Materials Processing
Derivation of Plunger Injection Input to Prevent Gas Defects by Algebraic Approach for Aluminum Alloy Diecasting
2024 年 65 巻 6 号 p. 657-664
詳細
2024 年 65 巻 6 号 p. 657-664
Die casting has many advantages, such as high precision, mass production, and excellent recyclability. However, gas defects in the product have become a problem. One countermeasure to this problem is to design the injection input of the plunger appropriately. Recently, CFD has been used to design plunger injection inputs, and by combining with optimization techniques, auto-design of plunger injection inputs has been possible. However, CFD is difficult to apply because of the high cost of CFD resources, the need to resource IT engineers, and the time required for its auto-design. Therefore, we propose a new injection input design method based on an algebraic approach, enabling anyone to design a plunger injection input that prevents gas defects with ease and at a low cost. Simulation and experimental results verified the effectiveness of the proposed method.
The die casting method has many advantages such as the mass production of castings with high surface accuracy and excellent recyclability [1]. However, gas in the sleeve can enter the molten metal, causing gas defects due to gas retention in the product [1]. One countermeasure to this problem is to design the injection input of the plunger appropriately, aiming to suppress the turbulence of molten metal and prevent gas confinement and entrainment, where gas becomes trapped within the molten metal [1]. However, as the behavior of molten metal during filling takes place inside the die casting machine and mold, direct observation of the molten metal’s behavior and the design of an appropriate injection input pose challenges [2].
Recently, computational fluid dynamics have begun to be used [3, 4]. Computational fluid dynamics allows for the visualization and quantitative evaluation of flow, facilitating the design of plunger injection inputs that take into account the behavior of molten metal during filling. Furthermore, the optimal design of plunger injection input in combination with optimization technology can prevent gas retention in molten metal and reduce gas defects [3, 4].
However, the introduction of an optimal design system has not progressed due to the cost of securing the resources necessary for fluid analysis, the difficulty of securing IT personnel, and the time and effort required for optimal design [5], since the design is performed by repeating fluid analysis in the conventional method. Therefore, there is a need to establish a design method for plunger injection input that can more easily prevent gas defects.
An easy method for designing injection input is to use a design formula that considers the filling time of molten metal [6, 7]. By taking into account the filling time of molten metal, it is possible to prevent the temperature drop during the filling process and avoid casting defects such as cold shuts and wrinkles resulting from poor flowability. Faura et al. also proposed a method to prevent molten metal from trapping gas in the sleeve during filling by formulating the relationship between the shape of the wave of molten metal and the plunger speed [8]. However, gas entrainment in the sleeve, which is a cause of casting defects, is not considered.
In this study, we propose a new injection input design method using an algebraic approach to design a plunger injection input that prevents gas retention in molten metal at a low cost and easily. Specifically, we clarify the relationship between the injection input and the flow phenomena that cause gas confinement and gas entrainment in the sleeve and develop a plunger injection input design system by constructing a mathematical model that expresses the relationship between the flow phenomena and the injection input. Finally, the effectiveness of the method is demonstrated by comparing it with the injection input derived by iterative optimization [4].
The injection input used in this study is the two-step injection method commonly used in production. In this section, the flow phenomena in the sleeve that must be considered in the design of the injection input are clarified. In order to obtain the appropriate injection input parameters, the optimal design of the injection input was performed using a genetic algorithm for the casting conditions. To identify the suitable injection inputs from the optimization results, clustering is performed using a neural network.
Specifically, the genetic algorithm is used to optimize the injection input for 28 casting conditions (sleeve diameter in 0.0100 [m] increments from 0.040 to 0.100 [m], sleeve fill rate in 5 [%] increments from 20 to 35 [%], and sleeve length constant at 0.347 [m]). In the CFD, viscosity, solidification, and heat conduction of the fluid were considered. The target fluid was an aluminum alloy ADC12, the molten metal temperature was 680°C, and the plunger tip shape was cylindrical. The material of the sleeve is SKD61. The physical properties of the fluid and sleeve material are shown in Table 1. The boundary conditions for the analysis were wall boundaries in all directions and mesh size is 3 [mm]. The shapes of the plunger and sleeve are shown in Fig. 1. The fluid analysis was performed using FLOW-3D [9] from Flow Science.
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Setup for analysis of flow behavior of molten metal in a sleeve.
In the optimization, gas confinement and gas entrainment are evaluated. First, define the objective function J1 to evaluate gas confinement. Gas confinement is evaluated by calculating the volume of air trapped in the sleeve. From previous studies, the volume of trapped air Ashut is obtained by eq. (1) [10], J1 represents the amount of volume confined within the sleeve, the larger the value, the larger the volume of trapped air.
| \begin{equation} J_{1} = A_{\text{shut}} = \sum_{i \in \omega}V_{\text{cell,${i}$}}F_{\text{space,${i}$}}(1 - F_{\text{fluid,${i}$}}(t_{\text{shut}})) \end{equation} | (1) |
Where tshut is the time when the sleeve outlet is filled with molten metal, ω is the number of mesh cells, Vcell is the mesh cell volume, Fspace is the volume fraction of space in the cell and Ffluid is the volume fraction of fluid in the space. Since it is possible to design injection inputs such that Ashut is completely zero when designing gas confinement phenomena, this study aimed to prevent gas confinement completely and set the evaluation value to 1000 as a penalty for inputs that cause gas confinement, the objective function of the optimization calculation is defined by eq. (2).
| \begin{equation} J_{1} = \begin{cases} 0 & \text{if $A_{\text{shut}} = 0.0$}\\ 1000 & \text{otherwise} \end{cases} \end{equation} | (2) |
Next, gas entrainment is evaluated by the amount of air entrainment due to the turbulence of molten metal. The amount of air entrainment is calculated using the air entrainment volume calculated by the air entrainment model [11]. To evaluate the ease of gas entrainment, the objective function J2 adds up the gas entrainment volume Vair until the filling is complete and obtains the evaluated value [10]. Where Vσ is the air entrainment volume fraction, tf is filling completion time.
| \begin{equation} J_{2} = V_{\text{air}} = \sum_{t = 0}^{t_{\text{f}}}\sum_{k = 1}^{\omega}V_{\sigma,k}F_{\text{fluid,${k}$}}F_{\text{space,${k}$}}V_{\text{cell,${k}$}} \end{equation} | (3) |
The design parameters were set as low injection speed vl, high injection speed vh, and high-speed switching position xsp. The optimization problem was formulated in eq. (4). The machine response time and variable constraints during speed switching were determined based on the mechanical properties of the die casting machine, subscripts min and max indicate the minimum and maximum possible values for each variable. In addition, a constraint on the filling time tf was set to prevent the failure of the molten metal around the molten metal due to a drop-in temperature. A multi-objective genetic algorithm, NSGA-II [12], was employed as the optimization method.
| \begin{equation} \begin{split} \text{Minimize} \quad & J_{1},J_{2}\\ \text{Subject to}\quad & v_{\text{I,min}} \leq v_{\text{l}} \leq v_{\text{l,max}},\\ & v_{\text{h,min}} \leq v_{\text{h}} \leq v_{\text{h,max}},\\ & x_{\text{sp}} \leq x_{\text{sp,max}},\\ & t_{\text{f}} \leq t_{\text{f,max}} \end{split} \end{equation} | (4) |
To visualize the injection input parameters, which prevent gas retention in the molten metal, vary with changes in casting conditions, a neural network was employed to extract the optimal solution set for 28 different conditions. To visualize the relationship between the casting conditions and the appropriate injection input, a neural network was used to extract the appropriate injection input for the optimization results of the injection input performed using a genetic algorithm. Self-Organization Map [13] was used for the neural network, SOM was used to extract appropriate injection input parameters from the optimization results. In the self-organizing map, the objective functions 1 and 2 of each individual are used as inputs, and classification is performed using the similarity of each individual. In the map, a square map is created, and the number of units in a piece is set to 13, which is the minimum number that can properly classify the optimization results, so that the results are classified into 169 units in total. The numbers in Fig. 2(a) indicate the number of individuals classified in each unit. The group that suppressed gas confinement and gas entrainment in the classification results was selected as the good solution set. For the classification results, the mean and variance of the objective function for each classified set were checked to confirm that proper classification was performed. The calculated average values are shown in Fig. 2(b) and Fig. 2(c).
Results of classification using SOM.
Next, a correlation analysis was conducted between the molten metal filling rate and sleeve diameter and the injection input parameters of the good solution set based on the extraction results of the good solution set for 28 casting conditions. The analysis showed the low speed, and the switching position became smaller as the filling rate increased. The switching position became smaller, and the low speed became larger as the sleeve diameter increased. The low-speed and the switching position have a significant effect on gas entrainment in the molten metal, and that the low-speed and the switching position change linearly with changes in the filling rate and sleeve diameter. As an example of the relationship, a visualization of the relationship is shown in Fig. 3. In Fig. 3, the legend indicates the molten metal filling rate.
Variation of the parameters of the set that prevent gas entrainment and gas entrapment under different casting condition.
For the obtained correlations, flow phenomena are analyzed to clarify the causes. The analysis results show that gas entrainment occurs depending on the wave collapse of the molten metal and the timing at which the wave contacts the top of the sleeve, and that the genetic algorithm design changes the shape of the wave generated in the sleeve in response to changes in casting conditions. Therefore, in this study, we focused on the waves generated by the plunger movement to define the conditions under which gas confinement or gas entrainment occurs and defined the three flow phenomena shown in Fig. 4 as the causes of gas entrainment or gas confinement.
Fluid behavior defined as a factor of gas entrainment or gas entrapment in this study.
The first is the gas entrainment due to top of the wave shown in Fig. 4(a). The faster the plunger speed, the larger the wave generated in the sleeve. When the wave is large, the wave tends to collapse or become turbulent when it contacts the sleeve wall or sprue section, and the molten metal tends to entrain gas.
When the waves are large enough not to be disturbed by contact, the molten metal waves are repeatedly reflected to prevent wave collapse when switching speeds, and the speed is switched at a position close to the sleeve outlet. In this case, the gas confinement due to the top of the wave shown in Fig. 4(b) becomes the second factor. If the top of the wave is large, the reflected wave will contact the top of the sleeve before switching speeds, and gas will be trapped.
The last factor is the gas confinement phenomenon due to the wave phase velocity shown in Fig. 4(c). After the plunger speed is switched, the top of the wave rapidly increases. Therefore, even when the reflected wave does not contact the top of the sleeve, gas is trapped between the plunger and the wave due to the increased top of the wave when the wave and plunger are far apart at the switch timing.
To clarify the changes in the top of the wave and phase velocity during molten metal filling, the top of the wave and phase velocity during molten metal filling were measured from analysis results.
When the top of the wave during molten-metal filling was measured by CFD, the top of the first wave generated at the reference point of the liquid surface at rest was assumed to be |Hy|, and the results of top and phase velocity changes over time (Fig. 5) show that the height of the top moves at |Hy| from the generation of the wave until reflection occurs, and furthermore, from the occurrence of reflection until the next reflection, the top of the wave increases by |Hy| from the top before reflection. The phase velocity of the wave was calculated by focusing on the sleeve length, the wave position, and the distance traveled by the plunger tip, and it was confirmed that the wave was traveling at a constant velocity. These phenomena were confirmed for different sleeve diameters and different molten metal filling rates.
Variation of top of the wave and wave phase velocity during molten metal filling.
In this study, it is assumed that the wave crest |Hy| before the reflection is constant and the phase velocity of the wave is constant between the onset of the reflection and the onset of the next reflection. Based on these assumptions, it is possible to calculate the position and top of the wave at any given time from the top of the wave and phase velocity at the time of wave generation, and to determine whether the wave will contact the top of the sleeve and the positional relationship between the plunger and the wave, and to determine whether the three conditions of gas entrainment defined in Fig. 4 are prevented. Therefore, it is possible to predict whether gas confinement and gas entrainment will occur by using the top of the wave and phase velocity at the timing when the wave is generated. Furthermore, by clarifying the relationship with the plunger speed, it is possible to design an injection input that prevents gas retention in the molten metal.
A model is constructed to derive the top of the wave from the casting conditions and the plunger injection speed. Here, we define the waves generated in the sleeve. Waves are classified into deep-water, shallow-water, and between waves [14]. Deep-water waves are waves where the water depth is larger than the wavelength, and shallow-water waves are waves where the wavelength is larger than the water depth.
The wave generated in the sleeve is assumed to be a shallow-water wave because the wavelength of the wave is sufficiently larger than the water depth at the time when the wave is generated from Fig. 5. Then, we assume that the work done by the plunger during a small time Δt [s] is converted into the mechanical energy of the shallow-water wave in a two-dimensional planar sleeve and construct a top of the wave model from the energy conversion equation. In constructing the model, the focus is on calculating the wave’s peak at the moment of wave generation. The model is built with the perspective of static energy conservation over a brief time span, specifically addressing the phenomenon from the initiation of plunger movement until the wave is generated. It does not encompass the calculation of the peaks of continuously changing waves.
The work of the plunger is the product of the force and the amount of movement. Therefore, if the amount of movement in Δt is the product of the minute time and the plunger speed, it can be expressed as eq. (5). The mechanical energy per unit width of shallow water waves is expressed as eq. (6) from previous studies [14]. Where Pp [Pa] is the plunger pressure, h [m] is the water depth, and vp [m/s] is the plunger velocity.
| \begin{equation} W = P_{p}hv_{p}\varDelta_{t} \end{equation} | (5) |
| \begin{equation} U = \frac{1}{4}\frac{\rho}{g}|gH_{\text{y}}|^{2} + \frac{1}{4}\frac{\rho}{g}|gH_{\text{y}}|^{2} \end{equation} | (6) |
Since it is difficult to measure Pp [Pa], the pressure term is expressed by eq. (7) [15], which expresses the relationship between the volumetric flow rate at the gate and the plunger pressure and eq. (8) [15], which is the continuity equation between the sleeve and the gate section. Cg [-] is the total flow coefficient, ρ [kg/m3] is the density of the fluid, A [m2] is the cross-sectional area, and V [m/s] is the velocity of the fluid, the subscripts p and g denote the plunger and gate, respectively.
| \begin{equation} Q_{\text{g}} = A_{\text{g}}C_{\text{g}}\sqrt{\frac{2gP_{\text{p}}}{\rho}} \end{equation} | (7) |
| \begin{equation} Q_{\text{g}} = A_{\text{g}}V_{\text{g}} = A_{\text{p}}V_{\text{p}} \end{equation} | (8) |
Substituting the pressure terms obtained by eq. (7) [15] and eq. (8) [15] into eq. (5), the energy conversion equation using eq. (5) and eq. (6) is expressed as eq. (9).
Equation (9) was converted to an expression for the top of the wave |Hy|, and since Cg is the flow coefficient, g is the acceleration of gravity, Δt is the micro-time, $A_{\text{p}}^{2}/A_{\text{g}}^{2}$ is the ratio of plunger to cross-sectional area, the ratio of gate cross-sectional area to plunger cross-sectional area is generally designed to be 1:1.15 [16], eq. (10), the model equation, was constructed by setting these constant parameters and the flow coefficient, which is an identification parameter, as the identification parameter k.
| \begin{equation} \frac{A_{\text{p}}^{2}\rho}{2gC_{\text{g}}^{2}A_{\text{g}}^{2}}v_{p}hv_{p}\varDelta_{t} = \frac{1}{2}\frac{\rho}{g}|gH_{\text{y}}|^{2} \end{equation} | (9) |
| \begin{equation} H_{\text{y}} = \sqrt{\frac{\varDelta_{t}}{gC_{\text{g}}^{2}}\frac{A_{\text{p}}^{2}}{A_{\text{g}}^{2}}hv_{\text{p}}^{3}} = \sqrt{khv_{\text{p}}^{3}} \end{equation} | (10) |
The parameters of the model were determined by fitting the model to the top of the wave measured by CFD. Since the water depth varies with the water depth and plunger speed, the sleeve diameter and sleeve length were set to 70 mm and 347 mm, respectively, and the parameters were fitted by changing the water depth and plunger speed and measuring the top of the wave. As a result of the fitting, k was set to 1.54. Figure 6 shows the identification results when k was set to 1.54.
Results of parameter identification for top of the wave.
Regarding the results in Fig. 6, the mesh cell size in the analysis is 3 [mm], so measurements can only be taken at intervals of 3 mm, which is the mesh size, and considering that the measurement results are in increments of 3 [mm], the model we built is valid.
3.2 Construction of phase velocity modelThe phase velocity of shallow-water waves has already been formulated in previous studies [14], and some studies have also considered the effect of the velocity of a moving object such as a plunger on the phase velocity of waves [17]. The phase velocity of a wave is expressed as eq. (11) in terms of a coordinate system moving with velocity v in the direction of wave motion and the absolute phase velocity when momentum is conserved.
| \begin{equation} c = \sqrt{g\left(h + \frac{g}{h}v\right) - 2\sqrt{gh}} \end{equation} | (11) |
The phase velocity of the waves in the sleeve cannot be obtained by eq. (11) because the waves in the sleeve are not strictly shallow water waves. Therefore, we multiply the phase velocity by a coefficient β as in eq. (12) and β is decided by fluid analysis results. The results are shown in Fig. 7. As before, the model is valid considering the mesh cell size error.
| \begin{equation} c' = \beta \sqrt{g\left(h + \frac{g}{h}v\right) - 2\sqrt{gh}} \end{equation} | (12) |
Results of parameter identification for wave phase velocity.
Using the constructed model, an injection formulation aimed at preventing gas retention is developed. First, to design the plunger injection from the constructed model, the top of the wave and wave position are used to formulate the conditions that prevent the flow phenomena defined in Fig. 4.
At the plunger speed to prevent gas confinement shown in Fig. 4(a), the wave reflects several times before switching the speed, and according to the CFD results, the molten metal is filled most quickly when the number of reflections is three in the number of reflections to prevent gas confinement shown in Fig. 4(b). Therefore, if the number of wave reflections is set to three, the height of the wave increases by |Hy| at each reflection, and the height of the wave at the plunger speed changeover timing is 4|Hy|. Therefore, if φ is the sleeve diameter and h is the water depth at rest, the condition for the wave not to contact the top of the sleeve after three wave reflections can be expressed by eq. (13).
| \begin{equation} 4H_{\text{y}} < \varphi - h \end{equation} | (13) |
Next, let Tk be the time when the wave contacts the sprue or plunger, and the condition that prevents Fig. 4(c) can be expressed by eq. (14). Where i is the number of wave reflections.
| \begin{equation} x_{\text{sp}} = v_{\text{p}} \sum_{\text{k} = 1}^{i + 1}T_{\text{k}} \end{equation} | (14) |
When vp is obtained, it is possible to design the speed switching timing when the plunger and wave contact each other by relative velocity from the phase velocity model, eq. (12). The plunger speed satisfying Fig. 4(a) and Fig. 4(b) can be designed by converting the inequality into an entity using a factor τ smaller than 1 for eq. (13) and using the top of the wave model, eq. (10), and the final design equation for the plunger speed can be obtained as eq. (15).
| \begin{equation} v_{\text{p}} = \root 3 \of{\frac{\{\tau(\varphi - h)\}^{2}}{4^{2}}\frac{1}{kh}} \end{equation} | (15) |
Since time T can be obtained by using the relative velocity between the plunger and the wave, the design equation for the velocity switching position is eq. (16).
| \begin{equation} \begin{split} & T_{1} = \frac{L}{c'},\quad T_{2} = \frac{L - v_{\text{p}}T_{1}}{v_{\text{p}} + c'},\\ & T_{3} = \frac{L - v_{\text{p}}(T_{1} + T_{2})}{v_{\text{p}} + c'},\quad T_{4} = \frac{L - v_{\text{p}}(T_{1} + T_{2} + T_{3})}{v_{\text{p}} + c'},\\ & x_{\text{sp}} = v_{\text{p}}(T_{1} + T_{2} + T_{3} + T_{4}) \end{split} \end{equation} | (16) |
The effectiveness of the proposed method is evaluated by designing injection inputs for the casting conditions shown in Table 2 and comparing them with the results of iterative optimization. The results of the design by the proposed method and optimization method are shown in Table 3. Note that the high speed in the proposed method is constant at 4.0 [m/s].
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CFD was performed using the designed injection input parameters to evaluate gas confinement and gas entrainment. As a result, it was confirmed that gas confinement did not occur in either the optimized method or the proposed method. The evaluation values of gas entrainment of the optimization and the proposed method were compared, and it was confirmed that that the values designed by the proposed method and those converged by optimization are equivalent, and that the optimal solution and the proposed method had the same degree of gas entrapment resistance. The individual with the lowest evaluation value among the evaluation values that converged in the optimization for each casting condition was selected as the optimal solution and compared with the evaluation value of the proposed method. Figure 8 shows the results of the comparison of the evaluation values.
Results of comparison of evaluation values between the optimization method and the proposed method for three casting conditions.
Figure 9 and Fig. 10 show the results of the fluid analysis of No. 1 designed by the proposed method and the optimization method, respectively.
Results of CFD for the injection input designed by optimization method.
Results of CFD for the injection input designed by optimization method.
The analysis results show that no wave collapse occurs for any of the inputs and that the speed is switched when the wave and plunger make contact without the wave contacting the top of the sleeve before switching speeds, thus satisfying the defined conditions for gas confinement and gas entrainment prevention.
Furthermore, from Fig. 11, which is the comparison result of the evaluation value between the optimization method and the proposed method, it can be said that the proposed method is as effective as the optimization method in preventing gas entrainment since the evaluation values converged by the optimization result are comparable to those by the proposed method.
Results of the comparison of evaluation values with optimization method.
Finally, casting experiments were conducted on a die casting machine using the designed injection inputs to compare product quality. In the experiment, four castings were manufactured for each product, and the cast defects in the products were evaluated after shot blasting and cutting. In the experiment, after the injection input was input, ten discarding runs were performed after the injection input was changed to stabilize the output, and the specimens were cast after the output was confirmed to be stable. Evaluation was performed by on-site engineers on casting porosity, cold shut, and peeling. The results of the experiment are shown in Table 4. Table 4 shows the sum of the defects of the four experiments on the three products. The evaluation results confirm that the quality was comparable.
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In the optimization, three computers with Intel Core i7-7700K processors were used for the design of one condition, and it took about 44 hours for the design. The proposed system can be designed by simply entering the casting condition parameters into spreadsheet software, and the design is instantaneous after the conditions are entered, indicating that the method can more easily design injection inputs that prevent gas retention.
This study uses an algebraic approach to develop a design system that can derive plunger injection inputs that prevent gas defects. The effectiveness of the proposed system was verified by iteratively optimizing the injection inputs for multiple casting conditions, deriving them using the proposed method, and comparing the results of computational fluid dynamics and casting experiments. The results show that the proposed method prevents gas defects as well as the optimal solution and has the same level of product quality, indicating that the proposed method can more easily design plunger injection inputs that prevent gas defects.
