二元共晶/包晶合金系微观偏析半解析数学模型的研究

Study on Semi-analytic Microsegregation Mathematical Model for Binaryeutectic/Peritectic Alloy System

基于典型微观控制单元体(通常指二次枝晶臂间距半长)内的溶质质量守恒关系,建立了适用于枝晶凝固方式的二元共晶/包晶合金微观偏析半解析数学模型,模型中充分考虑了固相反向扩散和枝晶结构粗化对液相溶质浓度的稀释效果。在引入适当的假设条件下,通过严格的数学推导,获得了模型的完整核心控制方程。在推导过程中应用了标准的粗化模型、二次方形式的固相溶质浓度分布以及抛物线固相生长方式等重要假设。其中,采用精度较高的四阶经典龙格-库塔数值微分方法,并结合具体的冷却条件,对模型的常微分核心控制方程来进行数值计算。为验证所建微观偏析模型的合理性和适用性,针对Al-4.9wt%Cu二元共晶合金进行模拟研究,通过将模型的计算结果与已有的实验测试数据以及其它特点各异的微观偏析半解析数学模型的预测结果进行对比分析,表明建立的微观偏析半解析数学模型具有相对较高的预测精度和能力,其预测结果最为接近于实测值。

A semi-analytic microsegregation mathematical model of dendrite solidification mode for binary eutectic/peritectic has been developed based on mass conservation in typical microscale control volume （normally refers to half of secondary dendrite arm space）. In the model effects of back-diffusion and coarsening on dilution of liquid concentration are considered comprehensively. Under appropriate assumptions, complete key control equations of the model have been obtained by strict mathematical derivation and using the standard coarsening model, quadratic solute concentration distribution in solid and parabolic growing mode of solid. The quartic classical Roger- Kutta numerical differential method, which has better precision in calculation, is used to solve the ordinary differential equation combined with actual cooling condition. For AI-4.9 wt% Cu eutectic alloy experiment was carried out to test rationality and applicability of present microsegregation model. By comparison of present and other well-known experimental results with the predictions of present model and preditions of the other semi-analytic microsegregation models which have different characteristic, it can be concluded that the present semi-analytic microsegregation model has best precision and shows best agreement with these experimental results.