influence the bulk mechani- cal properties, particularly the fracture ..... Twoother mechani.sms resulting from fluid flow wereinvestigated inearlier work by the ...
ISIJ International,
Qf Particles
Redistribution
O.
HANand J.
D.
Vol.
35 (1995), No,
693-699
6, pp.
during Solidification
HUNT
Department of Materials,
University
of Oxford, Parks Road. Oxford
(Received on December26. 1994; accepted in
final
OXI 3PH, England, UK.
form on February
17.
1995)
(inclusions and equiaxed grains) get pushed or engulfed by the freezing of particles in the solid is controlled by the forces acting on the particles and the flow in the liquid melt. The present paper describes recent work on determining someof the more important of particles during solidification. parameters which govern the redistribution
During solidification.
front.
particles
The redistribution
inclusions, equiaxed KEYWORDS:
1.
grains;
particle
concentration in the solid is illustrated in Fig. 1. If R is interthe growth velocity of the macroscopic solid-liquid face, and CL and Cs are the particle concentrations (in
Introduction
of a liquid containing disDuring the solidification persed second phase particles, the particles in the liquid melt can migrate towards or away from the freezing front; particles near the freezing front will either be engulfed or rejected. These two phenomenalead to redistribution of the particles during solidification. There are several processes of technological imporof particles tance which involves redistribution on freezing. The final distribution of particles is important in the processing of metal-matrix composites, the solidification of monotectic alloys, the growth of crystals from melt containing impurities, or the casting of metals of containing inclusions. An unfavourable distribution particles will detrimentally influence the bulk mechaniparticularly the fracture toughness.i~3) cal properties, of particles has the capacity to Controlled redistribution regions or layers, create novel materials with particle-rich particle concentration and therefore gradient of with or a of mechanical properties.4 ~ 6) This article is a discussion of the fundamental prinredistribution during solidification ciples of particle in the presence of convection. The paper covers three main parts: the forces on the particles in the liquid melt the velocity and concentration of during solidification;
number per unit volume at the solid-liquid interface), the numberof particles per unit area incorporated into the solid equals the numberof particle leaving the liquid. That
2.
and particle
The Formal Partition
...........(1) ........
of the particle normal to the interface and P a simple parameter that is either zero is or one. Whenthe interface pushes the particles P zero is it is velocity the of V~ otherwise one. The term
where
V,* is
the velocity is
R-
particles
relative
to the interface;
particles
tend to
moveawayfrom
(1) refers
to
a particular
local concentration
when V~ is positive the interface. Equation
particle
of particles
Cs=
size
and density. The
in the solid
P(R V~)CL
is
...........(2)
R
Theproblem is thus to find the particle velocity normal to the interface (V~), to calculate the local concentration of particles in the liquid (CL) whenthe interface passes and to determine whether the particles are pushed or not
(P= I
or
O).
Estimates of these terms
by considering the forces on the
near the freezing front and particle pushing by the freezing front. The treatments are confined to the non turbulent flow that will usually occur during the solidification of muchof a casting. Since the processes involved are very complex, this article concentrates on laying semi-quantitative foundations and presents models that can be refined by more sophisticated analysis. As presented the models refer to particle
is:
RCs=P(R-V~)CL
particles
a particular
flow.
fluid
redistribution,
pushing, solidificationj
maybe obtained
particles
and will be
considered in later sections.
si~e.
Equation
Whena solid-liquid interface grows, particles can be incorporated into the solid. Determination of the particle
Frg. 1.
Determination of the particle solid.
693
concentratron
m the
C 1995 iSIJ
ISIJ International,
Vol.
35 (1995), No.
6
For a cellular or dendritic freezing front, particles coming into the semi-solid zone maybe trapped by the
u",
freezing front. In the semi-solid zone, the particles may be pushed locally by the continued freezing of the solid Into the inter-dendritic regions. Equation (2) may in principle be used to treat both stages of the process. For the first V~ is the normal velocity and P is the particle pushlng parameter for the macroscopic dendrite front. For the latter case, V~ is the normal velocity and P is the particle pushing parameter for the local continued freezing of the solid. Clearly when V~ is greater than R the solid composition does not becomenegative as it would from Eq. (2). Under this condition P should be regarded as zero. The values of V~~-v~, P and CL in Eq. (2) can be estimated by considcring the forces acting on a particle.
The Forces Exerted on a Particle
3.
Suspension
in
Three forces on the particle are considered in the present work. Theseare gravity, viscous drag and a force due to a velocity gradient in the liquid. Clearly others could be included if required. The gravitational force is given by
Fg- 4 7ra3(pp 3 where a
is
the radius
p are the
force
motion of the
is
particle
••••••••••(3)
of a particle;
acceleration; pp and the liquid respectively.
The viscous
P)9
g is
in the opposite direction relative to the liquid
where p is the the particle
and
to the
and
viscosity
is
..........(4)
Vis
of the liquid;
the velocity
of
to the liquid. The right hand term particle in an unboundedliquid.7) As
surface. 0=alh8),
whenthe
particle
is
moving along
the
force Fd acts in a direction opposite to the motion of a particle relative to the liquid. The Saffman force FL acts normal to the interface and its sign is determined by the direction of fluid flow and the densities of the particles and the liquid. Whenthe interface is horizontal. FL is directed away from the interface (Fig. 2(b)); whenthe interface is vertical, FL acts towards or frorn the interface depending away on the relative densities of the particle and the liquid. These forces can be used to calculate particle velocities and concentrations for different geometries. Particle
Velocities
Fd=61Epav~e........
=-
Vp~s
the
normal velocity
..........(6)
is
v~
- -
""""""""(7)
67~e
2a2(pp-P)9 9/1
1
y2 (8)
H2
where y is the distance from the interface and is the half width of the flow channel. Equations (7) and (8) predict that particles less dense than the liquid moveawayfrom the interface and those denser than the liquid travel towards the interface during downwardflow. These predictions have been verified on single particles in a number of experiments. 18 ~ 23) The result of this motion on concentration is shownin Fig. 3and will be discussed later. For a horlzontal interface, the forces determining V~ are again Fg. FL and Fd, but whenthe density difference between the particle and the liquid is significant, FL can usually be neglected and the normal velosity V~ of the
H
..........(5)
local velocity gradient in the liquid: v is the kinematic viscosity and Vp is the velocity of the particle relative to the liquid in the direction parallel to the interface.
2showsthe directions
C 1995 ISIJ
of the
flow.
where Vp is governed by the local velocity of the liquid and the forces in the direction parallel to the interface. For a particle settling in a Poiseuille flow, Vpis given by8)
where Sis the
pending on
showing the direction
downwardfiow and horizontal
6.46afi7,Vp
...........
Figure
Horizontal flow
The viscous
V~
it is suggested that 0=1n(a/h).9) Forces are exerted on a particle near a surface due to the fluid velocity gradient. The force arises because of the different flow rate on the two sides of the particle. The force is accentuated when the particle movesat a velocity different from the average fluid flow rate. The difference can be the result of density differences between the solid and the liquid. Theproblem has been considered by a number of workers.12~i7) This Saffman force is here assumedto be of the forml5,16)
gravitational
forces in
By equating Fd and FL,
wall
FL=6.46,la2~l~Vp
Schematic illustration
O
one for a a particle approaches surface a more complicated expressions should be used.10,11) Whenthe separation h approaches zero and the particle is moving towards the is
2.
(b)
Generally the acceleration of a particle is small so that on the particle considered above may be assumedto be in equilibrium. The velocity of a particle can thus be estimated by equating the driving forces and the forcesl2) resisting motion. For a vertical interface with flow downthe interface, the driving force for the motion of the particle normal to the interface is FL given by Eq. (5); the resisting viscous force in the normal direction is given by
........
relative
Fig.
Downwardflow
the forces
considered to be of the form7 ~ 11)
Fd=67~,taVe
~~ (a)
4.
the gravity
density of the particle
FD~
of these three forces. The force Fg acts downwardsor upwards dethe densities of the particle and the liquid.
694
ISIJ International,
Vol,
35 (1995), No.
6
12
~ ~,
3cm/ sec 6cm/ sec
8
vn 1:vp
Experimental
Ana[ ytical
10
A
O1
6
--~ahN
4
CO
x
Ayo =
A
Xo
::
o
2 1 O1 (a)
0,3 0.4 0,5 0,6 07 0.8 0.9 Distance away from tube wail (mm)
0,2
2COumpolypropylene
particles,
xhl
Aym~- -
13
~.2
l.1
O90
density:
relative
0,l
Fig.
12
4.
Schematic illustration showing the terms used in the of the particle concentration in the liquid.
calculation Experimental
Analytical
10
3cm / sec 6cmI sec
8
V U 6
and x~ (see Fig. 4) parallel meansthat for small Ay~ at xo
~ a\
\
4
Vp"dy
a
A
2 02
0.1
0.3 0.4 0.5 06 07 Distance away from tube wall
particles, PMMA
100 um
relative
I
A
Co CmfAy
08 09 OI
1,1
l'
12 13
(mm)
Anal ytical
~
D
8
p
u
A
\
u 6 4
A\
a\
A A
\
Ol (c)
Fig.
1 L*-
0.2
E,
a
03 0.4 0.5 0.6 0.7 08 09 Distance away awa from tube wall (mm)
50 umalumina 3.
relative
particles,
density:
Ol
in
Particle
= -v~ - -
equation
is
(
2a2 pp P)9 -
9~O
"""""""""(9)
Stokes law.
shown in
The concentration near a surface can be calculated by following the path of the particles. Under the infiuence of the forces considered above, particles migrate across streamlines and can concentrate in different regions of the fluid. In the two dimensional situation, assuming particles start at A and B(separated by Ayo) then move to A' and B' (separated by Ay~ see Fig. 4), at steady state the particles entering and leaving A'B' must equal those entering AB. That is to conserve the numberof particles
CoVpody=
fAy
CmVp'mdy
'
(I
O)
.
local velocities
3.
and
The
give
lines
show the calculated con-
a reasonably good
fit
with ex-
In horizontal flow the normal velocity is mainly a force on the particle and balance of the gravitational viscous force acting normal to the interface. The normal velocity is thus constant except whenthe particle is very near the interface. This meansthe concentration varies little until the particle touches the interface. Similar considerations can be used to treat a free surface and to treat a free surface with Marangoni fiow. 6.
Criteria
for Particle
Pushing
In the present work it is assumedthat a solid-1iuqid interface will either capture or push a particle. When front, the particles solidification the pushed by are
~
where Vp', and Vp'~ are the
Fig.
centrations periment.24)
Concentration
fAy
..........(12)
where V is the local fluid velocity. These equations for Vp' and V~ maybe used to calculate the trajectories AA' and BB' and hence Ayo and Ay~. Assuming that the particle concentration is initially uniform and equal to Co at x=xo Eq. (1 l) maybe used to calculate C~. Experimental concentrations of particles were measured using a laser and a photographic time exposure. These experiments were reported in a previous paper.23) Examples for downwardflow over a vertical surface are
down-
the startthe density
can be obtained by equating Fg and Fd giving
When0= I this 5.
l3
12
Vp'=V+Vp .....
The concentration of particles near the wall ward flow of water. Here the dash dot line is
V~
Ay~ is taken
mm 378
ing composition, Co; the relative density is of the particles with respect to water.23,24)
particle
l,l
Vp'mdy
~
Experimental
3cm/ sec 6cm/ sec
.(11)
interface, C~will to be at the solid-liquid in particles the liquid at the be the concentration of interface, solid-liquid CL. In order to calculate Ayo from Ay~, the trajectories of the particles need to be determined. For a vertical interface with fluid flowing downwards, the normal velocity is given by Eq. (7) (with 6=1 Provided the particles are not too near the surface). The Vp in Eq. (7) is given by an expression of the form of Eq. (8). The velocity, Vp', of a particle relative to the interface is given by8) If
124
density:
12 10
This
fAy
\ IL
(b)
to the interface.
of the particles
695
C 1995 ISIJ
ISIJ International.
Vol.
35 (1995). No.
concentration in the solid will be zero no matter what the concentration of particles at the liquid side of the interface is. This means that when the particle is pushed the parameter P in Eq. (2) is zero. Three mechanismsfor particle pushing will be considered. In the past extensive work has been carried out on a model based on surface energy considerations.2s-35) Two other mechani.sms resulting from fluid flow were investigated in earlier work by the present
6
particle
LIQUID FR
VL
FD
authors 36, 37)
Frg.
6.1.
5.
to a flat The basic
.e
The forces on a particle at the momentwhenit starts moving. Thedriving force for the motion of the particle is
that pushing can occur when the solid particle greater than the sum of the solid-1iquid and energy the particle liquid. The driving force for the rejection of a particle arises from the surface energy difference, Aao
idea
/1L
FL
SOLID
,
Surface Energy Model This model can really only be applied horizontal surface in a freezing pure material.
Fwlf
the viscous drag
is
temperature and thus keep the surfaces apart. If the surfaces do not approach they cannot exert a force. This should meanthat the particle will be trapped with a layer of liquid left behind surrounding the particle. Whenparticle pushing occurs in alloy systems other mechanismsshould perhaps be considered. Despite these reservations, Eq. (14) and (1 5) could in principle be used growth front velocity so that P to determine the critical ca]culated. be may
is
'
A(T =(r -(ap,L+(rs,L);~O p,s o
where
...............(13)
L
the surface energy; the subscripts P, S. particle, the It is the solid and the liquid. represent suggested that, under this driving force, the particle is pushed by the interface while the solid grows at a rate smaller than a critical growth rate. Uhlmannet al.25) developed a model to predict the critical growth rate of cr
is
Pushing as a Result of Fluid Flow have been suggested by the mechanisms Two
6.2.
present authors for particle pushing resulting frorn fluid flow.36,37) In the first the particle is in contact with the solid and is movedover the surface by the fluid flow as the solid grows. In the second the particle near the front becomes trapped because of the roughness of the
the solid by considering the conservation of massin the region of contact between the particle and the interface and taking into account the viscous drag on the particle. Bolling et al.26) modified the surface energy treatment by solving the problem of the interface shape near a
particle,
obtaining an equation
solidification
(91(rs,L
R2= '
Particle
..........(14)
""""
p2a3(pp-P)
Q2A(T o
~a
kp
where 02 maybe obtained from Ref. the thermal conductivities
30);
kp and kL are and the liquid
more elaborate models have been presented The critical velocity obtained are of
in the literature.
manyforms depending on
the factors
included in the
physical
in
fluid
significant
an
alloy,
solute
flow at a rate
must diffuse against sufficient
to
freezing
front
mental
results
growing
fluid
The experiand indicate that a function of particle size and
at different
shown in
Fig.
5
N
this
prevent a
build up of solute in the gap. Any build up of solute in the gap will lead to a change in the liquidus
C 1995 ISIJ
Whenthe
rates.
6,
are the critical flow velocity is density, and the growth rate of the solid. Thecritical flow velocity required to put a particle into motion can be estimated by considering the forces on a particle. The forces on a particle at the momentit starts to moveare shownin Fig. where Fd is the viscous drag in the direction parallel to the interface, FRis the viscous drag in the direction normal to the interface when the particle is travelling at a rate R. and Ff are the normal force and the friction force due to the contact of the particle with the interface. By balancing forces in a horizontal and vertical direction it was found37,40) that a particle will slide over the freezing front whenthe local
model, and the approximations madein the calculations. Details can be 'found in references.32,38.39) It is difficult to see howthe simple models can be used for other than pure materials. The problem of solute transport in dilute alloys is considered in Ref. 32). A realistic model predicting particle pushing in an alloy should treat the solute diffusion as well as the physical fluid fiow. To push the particle liquid must be drawn into the gap between the solid and the particle. At the
sametime
in Fig. 5.
velocity occurs at the momentwhen the particle starts moving,37) then a small sideways motion of the liquid will be enough to keep the particle in motion. Experiments40) have been carried out by the present authors to measure the critical flow velocities for Polymethylmethecrylate and alumina particles over an ice
respectively.
Recently,
Mechanism
illustrated
ahead of the freezing front is large enoughthe particle can be rolled or slide over the growing local solid and thus not be engulfed. There is a critical liquid flow velocity above which a particle will be dislodged from a depression (see Fig. 5). The critical
5)
of the particle
Sliding/Rolling
flow in the liquid
(1
2_ kL
front.
The mechanismis
where (91 maybe obtained from Ref. 26). Stefanescu et al.30) took the different thermal conductivity In the particle and the solid into account obtaining
R.=
Particle
696
ISIJ International,
Vol.
35 (1995). No. Table
Analytical
101
•
R=68um/s
A
R=4.2um/s
~
~
100
~ L
Flow direction
(a)
Particle
Analytical
102
size
o
R=68 um/s
a
R=21um/s
A
R=4um/s
(l/m)
o
--r I
l:~
~tl:
L
L ~ AD,ri* ,_.~~A~t
~l
A~AI
H
,/:'
/-
A
x 10~3
11.33 2. 19
Alumina
r)
Remarks
Ball
Hole
O.41 0.41
O.02 0.05
Rejected Rejected
0.41
0.02
O.4 1
O.
Trapped Trapped
O5
A
muchmoredifficult
r 102
101
Particle
Fig.
1.35
-
series of experiments to analyse. front.42) artificial carried solidification using out were an The artificial front consisted of balls stuck together on a perspex base or an array of holes. Theartificial interface Other was arranged either horizontally or vertically. experiments were carried using ice salt solution. It was found that provided the particles were small compared with the dendrite spacing the results could be summarised by a function of the form Ref. 42)
i
100
(b)
V,,/Vp
-
PMMA
l OJ
Experimental
~3Q)
~
and dimension
ratio
Alumina particles
~~
~:
velocity
PMMAI.33 x lO~ 3
is
10]
Particles
Downwardflow Alumina
A
Horizontal flow
~O1
calculated
"/(D
ii
{
~~a)
c~
The
rati0.42)
Experimental
~~
~~
1.
6
size
l
(,lm)
OJ
particles PMMA
6.
The relationship the particle
size
fluid flow velocrty,
Eq
R
satisfies
the expression
VL>a( 2A f+tane (Pp-p)g+-A R 9,l h a l-ftane
-
.....(16)
..........(17)
Vp'
F
R
can be assumed to be one. The results for artificial interfaces were reported in detail in Ref. 42). The result for alumina and polymethylmethacrylate (PMMA)are listed in Table l. It was found that in a downwardfiow, the dimension ratio for both interfaces was larger than the velocity ratio, so particles were rejected by rough interfaces. In horizontal flow, the speed ratio was larger than the dimension ratio and the particles were trapped by the rough interfaces.
f
Here is the coefficient of friction; the angle ewasdefined as a roughness parameter and A is a constant.37,40) This is compared with the criterion measurementson the
fiow rate shownin Fig. 6. For a horizontal planar interface, the critical flow speed required to put a 100 /Im alumina particle into motion is about I mm/s. The mechanismmayalso be important for a particle located in the interdendritic region. It is critical
possible that the small amount of fluid flow resulting from the volume change on freezing may continually dislodge particles ensuring that they end up in the last liquid to freeze. This mechanism mayaccount for the observation that some particles are rejected by the dendrites in metal matrix composites.41) The mechanism may also explain particle pushing in a microgravity environment. In this case the gravity term disappears in Eq. (18) so the critical flow speed becomesvery small.
Any fluid
flow due to Marangoni flow on a free surface contraction or on freezing maybe sufficient to dislodge
particles.
Rejection by a RoughInterface The pushing mechanismsconsidered so far
V,,)
D
(16).40)
VL, at the centre of the particle
F(R
is the For a dendritic front r is the dendrite tip radius, dendrite spacing (these can be obtained from Ref. 43)). Trapping occurs when the interface dimension ratio r) is greater than rl(D times the ratio of the normal velocity V~ to the velocity of the particle over the is surface. The term - V~ the velocity of the particle relative to the interface in the normal direction. The term F is a parameter which can eventually take into account the complex flow over the rough interface42) but here
between the critical local flow speed, and the growth rate of the solid. The
curves are calculated with
D-r
>
For downwardflow on a
vertical front the velocity force pushes lighter particles away from the freezing front whereas denser particles are attracted to the front and are thus more likely to be trapped. This accounts for that lighter inclusions concentrate at the centre of an ingots.44 ~ 50) For horizontal flow, the density
gradient
difference
between the particle
and the liquid is usually the Saffman force. In the case of a microgravity environment, the force due to the velocity gradient always directs away from the freezing front, so that particle pushing is favoured (this is because Vp Will be small and negative in Eq. (5)).
muchmore important than
Particle
really
7.
only
apply to a planar front or at least to a front where the particle
size is large
Application
In the analysis of a real casting the velocity near the interface could be calculated using techniques similar to
comparedto the dendrite/cell
spacing. front will usually be For a casting the solidification dendritic. The trapping of particles by dendritic front a
those proposed by Flood and Davidson.5 1) It would then be necessary to calculate P, V~ and CL at each point on
697
C 1995 ISIJ
ISIJ Internationa[.
Vol,
interface. This maybe possible during the solid-liquid directional growth but for a real casting the task is still formidable. In certain situations the problem may be simplified. When the particles are small the sedimentation will negligible comparedwith the convective flow be rate in the liquid particle concentration the and rate maybe considered uniform except near freezing surfaces. Under these conditions it is of benifit to define a coefficient ko and an effective distribution distribution coefficient kE of particles and to proceed in muchthe as for solute. Rearranging Eq. (2) gives
35 (1995), No.
pointed out that the commonlyused model for particle pushing based on surface energy should not be used in alloy systems.
REFERENCES 1)
2) 3)
where ko tribution
is
P(R- V~)
Cs
= CL
5)
dis7)
coefficient,
8)
9)
CL
Cs
..........(19)
11)
particle in the in the liquid
Assumingthe average concentration Co and substituting Eq. (1 1) into Eq. (19)
liquid.
for
l 2) l 3) l4)
CLgives
fAy
l 5) l 6) l 7) l 8) 19)
Vp'ody
-k o fAy kE~
(20) .
V;dy
20)
m
2a2(pp-p)g
:~:
9'l
OR
p
21) 22)
.(21)
23) 24) 25)
=O
For a vertical freezing front, assuming CL= CLat x (the top free surface) and that the fluid flow approximately follows Hagen-Poiseuille law adjacent to the freezing front gives37) kE ko
(1
=
as
a function of ko
(1
=
+ x,
0'61
J~'7,a3(pp
26) 27) 28)
-
p)gx
~H(Vco + 2Vm)
J~:
+ o3
//
HR
p
(22)
D. M. Stefanescu, B. K. Dhindaw, S. A. Kacar and A. Moitra:
31)
Ch. K6rber, G, Rau, M. D. Cosmanand E. G. Cravalho: J. Crystal Growth, 72 (1985), 649. J. P6tschke and V. Rogge: J. Crysta/ Growth, 94 (1989), 726. G. Lipp, Ch. Kdrber and G. Rau: J. Crystal Growth, 99 (1990),
Metall.
Trans.,
19A (1988),
Mater., 39 (1991),
2847.
206. 34) 35)
36) 37)
proposed to describe the A partitioning at a freezing front between the solid and liquid. The local composition in the solid depends on the normal component of the particle velocity, the particle composition in the liquid and a pushing parameter. Estimates of these terms may be obtained by considering the forces acting on the particles. is
38)
J. Appl. Phys., 47 (1 976), 3956. S. N. Omenyiand A. W,Neumann: A. W. Neumann.S. N. Omenyiand C. J. van Oss: J. Phys. Chem., 86 (1982), 1267. Q. Han. J. P. Lindsay and J. D. Hunt: Cast Met., 6(1994), 237. Q. Han: D. Phil. Thesis, Department of Materials. University of Oxford, (1994). D. M. Stefanescu. S. Ahuja. R. Phaiinikar and B. K. Dhindaw:
Metall. 39)
Trans.,
24A (1993),
403.
D. Shangguan, S. Ahuja and D. M. Stefanescu: Metall.
23A (1992),
force, the viscous In the present work the gravitational force and the Saffman force are considered. Twonew mechanismsfor particle pushing are presented and it is
C 1995 ISIJ
3O)
33)
Conclusions
equation of particles
H. L. Goldsmith and S. G. Mason: Nature, 190 (1961), 1095. C. D. Denson, E. B. Chritansen and D. L. Salt: A.1.Ch.E. J., 12 (1966), 589. Q. Hanand J. D. Hunt: Mat. Sci. Eng.. A173 (1993), 221. Q. Hanand J. D. Hunt: J. Crystal Growth, 140 (1994), 398. D. R. Uhlmann, B. Chalmers and K. A. Jackson: J. Appl. Phys., 35 (1964), 2986. C. F. Bolling and J. Ciss6: J. Crystal Growth, 10 (1971), 56. and B. C. Pai: Acta Metall., R. Sasikumar. T. R. Ramamohan 37 (1989), 2805. A. A. Chernov, D. E. Temkin and A. M. Melnikova: Soviet 21 (1976), 369. R. Sasikumar and M. Kumar: Acta Metall. 2503.
32)
for kE can then be used to calculate the local solid composition and thus the change in average liquid composition.
formal
A. D. Maude: Brit. J. Appl. Phys., 12 (1961), 293. H. Brenner: Chem.Eng. Sci., 16 (1961), 242. S. L. Rubinowand J. B. Keller: J. Fluid Mech., Il (1961), 447. T. R. Auton: J. Fluid Mech., 183 (1987), 199. T. R. Auton, J. C. R, Hunt and M. Prud'homme:J. Fluid Mech., 197 (1988), 241. P. G. Saffman: J. Fluid Mech., 22 (1965), 385. P. G. Saffman: J. Fluid Mech., 31 (1968), 624. R. G. Cox and H. Brenner: Chem.Eng. Sci., 23 (1968), 147. G. Segr6 and A. Silberberg: J. Fluid Mech., 14 (1962), I 15. R. Eichhorn and S. Small: J. Fluid Mech., 20 (1964), 513. A. Karnis, H. L, Goldsmith and S. G. Mason: Can. J. Chem.
29)
(23) .
The expressions
8.
First
Phys.-Cryst., .
where
a4(pp-p)g
and I. Shiota, Proc.
Eng., (1966), 181.
For a horizontal front whenparticles are engulfed, kE (20).37) can be estimated using Eqs. (9), (12) and
kE( 1+
T. Hirai
Gradient Materials, Functionally Gradient Materials ForumJapan, (1990). D. Nath and P. K. Rohatgi: Composites, 4 (1981), 124. A. R. Kennedyand T. W. Clyne. Particle Pushing during the Solidification of Metal Matrix Composites, NewFrontiers in Cast Met,, Birmingham, England, (1991). G. G. Stokes: Trans. Camb.Phill. Soc., 9 (1851), 8, H. Brenner: Advances in Chem.Eng., 6(1966), 287. A. J. Goldman, R. G. Cox and H. Brenner: Chem.Eng. Sci., 22
Symp, on Functionally
(1967), 637.
lO)
where CL is the average concentration of is
6)
R
kE~ CL =ko CL
M. Yamanouchi,M. Koizumi, Int.
(18)
a function of P and V~. The effective kE, can be defined as
H. S. Co and J. A. Charles: GAIUS,203 (1965), 493. A. G. Franklin and D. H. Evans: ISIJ, 209 (1971), 369. B. P. Krishnan, H. R. Shetty and P. K. Rohatgi: Trans. AFS, 76 P.
(1976), 73. 4)
sameway
ko
6
Trans.,
669.
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R. Kiessling andN. Lange: Non-Metailic Inclusions in 3, Met. Soc,, London, (1978), 6.
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