电磁—凝固传输间接耦合过程的数值模拟
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摘要
“三传”是凝固过程中重要的传输现象，“热-动-质”各场量彼此紧密关联、相互影响，其传输行为决定着铸件的结晶质量和性能，电磁场的引入会使这些传输现象更加复杂。因此，对合金的电磁铸造过程建立一个普适的电磁凝固传输统一数值模型并采用一种高效的算法来求解，对于研究不同合金在不同电磁铸造工艺中的凝固传输行为，实现参数优化具有重要的理论和实际意义。本文通过建立电磁场与凝固传输间接耦合过程的统一数值模型，拓展并应用求解压力-速度(P-V)耦合过程的Direct-SIMPLE法(直接求解压力耦合方程的半隐方法)，着重解决电磁场与合金凝固传输间接耦合的宏观传输高效数值计算的相关问题。
采用ANSYS有限元(FEM)软件详尽的计算和分析了用于电磁连铸等电磁铸造加工中二维和三维数值计算的静磁场和行波/谐波磁场。发现物性参数假定为常数时其静磁场的磁感应强度幅值(|B|max)随电流密度载荷线性增加；相同电流密度载荷下一对极行波磁场的|B|max和感应电流密度都较二对极行波磁场的大，但二对极行波比一对极行波变化均匀。通过电磁模型的计算和分析，为电磁凝固传输间接耦合计算奠定了理论基础并提供了电磁载荷数据。
建立了能综合考虑铸件与铸模存在相对运动并适用于纯液/固相情况和有/无电磁场的电磁凝固耦合统一数学模型，基于交错网格并通过有限容积法(FVM)导出了求解二维/三维P-V耦合的Direct-SIMPLE算法。该方法导出的压力和速度场的分解式与修正式具有明确的物理意义，且只需对压力矩阵做迭代计算，而无需对速度和压力做反复迭代修正，因而可极大提高计算效率。
提出和编写了一套将二维/三维电磁场FEM格式数据转换成FVM数据格式的有效方法和程序，它可实现不同FEM和FVM软件间格式的相互转化。在三维模型中，将各种形状FEM单元统一分解为四面体单元再与FVM网格中心点查找和对应，然后用四面体形函数线性插值；二维模型则直接用三角形和四边形单元形函数插值。该方法简单可行，且精度高、耗时短。同时编写了用于二维和三维标/矢量场显示的通用后处理程序，该显示程序能实现坐标旋转和数据反馈等诸多功能。
将ANSYS计算的静磁场和行波磁场结果耦合到连铸过程的凝固传输二维计算，实现了电磁场与连铸凝固传输的间接耦合计算。结果表明，适当的静磁场能够有效抑制板坯连铸过程中的环流；随着磁场的增加，射流冲击和液面波动幅度减小、流股变弱。本文计算中大小为|B|max=5.5×10-3T的静磁场能起到很好的制动和降低宏观偏析的效果；施加一对极或二对极的行波磁场后，熔池中上下两股环流在洛仑兹力影响下被明显改变，加强了弯月面处和凝固界面前沿熔体的流动，本文中二对极行波磁场的综合作用优于一对极行波磁场。将本文模型和算法扩展至三维开放系统的薄板坯水平连铸计算，模拟了在不同外形和强度的虚拟和真实静磁场作用下单带式连铸的流动和传热。结果表明，静磁制动使熔体均匀的输送到水冷带上。本文考察的两类三种外形静磁场中，竖直条形及L形磁场的效果较佳，真实静磁场值|B|=0.4T时能使出口处冲击变小、单带上凝壳均匀生长，且流动形态达到最优。
基于本文提出的数学模型和导出的数值算法，采用FEM-FVM结合的间接耦合计算方案，对Ti-50Al合金直接和间接感应加热区熔定向凝固工艺进行了模拟、设计和实验，获得了质量完好的定向凝固试棒。有电磁场条件下的钛铝合金定向凝固工艺设计和无电磁场条件下的Al-Cu-Si共晶合金的模拟和实验表明，本文电磁凝固传输统一数值模型、算法及间接耦合方法是简单、准确和高效的。
In the coupled process of electromagnetic-solidification transport process(EM-STP) of multi-component alloys, the heat transport, mass transport and flowof alloy melt are the important transport phenomena in solidification process,which determines the crystalline quality and the ultimate performance of thecasting. The transport of various physical fields in solidification process areclosely interrelated and influenced each other, and those transport phenomenabecome even more complex once the EM-field was introduced. Therefore,establishing a universal unified numerical model for EM-STP during EM-casting(EMC), and proposing an efficient numerical methods for solving these processesare very important theoretically and practically for studying the laws of STP ofdifferent alloys and optimizing the parameters in various EM-casting. Theobjectives in this paper are solving the issues relating to macro-transportphenomena of EM-STP with efficient numerical calculation, by establishing a2D/3D uniform numerical model of EM-STP and expanding and applying theDirect-SIMPLE (Direct Semi-Implicit Method for Pressure Linked Equations)method that solve the coupled process of pressure field and velocity field.
Some EM models, including static magnetic field (SMF), traveling/harmonicmagnetic field (TMF/HMF) under two/three dimentional (2D/3D) were calculatedand analised by ANSYS finite element method (FEM) software in detailedly,which were prepared for2D and3D EM-STP numerical calculation such asEM-continous casting (EMCC) and etc. The rusults indicate that the maximummagnitude of magnetic flux denstity (|B|max) increase with current loads linearly in2D and3D EM models. The|B|maxand induced current density of1-pairs of poles(1-POP) TMF are larger than that of2-POP TMF, while it is not homogeneous andsmooth in comprison of2-POP TMF. All of these calculation results and analysisessever as groundwork and provid EM-data for indirect coupled EM-STP.
A uniform mathematical model for coupled EM-STP was proposed in thispaper, in which the relative movments between melt/solide phase and mold istaken into account, and it is suitable for the case of only liquid/solide phase existsin the model and that absence or with EM-field. The Direct-SIMPLE method forsolving pressure and velocity (P-V) in2D/3D models was deduced based onstagger grids system. The decompose and correction expressions of velocity andpressure formulas possess clear physical meanings in this method, and iterationcalculations only need to be carried out for pressure coefficient matrix instead ofrepeating iteration correction between pressure and velocity, thus the computational efficiency of numerical simulation be improved greatly.
A set of data format convertion method and program were proposed andwrited for2D and3D EM-field data, the data stored with FEM format and finitvolume/difference method (FVM/FDM) format can be converted between twodifferent softwares. All the elements with various shapes are divided intotetrahedron elements uniformly in3D model, and then the center points ofFVM/FDM grids mapping with these new sub-tetrahedron elements, once a centerpoint is confirm to be located in a tetrahedron element, it can be linear interpolatedwith the shape function of tetrahedron element. In a2D model, the shape functionof triangle or quadrilateral element would be directly used to linear interpolat. Thishandling method is not only simple and feasible, but also accurate and efficient.Subsequently, a universal post-process program for displaying scalar/vecter fieldsin2D and3D models was proposed, it can achieve coordinate rotation and datafeedback.
SMFs and TMFs exported by ANSYS were coupled to the CC transportcalculation, and an indirect-coupled process of STP with EMF was realized. Theresults show that the appropriate SMF can effectively suppress the circulationflows in the CC-process of slab, the impact of jet on the side wall and theamplitude of level fluctuation as well as the depth of stream decreases withincreasing magnetic flux density. A SMF with|B|max=5.5×10-3T plays a goodbraking effect and can decrease macrosegregtion remarkablely. Once1pair ofpoles (1-POP) or2-POP TMF was applied, the top and lower circulation flows aresignificantly transformed under the influence of the Lorentz force, the flow of meltat meniscus and the front of solidification interface are imforced reasonablely bylinear stirring, the comprehensive effects of2-POP TMF is better than that of1-POP TMF. Finally, the model and numerical method proposed in this paper wasused for the flow and heat transfer simulation in a3D open system of horizontalsingle-belt caster (HSBC), which under the actions of pseudo and the real SMFswith different shapes and intensities. The results show that the EMBr will producea uniform melt conveying to the water-cooled moving single-belt. Among the threecontours of SMF in this paper, a beneficial effect can be achived by the verticaland L-shaped magnetic field, and|B|max=0.4T can decrease the impact of outletflow, bring the solidification shell growth homogeneously on the single-belt andreach a optimal flow pattern in the mold.
The zone melting directional solidification (DS) process of Ti-50Al alloywith direct and indirect induction heating were numerical simulated respectively,where they were disigned and experimental verified using the numerical models,algorithms and FEM/FVM indirect-coupled calculation method proposed in thispaper, and directional solidification samples with good quality were obtained. The process design of Ti-Al alloy EM-DS under EM-field as well as the simulation andexperiment of Al-Cu-Si eutectic alloy show that the EM-STP unified numericalmodel, algorithm and indirect-coupled method are simple, accurate and efficient.
采用ANSYS有限元(FEM)软件详尽的计算和分析了用于电磁连铸等电磁铸造加工中二维和三维数值计算的静磁场和行波/谐波磁场。发现物性参数假定为常数时其静磁场的磁感应强度幅值(|B|max)随电流密度载荷线性增加；相同电流密度载荷下一对极行波磁场的|B|max和感应电流密度都较二对极行波磁场的大，但二对极行波比一对极行波变化均匀。通过电磁模型的计算和分析，为电磁凝固传输间接耦合计算奠定了理论基础并提供了电磁载荷数据。
建立了能综合考虑铸件与铸模存在相对运动并适用于纯液/固相情况和有/无电磁场的电磁凝固耦合统一数学模型，基于交错网格并通过有限容积法(FVM)导出了求解二维/三维P-V耦合的Direct-SIMPLE算法。该方法导出的压力和速度场的分解式与修正式具有明确的物理意义，且只需对压力矩阵做迭代计算，而无需对速度和压力做反复迭代修正，因而可极大提高计算效率。
提出和编写了一套将二维/三维电磁场FEM格式数据转换成FVM数据格式的有效方法和程序，它可实现不同FEM和FVM软件间格式的相互转化。在三维模型中，将各种形状FEM单元统一分解为四面体单元再与FVM网格中心点查找和对应，然后用四面体形函数线性插值；二维模型则直接用三角形和四边形单元形函数插值。该方法简单可行，且精度高、耗时短。同时编写了用于二维和三维标/矢量场显示的通用后处理程序，该显示程序能实现坐标旋转和数据反馈等诸多功能。
将ANSYS计算的静磁场和行波磁场结果耦合到连铸过程的凝固传输二维计算，实现了电磁场与连铸凝固传输的间接耦合计算。结果表明，适当的静磁场能够有效抑制板坯连铸过程中的环流；随着磁场的增加，射流冲击和液面波动幅度减小、流股变弱。本文计算中大小为|B|max=5.5×10-3T的静磁场能起到很好的制动和降低宏观偏析的效果；施加一对极或二对极的行波磁场后，熔池中上下两股环流在洛仑兹力影响下被明显改变，加强了弯月面处和凝固界面前沿熔体的流动，本文中二对极行波磁场的综合作用优于一对极行波磁场。将本文模型和算法扩展至三维开放系统的薄板坯水平连铸计算，模拟了在不同外形和强度的虚拟和真实静磁场作用下单带式连铸的流动和传热。结果表明，静磁制动使熔体均匀的输送到水冷带上。本文考察的两类三种外形静磁场中，竖直条形及L形磁场的效果较佳，真实静磁场值|B|=0.4T时能使出口处冲击变小、单带上凝壳均匀生长，且流动形态达到最优。
基于本文提出的数学模型和导出的数值算法，采用FEM-FVM结合的间接耦合计算方案，对Ti-50Al合金直接和间接感应加热区熔定向凝固工艺进行了模拟、设计和实验，获得了质量完好的定向凝固试棒。有电磁场条件下的钛铝合金定向凝固工艺设计和无电磁场条件下的Al-Cu-Si共晶合金的模拟和实验表明，本文电磁凝固传输统一数值模型、算法及间接耦合方法是简单、准确和高效的。
In the coupled process of electromagnetic-solidification transport process(EM-STP) of multi-component alloys, the heat transport, mass transport and flowof alloy melt are the important transport phenomena in solidification process,which determines the crystalline quality and the ultimate performance of thecasting. The transport of various physical fields in solidification process areclosely interrelated and influenced each other, and those transport phenomenabecome even more complex once the EM-field was introduced. Therefore,establishing a universal unified numerical model for EM-STP during EM-casting(EMC), and proposing an efficient numerical methods for solving these processesare very important theoretically and practically for studying the laws of STP ofdifferent alloys and optimizing the parameters in various EM-casting. Theobjectives in this paper are solving the issues relating to macro-transportphenomena of EM-STP with efficient numerical calculation, by establishing a2D/3D uniform numerical model of EM-STP and expanding and applying theDirect-SIMPLE (Direct Semi-Implicit Method for Pressure Linked Equations)method that solve the coupled process of pressure field and velocity field.
Some EM models, including static magnetic field (SMF), traveling/harmonicmagnetic field (TMF/HMF) under two/three dimentional (2D/3D) were calculatedand analised by ANSYS finite element method (FEM) software in detailedly,which were prepared for2D and3D EM-STP numerical calculation such asEM-continous casting (EMCC) and etc. The rusults indicate that the maximummagnitude of magnetic flux denstity (|B|max) increase with current loads linearly in2D and3D EM models. The|B|maxand induced current density of1-pairs of poles(1-POP) TMF are larger than that of2-POP TMF, while it is not homogeneous andsmooth in comprison of2-POP TMF. All of these calculation results and analysisessever as groundwork and provid EM-data for indirect coupled EM-STP.
A uniform mathematical model for coupled EM-STP was proposed in thispaper, in which the relative movments between melt/solide phase and mold istaken into account, and it is suitable for the case of only liquid/solide phase existsin the model and that absence or with EM-field. The Direct-SIMPLE method forsolving pressure and velocity (P-V) in2D/3D models was deduced based onstagger grids system. The decompose and correction expressions of velocity andpressure formulas possess clear physical meanings in this method, and iterationcalculations only need to be carried out for pressure coefficient matrix instead ofrepeating iteration correction between pressure and velocity, thus the computational efficiency of numerical simulation be improved greatly.
A set of data format convertion method and program were proposed andwrited for2D and3D EM-field data, the data stored with FEM format and finitvolume/difference method (FVM/FDM) format can be converted between twodifferent softwares. All the elements with various shapes are divided intotetrahedron elements uniformly in3D model, and then the center points ofFVM/FDM grids mapping with these new sub-tetrahedron elements, once a centerpoint is confirm to be located in a tetrahedron element, it can be linear interpolatedwith the shape function of tetrahedron element. In a2D model, the shape functionof triangle or quadrilateral element would be directly used to linear interpolat. Thishandling method is not only simple and feasible, but also accurate and efficient.Subsequently, a universal post-process program for displaying scalar/vecter fieldsin2D and3D models was proposed, it can achieve coordinate rotation and datafeedback.
SMFs and TMFs exported by ANSYS were coupled to the CC transportcalculation, and an indirect-coupled process of STP with EMF was realized. Theresults show that the appropriate SMF can effectively suppress the circulationflows in the CC-process of slab, the impact of jet on the side wall and theamplitude of level fluctuation as well as the depth of stream decreases withincreasing magnetic flux density. A SMF with|B|max=5.5×10-3T plays a goodbraking effect and can decrease macrosegregtion remarkablely. Once1pair ofpoles (1-POP) or2-POP TMF was applied, the top and lower circulation flows aresignificantly transformed under the influence of the Lorentz force, the flow of meltat meniscus and the front of solidification interface are imforced reasonablely bylinear stirring, the comprehensive effects of2-POP TMF is better than that of1-POP TMF. Finally, the model and numerical method proposed in this paper wasused for the flow and heat transfer simulation in a3D open system of horizontalsingle-belt caster (HSBC), which under the actions of pseudo and the real SMFswith different shapes and intensities. The results show that the EMBr will producea uniform melt conveying to the water-cooled moving single-belt. Among the threecontours of SMF in this paper, a beneficial effect can be achived by the verticaland L-shaped magnetic field, and|B|max=0.4T can decrease the impact of outletflow, bring the solidification shell growth homogeneously on the single-belt andreach a optimal flow pattern in the mold.
The zone melting directional solidification (DS) process of Ti-50Al alloywith direct and indirect induction heating were numerical simulated respectively,where they were disigned and experimental verified using the numerical models,algorithms and FEM/FVM indirect-coupled calculation method proposed in thispaper, and directional solidification samples with good quality were obtained. The process design of Ti-Al alloy EM-DS under EM-field as well as the simulation andexperiment of Al-Cu-Si eutectic alloy show that the EM-STP unified numericalmodel, algorithm and indirect-coupled method are simple, accurate and efficient.
引文
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