Estimation of the Critical Velocity in Pipeline Flow of Slurries

7/29/2019 Estimation of the Critical Velocity in Pipeline Flow of Slurries

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Powder Technology, 51 (1987 ) 35 - 47 35

E s t i m a t i o n o f t h e C r i t ic a l V e l o c i t y i n P i p e l i n e F l o w o f S l u r r i e s

R M TUR IAN, F. -L. H SU and T. -W. MADepartment of Chemical Engineering, University of Illinois at Chicago, Chicago, IL 60680 (U.S.A.)(Rece ived July 9, 1986; accepted Decemb er 16, 1986)

SUM M AR YA detailed examination of the critical

velocity in pipeline flow of non-colloidalslurries was carried out. Published criticalvelocity correlations were collected and wererecast into a standard form so that they couldbe compared with each other, and they werealso tested against a broad collection ofexpe rimen tal critical velocity data. Alt ogether,a total collection of 864 experimental criticalvelocity data, representing a broad variety ofsolid materials and pertaining to wide rangesof the variables involved, were used in thesetests. This rather substantial body of experi-mental data was also used as the basis fordeveloping a set of improved critical velocitycorrelations, w hich were established by fi ttingthe data to various forms of the standardequation.Am on g the principal results of this workare the following: The dependence of thecritical velocity on pipe diameter is verynearly equal to D 1/2, while its dependence onparticle size, for slurries of non-colloidalparticles, is very nearly equal to d Theanalytical result due to Oroskar and Turianindicates that v c d epends on pipe diameteras D 6, and on p arti cle size as d , whi le thebest empirical fits to the data suggest that thedependence on pipe diameter is approximatelyD 's , and that on parti cle size is at mo st d 6.In addition, both the newly establishedempirical correlations and the analyticalcorrelation due to Oroskar and Turian predicta max imu m in the v c vs. C curve, the max imu moccurring at solids volume concentration of0.25 to 0.30. The comparisons with experi-mental data further established that theanalytical result by Oroskar and Turian, andthe empirical correlations developed in thiswork, do a superior overall job of predictingthe data than othe r published correlations.

INTRODUCTIONT h i s p a p e r is c o n c e r n e d w i t h t h e c r it ic a lv e l o c i t y i n t h e f l o w t h r o u g h p i p e s o f s l u rr ie sc o m p o s e d o f n o n - c o l l o id a l s o l id s . T h e c r it ic a l

v e l o c i t y V c i s d e f i n e d a s t h e m i n i m u mv e l o c i t y d e m a r c a t i n g f l o w s i n w h i c h t h e s o l id sf o r m a b e d a t t h e b o t t o m o f th e p i p e f ro mf u l l y s u s p e n d e d f l o w s . I t i s a l s o r e f e r r e dt o a s t h e m i n i m u m c a r r y in g o r th e l i m i ti n gd e p o s i t v e lo c i t y , an d is p e r h a p s t h e m o s ti m p o r t a n t t r a n s i ti o n v e l o c i t y in s lu r r yt r a n s p o r t . I t is v e r y d i f f i c u l t t o d e t e r m i n ee x p e r i m e n t a l l y , b e c a u s e th e c r i ti c al c o n d i t i o nis d i f f i c u l t to d i s ce r n , a n d b e c a u s e t h e f l o wb e c o m e s u n s t a b l e n e a r t h e c r it ic a l c o n d i -t i o n .A m o n g t h e i m p o r t a n t t e c h n i c a l p r o b l e m si n p i p e l i n e f l o w o f s l u r r i e s a r e t h e d e t e r m i n a -t i o n o f t h e p r e ss u r e d r o p - t h r o u g h p u t r e la t i o n -s h ip , t h e p r e d i c t i o n o f t h e p r e v a il in g f l o wr e g im e , a n d t h e e s t i m a t i o n o f t h e c r i t ic a lv e l o c i t y . T h e s e p r o b l e m s h a v e b e e n t h es u b j e c t s o f e x t e n s iv e r e s ea r c h , a n d h a v e c o m et o a c q u i r e a c o n s i d e r a b l e l i t e r a t u r e . T h el i t e r a t u r e c o n t a i n s m a n y c o r r e l a t i o n s f o rp r e d i c t i o n o f p r es s u r e d r o p , a f e w s c h e m e sf o r d e l in e a t in g s l u rr y f l o w r e g im e s , a n d m a n yc o r r e l a t io n s f o r e s t i m a t i o n o f t h e c r it ic a lv e l o c i t y . T h e r e a r e w i d e d i s c r e p a n c i e s a m o n gt h e p r e d i c t i o n s f r o m t h e v a r i o u s c o r r e l a ti o n s .T h e s e d i f f i c u l t i e s w i l l b e e x a m i n e d i n d e t a i li n t h is p a p e r w i t h r e s p e c t t o c o r r e l a ti o n s f o rp r e d i c t i o n o f t h e c r i t ic a l v e l o c i t y . I n th i sp a p e r , w e r e v i e w p u b l i s h e d c o r r e l a t i o n s f o rt h e c r i t i c a l v e l o c i t y , w e c o m p a r e t h e m w i t he a c h o t h e r b y r e ca s t in g t h e m i n t o a s t a n d a r df o r m , w e t e s t t h e m a g a i n s t a b r o a d c o l l e c t i o no f p u b l i s h e d e x p e r i m e n t a l c r i t ic a l v e l o c i t yd a t a , a n d w e p r e s e n t a n e w s e t o f c r i ti c alv e l o c i t y c o r r e l a t io n s b a s e d o n f i t t in g v a r i o u sf o r m s o f t h e s t a n d a r d e q u a t i o n t o t h e e x p e r i-m e n t a l d a t a .

003 2-5 910 /87/ 3.50 Elsevier Sequoia/Printed in The Netherlands


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36P R E S S U R E D R O P C O R R E L T I O N S N D R E G I M ED E L I N E A T I O N

T h e r e h av e b e e n m a n y a t t e m p t s t o d e v e l o pc o r r e la t io n s f o r p r e d i c t in g t h e p r e s s ur e d r o pf o r t h e f l o w o f s l u r ri e s in p i p e l in e s . B e g i n n i n gw i t h t h e w o r k o f B l a tc h [ 1 ] , s o m e o f t h es u b s e q u e n t c o r r e l a t i o n s i n c l u d e t h o s e b yH o w a r d [ 2 ] , W i l s o n [ 3 ] , N e w i t t et al. [4 ] ,D u r a n d a n d c o - w o r k e r s [ 5 - 8 ] , Z a n d i a n dG o v a t o s [ 9 ] , a n d T u r i a n a n d Y u a n [ 1 0 ] .T h e w o r k o f D u r a n d a n d c o - w o r k e r s i n t h eea r l y f i f t i e s is s i g n i f i c an t b eca u se i t i n v o l v ede x a m i n a t i o n o f a v e r y w i d e r a n g e o f t h ep e r t i n e n t v ar i ab l e s in s l u r r y tr a n s p o r t . S a n dan d g rav e l sl u r ri e s , w i t h p a r t i c l e s i z e s r an g i n gf r o m 0 .2 t o 2 5 m m , i n p i p e s r a n g in g in d i a m -e t e r f r o m 3 . 8 t o 5 8 c m a n d s o l id s c o n c e n t ra -t i o n s u p to 6 0 b y v o l u m e w e r e i n v es t ig a t ed .D u r a n d ' s c o r r e l a t i o n i s g i v e n b yi - - i w _ K [ v2 CDO.S]miwC [g D (s - - 1 ) (1 )

T h e a d j u s t a b le c o n s t a n t s K a n d m w e r e t a k e nt o h a v e t h e v a l u e s 8 4 . 9 a n d - - 1 . 5 , r e s p e c t i v e ly ,f o r t h e s a n d a n d g r a v e l s l ur r ie s i n v e s t i g a te db y D u r a n d a n d c o - w o r k e r s . T h e d r a g c o e f f i -c i en t C D i s t h a t f o r s e t t l i n g o f t h e p a r t i c l e a ti t s t e r m i n a l v e l o c i t y i n t h e q u i e s c e n t ,u n b o u n d e d li q u id . F o r t h e e q u i v a le n ts p h e r i c a l p a r t i c l e o f d i a m e t e r d , i t i s g i v e n b y

4 g d ( s - - 1)C D - 3 v ~ 2 ( 2 )Z a n d i ' s a n d G o v a t o s ' [ 9 ] w o r k r e p r e s e n t st h e n e x t i m p o r t a n t c o n t r i b u t io n o n p r es s ur ed r o p p r e d ic t io n . T h e y p r o p o s e d t h e f o ll o w in gi m p r o v e d m o d i f i c a ti o n f o r p r e d i c ti o n o fs l u r r y p r e s s u r e d r o p :

i - - / w- 2804 - 1 93 f o r @ < 1 0 ( 3 )i ~ c

i -- iw _ 6.30@_0.3s 4~ , ci n w h i c h

v 2t~ CD 0 5g D ( s - - 1 )

fo r ~ > 1 0 (4 )

5)E q u a t i o n s ( 3 ) a n d ( 4 ) a r e v a l i d w h e n t h et r a n s i t i o n n u m b e r , d e s i g n a t e d b y Z a n d i a n dG o v a t o s b y N I , o b e y s t h e i n e q u a l i t y

V 2 C D 0 5N I - > 40 6)C D g ( s - - 1 )A c c o r d i n g to Z a n d i a n d G o v a t o s [ 9 ] , d a t ac h a r a c te r i z ed b y N~ < 4 0 d o n o t b e l o n g t o t h ef u l ly s u s p e n d e d r e g i m e o f f lo w .T u r i a n a n d Y u a n [ 1 0 ] d e v e l o p e d a ne x t e n d e d p r e s s u r e d r o p c o r r e l a t i o n s c h e m ew h i c h t a k e s i n t o a c c o u n t t h e f a c t t h a t v a r io u sf l o w r e g i m e s a r e o b s e r v e d t o p r e v a il i n s l u rr yt r a n s p o r t d e p e n d i n g u p o n f l ow c o n d i ti o n s .T h e i r c o r r e l a t i o n i s g i v e n b y t h e f o l l o w i n gre l a t i o n sh i p s :F l o w w i t h a s t a t i o n a r y b e d ( r e g i m e 0 ) :f - - fw = 12 .1 27 C 7389 w 7717CD-'4s4

1)1 (7)S a l t a t i o n f l o w ( r e g i m e 1 ) :f - - fw = 1 0 7 . 0 9 C 1 l S f w l ~ C v- ' 4 2 1 3

V 2 ] -1 .354 D g ( s - - I)] 8)

H e t e r o g e n e o u s f l o w ( r e g i m e 2 ) :f - - fw = 3 0 . 1 1 5 C 8 6 8 7 f w l 2 C D - 1 6 7 7

U 2 1 - 0 . 6 9 3 8 D g ~ : - - 1)J 9)

H o m o g e n e o u s f l o w ( r e g i m e 3 ):f - - f w = 8 . 5 3 8 C ' 5 2 4 f w l '4 2 8 C D 0 1516

v2 1-0.3s31 D g ( ~ - - 1 )J ( 1 0 )

i n w h i c h f a n d f w a r e t h e f r i c t i o n f a c t o r s f o rt h e s l u r r y a n d f o r w a t e r , r e s p e c t i v e l y , a t t h es a m e m e a n v e l o c i t y . B o t h f a n d f w a r ed e f i n e d i n t e r m s o f t h e c a r r i e r li q u i d d e n s i t y p .T h u s1 D (_~_~p) (11)f = 2 p v 2

w h e r e ( d p / d z ) i s t h e p r e s s u r e g r a d i e n t .A d e t a i l e d e v a l u a t i o n o f t h e v a r i o u s p r e s s u r ed r o p c o r r e l a t io n s , i n c l u d i n g t h o s e c i t e d a b o v e ,h a s b e e n g iv e n b y T u r i a n a n d Y u a n [ 1 0 ] .T h e n u m b e r o f p r e s s u r e d r o p c o r r e l a t i o n s


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for pipeline tran spor t o f slurries is quite large.Kazanskij [11] cites 37 such correlations inhis review. Virtually all these corre lations areempirical.Turian and Yuan [10] used their extendedpressure drop correlation, given by eqns. (7)to (10), to develop a comprehensive regimedelineation scheme. It is shown that maps ofthe various flow regimes may be drawn usingrelationships for what Turian and Yuan referto as the r e g im e t r a n s i t i o n n u m b e r R i i , whichpertains to transi tion be twee n regimes i and j.The expressions for the regime transitionnumber s R01, R02 and R0a, which perta in totransition between the flow with a stationarybed regime (0) and each of the followingthree: the saltation flow (1), heterogeneousflow (2) and homogeneou s flow (3) regimes,respectively, will be needed in our examina-tion of the problem relating to the criticalvelocity given below. The relationships forthese regime transition numbers are given bythe following equations:

v 2R01 = 4679.5 cl S3fw 1 5 4 C D - ' 6 1 6 D g ( s - - 1

(12)v

Ro2 = 0.1044 C - a22s fw- 1.o6s CD-O.o616Dg s _ I)1 3 )

V2R o 3 = 1 . 6 0 4 c O . 3 1 8 8 w _ O . ss 3 7C D - O . 7 4 ~ D g ( s _ 1)

(14)More detailed reviews of published pressuredrop correlations are given by Zandi andGovatos [10], Zandi [12], Turian and Yuan[10], and Kazanskij [11]. A review of flowregime delineation and a detailed descriptionof the extended scheme referred to in theforegoing is given by Tur ian and Yuan [10].

T H E CRI T I CA L V E L O CI T Y CO M PA RI SO N O FPU BL I SH E D CO RRE L A T I O N S

The critical velocity is a primary par ameterin the design of slurry pipelines. Many correla-tions fo r predicting the critical velocity havebeen proposed. Different researchers haveused dif ferent variables and variable groupings

37in their correlations. In order to compare andto evaluate these correlations, we will firstrecast them into a standard form.Under restricted conditions, which includeunif orml y sized spherical particles and smoothpipe walls, the critical velocity dependencecan be expressed byvc = f [d, D, C, (Ps -- P)g, P, Pl (15)Inclusion of th e variable grouping [(Ps -- P)g]implies that we view the eff ect of gravity tobe only manifested through net settling forceson the particles. In non-dimensional variableseqn. (15) can be written in the form

Vc

[2gD(s -- 1)] s= f [ ( d ) , D p[gD(s--1)] sp , C ] (16)

Use of the refe renc e veloci ty [2gD(s -- 1)] 0.sin eqn. (16) is motivated by the fact that it isgenerally assumed that the critical velocitydependenc e on pipe diameter is approximatelyaccording to D in.Using an analysis based on balancing theenergy required to suspend the particles withthat derived from dissipation of an appropriatefraction of the turbul ent eddies, Oroskar andTurian [13] derived the following criticalvelocity correlation:

VC 5 8/15- [ C 1 - - C):n -l ] 8/ls[2gD(s -- 1)] 1/2 21/2x [ D p [ g D ( s ~ - - 1 ) ] 1 / : ] 1 18

[ P(17)

where the ex ponen t n is the hindered settlingexpon ent in multi-particle sedimentation asgiven by Richardson and Zaki [14] and byGarside and A1-Dibouni [15]. The values of nfall approximately in the range 2 < n < 5,the upper limit corresponding to the lowReyno lds Stokes law region. The numericalcoefficient in eqn. (17) depends upon whatfraction of the turbu lent energy, intrinsic tothe flow, is assumed to be required in suspend-ing the particles at incipient suspension. Ifthis fraction is taken to be c~, then thenumerical coefficient would be [(5/6~)8/lS/21/2]. The value (58/lS/21/:) in eqn. (17)corresponds to ~ = 1/6.


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3 8

T A B L E 1C o m p a r i s o n o f p u b l i s h e d c r i t ic a l v e l o c i t y c o r r e l a t i o n s i n s l u r r y t r a n s p o r t

[ 2 g D ( s - - 1 ) ] ' s = f ( C ' s ) C D k [ p J \ D ]

A u t h o r f ( C , s ) k 1 m 1 5 a; R M S aW i l s o n [ 3 ]D u r a n d a n d

C o n d o l i o s [ 6 ]C r a v e n [ 1 9 [K n o r o z( r e p o r t e d b y

G o r j u n o v [ 2 0] )N e w i t t et al . [ 4 ]S p e l l s [ 2 1 ]C a i r n s et al . [ 22 ]T h o m a s [ 2 3 ]

T h o m a s [ 2 4 ]

1 . 1 3 2 2 [ C s / ( C s + 1 - - C ) ] ' 3 6 ~ ( s - - 1 ) - ' ~ 3 s - - 0 . 1 8 1 8 0 . 0 9 8 1 8 0 . 0 8 1 80 . 7 - 1 . 3 0 0 0

2 . 2 5 0 8 / ( 1 - - C ) 0 0 0 . 50 . 8 1 6 5 [ s C / ( 1 - - C ) ] 1 /6 - - 0 . 5 0 - - 1 / 1 2

1 3 . 8 8 0 4 - - 1 / 2 0 1 / 20 . 0 3 4 9 2 [ C ( s - - 1 ) * 1 ] 0 .6 32 7 0 0 . 6 3 2 7 0 . 8 1 6 32 . 7 0 7 4 C ' I 7 6 S / ( s 1 0.2353 0 0 . 1 7 6 5 0 . 5 8 8 22 1 . 3 6 3 5 - - 0 . 1 5 8 3 - - 0 . 6 8 3 4 - - 0 . 6 6 8 0

1 . 3 7 6 2 ( S - - 1 ) '1 87 8 - - 0 . 4 0 8 2 - - 0 . 1 8 3 7 - - 0 . 4 0 8 2

R e p o r t e d b y 1 . 3 6 2 2 ( s - - 1 ) 2/ 7 0 1 / 7 0S c h u l z [ 2 5 ]B r a u e r a n d 0 . 5 1 1 5 C '4 1 3 8 ( s - - 1 ) 0.2759 - - 0 . 6 2 0 7 0 . 5 1 7 2 0 . 6 2 0 7K r i e g e l [ 2 6 ]Z a n d i a n d ( 2 0 C ) 1 /2 - - 1 / 4 0 0G o v a t o s [ 9 ]W i e d e n r o t h [ 2 7 ] 0 . 4 5 5 9 ( s - - 1 ) - ' 2 s - - 0 . 2 5 0 0L a r s e n [ 2 8 ] 2 . 9 8 1 3 C I /4 - - 1 / 4 0 0

R o s e a n d 9 . 6 5 4 4 [ C / ( 1 - - C ) ] ' 4 ( s - - 1 ) ' s s - I - - 1 0 - - 0 . 2D u c k w o r t h[ 2 9 ]S h o o k [ 3 0 ] 4 . 4 7 C 1 /2 - - 0 . 2 5 0 0S h o o k [ 3 0 ] 2 . 4 3 C l / a - - 0 . 2 5 0 0B a b c o c k [ 3 1 ] 2 . 2 3 6 1 C 1 /2 - - 0 . 2 5 0 0B a i n a n d 2 . 4 6 0 7 C l / a - - 0 . 2 5 0 0B o n n i n g t o n[ 3 2 ]N o v a k a n d 1 . 3 1 3 7 0 0 - - 0 . 0 5N a l l u r i [ 3 3 ]

R o b i n s o n a n d 0 . 9 0 1 C A 6 0 0 0G r a f [ 3 4 ]K a o a n d W o o d ( 2 / 3 ) l / 2 g ( n ) b 0 0 1 / 2 - - 1 / n[ 3 5 ]C a r l e t o n a n d 3 . 0 1 3 7 1 1 6 . 7 ( 1 - - C )4 6 / K ] 4 IT 0 1 / 7 4 / 7C h e n g [ 1 6 ]

4 9 . 5 1 ; 0 . 6 2 0 63 6 . 5 3 ; 0 . 4 0 7 4f o r f ( C , s ) =1 . 2

7 7 . 5 9 ; 0 . 8 6 7 15 1 . 0 1 ; 0 . 6 6 9 5

9 3 . 4 3 ; 1 . 3 1 0 06 8 . 2 1 ; 1 . 5 0 1 25 8 . 1 7 ; 0 . 7 1 2 96 5 . 0 3 ; 0 . 7 4 6 1

d i l u t e ,f l o c c u l a t e dp a r t i c l e s2 9 . 9 5 ; 0 . 5 0 4 4d i l u t e , n o n -f l o c c u l a t e d

7 5 0 . 4 0 ; 7 . 0 8 7 72 2 4 . 6 4 ; 6 . 1 7 6 55 0 . 0 2 ; 0 . 6 5 2 16 9 . 9 5 ; 0 . 8 1 2 54 5 . 2 2 ; 0 . 6 0 9 01 7 2 . 3 5 ; 2 . 8 2 2 3

4 9 . 9 9 ; 0 . 6 5 1 83 4 . 5 0 ; 0 . 5 0 5 94 6 . 6 8 ; 0 . 6 1 0 03 4 . 4 8 ; 0 . 5 0 5 5

9 0 , 1 1 ; 0 . 8 2 5 42 6 . 4 9 ; 0 . 4 5 3 26 9 . 0 1 ; 0 . 7 9 3 9f o r n = 1 07 1 . 3 4 ; 0 . 9 8 1 0f o r < 0 . 2 4K = I

2 . 6 7 ( 1 - - C ) 4 6 l 4 / 73 . 0 1 3 7 K ~ - - - - T ] 2 ~ 0 1 / 7 4 / 7 f o r C > 0 . 2 4 ,C o = 0 . 6 5T u r i a n a n d 8 . 9 4 8 C '4 77 9 - - 0 . 0 2 7 2 - - 0 . 1 1 7 4 0 3 9 . 9 7 ; 0 . 4 6 3 6Y u a n [ 1 0 ] R o 2 , R o a < 1 c0 . 9 1 2 5 C - A s 6 - - 0 . 3 4 0 6 0 . 1 5 3 6 0 R o i , R o 3 < 13 . 2 5 1 C A Ts 9 - - 0 . 4 2 1 3 0 . 1 2 4 2 0 R o b R o : < 1W a s p e t a l . [ 3 6 ] 3 . 3 9 9 C '2 1 s6 0 0 1 / 6 2 6 . 6 8 ; 0 . 3 7 5 0

T h o m a s [ 1 7 ] 4 . 9 9 3 7 0 - - 5 / 2 1 0 6 0 . 1 4 ; 0 . 7 1 2 5T o d a e t a l . [ 3 7 ] 0 . 8 0 5 4 C -2 - - 0 . 2 5 0 - - 0 . 3 5 1 6 6 . 5 2 ; 1 . 7 4 8 6O r o s k a r a n d 2 - w 2 1 5 C ( 1 _ c ) 2 n - l ] a / l s 0 1 / 1 5 0 2 5 . 94 ; 0 . 4 3 3 1T u r i a n [ 1 3 ] f o r n = 2aT he absolute average per cent deviation/) and the root me an square deviation R M S are defined by eqns. 25)a n d ( 2 6 ) .

2 n 2 [ ( n + 1 ) ( n + 2 ) 1 1 / ~b g ( n ) = ( 2 n + l ) ( n + l ) [ n 2 2 / ( n + 2 ) JC U s e e q n s . ( 1 2 ) t o ( 1 4 ) t o c a l c u l a t e R 0 1 , R 0 2 a n d R o a .


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P u b l i s h e d c r i t i c a l v e l o c i t y r e l a t i o n s h i p s ,a n d e q n . ( 1 7 ) , s u g g e s t t h a t m o s t a v a i la b l ec o r r e la t io n s c a n b e s u b s u m e d u n d e r t h e f o r m

V c

[ 2 g D ( s - - 1 ) ] 1 /2= f ( C , s ) C D k l D p [ g D ( s - - 1 ) ] ' n l ' ( d ) m

18)i n w h i c h f ( C , s ) i s m a i n l y a f u n c t i o n o fp a r t ic l e c o n c e n t r a t i o n , a n d k , I a n d m t a k ed i f f e r e n t n u m e r i c a l v a l u e s f o r d i f f e r e n tc o r r e l at i o n s . I t n e e d s t o b e p o i n t e d o u t th a tt h e f o r m i n e q n . ( 1 8 ) i s a s p e c i a l c a s e o f e q n .( 1 6 ) e v e n t h o u g h t h e p a r t i c le d r a g c o e f f i c i e n tC D d o e s n o t a p p e a r e x p l i c i t ly a m o n g t h ed i m e n s i o n l e s s g r o u p s in e q n . ( 1 6 ) . F o r ap a r t ic l e s e tt li n g in a n u n b o u n d e d q u i e s c e n tl i q u i d , t h e q u a n t i t y A = C D l n N R e C a n beo b t a i n e d f r o m t h e d r a g c o e f f i c i e n t - p a r t i c l eR e y n o l d s n u m b e r r e la t io n s h i p . F u r t h e r m o r e ,A i s r e l a t e d t o t h e f i r s t t w o g r o u p s o n t h er i g h t .h a n d s id e o f e q n . ( 1 6 ) b yA = C D 1 2 N R e

_ 4 l n l D p [ g D ( s - - 1 ) ] l / 23 # D ] 3 n ( 1 9 )A c c o r d i n g l y , t h e d i m e n s i o n l e s s g r o u p s i ne q n . { 1 6 ) a re s u f f i c i e n t t o d e f i n e C D t h r o u g ht h e d r a g c o e f f i c i e n t r e l at i o n s h ip .

T a b l e 1 l is t s p u b l i s h e d c r i t i c a l v e l o c i t yc o r r e l a t i o n s w h i c h h a v e b e e n r e c a s t i n t h ef o r m a t o f e q n . ( 1 8 ) . T h e c r it ic a l v e l o c i t yc o r r e l a t io n s d u e to C a r l e t o n a n d C h e n g [ 1 6 ] ,T u r i a n an d Y u a n [ 1 0 ] a n d T h o m a s [ 1 7 ] , i nt h e i r o r i g i n a l f o r m s , c o n t a i n t h e F a n n i n gf r i c t i o n f a c t o r f o r t h e s u s p e n d i n g l i q u i d f wd e f i n e d b y e q n . ( 1 1 ) . I n o r d e r t o e l i m i n a t efw i n s u c h c a s e s, a n d r e c a s t t h e r e s u l t s i ns t a n d a r d f o r m a t , w e u s e t h e B l a s iu s [ 1 8 ]e x p r e s si o n f o r t h e f r i ct i o n f a c t o r b a s e d o nt h e ( 1 / 7 ) t h p o w e r t u r b u l e n t v e l o c i ty p ro f i l e :f w = 0 . 0 7 9 1 / ~ R e - 1 / 4 ( 2 0 )i n w h i c hI VR~ pn v 21)

PE q u a t i o n s ( 2 0 ) a n d ( 2 1 ) w i t h v = V c a r es u b s t i t u t e d f o r fw f o r t h e c r it ic a l v e l o c i t yc o r r e l a t i o n u n d e r c o n s i d e r a t i o n , a n d V c ist h e n s o l v e d f o r .

39T h e c r i t i c a l v e l o c i t y e x p r e s s i o n b a s e d o n

e q n . ( 1 2 ) o f T u r i a n a n d Y u a n [ 1 0 ] g i v e na b o v e i s

Vc[ 2 g D ( s - - 1 ) ] s

1- 20.s {4679 .5c l ' S3 fw1 64CD- ' 616} ' s( 2 2 )

S u b s t i t u t in g f o r fw f r o m e q n s . ( 2 0 ) a n d ( 2 1 )w i t h v = V c, w e o b t a i n a f t e r s o l v in g f o r V ct h e r e l a t i o n

v[ 2 g D ( s - - 1 ) ] s

= 8 .9 4 8 C 0.4779CD-0. 02n l D p [ g D ( s - - l )] 's ( 2 3 )

w h i c h i s o n e o f t h e e n t r ie s i n T a b l e 1 . I no r d e r t o u s e t h e c r i t i c a l v e l o c i t y c o r r e l a t i o nb a s e d o n T u r i a n s a n d Y u a n s [ 1 0 ] r e g im ed e l i n e a t i o n r e s u l t s , t h e v a l u e s o f t h e r e g i m et r a n s i t io n n u m b e r s R 0 1, R 0 : a n d R 0 3 n e e d t ob e c a l c u l a t e d u s i n g e q n s . ( 1 2 ) t o ( 1 4 ) .

M o s t o f t h e c r i t i c a l v e l o c i t y c o r r e l a t i o n sl i s t e d i n T a b l e 1 a r e e m p i r i c a l , a l t h o u g h s o m e ,l ik e t h o s e d u e t o K a o a n d W o o d [ 3 5 ] ,C a r l e t o n a n d C h e n g [ 1 6 ] , a n d O r o s k a r a n dT u r i an [ 1 3 ] , a r e b a s e d o n a n as s u m e d m o d e lf o r t h e p a r t i c l e s u s p e n s i o n p r o c e s s . A n u m b e ro f t h e c r i t i c a l v e l o c i t y c o r r e l a t i o n s w h i c hh a v e a p p e a r e d i n t h e l i t e ra t u r e c o u l d n o t b et r a n s f o r m e d i n t o th e s t a n d a r d f o r m g iv e n b ye q n . ( 1 8 ) . S o m e o f t h e s e a r e g iv e n in T a b l e 2 .T h e c o r r e la t io n d u e t o T h o m a s [ 2 4 ] i nT a b l e 2 c o m p r i s e s a s u m o f t w o t e r m s e a c hh a v i n g t h e s t a n d a r d f o r m .

R e s u l t s o f t e s ts o f t h e c o r r e l a t io n s l is t edi n T a b l e s 1 a n d 2 a g a i n s t e x p e r i m e n t a lc r it ic a l v e l o c i t y d a t a w i ll b e p r e s e n t e d i n t h en e x t s e c ti o n . I t is o b v i o u s u p o n c o m p a r i n gt h e v a r i o u s e x p r e s s i o n s i n T a b l es 1 a n d 2 t h a tt h e q u a l i t a t i v e d i s p a r i t i e s a m o n g t h e v a r i o u sc o r r e l a t i o n s w h i c h h a v e b e e n p r o p o s e d a r eq u i t e b r o a d . M o s t co r r e l a ti o n s , a n d t h ee x p e r i m e n t a l d a t a o n w h i c h t h e y a r e b a s e d ,i n d i c a t e t h a t t h e c r i t i c a l v e l o c i t y h a s a p p r o x i -m a t e l y a s q u a r e - r o o t d e p e n d e n c e o n p i p ed i a m e t e r ( ~ D 1 / 2 ) , a n d a r a t h e r w e a k d e p e n -d e n c e o n p a r t ic l e d i a m e t e r ( ~ d ) , e s p e c ia l lyf o r la r g e r n o n - c o l l o i d a l p a r t i c le s . A c c o r d i n g l y ,


6/13

4 0T A B L E 2C r i ti c al v e lo c i t y c o r r e l a ti o n s w h i c h c a n n o t b e c a st i n s t a n d a r d f o r m ( e q n . ( 1 8 ) )Y u f i n [ 3 8 ] : a

v c = 1 4 . 2 3 d ' 6 S D ' s4 e x p ( 1 . 3 6 [ 1 + C s - - 1 ) ] ' S d - 0 . 1 3 )/ 3 b = 3 5 . 7 7 ; R M S b = 0 . 4 4 7 0

T h o m a s [ 2 4 ] :v c

[ 2 g D ( s - 1 ) ] ' s p+ 6 . 9 8 7 7 c 0 . S T 1 4 C D _ ~ 4 6 2 5 1 D p [ g D s - 1) ] ' s -0 .07483

P/ ) = 7 5 . 0 2 ; R M S = 0 . 9 2 1 3

Y u f i n [ 3 9 ] : cv c = 9 . 8 D l i 3 V s l l 4 [ C s - - 1 ) + 0 . 6 ]/ ) = 5 1 . 2 6 ; R M S - - 0 . 7 8 4 4

Y u f i n a n d L o p a s i n [ 4 0 ] :dv c = 8 . 3 D 1 / 3 ( C ~ ) 1 /5 w i t h ~ = C - ' T s/ 9 -- 3 2 . 5 4 ; R M S = 0 . 4 4 0 6

B o n a p a c e [ 4 1 ] :

[ 2 g D ( s - - 1 ) ] 0 s = 2 ' s , ~ i D / \ S s a n d - -/ 9 = 4 2 4 . 7 5 ; R M S = 5 . 0 7 3 6

a Q u o t e d i n A S C E T a s k C o m m i t t e e R e p o r t [ 4 2 ] ; u n i t s i n f t a n d s.b T h e a b s o l u t e a v e r a g e p e r c e n t d e v i a t i o n / 9 a n d t h e r o o t m e a n s q u a r e d e v i a t i o n R M S a r e d e f in e d b y e q n s . ( 2 5 )a n d ( 2 6 ) .c Q u o t e d b y S a s i c a n d M a r j a n o v i c [ 4 3 ] ; u n i t s i n f t a n d s .d Q u o t e d b y W i e d e n r o t h a n d K i r c h n e r [ 4 4 ] ; u n i t s i n f t a ns s.

the correlations of Wilson [3], Newitt e t a l .[4], Spells [21], Cairns e t a l . [22], Thomas[24], Brauer and Kriegel [26], Rose andDuckworth [29], Carleton and Cheng [16],and Turian and Yuan [10] are in conflict tovarying degrees with exper imenta l evidence.Recently, Parzonka e t a l . [45] havepresented a substantial bod y of experimentaldata which suggests tha t th e critical velocityattains a maximum with concentration. Thecorrelation of Oroskar and Turian [13], givenby eqn. (17) and in Table 1, gives a maximumfor Vc at C = 1 / 2 n . It is clear from an examina-tion of details of the physical model uponwhich eqn. (17) is based, that the maximumin the critical velocity- conce ntrat ion relation-ship occurs because of increased particleinteraction with concentration. The critical

velocity initially increases with concentrat ion,but as more particles are added, hinderedsettling effects become increasingly influentialand counteract particle settling, whichexplains the reversal in the Vc vs. C relation-ship.The drag coeffi cient CD appears in a num berof the critical velocity correlations. For largeand heavy particles, which constitute themain type in coarse-particle slurry transport,CD is virtually independent of particleReynolds number N R e . In the high Reynoldsnumber, Newton s law, region N R e > 500),CD ~- 0.44. The value of the drag coeff icientfor intermediate and low values of th e particleReynolds number may be estimated fromcorrelations given by Bird e t a l . [46], byDavies [47] or by Turian and Yuan [10].


7/13

C O M P A R I S O N W I T H E X P E R I M E N TT h e c o r r e l a t io n s p r e s e n t e d i n T a b l e s 1 a n d 2

w e r e t e s t e d a g a i n s t e x p e r i m e n t a l c r i t i c a lv e l o c i t y d a t a . T h e d a t a u s e d w e r e c o l l e c t e df r o m t h e l i te r a t u r e . T a b l e 3 l is t s t h e s o u r c e so f t h e s e d a t a , t h e m a t e r i a l s c o m p r i s i n g t h es lu r ri es , a n d t h e r a n g e s o f t h e p e r t i n e n tv a r i a b l e s i n v o l v e d . T h e d a t a w e r e s c r e e n e df o r i n c o m p l e t e n e s s, r e d u n d a n c i e s a n d e v i d e n ti n a c c u r a c i e s , r e s u l t i n g i n a t o t a l c o l l e c t i o n o f8 6 4 d a t a p o i n t s a s d e p i c t e d i n T a b l e 3 . F o rm i x e d p a r t i c l e s iz e s, t h e p a r t i c le d i a m e t e rd s 0 , c o r r e s p o n d i n g t o t h a t a b o v e w h i c h 5 0o f t h e p a r t ic l e s b y w e i g h t l ie , w a s t a k e n a st h e e q u i v a l e n t p a r t i cl e d i a m e t e r .

I n o r d e r t o c o m p a r e t h e v a r i o u s c o r r e l a -t i o n s w i t h t h e e x p e r i m e n t a l d a t a , w e c a l c u l a t e dt h e p e r c e n t d e v i a t i o n f o r e a c h d a t a p o i n t a n dc o r r e l a t io n a s d e f i n e d b y

d e v = D v= ( V c (c a l c) - -V c (e x p ) ] X l O 0 ( 2 4 )

VC exp)F r o m t h e c a l c u l a t e d v a lu e s o f t h e p e r c e n td e v i a t i o n D v i f o r e a c h d a t a p o i n t a n d c o r re la -t i o n , a n o v e r a l l a b s o l u t e a v e r a g e p e r c e n td e v i a t i o n , a n d a n o v e r a ll r o o t m e a n s q u a r ed e v i a t i o n w e r e c a l c u l a t e d f o r e a c h c o r r e l a ti o n .T h e s e a r e d e fi n e d b y/ ~ - A b s . a v g. d e v .

N ]Dvi ]= ~ 2 5 )i= g

R M S _ t ~ - ~ [ V c ( ca l c) - - V C ( e x p ) ] 2 f sN 2 6 )

T h e r e s u l t s o f t h e s e c a l c u l a t i o n s a r e g i v e n i nt h e l a s t c o l u m n o f T a b le s 1 a n d 2 . T h e s ec o m p a r i s o n s w i t h e x p e r i m e n t a l d a t a i n d ic a t et h a t t h e c o r re l at io n s p r o p o s e d b y R o b i n s o na n d G r a f [ 3 4 ] , O r o s k a r a n d T u r i an [ 1 3 ] a n dW a s p et al . [ 3 6 ] d o a b e t t e r j o b o f p r e d ic t in gt h e c r i t i c a l v e l o c i t y t h a n t h e r e m a i n i n gr e l a t i o n s h i p s l i s t e d in T a b l e s 1 a n d 2 . T h ec o r r e la t io n s o f R o b i n s o n a n d G r a f a n d o fW a s p e t a l . are p u r e l y e m p i r ic a l a s s t a t e de a r l i e r . A d d i t i o n a l d e t a i l s r e l a t i n g t o t h ec o m p a r i s o n b e t w e e n e x p e r i m e n t s a n d c o rr el a-t i o n s w i ll b e p r e s e n t e d b e l o w .

4 1C O R R E L A T I O N S B Y R E G R E S S IO N

A s i d e f r o m t h e p r e v i o u s l y p u b l i s h e dc o r r e l a t i o n s l i s t e d in T a b l e s 1 a n d 2 , w ed e v e l o p e d a n a d d i t i o n a l s e t o f c o r r e l a t i o n su s in g t h e f o l l o w i n g f o r m o f e q n . ( 1 8 ) :

V c[ 2 g D ( s - - 1 ) ] 's

= x l C X ~ ( 1 - - C ) 3 1 D p[gD (s -- l )] s l X~(d( 2 7 )

V a r i o u s f o r m s o f e q n . ( 2 7 ) w e r e c o n s i d e r e d ,a n d t h e c o r r e s p o n d i n g v a l u e s o f t h e a d j u s t a b l ec o n s t a n t s X i w e r e d e t e r m i n e d b y f i t t in g t h e8 6 4 c r i t i c a l v e l o c i t y d a t a u s i n g m u l t i l i n e a rr e g r e s s i o n , in w h i c h t h e l i n e a r i ze d l o g f o r mo f t h e e q u a t i o n is f i t te d . T h e e s t i m a t e s o ft h e s e a d j u s t e d p a r a m e t e r s , t o g e t h e r w i t h t h ec o r r e s p o n d i n g s t a n d a r d e r r o r s o f t h e e s t i m a t e s ,a r e g i v e n i n T a b l e 4 . F i v e d i f f e r e n t c a s e s ,c o r r e s p o n d i n g t o e q u a t i o n s i n w h i c h v a r i o u sX i a r e a s s u m e d t o b e z e r o , a r e d e p i c t e d i nT a b l e 4 . F o r e a c h c a s e c o n s i d e r e d , a l l 8 6 4d a t a p o i n t s w e r e u s e d t o o b t a i n t h e e s t im a t e sf o r t h e p a r a m e t e r s i .

T h e r e s u l t s in T a b l e 4 le a d t o a n u m b e r o fc o n c l u s i o n s . B a s e d o n t h e v a l u e s o f t h ea b s o l u t e a v e ra g e p e r c e n t d e v i a t i o n s a n d a l sot h e p e r c e n t r o o t m e a n s q u a r e d e v i a t i o n sg i v e n i n t h e t a b l e , i t i s e v i d e n t t h a t t h ec o r r e l a t i o n s c o r r e s p o n d i n g t o a l l f i v e c a s e sa r e v i r t u a l l y e q u a l l y e f f e c t i v e i n p r e d i c t i n gt h e c r i t i c a l v e l o c i t y . T h e r e s u l t s f o r C a s e s 1a n d 3 su g g e st t h a t t h e d e p e n d e n c e o f t h ec r i t i c a l v e l o c i t y o n p a r t i c l e d i a m e t e r i s q u i t ew e a k , w h i c h is a p p r o x i m a t e l y d '6 . C o m -p a r i s o n o f t h e r e s u l t s f o r C a s e s 1 t o 3d e m o n s t r a t e s t h a t t h e o v e r al l e x p o n e n t o f Do n t h e r i g h t - h a n d s i d e o f e q n . ( 2 7 ) i s a b o u t- - 0 . 0 6 , w h i c h i s c l e a rl y v e r y s m a ll . A c c o r d -i n g l y , t h i s d e m o n s t r a t e s t h a t t h e c r i t i c a lv e l o c i ty d e p e n d e n c e o n p i p e d i a m e t e r is v e r yn e a r l y D 1/2. F o r O r o s k a r ' s a n d T u r i a n 's [ 1 3 ]c o r r e l a t i o n , g i v e n b y e q n . ( 1 7 ) , t h e c r i t i c a lv e l o c i t y d e p e n d e n c e o n p a r t ic l e d i a m e t e r i sg i v en b y d o a n d t h a t o n p i p e d i a m e t e r isD '6 . F i n a l ly , c o m p a r i s o n o f t h e r e s u l t s f o ra ll fi v e c a s es p r e s e n t e d i n T a b l e 4 d e m o n s t r a t e st h a t t h e c r i t ic a l v e l o c i t y d e p e n d e n c e o n t h ef a c t o r ( 1 - - C ) is i m p o r t a n t . I n d e e d , e x c l u s i o no f e i t h e r o f t h e t w o f a c t o r s C x2 a n d ( 1 - - C ) x3w o u l d p r e c l u d e t h e p o s s i b l e a t t a i n m e n t o f a


8/13

4

O

0 0 00 ~ 0 ~

, , ,0 0 0 0 0 0 0 0~ 0 0 ~ 0 0 ~ 0

0

0

~ ~

C -

CO

0 3

,

0 0 } ~ 0 0 0 0 0 0 0 0 0 0

~ m

00 0 0 0~ 0 0 0 ~ 0 ~

0 0 0 0 0 0 0 0 0

~ 0 0 ~ 0 0 0 0 0

0~ 0 ' 0 ~ k ' ~ ~ L O 0 ~ L O ~ ~ ' 0 C ~ L O ~ 0 L O L O L O L O ~ L O ~ 0 L ~ L O L O

~ ~ ~ ~ ~ ~ o ~ . ~ o o ~ ~ ~ ~ Z0 0 ~. ~ L~, -~ 0 ~ ~ ~ O~ ~ ~ ~ o~

[ - E--~ o rO ~

L O L O ~ t O L ' ' j , - - I 0 ' ~ - q D ~ . .~ . ,. .~ q D


9/13

4

0

. . . . . . ,

~ 0 ~ ~ . ~

L - -

0


10/13

44

TABLE 4Critical velocity cor relations using regression

Vc[2g D(s- 1)] s

= x 1 C X z ( 1 - - C ) x 3 1 D p [ g D ( s- -1 ) I 'S X ( dParameter Es t imate S tandard er ror of es t imateC ~ e l : E s t i m a t e a l l x iX1 1.79 51 1.08 78X2 0.1087 0.01 610X3 0.2501 0.0 9870X4 0.001 79 0.0 0767Xs 0.066 23 0.00 958D a = 20.53; RMS a= 0.34 16C~e 2: A~ume Xs = 0X1 1.84 71 1.090 1X2 0.1126 0.01 652X3 0.0342 1 0.100 4X4 --0. 0309 3 0.00621D = 21.54;RMS = 0 .3447C~e 3: A~ume X4 = 0X1 1.8176 1.0665X2 0.1086 0.01 608X3 0.2525 0.09 812)(_5 0. 06 48 6 0. 00 75 4D = 20.57;RMS = 0 .3412C~e 4: A~ume X4 = 0 and X s = 0X1 1.32 13 1.0 564X2 0.1182 0.016 71X3 0.3293 0.101 8D= 21.04;RMS = 0 .3552C~ e 5: A ss um e X3 = 0 X4 = 0 and Xs = 0X1 1.122 8 1.022 3X2 0 .07367 0 .00954D= 2 1 .3 5 ; R M S = 0 .3 5 5 9aThe absolu te average per cent devi at io n/) and theroot mean square deviation RMS are defined by eqns.(25) and (26).

maximum in the Vc vs C relationship, incontradictio n to a substantial body of experi-mental evidence which demonstrates thatsuch a maxi mum occurs [45].The occurrence of a maximum in theVc vs C relationship may , as previously stated,be explained on the basis of hindered settlingwhich becomes more pr onounc ed as concen-tration increases. For the critical velocitycorrelations corresp onding to Cases 1 thr ough 4in Table 4, the maximum in the vc vs Ccurve occurs at the conc entr atio n Cmax givenby

Cmax = C avc = o8C

2- - - 2 8 )X2 + X3

Based on the values of X2 and X3 given inTable 4 for Cases 1 throu gh 4, the value ofCmax ranges between 0.25 and 0.30. Accord-ing to Oroskar s and T urian s [13] relation,given by eqn. (17), the conc entr atio n Cmax =1/2n , and t he values of Cmax = 0.25 a nd 0.30correspond to n = 2 and 1.67, respectively.In order to provide a somewhat finer assess-ment of the relative predictive qualities ofthe empirical regression correlations in Table 4and also those by Robinson and Graf [34],Wasp et al [36], and Oroskar and Turian[13], which were found to be mos t promising,we present more detailed information ondeviations from experimental data. Table 5presents a listing of the maximum deviationand also the numbers of data points whichlie in various deviation bands for each of thesecorrelations. It must be emphasized that therelationships due to Robinson and Gr af [34]and Wasp et al [36], aside from being purelyempirical, are incapable of predicting amaximum in the Vc vs C relationship. Thislimitation, and the detailed comparisonspresented in Table 5 suggest that t he correla-tion due to Oroskar and Turian [13], givenby eqn. (17), and the empirical regressionresults correspondin g to Cases 1 and 3 in thiswork do th e best overall job of predicting thecritical velocity. It is, however, clear from anexaminati on of all cases that the dependen ceof Vc on particle size is quite weak.

CONCLUSIONSThe analysis presented in this work, andthe evaluations of published critical velocityrelationships fo r pipeline flow of slurries are

based on a substantial body of experimentaldata pertaining to a wide range of thepertinent variables. Critical velocity data arevery difficult to get, and are oft en uncertainand equivocal because the critical conditionis very difficult to ascertain. Indeed, theinstability in the flow, as the critical condit ionis approached, results in wide variations inmeasured variables, particularly the concen-tration (see, for example, Thomas [17]).


11/13

T A B L E 5D e ta i l e d c o mp a r i so n w i th e x p e r ime n ta l d a t a

VC[2gD(s - - 1 ) ] s

4 5

N o . o f p o in t s i n d e v i a t i o n b a n d< 2 0 2 0- 5 0 5 0 - 1 0 0 > 1 0 0

Maxdev.

R o b i n s o n a n dG r a f [ 3 4 ]Wasp e t a l .[ 3 6 ]O r o s k a r a n dTur ia n [ 13 ]R e g r e s s io n

results :Case 1Case 2Case 3Case 4Case 5

0.90 1C .16 339 458

3.399C 21s6 380 3881-6683C S333( 1 _ C ) L 6 t D p [ g D ( s - - 1)] s f 1/ls- 437 302X l C X 2 ( l _ C ) x 3 1 D p [ g D ( ; - - 1 ) ] S lX 4 ( d ) X

: X1 = 1 .795 1; X2 = 0 .1087 ;X3 = 0 .25 01; 535 260X,; = 0 .00 179 ; X5 = 0 .066 23;: X s = O ;X 1 = 1 . 8 4 7 1 ;X 2 = 0 . 1 1 2 6 ; 5 0 6 2 8 3X 3 = 0 . 3 4 2 1 ; X 4 = - - 0 . 0 3 0 9 3 ;: X4 = O;X1 = 1 .817 6; X2 = 0 .10 86; 533 262Xa = 0 .25 25; X4 = 0 .06 486 ;: X 4 = X s = O ;X 1 = 1 . 3 2 1 3 ;X 2 = 0 . 1 1 8 2 ; 5 2 2 2 7 0X3 = 0 .3293 ;: X a = X 4 = X s = O ; X l = l . 1 2 2 8 ; 511 281X2 = 0 .0 736 7;

67 0 80

84 12 206

118 7 164

65 4 12370 5 13065 4 12365 7 13264 8 132

T h i s is a n i n h e r e n t l i m i t a t i o n w h i c h m u s t b ew e i g h e d i n a ss e ss i n g t h e r e s u l t s o f t e s t sb e t w e e n c o r r e l a t i o n a n d e x p e r i m e n t . N o n e -t h e l es s , a n u m b e r o f c o n c l u s i o n s c a n b ed r a w n f r o m t h e a n a l y s i s c a r r i ed o u t h e r e ,a n d t h e s e a r e as f o l l o w s :

( 1) T h e d e p e n d e n c e o f t h e c r i t ic a l v e l o c i t yo n p i p e d i a m e t e r i s v e r y n e a r l y e q u a l t o D I n .

( 2 ) F o r s l u rr i es c o m p r i s e d o f la r g e n o n -c o l l o i d a l p a r t i c l e s , t h e c r i t i c a l v e l o c i t y i sv i r t u a l l y i n d e p e n d e n t o f p a r t i c l e si ze . T h ea n a l y t i c a l r e s u l t d u e t o O r o s k a r a n d T u r i a n[ 1 3 ] i n d i c a t e s t h a t V c i s i n d e p e n d e n t o f d,w h i l e a n a ly s i s o f t h e e x p e r i m e n t a l d a t as u g g e s ts t h a t Vc is a v e r y w e a k f u n c t i o n o f d .E m p i r i c a l f i t s t o t h e d a t a r e s u l t i n a c ri t ic a lv e l o c i t y - p a r t i c l e s i ze d e p e n d e n c e w h i c h i sa p p r o x i m a t e l y e q u a l t o d 6 .

( 3 ) A s u b s t a n t i a l b o d y o f e x p e r i m e n t a ld a t a s u g g e s ts t h a t t h e c r i t ic a l v e l o c i t y -c o n c e n t r a t i o n r e l a t i o n p o ss e s s es a m a x i m u m .A c c o r d i n g l y , c o r r e l a t i o n s w h i c h a c c o u n t f o rt h e o c c u r r e n c e o f a m a x i m u m i n t h e v c v s . Cr e l a t i o n s h i p a r e c o n s i s t e n t w i t h t h e e x p e r i -m e n t a l e v i d en c e . B o t h t h e a n a l y t i c a l r e s u l t o f

O r o s k a r a n d T u r i a n [ 1 3 ] a n d t h e e m p i r i c a l l yf i t t e d c o r r e l a t i o n s g i v e n in T a b l e 4 ( C a s e s 1t h r o u g h 4 ) i n d i c a te t h a t t h e c o n c e n t r a t i o n a tw h i c h t h e m a x i m u m i n Vc o c c u r s is b e t w e e n0 . 2 5 a n d 0 . 3 0 , w h i c h a r e s o m e w h a t h i g h e rt h a n t h e v a l u e p e r t a i n i n g t o th e e x p e r i m e n t a ld a t a p r e se n t e d b y P a r z o n k a e t a l . [ 4 5 ] .

A C K N O W L E D G M E N TT h i s w o r k w a s s u p p o r t e d b y t h e N a t i o n a l

S c ie n c e F o u n d a t i o n t h r o u g h G r a n t C P E8 1 1 1 2 5 8 , a n d b y t h e I n t e r n a t i o n a l F i n eP a r t i c l e R e s e a r c h I n s t i t u t e , I n c .

L I S T O F S Y M B O L SCC DD

c o n c e n t r a t i o n , v o l u m e f r a c t i o n( 4 / 3 ) g d ( s - - 1 ) / v ~ 2, d r a g c o e f f i c i e n t f o rf r e e -f ~ n i n g s p h e r ei n s id e d i a m e t e r o f p i p e , ma b s o l u t e a v e r ag e p e r c e n t d e v i a t i o n( e q n . ( 2 5 ) )


12/13

4 6~d

gi

LNNNR~~e--APR a bRMSvVcVsv~o

per cent deviation eqn. 24))diame ter of solid particle, m--AP/L) D/2pv:), Fanning frictionfactor for pipe flowgravitational acceleration, m/s:head loss for pipe flow, in feet of waterper foot of pipeconstant eqn. 1))pipe length, mnumber of data pointsV 2 C D 1 / : / [ C D g ( s - - 1 ) ] , eqn. 6))d p v ~ / p , p a r t i c l e R e y n o l d s n u m b e r f o rf r e e - f a l l in g s p h e r e sD p v / p , R e y n o l d s n u m b e r f o r p i p ep r e s s u r e d r o p i n l e n g t h L o f p i p e , P ar e g i m e n u m b e r e q n s . 1 2 ) , 1 3 ) , 1 4 ) )p e r c e n t r o o t m e a n s q u a r e d e v i a t i o ne q n . 2 6 ) )P s / P , s o l i d t o l i q u i d d e n s i t y r a t i om e a n v e l o c i t y , m / sc r i t i c a l v e l o c i t y o f s l u r r y , m / sh i n d e r e d s e t t l i n g v e l o c i t y o f p a r t i c l e s ,m / st e r m i n a l v e l o c i t y o f s p h e r e s e t t l i n g ina n u n b o u n d e d f l u id , m / s

GreekAP1)PPsPsand

symbolsNRcCD inviscosity of liquid, kg/ m s)kinemat ic viscosity of liquid, m2/sdensi ty o f liquid, kg/m 3density o f solid, kg/m 3densi ty of sand, kg/m 3constants i = 1, 2 .. . .. 5) eqn. 25))

Subscriptscalc calculatedexp experimentalw related to wate r or liquid0 regime of flow with a stationary bed1 ~ altation flow regime2 heterogeneous flow regime3 homogeneous flow regime

R E F E R E N C E S1 N . S . B l a t c h , T r a ns . A S C E , 5 7 1 9 0 6 ) 4 0 0 .2 G . W . H o w a r d , T r an s . A S C E , 1 0 4 1 9 3 9 ) 1 3 3 4 .3 W . E . W i l s o n , T r a ns . A S C E , 1 0 7 1 9 4 2 ) 1 5 7 6 .4 D . M . N e w i t t , J . F . R i c h a r d s o n , M . A b b o t t a n dR . B . T u r t l e , T r a n s . I n s t . C h e m . E n g r s . , 3 3 1 9 5 5 )9 3 .5 R . D u r a n d , L a H o u i l l e B l a n c h e , 6 1 9 5 1 ) 6 0 9 .6 R . D u r a n d a n d E . C o n d o l i o s , i n C o m p t e R e n d u

d e s D e u x i ~ m e s J o u r n d e s d e l H y d r a u l i q u e ,S o c i~ t ~ H y d r o t e c h n i q u e d e F r a n c e , P ar is , 1 9 5 2 )p p . 2 9 - 5 5 .7 R . D u r a n d , i n P r o c . M i n n e s o t a I n t e r n a t i o n a lH y d r a u l i c s C o n v e n t i o n ( 1 9 5 3 ) , p p . 8 9 - 1 0 3 .8 R . D u r a n d , P r o c . C o l l o q u i u m o n H y d r a u l i cT r a n s p o r t o f C o a l ( N o v . 5 - 6 , 1 9 5 2 ) , N a t i o n a lC o a l B o a r d , L o n d o n , E n g l a n d , 1 9 5 3 , p p . 3 9 - 5 2 .9 I . Z a n d i a n d G . G o v a t o s , P r o c . H y d r . D i v . , A S C E ,9 3 , H Y 3 , 1 9 6 7 ) 1 4 5 .1 0 R . M . T u r i a n a n d T . F . Y u a n , A I C h E J ., 2 3 1 9 7 7 )2 3 2 .1 1 I . K a z a n s k i j , P r o c . 5 t h I n t . C o n f . H y d r a u l i c T r a n s -p o r t o f S ol id s i n Pi pe s ( H Y D R O T R A N S P O R T 5 ),H a n o v e r , M a y 8 - 1 1 , 1 9 7 8 , p p . 4 7 - 7 9 .1 2 I . Z a n d i e d . ) , A d v a n c e s i n S o l i d - L i q u i d F l o w i nP i p e s a n d I t s A p p l i c a t i o n , P e r g a m o n , N e w Y o r k ,1 9 7 1 .1 3 A . R . O r o s k a r a n d R . M . T u r i a n , A I C h E J . , 2 61 9 8 0 ) 5 5 0 .1 4 J . F . R i c h a r d s o n a n d W . N . Z a k i , Tr ans . In s t .C h e m . E n g . , 3 2 1 9 5 4 ) 3 5 .1 5 J . G a r s i d e a n d M . R . A I - D i b o u n i , I n d . E n g . C h e m .P r o c e s s . R e s . D e v . , 1 6 1 9 7 7 ) 2 0 6 .

1 6 A . J . C a r l e t o n a n d D . C . - H . C h e n g , P r o c . 3 r d I n t .C o n f . H y d r a u l i c T r a n s p o r t o f S o l i d s i n P i p e s( H Y D R O T R A N S P O R T 3 ) , G o l d e n , C o l o r a d o ,U . S . A . , M a y 1 5 - 1 7 , 1 9 7 4 , p p . 5 7 - 7 4 .1 7 A . D . T h o m a s , I n t . J . M u l t i p h a s e F l o w , 5 1 9 7 9 )1 1 3 .18 H . B la s ius , Z . M a t h . u . P h y s i k , 5 6 1 9 0 8 ) 4 .

1 9 J . R . C r a v e n , P r o c . 5 t h H y d r a u l i c C o n f . , E n g i n e e r -i n g B u l l e t i n N o . 3 4 , S t a t e U n i v e r s i t y o f I o w a ,I o w a C i t y 1 9 5 3 ) .2 0 S . I . G o r j u n o v , G o s e n e r g o i z d a t 1 9 5 5 ) . S e e S a si ca n d M a r j a n o v i c [ 4 3 ] .2 1 K . E . S p e l l s , T r a n s . I n s t n . C h e m . E n g r s . , 3 31 9 5 5 ) 8 0 .2 2 R . C . C a i r n s , K . R . L a w t h e r a n d K . S . T u r n e r ,B r i t . C h e m . E n g n g . , 5 1 9 6 0 ) 8 4 9 .2 3 D . G . T h o m a s , A I C h E J . , 7 1 9 6 1 ) 4 2 3 .2 4 D . G . T h o m a s , A I C h E J . , 8 1 9 6 2 ) 3 7 3 .2 5 L . S c h u l z , B e r g b a u t e c h n i k , 1 2 J u l y 1 9 6 2 ) 3 5 3i n G e r m a n ) .2 6 H . B r a u e r a n d E . K r i e g e l , B a n d e r B l e c h e R o h r e , 61 9 6 5 ) 3 15 i n G e r m a n ) .2 7 W . W i e d e n r o t h , P h . D . D i s s e r t a t io n , T e c h n . H o c h -s c h u le H a n n o v e r 1 9 6 7 ) .2 8 I . L a r s e n , P r o c . A S C E , J . H y d r a u l i c s D i v . , 9 41 9 6 8 ) 3 3 2 .2 9 H . E . R o se a n d R . A . D u c k w o r t h , T h e E n g i n e e r ,2 2 7 1 9 6 9 ) 4 7 8 .3 0 C . A . S h o o k , P r o c . S y r u p . P i p e l i n e T r a n s p o r t o fS o l i d s , T h e C a n a d i a n S o c i e t y f o r C h e m i c a lE n g i n e e r i n g , T o r o n t o , 1 9 6 9 , 2 - 1 .3 1 H . A . B a b c o c k , i n I . Z a n d i e d . ) , A d v a n c e s i nS o l i d - L i q u i d F l o w i n P i p e s a n d i t s A p p l i c a t i o n ,P e r g a m o n , N e w Y o r k , 1 9 7 1 , p p . 1 2 5 - 1 4 8 .3 2 A . G . B a i n a n d S . T . B o n n i n g t o n , T h e H y d r a u l i cT r a n s p o r t o f S o l i d s b y P i p e li n e s , P e r g a m o n , N e wY o r k , 1 9 7 0 , p p . 1 9 - 2 3 .

3 3 P . N o v a k a n d C . N a l l u r i , P r o c . 2 n d I n t . C o n f .H y d r a u l ic T r a n s p o r t o f S o l id s i n P i p es ( H Y D R O -T R A N S P O R T 2 ), C o v e n t r y , U . K . , S e p t e m b e r2 0 - 2 2 , 1 9 7 2 , p p . 3 3 - 5 2 .


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34 M. P. Robi nso n and W. H. Graf, P r o c . A S C E ,J . H y d r a u l i c s D i v . , 7 8 1972) 1221.35 T.-Y. Kao and D. J. Wood, T r a n s . S o c . M i n .E n g r s. , A I M E , 2 5 5 1974) 39.36 E. J. Wasp, J. P. Kenn y and R. L. Gandhi, S o l i d -L i q u i d F l o w - - S l u r r y P i p e li n e T r a n s p o r t a t i o n ,Trans. Tech. Publ., Rockport, MA, 1977.37 M. Toda, J. Yonehara , T. Kimura and S. Maeda,I n t . C h e m . E n g . , 1 9 1979) 145.38 A. P. Yufin, I z v e s t i y a U . - S e r . T e k h . N a u k . 81949) 1146.39 A. P. Yufin, I z d a t e i s t v o Po S t r o i t e l s t v u . ,Moscow 1965) in Russian).40 A. P. Yufin and N. A. Lopasin, G i d r o t e c h -n i c e s k o e S t r o i t e l s t v o , 3 6 1966) 49 in Russian).

41 A. C. Bonapac e, P r o c . 3 r d I n t . C o n f . H y d r a u l i cT r a n sp o r t o f S o li d s i n P i pe s ( H Y D R O T R A N S -P O R T 3 ) , Gold en, CO, U.S.A., May 15 - 17, 197 4,pp. 29 - 43.42 ASCE Task Commi ttee, P r o c . A S C E , J . H y d r a u l i cD i v i s i o n , 9 6 1970) 1503.43 M. Sasic and P. Marjanovic , P r o c . 5 t h I n t . C o n f .H y d r a u l ic T r a n s p o r t o f S o l id s i n P ip e s ( H Y D R O -T R A N S P O R T 5 ) , Hanover, F.R.G., May 8 - 11,1978, pp. 61 - 69.44 W. Wiedenro th and M. Kirchner, P r o c . 2 n d I n t .C o n f . H y d r a u l i c T r a n s p o r t o f S o l i d s i n P i p e s( H Y D R O T R A N S P O R T 2 ) , Coventry, U.K.,Septe mber 20 - 22, 1972, pp. 1 - 22.45 W. Parzonka, J. M. Kenc hing ton and M. E. Charles,C a n a d i a n J . C h e m . E n g r . , 5 9 1981) 291.46 R. B. Bird, W. E. Stewart and E. N. Ligh tfoo t,T r a n s p o r t P h e n o m e n a , Wiley, New York, 1960.47 C. N. Davies, P r o c . P h y s . S o c. ( L o n d o n ) , 5 71945) 259.48 W. Schriek, L. G. Smi th, D. B. Haas and W. H. W.Husband, E x p e r i m e n t a l S t u d i e s o n t h e H y d ra u l icT r a n s p o r t o f C o a l , Report E73-17, SaskatchewanResearch Council, Saskatoon, Sask., Canada,October 1973.49 E. Goedd e, P r o c . 5 t h I n t . C o n f . H y d r a u l i c T r a n s -p o r t o f So li ds i n P i pe s ( H Y D R O T R A N S P O R T 5 ),

47Hanover, F.R.G., May 8 - 11, 1978, pp. 81 - 98.50 E. J. Wasp, T. C. Aude, J. P. Ken ny, R. H. Seiter,P. B. Williams and R . B. Jacques , P r o c . 1 s t I n t .C o n f . H y d r a u l i c T r a n s p o r t o f S o l i d s i n P i p e s,( H Y D R O T R A N S P O R T 1 ) , Coventry, U.K., Sep-temb er 1 - 4, 1970, pp. 53 76.51 W. Schriek, L. G. Smith , D. B. Haas and W. H. W.Husband, E x p e r i m e n t a l S t u d i e s o n t h e H y d ra u l i cT r a n s p o r t o f L i m e s t o n e , Report E73-10,Saskatchewan Research Council, Saskatoon,Sask., Canada, August 1973.52 W. Schriek, L. G. Smith, D. Haas and W. H. W.Husband, E x p e r i m e n t a l S t u d i e s o n t h e H y d ra u l i cT r a n s p o r t o f I r o n O r e , Report E73-12, Saskat-chewan Research Council, Saskatoon, Sask.,Canada, July 1973.53 L. G. Smit h, D. B. Haas, W. Schri ek and W. H. W.Husband, E x p e r i m e n t a l S t u d i e s o n t h e H y d ra u l i cT r a n s p o r t o f P o t a s h , Report E73-16, SaskatchewanResearch Council, Saskatoon, Sask., Canada,October 1973.54 C. A. Shook, W. Schriek, L. G. Smi th, D. B.Haas and W. H. W. Husband, E x p e r i m e n t a lS t u d i e s o f t h e T r a n s p o r t o f S a n d s i nL i q u i d s o f V a r y i n g P r o p e r t ie s i n 2 - a n d 4 - I n c hP i p e l i n e s , Report E73-20, Saskatchewan ResearchCouncil, Saskatoon, Sask., Canada, November1973.55 J. W. Hayd en and T. E. Stelson, in I. Zan di ed.),A d v a n c e s i n S o l i d - L i q u i d F l o w i n P i p e s a n d i tsA p p l i c a t i o n , Pergamon, New York, 1971, pp.149 - 164.56 W. Schriek, L. G. Smith , D. B. Haas and W. H. W.Husband, E x p e r i m e n t a l S t u d i e s o n t h e T r a n s p o r to f T w o D i f f e r e n t S a n d s i n W a t e r in 2- , 4- , 6- , 8 - ,1 0 - a n d 1 2 - I n c h P i p e l i n e s , Report E73-21,Saskatchewan Research Council, Saskatoon,Sask., Canada, July 1973.57 C. K. Wilson and E. W. Watt, P r o c . 3 r d I n t . C o n f .H y d r a u l ic T r a n s p o r t o f S o l id s i n P i p e s ( H Y D R O -T R A N S P O R T 3 ) , Golden, CO, U.S.A., May 15 -17, 1974, D1.

Estimation of the Critical Velocity in Pipeline Flow of Slurries

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