三维非等温聚合物熔体充填问题的相场方法

doi:10.19596/j.cnki.1001-246x.8659

摘要/Abstract

摘要：

针对三维非等温聚合物充填问题, 基于相场方法进行数值模拟研究。依据相场模型参数可以将整个区域内的流场控制方程写为统一形式, 并采用Cahn-Hilliard方程追踪牛顿和非牛顿流体之间的运动界面。此外, Cross-WLF模型用来描述非等温条件下, 聚合物熔体粘度的变化规律。依据特征线分裂格式对于统一的Navier-Stokes方程进行解耦, 并采用FEM和SUPG方法分别求解椭圆型、对流-扩散型子方程。以长方体薄壁型腔及带有圆柱嵌件的长方体薄壁型腔为例, 实现三维非等温聚合物熔体的充填过程, 并分析注射速度及浇口尺寸对于充填过程的影响。数值结果与实验结果吻合较好, 并且能够保证非牛顿流体较好的质量守恒性。

关键词: 非等温, 充填过程, Cross-WLF模型, 相场方法, 有限元

Abstract:

We investigate the non-isothermal non-Newtonian filling process according to the phase field method in three dimensions. The moving interface between the Newtonian and non-Newtonian fluid is captured by the Cahn-Hilliard equation. Based on phase field parameters, the governing equations of the flow field could be written in a unified form. After that, the Navier-Stokes equations are split into several sub-equations and the FEM and SUPG will be used to handle the appropriate equations. The variation of the viscosity of the polymer melt is described via the Cross-WLF model. The thin wall rectangular cavities with and without obstacles are considered to illustrate the convergence, robustness and accuracy of the numerical algorithm. The influences of the inlet velocity and the size of the injection gate on the filling process are analyzed. The numerical results agree well with the experimental data and also exhibit good mass conservation properties.

Key words: non-isothermal, filling process, Cross-WLF model, phase field method, finite element method

中图分类号:

O242.1

高普阳. 三维非等温聚合物熔体充填问题的相场方法[J]. 计算物理, 2023, 40(6): 689-698.

Puyang GAO. Non-isothermal Polymer Filling Process via Phase Field Method in Three Dimensions[J]. Chinese Journal of Computational Physics, 2023, 40(6): 689-698.

图/表 16

图1 整个计算过程的流程图

Fig.1 Schematic diagram of the whole calculation process

表1 聚丙烯的材料参数及其Cross-WLF模型中的参数

Table 1 Several parameters for the polypropylene (PP) and its Cross-WLF model

Parameter	Value	Parameter	Value
ρ_l/(kg·m^-3)	0.96 × 10³	D₁/(Pa·s)	6.93 × 10¹¹
C_l/(J·kg^-1·K^-1)	2.0 × 10³	D₂/K	263.15
κ_l/(W·m^-1·K^-1)	0.19	D₃/(K·Pa^-1)	0
τ^*/Pa	30 354	A₁	26.5
n	0.21	A₂/K	51.6

图2 长方体型腔轮廓示意图

Fig.2 Sketch map of the three dimensional rectangular cavity

图3 不同时刻熔体界面形态的实验结果(左列)和数值结果(右列) (a) t=1.1 s; (b) t=1.9 s; (c) t=3.5 s

Fig.3 Experimental data (left column) and numerical results (right column) of interface front development at different time instants (a) t=1.1 s; (b) t=1.9 s; (c) t=3.5 s

图4 (a) 聚合物熔体体积的数值结果(黑线)和真实值(红线); (b)聚合物熔体质量相对误差变化

Fig.4 (a) Exact solution (red line) and numerical result (black line) of polymer melt volume; (b) variation of relative mass error of polymer melt

图5 z=1.5 mm时，x-y平面上不同时刻温度的分布(a) t=1.1 s; (b) t=1.9s; (c) t=3.5 s

Fig.5 Distribution of temperature on the x-y plane with z=1.5 mm at different time instants (a) t=1.1 s; (b) t=1.9 s; (c) t=3.5 s

图6 t=3.5 s时温度场分布的三维视图

Fig.6 Three-dimensional view of distribution of temperature field at t=3.5 s

图7 v1和v2注射速度下，温度场的分布

Fig.7 Temperature field distribution with velocities v1 and v2

图8 带有圆柱形嵌件的长方体型腔示意图

Fig.8 Sketch of the rectangular cavity with a cylindrical obstacle

图9 实验结果(左列)和数值结果(右列)中自由界面在不同时刻的形态(a) t=0.8 s; (b) t=1.4 s

Fig.9 Experimental figures (left column) and numerical results (right column) of the free interface front at (a) t=0.8 s; (b) t=1.4 s

图10 (a) 聚合物熔体体积的精确结果(红线)和数值结果(黑线)；(b)聚合物熔体质量相对误差变化

Fig.10 (a) Exact solution (red line) and numerical result (black line) of polymer melt volume; (b) variation of relative mass error of polymer melt

图11 不同时刻温度场的分布(a) t=0.6 s；(b) t=0.8 s；(c) t=1.4 s

Fig.11 Distribution of temperature field at (a) t=0.6 s; (b) t=0.8 s; (c) t=1.4 s

图12 t=1.4 s时刻温度场分布的三维视图

Fig.12 Three-dimensional view of distribution of temperature field at t=1.4 s

图13 t=0.27 s时刻(a)熔体的前沿界面及(b)温度场的分布

Fig.13 (a) Interface front and (b) distribution of temperature at t=0.27 s

图14 t=0.275 s时刻(a)熔体的前沿界面及(b)温度场的分布

Fig.14 (a) Interface front and (b) distribution of temperature at t=0.275 s

图15 t=0.525 s时刻(a)熔体的前沿界面及(b)温度场的分布

Fig.15 (a) Interface front and (b) distribution of temperature field at t=0.525 s

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