FLUX AND THERMAL BEHAVIOR OF FLUX.pdf

8/10/2019 FLUX AND THERMAL BEHAVIOR OF FLUX.pdf

1/14

F lo w a n d T h e r m a l B e h a v i o r o f th e T o p S u r f a c e F l u x P o w d e r

L a y e rs i n C o n t i n u o u s C a s t i n g Mo l d s

R . M . M c D A V I D a n d B . G . T H O M A S

S t eady- s t a te f i n i t e - e lemen t mod e l s have been f o r m ul a t ed t o i nves t i ga t e the co up l ed f l u i d f l ow and

t he r ma l b ehav i o r o f t he t op- sur f ace f l ux l aye r s i n con t i nuous cas t i ng o f s t ee l s labs. T he t h r ee - d i -

mens i ona l ( 3 - D) F I D AP mode l i nc l udes the shea r s t r e s ses i mpose d on t he f l ux / s tee l in t e r f ace by f l ow

ve l oc i t i e s ca lcu l a t ed i n t he m ol t en s t ee l poo l . I t a l so i nc l udes d i f f e r en t t emper a t u r e - depen den t powd er

pr ope r t i e s f o r so l i d i f i ca t i on and me l t i ng . Good agr eement be t ween t he 3 - D mode l and expe r i ment a l

measur ement s was ob t a i ned . T he shea r f o r ces , i mposed by t he s t ee l su r f ace mot i on t owar d t he

submer ged en t r y nozz l e ( S E N) , c r ea t e a l a r ge r ec i r cu l a t i on zone i n t he l i qu i d f l ux poo l . I t s dep t h

i nc r eases w i t h i nc r eas i ng cas t i ng speed , i nc r eas i ng l i qu i d f lux con duc t i v i t y , and dec r eas i ng f l ux v i s -

cos i t y . F or t yp i ca l cond i t i ons , th i s zone con t a i ns a l mos t 4 kg o f f l ux , whi ch con t r i bu t e s t o an ave r ag e

r es i dence t i me o f abou t 2 mi nu t es . Addi t i ona l l y , because t he shea r f o r ces p r oduced by t he na r r ow f ace

consumpt i on and t he s t ee l f l ow oppose each o t he r , t he f l ow i n t he l i qu i d f l ux l aye r s epa r a t e s a t a

l oca t i on cen t e r ed 200 mm f r om t he na r r owf ace wa l l . T h i s f l ow sepa r a t i on dep l e t e s t he l i qu i d f l ux

poo l a t t h i s l oca t i on and ma y con t r i bu t e to gen e r i ca l l y poo r f eed i ng o f t he mol d- s t r and gap t he r e .

As a f u r t he r consequ ence , a r e l a t i ve l y co l d spo t deve l ops a t t he w i de f ace m ol d wa l l nea r t he s epa -

r a t i on po i n t . T h i s nonuni f o r mi t y i n t he t emper a t u r e d i s t r i bu t i on may r e su l t i n nonuni f o r m hea t r e -

mova l , and poss i b l y nonuni f o r m i n i t i a l she l l g r owt h i n t he meni scus r eg i on a l ong t he w i de f ace

of f - cor ne r r eg i on . I n t h i s way , p o t en t i a l s tee l qua l i t y p r ob l em s m ay be l i nked t o f l ow i n t he l i qu id

f lux pool .

I .

I N T R O D U T I O N

I N t he con t i nuous cas t i ng p r oces s , mol d f l ux i s added a s

a pow der o n t o t he su r f ace o f the l i qu i d s t ee l . T he f l ux

absor bs hea t f r om t he mol t en s t ee l , s i n t e r s , and me l t s i n t o

a poo l o f li qu i d , benea t h t he f l oa ti ng pow der . Be l ow t he

l i qu i d f l ux l aye r , s t ee l en t e r s t he mol d cav i t y t h r ough t he

submer ged en t r y nozz l e ( S E N) , r ec i r cu l a t e s up t he na r r ow-

f ace wa l l and a l ong t he t op su r f ace , and f l ows back t owar d

t he S E N ( F i gur e 1 ) . T he t op- sur f ace f l ux l aye r s p r ov i de

t he r ma l and chemi ca l i nsu l a t ion o f the t op su r f ace o f t he

l i qu i d s t ee l and a i d i n t he r em ova l o f nonmet a l l i c i nc lu -

s i ons . T he l i qu i d f l ux t hen i n f i l t r a t e s t he gap be t ween t he

mo l d w a l l and t he so l i d i f y i ng s t eel s t rand , wh er e i t ac ts a s

a l ubr i can t and p r om ot es s l ow , un i f o r m hea t t r ans f e r i n t he

mol d- s t r and gap . Co l l ec t i ve l y , these f unc t i ons wor k t o r e -

duce such p r ob l ems as su r f ace and subsur f ace i nc l us i ons ,

nonuni f o r m she l l g r owt h and b r eakou t s , su r f ace depr es -

s ions , and cracks .

G i ven t he i mpac t t ha t t he mol d f l ux has on s t ee l s l ab

qua l i t y , many p r ev i ous s t ud i e s have i nves t i ga t ed va r i ous

aspec t s o f powder / f l ux behav i o r . M os t have been ex pe r i -

ment a l o r i ndus t ri a l. F o r exampl e , s t ud i e s pe r f o r me d a t N i p-

pon S teel (Kim i tsu, Japan ) I~,21 and Inland S teel (East

Chica go, IN) ~31 hav e c onf i rm ed an inverse re la t ionship be-

t ween op t i mum l i qu i d f l ux v i scos i t y and cas t i ng speed t o

ach i eve un i f o r m hea t f l ow and e f f ec t i ve l ubr i ca t i on i n t he

m o l d - s t a n d g a p . O t h e r w o r k b y S a r d e m a n n a n d S c h r e w e~41

R . M . M c D A V I D G r a d u a te R e s e ar c h A s s i s t an t a n d B . G . T H O M A S

A s s o c i a t e P r o f e s s o r o f M e c h a n i c a l E n g i n e e r i n g a r e w i t h t h e D e p a r t m e n t

o f M e c h a n i c a l a n d I n d u s t r i a l E n g i n e e r i n g U n i v e r s i t y o f I l l in o i s a t

U r b a n a ~ h a m p a i g n U r b an a I L 6 1 80 1 .

M a n u s c r i p t s u b m i t t e d J u l y 2 5 1 9 95 .

has show n t he i mpor t ance o f ma i n t a i n i ng a su f f i c i en t ly

thick l iquid s lag layer . Speci f ical ly, they fo und that the for -

ma t i on o f c r acks i n t he s t ee l s l ab i s r educed by a t h i ck

l iquid f lux layer . Suff ic ient ly thick l iquid s lag layers are

a l so im p o r t a n t t o p r e v e n t c a r b o n p i c k u p b y th e st e el

f r om t he f l ux , a s r epor t ed by Naka t o e t a l t S ]

S ever a l r e sea r che r s have cha r ac t e r i zed t he f o r ma t i on o f

t he l iqu i d fl ux laye r us i ng the empi r i ca l p r ope r t y o f me l t -

ing ra te . t6 .v.s j Fo r ex am ple, BranionI71 com me nts that the

mel t i ng r a t e de t e r m i nes t he ab i l i ty o f t he f l ux t o ma i n t a i n

a s t ab l e l i qu id l aye r . Us i ng an appa r a t us de ve l ope d by L i n-

defel t and Hassels t rom,tgl several invest igators have meas-

ur ed t he r a t e a t whi ch m ol t en f l ux is p r oduced . T h i s me l t i ng

r a t e dec r eases w i t h i nc r eas ing ca r bon p r esen t i n t he f l ux , a s

dem on stra ted by Xie. E6J

F l ux behav i o r has a l so been i nves t i ga t ed us i ng ma t he -

ma t i ca l mode l s . M os t p r ev i ous mode l s a t t empt t o cha r ac -

ter ize f lux behavior in the thin mold-s t rand gap. t~o-~n

Bom ma r a j u and S aad ['~ mo de l ed f l ux f l ow as Co ue t t e f l ow

be t ween t he s t a t i ona r y coppe r mol d a nd t he s t ee l she ll mov-

ing a t the cas t ing speed. Anzai e t a L [ ~ 4 1 go a s t ep f u r t he r by

so l v i ng t he 1 - D Navi e r - S t o kes equa t i on whi ch i nc l udes t he

pr es sur e t e r m neg l ec t ed i n t he wor k by Bommar a j u . Anza i

p r ed i c t ed t ha t t he f l ow r a te o f f l ux i nc r eases a s v i s cos i t y i s

dec r eased an d t ha t t he t r ends i n f l ow r a t e and f l u id p r es sur e

f o l l ow t he s ame t empor a l pa t t e r n a s t he mol d- osc i l l a t i on

speed . S eve r a l hea t - t rans f e r mode l s o f the gap have be en

deve l ope d , wher e an e f f ec t i ve t he r ma l r e s i s tance o f t he gap

la y er s is so ug ht . I15,16,171

Onl y a f ew mode l s have ana l yzed t he i mpor t an t t op- sur -

fac e flux laye rs, ~3.~s.~gl Mo st of these are on e-d im en sion al

( l - D) hea t - conduc t i on ana l yses . T he s t udy by Naka t o e t

a l I I31 so l ved t he 1 -D hea t condu c t i on equa t i on t o f i nd t ha t

t he l i qu i d l aye r t h i ckness i nc r eases w i t h t he squa r e r oo t o f

672- -VOL UM E 27B AUGUST 1996 METALLURGICAL AND MATERIALS TRANSACTIONS B


2/14

SUBMERGEi ~ENTRY

POWDERED FLUX ~ NOZZLE SEN)

TOP-SURFACE

S IN T E RE D F L UX - - ~ \

L IQUID FLUX

RE-SOLID IF IED

N i t

~176

Fig. l --S che mat ic showing physica l interaction of flux and steel in a

continuous slab casting mold.

t i me . T he g r owt h cons t an t i n t he equa t i on was r e l a t ed t o

t he me l t i ng po i n t and t he r m a l cond uc t i v i t y o f t he f lux and

t o p r oces s pa r amet e r s such a s t he s t ee l su r f ace temper a t u r e .

Deha l l e et aLt~Sl showed , us i ng a s i mi l a r mode l , t ha t t he r e

i s no s i gn i f i can t change i n l i qu i d l aye r t h i ckness w i t h f u r -

t he r add i t ions o f powd er whe n t he su r f ace t emper a t u r e is

kep t be l ow 800 ~

A m or e de t a i l ed mod e l o f t he t op- sur f ace f l ux l aye r was

d e v e l o p e d b y N a k a n o e t a/ . t lg] In thei r model , the f lux do-

ma i n was d i sc r e t i zed i n t o a s e r i e s o f t h i n hor i zon t a l l aye r s ,

and t he 1 -D hea t condu c t i on equa t i on was so l ved ove r each

l aye r . T h i s mode l accur a t e l y accoun t ed f o r t he d i f f e r ence

i n ma t e r i a l p r ope r t i e s o f the va r i ous f o r m s o f t he f l ux a s i t

t r ans f o r ms f r om powder t o l i qu i d . F or s t eady- s t a t e cond i -

t i ons , t he mo de l r e su l t s ag r eed we l l w i t h expe r i ment . How -

eve r , t he l i qu id t he r ma l co nduc t i v i t y i n t he mo de l had t o

be a r b i t r a r i l y i nc r eased t o 4 t i mes t he powder va l ue t o ob-

t a i n th i s ag r eement . An i nc r ease o f 6 t imes was need t o

m a t c h t h e e x p e r i m e n t w h e n f lu x c o n s u m p t i o n w a s i n c o r -

por a t ed . T he i r conc l us i on was t ha t conv ec t i on in t he l i qu i d

poo l i s a s i gn i f i can t hea t - t r ans f e r mechan i sm, pa r t i cu l a r l y

when consumpt i on i s p r e sen t .

Based o n t h i s b r i e f r ev i ew , t he r e r ema i n seve r a l oppor -

t un i t i e s f o r i mpr oved ma t hemat i ca l mode l s t o he l p unde r -

s t and behav i o r o f t he f l ux . Re l a t i ve l y f ew mode l s f ocus on

behav i or o f t he t op- sur f ace f l ux l aye r s , a lt hough expe r i -

ment a l wor k has dem ons t r a t ed t he i mpor t ance o f the t h i ck-

nes s o f t he t op- sur f ace l i qu i d f l ux ava i lab l e t o sup p l y t he

gap . A l l o f the hea t - t r ans f e r mod e l s r ev i ewed co ns i de r on l y

condu c t i on o f hea t , desp i te conf i r ma t i on o f t he i mpo r t ance

o f ra diatio n thro ug h the l iquid flux 16,19-21] and the signifi-

cance o f convec t i on i n t he l i qu i d f l ux poo l t o hea t t r ans -

f e r . vg ,2zj F ur t he r mor e , none o f t he mode l s r ev i ewed cons i de r

t he e f f ec t s o f e i t he r st ee l mot i on o r f l ow i n t he t op su r f ace

l iquid f lux pool .

T he p r esen t wor k app l i e s a t h r ee - d i mens i ona l (3 - D) cou-

p l ed hea t - t r ans f e r and f l u i d -f l ow f i n i te - e l ement mode l t o an-

a l yze t he beh av i o r o f t he t op- sur f ace f l ux l ayer s . I t a i ms t o

m o r e a c c u r a t e l y q u a n ti f y th e p h e n o m e n a o f p o w d e r m e l ti n g

and l i qu i d f l ux behav i o r i n an ope r a t i ng cas t e r . T he com-

mer c i a l f i n it e e lemen t packag e F I D AP i s used t o so l ve t he

mode l equa t i ons . A 3- D f o r mul a t i on was needed because

pr e l i mi na r y 1 -D and t wo- d i men s i ona l ( 2 - D) mode l s d i d no t

p r oduce quan t i t a t i ve r e su l t s ( t hey neces sa r i l y neg l ec t t he

i m p o r t a n t p h e n o m e n a o f c o n v e c t i o n a n d 3 - D m a s s c o n -

sumpt i on f l ow) . T he r e su l ts o f the 3 - D m ode l a r e ve r i f i ed

exper i men t a l l y , usi ng da t a co l l ec t ed a t t he ope r a t i ng cas t e r

at LT V Steel (Cleveland, OH )J ~ -231 An ini tia l param et r ic

s t udy i s pe r f o r med t o a sce r t a i n t he e f f ec t o f p r oces s pa r a -

me t e r s such a s v i s cos i t y and t he r ma l co nduc t i v i t y . F i na l ly ,

a I - D t r ans i en t mode l i s deve l oped t o exp l o r e t he va l i d i t y

of t he s t eady- s t a te a s sumpt i on .

I I . M O D E L F O R M U L A T I O N

A M o d e l A s s u m p t i o n s

T he 3 - D coup l ed f l u id f l ow and hea t t r ans f e r w i t h change

of phase i n t he pow der , l i qu i d , and r e so l i d i f i ed fl ux l aye r s

was mode l ed i n t h i s s t udy . T he mode l accoun t s f o r t he

known l oca t i on and shape o f t he s t ee l / f l ux i n t e r f ace , mo-

ment um t r ans f e r be t ween t he l i qu i d s t ee l and f l ux , and r a -

d i a t ive and na t u r al co nvec t i ve hea t l os s a t t he f r ee su r f ace

of t he f lux . A sepa r at e 3 - D m ode l i s used t o ca l cu l a t e the

f low f ie ld in the l iquid s teel . E241 This s teel f lo w mo del and

t he f l ux f low mo de l p r esen t ed he r e a r e coup l ed t h r oug h t he

shear s t ress di s t r ibut ion a t thei r mutual inter face. Separate

t emper a t u r e - dependen t f unc t i ons f o r powder v i scos i t y and

t he r ma l conduc t i v i t y a r e app l i ed i n r eg i ons o f me l t i ng and

so l i d i f y ing powder .

M ode l a s sumpt i ons i nc l ude t he f o l l owi ng .

(1) F low is laminar . This i s consis ten t wi th Re ~- 70 (Table

I)

( 2 ) T he f l ow and t he r ma l f i e l ds a r e bo t h s t eady . T h i s a s -

sum pt ion i s invest igated in Sect ion VII wi th the 1-D

t r ans i en t mode l .

( 3 ) S t ee l su r f ace shape i s f i xed accor d i ng t o measur em ent s

made under s t eady ope r a t i ng cond i t i ons . T hus , su r f ace

waves and s l osh i ng a r e i gnor ed .

( 4 ) F l ux p r ope r t i e s a r e cha r ac t e r i zed so l e l y t h r ough t he

macr oscop i c ma t e r i a l p r ope r t i e s o f v i s cos i t y , t he r ma l

conduct ivi ty, speci f ic heat , l a tent heat , and densi ty. Ef-

f ec t s o f mi c r oscop i c k i ne t ic s such a s pa r t i c l e s i n t e r ing

a r e mani f e s t ed on l y i n t he va l ues o f t hese mac r oscop i c

proper t ies .

( 5 ) Dens i t y i s cons t an t and g r av i t y i s i gnor ed . T he mode l

neg l ec t s buoyan cy e f f ec t s due t o t he kn own dens i t y i n -

c r ease a s t he powder s i n t e r s and me l t s . T he e s t i ma t ed

R a l e ig h n u m b e r is - 1 0 5 t o 1 0 6 whi ch i s h i gh enough

t ha t na t u r a l convec t i on shou l d be i mpor t an t , bu t l ow

enough t ha t t he f l ow does no t become t u r bu l en t . T he

m o d i f i e d F r o u d e n u m b e r ~251 of on ly 0.07 7 (Tab le I)

shows t ha t even r e l a t i ve to t he d r i v i ng f o r ce o f the s t ee l

mo t ion, natural conv ect io n i s impo r tant . Thus , there i s

a s i gn i f i can t t endency f o r t he powd er t o f l oa t above t he

l i qu id f l ux l aye r whi l e g r av i t y i nduces f l ow o f t he l i qu id

f l ux downw ar d i n t o t he va l l eys i n the s t ee l su rf ace con-

tours .

( 6 ) P ow der f eed i ng is un i f o r m and s t eady ov e r the en t i r e

mol d a r ea . I n p r ac ti ce , th i s a s sumpt i on i s on l y appr ox-

i ma t ed by ve r y ca r e f u l ope r a t o r s o r au t omat i c powder

f eede r s .

( 7 ) M ol d osc i l l a t i on has no r o l e o t he r t han t o i mpose t he

cons t an t , un i f o r m, consum pt i on r a t e o f l i qu id f l ux i n t o

t he mol d- s t r and gap a r ou nd t he pe r i me t e r o f the mol d .

( 8 ) P owder and l i qu i d f l ux bo t h behave a s an i so t r op i c

New t on i an f l ui d w i t h t emper a t u r e - dep enden t v i s cos it y ,

t he r ma l con duc t i v i t y , and en t ha l py .

T her ma l con t ac t r e s i s t ance be t ween t he mol t en s t ee l

and mol ten f lux layers i s negl igible .

(9)

METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 27B AUGUST 1990---673


3/14

T a b l e I S i m u l a t i o n C o n d i t i o n s a n d N o m e n c l a t u r e

S y m b o l V a r i a b l e V a l u e

C ~ sp e c i f i c h e a t o f f l u x . J k g - t K - ~ F ig . 9

d u m o l d - s t r a n d g a p t h i c k n e s s i n s i m u l a t i o n d o m a i n , m 0 . 0 0 0 8

g a c c e l e r a t i o n d u e t o g r a v i t y , m s - 2 9 . 81

h~,~- ef fe c t iv e to ta l he at- t ran sfe r coeff icien t, I43 j W m -2 K ~ Eq . [A 3]

k p, k j p o w d e r f l u x , l i q u id f l u x t h e r m a l c o n d u c t iv i t y , W m - ~ K - ~ F ig . 8

l e m o l d - s t r a n d g a p l e n g t h i n s i m u l a t i o n d o m a i n , m 0 . 0 2 7

L j, E q u i v a l e n t H y d r a u l i c d i a m e t e r o f m o l d , m ( = \ / ( w ~ t ~ )) 0 . 2 8 2 9

m . f l u x s p e c i f i c c o n s u m p t i o n , k g m - ' - 0 . 6

m c f l u x c o n s u m p t i o n r a t e , k g s - ~ 0 . 0 3 2 6

M ,. f l u x c o n s u m p t i o n p e r m e t e r o f m o l d p e r i m e t e r , k g m -~ s -~ 0 . 0 1 0 0

N n u m b e r o f n o d e s 2 8 , 5 0 0

P p r e ssu r e , N m - 2

q , n o r m a l h e a t f l ux , W m - 2

t m h a l f - m o l d t h i c k n e s s , m O . 1 1 4 3

T t e m p e r a t u r e , ~

T ~ b a m b i e n t t e m p e r a t u r e , ~ 3 0

T fm o,, f l u x m e l t i n g t e m p e r a tu r e , ~ 1 0 0 0

~ f l u x s i n t e r i n g t e m p e r a t u r e , ~ 9 0 0

T~, m o l t e n s t e e l s u r f a c e t e m p e r a t u r e , ~ 1 5 5 0

T~u su r f a c e t e m p e r a tu r e o f f l u x , ~

u x, u y, u . v e lo c i t y c o m p o n e n t s i n x , y , a n d z d i r e c t i o n s , m s- ~

v c h a r a c t e r i s t i c v e l o c i t y o f l i q u i d f l u x , m s - '

Vc c a s t i n g sp e e d , m s - ~ 0 . 0 1 6 6

w p, w j p o w d e r f l u x, l i q u i d f l u x l a y e r t h i c k n e s s , m

w , h a l f - m o l d w i d t h , m 0 . 7

x c o o r d i n a t e i n m o l d w i d t h d i r e c t i o n , m

~ . t o t a l f l u x t h i c k n e s s , m

X j: m= m a x i m u m t o t a l f l u x t h i c k n e s s , m 0 . 0 3 5

Xj :~ ,, t o t a l f l u x t h i c k n e s s a t m e n i sc u s , m 0 . 0 1

Xt.SEN to t a l f l u x t h i c k n e ss a t SE N , m 0 . 0 2 7

y c o o r d i n a t e i n m o l d t h i c k n e s s d i r e c t i o n , m

z c o o r d i n a t e i n n e g a t i v e c a s t i n g d i r e c t i o n , m

2 d- / e n th a lp y o f f u s io n , k J k g - ~ 3 5 0

A T c h a r a c t e r i s t i c t e m p e r a t u r e d i f f e r e n c e f o r n a t u r a l c o n v e c t i o n , K 6 5 0

/ 3 V o l u m e t r i c e x p a n s i o n o f l i q u i d f l u x , K - ' 2 . 4 1 0 - 5

ej~ f l u x e m i ss iv i t y 0 . 7

3 (, su r f a c e t e n s io n f o r l i q u id s t e e l i n a i r , 261 N m ~ 1 . 6

/ z k in e t i c v i sc o s i t y o f f l u x , k g m - j s - ~ F ig . 7

/ z , k in e t i c v i sc o s i t y o f s t e e l a t l i q u id u s , k g m - ~ s - ~ 0 . 0 0 5 5

p d e n s i t y o f f l u x, k g m - 3 2 5 0 0

p . d e n s i t y o f s t e e l , k g m - 3 7 8 0 0

o S t e f a n - B o l t z r n a n n c o n s t a n t , W m - 2 K - 4 5 . 6 6 7 1 0 - 8

r sh e a r s t r e s s , N m - - '

R e

F r

B r

G r

W e

R e y n o l s n u m b e r = )

M o d i dF r o u d e u m b e r=

O n n k m a n n u m b e r =

Or a s h o f n um b e r=

tz ~ }

w e b e r n u m b e r = )

7 0

0 . 0 7 7

6 .0 10 -9

8 0

1.5 X 10 -3

674-- VOLUME 27B AUGUST 1996 METALLURGICAL AND MATERIALS TRANSACTIONS 13


4/14

S U B M E R G E D E N T R Y

N O Z Z L E S E N )

. o =

Fig . 2 - -S chem at ic o f con t inuous slab cast ing mold showing f lux

simula t ion domain .

N a n ~ w f ~

MoldWall

MoldWallConsumption egion

Widelace /

Mo/dWail / Plane:t~

'~1 .~ w i n=0 .70 0 - - ~ " ~ ~ " ~ ~ ~ ~ m

d g

0,001 SteeI-Rux Symmetry Xf,SEN= "027

Interlace

F i g . 3 - - D i m e n s i o n s o f s i m u l a t io n d o m a i n ( i n m e t e rs ) .

(1 0 ) V i sco u s d i s s ip a t io n en e rg y i s n eg l ig ib l e b ecau se th e

B r in k m a n n u m b er i s o n ly 1 0 -9 (T ab le I ) .

B . G o vern in g E q u a t io n s

B a s e d o n t h e p r o b l e m a n d a s s u m p t i o n s p o s e d i n S e c t io n

A , t h e fo l lo w in g f iv e co n t in u u m eq u a t io n s a r e so lv ed .

V. u -- 0 [1]

p u 9 V u = - V P + t z ( T ) V 9 Vu [2]

pCp [ , . V T ] = V . ( k ( r ) v r ) [3]

T h e y r e p r e s e n t m a s s , m o m e n t u m , a n d e n e r g y c o n s e r v a -

t i o n in t h ree d im en s io n s , r e sp ec t iv e ly . T h e N o m en c la tu re i s

g iv en in T ab le 1 .

C . G eo m et ry D e f in i t io n a n d B o u n d a ry C o n d i t io n s

S c h e m a t i c s o f t h e m o d e l d o m a i n w i t h d i m e n s i o n s a r e

g iv en in F ig u res 2 an d 3 . C o m p u ta t io n a l r eq u i r em en t s a r e

r e e S u r f a c e

U ~ = - 4 . ] X 1 0 5 m s - I

au~ = 0 Plane of Symmetry

Oz u~= 0

o o_~ o

az ax

aU~=o

ax

Wideface Wall

Widefaee onsumption ux, uy.u~ = 0 ,~

Plane of Symmetry

Uy =

t2* (tangential) Ouz

i l i ~ ~ ~ ~ In (normal) ~ y = 0

Flux-Steel Interface

xt

9

specifiedbasedon interpolation

with 3-D steel f low model

Narrowfaee

Wall ~2 .*= 0

u~

U y , U z =

0

Solidifying

S hell ~ / ~

Uz

=

0.017 ms~

y x

9 local coordinate y stem t steel/flux nterface

F i g . 4 - - - F l o w b o u n d a r y c o n d i t i o n s .

r e d u c e d b y i n v o k i n g b i f o ld s y m m e t r y t o m o d e l o n l y o n e -

q u a r t e r o f t h e p h y s i ca l d o m ain . T h e ex ac t sh ap e o f t h e s t ee l

f l ux i n t e r fa c e u s e d w a s i m p o s e d b a s e d o n s t e a d y m e a s u r e -

me n ts on an a ctual caster , U7,23] as desc r ibed la ter . The shap e

o f t h e m en i scu s p o r t i o n o f t h is i n t e r f ace i s d e t e rm in ed f ro m

a n a n a l y t i c al s o l u ti o n o f a m o d i f i e d v e rs i o n o f t he Y o u n g -

La pl ac e eq uat ion , tz6.27]

T h e f lo w b o u n d ary co n d i t i o n s a r e g iv en in F ig u re 4 , w i th

th e f i n i t e e l em en t m esh . A t t h e f l u x / s t ee l i n t e r f ace , a f i x ed

sh ea r s t r e s s , r :~ , co n d i t i o n w as im p o sed .

ux i . . x au, l [4]

T h e v a lu e o f ~ = w as ca l cu l a t ed b y i t e r a t i n g b e tw een th e

p r e s e n t 3 - D i s o t he r m a l f l o w m o d e l o f f lu x f l o w a n d a 3 - D

mo del o f tu rbu len t f low in the l iqu id s teel ,l :4 ] wi th the typ-

i ca l f l o w p a t t e rn sh o w n in F ig u re 5 . S h ea r s t r e s s f ro m th e

to p su r f ace o f th e s t ee l m o d e l i s ap p l i ed to th e b o t to m su r -

f ace o f t h e f l u x m o d e l . In t e r f ace v e lo c i t i e s f ro m th e f l u x

m o d e l a r e t h en ap p l i ed to t h e s t ee l f l o w m o d e l . I t e r a t i o n

co n t in u es u n t i l b o th s t r e s s an d v e lo c i ty a t t h e f l u x s t ee l

in t e r f ace ag ree . T h e sh ea r s t r e s s so lu t io n a lo n g th e f l u x / s t ee l

i n t e rf a c e i n c r e a s es f r o m z e r o a t t h e n a r r o w f a c e a n d S E N t o

a m a x i m u m o f a b o u t 0 . 2 6 N m - 2 n e a r t h e c e n t e r.

A u n i fo rm co n s u m p t io n r a t e o f l i q u id f l u x a ro u n d th e

m o ld p e r im e te r i s im p o sed b y f ix in g th e z v e lo c i t i e s i n to

th e t o p f r ee su r f ace an d th e y v e lo c i t i e s o u t o f t h e b o t to m

l a y e r o f e l e m e n t s a l o n g t h e w i d e f a c e ( A p p e n d ix ) . T h e m e -

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

th e w id e face , w h ere t h e d o m ain i s o f f se t f ro m th e w a l l .

S tu d ies u s in g 2 -D d o m ain s p a ra l l e l t o t h e n a r ro w face r e -

v ea l ed th i s s im p l i f i ca t io n d o es n o t s ig n i f i can t ly a f f ec t t h e

f lOW.[231

T h erm al b o u n d ary co n d i t i o n s a r e g iv en in F ig u re 6 . T h e

e f fec t iv e h ea t - t r an s fe r co e f f i c i en t f ro m th e t o p - su r f ace h o f f

M E T A L L U R G I C A L A N D M A T E R I A L S T R A N S A C T I O N S B V O L U M E 2 7B , A U G U S T 1 9 9 6 - -6 7 5


5/14

d 3

t -

E l

~ L

I

i

0 . 0

ft.

9 x 5 4 ( 0 . 2 3 x 1 . 4 0 m )

1 m / m i n

N o A r g o n

S E N

0 . 5

4

Z

Fig. 5--Velocities in steel calculated by a 3-D flow model. :4]

P l a n e o f

Symme t r y

ree

S m ' f a c e q ~ = 0

qn = heff(T Tsmbi~at /

F Planeof

Symmetry

qn=O

~ I ~ ~ . . . . .N F I u x - S t e e l I n . t e l ~ a c e

Z

arrowface

W al l So l id ify inghell y x

T= 1550 C

T = 300 C

Fig . 6 - -T herm al boundary condi t ions.

(E q . [A3 ] ) i s domi na t ed by r ad i a t i on bu t a l so i nc l udes na t -

u r a l con vec t i on i n a i r.

I II . M A T E R I A L P R O P E R T Y D A T A

A. Viscosity Tem perature R elationship

T he t h r ee fo rms o f f l ux (powder , l i qu i d , and g l a s sy /

c rys ta l l ine so l id ) w ere s i mul a t ed us i ng t wo d i f f e r en t ma t e -

r i a l mode l s , i n d i f f e r en t pa rt s o f t he dom ai n . O ne pa r t cha r -

ac t e r i zes t he f l ux a s l i qu i d coo l i ng t o a co he ren t so li d nea r

1 0 5

l o '

lO O O

l o o

l o

1

0 . 1

0 .01

. . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . . . . . . I . . . . I .

i i J . J -I ~ s o l i , ~ , ~ g c t . o . , . , . l ~ = a )

T T ~ 7 . . . . . i ] ~ ~ ( St ='m , ~r d- H i gh ~ .)

. .. .. . T - - T - - - 7 - - - ~ - . .. -- -- .. .. '- -- -T---Z . . . F . .. .. 7

i i "

. . . I . . . . I . . . . I . . . . I . . . . J . . . . I . . . . . . . . . . . i i

I I I I I I

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0

Fig . 7 - -Viscosi ty - tem perature models fo r mel t ing and so l id i fying o f two

different f luxes.

3 . 5

, . . , 3

' r 2 . 5

1 . 5

~ 0.5

0

I . f . I 1 I . I . I . . I I

' 1 ' 1 ' 1 ' , , ' I ' I ' [ ' I ' S

0 4 0 0 8 0 0 1 2 0 0 1 6 0 0 2 0 0 0

Tempe r a t u r e

( ~

Fig . 8 - -T herm al conduct iv i ty - temperature model

solidifying flux.

for mel t ing and

t he mol d wa l l s ( so l i d i fy i ng) . T he o t he r pa r t mode l s t he

pow der a s i t s in t e r s and me l t s t o fo rm t he l i qu id i n a l l o t he r

r e g io n s o f t h e m o l d ( m e l ti n g ). T h e v i s c o s i ty o f t h e p o w d e r

was e s t i ma t ed f rom rheo l og i ca l da t a used i n t he f i e l d o f

pa r t i c l e f l u i d iza t i on , 3~ a s sumi ng a me an pa r t i c le s ize o f

250 p .m and an ave rage dens i t y o f 1000 kg m -3. T he l i qu i d

v i scos i t y fo l l ows t he we l l -docu men t ed exponen t i a l f unc t i on

of t empera t u r e , 3.]~ T he maxi m um v i scos i t y o f r e so l i-

d i f ied f l ux was t r unca t ed a t 104 P as t o avo i d nu mer i ca l i n -

s t ab i l i t y . F i gure 7 shows t he da t a chosen t o approx i ma t e

t he v i sco s i t y - t empera t u r e r e l a ti onsh i ps o f f l uxes w i th t w o

t yp i ca l v i scos i t i e s - - s t anda rd ( case 1 ) and h i gh v i scos i t y

(case 2) .

B. Th erma l Co n d u c t i v i t y Temp era tu re Re la t i o n sh ip

T h e t h e r m a l c o n d u c t i v it y o f th e f l u x w a s a l s o m o d e l e d

d i f f e r en t l y fo r me l t i ng and so l id i f ica t ion . T he pa i r o f curves

i s g i ven i n F i gure 8 . T h e t he rma l cond uc t i v i t y o f s i n t e r ing

powder g r adua l l y i nc r eases a s a i r spaces d i sappea r . F or

bo t h m e l t i ng and so l i d i fy i ng f l ux , t he s t anda rd e f f ec t i ve

t he rma l condu c t i v i t y above t he me l t i ng t empera t u r e , ~ was

fixe d at 3 W m -t K -L Th is ty pica l va lue U9.22,37] assu me s no

conv ec t i on bu t i nc ludes bo t h co nduc t i on and r ad i a t i on

t h rough t he t r anspa ren t l iqu i d . Be l ow t he s i n t e r i ng temp er -

a t u r e (900 ~ t he conduc t i v i t i e s r epor t ed by T ay l or and

676---VOLUME 27B, AUGUS T 1996 METALLURGICAL AND MATERIALS TRANSACTIONS B


6/14

3 . 0 0 0 I 0 6 i . . . . . . . . ~ . . . . . . I . . . . . . . ~ . . . . I ~ . .. .

1 o o o 1 o i . .. . .. .. . .. . .. . . .. .. .. .. .. . .. .. i . . . . . . . . . . . . . . . . . . . . . . . . .

~ . o 0 o o ~ ~ ................ . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . ............

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0

T e m p e r a t u r e = C )

Fig. 9 E nthalpy temperature mo del for f lux.

Appl y Updated S heer S tress f rom S tee l

Flow Mode l el the Flux/Steel Interface

J

DEFINITION CONDITIONS MODELS

ISOTHERMALFLOW

MODEL

Ve~oehS, ~ Prcwammno~v~l or.

S hear S tressat

Flux/Steel Interface

Y E S ~ I n t e r f a c e

VeL ~ Flte~ ~te~ nte~ace

T ~ Ul S ~ Kd ~

3-D STEEL FLOW

MODEL

S hear S tressat

Flux/Steed

nterface

~lvod

or

f FLOWFIELD

. k i n

UNCOI~PLEDA D V E C T I O N I ~

D I F F U S I O N A N A L Y S I S

Temperatures solved for only.

~VECT10~N~FUSlON

TBIPI[RATURErlEt~ARI[Ut~.D

AhtlU.YSlS

FULLY COUPLEDFLUID

/ p

~fr)

FLOW HEAT TRANSFER

L ~

ANALYSIS I

Temp erature, Velocif~esand Pressure

solved or simultaneously,

~ C O U P L E D F L O W A N DLUID

TEMPERATURE FIELDS

Fig. 10--Flow chart of model solution methodology.

T a b l e II . S o l u t io n P a r a m e t e r s f o r 3 D M o d e l

Total Total

Relax- Disk Number Solution

Problem ation Space of Time

Type Factor RESCONV (MB) Iterations (CPUs)*

Isothermal

flow None 0.001 990 5 3500

Energy

(Advection-

Diffusion) 0.4 0.01 59 110 4320

Coupled 0.3 0.01 1,800 45 67 ,500

*CPU--seconds on the Cray Y-MP using 8 MW RAM.

Millst211 are used for the melting powder, assumed over

most o f the domain. In the totally solid state, a constant

value of 0.9 W m -] K -~ was adopted, according to Ta ylor

and Mills and Nagata

e t

a/. f3s.391

C . E n t h a t p y - T e m p e r a t u r e R e l a t i o n s h i p

An enthalpy-temperature model is used to simulate so-

lidification/melting of the flux. This is believed to be more

stable and accurate than an enh anced specific hea t

model. The assumed function, shown in Figure 9, is based

on sparse data from available literature. I37,4~ It includes a

latent heat of fusion of 350 kJ kg -~ at the melting point o f

1000 ~

IV. SOLUTION METHO DOLOG Y

The preceding governing equations, subject to the bound-

ary conditions discussed, were solved using the finite ele-

ment method with the commercial CFD code FIDAP. Full

details of the implementation of this model are found in the

F I D A P T h e o r e t ic a l M a n u a ~ 4 q and elsewhere.t23]

The Navier-Stokes and continuity equations were solved

using a mixed i .e . , u-P) formulation. Successive substitu-

tion proved to be more robust than the Newton-Raphson

method for solving the nonlinear algebraic finite element

equations. Oscillatory behavior was suppressed with an un-

der-relaxation coefficient between 0.3 and 0.4 (FIDAP ac-

celeration factor between 0.7 and 0.6), meaning that the

guess used to linearize the equations relied more heavily

on the past solution than on the most recent solution. A

solution was considered converged when the normalized

residual L-2 norm (RESCONV in FIDAP) fell below the

user-specified value of 0.01. Additionally, a smooth mono-

tonically decreasing residual norm was generally indicative

that a good solution was being achieved.

The inherent high nonlinearity in the momentum equa-

tions and the two-way coupling with the energy equation

together make this problem difficult to converge. Thus, a

good initial guess o f the flow field and temperature distri-

bution was required to avoid divergence. The solution strat-

egy developed to consistently and efficiently obtain a

converged solution is illustrated in Figure 10. The first step

solves an isothermal flow problem, assuming a constant

density o f 2500 kg m -3 and constant viscos ity of 0.03 Pas,

which is the correct value at the flux/steel interface.

The final mesh, shown in Figures 4 and 6, consisted of

3640 27-noded

i .e . ,

quadratic) brick elements and a total

of 28,500 nodes. Calculations were performed on a CRAY

Y-MP* supercomputer using a dire ct iterative solver in

*CRAY Y-MP s a trademark of Cray Research, Inc., Minneapolis, MN.

FIDAP. The computational requirements are summarized in

Table II. Initial attempts using the indire ct (segregated)

solv er in FIDAP were unsuccessful, as the 15-fold savings

in cost per iteration was more than offset by a 30-fold in-

crease in number of iterations required for convergence . All

pre- and postprocessing were performed on a Silicon

Graphics Personal Iris 4D/25 with 64 MB of RAM and over

400 MB of disk space.

V. TYPICA L RESULTS

The model was run to simulate behavior of the flux lay-

ers above a typical 0.23 1.40 m (9 54 in.) strand cast

METALLU RGICAL AND IVlATERIALS TRANSACTIONS B VOLUME 27B, AUG UST 1996--~77


7/14

...

Z

i j f . . . . - _ . . - .~D.~ s_

~ 1.7 mm ~ I

REFEREN EE TOR

1 . 7 m m s 1

J

F i g . I 1 - - C a l c u l a t e d v e l o c i t y d i s t r i b u t i o n i n f l u x l a y e r s a t t o p s u r f a c e a n d

a t m i d p l a n e s h o w i n g r ec i r c u la t i o n z o n e a n d f l o w s e p a ra t i o n s t a n d a r d

c o n d i t i o n s ) .

6

'~a 4

2

>~ o

.4

- l O

. . .. . . .. . .. . .. . . .. . .. . . i . . .. . .. . .. . .. . .. . .. . .. . T . . .. . .. . .. . .. . .. . .. . .. . i [ ~ ~ ' ~ ~ ' I , , - - ~ . w ~

i i i / I TopSudace 14 mm fromWF)

. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . .. . . . . . . . .. . . . . .. . . . . i 1 ~ ~ ' e ' ~ l ' - - ' ' l

I TopSteface I 14 mm rom WF)

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

Distance from Narrowface x m m )

F i g . 1 2 - - T o p s u r f a c e a n d f l u x / s t e e l i n t e r fa c e v e l o c i t y d i s t r i b u t i o n a s a

f u n c t i o n o f d i s t a n c e f r o m n a r r o w f a c e .

w i th o u t a rg o n a t 1 m m in -~ u n d e r t h e s t an d a rd co n d i t i o n s

g iv en in T ab le I . T h e f in a l co n v erg ed v e lo c i ty an d th e rm a l

so lu t io n s a r e p resen ted in F ig u res 1 1 th ro u g h 1 4 .

A . F l o w F i e l d

T h e m o s t o b v io u s f ea tu re o f t he f l o w f i e ld in F ig u re 1 1

i s t h e l a rg e r ec i r cu l a t i o n zo n e th a t ex t en d s f ro m th e S E N

to ab o u t 2 5 0 m m f ro m th e n a r ro w face . W h i l e t h e s t ee l f l o w

a t t em p t s t o d rag th e l iq u id f lu x to w ard th e S E N an o p p o s -

in g f lo w ca r r i e s t h e u p p e r l ay e r s o f f l u x an d p o w d er i n t h e

o p p o s i t e d i r ec t io n . B ase d o n th e ca l cu l a t ed s i ze o f th e l i q u id

reg io n th i s r ec i r cu l a t i o n zo n e co n ta in s ab o u t 3 . 65 k g o f

F i g . 1 3 - - C a l c u l a t e d v e l o c i t y d i s t r i b u t i o n a t f l u x / s t e e l i n t e r f a c e a n d a t

m i d p l a n e s h o w i n g f l o w s e p a r a t io n a n d r e c i r c u l at i o n s t a n d a r d c o n d it i o n s ).

8O

3 C

F i g . 1 4 - - - 3 - D t e m p e r a t u r e d i s t r i b u t i o n c a lc u l a t e d i n f l u x l a y e r ~

s t a n d a r d c o n d i t i o n s ) .

l i q u id f l u x . T h i s co r r e sp o n d s to a m ean r e s id en ce t im e o f

f lu x in t h e l i q u id p o o l o f 1 1 2 seco n d s .

P o w d e r f l u x a b o v e t h e r e c i r c u l a t i o n z o n e m o v e s s l o w l y

t o w a r d t h e n a r r o w f a c e a t a v e l o c i t y w h i c h i n c r e a se s w i t h

d i s t an ce f ro m th e w id e face . F ig u re 1 2 co m p are s t h e v e lo c -

i t y d i s t r i b u t io n a t tw o p o s i t i o n s a lo n g th e t o p su r f ace w i th

th e co r r e sp o n d in g v e lo c i ty a t t h e f l u x s t ee l i n t e r f ace . T h e

m a x i m u m v e l o c i t y a l o n g t he t o p s u r f a c e i s o n l y a b o u t 1 .5

m m s - L T h i s s lo w d r if t t o w a r d t he n a r r o w f a c e a w a y f r o m

th e S E N i s co n s i s t en t w i th t h e u su a l p l an t o b se rv a t io n th a t

p o w d e r m u s t b e a d d e d m o r e o f t e n n e a r t h e S E N t h a n n e a r

th e n a r ro w face .

A n o th e r s ig n i f i can t f ea tu re o f t h e f l o w f i e ld i s th e sep a -

r a t i o n p o in t d e t a i l ed in F ig u re 1 3. T h e ap p l i ed sh ea r s t r e s s

678- -V O L U ME 27B, A U G U S T 1996 META LLU RG ICA L A N D MA TERIA LS TRA N S A CTIO N S B


8/14

g

= =

~J

[ .

. =

2 5

2 0

1 5

1 0

5

0

1 2 0

D i s t a n c e f r o m ~ I

c o r n e r a l o n g

NF, y ra m)

1~ 2~ 3 4 5 6 7~

D i s t a n c e f r o m c o r n e r a l o n g

w i d e f a c e x r a m )

F i g . 1 5 - - L i q u i d l a y e r t h i c k n e s s a r o u n d m o l d p e r im e t e r s t a n d a r d

c o n d i t i o n s ) .

a lo n g th e f l u x s tee l i n t e r f ace i n c reases f ro m ze ro a t t h e n a r -

ro w face t o i t s m a x im u m o f 0 .2 6 N m -2, 0. 3 m f ro m th e

n a r ro w face , an d b ack d o w n to ze ro a t t h e S E N . T h i s sh ea r

s t r e s s a lw ay s d i r ec t s f l o w o f l i q u id f l u x t o w ard t h e S E N .

O n th e o th e r h an d , t h e co n su m p t io n o f li q u id fl u x i n to t h e

n a r r o w f a c e g a p t e n d s t o d r i v e l i q u i d f l u x a w a y f r o m t h e

S E N . A t t h e p o in t a lo n g th e f l u x / s tee l i n t e r f ace w h ere t h ese

d r iv in g fo rce s b a l an ce , a s ep a ra t i o n i n t h e f l u x f l o w f i e ld is

p r o d u c e d . T h i s s e p a r a t i on o c c u r s a b o u t 7 0 t o 8 0 m m f r o m

th e n a r ro w face .

T h e f l o w o f l i q u id f l u x i s p r ed o m in an t ly i n t h e x d i r ec -

t i o n . O v er m o s t o f t h e d o m ain , f l u x co n su m p t io n t o t h e

w id e face h as l i t t l e e f f ec t o n t h e f l o w p a t t e rn . C lo se t o t h e

m en i scu s , h o w ev er , t h e s t ee l f l o w an d co r r e sp o n d in g x d i -

r ec t i o n f l u x v e lo c i t i e s d im in i sh . H e re , th e co n su m p t io n f l o w

to w ard t h e m o ld w a l l s i s im p o r t an t . F ig u re 1 3 sh o w s th a t

f l o w to w ard t h e w id e face d o m in a t e s t h e f l o w p a t t e rn a t t h e

f lo w sep a ra t i o n p o in t . W i th o u t t h i s f l o w , a sh o r t ag e o f l i q -

u id f l u x co n su m p t io n an d co r r e sp o n d in g q u a l i t y p ro b l em s

are l i k e ly . T h es e r e su l t s sh o w th a t th e w o r s t p o t en t i a l p ro b -

l e m s s h o u ld e x i s t a t t h e o f f - c o m e r r e g io n o f th e w i d e f a c e ,

fo r t h e s t an d a rd co n d i t i o n s a s su m ed h e re .

r eg io n l eav es l e s s h ea t to m e l t t h e f l u x, w h ich l ead s t o b o th

a t h in n e r l i q u id l ay e r an d lo w er t o p - su r f ace t em p era tu re .

C. Flux Laver Thicknesses

T h e v a r i a t i o n i n t h i ck n ess o f t h e l i q uid f l u x l ay e r a ro u n d

th e m o ld p e r im e te r is sh o w n in F ig u re 1 5 . T h i s t h i ck n es s

i s v e ry im p o r t an t b ecau se i t co n t ro l s t h e am o u n t o f l i q u id

f lu x av a i l ab l e t o en t e r t h e g ap . T h e so l i d / l iq u id i n t e r f ace

w as i d en t if i ed u s in g t h e 1 0 0 0 ~ i so th e rm . C o n v ec t io n i n

th e r ec i r cu l a t i o n zo n e g en e ra t e s t h e t h i ck es t l i q u id f l u x

l a y e r n e a r th e S E N , b e t w e e n 4 0 0 a n d 7 0 0 m m f r o m t h e

n a r ro w face w a l l . N ea r t h e n a r ro w face , t h e l i q u id f l u x l ay e r

i s i n h e ren t ly t h in . H o w ev er , t h e t h in n es t l i q u id l ay e r i s

f o u n d a l o n g t h e w i d e f a c e w a l l b e t w e e n 1 50 a n d 2 5 0 m m

fro m th e co m er . T h i s r eg io n co r r e sp o n d s t o t h e f l o w sep -

a ra t i o n p o in t an d i s ex p ec t ed t o b e a g en e r i c l o ca t i o n a t

w h ich t h e g ap m ay b e s t a rv ed fo r l i q u id f l u x . I t s ex ac t l o -

ca t i o n sh o u ld v a ry w i th cas t i n g co n d i t i o n s , a s i t d ep en d s

d i r ec t l y o n t h e f l o w p a t t e rn d ev e lo p e d in t h e s t ee l an d o th e r

v a r i ab l e s .

V I. N U M E R I C A L V A L I D A T I O N

N u m e r i c a l s t u d i e s w e r e p e r f o r m e d t o o b t a i n a s u i t a b l e

g rad in g o f t h e 3 -D m esh . F u r th e r t e s t s t o en su re accu ra t e

m e s h - i n d e p e n d e n t r e s u l ts w e r e p e r f o r m e d u s i n g r e f i n e d 2 -

D m e s h e s , f o r c o m p u t a t i o n a l r e as o n s . T h e 2 - D m o d e l

s im u la t e s a s l i ce t h ro u g h th e x z cen te r p l an e so i n l e t z

v e lo c i t i e s a t t h e t o p su r f ace w ere r ed u ced to i n co rp o ra t e

f lu x co n su m p t io n o n ly t o t h e n a r ro w face . T h i s n ecessa ry

assu m p t io n fo r t h e 2 -D m o d e l m ak es i t o v e rp red i c t l i q u id

f lu x t h i ck n ess , p a r t i cu l a r ly a t t h e m en i scu s , w h ere h ea t l o s s

via f l u x t r an sp o r t t o f eed co n su m p t io n a lo n g th e w id e face

i s n eg l ec t ed . T h e 2 -D m o d e l w as o th e rw i se s im i l a r t o t h e

3 -D m o d e l .

U s in g a 2 -D m o d e l w i th t h e sam e m esh d i sc re t i za t i o n i n

th e x an d z d i r ec t i o n s a s t h e 3 -D m o d e l (4 4 0 n in e -n o d e

e l e m e n t s ) , t h e c o u p l e d p r o b l e m w a s s o l v e d Y 31 The mesh

w as t h en r e f in ed a lm o s t 1 6 - fo ld (6 4 0 0 n in e -n o d e e l em en t s )

a n d t h e a n a l y s i s r e - r a n . T h e m a x i m u m c h a n g e i n t h e m e l t

i n t e r f ace p o s i t i o n w as o n ly 1 5 p c t , so t h e 3 -D m o d e l m esh

r e f i n e m e n t w a s d e e m e d a d e q u at e .

B. Temperature Distribution

C o n v ec t iv e h ea t t r an sp o r t i n t h e r ec i r cu l a t i o n zo n e g en -

e ra t e s a m u ch th i ck e r l i q u id p o o l an d co r r e sp o n d in g h o t t e r

su r f ace c lo se r t o t h e S E N (F ig u re 1 4 ) . I n t h e m en i scu s r e -

g io n , w h ere t h e re i s n o v e r t i ca l ( z d i r ec t i o n ) co m p o n en t o f

f l o w , t h e l i q u id f l u x l ay e r i s t h in n e r . H o w ev er , t h e su r f ace

t em p era tu re c lo se t o t h e n a r ro w face i s r e l a t i v e ly h ig h b e -

cau se t h e s t ee l / f l u x i n t e r f ace i s c lo se t o t h e f r ee su r f ace

there .

A n o th e r p ro n o u n ced f ea tu re o f t h e t h e rm a l f i e ld i n F ig u re

1 4 i s t h e r e l a t i v e ly co o l t o p su r f ace n ea r t h e w id e face w a l l

i n t h e o f f - co rn e r r eg io n , ab o u t 1 5 0 to 2 5 0 m m f ro m th e

n a r ro w face w a l l . T h i s i s d u e t o t h e f l o w sep a ra t i o n p h e -

n o m en o n d i scu ssed p rev io u s ly . In e f f ec t , h ea t i s b e in g co n -

v ec t ed f ro m th a t l o ca t i o n t o w ard t h e n a r ro w face , w id e face ,

an d S E N fas t e r t h an i t can p ro p ag a t e t o t h e t o p o f t h e f l u x

b y c o n d u c ti o n . T h i s c o n v e c t i v e r e m o v a l o f h e a t f r o m t h is

V I I. E X P E R I M E N T A L V A L I D A T I O N

A. Experimental Procedure

T h e m o d e l p r e d i c ti o n s o f t h is w o r k w e r e v a l i d a t e d b y

m ea su r in g t h e d ep th s o f t h e p o w d er / l i q u id f l u x an d l i q u id

f lu x / st ee l i n t e r f aces a t an o p e ra t i n g s l ab cas t e r a t L T V S tee l.

T h e i n e x p e n s iv e m e t h o d e m p l o y e d ( F i gu r e 1 6) c o n s is t s o f

s i m p l y l o w e r i n g a w o o d e n n a i l b o a r d i n to t h e m o l d . T h e

n a i l b o a rd co n ta in s tw o ro w s o f s ev en s t ee l n a i l s an d a lu -

m in u m w i re . I t i s k ep t h o r i zo n ta l an d ca re fu l l y l o w ered

u n t il i t s u n d e r s id e j u s t t o u ch e s t h e t o p su r f ace o f th e m o ld

p o w d e r . E a c h r o w o f n ai ls i s t he n 8 0 m m f r o m t h e w i d e f a c e

m o ld w a l l s . L iq u id s t ee l co a t s t h e n a i l s u p t o t h e f l u x / s t ee l

i n t e r f ace . A t t h e sam e t im e , t h e l i q u id f l u x m e l t s t h e a lu -

m in u m w i res t o so m e p o in t ab o v e t h e s t ee l l ev e l . T h e n a i l

b o a rd i s r em o v ed a f t e r 1 t o 3 seco n d s . T h e d i s t an ce f ro m

th e en d o f th e a lu m in u m w i re t o t he s t ee l l ev e l o n t h e n a il

M E T A L L U R G I C A L A N D M A T E R I A L S T R A N S A C T I O N S B V O L U M E 2 7B , A U G U S T 1 99 6- -- 67 9


9/14

" N A I L . B O A P J D "

A L U M ]N IJ 'M W IR E ~ ~

S E N

- - . ~ r r i r -

s w ~ . _ _ _ _ ] - I _ .

f

Y

N A R R O W F A C E M O L D W A L L

[ ] P O W D E R F L U X [ ] U Q U I O F L U X [ ] S T E E L

F i g . 1 6 - - S c h e m a t i c o f l a y e r t h i c k n e s s m e a s u r e m e n t m e t h o d .

, - , - ,- , - , , , ,~ b b b ,~ d ~ , ,, ,, ,, -. ,, ,, ,.

. . . . . . . . . . . . .

, A l u m i n u m W i r e

Steel Nai l

I I D r e co 1

i q u i d S tee l

F l o w

m

= P o w d e r L a y e r

T h i c k n e s s

= L i qu id Lay er

T h i c k n e s s

S o l i d i f i ed S tee l

H e a d

f r o m M o l d

F i g . 1 7 ~ C l o s e - u p o f c r o s s s e c t i o n t h r o u g h n a i l b o a r d .

i n d i ca t e s t h e li q u id f l u x th i ck n ess F ig u re 1 7 ). T h e s lo p e o f

the so l id i f ied s teel ind icates f low d irect ion .

A l u m i n u m m e l t s a t a p p r o x i m a t e l y 6 6 0 ~ w h i l e t h e t y p -

i ca l f l u x m e l t i n g p o in t is ab o u t 1 00 0 ~ T h e a lu m in u m

w ires t h a t r em a in ed a f t e r ex t r ac t in g th e b o a rd a lw ay s h ad

p o in t ed en d s . T h i s i n d i ca t e s t h a t t h e im m ers io n t im e w as

t o o s h o r t t o a l l ow c o m p l e t e m e l t i ng o f t h e a l u m i n u m a n d

t h a t t he p o r t i o n o f w i r e t ha t w a s r e m o v e d w a s m e l t e d v e r y

q u ick ly . S u ch f a s t m e l t i n g w as d eem ed to b e t h e r e su l t o f

t r an s i en t m e l t i n g in a f l u id t em p era tu re co n s id e rab ly h o t t e r

th an th e l i qu id u s t em p era tu re o f a lu m in u m . H en ce , i t w as

- - - . . . . . . . . . ] 1

I DATA

F O R A P L A N E 8 0 m m [

25 ~ ........... -"

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . i

~l~ 15 ................................................................................................................................

d

10

.................................................: ...................

i i

.~ s

i

o l ~ I . . . . . . . . . . . . . . t

. . . .

0 100 200 300 400 500 600 700

D i s t a n c e f r o m N a r r o w F a c e

p a r a l l el to W i d e F a c e ) , x m m )

F i g . 1 8 - - C o m p a r i s o n o f m e a s u r e d a n d p r e d i c t e d l iq u i d l a y e r t h i c k n e s s e s .

N

t.

is

o

=

I Steel/Fluxnterface Experiment)

0 . . . . [ . . . . I . . . . [ , - - - .o - - - F lu xMe lt nterface Experiment)

i . . . . . . . . . Rux Melt nterface 3DModel,

- s -

~.;--.::--- ~...................

w D ~ - r ..................... .........................F....................... .....................

- 1 0 -

~ o ~ i . . . . . . . . . . . . . . . . . . i . . . . . . . . . . .. . . . . . . . . . . . . t . . . . . . . . . . .. . . . . . . . . . .. . . i . . . . . . . . . . . . . . . . . .

-3s . . . . I . . . . i . . . . . . . . . . . . . . . . . . . .

0 100 200 300 400 S 00 600 700

D i s ta n c e fr o m N a r r o w F a c e

p a r a l l e l t o w i d e f a e e ) , x r a m )

F i g . 1 9 ~ o m p a r i s o n o f m e a s u r e d a n d p r e d i c t e d m e l t - i n t e r f a c e p o s i t io n s .

a s s u m e d t h a t th e e n d o f t h e a l u m i n u m w i r e i n d i c a t es t h e

lo ca t io n o f t he i n t e r f ace b e tw een th e h ig h -c o n d u c t iv i ty l i q -

u id f l u x an d th e l o w -co n d u c t iv i ty p o w d er , r a th e r t h an th e

6 6 0 ~ i so th e rm .

B . C o m p a r i s o n w i t h M o d e l P r e d i c t i o n s

T h e s t an d a rd m o d e l co n d i t i o n s w ere ch o sen to m a tch th e

co n d i t i o n s o f th e s t ee l p l an t m easu re m en t s . F ig u re 1 8 co m -

p ares t h e l i q u id l ay e r t h i ck n ess f ro m th e m o d e l w i th t h e

m e a s u r e m e n t s . T h e m o d e l r e a s o n a b l y p r e d i c t s t h e o v e r a l l

t r en d o f l i q u id f l u x d ep th o v e r t h e en t i r e d o m ain . C o n s id -

e r in g th e c r u d e n a t ur e o f t h e m e a s u r i n g a p p a r a t u s a n d t h e

m a n y m o d e l i n g a s s u m p t i o n s , t h e a g r e e m e n t b e t w e e n t h e

m o d e l an d ex p e r im en t i s s ig n i f i can t .

F ig u re 1 9 co m p ares t h e m o d e l p red i c t i o n s o f th e l o ca t io n

o f t h e m e l t i n t e r f ace w i th t h o se o b ta in e d ex p e r im en ta l ly a t

t h e sam e p o s i t i o n . T h e ag reem en t im p l i e s t h a t t h e o b se rv ed

re l a t i v e ly f l a t i n t e r f ace o f t h e l i q u id f l u x l ay e r can b e ex -

p l a in ed w i th o u t g rav i ty , w h ich w as ig n o red in t h e m o d e l .

T h i s d em o n s t r a t e s t h e im p o r t an ce o f t h e r ec i r cu l a t i o n p a t -

t e rn g en e ra t ed b y th e f l o w in g s t ee l , a s d i scu ssed p rev io u s ly .

6 8 0 - - V O L U M E 2 7 B , A U G U S T 1 9 96 M E T A L L U R G I C A L A N D M A T E R I A L S T R A N S A C T I O N S B


10/14

/

le0

3

F i g . 2 0 - - 3 - D t e m p e r a t u r e d i s t r i b u ti o n c a l c u la t e d f o r f lu x w i t h l o w e r l i q u i d

v i s c o s i t y t e m p e r a t u r e s i n ~

V I I I . E F F E C T O F P R O C E S S V A R I A B L E S

D u e t o th e m a s s i v e c o m p u t e r r e q ui r e m e n t s o f th e 3 - D

m o d e l , i t w as o n ly ru n ag a in t o i n v es t ig a t e t h e e f f ec t o f

l i q u id f l u x v i sco s i ty . F u r th e r p a ram et r i c s tu d i e s w ere p e r -

fo rm ed u s in g 2 -D an d 1 -D s im p l i f i ca t io n s o f t h i s m o d e l .

S t a n d a r d c o n d it i o ns ( T a b l e I ) w e r e a s s u m e d , e x c e p t w h e r e

m en t io n ed o th e rw i se .

A Viscosity

T h e v i sc o s i ty - t em p era tu re cu rv e s ign i f i can t ly ch an g e s

w i th f l u x co m p o s i t i o n . T h e cu rv e fo r a g en e ra l ly l o w er v i s -

co s i ty f l u x i s i n c lu d ed in F ig u re 7 . F ig u re 2 0 sh o w s th e

e f f ec t o f u s in g th is l o w er v i sco s i ty f l u x o n th e t em p era tu re

co n to u r s . R e la t i v e t o t h a t w i th t h e s t an d a rd f l u x (F ig u re 1 4 ),

t h e su r f ace t em p era tu re fo r t h e l o w er v i sco s i ty f l u x i s g en -

e ra l ly h ig h e r , an d th e co ld r eg io n a s so c i a t ed w i th t h e f l o w

s e p a r a t i o n i s m o r e c o m p r e s s e d . T h e s e d i f f e r e n c e s a r e

cau sed b y ch an g es i n t h e f l o w f i e ld . T h e lo w er v i sco s i ty

p e rm i t s l a rg e r v e lo c i t i e s t o d ev e lo p in t h e l i q u id , fo r t h e

sam e s t ee l - f l o w d r iv in g fo rce a t t h e f l u x / s t ee l i n t e r f ace .

H i g h e r v e l o c i t ie s g e n e r a t e m o r e t h e r m a l c o n v e c t i v e m i x i n g ,

so m o re h ea t i s t r an s fe r r ed f ro m th e f l u x / s t ee l i n t e r f ace t o

th e u p p e r l ay e r s o f t h e f l u x .

F ig u res 2 1 an d 2 2 s h o w th e e f f ec t o f v i sco s i ty o n th e

p o s i t i o n o f t h e m e l t i n te r f ace a s g iv en b y th e 1 00 0 ~ t em -

p e ra tu re co n to u r . In g en e ra l , d ec reas in g f lu x v i sco s i ty sh if t s

th e m e l t i n t e r f ace u p w ard . T h e d eep e r l i q u id l ay e r i n t h e

cen t r a l r eg io n i s d u e to t h e s t ro n g e r r ec i r cu l a t i n g v e lo c i t i e s

a n d a s s o c i a t e d t h e r m a l c o n v e c t i o n t h a t a c c o m p a n y t h e

lo w er v i sco s i ty . T h e ch an g e n ea r t h e m en i scu s i s l e s s , h o w -

ev e r , an d th e ch an g e n ea r t h e n a r ro w face i s n eg l ig ib l e .

F e e d i n g t o t h e w i d e f a c e w o u l d b e e x p e c t e d t o i m p r o v e

w i th l o w er v i sco s i ty p o w d e r s , o w in g to t h e g en e ra l i n c rease

in l i qu id l ay e r d ep th s . T h i s f i n d in g i s i n ap p a ren t ag ree m e n t

w i th t h e f r eq u en t ly q u o ted em p i r i ca l r e l a t i o n sh ip o f /_ t Vc =

constan t,t~-3.421 wh ich s ta tes that a low er v isc osi ty shou ld be

u sed a t h ig h e r cas t i n g sp eed s t o ach iev e ad eq u a te l u b r i ca -

t i o n to av o id p o w d er - r e l a t ed q u a l i t y p ro b lem s .

B Thermal Condu ctiv i ty

T h e e f f ec t o f l i q u id f l u x th e rm a l co n d u c t iv i ty w as in v es -

t i g a t ed u s in g th e f i n e -m esh , 2 -D m o d e l , [231 des crib ed in Se c-

t i o n V I . B ecau se th i s m o d e l o v e rp red ic t s t h e l i q u id l ay e r

th i ck n ess , i t s p red i c t i o n s a r e i n t e rp re t ed r e l a t i v e ly .

S t e e V R u x I n t e r f a c e ( E x p e r i m e n t a l ) 1 i

0 ,

" M e l t I n t e r f a c e ( 3 D - C a s e 1 : H i g h e r ~ t ~ )

. . . . . . . . . M e l t I n t e r f a c e ( 3 D - C a .s e 2 : L o w e r / ~ ) | . , . . . .

. , . : . . -

5 - ; 2 . : : \ ;

. . . . . - - - _ ~

_ _

~ p O W O E R

F L U X

. . . . . ~ ~

-1o- ~ i ~ ..... ~... ... ... ... .... .... .. :::- ............> ~ . . .. .. .. .. . . . . . . .

-1 5 . . . . . . . . . . . . . ~ . . .. . . . .. . . . .. . . .. ~zZ: . . . . .. . . . .. . . .. . . . .. . . . .. . . .. . . . .. . . . .. . - ' - i . . . . . . . . . . . . . . . . . ~ . .. . . . .. . . .. . . . .. . . .. . . . .. . . . .. . . .. . . . .. .

. . . . . . . .. . . . . . . .. . . . . i . . . . . . . . . . . . . . . . . .

u o u t o ~ L O X . . . . . . . . . . . . . . . . . . .

:

9 -30

- - 3 s t ' ' ' I . . . . I . . . . I . . . . I . . . . I . . . . ~ . . . .

._~

~ , 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

D i s t a n c e f r o m N a r r o w f a c e

p a r a l l e l t o W i d e F a c e ) , x ( m m )

F i g, 2 1 - - E f f e c t o f v i s c o s i ty o n a v e r a g e p o s i t i o n o f m e l t i n t er f a c e.

I H I G H L I Q U I D V I S C O S I T Y I

1 L O W L I Q U ID VIS OSITY ~

F i g . 2 2 - - P o s i t i o n o f m e l t i n t e r f a c e a t s e v e r a l l o c a t i o n s i n t h e m o l d a s a

f u n c t i o n o f l i q u i d f l u x v i s c o s i t y .

F ig u re 2 3 co m p ares t h e m e l t i n t e r f aces fo r l i q u id t h e rm a l

cond uct iv i t i es o f 1 .0 W m -~ K -~ and the s tandard 3 .0 W

m -~ K - L A th ree fo ld i n c rease i n t h e rm a l co n d u c t iv i ty i n -

c reases t h e l i q u id l ay e r t h i ck n ess i n t h e cen t r a l r eg io n b y

o n ly 2 0 p c t . T h i s i s fu r th e r ev id en ce o f th e im p o r t an ce o f

co n v ec t io n th e re , H o w ev er , t h e re l a t i v e im p o r t an ce o f co n -

v ec t io n to co n d u c t io n v a r i e s w i th p o s i t i o n . C o n v ec t io n i s

l e s s im p o r t an t n ea r t h e m en i scu s , w h ere t h e t h ree fo ld i n -

c rease i n co n d u c t io n ro u g h ly d o u b les t h e l i q u id l ay e r th i ck -

ness .

M E T A L L U R G I C A L

A N D M A T E R I A L S T R A N S A C T I O N S B V O L U M E 2 7 B . A U G U S T 1 9 9 ~ - 6 8 1


11/14

s . . . . . . . i . . . I . . . . . . . [ . . . .

. . . . . . . : . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . , ' : . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N - 5 . . , . , . . . : . ~ , ~ - . . . . . . :. . . . . . . . - . ~ . . ..

"~ - lO . . . . .. . . . . , . . . i . . . . .. . . . . .. . . . . .. . . . .......~,.,,, ' :.......i ..................... ~ ........................ ....................... ...................

- = " ' - . . . . . ~ " i i i i 9

- = i \ i I . . . . . . . . M e l t l n t e c f a c e (2 O ' k l iQ = l O w m ' t K ' l I :

. . . . . . . . . . . . . . . . . . N I - - " 1 ' I :

. . .

- 3 0

. a s . . . . I . . . . . . . . ~ , . . . . , . . . . I . . . . I . . . .

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

D i s t a n c e f ro m N a r r o w f a c e , x m u m )

F i g . 2 3 - - E f f e c t o f l i q u i d t h e r m a l c o n d u c t i v i t y o n l o c a t i o n o f m e l t

i n t e r f a c e .

o . I . . . . . . . . . . . . [ . . . . / - - ~ - 1 -

. . . .

' - - - i , . - i - - - i i

j .

.5 - ~: : , - - , ; -~.- i . .. . . .. . . - - i . .. . . .. . . . ~ -~ . . . . . . . . ~ . . . . . . . . . . . . - . . . . .. . . .. . . .. . . . i .. .. .. .. .. .. .. .. .. .. . .

" U . ~ ' " . ~ " . - . - ' ~ . - " " i i . . .. . . . .. . ~ - - . ~

i i i

.

~ . . 2 0 T . . . . . . . . . . ~ '---~-1 ~ ' ~ ( ~ ' ~ 0 ~

~ - 3 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- 3 5 I . . . . I . . . . t I . . . . I . . . . . . . . I . . . .

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0

D i s t a n c e f r o m N a r r o w f a e e ,

x ( m m )

F i g . 2 4 - - C o m p a r i s o n o f p r e d i c t i o n o f m e l t i n te r f a c e l o c a t i o n f o r I -D , 2 -

D , a n d 3 - D m o d e l s .

0 03

. . . . . . . . . . . . I . . . . . . . . . . . . . . . . . .

0 . 0 2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . . .. . . . . ~ . . . . . .. . . . .. . i . . . . : ~ . ~ . . . . * . . . . . . . . ~ : . : - : . :: : : : L : z : : : . : : : L J . : 2 : . ' 2 2 L

E

2 . . . . . . . : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . 0 1 5 - - ? - " " - --~ ............ ..........T ............. ................ ............. .............

. . . . . . . . . . .. . . . T . . . . . . . . .. . . . . . . . . . . . . . . . . . .

: i . . . . . . . . . . - . . . . . . . . . - . . . . . . . . . . . . .

. ~ 0 . 0 1 . . . . . . . . . . . . . . . . . . - i . . . . .. . . . .. . . . . .. . . . .. . . . . i

0. 0 05 . . . .. . .. . . . . . . .. . . .. . . .. . ~ t . . . . . . . . . I f ,m o l I = ~ C , . m e l l = 1 1 0 0 * C

:,

I

T f ' r n d = l O 0 0 * C . . . . . .

T l m e 1 2 0 0 * C

i : , , , , , , , , , , J , , , , , , , , , , , , , , , , , , ,

o I I

0 . 0 5 0 , 1 0 . 1 5 0 . 2 0 , : > 5

P o w d e r L a y e r T h i c k n e ss

m

F i g . 2 5 - - 1 - D p r e d i c t i o n o f r e l a t i o n s h i p b e t w e e n p o w d e r l a y e r t h i c k n e s s

a n d l i q u id l a y e r t h i c k n e s s a s a f u n c t i o n o f t he f l u x m e l t i n g t e m p e r a t u r e .

C . 1 - D S t e a d y - S t a t e M o d e l

A s i mpl i f ied I - D mod e l was deve l op ed t o f u r the r inves -

t i ga te t he e f f ec t o f p r oces s and ma t e r i a l pa r amet e r s on l i q -

u i d f lux l aye r t h ickness . F l ux was a s sum ed t o f l ow

uni f o r m l y dow nwar d a t t he consump t i on r a te . Ana l y t i ca l

hea t ba l ances wer e pe r f o r med ove r t he f l ux domai n , ac -

coun t i ng f o r l a t en t hea t evo l u t i on a t t he l i qu i d / so l i d i n t e r -

f ace . T he r e su l t i ng equa t i ons ( Append i x) wer e so l ved on a

spreadsheet .

F i gur e 24 shows t he d i f f e r enc e be t wee n t he f l ux me l t

i n t e r f ace l oca t i ons p r ed i c t ed by t he 3 - D , 2 - D , and 1 - D

s t eady- s t a t e mode l s . Because t he I - D mode l does no t p r op-

e r l y accoun t f o r convec t i on i n t he l i qu i d poo l , i t

unde r pr ed i c t s t he l iqu i d l aye r t h ickness nea r t he S E N and

over pr ed i c t s i t nea r t he meni scus . T he 2 - D mode l i ncor -

por a t e s conve c t i on , bu t i t ove r pr ed i c t s the l i qu id l aye r

t h i ckness ove r mos t o f t he domai n becau se i t ignor es mass

consumpt i on t o t he w i de f ace a s d i s cus sed p r ev i ous l y . F i -

na l l y , i t can be s een t ha t t he 3 - D mode l does t he bes t j ob

of p r ed i c t i ng t he l oca t i on o f t he f l ux / me l t i n t e r face , pa r t i c -

u l a r l y a t t he meni scus w her e t he 3 - D e f f ec t s o f mass t r ans -

por t a r e mos t i mpor t an t .

Recog ni z i ng i t s li mi t a ti ons , t he 1 -D m ode l was f i rs t ap -

p l i ed t o i nves t i ga t e t he r e l a t i onsh i p be t ween powder l aye r

t h i ckness and l i qu i d l aye r t h i ckness . F or t o t a l f l ux l aye r

t h ic k n e s s l es s th a n 2 5 m m , a d d i n g m o r e p o w d e r i m p r o v e s

t he r ma l i nsu l a t i on and g r ea t l y i nc r eases t he s t eady- s t a t e

t h i ckness o f t he l i qu i d f lux l aye r . As t he p ow der l aye r i s

i nc r eased , i ts e f f ec t on l i qu i d t h i ckness l e s sens . F or po wd er

l aye r th i ckness g r ea t e r t han 75 m m, t he e f f ec t on l i qu i d

thickness i s smal l (F igure 25) .

T he m ode l was t hen used t o exami ne t he r e l a t i ve i mpo r -

t ance o f s eve r a l o t he r p r oces s and ma t e r i a l pa r amet e r s on

i nc r eas i ng t he l i qu i d f l ux dep t h . S pec i f i c f ind i ngs o f t h i s

mod e l , f o r t he s t anda r d ope r a t i ng cond i t i ons in T ab l e I and

f or a r e f e r ence powd er l aye r t h i ckness o f 5 cm , i nc l ude t he

f o l l owi ng .

( 1 ) A dec r ease in the me l t i ng po i n t f r om 1000 ~ t o 900

~ i nc r eases t he li qu i d l aye r t h i ckness by 20 pc t . T h i s

i s i l lus t ra ted in F igure 25.

( 2 ) Ha l v i ng t he consum pt i on r a t e doub l es the l i qu i d t h i ck-

ness .

( 3 ) I nc r eas i ng the l i qu i d the r ma l con duc t i v i t y f r om 2 t o 3

W m - l K - t i nc r eases t he l i qu i d t h i ckness b y 50 pc t .

T h i s p r opor t i ona l i nc r ease i s r ough l y cons i s t en t w i t h

t he e f f ec t o f t he rma l c ondu c t i v i t y i n t he men i scus r e -

g i on a s ca l cu l a t ed by t he 2 - D mode l .

( 4 ) Ha l v i ng t he pow der t he r ma l cond uc t i v i t y i nc r eases t he

l iquid thickness by less than 20 pct .

( 5 ) Dec r eas i ng t he l a t en t hea t by 67 pe t i nc r eases t he l i qu id

t h i ckness by 38 pc t .

( 6 ) P owder emi s s i v i t y a t t he t op su r f ace has a neg l i g i b l e

e f f ec t on t he l i qu id t h i ckness .

I X. 1 -D T R A N S I E N T M O D E L

A . M o d e l F o r m u l a t i o n

M a n y p r e v i o u s r e s e a r c h e r s h a v e m e a s u r e d a n d r e f e r r e d

t o t he me l t i ng r a t e o f a f l ux , t o cha r ac t e r i ze i t s me l t i ng

behav i or . T h i s wo r k i ns t ead cha r ac t e r i zes the m e l t i ng be -

hav i o r o f t he f lux us ing t he mo r e f undam ent a l p r ope r t i e s o f

t he r ma l conduc t i v i t y , en t ha l py , and v i scos i t y . Assumi ng

t ha t t he me l t i ng r a t e is s i mpl y a m ani f e s t a t i on o f t he mor e

f undament a l ma t e r i a l p r ope r t i e s , t he p r esen t mode l i ng ap-

pr oach shou l d a l so be capab l e o f ca lcu l a t i ng a me l t i ng r a te

f o r con t i nuous cas t i ng op e r a t i ng cond i t i ons .

6 8 2 - - V O L U M E 2 7B o A U G U S T 1 9 96 M E T A L L U R G I C A L A N D M A T E R I A L S T R A N S A C T I O N S B


12/14

0 , 0 2 5 L~ : ' . . . . . . . . N ~ l ~ C ~ w ~ n s p u l l2 t ~ . 4 1 < ~ J ( . . . . . . ' s t ~ l ) ~ - ~

, o . o 2

. . . . . . .- U : I I I T U 7 7 1 .. . . . . i i . . . . . . 1

:

: i T

0 . 0 1 : ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

.=

0 . 0 0 5 . , , , , : . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .: . . . . ' i:. . . . . . . . . . ., " . . . . . . ., , ' , , , ' ' , . , ,

0 r J I I i I I ~ I I I I

1 0 2 0 3 0 4 0 50 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0

Time minutes)

F i g . 2 6 E f f e c t o f l i q u i d f l ux c o n s u m p t i o n o n t i m e t a k e n to a c h i e v e s t e a d y

state.

0 . 0 3 5

0 . 0 3

y 0 . 0 2 5

E

0 . 0 2

0 . 0 1 5

~ O Ol

r~

0.005

. . . . . . . . . . . . . . . . . . . . . . . . ] . . . .

- - - o - - - F l ux I

.................. ~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . i . . . 9

Flux 11

i i i i i J F l u x I V

1 2 3 4 5 6

m o u n t o f C a r b o n

(wt )

F i g . 2 7 - - E f f e c t o f c a r b o n c o n t e n t o n m e l t i n g r a t e o f fl u x.

T o in v es t ig a t e t h e t r an s i en t m e l t i n g b eh av io r o f t h e f l u x ,

a 1 -D t r an s i en t f i n i t e e l em en t m o d e l w as fo rm u la t ed . T h e

m o d e l s im u la t e s a t h in s t r i p t h ro u g h th e f l u x . B y in c lu d in g

th e t r an s i en t t e rm an d n eg lec t in g th e x an d y t e rm s , t h e

e n e r g y c o n s e r v a t i o n e q u a t io n b e c o m e s

T h e m a s s a n d m o m e n t u m e q u a t i o ns s i m p l i f y t o g i v e a

co n s t an t v e lo c i ty i n t h e z di r ec t io n , w h ich m u s t b e im p o se d .

U s in g th e sam e s t an d a rd in p u t m a te r i a l p ro p e r t i e s an d ap -

p l i cab le b o u n d ary co n d i t i o n

(e .g . ,

1550 ~ at s teel / f lux in-

t e r f ace ) a s i n t h e 2 -D an d 3 -D m o d e l s , t h e p reced in g n o n -

l in ea r eq u a t io n w as so lv ed u s in g F ID A P . F lu x t em p era tu re

w as ca l cu l a t ed u s in g an im p l i c i t t im e in t eg ra t io n sch em e .

B . T y p i c a l R e s u l t s

P red ic t ed g ro w th o f t h e l iq u id l ay e r t h i ck n ess w i th t im e

i s sh o w n in F ig u re 2 6 . T h e m e l t i n t e r f ace ( i n d i ca t ed b y th e

1 0 00 ~ i so th e rm ) i s s een to r each eq u i l i b riu m a f t e r ab o u t

1 2 0 m in u tes w i th ze ro f l u x co n su m p t io n . T h e m e l t i n g

sp ee d d ec reases f ro m 0 .0 2 3 0 m m s -~ a t t = 1 m in u te ,

t o 0 .0 0 5 m m s - I a t t = 2 5 m in u tes , t o 0 . 0 0 05 m m s -~ a t t

= 6 0 m in u tes , acco rd in g to t h e t an g en t o f t h e n o -co n -

su m p t io n cu rv e in F ig u re 2 6 . T h ese v a lu es a r e co m p arab le

w i th t h e ex p e r im en ta l m e l t i n g r a t e s f ro m F ig u re 2 7 , b ased

o n d a t a o b ta in ed b y X ie

et al.V,

F o r t h e 3 . 8 p c t ca rb o n

co n ten t o f t h e f lu x s tu d ied in t h is p re sen t w o rk , t h e m e l t i n g

ra t e sh o u ld b e ap p ro x im a te ly 0 .0 1 k g m - : s -~ f ro m F ig u re

2 7 . T h i s m e l t i n g r a t e co r r e sp o n d s to a m e l t i n g sp eed o f

0 . 0 0 4 m m s -~ o r 0 . 0 8 k g / ( to n n e o f s t ee l) . T h i s ex p e r im en ta l

m e l t i n g sp eed co r re sp o n d s to th a t co m p u ted a t t = 2 7

minu tes in F igure 26 .

F u r th e r , F ig u re 2 7 sh o w s th a t m e l t i n g r a t e i s a s t ro n g

fu n c t io n o f ca rb o n co n ten t i n t h e f l u x . T h i s e f f ec t i s l ik e ly

d u e t o a n i n c r e a s e in t h e m a x i m u m o f t h e a p p a r e n t v i s c o s i t y

cu rv e (F ig u re 7 ) a t t h e f l u x /p o w d er i n t e r f ace , w i th i n c reas -

in g ca rb o n co n ten t . T h i s i s cau sed b y ca rb o n p a r t i c l e s f l o a t -

i n g o n th e l i q u id l ay e r an d co a t in g th e s in t e r in g f lu x an d i s

a f f ec t ed s t ro n g ly b y th e ca rb o n p a r t i c l e s i ze . T h e r e su l t i s

a d ec rease i n b o th m e l t i n g r a t e an d s t ead y l i q u id l ay e r

t h i c k n e s s w i t h i n c r e a s i n g c a r b o n . A s e c o n d p h e n o m e n o n

ex p la in in g th i s t r en d i s t h e i n c reased ap p a ren t l a t en t h ea t

o f t h e f l u x w i th i n c reas in g ca r b o n co n ten t , d u e to t h e en -

d o t h e r m i c r e a c t io n t o f o r m C O 2 a n d C a O f r o m C a C O 3 .

T h e t r an s i en t I - D m o d e l w as r e - ru n fo r a sp ec i f i c co n -

su m p t io n o f 0 . 4 k g / ( to n n e o f s t ee l) t o m a tch th e co n d i t i o n s

o f N a k a n o

e t at. [ 19J

fo r a cas t i n g sp eed o f 1 . 5 m /m in . F ig u re

2 6 i l l u s t r a t e s t h e e f f ec t o f im p o s in g th i s co n su m p t io n r a t e

o n th e t r an s i en t r e sp o n se . T h e t im e to ach iev e s t ead y s t a t e

i s s een to d ec rease t o 2 5 m in u tes , w h ich i s c lo se t o t h a t

p red ic t ed b y N ak an o . T h e d ro p in m e l t i n g r a t e w i th t im e i s

e v e n m o r e p r o n o u n c e d. I t sh o u l d b e n o t e d t h a t t he N a k a n o

m o d e l u ses a s in t e r in g m o d e l t o d e f in e t h e t h e rm a l co n d u c-

t i v i t y an d sp ec i f i c h ea t fo r t h e d i f f e r en t fo rm s o f t h e f l u x ,

w h i l e t h e m o d e l p r e s e n t e d h e r e u s e s a v e r a g e e m p i r i c a l d a t a

fo r t h ese p ro p e r t i e s a s fu n c t io n s o f t em p era tu re .

C . E v a h t a t io n o f S t e a d y - S t a t e A s s u m p t i o n

B ased o n th e r e su l t s o f S ec t io n B , i t s eem s r easo n ab le

t h a t q u a s i s t e a d y -s t a t e l iq u id le v e l s c a n b e a p p r o a c h e d

in p rac t i ce . N ak an o et al. t~91v a l id a t ed th i s co n cep t b y t ak in g

m e asu re m e n t s o f li q u id l ay e r d ep th a t an o p e ra t in g cas te r .

T h e i r r e su l t s sh o w th a t t h e p o o l t h i ck n ess ex h ib i t s a n ea r ly

co n s t an t v a lu e , w h ich v a r i e s w i th cas t i n g p a ram ete r s . T h e

i n t e rm i t t e n t a d di t io n o f p o w d e r w a s m o d e l e d b y D e h a l l e e t

aLt~S]

an d p red ic t ed to h av e n o s ig n i f i can t e f f ec t o n th e l i q -

u id l ay e r th i ck n ess , o n ce th e q u as i - s t ead y - s t a t e i s ach i ev ed .

H o w ev er , F ig u re 2 8 sch em at i ca l ly p resen t s t h e t y p i ca l v a r -

i a t i o n s i n th e l i q u id l ay e r t h ick n ess ex p ec t ed u n d e r t h ese

q u as i - s t ead y - s t a t e co n d i t i o n s .

T h e f lu x th i ck n ess d a t a p o in t s i n F ig u re 2 8 w ere m eas -

u red a t L T V S tee l a t t h e q u a r t e r w id th a lo n g th e m o ld cen -

t e r l i n e fo r a cas t i n g sp eed o f 1 . 07 m /m in . T h e p red ic t ed

1 -D s t ead y - s t a t e l i q u id f l u x th i ck n ess (2 . 1 cm ) w as t ak en

f ro m F ig u re 2 5 . F ig u re 2 8 sh o w s th a t w i th i n t e rm i t t en t

p o w d er ad d i t i o n , t h e l i q u id f l u x th i ck n ess p ro b ab ly v a r i e s

a b o u t a m e a n q u a s i - s t e a d y v a l u e , w h i c h m a y b e l e s s t h a n

th e s t ead y - s t a t e v a lu e p red ic t ed w i th a co n s t an t r a t e o f p o w -

d e r ad d i t i o n . T h e d i f f e ren ce b e tw een th e q u as i - s t ead y an d

s t ead y v a lu es a r i se s b ecau se th e t y p i ca l t im e in t e rv a l b e -

tw een p o w d er ad d i t i o n s (o n th e o rd e r o f 3 m in u tes ) i s so

m u ch l e s s t h an th e t im e n eed ed to r each s t ead y s t a t e ( ab o u t

2 5 m in u tes ) . T h e re i s n o t en o u g h t im e fo r t h e fu l l s t ead y -

s t a t e l i q u id t h i ck n ess t o d ev e lo p . B ased o n th e sam e a rg u -

ment , the t ime is su ff ic ien t ly shor t that the f luctuat ion in

METALLUR GIC AL AND MATER IALS TR ANSAC TIONS B VOLU ME 2 7 B , AUGUST 1 9 9 6- - -6 8 3


13/14

l iquid depth i s not s igni f icant so the quasi -s tead y-s ta te as-

sumpt i on appea r s t o be r easonab l e . T hese r e su l t s i mpl y t ha t

t he t r ue l i qu i d l aye r t h i ckness shou l d be sma l l e r t han t ha t

p r ed i c t ed by t he s t eady- s t a te mode l s .

X . C O N C L U S I O N S

A 3-D s t eady- s t a t e coup l ed f l u i d f l ow and hea t - t r ans f e r

f i n i te - e l ement mode l has been fo rmul a t ed t o ca l cu l a t e the

ve l oc i t y and t he rma l f i e lds deve l oped i n t he t op- sur f ace f l ux

l aye r s i n a con t i nuous s t ee l s lab-cas ti ng mo l d und er t yp i ca l

ope ra t i ng cond i t i ons . T he m ode l i nc l udes d i f f e r en t temper -

a t u r e -dependen t f unc t i ons fo r v i scos i t y and t he rma l con-

duc t i v i t y i n r eg i ons o f me l t i ng and so l i d i fy i ng o f t he f l ux .

T he mode l ca l cu l a t i ons rough l y agree wi t h measured f l ux

l aye r th i cknesses i n an ope ra t i ng s lab cas t i ng mol d fo r a

t yp i ca l r ec i r cu l a ti ng s t ee l f l ow pa t t e rn w i t h su r f ace f l ow d i -

r ec t ed back t ow ard t he S E N . S pec i f i c f i nd ings o f t he mod e l

i nc l ude t he fo l l owi ng .

1 . F l ow i n the meni scus r eg i on i s e ssen t i a l ly p lana r norma l

t o t he d i r ec ti on o f max i mu m hea t f l ux . T he re fore hea t

t r ans f e r i n t h i s r eg i on i s con t ro l l ed by conduc t i on and

latent heat of the f lux and the l iquid f lux laye r i s thin.

2 . Du e t o s t ee l f l ow dragg i ng f l ux a l ong t he i n t e r f ace a

l a rge r ec i r cu l a t i on zone deve l ops i n t he cen t r a l r eg i on

of t he f lux . T h i s i nc r eases convec t i on caus i ng a deepe r

l i qu i d l aye r nea r t he S E N. T he r e su l t i s an a l mos t f l a t

t op su r f ace o f the l i qu i d f lux l aye r desp i t e an uneven

s t ee l su r f ace and desp i t e i gnor i ng t he e f f e c t o f g r av i t y

i n t he mode l .

3 . Rec i r cu l a t i ng f l ow ca rr i e s pow der towa rd t he na r rowface

wa l l s oppos i t e t o t he d i r ec t i on o f s tee l f low. T h i s i m-

p l i e s t ha t more powd er mus t be added a t t he cen t e r nea r

t h e S E N e v e n i f c o n s u m p t i o n a ro u n d t h e m o l d p e r i m -

e t e r i s un i fo rm.

4 . F l ow sepa ra t i on in t he f l ux l aye r be t wee n 150 and 250

mm f rom t he na r rowface r e su lt s in a ve ry th i n l i qu i d

l aye r . T h i s may r e su l t in po or f eed i ng o f l iqu i d f l ux i n t o

t he mol d- s t r and gap pa r t i cu l a rl y a t t he o f f - corne r r eg i on

of t he w i de face . T h i n f l ux l aye r s and f eed i ng p rob l ems

are a l so gene r i c nea r t he na r rowface i n gene ra l .

Or)

UJ

Z

-I-

I -

X

- I

.J

I1.

6 . 5

4 . 0

1.2

Typical Measured

Values at Center p lane

and Quarter Mold W idth

Constant

P o w d e r

Intermittent Powder Addit ion )., ~, Ru x Depth

~

I

- 120 s Predicted 1-D S t e a d y

State L iquid Thickness

f U Q U I D

M E s )

F i g . 2 8 T r a n s i e n t s c h e m a t i c s h o w i n g a p o s s i b l e e f fe c t o f i n t e r m i t te n t

p o w d e r a d d i ti o n . T h e p o i n t s a re t y p i c a l m e a s u r e m e n t s a t t h e m o l d c e n t e r

p l a n e a n d q u a r t e r w i d t h .

5 . A co l d spo t deve l op s a l ong t he w i de face wa l l abou t 150

mm f rom t he na r rowface . T h i s i s a s soc i a t ed wi t h a t h i n

l i qu i d f lux l aye r and l ow hea t l os s t h rough t he f l ux . T h i s

sugges t s a t h i cke r r e so l i d i fi ed f l ux r im and g ene ra l l y

wor se qua l i t y p rob l ems mi gh t ex i s t t he r e .

6 . V i scos i t y p lays an i mpor t an t r o l e i n t he the rma l and f l ow

behav i or o f t he f l ux . L ower l i qu i d - f l ux v i scos i t y p ro -

duces t h i cke r l i qu i d l aye r s i n t he deep cen t r a l f l ux poo l

i n th e m o l d d u e t o b e t te r c o n v e c t iv e m i x i n g . T h i s m a y

con t r i bu t e t o t he h i ghe r consumpt i on and be t t e r qua l i t y

p e r f o r m a n c e o f l o w v i s c o s it y f lu x .

7 . Chang es i n l iqu i d f l ux v i scos i t y have l i tt l e e f f ec t on t he

l i qu i d l aye r t h i ckness nea r t he meni scus o f t he na r row-

face.

8 . F or t o t a l f l ux l aye r t h i cknesses le s s than 25 m m add i ng

mo re pow der s i gn i f i can t ly inc r eases t he dep t h o f the l i q -

u i d f l ux laye r . As t he pow der l aye r dep t h i nc r eases t h i s

e f f e

FLUX AND THERMAL BEHAVIOR OF FLUX.pdf

Documents

Transcript of FLUX AND THERMAL BEHAVIOR OF FLUX.pdf