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Redistribution of particles during casting of composite melts: Effects of buoyancy and particle pushing
Acta metall, mater. Vol. 39, No. 1I, pp. 2503-2508, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 :$3.00+ 0.00 Copyright ~ 1991 Pergamon Press pie
REDISTRIBUTION OF PARTICLES DURING CASTING OF COMPOSITE MELTS: EFFECTS OF BUOYANCY AND PARTICLE PUSHING R. SASIKUMAR and M. KUMAR Regional Research Laboratory (CSIR), Trivandrum 695 019, India (Received 24 September 1990; in revised form 8 May 1991)
Aid'act--When a melt containing a dispersion of second phase particles is solidified, the initial distribution of the particles can change due to three phenomena, namely, buoyant motion of the particles, pushing of the particles by the moving solidification front, and by convection currents in the melt. This paper presents a computer simulation model using which, the net redistribution due to the combined effect of the first two phenomena can be predicted. The existing theory for calculating the critical velocity for particle pushing is extended to include the effect of the buoyancy force and a numerical correlation is developed for easy calculation of the critical velocity. This correlation is incorporated into a computer programme which tracks the position, velocity and direction of the solidification front as well as the position of each particle in the melt as a function of time. The final positions of the particles describe the distribution of the particles in the solidified material. Predicted distributions for various heat extraction rates and particle sizes are presented for a system of silicon carbide particles in a pure aluminium melt solidifying unidirectionally as well as multidirectionally in cylindrical moulds. It is shown that for any heat extraction rate there is an optimum particle size which gives the maximum uniformity of distribution in the solidified material. R6sun~---Lorsqu'un mat6riau fondu contenant des particules dispers6es d'une seconde phase est solidifi6, la distribution initiale de particules peut changer ~i cause de trois ph6nom6nes qui sont: le mouvement flottant des particles, la pouss~e des particules sous raction du front de solidification et les courants de convexion darts la zone fondue. Cet article pr~sente un mod6le de simulation num6rique gr/ice auquel la redistribution effective due aux effets combings des deux premiers ph6nom~es peut &re pr~vue. La th6orie existante pour calculer la vitesse critique de pouss6e des particules est 6tendue pour tenir compte de reffet de la force de flottage et on d6veloppe une corr61ation num6rique permettant un calcul facile de la vitesse critique. Cette corr61ation est incorpor6e dans un programme de calcul qul donne la position, la vitesse, et la direction du front de solidification ainsi que la position de chaque partieule dans le bain en fonction du temps. Les positions finales des particules d6crivent la r6partition des particules dans le mat6riau solidifi6. Les r6partitions prevues pour diverses vitesses d'extraction de chaleur et pour diverses tailles de particules sont pr6sent6es pour un syst6me de particules de carbure de silicium darts un aluminium pur fondu d solidification unidirectionnelle ou multidirectionnelle dans des moules cylindriques. On montre que, pour toute vitesse d'extraction de chaleur, il existe une taille optimale de particules qui donne la r6partition la plus uniforme dans le mat6riau solidifi6. Ztm~mmeaf~mmg--Erstarrt eine Schmelze, die eine Dispersion von Teilchen einer zweiten Phase enthfilt, dann /indert sich die Anfangsverteilung der Teilchen wegen dreier Einfifisse, nimlich wegcn der schwimmenden Bewegung der Teilchen, wegen des Verschiebens der Teilchen durch die sich bewegende Erstarrungsfront und wegen Konvektionsstrfmungen in der Schmelze. Diese Arbeit stellt ein Modell vor, welches mit einer Computersimulation die resultierende Umverteilung dureh den kombinierten EinfluB der beiden ersten Effekte voraussagt. Die bestehende Theorie der kritischen Gesehwindigkeit ffir das Versehieben der Teilchen wird um den EinfluB der Schwimmkraft erweitert; for die leichte Bcreehmmg der kritisehen Geschwindigkeit wird eine numerische Korrelation entwickelt. D i m Korrelation wird in ein Computerprogram aufgenommen, welches Ort, Geschwindigkeit mad Richttmg der Erstarruagsfront wie auch den Ort eines jeden Teilchens in der Sehmelze in AbMngigkeit yon der Zeit veffolgt. Die Endpositionen der Teilchen b~chreiben deren Verteilung ira erstarrten Material. Vorausge~tgte Verteiltmgen f~r ver~hiedene Wirmeabfularraten und Teilchengr6~n werden f~r ein System von SiliziumkarbidTeilchen in reinen Alumiaiumsehmelzen, die gerichtet mad ungerichtet in zylindrisehen Formen emarren, vorgelegt. Es wird gezeigt, ~ es fur jede W/irmeabfuhrrate eine optimale Teilchengr6fle 8ibt, bei der sich die maximale Gleichm~Bigkeit der Verteilung im erstarrten Material einsteUt.
l. INTRODUCTION There are several processes of technological importance, which involve the solidification of liquids containing a dispersed second phase. These include
processing of metal-matrix composites, solidification of monotectic alloys, growth of crystals from melts containing impurities, castings of metals containing inclusions, etc. In all these processes, the final distribution of the second phase in the solidified material
2503
2504
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
is important; in some cases a uniform distribution of the second phase is essential and in others the particles should be removed from the material, for obtaining the requisite properties. Hence an understanding of the phenomena which contribute to the redistribution of the particles during solidification is very important in order to achieve the desired results. In an earlier paper [1], the redistribution of particles that occurs solely due to buoyant motion in a solidifying melt was studied using a computer simulation model. The phenomenon of particle pushing was studied more recently [2-4], ignoring the effects of buoyancy. This paper sets out to integrate the two studies into a model which can be used for studying the combined effect of the two phenomena on the final distribution of the particles in the solidified material. Theories for particle pushing [2-9] predict that there is a critical velocity of the solidification front, below which the front pushes the particles along with it. Above the critical velocity, the front grows around the particles and engulfs them. The critical velocity is a function of the particle size, the temperature gradient at the solidification front, and a number of material properties like the ratio of thermal conductivities of the particle and the melt, the coefficient of molecular surface forces between the particle and the solidification front, etc. In order to predict the final positions of the particles, it is necessary to know the velocity of buoyant motion, the velocity of the solidification front, and the critical velocity. Under non-steady state solidification conditions such as those which exist during the solidification of a casting, all the three velocities are functions of time because of the changing thermal profiles in the system. The thermal profiles as a function of time can be generated using a computer simulation of solidification. The velocity of buoyant motion is assumed to be equal to the terminal Stokes velocity of the particle, and the critical velocity is calculated using numerical correlations developed from the theory of particle pushing. The theory of particle pushing [3] is first modified to include the buoyancy force and the critical velocity is calculated over a wide range of values of the relevant parameters. The results are then fitted into a numerical correlation which facilitates easy calculation of the critical velocity for any experimental condition. This numerical correlation is incorporated into the computer simulation programme which yields the thermal conditions prevailing in the system, as well as the time and position at which each particle meets the solidification front, so that decisions can be taken as to whether the particle would be arrested at that position or pushed along with the solidification front. The model is demonstrated for a system consisting of silicon carbide particles dispersed in Aluminium melt, solidified vertically from the bottom as well as in cylindrical moulds. The predicted distributions for
various particle sizes and heat extraction rates, are presented. It is shown that, for any heat extraction rate, there is an optimum particle size which yields maximum volume fraction of the solidified material with uniform distribution of the particles. This result is offered as an explanation of some previous experimental observation [10] in the case of distribution of particles in cast metal matrix composites. 2. THE MODEL
2.1. Effect of buoyancy on the critical velocity of particle pushing The method for calculating the critical velocities has been described in previous papers [2, 3]. The first step is the calculation of the change in shape of the sofidification front as it approaches the particle. From this the dimensions of the gap between the particle and the front are calculated. The repulsive force exerted by the particle and the viscous drag on the particle depend on the gap and these are calculated and balanced to obtain the velocity of the particle Vp = F ~ D F A
(I)
where r dr h3(r )
~o "
FR=27rB
DFA = 12~tt/
'r dr
h3(p )
rp = radius of the particle
h(r) = the gap between the particle and the front B = t h e coefficient for molecular interaction between the front and the particle r / = viscosity of the melt. The maximum velocity under which steady state is possible is identified as the critical velocity. Numerically this is determined by calculating the Vp at different distances, H of the solidification front from the particle. As H decreases Vp increases and then goes through a maximum. It is concluded from this that if the solidification front were to have a velocity, V, equal to Vp(H), steady state pushing of the particle occurs when the front is at a distance H from the particle. If the V were to exceed the maximum possible value for Vp, then no steady state pushing is possible; the gap between the particle and the front continues to decrease and the particle is engulfed by the front. The maximum value of Vp is therefore identified as the critical velocity. The buoyancy force on the particle is given by ,
~
F i : ~nrp
Apg
(2)
where Ap : density difference between the melt and the particle, g = acceleration due to gravity.
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING The direction of the force is vertically upwards or downwards depending on whether the particle is lighter or heavier than the melt. Inclusion of the buoyancy force in the model does not bring about any change in the calculation of the shape of the solidification front or in the calculation of the repulsive molecular surface forces. However, the buoyancy comes into the calculation of the velocity of the particle as an additional force acting on it. Also, due to buoyancy, the particle has a velocity even when it is far away from the front unlike in the previous model where the particle is assumed to be at rest with respect to the melt till the repulsive force makes it move. This velocity is in the vertical direction and hence the particle approaches the front at an angle depending on the direction of motion of the front. Neglecting inertial effects [3], the velocity of the particle in the direction of motion of the front (i.e. normal to the front) is given by Vp. =
F R - Fg sin q~ DFA
(3)
where ~b = angle that the front makes with the vertical. Initially the particle is assumed to move with its terminal velocity given by
Vr _ 2nr~Apg
(4)
9~/ It is not imperative that the particle should under all circumstances approach close enough to the front for molecular surface forces to be effective. The particle interacts with the front only if the component of the terminal velocity in the direction of motion of the front is smaller than the velocity of the front. As the molecular surface forces start acting on the particle, the component of its velocity in the direction of motion of the front changes and is given by equation (3). The particle will continue to have a component of velocity parallel to the front due to the component of buoyancy in this direction. This velocity is given by v~ = vT cos ¢.
(5)
This implies that the particle is pushed by different regions of the front as it moves along the front. However, the same procedure that was applied in the case of no buoyancy can be applied here to determine the maximum possible value of Vpl and hence the critical velocity. The only change is the use of equation (3) instead of equation (1) for calculation of the particle velocity. Vpl is calculated at different distances of the front from the particle and the maximum value of Vp~ is identified as the critical velocity.
2.2. Numerical correlation depicting the correction for buoyancy In order to obtain a convenient expression for calculation of the critical velocity, the results
2505
from numerical evaluations for a wide range of the relevant parameters were correlated into the formula
vc = ~
B
[0..----2' ~ r0,,-00,,~3
04 - 0 . 1 IC2'004 Cy037 C4" - 0.725(C 8 sin ~b)°52] (6)
where B C2 = ~tr--~
C3 = ASGr 4p/B C, = [I - kp/km]/[2 + kp/km] Cg= 4~nr~gAp/B 4 ct = AS = G = km= kp =
solid liquid interfacial energy entropy change on fusion temperature gradient at the front thermal conductivity of melt thermal conductivity of particle.
This correlation was used in the computer simulation model for depicting the redistribution of particles in a solidifying melt. Strictly speaking, this correlation is not valid for application to a case of non-steady state solidification that occurs in a casting. However, it is assumed that the velocity of the solidification front changes sufficiently slowly to validate the assumption of quasi steady state pushing of particles by the front.
2.3. Computer simulation of redistribution of particles during solidification The first step in this is the simulation of the solidification process. This is performed by solving the heat balance equation by the Finite Volume Method. The heat balance across an element of volume V0 can be written as VoCop dT - ~ - VoL -d~f , = - fA k V T ' d A
(7)
where k = Cp = p = L = f = T= t=
thermal conductivity specific heat density latent heat fraction of solid temperature time
A= surface integral around the element. It is assumed that, f~ remains zero till the temperature reaches the freezing point. Once the freezing temperature Tm is reached, the solid starts forming and the temperature is assumed to remain constant (dT/dt = 0 ) until f, becomes equal to 1. After that the temperature decreases again and
2506
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
c2/'./c9tbecomes zero.
Equation (7) therefore splits
into
Liquid VF
l,'opCp ~-~ = -
VoL~=-
kVT.dA
T.~ I"=
kVr-dA
r=r~
(7a)
Horizontal
and 0
(7b)
In the case of multidirectional solidification as in the case of a cylindrical ingot, the angle that the solidification front makes with the vertical (~), at every location of the casting, also should be determined. For this it is assumed that in every volume element (I), the direction of the velocity of the solidification front is normal to the solidification front, and this in turn, is antiparallel to the direction of heat extraction from the element. In any element the direction of the net heat flux is calculated from the relative magnitudes of the heat fluxes in the radial and axial directions and is given by
o,.,=
L[f. .lj 0
I pl Vertical
Fig. 1. Inclination of the solidification front and the direction of the particle velocity under pushing conditions in an element.
It may be noted that the vertical velocity becomes equal to V and the horizontal velocity becomes zero in the case of being pushed by a horizontal front moving upwards.
tan_lf[;,4VT'dA~_n
3. RESULTS AND DISCUSSION
Initially, the particles are assumed to be uniformly distributed in the melt (i.e. every volume element has the same number of particles). The panicles then start moving with the terminal velocity and the position of each particle is followed in time. The dependence of the terminal velocity on the temperature of the melt is neglected. At the time a particle reaches the moving solidification front, the critical velocity for pushing the panicle under the thermal conditions existing at the solidification front at that time, is calculated using equation 6. It is checked whether the velocity of the solidification front at that time is greater than or less than the critical velocity. In the former case, the particle is assumed to come to rest (be engulfed) and in the latter case it is assumed to be pushed. It is assumed that, if the velocity of the front is below the critical velocity quasi steady state conditions with the panicle acquiring a velocity equal to that of the front in the direction of motion of the front, is instantaneously established. Since the particle continues to have a velocity Vrcos~ parallel to the direction of the front the vertical and horizontal velocities of the particle under conditions of pushing are given by (Fig. 1)
In order to demonstrate the model, numerical calculations based on the model were performed for a system consisting of silicon carbide particles disparsed in pure aluminium melt, solidified from the bottom upwards, and also in a cylindrical mould with heat extracted from the sides and the bottom. The simulation was performed for various particle sizes and heat extraction rates. The different heat extraction rates were simulated by assuming different values for the mould metal heat transfer coefficient, h. The fraction of the solidified material which has uniform distribution of the particles is calculated at the end of each simulation. The volume fraction of the material with uniform distribution is plotted against particle size for a range of heat extraction rates in Fig. 2. It is seen that, for very low particle sizes, the yield of the uniform material is zero; it suddenly increases to a maximum
,,.o
where
(9)
h:heat transfer coefficient
--~
80
.o'~
6O
~o
20
v,.=.,= x/v~ + v~ cosec cos(O,,+,~) Vho~= x/V 2 + V~ cos2 ~b sin (Op + ~b)
oos20
J I
o
0
2
4
6
8
10 12 14 16
18 20
Particle size (pro)
VF
Fig. 2. Variation of the volume fraction of the material containing uniform distribution of particles with the particle size.
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING Undirectional casting
C y l i n d r i c a l casting
rp -2p.m
rp -2p.m
D
rp =4p. nl
rp -4p.m
rp -401~m
rp,=4Op.m
Fig. 3. Distribution of particles of various sizes. (a) Unidirectional casting. (b) Cylindrical casting. at a certain particle size and then decreases with increasing particle size. The maximum shifts to larger values of the particle size as the heat extraction rate decreases. The type of distribution that occurs at different particle sizes is shown schematically in Fig. 3. At low particle sizes (case rp = 2/am), the critical velocity is high and the particles are pushed to the last solidifying liquid, As particle size increases (case rp = 4 pm), the critical velocity decreases and the solidification front engulfs the particles on encountering them. As the particle size further increases, (case rp = 40/~m) buoyant movement becomes the dominant phenomenon affecting the uniformity. The velocity of buoyant movement increases as the particle size increases, thereby decreasing the uniformity. Since the velocity of solidification decreases as distance from the chill increases, there can be, in some cases, engulfment of particles upto some distance, and then pushing which results in distributions of the type shown in Fig. 4. In these cases, the solidification velocity near the mould walls is higher than the critical velocity so that the particles near the walls are engulfed. Further away from the mould walls (chill end), the velocity of solidification becomes smaller than the critical velocity and this results in pushing of the particles into the last solidifying liquid. The above results show that for any heat extraction rate, there is an optimum size range for the particles for obtaining the maximum uniformity. Below this
2507
size, the particles are pushed by the solidification front; above it, the particles segregate due to buoyant movement. Some experimental observations on Aluminium graphite composites made more than a decade ago [10], show exactly similar trends in that, the volume fraction of the uniformly distributed region goes through a maximum [or equivalently the dimensions of the particle free region goes through a minimum (Fig. 5)], as particle size is changed. However, the particle sizes corresponding to the maximum are one order of magnitude higher than those predicted by our model. If the balance between particle pushing and buoyant movement is indeed the reason for the maximum, as is suggested in this paper, then it looks as though the critical velocities are actually much higher than the calculated ones. It should be borne in mind that the theoretical evaluation for critical velocity has been done assuming that the particle is a smooth sphere. In the case of rough particles, the molecular repulsive force is felt mostly by the rough projection that is nearest to the front and the radius of curvature that enters the analysis of critical velocity is the radius of (a) Pouring temperature 680°C Speed 680 rpm
-
-5o '=s
'~== o °
s
¢i a) ~
4
e-o. ~m
I
I
I
I
(b) 20 -- x
~
ts
m
•o
8
Pouring temperature 6800C Speed 680 rpm
e-.
iil e •
Q.
"|
. . . . . .
2O .|
39111-'~
I
I
I
I
80
100
140
180
Panicle size (~m)
::-" i •
•
Fig. 4. A case of initial engulfment and later pushing in the same casting. AMM
B
c~
............. ,
4
Fig. 5. Variation of the dimensions of the particle fre¢ region with particle size for Aluminium graphite composites [10]. (a) Along longitudinal direction. (b) Along radial direction.
2508
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
curvature of the projection which can be very different from the "particle radius". Thus it is possible for even larger particles to be pushed at normal solidification velocities, if they have very sharp projections on them. Also there is a lot of arbitrariness about the values of the parameters used in the calculations, especially, the coefficient of molecular interaction, B. Thus the results obtained in the above analysis may not have much quantitative significance; they should only be considered as showing qualitative trends. The above analysis deals with a pure metal advancing with a plane solidification front. In the case of alloy castings, the solidification front will be dendritic in nature and certain geometric criteria for entrapment of the particles in the dendritic network will also have to be considered in addition to the buoyant movement and particle pushing. These are being worked out at present and will be published separately [11].
4. CONCLUSIONS 1. The redistribution process of particles in solidifying melts under the combined action of buoyancy and particle pushing has been simulated. 2. The results show that the volume fraction of solidified material containing uniform distribution
of particles goes through a maximum as the particle size is changed. This is in qualitative confirmity with earlier experimental results. Acknowledgements--We express our sincere gratitude to Dr A. D. Damodaran, Director, Regional Research Laboratory, Trivandrum for his constant encouragement of the work. We thank Dr T. R. Ramamohan for helpful discussions, and also, Dr K. G. Satyanarayana and Dr B. C. Pai for their support of the work. REFERENCES
1. R. Sasikumar and B. C. Pal, Solidification Processing-1987 (edited by H. Jones), p. 451. The Institute of Metals, London (1989). 2. R. Sasikumar, T. R. Ramamohan and B. C. Pai, Principles of Solidification and Materials Processing, p. 859. Oxford & IBH (1989). 3. R. Sasikumar, T. R. Ramamohan and B. C. Pal, Acta metall. 37, 2805 (1989). 4. R. Sasikumar and T. R. Ramamohan, To be published. 5. D. R. Uhlmann, B. Chalmers and K. A. Jackson, J. appl. Phys. 35, 2986 (1964). 6. G. F. Boiling and J. Cisse, J. Cryst. Growth 10, 56 (1971). 7. R. R. Gilpin, J. Coll. Int. Sci. 74, 44 (1980). 8. A. A. Chemov, D. E. Temkin and A. M. Melnikova, Soviet Phys. Crystallogr. 21, 369 (1976). 9. D. M. Stefanescu, B. K. Dhindaw, S. A. Kacar and A. Moitra, Metall. Trans. 19.4,, 2847 (1988). 10. B. P. Krishnan, H. R. Shetty and P. K. Rohatgi, Trans. Am. Found. Soc. 76, 73 (1976). 11. L. Kurien and R. Sasikumar, To be published.
0956-7151/91 :$3.00+ 0.00 Copyright ~ 1991 Pergamon Press pie
REDISTRIBUTION OF PARTICLES DURING CASTING OF COMPOSITE MELTS: EFFECTS OF BUOYANCY AND PARTICLE PUSHING R. SASIKUMAR and M. KUMAR Regional Research Laboratory (CSIR), Trivandrum 695 019, India (Received 24 September 1990; in revised form 8 May 1991)
Aid'act--When a melt containing a dispersion of second phase particles is solidified, the initial distribution of the particles can change due to three phenomena, namely, buoyant motion of the particles, pushing of the particles by the moving solidification front, and by convection currents in the melt. This paper presents a computer simulation model using which, the net redistribution due to the combined effect of the first two phenomena can be predicted. The existing theory for calculating the critical velocity for particle pushing is extended to include the effect of the buoyancy force and a numerical correlation is developed for easy calculation of the critical velocity. This correlation is incorporated into a computer programme which tracks the position, velocity and direction of the solidification front as well as the position of each particle in the melt as a function of time. The final positions of the particles describe the distribution of the particles in the solidified material. Predicted distributions for various heat extraction rates and particle sizes are presented for a system of silicon carbide particles in a pure aluminium melt solidifying unidirectionally as well as multidirectionally in cylindrical moulds. It is shown that for any heat extraction rate there is an optimum particle size which gives the maximum uniformity of distribution in the solidified material. R6sun~---Lorsqu'un mat6riau fondu contenant des particules dispers6es d'une seconde phase est solidifi6, la distribution initiale de particules peut changer ~i cause de trois ph6nom6nes qui sont: le mouvement flottant des particles, la pouss~e des particules sous raction du front de solidification et les courants de convexion darts la zone fondue. Cet article pr~sente un mod6le de simulation num6rique gr/ice auquel la redistribution effective due aux effets combings des deux premiers ph6nom~es peut &re pr~vue. La th6orie existante pour calculer la vitesse critique de pouss6e des particules est 6tendue pour tenir compte de reffet de la force de flottage et on d6veloppe une corr61ation num6rique permettant un calcul facile de la vitesse critique. Cette corr61ation est incorpor6e dans un programme de calcul qul donne la position, la vitesse, et la direction du front de solidification ainsi que la position de chaque partieule dans le bain en fonction du temps. Les positions finales des particules d6crivent la r6partition des particules dans le mat6riau solidifi6. Les r6partitions prevues pour diverses vitesses d'extraction de chaleur et pour diverses tailles de particules sont pr6sent6es pour un syst6me de particules de carbure de silicium darts un aluminium pur fondu d solidification unidirectionnelle ou multidirectionnelle dans des moules cylindriques. On montre que, pour toute vitesse d'extraction de chaleur, il existe une taille optimale de particules qui donne la r6partition la plus uniforme dans le mat6riau solidifi6. Ztm~mmeaf~mmg--Erstarrt eine Schmelze, die eine Dispersion von Teilchen einer zweiten Phase enthfilt, dann /indert sich die Anfangsverteilung der Teilchen wegen dreier Einfifisse, nimlich wegcn der schwimmenden Bewegung der Teilchen, wegen des Verschiebens der Teilchen durch die sich bewegende Erstarrungsfront und wegen Konvektionsstrfmungen in der Schmelze. Diese Arbeit stellt ein Modell vor, welches mit einer Computersimulation die resultierende Umverteilung dureh den kombinierten EinfluB der beiden ersten Effekte voraussagt. Die bestehende Theorie der kritischen Gesehwindigkeit ffir das Versehieben der Teilchen wird um den EinfluB der Schwimmkraft erweitert; for die leichte Bcreehmmg der kritisehen Geschwindigkeit wird eine numerische Korrelation entwickelt. D i m Korrelation wird in ein Computerprogram aufgenommen, welches Ort, Geschwindigkeit mad Richttmg der Erstarruagsfront wie auch den Ort eines jeden Teilchens in der Sehmelze in AbMngigkeit yon der Zeit veffolgt. Die Endpositionen der Teilchen b~chreiben deren Verteilung ira erstarrten Material. Vorausge~tgte Verteiltmgen f~r ver~hiedene Wirmeabfularraten und Teilchengr6~n werden f~r ein System von SiliziumkarbidTeilchen in reinen Alumiaiumsehmelzen, die gerichtet mad ungerichtet in zylindrisehen Formen emarren, vorgelegt. Es wird gezeigt, ~ es fur jede W/irmeabfuhrrate eine optimale Teilchengr6fle 8ibt, bei der sich die maximale Gleichm~Bigkeit der Verteilung im erstarrten Material einsteUt.
l. INTRODUCTION There are several processes of technological importance, which involve the solidification of liquids containing a dispersed second phase. These include
processing of metal-matrix composites, solidification of monotectic alloys, growth of crystals from melts containing impurities, castings of metals containing inclusions, etc. In all these processes, the final distribution of the second phase in the solidified material
2503
2504
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
is important; in some cases a uniform distribution of the second phase is essential and in others the particles should be removed from the material, for obtaining the requisite properties. Hence an understanding of the phenomena which contribute to the redistribution of the particles during solidification is very important in order to achieve the desired results. In an earlier paper [1], the redistribution of particles that occurs solely due to buoyant motion in a solidifying melt was studied using a computer simulation model. The phenomenon of particle pushing was studied more recently [2-4], ignoring the effects of buoyancy. This paper sets out to integrate the two studies into a model which can be used for studying the combined effect of the two phenomena on the final distribution of the particles in the solidified material. Theories for particle pushing [2-9] predict that there is a critical velocity of the solidification front, below which the front pushes the particles along with it. Above the critical velocity, the front grows around the particles and engulfs them. The critical velocity is a function of the particle size, the temperature gradient at the solidification front, and a number of material properties like the ratio of thermal conductivities of the particle and the melt, the coefficient of molecular surface forces between the particle and the solidification front, etc. In order to predict the final positions of the particles, it is necessary to know the velocity of buoyant motion, the velocity of the solidification front, and the critical velocity. Under non-steady state solidification conditions such as those which exist during the solidification of a casting, all the three velocities are functions of time because of the changing thermal profiles in the system. The thermal profiles as a function of time can be generated using a computer simulation of solidification. The velocity of buoyant motion is assumed to be equal to the terminal Stokes velocity of the particle, and the critical velocity is calculated using numerical correlations developed from the theory of particle pushing. The theory of particle pushing [3] is first modified to include the buoyancy force and the critical velocity is calculated over a wide range of values of the relevant parameters. The results are then fitted into a numerical correlation which facilitates easy calculation of the critical velocity for any experimental condition. This numerical correlation is incorporated into the computer simulation programme which yields the thermal conditions prevailing in the system, as well as the time and position at which each particle meets the solidification front, so that decisions can be taken as to whether the particle would be arrested at that position or pushed along with the solidification front. The model is demonstrated for a system consisting of silicon carbide particles dispersed in Aluminium melt, solidified vertically from the bottom as well as in cylindrical moulds. The predicted distributions for
various particle sizes and heat extraction rates, are presented. It is shown that, for any heat extraction rate, there is an optimum particle size which yields maximum volume fraction of the solidified material with uniform distribution of the particles. This result is offered as an explanation of some previous experimental observation [10] in the case of distribution of particles in cast metal matrix composites. 2. THE MODEL
2.1. Effect of buoyancy on the critical velocity of particle pushing The method for calculating the critical velocities has been described in previous papers [2, 3]. The first step is the calculation of the change in shape of the sofidification front as it approaches the particle. From this the dimensions of the gap between the particle and the front are calculated. The repulsive force exerted by the particle and the viscous drag on the particle depend on the gap and these are calculated and balanced to obtain the velocity of the particle Vp = F ~ D F A
(I)
where r dr h3(r )
~o "
FR=27rB
DFA = 12~tt/
'r dr
h3(p )
rp = radius of the particle
h(r) = the gap between the particle and the front B = t h e coefficient for molecular interaction between the front and the particle r / = viscosity of the melt. The maximum velocity under which steady state is possible is identified as the critical velocity. Numerically this is determined by calculating the Vp at different distances, H of the solidification front from the particle. As H decreases Vp increases and then goes through a maximum. It is concluded from this that if the solidification front were to have a velocity, V, equal to Vp(H), steady state pushing of the particle occurs when the front is at a distance H from the particle. If the V were to exceed the maximum possible value for Vp, then no steady state pushing is possible; the gap between the particle and the front continues to decrease and the particle is engulfed by the front. The maximum value of Vp is therefore identified as the critical velocity. The buoyancy force on the particle is given by ,
~
F i : ~nrp
Apg
(2)
where Ap : density difference between the melt and the particle, g = acceleration due to gravity.
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING The direction of the force is vertically upwards or downwards depending on whether the particle is lighter or heavier than the melt. Inclusion of the buoyancy force in the model does not bring about any change in the calculation of the shape of the solidification front or in the calculation of the repulsive molecular surface forces. However, the buoyancy comes into the calculation of the velocity of the particle as an additional force acting on it. Also, due to buoyancy, the particle has a velocity even when it is far away from the front unlike in the previous model where the particle is assumed to be at rest with respect to the melt till the repulsive force makes it move. This velocity is in the vertical direction and hence the particle approaches the front at an angle depending on the direction of motion of the front. Neglecting inertial effects [3], the velocity of the particle in the direction of motion of the front (i.e. normal to the front) is given by Vp. =
F R - Fg sin q~ DFA
(3)
where ~b = angle that the front makes with the vertical. Initially the particle is assumed to move with its terminal velocity given by
Vr _ 2nr~Apg
(4)
9~/ It is not imperative that the particle should under all circumstances approach close enough to the front for molecular surface forces to be effective. The particle interacts with the front only if the component of the terminal velocity in the direction of motion of the front is smaller than the velocity of the front. As the molecular surface forces start acting on the particle, the component of its velocity in the direction of motion of the front changes and is given by equation (3). The particle will continue to have a component of velocity parallel to the front due to the component of buoyancy in this direction. This velocity is given by v~ = vT cos ¢.
(5)
This implies that the particle is pushed by different regions of the front as it moves along the front. However, the same procedure that was applied in the case of no buoyancy can be applied here to determine the maximum possible value of Vpl and hence the critical velocity. The only change is the use of equation (3) instead of equation (1) for calculation of the particle velocity. Vpl is calculated at different distances of the front from the particle and the maximum value of Vp~ is identified as the critical velocity.
2.2. Numerical correlation depicting the correction for buoyancy In order to obtain a convenient expression for calculation of the critical velocity, the results
2505
from numerical evaluations for a wide range of the relevant parameters were correlated into the formula
vc = ~
B
[0..----2' ~ r0,,-00,,~3
04 - 0 . 1 IC2'004 Cy037 C4" - 0.725(C 8 sin ~b)°52] (6)
where B C2 = ~tr--~
C3 = ASGr 4p/B C, = [I - kp/km]/[2 + kp/km] Cg= 4~nr~gAp/B 4 ct = AS = G = km= kp =
solid liquid interfacial energy entropy change on fusion temperature gradient at the front thermal conductivity of melt thermal conductivity of particle.
This correlation was used in the computer simulation model for depicting the redistribution of particles in a solidifying melt. Strictly speaking, this correlation is not valid for application to a case of non-steady state solidification that occurs in a casting. However, it is assumed that the velocity of the solidification front changes sufficiently slowly to validate the assumption of quasi steady state pushing of particles by the front.
2.3. Computer simulation of redistribution of particles during solidification The first step in this is the simulation of the solidification process. This is performed by solving the heat balance equation by the Finite Volume Method. The heat balance across an element of volume V0 can be written as VoCop dT - ~ - VoL -d~f , = - fA k V T ' d A
(7)
where k = Cp = p = L = f = T= t=
thermal conductivity specific heat density latent heat fraction of solid temperature time
A= surface integral around the element. It is assumed that, f~ remains zero till the temperature reaches the freezing point. Once the freezing temperature Tm is reached, the solid starts forming and the temperature is assumed to remain constant (dT/dt = 0 ) until f, becomes equal to 1. After that the temperature decreases again and
2506
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
c2/'./c9tbecomes zero.
Equation (7) therefore splits
into
Liquid VF
l,'opCp ~-~ = -
VoL~=-
kVT.dA
T.~ I"=
kVr-dA
r=r~
(7a)
Horizontal
and 0
(7b)
In the case of multidirectional solidification as in the case of a cylindrical ingot, the angle that the solidification front makes with the vertical (~), at every location of the casting, also should be determined. For this it is assumed that in every volume element (I), the direction of the velocity of the solidification front is normal to the solidification front, and this in turn, is antiparallel to the direction of heat extraction from the element. In any element the direction of the net heat flux is calculated from the relative magnitudes of the heat fluxes in the radial and axial directions and is given by
o,.,=
L[f. .lj 0
I pl Vertical
Fig. 1. Inclination of the solidification front and the direction of the particle velocity under pushing conditions in an element.
It may be noted that the vertical velocity becomes equal to V and the horizontal velocity becomes zero in the case of being pushed by a horizontal front moving upwards.
tan_lf[;,4VT'dA~_n
3. RESULTS AND DISCUSSION
Initially, the particles are assumed to be uniformly distributed in the melt (i.e. every volume element has the same number of particles). The panicles then start moving with the terminal velocity and the position of each particle is followed in time. The dependence of the terminal velocity on the temperature of the melt is neglected. At the time a particle reaches the moving solidification front, the critical velocity for pushing the panicle under the thermal conditions existing at the solidification front at that time, is calculated using equation 6. It is checked whether the velocity of the solidification front at that time is greater than or less than the critical velocity. In the former case, the particle is assumed to come to rest (be engulfed) and in the latter case it is assumed to be pushed. It is assumed that, if the velocity of the front is below the critical velocity quasi steady state conditions with the panicle acquiring a velocity equal to that of the front in the direction of motion of the front, is instantaneously established. Since the particle continues to have a velocity Vrcos~ parallel to the direction of the front the vertical and horizontal velocities of the particle under conditions of pushing are given by (Fig. 1)
In order to demonstrate the model, numerical calculations based on the model were performed for a system consisting of silicon carbide particles disparsed in pure aluminium melt, solidified from the bottom upwards, and also in a cylindrical mould with heat extracted from the sides and the bottom. The simulation was performed for various particle sizes and heat extraction rates. The different heat extraction rates were simulated by assuming different values for the mould metal heat transfer coefficient, h. The fraction of the solidified material which has uniform distribution of the particles is calculated at the end of each simulation. The volume fraction of the material with uniform distribution is plotted against particle size for a range of heat extraction rates in Fig. 2. It is seen that, for very low particle sizes, the yield of the uniform material is zero; it suddenly increases to a maximum
,,.o
where
(9)
h:heat transfer coefficient
--~
80
.o'~
6O
~o
20
v,.=.,= x/v~ + v~ cosec cos(O,,+,~) Vho~= x/V 2 + V~ cos2 ~b sin (Op + ~b)
oos20
J I
o
0
2
4
6
8
10 12 14 16
18 20
Particle size (pro)
VF
Fig. 2. Variation of the volume fraction of the material containing uniform distribution of particles with the particle size.
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING Undirectional casting
C y l i n d r i c a l casting
rp -2p.m
rp -2p.m
D
rp =4p. nl
rp -4p.m
rp -401~m
rp,=4Op.m
Fig. 3. Distribution of particles of various sizes. (a) Unidirectional casting. (b) Cylindrical casting. at a certain particle size and then decreases with increasing particle size. The maximum shifts to larger values of the particle size as the heat extraction rate decreases. The type of distribution that occurs at different particle sizes is shown schematically in Fig. 3. At low particle sizes (case rp = 2/am), the critical velocity is high and the particles are pushed to the last solidifying liquid, As particle size increases (case rp = 4 pm), the critical velocity decreases and the solidification front engulfs the particles on encountering them. As the particle size further increases, (case rp = 40/~m) buoyant movement becomes the dominant phenomenon affecting the uniformity. The velocity of buoyant movement increases as the particle size increases, thereby decreasing the uniformity. Since the velocity of solidification decreases as distance from the chill increases, there can be, in some cases, engulfment of particles upto some distance, and then pushing which results in distributions of the type shown in Fig. 4. In these cases, the solidification velocity near the mould walls is higher than the critical velocity so that the particles near the walls are engulfed. Further away from the mould walls (chill end), the velocity of solidification becomes smaller than the critical velocity and this results in pushing of the particles into the last solidifying liquid. The above results show that for any heat extraction rate, there is an optimum size range for the particles for obtaining the maximum uniformity. Below this
2507
size, the particles are pushed by the solidification front; above it, the particles segregate due to buoyant movement. Some experimental observations on Aluminium graphite composites made more than a decade ago [10], show exactly similar trends in that, the volume fraction of the uniformly distributed region goes through a maximum [or equivalently the dimensions of the particle free region goes through a minimum (Fig. 5)], as particle size is changed. However, the particle sizes corresponding to the maximum are one order of magnitude higher than those predicted by our model. If the balance between particle pushing and buoyant movement is indeed the reason for the maximum, as is suggested in this paper, then it looks as though the critical velocities are actually much higher than the calculated ones. It should be borne in mind that the theoretical evaluation for critical velocity has been done assuming that the particle is a smooth sphere. In the case of rough particles, the molecular repulsive force is felt mostly by the rough projection that is nearest to the front and the radius of curvature that enters the analysis of critical velocity is the radius of (a) Pouring temperature 680°C Speed 680 rpm
-
-5o '=s
'~== o °
s
¢i a) ~
4
e-o. ~m
I
I
I
I
(b) 20 -- x
~
ts
m
•o
8
Pouring temperature 6800C Speed 680 rpm
e-.
iil e •
Q.
"|
. . . . . .
2O .|
39111-'~
I
I
I
I
80
100
140
180
Panicle size (~m)
::-" i •
•
Fig. 4. A case of initial engulfment and later pushing in the same casting. AMM
B
c~
............. ,
4
Fig. 5. Variation of the dimensions of the particle fre¢ region with particle size for Aluminium graphite composites [10]. (a) Along longitudinal direction. (b) Along radial direction.
2508
SASIKUMAR and KUMAR: REDISTRIBUTION OF PARTICLES DURING CASTING
curvature of the projection which can be very different from the "particle radius". Thus it is possible for even larger particles to be pushed at normal solidification velocities, if they have very sharp projections on them. Also there is a lot of arbitrariness about the values of the parameters used in the calculations, especially, the coefficient of molecular interaction, B. Thus the results obtained in the above analysis may not have much quantitative significance; they should only be considered as showing qualitative trends. The above analysis deals with a pure metal advancing with a plane solidification front. In the case of alloy castings, the solidification front will be dendritic in nature and certain geometric criteria for entrapment of the particles in the dendritic network will also have to be considered in addition to the buoyant movement and particle pushing. These are being worked out at present and will be published separately [11].
4. CONCLUSIONS 1. The redistribution process of particles in solidifying melts under the combined action of buoyancy and particle pushing has been simulated. 2. The results show that the volume fraction of solidified material containing uniform distribution
of particles goes through a maximum as the particle size is changed. This is in qualitative confirmity with earlier experimental results. Acknowledgements--We express our sincere gratitude to Dr A. D. Damodaran, Director, Regional Research Laboratory, Trivandrum for his constant encouragement of the work. We thank Dr T. R. Ramamohan for helpful discussions, and also, Dr K. G. Satyanarayana and Dr B. C. Pai for their support of the work. REFERENCES
1. R. Sasikumar and B. C. Pal, Solidification Processing-1987 (edited by H. Jones), p. 451. The Institute of Metals, London (1989). 2. R. Sasikumar, T. R. Ramamohan and B. C. Pai, Principles of Solidification and Materials Processing, p. 859. Oxford & IBH (1989). 3. R. Sasikumar, T. R. Ramamohan and B. C. Pal, Acta metall. 37, 2805 (1989). 4. R. Sasikumar and T. R. Ramamohan, To be published. 5. D. R. Uhlmann, B. Chalmers and K. A. Jackson, J. appl. Phys. 35, 2986 (1964). 6. G. F. Boiling and J. Cisse, J. Cryst. Growth 10, 56 (1971). 7. R. R. Gilpin, J. Coll. Int. Sci. 74, 44 (1980). 8. A. A. Chemov, D. E. Temkin and A. M. Melnikova, Soviet Phys. Crystallogr. 21, 369 (1976). 9. D. M. Stefanescu, B. K. Dhindaw, S. A. Kacar and A. Moitra, Metall. Trans. 19.4,, 2847 (1988). 10. B. P. Krishnan, H. R. Shetty and P. K. Rohatgi, Trans. Am. Found. Soc. 76, 73 (1976). 11. L. Kurien and R. Sasikumar, To be published.