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u m e r i c a l c a lc u l a tio n s o ff l o w i n a h y d r o c y c l o n eope ra t i ng wi thou t an a i r co reM a l c o l m R D a v i d s on

CSIRO Division of Mineral Engineering, Lucas Heights Research Laboratories,Lucas Heights, N.S .W. , 2232, AustraliaReceived March 1987; revised October 1987)

S t e a d y f lo w in a h y d r o c y c l o n e o p e r a t i n g w i t h o u t a n a i r c o r e is m o d e l l e db y a f i n it e d i f f e r e n c e s o l u t i o n o f t h e N a v i e r - S t o k e s e q u a t i o n s , f o l l o w i n gt h e a p p r o a c h o f P e r ic l e o u s a n d R h o d e s , i n w h i c h t h e s h e a r s t r e ss d u e t ot a n g e n t i a l m o t i o n i s d e r iv e d fr o m t h e f a m i l i a r P r a n d tl m o m e n t u m t r a n s p o r tt h e o r y, a p p l ie d t o a n g u l a r m o m e n t u m . In t h i s a p p li c a ti o n , th e P r a n d t l m o d e lb r e a k s d o w n v e r y n e a r t h e a x is o f s y m m e t r y a n d a b o u n d a r y c o n d i t i o n ,

c o r r e s p o n d i n g t o z er o sh e a r , m u s t b e im p o s e d a t a r a d i u s o f a b o u t o n em i x i n g l e n g t h t o e n s u r e r e a l is t ic f l o w p r e d i c t i o n s . C a l c u l a t i o n s a r e b a s e do n a c o m m o n l y u s e d s o lu t i o n p r o c e d u r e f o r v e l o c it y a n d p r e s s u r e w h i c hu s e s t h e S I M P L E a l g o r i t h m o f P a t a n k a r a n d S p a l d i n g . H y b r i d f ir s t - o r d e ru p w i n d / c e n t r a l d i f f e r e n c i n g i s u s e d , a n d c a l c u la t e d f l o w v e l o c i ti e s a r e o b -t a i n e d w h i c h a g r e e w i t h b o t h p u b l i s h e d d a t a a n d a n a l y t i c a l p r e d i c t io n s .T h e c o r r e s p o n d i n g t r a n s p o r t e q u a t i o n fo r t h e d is p e r s e d p a r t ic l e ) p h a s eis s o l v e d s i m i la r l y, a n d t h e p r e d i c t e d e f f ic i e n c y c u r v e a n d d i s t r i b u t i o n s o fp a r t ic l e c o n c e n t r a t i o n a r e s h o w n . F i n a ll y, p r e d i c t e d v e l o c i t y a n d p a r t i c led i s t r ib u t i o n s a r e c o m p a r e d w i t h c o r r e s p o n d i n g r e s u lt s b a s ed o n q u a d r a t i cu p s t r e a m d i ff e r e n c in g o f t h e g o v e r n i n g e q u a t i o n s ; it is c o n c l u d e d t h a tn u m e r i c a l d i f fu s i o n h a s li tt le e f f e c t o n t h e p r e d i c t e d f l u i d o r p a r t ic l e m o v e -m e n t i n t h i s c a s e .

Keywords h y d r o c y c l o n e , n u m e r i c a l m o d e l , s w i r li n g f lo w, n u m e r i c a l d if -f u s i o n

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

A hyd roc yc lo ne (Brad ley ~, Sv aro vsk y 2) is a pa r t ic lesepara t ion dev ice , wide ly used in indus t ry, in which apar t i c le - f lu id mix ture i s in jec ted t angen t ia l ly, c rea t inga s t rong ly swi r l ing , r ec i rcu la t ing f low. When the hy-d r o c y c l o n e i s o p e n t o t h e a t m o s p h e r e , a n a ir c o r e fo r m sa long the cen t re l ine . Par t i c les move re la t ive to the

f lu id accord ing to a ba lance be tween cen t r i fuga l fo rceand v i scous d rag , wi th separa t ion depending on par-t i c le s i ze , dens i ty, and shape . For example , in c lass i -f i ca t ion accord ing to par t i c le s i ze , l a rge par t i c les t endt o m o v e t o t h e c y c l o n e p e r i p h e r y f r o m w h e r e t h e y p a s sto the under f low whi le smal l pa r t i c les a re ca r r i ed tothe cen t re and in to the over f low.

Recen t ly, Per ic leous , and Rho des 3 mode l led the f lowand par t i c le t r ansp or t in a hydro cyc lone us ing the com-

merc ia l ly ava i lab le f lu id f low compute r p rogramP H O E N I C S 4 t o s o l v e n u m e r i c a l ly t h e N a v i e r - S t o k e sequa t ions , incorpora t ing a modi f ied Prand t i mix ingleng th m ode l o f tu rbu len t v i sco s i ty. An undiscussedfea tu re o f the i r mod e l i s the ir equa t ion fo r t angen t ialmo t ion which i s based on Prand t l s theory app l ied toangula r mo m entum . I t impl ies , fo r cons tan t dens i ty,

tha t angu la r momentum i s t r anspor ted l ike a pass ivesca la r and no t accord ing to the cor responding laminarf low equa t ion modi f ied by tu rbu len t v i scos i ty. The ex-per ime nt o f Ke isa lP w as s imula ted , and sa t i s fac to rya g r e e m e n t b e t w e e n m o d e l a n d e x p e r i m e n t w a s o b -ta ined fo r the t angen tia l ve loc i ty d i s tr ibu t ion , p ressuredrop , and f low sp l i t .

Desp i te the favourab le compar i sons repor ted above ,fu r ther t e s t ing o f the Pe r ic leous -Rh odes m ode l is de -

© 988 Butterworth Publishers Appl . Math . Mo de l l ing , 1988 , Vol . 12 , Apr i l 119

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Numerical calculations of flo w in a hydrocyclone:

s i rab le to g ive con f idence in i ts genera l va l id i ty as ades ign too l , no t on ly beca use o f the unce r ta in ty sur-rounding the app l icab i l ity o f ind iv idua l tu rbu len cem o d e l s i n c o m p l e x f l o w s b u t a l s o b e c a u s e o f t h e p o s -s ib le impor tance o f numer ica l d i ffus ion in the f i r s t -o r d e r u p w i n d d i ff e r e n c in g u s e d b y P H O E N I C S . I n th isp a p e r w e c o n s i d e r t h e r e l a t e d , b u t c o m p u t a t i o n a l l ys i m p le r, c a s e o f a h y d r o c y c l o n e o p e r a t i n g w i t h o u t a na i r core (un l ike the Kelsa l l case ) . Th is c i rcumstanceo c c u r s w h e n t h e h y d r o c y c l o n e f o r m s p a r t o f a pi pi n gnetwork or in l iquid- l iquid separat ion (e .g . , water-oi l )where the a i r core i s ac t ive ly suppressed .

We s i m u l a t e t h e e x p e r i m e n t o f K n o w l e se t a l . 6usingt h e m o d e l o f P e r i c l e o u s a n d R h o d e s , a n d c o n f ir m t h ata mode l equa t ion fo r the t angen t ia l ve loc i ty, based onthe Prand t l theory, i s appropr ia te . S imply us ing thelaminar f low equa t ions , modi f ied by tu rbu len t v i scos -i ty, fo r the t angen t ia l a s we l l a s the o ther ve loc i tycomponents l eads to a p red ic ted t angen t ia l ve loc i tyd i s t r ibu t ion qu i te un l ike tha t found exper imenta l ly, aresu l t cons i s ten t wi th the exper ien ce o f o thers .

C a l c u l a ti o n s a r e b a s e d o n m e t h o d s e m b o d i e d i n th ec o m p u t e r p r o g r a m s T E A C H ( G o s m a n a n d I d e ri a h 7)a n d P H O E N I C S , a n d a r e p e r f o r m e d u s i n g t h e a u t h o r ' sown spec ia l -purpose p rogram. In par t i cu la r, ve loc i tyand pressure a re so lved on a s t aggered mesh us ing theSI M PL E a lgor i thm of Pa tankar and Spa ld ing. 8 As inT E A C H a n d P H O E N I C S , h y b r i d f i r s t - o r d e r u p -wind /cen t ra l d i ffe renc ing i s used fo r the f lu id f lowequa t ions and the t ranspor t equa t ion fo r the d i spersed(par ti c le ) phase . C a lcu la ted ver t i ca l and tangen tia l f lowveloc i t i e s and pre ssure d rop s , fo r the g iven f low sp l it ,a r e c o m p a r e d w i t h th e c o r r e s p o n d i n g d a t a o f K n o w l e se t a l . and ( fo r ve r t i ca l ve loc i t i e s ) wi th the theore t i ca lp red ic t ions o f Bloo r and Ingham. 9 The p red ic ted e ffi -c iency curv e and d i s t r ibu t ions o f pa r t i c le con cen t ra t ionare a l so shown. F ina l ly, to assess the impor tance o fnume r ica l d i ffus ion in th is app l ica t ion , the ca lcu la t ionsa r e c o m p a r e d w i t h c o r r e s p o n d i n g r e s u l ts b a s e d o n t h em o r e a c c u r a t e q u a d r a t i c u p s t r e a m d i f fe r e n c in g o f th egovern ing equa t ions .

Malcolm R DavidsonOVERFLOW

Vorl ex

finder

h e m o d e l

T h e d e v i c e u s e d b y K n o w l e se t a l . w a s a c o n v e n t i o n a ls m a ll -a n g le c y l i n d e r- o n - c o n e h y d r o c y c l o n e a n d isshown schemat ica l ly inF i g u r e 1 together with i ts re l -

a t ive d imens ions . T he d iam ete r o f the cy l indr ica l sec -t ion i s deno ted by D, i t s he igh t by H, and the resp ec t ived i a m e t e r s o f t h e i n fl o w, u n d e r f l o w, a n d o v e r f l o w b yd l , d , , and do . The l eng th o f the vo r tex f inder i s den o tedby h , and the overa l l he igh t o f the de v ice by L . Thefami l i a r cy lindr ica l coord ina tes ( r, x , 0 ) a re used wi ththe o r ig in cen t red a t the under f low.

To avo id so lv ing a fu l ly th ree -d im ens iona l p rob lem ,w e a s s u m e t h a t t h e f l o w is a x i sy m m e t r i c e v e n t h o u g hth i s wi l l c lea r ly no t be the case near the en t ry reg iona t the top . Ins tead o f the f lu id en te r ing the cyc lonetangentially through a pipe, i t is assumed to enter througha circular s t r ip o f width d; on the cy l indrical w al l . The

rad ia l ve loc i ty o f the incoming f lu id i s chosen to be

Retotive Dimensions

d~ID =0 2B

do/D = 0 3t

dJ D = 0163

hiD=Ok

HID=O 5

LID = 50

ct, 5.65

r_ _ _ _ u-Nd. b-

Under low

F i g u r e I S c h e m a t i c r e p r e s e n t a t io n o f t h e h y d r o c y c l o n e o fK n o w l e s e t a L 6 i n c l u d i n g r e l a t i v e d i m e n s i o n s

cons i s ten t wi th the spec i f i ed vo lumet r ic feed ra te , andi ts tangent ia l veloci ty is taken to be some fract ion (3 ' )o f the m ean ax ia l ve loc i ty in the in let p ipe ( to a l lowfor v i scous IossesJ ) .

F o l l o w i n g P e r i c l e o u s a n d R h o d e s 3 a n d P e r i c l e o u se ta l . , ~ ° we assu m e tha t the m ot ion o f the f luid par ti c lemix ture i s s t eady , incom press ib le , and sa t i s f i es the Na-v ie r-S to kes equa t ions wi th va r iab le v i scos i ty / zc rr ( tomod e l the e ffec t o f tu rbu len ce) and de ns i ty p ( to ac -count fo r pa r t i c le concen t ra t ion e ffec t s ) . Par t i c les o fe a c h g i v e n s i z e a r e c o n s i d e r e d a s a c o n t i n u o u s p h a s emoving re la t ive to the f lu id mix ture . The mode l equa-

t ions govern ing ax ia l , r ad ia l , and t angen t ia l ve loc i t i e s( u , v , w, respec t ive ly ) , the p ressure p , and par t i c leconcen t ra t ion c fo l low:

con t inu i ty :

l 0- (p U ) a x + ( p v ) = ( I )

m o m e n t u m :

O ( p u s h ) + 1 a7 g . , p v 6 )

a 2 )x tXerr + r Or

1 2 0 A p p l . M a t h . M o d e l l in g 1 9 8 8 Vo l. 1 2 A p r i l

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Num er i ca l c a l cu l a ti ons o f f low in a hyd rocyc lone : Ma lco lm R Dav idson

for ~b = u , v , W ( w h e r e W = r w i s t h e c i r c u l a t i o n a n d/ g W i s a n g u l a r m o m e n t u m ) , w h e r e

pS,, = - - - (3a)

Ox

Op w uS ~ = - + / 9 - - - / x ~ f r - g ( 3b )

Or r r ~

S w = 0 (3c)

p a r t i c l e c o n c e n t r a t i o n :

_ 1 OO (pUpC) + - - - ( rpvpc )

Ox r Or

°r)9 ac 1__0 rr~f r (4)= ~xx Fefr + r Orwh ere (up, vp) a re the (x , r ) co m po ne n t s o f pa r t i c l ev e l o c i t y, F~rr//9s t h e e f f e c t i v e t u r b u l e n t d i f f u s i o n co e f -f i c i e n t , a n d a x i a l s y m m e t r y ( o /a o = 0 ) i s a s s u m e d . A tm e s h p o i n t s a d j a c e n t t o a s o li d b o u n d a r y, w a l l f r i c t io n

m u s t b e i n c l u d e d i n e q u a t i o n ( 2 ). A d d i t i o n a l t e r m s i nt h e s o u r c e e q u a t i o n s ( 3 a ) a n d ( 3 b ) a s s o c i a t e d w i t h av a r y i n g / Z , rf ( a n d w h i c h b e c o m e z e r o w h e n / z e r r i s co n -s t a n t ) a r e f o u n d t o h a v e a n e g l i g i b l e e f f e c t a n d a r eo m i t t e d . F o r c o n v e n i e n c e w e n o w r e s t r ic t o u r a tt e n t i o nt o l o w p a r t i c l e c o n c e n t r a t i o n s s o t h a t / 9 c a n b e t a k e na s u n i f o r m , a n d p a r ti c l e t r a n s p o r t u n c o u p l e s f r o m t h eu n d e r l y i n g fl u id fl o w w h i c h c a n n o w b e d e t e r m i n e di n d e p e n d e n t l y. A l s o w e s e t F ~ r r = / ~ fr s i n c e t h e y a r eo f t h e s a m e o r d e r ( s e e , e . g . , L a u n d e r a n d S p a l d i n g I I,p. 28).

I f /z~r r i s r ep lace d b y the l am inar v i s cos i ty / zt , thee q u a t i o n s f o r u a n d v ( b u t n o t W ) a r e i d e n t i c a l t o t h ec o r r e s p o n d i n g l a m i n a r fl o w e q u a t i o n s . P e r ic l e o u s a n dR h o d e s 3 d o n o t d i s c u s s t h e i r m o d e l e q u a t i o n f o r W.Aw ay f rom so l id boundar ies , i t appears to be tha t above ,w h i c h i m p l i e s t h a t a n g u l a r m o m e n t u m i s t r a n s p o r t e da t in te r io r po in t s l ike a pass ive sca la r s inceS w = O ,u n l i ke a n e q u a t i o n b a s e d o n t h e c o r r e s p o n d i n g l a m i n a rf o r m f o r w h i c h S w = - ( 2 / r ) ( a / O r ) (/X~rrW). T hi s ise q u i v a l e n t to a s s u m i n g t h a t t h e s h e a r s t r es s c o m p o n e n tZrOd u e t o t a n g e n t i a l m o t i o n i s g i v e n b y

w~ m = / X ~ f r +

r a t h e r th a n t h e c o m m o n l y u s e d e x p r e s s i o n

: o w w

(5a)

(5b)

w h i c h i s b a s e d o n t h e c o r r e s p o n d i n g l a m i n a r f o r m . T h i sc a n b e s e e n m o r e c l e a r l y b y w r i t i n g t h e e q u a t i o n f o ra n g u l a r m o m e n t u m i n t e r m s o f c o m p o n e n t s o f th e s t r es st e n s o r r :

I 0 r v W )x p u W ) + 7 ~ /9

= r ( O ~ -r )z0.,.) + Ir_ (r 2v °) (6)

L i l l e y a n d C h i g i e r, 12 f o r e x a m p l e , g i v e t h e r e l e v a n t

e q u a t i o n f o r w, a n d a s i m p l e r e a r r a n g e m e n t y i e l d sequat ion (6) for W. Set t ing ~ox =I.~eff(Ow/Ox)and 1 reaccord ing to e i the r (5a ) o r (5b) r educes equa t ion (6 ) toa f o r m c o r r e s p o n d i n g t o e q u a t i o n ( 2 ) f o r tb = W, w i t hS w = 0 o r Sw = - (2 / r ) (O/Or)( I .~efrW),r e s p e c t i v e l y.E q u a t i o n ( 5 a) d e r iv e s f r o m P r a n d t l ' s m o m e n t u m t r an s -p o r t t h e o r y i n w h i c h t h e a n g u l a r m o m e n t u m o f fl u idp a r t i c l e s d i s p l a c e d p e r p e n d i c u l a r t o c i r c u l a r s t r e a m -l i n e s is a s s u m e d t o r e m a i n c o n s t a n t ( W a t te n d o rf ~ 3 ); th ed e r i v a t i o n i s a n a l o g o u s t o t h a t f o r u n i d i r e c t i o n a l f l o w(see, e .g . , Da vie s , ~4 p. 14) .

I f t h e e q u a t i o n f o r W is b a s e d o n e q u a t i o n ( 5b ), t h em o m e n t u m e q u a t i o n s f o r a ll v e lo c i t y c o m p o n e n t s h a v et h e s a m e f o r m a s t h e c o r r e s p o n d i n g l a m i n a r f l o w e q u a -t ions , modi f i ed by an i so t rop ic tu rbu len t v i scos i ty. Suchi s o t r o p i c t u r b u l e n c e c l o s u r e s b e c o m e l e s s v a li d in n o n -rec i rcu la t in g swi r l ing f low (e .g . , L i l l ey an d Chig ie r, ~2Li l l ey I~) a s th e l eve l o f swi r l inc reas es , and a re fou ndt o b e u n s u i t a b l e f o r t h e s t r o n g l y s w i r l in g r e c i r c u l a ti n gf l o w s f o u n d i n c y c l o n e s e p a r a t o r s ( Ay e r set a l .16) . In-d e e d , i t i s o u r e x p e r i e n c e t h a t s u c h a m o d e l p r e d i c t sa ta n g e n t i a l v e l o c i t y w h i c h i n c r e a s e s w i t h r ad i u s a c r o s st h e e n t i r e w i d t h o f t h e c y c l o n e , r a t h e r t h a n i n c r e a s i n go n l y n e a r t h e a x i s o f s y m m e t r y b e f o r e d e c r ea s i n gt o w a r d s t h e p e r i p h e r y, a s i s o b s e r v e d e x p e r i m e n t a l l y.I n v i e w o f t h i s q u a l i t a ti v e d i f f e r e n c e b e t w e e n t h e o r e t -i c al a n d e x p e r i m e n t a l b e h a v i o u r ~, a n d t h e s u c c e s sa c h i e v e d b y P e r i c l e ou s a n d R h o d e s u s i n g a m o d e l b a s e do n e q u a t i o n ( 5 a) , t h e r e s e e m s l i tt le p o i n t i n c o n s i d e r i n ga n a p p r o a c h b a s e d o n e q u a t i o n ( 5 b ) w i t h a n i s o t r o p i ct u r b u l e n t v i s c o s i t y. A h i g h e r - o r d e r t u r b u l e n c e c l o s u r eh a s a l s o b e e n s u c c e s s f u l l y u s e d t o m o d e l f l o w i n g a scyc lon es , 16 17 bu t i t i s no t con s id e red he re .

T h e P e r i c l e o u s - R h o d e s f o r m f o r /z e fr w a s b a s e d o na g e n e r a l i s a t i o n o f P r a n d t l ' s e x p r e s s i o n f o r / z e r r, c o r -r e s p o n d i n g t o u n i d i r e c t i o n a l f l o w, i n w h i c h t h e v e l o c i t yg r a d i e n t w a s r e p l a c e d b ya w / a r - w / r. H e r e w e u s e ar e p r e s e n t a t i o n c o n s i s t e n t w i t h t h e P r a n d t l t h e o r y a p -p l i e d t o a n g u l a r m o m e n t u m , o n w h i c h t h e m o d e l W-e q u a t i o n i s b a s e d , n a m e l y,

~ rr = I~i + pA z + (7)

w h e r e ~ i m o d e l s t h e t u r b u l e n c e e n t e r i n g t h e h y d r o -c y c l o n e ( s i n ceO w / a r + w / r = ( l / r ) ( O W / O r )i s smal l nea rthe in le t w her e th e ax ia l ve lo c i ty i s low) , and ,~ rep-

r e s e n t s a m i x i n g l e n g t h . H e r e w e h a v e i g n o r e d t h e( m u c h s m a l l e r ) u n d e r l y i n g l a m i n a r v i s c o s i t y / ~ t in e q u a -t i o n (7 ). T h e s a m e p ,~fr s u s e d i n e a c h m o d e l m o m e n t u me q u a t i o n . We s e t

la, = O.04pwiAi (8)

w h e r e t h e s u b s c r i p t i d e n o t e s a n i n l e t v a l u e . E q u a t i o n(8) is based on/zz =pA~w ( L a u n d e r a n d S p a l d i n g , jl p .25) wi th tu r bu l enc e ve l oc i ty w ' = 0 .04w,. fo r ax ia l f lowin a p ipe (Dav ies , j4 p . 28). F o l low ing Per ic le ous andRh od es , w e se t A; = 0 .1dg and A = /3d , wh ere d = Di n t h e c y c l o n e a n d d =d,, i n th e v o r t e x f i n d e r w i t h t h ef r a c t i o n /3 t o b e a d j u s t e d to g i v e b e s t a g r e e m e n t

b e t w e e n t h e c a l c u l a t e d a n d e x p e r i m e n t a l v e l o c i t y d i s -

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t r ibut ions. No at tempt is made here to account for thedifferen tial effect of sw irl on m ixing length, m m Sta-bi l i ty considerat ions indicate that turbulent mixing isinhibited or enhanced, depending on whether the an-gular momentum increases or decreases w ith radius.In hydrocyclones, the observed tangent ial veloci t iesare such that the angular mom entum increases w ithradius. 2

The veloci ty, re lat ive to the f luid, of par t ic les (as-sumed to be spherical) w ith d ens ity pp > p and diameterdp can be expressed as 3

Ure, Up U ; ~ I )g ~ D D ) 1/2. . . . (9a)

Vr~l = Vp -- V = -- 1 rC o ] (9b)

where g is the gravitational acceleration, C o is the dragcoefficient, and it is assumed that the centrifugal orgravitational force on a particle (modified by bouy-

ancy) is in equilibrium with the drag force. The dragcoefficient/particle Reynolds number relationship (see,e.g., Svarovsky 2) is approximated by

C o = 24/R Rep-< 57 (laminar, i.e., Stokes law)= 0.42 Rep -< 57 (tu rbulent) (10)

where

Rep = p UZrel + v~el)Z/2dp//xt

is the particle Reynolds number. To ensure continuityof the C D function, the Re, ranges of equation (10) arechanged slightly from those used by Pericleous andRhodes.

u m e r i c a l c o n s i d e r a t i o n s

We solve equations (1), (2), and (independently) (4)numerically on the nonuniform mesh shown in F i g u r e2. Note that the radial mesh spacing shown adjacentto the axis is further subdivided four times in an effortto resolve the rapid change in tangential velocity fromzero on the axis of symmetry to the peak values ob-served by Knowles e t a l . 6 immediately adjacent. (Thismatter will be discussed later in more detail.) VariablesW, p, and c are defined at the centre of each cell, andnormal velocities at cell walls are defined at their mid-points. Following the PHOENICS approach, the dis-cretised momen tum equations of variables whose con-trol volumes are intersected by the cone boundary areadjusted to accou nt for the fraction of the cell boundary

Malcolm R Davidson

and cell volume which is occupied by fluid. The con-tinuity equation is treated in the same way.

As discussed earlier, the radial and tangential ve-locities of fluid entering the computati onal domain arebased on the assumption of axial symmetry and takevalues consistent with the known volumetric feed rate(Q) (Here Q = 28.4 L/m , and we ignore viscous lossesat entry by setting 3' = 1). At the underf low and over-flow planes W is taken equal to its upstream value toapproximate (ass umed) zero-gradient conditions there.In the experiment of Knowles e t a / . , 80 of the in-coming flow passes to the overflow; we choose u niformaxial velocities at the underflow and overflow con-sistent with this flow split. The alternative of settingthe underflow and overflow pressures according to a p / a r= p w 2 / r and predicting the flow split, as was done byPericleous and Rhodes, is not considered here, sincethe absence of an air core results in an uncertain inter-pretation of the given pressure data at the exits. Fora rotating liquid core, radial (and axial) pressure gra-dients are much larger than for an air core, and thespecification by Knowles e t a / . of a single pressurevalue at each exit becomes ambiguous.

Flow normal to the boundary of the computationaldomain is set to zero, of course, except at those partsdefining the inlet, underflow, and overflow. FollowingPericleous and Rhodes , wall friction is applied at meshpoints adjacent to the walls of the vortex finder andthe roof the hydrocyclone (i.e., solid boundaries de-fined by c oordinat e lines) according to the familiar log-arithmic wall law; and at points adjacent to the coneboundary by adding a momentum sink to equation (2)given by

• 2~1/2mF 4 j = ½ f p u 2 + V 2 + w p - - ~ C ~ 11 )

where f is the friction factor, A is the cone surfacearea inters ectin g the cont rol volume V, and ~b = u , v, W.We choose f = 0.006 corresp onding to a value forsmoo th pipes (Davies, ~4 p. 33) when the Reynolds num-ber has the same orde r (104) as that in the Knowleshydrocyclone.

The region near the axis of symme try presents somedifficulty. Near the axis the tangential velocity distri-bution is expected to have the form corresponding tosolid -body rota tion (w = r and W ~ r 2) since w = 0 onr = 0. Such behaviour is typically observed in hydro-cyclones operating with an air core. However, whenthe air core is absent, this region of solid-body rotationappears t o be extre mely close to the axis since it is notevident in the data of Knowles e t a l 6 Despite mesh

AXIS OF SYMMETRYFigure N o n u n i f o r m c o m p u t a t i o n a l m e s h u s e d , b u t w i t h t h e r a d ia l m e s h s p a c in g s h o w n a d j a c e n t t o t h e a x is o f s y m m e t r y s u b d i v i d e d

a n a d d i t io n a l f o u r t i m e s . F o l l o w i n g P e r ic l e o u s a n d R h o d e s t h e c o m p u t a t i o n a l d o m a i n h a s b e e n e x t e n d e d a l o n g t h e v o r t e x f in d e r

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refinement near the axis, the expected behaviour therecould not be achieved without some additional con-straint on the solution for W other than that of impliedsymmetry. Indeed, the assumption (that fluid particlesin turbulent motion retain their angular momentumduring radial translation) underlying the model equa-tion for W breaks down near the axis of symmetry,since perturbed as well as mean values of angular mo-mentum are zero on r = 0 (otherwise the correspondi ngtangential velocities are infinite), whereas at a radiuscorrespon ding to the mixing length they are in generalnonzero. Ho wever, in contrast with the present work,Pericleous and Rhodes 3 do no t report any difficultynear the axis of symmetry or the need to apply specialconditions there. Presumably this is associated withthe presence of an air core in which air with low angularmomentum enters the hydrocyclone along the axis atthe underflow and overflow. Such counter- current flowat the underflow and overflow planes is not possiblein our model owing to the assumption of uniform ve-locity profiles at each exit; however there is no evi-dence in the data of Knowles e t a l . 6of downwards flowalong the axis of symmetry counter to the upwardsmoving overflow stream, which can occur under someconditions (Dabir and Petty2°). Furthermore, settingthe pressure at the underflow (to allow for the possi-bility of both inflow and outflow there), while retaininga uniform overflow velocity profile, results in a neg-ligible change in the predicted flow pattern.

In this paper we assume that the rate of strain

OWor Wr- rl (0~rW 2W)

is zero (consistent with solid-body rotation) near theaxis. In particular, we set

O W 2 W= 0 at r = rc (12)

Or r

where rc is a small nonzero radius. The width of cellsnearest the axis is set equal to re; equation (12) is thencomputationally equivalent, when linear interpolationis used, to setting W = 0 at nodes adjacent to the axis(i.e., at r -- ½r,.). (A tangential velocity which falls tozero before reaching the axis has been predicted forair cyclones using a higher-order turbulence model, t6.J7)We expect r,. to have the same order of magnitude as

the mixing length, since, roughly speaking, fluid par-ticles in the region r < r,. can cros s the axis o f sym-metry, whereas those elsewhere cannot. As was in-dicated earlier, the cells in F i g u r e 2 immediately adjacentto the axis are subdivided to give four computationalpoints within the experimentally unresolved region; inthat case 2 r , / D = 0.014. Doubl ing r,. (i.e ., a near-axialsubdivision by two rather than four) is found to havea negligible effect on the calculated flow velocities ex-cept for a reducti on of about 8% in the peak tangentialvelocity, and a consequent reduction of about 20% inthe pressure drop between the inlet and the axis ofsymmetry. The large variation in the pressure drop

occurs because the peak w value lies very near the axis

and Op/Or ~ pw2 / r.An alternative, more fundamental(e.g., second moment) turbulence model, which doesnot require the modelling parameter r,. may give betterresults near the axis; however, the predicted pressuredrop will still be sensitive to errors in the rapidly vary-ing w profile there.

The methods on which our calculations are basedare embodied in the computer programs TEACH andPHOENICS and are descr ibed in detail by Patankar. 2~Briefly, the governing flow equations are integratedover control volumes, and the resulting differenceequations are solved for velocity and pressure usingthe SIM PLE algorithm of Patank ar and Spalding. 8 Thediscretised momentum equations are first solved for agiven pressure field, which is then adjusted to ensuremass balance within cells. The process is repeated untilresidual errors in the momentu m and continuity equa-tions are s ufficiently small. False transient techn iquesare used here, incorporating recommendations ofRaith by and Schneider. 22 In par ticular, we set theirtime step factor E = 0.01 and choose the pressurerelaxation parameter according to their equation (30).

In this paper best agreement with the experimentalvelocity and pressure drop data of Knowles e t a l . 6isobtained by choosing the mixing length fraction/3 =0.005. Correspond ing comparisons with those data arediscussed in the next section. Increasing/3 increasesthe mixing length and predicts a flatter tangential ve-locity distribution across the hydr ocyclo ne with lowerpeak values, because of an increased flux of angularmoment um across the inner core bound ary at r = rc.Decreasing/3 has the converse effect.

omparison with theory and experimentCalculated profiles of tangential and vertical velocityare compared with the corresponding experimental dataof Knowles e t a l . 6in F i g u r e s 3 a )and 3b) . On theo-retical grounds the tangential velocity must initiallyincrease from zero away from the axis, and our axialboundary condition is chosen to ensure that such be-havior is predicted, even though the data does notresolve this near-axial region. Across most of the hy-drocy clone the tangential velocity decreases, with pre-dicted and experimental values in satisfactory agree-ment. Furth er, the general form of the predicted profile(increasing away from the axis before decreasing towards

the periphery) is consistent with that observed in hy-drocyclones operating with and without an air core(Kelsall 5 Dabir and Pett y 2°) and in air cyclones (Ayerset al . 16).

Knowles e t a l . obtained vertical velocity profilesfrom data averaged within the field of view, showingthe expected downward flow along the hydrocyclonewall and upward flow along the axis. These data areplotted in F i g u r e 3 b )and lie satisfactorily close to ourpredicted curves. The exception is the profile imme-diately below the vortex finder; we do not plot theview-averaged data there, since Knowles e t a l . useother data to obtain the velocity profile. This profile

is unlike the calculated shape, showing a striking un-

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Numerical calculations of flow in a hydrocyclone: Malcolm R Davidson

~ -9s

3.0 1.o

10 . -1.0L0 0 ~

750 0 0t~ 08 00 0t~ 08

2 r i D 2 r i D(o) (bl

F i g u r e C a l c u l a t e d p r o f i l e s ( s o l i d l i n e s ) o f t a n g e n t i a l ( a ) a n d

v e r ti c a l (b ) v e l o c i t y c o m p a r e d w i t h c o r r e s p o n d i n g e x p e r i m e n t a ld a t a o f K n o w l e se t a L 6 a t d i ff e r e n t h e i g h t s . I n ( b ), c o m p a r i s o ni s a l s o m a d e w i t h t h e o r e t i c a l v e r t i c a l v e l o c i t i e s ( d a s h e d l in e s )d e r i v e d f r o m t h e a n a l y s is o f B l o o r a n d I n g h a m s

dulation near the axis (interpreted in terms of a short-circuit flow direct from inlet to overflow). In deed, wesee from both the tangential and vertical velocity pro-files in Figure 3 that the model loses accuracy of de-scription near the vortex finder. This is not surprisingsince our assump tion of axial symmet ry is not valid inthe inlet region.

In Figures (3b) and (4) the vertical velocity profiles

are compared with theoretical predictions derived fromthe analysis of Bloor and Ingham. 9 These authors as-sumed that the flow in the (r, x) plane is inviscid witha prescribed vorticity distribution of fairly general form.The approximate expression

u = ½ Br- n (3a , . - 5r/z) (13)

for the vertical velocity was derived for hydrocy cloneswith small cone angle 2a,. and negligible underflowrate, where z is the height above the cone vertex (dif-fering from x, which is the height above the under flowplane) and B is a constant depending on the volumetricflow rate Q. In this paper the underflow rate is not

negligible; nevertheless, provided streamlines inter-secting the underflow lie close to the hydrocyclonewall, we still expect equation (13) to apply when Q isreplaced by the overf low rate (0.8Q here) ., This isdemonstrated in Figures (3b) and (4), in which thenumerical and analytical predictions agree well exceptin a region near the axis (equation (13) implies an in-finite velocity on the axis). This agreement with theanalysis of Bloor and Ingham confirms that turbulentmixing plays little role in the transfer of axial or radialmomentu m in the interior of the hydro cyclon e (in con-trast to its importance in the transport of angular mo-mentum) except, perhaps, near the axis. Indeed, nu-

merical predictions of u at the axial positions shown

in F i g u r e 3 b vary little with a fourfold change in mixinglength, except near the axis immediately below thevortex finder. Similar observations apply deeper withinthe hydrocyclone; the tt variation with mixing lengthbeing greatest in the upflowing fluid near the axis, butwith little variation on the axis itself.

Calculated values of/zerr are greatest adjacent to theaxis o f symmetry with values of 2lzerr/pwiDfalling rap-idly from a centra l value at r = r,. of about 2.2 x 10 -zto about 0.3 x 10 -2 at 2 r / D = 0.2 and thereafter de-creasing slowly toward s the inlet value of 0.2 x 10 -2(of course, very close to the wall it falls rapidly towardsthe laminar value). However, the turbulent viscosityshould decrease as r---~ 0 in the region of solid-bodyrotation (r < r,.), since from stability arguments weexpect radial mixing there to be suppressed relative tothat in the main body of the flow where angular mo-mentum increases less rapidly with radius. Further-more, radial stabilisation has been obse rved at the coreof vortex tubes (Escudier e t al.23 .In our calculations,we do not choose a reduced value of ~cfr on the axis;however, this will have little effect on the results, sincethe region r < r,. lies outs ide the calculation domain ofw and we have already shown that predicted values ofu in this region are not sensitive to the choice of ~crrfor r,. small. In cases for which r,. is not small, correctprediction of the flow in the core region is expected torequire proper modelling of the turbulent mixing there.

An assumption of Bioor and Ingham s model is thatthe tangential velocity (w) is independent of height x.Although inspection of F i g u r e 3 ( a ) would suggest thatthis is true of the numerical results, Figure 5 revealsthat, for 2 x / D < 7.0, the predicted tangential velocitydecreases towards the hydrocyclone wall more rapidlyas the underflow is approached. A fall in w, as the wallis approached along lines of constant radius, occursbecause of wall friction, and indeed the effect is lessif the friction factor f is reduced. Such beh aviour closeto the wall has been reported by Kelsall. 5 The degreeof axial nonun iformi ty in w which occurs in the presentcase is in contrast to the almost height-independent wdistribution in the Kelsall hydrocyclone, and is prob-

I J I I

0 B

04. ~

~ 0 0

-08- b =~I A I I

O 0 0 3 0 6 0 9

2 r i d

F i g u r e 4 P r o f il e s o f c a l c u l a t e d v e r t i c a l v e l o c i t y ( s o l id l in e s )c o m p a r e d w i t h c o r r e s p o n d i n g p r e d i c ti o n s ( d a s h e d l in e s ) b a s e do n t h e a n a l y s i s o f B l o o r a n d I n g h a m 9 a t a r a n g e o f h e i g h t s i n -

c l u d in g t h o s e f o r w h i c h d a t a a re u n a v a i l a b l e

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30 1 I I I

2 5 -

2 0

15

1.0

0.5

O 00 0 0 3 0 6 0 9

2 r i d

Figure P r o f i l e s o f c a l c u l a t e d t a n g e n t i a l v e l o c i t y a t a r a n g e o fh e i g h t s i n c l u d i n g t h o s e f o r w h i c h d a t a a r e u n a v a i l a b l e

25.0

200

15 0

5 0

0.0. I I I0 0 3 0 6 0 9

2 r i D

Figure R a d i a l p r e s s u r e v a r i a t i o n a c r o s s th e h y d r o c y c l o n e

ably the consequence of the smaller cone angle (lessspin-up, due to conservation of angular momentum, tooffset the effect of wall friction) and greater height ofthe Knowles hydrocyclone, together with the lowerfeed velocity (thicker wall boundary layer). The pre-dicted x dependence of w is expected to contribute tothe discre pancy bet ween numerical and analytical pre-dictions of u F igu re 4 ) away from the axis of sym-metry.

As discussed earlier, and we can see from F i g u r e6, radial pressure gradients in a region about the axisare high in the absence of an air core, and it is notclear to which radius the underfl ow and overflow pres-

sure values given by Knowles et a l . should be assigned.If we assume that they are the pressure values on theaxis of symmetry , then the dimensionless pressure drops(p, - p, , ) /pw~and pi - po) /pw~have calculated valuesof 5.7 and 25.2, respectively, compared with corre-sponding experimental values of 5.5 and 20.1. This

agreement is acceptable in view of the uncertainty sur-rounding the experimental p , and po values. We findlittle axial variation in the pressure, with the radialpressure distribution at positions below the vortex finderlying close to that shown in F i g u r e 6 . The greatestvariation in this direction occurs along the axis itself,with most of the pressure drop betw een underflow andoverflow occurring in the vortex finder.

Other results

Streamlines defined by

I a ~ I a ¢- v . . . . 1 4 )

r ar I ax

are shown in Figure 7 where the stream f unction ¢, onthe axis of symmetry is taken to be zero. In equation(14) velocities normal to cell faces intersected by thecone boundary (u only in this case)are multiplied bythe face fraction occupied by fluid; by analogy with aporous medium, the velocities used in equation (14)are Darcy velocities, not pore velocities. The slightwriggle in the streamlines close to the cone boundaryresults from imperfect modelling of this nonrectangu larboundary by the rectangular grid. The flow patternsexhibit the expected d eeply penetrating reversing flowcharacteristic of hydrocyc lones, with fluid flowing downalong the outer wall and up along the axis of symmetr y.Streamlines which intersect the underflow lie very closeto the cone boundary and are not shown. As in thePericleous-Rhodes simulation of the Kelsall experi-ment, the flow pattern contains a region of recircula-tion (closed streamlines) lying between the downflow-ing and upflowing streams.

A peculiarity of the predicted flow pattern is theabsence of any short-circuit flow from the inlet, acrossthe roof of the hydrocyclone, to the vortex finder.Physically, such a flow occurs in the top boundarylayer because the tangential velocity there is smallerthan that of the fluid below. This results in an outwardcentrifugal force on the boundary layer fluid which isless than the opp osing force of the radial pressure field,and so inwards radial flow occurs. Short-circuit flowdoes not occur in this simulation because we do notresolve the top boundary layer, relying instead on thelogarithmic wall law to determine the near-wall veloc-

ity. Furthermore, inspection of the predicted velocitydata reveals little axial varia tion in w in the inlet regionas the top boundary is approached. Calculated valuesof both u and v are small near the outer wall of thevortex finder; the detail of the predicted flow patternthere (s treamlines rising then reversing down along the

Figure 7

0.0 I I I I I T I I00 10 20 30 40 50 0 70 8.0 90 100

2 x / D

C a l c u l a t e d s t r e a m l i n e s i n t h e h y d r o c y c l o n e , w h e r e ~ * = ~ / ( 0. 2 5 D 2 w i) i s t h e d i m e n s i o n l e s s s t r e a m f u n c t i o n

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08 I i i I iF S , - . 1

0A.~ . ,

0 0 £ - - ~ I I

I I

1

I I

C I [ I : 1 0 ~0.S I

I I

0.8

O 0

0 B L . . .. . . - -

_ _ . _ 1 ~ ~ ~ - ~ _ ~0t~ ~

1.0 20 30 40 50 60 70 80 90 100

2 x / D

Figure C o n t o u r s o f c o n s t a n t p a r t ic l e c o n c e n t r a t i o n i n t h e h y d r o c y c l o n e f o r p a r t ic l e s h a v i n g d i a m e t e r s a ) 1 0 m i c r o n s , b ) 2 0 m i c r o n sc ) 3 0 m i c r o n s , a s s u m i n g t h e f e e d c o n c e n t r a t i o n i s l o w. T h e s o l id a n d d a s h e d c u r v e s a r e b a s e d o n t h e H Y B R I D a n d Q U I C K d i f f e r e n c i n

s c h e m e s , r e s p e c t i v e l y

vortex finder) is ther efore o f doubtful significance dueto loss of precision, and is consequently not shown inFigure 7 .

Curves o f constant particle concentration c/cl) areplotted in Figu re 8for particles with diameters d,) 10,20, and 30 microns, where c; is the feed concen trationof particles of given size, and c is assumed to be suf-ficiently small to have a negligible effect on the fluiddensity. Attention is focused on the solid lines in Fig-ure 8; the significance of the dashed curves will bediscussed later. We have set c = c; at upstream inletnodal points jus t ou tside the flow domain, and the par-ticle velocities relative to the fluid U r e l , / . ) r e l ) a r etakento be zero normal to solid boundaries. Furthermore,we have set vr¢~ = 0 on the bo und ary at the inlet, sincewe expect the actual geometry of the inlet region toprevent any backwards drift of particles.

Becau se the radial dr ift velocity Vr¢, is positive forparticles which are denser than w ater, the particle con-centration in Figu re 8 is greater near the periphery ofthe hydrocyclone than it is near the axis. Also, Vre,depends on the centrifugal acceleration w2/r, whereasthe negative) vertical drift veloc ity u~e, depen ds on thegravita tional acele ration g; u~¢~ s thus about three timessmaller than Vr¢~ nea r the in let radius but about 150times smaller near the axis, where it consequently hasa negligible effect on the predicted particle concentra-tion. The very small x variation in concentration nearthe axis is a reflection of similarly small variations inw Figure 5) ,and hence / - ) r e l ,there.

As dp increases , Vr~, also increases, and we can seefrom Figu re 8that the particle concent ration at a givenposition on the hydroc yclon e wall rises it also falls onthe axis); hence the fraction of particles passing throughthe underflow increases, as expected. The correspond-ing efficiency curve reflecting this behavi our is shownin Figure 9 .Unfor tunatel y, no efficiency data are avail-able for comparison in Figure 9 .At very low dp values

there is little buildup of particles along the wall and

10

z

~06

J0k

0zz

0.01 3 EU

I I I I

10 20 30 t,0PARTICLE DIAHETER HICRONS)

Figure 9 F r a c t i o n o f p a r ti c l e s e n t e r i n g t h e h y d r o c y c l o n e w h i c ha p p e a r a t t h e u n d e r f l o w a s a f u n c t i o n o f p a r t i c l e si z e( e f f i c i e n c yc u r v e )

c ~ c,. every wher e, so that the particle split betweenunderflow and overflow is approximately equal to theflow split. As dp and the concentration along the wallincrease, the concentration contours necessarily be-come more closely spaced.

T h e e f f ec t o f n u m e r i c a l d i f f u s io n

Our calculations thus far have been based on a hybriddifferencing scheme HYBRID), wi dely used to modelhighly convective flows, in which first-order upwinddifferencing is applied to stabilise the calculation, ex-cept when the cell Reynolds number is less than 2, inwhich case central differencing is used. However, theaccuracy of first-order upwind differencing is limited,and artificial numerical diffusion errors can be impor-tant. Such errors may be significant in the presence ofsource terms Leonard24), and when the streamlinesare skewed relative to the axes of the numerical gridde Vahl Davis and Mallinson25), as is the case here.

Despite the agreement with experimental data reported

here, the possibility remains that, in the absence of

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numerical diffusion, some other choice of mixing lengthwould be appropriate. Furt hermo re, the validity of ex-trapolated model behaviour must be in doubt unlessthe effects of false diffusion are minimal.

An alternative differencin g scheme (called QUICK ),proposed by Leonard 24 and involving an upstream shiftedquadratic interpolation at each control surface, has beenshown to be more accurate in general than the con-ventional HYBRID scheme for a variety of flows(Les chziner , 26 Han et a1. ,27Les chz ine r and Rodi, 28 Pa-tel and Markatos29), but it converges less readily.

We assess the importance of numerical diffusion inthe present application by solving the governing equa-tions for flow and particle transport using bothHYBRID and QUICK. The exception is the angularmomentum equation for which HYBRID is usedthroughout; when QUICK is used for this equation aswell, conver gence is unacceptab ly slow. To isolate theeffect of numerical diffusion on the solution of theparticle transport equation, we compare correspondingHYBRID and QUICK results based on the same un-derlying (HYBRID) flow calculation. In using QUI CK,we applied a source decomposition of the correspond-ing difference formulae tested by Han e t a l 2 7 (theirequation (8)).

The dashed contour lines of particle concentrationin F i g u r e 8 are based on QUICK differencing. We seethat numerical diffusion has little effect on the pre-dicted concentration pattern except in a region alongthe wall of the hydrocyclone where it results in con-centration s which are too low by about 1-7 . Thisoccurs because the down ward flow along the cone wallnecessarily cuts the x-axis at an angle, whereas theupward flow in the centre is almost parallel to the axis.Near the cone wall, the false diffusivity is calculated(see e.g. Leschziner 26) to be of similar order to thecorresponding turbulent diffusivity. Nevertheless, theeffect o f numerical diffusion there is small. Further -more, the integrated effect of false diffusion on theefficiency curve in F i g u r e 9 is found to be negligible.

Numerical diffusion is found to have little effect onthe predicted vertical velocity distribution. This is ex-pected since diffusion of axial momentum has alreadybeen shown to have a negligible role away from theaxis of symmetry, and hence false diffusion of similarorder will also be unimportant (in this considerationonly the region along the cone wall is relevant). Asdiscussed earlier, angular momentum decreases downalong the hydrocyclone wall; numerical diffusion isthus expected to result in tangential velocities therewhich are too high. However, as for the particle con-centration, the error is expected to be small.

o n c l u s i o n

The model o f Pericleous and Rhodes 3 is used to de-scribe the flow and particle transport in a hydrocy cioneoperating without an air core, and the experiment ofKnowles et a l . 6 is simulated. In this work the modelbreaks down to the axis of symmetry, and a specialbound ary condition (corresponding to zero shear) must

be imposed at a radius of similar magnitude to themixing length to ensur e realistic flow predictions. Ve-locity and pres sure are calculated on a staggered meshusing the SIMPLE algorithm of Patankar and Spald-ing; 8 also hybrid first-order upwind/central differenc-ing and false transient techniques are used, incorpo-rating r ecommend ations of Rait hby an d Schneider. 22

Calculated vertical and tangential flow velocities andpressure drops are shown to fit the corresponding dataof Knowles e t a l . for a suitable choice o f mixing length.Further , calculated vertical velocities are found to agreewith theoretical predictions based on the analysis ofBloor and Ingham, 9 except in a region near the axis.Comparison of results based on both hybrid and qua-dratic upwind differencing is used to assess the im-portance of numerical diffusion. False diffusion is foundto have little effect on the predicted particle concen-tration, except along the wall of the hydrocyclone whereit results in errors o f about 1-7 ; also the effect onthe predicted efficiency curve, and on the calculatedflow in the (r, x) plane, is negligible. A similarly smalleffect on the transport of angular momentum seemslikely.

N o t a t i o n

A

C

CiCoD

di do d.

d~EfgH

hLPpi, po, p, ,

Qr, x)

r{.

R e p

S , , , S , , , S wll.. U, 14

l i p , Up

/ I r c l , / - r e l

surface area of cone boundar y intersectinga control volume

constant in equation (13)particle concentrationfeed concentration of particlesdrag coefficientdiameter of cylindrical section of hydro-

cyclonediameters of inflow, overflow, and under-

flow, respectivelyparticle diametertime-step factor defined in Ref. 22friction factorgravitational accelerationheight of cylindrical section of hydrocy-

clonelength of vortex finderoverall height of hydrocyclonepressurepressure at inflow, overflow, and under-

flow, respectivelyvolumetric feed ratecylindrical coordinates as in F i g u r e Iwidth of computational cells nearest the

axisparticle Reynolds numbermoment um sources defined in equation (3)axial, radial, and tangential fluid veloci-

ties, respectivelyaxial and radial compon ents o f particle ve-

locity, respectivelyaxial and radial velocities, respectively, o f

particles relative to the fluid (drift ve-

locity)

Appl . Math. M odel l ing 1988 Vol. 12 Apri l2 7

8/18/2019 Davidson 1988

10/10

N u m e r i c a l c a lc u l a t io n s o f f lo w i n a h y d r o c y c l o n e : M a l c o l m R D a v i d s o n

VWW ~

c o n t r o l v o l u m e o f t h e c o m p u t a t io n a l g r id= r w w h e r e p W i s a n g u l a r m o m e n t u mturbu len t ve loc i ty f luc tua t ion in the in le t

p ipeh e i g h t a b o v e t h e c o n e v e r t e x

Greek symbolsl¢3

Feet

0A/Zeff, /zi

]./,Ib

P P.T

h a l f- a n g le o f t h e h y d r o c y c l o n emix ing l eng th f rac t ionreduc t ion fac to r o f tangen t i a l ve loc i ty a t

the inle ttu rbu len t exch ang e coeff i c ien t fo r pa r t i c le

t ransportcyl indr ical polar anglemix ing l eng thturbulent viscosi ty in the hydrocyclone and

a t the in le t r e spec t ive lylaminar v i scos i ty= U t 3 o r W

dens i ty o f f lu id and pa r t i c les r espec t ive lytu rbu len t shea r s t r esss t ream func t ion

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14 Dav ies , J . T. Turbulence Phenomena. A c a d e m i c P r e s s , L o n -don, 197215 L i l l ey, D . G . P red i c t i on o f i ne r t t u rbu l e n t sw i rl f l ows .

AIAA J. 1973, 11 7), 95 5-9 601 6 Ay e r s , W. H . , B o y s a n , F. , S w i t h e n b a n k , J . , a n d E w a n ,

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18 Brad shaw , P. E ff ec t s o f s t r eaml ine cu rva tu re on t u rbu l en t f l ow.AG AR Dog raph No . 169 , 1973

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p r o c e d u r e b a s e d o n q u a d r a t ic u p s t r e a m i n t e r p o l a t io n .Compat.Methods Appl. Mech. Eng. 1979, 19, 59-9825 de Vah l Dav i s , G . and Ma l l i n son , G . D . An eva lua t i on o f up -

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128 Ap p l . Ma th . Mo de l l i ng 1988 Vo l . 12 Apr i l