CHAPTER 3: POWDER FLOW DURING DIE FILLING
3.4. Die filling from a moving shoe
3.4.2. Mass flow rate and critical shoe velocity
In Figure 3.33, the average mass flow rate and fill ratio are plotted as a function of
shoe velocity for Case 1. The average mass flow rate M is defined in Eq. (3.1). The
fill ratio δ is the fraction of the die that is filled and it is defined as
total f
m m
δ
= (3.26)where mtotal is the total mass of particles deposited into the die at a given shoe velocity and mf is the mass of particles in a fully filled die.
As shown in Figure 3.33a, the average mass flow rate in air is about half of that in a
vacuum at a given shoe velocity due to the resistance of the entrapped air on the
powder flow. It is also observed that at lower shoe velocities the average mass flow
rate increases as the shoe velocity increases until it reaches a maximum, thereafter the
average mass flow rate decreases with increasing shoe velocity. During die filling
with a moving shoe, the effective discharging area (EDA) from which the powder
flows into the die increases from zero to the whole die opening. When the shoe
translates at a lower velocity, the filling process with a small EDA is prolonged
thereby leading to a smaller overall mass flow rate. As the translational velocity of the
shoe increases, the front of the powder mass moves rapidly across the cavity and the
mass flow rate increases due to the fact that the powder can be delivered into the die
through a larger area. However, the increase of shoe velocity can enhance the
interlocking of particles, which is caused by the lateral compressive stresses generated
by inertia effects (Wu et al., 2003c). Therefore, it can become more difficult for
particles to be detached from the bulk due to the enhanced interlocking of particles as
the shoe moves at a higher velocity, there is less time for the newly mobilised
particles to build up vertical speed above the die opening, thus, the particles flowing
into the die cavity have a lower vertical velocity and consequently a lower mass flow
rate is achieved.
(a)
(b)
Figure 3.33 The variation of (a) average mass flow rate and (b) fill ratio with shoe
It is observed from Figure 3.33b that the fill ratio δ is essentially equal to a value of
unity at lower shoe velocities, implying the die is completely filled, and it is less than
one at higher shoe velocities as the die is just partially filled. Obviously, there exists a
maximum shoe velocity at which the die can be completely filled, which is referred to
as critical shoe velocity (Wu et al., 2003c). It is also found from Figure 3.33b that the
fill ratio decreases with the increasing shoe velocity when the die can not be
completely filled (i.e., δ <1). For incomplete filling, experimental results (Wu and
Cocks, 2004; Sinka et al., 2004; Shneider et al., 2005, 2007) showed that the fill ratio
can be expressed in terms of the shoe velocity as
c s v v α
δ
= (3.27)where the shoe velocity vs should be higher than the critical shoe velocity vc, i.e.,
s
v > vc. Parameter
α
depends on the powder properties and process conditions. Based on Eq.(3.27), the best fit to the data in Figure 3.33b gives1.56 s 139.3 v
δ
= (3.28)for die filling in a vacuum, and
1.26 s 82.9 v
δ
= (3.29)for die filling in air. Therefore the critical shoe velocities are determined as 139.3
mm/s for die filling in a vacuum and 82.9 mm/s for die filling in air. It is evident that
the critical shoe velocity in air is much lower than that in a vacuum due to the fact that
the entrapped air resists the powder flowing into the die (Figures 3.22, 3.24 and 3.26).
By comparing Figures 3.33a and 3.33b, it is found that the average mass flow rate
appears to be highest when the shoe velocity is around the critical shoe velocity,
(a)
(b)
Figure 3.34 The variation of (a) average mass flow rate and (b) fill ratio with shoe
velocity for Cases 2 and 3.
Figure 3.34 shows the average mass flow rates and fill ratios at various shoe velocities
for Cases 2 and 3. As shown in Figure 3.34a, the trends are similar to those in Case 1
(Figure 3.33a): the average mass flow rate initially increases as the shoe velocity
increases, until it reaches a maximum and thereafter the flow rate decreases with
on the mass flow rate for die filling with these coarse particles, which are the air-inert
particles as classified in Figure 3.12. It is noted that for die filling with a deep die, a
higher powder bed consisting of more particles is introduced to the shoe in order to
fully fill the enlarged die cavity. As indicated by Wu and Cocks (2004) and Wu
(2008), a higher powder bed in the shoe can result in a higher powder flow rate due to
the increased filling intensity. Therefore, as shown in Figure 3.34a, the average mass
flow rate for die filling with a deep die is higher than that for die filling with a shallow
die at a given shoe velocity due to the higher powder bed in the shoe.
Using Eq. (3.27), the critical shoe velocities and the fill ratio-shoe velocity
relationships for incomplete filling can be determined by best fitting to the data in
Figure 3.34b, which gives
1.39 s 119.8 v
δ
= (3.30) and 1.39 s 118.4 vδ
= (3.31)for Case 2 in a vacuum and in air, respectively, and
1.28 s 87.9 v
δ
= (3.32) and 1.21 s 84.5 vδ
= (3.33)for Case 3 in a vacuum and in air, respectively. The critical shoe velocities for die
filling with a shallow die in a vacuum and in air are 119.4 mm/s and 118.4 mm/s,
and in air are 87.9 mm/s and 84.5 mm/s, respectively. Thus, for die filling with a
given die cavity, the critical shoe velocity in air is very close to that in a vacuum due
to the negligible effect of air on coarse particles. The critical shoe velocity is found to
be reduced by increasing the depth of the die due to the enlarged capability of the
cavity. By comparing Figures 3.34a and 3.34b, it is also observed that the average
mass flow rate achieves the maximum value at the critical shoe velocity for die filling
with coarse particles.
In Eq. (3.27),
α
generally has a value of 1.0-1.6 according to the extensive experimental data for a wide range of metal and pharmaceutical powders (Wu andCocks, 2004, 2006;Sinka et al., 2004; Shneider et al., 2005, 2007). From the Eqs.
(3.28)-(3.33),