aUniversity of Kassel, Department of Foundry Technology, Kurt-Wolters-Str. 3, 34125 Kassel, Germany
bWikki GmbH, Ziegelbergsweg 68, 38855 Wernigerode, Germany
cVolkswagen AG, Division of components manufacturing – Business Unit Casting Dr. Rudolf-Leiding-Platz 1, 34225 Baunatal, Germany
In this work, a toolchain for the solidification and the shrinkage of cast salt cores used in high-pressure die casting is implemented in the open-source computational fluid dynamics (CFD) library FOAM-extend to evaluate the displacements resulting from residual stresses. The toolchain consists of three stages, which first calculate the phase change, then generate the residual temperatures and finally calculate the residual stresses resulting in displacements. The calculated displacements were compared to 3D-Scans of casting trials of simple test geometries. The comparison showed acceptable agreement for engineering purposes.
이 작업에서는 고압 다이 캐스팅에 사용되는 캐스트 솔트 코어의 응고 및 수축을 위한 툴체인이 오픈 소스 CFD (전산 유체 역학) 라이브러리 FOAM-extend에서 구현되어 잔류 응력으로 인한 변위를 평가합니다. 툴체인은 먼저 위상 변화를 계산 한 다음 잔류 온도를 생성하고 마지막으로 변위를 유발하는 잔류 응력을 계산하는 세 단계로 구성됩니다. 계산 된 변위는 간단한 테스트 형상의 주조 시험의 3D 스캔과 비교되었습니다. 비교 결과 엔지니어링 목적에 대해 수용 가능한 동의가 나타났습니다.
잔류 온도 개념을 사용하여 솔트 코어의 응고 및 수축을 시뮬레이션 하는 도구 모음이 구현되었습니다. 경계 조건이 단순화 된 단순한 형상에서 테스트 되었습니다. 계산을 검증하기 위해 동일한 경계 조건을 사용하여 두 개의 중력 다이 형상으로 주조 시험을 수행했습니다.
계산하는 동안 개발된 솔버는 적절한 수준으로의 우수한 수치 수렴을 보여줍니다. 수축으로 인한 캐스트 솔트 코어의 변위 및 응고 중 응결력은 3D 스캔을 사용하여 조사되었습니다.
결과의 검증은 두 형상의 계산 된 변위와 실제 변위의 우수한 일관성을 보여줍니다. 변위는 엔지니어링 목적에 적합한 정확도로 예측됩니다. 계산 비용은 잔류 온도로부터의 잔류 응력 계산에 의해 좌우됩니다 (1 개의 정상 상태 고체 역학 시뮬레이션 비용). 매우 복잡한 초기 문제를 고려할 때 이것은 훌륭한 결과입니다.
응고 중에 잔류 응력을 계산하는 복잡한 문제는 모든 시간 단계에서 고체 역학의 과도 계산에 비해 훨씬 빠른 정상 상태 고체 역학 시뮬레이션으로 매핑 될 수 있습니다.
향후 작업에서는 다양한 경계 조건을 조사하여 솔버의 신뢰성을 테스트 해야 합니다. 이를 위해 문헌 데이터를 비교하기위한 추가 실험에서 특정 열 물리 데이터를 측정해야 합니다. 결합 열 전달 및 액체 수축을 고려하여 솔버의 추가 개선을 구현할 수 있습니다.
Cast salt cores, OpenFOAM FOAM-extend, Residual temperature concept, Residual stresses, Validation 3D-Scans
The uprising electric mobility has a high need for lightweight constructions to reduce car weights and increase the range per charge of the electric vehicle [1,2]. Especially non-iron lightweight castings from aluminum alloys are used in the body and the parts of the power train . To boost the weight savings potentials up to 50% it is efficient to integrate several components into a single piece of casting . The tooling technology of the traditional high pressure die casting process is reaching its limits on highly integrated lightweight parts including undercuts and internal cavities. These complex geometries require other tooling technologies than moving slides and tool sleeves, which are removed from the casting after each cycle . The alternatives for solid metal dies are lost cores to create the internal cavities . Lost cores made from salt can already be used in high pressure die casting to a limited extent [7,8].
Production of the cores is mainly separated in pressing salt powders and several casting processes from molten salts. The pressing of salt cores can be used for simple geometries but the process has geometrical limits and casting is the preferred process to be used for the complex geometrical shapes demanded by industry. The molten salt mixtures can be processed in different casting processes, such as die casting, hot and cold chamber die casting [9,10]. The solidification taking place in every casting process is the critical step producing salt cores. The high shrinkage of the molten salts (e.g. 18.5% shrinkage for mixture KCl 38 wt-% / Na2CO3 62 wt-% , over 15% shrinkage for mixture KCl-NaCl-Na2CO3-K2CO3 ) leads to cracking of the solidified salt core as a result of shrinking to the mold. As a consequence, core failure can be a problem due to the high dynamical forces in the following casting processes and possible weaknesses of the salt core resulting from the cracks. Different studies already describe the impact of casting on cores [13,14]. Simulations to consider the crack caused by solidification shrinkage have not been performed.
For engineering purposes, various commercial simulation tools are established in foundry technology, which allow the calculation of mold filling, solidification as well as the residual stresses in metal castings, molds and plastics technology. For this purpose, these tools use complex numerical methods in fluid and solid mechanics. Often there is a coupling of separate fluid and solid mechanics in each step, which was simplified by developments e.g. in Flow-3D, which provides a fully coupled fluid-structure interaction along with thermal stress evolution using a finite element approach . The modeling of stresses in casting usually results in a coupled 3-dimensional thermophysical problem including solidification with time-dependent boundary conditions . Studies and simulations mainly focus on the effects of solidification stresses in metallic castings [15,, , , ], but not on the stresses during casting of lost salt cores used in the tooling.
The present work, proves that a much simpler, especially regarding material properties and boundary conditions of these simulations, is able to yield adequate results for engineering purposes. The presented solution is easy to use and computationally inexpensive, focusing salt cores which have not been considered up to now.
In this study, the solidification and the resulting shrinkage of salt cores is simulated to obtain the residual stresses and shrinkage developed during the salt core casting process. In first place, the lost cores and dies used in simulation and validation are described, afterwards a description of the developed solver is provided. The computations are based on the CFD-toolbox OpenFOAM, in particular FOAM-extend 4.1. Based on this software framework, the solidification solver (solidificationFoam) and the calculation of the residual temperatures throughout the solidification (TResIntegrator) have been implemented. For the calculation of the residual stresses (TResStressFoam) the shipped solver (elasticThermalSolidFoam) is used with minor modifications. The provided tool-chain has been tested on simple geometries with simplified boundary conditions.
2. Experimental setup
This study aims for predicting the shrinkage and the displacement of cast salt cores used for creating cooling channels in die casting in an experimental numerical research. Appropriate simple geometries are chosen as cast components, which simplify the production of the dies, the casting experiments and the validation. Since the casting trials were carried out on a laboratory scale, gravity die casting of two different geometries is used. Both dies are designed to demold the casting as quickly as possible to allow free shrinkage. The used salt mixture consists of 50 wt-% sodium chloride and 50 wt-% sodium carbonate. This mixture is beneficial due to its nontoxic properties, its low cost and adequate bending strength of about 20 MPa .
2.1. Example structures
The hot-tear rod, Fig. 1, unifies the two testing geometries, bending rod and hot-tear sample, which are representative for salt core casting and can also be used for other testing. The geometry consists of three structural parts; the square cross section bar with an overall length of 180 mm and a side length of 22 mm, the cylindrical casting feeder and the two anchor bolts, which are placed over the corner of two side surfaces each, to avoid drafts. The mesh is generated with snappyHexMesh, an OpenFOAM internal meshing tool. The total cell count is 100k cells with an edge length of 1.3 mm.
The other geometry used is a closed ring structure, Fig. 2, with a combined sprue and feeder on top. The structure refers to the common use of salt cores as oil or cooling channels in castings. The outer diameter is 70 mm and the square cross-section has a side length of 12 mm. Similar to the hot-tear rod the mesh is created with snappyHexMesh. The cell count is approximately 120 k with an edge length of 0.73 mm.
The dies used in gravity casting, Fig. 3, consist of hot-work steel 1.234. The dies are divided into two halves, for the ring structure in a horizontal and the rod in a vertical dividing plane. Furthermore, the die of the ring structure has a movable center pin, which can be pulled early to avoid hot tears. In all tests, the dies are opened immediately after the formation of a stable surface layer and the castings are loosened to avoid cracking. For the ring structure it is possible after 10 s and for the rod after 20 s.
2.2. Boundary conditions
In the first approach, the boundary conditions are limited to the temperatures of the die and the molten salt mixture. In the experiment, the dies are fully heated in an oven each cycle to maintain constant and homogenous die temperatures to achieve an adjusted boundary condition for later validation. The temperature of the molds is set to 563.15 K. The salt mixture is provided in a laboratory furnace with a crucible made of a Ni-based alloy at a temperature of 963.15 K. The furnace and the dosing device are covered while melting and heating periods to maintain constant temperatures of the melt. The temperatures are controlled with type K thermocouples in each furnace, the dies and in the melt, respectively. The boundary conditions of the simulation are assumed according to the tests as follows, cf. Table 1, Where T is temperature, tsolidication is time passed until cell is frozen in, U is displacement and TRes is residual temperature. The introduction of heat transfer coefficients is avoided using the boundary condition of fixed value on the outer surfaces. Interfacial heat transfer is considered by linear mixing heat conductivities in liquid and solid state and averaged by volume of fluid.
Table 1. Boundary conditions of the simulation.
|value uniform 563.15||value uniform 0||traction uniform (0 0 0)|
p uniform 0
value uniform (0 0 0)
2.3. Material properties
The material properties used for the simulation are displayed in Table 2, where E is Young’s Modulus, ρ is density, ν is Poisson’s ratio and α is the thermal expansion coefficient. The Young’s Modulus was obtained from bending strength tests of the rods. To reduce complexity, the scope of data is focused according to the following assumptions of the simulation.
Table 2. Mechanical properties of salt used for the simulation.
The thermal properties are given in Table 3, where s and l refer to the solidus and liquidus state, respectively and c is the isobaric heat capacity, T is phase change temperature, λ is thermal conductivity and L is latent heat of fusion. The values are based on average literature references. The latent heat is approximated by a gradation of the values according to Jiang et al. . The main parameter set used for the first calculations is underlined.
Table 3. Thermal properties of salt used for the simulation.
|c [J/(kg•K)] ||T [K]||λ/ρ [W/(m•K)] ||L [J/kg] |
|1440||1440||908 ||930  942 ||6.4e-04||2.0e-04||283,300|
The validation is carried out using 3D-Scans of the example structures to evaluate the calculated displacements. Therefore, a Gom Atos Core 5 Scanner is used, Fig. 4. The examination of the computed and cast displacements is done using Gom Inspect 2019 software based on STL files. A number of 35 castings have been examined in this work.
3. Concept of the residual temperature
The simulations are carried out to assess the residual stresses in the casting process of salt cores used in aluminum high-pressure die casting. While casting the salt cores, the liquid salt mixture with a temperature above liquidus is poured into the preheated mold. During mold filling, partially solidified areas are formed as well as semi-solid areas in which both liquid and solid phases are present. In the state of the art the calculation of the filling is done with fluid mechanics and changes to solid mechanics during solidification to cope with appearing stresses. The solid phase expands as a moving front, which can numerically be represented in a coupled and moving field equation, but leads to high computational effort. A high complexity results from the coupled processes and the formation of mixed zones with unknown material properties, which cannot be calculated as well as the elasto-plastic behavior of these zones. The approach used has a significantly reduced computing effort and avoids these problems. Partly it originated in the calculation of plastics, but is modified for the first time to this new field of casting salt cores in foundry engineering.
The calculation strategy is obtained with the following assumptions as a sequential solution with advantageously low effort, whose reliability is checked with the subsequent validation. There are two stages of simulations:1
Simulation of the solidification process,2
Computation of the residual stresses.
Depending on the time when the salt core is removed from the mold, plastic deformation may play a role. This effect is important in metal casting, but it is unclear if it is important in the casting of salt. In order to capture this effect accurately, it would be necessary to calculate the elastic-plastic behavior of the salt material including a temperature-dependent yield stress together with the solidification process. This complexity was out of the scope of this study. In the current state, the following assumptions are used:•
At the beginning of the solidification process, the mold is completely filled with liquid salt at a given temperature. The liquid shrinkage shows no effect on later deformation and is not considered.•
The solidifying salt is demolded early enough so that its deformation can be considered fully elastic.
With these assumptions it is possible to employ the residual temperature concept proposed by Tropsa . The concept assumes two distinct materials existing at the solidification front:1
A fluid that relaxes freely without accumulating stresses.2
A solid that can sustain mechanical strains and can accumulate stresses.
During solidification, while reaching the solidus temperature Ts the strains are “frozen” into the material at the solidification front. The frozen-in strain reads:(1)Δεfr=α(Ts)ΔT
Where ∆εfr is the volumetric increment of frozen-in strain, α(Ts) the thermal expansion coefficient and ∆T the differential temperature between layers on opposite sides of the interface, respectively.
By expanding the temperature difference with a Taylor-series the following is obtained:(2)εfr(r)=αTres(r)(3)Tres(r)=∫C(r)∂T∂r|T=Ts.dr
Tres represents the residual temperature field. It is defined as the contour integral of the temperature gradients at the moment of solidification. It should be noted that in this case Tres is not a physical temperature but a gradient field representing the strength of the solidification shrinkage. This is depicted in Fig. 5. The residual stresses are then calculated by solving the thermo-elastic problem which reads:(4)∇·(μ(∇u+∇uT)−(λtr∇u−3KαTres)I)+ρfb=0where u is the displacement vector. This is done using existing solvers of the solid mechanics library .
4. Solution procedure in OpenFOAM
The overall sequence of the calculations is divided in three steps, shown in Fig. 6:
The solver provides the solution of the solidification process in the mold. The temperature gradient and time of solidification are recorded for each cell (frozen-in temperature gradient), cf. Fig. 5. For solidification, the simplified boundary conditions assume that the material properties for each phase are constant. During the phase change from liquid to solid the latent heat of fusion is taken into account.
The results of the first calculation are transferred to the next step to calculate the residual temperatures. In this step, the frozen-in temperature gradient is integrated along the solidification path using Eq. (3).
In his original work Tropsa  used an orthogonal line search procedure for this step. The present work applies a simpler approach to integrate the frozen-in temperature gradient along the solidification path to obtain the residual temperature.1
Interpolate ∇(Tsol) to cell faces.2
Search for the cell with highest solidification time [(6) in Fig. 7]. The residual temperature in this cell is set to zero.
Initiate a priority list and add the current cell to it. The priority list is ranked by the solidification time.4
Pick the cell with the smallest solidification time from the priority list (top entry of the priority list) [(5) in Fig. 7] and make it the current cell.5
Traverse all neighboring cells of the current cell:(a)
Skip the neighboring cell if its residual temperature has already been set.(b)
Update the residual temperature in the neighboring cell using the value in the current cell and applying an increment:i
Calculate the increment(5)ΔTres=|∇T·Δr|at the face where ∆r is the distance between the current and neighboring cell.ii
If the solidification time of the neighboring cell is smaller than that of the current cell then the update value is subtracted from the current cell’s e.g. (6)→(5), else it is added, e.g. (5)→(4).(c)
Add the neighboring cell to the priority list with its solidification time.6
Continue with Step 4. unless the priority list is empty.
With this algorithm the integration of the residual temperature is possible, even in the case of distinct local maxima in the solidification time. The complex weld line detection of Tropsa  is also eliminated.
The last step provides a solution of the thermo-elastic problem using Eq. (4). The solver calculates the displacements and the thermal stresses from the residual temperatures for an elastic solid reading from the integrated Tres-field. Here, a slightly modified solver derived from the shipped elasticThermalSolidFoam from the solid mechanics library included in FOAM-extend is used. The displacements resulting from this calculation cover the solidification shrinkage.
5. Results and discussion
All solvers showed excellent numerical stability and performance. The validation using 3D-Scans of the cast cores has been carried out. The predicted deformations due to the residual stresses calculated with the main parameter set accord well with the measured deformations.
5.1. Computational results
In Fig. 8 the residuals of the solver solidificationFoam over the number of iterations for the hot-tear rod are shown, Fig. 9 shows the same for the ring structure. Due to the transient nature of the solver every spike in the graph relates to a new timestep. As seen the solver is stable and shows strict convergence in every timestep for both geometries. After a short starting period, the solver accelerates and uses a nearly constant number of iterations in every time step, in total a number of less than 1600.
TResStressFoam on the other hand shows moderate instabilities with Aitken’s relaxation algorithm  with the ring case but eventually converges as well, Fig. 10. The residuals for the hot-tear rod show no abnormalities for a steady-state solver like this, Fig. 11.
The calculation of the Tres-field takes only seconds, followed by a calculation with the effort of a structural calculation for an elastic component in the last step, which results in a different magnitude of the calculation times compared to the state of the art.
5.2. Calculated temperature gradients
The results of the “frozen-in” temperature gradient for the example structures are shown in Figs. 12 and 13. As described in 4.2 the algorithm calculates the temperature gradients along the solidification path, which is indicated with the arrows. Both Tres-gradient fields represent the directions of the moving solidification fronts in the geometries.
They show a quite good accordance with the real solidification compared to sliced test samples, cf. Fig. 14. The sink marks, which in contrast to the trials do not occur in the simulation, can be easily recognized by the overlapping of the solidification direction markers, especially in the triangular shape in the upper part of the respective feeders. The agglomeration of arrows of different directions indicates the formation of cavities in the experiments.
5.3. Calculated displacements
The displacements resulting from the computation are displayed in the following Figs. 15 and 16. The displacements are amplified by a factor of 5 to illustrate the effects on the geometry color-scaled with the residual temperatures.
A slight bending of the ring as well as a retraction of the first solidifying corners resulting in a rounding of the feeder and the cross section of the ring is apparent. A rounding of the geometry can also be seen in the hot-tear rod. The TRes-fields in the geometries show the most recently solidified areas, which are located at places with high wall thickness and show a high potential for resulting stresses. If the temperature fields are considered as heat maps, they represent the areas of greatest liquid shrinkage, which lead to a decrease in volume. This decrease is not covered by the assumptions to simplify this first approach.
5.4. Validation results
The geometric comparison of both test geometries are in accordance with the behavior in the experiments. A shortening of the hot-tear rod can be seen, as well as a contraction of the entire geometry in relation to the geometrical origin in the feeder in all results, cf. Fig. 17. The validation is carried out comparing the determining dimensions of the example structures; the length of the rod in two positions between the feeder and the bottom face and the inner and outer diameter of the ring, as well as the average thickness of both geometries.
The measurements of the cast rods show a displacement of average – 6.52 mm resulting in a change of length of 3.76%. The calculated displacement results in an average of −7.56 mm and 4.20% related to the length. The thickness of the square rod is determined on average for the total length resulting in measured values of −0.72 mm (3.20%) compared to calculated −0.79 mm (3.53%). Minor deviations of the displacements are visible. The hot-tear rod under consideration shows a maximum deviation of 1.06 mm, which is less than 0.6% related to the total length of the rod and will be considered tolerable.
The geometric comparison of the ring structure shows a tendential accordance of the displacement in general with a slight deviation of the feeder area, cf. Fig. 18.
The measurements of the cast rings show a displacement of average −1.23 mm in the inner diameter (Pos. 1) resulting in a change of 2.70% and −2.26 mm (3.25%) related to the outer diameter (Pos. 2). The calculated displacements of the inner and outer diameter result in −2.04 mm (4.48%) and −2.68 mm (3.85%). The thickness of the ring cross section results in measured values of −0.25 mm (2.08%) compared to calculated −0.39 mm (3.23%). The total numbers of determining measurements for the structures used for comparison are shown in Tables 4 and 5.
Table 4. Average values of the displacements.
|Geometry||Thermal BC||Total average determined displacement [mm]|
|T salt [K]||T die [K]||Casting trials||Simulation|
Table 5. Percentage values of the displacements related to the original geometry.
|Geometry||Thermal BC||Total average determined displacement [%]|
|T salt [K]||T die [K]||Casting trials||Simulation|
A second boundary condition was introduced during the casting tests with the hot-tear rod and a colder die temperature of 463.15 K for further investigation. The simulation and validation has also been carried out with this boundary condition. The results are listed in Tables 4 and 5. The results show a slight tendency of decreasing displacements due to solidification shrinkage at a smaller temperature difference between melt and die. This tendency coincides with test results which show that a small temperature difference results in an improved processing of the molten salts. This is apparent in a lower susceptibility to hot tears before removal from the mold.
The influence of the missing liquid shrinkage to simplify the calculation model is visible in the more complex geometry of the ring and leads to the deviations shown. Especially the feeder in Fig. 18 shows a characteristic decrease in volume of the casting part in comparison. Nevertheless, this calculation also shows the tendency of the displacements in accordance with the experiment. The relative deviations in percent between simulation and measurement are shown in Table 6.
Table 6. Deviations from measured to simulated data.
|Geometry||Thermal BC||Deviation of displacement [%]|
|T salt [K]||T die [K]||Pos.1||Pos.2||Thk.|
For the same geometry, the deviations show an increase with rising temperature difference of the boundary conditions. For the same boundary condition, the deviations increase with the complexity respectively with decreasing volume of the geometry.
The deviation of the ring structure can indicate a geometry dependence of the displacement, because the smaller surface-area-to-volume ratio has a strong influence on the thermal boundary conditions in the experiment (su/vol: ring = 0,27 mm−1; rod = 0,13 mm−1). The cooling conditions therefore have a bigger impact on the displacement in the experiment. Nevertheless, the deviations of the ring structure are also significantly below a value of 2%. An improvement of the values of the displacements is likely to be achieved by an improved thermodynamics and implementation of the boundary conditions, which need to be investigated further, also with other geometries.
The variation of input data, in this case the liquidus temperature and the latent heat, results in the differences from the main parameter set shown in Table 7 at a salt temperature of 963.15 K and a die temperature of 563.15 K.
Table 7. Difference of the deviations of the varied parameters to the main parameter set.
|Geometry||Thermal BC||Difference of displacement [%]|
|L [J/kg]||liq. temp. [K]||Pos.1||Pos.2||Thk.|
It can be seen that the lower graded L, differing from , with the higher liquidus temperature provides better results for both geometries (shaded gray). It indicates that these values are closer to the real conditions, but that they also need to be validated by thermophysical measurements.
A toolchain for simulating the solidification and shrinkage of salt cores has been implemented using the residual temperature concept. It has been tested on simple geometries with simplified boundary conditions. In order to validate the calculations, casting trials with two geometries of gravity dies using the same boundary conditions have been performed. During computation, the developed solvers show good numerical convergence to adequate levels. The displacement of the cast salt cores resulting from the shrinkage and the stresses during solidification was investigated using 3D-Scans. The validations of the results show a good consistency of the calculated and the real displacements of both geometries. The displacements are predicted with adequate accuracy for engineering purposes. The computational cost is dominated by the calculation of the residual stresses from the residual temperature (cost of one steady-state solid mechanics simulation). Considering the very complex initial problem these are excellent results. The complex problem of calculating residual stresses during solidification can be mapped into a steady-state solid mechanics simulation, which is significantly faster compared to a transient calculation of the solid mechanics in every time step.
In future works the reliability of the solver has to be tested with investigations of different boundary conditions. For this purpose, specific thermophysical data should be measured in further experiments for comparison of the literature data. Additional improvements of the solver can be implemented to take into account conjugate heat transfer and liquid shrinkage.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors would like to thank Christian Schütz and Tim Suttner for their technical support.
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