M. Barkhudarov, Flow Science Inc., Santa Fe, New Mexico; S. Mascetti, XC Engineering, Italy; R. Pirovano, XC Engineering, Italy
Abstract
High pressure die casting is one of the most complex processes in the foundry world due to the wide range of physical phenomena and process parameters that control the outcome. A particular challenge is achieving optimal conditions in the shot sleeve from which metal is injected into the die cavity. The speed of the plunger in a horizontal shot sleeve must be carefully controlled to avoid unnecessary entrainment of air in the metal and, at the same time, minimize heat losses in the sleeve.
The paper presents a general solution for the flow of metal in a shot sleeve, based on the shallow water approximation of the interaction of the moving plunger and liquid metal. The derived analytical solution allows engineers to precisely control the behavior of metal in the shot sleeve during the slow-shot stage of the high pressure die casting process, minimizing the risk of air entrainment. Results are validated with three-dimensional numerical modeling of the process. Coupled with parametric optimization, the numerical model can improve the process conditions predicted by the analytical model.
Introduction
The speed of the plunger in a horizontal shot sleeve must be carefully controlled to avoid unnecessary entrainment of air in the metal and at the same time minimize heat losses in the sleeve. If the plunger moves too fast, large waves are created on the surface of the liquid metal that may overturn and entrain air into the metal, which will then be carried into the die cavity. A plunger moving too slow results in waves reflecting from the opposite end of the shot sleeve. The reflected waves prevent proper expulsion of air into the die cavity. In either case, the outcome is excessive porosity in the final casting.
Moreover, a slow plunger increases also oxidation of the free surface of liquid metal, and the heat losses because of the long contact time with the mold walls. In this article two approaches are used to limit these effects: a general solution for the plunger speed as a function of time and a full-physics, three-dimensional CFD optimization.
Mathematical model
The dynamics of waves in a horizontal shot sleeve can be analyzed by drawing an analogy with flow in an open channel. A detailed analysis is possible by modeling the flow of metal in a rectangular shot sleeve of length L and height H (justified for initial fill fractions in the range of 40-60% [1]) using the shallow water approximation [3]. In this approximation the flow in the vertical direction is neglected in comparison with the horizontal velocity component. The flow is modeled in two dimensions, with the x axis directed along the direction of motion of the plunger, and the z axis pointing upwards. If viscous forces are omitted, then the flow has only one velocity component, u, along the length of the channel.
The plunger speed in the positive x direction is given by dX/dt=X’(t), where X(t) defines the position of the plunger at time t>0. At the moving surface of the plunger, the velocity is defined as .
As the plunger moves along the length of the channel it sends waves traveling forward along the metal surface. Each wave is associated with a small segment of the metal free surface and the column of metal directly below it (Fig. 1). The location, metal speed and depth in a wave that separates from the surface of the plunger at time t=tp are given by [3]:
(1)
Where
According to Eq. (1), the metal speed, u, and depth, h, in each wave are constant and depend only on the time of the wave separation from the plunger, tp. They both increase with the speed of the plunger X’. Therefore, the first conclusion is that to maintain a monotonic slope of the metal surface in the direction away from the plunger, the latter must not decelerate. If this condition is not satisfied, then there will be waves sloped in both directions. When they reflect off the end of the sleeve and travel back towards the plunger, it creates unfavorable conditions for the evacuation of air from the sleeve and into the die cavity.
Controlling the Waves
Once a wave detaches from the plunger it travels at a constant speed given by
(2)
If the plunger accelerates, then each successive wave will move faster than the waves generated earlier. This will lead to a steepening of the surface slope as the waves travel further down the channel, and can potentialy result in overturning. If the speed of the plunger can be controlled as to limit the wave steepening during the slow shot stage, then the overturning can be avoided.
Let us analyze the evolution of the surface slope between two waves generated at the plunger at close instances, t2 > t1 (Fig. 2) [4]. The slope is given by
(3)
Using expression for x and h in Equation 1, the right-hand side, after linearizing with respect to Dt=t2-t1, can be rewritten as a function of the plunger speed at the time of the wave creation, t=t1, and time t:
(4)
Interestingly, if the plunger moves at a constant speed, i.e. X’’(t1)=0, then the right-hand side of Eq. (4) becomes zero and the slope of the free surface horizontal.
If the plunger accelerates, then the denominator on the right-hand side of Eq. (4) decreases and the slope increases with time. When the denominator reaches zero, the slope becomes vertical. The maximum slope in a wave, amax, is achieved when the wave reaches the end of the shot sleeve at t=tL. This time can be computed from the constant wave speed and the distance it has to travel from the point of its creation at the plunger surface to the end of the sleeve at x=L:
(5)
Replacing t in Eq. (4) with tL and rearranging terms yields
(6)
Equation (6) can now be used to calculate the velocity of the plunger as a function of time that maintains a certain slope of the metal surface during the slow shot stage. For example, if amax is set equal to 10 degrees, then the plunger acceleration given by Eq. (6) ensures that the slope of 10 degrees is not exceeded anywhere and anytime during the motion of the plunger. Note that the plunger velocity given by Eq. (6) is only a function of the initial amount of metal, h0, and the length of the sleeve, L, and not of the metal properties.
Equation (4) can be used to obtain the slope amin of the metal surface right at the plunger by setting t=t1:
(7)
Equation (7) gives the initial surface slope for a wave detaching from the plunger at time t=t1; it is a function of only the plunger’s acceleration and not its position or even velocity. As the wave propagates along the length of the channel it steepens reaching the maximum slope, amax, at the end of the channel at x=L, given by Eq. (6).
Equation (6) give a range of values for the plunger acceleration at any given time
(8)
Two things are achieved when the plunger acceleration stays within this range. Firstly, the slope of the metal surface is directed away from the plunger and towards the opposite end of the shot cylinder, helping to direct the air from the sleeve and into the runner system. Secondly, the slope will not exceed the angle defined by amax at any time during the slow shot process, preventing wave overturning and the entrainment of air in the metal.
Results
Equation (6) is an ordinary differential equation (ODE) that can be easily integrated numerically to obtain the solutions for X(t) and X’(t). The integration is done with respect to t1 using the initial values of the plunger location and speed at t=0: X(0)=0 and X’(0)=0.
Figure 3 shows numerical solutions for the plunger position, X(t), acceleration, X’’(t), and speed X’(t) (the latter is shown as a function of both time and distance along the channel length) for several values of amax. The integration was done for a shot cylinder of length L=0.7 m and height of H=0.1 m and the initial fill fraction of 40%, i.e., h0=0.04 m.
As expected, the plunger motion is slower for smaller values of amax. It takes the plunger 1.66 seconds to get to the end of the shot sleeve for the most conservative case considered with amax=5o, while for amax=90o the time is 0.83 seconds. However, these times will be longer if there is an additional constraint of the plunger velocity not to exceed the critical velocity at which the metal surface reaches the ceiling of the channel at h=H [1]. The critical velocity of the plunger can be derived from the solution for the metal depth h(t,x) given by Eq. (1) [5]:
(9)
and is shown in Fig. 3 by the horizontal dashed line. When the plunger reaches the critical velocity, the metal surface comes in contact with the ceiling of the shot cylinder. Beyond this point the shallow water theory used here becomes invalid and if the plunger continues to accelerate, then the potential for creating an overturning wave increases. It is usually recommended to limit the plunger velocity to the critical value during the slow shot stage.
Validation of the analytical approach using 3-D simulations
Three-dimensional simulations that include all the important physics of the die casting process remain, of course, the best way to validate the designs [4,6]. Simulation with the CFD code FLOW-3D was used to validate some of the predictions of the simplified model. In the simulation, more realistic conditions of viscous flow and a circular channel cross-section are used. Heat transfer and solidification are not included in the model based on the assumption that solidification in the shot sleeve is minimal, if any, and does not significantly affect the flow.
The length of the channel is L=0.7 m, the same as the one used to obtain solutions in Fig. 3. The shot diameter is D=0.1 m, and the initial fluid depth is h0=0.04 m. The velocity of the plunger is defined as a function of time using the solution for the maximum slope of the metal surface of amax=5o, given by Eq. (6) (curve 6 in Figure 3c). Two- and three-dimensional results are shown in Figs. 4 and 5.
There are several aspects of the numerical solution that match the analytical solution quite well. The slope of the wave largely stays within the 5o limit. The circular nature of the channel does not seem to affect much the profile of the free surface in the transverse direction. The critical point, at which the metal surfaces touches the top of the channel, is reached at t=1.37 sec, very close to where curve 6 crosses with the critical velocity on Fig. 3c. The velocity of the plunger at that time is 0.725 m/sec, close to the value of 0.73 m/sec given by Eq. (9) for the rectangular channel. Finally, the first wave arrives at the end of the shot sleeve at t=1.15 sec, while the theory predicts 1.12 sec.
Given the three-dimensional nature of the simulation and the fact that the flow is viscous, the agreement with the analytical solution is quite remarkable. Note that the same initial depth of metal, h0, was used for the round shot sleeve in the simulation and for the rectangular one in the analytical solution. In the present example, with h0=0.04 m, the initial fill fraction in the circular cylinder comes to 37.4%, as opposed to 40% for the rectangular channel. If the same initial fill fraction was used instead, the results of the two solutions would not be so close.
This is not to say that there are no differences in the two solutions. The velocity contours shown in the plane of symmetry in Fig. 5 indicate that one of the main assumptions used to obtain the analytical solution – that all flow variables vary only along the horizontal direction – does not quite hold. Firstly, a viscous boundary layer develops at the bottom of the shot sleeve. Secondly, the numerical results show that flow near the free surface moves faster than the bulk of the metal below it, resulting in a sort of a surge wave. The slope of the metal surface in this wave is about one and half to two times larger than 5o. It reaches the end of the channel at around 1.3 sec and then reflects back. As a result, air maybe entrained in the last stages of the process unless, for example, the reflected surge wave is redirected into the runner system.
Numerical Optimization 3-D Case Study
To overcome the limits of the analytical theory it is possible to perform a numerical optimization. In the present study, two different software were used: the three-dimensional fluid-dynamic solver FLOW-3D and the optimization software IMPROVEit. FLOW-3D is one of the best software for this analysis due to its capabilities to track accurately the free surface of the fluid, to evaluate the amount of air entrained and to manage moving objects coupled with the fluid. IMPROVEit is a numerical optimization technology, multi-parametric and multi-objective. Interacting with several software packages it runs simulations, obtains data and finds the optimal configuration with the lowest possible number of iterations.
The purpose is to find the best velocity profile for the piston during the first phase in order to minimize both the shot runtime and the air entrainment due to turbulence and surface breaking.
Optimization Setup
In collaboration with Bühler, a reference HPDC real machine was chosen to determine the number of parameters necessary for the optimization. The machine allows for twenty velocity data inputs, but for the slow shot there are tipically only four to six data points used, connected linearly to each other. In the present tests, six velocity values at six different locations were used, maintaining the initial and the final position with the possibilty to vary six velocities and four inner positions, with a total of ten design parameters (Figure 6).
To fix an upper limit for the velocity and to prevent from reversed initial run lengths (i.e. 3rd length < 2nd length) the design variables are defined as ratios of other quantities:
- velocity = ratio * maximum velocity (0.0<ratio<1.0)
- run length = ratio * remaining length (0.0<ratio<1.0)
The same length of the shot sleeve (L=0.7 m) as well as the same shot diameter (D=0.1 m) and initial fluid depth (h0=0.04 m) used in the 2-D analytical approach, were chosen to create the 3-D model for the optimization. During the optimization, IMPROVEit managed to set the ten design parameters, to run a CFD simulation, to receive the results of the amount of air entrained as well as the shot runtime and to calculate new and improved input values.
Results
IMPROVEit is not based on any specific algorithm (like gradients or genetic methods) but is a technology which adapts itself to the surface response of an approximate model. The strategy is not constant, but auto-adaptive in order to reach an optimal solution with the lowest possible number of iterations. After only few iterations a Pareto curve is calculated, able to improve significantly the standard setup: in Figure 7 the difference between 500 and 2000 optimization cycles is shown, underlining that the biggest improvements are already achieved within the first simulations. Every simulation is performed fast and within one day it is possible to complete the whole optimization.
Every point of the Pareto curve is an optimal solution and represents the best velocity profile of the piston with certain balances between runtime and air entrained. Note that these objectives relate to each other strongly, thus, the faster the shot is, the more air is entrained.
The most significant aspects noted are the following: firstly, the Pareto curves cover a wide range of values including two robust extremes, particularly important for an optimization process. Secondly, the slow shot runtime is lower than 0.9 sec. in any optimized simulation, about 30% faster than what is commonly used to prevent from surface breaking (runtimes are about 1.1 – 1.5 sec). As a result, any of these solutions allows also to reduce metal cooling and oxidations. Finally, corresponding to the slowest process the associated air entrained is practically null.
Nowadays, obtaining a smooth surface and minimum wave breaking with the slowest possible process, is one of the most important targets. To achieve this, users could choose the rightmost point of the Pareto curve for the setup of their machine. This particular solution is taken as a reference for the following analysis: its velocity profile is shown in Figure 10 and the metal surface simulated with FLOW-3D is shown in Figure 8 with a time sequence from the generation of the first wave until the end of the shot.
Interestingly, the velocity profile found by IMPROVEit is, between all the infinite combinations of accelerations and decelerations, very similar to what is actually set by qualified foundry users based on their experience: an initial slow and long acceleration followed by a constant step and a final steep acceleration. Again, the best result is achieved by a wave that stays next to the piston and rises until the liquid metal touches the ceiling of the cylinder. This requires a shot slow enough to prevent wave toppling ensuring that the air volume will be isolated near the end of the shot sleeve into the runner system.
Comparison between analytical and optimized results
The analytical theory and the optimized solution obtained using a 3-D CFD code, lead to similar conclusions, but only through a more detailed comparison it is possible to understand the accuracy and the limits of the theoretical model.
The analytical solution provides a precise acceleration of the piston for slow shot, comparable with the one calculated by using an optimization tool. As can be seen in Figure 9, the 2-D sliced images from both, the analytical and optimized results, show similar wave profiles where the wave prevents toppling prior to the entering of the runner system. In both cases, the piston controls the wave inside the domain by increasing its steepness without breaking it, rising the metal height at the piston face.
In Figure 10 a similar acceleration trend is shown for both the optimized design and the analytical results for different maximum slope angles from 0 sec to around 0.6 sec, where the liquid metal touches the ceiling of the internal cavity. The leveling off of acceleration happens almost at the same time. After this point, the two velocity profiles differ because the constant critical velocity in theory and the part where it stays constant is somewhat arbitrary. In the optimized velocity profile, instead, the acceleration is first reduced smoothly to delay as much as possible the increasing height of the wave. When the wave touches the top of the cylinder and starts to create a small upper running crest, the speed is increased fast to do not let it fall.
The wave generated by the piston is controlled by IMPROVEit in order to reach high angles of steepness without having the time to break or to touch the top of the geometry. It is followed and caught up by the piston as soon as the equilibrium cannot be maintained in the last time-frames, preventing the formation of big air bubbles. Moreover, the 3-D simulation is able to find the breaking angle of the running wave with a much more accurate and realistic solution, going closer to the physical limits than what can be predicted with simplified 1-D or 2-D theories.
Figure 9: Wave formation in 2-D slice from analytical and optimization case comparison; (a) analytical and (b) using the optimization tool.
Conclusion
The analytical and 3-D optimization results provided solutions that yielded low air entrainment. On one hand, the analytical method calculates a good acceleration curve that conservatively minimizes in most of the cases the amount of air entrained (this method is actually implemented as a simple calculator in FLOW-3D). On the other hand, with a numerical optimization it is possible to determine a more accurate curve that optimizes more than one objective simultaneously and improves the analytical solution in terms of runtime. In further analysis, the optimization tool allows for multiple parameterizations and can further optimize the fill height and the full casting simulation by varying locations and size of runner systems, gates, and/or overflows.
To proceed to further analysis of the Pareto curve for small values of air entrained, a Robust Design Optimization task (RDO) should be performed in order to identify which process time gives the most flexible and reliable solution for minimizing the air.
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