toda-1973-on the particle velocities in solid-.pdf

7/23/2019 Toda-1973-ON THE PARTICLE VELOCITIES IN SOLID-.pdf

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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



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



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



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 .

~à¬ '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



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 ]


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 ]


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



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


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 ]


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



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

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