1.
Chapter 5
Centrifugal Casting of Metal Matrix Composites
Abstract There are three major types of centrifugal casting used for the inﬁltration
of molten metal into ﬁbrous preforms. In this chapter, the inﬁltration of molten
metal into ﬁbrous preforms using centrifugal force is discussed theoretically and the
predictions of the theory are compared with experimental results. We discuss
the rotational speed necessary for inﬁltration to start, the pressure distribution in
the preforms, the velocity of the inﬁltration front and other important parameters.
When the volume fraction of ﬁbers is not high, the pressure necessary for the
inﬁltration of molten metal is low. This process is suitable for fabricating products
which are symmetrical around a rotational axis, and uses simple and economical
casting equipment. In addition to discussion of ﬁbrous preforms, centrifugal casting
of molten metal including ceramic particles is also discussed, focusing on the
theory of the behavior of a ceramic particle in molten metal in the centrifugal
force ﬁeld.
5.1 Inﬁltration of Molten Metal Using Centrifugal Force
If molten metal is rotated, centrifugal force will act on the molten metal. The force
increases radially from zero at the rotation center. This force generates pressure in
the molten metal, and the pressure increases with distance along the radius. This
pressure can be used to inﬁltrate molten metal into a porous preform made of
ceramic ﬁbers or particles and thus produce composites [1–4].
In this process, when the volume fraction Vf of the preform is relatively low, the
pressure needed for inﬁltration is not high. The apparatus is simple and economical,
although the process is only suitable for the symmetrical products. However, when
Vf of the preform is high, a high rotation number is needed, and the apparatus
required to produce the rate will be large.
Y. Nishida, Introduction to Metal Matrix Composites: Fabrication and Recycling,
DOI 10.1007/978-4-431-54237-7_5, # Springer Japan 2013
91
2.
5.1.1 Pressure Generated at the Surface of Preform
There are three cases for the production of composites using centrifugal force as
shown in Fig. 5.1. In Case 1, the preform is shaped like a pipe. Inﬁltration begins
from the inner surface of the pipe. The inﬁltration front spreads to the outer surface
of the pipe along the rotation radius. In Case 2, the volume of the preform is very
small compared with the total volume of molten metal, and therefore the displace-
ment of the inner surface of molten metal is negligible during inﬁltration. This
means that the pressure generated on the inner surface of the preform is almost
constant throughout inﬁltration. In addition, as the cross-section of the preform in
Case 2 is constant, it is possible to apply a one-dimensional inﬁltration model to this
case. In Case 3, the hatched area outside the fan-shaped molten metal also presses
the preform, which means that the pressure generated in Case 3 is higher than in the
other two cases. Therefore, the pressure which acts on the preform surface depends
on the type of inﬁltration.
A good starting point to develop the theory of centrifugal casting is to consider
the inﬁltration of molten metal in the small angle a at the rotation center during
preform
preform
preform
container
container
container
molten metal
molten metal
molten metal
rotation center
rotation
center
rotation
center
A
a
b
c
Fig. 5.1 Three major cases
for the inﬁltration of molten
metal by centrifugal casting
92 5 Centrifugal Casting of Metal Matrix Composites
3.
Case 1, as shown in Fig. 5.1a. The mass of this part is rmAdr and its acceleration is
o2
r. The centrifugal force, dFc, which acts on the region r to r + dr shown in
Fig. 5.1a, is given by:
dFc ¼ rmAo2
rdr; (5.1)
where rm is the density of the molten metal, A is the cross-sectional area of the fan-
shaped molten metal with small angle a, and o is the angular velocity (¼ 2pN,
where N is the number of revolutions per second or revolution number).
5.1.1.1 Case 1
Case 1 corresponds to Fig. 5.1a. By integration of Eq. (5.1) between r0 and r1, the
force, Fc, is obtained as:
Fc ¼
1
3
rmao2
ðr3
1 À r3
0Þ; (5.2)
where r0 and r1 are the locations of the inner surfaces of the molten metal and the
preform during rotation. The pressure P1, which acts on the inner surface of the
preform, is obtained by dividing Fc by the preform inner surface area ar1 at r ¼ r1.
When the inﬁltration of molten metal into the preform starts, the location of the
inner surface of the molten metal, rs(t), moves as shown in Fig. 5.2 and time-
dependent P1(t) during inﬁltration is given by:
P1ðtÞ ¼
1
3r1
rmo2
ðr3
1 À ðrsðtÞÞ3
Þ: (5.3)
We suppose that solidiﬁcation of the molten metal does not occur during
inﬁltration and that the molten metal is an incompressible liquid. When the location
of the inﬁltration front in the preform once inﬁltration starts is rf (t), rs(t) is given by:
rsðtÞ ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r2
0 þ f ðrf ðtÞÞ2
À r2
1
n or
; (5.4)
preform
front
container
molten metal
rotation
center
Fig. 5.2 Locations of the
inﬁltration front and inner
surface of molten metal
during inﬁltration
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 93
4.
where f is the porosity of the preform (¼ 1 À Vf). Substitution of Eq. (5.4) into
Eq. (5.3) yields:
P1ðtÞ ¼
rmo2
3r1
r3
1 À fr2
0 þ fððrf ðtÞÞ2
À r2
1Þg
3=2
h i
: (5.5)
This is the pressure which is applied to the inner surface of the preform during
inﬁltration.
5.1.1.2 Case 2
Case 2 is shown in Fig. 5.1b. The volume of the preform is so small compared with
the whole volume of the molten metal that the volume change of molten metal is
negligible during inﬁltration. In this case, the location of the inner surface of the
molten metal does not move during inﬁltration. Therefore, the pressure P1(t) in
Eq. (5.3) is constant throughout inﬁltration, and by putting rs(t) ¼ r0 in Eq. (5.3),
P1(t) becomes:
P1ðtÞ ¼
1
3r1
rmo2
ðr3
1 À r3
0Þ: (5.6)
5.1.1.3 Case 3
This case is shown in Fig. 5.1c. The cross-section of the molten metal column is
constant and does not change with increasing r. In other words, as well as the fan-
shaped metal with angle a, the hatched part of the molten metal also presses against
the inner surface of the preform. By putting A ¼ constant in Eq. (5.1), the force
which acts on the inner surface of the preform is given by:
Fc ¼
1
2
rmAo2
ðr2
1 À r2
0Þ: (5.7)
In this case, as inﬁltration proceeds, the location of the inner surface of the liquid
column moves. When the location is rs(t), the time-dependent P1(t) is given by:
P1ðtÞ ¼
1
2
rmo2
fr2
1 À ðrsðtÞÞ2
g; (5.8)
and
rsðtÞ ¼ r0 þ fðrf ðtÞ À r1Þ: (5.9)
94 5 Centrifugal Casting of Metal Matrix Composites
5.
Substituting Eq. (5.9) into Eq. (5.8), we obtain:
P1ðtÞ ¼
1
2
rmo2
½r2
1 À fr0 þ fðrf ðtÞ À r1Þg2
Š: (5.10)
5.1.2 Inﬁltration Start Pressure
The contact angle of a ceramic ﬁber with molten aluminum is usually higher than
90
. A certain pressure must be applied to the molten metal to induce inﬁltration. The
pressure is called the “threshold pressure”, and was discussed in detail in Chap. 3.
When the distribution of ﬁbers is random, the threshold pressure Pc is given by:
Pc ¼ À
4Vf g cos y
df ð1 À Vf Þ
; (5.11)
where g is the surface energy of the molten metal, y is the contact angle and df is the
diameter of ﬁbers. When P1(t) is smaller than Pc, inﬁltration cannot begin. The
minimum revolution number necessary for inﬁltration is given by Eqs. (5.12) and
(5.13):
N !
1
2p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3r1Pc
rmðr3
1 À r3
0Þ
s
ðfor Case 1 and Case 2Þ (5.12)
N !
1
2p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2Pc
rmðr2
1 À r2
0Þ
s
ðfor Case 3Þ (5.13)
5.1.3 Inﬁltration of Molten Metal into the Preform
Ignoring ﬂuid inertial contributions and preform compression, the ﬂow of the ﬂuid
(molten metal) in the preform in the radial direction is governed by Darcy’s law in a
centrifugal force ﬁeld [5, 6]:
@pðr; tÞ
@r
¼ À
mmuðr; tÞ
K
þ rmo2
r; (5.14)
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 95
6.
where p(r, t) is the pressure in the preform, mm is the viscosity of the molten metal, K
is the permeability, and u(r, t) is the radial superﬁcial velocity which is the mass of
molten metal which ﬂows into the preform per unit time and unit area.
5.1.3.1 Case 1
The continuity equation for the radial direction in cylindrical coordinates is given by:
rm
r
@fruðr; tÞg
@r
¼ 0: (5.15)
From Eq. (5.15), we obtain the following relationship:
ruðr; tÞ ¼ CðtÞ ði:e:; it is a function of time onlyÞ: (5.16)
Using Eq. (5.16), Eq. (5.14) is rewritten as:
@pðr; tÞ
@r
¼ À
mm
K
CðtÞ
r
þ rmo2
r: (5.17)
At a ﬁxed time, it is possible to regard Eq. (5.17) as an ordinary differential
equation with respect to r. Then, we can obtain the following solution:
pðr; tÞ À P1ðtÞ ¼ À
fmmCðtÞ
K
ln
r
r1
þ
1
2
rmo2
ðr2
À r2
1Þ; (5.18)
where P1(t) is the pressure at r ¼ r1. As the pressure at r ¼ rf (t) (inﬁltration front)
is Pc (the threshold pressure):
Pc À P1ðtÞ ¼ À
fmmCðtÞ
K
ln
rf ðtÞ
r1
þ
1
2
rmo2
fðrf ðtÞÞ2
À r2
1g: (5.19)
Alternatively, we have the following relationship between inﬁltration front rf (t)
and the superﬁcial velocity uf (t) at the front:
drf ðtÞ
dt
¼
uf ðtÞ
f
: (5.20)
If mm, K, rm and f are independent of t, substitution of Eqs. (5.16) and (5.19) into
Eq. (5.20) yields the following ordinary differential equation:
t ¼
ðrf ðtÞ
r1
mmf2
rf ðtÞ lnðrf ðtÞ=r1Þ
K½P1ðtÞ À Pc þ ð1=2Þrmo2fðrf ðtÞÞ2
À r2
1gŠ
drf ðtÞ: (5.21)
96 5 Centrifugal Casting of Metal Matrix Composites
7.
It is difﬁcult to obtain an analytical solution of this differential equation, but
numerical integration can be used.
The pressure distribution in the inﬁltrated region is obtained by substitution of C
(t) from Eq. (5.18) into Eq. (5.19):
pðr; tÞ ¼ P1ðtÞ À P1ðtÞ À Pc þ
1
2
rmo2
fðrf ðtÞÞ2
À r2
1g
lnðr=r1Þ
lnðrf ðtÞ=r1Þ
þ
1
2
rmo2
ðr2
À r2
1Þ (5.22)
Equation (5.22) gives the pressure distribution in the inﬁltrated region when the
inﬁltration front is located at position rf(t). This relationship shows that the pressure
is independent of f and K except for the threshold pressure Pc which depends on f.
When this process is applied industrially, the thickness of the preform is ﬁnite.
When the inﬁltration front reaches the outer surface of the preform (r ¼ rt), the
inﬁltration stops and the pressure distribution in the inﬁltrated region changes
instantaneously. Supposing that the framework of the ﬁbrous preform is sufﬁciently
rigid and does not deform in the inﬁltrated region, the centrifugal force, dFc, which
acts on the metal of the region r to r + dr in the preform, is given by:
dFc ¼ frmAo2
rdr: (5.23)
When the preform thickness is L (¼ rt À r1), the pressure in the preform p(r) at r
(r1 r rt) is given by:
pðrÞ ¼
1
3r1
rmo2
½r3
1 À fr2
0 þ fðr2
t À r2
1Þg3=2
Š þ
1
3r
frmo2
ðr3
À r3
1Þ: (5.24)
When the amount of molten metal is small and all molten metal enters the
preform before the inﬁltration front arrives at the outer surface of the preform, the
pressure curve is given only by the second term of Eq. (5.24).
5.1.3.2 Case 2
This is the case where P1(t) is constant (¼ P0). The cross section of the preform is
independent of r and constant. Therefore, the superﬁcial velocity u(r, t) in
Eq. (5.14) is also independent of r and expressed by u(t). The one-dimensional
form of the continuity equation, which applies in this situation, is:
@uðtÞ
@r
¼ 0: (5.25)
At a ﬁxed time, it is also possible to regard Eq. (5.14) as an ordinary differential
equation with respect to r. Then, we can obtain the following solution:
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 97
8.
pðr; tÞ ¼ P0 À
mmuðtÞ
K
ðr À r1Þ þ
1
2
rmo2
ðr2
À r2
1Þ: (5.26)
As the pressure at the inﬁltration front r ¼ rf (t) is Pc, we obtain:
P0 ¼
mmuðtÞ
K
ðrf ðtÞ À r1Þ À
1
2
rmo2
fðrf ðtÞÞ2
À r2
1g þ Pc: (5.27)
Alternatively, the velocity of the inﬁltration front is given by:
drf ðtÞ
dt
¼
uðtÞ
f
: (5.28)
Substitution of u(t) from Eq. (5.28) into Eq. (5.27) yields an ordinary differential
equation:
m3 þ m4rf ðtÞ
m1 þ m2frf ðtÞg2
drf ðtÞ ¼ dt; (5.29)
where
m1 ¼ K P0 À Pc À
rmo2
r2
1
2
; m2 ¼
Krmo2
2
; m3 ¼ Àmmfr1; m4 ¼ mmf:
The solution of Eq. (5.29) depends on the sign (negative or positive) of m1.
However, since we know P0 from Eq. (5.6), m1 is always negative. Then, the
following solution is obtained:
t ¼ n1 lnfðrf ðtÞÞ2
À n2
3g þ n2 ln
rf ðtÞ À n3
rf ðtÞ þ n3
þ n4; (5.30)
where
n1 ¼
m4
2m2
; n2 ¼
m3
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Àm2m1
p ; n3 ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À
m1
m2
r
; and
n4 ¼ Àn1 lnðr2
1 À n2
3Þ À n2 ln
r1 À n3
r1 þ n3
:
By substituting uðtÞ from Eq. (5.26) into Eq. (5.27), we obtain:
pðr; tÞ ¼
rf ðtÞ À r
rf ðtÞ À r1
P0 þ
1
2
rmo2
ðr À r1Þðr À rf ðtÞÞ þ Pc
r À r1
rf ðtÞ À r1
: (5.31)
This equation gives the pressure distribution in the inﬁltrated region when the
inﬁltration front is speciﬁed. The characteristic features of this equation are that the
equation does not include f, and that the pressure distribution pattern is
98 5 Centrifugal Casting of Metal Matrix Composites
9.
independent of the volume fraction of ﬁbers in the preform except for the depen-
dence of the threshold pressure, Pc, on f.
When the preform thickness is L, the pressure p(r) in the inﬁltrated region after
the inﬁltration front reaches the outer surface of the preform is given by:
pðrÞ ¼ P0 þ
1
2
frmo2
ðr2
À r2
1Þ; ðr1 r rtÞ: (5.32)
5.1.3.3 Case 3
This is the case shown in Fig. 5.1c. Since cross sectional variations can be neglected
because of the cylindrical conﬁguration, the superﬁcial velocity u(r, t) in Eq. (5.14)
is actually u(t) and the continuity equation in the one-dimensional form, Eq. (5.25),
should also be valid in this case. At a ﬁxed time, it is possible to regard Eq. (5.14) as
an ordinary differential equation with respect to r, if mm, K and rm are independent
of r. Then, we can get the following solution:
pðr; tÞ ¼ P1ðtÞ À
mmuðtÞ
K
ðr À r1Þ þ
1
2
rmo2
ðr2
À r2
1Þ; (5.33)
where P1(t) is the pressure at r ¼ r1. As the pressure p(r, t) at r ¼ rf (t) in Eq. (5.33)
is the threshold pressure Pc, we obtain:
P1ðtÞ ¼
mmuðtÞ
K
ðrf ðtÞ À r1Þ À
1
2
rmo2
ððrf ðtÞÞ2
À r2
1Þ þ Pc: (5.34)
Substitution of Eqs. (5.10) and (5.28) into Eq. (5.34) provides an ordinary
differential equation, if mm, K, rm and f are independent of t. The solution is:
t ¼
1
2m3
ln fðrf ðtÞÞ2
þ m1rf ðtÞ þ m2g
rf ðtÞ þ m6
rf ðtÞ þ m5
m4
þ m7; (5.35)
where
m1 ¼
2fðfr1 À r0Þ
1 À f2
; m2 ¼
Àr2
0 À f2
r2
1 þ 2fr0r1 À 2Pc=ðrmo2
Þ
1 À f2
;
m3 ¼
rmo2
Kð1 À f2
Þ
2mmf
; m4 ¼
m1 þ 2r1
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2
1=4 À m2
p ; m5 ¼
1
2
m1 À
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2
1
4
À m2
r
;
m6 ¼
1
2
m1 þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2
1
4
À m2
r
; and m7 ¼ À
1
2m3
ln ðr2
1 þ m1r1 þ m2Þ
r1 þ m6
r1 þ m5
m4
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 99
10.
Equation (5.35) gives the relationship between the location of the inﬁltration
front rf (t) and time t.
Alternatively, by substituting Eq. (5.34) into Eq. (5.33), we obtain the
pressure distribution in the inﬁltrated region when the inﬁltration front is speciﬁed
at rf (t):
pðr; tÞ ¼
ðrf ðtÞ À rÞP1ðtÞ þ ðr À r1ÞPc
rf ðtÞ À r1
þ
1
2
rmo2
ðr À r1Þðr À rf ðtÞÞ: (5.36)
When the preform thickness is L (¼ rt À r1), the pressure in the preform after
inﬁltration reaches the outer surface of the preform is:
pðrÞ ¼
1
2
rmo2
r2
1 À r0 þ fðrt À r1Þf g2
h i
þ
1
2
frmo2
ðr2
À r1
2
Þ: (5.37)
5.1.4 Example Calculations
The relationship between Vf and the permeability of the preform is needed for the
calculation of inﬁltration. There are many theoretical equations for the permeability
of porous media [7]. Among them, Langmuir’s equation [Eq. (5.38)] which applies
to ﬂow parallel to the ﬁber alignment and Happel’s equation [Eq. (5.39)] which
applies to ﬂow normal to ﬁbers can be used to obtain the permeability of the
preform. Generally the permeability calculated using these equations agrees well
with experimental results. The equations are:
K
R2
¼
1
4Vf
À ln Vf À
3
2
þ 2Vf À
1
2
Vf
2
; (5.38)
and
K
R2
¼
1
8Vf
À ln Vf þ
V2
f À 1
V2
f þ 1
!
; (5.39)
where R is the radius of the ﬁber. As the ﬁber distribution we used was random, the
permeability of the preform was obtained by averaging the values of permeability
calculated for ﬂows parallel and normal to the ﬁber alignment.
5.1.4.1 Case 1
The advance of the inﬁltration front of molten aluminum (without solidiﬁcation)
calculated using Eq. (5.21) is shown in Fig. 5.3. The calculation was performed for
100 5 Centrifugal Casting of Metal Matrix Composites
11.
a preform made from alumina short ﬁbers. The following data were also used in this
calculation: r0 ¼ 0.07 m, r1 ¼ 0.12 m, f (¼ 1 À Vf) ¼ 0.87, rm ¼ 2.38 Â 103
kg/
m3
, R (¼ df/2) ¼ 1.85 Â 10À6
m, g ¼ 0.893 Pa m, y ¼ 160
and mm ¼ 0.984 Â 10
À3
Pa s. The minimum revolution number N for the inﬁltration is 19.74 rev/s, as
calculated from Eq. (5.12). Figure 5.3 shows that if the revolution rate is slightly
higher than the minimum value, inﬁltration will be continuous. The pressure
distribution in the inﬁltrated region obtained from Eq. (5.22) is shown in Fig. 5.4
for N ¼ 50 rev/s. The pressure distribution curves are convex towards lower
pressures, because the second derivative of Eq. (5.22) is always positive and
Fig. 5.3 Relationship between inﬁltrated distance and time for a range of rotational speeds for
Case 1
Fig. 5.4 Change with time of pressure distribution in the inﬁltrated region for Case 1 when
N ¼ 50 rev/s
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 101
12.
proportional to o2
. The preform surface pressure decreases quickly as the inﬁltra-
tion advances, and the intermediate part of the pressure distribution curve falls
below the threshold pressure. Nevertheless, inﬁltration continues, because the
threshold pressure is needed only to wet ﬁbers with molten metal, and the pressure
at the inﬁltration front always remains constant at the threshold pressure. After
wetting, the molten metal does not separate from the preform, even if the pressure
becomes lower than the threshold pressure, unless enough work for the separation is
supplied to the molten metal/ﬁber interface.
When the preform thickness is L and the inﬁltration front reaches the location r
(t) (the outer surface of the preform), the pressure distribution curve shown in
Fig. 5.4 for N ¼ 50 rev/s is changed instantaneously to that of Fig. 5.5. Only the
preform surface pressure remains constant, and the pressure in the inﬁltrated region
rises quickly and becomes stable. In this calculation, in the cases of L ¼ 0.04 and
0.05 m, all liquid metal enters into the preform and a cavity is formed near the inner
surface of the preform, because of the lack of liquid metal.
5.1.4.2 Case 2
Figure 5.6 shows the relationship between time and the advance of the inﬁltration front
obtained from Eq. (5.30). The front advances faster than in Case 1, because the
preform surface pressure remains constant throughout the inﬁltration. The pressure
distribution curves are shown in Fig. 5.7 for N ¼ 50 rev/s. The pressure distribution
curves in this case are also convex toward lower pressure, because the second
derivative of Eq. (5.31) is also positive and proportional to o2
. The pressure distribu-
tion curves for ﬁnite preform thickness cases are shown in Fig. 5.8 for N ¼ 50 rev/s. In
Case 2, even if the preform thickness is different, the pressure distribution curve is the
same, and the curve depends only on the revolution number.
Fig. 5.5 Pressure distribution in the inﬁltrated region for Case 1 after the inﬁltration front reaches
the outer surface of the preform when N ¼ 50 rev/s
102 5 Centrifugal Casting of Metal Matrix Composites
13.
5.1.4.3 Case 3
Figure 5.9 shows the relationship between time and the advance of the inﬁltration
front obtained from Eq. (5.35) for Case 3. The front advances faster than in Case 1
and slower than in Case 2 for a particular revolution number. The pressure distri-
bution curves obtained from Eq. (5.36) are shown in Fig. 5.10 for N ¼ 50 rev/s.
These curves are also convex toward lower pressure. Regions having lower pressure
than the threshold pressure appeared at t ¼ 0.163 s in Fig. 5.10. The pressure
Fig. 5.6 Relationship between inﬁltrated distance and time for a range of rotational speeds for
Case 2
Fig. 5.7 Change with time of pressure distribution in the inﬁltrated region for Case 2 when
N ¼ 50 rev/s
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 103
14.
distribution curves obtained using Eq. (5.37) are shown in Fig. 5.11 for various
ﬁnite preform thickness at N ¼ 50 rev/s. Until the inﬁltration front arrives at the
outer surface of the preform, the preform surface pressure changes as given in
Fig. 5.10. When the front arrives at the preform outer surface, the change in the
preform surface pressure stops and the pressure distribution in the inﬁltrated region
becomes that shown in Fig. 5.11.
Fig. 5.8 Pressure distribution in the inﬁltrated region for Case 2 after the inﬁltration front reaches
the outer surface of the preform when N ¼ 50 rev/s
Fig. 5.9 Relationship between inﬁltrated distance and time for a range of rotational speeds for
Case 3
104 5 Centrifugal Casting of Metal Matrix Composites
15.
5.1.5 Examples of Composites Fabricated Using
Centrifugal Force
In this section we summarize a previous study where the feasibility of centrifugal
force for the inﬁltration of molten metal into preforms was examined and the
composites obtained were characterized [8]. A schematic of the apparatus used is
Fig. 5.10 Change with time of pressure distribution in the inﬁltrated region for Case 3 when
N ¼ 50 rev/s
Fig. 5.11 Pressure distribution in the inﬁltrated region for Case 3 after the inﬁltration front
reaches the outer surface of the preform when N ¼ 50 rev/s
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 105
16.
shown in Fig. 5.12. A preheated graphite container with a uniform cross section was
used. The container was balanced by a counterweight. A preheated alumina short
ﬁber preform was set on the bottom of the container which was rotated. When the
rotational speed reached the target value, molten pure aluminum was poured into a
pouring device concentric with the rotation shaft and a composite was produced.
The experimentally obtained relationship between the rotational speed and the
inﬁltrated distance for different volume fractions of ﬁber and initial preform
thickness is shown in Fig. 5.13. The diameter for the preform was 24 mm, and
the initial thicknesses of the preforms were 20 mm for Vf ¼ 6 %, 19 mm for
Vf ¼ 9 %, and 18 mm for Vf ¼ 13 %. The inﬁltrated distances for Vf ¼ 6 % and
preform
graphite container
graphite pipe
pouring device
Fig. 5.12 Schematic of the apparatus used to generate centrifugal force for the inﬁltration of
aluminum
Fig. 5.13 Relationship between the pressure generated and inﬁltrated distance. The initial
thicknesses of the preforms are 20 mm for Vf ¼ 6 %, 19 mm for Vf ¼ 9 %, and 18 mm for
Vf ¼ 13 %. The compressive deformation of preforms is neglected
106 5 Centrifugal Casting of Metal Matrix Composites
17.
9 % were either zero or 100 %. That is, in the case of Vf ¼ 6 %, inﬁltration did not
advance at 65 kPa (N ¼ 10 rev/s) but fully advanced at 70 kPa (N ¼ 11 rev/s). For
Vf ¼ 9 %, inﬁltration did not take place at 70 kPa (N ¼ 16 rev/s), but occurred fully
at 85 kPa (N ¼ 17 rev/s). For Vf ¼ 13 %, the molten metal advanced a little and
then stopped at 160 kPa (N ¼ 26 rev/s), but advanced much further at 230 kPa
(N ¼ 32 rev/s). Since the preheat temperatures of the preform and graphite con-
tainer were 703 and 673 K, respectively, the inﬁltration velocity was slow in this
case and the metal solidiﬁed during inﬁltration. Pressure higher than 300 kPa was
needed for full inﬁltration for the Vf ¼ 13 % preform.
The structural characteristics of samples are presented in Figs. 5.14 and 5.15.
A partially inﬁltrated sample with Vf ¼ 13 % is also shown in Fig. 5.14. The
preform also contained many particles, because the quality of the alumina/silica
ﬁbers was not high. The minimum pressures for the start of inﬁltration were
obtained from this experiment and are listed in Table 5.1, along with the threshold
pressure values calculated theoretically using Eq. (5.11). Both sets of values are in
agreement.
Fig. 5.14 Macrostructures of alumina/silica short ﬁber reinforced aluminum composite samples
fabricated using centrifugal force
5.1 Inﬁltration of Molten Metal Using Centrifugal Force 107
18.
5.2 Centrifugal Casting of Particle Dispersed Molten Metal
When molten metal which contains ceramic particles is cast in a centrifugal force
ﬁeld, the particles tend to segregate because of the density difference between the
molten metal and the ceramic particles. Here, we discuss the movement of the
particles and the forces which act on the particles. To simplify the model, solidiﬁ-
cation will be neglected.
A particle having density rp and radius Rp is present in molten metal having
density rm. The centrifugal force, which is proportional to the rotating radius r, is
acting on the particle. The equation for the force, Fc, which acts on the particle as a
result of the acceleration, o2
r is similar to Eq. (5.1):
Fc ¼
4
3
pR3
pðrp À rmÞo2
r: (5.40)
If the particle moves with very low velocity in the molten metal, as shown in
Fig. 5.16, Stokes’ drag formula will be applicable [9]. The viscous drag is given by:
Fv ¼ À6pmmRp
dr
dt
; (5.41)
where Fv acts on the particle in the opposite direction to the centrifugal force, and
has a negative sign. mm is the viscosity of the molten metal.
The fundamental expression of the movement of an object is: [force] ¼ [mass]
Â [acceleration]. For the particle, the total force is Fc + Fv. Thus, we obtain the
following differential equation:
Fig. 5.15 Microstructure of alumina/silica short ﬁber reinforced aluminum composites fabricated
using centrifugal force
Table 5.1 Comparison of threshold pressures
Vf (%) Pc (kPa) (theoretical) Pc (kPa) (experimental)
6 57.9 66
9 89.7 81
13 135.6 156
108 5 Centrifugal Casting of Metal Matrix Composites
19.
4
3
pR3
pðrp À rmÞo2
r À 6pmmRp
dr
dt
¼
4
3
pR3
prp
d2
r
dt2
: (5.42)
Gravity is neglected, because the gravity is much smaller than the centrifugal force.
The solution of Eq. (5.42), i.e., the location of the particle as a function of time, is:
r ¼ r0 exp
À9mm þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
81m2
m þ 16ðrp À rmÞrpR4
po2
q
4Rprp
8
:
9
=
;
t
2
4
3
5; (5.43)
where r0 is the initial location of the particle. The solution is a simple exponential
function. If the particle goes far away from the rotating center, the velocity of the
particle will increase and Stokes’ drag formula (equation (5.41)) will be invalid,
because the formula is only valid when the Reynolds number around the particle is
less than 0.5.
Examples of calculations using Eq. (5.43) are shown in Figs. 5.17 and 5.18. In these
calculations, the particle is a spherical SiC particle with density rp ¼ 3.15 Â 103
kg/m3
; the density rm of the molten aluminum is 2.38 Â 103
kg/m3
and the viscosity
of the molten aluminum mm ¼ 0.948 Â 10À3
Pa s. Figure 5.16 shows the movement
with time of particles with different radii, Rp, when the revolution number N is 10 rev/s.
Equation(5.43)givesthedimensionlessdisplacementratioofthelocationoftheparticle
to its initial location r/r0, which represents the movement of the particle. When the
particle radius is 1 mm (i.e., the diameter is 2 mm), the particle quickly moves away
from the rotating center. However, when the particle radius is 0.1 mm, the particle will
move slowly. Figure 5.17 shows the displacement ratio, r/r0 of a particle of radius
0.1 mm, for different revolution numbers N. When N ¼ 5 rev/s, the particle essentially
remainsatitsinitiallocationfortheentire300s.However,whenNislargerthan15rev/s
(900 rpm), the particle moves quickly.
However, Eq. (5.43) is quite different to “Stokes’ law” which gives:
dz
dt
¼
2R2
pðrp À rLÞg
9mL
; (5.44)
Fc
Fv
Rp
particle
Fig. 5.16 Viscous drag, Fv,
and centrifugal force, Fc,
which act on a particle during
centrifugal casting of particle
reinforced molten metal
5.2 Centrifugal Casting of Particle Dispersed Molten Metal 109
20.
where g is the gravitational acceleration, rL and mL are the density and viscosity of
the ﬂuid, respectively. The z-axis is taken as the gravity direction. In this equation,
the acceleration is constant, although it is o2
r in the centrifugal force ﬁeld.
In actual centrifugal casting, solidiﬁcation of metal occurs along with the
movement of particles. The solidiﬁcation in this case is very complicated and it is
very difﬁcult to analytically calculate the behavior, because solidiﬁcation depends
Fig. 5.18 Curves of displacement ratio r/r0 with time for various values of revolution number N;
particle radius is 0.1 mm
Fig. 5.17 Curves of displacement ratio r/r0 with time for various particle radii; revolution number
N is 10 rev/s
110 5 Centrifugal Casting of Metal Matrix Composites
21.
on the shape of container (mold), the composition of the alloy and on other factors.
In addition, Eq. (5.43) is the model of the movement for a single particle only.
When the number of particles is large, we should consider the interaction between
particles. Then, the apparent viscosity mapp must be used instead of the real viscosity
of the molten metal; the apparent viscosity increases when the number of particles
increases, and is given by [10]:
mapp ¼ mmð1 þ
5
2
Vf þ 10:05V2
f Þ: (5.45)
This mapp can then be used instead of mm in Eq. (5.43).
Centrifugal casting with solidiﬁcation in the presence of solid particles was
discussed numerically by Panda et al. [11]. They showed that the segregation of
particles in the alumina particles/aluminum or silicon carbide particles/aluminum
systems depends on the size of particle, rotational speed, heat transfer coefﬁcient at
the casting/mold interface and volume fraction of particles.
Examples of the application of centrifugal casting to composites include copper
alloy/graphite particle composites, which were developed by Kim et al. to improve
tribological properties of the inner surfaces of products [12]. Functionally graded
composites were also developed in TiB2 or TiC particles/aluminum alloy systems
by Kumar et al. [13].
References
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Al-11%Si alloy. J. Jpn. Foundry Eng. Soc. 57, 102–107 (1985)
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composites containing segregated alumina ﬁbers. J. Mater. Sci. Lett. 7, 830–832 (1988)
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aluminum alloy matrix composites containing short alumina ﬁbers by centrifugal force.
J. Jpn. Inst. Light Met. 40, 7–12 (1990)
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11. Panda, E., Mehrotra, S.P., Mazumdar, D.: Mathematical modeling of particle segregation
during centrifugal casting of metal matrix composites. Metall. Mater. Trans. 37A,
1675–1687 (2006)
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alloy-graphite particle composite. Metall. Mater. Trans. 31A, 1283–1293 (2000)
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112 5 Centrifugal Casting of Metal Matrix Composites
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