Fluid Velocities in Induction Melting Furnaces:
Part I. Theory and Laboratory Experiments
E R A C H D. T A R A P O R E
A N D J A M E S W. E V A N S
F l u i d flow in an induction f u r n a c e due to e l e c t r o m a g n e t i c s t i r r i n g f o r c e s i s p r e d i c t e d
t h e o r e t i c a l l y f r o m f u r n a c e d e s i g n p a r a m e t e r s by the s i m u l t a n e o u s s o l u t i o n of the M a x w e l l and N a v i e r S t o k e s e q u a t i o n s . S t r e a m l i n e p l o t s and v e l o c i t y p r o f i l e s a r e o b t a i n e d and
c o m p a r e d with s u r f a c e v e l o c i t i e s m e a s u r e d e x p e r i m e n t a l l y . T h e m e a s u r e m e n t s w e r e
m a d e on a m e r c u r y p o o l s t i r r e d i n d u c t i v e l y by a T o c c o 30 kW 3 kHz i n d u c t i o n m e l t i n g
unit. T h e a g r e e m e n t b e t w e e n the e x p e r i m e n t a l m e a s u r e m e n t s and t h e o r e t i c a l p r e d i c t i o n s
w a s good c o n s i d e r i n g t h a t no c u r v e f i t t i n g b y m a n i p u l a t i o n of a d j u s t a b l e p a r a m e t e r s w a s
i n v o l v e d . It i s b e l i e v e d t h a t s u c h a m o d e l would be of v a l u e in the d e s i g n and d e v e l o p m e n t of i n d u c t i o n f u r n a c e s .
IN
induction m e l t i n g f u r n a c e s the m e l t i s h e a t e d by
e l e c t r i c c u r r e n t s i n d u c e d by a f l u c t u a t i n g m a g n e t i c
f i e l d . T h e f l u c t u a t i n g m a g n e t i c f i e l d a r i s e s f r o m an
a l t e r n a t i n g c u r r e n t p a s s e d t h r o u g h a c o i l e x t e r n a l to
t h e m e l t by a g e n e r a t o r . The i n t e r a c t i o n b e t w e e n the
i n d u c e d c u r r e n t s and the m a g n e t i c f i e l d r e s u l t s in
e l e c t r o m a g n e t i c f o r c e s within t h e f l u i d . T h e s e f o r c e s ,
in t u r n , r e s u l t in a v i g o r o u s s t i r r i n g of the bath.
The stirring has three important consequences:
i) i m p r o v e d h o m o g e n i z a t i o n of t h e m e l t (following
t h e a d d i t i o n of a l l o y i n g a g e n t s , etc.),
ii) m a s s t r a n s f e r t h r o u g h the top s u r f a c e of the
m e l t (e.g. into a v a c u u m o r r e f i n i n g s l a g ) i s p r o m o t e d ,
and
i i i ) t h e l i f e t i m e of the r e f r a c t o r i e s c o n t a i n i n g t h e
m e l t i s s h o r t e n e d , due to e r o s i o n and e n h a n c e d m a s s
transfer.
D e s p i t e the i m p o r t a n c e of t h i s e l e c t r o m a g n e t i c s t i r r i n g it has r e c e i v e d v e r y l i t t l e i n v e s t i g a t i o n by p r o c e s s m e t a l l u r g i s t s and until now no m e t h o d of p r e d i c t i n g m e l t v e l o c i t i e s and r e f i n i n g r a t e s h a s b e e n
available, except via empirical or semiempirical
m e t h o d s . One s u c h s e m i e m p i r i c a l m e t h o d w a s p r o v i d e d by M a c h l i n 1 who u s e d a H i g b i e p e n e t r a t i o n
m o d e l 2 to p r e d i c t the r a t e s of m a s s t r a n s f e r t h r o u g h
the top s u r f a c e of a m e l t in an i n d u c t i o n m e l t i n g f u r n a c e . U n f o r t u n a t e l y , in o r d e r to u s e the M a c h l i n
m o d e l , m e l t v e l o c i t i e s at the top s u r f a c e m u s t f i r s t
be known and t h e s e v e l o c i t i e s had to be o b t a i n e d b y
m e a s u r e m e n t on the a c t u a l f u r n a c e . T h e v a l u e of the
M a c h l i n m o d e l l i e s in i t s p r e d i c t i o n of the relative
r a t e s of m a s s t r a n s f e r of s e v e r a l d i s s o l v e d s p e c i e s
a c r o s s t h e m e l t s u r f a c e . The m o d e l h a s b e e n u s e d by
s e v e r a l i n v e s t i g a t o r s to c o r r e l a t e e x p e r i m e n t a l d a t a
on m a s s t r a n s f e r (e.g. R e f s . 3 to 5).
A s e c o n d and m o r e s o p h i s t i c a t e d m o d e l h a s b e e n
p r o v i d e d by S z e k e l y and N a k a n i s h i . 6 T h e s e i n v e s t i g a t o r s s o l v e d the e l e c t r o m a g n e t i c f i e l d e q u a t i o n s f o r a n
A S E A - S K F f u r n a c e in o r d e r to c a l c u l a t e the s t i r r i n g
f o r c e s a c t i n g on the m e l t . T h e t u r b u l e n t fluid flow
e q u a t i o n s f o r the m e l t w e r e then s o l v e d u s i n g a two
e q u a t i o n m o d e l f o r t u r b u l e n c e and a l g o r i t h m s for the
ERACH D. TARAPORE and JAMES W. EVANS are Graduate
Student and Associate Professor of Metallurgy, Department of Materials
Science and Engineering, University of California, Berkeley, CA
94720.
Manuscript submitted December 19, 1975.
METALLURGICAL TRANSACTIONS B
s o l u t i o n of the e q u a t i o n s w h i c h have b e e n d e v e l o p e d
by S p a l d i n g . TM U n f o r t u n a t e l y the S z e k e l y - N a k a n i s h i
m o d e l i s not c o m p l e t e l y p r e d i c t i v e . A n unknown, A,
a p p e a r s in the s o l u t i o n of the e l e c t r o m a g n e t i c f i e l d
e q u a t i o n and the p a r a m e t e r m u s t be d e t e r m i n e d by
m e a s u r e m e n t of the m a g n e t i c f i e l d on t h e i n s i d e s u r f a c e of the c r u c i b l e c o n t a i n i n g the m e l t . T h i s m e a s urement is naturally extremely difficult under operating conditions. Furthermore a second parameter X
a r i s e s . In the c a s e of the A S E A - S K F f u r n a c e , o r o t h e r
f u r n a c e w i t h a p o l y p h a s e coil, X can b e r e a d i l y i d e n t i f i e d w i t h the w a v e l e n g t h of the i m p o s e d m a g n e t i c f i e l d .
H o w e v e r , in the c a s e of a f u r n a c e w i t h a s i n g l e p h a s e
c o i l ~ would have to be r e g a r d e d a s a s e c o n d unknown
r e q u i r i n g m e a s u r e m e n t on the a c t u a l f u r n a c e .
The r e s e a r c h d e s c r i b e d in the p r e s e n t p a p e r i s p a r t
of a continuing i n v e s t i g a t i o n a i m e d at p r o v i d i n g a c o m p l e t e l y p r e d i c t i v e m o d e l f o r fluid flow and m a s s t r a n s f e r in i n d u c t i o n m e l t i n g f u r n a c e s in t e r m s of v a r i a b l e s
w h i c h a r e u n d e r the c o n t r o l of the f u r n a c e d e s i g n e r
o r o p e r a t o r . Such v a r i a b l e s would be the g e o m e t r y
and s i z e of the c r u c i b l e and coil, and the c o i l p h a s i n g ,
c u r r e n t and f r e q u e n c y . In the p r e s e n t p a p e r e m p h a s i s
i s p l a c e d on the s o l u t i o n of t h e e l e c t r o m a g n e t i c f i e l d
e q u a t i o n s and the e x p e r i m e n t a l t e s t i n g of the c o m p u t e d
r e s u l t s in a 30 kW, 3 kHz f u r n a c e ; t h e s o l u t i o n of the
t u r b u l e n t f l u i d flow e q u a t i o n s i s d e s c r i b e d e l s e w h e r e 6-9
and i s t h e r e f o r e given l e s s a t t e n t i o n h e r e . A s u b s e quent p a p e r w i l l d e s c r i b e the t e s t i n g of t h e m o d e l on
a l a r g e i n d u s t r i a l f u r n a c e and p r e s e n t c o m p u t e r p r e d i c t i o n s of the r e s u l t s of v a r y i n g d e s i g n and o p e r a t i n g
v a r i a b l e s on l a r g e s c a l e f u r n a c e s .
THEORY
Solutions of the E l e c t r o m a g n e t i c
Field Equations
I n d u c t i o n m e l t i n g f u r n a c e s o p e r a t e at f r e q u e n c i e s
f o r w h i c h the w a v e l e n g t h i s l a r g e c o m p a r e d to the d i m e n s i o n s of the a p p a r a t u s . T h e M a x w e l l e q u a t i o n s
r e l a t i n g t h e e l e c t r i c f i e l d (E), the m a g n e t i c f i e l d (H),
the c u r r e n t d e n s i t y (J) and the d i s p l a c e m e n t c u r r e n t
m a y t h e r e f o r e be s i m p l i f i e d by a s s u m i n g t h a t the d i s p l a c e m e n t c u r r e n t can be n e g l e c t e d . T h e s i m p l i f i e d
equations are
v • E - - - ~ - ~aH
[1]
VOLUME 7B, SEPTEMBER 1976-343
VxH
=J
[2]
o.
[31
and
v .a
=
In a d d i t i o n it i s n e c e s s a r y to i n t r o d u c e O h m ' s law,
m o d i f i e d f o r t h e c a s e of t h e c o n d u c t o r m o v i n g with a
v e l o c i t y V.
J= a(E+ ~VXH).
[4]
Now c o n s i d e r the a p p l i c a t i o n of t h e s e e q u a t i o n s to t h e
a x i s y m m e t r i c m e l t and c o i l a r r a n g e m e n t d e p i c t e d in
F i g . 1. To a good a p p r o x i m a t i o n t h e s e c o n d t e r m in
p a r e n t h e s e s in Eq. [4] m a y b e n e g l e c t e d P '9 T h i s m e a n s
t h a t the e l e c t r o m a g n e t i c f i e l d e q u a t i o n s m a y b e s o l v e d
i n d e p e n d e n t l y of the fluid flow e q u a t i o n s . E, H, and J
a r e a l l v e c t o r s w h i c h a r e f u n c t i o n s of both p o s i t i o n in
t h e m e l t and t i m e . If the c u r r e n t i n t h e c o i l i s v a r y i n g s i n u s o i d a l l y then E, H and J w i l l a l s o v a r y s i n u s o i d a l l y and we m a y r e g a r d t h e m a s ' p h a s o r s ' . T h a t
i s , the p h y s i c a l l y r e a l E, H and J a r e the r e a l p a r t s
(in the m a t h e m a t i c a l s e n s e ) of t h e c o m p l e x functions
Eoej~t
[5]
It = Hoej ~ t
[61
Z
=
=
JoeJwt.
e q u a t i o n s i s to i n t r o d u c e the v e c t o r p o t e n t i a l A d e fined by
H =l--v • A.
[9]
T h i s l e a d s to Eq. [3] b e i n g a u t o m a t i c a l l y s a t i s f i e d .
A l s o s i n c e a c o m p l e t e s p e c i f i c a t i o n of a v e c t o r f i e l d
r e q u i r e s a s t a t e m e n t of b o t h i t s c u r l and i t s d i v e r g e n c e , l e t us choose
V . A = 0.
[10]
W i t h t h e s e d e f i n i t i o n s Eq. [2] m a y b e s i m p l i f i e d and
i n t e g r a t e d o v e r a f i n i t e v o l u m e that i n c l u d e s both the
m e l t and the c o i l s . 1~
A=~4o
f
J' dV'
[11]
I Ir'L
w h e r e I r ' l is the d i s t a n c e f r o m the c u r r e n t to the
point w h e r e the p o t e n t i a l i s b e i n g e v a l u a t e d , and dV'
i s the e l e m e n t of v o l u m e in the s o u r c e r e g i o n . J ' m u s t
i n c l u d e i n d u c e d c u r r e n t s within the m e l t a s w e l l a s the
c u r r e n t s within the c o i l .
Since we a r e i n t e r e s t e d in c u r r e n t s within the m e l t
we e v a l u a t e A at a l l p o i n t s within the m e l t . H o w e v e r
within the m e l t E i s due e n t i r e l y to e l e c t r o m a g n e t i c
i n d u c t i o n and p o s s e s s e s no s o u r c e s o r s i n k s so that
i t s d i v e r g e n c e i s z e r o , i.e., within the m e l t
[71
V . E = 0.
E q . [1] now b e c o m e s
[12]
F r o m Eq. [10]
V x E = -joJ~H.
[a]
T h e t i m e d e p e n d e n c e a p p e a r i n g in Eq. [1] h a s now
b e e n e l i m i n a t e d in E q . [8] at the p r i c e of the c o m p l i c a t i o n s t e m m i n g f r o m the f a c t t h a t E, H and J a r e now
complex.
A g e n e r a l m e t h o d of s i m p l i f i c a t i o n of t h i s s e t of
V . (-jo.,A) = O.
[13]
S u b s t i t u t i n g Eq. [9] in Eq. [8] we have
v
x
Z = v
(-jc,A).
x
[14]
Since the vectors E and-jcoA have everywhere the
same curl and divergence, we obtain
E
[15]
= -j~A.
Finally from [4], [II], and [15]
0
0
~'~
Jmelt =-
jo~at~ f
4~
ol ~
J'
dV'.
Eq. [16] i s the equation to be u s e d f o r c a l c u l a t i n g
t h e c u r r e n t d e n s i t i e s in t h e m e l t . E q s . [1] and [4] m a y
then be a p p l i e d to y i e l d the m a g n e t i c f i e l d . Eq. [4] m a y
a l s o b e u s e d to c a l c u l a t e the e l e c t r i c f i e l d , although
t h i s i s not r e q u i r e d in the c a l c u l a t i o n of the fluid v e locities.
F o r t h e s y s t e m d e p i c t e d in F i g . 1 Eq. [16] b e c o m e s :
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-jr
0
0
x f ( r , reoil , z, Zeoil)
0
I
0
Fig. 1--Schematic diagram of flow of molten metal in a crucible located axisyrametrically within an induction c o i l
344-VOLUME 7B, SEPTEMBER 1976
[16]
H R
Jo 0", z) = - j ~ , , , , fo fo Jo 0", ~ ') y 0", r', ~ , ~ ')ar'a~'
c~oil IOcoil(rCoil, Zeoil )
[17]
w h e r e I0coi 1 the coil c u r r e n t , i s a l s o a p h a s o r . The
g e o m e t r i c f a c t o r s f(r, r', z, z') in Eq. [17] a r e of the
f o r m of m u t u a l i n d u c t a n c e s b e t w e e n a c i r c u l a r c u r r e n t
loop at (r, z) and one at (r', z'). A l t e r n a t i v e l y they m a y
be r e g a r d e d a s G r e e n ' s f u n c t i o n s f o r t h e e l e c t r o m a g n e t i c f i e l d e q u a t i o n s . T h e s e i n d u c t a n c e s (including the
METALLURGICAL TRANSACTIONS B
ii) that t h e m e l t and c o i l f o r m an a x i s y m m e t r i c s y s t e m , so t h a t
s e l f i n d u c t a n c e f ( r , r, z, z)) can be found in s t a n d a r d
e l e c t r i c a l e n g i n e e r i n g t e x t b o o k s . 1~ T h e n u m e r i c a l
s o l u t i o n of t h i s i n t e g r a l equation and the n u m e r i c a l
c a l c u l a t i o n of the m a g n e t i c f i e l d i s d e s c r i b e d in the
Appendix.
a
T h e e l e c t r o m a g n e t i c f i e l d e q u a t i o n s having b e e n
s o l v e d n u m e r i c a l l y , t h i s s o l u t i o n can then be a p p l i e d
to s o l v e the fluid flow e q u a t i o n s . T h e e q u a t i o n s d e s c r i b i n g the fluid m o t i o n a r e the t i m e a v e r a g e d cont i n u i t y and N a v i e r - S t o k e s e q u a t i o n s w h i c h m a y be
found in t e x t s on t r a n s p o r t p h e n o m e n a (e.g. Ref. 12).
F o r c o m p u t a t i o n a l c o n v e n i e n c e , it i s b e s t to i n t r o d u c e
a s t r e a m function @ and a v o r t i c i t y ~ d e f i n e d by
[18]
ov r _ oyz
Oz
or
(O, Jo,O)
(H r, O,H z )
E : (o, E o , 0),
Solution of the F l u i d F l o w E q u a t i o n s
(a :
=
I'I :
iii) t h a t t h e e f f e c t s of n a t u r a l c o n v e c t i o n m a y b e i g nored,
iv) s t e a d y s t a t e ,
v) p o s i t i o n i n d e p e n d e n t p h y s i c a l p r o p e r t i e s , and
vi) that c u r v a t u r e of the m e l t m e n i s c u s m a y b e n e glected.
T h e s e a s s u m p t i o n s a r e b e l i e v e d to i n t r o d u c e only
s m a l l e r r o r s into the m o d e l although ii) would be i n a c c u r a t e f o r a c o i l with a p i t c h of a s i z e a p p r o a c h i n g
the c o i l d i a m e t e r . A d i s c u s s i o n of n a t u r a l c o n v e c t i o n
in i n d u c t i v e l y h e a t e d m e l t s h a s b e e n p r o v i d e d by
D a m a s k o s , Young and H u g h e s . ~3
and
E X P E R I M E N T A L PROCEDURE
Vr : _1.
0_..~.~
[19]
p r Oz
Vz
[20]
10~
-
p r Or"
The e q u a t i o n of continuity i s then a u t o m a t i c a l l y s a t i s fied and the N a v i e r - S t o k e s equation b e c o m e s
o
or
L
o__
_ wo,.
Or ~ r
= o.
[21]
/I
In t h i s e q u a t i o n W 0 i s the 0 - c o m p o n e n t of the ' m a g n e t i c v o r t i c i t y ' given b y
(O, Wo,O) = V
x
(J x H)
[22]
and a r i s e s f r o m the e l e c t r o m a g n e t i c s t i r r i n g f o r c e s .
M a n i p u l a t i o n of E q s . [18] t h r o u g h [20] y i e l d s t h e
s t r e a m function e q u a t i o n
F•177177
arLpr a r j
azLpr az j
-
~ = 0.
[23]
The v i s c o s i t y /~e a p p e a r i n g in Eq. [21] i s an e f f e c t i v e v i s c o s i t y given by the s u m of the l a m i n a r v i s c o s i t y
and a t u r b u l e n t c o m p o n e n t .
/~e = /~l + /~t"
[24]
A c c o r d i n g to the two e q u a t i o n m o d e l of t u r b u l e n c e the
t u r b u l e n t c o n t r i b u t i o n ktt i s given by
pk
~t : w~/2
Comero
[25]
w h e r e k and w a r e the ' s p e c i f i c t u r b u l e n c e e n e r g y ' and
' t u r b u l e n c e c h a r a c t e r i s t i c ' , k and w a r e t h e m s e l v e s
f u n c t i o n s of p o s i t i o n in the m e l t and m u s t be o b t a i n e d
by the s o l u t i o n of two a d d i t i o n a l p a r t i a l d i f f e r e n t i a l
e q u a t i o n s . F o r f u r t h e r d e t a i l s , p r e s e n t a t i o n of the
b o u n d a r y c o n d i t i o n s on {b, q~, k, and w, and a d i s c u s s i o n of n u m e r i c a l s o l u t i o n s , the r e a d e r i s r e f e r r e d
to the l i t e r a t u r e 7-9 and the A p p e n d i x .
It i s a p p r o p r i a t e to l i s t the a s s u m p t i o n s e n t a i l e d in
the m o d e l . T h e s e a r e :
i) t h a t the s e c o n d t e r m on the r i g h t of Eq. [4] m a y
be n e g l e c t e d ,
METALLURGICAL TRANSACTIONS B
T h e d i f f i c u l t i e s i n v o l v e d in t h e m e a s u r e m e n t of
v e l o c i t i e s within a high t e m p e r a t u r e m e l t a r e s e l f
e v i d e n t . T h e r e f o r e , it w a s d e c i d e d to t e s t t h e p r e d i c t i o n s of the c o m p u t e r c a l c u l a t i o n s by p e r f o r m i n g s u r f a c e v e l o c i t y m e a s u r e m e n t s on an i n d u c t i v e l y s t i r r e d
m e r c u r y m e l t at a p p r o x i m a t e l y r o o m t e m p e r a t u r e .
F i g . 2 i s a d i a g r a m of the a p p a r a t u s w h i c h c o n s i s t s of
a m e r c u r y p o o l l o c a t e d in a c y l i n d r i c a l g l a s s v e s s e l
within a w a t e r c o o l e d c o p p e r c o i l a t t a c h e d to a T o c c o
30 KW, 3 KHz i n d u c t i o n f u r n a c e p o w e r s u p p l y . T h e
p o o l r a n g e d in s i z e up to a p p r o x i m a t e l y 600 l b s . T h e
m e r c u r y w a s of t r i p l e d i s t i l l e d g r a d e . A d i m e n s i o n a l
a n a l y s i s of t h e e l e c t r o m a g n e t i c f i e l d e q u a t i o n s r e v e a l s
t h a t the d i s t r i b u t i o n of t h e s t i r r i n g f o r c e s a c r o s s the
m e l t i s d e t e r m i n e d by t h e r a t i o : (pool r a d i u s / s k i n
depth). T h e s i z e of the m e r c u r y p o o l w a s t h e r e f o r e
c h o s e n so t h a t t h i s r a t i o w a s a p p r o x i m a t e l y t h e s a m e
a s in a c t u a l c o m m e r c i a l i n d u c t i o n m e l t i n g f u r n a c e s
(~5, s a y ) . F u r t h e r m o r e , m e r c u r y at r o o m t e m p e r a t u r e h a s a v i s c o s i t y c l o s e to t h a t of m o l t e n s t e e l and
so t h e r o o m t e m p e r a t u r e m e r c u r y p o o l m a y be e x p e c t e d to show s i m i l a r b e h a v i o r to that of l a r g e high
temperature melts.
S u r f a c e v e l o c i t i e s w e r e m e a s u r e d by t a k i n g a t i m e
e x p o s u r e p h o t o g r a p h u n d e r s t r o b o s c o p i c l i g h t i n g of
a glass bead as it was swept across the surface. From
t h e d i s t a n c e s b e t w e e n a d j a c e n t i m a g e s on t h e r e s u l t ing p r i n t and the f r e q u e n c y of t h e s t r o b o s c o p e a v e l o c i t y m a p of the s u r f a c e could b e o b t a i n e d . T h i s
Stroboscopic ( ~
Flosh U ~ .
Melt
ControlUnit
3000 Hz
t/IOtar
Geblefator
60 Hz
Supply
0
0
0
0
0
0
Fig. 2--Schematic diagram of apparatus.
VOLUME 7B, SEPTEMBER 1976-345
was done at d i f f e r e n t c u r r e n t l e v e l s and for d i f f e r e n t
pool and coil g e o m e t r i e s . P r o b l e m s w e r e i n i t i a l l y enc o u n t e r e d due to the growth of a t h i n but c o h e r e n t c r u s t
on top of the m e r c u r y . T h i s was p r e s u m e d to be a n
oxide l a y e r and the p r o b l e m e l i m i n a t e d by m a i n t a i n i n g a t h i n l a y e r of dilute n i t r i c a c i d (5 pct) above the
m e r c u r y . M e a s u r e m e n t s with and without acid r e v e a l e d n e g l i g i b l e d i f f e r e n c e s i n s u r f a c e v e l o c i t y . The
g l a s s b e a d was weighted by m e a n s of a t u n g s t e n ' k e e l '
to e n s u r e it's s i n k i n g w e l l into the m e r c u r y s u r f a c e
and t h e r e b y t r u e l y r e f l e c t i n g the s u r f a c e v e l o c i t y .
T h e bead was dropped onto the s u r f a c e j u s t b e f o r e the
c a m e r a s h u t t e r was opened. Rough c a l c u l a t i o n s i n d i cate a ' r e l a x a t i o n t i m e ' of a p p r o x i m a t e l y 0.03 s f o r the
b e a d to a d j u s t i t s v e l o c i t y to that of the m e r c u r y s u r f a c e . It i s t h e r e f o r e b e l i e v e d that the bead a c c u r a t e l y
m e a s u r e d the v e l o c i t y of the top few m i l l i m e t e r s of
the m e r c u r y pool. The c u r r e n t t h r o u g h the coil w a s
m e a s u r e d b y m e a n s of a M a d i s o n E l e c t r i c 5 F P 7 5
400/5 c u r r e n t t r a n s f o r m e r c o n n e c t e d to a F l u k e Model
102 VAW m e t e r . Although no r e c o r d was kept of pool
t e m p e r a t u r e d u r i n g the r u n s , p o w e r was shut off and
the pool allowed to cool a f t e r i t s t e m p e r a t u r e had
r i s e n by 20~ or so.
E X P E R I M E N T A L RESULTS AND
COMPARISON WITH THEORY
F i g . 3 i s a plot of m e a s u r e d s u r f a c e v e l o c i t y a s a
f u n c t i o n of r a d i a l p o s i t i o n for s e v e r a l c u r r e n t l e v e l s
i n a 0.289 m d i a m pool, 0.2 m high, located at the m i d point of a 10 t u r n coil. The p r e c i s e g e o m e t r y c a n be
s e e n i n F i g . 4 which i s half of a c r o s s s e c t i o n . A l s o
shown i n F i g . 4 a r e the t h e o r e t i c a l s t r e a m l i n e s c a l c u lated u s i n g the m o d e l for a c u r r e n t of 144 a m p i n the
coil. The 'double loop ~ c i r c u l a t i o n p a t t e r n which t y p i -
c a l l y a r i s e s for a pool located at the c e n t e r of a s i n gle p h a s e coil can be c l e a r l y s e e n . A c l o s e i n s p e c t i o n
of F i g . 4 r e v e a l s that the c i r c u l a t i o n v e l o c i t i e s a r e
h i g h e r in the u p p e r loop ( s t r e a m f u n c t i o n g r a d i e n t s
g r e a t e r ) . T h i s is to b e expected on p h y s i c a l grounds
s i n c e t h e r e i s a l e s s e r liquid solid c o n t a c t a r e a and
t h e r e f o r e l e s s e r d r a g f o r c e s in the u p p e r half of the
pool t h a n i n the l o w e r half. F i g . 5 i s a plot of t h e o r e t i c a l s u r f a c e v e l o c i t y p r o f i l e s for the s a m e conditions
as the e x p e r i m e n t a l p r o f i l e s of F i g . 3. Both m e a s u r e d
and t h e o r e t i c a l v e l o c i t y p r o f i l e s show a m a x i m u m in
s u r f a c e v e l o c i t y . T h e r a d i u s of m a x i m u m velocity app e a r s to be i n d e p e n d e n t of c u r r e n t for the t h e o r e t i c a l
p r e d i c t i o n s and for the e x p e r i m e n t a l m e a s u r e m e n t s .
F o r a m o r e d i r e c t c o m p a r i s o n of the c u r v e s in F i g s .
3 and 5, a plot of the e x p e r i m e n t a l m a x i m u m velocity
a n d the t h e o r e t i c a l l y p r e d i c t e d v e l o c i t y at the s a m e
r a d i u s i s p r o v i d e d a s F i g . 6.
F i g s . 7, 8 and 9 a r e e x p e r i m e n t a l s u r f a c e v e l o c i t i e s , a s t r e a m l i n e plot and t h e o r e t i c a l s u r f a c e v e l o c i t i e s for a m e r c u r y pool of a s m a l l e r d i a m e t e r (0.21m)
a g a i n c e n t e r e d in the coil. A g r e e m e n t b e t w e e n e x p e r i m e n t a l and t h e o r e t i c a l s u r f a c e v e l o c i t i e s is p e r h a p s
b e s t s e e n i n Fig. 10 which is a plot of the e x p e r i m e n t a l
m a x i m u m s u r f a c e v e l o c i t y and the t h e o r e t i c a l v e l o c i t y
at the s a m e r a d i u s a g a i n s t coil c u r r e n t .
T h e effect of p l a c i n g the pool above the c e n t e r of
the coil i s s e e n i n F i g s . 11 t h r o u g h 14. The pool in
t h i s c a s e has the s a m e d i m e n s i o n s a s i n F i g s . 3
t h r o u g h 6 but now has b e e n d i s p l a c e d u p w a r d a d i s t a n c e of 0.085 m f r o m the bottom of a coil with an i n c r e a s e d pitch. Both e x p e r i m e n t a l l y m e a s u r e d and
theoretically predicted surface velocities are in-
9
[
.0
0
0
0
0
0
0
0
O
i 0
9
//
N IN
Rodius
= 0.1445m
Height = 0.204m
0
005
RQdi~l
O, lO
0J445
Distance (m)
Fig. 3--Measured surface velocity profiles for various current levels for a mercury pool (0.1445 m radius X 0.204 m
high) centered in a t 0 turn coil.
346-VOLUME 7B, SEPTEMBER 1976
Current = 1 4 4 omps
Fig. 4--Computed streamline plot for a mercury (viscosity
= 1.6145 • 10 -3 Nwt-s/m 2 and electrical conductivity = 1.07
X 106 mho/m) po01 located at the center of a single phase
coil. Geometry as in Fig. 3.
METALLURGICALTRANSACTIONSB
c r e a s e d t h e r e b y and the upper c i r c u l a t i o n loop has
grown at the expense of the lower one. The a g r e e ment between t h e o r y and e x p e r i m e n t d e t e r i o r a t e s
somewhat under t h e s e c i r c u m s t a n c e s . However, it
should be noted that the t h e o r e t i c a l l y p r e d i c t e d p r o p o r t i o n a l i t y between m e l t velocity and coil c u r r e n t
a p p a r e n t in Fig. 14 (and also in the c o r r e s p o n d i n g
F i g s . 6 and 10) is commonly r e p o r t e d to be t r u e (e.g.
Ref. 15), p r e s u m a b l y on the b a s i s of e x p e r i m e n t a l
m e a s u r e m e n t s which have not found t h e i r way into the
open l i t e r a t u r e .
F i g . 15 i s a computed s t r e a m l i n e plot for the c a s e
w h e r e the m e t a l pool i s p l a c e d well down in the f u r nace coil. F r o m a p r a c t i c a l vie~Tpoint this i s a p p r o x i O.IC
0.O~
0.10
O.O1
400
~ o.o4
"~ 0.01
004
0.02:
0.05
Rodiol Distonce(m)
OJ04
Fig. 7--Measured surface velocity profiles for various curr e n t l e v e l s f o r a m e r c u r y pool (0.104 m r a d i u s • 0.198 m
high) c e n t e r e d in a 10 t u r n coil.
O
0.05
0.10
Rodi01 Distonce (m)
O.1445
Q
Fig. 5--Computed surface velocity profiles for conditions
u n d e r w h i c h e x p e r i m e n t a l p r o f i l e s s h o w n in F i g . 3 w e r e
measured.
0.10
I
I
l
I
I
I
I
0
O
0
O
0
0
0
0
Q
I
Experimental
Slope= 1.39
_/O//Theor
0.05
ly /0"
\
etieol
Slope=1.0
\
,S
0.01
1(30
I
!
I
I
500
I
I
I
I
1000
Coil Current (omps)
Fig. 6--Experimental maximum velocity and theoretical vel o c i t y at s a m e r a d i u s c o m p a r e d a s f u n c t i o n s of coil c u r r e n t ,
f o r c o n d i t i o n s of F i g . 3.
METALLURGICAL TRANSACTIONS B
Rodius
= 0.104m
Height
= 0.198
Current
= 104omps
m
F i g . 8 - - C o m p u t e d s t r e a m l i n e plot f o r a m e r c u r y pool l o c a t e d
at t h e c e n t e r of a s i n g l e p h a s e coil. G e o m e t r y a s in F i g . 7.
VOLUME 7B, SEPTEMBER 1 9 7 6 - 3 4 7
m a t e l y the situation e n c o u n t e r e d when a f u r n a c e cont a i n s only a h e e l of m o l t e n m e t a l . A c c o r d i n g to F i g .
15 the l o w e r c i r c u l a t i o n loop grows at the e x p e n s e of
the upper loop when the m e l t i s d i s p l a c e d downwards
f r o m the c o i l c e n t e r . A t t e m p t s to m e a s u r e s u r f a c e
v e l o c i t i e s of the m e r c u r y for t h i s configuration did not
give r e l i a b l e r e s u l t s . T h e m e r c u r y s u r f a c e w a s h i g h l y
turbulent and o v e r s o m e r e g i o n s of the s u r f a c e the
m e r c u r y flow w a s a c t u a l l y the r e v e r s e of that p r e dicted in F i g . 15. It i s s u g g e s t e d that this d i s p a r i t y
b e t w e e n the m o d e l and e x p e r i m e n t m a y be due to the
m e n i s c u s c u r v a t u r e . The m o d e l a s s u m e s the m e n i s c u s
to be flat and it i s l i k e l y that this i s a poor a s s u m p -
I
0'101
I
592 omps
9
0.10
j
0.06
008
400
E"
omps
/
~
9 424
0.04
"G 0.06
-<
248
0.02:
_o
0.04
0
104
I
0.05
0
I
O. I 0
01445
Radial Distance (m)
0.02
Fig. 11--Measured surface velocity profiles for various c u r rent levels for a m e r c u r y pool (0.1445 m radius X 0.212 m
high) placed above the center of a 10 turn coil.
t-O~P "-v
0
I
0.05
R0di0l Distance (m)
1
0.104
Fig. 9 - - C o m p u t e d s u r f a c e v e l o c i t y p r o f i l e s f o r c o n d i t i o n s
under which experimental
profiles
s h o w n i n Fig. 7 w e r e
measured.
0.10,,
,
/Zf_.
Slope =1.37
Sbpe~l.O
0.05
//
9
11
9
~
9
9
9
, //EVxpe r i'menltol '
~ , , " Theoretical
u
_o
/ ~/ill
///e
Radius = 0.1445m
Height = 0 . 2 1 2 m
Current = 136amps
9
0.01
I
I00
I
I
I
500
Coil Current (ampa)
I
I
I
I
)(30
Fig. 1 0 - - E x p e r i m e n t a l m a x i m u m v e l o c i t y and t h e o r e t i c a l v e l o c i t y at s a m e r a d i u s c o m p a r e d a s f u n c t i o n s of coil c u r r e n t ,
f o r c o n d i t i o n s of Fig. 7.
348 VOLUME 7B, SEPTEMBER 1976
9
Fig. 12--Computed streamline plot for a m e r c u r y pool located
above the center of a single phase coil. Geometry as in Fig.
11.
METALLURGICAL TRANSACTIONS B
0
t i o n when the m a j o r p a r t of the s t i r r i n g f o r c e s and v e locity g r a d i e n t s a r e i n the m e n i s c u s r e g i o n .
Some c o m m e n t a r y on the fit b e t w e e n e x p e r i m e n t
and t h e o r y r e f l e c t e d i n F i g s . 3 t h r o u g h 15 is a p p r o p r i ate. T h e e x p e r i m e n t a l l y m e a s u r e d s u r f a c e v e l o c i t i e s
d e v i a t e f r o m the t h e o r e t i c a l l y p r e d i c t e d o n e s , p a r t i c u l a r l y at low and high r a d i i and i n the c a s e of a m e l t
d i s p l a c e d f r o m the coil c e n t e r . N e v e r t h e l e s s , it m u s t
be r e c o g n i z e d that the m o d e l p r e s e n t e d h e r e c o n t a i n s
o,ot
'
'
0
0
0
0
t
0
O.08r
0
5 04 a m p s
0
"G 0.06
0
0
o
> 004
Radius = 0.1445m
Height = 0.180 m
Current = ?..680rnlps
Fig. 15--Computed streamline plot for a mercury pool (0.1445
m radius • 0.212 m high) located below the center of a single
phase coil.
O.O2
0 ~
I
0
0.05
I
O.lO
0,1445
Radial Distance tin)
Fig. 13--Computed surface velocity profiles for conditions
under which experimental profiles shown in Fig. 11 were
measured.
O.lO
I
I
1
I
Experimental/
~if
1
!
1
CONCLUDING REMARKS
ii
Slope=2 ' V / /
~/
/ , ,/ ~ e
o.o
?,
/iX/! /
/
no a d j u s t a b l e p a r a m e t e r s which have b e e n m a n i p u l a t e d
to b r i n g about the fit. The fit i s even m o r e r e m a r k a b l e
when it i s r e c o g n i z e d that the K o l m o g o r o v - P r a n d t l
m o d e l f o r t u r b u l e n c e , as m o d i f i e d by Spalding and
u s e d h e r e , was developed to d e s c r i b e gas r e c i r c u l a t i o n i n f u e l f i r e d f u r n a c e s ; i t s a p p l i c a t i o n to m e t a l
systems has been vigorously pursued from a theoretical viewpoint z4 but it has b e e n t e s t e d e x p e r i m e n t a l l y
to only a l i m i t e d extent.
A c o m p l e t e l y p r e d i c t i v e m o d e l for fluid flow i n i n d u c t i o n m e l t i n g f u r n a c e s has b e e n d e v e l o p e d . I n p u t s
to the m o d e l c o n s i s t only of p h y s i c a l p r o p e r t y data
a n d p a r a m e t e r s u n d e r the c o n t r o l of the f u r n a c e d e s i g n e r and o p e r a t o r . No a d j u s t a b l e p a r a m e t e r s or
e x p e r i m e n t a l m e a s u r e m e n t s a r e e n t a i l e d i n the m o d e l .
When c o m p a r e d with a c t u a l s u r f a c e v e l o c i t y m e a s u r e m e n t s on a n i n d u c t i v e l y s t i r r e d m e r c u r y m e l t , the
m o d e l was found to show r e a s o n a b l e a g r e e m e n t with
e x p e r i m e n t . It i s s u g g e s t e d that the m o d e l m a y p r o v i d e
a u s e f u l tool i n the d e s i g n and d e v e l o p m e n t of i n d u c tion melting furnaces. Experiments are currently
u n d e r way to t e s t the m o d e l on a l a r g e v a c u u m i n d u c tion melting furnace.
Theoretical
Slope = 1,0
Igl~
I
APPENDIX
0.0
I
Ioo
I
I
I
500
coil Current (amps)
I
I
I
I
I000
Fig. 14--Experimental maximum velocity and theoretical velocity at same radius compared as functions of coil current,
for conditions of Fig. 11.
METALLURGICALTRANSACTIONSB
T h e c u r r e n t d i s t r i b u t i o n i n the m e l t has b e e n shown
to b e g i v e n by the i n t e g r a l e q u a t i o n
Jo(,',
: -J.,..
HR
.f~o.f-oJo(r', z')f
VOLUME 7B, SEPTEMBER 1976-349
--jr
c~oil I0coi 1 (rcoil , Zcoil )
xf(r,
r c o i l , z, Zcoil).
[17]
A s i m p l e a p p r o x i m a t e s o l u t i o n to this e q u a t i o n i s
a v a i l a b l e u s i n g a f i n i t e d i f f e r e n c e t e c h n i q u e to s o l v e
f o r the c u r r e n t d e n s i t y Jo at the n o d a l points of a g r i d
d i s t r i b u t e d o v e r the c r o s s - s e c t i o n of the m e l t as
s h o w n i n F i g . 16. The c u r r e n t d e n s i t y in the cell
a r o u n d each node i s then a p p r o x i m a t e d by the c u r r e n t
d e n s i t y at the node.
It i s known that the i n d u c e d c u r r e n t s d e c r e a s e
s h a r p l y within the c o r e of the m e l t , e s p e c i a l l y at high
f r e q u e n c y , a p h e n o m e n o n known a s the s k i n effect. In
o r d e r to be as a c c u r a t e a s p o s s i b l e i n the a p p r o x i m a t i o n of the c u r r e n t d e n s i t y d i s t r i b u t i o n by the v a l u e
of the c u r r e n t d e n s i t y at the nodes it was t h e r e f o r e
d e c i d e d to use G a u s s i a n q u a d r a t u r e to e v a l u a t e the
double i n t e g r a l on the r i g h t hand side of Eq. [17 ]. In
t h i s way t h e r e a r e a g r e a t e r n u m b e r of grid p o i n t s
n e a r the s u r f a c e w h e r e the g r a d i e n t i n c u r r e n t d e n s i t y
i s g r e a t e s t . Eq. [17] a f t e r t a k i n g finite d i f f e r e n c e s
b e c o m e s the set of M x N s i m u l t a n e o u s e q u a t i o n s .
M
N
co x Mutual i n d u c t a n c e b e t w e e n loops at m, n and p, q
R e s i s t a n c e of loop at m, n • a r e a Of loop at m, n
T h e m u t u a l i n d u c t a n c e was c a l c u l a t e d u s i n g s t a n d a r d
r i n g - r i n g f o r m u l a s [11] to yield
2a
k2
k~
[A2]
k2 =
[A3]
4ab
d2+ (a + b)2
w h e r e a and b a r e the r a d i i of the two c o n c e n t r i c coils
i n p a r a l l e l p l a n e s s e p a r a t e d by a d i s t a n c e d. K(k) and
E(k) a r e c o m p l e t e e l l i p t i c i n t e g r a l s of the f i r s t and
second kinds.
When p, q - m, n the self i n d u c t a n c e f o r m u l a has to
be u s e d .
~176'm, n, m, n -----o~L 'm, n
HR
"~4-~z
~z Vp UqJOP, qM ' m ' n ' p ' q
JOm, n + joJ
lot is the coil c u r r e n t ,
vp" and Vq a r e G a u s s i a n weight f a c t o r s , and
coM'rn, n,p,q is the r a t i o :
[A4]
T
= --Jc~
IotM'rn, n, t
[A1]
k2
4r(r-
a)
[A5]
where
M and N a r e the n u m b e r of r and z grid points,
T i s the n u m b e r of t u r n s of the coil,
I
i
M,N
I.Nb
0
1
I-
0
0
O
0
0
Q
I_
I
i
i
~
I
I
1~ -
I
1.3,-
Z.
n,q
indies i
P,~
Q
I. I ]i 2,1 3,1 4,1
M
.Imag ~~,
T
= -- co~ I0Imag M 'rn
t:l r
,n,t
and
M
N
H R p ~ _ q~_
.Real. ,
0 m, n + co ~
1 : 1 Vp VqdOp ' q iV1 m, n, p, q
jlmag
t turns
M,I
m,p indices
r
Fig. 16--Illustration of the finite difference grid showing nomenclature used.
350-VOLUME 7B, SEPTEMBER 1976
N
--aOm, nR' eal+ co ~HR ~
~ VpVqdOp, q iv1 m , n , p , q
--p
:1 q =1
9
1,2
w h e r e r is the r a d i u s of the coil of c r o s s s e c t i o n a l
r a d i u s a.
Also because these mutual inductances are accurate
only for coils with c i r c u l a r c r o s s - s e c t i o n s a c o r r e c tion has been made (Ref. 16) to allow the use of these
f o r m u l a s in the c a l c u l a t i o n of the m u t u a l i n d u c t a n c e
of coils with r e c t a n g u l a r c r o s s - s e c t i o n such as those
i n a pool with a G a u s s i a n grid s p a c i n g .
F i n a l l y to avoid d e a l i n g with c o m p l e x n u m b e r s we
s p l i t up the M • N c o m p l e x s i m u l t a n e o u s equations
into 2 • M • N r e a l s i m u l t a n e o u s l i n e a r a l g e b r a i c
equations.
=-
T
coS/Real M'
i:t
t
m, m t "
[A6]
T h e s e 2 • M • N equations can b e solved by m a t r i x
i n v e r s i o n to o b t a i n a c u r r e n t d e n s i t y d i s t r i b u t i o n
w i t h i n the m e l t . T h e s e v a l u e s of Jo a r e i n t e r p o l a t e d
u s i n g L a g r a n g i a n i n t e r p o l a t i o n to a s e t of u n i f o r m l y
s p a c e d m e s h points to f a c i l i t a t e d i f f e r e n t i a t i o n to obt a i n H r and Hz u s i n g E q s . [1] and [4].
METALLURGICALTRANSACTIONSB
T h e m a g n e t i c v o r t i c i t y W 0 i s t h e n g i v e n i n R e f . 16.
2/.t ( j 0 a e a l H r a e a l
W0 = _ -7
+ joImagHrImag)
- 2~t2coa(HzRealHrIrnag
-H zlmagH~.Real)
. [A7]
T h i s i s t h e f i r s t s t a g e of t h e s o l u t i o n . S o m e w h a t
similar, though less accurate, algorithms have been
described by Kolbe and Reiss.~7
The hydrodynamic equations to be solved to obtain
turbulent flow characteristics
are the vorticity and
s t r e a m f u n c t i o n e q u a t i o n s ( i . e . , E q s . [21] a n d [23])
together with the k and w equations. 7
0
0
o
o I
Ok\
0
[AS]
0
0(
07
Ok
0w)
I~ero-r
Ow
- rSw = 0
[Ag]
where S k and S w are source terms calculated using
parameters tabulated by Spalding. 7 The effective visc o s i t y m a y b e c a l c u l a t e d u s i n g E q s . [24] a n d [ 2 5 ] .
Finite differences are taken using the method of Spalding8 and solved by successive point iteration.
The boundary conditions for this geometry have been
d i s c u s s e d e l s e w h e r e ( R e f s . 6, 8) b u t a r e f i n e m e n t h a s
been included to take into account the high magnetic
vorticities at the surface grid points 9 When the
Navier-Stokes equation is written down and simplified
for points on the boundary (top, wall and bottom) an
i m p r o v e d e s t i m a t e i s o b t a i n e d f o r t h e g r a d i e n t 0ff//0n
where 9 = ~b/r and n is the normal to the surface.
T h i s v a l u e i s u s e d i n s t e a d of t h e c e n t r a l d i f f e r e n c e
gradient used by Spalding 8 and Szekely and Nakanishi 6
i n t h e e v a l u a t i o n of t h e d i f f u s i v e t e r m s o n e i n f r o m
the boundary. In this way the magnetic vorticity one
in from the boundary is increased by approximately
o n e h a l f of t h e m a g n e t i c v o r t i c i t y f o r t h e a d j a c e n t p o i n t
at the surface. Since the magnetic vorticity does not
a p p e a r i n t h e e v a l u a t i o n o f 9 - ~b//r a t t h e b o u n d a r i e s ,
this technique allows us to take into the calculation
t h e v a l u e s of t h e m a g n e t i c v o r t i c i t y a t t h e s u r f a c e
where it has its maximum value. This completes the
s e c o n d s t a g e of t h e s o l u t i o n .
T h e p l o t s s e e n i n F i g s . 4, 8 a n d 12 a r e t h e r e f o r e
generated in three stages, using three programs on the
CDC 6 4 0 0 .
FIELD: solves the electromagnetic field equations
to give W o . For a 9 • 9 mesh this solution takes app r o x i m a t e l y 62 s .
TURBFLO: is read in with W o from the FIELD solut i o n to s o l v e t h e f l u i d f l o w e q u a t i o n s t o y i e l d d i s t r i b u t i o n s of ~b, e p / r , k, w , v e l o c i t i e s V r a n d Vz a n d t h e
e f f e c t i v e v i s c o s i t y /~e" T h e t i m e t a k e n t o c o n v e r g e
METALLURGICAL TRANSACTIONS B
t o a s o l u t i o n u s i n g a 17 • 17 m e s h i s a p p r o x i m a t e l y 7
rain.
STREAM: is used to plot the resultant streamline
pattern using the CALCOMP plotter.
LIST OF SYMBOLS
A
= vector potential
d
= area
E
= electric field
H
= magnetic field
H
= h e i g h t of p o o l
I0coi 1 = coil current
J
= current density
j
=d:-I
k
= specific turbulence energy
L'
= self inductance - resistance
M'
= m u t u a l i n d u c t a n c e :- r e s i s t a n c e
R
= r a d i u s of p o o l
t
= time
V
= fluid velocity
We
= magnetic vorticity
w
= turbulence characteristic
p
= fluid density
tz
= permeability
/~e
= effective viscosity
ix l
= laminar viscosity
tz t
= turbulent viscosity
v
= Gaussian weight factors
r
= conductivity
q5
= vorticity
= stream function
= angular frequency
ACKNOWLEDGMENTS
The authors would like to express their gratitude
to the Committee on Research and to the Computer
C e n t e r , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , f o r s u p port which has made this investigation possible.
REFERENCES
1. E. S. Machlin: Trans. TMS-AIME, 1960, vol. 218, pp. 314-26.
2. R. Higbie:AIChEJ., 1935, vol. 31, p. 365.
3. A. G. Cowen and J. A. Charles: J. Iron Steellnst., 1971, vol. 209, pp. 37-45.
4. S. W. Graham and B. B. Argent: J. Iron Steellnst., 1967, vol. 205, pp. 1066-68.
5. R. D. Pehlke and J. F. Elliot: Trans. TMS-AIME, 1963, vol. 227, pp. 844-54.
6. J. Szekely and K. Nakanishi:Met. Trans. B, 1975, vol. 6B, pp. 245-56.
7. D. B. Spalding: VDI(Ver. Deut. lng.) Forschungsh, 1972, vol. 38, no. 549,
pp. 5-16.
8. D. B. Spalding,et al.: Heat and Mass Transfer in Recirculating Flows, Academic
Press, London, 1969.
9. E. D. Tarapore: Ph.D. Thesis, Universityof California, Berkeley, 1976.
10. J. D. Jackson: ClassicalElectrodynamics, John Wileyand Sons, Inc., New
York, 1962.
11. S. Ramo, J. R. Whinnery, and T. Van Duzer: Fields and Waves in Communications Electronics, John Wiley and Sons, Inc., New York, 1965.
12. R. B. Bird, W. E. Stewart, and E. N. Lightfood: Transport Phenomena, John
Wiley and Sons, Inc., New York, 1960.
13. N. J. Damaskos, F. J. Young, and W. F. Hughes:Proc. lEE, 1963, vol. 1lO,
no. 6, pp. 1089-95.
14. J. Szekely, S. Asai, and C.W. Chang: Process Engineering of Pyrometallurgy,
Inst. Mining and Metallurgy, London, 1974.
15. Tocco Meltmaster Bulletin MB-1012.
16. W. Vogt: Brown Boreri Rev., 1969, p. 25.
17. E. Kolbe and W. Reiss: Wiss. Z. Hochsch. Elektroteeh. Ilmenau, 1963, vol. 9,
pp. 311-17.
VOLUME 7B, SEPTEMBER 1976-351