三周期极小曲面多孔材料渗透率尺度特性研究

doi:10.16182/j.issn1004731x.joss.20-0966

摘要/Abstract

摘要：

三周期极小曲面（triply periodic minimal surface, TPMS）在多孔材料设计中的应用日益广泛，但其渗透性能的尺度特性缺乏研究。以4种常用的TPMS单元作为研究对象，在介绍其数学模型及孔隙率控制方法的基础上，建立基于CFD（computational fluid dynamics）的仿真分析模型；对不同孔隙率的TPMS单元和立方多孔结构进行数值仿真分析，明确了其尺度和渗透率之间定量关联规律，即在所选的尺度范围内，各类TPMS单元和立方多孔结构的渗透率是尺度的二次函数。对于TPMS多孔材料渗透性的设计，研究成果可以为TPMS单元类型和尺度的选择提供仿真方法支撑。

关键词: 三周期极小曲面, 渗透率, 尺度特性, CFD, 多孔材料

Abstract:

Triply Periodic Minimal Surface (TPMS) has been widely used in the design of porous materials, however, there is insufficient research on the scale characteristics of permeability. Four commonly used TPMS units are chosen as the research object, based on the introduction of their mathematical models and porosity control methods, a numerical simulation model based on CFD (computational fluid dynamics) is established; TPMS units and cubic porous structures with different porosities are analyzed, the quantitative correlation between their scales and permeability is clarified, i.e., within the selected scale range, the permeability of various TPMS units and cubic porous structures is the quadratic function of scaled. For the design of the permeability of TPMS porous materials, the research results can provide simulation method to support the selection of TPMS unit type and size.

Key words: TPMS, permeability, scale characteristics, computational fluid dynamics, porous material

中图分类号:

TP391.9

吴桐, 王清辉, 徐志佳. 三周期极小曲面多孔材料渗透率尺度特性研究[J]. 系统仿真学报, 2022, 34(5): 1015-1024.

Tong Wu, Qinghui Wang, Zhijia Xu. Study on the Scale Characteristics of Permeability of TPMS Porous Materials[J]. Journal of System Simulation, 2022, 34(5): 1015-1024.

图/表 23

表1

常用TPMS的简化隐式表达式*

TPMS	简化隐式表达式
P	$φ (r) = cos (X) + cos (Y) + cos (Z) = C$
G	$φ (r) = sin (X) cos (Y) + ? ? ? ? ? sin (Y) cos (Z) + sin (Z) cos (X) = C$
F-RD	$φ (r) = cos (X) cos (Y) cos (Z) + ? ? ? ? ? ? ? ? ? cos (2 X) cos (2 Y) cos (2 Z) ? ???? ? ? ? ? ? ? cos (2 X) cos (2 Y) ? cos (2 Y) cos (2 Z) ? ? ? ? ? ? ? ? ? ? ? cos (2 Z) cos (2 X) = C$
I-WP	$φ (r) = 2 [cos (X) cos (Y) + ? ? ? ? ? cos (Y) cos (Z) + cos (Z) cos (X)] ? ? ? ? ? ? [cos (2 X) + cos (2 Y) + cos (2 Z)] = C$

表1

图1

图2

表2

TPMS单元孔隙率与阈值拟合公式

TPMS	拟合公式
P	$P = 28.731 C + 49.997$
G	$P = 32.811 C +49 .969$
F-RD	$P = 13.300 C + 53.100$
I-WP	$P = 10.914 C +34 .689$

表2

表3

4种TPMS单元阈值取值范围及对应孔隙率范围

TPMS	阈值范围	对应孔隙率范围
P	$[? 0.7, 0.9]$	$[29.885, 75.855]$
G	$[? 0.7, 0.8]$	$[27.001, 76.218]$
F-RD	$[? 1.0, 1.7]$	$[39.800, 75.710]$
I-WP	$[? 0.9, 1.0]$	$[24.866, 45.603]$

表3

图3

表4

TPMS模型基本尺度参数范围表

模型种类	单元数量N	特征尺度?L_c/mm	?出口与入口间距?L/mm
P单元	1	$L 0$	$L 0$
立方多孔结构	1~5	$N ? L 0$	$N ? L 0$

表4

图4

图5

表5

网格独立性分析结果

网格数量	入口到出口之间的压降 $Δ P$ /Pa
??211 944	0.439 2
1 063 476	0.491 4
1 879 838	0.496 5

表5

图6

表6

图7

图8

图9

图10

图11

表7

TPMS单元渗透率与尺度之间的关系

TPMS	孔隙率/%	拟合表达式
P	50	$K = 10.066 L 2 + 0.207 L ? 0.409$
G	50	$K = 1.691 L 2 + 0.064 L ? 0.126$
F-RD	50	$K = 0.823 L 2 + 0.003 L ? 0.006$
I-WP	45	$K = 2.407 L 2 + 0.092 L ? 0.181$

表7

图12

图13

图14

图15

图16

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尺度/mm	入口速度/(mm/s)	计算雷诺数
1	5.000	5
5	1.000	5
10	0.500	5
15	0.300	4.5
20	0.250	5
30	0.140	4.2
40	0.100	4