Settling Velocity

8/3/2019 Settling Velocity

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Pergamon Chemical Engineering Science, V o l . 5 3 , N o . 2 , p p . 3 1 5 3 2 3 , 1 9 9 8~ ; 1 9 9 7 E l s e v i e r S c i e n c e L t d . A l l r i g h t s r e s e r v e dP r i n t e d i n G r e a t B r i t a i nPII: S0009-2509(97)00285-6 0 0 0 9 - 2 5 0 9 / 9 8 $ 1 9 . 0 0 + 0 . 0 0

A model of se t t l ing ve loc i tyT e r e n c e N . S m i t h

S c h o o l o f C h e m i c a l E n g i n e e r i n g , C u r t i n U n i v e r s i t y o f T e c h n o l o g y , P e r t h 6 00 1 , A u s t r a l i a( R e c e i v e d 3 0 J a n u a r y 1 9 97 ; a c c e p t e d 1 6 M a y 1 9 97 )

A b s t r a c t - - A t t e n t i o n i s d r a w n t o t h e l a c k o f a s u c c es s fu l t h e o r e ti c a l m o d e l f o r c a l c u l a t io n o f t h ee f fe c t o f s o li d s c o n c e n t r a t i o n o n t h e 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 f r o m s u s p e n s i o n . T h e p r i n c i p a lc h a l l e n g e t o f o r m u l a t i o n o f a m o d e l i s s o l u t i o n o f th e e q u a t i o n s o f f l ui d m o t i o n t h r o u g h a ne n s e m b l e o f s o l i d p a r t ic l e s w h i c h a r e r a n d o m l y d i s t r i b u t e d t o s p a c e . A n e w t h e o r e t i c a l m o d e lw h i c h a d d r e s s e s t h i s d i f fi c u lt y i s p r e s e n t e d . I t i s b a s e d o n p e r m e a t i o n o f th e f l u i d t h r o u g ha s t r u c t u r e c o m p o s e d o f ce l ls e a c h o f w h i c h c o n s i s t s o f a s i n g l e s p h e r i c a l p a r t i c l e a n d a na s s o c i a t e d b o d y o f fl u id . T h e q u a n t i t y o f f lu i d in e a c h c e l l i s a r a n d o m v a r i a b l e . U s i n g t h i sm o d e l , t h e s e t t li n g v e lo c i t y o f a s u s p e n s i o n a t s u b s t a n t i a l v o l u m e f r a c t i o n c a n b e c a l c u l a t e dw i t h o u t r e s o r t to e m p i r i c a l l y d e t e r m i n e d f a c t o r s. G o o d a g r e e m e n t w i t h e x p e r i m e n t a l s e t tl i n gv e l o c i t i e s i s o b t a i n e d . 1 9 9 7 E l s e v i e r S c i e n c e L t dKeywords: S e t t l i n g v e lo c i t y ; p a r t i c le s ; s u sp e n s i o n ; r a n d o m a r r a y .

INTRODUCTIONT h e s e t t l in g v e l o c i t y o f s o l id p a r t i c l e s f r o m s u s p e n s i o ni n f lu i d s i s a q u a n t i t y o f f u n d a m e n t a l i m p o r t a n c e t ot h e d e s i g n o f e n g i n e e r i n g p r o c e s s e s a n d p l a n t . R e g r e t -t a b l y , t h i s i m p o r t a n c e i s n o t m a t c h e d b y t h e a v a i l a b i l -i t y o f a n a l y t i c a l l y b a s e d f o r m u l a e f o r c a l c u l a t i o n o ft h e v e l o c i t y e x c e p t i n t h e s i m p l e s t o f c a se s .

C l a s s i c a l a n a l y s i s p r o v i d e s t h e r e s u l t f o r s e t tl i n g o fa s i n g l e s p h e r i c a l p a r t i c l e i n s l o w f l o w . I t d o e s n o t ,h o w e v e r , e x t e n d t o t h e e m e r g e n c e o f in e r t i a l e ff e ct s i nf lo w o f t h e f l u id a t h i g h e r R e y n o l d s N u m b e r . E s t i-m a t e s o f s e t t l in g v e l o c i t y m u s t b e o b t a i n e d f r o m c o r -r e l a ti o n s o f e x p e r i m e n t a l d a t a .N e i t h e r d o e s a n a l y s i s e x t e n d s u c c e s s f u l ly t o t h ee ff e ct o f s u b s t a n t i a l v o l u m e f r a c t i o n o f t h e s o l i d s o nt h e s e t t l i n g v e l o c i t y o f a p a r t i c l e . T o e s t i m a t e t h es e t t li n g v e l o c i t ie s o f p a r t ic l e s f r o m c o n c e n t r a t e d s u s -p e n s i o n s , i t is n e c e s s a r y , e v e n i n s l o w f l o w , t o r e s o r t t oe m p i r i c a l c o r r e l a t i o n s .

T h i s s i t u a t i o n d o e s n o t c o n s t i t u t e a s e r i o u s d e p r i v a -t i o n f o r s i m p l e a p p l i c a t i o n s s u c h a s t h e d e s i g n o fe q u i p m e n t f o r t h e s e t t l in g o f u n i f o r m l y s i z e d p a r t ic l e s .T h e c o r r e l a t i o n s a v a i l a b l e a r e q u i t e a d e q u a t e f o r t h isp u r p o s e .F o r m o r e c o m p l e x c a s es , h o w e v e r , s u c h a s t h e s e t t -l i n g o f m u l t i d i s p e r s e s u s p e n s i o n s , t h e l a c k o f a f u n d a -m e n t a l t h e o r y i s a s e r i o u s li m i t a t i o n . W h i l e s e v e r a lc o r r e l a ti o n s a n d p r o c e d u r e s h a v e b e e n p r o p o s e d , t h e yd o n o t e n j o y t h e s e c u r i t y o f a m e c h a n i s t i c b a s is a n d ,c o n s e q u e n t ly , m a y n o t b e a p p l i e d w i t h c o n f i d e n c eb e y o n d t h e r a n g e o f t h e p a r t i c u l a r e x p e r im e n t a l d a t af r o m w h i c h t h e y h a v e b e e n d e r i v e d .T h i s p a p e r p r e s e n t s a m o d e l f r o m w h i c h a s e t t l i n gv e l o c it y f o r a m o n o d i s p e r s e s u s p e n s i o n a t s u b s t a n t i a lv o l u m e f r a c t i o n c a n b e c a l c u l a t e d w i t h o u t r e s o r t t o

e m p i r i c a l l y d e t e r m i n e d f a c to r s . I t i s r e s t r i c t e d , in t h i sd e v e l o p m e n t , t o s l o w f lo w b u t i s, p e r h a p s , i l l u s t r a t iv eo f an a p p r o a c h w i th m o r e g e n e r a l a p p l i c a b il it y .

SIN G L E PA RT ICL E IN SL OW FL OWA s i n g l e p a r t i c l e o f s o l i d m a t e r i a l s e t t l i n g u n d e r

g r a v i t y th r o u g h a f lu i d re a c h e s a s t e a d y v a l u e o fv e l o c i t y d e t e r m i n e d b y t h e b a l a n c e b e t w e e n i t s w e i g h ta n d t h e r e s i s t a n c e t o i t s m o t i o n t h r o u g h t h e f l u i d .I n s l o w f l o w , w h e r e t h e m a g n i t u d e o f th e i n e r t i a le ff e ct s i s n e g l ig i b l e i n c o m p a r i s o n w i t h t h a t o f t h ev i s c o u s ef fe c ts , t h e e q u a t i o n o f m o t i o n o f a s i m p l e ,N e w t o n i a n f l ui d r e la t i v e t o t h e p a r t i c l e is

V P = #V 2u . (1 )S o l u t i o n o f t h i s e q u a t i o n f o r f l o w o f f lu i d r e l a t iv e t oa n i s o l a t e d s 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 y i e l d sa p a t t e r n o f n o r m a l a n d s h e a r s t r e ss e s o n t h e s u r f a c eo f t h e s p h e r e w h i c h m a y b e i n t e g r a t e d t o g i v e t h et r a c t i o n a s t h e f a m i l i a r S t o k e s f o r ce

F = 3nd#w (2 )i n w h i c h w i s t h e v e l o c i t y o f th e s p h e r e r e l a t i v e t o t h a to f th e u n d i s t u r b e d f l u id .

T h e t e r m i n a l s e t t l i n g v e l o c i t y o f t h e s p h e r e i s g i v e nb y t h e f o r m u l a

d2(pe -- PF)gWo - (3)18#B e y o n d t h e l i m i t i n g v a lu e , a b o u t 0 . 5, o f t h eR e y n o l d s N u m b e r o f f lo w t o w h i c h t h is a n a l y t i c a lr e s u l t a p p l ie s , t h e t e r m i n a l v e l o c i t y m a y b e r e c k o n e d

31 5


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316from the formula

[~ 1 d ( p e - P F ) g ] 1/2 (4)Wo : CD PF A

in which the drag coefficient, Co, is an empiricallydefined functi on of the Reynolds Num ber of the flowof fluid relative to the particle. For a particle which isnot spherical, CD is a function also of the shape of theparticle and of the orientation of the particle to thedirection of the flow of fluid.

EXPECTATIONS OF EFFECT OF CONCE NTRATI ONThe results in eqs (3) and (4) apply to the movement

of a single particle through a body of fluid of infiniteextent, undisturbed by any other motion.

Where particles of solid are situa ted in close prox-imity, as in the settling of a suspension or in thefluidization of a bed of particles, the patt ern of flow offluid about the individual particle is affected by thoseof its neighbours. In consequence, the settling velocityor, more generally, the relative velocity between thefluid and the particle differs from the result represent-ed by eq. (3) or eq. (4). It is changed by a factor whichdepends on the volume fraction of the solid particlesin the suspension.

It can be supposed that the factor might be ob-tained as a function of the volume fraction and theReynolds Number of flow of the single particle suchthat

W- - = fn[NRE, C] . (5)Wo

EMPIRICAL CORRELATIONS

Flu id i za t ion and se t t l ingSeveral empirical formulae have been derived to

describe the relationship between settling velocity andvolume fraction. Richardson an d Zaki (1954) offer therelatively simple form

w = w o ( 1 - c ) u ( 6 )

in which N is a function of the Reynolds Number offlow, d w o p / # .This correlation is commonly used for estimation of

velocities in settling and fluidization of solids at sub-stantial spatial concentrations.

It is appropriate, for developments which follow, toestablish the equivalence between the settling velocityof a suspension of solid particles and the superficialvelocity of the fluid through a bed of fluidized par-ticles. This is best shown by conside ration of the valueof the relative velocity between fluid and solid in eachcase.

In batch settling, that is, settling of a suspension ina vessel with a closed bot tom, there is, across any fixedhorizontal plane, an upward flow of fluid equal to thedownward flow of solids. The velocity, w n , of thesolids relative to the fluid is given by the differencebetween the downward velocity of the particles and

T. N. Smiththe velocity of displaced liquid as

- - W C WW R W 1 - c 1 - c (7 )This value is identical with the relative velocity offluid flowing at superficial velocity w through a bed ofsolids at concentra tion c.F l o w t h r o u g h p o r o u s b e d s

An alternative form of correlation of velocities infixed and fluidized beds of solid particles is based onthe formula of Kozeny (1927) for slow flow throughfine pores. Expressed in terms of the particle size of thesolids and of the porosity of the bed formed by them,the velocity of the fluid is related to the motive pres-sure gradient by

= ! 1 - . a _ 2 ] v Pv K L36(1 - e)2 7 (8)in which K is an experimentally determined constant.THEORETICAL MODELS

Pa r t ic le -par t ic le in te ract ionsThere have been many attempts to develop theore-

tical models which account for the effect of volumefraction on settling velocity. The approaches followedby various workers in fo rmulation of the model and insolution of the equations of moti on are discussed insome detail by Happel and Brenner (1965) and byBatchelor (1972).A particular complication in formulation ofa model is the necessity to incorporate a realisticspatial distribution of the particles. Settling is a dy-namic process. Freely settling particles move relative-ly to each other. Visual observations and statisticalanalysis by Smith (1968) of the locations of spheressettling in slow flow at a volume fraction of 0.025show random occupation of elements of space by thespheres.

Most of the models which deal with this complica-tion are restricted in application to sparse concentra-tions of particles. The flow of fluid relative to theparticles is visualized as a composite of the in teract ingfields of flow about the widely separated, individualspheres. Proceeding from a specified distr ibu tion ofthe particles to space, techn iques of linear superposi-tion of the flows about each particle are applied toexpress the interact ions and so to obtain approx imatesolutions. The limitation to very small concentrationsarises because the fundamental solution to the flowabout a particle in an infinite medium is unbo unded.In accounting for the intera ctions of many particles,the effects of those at great distances must eventuallybe discounted.

For a random dispersion of particles to space,Batchelor (1972) finds the resultw = Wo(1 - 6.55c) (9)

correct to the order of the concentrat ion, c.


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318 T. N. Smithresistance by a factor of 0.645. Wakiya (1957) obtains0.694 for mot ion across the line of centres. Resistancesof pairs of spheres separated by various distancesalong axes orientated at angles to the direction ofmotion are analysed and discussed by Happel andBrenner (1965). These authors also discuss the proced-ures for obtaining solutions for the resistances ofgroupings of more th an two spheres and refer to someapproximate results. The resistance of each spheredecreases with increasing number in the group.

The rate of flow of liquid through a pipe or pore ofradius r is given by the Hag en-Poiseuille formula

7[?"4Q = -~-p VP. (13)If the total sectional area of a porous medium withn uniformly sized pores, n~r2/4, is redistributed bydisplacement of the solid particles forming the wails ofthe pores so that the areas of some pores are de-creased and the areas of others are increased, theeffective value of the term r 4 in eq. (13) is increased.Consequently, the rate of flow through the mediu m isincreased by this disproportio nate contri bution of thelarger pores.Increase in ve loc i t y a t low concent ra t ions

Models such as tha t of Batchelor (1972) for settlingof particles at low concentra tions do account, to somedegree, for the effects of random dispersion to space.

There is evidence, however, that the scale of theallowance expressed by eq. (9) is insufficient. Experi-ments reported by Kaye an d Boardman (1962), byJohne (1966) and by Koglin (1971, 1973) show that thesettling velocity of particles at very low volume frac-tions may actually exceed the single-particle value.

The settling velocity of spherical particles in slowflow is found to rise from Wo for the single particle toreach a maxi mum value of about 1.5Wo at a volumefraction of about 0.01.

Koglin (1971) concludes that this behaviour arisesfrom dynamic effects which cause transitory forma-tions of particles such as those observed by Jayaweeraet al. (1964) and by Crowley (1971) rather than fromthe exercise of interparti cul ar forces which mightcause permanent aggregations.

The evidence shows that, as the volume fraction ofa suspension increases from zero, the settling velocityof particles first increases with volume fraction owingto a 'c lustering' effect but then decreases as the effectof resistance to flow of fluid between the particlesbecomes domin ant.

The arg ument support ing this observat ion of sett-ling velocities which exceed the value for a singleparticle is that, at very small concent rations , the effectof the prox imity of a neigh bour ing particle is to de-crease the resistance of the pair of particles toa greater degree than the increase in resistance appar-ent in a regular array at that concentration. It is therandom displacement or deviation from the regularposition and spacing which gives rise to this result.

A N E W M O D E LThe model proposed in this paper is applicable to

the settling of spherical particles of unifo rm size dis-persed r and omly in space. The flow of fluid relative tothe particles is a permeati on under slow flow throughthe interstitial volume.

Each solid particle is located at the centre ofa spherical cell in which it presents the resistance toflow of fluid through the cell. The flow of fluidthrough the cell is driven by the pressure gradientimposed on the suspension by the weight of the solidparticles.

The flow through each cell depends on the per-meability of the cell itself and also on that of thesurr ounding medium. The superficial velocity of fluidthrough the ensemble of cells is identified with thesettling velocity of the suspension of particles.

Sp at ial d istr ibution o f par t ic lesThe allocation of particles to space is based ona model of the probability of intrusion upon theprovince of a ne ighbouring particle as successivelymore distant elements of fluid in the space surround-ing an object particle are searched. Figure 2 depictsthe space su rrounding a particle as a series of shellscomposed of small, equal elements of volume.

While Fig. 2 shows a placement of elements of fluidspace around the object particle, there is no require-ment for a partic ular geometry. The probabi lity is thatof association of an element with the particle, not ofoccupation of a specific location.

The pr obability of no encount er of a second particlein successive inspections of t elemen ts of fluid volumeis the probability that the space between neighbour-ing particles consists of a volume at least as great asthat. It is given by the single-event Poisson distribu -tion or exponential distribution as

p( t ) = ex p( -2t ) (14)in which 2 is the mean probability of encounter ata single inspection.

Evidently,p= l for t= 0, p= 0 for t~o c.

The interp retation in this model of an encounter with aneighbouring particle after the addition of t elements

, , l l m l l , ,1 1 1 1 1 1 1 1 1 1 1 1 1

I I I I I I I I I I II I I I I I II l l

QFig. 2. Volume surrounding a particle.


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A model of settling velocityof volume is that these elements are shared equallywith the neighbour. Each of the particles is sur-rounded by t /2 elements of volume. This processcreates two cells of the same concentration simulta-neously. It does not, however, lead to bias since eachtrial is independ ent and the final result is the creationof two complete, identical populations.The maximum value of the volume fraction is thatat which the solid spheres make contact with eachother. The n umber of elements of volume occupied bythe particle and its associated fluid at this maximumconcen tra tion, CM, is m.

The vo lume fraction of solid in the cell consisting ofthe sphere, its associated fluid at ma ximum concentra-tion and the t /2 additional elements of volume is givenby

C M mc (15)m + t /2"The number of elements of volume, t, is related tovolume fraction by

t = 2 cM m - 2m. (16)C

The value of t corre spond ing to the mea n value, g, ofthe c oncentrati on of the solids in suspension providesthe value of 2, the frequency parameter in eq. (14),from the identity

1 C , If = 2 ""m 2m. (17)2The value of 2t to provide the probability asso-

ciated with the concentration c is then( C M / C ) - 12t - (18)( C M / g ) - 1

and that probability isp ( c ) = e x p [ (CM/g)(CM/C)--~]" (19)

The contact concentration, cu, in this formula mustbe given a value appropriate to the circumstances.The maximum value attainable is 0.7405, cor respond-ing to rhombohedral close packing of spheres.A simple cubic array gives 0.5236.

The probability p ( c ) is a cumulative frequency ofparticles with cell concentra tions less than or equal t othe value c. The frequency distribution is obtained bydifferentiation of this function. Equa tion (20) gives thefrequency of the individual particles or of the cellscont aini ng them which fall within the discrete intervalof concentration 6c centred on the value c as

f (c ) = p ( c + 6 c / 2 ) - p ( c - 6c / 2 ) . (20)The space in the suspension occupied by cells withinthis particular band of concentrat ion is found from

s(c) = f (c) /c (21)

319which is then used to obtain the fraction of spaceoccupied as

f ( c ) / cf (s) = Z [ f (c) /c~] " (22)The suspension is composed of particles of solid con-tained in cells distributed random ly to space with thefrequency distr ibut ion described by eq. (20).Th e in d iv idua l c e l l

The cell containing the individua l solid particle is ofthe kind developed by Happel (1958). The solid,spherical particle is at the centre of a spherical cell andthe fluid passes through the cell in the x-direction asillustrated in Fig. 3.

A condition of zero tangential shear stress is pre-scribed on the surface of the cell. This specificationprecludes interaction between adjacent cells by suchstresses and ensures hydrod ynami c balance betweenthe particle and the fluid within its conta ining cell.

The term Wo in eq. (11) is the single-particle settlingvelocity. For more general appli cation of this solu tionof this equation of motion, Wo may be substitutedfrom eq. (3) expressed in terms of the pressure gradient

V P = ( p v - p e ) g c (23)induced in the fluid to balance the weight of thesettling particles. This gives the single-particle velocityas

d 2 VPWo = - - - - . (24)18# c

Equati on (11) may then be written as3 - (9/2)7 + (9/2)75 - 376 d 2v = - - VP. (25)3 + 275 18#c

P e r m e a b i l i t y o f a c e llThe objective of this development is to obtain

a value of a permeability to flow of fluid of the me-dium formed by the r andom ly located solid particles.

. . . . . . . . . . . . . . .. . . . ' " " " . . , , ,y ' " " ' , , ,0 i. , , , , '. . . . . . . . . . . . . . .

, , , "J i . . . . . . . . . . . .ii S h e a r s t r e s s a t' . . , . s u r f a c e o f c e l l i s z e r o

" , " ' , ' . . . . ..........................Flow of fluid hrough cell

Fig. 3. Contiguous cells.


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320Restricting the model to slow flow, the permeability,k, is defined by the relationship

v = k V P / l ~ (26)in which v is the superficial velocity of the fluid of vis-cosity/~ induced by VP, the motive pressure gradient.

The permeability of the independent cell is thenobta ined from eqs (25) and (26) as3 - (9/2)7 + (9/2)7 s - 376 d 2k = 3 + 2y s 18~-~" (27)

The shape of the cell for which the permeability isobta ined is spherical. Spheres do not, of course, packto fill space entirely. It is assumed that, with onlya minor effect on the permeability, the spherical sur-face of any cell can be subjected to a degree of distor-tion sufficient to allow space to be filled by a group ofcontiguous cells.

Because of the specification of zero shear stress atthe fluid boundary of the cell, this requirement isconsidered to be reasonable.

To proceed with development of the model, a per-meability for each cell in the medium is required.These must then be combined so as to define thecomposite permeability of the whole medium.

The flow through an in dividual cell within a hetero-geneous medium composed of several types of cells isnot simply the p roduct of the permeabili ty of the celland the general pressure gradient in the medium. Itdepends also on the permeability of the surroundingmedium since fluid must pass to and from the cellthrough that medium. By analogy with the resultgiven by Carslaw and Jaeger (1959) for the thermal ormagnetic flux through a spherical cell of permeabilityk embedded in a medium of permeability k, the effec-tive permeability k' of a cell is obta ined as

3kk' - k. (28)2 k + kIt is evident from eq. (28) that the permeability of thesurr ounding medium restricts the flow which can beattained through the individual cell. The maxi mumvalue of the effective permeabil ity is 3k, three times thepermeability of the medium.F l o w t h r o u g h t h e m e d i u m

The permeability of a medium composed of anensemble of randomly placed cells with various valuesof permeability is obtained by summing the productsof the permeability and the fraction of space occupiedby each of the various species of cells so that

= Z f ( s ) k ' . (29)Since the value of k is required for calculation of eachof the individual effective permeabilities, eqs (28) and(29) must be solved simultaneously to obta in thisresult.

The superficial velocity of flow of fluid through themedium is found by insertion of the permeability fromeq. (29) into eq. (26).

T. N. SmithS e t t l i n 9 v e l o c i t y

It is appropriate now to review the model in termsof its validity as a reflection of the physical processesto which it is to be applied. Justification is soughtfor its use in the calculation of velocities in flowof fluids through porous media, in fluidization ofbeds of particles and in settling of particles fromsuspension.

Flow of a fluid through the cells of the medium isrepresented by an average value of the superficialvelocity. The velocity through any individual cell de-pends on its permeability. Larger cells, in which thefraction of volume occupied by the solid particle issmall, have larger permeabilities and allow greatervelocities. In smaller cells, where the solid occupiesa larger fraction of the volume, the permeabilities aresmaller and the velocities are smaller. The solid par-ticles remain stationary and the fluid passes througheach cell at a velocity which varies with the size of thecell. The model presented in this paper'is evidentlyapplicable to the case of flow through a porousmedium.

The solid particles in a homogeneously fluidizedbed are not stationary but are in constant motion.The movemen ts are, however, restricted to a relativelyshort scale of length by collisions with other particlesin the closely populated space of the bed. There is nosustained passage of individual particles through thebed. There is a dynamic equilibrium of short motion sof the particles which presents to the stream of fluida field of particles randomly allocated to space. Thiscorresponds with the nature of the suggested model.In settling of particles through a fluid, there is alsoa dyn amic equilibrium in the spatial arrangement. Asobserved by Kaye and Board man (1962), there isa transient formation of pairs and larger groups ofparticles which settle with greater velocities thanindividual particles. This effect leads to settling,at small volume concent ratio ns of solids, at velocitieswhich exceed the single-particle value. With increas-ing concentr ation , the effect is less significant becauseof the greater frequency of collisions with neig hbour -ing particles. Indeed, at concentrations approach-ing those of fluidized beds of solids, there can beno sustained differentiation of settling velocities be-tween particles. The interaction between fluid andparticles is of the kind embodied in the suggestedmodel.

To estimate the settling velocity using this model,the dis tribution of cell sizes for the par ticul ar concen-tration of the suspension and the correspondingvalues of f ( s ) and k must be calculated from eqs(18)-(20), (22) and (27).

Values of k' and k are then found using eqs (28)and (29).

Then the velocity is calculated from eq. (26) usingthe vaue of k and the pressure gradient from eq. (23)using the mean concentra tion to give

k ( P l " - - P ~ ) g cw -- (30)kt


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A model of settling velocityPERFORMANCE OF MODEL

Th eore t ica l se t t l in 9 ve loc i t ySettling velocities calculated from the model for

a range of concentrations up to 0.5, which may beregarded as the practical limit in settling, are present-ed in Table 1. These are obt ained using a value of cMin eq. (19) of 0.7405, the figure for rhombohe dra l closepacking, to generate the spat ial distribu tion. This fig-ure is chosen because it allows the probability thatneighbourin g particles might approach to ultimateproximity. Subst ituti on of smaller values, down to 0.5,has only a very slight effect on the shape of the low-concentration end of the spatial distribution. It is thispart of the distribution which presents the greatestpermeability to flow of fluid. Accordingly, the com-puted value of settling velocity has very little sensitiv-ity to the value of CM .

Also shown in Table 1 are velocities calculated fromthe theoretical model of Happel (1958) as presented in

321eq. (11). This model may be regarded as equiva lent tothe new model but with a regu lar spacing of particlesrather than a random dispersion to space.

The difference between the two models is obvious.The resistance to flow of fluid through a matrix ofsolid particles is maximized by spacing the particlesevenly.The velocities for the model of random placementreflect the reduction of the individual resistances ofparticles by the mechanisms previously discussed.C o m p a r i s o n w i t h e x p e r i m e n t s

Figure 4 shows a comparison of the calculatedsettling velocities with experimental values as repre-sented by the correlation of Richardson and Zaki(1954) given in eq. (6).

Correspondence of the results of the model withexperimental settling velocities is good for concen tra-tions of 0.05 and greater. This is the range of settlingand fluidization experiments from which the correl a-tion is derived.

Table 1. Theoretical settling velocitiesConcen- Settling ve loci ty Settling velocitytration model equation (11)

0.0 1.701 1.0000.01 1.222 0.6770.02 1.086 0.5940.05 0.841 0.4530.10 0.596 0.3220.15 0.432 0.2370.20 0.313 0.1770.25 0.225 0.1330.30 0.160 0.0990.35 0.111 0.0730.40 0.075 0.0530.45 0.050 0.0380.50 0.032 0.026

Velocity at small volume frac t ionFor concentr ations less than 0.05, the particular

experiments of Kaye and Boardman (1962), Johne(1966) and K ogl in (1971, 1973) must be accepted asa more appropriate depiction of the relationship be-tween settling velocity and concent ration. The resultsof Koglin (1973) are ind icated in Fig. 5 together withthe correlation of Richardson and Zaki from eq. (6)and those of the model.

As the volume frac tion of solids falls below 0.03, themodel shows an elevation of settling velocity abovethe single-particle value. The emergence of this effectat concentr ations at this level is in correspondencewith the experimental observations. However, as theconcentrat ion reduces to zero, the model maintainsa settling velocity of about 1.7 times greater t han thatof the single particle. This deviation is associated with

>,

>0t

1

1. 81 .61 .41 .2

10 .80 .60 . 40 .2

0

" - ' O - - R & Z ]M O D E L

0.1 0 .2 0 .3 0 .4 0 .5Conc e nt ra t ion

Fig. 4. Theoretical and experimental settling velocities.


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322 T.N. Smith1 .81 .61 ,41,2>,~

.S 0 .80. 60.40.2

I- -"O--" R & Z

M O D E L- - - O - - E X P T

0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5

C o n c e n t r a t i o n

Fig. 5. Settling velocity at very small concentration.

a departure of the model from reality at very smallconcentrations. Where particles are very widelyspaced, differential velocities between neighbouringparticles can be maintained over long distances andtimes. The model does not permit this. Fluid passesthrough a matrix of cells each of which containsa particle. The cells do not move relatively to oneanother. Accordingly, the model is appropriate onlywhen the concentr ation of solids is sufficiently great topresent impedi ment by collision to the sustained rela-tive movement of particles. At a co ncen trat ion of 0.1,the distance between centres of particles is 2.15particle diameters. At a concentration of 0.01, thedistance between centres of particles is 4.64 particlediameters and at a concentration of 0.001, the dis-tance is 10.00 particle diameters. It may be inferredfrom these figures that the model would be effectiveat a concentration of 0.1, approximate at 0.01 butineffective at 0.001.

The model does, however, clearly reflect the experi-mentally observed tendency to settling velocities greaterthan the single-particle value at small concentra tions.

CONCLUSIONA theoretical model which permits calculation of

the velocity of uniformly sized solid particles settlingin slow flow through fluids is presented. It is based onpermeation of the fluid through the interstitial spacebetween spherical particles randomly distributed tospace.

The model is construc ted with governing equat ionswhich require no substitution of empirically deter-mined factors.

Velocities obta ined from the model correspond wellwith correlations of experimental settling velocitiesand fluidization velocities in the practically import antrange of concentr ation from 0.1 to the contact concen-tration of the particles.

The model also reflects the observed tendency tosettling velocities greater t han the single-particle sett-ling velocity at concentrations of a bout 0.01. Themodel does, however, lose validity at very small con-centra tions where the mean distance between particlesis several times the particle diameter so that sustaineddifferential movement of neighbouring particles be-comes possible.

NOTATIONc volume fraction of solids in the suspens ion

mean volume fractioncM maxi mum value of volume fractionC~ drag coefficient of the particle in the fluidd diameter of the spherical particlef (c ) fraction of cells of concentra tion cf ( s ) fraction of spatial volume occupied by cells

with concentration cF drag force on a partic le9 gravitational accelerationk permeability to flowpermeability of the composite mediumk' effective permeabil ity of a cellK Kozeny constantm num ber of elements of fluid volumen numbe r of poresN exponent in Richardson and Zaki formulaNRE Reynolds of flow of a single particlep probabilityp(c) probabilit y that concent ration does not exceed cp(t) probability of no encounte r in t trialsV P gradient of pressureQ rate of flow through a porer radius of pores(c) volume of space occupied by cells of concen tra-

tion ct num ber of elements of fluid volumef mean value of t


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u v e c t o r v e l o c i t y o f f l u idv s upe r f i c i a l ve loc i ty o f f low of f lu idw s e t tl i n g v e l o c i t y o f p a r t i c l e sw0 s ing le -pa r t i c l e s e t t l ing ve loc i tyWR ve loc i ty o f s e t t l ing pa r t i c l e s re l a t ive to f lu idGreek letters7 r a t i o o f p a r t i c l e d i a m e t e r t o c e ll d i a m e t e re p o r o s i t y2 p a r a m e t e r i n p r o b a b i l i t y f u n c t io n# v i s cos i ty o f f lu idPF dens i ty o f f lu idp p d e n s i t y o f s o l i d p a r t i c l e

R E F E R E N C E SBatche lo r , G . K. (1972) J. Fluid Mech. 52, 245.Cars law, H. S . and Jaege r , J . C . (1959) Conduction ofHeat in Solids, 2 n d E d n , p . 4 2 6 . O x f o r d U n i v e r s i t y

P r e s s , L o n d o n , U . K .Crowley , J . M . (1971) J. Fluid Mech. 45, 151.

A mod el of settling velocity 323Happe l , J . (1958) A.I.Ch.E.J. 4, 197.H a p p e l , J. a n d B r e n n e r , H . ( 1 9 65 ) Low Reynolds Num-ber Hydrodynamics, P r e n t i c e - H a l l , E n g l e w o o dCliffs , NJ, U.S.A.J a y a w e e r a , K . O . L . F ., M a s o n , B . J . a n d S l a c k , G . W .(1964) J. Fluid Mech. 20, 121.Johne , R . (1966) Chem. lng. Technik 38, 428.K a y e , B . H . a n d B o a r d m a n , R . P . ( 19 6 2) Symposium onInteraction between Fluids and Particles, p. 17. Ins tnC h e m . E n g r s , L o n d o n , U . K .Kogl in , B . (1971) Chem. Ing. Technik 43, 761.Kogl in , B . (1973) In : Proc. 1st Int. Conf. ParticleTechnol., p . 2 6 6, I I T R e s e a r c h I n s t i tu t e , C h i c a g o .Ko zen y , J. (1927) Ber. Akad. Wiss. Wien Abt. Ila 136,271.R i c h a r d s o n , J . F . a n d Z a k i , W . N . (1 9 5 4) Trans. InstnChem. Engrs 32, 35.Smi th , T . N . (1968) J. Fluid Mech. 32, 203.S t ims on , M . and Je f f rey , G . B . (1926) Proc. Roy. Soc.A 111, 110.W a k i y a , S . ( 1 9 5 7 ) R e s . R e p o r t N o . 6 , N i i g a t a U n i v .Col l . E ng .

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