Contacting Problems Associated with
Aluminum and Ferro-AIIoy Additions in
Steelmaking-Hydrodynamic Aspects
R. I. L. GUTHRIE, R. C L I F T , AND H. HENEIN
It is c o m m o n p r a c t i c e to drop a l u m i n u m and f e r r o - a l l o y additions into t e e m i n g l a d l e s
e i t h e r p r i o r to, or d u r i n g f u r n a c e tapping o p e r a t i o n s . W a t e r model e x p e r i m e n t s a r e
d e s c r i b e d in which s p h e r e s of v a r i o u s d i a m e t e r s and specific g r a v i t i e s w e r e dropped f r o m
typical i n d u s t r i a l heights into w a t e r . M a x i m u m p e n e t r a t i o n d i s t a n c e s , t r a j e c t o r i e s and
r e t e n t i o n t i m e s w e r e m e a s u r e d and c o m p a r e d with t h e o r e t i c a l p r e d i c t i o n s b a s e d on
t r a n s i e n t fluid flow. The r e l a t i v e i m p o r t a n c e of steady drag, added m a s s and h i s t o r y
f o r c e s w e r e d e m o n s t r a t e d . R e s u l t s i n d i c a t e that i m m e r s i o n t i m e s a r e e x t r e m e l y s h o r t
(~1 s) for a l u m i n u m additions and low d e n s i t y f e r r o - a l l o y s . High d e n s i t y f e r r o - a l l o y s
r e m a i n i m m e r s e d c o n s i d e r a b l y longer and p e n e t r a t e much d e e p e r .
T HE
addition of solids to liquid s t e e l baths for a d j u s t i n g s t e e l c h e m i s t r y to r e q u i r e d s p e c i f i c a t i o n s or
for cooling p u r p o s e s is c o m m o n s t e e l m a k i n g p r a c t i c e .
Specific e x a m p l e s of solid additions to molten s t e e l
i n c l u d e the o c c a s i o n a l u s e of s c r a p m e t a l in B . O . F .
f u r n a c e s for cooling 'hot' heats, as well as the r e g u l a r
u s e of ladle additions of f e r r o - a l l o y s , c a r b o n and a l u m i n u m for alloying a n d / o r deoxidation p u r p o s e s . In
g e n e r a l , these alloying additions a r e made to the ladle
e i t h e r before or d u r i n g f u r n a c e tapping o p e r a t i o n s .
Although much i n f o r m a t i o n has b e e n gathered in
the p a s t on the way in which specific d i s s o l v e d a l l o y ing e l e m e n t s i n t e r a c t with oxygen and sulfur in the
s t e e l to f o r m o x i d e / s u l f i d e type i n c l u s i o n s (e.g., Ref.
1), the i n i t i a l d i s s o l u t i o n p r o c e s s e s t h e m s e l v e s have
b e e n l a r g e l y n e g l e c t e d . However, in o r d e r that o p t i m u m injection methods be identified and some of the
p r e s e n t e m p i r i c i s m in plant p r o c e d u r e s be r a t i o n a l i zed, it is n e c e s s a r y to u n d e r s t a n d the b a s i c p h e n o m e n a involved in these f i r s t s t e p s .
So far, it has b e e n d e m o n s t r a t e d 2,s that a solid s t e e l
s h e l l will i n i t i a l l y f o r m around any object that is
i m m e r s e d in a bath of molten s t e e l . Also, provided
the m e l t i n g ' p o i n t ' of the addition is lower than the
f r e e z i n g ' p o i n t ' of the steel, the object will n o r m a l l y
p r o c e e d to melt i n s i d e this shell. 4 Thus, one c o m m o n l y
ends up with the s i t u a t i o n of a m o l t e n core of f e r r o a l l o y or a l u m i n u m s u r r o u n d e d by a solid j a c k e t . The
j a c k e t then takes an a p p r e c i a b l e t i m e to melt back and
r e l e a s e its c o n t e n t s to the bath.
Much of this l a t t e r work has c o n c e n t r a t e d on t h e r m a l
a s p e c t s , and has supposed that the additions r e m a i n
i m m e r s e d during the c o u r s e of t h e i r m e l t i n g h i s t o r y .
T h i s will not n e c e s s a r i l y be the c a s e in actual p l a n t
p r a c t i c e s , s i n c e many of the additions c o m m o n l y made
to t e e m i n g l a d l e s , e t c . a r e l e s s d e n s e than m o l t e n
s t e e l and e x p e r i e n c e buoyancy f o r c e s . This is p a r t i c R. I. L. GUTHRIEis Associate Professor, Department of Miningand
Metallurgical Engineering,McGiIlUniversity, R. CLIFT is Associate
Professor, Department of Chemical Engineering,McGiflUniversity,
and H. HENEIN is Research Metallurgist,Sidbec-Dosco, Contrecoeur,
Quebec, and was formerly Graduate Student, Department of Mining
and MetallurgicalEngineering,McGillUniversity.
Manuscript submitted May 16, 1974.
METALLURGICALTRANSACTIONS B
u l a r l y true of a l u m i n u m whose d e n s i t y is 2.7 g m / c m 3
c o m p a r e d with steel of 7 to 7.2 g m / c m s.
PRESENT
WORK
The work presently described was undertaken to
elucidate the h y d r o d y n a m i c effects o c c u r r i n g when
alloy additions a r e i n j e c t e d into baths of m o l t e n
steel, and sought to e s t a b l i s h m a x i m u m likely depths
of p e n e t r a t i o n and r e t e n t i o n t i m e s . Such i n f o r m a t i o n
p r o v i d e s u s e f u l i n d i c a t i o n s on the extent or f e a s i b i l i t y
of s u b s u r f a c e m e l t i n g for a p a r t i c u l a r alloy additive.
In n o r m a l s t e e l m a k i n g p r a c t i c e , alloy additions to
a t e e m i n g ladle a r e made d u r i n g f u r n a c e tapping, and
e n t e r the s t e e l a f t e r p a s s a g e through alloy chutes
located on e i t h e r side of the t e e m i n g l a d l e . T h e s e
chutes a r e s t e e l tubes, about 0.3 m I.D., 3 m e t e r s long,
making an angle of about 45 deg to the h o r i z o n t a l so
as to d i r e c t the additions towards the c e n t e r of the
filling ladle. The alloy additions leaving the tube fall
in free flight about 2 to 4 m e t e r s before e n t e r i n g
the s t e e l .
In o r d e r to e x a m i n e the h y d r o d y n a m i c s of such
s i t u a t i o n s , some s i m p l i f i c a t i o n s had to be made in
the l a b o r a t o r y s c a l e s t u d i e s r e p o r t e d h e r e . In the f i r s t
i n s t a n c e , it was decided to c o n s i d e r the c a s e of
s p h e r i c a l additions, r a t h e r than i r r e g u l a r l y shaped
l u m p s or ingots, so as to s i m p l i f y the equations
d e s c r i b i n g s u b s u r f a c e motion and also to take a d v a n tage of the l i t e r a t u r e and theory that is a v a i l a b l e on
the motion of s p h e r e s through liquids. F o r s i m i l a r
r e a s o n s it was decided to t r e a t the c a s e of additions
p e n e t r a t i n g a s t e e l bath v e r t i c a l l y , r a t h e r than t h e i r
e n t e r i n g a t u r b u l e n t l y s t i r r e d bath at a slight angle as
o c c u r s in p r a c t i c e . F i n a l l y , due to the opacity of
liquid m e t a l s , a low t e m p e r a t u r e model was i n i t i a l l y
chosen. Thus, wooden s p h e r e s of v a r i o u s d i a m e t e r s
and specific g r a v i t i e s w e r e dropped through a i r into
a s t a g n a n t column of w a t e r . The r e s u l t i n g s u b s u r f a c e
t r a j e c t o r i e s were r e c o r d e d by cinephotography u s i n g
a 16mm Bolex Reflex C a m e r a . T h e s e r e s u l t s then
allowed an a p p r o p r i a t e h y d r o d y n a m i c m o d e l to be
chosen for p r e d i c t i n g t r a j e c t o r i e s , m a x i m u m i m m e r s i o n
depths and t i m e s for a wide r a n g e of conditions i n cluding those r e l a t i n g to s t e e l m a k i n g .
VOLUME 6B, JUNE 1975-321
D E V E L O P M E N T O F A P R E D I C T I V E MODEL
FOR SUBSURFACE MOTION
Equation of Motion
A n u m b e r of w o r k e r s (often a s s o c i a t e d with b a l l i s t i c
s t u d i e s ) have a l r e a d y i n v e s t i g a t e d the p h e n o m e n a o c c u r r i n g when high d e n s i t y s o l i d o b j e c t s e n t e r b a t h s of
s t a g n a n t w a t e r . (See r e c e n t r e v i e w s by Birkhoff 6 and
May6). F o l l o w i n g the c l a s s i c a l p h o t o g r a p h i c w o r k of
W o r t h i n g t o n and C o l e at the t u r n of the c e n t u r y , 7 much
of the s u b s e q u e n t r e s e a r c h h a s c o n c e n t r a t e d on s t u d y ing a) c a v i t y f o r m a t i o n when the s o l i d body f i r s t
e n t e r s the liquid (e.g., R e f s . 8, 9) and b) the b o d y ' s
s u d d e n d e c e l e r a t i o n during, and i m m e d i a t e l y following,
e n t r y (e.g., R e f s . 10, 11). T h o s e w o r k e r s (e.g., R e f s .
11, 12) who t r i e d e s t i m a t i n g the d r a g f o r c e s on the
body d u r i n g i t s s u b s e q u e n t d o w n w a r d motion t h r o u g h
the liquid e i t h e r i g n o r e d added m a s s e f f e c t s , o r c h o s e
to d i s r e g a r d s t a n d a r d d r a g c o e f f i c i e n t s a v a i l a b l e in
the l i t e r a t u r e in f a v o r of d e v e l o p i n g t h e i r own r a t h e r
specific correlations. As a result, their conclusions
l a c k g e n e r a l i t y and cannot be u s e f u l l y a p p l i e d to p r e d i c t i n g the motion of a body which is l e s s d e n s e than
the liquid into which i t is p r o j e c t e d .
A f r e s h a p p r o a c h was t h e r e f o r e t a k e n in the p r e s e n t
w o r k to d e t e r m i n e w h e t h e r s u b s u r f a c e motion could be
p r e d i c t e d s a t i s f a c t o r i l y on the b a s i s of N e w t o n ' s Second
Law of motion, by taking into a c c o u n t the v a r i o u s
f o r c e s shown s c h e m a t i c a l l y in F i g . 1, i.e.,
d (Ms U) = E F
[1 ]
dt
w h e r e Ms i s the m a s s of the s p h e r e , U i s i t s i n s t a n t a n e o u s v e l o c i t y , and F a r e the v a r i o u s f o r c e s a c t i n g
on the body. The d i r e c t i o n s of t h e s e f o r c e s a r e i n d i c a t e d in F i g . 1, and the v a r i o u s t e r m s a r e :
1) The weight of the s p h e r e , FG = Msg.
2} The buoyancy f o r c e , FB = Mg, w h e r e M i s the m a s s
of liquid d i s p l a c e d by the s p h e r e .
3) The d r a g d u e to the r e l a t i v e v e l o c i t y b e t w e e n
s p h e r e and liquid, FD = (CD~d~pUI UI ) / 8 w h e r e d is
the d i a m e t e r of the s p h e r e , U i s i t s v e l o c i t y t h r o u g h
the liquid, p i s the liquid d e n s i t y , and CD is the d r a g
c o e f f i c i e n t for s t e a d y motion t a b u l a t e d , for e x a m p l e ,
by L a p p l e and S h e p h e r d ? 3
4) The " a d d e d m a s s " t e r m , F A = CAM .(dU/dt),
w h i c h a l l o w s for the f a c t that a c c e l e r a t i o n of the
s p h e r e a l s o a c c e l e r a t e s liquid a r o u n d it. T h i s r e s u l t s
in a m o m e n t u m l o s s to the s u r r o u n d i n g fluid. Two a l t e r n a t i v e e s t i m a t e s a r e a v a i l a b l e for FA, depending on
w h e t h e r the a d d e d m a s s c o e f f i c i e n t , CA, i s e q u a t e d to
i t s c l a s s i c a l v a l u e of 1/214'1s o r i s a s s u m e d to v a r y
with p a r t i c l e v e l o c i t y and a c c e l e r a t i o n in the m a n n e r
d e s c r i b e d by O d a r and H a m i l t o n . 16
5) The " h i s t o r y " t e r m , FH = Ctt(d2/4)~d'-~fot(dU/dr)
(d'r/td'-t-Z-r-T),which a t t e m p t s to a c c o u n t for the d e p e n d ence of the i n s t a n t a n e o u s d r a g on the s t a t e of d e v e l o p m e n t of fluid motion a r o u n d the s p h e r e . The h i s t o r y
t e r m t h e r e f o r e d e p e n d s upon the p a s t a c c e l e r a t i o n o r
d e c e l e r a t i o n of the body. Two e s t i m a t e s for FH a r e
a v a i l a b l e , depending on w h e t h e r the h i s t o r y c o e f f i c i ent, CB, i s a s s i g n e d i t s c l a s s i c a l v a l u e of 6, 14,~s o r
a l l o w e d to v a r y with v e l o c i t y and a c c e l e r a t i o a f f
R e w r i t i n g Eq. [1 ],
dU = Msg Ms-~
Mg-FD
322-VOLUME 6B, JUNE 1975
-FA -FH
[21
Fig. 1--Forces on a sphere accelerating through a liquid
(schematic).
and i n s e r t i n g g e o m e t r i c f a c t o r s , s t e a d y d r a g , a d d e d
m a s s , and h i s t o r y c o e f f i c i e n t s ,
7rd3 d U - g ~__ (P - Ps)
--~" P s "-~ =
^
CD~d2pUfUl
d2 r"='--_, r t / d U \
8
dr
-c"~Tv~P~ Jo ~,-d-7-)
~-~.
~ ~d 3 d U
- t.A -~--p -~-
[3]
Also
U
dz
= d--t-
[4]
w h e r e z i s the d i s t a n c e of the Lowest point of the
s p h e r e below the s u r f a c e . The i n i t i a l conditions for
E q s . [3] and [4] a r e
t--o, z = 0 , U = U o
is]
w h e r e Uo is the s p h e r e e n t r y v e l o c i t y . Since Eq. [3]
i s too c o m p l e x f o r a n a l y t i c a l solution, n u m e r i c a l p r o c e d u r e s w e r e adopted to p r e d i c t v a l u e s of s p h e r e
v e l o c i t y , U, and i n s t a n t a n e o u s depth of i m m e r s i o n , z,
a s functions of i m m e r s i o n t i m e , t. (See Appendix).
It i s a p p r o p r i a t e to note h e r e that if the h i s t o r y t e r m
in Eq. [3] is d i s c a r d e d , the equation can be s i m p l i fied to r e a d :
dU
CDpUI Ul
(Ps + CAp) ~ = - ( P - Ps)g - 3
4d
[6]
or
METALLURGICAL TRANSACTIONS B
dU_ ( 1 - y ) g
3CDUrUF
dt - ( ~ - ~ - A - ~ - 4d(y + CA)
[7]
w h e r e ~, = Ps/P.
EXPERIMENTAL
PROCEDURES
In o r d e r to ~ s t the adequacy of the equations outlined above for d e s c r i b i n g p a r t i c l e motion through
a liquid, wooden s p h e r e s of v a r i o u s d i a m e t e r s ranging
b e t w e e n 0.95 and 5.08 c m s . , (3/8 to 2 in.), and v a r i o u s
s p e c i f i c g r a v i t i e s (0.35 to 0.8) w e r e dropped f r o m
h e i g h t s of 2.13 and 3.57 m e t e r s r e s p e c t i v e l y into a
1.37 m deep, 0.46m d i a m tank of P . V . C . filled with
w a t e r . A B o l e x R e f l e x C a m e r a o p e r a t i n g at 54 f r a m e s /
s r e c o r d e d the s e q u e n c e of e v e n t s as the s p h e r e p e n e t.rated the w a t e r , sank to its m a x i m u m depth and then
s t a r t e d to r i s e back to the s u r f a c e . No spin was i m -
p a r t e d to the s p h e r e s as a r e s u l t of the d r o p p i n g
m e c h a n i s m , which c o n s i s t e d of a s p r i n g - l o a d e d p l a t f o r m which quickly opened when the tension on the
s p r i n g w a s r e l e a s e d . A P . V . C . Guide Tube was u s e d
for the s m a l l e r b a l l s to a i m t h e i r e n t r y about 2 in.
away from, and slightly to one side of, a m e a s u r i n g
r u l e . Subsequent f r a m e by f r a m e a n a l y s i s of the f i l m
then p r o v i d e d the n e c e s s a r y data on t r a j e c t o r i e s for
the s e l e c t i o n of an a p p r o p r i a t e m a t h e m a t i c a l m o d e l .
The bottoms of the s p h e r e s w e r e u s e d to define the
l o c i of the t r a j e c t o r i e s .
E x p e r i m e n t a l R e s u l t s and S e l e c t i o n of M a t h e m a t i c a l
Model for T r a j e c t o r y P r e d i c t i o n s
F i g . 2 shows a t y p i c a l s e q u e n c e for a 3.65 c m d i a m
wooden s p h e r e having a density of 0.711 g c m -3, d r o p ped f r o m a height of 3.57 m into w a t e r . The b a c k -
Fig. 2--Typical series of hydrodynamic events for a 3.65 cm diam wooden sphere having a density of 0.711 g e m -z dropped
from a height of 3.57 meters into a 0.46 m diam tank of water. Subscripts denote frame number (Camera speed = 54 f.p.s.)
METALLURGICALTRANSACTIONS B
VOLUME 6B,JUNE 1975-323
ground scale is marked in intervals of 0.I feet (Surveyor's rule). F r a m e 0, corresponding to time zero,
shows the sphere just about to enter the water, its
reflection just below the water line being clearly
evident. Frames 2, 4 and 6 demonstrate the rapidity
of initial entry, as well as the formation and collapse
of a large entrained air cavity. F r a m e 16, 0.296 s
after initial entry, marks the m a x i m u m depth of penetration, (32 cms), while F r a m e 57 shows the sphere
about to resurface 1.05 s after initial entry. It is
interesting to note the spectacular entry period prior
to cavity collapse (which represents about 10 pct of
the total immersion time) and the evanescence of the
r e m a i n i n g s m a l l e n t r a i n e d a i r c a v i t y during the r e s t
of the d e s c e n t p e r i o d .
F i g . 3 shows s o m e t y p i c a l e x p e r i m e n t a l and t h e o r e t i c a l t r a j e c t o r i e s for two s p h e r e s d r o p p e d into w a t e r .
T h e t e r m ' t r a j e c t o r y ' i s u s e d h e r e in the s e n s e of the
depth of the o b j e c t below the s u r f a c e us t i m e ( i . e . , no
lateral motion is implied). The experimental results
shown in Fig. 3(a) refer to a 4.77 c m diam sphere
with a solid/liquid density ratio, y, of 0.351, while
those in Fig. 3(b) correspond to a sphere with a y
= 0.716 and diameter of 1.07 cm. A s seen, in both
cases, penetration into the liquid is rapid, as evidenced
by the steepness of the curves close to time zero. The
rapid entries are followed by decelerating penetrations until buoyancy forces finally reverse the direction of motion and the spheres accelerate towards
their terminal rising velocities. The m a x i m u m i m m e r sion time is considered reached when the top of the
sphere surfaces.
Five possible mathematical models were considered. They involved the following treatment of the
drag terms in Eq. [3]:
1) CA =
2) CA =
0.5, C A = 6.0, CD
v a r i a b l e , CH = v a r i a b l e ,
3) C A = 0.5, no h i s t o r y , CD
CD
4) CA = v a r i a b l e , no h i s t o r y , CD
5) No added m a s s , no h i s t o r y , CD
Fig. 3--Experimental and theoretical trajectories of spheres
dropped into water. Figures on theoretical trajectories indicate model number. (a) d = 4.77 cm. Ps = 0.351 gm/ce, U0
= 6.29 vs. s -l. (b) d = 1.07 cm. Ps = 0.716 gm/cc, U0 = 6.15
m. S-i.
324-VOLUME 6B, JUNE 1975
T h e c o r r e s p o n d i n g five p r e d i c t e d t r a j e c t o r i e s f o r
c o m p a r i s o n with the e x p e r i m e n t a l r e s u l t s have b e e n
l a b e l l e d , 1, 2, 3, 4 and 5, r e s p e c t i v e l y , in F i g s . 3(a)
and (b).
I t i s i m m e d i a t e l y e v i d e n t that c u r v e s 2 and 4, b a s e d
on O d a r and H a m i l t o n ' s v a r y i n g c o e f f i c i e n t s for CA
and C//, show s i g n i f i c a n t d i s c r e p a n c i e s , in that p r e d i c t e d m a x i m u m d e p t h s and i m m e r s i o n t i m e s a r e
o v e r e s t i m a t e d . S i m i l a r l y , c u r v e 5 b a s e d p u r e l y on
s t e a d y s t a t e d r a g with no h i s t o r y o r added m a s s
t e r m s s e r i o u s l y u n d e r e s t i m a t e s m a x i m u m d e p t h s and
immersion times.
H o w e v e r , u s e of the c o n s t a n t c l a s s i c a l added m a s s
c o e f f i c i e n t ( m o d e l s 1 and 3) g i v e s p e n e t r a t i o n d e p t h s
and i m m e r s i o n t i m e s within 10 p c t of t h o s e o b s e r v e d .
T h e e f f e c t of including h i s t o r y d r a g i s s e e n by c o m p a r Jag c u r v e s 1 and 3 in F i g s . 3(a) and (b) r e s p e c t i v e l y .
The r e a s o n s f o r the d i f f e r e n t e f f e c t of h i s t o r y d r a g
on s p h e r e s of low and high s p e c i f i c g r a v i t i e s a r e d i s c u s s e d in the A p p e n d i x . H o w e v e r , i t i s c l e a r f r o m
e a c h c a s e that even though the i n c l u s i o n of a h i s t o r y
d r a g t e r m m a y b r i n g the o v e r a l l s h a p e of the t r a j e c t o r y c l o s e r to that o b s e r v e d , the e f f e c t on p r e d i c t e d
m a x i m u m depth and i m m e r s i o n t i m e i s m i n i m a l .
T h e s e r u n s , and the o t h e r s e x a m i n e d , t h e r e f o r e , i n d i c a t e t h a t adequate p r e d i c t i o n s can be made on the b a s i s
of m o d e l 3, which i g n o r e s h i s t o r y e f f e c t s and t a k e s a
c o n s t a n t added m a s s c o e f f i c i e n t of 0.5. It thus c o r r e s ponds to Eq. [7], with CA : 1 / 2 .
F i g . 4 shows p r e d i c t e d and e x p e r i m e n t a l c u r v e s f o r
b a l l s of 1.07, 2.69 and 4.88 c m d i a m r e s p e c t i v e l y , and
= 0.71 d r o p p e d in f r e e f a l l f r o m a height of 3.57
m e t e r s . In c a l c u l a t i n g i n i t i a l e n t r y v e l o c i t i e s , account
w a s t a k e n of a i r r e s i s t a n c e d u r i n g the s p h e r e ' s d e s c e n t
to the liquid s u r f a c e . T h i s c o r r e c t i o n d e c r e a s e d e n t r y
v e l o c i t i e s below t h a t f o r u a r e s i s t e d motion (i.e., Uo
= ~ )
by 1 to 5 p c t depending on s p h e r e d i a m e t e r
and d e n s i t y . A s s e e n f r o m F i g . 4, l a r g e r b a l l s s i n k
d e e p e r and s t a y in l o n g e r . The a g r e e m e n t with m o d e l
3 i s a g a i n quite s a t i s f a c t o r y in view of v a r i a b i l i t y in
the e x p e r i m e n t a l d a t a .
F i g . 5 shows a p l o t of m a x i m u m depths of p e n e t r a tion v s s p h e r e d i a m e t e r for s o l i d / l i q u i d d e n s i t y r a t i o s
METALLURGICAL TRANSACTIONS B
of 0.72 and 0.365. The solid and broken c u r v e s r e p r e sent t h e o r e t i c a l p r e d i c t i o n s for spheres r e l e a s e d 2.13
and 3.57 m e t e r s above the liquid surface, and again
show good a g r e e m e n t with the experimental data. S i m i l a r l y , good a g r e e m e n t is achieved in Fig. 6 where
maximum i m m e r s i o n t i m e s a r e plotted v s s p h e r e
d i a m e t e r and c o m p a r e d with t h e o r e t i c a l p r e d i c t i o n s .
It may be noted that v e r y tittle effect of height of
drop on maximum depths or i m m e r s i o n times is either
observed or p r e d i c t e d .
Discussion of Mathematical Model
As seen f r o m the p r e c e d i n g section, h i s t o r y effects
a r e only of minor i m p o r t a n c e in determining p a r t i c l e
t r a j e c t o r i e s under p r e s e n t c i r c u m s t a n c e s . It is
equally c l e a r that added mass effects assume s i m i l a r
Fig. 4 - - T r a j e c t o r i e s of s p h e r e s d r o p p e d into w a t e r : e x p e r i m e n t a l , and p r e d i c t e d by model 3. (a) d = 1.07 cm, Ps = 0.716
gm,/cc, U 0 = 7.74 m s -1. ( b ) d = 2 . 6 9 cm, Ps = 0 . 7 1 1 g m / c c ,
U 0 = 8.09 m s -1. ( c ) d = 4.88 c m , Ps = 0 . 7 2 7 g m / c c , U 0
= 8.21 m s -i.
Fig. 5 - - M a x i m u m depth of p e n e t r a t i o n (cm) v s addition (cm);
e x p e r i m e n t a l v a l u e s c o m p a r e d with p r e d i c t e d c u r v e s b a s e d on
model 3.
METALLURGICAL TRANSACTIONS B
importance to steady d r a g f o r c e s in d e t e r m i n i n g the
depth and time of i m m e r s i o n , and cannot be ignored.
The fact that model 3 (CA = 0.5, CH =0, CD = CD)
gives such r e l i a b l e p r e d i c t i o n s is quite s u r p r i s i n g in
view of the complex phenomena o c c u r r i n g during a
s p h e r e ' s descent through a liquid. In all c a s e s , l a r g e
cavities of a i r w e r e e n t r a i n e d for about the f i r s t
half of their descent (i.e., Fig. 2). Consequently the
flow around their r e a r sections bore l i t t l e r e s e m b l a n c e
to experimental conditions for which steady d r a g data
have been obtained. The a g r e e m e n t between c u r v e s 3
and the e x p e r i m e n t a l t r a j e c t o r i e s in F i g s . 3(a) and (b)
may therefore be p a r t l y fortuitous and r e s u l t from
compensating e r r o r s in the steady d r a g and added
m a s s t e r m s , both of which a r e l a r g e during the
initial descent stage through the liquid. However, it
is interesting to note that the p r e s e n t work m o r e
c l e a r l y distinguishes the p r a c t i c a l m e r i t s between
assigning a constant added m a s s coefficient of 0.5 and
that of incorporating a v a r i a b l e added m a s s t e r m as
proposed by Odar an d Hamilton. 16 Odar subsequently
showed '7 that their v a r i a b l e coefficients gave r e l i a b l e
p r e d i c t i o n s for buoyant s p h e r e s a c c e l e r a t i n g f r o m
r e s t through a stagnant fluid. Clift e t al. ~a r e p e a t e d
s i m i l a r e x p e r i m e n t s and found that good t r a j e c t o r y
p r e d i c t i o n s could also be made by taking the standard
constant coefficients for CA and Ctt. Taken with the
findings of the p r e s e n t work, it would s e e m that the
c l a s s i c a l values for CA and CH a r e more a p p r o p r i a t e
and can be applied to more complex motions than has
p r e v i o u s l y been supposed.
Finally, it is a p p r o p r i a t e to d i s c u s s the r e a s o n s for
the s h o r t i m m e r s i o n depths and t i m e s p r e d i c t e d by
model 5 v s the longer, more c o r r e c t v a l u e s when
added m a s s is taken into account. In the l a t t e r case,
the sphere entering with a velocity Uo e s s e n t i a l l y
entrains an added m a s s of liquid whose volume is
equal to half that of the s p h e r e (i.e., M A = 1//2pVs,
Ms = PsVs). By imparting some of its momentum to
this liquid, the s p h e r e must slow down, its new v e l o c ity being
MsUo
U~ (gVls+ M A )
[8]
Since the steady d r a g t e r m is s m a l l e r at lower v e l o c i t i e s , the s p h e r e (and its a s s o c i a t e d liquid) is then
Fig. 6--Immersion times (s) vs addition size (cm); experimental values compared with predicted curves based on model
3.
VOLUME 6B, JUNE 1975-325
a b l e to p e n e t r a t e d e e p e r and s t a y in l o n g e r than a
s p h e r e with no added m a s s . In the l a t t e r e a s e , the
s p h e r e ' s v e l o c i t y does not d e c r e a s e on impact; s i n c e
d r a g f o r c e s a r e a p p r o x i m a t e l y p r o p o r t i o n a l to Uz in
t h i s high R e y n o l d s N u m b e r r a n g e (CD ~ constant), it
l o s e s i t s downward m o m e n t u m too r a p i d l y , r e s u l t i n g
in too s h o r t i m m e r s i o n d e p t h s and t i m e s . S i m i l a r
a r g u m e n t s exl~lain why p e n e t r a t i o n depths a r e not
much i n c r e a s e d by d r o p p i n g the s p h e r e s f r o m a
h e i g h t of 3.58 vs 2.12 m e t e r s .
EXTENSION O F W A T E R M O D E L TO
STEELMAKIN G CONDITIONS
Since the e x p e r i m e n t s u s i n g w a t e r i n d i c a t e that
r e l i a b l e t r a j e c t o r y p r e d i c t i o n s m a y be obtained on
the b a s i s of m o d e l 3, and s i n c e a l l the g o v e r n i n g
d i m e n s i o n l e s s g r o u p s for p a r t i c l e s p r o j e c t e d into
s t e e l c o i n c i d e with the r a n g e c o v e r e d by the p r e s e n t
e x p e r i m e n t s in w a t e r , good p r e d i c t i o n s c a n be m a d e
on the d e p t h s and i m m e r s i o n t i m e s of a l l o y a d d i t i o n s
d r o p p e d into s t e e l b a t h s . Thus, r e f e r r i n g to Eq. [7]
which d e s c r i b e s the s p h e r e ' s motion (CI-1 = O, CA
= 0.5, CD) it i s s e e n that the i m p o r t a n t d i m e n s i o n l e s s
p a r a m e t e r s a r e a) the s o l i d / l i q u i d d e n s i t y r a t i o n , Y,
and b) the s t a n d a r d d r a g coefficient, CD. Since CD is
s o l e l y a function of Re, the R e y n o l d s N u m b e r ( s e e A p pendix), and s i n c e the k i n e m a t i c v i s c o s i t y of s t e e l
[v = ( y / p ) ~ 0.064/7 = 0.00914] a l m o s t c o i n c i d e s with
that of w a t e r (~0.01), R e y n o l d s N u m b e r s and d r a g c o e f f i c i e n t s a r e a l m o s t the s a m e in both s y s t e m s (for
a given d and Uo). Thus, by d e l i b e r a t e l y c h o o s i n g
s p h e r e d i a m e t e r s and s o l i d A i q u i d d e n s i t y r a t i o s
c o v e r i n g the r a n g e of i n t e r e s t in s t e e l m a k i n g p r a c t i c e , c l o s e matching w a s a s s u r e d . S i m i l a r l y , F r o u d e
N u m b e r (UZ/gL) matching w a s a c h i e v e d by d r o p p i n g
the s p h e r e s f r o m the heights u s e d i n d u s t r i a l l y . A l s o ,
although the s u r f a c e t e n s i o n s of s t e e l and w a t e r a r e
m a r k e d l y d i f f e r e n t , s u r f a c e e n t r y e f f e c t s should be
r n i n i m a l in both i n s t a n c e s . T a k i n g the p a r t i c u l a r
e x a m p l e shown in F i g . 2 which c r e a t e s a c a v i t y h a v ing a m a x i m u m s u r f a c e a r e a about 38 t i m e s the c r o s s s e c t i o n a l a r e a of the s p h e r e , the r a t i o of the s u r f a c e
e n e r g y r e q u i r e m e n t s (38aTrd2/4) to the e n t r y k i n e t i c
e n e r g y (psU~ 97ra~/12) i s 114 r
Inserting approp r i a t e v a l u e s (114 • 71/0.711 • 3.65 • (815) 2) shows
the r a t i o to be 0.0047, which i s obviously n e g l i g i b l e .
One can m a k e s i m i l a r a r g u m e n t s for a l l o y s p r o j e c t e d
into s t e e l b a t h s even though i t s s u r f a c e t e n s i o n is
much g r e a t e r than w a t e r (~1000 d y n e s / c m ) s i n c e the
i m p o r t a n t r a t i o of p h y s i c a l ~ p a r a m e t e r s is r
and
Ps would be 4.98 g p e r cm- in this c a s e . One m a y
note that the d i m e n s i o n l e s s g r o u p i n g (PsLU2/a) is s i m i l a r to the W e b e r N o . (p/LU2/cr) which i n d i c a t e s the
r a t i o of i n e r t i a l to s u r f a c e t e n s i o n f o r c e s in a liquid
system.
It should be noted that the p r e s e n t w o r k d o e s not
t a k e into account t h e r m a l e f f e c t s , such a s the f o r m a tion of a s o l i d s t e e l s h e e l a r o u n d the o b j e c t . If t h e s e
a r e included (through s i m u l t a n e o u s solution of the
a p p r o p r i a t e p a r t i a l d i f f e r e n t i a l equations for h e a t
t r a n s f e r with the h y d r o d y n a m i c equations p r e s e n t e d
h e r e ; s e e Ref. 4), it t u r n s out t h a t the i n c r e a s e d s t e a d y
d r a g f o r c e r e s u l t i n g f r o m a s p h e r e ' s growth is m o r e
than c o m p e n s a t e d by an i n c r e a s e in the added m a s s
t e r m . The net r e s u l t , for a l u m i n u m , is s l i g h t l y
326-VOLUME 6B, JUNE 1975
l o n g e r i m m e r s i o n t i m e s and p e n e t r a t i o n depths about
4 p c t l e s s than those p r e s e n t l y p r e d i c t e d . R e c e n t
high t e m p e r a t u r e e x p e r i m e n t a l w o r k h a s c o n f i r m e d
t h i s l a t t e r point. ~9 F o r i n s t a n c e , a 2.5 c m d i a m s p h e r e
of a l u m i n u m d r o p p e d f r o m 3 m e t e r s w i l l r e m a i n
i m m e r s e d for 0.39 s vs 0.31 s for an e q u i v a l e n t
wooden s p h e r e .
DISCUSSION
OF RESULTS
OF APPLICABILITY
TO STEELMAKING
It i s r e a l i z e d that the p r e s e n t e x p e r i m e n t s i n v o l v ing s p h e r e s d r o p p e d f r o m v a r i o u s heights into s t a g nant liquids r e p r e s e n t an a p p r o x i m a t i o n of m a n y
h y d r o d y n a m i c e v e n t s in s t e e l m a k i n g p r a c t i c e s i n c e ,
a) the d e o x i d i z e r s may f i r s t p a s s through a s l a g
l a y e r in the filling l a d l e
b) a l a r g e n u m b e r of o b j e c t s m a y fall s i m u l t a n e ously
c) e n t r a i n m e n t in the p o u r i n g s t r e a m is p o s s i b l e
d) t u r b u l e n t liquid motions in the filling l a d l e m u s t
modify t r a j e c t o r i e s to v a r y i n g e x t e n t s depending
on the o b j e c t ' s 7 r a t i o .
It i s equally c l e a r that a full s c a l e w a t e r m o d e l of
the s y s t e m could a n s w e r many of t h e s e a s p e c t s . Howe v e r , in the m e a n t i m e , it is w o r t h noting t h a t r e c e n t
w o r k on the t r a j e c t o r i e s of a l u m i n u m s p h e r e s d r o p p e d
into s t e e l b a t h s , 19 showed that a 1 c m thick l a y e r of
s l a g had no effect in inhibiting s t e e l s h e l l f o r m a t i o n .
A l s o , although the p r e s e n t w o r k i s r e s t r i c t e d to s i n g l e s p h e r e s , i t i s known that for high R e y n o l d s n u m b e r s t y p i c a l of the p r e s e n t s i t u a t i o n , the o b j e c t s have
to be in e x t r e m e l y c l o s e p r o x i m i t y b e f o r e t h e r e i s
any a p p r e c i a b l e i n c r e a s e in d r a g . 2~ M o r e o v e r , such
e f f e c t s would s e r v e only to r e d u c e i m m e r s i o n t i m e s
s t i l l f u r t h e r . F i n a l l y , although an addition e n t r a i n e d
in a tapping s t r e a m should p e n e t r a t e much m o r e
d e e p l y than one e n t e r i n g a s t a g n a n t bath, q u a l i t a t i v e
w o r k 19 i n d i c a t e s that it should have a s t r o n g t e n d e n c y
to move away f r o m this l o c a l i z e d high v e l o c i t y r e g i o n
in the ladle and r e s u r f a c e in the n o r m a l way s h o r t l y
afterwards.
To conclude, the a u t h o r s c o n s i d e r that the p r e s e n t
m o d e l l i n g w o r k r e p r e s e n t s a good f i r s t a p p r o x i m a t i o n
to a c t u a l e v e n t s , and c l e a r l y d e m o n s t r a t e s the type
of h y d r o d y n a m i c contacting p r o b l e m s involved when
s o l i d a d d i t i o n s a r e made to s t e e l b a t h s .
Thus, r e f e r r i n g to F i g s . 4 and 5, it is s e e n t h a t a
s o l i d e n t e r i n g a liquid having a 7 r a t i o of 0.365 w i l l
s i n k about 13 c m s (5 in) and r e s u r f a c e 0.2 s a f t e r
entry, even when d r o p p e d through a i r f r o m a height of
3.58 m e t e r s . T h i s 7 c o r r e s p o n d s a p p r o x i m a t e l y to
an a l u m i n u m addition in s t e e l . Although s u b s u r f a c e
m e l t i n g is d e s i r a b l e for good r e c o v e r y and p r o c e s s
c o n t r o l , it is v e r y u n l i k e l y u n d e r such c o n d i t i o n s , and
one m u s t a n t i c i p a t e s e v e r e ( s l a g / a i r ) - a l u m i n u m i n t e r a c t i o n s for a l l n o r m a l p r o c e d u r e s , at any addition
s i z e s . S i m i l a r c o n s i d e r a t i o n s would apply to a l l o y
a d d i t i o n s such a s 25 p c t F e - S i (7 = 0.39 to 0.58),
Z r - S i (7 = 0.48 to 0.52) and 50 p c t F e - S i (7 = 0.58 to
0.67). A l l o y s such a s f e r r o m a n g a n e s e a r e s o m e w h a t
d i f f e r e n t s i n c e they have 7 r a t i o s in the r a n g e 0.9 to
1.04 and would p e n e t r a t e c o n s i d e r a b l y d e e p e r and
s t a y i m m e r s e d much l o n g e r .
In o r d e r to p r o v i d e a c o m p r e h e n s i v e s e t of p r e d i c METALLURGICAL TRANSACTIONSB
tions for steelmaking and to extrapolate outside the
ranges covered by the present water model experiments, computer predictions based on model 3 were
run, taking the density of liquid steel as 7.0 g per
cm-3 and its viscosity as 6.4 cP.
Fig. 7 demonstrates the effect of drop height (or
entry velocity) on immersion depths and times for 5
cm diam spheres of ferromanganese (apparent density
6.72 g per cm-3) and aluminum (p = 2.7) entering steel.
As seen, predictions have been made far beyond the
heights normally feasible (or desirable!) in practice.
The results indicate very clearly the difficulty of
maintaining low density alloy additions immersed,
even when they are subjected to very high entry
velocities. For example, a 5 cm diam aluminum
sphere, dropped from a height of 100 meters with an
entry velocity of 44 meters/s (145 ft/s) would only
penetrate 1 meter (3.3 ft) and would remain immersed
for 1.4 s. In the case of a ferromanganese addition
with a density of 6.72 g per cm-3 maximumdepths of
penetration and total immersion times would be considerably increased compared to aluminum, i.e., 2.4
meters and 11.2 s.
Striking confirmation of these predictions is provided by the recent work of Tanoue et al. at who have
developed a new method for adding aluminum to their
ladles at the Wakayama and Kashirna Works in Japan.
Known as the "Aluminumbullet shooting method",
their results showed that somewhat improved yields
and markedly better process control could be achieved
over previous standard practice by using a rotary
chamber type "shooter" to fire a continuousstream
of bullets intofillingladles. Takinga specific example from Fig. 20 of Ref. 21, they show that a 5 cm
diam bullet projected to adepth of i meter in still water
will remain immersed for 2.4 s. This time compares
closely with the 1.4 s immersion time predicted for
a 5 cm diam sphere. It also suggests that carefully
designed adding pieces can remain immersed almost
twice as long as spheres projected to equivalent
depths.
Fig. 7--Effect of height of drop (or entry velocity) on immersion times and maximum depth for 5 cm spheres of ferromanganese and aluminum in steel.
METALLURGICALTRANSACTIONS B
Fig. 8 p r e s e n t s m a x i m u m depths of p e n e t r a t i o n for
15 and 25 cm diam alloy additions v s alloy addition
density for m o r e n o r m a l e n t r y v e l o c i t i e s of 7.67
meters/s.
Fig. 9 p r e s e n t s s i m i l a r plots, giving m a x i m u m
i m m e r s i o n t i m e s v s alloy addition d e n s i t y . The e n t r y
v e l o c i t y quoted c o r r e s p o n d s to u n r e s i s t e d f r e e f a l l of
o b j e c t s f r o m a height of 3 m e t e r s above the bath. As
in the c a s e of Fig. 7, F i g s . 8 and 9 a r e p l o t t e d on a
s e m i - l o g a r i t h m i c b a s i s to c o v e r the wide r a n g e of
depths and i m m e r s i o n t i m e s p r e d i c t e d . As one might
expect, depths and i m m e r s i o n t i m e s i n c r e a s e r a p i d l y
for those additions having d e n s i t i e s c l o s e to m o l t e n
s t e e l . Thus, for alloy additions such as f e r r o m a n g a n e s e , with nominal or a p p a r e n t d e n s i t i e s in the r e g i o n
of 6.95 for instance, i m m e r s i o n t i m e s of 80 s and
depths of 90 c m a r e a c h i e v e d . T h i s i n d i c a t e s that
such additions will, in p r a c t i c e , have l i t t l e t e n d e n c y
to s u r f a c e and should g e n e r a l l y follow the liquid s t e e l
flow p a t t e r n s during the c o u r s e of t h e i r m e l t i n g h i s t o r y in a filling l a d l e .
Fig. 8--Effect of density ratio on maximum penetration for
spheres entering steel at 7.67 m per s -1.
VOLUME 6B,JUNE [975-327
APPENDIX
A s mentioned in the text, the full equation d e s c r i b ing the s p h e r e ' s s u b s u r f a c e m o t i o n (i.e., E q s . [3] and
[4 9 i s too c o m p l e x for a n a l y t i c a l solution, and n u m e r i c a l p r o c e d u r e s m u s t be adopted for the p r e d i c t i o n of
s p h e r e v e l o c i t y , U, and i n s t a n t a n e o u s depth, z, a s a
function of i m m e r s i o n t i m e . In v i e w of the c o m p l e x i t i e s involved in the full equation a r a t h e r high p o t e n t i a l for e r r o r s e x i s t e d . C o n s e q u e n t l y two i n d e p e n d e n t
n u m e r i c a l p r o c e d u r e s w e r e d e v e l o p e d for c r o s s - c h e c k ing p u r p o s e s ; a g r e e m e n t to b e t t e r than 5 p c t w a s
obtained f o r p r e d i c t e d t r a j e c t o r i e s .
The f i r s t p r o c e d u r e involved a s i m p l e n u m e r i c a l
s o l u t i o n of Eq. [7] (CA = 0.5, CH = 0, CD =CD), u s i n g
the s t a n d a r d v a l u e s of CD r e p o r t e d by L a p p l e and
S h e p h e r d ~a and A c h e n b a c h . aa Thus, a t any p a r t i c u l a r
t i m e i n s t a n t d u r i n g the s p h e r e ' s i m m e r s i o n , the
v e l o c i t y U w a s u s e d to c a l c u l a t e the R e y n o l d s N u m b e r . L i n e a r i n t e r p o l a t i o n u s i n g l i s t e d v a l u e s z3'22
of CD for the v a l u e s on e i t h e r s i d e of the R e y n o l d s
N u m b e r in q u e s t i o n then gave the r e q u i r e d v a l u e of
CD. In n u m e r i c a l f o r m , Eq. [7] thus r e a d s :
U'= u-(l-y)gat
3CDUIUIAt
[AI]
(T + 1/2) - 4 d (7 + 1/2)
with
z" = z + 05(U + U')At
[A2]
w h e r e U" and z" r e p r e s e n t the new v a l u e s of U and Z,
a f t e r At s . An i t e r a t i v e r o u t i n e t a k i n g a t = 10 "~ s e c o n d s
then p r o v e d s a t i s f a c t o r y in p r e d i c t i n g s p h e r e t r a j e c t o r i e s . F i n a l l y , initiale n t r y w a s taken into a c c o u n t v i a
E q . [8] and g r a d u a l e n t r y e f f e c t s i g n o r e d .
In the s e c o n d r o u t i n e , the following p r o c e d u r e s w e r e
a d o p t e d for the c a l c u l a t i o n of the d r a g c o e f f i c i e n t , CD
a d d e d m a s s c o e f f i c i e n t , CA, and h i s t o r y c o e f f i c i e n t , CH.
cv
Fig. 9--Effect of density ratio on immersion time for spheres
entering steel at 7.67 m per s "1.
By the s a m e token, it is a p p a r e n t that a l u m i n u m
a d d i t i o n s have l i t t l e chance of s u b s u r f a c e m e l t i n g
u n d e r normal p r a c t i c e .
1) The c o r r e l a t i n g p o l y n o m i a l s given by D a v i e s z~
w e r e u s e d for Re < 104. 2) F r o m R e = 104 to the c r i t i c a l R e y n o l d s N u m b e r of a p p r o x i m a t e l y 3 • l 0 S (Ref.
22), the equation of Clift and Gauvin a4 w a s u s e d , w h i l e
a b o v e the c r i t i c a l R e y n o l d s N u m b e r , c u r v e s w e r e
f i t t e d to the d a t a of A c h e n b a c h . aa
CA
CONCLUSIONS
1) The i n j e c t i o n of a l l o y a d d i t i o n s into s t a g n a n t
s t e e l b a t h s can be s i m u l a t e d with good a c c u r a c y by
d r o p p i n g wooden s p h e r e s of a p p r o p r i a t e s p e c i f i c
g r a v i t y f r o m the s a m e height into w a t e r .
2) Good t h e o r e t i c a l p r e d i c t i o n s can be m a d e on
s p h e r e t r a j e c t o r i e s and i m m e r s i o n t i m e s by taking
into a c c o u n t s t a n d a r d d r a g and a d d e d m a s s e f f e c t s .
3) D r a g f o r c e s r e l a t i n g to the p r e v i o u s h i s t o r y of
the o b j e c t ' s motion a r e s m a l l u n d e r p r e s e n t c i r c u m s t a n c e s and can be i g n o r e d .
4) A l l o y a d d i t i o n s of low d e n s i t y (e.g., A1, F e - S i ,
etc.) should e x h i b i t v e r y s h o r t i m m e r s i o n t i m e s (1 s),
even when i n j e c t e d at high v e l o c i t i e s (50 m p e r s-~),
on a c c o u n t of a r a p i d l o s s in m o m e n t u m and high
buoyancy f o r c e s .
5) A t y p i c a l high d e n s i t y a l l o y a d d i t i o n (e.g., 5 c m
d i a m F e - M n , p 6.96) should r e m a i n i m m e r s e d for
about a minute in a bath of s t a g n a n t s t e e l .
328-VOLUME 6B, JUNE 1975
The c l a s s i c a l v a l u e of 0.514 w a s t r i e d , and a l s o the
c o r r e l a t i o n s p r o p o s e d by O d a r and H a m i l t o n . z6
c~
T h e c l a s s i c a l v a l u e of 6.0 '4'z5 w a s t r i e d , and a l s o
the O d a r and H a m i l t o n c o r r e l a t i o n s . .6 The s o l u t i o n of
E q s . [3] and [4] then u s e d p r e v i o u s l y d e v e l o p e d n u m e r i c a l p r o c e d u r e s zs for c a l c u l a t i o n of U and z a s functions
of t allowing for the h i s t o r y t e r m . F o r the e a r l y
s t a g e s of m o t i o n c o r r e s p o n d i n g to i n c o m p l e t e s u b m e r s i o n ( i . e . , z -< d), the d r a g t e r m s w e r e r e d u c e d by
m u l t i p l y i n g by the f r a c t i o n of the s p h e r e v o l u m e i m mersed:
f-
z2(3d - 2)
d3
[A3]
ADowance w a s a l s o made for i n i t i a l e n t r y in the added
m a s s t e r m . Eq. [3] in the t e x t then b e c a m e (0 <-z<-d),
METALLURGICAL TRANSACTIONS B
LIST O F S Y M B O L S
~d ~
dU
ua~
--~- Ps-~-/-= - g - ~ - (fP - Os) - f C o
d.
- ~tfCA
pUI U I
7rd3
- ~ pU)
d2
( t (dg~ dr
- fcH--4~4-~-ffff ~o k - ~ / 4 7 =- T 9
[A4]
Multiplyingby 12/~d2p and rearranging then gives
2d (7 + f C A ) - ddU
- [ = - 2 g d ( f - 7) - 1.SfCD U[ UI
- 12CA~Z(d
- Z)
.['~" r t / d V \
dr
- 3 f C H ~ ~P jo ~-~-} ~ / t - r .
[A5]
To account for entry in the history integral, the veloci t y w a s a s s u m e d to c h a n g e i m p u l s i v e l y f r o m z e r o to
[7o a t t = 0; i . e . , d U / d t t o o k t h e f o r m of a D i r a c d e l t a f u n c t i o n a t t = 0. W h e n t h e h i s t o r y t e r m w a s i g n o r e d ,
the resulting equations were solved by the standard
R u n g e - K u t t a - M e r s o n p r o c e s s . 2~
Explanation for Predicted
History Effects in Fig. 3
R e f e r r i n g to E q . [3], i t m a y b e n o t e d t h a t d U / d r i s
negative so that the history drag is negative with res p e c t to t h e d i r e c t i o n i n d i c a t e d in F i g . 1; i.e., h i s t o r y
( l i k e a d d e d m a s s ) g e n e r a l l y a c t e d s o a s to c a r r y t h e
sphere further into the liquid.
F o r t h e low d e n s i t y r a t i o ( F i g . 3(a)), d U / d t i s v e r y
large following entry and the negative history drag is
s i g n i f i c a n t d u r i n g t h e e a r l y s t a g e s of t h e t r a j e c t o r y .
H o w e v e r , a t l o n g e r t i m e s t h e c o n t r i b u t i o n s of t h e d e celeration to the history integral decays so that the
initial impulsive acceleration term at t = 0 (corresp o n d i n g to i n i t i a l c o n t a c t w i t h t h e f l u i d a n d e q u a l to
Uo/,/T) t h e n t a k e s o v e r , m a k i n g t h e h i s t o r y d r a g
positive. Therefore the predicted maximum depths
o c c u r s o o n e r t h a n in t h e a b s e n c e of h i s t o r y . F o r t h e
l a T g e r d e n s i t y r a t i o h o w e v e r ( F i g . 3(b)), t h e d e c e l e r a tion was more gradual with time so that the history
drag remained negative except for very short times.
It should be noted that although the added mass
term has full theoretical justification both at high and
low R e y n o l d s n u m b e r s (i.e., ~ - i n v i s c i d ( p o t e n t i a l ) a n d
c r e e p i n g f l o w s , r e s p e c t i v e l y ) , t h e f o r m of t h e h i s t o r y
term is only strictly applicable for "creeping flow".
Nevertheless, for a sphere accelerating from rest,
i n c l u s i o n of b o t h FA a n d FH h a s b e e n s h o w n to g i v e
a c c u r a t e p r e d i c t i o n s of t h e m o t i o n u p to h i g h R e *7'la
and the history drag is then generally more important
than added mass. However, the present situation,
w h i c h i n v o l v e s t h e m o t i o n of a b o d y p r o j e c t e d i n t o a
liquid, is characterized by a large initial Reynolds
number. It was then uncertain whether the creeping
flow c o n c e p t of a h i s t o r y d r a g i s v a l i d o r e v e n a p p l i cable. The model experiments and predicted trajectories served to resolve this question for the present
case.
ACKNOWLEDGMENTS
T h e a u t h o r s a r e i n d e b t e d to t h e N a t i o n a l R e s e a r c h
of C a n a d a f o r f i n a n c i a l s u p p o r t of t h i s w o r k .
METALLURGICAL TRANSACTIONS B
CA
CD
CH
d
F
FA
FB
FD
FG
FH
g
h
M
Ms
t
U
Uo
Uo"
Y
~t
t,
p
Ps
a
r
Added mass coefficient
Steady state drag coefficient
History coefficient
Sphere diameter
Drag force(s)
Drag force from 'added mass'
Buoyancy force acting on sphere
Drag force from steady translation
F o r c e of g r a v i t y a c t i n g o n s p h e r e ( i . e . , w e i g h t )
Drag force resulting from history term
Gravitational constant
H e i g h t of d r o p t h r o u g h a i r
M a s s of w a t e r d i s p l a c e d by s p h e r e
M a s s of s p h e r e
Time
Velocity
Impact velocity
R e d u c e d i m p a c t v e l o c i t y t a k i n g a d d e d m a s s into
account
Solid/liquid density ratio
Viscosity
Kinematic viscosity
D e n s i t y of l i q u i d
D e n s i t y of s o l i d
Surface tension
Time (dummy variable in history term)
REFERENCES
1. W. Crafts and D. C. Hilty: ElectricIce Steel Proc.AIME, 1953, voL 11, pp.
121-50.
2. R. I. L. Guthiie and L. Gourtsoyannis: C.LM. Quarterly, 197l, vol. 10, no. 1,
pp. 37-46.
3. L. Gourtsoy~mnisand R. I. L. Guthde: 1972, C.I.M. Conference, Halifax,
Nova Scotia.
4. L. Gourtsoyannis,H. Henein,and R. t. L. Guthrie: PhysicalChemistry of
Production or Use of Alloy Additives, John Farrell,ed., pp. 45-67, T.M.S.,
A.I.M.E., 1974.
5. G. Birkhoff and E. H. ZarantoneUo:Jets, Wakesand Cavities, AcademicPress,
New York, 1959.
6. A. May:J. Hydronautics, 1970,vol. 4, p. 140.
7. A. M. Worthingtonand R. S. Cole: Phil Trans. Roy. Soc., 1900, vol. 194,
p. 175.
8. A. May:J. A ppt Phy&, 1951, vol. 22, p. 1219.
9. A. May: J. Appl Phys., 1952, vol. 23, p. 1362.
I0. M. Shiffman and D. C. Spencer: A.M.P. Report 42, 2R, National Defence
Research Committee, 1945.
11. E. G. Richardson:Pro~ Phys. Soe., 1948, vol. 61, p. 352.
12. A. May and J. C. WoodhuU:J. Appl Phys., 1948,vol. 19, p. 1109.
13. C. E. Lapple and C. B. Shepherd: Ind. Eng. Chem., 1940, vol. 32, pp. 605-17.
14. L. D. landau and E. M. Lifshitz: Fluid Mechanics, Pergamon Press, London,
1959.
15. A. B. Bassett: Phil Teans.Roy. Soc., 1888, vol. 179, p. 43.
16. F. Odar andW. S. Hamilton:Z FluidMedt, 1964, vol. 18, p. 302.
17. F. Odar: J. FluidMech., 1966, vol. 25, p. 591.
18. R. Clift, F. A. Adamji,and W. R. Richards: Int. Conf. on Particle Technology,
1973 (liT ResearchInstitute, Chicago), pp. 130-36.
19. H. Henein: McGiUUniversity,Montreal, P. Q., Unpublishedresearch, 1974.
20. L. B. Torobin and W. H. Gauvin: Can.Z Chem. Eng., 1961, vol. 39, p. 113.
21. T. Tanoue, Y. Umeda, H. Ichikawa, and T. Aoki: The Sumitomo Search
No. 9, May, 1973, pp. 74-87, also N.O.H~.C., Atlantic City, New Jersey,
May 1974.
22. E. Achenbach:J. FluidMeclt, 1972, voL 54, p. 565.
23. C. N. Davies:Proc.Roy. Soc., 1945, vol. 57, p. 259.
24. R. Clift and W. H. Gauvin: Can.J. o[Chen~ Eng., 197I, vol. 49, p. 439.
25. G. N. Lance:NumericalMethodsfor High Speed Computers, Iliffe and Sons,
London, 1960.
VOLUME 6B,JUNE 1975-329