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Particle pushing: critical flow rate required to put particles into motion
,. . . . . . . .
ELSEVIER
CRYSTAL GROWTH
Journal of Crystal Growth 152 (1995)221-227
Particle pushing: critical flow rate required to put particles into motion Q. Han
*, J . D . H u n t
Department of Materials, University of Oxford, Parks Roads, Oxford OX1 3PH, UK
Received 20 August 1994;manuscript received in final form 3 February 1995
Abstract
Particles can be rejected by the growing solid as a result of fluid flow. It has been observed that without fluid flow, particles are engulfed by the growing solid at very low growth rates. When the flow rate is higher than a critical value, a particle can be put into motion and thus prevented from being engulfed. The critical local flow rate required to put a particle into motion has been measured and modelled. This critical flow rate depends on the size and density of the particle, the roughness of the interface and the growth rate of the solid. For small particles, a very small sideways motion of the liquid is enough to put a particle into motion.
1. Introduction
Surface energy models have been widely used for predicting when particle pushing occurs ahead of an advancing solid-liquid interface 11-3]. These models predict critical growth rates as much as a thousand times smaller than those actually found in metal systems [4-6]. In addition the models usually neglect solute flow which if included, for other than pure materials, would lead to very much smaller pushing rates [7]. It has been found that, in hypoeutectic A1 alloy matrix systems, SiC particles are pushed by A1 dendrites into the interdendritic regions for all growth conditions so far examined [6]; A120 3, glass, mica, TiB 2 and ZrB2 particles are also pushed into the interdendritic regions [6]. Pushing at these high velocities cannot really be explained using surface * Corresponding author.
energy models. Thus different mechanisms need to be considered particularly when fluid flow is present. Recently, we have reported that almost all insoluble particles in the melt can be rejected by a very rapidly growing solid and that the solid can be kept particle free if the melt is vigorously stirred [8]. The experimental results show that under these conditions, particle pushing is much more likely to be a result of fluid flow rather than of direct surface energy interactions. A possible mechanism for particle pushing is that particles roll or slide over a freezing front, driven by the surrounding liquid [8]. If the fluid flow is too small to move the particle, the particle will tend to be overgrown by the growing solid. If the flow rate is large enough to move the particle, the particles will not be engulfed [8]. There is a critical flow rate above which particles that were previously stationary on an interface start moving.
0022-0248/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)00085-2
222
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
This p a p e r describes experimental work on the critical flow rate required to put a particle into motion. Based on the experimental results, a crude analysis has been made in which considers the basic principles involved.
S e c t i o n on A -- A
j l:leservolj r
Laser
~[~
AI
0
A~ ===:~ I
2. Experimental methods Two types of experiments were carried out. In the first, simple experiments were designed to measure the critical flow rate when particles start to move on a smooth artificial interface made of perspex. The second type of experiment was designed to measure the critical flow rate when particles start to move on an i c e - w a t e r freezing front.
2.1. Artificial interface The apparatus for measuring the critical flow rate required to put particles into motion over a flat stationary interface is very similar to that described in our previous p a p e r [9]. The apparatus consisted of a reservoir containing water, a viewing chamber in the form of a square perspex tube of about 20 m m deep, 20 m m wide and 1000 mm long, and a pump. The bottom wall of this tube was used as a horizontal solid-liquid interface. Particles were placed on the bottom wall through a port in the tube and the whole apparatus was filled with water which was then circulated around the apparatus. The mean flow rate of water in the tube was increased gradually by adjusting a valve until individual particles on the bottom wall began to move. The mean flow rate in the tube was then kept constant and 20 ~zm tracer particles of the same density as water were introduced into the liquid in order to show the local flow rate one particle radius from the wall. Videos of the tracer particles in the flowing water were taken using a video camera attached to a stereo-microscope. The local flow rate was assumed to be equal to the velocity of the tracer particles, obtained from frames of the video tape. The particles used in the experiments were P M M A (polymethylmethacrylate) particles, alumina particles and steel balls. P M M A particles
Gamota
n He~t sink
[
0
Fig. 1. Schematic diagram of the apparatus for measuring the critical flow rate required to put particles into motion over a freezing front.
were perfect spheres, the steel balls had small dents on the surface and the alumina particles were round but the surface was slightly rough, with irregularities.
2.2. Freezing front The second type of experiment was carried out to measure the critical flow rate required to put a particle into motion on an i c e - w a t e r freezing front. The schematic diagram of the experimental apparatus for measuring the critical flow rate is shown in Fig. 1. The apparatus consisted of a reservoir containing water and the suspended particles, a viewing chamber which was cooled at the bottom by a heat sink containing liquid nitrogen, a laser for illumination and a video camera attached to a microscope. The flow of the suspension through the viewing chamber was driven by gravity and the flow rate was adjusted by using a valve. In order to prevent the growing solid from being remelted when the liquid flowed over it, the liquid had to be kept at the temperature of the freezing front; to do this the water was mixed with ice in the reservoir. The reservoir, the viewing chamber and the connecting tubes were enclosed with ice in an insulating box m a d e of perspex to prevent heat loss from the liquid. The size of the rectangular viewing chamber was about 20 m m wide and 90 m m long and its side walls
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227 were made of perspex except for the bottom wall which was the heat sink. The depth of the chamber was either 40 m m or 20 m m in order to have different freezing rates. The heat sink, made of pure aluminium, was cooled with a continuous supply of liquid nitrogen. As the suspension passed over the heat sink, liquid froze on top of it and the freezing front advanced upwards. To prevent a dent forming under a stationary particle due to the advance of the solid-liquid interface, the flow rate was first slowed down to let the suspended particles settle out on the freezing front; immediately the flow rate was gradually increased until some of the particles started to move. The flow rate was again reduced and as soon as the particles settled out the flow rate was increased until some particles started moving again. At this moment, the growth rate of the solid, the flow rate of the liquid, and the behaviour of the particles on the freezing front were recorded using the video camera.
3. Experimental results
In the experiment the critical flow rate was found to be much higher for particles that were agglomerated than for single particles. The recorded critical flow rate refers to the latter case. Experimental results on an artificial interface are shown in Fig. 2. The critical flow rate required to put a particle into motion increases with increasing particle size for all the three kinds of particles. For alumina and P M M A parti102 Analytical
Experimental :
101 • ca,
8 -1
PMMA Alumina Steel
I0 °
223
Table 1 The cellular intervals 2× r (/zm)
R (~zm/s)
49 42 37
4.9 12 68.8
cles, which are several hundreds of microns in diameter, the critical flow rate was only a few millimetres per second. The density of particles also had an effect on the critical flow rate. Steel balls with the highest density of the three types of particles, started moving at the largest flow rates. Alumina particles, denser than P M M A particles, started moving at higher flow rates than the PMMA. Considering the i c e - w a t e r system, when water froze on the surface of the cooling jacket, the freezing front was cellular because of impurities. Table 1 lists the measured cell spacings at different freezing rates. The interstices between the cells were much smaller than the size of the particles tested so particles could move over the freezing front without falling into the interstices. In the absence of fluid flow, P M M A or alumina particles deposited on the freezing front were trapped by the solid which was growing at a rate of about 4 /xm/s. This was the lowest growth velocity obtainable with the present apparatus. When fluid flow was introduced, particles were not trapped by the solid if they were kept in motion. Fig. 3 shows the relationship between the critical flow rate of the liquid, the size of particles, and the freezing rate, R, for both P M M A and alumina particles. From Fig. 3 it can been seen that the critical flow rate increases with increasing size and density of the particles, and it increases slightly with increasing freezing rate. The particle size had a major effect on the critical flow rate. It was easier to put smaller particles in motion than larger ones.
lO-~
4. Discussion 102
103
104
4.1. Artificial interface
Particle size ~ m )
Fig. 2. Relationship between the critical flow rate and the particle size.
Fig. 4 shows the forces on a particle that settles on a stationary horizontal interface in the
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
224
The fluid flow surrounding the particle will apply a viscous force, FD, which tends to roll the particle over the solid-liquid interface. The viscous force is considered to be of the usual form
Analytical Experimental * R = 68 ~tm/s
lO t
- -
E
•
R = 4 . 2 ,um/s
t
*
FD=
t
,r 1oo
E
,="
b
Analytical Experimental ---• R = 6 8 p.mls -- -= R=21 #m/s
10 I
]03
102 Particle size ~ m }
10'
• .- =i ~ =
"
10o
|01
10 2 Particle size 0*m)
l0 J
Fig. 3. Relationship between the critical flow rate, the particle size and the growth rate of the solid. (a) Alumina particles, (b) P M M A particles.
6rr/xVLaE,
where/.t is the viscosity of the liquid. The term E is equal to 1 when the particles is in an unbounded liquid [10]; when the particle moves towards an interface • = a / h [11] where h is the distance from the interface (for small but non-zero h); when a particle moves over the interface it is suggested that E = 5 / 1 2 In(a/h) [12]. Clearly neither of the last two expressions can be used when h = 0; that is the case when the particle rests on the interface the situation to be considered in the present work. For this case • will be assumed to be a function of a to be determined by the present experiments. The velocity gradient of the liquid will produce a lift force, FL, on the particle which is defined by Saffman [13,14] as: F L = 6.46/zaE(S/t,)l/2up,
presence of fluid flow. There are two schematic particles on the interface. Particle a is a truncated sphere which settles on a flat smooth interface and particle b is a complete sphere which settles on a spherical depression of the same radius as the particle. In Fig. 4, VL is the velocity of the liquid, N is the normal force, Fy is the friction force given by Ff = fN, f is the coefficient of friction, and F w is the gravity force defined as: Fw = 4~ ( p p
-p)a3g,
(1)
where a is the radius of the particle, pp and p are the densities of the particle and the liquid.
Particle
Up = VL.
(4)
Because the particle is relatively small compared with the width of the tube, the velocity gradient in the liquid at the centre of the particle can be given by:
VL
-'~
~o SOLID
(5)
Thus the lift force can be described by:
Particle
a
(3)
where ~, is the kinematic viscosity, Up is the velocity of the particle relative to the surrounding liquid and S is the velocity gradient in the liquid. Since the particle is stationary on the interface before it starts moving, its velocity relative to the surrounding liquid is given by:
s = VL/a.
LIQUID
(2)
b
F L = 6.461~aZ(VL/av)1/2V L. The force balance gives:
FD'~ /
( 1 - f tan O ) = F w, FL + FD r f
Fig. 4. Forces on a particle that settles on a stationary horizontal interface during fluid flow.
(6)
(7)
f + tan 0
where for particle a in Fig. 4, tan 0 is given by: tan 0 = r / ( a z - rE) t/E,
(8)
Q. Han, J.D. Hunt/Journal of Crystal Growth152 (1995)221-227 Table 2 Values of f and tan 0
and for particle b in Fig. 4, tan 0 given by:
tanO=[a2-(a-d)2]l/E/(a-d).
(9)
If the interface is not flat, but instead is cellular, tan 0 will be determined by the size of the particle and the cellular spacing. Suppose that the cells are closely packed, the cell tip is hemispherical and the tip radius is equal to half of the cell spacing, the minimum value of tan 0 is given
by: tan 0 = r / [ 3 ( r + a) 2 - 4r E] 1/2,
(10)
where r is the radius of the cell. When the left-hand side of Eq. (7) is larger than the right-hand side, the particle will be pushed out of the dent and start moving. Combining Eqs. (1), (2), (6) and (7) gives:
,(1-ftanO)
]£VLa-2
= ~(pp
f-{- tan 0
/d, (_.~) 3/2
+AI v-~
-p)g.
(11)
Since the lift force, Fc, is much smaller than the force, F w, due to gravity, F L can be safely neglected and Eq. (11) can be simplified as: nO ) Id,VL-~E ( 1 2- f t+atnaO
=-~(pp-p)g.
(12)
When the left-hand side of Eq. (12) is larger than the right-hand side, particles will start to move, so the critical flow rate can be defined as: 2 a2
Pc= 9---~T (Po-P)g
f+ tan0 l-f tan0"
(13)
Experimental results in Fig. 2 show that the critical flow velocity is proportional to the radius of a particle. This size-dependence of the critical flow rate indicates that E should be in the form of
E = a/A,
(14)
where A is a proportional constant. Substituting Eq. (14) into (13) gives ) 2 v~= _~-
9~ a z ( p p - p ' g
f + tan 0 1 --ftan0
225
(15) "
In order to estimate the value of A using Eq. (15), f was measured by measuring the tilt angle,
Particles
tan 0 (/~m/a)
f
Steel balls PMMA particles Alumina particles
30 0 < 0.18
0.037 1.176 0.696
0, of the bottom tube wall at which particles started to rolling down due to gravity and tan ~b was taken as f. Noting that for PMMA and alumina particles, van der Waals' forces will have an effect on a particle in touch with the wall, so the measured friction coefficients are large. The value of tan 0 was evaluated using Eq. (8) because the interface was a smooth perspex surface. The results are listed in Table 2. For steel balls with small dents about 60 tzm in size, tan 0 will be roughly equal to 30 I~m/a, where a is the radius of the particle. For P M M A particles with a smooth surface, tan O is zero. The surface of the alumina particles was rough. By measuring the irregularities of the alumina particles, tan 0 was estimated to be smaller than 0.18. Other values used in the calculation were /.~ = 10 -3 k g / m . s , v = 10 -6 m2/s. The density of steel, PMMA, alumina, and water was taken to be 8000, 1240, 3780 and 1000 k g / m 3, respectively. By fitting the critical flow velocity with the data given above, A was estimated to be 2.62 × 10 -5 m for spherical particles (PMMA particles and steel balls), and 3.60 × 10 -6 m for particles with irregularity (alumina particles). By using this crude model, the effects of the particle size, a and tan 0 on the critical flow rate, V~, can be discussed. The critical flow rate increased linearly with increasing particle size. For given particles and an interface, VL/a is a constant if tan 0 tends to zero. This indicates that for small particles, only a small amount of fluid flow is needed to put them into motion. The critical flow rate first increased slightly with increasing tan 0 then increased rapidly as tan 0 approached 1/f. This implies that a small amount of liquid motion is needed to move a particle out of a small dent but a much larger liquid flow is required for a particle in a deep
226
Q. Han,J.D. Hunt/Journalof CrystalGrowth152 (1995)221-227
dent. When tan 0 is larger than l / f , the first term on the left-hand side of Eq. (11) is negative, so Vc would have to be unrealistically large to put a particle in motion. Therefore, 1 / f should be considered to be the upper roughness limit. This is expressed as: tan 0 < 1/f.
(16)
For a given cellular spacing, the minimum size of a particle that can be put into motion by laminar fluid flow can be determined from Eqs. (10) and (16); that is a > [~/(f2 + 4 ) / 3 - 1]r.
(17)
Similar to Eq. (7) we have from the force balance that
(18)
FL--FR+FD
When f = 0, a can be written as: a > (2/7cj - - 1)r.
Fig. 5. Drag force, F R, on a particle when the liquid is flowing at a critical velocity,Vc.
The right-hand side of the above equation is equal to the radius of the interstice between closely packed cells. When the size of a particle is smaller than the interstice, the particle will drop in and be trapped by the rough interface.
4.2. Freezing front Consider a spherical particle which settles on a flat solid-liquid interface which is advancing upwards at a rate, R, if the particle is not moved by the flowing liquid, the particle will remain stationary and be engulfed by the growing solid. Above a critical fluid flow rate the particle will move across the interface as it grows. For the critical flow velocity the forces on the particle will be in balance. The forces will be those considered earlier for the stationary front but in addition a vertical viscous drag should be included because in this case the pivot point will be moving vertically at the rate of the solid-liquid front. The vertical viscous drag arises because liquid must be drawn in behind the sphere (see Fig. 5). This will be given by Eq. (2) where in this case e is given approximately by a/h [1]. For the present simple model h has to be obtained empirically. It is assumed that the horizontal flow velocity is very much greater than the horizontal particle velocity; this means that the horizontal viscous drag is given by the same expression as before.
( l - f t a n O ) =F w. f+tan0
(19)
Substituting Eqs. (1), (2), (6) and (14) into Eq. (19) gives /z ( V L 1 - f t a n 0
R)
a
h
A
f+tan0
= 2(p v - p ) g ,
/x
(__~)3/2
+AI (20)
where tan 0 is given by Eq. (9), and A and h have to be determined by experiment. If the solid-liquid interface is not fiat, but instead is cellular or dendritic, tan 0 will be determined by the roughness of the interface given by Eq. (10) and the critical flow rate can be estimated using Eqs. (10) and (20). Comparison of the experimental results with the calculated values using Eqs. (10) and (20) is shown in Fig. 3. The data for the calculation is given in Tables 1 and 2, and the value of f and A are assumed to be the same as those obtained in the experiment on a stationary interface (it is difficult to measure f while the freezing front is growing). Without a more detailed analysis the value of h had to be determined by experiment. Assuming h to be 2 × 1 0 "6 m fits the experimental results reasonably well. Even better curve fitting could have been achieved by adjusting both A and h. The relationship between the critical flow rate can be discussed using the crude expression (Eq. (20)). For a planar solid-liquid interface, Eq. (20)
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
shows that the particle size has a major effect on the critical flow rate. For a particle of 100 ~zm in diameter, the critical flow rate is about 1 mm/s. When the interface is convex, the critical flow rate required to put a particle on the interface into motion will be very small. This cotdd be important in particle pushing by secondar.z dendrite arms. In this case any motion in the liquid will be able to dislodge the particle from a dendrite arm and thus push it into the remaining liquid between the dendrites. The small amount of liquid motion from feeding the contraction on freezing will probably be sufficient to dislodge particles. This may account for the fact that particles tend to segregate to the interdendritic regions under most conditions [6]. For a cellular freezing front, tan 0 is a function of particle size and the tip radius of the cells (which in turn depends on the growth rate of the solid, the temperature gradient ahead of the growing solid and the composition of the melt [15,16]. The models for the growth of cell arrays can be used to discuss the dependence of the critical flow rate on the growth rate of the solid.
5. Conclusions During solidification, fluid flow can prevent particles from being engulfed by the growing solid. For a given freezing front and type of particles, there is a critical flow rate above which particles will be pushed by the freezing front and below which particles will be trapped. The critical flow rate increases with increasing particle size and density, increasing roughness of the freezing: front and growth rate. Crude models have been developed and compared with experiment. For a known freezing
227
front and type of particles the models can be used for predicting the critical flow rate required to put a particle into motion, and thus prevent the particle from being engulfed by the growing solid.
Acknowledgements The authors would like to acknowledge the support of Brite-Euram grant No. BREU 0262 M: BE 3628-89 for the programme of "The Convective Effects in Solidification".
References [1] D.R. Uhlmann, B. Chalmers and K.A. Jackson, J. Appl. Phys. 35 (1964) 2986. [2] C.F. Boiling and J. Ciss6, J. Crystal Growth 10 (1971) 56. [3] S.N. Omenyi, A.W. Neumann and C.J. van Oss, J. Appl. Phys. 52 (1981) 789. [4] D.M. Stefanescu, B.K. Dhindaw and S.A. Kacar, Met. Trans. 19A (1988) 2847. [5] D.J. Lloyd, Composites Sci. Technol. 35 (1989) 159. [6] A. Mortensen and I. Jin, Intern. Mater. Rev. 37 (1992) 101. [7] J. P6tschke and V. Rogge, J. Crystal Growth 94 (1989) 726. [8] Q. Han, J.P. Lindsay and J.D. Hunt, Cast Met. 7 (1994) 237. [9] Q. Han and J.D. Hunt, J. Crystal Growth 140 (1994) 406. [10] G.G. Stokes, Trans. Cambridge Phil. Soc. 9 (1851) 8. [11] H. Brenner, Advan. Chem. Eng. 6 (1962) 287. [12] A.J. Goldman, R.G. Cox and H. Brenner, Chem. Eng. Sci. 22 (1967) 637. [13] P.G. Saffman, J. Fluid Mech. 22 (1965) 385. [14] P.G. Saffman, J. Fluid Mech. 31 (1968) 624 (Corrigendum). [15] J.D. Hunt and D.G. McCartney, Acta Met. 35 (1987) 89. [16] S. Lu and J.D. Hunt, J. Crystal Growth 123 (1992) 17.
ELSEVIER
CRYSTAL GROWTH
Journal of Crystal Growth 152 (1995)221-227
Particle pushing: critical flow rate required to put particles into motion Q. Han
*, J . D . H u n t
Department of Materials, University of Oxford, Parks Roads, Oxford OX1 3PH, UK
Received 20 August 1994;manuscript received in final form 3 February 1995
Abstract
Particles can be rejected by the growing solid as a result of fluid flow. It has been observed that without fluid flow, particles are engulfed by the growing solid at very low growth rates. When the flow rate is higher than a critical value, a particle can be put into motion and thus prevented from being engulfed. The critical local flow rate required to put a particle into motion has been measured and modelled. This critical flow rate depends on the size and density of the particle, the roughness of the interface and the growth rate of the solid. For small particles, a very small sideways motion of the liquid is enough to put a particle into motion.
1. Introduction
Surface energy models have been widely used for predicting when particle pushing occurs ahead of an advancing solid-liquid interface 11-3]. These models predict critical growth rates as much as a thousand times smaller than those actually found in metal systems [4-6]. In addition the models usually neglect solute flow which if included, for other than pure materials, would lead to very much smaller pushing rates [7]. It has been found that, in hypoeutectic A1 alloy matrix systems, SiC particles are pushed by A1 dendrites into the interdendritic regions for all growth conditions so far examined [6]; A120 3, glass, mica, TiB 2 and ZrB2 particles are also pushed into the interdendritic regions [6]. Pushing at these high velocities cannot really be explained using surface * Corresponding author.
energy models. Thus different mechanisms need to be considered particularly when fluid flow is present. Recently, we have reported that almost all insoluble particles in the melt can be rejected by a very rapidly growing solid and that the solid can be kept particle free if the melt is vigorously stirred [8]. The experimental results show that under these conditions, particle pushing is much more likely to be a result of fluid flow rather than of direct surface energy interactions. A possible mechanism for particle pushing is that particles roll or slide over a freezing front, driven by the surrounding liquid [8]. If the fluid flow is too small to move the particle, the particle will tend to be overgrown by the growing solid. If the flow rate is large enough to move the particle, the particles will not be engulfed [8]. There is a critical flow rate above which particles that were previously stationary on an interface start moving.
0022-0248/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0022-0248(95)00085-2
222
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
This p a p e r describes experimental work on the critical flow rate required to put a particle into motion. Based on the experimental results, a crude analysis has been made in which considers the basic principles involved.
S e c t i o n on A -- A
j l:leservolj r
Laser
~[~
AI
0
A~ ===:~ I
2. Experimental methods Two types of experiments were carried out. In the first, simple experiments were designed to measure the critical flow rate when particles start to move on a smooth artificial interface made of perspex. The second type of experiment was designed to measure the critical flow rate when particles start to move on an i c e - w a t e r freezing front.
2.1. Artificial interface The apparatus for measuring the critical flow rate required to put particles into motion over a flat stationary interface is very similar to that described in our previous p a p e r [9]. The apparatus consisted of a reservoir containing water, a viewing chamber in the form of a square perspex tube of about 20 m m deep, 20 m m wide and 1000 mm long, and a pump. The bottom wall of this tube was used as a horizontal solid-liquid interface. Particles were placed on the bottom wall through a port in the tube and the whole apparatus was filled with water which was then circulated around the apparatus. The mean flow rate of water in the tube was increased gradually by adjusting a valve until individual particles on the bottom wall began to move. The mean flow rate in the tube was then kept constant and 20 ~zm tracer particles of the same density as water were introduced into the liquid in order to show the local flow rate one particle radius from the wall. Videos of the tracer particles in the flowing water were taken using a video camera attached to a stereo-microscope. The local flow rate was assumed to be equal to the velocity of the tracer particles, obtained from frames of the video tape. The particles used in the experiments were P M M A (polymethylmethacrylate) particles, alumina particles and steel balls. P M M A particles
Gamota
n He~t sink
[
0
Fig. 1. Schematic diagram of the apparatus for measuring the critical flow rate required to put particles into motion over a freezing front.
were perfect spheres, the steel balls had small dents on the surface and the alumina particles were round but the surface was slightly rough, with irregularities.
2.2. Freezing front The second type of experiment was carried out to measure the critical flow rate required to put a particle into motion on an i c e - w a t e r freezing front. The schematic diagram of the experimental apparatus for measuring the critical flow rate is shown in Fig. 1. The apparatus consisted of a reservoir containing water and the suspended particles, a viewing chamber which was cooled at the bottom by a heat sink containing liquid nitrogen, a laser for illumination and a video camera attached to a microscope. The flow of the suspension through the viewing chamber was driven by gravity and the flow rate was adjusted by using a valve. In order to prevent the growing solid from being remelted when the liquid flowed over it, the liquid had to be kept at the temperature of the freezing front; to do this the water was mixed with ice in the reservoir. The reservoir, the viewing chamber and the connecting tubes were enclosed with ice in an insulating box m a d e of perspex to prevent heat loss from the liquid. The size of the rectangular viewing chamber was about 20 m m wide and 90 m m long and its side walls
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227 were made of perspex except for the bottom wall which was the heat sink. The depth of the chamber was either 40 m m or 20 m m in order to have different freezing rates. The heat sink, made of pure aluminium, was cooled with a continuous supply of liquid nitrogen. As the suspension passed over the heat sink, liquid froze on top of it and the freezing front advanced upwards. To prevent a dent forming under a stationary particle due to the advance of the solid-liquid interface, the flow rate was first slowed down to let the suspended particles settle out on the freezing front; immediately the flow rate was gradually increased until some of the particles started to move. The flow rate was again reduced and as soon as the particles settled out the flow rate was increased until some particles started moving again. At this moment, the growth rate of the solid, the flow rate of the liquid, and the behaviour of the particles on the freezing front were recorded using the video camera.
3. Experimental results
In the experiment the critical flow rate was found to be much higher for particles that were agglomerated than for single particles. The recorded critical flow rate refers to the latter case. Experimental results on an artificial interface are shown in Fig. 2. The critical flow rate required to put a particle into motion increases with increasing particle size for all the three kinds of particles. For alumina and P M M A parti102 Analytical
Experimental :
101 • ca,
8 -1
PMMA Alumina Steel
I0 °
223
Table 1 The cellular intervals 2× r (/zm)
R (~zm/s)
49 42 37
4.9 12 68.8
cles, which are several hundreds of microns in diameter, the critical flow rate was only a few millimetres per second. The density of particles also had an effect on the critical flow rate. Steel balls with the highest density of the three types of particles, started moving at the largest flow rates. Alumina particles, denser than P M M A particles, started moving at higher flow rates than the PMMA. Considering the i c e - w a t e r system, when water froze on the surface of the cooling jacket, the freezing front was cellular because of impurities. Table 1 lists the measured cell spacings at different freezing rates. The interstices between the cells were much smaller than the size of the particles tested so particles could move over the freezing front without falling into the interstices. In the absence of fluid flow, P M M A or alumina particles deposited on the freezing front were trapped by the solid which was growing at a rate of about 4 /xm/s. This was the lowest growth velocity obtainable with the present apparatus. When fluid flow was introduced, particles were not trapped by the solid if they were kept in motion. Fig. 3 shows the relationship between the critical flow rate of the liquid, the size of particles, and the freezing rate, R, for both P M M A and alumina particles. From Fig. 3 it can been seen that the critical flow rate increases with increasing size and density of the particles, and it increases slightly with increasing freezing rate. The particle size had a major effect on the critical flow rate. It was easier to put smaller particles in motion than larger ones.
lO-~
4. Discussion 102
103
104
4.1. Artificial interface
Particle size ~ m )
Fig. 2. Relationship between the critical flow rate and the particle size.
Fig. 4 shows the forces on a particle that settles on a stationary horizontal interface in the
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
224
The fluid flow surrounding the particle will apply a viscous force, FD, which tends to roll the particle over the solid-liquid interface. The viscous force is considered to be of the usual form
Analytical Experimental * R = 68 ~tm/s
lO t
- -
E
•
R = 4 . 2 ,um/s
t
*
FD=
t
,r 1oo
E
,="
b
Analytical Experimental ---• R = 6 8 p.mls -- -= R=21 #m/s
10 I
]03
102 Particle size ~ m }
10'
• .- =i ~ =
"
10o
|01
10 2 Particle size 0*m)
l0 J
Fig. 3. Relationship between the critical flow rate, the particle size and the growth rate of the solid. (a) Alumina particles, (b) P M M A particles.
6rr/xVLaE,
where/.t is the viscosity of the liquid. The term E is equal to 1 when the particles is in an unbounded liquid [10]; when the particle moves towards an interface • = a / h [11] where h is the distance from the interface (for small but non-zero h); when a particle moves over the interface it is suggested that E = 5 / 1 2 In(a/h) [12]. Clearly neither of the last two expressions can be used when h = 0; that is the case when the particle rests on the interface the situation to be considered in the present work. For this case • will be assumed to be a function of a to be determined by the present experiments. The velocity gradient of the liquid will produce a lift force, FL, on the particle which is defined by Saffman [13,14] as: F L = 6.46/zaE(S/t,)l/2up,
presence of fluid flow. There are two schematic particles on the interface. Particle a is a truncated sphere which settles on a flat smooth interface and particle b is a complete sphere which settles on a spherical depression of the same radius as the particle. In Fig. 4, VL is the velocity of the liquid, N is the normal force, Fy is the friction force given by Ff = fN, f is the coefficient of friction, and F w is the gravity force defined as: Fw = 4~ ( p p
-p)a3g,
(1)
where a is the radius of the particle, pp and p are the densities of the particle and the liquid.
Particle
Up = VL.
(4)
Because the particle is relatively small compared with the width of the tube, the velocity gradient in the liquid at the centre of the particle can be given by:
VL
-'~
~o SOLID
(5)
Thus the lift force can be described by:
Particle
a
(3)
where ~, is the kinematic viscosity, Up is the velocity of the particle relative to the surrounding liquid and S is the velocity gradient in the liquid. Since the particle is stationary on the interface before it starts moving, its velocity relative to the surrounding liquid is given by:
s = VL/a.
LIQUID
(2)
b
F L = 6.461~aZ(VL/av)1/2V L. The force balance gives:
FD'~ /
( 1 - f tan O ) = F w, FL + FD r f
Fig. 4. Forces on a particle that settles on a stationary horizontal interface during fluid flow.
(6)
(7)
f + tan 0
where for particle a in Fig. 4, tan 0 is given by: tan 0 = r / ( a z - rE) t/E,
(8)
Q. Han, J.D. Hunt/Journal of Crystal Growth152 (1995)221-227 Table 2 Values of f and tan 0
and for particle b in Fig. 4, tan 0 given by:
tanO=[a2-(a-d)2]l/E/(a-d).
(9)
If the interface is not flat, but instead is cellular, tan 0 will be determined by the size of the particle and the cellular spacing. Suppose that the cells are closely packed, the cell tip is hemispherical and the tip radius is equal to half of the cell spacing, the minimum value of tan 0 is given
by: tan 0 = r / [ 3 ( r + a) 2 - 4r E] 1/2,
(10)
where r is the radius of the cell. When the left-hand side of Eq. (7) is larger than the right-hand side, the particle will be pushed out of the dent and start moving. Combining Eqs. (1), (2), (6) and (7) gives:
,(1-ftanO)
]£VLa-2
= ~(pp
f-{- tan 0
/d, (_.~) 3/2
+AI v-~
-p)g.
(11)
Since the lift force, Fc, is much smaller than the force, F w, due to gravity, F L can be safely neglected and Eq. (11) can be simplified as: nO ) Id,VL-~E ( 1 2- f t+atnaO
=-~(pp-p)g.
(12)
When the left-hand side of Eq. (12) is larger than the right-hand side, particles will start to move, so the critical flow rate can be defined as: 2 a2
Pc= 9---~T (Po-P)g
f+ tan0 l-f tan0"
(13)
Experimental results in Fig. 2 show that the critical flow velocity is proportional to the radius of a particle. This size-dependence of the critical flow rate indicates that E should be in the form of
E = a/A,
(14)
where A is a proportional constant. Substituting Eq. (14) into (13) gives ) 2 v~= _~-
9~ a z ( p p - p ' g
f + tan 0 1 --ftan0
225
(15) "
In order to estimate the value of A using Eq. (15), f was measured by measuring the tilt angle,
Particles
tan 0 (/~m/a)
f
Steel balls PMMA particles Alumina particles
30 0 < 0.18
0.037 1.176 0.696
0, of the bottom tube wall at which particles started to rolling down due to gravity and tan ~b was taken as f. Noting that for PMMA and alumina particles, van der Waals' forces will have an effect on a particle in touch with the wall, so the measured friction coefficients are large. The value of tan 0 was evaluated using Eq. (8) because the interface was a smooth perspex surface. The results are listed in Table 2. For steel balls with small dents about 60 tzm in size, tan 0 will be roughly equal to 30 I~m/a, where a is the radius of the particle. For P M M A particles with a smooth surface, tan O is zero. The surface of the alumina particles was rough. By measuring the irregularities of the alumina particles, tan 0 was estimated to be smaller than 0.18. Other values used in the calculation were /.~ = 10 -3 k g / m . s , v = 10 -6 m2/s. The density of steel, PMMA, alumina, and water was taken to be 8000, 1240, 3780 and 1000 k g / m 3, respectively. By fitting the critical flow velocity with the data given above, A was estimated to be 2.62 × 10 -5 m for spherical particles (PMMA particles and steel balls), and 3.60 × 10 -6 m for particles with irregularity (alumina particles). By using this crude model, the effects of the particle size, a and tan 0 on the critical flow rate, V~, can be discussed. The critical flow rate increased linearly with increasing particle size. For given particles and an interface, VL/a is a constant if tan 0 tends to zero. This indicates that for small particles, only a small amount of fluid flow is needed to put them into motion. The critical flow rate first increased slightly with increasing tan 0 then increased rapidly as tan 0 approached 1/f. This implies that a small amount of liquid motion is needed to move a particle out of a small dent but a much larger liquid flow is required for a particle in a deep
226
Q. Han,J.D. Hunt/Journalof CrystalGrowth152 (1995)221-227
dent. When tan 0 is larger than l / f , the first term on the left-hand side of Eq. (11) is negative, so Vc would have to be unrealistically large to put a particle in motion. Therefore, 1 / f should be considered to be the upper roughness limit. This is expressed as: tan 0 < 1/f.
(16)
For a given cellular spacing, the minimum size of a particle that can be put into motion by laminar fluid flow can be determined from Eqs. (10) and (16); that is a > [~/(f2 + 4 ) / 3 - 1]r.
(17)
Similar to Eq. (7) we have from the force balance that
(18)
FL--FR+FD
When f = 0, a can be written as: a > (2/7cj - - 1)r.
Fig. 5. Drag force, F R, on a particle when the liquid is flowing at a critical velocity,Vc.
The right-hand side of the above equation is equal to the radius of the interstice between closely packed cells. When the size of a particle is smaller than the interstice, the particle will drop in and be trapped by the rough interface.
4.2. Freezing front Consider a spherical particle which settles on a flat solid-liquid interface which is advancing upwards at a rate, R, if the particle is not moved by the flowing liquid, the particle will remain stationary and be engulfed by the growing solid. Above a critical fluid flow rate the particle will move across the interface as it grows. For the critical flow velocity the forces on the particle will be in balance. The forces will be those considered earlier for the stationary front but in addition a vertical viscous drag should be included because in this case the pivot point will be moving vertically at the rate of the solid-liquid front. The vertical viscous drag arises because liquid must be drawn in behind the sphere (see Fig. 5). This will be given by Eq. (2) where in this case e is given approximately by a/h [1]. For the present simple model h has to be obtained empirically. It is assumed that the horizontal flow velocity is very much greater than the horizontal particle velocity; this means that the horizontal viscous drag is given by the same expression as before.
( l - f t a n O ) =F w. f+tan0
(19)
Substituting Eqs. (1), (2), (6) and (14) into Eq. (19) gives /z ( V L 1 - f t a n 0
R)
a
h
A
f+tan0
= 2(p v - p ) g ,
/x
(__~)3/2
+AI (20)
where tan 0 is given by Eq. (9), and A and h have to be determined by experiment. If the solid-liquid interface is not fiat, but instead is cellular or dendritic, tan 0 will be determined by the roughness of the interface given by Eq. (10) and the critical flow rate can be estimated using Eqs. (10) and (20). Comparison of the experimental results with the calculated values using Eqs. (10) and (20) is shown in Fig. 3. The data for the calculation is given in Tables 1 and 2, and the value of f and A are assumed to be the same as those obtained in the experiment on a stationary interface (it is difficult to measure f while the freezing front is growing). Without a more detailed analysis the value of h had to be determined by experiment. Assuming h to be 2 × 1 0 "6 m fits the experimental results reasonably well. Even better curve fitting could have been achieved by adjusting both A and h. The relationship between the critical flow rate can be discussed using the crude expression (Eq. (20)). For a planar solid-liquid interface, Eq. (20)
Q. Han, J.D. Hunt/Journal of Crystal Growth 152 (1995) 221-227
shows that the particle size has a major effect on the critical flow rate. For a particle of 100 ~zm in diameter, the critical flow rate is about 1 mm/s. When the interface is convex, the critical flow rate required to put a particle on the interface into motion will be very small. This cotdd be important in particle pushing by secondar.z dendrite arms. In this case any motion in the liquid will be able to dislodge the particle from a dendrite arm and thus push it into the remaining liquid between the dendrites. The small amount of liquid motion from feeding the contraction on freezing will probably be sufficient to dislodge particles. This may account for the fact that particles tend to segregate to the interdendritic regions under most conditions [6]. For a cellular freezing front, tan 0 is a function of particle size and the tip radius of the cells (which in turn depends on the growth rate of the solid, the temperature gradient ahead of the growing solid and the composition of the melt [15,16]. The models for the growth of cell arrays can be used to discuss the dependence of the critical flow rate on the growth rate of the solid.
5. Conclusions During solidification, fluid flow can prevent particles from being engulfed by the growing solid. For a given freezing front and type of particles, there is a critical flow rate above which particles will be pushed by the freezing front and below which particles will be trapped. The critical flow rate increases with increasing particle size and density, increasing roughness of the freezing: front and growth rate. Crude models have been developed and compared with experiment. For a known freezing
227
front and type of particles the models can be used for predicting the critical flow rate required to put a particle into motion, and thus prevent the particle from being engulfed by the growing solid.
Acknowledgements The authors would like to acknowledge the support of Brite-Euram grant No. BREU 0262 M: BE 3628-89 for the programme of "The Convective Effects in Solidification".
References [1] D.R. Uhlmann, B. Chalmers and K.A. Jackson, J. Appl. Phys. 35 (1964) 2986. [2] C.F. Boiling and J. Ciss6, J. Crystal Growth 10 (1971) 56. [3] S.N. Omenyi, A.W. Neumann and C.J. van Oss, J. Appl. Phys. 52 (1981) 789. [4] D.M. Stefanescu, B.K. Dhindaw and S.A. Kacar, Met. Trans. 19A (1988) 2847. [5] D.J. Lloyd, Composites Sci. Technol. 35 (1989) 159. [6] A. Mortensen and I. Jin, Intern. Mater. Rev. 37 (1992) 101. [7] J. P6tschke and V. Rogge, J. Crystal Growth 94 (1989) 726. [8] Q. Han, J.P. Lindsay and J.D. Hunt, Cast Met. 7 (1994) 237. [9] Q. Han and J.D. Hunt, J. Crystal Growth 140 (1994) 406. [10] G.G. Stokes, Trans. Cambridge Phil. Soc. 9 (1851) 8. [11] H. Brenner, Advan. Chem. Eng. 6 (1962) 287. [12] A.J. Goldman, R.G. Cox and H. Brenner, Chem. Eng. Sci. 22 (1967) 637. [13] P.G. Saffman, J. Fluid Mech. 22 (1965) 385. [14] P.G. Saffman, J. Fluid Mech. 31 (1968) 624 (Corrigendum). [15] J.D. Hunt and D.G. McCartney, Acta Met. 35 (1987) 89. [16] S. Lu and J.D. Hunt, J. Crystal Growth 123 (1992) 17.