i o u r is p r o b a b l y t o e n s u r e t h a t
m o d e l e s t im a t e s o f c e il in g - la y e r t e m p e r a t u r e a n d d e p t h a r e a c c o m p a n i e d b y
c l e a rl y s t a t e d l ik e l y e r r o r s a r i s in g f r o m t h e p o s s i b l e r a n g e o f v a r i a t i o n o f ~.
S e c o n d l y , i t c a n b e s e e n f r o m F i g s 9 - 1 2 t h a t v a r y i n g t h e d i s c h a r g e c o e f fi c ie n t
K r e su l ts i n a si m i la r f r a c ti o n a l c h a n g e in th e la y e r d e p t h ( ~ 4 0 ) , b u t a m u c h
s m a l l e r c h a n g e i n t e m p e r a t u r e , a s t h e l a y e r - d e p t h c h a n g e i s s t i l l s m a l l c o m -
p a r e d t o t h e p l u m e l e n g th . O p i m i z i n g K ( o r e v e n p o s s ib l y in t r o d u c i n g a f u n c-
t io n d e p e n d i n g o n o u t l e t a re a a n d / o r v e lo c it y, sa y) w ill u n d o u b t e d l y l e a d t o
a m o r e a c c u r a t e m o d e l .
F i n a ll y , a l t h o u g h w e h a v e s h o w n t h a t t h e n o n - B o u s s i n e s q r e g i o n o f a f ir e
p l u m e m a y e x t e n d s e v e ra l f la m e l e n g t h s i n t o t h e f ar f ie ld , t h e m o d e l s i m u l a t i o n
i n d ic a t e s t h a t t h e n o n - B o u s s i n e s q d e p a r t u r e s f r o m B o u s s i n e s q v e n ti l a ti o n
v a l u e s a r e s m a l l , a n d s o B o u s s i n e s q t h e o r y c a n b e a p p l i e d w i t h r e a s o n a b l e
c o n f i d e n c e w h e n s i m u l a t i n g t h e n a t u r a l v e n t i l a t i o n o f s m a l l fir es . A s s t a t e d
p r e v i o u s l y , th e t e r m ' s m a l l f ir e ' r e fe r s s o le l y t o t h e g e o m e t r y o f t h e f i re p l u m e ,
a n d i m p a r t s n o i n f o r m a t i o n a b o u t t h e p o w e r o u t p u t o f t h e fire . H o w e v e r , fo r
a f ir e w i t h a c o n v e c t i v e p o w e r o u t p u t s u c h t h a t n o n - B o u s s i n e s q e ff ec ts w o u l d
s i g n if i c a n tl y a l te r t h e v e n t i l a t i o n b e h a v i o u r , i t se e m s t h a t t h e l i k el y g e o m e t r i -
c a l si ze o f s u c h a f ir e w o u l d r e n d e r a ' s m a l l- f ir e ' m o d e l o f t h e t y p e u s e d h e r e
i n a p p r o p r i a t e .
A C K N O W L E D G E M E N T S
T h i s w o r k w a s s u p p o r t e d b y a r e se a rc h s t u d e n t s h i p f r o m t h e D e p a r t m e n t o f
E d u c a t i o n fo r N o r t h e r n I re la n d , a n d b y a C A S E s t u d e n t s h ip fr o m t h e H e a l t h
& S a f e t y E x e c u t i v e .
R E F E R E N C E S
1. Kramer , C. & Gerhard t , H . J . , Ven t i l a t ion and hea t smoke ex t rac t ion f rom
industr ial bui ld ings . J . W i n d E n g n 9 I n d . A e r o d y n . 29 (1988) 309-35.
2 . Th om as , P . H . , Hink ley , P . L ., The obald , C. R. & Simm s, D. L . , Inves t iga t ions in to
the f low of ho t gases in r oof ven t ing . F i r e R e s e a r c h T e c h n ic a l P a p e r N o . 7, Fire
Research Sta t ion , Wat fo rd , UK, 1963 .
3. Drysdale, D., A n I n t r o d u c t i o n t o F i r e D y n a m i c s . Wiley, New York, USA, 1985.
4 . Hink ley , P . L ., Ra tes o f 'p ro du ct io n ' o f ho t gases in roo f ven t ing exper iments .
F i r e
Sa f e t y J . 10 (1986) 57-65.
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5 6 G. G. Rooney P. F. Linden
5 . T h o m a s , P . H . , O n f o r m u l a e f o r t h e m o v e m e n t o f s m o k e i n f ir es . F ire Sci . Tech. 9
(1989) 1 3
6 . D e m b s e y , N . A ., P a g n i , P . J . W i l l i a m s o n , R . B ., C o m p a r t m e n t f ir e n e a r - f ie l d
m e a s u r e m e n t s . Fire Sa je ty J . 24 (1995) 383-419.
7 . Y i h , C . S., F r e e c o n v e c t i o n d u e t o a p o i n t s o u r c e o f h e a t. I n Proc . 1 s t US Na t .
Congr . App l . Mech . 1952, pp . 941-7 .
8 . M o r t o n , B . R ., T a y l o r , G . I . T u r n e r , J . S ., T u r b u l e n t g r a v i t a t i o n a l c o n v e c t i o n
f r o m m a i n t a i n e d a n d i n s t a n t a n e o u s s o u r c e s . Proc . Roy . Soc . A234 (1956) 1 23.
9 . M c C a f f r e y , B. J ., P u r e l y b u o y a n t d i f f u s i o n f l am e s : s o m e e x p e r i m e n t a l r e s u lt s ,
N a t i o n a l B u r e a u o f S t a n d a r d s , N S B I R 7 9- 19 10 , 1 97 9.
1 0. D e l i c h a t s i o s , M . A ., O n t h e s i m i l a r i ty o f v e l o c i ty a n d t e m p e r a t u r e p r o f i le s in
s t r o n g ( v a r ia b l e d e n s it y ) t u r b u l e n t b u o y a n t p l u m e s . Combust. Sci . Technol. 60
(1988) 253-66.
1 1. G u p t a , A . K . , K u m a r , S . S i n g h , B ., P l u m e a n a l y s i s a b o v e f i n it e -s i z e f ir e s o u r c e s
I n Fire Safe ty Sc ience Proc . 3rd In t . Syrup. 1991, pp . 445 54.
1 2 . G u p t a , A . K . , F i r e - p l u m e t h e o r i e s a n d t h e i r a n a l y s i s . J. Appl. Fire Sci . 2 (1993)
269 98.
1 3. C o x , G . C h i t t y , R ., S o m e s o u r c e - d e p e n d e n t e f fe c ts o f u n b o u n d e d fi re s Combus t .
F l a m e 6 (1985) 219 32.
1 4. R o o n e y , G . G . L i n d e n , P . F . , S i m i l a r i ty c o n s i d e r a t i o n s f o r n o n - B o s s i n e s q
p l u m e s i n a n u n s t r a t if i e d e n v i ro n m e n t . J . F lu id Mech . 318 (1996) 237 50.
15. H e s k e s t a d , G . , F i r e p l u m e a i r e n t r a i n m e n t a c c o r d i n g t o t w o c o m p e t i n g a s s u m p -
t ions . Proc . 21s t In t . Symp . on Combus t ion 1986, pp . 111 20.
1 6. M o r t o n , B . R ., M o d e l l i n g fi re p l u m e s . Proc . lOth In t . Symp. on Combus t ion 1965,
pp . 973-82 .
1 7. C e t e g e n , B . M . , Z u k o s k i , E . E . K u b o t a , T . , E n t r a i n m e n t i n t h e n e a r a n d f a r f ie l d
o f f i re p lum es . Combust. Sci . Technol. 39 (1984) 305 31.
18. T u r n e r , J . S., T u r b u l e n t e n t r a i n m e n t : t h e d e v e l o p m e n t o f t h e e n t r a i n m e n t a s s u m p -
t i o n , a n d i ts a p p l i c a t i o n t o g e o p h y s i c a l f lo w s . J . F lu id Mech . 173 (1986) 431 71.
1 9. R i c o u , F . P . S p a l d i n g , D . B ., M e a s u r e m e n t s o f e n t r a i n m e n t b y a x i s y m m e t r i c a l
t u r b u l e n t j e t s . J . F lu id Mech . 8 (1961 ) 21 32.
20 . L i nd en , P . F . , Lane -Se r i f , G . F . Sm eed , D . A ., E m pt y in g f i ll i ng boxes : t he f lu id
m e c h a n i c s o f n a t u r a l v e n t i la t io n . J . F l u i d M e c h . 212 (1990) 303 14.
2 1. C o o p e r , P . L i n d e n , P . F . , N a t u r a l v e n t i l a t i o n o f a n e n c l o s u r e c o n t a i n i n g t w o
b u o y a n c y s o u rc e s. J . F l u i d M e c h . 311 (1996) 153 76.
2 2. L i n d e n , P . F . C o o p e r , P ., M u l t i p l e s o u r c e s o f b u o y a n c y i n a n a t u r a l l y v e n t i l a t e d
e n c l o s u r e . J . F lu id Mech . 311 (1996) 177 92.
A P P E N D I X
W e r e f e r to F i g . 2 f o r t h e p o s i t i o n s A , B , C e t c . i n t h e v e n t i l a t i o n s y s t e m .
F o r t h e s t e a d y - s t a t e c a se w i t h t h e p r e s e n c e o f a p l u m e , w e m u s t r e p la c e t h e
c o n s e r v a t i o n o f v e r t i c a l v o l u m e f l u x w i t h t h e c o n s e r v a t i o n o f v e r ti c a l m a s s
f lu x , i.e . t h e v e r t ic a l m a s s f l u x W t h r o u g h a n y h o r i z o n t a l p l a n e m u s t b e
c o n s t a n t , a n d i n p a r t i c u l a r ,
W A = W B : W C A1)
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Strongly buoyant plum e similari~ 57
We note the difference that the presence of the plume makes. It provides
a mechanism for transferring fluid from the lower layer to the upper whilst
keeping the interface steady. Without the plume, the interface would have to
move, and we would be forced to have conservation of volume flux. We may
interpret the change to the conservation of mass flux in, e.g. the case of thermal
buoy ancy as showing that the ambient air is heated at the plume source and
expands by a significant amount, so that a grea ter volume flux is needed at the
exit to maintain the steady state.
Paper II uses the conservation of buoyancy flux between B and C to
demon strat e that the density of the upper layer is uniform. We may obtain the
same result here, either from the conservation of buoyancy flux or the
conservation of pseudo-buoyancy flux F,
F B = F c , W B = W c = ~ p B = p c
~P[h _z _H = PI, const (A2)
i.e. the upper layer is of uniform density P l equal to the plume density at B.
We may also use Bernoulli s theorem to obtain a relationship between the
entry and exit velocities and the hydrostatic head,
w 2 = 2 g ( H - h ) w 2 P o
(a3)
k Pl
where k( < 1) is a pressure loss coefficient to account for vent-edge effects upon
the inflow at A. This is almost identical to the Boussinesq case, except for the
factor of
P o / P l
on the r.h.s., which would be unity in that case.
If we denote the area of an opening by a, we have from eqn (A1) that
w ~ = w ~ \ K a c p ,a a p o ) 2
(a4)
where K is a discharge coefficient, as used in Paper I and by Cooper and
Linden. z~ Combin ing this mass balance with eqn (A3) gives
2 g ( H - - h)
(A5)
w 2 = ( 1 / k ) ( p o / p l ) + ( a A p o / K a c P x ) 2
We may now obtain the interface height from the plume similarity solution,
eqns (8)-(10) by matching the mass flux in the plume at B with the mass flux at
A obtained from the velocity at A.
Rear ranging eqn (10) gives the expression for the ratio of plume to ambient
densities,
p 1
P o 1 + ( N F Z / 3 / g ) z- 5/3 (A6)
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5 8 G. G. Rooney, P. F. Linden
and so the ratio of the densities in the upper and lower layers is given by
Pl _ P = 1
P o P o z = h 1 + ( N F Z / 3 / g ) h - s /3 = O ( h ) , say A7)
This is equivalent to obtaining the upper-layer temperatu re from conserva-
tion of heat flux in the plume, which is the method used in Paper I, or to
obtaining the reduced gravity of the upper layer from the reduced gravity of
the plume, as in Paper II.
We may also obtain the mass flux at B, which is given by
W n = 7 z w b 2 p [ z = h = N - l p o F 1 / 3 h S / 3 A8)
Hence, from eqns A1), 10), and using
W 2 p0 2 2 a9)
~-- W A a A
eqn A5) becomes the expression for the interface height,
~ 5 ) 1 / 2 N - 3 / 2 ~1
= A I O )
where
= h / H A 11)
is the fractional height of the interface, and
- - - - 0 1 / 2 K a A a c
A12)
[ ½ ( ( K 2 / k ) a 2 + ( a ~ / O ) ) ] / 2