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4.- Modeling the Spray Forming Process of Copper Alloys

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4.- Modeling the Spray Forming Process of Copper Alloys
Case 4
Modeling the Spray Forming Process of Copper Alloys
The purpose of research in this case was to help determine the feasibility of using the
spray forming process for the production of copper and copper alloys strips. In the spray
forming process, a molten metal jet discharged from a crucible through a pouring hole is
atomized by circumferential high speed gas jets. The atomized droplet dispersion cools and
partially solidifes as it falls and is collected by a suitably designed substrate. In this case,
the goal was to produce copper strip of a certain thickness so the horizontal substrate had
to move at controlled speed. By carefully adjusting the melt flow rate and substrate speed,
strips of any thickness could in principle be produced in a continuous manner.
The copper alloys produced must comply with very strict composition requirements.
Therefore, the producer was interested in determining whether strips with the required
compositional characteristics could indeed be produced. The producer acquired and installed
a pilot plant scale spray forming unit and began to perform empirical feasibility studies. As
this work was being carried out, interest was expressed in investigating the problem using
mathematical modeling methods.
A proposal was presented to the producer for the investigation of the problem using a
recently developed, semi-quantitative modeling methodology. The approach is similar to
the one used in molecular dynamics simulation methodology, on the celestial gravitational
mechanics of point masses used in astronomy and in charged particle plasma simulations.
The generic method is called the particle method. In particle methods, just as in molecular
dynamics one envisions a given material system as constituted by a collection of mutually
interacting material particles. While in molecular dynamics the particles are actual atoms
or molecules, in particle methods each particle is constituted by large numbers of atoms or
molecules. The interaction assumed to exist between particles in particle methods also resembles the one used in molecular dynamics studies, namely, when two particles are pushed
close together, they repel and when the particles are drawn appart, they attract. Moreover,
while the attraction force decays slowly with increasing separation, the repulsion force increases rapidly with decreasing separation. In symbols, for two particles i and j aligned with
the x axis and separated by distance r, the interaction force between them is given by
Fij = −
A B
+
rp rq
1
with A > 0, B > 0 and q > p > 0. Note that if B = 0 and p = 2 one obtains the gravitational
force acting between two point masses and if q = 12 and p = 6 (Lennard-Jones) molecular
forces result. The above equation also indicates that there is some distance r∗ for which the
force is zero, this is the equilibrium separation between the particles.
In a multiparticle system each particle interacts with all other particles in the assembly
and the net force acting on the particle must be computed by summing the interactions with
all particles, i.e. for a system containing n particles, the force on the i−th particle is
Fi =
N
X
Fij
j6=i=1
In practice, an assembly of particles placed initially at a selected set of individual locations
will in general not be in equilibrium. Particles will either be too close or too far from each
other and will tend to repel/attract. Therefore, proper initialization of a particle method
calculation requires a period of relaxation to allow interparticle separations to reach their
equilibrium values.
The numerical computation of the time evolution of the system of particles is based on
the (Newton) equations of motion for the particles, i.e.
mi
dri
= Fi
dt2
for i = 1, 2, ..., N .
Specifically, the calculation starts by computing the net force acting on each particle in
the assembly resulting from the interaction with all other particles. To reduce the expense of
computations an empirically determined cutoff radius is introduced and only the interaction
forces produced by those particles located relatively close to the given particle are accounted
for in the sum. For time advancement, the so called modified Verlet algorithm has proven
most stable and accurate. In this algorithm one considers a given particle of mass m at three
consecutive time levels, t − ∆t, t and t + ∆t, where ∆t is a small, suitably chosen time step
and then computes position at t + ∆t by
r(t + ∆t) = 2r(t) − r(t − ∆t) + ∆t2 F(t)/m
The velocity at t is computed by
v(t) =
r(t + ∆t) − r(t − ∆t)
2∆t
and finally, the velocity at t + ∆t is computed by
v(t + ∆t) = v(t) + ∆t[F(t + ∆t) + F(t)]/2
The above calculation is performed for all particles in the assembly to complete the
advancement of the computation by one time step from t to t + ∆t and the process is simply
repeated as many times as desired.
2
In the course of investigating the feasibility of producing quality copper preform strip by
spray forming the goal was to visualize the process by which the droplets impact and become incorporated into the preform. Furthermore, there was interest in determining whether
atomized gas entrained with the droplets was likely to become entrapped into the preform.
This was important since the strip must comply with stringent chemical composition requirements. Therefore, a model based on the particle method was created to simulate the
impingement of one or several droplets onto a partially solidified substrate. The droplets
and the substrate were simulated by collections of computational particles. Additional computational particles were added to represent the entrained gas. The assembly of particles
was given initial conditions representative of the impacting process and the computation was
carried out to estimate the behavior of the droplets as they impacted the substrate. Some
results of calculations are shown in the attached figures.
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