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E L S E V I E R Powder Technology 87 (1996) 203-210

POWDERTE HNOLOGY

heoretical and experimental study o f the transport of g ranular m aterialsby inclined vibratory conveyors

E.M. Sloot N.P. KruytDepartment o f Mechanical Engineering University of Twente PO Box 217 7500 AE Enschede Netherlands

Receive d 22 M arch 1995; revised 25 Octo ber 1995

A b s t r a c t

A theore t ica l and exper imen ta l s tudy was m ade of the conveying speed w i th which granular mater ia ls a re transpor ted by v ibra toryThe bas ic assum pt ion m ade i s tha t the layer of granular mater ia l can be cons idered as a poin t mass . The theory incorpora tes res tf l ight phase s of the material . Althoug h the em phasis o f this study is on the effect of the inclination (a nd de clination) of the convconveying speed, the effec ts of throw n umber, f r ic t ion coeffic ient and v ibra t ion angle on the con veying speed are a lso shown. A useis presented for measu r ing the coeff ic ient of f ric t ion be tween granular mater ia l/ rod v ibra tory conveyor. Exper iments w ere per formto ver i fy the poin t m ass theory. The agreem ent be tween theory and exper iment i s fa ir ly good for sl ide con veyors but for throw larger deviations are observed. Some possibil i t ies for improvement to the theory are briefly investigated.

Keywords: Friction; Granular material; Vibratory convey or

1 I n t r o d u c t i o n

Vi b r a t o r y c o n v e y o r s a r e o f t e n u s e d i n i n d u s t r y t o t r a n s p o r tg r a n u l a r m a t e r i a l s . T h e y c o n s i s t o f a t r o u g h w h i c h i s ( g e n -e r a l l y ) v i b r a t e d s i n u s o i d a l l y i n t i m e . T h i s v i b r a t i o n in d u c e st h e m o v e m e n t o f t h e g r a n u l a r m a t e r ia l a l o n g t he c o n v e y o rs u r f a c e . T h e d i r e c t io n , a m p l i t u d e a n d f r e q u e n c y o f t h e o sc i l -l a t io n s a r e d e s i g n p a r a m e t e r s o f th e c o n v e y o r, s e e F i g . 1 .

S o m e o f t h e m a i n a d v a n t a g e s o f v i b r a t o ry c o n v e y o r s a r ethe i r s imp le cons t ruc t i on , t he i r su i t ab i l i t y t o hand le ho t anda b r a s i v e m a t e r i a l s a n d t h e i r a p p l i c a b i l i t y a s d o s i n g e q u i p -m e n t . S i n c e t h e t r o u g h c a n b e t o t a ll y e n c l o s e d , t h e y a r e a ls ow e l l s u i t e d t o t h e t ra n s p o r t o f d u s t y m a t e r i a l s . S o m e d i s a d -v a n t a g e s o f v i b r a t o r y c o n v e y o r s a r e t h e ir n o i s y o p e r a t io n , t h ei n d u c e d v i b r a t i o n s o n t h e i r s u r r o u n d i n g s a n d t h e i r l i m i t e dt r a n s p o r t d is t a n c e . F u r t h e r m o r e , t h e g r a n u l a r m a t e r i a l m a y b ed a m a g e d w h e n i t i s s u b j e c t e d t o l a rg e a c c e l e r a t i o n s n o r m a lto t he t r ough .

A d i s t i n c ti o n c a n b e m a d e b e t w e e n s l i d e a n d t h r o w c o n -v e y o r s : f o r s li d e c o n v e y o r s t h e m a t e r i a l r e m a i n s i n c o n t a c tw i t h t h e t r o u g h s u r f a c e , w h i l e f o r t h r o w c o n v e y o r s t h e m a t e -r i al l o s e s c o n t a c t d u r i n g p a r t o f t h e c o n v e y i n g c y c l e . S o m e -t i m e s t h e c o n v e y o r s a r e p o s i t i o n e d u n d e r a s m a l l i n c l i n a t i o no r dec l i na t i on i n s t ead o f ho r i zon t a l l y. I t i s exp ec t e d t ha t t h i st r o u g h s l o p e w i l l h a v e a l a rg e i n f l u e n c e o n t h e c o n v e y i n gs p e e d , e s p e c i a l l y f o r s li d e c o n v e y o r s . T h i s i n f l u e n c e c o n s t i -t u t es a n i m p o r t a n t s u b j e c t o f th i s s t u d y.

0032-5910/96/ 15.00 © 1996 Elsevier Science S.A. All rights reservedP I I S 0 0 3 2 - 5 9 1 0 ( 9 6 ) 0 3 0 9 1 - 9

M a n y r e s e a r c h e r s i n v e s t i g a t e d t he t r a n s p o r t o f g r a n u l a rm a t e r i a l s b y v i b r a t o r y c o n v e y o r s . B o o t h a n d M c C a l l i o n [ 1 ]m a d e a t h e o r e t ic a l a n d e x p e r i m e n t a l s t u d y o f th e c o n v e y i n gs p e e d o f s li d e c o n v e y o r s . N e d d e r m a n a n d H a r d i n g [ 2 ]e x t e n d e d t h i s a n a ly s i s a n d p r e s e n t e d s o m e o p t i m i z a t i o n s t u d -i e s fo r ho r i zon t a l and i nc l i ned s l i d ing . Pa j e r e t a l . [ 3 ] dea l tw i t h p r a c t ic a l a s p e c t s o f sl i de a n d t h r o w c o n v e y o r s . E r d e s za n d S z a l a y [ 4 ] , E r d e s z a n d N 6 m e t h [ 5 ] m a d e a p r a c ti c a l a n dt h e o r e ti c a l s t u d y o f s l id e a n d t h r o w c o n v e y o r s . S o m e t h e o -r e ti c a l o p t i m i z a t i o n s t u d i e s w e r e p e r f o r m e d b y H o t a a n d K a r -m a k e r [ 6 ] . T h e m o s t e x t e n s iv e r e s e ar c h w a s r e p o r t e d b yR a d e m a c h e r a n d t e r B o rg [ 7 , 8 ] . T h e y p r e s e n t e d a p o i n t m a s st h e o r y f o r h o r i z o n t a l v i b r a t o r y c o n v e y o r s a s w e l l a s th e r e s u l tso f a la r g e n u m b e r o f e x p e r im e n t s t h a t w e r e p e r f o r m e d o ns e v e r a l t y p e s o f h o r i z o n t a l c o n v e y o r s t r a n s p o r t i n g m a n y d i f -f e r e n t g r a n u l a r m a t e r i a l s. S o m e p u r e l y n u m e r i c a l s t u d i e sw e r e d o n e b y N g e t a l. [ 9 ] a n d L i r a [ 1 0 ] . T h e e x p e r i m e n t a l

7 S

~ / / / / / / / / / I / / / / / / / / / / / / / / / / I / / / / / / / / / / / / / / / / / / / / / / >Fig. I. Lay-ou t of the vibratory convey or.

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204 E.M. Sloot, N.P. Kruyt / Pow der Technology 87 1996) 20 3-21 0

veri f icat ions in Ref . [9] were res t r ic ted to a s ingle vibrat ionangle . A special type of vibratory conveyor, a corrugatedv ib ra to ry conveyor, i s desc r ibed by Persson and Megnin[111.

The emphas i s o f the p resen t theore t ica l and exper imenta ls tudy is on incl ined and decl ined t ransport , both for s l ide andthrow con veyo rs . In contras t wi th Ng et al . [9] and Lim[ 10] , the the ory in this ar t ic le is largely analyt ical , in o rderto ga in more ins igh t in to the occur r ing phases o f m ot ion o fthe granular m ater ia l.

The basic assump tions in this theory are:( i ) the t rough is dr iven ful ly s inusoidal ly;( i i ) the granular mater ia l is assumed to behav e like a point

mass , i t moves l ike a r igid body with negl igiblerotat ions;

( i i i ) a ful ly plas t ic col l is ion is assumed when the layer ofgranular mater ia l h i ts the t rough surface af ter a f l ightphase;

( iv) the dis t inct ion betwee n the s ta t ic and kinet ic coeff ic ientof f r ic t ion is neglected, in contras t wi th Ref . [2] ;(v) f r ic t ion betw een the granular mater ia l and s ide wal ls of

the t rough is neglected,(vi) a i r drag is neglected.

2 . D e f i n i t i o n s

The x- and y-coordinates are def ined tangent ia l respec-t ive ly normal to the t rough (con vey or su r face ) , see a lso F ig .1 . Th e con vey or is or iented a t an incl inat ion angle a withrespect to the hor izontal ; a < 0 den otes a decl inat ion. Thecoord inate s is in the direct ion of vibrat ion; the angle betweenthe coord inate s and the t rough is the vibrat ion angle ]3. Thetrough is assumed to be dr iven s inusoidal ly in t ime withangular v eloci ty to and am pli tude r :

X tX t t ) = s i n t o t

r cos /3

)( t ( t ) = co s tot Je t (t ) = -s in tot

Y tYt(t) = = sin tot

r sin 13I ;'t (t ) =c os tot Yt( t) = - s in tot (1)

whe re xt and Yt are the tang ent ia l respect ively norm al t roughdisplacements . The dimensionless displacements , veloci t iesand accelera t ions are denoted by sy mb ols in capita ls.

The th row nu m be r / ' i s de f ined as the d imens ion less max-imum acce le ra tion normal to the t rough:

F = Yr. m.x toZrs in /3 (2 )g cos a g cos a

For F < 1 the mater ia l wi ll remain in contac t wi th the trough

surface; for F > 1 the mater ia l wi l l lose contact wi th the troughand a f l ight phase occurs . Thus s l ide con vey ors are charac-

ter ized by F < 1, whi le f l ight conv eyor s are character ized byF > 1 .

The mach ine numb er i s de f ined as the d imens ion less max-imum trough accelerat ion:

K = 'ma~ to2 r (3)g g

In pract ice the machine num ber wil l be res t r ic ted in order notto damag e the granular mater ia l .

Th e veloci ty eff ic iency, which is a dimen sionless measureof the conv eying speed, is def ined as the ra t io of the averagerela t ive speed and the maxim um tangen t ia l t rough veloci ty:

7- Xre lrel-q - - - = - - (4 )

car cos/3 27r

In this express ion Xrel = (Xm--Xt) denotes the re la t ive mate-r i al d i sp lacement dur ing one cy c le o fT = 2 r r / w . T h esub-scr ipt m is used to denote the point mass .

3 . T h e o r y

Dur ing one cyc le o f the mot ion o f the t rough , the po in tmass m ay b e in a res t, s l ide or f l ight phase, dep ending on theactual accelerat ions , veloci t ies and displacem ents of the p ointmass and t rough. In this sect ion these var ious phases ofmo tion are analysed.

Th e equat ions of mo tion of the point mass m are:

T Fw mgsin ct= m)c m

N m g c os o r- m y m (5 )

Here , the upper or low er s ign is used for s l id ing in the posi t iveor negat ive x-direct ion. The maximum fr ic t ion force isFw = p,N, wh ere /z is the coeff ic ient of f r ic t ion between gran-ular mater ia l and t rough.

3 . 1 . S l i d e a n d r e s t p h a s e s

A sl ide phase wil l occur when the tangent ia l accelerat ionof the t rough is larger than that of the poin t mass . Th e s l idephase is cal led posi t ive when the displacement of the pointmass is in the posi t ive tangent ia l d i rect ion re la t ive to thetrough and n egat ive otherwise . A res t phase occurs wh en therela t ive tangent ia l d isplacem ent is zero.

From Eq . (5 ) andym=Yt he dimensionless accelerat ionof the mass is obtained du ring a s l ide phase:

)(m(t) =tan/3[--~( T-/z- tan a)___/z s into t ] ( 6 )

Sl iding may s tart when the accelerat ions of the mass and thetrough are equal . This occu rs when:

1 /x 5:t an a ) t an /3sin tot= 5: ~ 1 5: /x tan/3 (7)

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E.M. Moot, N.P. Kruyt / Powder Technology 87 199 6)2 03 -21 0 205

where Eq. (1) was used. This impl ies that a posi t ive ornegat ive s l ide phase m ay occu r wi thin the in terval [ 61, 62],resp ectiv ely [ 63, 64] :

sin 6 1 2 = [ ~ ( /x + ta n a ) t a n / 3 ]' 1 + /x t an /3

[ 1 ( - t a n a ) t_an/3 ]sin 6 3 4 = C 1 - / x t a n /3 J ( 8 )

A s l ide phase w i l l ac tual ly begin a t 6 i ( i = 1 ,3) o nly i f thep rev ious phase has ended .

The veloci ty of the point mass i s obta ined by in tegra t ingEq . (6 ) i n t ime . The e mp loyed boundary cond i t ion is tha t theveloci t ies of point m ass and t roug h are equal a t the s tar t of aposi t ive or negat ive s l ide phase . This s tar t occurs a tw t =6p/N. The dimensionless veloci ty of the point mass i s thengiven by:

X m , v / N t ) = --T-/.L(cosw t - c o s 6p/N) tan/3

T-~tan /3( / .~ + tana ) t o t - 6 p / N )+ c o s t ~ p / N ( 9 )

Integra t ing Eq. ( 9) f rom 6 wy to ep/N and us ing Eqs . ( 1 and(8 ) , t he d imens ion les s r e l a tive d i sp lacemen t becomes :

XrcL p/N = (1 _+/~ tan /3) [sin 6- -s in •

+ cos 6 ( e - 6) - 1/ 2( • - 6) 2 sin6 ] (10 )

In th is equat ion the subscr ip ts P/N have been omit ted for 6and e . This ex press io n is used to ca lcula te the veloci ty eff i -c iency for posi t ive and negat ive s l ide phases .

The end of a s l ide phase , e , i s def ined as the t ime a t whichmass and t rough are moving wi th the same tangent ia l veloc-i ty. Using Eqs . (1 ) , (8) and (9) i t fo l lows:

c os • - c o s 6sin 6i ( 11 )

e - 6

In th is expres s ion i = 1 indicates a posi t ive s l ide phase andi = 3 a negat ive s l ide phase . The der ivat ion of th is equat ionis a lso show n in Refs . [ 1 ,2 and 8] .

A res t phase wi l l occur i f a s l ide phase has ended beforethe next phase starts.

3 . 2 . F l i g h t p h a s e

A f l igh t phase occur s i f t he no rma l fo rce N b ecomes ze ro .At the s tar t of a f l ight the mass and t rough hav e the same y-coord ina te and y -ve loc ity. F rom Eqs . (1 ) and (5 ) i t fo l lowsthat the f l ight phase s tar ts a t to t = ~ , where:

6F = arcs in ( 1 / F) (12 )

Dur ing the f l i gh t t he no rma l com ponen t o f t he acce le ra t ionac t ing on the po in t mass rema ins -g cos a . The norma ldisplacement can be determined by in tegra t ing wi th the

boundary con d i t ions o f equa l no rma l coord ina te s and ve loc -ities at o)t = 6F:

Ym(t) = - - 7 ~ w t - - f F ) Z + s i n 6 ~ + C O S g ~ o o t - - fF )( 1 3 )

The end o f the f l igh t is de r ived f rom the cond i t ionYm(EF) = Yt(EF), wh ich yields an imp licit eq uation in EF,us ing Eqs . (12) and ( 13) :

sin 8F -- sin eF + COS 6F( eF -- 6F)

-- 1/2 sin 6F( eF -- 6F) 2 = 0 (14 )

The end o f the f l i gh t can be expres sed in the num ber o f pe r iodsper fl ight, n: eF = 6F + 2~ 'n. C om binin g this e xpre ssion w ithEqs. (12) and (14) y ie lds :

E = [ _ \ 2 7 r n - s i n 2 7 r n l f + ( 1 5)

This equat ion w as a lso der ived in Refs . [ 3 ,5 and 8] . In orderto res t r ic t the machine number, n i s of ten kept below 1 inpract ice , or F< 3 .297.

Since a i r drag is neglected, the tangent ia l accelera t ion act -ing on the point mass remains equal to ~m= --g s in a , orXm = - ( 1 /F ) tan a tan/3 . Th e veloci ty of the point mass canbe determ ined by in tegra t ing th is express ion wi th the bound-ary cond i t ion of a kn own tangent ia l veloci ty V~ a t eF. Thisveloci ty i s determined f rom the occurr ing phase before thef l ight phase . The re la t ive d isplacement i s obta ined by in te-gra t ing f rom 6F to @ and by subtract ing the t rough displace-men t f rom that o f the point mass , accord ing to Eq. ( 1 ) . Thisresults in:

Xrel, F - - 2-F tan/3 tan a( eF -- 6F) 2

+ VF(ev -- 6F) --(sin eF -- sin 6F) (16 )

This equ at ion is used to ca lcula te the velo ci ty eff ic iency of af l ight phase acco rding to Eq. (4 ) .

The change o f the ve loc i ty o f t he po in t mass du r ing acol l i s ion wi th the t rough is re la ted to the to ta l l inear impulseat t ime aF by the l inear mom entum equat ion:

N d t = m [ 2 f m E ~ )- ~)m(eF ) ] (17 a)A t

T- t z j N d t = m [ . i m e ~ )- X m ( e F ) ]=mAfcm c ( 1 7 b )A t

Due to the assum pt ion o f a fu l ly p las t ic col l i s ion, the f i rs tterm on the r ight-hand s ide of Eq. (1 7a) equals the troughveloci ty a t EF. Th e s ign in ( 17b ) depen ds on the s ign of there la t ive veloci ty. Combining these two equat ions g ives thechange of veloci ty dur ing a col l i s ion:

A Xm . c = T-/z t a n / 3 [ c o s • v - c o s 6 F + F ( e F - - 6 F) ] ( 1 8 )

Here the value of the norm al veloci ty jus t b efore the col l i s ion(Pro( eF ) ) was obta ined f rom Eq. ( 1 ) and the der ivat ive o f

Eq . (13 ) . I t can be shown tha t t he to ta l change o f ve loc i ty o fthe point mass dur ing a f l ight and a col l i s ion is equal to the

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206 E.M. Sloot , N.P. Kruyt / Powd er Technology 87 1996) 203-21 0

c h a n g e o f v e l o c i t y d u r i n g a p o s i t iv e o r n e g a t i v e s l id e p h a s e[ 8 ] . I f t h e f l i g h t e n d s w i t h a t a n g e n t i a l v e l o c i t y o f t h e p o i n tm a s s t h a t i s n o t e q u a l t o t h a t o f th e t r o u g h , a p o s i t i v e o rn e g a t i v e s l i d e p h a s e o c c u r s , d e p e n d i n g o n t h e s ig n o f th i sd i f f e r e n c e , u n t i l t h e v e l o c i t ie s b e c o m e e q u a l .

3 3 Mod e an d velocity efficiency diagrams

I n o r d e r to c a l c u l a te t h e o c c u r r i n g m o d e s o f m o t i o n a n dt h e v e l o c i ty e f fi c ie n c i es , a c o m p u t e r p r o g r a m w a s d e v e l o p e dt o s o l v e t h e p r e s e n t e d e q u a t i o n s . T h e o c c u r r i n g p h a s e sa r e i n d i c a t e d w i t h P ( p o s i t i v e s l i d e ) , N ( n e g a t i v e s l i d e ) ,F ( f l i g h t ) a n d P ' a n d N ' f o r a p o s i t i v e o r n e g a t i v e s l i d e p h a s edi rec t ly a f te r a co l l i s ion .

T h e m a i n d i f f e r e n c e b e t w e e n t h is p r o g r a m a n d t h a t o f N g[ 9 ] i s t h a t t h e c u r r e n t p r o g r a m c a l c u l a t e s n u m e r i c a l l y o n l yt h e b e g i n n i n g a n d e n d o f e a c h p h a s e b y m e a n s o f th e e q u a -t i o n s d e v e l o p e d h e r e . T h e p r o g r a m c a l c u l a t e s t h e t i m ei n s t a n t s a t w h i c h t h e ( p o t e n t i a l l y o c c u r r i n g ) p h a s e s b e g i n

a n d e n d b y i t e r a t i n g u n t i l a p e r i o d i c s o l u t i o n w i t h p e r i o dT= 27r/w i s o b t a i n e d . A d e t a i l e d d e s c r i p t i o n o f th e p r o g r a mis g ive n in Ref . [ 12] .

I n th e c a s e o f h o r i z o n t a l tr a n s p o r t ( a = 0 ) , t h e o c c u r r i n gm o d e s a n d v e l o c i t y e f f i c ie n c y c a n b e e x p r e s s e d a s f u n c t i o n so f / x t a n /3 a n d / ( s e e R e f . [ 8 ] f o r r e s u l t s ) . I n F i g . 2t h e o c c u r r i n g m o d e s a r e s h o w n f o r a d e cl i n at i o n w i t ht a n a = - 0 . 1 0 a n d / z = 0 . 35 . I n F i g . 3 t h e v e lo c i t y e f f ic i e n c yi s p lo t t e d f o r t h e s a m e d e c l i n a t i o n a n d f r i c t i o n c o e f f ic i e n t . InF i g . 4 t h e v e l o c i t y e f f i c i e n c y i s p l o t t e d f o r a n i n c l i n a t i o n o ft a n a = + 0 . 1 0 a n d / z = 0 . 3 5 , a n d F i g . 5 s h o w s t h e e f f e c t o ft h e i n c li n a t io n o n t h e v e l o c i t y e ff i c ie n c y f o r c o m m o n v a l u e s

o f / x a n d / 3 .

3 4 Throw number fo r maximum velocity efficiency

I n t h i s s e c t i o n t h e t h r o w n u m b e r i s d e r i v e d f o r w h i c h t h ev e l o c i t y e f f i c i e n c y r e a c h e s i t s m a x i m u m . F o r a f u l l c y c l i cf l ig h t t h e f l ig h t n u m b e r n e q u a ls 1 ( [ ' = 3 . 2 9 7 ) . T h e n t h et a n g e n t i a l v e l o c i t y o f t h e p o i n t m a s s w i l l n o t c h a n g e i n ti m ei n th e c a s e o f h o r i z o n t a l t r a n s p o r t : i t f o l l o w s f r o m E q . ( 1 2 )t h a t t h e v e l o c i t y e f f i c i e n c y,Vv= c o s ( 6 F ) = 0 . 9 5 4 ( s e e t h e

~ - 0 . 3 5 ; t a n a - - 0 . 1 0

FPF

3 . 0 0 -

I 2.50-

2 .00-

~ N F R0 . 0 ( . ' ' : . . . . : . . . . ', . . . . ; . . . . ~ . . . . : . . . .

0 . 0 0 0 . 2 5 0 . . 5 0 0. 7 .' 5 1 . 0 0 1 . 2 5 1 . 5 0 1 . 7 5 2 . 0 0

> t a n p

Fig. 2. Mo de diagram of successive phases for declined transport (P =positive slide phase, N = negative slide phase, R =rest phase, F= flightphase, P '= positive slide p hase after a collision, N '= negative slide phaseafter a collision).

p - 0 . 3 5 ; t a n tt - - 0 . 1 0

1,2° t _ - ~ l g

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 1 .; 25 1 . 5 0 1 . 7 5 2 . 0 0

> t npFig . 3 . Veloci ty eff ic iency for dec l ined t ranspor t as a fu nct ion of v ibra t ionangle/3 and throw num ber F.

t~ 0 . 3 5 ; t a n a - + 0 . 1 0

1.00

I] 0,80

I 0 . 6 0 . 2 . 1 . 2

1. 1.0

0.20-

o ~ • ~0.00 0 . 5 0 1 . 0 0 1 . 5 0 2 . 0 0

> t a n I~

Fig. 4. Velocity efficiency for inclined transport as a function of vibrationangle/3 and throw num ber F.

£

3

2 . 5

2

1 . 5

1

0 . 5

0-0 .1

1 ,

o.6o.4

~

.O.2p

-O.O5 0 0.O5 0.1

: > t a n z

3

2 . 5

2

1

0 . 5

0

J

i / / / I I] 117

) J

: .6) A

-0.1 -0.05 0 0.05 0.1

~ t a r l a

Fig. 5. Iso-curvesof velocityefficiencyas a function of inclination angle aand throw number F. (a) /z = 0.2; (b )/z = 0.3; - - , / 3 = 30°; - - - , /3= 45 °.

h o r i z o n t a l l i n e f o r F = 3 . 2 9 7 i n F i g . 6 ) . I n o r d e r to i n c r e a s et h is e f f i c i e n c y, a s m a l l p o s i t i v e s l i d e p h a s e h a s t o b e p r e s e n tbefo re the f l igh t phase . Th e f l igh t then s ta r t s a t ep = 6~ ande n d s a t % = 6 p + 2 ~ . F o r a m a x i m u m d i m e n s i o n l e s s v e l o c i t yo f 1 a t t h e s t a rt o f th e f l i g h t , /3 m u s t b e e q u a l t o 0 a c c o r d i n gt o E q s . ( 8 ) a n d ( 9 ) w i t h 6 p = 61 ~ 0 . T h e c o r r e s p o n d i n gt h r o w n u m b e r i s d e r i v e d f r o m E q . ( 1 4 ) , u s i n g e F = 6 v + 2 7 r n :

27ten 2 - 1F - ( 1 9 )

2 ~ ' n c o s 2 ~ r n

To g e t h e r w i t h E q . ( 1 5 ) a n im p l i c i t r e l a t io n i n F i s o b t a i n e d ;

i ts s o lu t io n i s F = 2 . 9 7 5 f o r e a c h v a l u e o f a a n d t h e m a x i m u mv e l o c i t y e f f i c ie n c y i s o b t a i n e d f o r t h is t h r o w n u m b e r, s e e a l so

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E.M. Sloot, N.P. Kru yt / Powd er Technology 8 7 1996) 203-2 10 207

. X m in P F m o d e ( I = 2 .9 7 5 )

/ / ~ / ) ( m i n m o d e ( I = 3 . 2 9 7 )

tI ) - c 0 t

5p 8

-1

F i g . 6 . Ta n g e n t i a l v e l o c i ti e s o f t r o u g h a n d p o i n t m a s s a s a f u n c t i o n o f t im el a rg e t h r o w n u m b e r s ) .

Fig. 6 . The d epen den ce on tx and tan/3 seems to be v ery smal l,e spec ia lly fo r ho r i zon ta l t r anspor t [ 8 ] . S ince the max imumveloci ty eff ic iency is obta ined for F -- 2 .975, i t is not usefulto des ign a v ib ra to ry conveyo r wi th a th row num ber l a rge r

than 3.0.

3.5. Limiting case o f small vibration an gles fo r slidephases

In th is sect ion the case o f a smal l v ibra t ion angle i s s tudiedin order to der ive a re la t ion that descr ibes the inf luence ofincl inat ion o n the veloci ty eff ic iency for smal l v ibra t ionangles . Fur thermore , a method is descr ibed to measure thefr ic t ion coeff ic ient betw een g ranular mater ia l and t rough.

Equa t ions for the s tar t and end o f aP N mod e, which impl ies6N = •p and •N = 6p + 27r, are o btain ed f rom Eq. ( 11 ) :

( C - 2~r) s in 63s in (6p + 1 /2C ) -

2 s in ( 1 / 2 C )

27r• p = C + 6 p w it h C = ( 2 0 )

1 ( s in 81 ) / ( s in 83 )

This equa t ion is val id for each value o f /3 .For /3 ~ 0 , two cases are now dis t inguished:( i ) K<oo: for th is rea l is t ic case an e legant method was

deve loped to o bta in the pr im ar i ly k inet ic coeff ic ient of f r ic-t ion of the granular mater ia l . This method is a l ready men-t ioned in Ref . [ 8 ] ; here i t i s descr ibed in deta i l. T he t rough

has to move wi th a cer ta in ampl i tude in purely hor izonta ldi rec t ion ( /3 = 0) . In o rder to be cer ta in that a PN m odedevelop s for hor izonta l t ranspor t , K mu st be larger than 0 .67for a f r ic t ion coeff ic ient that i s smal ler than 0 .35, as can beobse rved in the mode d iag ram in F ig . 7 . By measur ing e i the ropt ica l ly, or by us ing a fe l t pen, the maximum rela t ive d is-p l acemen t be tween t rough and an open b ox o r cy l inde r f i ll edwith granular m ater ia l , the coeff ic ient of f r ic t ion can be ca l -cula ted impl ic i t ly f rom Eq. (10 ) . The resul t is shown inFig. 8.

( i i ) K ~ oc: th is hyp othet ica l case resul ts in a very s impleequat ion that dem onstra tes the inf luence of the incl inat ion of

the t rough on the veloci ty eff ic iency for smal l v ibra t ionangles . In case o f aP N mod e, wh ich wi l l a lway s be the case

r 0 . 1 0

I 0.05

0 I I I I0.05 0.10 0.15 0.20

> t a n pF i g . 7 . M o d e d i a g r a m f o r s m a l l v ib r a t io n a n g l e s a = 0 , t z = 0 . 3 5 ) .

2 K - - * o o

IX e llp ml K=1 0

1 . 5 -

K = 0 . 6 71-

0 I

0 , 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0. 35

F i g . 8 . M a x i m u m r e l a t iv e d is p l a c em e n t f o r a P N m o d e , a s a fu n c t i o n o ff r ic t ion coe ff ic ien t /~ and machine number K.

for inf in i te values of the machine number, i t fo l lows af tersome a lgebra f rom Eqs . (4 ) , (8 ) , (10 ) and (20 ) tha t t hel imi t ing value o f the eff ic iency for each incl inat ion or decl i -nat ion angle i s g iven by :

(C ) . / r r tan a~r /a _ ~ o : - c o s -~ : - s t n ~ ] ( 2 1 )

This exp ress ion is val id for F < I . Note that the s tart of theposi t ive s lide phase is g iven by 6p = 7r -1/2C according toEq. (2 0) for /3 --*0 and because 6p • [81,62] . This equat ionalso shows the ant isymmetry between posi t ive and negat ives lopes in case f l ~ 0 . Eq. (2 1) predic ts that the veloci ty eff i-c iency equals 0 .434 for /3 ~ 0 and /x = 0 .35; th is is consis tentwith the results presen ted in Fig. 3.

4 . E x p e r i m e n t a l

In o rde r to ve r i fy the deve loped po in t m ass theo ry, expe r-imen t s were pe r fo rmed wi th an e l ec t rohydrau l i ca l ly d rivenv ib ra to ry conveyor. The exac t d imens ions o f t h i s 3 m longconv eyor a re g iven in Ref . [8 ] . The inc l ina t ion o f the con-veyor was se t by l i f t ing i t a t one end over var ious heights .The t rough was d r iven s inuso ida lly in t ime by means o f onehor izonta l and two ver t ica l hydraul ic cyl inders wi th e lec-t ronic feed-back. By modifying the hor izonta l and ver t ica lampl i tudes and running f requency, the v ibra t ion angle andth row n umber were se t . The ve loc i ty e ff i c i ency was de te r-mined by measu r ing the t ime that e lapsed dur ing the t raverseof a m arker in the granu lar mater ia l over a f ixed dis tance .

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4 1 Verifications o f the assum ptions

With t he expe r imen ta l s e t -up , t he fo l l owing a s sumpt ionso f t he po in t mass t heo ry were ve r i fi ed :

( a ) by measu r ing t he d i sp l acemen t s i n time o f the t r oughwi th a d ig i t a l da t a acqu i s i ti on sys t em i t was ve r i f ied t ha t thed i sp l acem en t s o f t he t r ough we re s inuso ida l .

( b ) In o rde r t o i nves t i ga t e whe the r a ve r t ic a l g r ad i en t o fthe ve loc i t y was p re sen t , a p l a s t i c s t r aw was pu t i n to t heg ranu la r ma te r i a l . S ince t he s t r aw r ema ined i n i t s ve r t i c a lpos i t i on , i t was conc lude d tha t no such g rad i en t was p re sen t .

( c ) The e f f ec t o f t he wa l l f r i c ti on was i nves t i ga t ed byd i s t r ibu t ing a nu mb er o f s t r aws ove r t he w id th o f t he g r anu la rl aye r. On ly a sm a l l i n f luence o f t he wa l l f r i c ti on was ob se rvedwi th in a f ew mi l l ime t r e s d i s t ance f rom the wa l l s , comparedt o t he w i d t h o f t h e t r o u g h o f 2 0 0 m m .

( d ) F o r th r o w n u m b e r s F < 2 . 5 s t ab l e b e h a v i o u r w a s

o b s e r v e d . M e a s u r e m e n t s f o r l a rg e r th r o w n u m b e r s w e r e m o r ed i f f icu l t t o pe r fo rm due t o t he p r e sence o f s t rong wigg le s onthe su r f ace o f t he g r anu la r ma te r i a l .

4 2 Measurem ents

M e a s u r e m e n t s w e r e p e r f o r m e d w i t h s q u a r e p o l y -( v i n y l ) c h l o r i d e ( P V C ) g r a i n s ( / ~ = 0 . 2 3 ) a n d n e a r ly r o u n dgra in s o f sp inach s eed ( / z= 0 .24 ) , b o th fo r i nc l ina t i ons andd e c l in a t i o n s ( t a n a = + 0 . 0 2 a n d t an t~ = + 0 . 0 5 ) . T h e f r ic -

t i on coe ff i c i en t s we re measu red u s ing t he me thod desc r ibedin t he p r ev ious s ec t i on . S ince no l a rge d i f f e r ences we re foundbe twe en the ve loc i t y e f f i c i enc i e s o f t he se two ma te r i a l s , on lythe r e su lt s o f t he expe r ime n t s w i th PVC g ra in s a r e shownhere for tan c~ = + 0 .05 . In Figs . 9 and 10 the resul t s of theexpe r imen t s and deve loped theo ry a r e compared . The t o t a ls e t o f m e a s u r e m e n t s c a n b e o b t a i n e d f r o m t h e s e c o n d a u t h o r( N P K ) . A s a n e x t r a c h e c k , m e a s u r e m e n t s w e r e a l s o p e r -fo rmed wi th a s i ng l e wooden b lock w i th f r i c t i on e l emen t s( / x = 0 .18 ) i n s l id ing mo de .

P V C g r a i n s i n c l i n e d )t a n ,, - - * 0 . 0 4 6 p . - 0 . 2 3

t . 0 0 . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . P

4 ~ 1 = = = = = = : : = = : = : : : 2 : : = : : : : : : = l ~ , 1 5 1

11 + • v 1 , 9 6I 0 .715 • • • • • C I I ~ 1 5

6 1 . 0 6

o . 15 o o 0 . 6 0

0 . 2 5

o o o0 , 0 0 0 . 2 1 5 0 . 1 5 0 0 . 7 1 5 1 . 0 0 1 2 ,1 5 1 . ~ O 1 . 7 52 0 0

> t a n p

Fig. 10. Com parisonof theoretical and experimentalvelocityefficiency orinclined transport of PVC grains, tan a = + 0.046; the open symb olstandfor the theory and solid symbols or the experiments.

4 3 Com parison of theory and experiments

F a i r ly g o o d a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t i sfound fo r sl i de conv eyor s ( F < 1 .0 ) . Nega t ive ve loc i t y e f f i-c i enc i e s we re no t measu red fo r g r anu la r ma te r i a l s i n bu lk ,s ince t he expe r imen ta l s e t -up d id no t a l l ow th i s . A s ing l ewoo den b lock t hough d id show nega t ive e f f i c i enc i e s , a s p r e -d i c t ed by t he t heo ry. Ove ra l l , t he v e loc i t y e f f i c i enc ie s o f t h i ss ing l e mass ag ree s w i th in 20 w i th t he t heo re t i ca l p r ed i c -t i ons fo r F< 1 . Fo r t he g r anu la r ma te r i a l s , f o r l a rge t h rown u m b e r s ( F = 2 . 5) g o o d a g r e e m e n t is o b ta i n e d f o r n e g a t iv es lopes , bu t l a rge r dev i a t i ons a r e ob se rved fo r pos i t i ve s l opes .The i n f luence o f v ib ra t ion ang l e /3 on t he e f f i c i ency fo r t h rownum ber s a round 3 .0 appea r s the s am e a s a t 2 .5 , in ag reem en t

wi th t he t heo ry. He re r e l i ab l e measu remen t s we re ha rde r t ope r fo rm due t o t he p r e sence o f w igg l e s on t he su r f ace o f theg ranu la r ma te r i a l. Fo r F = 1 .5 - 2 t he t heo ry p red i c t s a l a rgein f luence o f t he v ib ra t i on ang l e on t he ve loc i t y e f f i c i ency.Howeve r, t he r e su l t s o f t he expe r imen t s do no t exh ib i t t h i sl a rge i n f luence . A s imi l a r o rde r o f d i s c r epancy was fo und byRad ema che r [ 8 ] and S loo t [ 12 ] fo r ho r i zon t a l tr anspo r t. Nget a l. [9] car r ied out ver i f ica t ions for a s ingle v ibra t io n angleo f 45° : f o r ho r i zon t a l t r anspo r t t he expe r im en t s ag ree bes twi th the theory for th is v ibra t ion angle .

P V C g r a i n s d e c l i n e d )t a n - , = - 0 . 0 5 1 I ~ = 0 . 2 3

t 2 5 [ r

t

11 t.o o vl 1253

• ÷ • Q 1 . 0 3

0 . 7 5 0 , 7 2• •

0 . 2 6 . . . . . . . . . ..0 . 0 0 C t ; 2 5 0 . ~ 0 0 . 7 5 1 . 0 0 1 . 2 5 I . ~ 0 1 . 7 5 2 . 0 0

> t a n 13

Fig . 9 . Co mpar ison of theore t ica l and exper im enta l ve loc i ty e ffic iency ford e c l i n ed t ra n s p o r t o f P V C g r a in s t a n c t = - 0 . 0 5 1 ; t h e o p e n s y m b o l ss t n d

for the theory and solid symbolsfor the experiments.

5 E x t e n s i o n s o f t h e p o i n t m a s s th e o r y

In an a t t emp t t o r e so lve t he obse rved d i s c r epanc i e sb e t w e e n e x p e r i m e n t s a n d t h e o r y f o r t h ro w n u m b e r s a r o u n d1.5 to 2 , the fo l lowing three modif ica t ions were br ief lyinves t iga ted:

( i ) By me asu r ing t he acce l e r a t i ons o f t he t r ough i t wasfound tha t they w ere no t pu re ly s i nuso ida l, a l t hough the d i s-p l acem en t s i gna l s we re s inuso ida l i n bo th d i r ec t ions . I n o rde rto i nves t i ga t e t he s ens i t iv i t y t o no i se i n t he acce l e r a t i ons o fthe p r ed i c ted ve loc i t y e f f i ci ency, a num er i ca l p rog ra m w as

deve loped , s imi l a r t o tha t o f Ng e t a l . [ 9 ] . I n t he s imu la t i onthe t rough w as sub j ec t ed t o an acce l e r a t i on w i th a h igh o rde r

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E.M. Sloot, N.P. Kruyt / Powd er Technology 8 7 1996) 203- 210 2 0 9

Ta b l e 1I n f l u e n c e o f t h e r e s t i t u t i o n c o e f f i c i e n t o n t h e v e l o c i t y e f f i c i e n c y ( t a n / 3

= 0 . 2 5 ; / 1 , = 0 . 2 3 )

t a n a e O f o r F = 1 .5 r / f o r F = 2 . 0

- 0 . 0 5 0 0 . 4 7 7 0 . 5 3 70 , l 0 ,477 0 .827

0 . 2 0 . 4 7 7 0 . 8 5 8

0 0 0 . 2 1 0 0 . 5 1 70 . 1 0 . 2 1 0 0 . 7 9 30 . 2 0 . 2 1 0 0 . 8 3 8

+ 0 .05 0 - 0 .065 0 .4980 .1 - 0 . 0 6 5 0 . 8 1 90 . 2 0 . 2 4 9 0 . 8 1 9

or iginal point mass theory forms a promis ing approach tofur ther research of the t ranspor t of granular mater ia ls byvibra tory conveyors . For fur ther research i t i s a lso recom-mended to include a i r drag in to the theory and to determinethe co eff ic ient of res t i tu t ion exper imen ta l ly•

The p resen ted po in t mass theo ry does no t t ake in to accoun t

the in teract ions between the par t ic les of the granular m ater ia l.By s imulat ing the behaviour of a l l par t ic les s imul taneously,a so-cal led discre te e lemen t method s imulat ion, these in ter-ac t ions can be p rope r ly accoun ted fo r. Hogue and Newland[ 15] g ive resul ts of such an approac h in thei r s tudy o f thes ieving process .

(k) ar t if ic ial noise wi th (d im ension less) am pl i tude a andpha se sh ift th:

. ~ = I 2 = - s i n t o t - a s i n k w t + ~r) ( 2 2 )

Calcula t ing som e eff ic iencies wi th d i fferent values of a andphase shi f ts ~b, the var ia t ion remained wi thin 10 for a < 0 .2and k--~25 (w hich is the order of measure d noise)• Thismeans tha t the l a rge d i f f e rence be tween theory and expe r i -men t s can no t be exp la ined by the dev ia t ion in t rough acce l -era t ion f rom the s inusoidal shape.

( i i ) The rota t ion o f the par t ic les could lead to an increaseof the re la t ive d isplacem ent due to thei r poss ibly turning ove rdur ing a col l i s ion of a par t ic le wi th the t rough. This wou ldlead to a m ax im um inc rease o f t he r e la t ive d i sp lacemen t inthe order of a par t ic le d iameter. Since t rough displacements

o f a round 12 mm pe r pe r iod were used in the expe rimen t s ,th is inf luence can be neglected for PVC grains and spinachs ee d ( d < 3 . 5 m m ) .

( i i i ) Van Kappe l [13 ] inves tiga t ed damping wi th in thelayer of granular mater ia l dur ing the t ranspor t in v ibra toryconvey ors f rom a theo re t ica l and expe r imen ta l po in t o f v i ew.In order to include a m ore soph is t ica ted mo del for the col li -s ion, a par t ly e las t ic col li s ion is assumed h ere . The re la t ionbe tween the no rma l ve loc i ti e s o f t he po in t mass be fo re andaf ter the co l l i s ion is def ined by mean s o f a coeff ic ient ofresti tution e [ 14]:

•

e - Yre l( ev ) (23 )

Note that a p ure ly p las t ic col l i s ion correspond s to e = 0 . Th emass t e rm in Eq . (17 a ) i s now rep laced by m(1 + e ) ; t heconse quen ce of th is i s tha t in Eq. (1 8) , p , i s replaced by/z ( 1 + e ) . Now the sys t em can be pe r iod ica l ove r a pe r iod o fmo re than 27r or even not per iodical a t al l. Th e veloci tye ff i c iency was de te rmined by ca l cu la t ing an ave rage d i sp lace -me nt ov er a long t im e in terval• The inf luence of a par t ly e las ticcol l i s ion on the ve loci ty eff ic iency is show n in Tab le 1 .

No te the l a rge inc rease o f t he ve loc i ty e ff ic i ency fo r a th rownum ber o f 2 .0 , even for a smal l co eff ic ient of res t itu t ion.

Al though the l a rge dev ia t ions fo r t h row numbers o f a round1.5 are no t fu l ly exp la ined, th is theoret ica l ref inemen t of the

6 Conclus ions

A point mass theory is presented for the res t , s l ide andf l ight phases that occ ur dur ing the t ranspo r t of granular mate-

r ia ls by incl ined vibra tory conv eyors . Plots are g iven, show-ing the inf luence of incl inat ion, throw number, f r ic t ioncoeff ic ient and v ibra t ion angle on the veloci ty eff ic iency.When the vibra t ion angle approaches 0 , the veloci ty eff i -c iency is ca lcula ted analyt ica l ly for s l ide conve yors .

A m ethod is descr ibed in deta i l for determin ing the coef-f ic ient of f r ic t ion betwe en gran ular mater ia l and t rou gh as afunc t ion o f the m easured max im um re l a tive d i sp lacemen t ona hor izonta l t rough.

Exper imen t s were pe r fo rmed w i th PVC gra ins and sp inachseed for incl ined and decl ined t ranspor t . For s l ide convey orsthe ag reemen t in ve loc i ty e ff ic i ency be tween theory and the

exper im ents was sa t is fac tory. For f l ight conve yor s the agree-men t was f a i r ly good fo r th row numbers o f abou t 2 .5 . Fo rth row num bers be tween 1 .5 and 2 .0 the expe r imen t s showeda veloci ty eff ic iency hat was a lmost indep enden t of the v ibra-t ion angle , contrary to what i s predic ted b y the theory.

In order to f ind an explanat ion fo r these devia t ions , som eposs ibi l it ies were inves t igated. This includ es the effe ct of apar t ly e las t ic col l is ion. Fo r sm al l coeff ic ients of res t i tu t ion(0 .1-0 .2) the theory coincides qui te wel l wi th the exper i -menta l data for throw numbers around 2 .0 .

7 L i s t o f symbol s

Cde

w

gkKmNn

r

cons tan t i n Eq . (20 ) ( - )d i ame te r (m)coeff ic ient of res t itu t ion ( - )f r ic t ion fo rce (N)acce le ra t ion due to g rav i ty (m s -2 )order of ar t if ic ia l noise ( - )m a c h i ne n u m b e r ( - )mass (kg )norma l fo rce (N )numb er o f pe r iods pe r f li gh t ( - )

radius , ampl i tude of d isplacem ent in d i rec t ion ofv ib ra t ion ( m)

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210 E.M. SIoot, N.P. Kruyt / Pow der Technology 8 7 1996) 203-2 10

s

t

T

UF

xX

YY

disp lacem ent in v ib ra tiona l d i r ec tion m )t i m e s )p e r i o d s )ave rage tangen t ia l ma te r ia l ve loc ity m s - ~tangent ia l mater ia l veloci ty a t the s tar t of thef l igh t m s -~ )coord ina te tangen t i al to t rough m )d imens ion less coo rd ina te t angen ti a l to trough

- )coord ina te normal to t rough m )d imens ion less coord ina te normal to t rough - )

G r e e k l e t t e r s

Ol

F

E

rl

tx

.0

incl inat ion angle °)v ib ra t ion ang le o )t h ro w n u m b e r - )s t a rt o f a phase - )

e n d o f a p h a s e - )end o f f l igh t phase jus t be fo re /a f t e r co ll i s ion - )ve loc i ty e ff i c i ency - )f r i ct ion coe ff i c i en t o f ma te r i a l on t rough su r face

- )angu la r ve loc i ty s - l)

S u b s c r i p t s

CF

imN

col l i s ionf l ight

index accord ing to Eq . 8 ) i = 1 ,3 )mate r i a l o r po in t massnega t ive s l ide phase

Pp ,

relt

negat ive s l ide phase af ter col l is ionpos i t ive s l ide phaseposi t ive s l ide phase af ter col l is ionre la t ive ma te r i a l - t rough)t r ou g h o r c o n v e y o r

c k n o w l e d g e m e n t s

The au thors wou ld l ike to thank Pro fesso r Dr. I r. F. J .C .R a d e m a c h e r R a d e m a c h e r E n g i n e er in g , B o r n e , N e t h e r-l ands ) , fo r the f ru i t fu l d i scuss ions and p leasan t coopera t ion .

R e f e r e n c e s

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