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The effects of heat released during fill on the deflections of die casting dies
Journal of Materials Processing Technology 142 (2003) 648–658
The effects of heat released during fill on the deflections of die casting dies Horacio Ahuett-Garza∗ , R. Allen Miller Department of Industrial Welding and Systems Engineering, The Ohio State University, Columbus, OH 43210, USA Received 29 August 2002; received in revised form 7 April 2003; accepted 15 May 2003
Abstract The quality of a die cast product is determined to a great extent by the mechanism of cavity fill. The evolution of this process has received attention both in industry and literature. Because of its effect on the cost of a numerical simulation, the question of whether or not the filling stage takes place under isothermal conditions needs to be addressed. This article analyzes the process of fill in die casting and presents relations that can be used to predict those conditions under which heat release may be significant. In order to put the relevance of fill conditions in context, the effects of heat released during fill on the prediction of die casting die deflections are introduced and discussed. © 2003 Elsevier B.V. All rights reserved. Keywords: Die casting; Process modeling; Die deflections
1. Introduction In the die casting process, non-ferrous parts of complex geometry are mass produced to near net shape. Aluminum, magnesium, zinc and copper alloys are commonly used in this process. Products range from small valve fittings and housings to large transmission casings. Die castings generally provide structural support of some form and in many cases must meet permeability requirements. Dimensional stability and porosity are two determining factors in the quality of a die casting. Thin walls are a common geometric feature of die castings. To a great extent, the nature of the process is determined by this characteristic. For example, relatively short solidification times, of the order of a second, are usually observed. The thinnest regions of the casting solidify much faster, in as little as a few tenths of a second. To prevent the solidification mechanisms from hindering metal flow into the cavity of a die, short fill times, in the order of tenths of a second, are required. Under these conditions, flow regimes at the gate are in the range of turbulence. Jet and atomized flows are not uncommon and even recommended for certain applications. The use of computer simulation codes to improve the quality of the process has received significant attention in ∗ Corresponding author. Present address: ITESM, Campus Monterrey, Mexico. E-mail address: [email protected] (H. Ahuett-Garza).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00685-X
the industry [1]. However, there are doubts about the ability of commercially available codes to model die casting conditions [2]. In particular, the prediction of fill patterns typically relies on gross simplifications to make the problem tractable. The most common simplification is the so called “instant fill assumption”, which establishes that due to the extremely short fill times encountered in die casting, there is no heat released during the stage of cavity fill. Clearly, this assumption inhibits the ability to predict thermal patterns of the casting at the end of fill as well as problems such as cold shuts. Lately, there has been an interest on the effect that the instant fill assumption has on the prediction of temperatures within the die [3]. In particular, there is a need to establish the conditions under which heat released during fill cannot be ignored. This article introduces relations that predict conditions under which heat released during fill may be significant. In order to put the importance of this issue in context, the effects of heat released during fill on the magnitudes of die casting die deflections are introduced and discussed.
2. Modeling cavity fill in die casting Numerous studies have looked at the process of cavity fill in die casting. Anzai and Uchida [4] carried out numerical simulations as well as experimental verification of filling patterns in flat plate die castings and compared their results with experimental data. Their main goal was to examine
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Nomenclature cp D h H k L n p r1 , r2 S t T u, v, w V x, y, z
specific heat at constant pressure characteristic size of die thermal resistance at the interface die-melting latent heat of solidification thermal conductivity characteristic length of the cavity direction normal to surface pressure difference between liquid and air principal radii of curvature distance from interface to solidification front time temperature velocity components characteristic velocity space coordinates
Greek symbols α thermal diffusivity (κ/ρcp ) θ temperature increment from ambient κ coefficient of thermal expansion µ viscosity ρ density σ surface tension Subscripts c conduction d die (steel) f fill i initial l liquid (casting) s solid (casting)
the influence of gate thickness upon filling patterns. Chen et al. [5] analyzed filling of cavities with cores using a two-dimensional, incompressible, isothermal, transient, free surface fluid flow, which is well suited to simulate water analog experiments. Frayce et al. [6] introduced two different finite element programs for the simulation of filling and solidification in die casting. These programs are independent and no indication is given as to the possibility of simulating non-isothermal flow, although the need to define initial temperatures for modeling solidification indicates that in fact only isothermal fill is handled. Hu et al. [7] performed numerical simulations of the filling flows in die casting and compared them with water analog experimental results. Their theoretical model was designed to predict the distribution of the mass fraction of the injected liquid, as opposed to determining the free surface. A good match is observed between the experimental patterns and the simulation results. The obvious drawback is that no temperature effects are accounted for. Iwata et al. [8] performed two-dimensional simulation of filling flows with two different formulations.
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One of these formulations is similar to Chen’s. In the other case, three-dimensional body flows in two dimensions with a prescribed thickness shape (Poiseuille flow) are developed. By comparing their results with experimental tests they were able to establish the kinematic viscosity of the molten metal. Kappel and Salloum [9] present a general description of the principles that govern the non-isothermal flow in die casting. They propose an approach in which the cavity is laid flat and discretized into small volumes. Their work focuses on the analysis of injection pressures as related to plunger velocities. Lindberg et al. [10] filmed the flow of aluminum into an experimental die. They identify three flow regimes: continuous jet flow, course particle jet flow and atomized jet flow. In spite of the limitation imposed by the filming speed, they were able to visualize the different regimes as well as some of the effects that the entrapped air has upon the flow. Lipinski et al. [11] present the numerical basis of Magmasoft,1 a commercial finite difference solver for the simulation of casting. In general, this software is used for gravity casting with more confidence. In an effort to characterize wear mechanisms in die casting, Venkatesan and Shivpuri [12] modeled the three-dimensional fill of an experimental cavity using Flow3D.2 They utilized water analog reports to fine tune their models. Simulation of fractions of a second of cavity fill required significant CPU time in a supercomputer. This work is interesting because it illustrates the great computational cost of performing filling simulations with any degree of accuracy. Common to all these studies is the realization of the complexity of the task of modeling cavity fill in die casting. In spite of all the simplifying assumptions, the computational expense is quite significant. As discussed in industry forums [2], commercial packages are better suited for the simulation of gravity casting. The basic mathematical formulation of such mechanisms under isothermal, non-turbulent conditions was presented by Eckert [13]. This formulation consists basically of the continuity and momentum equations for fluid flow with boundary conditions typical of die casting, namely, no velocity at the cavity surface (i.e. u = v = w = 0), and at the free surface of the flow: σ =p r1 + r 2 When non-isothermal fill is assumed, the energy equations would need to be solved simultaneously within the domain of the liquid and the die. The overall problem domain includes several interfaces for which equations need to be developed. At the interface solid–liquid within the casting, the energy equation takes the form ρH
1 2
∂S ∂ = k (Tl − Ts ) ∂t ∂n
MAGMA, Gmbh, Aachen, Germany. Flow Science, Inc., Santa Fe, NM.
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while at the liquid/die interfaces it takes the form −k
dT = h(Ts − Td ) dn
(1)
Initial conditions, such as casting temperature and die temperatures, are typically known. Given that the die is used continually, temperature fields within the die structure change as the casting operation continues, and therefore need to be carried from a given casting simulation cycle to establish initial conditions for the next. In view of the process of forced convection during fill, the heat transfer problem can be solved after the fluid flow problem is solved, i.e. these systems are uncoupled. Nevertheless, this is a simplification whose validity depends on the temperature variation in the casting during the filling process, given that the viscosity of the casting varies with the temperature. In the presence of turbulence, corrections must be made to the aforementioned equations. In the classical approach, the time averaging operation: 1 t0 f¯ = f dt t0 0 where f represents the variable in question, is used to decompose the actual flow variables into a mean plus a time-dependent fluctuation. The introduction of this approximation to the previous formulation generates more unknowns. For example, in a two-dimensional flow in a boundary layer the unknowns u v and v T are generated in the momentum and energy equations, giving the following system in two dimensions: ∂u¯ ∂¯v ∂u¯ ∂u ∂ µ ∂u¯ + = 0, u¯ + v¯ = − u v , ∂x ∂y ∂x ∂y ∂y ρ ∂y ∂T ∂T¯ ∂ ∂T ρcp u¯ + v¯ = α − vT ∂x ∂y ∂y ∂y Solution of this system may be obtained only after two more equations are introduced, a process known as turbulence modeling. The previous discussion shows that an accurate description of the filling processes in die casting is not a straight forward task. Modeling cavity fill requires the prediction of the transient, hydrodynamic behavior of a fluid in a regime that often surpasses the onset of turbulence while releasing heat. The fluid itself is a combination of solid particles suspended in a liquid that in some microscopic regions may flow through a partially solidified grid (mushy zone). At a macroscopic level, molten metal must displace air as it flows into the cavity. The air pressure varies as a function of both its temperature and the rate at which it is evacuated from the cavity. Analytical solution of these systems of equations under typical die casting conditions is not an option. The alternative is to look for numerical tools. Even with this approach, the task is extremely challenging. The specific issues that must be faced when trying to obtain a solution are:
• the inherent difficulty in solving the Navier Stokes equations, particularly the problems associated with the nonlinear convective terms of the momentum and energy equations; • the need to provide a time averaging solution of the equations in the turbulent regime; • the need to predict fill fronts simultaneously with other primary variables: velocity, pressures, etc. in a threedimensional space; • the iterative nature of the numerical process needed to solve the transient problem. In view of these problems, it is not surprising that important simplifications are made to make the problems tractable.
3. Heat released during fill and solidification The mechanisms of post-cavity fill heat conduction in die casting have been analyzed in several studies. Barone and Caulk [14,15] summarize the theory that led to the development of DieCAS, a boundary element based computer code with customized utilities for the thermal analysis of the die casting process used by General Motors. Their formulation is based on three simplifying assumptions: the nominal thickness of the casting is small compared with its size, the thermal conductivity of the casting is greater than the conductivity of the die steel, and the casting cycle is short compared with the start up transient in the die. They also establish that the periodic transients associated with the repeated injection cycle penetrate only a short distance into the die. The temperature of the bulk therefore maintains a steady pattern in time. This program predicts thermal patterns in the die and cavity after thermal periodicity in the die structure is reached. In essence, with this approach there is no need to simulate multiple cycles to reach quasi-steady state, which is the normal practice for simulating die casting. Early versions of the software were incapable of predicting heat loss during fill. However, later work by Barone and Kock [3] focused on developing a similar formulation to predict temperatures within the die and the molten metal immediately after fill. Davey and Hinduja [16,17] also developed a BE based program to simulate heat transfer conditions for two different stages of the process, namely, during and after fill. Although they recognize the existence of two temperature regions in the die, their model does not specifically deal with the solution in either one of them as separate entities in the manner presented by Caulk. They recognized the need for accurate data, in particular regarding heat transfer coefficients, in order to match field observations, as results from their analysis had difficulty in matching empirical data. Frayce and Loong [18] describe the difficulties in performing numerical simulation of the die casting process and introduced Prometheus-3D for the prediction of filling patterns during casting. They point out, just as Lipinski does, that for the same number of degrees of freedom, finite
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difference schemes require less computational times than finite element approaches for thermal analysis of the casting process. The work by Granchi et al. [19] represents one of the early attempts at simulating heat transfer within the structure of the die. A finite difference formulation was used for this purpose. While no attempts were made to model a real die, a simple geometry was modeled using the instant fill assumption in order to reduce computational efforts. Also of great relevance are the work by Nelson [20] and Papai and Mobley [21], who made detailed temperature measurements in die casting dies, from which the values of parameters like the heat transfer coefficient at the interface casting-die were calculated. The issue of determining the conditions under which heat released during fill is relevant is now addressed. Following a scale analysis similar to what Eckert [13] proposed, it can be shown that the ratio of ts , the time scale for solidification, to tc , the time scale for conduction within the die, is given by Hm ts = tc cpd Tm−d where Tm−d represents the characteristic temperature difference between die and molten metal. Using values from Table 1, this equation results in ts = 1.9 tc or the time scale for conduction is of about the same order of magnitude as the time scale for solidification. This fact illustrates the point that the heat transfer characteristics of the die do not hinder the solidification process. Instead, die casting belongs in the class of processes whose solidification time is governed by the characteristics of the interface between molten metal and die. Therefore, the amount of heat released during fill depends on the heat transfer characteristics of this interface. Through a scale analysis, the conditions under which heat released during fill is significant can be established to a first order of magnitude. Referring to Fig. 1, maximum heat release occurs at the moment when the stream fills the cavity’s cross-section. Once this occurs, it is reasonable to assume that all heat released by the casting is absorbed by the die through the interface’s finite conductance. If metal solidifies across this thickness prior to the completion of cavity fill, a cold shut will occur. Table 1 Typical process parameters and material properties for aluminum die casting as reported in the literature [20,21] (W/m2
h K) H (aluminum) (J/kg) T (die cavity surface) (◦ C) cpd (steel) (J/kg K) cpm (solid aluminum) (J/kg K) ρm (aluminum) (kg/m3 ) ρd (steel) (kg/m3 ) κ (steel) (m/mK)
20000–66500 39000 200–500 590 200 2570 7600 14.4E−06
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Fig. 1. Schematic illustration of casting-die interface and typical thermal patterns in die structures. Dotted lines represent the middle of the casting wall. Solidification front advances toward center line.
Field observations have shown that a significant amount of solidified material is present as the flow fills the cavity. Similarly, short shots routinely show that a thin layer of solidified metal forms rapidly as the cavity fills. The superheat that is typically applied to the melt prior to casting is greatly reduced by the time the cavity begins to fill, and the dominant source of thermal loads is the latent heat. It follows that, for solidification to occur, the total heat transferred into the die per unit area is given by ρH
∂S = h(Ts − Td ) = hTs−d ∂t
(2)
where S is the length of solidified metal measured from the casting-die interface across the thickness of the part, H the latent heat of the solidifying material and h the thermal resistance of the interface. Solving for S: hTs−d ts dt S= ρH 0 Assuming that the time scale for solidification is defined as the time in which the solidification front reaches the middle of the casting (refer to Fig. 1), then S = 21 where is the thickness of the part3 ts =
ρH 2hTs−d
(3)
the time scale for fill is given by tf =
L V
(4)
3 It is assumed that only half of the total heat released by the part is absorbed by either die half.
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where V is the velocity with which the fluid flows into the cavity and L the length from the gate to the farthest point in the cavity. Combining Eqs. (3) and (4): ts ρH V = tf 2LhTs−d At the interface between solidified part and die, terms on either side of Eq. (1) should be of the same order, or Tl−s k (5) x ≈ |h(Ts − Td )| The literature reports experimental measurements of some of these parameters. Selected values have been summarized in Table 1. For this equation to hold using values from this table, the factors on either side of Eq. (5) are of the order of 12.5E06 W/m2 , and consequently ts ∼ 80 = tf 2 tf It would be expected that heat released during fill would not be significant as long as ts /tf ≥ 10, which requires 1 ≥ tf 4.0
(relation (1))
where is in meters and tf is in seconds. In this case, fill time is at least an order of magnitude faster than the time required for solidification. In other words, for all practical purposes a thermal analysis can assume that the molten metal does not lose heat as it flows into the cavity. Similarly, fill should
proceed without thermally induced problems (such as cold shuts) whenever ts /tf ≥ 1 or 1 ≥ tf 40
(relation (2))
In essence, if solidification takes at least as long as it takes to fill the cavity, the filling process should not be hindered by the mechanisms of heat release. Finally, safe practice would dictate ts /tf ≥ 2 which in turn results in 1 ≥ tf 20
(relation (3))
Relations (1)–(3) are consistent with the field observations reported by Chen [22], who analyzed casting conditions of several good-quality aluminum die castings in an effort to establish rules for the selection of process conditions. Chen understood the importance that fill times have upon the quality of the part, and studied the relationship between this process condition and the average thickness of the part. The data he reported has been reproduced in Fig. 2. Chen fitted this data with a curve that could be used by casting process designers as a guideline for defining fill times. Lines that represent relations (1)–(3) have been superimposed on his data in the same figure. As shown in Fig. 2, relation (2) establishes an upper ideal limit for fill to proceed without heat related problems. Relation (3) provides a conservative value for the maximum fill time that can be allowed without thermal problems. As shown in the figure, in most practical applications fill
Fig. 2. Comparison of results of scale analysis with field data. Field data reported by Chen [27]. Equations were adjusted to account for difference in units.
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proceeds in a regime in which heat release is not negligible, as shown by the line that represents relation (1). Interestingly, almost all cases fill in the range defined by these relations, and the majority fills under conditions that would be considered safe (region between relations (1) and (3)).
4. The relevance of heat released during fill in the context of die deflections Die casting die distortions are caused by the application of thermal loads from the solidification process and by the mechanical stresses produced by the injection pressure and the closing action of the machine. While the prediction of stresses and distortions is easier to track in a simulation than the fluid and heat transfer problems described in a previous section, current simulation practices do not help improve the ability to predict or establish product tolerances. This is the result of a combination of factors: inaccurate models for loads, a limited ability to reproduce accurately the geometry of the casting with a mesh, and a wide range of dimensional and temporal scales where loads and deflections need to be tracked. Due to the need for better dimensional control of castings the prediction and attenuation of die deflections during the casting operation has become an important issue. In particular, computer modeling has received attention as a tool to predict deflections. Barone and Caulk [14,23,24] report the use of DieCAS for the prediction of thermally induced die deflections in transmission casing dies. For this purpose, temperature data from their thermal simulation is transferred to a thermoelastic model of the die. Miller et al. [25,26] developed die deflection simulation techniques based on ABAQUS4 to predict the response of a die under both, thermal loads caused by the solidification process and mechanical loads caused by the injection pressure and the die casting machine’s closing action. Paliani and Brevick [27] used the same code to predict thermally induced deflections in cold chamber shot sleeves. Motivated by the need to predict and control deflections, and save manufacturing costs associated with the removal of excess material in castings, Prince and Ramsey [28] measured slide blowback in large transmission casing dies. They report blowback magnitudes of the order of 1 mm. These studies indicate that thermally induced deflections account for most of the deviations from die dimensions at room temperature. Cavity deflections caused by mechanical loads are typically smaller (about an order of magnitude), except in the case of slide blowback in large dies, which can also reach 1 mm at the moment that the intensification pressure is applied. In addition, the simulation studies have shown that proper modeling of loads becomes a critical issue in order to reduce computational effort while maintaining
4
ABAQUS, Inc., Pawtucket, RI.
653
a certain degree of accuracy in the predictions of simulations. The nature of thermal fields in the die has an important effect on the magnitudes of die deflections. In those conditions in which a significant amount of heat is released by the flowing material, the application of the instant fill assumption will cause an error on the predicted thermal fields in the structure of the die and the casting. Given the expense of predicting flow patterns under non-isothermal conditions, it is necessary to establish the magnitude of the deviation caused by assuming instant fill. In general, heat released during fill is significant whenever ts /tf ∼ = 1. Heat released during fill under those conditions may affect predicted deflection patterns in two ways: • Within the time scale of a single cycle, die deflection patterns may differ significantly from those found under instant fill conditions in the regions in the immediate vicinity of the cavity. • Within the time scale of the whole casting operation, the quasi-steady state-temperature fields in the bulk of the die may be affected, resulting in variations of the deflection patterns and the stiffness of the die. These two statements depict the existence of different time and size scales that characterizes the die casting process. That is, the differences in predicted values can vary from highly localized, temporal conditions to deviations that can span the structure of the die over the longer run. To estimate the effect that heat released during fill may have on die deflections, temperature patterns and their associated die growths need to be analyzed. The magnitude of growth as a function of variations in the thermal fields of the die can be established from the equation: D = κθD
(6)
where D indicates the linear growth due to an increase of temperature θ with respect to some reference temperature, given an initial size of D. Clearly, the sensitivity of thermally induced deflections to variations in temperature magnitudes is simply ∂D = κD ∂θ As already explained, die casting is characterized by the existence of two different regions in which thermal fields in the die vary markedly. Fig. 1 presents a schematic representation of the different thermal regions within the die structure during the casting operation, once quasi-steady state conditions are reached, i.e. after the die has been heated up to normal cyclic operating conditions. The region in the immediate vicinity of the casting is subject to cyclic variations whose period is proportional to the duration of a single cycle. Barone has indicated that the depth of this region is given by depth = 1.5(αt)1/2
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For steel dies (α = 0.04 cm2 /s) and cycle times typical of die casting (i.e. 1 min), this depth is of the order of 0.02 m, or about 5–10% of the total thickness of a typical die half. The temperature variation in this region is roughly inversely proportional to the distance from the surface. Temperatures in the bulk of the die are less sensitive to instantaneous changes in process conditions, and are determined by the heat given up by the part as well as the heat removed by the cooling lines within the structure of the die. The response time in the latter region is comparable to the time it takes to reach quasi-steady state conditions. Under those assumptions, temperatures in these two regions of the die are given by 10x D for 0 ≤ x ≤ , θ + θ 1 − D 10 T = (7) D θ for
(8)
for typical applications reported in the literature, θ ≈ 2θ or 3θ and therefore the sensitivity of the deflections to variations in the thermal fields in the vicinity of the cavity for conditions typical of die casting is, within a 10–15% error: ∂D ∼ = κD ∂θ
(9)
The temperature of the bulk of the die (θ) is necessarily defined by the total heat supplied by the casting process, combined with the heat transfer characteristics of the die (geometry, design of cooling lines). On average, it is unlikely to vary drastically as a result of the redistribution of heat within the casting. On the other hand, thermal fields in the region that surrounds the cavity would be much more sensitive to variations in the temperature of the casting. The term κθD in Eq. (8) accounts for the contribution of the temperature of the bulk of the die to the overall deflections, and can be assumed to be relatively insensitive to small variations in process parameters. The term κθD/20 accounts for the deflections induced by thermal conditions in the vicinity of the cavity, which are known to vary within a casting cycle. Typical values for κ (14.4E−06 m/mK), θ (100–200 ◦ C), D (0.5–1 m), applied to Eq. (9) suggest deflection magnitudes due to thermal growth in the range of 0.7–2.8 mm (0.03–0.110 in.) in the structure of the die, with an increase of 10–15% from these values (anywhere from 0.07 to 0.4 mm more) in the vicinity of the cavity. The previous analysis suggests that normal variations in the temperature fields in the vicinity of the casting (θ)
produce deflections whose magnitudes lie in the range of achievable tolerances in commercial applications of die casting. If a simulation could predict variations of thermal fields in this region, corrective actions could be taken to improve achievable tolerances. In addition, heat released during fill can be expected to affect the nature of the temperature field around the cavity. Consequently, the cost of simulating the evolution of temperatures in this region may be justified if the ability to predict and control tolerances is also improved.
5. A numerical example Results from a thermo-elastic 2D simulation are used to support the conclusions of the previous analysis. The case is based on an exercise designed by Barone and Caulk [6]. While simple in conception, this exercise presents the same difficulties encountered in any die casting model. ABAQUS is designed to simulate structural problems in mechanics, and large deformation processes such as those encountered in metal forming. However, it does not model problems typical of fluid mechanics, and there is no interface with systems that model casting. Consequently, simplifications need to be made when developing the models for loads and boundary conditions. Fig. 3 summarizes the dimensions of the idealized twodimensional die, while Table 2 presents the conditions under which the simulation is conducted. The geometry represents a small die, and the process conditions are typical of die casting. The cavity is rectangular in shape and located at the center of the die. The filling stage is not modeled explicitly. In one case, instant fill conditions are assumed and therefore the initial temperature of the casting is uniform. The casting is allowed to release heat, including latent heat, for 15 s, through a typical interface conductance of 10,000 W/m2 k, the same value used by Barone and Caulk. After the solidification period, heat is removed from the die in the cavity region to represent the application of lubricant spray. Non-instant fill conditions are modeled with the same geometry and processing conditions in order to compare with the previous case. In this model, heat released during fill is accounted for by assuming a redistribution of the heat within the casting. At the beginning of the casting, the heat content of the part is the same as the instant fill case. However, the volume of the casting that is assumed to travel the whole Table 2 Process conditions for simulation exercise Casting material Casting thickness Tl , Ts cpm (liquid aluminum) Tinjection (instant fill) Heat removed by spray TA (non-instant) TB (non-instant)
Aluminum 0.127 mm (0.005 in.) 580 ◦ C, 520 ◦ C 963 J/kg K 630 ◦ C 625E03 J/s m3 for 3 s (20% of input heat) 830 ◦ C 630 ◦ C
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655
Fig. 3. Model for testing the effect of instant fill vs. non-instant fill conditions. Description of geometry and conditions. Dimensions in meters.
length of the cavity looses about 50% of the latent heat. This heat is added to the volume that represents the region that ends up at the gate, in such a way that the die in this region absorbs more heat. The initial heat content of the casting varies linearly within these extremes. The middle of the casting is in the same conditions in both cases. At points along the casting that gain heat, their initial temperature is defined by Ti = 630 +
F cp
where F is the fraction of heat to be added from the corresponding volume in the casting. The portion of the material that travels the length of the cavity is therefore half-solidified at the time it reaches the farthest point from the gate. This condition roughly represents the limiting line defined by relation (2) in Fig. 2. Again, this model does not represent the filling process. Instead, the model only tries to represent the heat load that the die would be subject under non-isothermal fill conditions. Other conditions such as cycle time and heat removal due to lubricant spray were the same as in the isothermal fill case. The maximum temperature difference at any node between consecutive heat cycles was recorded for both cases. Fig. 4 presents the trend in the variation of temperatures between consecutive cycles for the instant fill case. The non-instant fill case results behaved in a very similar manner. After 15 cycles the maximum difference was down to about 5 ◦ C, or roughly 1% of the maximum temperatures in the structure of the die. At this point, it was assumed that quasi-steady state conditions had been reached. Nodal temperatures were transferred to a thermoelastic deflection model to estimate the magnitudes of the deflections. An interesting point is the fact that a trend line predicts that, strictly speaking, periodicity conditions would be reached after approximately 275 cycles for both cases. In their work with a similar model, Barone and Caulk report that these conditions are reached after about 300 cycles. This raises the
issue of whether or not it is feasible to reach quasi steady state conditions in the field. This exercise suggests that, in the field, the casting cycle is likely to occur within a range of conditions and not at fixed values of temperature. Maximum temperatures during a casting cycle reach 415 ◦ C in the instant fill case and 466 ◦ C in the non-instant fill case. Maximum deflections reach 0.57 mm (0.022 in.) and 0.58 mm (0.023 in.) from dimensions at room temperature, respectively. Fig. 5 shows the differences between both cases. The maximum difference in temperatures for these cases ranges from +54 to −49 ◦ C, with the hottest location being on the gate side, and the coldest point being at the end of the cavity, both on the cavity surface, non-instant fill case. As expected, about 80% of the die structure maintains a temperature that lies within approximately ±7 ◦ C in both cases. The maximum difference in deflections between the instant fill and non-instant fill cases lies within a 0.08 mm (0.003 in.) range. This difference is about equal in magnitude
Fig. 4. Rate of convergence of temperature profiles towards periodicity conditions. Maximum difference in temperature between consecutive cycles, instant fill case.
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Fig. 5. Comparison of results, instant fill vs. non-instant fill models: (a) difference in temperatures (◦ C); (b) difference in magnitudes of deflections along the vertical direction (m).
and opposite in direction about the symmetry plane (about ±0.04 mm or ±0.0015 in., taking as reference the instant fill case). These deflection magnitudes are consistent with the analysis presented in the previous section, specifically with what Eqs. (8) and (9) indicate. It is interesting to note that the magnitude of the variation between both cases lies in the range of the resolution of typical simulation capabilities, i.e. experience has shown that deflection patterns predicted by a simulation for commercial applications are accurate within ±0.05 mm (0.002 in.). The decision to include or not heat released during fill in the
analysis may thus account for a significant part of the error of a typical simulation. Also, as predicted by Eqs. (8) and (9) the differences in results from both cases are in the range of manufacturing/assembly allowances for details of the cavity in commercial applications. The results suggest that the prediction of heat released during fill might have only marginal effects on the quality of the simulation results for average filling conditions (fill times recommended by Chen) or when cavities fill rapidly. These situations correspond to the region delimited by relations (1) and (3) (see Fig. 2). The expense of predicting the true
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evolution of temperature as the cavity fills is probably not justified under those conditions. However, for those applications in which the filling regime lies in the region delimited by relations (2) and (3) errors in the predictions of the magnitudes of cavity deflections caused by the instant fill assumption are relevant, about the same size of achievable tolerances, and consequently the cost of modeling this phenomena may be justified. One of the trends in the industry is towards higher precision applications, and tight tolerance control is desirable. Such applications represent an ideal situation in which tracking heat release during fill may be relevant. The results also suggest that the maximum differences in temperature predicted by these models within the part and the structure of the die in the near vicinity of the casting are indeed significant for the same conditions (delimited by relations (2) and (3)). In this case, heat released during fill can result in thermal variations that may affect design decisions or problem diagnostics. As an example, variations in thermal patterns may have an impact on the ability to predict soldering. In this context, the need to establish accurate thermal fields may justify the cost of predicting heat release during fill.
6. Conclusions This article analyzed the process of die fill and introduced relations that establish the conditions under which heat released during this process is significant. These relations are given in terms of the thickness of the casting and the fill times. Comparison with field data indicates that under typical casting conditions the molten metal does release heat as the cavity fills. The analysis of this work suggests that, for average commercial applications, the magnitude of the deviations caused by assuming instant fill conditions is relatively small in the context of die deflections. This is due to the fact that deflections depend much more heavily on the temperature of the bulk of the die, which is fairly insensitive to the redistribution of heat within the casting. On the other hand, the prediction of thermal fields in the presence of significant heat release is relevant because the induced variations in deflections are of the same order of magnitude as achievable tolerances in die castings. Field data indicates that such filling conditions are not uncommon. The relations introduced by this work can be used to identify these conditions. Finally, simulation results suggest that the prediction of heat release is indeed relevant in the context of thermal behavior of the die. The nature of the thermal fields in the structure of the die has an important effect on the quality of the casting operation. The cost of predicting thermal patterns when significant heat release during fill occurs may be justified by an improvement in the quality of the casting operation.
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As the industry evolves towards high precision applications, a trend that is being observed in competing industries and which requires very accurate-dimensional control of mass produced components, modeling of heat released during fill should become more relevant. Accurate prediction of die deflections can help control dimensional variability of the cavity and, in turn, improve achievable tolerances in die casting.
Acknowledgements This work was done as part of a research program funded by the US Department of Energy. Close monitoring and guidance was provided by the North American Die Casting Association, through its Computer Modeling Task Group.
References [1] R.L. Smith, D.E. Phenicie, A.K. Agarwal, H.L. Kallien, Optimizing production of a die casting by numerical simulation of die filling and solidification, Die Cast. Eng. 37 (1993) 40–48. [2] D. Caulk, Keynote Speech: Opportunities and Challenges in Die Casting Analysis, NADCA Flow Modeling Thermal Simulation for Die Casting Forum, Chicago, IL, 1996. [3] M.R. Barone, E. Kock, A method for analysing the effect of flow on heat transfer in die casting, Int. J. Numer. Meth. Heat Fluid Flow 3 (1993) 457–472. [4] K. Anzai, T. Uchida, Mold filling patterns of flat plate die castings, in: Transactions of the Modeling of Casting, Welding and Advanced Solidification Processes V, 1991. [5] S. Chen, D.B. Johnson, P.E. Raad, Simulated filling of dies with cores, in: Transactions of the NADCA Congress and Exposition, Detroit, MI, 1991. [6] D. Frayce, J.F. Hétu, C.A. Loong, Numerical modeling of filling and solidification in die casting, in: Transactions of the NADCA International Congress and Exposition, Cleveland, OH, 1993. [7] J. Hu, S. Ramlingam, G. Meyerson, E.R.G. Eckert, R.J. Goldstein, Experiment and computer modelling of the filling flows in pressure die casting, Comput. Eng. 1 (1992) 401–410. [8] Y. Iwata, Y. Yamamoto, K. Yonekura, Y. Kagami, Computer simulation of molten metal flow in thin plate die castings, in: Transactions of the NADCA Congress and Exposition, Detroit, MI, 1991. [9] V.V. Kappel, G. Salloum, Modelling of the metal flow in the die casting process, in: Transactions of the 15th NADCA International Die Casting Congress and Exposition, St. Louis, MO, 1989. [10] B. Lindberg, P.N. Hansen, S.F. Hansen, High speed filming of mold filling process of al alloys in high pressure die casting, in: Modeling of Casting, Welding and Advanced Solidification Processes V, 1991. [11] M. Lipinski, W. Schaefer, S. Andersen, Modeling of combined heat and fluid flow for determination of filling sequence for real complex shaped castings, in: Modeling of Casting, Welding and Advanced Solidification Processes V, 1991. [12] K. Venkatesan, R. Shivpuri, Numerical simulation of die cavity filling in die castings and an evaluation of process parameters on die wear, in: Transactions of the NADCA International Die Casting Congress and Exposition, Cleveland, OH, 1993. [13] E.R.G. Eckert, Similarity analysis applied to the die casting process, ASME J. Eng. Mater. Technol. 111 (1989) 393–398. [14] M.R. Barone, D.A. Caulk, A new method for thermal analysis of die casting, Trans. ASME, J. Heat Transfer 115 (1993) 284–293.
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[15] D.A. Caulk, A method for analyzing heat conduction with high frequency periodic boundary conditions, Trans. ASME, J. Heat Transfer 112 (1990) 280–287. [16] K. Davey, S. Hinduja, Modeling the transient thermal behavior of the pressure die-casting process with the BEM, Appl. Math. Model. 14 (1990) 394–409. [17] K. Davey, S. Hinduja, Modeling of the pressure die casting process with the boundary element method: steady state approximation, Int. J. Numer. Meth. Eng. 30 (1990) 1275–1299. [18] D. Frayce, C.A. Loong, Mathematical modeling of the die casting process, in: Light Metals, 1991. [19] M. Granchi, E. Vettori, E. Cerini, R. Remorini, Computer thermal analysis of die casting dies, in: Transactions of the 12th SDCE International Die Casting Congress and Exposition, Minneapolis, MN, 1983. [20] C.W. Nelson, Nature of heat transfer at the die face, in: Transactions of the Sixth SDCE International Die Casting Congress, Cleveland, OH, 1970. [21] J. Papai, C. Mobley Die thermal fields and heat fluxes during die casting of 380 aluminum alloy in H-13 steels, in: Transactions of the NADCA Congress and Exposition, Detroit, MI, 1991. [22] P. Chen, Relationships between die cavity fill time and die casting average thickness, surface area, and outside diameter, in: Transactions of the NADCA Congress and Exposition, Detroit, MI, 1991.
[23] M.R. Barone, D.A. Caulk, dieCAS—thermal analysis software for die casting: modeling approach, in: Transactions of the NADCA International Congress and Exposition, Cleveland, OH, 1993. [24] M.R. Barone, D.A. Caulk, D.E. Siefker, dieCAS—thermal analysis software for die casting: results and application, in: Transactions of the NADCA International Die Casting Congress and Exposition, Cleveland, OH, 1993. [25] H. Ahuett-Garza, K. Hegde, R.A. Miller, G. Padiyar, NADCA sponsored research: FEM analysis and simulation of die deflections, Part II, Results, in: Transactions of the NADCA International Die Casting Congress. [26] K. Hegde, H. Ahuett-Garza, R.A. Miller, G. Padiyar, NADCA sponsored research: FEM analysis and simulation of die deflections, Part I, Modeling and simulation, in: Transactions of the NADCA International Die Casting Congress and Exposition, Indianapolis, IN, USA, 1995. [27] P.A. Paliani, J.R. Brevick, Thermoelastic Deformation of Cold Chamber Die Casting Machine Shot Sleeves, Engineering Research Center for Net Shape Manufacturing, The Ohio State University ERC/NSM-C-93-24, July 1993. [28] G.A. Prince, C.R. Ramsey, Introduction to die movement and distortion: a case study, in: Transactions of the NADCA International Die Casting Congress and Exposition, Indianapolis, IN, 1995.
The effects of heat released during fill on the deflections of die casting dies Horacio Ahuett-Garza∗ , R. Allen Miller Department of Industrial Welding and Systems Engineering, The Ohio State University, Columbus, OH 43210, USA Received 29 August 2002; received in revised form 7 April 2003; accepted 15 May 2003
Abstract The quality of a die cast product is determined to a great extent by the mechanism of cavity fill. The evolution of this process has received attention both in industry and literature. Because of its effect on the cost of a numerical simulation, the question of whether or not the filling stage takes place under isothermal conditions needs to be addressed. This article analyzes the process of fill in die casting and presents relations that can be used to predict those conditions under which heat release may be significant. In order to put the relevance of fill conditions in context, the effects of heat released during fill on the prediction of die casting die deflections are introduced and discussed. © 2003 Elsevier B.V. All rights reserved. Keywords: Die casting; Process modeling; Die deflections
1. Introduction In the die casting process, non-ferrous parts of complex geometry are mass produced to near net shape. Aluminum, magnesium, zinc and copper alloys are commonly used in this process. Products range from small valve fittings and housings to large transmission casings. Die castings generally provide structural support of some form and in many cases must meet permeability requirements. Dimensional stability and porosity are two determining factors in the quality of a die casting. Thin walls are a common geometric feature of die castings. To a great extent, the nature of the process is determined by this characteristic. For example, relatively short solidification times, of the order of a second, are usually observed. The thinnest regions of the casting solidify much faster, in as little as a few tenths of a second. To prevent the solidification mechanisms from hindering metal flow into the cavity of a die, short fill times, in the order of tenths of a second, are required. Under these conditions, flow regimes at the gate are in the range of turbulence. Jet and atomized flows are not uncommon and even recommended for certain applications. The use of computer simulation codes to improve the quality of the process has received significant attention in ∗ Corresponding author. Present address: ITESM, Campus Monterrey, Mexico. E-mail address: [email protected] (H. Ahuett-Garza).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00685-X
the industry [1]. However, there are doubts about the ability of commercially available codes to model die casting conditions [2]. In particular, the prediction of fill patterns typically relies on gross simplifications to make the problem tractable. The most common simplification is the so called “instant fill assumption”, which establishes that due to the extremely short fill times encountered in die casting, there is no heat released during the stage of cavity fill. Clearly, this assumption inhibits the ability to predict thermal patterns of the casting at the end of fill as well as problems such as cold shuts. Lately, there has been an interest on the effect that the instant fill assumption has on the prediction of temperatures within the die [3]. In particular, there is a need to establish the conditions under which heat released during fill cannot be ignored. This article introduces relations that predict conditions under which heat released during fill may be significant. In order to put the importance of this issue in context, the effects of heat released during fill on the magnitudes of die casting die deflections are introduced and discussed.
2. Modeling cavity fill in die casting Numerous studies have looked at the process of cavity fill in die casting. Anzai and Uchida [4] carried out numerical simulations as well as experimental verification of filling patterns in flat plate die castings and compared their results with experimental data. Their main goal was to examine
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Nomenclature cp D h H k L n p r1 , r2 S t T u, v, w V x, y, z
specific heat at constant pressure characteristic size of die thermal resistance at the interface die-melting latent heat of solidification thermal conductivity characteristic length of the cavity direction normal to surface pressure difference between liquid and air principal radii of curvature distance from interface to solidification front time temperature velocity components characteristic velocity space coordinates
Greek symbols α thermal diffusivity (κ/ρcp ) θ temperature increment from ambient κ coefficient of thermal expansion µ viscosity ρ density σ surface tension Subscripts c conduction d die (steel) f fill i initial l liquid (casting) s solid (casting)
the influence of gate thickness upon filling patterns. Chen et al. [5] analyzed filling of cavities with cores using a two-dimensional, incompressible, isothermal, transient, free surface fluid flow, which is well suited to simulate water analog experiments. Frayce et al. [6] introduced two different finite element programs for the simulation of filling and solidification in die casting. These programs are independent and no indication is given as to the possibility of simulating non-isothermal flow, although the need to define initial temperatures for modeling solidification indicates that in fact only isothermal fill is handled. Hu et al. [7] performed numerical simulations of the filling flows in die casting and compared them with water analog experimental results. Their theoretical model was designed to predict the distribution of the mass fraction of the injected liquid, as opposed to determining the free surface. A good match is observed between the experimental patterns and the simulation results. The obvious drawback is that no temperature effects are accounted for. Iwata et al. [8] performed two-dimensional simulation of filling flows with two different formulations.
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One of these formulations is similar to Chen’s. In the other case, three-dimensional body flows in two dimensions with a prescribed thickness shape (Poiseuille flow) are developed. By comparing their results with experimental tests they were able to establish the kinematic viscosity of the molten metal. Kappel and Salloum [9] present a general description of the principles that govern the non-isothermal flow in die casting. They propose an approach in which the cavity is laid flat and discretized into small volumes. Their work focuses on the analysis of injection pressures as related to plunger velocities. Lindberg et al. [10] filmed the flow of aluminum into an experimental die. They identify three flow regimes: continuous jet flow, course particle jet flow and atomized jet flow. In spite of the limitation imposed by the filming speed, they were able to visualize the different regimes as well as some of the effects that the entrapped air has upon the flow. Lipinski et al. [11] present the numerical basis of Magmasoft,1 a commercial finite difference solver for the simulation of casting. In general, this software is used for gravity casting with more confidence. In an effort to characterize wear mechanisms in die casting, Venkatesan and Shivpuri [12] modeled the three-dimensional fill of an experimental cavity using Flow3D.2 They utilized water analog reports to fine tune their models. Simulation of fractions of a second of cavity fill required significant CPU time in a supercomputer. This work is interesting because it illustrates the great computational cost of performing filling simulations with any degree of accuracy. Common to all these studies is the realization of the complexity of the task of modeling cavity fill in die casting. In spite of all the simplifying assumptions, the computational expense is quite significant. As discussed in industry forums [2], commercial packages are better suited for the simulation of gravity casting. The basic mathematical formulation of such mechanisms under isothermal, non-turbulent conditions was presented by Eckert [13]. This formulation consists basically of the continuity and momentum equations for fluid flow with boundary conditions typical of die casting, namely, no velocity at the cavity surface (i.e. u = v = w = 0), and at the free surface of the flow: σ =p r1 + r 2 When non-isothermal fill is assumed, the energy equations would need to be solved simultaneously within the domain of the liquid and the die. The overall problem domain includes several interfaces for which equations need to be developed. At the interface solid–liquid within the casting, the energy equation takes the form ρH
1 2
∂S ∂ = k (Tl − Ts ) ∂t ∂n
MAGMA, Gmbh, Aachen, Germany. Flow Science, Inc., Santa Fe, NM.
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while at the liquid/die interfaces it takes the form −k
dT = h(Ts − Td ) dn
(1)
Initial conditions, such as casting temperature and die temperatures, are typically known. Given that the die is used continually, temperature fields within the die structure change as the casting operation continues, and therefore need to be carried from a given casting simulation cycle to establish initial conditions for the next. In view of the process of forced convection during fill, the heat transfer problem can be solved after the fluid flow problem is solved, i.e. these systems are uncoupled. Nevertheless, this is a simplification whose validity depends on the temperature variation in the casting during the filling process, given that the viscosity of the casting varies with the temperature. In the presence of turbulence, corrections must be made to the aforementioned equations. In the classical approach, the time averaging operation: 1 t0 f¯ = f dt t0 0 where f represents the variable in question, is used to decompose the actual flow variables into a mean plus a time-dependent fluctuation. The introduction of this approximation to the previous formulation generates more unknowns. For example, in a two-dimensional flow in a boundary layer the unknowns u v and v T are generated in the momentum and energy equations, giving the following system in two dimensions: ∂u¯ ∂¯v ∂u¯ ∂u ∂ µ ∂u¯ + = 0, u¯ + v¯ = − u v , ∂x ∂y ∂x ∂y ∂y ρ ∂y ∂T ∂T¯ ∂ ∂T ρcp u¯ + v¯ = α − vT ∂x ∂y ∂y ∂y Solution of this system may be obtained only after two more equations are introduced, a process known as turbulence modeling. The previous discussion shows that an accurate description of the filling processes in die casting is not a straight forward task. Modeling cavity fill requires the prediction of the transient, hydrodynamic behavior of a fluid in a regime that often surpasses the onset of turbulence while releasing heat. The fluid itself is a combination of solid particles suspended in a liquid that in some microscopic regions may flow through a partially solidified grid (mushy zone). At a macroscopic level, molten metal must displace air as it flows into the cavity. The air pressure varies as a function of both its temperature and the rate at which it is evacuated from the cavity. Analytical solution of these systems of equations under typical die casting conditions is not an option. The alternative is to look for numerical tools. Even with this approach, the task is extremely challenging. The specific issues that must be faced when trying to obtain a solution are:
• the inherent difficulty in solving the Navier Stokes equations, particularly the problems associated with the nonlinear convective terms of the momentum and energy equations; • the need to provide a time averaging solution of the equations in the turbulent regime; • the need to predict fill fronts simultaneously with other primary variables: velocity, pressures, etc. in a threedimensional space; • the iterative nature of the numerical process needed to solve the transient problem. In view of these problems, it is not surprising that important simplifications are made to make the problems tractable.
3. Heat released during fill and solidification The mechanisms of post-cavity fill heat conduction in die casting have been analyzed in several studies. Barone and Caulk [14,15] summarize the theory that led to the development of DieCAS, a boundary element based computer code with customized utilities for the thermal analysis of the die casting process used by General Motors. Their formulation is based on three simplifying assumptions: the nominal thickness of the casting is small compared with its size, the thermal conductivity of the casting is greater than the conductivity of the die steel, and the casting cycle is short compared with the start up transient in the die. They also establish that the periodic transients associated with the repeated injection cycle penetrate only a short distance into the die. The temperature of the bulk therefore maintains a steady pattern in time. This program predicts thermal patterns in the die and cavity after thermal periodicity in the die structure is reached. In essence, with this approach there is no need to simulate multiple cycles to reach quasi-steady state, which is the normal practice for simulating die casting. Early versions of the software were incapable of predicting heat loss during fill. However, later work by Barone and Kock [3] focused on developing a similar formulation to predict temperatures within the die and the molten metal immediately after fill. Davey and Hinduja [16,17] also developed a BE based program to simulate heat transfer conditions for two different stages of the process, namely, during and after fill. Although they recognize the existence of two temperature regions in the die, their model does not specifically deal with the solution in either one of them as separate entities in the manner presented by Caulk. They recognized the need for accurate data, in particular regarding heat transfer coefficients, in order to match field observations, as results from their analysis had difficulty in matching empirical data. Frayce and Loong [18] describe the difficulties in performing numerical simulation of the die casting process and introduced Prometheus-3D for the prediction of filling patterns during casting. They point out, just as Lipinski does, that for the same number of degrees of freedom, finite
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difference schemes require less computational times than finite element approaches for thermal analysis of the casting process. The work by Granchi et al. [19] represents one of the early attempts at simulating heat transfer within the structure of the die. A finite difference formulation was used for this purpose. While no attempts were made to model a real die, a simple geometry was modeled using the instant fill assumption in order to reduce computational efforts. Also of great relevance are the work by Nelson [20] and Papai and Mobley [21], who made detailed temperature measurements in die casting dies, from which the values of parameters like the heat transfer coefficient at the interface casting-die were calculated. The issue of determining the conditions under which heat released during fill is relevant is now addressed. Following a scale analysis similar to what Eckert [13] proposed, it can be shown that the ratio of ts , the time scale for solidification, to tc , the time scale for conduction within the die, is given by Hm ts = tc cpd Tm−d where Tm−d represents the characteristic temperature difference between die and molten metal. Using values from Table 1, this equation results in ts = 1.9 tc or the time scale for conduction is of about the same order of magnitude as the time scale for solidification. This fact illustrates the point that the heat transfer characteristics of the die do not hinder the solidification process. Instead, die casting belongs in the class of processes whose solidification time is governed by the characteristics of the interface between molten metal and die. Therefore, the amount of heat released during fill depends on the heat transfer characteristics of this interface. Through a scale analysis, the conditions under which heat released during fill is significant can be established to a first order of magnitude. Referring to Fig. 1, maximum heat release occurs at the moment when the stream fills the cavity’s cross-section. Once this occurs, it is reasonable to assume that all heat released by the casting is absorbed by the die through the interface’s finite conductance. If metal solidifies across this thickness prior to the completion of cavity fill, a cold shut will occur. Table 1 Typical process parameters and material properties for aluminum die casting as reported in the literature [20,21] (W/m2
h K) H (aluminum) (J/kg) T (die cavity surface) (◦ C) cpd (steel) (J/kg K) cpm (solid aluminum) (J/kg K) ρm (aluminum) (kg/m3 ) ρd (steel) (kg/m3 ) κ (steel) (m/mK)
20000–66500 39000 200–500 590 200 2570 7600 14.4E−06
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Fig. 1. Schematic illustration of casting-die interface and typical thermal patterns in die structures. Dotted lines represent the middle of the casting wall. Solidification front advances toward center line.
Field observations have shown that a significant amount of solidified material is present as the flow fills the cavity. Similarly, short shots routinely show that a thin layer of solidified metal forms rapidly as the cavity fills. The superheat that is typically applied to the melt prior to casting is greatly reduced by the time the cavity begins to fill, and the dominant source of thermal loads is the latent heat. It follows that, for solidification to occur, the total heat transferred into the die per unit area is given by ρH
∂S = h(Ts − Td ) = hTs−d ∂t
(2)
where S is the length of solidified metal measured from the casting-die interface across the thickness of the part, H the latent heat of the solidifying material and h the thermal resistance of the interface. Solving for S: hTs−d ts dt S= ρH 0 Assuming that the time scale for solidification is defined as the time in which the solidification front reaches the middle of the casting (refer to Fig. 1), then S = 21 where is the thickness of the part3 ts =
ρH 2hTs−d
(3)
the time scale for fill is given by tf =
L V
(4)
3 It is assumed that only half of the total heat released by the part is absorbed by either die half.
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where V is the velocity with which the fluid flows into the cavity and L the length from the gate to the farthest point in the cavity. Combining Eqs. (3) and (4): ts ρH V = tf 2LhTs−d At the interface between solidified part and die, terms on either side of Eq. (1) should be of the same order, or Tl−s k (5) x ≈ |h(Ts − Td )| The literature reports experimental measurements of some of these parameters. Selected values have been summarized in Table 1. For this equation to hold using values from this table, the factors on either side of Eq. (5) are of the order of 12.5E06 W/m2 , and consequently ts ∼ 80 = tf 2 tf It would be expected that heat released during fill would not be significant as long as ts /tf ≥ 10, which requires 1 ≥ tf 4.0
(relation (1))
where is in meters and tf is in seconds. In this case, fill time is at least an order of magnitude faster than the time required for solidification. In other words, for all practical purposes a thermal analysis can assume that the molten metal does not lose heat as it flows into the cavity. Similarly, fill should
proceed without thermally induced problems (such as cold shuts) whenever ts /tf ≥ 1 or 1 ≥ tf 40
(relation (2))
In essence, if solidification takes at least as long as it takes to fill the cavity, the filling process should not be hindered by the mechanisms of heat release. Finally, safe practice would dictate ts /tf ≥ 2 which in turn results in 1 ≥ tf 20
(relation (3))
Relations (1)–(3) are consistent with the field observations reported by Chen [22], who analyzed casting conditions of several good-quality aluminum die castings in an effort to establish rules for the selection of process conditions. Chen understood the importance that fill times have upon the quality of the part, and studied the relationship between this process condition and the average thickness of the part. The data he reported has been reproduced in Fig. 2. Chen fitted this data with a curve that could be used by casting process designers as a guideline for defining fill times. Lines that represent relations (1)–(3) have been superimposed on his data in the same figure. As shown in Fig. 2, relation (2) establishes an upper ideal limit for fill to proceed without heat related problems. Relation (3) provides a conservative value for the maximum fill time that can be allowed without thermal problems. As shown in the figure, in most practical applications fill
Fig. 2. Comparison of results of scale analysis with field data. Field data reported by Chen [27]. Equations were adjusted to account for difference in units.
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proceeds in a regime in which heat release is not negligible, as shown by the line that represents relation (1). Interestingly, almost all cases fill in the range defined by these relations, and the majority fills under conditions that would be considered safe (region between relations (1) and (3)).
4. The relevance of heat released during fill in the context of die deflections Die casting die distortions are caused by the application of thermal loads from the solidification process and by the mechanical stresses produced by the injection pressure and the closing action of the machine. While the prediction of stresses and distortions is easier to track in a simulation than the fluid and heat transfer problems described in a previous section, current simulation practices do not help improve the ability to predict or establish product tolerances. This is the result of a combination of factors: inaccurate models for loads, a limited ability to reproduce accurately the geometry of the casting with a mesh, and a wide range of dimensional and temporal scales where loads and deflections need to be tracked. Due to the need for better dimensional control of castings the prediction and attenuation of die deflections during the casting operation has become an important issue. In particular, computer modeling has received attention as a tool to predict deflections. Barone and Caulk [14,23,24] report the use of DieCAS for the prediction of thermally induced die deflections in transmission casing dies. For this purpose, temperature data from their thermal simulation is transferred to a thermoelastic model of the die. Miller et al. [25,26] developed die deflection simulation techniques based on ABAQUS4 to predict the response of a die under both, thermal loads caused by the solidification process and mechanical loads caused by the injection pressure and the die casting machine’s closing action. Paliani and Brevick [27] used the same code to predict thermally induced deflections in cold chamber shot sleeves. Motivated by the need to predict and control deflections, and save manufacturing costs associated with the removal of excess material in castings, Prince and Ramsey [28] measured slide blowback in large transmission casing dies. They report blowback magnitudes of the order of 1 mm. These studies indicate that thermally induced deflections account for most of the deviations from die dimensions at room temperature. Cavity deflections caused by mechanical loads are typically smaller (about an order of magnitude), except in the case of slide blowback in large dies, which can also reach 1 mm at the moment that the intensification pressure is applied. In addition, the simulation studies have shown that proper modeling of loads becomes a critical issue in order to reduce computational effort while maintaining
4
ABAQUS, Inc., Pawtucket, RI.
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a certain degree of accuracy in the predictions of simulations. The nature of thermal fields in the die has an important effect on the magnitudes of die deflections. In those conditions in which a significant amount of heat is released by the flowing material, the application of the instant fill assumption will cause an error on the predicted thermal fields in the structure of the die and the casting. Given the expense of predicting flow patterns under non-isothermal conditions, it is necessary to establish the magnitude of the deviation caused by assuming instant fill. In general, heat released during fill is significant whenever ts /tf ∼ = 1. Heat released during fill under those conditions may affect predicted deflection patterns in two ways: • Within the time scale of a single cycle, die deflection patterns may differ significantly from those found under instant fill conditions in the regions in the immediate vicinity of the cavity. • Within the time scale of the whole casting operation, the quasi-steady state-temperature fields in the bulk of the die may be affected, resulting in variations of the deflection patterns and the stiffness of the die. These two statements depict the existence of different time and size scales that characterizes the die casting process. That is, the differences in predicted values can vary from highly localized, temporal conditions to deviations that can span the structure of the die over the longer run. To estimate the effect that heat released during fill may have on die deflections, temperature patterns and their associated die growths need to be analyzed. The magnitude of growth as a function of variations in the thermal fields of the die can be established from the equation: D = κθD
(6)
where D indicates the linear growth due to an increase of temperature θ with respect to some reference temperature, given an initial size of D. Clearly, the sensitivity of thermally induced deflections to variations in temperature magnitudes is simply ∂D = κD ∂θ As already explained, die casting is characterized by the existence of two different regions in which thermal fields in the die vary markedly. Fig. 1 presents a schematic representation of the different thermal regions within the die structure during the casting operation, once quasi-steady state conditions are reached, i.e. after the die has been heated up to normal cyclic operating conditions. The region in the immediate vicinity of the casting is subject to cyclic variations whose period is proportional to the duration of a single cycle. Barone has indicated that the depth of this region is given by depth = 1.5(αt)1/2
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For steel dies (α = 0.04 cm2 /s) and cycle times typical of die casting (i.e. 1 min), this depth is of the order of 0.02 m, or about 5–10% of the total thickness of a typical die half. The temperature variation in this region is roughly inversely proportional to the distance from the surface. Temperatures in the bulk of the die are less sensitive to instantaneous changes in process conditions, and are determined by the heat given up by the part as well as the heat removed by the cooling lines within the structure of the die. The response time in the latter region is comparable to the time it takes to reach quasi-steady state conditions. Under those assumptions, temperatures in these two regions of the die are given by 10x D for 0 ≤ x ≤ , θ + θ 1 − D 10 T = (7) D θ for
(8)
for typical applications reported in the literature, θ ≈ 2θ or 3θ and therefore the sensitivity of the deflections to variations in the thermal fields in the vicinity of the cavity for conditions typical of die casting is, within a 10–15% error: ∂D ∼ = κD ∂θ
(9)
The temperature of the bulk of the die (θ) is necessarily defined by the total heat supplied by the casting process, combined with the heat transfer characteristics of the die (geometry, design of cooling lines). On average, it is unlikely to vary drastically as a result of the redistribution of heat within the casting. On the other hand, thermal fields in the region that surrounds the cavity would be much more sensitive to variations in the temperature of the casting. The term κθD in Eq. (8) accounts for the contribution of the temperature of the bulk of the die to the overall deflections, and can be assumed to be relatively insensitive to small variations in process parameters. The term κθD/20 accounts for the deflections induced by thermal conditions in the vicinity of the cavity, which are known to vary within a casting cycle. Typical values for κ (14.4E−06 m/mK), θ (100–200 ◦ C), D (0.5–1 m), applied to Eq. (9) suggest deflection magnitudes due to thermal growth in the range of 0.7–2.8 mm (0.03–0.110 in.) in the structure of the die, with an increase of 10–15% from these values (anywhere from 0.07 to 0.4 mm more) in the vicinity of the cavity. The previous analysis suggests that normal variations in the temperature fields in the vicinity of the casting (θ)
produce deflections whose magnitudes lie in the range of achievable tolerances in commercial applications of die casting. If a simulation could predict variations of thermal fields in this region, corrective actions could be taken to improve achievable tolerances. In addition, heat released during fill can be expected to affect the nature of the temperature field around the cavity. Consequently, the cost of simulating the evolution of temperatures in this region may be justified if the ability to predict and control tolerances is also improved.
5. A numerical example Results from a thermo-elastic 2D simulation are used to support the conclusions of the previous analysis. The case is based on an exercise designed by Barone and Caulk [6]. While simple in conception, this exercise presents the same difficulties encountered in any die casting model. ABAQUS is designed to simulate structural problems in mechanics, and large deformation processes such as those encountered in metal forming. However, it does not model problems typical of fluid mechanics, and there is no interface with systems that model casting. Consequently, simplifications need to be made when developing the models for loads and boundary conditions. Fig. 3 summarizes the dimensions of the idealized twodimensional die, while Table 2 presents the conditions under which the simulation is conducted. The geometry represents a small die, and the process conditions are typical of die casting. The cavity is rectangular in shape and located at the center of the die. The filling stage is not modeled explicitly. In one case, instant fill conditions are assumed and therefore the initial temperature of the casting is uniform. The casting is allowed to release heat, including latent heat, for 15 s, through a typical interface conductance of 10,000 W/m2 k, the same value used by Barone and Caulk. After the solidification period, heat is removed from the die in the cavity region to represent the application of lubricant spray. Non-instant fill conditions are modeled with the same geometry and processing conditions in order to compare with the previous case. In this model, heat released during fill is accounted for by assuming a redistribution of the heat within the casting. At the beginning of the casting, the heat content of the part is the same as the instant fill case. However, the volume of the casting that is assumed to travel the whole Table 2 Process conditions for simulation exercise Casting material Casting thickness Tl , Ts cpm (liquid aluminum) Tinjection (instant fill) Heat removed by spray TA (non-instant) TB (non-instant)
Aluminum 0.127 mm (0.005 in.) 580 ◦ C, 520 ◦ C 963 J/kg K 630 ◦ C 625E03 J/s m3 for 3 s (20% of input heat) 830 ◦ C 630 ◦ C
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Fig. 3. Model for testing the effect of instant fill vs. non-instant fill conditions. Description of geometry and conditions. Dimensions in meters.
length of the cavity looses about 50% of the latent heat. This heat is added to the volume that represents the region that ends up at the gate, in such a way that the die in this region absorbs more heat. The initial heat content of the casting varies linearly within these extremes. The middle of the casting is in the same conditions in both cases. At points along the casting that gain heat, their initial temperature is defined by Ti = 630 +
F cp
where F is the fraction of heat to be added from the corresponding volume in the casting. The portion of the material that travels the length of the cavity is therefore half-solidified at the time it reaches the farthest point from the gate. This condition roughly represents the limiting line defined by relation (2) in Fig. 2. Again, this model does not represent the filling process. Instead, the model only tries to represent the heat load that the die would be subject under non-isothermal fill conditions. Other conditions such as cycle time and heat removal due to lubricant spray were the same as in the isothermal fill case. The maximum temperature difference at any node between consecutive heat cycles was recorded for both cases. Fig. 4 presents the trend in the variation of temperatures between consecutive cycles for the instant fill case. The non-instant fill case results behaved in a very similar manner. After 15 cycles the maximum difference was down to about 5 ◦ C, or roughly 1% of the maximum temperatures in the structure of the die. At this point, it was assumed that quasi-steady state conditions had been reached. Nodal temperatures were transferred to a thermoelastic deflection model to estimate the magnitudes of the deflections. An interesting point is the fact that a trend line predicts that, strictly speaking, periodicity conditions would be reached after approximately 275 cycles for both cases. In their work with a similar model, Barone and Caulk report that these conditions are reached after about 300 cycles. This raises the
issue of whether or not it is feasible to reach quasi steady state conditions in the field. This exercise suggests that, in the field, the casting cycle is likely to occur within a range of conditions and not at fixed values of temperature. Maximum temperatures during a casting cycle reach 415 ◦ C in the instant fill case and 466 ◦ C in the non-instant fill case. Maximum deflections reach 0.57 mm (0.022 in.) and 0.58 mm (0.023 in.) from dimensions at room temperature, respectively. Fig. 5 shows the differences between both cases. The maximum difference in temperatures for these cases ranges from +54 to −49 ◦ C, with the hottest location being on the gate side, and the coldest point being at the end of the cavity, both on the cavity surface, non-instant fill case. As expected, about 80% of the die structure maintains a temperature that lies within approximately ±7 ◦ C in both cases. The maximum difference in deflections between the instant fill and non-instant fill cases lies within a 0.08 mm (0.003 in.) range. This difference is about equal in magnitude
Fig. 4. Rate of convergence of temperature profiles towards periodicity conditions. Maximum difference in temperature between consecutive cycles, instant fill case.
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Fig. 5. Comparison of results, instant fill vs. non-instant fill models: (a) difference in temperatures (◦ C); (b) difference in magnitudes of deflections along the vertical direction (m).
and opposite in direction about the symmetry plane (about ±0.04 mm or ±0.0015 in., taking as reference the instant fill case). These deflection magnitudes are consistent with the analysis presented in the previous section, specifically with what Eqs. (8) and (9) indicate. It is interesting to note that the magnitude of the variation between both cases lies in the range of the resolution of typical simulation capabilities, i.e. experience has shown that deflection patterns predicted by a simulation for commercial applications are accurate within ±0.05 mm (0.002 in.). The decision to include or not heat released during fill in the
analysis may thus account for a significant part of the error of a typical simulation. Also, as predicted by Eqs. (8) and (9) the differences in results from both cases are in the range of manufacturing/assembly allowances for details of the cavity in commercial applications. The results suggest that the prediction of heat released during fill might have only marginal effects on the quality of the simulation results for average filling conditions (fill times recommended by Chen) or when cavities fill rapidly. These situations correspond to the region delimited by relations (1) and (3) (see Fig. 2). The expense of predicting the true
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evolution of temperature as the cavity fills is probably not justified under those conditions. However, for those applications in which the filling regime lies in the region delimited by relations (2) and (3) errors in the predictions of the magnitudes of cavity deflections caused by the instant fill assumption are relevant, about the same size of achievable tolerances, and consequently the cost of modeling this phenomena may be justified. One of the trends in the industry is towards higher precision applications, and tight tolerance control is desirable. Such applications represent an ideal situation in which tracking heat release during fill may be relevant. The results also suggest that the maximum differences in temperature predicted by these models within the part and the structure of the die in the near vicinity of the casting are indeed significant for the same conditions (delimited by relations (2) and (3)). In this case, heat released during fill can result in thermal variations that may affect design decisions or problem diagnostics. As an example, variations in thermal patterns may have an impact on the ability to predict soldering. In this context, the need to establish accurate thermal fields may justify the cost of predicting heat release during fill.
6. Conclusions This article analyzed the process of die fill and introduced relations that establish the conditions under which heat released during this process is significant. These relations are given in terms of the thickness of the casting and the fill times. Comparison with field data indicates that under typical casting conditions the molten metal does release heat as the cavity fills. The analysis of this work suggests that, for average commercial applications, the magnitude of the deviations caused by assuming instant fill conditions is relatively small in the context of die deflections. This is due to the fact that deflections depend much more heavily on the temperature of the bulk of the die, which is fairly insensitive to the redistribution of heat within the casting. On the other hand, the prediction of thermal fields in the presence of significant heat release is relevant because the induced variations in deflections are of the same order of magnitude as achievable tolerances in die castings. Field data indicates that such filling conditions are not uncommon. The relations introduced by this work can be used to identify these conditions. Finally, simulation results suggest that the prediction of heat release is indeed relevant in the context of thermal behavior of the die. The nature of the thermal fields in the structure of the die has an important effect on the quality of the casting operation. The cost of predicting thermal patterns when significant heat release during fill occurs may be justified by an improvement in the quality of the casting operation.
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As the industry evolves towards high precision applications, a trend that is being observed in competing industries and which requires very accurate-dimensional control of mass produced components, modeling of heat released during fill should become more relevant. Accurate prediction of die deflections can help control dimensional variability of the cavity and, in turn, improve achievable tolerances in die casting.
Acknowledgements This work was done as part of a research program funded by the US Department of Energy. Close monitoring and guidance was provided by the North American Die Casting Association, through its Computer Modeling Task Group.
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