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te n so r a n d o f th e s tre ss te n so r d e fin e d b y E q . 3 )
[ g / c m .s e c 2 ] , [ 1 / s e c ]
L it e ra t u re C i t e d
1 ) F o s t e r , R . D . a n d J . G . S l a t t e r y : A p p l . S c i . R e s . , A 1 2 , 2 1 3
1 9 6 2 )
2 ) H i l l , R . a n d G . P o w e r : Q u a t . J . M ec h . A p p l . M at h . , 9 , 3 1 3
1 9 5 6 )
3 H o p k e , S . W . a n d J . C . S l a t t e r y : A I C h E J o u r n a l , 1 6 , 2 2 4 1 9 7 0
4 J o h n s o n , M . W . J r . : T r a n s . S o c . R h e o l , 5 , 9 1 9 6 1
. 5 ) K a t o , H ., N . T a c h i b a n a a n d K . O i k a w a : T r a n s . J S M E , 3 8 ,
8 2 1 1 9 7 2
6 T u r i a n , R . M . : A IC h E J o u r n a l , 1 3 , 1 0 0 0 1 9 6 7
7 ) W a s s e r m a n , M . L . a n d J . G . S l a t t e r y : A I C h E J o u r n a l , 1 0 , 3 8 3
1 9 6 4 )
8
Y o s h i o k a , N .
a n d K. A d a c h i :
J. C h e m .
E n g .
J a p a n , 4 , 2 1 7
1 9 7 1 )
9
Y o s h i o k a ,
N.
a n d
K. A d a c h i :
J. C h e m . E n g . Japan, 4 , 2 2 1
1 9 7 1 )
1 0 ) Y o s h i o k a , N . , K . A d a c h i a n d H . I s h i m u r a : K a g a k u K o g a k u , 3 5 ,
1 1 4 4 1 9 7 1
Y o s h i o k a , N . a n d R. Nakamura:
K a g a k u
Kogaku, 2 9 , 7 9 1
1 9 6 5 )
1 2 Z i e g e n h a g e n , A .: A p p l . S c i . R e s . , A 1 4 , 4 3 1 9 6 4
O N
T H E
P A R T I C L E V E L O C I T I E S I N S O L I D - L I Q U I D
T W O -P H A S E F L O W T H R O U G H S T R A IG H T P IP E S
A N D B E N D S*
M as a y uk i T O DA, T o ich i ISH IK AW A,
Sh o z a b v ro SAIT O a n d Siro M AE DA
D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g , T o h o k u U n i v e r s i t y ,
S e n d a i J a p a n
T h e m e a n p a r t i c l e v e l o c i t i e s i n h o r i z o n t a l p i p e , v e r t i c a l p i p e a n d p i p e b e n d s m a d e o f
t r a n s p a r e n t p o l y a c r y l a t e p i p e 3 0 . 2 m m i n i n s i d e d i a m e t e r w e r e i n v e s t i g a t e d e x p e r i m e n -
t a l l y . T h e r a d i i o f c u r v a t u r e o f t h e b e n d s w e r e 1 2 , 2 4 a n d 4 8 c m , T h e s o l i d p a r t i c l e s u s e d
w e re gla ss b e a d s w hich h a d a m e a n p a r tic le d ia m e te r o f 0 .1 8 9 cm a n d a d e n s ity o f 2 .5
g/cm 3 . Ra dio ac tiv e p ar t ic le s w e re i n tro duce d a s tr a ce r a nd th e p ar tic le v elo ci tie s w e r e
d e t e r m i n e d b y s c i n ti l l a t i o n p r o b e s .
T h e p a rt icle v e lo c itie s in b o th th e s tra igh t p ip e s a n d th e b e n d s a re d is tr ib ute d in
w i d e r a n g e s , b e c a u s e t h e f l o w p a t h o f e a c h p a r t i c l e i n p i p e i s d i f f e r e n t . T h e p a r t i c l e
v e l o c i t y i n v e r t i c a l p i p e i s g r e a t e r t h a n t h a t i n h o r i z o n t a l p i p e . T h e p a r t i c l e v e l o c i t i e s i n
v e rtica l b e n d s w ith h o riz o n ta l a p p ro a ch flo w a re in ge n e ra l sm a lle r th a n th o se in th e
o th e r b e n ds . T he e f fe c t o f th e r a d ius o f cu rv a tu re o n th e p a rt ic le v e lo ci ty c o m es to b e
la rge r w he n th e m ea n flo w ra te o f s lurry in cr e as e s.
I n t r o d u ct i o n
I n s t u d y i n g h y d r a u l i c t r a n s p o r t a t i o n o f s o l i d m a t e -
r i a l i n a p i p e , i t i s n e c e s s a r y t o k n o w t h e v e l o c i t i e s o f
t h e s o li d s a n d t h e f l u i d s a t v a r i o u s l o c a t i o n s a l o n g t h e
P i p e -
T h e v e l o c i t y d i s t r i b u t i o n o f t h e f l u i d i n t w o - p h a s e
f l o w
h a s
b e e n s t u d i e d b y m a ny a ut h o r s 1 2 *4 5 6) .
H o w e v e r , t h e v e l o c i t i e s o f t h e s o l i d p a r t i c l e s h a v e n o t
b ee n ex am in e d to a s ufficien t e x te n t a s y e t, m a in ly
b e c a u s e o f t h e d i f f i c u l t i e s i n t h e i r m e a s u r e m e n t . O n l y a
f e w s t u d i e s 3 6 ) h a v e b e e n m a d e o f t h e v e l o c i t y o f a
s i n g l e p a r t i c l e i n f l u i d f l o w .
I n t h e p r e s e n t s t u d y , t h e v e l o c i t i e s o f t h e p a r t i c l e s
i n h o r i z o n t l p i p e v e r t i c l p i p e n d b e n d s w i t h v r i o u s
r a d ii o f c u r v a t u r e w e r e m e a s u r e d .
1 E x p e r im en ta l Ap p a r a tu s a n d Pro ce d ure
T h e e x pe rim e nta l a pp ar atus is s ho w ns ch e ma tic ally
i n F i g . 1 . A t r a n s p a r e n t p o l y a c r y l a t e p i p e o f 3 0 . 2 m m
i n s i d e d i a m e te r w a s u s e d . T h r e e k i n d s o f r i g h t - a n g l e
b e n d s 7 } w e r e u s e d , o f w h i c h t h e r a d i i o f c u r v a t u r e
w e r e 1 2 , 2 4 a n d 4 8 c m . T h e s o l i d p a r t i c l e s u s e d i n t h i s
e x p e r i m e n t were
- g l a s s b e a d s ,
w h o s e mean particle
d i a m e t e r a n d d e n s i t y w e r e 0 . 1 8 9 c m a n d 2 . 5 g / c m 3 ,
r e s p e c t i v e l y .
T h e m e a n flo w ra te o f th e tw o-p h a s e f lo w a n d th e
c o n c e n tr a t i o n o f s u p p l i e d p a r t i c l e s w e r e e v a l u a t e d b y
m e a s u r i n g t h e f l o w r a t e a t t h e o u t l e t d i r e c t l y . T o d e t e r -
Received on September 7, 1972
Presented at the Local Meeting of The Soc. of Chem. Engrs.,
Japan Akita, Sept. 1971)
•§980å‘ä‘å‰ÔŠªŽšÂ—t
“ Œ – k ‘ å Š w H Š w ” ‰ » Š w H Š w ‰ È • @ “ s “ c ¹ ” V
1 40
J O UR N ALO F C H E M IC AL E NGIN EE RING O FJ A PA N
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F ig . 1 Schem atic d iag ram of experim en ta l appara tu s
f
/f
\ R
= 2 4 f t m 3
Jeff i ds« 0.189CcnO
1g /
I
P ss
2 . 5 C g / c m 3 3
J f f / ' 0 m = 0 .7 6à C m /s ea )
li J j)
\
m e= 5 . 0 1 ' C % 3
= a5 Eft J
å h1
y JI o
H o r i z o n ta l p ip e
mY' I o- Horizontal bend
dJ/J j I c^ V ertica l bend(H -V )
r
I à
V er t i c a l p i p e
J jJ à | 4
V e r t i c a l b en d( V - H )
y ^ ^ ^ /
j M e a n
v e l o c i t y o f s l u r r y
0 : 0. 5 1. 0 1 : 5 2 . 0
U s C m / s l
F i g . 2 C um u l a t i v e d i s t r i b u t i o n f u n c t i o n o f p a r t i c l e
v e l o c i t y ( R un N o . 1 4 )
i .0 |- ,-= i -I^> -T-^ -å ^ sp rt f^ ff0^ - ^ -+ n
R = 2 4 [ c r r o x s ^ J I S ^ t / *
d s = 0 J8 9c c m D /V
| j f - & Y à
Ps= 2.5 rg/cm ] / V -i/Y /
m= 2 . 6 7
C m / s e c
> P
/j // /*
m c= 3 . 8 3 C % D
/
/ \ i > i /
_«
I \ f l //
-O -i/ à
~ 0.5 -/ ~/ -fl ~/ ° Horizontal pipe
6
P y/l
©
-o H o r i z o n ta l b e n d
/ / / / I / < )
V e r t i c a l b e n d ( H -V )
/
V cr - ^ à à V er t i c a l p i p e
f
f
^ -dr A ^ V e r t i c al b e n d ( V - H )
/f >/ t O ^ I
/* | M e a n
v e l o c i t y o f s l u r r y
o\j£LooS^v+a*^ l_J I 1 u
1. 5
2 0 2 . 5
3. 0
3 . 5 4 . 0
U s C m / s ]
F i g . 3 C u m u la t i v e d i s t r i b u t i o n f u n c t i o n o f p a r t i c l e
v e l o c i ty (R u n N o. 2 2 )
a* ~
M W I ds= 0.I89lcto
ff If />s= 2.5 cg/cra'3
9/ V Dm» 1.03 Cm/seci)
- 6? / ' mc= 0.81 C%3
** i/ » ' ° Horizontal P'Pe
9/ ff I ° Horizontal bend
AO -J-1 \ <> Vertical bend(H-V)
yj If i à . Vertical pipe
$6 j}i I æf Vertical bend(V-H) J$-1$9 I MMn velocity of slurry
0 5*5 *^ I.O 1.5 2.0
Us Cm/s^
F i g . 4 C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n o f p a r t i c l e
v e l o c i t y ( R u n N o . 2 3 )
1.0, 1 1 Jr^Y^^ '^^^^^
d s =
O .I 8 9 C c n O >S
I A S & Z r
Ps - 2. 5
C g / c m
/ ^ < T / /
0ro= 2 . 7 6
C m / s e e 3
/ / t / l '
mc= 0.37 C J / \jf//
5 , ^ r. - / p : / M e a n v e l o c i t y o f s l u r r y
# /// o
H o r i z o n t a l p ip e
/ yV> y | -o- Horizontal bend
y
A/ jM
V e r t i c a l b e n d ( H - V )
> >
y ^ ^
I
à
V e r t i c a l p i p e
» ^ t
I
4
V e r t i c a l b e n d ( V - H )
oL^i . f«^J I L _ 1
0 2.0 2 . 5 3 . 0 3 . 5 4 . 0
F ig . 5 C um u la t i v e d i s t r i b u t i o n f u n c t i o n o f p a r t i c l e
v e l o c i t y (R u n N o . 2 4 )
3 . o D -
3 . 0 2
c c n o ' / _
d s=
0 . I 8 9 c c m 3 A /
p % - -
2. 5 C g / c m 3 ] / ~ %
mc[ ] / cf
® 0.31-0.53 / ft
o0 72 1 22 / /
2. o ->
1 .6 7 - 5 . 0 1 / f - 7 7 ^ ^
3 -OSN.o. /y
£ - /A
/ c/^~
1.0- /--p- C ___ L_
0
1. 0
2 . 0 3 . 0
U m C m / s D
F i g . 6 P a r t i c l e v e l o c i t y i n h o r i z o n t a l p i p e
m i n e t h e v e l o c i t i e s o f t h e p a r t i c l e s i n t w o - p h a s e f l o w ,
r a d i o a c t i v e p a r t i c l e s w e r e i n t r o d u c e d a s t r a c e r . I n t h e
s t r a i g h t p i p e s , tw o s c i n t i l l a t i o n p r o b e s w e r e p l a c e d t o
d e t e c t th e i n t e n s i ty o f ^ - r a y a t th e m e a s u r i n g s e c t i o n
s h o w n i n F i g . 1 , a t a d i s t a n c e o f a b o u t 0 . 6 m . I n t h i s
m e a s u ri n g r e g i o n , t h e p re s s u re d ro p w a s d e t e rm i n e d
along the direction offlow . It w as confirm ed that the
f l o w i n v o l v e d n o a c c e l e r a t i o n . O n t h e o t h e r h a n d , i n
t h e b e n d s t w o s c i n t i l l a t i o n p r o b e s w e r e p l a c e d a t t h e
i n l e t a n d t h e o u t l e t o f a b e n d . F o r m a k i n g t h e t r a c e r , a
sm all am ount of glass beads w hose diam eter w as very
s i m i l a r t o t h e a v e r a g e p a r t i c l e d ia m e te r w a s i r r a d ia t e d
w i t h f - r a y . T h e i r r a d i a t i o n w a s c a r r i e d o u t b y t h e
3 0 0 -M e V L i n a c o f T o h o k u U n i v e r s i t y f o r t h r e e h o u r s ,
a n d t h e h a l f - d e c a y t i m e w a s a b o u t t h r e e h o u r s . T h i s
w a s e n o u g h f o r t h e p u r p o s e o f t h e p r e se n t e x p e r i m e n t .
T h e p a r t i c l e v e l o c i t i e s w e r e d e t e r m i n e d b y d i v i d i n g
th e t r a j e c t o r y l e n g th b e tw e e n tw o s c i n t i l l a t i o n p r o b e s
b y
th e
r e s i d e n c e
t ime .
I n
t h i s c a s e , i t i s
n e c e s s a r y t o
k n o w th e t r a j e c to r y l e n g th o f p a r t i c l e s i n th e p i p e o r
b e n d . I n t h i s w o r k , i t w a s f o u n d f r o m p h o t o g r a p h i c
o b s e r v a t i o n t h a t t h e l o c i o f p a r t i c l e m o t i o n s w e r e c o m -
p a r a t i v e l y p a r a l l e l t o th e a x i s o f s t r a i g h t p i p e s o r b e n d s .
Th u s , t h e a c t u a l l e n g t h o f t h e t r a j e c t o ry f o r s t r a ig h t
p i p e w a s a s s u m e d t o b e r e p r e s e n t e d b y t h e l o n g i t u d i n a l
d i s t a n c e a l o n g t h e a x i s . I n t h e b e n d s , i t w a s d e f i n e d a s
t h e a x i a l l e n g t h f r o m t h e i n l e t t o t h e o u t l e t o f a b e n d .
A l t h o u g h t h i s d e f i n i t i o n i s n o t g rounded i n t h e o r y , i t
s e e m s a lmos t
correct
because t h e d i f f e r e n c e b e tw e e n
t h e d i s t a n c e m o v e d a l o n g t h e a x i s a n d t h a t a l o n g t h e
i n s i d e o r o u t s i d e w a l l o f a b e n d i s a t m o s t ± 1 2 % .
V O L . 6 N O . 2 1 9 7 3
1 4 1
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T ab le 1 M e a n p a r t i c l e v e l o c i t y ( R = 1 2 c x n )
R u n N o . U m [ n a / g f ] m c [ v o l% ] 1 7 « i r [ m / s ] t f .M [ m / s ] . * 7 W i? [ m / s ] * 7 s f [ m / s ] * 7 S F H i? [ m / s ]
1
2 . 1 9 0
0 . 4 3 8 L902 L 6 4 5 L918
2 2 . 1 2 8
0 . 9 8 7
2.362 1 3 7 3
2172
3 1 . 8 1 5
3 . 4 8 9 1.493 1 . 3 5 4 1 734
4 0 . 9 1
1 1 . 4 0 5
0.627
0 . 4 6 4 0731
5 2 .7 4 0 0 . 4 56
2.363
2 . 0 7 4 2490
6 2 . 7 1 3 0 . 7 4 9
2.243 1 . 8 4 8
2212
7
2 . 5 7 2
2 . 8 7 0
2.030
1 .65 7
2340
8 0 . 9 1 5 0 . 4 7 1
0.887
0 . 5 8 9
0876
9
1 . 0 6 6
3 . 3 2 4 0.802 0 . 4 8 3
1 092
1 0 1 . 7 9 8 9 9 2 1.445 1 . 1 4 1 1 460
T a b l e 2 M e a n p a r t i c l e v e l o c i t y ( / ? = 2 4 cm )
R u n N o . U m [ m / s ] m c [ v o l% ] U s H [ m / s ] t / S j f f j B [ m / s ] £ 7 W i? [ m / s ] C 7 5 f [ m / s ] * 7 S F _ 7 i? [ m / s ]
H 0 862 6 580 0 703 0 81 7 0 834 X853 0 815
12 0 . 9 7 1 3 .0 7 0 0 . 7 9 7 1 . 0 2 4 0 .7 7 9 0 .9 2 2 0 .8 7 9
13 1 . 1 8 8 1 . 2 1 7 0 . 8 3 0
1 . 0 4 0 0 . 8 1 5 1 . 0 0 1 1 . 8 0 0
1 4 0 . 7 5 6
5 1
6 . 4 0 1 0 . 4 0 0
0 . 5 0 3 5 1 9 4 4
15 2 . 9 6 0
1 . 0 7 5 2 . 7 2 3
2 . 8 8 5
2 . 4 1 0
3 . 1 2 1 2 . 8 5 0
1 6 2 . 9 2 0 2 . 3 0 0 2 . 8 1 4 2 . 9 7 1 _
17 1 . 7 7 4 3 . 6 5 3
1 . 4 1 9
1 . 6 1 5
1 . 3 4 0
1 . 6 4 3 1 . 7 2 8
1 8
1 . 9 5 2 0 . 7 1 5
1 . 6 4 5 1 . 6 9 0 1 . 6 3 9 2 . 1 5 1 1 . 7 3 7
19
1 . 8 6 6
1 1 7 1 1 . 5 2 0 1 . 7 1 5 1 . 4 8 6 2 . 0 4 5 1 . 7 3 8
20
1 . 1 2 2
3 . 2 4 2 0 . 6 7 0
0 . 9 7 4
0 . 7 1 0 0 . 9 1 2 0 . 9 2 9
2 1 1 . 3 8 2 1 . 6 7 3 1 . 0 4 2 1 . 3 0 4 0 . 9 6 3 1 . 3 2 7 1 . 3 3 7
2 2
2 6 7
3 .8 3 0 2 . 4 3 2 2 . 62 6 1 . 9 9 1
2 8 3 6 2 7 5
2 3 1 . 0 3
0 . 8 1 0 0 . 6 7 6
0 . 8 2 1
0 . 6 4 5 0 . 8 4 8 0 . 8 3 4
24
2 . 7 5 5
0 .3 7 4 2 . 7 5 4
2 . 7 8 5 2 . 1 8 6 2 . 7 1 7 2 . 8 8 0
2 5 2 7 2 0 . 9 2 6
2 . 5 4 1
2 .69 2 2 .0 8 8 2 7 5 9 2 6 2 1
T ab le 3 M ean partic le v e lo c ity U =48cm
R u n N o . U m [ m / s ] m c [ v o l% ] * 7 S iy [ m / s ] E / , M [ m / s ] U s h v b [ m / s ] J 7 s f [ m / s ] C / « V f l - s [ m / s ]
2 6 1 . 2 9 3 1 . 0 1 0 1.195 1 . 1 1 2
1 186
2 7 0 .9 8 6 3 .4 6 0 0.688 0 . 6 3 8 0780
2 8
1 . 8 6 0 0 . 3 0 8
1 4 8 6
1 . 8 5 4 1 . 9 3 0 1 . 9 9 5 2 . 0 2 0
2 9 1 . 7 5 0 1 . 0 0 7 1.690
1 . 6 5 2
1 909
3 0
1 . 5 5 0
3 7 2
1.315
1 . 2 2 1 1 579
31
2 . 8 8 0 0 . 5 8 3
2.905
3 . 1 3 5 3191
3 2
2 . 8 4 0
1 . 1 7 1
2.760 2 . 8 3 1 3055
3 3
2 7 5
3 . 2 2 3 2.640
2 . 5 4 5
3010
3 4 0 . 9 65 0 .5 2 8 0 .6 6 8
0 . 7 7 4
0 . 8 4 9
0 . 8 3 7 0 . 6 4 0
2 E xp erim en ta l R esu lts an d D iscussio n
S om eexam ples o f th e cum ula tiv e distr ib utio n fun c-
t io n s o f th e p a r t ic le v e lo c i t ie s a re s h o w n in F ig s . 2 to 5 .
I n t h i s e x p e r im e n t , t h e m e a n p a r t i c l e v e l o c i t y U s i n
t h e tw o - p h a s e f l o w i s t h e v e lo c i t y a t w h ic h t h e c u m u -
la t iv e d is t r ib u t i o n f u n c t i o n is 0 .5 . T h e m e a n p a r t ic le
v e lo c i t i e s o b t a in e d a r e s u m m a r i z e d in T ab l e s 1 to 3 .
T h e s e f i g u r e s s h o w t h a t t h e p a r t i c l e v e l o c i t i e s a r e d is -
tributed over a w ide range because the flow path o f
e a c h p a r t ic le in a p ip e is d i f f e re n t . I n a d d i t io n , th e
m e a n p a r t i c le v e l o c i t y a t e a c h t e s t s e c t io n i s s t r o n g ly
a f f e c te d b y th e o p e ra t in g c o n d it io n .
2 .
1
P a r t i c l e
v e l o c i t i e s
in stra ig h t p ipe
a
H o r i z o n t a l p ip e
F ig . 6 s h o w s th e r e la t io n s h ip b e tw e en t h e m e a n p a r -
t ic le v e lo c ity in h o r iz o n ta l p ip e , U sh a n d th e m ean
v e l o c i t y o f s l u r r y J J m - U s h w h i c h i s n o t a f f e c t e d b y t h e
s m a l l c h a n g e o f c o nc e n tr a t i o n m c a s f o u n d in th is e x p er i -
m e n t , i s a lw a y s s m a ll e r t h an U m , a n d t e n d s t o a p p r o a c h
t h e v a lu e o f U m w i th in c r e a s in g f lo w r a te . A t lo w f lo w
r a te , a lm os t a l l p a r t ic le s f lo w a lo n g th e p ip e b o tto m
w i t h s m a l l e r v e lo c i ty t h a n U m . O n th e o th e r h a n d , a t
h i g h e r f lo w r a te U s h a p p r o a c h e s U m , b e c a u s e a lm o s t
a l l
par t i c l e s f l o w
t h r o u g h
th e
p ip e in s u s p e n s io n .
I n g e n e r a l , i t i s s a i d 8 ) th a t , i n th e f lo w o f s e t t l in g
s l u rr i e s , p a r t i c l e m o t i o n i s g ov e r n e d t o a l a r g e e x t e n t
b y th e d im e n s io n le s s t e r m U ll l g D { p s j p w - \ ) . I n th i s
c a s e a l s o , a s s h o w n i n F ig . 7 , t h e e x p e r im e n t a l d a t a a r e
c o r r e l a te d r e l a t i v e ly w e ll b y th e a b o v e d im e n s io n l e s s
t e r m th ro u g h t h e f o l l o w i n g e m p ir i c a l e q u at io n .
(U s H I U m ) = 0 .5 3 (U H g D (p s lP w - ) ) < » . « ( 1 )
b V e r t ic a l p ip e
F ig . 8 s h ow s t h e r e l a t i o n s h ip b e tw e e n t h e m e a n p a r -
t ic le v e lo c ity in v e r tic a l p ip e , U sy an d U m . T h e e f f e c t
o f m c o n U s V c a n b e n e g l e c t e d , s im i l a r l y t o t h e c a s e o f
t h e h o r i z o n t a l p i p e . H o w e v e r i t i s d i f f e r e n t f r o m t h a t
o f th e h o r iz o n ta l p ip e in th e c a s e w h en U m is g re a te r
th an a b o u t 2 m / s e c . T h at is , U s V b ec o m e s l a rg e r th an
[ 7 m , a s s h o w n in
F i g .
8 . I t w as obse rved
t h a t , w it h
in c re a s in g f lo w ra te o f s lu r ry , th e p a r t ic le s in th e v e r ti -
c a l
p i p e
t e n d to cong r ega t e n e a r t h e a x i s ,
w h e re th e
fluid flow s w ith h ig her ve loc ity . T h is show s th e reason
f o r t h e p a r t i c u la r p h e n o m e na m e n t io n e d a b o v e .
T h e e x p e r im e n t a l d a t a c o u l d b e c o r r e l a t e d a s w e l l a s
i n th e c a s e o f h o r iz o n t a l f l o w , a s s h o w n i n F i g . 9 , a n d
th e f ollo w in g e qu atio n w a s o b ta in ed .
( U , r I U m ) = 0 . 7 2 ( U l l g D ( P s lp w - I ) ) - . (2 )
14 2
JO U RN ALO F C HE M IC ALE NG IN EE RIN G O FJA PA N
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2
Q . å å .
å
å .
å å
å
i
' å å å ' å i
å å
0 à" 3 .0 2 r c iw
d ,* 0 . 1 8 9 a i m
/ > « 2 . 5 l ^ / c m * }
In o 0 . 3 1 - 0 . 5 3
_ _ ^ ^ -
10 o 0 . 7 2 - 1 . 2 2
^
'I l ^J-% ^^
f e l 0 . 6 -
^ ~~~-Si* \
0.4 - fmpiricfll E<j
D s h .0 , 3 - [ 'V '0 & j ° 2
Q pl 1
1
1 1 1 > « - 1
1 1
å I 2 4
6 8
10
2 0 4 0
g D ( W I )
F i g . 7 R e l a t i o n s h i p b e t w e e n ( U s H /U m ) a n d
UllgD(Pslpw -l)
30
D = 3 . 0 2
c c m D / / -
d s
=
0 . 1 8 9
c c n o - f y s
ft= 2 .5 C g / c r r f r J ^ >
mc C 3 //
0 .3 1 - 1 .0 8 /
o I 1 7 2 5 0 q ® /
20
k > -3 . 0 0 - 5 .0 1 " y
1 - ^ - u m ^
, , _ ^ 1
^
,
1
,
1 1
0
1. 0
2 . 0 3 . 0
0 m C m / s 3
F i g . 8 P a r t i c l e v e l o c i t y i n v e r t i c a l p i p e
0 * 3 . 0 2 c c m }
d < =
O .I 8 9 c c n o
o 0 .3 1- 0. 53
© 0.72-1.22
à 1.67-5.01
f m p i r i c o l E ( j .
~ଠ'0-72 \ q D ( P J P m - \ ) \
2 0 3 0
F i g . 9 R e l a t i o n s h i p b e t w e e n ( U s v l U m ) a n d
V ilgD lp.lpw - l)
30 0 = 3 . 0 2 a m : / ~ ~
R = 12 a i m a /
d s = O . I 8 9 a m 3
.
/ *
^ = 2 . 5
C g / c n f t ^ ' >^
mcC ^ / y^
.0 ® 0.44-0.47 /~^S®
rn O 0 .7 5 - 1 .4 I / / /
S © 2 . 8 7 - 3 . 5 0 / å > i
J n /y/
å /
o^ j I 1 1 1 L I
0
1.0
2 . 0 , 3 . 0
U m C m/sD
F i g . 1 0 P a r t i c l e v e l o c i t y i n h o r i z o n t a l b e n d
3.o-
D s 3 . 0 2 c c r m . / _
R = 2 4
c c r m ) ®
d sn 0 . 1 8 9 c c r m ^ ^
t=
2 .5
C g / c m 3 ] > ^
m e C % ) I / /
®
0 .3 7 ^ - 1 .0 8 \ , y
2. 0 //
-j
o 1 .1 7 - ^2 .5 0 / /
e à < { > å
3 .0 0 - - 5 .0 I / 4 r ® l
J -
D s H B = U m / / ?
i.o k å
l / X
i
0 1. 0 2 . 0 3 . 0
Um
C m / s D ^
F i g . l l P a r t i c l e v e l o c i t y i n h o r i z o n t a l b e n d
30~ D=3.02 CcmD" ~~¥~
R-48 ccrru j£
ds- å0.1 89ccm3 /©
Ps* 2.5 Gg/cm3] ^/^
mc C%] //
2.0- ® 0.3I-*-0.53 ~~~//
« o 1.00-I.17 /(/
\ © 3.22-3.72 /7
10-^
//
/ , 1 , 1 L
0 1.0 2.0 3.0
UmCm/s]
F i g . 1 2 P a r t i c l e v e l o c i t y i n h o r i z o n t a l b e n d
3 . 0 . < - 1
D - 3 . 0 2 c c n o
d s = 0 . I 8 9 c c m n
me
«
3 .
23-5.00^-^^ ~^x^.
ro/-lI /^ ~~ 0
^X Dm-
. 5 ( m /s l
2 . 0
c^ - ^ _ _ J
^ ^ ^ | -
I
m ^
Um=2.0
C m / s i '
m ^
l0 °_ , D m * -
1 . 5 O n / s j
®.
® ^_
Um =
1 . 0 I m /s . 1
å t 1
0 10
2 0 3 0 « »
R / a C - :
F i g . 1 3 E f f e c t o f R ja o n p a r t i c l e v e l o c i t y i n h o r i z o n t a l b e n d
r
~
~
[7 1
3.0-à
D=
3 .0 2 c c r m ~ >
R
*1 2 c c m ; > /
ds = 0 .I 8 9 a r m /
9% -
2. 5
C g / c m * ] /
/
2. 0
-90.44-0.47 / >/^' ~
5 o 0 . 7 5 - 1 . 4 1 / j/s fs '
^ -< ^ 2 .8 7 - 3 . 4 9 / X X y
J IW U ny/ //^
\ / , L _ . L
1. 0
2 . 0 3 . 0
U m C r n /s 3
F i g . 1 4 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w it h
h o r i z o n t a l a p p r o a c h f l o w
V O L . 6
N O .2 1 9 7 3
1 4 3
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30 D = 3 0 2 ccm:- , 7
R = 2 4 [ c m : /
d s= 0 . I 8 9 c c m 3 /
P ss 2. 5 C g / c m 3 3 /
«JX> / X
® 0 / 3 7 - 1 . 0 8 / yf
,:2-°-o..7250 /-/ -
| ^ 3 . 0 0 - 5 . 0 1 / @ / ^
i ~ D m - - D s H V B / ®-^K
0 1 0
2. 0
3 . 0 .
U m C m /s 3
F i g . 1 5 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w i t h
h o r i z o n ta l a p p ro a c h f l o w
30--
D = 3 . 0 2
c c m ]
/ //.
R=48 Lcmj /P]/
ds = 0 . 1 8 9 C c m ] // /
ps~- 2.5 Cg/cm3] / K
mc[%] /// /
0 . 3 1 - 0 . 5 3 / / / /
2.o /// y
-V°
1-0 I-1 .17 //A/
I < > - 3 . 2 2 - 3 . 7 2 J y y
1
U SH V b - U m / / /
/ //
/ I I J
I I I
0 1. 0
2 . 0 3 . 0
D m C m /s ]
F i g . 1 6 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w it h
h o r i z o n ta l a p p ro a c h f l o w
_ i r j
D * 3 . 0 2 rc n u
s - 0 . I 8 9 c c i m
f t = 2 . 5 l e g / c m 3 ] ' _----<> : °
m s = 3 . 0 0 ^ 3 . 7 2 C % 3 ^~~^ H b-2.5 C m /s D
J °^ e ' Um=2.0Cm/a]
1 . 0 - ' å å å ' å L ^ | ° ' - - 1 . 5 C m / s i .
er- Dm ' I.O On /s:
t - ^ > J
1
1 J
10
2 0 3 0 « >
R / a c - 3
F ig . 17 E ffec t o f R ja on pa rtic le ve lo c ity in v e rtic a l
b e n d w i t h h o r i z o n t a l a p p r r o a c h f l o w
3.o- D=3.02 ccrro [ . y -
R = 1 2 å à c c m n
. /\
d s = 0 . I 8 9 c c i m
/
ps = 2 . 5 ;
C g / c m 3 }
. / ^ /
® 0.44 0.47 ^y?< /
rn 2 .0 å
o
0 . 7 5 - à 1 . 4 1
~//s®/ ^^
^
-$ -
2 . 8 7 - 3 . 4 9 / 4 y /
/
/
/
/
Z I
: I
I 1 1 1
0 1 0 2 . 0 3 . 0
D m C m / s ]
F i g . 1 8 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w it h
v e r t i c a l a p p r o ac h f l o w
2 . 2 P a r t i c l e v e l o c i t i e s i n b e n d s
a ) H o r i z o n t a l b e n d s
T h e m e a n p a r t i c l e v e l o c i t i e s i n h o r i z o n t a l b e n d s ,
U shb> are show n in F ig s . 10 to 12. In any rad iu s o f
c u rv a tu re , U s H Bis n o t a f f e c t e d b y m c u n d e r th i s e x pe r i -
m e n t a l c o n d i t i o n , a n d i t i s n e a r l y p r o p o r t io n a l to U m .
T h e s l o p e o f U s h b v s - U m i n c r e a s e s g r a d u a l l y w it h t h e
in c re a se o f rad iu s o f cu rv a tu re . In th e case o f R =24
c m , th e s lo p e is n e a r ly e q u a l to th a t in h o r iz o n ta l p ip e ,
and in th e ca se o fR =48 cm th e s lo p e becom es g rea te r
t h a n th a t in h o r iz o n ta l p ip e . T h e s e p h e n o m en a c o u ld
b e c a u s e d b y m a n y f a c t o r s , s u ch a s s e c o n d a r y f l o w o f
f lu id , c e n t r i f u g a l f o r c e a n d th e f r ic t io n f o r c e b e t w e e n
p ar t i c l e s a n d p ip e w a l l .
F ig . 1 3 s h o w s t h e e f f e c t o f R j a o n U s H b m h o r i z o n t a l
b e n d s . I t is f o u nd th a t U s H B in c r e a s e s w ith in c re a s in g
R ja
u p t o abou t Rja=20 , b u t t h e n
i t d e c r e a s e s
g r a d u a l l y . T h i s i s p r e s um e d to b e d u e to t h e f o l l o w -
i n g r e a s o n s ; i ) A t h i g h f l o w r a t e , t h e f r i c t i o n f o r c e b e -
tw
p a r t i c l e s
a n d
p i p e w a l l
i n c r e a s e s be c a u se th e
c e n t r i f u g a l f o r c e . a c t i n g o n t h e p a r t i c l e i n c r e a s e s w it h
d ec r e a s e o f R j a . i i ) . A s R j a b e c o m e s s m a l l , t h e c h a n g e
o f d ir e c t io n o f th e p a r t i c le m ot io n p e r u ni t l e n g th o f th e
p ip e b e n d b e c o m es la rg e a n d th e in e r t ia o f th e p a r t ic le s
d e c r e a s e s . W h e n R [ a b e c o m e s in f in i te , U .s h - b w ^ a P
p r o a c h U s h i n t h e h o r i z o n t a l p i p e ,
b ) V e r t i c a l b e n d s w ith h o r i z o n ta l a p p r o a c h f lo w
Th e
mean p a r t i c l e
v e l o c i t ie s in
v e r t i c a l b e n d s ,
U S H V B > a r e s h o w n v s . U f a i n F i g s . 1 4 t o 1 6 . I n a n y
r a d iu s o f c u rv a t u r e , U sh v b 1 8 n e a r l y p r o p o r t io n a l to U m
a n d t h e e f f e c t o f m c o n U s h v b a p p e a r s i n t h e c a s e s o f
i ^ = 1 2 a n d 4 8 c m . I n t h e s e f i g u r e s , t h e s l o p e o f U s h v b
v s . Um i nc re a se s w i t h i n cr e a s i n g R . T h i s t e n d en c y
a g r e e s w i th th a t in th e h o r i z o n ta l p ip e b e n d s . I n th e
c a s e s o f R = 1 2 a n d 2 4 c m , t h e s l o p e o f U s h v b ^ s s m al l e r
th a n un ity. O n t h e o th e r h an d , in th e c ase o f R = 48
c m t h e s l o p e i s n e a r l y e q u a l t o t h a t i n t h e h o r i z o n t a l
p i p e . _
F ig . 17 sh ow s th e e ffe c t o fR ja o n U sHVB . At h igh
f l ow r a t e ,
UsHVB
in c rea se s
w i t h i n c r e a s in g
R ja a n d
app roache s th e va lu e o f U sh-
c ) V e r t ic a l b e n d s w ith v e r t ic a l a p p ro a c h f lo w
T h e b e h a v i o r o f p a r t i c l e s i n b e n d s i s m u c h c o m p l i -
c a t e d
b y
th e effect o f g ra vi t y , c e n tr i f u g a l f o r c e a n d
s e c o n d ar y f l o w o f f l u id . T h e f l o w s t a t e s o f h o r i z o n t a l
a n d v e r t i c a l b e n d s w it h h o r i z o n t a l a p p r o a c h f l o w a r e
d e t a i l e d in
th e
pr ev ious p a p e r 7 > .
P h o t o s . 1 a n d 2
s h o w t h e
f l ow
s t a t e s in v er t i c al b e n d s w i t h v e r t i c a l
a p p r o a c h f l o w . At low f l ow
r a t e ,
th e p a r t ic l e s a re
t r anspor t ed
in a
s t a t e o f
suspens ion in th e ben d . A t
h ig h f l o w r a te , a s s h o w n i n P ho t o . 2 , a lm os t a l l p a r t ic l e s
a r e t r a n s p o r te d a l o n g t h e o u ts i d e w a l l o f t h e b e n d , a s
w e l l a s i n h o r i z o n t a l a n d v e r t i c a l b e n d s 7 } b e c a u s e t h e
14 4
JO U R NA LO F C H EM IC A LE NG IN EE R IN G O F JA PA N
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F lo w d ir ec tio n
R=24cm, U =O .i
Sm/s,
2.20vol^
P hoto. 1 Flow state for low flow rate in vertical
b en d w i t h v e r t i c a l a p p r o a c h f l o w
F lo w d ire ctio n
R=24cm,
U =2.54m/s,
Photo. 2 Flow state for h igh flow rate in vertical
b en d w i t h v er t i c a l a p p r o ac h f l o w
3.o--D
= 3 . 0 2 C cm : V -
R = 2 4 c c n o / ®
ds = 0 . 1 8 9 c cm ) - ^ f
i ° s = 2 .5 C g / c m 3 ] ^ /
mc % ] / ^
20
à 0 . 3 7 - 1 . 0 8 y /
K
o I . 7 - - 2 . 3 0 /^ J
I
> 3 . 0 0 - 5 . 0 1 / < /
|
-
-0 S VHB-U m //
zf
/A
0 1. 0
2 . 0 3 .0
D m C m / sD
F i g . 1 9 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w it h
vertica l ap proach flow
~ p P T~ ?
3-° D-3.02W //~
R
= 4 8 K m ] / /
ds = 0 . I 8 9 c c m : / /
Ps = 2 5
[ g / c m 3 ] / /
m c C % ) ' //
® 0.31 .0.53 ^ //
0 å ®//- å
< o 1 . 0 0 - - - . 1 7 c ) / /
1 ^ 3 .2 2 - - 3 . 7 2 j /
^ UsVHB=D m //
/Y
/f t
/ | ; , . 1
0
1. 0
2 . 0 3 . 0
U m C m / s ]
F i g . 2 0 P a r t i c l e v e l o c i t y i n v e r t i c a l b e n d w it h
v e r t i c al a pp ro ac h f lo w
fD
« 3 . 0 2
C c n t f 1 f
( f 9 *
0 . 1 8 9 o t r r o
_ _ - - - Q
-me'
3 2 2 - 3 . 7 i%l^^^^ u»°2-5 ^^
^ 2.0 j ^^ {^=2.0 Crn/^
j- T ^ à å - ^ . -
O t n = 1 .5 t m /s ]
r 0 , . _ - i 4~- ;- å å -
0ffl=
1 . 0 C m / s ]
X* I .I
I
i [ . t i _ T
0
10
2 0 3 0 ° °
R /a C - 3 . 4 ^ .
F i g . 2 1 E f f e c t o f R ja o n p a r t i c l e v e l o c i t y i n v e r t i c a l
b en d w i t h v e r t i c a l a p p ro ac h f l o w
D = 3 .0 2 c c /m
3l ° d s = 0 . I 8 9 c c r r n , / » , « 2 . 5 C g / c m 3 ]
mc=3. 2 2 - 3 . 7 2 C % 3 /- R «no
2
.o - ' -o ' KyJ D m = 2 - 5 C m / £
\
/ , . H . P I
N--o^
U «
» i o ^ T
f
i
H.P H-H.B H.P H-V.B V.P V-H.B H.P
F i g . 2 2 C h a n g e o f p a r t i c l e v e l o c i t y a t e a c h f l o w p a t h
p a r t ic le s a r e s t r o n g ly a f f e c te d b y in e r t ia l a n d c e n t r i f -
u g al f o rc es .
F i g s .
1 8 to 2 0
show
th e
r e l a t i o n s h i p
between th e
mean
p a r t i c l e
v e l o c i t i e s UsVHB
a nd
Um
in v ertica l
b e n d s . T h e e f f e c t o f m c o n U s v h b d o e s n o t a p p e a r w it h
t h e e x c e p t i o n o f R = 1 2 c m , a n d i n a n y r a d i u s o f c u r v a -
tu r e U sv h b is n e a r ly p r o p o r t i o n a l to U m >I t i s c l e a r ly
s e e n f r o m th e s e f i g u r e s t h a t th e s lo p e o f U sv h b v s - U m
in c r e a s e s w ith in c r e a s in g R . T h is te n d e n c y i s s im i la r to
t h e c a s e o f h o ri z o n t a l a n d v e r t i c a l b en d s w i t h h o ri -
z o n t a l a p p r o a c h f l o w . I n t h e c a s e o f R = 4 8 c m , t h e
s l o p e i s
g r e a t e r than t h a t in
t h e
v e r t i c a l p ip e . T h e s e
phenomena are similar to the case of horizontal bends.
I t
is
c o n s i d e r e d t h a t , a t h i g h f l o w r a t e ,
almost a ll
p a r t i c l e s a r e t r a n s p o r t e d a l o n g t h e o u t s i d e w a l l o f t h e
b e n d b y h ig h e r f lu id v e lo c i ty n e a r th e o u ts id e w a l l .
F ig . 2 1 s h o w s th e e f f e c t o fR /a o n U s v h b - A t l o w f l o w
r a t e , U s v h b
tends to i n c r e a s e w i t h
d e c r e a s i n g R ja .
O n th e o th e r h a n d , a t h ig h f lo w r a t e C / s v h b d e c r e a s e s
w it h d e c r e a s i n g R \a a s w e l l a s t h e o t h e r b e n d s d e -
s c r i b e d i n t h e p r e v i o u s s e c t i o n s . A s t h e r a t i o o f R ja
b e c o m e s in f in i t e , U s v h b w ^ a p p r o a c h U s v in th e v e r -
t i c a l p i p e .
2 . 3 Comparison of th e velocities in each test
section
T h e m e a n p a r t i c l e v e l o c i t y i n e a c h t e s t s e c t i o n u n d e r
c o n s t a n t m c i s s h o w n in F ig . 2 2 . T h e v e l o c i t y in t h e
v e r t ic a l p ip e is a lw a y s g re a te r th a n th a t in th e h o r iz o n -
t a l p i p e . T h i s i s d u e t o t h e f a c t t h a t i n t h e v e r t i c a l p i p e
t h e p a r t i c l e s t e n d to m o v e p r e f e r e n t i a l l y n e a r t h e a x i s
V O L . 6 N O .2 : 1 9 7 3 -
14 5
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http://slidepdf.com/reader/full/toda-1973-on-the-particle-velocities-in-solid-pdf 7/7
in acco rd ance w ith an in c rea se o f U m w hile in th e
h o r i z o n t a l p i p e m o s t o f t h e p a r t i c l e s a r e a lw a y s t r a n s -
p o r t e d n e a r t h e p i p e b o t t o m b y t h e f l u i d a t a r a t e
m uch s low er th an th a t o f m ean f low 6 . M oreo ve r
F ig . 2 2 s h o w s t h a t t h e m e a n p a r t i c l e v e l o c i t y i n v e r t i c a l
b e nd s
w i th h o ri z o n ta l a pp r o a c h f l ow is t h e s m a ll e s t
a m on g th o s e i n a l l p i p e b e n d s .
3 C o n cl u si o n
T h e f l o w s t a t e s i n t h e b e n d s w e r e o b s e r v e d f o r s o l i d -
l i q u i d tw o - ph a s e f l o w . Th e
p a r t i c l e v e l o c i t ie s i n
s t r a i g h t p i p e s a n d b e n d s w e r e m e a s u r e d a n d t h e f o l l o w -
in g re sults w ere o bta in ed .
i ) U n d e r t h e e x p e r im en t a l c o n d i t i o n s o f f 7 m = 0 . 7 to
3 . 0 m /s a n d m c = 0 . 3 t o 5 . 0 v o l , t h e m e a n p a r t i c l e
v e l o c i t i e s i n s t r a i g h t p i p e s a n d b e n d s a r e n e a r l y p r o -
p o r t i o n a l t o U m .
i i ) T h e m ea n p a r t i c l e v e l o c i t i e s i n t h e v e r t i c a l p i p e
a r e g r e a t e r t h a n t h o s e i n t h e h o r i z o n t a l p i p e ,
i i i ) E m p i r i c a l e q ua t io n s e x p r e s s in g th e m e a n p a r t i c l e
v e l o c i t i e s i n h o r i z o n t a l a n d v e r t i c a l p i p e s a r e p r o p o s e d ,
a s g i v e n b y E q s . l ) a n d 2 ) . T h e s e e q u a t i o n s g iv e th e
m ea n p a r t ic l e v e lo c i t y w ith in 1 0 e r r o r .
i v ) T h e m e a n p a r t i c l e v e l o c i t i e s i n v e r t i c a l b e n d s w it h
h o riz o nta l a pp ro a ch flo w a re a lw a y s sm a l le r th a n th o se
i n t h e o t h e r b e n d s .
v ) T h e e ff e c t o f r a d i u s o f c u r v a t u r e o n p a r t i c l e v e l o c -
i t i e s b ec o m e s s i g n i f i c a n t a s t h e m e an f l o w r a t e i n c r e a s e s
a n d /o r t h e r a d i u s o f c u r v a t u r e d e c re a s e s .
A c kn ow le d ge m e nt
T h i s w o r k w a s s u p p o r t e d b y t h e S c i e n c e R e s e a r c h F o u n d a t i o n
o f E du c a t io n a l M i n i s t r y , J a p a n , G ra n t N o . 5 0 1 6 5 . T h e a u t h o r s
a p pr e c i a t e t h e s u pp o r t l e a d in g to t h e p ub li c a t io n o f th is a r t i c le .
N o m e n c l a t u r e
radius of pipe m]
= diameter of pipe [m]
= cum u la t iv e d is t r ib u t io n fu n c t io n [ -]
= acceleration due to gravity [m s2]
= c o nc e ntr a t io n o f s o lid s in m ixtu re d is c ha rg ed
from end of pipe vol ]
= radius of curvature of bend [cm]
= mean velocity of slurry [m s]
= particle velocity [m s]
= mean particle velocity [m s]
= m ean pa rtic le ve lo c ity in h o rizon ta l pipe [m /s]
= m e a n p a r t ic le v e lo c i ty in ve r tic a l p ip e [m /s ]
= m ean pa r t ic le v e lo c i ty in h o r iz o n ta l b e n d [m /s J
= m e an pa r t ic l e v e lo ci ty in v e r t ic a l b en d
w ith h o r iz o n ta l a pp ro a c h f lo w [m /s ]
= m ea n p a r t i c le v e lo c i ty in v e r t i c a l b e n d
w ith v e r t ic a l a p p ro a c h f lo w [m /s ]
= d e n s i ty o f p a r t i c l e [g /c m 3]
= density of water [g crn3]
L i t e r a t u re C i t e d
1 ) A yu k a w a , K . : P r e p r in t s f o r 4 7 t h A nn u a l M ee t in g o f J . S - M . E .
No
2 1 6 1 9 6 9
2 D u r a n d , R . : L a H o u i l l e B l a n c h e , 8 , 1 2 4 1 9 5 3
3 I k i , S . a n d Z . H o k a o : M M I J , 8 4 , N o . 9 5 7 , 1 5 1 9 6 8
4 M u r o t a , A . : J S C E , 3 8 , 4 7 8 1 9 5 3
5 ) N ew it t , D . M ., J . F . R i c h a r d s o n a n d C . A . S h o c k : P r o c e e d i n g s
o f th e Sym po s iumo n th e In te ra c tio n b e tw een F lu id s a n d
P a r t i c l e s 8 7 , L o n d o n , I n s t . G h e m . E n g r s . 1 9 6 2 )
6 ) T o d a , M ., H . K o n n o , S . S a i t o a n d S . M ae d a : K a g a k u K o g a k u ,
3 3 , N o . 1 , 6 7 1 9 6 9 )
7 ) T o d a , M ., N . K o m o r i , S . S a i t o a n d S . M ae d a : J . C h e m . E n g .
J a p a n , 5 , N o . 1 , 4 1 9 7 2
8 ) T e r a d a , S . : P r e p r i n ts fo r 4 0 th A nn u a l M ee t i n g o f J . S . M . E . ,
No
8 6 ,
1 3 7 1 9 6 3
1 4 6
JO UR NA LO F C HE MIC AL E NG IN EER IN G O FJA PA N